, we compute this value in the case where the options of $S$ are in ${\mathcal D_0^{\Box}}$ and ${\mathcal D_0^{\Box}}$:    ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------    ${\mathcal{G}}(S {{\tikz[baseline=-0]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=0]{} -- (0.0,0) node[circle,fill=black, minimum size=0, inner sep=0] {} node[midway,above,scale=0.0] {$0$};}}}S')$   $=$  $\operatorname{mex}( {\mathcal{G}}( \mathcal D_0 {{\tikz[baseline=-0]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=0]{} -- (0.0,0) node[circle,fill=black, minimum size=0, inner sep=0] {} node[midway,above,scale=0.0] {$0$};}}}\mathcal D_0 ) , {\mathcal{G}}( {\mathcal D_0^{\Box}}{{\tikz[baseline=-0]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=0]{} -- (0.0,0) node[circle,fill=black, minimum size=0, inner sep=0] {} node[midway,above,scale=0.0] {$0$};}}}\mathcal D_0 ) , {\mathcal{G}}( {\mathcal D_0^{\Box}}{{\tikz[baseline=-0]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=0]{} -- (0.0,0) node[circle,fill=black, minimum size=0, inner sep=0] {} node[midway,above,scale=0.0] {$0$};}}}\mathcal D_0 ) , {\mathcal{G}}( \mathcal D_0 {{\tikz[baseline=-0]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=0]{} -- (0.0,0) node[circle,fill=black, minimum size=0, inner sep=0] {} node[midway,above,scale=0.0] {$0$};}}}\mathcal D_0 ) , {\mathcal{G}}( \mathcal D_0 {{\tikz[baseline=-0]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=0]{} -- (0.0,0) node[circle,fill=black, minimum size=0, inner sep=0] {} node[midway,above,scale=0.0] {$0$};}}}\mathcal D_0 ) , {\mathcal{G}}( \mathcal D_0 {{\tikz[baseline=-0]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=0]{} -- (0.0,0) node[circle,fill=black, minimum size=0, inner sep=0] {} node[midway,above,scale=0.0] {$0$};}}}\mathcal D_0 ) )$                                                                                                                                                                                                                           $=$  $\operatorname{mex}( 0,0,0,0,0,0 )$ (by induction hypothesis)                                                                                                                                                                                                                           $=$  0   ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------  Now, we compute this value in the case where the options of $S$ are in $