1.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
1.2 +++ b/doc-src/TutorialI/CTL/document/CTL.tex Mon Oct 02 14:32:33 2000 +0200
1.3 @@ -0,0 +1,228 @@
1.4 +%
1.5 +\begin{isabellebody}%
1.6 +\def\isabellecontext{CTL}%
1.7 +\isacommand{theory}\ CTL\ {\isacharequal}\ Main{\isacharcolon}\isanewline
1.8 +\isanewline
1.9 +\isacommand{typedecl}\ atom\isanewline
1.10 +\isacommand{types}\ state\ {\isacharequal}\ {\isachardoublequote}atom\ set{\isachardoublequote}\isanewline
1.11 +\isanewline
1.12 +\isacommand{datatype}\ ctl{\isacharunderscore}form\ {\isacharequal}\ Atom\ atom\isanewline
1.13 +\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharbar}\ NOT\ ctl{\isacharunderscore}form\isanewline
1.14 +\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharbar}\ And\ ctl{\isacharunderscore}form\ ctl{\isacharunderscore}form\isanewline
1.15 +\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharbar}\ AX\ ctl{\isacharunderscore}form\isanewline
1.16 +\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharbar}\ EF\ ctl{\isacharunderscore}form\isanewline
1.17 +\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharbar}\ AF\ ctl{\isacharunderscore}form\isanewline
1.18 +\isanewline
1.19 +\isacommand{consts}\ valid\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}state\ {\isasymRightarrow}\ ctl{\isacharunderscore}form\ {\isasymRightarrow}\ bool{\isachardoublequote}\ {\isacharparenleft}{\isachardoublequote}{\isacharparenleft}{\isacharunderscore}\ {\isasymTurnstile}\ {\isacharunderscore}{\isacharparenright}{\isachardoublequote}\ {\isacharbrackleft}\isadigit{8}\isadigit{0}{\isacharcomma}\isadigit{8}\isadigit{0}{\isacharbrackright}\ \isadigit{8}\isadigit{0}{\isacharparenright}\isanewline
1.20 +\ \ \ \ \ \ \ M\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharparenleft}state\ {\isasymtimes}\ state{\isacharparenright}set{\isachardoublequote}\isanewline
1.21 +\isanewline
1.22 +\isacommand{constdefs}\ Paths\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}state\ {\isasymRightarrow}\ {\isacharparenleft}nat\ {\isasymRightarrow}\ state{\isacharparenright}set{\isachardoublequote}\isanewline
1.23 +{\isachardoublequote}Paths\ s\ {\isasymequiv}\ {\isacharbraceleft}p{\isachardot}\ s\ {\isacharequal}\ p\ \isadigit{0}\ {\isasymand}\ {\isacharparenleft}{\isasymforall}i{\isachardot}\ {\isacharparenleft}p\ i{\isacharcomma}\ p{\isacharparenleft}i{\isacharplus}\isadigit{1}{\isacharparenright}{\isacharparenright}\ {\isasymin}\ M{\isacharparenright}{\isacharbraceright}{\isachardoublequote}\isanewline
1.24 +\isanewline
1.25 +\isacommand{primrec}\isanewline
1.26 +{\isachardoublequote}s\ {\isasymTurnstile}\ Atom\ a\ \ {\isacharequal}\ \ {\isacharparenleft}a{\isasymin}s{\isacharparenright}{\isachardoublequote}\isanewline
1.27 +{\isachardoublequote}s\ {\isasymTurnstile}\ NOT\ f\ \ \ {\isacharequal}\ {\isacharparenleft}{\isachartilde}{\isacharparenleft}s\ {\isasymTurnstile}\ f{\isacharparenright}{\isacharparenright}{\isachardoublequote}\isanewline
1.28 +{\isachardoublequote}s\ {\isasymTurnstile}\ And\ f\ g\ {\isacharequal}\ {\isacharparenleft}s\ {\isasymTurnstile}\ f\ {\isasymand}\ s\ {\isasymTurnstile}\ g{\isacharparenright}{\isachardoublequote}\isanewline
1.29 +{\isachardoublequote}s\ {\isasymTurnstile}\ AX\ f\ \ \ \ {\isacharequal}\ {\isacharparenleft}{\isasymforall}t{\isachardot}\ {\isacharparenleft}s{\isacharcomma}t{\isacharparenright}\ {\isasymin}\ M\ {\isasymlongrightarrow}\ t\ {\isasymTurnstile}\ f{\isacharparenright}{\isachardoublequote}\isanewline
