
diff -r 0e7b145c3a89 -r 3f2a9f400168 doc-src/TutorialI/CTL/document/PDL.tex
--- a/doc-src/TutorialI/CTL/document/PDL.tex	Wed May 25 09:03:53 2005 +0200
+++ b/doc-src/TutorialI/CTL/document/PDL.tex	Wed May 25 09:04:24 2005 +0200
@@ -87,9 +87,11 @@
 \isamarkuptrue%
 \isacommand{lemma}\ mono{\isacharunderscore}ef{\isacharcolon}\ {\isachardoublequote}mono{\isacharparenleft}{\isasymlambda}T{\isachardot}\ A\ {\isasymunion}\ {\isacharparenleft}M{\isasyminverse}\ {\isacharbackquote}{\isacharbackquote}\ T{\isacharparenright}{\isacharparenright}{\isachardoublequote}\isanewline
 \isamarkupfalse%
-\isamarkupfalse%
+\isacommand{apply}{\isacharparenleft}rule\ monoI{\isacharparenright}\isanewline
 \isamarkupfalse%
+\isacommand{apply}\ blast\isanewline
 \isamarkupfalse%
+\isacommand{done}\isamarkupfalse%
 %
 \begin{isamarkuptext}%
 \noindent
@@ -99,30 +101,112 @@
 \isamarkuptrue%
 \isacommand{lemma}\ EF{\isacharunderscore}lemma{\isacharcolon}\isanewline
 \ \ {\isachardoublequote}lfp{\isacharparenleft}{\isasymlambda}T{\isachardot}\ A\ {\isasymunion}\ {\isacharparenleft}M{\isasyminverse}\ {\isacharbackquote}{\isacharbackquote}\ T{\isacharparenright}{\isacharparenright}\ {\isacharequal}\ {\isacharbraceleft}s{\isachardot}\ {\isasymexists}t{\isachardot}\ {\isacharparenleft}s{\isacharcomma}t{\isacharparenright}\ {\isasymin}\ M\isactrlsup {\isacharasterisk}\ {\isasymand}\ t\ {\isasymin}\ A{\isacharbraceright}{\isachardoublequote}\isamarkupfalse%
-\isamarkuptrue%
-\isamarkupfalse%
-\isamarkupfalse%
-\isamarkupfalse%
-\isamarkupfalse%
+%
+\begin{isamarkuptxt}%
+\noindent
+The equality is proved in the canonical fashion by proving that each set
+includes the other; the inclusion is shown pointwise:%
+\end{isamarkuptxt}%
 \isamarkuptrue%
-\isamarkupfalse%
-\isamarkupfalse%
+\isacommand{apply}{\isacharparenleft}rule\ equalityI{\isacharparenright}\isanewline
+\ \isamarkupfalse%
+\isacommand{apply}{\isacharparenleft}rule\ subsetI{\isacharparenright}\isanewline
+\ \isamarkupfalse%
+\isacommand{apply}{\isacharparenleft}simp{\isacharparenright}\isamarkupfalse%
 \isamarkupfalse%
-\isamarkuptrue%
-\isamarkupfalse%
+%
+\begin{isamarkuptxt}%
+\noindent
+Simplification leaves us with the following first subgoal
+\begin{isabelle}%
+\ {\isadigit{1}}{\isachardot}\ {\isasymAnd}s{\isachardot}\ s\ {\isasymin}\ lfp\ {\isacharparenleft}{\isasymlambda}T{\isachardot}\ A\ {\isasymunion}\ M{\isasyminverse}\ {\isacharbackquote}{\isacharbackquote}\ T{\isacharparenright}\ {\isasymLongrightarrow}\ {\isasymexists}t{\isachardot}\ {\isacharparenleft}s{\isacharcomma}\ t{\isacharparenright}\ {\isasymin}\ M\isactrlsup {\isacharasterisk}\ {\isasymand}\ t\ {\isasymin}\ A%
+\end{isabelle}
+which is proved by \isa{lfp}-induction:%
+\end{isamarkuptxt}%
+\ \isamarkuptrue%
+\isacommand{apply}{\isacharparenleft}erule\ lfp{\isacharunderscore}induct{\isacharparenright}\isanewline
+\ \ \isamarkupfalse%
+\isacommand{apply}{\isacharparenleft}rule\ mono{\isacharunderscore}ef{\isacharparenright}\isanewline
+\ \isamarkupfalse%
+\isacommand{apply}{\isacharparenleft}simp{\isacharparenright}\isamarkupfalse%
+%
+\begin{isamarkuptxt}%
+\noindent
+Having disposed of the monotonicity subgoal,
+simplification leaves us with the following goal:
+\begin{isabelle}
+\ {\isadigit{1}}{\isachardot}\ {\isasymAnd}x{\isachardot}\ x\ {\isasymin}\ A\ {\isasymor}\isanewline
+\ \ \ \ \ \ \ \ \ x\ {\isasymin}\ M{\isasyminverse}\ {\isacharbackquote}{\isacharbackquote}\ {\isacharparenleft}lfp\ {\isacharparenleft}\dots{\isacharparenright}\ {\isasyminter}\ {\isacharbraceleft}x{\isachardot}\ {\isasymexists}t{\isachardot}\ {\isacharparenleft}x{\isacharcomma}\ t{\isacharparenright}\ {\isasymin}\ M\isactrlsup {\isacharasterisk}\ {\isasymand}\ t\ {\isasymin}\ A{\isacharbraceright}{\isacharparenright}\isanewline
+\ \ \ \ \ \ \ \ {\isasymLongrightarrow}\ {\isasymexists}t{\isachardot}\ {\isacharparenleft}x{\isacharcomma}\ t{\isacharparenright}\ {\isasymin}\ M\isactrlsup {\isacharasterisk}\ {\isasymand}\ t\ {\isasymin}\ A
+\end{isabelle}
+It is proved by \isa{blast}, using the transitivity of 
+\isa{M\isactrlsup {\isacharasterisk}}.%
+\end{isamarkuptxt}%
+\ \isamarkuptrue%
+\isacommand{apply}{\isacharparenleft}blast\ intro{\isacharcolon}\ rtrancl{\isacharunderscore}trans{\isacharparenright}\isamarkupfalse%
+%
+\begin{isamarkuptxt}%
+We now return to the second set inclusion subgoal, which is again proved
+pointwise:%
+\end{isamarkuptxt}%
 \isamarkuptrue%
-\isamarkupfalse%
-\isamarkupfalse%
-\isamarkuptrue%
-\isamarkupfalse%
-\isamarkuptrue%
+\isacommand{apply}{\isacharparenleft}rule\ subsetI{\isacharparenright}\isanewline
 \isamarkupfalse%
-\isamarkuptrue%
-\isamarkupfalse%
+\isacommand{apply}{\isacharparenleft}simp{\isacharcomma}\ clarify{\isacharparenright}\isamarkupfalse%
+%
+\begin{isamarkuptxt}%
+\noindent
+After simplification and clarification we are left with
+\begin{isabelle}%
+\ {\isadigit{1}}{\isachardot}\ {\isasymAnd}x\ t{\isachardot}\ {\isasymlbrakk}{\isacharparenleft}x{\isacharcomma}\ t{\isacharparenright}\ {\isasymin}\ M\isactrlsup {\isacharasterisk}{\isacharsemicolon}\ t\ {\isasymin}\ A{\isasymrbrakk}\ {\isasymLongrightarrow}\ x\ {\isasymin}\ lfp\ {\isacharparenleft}{\isasymlambda}T{\isachardot}\ A\ {\isasymunion}\ M{\isasyminverse}\ {\isacharbackquote}{\isacharbackquote}\ T{\isacharparenright}%
+\end{isabelle}
+This goal is proved by induction on \isa{{\isacharparenleft}s{\isacharcomma}\ t{\isacharparenright}\ {\isasymin}\ M\isactrlsup {\isacharasterisk}}. But since the model
+checker works backwards (from \isa{t} to \isa{s}), we cannot use the
+induction theorem \isa{rtrancl{\isacharunderscore}induct}: it works in the
+forward direction. Fortunately the converse induction theorem
+\isa{converse{\isacharunderscore}rtrancl{\isacharunderscore}induct} already exists:
+\begin{isabelle}%
+\ \ \ \ \ {\isasymlbrakk}{\isacharparenleft}a{\isacharcomma}\ b{\isacharparenright}\ {\isasymin}\ r\isactrlsup {\isacharasterisk}{\isacharsemicolon}\ P\ b{\isacharsemicolon}\isanewline
+\isaindent{\ \ \ \ \ \ }{\isasymAnd}y\ z{\isachardot}\ {\isasymlbrakk}{\isacharparenleft}y{\isacharcomma}\ z{\isacharparenright}\ {\isasymin}\ r{\isacharsemicolon}\ {\isacharparenleft}z{\isacharcomma}\ b{\isacharparenright}\ {\isasymin}\ r\isactrlsup {\isacharasterisk}{\isacharsemicolon}\ P\ z{\isasymrbrakk}\ {\isasymLongrightarrow}\ P\ y{\isasymrbrakk}\isanewline
+\isaindent{\ \ \ \ \ }{\isasymLongrightarrow}\ P\ a%
+\end{isabelle}
