doc-src/TutorialI/CTL/PDL.thy

 propositional connectives of 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:
 *}
 
-datatype pdl_form = Atom atom
-                  | NOT pdl_form
-                  | And pdl_form pdl_form
-                  | AX pdl_form
-                  | EF pdl_form;
+datatype formula = Atom atom
+                  | Neg formula
+                  | And formula formula
+                  | AX formula
+                  | EF formula
 
 text{*\noindent
+This is almost the same as in the boolean expression case study in
+\S\ref{sec:boolex}, except that what used to be called @{text Var} is now
+called @{term Atom}.
+
 The meaning of these formulae is given by saying which formula is true in
 which state:
 *}
 
-consts valid :: "state \<Rightarrow> pdl_form \<Rightarrow> bool" ("(_ \<Turnstile> _)" [80,80] 80)
+consts valid :: "state \<Rightarrow> formula \<Rightarrow> bool"   ("(_ \<Turnstile> _)" [80,80] 80)
+
+text{*\noindent
+The concrete syntax annotation allows us to write @{term"s \<Turnstile> f"} instead of
+@{text"valid s f"}.
+
+The definition of @{text"\<Turnstile>"} is by recursion over the syntax:
+*}
 
 primrec
 "s \<Turnstile> Atom a  = (a \<in> L s)"
-"s \<Turnstile> NOT f   = (\<not>(s \<Turnstile> f))"
+"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^* \<and> t \<Turnstile> f)";
 
-text{*
+text{*\noindent
+The first three equations should be self-explanatory. The temporal formula
+@{term"AX f"} means that @{term f} is true in all next states whereas
+@{term"EF f"} means that there exists some future state in which @{term f} is
+true. The future is expressed via @{text"^*"}, the transitive reflexive
+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 and is defined by recursion over the syntax:
+states where the formula is true and is defined by recursion over the syntax,
+too:
 *}
 
-consts mc :: "pdl_form \<Rightarrow> state set";
+consts mc :: "formula \<Rightarrow> state set";
 primrec
 "mc(Atom a)  = {s. a \<in> L s}"
-"mc(NOT f)   = -mc f"
+"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^-1 ^^ T)"
 
 
-text{*
-Only the equation for @{term EF} deserves a comment: the postfix @{text"^-1"}
-and the infix @{text"^^"} are predefined and denote the converse of a
-relation and the application of a relation to a set. Thus @{term "M^-1 ^^ T"}
-is the set of all predecessors of @{term T} and @{term"mc(EF f)"} is the least
-set @{term T} containing @{term"mc f"} and all predecessors of @{term T}.
+text{*\noindent
+Only the equation for @{term EF} deserves some comments. Remember that the
+postfix @{text"^-1"} and the infix @{text"^^"} are predefined and denote the
+converse of a relation and the application of a relation to a set. Thus
+@{term "M^-1 ^^ T"} is the set of all predecessors of @{term T} and the least
+fixpoint (@{term lfp}) of @{term"\<lambda>T. mc f \<union> M^-1 ^^ 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---that
+will be proved in a moment.
+
+First we prove monotonicity of the function inside @{term lfp}
 *}
 
-lemma mono_lemma: "mono(\<lambda>T. A \<union> M^-1 ^^ T)"
-apply(rule monoI);
-by(blast);
+lemma mono_ef: "mono(\<lambda>T. A \<union> M^-1 ^^ T)"
+apply(rule monoI)
+by(blast)
 
