3 \def\isabellecontext{CTLind}%
6 \isamarkupsubsection{CTL Revisited%
10 \begin{isamarkuptext}%
11 \label{sec:CTL-revisited}
13 The purpose of this section is twofold: to demonstrate
14 some of the induction principles and heuristics discussed above and to
15 show how inductive definitions can simplify proofs.
16 In \S\ref{sec:CTL} we gave a fairly involved proof of the correctness of a
17 model checker for CTL\@. In particular the proof of the
18 \isa{infinity{\isacharunderscore}lemma} on the way to \isa{AF{\isacharunderscore}lemma{\isadigit{2}}} is not as
19 simple as one might expect, due to the \isa{SOME} operator
20 involved. Below we give a simpler proof of \isa{AF{\isacharunderscore}lemma{\isadigit{2}}}
21 based on an auxiliary inductive definition.
23 Let us call a (finite or infinite) path \emph{\isa{A}-avoiding} if it does
24 not touch any node in the set \isa{A}. Then \isa{AF{\isacharunderscore}lemma{\isadigit{2}}} says
25 that if no infinite path from some state \isa{s} is \isa{A}-avoiding,
26 then \isa{s\ {\isasymin}\ lfp\ {\isacharparenleft}af\ A{\isacharparenright}}. We prove this by inductively defining the set
27 \isa{Avoid\ s\ A} of states reachable from \isa{s} by a finite \isa{A}-avoiding path:
28 % Second proof of opposite direction, directly by well-founded induction
29 % on the initial segment of M that avoids A.%
32 \isacommand{consts}\ Avoid\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}state\ {\isasymRightarrow}\ state\ set\ {\isasymRightarrow}\ state\ set{\isachardoublequote}\isanewline
34 \isacommand{inductive}\ {\isachardoublequote}Avoid\ s\ A{\isachardoublequote}\isanewline
35 \isakeyword{intros}\ {\isachardoublequote}s\ {\isasymin}\ Avoid\ s\ A{\isachardoublequote}\isanewline
36 \ \ \ \ \ \ \ {\isachardoublequote}{\isasymlbrakk}\ t\ {\isasymin}\ Avoid\ s\ A{\isacharsemicolon}\ t\ {\isasymnotin}\ A{\isacharsemicolon}\ {\isacharparenleft}t{\isacharcomma}u{\isacharparenright}\ {\isasymin}\ M\ {\isasymrbrakk}\ {\isasymLongrightarrow}\ u\ {\isasymin}\ Avoid\ s\ A{\isachardoublequote}\isamarkupfalse%
38 \begin{isamarkuptext}%
39 It is easy to see that for any infinite \isa{A}-avoiding path \isa{f}
40 with \isa{f\ {\isadigit{0}}\ {\isasymin}\ Avoid\ s\ A} there is an infinite \isa{A}-avoiding path
41 starting with \isa{s} because (by definition of \isa{Avoid}) there is a
42 finite \isa{A}-avoiding path from \isa{s} to \isa{f\ {\isadigit{0}}}.
43 The proof is by induction on \isa{f\ {\isadigit{0}}\ {\isasymin}\ Avoid\ s\ A}. However,
44 this requires the following
45 reformulation, as explained in \S\ref{sec:ind-var-in-prems} above;
46 the \isa{rule{\isacharunderscore}format} directive undoes the reformulation after the proof.%
49 \isacommand{lemma}\ ex{\isacharunderscore}infinite{\isacharunderscore}path{\isacharbrackleft}rule{\isacharunderscore}format{\isacharbrackright}{\isacharcolon}\isanewline
50 \ \ {\isachardoublequote}t\ {\isasymin}\ Avoid\ s\ A\ \ {\isasymLongrightarrow}\isanewline
51 \ \ \ {\isasymforall}f{\isasymin}Paths\ t{\isachardot}\ {\isacharparenleft}{\isasymforall}i{\isachardot}\ f\ i\ {\isasymnotin}\ A{\isacharparenright}\ {\isasymlongrightarrow}\ {\isacharparenleft}{\isasymexists}p{\isasymin}Paths\ s{\isachardot}\ {\isasymforall}i{\isachardot}\ p\ i\ {\isasymnotin}\ A{\isacharparenright}{\isachardoublequote}\isanewline
53 \isacommand{apply}{\isacharparenleft}erule\ Avoid{\isachardot}induct{\isacharparenright}\isanewline
55 \isacommand{apply}{\isacharparenleft}blast{\isacharparenright}\isanewline
57 \isacommand{apply}{\isacharparenleft}clarify{\isacharparenright}\isanewline
59 \isacommand{apply}{\isacharparenleft}drule{\isacharunderscore}tac\ x\ {\isacharequal}\ {\isachardoublequote}{\isasymlambda}i{\isachardot}\ case\ i\ of\ {\isadigit{0}}\ {\isasymRightarrow}\ t\ {\isacharbar}\ Suc\ i\ {\isasymRightarrow}\ f\ i{\isachardoublequote}\ \isakeyword{in}\ bspec{\isacharparenright}\isanewline
61 \isacommand{apply}{\isacharparenleft}simp{\isacharunderscore}all\ add{\isacharcolon}\ Paths{\isacharunderscore}def\ split{\isacharcolon}\ nat{\isachardot}split{\isacharparenright}\isanewline
63 \isacommand{done}\isamarkupfalse%
65 \begin{isamarkuptext}%
67 The base case (\isa{t\ {\isacharequal}\ s}) is trivial and proved by \isa{blast}.
68 In the induction step, we have an infinite \isa{A}-avoiding path \isa{f}
69 starting from \isa{u}, a successor of \isa{t}. Now we simply instantiate
70 the \isa{{\isasymforall}f{\isasymin}Paths\ t} in the induction hypothesis by the path starting with
71 \isa{t} and continuing with \isa{f}. That is what the above $\lambda$-term
72 expresses. Simplification shows that this is a path starting with \isa{t}
73 and that the instantiated induction hypothesis implies the conclusion.
75 Now we come to the key lemma. Assuming that no infinite \isa{A}-avoiding
76 path starts from \isa{s}, we want to show \isa{s\ {\isasymin}\ lfp\ {\isacharparenleft}af\ A{\isacharparenright}}. For the
77 inductive proof this must be generalized to the statement that every point \isa{t}
78 ``between'' \isa{s} and \isa{A}, in other words all of \isa{Avoid\ s\ A},
79 is contained in \isa{lfp\ {\isacharparenleft}af\ A{\isacharparenright}}:%
82 \isacommand{lemma}\ Avoid{\isacharunderscore}in{\isacharunderscore}lfp{\isacharbrackleft}rule{\isacharunderscore}format{\isacharparenleft}no{\isacharunderscore}asm{\isacharparenright}{\isacharbrackright}{\isacharcolon}\isanewline
83 \ \ {\isachardoublequote}{\isasymforall}p{\isasymin}Paths\ s{\isachardot}\ {\isasymexists}i{\isachardot}\ p\ i\ {\isasymin}\ A\ {\isasymLongrightarrow}\ t\ {\isasymin}\ Avoid\ s\ A\ {\isasymlongrightarrow}\ t\ {\isasymin}\ lfp{\isacharparenleft}af\ A{\isacharparenright}{\isachardoublequote}\isamarkupfalse%
87 The proof is by induction on the ``distance'' between \isa{t} and \isa{A}. Remember that \isa{lfp\ {\isacharparenleft}af\ A{\isacharparenright}\ {\isacharequal}\ A\ {\isasymunion}\ M{\isasyminverse}\ {\isacharbackquote}{\isacharbackquote}\ lfp\ {\isacharparenleft}af\ A{\isacharparenright}}.
88 If \isa{t} is already in \isa{A}, then \isa{t\ {\isasymin}\ lfp\ {\isacharparenleft}af\ A{\isacharparenright}} is
89 trivial. If \isa{t} is not in \isa{A} but all successors are in
90 \isa{lfp\ {\isacharparenleft}af\ A{\isacharparenright}} (induction hypothesis), then \isa{t\ {\isasymin}\ lfp\ {\isacharparenleft}af\ A{\isacharparenright}} is
93 The formal counterpart of this proof sketch is a well-founded induction
94 on~\isa{M} restricted to \isa{Avoid\ s\ A\ {\isacharminus}\ A}, roughly speaking:
96 \ \ \ \ \ {\isacharbraceleft}{\isacharparenleft}y{\isacharcomma}\ x{\isacharparenright}{\isachardot}\ {\isacharparenleft}x{\isacharcomma}\ y{\isacharparenright}\ {\isasymin}\ M\ {\isasymand}\ x\ {\isasymin}\ Avoid\ s\ A\ {\isasymand}\ x\ {\isasymnotin}\ A{\isacharbraceright}%
98 As we shall see presently, the absence of infinite \isa{A}-avoiding paths
99 starting from \isa{s} implies well-foundedness of this relation. For the
100 moment we assume this and proceed with the induction:%
103 \isacommand{apply}{\isacharparenleft}subgoal{\isacharunderscore}tac\ {\isachardoublequote}wf{\isacharbraceleft}{\isacharparenleft}y{\isacharcomma}x{\isacharparenright}{\isachardot}\ {\isacharparenleft}x{\isacharcomma}y{\isacharparenright}\ {\isasymin}\ M\ {\isasymand}\ x\ {\isasymin}\ Avoid\ s\ A\ {\isasymand}\ x\ {\isasymnotin}\ A{\isacharbraceright}{\isachardoublequote}{\isacharparenright}\isanewline
105 \isacommand{apply}{\isacharparenleft}erule{\isacharunderscore}tac\ a\ {\isacharequal}\ t\ \isakeyword{in}\ wf{\isacharunderscore}induct{\isacharparenright}\isanewline
107 \isacommand{apply}{\isacharparenleft}clarsimp{\isacharparenright}\isamarkupfalse%
110 \begin{isamarkuptxt}%
113 \ {\isadigit{1}}{\isachardot}\ {\isasymAnd}t{\isachardot}\ {\isasymlbrakk}{\isasymforall}p{\isasymin}Paths\ s{\isachardot}\ {\isasymexists}i{\isachardot}\ p\ i\ {\isasymin}\ A{\isacharsemicolon}\isanewline
114 \isaindent{\ {\isadigit{1}}{\isachardot}\ {\isasymAnd}t{\isachardot}\ \ }{\isasymforall}y{\isachardot}\ {\isacharparenleft}t{\isacharcomma}\ y{\isacharparenright}\ {\isasymin}\ M\ {\isasymand}\ t\ {\isasymnotin}\ A\ {\isasymlongrightarrow}\isanewline
115 \isaindent{\ {\isadigit{1}}{\isachardot}\ {\isasymAnd}t{\isachardot}\ \ {\isasymforall}y{\isachardot}\ }y\ {\isasymin}\ Avoid\ s\ A\ {\isasymlongrightarrow}\ y\ {\isasymin}\ lfp\ {\isacharparenleft}af\ A{\isacharparenright}{\isacharsemicolon}\isanewline
116 \isaindent{\ {\isadigit{1}}{\isachardot}\ {\isasymAnd}t{\isachardot}\ \ }t\ {\isasymin}\ Avoid\ s\ A{\isasymrbrakk}\isanewline
117 \isaindent{\ {\isadigit{1}}{\isachardot}\ {\isasymAnd}t{\isachardot}\ }{\isasymLongrightarrow}\ t\ {\isasymin}\ lfp\ {\isacharparenleft}af\ A{\isacharparenright}\isanewline
118 \ {\isadigit{2}}{\isachardot}\ {\isasymforall}p{\isasymin}Paths\ s{\isachardot}\ {\isasymexists}i{\isachardot}\ p\ i\ {\isasymin}\ A\ {\isasymLongrightarrow}\isanewline
119 \isaindent{\ {\isadigit{2}}{\isachardot}\ }wf\ {\isacharbraceleft}{\isacharparenleft}y{\isacharcomma}\ x{\isacharparenright}{\isachardot}\ {\isacharparenleft}x{\isacharcomma}\ y{\isacharparenright}\ {\isasymin}\ M\ {\isasymand}\ x\ {\isasymin}\ Avoid\ s\ A\ {\isasymand}\ x\ {\isasymnotin}\ A{\isacharbraceright}%
121 Now the induction hypothesis states that if \isa{t\ {\isasymnotin}\ A}
122 then all successors of \isa{t} that are in \isa{Avoid\ s\ A} are in
123 \isa{lfp\ {\isacharparenleft}af\ A{\isacharparenright}}. Unfolding \isa{lfp} in the conclusion of the first
124 subgoal once, we have to prove that \isa{t} is in \isa{A} or all successors
125 of \isa{t} are in \isa{lfp\ {\isacharparenleft}af\ A{\isacharparenright}}. But if \isa{t} is not in \isa{A},
127 \isa{Avoid}-rule implies that all successors of \isa{t} are in
128 \isa{Avoid\ s\ A}, because we also assume \isa{t\ {\isasymin}\ Avoid\ s\ A}.
129 Hence, by the induction hypothesis, all successors of \isa{t} are indeed in
130 \isa{lfp\ {\isacharparenleft}af\ A{\isacharparenright}}. Mechanically:%
133 \isacommand{apply}{\isacharparenleft}subst\ lfp{\isacharunderscore}unfold{\isacharbrackleft}OF\ mono{\isacharunderscore}af{\isacharbrackright}{\isacharparenright}\isanewline
135 \isacommand{apply}{\isacharparenleft}simp\ {\isacharparenleft}no{\isacharunderscore}asm{\isacharparenright}\ add{\isacharcolon}\ af{\isacharunderscore}def{\isacharparenright}\isanewline
137 \isacommand{apply}{\isacharparenleft}blast\ intro{\isacharcolon}\ Avoid{\isachardot}intros{\isacharparenright}\isamarkupfalse%
139 \begin{isamarkuptxt}%
140 Having proved the main goal, we return to the proof obligation that the
141 relation used above is indeed well-founded. This is proved by contradiction: if
142 the relation is not well-founded then there exists an infinite \isa{A}-avoiding path all in \isa{Avoid\ s\ A}, by theorem
143 \isa{wf{\isacharunderscore}iff{\isacharunderscore}no{\isacharunderscore}infinite{\isacharunderscore}down{\isacharunderscore}chain}:
145 \ \ \ \ \ wf\ r\ {\isacharequal}\ {\isacharparenleft}{\isasymnot}\ {\isacharparenleft}{\isasymexists}f{\isachardot}\ {\isasymforall}i{\isachardot}\ {\isacharparenleft}f\ {\isacharparenleft}Suc\ i{\isacharparenright}{\isacharcomma}\ f\ i{\isacharparenright}\ {\isasymin}\ r{\isacharparenright}{\isacharparenright}%
147 From lemma \isa{ex{\isacharunderscore}infinite{\isacharunderscore}path} the existence of an infinite
148 \isa{A}-avoiding path starting in \isa{s} follows, contradiction.%
151 \isacommand{apply}{\isacharparenleft}erule\ contrapos{\isacharunderscore}pp{\isacharparenright}\isanewline
153 \isacommand{apply}{\isacharparenleft}simp\ add{\isacharcolon}\ wf{\isacharunderscore}iff{\isacharunderscore}no{\isacharunderscore}infinite{\isacharunderscore}down{\isacharunderscore}chain{\isacharparenright}\isanewline
155 \isacommand{apply}{\isacharparenleft}erule\ exE{\isacharparenright}\isanewline
157 \isacommand{apply}{\isacharparenleft}rule\ ex{\isacharunderscore}infinite{\isacharunderscore}path{\isacharparenright}\isanewline
159 \isacommand{apply}{\isacharparenleft}auto\ simp\ add{\isacharcolon}\ Paths{\isacharunderscore}def{\isacharparenright}\isanewline
161 \isacommand{done}\isamarkupfalse%
163 \begin{isamarkuptext}%
164 The \isa{{\isacharparenleft}no{\isacharunderscore}asm{\isacharparenright}} modifier of the \isa{rule{\isacharunderscore}format} directive in the
165 statement of the lemma means
166 that the assumption is left unchanged; otherwise the \isa{{\isasymforall}p}
168 into a \isa{{\isasymAnd}p}, which would complicate matters below. As it is,
169 \isa{Avoid{\isacharunderscore}in{\isacharunderscore}lfp} is now
171 \ \ \ \ \ {\isasymlbrakk}{\isasymforall}p{\isasymin}Paths\ s{\isachardot}\ {\isasymexists}i{\isachardot}\ p\ i\ {\isasymin}\ A{\isacharsemicolon}\ t\ {\isasymin}\ Avoid\ s\ A{\isasymrbrakk}\ {\isasymLongrightarrow}\ t\ {\isasymin}\ lfp\ {\isacharparenleft}af\ A{\isacharparenright}%
173 The main theorem is simply the corollary where \isa{t\ {\isacharequal}\ s},
174 when the assumption \isa{t\ {\isasymin}\ Avoid\ s\ A} is trivially true
175 by the first \isa{Avoid}-rule. Isabelle confirms this:%
179 \isacommand{theorem}\ AF{\isacharunderscore}lemma{\isadigit{2}}{\isacharcolon}\ \ {\isachardoublequote}{\isacharbraceleft}s{\isachardot}\ {\isasymforall}p\ {\isasymin}\ Paths\ s{\isachardot}\ {\isasymexists}\ i{\isachardot}\ p\ i\ {\isasymin}\ A{\isacharbraceright}\ {\isasymsubseteq}\ lfp{\isacharparenleft}af\ A{\isacharparenright}{\isachardoublequote}\isanewline
181 \isacommand{by}{\isacharparenleft}auto\ elim{\isacharcolon}\ Avoid{\isacharunderscore}in{\isacharunderscore}lfp\ intro{\isacharcolon}\ Avoid{\isachardot}intros{\isacharparenright}\isanewline
188 %%% TeX-master: "root"