%
\begin{isabellebody}%
\def\isabellecontext{Overloading}%
\isacommand{instance}\ list\ {\isacharcolon}{\isacharcolon}\ {\isacharparenleft}{\isachardoublequote}term{\isachardoublequote}{\isacharparenright}ordrel\isanewline
\isacommand{by}\ intro{\isacharunderscore}classes%
\begin{isamarkuptext}%
\noindent
This \isacommand{instance} declaration can be read like the declaration of
a function on types: the constructor \isa{list} maps types of class \isa{term}, i.e.\ all HOL types, to types of class \isa{ordrel}, i.e.\
if \isa{ty\ {\isacharcolon}{\isacharcolon}\ term} then \isa{ty\ list\ {\isacharcolon}{\isacharcolon}\ ordrel}.
Of course we should also define the meaning of \isa{{\isacharless}{\isacharless}{\isacharequal}} and
\isa{{\isacharless}{\isacharless}} on lists:%
\end{isamarkuptext}%
\isacommand{defs}\ {\isacharparenleft}\isakeyword{overloaded}{\isacharparenright}\isanewline
prefix{\isacharunderscore}def{\isacharcolon}\isanewline
\ \ {\isachardoublequote}xs\ {\isacharless}{\isacharless}{\isacharequal}\ {\isacharparenleft}ys{\isacharcolon}{\isacharcolon}{\isacharprime}a{\isacharcolon}{\isacharcolon}ordrel\ list{\isacharparenright}\ \ {\isasymequiv}\ \ {\isasymexists}zs{\isachardot}\ ys\ {\isacharequal}\ xs{\isacharat}zs{\isachardoublequote}\isanewline
strict{\isacharunderscore}prefix{\isacharunderscore}def{\isacharcolon}\isanewline
\ \ {\isachardoublequote}xs\ {\isacharless}{\isacharless}\ {\isacharparenleft}ys{\isacharcolon}{\isacharcolon}{\isacharprime}a{\isacharcolon}{\isacharcolon}ordrel\ list{\isacharparenright}\ \ \ {\isasymequiv}\ \ xs\ {\isacharless}{\isacharless}{\isacharequal}\ ys\ {\isasymand}\ xs\ {\isasymnoteq}\ ys{\isachardoublequote}%
\begin{isamarkuptext}%
We could also have made the second definition non-overloaded once and for
all: \isa{x\ {\isacharless}{\isacharless}\ y\ {\isasymequiv}\ x\ {\isacharless}{\isacharless}{\isacharequal}\ y\ {\isasymand}\ x\ {\isasymnoteq}\ y}. This would have saved us writing
many similar definitions at different types, but it would also have fixed
that \isa{{\isacharless}{\isacharless}} is defined in terms of \isa{{\isacharless}{\isacharless}{\isacharequal}} and never the other way
around. Below you will see why we want to avoid this asymmetry.%
\end{isamarkuptext}%
\end{isabellebody}%
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