doc-src/TutorialI/Types/document/Numbers.tex

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%
\begin{isabellebody}%
\def\isabellecontext{Numbers}%
\isanewline
\isacommand{theory}\ Numbers\ {\isacharequal}\ Real{\isacharcolon}\isanewline
\isanewline
\isacommand{ML}\ {\isachardoublequote}Pretty{\isachardot}setmargin\ {\isadigit{6}}{\isadigit{4}}{\isachardoublequote}\isanewline
\isacommand{ML}\ {\isachardoublequote}IsarOutput{\isachardot}indent\ {\isacharcolon}{\isacharequal}\ {\isadigit{0}}{\isachardoublequote}%
\begin{isamarkuptext}%
numeric literals; default simprules; can re-orient%
\end{isamarkuptext}%
\isacommand{lemma}\ {\isachardoublequote}{\isadigit{2}}\ {\isacharasterisk}\ m\ {\isacharequal}\ m\ {\isacharplus}\ m{\isachardoublequote}%
\begin{isamarkuptxt}%
\begin{isabelle}%
\ {\isadigit{1}}{\isachardot}\ {\isacharparenleft}{\isadigit{2}}{\isasymColon}{\isacharprime}a{\isacharparenright}\ {\isacharasterisk}\ m\ {\isacharequal}\ m\ {\isacharplus}\ m%
\end{isabelle}%
\end{isamarkuptxt}%
\isacommand{oops}\isanewline
\isanewline
\isacommand{consts}\ h\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}nat\ {\isasymRightarrow}\ nat{\isachardoublequote}\isanewline
\isacommand{recdef}\ h\ {\isachardoublequote}{\isacharbraceleft}{\isacharbraceright}{\isachardoublequote}\isanewline
{\isachardoublequote}h\ i\ {\isacharequal}\ {\isacharparenleft}if\ i\ {\isacharequal}\ {\isadigit{3}}\ then\ {\isadigit{2}}\ else\ i{\isacharparenright}{\isachardoublequote}%
\begin{isamarkuptext}%
\isa{h\ {\isadigit{3}}\ {\isacharequal}\ {\isadigit{2}}}
\isa{h\ i\ {\isacharequal}\ i}%
\end{isamarkuptext}%
%
\begin{isamarkuptext}%
\begin{isabelle}%
Numeral{\isadigit{0}}\ {\isacharequal}\ {\isadigit{0}}%
\end{isabelle}
\rulename{numeral_0_eq_0}

\begin{isabelle}%
Numeral{\isadigit{1}}\ {\isacharequal}\ {\isadigit{1}}%
\end{isabelle}
\rulename{numeral_1_eq_1}

\begin{isabelle}%
{\isadigit{2}}\ {\isacharplus}\ n\ {\isacharequal}\ Suc\ {\isacharparenleft}Suc\ n{\isacharparenright}%
\end{isabelle}
\rulename{add_2_eq_Suc}

\begin{isabelle}%
n\ {\isacharplus}\ {\isadigit{2}}\ {\isacharequal}\ Suc\ {\isacharparenleft}Suc\ n{\isacharparenright}%
\end{isabelle}
\rulename{add_2_eq_Suc'}

\begin{isabelle}%
m\ {\isacharplus}\ n\ {\isacharplus}\ k\ {\isacharequal}\ m\ {\isacharplus}\ {\isacharparenleft}n\ {\isacharplus}\ k{\isacharparenright}%
\end{isabelle}
\rulename{add_assoc}

\begin{isabelle}%
m\ {\isacharplus}\ n\ {\isacharequal}\ n\ {\isacharplus}\ m%
\end{isabelle}
\rulename{add_commute}

\begin{isabelle}%
x\ {\isacharplus}\ {\isacharparenleft}y\ {\isacharplus}\ z{\isacharparenright}\ {\isacharequal}\ y\ {\isacharplus}\ {\isacharparenleft}x\ {\isacharplus}\ z{\isacharparenright}%
\end{isabelle}
\rulename{add_left_commute}

these form add_ac; similarly there is mult_ac%
\end{isamarkuptext}%
\isacommand{lemma}\ {\isachardoublequote}Suc{\isacharparenleft}i\ {\isacharplus}\ j{\isacharasterisk}l{\isacharasterisk}k\ {\isacharplus}\ m{\isacharasterisk}n{\isacharparenright}\ {\isacharequal}\ f\ {\isacharparenleft}n{\isacharasterisk}m\ {\isacharplus}\ i\ {\isacharplus}\ k{\isacharasterisk}j{\isacharasterisk}l{\isacharparenright}{\isachardoublequote}%
\begin{isamarkuptxt}%
\begin{isabelle}%
\ {\isadigit{1}}{\isachardot}\ Suc\ {\isacharparenleft}i\ {\isacharplus}\ j\ {\isacharasterisk}\ l\ {\isacharasterisk}\ k\ {\isacharplus}\ m\ {\isacharasterisk}\ n{\isacharparenright}\ {\isacharequal}\ f\ {\isacharparenleft}n\ {\isacharasterisk}\ m\ {\isacharplus}\ i\ {\isacharplus}\ k\ {\isacharasterisk}\ j\ {\isacharasterisk}\ l{\isacharparenright}%
\end{isabelle}%
\end{isamarkuptxt}%
\isacommand{apply}\ {\isacharparenleft}simp\ add{\isacharcolon}\ add{\isacharunderscore}ac\ mult{\isacharunderscore}ac{\isacharparenright}%
\begin{isamarkuptxt}%
\begin{isabelle}%
\ {\isadigit{1}}{\isachardot}\ Suc\ {\isacharparenleft}i\ {\isacharplus}\ {\isacharparenleft}m\ {\isacharasterisk}\ n\ {\isacharplus}\ j\ {\isacharasterisk}\ {\isacharparenleft}k\ {\isacharasterisk}\ l{\isacharparenright}{\isacharparenright}{\isacharparenright}\ {\isacharequal}\isanewline
\isaindent{\ {\isadigit{1}}{\isachardot}\ }f\ {\isacharparenleft}i\ {\isacharplus}\ {\isacharparenleft}m\ {\isacharasterisk}\ n\ {\isacharplus}\ j\ {\isacharasterisk}\ {\isacharparenleft}k\ {\isacharasterisk}\ l{\isacharparenright}{\isacharparenright}{\isacharparenright}%
\end{isabelle}%
\end{isamarkuptxt}%
\isacommand{oops}%
\begin{isamarkuptext}%
\begin{isabelle}%
{\isasymlbrakk}i\ {\isasymle}\ j{\isacharsemicolon}\ k\ {\isasymle}\ l{\isasymrbrakk}\ {\isasymLongrightarrow}\ i\ {\isacharasterisk}\ k\ {\isasymle}\ j\ {\isacharasterisk}\ l%
\end{isabelle}
\rulename{mult_le_mono}

\begin{isabelle}%
{\isasymlbrakk}i\ {\isacharless}\ j{\isacharsemicolon}\ {\isadigit{0}}\ {\isacharless}\ k{\isasymrbrakk}\ {\isasymLongrightarrow}\ i\ {\isacharasterisk}\ k\ {\isacharless}\ j\ {\isacharasterisk}\ k%
\end{isabelle}
\rulename{mult_less_mono1}

\begin{isabelle}%
m\ {\isasymle}\ n\ {\isasymLongrightarrow}\ m\ div\ k\ {\isasymle}\ n\ div\ k%
\end{isabelle}
\rulename{div_le_mono}

\begin{isabelle}%
{\isacharparenleft}m\ {\isacharplus}\ n{\isacharparenright}\ {\isacharasterisk}\ k\ {\isacharequal}\ m\ {\isacharasterisk}\ k\ {\isacharplus}\ n\ {\isacharasterisk}\ k%
\end{isabelle}
\rulename{add_mult_distrib}

\begin{isabelle}%
{\isacharparenleft}m\ {\isacharminus}\ n{\isacharparenright}\ {\isacharasterisk}\ k\ {\isacharequal}\ m\ {\isacharasterisk}\ k\ {\isacharminus}\ n\ {\isacharasterisk}\ k%
\end{isabelle}
\rulename{diff_mult_distrib}

\begin{isabelle}%
m\ mod\ n\ {\isacharasterisk}\ k\ {\isacharequal}\ m\ {\isacharasterisk}\ k\ mod\ {\isacharparenleft}n\ {\isacharasterisk}\ k{\isacharparenright}%
\end{isabelle}
\rulename{mod_mult_distrib}

