to10.96pt{\pgfpicture\makeatletter\hbox{\hskip 5.4793pt\lower-2.25708pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.2793pt}{3.22221pt}\pgfsys@curveto{5.2793pt}{6.13792pt}{2.91571pt}{8.50151pt}{0.0pt}{8.50151pt}\pgfsys@curveto{-2.91571pt}{8.50151pt}{-5.2793pt}{6.13792pt}{-5.2793pt}{3.22221pt}\pgfsys@curveto{-5.2793pt}{0.3065pt}{-2.91571pt}{-2.05708pt}{0.0pt}{-2.05708pt}\pgfsys@curveto{2.91571pt}{-2.05708pt}{5.2793pt}{0.3065pt}{5.2793pt}{3.22221pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{3.22221pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\displaystyle 4$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to10.96pt{\vbox to10.96pt{\pgfpicture\makeatletter\hbox{\hskip
5.4793pt\lower-2.25708pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.2793pt}{3.22221pt}\pgfsys@curveto{5.2793pt}{6.13792pt}{2.91571pt}{8.50151pt}{0.0pt}{8.50151pt}\pgfsys@curveto{-2.91571pt}{8.50151pt}{-5.2793pt}{6.13792pt}{-5.2793pt}{3.22221pt}\pgfsys@curveto{-5.2793pt}{0.3065pt}{-2.91571pt}{-2.05708pt}{0.0pt}{-2.05708pt}\pgfsys@curveto{2.91571pt}{-2.05708pt}{5.2793pt}{0.3065pt}{5.2793pt}{3.22221pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{3.22221pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\textstyle 4$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to8.5pt{\vbox to8.5pt{\pgfpicture\makeatletter\hbox{\hskip
4.25005pt\lower-1.99449pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{4.05005pt}{2.25555pt}\pgfsys@curveto{4.05005pt}{4.49236pt}{2.2368pt}{6.3056pt}{0.0pt}{6.3056pt}\pgfsys@curveto{-2.2368pt}{6.3056pt}{-4.05005pt}{4.49236pt}{-4.05005pt}{2.25555pt}\pgfsys@curveto{-4.05005pt}{0.01875pt}{-2.2368pt}{-1.7945pt}{0.0pt}{-1.7945pt}\pgfsys@curveto{2.2368pt}{-1.7945pt}{4.05005pt}{0.01875pt}{4.05005pt}{2.25555pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{2.25555pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.75pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptstyle 4$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to6.89pt{\vbox to6.89pt{\pgfpicture\makeatletter\hbox{\hskip
3.44536pt\lower-1.83426pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{3.24536pt}{1.6111pt}\pgfsys@curveto{3.24536pt}{3.40347pt}{1.79237pt}{4.85646pt}{0.0pt}{4.85646pt}\pgfsys@curveto{-1.79237pt}{4.85646pt}{-3.24536pt}{3.40347pt}{-3.24536pt}{1.6111pt}\pgfsys@curveto{-3.24536pt}{-0.18127pt}{-1.79237pt}{-1.63426pt}{0.0pt}{-1.63426pt}\pgfsys@curveto{1.79237pt}{-1.63426pt}{3.24536pt}{-0.18127pt}{3.24536pt}{1.6111pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{1.6111pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.25pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptscriptstyle
4$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}&\geq\left[\sum\frac{f^{p}(k,t+1,0)c^{p}(k,t+1,0)}{\sum
f^{p}(k,t+1,0)}\right]\overbrace{\left[\frac{\sum f^{p}(k+1,t+1,0)}{\sum
f^{p}(k+1,t,0)}-\frac{\sum f^{p}(k,t+1,0)}{\sum
f^{p}(k,t,0)}\right]}^{\mathchoice{\leavevmode\hbox to10.22pt{\vbox
to10.22pt{\pgfpicture\makeatletter\hbox{\hskip 5.10966pt\lower-2.67912pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{4.90967pt}{2.43054pt}\pgfsys@curveto{4.90967pt}{5.1421pt}{2.71156pt}{7.34021pt}{0.0pt}{7.34021pt}\pgfsys@curveto{-2.71156pt}{7.34021pt}{-4.90967pt}{5.1421pt}{-4.90967pt}{2.43054pt}\pgfsys@curveto{-4.90967pt}{-0.28102pt}{-2.71156pt}{-2.47913pt}{0.0pt}{-2.47913pt}\pgfsys@curveto{2.71156pt}{-2.47913pt}{4.90967pt}{-0.28102pt}{4.90967pt}{2.43054pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{2.43054pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.25208pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\displaystyle 4b$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to10.22pt{\vbox to10.22pt{\pgfpicture\makeatletter\hbox{\hskip
5.10966pt\lower-2.67912pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{4.90967pt}{2.43054pt}\pgfsys@curveto{4.90967pt}{5.1421pt}{2.71156pt}{7.34021pt}{0.0pt}{7.34021pt}\pgfsys@curveto{-2.71156pt}{7.34021pt}{-4.90967pt}{5.1421pt}{-4.90967pt}{2.43054pt}\pgfsys@curveto{-4.90967pt}{-0.28102pt}{-2.71156pt}{-2.47913pt}{0.0pt}{-2.47913pt}\pgfsys@curveto{2.71156pt}{-2.47913pt}{4.90967pt}{-0.28102pt}{4.90967pt}{2.43054pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{2.43054pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.25208pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\textstyle 4b$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to7.77pt{\vbox to7.77pt{\pgfpicture\makeatletter\hbox{\hskip
3.8853pt\lower-2.18391pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{3.6853pt}{1.70139pt}\pgfsys@curveto{3.6853pt}{3.73674pt}{2.03535pt}{5.38669pt}{0.0pt}{5.38669pt}\pgfsys@curveto{-2.03535pt}{5.38669pt}{-3.6853pt}{3.73674pt}{-3.6853pt}{1.70139pt}\pgfsys@curveto{-3.6853pt}{-0.33397pt}{-2.03535pt}{-1.98392pt}{0.0pt}{-1.98392pt}\pgfsys@curveto{2.03535pt}{-1.98392pt}{3.6853pt}{-0.33397pt}{3.6853pt}{1.70139pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{1.70139pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.27644pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptstyle 4b$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to6.13pt{\vbox to6.13pt{\pgfpicture\makeatletter\hbox{\hskip
3.06718pt\lower-1.85191pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{2.86719pt}{1.21527pt}\pgfsys@curveto{2.86719pt}{2.79878pt}{1.58351pt}{4.08246pt}{0.0pt}{4.08246pt}\pgfsys@curveto{-1.58351pt}{4.08246pt}{-2.86719pt}{2.79878pt}{-2.86719pt}{1.21527pt}\pgfsys@curveto{-2.86719pt}{-0.36824pt}{-1.58351pt}{-1.65192pt}{0.0pt}{-1.65192pt}\pgfsys@curveto{1.58351pt}{-1.65192pt}{2.86719pt}{-0.36824pt}{2.86719pt}{1.21527pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{1.21527pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.62604pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptscriptstyle
4b$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}\end{split}$

Analyzing the term $\textstyle 4b$, we have

$\begin{split}\mathchoice{\leavevmode\hbox to14.43pt{\vbox
to14.43pt{\pgfpicture\makeatletter\hbox{\hskip 7.21367pt\lower-3.74146pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{7.01367pt}{3.47221pt}\pgfsys@curveto{7.01367pt}{7.3458pt}{3.87358pt}{10.48589pt}{0.0pt}{10.48589pt}\pgfsys@curveto{-3.87358pt}{10.48589pt}{-7.01367pt}{7.3458pt}{-7.01367pt}{3.47221pt}\pgfsys@curveto{-7.01367pt}{-0.40137pt}{-3.87358pt}{-3.54146pt}{0.0pt}{-3.54146pt}\pgfsys@curveto{3.87358pt}{-3.54146pt}{7.01367pt}{-0.40137pt}{7.01367pt}{3.47221pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{3.47221pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-4.64583pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\displaystyle 4b$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to14.43pt{\vbox to14.43pt{\pgfpicture\makeatletter\hbox{\hskip
7.21367pt\lower-3.74146pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{7.01367pt}{3.47221pt}\pgfsys@curveto{7.01367pt}{7.3458pt}{3.87358pt}{10.48589pt}{0.0pt}{10.48589pt}\pgfsys@curveto{-3.87358pt}{10.48589pt}{-7.01367pt}{7.3458pt}{-7.01367pt}{3.47221pt}\pgfsys@curveto{-7.01367pt}{-0.40137pt}{-3.87358pt}{-3.54146pt}{0.0pt}{-3.54146pt}\pgfsys@curveto{3.87358pt}{-3.54146pt}{7.01367pt}{-0.40137pt}{7.01367pt}{3.47221pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{3.47221pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-4.64583pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\textstyle 4b$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to10.93pt{\vbox to10.93pt{\pgfpicture\makeatletter\hbox{\hskip
5.46465pt\lower-3.0341pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.26465pt}{2.43054pt}\pgfsys@curveto{5.26465pt}{5.33815pt}{2.90761pt}{7.69519pt}{0.0pt}{7.69519pt}\pgfsys@curveto{-2.90761pt}{7.69519pt}{-5.26465pt}{5.33815pt}{-5.26465pt}{2.43054pt}\pgfsys@curveto{-5.26465pt}{-0.47707pt}{-2.90761pt}{-2.8341pt}{0.0pt}{-2.8341pt}\pgfsys@curveto{2.90761pt}{-2.8341pt}{5.26465pt}{-0.47707pt}{5.26465pt}{2.43054pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{2.43054pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.25208pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptstyle 4b$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to8.59pt{\vbox to8.59pt{\pgfpicture\makeatletter\hbox{\hskip
4.29594pt\lower-2.55984pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{4.09595pt}{1.7361pt}\pgfsys@curveto{4.09595pt}{3.99825pt}{2.26215pt}{5.83205pt}{0.0pt}{5.83205pt}\pgfsys@curveto{-2.26215pt}{5.83205pt}{-4.09595pt}{3.99825pt}{-4.09595pt}{1.7361pt}\pgfsys@curveto{-4.09595pt}{-0.52605pt}{-2.26215pt}{-2.35985pt}{0.0pt}{-2.35985pt}\pgfsys@curveto{2.26215pt}{-2.35985pt}{4.09595pt}{-0.52605pt}{4.09595pt}{1.7361pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{1.7361pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.3229pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptscriptstyle
4b$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}&=\frac{\left(\sum
f^{p}(k+1,t+1,0)\right)\left(\sum f^{p}(k,t,0)\right)-\left(\sum
f^{p}(k,t+1,0)\right)\left(\sum f^{p}(k+1,t,0)\right)}{\left(\sum
f^{p}(k+1,t,0)\right)\left(\sum f^{p}(k,t,0)\right)}\\\
&=-\frac{\sum_{\\{(p_{1},p_{2}):p_{1}<p_{2},p_{2}\neq
1\\}}\frac{Pr[P=p_{1}]Pr[P=p_{2}]}{OPT(p_{1})OPT(p_{2})}(p_{1}p_{2})^{k}\left[(1-p_{1})(1-p_{2})\right]^{t-k-1}(p_{1}-p_{2})^{2}}{\left(\sum
f^{p}(k+1,t,1)\right)\left(\sum f^{p}(k,t,1)\right)}\\\
&\geq-\mathchoice{\leavevmode\hbox to14.43pt{\vbox
to14.43pt{\pgfpicture\makeatletter\hbox{\hskip 7.21367pt\lower-3.74146pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{7.01367pt}{3.47221pt}\pgfsys@curveto{7.01367pt}{7.3458pt}{3.87358pt}{10.48589pt}{0.0pt}{10.48589pt}\pgfsys@curveto{-3.87358pt}{10.48589pt}{-7.01367pt}{7.3458pt}{-7.01367pt}{3.47221pt}\pgfsys@curveto{-7.01367pt}{-0.40137pt}{-3.87358pt}{-3.54146pt}{0.0pt}{-3.54146pt}\pgfsys@curveto{3.87358pt}{-3.54146pt}{7.01367pt}{-0.40137pt}{7.01367pt}{3.47221pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{3.47221pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-4.64583pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\displaystyle 3b$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to14.43pt{\vbox to14.43pt{\pgfpicture\makeatletter\hbox{\hskip
7.21367pt\lower-3.74146pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{7.01367pt}{3.47221pt}\pgfsys@curveto{7.01367pt}{7.3458pt}{3.87358pt}{10.48589pt}{0.0pt}{10.48589pt}\pgfsys@curveto{-3.87358pt}{10.48589pt}{-7.01367pt}{7.3458pt}{-7.01367pt}{3.47221pt}\pgfsys@curveto{-7.01367pt}{-0.40137pt}{-3.87358pt}{-3.54146pt}{0.0pt}{-3.54146pt}\pgfsys@curveto{3.87358pt}{-3.54146pt}{7.01367pt}{-0.40137pt}{7.01367pt}{3.47221pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{3.47221pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-4.64583pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\textstyle 3b$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to10.93pt{\vbox to10.93pt{\pgfpicture\makeatletter\hbox{\hskip
5.46465pt\lower-3.0341pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.26465pt}{2.43054pt}\pgfsys@curveto{5.26465pt}{5.33815pt}{2.90761pt}{7.69519pt}{0.0pt}{7.69519pt}\pgfsys@curveto{-2.90761pt}{7.69519pt}{-5.26465pt}{5.33815pt}{-5.26465pt}{2.43054pt}\pgfsys@curveto{-5.26465pt}{-0.47707pt}{-2.90761pt}{-2.8341pt}{0.0pt}{-2.8341pt}\pgfsys@curveto{2.90761pt}{-2.8341pt}{5.26465pt}{-0.47707pt}{5.26465pt}{2.43054pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{2.43054pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.25208pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptstyle 3b$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to8.59pt{\vbox to8.59pt{\pgfpicture\makeatletter\hbox{\hskip
4.29594pt\lower-2.55984pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{4.09595pt}{1.7361pt}\pgfsys@curveto{4.09595pt}{3.99825pt}{2.26215pt}{5.83205pt}{0.0pt}{5.83205pt}\pgfsys@curveto{-2.26215pt}{5.83205pt}{-4.09595pt}{3.99825pt}{-4.09595pt}{1.7361pt}\pgfsys@curveto{-4.09595pt}{-0.52605pt}{-2.26215pt}{-2.35985pt}{0.0pt}{-2.35985pt}\pgfsys@curveto{2.26215pt}{-2.35985pt}{4.09595pt}{-0.52605pt}{4.09595pt}{1.7361pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{1.7361pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.3229pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptscriptstyle
3b$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\end{split}$

Thus combining $\textstyle 3$ and $\textstyle 4$ to give a bound on
$l_{t}(k+1)-l_{t}(k)$, we have

$\begin{split}l_{t}(k+1)-l_{t}(k)&\geq\mathchoice{\leavevmode\hbox
to14.43pt{\vbox to14.43pt{\pgfpicture\makeatletter\hbox{\hskip
7.21367pt\lower-3.74146pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{7.01367pt}{3.47221pt}\pgfsys@curveto{7.01367pt}{7.3458pt}{3.87358pt}{10.48589pt}{0.0pt}{10.48589pt}\pgfsys@curveto{-3.87358pt}{10.48589pt}{-7.01367pt}{7.3458pt}{-7.01367pt}{3.47221pt}\pgfsys@curveto{-7.01367pt}{-0.40137pt}{-3.87358pt}{-3.54146pt}{0.0pt}{-3.54146pt}\pgfsys@curveto{3.87358pt}{-3.54146pt}{7.01367pt}{-0.40137pt}{7.01367pt}{3.47221pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{3.47221pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-4.64583pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\displaystyle 3b$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to14.43pt{\vbox to14.43pt{\pgfpicture\makeatletter\hbox{\hskip
7.21367pt\lower-3.74146pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{7.01367pt}{3.47221pt}\pgfsys@curveto{7.01367pt}{7.3458pt}{3.87358pt}{10.48589pt}{0.0pt}{10.48589pt}\pgfsys@curveto{-3.87358pt}{10.48589pt}{-7.01367pt}{7.3458pt}{-7.01367pt}{3.47221pt}\pgfsys@curveto{-7.01367pt}{-0.40137pt}{-3.87358pt}{-3.54146pt}{0.0pt}{-3.54146pt}\pgfsys@curveto{3.87358pt}{-3.54146pt}{7.01367pt}{-0.40137pt}{7.01367pt}{3.47221pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{3.47221pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-4.64583pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\textstyle 3b$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to10.93pt{\vbox to10.93pt{\pgfpicture\makeatletter\hbox{\hskip
5.46465pt\lower-3.0341pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.26465pt}{2.43054pt}\pgfsys@curveto{5.26465pt}{5.33815pt}{2.90761pt}{7.69519pt}{0.0pt}{7.69519pt}\pgfsys@curveto{-2.90761pt}{7.69519pt}{-5.26465pt}{5.33815pt}{-5.26465pt}{2.43054pt}\pgfsys@curveto{-5.26465pt}{-0.47707pt}{-2.90761pt}{-2.8341pt}{0.0pt}{-2.8341pt}\pgfsys@curveto{2.90761pt}{-2.8341pt}{5.26465pt}{-0.47707pt}{5.26465pt}{2.43054pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{2.43054pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.25208pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptstyle 3b$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to8.59pt{\vbox to8.59pt{\pgfpicture\makeatletter\hbox{\hskip
4.29594pt\lower-2.55984pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{4.09595pt}{1.7361pt}\pgfsys@curveto{4.09595pt}{3.99825pt}{2.26215pt}{5.83205pt}{0.0pt}{5.83205pt}\pgfsys@curveto{-2.26215pt}{5.83205pt}{-4.09595pt}{3.99825pt}{-4.09595pt}{1.7361pt}\pgfsys@curveto{-4.09595pt}{-0.52605pt}{-2.26215pt}{-2.35985pt}{0.0pt}{-2.35985pt}\pgfsys@curveto{2.26215pt}{-2.35985pt}{4.09595pt}{-0.52605pt}{4.09595pt}{1.7361pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{1.7361pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.3229pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptscriptstyle
3b$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\left[\sum\frac{f^{p}(k,t+1,1)c^{p}(k,t+1,1)}{\sum
f^{p}(k,t+1,1)}-\sum\frac{f^{p}(k,t+1,0)c^{p}(k,t+1,0)}{\sum
f^{p}(k,t+1,0)}\right]\\\ &\geq 0\end{split}$

which follows from $\textstyle 3b$ being non-negative and the inductive
assumption that $u_{t+1}(k)\geq l_{t+1}(k)$.

