--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/TutorialI/Types/document/Numbers.tex Wed Dec 06 11:47:01 2000 +0100
@@ -0,0 +1,239 @@
+%
+\begin{isabellebody}%
+\def\isabellecontext{Numbers}%
+\isanewline
+\isacommand{theory}\ Numbers\ {\isacharequal}\ Main{\isacharcolon}\isanewline
+\isanewline
+\isacommand{ML}\ {\isachardoublequote}Pretty{\isachardot}setmargin\ {\isadigit{6}}{\isadigit{4}}{\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}\isanewline
+\isacommand{oops}%
+\begin{isamarkuptext}%
+proof\ {\isacharparenleft}prove{\isacharparenright}{\isacharcolon}\ step\ {\isadigit{0}}\isanewline
+\isanewline
+goal\ {\isacharparenleft}lemma{\isacharparenright}{\isacharcolon}\isanewline
+{\isacharparenleft}{\isacharhash}{\isadigit{2}}{\isasymColon}{\isacharprime}a{\isacharparenright}\ {\isacharasterisk}\ m\ {\isacharequal}\ m\ {\isacharplus}\ m\isanewline
+\ {\isadigit{1}}{\isachardot}\ {\isacharparenleft}{\isacharhash}{\isadigit{2}}{\isasymColon}{\isacharprime}a{\isacharparenright}\ {\isacharasterisk}\ m\ {\isacharequal}\ m\ {\isacharplus}\ m
+
+
+\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}\isanewline
+\isacommand{apply}\ {\isacharparenleft}simp\ add{\isacharcolon}\ add{\isacharunderscore}ac\ mult{\isacharunderscore}ac{\isacharparenright}\isanewline
+\isacommand{oops}%
+\begin{isamarkuptext}%
+proof\ {\isacharparenleft}prove{\isacharparenright}{\isacharcolon}\ step\ {\isadigit{0}}\isanewline
+\isanewline
+goal\ {\isacharparenleft}lemma{\isacharparenright}{\isacharcolon}\isanewline
+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}\isanewline
+\ {\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}
+
+proof\ {\isacharparenleft}prove{\isacharparenright}{\isacharcolon}\ step\ {\isadigit{1}}\isanewline
+\isanewline
+goal\ {\isacharparenleft}lemma{\isacharparenright}{\isacharcolon}\isanewline
+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}\isanewline
+\ {\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{isamarkuptext}%
+%
+\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\ {\isacharparenleft}m\ {\isacharplus}\ n{\isacharparenright}%
+\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}\isanewline
+\ \ \ \ \ 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{end}\isanewline
+\end{isabellebody}%
+%%% Local Variables:
+%%% mode: latex
+%%% TeX-master: "root"
+%%% End: