\begin{isabelle}%
%
\isamarkupheader{Syntactic classes}
\isacommand{theory}~Product~=~Main:%
\begin{isamarkuptext}%
\medskip\noindent There is still a feature of Isabelle's type system
left that we have not yet discussed. When declaring polymorphic
constants $c :: \sigma$, the type variables occurring in $\sigma$ may
be constrained by type classes (or even general sorts) in an
arbitrary way. Note that by default, in Isabelle/HOL the declaration
$\TIMES :: \alpha \To \alpha \To \alpha$ is actually an abbreviation
for $\TIMES :: (\alpha::term) \To \alpha \To \alpha$. Since class
$term$ is the universal class of HOL, this is not really a constraint
at all.
The $product$ class below provides a less degenerate example of
syntactic type classes.%
\end{isamarkuptext}%
\isacommand{axclass}\isanewline
~~product~<~{"}term{"}\isanewline
\isacommand{consts}\isanewline
~~product~::~{"}'a::product~{\isasymRightarrow}~'a~{\isasymRightarrow}~'a{"}~~~~(\isakeyword{infixl}~{"}{\isasymOtimes}{"}~70)%
\begin{isamarkuptext}%
Here class $product$ is defined as subclass of $term$ without any
additional axioms. This effects in logical equivalence of $product$
and $term$, as is reflected by the trivial introduction rule
generated for this definition.
\medskip So what is the difference of declaring $\TIMES :: (\alpha ::
product) \To \alpha \To \alpha$ vs.\ declaring $\TIMES :: (\alpha ::
term) \To \alpha \To \alpha$ anyway? In this particular case where
$product \equiv term$, it should be obvious that both declarations
are the same from the logic's point of view. It even makes the most
sense to remove sort constraints from constant declarations, as far
as the purely logical meaning is concerned \cite{Wenzel:1997:TPHOL}.
On the other hand there are syntactic differences, of course.
Constants $\TIMES^\tau$ are rejected by the type-checker, unless the
arity $\tau :: product$ is part of the type signature. In our
example, this arity may be always added when required by means of an
$\isarkeyword{instance}$ with the trivial proof $\BY{intro_classes}$.
\medskip Thus, we may observe the following discipline of using
syntactic classes. Overloaded polymorphic constants have their type
arguments restricted to an associated (logically trivial) class $c$.
Only immediately before \emph{specifying} these constants on a
certain type $\tau$ do we instantiate $\tau :: c$.
This is done for class $product$ and type $bool$ as follows.%
\end{isamarkuptext}%
\isacommand{instance}~bool~::~product\isanewline
~~\isacommand{by}~intro\_classes\isanewline
\isacommand{defs}~(\isakeyword{overloaded})\isanewline
~~product\_bool\_def:~{"}x~{\isasymOtimes}~y~{\isasymequiv}~x~{\isasymand}~y{"}%
\begin{isamarkuptext}%
The definition $prod_bool_def$ becomes syntactically well-formed only
after the arity $bool :: product$ is made known to the type checker.
\medskip It is very important to see that above $\DEFS$ are not
directly connected with $\isarkeyword{instance}$ at all! We were
just following our convention to specify $\TIMES$ on $bool$ after
having instantiated $bool :: product$. Isabelle does not require
these definitions, which is in contrast to programming languages like
Haskell \cite{haskell-report}.
\medskip While Isabelle type classes and those of Haskell are almost
the same as far as type-checking and type inference are concerned,
there are important semantic differences. Haskell classes require
their instances to \emph{provide operations} of certain \emph{names}.
Therefore, its \texttt{instance} has a \texttt{where} part that tells
the system what these ``member functions'' should be.
This style of \texttt{instance} won't make much sense in Isabelle's
meta-logic, because there is no internal notion of ``providing
operations'' or even ``names of functions''.%
\end{isamarkuptext}%
\isacommand{end}\end{isabelle}%
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