(*<*) theory pairs2 imports Main begin (*>*) text\<open>\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 isalso 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>\<open>nat\<close>, are datatypes, which means in particular that \<open>induct_tac\<close> and \<open>case_tac\<close> 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}. \<close> (*<*) end (*>*)
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