introduce attribute case_prod for combining case rules
* * *
[PATCH 1/9] Make tac independent of consumes
From 1a53ef4c070f924ff8e69529b0c1b13fa2844722 Mon Sep 17 00:00:00 2001
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case_product.ML | 11 ++++++-----
1 files changed, 6 insertions(+), 5 deletions(-)
* * *
[PATCH 2/9] Replace MRS by OF
From da55d08ae287bfdc18dec554067b45a3e298c243 Mon Sep 17 00:00:00 2001
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case_product.ML | 4 ++--
1 files changed, 2 insertions(+), 2 deletions(-)
* * *
[PATCH 3/9] Clean up tactic
From d26d50caaa3526c8c0625d7395467c6914f1a8d9 Mon Sep 17 00:00:00 2001
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case_product.ML | 13 +++++--------
1 files changed, 5 insertions(+), 8 deletions(-)
* * *
[PATCH 4/9] Move out get_consumes a bit more
From 6ed5415f29cc43cea30c216edb1e74ec6c0f6c33 Mon Sep 17 00:00:00 2001
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case_product.ML | 4 +++-
1 files changed, 3 insertions(+), 1 deletions(-)
* * *
[PATCH 5/9] Clean up tactic
From d301cfe6377e9f7db74b190fb085e0e66eeadfb5 Mon Sep 17 00:00:00 2001
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case_product.ML | 3 +--
1 files changed, 1 insertions(+), 2 deletions(-)
* * *
[PATCH 6/9] Cleanup build_concl_prems
From c30889e131e74a92caac5b1e6d3d816890406ca7 Mon Sep 17 00:00:00 2001
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case_product.ML | 18 ++++++++----------
1 files changed, 8 insertions(+), 10 deletions(-)
* * *
[PATCH 7/9] Split logic & annotation a bit
From e065606118308c96e013fad72ab9ad89b5a4b945 Mon Sep 17 00:00:00 2001
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case_product.ML | 26 ++++++++++++++++++--------
1 files changed, 18 insertions(+), 8 deletions(-)
* * *
[PATCH 8/9] Remove dependency on consumes attribute
From e2a4de044481586d6127bbabd0ef48d0e3ab57c0 Mon Sep 17 00:00:00 2001
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case_product.ML | 46 ++++++++++++++++++++++------------------------
1 files changed, 22 insertions(+), 24 deletions(-)
* * *
[PATCH 9/9] whitespace
From 59f5036bd8f825da4a362970292a34ec06c0f1a2 Mon Sep 17 00:00:00 2001
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case_product.ML | 2 +-
1 files changed, 1 insertions(+), 1 deletions(-)
%
\begin{isabellebody}%
\def\isabellecontext{Itrev}%
%
\isadelimtheory
%
\endisadelimtheory
%
\isatagtheory
%
\endisatagtheory
{\isafoldtheory}%
%
\isadelimtheory
%
\endisadelimtheory
%
\isadelimML
%
\endisadelimML
%
\isatagML
%
\endisatagML
{\isafoldML}%
%
\isadelimML
%
\endisadelimML
%
\isamarkupsection{Induction Heuristics%
}
\isamarkuptrue%
%
\begin{isamarkuptext}%
\label{sec:InductionHeuristics}
\index{induction heuristics|(}%
The purpose of this section is to illustrate some simple heuristics for
inductive proofs. The first one we have already mentioned in our initial
example:
\begin{quote}
\emph{Theorems about recursive functions are proved by induction.}
\end{quote}
In case the function has more than one argument
\begin{quote}
\emph{Do induction on argument number $i$ if the function is defined by
recursion in argument number $i$.}
\end{quote}
When we look at the proof of \isa{{\isaliteral{28}{\isacharparenleft}}xs{\isaliteral{40}{\isacharat}}ys{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{40}{\isacharat}}\ zs\ {\isaliteral{3D}{\isacharequal}}\ xs\ {\isaliteral{40}{\isacharat}}\ {\isaliteral{28}{\isacharparenleft}}ys{\isaliteral{40}{\isacharat}}zs{\isaliteral{29}{\isacharparenright}}}
in \S\ref{sec:intro-proof} we find
\begin{itemize}
\item \isa{{\isaliteral{40}{\isacharat}}} is recursive in
the first argument
\item \isa{xs} occurs only as the first argument of
\isa{{\isaliteral{40}{\isacharat}}}
\item both \isa{ys} and \isa{zs} occur at least once as
the second argument of \isa{{\isaliteral{40}{\isacharat}}}
\end{itemize}
Hence it is natural to perform induction on~\isa{xs}.
