(*<*)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(*>*)