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\begin{isabellebody}%
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\def\isabellecontext{Itrev}%
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%
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\isamarkupsection{Induction heuristics}
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%
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\begin{isamarkuptext}%
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\label{sec:InductionHeuristics}
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The purpose of this section is to illustrate some simple heuristics for
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inductive proofs. The first one we have already mentioned in our initial
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example:
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\begin{quote}
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\emph{Theorems about recursive functions are proved by induction.}
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\end{quote}
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In case the function has more than one argument
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\begin{quote}
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\emph{Do induction on argument number $i$ if the function is defined by
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recursion in argument number $i$.}
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\end{quote}
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When we look at the proof of \isa{{\isachardoublequote}{\isacharparenleft}xs\ {\isacharat}\ ys{\isacharparenright}\ {\isacharat}\ zs\ {\isacharequal}\ xs\ {\isacharat}\ {\isacharparenleft}ys\ {\isacharat}\ zs{\isacharparenright}{\isachardoublequote}}
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in \S\ref{sec:intro-proof} we find (a) \isa{{\isacharat}} is recursive in
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the first argument, (b) \isa{xs} occurs only as the first argument of
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\isa{{\isacharat}}, and (c) both \isa{ys} and \isa{zs} occur at least once as
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the second argument of \isa{{\isacharat}}. Hence it is natural to perform induction
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on \isa{xs}.
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The key heuristic, and the main point of this section, is to
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generalize the goal before induction. The reason is simple: if the goal is
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too specific, the induction hypothesis is too weak to allow the induction
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step to go through. Let us now illustrate the idea with an example.
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Function \isa{rev} has quadratic worst-case running time
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because it calls function \isa{{\isacharat}} for each element of the list and
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\isa{{\isacharat}} is linear in its first argument. A linear time version of
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\isa{rev} reqires an extra argument where the result is accumulated
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gradually, using only \isa{{\isacharhash}}:%
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\end{isamarkuptext}%
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\isacommand{consts}\ itrev\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharprime}a\ list\ {\isasymRightarrow}\ {\isacharprime}a\ list\ {\isasymRightarrow}\ {\isacharprime}a\ list{\isachardoublequote}\isanewline
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\isacommand{primrec}\isanewline
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{\isachardoublequote}itrev\ {\isacharbrackleft}{\isacharbrackright}\ \ \ \ \ ys\ {\isacharequal}\ ys{\isachardoublequote}\isanewline
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{\isachardoublequote}itrev\ {\isacharparenleft}x{\isacharhash}xs{\isacharparenright}\ ys\ {\isacharequal}\ itrev\ xs\ {\isacharparenleft}x{\isacharhash}ys{\isacharparenright}{\isachardoublequote}%
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\begin{isamarkuptext}%
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\noindent
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The behaviour of \isa{itrev} is simple: it reverses
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its first argument by stacking its elements onto the second argument,
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and returning that second argument when the first one becomes
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empty. Note that \isa{itrev} is tail-recursive, i.e.\ it can be
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compiled into a loop.
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Naturally, we would like to show that \isa{itrev} does indeed reverse
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its first argument provided the second one is empty:%
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\end{isamarkuptext}%
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\isacommand{lemma}\ {\isachardoublequote}itrev\ xs\ {\isacharbrackleft}{\isacharbrackright}\ {\isacharequal}\ rev\ xs{\isachardoublequote}%
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\begin{isamarkuptxt}%
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\noindent
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There is no choice as to the induction variable, and we immediately simplify:%
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\end{isamarkuptxt}%
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\isacommand{apply}{\isacharparenleft}induct{\isacharunderscore}tac\ xs{\isacharcomma}\ simp{\isacharunderscore}all{\isacharparenright}%
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\begin{isamarkuptxt}%
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\noindent
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Unfortunately, this is not a complete success:
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\begin{isabelle}\makeatother
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~1.~\dots~itrev~list~[]~=~rev~list~{\isasymLongrightarrow}~itrev~list~[a]~=~rev~list~@~[a]%
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\end{isabelle}
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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{{\isacharbrackleft}{\isacharbrackright}}. The corresponding heuristic:
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\begin{quote}
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\emph{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\ {\isacharequal}\ rev\ xs} is
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just not true---the correct generalization is%
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\end{isamarkuptxt}%
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\isacommand{lemma}\ {\isachardoublequote}itrev\ xs\ ys\ {\isacharequal}\ rev\ xs\ {\isacharat}\ ys{\isachardoublequote}%
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\begin{isamarkuptxt}%
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\noindent
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If \isa{ys} is replaced by \isa{{\isacharbrackleft}{\isacharbrackright}}, 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{isabelle}%
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\ {\isadigit{1}}{\isachardot}\ {\isasymAnd}a\ list{\isachardot}\isanewline
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\ \ \ \ \ \ \ itrev\ list\ ys\ {\isacharequal}\ rev\ list\ {\isacharat}\ ys\ {\isasymLongrightarrow}\isanewline
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\ \ \ \ \ \ \ itrev\ list\ {\isacharparenleft}a\ {\isacharhash}\ ys{\isacharparenright}\ {\isacharequal}\ rev\ list\ {\isacharat}\ a\ {\isacharhash}\ ys%
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\end{isabelle}
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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\ {\isacharhash}\ 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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\end{isamarkuptxt}%
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\isacommand{lemma}\ {\isachardoublequote}{\isasymforall}ys{\isachardot}\ itrev\ xs\ ys\ {\isacharequal}\ rev\ xs\ {\isacharat}\ ys{\isachardoublequote}%
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\begin{isamarkuptext}%
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\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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\emph{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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In general, if you have tried the above heuristics and still find your
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induction does not go through, and no obvious lemma suggests itself, you may
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need to generalize your proposition even further. This requires insight into
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the problem at hand and is beyond simple rules of thumb. In a nutshell: you
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will need to be creative. Additionally, you can read \S\ref{sec:advanced-ind}
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to learn about some advanced techniques for inductive proofs.
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A final point worth mentioning is the orientation of the equation we just
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proved: the more complex notion (\isa{itrev}) is on the left-hand
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side, the simpler one (\isa{rev}) on the right-hand side. This constitutes
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another, albeit weak heuristic that is not restricted to induction:
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\begin{quote}
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\emph{The right-hand side of an equation should (in some sense) be simpler
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than the left-hand side.}
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\end{quote}
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This heuristic is tricky to apply because it is not obvious that
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\isa{rev\ xs\ {\isacharat}\ ys} is simpler than \isa{itrev\ xs\ ys}. But see what
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happens if you try to prove \isa{rev\ xs\ {\isacharat}\ ys\ {\isacharequal}\ itrev\ xs\ ys}!%
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\end{isamarkuptext}%
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\end{isabellebody}%
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%%% Local Variables:
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%%% mode: latex
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%%% TeX-master: "root"
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%%% End:
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