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

(adjusted)

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\begin{isabellebody}%
\def\isabellecontext{Numbers}%
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\isadelimtheory
\isanewline
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\endisadelimtheory
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\isatagtheory
\isacommand{theory}\isamarkupfalse%
\ Numbers\isanewline
\isakeyword{imports}\ Complex{\isacharunderscore}Main\isanewline
\isakeyword{begin}%
\endisatagtheory
{\isafoldtheory}%
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\isadelimtheory
\isanewline
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\endisadelimtheory
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\isadelimML
\isanewline
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\endisadelimML
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\isatagML
\isacommand{ML}\isamarkupfalse%
\ {\isachardoublequoteopen}Pretty{\isachardot}setmargin\ {\isadigit{6}}{\isadigit{4}}{\isachardoublequoteclose}\isanewline
\isacommand{ML}\isamarkupfalse%
\ {\isachardoublequoteopen}ThyOutput{\isachardot}indent\ {\isacharcolon}{\isacharequal}\ {\isadigit{0}}{\isachardoublequoteclose}%
\endisatagML
{\isafoldML}%
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\isadelimML
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\endisadelimML
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\begin{isamarkuptext}%
numeric literals; default simprules; can re-orient%
\end{isamarkuptext}%
\isamarkuptrue%
\isacommand{lemma}\isamarkupfalse%
\ {\isachardoublequoteopen}{\isadigit{2}}\ {\isacharasterisk}\ m\ {\isacharequal}\ m\ {\isacharplus}\ m{\isachardoublequoteclose}%
\isadelimproof
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\endisadelimproof
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\isatagproof
%
\begin{isamarkuptxt}%
\begin{isabelle}%
\ {\isadigit{1}}{\isachardot}\ {\isacharparenleft}{\isadigit{2}}{\isasymColon}{\isacharprime}a{\isacharparenright}\ {\isacharasterisk}\ m\ {\isacharequal}\ m\ {\isacharplus}\ m%
\end{isabelle}%
\end{isamarkuptxt}%
\isamarkuptrue%
\isacommand{oops}\isamarkupfalse%
%
\endisatagproof
{\isafoldproof}%
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\isadelimproof
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\endisadelimproof
\isanewline
\isanewline
\isacommand{consts}\isamarkupfalse%
\ h\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}nat\ {\isasymRightarrow}\ nat{\isachardoublequoteclose}\isanewline
\isacommand{recdef}\isamarkupfalse%
\ h\ {\isachardoublequoteopen}{\isacharbraceleft}{\isacharbraceright}{\isachardoublequoteclose}\isanewline
{\isachardoublequoteopen}h\ i\ {\isacharequal}\ {\isacharparenleft}if\ i\ {\isacharequal}\ {\isadigit{3}}\ then\ {\isadigit{2}}\ else\ i{\isacharparenright}{\isachardoublequoteclose}%
\begin{isamarkuptext}%
\isa{h\ {\isadigit{3}}\ {\isacharequal}\ {\isadigit{2}}}
\isa{h\ i\ {\isacharequal}\ i}%
\end{isamarkuptext}%
\isamarkuptrue%
%
\begin{isamarkuptext}%
\begin{isabelle}%
Numeral{\isadigit{0}}\ {\isacharequal}\ {\isacharparenleft}{\isadigit{0}}{\isasymColon}{\isacharprime}a{\isacharparenright}%
\end{isabelle}
\rulename{numeral_0_eq_0}

\begin{isabelle}%
Numeral{\isadigit{1}}\ {\isacharequal}\ {\isacharparenleft}{\isadigit{1}}{\isasymColon}{\isacharprime}a{\isacharparenright}%
\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}%
a\ {\isacharplus}\ b\ {\isacharplus}\ c\ {\isacharequal}\ a\ {\isacharplus}\ {\isacharparenleft}b\ {\isacharplus}\ c{\isacharparenright}%
\end{isabelle}
\rulename{add_assoc}

\begin{isabelle}%
a\ {\isacharplus}\ b\ {\isacharequal}\ b\ {\isacharplus}\ a%
\end{isabelle}
\rulename{add_commute}

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

these form add_ac; similarly there is mult_ac%
\end{isamarkuptext}%
\isamarkuptrue%
\isacommand{lemma}\isamarkupfalse%
\ {\isachardoublequoteopen}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}{\isachardoublequoteclose}%
\isadelimproof
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\endisadelimproof
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\isatagproof
%
\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}%
\isamarkuptrue%
\isacommand{apply}\isamarkupfalse%
\ {\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}%
\isamarkuptrue%
\isacommand{oops}\isamarkupfalse%
%
\endisatagproof
{\isafoldproof}%
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\isadelimproof
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\endisadelimproof
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\begin{isamarkuptext}%
\begin{isabelle}%
m\ {\isasymle}\ n\ {\isasymLongrightarrow}\ m\ div\ k\ {\isasymle}\ n\ div\ k%
\end{isabelle}
\rulename{div_le_mono}

\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}%
\isamarkuptrue%
\isacommand{lemma}\isamarkupfalse%
\ {\isachardoublequoteopen}{\isacharparenleft}n\ {\isacharminus}\ {\isadigit{1}}{\isacharparenright}\ {\isacharasterisk}\ {\isacharparenleft}n\ {\isacharplus}\ {\isadigit{1}}{\isacharparenright}\ {\isacharequal}\ n\ {\isacharasterisk}\ n\ {\isacharminus}\ {\isacharparenleft}{\isadigit{1}}{\isacharcolon}{\isacharcolon}nat{\isacharparenright}{\isachardoublequoteclose}\isanewline
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\isadelimproof
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\endisadelimproof
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\isatagproof
\isacommand{apply}\isamarkupfalse%
\ {\isacharparenleft}clarsimp\ split{\isacharcolon}\ nat{\isacharunderscore}diff{\isacharunderscore}split\ iff\ del{\isacharcolon}\ less{\isacharunderscore}Suc{\isadigit{0}}{\isacharparenright}\isanewline
\ %
\isamarkupcmt{\begin{isabelle}%
\ {\isadigit{1}}{\isachardot}\ {\isasymAnd}d{\isachardot}\ {\isasymlbrakk}n\ {\isacharless}\ Suc\ {\isadigit{0}}{\isacharsemicolon}\ n\ {\isacharasterisk}\ n\ {\isacharequal}\ Suc\ d{\isasymrbrakk}\ {\isasymLongrightarrow}\ d\ {\isacharequal}\ {\isadigit{0}}%
\end{isabelle}%
}
\isanewline
\isacommand{apply}\isamarkupfalse%
\ {\isacharparenleft}subgoal{\isacharunderscore}tac\ {\isachardoublequoteopen}n{\isacharequal}{\isadigit{0}}{\isachardoublequoteclose}{\isacharcomma}\ force{\isacharcomma}\ arith{\isacharparenright}\isanewline
\isacommand{done}\isamarkupfalse%
%
\endisatagproof
{\isafoldproof}%
%
\isadelimproof
\isanewline
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\endisadelimproof
\isanewline
\isanewline
\isacommand{lemma}\isamarkupfalse%
\ {\isachardoublequoteopen}{\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}{\isachardoublequoteclose}\isanewline
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\isadelimproof
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\endisadelimproof
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\isatagproof
\isacommand{apply}\isamarkupfalse%
\ {\isacharparenleft}simp\ split{\isacharcolon}\ nat{\isacharunderscore}diff{\isacharunderscore}split{\isacharcomma}\ clarify{\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}\isamarkupfalse%
\ {\isacharparenleft}subgoal{\isacharunderscore}tac\ {\isachardoublequoteopen}n{\isacharequal}{\isadigit{0}}\ {\isacharbar}\ n{\isacharequal}{\isadigit{1}}{\isachardoublequoteclose}{\isacharcomma}\ force{\isacharcomma}\ arith{\isacharparenright}\isanewline
\isacommand{done}\isamarkupfalse%
%
\endisatagproof
{\isafoldproof}%
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\isadelimproof
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\endisadelimproof
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\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}%
a\ div\ b\ {\isacharasterisk}\ b\ {\isacharplus}\ a\ mod\ b\ {\isacharequal}\ a%
\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\ {\isacharparenleft}{\isadigit{0}}{\isasymColon}{\isacharprime}a{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}{\isadigit{0}}{\isasymColon}{\isacharprime}a{\isacharparenright}%
\end{isabelle}
\rulename{div_by_0}

\begin{isabelle}%
a\ mod\ {\isacharparenleft}{\isadigit{0}}{\isasymColon}{\isacharprime}a{\isacharparenright}\ {\isacharequal}\ a%
\end{isabelle}
\rulename{mod_by_0}