1.30 +{\isachardoublequote}s\ {\isasymTurnstile}\ EF\ f\ \ \ \ {\isacharequal}\ {\isacharparenleft}{\isasymexists}t{\isachardot}\ {\isacharparenleft}s{\isacharcomma}t{\isacharparenright}\ {\isasymin}\ M{\isacharcircum}{\isacharasterisk}\ {\isasymand}\ t\ {\isasymTurnstile}\ f{\isacharparenright}{\isachardoublequote}\isanewline
1.31 +{\isachardoublequote}s\ {\isasymTurnstile}\ AF\ f\ \ \ \ {\isacharequal}\ {\isacharparenleft}{\isasymforall}p\ {\isasymin}\ Paths\ s{\isachardot}\ {\isasymexists}i{\isachardot}\ p\ i\ {\isasymTurnstile}\ f{\isacharparenright}{\isachardoublequote}\isanewline
1.32 +\isanewline
1.33 +\isacommand{constdefs}\ af\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}state\ set\ {\isasymRightarrow}\ state\ set\ {\isasymRightarrow}\ state\ set{\isachardoublequote}\isanewline
1.34 +{\isachardoublequote}af\ A\ T\ {\isasymequiv}\ A\ {\isasymunion}\ {\isacharbraceleft}s{\isachardot}\ {\isasymforall}t{\isachardot}\ {\isacharparenleft}s{\isacharcomma}\ t{\isacharparenright}\ {\isasymin}\ M\ {\isasymlongrightarrow}\ t\ {\isasymin}\ T{\isacharbraceright}{\isachardoublequote}\isanewline
1.35 +\isanewline
1.36 +\isacommand{lemma}\ mono{\isacharunderscore}af{\isacharcolon}\ {\isachardoublequote}mono{\isacharparenleft}af\ A{\isacharparenright}{\isachardoublequote}\isanewline
1.37 +\isacommand{by}{\isacharparenleft}force\ simp\ add{\isacharcolon}\ af{\isacharunderscore}def\ intro{\isacharcolon}monoI{\isacharparenright}\isanewline
1.38 +\isanewline
1.39 +\isacommand{consts}\ mc\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}ctl{\isacharunderscore}form\ {\isasymRightarrow}\ state\ set{\isachardoublequote}\isanewline
1.40 +\isacommand{primrec}\isanewline
1.41 +{\isachardoublequote}mc{\isacharparenleft}Atom\ a{\isacharparenright}\ \ {\isacharequal}\ {\isacharbraceleft}s{\isachardot}\ a{\isasymin}s{\isacharbraceright}{\isachardoublequote}\isanewline
1.42 +{\isachardoublequote}mc{\isacharparenleft}NOT\ f{\isacharparenright}\ \ \ {\isacharequal}\ {\isacharminus}mc\ f{\isachardoublequote}\isanewline
1.43 +{\isachardoublequote}mc{\isacharparenleft}And\ f\ g{\isacharparenright}\ {\isacharequal}\ mc\ f\ {\isasyminter}\ mc\ g{\isachardoublequote}\isanewline
1.44 +{\isachardoublequote}mc{\isacharparenleft}AX\ f{\isacharparenright}\ \ \ \ {\isacharequal}\ {\isacharbraceleft}s{\isachardot}\ {\isasymforall}t{\isachardot}\ {\isacharparenleft}s{\isacharcomma}t{\isacharparenright}\ {\isasymin}\ M\ \ {\isasymlongrightarrow}\ t\ {\isasymin}\ mc\ f{\isacharbraceright}{\isachardoublequote}\isanewline
1.45 +{\isachardoublequote}mc{\isacharparenleft}EF\ f{\isacharparenright}\ \ \ \ {\isacharequal}\ lfp{\isacharparenleft}{\isasymlambda}T{\isachardot}\ mc\ f\ {\isasymunion}\ {\isacharbraceleft}s{\isachardot}\ {\isasymexists}t{\isachardot}\ {\isacharparenleft}s{\isacharcomma}t{\isacharparenright}{\isasymin}M\ {\isasymand}\ t{\isasymin}T{\isacharbraceright}{\isacharparenright}{\isachardoublequote}\isanewline
1.46 +{\isachardoublequote}mc{\isacharparenleft}AF\ f{\isacharparenright}\ \ \ \ {\isacharequal}\ lfp{\isacharparenleft}af{\isacharparenleft}mc\ f{\isacharparenright}{\isacharparenright}{\isachardoublequote}\isanewline
1.47 +\isanewline
1.48 +\isacommand{lemma}\ mono{\isacharunderscore}ef{\isacharcolon}\ {\isachardoublequote}mono{\isacharparenleft}{\isasymlambda}T{\isachardot}\ A\ {\isasymunion}\ {\isacharbraceleft}s{\isachardot}\ {\isasymexists}t{\isachardot}\ {\isacharparenleft}s{\isacharcomma}t{\isacharparenright}{\isasymin}M\ {\isasymand}\ t{\isasymin}T{\isacharbraceright}{\isacharparenright}{\isachardoublequote}\isanewline