-lemma lfp_conv_EF:
-"lfp(\<lambda>T. A \<union> M^-1 ^^ T) = {s. \<exists>t. (s,t) \<in> M^* \<and> t \<in> A}";
+text{*\noindent
+in order to make sure it really has a least fixpoint.
+
+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^-1 ^^ T) = {s. \<exists>t. (s,t) \<in> M^* \<and> t \<in> A}"
+
+txt{*\noindent
+The equality is proved in the canonical fashion by proving that each set
+contains the other; the containment is shown pointwise:
+*}
+
 apply(rule equalityI);
  apply(rule subsetI);
- apply(simp);
- apply(erule Lfp.induct);
-  apply(rule mono_lemma);
- apply(simp);
+ apply(simp)
+(*pr(latex xsymbols symbols);*)
+txt{*\noindent
+Simplification leaves us with the following first subgoal
+\begin{isabelle}
+\ \isadigit{1}{\isachardot}\ {\isasymAnd}s{\isachardot}\ s\ {\isasymin}\ lfp\ {\isacharparenleft}{\isasymlambda}T{\isachardot}\ A\ {\isasymunion}\ M{\isacharcircum}{\isacharminus}\isadigit{1}\ {\isacharcircum}{\isacharcircum}\ T{\isacharparenright}\ {\isasymLongrightarrow}\ {\isasymexists}t{\isachardot}\ {\isacharparenleft}s{\isacharcomma}\ t{\isacharparenright}\ {\isasymin}\ M{\isacharcircum}{\isacharasterisk}\ {\isasymand}\ t\ {\isasymin}\ A
+\end{isabelle}
+which is proved by @{term lfp}-induction:
+*}
+
+ apply(erule Lfp.induct)
+  apply(rule mono_ef)
+ apply(simp)
+(*pr(latex xsymbols symbols);*)
+txt{*\noindent
+Having disposed of the monotonicity subgoal,
+simplification leaves us with the following main goal
+\begin{isabelle}
+\ \isadigit{1}{\isachardot}\ {\isasymAnd}s{\isachardot}\ s\ {\isasymin}\ A\ {\isasymor}\isanewline
+\ \ \ \ \ \ \ \ \ 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
+\ \ \ \ \ \ \ \ \ {\isasymLongrightarrow}\ {\isasymexists}t{\isachardot}\ {\isacharparenleft}s{\isacharcomma}\ t{\isacharparenright}\ {\isasymin}\ M{\isacharcircum}{\isacharasterisk}\ {\isasymand}\ t\ {\isasymin}\ A
+\end{isabelle}
+which is proved by @{text blast} with the help of a few lemmas about
+@{text"^*"}:
+*}
+
  apply(blast intro: r_into_rtrancl rtrancl_trans);
-apply(rule subsetI);
-apply(simp);
-apply(erule exE);
-apply(erule conjE);
-apply(erule_tac P = "t\<in>A" in rev_mp);
-apply(erule converse_rtrancl_induct);
- apply(rule ssubst[OF lfp_Tarski[OF mono_lemma]]);
- apply(blast);
-apply(rule ssubst[OF lfp_Tarski[OF mono_lemma]]);
-by(blast);
+
+txt{*
+We now return to the second set containment subgoal, which is again proved
+pointwise:
+*}
+
+apply(rule subsetI)
+apply(simp, clarify)
+(*pr(latex xsymbols symbols);*)
+txt{*\noindent
+After simplification and clarification we are left with
+\begin{isabelle}
+\ \isadigit{1}{\isachardot}\ {\isasymAnd}s\ t{\isachardot}\ {\isasymlbrakk}{\isacharparenleft}s{\isacharcomma}\ t{\isacharparenright}\ {\isasymin}\ M{\isacharcircum}{\isacharasterisk}{\isacharsemicolon}\ t\ {\isasymin}\ A{\isasymrbrakk}\ {\isasymLongrightarrow}\ s\ {\isasymin}\ lfp\ {\isacharparenleft}{\isasymlambda}T{\isachardot}\ A\ {\isasymunion}\ M{\isacharcircum}{\isacharminus}\isadigit{1}\ {\isacharcircum}{\isacharcircum}\ T{\isacharparenright}
+\end{isabelle}
+This goal is proved by induction on @{term"(s,t)\<in>M^*"}. But since the model
+checker works backwards (from @{term t} to @{term s}), we cannot use the
+induction theorem @{thm[source]rtrancl_induct} because 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^*"} 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)
+(*pr(latex xsymbols symbols);*)
+txt{*\noindent
+The base case
+\begin{isabelle}
+\ \isadigit{1}{\isachardot}\ {\isasymAnd}t{\isachardot}\ t\ {\isasymin}\ A\ {\isasymLongrightarrow}\ t\ {\isasymin}\ lfp\ {\isacharparenleft}{\isasymlambda}T{\isachardot}\ A\ {\isasymunion}\ M{\isacharcircum}{\isacharminus}\isadigit{1}\ {\isacharcircum}{\isacharcircum}\ T{\isacharparenright}
+\end{isabelle}
+is solved by unrolling @{term lfp} once
+*}
+
+ apply(rule ssubst[OF lfp_Tarski[OF mono_ef]])
+(*pr(latex xsymbols symbols);*)
+txt{*
+\begin{isabelle}
+\ \isadigit{1}{\isachardot}\ {\isasymAnd}t{\isachardot}\ t\ {\isasymin}\ A\ {\isasymLongrightarrow}\ t\ {\isasymin}\ A\ {\isasymunion}\ M{\isacharcircum}{\isacharminus}\isadigit{1}\ {\isacharcircum}{\isacharcircum}\ lfp\ {\isacharparenleft}{\isasymlambda}T{\isachardot}\ A\ {\isasymunion}\ M{\isacharcircum}{\isacharminus}\isadigit{1}\ {\isacharcircum}{\isacharcircum}\ T{\isacharparenright}
+\end{isabelle}
+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(rule ssubst[OF lfp_Tarski[OF mono_ef]])
+by(blast)
+
+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);
-by(auto simp add:lfp_conv_EF);
+by(auto simp add:EF_lemma);
 (*<*)end(*>*)