+It says that if \isa{{\isacharparenleft}a{\isacharcomma}\ b{\isacharparenright}\ {\isasymin}\ r\isactrlsup {\isacharasterisk}} and we know \isa{P\ b} then we can infer
+\isa{P\ a} provided each step backwards from a predecessor \isa{z} of
+\isa{b} preserves \isa{P}.%
+\end{isamarkuptxt}%
 \isamarkuptrue%
-\isamarkupfalse%
+\isacommand{apply}{\isacharparenleft}erule\ converse{\isacharunderscore}rtrancl{\isacharunderscore}induct{\isacharparenright}\isamarkupfalse%
+%
+\begin{isamarkuptxt}%
+\noindent
+The base case
+\begin{isabelle}%
+\ {\isadigit{1}}{\isachardot}\ {\isasymAnd}x\ t{\isachardot}\ t\ {\isasymin}\ A\ {\isasymLongrightarrow}\ t\ {\isasymin}\ lfp\ {\isacharparenleft}{\isasymlambda}T{\isachardot}\ A\ {\isasymunion}\ M{\isasyminverse}\ {\isacharbackquote}{\isacharbackquote}\ T{\isacharparenright}%
+\end{isabelle}
+is solved by unrolling \isa{lfp} once%
+\end{isamarkuptxt}%
+\ \isamarkuptrue%
+\isacommand{apply}{\isacharparenleft}subst\ lfp{\isacharunderscore}unfold{\isacharbrackleft}OF\ mono{\isacharunderscore}ef{\isacharbrackright}{\isacharparenright}\isamarkupfalse%
+%
+\begin{isamarkuptxt}%
+\begin{isabelle}%
+\ {\isadigit{1}}{\isachardot}\ {\isasymAnd}x\ t{\isachardot}\ t\ {\isasymin}\ A\ {\isasymLongrightarrow}\ t\ {\isasymin}\ A\ {\isasymunion}\ M{\isasyminverse}\ {\isacharbackquote}{\isacharbackquote}\ lfp\ {\isacharparenleft}{\isasymlambda}T{\isachardot}\ A\ {\isasymunion}\ M{\isasyminverse}\ {\isacharbackquote}{\isacharbackquote}\ T{\isacharparenright}%
+\end{isabelle}
+and disposing of the resulting trivial subgoal automatically:%
+\end{isamarkuptxt}%
+\ \isamarkuptrue%
+\isacommand{apply}{\isacharparenleft}blast{\isacharparenright}\isamarkupfalse%
+%
+\begin{isamarkuptxt}%
+\noindent
+The proof of the induction step is identical to the one for the base case:%
+\end{isamarkuptxt}%
+\isamarkuptrue%
+\isacommand{apply}{\isacharparenleft}subst\ lfp{\isacharunderscore}unfold{\isacharbrackleft}OF\ mono{\isacharunderscore}ef{\isacharbrackright}{\isacharparenright}\isanewline
 \isamarkupfalse%
+\isacommand{apply}{\isacharparenleft}blast{\isacharparenright}\isanewline
 \isamarkupfalse%
+\isacommand{done}\isamarkupfalse%
 %
 \begin{isamarkuptext}%
 The main theorem is proved in the familiar manner: induction followed by
@@ -131,9 +215,11 @@
 \isamarkuptrue%
 \isacommand{theorem}\ {\isachardoublequote}mc\ f\ {\isacharequal}\ {\isacharbraceleft}s{\isachardot}\ s\ {\isasymTurnstile}\ f{\isacharbraceright}{\isachardoublequote}\isanewline
 \isamarkupfalse%
-\isamarkupfalse%
+\isacommand{apply}{\isacharparenleft}induct{\isacharunderscore}tac\ f{\isacharparenright}\isanewline
 \isamarkupfalse%
+\isacommand{apply}{\isacharparenleft}auto\ simp\ add{\isacharcolon}\ EF{\isacharunderscore}lemma{\isacharparenright}\isanewline
 \isamarkupfalse%
+\isacommand{done}\isamarkupfalse%
 %
 \begin{isamarkuptext}%
 \begin{exercise}
@@ -158,17 +244,14 @@
 \isamarkupfalse%
 \isamarkupfalse%
 \isamarkupfalse%
-\isanewline
-\isamarkupfalse%
-\isamarkupfalse%
-\isamarkupfalse%
-\isanewline
 \isamarkupfalse%
 \isamarkupfalse%
 \isamarkupfalse%
 \isamarkupfalse%
 \isamarkupfalse%
-\isanewline
+\isamarkupfalse%
+\isamarkupfalse%
+\isamarkupfalse%
 \isamarkupfalse%
 \isamarkupfalse%
 \end{isabellebody}%