1.49 +\isacommand{apply}{\isacharparenleft}rule\ monoI{\isacharparenright}\isanewline
1.50 +\isacommand{by}{\isacharparenleft}blast{\isacharparenright}\isanewline
1.51 +\isanewline
1.52 +\isacommand{lemma}\ lfp{\isacharunderscore}conv{\isacharunderscore}EF{\isacharcolon}\isanewline
1.53 +{\isachardoublequote}lfp{\isacharparenleft}{\isasymlambda}T{\isachardot}\ A\ {\isasymunion}\ {\isacharbraceleft}s{\isachardot}\ {\isasymexists}t{\isachardot}\ {\isacharparenleft}s{\isacharcomma}t{\isacharparenright}{\isasymin}M\ {\isasymand}\ t{\isasymin}T{\isacharbraceright}{\isacharparenright}\ {\isacharequal}\ {\isacharbraceleft}s{\isachardot}\ {\isasymexists}t{\isachardot}\ {\isacharparenleft}s{\isacharcomma}t{\isacharparenright}\ {\isasymin}\ M{\isacharcircum}{\isacharasterisk}\ {\isasymand}\ t\ {\isasymin}\ A{\isacharbraceright}{\isachardoublequote}\isanewline
1.54 +\isacommand{apply}{\isacharparenleft}rule\ equalityI{\isacharparenright}\isanewline
1.55 +\ \isacommand{apply}{\isacharparenleft}rule\ subsetI{\isacharparenright}\isanewline
1.56 +\ \isacommand{apply}{\isacharparenleft}simp{\isacharparenright}\isanewline
1.57 +\ \isacommand{apply}{\isacharparenleft}erule\ Lfp{\isachardot}induct{\isacharparenright}\isanewline
1.58 +\ \ \isacommand{apply}{\isacharparenleft}rule\ mono{\isacharunderscore}ef{\isacharparenright}\isanewline
1.59 +\ \isacommand{apply}{\isacharparenleft}simp{\isacharparenright}\isanewline
1.60 +\ \isacommand{apply}{\isacharparenleft}blast\ intro{\isacharcolon}\ r{\isacharunderscore}into{\isacharunderscore}rtrancl\ rtrancl{\isacharunderscore}trans{\isacharparenright}\isanewline
1.61 +\isacommand{apply}{\isacharparenleft}rule\ subsetI{\isacharparenright}\isanewline
1.62 +\isacommand{apply}{\isacharparenleft}simp{\isacharparenright}\isanewline
1.63 +\isacommand{apply}{\isacharparenleft}erule\ exE{\isacharparenright}\isanewline
1.64 +\isacommand{apply}{\isacharparenleft}erule\ conjE{\isacharparenright}\isanewline
1.65 +\isacommand{apply}{\isacharparenleft}erule{\isacharunderscore}tac\ P\ {\isacharequal}\ {\isachardoublequote}t{\isasymin}A{\isachardoublequote}\ \isakeyword{in}\ rev{\isacharunderscore}mp{\isacharparenright}\isanewline
1.66 +\isacommand{apply}{\isacharparenleft}erule\ converse{\isacharunderscore}rtrancl{\isacharunderscore}induct{\isacharparenright}\isanewline
1.67 +\ \isacommand{apply}{\isacharparenleft}rule\ ssubst\ {\isacharbrackleft}OF\ lfp{\isacharunderscore}Tarski{\isacharbrackleft}OF\ mono{\isacharunderscore}ef{\isacharbrackright}{\isacharbrackright}{\isacharparenright}\isanewline
1.68 +\ \isacommand{apply}{\isacharparenleft}blast{\isacharparenright}\isanewline
1.69 +\isacommand{apply}{\isacharparenleft}rule\ ssubst\ {\isacharbrackleft}OF\ lfp{\isacharunderscore}Tarski{\isacharbrackleft}OF\ mono{\isacharunderscore}ef{\isacharbrackright}{\isacharbrackright}{\isacharparenright}\isanewline
1.70 +\isacommand{by}{\isacharparenleft}blast{\isacharparenright}\isanewline
1.71 +\isanewline
1.72 +\isacommand{theorem}\ lfp{\isacharunderscore}subset{\isacharunderscore}AF{\isacharcolon}\isanewline
1.73 +{\isachardoublequote}lfp{\isacharparenleft}af\ A{\isacharparenright}\ {\isasymsubseteq}\ {\isacharbraceleft}s{\isachardot}\ {\isasymforall}\ p\ {\isasymin}\ Paths\ s{\isachardot}\ {\isasymexists}\ i{\isachardot}\ p\ i\ {\isasymin}\ A{\isacharbraceright}{\isachardoublequote}\isanewline
1.74 +\isacommand{apply}{\isacharparenleft}rule\ subsetI{\isacharparenright}\isanewline