All that is left to prove that the third induction assumption is maintained
i.e $u_{t}(k)\geq l_{t}(k)\forall k\in\\{0\\}\cup[t-1]$. First note that if at
some $k^{*}\in\\{0\\}\cup[t-1]$, the algorithm stops, then we clearly have for
all $k\geq k^{*}$,

$u_{t}(k)=\frac{\sum_{p\in
supp(P)}\frac{Pr[(k,t,1)|p]Pr[P=p]}{OPT(p)}B}{\sum_{p\in
supp(P)}\frac{Pr[(k,t,1)|p]Pr[P=p]}{OPT(p)}}=B>l_{t}(k)$

where the first equality follows from the fact that
$c^{p}(k,t,1)=B\quad\forall p\in supp(P),\forall k\geq k^{*}$ and the
inequality from the fact that the agent never stops on seeing a bad day in the
Backwards Induction algorithm and thus it must have been that

$\begin{split}B\sum_{p\in
supp(P)}\frac{Pr[(k,t,0)|p]Pr[P=p]}{OPT(p)}&>\sum_{p\in
supp(P)}\frac{Pr[(k,t,0)|p]Pr[P=p]}{OPT(p)}[pc^{p}(k,t+1,1)+(1-p)c^{p}(k,t+1,0)]\\\
&=\sum_{p\in
supp(P)}\frac{Pr[(k,t,0)|p]Pr[P=p]}{OPT(p)}c^{p}(k,t,0)\end{split}$

For $k<k^{*}$, i.e those count of days where the agent continues, we have that

$u_{t}(k)-l_{t}(k)=1+\overbrace{\sum p\left[\frac{f^{p}(k,t,1)}{\sum
f^{p}(k,t,1)}c^{p}(k+1,t+1,1)-\frac{f^{p}(k,t,0)}{\sum
f^{p}(k,t,0)}c^{p}(k,t+1,1)\right]}^{\mathchoice{\leavevmode\hbox
to7.79pt{\vbox to7.79pt{\pgfpicture\makeatletter\hbox{\hskip
3.89555pt\lower-1.64pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{3.69556pt}{2.25555pt}\pgfsys@curveto{3.69556pt}{4.29657pt}{2.04102pt}{5.95111pt}{0.0pt}{5.95111pt}\pgfsys@curveto{-2.04102pt}{5.95111pt}{-3.69556pt}{4.29657pt}{-3.69556pt}{2.25555pt}\pgfsys@curveto{-3.69556pt}{0.21454pt}{-2.04102pt}{-1.44pt}{0.0pt}{-1.44pt}\pgfsys@curveto{2.04102pt}{-1.44pt}{3.69556pt}{0.21454pt}{3.69556pt}{2.25555pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{2.25555pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.75pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\displaystyle 5$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to7.79pt{\vbox to7.79pt{\pgfpicture\makeatletter\hbox{\hskip
3.89555pt\lower-1.64pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{3.69556pt}{2.25555pt}\pgfsys@curveto{3.69556pt}{4.29657pt}{2.04102pt}{5.95111pt}{0.0pt}{5.95111pt}\pgfsys@curveto{-2.04102pt}{5.95111pt}{-3.69556pt}{4.29657pt}{-3.69556pt}{2.25555pt}\pgfsys@curveto{-3.69556pt}{0.21454pt}{-2.04102pt}{-1.44pt}{0.0pt}{-1.44pt}\pgfsys@curveto{2.04102pt}{-1.44pt}{3.69556pt}{0.21454pt}{3.69556pt}{2.25555pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{2.25555pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.75pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\textstyle 5$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to6.07pt{\vbox to6.07pt{\pgfpicture\makeatletter\hbox{\hskip
3.03496pt\lower-1.45607pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{2.83496pt}{1.57889pt}\pgfsys@curveto{2.83496pt}{3.1446pt}{1.56572pt}{4.41385pt}{0.0pt}{4.41385pt}\pgfsys@curveto{-1.56572pt}{4.41385pt}{-2.83496pt}{3.1446pt}{-2.83496pt}{1.57889pt}\pgfsys@curveto{-2.83496pt}{0.01317pt}{-1.56572pt}{-1.25607pt}{0.0pt}{-1.25607pt}\pgfsys@curveto{1.56572pt}{-1.25607pt}{2.83496pt}{0.01317pt}{2.83496pt}{1.57889pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{1.57889pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.22499pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptstyle 5$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to4.94pt{\vbox to4.94pt{\pgfpicture\makeatletter\hbox{\hskip
2.47173pt\lower-1.34395pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{2.27173pt}{1.12778pt}\pgfsys@curveto{2.27173pt}{2.38243pt}{1.25465pt}{3.3995pt}{0.0pt}{3.3995pt}\pgfsys@curveto{-1.25465pt}{3.3995pt}{-2.27173pt}{2.38243pt}{-2.27173pt}{1.12778pt}\pgfsys@curveto{-2.27173pt}{-0.12688pt}{-1.25465pt}{-1.14395pt}{0.0pt}{-1.14395pt}\pgfsys@curveto{1.25465pt}{-1.14395pt}{2.27173pt}{-0.12688pt}{2.27173pt}{1.12778pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{1.12778pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-0.875pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptscriptstyle
5$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}+\\\
\underbrace{\sum(1-p)\left[\frac{f^{p}(k,t,1)}{\sum
f^{p}(k,t,1)}c^{p}(k+1,t+1,0)-\frac{f^{p}(k,t,0)}{\sum
f^{p}(k,t,0)}c^{p}(k,t+1,0)\right]}_{\mathchoice{\leavevmode\hbox
to7.79pt{\vbox to7.79pt{\pgfpicture\makeatletter\hbox{\hskip
3.89555pt\lower-1.64pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{3.69556pt}{2.25555pt}\pgfsys@curveto{3.69556pt}{4.29657pt}{2.04102pt}{5.95111pt}{0.0pt}{5.95111pt}\pgfsys@curveto{-2.04102pt}{5.95111pt}{-3.69556pt}{4.29657pt}{-3.69556pt}{2.25555pt}\pgfsys@curveto{-3.69556pt}{0.21454pt}{-2.04102pt}{-1.44pt}{0.0pt}{-1.44pt}\pgfsys@curveto{2.04102pt}{-1.44pt}{3.69556pt}{0.21454pt}{3.69556pt}{2.25555pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{2.25555pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.75pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\displaystyle 6$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to7.79pt{\vbox to7.79pt{\pgfpicture\makeatletter\hbox{\hskip
3.89555pt\lower-1.64pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{3.69556pt}{2.25555pt}\pgfsys@curveto{3.69556pt}{4.29657pt}{2.04102pt}{5.95111pt}{0.0pt}{5.95111pt}\pgfsys@curveto{-2.04102pt}{5.95111pt}{-3.69556pt}{4.29657pt}{-3.69556pt}{2.25555pt}\pgfsys@curveto{-3.69556pt}{0.21454pt}{-2.04102pt}{-1.44pt}{0.0pt}{-1.44pt}\pgfsys@curveto{2.04102pt}{-1.44pt}{3.69556pt}{0.21454pt}{3.69556pt}{2.25555pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{2.25555pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.75pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\textstyle 6$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to6.07pt{\vbox to6.07pt{\pgfpicture\makeatletter\hbox{\hskip
3.03496pt\lower-1.45607pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{2.83496pt}{1.57889pt}\pgfsys@curveto{2.83496pt}{3.1446pt}{1.56572pt}{4.41385pt}{0.0pt}{4.41385pt}\pgfsys@curveto{-1.56572pt}{4.41385pt}{-2.83496pt}{3.1446pt}{-2.83496pt}{1.57889pt}\pgfsys@curveto{-2.83496pt}{0.01317pt}{-1.56572pt}{-1.25607pt}{0.0pt}{-1.25607pt}\pgfsys@curveto{1.56572pt}{-1.25607pt}{2.83496pt}{0.01317pt}{2.83496pt}{1.57889pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{1.57889pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.22499pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptstyle 6$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to4.94pt{\vbox to4.94pt{\pgfpicture\makeatletter\hbox{\hskip
2.47173pt\lower-1.34395pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{2.27173pt}{1.12778pt}\pgfsys@curveto{2.27173pt}{2.38243pt}{1.25465pt}{3.3995pt}{0.0pt}{3.3995pt}\pgfsys@curveto{-1.25465pt}{3.3995pt}{-2.27173pt}{2.38243pt}{-2.27173pt}{1.12778pt}\pgfsys@curveto{-2.27173pt}{-0.12688pt}{-1.25465pt}{-1.14395pt}{0.0pt}{-1.14395pt}\pgfsys@curveto{1.25465pt}{-1.14395pt}{2.27173pt}{-0.12688pt}{2.27173pt}{1.12778pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{1.12778pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-0.875pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptscriptstyle
6$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}$

Considering 5, using the i.i.d assumption on the class of distributions we
consider,

