(*<*)theory PDL imports Base begin(*>*)

subsection{*Propositional Dynamic Logic --- PDL*}

text{*\index{PDL|(}
The formulae of PDL are built up from atomic propositions via
negation and conjunction and the two temporal
connectives @{text AX} and @{text EF}\@. Since formulae are essentially
syntax trees, they are naturally modelled as a datatype:%
\footnote{The customary definition of PDL
@{cite "HarelKT-DL"} looks quite different from ours, but the two are easily
shown to be equivalent.}
*}

datatype formula = Atom "atom"
                  | Neg formula
                  | And formula formula
                  | AX formula
                  | EF formula

text{*\noindent
This resembles the boolean expression case study in
\S\ref{sec:boolex}.
A validity relation between states and formulae specifies the semantics.
The syntax annotation allows us to write @{text"s \<Turnstile> f"} instead of
\hbox{@{text"valid s f"}}. The definition is by recursion over the syntax:
*}

primrec valid :: "state \<Rightarrow> formula \<Rightarrow> bool"   ("(_ \<Turnstile> _)" [80,80] 80)
where
"s \<Turnstile> Atom a  = (a \<in> L s)" |
"s \<Turnstile> Neg f   = (\<not>(s \<Turnstile> f))" |
"s \<Turnstile> And f g = (s \<Turnstile> f \<and> s \<Turnstile> g)" |
"s \<Turnstile> AX f    = (\<forall>t. (s,t) \<in> M \<longrightarrow> t \<Turnstile> f)" |
"s \<Turnstile> EF f    = (\<exists>t. (s,t) \<in> M\<^sup>* \<and> t \<Turnstile> f)"

text{*\noindent
The first three equations should be self-explanatory. The temporal formula
@{term"AX f"} means that @{term f} is true in \emph{A}ll ne\emph{X}t states whereas
@{term"EF f"} means that there \emph{E}xists some \emph{F}uture state in which @{term f} is
true. The future is expressed via @{text"\<^sup>*"}, the reflexive transitive
closure. Because of reflexivity, the future includes the present.

Now we come to the model checker itself. It maps a formula into the
set of states where the formula is true.  It too is defined by
recursion over the syntax: *}

primrec mc :: "formula \<Rightarrow> state set" where
"mc(Atom a)  = {s. a \<in> L s}" |
"mc(Neg f)   = -mc f" |
"mc(And f g) = mc f \<inter> mc g" |
"mc(AX f)    = {s. \<forall>t. (s,t) \<in> M  \<longrightarrow> t \<in> mc f}" |
"mc(EF f)    = lfp(\<lambda>T. mc f \<union> (M\<inverse> `` T))"

text{*\noindent
Only the equation for @{term EF} deserves some comments. Remember that the
postfix @{text"\<inverse>"} and the infix @{text"``"} are predefined and denote the
converse of a relation and the image of a set under a relation.  Thus
@{term "M\<inverse> `` T"} is the set of all predecessors of @{term T} and the least
fixed point (@{term lfp}) of @{term"\<lambda>T. mc f \<union> M\<inverse> `` T"} is the least set
@{term T} containing @{term"mc f"} and all predecessors of @{term T}. If you
find it hard to see that @{term"mc(EF f)"} contains exactly those states from
which there is a path to a state where @{term f} is true, do not worry --- this
will be proved in a moment.

First we prove monotonicity of the function inside @{term lfp}
in order to make sure it really has a least fixed point.
*}

lemma mono_ef: "mono(\<lambda>T. A \<union> (M\<inverse> `` T))"
apply(rule monoI)
apply blast
done

text{*\noindent
Now we can relate model checking and semantics. For the @{text EF} case we need
a separate lemma:
*}

lemma EF_lemma:
  "lfp(\<lambda>T. A \<union> (M\<inverse> `` T)) = {s. \<exists>t. (s,t) \<in> M\<^sup>* \<and> t \<in> A}"

txt{*\noindent
The equality is proved in the canonical fashion by proving that each set
includes the other; the inclusion is shown pointwise:
*}

apply(rule equalityI)
 apply(rule subsetI)
 apply(simp)(*<*)apply(rename_tac s)(*>*)

txt{*\noindent
Simplification leaves us with the following first subgoal
@{subgoals[display,indent=0,goals_limit=1]}
which is proved by @{term lfp}-induction:
*}

 apply(erule lfp_induct_set)
  apply(rule mono_ef)
 apply(simp)
(*pr(latex xsymbols symbols);*)
txt{*\noindent
Having disposed of the monotonicity subgoal,
simplification leaves us with the following goal:
\begin{isabelle}
\ {\isadigit{1}}{\isachardot}\ {\isasymAnd}x{\isachardot}\ x\ {\isasymin}\ A\ {\isasymor}\isanewline
\ \ \ \ \ \ \ \ \ 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
\ \ \ \ \ \ \ \ {\isasymLongrightarrow}\ {\isasymexists}t{\isachardot}\ {\isacharparenleft}x{\isacharcomma}\ t{\isacharparenright}\ {\isasymin}\ M\isactrlsup {\isacharasterisk}\ {\isasymand}\ t\ {\isasymin}\ A
\end{isabelle}
It is proved by @{text blast}, using the transitivity of 
\isa{M\isactrlsup {\isacharasterisk}}.
*}

 apply(blast intro: rtrancl_trans)

txt{*
We now return to the second set inclusion subgoal, which is again proved
pointwise:
*}

apply(rule subsetI)
apply(simp, clarify)

txt{*\noindent
After simplification and clarification we are left with
@{subgoals[display,indent=0,goals_limit=1]}
This goal is proved by induction on @{term"(s,t)\<in>M\<^sup>*"}. But since the model
checker works backwards (from @{term t} to @{term s}), we cannot use the
induction theorem @{thm[source]rtrancl_induct}: it works in the
forward direction. Fortunately the converse induction theorem
@{thm[source]converse_rtrancl_induct} already exists:
@{thm[display,margin=60]converse_rtrancl_induct[no_vars]}
It says that if @{prop"(a,b):r\<^sup>*"} and we know @{prop"P b"} then we can infer
@{prop"P a"} provided each step backwards from a predecessor @{term z} of
@{term b} preserves @{term P}.
*}

apply(erule converse_rtrancl_induct)

txt{*\noindent
The base case
@{subgoals[display,indent=0,goals_limit=1]}
is solved by unrolling @{term lfp} once
*}

 apply(subst lfp_unfold[OF mono_ef])

txt{*
@{subgoals[display,indent=0,goals_limit=1]}
and disposing of the resulting trivial subgoal automatically:
*}

 apply(blast)

txt{*\noindent
The proof of the induction step is identical to the one for the base case:
*}

apply(subst lfp_unfold[OF mono_ef])
apply(blast)
done

text{*
The main theorem is proved in the familiar manner: induction followed by
@{text auto} augmented with the lemma as a simplification rule.
*}

theorem "mc f = {s. s \<Turnstile> f}"
apply(induct_tac f)
apply(auto simp add: EF_lemma)
done

text{*
\begin{exercise}
@{term AX} has a dual operator @{term EN} 
(``there exists a next state such that'')%
\footnote{We cannot use the customary @{text EX}: it is reserved
as the \textsc{ascii}-equivalent of @{text"\<exists>"}.}
with the intended semantics
@{prop[display]"(s \<Turnstile> EN f) = (EX t. (s,t) : M & t \<Turnstile> f)"}
Fortunately, @{term"EN f"} can already be expressed as a PDL formula. How?

Show that the semantics for @{term EF} satisfies the following recursion equation:
@{prop[display]"(s \<Turnstile> EF f) = (s \<Turnstile> f | s \<Turnstile> EN(EF f))"}
\end{exercise}
\index{PDL|)}
*}
(*<*)
theorem main: "mc f = {s. s \<Turnstile> f}"
apply(induct_tac f)
apply(auto simp add: EF_lemma)
done

lemma aux: "s \<Turnstile> f = (s : mc f)"
apply(simp add: main)
done

lemma "(s \<Turnstile> EF f) = (s \<Turnstile> f | s \<Turnstile> Neg(AX(Neg(EF f))))"
apply(simp only: aux)
apply(simp)
apply(subst lfp_unfold[OF mono_ef], fast)
done

end
(*>*)
