
nipkow@10225: %
nipkow@10225: \begin{isabellebody}%
nipkow@10225: \def\isabellecontext{Star}%
wenzelm@11866: \isamarkupfalse%
nipkow@10225: %
paulson@10878: \isamarkupsection{The Reflexive Transitive Closure%
wenzelm@10395: }
wenzelm@11866: \isamarkuptrue%
nipkow@10225: %
nipkow@10225: \begin{isamarkuptext}%
nipkow@10242: \label{sec:rtc}
paulson@11494: \index{reflexive transitive closure!defining inductively|(}%
paulson@10878: An inductive definition may accept parameters, so it can express 
paulson@10878: functions that yield sets.
paulson@10878: Relations too can be defined inductively, since they are just sets of pairs.
paulson@10878: A perfect example is the function that maps a relation to its
paulson@10878: reflexive transitive closure.  This concept was already
nipkow@11147: introduced in \S\ref{sec:Relations}, where the operator \isa{\isactrlsup {\isacharasterisk}} was
nipkow@10520: defined as a least fixed point because inductive definitions were not yet
nipkow@10520: available. But now they are:%
nipkow@10225: \end{isamarkuptext}%
wenzelm@11866: \isamarkuptrue%
nipkow@10242: \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
wenzelm@11866: \isamarkupfalse%
nipkow@10225: \isacommand{inductive}\ {\isachardoublequote}r{\isacharasterisk}{\isachardoublequote}\isanewline
nipkow@10225: \isakeyword{intros}\isanewline
nipkow@10242: rtc{\isacharunderscore}refl{\isacharbrackleft}iff{\isacharbrackright}{\isacharcolon}\ \ {\isachardoublequote}{\isacharparenleft}x{\isacharcomma}x{\isacharparenright}\ {\isasymin}\ r{\isacharasterisk}{\isachardoublequote}\isanewline
wenzelm@11866: 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%
wenzelm@11866: %
nipkow@10242: \begin{isamarkuptext}%
nipkow@10242: \noindent
nipkow@10242: The function \isa{rtc} is annotated with concrete syntax: instead of
paulson@11494: \isa{rtc\ r} we can write \isa{r{\isacharasterisk}}. The actual definition
nipkow@10520: consists of two rules. Reflexivity is obvious and is immediately given the
nipkow@10520: \isa{iff} attribute to increase automation. The
nipkow@10363: second rule, \isa{rtc{\isacharunderscore}step}, says that we can always add one more
nipkow@10363: \isa{r}-step to the left. Although we could make \isa{rtc{\isacharunderscore}step} an
nipkow@10520: introduction rule, this is dangerous: the recursion in the second premise
nipkow@10520: slows down and may even kill the automatic tactics.
nipkow@10242: 
nipkow@10242: The above definition of the concept of reflexive transitive closure may
nipkow@10242: be sufficiently intuitive but it is certainly not the only possible one:
paulson@10878: for a start, it does not even mention transitivity.
nipkow@10242: The rest of this section is devoted to proving that it is equivalent to
paulson@10878: the standard definition. We start with a simple lemma:%
nipkow@10242: \end{isamarkuptext}%
wenzelm@11866: \isamarkuptrue%
nipkow@11308: \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
wenzelm@11866: \isamarkupfalse%
wenzelm@11866: \isacommand{by}{\isacharparenleft}blast\ intro{\isacharcolon}\ rtc{\isacharunderscore}step{\isacharparenright}\isamarkupfalse%
wenzelm@11866: %
nipkow@10242: \begin{isamarkuptext}%
nipkow@10242: \noindent
nipkow@10242: Although the lemma itself is an unremarkable consequence of the basic rules,
nipkow@10242: it has the advantage that it can be declared an introduction rule without the
nipkow@10242: danger of killing the automatic tactics because \isa{r{\isacharasterisk}} occurs only in
nipkow@10242: the conclusion and not in the premise. Thus some proofs that would otherwise
nipkow@10242: need \isa{rtc{\isacharunderscore}step} can now be found automatically. The proof also
paulson@10878: shows that \isa{blast} is able to handle \isa{rtc{\isacharunderscore}step}. But
nipkow@10242: 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.
nipkow@10242: 
nipkow@10520: To prove transitivity, we need rule induction, i.e.\ theorem
nipkow@10520: \isa{rtc{\isachardot}induct}:
nipkow@10520: \begin{isabelle}%
nipkow@10696: \ \ \ \ \ {\isasymlbrakk}{\isacharparenleft}{\isacharquery}xb{\isacharcomma}\ {\isacharquery}xa{\isacharparenright}\ {\isasymin}\ {\isacharquery}r{\isacharasterisk}{\isacharsemicolon}\ {\isasymAnd}x{\isachardot}\ {\isacharquery}P\ x\ x{\isacharsemicolon}\isanewline
paulson@14379: \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
wenzelm@10950: \isaindent{\ \ \ \ \ }{\isasymLongrightarrow}\ {\isacharquery}P\ {\isacharquery}xb\ {\isacharquery}xa%
nipkow@10520: \end{isabelle}
nipkow@10520: 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,
nipkow@10520: i.e.\ if \isa{{\isacharquery}P} holds for the conclusion provided it holds for the
nipkow@10520: premises. In general, rule induction for an $n$-ary inductive relation $R$
nipkow@10520: expects a premise of the form $(x@1,\dots,x@n) \in R$.
nipkow@10520: 
nipkow@10520: Now we turn to the inductive proof of transitivity:%
nipkow@10242: \end{isamarkuptext}%
wenzelm@11866: \isamarkuptrue%
nipkow@10520: \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
wenzelm@11866: \isamarkupfalse%
wenzelm@11866: \isacommand{apply}{\isacharparenleft}erule\ rtc{\isachardot}induct{\isacharparenright}\isamarkupfalse%
wenzelm@11866: %
nipkow@10363: \begin{isamarkuptxt}%
nipkow@10242: \noindent
paulson@11494: Unfortunately, even the base case is a problem:
nipkow@10363: \begin{isabelle}%
nipkow@10363: \ {\isadigit{1}}{\isachardot}\ {\isasymAnd}x{\isachardot}\ {\isacharparenleft}y{\isacharcomma}\ z{\isacharparenright}\ {\isasymin}\ r{\isacharasterisk}\ {\isasymLongrightarrow}\ {\isacharparenleft}x{\isacharcomma}\ z{\isacharparenright}\ {\isasymin}\ r{\isacharasterisk}%
nipkow@10242: \end{isabelle}
paulson@11494: We have to abandon this proof attempt.
nipkow@10520: To understand what is going on, let us look again at \isa{rtc{\isachardot}induct}.
nipkow@10520: In the above application of \isa{erule}, the first premise of
nipkow@10520: \isa{rtc{\isachardot}induct} is unified with the first suitable assumption, which
nipkow@10520: is \isa{{\isacharparenleft}x{\isacharcomma}\ y{\isacharparenright}\ {\isasymin}\ r{\isacharasterisk}} rather than \isa{{\isacharparenleft}y{\isacharcomma}\ z{\isacharparenright}\ {\isasymin}\ r{\isacharasterisk}}. Although that
nipkow@10520: is what we want, it is merely due to the order in which the assumptions occur
nipkow@10520: in the subgoal, which it is not good practice to rely on. As a result,
nipkow@10520: \isa{{\isacharquery}xb} becomes \isa{x}, \isa{{\isacharquery}xa} becomes
nipkow@10520: \isa{y} and \isa{{\isacharquery}P} becomes \isa{{\isasymlambda}u\ v{\isachardot}\ {\isacharparenleft}u{\isacharcomma}\ z{\isacharparenright}\ {\isasymin}\ r{\isacharasterisk}}, thus
nipkow@10242: yielding the above subgoal. So what went wrong?
nipkow@10242: 
nipkow@10520: When looking at the instantiation of \isa{{\isacharquery}P} we see that it does not
nipkow@10520: depend on its second parameter at all. The reason is that in our original
nipkow@10520: goal, of the pair \isa{{\isacharparenleft}x{\isacharcomma}\ y{\isacharparenright}} only \isa{x} appears also in the
nipkow@10520: conclusion, but not \isa{y}. Thus our induction statement is too
nipkow@10520: weak. Fortunately, it can easily be strengthened:
nipkow@10242: transfer the additional premise \isa{{\isacharparenleft}y{\isacharcomma}\ z{\isacharparenright}\ {\isasymin}\ r{\isacharasterisk}} into the conclusion:%
nipkow@10363: \end{isamarkuptxt}%
wenzelm@11866: \isamarkuptrue%
wenzelm@11866: \isamarkupfalse%
nipkow@10242: \isacommand{lemma}\ rtc{\isacharunderscore}trans{\isacharbrackleft}rule{\isacharunderscore}format{\isacharbrackright}{\isacharcolon}\isanewline
wenzelm@11866: \ \ {\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%
wenzelm@11866: %
nipkow@10242: \begin{isamarkuptxt}%
nipkow@10242: \noindent
nipkow@10242: This is not an obscure trick but a generally applicable heuristic:
nipkow@10242: \begin{quote}\em
nipkow@11257: When proving a statement by rule induction on $(x@1,\dots,x@n) \in R$,
nipkow@10242: pull all other premises containing any of the $x@i$ into the conclusion
nipkow@10242: using $\longrightarrow$.
nipkow@10242: \end{quote}
nipkow@10242: A similar heuristic for other kinds of inductions is formulated in
nipkow@10242: \S\ref{sec:ind-var-in-prems}. The \isa{rule{\isacharunderscore}format} directive turns
nipkow@11147: \isa{{\isasymlongrightarrow}} back into \isa{{\isasymLongrightarrow}}: in the end we obtain the original
nipkow@10363: statement of our lemma.%
nipkow@10363: \end{isamarkuptxt}%
wenzelm@11866: \isamarkuptrue%
wenzelm@11866: \isacommand{apply}{\isacharparenleft}erule\ rtc{\isachardot}induct{\isacharparenright}\isamarkupfalse%
wenzelm@11866: %
nipkow@10363: \begin{isamarkuptxt}%
nipkow@10363: \noindent
nipkow@10242: Now induction produces two subgoals which are both proved automatically:
nipkow@10363: \begin{isabelle}%
nipkow@10242: \ {\isadigit{1}}{\isachardot}\ {\isasymAnd}x{\isachardot}\ {\isacharparenleft}x{\isacharcomma}\ z{\isacharparenright}\ {\isasymin}\ r{\isacharasterisk}\ {\isasymlongrightarrow}\ {\isacharparenleft}x{\isacharcomma}\ z{\isacharparenright}\ {\isasymin}\ r{\isacharasterisk}\isanewline
nipkow@10242: \ {\isadigit{2}}{\isachardot}\ {\isasymAnd}x\ y\ za{\isachardot}\isanewline
wenzelm@10950: \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
wenzelm@10950: \isaindent{\ {\isadigit{2}}{\isachardot}\ \ \ \ }{\isasymLongrightarrow}\ {\isacharparenleft}za{\isacharcomma}\ z{\isacharparenright}\ {\isasymin}\ r{\isacharasterisk}\ {\isasymlongrightarrow}\ {\isacharparenleft}x{\isacharcomma}\ z{\isacharparenright}\ {\isasymin}\ r{\isacharasterisk}%
nipkow@10242: \end{isabelle}%
nipkow@10242: \end{isamarkuptxt}%
wenzelm@11866: \ \isamarkuptrue%
wenzelm@11866: \isacommand{apply}{\isacharparenleft}blast{\isacharparenright}\isanewline
wenzelm@11866: \isamarkupfalse%
nipkow@10237: \isacommand{apply}{\isacharparenleft}blast\ intro{\isacharcolon}\ rtc{\isacharunderscore}step{\isacharparenright}\isanewline
wenzelm@11866: \isamarkupfalse%
wenzelm@11866: \isacommand{done}\isamarkupfalse%
wenzelm@11866: %
nipkow@10242: \begin{isamarkuptext}%
nipkow@10242: Let us now prove that \isa{r{\isacharasterisk}} is really the reflexive transitive closure
nipkow@10242: of \isa{r}, i.e.\ the least reflexive and transitive
nipkow@10242: relation containing \isa{r}. The latter is easily formalized%
nipkow@10242: \end{isamarkuptext}%
wenzelm@11866: \isamarkuptrue%
nipkow@10237: \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
wenzelm@11866: \isamarkupfalse%
nipkow@10237: \isacommand{inductive}\ {\isachardoublequote}rtc{\isadigit{2}}\ r{\isachardoublequote}\isanewline
nipkow@10225: \isakeyword{intros}\isanewline
nipkow@10237: {\isachardoublequote}{\isacharparenleft}x{\isacharcomma}y{\isacharparenright}\ {\isasymin}\ r\ {\isasymLongrightarrow}\ {\isacharparenleft}x{\isacharcomma}y{\isacharparenright}\ {\isasymin}\ rtc{\isadigit{2}}\ r{\isachardoublequote}\isanewline
nipkow@10237: {\isachardoublequote}{\isacharparenleft}x{\isacharcomma}x{\isacharparenright}\ {\isasymin}\ rtc{\isadigit{2}}\ r{\isachardoublequote}\isanewline
wenzelm@11866: {\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%
wenzelm@11866: %
nipkow@10237: \begin{isamarkuptext}%
nipkow@10242: \noindent
nipkow@10242: and the equivalence of the two definitions is easily shown by the obvious rule
nipkow@10237: inductions:%
nipkow@10237: \end{isamarkuptext}%
wenzelm@11866: \isamarkuptrue%
nipkow@10237: \isacommand{lemma}\ {\isachardoublequote}{\isacharparenleft}x{\isacharcomma}y{\isacharparenright}\ {\isasymin}\ rtc{\isadigit{2}}\ r\ {\isasymLongrightarrow}\ {\isacharparenleft}x{\isacharcomma}y{\isacharparenright}\ {\isasymin}\ r{\isacharasterisk}{\isachardoublequote}\isanewline
wenzelm@11866: \isamarkupfalse%
nipkow@10237: \isacommand{apply}{\isacharparenleft}erule\ rtc{\isadigit{2}}{\isachardot}induct{\isacharparenright}\isanewline
wenzelm@11866: \ \ \isamarkupfalse%
wenzelm@11866: \isacommand{apply}{\isacharparenleft}blast{\isacharparenright}\isanewline
wenzelm@11866: \ \isamarkupfalse%
wenzelm@11866: \isacommand{apply}{\isacharparenleft}blast{\isacharparenright}\isanewline
wenzelm@11866: \isamarkupfalse%
nipkow@10237: \isacommand{apply}{\isacharparenleft}blast\ intro{\isacharcolon}\ rtc{\isacharunderscore}trans{\isacharparenright}\isanewline
wenzelm@11866: \isamarkupfalse%
nipkow@10237: \isacommand{done}\isanewline
nipkow@10225: \isanewline
wenzelm@11866: \isamarkupfalse%
nipkow@10237: \isacommand{lemma}\ {\isachardoublequote}{\isacharparenleft}x{\isacharcomma}y{\isacharparenright}\ {\isasymin}\ r{\isacharasterisk}\ {\isasymLongrightarrow}\ {\isacharparenleft}x{\isacharcomma}y{\isacharparenright}\ {\isasymin}\ rtc{\isadigit{2}}\ r{\isachardoublequote}\isanewline
wenzelm@11866: \isamarkupfalse%
nipkow@10237: \isacommand{apply}{\isacharparenleft}erule\ rtc{\isachardot}induct{\isacharparenright}\isanewline
wenzelm@11866: \ \isamarkupfalse%
nipkow@10237: \isacommand{apply}{\isacharparenleft}blast\ intro{\isacharcolon}\ rtc{\isadigit{2}}{\isachardot}intros{\isacharparenright}\isanewline
wenzelm@11866: \isamarkupfalse%
wenzelm@11866: \isacommand{apply}{\isacharparenleft}blast\ intro{\isacharcolon}\ rtc{\isadigit{2}}{\isachardot}intros{\isacharparenright}\isanewline
wenzelm@11866: \isamarkupfalse%
wenzelm@11866: \isacommand{done}\isamarkupfalse%
wenzelm@11866: %
nipkow@10242: \begin{isamarkuptext}%
nipkow@10242: So why did we start with the first definition? Because it is simpler. It
nipkow@10242: contains only two rules, and the single step rule is simpler than
nipkow@10242: transitivity.  As a consequence, \isa{rtc{\isachardot}induct} is simpler than
paulson@10878: \isa{rtc{\isadigit{2}}{\isachardot}induct}. Since inductive proofs are hard enough
nipkow@11147: anyway, we should always pick the simplest induction schema available.
nipkow@10242: Hence \isa{rtc} is the definition of choice.
paulson@11494: \index{reflexive transitive closure!defining inductively|)}
nipkow@10242: 
nipkow@10520: \begin{exercise}\label{ex:converse-rtc-step}
nipkow@10242: Show that the converse of \isa{rtc{\isacharunderscore}step} also holds:
nipkow@10242: \begin{isabelle}%
nipkow@10696: \ \ \ \ \ {\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}%
nipkow@10242: \end{isabelle}
nipkow@10520: \end{exercise}
nipkow@10520: \begin{exercise}
nipkow@10520: Repeat the development of this section, but starting with a definition of
nipkow@10520: \isa{rtc} where \isa{rtc{\isacharunderscore}step} is replaced by its converse as shown
nipkow@10520: in exercise~\ref{ex:converse-rtc-step}.
nipkow@10242: \end{exercise}%
nipkow@10242: \end{isamarkuptext}%
wenzelm@11866: \isamarkuptrue%
wenzelm@11866: \isamarkupfalse%
wenzelm@11866: \isamarkupfalse%
wenzelm@11866: \isamarkupfalse%
wenzelm@11866: \isamarkupfalse%
wenzelm@11866: \isamarkupfalse%
wenzelm@11866: \isamarkupfalse%
nipkow@10225: \end{isabellebody}%
nipkow@10225: %%% Local Variables:
nipkow@10225: %%% mode: latex
nipkow@10225: %%% TeX-master: "root"
nipkow@10225: %%% End:




