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\begin{verbatim}#|

 Copyright (C) 1994 by Computational Logic, Inc.  All Rights Reserved.

 This script is hereby placed in the public domain, and therefore unlimited
 editing and redistribution is permitted.

 NO WARRANTY

 Computational Logic, Inc. PROVIDES ABSOLUTELY NO WARRANTY.  THE EVENT SCRIPT
 IS PROVIDED "AS IS" WITHOUT WARRANTY OF ANY KIND, EITHER EXPRESS OR IMPLIED,
 INCLUDING, BUT NOT LIMITED TO, ANY IMPLIED WARRANTIES OF MERCHANTABILITY AND
 FITNESS FOR A PARTICULAR PURPOSE.  THE ENTIRE RISK AS TO THE QUALITY AND
 PERFORMANCE OF THE SCRIPT IS WITH YOU.  SHOULD THE SCRIPT PROVE DEFECTIVE, YOU
 ASSUME THE COST OF ALL NECESSARY SERVICING, REPAIR OR CORRECTION.

 IN NO EVENT WILL Computational Logic, Inc. BE LIABLE TO YOU FOR ANY DAMAGES,
 ANY LOST PROFITS, LOST MONIES, OR OTHER SPECIAL, INCIDENTAL OR CONSEQUENTIAL
 DAMAGES ARISING OUT OF THE USE OR INABILITY TO USE THIS SCRIPT (INCLUDING BUT
 NOT LIMITED TO LOSS OF DATA OR DATA BEING RENDERED INACCURATE OR LOSSES
 SUSTAINED BY THIRD PARTIES), EVEN IF YOU HAVE ADVISED US OF THE POSSIBILITY OF
 SUCH DAMAGES, OR FOR ANY CLAIM BY ANY OTHER PARTY.

|#

\end{verbatim}
\noindent{\sc Event}:   Start with the initial {\bf nqthm} theory.

\addvspace{18pt}

\begin{tabbing}{\sc Definition}:\index{boardp} \\  
\=\+{\rm{boardp}}\,({\it{x\/}})\index{boardp} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{x\/}} $\simeq\;${{\bf{nil}}}${}$$\;\;${\bf then }{\it{x\/}} $=$ {\bf{nil}} \\ 
{\bf else }\=\+(({\rm{car}}\,({\it{x\/}}) $=$ {\tt{'}}{\tt{x}}) $\vee$ ({\rm{car}}\,({\it{x\/}}) $=$ {\tt{'}}{\tt{o}}) $\vee$ ({\rm{car}}\,({\it{x\/}}) $=$ {\bf{f}})) \\ 
$\wedge$$\;\;\;\;${\rm{boardp}}\,({\rm{cdr}}\,({\it{x\/}}))\index{boardp}\-$\;${\bf  endif}\-\-
\end{tabbing}

\begin{tabbing}{\sc Definition}:\index{endp} \\  
\=\+{\rm{endp}}\,({\it{board\/}})\index{endp} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{board\/}} $\simeq\;${{\bf{nil}}}${}$$\;\;${\bf then }{\bf{t}} \\ 
{\bf elseif }{\rm{car}}\,({\it{board\/}})$\;\;${\bf then }{\rm{endp}}\,({\rm{cdr}}\,({\it{board\/}}))\index{endp} \\ 
{\bf else }{\bf{f}}$\;${\bf  endif}\-\-
\end{tabbing}

\begin{tabbing}{\sc Definition}:\index{legal-movep} \\  
\=\+{\rm{legal-movep}}\,({\it{board1\/}}, {\it{player\/}}, {\it{board2\/}})\index{legal-movep} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{board1\/}} $\simeq\;${{\bf{nil}}}${}$$\;\;${\bf then }{\bf{f}} \\ 
{\bf elseif }{\rm{car}}\,({\it{board1\/}}) $=$ {\rm{car}}\,({\it{board2\/}}) \\ 
{\bf then }{\rm{legal-movep}}\,({\rm{cdr}}\,({\it{board1\/}}), {\it{player\/}}, {\rm{cdr}}\,({\it{board2\/}}))\index{legal-movep} \\ 
{\bf elseif }({\rm{car}}\,({\it{board1\/}}) $=$ {\bf{f}}) $\wedge$ ({\rm{car}}\,({\it{board2\/}}) $=$ {\it{player\/}}) \\ 
{\bf then }{\rm{cdr}}\,({\it{board1\/}}) $=$ {\rm{cdr}}\,({\it{board2\/}}) \\ 
{\bf else }{\bf{f}}$\;${\bf  endif}\-\-
\end{tabbing}

\begin{tabbing}{\sc Definition}:\index{nth} \\  
\=\+{\rm{nth}}\,({\it{i\/}}, {\it{board\/}})\index{nth} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{i\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\rm{car}}\,({\it{board\/}}) \\ 
{\bf else }{\rm{nth}}\,({\it{i\/}} $-\;1$, {\rm{cdr}}\,({\it{board\/}}))\index{nth}$\;${\bf  endif}\-\-
\end{tabbing}

\begin{tabbing}{\sc Definition}:\index{winp1} \\  
\=\+{\rm{winp1}}\,({\it{x\/}}, {\it{line\/}}, {\it{board\/}})\index{winp1} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{line\/}} $\simeq\;${{\bf{nil}}}${}$$\;\;${\bf then }{\bf{t}} \\ 
{\bf elseif }{\it{x\/}} $=$ {\rm{nth}}\,({\rm{car}}\,({\it{line\/}}), {\it{board\/}})\index{nth}$\;\;${\bf then }{\rm{winp1}}\,({\it{x\/}}, {\rm{cdr}}\,({\it{line\/}}), {\it{board\/}})\index{winp1} \\ 
{\bf else }{\bf{f}}$\;${\bf  endif}\-\-
\end{tabbing}

\begin{tabbing}{\sc Definition}:\index{winp} \\  
\=\+{\rm{winp}}\,({\it{x\/}}, {\it{board\/}})\index{winp} \\ 
$=$$\;\;\;\;$(\=\+{\rm{winp1}}\,({\it{x\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{)}}, {\it{board\/}})\index{winp1} \\ 
$\vee$$\;\;\;\;${\rm{winp1}}\,({\it{x\/}}, {\tt{'}}{\tt{(}}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{ }}{\tt{5}}{\tt{)}}, {\it{board\/}})\index{winp1} \\ 
$\vee$$\;\;\;\;${\rm{winp1}}\,({\it{x\/}}, {\tt{'}}{\tt{(}}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{)}}, {\it{board\/}})\index{winp1} \\ 
$\vee$$\;\;\;\;${\rm{winp1}}\,({\it{x\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{3}}{\tt{ }}{\tt{6}}{\tt{)}}, {\it{board\/}})\index{winp1} \\ 
$\vee$$\;\;\;\;${\rm{winp1}}\,({\it{x\/}}, {\tt{'}}{\tt{(}}{\tt{1}}{\tt{ }}{\tt{4}}{\tt{ }}{\tt{7}}{\tt{)}}, {\it{board\/}})\index{winp1} \\ 
$\vee$$\;\;\;\;${\rm{winp1}}\,({\it{x\/}}, {\tt{'}}{\tt{(}}{\tt{2}}{\tt{ }}{\tt{5}}{\tt{ }}{\tt{8}}{\tt{)}}, {\it{board\/}})\index{winp1} \\ 
$\vee$$\;\;\;\;${\rm{winp1}}\,({\it{x\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{4}}{\tt{ }}{\tt{8}}{\tt{)}}, {\it{board\/}})\index{winp1} \\ 
$\vee$$\;\;\;\;${\rm{winp1}}\,({\it{x\/}}, {\tt{'}}{\tt{(}}{\tt{2}}{\tt{ }}{\tt{4}}{\tt{ }}{\tt{6}}{\tt{)}}, {\it{board\/}})\index{winp1})\-\-
\end{tabbing}


\noindent{\sc Event}:   Disable winp.

\addvspace{18pt}

\begin{tabbing}{\sc Definition}:\index{other-player} \\  
\=\+{\rm{other-player}}\,({\it{player\/}})\index{other-player} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{player\/}} $=$ {\tt{'}}{\tt{x}}$\;\;${\bf then }{\tt{'}}{\tt{o}} \\ 
{\bf else }{\tt{'}}{\tt{x}}$\;${\bf  endif}\-\-
\end{tabbing}

\begin{tabbing}{\sc Definition}:\index{end} \\  
\=\+{\rm{end}}\,({\it{outcome\/}}, {\it{player\/}}, {\it{lst\/}})\index{end} \\ 
$=$$\;\;\;\;$\=\+{\bf let }\=\+{\it{outcome\/}}{\bf $\;$ be$\;\;$}\=\+{\bf if }{\it{outcome\/}} $=$ {\tt{'}}{\tt{win}} \\ 
{\bf then }\=\+{\bf if }{\it{player\/}} $=$ {\tt{'}}{\tt{o}}$\;\;${\bf then }{\tt{'}}{\tt{o-win}} \\ 
{\bf else }{\tt{'}}{\tt{x-win}}$\;${\bf  endif}\- \\ 
{\bf else }{\tt{'}}{\tt{draw}}$\;${\bf  endif}\-\- \\ 
{\bf in} \\ 
\=\+{\bf if }{\it{lst\/}} $=$ {\rm{list}}\,({\it{outcome\/}})$\;\;${\bf then }{\it{outcome\/}} \\ 
{\bf else }{\bf{f}}$\;${\bf  endif}\-$\;${\bf  endlet}\-\-
\end{tabbing}

\begin{tabbing}{\sc Definition}:\index{tic-tac-toep1} \\  
\=\+{\rm{tic-tac-toep1}}\,({\it{board\/}}, {\it{player\/}}, {\it{lst\/}})\index{tic-tac-toep1} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{lst\/}} $\simeq\;${{\bf{nil}}}${}$$\;\;${\bf then }{\rm{end}}\,({\tt{'}}{\tt{win}}, {\rm{other-player}}\,({\it{player\/}})\index{other-player}, {\it{lst\/}})\index{end} \\ 
{\bf elseif }$\neg$ {\rm{boardp}}\,({\rm{car}}\,({\it{lst\/}}))\index{boardp}$\;\;${\bf then }{\bf{f}} \\ 
{\bf elseif }$\neg$ {\rm{legal-movep}}\,({\it{board\/}}, {\it{player\/}}, {\rm{car}}\,({\it{lst\/}}))\index{legal-movep} \\ 
{\bf then }{\rm{end}}\,({\tt{'}}{\tt{win}}, {\rm{other-player}}\,({\it{player\/}})\index{other-player}, {\rm{cdr}}\,({\it{lst\/}}))\index{end} \\ 
{\bf elseif }{\rm{winp}}\,({\it{player\/}}, {\rm{car}}\,({\it{lst\/}}))\index{winp}$\;\;${\bf then }{\rm{end}}\,({\tt{'}}{\tt{win}}, {\it{player\/}}, {\rm{cdr}}\,({\it{lst\/}}))\index{end} \\ 
{\bf elseif }{\rm{endp}}\,({\rm{car}}\,({\it{lst\/}}))\index{endp}$\;\;${\bf then }{\rm{end}}\,({\tt{'}}{\tt{draw}}, {\it{player\/}}, {\rm{cdr}}\,({\it{lst\/}}))\index{end} \\ 
{\bf else }{\rm{tic-tac-toep1}}\,({\rm{car}}\,({\it{lst\/}}), {\rm{other-player}}\,({\it{player\/}})\index{other-player}, {\rm{cdr}}\,({\it{lst\/}}))\index{tic-tac-toep1}$\;${\bf  endif}\-\-
\end{tabbing}

\begin{tabbing}{\sc Definition}:\index{tic-tac-toep} \\  
\=\+{\rm{tic-tac-toep}}\,({\it{lst\/}})\index{tic-tac-toep} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{listp}}\,({\it{lst\/}}) $\wedge$ ({\rm{car}}\,({\it{lst\/}}) $=$ {\rm{list}}\,({\bf{f}}, {\bf{f}}, {\bf{f}}, {\bf{f}}, {\bf{f}}, {\bf{f}}, {\bf{f}}, {\bf{f}}, {\bf{f}})) \\ 
{\bf then }{\rm{tic-tac-toep1}}\,({\rm{car}}\,({\it{lst\/}}), {\tt{'}}{\tt{x}}, {\rm{cdr}}\,({\it{lst\/}}))\index{tic-tac-toep1} \\ 
{\bf else }{\bf{f}}$\;${\bf  endif}\-\-
\end{tabbing}


\begin{verbatim}; We now develop a function, tic-tac-toe, that plays a game of
; tic-tac-toe against a given list of opponent moves.  We prove
; two things about it:  that it plays tic-tac-toe and that it
; never loses.  Something else we could prove about it is that
; it plays "fair", i.e., it respects the user's moves and it
; doesn't cheat by looking ahead.  For example, we could arrange
; for the program to win by ignoring the user's request to put his
; 'x in a winning spot!  Or we could look ahead to his second move
; and put our first 'o on that spot, guaranteeing that the poor
; user always makes an illegal second move!  We'll return to these
; issues after exhibiting the definition.

\end{verbatim}\begin{tabbing}{\sc Definition}:\index{move} \\  
\=\+{\rm{move}}\,({\it{x\/}}, {\it{i\/}}, {\it{board\/}})\index{move} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{board\/}} $\simeq\;${{\bf{nil}}}${}$$\;\;${\bf then }{\it{board\/}} \\ 
{\bf elseif }{\it{i\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\rm{cons}}\,({\it{x\/}}, {\rm{cdr}}\,({\it{board\/}})) \\ 
{\bf else }{\rm{cons}}\,({\rm{car}}\,({\it{board\/}}), {\rm{move}}\,({\it{x\/}}, {\it{i\/}} $-\;1$, {\rm{cdr}}\,({\it{board\/}}))\index{move})$\;${\bf  endif}\-\-
\end{tabbing}

\begin{tabbing}{\sc Definition}:\index{legalp} \\  
{\rm{legalp}}\,({\it{i\/}}, {\it{board\/}})\index{legalp} $=$ (({\it{i\/}} $<$ {\tt{'}}{\tt{9}}) $\wedge$ ({\rm{nth}}\,({\it{i\/}}, {\it{board\/}})\index{nth} $=$ {\bf{f}}))
\end{tabbing}

\begin{tabbing}{\sc Definition}:\index{open-squares} \\  
\=\+{\rm{open-squares}}\,({\it{board\/}})\index{open-squares} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{board\/}} $\simeq\;${{\bf{nil}}}${}$$\;\;${\bf then }{\tt{0}} \\ 
{\bf elseif }{\rm{car}}\,({\it{board\/}})$\;\;${\bf then }{\rm{open-squares}}\,({\rm{cdr}}\,({\it{board\/}}))\index{open-squares} \\ 
{\bf else }$1\;+$ {\rm{open-squares}}\,({\rm{cdr}}\,({\it{board\/}}))\index{open-squares}$\;${\bf  endif}\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  legal-moves-reduce-open-squares \\ \index{legal-moves-reduce-open-squares}
\=\+({\it{x\/}} $\wedge$ {\rm{legalp}}\,({\it{i\/}}, {\it{board\/}})\index{legalp}) \\ 
$\rightarrow$$\;\;\;\;$({\rm{open-squares}}\,({\rm{move}}\,({\it{x\/}}, {\it{i\/}}, {\it{board\/}})\index{move})\index{open-squares} $<$ {\rm{open-squares}}\,({\it{board\/}})\index{open-squares})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  move-never-increases-open-squares \\ \index{move-never-increases-open-square!s}
{\it{x\/}} $\rightarrow$ ({\rm{open-squares}}\,({\it{board\/}})\index{open-squares} $\not<$ {\rm{open-squares}}\,({\rm{move}}\,({\it{x\/}}, {\it{i\/}}, {\it{board\/}})\index{move})\index{open-squares})
\end{tabbing}

\begin{verbatim}; A carefully crafted aspect of pick-move, defined below, is that
; it *always* returns a number less than 9.  This required a
; subtle coding of pick-move1, where we return 0 when we fail to
; find an open square.  The reason we want pick-move always to
; return a number less than 9 has nothing to do with its
; correctness vis-a-vis tic-tac-toep.  Rather, it is concerned
; with its easy implementation.  If pick-move sometimes returns
; 9 (say), then the implementation may violate certain bounds
; (on say the size of numerically represented boards when a 
; moven is carried out on 9).  The only way we can prevent it
; is to carry along the invariant that the board is not at
; and endp.  But we don't want endp in the implementation -- it
; suffices just to count the moves.  Since we just count moves, the
; legality of pick-move's answer depends on the invariant that
; the number of moves is related in a certain way to the number of
; open squares and hence to endp.  Rather than maintain this invariant,
; I am just defining pick-move1 to return 0 when it is tempted to
; return 9.

\end{verbatim}\begin{tabbing}{\sc Definition}:\index{pick-move1} \\  
\=\+{\rm{pick-move1}}\,({\it{board\/}}, {\it{i\/}})\index{pick-move1} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{board\/}} $\simeq\;${{\bf{nil}}}${}$$\;\;${\bf then }{\tt{0}} \\ 
{\bf elseif }{\rm{car}}\,({\it{board\/}}) $=$ {\bf{f}}$\;\;${\bf then }{\it{i\/}} \\ 
{\bf else }{\rm{pick-move1}}\,({\rm{cdr}}\,({\it{board\/}}), $1\;+$ {\it{i\/}})\index{pick-move1}$\;${\bf  endif}\-\-
\end{tabbing}

\begin{tabbing}{\sc Definition}:\index{pick-move} \\  
\=\+{\rm{pick-move}}\,({\it{board\/}})\index{pick-move} \\ 
$=$$\;\;\;\;$\=\+{\bf let }\=\+{\it{book-move\/}}{\bf $\;$ be$\;\;$}{\rm{assoc}}\,(\=\+{\it{board\/}}, \\ 
{\tt{'}}{\tt{(}}\=\+{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{2}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{2}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}} \\ 
{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{3}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}} \\ 
{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{3}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{5}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{2}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{2}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}} \\ 
{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{6}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{1}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{1}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{1}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}} \\ 
{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{5}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{7}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}} \\ 
{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{6}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}} \\ 
{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{1}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}} \\ 
{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{7}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{6}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{6}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{6}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{2}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{2}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}} \\ 
{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{2}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{5}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{2}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{2}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{6}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}} \\ 
{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{3}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}} \\ 
{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{3}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}} \\ 
{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{6}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{7}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{o}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{3}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{o}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{3}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{5}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{6}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{7}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{6}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{6}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{6}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{2}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}} \\ 
{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{5}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{6}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{7}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{7}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{7}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{7}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{4}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{2}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{2}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}} \\ 
{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{3}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}} \\ 
{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{3}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{5}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{2}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{2}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}} \\ 
{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{6}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}} \\ 
{\tt{o}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{5}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}} \\ 
{\tt{o}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{5}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{3}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}} \\ 
{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{8}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}} \\ 
{\tt{o}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{3}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}} \\ 
{\tt{o}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{6}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}} \\ 
{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{8}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{2}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}} \\ 
{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{2}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}} \\ 
{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{5}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{8}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{8}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{2}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{2}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{6}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}} \\ 
{\tt{o}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{6}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{8}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{6}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}} \\ 
{\tt{o}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{5}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}} \\ 
{\tt{o}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{5}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{8}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{6}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{6}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{6}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{2}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{2}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}} \\ 
{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{2}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{2}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}} \\ 
{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{5}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{8}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{6}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{4}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{1}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{1}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{1}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}} \\ 
{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{5}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{7}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}} \\ 
{\tt{o}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{5}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}} \\ 
{\tt{o}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{5}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{3}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}} \\ 
{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{8}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}} \\ 
{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{3}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{8}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{7}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{8}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{8}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{8}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{8}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{7}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{8}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{7}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{7}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{7}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{1}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}} \\ 
{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{5}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{7}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{8}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{8}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{8}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{8}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{o}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{5}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{o}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{5}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{7}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{8}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{5}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{5}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{3}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{4}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}} \\ 
{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{6}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}} \\ 
{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{1}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}} \\ 
{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{7}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{6}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{6}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{6}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{2}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}} \\ 
{\tt{o}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{3}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}} \\ 
{\tt{o}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{6}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}} \\ 
{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{8}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}} \\ 
{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{3}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{8}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{2}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{8}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{8}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{8}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{8}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{o}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{7}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{o}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{7}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}} \\ 
{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{8}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{6}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{6}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{7}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{8}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{8}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{6}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{6}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}} \\ 
{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{2}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{6}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{6}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{7}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{8}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{3}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{6}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{6}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}} \\ 
{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{2}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{4}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{7}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{2}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{6}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{7}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{2}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{6}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{6}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{2}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{6}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{6}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}} \\ 
{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{3}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{8}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{2}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{6}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{7}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{5}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{3}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{3}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{3}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{5}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{6}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}} \\ 
{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{6}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{3}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{5}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{8}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{8}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{6}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{7}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{2}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}} \\ 
{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{2}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{5}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{2}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{2}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{6}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{2}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}} \\ 
{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{2}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}} \\ 
{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{5}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{8}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{8}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{2}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{2}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{6}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{7}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{8}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{8}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{8}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{8}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{2}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{8}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{8}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{8}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{8}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}} \\ 
{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{6}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{7}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{5}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{8}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{8}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{8}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{8}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}} \\ 
{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{6}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{7}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{8}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{8}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{8}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{8}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{2}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{2}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{2}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{7}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{6}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{4}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}} \\ 
{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{3}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}} \\ 
{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{3}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}} \\ 
{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{6}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{7}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{o}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{3}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{o}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{3}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{5}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}} \\ 
{\tt{o}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{6}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{8}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{7}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{8}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{7}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{7}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{7}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{1}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{o}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{7}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{o}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{7}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}} \\ 
{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{8}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}} \\ 
{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{6}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{7}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{5}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{8}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{8}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{8}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{8}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{5}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{8}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{8}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{8}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{8}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{7}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{8}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{7}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{7}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{7}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{4}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{6}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{7}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{6}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{6}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{6}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{2}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{6}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}} \\ 
{\tt{o}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{5}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}} \\ 
{\tt{o}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{5}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{8}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{6}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{6}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{6}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{2}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}} \\ 
{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{5}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{7}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{8}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{8}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{8}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{8}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{6}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{6}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{7}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{8}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{8}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{6}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{6}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}} \\ 
{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{2}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}} \\ 
{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{6}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{7}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{8}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{8}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{8}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{8}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{5}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{8}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{8}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{8}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{8}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{6}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{6}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{6}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{6}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{4}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}} \\ 
{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{5}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{6}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{7}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{7}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{7}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{7}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{2}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}} \\ 
{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{2}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{2}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}} \\ 
{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{5}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{8}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{6}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{o}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{5}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{o}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{5}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{7}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{8}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{5}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{5}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{3}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{6}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{6}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{7}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{8}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{3}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{6}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{6}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}} \\ 
{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{2}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{2}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{2}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{2}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{7}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{6}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{7}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{8}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{7}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{7}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{7}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{6}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{6}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{6}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{o}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{6}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{(}}\=\+{\tt{x}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{ }}{\tt{*1*false}} \\ 
{\tt{*1*false}}{\tt{ }}{\tt{*1*false}}{\tt{)}}\- \\ 
{\tt{.\ }}{\tt{4}}{\tt{)}}\-{\tt{)}}\-)\-\- \\ 
{\bf in} \\ 
\=\+{\bf if }{\it{book-move\/}}$\;\;${\bf then }{\rm{cdr}}\,({\it{book-move\/}}) \\ 
{\bf else }{\rm{pick-move1}}\,({\it{board\/}}, {\tt{0}})\index{pick-move1}$\;${\bf  endif}\-$\;${\bf  endlet}\-\-
\end{tabbing}


\noindent{\sc Event}:   Disable pick-move.

\addvspace{18pt}

\begin{verbatim}; Note that in the following automation of the game, we do not even
; check that the user might have won nor do we check that our moves
; are legal.  We know these checks are redundant.  It is worth noting
; that our formal spec of the game will force us to prove this
; redundancy.  Suppose the user did win but we then replied with another
; move.  Then the game would be ill-formed.  Similarly, if our move
; is illegal, either because our 'o replaces an 'x or because our 'o
; goes into a nonexistent square, the resulting board will not pass
; legal-movep.

\end{verbatim}\begin{tabbing}{\sc Definition}:\index{tic-tac-toe1} \\  
\=\+{\rm{tic-tac-toe1}}\,({\it{xmoves\/}}, {\it{board\/}})\index{tic-tac-toe1} \\ 
$=$$\;\;\;\;$\=\+{\bf let }\=\+{\it{board1\/}}{\bf $\;$ be$\;\;$}{\rm{move}}\,({\tt{'}}{\tt{x}}, {\rm{car}}\,({\it{xmoves\/}}), {\it{board\/}})\index{move}\- \\ 
{\bf in} \\ 
\=\+{\bf if }{\rm{legalp}}\,({\rm{car}}\,({\it{xmoves\/}}), {\it{board\/}})\index{legalp} \\ 
{\bf then }\=\+{\bf if }{\rm{endp}}\,({\it{board1\/}})\index{endp}$\;\;${\bf then }{\rm{list}}\,({\it{board1\/}}, {\tt{'}}{\tt{draw}}) \\ 
{\bf else }\=\+{\bf let }\=\+{\it{board2\/}}{\bf $\;$ be$\;\;$}{\rm{move}}\,(\=\+{\tt{'}}{\tt{o}}, \\ 
{\rm{pick-move}}\,({\it{board1\/}})\index{pick-move}, \\ 
{\it{board1\/}})\-\index{move}\- \\ 
{\bf in} \\ 
\=\+{\bf if }{\rm{winp}}\,({\tt{'}}{\tt{o}}, {\it{board2\/}})\index{winp} \\ 
{\bf then }{\rm{list}}\,({\it{board1\/}}, {\it{board2\/}}, {\tt{'}}{\tt{o-win}}) \\ 
{\bf else }{\rm{cons}}\,(\=\+{\it{board1\/}}, \\ 
{\rm{cons}}\,(\=\+{\it{board2\/}}, \\ 
{\rm{tic-tac-toe1}}\,(\=\+{\rm{cdr}}\,({\it{xmoves\/}}), \\ 
{\it{board2\/}})\-\index{tic-tac-toe1})\-)\-$\;${\bf  endif}\-$\;${\bf  endlet}\-$\;${\bf  endif}\- \\ 
{\bf else }{\rm{list}}\,({\it{board1\/}}, {\tt{'}}{\tt{o-win}})$\;${\bf  endif}\-$\;${\bf  endlet}\-\-
\end{tabbing}

\begin{tabbing}{\sc Definition}:\index{tic-tac-toe} \\  
\=\+{\rm{tic-tac-toe}}\,({\it{xmoves\/}})\index{tic-tac-toe} \\ 
$=$$\;\;\;\;$\=\+{\bf let }\=\+{\it{board0\/}}{\bf $\;$ be$\;\;$}{\rm{list}}\,({\bf{f}}, {\bf{f}}, {\bf{f}}, {\bf{f}}, {\bf{f}}, {\bf{f}}, {\bf{f}}, {\bf{f}}, {\bf{f}})\- \\ 
{\bf in} \\ 
{\rm{cons}}\,({\it{board0\/}}, {\rm{tic-tac-toe1}}\,({\it{xmoves\/}}, {\it{board0\/}})\index{tic-tac-toe1})$\;${\bf  endlet}\-\-
\end{tabbing}

\begin{verbatim}; We now set about trying to prove that tic-tac-toe plays the game.
; We start with some fundamentals that I know will be needed even
; though I haven't seen their explicit use.

\end{verbatim}\begin{tabbing}{\sc Definition}:\index{length} \\  
\=\+{\rm{length}}\,({\it{x\/}})\index{length} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{x\/}} $\simeq\;${{\bf{nil}}}${}$$\;\;${\bf then }{\tt{0}} \\ 
{\bf else }$1\;+$ {\rm{length}}\,({\rm{cdr}}\,({\it{x\/}}))\index{length}$\;${\bf  endif}\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  length-move \\ \index{length-move}
{\rm{length}}\,({\rm{move}}\,({\it{x\/}}, {\it{i\/}}, {\it{board\/}})\index{move})\index{length} $=$ {\rm{length}}\,({\it{board\/}})\index{length}
\end{tabbing}

\begin{tabbing}{\sc Definition}:\index{legal-bookp} \\  
\=\+{\rm{legal-bookp}}\,({\it{alist\/}})\index{legal-bookp} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{alist\/}} $\simeq\;${{\bf{nil}}}${}$$\;\;${\bf then }{\bf{t}} \\ 
{\bf else }\=\+{\rm{legalp}}\,({\rm{cdar}}\,({\it{alist\/}}), {\rm{caar}}\,({\it{alist\/}}))\index{legalp} \\ 
$\wedge$$\;\;\;\;${\rm{legal-bookp}}\,({\rm{cdr}}\,({\it{alist\/}}))\index{legal-bookp}\-$\;${\bf  endif}\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  legal-bookp-implies-legalp-cdr-assoc \\ \index{legal-bookp-implies-legalp-cdr-!assoc}
\=\+({\rm{legal-bookp}}\,({\it{alist\/}})\index{legal-bookp} $\wedge$ {\rm{assoc}}\,({\it{board\/}}, {\it{alist\/}})) \\ 
$\rightarrow$$\;\;\;\;${\rm{legalp}}\,({\rm{cdr}}\,({\rm{assoc}}\,({\it{board\/}}, {\it{alist\/}})), {\it{board\/}})\index{legalp}\-
\end{tabbing}

\begin{tabbing}{\sc Definition}:\index{pick-move2} \\  
\=\+{\rm{pick-move2}}\,({\it{board\/}})\index{pick-move2} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{board\/}} $\simeq\;${{\bf{nil}}}${}$$\;\;${\bf then }{\tt{0}} \\ 
{\bf elseif }{\rm{car}}\,({\it{board\/}}) $=$ {\bf{f}}$\;\;${\bf then }{\tt{0}} \\ 
{\bf else }$1\;+$ {\rm{pick-move2}}\,({\rm{cdr}}\,({\it{board\/}}))\index{pick-move2}$\;${\bf  endif}\-\-
\end{tabbing}


\begin{verbatim}; The following theorem probably holds a record for successfully
; completed subgoals:

;     That finishes the proof of *1.1.1.1.1.1.1.1.1.1, which, in turn, finishes 
;the proof of *1.1.1.1.1.1.1.1.1, which finishes the proof of *1.1.1.1.1.1.1.1, 
;which finishes the proof of *1.1.1.1.1.1.1, which, consequently, finishes the 
;proof of *1.1.1.1.1.1, which finishes the proof of *1.1.1.1.1, which finishes 
;the proof of *1.1.1.1, which finishes the proof of *1.1.1, which, in turn, 
;finishes the proof of *1.1, which finishes the proof of *1.  Q.E.D. 

;[ 0.0 6.5 3.4 ]   (On a Sparc 2)

; This would be even more impressive if boards had length 1000!

\end{verbatim}\begin{tabbing}{\sc Theorem}:  not-endp-implies-legalp-pick-move2 \\ \index{not-endp-implies-legalp-pick-move!2}
\=\+(($\neg$ {\rm{endp}}\,({\it{board\/}})\index{endp}) $\wedge$ ({\rm{length}}\,({\it{board\/}})\index{length} $<$ {\tt{10}})) \\ 
$\rightarrow$$\;\;\;\;${\rm{legalp}}\,({\rm{pick-move2}}\,({\it{board\/}})\index{pick-move2}, {\it{board\/}})\index{legalp}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  pick-move1-is-pick-move2 \\ \index{pick-move1-is-pick-move2}
\=\+(($\neg$ {\rm{endp}}\,({\it{board\/}})\index{endp}) $\wedge$ ({\it{i\/}} $\in\;${\bf{N}}${}$)) \\ 
$\rightarrow$$\;\;\;\;$({\rm{pick-move1}}\,({\it{board\/}}, {\it{i\/}})\index{pick-move1} $=$ ({\it{i\/}} $+$ {\rm{pick-move2}}\,({\it{board\/}})\index{pick-move2}))\-
\end{tabbing}

\begin{verbatim}; If you don't do the following we get stack overflow while trying to confirm
; that the alist built into pick-moves is a legal-bookp.

\end{verbatim}
\noindent{\sc Event}:   For efficiency, compile those definitions not yet compiled.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  legalp-pick-move \\ \index{legalp-pick-move}
\=\+(($\neg$ {\rm{endp}}\,({\it{board\/}})\index{endp}) $\wedge$ ({\rm{length}}\,({\it{board\/}})\index{length} $<$ {\tt{10}})) \\ 
$\rightarrow$$\;\;\;\;${\rm{legalp}}\,({\rm{pick-move}}\,({\it{board\/}})\index{pick-move}, {\it{board\/}})\index{legalp}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  boardp-move \\ \index{boardp-move}
\=\+({\rm{boardp}}\,({\it{board\/}})\index{boardp} $\wedge$ (({\it{player\/}} $=$ {\tt{'}}{\tt{x}}) $\vee$ ({\it{player\/}} $=$ {\tt{'}}{\tt{o}}))) \\ 
$\rightarrow$$\;\;\;\;${\rm{boardp}}\,({\rm{move}}\,({\it{player\/}}, {\it{i\/}}, {\it{board\/}})\index{move})\index{boardp}\-
\end{tabbing}

\begin{verbatim}; We now establish that pick-move indeed has our "carefully crafted
; aspect" of less than 9, when (endp board)!

\end{verbatim}\begin{tabbing}{\sc Theorem}:  legalp-implies-lessp-9 \\ \index{legalp-implies-lessp-9}
\=\+({\rm{legalp}}\,({\rm{cdr}}\,({\rm{assoc}}\,({\it{board\/}}, {\it{alist\/}})), {\it{board\/}})\index{legalp} $\wedge$ ({\rm{length}}\,({\it{board\/}})\index{length} $=$ {\tt{9}})) \\ 
$\rightarrow$$\;\;\;\;$({\rm{cdr}}\,({\rm{assoc}}\,({\it{board\/}}, {\it{alist\/}})) $<$ {\tt{9}})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  endp-implies-pick-move1-0 \\ \index{endp-implies-pick-move1-0}
{\rm{endp}}\,({\it{board\/}})\index{endp} $\rightarrow$ ({\rm{pick-move1}}\,({\it{board\/}}, {\it{i\/}})\index{pick-move1} $=$ {\tt{0}})
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  lessp-pick-move \\ \index{lessp-pick-move}
({\rm{length}}\,({\it{board\/}})\index{length} $=$ {\tt{9}}) $\rightarrow$ ({\rm{pick-move}}\,({\it{board\/}})\index{pick-move} $<$ {\tt{9}})
\end{tabbing}

\begin{verbatim}; The strategy of our proof that tic-tac-toe never loses is to arrange
; for tic-tac-toe to expand so as to play all possible games!  The first
; step is the lemma below, which forces the function open provided the board
; is given as an explicit list of 9 elements.  The lemma below is just
; the equation of (tic-tac-toe xmoves board) with its body, for board
; replaced by an explicit list.

