%
\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}\ intro{\isacharunderscore}classes\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}\ intro{\isacharunderscore}classes\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 $\isakeyword{instance}$ 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 \isa{intro{\isacharunderscore}classes} proof method
does 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
$\isakeyword{instance}~c@1 < c@2$, namely \emph{abstract
instantiation} --- $c@1$ is more special than $c@2$ and thus an
instance of $c@2$. Even more interesting for practical applications
are \emph{concrete instantiations} of axiomatic type classes. That
is, certain simple schemes $(\alpha@1, \ldots, \alpha@n)t :: c$ 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 $\isakeyword{instance}$ 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 $c^\tau \equiv t$ may also contain
constants of name $c$ on the right-hand side --- if these have types
that are structurally simpler than $\tau$.
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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