%

\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}{\isacharhash}{\isadigit{2}}\ {\isacharasterisk}\ m\ {\isacharequal}\ m\ {\isacharplus}\ m{\isachardoublequote}%

\begin{isamarkuptxt}%

\begin{isabelle}%

\ {\isadigit{1}}{\isachardot}\ {\isacharparenleft}{\isacharhash}{\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}\ {\isacharhash}{\isadigit{3}}\ then\ {\isacharhash}{\isadigit{2}}\ else\ i{\isacharparenright}{\isachardoublequote}%

\begin{isamarkuptext}%

\isa{h\ {\isacharhash}{\isadigit{3}}\ {\isacharequal}\ {\isacharhash}{\isadigit{2}}}

\isa{h\ i\ {\isacharequal}\ i}%

\end{isamarkuptext}%

%

\begin{isamarkuptext}%

\begin{isabelle}%

{\isacharhash}{\isadigit{0}}\ {\isacharequal}\ {\isadigit{0}}%

\end{isabelle}

\rulename{numeral_0_eq_0}

\begin{isabelle}%

{\isacharhash}{\isadigit{1}}\ {\isacharequal}\ {\isadigit{1}}%

\end{isabelle}

\rulename{numeral_1_eq_1}

\begin{isabelle}%

{\isacharhash}{\isadigit{2}}\ {\isacharplus}\ n\ {\isacharequal}\ Suc\ {\isacharparenleft}Suc\ n{\isacharparenright}%

\end{isabelle}

\rulename{add_2_eq_Suc}

\begin{isabelle}%

n\ {\isacharplus}\ {\isacharhash}{\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

\ \ \ \ 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{1}}{\isacharparenright}{\isacharasterisk}{\isacharparenleft}n{\isacharplus}{\isadigit{1}}{\isacharparenright}\ {\isacharequal}\ n{\isacharasterisk}n\ {\isacharminus}\ {\isadigit{1}}{\isachardoublequote}\isanewline

\isacommand{apply}\ {\isacharparenleft}simp\ split{\isacharcolon}\ nat{\isacharunderscore}diff{\isacharunderscore}split{\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}%

{\isacharhash}{\isadigit{0}}\ {\isacharless}\ b\ {\isasymLongrightarrow}\ {\isacharhash}{\isadigit{0}}\ {\isasymle}\ a\ mod\ b%

\end{isabelle}

\rulename{pos_mod_sign}

\begin{isabelle}%

{\isacharhash}{\isadigit{0}}\ {\isacharless}\ b\ {\isasymLongrightarrow}\ a\ mod\ b\ {\isacharless}\ b%

\end{isabelle}

\rulename{pos_mod_bound}

\begin{isabelle}%

b\ {\isacharless}\ {\isacharhash}{\isadigit{0}}\ {\isasymLongrightarrow}\ a\ mod\ b\ {\isasymle}\ {\isacharhash}{\isadigit{0}}%

\end{isabelle}

\rulename{neg_mod_sign}

\begin{isabelle}%

b\ {\isacharless}\ {\isacharhash}{\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}%

{\isacharhash}{\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}%

{\isacharhash}{\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}{\isasymlbrakk}abs\ x\ {\isacharless}\ a{\isacharsemicolon}\ abs\ y\ {\isacharless}\ b{\isasymrbrakk}\ {\isasymLongrightarrow}\ abs\ x\ {\isacharplus}\ abs\ y\ {\isacharless}\ {\isacharparenleft}a\ {\isacharplus}\ b\ {\isacharcolon}{\isacharcolon}\ int{\isacharparenright}{\isachardoublequote}\isanewline

\isacommand{by}\ arith%

\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{end}\isanewline

\end{isabellebody}%

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