\end{verbatim}\begin{tabbing}{\sc Theorem}:  tic-tac-toe1-opener \\ \index{tic-tac-toe1-opener}
\=\+{\rm{tic-tac-toe1}}\,({\it{xmoves\/}}, {\rm{list}}\,({\it{s0\/}}, {\it{s1\/}}, {\it{s2\/}}, {\it{s3\/}}, {\it{s4\/}}, {\it{s5\/}}, {\it{s6\/}}, {\it{s7\/}}, {\it{s8\/}}))\index{tic-tac-toe1} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{legalp}}\,({\rm{car}}\,({\it{xmoves\/}}), {\rm{list}}\,({\it{s0\/}}, {\it{s1\/}}, {\it{s2\/}}, {\it{s3\/}}, {\it{s4\/}}, {\it{s5\/}}, {\it{s6\/}}, {\it{s7\/}}, {\it{s8\/}}))\index{legalp} \\ 
{\bf then }\=\+{\bf if }{\rm{endp}}\,({\rm{move}}\,({\tt{'}}{\tt{x}}, {\rm{car}}\,({\it{xmoves\/}}), {\rm{list}}\,({\it{s0\/}}, {\it{s1\/}}, {\it{s2\/}}, {\it{s3\/}}, {\it{s4\/}}, {\it{s5\/}}, {\it{s6\/}}, {\it{s7\/}}, {\it{s8\/}}))\index{move})\index{endp} \\ 
{\bf then }{\rm{list}}\,(\=\+{\rm{move}}\,({\tt{'}}{\tt{x}}, {\rm{car}}\,({\it{xmoves\/}}), {\rm{list}}\,({\it{s0\/}}, {\it{s1\/}}, {\it{s2\/}}, {\it{s3\/}}, {\it{s4\/}}, {\it{s5\/}}, {\it{s6\/}}, {\it{s7\/}}, {\it{s8\/}}))\index{move}, \\ 
{\tt{'}}{\tt{draw}})\- \\ 
{\bf elseif }{\rm{winp}}\,(\=\+{\tt{'}}{\tt{o}}, \\ 
{\rm{move}}\,(\=\+{\tt{'}}{\tt{o}}, \\ 
{\rm{pick-move}}\,({\rm{move}}\,(\=\+{\tt{'}}{\tt{x}}, \\ 
{\rm{car}}\,({\it{xmoves\/}}), \\ 
{\rm{list}}\,({\it{s0\/}}, {\it{s1\/}}, {\it{s2\/}}, {\it{s3\/}}, {\it{s4\/}}, {\it{s5\/}}, {\it{s6\/}}, {\it{s7\/}}, {\it{s8\/}}))\-\index{move})\index{pick-move}, \\ 
{\rm{move}}\,(\=\+{\tt{'}}{\tt{x}}, \\ 
{\rm{car}}\,({\it{xmoves\/}}), \\ 
{\rm{list}}\,({\it{s0\/}}, {\it{s1\/}}, {\it{s2\/}}, {\it{s3\/}}, {\it{s4\/}}, {\it{s5\/}}, {\it{s6\/}}, {\it{s7\/}}, {\it{s8\/}}))\-\index{move})\-\index{move})\-\index{winp} \\ 
{\bf then }{\rm{list}}\,(\=\+{\rm{move}}\,({\tt{'}}{\tt{x}}, {\rm{car}}\,({\it{xmoves\/}}), {\rm{list}}\,({\it{s0\/}}, {\it{s1\/}}, {\it{s2\/}}, {\it{s3\/}}, {\it{s4\/}}, {\it{s5\/}}, {\it{s6\/}}, {\it{s7\/}}, {\it{s8\/}}))\index{move}, \\ 
{\rm{move}}\,(\=\+{\tt{'}}{\tt{o}}, \\ 
{\rm{pick-move}}\,({\rm{move}}\,(\=\+{\tt{'}}{\tt{x}}, \\ 
{\rm{car}}\,({\it{xmoves\/}}), \\ 
{\rm{list}}\,({\it{s0\/}}, {\it{s1\/}}, {\it{s2\/}}, {\it{s3\/}}, {\it{s4\/}}, {\it{s5\/}}, {\it{s6\/}}, {\it{s7\/}}, {\it{s8\/}}))\-\index{move})\index{pick-move}, \\ 
{\rm{move}}\,(\=\+{\tt{'}}{\tt{x}}, \\ 
{\rm{car}}\,({\it{xmoves\/}}), \\ 
{\rm{list}}\,({\it{s0\/}}, {\it{s1\/}}, {\it{s2\/}}, {\it{s3\/}}, {\it{s4\/}}, {\it{s5\/}}, {\it{s6\/}}, {\it{s7\/}}, {\it{s8\/}}))\-\index{move})\-\index{move}, \\ 
{\tt{'}}{\tt{o-win}})\- \\ 
{\bf else }{\rm{cons}}\,(\=\+{\rm{move}}\,(\=\+{\tt{'}}{\tt{x}}, \\ 
{\rm{car}}\,({\it{xmoves\/}}), \\ 
{\rm{list}}\,({\it{s0\/}}, {\it{s1\/}}, {\it{s2\/}}, {\it{s3\/}}, {\it{s4\/}}, {\it{s5\/}}, {\it{s6\/}}, {\it{s7\/}}, {\it{s8\/}}))\-\index{move}, \\ 
{\rm{cons}}\,(\=\+{\rm{move}}\,(\=\+{\tt{'}}{\tt{o}}, \\ 
{\rm{pick-move}}\,({\rm{move}}\,(\=\+{\tt{'}}{\tt{x}}, \\ 
{\rm{car}}\,({\it{xmoves\/}}), \\ 
{\rm{list}}\,(\=\+{\it{s0\/}}, \\ 
{\it{s1\/}}, \\ 
{\it{s2\/}}, \\ 
{\it{s3\/}}, \\ 
{\it{s4\/}}, \\ 
{\it{s5\/}}, \\ 
{\it{s6\/}}, \\ 
{\it{s7\/}}, \\ 
{\it{s8\/}})\-)\-\index{move})\index{pick-move}, \\ 
{\rm{move}}\,(\=\+{\tt{'}}{\tt{x}}, \\ 
{\rm{car}}\,({\it{xmoves\/}}), \\ 
{\rm{list}}\,({\it{s0\/}}, {\it{s1\/}}, {\it{s2\/}}, {\it{s3\/}}, {\it{s4\/}}, {\it{s5\/}}, {\it{s6\/}}, {\it{s7\/}}, {\it{s8\/}}))\-\index{move})\-\index{move}, \\ 
{\rm{tic-tac-toe1}}\,(\=\+{\rm{cdr}}\,({\it{xmoves\/}}), \\ 
{\rm{move}}\,(\=\+{\tt{'}}{\tt{o}}, \\ 
{\rm{pick-move}}\,({\rm{move}}\,(\=\+{\tt{'}}{\tt{x}}, \\ 
{\rm{car}}\,({\it{xmoves\/}}), \\ 
{\rm{list}}\,(\=\+{\it{s0\/}}, \\ 
{\it{s1\/}}, \\ 
{\it{s2\/}}, \\ 
{\it{s3\/}}, \\ 
{\it{s4\/}}, \\ 
{\it{s5\/}}, \\ 
{\it{s6\/}}, \\ 
{\it{s7\/}}, \\ 
{\it{s8\/}})\-)\-\index{move})\index{pick-move}, \\ 
{\rm{move}}\,(\=\+{\tt{'}}{\tt{x}}, \\ 
{\rm{car}}\,({\it{xmoves\/}}), \\ 
{\rm{list}}\,(\=\+{\it{s0\/}}, \\ 
{\it{s1\/}}, \\ 
{\it{s2\/}}, \\ 
{\it{s3\/}}, \\ 
{\it{s4\/}}, \\ 
{\it{s5\/}}, \\ 
{\it{s6\/}}, \\ 
{\it{s7\/}}, \\ 
{\it{s8\/}})\-)\-\index{move})\-\index{move})\-\index{tic-tac-toe1})\-)\-$\;${\bf  endif}\- \\ 
{\bf else }{\rm{list}}\,(\=\+{\rm{move}}\,({\tt{'}}{\tt{x}}, {\rm{car}}\,({\it{xmoves\/}}), {\rm{list}}\,({\it{s0\/}}, {\it{s1\/}}, {\it{s2\/}}, {\it{s3\/}}, {\it{s4\/}}, {\it{s5\/}}, {\it{s6\/}}, {\it{s7\/}}, {\it{s8\/}}))\index{move}, \\ 
{\tt{'}}{\tt{o-win}})\-$\;${\bf  endif}\-\-
\end{tabbing}


\noindent{\sc Event}:   Disable tic-tac-toe1.

\addvspace{18pt}

\begin{verbatim}; With that lemma in place, the function will expand repeatedly, as soon as
; board2 is reduced to normal form.

; The next problem is to force the first legalp expression above to cause us
; to case split on all the legal moves on the given board.  Our first attempt
; at this proof just let legalp expand and we considered all 9 possible moves.
; But this caused a tremendous explosion.

; The sequence of lemmas below culminate in the following wondrous
; effect.  (legalp x (list s0 s1 ... s8)) expands to (or ...(equal x
; i)...)  for just those i such that si is f.  To achieve this we
; transform legalp in two steps.  First we show that it is equal to
; legalp1, below, which does not use NTH and instead decrements x and
; cdrs board.  Then we transform legalp1 into legalp2, which keeps x
; unchanged but cdrs board as it increments an accumulator i.
; Finally, we prove the opener lemmas that let the cdring of board
; drive the production of a bunch of disjuncts against the running i.
; This development was fairly subtle for me.

\end{verbatim}\begin{tabbing}{\sc Definition}:\index{legalp1} \\  
\=\+{\rm{legalp1}}\,({\it{x\/}}, {\it{board\/}})\index{legalp1} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{x\/}} $\simeq {\tt{0}}$$\;\;${\bf then }$\neg$ {\rm{car}}\,({\it{board\/}}) \\ 
{\bf elseif }{\it{board\/}} $\simeq\;${{\bf{nil}}}${}$$\;\;${\bf then }{\bf{f}} \\ 
{\bf else }{\rm{legalp1}}\,({\it{x\/}} $-\;1$, {\rm{cdr}}\,({\it{board\/}}))\index{legalp1}$\;${\bf  endif}\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  legalp1-fact1 \\ \index{legalp1-fact1}
({\it{x\/}} $\not<$ {\rm{length}}\,({\it{board\/}})\index{length}) $\rightarrow$ ($\neg$ {\rm{legalp1}}\,({\it{x\/}}, {\it{board\/}})\index{legalp1})
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  legalp1-fact2 \\ \index{legalp1-fact2}
{\rm{nth}}\,({\it{x\/}}, {\it{board\/}})\index{nth} $\rightarrow$ ($\neg$ {\rm{legalp1}}\,({\it{x\/}}, {\it{board\/}})\index{legalp1})
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  legalp1-fact3 \\ \index{legalp1-fact3}
(({\it{x\/}} $<$ {\rm{length}}\,({\it{board\/}})\index{length}) $\wedge$ ($\neg$ {\rm{nth}}\,({\it{x\/}}, {\it{board\/}})\index{nth})) $\rightarrow$ {\rm{legalp1}}\,({\it{x\/}}, {\it{board\/}})\index{legalp1}
\end{tabbing}

\begin{verbatim}; So here is the first half...

\end{verbatim}\begin{tabbing}{\sc Theorem}:  legalp-is-legalp1 \\ \index{legalp-is-legalp1}
({\rm{length}}\,({\it{board\/}})\index{length} $=$ {\tt{'}}{\tt{9}}) $\rightarrow$ ({\rm{legalp}}\,({\it{x\/}}, {\it{board\/}})\index{legalp} $=$ {\rm{legalp1}}\,({\it{x\/}}, {\it{board\/}})\index{legalp1})
\end{tabbing}

\begin{verbatim}; Now we introduce legalp2.

\end{verbatim}\begin{tabbing}{\sc Definition}:\index{legalp2} \\  
\=\+{\rm{legalp2}}\,({\it{x\/}}, {\it{board\/}}, {\it{i\/}})\index{legalp2} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{board\/}} $\simeq\;${{\bf{nil}}}${}$$\;\;${\bf then }{\bf{f}} \\ 
{\bf elseif }($\neg$ {\rm{car}}\,({\it{board\/}})) $\wedge$ ({\rm{fix}}\,({\it{x\/}}) $=$ {\it{i\/}})$\;\;${\bf then }{\bf{t}} \\ 
{\bf else }{\rm{legalp2}}\,({\it{x\/}}, {\rm{cdr}}\,({\it{board\/}}), $1\;+$ {\it{i\/}})\index{legalp2}$\;${\bf  endif}\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  legalp2-fact1 \\ \index{legalp2-fact1}
({\it{x\/}} $<$ {\it{i\/}}) $\rightarrow$ ($\neg$ {\rm{legalp2}}\,({\it{x\/}}, {\it{board\/}}, {\it{i\/}})\index{legalp2})
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  sub1-difference \\ \index{sub1-difference}
(({\it{x\/}} $-$ {\it{y\/}}) $-\;1$) $=$ (({\it{x\/}} $-\;1$) $-$ {\it{y\/}})
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  legalp1-is-legalp2-gen \\ \index{legalp1-is-legalp2-gen}
(({\it{i\/}} $\in\;${\bf{N}}${}$) $\wedge$ ({\it{x\/}} $\not<$ {\it{i\/}})) $\rightarrow$ ({\rm{legalp1}}\,({\it{x\/}} $-$ {\it{i\/}}, {\it{board\/}})\index{legalp1} $=$ {\rm{legalp2}}\,({\it{x\/}}, {\it{board\/}}, {\it{i\/}})\index{legalp2})
\end{tabbing}


\noindent{\sc Event}:   Disable sub1-difference.

\addvspace{18pt}

\begin{verbatim}; Here is the second half...

\end{verbatim}\begin{tabbing}{\sc Theorem}:  legalp1-is-legalp2 \\ \index{legalp1-is-legalp2}
{\rm{legalp1}}\,({\it{x\/}}, {\it{board\/}})\index{legalp1} $=$ {\rm{legalp2}}\,({\it{x\/}}, {\it{board\/}}, {\tt{'}}{\tt{0}})\index{legalp2}
\end{tabbing}

\begin{verbatim}; Now we'll show how to use semi-explicit boards to drive the production
; of disjuncts.

\end{verbatim}\begin{tabbing}{\sc Theorem}:  legalp2-opener-f \\ \index{legalp2-opener-f}
\=\+({\it{i\/}} $\in\;${\bf{N}}${}$) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{legalp2}}\,({\it{x\/}}, {\rm{cons}}\,({\bf{f}}, {\it{board\/}}), {\it{i\/}})\index{legalp2} \\ 
$=$$\;\;\;\;$(\=\+\=\+{\bf if }{\it{i\/}} $=$ {\tt{0}}$\;\;${\bf then }{\it{x\/}} $\simeq {\tt{0}}$ \\ 
{\bf else }{\it{x\/}} $=$ {\it{i\/}}$\;${\bf  endif}\- \\ 
$\vee$$\;\;\;\;${\rm{legalp2}}\,({\it{x\/}}, {\it{board\/}}, $1\;+$ {\it{i\/}})\index{legalp2})\-)\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  legalp2-opener-x \\ \index{legalp2-opener-x}
{\rm{legalp2}}\,({\it{x\/}}, {\rm{cons}}\,({\tt{'}}{\tt{x}}, {\it{board\/}}), {\it{i\/}})\index{legalp2} $=$ {\rm{legalp2}}\,({\it{x\/}}, {\it{board\/}}, $1\;+$ {\it{i\/}})\index{legalp2}
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  legalp2-opener-o \\ \index{legalp2-opener-o}
{\rm{legalp2}}\,({\it{x\/}}, {\rm{cons}}\,({\tt{'}}{\tt{o}}, {\it{board\/}}), {\it{i\/}})\index{legalp2} $=$ {\rm{legalp2}}\,({\it{x\/}}, {\it{board\/}}, $1\;+$ {\it{i\/}})\index{legalp2}
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  legalp2-opener-nil \\ \index{legalp2-opener-nil}
{\rm{legalp2}}\,({\it{x\/}}, {\tt{'}}{\tt{nil}}, {\it{i\/}})\index{legalp2} $=$ {\bf{f}}
\end{tabbing}


\noindent{\sc Event}:   Disable legalp2.

\addvspace{18pt}

\noindent{\sc Event}:   Disable legalp.

\addvspace{18pt}

\begin{verbatim}; And now we put it all together and cut out the middlemen:

\end{verbatim}\begin{tabbing}{\sc Theorem}:  legalp-simplifier \\ \index{legalp-simplifier}
\=\+{\rm{legalp}}\,({\it{x\/}}, {\rm{list}}\,({\it{s0\/}}, {\it{s1\/}}, {\it{s2\/}}, {\it{s3\/}}, {\it{s4\/}}, {\it{s5\/}}, {\it{s6\/}}, {\it{s7\/}}, {\it{s8\/}}))\index{legalp} \\ 
$=$$\;\;\;\;${\rm{legalp2}}\,({\it{x\/}}, {\rm{list}}\,({\it{s0\/}}, {\it{s1\/}}, {\it{s2\/}}, {\it{s3\/}}, {\it{s4\/}}, {\it{s5\/}}, {\it{s6\/}}, {\it{s7\/}}, {\it{s8\/}}), {\tt{0}})\index{legalp2}\-
\end{tabbing}


\noindent{\sc Event}:   Disable legalp-is-legalp1.

\addvspace{18pt}

\noindent{\sc Event}:   Disable legalp1-is-legalp2.

\addvspace{18pt}

\noindent{\sc Event}:   Disable member.

\addvspace{18pt}

\noindent{\sc Event}:   Disable move.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  noop-move-gen \\ \index{noop-move-gen}
({\it{i\/}} $\not<$ {\rm{length}}\,({\it{board\/}})\index{length}) $\rightarrow$ ({\rm{move}}\,({\it{x\/}}, {\it{i\/}}, {\it{board\/}})\index{move} $=$ {\it{board\/}})
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  noop-move \\ \index{noop-move}
\=\+(\=\+({\rm{length}}\,({\it{board\/}})\index{length} $=$ {\tt{9}}) \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\not=$ {\tt{0}}) \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\not=$ {\tt{1}}) \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\not=$ {\tt{2}}) \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\not=$ {\tt{3}}) \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\not=$ {\tt{4}}) \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\not=$ {\tt{5}}) \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\not=$ {\tt{6}}) \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\not=$ {\tt{7}}) \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\not=$ {\tt{8}}))\- \\ 
$\rightarrow$$\;\;\;\;$({\rm{move}}\,({\it{x\/}}, {\it{i\/}}, {\it{board\/}})\index{move} $=$ {\it{board\/}})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  move-zerop \\ \index{move-zerop}
\=\+(({\rm{length}}\,({\it{board\/}})\index{length} $=$ {\tt{9}}) $\wedge$ ({\it{i\/}} $\not\in\;${\bf{N}}${}$)) \\ 
$\rightarrow$$\;\;\;\;$({\rm{move}}\,({\it{x\/}}, {\it{i\/}}, {\it{board\/}})\index{move} $=$ {\rm{cons}}\,({\it{x\/}}, {\rm{cdr}}\,({\it{board\/}})))\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  not-legal-movep-x-x \\ \index{not-legal-movep-x-x}
$\neg$ {\rm{legal-movep}}\,({\it{board\/}}, {\tt{'}}{\tt{x}}, {\it{board\/}})\index{legal-movep}
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  not-legal-movep-if-occupied \\ \index{not-legal-movep-if-occupied}
{\rm{nth}}\,({\it{i\/}}, {\it{board\/}})\index{nth} $\rightarrow$ ($\neg$ {\rm{legal-movep}}\,({\it{board\/}}, {\tt{'}}{\tt{x}}, {\rm{move}}\,({\tt{'}}{\tt{x}}, {\it{i\/}}, {\it{board\/}})\index{move})\index{legal-movep})
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  not-legal-movep-if-too-big \\ \index{not-legal-movep-if-too-big}
({\it{i\/}} $\not<$ {\rm{length}}\,({\it{board\/}})\index{length}) $\rightarrow$ ($\neg$ {\rm{legal-movep}}\,({\it{board\/}}, {\tt{'}}{\tt{x}}, {\rm{move}}\,({\tt{'}}{\tt{x}}, {\it{i\/}}, {\it{board\/}})\index{move})\index{legal-movep})
\end{tabbing}

\begin{verbatim}; This is another hideously silly descent from 9 to 0 in subgoals.  But, hey,
; it works...

