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theory pairs = Main:;

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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 replace the term by a pair of variables.

\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}.

*}

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end

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