3 \def\isabellecontext{PDL}%
5 \isamarkupsubsection{Propositional dynmic logic---PDL}
6 \isacommand{datatype}\ ctl{\isacharunderscore}form\ {\isacharequal}\ Atom\ atom\isanewline
7 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharbar}\ NOT\ ctl{\isacharunderscore}form\isanewline
8 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharbar}\ And\ ctl{\isacharunderscore}form\ ctl{\isacharunderscore}form\isanewline
9 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharbar}\ AX\ ctl{\isacharunderscore}form\isanewline
10 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharbar}\ EF\ ctl{\isacharunderscore}form\isanewline
12 \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
13 \ \ \ \ \ \ \ M\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharparenleft}state\ {\isasymtimes}\ state{\isacharparenright}set{\isachardoublequote}\isanewline
15 \isacommand{primrec}\isanewline
16 {\isachardoublequote}s\ {\isasymTurnstile}\ Atom\ a\ \ {\isacharequal}\ {\isacharparenleft}a{\isasymin}s{\isacharparenright}{\isachardoublequote}\isanewline
17 {\isachardoublequote}s\ {\isasymTurnstile}\ NOT\ f\ \ \ {\isacharequal}\ {\isacharparenleft}{\isasymnot}{\isacharparenleft}s\ {\isasymTurnstile}\ f{\isacharparenright}{\isacharparenright}{\isachardoublequote}\isanewline
18 {\isachardoublequote}s\ {\isasymTurnstile}\ And\ f\ g\ {\isacharequal}\ {\isacharparenleft}s\ {\isasymTurnstile}\ f\ {\isasymand}\ s\ {\isasymTurnstile}\ g{\isacharparenright}{\isachardoublequote}\isanewline
19 {\isachardoublequote}s\ {\isasymTurnstile}\ AX\ f\ \ \ \ {\isacharequal}\ {\isacharparenleft}{\isasymforall}t{\isachardot}\ {\isacharparenleft}s{\isacharcomma}t{\isacharparenright}\ {\isasymin}\ M\ {\isasymlongrightarrow}\ t\ {\isasymTurnstile}\ f{\isacharparenright}{\isachardoublequote}\isanewline
20 {\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
22 \isacommand{consts}\ mc\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}ctl{\isacharunderscore}form\ {\isasymRightarrow}\ state\ set{\isachardoublequote}\isanewline
23 \isacommand{primrec}\isanewline
24 {\isachardoublequote}mc{\isacharparenleft}Atom\ a{\isacharparenright}\ \ {\isacharequal}\ {\isacharbraceleft}s{\isachardot}\ a{\isasymin}s{\isacharbraceright}{\isachardoublequote}\isanewline
25 {\isachardoublequote}mc{\isacharparenleft}NOT\ f{\isacharparenright}\ \ \ {\isacharequal}\ {\isacharminus}mc\ f{\isachardoublequote}\isanewline
26 {\isachardoublequote}mc{\isacharparenleft}And\ f\ g{\isacharparenright}\ {\isacharequal}\ mc\ f\ Int\ mc\ g{\isachardoublequote}\isanewline
27 {\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
28 {\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
30 \isacommand{lemma}\ mono{\isacharunderscore}lemma{\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
31 \isacommand{apply}{\isacharparenleft}rule\ monoI{\isacharparenright}\isanewline
32 \isacommand{by}{\isacharparenleft}blast{\isacharparenright}\isanewline
34 \isacommand{lemma}\ lfp{\isacharunderscore}conv{\isacharunderscore}EF{\isacharcolon}\isanewline
35 {\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
36 \isacommand{apply}{\isacharparenleft}rule\ equalityI{\isacharparenright}\isanewline
37 \ \isacommand{apply}{\isacharparenleft}rule\ subsetI{\isacharparenright}\isanewline
38 \ \isacommand{apply}{\isacharparenleft}simp{\isacharparenright}\isanewline
39 \ \isacommand{apply}{\isacharparenleft}erule\ Lfp{\isachardot}induct{\isacharparenright}\isanewline
40 \ \ \isacommand{apply}{\isacharparenleft}rule\ mono{\isacharunderscore}lemma{\isacharparenright}\isanewline
41 \ \isacommand{apply}{\isacharparenleft}simp{\isacharparenright}\isanewline
42 \ \isacommand{apply}{\isacharparenleft}blast\ intro{\isacharcolon}\ r{\isacharunderscore}into{\isacharunderscore}rtrancl\ rtrancl{\isacharunderscore}trans{\isacharparenright}\isanewline
43 \isacommand{apply}{\isacharparenleft}rule\ subsetI{\isacharparenright}\isanewline
44 \isacommand{apply}{\isacharparenleft}simp{\isacharparenright}\isanewline
45 \isacommand{apply}{\isacharparenleft}erule\ exE{\isacharparenright}\isanewline
46 \isacommand{apply}{\isacharparenleft}erule\ conjE{\isacharparenright}\isanewline
47 \isacommand{apply}{\isacharparenleft}erule{\isacharunderscore}tac\ P\ {\isacharequal}\ {\isachardoublequote}t{\isasymin}A{\isachardoublequote}\ \isakeyword{in}\ rev{\isacharunderscore}mp{\isacharparenright}\isanewline
48 \isacommand{apply}{\isacharparenleft}erule\ converse{\isacharunderscore}rtrancl{\isacharunderscore}induct{\isacharparenright}\isanewline
49 \ \isacommand{apply}{\isacharparenleft}rule\ ssubst{\isacharbrackleft}OF\ lfp{\isacharunderscore}Tarski{\isacharbrackleft}OF\ mono{\isacharunderscore}lemma{\isacharbrackright}{\isacharbrackright}{\isacharparenright}\isanewline
50 \ \isacommand{apply}{\isacharparenleft}blast{\isacharparenright}\isanewline
51 \isacommand{apply}{\isacharparenleft}rule\ ssubst{\isacharbrackleft}OF\ lfp{\isacharunderscore}Tarski{\isacharbrackleft}OF\ mono{\isacharunderscore}lemma{\isacharbrackright}{\isacharbrackright}{\isacharparenright}\isanewline
52 \isacommand{by}{\isacharparenleft}blast{\isacharparenright}\isanewline
54 \isacommand{theorem}\ {\isachardoublequote}mc\ f\ {\isacharequal}\ {\isacharbraceleft}s{\isachardot}\ s\ {\isasymTurnstile}\ f{\isacharbraceright}{\isachardoublequote}\isanewline
55 \isacommand{apply}{\isacharparenleft}induct{\isacharunderscore}tac\ f{\isacharparenright}\isanewline
56 \isacommand{by}{\isacharparenleft}auto\ simp\ add{\isacharcolon}lfp{\isacharunderscore}conv{\isacharunderscore}EF{\isacharparenright}\isanewline
58 \isacommand{end}\isanewline
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