(*<*)

 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

 @{text"case"}-expression analyzing $e$ is

 \[

 \begin{array}{rrcl}

 @{text"case"}~e~@{text"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 @{text"case"}-expressions: instead

 of

 % 

 \begin{quote}

 @{text"case xs of [] => 1 | [x] => x | x#(y#zs) => y"}

 %~~~case~xs~of~[]~{\isasymRightarrow}~1~|~[x]~{\isasymRightarrow}~x~|~x~\#~y~\#~zs~{\isasymRightarrow}~y

 \end{quote}

 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 @{text"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

 (*>*)