1.75 +\isacommand{apply}{\isacharparenleft}erule\ Lfp{\isachardot}induct{\isacharbrackleft}OF\ {\isacharunderscore}\ mono{\isacharunderscore}af{\isacharbrackright}{\isacharparenright}\isanewline
1.76 +\isacommand{apply}{\isacharparenleft}simp\ add{\isacharcolon}\ af{\isacharunderscore}def\ Paths{\isacharunderscore}def{\isacharparenright}\isanewline
1.77 +\isacommand{apply}{\isacharparenleft}erule\ disjE{\isacharparenright}\isanewline
1.78 +\ \isacommand{apply}{\isacharparenleft}blast{\isacharparenright}\isanewline
1.79 +\isacommand{apply}{\isacharparenleft}clarify{\isacharparenright}\isanewline
1.80 +\isacommand{apply}{\isacharparenleft}erule{\isacharunderscore}tac\ x\ {\isacharequal}\ {\isachardoublequote}p\ \isadigit{1}{\isachardoublequote}\ \isakeyword{in}\ allE{\isacharparenright}\isanewline
1.81 +\isacommand{apply}{\isacharparenleft}clarsimp{\isacharparenright}\isanewline
1.82 +\isacommand{apply}{\isacharparenleft}erule{\isacharunderscore}tac\ x\ {\isacharequal}\ {\isachardoublequote}{\isasymlambda}i{\isachardot}\ p{\isacharparenleft}i{\isacharplus}\isadigit{1}{\isacharparenright}{\isachardoublequote}\ \isakeyword{in}\ allE{\isacharparenright}\isanewline
1.83 +\isacommand{apply}{\isacharparenleft}simp{\isacharparenright}\isanewline
1.84 +\isacommand{by}{\isacharparenleft}blast{\isacharparenright}%
1.85 +\begin{isamarkuptext}%
1.86 +The opposite direction is proved by contradiction: if some state
1.87 +{term s} is not in \isa{lfp\ {\isacharparenleft}af\ A{\isacharparenright}}, then we can construct an
1.88 +infinite \isa{A}-avoiding path starting from \isa{s}. The reason is
1.89 +that by unfolding \isa{lfp} we find that if \isa{s} is not in
1.90 +\isa{lfp\ {\isacharparenleft}af\ A{\isacharparenright}}, then \isa{s} is not in \isa{A} and there is a
1.91 +direct successor of \isa{s} that is again not in \isa{lfp\ {\isacharparenleft}af\ A{\isacharparenright}}. Iterating this argument yields the promised infinite
1.92 +\isa{A}-avoiding path. Let us formalize this sketch.
1.93 +
1.94 +The one-step argument in the above sketch%
1.95 +\end{isamarkuptext}%
1.96 +\isacommand{lemma}\ not{\isacharunderscore}in{\isacharunderscore}lfp{\isacharunderscore}afD{\isacharcolon}\isanewline
1.97 +\ {\isachardoublequote}s\ {\isasymnotin}\ lfp{\isacharparenleft}af\ A{\isacharparenright}\ {\isasymLongrightarrow}\ s\ {\isasymnotin}\ A\ {\isasymand}\ {\isacharparenleft}{\isasymexists}\ t{\isachardot}\ {\isacharparenleft}s{\isacharcomma}t{\isacharparenright}{\isasymin}M\ {\isasymand}\ t\ {\isasymnotin}\ lfp{\isacharparenleft}af\ A{\isacharparenright}{\isacharparenright}{\isachardoublequote}\isanewline
1.98 +\isacommand{apply}{\isacharparenleft}erule\ swap{\isacharparenright}\isanewline
1.99 +\isacommand{apply}{\isacharparenleft}rule\ ssubst{\isacharbrackleft}OF\ lfp{\isacharunderscore}Tarski{\isacharbrackleft}OF\ mono{\isacharunderscore}af{\isacharbrackright}{\isacharbrackright}{\isacharparenright}\isanewline
1.100 +\isacommand{by}{\isacharparenleft}simp\ add{\isacharcolon}af{\isacharunderscore}def{\isacharparenright}%
1.101 +\begin{isamarkuptext}%
1.102 +\noindent
1.103 +is proved by a variant of contraposition (\isa{swap}:
1.104 +\isa{{\isasymlbrakk}{\isasymnot}\ Pa{\isacharsemicolon}\ {\isasymnot}\ P\ {\isasymLongrightarrow}\ Pa{\isasymrbrakk}\ {\isasymLongrightarrow}\ P}), i.e.\ assuming the negation of the conclusion
1.105 +and proving \isa{s\ {\isasymin}\ lfp\ {\isacharparenleft}af\ A{\isacharparenright}}. Unfolding \isa{lfp} once and
1.106 +simplifying with the definition of \isa{af} finishes the proof.