\begin{isabelle}%
P\ {\isacharparenleft}a\ {\isacharminus}\ b{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}{\isacharparenleft}a\ {\isacharless}\ b\ {\isasymlongrightarrow}\ P\ {\isadigit{0}}{\isacharparenright}\ {\isasymand}\ {\isacharparenleft}{\isasymforall}d{\isachardot}\ a\ {\isacharequal}\ b\ {\isacharplus}\ d\ {\isasymlongrightarrow}\ P\ d{\isacharparenright}{\isacharparenright}%
\end{isabelle}
\rulename{nat_diff_split}%
\end{isamarkuptext}%
\isacommand{lemma}\ {\isachardoublequote}{\isacharparenleft}n\ {\isacharminus}\ {\isadigit{2}}{\isacharparenright}\ {\isacharasterisk}\ {\isacharparenleft}n\ {\isacharplus}\ {\isadigit{2}}{\isacharparenright}\ {\isacharequal}\ n\ {\isacharasterisk}\ n\ {\isacharminus}\ {\isacharparenleft}{\isadigit{4}}{\isacharcolon}{\isacharcolon}nat{\isacharparenright}{\isachardoublequote}\isanewline
\isacommand{apply}\ {\isacharparenleft}clarsimp\ split{\isacharcolon}\ nat{\isacharunderscore}diff{\isacharunderscore}split{\isacharparenright}\isanewline
\ %
\isamarkupcmt{\begin{isabelle}%
\ {\isadigit{1}}{\isachardot}\ {\isasymAnd}d{\isachardot}\ {\isasymlbrakk}n\ {\isacharless}\ {\isadigit{2}}{\isacharsemicolon}\ n\ {\isacharasterisk}\ n\ {\isacharequal}\ {\isadigit{4}}\ {\isacharplus}\ d{\isasymrbrakk}\ {\isasymLongrightarrow}\ d\ {\isacharequal}\ {\isadigit{0}}%
\end{isabelle}%
}
\isanewline
\isacommand{apply}\ {\isacharparenleft}subgoal{\isacharunderscore}tac\ {\isachardoublequote}n{\isacharequal}{\isadigit{0}}\ {\isacharbar}\ n{\isacharequal}{\isadigit{1}}{\isachardoublequote}{\isacharcomma}\ force{\isacharcomma}\ arith{\isacharparenright}\isanewline
\isacommand{done}%
\begin{isamarkuptext}%
\begin{isabelle}%
m\ mod\ n\ {\isacharequal}\ {\isacharparenleft}if\ m\ {\isacharless}\ n\ then\ m\ else\ {\isacharparenleft}m\ {\isacharminus}\ n{\isacharparenright}\ mod\ n{\isacharparenright}%
\end{isabelle}
\rulename{mod_if}

\begin{isabelle}%
m\ div\ n\ {\isacharasterisk}\ n\ {\isacharplus}\ m\ mod\ n\ {\isacharequal}\ m%
\end{isabelle}
\rulename{mod_div_equality}


\begin{isabelle}%
a\ {\isacharasterisk}\ b\ div\ c\ {\isacharequal}\ a\ {\isacharasterisk}\ {\isacharparenleft}b\ div\ c{\isacharparenright}\ {\isacharplus}\ a\ {\isacharasterisk}\ {\isacharparenleft}b\ mod\ c{\isacharparenright}\ div\ c%
\end{isabelle}
\rulename{div_mult1_eq}

\begin{isabelle}%
a\ {\isacharasterisk}\ b\ mod\ c\ {\isacharequal}\ a\ {\isacharasterisk}\ {\isacharparenleft}b\ mod\ c{\isacharparenright}\ mod\ c%
\end{isabelle}
\rulename{mod_mult1_eq}

\begin{isabelle}%
a\ div\ {\isacharparenleft}b\ {\isacharasterisk}\ c{\isacharparenright}\ {\isacharequal}\ a\ div\ b\ div\ c%
\end{isabelle}
\rulename{div_mult2_eq}

\begin{isabelle}%
a\ mod\ {\isacharparenleft}b\ {\isacharasterisk}\ c{\isacharparenright}\ {\isacharequal}\ b\ {\isacharasterisk}\ {\isacharparenleft}a\ div\ b\ mod\ c{\isacharparenright}\ {\isacharplus}\ a\ mod\ b%
\end{isabelle}
\rulename{mod_mult2_eq}

\begin{isabelle}%
{\isadigit{0}}\ {\isacharless}\ c\ {\isasymLongrightarrow}\ c\ {\isacharasterisk}\ a\ div\ {\isacharparenleft}c\ {\isacharasterisk}\ b{\isacharparenright}\ {\isacharequal}\ a\ div\ b%
\end{isabelle}
\rulename{div_mult_mult1}

