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\begin{isabellebody}%

\def\isabellecontext{Star}%

\isamarkupfalse%

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\isamarkupsection{The Reflexive Transitive Closure%

}

\isamarkuptrue%

%

\begin{isamarkuptext}%

\label{sec:rtc}

\index{reflexive transitive closure!defining inductively|(}%

An inductive definition may accept parameters, so it can express 

functions that yield sets.

Relations too can be defined inductively, since they are just sets of pairs.

A perfect example is the function that maps a relation to its

reflexive transitive closure.  This concept was already

introduced in \S\ref{sec:Relations}, where the operator \isa{\isactrlsup {\isacharasterisk}} was

defined as a least fixed point because inductive definitions were not yet

available. But now they are:%

\end{isamarkuptext}%

\isamarkuptrue%

\isacommand{consts}\ rtc\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharparenleft}{\isacharprime}a\ {\isasymtimes}\ {\isacharprime}a{\isacharparenright}set\ {\isasymRightarrow}\ {\isacharparenleft}{\isacharprime}a\ {\isasymtimes}\ {\isacharprime}a{\isacharparenright}set{\isachardoublequote}\ \ \ {\isacharparenleft}{\isachardoublequote}{\isacharunderscore}{\isacharasterisk}{\isachardoublequote}\ {\isacharbrackleft}{\isadigit{1}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isacharbrackright}\ {\isadigit{9}}{\isadigit{9}}{\isadigit{9}}{\isacharparenright}\isanewline

\isamarkupfalse%

\isacommand{inductive}\ {\isachardoublequote}r{\isacharasterisk}{\isachardoublequote}\isanewline

\isakeyword{intros}\isanewline

rtc{\isacharunderscore}refl{\isacharbrackleft}iff{\isacharbrackright}{\isacharcolon}\ \ {\isachardoublequote}{\isacharparenleft}x{\isacharcomma}x{\isacharparenright}\ {\isasymin}\ r{\isacharasterisk}{\isachardoublequote}\isanewline

rtc{\isacharunderscore}step{\isacharcolon}\ \ \ \ \ \ \ {\isachardoublequote}{\isasymlbrakk}\ {\isacharparenleft}x{\isacharcomma}y{\isacharparenright}\ {\isasymin}\ r{\isacharsemicolon}\ {\isacharparenleft}y{\isacharcomma}z{\isacharparenright}\ {\isasymin}\ r{\isacharasterisk}\ {\isasymrbrakk}\ {\isasymLongrightarrow}\ {\isacharparenleft}x{\isacharcomma}z{\isacharparenright}\ {\isasymin}\ r{\isacharasterisk}{\isachardoublequote}\isamarkupfalse%

%

\begin{isamarkuptext}%

\noindent

The function \isa{rtc} is annotated with concrete syntax: instead of

\isa{rtc\ r} we can write \isa{r{\isacharasterisk}}. The actual definition

consists of two rules. Reflexivity is obvious and is immediately given the

\isa{iff} attribute to increase automation. The

second rule, \isa{rtc{\isacharunderscore}step}, says that we can always add one more

\isa{r}-step to the left. Although we could make \isa{rtc{\isacharunderscore}step} an

introduction rule, this is dangerous: the recursion in the second premise

slows down and may even kill the automatic tactics.

The above definition of the concept of reflexive transitive closure may

be sufficiently intuitive but it is certainly not the only possible one:

for a start, it does not even mention transitivity.

The rest of this section is devoted to proving that it is equivalent to

the standard definition. We start with a simple lemma:%

\end{isamarkuptext}%

\isamarkuptrue%

\isacommand{lemma}\ {\isacharbrackleft}intro{\isacharbrackright}{\isacharcolon}\ {\isachardoublequote}{\isacharparenleft}x{\isacharcomma}y{\isacharparenright}\ {\isasymin}\ r\ {\isasymLongrightarrow}\ {\isacharparenleft}x{\isacharcomma}y{\isacharparenright}\ {\isasymin}\ r{\isacharasterisk}{\isachardoublequote}\isanewline

\isamarkupfalse%

\isacommand{by}{\isacharparenleft}blast\ intro{\isacharcolon}\ rtc{\isacharunderscore}step{\isacharparenright}\isamarkupfalse%

%

\begin{isamarkuptext}%

\noindent

Although the lemma itself is an unremarkable consequence of the basic rules,

it has the advantage that it can be declared an introduction rule without the

danger of killing the automatic tactics because \isa{r{\isacharasterisk}} occurs only in

the conclusion and not in the premise. Thus some proofs that would otherwise

need \isa{rtc{\isacharunderscore}step} can now be found automatically. The proof also

shows that \isa{blast} is able to handle \isa{rtc{\isacharunderscore}step}. But

some of the other automatic tactics are more sensitive, and even \isa{blast} can be lead astray in the presence of large numbers of rules.

To prove transitivity, we need rule induction, i.e.\ theorem

\isa{rtc{\isachardot}induct}:

\begin{isabelle}%

\ \ \ \ \ {\isasymlbrakk}{\isacharparenleft}{\isacharquery}xb{\isacharcomma}\ {\isacharquery}xa{\isacharparenright}\ {\isasymin}\ {\isacharquery}r{\isacharasterisk}{\isacharsemicolon}\ {\isasymAnd}x{\isachardot}\ {\isacharquery}P\ x\ x{\isacharsemicolon}\isanewline

\isaindent{\ \ \ \ \ \ }{\isasymAnd}x\ y\ z{\isachardot}\ {\isasymlbrakk}{\isacharparenleft}x{\isacharcomma}\ y{\isacharparenright}\ {\isasymin}\ {\isacharquery}r{\isacharsemicolon}\ {\isacharparenleft}y{\isacharcomma}\ z{\isacharparenright}\ {\isasymin}\ {\isacharquery}r{\isacharasterisk}{\isacharsemicolon}\ {\isacharquery}P\ y\ z{\isasymrbrakk}\ {\isasymLongrightarrow}\ {\isacharquery}P\ x\ z{\isasymrbrakk}\isanewline

\isaindent{\ \ \ \ \ }{\isasymLongrightarrow}\ {\isacharquery}P\ {\isacharquery}xb\ {\isacharquery}xa%

\end{isabelle}

It says that \isa{{\isacharquery}P} holds for an arbitrary pair \isa{{\isacharparenleft}{\isacharquery}xb{\isacharcomma}{\isacharquery}xa{\isacharparenright}\ {\isasymin}\ {\isacharquery}r{\isacharasterisk}} if \isa{{\isacharquery}P} is preserved by all rules of the inductive definition,

i.e.\ if \isa{{\isacharquery}P} holds for the conclusion provided it holds for the

premises. In general, rule induction for an $n$-ary inductive relation $R$

expects a premise of the form $(x@1,\dots,x@n) \in R$.

Now we turn to the inductive proof of transitivity:%

\end{isamarkuptext}%

\isamarkuptrue%

\isacommand{lemma}\ rtc{\isacharunderscore}trans{\isacharcolon}\ {\isachardoublequote}{\isasymlbrakk}\ {\isacharparenleft}x{\isacharcomma}y{\isacharparenright}\ {\isasymin}\ r{\isacharasterisk}{\isacharsemicolon}\ {\isacharparenleft}y{\isacharcomma}z{\isacharparenright}\ {\isasymin}\ r{\isacharasterisk}\ {\isasymrbrakk}\ {\isasymLongrightarrow}\ {\isacharparenleft}x{\isacharcomma}z{\isacharparenright}\ {\isasymin}\ r{\isacharasterisk}{\isachardoublequote}\isanewline

\isamarkupfalse%

\isacommand{apply}{\isacharparenleft}erule\ rtc{\isachardot}induct{\isacharparenright}\isamarkupfalse%

%

\begin{isamarkuptxt}%

\noindent

Unfortunately, even the base case is a problem:

\begin{isabelle}%

\ {\isadigit{1}}{\isachardot}\ {\isasymAnd}x{\isachardot}\ {\isacharparenleft}y{\isacharcomma}\ z{\isacharparenright}\ {\isasymin}\ r{\isacharasterisk}\ {\isasymLongrightarrow}\ {\isacharparenleft}x{\isacharcomma}\ z{\isacharparenright}\ {\isasymin}\ r{\isacharasterisk}%

\end{isabelle}

We have to abandon this proof attempt.

To understand what is going on, let us look again at \isa{rtc{\isachardot}induct}.

In the above application of \isa{erule}, the first premise of

\isa{rtc{\isachardot}induct} is unified with the first suitable assumption, which

is \isa{{\isacharparenleft}x{\isacharcomma}\ y{\isacharparenright}\ {\isasymin}\ r{\isacharasterisk}} rather than \isa{{\isacharparenleft}y{\isacharcomma}\ z{\isacharparenright}\ {\isasymin}\ r{\isacharasterisk}}. Although that

is what we want, it is merely due to the order in which the assumptions occur

in the subgoal, which it is not good practice to rely on. As a result,

\isa{{\isacharquery}xb} becomes \isa{x}, \isa{{\isacharquery}xa} becomes

\isa{y} and \isa{{\isacharquery}P} becomes \isa{{\isasymlambda}u\ v{\isachardot}\ {\isacharparenleft}u{\isacharcomma}\ z{\isacharparenright}\ {\isasymin}\ r{\isacharasterisk}}, thus

yielding the above subgoal. So what went wrong?

When looking at the instantiation of \isa{{\isacharquery}P} we see that it does not

depend on its second parameter at all. The reason is that in our original

goal, of the pair \isa{{\isacharparenleft}x{\isacharcomma}\ y{\isacharparenright}} only \isa{x} appears also in the

conclusion, but not \isa{y}. Thus our induction statement is too

weak. Fortunately, it can easily be strengthened:

transfer the additional premise \isa{{\isacharparenleft}y{\isacharcomma}\ z{\isacharparenright}\ {\isasymin}\ r{\isacharasterisk}} into the conclusion:%

\end{isamarkuptxt}%

\isamarkuptrue%

\isamarkupfalse%

\isacommand{lemma}\ rtc{\isacharunderscore}trans{\isacharbrackleft}rule{\isacharunderscore}format{\isacharbrackright}{\isacharcolon}\isanewline

\ \ {\isachardoublequote}{\isacharparenleft}x{\isacharcomma}y{\isacharparenright}\ {\isasymin}\ r{\isacharasterisk}\ {\isasymLongrightarrow}\ {\isacharparenleft}y{\isacharcomma}z{\isacharparenright}\ {\isasymin}\ r{\isacharasterisk}\ {\isasymlongrightarrow}\ {\isacharparenleft}x{\isacharcomma}z{\isacharparenright}\ {\isasymin}\ r{\isacharasterisk}{\isachardoublequote}\isamarkupfalse%

%

\begin{isamarkuptxt}%

\noindent

This is not an obscure trick but a generally applicable heuristic:

\begin{quote}\em

When proving a statement by rule induction on $(x@1,\dots,x@n) \in R$,

pull all other premises containing any of the $x@i$ into the conclusion

using $\longrightarrow$.

\end{quote}

A similar heuristic for other kinds of inductions is formulated in

\S\ref{sec:ind-var-in-prems}. The \isa{rule{\isacharunderscore}format} directive turns

\isa{{\isasymlongrightarrow}} back into \isa{{\isasymLongrightarrow}}: in the end we obtain the original

statement of our lemma.%

\end{isamarkuptxt}%

\isamarkuptrue%

\isacommand{apply}{\isacharparenleft}erule\ rtc{\isachardot}induct{\isacharparenright}\isamarkupfalse%

%

\begin{isamarkuptxt}%

\noindent

Now induction produces two subgoals which are both proved automatically:

\begin{isabelle}%

\ {\isadigit{1}}{\isachardot}\ {\isasymAnd}x{\isachardot}\ {\isacharparenleft}x{\isacharcomma}\ z{\isacharparenright}\ {\isasymin}\ r{\isacharasterisk}\ {\isasymlongrightarrow}\ {\isacharparenleft}x{\isacharcomma}\ z{\isacharparenright}\ {\isasymin}\ r{\isacharasterisk}\isanewline

\ {\isadigit{2}}{\isachardot}\ {\isasymAnd}x\ y\ za{\isachardot}\isanewline

\isaindent{\ {\isadigit{2}}{\isachardot}\ \ \ \ }{\isasymlbrakk}{\isacharparenleft}x{\isacharcomma}\ y{\isacharparenright}\ {\isasymin}\ r{\isacharsemicolon}\ {\isacharparenleft}y{\isacharcomma}\ za{\isacharparenright}\ {\isasymin}\ r{\isacharasterisk}{\isacharsemicolon}\ {\isacharparenleft}za{\isacharcomma}\ z{\isacharparenright}\ {\isasymin}\ r{\isacharasterisk}\ {\isasymlongrightarrow}\ {\isacharparenleft}y{\isacharcomma}\ z{\isacharparenright}\ {\isasymin}\ r{\isacharasterisk}{\isasymrbrakk}\isanewline

\isaindent{\ {\isadigit{2}}{\isachardot}\ \ \ \ }{\isasymLongrightarrow}\ {\isacharparenleft}za{\isacharcomma}\ z{\isacharparenright}\ {\isasymin}\ r{\isacharasterisk}\ {\isasymlongrightarrow}\ {\isacharparenleft}x{\isacharcomma}\ z{\isacharparenright}\ {\isasymin}\ r{\isacharasterisk}%

\end{isabelle}%

\end{isamarkuptxt}%

\ \isamarkuptrue%

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

\isamarkupfalse%

\isacommand{apply}{\isacharparenleft}blast\ intro{\isacharcolon}\ rtc{\isacharunderscore}step{\isacharparenright}\isanewline

\isamarkupfalse%

\isacommand{done}\isamarkupfalse%

%

\begin{isamarkuptext}%

Let us now prove that \isa{r{\isacharasterisk}} is really the reflexive transitive closure

of \isa{r}, i.e.\ the least reflexive and transitive

relation containing \isa{r}. The latter is easily formalized%

\end{isamarkuptext}%

\isamarkuptrue%

\isacommand{consts}\ rtc{\isadigit{2}}\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharparenleft}{\isacharprime}a\ {\isasymtimes}\ {\isacharprime}a{\isacharparenright}set\ {\isasymRightarrow}\ {\isacharparenleft}{\isacharprime}a\ {\isasymtimes}\ {\isacharprime}a{\isacharparenright}set{\isachardoublequote}\isanewline

\isamarkupfalse%

\isacommand{inductive}\ {\isachardoublequote}rtc{\isadigit{2}}\ r{\isachardoublequote}\isanewline

\isakeyword{intros}\isanewline

{\isachardoublequote}{\isacharparenleft}x{\isacharcomma}y{\isacharparenright}\ {\isasymin}\ r\ {\isasymLongrightarrow}\ {\isacharparenleft}x{\isacharcomma}y{\isacharparenright}\ {\isasymin}\ rtc{\isadigit{2}}\ r{\isachardoublequote}\isanewline

{\isachardoublequote}{\isacharparenleft}x{\isacharcomma}x{\isacharparenright}\ {\isasymin}\ rtc{\isadigit{2}}\ r{\isachardoublequote}\isanewline

{\isachardoublequote}{\isasymlbrakk}\ {\isacharparenleft}x{\isacharcomma}y{\isacharparenright}\ {\isasymin}\ rtc{\isadigit{2}}\ r{\isacharsemicolon}\ {\isacharparenleft}y{\isacharcomma}z{\isacharparenright}\ {\isasymin}\ rtc{\isadigit{2}}\ r\ {\isasymrbrakk}\ {\isasymLongrightarrow}\ {\isacharparenleft}x{\isacharcomma}z{\isacharparenright}\ {\isasymin}\ rtc{\isadigit{2}}\ r{\isachardoublequote}\isamarkupfalse%

%

\begin{isamarkuptext}%

\noindent

and the equivalence of the two definitions is easily shown by the obvious rule

inductions:%

\end{isamarkuptext}%

\isamarkuptrue%

\isacommand{lemma}\ {\isachardoublequote}{\isacharparenleft}x{\isacharcomma}y{\isacharparenright}\ {\isasymin}\ rtc{\isadigit{2}}\ r\ {\isasymLongrightarrow}\ {\isacharparenleft}x{\isacharcomma}y{\isacharparenright}\ {\isasymin}\ r{\isacharasterisk}{\isachardoublequote}\isanewline

\isamarkupfalse%

\isacommand{apply}{\isacharparenleft}erule\ rtc{\isadigit{2}}{\isachardot}induct{\isacharparenright}\isanewline

\ \ \isamarkupfalse%

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

\ \isamarkupfalse%

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

\isamarkupfalse%

\isacommand{apply}{\isacharparenleft}blast\ intro{\isacharcolon}\ rtc{\isacharunderscore}trans{\isacharparenright}\isanewline

\isamarkupfalse%

\isacommand{done}\isanewline

\isanewline

\isamarkupfalse%

\isacommand{lemma}\ {\isachardoublequote}{\isacharparenleft}x{\isacharcomma}y{\isacharparenright}\ {\isasymin}\ r{\isacharasterisk}\ {\isasymLongrightarrow}\ {\isacharparenleft}x{\isacharcomma}y{\isacharparenright}\ {\isasymin}\ rtc{\isadigit{2}}\ r{\isachardoublequote}\isanewline

\isamarkupfalse%

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

\ \isamarkupfalse%

\isacommand{apply}{\isacharparenleft}blast\ intro{\isacharcolon}\ rtc{\isadigit{2}}{\isachardot}intros{\isacharparenright}\isanewline

\isamarkupfalse%

\isacommand{apply}{\isacharparenleft}blast\ intro{\isacharcolon}\ rtc{\isadigit{2}}{\isachardot}intros{\isacharparenright}\isanewline

\isamarkupfalse%

\isacommand{done}\isamarkupfalse%

%

\begin{isamarkuptext}%

So why did we start with the first definition? Because it is simpler. It

contains only two rules, and the single step rule is simpler than

transitivity.  As a consequence, \isa{rtc{\isachardot}induct} is simpler than

\isa{rtc{\isadigit{2}}{\isachardot}induct}. Since inductive proofs are hard enough

anyway, we should always pick the simplest induction schema available.

Hence \isa{rtc} is the definition of choice.

\index{reflexive transitive closure!defining inductively|)}

\begin{exercise}\label{ex:converse-rtc-step}

Show that the converse of \isa{rtc{\isacharunderscore}step} also holds:

\begin{isabelle}%

\ \ \ \ \ {\isasymlbrakk}{\isacharparenleft}x{\isacharcomma}\ y{\isacharparenright}\ {\isasymin}\ r{\isacharasterisk}{\isacharsemicolon}\ {\isacharparenleft}y{\isacharcomma}\ z{\isacharparenright}\ {\isasymin}\ r{\isasymrbrakk}\ {\isasymLongrightarrow}\ {\isacharparenleft}x{\isacharcomma}\ z{\isacharparenright}\ {\isasymin}\ r{\isacharasterisk}%

\end{isabelle}

\end{exercise}

\begin{exercise}

Repeat the development of this section, but starting with a definition of

\isa{rtc} where \isa{rtc{\isacharunderscore}step} is replaced by its converse as shown

in exercise~\ref{ex:converse-rtc-step}.

\end{exercise}%

\end{isamarkuptext}%

\isamarkuptrue%

\isamarkupfalse%

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\isamarkupfalse%

\isamarkupfalse%

\isamarkupfalse%

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\end{isabellebody}%

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