%

\begin{isabellebody}%

\def\isabellecontext{Overloading{\isadigit{2}}}%

\isamarkupfalse%

%

\begin{isamarkuptext}%

Of course this is not the only possible definition of the two relations.

Componentwise comparison of lists of equal length also makes sense. This time

the elements of the list must also be of class \isa{ordrel} to permit their

comparison:%

\end{isamarkuptext}%

\isamarkuptrue%

\isacommand{instance}\ list\ {\isacharcolon}{\isacharcolon}\ {\isacharparenleft}ordrel{\isacharparenright}ordrel\isanewline

\isamarkupfalse%

\isacommand{by}\ intro{\isacharunderscore}classes\isanewline

\isanewline

\isamarkupfalse%

\isacommand{defs}\ {\isacharparenleft}\isakeyword{overloaded}{\isacharparenright}\isanewline

le{\isacharunderscore}list{\isacharunderscore}def{\isacharcolon}\ {\isachardoublequote}xs\ {\isacharless}{\isacharless}{\isacharequal}\ {\isacharparenleft}ys{\isacharcolon}{\isacharcolon}{\isacharprime}a{\isacharcolon}{\isacharcolon}ordrel\ list{\isacharparenright}\ {\isasymequiv}\isanewline

\ \ \ \ \ \ \ \ \ \ \ \ \ \ size\ xs\ {\isacharequal}\ size\ ys\ {\isasymand}\ {\isacharparenleft}{\isasymforall}i{\isacharless}size\ xs{\isachardot}\ xs{\isacharbang}i\ {\isacharless}{\isacharless}{\isacharequal}\ ys{\isacharbang}i{\isacharparenright}{\isachardoublequote}\isamarkupfalse%

%

\begin{isamarkuptext}%

\noindent

The infix function \isa{{\isacharbang}} yields the nth element of a list.

\begin{warn}

A type constructor can be instantiated in only one way to

a given type class.  For example, our two instantiations of \isa{list} must

reside in separate theories with disjoint scopes.

\end{warn}%

\end{isamarkuptext}%

\isamarkuptrue%

%

\isamarkupsubsubsection{Predefined Overloading%

}

\isamarkuptrue%

%

\begin{isamarkuptext}%

HOL comes with a number of overloaded constants and corresponding classes.

The most important ones are listed in Table~\ref{tab:overloading} in the appendix. They are

defined on all numeric types and sometimes on other types as well, for example

$-$ and \isa{{\isasymle}} on sets.

In addition there is a special syntax for bounded quantifiers:

\begin{center}

\begin{tabular}{lcl}

\isa{{\isasymforall}x{\isasymle}y{\isachardot}\ P\ x} & \isa{{\isasymrightleftharpoons}} & \isa{{\isachardoublequote}{\isasymforall}x{\isachardot}\ x\ {\isasymle}\ y\ {\isasymlongrightarrow}\ P\ x{\isachardoublequote}} \\

\isa{{\isasymexists}x{\isasymle}y{\isachardot}\ P\ x} & \isa{{\isasymrightleftharpoons}} & \isa{{\isachardoublequote}{\isasymexists}x{\isachardot}\ x\ {\isasymle}\ y\ {\isasymand}\ P\ x{\isachardoublequote}}

\end{tabular}

\end{center}

And analogously for \isa{{\isacharless}} instead of \isa{{\isasymle}}.%

\end{isamarkuptext}%

\isamarkuptrue%

\isamarkupfalse%

\end{isabellebody}%

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