$\mathchoice{\leavevmode\hbox to10.96pt{\vbox
to10.96pt{\pgfpicture\makeatletter\hbox{\hskip 5.4793pt\lower-2.25708pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.2793pt}{3.22221pt}\pgfsys@curveto{5.2793pt}{6.13792pt}{2.91571pt}{8.50151pt}{0.0pt}{8.50151pt}\pgfsys@curveto{-2.91571pt}{8.50151pt}{-5.2793pt}{6.13792pt}{-5.2793pt}{3.22221pt}\pgfsys@curveto{-5.2793pt}{0.3065pt}{-2.91571pt}{-2.05708pt}{0.0pt}{-2.05708pt}\pgfsys@curveto{2.91571pt}{-2.05708pt}{5.2793pt}{0.3065pt}{5.2793pt}{3.22221pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{3.22221pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\displaystyle 5$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to10.96pt{\vbox to10.96pt{\pgfpicture\makeatletter\hbox{\hskip
5.4793pt\lower-2.25708pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.2793pt}{3.22221pt}\pgfsys@curveto{5.2793pt}{6.13792pt}{2.91571pt}{8.50151pt}{0.0pt}{8.50151pt}\pgfsys@curveto{-2.91571pt}{8.50151pt}{-5.2793pt}{6.13792pt}{-5.2793pt}{3.22221pt}\pgfsys@curveto{-5.2793pt}{0.3065pt}{-2.91571pt}{-2.05708pt}{0.0pt}{-2.05708pt}\pgfsys@curveto{2.91571pt}{-2.05708pt}{5.2793pt}{0.3065pt}{5.2793pt}{3.22221pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{3.22221pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\textstyle 5$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to8.5pt{\vbox to8.5pt{\pgfpicture\makeatletter\hbox{\hskip
4.25005pt\lower-1.99449pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{4.05005pt}{2.25555pt}\pgfsys@curveto{4.05005pt}{4.49236pt}{2.2368pt}{6.3056pt}{0.0pt}{6.3056pt}\pgfsys@curveto{-2.2368pt}{6.3056pt}{-4.05005pt}{4.49236pt}{-4.05005pt}{2.25555pt}\pgfsys@curveto{-4.05005pt}{0.01875pt}{-2.2368pt}{-1.7945pt}{0.0pt}{-1.7945pt}\pgfsys@curveto{2.2368pt}{-1.7945pt}{4.05005pt}{0.01875pt}{4.05005pt}{2.25555pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{2.25555pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.75pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptstyle 5$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to6.89pt{\vbox to6.89pt{\pgfpicture\makeatletter\hbox{\hskip
3.44536pt\lower-1.83426pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{3.24536pt}{1.6111pt}\pgfsys@curveto{3.24536pt}{3.40347pt}{1.79237pt}{4.85646pt}{0.0pt}{4.85646pt}\pgfsys@curveto{-1.79237pt}{4.85646pt}{-3.24536pt}{3.40347pt}{-3.24536pt}{1.6111pt}\pgfsys@curveto{-3.24536pt}{-0.18127pt}{-1.79237pt}{-1.63426pt}{0.0pt}{-1.63426pt}\pgfsys@curveto{1.79237pt}{-1.63426pt}{3.24536pt}{-0.18127pt}{3.24536pt}{1.6111pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{1.6111pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.25pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptscriptstyle
5$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}=\frac{\sum
f^{p}(k+1,t+1,1)}{\sum
f^{p}(k,t,1)}\left[\sum\frac{f^{p}(k+1,t+1,1)c^{p}(k+1,t+1,1)}{\sum
f^{p}(k+1,t+1,1)}\right]-\\\ \frac{\sum f^{p}(k,t+1,1)}{\sum
f^{p}(k,t,0)}\left[\sum\frac{f^{p}(k,t+1,1)c^{p}(k,t+1,1)}{\sum
f^{p}(k,t+1,1)}\right]$ (12)

Utilizing the inductive assumption at stage $(t+1)$ on the monotonicity of
$u_{t+1}(k)$, we have that

$\begin{split}\mathchoice{\leavevmode\hbox to10.96pt{\vbox
to10.96pt{\pgfpicture\makeatletter\hbox{\hskip 5.4793pt\lower-2.25708pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.2793pt}{3.22221pt}\pgfsys@curveto{5.2793pt}{6.13792pt}{2.91571pt}{8.50151pt}{0.0pt}{8.50151pt}\pgfsys@curveto{-2.91571pt}{8.50151pt}{-5.2793pt}{6.13792pt}{-5.2793pt}{3.22221pt}\pgfsys@curveto{-5.2793pt}{0.3065pt}{-2.91571pt}{-2.05708pt}{0.0pt}{-2.05708pt}\pgfsys@curveto{2.91571pt}{-2.05708pt}{5.2793pt}{0.3065pt}{5.2793pt}{3.22221pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{3.22221pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\displaystyle 5$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to10.96pt{\vbox to10.96pt{\pgfpicture\makeatletter\hbox{\hskip
5.4793pt\lower-2.25708pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.2793pt}{3.22221pt}\pgfsys@curveto{5.2793pt}{6.13792pt}{2.91571pt}{8.50151pt}{0.0pt}{8.50151pt}\pgfsys@curveto{-2.91571pt}{8.50151pt}{-5.2793pt}{6.13792pt}{-5.2793pt}{3.22221pt}\pgfsys@curveto{-5.2793pt}{0.3065pt}{-2.91571pt}{-2.05708pt}{0.0pt}{-2.05708pt}\pgfsys@curveto{2.91571pt}{-2.05708pt}{5.2793pt}{0.3065pt}{5.2793pt}{3.22221pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{3.22221pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\textstyle 5$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to8.5pt{\vbox to8.5pt{\pgfpicture\makeatletter\hbox{\hskip
4.25005pt\lower-1.99449pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{4.05005pt}{2.25555pt}\pgfsys@curveto{4.05005pt}{4.49236pt}{2.2368pt}{6.3056pt}{0.0pt}{6.3056pt}\pgfsys@curveto{-2.2368pt}{6.3056pt}{-4.05005pt}{4.49236pt}{-4.05005pt}{2.25555pt}\pgfsys@curveto{-4.05005pt}{0.01875pt}{-2.2368pt}{-1.7945pt}{0.0pt}{-1.7945pt}\pgfsys@curveto{2.2368pt}{-1.7945pt}{4.05005pt}{0.01875pt}{4.05005pt}{2.25555pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{2.25555pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.75pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptstyle 5$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to6.89pt{\vbox to6.89pt{\pgfpicture\makeatletter\hbox{\hskip
3.44536pt\lower-1.83426pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{3.24536pt}{1.6111pt}\pgfsys@curveto{3.24536pt}{3.40347pt}{1.79237pt}{4.85646pt}{0.0pt}{4.85646pt}\pgfsys@curveto{-1.79237pt}{4.85646pt}{-3.24536pt}{3.40347pt}{-3.24536pt}{1.6111pt}\pgfsys@curveto{-3.24536pt}{-0.18127pt}{-1.79237pt}{-1.63426pt}{0.0pt}{-1.63426pt}\pgfsys@curveto{1.79237pt}{-1.63426pt}{3.24536pt}{-0.18127pt}{3.24536pt}{1.6111pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{1.6111pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.25pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptscriptstyle
5$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}&\geq\left[\sum\frac{f^{p}(k,t+1,1)c^{p}(k,t+1,1)}{\sum
f^{p}(k,t+1,1)}\right]\overbrace{\left[\frac{\sum f^{p}(k+1,t+1,1)}{\sum
f^{p}(k,t,1)}-\frac{\sum f^{p}(k,t+1,1)}{\sum
f^{p}(k,t,0)}\right]}^{\mathchoice{\leavevmode\hbox to10.22pt{\vbox
to10.22pt{\pgfpicture\makeatletter\hbox{\hskip 5.10966pt\lower-2.67912pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{4.90967pt}{2.43054pt}\pgfsys@curveto{4.90967pt}{5.1421pt}{2.71156pt}{7.34021pt}{0.0pt}{7.34021pt}\pgfsys@curveto{-2.71156pt}{7.34021pt}{-4.90967pt}{5.1421pt}{-4.90967pt}{2.43054pt}\pgfsys@curveto{-4.90967pt}{-0.28102pt}{-2.71156pt}{-2.47913pt}{0.0pt}{-2.47913pt}\pgfsys@curveto{2.71156pt}{-2.47913pt}{4.90967pt}{-0.28102pt}{4.90967pt}{2.43054pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{2.43054pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.25208pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\displaystyle 5b$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to10.22pt{\vbox to10.22pt{\pgfpicture\makeatletter\hbox{\hskip
5.10966pt\lower-2.67912pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{4.90967pt}{2.43054pt}\pgfsys@curveto{4.90967pt}{5.1421pt}{2.71156pt}{7.34021pt}{0.0pt}{7.34021pt}\pgfsys@curveto{-2.71156pt}{7.34021pt}{-4.90967pt}{5.1421pt}{-4.90967pt}{2.43054pt}\pgfsys@curveto{-4.90967pt}{-0.28102pt}{-2.71156pt}{-2.47913pt}{0.0pt}{-2.47913pt}\pgfsys@curveto{2.71156pt}{-2.47913pt}{4.90967pt}{-0.28102pt}{4.90967pt}{2.43054pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{2.43054pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.25208pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\textstyle 5b$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to7.77pt{\vbox to7.77pt{\pgfpicture\makeatletter\hbox{\hskip
3.8853pt\lower-2.18391pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{3.6853pt}{1.70139pt}\pgfsys@curveto{3.6853pt}{3.73674pt}{2.03535pt}{5.38669pt}{0.0pt}{5.38669pt}\pgfsys@curveto{-2.03535pt}{5.38669pt}{-3.6853pt}{3.73674pt}{-3.6853pt}{1.70139pt}\pgfsys@curveto{-3.6853pt}{-0.33397pt}{-2.03535pt}{-1.98392pt}{0.0pt}{-1.98392pt}\pgfsys@curveto{2.03535pt}{-1.98392pt}{3.6853pt}{-0.33397pt}{3.6853pt}{1.70139pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{1.70139pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.27644pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptstyle 5b$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to6.13pt{\vbox to6.13pt{\pgfpicture\makeatletter\hbox{\hskip
3.06718pt\lower-1.85191pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{2.86719pt}{1.21527pt}\pgfsys@curveto{2.86719pt}{2.79878pt}{1.58351pt}{4.08246pt}{0.0pt}{4.08246pt}\pgfsys@curveto{-1.58351pt}{4.08246pt}{-2.86719pt}{2.79878pt}{-2.86719pt}{1.21527pt}\pgfsys@curveto{-2.86719pt}{-0.36824pt}{-1.58351pt}{-1.65192pt}{0.0pt}{-1.65192pt}\pgfsys@curveto{1.58351pt}{-1.65192pt}{2.86719pt}{-0.36824pt}{2.86719pt}{1.21527pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{1.21527pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.62604pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptscriptstyle
5b$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}\end{split}$