1.107 +
1.108 +Now we iterate this process. The following construction of the desired
1.109 +path is parameterized by a predicate \isa{P} that should hold along the path:%
1.110 +\end{isamarkuptext}%
1.111 +\isacommand{consts}\ path\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}state\ {\isasymRightarrow}\ {\isacharparenleft}state\ {\isasymRightarrow}\ bool{\isacharparenright}\ {\isasymRightarrow}\ {\isacharparenleft}nat\ {\isasymRightarrow}\ state{\isacharparenright}{\isachardoublequote}\isanewline
1.112 +\isacommand{primrec}\isanewline
1.113 +{\isachardoublequote}path\ s\ P\ \isadigit{0}\ {\isacharequal}\ s{\isachardoublequote}\isanewline
1.114 +{\isachardoublequote}path\ s\ P\ {\isacharparenleft}Suc\ n{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}SOME\ t{\isachardot}\ {\isacharparenleft}path\ s\ P\ n{\isacharcomma}t{\isacharparenright}\ {\isasymin}\ M\ {\isasymand}\ P\ t{\isacharparenright}{\isachardoublequote}%
1.115 +\begin{isamarkuptext}%
1.116 +\noindent
1.117 +Element \isa{n\ {\isacharplus}\ \isadigit{1}} on this path is some arbitrary successor
1.118 +\isa{t} of element \isa{n} such that \isa{P\ t} holds. Of
1.119 +course, such a \isa{t} may in general not exist, but that is of no
1.120 +concern to us since we will only use \isa{path} in such cases where a
1.121 +suitable \isa{t} does exist.
1.122 +
1.123 +Now we prove that if each state \isa{s} that satisfies \isa{P}
1.124 +has a successor that again satisfies \isa{P}, then there exists an infinite \isa{P}-path.%
1.125 +\end{isamarkuptext}%
1.126 +\isacommand{lemma}\ seq{\isacharunderscore}lemma{\isacharcolon}\isanewline
1.127 +{\isachardoublequote}{\isasymlbrakk}\ P\ s{\isacharsemicolon}\ {\isasymforall}s{\isachardot}\ P\ s\ {\isasymlongrightarrow}\ {\isacharparenleft}{\isasymexists}\ t{\isachardot}\ {\isacharparenleft}s{\isacharcomma}t{\isacharparenright}{\isasymin}M\ {\isasymand}\ P\ t{\isacharparenright}\ {\isasymrbrakk}\ {\isasymLongrightarrow}\ {\isasymexists}p{\isasymin}Paths\ s{\isachardot}\ {\isasymforall}i{\isachardot}\ P{\isacharparenleft}p\ i{\isacharparenright}{\isachardoublequote}%
1.128 +\begin{isamarkuptxt}%
1.129 +\noindent
1.130 +First we rephrase the conclusion slightly because we need to prove both the path property
1.131 +and the fact that \isa{P} holds simultaneously:%
1.132 +\end{isamarkuptxt}%
1.133 +\isacommand{apply}{\isacharparenleft}subgoal{\isacharunderscore}tac\ {\isachardoublequote}{\isasymexists}\ p{\isachardot}\ s\ {\isacharequal}\ p\ \isadigit{0}\ {\isasymand}\ {\isacharparenleft}{\isasymforall}\ i{\isachardot}\ {\isacharparenleft}p\ i{\isacharcomma}p{\isacharparenleft}i{\isacharplus}\isadigit{1}{\isacharparenright}{\isacharparenright}\ {\isasymin}\ M\ {\isasymand}\ P{\isacharparenleft}p\ i{\isacharparenright}{\isacharparenright}{\isachardoublequote}{\isacharparenright}%
1.134 +\begin{isamarkuptxt}%
1.135 +\noindent
1.136 +From this proposition the original goal follows easily%
1.137 +\end{isamarkuptxt}%
1.138 +\ \isacommand{apply}{\isacharparenleft}simp\ add{\isacharcolon}Paths{\isacharunderscore}def{\isacharcomma}\ blast{\isacharparenright}\isanewline
1.139 +\isacommand{apply}{\isacharparenleft}rule{\isacharunderscore}tac\ x\ {\isacharequal}\ {\isachardoublequote}path\ s\ P{\isachardoublequote}\ \isakeyword{in}\ exI{\isacharparenright}\isanewline
1.140 +\isacommand{apply}{\isacharparenleft}simp{\isacharparenright}\isanewline
