doc-src/AxClass/generated/Semigroups.tex

 \isamarkuptrue%
 \isacommand{theory}\ Semigroups\ {\isacharequal}\ Main{\isacharcolon}\isamarkupfalse%
 %
 \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}.
+  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.%
+  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}\ {\isachardoublequote}term{\isachardoublequote}\isanewline
+\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.
+  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}}.%
+  \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}\ {\isachardoublequote}term{\isachardoublequote}\isanewline
+\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.%
+  not quite the same.%
 \end{isamarkuptext}%
 \isamarkuptrue%
-\isacommand{end}\isamarkupfalse%
+\isacommand{end}\isanewline
+\isamarkupfalse%
 \end{isabellebody}%
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