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\chapter{Introduction}


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A Haskellstyle typesystem \cite{haskellreport} with ordered typeclasses

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has been present in Isabelle since 1991 already \cite{nipkowsorts93}.


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Initially, classes have mainly served as a \emph{purely syntactic} tool to


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formulate polymorphic objectlogics in a clean way, such as the standard


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Isabelle formulation of manysorted FOL \cite{paulsonisabook}.

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Applying classes at the \emph{logical level} to provide a simple notion of


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abstract theories and instantiations to concrete ones, has been long proposed

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as well \cite{nipkowtypes93,nipkowsorts93}. At that time, Isabelle still

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lacked builtin support for these \emph{axiomatic type classes}. More


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importantly, their semantics was not yet fully fleshed out (and unnecessarily


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complicated, too).


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Since Isabelle94, actual axiomatic type classes have been an integral part of


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Isabelle's metalogic. This very simple implementation is based on a


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straightforward extension of traditional simplytyped HigherOrder Logic, by


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including types qualified by logical predicates and overloaded constant


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definitions (see \cite{Wenzel:1997:TPHOL} for further details).

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Yet even until Isabelle99, there used to be still a fundamental methodological

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problem in using axiomatic type classes conveniently, due to the traditional


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distinction of Isabelle theory files vs.\ ML proof scripts. This has been


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finally overcome with the advent of Isabelle/Isar theories


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\cite{isabelleisarref}: now definitions and proofs may be freely intermixed.


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This nicely accommodates the usual procedure of defining axiomatic type


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classes, proving abstract properties, defining operations on concrete types,


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proving concrete properties for instantiation of classes etc.


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\medskip


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So to cut a long story short, the present version of axiomatic type classes

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now provides an even more useful and convenient mechanism for lightweight


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abstract theories, without any special technical provisions to be observed by


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the user.

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\chapter{Examples}\label{sec:ex}


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Axiomatic type classes are a concept of Isabelle's metalogic


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\cite{paulsonisabook,Wenzel:1997:TPHOL}. They may be applied to any


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objectlogic that directly uses the meta type system, such as Isabelle/HOL


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\cite{isabelleHOL}. Subsequently, we present various examples that are all


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formulated within HOL, except the one of \secref{sec:exnatclass} which is in

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FOL. See also \url{http://isabelle.in.tum.de/library/HOL/AxClasses/} and

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\url{http://isabelle.in.tum.de/library/FOL/ex/NatClass.html}.

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\input{generated/Semigroups}


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\input{generated/Group}


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\input{generated/Product}

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\input{generated/NatClass}

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%% FIXME move some parts to ref or isarref manual (!?);


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% \chapter{The user interface of Isabelle's axclass package}


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% The actual axiomatic type class package of Isabelle/Pure mainly consists


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% of two new theory sections: \texttt{axclass} and \texttt{instance}. Some


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% typical applications of these have already been demonstrated in


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% \secref{sec:ex}, below their syntax and semantics are presented more


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% completely.


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% \section{The axclass section}


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% Within theory files, \texttt{axclass} introduces an axiomatic type class


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% definition. Its concrete syntax is:


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% \begin{matharray}{l}


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% \texttt{axclass} \\


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% \ \ c \texttt{ < } c@1\texttt, \ldots\texttt, c@n \\


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% \ \ id@1\ axm@1 \\


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% \ \ \vdots \\


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% \ \ id@m\ axm@m


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% \emphnd{matharray}


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% Where $c, c@1, \ldots, c@n$ are classes (category $id$ or

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% $string$) and $axm@1, \ldots, axm@m$ (with $m \geq

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% 0$) are formulas (category $string$).


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% Class $c$ has to be new, and sort $\{c@1, \ldots, c@n\}$ a subsort of


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% \texttt{logic}. Each class axiom $axm@j$ may contain any term


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% variables, but at most one type variable (which need not be the same


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% for all axioms). The sort of this type variable has to be a supersort


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% of $\{c@1, \ldots, c@n\}$.


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% \medskip


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% The \texttt{axclass} section declares $c$ as subclass of $c@1, \ldots,


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% c@n$ to the type signature.


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% Furthermore, $axm@1, \ldots, axm@m$ are turned into the


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% ``abstract axioms'' of $c$ with names $id@1, \ldots,


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% id@m$. This is done by replacing all occurring type variables


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% by $\alpha :: c$. Original axioms that do not contain any type


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% variable will be prefixed by the logical precondition


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% $\texttt{OFCLASS}(\alpha :: \texttt{logic}, c\texttt{_class})$.


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% Another axiom of name $c\texttt{I}$  the ``class $c$ introduction


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% rule''  is built from the respective universal closures of


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% $axm@1, \ldots, axm@m$ appropriately.


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% \section{The instance section}


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% Section \texttt{instance} proves class inclusions or type arities at the


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% logical level and then transfers these into the type signature.


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% Its concrete syntax is:


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% \begin{matharray}{l}


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% \texttt{instance} \\


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% \ \ [\ c@1 \texttt{ < } c@2 \ \


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% t \texttt{ ::\ (}sort@1\texttt, \ldots \texttt, sort@n\texttt) sort\ ] \\


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% \ \ [\ \texttt(name@1 \texttt, \ldots\texttt, name@m\texttt)\ ] \\


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% \ \ [\ \texttt{\{} text \texttt{\}}\ ]


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% \emphnd{matharray}


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% Where $c@1, c@2$ are classes and $t$ is an $n$place type constructor


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% (all of category $id$ or $string)$. Furthermore,


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% $sort@i$ are sorts in the usual Isabellesyntax.


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% \medskip


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% Internally, \texttt{instance} first sets up an appropriate goal that


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% expresses the class inclusion or type arity as a metaproposition.


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% Then tactic \texttt{AxClass.axclass_tac} is applied with all preceding


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% metadefinitions of the current theory file and the usersupplied


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% witnesses. The latter are $name@1, \ldots, name@m$, where


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% $id$ refers to an \MLname of a theorem, and $string$ to an


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% axiom of the current theory node\footnote{Thus, the user may reference


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% axioms from above this \texttt{instance} in the theory file. Note


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% that new axioms appear at the \MLtoplevel only after the file is


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% processed completely.}.


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% Tactic \texttt{AxClass.axclass_tac} first unfolds the class definition by


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% resolving with rule $c\texttt\texttt{I}$, and then applies the witnesses


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% according to their form: Metadefinitions are unfolded, all other


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% formulas are repeatedly resolved\footnote{This is done in a way that


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% enables proper object\emph{rules} to be used as witnesses for


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% corresponding class axioms.} with.


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% The final optional argument $text$ is \MLcode of an arbitrary


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% user tactic which is applied last to any remaining goals.


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% \medskip


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% Because of the complexity of \texttt{instance}'s witnessing mechanisms,


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% new users of the axclass package are advised to only use the simple


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% form $\texttt{instance}\ \ldots\ (id@1, \ldots, id@!m)$, where


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% the identifiers refer to theorems that are appropriate type instances


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% of the class axioms. This typically requires an auxiliary theory,


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% though, which defines some constants and then proves these witnesses.


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%%% Local Variables:


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%%% mode: latex


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%%% TeXmaster: "axclass"


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%%% End:


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% LocalWords: Isabelle FOL
