--- a/doc-src/TutorialI/CTL/document/CTLind.tex Wed May 25 09:03:53 2005 +0200
+++ b/doc-src/TutorialI/CTL/document/CTLind.tex Wed May 25 09:04:24 2005 +0200
@@ -50,12 +50,17 @@
\ \ {\isachardoublequote}t\ {\isasymin}\ Avoid\ s\ A\ \ {\isasymLongrightarrow}\isanewline
\ \ \ {\isasymforall}f{\isasymin}Paths\ t{\isachardot}\ {\isacharparenleft}{\isasymforall}i{\isachardot}\ f\ i\ {\isasymnotin}\ A{\isacharparenright}\ {\isasymlongrightarrow}\ {\isacharparenleft}{\isasymexists}p{\isasymin}Paths\ s{\isachardot}\ {\isasymforall}i{\isachardot}\ p\ i\ {\isasymnotin}\ A{\isacharparenright}{\isachardoublequote}\isanewline
\isamarkupfalse%
+\isacommand{apply}{\isacharparenleft}erule\ Avoid{\isachardot}induct{\isacharparenright}\isanewline
+\ \isamarkupfalse%
+\isacommand{apply}{\isacharparenleft}blast{\isacharparenright}\isanewline
\isamarkupfalse%
-\isamarkupfalse%
+\isacommand{apply}{\isacharparenleft}clarify{\isacharparenright}\isanewline
\isamarkupfalse%
-\isamarkupfalse%
+\isacommand{apply}{\isacharparenleft}drule{\isacharunderscore}tac\ x\ {\isacharequal}\ {\isachardoublequote}{\isasymlambda}i{\isachardot}\ case\ i\ of\ {\isadigit{0}}\ {\isasymRightarrow}\ t\ {\isacharbar}\ Suc\ i\ {\isasymRightarrow}\ f\ i{\isachardoublequote}\ \isakeyword{in}\ bspec{\isacharparenright}\isanewline
\isamarkupfalse%
+\isacommand{apply}{\isacharparenleft}simp{\isacharunderscore}all\ add{\isacharcolon}\ Paths{\isacharunderscore}def\ split{\isacharcolon}\ nat{\isachardot}split{\isacharparenright}\isanewline
\isamarkupfalse%
+\isacommand{done}\isamarkupfalse%
%
\begin{isamarkuptext}%
\noindent
@@ -76,22 +81,84 @@
\isamarkuptrue%
\isacommand{lemma}\ Avoid{\isacharunderscore}in{\isacharunderscore}lfp{\isacharbrackleft}rule{\isacharunderscore}format{\isacharparenleft}no{\isacharunderscore}asm{\isacharparenright}{\isacharbrackright}{\isacharcolon}\isanewline
\ \ {\isachardoublequote}{\isasymforall}p{\isasymin}Paths\ s{\isachardot}\ {\isasymexists}i{\isachardot}\ p\ i\ {\isasymin}\ A\ {\isasymLongrightarrow}\ t\ {\isasymin}\ Avoid\ s\ A\ {\isasymlongrightarrow}\ t\ {\isasymin}\ lfp{\isacharparenleft}af\ A{\isacharparenright}{\isachardoublequote}\isamarkupfalse%
+%
+\begin{isamarkuptxt}%
+\noindent
+The proof is by induction on the ``distance'' between \isa{t} and \isa{A}. Remember that \isa{lfp\ {\isacharparenleft}af\ A{\isacharparenright}\ {\isacharequal}\ A\ {\isasymunion}\ M{\isasyminverse}\ {\isacharbackquote}{\isacharbackquote}\ lfp\ {\isacharparenleft}af\ A{\isacharparenright}}.
+If \isa{t} is already in \isa{A}, then \isa{t\ {\isasymin}\ lfp\ {\isacharparenleft}af\ A{\isacharparenright}} is
+trivial. If \isa{t} is not in \isa{A} but all successors are in
+\isa{lfp\ {\isacharparenleft}af\ A{\isacharparenright}} (induction hypothesis), then \isa{t\ {\isasymin}\ lfp\ {\isacharparenleft}af\ A{\isacharparenright}} is
+again trivial.
+
+The formal counterpart of this proof sketch is a well-founded induction
+on~\isa{M} restricted to \isa{Avoid\ s\ A\ {\isacharminus}\ A}, roughly speaking:
+\begin{isabelle}%
+\ \ \ \ \ {\isacharbraceleft}{\isacharparenleft}y{\isacharcomma}\ x{\isacharparenright}{\isachardot}\ {\isacharparenleft}x{\isacharcomma}\ y{\isacharparenright}\ {\isasymin}\ M\ {\isasymand}\ x\ {\isasymin}\ Avoid\ s\ A\ {\isasymand}\ x\ {\isasymnotin}\ A{\isacharbraceright}%
+\end{isabelle}
+As we shall see presently, the absence of infinite \isa{A}-avoiding paths
+starting from \isa{s} implies well-foundedness of this relation. For the
+moment we assume this and proceed with the induction:%
+\end{isamarkuptxt}%
\isamarkuptrue%
-\isamarkupfalse%
-\isamarkupfalse%
-\isamarkupfalse%
-\isamarkupfalse%
-\isamarkuptrue%
-\isamarkupfalse%
+\isacommand{apply}{\isacharparenleft}subgoal{\isacharunderscore}tac\ {\isachardoublequote}wf{\isacharbraceleft}{\isacharparenleft}y{\isacharcomma}x{\isacharparenright}{\isachardot}\ {\isacharparenleft}x{\isacharcomma}y{\isacharparenright}\ {\isasymin}\ M\ {\isasymand}\ x\ {\isasymin}\ Avoid\ s\ A\ {\isasymand}\ x\ {\isasymnotin}\ A{\isacharbraceright}{\isachardoublequote}{\isacharparenright}\isanewline
+\ \isamarkupfalse%
+\isacommand{apply}{\isacharparenleft}erule{\isacharunderscore}tac\ a\ {\isacharequal}\ t\ \isakeyword{in}\ wf{\isacharunderscore}induct{\isacharparenright}\isanewline
+\ \isamarkupfalse%
+\isacommand{apply}{\isacharparenleft}clarsimp{\isacharparenright}\isamarkupfalse%
\isamarkupfalse%
-\isamarkupfalse%
+%
+\begin{isamarkuptxt}%
+\noindent
+\begin{isabelle}%
+\ {\isadigit{1}}{\isachardot}\ {\isasymAnd}t{\isachardot}\ {\isasymlbrakk}{\isasymforall}p{\isasymin}Paths\ s{\isachardot}\ {\isasymexists}i{\isachardot}\ p\ i\ {\isasymin}\ A{\isacharsemicolon}\isanewline
+\isaindent{\ {\isadigit{1}}{\isachardot}\ {\isasymAnd}t{\isachardot}\ \ }{\isasymforall}y{\isachardot}\ {\isacharparenleft}t{\isacharcomma}\ y{\isacharparenright}\ {\isasymin}\ M\ {\isasymand}\ t\ {\isasymnotin}\ A\ {\isasymlongrightarrow}\isanewline
+\isaindent{\ {\isadigit{1}}{\isachardot}\ {\isasymAnd}t{\isachardot}\ \ {\isasymforall}y{\isachardot}\ }y\ {\isasymin}\ Avoid\ s\ A\ {\isasymlongrightarrow}\ y\ {\isasymin}\ lfp\ {\isacharparenleft}af\ A{\isacharparenright}{\isacharsemicolon}\isanewline
+\isaindent{\ {\isadigit{1}}{\isachardot}\ {\isasymAnd}t{\isachardot}\ \ }t\ {\isasymin}\ Avoid\ s\ A{\isasymrbrakk}\isanewline
+\isaindent{\ {\isadigit{1}}{\isachardot}\ {\isasymAnd}t{\isachardot}\ }{\isasymLongrightarrow}\ t\ {\isasymin}\ lfp\ {\isacharparenleft}af\ A{\isacharparenright}\isanewline
+\ {\isadigit{2}}{\isachardot}\ {\isasymforall}p{\isasymin}Paths\ s{\isachardot}\ {\isasymexists}i{\isachardot}\ p\ i\ {\isasymin}\ A\ {\isasymLongrightarrow}\isanewline
+\isaindent{\ {\isadigit{2}}{\isachardot}\ }wf\ {\isacharbraceleft}{\isacharparenleft}y{\isacharcomma}\ x{\isacharparenright}{\isachardot}\ {\isacharparenleft}x{\isacharcomma}\ y{\isacharparenright}\ {\isasymin}\ M\ {\isasymand}\ x\ {\isasymin}\ Avoid\ s\ A\ {\isasymand}\ x\ {\isasymnotin}\ A{\isacharbraceright}%
+\end{isabelle}
+Now the induction hypothesis states that if \isa{t\ {\isasymnotin}\ A}
+then all successors of \isa{t} that are in \isa{Avoid\ s\ A} are in
+\isa{lfp\ {\isacharparenleft}af\ A{\isacharparenright}}. Unfolding \isa{lfp} in the conclusion of the first
+subgoal once, we have to prove that \isa{t} is in \isa{A} or all successors
+of \isa{t} are in \isa{lfp\ {\isacharparenleft}af\ A{\isacharparenright}}. But if \isa{t} is not in \isa{A},
+the second
+\isa{Avoid}-rule implies that all successors of \isa{t} are in
+\isa{Avoid\ s\ A}, because we also assume \isa{t\ {\isasymin}\ Avoid\ s\ A}.
+Hence, by the induction hypothesis, all successors of \isa{t} are indeed in
+\isa{lfp\ {\isacharparenleft}af\ A{\isacharparenright}}. Mechanically:%
+\end{isamarkuptxt}%
+\ \isamarkuptrue%
+\isacommand{apply}{\isacharparenleft}subst\ lfp{\isacharunderscore}unfold{\isacharbrackleft}OF\ mono{\isacharunderscore}af{\isacharbrackright}{\isacharparenright}\isanewline
+\ \isamarkupfalse%
+\isacommand{apply}{\isacharparenleft}simp\ {\isacharparenleft}no{\isacharunderscore}asm{\isacharparenright}\ add{\isacharcolon}\ af{\isacharunderscore}def{\isacharparenright}\isanewline
+\ \isamarkupfalse%
+\isacommand{apply}{\isacharparenleft}blast\ intro{\isacharcolon}\ Avoid{\isachardot}intros{\isacharparenright}\isamarkupfalse%
+%
+\begin{isamarkuptxt}%
+Having proved the main goal, we return to the proof obligation that the
+relation used above is indeed well-founded. This is proved by contradiction: if
+the relation is not well-founded then there exists an infinite \isa{A}-avoiding path all in \isa{Avoid\ s\ A}, by theorem
+\isa{wf{\isacharunderscore}iff{\isacharunderscore}no{\isacharunderscore}infinite{\isacharunderscore}down{\isacharunderscore}chain}:
+\begin{isabelle}%
+\ \ \ \ \ wf\ r\ {\isacharequal}\ {\isacharparenleft}{\isasymnot}\ {\isacharparenleft}{\isasymexists}f{\isachardot}\ {\isasymforall}i{\isachardot}\ {\isacharparenleft}f\ {\isacharparenleft}Suc\ i{\isacharparenright}{\isacharcomma}\ f\ i{\isacharparenright}\ {\isasymin}\ r{\isacharparenright}{\isacharparenright}%
+\end{isabelle}
+From lemma \isa{ex{\isacharunderscore}infinite{\isacharunderscore}path} the existence of an infinite
+\isa{A}-avoiding path starting in \isa{s} follows, contradiction.%
+\end{isamarkuptxt}%
\isamarkuptrue%
-\isamarkupfalse%
+\isacommand{apply}{\isacharparenleft}erule\ contrapos{\isacharunderscore}pp{\isacharparenright}\isanewline
\isamarkupfalse%
+\isacommand{apply}{\isacharparenleft}simp\ add{\isacharcolon}\ wf{\isacharunderscore}iff{\isacharunderscore}no{\isacharunderscore}infinite{\isacharunderscore}down{\isacharunderscore}chain{\isacharparenright}\isanewline
\isamarkupfalse%
+\isacommand{apply}{\isacharparenleft}erule\ exE{\isacharparenright}\isanewline
\isamarkupfalse%
+\isacommand{apply}{\isacharparenleft}rule\ ex{\isacharunderscore}infinite{\isacharunderscore}path{\isacharparenright}\isanewline
\isamarkupfalse%
+\isacommand{apply}{\isacharparenleft}auto\ simp\ add{\isacharcolon}\ Paths{\isacharunderscore}def{\isacharparenright}\isanewline
\isamarkupfalse%
+\isacommand{done}\isamarkupfalse%
%
\begin{isamarkuptext}%
The \isa{{\isacharparenleft}no{\isacharunderscore}asm{\isacharparenright}} modifier of the \isa{rule{\isacharunderscore}format} directive in the
@@ -111,6 +178,7 @@
\isamarkuptrue%
\isacommand{theorem}\ AF{\isacharunderscore}lemma{\isadigit{2}}{\isacharcolon}\ \ {\isachardoublequote}{\isacharbraceleft}s{\isachardot}\ {\isasymforall}p\ {\isasymin}\ Paths\ s{\isachardot}\ {\isasymexists}\ i{\isachardot}\ p\ i\ {\isasymin}\ A{\isacharbraceright}\ {\isasymsubseteq}\ lfp{\isacharparenleft}af\ A{\isacharparenright}{\isachardoublequote}\isanewline
\isamarkupfalse%
+\isacommand{by}{\isacharparenleft}auto\ elim{\isacharcolon}\ Avoid{\isacharunderscore}in{\isacharunderscore}lfp\ intro{\isacharcolon}\ Avoid{\isachardot}intros{\isacharparenright}\isanewline
\isanewline
\isamarkupfalse%
\isamarkupfalse%