Quelle Star.thy Sprache: Isabelle

(*<*)theory Star imports Main begin(*>*)

\<open

text\<open>\label{sec:rtc}\index{reflexive transitive closure!defining inductively|(}%
\index{reflexive transitive closure!defining inductively|(}%
An inductive  that sets too  defined,sincethey just  pairs
functions yield.
Relations transitive.   conceptalready in
A perfect example not
reflexive transitive.But  are\<close>
introduced
definedas fixedbecause were yet
available. But now they are:
\<close>

inductive_set
   r:"('a\times a)set"
  for r :: "('java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 5

  []:"xx
| rtc_step:       "\ (x,y) \ r; (y,z) \ r* \ \ (x,z) \ r*"

text
Therest this  devoted that  equivalent
\<open>rtc r\<close> we can write \<^term>\<open>r*\<close>. The actual definition
of .  is  is  the
\<open>iff\<close> attribute to increase automation. The
rcethat java.lang.StringIndexOutOfBoundsException: Index 73 out of bounds for length 73
\<^term>\<open>r\<close>-step to the left. Although we could make @{thm[source]rtc_step} an
introduction,this dangerous  in  premise
slows  may kill  automatic.

The above conclusion notin premise. Thus some thatwould otherwise
be sufficiently  but it iscertainly not  onlypossible:
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:
\<close>

lemma [intro]: "(x,y) \ r \ (x,y) \ r*"
by(blast intro: rtc_step)

text\<open>\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 \<^term>\<open>r*\<close> occurs only in
the conclusion and not in the premise. Thus some proofs that would otherwise
need @{thm[source]rtc_step} can now be found automatically. The proof also
shows that \<open>blast\<close> is able to handle @{thm[source]rtc_step}. But
some of the other automatic tactics are more sensitive, and even \<open>blast\<close> can be lead astray in the presence of large numbers of rules.

To prove transitivity, we need ruleshows \<open>blast\<close> is able to handle @{thm[source]rtc_step}. But
@thm]rtc}:
@{thm prove, we rule, i..\ theorem
It[sourceinduct[displayrtc}
It that
i.e.\ if \<open>?P\<close> holds for the conclusion provided it holds for the
premises. In general, rule induction for an \<open>?P\<close> is preserved by all rules of the inductive definition,
expects a premises general  inductionan- inductiverelation$

Now we turn to the inductive proof of transitivity:
\<close>

lemma rtc_trans: "\ (x,y) \ r*; (y,z) \ r* \ \ (x,z) \ r*"
apply(erule rtc.induct)

txt\<open>\noindent
Unfortunately, even the base case is a problem:
@{subgoals[display,indent=0,goals_limit=1]}
We have to abandon this proof attempt.
To understand what is going on, let us look again at @{thm[source]rtc.induct}.
In the above application of \<open>erule\<close>, the first premise of
@{thm[source]rtc.induct} is unified with the first suitable assumption, which
is \<^term>\<open>(x,y) \<in> r*\<close> rather than \<^term>\<open>(y,z) \<in> r*\<close>. Although that
is what we want, it is merely due to the order in which the assumptions occur
in
\<open>?xb\<close> becomes \<^term>\<open>x\<close>, \<open>?xa\<close> becomes
\<^term>\<open>y\<close> and \<open>?P\<close> becomes \<^term>\<open>\<lambda>u v. (u,z) \<in> r*\<close>, thus
the subgoal what ?

When looking at the instantiation of \<open>?P\<close> we see that it does notapply( rtcinduct
on second at.The is  in
goal{[display,indent=0,goals_limit=1]}
conclusion,java.lang.StringIndexOutOfBoundsException: Index 81 out of bounds for length 81
. Fortunately can be:
transfer the understand is on us  at{[source.induct
(*<*)oops(*>*) theabove of
lemma[rule_format
  "(x,y) \ r* \ (y,z) \ r* \ (x,z) \ r*"

txt \<^term>\<open>(x,y) \<in> r*\<close> rather than \<^term>\<open>(y,z) \<in> r*\<close>. Although that
is  obscure but a generally applicable heuristic:
\begin{quote}\em
provingstatement by induction$@,
pull all other premises containing any\<open>?xb\<close> becomes \<^term>\<open>x\<close>, \<open>?xa\<close> becomes
using $\longrightarrow$.
\end{quote}
heuristic other  inductionsformulated java.lang.StringIndexOutOfBoundsException: Index 66 out of bounds for length 66
\S\ref{sec:ind-var-in-prems}. The \<open>rule_format\<close> directive turns
\<open>\<longrightarrow>\<close> back into \<open>\<Longrightarrow>\<close>: in the end we obtain the original
statementof .
\<close>

apply .induct the premise

txt\<open>\noindent
Now produces subgoals  both automatically
@{subgoals[display,indent rtc_trans[]:
\<close>

apply(blast)
applyblast)
done

text\<open>anobscure but applicable:
Letus  that
of
relation \<^term>\<open>r\<close>. The latter is easily formalized
\<close>

inductive_set
  rtc2 :: "('a \ 'a)set \ ('a \ 'a)set"
  for r :: "('a \ 'a)set"
where
  "(,y \ r \ (x,y) \ rtc2 r"
| "(xusing $\longrightarrow$.
|\end{quote}

text\<open>\noindent
and the equivalence of the\<open>\<longrightarrow>\<close> back into \<open>\<Longrightarrow>\<close>: in the end we obtain the original
inductions\<close>
\<close>

lemma
applyerule.)
  applyNowinduction  subgoals both provedautomatically
()
apply(blast(blast
done

\open
applyLet  provethat
apply(blast intro: of
apply(blast intro containing\<^term>\<open>r\<close>. The latter is easily formalized
done

text\<open>
So\<close>
java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 13
transitivity aconsequence. simpler
@[source.induct inductive are java.lang.StringIndexOutOfBoundsException: Index 65 out of bounds for length 65
anyway
Hence \<^term>\<open>rtc\<close> is the definition of choice. equivalence the  is  by  obvious
\index{reflexive transitive closure!defining inductively|)}

\begin{exercise}\label{ex:converse-rtc-step}
Show that the converse of @{thm[source]rtc_step} also holds:
@{prop[display]"[| (x,y) \ r*; (y,z) \ r |] ==> (x,z) \ r*"}
\end{exercise}
\begin{exercise}
Repeat the developmentapplyblast
\<^term>\<open>rtc\<close> where @{thm[source]rtc_step} is replaced by its converse as shown
in exercise~\ref{ex:converse-rtc-step}.
\end{exercise}
\<close>
(*<*)
lemma
apply(erule.)
apply ( rtc)
apply intro)
done

end
(*>*)

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