diff -r 153fe83804c9 -r 096792bdc58e doc-src/AxClass/generated/Semigroups.tex --- a/doc-src/AxClass/generated/Semigroups.tex Fri Aug 19 22:28:23 2005 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,88 +0,0 @@ -% -\begin{isabellebody}% -\def\isabellecontext{Semigroups}% -\isamarkuptrue% -% -\isamarkupheader{Semigroups% -} -% -\isadelimtheory -% -\endisadelimtheory -% -\isatagtheory -\isamarkupfalse% -\isacommand{theory}\ Semigroups\ \isakeyword{imports}\ Main\ \isakeyword{begin}% -\endisatagtheory -{\isafoldtheory}% -% -\isadelimtheory -% -\endisadelimtheory -\isamarkuptrue% -% -\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}% -\isamarkupfalse% -\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}\isamarkuptrue% -% -\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}% -\isamarkupfalse% -\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}\isamarkuptrue% -% -\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}% -% -\isadelimtheory -% -\endisadelimtheory -% -\isatagtheory -\isamarkupfalse% -\isacommand{end}% -\endisatagtheory -{\isafoldtheory}% -% -\isadelimtheory -% -\endisadelimtheory -\isanewline -\end{isabellebody}% -%%% Local Variables: -%%% mode: latex -%%% TeX-master: "root" -%%% End: