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\begin{isabellebody}%
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\def\isabellecontext{PDL}%
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%
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\isadelimtheory
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%
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\endisadelimtheory
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%
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\isatagtheory
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%
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\endisatagtheory
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{\isafoldtheory}%
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%
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\isadelimtheory
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%
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\endisadelimtheory
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%
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\isamarkupsubsection{Propositional Dynamic Logic --- PDL%
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}
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\isamarkuptrue%
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%
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\begin{isamarkuptext}%
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\index{PDL|(}
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The formulae of PDL are built up from atomic propositions via
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negation and conjunction and the two temporal
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connectives \isa{AX} and \isa{EF}\@. Since formulae are essentially
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syntax trees, they are naturally modelled as a datatype:%
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\footnote{The customary definition of PDL
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\cite{HarelKT-DL} looks quite different from ours, but the two are easily
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shown to be equivalent.}%
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\end{isamarkuptext}%
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\isamarkuptrue%
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\isacommand{datatype}\isamarkupfalse%
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\ formula\ {\isaliteral{3D}{\isacharequal}}\ Atom\ {\isaliteral{22}{\isachardoublequoteopen}}atom{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
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\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isaliteral{7C}{\isacharbar}}\ Neg\ formula\isanewline
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\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isaliteral{7C}{\isacharbar}}\ And\ formula\ formula\isanewline
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\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isaliteral{7C}{\isacharbar}}\ AX\ formula\isanewline
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\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isaliteral{7C}{\isacharbar}}\ EF\ formula%
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\begin{isamarkuptext}%
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\noindent
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This resembles the boolean expression case study in
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\S\ref{sec:boolex}.
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A validity relation between states and formulae specifies the semantics.
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The syntax annotation allows us to write \isa{s\ {\isaliteral{5C3C5475726E7374696C653E}{\isasymTurnstile}}\ f} instead of
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\hbox{\isa{valid\ s\ f}}. The definition is by recursion over the syntax:%
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\end{isamarkuptext}%
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\isamarkuptrue%
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\isacommand{primrec}\isamarkupfalse%
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\ valid\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}state\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ formula\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ bool{\isaliteral{22}{\isachardoublequoteclose}}\ \ \ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5F}{\isacharunderscore}}\ {\isaliteral{5C3C5475726E7374696C653E}{\isasymTurnstile}}\ {\isaliteral{5F}{\isacharunderscore}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{5B}{\isacharbrackleft}}{\isadigit{8}}{\isadigit{0}}{\isaliteral{2C}{\isacharcomma}}{\isadigit{8}}{\isadigit{0}}{\isaliteral{5D}{\isacharbrackright}}\ {\isadigit{8}}{\isadigit{0}}{\isaliteral{29}{\isacharparenright}}\isanewline
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\isakeyword{where}\isanewline
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{\isaliteral{22}{\isachardoublequoteopen}}s\ {\isaliteral{5C3C5475726E7374696C653E}{\isasymTurnstile}}\ Atom\ a\ \ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}a\ {\isaliteral{5C3C696E3E}{\isasymin}}\ L\ s{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{7C}{\isacharbar}}\isanewline
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{\isaliteral{22}{\isachardoublequoteopen}}s\ {\isaliteral{5C3C5475726E7374696C653E}{\isasymTurnstile}}\ Neg\ f\ \ \ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6E6F743E}{\isasymnot}}{\isaliteral{28}{\isacharparenleft}}s\ {\isaliteral{5C3C5475726E7374696C653E}{\isasymTurnstile}}\ f{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{7C}{\isacharbar}}\isanewline
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{\isaliteral{22}{\isachardoublequoteopen}}s\ {\isaliteral{5C3C5475726E7374696C653E}{\isasymTurnstile}}\ And\ f\ g\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}s\ {\isaliteral{5C3C5475726E7374696C653E}{\isasymTurnstile}}\ f\ {\isaliteral{5C3C616E643E}{\isasymand}}\ s\ {\isaliteral{5C3C5475726E7374696C653E}{\isasymTurnstile}}\ g{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{7C}{\isacharbar}}\isanewline
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{\isaliteral{22}{\isachardoublequoteopen}}s\ {\isaliteral{5C3C5475726E7374696C653E}{\isasymTurnstile}}\ AX\ f\ \ \ \ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}t{\isaliteral{2E}{\isachardot}}\ {\isaliteral{28}{\isacharparenleft}}s{\isaliteral{2C}{\isacharcomma}}t{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C696E3E}{\isasymin}}\ M\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ t\ {\isaliteral{5C3C5475726E7374696C653E}{\isasymTurnstile}}\ f{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{7C}{\isacharbar}}\isanewline
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{\isaliteral{22}{\isachardoublequoteopen}}s\ {\isaliteral{5C3C5475726E7374696C653E}{\isasymTurnstile}}\ EF\ f\ \ \ \ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6578697374733E}{\isasymexists}}t{\isaliteral{2E}{\isachardot}}\ {\isaliteral{28}{\isacharparenleft}}s{\isaliteral{2C}{\isacharcomma}}t{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C696E3E}{\isasymin}}\ M\isaliteral{5C3C5E7375703E}{}\isactrlsup {\isaliteral{2A}{\isacharasterisk}}\ {\isaliteral{5C3C616E643E}{\isasymand}}\ t\ {\isaliteral{5C3C5475726E7374696C653E}{\isasymTurnstile}}\ f{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}%
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\begin{isamarkuptext}%
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\noindent
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The first three equations should be self-explanatory. The temporal formula
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\isa{AX\ f} means that \isa{f} is true in \emph{A}ll ne\emph{X}t states whereas
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\isa{EF\ f} means that there \emph{E}xists some \emph{F}uture state in which \isa{f} is
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true. The future is expressed via \isa{\isaliteral{5C3C5E7375703E}{}\isactrlsup {\isaliteral{2A}{\isacharasterisk}}}, the reflexive transitive
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closure. Because of reflexivity, the future includes the present.
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Now we come to the model checker itself. It maps a formula into the
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set of states where the formula is true. It too is defined by
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recursion over the syntax:%
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\end{isamarkuptext}%
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\isamarkuptrue%
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\isacommand{primrec}\isamarkupfalse%
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\ mc\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}formula\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ state\ set{\isaliteral{22}{\isachardoublequoteclose}}\ \isakeyword{where}\isanewline
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{\isaliteral{22}{\isachardoublequoteopen}}mc{\isaliteral{28}{\isacharparenleft}}Atom\ a{\isaliteral{29}{\isacharparenright}}\ \ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{7B}{\isacharbraceleft}}s{\isaliteral{2E}{\isachardot}}\ a\ {\isaliteral{5C3C696E3E}{\isasymin}}\ L\ s{\isaliteral{7D}{\isacharbraceright}}{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{7C}{\isacharbar}}\isanewline
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{\isaliteral{22}{\isachardoublequoteopen}}mc{\isaliteral{28}{\isacharparenleft}}Neg\ f{\isaliteral{29}{\isacharparenright}}\ \ \ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{2D}{\isacharminus}}mc\ f{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{7C}{\isacharbar}}\isanewline
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{\isaliteral{22}{\isachardoublequoteopen}}mc{\isaliteral{28}{\isacharparenleft}}And\ f\ g{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ mc\ f\ {\isaliteral{5C3C696E7465723E}{\isasyminter}}\ mc\ g{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{7C}{\isacharbar}}\isanewline
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{\isaliteral{22}{\isachardoublequoteopen}}mc{\isaliteral{28}{\isacharparenleft}}AX\ f{\isaliteral{29}{\isacharparenright}}\ \ \ \ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{7B}{\isacharbraceleft}}s{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}t{\isaliteral{2E}{\isachardot}}\ {\isaliteral{28}{\isacharparenleft}}s{\isaliteral{2C}{\isacharcomma}}t{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C696E3E}{\isasymin}}\ M\ \ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ t\ {\isaliteral{5C3C696E3E}{\isasymin}}\ mc\ f{\isaliteral{7D}{\isacharbraceright}}{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{7C}{\isacharbar}}\isanewline
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{\isaliteral{22}{\isachardoublequoteopen}}mc{\isaliteral{28}{\isacharparenleft}}EF\ f{\isaliteral{29}{\isacharparenright}}\ \ \ \ {\isaliteral{3D}{\isacharequal}}\ lfp{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}T{\isaliteral{2E}{\isachardot}}\ mc\ f\ {\isaliteral{5C3C756E696F6E3E}{\isasymunion}}\ {\isaliteral{28}{\isacharparenleft}}M{\isaliteral{5C3C696E76657273653E}{\isasyminverse}}\ {\isaliteral{60}{\isacharbackquote}}{\isaliteral{60}{\isacharbackquote}}\ T{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}%
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\begin{isamarkuptext}%
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\noindent
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Only the equation for \isa{EF} deserves some comments. Remember that the
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postfix \isa{{\isaliteral{5C3C696E76657273653E}{\isasyminverse}}} and the infix \isa{{\isaliteral{60}{\isacharbackquote}}{\isaliteral{60}{\isacharbackquote}}} are predefined and denote the
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converse of a relation and the image of a set under a relation. Thus
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\isa{M{\isaliteral{5C3C696E76657273653E}{\isasyminverse}}\ {\isaliteral{60}{\isacharbackquote}}{\isaliteral{60}{\isacharbackquote}}\ T} is the set of all predecessors of \isa{T} and the least
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fixed point (\isa{lfp}) of \isa{{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}T{\isaliteral{2E}{\isachardot}}\ mc\ f\ {\isaliteral{5C3C756E696F6E3E}{\isasymunion}}\ M{\isaliteral{5C3C696E76657273653E}{\isasyminverse}}\ {\isaliteral{60}{\isacharbackquote}}{\isaliteral{60}{\isacharbackquote}}\ T} is the least set
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\isa{T} containing \isa{mc\ f} and all predecessors of \isa{T}. If you
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find it hard to see that \isa{mc\ {\isaliteral{28}{\isacharparenleft}}EF\ f{\isaliteral{29}{\isacharparenright}}} contains exactly those states from
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which there is a path to a state where \isa{f} is true, do not worry --- this
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will be proved in a moment.
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First we prove monotonicity of the function inside \isa{lfp}
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in order to make sure it really has a least fixed point.%
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\end{isamarkuptext}%
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\isamarkuptrue%
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\isacommand{lemma}\isamarkupfalse%
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\ mono{\isaliteral{5F}{\isacharunderscore}}ef{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}mono{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}T{\isaliteral{2E}{\isachardot}}\ A\ {\isaliteral{5C3C756E696F6E3E}{\isasymunion}}\ {\isaliteral{28}{\isacharparenleft}}M{\isaliteral{5C3C696E76657273653E}{\isasyminverse}}\ {\isaliteral{60}{\isacharbackquote}}{\isaliteral{60}{\isacharbackquote}}\ T{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
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%
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\isadelimproof
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%
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\endisadelimproof
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%
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\isatagproof
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\isacommand{apply}\isamarkupfalse%
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{\isaliteral{28}{\isacharparenleft}}rule\ monoI{\isaliteral{29}{\isacharparenright}}\isanewline
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\isacommand{apply}\isamarkupfalse%
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\ blast\isanewline
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\isacommand{done}\isamarkupfalse%
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%
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\endisatagproof
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{\isafoldproof}%
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%
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\isadelimproof
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%
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\endisadelimproof
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%
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\begin{isamarkuptext}%
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\noindent
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Now we can relate model checking and semantics. For the \isa{EF} case we need
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a separate lemma:%
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\end{isamarkuptext}%
