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\begin{isabellebody}%
\def\isabellecontext{Semigroups}%
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\isamarkupheader{Semigroups%
}
\isamarkuptrue%
\isacommand{theory}\ Semigroups\ {\isacharequal}\ Main{\isacharcolon}\isamarkupfalse%
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\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 \isa{{\isacharprime}a}. Class axioms typically contain polymorphic constants that
depend on this type \isa{{\isacharprime}a}. These \emph{characteristic
constants} behave like operations associated with the ``carrier''
type \isa{{\isacharprime}a}.
We illustrate these basic concepts by the following formulation of
semigroups.%
\end{isamarkuptext}%
\isamarkuptrue%
\isacommand{consts}\isanewline
\ \ times\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a{\isachardoublequote}\ \ \ \ {\isacharparenleft}\isakeyword{infixl}\ {\isachardoublequote}{\isasymodot}{\isachardoublequote}\ {\isadigit{7}}{\isadigit{0}}{\isacharparenright}\isanewline
\isamarkupfalse%
\isacommand{axclass}\ semigroup\ {\isasymsubseteq}\ type\isanewline
\ \ assoc{\isacharcolon}\ {\isachardoublequote}{\isacharparenleft}x\ {\isasymodot}\ y{\isacharparenright}\ {\isasymodot}\ z\ {\isacharequal}\ x\ {\isasymodot}\ {\isacharparenleft}y\ {\isasymodot}\ z{\isacharparenright}{\isachardoublequote}\isamarkupfalse%
%
\begin{isamarkuptext}%
\noindent Above we have first declared a polymorphic constant \isa{{\isasymodot}\ {\isasymColon}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a} and then defined the class \isa{semigroup} of
all types \isa{{\isasymtau}} such that \isa{{\isasymodot}\ {\isasymColon}\ {\isasymtau}\ {\isasymRightarrow}\ {\isasymtau}\ {\isasymRightarrow}\ {\isasymtau}} is indeed an
associative operator. The \isa{assoc} axiom contains exactly one
type variable, which is invisible in the above presentation, though.
Also note that free term variables (like \isa{x}, \isa{y},
\isa{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 \isa{{\isacharprime}a} and a fixed
\emph{signature}. Different signatures require different classes.
Below, class \isa{plus{\isacharunderscore}semigroup} represents semigroups \isa{{\isacharparenleft}{\isasymtau}{\isacharcomma}\ {\isasymoplus}\isactrlsup {\isasymtau}{\isacharparenright}}, while the original \isa{semigroup} would
correspond to semigroups of the form \isa{{\isacharparenleft}{\isasymtau}{\isacharcomma}\ {\isasymodot}\isactrlsup {\isasymtau}{\isacharparenright}}.%
\end{isamarkuptext}%
\isamarkuptrue%
\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
\isamarkupfalse%
\isacommand{axclass}\ plus{\isacharunderscore}semigroup\ {\isasymsubseteq}\ type\isanewline
\ \ assoc{\isacharcolon}\ {\isachardoublequote}{\isacharparenleft}x\ {\isasymoplus}\ y{\isacharparenright}\ {\isasymoplus}\ z\ {\isacharequal}\ x\ {\isasymoplus}\ {\isacharparenleft}y\ {\isasymoplus}\ z{\isacharparenright}{\isachardoublequote}\isamarkupfalse%
%
\begin{isamarkuptext}%
\noindent Even if classes \isa{plus{\isacharunderscore}semigroup} and \isa{semigroup} both represent semigroups in a sense, they are certainly
not quite the same.%
\end{isamarkuptext}%
\isamarkuptrue%
\isacommand{end}\isanewline
\isamarkupfalse%
\end{isabellebody}%
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