\documentstyle[makeidx]{article}
\makeindex
%\setlength{\oddsidemargin}{.5in}
%\setlength{\evensidemargin}{.5in}
%\setlength{\textwidth}{5.8in}
\begin{document}
%\setlength{\parindent}{0pt}
%\newcounter{bean}
%\begin{list}{\arabic{bean}.}{\usecounter{bean}\setlength{\leftmargin}{0pt}\setlength{\rightmargin}{0pt}\setlength{\listparindent}{20pt}\setlength{\parsep}{5pt}}
%\item[]

\begin{verbatim}#|

 Copyright (C) 1994 by Misao Nagayama and Carolyn Talcott.  All Rights Reserved.

 You may copy and distribute verbatim copies of this Nqthm-1992 event script as
 you receive it, in any medium, including embedding it verbatim in derivative
 works, provided that you conspicuously and appropriately publish on each copy
 a valid copyright notice "Copyright (C) 1994 by Misao Nagayama and Carolyn
 Talcott.  All Rights Reserved."

 NO WARRANTY

 Misao Nagayama and Carolyn Talcott PROVIDE 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 Misao Nagayama or Carolyn Talcott 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{verbatim};;;;;;;;;;;;;;;;;;;;;;   mutex-atomic.ev    ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;

;;;;;;;;;;;;;;;;;;;;;;;;;;;;;  com.ev  ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;

;*sequence and finite set utilities

;;;The ith entry in l.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{nth} \\  
\=\+{\rm{nth}}\,({\it{l\/}}, {\it{i\/}})\index{nth} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{listp}}\,({\it{l\/}}) \\ 
{\bf then }\=\+{\bf if }{\it{i\/}} $=$ {\tt{1}}$\;\;${\bf then }{\rm{car}}\,({\it{l\/}}) \\ 
{\bf else }{\rm{nth}}\,({\rm{cdr}}\,({\it{l\/}}), {\it{i\/}} $-\;1$)\index{nth}$\;${\bf  endif}\- \\ 
{\bf elseif }{\it{l\/}} $\in\;${\bf{N}}${}$ \\ 
{\bf then }\=\+{\bf if }{\it{i\/}} $=$ {\tt{1}}$\;\;${\bf then }{\it{l\/}} \\ 
{\bf else }{\bf{f}}$\;${\bf  endif}\- \\ 
{\bf else }{\bf{f}}$\;${\bf  endif}\-\-
\end{tabbing}

\noindent{\sc Event}:   Disable nth.

\addvspace{18pt}

\begin{verbatim};;;update ith entry of l to be k
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{move} \\  
\=\+{\rm{move}}\,({\it{l\/}}, {\it{i\/}}, {\it{k\/}})\index{move} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{i\/}} $=$ {\tt{0}}$\;\;${\bf then }{\it{l\/}} \\ 
{\bf elseif }{\it{l\/}} $\simeq\;${{\bf{nil}}}${}$ \\ 
{\bf then }\=\+{\bf if }{\it{i\/}} $=$ {\tt{1}}$\;\;${\bf then }{\it{k\/}} \\ 
{\bf else }{\it{l\/}}$\;${\bf  endif}\- \\ 
{\bf elseif }{\it{i\/}} $=$ {\tt{1}}$\;\;${\bf then }{\rm{cons}}\,({\it{k\/}}, {\rm{cdr}}\,({\it{l\/}})) \\ 
{\bf else }{\rm{cons}}\,({\rm{car}}\,({\it{l\/}}), {\rm{move}}\,({\rm{cdr}}\,({\it{l\/}}), {\it{i\/}} $-\;1$, {\it{k\/}})\index{move})$\;${\bf  endif}\-\-
\end{tabbing}

\noindent{\sc Event}:   Disable move.

\addvspace{18pt}

\begin{tabbing}{\sc Definition}:\index{at}$\;\;$
{\rm{at}}\,({\it{l\/}}, {\it{i\/}}, {\it{k\/}})\index{at} $=$ ({\rm{nth}}\,({\it{l\/}}, {\it{i\/}})\index{nth} $=$ {\it{k\/}})
\end{tabbing}

\noindent{\sc Event}:   Disable at.

\addvspace{18pt}

\begin{tabbing}{\sc Definition}:\index{length} \\  
\=\+{\rm{length}}\,({\it{l\/}})\index{length} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{listp}}\,({\it{l\/}})$\;\;${\bf then }$1\;+$ {\rm{length}}\,({\rm{cdr}}\,({\it{l\/}}))\index{length} \\ 
{\bf else }{\sc{zero}}$\;${\bf  endif}\-\-
\end{tabbing}

\noindent{\sc Event}:   Disable length.

\addvspace{18pt}

\begin{verbatim};;;The nth entry in l is in the list i.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{union-at-n}$\;\;$
{\rm{union-at-n}}\,({\it{l\/}}, {\it{n\/}}, {\it{i\/}})\index{union-at-n} $=$ ({\rm{nth}}\,({\it{l\/}}, {\it{n\/}})\index{nth} $\in$ {\it{i\/}})
\end{tabbing}

\noindent{\sc Event}:   Disable union-at-n.

\addvspace{18pt}

\begin{verbatim};;;Any entry in l is in the list i.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{all-union} \\  
\=\+{\rm{all-union}}\,({\it{l\/}}, {\it{n\/}}, {\it{i\/}})\index{all-union} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{n\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\bf{t}} \\ 
{\bf else }{\rm{union-at-n}}\,({\it{l\/}}, {\it{n\/}}, {\it{i\/}})\index{union-at-n} $\wedge$ {\rm{all-union}}\,({\it{l\/}}, {\it{n\/}} $-\;1$, {\it{i\/}})\index{all-union}$\;${\bf  endif}\-\-
\end{tabbing}

\noindent{\sc Event}:   Disable all-union.

\addvspace{18pt}

\begin{verbatim};;;There exists an entry in l which belongs to 
;;;the list i, moreover when exists, some such
;;;j is returned.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{exist-union} \\  
\=\+{\rm{exist-union}}\,({\it{l\/}}, {\it{n\/}}, {\it{i\/}})\index{exist-union} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{n\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\bf{f}} \\ 
{\bf elseif }{\rm{union-at-n}}\,({\it{l\/}}, {\it{n\/}}, {\it{i\/}})\index{union-at-n}$\;\;${\bf then }{\it{n\/}} \\ 
{\bf else }{\rm{exist-union}}\,({\it{l\/}}, {\it{n\/}} $-\;1$, {\it{i\/}})\index{exist-union}$\;${\bf  endif}\-\-
\end{tabbing}

\noindent{\sc Event}:   Disable exist-union.

\addvspace{18pt}

\begin{verbatim};;;n is in the intersection of l8-12 and g34.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{intersect-8-12-3-4-at-n} \\  
\=\+{\rm{intersect-8-12-3-4-at-n}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{intersect-8-12-3-4-at-n} \\ 
$=$$\;\;\;\;$({\rm{union-at-n}}\,({\it{l\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n} $\wedge$ {\rm{union-at-n}}\,({\it{g\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{)}})\index{union-at-n})\-
\end{tabbing}


\noindent{\sc Event}:   Disable intersect-8-12-3-4-at-n.

\addvspace{18pt}

\begin{verbatim};;;There exists n in the intersection of l8-12 and g34.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{exist-intersect-8-12-3-4} \\  
\=\+{\rm{exist-intersect-8-12-3-4}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{exist-intersect-8-12-3-4} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{n\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\bf{f}} \\ 
{\bf elseif }{\rm{intersect-8-12-3-4-at-n}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{intersect-8-12-3-4-at-n}$\;\;${\bf then }{\it{n\/}} \\ 
{\bf else }{\rm{exist-intersect-8-12-3-4}}\,({\it{n\/}} $-\;1$, {\it{l\/}}, {\it{g\/}})\index{exist-intersect-8-12-3-4}$\;${\bf  endif}\-\-
\end{tabbing}

\noindent{\sc Event}:   Disable exist-intersect-8-12-3-4.

\addvspace{18pt}

\begin{verbatim};*Flag invariant.

\end{verbatim}\begin{tabbing}{\sc Definition}:\index{lg-1-at-n} \\  
\=\+{\rm{lg-1-at-n}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-1-at-n} \\ 
$=$$\;\;\;\;$(\=\+({\rm{at}}\,({\it{l\/}}, {\it{n\/}}, {\tt{0}})\index{at} $\wedge$ {\rm{at}}\,({\it{g\/}}, {\it{n\/}}, {\tt{0}})\index{at}) \\ 
$\vee$$\;\;\;\;$({\rm{at}}\,({\it{l\/}}, {\it{n\/}}, {\tt{1}})\index{at} $\wedge$ {\rm{at}}\,({\it{g\/}}, {\it{n\/}}, {\tt{0}})\index{at}) \\ 
$\vee$$\;\;\;\;$({\rm{at}}\,({\it{l\/}}, {\it{n\/}}, {\tt{2}})\index{at} $\wedge$ {\rm{at}}\,({\it{g\/}}, {\it{n\/}}, {\tt{0}})\index{at}) \\ 
$\vee$$\;\;\;\;$({\rm{at}}\,({\it{l\/}}, {\it{n\/}}, {\tt{3}})\index{at} $\wedge$ {\rm{at}}\,({\it{g\/}}, {\it{n\/}}, {\tt{1}})\index{at}) \\ 
$\vee$$\;\;\;\;$({\rm{at}}\,({\it{l\/}}, {\it{n\/}}, {\tt{4}})\index{at} $\wedge$ {\rm{at}}\,({\it{g\/}}, {\it{n\/}}, {\tt{1}})\index{at}))\-\-
\end{tabbing}


\noindent{\sc Event}:   Disable lg-1-at-n.

\addvspace{18pt}

\begin{tabbing}{\sc Definition}:\index{lg-2-at-n} \\  
\=\+{\rm{lg-2-at-n}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-2-at-n} \\ 
$=$$\;\;\;\;$(\=\+({\rm{at}}\,({\it{l\/}}, {\it{n\/}}, {\tt{5}})\index{at} $\wedge$ {\rm{at}}\,({\it{g\/}}, {\it{n\/}}, {\tt{3}})\index{at}) \\ 
$\vee$$\;\;\;\;$({\rm{at}}\,({\it{l\/}}, {\it{n\/}}, {\tt{6}})\index{at} $\wedge$ {\rm{at}}\,({\it{g\/}}, {\it{n\/}}, {\tt{3}})\index{at}) \\ 
$\vee$$\;\;\;\;$({\rm{at}}\,({\it{l\/}}, {\it{n\/}}, {\tt{7}})\index{at} $\wedge$ {\rm{at}}\,({\it{g\/}}, {\it{n\/}}, {\tt{2}})\index{at}) \\ 
$\vee$$\;\;\;\;$({\rm{at}}\,({\it{l\/}}, {\it{n\/}}, {\tt{8}})\index{at} $\wedge$ {\rm{at}}\,({\it{g\/}}, {\it{n\/}}, {\tt{3}})\index{at}) \\ 
$\vee$$\;\;\;\;$({\rm{at}}\,({\it{l\/}}, {\it{n\/}}, {\tt{8}})\index{at} $\wedge$ {\rm{at}}\,({\it{g\/}}, {\it{n\/}}, {\tt{2}})\index{at}))\-\-
\end{tabbing}     

\noindent{\sc Event}:   Disable lg-2-at-n.

\addvspace{18pt}       

\begin{tabbing}{\sc Definition}:\index{lg-3-at-n} \\  
\=\+{\rm{lg-3-at-n}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-3-at-n} \\ 
$=$$\;\;\;\;$(\=\+({\rm{at}}\,({\it{l\/}}, {\it{n\/}}, {\tt{9}})\index{at} $\wedge$ {\rm{at}}\,({\it{g\/}}, {\it{n\/}}, {\tt{4}})\index{at}) \\ 
$\vee$$\;\;\;\;$({\rm{at}}\,({\it{l\/}}, {\it{n\/}}, {\tt{10}})\index{at} $\wedge$ {\rm{at}}\,({\it{g\/}}, {\it{n\/}}, {\tt{4}})\index{at}) \\ 
$\vee$$\;\;\;\;$({\rm{at}}\,({\it{l\/}}, {\it{n\/}}, {\tt{11}})\index{at} $\wedge$ {\rm{at}}\,({\it{g\/}}, {\it{n\/}}, {\tt{4}})\index{at}) \\ 
$\vee$$\;\;\;\;$({\rm{at}}\,({\it{l\/}}, {\it{n\/}}, {\tt{12}})\index{at} $\wedge$ {\rm{at}}\,({\it{g\/}}, {\it{n\/}}, {\tt{4}})\index{at}))\-\-
\end{tabbing}

\noindent{\sc Event}:   Disable lg-3-at-n.

\addvspace{18pt}

\begin{tabbing}{\sc Definition}:\index{lg-at-n} \\  
\=\+{\rm{lg-at-n}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-at-n} \\ 
$=$$\;\;\;\;$({\rm{lg-1-at-n}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-1-at-n} $\wedge$ {\rm{lg-2-at-n}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-2-at-n} $\wedge$ {\rm{lg-3-at-n}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-3-at-n})\-
\end{tabbing}

\noindent{\sc Event}:   Disable lg-at-n.

\addvspace{18pt}

\begin{tabbing}{\sc Definition}:\index{lg} \\  
\=\+{\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{n\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\bf{t}} \\ 
{\bf else }{\rm{lg-at-n}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-at-n} $\wedge$ {\rm{lg}}\,({\it{n\/}} $-\;1$, {\it{l\/}}, {\it{g\/}})\index{lg}$\;${\bf  endif}\-\-
\end{tabbing}

\noindent{\sc Event}:   Disable lg.

\addvspace{18pt}


\begin{verbatim};*The set {1...n}.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{nset} \\  
\=\+{\rm{nset}}\,({\it{n\/}})\index{nset} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{n\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\bf{nil}} \\ 
{\bf else }{\rm{cons}}\,({\it{n\/}}, {\rm{nset}}\,({\it{n\/}} $-\;1$)\index{nset})$\;${\bf  endif}\-\-
\end{tabbing}

\noindent{\sc Event}:   Disable nset.

\addvspace{18pt}

\begin{verbatim};;;n belongs to nset.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  n-in-nset \\ \index{n-in-nset}
({\it{n\/}} $\not\simeq {\tt{0}}$) $\rightarrow$ ({\it{n\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset})
\end{tabbing}

\begin{verbatim};;;Any element in nset is a number.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  nset-number \\ \index{nset-number}
({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\rightarrow$ ({\it{k\/}} $\in\;${\bf{N}}${}$)
\end{tabbing}

\begin{verbatim};;;If a nonzero number plus one belongs to nset, 
;;;then so does the nonzero number itself.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  add1-nset \\ \index{add1-nset}
(({\it{k\/}} $\not\simeq {\tt{0}}$) $\wedge$ (($1\;+$ {\it{k\/}}) $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset})) $\rightarrow$ ({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset})
\end{tabbing}                 

\begin{verbatim};;;Any list has its length at least nonzero.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  list-ln \\ \index{list-ln}
{\rm{listp}}\,({\it{l\/}}) $\rightarrow$ ({\rm{length}}\,({\it{l\/}})\index{length} $\not=$ {\tt{0}})
\end{tabbing}

\begin{verbatim};;;(move l k i) is again a list if l is a list.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  move-is-list \\ \index{move-is-list}
{\rm{listp}}\,({\it{l\/}}) $\rightarrow$ {\rm{listp}}\,({\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\it{i\/}})\index{move})
\end{tabbing}


\noindent{\sc Event}:   Enable length.

\addvspace{18pt}

\begin{verbatim};;;(move l k i) has i as its kth entry.
;;;(enable length) is critical to prove this lemma.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  move-nth \\ \index{move-nth}
({\rm{listp}}\,({\it{l\/}}) $\wedge$ ({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset})) $\rightarrow$ ({\rm{nth}}\,({\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\it{i\/}})\index{move}, {\it{k\/}})\index{nth} $=$ {\it{i\/}})
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  zero-not-member-nset \\ \index{zero-not-member-nset}
{\tt{0}} $\not\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}
\end{tabbing}

\begin{verbatim};;;Lists l and (move l k i) have the same length.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  move-unchange-length \\ \index{move-unchange-length}
\=\+({\rm{listp}}\,({\it{l\/}}) $\wedge$ ({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset})) \\ 
$\rightarrow$$\;\;\;\;$({\rm{length}}\,({\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\it{i\/}})\index{move})\index{length} $=$ {\rm{length}}\,({\it{l\/}})\index{length})\-
\end{tabbing}

\begin{verbatim};;;Lists l and (move l k i) have the same entries 
;;;except kth one.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  move-unchange-other-than-nth \\ \index{move-unchange-other-than-nth}
\=\+({\rm{listp}}\,({\it{l\/}}) $\wedge$ ({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) $\wedge$ ({\it{j\/}} $\not=$ {\it{k\/}})) \\ 
$\rightarrow$$\;\;\;\;$({\rm{nth}}\,({\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\it{i\/}})\index{move}, {\it{j\/}})\index{nth} $=$ {\rm{nth}}\,({\it{l\/}}, {\it{j\/}})\index{nth})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  member-ex-union \\ \index{member-ex-union}
{\rm{exist-union}}\,({\it{l\/}}, {\it{n\/}}, {\it{i\/}})\index{exist-union} $\rightarrow$ ({\rm{exist-union}}\,({\it{l\/}}, {\it{n\/}}, {\it{i\/}})\index{exist-union} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset})
\end{tabbing}

\begin{verbatim};;;(exist-union l n i) is a number.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  number-ex-union \\ \index{number-ex-union}
{\rm{exist-union}}\,({\it{l\/}}, {\it{n\/}}, {\it{i\/}})\index{exist-union} $\rightarrow$ ({\rm{exist-union}}\,({\it{l\/}}, {\it{n\/}}, {\it{i\/}})\index{exist-union} $\in\;${\bf{N}}${}$)
\end{tabbing}

\begin{verbatim};;;(exist-intersect-8-12-3-4 n l g) belongs to nset.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  member-intersect \\ \index{member-intersect}
\=\+{\rm{exist-intersect-8-12-3-4}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{exist-intersect-8-12-3-4} \\ 
$\rightarrow$$\;\;\;\;$({\rm{exist-intersect-8-12-3-4}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{exist-intersect-8-12-3-4} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset})\-
\end{tabbing}

\begin{verbatim};;;(exist-intersect-8-12-3-4 n l g) is a number.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  number-intersect \\ \index{number-intersect}
{\rm{exist-intersect-8-12-3-4}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{exist-intersect-8-12-3-4} $\rightarrow$ ({\rm{exist-intersect-8-12-3-4}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{exist-intersect-8-12-3-4} $\in\;${\bf{N}}${}$)
\end{tabbing}

\begin{verbatim};;;any member of nset is nonzero.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  k-not-0 \\ \index{k-not-0}
({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\rightarrow$ ({\it{k\/}} $\not\simeq {\tt{0}}$)
\end{tabbing}

\begin{verbatim};*lemmas for a0 

;;;If j's entry in l is between 8..12 then  
;;;(exist-union l n '(8 9 10 11 12)) holds.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  j-ex-l8-12 \\ \index{j-ex-l8-12}
\=\+(({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{union-at-n}}\,({\it{l\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n}) \\ 
$\rightarrow$$\;\;\;\;${\rm{exist-union}}\,({\it{l\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{exist-union}\-
\end{tabbing}

\begin{verbatim};;;Witness of (exist-union lp n '(8 9 10 11 12)) 
;;;has in lp its entry between 8...12.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  ex-lp8-12-in-lp8-12 \\ \index{ex-lp8-12-in-lp8-12}
\=\+{\rm{exist-union}}\,({\it{lp\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{exist-union} \\ 
$\rightarrow$$\;\;\;\;${\rm{union-at-n}}\,(\=\+{\it{lp\/}}, \\ 
{\rm{exist-union}}\,({\it{lp\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{exist-union}, \\ 
{\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\-\index{union-at-n}\-
\end{tabbing}

\begin{verbatim};;;If (not (exist-union l n '(8 9 10 11 12))) 
;;;holds, then (not (exist-union g n '(4))) by lg.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  ex-if4 \\ \index{ex-if4}
\=\+(($\neg$ {\rm{exist-union}}\,({\it{l\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{exist-union}) $\wedge$ {\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg}) \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{exist-union}}\,({\it{g\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{4}}{\tt{)}})\index{exist-union})\-
\end{tabbing}

\begin{verbatim};;;If (not (exist-union g n '(1))) holds,
;;; then there is no entry either 3 or 4.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  l34-empty \\ \index{l34-empty}
\=\+(({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} $\wedge$ ($\neg$ {\rm{exist-union}}\,({\it{g\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{1}}{\tt{)}})\index{exist-union})) \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{union-at-n}}\,({\it{l\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{)}})\index{union-at-n})\-
\end{tabbing}

\begin{verbatim};;;If j's entry in lp is 4, then (certainly)
;;;it is either 3 or 4.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  lp4-then-un34 \\ \index{lp4-then-un34}
{\rm{at}}\,({\it{lp\/}}, {\it{j\/}}, {\tt{4}})\index{at} $\rightarrow$ {\rm{union-at-n}}\,({\it{lp\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{)}})\index{union-at-n}
\end{tabbing}

\begin{verbatim};;;If (exist-intersect-8-12-3-4 n l g) holds, 
;;;then so does (exist-union g n '(3 4)).
\end{verbatim}\begin{tabbing}{\sc Theorem}:  int-8-12-3-4-then-un34 \\ \index{int-8-12-3-4-then-un34}
{\rm{exist-intersect-8-12-3-4}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{exist-intersect-8-12-3-4} $\rightarrow$ {\rm{exist-union}}\,({\it{g\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{)}})\index{exist-union}
\end{tabbing}

\begin{verbatim};*lemmas for a1

;;;i is the witness of 
;;;(exist-intersect-8-12-3-4 n lp gp).
\end{verbatim}\begin{tabbing}{\sc Theorem}:  int-wtn \\ \index{int-wtn}
\=\+(({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{intersect-8-12-3-4-at-n}}\,({\it{j\/}}, {\it{lp\/}}, {\it{gp\/}})\index{intersect-8-12-3-4-at-n}) \\ 
$\rightarrow$$\;\;\;\;${\rm{exist-intersect-8-12-3-4}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}})\index{exist-intersect-8-12-3-4}\-
\end{tabbing}

\begin{verbatim};;;If there exists j such that j's entry in lp 
;;;is between 8..12 and entry in gp is either 3 or 4
;;;then (intersect-8-12-3-4-at-n j lp gp) holds.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  un8-12-and-un34-then-int \\ \index{un8-12-and-un34-then-int}
\=\+({\rm{union-at-n}}\,({\it{lp\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n} $\wedge$ {\rm{union-at-n}}\,({\it{gp\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{)}})\index{union-at-n}) \\ 
$\rightarrow$$\;\;\;\;${\rm{intersect-8-12-3-4-at-n}}\,({\it{j\/}}, {\it{lp\/}}, {\it{gp\/}})\index{intersect-8-12-3-4-at-n}\-
\end{tabbing}

\begin{verbatim};;;By the two lemmas above, 
;;;(exist-intersect-8-12-3-4 n lp gp) holds provided 
;;;that there exists j such that j's entry in lp is
;;;between 8..12 and entry in gp is either 3 or 4.

;* ep-l8-12

;;;If the k's entry in l is 5, then the k's entry 
;;;in g is 3 by lg.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  lg-l5-g3 \\ \index{lg-l5-g3}
(({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} $\wedge$ {\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{5}})\index{at}) $\rightarrow$ {\rm{at}}\,({\it{g\/}}, {\it{k\/}}, {\tt{3}})\index{at}
\end{tabbing}

\begin{verbatim};;;If the k's entry in gp is 3 then certainly 
;;;it is either 3 or 4.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  gp3-then-un34 \\ \index{gp3-then-un34}
{\rm{at}}\,({\it{gp\/}}, {\it{k\/}}, {\tt{3}})\index{at} $\rightarrow$ {\rm{union-at-n}}\,({\it{gp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{)}})\index{union-at-n}
\end{tabbing}

\begin{verbatim};;;nep-l8-12

;;;If the k's entry in l is between 8..12 then 
;;;it is either between 8..11 or equal to 12.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  case-k \\ \index{case-k}
\=\+(\=\+{\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{)}})\index{union-at-n}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{12}})\index{at}\-
\end{tabbing}

\begin{verbatim};;;;;k-not-l8-12

;;;If (exist-intersect-8-12-3-4 n l g) holds
;;;then the witness has its entry in g either equal 
;;;to 3 or 4.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  intersect-8-12-3-4-then-3-4 \\ \index{intersect-8-12-3-4-then-3-4}
\=\+{\rm{exist-intersect-8-12-3-4}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{exist-intersect-8-12-3-4} \\ 
$\rightarrow$$\;\;\;\;${\rm{union-at-n}}\,({\it{g\/}}, {\rm{exist-intersect-8-12-3-4}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{exist-intersect-8-12-3-4}, {\tt{'}}{\tt{(}}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{)}})\index{union-at-n}\-
\end{tabbing}

\begin{verbatim};;;If (exist-intersect-8-12-3-4 n l g) holds,
;;; then the witness has its entry in g between 8 and 12.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  intersect-8-12-3-4-then-8-12 \\ \index{intersect-8-12-3-4-then-8-12}
\=\+{\rm{exist-intersect-8-12-3-4}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{exist-intersect-8-12-3-4} \\ 
$\rightarrow$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\rm{exist-intersect-8-12-3-4}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{exist-intersect-8-12-3-4}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n}\-
\end{tabbing}

\begin{verbatim};;;k-in-l8-11

;;;If k's entry in lp is between 9 and 12,
;;;then it is certainly between 8 and 12.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  un9-12-then-un8-12 \\ \index{un9-12-then-un8-12}
\=\+{\rm{union-at-n}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n} \\ 
$\rightarrow$$\;\;\;\;${\rm{union-at-n}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n}\-
\end{tabbing}

\begin{verbatim};;;If the i's entry in l is between 9 and 12,
;;;then the k's entry in g is 4.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  if4 \\ \index{if4}
\=\+(({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} $\wedge$ {\rm{union-at-n}}\,({\it{l\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n}) \\ 
$\rightarrow$$\;\;\;\;${\rm{at}}\,({\it{g\/}}, {\it{j\/}}, {\tt{4}})\index{at}\-
\end{tabbing}

\begin{verbatim};;;k-in-l12

;;;If (exist-union lp n '(8 9 10 11 12)) holds then
;;;its witness does not have its entry in lp equal to 1.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  ex-lp8-12-not-in-lp0 \\ \index{ex-lp8-12-not-in-lp0}
\=\+{\rm{exist-union}}\,({\it{lp\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{exist-union} \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{at}}\,({\it{lp\/}}, {\rm{exist-union}}\,({\it{lp\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{exist-union}, {\tt{0}})\index{at})\-
\end{tabbing}

\begin{verbatim};;;If k's entry in lp is between 8 and 12,
;;; then it is either between 8 and 11 or 12.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  k-in-lp9-12-or-lp8 \\ \index{k-in-lp9-12-or-lp8}
\=\+(\=\+{\rm{union-at-n}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{union-at-n}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{at}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{8}})\index{at}\-
\end{tabbing}

\begin{verbatim};;;If the k's entry is either 5 or 7,
;;;then it is between 5 and 7.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  un57-then-un5-12 \\ \index{un57-then-un5-12}
{\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{7}}{\tt{)}})\index{union-at-n} $\rightarrow$ {\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n}
\end{tabbing}

\begin{verbatim};;;If the k's entry in l is between 8 and 11,
;;;then it is between 5 and 12.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  un8-11-then-un5-12 \\ \index{un8-11-then-un5-12}
\=\+{\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{)}})\index{union-at-n} \\ 
$\rightarrow$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n}\-
\end{tabbing}

\begin{verbatim};;;If the k's entry in l is between 8 and 12,
;;;then it is between 5 and 12.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  un8-12-then-un5-12 \\ \index{un8-12-then-un5-12}
\=\+{\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n} \\ 
$\rightarrow$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n}\-
\end{tabbing}


\begin{verbatim};*lemmas for a2

;;;i-eq-k-j-neq-k

;;;If the k's entry in l is either 10 or 11, 
;;;then the k's entry in l is between 10 and 12.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  un10-11-then-un10-12 \\ \index{un10-11-then-un10-12}
{\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{)}})\index{union-at-n} $\rightarrow$ {\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n}
\end{tabbing}

\begin{verbatim};;;If the j's entry in g is either 0 or 1 then
;;;the j's entry in l is not between 5 and 12.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  if1 \\ \index{if1}
\=\+(({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} $\wedge$ {\rm{union-at-n}}\,({\it{g\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{)}})\index{union-at-n}) \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{union-at-n}}\,({\it{l\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n})\-
\end{tabbing}

\begin{verbatim};;;j-eq-k-i-neq-k

;;;If the k's entry in l is between 5 and 7,
;;;then it is certainly between 5 and 12.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  un5-7-then-un5-11 \\ \index{un5-7-then-un5-11}
{\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{)}})\index{union-at-n} $\rightarrow$ {\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{)}})\index{union-at-n}
\end{tabbing}

\begin{verbatim};;;If the k's entry in lp is between 5 and 7 then 
;;;it is certain between 5 and 11.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  un57-then-un5-11 \\ \index{un57-then-un5-11}
{\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{7}}{\tt{)}})\index{union-at-n} $\rightarrow$ {\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{)}})\index{union-at-n}
\end{tabbing}

\begin{verbatim};;;If the k's entry in l is between 8 and 11,
;;;then it is certainly between 5 and 11.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  un8-11-then-un5-11 \\ \index{un8-11-then-un5-11}
\=\+{\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{)}})\index{union-at-n} \\ 
$\rightarrow$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{)}})\index{union-at-n}\-
\end{tabbing}

\begin{verbatim};;;If the k's entry in lp is between 5 and 12 and
;;;the k's entry in lp is between 5 and 7, then
;;;the k's entry in lp in fact is between 9 and 12.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  k-in-lp5-7-or-lp8-or-lp9-12 \\ \index{k-in-lp5-7-or-lp8-or-lp9-12}
\=\+(\=\+{\rm{union-at-n}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{union-at-n}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{)}})\index{union-at-n}) \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{at}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{8}})\index{at}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{union-at-n}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n}\-
\end{tabbing}

\begin{verbatim};;;If the k's entry in l is between 5 and 11,
;;; then it is certainly between 5 and 12.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  un5-11-then-un5-12 \\ \index{un5-11-then-un5-12}
\=\+{\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{)}})\index{union-at-n} \\ 
$\rightarrow$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n}\-
\end{tabbing}

\begin{verbatim};;;If the k's entry in l is between 10 and 12,
;;; then it is certainly between 8 and 12.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  un10-12-then-un8-12 \\ \index{un10-12-then-un8-12}
{\rm{union-at-n}}\,({\it{l\/}}, {\it{i\/}}, {\tt{'}}{\tt{(}}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n} $\rightarrow$ {\rm{union-at-n}}\,({\it{l\/}}, {\it{i\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n}
\end{tabbing}

\begin{verbatim};;;j-eq-k-i-neq-k

;;;If (exist-union l n '(8 9 10 11 12)) does not hold,
;;;then the i's entry in l is not between 10 and 12.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  i-not-l10-12 \\ \index{i-not-l10-12}
\=\+(({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ ($\neg$ {\rm{exist-union}}\,({\it{l\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{exist-union})) \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{union-at-n}}\,({\it{l\/}}, {\it{i\/}}, {\tt{'}}{\tt{(}}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n})\-
\end{tabbing}


\begin{verbatim};*lemmas for a3

;;;j-eq-k-i-neq-k

;;;If the k's entry in l is between 5 and 11,
;;;then the k's entry in l is between 9 and 11.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  un5-11-eq-un58-or-un8-11 \\ \index{un5-11-eq-un58-or-un8-11}
\=\+(\=\+{\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{)}})\index{union-at-n} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{)}})\index{union-at-n}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{)}})\index{union-at-n}\-
\end{tabbing}

\begin{verbatim};;;If the k's entry in g is 4,
;;;then the k's entry in l is between 5 and 8.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  a3-if4 \\ \index{a3-if4}
\=\+(({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} $\wedge$ {\rm{at}}\,({\it{g\/}}, {\it{k\/}}, {\tt{4}})\index{at}) \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{)}})\index{union-at-n})\-
\end{tabbing}

\begin{verbatim};;;If the k's entry in l is between 5 and 11,
;;;and the k's entry in l is between 5 and 12,
;;;then the k's entry in l is 9 and 11.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  k-in-l5-11-g4-then-l9-11 \\ \index{k-in-l5-11-g4-then-l9-11}
\=\+(\=\+({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{)}})\index{union-at-n} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{g\/}}, {\it{k\/}}, {\tt{4}})\index{at})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{)}})\index{union-at-n}\-
\end{tabbing}  \begin{verbatim}; dropped second use hint -- rsb

;;;If the i's entry in l is 12, 
;;;then the i's entry in l is between 8 and 12.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  l12-then-un8-12 \\ \index{l12-then-un8-12}
{\rm{at}}\,({\it{l\/}}, {\it{i\/}}, {\tt{12}})\index{at} $\rightarrow$ {\rm{union-at-n}}\,({\it{l\/}}, {\it{i\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n}
\end{tabbing}

\begin{verbatim};;;If (exist-union l n '(8 9 10 11 12)) does not hold,
;;;then the i's entry in l is 12.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  i-not-in-l12 \\ \index{i-not-in-l12}
\=\+(({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ ($\neg$ {\rm{exist-union}}\,({\it{l\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{exist-union})) \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{at}}\,({\it{l\/}}, {\it{i\/}}, {\tt{12}})\index{at})\-
\end{tabbing}

\begin{verbatim};;;j-neq-k-i-eq-k

;;;If the k's entry in l is 11,
;;; then the k's entry in l is between 10 and 12.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  l11-then-un10-12 \\ \index{l11-then-un10-12}
{\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{11}})\index{at} $\rightarrow$ {\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n}
\end{tabbing}

\begin{verbatim};;;If the j's entry in g is either 2 or 3,
;;;then the j's entry in l is between 5 and 8 by lg.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  if3 \\ \index{if3}
\=\+(({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} $\wedge$ ($\neg$ {\rm{union-at-n}}\,({\it{g\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{)}})\index{union-at-n})) \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{union-at-n}}\,({\it{l\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{)}})\index{union-at-n})\-
\end{tabbing}


\begin{verbatim};;;If the j's entry in l is between 5 and 12 and
;;;the j's entry in l is between 5 and 8, then
;;;the j's entry in l is 9 and 12.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  l5-12-eq-l5-8-or-l9-12 \\ \index{l5-12-eq-l5-8-or-l9-12}
\=\+(\=\+{\rm{union-at-n}}\,({\it{l\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{union-at-n}}\,({\it{l\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{)}})\index{union-at-n}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n}\-
\end{tabbing}

\begin{verbatim};;;i-j-eq-k

;;;If the k's entry in lp is 12,
;;;then it is certainly between 5 and 12.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  l12-then-un9-12 \\ \index{l12-then-un9-12}
{\rm{at}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{12}})\index{at} $\rightarrow$ {\rm{union-at-n}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n}
\end{tabbing}

\begin{verbatim};*lemmas for b1a

;;;If the u's entry in g is 4,
;;;then the u's entry in l is between 8 and 12 by lg.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  b1a-if4 \\ \index{b1a-if4}
\=\+(({\it{u\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} $\wedge$ {\rm{at}}\,({\it{g\/}}, {\it{u\/}}, {\tt{4}})\index{at}) \\ 
$\rightarrow$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{u\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n}\-
\end{tabbing}

\begin{verbatim};*lemmas for b1b

;;;If the k's entry in lp is between 9 and 12,
;;;then the k's entry in gp is iether 3 or 4 by lg.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  lp9-12-then-k-in-g34 \\ \index{lp9-12-then-k-in-g34}
\=\+(({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{union-at-n}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n} $\wedge$ {\rm{lg}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}})\index{lg}) \\ 
$\rightarrow$$\;\;\;\;${\rm{union-at-n}}\,({\it{gp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{)}})\index{union-at-n}\-
\end{tabbing}

\begin{verbatim};;;If the k's entry in lp is between 8 and 12, and
;;;it is not 8, then it is certainly between 9 and 12.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  un8-12-then-l8-or-l9-12 \\ \index{un8-12-then-l8-or-l9-12}
\=\+({\rm{union-at-n}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n} $\wedge$ ($\neg$ {\rm{at}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{8}})\index{at})) \\ 
$\rightarrow$$\;\;\;\;${\rm{union-at-n}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n}\-
\end{tabbing}


\begin{verbatim};;;;;;;;;;;;;;;;;;;;;;;;;; defn.ev ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;;;Well-formed-state.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{ws} \\  
\=\+{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} \\ 
$=$$\;\;\;\;$(\=\+({\it{n\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\rm{length}}\,({\it{l\/}})\index{length} $=$ {\it{n\/}}) \\ 
$\wedge$$\;\;\;\;$({\rm{length}}\,({\it{g\/}})\index{length} $=$ {\it{n\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{all-union}}\,({\it{l\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{ }}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{all-union} \\ 
$\wedge$$\;\;\;\;${\rm{all-union}}\,({\it{g\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{)}})\index{all-union})\-\-
\end{tabbing}

\noindent{\sc Event}:   Disable ws.

\addvspace{18pt}

\begin{verbatim};;;Transitions.

\end{verbatim}\begin{tabbing}{\sc Definition}:\index{rhoi0} \\  
\=\+{\rm{rhoi0}}\,({\it{n\/}}, {\it{i\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi0} \\ 
$=$$\;\;\;\;$({\rm{at}}\,({\it{l\/}}, {\it{i\/}}, {\tt{0}})\index{at} $\wedge$ ({\it{gp\/}} $=$ {\it{g\/}}) $\wedge$ ({\it{lp\/}} $=$ {\rm{move}}\,({\it{l\/}}, {\it{i\/}}, {\tt{1}})\index{move}))\-
\end{tabbing}

\begin{tabbing}{\sc Definition}:\index{rhoi1a} \\  
\=\+{\rm{rhoi1a}}\,({\it{n\/}}, {\it{i\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi1a} \\ 
$=$$\;\;\;\;$({\rm{at}}\,({\it{l\/}}, {\it{i\/}}, {\tt{1}})\index{at} $\wedge$ ({\it{gp\/}} $=$ {\it{g\/}}) $\wedge$ ({\it{lp\/}} $=$ {\rm{move}}\,({\it{l\/}}, {\it{i\/}}, {\tt{2}})\index{move}))\-
\end{tabbing}

\begin{tabbing}{\sc Definition}:\index{rhoi1b} \\  
{\rm{rhoi1b}}\,({\it{n\/}}, {\it{i\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi1b} $=$ ({\rm{at}}\,({\it{l\/}}, {\it{i\/}}, {\tt{1}})\index{at} $\wedge$ ({\it{g\/}} $=$ {\it{gp\/}}) $\wedge$ ({\it{lp\/}} $=$ {\it{l\/}}))
\end{tabbing}

\begin{tabbing}{\sc Definition}:\index{rhoi2} \\  
\=\+{\rm{rhoi2}}\,({\it{n\/}}, {\it{i\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi2} \\ 
$=$$\;\;\;\;$({\rm{at}}\,({\it{l\/}}, {\it{i\/}}, {\tt{2}})\index{at} $\wedge$ ({\it{lp\/}} $=$ {\rm{move}}\,({\it{l\/}}, {\it{i\/}}, {\tt{3}})\index{move}) $\wedge$ ({\it{gp\/}} $=$ {\rm{move}}\,({\it{g\/}}, {\it{i\/}}, {\tt{1}})\index{move}))\-
\end{tabbing}

\begin{tabbing}{\sc Definition}:\index{rhoi3a} \\  
\=\+{\rm{rhoi3a}}\,({\it{n\/}}, {\it{i\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi3a} \\ 
$=$$\;\;\;\;$(\=\+{\rm{at}}\,({\it{l\/}}, {\it{i\/}}, {\tt{3}})\index{at} \\ 
$\wedge$$\;\;\;\;$({\it{gp\/}} $=$ {\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{lp\/}} $=$ {\rm{move}}\,({\it{l\/}}, {\it{i\/}}, {\tt{4}})\index{move}) \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{exist-union}}\,({\it{g\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{)}})\index{exist-union}))\-\-
\end{tabbing}

\begin{tabbing}{\sc Definition}:\index{rhoi3b} \\  
\=\+{\rm{rhoi3b}}\,({\it{n\/}}, {\it{i\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi3b} \\ 
$=$$\;\;\;\;$(\=\+{\rm{at}}\,({\it{l\/}}, {\it{i\/}}, {\tt{3}})\index{at} \\ 
$\wedge$$\;\;\;\;$({\it{gp\/}} $=$ {\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{lp\/}} $=$ {\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{exist-union}}\,({\it{g\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{)}})\index{exist-union})\-\-
\end{tabbing}

\begin{tabbing}{\sc Definition}:\index{rhoi4} \\  
\=\+{\rm{rhoi4}}\,({\it{n\/}}, {\it{i\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi4} \\ 
$=$$\;\;\;\;$({\rm{at}}\,({\it{l\/}}, {\it{i\/}}, {\tt{4}})\index{at} $\wedge$ ({\it{gp\/}} $=$ {\rm{move}}\,({\it{g\/}}, {\it{i\/}}, {\tt{3}})\index{move}) $\wedge$ ({\it{lp\/}} $=$ {\rm{move}}\,({\it{l\/}}, {\it{i\/}}, {\tt{5}})\index{move}))\-
\end{tabbing}

\begin{tabbing}{\sc Definition}:\index{rhoi5a} \\  
\=\+{\rm{rhoi5a}}\,({\it{n\/}}, {\it{i\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi5a} \\ 
$=$$\;\;\;\;$(\=\+{\rm{at}}\,({\it{l\/}}, {\it{i\/}}, {\tt{5}})\index{at} \\ 
$\wedge$$\;\;\;\;$({\it{gp\/}} $=$ {\it{g\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{exist-union}}\,({\it{g\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{1}}{\tt{)}})\index{exist-union} \\ 
$\wedge$$\;\;\;\;$({\it{lp\/}} $=$ {\rm{move}}\,({\it{l\/}}, {\it{i\/}}, {\tt{6}})\index{move}))\-\-
\end{tabbing}

\begin{tabbing}{\sc Definition}:\index{rhoi5b} \\  
\=\+{\rm{rhoi5b}}\,({\it{n\/}}, {\it{i\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi5b} \\ 
$=$$\;\;\;\;$(\=\+{\rm{at}}\,({\it{l\/}}, {\it{i\/}}, {\tt{5}})\index{at} \\ 
$\wedge$$\;\;\;\;$({\it{gp\/}} $=$ {\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{exist-union}}\,({\it{g\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{1}}{\tt{)}})\index{exist-union}) \\ 
$\wedge$$\;\;\;\;$({\it{lp\/}} $=$ {\rm{move}}\,({\it{l\/}}, {\it{i\/}}, {\tt{8}})\index{move}))\-\-
\end{tabbing}

\begin{tabbing}{\sc Definition}:\index{rhoi6} \\  
\=\+{\rm{rhoi6}}\,({\it{n\/}}, {\it{i\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi6} \\ 
$=$$\;\;\;\;$({\rm{at}}\,({\it{l\/}}, {\it{i\/}}, {\tt{6}})\index{at} $\wedge$ ({\it{gp\/}} $=$ {\rm{move}}\,({\it{g\/}}, {\it{i\/}}, {\tt{2}})\index{move}) $\wedge$ ({\it{lp\/}} $=$ {\rm{move}}\,({\it{l\/}}, {\it{i\/}}, {\tt{7}})\index{move}))\-
\end{tabbing}

\begin{tabbing}{\sc Definition}:\index{rhoi7a} \\  
\=\+{\rm{rhoi7a}}\,({\it{n\/}}, {\it{i\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi7a} \\ 
$=$$\;\;\;\;$(\=\+{\rm{at}}\,({\it{l\/}}, {\it{i\/}}, {\tt{7}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{exist-union}}\,({\it{g\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{4}}{\tt{)}})\index{exist-union} \\ 
$\wedge$$\;\;\;\;$({\it{lp\/}} $=$ {\rm{move}}\,({\it{l\/}}, {\it{i\/}}, {\tt{8}})\index{move}) \\ 
$\wedge$$\;\;\;\;$({\it{gp\/}} $=$ {\it{g\/}}))\-\-
\end{tabbing}

\begin{tabbing}{\sc Definition}:\index{rhoi7b} \\  
\=\+{\rm{rhoi7b}}\,({\it{n\/}}, {\it{i\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi7b} \\ 
$=$$\;\;\;\;$(\=\+{\rm{at}}\,({\it{l\/}}, {\it{i\/}}, {\tt{7}})\index{at} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{exist-union}}\,({\it{g\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{4}}{\tt{)}})\index{exist-union}) \\ 
$\wedge$$\;\;\;\;$({\it{lp\/}} $=$ {\it{l\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{gp\/}} $=$ {\it{g\/}}))\-\-
\end{tabbing}

\begin{tabbing}{\sc Definition}:\index{rhoi8} \\  
\=\+{\rm{rhoi8}}\,({\it{n\/}}, {\it{i\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi8} \\ 
$=$$\;\;\;\;$({\rm{at}}\,({\it{l\/}}, {\it{i\/}}, {\tt{8}})\index{at} $\wedge$ ({\it{gp\/}} $=$ {\rm{move}}\,({\it{g\/}}, {\it{i\/}}, {\tt{4}})\index{move}) $\wedge$ ({\it{lp\/}} $=$ {\rm{move}}\,({\it{l\/}}, {\it{i\/}}, {\tt{9}})\index{move}))\-
\end{tabbing}

\begin{tabbing}{\sc Definition}:\index{phi9} \\  
\=\+{\rm{phi9}}\,({\it{i\/}}, {\it{n\/}}, {\it{g\/}})\index{phi9} \\ 
$=$$\;\;\;\;$\=\+{\bf if }({\it{g\/}} $\simeq\;${{\bf{nil}}}${}$) $\vee$ ({\it{i\/}} $\not\in\;${\bf{N}}${}$) $\vee$ ({\it{n\/}} $\not\in\;${\bf{N}}${}$)$\;\;${\bf then }{\bf{f}} \\ 
{\bf elseif }{\it{n\/}} $=$ {\tt{0}}$\;\;${\bf then }{\bf{t}} \\ 
{\bf else }\=\+(({\it{n\/}} $\not<$ {\it{i\/}}) $\wedge$ {\rm{phi9}}\,({\it{i\/}}, {\it{n\/}} $-\;1$, {\it{g\/}})\index{phi9}) \\ 
$\vee$$\;\;\;\;$({\rm{union-at-n}}\,({\it{g\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{)}})\index{union-at-n} $\wedge$ {\rm{phi9}}\,({\it{i\/}}, {\it{n\/}} $-\;1$, {\it{g\/}})\index{phi9})\-$\;${\bf  endif}\-\-
\end{tabbing}


\noindent{\sc Event}:   Disable phi9.

