%
\begin{isabellebody}%
\setisabellecontext{MoreNat}%
%
\isamarkupsection{Auxiliary lemmas about natural numbers%
}
\isamarkuptrue%
%
\isadelimtheory
%
\endisadelimtheory
%
\isatagtheory
\isacommand{theory}\isamarkupfalse%
\ MoreNat\isanewline
\isakeyword{imports}\ Main\isanewline
\isakeyword{begin}%
\endisatagtheory
{\isafoldtheory}%
%
\isadelimtheory
%
\endisadelimtheory
\isanewline
\isanewline
\isacommand{lemma}\isamarkupfalse%
\ power{\isacharunderscore}mod{\isacharcolon}\ {\isachardoublequoteopen}{\isasymlbrakk}k\ {\isachargreater}\ {\isacharparenleft}{\isadigit{1}}{\isacharcolon}{\isacharcolon}nat{\isacharparenright}{\isacharsemicolon}\ x\ {\isachargreater}\ {\isadigit{0}}\ {\isasymrbrakk}\ {\isasymLongrightarrow}\ {\isacharparenleft}k\ {\isacharcircum}\ x{\isacharparenright}\ mod\ k\ {\isacharequal}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
%
\isadelimproof
%
\endisadelimproof
%
\isatagproof
\isacommand{by}\isamarkupfalse%
\ {\isacharparenleft}induct\ x{\isacharparenright}\ auto%
\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:
