\begin{isabelle}%
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\isamarkupheader{Semigroups}
\isacommand{theory}~Semigroups~=~Main:%
\begin{isamarkuptext}%
\medskip\noindent An axiomatic type class is simply a class of types
that all meet certain \emph{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~::~{"}'a~{\isasymRightarrow}~'a~{\isasymRightarrow}~'a{"}~~~~(\isakeyword{infixl}~{"}{\isasymOtimes}{"}~70)\isanewline
\isacommand{axclass}\isanewline
~~semigroup~<~{"}term{"}\isanewline
~~assoc:~{"}(x~{\isasymOtimes}~y)~{\isasymOtimes}~z~=~x~{\isasymOtimes}~(y~{\isasymOtimes}~z){"}%
\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.
Subsequently, class $plus_semigroup$ represents semigroups of the
form $(\tau, \PLUS^\tau)$, while $semigroup$ above would correspond
to semigroups $(\tau, \TIMES^\tau)$.%
\end{isamarkuptext}%
\isacommand{consts}\isanewline
~~plus~::~{"}'a~{\isasymRightarrow}~'a~{\isasymRightarrow}~'a{"}~~~~(\isakeyword{infixl}~{"}{\isasymOplus}{"}~70)\isanewline
\isacommand{axclass}\isanewline
~~plus\_semigroup~<~{"}term{"}\isanewline
~~assoc:~{"}(x~{\isasymOplus}~y)~{\isasymOplus}~z~=~x~{\isasymOplus}~(y~{\isasymOplus}~z){"}%
\begin{isamarkuptext}%
\noindent Even if classes $plus_semigroup$ and $times_semigroup$ both
represent semigroups in a sense, they are certainly not the same!%
\end{isamarkuptext}%
\isacommand{end}\end{isabelle}%