tuned generated stuff;

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/AxClass/Group/document/Group.tex	Fri Aug 19 22:44:01 2005 +0200
@@ -0,0 +1,511 @@
+%
+\begin{isabellebody}%
+\def\isabellecontext{Group}%
+\isamarkuptrue%
+%
+\isamarkupheader{Basic group theory%
+}
+%
+\isadelimtheory
+%
+\endisadelimtheory
+%
+\isatagtheory
+\isamarkupfalse%
+\isacommand{theory}\ Group\ \isakeyword{imports}\ Main\ \isakeyword{begin}%
+\endisatagtheory
+{\isafoldtheory}%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+\medskip\noindent The meta-level type system of Isabelle supports
+  \emph{intersections} and \emph{inclusions} of type classes. These
+  directly correspond to intersections and inclusions of type
+  predicates in a purely set theoretic sense. This is sufficient as a
+  means to describe simple hierarchies of structures.  As an
+  illustration, we use the well-known example of semigroups, monoids,
+  general groups and Abelian groups.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsubsection{Monoids and Groups%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+First we declare some polymorphic constants required later for the
+  signature parts of our structures.%
+\end{isamarkuptext}%
+\isamarkupfalse%
+\isacommand{consts}\isanewline
+\ \ times\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a{\isachardoublequote}\ \ \ \ {\isacharparenleft}\isakeyword{infixl}\ {\isachardoublequote}{\isasymodot}{\isachardoublequote}\ {\isadigit{7}}{\isadigit{0}}{\isacharparenright}\isanewline
+\ \ invers\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a{\isachardoublequote}\ \ \ \ {\isacharparenleft}{\isachardoublequote}{\isacharparenleft}{\isacharunderscore}{\isasyminv}{\isacharparenright}{\isachardoublequote}\ {\isacharbrackleft}{\isadigit{1}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isacharbrackright}\ {\isadigit{9}}{\isadigit{9}}{\isadigit{9}}{\isacharparenright}\isanewline
+\ \ one\ {\isacharcolon}{\isacharcolon}\ {\isacharprime}a\ \ \ \ {\isacharparenleft}{\isachardoublequote}{\isasymone}{\isachardoublequote}{\isacharparenright}\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+\noindent Next we define class \isa{monoid} of monoids with
+  operations \isa{{\isasymodot}} and \isa{{\isasymone}}.  Note that multiple class
+  axioms are allowed for user convenience --- they simply represent
+  the conjunction of their respective universal closures.%
+\end{isamarkuptext}%
+\isamarkupfalse%
+\isacommand{axclass}\ monoid\ {\isasymsubseteq}\ type\isanewline
+\ \ assoc{\isacharcolon}\ {\isachardoublequote}{\isacharparenleft}x\ {\isasymodot}\ y{\isacharparenright}\ {\isasymodot}\ z\ {\isacharequal}\ x\ {\isasymodot}\ {\isacharparenleft}y\ {\isasymodot}\ z{\isacharparenright}{\isachardoublequote}\isanewline
+\ \ left{\isacharunderscore}unit{\isacharcolon}\ {\isachardoublequote}{\isasymone}\ {\isasymodot}\ x\ {\isacharequal}\ x{\isachardoublequote}\isanewline
+\ \ right{\isacharunderscore}unit{\isacharcolon}\ {\isachardoublequote}x\ {\isasymodot}\ {\isasymone}\ {\isacharequal}\ x{\isachardoublequote}\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+\noindent So class \isa{monoid} contains exactly those types
+  \isa{{\isasymtau}} where \isa{{\isasymodot}\ {\isasymColon}\ {\isasymtau}\ {\isasymRightarrow}\ {\isasymtau}\ {\isasymRightarrow}\ {\isasymtau}} and \isa{{\isasymone}\ {\isasymColon}\ {\isasymtau}}
+  are specified appropriately, such that \isa{{\isasymodot}} is associative and
+  \isa{{\isasymone}} is a left and right unit element for the \isa{{\isasymodot}}
+  operation.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+\medskip Independently of \isa{monoid}, we now define a linear
+  hierarchy of semigroups, general groups and Abelian groups.  Note
+  that the names of class axioms are automatically qualified with each
+  class name, so we may re-use common names such as \isa{assoc}.%
+\end{isamarkuptext}%
+\isamarkupfalse%
+\isacommand{axclass}\ semigroup\ {\isasymsubseteq}\ type\isanewline
+\ \ assoc{\isacharcolon}\ {\isachardoublequote}{\isacharparenleft}x\ {\isasymodot}\ y{\isacharparenright}\ {\isasymodot}\ z\ {\isacharequal}\ x\ {\isasymodot}\ {\isacharparenleft}y\ {\isasymodot}\ z{\isacharparenright}{\isachardoublequote}\isanewline
+\isanewline
+\isamarkupfalse%
+\isacommand{axclass}\ group\ {\isasymsubseteq}\ semigroup\isanewline
+\ \ left{\isacharunderscore}unit{\isacharcolon}\ {\isachardoublequote}{\isasymone}\ {\isasymodot}\ x\ {\isacharequal}\ x{\isachardoublequote}\isanewline
+\ \ left{\isacharunderscore}inverse{\isacharcolon}\ {\isachardoublequote}x{\isasyminv}\ {\isasymodot}\ x\ {\isacharequal}\ {\isasymone}{\isachardoublequote}\isanewline
+\isanewline
+\isamarkupfalse%
+\isacommand{axclass}\ agroup\ {\isasymsubseteq}\ group\isanewline
+\ \ commute{\isacharcolon}\ {\isachardoublequote}x\ {\isasymodot}\ y\ {\isacharequal}\ y\ {\isasymodot}\ x{\isachardoublequote}\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+\noindent Class \isa{group} inherits associativity of \isa{{\isasymodot}}
+  from \isa{semigroup} and adds two further group axioms. Similarly,
+  \isa{agroup} is defined as the subset of \isa{group} such that
+  for all of its elements \isa{{\isasymtau}}, the operation \isa{{\isasymodot}\ {\isasymColon}\ {\isasymtau}\ {\isasymRightarrow}\ {\isasymtau}\ {\isasymRightarrow}\ {\isasymtau}} is even commutative.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsubsection{Abstract reasoning%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+In a sense, axiomatic type classes may be viewed as \emph{abstract
+  theories}.  Above class definitions gives rise to abstract axioms
+  \isa{assoc}, \isa{left{\isacharunderscore}unit}, \isa{left{\isacharunderscore}inverse}, \isa{commute}, where any of these contain a type variable \isa{{\isacharprime}a\ {\isasymColon}\ c} that is restricted to types of the corresponding class \isa{c}.  \emph{Sort constraints} like this express a logical
+  precondition for the whole formula.  For example, \isa{assoc}
+  states that for all \isa{{\isasymtau}}, provided that \isa{{\isasymtau}\ {\isasymColon}\ semigroup}, the operation \isa{{\isasymodot}\ {\isasymColon}\ {\isasymtau}\ {\isasymRightarrow}\ {\isasymtau}\ {\isasymRightarrow}\ {\isasymtau}} is associative.
+
+  \medskip From a technical point of view, abstract axioms are just
+  ordinary Isabelle theorems, which may be used in proofs without
+  special treatment.  Such ``abstract proofs'' usually yield new
+  ``abstract theorems''.  For example, we may now derive the following
+  well-known laws of general groups.%
+\end{isamarkuptext}%
+\isamarkupfalse%
+\isacommand{theorem}\ group{\isacharunderscore}right{\isacharunderscore}inverse{\isacharcolon}\ {\isachardoublequote}x\ {\isasymodot}\ x{\isasyminv}\ {\isacharequal}\ {\isacharparenleft}{\isasymone}{\isasymColon}{\isacharprime}a{\isasymColon}group{\isacharparenright}{\isachardoublequote}\isanewline
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+\isamarkupfalse%
+\isacommand{proof}\ {\isacharminus}\isanewline
+\ \ \isamarkupfalse%
+\isacommand{have}\ {\isachardoublequote}x\ {\isasymodot}\ x{\isasyminv}\ {\isacharequal}\ {\isasymone}\ {\isasymodot}\ {\isacharparenleft}x\ {\isasymodot}\ x{\isasyminv}{\isacharparenright}{\isachardoublequote}\isanewline
+\ \ \ \ \isamarkupfalse%
+\isacommand{by}\ {\isacharparenleft}simp\ only{\isacharcolon}\ group{\isachardot}left{\isacharunderscore}unit{\isacharparenright}\isanewline
+\ \ \isamarkupfalse%
+\isacommand{also}\ \isamarkupfalse%
+\isacommand{have}\ {\isachardoublequote}{\isachardot}{\isachardot}{\isachardot}\ {\isacharequal}\ {\isasymone}\ {\isasymodot}\ x\ {\isasymodot}\ x{\isasyminv}{\isachardoublequote}\isanewline
+\ \ \ \ \isamarkupfalse%
+\isacommand{by}\ {\isacharparenleft}simp\ only{\isacharcolon}\ semigroup{\isachardot}assoc{\isacharparenright}\isanewline
+\ \ \isamarkupfalse%
+\isacommand{also}\ \isamarkupfalse%
+\isacommand{have}\ {\isachardoublequote}{\isachardot}{\isachardot}{\isachardot}\ {\isacharequal}\ {\isacharparenleft}x{\isasyminv}{\isacharparenright}{\isasyminv}\ {\isasymodot}\ x{\isasyminv}\ {\isasymodot}\ x\ {\isasymodot}\ x{\isasyminv}{\isachardoublequote}\isanewline
+\ \ \ \ \isamarkupfalse%
+\isacommand{by}\ {\isacharparenleft}simp\ only{\isacharcolon}\ group{\isachardot}left{\isacharunderscore}inverse{\isacharparenright}\isanewline
+\ \ \isamarkupfalse%
+\isacommand{also}\ \isamarkupfalse%
+\isacommand{have}\ {\isachardoublequote}{\isachardot}{\isachardot}{\isachardot}\ {\isacharequal}\ {\isacharparenleft}x{\isasyminv}{\isacharparenright}{\isasyminv}\ {\isasymodot}\ {\isacharparenleft}x{\isasyminv}\ {\isasymodot}\ x{\isacharparenright}\ {\isasymodot}\ x{\isasyminv}{\isachardoublequote}\isanewline
+\ \ \ \ \isamarkupfalse%
+\isacommand{by}\ {\isacharparenleft}simp\ only{\isacharcolon}\ semigroup{\isachardot}assoc{\isacharparenright}\isanewline
+\ \ \isamarkupfalse%
+\isacommand{also}\ \isamarkupfalse%
+\isacommand{have}\ {\isachardoublequote}{\isachardot}{\isachardot}{\isachardot}\ {\isacharequal}\ {\isacharparenleft}x{\isasyminv}{\isacharparenright}{\isasyminv}\ {\isasymodot}\ {\isasymone}\ {\isasymodot}\ x{\isasyminv}{\isachardoublequote}\isanewline
+\ \ \ \ \isamarkupfalse%
+\isacommand{by}\ {\isacharparenleft}simp\ only{\isacharcolon}\ group{\isachardot}left{\isacharunderscore}inverse{\isacharparenright}\isanewline
+\ \ \isamarkupfalse%
+\isacommand{also}\ \isamarkupfalse%
+\isacommand{have}\ {\isachardoublequote}{\isachardot}{\isachardot}{\isachardot}\ {\isacharequal}\ {\isacharparenleft}x{\isasyminv}{\isacharparenright}{\isasyminv}\ {\isasymodot}\ {\isacharparenleft}{\isasymone}\ {\isasymodot}\ x{\isasyminv}{\isacharparenright}{\isachardoublequote}\isanewline
+\ \ \ \ \isamarkupfalse%
+\isacommand{by}\ {\isacharparenleft}simp\ only{\isacharcolon}\ semigroup{\isachardot}assoc{\isacharparenright}\isanewline
+\ \ \isamarkupfalse%
+\isacommand{also}\ \isamarkupfalse%
+\isacommand{have}\ {\isachardoublequote}{\isachardot}{\isachardot}{\isachardot}\ {\isacharequal}\ {\isacharparenleft}x{\isasyminv}{\isacharparenright}{\isasyminv}\ {\isasymodot}\ x{\isasyminv}{\isachardoublequote}\isanewline
+\ \ \ \ \isamarkupfalse%
+\isacommand{by}\ {\isacharparenleft}simp\ only{\isacharcolon}\ group{\isachardot}left{\isacharunderscore}unit{\isacharparenright}\isanewline
+\ \ \isamarkupfalse%
+\isacommand{also}\ \isamarkupfalse%
+\isacommand{have}\ {\isachardoublequote}{\isachardot}{\isachardot}{\isachardot}\ {\isacharequal}\ {\isasymone}{\isachardoublequote}\isanewline
+\ \ \ \ \isamarkupfalse%
+\isacommand{by}\ {\isacharparenleft}simp\ only{\isacharcolon}\ group{\isachardot}left{\isacharunderscore}inverse{\isacharparenright}\isanewline
+\ \ \isamarkupfalse%
+\isacommand{finally}\ \isamarkupfalse%
+\isacommand{show}\ {\isacharquery}thesis\ \isamarkupfalse%
+\isacommand{{\isachardot}}\isanewline
+\isamarkupfalse%
+\isacommand{qed}%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+\noindent With \isa{group{\isacharunderscore}right{\isacharunderscore}inverse} already available, \isa{group{\isacharunderscore}right{\isacharunderscore}unit}\label{thm:group-right-unit} is now established
+  much easier.%
+\end{isamarkuptext}%
+\isamarkupfalse%
+\isacommand{theorem}\ group{\isacharunderscore}right{\isacharunderscore}unit{\isacharcolon}\ {\isachardoublequote}x\ {\isasymodot}\ {\isasymone}\ {\isacharequal}\ {\isacharparenleft}x{\isasymColon}{\isacharprime}a{\isasymColon}group{\isacharparenright}{\isachardoublequote}\isanewline
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+\isamarkupfalse%
+\isacommand{proof}\ {\isacharminus}\isanewline
+\ \ \isamarkupfalse%
+\isacommand{have}\ {\isachardoublequote}x\ {\isasymodot}\ {\isasymone}\ {\isacharequal}\ x\ {\isasymodot}\ {\isacharparenleft}x{\isasyminv}\ {\isasymodot}\ x{\isacharparenright}{\isachardoublequote}\isanewline
+\ \ \ \ \isamarkupfalse%
+\isacommand{by}\ {\isacharparenleft}simp\ only{\isacharcolon}\ group{\isachardot}left{\isacharunderscore}inverse{\isacharparenright}\isanewline
+\ \ \isamarkupfalse%
+\isacommand{also}\ \isamarkupfalse%
+\isacommand{have}\ {\isachardoublequote}{\isachardot}{\isachardot}{\isachardot}\ {\isacharequal}\ x\ {\isasymodot}\ x{\isasyminv}\ {\isasymodot}\ x{\isachardoublequote}\isanewline
+\ \ \ \ \isamarkupfalse%
+\isacommand{by}\ {\isacharparenleft}simp\ only{\isacharcolon}\ semigroup{\isachardot}assoc{\isacharparenright}\isanewline
+\ \ \isamarkupfalse%
+\isacommand{also}\ \isamarkupfalse%
+\isacommand{have}\ {\isachardoublequote}{\isachardot}{\isachardot}{\isachardot}\ {\isacharequal}\ {\isasymone}\ {\isasymodot}\ x{\isachardoublequote}\isanewline
+\ \ \ \ \isamarkupfalse%
+\isacommand{by}\ {\isacharparenleft}simp\ only{\isacharcolon}\ group{\isacharunderscore}right{\isacharunderscore}inverse{\isacharparenright}\isanewline
+\ \ \isamarkupfalse%
+\isacommand{also}\ \isamarkupfalse%
+\isacommand{have}\ {\isachardoublequote}{\isachardot}{\isachardot}{\isachardot}\ {\isacharequal}\ x{\isachardoublequote}\isanewline
+\ \ \ \ \isamarkupfalse%
+\isacommand{by}\ {\isacharparenleft}simp\ only{\isacharcolon}\ group{\isachardot}left{\isacharunderscore}unit{\isacharparenright}\isanewline
+\ \ \isamarkupfalse%
+\isacommand{finally}\ \isamarkupfalse%
+\isacommand{show}\ {\isacharquery}thesis\ \isamarkupfalse%
+\isacommand{{\isachardot}}\isanewline
+\isamarkupfalse%
+\isacommand{qed}%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+\medskip Abstract theorems may be instantiated to only those types
+  \isa{{\isasymtau}} where the appropriate class membership \isa{{\isasymtau}\ {\isasymColon}\ c} is
+  known at Isabelle's type signature level.  Since we have \isa{agroup\ {\isasymsubseteq}\ group\ {\isasymsubseteq}\ semigroup} by definition, all theorems of \isa{semigroup} and \isa{group} are automatically inherited by \isa{group} and \isa{agroup}.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsubsection{Abstract instantiation%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+From the definition, the \isa{monoid} and \isa{group} classes
+  have been independent.  Note that for monoids, \isa{right{\isacharunderscore}unit}
+  had to be included as an axiom, but for groups both \isa{right{\isacharunderscore}unit} and \isa{right{\isacharunderscore}inverse} are derivable from the other
+  axioms.  With \isa{group{\isacharunderscore}right{\isacharunderscore}unit} derived as a theorem of group
+  theory (see page~\pageref{thm:group-right-unit}), we may now
+  instantiate \isa{monoid\ {\isasymsubseteq}\ semigroup} and \isa{group\ {\isasymsubseteq}\ monoid} properly as follows (cf.\ \figref{fig:monoid-group}).
+
+ \begin{figure}[htbp]
+   \begin{center}
+     \small
+     \unitlength 0.6mm
+     \begin{picture}(65,90)(0,-10)
+       \put(15,10){\line(0,1){10}} \put(15,30){\line(0,1){10}}
+       \put(15,50){\line(1,1){10}} \put(35,60){\line(1,-1){10}}
+       \put(15,5){\makebox(0,0){\isa{agroup}}}
+       \put(15,25){\makebox(0,0){\isa{group}}}
+       \put(15,45){\makebox(0,0){\isa{semigroup}}}
+       \put(30,65){\makebox(0,0){\isa{type}}} \put(50,45){\makebox(0,0){\isa{monoid}}}
+     \end{picture}
+     \hspace{4em}
+     \begin{picture}(30,90)(0,0)
+       \put(15,10){\line(0,1){10}} \put(15,30){\line(0,1){10}}
+       \put(15,50){\line(0,1){10}} \put(15,70){\line(0,1){10}}
+       \put(15,5){\makebox(0,0){\isa{agroup}}}
+       \put(15,25){\makebox(0,0){\isa{group}}}
+       \put(15,45){\makebox(0,0){\isa{monoid}}}
+       \put(15,65){\makebox(0,0){\isa{semigroup}}}
+       \put(15,85){\makebox(0,0){\isa{type}}}
+     \end{picture}
+     \caption{Monoids and groups: according to definition, and by proof}
+     \label{fig:monoid-group}
+   \end{center}
+ \end{figure}%
+\end{isamarkuptext}%
+\isamarkupfalse%
+\isacommand{instance}\ monoid\ {\isasymsubseteq}\ semigroup\isanewline
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+\isamarkupfalse%
+\isacommand{proof}\isanewline
+\ \ \isamarkupfalse%
+\isacommand{fix}\ x\ y\ z\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharprime}a{\isasymColon}monoid{\isachardoublequote}\isanewline
+\ \ \isamarkupfalse%
+\isacommand{show}\ {\isachardoublequote}x\ {\isasymodot}\ y\ {\isasymodot}\ z\ {\isacharequal}\ x\ {\isasymodot}\ {\isacharparenleft}y\ {\isasymodot}\ z{\isacharparenright}{\isachardoublequote}\isanewline
+\ \ \ \ \isamarkupfalse%
+\isacommand{by}\ {\isacharparenleft}rule\ monoid{\isachardot}assoc{\isacharparenright}\isanewline
+\isamarkupfalse%
+\isacommand{qed}%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+\isanewline
+%
+\endisadelimproof
+\isanewline
+\isamarkupfalse%
+\isacommand{instance}\ group\ {\isasymsubseteq}\ monoid\isanewline
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+\isamarkupfalse%
+\isacommand{proof}\isanewline
+\ \ \isamarkupfalse%
+\isacommand{fix}\ x\ y\ z\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharprime}a{\isasymColon}group{\isachardoublequote}\isanewline
+\ \ \isamarkupfalse%
+\isacommand{show}\ {\isachardoublequote}x\ {\isasymodot}\ y\ {\isasymodot}\ z\ {\isacharequal}\ x\ {\isasymodot}\ {\isacharparenleft}y\ {\isasymodot}\ z{\isacharparenright}{\isachardoublequote}\isanewline
+\ \ \ \ \isamarkupfalse%
+\isacommand{by}\ {\isacharparenleft}rule\ semigroup{\isachardot}assoc{\isacharparenright}\isanewline
+\ \ \isamarkupfalse%
+\isacommand{show}\ {\isachardoublequote}{\isasymone}\ {\isasymodot}\ x\ {\isacharequal}\ x{\isachardoublequote}\isanewline
+\ \ \ \ \isamarkupfalse%
+\isacommand{by}\ {\isacharparenleft}rule\ group{\isachardot}left{\isacharunderscore}unit{\isacharparenright}\isanewline
+\ \ \isamarkupfalse%
+\isacommand{show}\ {\isachardoublequote}x\ {\isasymodot}\ {\isasymone}\ {\isacharequal}\ x{\isachardoublequote}\isanewline
+\ \ \ \ \isamarkupfalse%
+\isacommand{by}\ {\isacharparenleft}rule\ group{\isacharunderscore}right{\isacharunderscore}unit{\isacharparenright}\isanewline
+\isamarkupfalse%
+\isacommand{qed}%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+\medskip The $\INSTANCE$ command sets up an appropriate goal that
+  represents the class inclusion (or type arity, see
+  \secref{sec:inst-arity}) to be proven (see also
+  \cite{isabelle-isar-ref}).  The initial proof step causes
+  back-chaining of class membership statements wrt.\ the hierarchy of
+  any classes defined in the current theory; the effect is to reduce
+  to the initial statement to a number of goals that directly
+  correspond to any class axioms encountered on the path upwards
+  through the class hierarchy.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsubsection{Concrete instantiation \label{sec:inst-arity}%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+So far we have covered the case of the form $\INSTANCE$~\isa{c\isactrlsub {\isadigit{1}}\ {\isasymsubseteq}\ c\isactrlsub {\isadigit{2}}}, namely \emph{abstract instantiation} ---
+  $c@1$ is more special than \isa{c\isactrlsub {\isadigit{1}}} and thus an instance
+  of \isa{c\isactrlsub {\isadigit{2}}}.  Even more interesting for practical
+  applications are \emph{concrete instantiations} of axiomatic type
+  classes.  That is, certain simple schemes \isa{{\isacharparenleft}{\isasymalpha}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isasymalpha}\isactrlsub n{\isacharparenright}\ t\ {\isasymColon}\ c} of class membership may be established at the
+  logical level and then transferred to Isabelle's type signature
+  level.
+
+  \medskip As a typical example, we show that type \isa{bool} with
+  exclusive-or as \isa{{\isasymodot}} operation, identity as \isa{{\isasyminv}}, and
+  \isa{False} as \isa{{\isasymone}} forms an Abelian group.%
+\end{isamarkuptext}%
+\isamarkupfalse%
+\isacommand{defs}\ {\isacharparenleft}\isakeyword{overloaded}{\isacharparenright}\isanewline
+\ \ times{\isacharunderscore}bool{\isacharunderscore}def{\isacharcolon}\ {\isachardoublequote}x\ {\isasymodot}\ y\ {\isasymequiv}\ x\ {\isasymnoteq}\ {\isacharparenleft}y{\isasymColon}bool{\isacharparenright}{\isachardoublequote}\isanewline
+\ \ inverse{\isacharunderscore}bool{\isacharunderscore}def{\isacharcolon}\ {\isachardoublequote}x{\isasyminv}\ {\isasymequiv}\ x{\isasymColon}bool{\isachardoublequote}\isanewline
+\ \ unit{\isacharunderscore}bool{\isacharunderscore}def{\isacharcolon}\ {\isachardoublequote}{\isasymone}\ {\isasymequiv}\ False{\isachardoublequote}\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+\medskip It is important to note that above $\DEFS$ are just
+  overloaded meta-level constant definitions, where type classes are
+  not yet involved at all.  This form of constant definition with
+  overloading (and optional recursion over the syntactic structure of
+  simple types) are admissible as definitional extensions of plain HOL
+  \cite{Wenzel:1997:TPHOL}.  The Haskell-style type system is not
+  required for overloading.  Nevertheless, overloaded definitions are
+  best applied in the context of type classes.
+
+  \medskip Since we have chosen above $\DEFS$ of the generic group
+  operations on type \isa{bool} appropriately, the class membership
+  \isa{bool\ {\isasymColon}\ agroup} may be now derived as follows.%
+\end{isamarkuptext}%
+\isamarkupfalse%
+\isacommand{instance}\ bool\ {\isacharcolon}{\isacharcolon}\ agroup\isanewline
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+\isamarkupfalse%
+\isacommand{proof}\ {\isacharparenleft}intro{\isacharunderscore}classes{\isacharcomma}\isanewline
+\ \ \ \ unfold\ times{\isacharunderscore}bool{\isacharunderscore}def\ inverse{\isacharunderscore}bool{\isacharunderscore}def\ unit{\isacharunderscore}bool{\isacharunderscore}def{\isacharparenright}\isanewline
+\ \ \isamarkupfalse%
+\isacommand{fix}\ x\ y\ z\isanewline
+\ \ \isamarkupfalse%
+\isacommand{show}\ {\isachardoublequote}{\isacharparenleft}{\isacharparenleft}x\ {\isasymnoteq}\ y{\isacharparenright}\ {\isasymnoteq}\ z{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}x\ {\isasymnoteq}\ {\isacharparenleft}y\ {\isasymnoteq}\ z{\isacharparenright}{\isacharparenright}{\isachardoublequote}\ \isamarkupfalse%
+\isacommand{by}\ blast\isanewline
+\ \ \isamarkupfalse%
+\isacommand{show}\ {\isachardoublequote}{\isacharparenleft}False\ {\isasymnoteq}\ x{\isacharparenright}\ {\isacharequal}\ x{\isachardoublequote}\ \isamarkupfalse%
+\isacommand{by}\ blast\isanewline
+\ \ \isamarkupfalse%
+\isacommand{show}\ {\isachardoublequote}{\isacharparenleft}x\ {\isasymnoteq}\ x{\isacharparenright}\ {\isacharequal}\ False{\isachardoublequote}\ \isamarkupfalse%
+\isacommand{by}\ blast\isanewline
+\ \ \isamarkupfalse%
+\isacommand{show}\ {\isachardoublequote}{\isacharparenleft}x\ {\isasymnoteq}\ y{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}y\ {\isasymnoteq}\ x{\isacharparenright}{\isachardoublequote}\ \isamarkupfalse%
+\isacommand{by}\ blast\isanewline
+\isamarkupfalse%
+\isacommand{qed}%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+The result of an $\INSTANCE$ statement is both expressed as a
+  theorem of Isabelle's meta-logic, and as a type arity of the type
+  signature.  The latter enables type-inference system to take care of
+  this new instance automatically.
+
+  \medskip We could now also instantiate our group theory classes to
+  many other concrete types.  For example, \isa{int\ {\isasymColon}\ agroup}
+  (e.g.\ by defining \isa{{\isasymodot}} as addition, \isa{{\isasyminv}} as negation
+  and \isa{{\isasymone}} as zero) or \isa{list\ {\isasymColon}\ {\isacharparenleft}type{\isacharparenright}\ semigroup}
+  (e.g.\ if \isa{{\isasymodot}} is defined as list append).  Thus, the
+  characteristic constants \isa{{\isasymodot}}, \isa{{\isasyminv}}, \isa{{\isasymone}}
+  really become overloaded, i.e.\ have different meanings on different
+  types.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsubsection{Lifting and Functors%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+As already mentioned above, overloading in the simply-typed HOL
+  systems may include recursion over the syntactic structure of types.
+  That is, definitional equations \isa{c\isactrlsup {\isasymtau}\ {\isasymequiv}\ t} may also
+  contain constants of name \isa{c} on the right-hand side --- if
+  these have types that are structurally simpler than \isa{{\isasymtau}}.
+
+  This feature enables us to \emph{lift operations}, say to Cartesian
+  products, direct sums or function spaces.  Subsequently we lift
+  \isa{{\isasymodot}} component-wise to binary products \isa{{\isacharprime}a\ {\isasymtimes}\ {\isacharprime}b}.%
+\end{isamarkuptext}%
+\isamarkupfalse%
+\isacommand{defs}\ {\isacharparenleft}\isakeyword{overloaded}{\isacharparenright}\isanewline
+\ \ times{\isacharunderscore}prod{\isacharunderscore}def{\isacharcolon}\ {\isachardoublequote}p\ {\isasymodot}\ q\ {\isasymequiv}\ {\isacharparenleft}fst\ p\ {\isasymodot}\ fst\ q{\isacharcomma}\ snd\ p\ {\isasymodot}\ snd\ q{\isacharparenright}{\isachardoublequote}\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+It is very easy to see that associativity of \isa{{\isasymodot}} on \isa{{\isacharprime}a}
+  and \isa{{\isasymodot}} on \isa{{\isacharprime}b} transfers to \isa{{\isasymodot}} on \isa{{\isacharprime}a\ {\isasymtimes}\ {\isacharprime}b}.  Hence the binary type constructor \isa{{\isasymodot}} maps semigroups
+  to semigroups.  This may be established formally as follows.%
+\end{isamarkuptext}%
+\isamarkupfalse%
+\isacommand{instance}\ {\isacharasterisk}\ {\isacharcolon}{\isacharcolon}\ {\isacharparenleft}semigroup{\isacharcomma}\ semigroup{\isacharparenright}\ semigroup\isanewline
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+\isamarkupfalse%
+\isacommand{proof}\ {\isacharparenleft}intro{\isacharunderscore}classes{\isacharcomma}\ unfold\ times{\isacharunderscore}prod{\isacharunderscore}def{\isacharparenright}\isanewline
+\ \ \isamarkupfalse%
+\isacommand{fix}\ p\ q\ r\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharprime}a{\isasymColon}semigroup\ {\isasymtimes}\ {\isacharprime}b{\isasymColon}semigroup{\isachardoublequote}\isanewline
+\ \ \isamarkupfalse%
+\isacommand{show}\isanewline
+\ \ \ \ {\isachardoublequote}{\isacharparenleft}fst\ {\isacharparenleft}fst\ p\ {\isasymodot}\ fst\ q{\isacharcomma}\ snd\ p\ {\isasymodot}\ snd\ q{\isacharparenright}\ {\isasymodot}\ fst\ r{\isacharcomma}\isanewline
+\ \ \ \ \ \ snd\ {\isacharparenleft}fst\ p\ {\isasymodot}\ fst\ q{\isacharcomma}\ snd\ p\ {\isasymodot}\ snd\ q{\isacharparenright}\ {\isasymodot}\ snd\ r{\isacharparenright}\ {\isacharequal}\isanewline
+\ \ \ \ \ \ \ {\isacharparenleft}fst\ p\ {\isasymodot}\ fst\ {\isacharparenleft}fst\ q\ {\isasymodot}\ fst\ r{\isacharcomma}\ snd\ q\ {\isasymodot}\ snd\ r{\isacharparenright}{\isacharcomma}\isanewline
+\ \ \ \ \ \ \ \ snd\ p\ {\isasymodot}\ snd\ {\isacharparenleft}fst\ q\ {\isasymodot}\ fst\ r{\isacharcomma}\ snd\ q\ {\isasymodot}\ snd\ r{\isacharparenright}{\isacharparenright}{\isachardoublequote}\isanewline
+\ \ \ \ \isamarkupfalse%
+\isacommand{by}\ {\isacharparenleft}simp\ add{\isacharcolon}\ semigroup{\isachardot}assoc{\isacharparenright}\isanewline
+\isamarkupfalse%
+\isacommand{qed}%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+Thus, if we view class instances as ``structures'', then overloaded
+  constant definitions with recursion over types indirectly provide
+  some kind of ``functors'' --- i.e.\ mappings between abstract
+  theories.%
+\end{isamarkuptext}%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+%
+\isatagtheory
+\isamarkupfalse%
+\isacommand{end}%
+\endisatagtheory
+{\isafoldtheory}%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+\isanewline
+\end{isabellebody}%
+%%% Local Variables:
+%%% mode: latex
+%%% TeX-master: "root"
+%%% End:

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/AxClass/Group/document/Product.tex	Fri Aug 19 22:44:01 2005 +0200
@@ -0,0 +1,132 @@
+%
+\begin{isabellebody}%
+\def\isabellecontext{Product}%
+\isamarkuptrue%
+%
+\isamarkupheader{Syntactic classes%
+}
+%
+\isadelimtheory
+%
+\endisadelimtheory
+%
+\isatagtheory
+\isamarkupfalse%
+\isacommand{theory}\ Product\ \isakeyword{imports}\ Main\ \isakeyword{begin}%
+\endisatagtheory
+{\isafoldtheory}%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+\isamarkuptrue%
+%
+\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 \isa{c\ {\isasymColon}\ {\isasymsigma}}, the type variables occurring in \isa{{\isasymsigma}}
+  may be constrained by type classes (or even general sorts) in an
+  arbitrary way.  Note that by default, in Isabelle/HOL the
+  declaration \isa{{\isasymodot}\ {\isasymColon}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a} is actually an abbreviation
+  for \isa{{\isasymodot}\ {\isasymColon}\ {\isacharprime}a{\isasymColon}type\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a} Since class \isa{type} is the
+  universal class of HOL, this is not really a constraint at all.
+
+ The \isa{product} class below provides a less degenerate example of
+ syntactic type classes.%
+\end{isamarkuptext}%
+\isamarkupfalse%
+\isacommand{axclass}\isanewline
+\ \ product\ {\isasymsubseteq}\ type\isanewline
+\isamarkupfalse%
+\isacommand{consts}\isanewline
+\ \ 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}\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+Here class \isa{product} is defined as subclass of \isa{type}
+  without any additional axioms.  This effects in logical equivalence
+  of \isa{product} and \isa{type}, as is reflected by the trivial
+  introduction rule generated for this definition.
+
+  \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
+  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 \isa{{\isasymodot}} on some type \isa{{\isasymtau}} are rejected by the
+  type-checker, unless the arity \isa{{\isasymtau}\ {\isasymColon}\ product} is part of the
+  type signature.  In our example, this arity may be always added when
+  required by means of an $\INSTANCE$ with the default proof $\DDOT$.
+
+  \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
+  \isa{c}.  Only immediately before \emph{specifying} these
+  constants on a certain type \isa{{\isasymtau}} do we instantiate \isa{{\isasymtau}\ {\isasymColon}\ c}.
+
+  This is done for class \isa{product} and type \isa{bool} as
+  follows.%
+\end{isamarkuptext}%
+\isamarkupfalse%
+\isacommand{instance}\ bool\ {\isacharcolon}{\isacharcolon}\ product%
+\isadelimproof
+\ %
+\endisadelimproof
+%
+\isatagproof
+\isamarkupfalse%
+\isacommand{{\isachardot}{\isachardot}}%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+\isanewline
+\isamarkupfalse%
+\isacommand{defs}\ {\isacharparenleft}\isakeyword{overloaded}{\isacharparenright}\isanewline
+\ \ product{\isacharunderscore}bool{\isacharunderscore}def{\isacharcolon}\ {\isachardoublequote}x\ {\isasymodot}\ y\ {\isasymequiv}\ x\ {\isasymand}\ y{\isachardoublequote}\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+The definition \isa{prod{\isacharunderscore}bool{\isacharunderscore}def} becomes syntactically
+ well-formed only after the arity \isa{bool\ {\isasymColon}\ product} is made
+ known to the type checker.
+
+ \medskip It is very important to see that above $\DEFS$ are not
+ directly connected with $\INSTANCE$ at all!  We were just following
+ our convention to specify \isa{{\isasymodot}} on \isa{bool} after having
+ instantiated \isa{bool\ {\isasymColon}\ 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} would not make much sense in
+ Isabelle's meta-logic, because there is no internal notion of
+ ``providing operations'' or even ``names of functions''.%
+\end{isamarkuptext}%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+%
+\isatagtheory
+\isamarkupfalse%
+\isacommand{end}%
+\endisatagtheory
+{\isafoldtheory}%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+\isanewline
+\end{isabellebody}%
+%%% Local Variables:
+%%% mode: latex
+%%% TeX-master: "root"
+%%% End:

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/AxClass/Group/document/Semigroups.tex	Fri Aug 19 22:44:01 2005 +0200
@@ -0,0 +1,88 @@
+%
+\begin{isabellebody}%
+\def\isabellecontext{Semigroups}%
+\isamarkuptrue%
+%
+\isamarkupheader{Semigroups%
+}
+%
+\isadelimtheory
+%
+\endisadelimtheory
+%
+\isatagtheory
+\isamarkupfalse%
+\isacommand{theory}\ Semigroups\ \isakeyword{imports}\ Main\ \isakeyword{begin}%
+\endisatagtheory
+{\isafoldtheory}%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+\medskip\noindent An axiomatic type class is simply a class of types
+  that all meet certain properties, which are also called \emph{class
+  axioms}. Thus, type classes may be also understood as type
+  predicates --- i.e.\ abstractions over a single type argument \isa{{\isacharprime}a}.  Class axioms typically contain polymorphic constants that
+  depend on this type \isa{{\isacharprime}a}.  These \emph{characteristic
+  constants} behave like operations associated with the ``carrier''
+  type \isa{{\isacharprime}a}.
+
+  We illustrate these basic concepts by the following formulation of
+  semigroups.%
+\end{isamarkuptext}%
+\isamarkupfalse%
+\isacommand{consts}\isanewline
+\ \ times\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a{\isachardoublequote}\ \ \ \ {\isacharparenleft}\isakeyword{infixl}\ {\isachardoublequote}{\isasymodot}{\isachardoublequote}\ {\isadigit{7}}{\isadigit{0}}{\isacharparenright}\isanewline
+\isamarkupfalse%
+\isacommand{axclass}\ semigroup\ {\isasymsubseteq}\ type\isanewline
+\ \ assoc{\isacharcolon}\ {\isachardoublequote}{\isacharparenleft}x\ {\isasymodot}\ y{\isacharparenright}\ {\isasymodot}\ z\ {\isacharequal}\ x\ {\isasymodot}\ {\isacharparenleft}y\ {\isasymodot}\ z{\isacharparenright}{\isachardoublequote}\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+\noindent Above we have first declared a polymorphic constant \isa{{\isasymodot}\ {\isasymColon}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a} and then defined the class \isa{semigroup} of
+  all types \isa{{\isasymtau}} such that \isa{{\isasymodot}\ {\isasymColon}\ {\isasymtau}\ {\isasymRightarrow}\ {\isasymtau}\ {\isasymRightarrow}\ {\isasymtau}} is indeed an
+  associative operator.  The \isa{assoc} axiom contains exactly one
+  type variable, which is invisible in the above presentation, though.
+  Also note that free term variables (like \isa{x}, \isa{y},
+  \isa{z}) are allowed for user convenience --- conceptually all of
+  these are bound by outermost universal quantifiers.
+
+  \medskip In general, type classes may be used to describe
+  \emph{structures} with exactly one carrier \isa{{\isacharprime}a} and a fixed
+  \emph{signature}.  Different signatures require different classes.
+  Below, class \isa{plus{\isacharunderscore}semigroup} represents semigroups \isa{{\isacharparenleft}{\isasymtau}{\isacharcomma}\ {\isasymoplus}\isactrlsup {\isasymtau}{\isacharparenright}}, while the original \isa{semigroup} would
+  correspond to semigroups of the form \isa{{\isacharparenleft}{\isasymtau}{\isacharcomma}\ {\isasymodot}\isactrlsup {\isasymtau}{\isacharparenright}}.%
+\end{isamarkuptext}%
+\isamarkupfalse%
+\isacommand{consts}\isanewline
+\ \ plus\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a{\isachardoublequote}\ \ \ \ {\isacharparenleft}\isakeyword{infixl}\ {\isachardoublequote}{\isasymoplus}{\isachardoublequote}\ {\isadigit{7}}{\isadigit{0}}{\isacharparenright}\isanewline
+\isamarkupfalse%
+\isacommand{axclass}\ plus{\isacharunderscore}semigroup\ {\isasymsubseteq}\ type\isanewline
+\ \ assoc{\isacharcolon}\ {\isachardoublequote}{\isacharparenleft}x\ {\isasymoplus}\ y{\isacharparenright}\ {\isasymoplus}\ z\ {\isacharequal}\ x\ {\isasymoplus}\ {\isacharparenleft}y\ {\isasymoplus}\ z{\isacharparenright}{\isachardoublequote}\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+\noindent Even if classes \isa{plus{\isacharunderscore}semigroup} and \isa{semigroup} both represent semigroups in a sense, they are certainly
+  not quite the same.%
+\end{isamarkuptext}%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+%
+\isatagtheory
+\isamarkupfalse%
+\isacommand{end}%
+\endisatagtheory
+{\isafoldtheory}%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+\isanewline
+\end{isabellebody}%
+%%% Local Variables:
+%%% mode: latex
+%%% TeX-master: "root"
+%%% End:

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/AxClass/Group/document/isabelle.sty	Fri Aug 19 22:44:01 2005 +0200
@@ -0,0 +1,191 @@
+%%
+%% Author: Markus Wenzel, TU Muenchen
+%%
+%% macros for Isabelle generated LaTeX output
+%%
+%% $Id$
+
+%%% Simple document preparation (based on theory token language and symbols)
+
+% isabelle environments
+
+\newcommand{\isabellecontext}{UNKNOWN}
+
+\newcommand{\isastyle}{\small\tt\slshape}
+\newcommand{\isastyleminor}{\small\tt\slshape}
+\newcommand{\isastylescript}{\footnotesize\tt\slshape}
+\newcommand{\isastyletext}{\normalsize\rm}
+\newcommand{\isastyletxt}{\rm}
+\newcommand{\isastylecmt}{\rm}
+
+%symbol markup -- \emph achieves decent spacing via italic corrections
+\newcommand{\isamath}[1]{\emph{$#1$}}
+\newcommand{\isatext}[1]{\emph{#1}}
+\newcommand{\isascriptstyle}{\def\isamath##1{##1}\def\isatext##1{\mbox{\isastylescript##1}}}
+\newcommand{\isactrlsub}[1]{\emph{${}\sb{\mbox{\isastylescript{#1}}}$}}
+\newcommand{\isactrlsup}[1]{\emph{${}\sp{\mbox{\isastylescript{#1}}}$}}
+\newcommand{\isactrlisub}[1]{\emph{${}\sb{\mbox{\isastylescript{#1}}}$}}
+\newcommand{\isactrlisup}[1]{\emph{${}\sp{\mbox{\isastylescript{#1}}}$}}
+\newcommand{\isactrlbsub}{\emph\bgroup\begin{math}{}\sb\bgroup\mbox\bgroup\isastylescript}
+\newcommand{\isactrlesub}{\egroup\egroup\end{math}\egroup}
+\newcommand{\isactrlbsup}{\emph\bgroup\begin{math}{}\sp\bgroup\mbox\bgroup\isastylescript}
+\newcommand{\isactrlesup}{\egroup\egroup\end{math}\egroup}
+\newcommand{\isactrlbold}[1]{{\bfseries\upshape\boldmath#1}}
+
+
+\newdimen\isa@parindent\newdimen\isa@parskip
+
+\newenvironment{isabellebody}{%
+\isamarkuptrue\par%
+\isa@parindent\parindent\parindent0pt%
+\isa@parskip\parskip\parskip0pt%
+\isastyle}{\par}
+
+\newenvironment{isabelle}
+{\begin{trivlist}\begin{isabellebody}\item\relax}
+{\end{isabellebody}\end{trivlist}}
+
+\newcommand{\isa}[1]{\emph{\isastyleminor #1}}
+
+\newcommand{\isaindent}[1]{\hphantom{#1}}
+\newcommand{\isanewline}{\mbox{}\par\mbox{}}
+\newcommand{\isasep}{} % override with e.g. \renewcommand{\isasep}{\vspace{1ex}}
+\newcommand{\isadigit}[1]{#1}
+
+\chardef\isacharbang=`\!
+\chardef\isachardoublequote=`\"
+\chardef\isacharhash=`\#
+\chardef\isachardollar=`\$
+\chardef\isacharpercent=`\%
+\chardef\isacharampersand=`\&
+\chardef\isacharprime=`\'
+\chardef\isacharparenleft=`\(
+\chardef\isacharparenright=`\)
+\chardef\isacharasterisk=`\*
+\chardef\isacharplus=`\+
+\chardef\isacharcomma=`\,
+\chardef\isacharminus=`\-
+\chardef\isachardot=`\.
+\chardef\isacharslash=`\/
+\chardef\isacharcolon=`\:
+\chardef\isacharsemicolon=`\;
+\chardef\isacharless=`\<
+\chardef\isacharequal=`\=
+\chardef\isachargreater=`\>
+\chardef\isacharquery=`\?
+\chardef\isacharat=`\@
+\chardef\isacharbrackleft=`\[
+\chardef\isacharbackslash=`\\
+\chardef\isacharbrackright=`\]
+\chardef\isacharcircum=`\^
+\chardef\isacharunderscore=`\_
+\chardef\isacharbackquote=`\`
+\chardef\isacharbraceleft=`\{
+\chardef\isacharbar=`\|
+\chardef\isacharbraceright=`\}
+\chardef\isachartilde=`\~
+
+
+% keyword and section markup
+
+\newcommand{\isakeywordcharunderscore}{\_}
+\newcommand{\isakeyword}[1]
+{\emph{\bf\def\isachardot{.}\def\isacharunderscore{\isakeywordcharunderscore}%
+\def\isacharbraceleft{\{}\def\isacharbraceright{\}}#1}}
+\newcommand{\isacommand}[1]{\isakeyword{#1}}
+
+\newcommand{\isamarkupheader}[1]{\section{#1}}
+\newcommand{\isamarkupchapter}[1]{\chapter{#1}}
+\newcommand{\isamarkupsection}[1]{\section{#1}}
+\newcommand{\isamarkupsubsection}[1]{\subsection{#1}}
+\newcommand{\isamarkupsubsubsection}[1]{\subsubsection{#1}}
+\newcommand{\isamarkupsect}[1]{\section{#1}}
+\newcommand{\isamarkupsubsect}[1]{\subsection{#1}}
+\newcommand{\isamarkupsubsubsect}[1]{\subsubsection{#1}}
+
+\newif\ifisamarkup
+\newcommand{\isabeginpar}{\par\ifisamarkup\relax\else\medskip\fi}
+\newcommand{\isaendpar}{\par\medskip}
+\newenvironment{isapar}{\parindent\isa@parindent\parskip\isa@parskip\isabeginpar}{\isaendpar}
+\newenvironment{isamarkuptext}{\isastyletext\begin{isapar}}{\end{isapar}}
+\newenvironment{isamarkuptxt}{\isastyletxt\begin{isapar}}{\end{isapar}}
+\newcommand{\isamarkupcmt}[1]{{\isastylecmt--- #1}}
+
+
+% alternative styles
+
+\newcommand{\isabellestyle}{}
+\def\isabellestyle#1{\csname isabellestyle#1\endcsname}
+
+\newcommand{\isabellestylett}{%
+\renewcommand{\isastyle}{\small\tt}%
+\renewcommand{\isastyleminor}{\small\tt}%
+\renewcommand{\isastylescript}{\footnotesize\tt}%
+}
+\newcommand{\isabellestyleit}{%
+\renewcommand{\isastyle}{\small\it}%
+\renewcommand{\isastyleminor}{\it}%
+\renewcommand{\isastylescript}{\footnotesize\it}%
+\renewcommand{\isakeywordcharunderscore}{\mbox{-}}%
+\renewcommand{\isacharbang}{\isamath{!}}%
+\renewcommand{\isachardoublequote}{}%
+\renewcommand{\isacharhash}{\isamath{\#}}%
+\renewcommand{\isachardollar}{\isamath{\$}}%
+\renewcommand{\isacharpercent}{\isamath{\%}}%
+\renewcommand{\isacharampersand}{\isamath{\&}}%
+\renewcommand{\isacharprime}{\isamath{\mskip2mu{'}\mskip-2mu}}%
+\renewcommand{\isacharparenleft}{\isamath{(}}%
+\renewcommand{\isacharparenright}{\isamath{)}}%
+\renewcommand{\isacharasterisk}{\isamath{*}}%
+\renewcommand{\isacharplus}{\isamath{+}}%
+\renewcommand{\isacharcomma}{\isamath{\mathord,}}%
+\renewcommand{\isacharminus}{\isamath{-}}%
+\renewcommand{\isachardot}{\isamath{\mathord.}}%
+\renewcommand{\isacharslash}{\isamath{/}}%
+\renewcommand{\isacharcolon}{\isamath{\mathord:}}%
+\renewcommand{\isacharsemicolon}{\isamath{\mathord;}}%
+\renewcommand{\isacharless}{\isamath{<}}%
+\renewcommand{\isacharequal}{\isamath{=}}%
+\renewcommand{\isachargreater}{\isamath{>}}%
+\renewcommand{\isacharat}{\isamath{@}}%
+\renewcommand{\isacharbrackleft}{\isamath{[}}%
+\renewcommand{\isacharbackslash}{\isamath{\backslash}}%
+\renewcommand{\isacharbrackright}{\isamath{]}}%
+\renewcommand{\isacharunderscore}{\mbox{-}}%
+\renewcommand{\isacharbraceleft}{\isamath{\{}}%
+\renewcommand{\isacharbar}{\isamath{\mid}}%
+\renewcommand{\isacharbraceright}{\isamath{\}}}%
+\renewcommand{\isachartilde}{\isamath{{}\sp{\sim}}}%
+}
+
+\newcommand{\isabellestylesl}{%
+\isabellestyleit%
+\renewcommand{\isastyle}{\small\sl}%
+\renewcommand{\isastyleminor}{\sl}%
+\renewcommand{\isastylescript}{\footnotesize\sl}%
+}
+
+
+% tagged regions
+
+%plain TeX version of comment package -- much faster!
+\let\isafmtname\fmtname\def\fmtname{plain}
+\usepackage{comment}
+\let\fmtname\isafmtname
+
+\newcommand{\isafold}[1]{\emph{$\langle\mathord{\mathit{#1}}\rangle$}}
+
+\newcommand{\isakeeptag}[1]%
+{\includecomment{isadelim#1}\includecomment{isatag#1}\csarg\def{isafold#1}{}}
+\newcommand{\isadroptag}[1]%
+{\excludecomment{isadelim#1}\excludecomment{isatag#1}\csarg\def{isafold#1}{}}
+\newcommand{\isafoldtag}[1]%
+{\includecomment{isadelim#1}\excludecomment{isatag#1}\csarg\def{isafold#1}{\isafold{#1}}}
+
+\isakeeptag{theory}
+\isakeeptag{proof}
+\isakeeptag{ML}
+\isakeeptag{visible}
+\isadroptag{invisible}
+
+\IfFileExists{isabelletags.sty}{\usepackage{isabelletags}}{}

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/AxClass/Group/document/isabellesym.sty	Fri Aug 19 22:44:01 2005 +0200
@@ -0,0 +1,362 @@
+%%
+%% Author: Markus Wenzel, TU Muenchen
+%%
+%% definitions of standard Isabelle symbols
+%%
+%% $Id$
+
+% symbol definitions
+
+\newcommand{\isasymzero}{\isamath{\mathbf{0}}}  %requires amssymb
+\newcommand{\isasymone}{\isamath{\mathbf{1}}}   %requires amssymb
+\newcommand{\isasymtwo}{\isamath{\mathbf{2}}}   %requires amssymb
+\newcommand{\isasymthree}{\isamath{\mathbf{3}}} %requires amssymb
+\newcommand{\isasymfour}{\isamath{\mathbf{4}}}  %requires amssymb
+\newcommand{\isasymfive}{\isamath{\mathbf{5}}}  %requires amssymb
+\newcommand{\isasymsix}{\isamath{\mathbf{6}}}   %requires amssymb
+\newcommand{\isasymseven}{\isamath{\mathbf{7}}} %requires amssymb
+\newcommand{\isasymeight}{\isamath{\mathbf{8}}} %requires amssymb
+\newcommand{\isasymnine}{\isamath{\mathbf{9}}}  %requires amssymb
+\newcommand{\isasymA}{\isamath{\mathcal{A}}}
+\newcommand{\isasymB}{\isamath{\mathcal{B}}}
+\newcommand{\isasymC}{\isamath{\mathcal{C}}}
+\newcommand{\isasymD}{\isamath{\mathcal{D}}}
+\newcommand{\isasymE}{\isamath{\mathcal{E}}}
+\newcommand{\isasymF}{\isamath{\mathcal{F}}}
+\newcommand{\isasymG}{\isamath{\mathcal{G}}}
+\newcommand{\isasymH}{\isamath{\mathcal{H}}}
+\newcommand{\isasymI}{\isamath{\mathcal{I}}}
+\newcommand{\isasymJ}{\isamath{\mathcal{J}}}
+\newcommand{\isasymK}{\isamath{\mathcal{K}}}
+\newcommand{\isasymL}{\isamath{\mathcal{L}}}
+\newcommand{\isasymM}{\isamath{\mathcal{M}}}
+\newcommand{\isasymN}{\isamath{\mathcal{N}}}
+\newcommand{\isasymO}{\isamath{\mathcal{O}}}
+\newcommand{\isasymP}{\isamath{\mathcal{P}}}
+\newcommand{\isasymQ}{\isamath{\mathcal{Q}}}
+\newcommand{\isasymR}{\isamath{\mathcal{R}}}
+\newcommand{\isasymS}{\isamath{\mathcal{S}}}
+\newcommand{\isasymT}{\isamath{\mathcal{T}}}
+\newcommand{\isasymU}{\isamath{\mathcal{U}}}
+\newcommand{\isasymV}{\isamath{\mathcal{V}}}
+\newcommand{\isasymW}{\isamath{\mathcal{W}}}
+\newcommand{\isasymX}{\isamath{\mathcal{X}}}
+\newcommand{\isasymY}{\isamath{\mathcal{Y}}}
+\newcommand{\isasymZ}{\isamath{\mathcal{Z}}}
+\newcommand{\isasyma}{\isamath{\mathrm{a}}}
+\newcommand{\isasymb}{\isamath{\mathrm{b}}}
+\newcommand{\isasymc}{\isamath{\mathrm{c}}}
+\newcommand{\isasymd}{\isamath{\mathrm{d}}}
+\newcommand{\isasyme}{\isamath{\mathrm{e}}}
+\newcommand{\isasymf}{\isamath{\mathrm{f}}}
+\newcommand{\isasymg}{\isamath{\mathrm{g}}}
+\newcommand{\isasymh}{\isamath{\mathrm{h}}}
+\newcommand{\isasymi}{\isamath{\mathrm{i}}}
+\newcommand{\isasymj}{\isamath{\mathrm{j}}}
+\newcommand{\isasymk}{\isamath{\mathrm{k}}}
+\newcommand{\isasyml}{\isamath{\mathrm{l}}}
+\newcommand{\isasymm}{\isamath{\mathrm{m}}}
+\newcommand{\isasymn}{\isamath{\mathrm{n}}}
+\newcommand{\isasymo}{\isamath{\mathrm{o}}}
+\newcommand{\isasymp}{\isamath{\mathrm{p}}}
+\newcommand{\isasymq}{\isamath{\mathrm{q}}}
+\newcommand{\isasymr}{\isamath{\mathrm{r}}}
+\newcommand{\isasyms}{\isamath{\mathrm{s}}}
+\newcommand{\isasymt}{\isamath{\mathrm{t}}}
+\newcommand{\isasymu}{\isamath{\mathrm{u}}}
+\newcommand{\isasymv}{\isamath{\mathrm{v}}}
+\newcommand{\isasymw}{\isamath{\mathrm{w}}}
+\newcommand{\isasymx}{\isamath{\mathrm{x}}}
+\newcommand{\isasymy}{\isamath{\mathrm{y}}}
+\newcommand{\isasymz}{\isamath{\mathrm{z}}}
+\newcommand{\isasymAA}{\isamath{\mathfrak{A}}}  %requires eufrak
+\newcommand{\isasymBB}{\isamath{\mathfrak{B}}}  %requires eufrak
+\newcommand{\isasymCC}{\isamath{\mathfrak{C}}}  %requires eufrak
+\newcommand{\isasymDD}{\isamath{\mathfrak{D}}}  %requires eufrak
+\newcommand{\isasymEE}{\isamath{\mathfrak{E}}}  %requires eufrak
+\newcommand{\isasymFF}{\isamath{\mathfrak{F}}}  %requires eufrak
+\newcommand{\isasymGG}{\isamath{\mathfrak{G}}}  %requires eufrak
+\newcommand{\isasymHH}{\isamath{\mathfrak{H}}}  %requires eufrak
+\newcommand{\isasymII}{\isamath{\mathfrak{I}}}  %requires eufrak
+\newcommand{\isasymJJ}{\isamath{\mathfrak{J}}}  %requires eufrak
+\newcommand{\isasymKK}{\isamath{\mathfrak{K}}}  %requires eufrak
+\newcommand{\isasymLL}{\isamath{\mathfrak{L}}}  %requires eufrak
+\newcommand{\isasymMM}{\isamath{\mathfrak{M}}}  %requires eufrak
+\newcommand{\isasymNN}{\isamath{\mathfrak{N}}}  %requires eufrak
+\newcommand{\isasymOO}{\isamath{\mathfrak{O}}}  %requires eufrak
+\newcommand{\isasymPP}{\isamath{\mathfrak{P}}}  %requires eufrak
+\newcommand{\isasymQQ}{\isamath{\mathfrak{Q}}}  %requires eufrak
+\newcommand{\isasymRR}{\isamath{\mathfrak{R}}}  %requires eufrak
+\newcommand{\isasymSS}{\isamath{\mathfrak{S}}}  %requires eufrak
+\newcommand{\isasymTT}{\isamath{\mathfrak{T}}}  %requires eufrak
+\newcommand{\isasymUU}{\isamath{\mathfrak{U}}}  %requires eufrak
+\newcommand{\isasymVV}{\isamath{\mathfrak{V}}}  %requires eufrak
+\newcommand{\isasymWW}{\isamath{\mathfrak{W}}}  %requires eufrak
+\newcommand{\isasymXX}{\isamath{\mathfrak{X}}}  %requires eufrak
+\newcommand{\isasymYY}{\isamath{\mathfrak{Y}}}  %requires eufrak
+\newcommand{\isasymZZ}{\isamath{\mathfrak{Z}}}  %requires eufrak
+\newcommand{\isasymaa}{\isamath{\mathfrak{a}}}  %requires eufrak
+\newcommand{\isasymbb}{\isamath{\mathfrak{b}}}  %requires eufrak
+\newcommand{\isasymcc}{\isamath{\mathfrak{c}}}  %requires eufrak
+\newcommand{\isasymdd}{\isamath{\mathfrak{d}}}  %requires eufrak
+\newcommand{\isasymee}{\isamath{\mathfrak{e}}}  %requires eufrak
+\newcommand{\isasymff}{\isamath{\mathfrak{f}}}  %requires eufrak
+\newcommand{\isasymgg}{\isamath{\mathfrak{g}}}  %requires eufrak
+\newcommand{\isasymhh}{\isamath{\mathfrak{h}}}  %requires eufrak
+\newcommand{\isasymii}{\isamath{\mathfrak{i}}}  %requires eufrak
+\newcommand{\isasymjj}{\isamath{\mathfrak{j}}}  %requires eufrak
+\newcommand{\isasymkk}{\isamath{\mathfrak{k}}}  %requires eufrak
+\newcommand{\isasymll}{\isamath{\mathfrak{l}}}  %requires eufrak
+\newcommand{\isasymmm}{\isamath{\mathfrak{m}}}  %requires eufrak
+\newcommand{\isasymnn}{\isamath{\mathfrak{n}}}  %requires eufrak
+\newcommand{\isasymoo}{\isamath{\mathfrak{o}}}  %requires eufrak
+\newcommand{\isasympp}{\isamath{\mathfrak{p}}}  %requires eufrak
+\newcommand{\isasymqq}{\isamath{\mathfrak{q}}}  %requires eufrak
+\newcommand{\isasymrr}{\isamath{\mathfrak{r}}}  %requires eufrak
+\newcommand{\isasymss}{\isamath{\mathfrak{s}}}  %requires eufrak
+\newcommand{\isasymtt}{\isamath{\mathfrak{t}}}  %requires eufrak
+\newcommand{\isasymuu}{\isamath{\mathfrak{u}}}  %requires eufrak
+\newcommand{\isasymvv}{\isamath{\mathfrak{v}}}  %requires eufrak
+\newcommand{\isasymww}{\isamath{\mathfrak{w}}}  %requires eufrak
+\newcommand{\isasymxx}{\isamath{\mathfrak{x}}}  %requires eufrak
+\newcommand{\isasymyy}{\isamath{\mathfrak{y}}}  %requires eufrak
+\newcommand{\isasymzz}{\isamath{\mathfrak{z}}}  %requires eufrak
+\newcommand{\isasymalpha}{\isamath{\alpha}}
+\newcommand{\isasymbeta}{\isamath{\beta}}
+\newcommand{\isasymgamma}{\isamath{\gamma}}
+\newcommand{\isasymdelta}{\isamath{\delta}}
+\newcommand{\isasymepsilon}{\isamath{\varepsilon}}
+\newcommand{\isasymzeta}{\isamath{\zeta}}
+\newcommand{\isasymeta}{\isamath{\eta}}
+\newcommand{\isasymtheta}{\isamath{\vartheta}}
+\newcommand{\isasymiota}{\isamath{\iota}}
+\newcommand{\isasymkappa}{\isamath{\kappa}}
+\newcommand{\isasymlambda}{\isamath{\lambda}}
+\newcommand{\isasymmu}{\isamath{\mu}}
+\newcommand{\isasymnu}{\isamath{\nu}}
+\newcommand{\isasymxi}{\isamath{\xi}}
+\newcommand{\isasympi}{\isamath{\pi}}
+\newcommand{\isasymrho}{\isamath{\varrho}}
+\newcommand{\isasymsigma}{\isamath{\sigma}}
+\newcommand{\isasymtau}{\isamath{\tau}}
+\newcommand{\isasymupsilon}{\isamath{\upsilon}}
+\newcommand{\isasymphi}{\isamath{\varphi}}
+\newcommand{\isasymchi}{\isamath{\chi}}
+\newcommand{\isasympsi}{\isamath{\psi}}
+\newcommand{\isasymomega}{\isamath{\omega}}
+\newcommand{\isasymGamma}{\isamath{\Gamma}}
+\newcommand{\isasymDelta}{\isamath{\Delta}}
+\newcommand{\isasymTheta}{\isamath{\Theta}}
+\newcommand{\isasymLambda}{\isamath{\Lambda}}
+\newcommand{\isasymXi}{\isamath{\Xi}}
+\newcommand{\isasymPi}{\isamath{\Pi}}
+\newcommand{\isasymSigma}{\isamath{\Sigma}}
+\newcommand{\isasymUpsilon}{\isamath{\Upsilon}}
+\newcommand{\isasymPhi}{\isamath{\Phi}}
+\newcommand{\isasymPsi}{\isamath{\Psi}}
+\newcommand{\isasymOmega}{\isamath{\Omega}}
+\newcommand{\isasymbool}{\isamath{\mathrm{I}\mkern-3.8mu\mathrm{B}}}
+\newcommand{\isasymcomplex}{\isamath{\mathrm{C}\mkern-15mu{\phantom{\mathrm{t}}\vrule}\mkern9mu}}
+\newcommand{\isasymnat}{\isamath{\mathrm{I}\mkern-3.8mu\mathrm{N}}}
+\newcommand{\isasymrat}{\isamath{\mathrm{Q}\mkern-16mu{\phantom{\mathrm{t}}\vrule}\mkern10mu}}
+\newcommand{\isasymreal}{\isamath{\mathrm{I}\mkern-3.8mu\mathrm{R}}}
+\newcommand{\isasymint}{\isamath{\mathsf{Z}\mkern-7.5mu\mathsf{Z}}}
+\newcommand{\isasymleftarrow}{\isamath{\leftarrow}}
+\newcommand{\isasymlongleftarrow}{\isamath{\longleftarrow}}
+\newcommand{\isasymrightarrow}{\isamath{\rightarrow}}
+\newcommand{\isasymlongrightarrow}{\isamath{\longrightarrow}}
+\newcommand{\isasymLeftarrow}{\isamath{\Leftarrow}}
+\newcommand{\isasymLongleftarrow}{\isamath{\Longleftarrow}}
+\newcommand{\isasymRightarrow}{\isamath{\Rightarrow}}
+\newcommand{\isasymLongrightarrow}{\isamath{\Longrightarrow}}
+\newcommand{\isasymleftrightarrow}{\isamath{\leftrightarrow}}
+\newcommand{\isasymlongleftrightarrow}{\isamath{\longleftrightarrow}}
+\newcommand{\isasymLeftrightarrow}{\isamath{\Leftrightarrow}}
+\newcommand{\isasymLongleftrightarrow}{\isamath{\Longleftrightarrow}}
+\newcommand{\isasymmapsto}{\isamath{\mapsto}}
+\newcommand{\isasymlongmapsto}{\isamath{\longmapsto}}
+\newcommand{\isasymmidarrow}{\isamath{\relbar}}
+\newcommand{\isasymMidarrow}{\isamath{\Relbar}}
+\newcommand{\isasymhookleftarrow}{\isamath{\hookleftarrow}}
+\newcommand{\isasymhookrightarrow}{\isamath{\hookrightarrow}}
+\newcommand{\isasymleftharpoondown}{\isamath{\leftharpoondown}}
+\newcommand{\isasymrightharpoondown}{\isamath{\rightharpoondown}}
+\newcommand{\isasymleftharpoonup}{\isamath{\leftharpoonup}}
+\newcommand{\isasymrightharpoonup}{\isamath{\rightharpoonup}}
+\newcommand{\isasymrightleftharpoons}{\isamath{\rightleftharpoons}}
+\newcommand{\isasymdownharpoonleft}{\isamath{\downharpoonleft}}  %requires amssymb
+\newcommand{\isasymdownharpoonright}{\isamath{\downharpoonright}}  %requires amssymb
+\newcommand{\isasymupharpoonleft}{\isamath{\upharpoonleft}}  %requires amssymb
+\newcommand{\isasymupharpoonright}{\isamath{\upharpoonright}}  %requires amssymb
+\newcommand{\isasymrestriction}{\isamath{\restriction}}  %requires amssymb
+\newcommand{\isasymleadsto}{\isamath{\leadsto}}  %requires amssymb
+\newcommand{\isasymup}{\isamath{\uparrow}}
+\newcommand{\isasymUp}{\isamath{\Uparrow}}
+\newcommand{\isasymdown}{\isamath{\downarrow}}
+\newcommand{\isasymDown}{\isamath{\Downarrow}}
+\newcommand{\isasymupdown}{\isamath{\updownarrow}}
+\newcommand{\isasymUpdown}{\isamath{\Updownarrow}}
+\newcommand{\isasymlangle}{\isamath{\langle}}
+\newcommand{\isasymrangle}{\isamath{\rangle}}
+\newcommand{\isasymlceil}{\isamath{\lceil}}
+\newcommand{\isasymrceil}{\isamath{\rceil}}
+\newcommand{\isasymlfloor}{\isamath{\lfloor}}
+\newcommand{\isasymrfloor}{\isamath{\rfloor}}
+\newcommand{\isasymlparr}{\isamath{\mathopen{(\mkern-3mu\mid}}}
+\newcommand{\isasymrparr}{\isamath{\mathclose{\mid\mkern-3mu)}}}
+\newcommand{\isasymlbrakk}{\isamath{\mathopen{\lbrack\mkern-3mu\lbrack}}}
+\newcommand{\isasymrbrakk}{\isamath{\mathclose{\rbrack\mkern-3mu\rbrack}}}
+\newcommand{\isasymlbrace}{\isamath{\mathopen{\lbrace\mkern-4.5mu\mid}}}
+\newcommand{\isasymrbrace}{\isamath{\mathclose{\mid\mkern-4.5mu\rbrace}}}
+\newcommand{\isasymguillemotleft}{\isatext{\flqq}}  %requires babel 
+\newcommand{\isasymguillemotright}{\isatext{\frqq}}  %requires babel 
+\newcommand{\isasymColon}{\isamath{\mathrel{::}}}
+\newcommand{\isasymnot}{\isamath{\neg}}
+\newcommand{\isasymbottom}{\isamath{\bot}}
+\newcommand{\isasymtop}{\isamath{\top}}
+\newcommand{\isasymand}{\isamath{\wedge}}
+\newcommand{\isasymAnd}{\isamath{\bigwedge}}
+\newcommand{\isasymor}{\isamath{\vee}}
+\newcommand{\isasymOr}{\isamath{\bigvee}}
+\newcommand{\isasymforall}{\isamath{\forall\,}}
+\newcommand{\isasymexists}{\isamath{\exists\,}}
+\newcommand{\isasymnexists}{\isamath{\nexists\,}}  %requires amssymb
+\newcommand{\isasymbox}{\isamath{\Box}}  %requires amssymb
+\newcommand{\isasymdiamond}{\isamath{\Diamond}}  %requires amssymb
+\newcommand{\isasymturnstile}{\isamath{\vdash}}
+\newcommand{\isasymTurnstile}{\isamath{\models}}
+\newcommand{\isasymtturnstile}{\isamath{\vdash\!\!\!\vdash}}
+\newcommand{\isasymTTurnstile}{\isamath{\mid\!\models}}
+\newcommand{\isasymstileturn}{\isamath{\dashv}}
+\newcommand{\isasymsurd}{\isamath{\surd}}
+\newcommand{\isasymle}{\isamath{\le}}
+\newcommand{\isasymge}{\isamath{\ge}}
+\newcommand{\isasymlless}{\isamath{\ll}}
+\newcommand{\isasymggreater}{\isamath{\gg}}
+\newcommand{\isasymlesssim}{\isamath{\lesssim}}  %requires amssymb
+\newcommand{\isasymgreatersim}{\isamath{\gtrsim}}  %requires amssymb
+\newcommand{\isasymlessapprox}{\isamath{\lessapprox}}  %requires amssymb
+\newcommand{\isasymgreaterapprox}{\isamath{\gtrapprox}}  %requires amssymb
+\newcommand{\isasymin}{\isamath{\in}}
+\newcommand{\isasymnotin}{\isamath{\notin}}
+\newcommand{\isasymsubset}{\isamath{\subset}}
+\newcommand{\isasymsupset}{\isamath{\supset}}
+\newcommand{\isasymsubseteq}{\isamath{\subseteq}}
+\newcommand{\isasymsupseteq}{\isamath{\supseteq}}
+\newcommand{\isasymsqsubset}{\isamath{\sqsubset}}  %requires amssymb
+\newcommand{\isasymsqsupset}{\isamath{\sqsupset}}  %requires amssymb
+\newcommand{\isasymsqsubseteq}{\isamath{\sqsubseteq}}
+\newcommand{\isasymsqsupseteq}{\isamath{\sqsupseteq}}
+\newcommand{\isasyminter}{\isamath{\cap}}
+\newcommand{\isasymInter}{\isamath{\bigcap\,}}
+\newcommand{\isasymunion}{\isamath{\cup}}
+\newcommand{\isasymUnion}{\isamath{\bigcup\,}}
+\newcommand{\isasymsqunion}{\isamath{\sqcup}}
+\newcommand{\isasymSqunion}{\isamath{\bigsqcup\,}}
+\newcommand{\isasymsqinter}{\isamath{\sqcap}}
+\newcommand{\isasymSqinter}{\isamath{\bigsqcap\,}}  %requires amsmath
+\newcommand{\isasymuplus}{\isamath{\uplus}}
+\newcommand{\isasymUplus}{\isamath{\biguplus\,}}
+\newcommand{\isasymnoteq}{\isamath{\not=}}
+\newcommand{\isasymsim}{\isamath{\sim}}
+\newcommand{\isasymdoteq}{\isamath{\doteq}}
+\newcommand{\isasymsimeq}{\isamath{\simeq}}
+\newcommand{\isasymapprox}{\isamath{\approx}}
+\newcommand{\isasymasymp}{\isamath{\asymp}}
+\newcommand{\isasymcong}{\isamath{\cong}}
+\newcommand{\isasymsmile}{\isamath{\smile}}
+\newcommand{\isasymequiv}{\isamath{\equiv}}
+\newcommand{\isasymfrown}{\isamath{\frown}}
+\newcommand{\isasympropto}{\isamath{\propto}}
+\newcommand{\isasymsome}{\isamath{\epsilon\,}}
+\newcommand{\isasymJoin}{\isamath{\Join}}  %requires amssymb
+\newcommand{\isasymbowtie}{\isamath{\bowtie}}
+\newcommand{\isasymprec}{\isamath{\prec}}
+\newcommand{\isasymsucc}{\isamath{\succ}}
+\newcommand{\isasympreceq}{\isamath{\preceq}}
+\newcommand{\isasymsucceq}{\isamath{\succeq}}
+\newcommand{\isasymparallel}{\isamath{\parallel}}
+\newcommand{\isasymbar}{\isamath{\mid}}
+\newcommand{\isasymplusminus}{\isamath{\pm}}
+\newcommand{\isasymminusplus}{\isamath{\mp}}
+\newcommand{\isasymtimes}{\isamath{\times}}
+\newcommand{\isasymdiv}{\isamath{\div}}
+\newcommand{\isasymcdot}{\isamath{\cdot}}
+\newcommand{\isasymstar}{\isamath{\star}}
+\newcommand{\isasymbullet}{\boldmath\isamath{\mathchoice{\displaystyle{\cdot}}{\textstyle{\cdot}}{\scriptstyle{\bullet}}{\scriptscriptstyle{\bullet}}}}
+\newcommand{\isasymcirc}{\isamath{\circ}}
+\newcommand{\isasymdagger}{\isamath{\dagger}}
+\newcommand{\isasymddagger}{\isamath{\ddagger}}
+\newcommand{\isasymlhd}{\isamath{\lhd}}  %requires amssymb
+\newcommand{\isasymrhd}{\isamath{\rhd}}  %requires amssymb
+\newcommand{\isasymunlhd}{\isamath{\unlhd}}  %requires amssymb
+\newcommand{\isasymunrhd}{\isamath{\unrhd}}  %requires amssymb
+\newcommand{\isasymtriangleleft}{\isamath{\triangleleft}}
+\newcommand{\isasymtriangleright}{\isamath{\triangleright}}
+\newcommand{\isasymtriangle}{\isamath{\triangle}}
+\newcommand{\isasymtriangleq}{\isamath{\triangleq}}  %requires amssymb
+\newcommand{\isasymoplus}{\isamath{\oplus}}
+\newcommand{\isasymOplus}{\isamath{\bigoplus\,}}
+\newcommand{\isasymotimes}{\isamath{\otimes}}
+\newcommand{\isasymOtimes}{\isamath{\bigotimes\,}}
+\newcommand{\isasymodot}{\isamath{\odot}}
+\newcommand{\isasymOdot}{\isamath{\bigodot\,}}
+\newcommand{\isasymominus}{\isamath{\ominus}}
+\newcommand{\isasymoslash}{\isamath{\oslash}}
+\newcommand{\isasymdots}{\isamath{\dots}}
+\newcommand{\isasymcdots}{\isamath{\cdots}}
+\newcommand{\isasymSum}{\isamath{\sum\,}}
+\newcommand{\isasymProd}{\isamath{\prod\,}}
+\newcommand{\isasymCoprod}{\isamath{\coprod\,}}
+\newcommand{\isasyminfinity}{\isamath{\infty}}
+\newcommand{\isasymintegral}{\isamath{\int\,}}
+\newcommand{\isasymointegral}{\isamath{\oint\,}}
+\newcommand{\isasymclubsuit}{\isamath{\clubsuit}}
+\newcommand{\isasymdiamondsuit}{\isamath{\diamondsuit}}
+\newcommand{\isasymheartsuit}{\isamath{\heartsuit}}
+\newcommand{\isasymspadesuit}{\isamath{\spadesuit}}
+\newcommand{\isasymaleph}{\isamath{\aleph}}
+\newcommand{\isasymemptyset}{\isamath{\emptyset}}
+\newcommand{\isasymnabla}{\isamath{\nabla}}
+\newcommand{\isasympartial}{\isamath{\partial}}
+\newcommand{\isasymRe}{\isamath{\Re}}
+\newcommand{\isasymIm}{\isamath{\Im}}
+\newcommand{\isasymflat}{\isamath{\flat}}
+\newcommand{\isasymnatural}{\isamath{\natural}}
+\newcommand{\isasymsharp}{\isamath{\sharp}}
+\newcommand{\isasymangle}{\isamath{\angle}}
+\newcommand{\isasymcopyright}{\isatext{\rm\copyright}}
+\newcommand{\isasymregistered}{\isatext{\rm\textregistered}}
+\newcommand{\isasymhyphen}{\isatext{\rm-}}
+\newcommand{\isasyminverse}{\isamath{{}^{-1}}}
+\newcommand{\isasymonesuperior}{\isamath{\mathonesuperior}}  %requires latin1
+\newcommand{\isasymonequarter}{\isatext{\rm\textonequarter}}  %requires latin1
+\newcommand{\isasymtwosuperior}{\isamath{\mathtwosuperior}}  %requires latin1
+\newcommand{\isasymonehalf}{\isatext{\rm\textonehalf}}  %requires latin1
+\newcommand{\isasymthreesuperior}{\isamath{\maththreesuperior}}  %requires latin1
+\newcommand{\isasymthreequarters}{\isatext{\rm\textthreequarters}}  %requires latin1
+\newcommand{\isasymordfeminine}{\isatext{\rm\textordfeminine}}
+\newcommand{\isasymordmasculine}{\isatext{\rm\textordmasculine}}
+\newcommand{\isasymsection}{\isatext{\rm\S}}
+\newcommand{\isasymparagraph}{\isatext{\rm\P}}
+\newcommand{\isasymexclamdown}{\isatext{\rm\textexclamdown}}
+\newcommand{\isasymquestiondown}{\isatext{\rm\textquestiondown}}
+\newcommand{\isasymeuro}{\isatext{\textgreek{\euro}}}  %requires greek babel
+\newcommand{\isasympounds}{\isamath{\pounds}}
+\newcommand{\isasymyen}{\isatext{\yen}}  %requires amssymb
+\newcommand{\isasymcent}{\isatext{\textcent}}  %requires textcomp
+\newcommand{\isasymcurrency}{\isatext{\textcurrency}} %requires textcomp
+\newcommand{\isasymdegree}{\isatext{\rm\textdegree}}  %requires latin1
+\newcommand{\isasymamalg}{\isamath{\amalg}}
+\newcommand{\isasymmho}{\isamath{\mho}}  %requires amssymb
+\newcommand{\isasymlozenge}{\isamath{\lozenge}}  %requires amssymb
+\newcommand{\isasymwp}{\isamath{\wp}}
+\newcommand{\isasymwrong}{\isamath{\wr}}
+\newcommand{\isasymstruct}{\isamath{\diamond}}
+\newcommand{\isasymacute}{\isatext{\'\relax}}
+\newcommand{\isasymindex}{\isatext{\i}}
+\newcommand{\isasymdieresis}{\isatext{\"\relax}}
+\newcommand{\isasymcedilla}{\isatext{\c\relax}}
+\newcommand{\isasymhungarumlaut}{\isatext{\H\relax}}
+\newcommand{\isasymspacespace}{\isamath{~~}}
+\newcommand{\isasymmodule}{\isamath{\langle}\isakeyword{module}\isamath{\rangle}}

--- a/doc-src/AxClass/IsaMakefile	Fri Aug 19 22:28:23 2005 +0200
+++ b/doc-src/AxClass/IsaMakefile	Fri Aug 19 22:44:01 2005 +0200
@@ -13,7 +13,7 @@
 SRC = $(ISABELLE_HOME)/src
 OUT = $(ISABELLE_OUTPUT)
 LOG = $(OUT)/log
-USEDIR = $(ISATOOL) usedir -d "" -D ../generated
+USEDIR = $(ISATOOL) usedir -d false -D document
 
 
 ## Group
@@ -26,7 +26,7 @@
 $(LOG)/HOL-Group.gz: $(OUT)/HOL Group/ROOT.ML Group/Group.thy \
   Group/Product.thy Group/Semigroups.thy
 	@$(USEDIR) $(OUT)/HOL Group
-	@rm -f generated/pdfsetup.sty generated/session.tex
+	@rm -f Group/document/pdfsetup.sty Group/document/session.tex
 
 
 ## Nat
@@ -38,7 +38,7 @@
 
 $(LOG)/FOL-Nat.gz: $(OUT)/FOL Nat/ROOT.ML Nat/NatClass.ML Nat/NatClass.thy
 	@$(USEDIR) $(OUT)/FOL Nat
-	@rm -f generated/pdfsetup.sty generated/session.tex
+	@rm -f Nat/document/*.sty Nat/document/session.tex
 
 
 ## clean

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/AxClass/Nat/document/NatClass.tex	Fri Aug 19 22:44:01 2005 +0200
@@ -0,0 +1,96 @@
+%
+\begin{isabellebody}%
+\def\isabellecontext{NatClass}%
+\isamarkuptrue%
+%
+\isamarkupheader{Defining natural numbers in FOL \label{sec:ex-natclass}%
+}
+%
+\isadelimtheory
+%
+\endisadelimtheory
+%
+\isatagtheory
+\isamarkupfalse%
+\isacommand{theory}\ NatClass\ \isakeyword{imports}\ FOL\ \isakeyword{begin}%
+\endisatagtheory
+{\isafoldtheory}%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+\medskip\noindent Axiomatic type classes abstract over exactly one
+ type argument. Thus, any \emph{axiomatic} theory extension where each
+ axiom refers to at most one type variable, may be trivially turned
+ into a \emph{definitional} one.
+
+ We illustrate this with the natural numbers in
+ Isabelle/FOL.\footnote{See also
+ \url{http://isabelle.in.tum.de/library/FOL/ex/NatClass.html}}%
+\end{isamarkuptext}%
+\isamarkupfalse%
+\isacommand{consts}\isanewline
+\ \ zero\ {\isacharcolon}{\isacharcolon}\ {\isacharprime}a\ \ \ \ {\isacharparenleft}{\isachardoublequote}{\isasymzero}{\isachardoublequote}{\isacharparenright}\isanewline
+\ \ Suc\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a{\isachardoublequote}\isanewline
+\ \ rec\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharparenleft}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a{\isacharparenright}\ {\isasymRightarrow}\ {\isacharprime}a{\isachardoublequote}\isanewline
+\isanewline
+\isamarkupfalse%
+\isacommand{axclass}\ nat\ {\isasymsubseteq}\ {\isachardoublequote}term{\isachardoublequote}\isanewline
+\ \ induct{\isacharcolon}\ {\isachardoublequote}P{\isacharparenleft}{\isasymzero}{\isacharparenright}\ {\isasymLongrightarrow}\ {\isacharparenleft}{\isasymAnd}x{\isachardot}\ P{\isacharparenleft}x{\isacharparenright}\ {\isasymLongrightarrow}\ P{\isacharparenleft}Suc{\isacharparenleft}x{\isacharparenright}{\isacharparenright}{\isacharparenright}\ {\isasymLongrightarrow}\ P{\isacharparenleft}n{\isacharparenright}{\isachardoublequote}\isanewline
+\ \ Suc{\isacharunderscore}inject{\isacharcolon}\ {\isachardoublequote}Suc{\isacharparenleft}m{\isacharparenright}\ {\isacharequal}\ Suc{\isacharparenleft}n{\isacharparenright}\ {\isasymLongrightarrow}\ m\ {\isacharequal}\ n{\isachardoublequote}\isanewline
+\ \ Suc{\isacharunderscore}neq{\isacharunderscore}{\isadigit{0}}{\isacharcolon}\ {\isachardoublequote}Suc{\isacharparenleft}m{\isacharparenright}\ {\isacharequal}\ {\isasymzero}\ {\isasymLongrightarrow}\ R{\isachardoublequote}\isanewline
+\ \ rec{\isacharunderscore}{\isadigit{0}}{\isacharcolon}\ {\isachardoublequote}rec{\isacharparenleft}{\isasymzero}{\isacharcomma}\ a{\isacharcomma}\ f{\isacharparenright}\ {\isacharequal}\ a{\isachardoublequote}\isanewline
+\ \ rec{\isacharunderscore}Suc{\isacharcolon}\ {\isachardoublequote}rec{\isacharparenleft}Suc{\isacharparenleft}m{\isacharparenright}{\isacharcomma}\ a{\isacharcomma}\ f{\isacharparenright}\ {\isacharequal}\ f{\isacharparenleft}m{\isacharcomma}\ rec{\isacharparenleft}m{\isacharcomma}\ a{\isacharcomma}\ f{\isacharparenright}{\isacharparenright}{\isachardoublequote}\isanewline
+\isanewline
+\isamarkupfalse%
+\isacommand{constdefs}\isanewline
+\ \ add\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharprime}a{\isacharcolon}{\isacharcolon}nat\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a{\isachardoublequote}\ \ \ \ {\isacharparenleft}\isakeyword{infixl}\ {\isachardoublequote}{\isacharplus}{\isachardoublequote}\ {\isadigit{6}}{\isadigit{0}}{\isacharparenright}\isanewline
+\ \ {\isachardoublequote}m\ {\isacharplus}\ n\ {\isasymequiv}\ rec{\isacharparenleft}m{\isacharcomma}\ n{\isacharcomma}\ {\isasymlambda}x\ y{\isachardot}\ Suc{\isacharparenleft}y{\isacharparenright}{\isacharparenright}{\isachardoublequote}\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+This is an abstract version of the plain \isa{Nat} theory in
+ FOL.\footnote{See
+ \url{http://isabelle.in.tum.de/library/FOL/ex/Nat.html}} Basically,
+ we have just replaced all occurrences of type \isa{nat} by \isa{{\isacharprime}a} and used the natural number axioms to define class \isa{nat}.
+ There is only a minor snag, that the original recursion operator
+ \isa{rec} had to be made monomorphic.
+
+ Thus class \isa{nat} contains exactly those types \isa{{\isasymtau}} that
+ are isomorphic to ``the'' natural numbers (with signature \isa{{\isasymzero}}, \isa{Suc}, \isa{rec}).
+
+ \medskip What we have done here can be also viewed as \emph{type
+ specification}.  Of course, it still remains open if there is some
+ type at all that meets the class axioms.  Now a very nice property of
+ axiomatic type classes is that abstract reasoning is always possible
+ --- independent of satisfiability.  The meta-logic won't break, even
+ if some classes (or general sorts) turns out to be empty later ---
+ ``inconsistent'' class definitions may be useless, but do not cause
+ any harm.
+
+ Theorems of the abstract natural numbers may be derived in the same
+ way as for the concrete version.  The original proof scripts may be
+ re-used with some trivial changes only (mostly adding some type
+ constraints).%
+\end{isamarkuptext}%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+%
+\isatagtheory
+\isamarkupfalse%
+\isacommand{end}%
+\endisatagtheory
+{\isafoldtheory}%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+\end{isabellebody}%
+%%% Local Variables:
+%%% mode: latex
+%%% TeX-master: "root"
+%%% End:

--- a/doc-src/AxClass/axclass.tex	Fri Aug 19 22:28:23 2005 +0200
+++ b/doc-src/AxClass/axclass.tex	Fri Aug 19 22:44:01 2005 +0200
@@ -1,7 +1,7 @@
 
 \documentclass[12pt,a4paper,fleqn]{report}
 \usepackage{graphicx,../iman,../extra,../isar}
-\usepackage{generated/isabelle,generated/isabellesym}
+\usepackage{Group/document/isabelle,Group/document/isabellesym}
 \usepackage{../pdfsetup}  % last one!
 
 \isabellestyle{it}

--- a/doc-src/AxClass/body.tex	Fri Aug 19 22:28:23 2005 +0200
+++ b/doc-src/AxClass/body.tex	Fri Aug 19 22:44:01 2005 +0200
@@ -47,13 +47,13 @@
 FOL.  See also \url{http://isabelle.in.tum.de/library/HOL/AxClasses/} and
 \url{http://isabelle.in.tum.de/library/FOL/ex/NatClass.html}.
 
-\input{generated/Semigroups}
+\input{Group/document/Semigroups}
 
-\input{generated/Group}
+\input{Group/document/Group}
 
-\input{generated/Product}
+\input{Group/document/Product}
 
-\input{generated/NatClass}
+\input{Nat/document/NatClass}
 
 
 %% FIXME move some parts to ref or isar-ref manual (!?);

--- a/doc-src/AxClass/generated/Group.tex	Fri Aug 19 22:28:23 2005 +0200
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,511 +0,0 @@
-%
-\begin{isabellebody}%
-\def\isabellecontext{Group}%
-\isamarkuptrue%
-%
-\isamarkupheader{Basic group theory%
-}
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isamarkupfalse%
-\isacommand{theory}\ Group\ \isakeyword{imports}\ Main\ \isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-\medskip\noindent The meta-level type system of Isabelle supports
-  \emph{intersections} and \emph{inclusions} of type classes. These
-  directly correspond to intersections and inclusions of type
-  predicates in a purely set theoretic sense. This is sufficient as a
-  means to describe simple hierarchies of structures.  As an
-  illustration, we use the well-known example of semigroups, monoids,
-  general groups and Abelian groups.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsubsection{Monoids and Groups%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-First we declare some polymorphic constants required later for the
-  signature parts of our structures.%
-\end{isamarkuptext}%
-\isamarkupfalse%
-\isacommand{consts}\isanewline
-\ \ times\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a{\isachardoublequote}\ \ \ \ {\isacharparenleft}\isakeyword{infixl}\ {\isachardoublequote}{\isasymodot}{\isachardoublequote}\ {\isadigit{7}}{\isadigit{0}}{\isacharparenright}\isanewline
-\ \ invers\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a{\isachardoublequote}\ \ \ \ {\isacharparenleft}{\isachardoublequote}{\isacharparenleft}{\isacharunderscore}{\isasyminv}{\isacharparenright}{\isachardoublequote}\ {\isacharbrackleft}{\isadigit{1}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isacharbrackright}\ {\isadigit{9}}{\isadigit{9}}{\isadigit{9}}{\isacharparenright}\isanewline
-\ \ one\ {\isacharcolon}{\isacharcolon}\ {\isacharprime}a\ \ \ \ {\isacharparenleft}{\isachardoublequote}{\isasymone}{\isachardoublequote}{\isacharparenright}\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-\noindent Next we define class \isa{monoid} of monoids with
-  operations \isa{{\isasymodot}} and \isa{{\isasymone}}.  Note that multiple class
-  axioms are allowed for user convenience --- they simply represent
-  the conjunction of their respective universal closures.%
-\end{isamarkuptext}%
-\isamarkupfalse%
-\isacommand{axclass}\ monoid\ {\isasymsubseteq}\ type\isanewline
-\ \ assoc{\isacharcolon}\ {\isachardoublequote}{\isacharparenleft}x\ {\isasymodot}\ y{\isacharparenright}\ {\isasymodot}\ z\ {\isacharequal}\ x\ {\isasymodot}\ {\isacharparenleft}y\ {\isasymodot}\ z{\isacharparenright}{\isachardoublequote}\isanewline
-\ \ left{\isacharunderscore}unit{\isacharcolon}\ {\isachardoublequote}{\isasymone}\ {\isasymodot}\ x\ {\isacharequal}\ x{\isachardoublequote}\isanewline
-\ \ right{\isacharunderscore}unit{\isacharcolon}\ {\isachardoublequote}x\ {\isasymodot}\ {\isasymone}\ {\isacharequal}\ x{\isachardoublequote}\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-\noindent So class \isa{monoid} contains exactly those types
-  \isa{{\isasymtau}} where \isa{{\isasymodot}\ {\isasymColon}\ {\isasymtau}\ {\isasymRightarrow}\ {\isasymtau}\ {\isasymRightarrow}\ {\isasymtau}} and \isa{{\isasymone}\ {\isasymColon}\ {\isasymtau}}
-  are specified appropriately, such that \isa{{\isasymodot}} is associative and
-  \isa{{\isasymone}} is a left and right unit element for the \isa{{\isasymodot}}
-  operation.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-\medskip Independently of \isa{monoid}, we now define a linear
-  hierarchy of semigroups, general groups and Abelian groups.  Note
-  that the names of class axioms are automatically qualified with each
-  class name, so we may re-use common names such as \isa{assoc}.%
-\end{isamarkuptext}%
-\isamarkupfalse%
-\isacommand{axclass}\ semigroup\ {\isasymsubseteq}\ type\isanewline
-\ \ assoc{\isacharcolon}\ {\isachardoublequote}{\isacharparenleft}x\ {\isasymodot}\ y{\isacharparenright}\ {\isasymodot}\ z\ {\isacharequal}\ x\ {\isasymodot}\ {\isacharparenleft}y\ {\isasymodot}\ z{\isacharparenright}{\isachardoublequote}\isanewline
-\isanewline
-\isamarkupfalse%
-\isacommand{axclass}\ group\ {\isasymsubseteq}\ semigroup\isanewline
-\ \ left{\isacharunderscore}unit{\isacharcolon}\ {\isachardoublequote}{\isasymone}\ {\isasymodot}\ x\ {\isacharequal}\ x{\isachardoublequote}\isanewline
-\ \ left{\isacharunderscore}inverse{\isacharcolon}\ {\isachardoublequote}x{\isasyminv}\ {\isasymodot}\ x\ {\isacharequal}\ {\isasymone}{\isachardoublequote}\isanewline
-\isanewline
-\isamarkupfalse%
-\isacommand{axclass}\ agroup\ {\isasymsubseteq}\ group\isanewline
-\ \ commute{\isacharcolon}\ {\isachardoublequote}x\ {\isasymodot}\ y\ {\isacharequal}\ y\ {\isasymodot}\ x{\isachardoublequote}\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-\noindent Class \isa{group} inherits associativity of \isa{{\isasymodot}}
-  from \isa{semigroup} and adds two further group axioms. Similarly,
-  \isa{agroup} is defined as the subset of \isa{group} such that
-  for all of its elements \isa{{\isasymtau}}, the operation \isa{{\isasymodot}\ {\isasymColon}\ {\isasymtau}\ {\isasymRightarrow}\ {\isasymtau}\ {\isasymRightarrow}\ {\isasymtau}} is even commutative.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsubsection{Abstract reasoning%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-In a sense, axiomatic type classes may be viewed as \emph{abstract
-  theories}.  Above class definitions gives rise to abstract axioms
-  \isa{assoc}, \isa{left{\isacharunderscore}unit}, \isa{left{\isacharunderscore}inverse}, \isa{commute}, where any of these contain a type variable \isa{{\isacharprime}a\ {\isasymColon}\ c} that is restricted to types of the corresponding class \isa{c}.  \emph{Sort constraints} like this express a logical
-  precondition for the whole formula.  For example, \isa{assoc}
-  states that for all \isa{{\isasymtau}}, provided that \isa{{\isasymtau}\ {\isasymColon}\ semigroup}, the operation \isa{{\isasymodot}\ {\isasymColon}\ {\isasymtau}\ {\isasymRightarrow}\ {\isasymtau}\ {\isasymRightarrow}\ {\isasymtau}} is associative.
-
-  \medskip From a technical point of view, abstract axioms are just
-  ordinary Isabelle theorems, which may be used in proofs without
-  special treatment.  Such ``abstract proofs'' usually yield new
-  ``abstract theorems''.  For example, we may now derive the following
-  well-known laws of general groups.%
-\end{isamarkuptext}%
-\isamarkupfalse%
-\isacommand{theorem}\ group{\isacharunderscore}right{\isacharunderscore}inverse{\isacharcolon}\ {\isachardoublequote}x\ {\isasymodot}\ x{\isasyminv}\ {\isacharequal}\ {\isacharparenleft}{\isasymone}{\isasymColon}{\isacharprime}a{\isasymColon}group{\isacharparenright}{\isachardoublequote}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isamarkupfalse%
-\isacommand{proof}\ {\isacharminus}\isanewline
-\ \ \isamarkupfalse%
-\isacommand{have}\ {\isachardoublequote}x\ {\isasymodot}\ x{\isasyminv}\ {\isacharequal}\ {\isasymone}\ {\isasymodot}\ {\isacharparenleft}x\ {\isasymodot}\ x{\isasyminv}{\isacharparenright}{\isachardoublequote}\isanewline
-\ \ \ \ \isamarkupfalse%
-\isacommand{by}\ {\isacharparenleft}simp\ only{\isacharcolon}\ group{\isachardot}left{\isacharunderscore}unit{\isacharparenright}\isanewline
-\ \ \isamarkupfalse%
-\isacommand{also}\ \isamarkupfalse%
-\isacommand{have}\ {\isachardoublequote}{\isachardot}{\isachardot}{\isachardot}\ {\isacharequal}\ {\isasymone}\ {\isasymodot}\ x\ {\isasymodot}\ x{\isasyminv}{\isachardoublequote}\isanewline
-\ \ \ \ \isamarkupfalse%
-\isacommand{by}\ {\isacharparenleft}simp\ only{\isacharcolon}\ semigroup{\isachardot}assoc{\isacharparenright}\isanewline
-\ \ \isamarkupfalse%
-\isacommand{also}\ \isamarkupfalse%
-\isacommand{have}\ {\isachardoublequote}{\isachardot}{\isachardot}{\isachardot}\ {\isacharequal}\ {\isacharparenleft}x{\isasyminv}{\isacharparenright}{\isasyminv}\ {\isasymodot}\ x{\isasyminv}\ {\isasymodot}\ x\ {\isasymodot}\ x{\isasyminv}{\isachardoublequote}\isanewline
-\ \ \ \ \isamarkupfalse%
-\isacommand{by}\ {\isacharparenleft}simp\ only{\isacharcolon}\ group{\isachardot}left{\isacharunderscore}inverse{\isacharparenright}\isanewline
-\ \ \isamarkupfalse%
-\isacommand{also}\ \isamarkupfalse%
-\isacommand{have}\ {\isachardoublequote}{\isachardot}{\isachardot}{\isachardot}\ {\isacharequal}\ {\isacharparenleft}x{\isasyminv}{\isacharparenright}{\isasyminv}\ {\isasymodot}\ {\isacharparenleft}x{\isasyminv}\ {\isasymodot}\ x{\isacharparenright}\ {\isasymodot}\ x{\isasyminv}{\isachardoublequote}\isanewline
-\ \ \ \ \isamarkupfalse%
-\isacommand{by}\ {\isacharparenleft}simp\ only{\isacharcolon}\ semigroup{\isachardot}assoc{\isacharparenright}\isanewline
-\ \ \isamarkupfalse%
-\isacommand{also}\ \isamarkupfalse%
-\isacommand{have}\ {\isachardoublequote}{\isachardot}{\isachardot}{\isachardot}\ {\isacharequal}\ {\isacharparenleft}x{\isasyminv}{\isacharparenright}{\isasyminv}\ {\isasymodot}\ {\isasymone}\ {\isasymodot}\ x{\isasyminv}{\isachardoublequote}\isanewline
-\ \ \ \ \isamarkupfalse%
-\isacommand{by}\ {\isacharparenleft}simp\ only{\isacharcolon}\ group{\isachardot}left{\isacharunderscore}inverse{\isacharparenright}\isanewline
-\ \ \isamarkupfalse%
-\isacommand{also}\ \isamarkupfalse%
-\isacommand{have}\ {\isachardoublequote}{\isachardot}{\isachardot}{\isachardot}\ {\isacharequal}\ {\isacharparenleft}x{\isasyminv}{\isacharparenright}{\isasyminv}\ {\isasymodot}\ {\isacharparenleft}{\isasymone}\ {\isasymodot}\ x{\isasyminv}{\isacharparenright}{\isachardoublequote}\isanewline
-\ \ \ \ \isamarkupfalse%
-\isacommand{by}\ {\isacharparenleft}simp\ only{\isacharcolon}\ semigroup{\isachardot}assoc{\isacharparenright}\isanewline
-\ \ \isamarkupfalse%
-\isacommand{also}\ \isamarkupfalse%
-\isacommand{have}\ {\isachardoublequote}{\isachardot}{\isachardot}{\isachardot}\ {\isacharequal}\ {\isacharparenleft}x{\isasyminv}{\isacharparenright}{\isasyminv}\ {\isasymodot}\ x{\isasyminv}{\isachardoublequote}\isanewline
-\ \ \ \ \isamarkupfalse%
-\isacommand{by}\ {\isacharparenleft}simp\ only{\isacharcolon}\ group{\isachardot}left{\isacharunderscore}unit{\isacharparenright}\isanewline
-\ \ \isamarkupfalse%
-\isacommand{also}\ \isamarkupfalse%
-\isacommand{have}\ {\isachardoublequote}{\isachardot}{\isachardot}{\isachardot}\ {\isacharequal}\ {\isasymone}{\isachardoublequote}\isanewline
-\ \ \ \ \isamarkupfalse%
-\isacommand{by}\ {\isacharparenleft}simp\ only{\isacharcolon}\ group{\isachardot}left{\isacharunderscore}inverse{\isacharparenright}\isanewline
-\ \ \isamarkupfalse%
-\isacommand{finally}\ \isamarkupfalse%
-\isacommand{show}\ {\isacharquery}thesis\ \isamarkupfalse%
-\isacommand{{\isachardot}}\isanewline
-\isamarkupfalse%
-\isacommand{qed}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-\noindent With \isa{group{\isacharunderscore}right{\isacharunderscore}inverse} already available, \isa{group{\isacharunderscore}right{\isacharunderscore}unit}\label{thm:group-right-unit} is now established
-  much easier.%
-\end{isamarkuptext}%
-\isamarkupfalse%
-\isacommand{theorem}\ group{\isacharunderscore}right{\isacharunderscore}unit{\isacharcolon}\ {\isachardoublequote}x\ {\isasymodot}\ {\isasymone}\ {\isacharequal}\ {\isacharparenleft}x{\isasymColon}{\isacharprime}a{\isasymColon}group{\isacharparenright}{\isachardoublequote}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isamarkupfalse%
-\isacommand{proof}\ {\isacharminus}\isanewline
-\ \ \isamarkupfalse%
-\isacommand{have}\ {\isachardoublequote}x\ {\isasymodot}\ {\isasymone}\ {\isacharequal}\ x\ {\isasymodot}\ {\isacharparenleft}x{\isasyminv}\ {\isasymodot}\ x{\isacharparenright}{\isachardoublequote}\isanewline
-\ \ \ \ \isamarkupfalse%
-\isacommand{by}\ {\isacharparenleft}simp\ only{\isacharcolon}\ group{\isachardot}left{\isacharunderscore}inverse{\isacharparenright}\isanewline
-\ \ \isamarkupfalse%
-\isacommand{also}\ \isamarkupfalse%
-\isacommand{have}\ {\isachardoublequote}{\isachardot}{\isachardot}{\isachardot}\ {\isacharequal}\ x\ {\isasymodot}\ x{\isasyminv}\ {\isasymodot}\ x{\isachardoublequote}\isanewline
-\ \ \ \ \isamarkupfalse%
-\isacommand{by}\ {\isacharparenleft}simp\ only{\isacharcolon}\ semigroup{\isachardot}assoc{\isacharparenright}\isanewline
-\ \ \isamarkupfalse%
-\isacommand{also}\ \isamarkupfalse%
-\isacommand{have}\ {\isachardoublequote}{\isachardot}{\isachardot}{\isachardot}\ {\isacharequal}\ {\isasymone}\ {\isasymodot}\ x{\isachardoublequote}\isanewline
-\ \ \ \ \isamarkupfalse%
-\isacommand{by}\ {\isacharparenleft}simp\ only{\isacharcolon}\ group{\isacharunderscore}right{\isacharunderscore}inverse{\isacharparenright}\isanewline
-\ \ \isamarkupfalse%
-\isacommand{also}\ \isamarkupfalse%
-\isacommand{have}\ {\isachardoublequote}{\isachardot}{\isachardot}{\isachardot}\ {\isacharequal}\ x{\isachardoublequote}\isanewline
-\ \ \ \ \isamarkupfalse%
-\isacommand{by}\ {\isacharparenleft}simp\ only{\isacharcolon}\ group{\isachardot}left{\isacharunderscore}unit{\isacharparenright}\isanewline
-\ \ \isamarkupfalse%
-\isacommand{finally}\ \isamarkupfalse%
-\isacommand{show}\ {\isacharquery}thesis\ \isamarkupfalse%
-\isacommand{{\isachardot}}\isanewline
-\isamarkupfalse%
-\isacommand{qed}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-\medskip Abstract theorems may be instantiated to only those types
-  \isa{{\isasymtau}} where the appropriate class membership \isa{{\isasymtau}\ {\isasymColon}\ c} is
-  known at Isabelle's type signature level.  Since we have \isa{agroup\ {\isasymsubseteq}\ group\ {\isasymsubseteq}\ semigroup} by definition, all theorems of \isa{semigroup} and \isa{group} are automatically inherited by \isa{group} and \isa{agroup}.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsubsection{Abstract instantiation%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-From the definition, the \isa{monoid} and \isa{group} classes
-  have been independent.  Note that for monoids, \isa{right{\isacharunderscore}unit}
-  had to be included as an axiom, but for groups both \isa{right{\isacharunderscore}unit} and \isa{right{\isacharunderscore}inverse} are derivable from the other
-  axioms.  With \isa{group{\isacharunderscore}right{\isacharunderscore}unit} derived as a theorem of group
-  theory (see page~\pageref{thm:group-right-unit}), we may now
-  instantiate \isa{monoid\ {\isasymsubseteq}\ semigroup} and \isa{group\ {\isasymsubseteq}\ monoid} properly as follows (cf.\ \figref{fig:monoid-group}).
-
- \begin{figure}[htbp]
-   \begin{center}
-     \small
-     \unitlength 0.6mm
-     \begin{picture}(65,90)(0,-10)
-       \put(15,10){\line(0,1){10}} \put(15,30){\line(0,1){10}}
-       \put(15,50){\line(1,1){10}} \put(35,60){\line(1,-1){10}}
-       \put(15,5){\makebox(0,0){\isa{agroup}}}
-       \put(15,25){\makebox(0,0){\isa{group}}}
-       \put(15,45){\makebox(0,0){\isa{semigroup}}}
-       \put(30,65){\makebox(0,0){\isa{type}}} \put(50,45){\makebox(0,0){\isa{monoid}}}
-     \end{picture}
-     \hspace{4em}
-     \begin{picture}(30,90)(0,0)
-       \put(15,10){\line(0,1){10}} \put(15,30){\line(0,1){10}}
-       \put(15,50){\line(0,1){10}} \put(15,70){\line(0,1){10}}
-       \put(15,5){\makebox(0,0){\isa{agroup}}}
-       \put(15,25){\makebox(0,0){\isa{group}}}
-       \put(15,45){\makebox(0,0){\isa{monoid}}}
-       \put(15,65){\makebox(0,0){\isa{semigroup}}}
-       \put(15,85){\makebox(0,0){\isa{type}}}
-     \end{picture}
-     \caption{Monoids and groups: according to definition, and by proof}
-     \label{fig:monoid-group}
-   \end{center}
- \end{figure}%
-\end{isamarkuptext}%
-\isamarkupfalse%
-\isacommand{instance}\ monoid\ {\isasymsubseteq}\ semigroup\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isamarkupfalse%
-\isacommand{proof}\isanewline
-\ \ \isamarkupfalse%
-\isacommand{fix}\ x\ y\ z\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharprime}a{\isasymColon}monoid{\isachardoublequote}\isanewline
-\ \ \isamarkupfalse%
-\isacommand{show}\ {\isachardoublequote}x\ {\isasymodot}\ y\ {\isasymodot}\ z\ {\isacharequal}\ x\ {\isasymodot}\ {\isacharparenleft}y\ {\isasymodot}\ z{\isacharparenright}{\isachardoublequote}\isanewline
-\ \ \ \ \isamarkupfalse%
-\isacommand{by}\ {\isacharparenleft}rule\ monoid{\isachardot}assoc{\isacharparenright}\isanewline
-\isamarkupfalse%
-\isacommand{qed}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isamarkupfalse%
-\isacommand{instance}\ group\ {\isasymsubseteq}\ monoid\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isamarkupfalse%
-\isacommand{proof}\isanewline
-\ \ \isamarkupfalse%
-\isacommand{fix}\ x\ y\ z\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharprime}a{\isasymColon}group{\isachardoublequote}\isanewline
-\ \ \isamarkupfalse%
-\isacommand{show}\ {\isachardoublequote}x\ {\isasymodot}\ y\ {\isasymodot}\ z\ {\isacharequal}\ x\ {\isasymodot}\ {\isacharparenleft}y\ {\isasymodot}\ z{\isacharparenright}{\isachardoublequote}\isanewline
-\ \ \ \ \isamarkupfalse%
-\isacommand{by}\ {\isacharparenleft}rule\ semigroup{\isachardot}assoc{\isacharparenright}\isanewline
-\ \ \isamarkupfalse%
-\isacommand{show}\ {\isachardoublequote}{\isasymone}\ {\isasymodot}\ x\ {\isacharequal}\ x{\isachardoublequote}\isanewline
-\ \ \ \ \isamarkupfalse%
-\isacommand{by}\ {\isacharparenleft}rule\ group{\isachardot}left{\isacharunderscore}unit{\isacharparenright}\isanewline
-\ \ \isamarkupfalse%
-\isacommand{show}\ {\isachardoublequote}x\ {\isasymodot}\ {\isasymone}\ {\isacharequal}\ x{\isachardoublequote}\isanewline
-\ \ \ \ \isamarkupfalse%
-\isacommand{by}\ {\isacharparenleft}rule\ group{\isacharunderscore}right{\isacharunderscore}unit{\isacharparenright}\isanewline
-\isamarkupfalse%
-\isacommand{qed}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-\medskip The $\INSTANCE$ command sets up an appropriate goal that
-  represents the class inclusion (or type arity, see
-  \secref{sec:inst-arity}) to be proven (see also
-  \cite{isabelle-isar-ref}).  The initial proof step causes
-  back-chaining of class membership statements wrt.\ the hierarchy of
-  any classes defined in the current theory; the effect is to reduce
-  to the initial statement to a number of goals that directly
-  correspond to any class axioms encountered on the path upwards
-  through the class hierarchy.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsubsection{Concrete instantiation \label{sec:inst-arity}%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-So far we have covered the case of the form $\INSTANCE$~\isa{c\isactrlsub {\isadigit{1}}\ {\isasymsubseteq}\ c\isactrlsub {\isadigit{2}}}, namely \emph{abstract instantiation} ---
-  $c@1$ is more special than \isa{c\isactrlsub {\isadigit{1}}} and thus an instance
-  of \isa{c\isactrlsub {\isadigit{2}}}.  Even more interesting for practical
-  applications are \emph{concrete instantiations} of axiomatic type
-  classes.  That is, certain simple schemes \isa{{\isacharparenleft}{\isasymalpha}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isasymalpha}\isactrlsub n{\isacharparenright}\ t\ {\isasymColon}\ c} of class membership may be established at the
-  logical level and then transferred to Isabelle's type signature
-  level.
-
-  \medskip As a typical example, we show that type \isa{bool} with
-  exclusive-or as \isa{{\isasymodot}} operation, identity as \isa{{\isasyminv}}, and
-  \isa{False} as \isa{{\isasymone}} forms an Abelian group.%
-\end{isamarkuptext}%
-\isamarkupfalse%
-\isacommand{defs}\ {\isacharparenleft}\isakeyword{overloaded}{\isacharparenright}\isanewline
-\ \ times{\isacharunderscore}bool{\isacharunderscore}def{\isacharcolon}\ {\isachardoublequote}x\ {\isasymodot}\ y\ {\isasymequiv}\ x\ {\isasymnoteq}\ {\isacharparenleft}y{\isasymColon}bool{\isacharparenright}{\isachardoublequote}\isanewline
-\ \ inverse{\isacharunderscore}bool{\isacharunderscore}def{\isacharcolon}\ {\isachardoublequote}x{\isasyminv}\ {\isasymequiv}\ x{\isasymColon}bool{\isachardoublequote}\isanewline
-\ \ unit{\isacharunderscore}bool{\isacharunderscore}def{\isacharcolon}\ {\isachardoublequote}{\isasymone}\ {\isasymequiv}\ False{\isachardoublequote}\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-\medskip It is important to note that above $\DEFS$ are just
-  overloaded meta-level constant definitions, where type classes are
-  not yet involved at all.  This form of constant definition with
-  overloading (and optional recursion over the syntactic structure of
-  simple types) are admissible as definitional extensions of plain HOL
-  \cite{Wenzel:1997:TPHOL}.  The Haskell-style type system is not
-  required for overloading.  Nevertheless, overloaded definitions are
-  best applied in the context of type classes.
-
-  \medskip Since we have chosen above $\DEFS$ of the generic group
-  operations on type \isa{bool} appropriately, the class membership
-  \isa{bool\ {\isasymColon}\ agroup} may be now derived as follows.%
-\end{isamarkuptext}%
-\isamarkupfalse%
-\isacommand{instance}\ bool\ {\isacharcolon}{\isacharcolon}\ agroup\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isamarkupfalse%
-\isacommand{proof}\ {\isacharparenleft}intro{\isacharunderscore}classes{\isacharcomma}\isanewline
-\ \ \ \ unfold\ times{\isacharunderscore}bool{\isacharunderscore}def\ inverse{\isacharunderscore}bool{\isacharunderscore}def\ unit{\isacharunderscore}bool{\isacharunderscore}def{\isacharparenright}\isanewline
-\ \ \isamarkupfalse%
-\isacommand{fix}\ x\ y\ z\isanewline
-\ \ \isamarkupfalse%
-\isacommand{show}\ {\isachardoublequote}{\isacharparenleft}{\isacharparenleft}x\ {\isasymnoteq}\ y{\isacharparenright}\ {\isasymnoteq}\ z{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}x\ {\isasymnoteq}\ {\isacharparenleft}y\ {\isasymnoteq}\ z{\isacharparenright}{\isacharparenright}{\isachardoublequote}\ \isamarkupfalse%
-\isacommand{by}\ blast\isanewline
-\ \ \isamarkupfalse%
-\isacommand{show}\ {\isachardoublequote}{\isacharparenleft}False\ {\isasymnoteq}\ x{\isacharparenright}\ {\isacharequal}\ x{\isachardoublequote}\ \isamarkupfalse%
-\isacommand{by}\ blast\isanewline
-\ \ \isamarkupfalse%
-\isacommand{show}\ {\isachardoublequote}{\isacharparenleft}x\ {\isasymnoteq}\ x{\isacharparenright}\ {\isacharequal}\ False{\isachardoublequote}\ \isamarkupfalse%
-\isacommand{by}\ blast\isanewline
-\ \ \isamarkupfalse%
-\isacommand{show}\ {\isachardoublequote}{\isacharparenleft}x\ {\isasymnoteq}\ y{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}y\ {\isasymnoteq}\ x{\isacharparenright}{\isachardoublequote}\ \isamarkupfalse%
-\isacommand{by}\ blast\isanewline
-\isamarkupfalse%
-\isacommand{qed}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-The result of an $\INSTANCE$ statement is both expressed as a
-  theorem of Isabelle's meta-logic, and as a type arity of the type
-  signature.  The latter enables type-inference system to take care of
-  this new instance automatically.
-
-  \medskip We could now also instantiate our group theory classes to
-  many other concrete types.  For example, \isa{int\ {\isasymColon}\ agroup}
-  (e.g.\ by defining \isa{{\isasymodot}} as addition, \isa{{\isasyminv}} as negation
-  and \isa{{\isasymone}} as zero) or \isa{list\ {\isasymColon}\ {\isacharparenleft}type{\isacharparenright}\ semigroup}
-  (e.g.\ if \isa{{\isasymodot}} is defined as list append).  Thus, the
-  characteristic constants \isa{{\isasymodot}}, \isa{{\isasyminv}}, \isa{{\isasymone}}
-  really become overloaded, i.e.\ have different meanings on different
-  types.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsubsection{Lifting and Functors%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-As already mentioned above, overloading in the simply-typed HOL
-  systems may include recursion over the syntactic structure of types.
-  That is, definitional equations \isa{c\isactrlsup {\isasymtau}\ {\isasymequiv}\ t} may also
-  contain constants of name \isa{c} on the right-hand side --- if
-  these have types that are structurally simpler than \isa{{\isasymtau}}.
-
-  This feature enables us to \emph{lift operations}, say to Cartesian
-  products, direct sums or function spaces.  Subsequently we lift
-  \isa{{\isasymodot}} component-wise to binary products \isa{{\isacharprime}a\ {\isasymtimes}\ {\isacharprime}b}.%
-\end{isamarkuptext}%
-\isamarkupfalse%
-\isacommand{defs}\ {\isacharparenleft}\isakeyword{overloaded}{\isacharparenright}\isanewline
-\ \ times{\isacharunderscore}prod{\isacharunderscore}def{\isacharcolon}\ {\isachardoublequote}p\ {\isasymodot}\ q\ {\isasymequiv}\ {\isacharparenleft}fst\ p\ {\isasymodot}\ fst\ q{\isacharcomma}\ snd\ p\ {\isasymodot}\ snd\ q{\isacharparenright}{\isachardoublequote}\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-It is very easy to see that associativity of \isa{{\isasymodot}} on \isa{{\isacharprime}a}
-  and \isa{{\isasymodot}} on \isa{{\isacharprime}b} transfers to \isa{{\isasymodot}} on \isa{{\isacharprime}a\ {\isasymtimes}\ {\isacharprime}b}.  Hence the binary type constructor \isa{{\isasymodot}} maps semigroups
-  to semigroups.  This may be established formally as follows.%
-\end{isamarkuptext}%
-\isamarkupfalse%
-\isacommand{instance}\ {\isacharasterisk}\ {\isacharcolon}{\isacharcolon}\ {\isacharparenleft}semigroup{\isacharcomma}\ semigroup{\isacharparenright}\ semigroup\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isamarkupfalse%
-\isacommand{proof}\ {\isacharparenleft}intro{\isacharunderscore}classes{\isacharcomma}\ unfold\ times{\isacharunderscore}prod{\isacharunderscore}def{\isacharparenright}\isanewline
-\ \ \isamarkupfalse%
-\isacommand{fix}\ p\ q\ r\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharprime}a{\isasymColon}semigroup\ {\isasymtimes}\ {\isacharprime}b{\isasymColon}semigroup{\isachardoublequote}\isanewline
-\ \ \isamarkupfalse%
-\isacommand{show}\isanewline
-\ \ \ \ {\isachardoublequote}{\isacharparenleft}fst\ {\isacharparenleft}fst\ p\ {\isasymodot}\ fst\ q{\isacharcomma}\ snd\ p\ {\isasymodot}\ snd\ q{\isacharparenright}\ {\isasymodot}\ fst\ r{\isacharcomma}\isanewline
-\ \ \ \ \ \ snd\ {\isacharparenleft}fst\ p\ {\isasymodot}\ fst\ q{\isacharcomma}\ snd\ p\ {\isasymodot}\ snd\ q{\isacharparenright}\ {\isasymodot}\ snd\ r{\isacharparenright}\ {\isacharequal}\isanewline
-\ \ \ \ \ \ \ {\isacharparenleft}fst\ p\ {\isasymodot}\ fst\ {\isacharparenleft}fst\ q\ {\isasymodot}\ fst\ r{\isacharcomma}\ snd\ q\ {\isasymodot}\ snd\ r{\isacharparenright}{\isacharcomma}\isanewline
-\ \ \ \ \ \ \ \ snd\ p\ {\isasymodot}\ snd\ {\isacharparenleft}fst\ q\ {\isasymodot}\ fst\ r{\isacharcomma}\ snd\ q\ {\isasymodot}\ snd\ r{\isacharparenright}{\isacharparenright}{\isachardoublequote}\isanewline
-\ \ \ \ \isamarkupfalse%
-\isacommand{by}\ {\isacharparenleft}simp\ add{\isacharcolon}\ semigroup{\isachardot}assoc{\isacharparenright}\isanewline
-\isamarkupfalse%
-\isacommand{qed}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-Thus, if we view class instances as ``structures'', then overloaded
-  constant definitions with recursion over types indirectly provide
-  some kind of ``functors'' --- i.e.\ mappings between abstract
-  theories.%
-\end{isamarkuptext}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isamarkupfalse%
-\isacommand{end}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-\isanewline
-\end{isabellebody}%
-%%% Local Variables:
-%%% mode: latex
-%%% TeX-master: "root"
-%%% End:

--- a/doc-src/AxClass/generated/NatClass.tex	Fri Aug 19 22:28:23 2005 +0200
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,96 +0,0 @@
-%
-\begin{isabellebody}%
-\def\isabellecontext{NatClass}%
-\isamarkuptrue%
-%
-\isamarkupheader{Defining natural numbers in FOL \label{sec:ex-natclass}%
-}
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isamarkupfalse%
-\isacommand{theory}\ NatClass\ \isakeyword{imports}\ FOL\ \isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-\medskip\noindent Axiomatic type classes abstract over exactly one
- type argument. Thus, any \emph{axiomatic} theory extension where each
- axiom refers to at most one type variable, may be trivially turned
- into a \emph{definitional} one.
-
- We illustrate this with the natural numbers in
- Isabelle/FOL.\footnote{See also
- \url{http://isabelle.in.tum.de/library/FOL/ex/NatClass.html}}%
-\end{isamarkuptext}%
-\isamarkupfalse%
-\isacommand{consts}\isanewline
-\ \ zero\ {\isacharcolon}{\isacharcolon}\ {\isacharprime}a\ \ \ \ {\isacharparenleft}{\isachardoublequote}{\isasymzero}{\isachardoublequote}{\isacharparenright}\isanewline
-\ \ Suc\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a{\isachardoublequote}\isanewline
-\ \ rec\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharparenleft}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a{\isacharparenright}\ {\isasymRightarrow}\ {\isacharprime}a{\isachardoublequote}\isanewline
-\isanewline
-\isamarkupfalse%
-\isacommand{axclass}\ nat\ {\isasymsubseteq}\ {\isachardoublequote}term{\isachardoublequote}\isanewline
-\ \ induct{\isacharcolon}\ {\isachardoublequote}P{\isacharparenleft}{\isasymzero}{\isacharparenright}\ {\isasymLongrightarrow}\ {\isacharparenleft}{\isasymAnd}x{\isachardot}\ P{\isacharparenleft}x{\isacharparenright}\ {\isasymLongrightarrow}\ P{\isacharparenleft}Suc{\isacharparenleft}x{\isacharparenright}{\isacharparenright}{\isacharparenright}\ {\isasymLongrightarrow}\ P{\isacharparenleft}n{\isacharparenright}{\isachardoublequote}\isanewline
-\ \ Suc{\isacharunderscore}inject{\isacharcolon}\ {\isachardoublequote}Suc{\isacharparenleft}m{\isacharparenright}\ {\isacharequal}\ Suc{\isacharparenleft}n{\isacharparenright}\ {\isasymLongrightarrow}\ m\ {\isacharequal}\ n{\isachardoublequote}\isanewline
-\ \ Suc{\isacharunderscore}neq{\isacharunderscore}{\isadigit{0}}{\isacharcolon}\ {\isachardoublequote}Suc{\isacharparenleft}m{\isacharparenright}\ {\isacharequal}\ {\isasymzero}\ {\isasymLongrightarrow}\ R{\isachardoublequote}\isanewline
-\ \ rec{\isacharunderscore}{\isadigit{0}}{\isacharcolon}\ {\isachardoublequote}rec{\isacharparenleft}{\isasymzero}{\isacharcomma}\ a{\isacharcomma}\ f{\isacharparenright}\ {\isacharequal}\ a{\isachardoublequote}\isanewline
-\ \ rec{\isacharunderscore}Suc{\isacharcolon}\ {\isachardoublequote}rec{\isacharparenleft}Suc{\isacharparenleft}m{\isacharparenright}{\isacharcomma}\ a{\isacharcomma}\ f{\isacharparenright}\ {\isacharequal}\ f{\isacharparenleft}m{\isacharcomma}\ rec{\isacharparenleft}m{\isacharcomma}\ a{\isacharcomma}\ f{\isacharparenright}{\isacharparenright}{\isachardoublequote}\isanewline
-\isanewline
-\isamarkupfalse%
-\isacommand{constdefs}\isanewline
-\ \ add\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharprime}a{\isacharcolon}{\isacharcolon}nat\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a{\isachardoublequote}\ \ \ \ {\isacharparenleft}\isakeyword{infixl}\ {\isachardoublequote}{\isacharplus}{\isachardoublequote}\ {\isadigit{6}}{\isadigit{0}}{\isacharparenright}\isanewline
-\ \ {\isachardoublequote}m\ {\isacharplus}\ n\ {\isasymequiv}\ rec{\isacharparenleft}m{\isacharcomma}\ n{\isacharcomma}\ {\isasymlambda}x\ y{\isachardot}\ Suc{\isacharparenleft}y{\isacharparenright}{\isacharparenright}{\isachardoublequote}\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-This is an abstract version of the plain \isa{Nat} theory in
- FOL.\footnote{See
- \url{http://isabelle.in.tum.de/library/FOL/ex/Nat.html}} Basically,
- we have just replaced all occurrences of type \isa{nat} by \isa{{\isacharprime}a} and used the natural number axioms to define class \isa{nat}.
- There is only a minor snag, that the original recursion operator
- \isa{rec} had to be made monomorphic.
-
- Thus class \isa{nat} contains exactly those types \isa{{\isasymtau}} that
- are isomorphic to ``the'' natural numbers (with signature \isa{{\isasymzero}}, \isa{Suc}, \isa{rec}).
-
- \medskip What we have done here can be also viewed as \emph{type
- specification}.  Of course, it still remains open if there is some
- type at all that meets the class axioms.  Now a very nice property of
- axiomatic type classes is that abstract reasoning is always possible
- --- independent of satisfiability.  The meta-logic won't break, even
- if some classes (or general sorts) turns out to be empty later ---
- ``inconsistent'' class definitions may be useless, but do not cause
- any harm.
-
- Theorems of the abstract natural numbers may be derived in the same
- way as for the concrete version.  The original proof scripts may be
- re-used with some trivial changes only (mostly adding some type
- constraints).%
-\end{isamarkuptext}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isamarkupfalse%
-\isacommand{end}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-\end{isabellebody}%
-%%% Local Variables:
-%%% mode: latex
-%%% TeX-master: "root"
-%%% End:

--- a/doc-src/AxClass/generated/Product.tex	Fri Aug 19 22:28:23 2005 +0200
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,132 +0,0 @@
-%
-\begin{isabellebody}%
-\def\isabellecontext{Product}%
-\isamarkuptrue%
-%
-\isamarkupheader{Syntactic classes%
-}
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isamarkupfalse%
-\isacommand{theory}\ Product\ \isakeyword{imports}\ Main\ \isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-\isamarkuptrue%
-%
-\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 \isa{c\ {\isasymColon}\ {\isasymsigma}}, the type variables occurring in \isa{{\isasymsigma}}
-  may be constrained by type classes (or even general sorts) in an
-  arbitrary way.  Note that by default, in Isabelle/HOL the
-  declaration \isa{{\isasymodot}\ {\isasymColon}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a} is actually an abbreviation
-  for \isa{{\isasymodot}\ {\isasymColon}\ {\isacharprime}a{\isasymColon}type\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a} Since class \isa{type} is the
-  universal class of HOL, this is not really a constraint at all.
-
- The \isa{product} class below provides a less degenerate example of
- syntactic type classes.%
-\end{isamarkuptext}%
-\isamarkupfalse%
-\isacommand{axclass}\isanewline
-\ \ product\ {\isasymsubseteq}\ type\isanewline
-\isamarkupfalse%
-\isacommand{consts}\isanewline
-\ \ 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}\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-Here class \isa{product} is defined as subclass of \isa{type}
-  without any additional axioms.  This effects in logical equivalence
-  of \isa{product} and \isa{type}, as is reflected by the trivial
-  introduction rule generated for this definition.
-
-  \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
-  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 \isa{{\isasymodot}} on some type \isa{{\isasymtau}} are rejected by the
-  type-checker, unless the arity \isa{{\isasymtau}\ {\isasymColon}\ product} is part of the
-  type signature.  In our example, this arity may be always added when
-  required by means of an $\INSTANCE$ with the default proof $\DDOT$.
-
-  \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
-  \isa{c}.  Only immediately before \emph{specifying} these
-  constants on a certain type \isa{{\isasymtau}} do we instantiate \isa{{\isasymtau}\ {\isasymColon}\ c}.
-
-  This is done for class \isa{product} and type \isa{bool} as
-  follows.%
-\end{isamarkuptext}%
-\isamarkupfalse%
-\isacommand{instance}\ bool\ {\isacharcolon}{\isacharcolon}\ product%
-\isadelimproof
-\ %
-\endisadelimproof
-%
-\isatagproof
-\isamarkupfalse%
-\isacommand{{\isachardot}{\isachardot}}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-\isanewline
-\isamarkupfalse%
-\isacommand{defs}\ {\isacharparenleft}\isakeyword{overloaded}{\isacharparenright}\isanewline
-\ \ product{\isacharunderscore}bool{\isacharunderscore}def{\isacharcolon}\ {\isachardoublequote}x\ {\isasymodot}\ y\ {\isasymequiv}\ x\ {\isasymand}\ y{\isachardoublequote}\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-The definition \isa{prod{\isacharunderscore}bool{\isacharunderscore}def} becomes syntactically
- well-formed only after the arity \isa{bool\ {\isasymColon}\ product} is made
- known to the type checker.
-
- \medskip It is very important to see that above $\DEFS$ are not
- directly connected with $\INSTANCE$ at all!  We were just following
- our convention to specify \isa{{\isasymodot}} on \isa{bool} after having
- instantiated \isa{bool\ {\isasymColon}\ 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} would not make much sense in
- Isabelle's meta-logic, because there is no internal notion of
- ``providing operations'' or even ``names of functions''.%
-\end{isamarkuptext}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isamarkupfalse%
-\isacommand{end}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-\isanewline
-\end{isabellebody}%
-%%% Local Variables:
-%%% mode: latex
-%%% TeX-master: "root"
-%%% End:

--- a/doc-src/AxClass/generated/Semigroups.tex	Fri Aug 19 22:28:23 2005 +0200
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,88 +0,0 @@
-%
-\begin{isabellebody}%
-\def\isabellecontext{Semigroups}%
-\isamarkuptrue%
-%
-\isamarkupheader{Semigroups%
-}
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isamarkupfalse%
-\isacommand{theory}\ Semigroups\ \isakeyword{imports}\ Main\ \isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-\medskip\noindent An axiomatic type class is simply a class of types
-  that all meet certain properties, which are also called \emph{class
-  axioms}. Thus, type classes may be also understood as type
-  predicates --- i.e.\ abstractions over a single type argument \isa{{\isacharprime}a}.  Class axioms typically contain polymorphic constants that
-  depend on this type \isa{{\isacharprime}a}.  These \emph{characteristic
-  constants} behave like operations associated with the ``carrier''
-  type \isa{{\isacharprime}a}.
-
-  We illustrate these basic concepts by the following formulation of
-  semigroups.%
-\end{isamarkuptext}%
-\isamarkupfalse%
-\isacommand{consts}\isanewline
-\ \ times\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a{\isachardoublequote}\ \ \ \ {\isacharparenleft}\isakeyword{infixl}\ {\isachardoublequote}{\isasymodot}{\isachardoublequote}\ {\isadigit{7}}{\isadigit{0}}{\isacharparenright}\isanewline
-\isamarkupfalse%
-\isacommand{axclass}\ semigroup\ {\isasymsubseteq}\ type\isanewline
-\ \ assoc{\isacharcolon}\ {\isachardoublequote}{\isacharparenleft}x\ {\isasymodot}\ y{\isacharparenright}\ {\isasymodot}\ z\ {\isacharequal}\ x\ {\isasymodot}\ {\isacharparenleft}y\ {\isasymodot}\ z{\isacharparenright}{\isachardoublequote}\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-\noindent Above we have first declared a polymorphic constant \isa{{\isasymodot}\ {\isasymColon}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a} and then defined the class \isa{semigroup} of
-  all types \isa{{\isasymtau}} such that \isa{{\isasymodot}\ {\isasymColon}\ {\isasymtau}\ {\isasymRightarrow}\ {\isasymtau}\ {\isasymRightarrow}\ {\isasymtau}} is indeed an
-  associative operator.  The \isa{assoc} axiom contains exactly one
-  type variable, which is invisible in the above presentation, though.
-  Also note that free term variables (like \isa{x}, \isa{y},
-  \isa{z}) are allowed for user convenience --- conceptually all of
-  these are bound by outermost universal quantifiers.
-
-  \medskip In general, type classes may be used to describe
-  \emph{structures} with exactly one carrier \isa{{\isacharprime}a} and a fixed
-  \emph{signature}.  Different signatures require different classes.
-  Below, class \isa{plus{\isacharunderscore}semigroup} represents semigroups \isa{{\isacharparenleft}{\isasymtau}{\isacharcomma}\ {\isasymoplus}\isactrlsup {\isasymtau}{\isacharparenright}}, while the original \isa{semigroup} would
-  correspond to semigroups of the form \isa{{\isacharparenleft}{\isasymtau}{\isacharcomma}\ {\isasymodot}\isactrlsup {\isasymtau}{\isacharparenright}}.%
-\end{isamarkuptext}%
-\isamarkupfalse%
-\isacommand{consts}\isanewline
-\ \ plus\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a{\isachardoublequote}\ \ \ \ {\isacharparenleft}\isakeyword{infixl}\ {\isachardoublequote}{\isasymoplus}{\isachardoublequote}\ {\isadigit{7}}{\isadigit{0}}{\isacharparenright}\isanewline
-\isamarkupfalse%
-\isacommand{axclass}\ plus{\isacharunderscore}semigroup\ {\isasymsubseteq}\ type\isanewline
-\ \ assoc{\isacharcolon}\ {\isachardoublequote}{\isacharparenleft}x\ {\isasymoplus}\ y{\isacharparenright}\ {\isasymoplus}\ z\ {\isacharequal}\ x\ {\isasymoplus}\ {\isacharparenleft}y\ {\isasymoplus}\ z{\isacharparenright}{\isachardoublequote}\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-\noindent Even if classes \isa{plus{\isacharunderscore}semigroup} and \isa{semigroup} both represent semigroups in a sense, they are certainly
-  not quite the same.%
-\end{isamarkuptext}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isamarkupfalse%
-\isacommand{end}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-\isanewline
-\end{isabellebody}%
-%%% Local Variables:
-%%% mode: latex
-%%% TeX-master: "root"
-%%% End:

--- a/doc-src/AxClass/generated/isabelle.sty	Fri Aug 19 22:28:23 2005 +0200
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,191 +0,0 @@
-%%
-%% Author: Markus Wenzel, TU Muenchen
-%%
-%% macros for Isabelle generated LaTeX output
-%%
-%% 
-
-%%% Simple document preparation (based on theory token language and symbols)
-
-% isabelle environments
-
-\newcommand{\isabellecontext}{UNKNOWN}
-
-\newcommand{\isastyle}{\small\tt\slshape}
-\newcommand{\isastyleminor}{\small\tt\slshape}
-\newcommand{\isastylescript}{\footnotesize\tt\slshape}
-\newcommand{\isastyletext}{\normalsize\rm}
-\newcommand{\isastyletxt}{\rm}
-\newcommand{\isastylecmt}{\rm}
-
-%symbol markup -- \emph achieves decent spacing via italic corrections
-\newcommand{\isamath}[1]{\emph{$#1$}}
-\newcommand{\isatext}[1]{\emph{#1}}
-\newcommand{\isascriptstyle}{\def\isamath##1{##1}\def\isatext##1{\mbox{\isastylescript##1}}}
-\newcommand{\isactrlsub}[1]{\emph{${}\sb{\mbox{\isastylescript{#1}}}$}}
-\newcommand{\isactrlsup}[1]{\emph{${}\sp{\mbox{\isastylescript{#1}}}$}}
-\newcommand{\isactrlisub}[1]{\emph{${}\sb{\mbox{\isastylescript{#1}}}$}}
-\newcommand{\isactrlisup}[1]{\emph{${}\sp{\mbox{\isastylescript{#1}}}$}}
-\newcommand{\isactrlbsub}{\emph\bgroup\begin{math}{}\sb\bgroup\mbox\bgroup\isastylescript}
-\newcommand{\isactrlesub}{\egroup\egroup\end{math}\egroup}
-\newcommand{\isactrlbsup}{\emph\bgroup\begin{math}{}\sp\bgroup\mbox\bgroup\isastylescript}
-\newcommand{\isactrlesup}{\egroup\egroup\end{math}\egroup}
-\newcommand{\isactrlbold}[1]{{\bfseries\upshape\boldmath#1}}
-
-
-\newdimen\isa@parindent\newdimen\isa@parskip
-
-\newenvironment{isabellebody}{%
-\isamarkuptrue\par%
-\isa@parindent\parindent\parindent0pt%
-\isa@parskip\parskip\parskip0pt%
-\isastyle}{\par}
-
-\newenvironment{isabelle}
-{\begin{trivlist}\begin{isabellebody}\item\relax}
-{\end{isabellebody}\end{trivlist}}
-
-\newcommand{\isa}[1]{\emph{\isastyleminor #1}}
-
-\newcommand{\isaindent}[1]{\hphantom{#1}}
-\newcommand{\isanewline}{\mbox{}\par\mbox{}}
-\newcommand{\isasep}{} % override with e.g. \renewcommand{\isasep}{\vspace{1ex}}
-\newcommand{\isadigit}[1]{#1}
-
-\chardef\isacharbang=`\!
-\chardef\isachardoublequote=`\"
-\chardef\isacharhash=`\#
-\chardef\isachardollar=`\$
-\chardef\isacharpercent=`\%
-\chardef\isacharampersand=`\&
-\chardef\isacharprime=`\'
-\chardef\isacharparenleft=`\(
-\chardef\isacharparenright=`\)
-\chardef\isacharasterisk=`\*
-\chardef\isacharplus=`\+
-\chardef\isacharcomma=`\,
-\chardef\isacharminus=`\-
-\chardef\isachardot=`\.
-\chardef\isacharslash=`\/
-\chardef\isacharcolon=`\:
-\chardef\isacharsemicolon=`\;
-\chardef\isacharless=`\<
-\chardef\isacharequal=`\=
-\chardef\isachargreater=`\>
-\chardef\isacharquery=`\?
-\chardef\isacharat=`\@
-\chardef\isacharbrackleft=`\[
-\chardef\isacharbackslash=`\\
-\chardef\isacharbrackright=`\]
-\chardef\isacharcircum=`\^
-\chardef\isacharunderscore=`\_
-\chardef\isacharbackquote=`\`
-\chardef\isacharbraceleft=`\{
-\chardef\isacharbar=`\|
-\chardef\isacharbraceright=`\}
-\chardef\isachartilde=`\~
-
-
-% keyword and section markup
-
-\newcommand{\isakeywordcharunderscore}{\_}
-\newcommand{\isakeyword}[1]
-{\emph{\bf\def\isachardot{.}\def\isacharunderscore{\isakeywordcharunderscore}%
-\def\isacharbraceleft{\{}\def\isacharbraceright{\}}#1}}
-\newcommand{\isacommand}[1]{\isakeyword{#1}}
-
-\newcommand{\isamarkupheader}[1]{\section{#1}}
-\newcommand{\isamarkupchapter}[1]{\chapter{#1}}
-\newcommand{\isamarkupsection}[1]{\section{#1}}
-\newcommand{\isamarkupsubsection}[1]{\subsection{#1}}
-\newcommand{\isamarkupsubsubsection}[1]{\subsubsection{#1}}
-\newcommand{\isamarkupsect}[1]{\section{#1}}
-\newcommand{\isamarkupsubsect}[1]{\subsection{#1}}
-\newcommand{\isamarkupsubsubsect}[1]{\subsubsection{#1}}
-
-\newif\ifisamarkup
-\newcommand{\isabeginpar}{\par\ifisamarkup\relax\else\medskip\fi}
-\newcommand{\isaendpar}{\par\medskip}
-\newenvironment{isapar}{\parindent\isa@parindent\parskip\isa@parskip\isabeginpar}{\isaendpar}
-\newenvironment{isamarkuptext}{\isastyletext\begin{isapar}}{\end{isapar}}
-\newenvironment{isamarkuptxt}{\isastyletxt\begin{isapar}}{\end{isapar}}
-\newcommand{\isamarkupcmt}[1]{{\isastylecmt--- #1}}
-
-
-% alternative styles
-
-\newcommand{\isabellestyle}{}
-\def\isabellestyle#1{\csname isabellestyle#1\endcsname}
-
-\newcommand{\isabellestylett}{%
-\renewcommand{\isastyle}{\small\tt}%
-\renewcommand{\isastyleminor}{\small\tt}%
-\renewcommand{\isastylescript}{\footnotesize\tt}%
-}
-\newcommand{\isabellestyleit}{%
-\renewcommand{\isastyle}{\small\it}%
-\renewcommand{\isastyleminor}{\it}%
-\renewcommand{\isastylescript}{\footnotesize\it}%
-\renewcommand{\isakeywordcharunderscore}{\mbox{-}}%
-\renewcommand{\isacharbang}{\isamath{!}}%
-\renewcommand{\isachardoublequote}{}%
-\renewcommand{\isacharhash}{\isamath{\#}}%
-\renewcommand{\isachardollar}{\isamath{\$}}%
-\renewcommand{\isacharpercent}{\isamath{\%}}%
-\renewcommand{\isacharampersand}{\isamath{\&}}%
-\renewcommand{\isacharprime}{\isamath{\mskip2mu{'}\mskip-2mu}}%
-\renewcommand{\isacharparenleft}{\isamath{(}}%
-\renewcommand{\isacharparenright}{\isamath{)}}%
-\renewcommand{\isacharasterisk}{\isamath{*}}%
-\renewcommand{\isacharplus}{\isamath{+}}%
-\renewcommand{\isacharcomma}{\isamath{\mathord,}}%
-\renewcommand{\isacharminus}{\isamath{-}}%
-\renewcommand{\isachardot}{\isamath{\mathord.}}%
-\renewcommand{\isacharslash}{\isamath{/}}%
-\renewcommand{\isacharcolon}{\isamath{\mathord:}}%
-\renewcommand{\isacharsemicolon}{\isamath{\mathord;}}%
-\renewcommand{\isacharless}{\isamath{<}}%
-\renewcommand{\isacharequal}{\isamath{=}}%
-\renewcommand{\isachargreater}{\isamath{>}}%
-\renewcommand{\isacharat}{\isamath{@}}%
-\renewcommand{\isacharbrackleft}{\isamath{[}}%
-\renewcommand{\isacharbackslash}{\isamath{\backslash}}%
-\renewcommand{\isacharbrackright}{\isamath{]}}%
-\renewcommand{\isacharunderscore}{\mbox{-}}%
-\renewcommand{\isacharbraceleft}{\isamath{\{}}%
-\renewcommand{\isacharbar}{\isamath{\mid}}%
-\renewcommand{\isacharbraceright}{\isamath{\}}}%
-\renewcommand{\isachartilde}{\isamath{{}\sp{\sim}}}%
-}
-
-\newcommand{\isabellestylesl}{%
-\isabellestyleit%
-\renewcommand{\isastyle}{\small\sl}%
-\renewcommand{\isastyleminor}{\sl}%
-\renewcommand{\isastylescript}{\footnotesize\sl}%
-}
-
-
-% tagged regions
-
-%plain TeX version of comment package -- much faster!
-\let\isafmtname\fmtname\def\fmtname{plain}
-\usepackage{comment}
-\let\fmtname\isafmtname
-
-\newcommand{\isafold}[1]{\emph{$\langle\mathord{\mathit{#1}}\rangle$}}
-
-\newcommand{\isakeeptag}[1]%
-{\includecomment{isadelim#1}\includecomment{isatag#1}\csarg\def{isafold#1}{}}
-\newcommand{\isadroptag}[1]%
-{\excludecomment{isadelim#1}\excludecomment{isatag#1}\csarg\def{isafold#1}{}}
-\newcommand{\isafoldtag}[1]%
-{\includecomment{isadelim#1}\excludecomment{isatag#1}\csarg\def{isafold#1}{\isafold{#1}}}
-
-\isakeeptag{theory}
-\isakeeptag{proof}
-\isakeeptag{ML}
-\isakeeptag{visible}
-\isadroptag{invisible}
-
-\IfFileExists{isabelletags.sty}{\usepackage{isabelletags}}{}

--- a/doc-src/AxClass/generated/isabellesym.sty	Fri Aug 19 22:28:23 2005 +0200
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,362 +0,0 @@
-%%
-%% Author: Markus Wenzel, TU Muenchen
-%%
-%% definitions of standard Isabelle symbols
-%%
-%% 
-
-% symbol definitions
-
-\newcommand{\isasymzero}{\isamath{\mathbf{0}}}  %requires amssymb
-\newcommand{\isasymone}{\isamath{\mathbf{1}}}   %requires amssymb
-\newcommand{\isasymtwo}{\isamath{\mathbf{2}}}   %requires amssymb
-\newcommand{\isasymthree}{\isamath{\mathbf{3}}} %requires amssymb
-\newcommand{\isasymfour}{\isamath{\mathbf{4}}}  %requires amssymb
-\newcommand{\isasymfive}{\isamath{\mathbf{5}}}  %requires amssymb
-\newcommand{\isasymsix}{\isamath{\mathbf{6}}}   %requires amssymb
-\newcommand{\isasymseven}{\isamath{\mathbf{7}}} %requires amssymb
-\newcommand{\isasymeight}{\isamath{\mathbf{8}}} %requires amssymb
-\newcommand{\isasymnine}{\isamath{\mathbf{9}}}  %requires amssymb
-\newcommand{\isasymA}{\isamath{\mathcal{A}}}
-\newcommand{\isasymB}{\isamath{\mathcal{B}}}
-\newcommand{\isasymC}{\isamath{\mathcal{C}}}
-\newcommand{\isasymD}{\isamath{\mathcal{D}}}
-\newcommand{\isasymE}{\isamath{\mathcal{E}}}
-\newcommand{\isasymF}{\isamath{\mathcal{F}}}
-\newcommand{\isasymG}{\isamath{\mathcal{G}}}
-\newcommand{\isasymH}{\isamath{\mathcal{H}}}
-\newcommand{\isasymI}{\isamath{\mathcal{I}}}
-\newcommand{\isasymJ}{\isamath{\mathcal{J}}}
-\newcommand{\isasymK}{\isamath{\mathcal{K}}}
-\newcommand{\isasymL}{\isamath{\mathcal{L}}}
-\newcommand{\isasymM}{\isamath{\mathcal{M}}}
-\newcommand{\isasymN}{\isamath{\mathcal{N}}}
-\newcommand{\isasymO}{\isamath{\mathcal{O}}}
-\newcommand{\isasymP}{\isamath{\mathcal{P}}}
-\newcommand{\isasymQ}{\isamath{\mathcal{Q}}}
-\newcommand{\isasymR}{\isamath{\mathcal{R}}}
-\newcommand{\isasymS}{\isamath{\mathcal{S}}}
-\newcommand{\isasymT}{\isamath{\mathcal{T}}}
-\newcommand{\isasymU}{\isamath{\mathcal{U}}}
-\newcommand{\isasymV}{\isamath{\mathcal{V}}}
-\newcommand{\isasymW}{\isamath{\mathcal{W}}}
-\newcommand{\isasymX}{\isamath{\mathcal{X}}}
-\newcommand{\isasymY}{\isamath{\mathcal{Y}}}
-\newcommand{\isasymZ}{\isamath{\mathcal{Z}}}
-\newcommand{\isasyma}{\isamath{\mathrm{a}}}
-\newcommand{\isasymb}{\isamath{\mathrm{b}}}
-\newcommand{\isasymc}{\isamath{\mathrm{c}}}
-\newcommand{\isasymd}{\isamath{\mathrm{d}}}
-\newcommand{\isasyme}{\isamath{\mathrm{e}}}
-\newcommand{\isasymf}{\isamath{\mathrm{f}}}
-\newcommand{\isasymg}{\isamath{\mathrm{g}}}
-\newcommand{\isasymh}{\isamath{\mathrm{h}}}
-\newcommand{\isasymi}{\isamath{\mathrm{i}}}
-\newcommand{\isasymj}{\isamath{\mathrm{j}}}
-\newcommand{\isasymk}{\isamath{\mathrm{k}}}
-\newcommand{\isasyml}{\isamath{\mathrm{l}}}
-\newcommand{\isasymm}{\isamath{\mathrm{m}}}
-\newcommand{\isasymn}{\isamath{\mathrm{n}}}
-\newcommand{\isasymo}{\isamath{\mathrm{o}}}
-\newcommand{\isasymp}{\isamath{\mathrm{p}}}
-\newcommand{\isasymq}{\isamath{\mathrm{q}}}
-\newcommand{\isasymr}{\isamath{\mathrm{r}}}
-\newcommand{\isasyms}{\isamath{\mathrm{s}}}
-\newcommand{\isasymt}{\isamath{\mathrm{t}}}
-\newcommand{\isasymu}{\isamath{\mathrm{u}}}
-\newcommand{\isasymv}{\isamath{\mathrm{v}}}
-\newcommand{\isasymw}{\isamath{\mathrm{w}}}
-\newcommand{\isasymx}{\isamath{\mathrm{x}}}
-\newcommand{\isasymy}{\isamath{\mathrm{y}}}
-\newcommand{\isasymz}{\isamath{\mathrm{z}}}
-\newcommand{\isasymAA}{\isamath{\mathfrak{A}}}  %requires eufrak
-\newcommand{\isasymBB}{\isamath{\mathfrak{B}}}  %requires eufrak
-\newcommand{\isasymCC}{\isamath{\mathfrak{C}}}  %requires eufrak
-\newcommand{\isasymDD}{\isamath{\mathfrak{D}}}  %requires eufrak
-\newcommand{\isasymEE}{\isamath{\mathfrak{E}}}  %requires eufrak
-\newcommand{\isasymFF}{\isamath{\mathfrak{F}}}  %requires eufrak
-\newcommand{\isasymGG}{\isamath{\mathfrak{G}}}  %requires eufrak
-\newcommand{\isasymHH}{\isamath{\mathfrak{H}}}  %requires eufrak
-\newcommand{\isasymII}{\isamath{\mathfrak{I}}}  %requires eufrak
-\newcommand{\isasymJJ}{\isamath{\mathfrak{J}}}  %requires eufrak
-\newcommand{\isasymKK}{\isamath{\mathfrak{K}}}  %requires eufrak
-\newcommand{\isasymLL}{\isamath{\mathfrak{L}}}  %requires eufrak
-\newcommand{\isasymMM}{\isamath{\mathfrak{M}}}  %requires eufrak
-\newcommand{\isasymNN}{\isamath{\mathfrak{N}}}  %requires eufrak
-\newcommand{\isasymOO}{\isamath{\mathfrak{O}}}  %requires eufrak
-\newcommand{\isasymPP}{\isamath{\mathfrak{P}}}  %requires eufrak
-\newcommand{\isasymQQ}{\isamath{\mathfrak{Q}}}  %requires eufrak
-\newcommand{\isasymRR}{\isamath{\mathfrak{R}}}  %requires eufrak
-\newcommand{\isasymSS}{\isamath{\mathfrak{S}}}  %requires eufrak
-\newcommand{\isasymTT}{\isamath{\mathfrak{T}}}  %requires eufrak
-\newcommand{\isasymUU}{\isamath{\mathfrak{U}}}  %requires eufrak
-\newcommand{\isasymVV}{\isamath{\mathfrak{V}}}  %requires eufrak
-\newcommand{\isasymWW}{\isamath{\mathfrak{W}}}  %requires eufrak
-\newcommand{\isasymXX}{\isamath{\mathfrak{X}}}  %requires eufrak
-\newcommand{\isasymYY}{\isamath{\mathfrak{Y}}}  %requires eufrak
-\newcommand{\isasymZZ}{\isamath{\mathfrak{Z}}}  %requires eufrak
-\newcommand{\isasymaa}{\isamath{\mathfrak{a}}}  %requires eufrak
-\newcommand{\isasymbb}{\isamath{\mathfrak{b}}}  %requires eufrak
-\newcommand{\isasymcc}{\isamath{\mathfrak{c}}}  %requires eufrak
-\newcommand{\isasymdd}{\isamath{\mathfrak{d}}}  %requires eufrak
-\newcommand{\isasymee}{\isamath{\mathfrak{e}}}  %requires eufrak
-\newcommand{\isasymff}{\isamath{\mathfrak{f}}}  %requires eufrak
-\newcommand{\isasymgg}{\isamath{\mathfrak{g}}}  %requires eufrak
-\newcommand{\isasymhh}{\isamath{\mathfrak{h}}}  %requires eufrak
-\newcommand{\isasymii}{\isamath{\mathfrak{i}}}  %requires eufrak
-\newcommand{\isasymjj}{\isamath{\mathfrak{j}}}  %requires eufrak
-\newcommand{\isasymkk}{\isamath{\mathfrak{k}}}  %requires eufrak
-\newcommand{\isasymll}{\isamath{\mathfrak{l}}}  %requires eufrak
-\newcommand{\isasymmm}{\isamath{\mathfrak{m}}}  %requires eufrak
-\newcommand{\isasymnn}{\isamath{\mathfrak{n}}}  %requires eufrak
-\newcommand{\isasymoo}{\isamath{\mathfrak{o}}}  %requires eufrak
-\newcommand{\isasympp}{\isamath{\mathfrak{p}}}  %requires eufrak
-\newcommand{\isasymqq}{\isamath{\mathfrak{q}}}  %requires eufrak
-\newcommand{\isasymrr}{\isamath{\mathfrak{r}}}  %requires eufrak
-\newcommand{\isasymss}{\isamath{\mathfrak{s}}}  %requires eufrak
-\newcommand{\isasymtt}{\isamath{\mathfrak{t}}}  %requires eufrak
-\newcommand{\isasymuu}{\isamath{\mathfrak{u}}}  %requires eufrak
-\newcommand{\isasymvv}{\isamath{\mathfrak{v}}}  %requires eufrak
-\newcommand{\isasymww}{\isamath{\mathfrak{w}}}  %requires eufrak
-\newcommand{\isasymxx}{\isamath{\mathfrak{x}}}  %requires eufrak
-\newcommand{\isasymyy}{\isamath{\mathfrak{y}}}  %requires eufrak
-\newcommand{\isasymzz}{\isamath{\mathfrak{z}}}  %requires eufrak
-\newcommand{\isasymalpha}{\isamath{\alpha}}
-\newcommand{\isasymbeta}{\isamath{\beta}}
-\newcommand{\isasymgamma}{\isamath{\gamma}}
-\newcommand{\isasymdelta}{\isamath{\delta}}
-\newcommand{\isasymepsilon}{\isamath{\varepsilon}}
-\newcommand{\isasymzeta}{\isamath{\zeta}}
-\newcommand{\isasymeta}{\isamath{\eta}}
-\newcommand{\isasymtheta}{\isamath{\vartheta}}
-\newcommand{\isasymiota}{\isamath{\iota}}
-\newcommand{\isasymkappa}{\isamath{\kappa}}
-\newcommand{\isasymlambda}{\isamath{\lambda}}
-\newcommand{\isasymmu}{\isamath{\mu}}
-\newcommand{\isasymnu}{\isamath{\nu}}
-\newcommand{\isasymxi}{\isamath{\xi}}
-\newcommand{\isasympi}{\isamath{\pi}}
-\newcommand{\isasymrho}{\isamath{\varrho}}
-\newcommand{\isasymsigma}{\isamath{\sigma}}
-\newcommand{\isasymtau}{\isamath{\tau}}
-\newcommand{\isasymupsilon}{\isamath{\upsilon}}
-\newcommand{\isasymphi}{\isamath{\varphi}}
-\newcommand{\isasymchi}{\isamath{\chi}}
-\newcommand{\isasympsi}{\isamath{\psi}}
-\newcommand{\isasymomega}{\isamath{\omega}}
-\newcommand{\isasymGamma}{\isamath{\Gamma}}
-\newcommand{\isasymDelta}{\isamath{\Delta}}
-\newcommand{\isasymTheta}{\isamath{\Theta}}
-\newcommand{\isasymLambda}{\isamath{\Lambda}}
-\newcommand{\isasymXi}{\isamath{\Xi}}
-\newcommand{\isasymPi}{\isamath{\Pi}}
-\newcommand{\isasymSigma}{\isamath{\Sigma}}
-\newcommand{\isasymUpsilon}{\isamath{\Upsilon}}
-\newcommand{\isasymPhi}{\isamath{\Phi}}
-\newcommand{\isasymPsi}{\isamath{\Psi}}
-\newcommand{\isasymOmega}{\isamath{\Omega}}
-\newcommand{\isasymbool}{\isamath{\mathrm{I}\mkern-3.8mu\mathrm{B}}}
-\newcommand{\isasymcomplex}{\isamath{\mathrm{C}\mkern-15mu{\phantom{\mathrm{t}}\vrule}\mkern9mu}}
-\newcommand{\isasymnat}{\isamath{\mathrm{I}\mkern-3.8mu\mathrm{N}}}
-\newcommand{\isasymrat}{\isamath{\mathrm{Q}\mkern-16mu{\phantom{\mathrm{t}}\vrule}\mkern10mu}}
-\newcommand{\isasymreal}{\isamath{\mathrm{I}\mkern-3.8mu\mathrm{R}}}
-\newcommand{\isasymint}{\isamath{\mathsf{Z}\mkern-7.5mu\mathsf{Z}}}
-\newcommand{\isasymleftarrow}{\isamath{\leftarrow}}
-\newcommand{\isasymlongleftarrow}{\isamath{\longleftarrow}}
-\newcommand{\isasymrightarrow}{\isamath{\rightarrow}}
-\newcommand{\isasymlongrightarrow}{\isamath{\longrightarrow}}
-\newcommand{\isasymLeftarrow}{\isamath{\Leftarrow}}
-\newcommand{\isasymLongleftarrow}{\isamath{\Longleftarrow}}
-\newcommand{\isasymRightarrow}{\isamath{\Rightarrow}}
-\newcommand{\isasymLongrightarrow}{\isamath{\Longrightarrow}}
-\newcommand{\isasymleftrightarrow}{\isamath{\leftrightarrow}}
-\newcommand{\isasymlongleftrightarrow}{\isamath{\longleftrightarrow}}
-\newcommand{\isasymLeftrightarrow}{\isamath{\Leftrightarrow}}
-\newcommand{\isasymLongleftrightarrow}{\isamath{\Longleftrightarrow}}
-\newcommand{\isasymmapsto}{\isamath{\mapsto}}
-\newcommand{\isasymlongmapsto}{\isamath{\longmapsto}}
-\newcommand{\isasymmidarrow}{\isamath{\relbar}}
-\newcommand{\isasymMidarrow}{\isamath{\Relbar}}
-\newcommand{\isasymhookleftarrow}{\isamath{\hookleftarrow}}
-\newcommand{\isasymhookrightarrow}{\isamath{\hookrightarrow}}
-\newcommand{\isasymleftharpoondown}{\isamath{\leftharpoondown}}
-\newcommand{\isasymrightharpoondown}{\isamath{\rightharpoondown}}
-\newcommand{\isasymleftharpoonup}{\isamath{\leftharpoonup}}
-\newcommand{\isasymrightharpoonup}{\isamath{\rightharpoonup}}
-\newcommand{\isasymrightleftharpoons}{\isamath{\rightleftharpoons}}
-\newcommand{\isasymdownharpoonleft}{\isamath{\downharpoonleft}}  %requires amssymb
-\newcommand{\isasymdownharpoonright}{\isamath{\downharpoonright}}  %requires amssymb
-\newcommand{\isasymupharpoonleft}{\isamath{\upharpoonleft}}  %requires amssymb
-\newcommand{\isasymupharpoonright}{\isamath{\upharpoonright}}  %requires amssymb
-\newcommand{\isasymrestriction}{\isamath{\restriction}}  %requires amssymb
-\newcommand{\isasymleadsto}{\isamath{\leadsto}}  %requires amssymb
-\newcommand{\isasymup}{\isamath{\uparrow}}
-\newcommand{\isasymUp}{\isamath{\Uparrow}}
-\newcommand{\isasymdown}{\isamath{\downarrow}}
-\newcommand{\isasymDown}{\isamath{\Downarrow}}
-\newcommand{\isasymupdown}{\isamath{\updownarrow}}
-\newcommand{\isasymUpdown}{\isamath{\Updownarrow}}
-\newcommand{\isasymlangle}{\isamath{\langle}}
-\newcommand{\isasymrangle}{\isamath{\rangle}}
-\newcommand{\isasymlceil}{\isamath{\lceil}}
-\newcommand{\isasymrceil}{\isamath{\rceil}}
-\newcommand{\isasymlfloor}{\isamath{\lfloor}}
-\newcommand{\isasymrfloor}{\isamath{\rfloor}}
-\newcommand{\isasymlparr}{\isamath{\mathopen{(\mkern-3mu\mid}}}
-\newcommand{\isasymrparr}{\isamath{\mathclose{\mid\mkern-3mu)}}}
-\newcommand{\isasymlbrakk}{\isamath{\mathopen{\lbrack\mkern-3mu\lbrack}}}
-\newcommand{\isasymrbrakk}{\isamath{\mathclose{\rbrack\mkern-3mu\rbrack}}}
-\newcommand{\isasymlbrace}{\isamath{\mathopen{\lbrace\mkern-4.5mu\mid}}}
-\newcommand{\isasymrbrace}{\isamath{\mathclose{\mid\mkern-4.5mu\rbrace}}}
-\newcommand{\isasymguillemotleft}{\isatext{\flqq}}  %requires babel 
-\newcommand{\isasymguillemotright}{\isatext{\frqq}}  %requires babel 
-\newcommand{\isasymColon}{\isamath{\mathrel{::}}}
-\newcommand{\isasymnot}{\isamath{\neg}}
-\newcommand{\isasymbottom}{\isamath{\bot}}
-\newcommand{\isasymtop}{\isamath{\top}}
-\newcommand{\isasymand}{\isamath{\wedge}}
-\newcommand{\isasymAnd}{\isamath{\bigwedge}}
-\newcommand{\isasymor}{\isamath{\vee}}
-\newcommand{\isasymOr}{\isamath{\bigvee}}
-\newcommand{\isasymforall}{\isamath{\forall\,}}
-\newcommand{\isasymexists}{\isamath{\exists\,}}
-\newcommand{\isasymnexists}{\isamath{\nexists\,}}  %requires amssymb
-\newcommand{\isasymbox}{\isamath{\Box}}  %requires amssymb
-\newcommand{\isasymdiamond}{\isamath{\Diamond}}  %requires amssymb
-\newcommand{\isasymturnstile}{\isamath{\vdash}}
-\newcommand{\isasymTurnstile}{\isamath{\models}}
-\newcommand{\isasymtturnstile}{\isamath{\vdash\!\!\!\vdash}}
-\newcommand{\isasymTTurnstile}{\isamath{\mid\!\models}}
-\newcommand{\isasymstileturn}{\isamath{\dashv}}
-\newcommand{\isasymsurd}{\isamath{\surd}}
-\newcommand{\isasymle}{\isamath{\le}}
-\newcommand{\isasymge}{\isamath{\ge}}
-\newcommand{\isasymlless}{\isamath{\ll}}
-\newcommand{\isasymggreater}{\isamath{\gg}}
-\newcommand{\isasymlesssim}{\isamath{\lesssim}}  %requires amssymb
-\newcommand{\isasymgreatersim}{\isamath{\gtrsim}}  %requires amssymb
-\newcommand{\isasymlessapprox}{\isamath{\lessapprox}}  %requires amssymb
-\newcommand{\isasymgreaterapprox}{\isamath{\gtrapprox}}  %requires amssymb
-\newcommand{\isasymin}{\isamath{\in}}
-\newcommand{\isasymnotin}{\isamath{\notin}}
-\newcommand{\isasymsubset}{\isamath{\subset}}
-\newcommand{\isasymsupset}{\isamath{\supset}}
-\newcommand{\isasymsubseteq}{\isamath{\subseteq}}
-\newcommand{\isasymsupseteq}{\isamath{\supseteq}}
-\newcommand{\isasymsqsubset}{\isamath{\sqsubset}}  %requires amssymb
-\newcommand{\isasymsqsupset}{\isamath{\sqsupset}}  %requires amssymb
-\newcommand{\isasymsqsubseteq}{\isamath{\sqsubseteq}}
-\newcommand{\isasymsqsupseteq}{\isamath{\sqsupseteq}}
-\newcommand{\isasyminter}{\isamath{\cap}}
-\newcommand{\isasymInter}{\isamath{\bigcap\,}}
-\newcommand{\isasymunion}{\isamath{\cup}}
-\newcommand{\isasymUnion}{\isamath{\bigcup\,}}
-\newcommand{\isasymsqunion}{\isamath{\sqcup}}
-\newcommand{\isasymSqunion}{\isamath{\bigsqcup\,}}
-\newcommand{\isasymsqinter}{\isamath{\sqcap}}
-\newcommand{\isasymSqinter}{\isamath{\bigsqcap\,}}  %requires amsmath
-\newcommand{\isasymuplus}{\isamath{\uplus}}
-\newcommand{\isasymUplus}{\isamath{\biguplus\,}}
-\newcommand{\isasymnoteq}{\isamath{\not=}}
-\newcommand{\isasymsim}{\isamath{\sim}}
-\newcommand{\isasymdoteq}{\isamath{\doteq}}
-\newcommand{\isasymsimeq}{\isamath{\simeq}}
-\newcommand{\isasymapprox}{\isamath{\approx}}
-\newcommand{\isasymasymp}{\isamath{\asymp}}
-\newcommand{\isasymcong}{\isamath{\cong}}
-\newcommand{\isasymsmile}{\isamath{\smile}}
-\newcommand{\isasymequiv}{\isamath{\equiv}}
-\newcommand{\isasymfrown}{\isamath{\frown}}
-\newcommand{\isasympropto}{\isamath{\propto}}
-\newcommand{\isasymsome}{\isamath{\epsilon\,}}
-\newcommand{\isasymJoin}{\isamath{\Join}}  %requires amssymb
-\newcommand{\isasymbowtie}{\isamath{\bowtie}}
-\newcommand{\isasymprec}{\isamath{\prec}}
-\newcommand{\isasymsucc}{\isamath{\succ}}
-\newcommand{\isasympreceq}{\isamath{\preceq}}
-\newcommand{\isasymsucceq}{\isamath{\succeq}}
-\newcommand{\isasymparallel}{\isamath{\parallel}}
-\newcommand{\isasymbar}{\isamath{\mid}}
-\newcommand{\isasymplusminus}{\isamath{\pm}}
-\newcommand{\isasymminusplus}{\isamath{\mp}}
-\newcommand{\isasymtimes}{\isamath{\times}}
-\newcommand{\isasymdiv}{\isamath{\div}}
-\newcommand{\isasymcdot}{\isamath{\cdot}}
-\newcommand{\isasymstar}{\isamath{\star}}
-\newcommand{\isasymbullet}{\boldmath\isamath{\mathchoice{\displaystyle{\cdot}}{\textstyle{\cdot}}{\scriptstyle{\bullet}}{\scriptscriptstyle{\bullet}}}}
-\newcommand{\isasymcirc}{\isamath{\circ}}
-\newcommand{\isasymdagger}{\isamath{\dagger}}
-\newcommand{\isasymddagger}{\isamath{\ddagger}}
-\newcommand{\isasymlhd}{\isamath{\lhd}}  %requires amssymb
-\newcommand{\isasymrhd}{\isamath{\rhd}}  %requires amssymb
-\newcommand{\isasymunlhd}{\isamath{\unlhd}}  %requires amssymb
-\newcommand{\isasymunrhd}{\isamath{\unrhd}}  %requires amssymb
-\newcommand{\isasymtriangleleft}{\isamath{\triangleleft}}
-\newcommand{\isasymtriangleright}{\isamath{\triangleright}}
-\newcommand{\isasymtriangle}{\isamath{\triangle}}
-\newcommand{\isasymtriangleq}{\isamath{\triangleq}}  %requires amssymb
-\newcommand{\isasymoplus}{\isamath{\oplus}}
-\newcommand{\isasymOplus}{\isamath{\bigoplus\,}}
-\newcommand{\isasymotimes}{\isamath{\otimes}}
-\newcommand{\isasymOtimes}{\isamath{\bigotimes\,}}
-\newcommand{\isasymodot}{\isamath{\odot}}
-\newcommand{\isasymOdot}{\isamath{\bigodot\,}}
-\newcommand{\isasymominus}{\isamath{\ominus}}
-\newcommand{\isasymoslash}{\isamath{\oslash}}
-\newcommand{\isasymdots}{\isamath{\dots}}
-\newcommand{\isasymcdots}{\isamath{\cdots}}
-\newcommand{\isasymSum}{\isamath{\sum\,}}
-\newcommand{\isasymProd}{\isamath{\prod\,}}
-\newcommand{\isasymCoprod}{\isamath{\coprod\,}}
-\newcommand{\isasyminfinity}{\isamath{\infty}}
-\newcommand{\isasymintegral}{\isamath{\int\,}}
-\newcommand{\isasymointegral}{\isamath{\oint\,}}
-\newcommand{\isasymclubsuit}{\isamath{\clubsuit}}
-\newcommand{\isasymdiamondsuit}{\isamath{\diamondsuit}}
-\newcommand{\isasymheartsuit}{\isamath{\heartsuit}}
-\newcommand{\isasymspadesuit}{\isamath{\spadesuit}}
-\newcommand{\isasymaleph}{\isamath{\aleph}}
-\newcommand{\isasymemptyset}{\isamath{\emptyset}}
-\newcommand{\isasymnabla}{\isamath{\nabla}}
-\newcommand{\isasympartial}{\isamath{\partial}}
-\newcommand{\isasymRe}{\isamath{\Re}}
-\newcommand{\isasymIm}{\isamath{\Im}}
-\newcommand{\isasymflat}{\isamath{\flat}}
-\newcommand{\isasymnatural}{\isamath{\natural}}
-\newcommand{\isasymsharp}{\isamath{\sharp}}
-\newcommand{\isasymangle}{\isamath{\angle}}
-\newcommand{\isasymcopyright}{\isatext{\rm\copyright}}
-\newcommand{\isasymregistered}{\isatext{\rm\textregistered}}
-\newcommand{\isasymhyphen}{\isatext{\rm-}}
-\newcommand{\isasyminverse}{\isamath{{}^{-1}}}
-\newcommand{\isasymonesuperior}{\isamath{\mathonesuperior}}  %requires latin1
-\newcommand{\isasymonequarter}{\isatext{\rm\textonequarter}}  %requires latin1
-\newcommand{\isasymtwosuperior}{\isamath{\mathtwosuperior}}  %requires latin1
-\newcommand{\isasymonehalf}{\isatext{\rm\textonehalf}}  %requires latin1
-\newcommand{\isasymthreesuperior}{\isamath{\maththreesuperior}}  %requires latin1
-\newcommand{\isasymthreequarters}{\isatext{\rm\textthreequarters}}  %requires latin1
-\newcommand{\isasymordfeminine}{\isatext{\rm\textordfeminine}}
-\newcommand{\isasymordmasculine}{\isatext{\rm\textordmasculine}}
-\newcommand{\isasymsection}{\isatext{\rm\S}}
-\newcommand{\isasymparagraph}{\isatext{\rm\P}}
-\newcommand{\isasymexclamdown}{\isatext{\rm\textexclamdown}}
-\newcommand{\isasymquestiondown}{\isatext{\rm\textquestiondown}}
-\newcommand{\isasymeuro}{\isatext{\textgreek{\euro}}}  %requires greek babel
-\newcommand{\isasympounds}{\isamath{\pounds}}
-\newcommand{\isasymyen}{\isatext{\yen}}  %requires amssymb
-\newcommand{\isasymcent}{\isatext{\textcent}}  %requires textcomp
-\newcommand{\isasymcurrency}{\isatext{\textcurrency}} %requires textcomp
-\newcommand{\isasymdegree}{\isatext{\rm\textdegree}}  %requires latin1
-\newcommand{\isasymamalg}{\isamath{\amalg}}
-\newcommand{\isasymmho}{\isamath{\mho}}  %requires amssymb
-\newcommand{\isasymlozenge}{\isamath{\lozenge}}  %requires amssymb
-\newcommand{\isasymwp}{\isamath{\wp}}
-\newcommand{\isasymwrong}{\isamath{\wr}}
-\newcommand{\isasymstruct}{\isamath{\diamond}}
-\newcommand{\isasymacute}{\isatext{\'\relax}}
-\newcommand{\isasymindex}{\isatext{\i}}
-\newcommand{\isasymdieresis}{\isatext{\"\relax}}
-\newcommand{\isasymcedilla}{\isatext{\c\relax}}
-\newcommand{\isasymhungarumlaut}{\isatext{\H\relax}}
-\newcommand{\isasymspacespace}{\isamath{~~}}
-\newcommand{\isasymmodule}{\isamath{\langle}\isakeyword{module}\isamath{\rangle}}