\addvspace{18pt}

\begin{tabbing}{\sc Definition}:\index{rhoi9a} \\  
\=\+{\rm{rhoi9a}}\,({\it{n\/}}, {\it{i\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi9a} \\ 
$=$$\;\;\;\;$({\rm{at}}\,({\it{l\/}}, {\it{i\/}}, {\tt{9}})\index{at} $\wedge$ ({\it{gp\/}} $=$ {\it{g\/}}) $\wedge$ {\rm{phi9}}\,({\it{i\/}}, {\it{n\/}}, {\it{g\/}})\index{phi9} $\wedge$ ({\it{lp\/}} $=$ {\rm{move}}\,({\it{l\/}}, {\it{i\/}}, {\tt{10}})\index{move}))\-
\end{tabbing}

\begin{tabbing}{\sc Definition}:\index{rhoi9b} \\  
\=\+{\rm{rhoi9b}}\,({\it{n\/}}, {\it{i\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi9b} \\ 
$=$$\;\;\;\;$({\rm{at}}\,({\it{l\/}}, {\it{i\/}}, {\tt{9}})\index{at} $\wedge$ ({\it{gp\/}} $=$ {\it{g\/}}) $\wedge$ ($\neg$ {\rm{phi9}}\,({\it{i\/}}, {\it{n\/}}, {\it{g\/}})\index{phi9}) $\wedge$ ({\it{lp\/}} $=$ {\it{l\/}}))\-
\end{tabbing}

\begin{tabbing}{\sc Definition}:\index{rhoi10} \\  
\=\+{\rm{rhoi10}}\,({\it{n\/}}, {\it{i\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi10} \\ 
$=$$\;\;\;\;$({\rm{at}}\,({\it{l\/}}, {\it{i\/}}, {\tt{10}})\index{at} $\wedge$ ({\it{lp\/}} $=$ {\rm{move}}\,({\it{l\/}}, {\it{i\/}}, {\tt{11}})\index{move}) $\wedge$ ({\it{gp\/}} $=$ {\it{g\/}}))\-
\end{tabbing}

\begin{tabbing}{\sc Definition}:\index{phi11} \\  
\=\+{\rm{phi11}}\,({\it{i\/}}, {\it{n\/}}, {\it{g\/}})\index{phi11} \\ 
$=$$\;\;\;\;$\=\+{\bf if }({\it{g\/}} $\simeq\;${{\bf{nil}}}${}$) $\vee$ ({\it{i\/}} $\not\in\;${\bf{N}}${}$) $\vee$ ({\it{n\/}} $\not\in\;${\bf{N}}${}$)$\;\;${\bf then }{\bf{f}} \\ 
{\bf elseif }{\it{n\/}} $=$ {\tt{0}}$\;\;${\bf then }{\bf{t}} \\ 
{\bf else }\=\+(({\it{i\/}} $\not<$ {\it{n\/}}) $\wedge$ {\rm{phi11}}\,({\it{i\/}}, {\it{n\/}} $-\;1$, {\it{g\/}})\index{phi11}) \\ 
$\vee$$\;\;\;\;$(\=\+($\neg$ {\rm{union-at-n}}\,({\it{g\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{)}})\index{union-at-n}) \\ 
$\wedge$$\;\;\;\;${\rm{phi11}}\,({\it{i\/}}, {\it{n\/}} $-\;1$, {\it{g\/}})\index{phi11})\-\-$\;${\bf  endif}\-\-
\end{tabbing}


\noindent{\sc Event}:   Disable phi11.

\addvspace{18pt}

\begin{tabbing}{\sc Definition}:\index{rhoi11a} \\  
\=\+{\rm{rhoi11a}}\,({\it{n\/}}, {\it{i\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi11a} \\ 
$=$$\;\;\;\;$(\=\+{\rm{at}}\,({\it{l\/}}, {\it{i\/}}, {\tt{11}})\index{at} \\ 
$\wedge$$\;\;\;\;$({\it{gp\/}} $=$ {\it{g\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{phi11}}\,({\it{i\/}}, {\it{n\/}}, {\it{g\/}})\index{phi11} \\ 
$\wedge$$\;\;\;\;$({\it{lp\/}} $=$ {\rm{move}}\,({\it{l\/}}, {\it{i\/}}, {\tt{12}})\index{move}))\-\-
\end{tabbing}

\begin{tabbing}{\sc Definition}:\index{rhoi11b} \\  
\=\+{\rm{rhoi11b}}\,({\it{n\/}}, {\it{i\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi11b} \\ 
$=$$\;\;\;\;$({\rm{at}}\,({\it{l\/}}, {\it{i\/}}, {\tt{11}})\index{at} $\wedge$ ({\it{gp\/}} $=$ {\it{g\/}}) $\wedge$ ($\neg$ {\rm{phi11}}\,({\it{i\/}}, {\it{n\/}}, {\it{g\/}})\index{phi11}) $\wedge$ ({\it{lp\/}} $=$ {\it{l\/}}))\-
\end{tabbing}

\begin{tabbing}{\sc Definition}:\index{rhoi12} \\  
\=\+{\rm{rhoi12}}\,({\it{n\/}}, {\it{i\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi12} \\ 
$=$$\;\;\;\;$({\rm{at}}\,({\it{l\/}}, {\it{i\/}}, {\tt{12}})\index{at} $\wedge$ ({\it{gp\/}} $=$ {\rm{move}}\,({\it{g\/}}, {\it{i\/}}, {\tt{0}})\index{move}) $\wedge$ ({\it{lp\/}} $=$ {\rm{move}}\,({\it{l\/}}, {\it{i\/}}, {\tt{0}})\index{move}))\-
\end{tabbing}

\begin{verbatim};;;The transition operates on i'th.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{rhoi} \\  
\=\+{\rm{rhoi}}\,({\it{n\/}}, {\it{i\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} \\ 
$=$$\;\;\;\;$(\=\+{\rm{rhoi0}}\,({\it{n\/}}, {\it{i\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi0} \\ 
$\vee$$\;\;\;\;${\rm{rhoi1a}}\,({\it{n\/}}, {\it{i\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi1a} \\ 
$\vee$$\;\;\;\;${\rm{rhoi1b}}\,({\it{n\/}}, {\it{i\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi1b} \\ 
$\vee$$\;\;\;\;${\rm{rhoi2}}\,({\it{n\/}}, {\it{i\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi2} \\ 
$\vee$$\;\;\;\;${\rm{rhoi3a}}\,({\it{n\/}}, {\it{i\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi3a} \\ 
$\vee$$\;\;\;\;${\rm{rhoi3b}}\,({\it{n\/}}, {\it{i\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi3b} \\ 
$\vee$$\;\;\;\;${\rm{rhoi4}}\,({\it{n\/}}, {\it{i\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi4} \\ 
$\vee$$\;\;\;\;${\rm{rhoi5a}}\,({\it{n\/}}, {\it{i\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi5a} \\ 
$\vee$$\;\;\;\;${\rm{rhoi5b}}\,({\it{n\/}}, {\it{i\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi5b} \\ 
$\vee$$\;\;\;\;${\rm{rhoi6}}\,({\it{n\/}}, {\it{i\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi6} \\ 
$\vee$$\;\;\;\;${\rm{rhoi7a}}\,({\it{n\/}}, {\it{i\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi7a} \\ 
$\vee$$\;\;\;\;${\rm{rhoi7b}}\,({\it{n\/}}, {\it{i\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi7b} \\ 
$\vee$$\;\;\;\;${\rm{rhoi8}}\,({\it{n\/}}, {\it{i\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi8} \\ 
$\vee$$\;\;\;\;${\rm{rhoi9a}}\,({\it{n\/}}, {\it{i\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi9a} \\ 
$\vee$$\;\;\;\;${\rm{rhoi9b}}\,({\it{n\/}}, {\it{i\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi9b} \\ 
$\vee$$\;\;\;\;${\rm{rhoi10}}\,({\it{n\/}}, {\it{i\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi10} \\ 
$\vee$$\;\;\;\;${\rm{rhoi11a}}\,({\it{n\/}}, {\it{i\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi11a} \\ 
$\vee$$\;\;\;\;${\rm{rhoi11b}}\,({\it{n\/}}, {\it{i\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi11b} \\ 
$\vee$$\;\;\;\;${\rm{rhoi12}}\,({\it{n\/}}, {\it{i\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi12})\-\-
\end{tabbing}

\noindent{\sc Event}:   Disable rhoi.

\addvspace{18pt}

\begin{verbatim};;;  Propositions
;;;a0

\end{verbatim}\begin{tabbing}{\sc Definition}:\index{a0} \\  
\=\+{\rm{a0}}\,({\it{n\/}}, {\it{l\/}}, {\it{k\/}})\index{a0} \\ 
$=$$\;\;\;\;$(\=\+(({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{exist-union}}\,({\it{l\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{exist-union}) \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{4}})\index{at}))\-\-
\end{tabbing}

\noindent{\sc Event}:   Disable a0.

\addvspace{18pt}

\begin{verbatim};;;a1

\end{verbatim}\begin{tabbing}{\sc Definition}:\index{a1} \\  
\=\+{\rm{a1}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{a1} \\ 
$=$$\;\;\;\;$(\=\+{\rm{exist-union}}\,({\it{l\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{exist-union} \\ 
$\rightarrow$$\;\;\;\;${\rm{exist-intersect-8-12-3-4}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{exist-intersect-8-12-3-4})\-\-
\end{tabbing}

\noindent{\sc Event}:   Disable a1.

\addvspace{18pt}

\begin{verbatim};;;; a2

\end{verbatim}\begin{tabbing}{\sc Definition}:\index{a2-at-n1-n2} \\  
\=\+{\rm{a2-at-n1-n2}}\,({\it{n1\/}}, {\it{n2\/}}, {\it{l\/}})\index{a2-at-n1-n2} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{union-at-n}}\,({\it{l\/}}, {\it{n1\/}}, {\tt{'}}{\tt{(}}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n} \\ 
{\bf then }$\neg$ {\rm{union-at-n}}\,({\it{l\/}}, {\it{n2\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n} \\ 
{\bf else }{\bf{t}}$\;${\bf  endif}\-\-
\end{tabbing}


\noindent{\sc Event}:   Disable a2-at-n1-n2.

\addvspace{18pt}

\begin{tabbing}{\sc Definition}:\index{a2-at-n2} \\  
\=\+{\rm{a2-at-n2}}\,({\it{n1\/}}, {\it{n2\/}}, {\it{l\/}})\index{a2-at-n2} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{n2\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\bf{t}} \\ 
{\bf elseif }{\it{n2\/}} $\not<$ {\it{n1\/}}$\;\;${\bf then }{\rm{a2-at-n2}}\,({\it{n1\/}}, {\it{n2\/}} $-\;1$, {\it{l\/}})\index{a2-at-n2} \\ 
{\bf else }{\rm{a2-at-n1-n2}}\,({\it{n1\/}}, {\it{n2\/}}, {\it{l\/}})\index{a2-at-n1-n2} $\wedge$ {\rm{a2-at-n2}}\,({\it{n1\/}}, {\it{n2\/}} $-\;1$, {\it{l\/}})\index{a2-at-n2}$\;${\bf  endif}\-\-
\end{tabbing}


\noindent{\sc Event}:   Disable a2-at-n2.

\addvspace{18pt}

\begin{tabbing}{\sc Definition}:\index{a2} \\  
\=\+{\rm{a2}}\,({\it{n1\/}}, {\it{n2\/}}, {\it{l\/}})\index{a2} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{n1\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\bf{t}} \\ 
{\bf else }{\rm{a2-at-n2}}\,({\it{n1\/}}, {\it{n2\/}}, {\it{l\/}})\index{a2-at-n2} $\wedge$ {\rm{a2}}\,({\it{n1\/}} $-\;1$, {\it{n2\/}}, {\it{l\/}})\index{a2}$\;${\bf  endif}\-\-
\end{tabbing}

\noindent{\sc Event}:   Disable a2.

\addvspace{18pt}

\begin{verbatim};;;a3

\end{verbatim}\begin{tabbing}{\sc Definition}:\index{a3-at-n1-n2} \\  
\=\+{\rm{a3-at-n1-n2}}\,({\it{n1\/}}, {\it{n2\/}}, {\it{l\/}}, {\it{g\/}})\index{a3-at-n1-n2} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{at}}\,({\it{l\/}}, {\it{n1\/}}, {\tt{12}})\index{at} $\wedge$ {\rm{union-at-n}}\,({\it{l\/}}, {\it{n2\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n} \\ 
{\bf then }{\rm{at}}\,({\it{g\/}}, {\it{n2\/}}, {\tt{4}})\index{at} \\ 
{\bf else }{\bf{t}}$\;${\bf  endif}\-\-
\end{tabbing}


\noindent{\sc Event}:   Disable a3-at-n1-n2.

\addvspace{18pt}

\begin{tabbing}{\sc Definition}:\index{a3-at-n2} \\  
\=\+{\rm{a3-at-n2}}\,({\it{n1\/}}, {\it{n2\/}}, {\it{l\/}}, {\it{g\/}})\index{a3-at-n2} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{n2\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\bf{t}} \\ 
{\bf else }{\rm{a3-at-n1-n2}}\,({\it{n1\/}}, {\it{n2\/}}, {\it{l\/}}, {\it{g\/}})\index{a3-at-n1-n2} $\wedge$ {\rm{a3-at-n2}}\,({\it{n1\/}}, {\it{n2\/}} $-\;1$, {\it{l\/}}, {\it{g\/}})\index{a3-at-n2}$\;${\bf  endif}\-\-
\end{tabbing}


\noindent{\sc Event}:   Disable a3-at-n2.

\addvspace{18pt}

\begin{tabbing}{\sc Definition}:\index{a3} \\  
\=\+{\rm{a3}}\,({\it{n1\/}}, {\it{n2\/}}, {\it{l\/}}, {\it{g\/}})\index{a3} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{n1\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\bf{t}} \\ 
{\bf else }{\rm{a3-at-n2}}\,({\it{n1\/}}, {\it{n2\/}}, {\it{l\/}}, {\it{g\/}})\index{a3-at-n2} $\wedge$ {\rm{a3}}\,({\it{n1\/}} $-\;1$, {\it{n2\/}}, {\it{l\/}}, {\it{g\/}})\index{a3}$\;${\bf  endif}\-\-
\end{tabbing}

\noindent{\sc Event}:   Disable a3.

\addvspace{18pt}


\begin{verbatim};;;;;;;;;;;;;;;;;;;;;;;;;; basic.ev ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;;;ws implies that n is a number.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  ws-num-n \\ \index{ws-num-n}
{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} $\rightarrow$ ({\it{n\/}} $\in\;${\bf{N}}${}$)
\end{tabbing}

\begin{verbatim};;;ws implies that l is a list.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  ws-list-l \\ \index{ws-list-l}
{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} $\rightarrow$ {\rm{listp}}\,({\it{l\/}})
\end{tabbing}

\begin{verbatim};;;ws implies that g is a list.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  ws-list-g \\ \index{ws-list-g}
{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} $\rightarrow$ {\rm{listp}}\,({\it{g\/}})
\end{tabbing}

\begin{verbatim};;;ws implies that length of l is n.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  ws-ln-l \\ \index{ws-ln-l}
{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} $\rightarrow$ ({\rm{length}}\,({\it{l\/}})\index{length} $=$ {\it{n\/}})
\end{tabbing}

\begin{verbatim};;;ws implies that length of g is n.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  ws-ln-g \\ \index{ws-ln-g}
{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} $\rightarrow$ ({\rm{length}}\,({\it{g\/}})\index{length} $=$ {\it{n\/}})
\end{tabbing}

\begin{verbatim};;;ws and rho imply that lp is a list.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  ws-ln-lp \\ \index{ws-ln-lp}
({\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} $\wedge$ ({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi}) $\rightarrow$ {\rm{listp}}\,({\it{lp\/}})
\end{tabbing}

\begin{verbatim};;;ws and rho imply that gp is a list.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  ws-ln-gp \\ \index{ws-ln-gp}
({\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} $\wedge$ ({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi}) $\rightarrow$ {\rm{listp}}\,({\it{gp\/}})
\end{tabbing}

\begin{verbatim};;;ws implies that n is nonzero.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  ws-n-not-0 \\ \index{ws-n-not-0}
{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} $\rightarrow$ ({\it{n\/}} $\not\simeq {\tt{0}}$)
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  n-not-0 \\ \index{n-not-0}
{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} $\rightarrow$ ({\it{n\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset})
\end{tabbing}

\begin{verbatim};*the rho! lemmas

;;;Auxiliary lemma.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  lm-l-rholemma \\ \index{lm-l-rholemma}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\not=$ {\it{j\/}}))\- \\ 
$\rightarrow$$\;\;\;\;$({\rm{nth}}\,({\it{l\/}}, {\it{j\/}})\index{nth} $=$ {\rm{nth}}\,({\it{lp\/}}, {\it{j\/}})\index{nth})\-
\end{tabbing}


\noindent{\sc Event}:   Disable lm-l-rholemma.

\addvspace{18pt}

\begin{verbatim};;;Rholemma for list l.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  l-rholemma \\ \index{l-rholemma}
\=\+(\=\+{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\not=$ {\it{j\/}}))\- \\ 
$\rightarrow$$\;\;\;\;$({\rm{nth}}\,({\it{l\/}}, {\it{j\/}})\index{nth} $=$ {\rm{nth}}\,({\it{lp\/}}, {\it{j\/}})\index{nth})\-
\end{tabbing}

\begin{verbatim};;;Auxiliary lemma.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  lm-g-rholemma \\ \index{lm-g-rholemma}
\=\+(\=\+{\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{g\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{g\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\not=$ {\it{j\/}}))\- \\ 
$\rightarrow$$\;\;\;\;$({\rm{nth}}\,({\it{g\/}}, {\it{j\/}})\index{nth} $=$ {\rm{nth}}\,({\it{gp\/}}, {\it{j\/}})\index{nth})\-
\end{tabbing}
 

\noindent{\sc Event}:   Disable lm-g-rholemma.

\addvspace{18pt}

\begin{verbatim};;;Rholemma for list g.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  g-rholemma \\ \index{g-rholemma}
\=\+(\=\+{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\not=$ {\it{j\/}}))\- \\ 
$\rightarrow$$\;\;\;\;$({\rm{nth}}\,({\it{g\/}}, {\it{j\/}})\index{nth} $=$ {\rm{nth}}\,({\it{gp\/}}, {\it{j\/}})\index{nth})\-
\end{tabbing}

\begin{verbatim};;; lp-gp-same-l-g

;;;Another version of Rholemma for l.
;;;It applies to (union-at-n l j m) in stead of 
;;;(nth l j).
\end{verbatim}\begin{tabbing}{\sc Theorem}:  lp-same-l \\ \index{lp-same-l}
\=\+(\=\+{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{m\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{j\/}}, {\it{m\/}})\index{union-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{union-at-n}}\,({\it{lp\/}}, {\it{j\/}}, {\it{m\/}})\index{union-at-n}\-
\end{tabbing}

\begin{verbatim};;;Contrast to the one above, 
;;;the order of l and lp is reversed.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  l-same-lp \\ \index{l-same-lp}
\=\+(\=\+{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{m\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{lp\/}}, {\it{j\/}}, {\it{m\/}})\index{union-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{j\/}}, {\it{m\/}})\index{union-at-n}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  lp-same-l-not \\ \index{lp-same-l-not}
\=\+(\=\+{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{m\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{union-at-n}}\,({\it{l\/}}, {\it{j\/}}, {\it{m\/}})\index{union-at-n}))\- \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{union-at-n}}\,({\it{lp\/}}, {\it{j\/}}, {\it{m\/}})\index{union-at-n})\-
\end{tabbing}

\begin{verbatim};;;Another version of Rholemma for g.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  gp-same-g \\ \index{gp-same-g}
\=\+(\=\+{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{m\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{g\/}}, {\it{j\/}}, {\it{m\/}})\index{union-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{union-at-n}}\,({\it{gp\/}}, {\it{j\/}}, {\it{m\/}})\index{union-at-n}\-
\end{tabbing}

\begin{verbatim};;;Contrast to the one above, 
;;;the order of g and gp is reversed.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  g-same-gp \\ \index{g-same-gp}
\=\+(\=\+{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{m\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{gp\/}}, {\it{j\/}}, {\it{m\/}})\index{union-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{union-at-n}}\,({\it{g\/}}, {\it{j\/}}, {\it{m\/}})\index{union-at-n}\-
\end{tabbing}


\begin{verbatim};;;It applies to (at l j m) in stead of 
;;;(nth l j).
\end{verbatim}\begin{tabbing}{\sc Theorem}:  l-same-lp-at \\ \index{l-same-lp-at}
\=\+(\=\+{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{m\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;${\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{lp\/}}, {\it{j\/}}, {\it{m\/}})\index{at})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{j\/}}, {\it{m\/}})\index{at}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  gp-same-g-at \\ \index{gp-same-g-at}
\=\+(\=\+{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{m\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;${\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{g\/}}, {\it{j\/}}, {\it{m\/}})\index{at})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{at}}\,({\it{gp\/}}, {\it{j\/}}, {\it{m\/}})\index{at}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  l-same-lp-at-not \\ \index{l-same-lp-at-not}
\=\+(\=\+{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} \\ 
$\wedge$$\;\;\;\;$({\it{m\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{at}}\,({\it{l\/}}, {\it{j\/}}, {\it{m\/}})\index{at}))\- \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{at}}\,({\it{lp\/}}, {\it{j\/}}, {\it{m\/}})\index{at})\-
\end{tabbing}

\begin{verbatim};*basic properties of a2

;;;Auxiliary lemma.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  lm-a2-at-n2-a2-at-n1-n2 \\ \index{lm-a2-at-n2-a2-at-n1-n2}
\=\+(({\it{n\/}} $\in\;${\bf{N}}${}$) $\wedge$ ({\it{k\/}} $\in\;${\bf{N}}${}$) $\wedge$ ({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ ({\it{j\/}} $<$ {\it{k\/}}) $\wedge$ {\rm{a2-at-n2}}\,({\it{k\/}}, {\it{n\/}}, {\it{l\/}})\index{a2-at-n2}) \\ 
$\rightarrow$$\;\;\;\;${\rm{a2-at-n1-n2}}\,({\it{k\/}}, {\it{j\/}}, {\it{l\/}})\index{a2-at-n1-n2}\-
\end{tabbing}