If the prior distribution had no mass on $p=1$, then observe that
$\mathchoice{\leavevmode\hbox to14.43pt{\vbox
to14.43pt{\pgfpicture\makeatletter\hbox{\hskip 7.21367pt\lower-3.74146pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{7.01367pt}{3.47221pt}\pgfsys@curveto{7.01367pt}{7.3458pt}{3.87358pt}{10.48589pt}{0.0pt}{10.48589pt}\pgfsys@curveto{-3.87358pt}{10.48589pt}{-7.01367pt}{7.3458pt}{-7.01367pt}{3.47221pt}\pgfsys@curveto{-7.01367pt}{-0.40137pt}{-3.87358pt}{-3.54146pt}{0.0pt}{-3.54146pt}\pgfsys@curveto{3.87358pt}{-3.54146pt}{7.01367pt}{-0.40137pt}{7.01367pt}{3.47221pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{3.47221pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-4.64583pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\displaystyle 5b$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to14.43pt{\vbox to14.43pt{\pgfpicture\makeatletter\hbox{\hskip
7.21367pt\lower-3.74146pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{7.01367pt}{3.47221pt}\pgfsys@curveto{7.01367pt}{7.3458pt}{3.87358pt}{10.48589pt}{0.0pt}{10.48589pt}\pgfsys@curveto{-3.87358pt}{10.48589pt}{-7.01367pt}{7.3458pt}{-7.01367pt}{3.47221pt}\pgfsys@curveto{-7.01367pt}{-0.40137pt}{-3.87358pt}{-3.54146pt}{0.0pt}{-3.54146pt}\pgfsys@curveto{3.87358pt}{-3.54146pt}{7.01367pt}{-0.40137pt}{7.01367pt}{3.47221pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{3.47221pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-4.64583pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\textstyle 5b$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to10.93pt{\vbox to10.93pt{\pgfpicture\makeatletter\hbox{\hskip
5.46465pt\lower-3.0341pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.26465pt}{2.43054pt}\pgfsys@curveto{5.26465pt}{5.33815pt}{2.90761pt}{7.69519pt}{0.0pt}{7.69519pt}\pgfsys@curveto{-2.90761pt}{7.69519pt}{-5.26465pt}{5.33815pt}{-5.26465pt}{2.43054pt}\pgfsys@curveto{-5.26465pt}{-0.47707pt}{-2.90761pt}{-2.8341pt}{0.0pt}{-2.8341pt}\pgfsys@curveto{2.90761pt}{-2.8341pt}{5.26465pt}{-0.47707pt}{5.26465pt}{2.43054pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{2.43054pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.25208pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptstyle 5b$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to8.59pt{\vbox to8.59pt{\pgfpicture\makeatletter\hbox{\hskip
4.29594pt\lower-2.55984pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{4.09595pt}{1.7361pt}\pgfsys@curveto{4.09595pt}{3.99825pt}{2.26215pt}{5.83205pt}{0.0pt}{5.83205pt}\pgfsys@curveto{-2.26215pt}{5.83205pt}{-4.09595pt}{3.99825pt}{-4.09595pt}{1.7361pt}\pgfsys@curveto{-4.09595pt}{-0.52605pt}{-2.26215pt}{-2.35985pt}{0.0pt}{-2.35985pt}\pgfsys@curveto{2.26215pt}{-2.35985pt}{4.09595pt}{-0.52605pt}{4.09595pt}{1.7361pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{1.7361pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.3229pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptscriptstyle
5b$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}=\mathchoice{\leavevmode\hbox
to14.43pt{\vbox to14.43pt{\pgfpicture\makeatletter\hbox{\hskip
7.21367pt\lower-3.74146pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{7.01367pt}{3.47221pt}\pgfsys@curveto{7.01367pt}{7.3458pt}{3.87358pt}{10.48589pt}{0.0pt}{10.48589pt}\pgfsys@curveto{-3.87358pt}{10.48589pt}{-7.01367pt}{7.3458pt}{-7.01367pt}{3.47221pt}\pgfsys@curveto{-7.01367pt}{-0.40137pt}{-3.87358pt}{-3.54146pt}{0.0pt}{-3.54146pt}\pgfsys@curveto{3.87358pt}{-3.54146pt}{7.01367pt}{-0.40137pt}{7.01367pt}{3.47221pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{3.47221pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-4.64583pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\displaystyle 3b$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to14.43pt{\vbox to14.43pt{\pgfpicture\makeatletter\hbox{\hskip
7.21367pt\lower-3.74146pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{7.01367pt}{3.47221pt}\pgfsys@curveto{7.01367pt}{7.3458pt}{3.87358pt}{10.48589pt}{0.0pt}{10.48589pt}\pgfsys@curveto{-3.87358pt}{10.48589pt}{-7.01367pt}{7.3458pt}{-7.01367pt}{3.47221pt}\pgfsys@curveto{-7.01367pt}{-0.40137pt}{-3.87358pt}{-3.54146pt}{0.0pt}{-3.54146pt}\pgfsys@curveto{3.87358pt}{-3.54146pt}{7.01367pt}{-0.40137pt}{7.01367pt}{3.47221pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{3.47221pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-4.64583pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\textstyle 3b$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to10.93pt{\vbox to10.93pt{\pgfpicture\makeatletter\hbox{\hskip
5.46465pt\lower-3.0341pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.26465pt}{2.43054pt}\pgfsys@curveto{5.26465pt}{5.33815pt}{2.90761pt}{7.69519pt}{0.0pt}{7.69519pt}\pgfsys@curveto{-2.90761pt}{7.69519pt}{-5.26465pt}{5.33815pt}{-5.26465pt}{2.43054pt}\pgfsys@curveto{-5.26465pt}{-0.47707pt}{-2.90761pt}{-2.8341pt}{0.0pt}{-2.8341pt}\pgfsys@curveto{2.90761pt}{-2.8341pt}{5.26465pt}{-0.47707pt}{5.26465pt}{2.43054pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{2.43054pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.25208pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptstyle 3b$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to8.59pt{\vbox to8.59pt{\pgfpicture\makeatletter\hbox{\hskip
4.29594pt\lower-2.55984pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{4.09595pt}{1.7361pt}\pgfsys@curveto{4.09595pt}{3.99825pt}{2.26215pt}{5.83205pt}{0.0pt}{5.83205pt}\pgfsys@curveto{-2.26215pt}{5.83205pt}{-4.09595pt}{3.99825pt}{-4.09595pt}{1.7361pt}\pgfsys@curveto{-4.09595pt}{-0.52605pt}{-2.26215pt}{-2.35985pt}{0.0pt}{-2.35985pt}\pgfsys@curveto{2.26215pt}{-2.35985pt}{4.09595pt}{-0.52605pt}{4.09595pt}{1.7361pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{1.7361pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.3229pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptscriptstyle
3b$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}$. In the case the
adversary put some mass on $p=1$,

