(*<*)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

 (*>*)