author | huffman |
Thu, 14 Jul 2005 01:04:30 +0200 | |
changeset 16823 | 13f3768a6f14 |
parent 12338 | de0f4a63baa5 |
child 17132 | 153fe83804c9 |
permissions | -rw-r--r-- |
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\begin{isabellebody}% |
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\def\isabellecontext{Product}% |
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\isamarkupheader{Syntactic classes% |
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} |
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\isamarkuptrue% |
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\isacommand{theory}\ Product\ {\isacharequal}\ Main{\isacharcolon}\isamarkupfalse% |
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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}\isanewline |
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\ \ product\ {\isasymsubseteq}\ type\isanewline |
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\isamarkupfalse% |
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\isacommand{consts}\isanewline |
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\ \ product\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharprime}a{\isasymColon}product\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a{\isachardoublequote}\ \ \ \ {\isacharparenleft}\isakeyword{infixl}\ {\isachardoublequote}{\isasymodot}{\isachardoublequote}\ {\isadigit{7}}{\isadigit{0}}{\isacharparenright}\isamarkupfalse% |
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% |
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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 $\INSTANCE$ with the default proof $\DDOT$. |
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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}\ bool\ {\isacharcolon}{\isacharcolon}\ product\ \isamarkupfalse% |
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\isacommand{{\isachardot}{\isachardot}}\isanewline |
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\isamarkupfalse% |
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\isacommand{defs}\ {\isacharparenleft}\isakeyword{overloaded}{\isacharparenright}\isanewline |
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\ \ product{\isacharunderscore}bool{\isacharunderscore}def{\isacharcolon}\ {\isachardoublequote}x\ {\isasymodot}\ y\ {\isasymequiv}\ x\ {\isasymand}\ y{\isachardoublequote}\isamarkupfalse% |
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% |
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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 $\DEFS$ are not |
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directly connected with $\INSTANCE$ at all! We were just following |
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our convention to specify \isa{{\isasymodot}} on \isa{bool} after having |
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instantiated \isa{bool\ {\isasymColon}\ product}. Isabelle does not require |
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these definitions, which is in contrast to programming languages like |
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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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\isacommand{end}\isanewline |
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\isamarkupfalse% |
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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: |