2 theory pairs imports Main begin;
4 text{*\label{sec:pairs}\index{pairs and tuples}
5 HOL also has ordered pairs: \isa{($a@1$,$a@2$)} is of type $\tau@1$
6 \indexboldpos{\isasymtimes}{$Isatype} $\tau@2$ provided each $a@i$ is of type
7 $\tau@i$. The functions \cdx{fst} and
8 \cdx{snd} extract the components of a pair:
9 \isa{fst($x$,$y$) = $x$} and \isa{snd($x$,$y$) = $y$}. Tuples
10 are simulated by pairs nested to the right: \isa{($a@1$,$a@2$,$a@3$)} stands
11 for \isa{($a@1$,($a@2$,$a@3$))} and $\tau@1 \times \tau@2 \times \tau@3$ for
12 $\tau@1 \times (\tau@2 \times \tau@3)$. Therefore we have
13 \isa{fst(snd($a@1$,$a@2$,$a@3$)) = $a@2$}.
18 There is also the type \tydx{unit}, which contains exactly one
19 element denoted by~\cdx{()}. This type can be viewed
20 as a degenerate product with 0 components.
22 Products, like type @{typ nat}, are datatypes, which means
23 in particular that @{text induct_tac} and @{text case_tac} are applicable to
24 terms of product type.
25 Both split the term into a number of variables corresponding to the tuple structure
28 Tuples with more than two or three components become unwieldy;
29 records are preferable.
31 For more information on pairs and records see Chapter~\ref{ch:more-types}.