1.141 +\isacommand{apply}{\isacharparenleft}intro\ strip{\isacharparenright}\isanewline
1.142 +\isacommand{apply}{\isacharparenleft}induct{\isacharunderscore}tac\ i{\isacharparenright}\isanewline
1.143 +\ \isacommand{apply}{\isacharparenleft}simp{\isacharparenright}\isanewline
1.144 +\ \isacommand{apply}{\isacharparenleft}fast\ intro{\isacharcolon}someI\isadigit{2}EX{\isacharparenright}\isanewline
1.145 +\isacommand{apply}{\isacharparenleft}simp{\isacharparenright}\isanewline
1.146 +\isacommand{apply}{\isacharparenleft}rule\ someI\isadigit{2}EX{\isacharparenright}\isanewline
1.147 +\ \isacommand{apply}{\isacharparenleft}blast{\isacharparenright}\isanewline
1.148 +\isacommand{apply}{\isacharparenleft}rule\ someI\isadigit{2}EX{\isacharparenright}\isanewline
1.149 +\ \isacommand{apply}{\isacharparenleft}blast{\isacharparenright}\isanewline
1.150 +\isacommand{by}{\isacharparenleft}blast{\isacharparenright}\isanewline
1.151 +\isanewline
1.152 +\isacommand{lemma}\ seq{\isacharunderscore}lemma{\isacharcolon}\isanewline
1.153 +{\isachardoublequote}{\isasymlbrakk}\ P\ s{\isacharsemicolon}\ {\isasymforall}\ s{\isachardot}\ P\ s\ {\isasymlongrightarrow}\ {\isacharparenleft}{\isasymexists}\ t{\isachardot}\ {\isacharparenleft}s{\isacharcomma}t{\isacharparenright}{\isasymin}M\ {\isasymand}\ P\ t{\isacharparenright}\ {\isasymrbrakk}\ {\isasymLongrightarrow}\isanewline
1.154 +\ {\isasymexists}\ p{\isasymin}Paths\ s{\isachardot}\ {\isasymforall}\ i{\isachardot}\ P{\isacharparenleft}p\ i{\isacharparenright}{\isachardoublequote}\isanewline
1.155 +\isacommand{apply}{\isacharparenleft}subgoal{\isacharunderscore}tac\isanewline
1.156 +\ {\isachardoublequote}{\isasymexists}\ p{\isachardot}\ s\ {\isacharequal}\ p\ \isadigit{0}\ {\isasymand}\ {\isacharparenleft}{\isasymforall}\ i{\isachardot}\ {\isacharparenleft}p\ i{\isacharcomma}p{\isacharparenleft}Suc\ i{\isacharparenright}{\isacharparenright}{\isasymin}M\ {\isasymand}\ P{\isacharparenleft}p\ i{\isacharparenright}{\isacharparenright}{\isachardoublequote}{\isacharparenright}\isanewline
1.157 +\ \isacommand{apply}{\isacharparenleft}simp\ add{\isacharcolon}Paths{\isacharunderscore}def{\isacharparenright}\isanewline
1.158 +\ \isacommand{apply}{\isacharparenleft}blast{\isacharparenright}\isanewline
1.159 +\isacommand{apply}{\isacharparenleft}rule{\isacharunderscore}tac\ x\ {\isacharequal}\ {\isachardoublequote}nat{\isacharunderscore}rec\ s\ {\isacharparenleft}{\isasymlambda}n\ t{\isachardot}\ SOME\ u{\isachardot}\ {\isacharparenleft}t{\isacharcomma}u{\isacharparenright}{\isasymin}M\ {\isasymand}\ P\ u{\isacharparenright}{\isachardoublequote}\ \isakeyword{in}\ exI{\isacharparenright}\isanewline
1.160 +\isacommand{apply}{\isacharparenleft}simp{\isacharparenright}\isanewline
1.161 +\isacommand{apply}{\isacharparenleft}intro\ strip{\isacharparenright}\isanewline
1.162 +\isacommand{apply}{\isacharparenleft}induct{\isacharunderscore}tac\ i{\isacharparenright}\isanewline
1.163 +\ \isacommand{apply}{\isacharparenleft}simp{\isacharparenright}\isanewline
1.164 +\ \isacommand{apply}{\isacharparenleft}fast\ intro{\isacharcolon}someI\isadigit{2}EX{\isacharparenright}\isanewline
1.165 +\isacommand{apply}{\isacharparenleft}simp{\isacharparenright}\isanewline
1.166 +\isacommand{apply}{\isacharparenleft}rule\ someI\isadigit{2}EX{\isacharparenright}\isanewline
1.167 +\ \isacommand{apply}{\isacharparenleft}blast{\isacharparenright}\isanewline