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

\begin{isabelle}%
{\isasymlbrakk}a\ dvd\ b{\isacharsemicolon}\ a\ dvd\ c{\isasymrbrakk}\ {\isasymLongrightarrow}\ a\ dvd\ b\ {\isacharplus}\ c%
\end{isabelle}
\rulename{dvd_add}

For the integers, I'd list a few theorems that somehow involve negative 
numbers.%
\end{isamarkuptext}%
\isamarkuptrue%
%
\begin{isamarkuptext}%
Division, remainder of negatives


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

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

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

\begin{isabelle}%
b\ {\isacharless}\ {\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}%
{\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}%
{\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}%
\end{isamarkuptext}%
\isamarkuptrue%
\isacommand{lemma}\isamarkupfalse%
\ {\isachardoublequoteopen}abs\ {\isacharparenleft}x{\isacharplus}y{\isacharparenright}\ {\isasymle}\ abs\ x\ {\isacharplus}\ abs\ {\isacharparenleft}y\ {\isacharcolon}{\isacharcolon}\ int{\isacharparenright}{\isachardoublequoteclose}\isanewline
%
\isadelimproof
%
\endisadelimproof
%
\isatagproof
\isacommand{by}\isamarkupfalse%
\ arith%
\endisatagproof
{\isafoldproof}%
%
\isadelimproof
\isanewline
%
\endisadelimproof
\isanewline
\isacommand{lemma}\isamarkupfalse%
\ {\isachardoublequoteopen}abs\ {\isacharparenleft}{\isadigit{2}}{\isacharasterisk}x{\isacharparenright}\ {\isacharequal}\ {\isadigit{2}}\ {\isacharasterisk}\ abs\ {\isacharparenleft}x\ {\isacharcolon}{\isacharcolon}\ int{\isacharparenright}{\isachardoublequoteclose}\isanewline
%
\isadelimproof
%
\endisadelimproof
%
\isatagproof
\isacommand{by}\isamarkupfalse%
\ {\isacharparenleft}simp\ add{\isacharcolon}\ abs{\isacharunderscore}if{\isacharparenright}%
\endisatagproof
{\isafoldproof}%
%
\isadelimproof
%
\endisadelimproof
%
\begin{isamarkuptext}%
Induction rules for the Integers

\begin{isabelle}%
{\isasymlbrakk}k\ {\isasymle}\ i{\isacharsemicolon}\ P\ k{\isacharsemicolon}\ {\isasymAnd}i{\isachardot}\ {\isasymlbrakk}k\ {\isasymle}\ i{\isacharsemicolon}\ P\ i{\isasymrbrakk}\ {\isasymLongrightarrow}\ P\ {\isacharparenleft}i\ {\isacharplus}\ {\isadigit{1}}{\isacharparenright}{\isasymrbrakk}\ {\isasymLongrightarrow}\ P\ i%
\end{isabelle}
\rulename{int_ge_induct}

\begin{isabelle}%
{\isasymlbrakk}k\ {\isacharless}\ i{\isacharsemicolon}\ P\ {\isacharparenleft}k\ {\isacharplus}\ {\isadigit{1}}{\isacharparenright}{\isacharsemicolon}\ {\isasymAnd}i{\isachardot}\ {\isasymlbrakk}k\ {\isacharless}\ i{\isacharsemicolon}\ P\ i{\isasymrbrakk}\ {\isasymLongrightarrow}\ P\ {\isacharparenleft}i\ {\isacharplus}\ {\isadigit{1}}{\isacharparenright}{\isasymrbrakk}\ {\isasymLongrightarrow}\ P\ i%
\end{isabelle}
\rulename{int_gr_induct}

\begin{isabelle}%
{\isasymlbrakk}i\ {\isasymle}\ k{\isacharsemicolon}\ P\ k{\isacharsemicolon}\ {\isasymAnd}i{\isachardot}\ {\isasymlbrakk}i\ {\isasymle}\ k{\isacharsemicolon}\ P\ i{\isasymrbrakk}\ {\isasymLongrightarrow}\ P\ {\isacharparenleft}i\ {\isacharminus}\ {\isadigit{1}}{\isacharparenright}{\isasymrbrakk}\ {\isasymLongrightarrow}\ P\ i%
\end{isabelle}
\rulename{int_le_induct}

