3 \def\isabellecontext{PDL}%
5 \isamarkupsubsection{Propositional dynmic logic---PDL}
8 The formulae of PDL are built up from atomic propositions via the customary
9 propositional connectives of negation and conjunction and the two temporal
10 connectives \isa{AX} and \isa{EF}. Since formulae are essentially
11 (syntax) trees, they are naturally modelled as a datatype:%
13 \isacommand{datatype}\ pdl{\isacharunderscore}form\ {\isacharequal}\ Atom\ atom\isanewline
14 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharbar}\ NOT\ pdl{\isacharunderscore}form\isanewline
15 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharbar}\ And\ pdl{\isacharunderscore}form\ pdl{\isacharunderscore}form\isanewline
16 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharbar}\ AX\ pdl{\isacharunderscore}form\isanewline
17 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharbar}\ EF\ pdl{\isacharunderscore}form%
18 \begin{isamarkuptext}%
20 The meaning of these formulae is given by saying which formula is true in
23 \isacommand{consts}\ valid\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}state\ {\isasymRightarrow}\ pdl{\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
25 \isacommand{primrec}\isanewline
26 {\isachardoublequote}s\ {\isasymTurnstile}\ Atom\ a\ \ {\isacharequal}\ {\isacharparenleft}a\ {\isasymin}\ L\ s{\isacharparenright}{\isachardoublequote}\isanewline
27 {\isachardoublequote}s\ {\isasymTurnstile}\ NOT\ f\ \ \ {\isacharequal}\ {\isacharparenleft}{\isasymnot}{\isacharparenleft}s\ {\isasymTurnstile}\ f{\isacharparenright}{\isacharparenright}{\isachardoublequote}\isanewline
28 {\isachardoublequote}s\ {\isasymTurnstile}\ And\ f\ g\ {\isacharequal}\ {\isacharparenleft}s\ {\isasymTurnstile}\ f\ {\isasymand}\ s\ {\isasymTurnstile}\ g{\isacharparenright}{\isachardoublequote}\isanewline
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
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}%
31 \begin{isamarkuptext}%
32 Now we come to the model checker itself. It maps a formula into the set of
33 states where the formula is true and is defined by recursion over the syntax:%
35 \isacommand{consts}\ mc\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}pdl{\isacharunderscore}form\ {\isasymRightarrow}\ state\ set{\isachardoublequote}\isanewline
36 \isacommand{primrec}\isanewline
37 {\isachardoublequote}mc{\isacharparenleft}Atom\ a{\isacharparenright}\ \ {\isacharequal}\ {\isacharbraceleft}s{\isachardot}\ a\ {\isasymin}\ L\ s{\isacharbraceright}{\isachardoublequote}\isanewline
38 {\isachardoublequote}mc{\isacharparenleft}NOT\ f{\isacharparenright}\ \ \ {\isacharequal}\ {\isacharminus}mc\ f{\isachardoublequote}\isanewline
39 {\isachardoublequote}mc{\isacharparenleft}And\ f\ g{\isacharparenright}\ {\isacharequal}\ mc\ f\ {\isasyminter}\ mc\ g{\isachardoublequote}\isanewline
40 {\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
41 {\isachardoublequote}mc{\isacharparenleft}EF\ f{\isacharparenright}\ \ \ \ {\isacharequal}\ lfp{\isacharparenleft}{\isasymlambda}T{\isachardot}\ mc\ f\ {\isasymunion}\ M{\isacharcircum}{\isacharminus}\isadigit{1}\ {\isacharcircum}{\isacharcircum}\ T{\isacharparenright}{\isachardoublequote}%
42 \begin{isamarkuptext}%
43 Only the equation for \isa{EF} deserves a comment: the postfix \isa{{\isacharcircum}{\isacharminus}\isadigit{1}}
44 and the infix \isa{{\isacharcircum}{\isacharcircum}} are predefined and denote the converse of a
45 relation and the application of a relation to a set. Thus \isa{M{\isacharcircum}{\isacharminus}\isadigit{1}\ {\isacharcircum}{\isacharcircum}\ T}
46 is the set of all predecessors of \isa{T} and \isa{mc\ {\isacharparenleft}EF\ f{\isacharparenright}} is the least