\noindent{\sc Event}:   Disable lm-a2-at-n2-a2-at-n1-n2.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  a2-at-n2-a2-at-n1-n2 \\ \index{a2-at-n2-a2-at-n1-n2}
\=\+(\=\+{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $<$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{a2-at-n2}}\,({\it{k\/}}, {\it{n\/}}, {\it{l\/}})\index{a2-at-n2})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{a2-at-n1-n2}}\,({\it{k\/}}, {\it{j\/}}, {\it{l\/}})\index{a2-at-n1-n2}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  lm-a2-a2-at-n2 \\ \index{lm-a2-a2-at-n2}
(({\it{n\/}} $\in\;${\bf{N}}${}$) $\wedge$ ({\it{i\/}} $\in\;${\bf{N}}${}$) $\wedge$ ({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{a2}}\,({\it{n\/}}, {\it{i\/}}, {\it{l\/}})\index{a2}) $\rightarrow$ {\rm{a2-at-n2}}\,({\it{k\/}}, {\it{i\/}}, {\it{l\/}})\index{a2-at-n2}
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  a2-a2-at-n2 \\ \index{a2-a2-at-n2}
\=\+({\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} $\wedge$ ({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ ({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{a2}}\,({\it{n\/}}, {\it{i\/}}, {\it{l\/}})\index{a2}) \\ 
$\rightarrow$$\;\;\;\;${\rm{a2-at-n2}}\,({\it{k\/}}, {\it{i\/}}, {\it{l\/}})\index{a2-at-n2}\-
\end{tabbing}

\begin{verbatim};*basic properties of a3

\end{verbatim}\begin{tabbing}{\sc Theorem}:  lm-a3-at-n2-a3-at-n1-n2 \\ \index{lm-a3-at-n2-a3-at-n1-n2}
\=\+(({\it{n\/}} $\in\;${\bf{N}}${}$) $\wedge$ ({\it{u\/}} $\in\;${\bf{N}}${}$) $\wedge$ ({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{a3-at-n2}}\,({\it{u\/}}, {\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{a3-at-n2}) \\ 
$\rightarrow$$\;\;\;\;${\rm{a3-at-n1-n2}}\,({\it{u\/}}, {\it{j\/}}, {\it{l\/}}, {\it{g\/}})\index{a3-at-n1-n2}\-
\end{tabbing}


\noindent{\sc Event}:   Disable lm-a3-at-n2-a3-at-n1-n2.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  a3-at-n2-a3-at-n1-n2 \\ \index{a3-at-n2-a3-at-n1-n2}
\=\+({\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} $\wedge$ ({\it{u\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ ({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{a3-at-n2}}\,({\it{u\/}}, {\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{a3-at-n2}) \\ 
$\rightarrow$$\;\;\;\;${\rm{a3-at-n1-n2}}\,({\it{u\/}}, {\it{j\/}}, {\it{l\/}}, {\it{g\/}})\index{a3-at-n1-n2}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  lm-a3-a3-at-n2 \\ \index{lm-a3-a3-at-n2}
\=\+(({\it{n\/}} $\in\;${\bf{N}}${}$) $\wedge$ ({\it{i\/}} $\in\;${\bf{N}}${}$) $\wedge$ ({\it{u\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{a3}}\,({\it{n\/}}, {\it{i\/}}, {\it{l\/}}, {\it{g\/}})\index{a3}) \\ 
$\rightarrow$$\;\;\;\;${\rm{a3-at-n2}}\,({\it{u\/}}, {\it{i\/}}, {\it{l\/}}, {\it{g\/}})\index{a3-at-n2}\-
\end{tabbing}


\noindent{\sc Event}:   Disable lm-a3-a3-at-n2.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  a3-a3-at-n2 \\ \index{a3-a3-at-n2}
\=\+({\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} $\wedge$ ({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ ({\it{u\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{a3}}\,({\it{n\/}}, {\it{i\/}}, {\it{l\/}}, {\it{g\/}})\index{a3}) \\ 
$\rightarrow$$\;\;\;\;${\rm{a3-at-n2}}\,({\it{u\/}}, {\it{i\/}}, {\it{l\/}}, {\it{g\/}})\index{a3-at-n2}\-
\end{tabbing}

\begin{verbatim};;;;;;;Instances used in the proofs of 
;;;;;;;a1 a2 and a3.

;;;(a2 n n l) and (a3 n n l g) are involved 
;;;in double bounded quantifiers 
;;;\forall i \leq n \forall j \leq n, 
;;;with their quantifier-free formulas 
;;;(a3-at-n1-n2 i j l g) and (a2-at-n1-n2 i j l) 
;;;respectively. What follows are all instances of 
;;;the following type: If (a3 n n l g) holds, then 
;;;in particular, so its instance 
;;;(a3-at-n1-n2 i j l g) does.

;;;The instances are i and j.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  a3-i-j-a3-at-n1-n2 \\ \index{a3-i-j-a3-at-n1-n2}
\=\+({\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} $\wedge$ ({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ ({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{a3}}\,({\it{n\/}}, {\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{a3}) \\ 
$\rightarrow$$\;\;\;\;${\rm{a3-at-n1-n2}}\,({\it{i\/}}, {\it{j\/}}, {\it{l\/}}, {\it{g\/}})\index{a3-at-n1-n2}\-
\end{tabbing}

\begin{verbatim};;;The instances are k and 
;;;(exist-union lp n '(8 9 10 11 12)).
\end{verbatim}\begin{tabbing}{\sc Theorem}:  a3-ex-a3-at-n1-n2 \\ \index{a3-ex-a3-at-n1-n2}
\=\+(\=\+{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{a3}}\,({\it{n\/}}, {\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{a3} \\ 
$\wedge$$\;\;\;\;${\rm{exist-union}}\,({\it{lp\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{exist-union})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{a3-at-n1-n2}}\,({\it{k\/}}, {\rm{exist-union}}\,({\it{lp\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{exist-union}, {\it{l\/}}, {\it{g\/}})\index{a3-at-n1-n2}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  a2-n-a2-at-n2 \\ \index{a2-n-a2-at-n2}
({\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} $\wedge$ ({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{a2}}\,({\it{n\/}}, {\it{n\/}}, {\it{l\/}})\index{a2}) $\rightarrow$ {\rm{a2-at-n2}}\,({\it{k\/}}, {\it{n\/}}, {\it{l\/}})\index{a2-at-n2}
\end{tabbing}

\begin{verbatim};;;The instances are i and j.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  a2-i-j-a2-at-n1-n2 \\ \index{a2-i-j-a2-at-n1-n2}
\=\+(\=\+{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{a2}}\,({\it{n\/}}, {\it{n\/}}, {\it{l\/}})\index{a2} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $<$ {\it{i\/}}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{a2-at-n1-n2}}\,({\it{i\/}}, {\it{j\/}}, {\it{l\/}})\index{a2-at-n1-n2}\-
\end{tabbing}

\begin{verbatim};;;;;;;;;;;;;;;;;;;;;;;;;;; ws.ev ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
\end{verbatim}\begin{tabbing}{\sc Theorem}:  j-eq-k-move-member-g \\ \index{j-eq-k-move-member-g}
\=\+({\rm{listp}}\,({\it{g\/}}) $\wedge$ {\rm{listp}}\,({\it{m\/}}) $\wedge$ ({\it{i\/}} $\in$ {\it{m\/}}) $\wedge$ ({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{g\/}})\index{length})\index{nset})) \\ 
$\rightarrow$$\;\;\;\;$({\rm{nth}}\,({\rm{move}}\,({\it{g\/}}, {\it{k\/}}, {\it{i\/}})\index{move}, {\it{k\/}})\index{nth} $\in$ {\it{m\/}})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  j-neq-k-move-member-g \\ \index{j-neq-k-move-member-g}
\=\+(\=\+{\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{m\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{g\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\it{m\/}}) \\ 
$\wedge$$\;\;\;\;$({\rm{nth}}\,({\it{g\/}}, {\it{j\/}})\index{nth} $\in$ {\it{m\/}}))\- \\ 
$\rightarrow$$\;\;\;\;$({\rm{nth}}\,({\rm{move}}\,({\it{g\/}}, {\it{k\/}}, {\it{i\/}})\index{move}, {\it{j\/}})\index{nth} $\in$ {\it{m\/}})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  move-member-g \\ \index{move-member-g}
\=\+(\=\+{\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{m\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\it{m\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{g\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\rm{nth}}\,({\it{g\/}}, {\it{j\/}})\index{nth} $\in$ {\it{m\/}}))\- \\ 
$\rightarrow$$\;\;\;\;$({\rm{nth}}\,({\rm{move}}\,({\it{g\/}}, {\it{k\/}}, {\it{i\/}})\index{move}, {\it{j\/}})\index{nth} $\in$ {\it{m\/}})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  move-member-l \\ \index{move-member-l}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{m\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\it{m\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\rm{nth}}\,({\it{l\/}}, {\it{j\/}})\index{nth} $\in$ {\it{m\/}}))\- \\ 
$\rightarrow$$\;\;\;\;$({\rm{nth}}\,({\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\it{i\/}})\index{move}, {\it{j\/}})\index{nth} $\in$ {\it{m\/}})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  ws-union-g \\ \index{ws-union-g}
{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} $\rightarrow$ {\rm{all-union}}\,({\it{g\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{)}})\index{all-union}
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  ws-union-l \\ \index{ws-union-l}
{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} $\rightarrow$ {\rm{all-union}}\,({\it{l\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{ }}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{all-union}
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  rho0-preserves-union-g \\ \index{rho0-preserves-union-g}
\=\+({\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} $\wedge$ ({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{rhoi0}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi0}) \\ 
$\rightarrow$$\;\;\;\;${\rm{all-union}}\,({\it{gp\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{)}})\index{all-union}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  rho1a-preserves-union-g \\ \index{rho1a-preserves-union-g}
\=\+({\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} $\wedge$ ({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{rhoi1a}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi1a}) \\ 
$\rightarrow$$\;\;\;\;${\rm{all-union}}\,({\it{gp\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{)}})\index{all-union}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  rho1b-preserves-union-g \\ \index{rho1b-preserves-union-g}
\=\+({\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} $\wedge$ ({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{rhoi1b}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi1b}) \\ 
$\rightarrow$$\;\;\;\;${\rm{all-union}}\,({\it{gp\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{)}})\index{all-union}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  lm-rho2-preserves-union-g \\ \index{lm-rho2-preserves-union-g}
\=\+(\=\+{\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\rm{length}}\,({\it{g\/}})\index{length} $=$ {\it{n\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{all-union}}\,({\it{g\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{)}})\index{all-union} \\ 
$\wedge$$\;\;\;\;${\rm{rhoi2}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi2})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{all-union}}\,({\it{gp\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{)}})\index{all-union}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  rho2-preserves-union-g \\ \index{rho2-preserves-union-g}
\=\+({\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} $\wedge$ ({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{rhoi2}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi2}) \\ 
$\rightarrow$$\;\;\;\;${\rm{all-union}}\,({\it{gp\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{)}})\index{all-union}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  rho3a-preserves-union-g \\ \index{rho3a-preserves-union-g}
\=\+({\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} $\wedge$ ({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{rhoi3a}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi3a}) \\ 
$\rightarrow$$\;\;\;\;${\rm{all-union}}\,({\it{gp\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{)}})\index{all-union}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  rho3b-preserves-union-g \\ \index{rho3b-preserves-union-g}
\=\+({\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} $\wedge$ ({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{rhoi3b}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi3b}) \\ 
$\rightarrow$$\;\;\;\;${\rm{all-union}}\,({\it{gp\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{)}})\index{all-union}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  lm-rho4-preserves-union-g \\ \index{lm-rho4-preserves-union-g}
\=\+(\=\+{\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\rm{length}}\,({\it{g\/}})\index{length} $=$ {\it{n\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{all-union}}\,({\it{g\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{)}})\index{all-union} \\ 
$\wedge$$\;\;\;\;${\rm{rhoi4}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi4})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{all-union}}\,({\it{gp\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{)}})\index{all-union}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  rho4-preserves-union-g \\ \index{rho4-preserves-union-g}
\=\+({\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} $\wedge$ ({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{rhoi4}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi4}) \\ 
$\rightarrow$$\;\;\;\;${\rm{all-union}}\,({\it{gp\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{)}})\index{all-union}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  rho5a-preserves-union-g \\ \index{rho5a-preserves-union-g}
\=\+({\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} $\wedge$ ({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{rhoi5a}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi5a}) \\ 
$\rightarrow$$\;\;\;\;${\rm{all-union}}\,({\it{gp\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{)}})\index{all-union}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  rho5b-preserves-union-g \\ \index{rho5b-preserves-union-g}
\=\+({\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} $\wedge$ ({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{rhoi5b}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi5b}) \\ 
$\rightarrow$$\;\;\;\;${\rm{all-union}}\,({\it{gp\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{)}})\index{all-union}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  lm-rho6-preserves-union-g \\ \index{lm-rho6-preserves-union-g}
\=\+(\=\+{\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\rm{length}}\,({\it{g\/}})\index{length} $=$ {\it{n\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{all-union}}\,({\it{g\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{)}})\index{all-union} \\ 
$\wedge$$\;\;\;\;${\rm{rhoi6}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi6})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{all-union}}\,({\it{gp\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{)}})\index{all-union}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  rho6-preserves-union-g \\ \index{rho6-preserves-union-g}
\=\+({\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} $\wedge$ ({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{rhoi6}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi6}) \\ 
$\rightarrow$$\;\;\;\;${\rm{all-union}}\,({\it{gp\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{)}})\index{all-union}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  rho7a-preserves-union-g \\ \index{rho7a-preserves-union-g}
\=\+({\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} $\wedge$ ({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{rhoi7a}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi7a}) \\ 
$\rightarrow$$\;\;\;\;${\rm{all-union}}\,({\it{gp\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{)}})\index{all-union}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  rho7b-preserves-union-g \\ \index{rho7b-preserves-union-g}
\=\+({\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} $\wedge$ ({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{rhoi7b}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi7b}) \\ 
$\rightarrow$$\;\;\;\;${\rm{all-union}}\,({\it{gp\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{)}})\index{all-union}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  lm-rho8-preserves-union-g \\ \index{lm-rho8-preserves-union-g}
\=\+(\=\+{\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\rm{length}}\,({\it{g\/}})\index{length} $=$ {\it{n\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{all-union}}\,({\it{g\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{)}})\index{all-union} \\ 
$\wedge$$\;\;\;\;${\rm{rhoi8}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi8})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{all-union}}\,({\it{gp\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{)}})\index{all-union}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  rho8-preserves-union-g \\ \index{rho8-preserves-union-g}
\=\+({\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} $\wedge$ ({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{rhoi8}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi8}) \\ 
$\rightarrow$$\;\;\;\;${\rm{all-union}}\,({\it{gp\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{)}})\index{all-union}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  rho9a-preserves-union-g \\ \index{rho9a-preserves-union-g}
\=\+({\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} $\wedge$ ({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{rhoi9a}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi9a}) \\ 
$\rightarrow$$\;\;\;\;${\rm{all-union}}\,({\it{gp\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{)}})\index{all-union}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  rho9b-preserves-union-g \\ \index{rho9b-preserves-union-g}
\=\+({\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} $\wedge$ ({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{rhoi9b}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi9b}) \\ 
$\rightarrow$$\;\;\;\;${\rm{all-union}}\,({\it{gp\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{)}})\index{all-union}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  lm-rho10-preserves-union-g \\ \index{lm-rho10-preserves-union-g}
\=\+(\=\+{\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\rm{length}}\,({\it{g\/}})\index{length} $=$ {\it{n\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{all-union}}\,({\it{g\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{)}})\index{all-union} \\ 
$\wedge$$\;\;\;\;${\rm{rhoi10}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi10})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{all-union}}\,({\it{gp\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{)}})\index{all-union}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  rho10-preserves-union-g \\ \index{rho10-preserves-union-g}
\=\+({\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} $\wedge$ ({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{rhoi10}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi10}) \\ 
$\rightarrow$$\;\;\;\;${\rm{all-union}}\,({\it{gp\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{)}})\index{all-union}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  rho11a-preserves-union-g \\ \index{rho11a-preserves-union-g}
\=\+({\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} $\wedge$ ({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{rhoi11a}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi11a}) \\ 
$\rightarrow$$\;\;\;\;${\rm{all-union}}\,({\it{gp\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{)}})\index{all-union}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  rho11b-preserves-union-g \\ \index{rho11b-preserves-union-g}
\=\+({\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} $\wedge$ ({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{rhoi11b}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi11b}) \\ 
$\rightarrow$$\;\;\;\;${\rm{all-union}}\,({\it{gp\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{)}})\index{all-union}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  lm-rho12-preserves-union-g \\ \index{lm-rho12-preserves-union-g}
\=\+(\=\+{\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\rm{length}}\,({\it{g\/}})\index{length} $=$ {\it{n\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{all-union}}\,({\it{g\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{)}})\index{all-union} \\ 
$\wedge$$\;\;\;\;${\rm{rhoi12}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi12})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{all-union}}\,({\it{gp\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{)}})\index{all-union}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  rho12-preserves-union-g \\ \index{rho12-preserves-union-g}
\=\+({\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} $\wedge$ ({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{rhoi12}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi12}) \\ 
$\rightarrow$$\;\;\;\;${\rm{all-union}}\,({\it{gp\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{)}})\index{all-union}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  rho-preserves-union-g \\ \index{rho-preserves-union-g}
\=\+({\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} $\wedge$ ({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi}) \\ 
$\rightarrow$$\;\;\;\;${\rm{all-union}}\,({\it{gp\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{)}})\index{all-union}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  lm-rho0-preserves-union-l \\ \index{lm-rho0-preserves-union-l}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;$({\rm{length}}\,({\it{l\/}})\index{length} $=$ {\it{n\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{all-union}}\,({\it{l\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{ }}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{all-union} \\ 
$\wedge$$\;\;\;\;${\rm{rhoi0}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi0})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{all-union}}\,({\it{lp\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{ }}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{all-union}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  rho0-preserves-union-l \\ \index{rho0-preserves-union-l}
\=\+({\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} $\wedge$ ({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{rhoi0}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi0}) \\ 
$\rightarrow$$\;\;\;\;${\rm{all-union}}\,({\it{lp\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{ }}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{all-union}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  lm-rho1a-preserves-union-l \\ \index{lm-rho1a-preserves-union-l}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;$({\rm{length}}\,({\it{l\/}})\index{length} $=$ {\it{n\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{all-union}}\,({\it{l\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{ }}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{all-union} \\ 
$\wedge$$\;\;\;\;${\rm{rhoi1a}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi1a})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{all-union}}\,({\it{lp\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{ }}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{all-union}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  rho1a-preserves-union-l \\ \index{rho1a-preserves-union-l}
\=\+({\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} $\wedge$ ({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{rhoi1a}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi1a}) \\ 
$\rightarrow$$\;\;\;\;${\rm{all-union}}\,({\it{lp\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{ }}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{all-union}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  lm-rho1b-preserves-union-l \\ \index{lm-rho1b-preserves-union-l}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;$({\rm{length}}\,({\it{l\/}})\index{length} $=$ {\it{n\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{all-union}}\,({\it{l\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{ }}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{all-union} \\ 
$\wedge$$\;\;\;\;${\rm{rhoi1b}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi1b})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{all-union}}\,({\it{lp\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{ }}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{all-union}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  rho1b-preserves-union-l \\ \index{rho1b-preserves-union-l}
\=\+({\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} $\wedge$ ({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{rhoi1b}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi1b}) \\ 
$\rightarrow$$\;\;\;\;${\rm{all-union}}\,({\it{lp\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{ }}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{all-union}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  lm-rho2-preserves-union-l \\ \index{lm-rho2-preserves-union-l}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;$({\rm{length}}\,({\it{l\/}})\index{length} $=$ {\it{n\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{all-union}}\,({\it{l\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{ }}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{all-union} \\ 
$\wedge$$\;\;\;\;${\rm{rhoi2}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi2})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{all-union}}\,({\it{lp\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{ }}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{all-union}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  rho2-preserves-union-l \\ \index{rho2-preserves-union-l}
\=\+({\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} $\wedge$ ({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{rhoi2}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi2}) \\ 
$\rightarrow$$\;\;\;\;${\rm{all-union}}\,({\it{lp\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{ }}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{all-union}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  lm-rho3a-preserves-union-l \\ \index{lm-rho3a-preserves-union-l}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;$({\rm{length}}\,({\it{l\/}})\index{length} $=$ {\it{n\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{all-union}}\,({\it{l\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{ }}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{all-union} \\ 
$\wedge$$\;\;\;\;${\rm{rhoi3a}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi3a})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{all-union}}\,({\it{lp\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{ }}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{all-union}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  rho3a-preserves-union-l \\ \index{rho3a-preserves-union-l}
\=\+({\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} $\wedge$ ({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{rhoi3a}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi3a}) \\ 
$\rightarrow$$\;\;\;\;${\rm{all-union}}\,({\it{lp\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{ }}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{all-union}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  lm-rho3b-preserves-union-l \\ \index{lm-rho3b-preserves-union-l}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;$({\rm{length}}\,({\it{l\/}})\index{length} $=$ {\it{n\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{all-union}}\,({\it{l\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{ }}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{all-union} \\ 
$\wedge$$\;\;\;\;${\rm{rhoi3b}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi3b})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{all-union}}\,({\it{lp\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{ }}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{all-union}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  rho3b-preserves-union-l \\ \index{rho3b-preserves-union-l}
\=\+({\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} $\wedge$ ({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{rhoi3b}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi3b}) \\ 
$\rightarrow$$\;\;\;\;${\rm{all-union}}\,({\it{lp\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{ }}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{all-union}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  lm-rho4-preserves-union-l \\ \index{lm-rho4-preserves-union-l}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;$({\rm{length}}\,({\it{l\/}})\index{length} $=$ {\it{n\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{all-union}}\,({\it{l\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{ }}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{all-union} \\ 
$\wedge$$\;\;\;\;${\rm{rhoi4}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi4})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{all-union}}\,({\it{lp\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{ }}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{all-union}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  rho4-preserves-union-l \\ \index{rho4-preserves-union-l}
\=\+({\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} $\wedge$ ({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{rhoi4}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi4}) \\ 
$\rightarrow$$\;\;\;\;${\rm{all-union}}\,({\it{lp\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{ }}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{all-union}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  lm-rho5a-preserves-union-l \\ \index{lm-rho5a-preserves-union-l}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;$({\rm{length}}\,({\it{l\/}})\index{length} $=$ {\it{n\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{all-union}}\,({\it{l\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{ }}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{all-union} \\ 
$\wedge$$\;\;\;\;${\rm{rhoi5a}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi5a})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{all-union}}\,({\it{lp\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{ }}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{all-union}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  rho5a-preserves-union-l \\ \index{rho5a-preserves-union-l}
\=\+({\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} $\wedge$ ({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{rhoi5a}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi5a}) \\ 
$\rightarrow$$\;\;\;\;${\rm{all-union}}\,({\it{lp\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{ }}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{all-union}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  lm-rho5b-preserves-union-l \\ \index{lm-rho5b-preserves-union-l}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;$({\rm{length}}\,({\it{l\/}})\index{length} $=$ {\it{n\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{all-union}}\,({\it{l\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{ }}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{all-union} \\ 
$\wedge$$\;\;\;\;${\rm{rhoi5b}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi5b})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{all-union}}\,({\it{lp\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{ }}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{all-union}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  rho5b-preserves-union-l \\ \index{rho5b-preserves-union-l}
\=\+({\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} $\wedge$ ({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{rhoi5b}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi5b}) \\ 
$\rightarrow$$\;\;\;\;${\rm{all-union}}\,({\it{lp\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{ }}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{all-union}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  lm-rho6-preserves-union-l \\ \index{lm-rho6-preserves-union-l}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;$({\rm{length}}\,({\it{l\/}})\index{length} $=$ {\it{n\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{all-union}}\,({\it{l\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{ }}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{all-union} \\ 
$\wedge$$\;\;\;\;${\rm{rhoi6}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi6})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{all-union}}\,({\it{lp\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{ }}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{all-union}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  rho6-preserves-union-l \\ \index{rho6-preserves-union-l}
\=\+({\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} $\wedge$ ({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{rhoi6}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi6}) \\ 
$\rightarrow$$\;\;\;\;${\rm{all-union}}\,({\it{lp\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{ }}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{all-union}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  lm-rho7a-preserves-union-l \\ \index{lm-rho7a-preserves-union-l}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;$({\rm{length}}\,({\it{l\/}})\index{length} $=$ {\it{n\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{all-union}}\,({\it{l\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{ }}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{all-union} \\ 
$\wedge$$\;\;\;\;${\rm{rhoi7a}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi7a})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{all-union}}\,({\it{lp\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{ }}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{all-union}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  rho7a-preserves-union-l \\ \index{rho7a-preserves-union-l}
\=\+({\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} $\wedge$ ({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{rhoi7a}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi7a}) \\ 
$\rightarrow$$\;\;\;\;${\rm{all-union}}\,({\it{lp\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{ }}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{all-union}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  lm-rho7b-preserves-union-l \\ \index{lm-rho7b-preserves-union-l}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;$({\rm{length}}\,({\it{l\/}})\index{length} $=$ {\it{n\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{all-union}}\,({\it{l\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{ }}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{all-union} \\ 
$\wedge$$\;\;\;\;${\rm{rhoi7b}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi7b})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{all-union}}\,({\it{lp\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{ }}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{all-union}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  rho7b-preserves-union-l \\ \index{rho7b-preserves-union-l}
\=\+({\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} $\wedge$ ({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{rhoi7b}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi7b}) \\ 
$\rightarrow$$\;\;\;\;${\rm{all-union}}\,({\it{lp\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{ }}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{all-union}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  lm-rho8-preserves-union-l \\ \index{lm-rho8-preserves-union-l}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;$({\rm{length}}\,({\it{l\/}})\index{length} $=$ {\it{n\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{all-union}}\,({\it{l\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{ }}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{all-union} \\ 
$\wedge$$\;\;\;\;${\rm{rhoi8}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi8})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{all-union}}\,({\it{lp\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{ }}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{all-union}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  rho8-preserves-union-l \\ \index{rho8-preserves-union-l}
\=\+({\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} $\wedge$ ({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{rhoi8}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi8}) \\ 
$\rightarrow$$\;\;\;\;${\rm{all-union}}\,({\it{lp\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{ }}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{all-union}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  lm-rho9a-preserves-union-l \\ \index{lm-rho9a-preserves-union-l}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;$({\rm{length}}\,({\it{l\/}})\index{length} $=$ {\it{n\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{all-union}}\,({\it{l\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{ }}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{all-union} \\ 
$\wedge$$\;\;\;\;${\rm{rhoi9a}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi9a})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{all-union}}\,({\it{lp\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{ }}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{all-union}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  rho9a-preserves-union-l \\ \index{rho9a-preserves-union-l}
\=\+({\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} $\wedge$ ({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{rhoi9a}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi9a}) \\ 
$\rightarrow$$\;\;\;\;${\rm{all-union}}\,({\it{lp\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{ }}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{all-union}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  lm-rho9b-preserves-union-l \\ \index{lm-rho9b-preserves-union-l}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;$({\rm{length}}\,({\it{l\/}})\index{length} $=$ {\it{n\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{all-union}}\,({\it{l\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{ }}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{all-union} \\ 
$\wedge$$\;\;\;\;${\rm{rhoi9b}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi9b})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{all-union}}\,({\it{lp\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{ }}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{all-union}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  rho9b-preserves-union-l \\ \index{rho9b-preserves-union-l}
\=\+({\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} $\wedge$ ({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{rhoi9b}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi9b}) \\ 
$\rightarrow$$\;\;\;\;${\rm{all-union}}\,({\it{lp\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{ }}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{all-union}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  lm-rho10-preserves-union-l \\ \index{lm-rho10-preserves-union-l}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;$({\rm{length}}\,({\it{l\/}})\index{length} $=$ {\it{n\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{all-union}}\,({\it{l\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{ }}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{all-union} \\ 
$\wedge$$\;\;\;\;${\rm{rhoi10}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi10})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{all-union}}\,({\it{lp\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{ }}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{all-union}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  rho10-preserves-union-l \\ \index{rho10-preserves-union-l}
\=\+({\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} $\wedge$ ({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{rhoi10}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi10}) \\ 
$\rightarrow$$\;\;\;\;${\rm{all-union}}\,({\it{lp\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{ }}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{all-union}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  lm-rho11a-preserves-union-l \\ \index{lm-rho11a-preserves-union-l}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;$({\rm{length}}\,({\it{l\/}})\index{length} $=$ {\it{n\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{all-union}}\,({\it{l\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{ }}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{all-union} \\ 
$\wedge$$\;\;\;\;${\rm{rhoi11a}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi11a})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{all-union}}\,({\it{lp\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{ }}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{all-union}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  rho11a-preserves-union-l \\ \index{rho11a-preserves-union-l}
\=\+({\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} $\wedge$ ({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{rhoi11a}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi11a}) \\ 
$\rightarrow$$\;\;\;\;${\rm{all-union}}\,({\it{lp\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{ }}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{all-union}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  lm-rho11b-preserves-union-l \\ \index{lm-rho11b-preserves-union-l}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;$({\rm{length}}\,({\it{l\/}})\index{length} $=$ {\it{n\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{all-union}}\,({\it{l\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{ }}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{all-union} \\ 
$\wedge$$\;\;\;\;${\rm{rhoi11b}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi11b})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{all-union}}\,({\it{lp\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{ }}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{all-union}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  rho11b-preserves-union-l \\ \index{rho11b-preserves-union-l}
\=\+({\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} $\wedge$ ({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{rhoi11b}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi11b}) \\ 
$\rightarrow$$\;\;\;\;${\rm{all-union}}\,({\it{lp\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{ }}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{all-union}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  lm-rho12-preserves-union-l \\ \index{lm-rho12-preserves-union-l}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;$({\rm{length}}\,({\it{l\/}})\index{length} $=$ {\it{n\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{all-union}}\,({\it{l\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{ }}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{all-union} \\ 
$\wedge$$\;\;\;\;${\rm{rhoi12}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi12})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{all-union}}\,({\it{lp\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{ }}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{all-union}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  rho12-preserves-union-l \\ \index{rho12-preserves-union-l}
\=\+({\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} $\wedge$ ({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{rhoi12}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi12}) \\ 
$\rightarrow$$\;\;\;\;${\rm{all-union}}\,({\it{lp\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{ }}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{all-union}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  rho-preserves-union-l \\ \index{rho-preserves-union-l}
\=\+({\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} $\wedge$ ({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi}) \\ 
$\rightarrow$$\;\;\;\;${\rm{all-union}}\,({\it{lp\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{ }}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{all-union}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  lm-rho-preserves-ln-l \\ \index{lm-rho-preserves-ln-l}
\=\+({\rm{listp}}\,({\it{l\/}}) $\wedge$ ({\rm{length}}\,({\it{l\/}})\index{length} $=$ {\it{n\/}}) $\wedge$ ({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi}) \\ 
$\rightarrow$$\;\;\;\;$({\rm{length}}\,({\it{lp\/}})\index{length} $=$ {\it{n\/}})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  rho-preserves-ln-l \\ \index{rho-preserves-ln-l}
\=\+({\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} $\wedge$ ({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi}) \\ 
$\rightarrow$$\;\;\;\;$({\rm{length}}\,({\it{lp\/}})\index{length} $=$ {\it{n\/}})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  lm-rho-preserves-ln-g \\ \index{lm-rho-preserves-ln-g}
\=\+({\rm{listp}}\,({\it{g\/}}) $\wedge$ ({\rm{length}}\,({\it{g\/}})\index{length} $=$ {\it{n\/}}) $\wedge$ ({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi}) \\ 
$\rightarrow$$\;\;\;\;$({\rm{length}}\,({\it{gp\/}})\index{length} $=$ {\it{n\/}})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  rho-preserves-ln-g \\ \index{rho-preserves-ln-g}
\=\+({\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} $\wedge$ ({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi}) \\ 
$\rightarrow$$\;\;\;\;$({\rm{length}}\,({\it{gp\/}})\index{length} $=$ {\it{n\/}})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  lm-rho-preserves-ws \\ \index{lm-rho-preserves-ws}
\=\+(\=\+({\it{n\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{lp\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{gp\/}}) \\ 
$\wedge$$\;\;\;\;$({\rm{length}}\,({\it{lp\/}})\index{length} $=$ {\it{n\/}}) \\ 
$\wedge$$\;\;\;\;$({\rm{length}}\,({\it{gp\/}})\index{length} $=$ {\it{n\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{all-union}}\,({\it{lp\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{ }}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{all-union} \\ 
$\wedge$$\;\;\;\;${\rm{all-union}}\,({\it{gp\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{)}})\index{all-union})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{ws}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}})\index{ws}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  rho-preserves-ws \\ \index{rho-preserves-ws}
({\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} $\wedge$ ({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi}) $\rightarrow$ {\rm{ws}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}})\index{ws}
\end{tabbing}

\begin{verbatim};;;;;;;;;;;;;;;;;;;;;;;;;;; lg.ev ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;;;rhoi0
\end{verbatim}\begin{tabbing}{\sc Theorem}:  n-neq-k-rhoi0 \\ \index{n-neq-k-rhoi0}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\not=$ {\it{n\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{0}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{1}})\index{move}, {\it{g\/}})\index{lg-at-n}\-
\end{tabbing}


\noindent{\sc Event}:   Disable n-neq-k-rhoi0.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  n-eq-k-rhoi0 \\ \index{n-eq-k-rhoi0}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{0}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg-at-n}}\,({\it{k\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg-at-n}}\,({\it{k\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{1}})\index{move}, {\it{g\/}})\index{lg-at-n}\-
\end{tabbing}


\noindent{\sc Event}:   Disable n-eq-k-rhoi0.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  lg-at-rhoi0 \\ \index{lg-at-rhoi0}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{0}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{1}})\index{move}, {\it{g\/}})\index{lg-at-n}\-
\end{tabbing}


\noindent{\sc Event}:   Disable lg-at-rhoi0.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  lg-rhoi0 \\ \index{lg-rhoi0}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{0}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{1}})\index{move}, {\it{g\/}})\index{lg}\-
\end{tabbing}


\noindent{\sc Event}:   Disable lg-rhoi0.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  rhoi0-preserves-lg \\ \index{rhoi0-preserves-lg}
\=\+({\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} $\wedge$ ({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{rhoi0}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi0} $\wedge$ {\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg}) \\ 
$\rightarrow$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}})\index{lg}\-
\end{tabbing}

\begin{verbatim};;;rhoi1a
\end{verbatim}\begin{tabbing}{\sc Theorem}:  n-neq-k-rhoi1a \\ \index{n-neq-k-rhoi1a}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\not=$ {\it{n\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{1}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{2}})\index{move}, {\it{g\/}})\index{lg-at-n}\-
\end{tabbing}


\noindent{\sc Event}:   Disable n-neq-k-rhoi1a.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  n-eq-k-rhoi1a \\ \index{n-eq-k-rhoi1a}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{1}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg-at-n}}\,({\it{k\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{2}})\index{move}, {\it{g\/}})\index{lg-at-n}\-
\end{tabbing}


\noindent{\sc Event}:   Disable n-eq-k-rhoi1a.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  lg-at-rhoi1a \\ \index{lg-at-rhoi1a}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{1}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{2}})\index{move}, {\it{g\/}})\index{lg-at-n}\-
\end{tabbing}


\noindent{\sc Event}:   Disable lg-at-rhoi1a.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  lg-rhoi1a \\ \index{lg-rhoi1a}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{1}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{2}})\index{move}, {\it{g\/}})\index{lg}\-
\end{tabbing}


\noindent{\sc Event}:   Disable lg-rhoi1a.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  rhoi1a-preserves-lg \\ \index{rhoi1a-preserves-lg}
\=\+({\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} $\wedge$ ({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{rhoi1a}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi1a} $\wedge$ {\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg}) \\ 
$\rightarrow$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}})\index{lg}\-
\end{tabbing}

\begin{verbatim};;;rhoi1b
\end{verbatim}\begin{tabbing}{\sc Theorem}:  rhoi1b-preserves-lg \\ \index{rhoi1b-preserves-lg}
\=\+({\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} $\wedge$ ({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{rhoi1b}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi1b} $\wedge$ {\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg}) \\ 
$\rightarrow$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}})\index{lg}\-
\end{tabbing}

\begin{verbatim};;;rhoi2
\end{verbatim}\begin{tabbing}{\sc Theorem}:  n-neq-k-rhoi2 \\ \index{n-neq-k-rhoi2}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\not=$ {\it{n\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{2}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{3}})\index{move}, {\rm{move}}\,({\it{g\/}}, {\it{k\/}}, {\tt{1}})\index{move})\index{lg-at-n}\-
\end{tabbing}


\noindent{\sc Event}:   Disable n-neq-k-rhoi2.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  n-eq-k-rhoi2 \\ \index{n-eq-k-rhoi2}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{2}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg-at-n}}\,({\it{k\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{3}})\index{move}, {\rm{move}}\,({\it{g\/}}, {\it{k\/}}, {\tt{1}})\index{move})\index{lg-at-n}\-
\end{tabbing}


\noindent{\sc Event}:   Disable n-eq-k-rhoi2.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  lg-at-rhoi2 \\ \index{lg-at-rhoi2}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{2}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{3}})\index{move}, {\rm{move}}\,({\it{g\/}}, {\it{k\/}}, {\tt{1}})\index{move})\index{lg-at-n}\-
\end{tabbing}


\noindent{\sc Event}:   Disable lg-at-rhoi2.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  lg-rhoi2 \\ \index{lg-rhoi2}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{2}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{3}})\index{move}, {\rm{move}}\,({\it{g\/}}, {\it{k\/}}, {\tt{1}})\index{move})\index{lg}\-
\end{tabbing}


\noindent{\sc Event}:   Disable lg-rhoi2.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  rhoi2-preserves-lg \\ \index{rhoi2-preserves-lg}
\=\+({\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} $\wedge$ ({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{rhoi2}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi2} $\wedge$ {\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg}) \\ 
$\rightarrow$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}})\index{lg}\-
\end{tabbing}

\begin{verbatim};;;rhoi3a
\end{verbatim}\begin{tabbing}{\sc Theorem}:  n-neq-k-rhoi3a \\ \index{n-neq-k-rhoi3a}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\not=$ {\it{n\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{3}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{4}})\index{move}, {\it{g\/}})\index{lg-at-n}\-
\end{tabbing}


\noindent{\sc Event}:   Disable n-neq-k-rhoi3a.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  n-eq-k-rhoi3a \\ \index{n-eq-k-rhoi3a}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{3}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg-at-n}}\,({\it{k\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg-at-n}}\,({\it{k\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{4}})\index{move}, {\it{g\/}})\index{lg-at-n}\-
\end{tabbing}


\noindent{\sc Event}:   Disable n-eq-k-rhoi3a.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  lg-at-rhoi3a \\ \index{lg-at-rhoi3a}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{3}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{4}})\index{move}, {\it{g\/}})\index{lg-at-n}\-
\end{tabbing}


\noindent{\sc Event}:   Disable lg-at-rhoi3a.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  lg-rhoi3a \\ \index{lg-rhoi3a}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{3}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{4}})\index{move}, {\it{g\/}})\index{lg}\-
\end{tabbing}


\noindent{\sc Event}:   Disable lg-rhoi3a.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  rhoi3a-preserves-lg \\ \index{rhoi3a-preserves-lg}
\=\+({\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} $\wedge$ ({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{rhoi3a}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi3a} $\wedge$ {\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg}) \\ 
$\rightarrow$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}})\index{lg}\-
\end{tabbing}

\begin{verbatim};;;rhoi3b
\end{verbatim}\begin{tabbing}{\sc Theorem}:  rhoi3b-preserves-lg \\ \index{rhoi3b-preserves-lg}
\=\+({\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} $\wedge$ ({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{rhoi3b}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi3b} $\wedge$ {\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg}) \\ 
$\rightarrow$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}})\index{lg}\-
\end{tabbing}

\begin{verbatim};;;rhoi4
\end{verbatim}\begin{tabbing}{\sc Theorem}:  n-neq-k-rhoi4 \\ \index{n-neq-k-rhoi4}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\not=$ {\it{n\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{4}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{5}})\index{move}, {\rm{move}}\,({\it{g\/}}, {\it{k\/}}, {\tt{3}})\index{move})\index{lg-at-n}\-
\end{tabbing}


\noindent{\sc Event}:   Disable n-neq-k-rhoi4.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  n-eq-k-rhoi4 \\ \index{n-eq-k-rhoi4}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{4}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg-at-n}}\,({\it{k\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{5}})\index{move}, {\rm{move}}\,({\it{g\/}}, {\it{k\/}}, {\tt{3}})\index{move})\index{lg-at-n}\-
\end{tabbing}


\noindent{\sc Event}:   Disable n-eq-k-rhoi4.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  lg-at-rhoi4 \\ \index{lg-at-rhoi4}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{4}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{5}})\index{move}, {\rm{move}}\,({\it{g\/}}, {\it{k\/}}, {\tt{3}})\index{move})\index{lg-at-n}\-
\end{tabbing}


\noindent{\sc Event}:   Disable lg-at-rhoi4.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  lg-rhoi4 \\ \index{lg-rhoi4}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{4}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{5}})\index{move}, {\rm{move}}\,({\it{g\/}}, {\it{k\/}}, {\tt{3}})\index{move})\index{lg}\-
\end{tabbing}


\noindent{\sc Event}:   Disable lg-rhoi4.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  rhoi4-preserves-lg \\ \index{rhoi4-preserves-lg}
\=\+({\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} $\wedge$ ({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{rhoi4}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi4} $\wedge$ {\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg}) \\ 
$\rightarrow$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}})\index{lg}\-
\end{tabbing}

\begin{verbatim};;;rhoi5a
\end{verbatim}\begin{tabbing}{\sc Theorem}:  n-neq-k-rhoi5a \\ \index{n-neq-k-rhoi5a}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\not=$ {\it{n\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{5}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{6}})\index{move}, {\it{g\/}})\index{lg-at-n}\-
\end{tabbing}


\noindent{\sc Event}:   Disable n-neq-k-rhoi5a.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  n-eq-k-rhoi5a \\ \index{n-eq-k-rhoi5a}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{5}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg-at-n}}\,({\it{k\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg-at-n}}\,({\it{k\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{6}})\index{move}, {\it{g\/}})\index{lg-at-n}\-
\end{tabbing}


\noindent{\sc Event}:   Disable n-eq-k-rhoi5a.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  lg-at-rhoi5a \\ \index{lg-at-rhoi5a}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{5}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{6}})\index{move}, {\it{g\/}})\index{lg-at-n}\-
\end{tabbing}


\noindent{\sc Event}:   Disable lg-at-rhoi5a.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  lg-rhoi5a \\ \index{lg-rhoi5a}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{5}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{6}})\index{move}, {\it{g\/}})\index{lg}\-
\end{tabbing}


\noindent{\sc Event}:   Disable lg-rhoi5a.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  rhoi5a-preserves-lg \\ \index{rhoi5a-preserves-lg}
\=\+({\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} $\wedge$ ({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{rhoi5a}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi5a} $\wedge$ {\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg}) \\ 
$\rightarrow$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}})\index{lg}\-
\end{tabbing}

\begin{verbatim};;;rhoi5b
\end{verbatim}\begin{tabbing}{\sc Theorem}:  n-neq-k-rhoi5b \\ \index{n-neq-k-rhoi5b}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\not=$ {\it{n\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{5}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{8}})\index{move}, {\it{g\/}})\index{lg-at-n}\-
\end{tabbing}


\noindent{\sc Event}:   Disable n-neq-k-rhoi5b.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  n-eq-k-rhoi5b \\ \index{n-eq-k-rhoi5b}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{5}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg-at-n}}\,({\it{k\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg-at-n}}\,({\it{k\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{8}})\index{move}, {\it{g\/}})\index{lg-at-n}\-
\end{tabbing}


\noindent{\sc Event}:   Disable n-eq-k-rhoi5b.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  lg-at-rhoi5b \\ \index{lg-at-rhoi5b}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{5}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{8}})\index{move}, {\it{g\/}})\index{lg-at-n}\-
\end{tabbing}


\noindent{\sc Event}:   Disable lg-at-rhoi5b.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  lg-rhoi5b \\ \index{lg-rhoi5b}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{5}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{8}})\index{move}, {\it{g\/}})\index{lg}\-
\end{tabbing}


\noindent{\sc Event}:   Disable lg-rhoi5b.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  rhoi5b-preserves-lg \\ \index{rhoi5b-preserves-lg}
\=\+({\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} $\wedge$ ({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{rhoi5b}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi5b} $\wedge$ {\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg}) \\ 
$\rightarrow$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}})\index{lg}\-
\end{tabbing}

\begin{verbatim};;;rhoi6
\end{verbatim}\begin{tabbing}{\sc Theorem}:  n-neq-k-rhoi6 \\ \index{n-neq-k-rhoi6}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\not=$ {\it{n\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{6}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{7}})\index{move}, {\rm{move}}\,({\it{g\/}}, {\it{k\/}}, {\tt{2}})\index{move})\index{lg-at-n}\-
\end{tabbing}


\noindent{\sc Event}:   Disable n-neq-k-rhoi6.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  n-eq-k-rhoi6 \\ \index{n-eq-k-rhoi6}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{6}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg-at-n}}\,({\it{k\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{7}})\index{move}, {\rm{move}}\,({\it{g\/}}, {\it{k\/}}, {\tt{2}})\index{move})\index{lg-at-n}\-
\end{tabbing}


\noindent{\sc Event}:   Disable n-eq-k-rhoi6.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  lg-at-rhoi6 \\ \index{lg-at-rhoi6}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{6}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{7}})\index{move}, {\rm{move}}\,({\it{g\/}}, {\it{k\/}}, {\tt{2}})\index{move})\index{lg-at-n}\-
\end{tabbing}


\noindent{\sc Event}:   Disable lg-at-rhoi6.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  lg-rhoi6 \\ \index{lg-rhoi6}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{6}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{7}})\index{move}, {\rm{move}}\,({\it{g\/}}, {\it{k\/}}, {\tt{2}})\index{move})\index{lg}\-
\end{tabbing}


\noindent{\sc Event}:   Disable lg-rhoi6.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  rhoi6-preserves-lg \\ \index{rhoi6-preserves-lg}
\=\+({\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} $\wedge$ ({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{rhoi6}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi6} $\wedge$ {\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg}) \\ 
$\rightarrow$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}})\index{lg}\-
\end{tabbing}

\begin{verbatim};;;rhoi7a
\end{verbatim}\begin{tabbing}{\sc Theorem}:  n-neq-k-rhoi7a \\ \index{n-neq-k-rhoi7a}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\not=$ {\it{n\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{7}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{8}})\index{move}, {\it{g\/}})\index{lg-at-n}\-
\end{tabbing}


\noindent{\sc Event}:   Disable n-neq-k-rhoi7a.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  n-eq-k-rhoi7a \\ \index{n-eq-k-rhoi7a}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{7}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg-at-n}}\,({\it{k\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg-at-n}}\,({\it{k\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{8}})\index{move}, {\it{g\/}})\index{lg-at-n}\-
\end{tabbing}


\noindent{\sc Event}:   Disable n-eq-k-rhoi7a.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  lg-at-rhoi7a \\ \index{lg-at-rhoi7a}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{7}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{8}})\index{move}, {\it{g\/}})\index{lg-at-n}\-
\end{tabbing}


\noindent{\sc Event}:   Disable lg-at-rhoi7a.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  lg-rhoi7a \\ \index{lg-rhoi7a}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{7}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{8}})\index{move}, {\it{g\/}})\index{lg}\-
\end{tabbing}


\noindent{\sc Event}:   Disable lg-rhoi7a.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  rhoi7a-preserves-lg \\ \index{rhoi7a-preserves-lg}
\=\+({\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} $\wedge$ ({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{rhoi7a}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi7a} $\wedge$ {\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg}) \\ 
$\rightarrow$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}})\index{lg}\-
\end{tabbing}

\begin{verbatim};;;rhoi7b
\end{verbatim}\begin{tabbing}{\sc Theorem}:  rhoi7b-preserves-lg \\ \index{rhoi7b-preserves-lg}
\=\+({\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} $\wedge$ ({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{rhoi7b}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi7b} $\wedge$ {\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg}) \\ 
$\rightarrow$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}})\index{lg}\-
\end{tabbing}

\begin{verbatim};;;rhoi8
\end{verbatim}\begin{tabbing}{\sc Theorem}:  n-neq-k-rhoi8 \\ \index{n-neq-k-rhoi8}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\not=$ {\it{n\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{8}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{9}})\index{move}, {\rm{move}}\,({\it{g\/}}, {\it{k\/}}, {\tt{4}})\index{move})\index{lg-at-n}\-
\end{tabbing}


\noindent{\sc Event}:   Disable n-neq-k-rhoi8.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  n-eq-k-rhoi8 \\ \index{n-eq-k-rhoi8}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{8}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg-at-n}}\,({\it{k\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{9}})\index{move}, {\rm{move}}\,({\it{g\/}}, {\it{k\/}}, {\tt{4}})\index{move})\index{lg-at-n}\-
\end{tabbing}


\noindent{\sc Event}:   Disable n-eq-k-rhoi8.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  lg-at-rhoi8 \\ \index{lg-at-rhoi8}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{8}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{9}})\index{move}, {\rm{move}}\,({\it{g\/}}, {\it{k\/}}, {\tt{4}})\index{move})\index{lg-at-n}\-
\end{tabbing}


\noindent{\sc Event}:   Disable lg-at-rhoi8.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  lg-rhoi8 \\ \index{lg-rhoi8}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{8}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{9}})\index{move}, {\rm{move}}\,({\it{g\/}}, {\it{k\/}}, {\tt{4}})\index{move})\index{lg}\-
\end{tabbing}


\noindent{\sc Event}:   Disable lg-rhoi8.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  rhoi8-preserves-lg \\ \index{rhoi8-preserves-lg}
\=\+({\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} $\wedge$ ({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{rhoi8}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi8} $\wedge$ {\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg}) \\ 
$\rightarrow$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}})\index{lg}\-
\end{tabbing}

\begin{verbatim};;;rhoi9a
\end{verbatim}\begin{tabbing}{\sc Theorem}:  n-neq-k-rhoi9a \\ \index{n-neq-k-rhoi9a}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\not=$ {\it{n\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{9}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{10}})\index{move}, {\it{g\/}})\index{lg-at-n}\-
\end{tabbing}


\noindent{\sc Event}:   Disable n-neq-k-rhoi9a.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  n-eq-k-rhoi9a \\ \index{n-eq-k-rhoi9a}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{9}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg-at-n}}\,({\it{k\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg-at-n}}\,({\it{k\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{10}})\index{move}, {\it{g\/}})\index{lg-at-n}\-
\end{tabbing}


\noindent{\sc Event}:   Disable n-eq-k-rhoi9a.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  lg-at-rhoi9a \\ \index{lg-at-rhoi9a}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{9}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{10}})\index{move}, {\it{g\/}})\index{lg-at-n}\-
\end{tabbing}


\noindent{\sc Event}:   Disable lg-at-rhoi9a.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  lg-rhoi9a \\ \index{lg-rhoi9a}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{9}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{10}})\index{move}, {\it{g\/}})\index{lg}\-
\end{tabbing}


\noindent{\sc Event}:   Disable lg-rhoi9a.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  rhoi9a-preserves-lg \\ \index{rhoi9a-preserves-lg}
\=\+({\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} $\wedge$ ({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{rhoi9a}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi9a} $\wedge$ {\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg}) \\ 
$\rightarrow$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}})\index{lg}\-
\end{tabbing}

\begin{verbatim};;;rhoi9b
\end{verbatim}\begin{tabbing}{\sc Theorem}:  rhoi9b-preserves-lg \\ \index{rhoi9b-preserves-lg}
\=\+({\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} $\wedge$ ({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{rhoi9b}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi9b} $\wedge$ {\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg}) \\ 
$\rightarrow$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}})\index{lg}\-
\end{tabbing}

\begin{verbatim};;;rhoi10
\end{verbatim}\begin{tabbing}{\sc Theorem}:  n-neq-k-rhoi10 \\ \index{n-neq-k-rhoi10}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\not=$ {\it{n\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{10}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{11}})\index{move}, {\it{g\/}})\index{lg-at-n}\-
\end{tabbing}


\noindent{\sc Event}:   Disable n-neq-k-rhoi10.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  n-eq-k-rhoi10 \\ \index{n-eq-k-rhoi10}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{10}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg-at-n}}\,({\it{k\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg-at-n}}\,({\it{k\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{11}})\index{move}, {\it{g\/}})\index{lg-at-n}\-
\end{tabbing}


\noindent{\sc Event}:   Disable n-eq-k-rhoi10.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  lg-at-rhoi10 \\ \index{lg-at-rhoi10}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{10}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{11}})\index{move}, {\it{g\/}})\index{lg-at-n}\-
\end{tabbing}


\noindent{\sc Event}:   Disable lg-at-rhoi10.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  lg-rhoi10 \\ \index{lg-rhoi10}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{10}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{11}})\index{move}, {\it{g\/}})\index{lg}\-
\end{tabbing}


\noindent{\sc Event}:   Disable lg-rhoi10.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  rhoi10-preserves-lg \\ \index{rhoi10-preserves-lg}
\=\+({\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} $\wedge$ ({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{rhoi10}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi10} $\wedge$ {\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg}) \\ 
$\rightarrow$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}})\index{lg}\-
\end{tabbing}

\begin{verbatim};;;rhoi11a
\end{verbatim}\begin{tabbing}{\sc Theorem}:  n-neq-k-rhoi11a \\ \index{n-neq-k-rhoi11a}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\not=$ {\it{n\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{11}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{12}})\index{move}, {\it{g\/}})\index{lg-at-n}\-
\end{tabbing}


\noindent{\sc Event}:   Disable n-neq-k-rhoi11a.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  n-eq-k-rhoi11a \\ \index{n-eq-k-rhoi11a}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{11}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg-at-n}}\,({\it{k\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg-at-n}}\,({\it{k\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{12}})\index{move}, {\it{g\/}})\index{lg-at-n}\-
\end{tabbing}


\noindent{\sc Event}:   Disable n-eq-k-rhoi11a.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  lg-at-rhoi11a \\ \index{lg-at-rhoi11a}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{11}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{12}})\index{move}, {\it{g\/}})\index{lg-at-n}\-
\end{tabbing}


\noindent{\sc Event}:   Disable lg-at-rhoi11a.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  lg-rhoi11a \\ \index{lg-rhoi11a}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{11}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{12}})\index{move}, {\it{g\/}})\index{lg}\-
\end{tabbing}


\noindent{\sc Event}:   Disable lg-rhoi11a.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  rhoi11a-preserves-lg \\ \index{rhoi11a-preserves-lg}
\=\+({\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} $\wedge$ ({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{rhoi11a}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi11a} $\wedge$ {\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg}) \\ 
$\rightarrow$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}})\index{lg}\-
\end{tabbing}

\begin{verbatim};;;rhoi11b
\end{verbatim}\begin{tabbing}{\sc Theorem}:  rhoi11b-preserves-lg \\ \index{rhoi11b-preserves-lg}
\=\+({\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} $\wedge$ ({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{rhoi11b}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi11b} $\wedge$ {\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg}) \\ 
$\rightarrow$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}})\index{lg}\-
\end{tabbing}

\begin{verbatim};;;rhoi12
\end{verbatim}\begin{tabbing}{\sc Theorem}:  n-neq-k-rhoi12 \\ \index{n-neq-k-rhoi12}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\not=$ {\it{n\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{12}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{0}})\index{move}, {\rm{move}}\,({\it{g\/}}, {\it{k\/}}, {\tt{0}})\index{move})\index{lg-at-n}\-
\end{tabbing}
 

\noindent{\sc Event}:   Disable n-neq-k-rhoi12.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  n-eq-k-rhoi12 \\ \index{n-eq-k-rhoi12}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{12}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg-at-n}}\,({\it{k\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{0}})\index{move}, {\rm{move}}\,({\it{g\/}}, {\it{k\/}}, {\tt{0}})\index{move})\index{lg-at-n}\-
\end{tabbing}


\noindent{\sc Event}:   Disable n-eq-k-rhoi12.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  lg-at-rhoi12 \\ \index{lg-at-rhoi12}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{12}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{0}})\index{move}, {\rm{move}}\,({\it{g\/}}, {\it{k\/}}, {\tt{0}})\index{move})\index{lg-at-n}\-
\end{tabbing}


\noindent{\sc Event}:   Disable lg-at-rhoi12.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  lg-rhoi12 \\ \index{lg-rhoi12}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{12}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{0}})\index{move}, {\rm{move}}\,({\it{g\/}}, {\it{k\/}}, {\tt{0}})\index{move})\index{lg}\-
\end{tabbing}


\noindent{\sc Event}:   Disable lg-rhoi12.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  rhoi12-preserves-lg \\ \index{rhoi12-preserves-lg}
\=\+({\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} $\wedge$ ({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{rhoi12}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi12} $\wedge$ {\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg}) \\ 
$\rightarrow$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}})\index{lg}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  rho-preserves-lg \\ \index{rho-preserves-lg}
\=\+({\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} $\wedge$ ({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} $\wedge$ {\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg}) \\ 
$\rightarrow$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}})\index{lg}\-
\end{tabbing}


\noindent{\sc Event}:   Disable rhoi0-preserves-lg.

\addvspace{18pt}

\noindent{\sc Event}:   Disable rhoi1a-preserves-lg.

\addvspace{18pt}

\noindent{\sc Event}:   Disable rhoi1b-preserves-lg.

\addvspace{18pt}

\noindent{\sc Event}:   Disable rhoi2-preserves-lg.

\addvspace{18pt}

\noindent{\sc Event}:   Disable rhoi3a-preserves-lg.

\addvspace{18pt}

\noindent{\sc Event}:   Disable rhoi3b-preserves-lg.

\addvspace{18pt}

\noindent{\sc Event}:   Disable rhoi4-preserves-lg.

\addvspace{18pt}

\noindent{\sc Event}:   Disable rhoi5a-preserves-lg.

\addvspace{18pt}

\noindent{\sc Event}:   Disable rhoi5b-preserves-lg.

\addvspace{18pt}

\noindent{\sc Event}:   Disable rhoi6-preserves-lg.

\addvspace{18pt}

\noindent{\sc Event}:   Disable rhoi7a-preserves-lg.

\addvspace{18pt}

\noindent{\sc Event}:   Disable rhoi7b-preserves-lg.

\addvspace{18pt}

\noindent{\sc Event}:   Disable rhoi8-preserves-lg.

\addvspace{18pt}

\noindent{\sc Event}:   Disable rhoi9a-preserves-lg.

\addvspace{18pt}

\noindent{\sc Event}:   Disable rhoi9b-preserves-lg.

\addvspace{18pt}

\noindent{\sc Event}:   Disable rhoi10-preserves-lg.

\addvspace{18pt}

\noindent{\sc Event}:   Disable rhoi11a-preserves-lg.

\addvspace{18pt}

\noindent{\sc Event}:   Disable rhoi11b-preserves-lg.

\addvspace{18pt}

\noindent{\sc Event}:   Disable rhoi12-preserves-lg.

\addvspace{18pt}


\begin{verbatim};;;;;;;;;;;;;;;;;;;;;;;;;;; a0.ev ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;;;(exist-union lp n '(8 9 10 11 12))  and 
;;;(not (exist-union l n '(8 9 10 11 12))) 
;;;implies that k is the witness of 
;;;(exist-union lp n '(8 9 10 11 12)). This 
;;;proposition would have been more natural 
;;;if we had been able to prove:
;;;(prove-lemma exist-l8-12 (rewrite)
;;;   (implies (and (ws n l g)
;;;                 (member k (nset n))
;;;                 (rhoi n k l g lp gp)
;;;                 (exist-union lp n '(8 9 10 11 12))
;;;                 (not (exist-union l n '(8 9 10 11 12))))
;;;            (equal k (exist-union lp n '(8 9 10 11 12))))).
;;;However Bmp refused to rewrite equal clause. 

\end{verbatim}\begin{tabbing}{\sc Theorem}:  exist-l8-12 \\ \index{exist-l8-12}
\=\+(\=\+{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} \\ 
$\wedge$$\;\;\;\;${\rm{exist-union}}\,({\it{lp\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{exist-union} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\not=$ {\rm{exist-union}}\,({\it{lp\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{exist-union}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{exist-union}}\,({\it{l\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{exist-union}\-
\end{tabbing}

\begin{verbatim};;;If (exist-union lp n '(8 9 10 11 12)) and 
;;;(not (exist-union l n '(8 9 10 11 12))) hold,
;;;then the k's entry of lp is between 8..12.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  k-in-lp8-12 \\ \index{k-in-lp8-12}
\=\+(\=\+{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} \\ 
$\wedge$$\;\;\;\;${\rm{exist-union}}\,({\it{lp\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{exist-union} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{exist-union}}\,({\it{l\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{exist-union}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{union-at-n}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n}\-
\end{tabbing}

\begin{verbatim};;;If k's entry in lp is between 8..12 and 
;;;k's entry of l is not between 8..12,
;;;then k's entry of l is either 5 or 7.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  k-not-in-l8-12-then-l57 \\ \index{k-not-in-l8-12-then-l57}
\=\+(\=\+{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n}) \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{7}})\index{at}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{5}})\index{at}\-
\end{tabbing}

\begin{verbatim};;;If k's entry in lp is between 8..12 and
;;;(not (exist-union l n '(8 9 10 11 12))) holds,
;;;then k's entry of l is either 5 or 7.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  k-in-l57 \\ \index{k-in-l57}
\=\+(\=\+{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{exist-union}}\,({\it{l\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{exist-union}) \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{7}})\index{at}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{5}})\index{at}\-
\end{tabbing}

\begin{verbatim};;;If (exist-union lp n '(8 9 10 11 12)) and 
;;;(not (exist-union l n '(8 9 10 11 12))) hold,
;;;then the k's entry of l is between either 5 or 7.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  ex-k-in-l57 \\ \index{ex-k-in-l57}
\=\+(\=\+{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} \\ 
$\wedge$$\;\;\;\;${\rm{exist-union}}\,({\it{lp\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{exist-union} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{exist-union}}\,({\it{l\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{exist-union}) \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{7}})\index{at}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{5}})\index{at}\-
\end{tabbing}

\begin{verbatim};;;Auxiliary lemma for ex-cond-rhoi5.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  cond-rhoi5 \\ \index{cond-rhoi5}
\=\+(\=\+{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{5}})\index{at})\- \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{exist-union}}\,({\it{g\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{1}}{\tt{)}})\index{exist-union})\-
\end{tabbing}

\begin{verbatim};;;If (exist-union lp n '(8 9 10 11 12)) and 
;;;(not (exist-union l n '(8 9 10 11 12))) and
;;;the k's entry in l is 5 then 
;;;(not (exist-union g n '(1))) holds.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  ex-cond-rhoi5 \\ \index{ex-cond-rhoi5}
\=\+(\=\+{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} \\ 
$\wedge$$\;\;\;\;${\rm{exist-union}}\,({\it{lp\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{exist-union} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{exist-union}}\,({\it{l\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{exist-union}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{5}})\index{at})\- \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{exist-union}}\,({\it{g\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{1}}{\tt{)}})\index{exist-union})\-
\end{tabbing}

\begin{verbatim};;;Auxiliary lemma for ex-cond-rhoi7.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  cond-rhoi7 \\ \index{cond-rhoi7}
\=\+(\=\+{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{7}})\index{at})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{exist-union}}\,({\it{g\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{4}}{\tt{)}})\index{exist-union}\-
\end{tabbing}

\begin{verbatim};;;If (exist-union lp n '(8 9 10 11 12)) and 
;;;(not (exist-union l n '(8 9 10 11 12))) and
;;;the k's entry in l is 7, then 
;;;(exist-union g n '(4)) holds.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  ex-cond-rhoi7 \\ \index{ex-cond-rhoi7}
\=\+(\=\+{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} \\ 
$\wedge$$\;\;\;\;${\rm{exist-union}}\,({\it{lp\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{exist-union} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{exist-union}}\,({\it{l\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{exist-union}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{7}})\index{at})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{exist-union}}\,({\it{g\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{4}}{\tt{)}})\index{exist-union}\-
\end{tabbing}

\begin{verbatim};;;If (exist-union lp n '(8 9 10 11 12)) 
;;;and (not (exist-union l n '(8 9 10 11 12))),
;;;then (not (exist-union g n '(1))) holds.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  l5-only-lp8 \\ \index{l5-only-lp8}
\=\+(\=\+{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{exist-union}}\,({\it{l\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{exist-union}) \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{exist-union}}\,({\it{lp\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{exist-union})\- \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{exist-union}}\,({\it{g\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{1}}{\tt{)}})\index{exist-union})\-
\end{tabbing}

\begin{verbatim};;;If j is not equal to k and j's entry of l
;;;is neither 3 or 4, then j's entry of lp
;;;is not 4.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  j-neq-k-j-not-in-lp4 \\ \index{j-neq-k-j-not-in-lp4}
\=\+(\=\+{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{union-at-n}}\,({\it{l\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{)}})\index{union-at-n}))\- \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{at}}\,({\it{lp\/}}, {\it{j\/}}, {\tt{4}})\index{at})\-
\end{tabbing}

\begin{verbatim};;;If k's entry of l is neither 3 or 4, then
;;;k's entry of lp is not 4.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  j-eq-k-j-not-in-lp4 \\ \index{j-eq-k-j-not-in-lp4}
\=\+(\=\+{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{)}})\index{union-at-n}))\- \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{at}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{4}})\index{at})\-
\end{tabbing}

\begin{verbatim};;;If j's entry of l is neither 3 or 4, then 
;;;j's entry of lp is not 4.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  lp4-empty \\ \index{lp4-empty}
\=\+(\=\+{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{union-at-n}}\,({\it{l\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{)}})\index{union-at-n}))\- \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{at}}\,({\it{lp\/}}, {\it{j\/}}, {\tt{4}})\index{at})\-
\end{tabbing}

\begin{verbatim};;;If (not (exist-union l n '(8 9 10 11 12)))
;;; and (exist-union lp n '(8 9 10 11 12)) hold,
;;;then there is no entry 4 in l.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  l8-l12-empty \\ \index{l8-l12-empty}
\=\+(\=\+{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{exist-union}}\,({\it{l\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{exist-union}) \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{a0}}\,({\it{n\/}}, {\it{lp\/}}, {\it{j\/}})\index{a0}\-
\end{tabbing}

\begin{verbatim};;;If (exist-union g n '(3 4)) holds and 
;;;the k's entry in l is not 4, then
;;;the k's entry in lp is not 4 either. 
;;;(Doorway is locked.)
\end{verbatim}\begin{tabbing}{\sc Theorem}:  dwy-lckd \\ \index{dwy-lckd}
\=\+(\=\+{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} \\ 
$\wedge$$\;\;\;\;${\rm{exist-union}}\,({\it{g\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{)}})\index{exist-union} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{4}})\index{at}))\- \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{at}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{4}})\index{at})\-
\end{tabbing}

\begin{verbatim};;;If (exist-union l n '(8 9 10 11 12))
;;;holds and j is equal to k, then 
;;;j's entry in lp is not 4.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  j-eq-k-l8-l12-nonemp \\ \index{j-eq-k-l8-l12-nonemp}
\=\+(\=\+{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} \\ 
$\wedge$$\;\;\;\;${\rm{exist-union}}\,({\it{l\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{exist-union} \\ 
$\wedge$$\;\;\;\;${\rm{a0}}\,({\it{n\/}}, {\it{l\/}}, {\it{k\/}})\index{a0} \\ 
$\wedge$$\;\;\;\;${\rm{a1}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{a1})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{a0}}\,({\it{n\/}}, {\it{lp\/}}, {\it{k\/}})\index{a0}\-
\end{tabbing}

\begin{verbatim};;;If (exist-union l n '(8 9 10 11 12))
;;; holds and j is not equal to k, then 
;;;the j's entry in lp is not 4.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  j-neq-k-l8-l12-nonemp \\ \index{j-neq-k-l8-l12-nonemp}
\=\+(\=\+{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{a0}}\,({\it{n\/}}, {\it{l\/}}, {\it{j\/}})\index{a0} \\ 
$\wedge$$\;\;\;\;${\rm{exist-union}}\,({\it{l\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{exist-union})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{a0}}\,({\it{n\/}}, {\it{lp\/}}, {\it{j\/}})\index{a0}\-
\end{tabbing}

\begin{verbatim};;;If (exist-union l n '(8 9 10 11 12))
;;;holds then there is no entry 4 in lp.
;;;The order of the use hints is critical. 
;;;Change the order and we fail.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  l8-l12-nonemp \\ \index{l8-l12-nonemp}
\=\+(\=\+{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} \\ 
$\wedge$$\;\;\;\;${\rm{exist-union}}\,({\it{l\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{exist-union} \\ 
$\wedge$$\;\;\;\;${\rm{a0}}\,({\it{n\/}}, {\it{l\/}}, {\it{j\/}})\index{a0} \\ 
$\wedge$$\;\;\;\;${\rm{a1}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{a1})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{a0}}\,({\it{n\/}}, {\it{lp\/}}, {\it{j\/}})\index{a0}\-
\end{tabbing}

\begin{verbatim};;;If (exist-union lp n '(8 9 10 11 12))
;;;holds, then there is no entry 4 in lp.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  rho-preserves-a0 \\ \index{rho-preserves-a0}
\=\+(\=\+{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{a0}}\,({\it{n\/}}, {\it{l\/}}, {\it{j\/}})\index{a0} \\ 
$\wedge$$\;\;\;\;${\rm{a1}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{a1})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{a0}}\,({\it{n\/}}, {\it{lp\/}}, {\it{j\/}})\index{a0}\-
\end{tabbing}

\begin{verbatim};;;;;;;;;;;;;;;;;;;;;;;;;;; a1.ev ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;* ep-l8-12