The key heuristic, and the main point of this section, is to
\emph{generalize the goal before induction}.
The reason is simple: if the goal is
too specific, the induction hypothesis is too weak to allow the induction
step to go through. Let us illustrate the idea with an example.
Function \cdx{rev} has quadratic worst-case running time
because it calls function \isa{{\isaliteral{40}{\isacharat}}} for each element of the list and
\isa{{\isaliteral{40}{\isacharat}}} is linear in its first argument. A linear time version of
\isa{rev} reqires an extra argument where the result is accumulated
gradually, using only~\isa{{\isaliteral{23}{\isacharhash}}}:%
\end{isamarkuptext}%
\isamarkuptrue%
\isacommand{primrec}\isamarkupfalse%
\ itrev\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{27}{\isacharprime}}a\ list\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{27}{\isacharprime}}a\ list\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{27}{\isacharprime}}a\ list{\isaliteral{22}{\isachardoublequoteclose}}\ \isakeyword{where}\isanewline
{\isaliteral{22}{\isachardoublequoteopen}}itrev\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}\ \ \ \ \ ys\ {\isaliteral{3D}{\isacharequal}}\ ys{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{7C}{\isacharbar}}\isanewline
{\isaliteral{22}{\isachardoublequoteopen}}itrev\ {\isaliteral{28}{\isacharparenleft}}x{\isaliteral{23}{\isacharhash}}xs{\isaliteral{29}{\isacharparenright}}\ ys\ {\isaliteral{3D}{\isacharequal}}\ itrev\ xs\ {\isaliteral{28}{\isacharparenleft}}x{\isaliteral{23}{\isacharhash}}ys{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}%
\begin{isamarkuptext}%
\noindent
The behaviour of \cdx{itrev} is simple: it reverses
its first argument by stacking its elements onto the second argument,
and returning that second argument when the first one becomes
empty. Note that \isa{itrev} is tail-recursive: it can be
compiled into a loop.
Naturally, we would like to show that \isa{itrev} does indeed reverse
its first argument provided the second one is empty:%
\end{isamarkuptext}%
\isamarkuptrue%
\isacommand{lemma}\isamarkupfalse%
\ {\isaliteral{22}{\isachardoublequoteopen}}itrev\ xs\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{3D}{\isacharequal}}\ rev\ xs{\isaliteral{22}{\isachardoublequoteclose}}%
\isadelimproof
%
\endisadelimproof
%
\isatagproof
%
\begin{isamarkuptxt}%
\noindent
There is no choice as to the induction variable, and we immediately simplify:%
\end{isamarkuptxt}%
\isamarkuptrue%
\isacommand{apply}\isamarkupfalse%
{\isaliteral{28}{\isacharparenleft}}induct{\isaliteral{5F}{\isacharunderscore}}tac\ xs{\isaliteral{2C}{\isacharcomma}}\ simp{\isaliteral{5F}{\isacharunderscore}}all{\isaliteral{29}{\isacharparenright}}%
\begin{isamarkuptxt}%
\noindent
Unfortunately, this attempt does not prove
the induction step:
\begin{isabelle}%
\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}a\ list{\isaliteral{2E}{\isachardot}}\isanewline
\isaindent{\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ \ \ \ }itrev\ list\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{3D}{\isacharequal}}\ rev\ list\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ itrev\ list\ {\isaliteral{5B}{\isacharbrackleft}}a{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{3D}{\isacharequal}}\ rev\ list\ {\isaliteral{40}{\isacharat}}\ {\isaliteral{5B}{\isacharbrackleft}}a{\isaliteral{5D}{\isacharbrackright}}%
\end{isabelle}
The induction hypothesis is too weak. The fixed
argument,~\isa{{\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}}, prevents it from rewriting the conclusion.
This example suggests a heuristic:
\begin{quote}\index{generalizing induction formulae}%
\emph{Generalize goals for induction by replacing constants by variables.}
\end{quote}
Of course one cannot do this na\"{\i}vely: \isa{itrev\ xs\ ys\ {\isaliteral{3D}{\isacharequal}}\ rev\ xs} is
just not true. The correct generalization is%
\end{isamarkuptxt}%
\isamarkuptrue%
%
\endisatagproof
{\isafoldproof}%
%
\isadelimproof
%
\endisadelimproof
\isacommand{lemma}\isamarkupfalse%
\ {\isaliteral{22}{\isachardoublequoteopen}}itrev\ xs\ ys\ {\isaliteral{3D}{\isacharequal}}\ rev\ xs\ {\isaliteral{40}{\isacharat}}\ ys{\isaliteral{22}{\isachardoublequoteclose}}%
\isadelimproof
%
\endisadelimproof
%
\isatagproof
%
\begin{isamarkuptxt}%
\noindent
If \isa{ys} is replaced by \isa{{\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}}, the right-hand side simplifies to
\isa{rev\ xs}, as required.
In this instance it was easy to guess the right generalization.
Other situations can require a good deal of creativity.
Although we now have two variables, only \isa{xs} is suitable for
induction, and we repeat our proof attempt. Unfortunately, we are still
not there:
\begin{isabelle}%
\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}a\ list{\isaliteral{2E}{\isachardot}}\isanewline
\isaindent{\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ \ \ \ }itrev\ list\ ys\ {\isaliteral{3D}{\isacharequal}}\ rev\ list\ {\isaliteral{40}{\isacharat}}\ ys\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\isanewline
\isaindent{\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ \ \ \ }itrev\ list\ {\isaliteral{28}{\isacharparenleft}}a\ {\isaliteral{23}{\isacharhash}}\ ys{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ rev\ list\ {\isaliteral{40}{\isacharat}}\ a\ {\isaliteral{23}{\isacharhash}}\ ys%
\end{isabelle}
The induction hypothesis is still too weak, but this time it takes no
intuition to generalize: the problem is that \isa{ys} is fixed throughout
the subgoal, but the induction hypothesis needs to be applied with
\isa{a\ {\isaliteral{23}{\isacharhash}}\ ys} instead of \isa{ys}. Hence we prove the theorem
for all \isa{ys} instead of a fixed one:%
\end{isamarkuptxt}%
\isamarkuptrue%
%
\endisatagproof
{\isafoldproof}%
%
\isadelimproof
%
\endisadelimproof
\isacommand{lemma}\isamarkupfalse%
\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}ys{\isaliteral{2E}{\isachardot}}\ itrev\ xs\ ys\ {\isaliteral{3D}{\isacharequal}}\ rev\ xs\ {\isaliteral{40}{\isacharat}}\ ys{\isaliteral{22}{\isachardoublequoteclose}}%
\isadelimproof
%
\endisadelimproof
%
\isatagproof
%
\endisatagproof
{\isafoldproof}%
%
\isadelimproof
%
\endisadelimproof
%
\begin{isamarkuptext}%
\noindent
This time induction on \isa{xs} followed by simplification succeeds. This
leads to another heuristic for generalization:
\begin{quote}
\emph{Generalize goals for induction by universally quantifying all free
variables {\em(except the induction variable itself!)}.}
\end{quote}
This prevents trivial failures like the one above and does not affect the
validity of the goal. However, this heuristic should not be applied blindly.
It is not always required, and the additional quantifiers can complicate
matters in some cases. The variables that should be quantified are typically
those that change in recursive calls.
A final point worth mentioning is the orientation of the equation we just
proved: the more complex notion (\isa{itrev}) is on the left-hand
side, the simpler one (\isa{rev}) on the right-hand side. This constitutes
another, albeit weak heuristic that is not restricted to induction:
\begin{quote}
\emph{The right-hand side of an equation should (in some sense) be simpler
than the left-hand side.}
\end{quote}
This heuristic is tricky to apply because it is not obvious that
\isa{rev\ xs\ {\isaliteral{40}{\isacharat}}\ ys} is simpler than \isa{itrev\ xs\ ys}. But see what
happens if you try to prove \isa{rev\ xs\ {\isaliteral{40}{\isacharat}}\ ys\ {\isaliteral{3D}{\isacharequal}}\ itrev\ xs\ ys}!
If you have tried these heuristics and still find your
induction does not go through, and no obvious lemma suggests itself, you may
need to generalize your proposition even further. This requires insight into
the problem at hand and is beyond simple rules of thumb.
Additionally, you can read \S\ref{sec:advanced-ind}
to learn about some advanced techniques for inductive proofs.%
\index{induction heuristics|)}%
\end{isamarkuptext}%
\isamarkuptrue%
%
\isadelimML
%
\endisadelimML
%
\isatagML
%
\endisatagML
{\isafoldML}%
%
\isadelimML
%
\endisadelimML
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\isadelimtheory
%
\endisadelimtheory
%
\isatagtheory
%
\endisatagtheory
{\isafoldtheory}%
%
\isadelimtheory
%
\endisadelimtheory
\end{isabellebody}%
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