\begin{isabelle}%
{\isasymlbrakk}i\ {\isacharless}\ k{\isacharsemicolon}\ P\ {\isacharparenleft}k\ {\isacharminus}\ {\isadigit{1}}{\isacharparenright}{\isacharsemicolon}\ {\isasymAnd}i{\isachardot}\ {\isasymlbrakk}i\ {\isacharless}\ k{\isacharsemicolon}\ P\ i{\isasymrbrakk}\ {\isasymLongrightarrow}\ P\ {\isacharparenleft}i\ {\isacharminus}\ {\isadigit{1}}{\isacharparenright}{\isasymrbrakk}\ {\isasymLongrightarrow}\ P\ i%
\end{isabelle}
\rulename{int_less_induct}%
\end{isamarkuptext}%
\isamarkuptrue%
%
\begin{isamarkuptext}%
FIELDS

\begin{isabelle}%
x\ {\isacharless}\ y\ {\isasymLongrightarrow}\ {\isasymexists}z{\isachargreater}x{\isachardot}\ z\ {\isacharless}\ y%
\end{isabelle}
\rulename{dense}

\begin{isabelle}%
a\ {\isacharasterisk}\ {\isacharparenleft}b\ {\isacharslash}\ c{\isacharparenright}\ {\isacharequal}\ a\ {\isacharasterisk}\ b\ {\isacharslash}\ c%
\end{isabelle}
\rulename{times_divide_eq_right}

\begin{isabelle}%
b\ {\isacharslash}\ c\ {\isacharasterisk}\ a\ {\isacharequal}\ b\ {\isacharasterisk}\ a\ {\isacharslash}\ c%
\end{isabelle}
\rulename{times_divide_eq_left}

\begin{isabelle}%
a\ {\isacharslash}\ {\isacharparenleft}b\ {\isacharslash}\ c{\isacharparenright}\ {\isacharequal}\ a\ {\isacharasterisk}\ c\ {\isacharslash}\ b%
\end{isabelle}
\rulename{divide_divide_eq_right}

\begin{isabelle}%
a\ {\isacharslash}\ b\ {\isacharslash}\ c\ {\isacharequal}\ a\ {\isacharslash}\ {\isacharparenleft}b\ {\isacharasterisk}\ c{\isacharparenright}%
\end{isabelle}
\rulename{divide_divide_eq_left}

\begin{isabelle}%
{\isacharminus}\ {\isacharparenleft}a\ {\isacharslash}\ b{\isacharparenright}\ {\isacharequal}\ {\isacharminus}\ a\ {\isacharslash}\ b%
\end{isabelle}
\rulename{minus_divide_left}

\begin{isabelle}%
{\isacharminus}\ {\isacharparenleft}a\ {\isacharslash}\ b{\isacharparenright}\ {\isacharequal}\ a\ {\isacharslash}\ {\isacharminus}\ b%
\end{isabelle}
\rulename{minus_divide_right}

This last NOT a simprule

\begin{isabelle}%
{\isacharparenleft}a\ {\isacharplus}\ b{\isacharparenright}\ {\isacharslash}\ c\ {\isacharequal}\ a\ {\isacharslash}\ c\ {\isacharplus}\ b\ {\isacharslash}\ c%
\end{isabelle}
\rulename{add_divide_distrib}%
\end{isamarkuptext}%
\isamarkuptrue%
\isacommand{lemma}\isamarkupfalse%
\ {\isachardoublequoteopen}{\isadigit{3}}{\isacharslash}{\isadigit{4}}\ {\isacharless}\ {\isacharparenleft}{\isadigit{7}}{\isacharslash}{\isadigit{8}}\ {\isacharcolon}{\isacharcolon}\ real{\isacharparenright}{\isachardoublequoteclose}\isanewline
%
\isadelimproof
%
\endisadelimproof
%
\isatagproof
\isacommand{by}\isamarkupfalse%
\ simp%
\endisatagproof
{\isafoldproof}%
%
\isadelimproof
\ \isanewline
%
\endisadelimproof
\isanewline
\isacommand{lemma}\isamarkupfalse%
\ {\isachardoublequoteopen}P\ {\isacharparenleft}{\isacharparenleft}{\isadigit{3}}{\isacharslash}{\isadigit{4}}{\isacharparenright}\ {\isacharasterisk}\ {\isacharparenleft}{\isadigit{8}}{\isacharslash}{\isadigit{1}}{\isadigit{5}}\ {\isacharcolon}{\isacharcolon}\ real{\isacharparenright}{\isacharparenright}{\isachardoublequoteclose}%
\isadelimproof
%
\endisadelimproof
%
\isatagproof
%
\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}%
\isamarkuptrue%
\isacommand{apply}\isamarkupfalse%
\ simp%
\begin{isamarkuptxt}%
\begin{isabelle}%
\ {\isadigit{1}}{\isachardot}\ P\ {\isacharparenleft}{\isadigit{2}}\ {\isacharslash}\ {\isadigit{5}}{\isacharparenright}%
\end{isabelle}%
\end{isamarkuptxt}%
\isamarkuptrue%
\isacommand{oops}\isamarkupfalse%
%
\endisatagproof
{\isafoldproof}%
%
\isadelimproof
%
\endisadelimproof
\isanewline
\isanewline
\isacommand{lemma}\isamarkupfalse%
\ {\isachardoublequoteopen}{\isacharparenleft}{\isadigit{3}}{\isacharslash}{\isadigit{4}}{\isacharparenright}\ {\isacharasterisk}\ {\isacharparenleft}{\isadigit{8}}{\isacharslash}{\isadigit{1}}{\isadigit{5}}{\isacharparenright}\ {\isacharless}\ {\isacharparenleft}x\ {\isacharcolon}{\isacharcolon}\ real{\isacharparenright}{\isachardoublequoteclose}%
\isadelimproof
%
\endisadelimproof
%
\isatagproof
%
\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}%
\isamarkuptrue%
\isacommand{apply}\isamarkupfalse%
\ simp%
\begin{isamarkuptxt}%
\begin{isabelle}%
\ {\isadigit{1}}{\isachardot}\ {\isadigit{2}}\ {\isacharless}\ x\ {\isacharasterisk}\ {\isadigit{5}}%
\end{isabelle}%
\end{isamarkuptxt}%
\isamarkuptrue%
\isacommand{oops}\isamarkupfalse%
%
\endisatagproof
{\isafoldproof}%
%
\isadelimproof
%
\endisadelimproof
%
\begin{isamarkuptext}%
Ring and Field

Requires a field, or else an ordered ring

\begin{isabelle}%
{\isacharparenleft}a\ {\isacharasterisk}\ b\ {\isacharequal}\ {\isacharparenleft}{\isadigit{0}}{\isasymColon}{\isacharprime}a{\isacharparenright}{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}a\ {\isacharequal}\ {\isacharparenleft}{\isadigit{0}}{\isasymColon}{\isacharprime}a{\isacharparenright}\ {\isasymor}\ b\ {\isacharequal}\ {\isacharparenleft}{\isadigit{0}}{\isasymColon}{\isacharprime}a{\isacharparenright}{\isacharparenright}%
\end{isabelle}
\rulename{mult_eq_0_iff}

\begin{isabelle}%
{\isacharparenleft}a\ {\isacharasterisk}\ c\ {\isacharequal}\ b\ {\isacharasterisk}\ c{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}c\ {\isacharequal}\ {\isacharparenleft}{\isadigit{0}}{\isasymColon}{\isacharprime}a{\isacharparenright}\ {\isasymor}\ a\ {\isacharequal}\ b{\isacharparenright}%
\end{isabelle}
\rulename{mult_cancel_right}

\begin{isabelle}%
{\isacharparenleft}c\ {\isacharasterisk}\ a\ {\isacharequal}\ c\ {\isacharasterisk}\ b{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}c\ {\isacharequal}\ {\isacharparenleft}{\isadigit{0}}{\isasymColon}{\isacharprime}a{\isacharparenright}\ {\isasymor}\ a\ {\isacharequal}\ b{\isacharparenright}%
\end{isabelle}
\rulename{mult_cancel_left}%
\end{isamarkuptext}%
\isamarkuptrue%
%
\isadelimML
%
\endisadelimML
%
\isatagML
\isacommand{ML}\isamarkupfalse%
{\isacharverbatimopen}set\ show{\isacharunderscore}sorts{\isacharverbatimclose}%
\endisatagML
{\isafoldML}%
%
\isadelimML
%
\endisadelimML
%
\begin{isamarkuptext}%
effect of show sorts on the above

