doc-src/TutorialI/Misc/case_exprs.thy

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/TutorialI/Misc/case_exprs.thy	Wed Aug 30 14:38:48 2000 +0200
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+(*<*)
+theory case_exprs = Main:
+(*>*)
+
+subsection{*Case expressions*}
+
+text{*\label{sec:case-expressions}
+HOL also features \isaindexbold{case}-expressions for analyzing
+elements of a datatype. For example,
+\begin{quote}
+@{term[display]"case xs of [] => 1 | y#ys => y"}
+\end{quote}
+evaluates to @{term"1"} if @{term"xs"} is @{term"[]"} and to @{term"y"} if 
+@{term"xs"} is @{term"y#ys"}. (Since the result in both branches must be of
+the same type, it follows that @{term"y"} is of type @{typ"nat"} and hence
+that @{term"xs"} is of type @{typ"nat list"}.)
+
+In general, if $e$ is a term of the datatype $t$ defined in
+\S\ref{sec:general-datatype} above, the corresponding
+\isa{case}-expression analyzing $e$ is
+\[
+\begin{array}{rrcl}
+\isa{case}~e~\isa{of} & C@1~x@ {11}~\dots~x@ {1k@1} & \To & e@1 \\
+                           \vdots \\
+                           \mid & C@m~x@ {m1}~\dots~x@ {mk@m} & \To & e@m
+\end{array}
+\]
+
+\begin{warn}
+\emph{All} constructors must be present, their order is fixed, and nested
+patterns are not supported.  Violating these restrictions results in strange
+error messages.
+\end{warn}
+\noindent
+Nested patterns can be simulated by nested \isa{case}-expressions: instead
+of
+% case xs of [] => 1 | [x] => x | x#(y#zs) => y
+\begin{isabelle}
+~~~case~xs~of~[]~{\isasymRightarrow}~1~|~[x]~{\isasymRightarrow}~x~|~x~\#~y~\#~zs~{\isasymRightarrow}~y
+\end{isabelle}
+write
+\begin{quote}
+@{term[display,eta_contract=false,margin=50]"case xs of [] => 1 | x#ys => (case ys of [] => x | y#zs => y)"}
+\end{quote}
+
+Note that \isa{case}-expressions may need to be enclosed in parentheses to
+indicate their scope
+*}
+
+subsection{*Structural induction and case distinction*}
+
+text{*
+\indexbold{structural induction}
+\indexbold{induction!structural}
+\indexbold{case distinction}
+Almost all the basic laws about a datatype are applied automatically during
+simplification. Only induction is invoked by hand via \isaindex{induct_tac},
+which works for any datatype. In some cases, induction is overkill and a case
+distinction over all constructors of the datatype suffices. This is performed
+by \isaindexbold{case_tac}. A trivial example:
+*}
+
+lemma "(case xs of [] \<Rightarrow> [] | y#ys \<Rightarrow> xs) = xs";
+apply(case_tac xs);
+
+txt{*\noindent
+results in the proof state
+\begin{isabelle}
+~1.~xs~=~[]~{\isasymLongrightarrow}~(case~xs~of~[]~{\isasymRightarrow}~[]~|~y~\#~ys~{\isasymRightarrow}~xs)~=~xs\isanewline
+~2.~{\isasymAnd}a~list.~xs=a\#list~{\isasymLongrightarrow}~(case~xs~of~[]~{\isasymRightarrow}~[]~|~y\#ys~{\isasymRightarrow}~xs)~=~xs%
+\end{isabelle}
+which is solved automatically:
+*}
+
+by(auto)
+
+text{*
+Note that we do not need to give a lemma a name if we do not intend to refer
+to it explicitly in the future.
+*}
+
+(*<*)
+end
+(*>*)