doc-src/TutorialI/Misc/pairs2.thy

+(*<*)
+theory pairs2 imports Main begin;
+(*>*)
+text{*\label{sec:pairs}\index{pairs and tuples}
+HOL also has ordered pairs: \isa{($a@1$,$a@2$)} is of type $\tau@1$
+\indexboldpos{\isasymtimes}{$Isatype} $\tau@2$ provided each $a@i$ is of type
+$\tau@i$. The functions \cdx{fst} and
+\cdx{snd} extract the components of a pair:
+ \isa{fst($x$,$y$) = $x$} and \isa{snd($x$,$y$) = $y$}. Tuples
+are simulated by pairs nested to the right: \isa{($a@1$,$a@2$,$a@3$)} stands
+for \isa{($a@1$,($a@2$,$a@3$))} and $\tau@1 \times \tau@2 \times \tau@3$ for
+$\tau@1 \times (\tau@2 \times \tau@3)$. Therefore we have
+\isa{fst(snd($a@1$,$a@2$,$a@3$)) = $a@2$}.
+
+Remarks:
+\begin{itemize}
+\item
+There is also the type \tydx{unit}, which contains exactly one
+element denoted by~\cdx{()}.  This type can be viewed
+as a degenerate product with 0 components.
+\item
+Products, like type @{typ nat}, are datatypes, which means
+in particular that @{text induct_tac} and @{text case_tac} are applicable to
+terms of product type.
+Both split the term into a number of variables corresponding to the tuple structure
+(up to 7 components).
+\item
+Tuples with more than two or three components become unwieldy;
+records are preferable.
+\end{itemize}
+For more information on pairs and records see Chapter~\ref{ch:more-types}.
+*}
+(*<*)
+end
+(*>*)