%
\begin{isabellebody}%
\setisabellecontext{Family}%
%
\isamarkupsection{Families of sets%
}
\isamarkuptrue%
%
\isadelimtheory
%
\endisadelimtheory
%
\isatagtheory
\isacommand{theory}\isamarkupfalse%
\ Family\isanewline
\isakeyword{imports}\ Main\isanewline
\isakeyword{begin}%
\endisatagtheory
{\isafoldtheory}%
%
\isadelimtheory
%
\endisadelimtheory
\isanewline
\isanewline
\isacommand{definition}\isamarkupfalse%
\ count\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a\ set\ set\ {\isasymRightarrow}\ nat{\isachardoublequoteclose}\ \isakeyword{where}\ \isanewline
\ \ {\isachardoublequoteopen}count\ a\ F\ {\isasymequiv}\ card\ {\isacharbraceleft}S\ {\isasymin}\ F{\isachardot}\ a\ {\isasymin}\ S{\isacharbraceright}{\isachardoublequoteclose}\isanewline
\isanewline
\isacommand{lemma}\isamarkupfalse%
\ count{\isacharunderscore}Un{\isacharunderscore}disjoint{\isacharcolon}\isanewline
\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}A\ {\isasyminter}\ B\ {\isacharequal}\ {\isacharbraceleft}{\isacharbraceright}{\isachardoublequoteclose}\ \isakeyword{and}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\ {\isachardoublequoteopen}finite\ B{\isachardoublequoteclose}\isanewline
\ \ \isakeyword{shows}\ {\isachardoublequoteopen}count\ a\ {\isacharparenleft}A\ {\isasymunion}\ B{\isacharparenright}\ {\isacharequal}\ count\ a\ A\ {\isacharplus}\ count\ a\ B{\isachardoublequoteclose}\isanewline
%
\isadelimproof
%
\endisadelimproof
%
\isatagproof
\isacommand{proof}\isamarkupfalse%
{\isacharminus}\isanewline
\ \ \isacommand{have}\isamarkupfalse%
\ {\isachardoublequoteopen}{\isacharbraceleft}S\ {\isasymin}\ A{\isachardot}\ a\ {\isasymin}\ S{\isacharbraceright}\ {\isasyminter}\ {\isacharbraceleft}S\ {\isasymin}\ B{\isachardot}\ a\ {\isasymin}\ S{\isacharbraceright}\ {\isacharequal}\ {\isacharbraceleft}{\isacharbraceright}{\isachardoublequoteclose}\isanewline
\ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}S\ {\isasymin}\ A\ {\isasymunion}\ B{\isachardot}\ a\ {\isasymin}\ S{\isacharbraceright}\ {\isacharequal}\ {\isacharbraceleft}S\ {\isasymin}\ A{\isachardot}\ a\ {\isasymin}\ S{\isacharbraceright}\ {\isasymunion}\ {\isacharbraceleft}S\ {\isasymin}\ B{\isachardot}\ a\ {\isasymin}\ S{\isacharbraceright}{\isachardoublequoteclose}\isanewline
\ \ \ \ \isacommand{using}\isamarkupfalse%
\ assms\isanewline
\ \ \ \ \isacommand{by}\isamarkupfalse%
\ auto\isanewline
\ \ \isacommand{thus}\isamarkupfalse%
\ {\isacharquery}thesis\isanewline
\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
\ count{\isacharunderscore}def\isanewline
\ \ \ \ \isacommand{using}\isamarkupfalse%
\ assms\ card{\isacharunderscore}Un{\isacharunderscore}disjoint{\isacharbrackleft}of\ {\isachardoublequoteopen}{\isacharbraceleft}S\ {\isasymin}\ A{\isachardot}\ a\ {\isasymin}\ S{\isacharbraceright}{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isacharbraceleft}S\ {\isasymin}\ B{\isachardot}\ a\ {\isasymin}\ S{\isacharbraceright}{\isachardoublequoteclose}{\isacharbrackright}\isanewline
\ \ \ \ \isacommand{by}\isamarkupfalse%
\ auto\isanewline
\isacommand{qed}\isamarkupfalse%
%
\endisatagproof
{\isafoldproof}%
%
\isadelimproof
\isanewline
%
\endisadelimproof
%
\isadelimtheory
\isanewline
%
\endisadelimtheory
%
\isatagtheory
\isacommand{end}\isamarkupfalse%
%
\endisatagtheory
{\isafoldtheory}%
%
\isadelimtheory
%
\endisadelimtheory
%
\end{isabellebody}%
%%% Local Variables:
%%% mode: latex
%%% TeX-master: "root"
%%% End:
