doc-src/TutorialI/Inductive/document/AB.tex

 \begin{isabelle}%
 \ \ \ \ \ {\isasymlbrakk}{\isasymforall}i{\isachardot}\ i\ {\isacharless}\ n\ {\isasymlongrightarrow}\ {\isasymbar}f\ {\isacharparenleft}i\ {\isacharplus}\ {\isadigit{1}}{\isacharparenright}\ {\isacharminus}\ f\ i{\isasymbar}\ {\isasymle}\ {\isacharhash}{\isadigit{1}}{\isacharsemicolon}\ f\ {\isadigit{0}}\ {\isasymle}\ k{\isacharsemicolon}\ k\ {\isasymle}\ f\ n{\isasymrbrakk}\isanewline
 \ \ \ \ \ {\isasymLongrightarrow}\ {\isasymexists}i{\isachardot}\ i\ {\isasymle}\ n\ {\isasymand}\ f\ i\ {\isacharequal}\ k%
 \end{isabelle}
 where \isa{f} is of type \isa{nat\ {\isasymRightarrow}\ int}, \isa{int} are the integers,
-\isa{abs} is the absolute value function, and \isa{{\isacharhash}{\isadigit{1}}} is the
+\isa{{\isasymbar}{\isachardot}{\isasymbar}} is the absolute value function, and \isa{{\isacharhash}{\isadigit{1}}} is the
 integer 1 (see \S\ref{sec:numbers}).
 
 First we show that the our specific function, the difference between the
 numbers of \isa{a}'s and \isa{b}'s, does indeed only change by 1 in every
 move to the right. At this point we also start generalizing from \isa{a}'s
 and \isa{b}'s to an arbitrary property \isa{P}. Otherwise we would have
 to prove the desired lemma twice, once as stated above and once with the
 roles of \isa{a}'s and \isa{b}'s interchanged.%
 \end{isamarkuptext}%
 \isacommand{lemma}\ step{\isadigit{1}}{\isacharcolon}\ {\isachardoublequote}{\isasymforall}i\ {\isacharless}\ size\ w{\isachardot}\isanewline
-\ \ abs{\isacharparenleft}{\isacharparenleft}int{\isacharparenleft}size{\isacharbrackleft}x{\isasymin}take\ {\isacharparenleft}i{\isacharplus}{\isadigit{1}}{\isacharparenright}\ w{\isachardot}\ \ P\ x{\isacharbrackright}{\isacharparenright}\ {\isacharminus}\isanewline
-\ \ \ \ \ \ \ int{\isacharparenleft}size{\isacharbrackleft}x{\isasymin}take\ {\isacharparenleft}i{\isacharplus}{\isadigit{1}}{\isacharparenright}\ w{\isachardot}\ {\isasymnot}P\ x{\isacharbrackright}{\isacharparenright}{\isacharparenright}\isanewline
-\ \ \ \ \ \ {\isacharminus}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}int{\isacharparenleft}size{\isacharbrackleft}x{\isasymin}take\ i\ w{\isachardot}\ \ P\ x{\isacharbrackright}{\isacharparenright}\ {\isacharminus}\isanewline
-\ \ \ \ \ \ \ int{\isacharparenleft}size{\isacharbrackleft}x{\isasymin}take\ i\ w{\isachardot}\ {\isasymnot}P\ x{\isacharbrackright}{\isacharparenright}{\isacharparenright}{\isacharparenright}\ {\isasymle}\ {\isacharhash}{\isadigit{1}}{\isachardoublequote}%
+\ \ {\isasymbar}{\isacharparenleft}int{\isacharparenleft}size{\isacharbrackleft}x{\isasymin}take\ {\isacharparenleft}i{\isacharplus}{\isadigit{1}}{\isacharparenright}\ w{\isachardot}\ P\ x{\isacharbrackright}{\isacharparenright}{\isacharminus}int{\isacharparenleft}size{\isacharbrackleft}x{\isasymin}take\ {\isacharparenleft}i{\isacharplus}{\isadigit{1}}{\isacharparenright}\ w{\isachardot}\ {\isasymnot}P\ x{\isacharbrackright}{\isacharparenright}{\isacharparenright}\isanewline
+\ \ \ {\isacharminus}\ {\isacharparenleft}int{\isacharparenleft}size{\isacharbrackleft}x{\isasymin}take\ i\ w{\isachardot}\ P\ x{\isacharbrackright}{\isacharparenright}{\isacharminus}int{\isacharparenleft}size{\isacharbrackleft}x{\isasymin}take\ i\ w{\isachardot}\ {\isasymnot}P\ x{\isacharbrackright}{\isacharparenright}{\isacharparenright}{\isasymbar}\ {\isasymle}\ {\isacharhash}{\isadigit{1}}{\isachardoublequote}%
 \begin{isamarkuptxt}%
 \noindent
 The lemma is a bit hard to read because of the coercion function
-\isa{{\isachardoublequote}int{\isacharcolon}{\isacharcolon}nat\ {\isasymRightarrow}\ int{\isachardoublequote}}. It is required because \isa{size} returns
+\isa{{\isachardoublequote}int\ {\isacharcolon}{\isacharcolon}\ nat\ {\isasymRightarrow}\ int{\isachardoublequote}}. It is required because \isa{size} returns
 a natural number, but \isa{{\isacharminus}} on \isa{nat} will do the wrong thing.
 Function \isa{take} is predefined and \isa{take\ i\ xs} is the prefix of
 length \isa{i} of \isa{xs}; below we als need \isa{drop\ i\ xs}, which
 is what remains after that prefix has been dropped from \isa{xs}.
 