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\isamarkuptrue%
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\isacommand{lemma}\isamarkupfalse%
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\ EF{\isaliteral{5F}{\isacharunderscore}}lemma{\isaliteral{3A}{\isacharcolon}}\isanewline
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\ \ {\isaliteral{22}{\isachardoublequoteopen}}lfp{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}T{\isaliteral{2E}{\isachardot}}\ A\ {\isaliteral{5C3C756E696F6E3E}{\isasymunion}}\ {\isaliteral{28}{\isacharparenleft}}M{\isaliteral{5C3C696E76657273653E}{\isasyminverse}}\ {\isaliteral{60}{\isacharbackquote}}{\isaliteral{60}{\isacharbackquote}}\ T{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{7B}{\isacharbraceleft}}s{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C6578697374733E}{\isasymexists}}t{\isaliteral{2E}{\isachardot}}\ {\isaliteral{28}{\isacharparenleft}}s{\isaliteral{2C}{\isacharcomma}}t{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C696E3E}{\isasymin}}\ M\isaliteral{5C3C5E7375703E}{}\isactrlsup {\isaliteral{2A}{\isacharasterisk}}\ {\isaliteral{5C3C616E643E}{\isasymand}}\ t\ {\isaliteral{5C3C696E3E}{\isasymin}}\ A{\isaliteral{7D}{\isacharbraceright}}{\isaliteral{22}{\isachardoublequoteclose}}%
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\isadelimproof
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%
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\endisadelimproof
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%
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\isatagproof
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%
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\begin{isamarkuptxt}%
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\noindent
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The equality is proved in the canonical fashion by proving that each set
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includes the other; the inclusion is shown pointwise:%
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\end{isamarkuptxt}%
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\isamarkuptrue%
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\isacommand{apply}\isamarkupfalse%
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{\isaliteral{28}{\isacharparenleft}}rule\ equalityI{\isaliteral{29}{\isacharparenright}}\isanewline
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\ \isacommand{apply}\isamarkupfalse%
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{\isaliteral{28}{\isacharparenleft}}rule\ subsetI{\isaliteral{29}{\isacharparenright}}\isanewline
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\ \isacommand{apply}\isamarkupfalse%
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{\isaliteral{28}{\isacharparenleft}}simp{\isaliteral{29}{\isacharparenright}}%
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\begin{isamarkuptxt}%
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\noindent
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Simplification leaves us with the following first subgoal
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\begin{isabelle}%
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\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}s{\isaliteral{2E}{\isachardot}}\ s\ {\isaliteral{5C3C696E3E}{\isasymin}}\ lfp\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}T{\isaliteral{2E}{\isachardot}}\ A\ {\isaliteral{5C3C756E696F6E3E}{\isasymunion}}\ M{\isaliteral{5C3C696E76657273653E}{\isasyminverse}}\ {\isaliteral{60}{\isacharbackquote}}{\isaliteral{60}{\isacharbackquote}}\ T{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{5C3C6578697374733E}{\isasymexists}}t{\isaliteral{2E}{\isachardot}}\ {\isaliteral{28}{\isacharparenleft}}s{\isaliteral{2C}{\isacharcomma}}\ t{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C696E3E}{\isasymin}}\ M\isaliteral{5C3C5E7375703E}{}\isactrlsup {\isaliteral{2A}{\isacharasterisk}}\ {\isaliteral{5C3C616E643E}{\isasymand}}\ t\ {\isaliteral{5C3C696E3E}{\isasymin}}\ A%
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\end{isabelle}
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which is proved by \isa{lfp}-induction:%
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\end{isamarkuptxt}%
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\isamarkuptrue%
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\ \isacommand{apply}\isamarkupfalse%
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{\isaliteral{28}{\isacharparenleft}}erule\ lfp{\isaliteral{5F}{\isacharunderscore}}induct{\isaliteral{5F}{\isacharunderscore}}set{\isaliteral{29}{\isacharparenright}}\isanewline
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\ \ \isacommand{apply}\isamarkupfalse%
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{\isaliteral{28}{\isacharparenleft}}rule\ mono{\isaliteral{5F}{\isacharunderscore}}ef{\isaliteral{29}{\isacharparenright}}\isanewline
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\ \isacommand{apply}\isamarkupfalse%
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{\isaliteral{28}{\isacharparenleft}}simp{\isaliteral{29}{\isacharparenright}}%
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\begin{isamarkuptxt}%
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\noindent
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Having disposed of the monotonicity subgoal,
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simplification leaves us with the following goal:
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\begin{isabelle}
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\ {\isadigit{1}}{\isachardot}\ {\isasymAnd}x{\isachardot}\ x\ {\isasymin}\ A\ {\isasymor}\isanewline
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\ \ \ \ \ \ \ \ \ x\ {\isasymin}\ M{\isasyminverse}\ {\isacharbackquote}{\isacharbackquote}\ {\isacharparenleft}lfp\ {\isacharparenleft}\dots{\isacharparenright}\ {\isasyminter}\ {\isacharbraceleft}x{\isachardot}\ {\isasymexists}t{\isachardot}\ {\isacharparenleft}x{\isacharcomma}\ t{\isacharparenright}\ {\isasymin}\ M\isactrlsup {\isacharasterisk}\ {\isasymand}\ t\ {\isasymin}\ A{\isacharbraceright}{\isacharparenright}\isanewline
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\ \ \ \ \ \ \ \ {\isasymLongrightarrow}\ {\isasymexists}t{\isachardot}\ {\isacharparenleft}x{\isacharcomma}\ t{\isacharparenright}\ {\isasymin}\ M\isactrlsup {\isacharasterisk}\ {\isasymand}\ t\ {\isasymin}\ A
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\end{isabelle}
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It is proved by \isa{blast}, using the transitivity of
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\isa{M\isactrlsup {\isacharasterisk}}.%
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\end{isamarkuptxt}%
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\isamarkuptrue%
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\ \isacommand{apply}\isamarkupfalse%
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{\isaliteral{28}{\isacharparenleft}}blast\ intro{\isaliteral{3A}{\isacharcolon}}\ rtrancl{\isaliteral{5F}{\isacharunderscore}}trans{\isaliteral{29}{\isacharparenright}}%
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\begin{isamarkuptxt}%
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We now return to the second set inclusion subgoal, which is again proved
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pointwise:%
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\end{isamarkuptxt}%
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\isamarkuptrue%
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\isacommand{apply}\isamarkupfalse%
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{\isaliteral{28}{\isacharparenleft}}rule\ subsetI{\isaliteral{29}{\isacharparenright}}\isanewline
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\isacommand{apply}\isamarkupfalse%
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{\isaliteral{28}{\isacharparenleft}}simp{\isaliteral{2C}{\isacharcomma}}\ clarify{\isaliteral{29}{\isacharparenright}}%
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\begin{isamarkuptxt}%
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\noindent
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After simplification and clarification we are left with
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\begin{isabelle}%
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\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}x\ t{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}{\isaliteral{28}{\isacharparenleft}}x{\isaliteral{2C}{\isacharcomma}}\ t{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C696E3E}{\isasymin}}\ M\isaliteral{5C3C5E7375703E}{}\isactrlsup {\isaliteral{2A}{\isacharasterisk}}{\isaliteral{3B}{\isacharsemicolon}}\ t\ {\isaliteral{5C3C696E3E}{\isasymin}}\ A{\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ x\ {\isaliteral{5C3C696E3E}{\isasymin}}\ lfp\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}T{\isaliteral{2E}{\isachardot}}\ A\ {\isaliteral{5C3C756E696F6E3E}{\isasymunion}}\ M{\isaliteral{5C3C696E76657273653E}{\isasyminverse}}\ {\isaliteral{60}{\isacharbackquote}}{\isaliteral{60}{\isacharbackquote}}\ T{\isaliteral{29}{\isacharparenright}}%
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\end{isabelle}
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This goal is proved by induction on \isa{{\isaliteral{28}{\isacharparenleft}}s{\isaliteral{2C}{\isacharcomma}}\ t{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C696E3E}{\isasymin}}\ M\isaliteral{5C3C5E7375703E}{}\isactrlsup {\isaliteral{2A}{\isacharasterisk}}}. But since the model
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checker works backwards (from \isa{t} to \isa{s}), we cannot use the
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induction theorem \isa{rtrancl{\isaliteral{5F}{\isacharunderscore}}induct}: it works in the
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forward direction. Fortunately the converse induction theorem
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\isa{converse{\isaliteral{5F}{\isacharunderscore}}rtrancl{\isaliteral{5F}{\isacharunderscore}}induct} already exists:
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\begin{isabelle}%
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\ \ \ \ \ {\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}{\isaliteral{28}{\isacharparenleft}}a{\isaliteral{2C}{\isacharcomma}}\ b{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C696E3E}{\isasymin}}\ r\isaliteral{5C3C5E7375703E}{}\isactrlsup {\isaliteral{2A}{\isacharasterisk}}{\isaliteral{3B}{\isacharsemicolon}}\ P\ b{\isaliteral{3B}{\isacharsemicolon}}\isanewline
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\isaindent{\ \ \ \ \ \ }{\isaliteral{5C3C416E643E}{\isasymAnd}}y\ z{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}{\isaliteral{28}{\isacharparenleft}}y{\isaliteral{2C}{\isacharcomma}}\ z{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C696E3E}{\isasymin}}\ r{\isaliteral{3B}{\isacharsemicolon}}\ {\isaliteral{28}{\isacharparenleft}}z{\isaliteral{2C}{\isacharcomma}}\ b{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C696E3E}{\isasymin}}\ r\isaliteral{5C3C5E7375703E}{}\isactrlsup {\isaliteral{2A}{\isacharasterisk}}{\isaliteral{3B}{\isacharsemicolon}}\ P\ z{\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ P\ y{\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\isanewline
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\isaindent{\ \ \ \ \ }{\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ P\ a%
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\end{isabelle}
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It says that if \isa{{\isaliteral{28}{\isacharparenleft}}a{\isaliteral{2C}{\isacharcomma}}\ b{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C696E3E}{\isasymin}}\ r\isaliteral{5C3C5E7375703E}{}\isactrlsup {\isaliteral{2A}{\isacharasterisk}}} and we know \isa{P\ b} then we can infer
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\isa{P\ a} provided each step backwards from a predecessor \isa{z} of
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\isa{b} preserves \isa{P}.%
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\end{isamarkuptxt}%
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\isamarkuptrue%
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\isacommand{apply}\isamarkupfalse%
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{\isaliteral{28}{\isacharparenleft}}erule\ converse{\isaliteral{5F}{\isacharunderscore}}rtrancl{\isaliteral{5F}{\isacharunderscore}}induct{\isaliteral{29}{\isacharparenright}}%
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|
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\begin{isamarkuptxt}%
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|
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\noindent
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|
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The base case
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\begin{isabelle}%
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|
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\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}x\ t{\isaliteral{2E}{\isachardot}}\ t\ {\isaliteral{5C3C696E3E}{\isasymin}}\ A\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ t\ {\isaliteral{5C3C696E3E}{\isasymin}}\ lfp\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}T{\isaliteral{2E}{\isachardot}}\ A\ {\isaliteral{5C3C756E696F6E3E}{\isasymunion}}\ M{\isaliteral{5C3C696E76657273653E}{\isasyminverse}}\ {\isaliteral{60}{\isacharbackquote}}{\isaliteral{60}{\isacharbackquote}}\ T{\isaliteral{29}{\isacharparenright}}%
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\end{isabelle}
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|
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is solved by unrolling \isa{lfp} once%
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\end{isamarkuptxt}%
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\isamarkuptrue%
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\ \isacommand{apply}\isamarkupfalse%
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{\isaliteral{28}{\isacharparenleft}}subst\ lfp{\isaliteral{5F}{\isacharunderscore}}unfold{\isaliteral{5B}{\isacharbrackleft}}OF\ mono{\isaliteral{5F}{\isacharunderscore}}ef{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{29}{\isacharparenright}}%
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|
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\begin{isamarkuptxt}%
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\begin{isabelle}%
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|
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\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}x\ t{\isaliteral{2E}{\isachardot}}\ t\ {\isaliteral{5C3C696E3E}{\isasymin}}\ A\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ t\ {\isaliteral{5C3C696E3E}{\isasymin}}\ A\ {\isaliteral{5C3C756E696F6E3E}{\isasymunion}}\ M{\isaliteral{5C3C696E76657273653E}{\isasyminverse}}\ {\isaliteral{60}{\isacharbackquote}}{\isaliteral{60}{\isacharbackquote}}\ lfp\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}T{\isaliteral{2E}{\isachardot}}\ A\ {\isaliteral{5C3C756E696F6E3E}{\isasymunion}}\ M{\isaliteral{5C3C696E76657273653E}{\isasyminverse}}\ {\isaliteral{60}{\isacharbackquote}}{\isaliteral{60}{\isacharbackquote}}\ T{\isaliteral{29}{\isacharparenright}}%
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\end{isabelle}
|
|
217 |
and disposing of the resulting trivial subgoal automatically:%
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|
218 |
\end{isamarkuptxt}%
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|
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\isamarkuptrue%
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|
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\ \isacommand{apply}\isamarkupfalse%
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|