1.168 +\isacommand{apply}{\isacharparenleft}rule\ someI\isadigit{2}EX{\isacharparenright}\isanewline
1.169 +\ \isacommand{apply}{\isacharparenleft}blast{\isacharparenright}\isanewline
1.170 +\isacommand{by}{\isacharparenleft}blast{\isacharparenright}\isanewline
1.171 +\isanewline
1.172 +\isacommand{theorem}\ AF{\isacharunderscore}subset{\isacharunderscore}lfp{\isacharcolon}\isanewline
1.173 +{\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
1.174 +\isacommand{apply}{\isacharparenleft}rule\ subsetI{\isacharparenright}\isanewline
1.175 +\isacommand{apply}{\isacharparenleft}erule\ contrapos\isadigit{2}{\isacharparenright}\isanewline
1.176 +\isacommand{apply}\ simp\isanewline
1.177 +\isacommand{apply}{\isacharparenleft}drule\ seq{\isacharunderscore}lemma{\isacharparenright}\isanewline
1.178 +\isacommand{by}{\isacharparenleft}auto\ dest{\isacharcolon}not{\isacharunderscore}in{\isacharunderscore}lfp{\isacharunderscore}afD{\isacharparenright}\isanewline
1.179 +\isanewline
1.180 +\isanewline
1.181 +\isanewline
1.182 +\isanewline
1.183 +\isacommand{consts}\ Avoid\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}state\ {\isasymRightarrow}\ state\ set\ {\isasymRightarrow}\ state\ set{\isachardoublequote}\isanewline
1.184 +\isacommand{inductive}\ {\isachardoublequote}Avoid\ s\ A{\isachardoublequote}\isanewline
1.185 +\isakeyword{intros}\ {\isachardoublequote}s\ {\isasymin}\ Avoid\ s\ A{\isachardoublequote}\isanewline
1.186 +\ \ \ \ \ \ \ {\isachardoublequote}{\isasymlbrakk}\ t\ {\isasymin}\ Avoid\ s\ A{\isacharsemicolon}\ t\ {\isasymnotin}\ A{\isacharsemicolon}\ {\isacharparenleft}t{\isacharcomma}u{\isacharparenright}\ {\isasymin}\ M\ {\isasymrbrakk}\ {\isasymLongrightarrow}\ u\ {\isasymin}\ Avoid\ s\ A{\isachardoublequote}\isanewline
1.187 +\isanewline
1.188 +\isanewline
1.189 +\isacommand{lemma}\ ex{\isacharunderscore}infinite{\isacharunderscore}path{\isacharbrackleft}rule{\isacharunderscore}format{\isacharbrackright}{\isacharcolon}\isanewline
1.190 +{\isachardoublequote}t\ {\isasymin}\ Avoid\ s\ A\ \ {\isasymLongrightarrow}\isanewline
1.191 +\ {\isasymforall}f{\isachardot}\ t\ {\isacharequal}\ f\ \isadigit{0}\ {\isasymlongrightarrow}\ {\isacharparenleft}{\isasymforall}i{\isachardot}\ {\isacharparenleft}f\ i{\isacharcomma}\ f\ {\isacharparenleft}Suc\ i{\isacharparenright}{\isacharparenright}\ {\isasymin}\ M\ {\isasymand}\ f\ i\ {\isasymin}\ Avoid\ s\ A\ {\isasymand}\ f\ i\ {\isasymnotin}\ A{\isacharparenright}\isanewline
1.192 +\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isasymlongrightarrow}\ {\isacharparenleft}{\isasymexists}\ p{\isasymin}Paths\ s{\isachardot}\ {\isasymforall}i{\isachardot}\ p\ i\ {\isasymnotin}\ A{\isacharparenright}{\isachardoublequote}\isanewline
1.193 +\isacommand{apply}{\isacharparenleft}simp\ add{\isacharcolon}Paths{\isacharunderscore}def{\isacharparenright}\isanewline
1.194 +\isacommand{apply}{\isacharparenleft}erule\ Avoid{\isachardot}induct{\isacharparenright}\isanewline
1.195 +\ \isacommand{apply}{\isacharparenleft}blast{\isacharparenright}\isanewline
1.196 +\isacommand{apply}{\isacharparenleft}rule\ allI{\isacharparenright}\isanewline
1.197 +\isacommand{apply}{\isacharparenleft}erule{\isacharunderscore}tac\ x\ {\isacharequal}\ {\isachardoublequote}{\isasymlambda}i{\isachardot}\ case\ i\ of\ \isadigit{0}\ {\isasymRightarrow}\ t\ {\isacharbar}\ Suc\ i\ {\isasymRightarrow}\ f\ i{\isachardoublequote}\ \isakeyword{in}\ allE{\isacharparenright}\isanewline
1.198 +\isacommand{by}{\isacharparenleft}force\ split{\isacharcolon}nat{\isachardot}split{\isacharparenright}\isanewline
1.199 +\isanewline
1.200 +\isacommand{lemma}\ Avoid{\isacharunderscore}in{\isacharunderscore}lfp{\isacharbrackleft}rule{\isacharunderscore}format{\isacharparenleft}no{\isacharunderscore}asm{\isacharparenright}{\isacharbrackright}{\isacharcolon}\isanewline
1.201 +{\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}\isanewline
1.202 +\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}\ y\ {\isasymin}\ Avoid\ s\ A\ {\isasymand}\ x\ {\isasymnotin}\ A{\isacharbraceright}{\isachardoublequote}{\isacharparenright}\isanewline
1.203 +\ \isacommand{apply}{\isacharparenleft}erule{\isacharunderscore}tac\ a\ {\isacharequal}\ t\ \isakeyword{in}\ wf{\isacharunderscore}induct{\isacharparenright}\isanewline
1.204 +\ \isacommand{apply}{\isacharparenleft}clarsimp{\isacharparenright}\isanewline
1.205 +\ \isacommand{apply}{\isacharparenleft}rule\ ssubst\ {\isacharbrackleft}OF\ lfp{\isacharunderscore}Tarski{\isacharbrackleft}OF\ mono{\isacharunderscore}af{\isacharbrackright}{\isacharbrackright}{\isacharparenright}\isanewline
1.206 +\ \isacommand{apply}{\isacharparenleft}unfold\ af{\isacharunderscore}def{\isacharparenright}\isanewline
1.207 +\ \isacommand{apply}{\isacharparenleft}blast\ intro{\isacharcolon}Avoid{\isachardot}intros{\isacharparenright}\isanewline
1.208 +\isacommand{apply}{\isacharparenleft}erule\ contrapos\isadigit{2}{\isacharparenright}\isanewline
1.209 +\isacommand{apply}{\isacharparenleft}simp\ add{\isacharcolon}wf{\isacharunderscore}iff{\isacharunderscore}no{\isacharunderscore}infinite{\isacharunderscore}down{\isacharunderscore}chain{\isacharparenright}\isanewline
1.210 +\isacommand{apply}{\isacharparenleft}erule\ exE{\isacharparenright}\isanewline
1.211 +\isacommand{apply}{\isacharparenleft}rule\ ex{\isacharunderscore}infinite{\isacharunderscore}path{\isacharparenright}\isanewline
1.212 +\isacommand{by}{\isacharparenleft}auto{\isacharparenright}\isanewline
1.213 +\isanewline
1.214 +\isacommand{theorem}\ AF{\isacharunderscore}subset{\isacharunderscore}lfp{\isacharcolon}\isanewline
1.215 +{\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
1.216 +\isacommand{apply}{\isacharparenleft}rule\ subsetI{\isacharparenright}\isanewline
1.217 +\isacommand{apply}{\isacharparenleft}simp{\isacharparenright}\isanewline
1.218 +\isacommand{apply}{\isacharparenleft}erule\ Avoid{\isacharunderscore}in{\isacharunderscore}lfp{\isacharparenright}\isanewline
1.219 +\isacommand{by}{\isacharparenleft}rule\ Avoid{\isachardot}intros{\isacharparenright}\isanewline
1.220 +\isanewline
1.221 +\isanewline
1.222 +\isacommand{theorem}\ {\isachardoublequote}mc\ f\ {\isacharequal}\ {\isacharbraceleft}s{\isachardot}\ s\ {\isasymTurnstile}\ f{\isacharbraceright}{\isachardoublequote}\isanewline
1.223 +\isacommand{apply}{\isacharparenleft}induct{\isacharunderscore}tac\ f{\isacharparenright}\isanewline
1.224 +\isacommand{by}{\isacharparenleft}auto\ simp\ add{\isacharcolon}\ lfp{\isacharunderscore}conv{\isacharunderscore}EF\ equalityI{\isacharbrackleft}OF\ lfp{\isacharunderscore}subset{\isacharunderscore}AF\ AF{\isacharunderscore}subset{\isacharunderscore}lfp{\isacharbrackright}{\isacharparenright}\isanewline
1.225 +\isanewline
1.226 +\isacommand{end}\isanewline
1.227 +\end{isabellebody}%
1.228 +%%% Local Variables:
1.229 +%%% mode: latex
1.230 +%%% TeX-master: "root"
1.231 +%%% End: