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\begin{isabellebody}%
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\def\isabellecontext{Product}%
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%
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\isamarkupheader{Syntactic classes%
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}
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\isamarkuptrue%
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%
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\isadelimtheory
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%
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\endisadelimtheory
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%
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\isatagtheory
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\isacommand{theory}\isamarkupfalse%
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\ Product\ \isakeyword{imports}\ Main\ \isakeyword{begin}%
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\endisatagtheory
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{\isafoldtheory}%
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%
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\isadelimtheory
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%
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\endisadelimtheory
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%
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\begin{isamarkuptext}%
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\medskip\noindent There is still a feature of Isabelle's type system
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left that we have not yet discussed. When declaring polymorphic
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constants \isa{c\ {\isasymColon}\ {\isasymsigma}}, the type variables occurring in \isa{{\isasymsigma}}
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may be constrained by type classes (or even general sorts) in an
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arbitrary way. Note that by default, in Isabelle/HOL the
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declaration \isa{{\isasymodot}\ {\isasymColon}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a} is actually an abbreviation
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for \isa{{\isasymodot}\ {\isasymColon}\ {\isacharprime}a{\isasymColon}type\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a} Since class \isa{type} is the
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universal class of HOL, this is not really a constraint at all.
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The \isa{product} class below provides a less degenerate example of
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syntactic type classes.%
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\end{isamarkuptext}%
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\isamarkuptrue%
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\isacommand{axclass}\isamarkupfalse%
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\isanewline
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\ \ product\ {\isasymsubseteq}\ type\isanewline
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\isacommand{consts}\isamarkupfalse%
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\isanewline
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\ \ product\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharprime}a{\isasymColon}product\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a{\isachardoublequoteclose}\ \ \ \ {\isacharparenleft}\isakeyword{infixl}\ {\isachardoublequoteopen}{\isasymodot}{\isachardoublequoteclose}\ {\isadigit{7}}{\isadigit{0}}{\isacharparenright}%
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\begin{isamarkuptext}%
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Here class \isa{product} is defined as subclass of \isa{type}
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without any additional axioms. This effects in logical equivalence
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of \isa{product} and \isa{type}, as is reflected by the trivial
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introduction rule generated for this definition.
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\medskip So what is the difference of declaring \isa{{\isasymodot}\ {\isasymColon}\ {\isacharprime}a{\isasymColon}product\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a} vs.\ declaring \isa{{\isasymodot}\ {\isasymColon}\ {\isacharprime}a{\isasymColon}type\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a} anyway? In this particular case where \isa{product\ {\isasymequiv}\ type}, it should be obvious that both declarations are the same
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from the logic's point of view. It even makes the most sense to
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remove sort constraints from constant declarations, as far as the
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purely logical meaning is concerned \cite{Wenzel:1997:TPHOL}.
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On the other hand there are syntactic differences, of course.
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Constants \isa{{\isasymodot}} on some type \isa{{\isasymtau}} are rejected by the
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type-checker, unless the arity \isa{{\isasymtau}\ {\isasymColon}\ product} is part of the
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type signature. In our example, this arity may be always added when
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required by means of an \isakeyword{instance} with the default proof
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(double-dot).
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\medskip Thus, we may observe the following discipline of using
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syntactic classes. Overloaded polymorphic constants have their type
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arguments restricted to an associated (logically trivial) class
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\isa{c}. Only immediately before \emph{specifying} these
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constants on a certain type \isa{{\isasymtau}} do we instantiate \isa{{\isasymtau}\ {\isasymColon}\ c}.
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This is done for class \isa{product} and type \isa{bool} as
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follows.%
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\end{isamarkuptext}%
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\isamarkuptrue%
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\isacommand{instance}\isamarkupfalse%
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\ bool\ {\isacharcolon}{\isacharcolon}\ product%
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\isadelimproof
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\ %
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\endisadelimproof
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%
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\isatagproof
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\isacommand{{\isachardot}{\isachardot}}\isamarkupfalse%
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%
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\endisatagproof
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{\isafoldproof}%
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%
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\isadelimproof
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%
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\endisadelimproof
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\isanewline
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\isacommand{defs}\isamarkupfalse%
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\ {\isacharparenleft}\isakeyword{overloaded}{\isacharparenright}\isanewline
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\ \ product{\isacharunderscore}bool{\isacharunderscore}def{\isacharcolon}\ {\isachardoublequoteopen}x\ {\isasymodot}\ y\ {\isasymequiv}\ x\ {\isasymand}\ y{\isachardoublequoteclose}%
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\begin{isamarkuptext}%
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The definition \isa{prod{\isacharunderscore}bool{\isacharunderscore}def} becomes syntactically
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well-formed only after the arity \isa{bool\ {\isasymColon}\ product} is made
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known to the type checker.
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\medskip It is very important to see that above \isakeyword{defs} are
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not directly connected with \isakeyword{instance} at all! We were
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just following our convention to specify \isa{{\isasymodot}} on \isa{bool}
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after having instantiated \isa{bool\ {\isasymColon}\ product}. Isabelle does
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not require these definitions, which is in contrast to programming
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languages like Haskell \cite{haskell-report}.
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\medskip While Isabelle type classes and those of Haskell are almost
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the same as far as type-checking and type inference are concerned,
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there are important semantic differences. Haskell classes require
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their instances to \emph{provide operations} of certain \emph{names}.
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Therefore, its \texttt{instance} has a \texttt{where} part that tells
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the system what these ``member functions'' should be.
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This style of \texttt{instance} would not make much sense in
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Isabelle's meta-logic, because there is no internal notion of
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``providing operations'' or even ``names of functions''.%
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\end{isamarkuptext}%
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\isamarkuptrue%
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%
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\isadelimtheory
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%
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\endisadelimtheory
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%
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\isatagtheory
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\isacommand{end}\isamarkupfalse%
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%
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\endisatagtheory
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{\isafoldtheory}%
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%
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\isadelimtheory
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%
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\endisadelimtheory
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\isanewline
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\end{isabellebody}%
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%%% Local Variables:
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%%% mode: latex
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%%% TeX-master: "root"
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%%% End:
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