\end{verbatim}\begin{tabbing}{\sc Theorem}:  legal-movep-move \\ \index{legal-movep-move}
\=\+({\rm{boardp}}\,({\it{board\/}})\index{boardp} $\wedge$ ({\rm{length}}\,({\it{board\/}})\index{length} $=$ {\tt{9}})) \\ 
$\rightarrow$$\;\;\;\;$({\rm{legal-movep}}\,({\it{board\/}}, {\tt{'}}{\tt{x}}, {\rm{move}}\,({\tt{'}}{\tt{x}}, {\it{i\/}}, {\it{board\/}})\index{move})\index{legal-movep} $=$ {\rm{legalp}}\,({\it{i\/}}, {\it{board\/}})\index{legalp})\-
\end{tabbing}


\noindent{\sc Event}:   Disable legal-movep.

\addvspace{18pt}

\begin{verbatim}; The following theorem produces 4.8 Mb of .proofs output.  The proof
; takes roughly 7.5 minutes and generates 2930 cases.

\end{verbatim}\begin{tabbing}{\sc Theorem}:  tic-tac-toe-correct1 \\ \index{tic-tac-toe-correct1}
{\rm{tic-tac-toep}}\,({\rm{tic-tac-toe}}\,({\it{xmoves\/}})\index{tic-tac-toe})\index{tic-tac-toep} $\in$ {\tt{'}}{\tt{(}}{\tt{draw}}{\tt{ }}{\tt{o-win}}{\tt{)}}
\end{tabbing}


\noindent{\sc Event}:   Disable tic-tac-toe1-opener.

\addvspace{18pt}

\noindent{\sc Event}:   Disable legalp-simplifier.

\addvspace{18pt}

\begin{verbatim}; Comments on What is Not Proved

; This is not a very satisfying specification for an implementation of tic-tac-toe.
; I can think of two ways to cheat while satisfying this spec.

; (1) Return a fixed game or, more generally, ignore the user's moves
;     when they get in your way of winning.  One could still prove the
;     above theorem.

; (2) Look ahead into the user's moves and always make your first move
;     to the square to which he'll make his second.  This guarantees
;     that he will always play illegal games and lose by default.

; We could avoid (1) by proving another theorem that says "the user's
; actual moves (extracted from the sequence of boards returned by
; tic-tac-toe) is an initial subsequence of the specified moves."
; I think this would not be hard to prove.

; We could avoid (2) by proving a theorem that says "If two lists of user
; moves, xmoves1 and xmoves2, agree on the first i moves, then 
; (tic-tac-toe xmoves1) and (tic-tac-toe xmoves2) agree on the first
; 2i boards."  I believe this theorem rules out the possibility that
; the machine is looking ahead.

; We address these two concerns below, in slightly reformulated ways.

; We now turn to the next problem, proving that the function honors
; the requested moves by 'x.

\end{verbatim}\begin{tabbing}{\sc Definition}:\index{x-moves-honored1} \\  
\=\+{\rm{x-moves-honored1}}\,({\it{board\/}}, {\it{xmoves\/}}, {\it{lst\/}})\index{x-moves-honored1} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{lst\/}} $\simeq\;${{\bf{nil}}}${}$$\;\;${\bf then }{\bf{t}} \\ 
{\bf elseif }{\rm{cdr}}\,({\it{lst\/}}) $\simeq\;${{\bf{nil}}}${}$$\;\;${\bf then }{\bf{t}} \\ 
{\bf else }\=\+({\rm{car}}\,({\it{lst\/}}) $=$ {\rm{move}}\,({\tt{'}}{\tt{x}}, {\rm{car}}\,({\it{xmoves\/}}), {\it{board\/}})\index{move}) \\ 
$\wedge$$\;\;\;\;${\rm{x-moves-honored1}}\,({\rm{cadr}}\,({\it{lst\/}}), {\rm{cdr}}\,({\it{xmoves\/}}), {\rm{cddr}}\,({\it{lst\/}}))\index{x-moves-honored1}\-$\;${\bf  endif}\-\-
\end{tabbing}

\begin{tabbing}{\sc Definition}:\index{x-moves-honored} \\  
{\rm{x-moves-honored}}\,({\it{xmoves\/}}, {\it{lst\/}})\index{x-moves-honored} $=$ {\rm{x-moves-honored1}}\,({\rm{car}}\,({\it{lst\/}}), {\it{xmoves\/}}, {\rm{cdr}}\,({\it{lst\/}}))\index{x-moves-honored1}
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  tic-tac-toe-correct2-lemma \\ \index{tic-tac-toe-correct2-lemma}
{\rm{boardp}}\,({\it{board\/}})\index{boardp} $\rightarrow$ {\rm{x-moves-honored1}}\,({\it{board\/}}, {\it{xmoves\/}}, {\rm{tic-tac-toe1}}\,({\it{xmoves\/}}, {\it{board\/}})\index{tic-tac-toe1})\index{x-moves-honored1}
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  tic-tac-toe-correct2 \\ \index{tic-tac-toe-correct2}
{\rm{x-moves-honored}}\,({\it{xmoves\/}}, {\rm{tic-tac-toe}}\,({\it{xmoves\/}})\index{tic-tac-toe})\index{x-moves-honored}
\end{tabbing}

\begin{verbatim}; The final conjunction of the specification is that the game playing
; program does not look-ahead into xmoves!

\end{verbatim}\begin{tabbing}{\sc Definition}:\index{init-game1} \\  
\=\+{\rm{init-game1}}\,({\it{n\/}}, {\it{lst\/}})\index{init-game1} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{n\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\bf{nil}} \\ 
{\bf elseif }{\it{lst\/}} $\simeq\;${{\bf{nil}}}${}$$\;\;${\bf then }{\bf{nil}} \\ 
{\bf elseif }{\rm{cdr}}\,({\it{lst\/}}) $\simeq\;${{\bf{nil}}}${}$$\;\;${\bf then }{\rm{list}}\,({\rm{car}}\,({\it{lst\/}})) \\ 
{\bf else }{\rm{cons}}\,({\rm{car}}\,({\it{lst\/}}), {\rm{cons}}\,({\rm{cadr}}\,({\it{lst\/}}), {\rm{init-game1}}\,({\it{n\/}} $-\;1$, {\rm{cddr}}\,({\it{lst\/}}))\index{init-game1}))$\;${\bf  endif}\-\-
\end{tabbing}

\begin{tabbing}{\sc Definition}:\index{init-game} \\  
{\rm{init-game}}\,({\it{n\/}}, {\it{lst\/}})\index{init-game} $=$ {\rm{cons}}\,({\rm{car}}\,({\it{lst\/}}), {\rm{init-game1}}\,({\it{n\/}}, {\rm{cdr}}\,({\it{lst\/}}))\index{init-game1})
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  tic-tac-toe-correct3-lemma \\ \index{tic-tac-toe-correct3-lemma}
\=\+(\=\+{\rm{init-game1}}\,({\rm{length}}\,({\it{opening\/}})\index{length}, {\rm{tic-tac-toe1}}\,({\rm{append}}\,({\it{opening\/}}, {\it{xmoves1\/}}), {\it{board\/}})\index{tic-tac-toe1})\index{init-game1} \\ 
$=$$\;\;\;\;${\rm{init-game1}}\,(\=\+{\rm{length}}\,({\it{opening\/}})\index{length}, \\ 
{\rm{tic-tac-toe1}}\,({\rm{append}}\,({\it{opening\/}}, {\it{xmoves2\/}}), {\it{board\/}})\index{tic-tac-toe1})\-\index{init-game1})\- \\ 
$=$$\;\;\;\;${\bf{t}}\-
\end{tabbing}  

\begin{tabbing}{\sc Theorem}:  tic-tac-toe-correct3 \\ \index{tic-tac-toe-correct3}
\=\+(\=\+{\rm{init-game}}\,({\rm{length}}\,({\it{opening\/}})\index{length}, {\rm{tic-tac-toe}}\,({\rm{append}}\,({\it{opening\/}}, {\it{xmoves1\/}}))\index{tic-tac-toe})\index{init-game} \\ 
$=$$\;\;\;\;${\rm{init-game}}\,({\rm{length}}\,({\it{opening\/}})\index{length}, {\rm{tic-tac-toe}}\,({\rm{append}}\,({\it{opening\/}}, {\it{xmoves2\/}}))\index{tic-tac-toe})\index{init-game})\- \\ 
$=$$\;\;\;\;${\bf{t}}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  tic-tac-toe-correct \\ \index{tic-tac-toe-correct}
\=\+({\rm{tic-tac-toep}}\,({\rm{tic-tac-toe}}\,({\it{xmoves\/}})\index{tic-tac-toe})\index{tic-tac-toep} $\in$ {\tt{'}}{\tt{(}}{\tt{draw}}{\tt{ }}{\tt{o-win}}{\tt{)}}) \\ 
$\wedge$$\;\;\;\;${\rm{x-moves-honored}}\,({\it{xmoves\/}}, {\rm{tic-tac-toe}}\,({\it{xmoves\/}})\index{tic-tac-toe})\index{x-moves-honored} \\ 
$\wedge$$\;\;\;\;$(\=\+{\rm{init-game}}\,({\rm{length}}\,({\it{opening\/}})\index{length}, {\rm{tic-tac-toe}}\,({\rm{append}}\,({\it{opening\/}}, {\it{xmoves1\/}}))\index{tic-tac-toe})\index{init-game} \\ 
$=$$\;\;\;\;${\rm{init-game}}\,(\=\+{\rm{length}}\,({\it{opening\/}})\index{length}, \\ 
{\rm{tic-tac-toe}}\,({\rm{append}}\,({\it{opening\/}}, {\it{xmoves2\/}}))\index{tic-tac-toe})\-\index{init-game})\-\-
\end{tabbing}
       
\begin{verbatim}; The following theorem establishes that the outcome reported by a
; legal game is in fact the outcome reported by the definition of the
; game.  Note that this is a fact about legal games, not about our
; particular strategy.

\end{verbatim}\begin{tabbing}{\sc Theorem}:  tic-tac-toep1-outcome \\ \index{tic-tac-toep1-outcome}
\=\+{\rm{tic-tac-toep1}}\,({\it{board\/}}, {\it{player\/}}, {\it{lst\/}})\index{tic-tac-toep1} \\ 
$\rightarrow$$\;\;\;\;$({\rm{tic-tac-toep1}}\,({\it{board\/}}, {\it{player\/}}, {\it{lst\/}})\index{tic-tac-toep1} $=$ {\rm{nth}}\,({\rm{length}}\,({\it{lst\/}})\index{length} $-\;1$, {\it{lst\/}})\index{nth})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  tic-tac-toep-outcome \\ \index{tic-tac-toep-outcome}
{\rm{tic-tac-toep}}\,({\it{lst\/}})\index{tic-tac-toep} $\rightarrow$ ({\rm{tic-tac-toep}}\,({\it{lst\/}})\index{tic-tac-toep} $=$ {\rm{nth}}\,({\rm{length}}\,({\it{lst\/}})\index{length} $-\;1$, {\it{lst\/}})\index{nth})
\end{tabbing}

\begin{verbatim}#|
Historical Plaque:

My original formulation of the problem did not contain tic-tac-toep.
Instead I had a tic-tac-toe function which allegedly played a game
against the user.  The intent was that the buyer read this function
and was convinced that it played the game we call tic-tac-toe.  Then
I proved that it never lost, and the buyer would have been surprised
and pleased.  Here is the definition and the main theorem:

(defn tic-tac-toe (xmoves board)
  (if (legalp (car xmoves) board)
      (if (winp 'x
                (move 'x (car xmoves) board))
          'x-won
          (if (endp (move 'x (car xmoves) board))
              'draw
              (if (legalp (pick-move (move 'x (car xmoves) board))
                          (move 'x (car xmoves) board))
                  (if (winp 'o
                            (move 'o
                                  (pick-move (move 'x (car xmoves) board))
                                  (move 'x (car xmoves) board)))
                      'o-won
                      (tic-tac-toe (cdr xmoves)
                                   (move 'o
                                         (pick-move (move 'x (car xmoves) board))
                                         (move 'x (car xmoves) board))))
                  'o-illegal)))
      'x-illegal)
  ((lessp (open-squares board))))

(prove-lemma pick-move-never-loses nil
  (member (tic-tac-toe xmoves
                       (list f f f
                             f f f
                             f f f))
          '(x-illegal draw o-won)))

I found this an unsatisfying formalization when I noted to Bill Young
that, as the implementor of this program, he could omit the test
that the user won and the test that the program's move is legal.
That made me realize that tic-tac-toe, as defined above, is equivalent
to the version that makes no such tests.  But when that shorter
version is taken as a spec, it is not credible.  Simple syntactic inspection
reveals the user can't win (that token is never mentioned by the program)
and hence, without complete analysis of pick-move, the buyer can't
come to the opinion that tic-tac-toe is even the game implemented.

|#

; Before we start the implementation, we will introduce the standard
; small set of arithmetic facts.  We'll prove more as needed.

\end{verbatim}\begin{tabbing}{\sc Theorem}:  plus-add1 \\ \index{plus-add1}
({\it{x\/}} $+$ ($1\;+$ {\it{y\/}})) $=$ ($1\;+$ ({\it{x\/}} $+$ {\it{y\/}}))
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  plus-commutes1 \\ \index{plus-commutes1}
({\it{x\/}} $+$ {\it{y\/}}) $=$ ({\it{y\/}} $+$ {\it{x\/}})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  plus-commutes2 \\ \index{plus-commutes2}
({\it{x\/}} $+$ {\it{y\/}} $+$ {\it{z\/}}) $=$ ({\it{y\/}} $+$ {\it{x\/}} $+$ {\it{z\/}})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  plus-associates \\ \index{plus-associates}
(({\it{x\/}} $+$ {\it{y\/}}) $+$ {\it{z\/}}) $=$ ({\it{x\/}} $+$ {\it{y\/}} $+$ {\it{z\/}})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  times-0 \\ \index{times-0}
({\it{x\/}} $*$ {\tt{0}}) $=$ {\tt{0}}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  times-non-numberp \\ \index{times-non-numberp}
({\it{z\/}} $\not\in\;${\bf{N}}${}$) $\rightarrow$ (({\it{x\/}} $*$ {\it{z\/}}) $=$ {\tt{0}})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  times-add1 \\ \index{times-add1}
({\it{x\/}} $*$ ($1\;+$ {\it{y\/}})) $=$ ({\it{x\/}} $+$ ({\it{x\/}} $*$ {\it{y\/}}))
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  times-distributes1 \\ \index{times-distributes1}
({\it{x\/}} $*$ ({\it{y\/}} $+$ {\it{z\/}})) $=$ (({\it{x\/}} $*$ {\it{y\/}}) $+$ ({\it{x\/}} $*$ {\it{z\/}}))
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  times-commutes1 \\ \index{times-commutes1}
({\it{x\/}} $*$ {\it{y\/}}) $=$ ({\it{y\/}} $*$ {\it{x\/}})
\end{tabbing} 
\begin{tabbing}{\sc Theorem}:  times-commutes2 \\ \index{times-commutes2}
({\it{x\/}} $*$ {\it{y\/}} $*$ {\it{z\/}}) $=$ ({\it{y\/}} $*$ {\it{x\/}} $*$ {\it{z\/}})
\end{tabbing} 
\begin{tabbing}{\sc Theorem}:  times-associates \\ \index{times-associates}
(({\it{x\/}} $*$ {\it{y\/}}) $*$ {\it{z\/}}) $=$ ({\it{x\/}} $*$ {\it{y\/}} $*$ {\it{z\/}})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  times-distributes2 \\ \index{times-distributes2}
(({\it{x\/}} $+$ {\it{y\/}}) $*$ {\it{z\/}}) $=$ (({\it{x\/}} $*$ {\it{z\/}}) $+$ ({\it{y\/}} $*$ {\it{z\/}}))
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  difference-is-0 \\ \index{difference-is-0}
({\it{y\/}} $\not<$ {\it{x\/}}) $\rightarrow$ (({\it{x\/}} $-$ {\it{y\/}}) $=$ {\tt{0}})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  difference-plus-cancellation1 \\ \index{difference-plus-cancellation1}
(({\it{i\/}} $+$ {\it{x\/}}) $-$ {\it{i\/}}) $=$ {\rm{fix}}\,({\it{x\/}})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  difference-plus-cancellation2 \\ \index{difference-plus-cancellation2}
(({\it{i\/}} $+$ {\it{x\/}}) $-$ ({\it{i\/}} $+$ {\it{y\/}})) $=$ ({\it{x\/}} $-$ {\it{y\/}})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  difference-plus-cancellation3 \\ \index{difference-plus-cancellation3}
(({\it{i\/}} $+$ {\it{j\/}} $+$ {\it{x\/}}) $-$ {\it{j\/}}) $=$ ({\it{i\/}} $+$ {\it{x\/}})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  lessp-remainder \\ \index{lessp-remainder}
(({\it{x\/}} ${}${\bf{mod}}${}$ {\it{y\/}}) $<$ {\it{y\/}}) $=$ ({\it{y\/}} $\not\simeq {\tt{0}}$)
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  remainder-quotient-elim \\ \index{remainder-quotient-elim}
({\it{x\/}} $\in\;${\bf{N}}${}$) $\rightarrow$ ((({\it{x\/}} ${}${\bf{mod}}${}$ {\it{y\/}}) $+$ ({\it{y\/}} $*$ ({\it{x\/}} $\div$ {\it{y\/}}))) $=$ {\it{x\/}})
\end{tabbing}

\begin{verbatim}; Our strategy for implementing tic-tac-toe is to transform the
; function in several steps to one that has a very straightforward
; micro-Gypsy implementation.  Each of the steps will produce a
; function with the name tic-tac-toe-x, where x = a, b, c, etc.  Each
; time we define such a function we will prove that it is equivalent
; to the previous such function, starting with tic-tac-toe at the top.
; When the sequence of steps is done, we will combine the results into
; a single equation relating tic-tac-toe to the (final) implementation
; function.

; We first eliminate the need for the endp test by just counting the
; moves.

