obsolete;

--- a/doc-src/AxClass/Group/document/root.tex	Fri Aug 19 22:25:14 2005 +0200
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,6 +0,0 @@
-
-\documentclass{article}
-
-\begin{document}
---- dummy ---
-\end{document}

--- a/doc-src/AxClass/Nat/document/Group.tex	Fri Aug 19 22:25:14 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/Nat/document/NatClass.tex	Fri Aug 19 22:25:14 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/Nat/document/Product.tex	Fri Aug 19 22:25:14 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/Nat/document/Semigroups.tex	Fri Aug 19 22:25:14 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/Nat/document/root.tex	Fri Aug 19 22:25:14 2005 +0200
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,6 +0,0 @@
-
-\documentclass{article}
-
-\begin{document}
---- dummy ---
-\end{document}