;;;Auxiliary lemma for at-gp-rhoi5
\end{verbatim}\begin{tabbing}{\sc Theorem}:  gp-rhoi5 \\ \index{gp-rhoi5}
\=\+(\=\+{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{5}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{g\/}}, {\it{k\/}}, {\tt{3}})\index{at})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{at}}\,({\it{gp\/}}, {\it{k\/}}, {\tt{3}})\index{at}\-
\end{tabbing}

\begin{verbatim};;;If (not (exist-union l n '(8 9 10 11 12))), 
;;;(exist-union lp n '(8 9 10 11 12)) and the k's
;;;entry in l is 5 then the k's entry in gp is 3.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  ex-gp-rhoi5 \\ \index{ex-gp-rhoi5}
\=\+(\=\+{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{exist-union}}\,({\it{l\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{exist-union}) \\ 
$\wedge$$\;\;\;\;${\rm{exist-union}}\,({\it{lp\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{exist-union} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{5}})\index{at})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{at}}\,({\it{gp\/}}, {\it{k\/}}, {\tt{3}})\index{at}\-
\end{tabbing}

\begin{verbatim};;;If (not (exist-union l n '(8 9 10 11 12)))
;;;and (exist-union lp n '(8 9 10 11 12)) holds,
;;;then the k's entry is either 3 or 4.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  k-in-gp34 \\ \index{k-in-gp34}
\=\+(\=\+{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{exist-union}}\,({\it{l\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{exist-union}) \\ 
$\wedge$$\;\;\;\;${\rm{exist-union}}\,({\it{lp\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{exist-union})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{union-at-n}}\,({\it{gp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{)}})\index{union-at-n}\-
\end{tabbing}

\begin{verbatim};;;If (exist-union lp n '(8 9 10 11 12)) and
;;;(not (exist-union l n '(8 9 10 11 12))) holds,
;;;then so does (exist-intersect-8-12-3-4 n lp gp).
\end{verbatim}\begin{tabbing}{\sc Theorem}:  lm-a1-ep-l8-12 \\ \index{lm-a1-ep-l8-12}
\=\+(\=\+{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} \\ 
$\wedge$$\;\;\;\;${\rm{exist-union}}\,({\it{lp\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{exist-union} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{exist-union}}\,({\it{l\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{exist-union}) \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{exist-intersect-8-12-3-4}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}})\index{exist-intersect-8-12-3-4}\-
\end{tabbing}

\begin{verbatim};;;If (not (exist-union l n '(8 9 10 11 12))) holds,
;;;then so does a1.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  a1-ep-l8-12 \\ \index{a1-ep-l8-12}
\=\+(\=\+{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{exist-union}}\,({\it{l\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{exist-union}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{a1}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}})\index{a1}\-
\end{tabbing}

\begin{verbatim};* nep-l8-12

;;;If (exist-intersect-8-12-3-4 n l g) holds and
;;;k is not its witness then 
;;;(exist-intersect-8-12-3-4 n lp gp) holds.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  int-k-not-ex-int \\ \index{int-k-not-ex-int}
\=\+(\=\+{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\not=$ {\rm{exist-intersect-8-12-3-4}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{exist-intersect-8-12-3-4}) \\ 
$\wedge$$\;\;\;\;${\rm{exist-intersect-8-12-3-4}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{exist-intersect-8-12-3-4})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{exist-intersect-8-12-3-4}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}})\index{exist-intersect-8-12-3-4}\-
\end{tabbing}

\begin{verbatim};;;If (exist-union l n '(8 9 10 11 12)) holds and 
;;;k's enrty is not between 8 and 12 then
;;;(exist-intersect-8-12-3-4 n lp gp) holds.
;;; j \neq k
\end{verbatim}\begin{tabbing}{\sc Theorem}:  a1-k-not-in-l8-12-nep-l8-12 \\ \index{a1-k-not-in-l8-12-nep-l8-12}
\=\+(\=\+{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} \\ 
$\wedge$$\;\;\;\;${\rm{exist-union}}\,({\it{l\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{exist-union} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n}) \\ 
$\wedge$$\;\;\;\;${\rm{a1}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{a1})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{exist-intersect-8-12-3-4}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}})\index{exist-intersect-8-12-3-4}\-
\end{tabbing}

\begin{verbatim};* k-in-l8-11
;;;If the k's entry in l is between 8 and 11,
;;;then the k's entry in lp is between 9 and 12.
;;;We need rho-preserves-lg.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  l8-11-k-in-lp9-12 \\ \index{l8-11-k-in-lp9-12}
\=\+(\=\+{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{)}})\index{union-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{union-at-n}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n}\-
\end{tabbing}

\begin{verbatim};;;If the k's entry in l is between 8 and 11,
;;;then the k's entry in lp is between 8 and 12
;;;and the entry in gp is either 3 or 4.
;;;l8-11-k-in-lp9-12, un9-12-then-un8-12 and 
;;;rho-preserves-lg are used.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  lm-a1-k-in-l8-11-nep-l8-12 \\ \index{lm-a1-k-in-l8-11-nep-l8-12}
\=\+(\=\+{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{)}})\index{union-at-n} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg})\- \\ 
$\rightarrow$$\;\;\;\;$({\rm{union-at-n}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n} $\wedge$ {\rm{union-at-n}}\,({\it{gp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{)}})\index{union-at-n})\-
\end{tabbing}

\begin{verbatim};;;If (exist-union lp n '(8 9 10 11 12)) holds,
;;;and the k's entry in l is between 8 and 11 then
;;;(exist-intersect-8-12-3-4 n lp gp) holds.
;;; j \eq k and k \in l8-11
\end{verbatim}\begin{tabbing}{\sc Theorem}:  a1-k-in-l8-11-nep-l8-12 \\ \index{a1-k-in-l8-11-nep-l8-12}
\=\+(\=\+{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{exist-union}}\,({\it{lp\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{exist-union} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{)}})\index{union-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{exist-intersect-8-12-3-4}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}})\index{exist-intersect-8-12-3-4}\-
\end{tabbing}

\begin{verbatim};;;If the k's entry in l is 12 then the k's entry in l is 0.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  k-in-lp0 \\ \index{k-in-lp0}
\=\+({\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} $\wedge$ ({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} $\wedge$ {\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{12}})\index{at}) \\ 
$\rightarrow$$\;\;\;\;${\rm{at}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{0}})\index{at}\-
\end{tabbing}

\begin{verbatim};;;If (exist-union lp n '(8 9 10 11 12)) holds
;;;and k's entry in l is 12, then k is not the 
;;;witness of (exist-union lp n '(8 9 10 11 12)).
\end{verbatim}\begin{tabbing}{\sc Theorem}:  k-not-ex-lp8-12 \\ \index{k-not-ex-lp8-12}
\=\+(\=\+{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} \\ 
$\wedge$$\;\;\;\;${\rm{exist-union}}\,({\it{lp\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{exist-union} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{12}})\index{at})\- \\ 
$\rightarrow$$\;\;\;\;$({\it{k\/}} $\not=$ {\rm{exist-union}}\,({\it{lp\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{exist-union})\-
\end{tabbing}

\begin{verbatim};;;If the k's entry in lp is 8,
;;;then k's entry in l is either 5 or 7.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  lp8-k-in-l57 \\ \index{lp8-k-in-l57}
\=\+({\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} $\wedge$ ({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} $\wedge$ {\rm{at}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{8}})\index{at}) \\ 
$\rightarrow$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{7}}{\tt{)}})\index{union-at-n}\-
\end{tabbing}

\begin{verbatim};;;If the k's entry in lp is 8,
;;;then k's entry in l is between 5 and 12.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  k-in-lp8-then-l5-12 \\ \index{k-in-lp8-then-l5-12}
\=\+({\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} $\wedge$ {\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} $\wedge$ ({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{at}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{8}})\index{at}) \\ 
$\rightarrow$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n}\-
\end{tabbing}

\begin{verbatim};;;If the k's entry in lp is between 9 and 12,
;;;then  the k's entry in l is between 8 and 11.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  lp9-12-k-in-l8-11 \\ \index{lp9-12-k-in-l8-11}
\=\+(\=\+{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{)}})\index{union-at-n}\-
\end{tabbing}

\begin{verbatim};;;If the k's entry in lp is between 9 and 12,
;;;then the k's entry in l is between 5 and 12.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  k-in-lp9-12-then-l5-12 \\ \index{k-in-lp9-12-then-l5-12}
\=\+(\=\+{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} \\ 
$\wedge$$\;\;\;\;${\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n}\-
\end{tabbing}

\begin{verbatim};;;If the k's entry in lp is between 8 and 12,
;;;then the k's entry in l is between 5 and 12.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  k-in-l5-12 \\ \index{k-in-l5-12}
\=\+(\=\+{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} \\ 
$\wedge$$\;\;\;\;${\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n}\-
\end{tabbing}

\begin{verbatim};;;If (exist-union lp n '(8 9 10 11 12)) holds
;;;and k is not its witness, then the witness has 
;;;its entry in l between 5 and 12.
;;;ex-lp8-12-in-lp8-12, member-ex-union used.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  k-neq-ex-lp8-12-in-l5-12 \\ \index{k-neq-ex-lp8-12-in-l5-12}
\=\+(\=\+{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} \\ 
$\wedge$$\;\;\;\;${\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{exist-union}}\,({\it{lp\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{exist-union} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\not=$ {\rm{exist-union}}\,({\it{lp\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{exist-union}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{union-at-n}}\,(\=\+{\it{l\/}}, \\ 
{\rm{exist-union}}\,({\it{lp\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{exist-union}, \\ 
{\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\-\index{union-at-n}\-
\end{tabbing}

\begin{verbatim};;;If (exist-union lp n '(8 9 10 11 12)) holds and
;;;the witness has its entry in lp between 8 and 12,
;;;then its entry in l is between 5 and 12.
;;;ex-lp8-12-in-lp8-12, member-ex-union used.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  ex-lp8-12-then-l5-12 \\ \index{ex-lp8-12-then-l5-12}
\=\+(\=\+{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} \\ 
$\wedge$$\;\;\;\;${\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{exist-union}}\,({\it{lp\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{exist-union})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{union-at-n}}\,(\=\+{\it{l\/}}, \\ 
{\rm{exist-union}}\,({\it{lp\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{exist-union}, \\ 
{\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\-\index{union-at-n}\-
\end{tabbing}

\begin{verbatim};;;If (exist-union lp n '(8 9 10 11 12)) holds
;;;and k is not the witness of 
;;;(exist-union lp n '(8 9 10 11 12)), then 
;;;the witness has its entry 4 in gp.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  ex-lp8-12-in-gp4 \\ \index{ex-lp8-12-in-gp4}
\=\+(\=\+{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{12}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{a3-at-n1-n2}}\,({\it{k\/}}, {\rm{exist-union}}\,({\it{lp\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{exist-union}, {\it{l\/}}, {\it{g\/}})\index{a3-at-n1-n2} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\not=$ {\rm{exist-union}}\,({\it{lp\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{exist-union}) \\ 
$\wedge$$\;\;\;\;${\rm{exist-union}}\,({\it{lp\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{exist-union})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{at}}\,({\it{gp\/}}, {\rm{exist-union}}\,({\it{lp\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{exist-union}, {\tt{4}})\index{at}\-
\end{tabbing}

\begin{verbatim};;;If (exist-union lp n '(8 9 10 11 12)) holds and
;;;k's entry in l is 12 then the witness has its 
;;;entry in lp between 8 and 12 and in gp either 3 or 4.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  lm-a1-k-in-l12-nep-l8-12 \\ \index{lm-a1-k-in-l12-nep-l8-12}
\=\+(\=\+{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} \\ 
$\wedge$$\;\;\;\;${\rm{exist-union}}\,({\it{lp\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{exist-union} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{12}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{a3-at-n1-n2}}\,({\it{k\/}}, {\rm{exist-union}}\,({\it{lp\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{exist-union}, {\it{l\/}}, {\it{g\/}})\index{a3-at-n1-n2})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{union-at-n}}\,({\it{gp\/}}, {\rm{exist-union}}\,({\it{lp\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{exist-union}, {\tt{'}}{\tt{(}}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{)}})\index{union-at-n}\-
\end{tabbing}

\begin{verbatim};;;If (exist-union lp n '(8 9 10 11 12)) holds
;;;and k's entry in l is 12, then 
;;;(exist-intersect-8-12-3-4 n lp gp) holds.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  a1-k-in-l12-nep-l8-12 \\ \index{a1-k-in-l12-nep-l8-12}
\=\+(\=\+{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} \\ 
$\wedge$$\;\;\;\;${\rm{exist-union}}\,({\it{lp\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{exist-union} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{12}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{a3-at-n1-n2}}\,({\it{k\/}}, {\rm{exist-union}}\,({\it{lp\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{exist-union}, {\it{l\/}}, {\it{g\/}})\index{a3-at-n1-n2})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{exist-intersect-8-12-3-4}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}})\index{exist-intersect-8-12-3-4}\-
\end{tabbing}

\begin{verbatim};;;Auxiliary lemma for a1-nep-l8-12.
;;;We have an instance of a3 in the lemma.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  lm1-a1-nep-l8-12 \\ \index{lm1-a1-nep-l8-12}
\=\+(\=\+{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{a1}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{a1} \\ 
$\wedge$$\;\;\;\;${\rm{a3-at-n1-n2}}\,({\it{k\/}}, {\rm{exist-union}}\,({\it{lp\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{exist-union}, {\it{l\/}}, {\it{g\/}})\index{a3-at-n1-n2} \\ 
$\wedge$$\;\;\;\;${\rm{exist-union}}\,({\it{lp\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{exist-union} \\ 
$\wedge$$\;\;\;\;${\rm{exist-union}}\,({\it{l\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{exist-union})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{exist-intersect-8-12-3-4}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}})\index{exist-intersect-8-12-3-4}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  lm-a1-nep-l8-12 \\ \index{lm-a1-nep-l8-12}
\=\+(\=\+{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{a1}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{a1} \\ 
$\wedge$$\;\;\;\;${\rm{a3}}\,({\it{n\/}}, {\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{a3} \\ 
$\wedge$$\;\;\;\;${\rm{exist-union}}\,({\it{l\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{exist-union} \\ 
$\wedge$$\;\;\;\;${\rm{exist-union}}\,({\it{lp\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{exist-union})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{exist-intersect-8-12-3-4}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}})\index{exist-intersect-8-12-3-4}\-
\end{tabbing}

\begin{verbatim};;;If (exist-union lp n '(8 9 10 11 12)) and
;;;(exist-union l n '(8 9 10 11 12)) hold,
;;;then so does (exist-intersect-8-12-3-4 n lp gp).
\end{verbatim}\begin{tabbing}{\sc Theorem}:  a1-nep-l8-12 \\ \index{a1-nep-l8-12}
\=\+(\=\+{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{a1}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{a1} \\ 
$\wedge$$\;\;\;\;${\rm{a3}}\,({\it{n\/}}, {\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{a3} \\ 
$\wedge$$\;\;\;\;${\rm{exist-union}}\,({\it{l\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{exist-union})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{a1}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}})\index{a1}\-
\end{tabbing}

\begin{verbatim};;;If (exist-union lp n '(8 9 10 11 12)) holds,
;;;then so does (exist-intersect-8-12-3-4 n lp gp).
\end{verbatim}\begin{tabbing}{\sc Theorem}:  rho-preserves-a1 \\ \index{rho-preserves-a1}
\=\+(\=\+{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{a1}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{a1} \\ 
$\wedge$$\;\;\;\;${\rm{a3}}\,({\it{n\/}}, {\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{a3})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{a1}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}})\index{a1}\-
\end{tabbing}

\begin{verbatim};;;;;;;;;;;;;;;;;;;;;;;;;;; a2.ev ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;* i-eq-k-j-neq-k

;;;If the k's entry in lp is between 10 and 12
;;;and the k's entry in l is between 10 and 12,
;;; then (phi9 k n g) holds.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  k-in-l10-11-or-phi9 \\ \index{k-in-l10-11-or-phi9}
\=\+(\=\+{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{)}})\index{union-at-n}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{phi9}}\,({\it{k\/}}, {\it{n\/}}, {\it{g\/}})\index{phi9}\-
\end{tabbing}

\begin{verbatim};;;If j is less than k and (phi9 k n g) holds,
;;;then the j's entry in g is either 0 or 1.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  phi9-j-in-g01 \\ \index{phi9-j-in-g01}
\=\+(({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ ({\it{j\/}} $<$ {\it{k\/}}) $\wedge$ {\rm{phi9}}\,({\it{k\/}}, {\it{n\/}}, {\it{g\/}})\index{phi9}) \\ 
$\rightarrow$$\;\;\;\;${\rm{union-at-n}}\,({\it{g\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{)}})\index{union-at-n}\-
\end{tabbing}

\begin{verbatim};;;If j is less than k and (phi9 k n g) holds,
;;;then the j's entry in lp is not between 5 and 12.
;;;lp-same-l-not is used.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  case-k-in-phi9 \\ \index{case-k-in-phi9}
\=\+(\=\+{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $<$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{phi9}}\,({\it{k\/}}, {\it{n\/}}, {\it{g\/}})\index{phi9})\- \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{union-at-n}}\,({\it{lp\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n})\-
\end{tabbing}

\begin{verbatim};;;If j is not equal to k and the k's entry in l is
;;;either 10 or 11, then the j's entry in lp is not
;;;between 5 and 12.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  case-k-in-l10-11 \\ \index{case-k-in-l10-11}
\=\+(\=\+{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} \\ 
$\wedge$$\;\;\;\;${\rm{a2-at-n1-n2}}\,({\it{k\/}}, {\it{j\/}}, {\it{l\/}})\index{a2-at-n1-n2} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{)}})\index{union-at-n})\- \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{union-at-n}}\,({\it{lp\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n})\-
\end{tabbing}

\begin{verbatim};;;Auxiliary lemma for lm-i-eq-k-j-neq-k with 
;;;(a2-at-n1-n2 k j l).
\end{verbatim}\begin{tabbing}{\sc Theorem}:  lm1-i-eq-k-j-neq-k \\ \index{lm1-i-eq-k-j-neq-k}
\=\+(\=\+{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $<$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{a2-at-n1-n2}}\,({\it{k\/}}, {\it{j\/}}, {\it{l\/}})\index{a2-at-n1-n2} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n})\- \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{union-at-n}}\,({\it{lp\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n})\-
\end{tabbing}

\begin{verbatim};;;If j is less then k and the k's entry in lp is
;;;between 10 and 12, then the j's entry in lp is
;;;not between 5 and 12.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  lm-i-eq-k-j-neq-k \\ \index{lm-i-eq-k-j-neq-k}
\=\+(\=\+{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $<$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{a2-at-n1-n2}}\,({\it{k\/}}, {\it{j\/}}, {\it{l\/}})\index{a2-at-n1-n2})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{a2-at-n1-n2}}\,({\it{k\/}}, {\it{j\/}}, {\it{lp\/}})\index{a2-at-n1-n2}\-
\end{tabbing}

\begin{verbatim};;;If j is less than k,
;;;then (a2-at-n1-n2 k j lp) holds.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  i-eq-k-j-neq-k \\ \index{i-eq-k-j-neq-k}
\=\+(\=\+{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $<$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{a2}}\,({\it{n\/}}, {\it{n\/}}, {\it{l\/}})\index{a2})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{a2-at-n1-n2}}\,({\it{k\/}}, {\it{j\/}}, {\it{lp\/}})\index{a2-at-n1-n2}\-
\end{tabbing}

\begin{verbatim};* j-eq-k-i-neq-k

;;;If the k's entry in l is not 4 and the k's entry in lp
;;;is between 5 and 7, then the k's entry in l is 
;;;between 5 and 7.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  k-in-lp5-7-not-l4-then-l5-7 \\ \index{k-in-lp5-7-not-l4-then-l5-7}
\=\+(\=\+{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{4}})\index{at}) \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{)}})\index{union-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{)}})\index{union-at-n}\-
\end{tabbing}

\begin{verbatim};;;If the k's entry in lp is between 5 and 7 then 
;;;the k's entry in l is  certainly between 5 and 12.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  k-in-lp5-7-then-l5-11 \\ \index{k-in-lp5-7-then-l5-11}
\=\+(\=\+{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{4}})\index{at}) \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{)}})\index{union-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{)}})\index{union-at-n}\-
\end{tabbing}