\end{verbatim}\begin{tabbing}{\sc Definition}:\index{tic-tac-toe1-a} \\  
\=\+{\rm{tic-tac-toe1-a}}\,({\it{xmoves\/}}, {\it{board\/}}, {\it{i\/}})\index{tic-tac-toe1-a} \\ 
$=$$\;\;\;\;$\=\+{\bf let }\=\+{\it{board1\/}}{\bf $\;$ be$\;\;$}{\rm{move}}\,({\tt{'}}{\tt{x}}, {\rm{car}}\,({\it{xmoves\/}}), {\it{board\/}})\index{move}\- \\ 
{\bf in} \\ 
\=\+{\bf if }{\rm{legalp}}\,({\rm{car}}\,({\it{xmoves\/}}), {\it{board\/}})\index{legalp} \\ 
{\bf then }\=\+{\bf if }{\it{i\/}} $<$ {\tt{9}} \\ 
{\bf then }\=\+{\bf let }\=\+{\it{board2\/}}{\bf $\;$ be$\;\;$}{\rm{move}}\,({\tt{'}}{\tt{o}}, {\rm{pick-move}}\,({\it{board1\/}})\index{pick-move}, {\it{board1\/}})\index{move}\- \\ 
{\bf in} \\ 
\=\+{\bf if }{\rm{winp}}\,({\tt{'}}{\tt{o}}, {\it{board2\/}})\index{winp} \\ 
{\bf then }{\rm{list}}\,({\it{board1\/}}, {\it{board2\/}}, {\tt{'}}{\tt{o-win}}) \\ 
{\bf else }{\rm{cons}}\,(\=\+{\it{board1\/}}, \\ 
{\rm{cons}}\,(\=\+{\it{board2\/}}, \\ 
{\rm{tic-tac-toe1-a}}\,(\=\+{\rm{cdr}}\,({\it{xmoves\/}}), \\ 
{\it{board2\/}}, \\ 
$1\;+$ ($1\;+$ {\it{i\/}}))\-\index{tic-tac-toe1-a})\-)\-$\;${\bf  endif}\-$\;${\bf  endlet}\- \\ 
{\bf else }{\rm{list}}\,({\it{board1\/}}, {\tt{'}}{\tt{draw}})$\;${\bf  endif}\- \\ 
{\bf else }{\rm{list}}\,({\it{board1\/}}, {\tt{'}}{\tt{o-win}})$\;${\bf  endif}\-$\;${\bf  endlet}\-\-
\end{tabbing}

\begin{tabbing}{\sc Definition}:\index{tic-tac-toe-a} \\  
\=\+{\rm{tic-tac-toe-a}}\,({\it{xmoves\/}})\index{tic-tac-toe-a} \\ 
$=$$\;\;\;\;$\=\+{\bf let }\=\+{\it{board0\/}}{\bf $\;$ be$\;\;$}{\rm{list}}\,({\bf{f}}, {\bf{f}}, {\bf{f}}, {\bf{f}}, {\bf{f}}, {\bf{f}}, {\bf{f}}, {\bf{f}}, {\bf{f}})\- \\ 
{\bf in} \\ 
{\rm{cons}}\,({\it{board0\/}}, {\rm{tic-tac-toe1-a}}\,({\it{xmoves\/}}, {\it{board0\/}}, {\tt{1}})\index{tic-tac-toe1-a})$\;${\bf  endlet}\-\-
\end{tabbing}


\noindent{\sc Event}:   Enable tic-tac-toe1.

\addvspace{18pt}

\noindent{\sc Event}:   Enable legalp.

\addvspace{18pt}

\noindent{\sc Event}:   Enable move.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  endp-open-squares \\ \index{endp-open-squares}
{\rm{endp}}\,({\it{board\/}})\index{endp} $=$ ({\rm{open-squares}}\,({\it{board\/}})\index{open-squares} $\simeq {\tt{0}}$)
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  open-squares-move-rewrite \\ \index{open-squares-move-rewrite}
\=\+({\it{x\/}} $\wedge$ {\rm{legalp}}\,({\it{i\/}}, {\it{board\/}})\index{legalp}) \\ 
$\rightarrow$$\;\;\;\;$({\rm{open-squares}}\,({\rm{move}}\,({\it{x\/}}, {\it{i\/}}, {\it{board\/}})\index{move})\index{open-squares} $=$ ({\rm{open-squares}}\,({\it{board\/}})\index{open-squares} $-\;1$))\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  add1-difference \\ \index{add1-difference}
({\it{x\/}} $\not<$ {\it{y\/}}) $\rightarrow$ (($1\;+$ ({\it{x\/}} $-$ {\it{y\/}})) $=$ (($1\;+$ {\it{x\/}}) $-$ {\it{y\/}}))
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  lessp-open-squares-length \\ \index{lessp-open-squares-length}
{\rm{length}}\,({\it{board\/}})\index{length} $\not<$ {\rm{open-squares}}\,({\it{board\/}})\index{open-squares}
\end{tabbing}

\begin{verbatim}; The proof of the following lemma is tedious.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  lemma-a1 \\ \index{lemma-a1}
\=\+({\rm{length}}\,({\it{board\/}})\index{length} $=$ {\tt{9}}) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{tic-tac-toe1}}\,({\it{xmoves\/}}, {\it{board\/}})\index{tic-tac-toe1} \\ 
$=$$\;\;\;\;${\rm{tic-tac-toe1-a}}\,({\it{xmoves\/}}, {\it{board\/}}, {\tt{10}} $-$ {\rm{open-squares}}\,({\it{board\/}})\index{open-squares})\index{tic-tac-toe1-a})\-\-
\end{tabbing}


\noindent{\sc Event}:   Disable add1-difference.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  lemma-a \\ \index{lemma-a}
{\rm{tic-tac-toe}}\,({\it{xmoves\/}})\index{tic-tac-toe} $=$ {\rm{tic-tac-toe-a}}\,({\it{xmoves\/}})\index{tic-tac-toe-a}
\end{tabbing}

\begin{verbatim}; Next, we will convert the board from a list representation to a
; numeric interpretation.  Suppose that tic-tac-toe-b is defined
; analogously to tic-tac-toe-a but deals with numeric boards.
; Then we will want to prove:

; (equal (tic-tac-toe-a xmoves)
;        (number->board* (tic-tac-toe-b xmoves)))

\end{verbatim}\begin{tabbing}{\sc Definition}:\index{piece-$>$number} \\  
\=\+{\rm{piece-$>$number}}\,({\it{x\/}})\index{piece-$>$number} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{x\/}} $=$ {\bf{f}}$\;\;${\bf then }{\tt{0}} \\ 
{\bf elseif }{\it{x\/}} $=$ {\tt{'}}{\tt{x}}$\;\;${\bf then }{\tt{1}} \\ 
{\bf else }{\tt{2}}$\;${\bf  endif}\-\-
\end{tabbing}

\begin{tabbing}{\sc Definition}:\index{number-$>$piece} \\  
\=\+{\rm{number-$>$piece}}\,({\it{j\/}})\index{number-$>$piece} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{j\/}} $=$ {\tt{0}}$\;\;${\bf then }{\bf{f}} \\ 
{\bf elseif }{\it{j\/}} $=$ {\tt{1}}$\;\;${\bf then }{\tt{'}}{\tt{x}} \\ 
{\bf else }{\tt{'}}{\tt{o}}$\;${\bf  endif}\-\-
\end{tabbing}

\begin{tabbing}{\sc Definition}:\index{board-$>$number} \\  
\=\+{\rm{board-$>$number}}\,({\it{board\/}})\index{board-$>$number} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{board\/}} $\simeq\;${{\bf{nil}}}${}$$\;\;${\bf then }{\tt{0}} \\ 
{\bf else }\=\+{\rm{piece-$>$number}}\,({\rm{car}}\,({\it{board\/}}))\index{piece-$>$number} \\ 
$+$$\;\;\;\;$({\tt{4}} $*$ {\rm{board-$>$number}}\,({\rm{cdr}}\,({\it{board\/}}))\index{board-$>$number})\-$\;${\bf  endif}\-\-
\end{tabbing}

\begin{tabbing}{\sc Definition}:\index{number-$>$board} \\  
\=\+{\rm{number-$>$board}}\,({\it{n\/}}, {\it{len\/}})\index{number-$>$board} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{len\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\bf{nil}} \\ 
{\bf else }{\rm{cons}}\,(\=\+{\rm{number-$>$piece}}\,({\it{n\/}} ${}${\bf{mod}}${}$ {\tt{4}})\index{number-$>$piece}, \\ 
{\rm{number-$>$board}}\,({\it{n\/}} $\div$ {\tt{4}}, {\it{len\/}} $-\;1$)\index{number-$>$board})\-$\;${\bf  endif}\-\-
\end{tabbing}

\begin{tabbing}{\sc Definition}:\index{boardp*} \\  
\=\+{\rm{boardp*}}\,({\it{lst\/}}, {\it{len\/}})\index{boardp*} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{lst\/}} $\simeq\;${{\bf{nil}}}${}$$\;\;${\bf then }{\it{lst\/}} $=$ {\bf{nil}} \\ 
{\bf else }\=\+{\rm{boardp}}\,({\rm{car}}\,({\it{lst\/}}))\index{boardp} \\ 
$\wedge$$\;\;\;\;$({\rm{length}}\,({\rm{car}}\,({\it{lst\/}}))\index{length} $=$ {\it{len\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{boardp*}}\,({\rm{cdr}}\,({\it{lst\/}}), {\it{len\/}})\index{boardp*}\-$\;${\bf  endif}\-\-
\end{tabbing}

\begin{tabbing}{\sc Definition}:\index{number-$>$board*} \\  
\=\+{\rm{number-$>$board*}}\,({\it{lst\/}}, {\it{len\/}})\index{number-$>$board*} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{lst\/}} $\simeq\;${{\bf{nil}}}${}$$\;\;${\bf then }{\bf{nil}} \\ 
{\bf else }{\rm{cons}}\,(\=\+\=\+{\bf if }{\rm{car}}\,({\it{lst\/}}) $=$ {\tt{-2}}$\;\;${\bf then }{\tt{'}}{\tt{o-win}} \\ 
{\bf elseif }{\rm{car}}\,({\it{lst\/}}) $=$ {\tt{-1}}$\;\;${\bf then }{\tt{'}}{\tt{draw}} \\ 
{\bf else }{\rm{number-$>$board}}\,({\rm{car}}\,({\it{lst\/}}), {\it{len\/}})\index{number-$>$board}$\;${\bf  endif}\-, \\ 
{\rm{number-$>$board*}}\,({\rm{cdr}}\,({\it{lst\/}}), {\it{len\/}})\index{number-$>$board*})\-$\;${\bf  endif}\-\-
\end{tabbing}

\begin{verbatim}; Some more arithmetic...

\end{verbatim}\begin{tabbing}{\sc Theorem}:  remainder-plus-times \\ \index{remainder-plus-times}
(({\it{x\/}} $\in\;${\bf{N}}${}$) $\wedge$ ({\it{x\/}} $<$ {\it{y\/}})) $\rightarrow$ ((({\it{x\/}} $+$ ({\it{y\/}} $*$ {\it{z\/}})) ${}${\bf{mod}}${}$ {\it{y\/}}) $=$ {\it{x\/}})
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  remainder-plus-times-instance \\ \index{remainder-plus-times-instance}
({\it{y\/}} $\not\simeq {\tt{0}}$) $\rightarrow$ ((({\it{y\/}} $*$ {\it{z\/}}) ${}${\bf{mod}}${}$ {\it{y\/}}) $=$ {\tt{0}})
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  quotient-plus-times \\ \index{quotient-plus-times}
(({\it{z\/}} $\in\;${\bf{N}}${}$) $\wedge$ ({\it{x\/}} $<$ {\it{y\/}})) $\rightarrow$ ((({\it{x\/}} $+$ ({\it{y\/}} $*$ {\it{z\/}})) $\div$ {\it{y\/}}) $=$ {\it{z\/}})
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  quotient-plus-times-instance \\ \index{quotient-plus-times-instance}
(({\it{z\/}} $\in\;${\bf{N}}${}$) $\wedge$ ({\it{y\/}} $\not\simeq {\tt{0}}$)) $\rightarrow$ ((({\it{y\/}} $*$ {\it{z\/}}) $\div$ {\it{y\/}}) $=$ {\it{z\/}})
\end{tabbing}

\begin{verbatim}; The crucial fact about this representation is:

\end{verbatim}\begin{tabbing}{\sc Theorem}:  board-$>$number-$>$board \\ \index{board-$>$number-$>$board}
\=\+{\rm{boardp}}\,({\it{board\/}})\index{boardp} \\ 
$\rightarrow$$\;\;\;\;$({\rm{number-$>$board}}\,({\rm{board-$>$number}}\,({\it{board\/}})\index{board-$>$number}, {\rm{length}}\,({\it{board\/}})\index{length})\index{number-$>$board} $=$ {\it{board\/}})\-
\end{tabbing}

\begin{verbatim}; Now we begin a litany of definitions of the numeric counterparts of
; of the subroutines of tic-tac-toe-a.

\end{verbatim}\begin{tabbing}{\sc Definition}:\index{moven} \\  
\=\+{\rm{moven}}\,({\it{x\/}}, {\it{i\/}}, {\it{board\/}})\index{moven} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{i\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\it{x\/}} $+$ ({\tt{4}} $*$ ({\it{board\/}} $\div$ {\tt{4}})) \\ 
{\bf else }({\it{board\/}} ${}${\bf{mod}}${}$ {\tt{4}}) $+$ ({\tt{4}} $*$ {\rm{moven}}\,({\it{x\/}}, {\it{i\/}} $-\;1$, {\it{board\/}} $\div$ {\tt{4}})\index{moven})$\;${\bf  endif}\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  move-is-moven \\ \index{move-is-moven}
\=\+((({\it{x\/}} $=$ {\tt{'}}{\tt{x}}) $\vee$ ({\it{x\/}} $=$ {\tt{'}}{\tt{o}})) $\wedge$ {\rm{boardp}}\,({\it{board\/}})\index{boardp}) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{move}}\,({\it{x\/}}, {\it{i\/}}, {\it{board\/}})\index{move} \\ 
$=$$\;\;\;\;${\rm{number-$>$board}}\,(\=\+{\rm{moven}}\,({\rm{piece-$>$number}}\,({\it{x\/}})\index{piece-$>$number}, {\it{i\/}}, {\rm{board-$>$number}}\,({\it{board\/}})\index{board-$>$number})\index{moven}, \\ 
{\rm{length}}\,({\it{board\/}})\index{length})\-\index{number-$>$board})\-\-
\end{tabbing}

\begin{tabbing}{\sc Definition}:\index{nthn} \\  
\=\+{\rm{nthn}}\,({\it{i\/}}, {\it{board\/}})\index{nthn} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{i\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\it{board\/}} ${}${\bf{mod}}${}$ {\tt{4}} \\ 
{\bf else }{\rm{nthn}}\,({\it{i\/}} $-\;1$, {\it{board\/}} $\div$ {\tt{4}})\index{nthn}$\;${\bf  endif}\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  nth-is-nthn \\ \index{nth-is-nthn}
\=\+(({\it{i\/}} $<$ {\rm{length}}\,({\it{board\/}})\index{length}) $\wedge$ {\rm{boardp}}\,({\it{board\/}})\index{boardp}) \\ 
$\rightarrow$$\;\;\;\;$({\rm{nth}}\,({\it{i\/}}, {\it{board\/}})\index{nth} $=$ {\rm{number-$>$piece}}\,({\rm{nthn}}\,({\it{i\/}}, {\rm{board-$>$number}}\,({\it{board\/}})\index{board-$>$number})\index{nthn})\index{number-$>$piece})\-
\end{tabbing}

\begin{tabbing}{\sc Definition}:\index{legalpn} \\  
{\rm{legalpn}}\,({\it{i\/}}, {\it{board\/}})\index{legalpn} $=$ (({\it{i\/}} $<$ {\tt{9}}) $\wedge$ ({\rm{nthn}}\,({\it{i\/}}, {\it{board\/}})\index{nthn} $=$ {\tt{0}}))
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  legalp-is-legalpn \\ \index{legalp-is-legalpn}
\=\+({\rm{boardp}}\,({\it{board\/}})\index{boardp} $\wedge$ ({\rm{length}}\,({\it{board\/}})\index{length} $=$ {\tt{9}})) \\ 
$\rightarrow$$\;\;\;\;$({\rm{legalp}}\,({\it{i\/}}, {\it{board\/}})\index{legalp} $=$ {\rm{legalpn}}\,({\it{i\/}}, {\rm{board-$>$number}}\,({\it{board\/}})\index{board-$>$number})\index{legalpn})\-
\end{tabbing}

\begin{tabbing}{\sc Definition}:\index{winp1n} \\  
\=\+{\rm{winp1n}}\,({\it{x\/}}, {\it{i\/}}, {\it{j\/}}, {\it{k\/}}, {\it{board\/}})\index{winp1n} \\ 
$=$$\;\;\;\;$(\=\+({\it{x\/}} $=$ {\rm{nthn}}\,({\it{i\/}}, {\it{board\/}})\index{nthn}) \\ 
$\wedge$$\;\;\;\;$({\it{x\/}} $=$ {\rm{nthn}}\,({\it{j\/}}, {\it{board\/}})\index{nthn}) \\ 
$\wedge$$\;\;\;\;$({\it{x\/}} $=$ {\rm{nthn}}\,({\it{k\/}}, {\it{board\/}})\index{nthn}))\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  lessp-nthn-3-lemma \\ \index{lessp-nthn-3-lemma}
\=\+(({\it{i\/}} $<$ {\rm{length}}\,({\it{board\/}})\index{length}) $\wedge$ {\rm{boardp}}\,({\it{board\/}})\index{boardp}) \\ 
$\rightarrow$$\;\;\;\;$({\rm{piece-$>$number}}\,({\rm{nth}}\,({\it{i\/}}, {\it{board\/}})\index{nth})\index{piece-$>$number} $=$ {\rm{nthn}}\,({\it{i\/}}, {\rm{board-$>$number}}\,({\it{board\/}})\index{board-$>$number})\index{nthn})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  lessp-nthn-3 \\ \index{lessp-nthn-3}
\=\+(({\it{i\/}} $<$ {\rm{length}}\,({\it{board\/}})\index{length}) $\wedge$ {\rm{boardp}}\,({\it{board\/}})\index{boardp}) \\ 
$\rightarrow$$\;\;\;\;$({\rm{nthn}}\,({\it{i\/}}, {\rm{board-$>$number}}\,({\it{board\/}})\index{board-$>$number})\index{nthn} $<$ {\tt{3}})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  winp1-is-winp1n \\ \index{winp1-is-winp1n}
\=\+(\=\+{\rm{boardp}}\,({\it{board\/}})\index{boardp} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $<$ {\rm{length}}\,({\it{board\/}})\index{length}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $<$ {\rm{length}}\,({\it{board\/}})\index{length}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $<$ {\rm{length}}\,({\it{board\/}})\index{length}) \\ 
$\wedge$$\;\;\;\;$(({\it{x\/}} $=$ {\tt{'}}{\tt{x}}) $\vee$ ({\it{x\/}} $=$ {\tt{'}}{\tt{o}})))\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{winp1}}\,({\it{x\/}}, {\rm{list}}\,({\it{i\/}}, {\it{j\/}}, {\it{k\/}}), {\it{board\/}})\index{winp1} \\ 
$=$$\;\;\;\;${\rm{winp1n}}\,({\rm{piece-$>$number}}\,({\it{x\/}})\index{piece-$>$number}, {\it{i\/}}, {\it{j\/}}, {\it{k\/}}, {\rm{board-$>$number}}\,({\it{board\/}})\index{board-$>$number})\index{winp1n})\-\-
\end{tabbing}

