
diff -r 65ede8dfe304 -r bb4ede27fcb7 doc-src/TutorialI/CTL/document/CTLind.tex
--- a/doc-src/TutorialI/CTL/document/CTLind.tex	Mon Mar 05 15:47:11 2001 +0100
+++ b/doc-src/TutorialI/CTL/document/CTLind.tex	Wed Mar 07 15:54:11 2001 +0100
@@ -58,56 +58,57 @@
 expresses.  Simplification shows that this is a path starting with \isa{t} 
 and that the instantiated induction hypothesis implies the conclusion.
 
-Now we come to the key lemma. It says that if \isa{t} can be reached by a
-finite \isa{A}-avoiding path from \isa{s}, then \isa{t\ {\isasymin}\ lfp\ {\isacharparenleft}af\ A{\isacharparenright}},
-provided there is no infinite \isa{A}-avoiding path starting from \isa{s}.%
+Now we come to the key lemma. Assuming that no infinite \isa{A}-avoiding
+path starts from \isa{s}, we want to show \isa{s\ {\isasymin}\ lfp\ {\isacharparenleft}af\ A{\isacharparenright}}. This
+can be generalized by proving that every point \isa{t} ``between''
+\isa{s} and \isa{A}, i.e.\ all of \isa{Avoid\ s\ A},
+is contained in \isa{lfp\ {\isacharparenleft}af\ A{\isacharparenright}}:%
 \end{isamarkuptext}%
 \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}%
 \begin{isamarkuptxt}%
 \noindent
-The trick is not to induct on \isa{t\ {\isasymin}\ Avoid\ s\ A}, as even the base
-case would be a problem, but to proceed by well-founded induction on~\isa{t}. Hence\hbox{ \isa{t\ {\isasymin}\ Avoid\ s\ A}} must be brought into the conclusion as
-well, which the directive \isa{rule{\isacharunderscore}format} undoes at the end (see below).
-But induction with respect to which well-founded relation? The
-one we want is the restriction
-of \isa{M} to \isa{Avoid\ s\ A}:
+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
+w.r.t.\ \isa{M} restricted to (roughly speaking) \isa{Avoid\ s\ A\ {\isacharminus}\ A}:
 \begin{isabelle}%
-\ \ \ \ \ {\isacharbraceleft}{\isacharparenleft}y{\isacharcomma}\ x{\isacharparenright}{\isachardot}\ {\isacharparenleft}x{\isacharcomma}\ y{\isacharparenright}\ {\isasymin}\ M\ {\isasymand}\ x\ {\isasymin}\ Avoid\ s\ A\ {\isasymand}\ y\ {\isasymin}\ Avoid\ s\ A\ {\isasymand}\ x\ {\isasymnotin}\ A{\isacharbraceright}%
+\ \ \ \ \ {\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 in a moment, 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}%
-\isacommand{apply}{\isacharparenleft}subgoal{\isacharunderscore}tac\isanewline
-\ \ {\isachardoublequote}wf{\isacharbraceleft}{\isacharparenleft}y{\isacharcomma}x{\isacharparenright}{\isachardot}\ {\isacharparenleft}x{\isacharcomma}y{\isacharparenright}{\isasymin}M\ {\isasymand}\ x\ {\isasymin}\ Avoid\ s\ A\ {\isasymand}\ y\ {\isasymin}\ Avoid\ s\ A\ {\isasymand}\ x\ {\isasymnotin}\ A{\isacharbraceright}{\isachardoublequote}{\isacharparenright}\isanewline
+\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
 \ \isacommand{apply}{\isacharparenleft}erule{\isacharunderscore}tac\ a\ {\isacharequal}\ t\ \isakeyword{in}\ wf{\isacharunderscore}induct{\isacharparenright}\isanewline
 \ \isacommand{apply}{\isacharparenleft}clarsimp{\isacharparenright}%
 \begin{isamarkuptxt}%
 \noindent
 \begin{isabelle}%
-\ {\isadigit{1}}{\isachardot}\ {\isasymAnd}x{\isachardot}\ {\isasymlbrakk}{\isasymforall}p{\isasymin}Paths\ s{\isachardot}\ {\isasymexists}i{\isachardot}\ p\ i\ {\isasymin}\ A{\isacharsemicolon}\ x\ {\isasymin}\ Avoid\ s\ A{\isacharsemicolon}\isanewline
-\isaindent{\ {\isadigit{1}}{\isachardot}\ {\isasymAnd}x{\isachardot}\ \ \ \ }{\isasymforall}y{\isachardot}\ {\isacharparenleft}x{\isacharcomma}\ y{\isacharparenright}\ {\isasymin}\ M\ {\isasymand}\ y\ {\isasymin}\ Avoid\ s\ A\ {\isasymand}\ x\ {\isasymnotin}\ A\ {\isasymlongrightarrow}\isanewline
-\isaindent{\ {\isadigit{1}}{\isachardot}\ {\isasymAnd}x{\isachardot}\ \ \ \ {\isasymforall}y{\isachardot}\ }y\ {\isasymin}\ lfp\ {\isacharparenleft}af\ A{\isacharparenright}{\isasymrbrakk}\isanewline
-\isaindent{\ {\isadigit{1}}{\isachardot}\ {\isasymAnd}x{\isachardot}\ }{\isasymLongrightarrow}\ x\ {\isasymin}\ lfp\ {\isacharparenleft}af\ A{\isacharparenright}\isanewline
+\ {\isadigit{1}}{\isachardot}\ {\isasymAnd}t{\isachardot}\ {\isasymlbrakk}{\isasymforall}p{\isasymin}Paths\ s{\isachardot}\ {\isasymexists}i{\isachardot}\ p\ i\ {\isasymin}\ A{\isacharsemicolon}\ t\ {\isasymin}\ Avoid\ s\ 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}{\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}\isanewline
-\isaindent{\ {\isadigit{2}}{\isachardot}\ wf\ \ }{\isacharparenleft}x{\isacharcomma}\ y{\isacharparenright}\ {\isasymin}\ M\ {\isasymand}\ x\ {\isasymin}\ Avoid\ s\ A\ {\isasymand}\ y\ {\isasymin}\ Avoid\ s\ A\ {\isasymand}\ x\ {\isasymnotin}\ A{\isacharbraceright}%
+\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}
-\REMARK{I put in this proof state but I wonder if readers will be able to
-follow this proof. You could prove that your relation is WF in a 
-lemma beforehand, maybe omitting that proof.}
 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}}. To prove the actual goal we unfold \isa{lfp} once. Now
-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}}. If \isa{t} is not in \isa{A}, the second
+\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}}: 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}), and
 hence, by the induction hypothesis, all successors of \isa{t} are indeed in
 \isa{lfp\ {\isacharparenleft}af\ A{\isacharparenright}}. Mechanically:%
 \end{isamarkuptxt}%
-\ \isacommand{apply}{\isacharparenleft}rule\ ssubst\ {\isacharbrackleft}OF\ lfp{\isacharunderscore}unfold{\isacharbrackleft}OF\ mono{\isacharunderscore}af{\isacharbrackright}{\isacharbrackright}{\isacharparenright}\isanewline
-\ \isacommand{apply}{\isacharparenleft}simp\ only{\isacharcolon}\ af{\isacharunderscore}def{\isacharparenright}\isanewline
+\ \isacommand{apply}{\isacharparenleft}subst\ lfp{\isacharunderscore}unfold{\isacharbrackleft}OF\ mono{\isacharunderscore}af{\isacharbrackright}{\isacharparenright}\isanewline
+\ \isacommand{apply}{\isacharparenleft}simp\ {\isacharparenleft}no{\isacharunderscore}asm{\isacharparenright}\ add{\isacharcolon}\ af{\isacharunderscore}def{\isacharparenright}\isanewline
 \ \isacommand{apply}{\isacharparenleft}blast\ intro{\isacharcolon}Avoid{\isachardot}intros{\isacharparenright}%
 \begin{isamarkuptxt}%
 Having proved the main goal we return to the proof obligation that the above
@@ -127,7 +128,8 @@
 \isacommand{apply}{\isacharparenleft}auto\ simp\ add{\isacharcolon}Paths{\isacharunderscore}def{\isacharparenright}\isanewline
 \isacommand{done}%
 \begin{isamarkuptext}%
-The \isa{{\isacharparenleft}no{\isacharunderscore}asm{\isacharparenright}} modifier of the \isa{rule{\isacharunderscore}format} directive means
+The \isa{{\isacharparenleft}no{\isacharunderscore}asm{\isacharparenright}} modifier of the \isa{rule{\isacharunderscore}format} directive in the
+statement of the lemma means
 that the assumption is left unchanged --- otherwise the \isa{{\isasymforall}p} is turned
 into a \isa{{\isasymAnd}p}, which would complicate matters below. As it is,
 \isa{Avoid{\isacharunderscore}in{\isacharunderscore}lfp} is now




