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\begin{isabellebody}%

\def\isabellecontext{CTLind}%

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\isamarkupsubsection{CTL revisited}

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\begin{isamarkuptext}%

\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

\isa{infinity{\isacharunderscore}lemma} on the way to \isa{AF{\isacharunderscore}lemma{\isadigit{2}}} is not as

simple as one might intuitively expect, due to the \isa{SOME} operator

involved. Below we give a simpler proof of \isa{AF{\isacharunderscore}lemma{\isadigit{2}}}

based on an auxiliary inductive definition.

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

not touch any node in the set \isa{A}. Then \isa{AF{\isacharunderscore}lemma{\isadigit{2}}} says

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

then \isa{s\ {\isasymin}\ lfp\ {\isacharparenleft}af\ A{\isacharparenright}}. We prove this by inductively defining the set

\isa{Avoid\ s\ A} of states reachable from \isa{s} by a finite \isa{A}-avoiding path:

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

% on the initial segment of M that avoids A.%

\end{isamarkuptext}%

\isacommand{consts}\ Avoid\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}state\ {\isasymRightarrow}\ state\ set\ {\isasymRightarrow}\ state\ set{\isachardoublequote}\isanewline

\isacommand{inductive}\ {\isachardoublequote}Avoid\ s\ A{\isachardoublequote}\isanewline

\isakeyword{intros}\ {\isachardoublequote}s\ {\isasymin}\ Avoid\ s\ A{\isachardoublequote}\isanewline

\ \ \ \ \ \ \ {\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}%

\begin{isamarkuptext}%

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

with \isa{f\ {\isadigit{0}}\ {\isasymin}\ Avoid\ s\ A} there is an infinite \isa{A}-avoiding path

starting with \isa{s} because (by definition of \isa{Avoid}) there is a

finite \isa{A}-avoiding path from \isa{s} to \isa{f\ {\isadigit{0}}}.

The proof is by induction on \isa{f\ {\isadigit{0}}\ {\isasymin}\ Avoid\ s\ A}. However,

this requires the following

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

the \isa{rule{\isacharunderscore}format} directive undoes the reformulation after the proof.%

\end{isamarkuptext}%

\isacommand{lemma}\ ex{\isacharunderscore}infinite{\isacharunderscore}path{\isacharbrackleft}rule{\isacharunderscore}format{\isacharbrackright}{\isacharcolon}\isanewline

\ \ {\isachardoublequote}t\ {\isasymin}\ Avoid\ s\ A\ \ {\isasymLongrightarrow}\isanewline

\ \ \ {\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

\isacommand{apply}{\isacharparenleft}erule\ Avoid{\isachardot}induct{\isacharparenright}\isanewline

\ \isacommand{apply}{\isacharparenleft}blast{\isacharparenright}\isanewline

\isacommand{apply}{\isacharparenleft}clarify{\isacharparenright}\isanewline

\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

\isacommand{apply}{\isacharparenleft}simp{\isacharunderscore}all\ add{\isacharcolon}Paths{\isacharunderscore}def\ split{\isacharcolon}nat{\isachardot}split{\isacharparenright}\isanewline

\isacommand{done}%

\begin{isamarkuptext}%

\noindent

The base case (\isa{t\ {\isacharequal}\ s}) is trivial (\isa{blast}).

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

starting from \isa{u}, a successor of \isa{t}. Now we simply instantiate

the \isa{{\isasymforall}f{\isasymin}Paths\ t} in the induction hypothesis by the path starting with

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

expresses. That fact that this is a path starting with \isa{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 \isa{t} can be reached by a

finite \isa{A}-avoiding path from \isa{s}, then \isa{t\ {\isasymin}\ lfp\ {\isacharparenleft}af\ A{\isacharparenright}},

provided there is no infinite \isa{A}-avoiding path starting from \isa{s}.%

\end{isamarkuptext}%

\isacommand{lemma}\ Avoid{\isacharunderscore}in{\isacharunderscore}lfp{\isacharbrackleft}rule{\isacharunderscore}format{\isacharparenleft}no{\isacharunderscore}asm{\isacharparenright}{\isacharbrackright}{\isacharcolon}\isanewline

\ \ {\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}%

\begin{isamarkuptxt}%

\noindent

The trick is not to induct on \isa{t\ {\isasymin}\ Avoid\ s\ A}, as already the base

case would be a problem, but to proceed by well-founded induction \isa{t}. Hence \isa{t\ {\isasymin}\ Avoid\ s\ A} needs to be brought into the conclusion as

well, which the directive \isa{rule{\isacharunderscore}format} undoes at the end (see below).

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

of \isa{M} to \isa{Avoid\ s\ A}:

\begin{isabelle}%

\ \ \ \ \ {\isacharbraceleft}{\isacharparenleft}y{\isacharcomma}\ x{\isacharparenright}{\isachardot}\ {\isacharparenleft}x{\isacharcomma}\ y{\isacharparenright}\ {\isasymin}\ M\ {\isasymand}\ x\ {\isasymin}\ Avoid\ s\ A\ {\isasymand}\ y\ {\isasymin}\ Avoid\ s\ A\ {\isasymand}\ x\ {\isasymnotin}\ A{\isacharbraceright}%

\end{isabelle}

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

starting from \isa{s} implies well-foundedness of this relation. For the

moment we assume this and proceed with the induction:%

\end{isamarkuptxt}%

\isacommand{apply}{\isacharparenleft}subgoal{\isacharunderscore}tac\isanewline

\ \ {\isachardoublequote}wf{\isacharbraceleft}{\isacharparenleft}y{\isacharcomma}x{\isacharparenright}{\isachardot}\ {\isacharparenleft}x{\isacharcomma}y{\isacharparenright}{\isasymin}M\ {\isasymand}\ x\ {\isasymin}\ Avoid\ s\ A\ {\isasymand}\ y\ {\isasymin}\ Avoid\ s\ A\ {\isasymand}\ x\ {\isasymnotin}\ A{\isacharbraceright}{\isachardoublequote}{\isacharparenright}\isanewline

\ \isacommand{apply}{\isacharparenleft}erule{\isacharunderscore}tac\ a\ {\isacharequal}\ t\ \isakeyword{in}\ wf{\isacharunderscore}induct{\isacharparenright}\isanewline

\ \isacommand{apply}{\isacharparenleft}clarsimp{\isacharparenright}%

\begin{isamarkuptxt}%

\noindent

Now can assume additionally (induction hypothesis) that if \isa{t\ {\isasymnotin}\ A}

then all successors of \isa{t} that are in \isa{Avoid\ s\ A} are in

\isa{lfp\ {\isacharparenleft}af\ A{\isacharparenright}}. To prove the actual goal we unfold \isa{lfp} once. Now

we have to prove that \isa{t} is in \isa{A} or all successors of \isa{t} are in \isa{lfp\ {\isacharparenleft}af\ A{\isacharparenright}}. If \isa{t} is not in \isa{A}, the second

\isa{Avoid}-rule implies that all successors of \isa{t} are in

\isa{Avoid\ s\ A} (because we also assume \isa{t\ {\isasymin}\ Avoid\ s\ A}), and

hence, by the induction hypothesis, all successors of \isa{t} are indeed in

\isa{lfp\ {\isacharparenleft}af\ A{\isacharparenright}}. Mechanically:%

\end{isamarkuptxt}%

\ \isacommand{apply}{\isacharparenleft}rule\ ssubst\ {\isacharbrackleft}OF\ lfp{\isacharunderscore}unfold{\isacharbrackleft}OF\ mono{\isacharunderscore}af{\isacharbrackright}{\isacharbrackright}{\isacharparenright}\isanewline

\ \isacommand{apply}{\isacharparenleft}simp\ only{\isacharcolon}\ af{\isacharunderscore}def{\isacharparenright}\isanewline

\ \isacommand{apply}{\isacharparenleft}blast\ intro{\isacharcolon}Avoid{\isachardot}intros{\isacharparenright}%

\begin{isamarkuptxt}%

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 \isa{A}-avoiding path all in \isa{Avoid\ s\ A}, by theorem

\isa{wf{\isacharunderscore}iff{\isacharunderscore}no{\isacharunderscore}infinite{\isacharunderscore}down{\isacharunderscore}chain}:

\begin{isabelle}%

\ \ \ \ \ 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}%

\end{isabelle}

From lemma \isa{ex{\isacharunderscore}infinite{\isacharunderscore}path} the existence of an infinite

\isa{A}-avoiding path starting in \isa{s} follows, just as required for

the contraposition.%

\end{isamarkuptxt}%

\isacommand{apply}{\isacharparenleft}erule\ contrapos{\isacharunderscore}pp{\isacharparenright}\isanewline

\isacommand{apply}{\isacharparenleft}simp\ add{\isacharcolon}wf{\isacharunderscore}iff{\isacharunderscore}no{\isacharunderscore}infinite{\isacharunderscore}down{\isacharunderscore}chain{\isacharparenright}\isanewline

\isacommand{apply}{\isacharparenleft}erule\ exE{\isacharparenright}\isanewline

\isacommand{apply}{\isacharparenleft}rule\ ex{\isacharunderscore}infinite{\isacharunderscore}path{\isacharparenright}\isanewline

\isacommand{apply}{\isacharparenleft}auto\ simp\ add{\isacharcolon}Paths{\isacharunderscore}def{\isacharparenright}\isanewline

\isacommand{done}%

\begin{isamarkuptext}%

The \isa{{\isacharparenleft}no{\isacharunderscore}asm{\isacharparenright}} modifier of the \isa{rule{\isacharunderscore}format} directive means

that the assumption is left unchanged---otherwise the \isa{{\isasymforall}p} is turned

into a \isa{{\isasymAnd}p}, which would complicate matters below. As it is,

\isa{Avoid{\isacharunderscore}in{\isacharunderscore}lfp} is now

\begin{isabelle}%

\ \ \ \ \ {\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}%

\end{isabelle}

The main theorem is simply the corollary where \isa{t\ {\isacharequal}\ s},

in which case the assumption \isa{t\ {\isasymin}\ Avoid\ s\ A} is trivially true

by the first \isa{Avoid}-rule). Isabelle confirms this:%

\end{isamarkuptext}%

\isacommand{theorem}\ AF{\isacharunderscore}lemma{\isadigit{2}}{\isacharcolon}\isanewline

\ \ {\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

\isacommand{by}{\isacharparenleft}auto\ elim{\isacharcolon}Avoid{\isacharunderscore}in{\isacharunderscore}lfp\ intro{\isacharcolon}Avoid{\isachardot}intros{\isacharparenright}\isanewline

\isanewline

\end{isabellebody}%

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