\begin{isabelle}%
a\ div\ {\isadigit{0}}\ {\isacharequal}\ {\isadigit{0}}%
\end{isabelle}
\rulename{DIVISION_BY_ZERO_DIV}

\begin{isabelle}%
a\ mod\ {\isadigit{0}}\ {\isacharequal}\ a%
\end{isabelle}
\rulename{DIVISION_BY_ZERO_MOD}

\begin{isabelle}%
{\isasymlbrakk}m\ dvd\ n{\isacharsemicolon}\ n\ dvd\ m{\isasymrbrakk}\ {\isasymLongrightarrow}\ m\ {\isacharequal}\ n%
\end{isabelle}
\rulename{dvd_anti_sym}

\begin{isabelle}%
{\isasymlbrakk}k\ dvd\ m{\isacharsemicolon}\ k\ dvd\ n{\isasymrbrakk}\ {\isasymLongrightarrow}\ k\ dvd\ m\ {\isacharplus}\ n%
\end{isabelle}
\rulename{dvd_add}

For the integers, I'd list a few theorems that somehow involve negative 
numbers.  

Division, remainder of negatives


\begin{isabelle}%
Numeral{\isadigit{0}}\ {\isacharless}\ b\ {\isasymLongrightarrow}\ Numeral{\isadigit{0}}\ {\isasymle}\ a\ mod\ b%
\end{isabelle}
\rulename{pos_mod_sign}

\begin{isabelle}%
Numeral{\isadigit{0}}\ {\isacharless}\ b\ {\isasymLongrightarrow}\ a\ mod\ b\ {\isacharless}\ b%
\end{isabelle}
\rulename{pos_mod_bound}

\begin{isabelle}%
b\ {\isacharless}\ Numeral{\isadigit{0}}\ {\isasymLongrightarrow}\ a\ mod\ b\ {\isasymle}\ Numeral{\isadigit{0}}%
\end{isabelle}
\rulename{neg_mod_sign}

\begin{isabelle}%
b\ {\isacharless}\ Numeral{\isadigit{0}}\ {\isasymLongrightarrow}\ b\ {\isacharless}\ a\ mod\ b%
\end{isabelle}
\rulename{neg_mod_bound}

\begin{isabelle}%
{\isacharparenleft}a\ {\isacharplus}\ b{\isacharparenright}\ div\ c\ {\isacharequal}\ a\ div\ c\ {\isacharplus}\ b\ div\ c\ {\isacharplus}\ {\isacharparenleft}a\ mod\ c\ {\isacharplus}\ b\ mod\ c{\isacharparenright}\ div\ c%
\end{isabelle}
\rulename{zdiv_zadd1_eq}

\begin{isabelle}%
{\isacharparenleft}a\ {\isacharplus}\ b{\isacharparenright}\ mod\ c\ {\isacharequal}\ {\isacharparenleft}a\ mod\ c\ {\isacharplus}\ b\ mod\ c{\isacharparenright}\ mod\ c%
\end{isabelle}
\rulename{zmod_zadd1_eq}

\begin{isabelle}%
a\ {\isacharasterisk}\ b\ div\ c\ {\isacharequal}\ a\ {\isacharasterisk}\ {\isacharparenleft}b\ div\ c{\isacharparenright}\ {\isacharplus}\ a\ {\isacharasterisk}\ {\isacharparenleft}b\ mod\ c{\isacharparenright}\ div\ c%
\end{isabelle}
\rulename{zdiv_zmult1_eq}

\begin{isabelle}%
a\ {\isacharasterisk}\ b\ mod\ c\ {\isacharequal}\ a\ {\isacharasterisk}\ {\isacharparenleft}b\ mod\ c{\isacharparenright}\ mod\ c%
\end{isabelle}
\rulename{zmod_zmult1_eq}

\begin{isabelle}%
Numeral{\isadigit{0}}\ {\isacharless}\ c\ {\isasymLongrightarrow}\ a\ div\ {\isacharparenleft}b\ {\isacharasterisk}\ c{\isacharparenright}\ {\isacharequal}\ a\ div\ b\ div\ c%
\end{isabelle}
\rulename{zdiv_zmult2_eq}

\begin{isabelle}%
Numeral{\isadigit{0}}\ {\isacharless}\ c\ {\isasymLongrightarrow}\ a\ mod\ {\isacharparenleft}b\ {\isacharasterisk}\ c{\isacharparenright}\ {\isacharequal}\ b\ {\isacharasterisk}\ {\isacharparenleft}a\ div\ b\ mod\ c{\isacharparenright}\ {\isacharplus}\ a\ mod\ b%
\end{isabelle}
\rulename{zmod_zmult2_eq}

\begin{isabelle}%
{\isasymbar}x\ {\isacharasterisk}\ y{\isasymbar}\ {\isacharequal}\ {\isasymbar}x{\isasymbar}\ {\isacharasterisk}\ {\isasymbar}y{\isasymbar}%
\end{isabelle}
\rulename{abs_mult}%
\end{isamarkuptext}%
\isacommand{lemma}\ {\isachardoublequote}abs\ {\isacharparenleft}x{\isacharplus}y{\isacharparenright}\ {\isasymle}\ abs\ x\ {\isacharplus}\ abs\ {\isacharparenleft}y\ {\isacharcolon}{\isacharcolon}\ int{\isacharparenright}{\isachardoublequote}\isanewline
\isacommand{by}\ arith\isanewline
\isanewline
\isacommand{lemma}\ {\isachardoublequote}abs\ {\isacharparenleft}{\isadigit{2}}{\isacharasterisk}x{\isacharparenright}\ {\isacharequal}\ {\isadigit{2}}\ {\isacharasterisk}\ abs\ {\isacharparenleft}x\ {\isacharcolon}{\isacharcolon}\ int{\isacharparenright}{\isachardoublequote}\isanewline
\isacommand{by}\ {\isacharparenleft}simp\ add{\isacharcolon}\ zabs{\isacharunderscore}def{\isacharparenright}%
\begin{isamarkuptext}%
REALS

\begin{isabelle}%
{\isasymbar}r{\isasymbar}\ {\isacharcircum}\ n\ {\isacharequal}\ {\isasymbar}r\ {\isacharcircum}\ n{\isasymbar}%
\end{isabelle}
\rulename{realpow_abs}

\begin{isabelle}%
x\ {\isacharless}\ y\ {\isasymLongrightarrow}\ {\isasymexists}r{\isachardot}\ x\ {\isacharless}\ r\ {\isasymand}\ r\ {\isacharless}\ y%
\end{isabelle}
\rulename{real_dense}

\begin{isabelle}%
{\isasymbar}r{\isasymbar}\ {\isacharcircum}\ n\ {\isacharequal}\ {\isasymbar}r\ {\isacharcircum}\ n{\isasymbar}%
\end{isabelle}
\rulename{realpow_abs}

\begin{isabelle}%
x\ {\isacharasterisk}\ {\isacharparenleft}y\ {\isacharslash}\ z{\isacharparenright}\ {\isacharequal}\ x\ {\isacharasterisk}\ y\ {\isacharslash}\ z%
\end{isabelle}
\rulename{real_times_divide1_eq}

\begin{isabelle}%
y\ {\isacharslash}\ z\ {\isacharasterisk}\ x\ {\isacharequal}\ y\ {\isacharasterisk}\ x\ {\isacharslash}\ z%
\end{isabelle}
\rulename{real_times_divide2_eq}

\begin{isabelle}%
x\ {\isacharslash}\ {\isacharparenleft}y\ {\isacharslash}\ z{\isacharparenright}\ {\isacharequal}\ x\ {\isacharasterisk}\ z\ {\isacharslash}\ y%
\end{isabelle}
\rulename{real_divide_divide1_eq}

\begin{isabelle}%
x\ {\isacharslash}\ y\ {\isacharslash}\ z\ {\isacharequal}\ x\ {\isacharslash}\ {\isacharparenleft}y\ {\isacharasterisk}\ z{\isacharparenright}%
\end{isabelle}
\rulename{real_divide_divide2_eq}

\begin{isabelle}%
{\isacharminus}\ x\ {\isacharslash}\ y\ {\isacharequal}\ {\isacharminus}\ {\isacharparenleft}x\ {\isacharslash}\ y{\isacharparenright}%
\end{isabelle}
\rulename{real_minus_divide_eq}

\begin{isabelle}%
x\ {\isacharslash}\ {\isacharminus}\ y\ {\isacharequal}\ {\isacharminus}\ {\isacharparenleft}x\ {\isacharslash}\ y{\isacharparenright}%
\end{isabelle}
\rulename{real_divide_minus_eq}

This last NOT a simprule

\begin{isabelle}%
{\isacharparenleft}x\ {\isacharplus}\ y{\isacharparenright}\ {\isacharslash}\ z\ {\isacharequal}\ x\ {\isacharslash}\ z\ {\isacharplus}\ y\ {\isacharslash}\ z%
\end{isabelle}
\rulename{real_add_divide_distrib}%
\end{isamarkuptext}%
\isacommand{lemma}\ {\isachardoublequote}{\isadigit{3}}{\isacharslash}{\isadigit{4}}\ {\isacharless}\ {\isacharparenleft}{\isadigit{7}}{\isacharslash}{\isadigit{8}}\ {\isacharcolon}{\isacharcolon}\ real{\isacharparenright}{\isachardoublequote}\isanewline
\isacommand{by}\ simp\ \isanewline
\isanewline
\isacommand{lemma}\ {\isachardoublequote}P\ {\isacharparenleft}{\isacharparenleft}{\isadigit{3}}{\isacharslash}{\isadigit{4}}{\isacharparenright}\ {\isacharasterisk}\ {\isacharparenleft}{\isadigit{8}}{\isacharslash}{\isadigit{1}}{\isadigit{5}}\ {\isacharcolon}{\isacharcolon}\ real{\isacharparenright}{\isacharparenright}{\isachardoublequote}%
\begin{isamarkuptxt}%
\begin{isabelle}%
\ {\isadigit{1}}{\isachardot}\ P\ {\isacharparenleft}{\isadigit{3}}\ {\isacharslash}\ {\isadigit{4}}\ {\isacharasterisk}\ {\isacharparenleft}{\isadigit{8}}\ {\isacharslash}\ {\isadigit{1}}{\isadigit{5}}{\isacharparenright}{\isacharparenright}%
\end{isabelle}%
\end{isamarkuptxt}%
\isacommand{apply}\ simp%
\begin{isamarkuptxt}%
\begin{isabelle}%
\ {\isadigit{1}}{\isachardot}\ P\ {\isacharparenleft}{\isadigit{2}}\ {\isacharslash}\ {\isadigit{5}}{\isacharparenright}%
\end{isabelle}%
\end{isamarkuptxt}%
\isacommand{oops}\isanewline
\isanewline
\isacommand{lemma}\ {\isachardoublequote}{\isacharparenleft}{\isadigit{3}}{\isacharslash}{\isadigit{4}}{\isacharparenright}\ {\isacharasterisk}\ {\isacharparenleft}{\isadigit{8}}{\isacharslash}{\isadigit{1}}{\isadigit{5}}{\isacharparenright}\ {\isacharless}\ {\isacharparenleft}x\ {\isacharcolon}{\isacharcolon}\ real{\isacharparenright}{\isachardoublequote}%
\begin{isamarkuptxt}%
\begin{isabelle}%
\ {\isadigit{1}}{\isachardot}\ {\isadigit{3}}\ {\isacharslash}\ {\isadigit{4}}\ {\isacharasterisk}\ {\isacharparenleft}{\isadigit{8}}\ {\isacharslash}\ {\isadigit{1}}{\isadigit{5}}{\isacharparenright}\ {\isacharless}\ x%
\end{isabelle}%
\end{isamarkuptxt}%
\isacommand{apply}\ simp%
\begin{isamarkuptxt}%
\begin{isabelle}%
\ {\isadigit{1}}{\isachardot}\ {\isadigit{2}}\ {\isacharless}\ x\ {\isacharasterisk}\ {\isadigit{5}}%
\end{isabelle}%
\end{isamarkuptxt}%
\isacommand{oops}\isanewline
\isanewline
\isacommand{lemma}\ {\isachardoublequote}{\isacharparenleft}{\isadigit{3}}{\isacharslash}{\isadigit{4}}{\isacharparenright}\ {\isacharasterisk}\ {\isacharparenleft}{\isadigit{1}}{\isadigit{0}}{\isacharcircum}{\isadigit{1}}{\isadigit{5}}{\isacharparenright}\ {\isacharless}\ {\isacharparenleft}x\ {\isacharcolon}{\isacharcolon}\ real{\isacharparenright}{\isachardoublequote}\isanewline
\isacommand{apply}\ simp\ \isanewline
\isacommand{oops}\isanewline
\isanewline
\isanewline
\isanewline
\isacommand{end}\isanewline
\end{isabellebody}%
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