\begin{tabbing}{\sc Definition}:\index{winpn} \\  
\=\+{\rm{winpn}}\,({\it{x\/}}, {\it{board\/}})\index{winpn} \\ 
$=$$\;\;\;\;$(\=\+{\rm{winp1n}}\,({\it{x\/}}, {\tt{0}}, {\tt{1}}, {\tt{2}}, {\it{board\/}})\index{winp1n} \\ 
$\vee$$\;\;\;\;${\rm{winp1n}}\,({\it{x\/}}, {\tt{3}}, {\tt{4}}, {\tt{5}}, {\it{board\/}})\index{winp1n} \\ 
$\vee$$\;\;\;\;${\rm{winp1n}}\,({\it{x\/}}, {\tt{6}}, {\tt{7}}, {\tt{8}}, {\it{board\/}})\index{winp1n} \\ 
$\vee$$\;\;\;\;${\rm{winp1n}}\,({\it{x\/}}, {\tt{0}}, {\tt{3}}, {\tt{6}}, {\it{board\/}})\index{winp1n} \\ 
$\vee$$\;\;\;\;${\rm{winp1n}}\,({\it{x\/}}, {\tt{1}}, {\tt{4}}, {\tt{7}}, {\it{board\/}})\index{winp1n} \\ 
$\vee$$\;\;\;\;${\rm{winp1n}}\,({\it{x\/}}, {\tt{2}}, {\tt{5}}, {\tt{8}}, {\it{board\/}})\index{winp1n} \\ 
$\vee$$\;\;\;\;${\rm{winp1n}}\,({\it{x\/}}, {\tt{0}}, {\tt{4}}, {\tt{8}}, {\it{board\/}})\index{winp1n} \\ 
$\vee$$\;\;\;\;${\rm{winp1n}}\,({\it{x\/}}, {\tt{2}}, {\tt{4}}, {\tt{6}}, {\it{board\/}})\index{winp1n})\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  winp-is-winpn \\ \index{winp-is-winpn}
\=\+(({\rm{length}}\,({\it{board\/}})\index{length} $=$ {\tt{9}}) $\wedge$ {\rm{boardp}}\,({\it{board\/}})\index{boardp} $\wedge$ (({\it{x\/}} $=$ {\tt{'}}{\tt{x}}) $\vee$ ({\it{x\/}} $=$ {\tt{'}}{\tt{o}}))) \\ 
$\rightarrow$$\;\;\;\;$({\rm{winp}}\,({\it{x\/}}, {\it{board\/}})\index{winp} $=$ {\rm{winpn}}\,({\rm{piece-$>$number}}\,({\it{x\/}})\index{piece-$>$number}, {\rm{board-$>$number}}\,({\it{board\/}})\index{board-$>$number})\index{winpn})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  quotient-4-decreases \\ \index{quotient-4-decreases}
({\it{n\/}} $\not\simeq {\tt{0}}$) $\rightarrow$ (({\it{n\/}} $\div$ {\tt{4}}) $<$ {\it{n\/}})
\end{tabbing}

\begin{verbatim}; I originally defined pick-move1n as follows:
; (defn pick-move1n (board i)
;   (cond ((zerop board) 0)
;         ((zerop (remainder board 4)) i)
;         (t (pick-move1n (quotient board 4) (add1 i)))))
; And to my great surprise, this function isn't anything
; like pick-move1!  Consider the fact that
; on (list 'x 'o f f f f f f f) pick-move1 returns 2 but
; (on the numeric image of the above board) pick-move1n
; returns 0.  This seems to me a likely bug to have occurred
; in a program attempting to implement this.

\end{verbatim}\begin{tabbing}{\sc Definition}:\index{pick-move1n} \\  
\=\+{\rm{pick-move1n}}\,({\it{board\/}}, {\it{i\/}}, {\it{len\/}})\index{pick-move1n} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{len\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\tt{0}} \\ 
{\bf elseif }({\it{board\/}} ${}${\bf{mod}}${}$ {\tt{4}}) $\simeq {\tt{0}}$$\;\;${\bf then }{\it{i\/}} \\ 
{\bf else }{\rm{pick-move1n}}\,({\it{board\/}} $\div$ {\tt{4}}, $1\;+$ {\it{i\/}}, {\it{len\/}} $-\;1$)\index{pick-move1n}$\;${\bf  endif}\-\-
\end{tabbing}

\begin{tabbing}{\sc Definition}:\index{board-$>$number-alist} \\  
\=\+{\rm{board-$>$number-alist}}\,({\it{alist\/}})\index{board-$>$number-alist} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{alist\/}} $\simeq\;${{\bf{nil}}}${}$$\;\;${\bf then }{\bf{nil}} \\ 
{\bf else }{\rm{cons}}\,(\=\+{\rm{cons}}\,({\rm{board-$>$number}}\,({\rm{caar}}\,({\it{alist\/}}))\index{board-$>$number}, {\rm{cdar}}\,({\it{alist\/}})), \\ 
{\rm{board-$>$number-alist}}\,({\rm{cdr}}\,({\it{alist\/}}))\index{board-$>$number-alist})\-$\;${\bf  endif}\-\-
\end{tabbing}

\begin{tabbing}{\sc Definition}:\index{boardp-alistp} \\  
\=\+{\rm{boardp-alistp}}\,({\it{alist\/}})\index{boardp-alistp} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{alist\/}} $\simeq\;${{\bf{nil}}}${}$$\;\;${\bf then }{\it{alist\/}} $=$ {\bf{nil}} \\ 
{\bf else }\=\+{\rm{boardp}}\,({\rm{caar}}\,({\it{alist\/}}))\index{boardp} \\ 
$\wedge$$\;\;\;\;$({\rm{length}}\,({\rm{caar}}\,({\it{alist\/}}))\index{length} $=$ {\tt{9}}) \\ 
$\wedge$$\;\;\;\;${\rm{boardp-alistp}}\,({\rm{cdr}}\,({\it{alist\/}}))\index{boardp-alistp}\-$\;${\bf  endif}\-\-
\end{tabbing}

\begin{tabbing}{\sc Definition}:\index{pick-moven} \\  
\=\+{\rm{pick-moven}}\,({\it{board\/}})\index{pick-moven} \\ 
$=$$\;\;\;\;$\=\+{\bf let }\=\+{\it{book-move\/}}{\bf $\;$ be$\;\;$}{\rm{assoc}}\,(\=\+{\it{board\/}}, \\ 
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{\tt{(}}{\tt{5656}}{\tt{ .\ }}{\tt{7}}{\tt{)}} \\ 
{\tt{(}}{\tt{4696}}{\tt{ .\ }}{\tt{7}}{\tt{)}} \\ 
{\tt{(}}{\tt{4633}}{\tt{ .\ }}{\tt{7}}{\tt{)}} \\ 
{\tt{(}}{\tt{4624}}{\tt{ .\ }}{\tt{1}}{\tt{)}} \\ 
{\tt{(}}{\tt{136729}}{\tt{ .\ }}{\tt{7}}{\tt{)}} \\ 
{\tt{(}}{\tt{132697}}{\tt{ .\ }}{\tt{7}}{\tt{)}} \\ 
{\tt{(}}{\tt{1552}}{\tt{ .\ }}{\tt{8}}{\tt{)}} \\ 
{\tt{(}}{\tt{84562}}{\tt{ .\ }}{\tt{6}}{\tt{)}} \\ 
{\tt{(}}{\tt{72274}}{\tt{ .\ }}{\tt{7}}{\tt{)}} \\ 
{\tt{(}}{\tt{66130}}{\tt{ .\ }}{\tt{5}}{\tt{)}} \\ 
{\tt{(}}{\tt{16978}}{\tt{ .\ }}{\tt{8}}{\tt{)}} \\ 
{\tt{(}}{\tt{4690}}{\tt{ .\ }}{\tt{8}}{\tt{)}} \\ 
{\tt{(}}{\tt{1618}}{\tt{ .\ }}{\tt{8}}{\tt{)}} \\ 
{\tt{(}}{\tt{598}}{\tt{ .\ }}{\tt{8}}{\tt{)}} \\ 
{\tt{(}}{\tt{66070}}{\tt{ .\ }}{\tt{5}}{\tt{)}} \\ 
{\tt{(}}{\tt{16918}}{\tt{ .\ }}{\tt{8}}{\tt{)}} \\ 
{\tt{(}}{\tt{4630}}{\tt{ .\ }}{\tt{8}}{\tt{)}} \\ 
{\tt{(}}{\tt{1558}}{\tt{ .\ }}{\tt{8}}{\tt{)}} \\ 
{\tt{(}}{\tt{598}}{\tt{ .\ }}{\tt{8}}{\tt{)}} \\ 
{\tt{(}}{\tt{66073}}{\tt{ .\ }}{\tt{7}}{\tt{)}} \\ 
{\tt{(}}{\tt{18073}}{\tt{ .\ }}{\tt{8}}{\tt{)}} \\ 
{\tt{(}}{\tt{4633}}{\tt{ .\ }}{\tt{7}}{\tt{)}} \\ 
{\tt{(}}{\tt{1561}}{\tt{ .\ }}{\tt{7}}{\tt{)}} \\ 
{\tt{(}}{\tt{601}}{\tt{ .\ }}{\tt{7}}{\tt{)}} \\ 
{\tt{(}}{\tt{16}}{\tt{ .\ }}{\tt{4}}{\tt{)}} \\ 
{\tt{(}}{\tt{82468}}{\tt{ .\ }}{\tt{6}}{\tt{)}} \\ 
{\tt{(}}{\tt{70180}}{\tt{ .\ }}{\tt{7}}{\tt{)}} \\ 
{\tt{(}}{\tt{67108}}{\tt{ .\ }}{\tt{6}}{\tt{)}} \\ 
{\tt{(}}{\tt{66148}}{\tt{ .\ }}{\tt{6}}{\tt{)}} \\ 
{\tt{(}}{\tt{66085}}{\tt{ .\ }}{\tt{6}}{\tt{)}} \\ 
{\tt{(}}{\tt{66052}}{\tt{ .\ }}{\tt{2}}{\tt{)}} \\ 
{\tt{(}}{\tt{82468}}{\tt{ .\ }}{\tt{6}}{\tt{)}} \\ 
{\tt{(}}{\tt{152164}}{\tt{ .\ }}{\tt{5}}{\tt{)}} \\ 
{\tt{(}}{\tt{152101}}{\tt{ .\ }}{\tt{5}}{\tt{)}} \\ 
{\tt{(}}{\tt{21028}}{\tt{ .\ }}{\tt{8}}{\tt{)}} \\ 
{\tt{(}}{\tt{17956}}{\tt{ .\ }}{\tt{6}}{\tt{)}} \\ 
{\tt{(}}{\tt{16996}}{\tt{ .\ }}{\tt{6}}{\tt{)}} \\ 
{\tt{(}}{\tt{16933}}{\tt{ .\ }}{\tt{6}}{\tt{)}} \\ 
{\tt{(}}{\tt{16900}}{\tt{ .\ }}{\tt{2}}{\tt{)}} \\ 
{\tt{(}}{\tt{102934}}{\tt{ .\ }}{\tt{5}}{\tt{)}} \\ 
{\tt{(}}{\tt{70150}}{\tt{ .\ }}{\tt{7}}{\tt{)}} \\ 
{\tt{(}}{\tt{20998}}{\tt{ .\ }}{\tt{8}}{\tt{)}} \\ 
{\tt{(}}{\tt{5638}}{\tt{ .\ }}{\tt{8}}{\tt{)}} \\ 
{\tt{(}}{\tt{4678}}{\tt{ .\ }}{\tt{8}}{\tt{)}} \\ 
{\tt{(}}{\tt{4630}}{\tt{ .\ }}{\tt{8}}{\tt{)}} \\ 
{\tt{(}}{\tt{67108}}{\tt{ .\ }}{\tt{6}}{\tt{)}} \\ 
{\tt{(}}{\tt{17956}}{\tt{ .\ }}{\tt{6}}{\tt{)}} \\ 
{\tt{(}}{\tt{71206}}{\tt{ .\ }}{\tt{7}}{\tt{)}} \\ 
{\tt{(}}{\tt{22054}}{\tt{ .\ }}{\tt{8}}{\tt{)}} \\ 
{\tt{(}}{\tt{5734}}{\tt{ .\ }}{\tt{8}}{\tt{)}} \\ 
{\tt{(}}{\tt{1636}}{\tt{ .\ }}{\tt{6}}{\tt{)}} \\ 
{\tt{(}}{\tt{1573}}{\tt{ .\ }}{\tt{6}}{\tt{)}} \\ 
{\tt{(}}{\tt{1540}}{\tt{ .\ }}{\tt{2}}{\tt{)}} \\ 
{\tt{(}}{\tt{82534}}{\tt{ .\ }}{\tt{6}}{\tt{)}} \\ 
{\tt{(}}{\tt{70246}}{\tt{ .\ }}{\tt{7}}{\tt{)}} \\ 
{\tt{(}}{\tt{16966}}{\tt{ .\ }}{\tt{8}}{\tt{)}} \\ 
{\tt{(}}{\tt{4678}}{\tt{ .\ }}{\tt{8}}{\tt{)}} \\ 
{\tt{(}}{\tt{1606}}{\tt{ .\ }}{\tt{8}}{\tt{)}} \\ 
{\tt{(}}{\tt{598}}{\tt{ .\ }}{\tt{8}}{\tt{)}} \\ 
{\tt{(}}{\tt{66070}}{\tt{ .\ }}{\tt{5}}{\tt{)}} \\ 
{\tt{(}}{\tt{16918}}{\tt{ .\ }}{\tt{8}}{\tt{)}} \\ 
{\tt{(}}{\tt{4630}}{\tt{ .\ }}{\tt{8}}{\tt{)}} \\ 
{\tt{(}}{\tt{1558}}{\tt{ .\ }}{\tt{8}}{\tt{)}} \\ 
{\tt{(}}{\tt{598}}{\tt{ .\ }}{\tt{8}}{\tt{)}} \\ 
{\tt{(}}{\tt{66085}}{\tt{ .\ }}{\tt{6}}{\tt{)}} \\ 
{\tt{(}}{\tt{16933}}{\tt{ .\ }}{\tt{6}}{\tt{)}} \\ 
{\tt{(}}{\tt{1573}}{\tt{ .\ }}{\tt{6}}{\tt{)}} \\ 
{\tt{(}}{\tt{613}}{\tt{ .\ }}{\tt{6}}{\tt{)}} \\ 
{\tt{(}}{\tt{4}}{\tt{ .\ }}{\tt{4}}{\tt{)}} \\ 
{\tt{(}}{\tt{90649}}{\tt{ .\ }}{\tt{5}}{\tt{)}} \\ 
{\tt{(}}{\tt{82441}}{\tt{ .\ }}{\tt{6}}{\tt{)}} \\ 
{\tt{(}}{\tt{70153}}{\tt{ .\ }}{\tt{7}}{\tt{)}} \\ 
{\tt{(}}{\tt{67081}}{\tt{ .\ }}{\tt{7}}{\tt{)}} \\ 
{\tt{(}}{\tt{66121}}{\tt{ .\ }}{\tt{7}}{\tt{)}} \\ 
{\tt{(}}{\tt{66073}}{\tt{ .\ }}{\tt{7}}{\tt{)}} \\ 
{\tt{(}}{\tt{90625}}{\tt{ .\ }}{\tt{2}}{\tt{)}} \\ 
{\tt{(}}{\tt{26113}}{\tt{ .\ }}{\tt{2}}{\tt{)}} \\ 
{\tt{(}}{\tt{25153}}{\tt{ .\ }}{\tt{2}}{\tt{)}} \\ 
{\tt{(}}{\tt{90649}}{\tt{ .\ }}{\tt{5}}{\tt{)}} \\ 
{\tt{(}}{\tt{26137}}{\tt{ .\ }}{\tt{8}}{\tt{)}} \\ 
{\tt{(}}{\tt{16897}}{\tt{ .\ }}{\tt{6}}{\tt{)}} \\ 
{\tt{(}}{\tt{70273}}{\tt{ .\ }}{\tt{5}}{\tt{)}} \\ 
{\tt{(}}{\tt{21121}}{\tt{ .\ }}{\tt{5}}{\tt{)}} \\ 
{\tt{(}}{\tt{71305}}{\tt{ .\ }}{\tt{7}}{\tt{)}} \\ 
{\tt{(}}{\tt{22153}}{\tt{ .\ }}{\tt{8}}{\tt{)}} \\ 
{\tt{(}}{\tt{4753}}{\tt{ .\ }}{\tt{5}}{\tt{)}} \\ 
{\tt{(}}{\tt{4741}}{\tt{ .\ }}{\tt{5}}{\tt{)}} \\ 
{\tt{(}}{\tt{4609}}{\tt{ .\ }}{\tt{3}}{\tt{)}} \\ 
{\tt{(}}{\tt{67105}}{\tt{ .\ }}{\tt{6}}{\tt{)}} \\ 
{\tt{(}}{\tt{17953}}{\tt{ .\ }}{\tt{6}}{\tt{)}} \\ 
{\tt{(}}{\tt{71329}}{\tt{ .\ }}{\tt{7}}{\tt{)}} \\ 
{\tt{(}}{\tt{22177}}{\tt{ .\ }}{\tt{8}}{\tt{)}} \\ 
{\tt{(}}{\tt{5665}}{\tt{ .\ }}{\tt{3}}{\tt{)}} \\ 
{\tt{(}}{\tt{1633}}{\tt{ .\ }}{\tt{6}}{\tt{)}} \\ 
{\tt{(}}{\tt{1573}}{\tt{ .\ }}{\tt{6}}{\tt{)}} \\ 
{\tt{(}}{\tt{1537}}{\tt{ .\ }}{\tt{2}}{\tt{)}} \\ 
{\tt{(}}{\tt{74305}}{\tt{ .\ }}{\tt{2}}{\tt{)}} \\ 
{\tt{(}}{\tt{25153}}{\tt{ .\ }}{\tt{2}}{\tt{)}} \\ 
{\tt{(}}{\tt{9793}}{\tt{ .\ }}{\tt{2}}{\tt{)}} \\ 
{\tt{(}}{\tt{74329}}{\tt{ .\ }}{\tt{7}}{\tt{)}} \\ 
{\tt{(}}{\tt{577}}{\tt{ .\ }}{\tt{6}}{\tt{)}} \\ 
{\tt{(}}{\tt{66073}}{\tt{ .\ }}{\tt{7}}{\tt{)}} \\ 
{\tt{(}}{\tt{18073}}{\tt{ .\ }}{\tt{8}}{\tt{)}} \\ 
{\tt{(}}{\tt{4633}}{\tt{ .\ }}{\tt{7}}{\tt{)}} \\ 
{\tt{(}}{\tt{1561}}{\tt{ .\ }}{\tt{7}}{\tt{)}} \\ 
{\tt{(}}{\tt{601}}{\tt{ .\ }}{\tt{7}}{\tt{)}} \\ 
{\tt{(}}{\tt{66085}}{\tt{ .\ }}{\tt{6}}{\tt{)}} \\ 
{\tt{(}}{\tt{16933}}{\tt{ .\ }}{\tt{6}}{\tt{)}} \\ 
{\tt{(}}{\tt{1573}}{\tt{ .\ }}{\tt{6}}{\tt{)}} \\ 
{\tt{(}}{\tt{613}}{\tt{ .\ }}{\tt{6}}{\tt{)}} \\ 
{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{4}}{\tt{)}}{\tt{)}}\-)\-\- \\ 
{\bf in} \\ 
\=\+{\bf if }{\it{book-move\/}}$\;\;${\bf then }{\rm{cdr}}\,({\it{book-move\/}}) \\ 
{\bf else }{\rm{pick-move1n}}\,({\it{board\/}}, {\tt{0}}, {\tt{9}})\index{pick-move1n}$\;${\bf  endif}\-$\;${\bf  endlet}\-\-
\end{tabbing}


\noindent{\sc Event}:   For efficiency, compile those definitions not yet compiled.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  board-$>$number-one-to-one \\ \index{board-$>$number-one-to-one}
\=\+(({\rm{length}}\,({\it{board1\/}})\index{length} $=$ {\rm{length}}\,({\it{board2\/}})\index{length}) $\wedge$ {\rm{boardp}}\,({\it{board1\/}})\index{boardp} $\wedge$ {\rm{boardp}}\,({\it{board2\/}})\index{boardp}) \\ 
$\rightarrow$$\;\;\;\;$(\=\+({\rm{board-$>$number}}\,({\it{board1\/}})\index{board-$>$number} $=$ {\rm{board-$>$number}}\,({\it{board2\/}})\index{board-$>$number}) \\ 
$=$$\;\;\;\;$({\it{board1\/}} $=$ {\it{board2\/}}))\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  assoc-board-$>$number \\ \index{assoc-board-$>$number}
\=\+(({\rm{length}}\,({\it{board\/}})\index{length} $=$ {\tt{9}}) $\wedge$ {\rm{boardp}}\,({\it{board\/}})\index{boardp} $\wedge$ {\rm{boardp-alistp}}\,({\it{alist\/}})\index{boardp-alistp}) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{assoc}}\,({\it{board\/}}, {\it{alist\/}}) \\ 
$\leftrightarrow$$\;\;\;\;${\rm{assoc}}\,({\rm{board-$>$number}}\,({\it{board\/}})\index{board-$>$number}, {\rm{board-$>$number-alist}}\,({\it{alist\/}})\index{board-$>$number-alist}))\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  cdr-assoc-board-$>$number \\ \index{cdr-assoc-board-$>$number}
\=\+(({\rm{length}}\,({\it{board\/}})\index{length} $=$ {\tt{9}}) $\wedge$ {\rm{boardp}}\,({\it{board\/}})\index{boardp} $\wedge$ {\rm{boardp-alistp}}\,({\it{alist\/}})\index{boardp-alistp}) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{cdr}}\,({\rm{assoc}}\,({\it{board\/}}, {\it{alist\/}})) \\ 
$=$$\;\;\;\;${\rm{cdr}}\,({\rm{assoc}}\,({\rm{board-$>$number}}\,({\it{board\/}})\index{board-$>$number}, {\rm{board-$>$number-alist}}\,({\it{alist\/}})\index{board-$>$number-alist})))\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  pick-move1-is-pick-move1n \\ \index{pick-move1-is-pick-move1n}
\=\+{\rm{boardp}}\,({\it{board\/}})\index{boardp} \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{pick-move1}}\,({\it{board\/}}, {\it{i\/}})\index{pick-move1} \\ 
$=$$\;\;\;\;${\rm{pick-move1n}}\,({\rm{board-$>$number}}\,({\it{board\/}})\index{board-$>$number}, {\it{i\/}}, {\rm{length}}\,({\it{board\/}})\index{length})\index{pick-move1n})\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  pick-move-is-pick-moven \\ \index{pick-move-is-pick-moven}
\=\+(({\rm{length}}\,({\it{board\/}})\index{length} $=$ {\tt{9}}) $\wedge$ {\rm{boardp}}\,({\it{board\/}})\index{boardp}) \\ 
$\rightarrow$$\;\;\;\;$({\rm{pick-move}}\,({\it{board\/}})\index{pick-move} $=$ {\rm{pick-moven}}\,({\rm{board-$>$number}}\,({\it{board\/}})\index{board-$>$number})\index{pick-moven})\-
\end{tabbing}


\noindent{\sc Event}:   Disable pick-moven.

\addvspace{18pt}

\begin{tabbing}{\sc Definition}:\index{tic-tac-toe1-b} \\  
\=\+{\rm{tic-tac-toe1-b}}\,({\it{xmoves\/}}, {\it{board\/}}, {\it{i\/}})\index{tic-tac-toe1-b} \\ 
$=$$\;\;\;\;$\=\+{\bf let }\=\+{\it{board1\/}}{\bf $\;$ be$\;\;$}{\rm{moven}}\,({\tt{1}}, {\rm{car}}\,({\it{xmoves\/}}), {\it{board\/}})\index{moven}\- \\ 
{\bf in} \\ 
\=\+{\bf if }{\rm{legalpn}}\,({\rm{car}}\,({\it{xmoves\/}}), {\it{board\/}})\index{legalpn} \\ 
{\bf then }\=\+{\bf if }{\it{i\/}} $<$ {\tt{9}} \\ 
{\bf then }\=\+{\bf let }\=\+{\it{board2\/}}{\bf $\;$ be$\;\;$}{\rm{moven}}\,({\tt{2}}, {\rm{pick-moven}}\,({\it{board1\/}})\index{pick-moven}, {\it{board1\/}})\index{moven}\- \\ 
{\bf in} \\ 
\=\+{\bf if }{\rm{winpn}}\,({\tt{2}}, {\it{board2\/}})\index{winpn} \\ 
{\bf then }{\rm{list}}\,({\it{board1\/}}, {\it{board2\/}}, {\tt{-2}}) \\ 
{\bf else }{\rm{cons}}\,(\=\+{\it{board1\/}}, \\ 
{\rm{cons}}\,(\=\+{\it{board2\/}}, \\ 
{\rm{tic-tac-toe1-b}}\,(\=\+{\rm{cdr}}\,({\it{xmoves\/}}), \\ 
{\it{board2\/}}, \\ 
$1\;+$ ($1\;+$ {\it{i\/}}))\-\index{tic-tac-toe1-b})\-)\-$\;${\bf  endif}\-$\;${\bf  endlet}\- \\ 
{\bf else }{\rm{list}}\,({\it{board1\/}}, {\tt{-1}})$\;${\bf  endif}\- \\ 
{\bf else }{\rm{list}}\,({\it{board1\/}}, {\tt{-2}})$\;${\bf  endif}\-$\;${\bf  endlet}\-\-
\end{tabbing}

\begin{tabbing}{\sc Definition}:\index{tic-tac-toe-b} \\  
{\rm{tic-tac-toe-b}}\,({\it{xmoves\/}})\index{tic-tac-toe-b} $=$ {\rm{cons}}\,({\tt{0}}, {\rm{tic-tac-toe1-b}}\,({\it{xmoves\/}}, {\tt{0}}, {\tt{1}})\index{tic-tac-toe1-b})
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  legalpn-implies-lessp \\ \index{legalpn-implies-lessp}
\=\+({\rm{legalpn}}\,({\it{i\/}}, {\rm{board-$>$number}}\,({\it{board\/}})\index{board-$>$number})\index{legalpn} $\wedge$ ({\rm{length}}\,({\it{board\/}})\index{length} $=$ {\tt{9}})) \\ 
$\rightarrow$$\;\;\;\;$(({\it{i\/}} $<$ {\tt{9}}) $=$ {\bf{t}})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  length-number-$>$board \\ \index{length-number-$>$board}
{\rm{length}}\,({\rm{number-$>$board}}\,({\it{n\/}}, {\it{len\/}})\index{number-$>$board})\index{length} $=$ {\rm{fix}}\,({\it{len\/}})
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  boardp-number-board \\ \index{boardp-number-board}
{\rm{boardp}}\,({\rm{number-$>$board}}\,({\it{n\/}}, {\it{len\/}})\index{number-$>$board})\index{boardp}
\end{tabbing}

\begin{tabbing}{\sc Definition}:\index{boardpn} \\  
\=\+{\rm{boardpn}}\,({\it{n\/}})\index{boardpn} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{n\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\bf{t}} \\ 
{\bf elseif }({\it{n\/}} ${}${\bf{mod}}${}$ {\tt{4}}) $<$ {\tt{3}}$\;\;${\bf then }{\rm{boardpn}}\,({\it{n\/}} $\div$ {\tt{4}})\index{boardpn} \\ 
{\bf else }{\bf{f}}$\;${\bf  endif}\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  boardpn-board-$>$number \\ \index{boardpn-board-$>$number}
{\rm{boardp}}\,({\it{board\/}})\index{boardp} $\rightarrow$ {\rm{boardpn}}\,({\rm{board-$>$number}}\,({\it{board\/}})\index{board-$>$number})\index{boardpn}
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  boardpn-moven \\ \index{boardpn-moven}
\=\+(({\it{board\/}} $\in\;${\bf{N}}${}$) $\wedge$ {\rm{boardpn}}\,({\it{board\/}})\index{boardpn} $\wedge$ (({\it{x\/}} $=$ {\tt{1}}) $\vee$ ({\it{x\/}} $=$ {\tt{2}}))) \\ 
$\rightarrow$$\;\;\;\;${\rm{boardpn}}\,({\rm{moven}}\,({\it{x\/}}, {\it{i\/}}, {\it{board\/}})\index{moven})\index{boardpn}\-
\end{tabbing}

\begin{verbatim}; The following function is not the length of the board because the
; high numbered squares might be empty!

\end{verbatim}\begin{tabbing}{\sc Definition}:\index{lengthn} \\  
\=\+{\rm{lengthn}}\,({\it{board\/}})\index{lengthn} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{board\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\tt{0}} \\ 
{\bf else }$1\;+$ {\rm{lengthn}}\,({\it{board\/}} $\div$ {\tt{4}})\index{lengthn}$\;${\bf  endif}\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  number-$>$board-$>$number \\ \index{number-$>$board-$>$number}
\=\+(({\it{board\/}} $\in\;${\bf{N}}${}$) $\wedge$ {\rm{boardpn}}\,({\it{board\/}})\index{boardpn} $\wedge$ ({\it{len\/}} $\not<$ {\rm{lengthn}}\,({\it{board\/}})\index{lengthn})) \\ 
$\rightarrow$$\;\;\;\;$({\rm{board-$>$number}}\,({\rm{number-$>$board}}\,({\it{board\/}}, {\it{len\/}})\index{number-$>$board})\index{board-$>$number} $=$ {\it{board\/}})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  lengthn-board-$>$number \\ \index{lengthn-board-$>$number}
{\rm{length}}\,({\it{board\/}})\index{length} $\not<$ {\rm{lengthn}}\,({\rm{board-$>$number}}\,({\it{board\/}})\index{board-$>$number})\index{lengthn}
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  lengthn-moven \\ \index{lengthn-moven}
\=\+(({\it{x\/}} $=$ {\tt{1}}) $\vee$ ({\it{x\/}} $=$ {\tt{2}})) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{lengthn}}\,({\rm{moven}}\,({\it{x\/}}, {\it{i\/}}, {\it{board\/}})\index{moven})\index{lengthn} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{i\/}} $<$ {\rm{lengthn}}\,({\it{board\/}})\index{lengthn}$\;\;${\bf then }{\rm{lengthn}}\,({\it{board\/}})\index{lengthn} \\ 
{\bf else }$1\;+$ {\it{i\/}}$\;${\bf  endif}\-)\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  lengthn-moven-hack-1 \\ \index{lengthn-moven-hack-1}
\=\+(({\it{i\/}} $<$ {\tt{9}}) $\wedge$ {\rm{boardp}}\,({\it{board\/}})\index{boardp} $\wedge$ ({\rm{length}}\,({\it{board\/}})\index{length} $=$ {\tt{9}})) \\ 
$\rightarrow$$\;\;\;\;$({\tt{9}} $\not<$ {\rm{lengthn}}\,({\rm{moven}}\,({\tt{1}}, {\it{i\/}}, {\rm{board-$>$number}}\,({\it{board\/}})\index{board-$>$number})\index{moven})\index{lengthn})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  lessp-pick-moven \\ \index{lessp-pick-moven}
\=\+(({\it{i\/}} $<$ {\tt{9}}) $\wedge$ {\rm{boardp}}\,({\it{board\/}})\index{boardp} $\wedge$ ({\rm{length}}\,({\it{board\/}})\index{length} $=$ {\tt{9}})) \\ 
$\rightarrow$$\;\;\;\;$({\rm{pick-moven}}\,({\rm{moven}}\,({\tt{1}}, {\it{i\/}}, {\rm{board-$>$number}}\,({\it{board\/}})\index{board-$>$number})\index{moven})\index{pick-moven} $<$ {\tt{9}})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  lengthn-moven-hack-2 \\ \index{lengthn-moven-hack-2}
\=\+(({\it{i\/}} $<$ {\tt{9}}) $\wedge$ {\rm{boardp}}\,({\it{board\/}})\index{boardp} $\wedge$ ({\rm{length}}\,({\it{board\/}})\index{length} $=$ {\tt{9}})) \\ 
$\rightarrow$$\;\;\;\;$({\tt{9}} $\not<$ {\rm{lengthn}}\,({\rm{moven}}\,(\=\+{\tt{2}}, \\ 
{\rm{pick-moven}}\,({\rm{moven}}\,({\tt{1}}, {\it{i\/}}, {\rm{board-$>$number}}\,({\it{board\/}})\index{board-$>$number})\index{moven})\index{pick-moven}, \\ 
{\rm{moven}}\,({\tt{1}}, {\it{i\/}}, {\rm{board-$>$number}}\,({\it{board\/}})\index{board-$>$number})\index{moven})\-\index{moven})\index{lengthn})\-
\end{tabbing}

\begin{verbatim}; The theorem prover does the wrong induction below.  It too chooses
; the one suggested by tic-tac-toe1-a, but it throws away the board
; instantiation because of the competition from board->number.

\end{verbatim}\begin{tabbing}{\sc Theorem}:  lemma-b1 \\ \index{lemma-b1}
\=\+({\rm{boardp}}\,({\it{board\/}})\index{boardp} $\wedge$ ({\rm{length}}\,({\it{board\/}})\index{length} $=$ {\tt{9}})) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{tic-tac-toe1-a}}\,({\it{xmoves\/}}, {\it{board\/}}, {\it{i\/}})\index{tic-tac-toe1-a} \\ 
$=$$\;\;\;\;${\rm{number-$>$board*}}\,(\=\+{\rm{tic-tac-toe1-b}}\,({\it{xmoves\/}}, {\rm{board-$>$number}}\,({\it{board\/}})\index{board-$>$number}, {\it{i\/}})\index{tic-tac-toe1-b}, \\ 
{\tt{9}})\-\index{number-$>$board*})\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  lemma-b \\ \index{lemma-b}
{\rm{tic-tac-toe-a}}\,({\it{xmoves\/}})\index{tic-tac-toe-a} $=$ {\rm{number-$>$board*}}\,({\rm{tic-tac-toe-b}}\,({\it{xmoves\/}})\index{tic-tac-toe-b}, {\tt{9}})\index{number-$>$board*}
\end{tabbing}

\begin{verbatim}; The next transformation will remove the consing of the list of
; boards and replace it by an accumulator treated as an array.
; Because of the explicit test that i is less than 9 -- and the
; knowledge that i starts at 0 and so never gets larger than 9 itself
; -- we know that an array of length 10 will suffice to hold all the
; boards we might generate and thus an array of length 11 will hold all
; the boards plus the final outcome.  We call the array t3.  Thus, in t3(0)
; through t3(9) one will find boards generated.  But we may not fill
; it because the game may end with a win.  We store the outcome in the
; last array element filled.  Thus, to recreate the game, make a list of
; boards form the elements of the array, starting at t3(0) and going through
; the one containing -2 or -1.