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{\isaliteral{28}{\isacharparenleft}}blast{\isaliteral{29}{\isacharparenright}}%
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|
222 |
\begin{isamarkuptxt}%
|
|
223 |
\noindent
|
|
224 |
The proof of the induction step is identical to the one for the base case:%
|
|
225 |
\end{isamarkuptxt}%
|
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|
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\isamarkuptrue%
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|
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\isacommand{apply}\isamarkupfalse%
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|
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{\isaliteral{28}{\isacharparenleft}}subst\ lfp{\isaliteral{5F}{\isacharunderscore}}unfold{\isaliteral{5B}{\isacharbrackleft}}OF\ mono{\isaliteral{5F}{\isacharunderscore}}ef{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{29}{\isacharparenright}}\isanewline
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|
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\isacommand{apply}\isamarkupfalse%
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|
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{\isaliteral{28}{\isacharparenleft}}blast{\isaliteral{29}{\isacharparenright}}\isanewline
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|
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\isacommand{done}\isamarkupfalse%
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|
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%
|
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|
233 |
\endisatagproof
|
|
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{\isafoldproof}%
|
|
235 |
%
|
|
236 |
\isadelimproof
|
|
237 |
%
|
|
238 |
\endisadelimproof
|
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|
239 |
%
|
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|
240 |
\begin{isamarkuptext}%
|
|
241 |
The main theorem is proved in the familiar manner: induction followed by
|
|
242 |
\isa{auto} augmented with the lemma as a simplification rule.%
|
|
243 |
\end{isamarkuptext}%
|
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|
244 |
\isamarkuptrue%
|
|
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\isacommand{theorem}\isamarkupfalse%
|
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|
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\ {\isaliteral{22}{\isachardoublequoteopen}}mc\ f\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{7B}{\isacharbraceleft}}s{\isaliteral{2E}{\isachardot}}\ s\ {\isaliteral{5C3C5475726E7374696C653E}{\isasymTurnstile}}\ f{\isaliteral{7D}{\isacharbraceright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
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|
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%
|
|
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\isadelimproof
|
|
249 |
%
|
|
250 |
\endisadelimproof
|
|
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%
|
|
252 |
\isatagproof
|
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|
253 |
\isacommand{apply}\isamarkupfalse%
|
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|
254 |
{\isaliteral{28}{\isacharparenleft}}induct{\isaliteral{5F}{\isacharunderscore}}tac\ f{\isaliteral{29}{\isacharparenright}}\isanewline
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|
255 |
\isacommand{apply}\isamarkupfalse%
|
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|
256 |
{\isaliteral{28}{\isacharparenleft}}auto\ simp\ add{\isaliteral{3A}{\isacharcolon}}\ EF{\isaliteral{5F}{\isacharunderscore}}lemma{\isaliteral{29}{\isacharparenright}}\isanewline
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|
257 |
\isacommand{done}\isamarkupfalse%
|
|
258 |
%
|
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|
259 |
\endisatagproof
|
|
260 |
{\isafoldproof}%
|
|
261 |
%
|
|
262 |
\isadelimproof
|
|
263 |
%
|
|
264 |
\endisadelimproof
|
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|
265 |
%
|
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|
266 |
\begin{isamarkuptext}%
|
|
267 |
\begin{exercise}
|
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|
268 |
\isa{AX} has a dual operator \isa{EN}
|
|
269 |
(``there exists a next state such that'')%
|
|
270 |
\footnote{We cannot use the customary \isa{EX}: it is reserved
|
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|
271 |
as the \textsc{ascii}-equivalent of \isa{{\isaliteral{5C3C6578697374733E}{\isasymexists}}}.}
|
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|
272 |
with the intended semantics
|
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|
273 |
\begin{isabelle}%
|
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|
274 |
\ \ \ \ \ s\ {\isaliteral{5C3C5475726E7374696C653E}{\isasymTurnstile}}\ EN\ f\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6578697374733E}{\isasymexists}}t{\isaliteral{2E}{\isachardot}}\ {\isaliteral{28}{\isacharparenleft}}s{\isaliteral{2C}{\isacharcomma}}\ t{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C696E3E}{\isasymin}}\ M\ {\isaliteral{5C3C616E643E}{\isasymand}}\ t\ {\isaliteral{5C3C5475726E7374696C653E}{\isasymTurnstile}}\ f{\isaliteral{29}{\isacharparenright}}%
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|
275 |
\end{isabelle}
|
|
276 |
Fortunately, \isa{EN\ f} can already be expressed as a PDL formula. How?
|
|
277 |
|
|
278 |
Show that the semantics for \isa{EF} satisfies the following recursion equation:
|
|
279 |
\begin{isabelle}%
|
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|
280 |
\ \ \ \ \ s\ {\isaliteral{5C3C5475726E7374696C653E}{\isasymTurnstile}}\ EF\ f\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}s\ {\isaliteral{5C3C5475726E7374696C653E}{\isasymTurnstile}}\ f\ {\isaliteral{5C3C6F723E}{\isasymor}}\ s\ {\isaliteral{5C3C5475726E7374696C653E}{\isasymTurnstile}}\ EN\ {\isaliteral{28}{\isacharparenleft}}EF\ f{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}%
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|
281 |
\end{isabelle}
|
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|
282 |
\end{exercise}
|
|
283 |
\index{PDL|)}%
|
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|
284 |
\end{isamarkuptext}%
|
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|
285 |
\isamarkuptrue%
|
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|
286 |
%
|
|
287 |
\isadelimproof
|
|
288 |
%
|
|
289 |
\endisadelimproof
|
|
290 |
%
|
|
291 |
\isatagproof
|
|
292 |
%
|
|
293 |
\endisatagproof
|
|
294 |
{\isafoldproof}%
|
|
295 |
%
|
|
296 |
\isadelimproof
|
|
297 |
%
|
|
298 |
\endisadelimproof
|
|
299 |
%
|
|
300 |
\isadelimproof
|
|
301 |
%
|
|
302 |
\endisadelimproof
|
|
303 |
%
|
|
304 |
\isatagproof
|
|
305 |
%
|
|
306 |
\endisatagproof
|
|
307 |
{\isafoldproof}%
|
|
308 |
%
|
|
309 |
\isadelimproof
|
|
310 |
%
|
|
311 |
\endisadelimproof
|
|
312 |
%
|
|
313 |
\isadelimproof
|
|
314 |
%
|
|
315 |
\endisadelimproof
|
|
316 |
%
|
|
317 |
\isatagproof
|
|
318 |
%
|
|
319 |
\endisatagproof
|
|
320 |
{\isafoldproof}%
|
|
321 |
%
|
|
322 |
\isadelimproof
|
|
323 |
%
|
|
324 |
\endisadelimproof
|
|
325 |
%
|
|
326 |
\isadelimtheory
|
|
327 |
%
|
|
328 |
\endisadelimtheory
|
|
329 |
%
|
|
330 |
\isatagtheory
|
|
331 |
%
|
|
332 |
\endisatagtheory
|
|
333 |
{\isafoldtheory}%
|
|
334 |
%
|
|
335 |
\isadelimtheory
|
|
336 |
%
|
|
337 |
\endisadelimtheory
|
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|
338 |
\end{isabellebody}%
|
10123
|
339 |
%%% Local Variables:
|
|
340 |
%%% mode: latex
|
|
341 |
%%% TeX-master: "root"
|
|
342 |
%%% End:
|