doc-src/TutorialI/Misc/document/case_splits.tex

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/TutorialI/Misc/document/case_splits.tex	Wed Apr 19 12:59:38 2000 +0200
@@ -0,0 +1,65 @@
+\begin{isabelle}%
+%
+\begin{isamarkuptext}%
+Goals containing \isaindex{if}-expressions are usually proved by case
+distinction on the condition of the \isa{if}. For example the goal%
+\end{isamarkuptext}%
+\isacommand{lemma}~{"}{\isasymforall}xs.~if~xs~=~[]~then~rev~xs~=~[]~else~rev~xs~{\isasymnoteq}~[]{"}%
+\begin{isamarkuptxt}%
+\noindent
+can be split into
+\begin{isabellepar}%
+~1.~{\isasymforall}xs.~(xs~=~[]~{\isasymlongrightarrow}~rev~xs~=~[])~{\isasymand}~(xs~{\isasymnoteq}~[]~{\isasymlongrightarrow}~rev~xs~{\isasymnoteq}~[])%
+\end{isabellepar}%
+by a degenerate form of simplification%
+\end{isamarkuptxt}%
+\isacommand{apply}(simp~only:~split:~split\_if)%
+\begin{isamarkuptext}%
+\noindent
+where no simplification rules are included (\isa{only:} is followed by the
+empty list of theorems) but the rule \isaindexbold{split_if} for
+splitting \isa{if}s is added (via the modifier \isa{split:}). Because
+case-splitting on \isa{if}s is almost always the right proof strategy, the
+simplifier performs it automatically. Try \isacommand{apply}\isa{(simp)}
+on the initial goal above.
+
+This splitting idea generalizes from \isa{if} to \isaindex{case}:%
+\end{isamarkuptext}%
+\isacommand{lemma}~{"}(case~xs~of~[]~{\isasymRightarrow}~zs~|~y\#ys~{\isasymRightarrow}~y\#(ys@zs))~=~xs@zs{"}%
+\begin{isamarkuptxt}%
+\noindent
+becomes
+\begin{isabellepar}%
+~1.~(xs~=~[]~{\isasymlongrightarrow}~zs~=~xs~@~zs)~{\isasymand}\isanewline
+~~~~({\isasymforall}a~list.~xs~=~a~\#~list~{\isasymlongrightarrow}~a~\#~list~@~zs~=~xs~@~zs)%
+\end{isabellepar}%
+by typing%
+\end{isamarkuptxt}%
+\isacommand{apply}(simp~only:~split:~list.split)%
+\begin{isamarkuptext}%
+\noindent
+In contrast to \isa{if}-expressions, the simplifier does not split
+\isa{case}-expressions by default because this can lead to nontermination
+in case of recursive datatypes. Again, if the \isa{only:} modifier is
+dropped, the above goal is solved, which \isacommand{apply}\isa{(simp)}
+alone will not do:%
+\end{isamarkuptext}%
+\isacommand{apply}(simp~split:~list.split)\isacommand{.}%
+\begin{isamarkuptext}%
+In general, every datatype $t$ comes with a theorem
+\isa{$t$.split} which can be declared to be a \bfindex{split rule} either
+locally as above, or by giving it the \isa{split} attribute globally:%
+\end{isamarkuptext}%
+\isacommand{theorems}~[split]~=~list.split%
+\begin{isamarkuptext}%
+\noindent
+The \isa{split} attribute can be removed with the \isa{del} modifier,
+either locally%
+\end{isamarkuptext}%
+\isacommand{apply}(simp~split~del:~split\_if)%
+\begin{isamarkuptext}%
+\noindent
+or globally:%
+\end{isamarkuptext}%
+\isacommand{theorems}~[split~del]~=~list.split\isanewline
+\end{isabelle}%