doc-src/TutorialI/CTL/PDL.thy
author nipkow
Wed Dec 06 13:22:58 2000 +0100 (2000-12-06)
changeset 10608 620647438780
parent 10524 270b285d48ee
child 10800 2d4c058749a7
permissions -rw-r--r--
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(*<*)theory PDL = Base:(*>*)
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subsection{*Propositional dynamic logic---PDL*}
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text{*\index{PDL|(}
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The formulae of PDL are built up from atomic propositions via the customary
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propositional connectives of negation and conjunction and the two temporal
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connectives @{text AX} and @{text EF}. Since formulae are essentially
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(syntax) trees, they are naturally modelled as a datatype:
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*}
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datatype formula = Atom atom
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                  | Neg formula
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                  | And formula formula
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                  | AX formula
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                  | EF formula
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text{*\noindent
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This is almost the same as in the boolean expression case study in
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\S\ref{sec:boolex}, except that what used to be called @{text Var} is now
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called @{term Atom}.
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The meaning of these formulae is given by saying which formula is true in
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which state:
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*}
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consts valid :: "state \<Rightarrow> formula \<Rightarrow> bool"   ("(_ \<Turnstile> _)" [80,80] 80)
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text{*\noindent
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The concrete syntax annotation allows us to write @{term"s \<Turnstile> f"} instead of
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@{text"valid s f"}.
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The definition of @{text"\<Turnstile>"} is by recursion over the syntax:
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*}
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primrec
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"s \<Turnstile> Atom a  = (a \<in> L s)"
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"s \<Turnstile> Neg f   = (\<not>(s \<Turnstile> f))"
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"s \<Turnstile> And f g = (s \<Turnstile> f \<and> s \<Turnstile> g)"
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"s \<Turnstile> AX f    = (\<forall>t. (s,t) \<in> M \<longrightarrow> t \<Turnstile> f)"
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"s \<Turnstile> EF f    = (\<exists>t. (s,t) \<in> M^* \<and> t \<Turnstile> f)";
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text{*\noindent
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The first three equations should be self-explanatory. The temporal formula
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@{term"AX f"} means that @{term f} is true in all next states whereas
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@{term"EF f"} means that there exists some future state in which @{term f} is
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true. The future is expressed via @{text"^*"}, the transitive reflexive
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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 set of
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states where the formula is true and is defined by recursion over the syntax,
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too:
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*}
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consts mc :: "formula \<Rightarrow> state set";
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primrec
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"mc(Atom a)  = {s. a \<in> L s}"
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"mc(Neg f)   = -mc f"
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"mc(And f g) = mc f \<inter> mc g"
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"mc(AX f)    = {s. \<forall>t. (s,t) \<in> M  \<longrightarrow> t \<in> mc f}"
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"mc(EF f)    = lfp(\<lambda>T. mc f \<union> M^-1 ^^ T)"
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text{*\noindent
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Only the equation for @{term EF} deserves some comments. Remember that the
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postfix @{text"^-1"} and the infix @{text"^^"} are predefined and denote the
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converse of a relation and the application of a relation to a set. Thus
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@{term "M^-1 ^^ T"} is the set of all predecessors of @{term T} and the least
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fixed point (@{term lfp}) of @{term"\<lambda>T. mc f \<union> M^-1 ^^ T"} is the least set
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@{term T} containing @{term"mc f"} and all predecessors of @{term T}. If you
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find it hard to see that @{term"mc(EF f)"} contains exactly those states from
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which there is a path to a state where @{term f} is true, do not worry---that
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will be proved in a moment.
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First we prove monotonicity of the function inside @{term lfp}
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*}
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lemma mono_ef: "mono(\<lambda>T. A \<union> M^-1 ^^ T)"
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apply(rule monoI)
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apply blast
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done
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text{*\noindent
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in order to make sure it really has a least fixed point.
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Now we can relate model checking and semantics. For the @{text EF} case we need
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a separate lemma:
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*}
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lemma EF_lemma:
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  "lfp(\<lambda>T. A \<union> M^-1 ^^ T) = {s. \<exists>t. (s,t) \<in> M^* \<and> t \<in> A}"
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txt{*\noindent
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The equality is proved in the canonical fashion by proving that each set
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contains the other; the containment is shown pointwise:
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*}
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apply(rule equalityI);
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 apply(rule subsetI);
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 apply(simp)(*<*)apply(rename_tac s)(*>*)
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txt{*\noindent
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Simplification leaves us with the following first subgoal
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@{subgoals[display,indent=0,goals_limit=1]}
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which is proved by @{term lfp}-induction:
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*}
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 apply(erule lfp_induct)
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  apply(rule mono_ef)
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 apply(simp)
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(*pr(latex xsymbols symbols);*)
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txt{*\noindent
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Having disposed of the monotonicity subgoal,
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simplification leaves us with the following main goal
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\begin{isabelle}
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\ \isadigit{1}{\isachardot}\ {\isasymAnd}s{\isachardot}\ s\ {\isasymin}\ A\ {\isasymor}\isanewline
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\ \ \ \ \ \ \ \ \ s\ {\isasymin}\ M{\isacharcircum}{\isacharminus}\isadigit{1}\ {\isacharcircum}{\isacharcircum}\ {\isacharparenleft}lfp\ {\isacharparenleft}{\dots}{\isacharparenright}\ {\isasyminter}\ {\isacharbraceleft}x{\isachardot}\ {\isasymexists}t{\isachardot}\ {\isacharparenleft}x{\isacharcomma}\ t{\isacharparenright}\ {\isasymin}\ M{\isacharcircum}{\isacharasterisk}\ {\isasymand}\ t\ {\isasymin}\ A{\isacharbraceright}{\isacharparenright}\isanewline
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\ \ \ \ \ \ \ \ \ {\isasymLongrightarrow}\ {\isasymexists}t{\isachardot}\ {\isacharparenleft}s{\isacharcomma}\ t{\isacharparenright}\ {\isasymin}\ M{\isacharcircum}{\isacharasterisk}\ {\isasymand}\ t\ {\isasymin}\ A
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\end{isabelle}
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which is proved by @{text blast} with the help of transitivity of @{text"^*"}:
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*}
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 apply(blast intro: rtrancl_trans);
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txt{*
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We now return to the second set containment subgoal, which is again proved
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pointwise:
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*}
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apply(rule subsetI)
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apply(simp, clarify)
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txt{*\noindent
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After simplification and clarification we are left with
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@{subgoals[display,indent=0,goals_limit=1]}
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This goal is proved by induction on @{term"(s,t)\<in>M^*"}. But since the model
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checker works backwards (from @{term t} to @{term s}), we cannot use the
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induction theorem @{thm[source]rtrancl_induct} because it works in the
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forward direction. Fortunately the converse induction theorem
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@{thm[source]converse_rtrancl_induct} already exists:
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@{thm[display,margin=60]converse_rtrancl_induct[no_vars]}
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It says that if @{prop"(a,b):r^*"} and we know @{prop"P b"} then we can infer
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@{prop"P a"} provided each step backwards from a predecessor @{term z} of
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@{term b} preserves @{term P}.
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*}
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apply(erule converse_rtrancl_induct)
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txt{*\noindent
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The base case
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@{subgoals[display,indent=0,goals_limit=1]}
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is solved by unrolling @{term lfp} once
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*}
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 apply(rule ssubst[OF lfp_unfold[OF mono_ef]])
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txt{*
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@{subgoals[display,indent=0,goals_limit=1]}
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and disposing of the resulting trivial subgoal automatically:
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*}
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 apply(blast)
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txt{*\noindent
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The proof of the induction step is identical to the one for the base case:
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*}
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apply(rule ssubst[OF lfp_unfold[OF mono_ef]])
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apply(blast)
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done
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text{*
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The main theorem is proved in the familiar manner: induction followed by
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@{text auto} augmented with the lemma as a simplification rule.
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*}
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theorem "mc f = {s. s \<Turnstile> f}";
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apply(induct_tac f);
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apply(auto simp add:EF_lemma);
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done;
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text{*
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\begin{exercise}
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@{term AX} has a dual operator @{term EN}\footnote{We cannot use the customary @{text EX}
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as that is the ASCII equivalent of @{text"\<exists>"}}
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(``there exists a next state such that'') with the intended semantics
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@{prop[display]"(s \<Turnstile> EN f) = (EX t. (s,t) : M & t \<Turnstile> f)"}
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Fortunately, @{term"EN f"} can already be expressed as a PDL formula. How?
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Show that the semantics for @{term EF} satisfies the following recursion equation:
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@{prop[display]"(s \<Turnstile> EF f) = (s \<Turnstile> f | s \<Turnstile> EN(EF f))"}
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\end{exercise}
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\index{PDL|)}
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*}
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(*<*)
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theorem main: "mc f = {s. s \<Turnstile> f}";
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apply(induct_tac f);
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apply(auto simp add:EF_lemma);
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done;
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lemma aux: "s \<Turnstile> f = (s : mc f)";
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apply(simp add:main);
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done;
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lemma "(s \<Turnstile> EF f) = (s \<Turnstile> f | s \<Turnstile> Neg(AX(Neg(EF f))))";
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apply(simp only:aux);
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apply(simp);
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apply(rule ssubst[OF lfp_unfold[OF mono_ef]], fast);
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done
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end
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(*>*)