47 set \isa{T} containing \isa{mc\ f} and all predecessors of \isa{T}.%
49 \isacommand{lemma}\ mono{\isacharunderscore}lemma{\isacharcolon}\ {\isachardoublequote}mono{\isacharparenleft}{\isasymlambda}T{\isachardot}\ A\ {\isasymunion}\ M{\isacharcircum}{\isacharminus}\isadigit{1}\ {\isacharcircum}{\isacharcircum}\ T{\isacharparenright}{\isachardoublequote}\isanewline
50 \isacommand{apply}{\isacharparenleft}rule\ monoI{\isacharparenright}\isanewline
51 \isacommand{by}{\isacharparenleft}blast{\isacharparenright}\isanewline
53 \isacommand{lemma}\ lfp{\isacharunderscore}conv{\isacharunderscore}EF{\isacharcolon}\isanewline
54 {\isachardoublequote}lfp{\isacharparenleft}{\isasymlambda}T{\isachardot}\ A\ {\isasymunion}\ M{\isacharcircum}{\isacharminus}\isadigit{1}\ {\isacharcircum}{\isacharcircum}\ T{\isacharparenright}\ {\isacharequal}\ {\isacharbraceleft}s{\isachardot}\ {\isasymexists}t{\isachardot}\ {\isacharparenleft}s{\isacharcomma}t{\isacharparenright}\ {\isasymin}\ M{\isacharcircum}{\isacharasterisk}\ {\isasymand}\ t\ {\isasymin}\ A{\isacharbraceright}{\isachardoublequote}\isanewline
55 \isacommand{apply}{\isacharparenleft}rule\ equalityI{\isacharparenright}\isanewline
56 \ \isacommand{apply}{\isacharparenleft}rule\ subsetI{\isacharparenright}\isanewline
57 \ \isacommand{apply}{\isacharparenleft}simp{\isacharparenright}\isanewline
58 \ \isacommand{apply}{\isacharparenleft}erule\ Lfp{\isachardot}induct{\isacharparenright}\isanewline
59 \ \ \isacommand{apply}{\isacharparenleft}rule\ mono{\isacharunderscore}lemma{\isacharparenright}\isanewline
60 \ \isacommand{apply}{\isacharparenleft}simp{\isacharparenright}\isanewline
61 \ \isacommand{apply}{\isacharparenleft}blast\ intro{\isacharcolon}\ r{\isacharunderscore}into{\isacharunderscore}rtrancl\ rtrancl{\isacharunderscore}trans{\isacharparenright}\isanewline
62 \isacommand{apply}{\isacharparenleft}rule\ subsetI{\isacharparenright}\isanewline
63 \isacommand{apply}{\isacharparenleft}simp{\isacharparenright}\isanewline
64 \isacommand{apply}{\isacharparenleft}erule\ exE{\isacharparenright}\isanewline
65 \isacommand{apply}{\isacharparenleft}erule\ conjE{\isacharparenright}\isanewline
66 \isacommand{apply}{\isacharparenleft}erule{\isacharunderscore}tac\ P\ {\isacharequal}\ {\isachardoublequote}t{\isasymin}A{\isachardoublequote}\ \isakeyword{in}\ rev{\isacharunderscore}mp{\isacharparenright}\isanewline
67 \isacommand{apply}{\isacharparenleft}erule\ converse{\isacharunderscore}rtrancl{\isacharunderscore}induct{\isacharparenright}\isanewline
68 \ \isacommand{apply}{\isacharparenleft}rule\ ssubst{\isacharbrackleft}OF\ lfp{\isacharunderscore}Tarski{\isacharbrackleft}OF\ mono{\isacharunderscore}lemma{\isacharbrackright}{\isacharbrackright}{\isacharparenright}\isanewline
69 \ \isacommand{apply}{\isacharparenleft}blast{\isacharparenright}\isanewline
70 \isacommand{apply}{\isacharparenleft}rule\ ssubst{\isacharbrackleft}OF\ lfp{\isacharunderscore}Tarski{\isacharbrackleft}OF\ mono{\isacharunderscore}lemma{\isacharbrackright}{\isacharbrackright}{\isacharparenright}\isanewline
71 \isacommand{by}{\isacharparenleft}blast{\isacharparenright}\isanewline
73 \isacommand{theorem}\ {\isachardoublequote}mc\ f\ {\isacharequal}\ {\isacharbraceleft}s{\isachardot}\ s\ {\isasymTurnstile}\ f{\isacharbraceright}{\isachardoublequote}\isanewline
74 \isacommand{apply}{\isacharparenleft}induct{\isacharunderscore}tac\ f{\isacharparenright}\isanewline
75 \isacommand{by}{\isacharparenleft}auto\ simp\ add{\isacharcolon}lfp{\isacharunderscore}conv{\isacharunderscore}EF{\isacharparenright}\end{isabellebody}%
78 %%% TeX-master: "root"