\end{verbatim}\begin{tabbing}{\sc Definition}:\index{array-$>$lst} \\  
\=\+{\rm{array-$>$lst}}\,({\it{i\/}}, {\it{array\/}})\index{array-$>$lst} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{i\/}} $\not<$ {\rm{length}}\,({\it{array\/}})\index{length}$\;\;${\bf then }{\bf{nil}} \\ 
{\bf elseif }{\rm{nth}}\,({\it{i\/}}, {\it{array\/}})\index{nth} $\in\;${\bf{N}}${}$ \\ 
{\bf then }{\rm{cons}}\,({\rm{nth}}\,({\it{i\/}}, {\it{array\/}})\index{nth}, {\rm{array-$>$lst}}\,($1\;+$ {\it{i\/}}, {\it{array\/}})\index{array-$>$lst}) \\ 
{\bf else }{\rm{list}}\,({\rm{nth}}\,({\it{i\/}}, {\it{array\/}})\index{nth})$\;${\bf  endif}\-\-
\end{tabbing}

\begin{tabbing}{\sc Definition}:\index{put} \\  
\=\+{\rm{put}}\,({\it{val\/}}, {\it{n\/}}, {\it{lst\/}})\index{put} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{n\/}} $\simeq {\tt{0}}$ \\ 
{\bf then }\=\+{\bf if }{\rm{listp}}\,({\it{lst\/}})$\;\;${\bf then }{\rm{cons}}\,({\it{val\/}}, {\rm{cdr}}\,({\it{lst\/}})) \\ 
{\bf else }{\rm{list}}\,({\it{val\/}})$\;${\bf  endif}\- \\ 
{\bf else }{\rm{cons}}\,({\rm{car}}\,({\it{lst\/}}), {\rm{put}}\,({\it{val\/}}, {\it{n\/}} $-\;1$, {\rm{cdr}}\,({\it{lst\/}}))\index{put})$\;${\bf  endif}\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  nth-put \\ \index{nth-put}
\=\+(({\it{i\/}} $\in\;${\bf{N}}${}$) $\wedge$ ({\it{j\/}} $\in\;${\bf{N}}${}$) $\wedge$ ({\it{j\/}} $<$ {\rm{length}}\,({\it{array\/}})\index{length})) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{nth}}\,({\it{i\/}}, {\rm{put}}\,({\it{val\/}}, {\it{j\/}}, {\it{array\/}})\index{put})\index{nth} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{i\/}} $=$ {\it{j\/}}$\;\;${\bf then }{\it{val\/}} \\ 
{\bf else }{\rm{nth}}\,({\it{i\/}}, {\it{array\/}})\index{nth}$\;${\bf  endif}\-)\-\-
\end{tabbing}

\begin{tabbing}{\sc Definition}:\index{tic-tac-toe1-c} \\  
\=\+{\rm{tic-tac-toe1-c}}\,({\it{xmoves\/}}, {\it{board\/}}, {\it{i\/}}, {\it{t3\/}})\index{tic-tac-toe1-c} \\ 
$=$$\;\;\;\;$\=\+{\bf let }\=\+{\it{board1\/}}{\bf $\;$ be$\;\;$}{\rm{moven}}\,({\tt{1}}, {\rm{car}}\,({\it{xmoves\/}}), {\it{board\/}})\index{moven}\- \\ 
{\bf in} \\ 
\=\+{\bf if }{\rm{legalpn}}\,({\rm{car}}\,({\it{xmoves\/}}), {\it{board\/}})\index{legalpn} \\ 
{\bf then }\=\+{\bf if }{\it{i\/}} $<$ {\tt{9}} \\ 
{\bf then }\=\+{\bf let }\=\+{\it{board2\/}}{\bf $\;$ be$\;\;$}{\rm{moven}}\,({\tt{2}}, {\rm{pick-moven}}\,({\it{board1\/}})\index{pick-moven}, {\it{board1\/}})\index{moven}\- \\ 
{\bf in} \\ 
\=\+{\bf if }{\rm{winpn}}\,({\tt{2}}, {\it{board2\/}})\index{winpn} \\ 
{\bf then }{\rm{put}}\,(\=\+{\tt{-2}}, \\ 
$1\;+$ ($1\;+$ {\it{i\/}}), \\ 
{\rm{put}}\,({\it{board2\/}}, $1\;+$ {\it{i\/}}, {\rm{put}}\,({\it{board1\/}}, {\it{i\/}}, {\it{t3\/}})\index{put})\index{put})\-\index{put} \\ 
{\bf else }{\rm{tic-tac-toe1-c}}\,(\=\+{\rm{cdr}}\,({\it{xmoves\/}}), \\ 
{\it{board2\/}}, \\ 
$1\;+$ ($1\;+$ {\it{i\/}}), \\ 
{\rm{put}}\,(\=\+{\it{board2\/}}, \\ 
$1\;+$ {\it{i\/}}, \\ 
{\rm{put}}\,({\it{board1\/}}, {\it{i\/}}, {\it{t3\/}})\index{put})\-\index{put})\-\index{tic-tac-toe1-c}$\;${\bf  endif}\-$\;${\bf  endlet}\- \\ 
{\bf else }{\rm{put}}\,({\tt{-1}}, $1\;+$ {\it{i\/}}, {\rm{put}}\,({\it{board1\/}}, {\it{i\/}}, {\it{t3\/}})\index{put})\index{put}$\;${\bf  endif}\- \\ 
{\bf else }{\rm{put}}\,({\tt{-2}}, $1\;+$ {\it{i\/}}, {\rm{put}}\,({\it{board1\/}}, {\it{i\/}}, {\it{t3\/}})\index{put})\index{put}$\;${\bf  endif}\-$\;${\bf  endlet}\-\-
\end{tabbing}

\begin{tabbing}{\sc Definition}:\index{tic-tac-toe-c} \\  
\=\+{\rm{tic-tac-toe-c}}\,({\it{xmoves\/}})\index{tic-tac-toe-c} \\ 
$=$$\;\;\;\;${\rm{tic-tac-toe1-c}}\,({\it{xmoves\/}}, {\tt{0}}, {\tt{1}}, {\rm{list}}\,({\tt{0}}, {\tt{0}}, {\tt{0}}, {\tt{0}}, {\tt{0}}, {\tt{0}}, {\tt{0}}, {\tt{0}}, {\tt{0}}, {\tt{0}}, {\tt{0}}))\index{tic-tac-toe1-c}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  length-put \\ \index{length-put}
({\it{i\/}} $<$ {\rm{length}}\,({\it{array\/}})\index{length}) $\rightarrow$ ({\rm{length}}\,({\rm{put}}\,({\it{val\/}}, {\it{i\/}}, {\it{array\/}})\index{put})\index{length} $=$ {\rm{length}}\,({\it{array\/}})\index{length})
\end{tabbing}

\begin{verbatim}; The following is just the definition of array->lst and forces us to
; open it.  This would run forever if given the chance.  But there is
; another rule coming...

\end{verbatim}\begin{tabbing}{\sc Theorem}:  array-$>$lst-opener \\ \index{array-$>$lst-opener}
\=\+(({\it{j\/}} $<$ {\rm{length}}\,({\it{array\/}})\index{length}) $\wedge$ ({\it{i\/}} $<$ {\rm{length}}\,({\it{array\/}})\index{length})) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{array-$>$lst}}\,({\it{i\/}}, {\rm{put}}\,({\it{val\/}}, {\it{j\/}}, {\it{array\/}})\index{put})\index{array-$>$lst} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{nth}}\,({\it{i\/}}, {\rm{put}}\,({\it{val\/}}, {\it{j\/}}, {\it{array\/}})\index{put})\index{nth} $\in\;${\bf{N}}${}$ \\ 
{\bf then }{\rm{cons}}\,(\=\+{\rm{nth}}\,({\it{i\/}}, {\rm{put}}\,({\it{val\/}}, {\it{j\/}}, {\it{array\/}})\index{put})\index{nth}, \\ 
{\rm{array-$>$lst}}\,($1\;+$ {\it{i\/}}, {\rm{put}}\,({\it{val\/}}, {\it{j\/}}, {\it{array\/}})\index{put})\index{array-$>$lst})\- \\ 
{\bf else }{\rm{list}}\,({\rm{nth}}\,({\it{i\/}}, {\rm{put}}\,({\it{val\/}}, {\it{j\/}}, {\it{array\/}})\index{put})\index{nth})$\;${\bf  endif}\-)\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  array-$>$lst-opener-stopper \\ \index{array-$>$lst-opener-stopper}
\=\+(({\it{i\/}} $\in\;${\bf{N}}${}$) $\wedge$ ({\it{j\/}} $\in\;${\bf{N}}${}$) $\wedge$ ({\it{j\/}} $<$ {\rm{length}}\,({\it{array\/}})\index{length}) $\wedge$ ({\it{j\/}} $<$ {\it{i\/}})) \\ 
$\rightarrow$$\;\;\;\;$({\rm{array-$>$lst}}\,({\it{i\/}}, {\rm{put}}\,({\it{val\/}}, {\it{j\/}}, {\it{array\/}})\index{put})\index{array-$>$lst} $=$ {\rm{array-$>$lst}}\,({\it{i\/}}, {\it{array\/}})\index{array-$>$lst})\-
\end{tabbing}

\begin{verbatim}; It surprised me that I had to state that i is always odd below.  I mean,
; it shouldn't have surprised me, but it did.

\end{verbatim}\begin{tabbing}{\sc Theorem}:  length-tic-tac-toe1-c \\ \index{length-tic-tac-toe1-c}
\=\+(\=\+({\it{board\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;$(({\it{i\/}} ${}${\bf{mod}}${}$ {\tt{2}}) $=$ {\tt{1}}) \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $<$ {\tt{10}}) \\ 
$\wedge$$\;\;\;\;$({\rm{length}}\,({\it{t3\/}})\index{length} $=$ {\tt{11}}))\- \\ 
$\rightarrow$$\;\;\;\;$({\rm{length}}\,({\rm{tic-tac-toe1-c}}\,({\it{xmoves\/}}, {\it{board\/}}, {\it{i\/}}, {\it{t3\/}})\index{tic-tac-toe1-c})\index{length} $=$ {\tt{11}})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  nth-tic-tac-toe1-c \\ \index{nth-tic-tac-toe1-c}
\=\+(\=\+({\it{board\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;$(({\it{i\/}} ${}${\bf{mod}}${}$ {\tt{2}}) $=$ {\tt{1}}) \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $<$ {\tt{10}}) \\ 
$\wedge$$\;\;\;\;$({\rm{length}}\,({\it{t3\/}})\index{length} $=$ {\tt{11}}) \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $<$ {\it{i\/}}))\- \\ 
$\rightarrow$$\;\;\;\;$({\rm{nth}}\,({\it{n\/}}, {\rm{tic-tac-toe1-c}}\,({\it{xmoves\/}}, {\it{board\/}}, {\it{i\/}}, {\it{t3\/}})\index{tic-tac-toe1-c})\index{nth} $=$ {\rm{nth}}\,({\it{n\/}}, {\it{t3\/}})\index{nth})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  remainder-add1-add1-2 \\ \index{remainder-add1-add1-2}
(($1\;+$ ($1\;+$ {\it{i\/}})) ${}${\bf{mod}}${}$ {\tt{2}}) $=$ ({\it{i\/}} ${}${\bf{mod}}${}$ {\tt{2}})
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  odd-lessp-9-is-lessp-8 \\ \index{odd-lessp-9-is-lessp-8}
((({\it{i\/}} ${}${\bf{mod}}${}$ {\tt{2}}) $=$ {\tt{1}}) $\wedge$ ({\it{i\/}} $<$ {\tt{9}})) $\rightarrow$ (({\it{i\/}} $<$ {\tt{8}}) $=$ {\bf{t}})
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  lemma-c1 \\ \index{lemma-c1}
\=\+(\=\+({\it{board\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;$(({\it{i\/}} ${}${\bf{mod}}${}$ {\tt{2}}) $=$ {\tt{1}}) \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $<$ {\tt{10}}) \\ 
$\wedge$$\;\;\;\;$({\rm{length}}\,({\it{t3\/}})\index{length} $=$ {\tt{11}}))\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{tic-tac-toe1-b}}\,({\it{xmoves\/}}, {\it{board\/}}, {\it{i\/}})\index{tic-tac-toe1-b} \\ 
$=$$\;\;\;\;${\rm{array-$>$lst}}\,({\it{i\/}}, {\rm{tic-tac-toe1-c}}\,({\it{xmoves\/}}, {\it{board\/}}, {\it{i\/}}, {\it{t3\/}})\index{tic-tac-toe1-c})\index{array-$>$lst})\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  lemma-c \\ \index{lemma-c}
{\rm{tic-tac-toe-b}}\,({\it{xmoves\/}})\index{tic-tac-toe-b} $=$ {\rm{array-$>$lst}}\,({\tt{0}}, {\rm{tic-tac-toe-c}}\,({\it{xmoves\/}})\index{tic-tac-toe-c})\index{array-$>$lst}
\end{tabbing}

\begin{verbatim}; I should have thought of this earlier but it escaped me:  I need
; to treat xmoves as an array, not a list.  This means using another
; index, j, to get its elements.

\end{verbatim}\begin{tabbing}{\sc Definition}:\index{tic-tac-toe1-d} \\  
\=\+{\rm{tic-tac-toe1-d}}\,({\it{xmoves\/}}, {\it{j\/}}, {\it{board\/}}, {\it{i\/}}, {\it{t3\/}})\index{tic-tac-toe1-d} \\ 
$=$$\;\;\;\;$\=\+{\bf let }\=\+{\it{board1\/}}{\bf $\;$ be$\;\;$}{\rm{moven}}\,({\tt{1}}, {\rm{nth}}\,({\it{j\/}}, {\it{xmoves\/}})\index{nth}, {\it{board\/}})\index{moven}\- \\ 
{\bf in} \\ 
\=\+{\bf if }{\rm{legalpn}}\,({\rm{nth}}\,({\it{j\/}}, {\it{xmoves\/}})\index{nth}, {\it{board\/}})\index{legalpn} \\ 
{\bf then }\=\+{\bf if }{\it{i\/}} $<$ {\tt{9}} \\ 
{\bf then }\=\+{\bf let }\=\+{\it{board2\/}}{\bf $\;$ be$\;\;$}{\rm{moven}}\,({\tt{2}}, {\rm{pick-moven}}\,({\it{board1\/}})\index{pick-moven}, {\it{board1\/}})\index{moven}\- \\ 
{\bf in} \\ 
\=\+{\bf if }{\rm{winpn}}\,({\tt{2}}, {\it{board2\/}})\index{winpn} \\ 
{\bf then }{\rm{put}}\,(\=\+{\tt{-2}}, \\ 
$1\;+$ ($1\;+$ {\it{i\/}}), \\ 
{\rm{put}}\,({\it{board2\/}}, $1\;+$ {\it{i\/}}, {\rm{put}}\,({\it{board1\/}}, {\it{i\/}}, {\it{t3\/}})\index{put})\index{put})\-\index{put} \\ 
{\bf else }{\rm{tic-tac-toe1-d}}\,(\=\+{\it{xmoves\/}}, \\ 
$1\;+$ {\it{j\/}}, \\ 
{\it{board2\/}}, \\ 
$1\;+$ ($1\;+$ {\it{i\/}}), \\ 
{\rm{put}}\,(\=\+{\it{board2\/}}, \\ 
$1\;+$ {\it{i\/}}, \\ 
{\rm{put}}\,({\it{board1\/}}, {\it{i\/}}, {\it{t3\/}})\index{put})\-\index{put})\-\index{tic-tac-toe1-d}$\;${\bf  endif}\-$\;${\bf  endlet}\- \\ 
{\bf else }{\rm{put}}\,({\tt{-1}}, $1\;+$ {\it{i\/}}, {\rm{put}}\,({\it{board1\/}}, {\it{i\/}}, {\it{t3\/}})\index{put})\index{put}$\;${\bf  endif}\- \\ 
{\bf else }{\rm{put}}\,({\tt{-2}}, $1\;+$ {\it{i\/}}, {\rm{put}}\,({\it{board1\/}}, {\it{i\/}}, {\it{t3\/}})\index{put})\index{put}$\;${\bf  endif}\-$\;${\bf  endlet}\-\-
\end{tabbing}

\begin{tabbing}{\sc Definition}:\index{tic-tac-toe-d} \\  
\=\+{\rm{tic-tac-toe-d}}\,({\it{xmoves\/}})\index{tic-tac-toe-d} \\ 
$=$$\;\;\;\;${\rm{tic-tac-toe1-d}}\,({\it{xmoves\/}}, {\tt{0}}, {\tt{0}}, {\tt{1}}, {\rm{list}}\,({\tt{0}}, {\tt{0}}, {\tt{0}}, {\tt{0}}, {\tt{0}}, {\tt{0}}, {\tt{0}}, {\tt{0}}, {\tt{0}}, {\tt{0}}, {\tt{0}}))\index{tic-tac-toe1-d}\-
\end{tabbing}

\begin{tabbing}{\sc Definition}:\index{nth-cdr} \\  
\=\+{\rm{nth-cdr}}\,({\it{j\/}}, {\it{lst\/}})\index{nth-cdr} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{j\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\it{lst\/}} \\ 
{\bf else }{\rm{nth-cdr}}\,({\it{j\/}} $-\;1$, {\rm{cdr}}\,({\it{lst\/}}))\index{nth-cdr}$\;${\bf  endif}\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  car-nth-cdr \\ \index{car-nth-cdr}
{\rm{car}}\,({\rm{nth-cdr}}\,({\it{j\/}}, {\it{lst\/}})\index{nth-cdr}) $=$ {\rm{nth}}\,({\it{j\/}}, {\it{lst\/}})\index{nth}
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  cdr-nth-cdr \\ \index{cdr-nth-cdr}
{\rm{cdr}}\,({\rm{nth-cdr}}\,({\it{j\/}}, {\it{lst\/}})\index{nth-cdr}) $=$ {\rm{nth-cdr}}\,({\it{j\/}}, {\rm{cdr}}\,({\it{lst\/}}))\index{nth-cdr}
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  lemma-d1 \\ \index{lemma-d1}
\=\+{\rm{tic-tac-toe1-c}}\,({\rm{nth-cdr}}\,({\it{j\/}}, {\it{xmoves\/}})\index{nth-cdr}, {\it{board\/}}, {\it{i\/}}, {\it{t3\/}})\index{tic-tac-toe1-c} \\ 
$=$$\;\;\;\;${\rm{tic-tac-toe1-d}}\,({\it{xmoves\/}}, {\it{j\/}}, {\it{board\/}}, {\it{i\/}}, {\it{t3\/}})\index{tic-tac-toe1-d}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  lemma-d \\ \index{lemma-d}
{\rm{tic-tac-toe-c}}\,({\it{xmoves\/}})\index{tic-tac-toe-c} $=$ {\rm{tic-tac-toe-d}}\,({\it{xmoves\/}})\index{tic-tac-toe-d}
\end{tabbing}

\begin{tabbing}{\sc Definition}:\index{up}$\;\;$
{\rm{up}}\,({\it{t3\/}})\index{up} $=$ {\rm{number-$>$board*}}\,({\rm{array-$>$lst}}\,({\tt{0}}, {\it{t3\/}})\index{array-$>$lst}, {\tt{9}})\index{number-$>$board*}
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  tic-tac-toe-d-correct \\ \index{tic-tac-toe-d-correct}
\=\+({\rm{tic-tac-toep}}\,({\rm{up}}\,({\rm{tic-tac-toe-d}}\,({\it{xmoves\/}})\index{tic-tac-toe-d})\index{up})\index{tic-tac-toep} $\in$ {\tt{'}}{\tt{(}}{\tt{draw}}{\tt{ }}{\tt{o-win}}{\tt{)}}) \\ 
$\wedge$$\;\;\;\;${\rm{x-moves-honored}}\,({\it{xmoves\/}}, {\rm{up}}\,({\rm{tic-tac-toe-d}}\,({\it{xmoves\/}})\index{tic-tac-toe-d})\index{up})\index{x-moves-honored} \\ 
$\wedge$$\;\;\;\;$(\=\+{\rm{init-game}}\,({\rm{length}}\,({\it{opening\/}})\index{length}, {\rm{up}}\,({\rm{tic-tac-toe-d}}\,({\rm{append}}\,({\it{opening\/}}, {\it{xmoves1\/}}))\index{tic-tac-toe-d})\index{up})\index{init-game} \\ 
$=$$\;\;\;\;${\rm{init-game}}\,(\=\+{\rm{length}}\,({\it{opening\/}})\index{length}, \\ 
{\rm{up}}\,({\rm{tic-tac-toe-d}}\,({\rm{append}}\,({\it{opening\/}}, {\it{xmoves2\/}}))\index{tic-tac-toe-d})\index{up})\-\index{init-game})\-\-
\end{tabbing}




%\end{list}
\printindex
\end{document}
