\begin{isabelle}%
%
\isamarkupheader{Semigroups}
\isacommand{theory}\ Semigroups\ {\isacharequal}\ Main{\isacharcolon}%
\begin{isamarkuptext}%
\medskip\noindent An axiomatic type class is simply a class of types
that all meet certain properties, which are also called \emph{class
axioms}. Thus, type classes may be also understood as type predicates
--- i.e.\ abstractions over a single type argument $\alpha$. Class
axioms typically contain polymorphic constants that depend on this
type $\alpha$. These \emph{characteristic constants} behave like
operations associated with the ``carrier'' type $\alpha$.
We illustrate these basic concepts by the following formulation of
semigroups.%
\end{isamarkuptext}%
\isacommand{consts}\isanewline
\ \ times\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a{\isachardoublequote}\ \ \ \ {\isacharparenleft}\isakeyword{infixl}\ {\isachardoublequote}{\isasymOtimes}{\isachardoublequote}\ \isadigit{7}\isadigit{0}{\isacharparenright}\isanewline
\isacommand{axclass}\isanewline
\ \ semigroup\ {\isacharless}\ {\isachardoublequote}term{\isachardoublequote}\isanewline
\ \ assoc{\isacharcolon}\ {\isachardoublequote}{\isacharparenleft}x\ {\isasymOtimes}\ y{\isacharparenright}\ {\isasymOtimes}\ z\ {\isacharequal}\ x\ {\isasymOtimes}\ {\isacharparenleft}y\ {\isasymOtimes}\ z{\isacharparenright}{\isachardoublequote}%
\begin{isamarkuptext}%
\noindent Above we have first declared a polymorphic constant $\TIMES
:: \alpha \To \alpha \To \alpha$ and then defined the class
$semigroup$ of all types $\tau$ such that $\TIMES :: \tau \To \tau
\To \tau$ is indeed an associative operator. The $assoc$ axiom
contains exactly one type variable, which is invisible in the above
presentation, though. Also note that free term variables (like $x$,
$y$, $z$) are allowed for user convenience --- conceptually all of
these are bound by outermost universal quantifiers.
\medskip In general, type classes may be used to describe
\emph{structures} with exactly one carrier $\alpha$ and a fixed
\emph{signature}. Different signatures require different classes.
Below, class $plus_semigroup$ represents semigroups of the form
$(\tau, \PLUS^\tau)$, while the original $semigroup$ would correspond
to semigroups $(\tau, \TIMES^\tau)$.%
\end{isamarkuptext}%
\isacommand{consts}\isanewline
\ \ plus\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a{\isachardoublequote}\ \ \ \ {\isacharparenleft}\isakeyword{infixl}\ {\isachardoublequote}{\isasymOplus}{\isachardoublequote}\ \isadigit{7}\isadigit{0}{\isacharparenright}\isanewline
\isacommand{axclass}\isanewline
\ \ plus{\isacharunderscore}semigroup\ {\isacharless}\ {\isachardoublequote}term{\isachardoublequote}\isanewline
\ \ assoc{\isacharcolon}\ {\isachardoublequote}{\isacharparenleft}x\ {\isasymOplus}\ y{\isacharparenright}\ {\isasymOplus}\ z\ {\isacharequal}\ x\ {\isasymOplus}\ {\isacharparenleft}y\ {\isasymOplus}\ z{\isacharparenright}{\isachardoublequote}%
\begin{isamarkuptext}%
\noindent Even if classes $plus_semigroup$ and $semigroup$ both
represent semigroups in a sense, they are certainly not quite the
same.%
\end{isamarkuptext}%
\isacommand{end}\end{isabelle}%
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