%
\begin{isabellebody}%
\def\isabellecontext{Group}%
%
\isamarkupheader{Basic group theory%
}
\isacommand{theory}\ Group\ {\isacharequal}\ Main{\isacharcolon}%
\begin{isamarkuptext}%
\medskip\noindent The meta-level type system of Isabelle supports
 \emph{intersections} and \emph{inclusions} of type classes. These
 directly correspond to intersections and inclusions of type
 predicates in a purely set theoretic sense. This is sufficient as a
 means to describe simple hierarchies of structures.  As an
 illustration, we use the well-known example of semigroups, monoids,
 general groups and Abelian groups.%
\end{isamarkuptext}%
%
\isamarkupsubsection{Monoids and Groups%
}
%
\begin{isamarkuptext}%
First we declare some polymorphic constants required later for the
 signature parts of our structures.%
\end{isamarkuptext}%
\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
\ \ inverse\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a{\isachardoublequote}\ \ \ \ {\isacharparenleft}{\isachardoublequote}{\isacharparenleft}{\isacharunderscore}{\isasyminv}{\isacharparenright}{\isachardoublequote}\ {\isacharbrackleft}{\isadigit{1}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isacharbrackright}\ {\isadigit{9}}{\isadigit{9}}{\isadigit{9}}{\isacharparenright}\isanewline
\ \ one\ {\isacharcolon}{\isacharcolon}\ {\isacharprime}a\ \ \ \ {\isacharparenleft}{\isachardoublequote}{\isasymunit}{\isachardoublequote}{\isacharparenright}%
\begin{isamarkuptext}%
\noindent Next we define class \isa{monoid} of monoids with
 operations \isa{{\isasymodot}} and \isa{{\isasymunit}}.  Note that multiple class
 axioms are allowed for user convenience --- they simply represent the
 conjunction of their respective universal closures.%
\end{isamarkuptext}%
\isacommand{axclass}\ monoid\ {\isacharless}\ {\isachardoublequote}term{\isachardoublequote}\isanewline
\ \ assoc{\isacharcolon}\ {\isachardoublequote}{\isacharparenleft}x\ {\isasymodot}\ y{\isacharparenright}\ {\isasymodot}\ z\ {\isacharequal}\ x\ {\isasymodot}\ {\isacharparenleft}y\ {\isasymodot}\ z{\isacharparenright}{\isachardoublequote}\isanewline
\ \ left{\isacharunderscore}unit{\isacharcolon}\ {\isachardoublequote}{\isasymunit}\ {\isasymodot}\ x\ {\isacharequal}\ x{\isachardoublequote}\isanewline
\ \ right{\isacharunderscore}unit{\isacharcolon}\ {\isachardoublequote}x\ {\isasymodot}\ {\isasymunit}\ {\isacharequal}\ x{\isachardoublequote}%
\begin{isamarkuptext}%
\noindent So class \isa{monoid} contains exactly those types \isa{{\isasymtau}} where \isa{{\isasymodot}\ {\isasymColon}\ {\isasymtau}\ {\isasymRightarrow}\ {\isasymtau}\ {\isasymRightarrow}\ {\isasymtau}} and \isa{{\isasymunit}\ {\isasymColon}\ {\isasymtau}} are
 specified appropriately, such that \isa{{\isasymodot}} is associative and
 \isa{{\isasymunit}} is a left and right unit element for the \isa{{\isasymodot}}
 operation.%
\end{isamarkuptext}%
%
\begin{isamarkuptext}%
\medskip Independently of \isa{monoid}, we now define a linear
 hierarchy of semigroups, general groups and Abelian groups.  Note
 that the names of class axioms are automatically qualified with each
 class name, so we may re-use common names such as \isa{assoc}.%
\end{isamarkuptext}%
\isacommand{axclass}\ semigroup\ {\isacharless}\ {\isachardoublequote}term{\isachardoublequote}\isanewline
\ \ assoc{\isacharcolon}\ {\isachardoublequote}{\isacharparenleft}x\ {\isasymodot}\ y{\isacharparenright}\ {\isasymodot}\ z\ {\isacharequal}\ x\ {\isasymodot}\ {\isacharparenleft}y\ {\isasymodot}\ z{\isacharparenright}{\isachardoublequote}\isanewline
\isanewline
\isacommand{axclass}\ group\ {\isacharless}\ semigroup\isanewline
\ \ left{\isacharunderscore}unit{\isacharcolon}\ {\isachardoublequote}{\isasymunit}\ {\isasymodot}\ x\ {\isacharequal}\ x{\isachardoublequote}\isanewline
\ \ left{\isacharunderscore}inverse{\isacharcolon}\ {\isachardoublequote}x{\isasyminv}\ {\isasymodot}\ x\ {\isacharequal}\ {\isasymunit}{\isachardoublequote}\isanewline
\isanewline
\isacommand{axclass}\ agroup\ {\isacharless}\ group\isanewline
\ \ commute{\isacharcolon}\ {\isachardoublequote}x\ {\isasymodot}\ y\ {\isacharequal}\ y\ {\isasymodot}\ x{\isachardoublequote}%
\begin{isamarkuptext}%
\noindent Class \isa{group} inherits associativity of \isa{{\isasymodot}}
 from \isa{semigroup} and adds two further group axioms. Similarly,
 \isa{agroup} is defined as the subset of \isa{group} such that
 for all of its elements \isa{{\isasymtau}}, the operation \isa{{\isasymodot}\ {\isasymColon}\ {\isasymtau}\ {\isasymRightarrow}\ {\isasymtau}\ {\isasymRightarrow}\ {\isasymtau}}
 is even commutative.%
\end{isamarkuptext}%
%
\isamarkupsubsection{Abstract reasoning%
}
%
\begin{isamarkuptext}%
In a sense, axiomatic type classes may be viewed as \emph{abstract
 theories}.  Above class definitions gives rise to abstract axioms
 \isa{assoc}, \isa{left{\isacharunderscore}unit}, \isa{left{\isacharunderscore}inverse}, \isa{commute}, where any of these contain a type variable \isa{{\isacharprime}a\ {\isasymColon}\ c}
 that is restricted to types of the corresponding class \isa{c}.
 \emph{Sort constraints} like this express a logical precondition for
 the whole formula.  For example, \isa{assoc} states that for all
 \isa{{\isasymtau}}, provided that \isa{{\isasymtau}\ {\isasymColon}\ semigroup}, the operation
 \isa{{\isasymodot}\ {\isasymColon}\ {\isasymtau}\ {\isasymRightarrow}\ {\isasymtau}\ {\isasymRightarrow}\ {\isasymtau}} is associative.

 \medskip From a technical point of view, abstract axioms are just
 ordinary Isabelle theorems, which may be used in proofs without
 special treatment.  Such ``abstract proofs'' usually yield new
 ``abstract theorems''.  For example, we may now derive the following
 well-known laws of general groups.%
\end{isamarkuptext}%
\isacommand{theorem}\ group{\isacharunderscore}right{\isacharunderscore}inverse{\isacharcolon}\ {\isachardoublequote}x\ {\isasymodot}\ x{\isasyminv}\ {\isacharequal}\ {\isacharparenleft}{\isasymunit}{\isasymColon}{\isacharprime}a{\isasymColon}group{\isacharparenright}{\isachardoublequote}\isanewline
\isacommand{proof}\ {\isacharminus}\isanewline
\ \ \isacommand{have}\ {\isachardoublequote}x\ {\isasymodot}\ x{\isasyminv}\ {\isacharequal}\ {\isasymunit}\ {\isasymodot}\ {\isacharparenleft}x\ {\isasymodot}\ x{\isasyminv}{\isacharparenright}{\isachardoublequote}\isanewline
\ \ \ \ \isacommand{by}\ {\isacharparenleft}simp\ only{\isacharcolon}\ group{\isachardot}left{\isacharunderscore}unit{\isacharparenright}\isanewline
\ \ \isacommand{also}\ \isacommand{have}\ {\isachardoublequote}{\isachardot}{\isachardot}{\isachardot}\ {\isacharequal}\ {\isasymunit}\ {\isasymodot}\ x\ {\isasymodot}\ x{\isasyminv}{\isachardoublequote}\isanewline
\ \ \ \ \isacommand{by}\ {\isacharparenleft}simp\ only{\isacharcolon}\ semigroup{\isachardot}assoc{\isacharparenright}\isanewline
\ \ \isacommand{also}\ \isacommand{have}\ {\isachardoublequote}{\isachardot}{\isachardot}{\isachardot}\ {\isacharequal}\ {\isacharparenleft}x{\isasyminv}{\isacharparenright}{\isasyminv}\ {\isasymodot}\ x{\isasyminv}\ {\isasymodot}\ x\ {\isasymodot}\ x{\isasyminv}{\isachardoublequote}\isanewline
\ \ \ \ \isacommand{by}\ {\isacharparenleft}simp\ only{\isacharcolon}\ group{\isachardot}left{\isacharunderscore}inverse{\isacharparenright}\isanewline
\ \ \isacommand{also}\ \isacommand{have}\ {\isachardoublequote}{\isachardot}{\isachardot}{\isachardot}\ {\isacharequal}\ {\isacharparenleft}x{\isasyminv}{\isacharparenright}{\isasyminv}\ {\isasymodot}\ {\isacharparenleft}x{\isasyminv}\ {\isasymodot}\ x{\isacharparenright}\ {\isasymodot}\ x{\isasyminv}{\isachardoublequote}\isanewline
\ \ \ \ \isacommand{by}\ {\isacharparenleft}simp\ only{\isacharcolon}\ semigroup{\isachardot}assoc{\isacharparenright}\isanewline
\ \ \isacommand{also}\ \isacommand{have}\ {\isachardoublequote}{\isachardot}{\isachardot}{\isachardot}\ {\isacharequal}\ {\isacharparenleft}x{\isasyminv}{\isacharparenright}{\isasyminv}\ {\isasymodot}\ {\isasymunit}\ {\isasymodot}\ x{\isasyminv}{\isachardoublequote}\isanewline
\ \ \ \ \isacommand{by}\ {\isacharparenleft}simp\ only{\isacharcolon}\ group{\isachardot}left{\isacharunderscore}inverse{\isacharparenright}\isanewline
\ \ \isacommand{also}\ \isacommand{have}\ {\isachardoublequote}{\isachardot}{\isachardot}{\isachardot}\ {\isacharequal}\ {\isacharparenleft}x{\isasyminv}{\isacharparenright}{\isasyminv}\ {\isasymodot}\ {\isacharparenleft}{\isasymunit}\ {\isasymodot}\ x{\isasyminv}{\isacharparenright}{\isachardoublequote}\isanewline
\ \ \ \ \isacommand{by}\ {\isacharparenleft}simp\ only{\isacharcolon}\ semigroup{\isachardot}assoc{\isacharparenright}\isanewline
\ \ \isacommand{also}\ \isacommand{have}\ {\isachardoublequote}{\isachardot}{\isachardot}{\isachardot}\ {\isacharequal}\ {\isacharparenleft}x{\isasyminv}{\isacharparenright}{\isasyminv}\ {\isasymodot}\ x{\isasyminv}{\isachardoublequote}\isanewline
\ \ \ \ \isacommand{by}\ {\isacharparenleft}simp\ only{\isacharcolon}\ group{\isachardot}left{\isacharunderscore}unit{\isacharparenright}\isanewline
\ \ \isacommand{also}\ \isacommand{have}\ {\isachardoublequote}{\isachardot}{\isachardot}{\isachardot}\ {\isacharequal}\ {\isasymunit}{\isachardoublequote}\isanewline
\ \ \ \ \isacommand{by}\ {\isacharparenleft}simp\ only{\isacharcolon}\ group{\isachardot}left{\isacharunderscore}inverse{\isacharparenright}\isanewline
\ \ \isacommand{finally}\ \isacommand{show}\ {\isacharquery}thesis\ \isacommand{{\isachardot}}\isanewline
\isacommand{qed}%
\begin{isamarkuptext}%
\noindent With \isa{group{\isacharunderscore}right{\isacharunderscore}inverse} already available, \isa{group{\isacharunderscore}right{\isacharunderscore}unit}\label{thm:group-right-unit} is now established much
 easier.%
\end{isamarkuptext}%
\isacommand{theorem}\ group{\isacharunderscore}right{\isacharunderscore}unit{\isacharcolon}\ {\isachardoublequote}x\ {\isasymodot}\ {\isasymunit}\ {\isacharequal}\ {\isacharparenleft}x{\isasymColon}{\isacharprime}a{\isasymColon}group{\isacharparenright}{\isachardoublequote}\isanewline
\isacommand{proof}\ {\isacharminus}\isanewline
\ \ \isacommand{have}\ {\isachardoublequote}x\ {\isasymodot}\ {\isasymunit}\ {\isacharequal}\ x\ {\isasymodot}\ {\isacharparenleft}x{\isasyminv}\ {\isasymodot}\ x{\isacharparenright}{\isachardoublequote}\isanewline
\ \ \ \ \isacommand{by}\ {\isacharparenleft}simp\ only{\isacharcolon}\ group{\isachardot}left{\isacharunderscore}inverse{\isacharparenright}\isanewline
\ \ \isacommand{also}\ \isacommand{have}\ {\isachardoublequote}{\isachardot}{\isachardot}{\isachardot}\ {\isacharequal}\ x\ {\isasymodot}\ x{\isasyminv}\ {\isasymodot}\ x{\isachardoublequote}\isanewline
\ \ \ \ \isacommand{by}\ {\isacharparenleft}simp\ only{\isacharcolon}\ semigroup{\isachardot}assoc{\isacharparenright}\isanewline
\ \ \isacommand{also}\ \isacommand{have}\ {\isachardoublequote}{\isachardot}{\isachardot}{\isachardot}\ {\isacharequal}\ {\isasymunit}\ {\isasymodot}\ x{\isachardoublequote}\isanewline
\ \ \ \ \isacommand{by}\ {\isacharparenleft}simp\ only{\isacharcolon}\ group{\isacharunderscore}right{\isacharunderscore}inverse{\isacharparenright}\isanewline
\ \ \isacommand{also}\ \isacommand{have}\ {\isachardoublequote}{\isachardot}{\isachardot}{\isachardot}\ {\isacharequal}\ x{\isachardoublequote}\isanewline
\ \ \ \ \isacommand{by}\ {\isacharparenleft}simp\ only{\isacharcolon}\ group{\isachardot}left{\isacharunderscore}unit{\isacharparenright}\isanewline
\ \ \isacommand{finally}\ \isacommand{show}\ {\isacharquery}thesis\ \isacommand{{\isachardot}}\isanewline
\isacommand{qed}%
\begin{isamarkuptext}%
\medskip Abstract theorems may be instantiated to only those types
 \isa{{\isasymtau}} where the appropriate class membership \isa{{\isasymtau}\ {\isasymColon}\ c} is
 known at Isabelle's type signature level.  Since we have \isa{agroup\ {\isasymsubseteq}\ group\ {\isasymsubseteq}\ semigroup} by definition, all theorems of \isa{semigroup} and \isa{group} are automatically inherited by \isa{group} and \isa{agroup}.%
\end{isamarkuptext}%
%
\isamarkupsubsection{Abstract instantiation%
}
%
\begin{isamarkuptext}%
From the definition, the \isa{monoid} and \isa{group} classes
 have been independent.  Note that for monoids, \isa{right{\isacharunderscore}unit} had
 to be included as an axiom, but for groups both \isa{right{\isacharunderscore}unit}
 and \isa{right{\isacharunderscore}inverse} are derivable from the other axioms.  With
 \isa{group{\isacharunderscore}right{\isacharunderscore}unit} derived as a theorem of group theory (see
 page~\pageref{thm:group-right-unit}), we may now instantiate \isa{monoid\ {\isasymsubseteq}\ semigroup} and \isa{group\ {\isasymsubseteq}\ monoid} properly as
 follows (cf.\ \figref{fig:monoid-group}).

 \begin{figure}[htbp]
   \begin{center}
     \small
     \unitlength 0.6mm
     \begin{picture}(65,90)(0,-10)
       \put(15,10){\line(0,1){10}} \put(15,30){\line(0,1){10}}
       \put(15,50){\line(1,1){10}} \put(35,60){\line(1,-1){10}}
       \put(15,5){\makebox(0,0){\isa{agroup}}}
       \put(15,25){\makebox(0,0){\isa{group}}}
       \put(15,45){\makebox(0,0){\isa{semigroup}}}
       \put(30,65){\makebox(0,0){\isa{term}}} \put(50,45){\makebox(0,0){\isa{monoid}}}
     \end{picture}
     \hspace{4em}
     \begin{picture}(30,90)(0,0)
       \put(15,10){\line(0,1){10}} \put(15,30){\line(0,1){10}}
       \put(15,50){\line(0,1){10}} \put(15,70){\line(0,1){10}}
       \put(15,5){\makebox(0,0){\isa{agroup}}}
       \put(15,25){\makebox(0,0){\isa{group}}}
       \put(15,45){\makebox(0,0){\isa{monoid}}}
       \put(15,65){\makebox(0,0){\isa{semigroup}}}
       \put(15,85){\makebox(0,0){\isa{term}}}
     \end{picture}
     \caption{Monoids and groups: according to definition, and by proof}
     \label{fig:monoid-group}
   \end{center}
 \end{figure}%
\end{isamarkuptext}%
\isacommand{instance}\ monoid\ {\isacharless}\ semigroup\isanewline
\isacommand{proof}\isanewline
\ \ \isacommand{fix}\ x\ y\ z\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharprime}a{\isasymColon}monoid{\isachardoublequote}\isanewline
\ \ \isacommand{show}\ {\isachardoublequote}x\ {\isasymodot}\ y\ {\isasymodot}\ z\ {\isacharequal}\ x\ {\isasymodot}\ {\isacharparenleft}y\ {\isasymodot}\ z{\isacharparenright}{\isachardoublequote}\isanewline
\ \ \ \ \isacommand{by}\ {\isacharparenleft}rule\ monoid{\isachardot}assoc{\isacharparenright}\isanewline
\isacommand{qed}\isanewline
\isanewline
\isacommand{instance}\ group\ {\isacharless}\ monoid\isanewline
\isacommand{proof}\isanewline
\ \ \isacommand{fix}\ x\ y\ z\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharprime}a{\isasymColon}group{\isachardoublequote}\isanewline
\ \ \isacommand{show}\ {\isachardoublequote}x\ {\isasymodot}\ y\ {\isasymodot}\ z\ {\isacharequal}\ x\ {\isasymodot}\ {\isacharparenleft}y\ {\isasymodot}\ z{\isacharparenright}{\isachardoublequote}\isanewline
\ \ \ \ \isacommand{by}\ {\isacharparenleft}rule\ semigroup{\isachardot}assoc{\isacharparenright}\isanewline
\ \ \isacommand{show}\ {\isachardoublequote}{\isasymunit}\ {\isasymodot}\ x\ {\isacharequal}\ x{\isachardoublequote}\isanewline
\ \ \ \ \isacommand{by}\ {\isacharparenleft}rule\ group{\isachardot}left{\isacharunderscore}unit{\isacharparenright}\isanewline
\ \ \isacommand{show}\ {\isachardoublequote}x\ {\isasymodot}\ {\isasymunit}\ {\isacharequal}\ x{\isachardoublequote}\isanewline
\ \ \ \ \isacommand{by}\ {\isacharparenleft}rule\ group{\isacharunderscore}right{\isacharunderscore}unit{\isacharparenright}\isanewline
\isacommand{qed}%
\begin{isamarkuptext}%
\medskip The $\INSTANCE$ command sets up an appropriate goal that
 represents the class inclusion (or type arity, see
 \secref{sec:inst-arity}) to be proven (see also
 \cite{isabelle-isar-ref}).  The initial proof step causes
 back-chaining of class membership statements wrt.\ the hierarchy of
 any classes defined in the current theory; the effect is to reduce to
 the initial statement to a number of goals that directly correspond
 to any class axioms encountered on the path upwards through the class
 hierarchy.%
\end{isamarkuptext}%
%
\isamarkupsubsection{Concrete instantiation \label{sec:inst-arity}%
}
%
\begin{isamarkuptext}%
So far we have covered the case of the form $\INSTANCE$~\isa{c\isactrlsub {\isadigit{1}}\ {\isacharless}\ c\isactrlsub {\isadigit{2}}}, namely \emph{abstract instantiation} ---
 $c@1$ is more special than \isa{c\isactrlsub {\isadigit{1}}} and thus an instance
 of \isa{c\isactrlsub {\isadigit{2}}}.  Even more interesting for practical
 applications are \emph{concrete instantiations} of axiomatic type
 classes.  That is, certain simple schemes
 \isa{{\isacharparenleft}{\isasymalpha}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isasymalpha}\isactrlsub n{\isacharparenright}\ t\ {\isasymColon}\ c} of class membership may be
 established at the logical level and then transferred to Isabelle's
 type signature level.

 \medskip As a typical example, we show that type \isa{bool} with
 exclusive-or as \isa{{\isasymodot}} operation, identity as \isa{{\isasyminv}}, and
 \isa{False} as \isa{{\isasymunit}} forms an Abelian group.%
\end{isamarkuptext}%
\isacommand{defs}\ {\isacharparenleft}\isakeyword{overloaded}{\isacharparenright}\isanewline
\ \ times{\isacharunderscore}bool{\isacharunderscore}def{\isacharcolon}\ {\isachardoublequote}x\ {\isasymodot}\ y\ {\isasymequiv}\ x\ {\isasymnoteq}\ {\isacharparenleft}y{\isasymColon}bool{\isacharparenright}{\isachardoublequote}\isanewline
\ \ inverse{\isacharunderscore}bool{\isacharunderscore}def{\isacharcolon}\ {\isachardoublequote}x{\isasyminv}\ {\isasymequiv}\ x{\isasymColon}bool{\isachardoublequote}\isanewline
\ \ unit{\isacharunderscore}bool{\isacharunderscore}def{\isacharcolon}\ {\isachardoublequote}{\isasymunit}\ {\isasymequiv}\ False{\isachardoublequote}%
\begin{isamarkuptext}%
\medskip It is important to note that above $\DEFS$ are just
 overloaded meta-level constant definitions, where type classes are
 not yet involved at all.  This form of constant definition with
 overloading (and optional recursion over the syntactic structure of
 simple types) are admissible as definitional extensions of plain HOL
 \cite{Wenzel:1997:TPHOL}.  The Haskell-style type system is not
 required for overloading.  Nevertheless, overloaded definitions are
 best applied in the context of type classes.

 \medskip Since we have chosen above $\DEFS$ of the generic group
 operations on type \isa{bool} appropriately, the class membership
 \isa{bool\ {\isasymColon}\ agroup} may be now derived as follows.%
\end{isamarkuptext}%
\isacommand{instance}\ bool\ {\isacharcolon}{\isacharcolon}\ agroup\isanewline
\isacommand{proof}\ {\isacharparenleft}intro{\isacharunderscore}classes{\isacharcomma}\isanewline
\ \ \ \ unfold\ times{\isacharunderscore}bool{\isacharunderscore}def\ inverse{\isacharunderscore}bool{\isacharunderscore}def\ unit{\isacharunderscore}bool{\isacharunderscore}def{\isacharparenright}\isanewline
\ \ \isacommand{fix}\ x\ y\ z\isanewline
\ \ \isacommand{show}\ {\isachardoublequote}{\isacharparenleft}{\isacharparenleft}x\ {\isasymnoteq}\ y{\isacharparenright}\ {\isasymnoteq}\ z{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}x\ {\isasymnoteq}\ {\isacharparenleft}y\ {\isasymnoteq}\ z{\isacharparenright}{\isacharparenright}{\isachardoublequote}\ \isacommand{by}\ blast\isanewline
\ \ \isacommand{show}\ {\isachardoublequote}{\isacharparenleft}False\ {\isasymnoteq}\ x{\isacharparenright}\ {\isacharequal}\ x{\isachardoublequote}\ \isacommand{by}\ blast\isanewline
\ \ \isacommand{show}\ {\isachardoublequote}{\isacharparenleft}x\ {\isasymnoteq}\ x{\isacharparenright}\ {\isacharequal}\ False{\isachardoublequote}\ \isacommand{by}\ blast\isanewline
\ \ \isacommand{show}\ {\isachardoublequote}{\isacharparenleft}x\ {\isasymnoteq}\ y{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}y\ {\isasymnoteq}\ x{\isacharparenright}{\isachardoublequote}\ \isacommand{by}\ blast\isanewline
\isacommand{qed}%
\begin{isamarkuptext}%
The result of an $\INSTANCE$ statement is both expressed as a theorem
 of Isabelle's meta-logic, and as a type arity of the type signature.
 The latter enables type-inference system to take care of this new
 instance automatically.

 \medskip We could now also instantiate our group theory classes to
 many other concrete types.  For example, \isa{int\ {\isasymColon}\ agroup}
 (e.g.\ by defining \isa{{\isasymodot}} as addition, \isa{{\isasyminv}} as negation
 and \isa{{\isasymunit}} as zero) or \isa{list\ {\isasymColon}\ {\isacharparenleft}term{\isacharparenright}\ semigroup}
 (e.g.\ if \isa{{\isasymodot}} is defined as list append).  Thus, the
 characteristic constants \isa{{\isasymodot}}, \isa{{\isasyminv}}, \isa{{\isasymunit}}
 really become overloaded, i.e.\ have different meanings on different
 types.%
\end{isamarkuptext}%
%
\isamarkupsubsection{Lifting and Functors%
}
%
\begin{isamarkuptext}%
As already mentioned above, overloading in the simply-typed HOL
 systems may include recursion over the syntactic structure of types.
 That is, definitional equations \isa{c\isactrlsup {\isasymtau}\ {\isasymequiv}\ t} may also
 contain constants of name \isa{c} on the right-hand side --- if
 these have types that are structurally simpler than \isa{{\isasymtau}}.

 This feature enables us to \emph{lift operations}, say to Cartesian
 products, direct sums or function spaces.  Subsequently we lift
 \isa{{\isasymodot}} component-wise to binary products \isa{{\isacharprime}a\ {\isasymtimes}\ {\isacharprime}b}.%
\end{isamarkuptext}%
\isacommand{defs}\ {\isacharparenleft}\isakeyword{overloaded}{\isacharparenright}\isanewline
\ \ times{\isacharunderscore}prod{\isacharunderscore}def{\isacharcolon}\ {\isachardoublequote}p\ {\isasymodot}\ q\ {\isasymequiv}\ {\isacharparenleft}fst\ p\ {\isasymodot}\ fst\ q{\isacharcomma}\ snd\ p\ {\isasymodot}\ snd\ q{\isacharparenright}{\isachardoublequote}%
\begin{isamarkuptext}%
It is very easy to see that associativity of \isa{{\isasymodot}} on \isa{{\isacharprime}a}
 and \isa{{\isasymodot}} on \isa{{\isacharprime}b} transfers to \isa{{\isasymodot}} on \isa{{\isacharprime}a\ {\isasymtimes}\ {\isacharprime}b}.
 Hence the binary type constructor \isa{{\isasymodot}} maps semigroups to
 semigroups.  This may be established formally as follows.%
\end{isamarkuptext}%
\isacommand{instance}\ {\isacharasterisk}\ {\isacharcolon}{\isacharcolon}\ {\isacharparenleft}semigroup{\isacharcomma}\ semigroup{\isacharparenright}\ semigroup\isanewline
\isacommand{proof}\ {\isacharparenleft}intro{\isacharunderscore}classes{\isacharcomma}\ unfold\ times{\isacharunderscore}prod{\isacharunderscore}def{\isacharparenright}\isanewline
\ \ \isacommand{fix}\ p\ q\ r\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharprime}a{\isasymColon}semigroup\ {\isasymtimes}\ {\isacharprime}b{\isasymColon}semigroup{\isachardoublequote}\isanewline
\ \ \isacommand{show}\isanewline
\ \ \ \ {\isachardoublequote}{\isacharparenleft}fst\ {\isacharparenleft}fst\ p\ {\isasymodot}\ fst\ q{\isacharcomma}\ snd\ p\ {\isasymodot}\ snd\ q{\isacharparenright}\ {\isasymodot}\ fst\ r{\isacharcomma}\isanewline
\ \ \ \ \ \ snd\ {\isacharparenleft}fst\ p\ {\isasymodot}\ fst\ q{\isacharcomma}\ snd\ p\ {\isasymodot}\ snd\ q{\isacharparenright}\ {\isasymodot}\ snd\ r{\isacharparenright}\ {\isacharequal}\isanewline
\ \ \ \ \ \ \ {\isacharparenleft}fst\ p\ {\isasymodot}\ fst\ {\isacharparenleft}fst\ q\ {\isasymodot}\ fst\ r{\isacharcomma}\ snd\ q\ {\isasymodot}\ snd\ r{\isacharparenright}{\isacharcomma}\isanewline
\ \ \ \ \ \ \ \ snd\ p\ {\isasymodot}\ snd\ {\isacharparenleft}fst\ q\ {\isasymodot}\ fst\ r{\isacharcomma}\ snd\ q\ {\isasymodot}\ snd\ r{\isacharparenright}{\isacharparenright}{\isachardoublequote}\isanewline
\ \ \ \ \isacommand{by}\ {\isacharparenleft}simp\ add{\isacharcolon}\ semigroup{\isachardot}assoc{\isacharparenright}\isanewline
\isacommand{qed}%
\begin{isamarkuptext}%
Thus, if we view class instances as ``structures'', then overloaded
 constant definitions with recursion over types indirectly provide
 some kind of ``functors'' --- i.e.\ mappings between abstract
 theories.%
\end{isamarkuptext}%
\isacommand{end}\end{isabellebody}%
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