1 (*<*)theory CTLind = CTL:(*>*)
3 subsection{*CTL Revisited*}
5 text{*\label{sec:CTL-revisited}
6 The purpose of this section is twofold: we want to demonstrate
7 some of the induction principles and heuristics discussed above and we want to
8 show how inductive definitions can simplify proofs.
9 In \S\ref{sec:CTL} we gave a fairly involved proof of the correctness of a
10 model checker for CTL\@. In particular the proof of the
11 @{thm[source]infinity_lemma} on the way to @{thm[source]AF_lemma2} is not as
12 simple as one might intuitively expect, due to the @{text SOME} operator
13 involved. Below we give a simpler proof of @{thm[source]AF_lemma2}
14 based on an auxiliary inductive definition.
16 Let us call a (finite or infinite) path \emph{@{term A}-avoiding} if it does
17 not touch any node in the set @{term A}. Then @{thm[source]AF_lemma2} says
18 that if no infinite path from some state @{term s} is @{term A}-avoiding,
19 then @{prop"s \<in> lfp(af A)"}. We prove this by inductively defining the set
20 @{term"Avoid s A"} of states reachable from @{term s} by a finite @{term
22 % Second proof of opposite direction, directly by well-founded induction
23 % on the initial segment of M that avoids A.
26 consts Avoid :: "state \<Rightarrow> state set \<Rightarrow> state set";
28 intros "s \<in> Avoid s A"
29 "\<lbrakk> t \<in> Avoid s A; t \<notin> A; (t,u) \<in> M \<rbrakk> \<Longrightarrow> u \<in> Avoid s A";
32 It is easy to see that for any infinite @{term A}-avoiding path @{term f}
33 with @{prop"f 0 \<in> Avoid s A"} there is an infinite @{term A}-avoiding path
34 starting with @{term s} because (by definition of @{term Avoid}) there is a
35 finite @{term A}-avoiding path from @{term s} to @{term"f 0"}.
36 The proof is by induction on @{prop"f 0 \<in> Avoid s A"}. However,
37 this requires the following
38 reformulation, as explained in \S\ref{sec:ind-var-in-prems} above;
39 the @{text rule_format} directive undoes the reformulation after the proof.
42 lemma ex_infinite_path[rule_format]:
43 "t \<in> Avoid s A \<Longrightarrow>
44 \<forall>f\<in>Paths t. (\<forall>i. f i \<notin> A) \<longrightarrow> (\<exists>p\<in>Paths s. \<forall>i. p i \<notin> A)";
45 apply(erule Avoid.induct);
48 apply(drule_tac x = "\<lambda>i. case i of 0 \<Rightarrow> t | Suc i \<Rightarrow> f i" in bspec);
49 apply(simp_all add:Paths_def split:nat.split);
53 The base case (@{prop"t = s"}) is trivial (@{text blast}).
54 In the induction step, we have an infinite @{term A}-avoiding path @{term f}
55 starting from @{term u}, a successor of @{term t}. Now we simply instantiate
56 the @{text"\<forall>f\<in>Paths t"} in the induction hypothesis by the path starting with
57 @{term t} and continuing with @{term f}. That is what the above $\lambda$-term
58 expresses. Simplification shows that this is a path starting with @{term t}
59 and that the instantiated induction hypothesis implies the conclusion.
61 Now we come to the key lemma. It says that if @{term t} can be reached by a
62 finite @{term A}-avoiding path from @{term s}, then @{prop"t \<in> lfp(af A)"},
63 provided there is no infinite @{term A}-avoiding path starting from @{term
67 lemma Avoid_in_lfp[rule_format(no_asm)]:
68 "\<forall>p\<in>Paths s. \<exists>i. p i \<in> A \<Longrightarrow> t \<in> Avoid s A \<longrightarrow> t \<in> lfp(af A)";
70 The trick is not to induct on @{prop"t \<in> Avoid s A"}, as even the base
71 case would be a problem, but to proceed by well-founded induction on~@{term
72 t}. Hence\hbox{ @{prop"t \<in> Avoid s A"}} must be brought into the conclusion as
73 well, which the directive @{text rule_format} undoes at the end (see below).
74 But induction with respect to which well-founded relation? The
75 one we want is the restriction
76 of @{term M} to @{term"Avoid s A"}:
77 @{term[display]"{(y,x). (x,y) \<in> M \<and> x \<in> Avoid s A \<and> y \<in> Avoid s A \<and> x \<notin> A}"}
78 As we shall see in a moment, the absence of infinite @{term A}-avoiding paths
79 starting from @{term s} implies well-foundedness of this relation. For the
80 moment we assume this and proceed with the induction:
84 "wf{(y,x). (x,y)\<in>M \<and> x \<in> Avoid s A \<and> y \<in> Avoid s A \<and> x \<notin> A}");
85 apply(erule_tac a = t in wf_induct);
89 @{subgoals[display,indent=0,margin=65]}
90 \REMARK{I put in this proof state but I wonder if readers will be able to
91 follow this proof. You could prove that your relation is WF in a
92 lemma beforehand, maybe omitting that proof.}
93 Now the induction hypothesis states that if @{prop"t \<notin> A"}
94 then all successors of @{term t} that are in @{term"Avoid s A"} are in
95 @{term"lfp (af A)"}. To prove the actual goal we unfold @{term lfp} once. Now
96 we have to prove that @{term t} is in @{term A} or all successors of @{term
97 t} are in @{term"lfp (af A)"}. If @{term t} is not in @{term A}, the second
98 @{term Avoid}-rule implies that all successors of @{term t} are in
99 @{term"Avoid s A"} (because we also assume @{prop"t \<in> Avoid s A"}), and
100 hence, by the induction hypothesis, all successors of @{term t} are indeed in
101 @{term"lfp(af A)"}. Mechanically:
104 apply(rule ssubst [OF lfp_unfold[OF mono_af]]);
105 apply(simp only: af_def);
106 apply(blast intro:Avoid.intros);
109 Having proved the main goal we return to the proof obligation that the above
110 relation is indeed well-founded. This is proved by contradiction: if
111 the relation is not well-founded then there exists an infinite @{term
112 A}-avoiding path all in @{term"Avoid s A"}, by theorem
113 @{thm[source]wf_iff_no_infinite_down_chain}:
114 @{thm[display]wf_iff_no_infinite_down_chain[no_vars]}
115 From lemma @{thm[source]ex_infinite_path} the existence of an infinite
116 @{term A}-avoiding path starting in @{term s} follows, contradiction.
119 apply(erule contrapos_pp);
120 apply(simp add:wf_iff_no_infinite_down_chain);
122 apply(rule ex_infinite_path);
123 apply(auto simp add:Paths_def);
127 The @{text"(no_asm)"} modifier of the @{text"rule_format"} directive means
128 that the assumption is left unchanged --- otherwise the @{text"\<forall>p"} is turned
129 into a @{text"\<And>p"}, which would complicate matters below. As it is,
130 @{thm[source]Avoid_in_lfp} is now
131 @{thm[display]Avoid_in_lfp[no_vars]}
132 The main theorem is simply the corollary where @{prop"t = s"},
133 in which case the assumption @{prop"t \<in> Avoid s A"} is trivially true
134 by the first @{term Avoid}-rule. Isabelle confirms this:
137 theorem AF_lemma2: "{s. \<forall>p \<in> Paths s. \<exists> i. p i \<in> A} \<subseteq> lfp(af A)";
138 by(auto elim:Avoid_in_lfp intro:Avoid.intros);