(*<*)theory CTLind imports CTL begin(*>*)

 subsection{*CTL Revisited*}

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

 \index{CTL|(}%

 The purpose of this section is twofold: to demonstrate

 some of the induction principles and heuristics discussed above and 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 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.

 *}

 inductive_set

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

   for s :: state and A :: "state set"

 where

     "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::nat) \<in> Avoid s A"} there is an infinite @{term A}-avoiding path

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

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

 The proof is by induction on @{prop"f(0::nat) \<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 and proved by @{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.  Simplification shows that this is a path starting with @{term t} 

 and that the instantiated induction hypothesis implies the conclusion.

 Now we come to the key lemma. Assuming that no infinite @{term A}-avoiding

 path starts from @{term s}, we want to show @{prop"s \<in> lfp(af A)"}. For the

 inductive proof this must be generalized to the statement that every point @{term t}

 ``between'' @{term s} and @{term A}, in other words all of @{term"Avoid s A"},

 is contained in @{term"lfp(af A)"}:

 *}

 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 proof is by induction on the ``distance'' between @{term t} and @{term

 A}. Remember that @{prop"lfp(af A) = A \<union> M\<inverse> `` lfp(af A)"}.

 If @{term t} is already in @{term A}, then @{prop"t \<in> lfp(af A)"} is

 trivial. If @{term t} is not in @{term A} but all successors are in

 @{term"lfp(af A)"} (induction hypothesis), then @{prop"t \<in> lfp(af A)"} is

 again trivial.

 The formal counterpart of this proof sketch is a well-founded induction

 on~@{term M} restricted to @{term"Avoid s A - A"}, roughly speaking:

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

 As we shall see presently, 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> x \<notin> A}");

  apply(erule_tac a = t in wf_induct);

  apply(clarsimp);

 (*<*)apply(rename_tac t)(*>*)

 txt{*\noindent

 @{subgoals[display,indent=0,margin=65]}

 Now the induction hypothesis states 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)"}. Unfolding @{term lfp} in the conclusion of the first

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

 of @{term t} are in @{term"lfp (af A)"}.  But if @{term t} is not in @{term A},

 the second 

 @{const 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"}.

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

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

 *}

  apply(subst lfp_unfold[OF mono_af]);

  apply(simp (no_asm) add: af_def);

  apply(blast intro: Avoid.intros);

 txt{*

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

 relation used above is indeed well-founded. This is proved by contradiction: if

 the relation is not well-founded then 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, contradiction.

 *}

 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 in the

 statement of the lemma means

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

 would be 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"},

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

 by the first @{const Avoid}-rule. Isabelle confirms this:%

 \index{CTL|)}*}

 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(*>*)