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koteitan @koteitan

\begin{eqnarray}
f_0(n)&=&n+1\\
f_{\alpha+1}(n)&=&f_{\alpha}^n(n)\\
K&=&f_{(0,0)(1,1)\cdots(10^{100},10^{100})[10^{100}~]}(10^{100})\\
S&=&(S_{00},S_{01})(S_{11},S_{11})\cdots (S_{k0},S_{k1})\\
S[n]&=&(S_{00},S_{01})(S_{11},S_{11})\cdots (S_{(k-1)0},S_{(k-1)1})[n+1]~(\mathrm{if}~ S_{k0}=0 \land S_{k1}=0)\\
S[n]&=&GB_0B_1B_2 \cdots B_n[n+1]~(\rm{otherwise})\\
G&=&(S_{00},S_{01})(S_{11},S_{11})\cdots (S_{(i-1)0},S_{(r-1)1})\\
\end{eqnarray}