 \isacommand{lemma}\ part{\isadigit{1}}{\isacharcolon}\isanewline
 \ {\isachardoublequote}size{\isacharbrackleft}x{\isasymin}w{\isachardot}\ P\ x{\isacharbrackright}\ {\isacharequal}\ size{\isacharbrackleft}x{\isasymin}w{\isachardot}\ {\isasymnot}P\ x{\isacharbrackright}{\isacharplus}{\isadigit{2}}\ {\isasymLongrightarrow}\isanewline
 \ \ {\isasymexists}i{\isasymle}size\ w{\isachardot}\ size{\isacharbrackleft}x{\isasymin}take\ i\ w{\isachardot}\ P\ x{\isacharbrackright}\ {\isacharequal}\ size{\isacharbrackleft}x{\isasymin}take\ i\ w{\isachardot}\ {\isasymnot}P\ x{\isacharbrackright}{\isacharplus}{\isadigit{1}}{\isachardoublequote}%
 \begin{isamarkuptxt}%
 \noindent
-This is proved with the help of the intermediate value theorem, instantiated
-appropriately and with its first premise disposed of by lemma
-\isa{step{\isadigit{1}}}.%
+This is proved by force with the help of the intermediate value theorem,
+instantiated appropriately and with its first premise disposed of by lemma
+\isa{step{\isadigit{1}}}:%
 \end{isamarkuptxt}%
 \isacommand{apply}{\isacharparenleft}insert\ nat{\isadigit{0}}{\isacharunderscore}intermed{\isacharunderscore}int{\isacharunderscore}val{\isacharbrackleft}OF\ step{\isadigit{1}}{\isacharcomma}\ of\ {\isachardoublequote}P{\isachardoublequote}\ {\isachardoublequote}w{\isachardoublequote}\ {\isachardoublequote}{\isacharhash}{\isadigit{1}}{\isachardoublequote}{\isacharbrackright}{\isacharparenright}\isanewline
-\isacommand{apply}\ simp\isanewline
-\isacommand{by}{\isacharparenleft}simp\ del{\isacharcolon}int{\isacharunderscore}Suc\ add{\isacharcolon}zdiff{\isacharunderscore}eq{\isacharunderscore}eq\ sym{\isacharbrackleft}OF\ int{\isacharunderscore}Suc{\isacharbrackright}{\isacharparenright}%
-\begin{isamarkuptext}%
-\noindent
-The additional lemmas are needed to mediate between \isa{nat} and \isa{int}.
+\isacommand{by}\ force%
+\begin{isamarkuptext}%
+\noindent
 
 Lemma \isa{part{\isadigit{1}}} tells us only about the prefix \isa{take\ i\ w}.
 The suffix \isa{drop\ i\ w} is dealt with in the following easy lemma:%
 \end{isamarkuptext}%
 \isacommand{lemma}\ part{\isadigit{2}}{\isacharcolon}\isanewline