\begin{isabelle}%
{\isacharparenleft}{\isacharparenleft}c{\isasymColon}{\isacharprime}a{\isasymColon}ring{\isacharunderscore}no{\isacharunderscore}zero{\isacharunderscore}divisors{\isacharparenright}\ {\isacharasterisk}\ {\isacharparenleft}a{\isasymColon}{\isacharprime}a{\isasymColon}ring{\isacharunderscore}no{\isacharunderscore}zero{\isacharunderscore}divisors{\isacharparenright}\ {\isacharequal}\isanewline
\isaindent{{\isacharparenleft}}c\ {\isacharasterisk}\ {\isacharparenleft}b{\isasymColon}{\isacharprime}a{\isasymColon}ring{\isacharunderscore}no{\isacharunderscore}zero{\isacharunderscore}divisors{\isacharparenright}{\isacharparenright}\ {\isacharequal}\isanewline
{\isacharparenleft}c\ {\isacharequal}\ {\isacharparenleft}{\isadigit{0}}{\isasymColon}{\isacharprime}a{\isasymColon}ring{\isacharunderscore}no{\isacharunderscore}zero{\isacharunderscore}divisors{\isacharparenright}\ {\isasymor}\ a\ {\isacharequal}\ b{\isacharparenright}%
\end{isabelle}
\rulename{mult_cancel_left}%
\end{isamarkuptext}%
\isamarkuptrue%
%
\isadelimML
%
\endisadelimML
%
\isatagML
\isacommand{ML}\isamarkupfalse%
{\isacharverbatimopen}reset\ show{\isacharunderscore}sorts{\isacharverbatimclose}%
\endisatagML
{\isafoldML}%
%
\isadelimML
%
\endisadelimML
%
\begin{isamarkuptext}%
absolute value

\begin{isabelle}%
{\isasymbar}a\ {\isacharasterisk}\ b{\isasymbar}\ {\isacharequal}\ {\isasymbar}a{\isasymbar}\ {\isacharasterisk}\ {\isasymbar}b{\isasymbar}%
\end{isabelle}
\rulename{abs_mult}

\begin{isabelle}%
{\isacharparenleft}{\isasymbar}a{\isasymbar}\ {\isasymle}\ b{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}a\ {\isasymle}\ b\ {\isasymand}\ {\isacharminus}\ a\ {\isasymle}\ b{\isacharparenright}%
\end{isabelle}
\rulename{abs_le_iff}

\begin{isabelle}%
{\isasymbar}a\ {\isacharplus}\ b{\isasymbar}\ {\isasymle}\ {\isasymbar}a{\isasymbar}\ {\isacharplus}\ {\isasymbar}b{\isasymbar}%
\end{isabelle}
\rulename{abs_triangle_ineq}

\begin{isabelle}%
a\ {\isacharcircum}\ {\isacharparenleft}m\ {\isacharplus}\ n{\isacharparenright}\ {\isacharequal}\ a\ {\isacharcircum}\ m\ {\isacharasterisk}\ a\ {\isacharcircum}\ n%
\end{isabelle}
\rulename{power_add}

\begin{isabelle}%
a\ {\isacharcircum}\ {\isacharparenleft}m\ {\isacharasterisk}\ n{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}a\ {\isacharcircum}\ m{\isacharparenright}\ {\isacharcircum}\ n%
\end{isabelle}
\rulename{power_mult}

\begin{isabelle}%
{\isasymbar}a\ {\isacharcircum}\ n{\isasymbar}\ {\isacharequal}\ {\isasymbar}a{\isasymbar}\ {\isacharcircum}\ n%
\end{isabelle}
\rulename{power_abs}%
\end{isamarkuptext}%
\isamarkuptrue%
%
\isadelimtheory
%
\endisadelimtheory
%
\isatagtheory
\isacommand{end}\isamarkupfalse%
%
\endisatagtheory
{\isafoldtheory}%
%
\isadelimtheory
%
\endisadelimtheory
\isanewline
\end{isabellebody}%
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