$\left\\{\begin{array}[]{ll}$ \leavevmode\hbox to15.44pt{\vbox
to15.44pt{\pgfpicture\makeatletter\hbox{\hskip 7.71928pt\lower-4.24707pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{7.51929pt}{3.47221pt}\pgfsys@curveto{7.51929pt}{7.62505pt}{4.15283pt}{10.9915pt}{0.0pt}{10.9915pt}\pgfsys@curveto{-4.15283pt}{10.9915pt}{-7.51929pt}{7.62505pt}{-7.51929pt}{3.47221pt}\pgfsys@curveto{-7.51929pt}{-0.68062pt}{-4.15283pt}{-4.04707pt}{0.0pt}{-4.04707pt}\pgfsys@curveto{4.15283pt}{-4.04707pt}{7.51929pt}{-0.68062pt}{7.51929pt}{3.47221pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{3.47221pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-5.27779pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{5b}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}} = \leavevmode\hbox
to15.44pt{\vbox to15.44pt{\pgfpicture\makeatletter\hbox{\hskip
7.71928pt\lower-4.24707pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{7.51929pt}{3.47221pt}\pgfsys@curveto{7.51929pt}{7.62505pt}{4.15283pt}{10.9915pt}{0.0pt}{10.9915pt}\pgfsys@curveto{-4.15283pt}{10.9915pt}{-7.51929pt}{7.62505pt}{-7.51929pt}{3.47221pt}\pgfsys@curveto{-7.51929pt}{-0.68062pt}{-4.15283pt}{-4.04707pt}{0.0pt}{-4.04707pt}\pgfsys@curveto{4.15283pt}{-4.04707pt}{7.51929pt}{-0.68062pt}{7.51929pt}{3.47221pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{3.47221pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-5.27779pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{3b}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$&\mbox{if }k<t-1\\\
\mathchoice{\leavevmode\hbox to14.43pt{\vbox
to14.43pt{\pgfpicture\makeatletter\hbox{\hskip 7.21367pt\lower-3.74146pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{7.01367pt}{3.47221pt}\pgfsys@curveto{7.01367pt}{7.3458pt}{3.87358pt}{10.48589pt}{0.0pt}{10.48589pt}\pgfsys@curveto{-3.87358pt}{10.48589pt}{-7.01367pt}{7.3458pt}{-7.01367pt}{3.47221pt}\pgfsys@curveto{-7.01367pt}{-0.40137pt}{-3.87358pt}{-3.54146pt}{0.0pt}{-3.54146pt}\pgfsys@curveto{3.87358pt}{-3.54146pt}{7.01367pt}{-0.40137pt}{7.01367pt}{3.47221pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{3.47221pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-4.64583pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\displaystyle 5b$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to14.43pt{\vbox to14.43pt{\pgfpicture\makeatletter\hbox{\hskip
7.21367pt\lower-3.74146pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{7.01367pt}{3.47221pt}\pgfsys@curveto{7.01367pt}{7.3458pt}{3.87358pt}{10.48589pt}{0.0pt}{10.48589pt}\pgfsys@curveto{-3.87358pt}{10.48589pt}{-7.01367pt}{7.3458pt}{-7.01367pt}{3.47221pt}\pgfsys@curveto{-7.01367pt}{-0.40137pt}{-3.87358pt}{-3.54146pt}{0.0pt}{-3.54146pt}\pgfsys@curveto{3.87358pt}{-3.54146pt}{7.01367pt}{-0.40137pt}{7.01367pt}{3.47221pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{3.47221pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-4.64583pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\textstyle 5b$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to10.93pt{\vbox to10.93pt{\pgfpicture\makeatletter\hbox{\hskip
5.46465pt\lower-3.0341pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.26465pt}{2.43054pt}\pgfsys@curveto{5.26465pt}{5.33815pt}{2.90761pt}{7.69519pt}{0.0pt}{7.69519pt}\pgfsys@curveto{-2.90761pt}{7.69519pt}{-5.26465pt}{5.33815pt}{-5.26465pt}{2.43054pt}\pgfsys@curveto{-5.26465pt}{-0.47707pt}{-2.90761pt}{-2.8341pt}{0.0pt}{-2.8341pt}\pgfsys@curveto{2.90761pt}{-2.8341pt}{5.26465pt}{-0.47707pt}{5.26465pt}{2.43054pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{2.43054pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.25208pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptstyle 5b$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to8.59pt{\vbox to8.59pt{\pgfpicture\makeatletter\hbox{\hskip
4.29594pt\lower-2.55984pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{4.09595pt}{1.7361pt}\pgfsys@curveto{4.09595pt}{3.99825pt}{2.26215pt}{5.83205pt}{0.0pt}{5.83205pt}\pgfsys@curveto{-2.26215pt}{5.83205pt}{-4.09595pt}{3.99825pt}{-4.09595pt}{1.7361pt}\pgfsys@curveto{-4.09595pt}{-0.52605pt}{-2.26215pt}{-2.35985pt}{0.0pt}{-2.35985pt}\pgfsys@curveto{2.26215pt}{-2.35985pt}{4.09595pt}{-0.52605pt}{4.09595pt}{1.7361pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{1.7361pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.3229pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptscriptstyle
5b$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}=\mathchoice{\leavevmode\hbox
to14.43pt{\vbox to14.43pt{\pgfpicture\makeatletter\hbox{\hskip
7.21367pt\lower-3.74146pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{7.01367pt}{3.47221pt}\pgfsys@curveto{7.01367pt}{7.3458pt}{3.87358pt}{10.48589pt}{0.0pt}{10.48589pt}\pgfsys@curveto{-3.87358pt}{10.48589pt}{-7.01367pt}{7.3458pt}{-7.01367pt}{3.47221pt}\pgfsys@curveto{-7.01367pt}{-0.40137pt}{-3.87358pt}{-3.54146pt}{0.0pt}{-3.54146pt}\pgfsys@curveto{3.87358pt}{-3.54146pt}{7.01367pt}{-0.40137pt}{7.01367pt}{3.47221pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{3.47221pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-4.64583pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\displaystyle 3b$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to14.43pt{\vbox to14.43pt{\pgfpicture\makeatletter\hbox{\hskip
7.21367pt\lower-3.74146pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{7.01367pt}{3.47221pt}\pgfsys@curveto{7.01367pt}{7.3458pt}{3.87358pt}{10.48589pt}{0.0pt}{10.48589pt}\pgfsys@curveto{-3.87358pt}{10.48589pt}{-7.01367pt}{7.3458pt}{-7.01367pt}{3.47221pt}\pgfsys@curveto{-7.01367pt}{-0.40137pt}{-3.87358pt}{-3.54146pt}{0.0pt}{-3.54146pt}\pgfsys@curveto{3.87358pt}{-3.54146pt}{7.01367pt}{-0.40137pt}{7.01367pt}{3.47221pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{3.47221pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-4.64583pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\textstyle 3b$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to10.93pt{\vbox to10.93pt{\pgfpicture\makeatletter\hbox{\hskip
5.46465pt\lower-3.0341pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.26465pt}{2.43054pt}\pgfsys@curveto{5.26465pt}{5.33815pt}{2.90761pt}{7.69519pt}{0.0pt}{7.69519pt}\pgfsys@curveto{-2.90761pt}{7.69519pt}{-5.26465pt}{5.33815pt}{-5.26465pt}{2.43054pt}\pgfsys@curveto{-5.26465pt}{-0.47707pt}{-2.90761pt}{-2.8341pt}{0.0pt}{-2.8341pt}\pgfsys@curveto{2.90761pt}{-2.8341pt}{5.26465pt}{-0.47707pt}{5.26465pt}{2.43054pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{2.43054pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.25208pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptstyle 3b$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to8.59pt{\vbox to8.59pt{\pgfpicture\makeatletter\hbox{\hskip
4.29594pt\lower-2.55984pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{4.09595pt}{1.7361pt}\pgfsys@curveto{4.09595pt}{3.99825pt}{2.26215pt}{5.83205pt}{0.0pt}{5.83205pt}\pgfsys@curveto{-2.26215pt}{5.83205pt}{-4.09595pt}{3.99825pt}{-4.09595pt}{1.7361pt}\pgfsys@curveto{-4.09595pt}{-0.52605pt}{-2.26215pt}{-2.35985pt}{0.0pt}{-2.35985pt}\pgfsys@curveto{2.26215pt}{-2.35985pt}{4.09595pt}{-0.52605pt}{4.09595pt}{1.7361pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{1.7361pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.3229pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptscriptstyle
3b$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}+\frac{\frac{Pr[P=1]}{B}\sum_{p\in
supp(P)\setminus\\{1\\}}\frac{Pr[P=p]}{OPT(p)}p^{k}(1-p)^{t-k+1}}{(\sum
f^{p}(k,t,1))(\sum f^{p}(k,t,0))}&\mbox{if }t=k-1\end{array}\right.$

Analyzing term $\textstyle 6$,

$\mathchoice{\leavevmode\hbox to10.96pt{\vbox
to10.96pt{\pgfpicture\makeatletter\hbox{\hskip 5.4793pt\lower-2.25708pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.2793pt}{3.22221pt}\pgfsys@curveto{5.2793pt}{6.13792pt}{2.91571pt}{8.50151pt}{0.0pt}{8.50151pt}\pgfsys@curveto{-2.91571pt}{8.50151pt}{-5.2793pt}{6.13792pt}{-5.2793pt}{3.22221pt}\pgfsys@curveto{-5.2793pt}{0.3065pt}{-2.91571pt}{-2.05708pt}{0.0pt}{-2.05708pt}\pgfsys@curveto{2.91571pt}{-2.05708pt}{5.2793pt}{0.3065pt}{5.2793pt}{3.22221pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{3.22221pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\displaystyle 6$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to10.96pt{\vbox to10.96pt{\pgfpicture\makeatletter\hbox{\hskip
5.4793pt\lower-2.25708pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.2793pt}{3.22221pt}\pgfsys@curveto{5.2793pt}{6.13792pt}{2.91571pt}{8.50151pt}{0.0pt}{8.50151pt}\pgfsys@curveto{-2.91571pt}{8.50151pt}{-5.2793pt}{6.13792pt}{-5.2793pt}{3.22221pt}\pgfsys@curveto{-5.2793pt}{0.3065pt}{-2.91571pt}{-2.05708pt}{0.0pt}{-2.05708pt}\pgfsys@curveto{2.91571pt}{-2.05708pt}{5.2793pt}{0.3065pt}{5.2793pt}{3.22221pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{3.22221pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\textstyle 6$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to8.5pt{\vbox to8.5pt{\pgfpicture\makeatletter\hbox{\hskip
4.25005pt\lower-1.99449pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{4.05005pt}{2.25555pt}\pgfsys@curveto{4.05005pt}{4.49236pt}{2.2368pt}{6.3056pt}{0.0pt}{6.3056pt}\pgfsys@curveto{-2.2368pt}{6.3056pt}{-4.05005pt}{4.49236pt}{-4.05005pt}{2.25555pt}\pgfsys@curveto{-4.05005pt}{0.01875pt}{-2.2368pt}{-1.7945pt}{0.0pt}{-1.7945pt}\pgfsys@curveto{2.2368pt}{-1.7945pt}{4.05005pt}{0.01875pt}{4.05005pt}{2.25555pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{2.25555pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.75pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptstyle 6$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to6.89pt{\vbox to6.89pt{\pgfpicture\makeatletter\hbox{\hskip
3.44536pt\lower-1.83426pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{3.24536pt}{1.6111pt}\pgfsys@curveto{3.24536pt}{3.40347pt}{1.79237pt}{4.85646pt}{0.0pt}{4.85646pt}\pgfsys@curveto{-1.79237pt}{4.85646pt}{-3.24536pt}{3.40347pt}{-3.24536pt}{1.6111pt}\pgfsys@curveto{-3.24536pt}{-0.18127pt}{-1.79237pt}{-1.63426pt}{0.0pt}{-1.63426pt}\pgfsys@curveto{1.79237pt}{-1.63426pt}{3.24536pt}{-0.18127pt}{3.24536pt}{1.6111pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{1.6111pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.25pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptscriptstyle
6$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}=\frac{\sum
f^{p}(k+1,t+1,0)}{\sum
f^{p}(k,t,1)}\left[\sum\frac{f^{p}(k+1,t+1,0)c^{p}(k+1,t+1,0)}{\sum
f^{p}(k+1,t+1,0)}\right]-\\\ \frac{\sum f^{p}(k,t+1,0)}{\sum
f^{p}(k,t,0)}\left[\sum\frac{f^{p}(k,t+1,0)c^{p}(k,t+1,0)}{\sum
f^{p}(k,t+1,0)}\right]$ (13)

Utilizing the inductive assumption corresponding to the non-decreasing
property of $l_{t+1}(k)$, we have that

$\begin{split}\mathchoice{\leavevmode\hbox to10.96pt{\vbox
to10.96pt{\pgfpicture\makeatletter\hbox{\hskip 5.4793pt\lower-2.25708pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.2793pt}{3.22221pt}\pgfsys@curveto{5.2793pt}{6.13792pt}{2.91571pt}{8.50151pt}{0.0pt}{8.50151pt}\pgfsys@curveto{-2.91571pt}{8.50151pt}{-5.2793pt}{6.13792pt}{-5.2793pt}{3.22221pt}\pgfsys@curveto{-5.2793pt}{0.3065pt}{-2.91571pt}{-2.05708pt}{0.0pt}{-2.05708pt}\pgfsys@curveto{2.91571pt}{-2.05708pt}{5.2793pt}{0.3065pt}{5.2793pt}{3.22221pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{3.22221pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\displaystyle 6$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to10.96pt{\vbox to10.96pt{\pgfpicture\makeatletter\hbox{\hskip
5.4793pt\lower-2.25708pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.2793pt}{3.22221pt}\pgfsys@curveto{5.2793pt}{6.13792pt}{2.91571pt}{8.50151pt}{0.0pt}{8.50151pt}\pgfsys@curveto{-2.91571pt}{8.50151pt}{-5.2793pt}{6.13792pt}{-5.2793pt}{3.22221pt}\pgfsys@curveto{-5.2793pt}{0.3065pt}{-2.91571pt}{-2.05708pt}{0.0pt}{-2.05708pt}\pgfsys@curveto{2.91571pt}{-2.05708pt}{5.2793pt}{0.3065pt}{5.2793pt}{3.22221pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{3.22221pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\textstyle 6$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to8.5pt{\vbox to8.5pt{\pgfpicture\makeatletter\hbox{\hskip
4.25005pt\lower-1.99449pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{4.05005pt}{2.25555pt}\pgfsys@curveto{4.05005pt}{4.49236pt}{2.2368pt}{6.3056pt}{0.0pt}{6.3056pt}\pgfsys@curveto{-2.2368pt}{6.3056pt}{-4.05005pt}{4.49236pt}{-4.05005pt}{2.25555pt}\pgfsys@curveto{-4.05005pt}{0.01875pt}{-2.2368pt}{-1.7945pt}{0.0pt}{-1.7945pt}\pgfsys@curveto{2.2368pt}{-1.7945pt}{4.05005pt}{0.01875pt}{4.05005pt}{2.25555pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{2.25555pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.75pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptstyle 6$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to6.89pt{\vbox to6.89pt{\pgfpicture\makeatletter\hbox{\hskip
3.44536pt\lower-1.83426pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{3.24536pt}{1.6111pt}\pgfsys@curveto{3.24536pt}{3.40347pt}{1.79237pt}{4.85646pt}{0.0pt}{4.85646pt}\pgfsys@curveto{-1.79237pt}{4.85646pt}{-3.24536pt}{3.40347pt}{-3.24536pt}{1.6111pt}\pgfsys@curveto{-3.24536pt}{-0.18127pt}{-1.79237pt}{-1.63426pt}{0.0pt}{-1.63426pt}\pgfsys@curveto{1.79237pt}{-1.63426pt}{3.24536pt}{-0.18127pt}{3.24536pt}{1.6111pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{1.6111pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.25pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptscriptstyle
6$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}&\geq\left[\sum\frac{f^{p}(k,t+1,0)c^{p}(k,t+1,0)}{\sum
f^{p}(k,t+1,0)}\right]\overbrace{\left[\frac{\sum f^{p}(k+1,t+1,0)}{\sum
f^{p}(k,t,1)}-\frac{\sum f^{p}(k,t+1,0)}{\sum
f^{p}(k,t,0)}\right]}^{\mathchoice{\leavevmode\hbox to10.22pt{\vbox
to10.22pt{\pgfpicture\makeatletter\hbox{\hskip 5.10966pt\lower-2.67912pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{4.90967pt}{2.43054pt}\pgfsys@curveto{4.90967pt}{5.1421pt}{2.71156pt}{7.34021pt}{0.0pt}{7.34021pt}\pgfsys@curveto{-2.71156pt}{7.34021pt}{-4.90967pt}{5.1421pt}{-4.90967pt}{2.43054pt}\pgfsys@curveto{-4.90967pt}{-0.28102pt}{-2.71156pt}{-2.47913pt}{0.0pt}{-2.47913pt}\pgfsys@curveto{2.71156pt}{-2.47913pt}{4.90967pt}{-0.28102pt}{4.90967pt}{2.43054pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{2.43054pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.25208pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\displaystyle 6b$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to10.22pt{\vbox to10.22pt{\pgfpicture\makeatletter\hbox{\hskip
5.10966pt\lower-2.67912pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{4.90967pt}{2.43054pt}\pgfsys@curveto{4.90967pt}{5.1421pt}{2.71156pt}{7.34021pt}{0.0pt}{7.34021pt}\pgfsys@curveto{-2.71156pt}{7.34021pt}{-4.90967pt}{5.1421pt}{-4.90967pt}{2.43054pt}\pgfsys@curveto{-4.90967pt}{-0.28102pt}{-2.71156pt}{-2.47913pt}{0.0pt}{-2.47913pt}\pgfsys@curveto{2.71156pt}{-2.47913pt}{4.90967pt}{-0.28102pt}{4.90967pt}{2.43054pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{2.43054pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.25208pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\textstyle 6b$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to7.77pt{\vbox to7.77pt{\pgfpicture\makeatletter\hbox{\hskip
3.8853pt\lower-2.18391pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{3.6853pt}{1.70139pt}\pgfsys@curveto{3.6853pt}{3.73674pt}{2.03535pt}{5.38669pt}{0.0pt}{5.38669pt}\pgfsys@curveto{-2.03535pt}{5.38669pt}{-3.6853pt}{3.73674pt}{-3.6853pt}{1.70139pt}\pgfsys@curveto{-3.6853pt}{-0.33397pt}{-2.03535pt}{-1.98392pt}{0.0pt}{-1.98392pt}\pgfsys@curveto{2.03535pt}{-1.98392pt}{3.6853pt}{-0.33397pt}{3.6853pt}{1.70139pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{1.70139pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.27644pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptstyle 6b$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to6.13pt{\vbox to6.13pt{\pgfpicture\makeatletter\hbox{\hskip
3.06718pt\lower-1.85191pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{2.86719pt}{1.21527pt}\pgfsys@curveto{2.86719pt}{2.79878pt}{1.58351pt}{4.08246pt}{0.0pt}{4.08246pt}\pgfsys@curveto{-1.58351pt}{4.08246pt}{-2.86719pt}{2.79878pt}{-2.86719pt}{1.21527pt}\pgfsys@curveto{-2.86719pt}{-0.36824pt}{-1.58351pt}{-1.65192pt}{0.0pt}{-1.65192pt}\pgfsys@curveto{1.58351pt}{-1.65192pt}{2.86719pt}{-0.36824pt}{2.86719pt}{1.21527pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{1.21527pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.62604pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptscriptstyle
6b$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}\end{split}$

Analyzing the term $\textstyle 6b$, again we observe that if the prior
distribution had no mass on $p=1$, then $\mathchoice{\leavevmode\hbox
to14.43pt{\vbox to14.43pt{\pgfpicture\makeatletter\hbox{\hskip
7.21367pt\lower-3.74146pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{7.01367pt}{3.47221pt}\pgfsys@curveto{7.01367pt}{7.3458pt}{3.87358pt}{10.48589pt}{0.0pt}{10.48589pt}\pgfsys@curveto{-3.87358pt}{10.48589pt}{-7.01367pt}{7.3458pt}{-7.01367pt}{3.47221pt}\pgfsys@curveto{-7.01367pt}{-0.40137pt}{-3.87358pt}{-3.54146pt}{0.0pt}{-3.54146pt}\pgfsys@curveto{3.87358pt}{-3.54146pt}{7.01367pt}{-0.40137pt}{7.01367pt}{3.47221pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{3.47221pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-4.64583pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\displaystyle 6b$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to14.43pt{\vbox to14.43pt{\pgfpicture\makeatletter\hbox{\hskip
7.21367pt\lower-3.74146pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{7.01367pt}{3.47221pt}\pgfsys@curveto{7.01367pt}{7.3458pt}{3.87358pt}{10.48589pt}{0.0pt}{10.48589pt}\pgfsys@curveto{-3.87358pt}{10.48589pt}{-7.01367pt}{7.3458pt}{-7.01367pt}{3.47221pt}\pgfsys@curveto{-7.01367pt}{-0.40137pt}{-3.87358pt}{-3.54146pt}{0.0pt}{-3.54146pt}\pgfsys@curveto{3.87358pt}{-3.54146pt}{7.01367pt}{-0.40137pt}{7.01367pt}{3.47221pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{3.47221pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-4.64583pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\textstyle 6b$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to10.93pt{\vbox to10.93pt{\pgfpicture\makeatletter\hbox{\hskip
5.46465pt\lower-3.0341pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.26465pt}{2.43054pt}\pgfsys@curveto{5.26465pt}{5.33815pt}{2.90761pt}{7.69519pt}{0.0pt}{7.69519pt}\pgfsys@curveto{-2.90761pt}{7.69519pt}{-5.26465pt}{5.33815pt}{-5.26465pt}{2.43054pt}\pgfsys@curveto{-5.26465pt}{-0.47707pt}{-2.90761pt}{-2.8341pt}{0.0pt}{-2.8341pt}\pgfsys@curveto{2.90761pt}{-2.8341pt}{5.26465pt}{-0.47707pt}{5.26465pt}{2.43054pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{2.43054pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.25208pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptstyle 6b$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to8.59pt{\vbox to8.59pt{\pgfpicture\makeatletter\hbox{\hskip
4.29594pt\lower-2.55984pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{4.09595pt}{1.7361pt}\pgfsys@curveto{4.09595pt}{3.99825pt}{2.26215pt}{5.83205pt}{0.0pt}{5.83205pt}\pgfsys@curveto{-2.26215pt}{5.83205pt}{-4.09595pt}{3.99825pt}{-4.09595pt}{1.7361pt}\pgfsys@curveto{-4.09595pt}{-0.52605pt}{-2.26215pt}{-2.35985pt}{0.0pt}{-2.35985pt}\pgfsys@curveto{2.26215pt}{-2.35985pt}{4.09595pt}{-0.52605pt}{4.09595pt}{1.7361pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{1.7361pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.3229pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptscriptstyle
6b$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}=\mathchoice{\leavevmode\hbox
to14.43pt{\vbox to14.43pt{\pgfpicture\makeatletter\hbox{\hskip
7.21367pt\lower-3.74146pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{7.01367pt}{3.47221pt}\pgfsys@curveto{7.01367pt}{7.3458pt}{3.87358pt}{10.48589pt}{0.0pt}{10.48589pt}\pgfsys@curveto{-3.87358pt}{10.48589pt}{-7.01367pt}{7.3458pt}{-7.01367pt}{3.47221pt}\pgfsys@curveto{-7.01367pt}{-0.40137pt}{-3.87358pt}{-3.54146pt}{0.0pt}{-3.54146pt}\pgfsys@curveto{3.87358pt}{-3.54146pt}{7.01367pt}{-0.40137pt}{7.01367pt}{3.47221pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{3.47221pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-4.64583pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\displaystyle 4b$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to14.43pt{\vbox to14.43pt{\pgfpicture\makeatletter\hbox{\hskip
7.21367pt\lower-3.74146pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{7.01367pt}{3.47221pt}\pgfsys@curveto{7.01367pt}{7.3458pt}{3.87358pt}{10.48589pt}{0.0pt}{10.48589pt}\pgfsys@curveto{-3.87358pt}{10.48589pt}{-7.01367pt}{7.3458pt}{-7.01367pt}{3.47221pt}\pgfsys@curveto{-7.01367pt}{-0.40137pt}{-3.87358pt}{-3.54146pt}{0.0pt}{-3.54146pt}\pgfsys@curveto{3.87358pt}{-3.54146pt}{7.01367pt}{-0.40137pt}{7.01367pt}{3.47221pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{3.47221pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-4.64583pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\textstyle 4b$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to10.93pt{\vbox to10.93pt{\pgfpicture\makeatletter\hbox{\hskip
5.46465pt\lower-3.0341pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.26465pt}{2.43054pt}\pgfsys@curveto{5.26465pt}{5.33815pt}{2.90761pt}{7.69519pt}{0.0pt}{7.69519pt}\pgfsys@curveto{-2.90761pt}{7.69519pt}{-5.26465pt}{5.33815pt}{-5.26465pt}{2.43054pt}\pgfsys@curveto{-5.26465pt}{-0.47707pt}{-2.90761pt}{-2.8341pt}{0.0pt}{-2.8341pt}\pgfsys@curveto{2.90761pt}{-2.8341pt}{5.26465pt}{-0.47707pt}{5.26465pt}{2.43054pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{2.43054pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.25208pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptstyle 4b$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to8.59pt{\vbox to8.59pt{\pgfpicture\makeatletter\hbox{\hskip
4.29594pt\lower-2.55984pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{4.09595pt}{1.7361pt}\pgfsys@curveto{4.09595pt}{3.99825pt}{2.26215pt}{5.83205pt}{0.0pt}{5.83205pt}\pgfsys@curveto{-2.26215pt}{5.83205pt}{-4.09595pt}{3.99825pt}{-4.09595pt}{1.7361pt}\pgfsys@curveto{-4.09595pt}{-0.52605pt}{-2.26215pt}{-2.35985pt}{0.0pt}{-2.35985pt}\pgfsys@curveto{2.26215pt}{-2.35985pt}{4.09595pt}{-0.52605pt}{4.09595pt}{1.7361pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{1.7361pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.3229pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptscriptstyle
4b$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}$. In the case the
adversary put some mass on $p=1$,

$\left\\{\begin{array}[]{ll}$ \leavevmode\hbox to15.44pt{\vbox
to15.44pt{\pgfpicture\makeatletter\hbox{\hskip 7.71928pt\lower-4.24707pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{7.51929pt}{3.47221pt}\pgfsys@curveto{7.51929pt}{7.62505pt}{4.15283pt}{10.9915pt}{0.0pt}{10.9915pt}\pgfsys@curveto{-4.15283pt}{10.9915pt}{-7.51929pt}{7.62505pt}{-7.51929pt}{3.47221pt}\pgfsys@curveto{-7.51929pt}{-0.68062pt}{-4.15283pt}{-4.04707pt}{0.0pt}{-4.04707pt}\pgfsys@curveto{4.15283pt}{-4.04707pt}{7.51929pt}{-0.68062pt}{7.51929pt}{3.47221pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{3.47221pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-5.27779pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{6b}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}} = \leavevmode\hbox
to15.44pt{\vbox to15.44pt{\pgfpicture\makeatletter\hbox{\hskip
7.71928pt\lower-4.24707pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{7.51929pt}{3.47221pt}\pgfsys@curveto{7.51929pt}{7.62505pt}{4.15283pt}{10.9915pt}{0.0pt}{10.9915pt}\pgfsys@curveto{-4.15283pt}{10.9915pt}{-7.51929pt}{7.62505pt}{-7.51929pt}{3.47221pt}\pgfsys@curveto{-7.51929pt}{-0.68062pt}{-4.15283pt}{-4.04707pt}{0.0pt}{-4.04707pt}\pgfsys@curveto{4.15283pt}{-4.04707pt}{7.51929pt}{-0.68062pt}{7.51929pt}{3.47221pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{3.47221pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-5.27779pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{4b}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$&\mbox{if }k<t-1\\\
\mathchoice{\leavevmode\hbox to14.43pt{\vbox
to14.43pt{\pgfpicture\makeatletter\hbox{\hskip 7.21367pt\lower-3.74146pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{7.01367pt}{3.47221pt}\pgfsys@curveto{7.01367pt}{7.3458pt}{3.87358pt}{10.48589pt}{0.0pt}{10.48589pt}\pgfsys@curveto{-3.87358pt}{10.48589pt}{-7.01367pt}{7.3458pt}{-7.01367pt}{3.47221pt}\pgfsys@curveto{-7.01367pt}{-0.40137pt}{-3.87358pt}{-3.54146pt}{0.0pt}{-3.54146pt}\pgfsys@curveto{3.87358pt}{-3.54146pt}{7.01367pt}{-0.40137pt}{7.01367pt}{3.47221pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{3.47221pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-4.64583pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\displaystyle 6b$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to14.43pt{\vbox to14.43pt{\pgfpicture\makeatletter\hbox{\hskip
7.21367pt\lower-3.74146pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{7.01367pt}{3.47221pt}\pgfsys@curveto{7.01367pt}{7.3458pt}{3.87358pt}{10.48589pt}{0.0pt}{10.48589pt}\pgfsys@curveto{-3.87358pt}{10.48589pt}{-7.01367pt}{7.3458pt}{-7.01367pt}{3.47221pt}\pgfsys@curveto{-7.01367pt}{-0.40137pt}{-3.87358pt}{-3.54146pt}{0.0pt}{-3.54146pt}\pgfsys@curveto{3.87358pt}{-3.54146pt}{7.01367pt}{-0.40137pt}{7.01367pt}{3.47221pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{3.47221pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-4.64583pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\textstyle 6b$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to10.93pt{\vbox to10.93pt{\pgfpicture\makeatletter\hbox{\hskip
5.46465pt\lower-3.0341pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.26465pt}{2.43054pt}\pgfsys@curveto{5.26465pt}{5.33815pt}{2.90761pt}{7.69519pt}{0.0pt}{7.69519pt}\pgfsys@curveto{-2.90761pt}{7.69519pt}{-5.26465pt}{5.33815pt}{-5.26465pt}{2.43054pt}\pgfsys@curveto{-5.26465pt}{-0.47707pt}{-2.90761pt}{-2.8341pt}{0.0pt}{-2.8341pt}\pgfsys@curveto{2.90761pt}{-2.8341pt}{5.26465pt}{-0.47707pt}{5.26465pt}{2.43054pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{2.43054pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.25208pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptstyle 6b$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }