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(*<*)
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theory Itrev = Main:;
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(*>*)
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text{*
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We define a tail-recursive version of list-reversal,
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i.e.\ one that can be compiled into a loop:
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*};
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consts itrev :: "'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list";
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primrec
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"itrev [] ys = ys"
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"itrev (x#xs) ys = itrev xs (x#ys)";
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text{*\noindent
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The behaviour of \isa{itrev} is simple: it reverses its first argument by
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stacking its elements onto the second argument, and returning that second
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argument when the first one becomes empty.
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We need to show that \isa{itrev} does indeed reverse its first argument
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provided the second one is empty:
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*};
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lemma "itrev xs [] = rev xs";
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txt{*\noindent
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There is no choice as to the induction variable, and we immediately simplify:
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*};
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apply(induct_tac xs, auto);
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txt{*\noindent
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Unfortunately, this is not a complete success:
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\begin{isabellepar}%
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~1.~\dots~itrev~list~[]~=~rev~list~{\isasymLongrightarrow}~itrev~list~[a]~=~rev~list~@~[a]%
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\end{isabellepar}%
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Just as predicted above, the overall goal, and hence the induction
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hypothesis, is too weak to solve the induction step because of the fixed
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\isa{[]}. The corresponding heuristic:
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\begin{quote}
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{\em 3. Generalize goals for induction by replacing constants by variables.}
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\end{quote}
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Of course one cannot do this na\"{\i}vely: \isa{itrev xs ys = rev xs} is
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just not true---the correct generalization is
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*};
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(*<*)oops;(*>*)
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lemma "itrev xs ys = rev xs @ ys";
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txt{*\noindent
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If \isa{ys} is replaced by \isa{[]}, the right-hand side simplifies to
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\isa{rev xs}, just as required.
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In this particular instance it was easy to guess the right generalization,
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but in more complex situations a good deal of creativity is needed. This is
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the main source of complications in inductive proofs.
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Although we now have two variables, only \isa{xs} is suitable for
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induction, and we repeat our above proof attempt. Unfortunately, we are still
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not there:
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\begin{isabellepar}%
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~1.~{\isasymAnd}a~list.\isanewline
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~~~~~~~itrev~list~ys~=~rev~list~@~ys~{\isasymLongrightarrow}\isanewline
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~~~~~~~itrev~list~(a~\#~ys)~=~rev~list~@~a~\#~ys%
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\end{isabellepar}%
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The induction hypothesis is still too weak, but this time it takes no
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intuition to generalize: the problem is that \isa{ys} is fixed throughout
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the subgoal, but the induction hypothesis needs to be applied with
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\isa{a \# ys} instead of \isa{ys}. Hence we prove the theorem
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for all \isa{ys} instead of a fixed one:
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*};
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(*<*)oops;(*>*)
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lemma "\\<forall>ys. itrev xs ys = rev xs @ ys";
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txt{*\noindent
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This time induction on \isa{xs} followed by simplification succeeds. This
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leads to another heuristic for generalization:
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\begin{quote}
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{\em 4. Generalize goals for induction by universally quantifying all free
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variables {\em(except the induction variable itself!)}.}
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\end{quote}
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This prevents trivial failures like the above and does not change the
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provability of the goal. Because it is not always required, and may even
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complicate matters in some cases, this heuristic is often not
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applied blindly.
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*};
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(*<*)
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by(induct_tac xs, simp, simp);
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end
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(*>*)
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