\begin{verbatim};;;If the k's entry in lp is 8,
;;;then the k's entry in l is between 5 and 11.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  k-in-lp8-then-l5-11 \\ \index{k-in-lp8-then-l5-11}
\=\+({\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} $\wedge$ ({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} $\wedge$ {\rm{at}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{8}})\index{at}) \\ 
$\rightarrow$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{)}})\index{union-at-n}\-
\end{tabbing}

\begin{verbatim};;;If the k's entry in lp is between 9 and 12,
;;;then the k's entry in l is between 5 and 12.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  k-in-lp9-12-then-l5-11 \\ \index{k-in-lp9-12-then-l5-11}
\=\+(\=\+{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{)}})\index{union-at-n}\-
\end{tabbing}

\begin{verbatim};;;If the k's entry in l is not 4 an the k's entry in lp is 
;;;between 5 and 12, then the k's entry in l is 
;;;between 5 and 11.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  k-in-l5-11 \\ \index{k-in-l5-11}
\=\+(\=\+{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{4}})\index{at}) \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{)}})\index{union-at-n}\-
\end{tabbing}

\begin{verbatim};;;If the k's entry in l is not 4, and the k's entry 
;;;in lp is not between 5 and 12, then the k's entry 
;;;in lp is not between 5 and 12.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  k-not-in-l4 \\ \index{k-not-in-l4}
\=\+(\=\+{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{4}})\index{at}) \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n}))\- \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{union-at-n}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n})\-
\end{tabbing}

\begin{verbatim};;;If a0 holds, and the k's entry in l is not 
;;;between 5 and 12, then the k's entry in lp is not
;;;between 5 and 12.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  k-not-in-lp5-12 \\ \index{k-not-in-lp5-12}
\=\+(\=\+{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} \\ 
$\wedge$$\;\;\;\;${\rm{a0}}\,({\it{n\/}}, {\it{l\/}}, {\it{k\/}})\index{a0} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{i\/}}, {\tt{'}}{\tt{(}}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n}))\- \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{union-at-n}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n})\-
\end{tabbing}

\begin{verbatim};;;Auxiliary lemma for lm-i-neq-k-j-eq-k.
;;;There is (a2-at-n1-n2 i k l) in the lemma.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  lm1-i-neq-k-j-eq-k \\ \index{lm1-i-neq-k-j-eq-k}
\=\+(\=\+{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $<$ {\it{i\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{a0}}\,({\it{n\/}}, {\it{l\/}}, {\it{k\/}})\index{a0} \\ 
$\wedge$$\;\;\;\;${\rm{a2-at-n1-n2}}\,({\it{i\/}}, {\it{k\/}}, {\it{l\/}})\index{a2-at-n1-n2} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{lp\/}}, {\it{i\/}}, {\tt{'}}{\tt{(}}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n})\- \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{union-at-n}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n})\-
\end{tabbing}

\begin{verbatim};;;If k is less than i and the i's entry in lp is 
;;;between 10 and 12, then the k's entry in lp is 
;;;between 5 and 12.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  lm-i-neq-k-j-eq-k \\ \index{lm-i-neq-k-j-eq-k}
\=\+(\=\+{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $<$ {\it{i\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{a0}}\,({\it{n\/}}, {\it{l\/}}, {\it{k\/}})\index{a0} \\ 
$\wedge$$\;\;\;\;${\rm{a2-at-n1-n2}}\,({\it{i\/}}, {\it{k\/}}, {\it{l\/}})\index{a2-at-n1-n2})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{a2-at-n1-n2}}\,({\it{i\/}}, {\it{k\/}}, {\it{lp\/}})\index{a2-at-n1-n2}\-
\end{tabbing}
 
\begin{verbatim};;;If k is less than i then (a2-at-n1-n2 i k lp) holds.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  i-neq-k-j-eq-k \\ \index{i-neq-k-j-eq-k}
\=\+(\=\+{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $<$ {\it{i\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{a0}}\,({\it{n\/}}, {\it{l\/}}, {\it{k\/}})\index{a0} \\ 
$\wedge$$\;\;\;\;${\rm{a2}}\,({\it{n\/}}, {\it{n\/}}, {\it{l\/}})\index{a2})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{a2-at-n1-n2}}\,({\it{i\/}}, {\it{k\/}}, {\it{lp\/}})\index{a2-at-n1-n2}\-
\end{tabbing}

\begin{verbatim};* i-j-neq-k

;;;If i and j are not equal to k and the i's entry in lp is
;;;between 10 and 12, then the j's entry in lp is 
;;;between 5 and 12.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  lm-i-j-neq-k \\ \index{lm-i-j-neq-k}
\=\+(\=\+{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $<$ {\it{i\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{a2-at-n1-n2}}\,({\it{i\/}}, {\it{j\/}}, {\it{l\/}})\index{a2-at-n1-n2})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{a2-at-n1-n2}}\,({\it{i\/}}, {\it{j\/}}, {\it{lp\/}})\index{a2-at-n1-n2}\-
\end{tabbing}

\begin{verbatim};;;If i and j are not equal to k,
;;; then (a2-at-n1-n2 i j lp) holds.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  i-j-neq-k \\ \index{i-j-neq-k}
\=\+(\=\+{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $<$ {\it{i\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{a2}}\,({\it{n\/}}, {\it{n\/}}, {\it{l\/}})\index{a2})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{a2-at-n1-n2}}\,({\it{i\/}}, {\it{j\/}}, {\it{lp\/}})\index{a2-at-n1-n2}\-
\end{tabbing}

\begin{verbatim};;;If i is not equal to k and j is less than i,
;;;then (a2-at-n1-n2 i j lp) holds.
;;;The order of the hints is crucial.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  i-neq-k \\ \index{i-neq-k}
\=\+(\=\+{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $<$ {\it{i\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{a0}}\,({\it{n\/}}, {\it{l\/}}, {\it{k\/}})\index{a0} \\ 
$\wedge$$\;\;\;\;${\rm{a2}}\,({\it{n\/}}, {\it{n\/}}, {\it{l\/}})\index{a2})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{a2-at-n1-n2}}\,({\it{i\/}}, {\it{j\/}}, {\it{lp\/}})\index{a2-at-n1-n2}\-
\end{tabbing}

\begin{verbatim};;;If j is less than k then (a2-at-n1-n2 k j lp) holds.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  i-eq-k \\ \index{i-eq-k}
\=\+(\=\+{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $<$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{a2}}\,({\it{n\/}}, {\it{n\/}}, {\it{l\/}})\index{a2})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{a2-at-n1-n2}}\,({\it{k\/}}, {\it{j\/}}, {\it{lp\/}})\index{a2-at-n1-n2}\-
\end{tabbing}

\begin{verbatim};;;If i is less than j then (a2-at-n1-n2 k j lp) holds.
;;;Again the order of the hints is crucial.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  rho-preserves-a2 \\ \index{rho-preserves-a2}
\=\+(\=\+{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $<$ {\it{i\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{a0}}\,({\it{n\/}}, {\it{l\/}}, {\it{k\/}})\index{a0} \\ 
$\wedge$$\;\;\;\;${\rm{a2}}\,({\it{n\/}}, {\it{n\/}}, {\it{l\/}})\index{a2})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{a2-at-n1-n2}}\,({\it{i\/}}, {\it{j\/}}, {\it{lp\/}})\index{a2-at-n1-n2}\-
\end{tabbing}

\begin{verbatim};;;;;;;;;;;;;;;;;;;;;;;;;;; a3.ev ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;* j-eq-k-i-neq-k

;;;If the i's entry in l is 12 and the k's entry in lp is
;;;between 5 and 12 then the k's entry in l is between 9 
;;;and 11.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  lm-k-in-l9-11 \\ \index{lm-k-in-l9-11}
\=\+(\=\+{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{a0}}\,({\it{n\/}}, {\it{l\/}}, {\it{k\/}})\index{a0} \\ 
$\wedge$$\;\;\;\;${\rm{a3-at-n1-n2}}\,({\it{i\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}})\index{a3-at-n1-n2} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{i\/}}, {\tt{12}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{)}})\index{union-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{)}})\index{union-at-n}\-
\end{tabbing}

\begin{verbatim};;;If i is not equal to k, the i's entry in l is 12, 
;;;and the k's entry in lp is between 5 and 12,
;;;then the k's entry in l is between 9 and 11.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  k-in-l9-11 \\ \index{k-in-l9-11}
\=\+(\=\+{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{a0}}\,({\it{n\/}}, {\it{l\/}}, {\it{k\/}})\index{a0} \\ 
$\wedge$$\;\;\;\;${\rm{a3-at-n1-n2}}\,({\it{i\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}})\index{a3-at-n1-n2} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{i\/}}, {\tt{12}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{)}})\index{union-at-n}\-
\end{tabbing}

\begin{verbatim};;;If the k's entry in lp is between 9 and 11
;;;then the k's entry in lp is between 9 and 12.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  k-in-lp9-12 \\ \index{k-in-lp9-12}
\=\+(\=\+{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{)}})\index{union-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{union-at-n}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n}\-
\end{tabbing}

\begin{verbatim};;Auxiliary lemma for lm-a3-i-neq-k-j-eq-k.
;;;There is (a3-at-n1-n2 i k l g) in the lemma.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  lm1-a3-i-neq-k-j-eq-k \\ \index{lm1-a3-i-neq-k-j-eq-k}
\=\+(\=\+{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{a0}}\,({\it{n\/}}, {\it{l\/}}, {\it{k\/}})\index{a0} \\ 
$\wedge$$\;\;\;\;${\rm{a3-at-n1-n2}}\,({\it{i\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}})\index{a3-at-n1-n2} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{i\/}}, {\tt{12}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{at}}\,({\it{gp\/}}, {\it{k\/}}, {\tt{4}})\index{at}\-
\end{tabbing}

\begin{verbatim};;;If i is not equal to k, the i's entry in lp is 12,
;;;and the k's entry in lp is between 5 and 12,
;;;then the k's entry in gp is 4.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  lm-a3-i-neq-k-j-eq-k \\ \index{lm-a3-i-neq-k-j-eq-k}
\=\+(\=\+{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{a0}}\,({\it{n\/}}, {\it{l\/}}, {\it{k\/}})\index{a0} \\ 
$\wedge$$\;\;\;\;${\rm{a3-at-n1-n2}}\,({\it{i\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}})\index{a3-at-n1-n2})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{a3-at-n1-n2}}\,({\it{i\/}}, {\it{k\/}}, {\it{lp\/}}, {\it{gp\/}})\index{a3-at-n1-n2}\-
\end{tabbing}

\begin{verbatim};;;If i is not equal to k,
;;;then (a3-at-n1-n2 i k lp gp) holds.
;;;The order of the hypotheses is crucial.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  a3-i-neq-k-j-eq-k \\ \index{a3-i-neq-k-j-eq-k}
\=\+(\=\+{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{a0}}\,({\it{n\/}}, {\it{l\/}}, {\it{k\/}})\index{a0} \\ 
$\wedge$$\;\;\;\;${\rm{a3}}\,({\it{n\/}}, {\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{a3} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\not=$ {\it{k\/}}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{a3-at-n1-n2}}\,({\it{i\/}}, {\it{k\/}}, {\it{lp\/}}, {\it{gp\/}})\index{a3-at-n1-n2}\-
\end{tabbing}

\begin{verbatim};* i-eq-k-j-neq-k

;;;If the k's entry in lp is 12 then (phi11 k n g) holds.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  cond-rhoi11 \\ \index{cond-rhoi11}
\=\+({\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} $\wedge$ ({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} $\wedge$ {\rm{at}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{12}})\index{at}) \\ 
$\rightarrow$$\;\;\;\;${\rm{phi11}}\,({\it{k\/}}, {\it{n\/}}, {\it{g\/}})\index{phi11}\-
\end{tabbing}

\begin{verbatim};;;If the k's entry in l is between 10 and 12, 
;;;the j's entry in l is between 5 and 12, and
;;;(a2-at-n2 k n l) holds, then k is less than j.
;;;Because Bmp does not rewrite the clause 
;;;(lessp k j), we take its contrapositive.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  k-lt-j \\ \index{k-lt-j}
\=\+(\=\+({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\not<$ {\it{j\/}}))\- \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{a2-at-n2}}\,({\it{k\/}}, {\it{n\/}}, {\it{l\/}})\index{a2-at-n2})\-
\end{tabbing}


\begin{verbatim};;;If k is less than j and (phi11 k n g) holds,
;;;then the j's entry in g is either 2 or 3.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  phi11-j-not-in-g23 \\ \index{phi11-j-not-in-g23}
\=\+(({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ ({\it{k\/}} $<$ {\it{j\/}}) $\wedge$ {\rm{phi11}}\,({\it{k\/}}, {\it{n\/}}, {\it{g\/}})\index{phi11}) \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{union-at-n}}\,({\it{g\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{)}})\index{union-at-n})\-
\end{tabbing}

\begin{verbatim};;;If j is not equal to k, (a2-at-n2 k n l), (phi11 k n g)
;;;the k's entry in l is between 10 and 12 and
;;;the j's entry in l is between 5 and 12,
;;;then the j's entry in g is either 2 or 3.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  lm1-j-not-in-g23 \\ \index{lm1-j-not-in-g23}
\=\+(\=\+({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{a2-at-n2}}\,({\it{k\/}}, {\it{n\/}}, {\it{l\/}})\index{a2-at-n2} \\ 
$\wedge$$\;\;\;\;${\rm{phi11}}\,({\it{k\/}}, {\it{n\/}}, {\it{g\/}})\index{phi11} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n})\- \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{union-at-n}}\,({\it{g\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{)}})\index{union-at-n})\-
\end{tabbing}

\begin{verbatim};;;If j is not equal to k, the k's entry in l is 
;;;between 10 and 12, the k's entry in lp is 12 and 
;;;the j's entry in l is between 5 and 12,
;;;then the j's entry in g is either 2 or 3.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  lm2-j-not-in-g23 \\ \index{lm2-j-not-in-g23}
\=\+(\=\+{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{a2-at-n2}}\,({\it{k\/}}, {\it{n\/}}, {\it{l\/}})\index{a2-at-n2} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{12}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{11}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n})\- \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{union-at-n}}\,({\it{g\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{)}})\index{union-at-n})\-
\end{tabbing}

\begin{verbatim};;;If j is not equal to k, the k's entry in lp is 12,
;;;and the j's entry in l is between 5 and 12,
;;;then the j's entry in g is either 2 or 3.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  j-not-in-g23 \\ \index{j-not-in-g23}
\=\+(\=\+{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{a2}}\,({\it{n\/}}, {\it{n\/}}, {\it{l\/}})\index{a2} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{11}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{12}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n})\- \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{union-at-n}}\,({\it{g\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{)}})\index{union-at-n})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  j-in-g4 \\ \index{j-in-g4}
\=\+(\=\+({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{union-at-n}}\,({\it{g\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{)}})\index{union-at-n}) \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{at}}\,({\it{g\/}}, {\it{j\/}}, {\tt{4}})\index{at}\-
\end{tabbing}

\begin{verbatim};;;un9-12-then-un8-12, if3, j-not-in-g23,
;;;l5-12-eq-l5-8-or-l9-12, un8-12-then-un5-12, 
;;;and j-in-g4 are used.
;;;If j is not equal to k, the k's entry in lp is 12, 
;;;the j's entry in l is between 5 and 12,
;;;then the j's entry in g is 4.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  a3-j-in-l5-12 \\ \index{a3-j-in-l5-12}
\=\+(\=\+{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{a2}}\,({\it{n\/}}, {\it{n\/}}, {\it{l\/}})\index{a2} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{11}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{12}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{at}}\,({\it{g\/}}, {\it{j\/}}, {\tt{4}})\index{at}\-
\end{tabbing}

\begin{verbatim};;;If the k's entry in lp is 12,
;;;then the k's entry in l is 11.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  k-in-l11 \\ \index{k-in-l11}
\=\+({\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} $\wedge$ ({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} $\wedge$ {\rm{at}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{12}})\index{at}) \\ 
$\rightarrow$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{11}})\index{at}\-
\end{tabbing}

\begin{verbatim};;;If k is not equal to j and the j's entry in g is 4,
;;;then the j's entry in gp is 4.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  lm1-a3-i-eq-k-j-neq-k \\ \index{lm1-a3-i-eq-k-j-neq-k}
\=\+(\=\+{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{a2}}\,({\it{n\/}}, {\it{n\/}}, {\it{l\/}})\index{a2} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{12}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{lp\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{at}}\,({\it{g\/}}, {\it{j\/}}, {\tt{4}})\index{at}\-
\end{tabbing}

\begin{verbatim};;;If j is not equal to k, the k's entry in lp is 12, 
;;;and the j's entry in lp is between 5 and 12,
;;;then the j's entry in gp is 4.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  lm-a3-i-eq-k-j-neq-k \\ \index{lm-a3-i-eq-k-j-neq-k}
\=\+(\=\+{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{a2}}\,({\it{n\/}}, {\it{n\/}}, {\it{l\/}})\index{a2} \\ 
$\wedge$$\;\;\;\;${\rm{a3-at-n1-n2}}\,({\it{k\/}}, {\it{j\/}}, {\it{l\/}}, {\it{g\/}})\index{a3-at-n1-n2})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{a3-at-n1-n2}}\,({\it{k\/}}, {\it{j\/}}, {\it{lp\/}}, {\it{gp\/}})\index{a3-at-n1-n2}\-
\end{tabbing}

\begin{verbatim};;;If j is not equal to k then (a3-at-n1-n2 k j lp gp).
\end{verbatim}\begin{tabbing}{\sc Theorem}:  a3-i-eq-k-j-neq-k \\ \index{a3-i-eq-k-j-neq-k}
\=\+(\=\+{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{a2}}\,({\it{n\/}}, {\it{n\/}}, {\it{l\/}})\index{a2} \\ 
$\wedge$$\;\;\;\;${\rm{a3}}\,({\it{n\/}}, {\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{a3})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{a3-at-n1-n2}}\,({\it{k\/}}, {\it{j\/}}, {\it{lp\/}}, {\it{gp\/}})\index{a3-at-n1-n2}\-
\end{tabbing}

\begin{verbatim};* i-j-neq-k

;;;If i,j are not equal to k, the i's entry in lp is 12
;;;and the j's entry in lp between 5 and 12
;;;then the j's entry in gp is 4.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  lm-a3-i-j-neq-k \\ \index{lm-a3-i-j-neq-k}
\=\+(\=\+{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{a3-at-n1-n2}}\,({\it{i\/}}, {\it{j\/}}, {\it{l\/}}, {\it{g\/}})\index{a3-at-n1-n2})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{a3-at-n1-n2}}\,({\it{i\/}}, {\it{j\/}}, {\it{lp\/}}, {\it{gp\/}})\index{a3-at-n1-n2}\-
\end{tabbing}

\begin{verbatim};;;If i,j are not equal to k, 
;;;then (a3-at-n1-n2 i j lp gp).
\end{verbatim}\begin{tabbing}{\sc Theorem}:  a3-i-j-neq-k \\ \index{a3-i-j-neq-k}
\=\+(\=\+{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} \\ 
$\wedge$$\;\;\;\;${\rm{a3}}\,({\it{n\/}}, {\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{a3} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\not=$ {\it{k\/}}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{a3-at-n1-n2}}\,({\it{i\/}}, {\it{j\/}}, {\it{lp\/}}, {\it{gp\/}})\index{a3-at-n1-n2}\-
\end{tabbing}

\begin{verbatim};* i-j-eq-k

\end{verbatim}\begin{tabbing}{\sc Theorem}:  lm-a3-i-j-eq-k \\ \index{lm-a3-i-j-eq-k}
\=\+(\=\+{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{a2}}\,({\it{n\/}}, {\it{n\/}}, {\it{l\/}})\index{a2} \\ 
$\wedge$$\;\;\;\;${\rm{a3-at-n1-n2}}\,({\it{k\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}})\index{a3-at-n1-n2})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{a3-at-n1-n2}}\,({\it{k\/}}, {\it{k\/}}, {\it{lp\/}}, {\it{gp\/}})\index{a3-at-n1-n2}\-
\end{tabbing}

\begin{verbatim};;;(a3-at-n1-n2 k k lp gp) holds by lg.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  a3-i-j-eq-k \\ \index{a3-i-j-eq-k}
\=\+(\=\+{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{a2}}\,({\it{n\/}}, {\it{n\/}}, {\it{l\/}})\index{a2} \\ 
$\wedge$$\;\;\;\;${\rm{a3}}\,({\it{n\/}}, {\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{a3})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{a3-at-n1-n2}}\,({\it{k\/}}, {\it{k\/}}, {\it{lp\/}}, {\it{gp\/}})\index{a3-at-n1-n2}\-
\end{tabbing}

\begin{verbatim};;;(a3-at-n1-n2 k j lp gp) holds.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  a3-i-eq-k \\ \index{a3-i-eq-k}
\=\+(\=\+{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{a2}}\,({\it{n\/}}, {\it{n\/}}, {\it{l\/}})\index{a2} \\ 
$\wedge$$\;\;\;\;${\rm{a3}}\,({\it{n\/}}, {\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{a3})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{a3-at-n1-n2}}\,({\it{k\/}}, {\it{j\/}}, {\it{lp\/}}, {\it{gp\/}})\index{a3-at-n1-n2}\-
\end{tabbing}

\begin{verbatim};;;If i is not equal to k,
;;;then (a3-at-n1-n2 i j lp gp) holds.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  a3-i-neq-k \\ \index{a3-i-neq-k}
\=\+(\=\+{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{a0}}\,({\it{n\/}}, {\it{l\/}}, {\it{k\/}})\index{a0} \\ 
$\wedge$$\;\;\;\;${\rm{a2}}\,({\it{n\/}}, {\it{n\/}}, {\it{l\/}})\index{a2} \\ 
$\wedge$$\;\;\;\;${\rm{a3}}\,({\it{n\/}}, {\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{a3} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\not=$ {\it{k\/}}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{a3-at-n1-n2}}\,({\it{i\/}}, {\it{j\/}}, {\it{lp\/}}, {\it{gp\/}})\index{a3-at-n1-n2}\-
\end{tabbing}

\begin{verbatim};;;(a3-at-n1-n2 i j lp gp) holds.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  rho-preserves-a3 \\ \index{rho-preserves-a3}
\=\+(\=\+{\rm{ws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{ws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{rhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{lp\/}}, {\it{gp\/}})\index{rhoi} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{a0}}\,({\it{n\/}}, {\it{l\/}}, {\it{k\/}})\index{a0} \\ 
$\wedge$$\;\;\;\;${\rm{a2}}\,({\it{n\/}}, {\it{n\/}}, {\it{l\/}})\index{a2} \\ 
$\wedge$$\;\;\;\;${\rm{a3}}\,({\it{n\/}}, {\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{a3})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{a3-at-n1-n2}}\,({\it{i\/}}, {\it{j\/}}, {\it{lp\/}}, {\it{gp\/}})\index{a3-at-n1-n2}\-
\end{tabbing}


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