(*<*)theory CTLind = CTL:(*>*)

 subsection{*CTL revisited*}

 text{*\label{sec:CTL-revisited}

 The purpose of this section is twofold: we want to demonstrate

 some of the induction principles and heuristics discussed above and we want to

 show how inductive definitions can simplify proofs.

 In \S\ref{sec:CTL} we gave a fairly involved proof of the correctness of a

 model checker for CTL. In particular the proof of the

 @{thm[source]infinity_lemma} on the way to @{thm[source]AF_lemma2} is not as

 simple as one might intuitively expect, due to the @{text SOME} operator

 involved. Below we give a simpler proof of @{thm[source]AF_lemma2}

 based on an auxiliary inductive definition.

 Let us call a (finite or infinite) path \emph{@{term A}-avoiding} if it does

 not touch any node in the set @{term A}. Then @{thm[source]AF_lemma2} says

 that if no infinite path from some state @{term s} is @{term A}-avoiding,

 then @{prop"s \<in> lfp(af A)"}. We prove this by inductively defining the set

 @{term"Avoid s A"} of states reachable from @{term s} by a finite @{term

 A}-avoiding path:

 % Second proof of opposite direction, directly by well-founded induction

 % on the initial segment of M that avoids A.

 *}

 consts Avoid :: "state \<Rightarrow> state set \<Rightarrow> state set";

 inductive "Avoid s A"

 intros "s \<in> Avoid s A"

        "\<lbrakk> t \<in> Avoid s A; t \<notin> A; (t,u) \<in> M \<rbrakk> \<Longrightarrow> u \<in> Avoid s A";

 text{*

 It is easy to see that for any infinite @{term A}-avoiding path @{term f}

 with @{prop"f 0 \<in> Avoid s A"} there is an infinite @{term A}-avoiding path

 starting with @{term s} because (by definition of @{term Avoid}) there is a

 finite @{term A}-avoiding path from @{term s} to @{term"f 0"}.

 The proof is by induction on @{prop"f 0 \<in> Avoid s A"}. However,

 this requires the following

 reformulation, as explained in \S\ref{sec:ind-var-in-prems} above;

 the @{text rule_format} directive undoes the reformulation after the proof.

 *}

 lemma ex_infinite_path[rule_format]:

   "t \<in> Avoid s A  \<Longrightarrow>

    \<forall>f\<in>Paths t. (\<forall>i. f i \<notin> A) \<longrightarrow> (\<exists>p\<in>Paths s. \<forall>i. p i \<notin> A)";

 apply(erule Avoid.induct);

  apply(blast);

 apply(clarify);

 apply(drule_tac x = "\<lambda>i. case i of 0 \<Rightarrow> t | Suc i \<Rightarrow> f i" in bspec);

 apply(simp_all add:Paths_def split:nat.split);

 done

 text{*\noindent

 The base case (@{prop"t = s"}) is trivial (@{text blast}).

 In the induction step, we have an infinite @{term A}-avoiding path @{term f}

 starting from @{term u}, a successor of @{term t}. Now we simply instantiate

 the @{text"\<forall>f\<in>Paths t"} in the induction hypothesis by the path starting with

 @{term t} and continuing with @{term f}. That is what the above $\lambda$-term

 expresses. That fact that this is a path starting with @{term t} and that

 the instantiated induction hypothesis implies the conclusion is shown by

 simplification.

 Now we come to the key lemma. It says that if @{term t} can be reached by a

 finite @{term A}-avoiding path from @{term s}, then @{prop"t \<in> lfp(af A)"},

 provided there is no infinite @{term A}-avoiding path starting from @{term

 s}.

 *}

 lemma Avoid_in_lfp[rule_format(no_asm)]:

   "\<forall>p\<in>Paths s. \<exists>i. p i \<in> A \<Longrightarrow> t \<in> Avoid s A \<longrightarrow> t \<in> lfp(af A)";

 txt{*\noindent

 The trick is not to induct on @{prop"t \<in> Avoid s A"}, as already the base

 case would be a problem, but to proceed by well-founded induction @{term

 t}. Hence @{prop"t \<in> Avoid s A"} needs to be brought into the conclusion as

 well, which the directive @{text rule_format} undoes at the end (see below).

 But induction with respect to which well-founded relation? The restriction

 of @{term M} to @{term"Avoid s A"}:

 @{term[display]"{(y,x). (x,y) \<in> M \<and> x \<in> Avoid s A \<and> y \<in> Avoid s A \<and> x \<notin> A}"}

 As we shall see in a moment, the absence of infinite @{term A}-avoiding paths

 starting from @{term s} implies well-foundedness of this relation. For the

 moment we assume this and proceed with the induction:

 *}

 apply(subgoal_tac

   "wf{(y,x). (x,y)\<in>M \<and> x \<in> Avoid s A \<and> y \<in> Avoid s A \<and> x \<notin> A}");

  apply(erule_tac a = t in wf_induct);

  apply(clarsimp);

 txt{*\noindent

 Now can assume additionally (induction hypothesis) that if @{prop"t \<notin> A"}

 then all successors of @{term t} that are in @{term"Avoid s A"} are in

 @{term"lfp (af A)"}. To prove the actual goal we unfold @{term lfp} once. Now

 we have to prove that @{term t} is in @{term A} or all successors of @{term

 t} are in @{term"lfp (af A)"}. If @{term t} is not in @{term A}, the second

 @{term Avoid}-rule implies that all successors of @{term t} are in

 @{term"Avoid s A"} (because we also assume @{prop"t \<in> Avoid s A"}), and

 hence, by the induction hypothesis, all successors of @{term t} are indeed in

 @{term"lfp(af A)"}. Mechanically:

 *}

  apply(rule ssubst [OF lfp_unfold[OF mono_af]]);

  apply(simp only: af_def);

  apply(blast intro:Avoid.intros);

 txt{*

 Having proved the main goal we return to the proof obligation that the above

 relation is indeed well-founded. This is proved by contraposition: we assume

 the relation is not well-founded. Thus there exists an infinite @{term

 A}-avoiding path all in @{term"Avoid s A"}, by theorem

 @{thm[source]wf_iff_no_infinite_down_chain}:

 @{thm[display]wf_iff_no_infinite_down_chain[no_vars]}

 From lemma @{thm[source]ex_infinite_path} the existence of an infinite

 @{term A}-avoiding path starting in @{term s} follows, just as required for

 the contraposition.

 *}

 apply(erule contrapos_pp);

 apply(simp add:wf_iff_no_infinite_down_chain);

 apply(erule exE);

 apply(rule ex_infinite_path);

 apply(auto simp add:Paths_def);

 done

 text{*

 The @{text"(no_asm)"} modifier of the @{text"rule_format"} directive means

 that the assumption is left unchanged---otherwise the @{text"\<forall>p"} is turned

 into a @{text"\<And>p"}, which would complicate matters below. As it is,

 @{thm[source]Avoid_in_lfp} is now

 @{thm[display]Avoid_in_lfp[no_vars]}

 The main theorem is simply the corollary where @{prop"t = s"},

 in which case the assumption @{prop"t \<in> Avoid s A"} is trivially true

 by the first @{term Avoid}-rule). Isabelle confirms this:

 *}

 theorem AF_lemma2:

   "{s. \<forall>p \<in> Paths s. \<exists> i. p i \<in> A} \<subseteq> lfp(af A)";

 by(auto elim:Avoid_in_lfp intro:Avoid.intros);

 (*<*)end(*>*)