doc-src/AxClass/generated/Group.tex

 \begin{isabellebody}%
 \def\isabellecontext{Group}%
 %
 \isamarkupheader{Basic group theory%
 }
-\isacommand{theory}\ Group\ {\isacharequal}\ Main{\isacharcolon}%
+\isamarkuptrue%
+\isacommand{theory}\ Group\ {\isacharequal}\ Main{\isacharcolon}\isamarkupfalse%
+%
 \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}%
+\isamarkuptrue%
 \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}{\isasymunit}{\isachardoublequote}{\isacharparenright}%
+\ \ one\ {\isacharcolon}{\isacharcolon}\ {\isacharprime}a\ \ \ \ {\isacharparenleft}{\isachardoublequote}{\isasymunit}{\isachardoublequote}{\isacharparenright}\isamarkupfalse%
+%
 \begin{isamarkuptext}%
 \noindent Next we define class \isa{monoid} of monoids with
  operations \isa{{\isasymodot}} and \isa{{\isasymunit}}.  Note that multiple class
  axioms are allowed for user convenience --- they simply represent the
  conjunction of their respective universal closures.%
 \end{isamarkuptext}%
+\isamarkuptrue%
 \isacommand{axclass}\ monoid\ {\isasymsubseteq}\ {\isachardoublequote}term{\isachardoublequote}\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}{\isasymunit}\ {\isasymodot}\ x\ {\isacharequal}\ x{\isachardoublequote}\isanewline
-\ \ right{\isacharunderscore}unit{\isacharcolon}\ {\isachardoublequote}x\ {\isasymodot}\ {\isasymunit}\ {\isacharequal}\ x{\isachardoublequote}%
+\ \ right{\isacharunderscore}unit{\isacharcolon}\ {\isachardoublequote}x\ {\isasymodot}\ {\isasymunit}\ {\isacharequal}\ x{\isachardoublequote}\isamarkupfalse%
+%
 \begin{isamarkuptext}%
 \noindent So class \isa{monoid} contains exactly those types \isa{{\isasymtau}} where \isa{{\isasymodot}\ {\isasymColon}\ {\isasymtau}\ {\isasymRightarrow}\ {\isasymtau}\ {\isasymRightarrow}\ {\isasymtau}} and \isa{{\isasymunit}\ {\isasymColon}\ {\isasymtau}} are
  specified appropriately, such that \isa{{\isasymodot}} is associative and
  \isa{{\isasymunit}} 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}%
+\isamarkuptrue%
 \isacommand{axclass}\ semigroup\ {\isasymsubseteq}\ {\isachardoublequote}term{\isachardoublequote}\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}{\isasymunit}\ {\isasymodot}\ x\ {\isacharequal}\ x{\isachardoublequote}\isanewline
 \ \ left{\isacharunderscore}inverse{\isacharcolon}\ {\isachardoublequote}x{\isasyminv}\ {\isasymodot}\ x\ {\isacharequal}\ {\isasymunit}{\isachardoublequote}\isanewline
 \isanewline
+\isamarkupfalse%
 \isacommand{axclass}\ agroup\ {\isasymsubseteq}\ group\isanewline
-\ \ commute{\isacharcolon}\ {\isachardoublequote}x\ {\isasymodot}\ y\ {\isacharequal}\ y\ {\isasymodot}\ x{\isachardoublequote}%
+\ \ commute{\isacharcolon}\ {\isachardoublequote}x\ {\isasymodot}\ y\ {\isacharequal}\ y\ {\isasymodot}\ x{\isachardoublequote}\isamarkupfalse%
+%
 \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}

  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}%
+\isamarkuptrue%
 \isacommand{theorem}\ group{\isacharunderscore}right{\isacharunderscore}inverse{\isacharcolon}\ {\isachardoublequote}x\ {\isasymodot}\ x{\isasyminv}\ {\isacharequal}\ {\isacharparenleft}{\isasymunit}{\isasymColon}{\isacharprime}a{\isasymColon}group{\isacharparenright}{\isachardoublequote}\isanewline
+\isamarkupfalse%
 \isacommand{proof}\ {\isacharminus}\isanewline
-\ \ \isacommand{have}\ {\isachardoublequote}x\ {\isasymodot}\ x{\isasyminv}\ {\isacharequal}\ {\isasymunit}\ {\isasymodot}\ {\isacharparenleft}x\ {\isasymodot}\ x{\isasyminv}{\isacharparenright}{\isachardoublequote}\isanewline
-\ \ \ \ \isacommand{by}\ {\isacharparenleft}simp\ only{\isacharcolon}\ group{\isachardot}left{\isacharunderscore}unit{\isacharparenright}\isanewline
-\ \ \isacommand{also}\ \isacommand{have}\ {\isachardoublequote}{\isachardot}{\isachardot}{\isachardot}\ {\isacharequal}\ {\isasymunit}\ {\isasymodot}\ x\ {\isasymodot}\ x{\isasyminv}{\isachardoublequote}\isanewline
-\ \ \ \ \isacommand{by}\ {\isacharparenleft}simp\ only{\isacharcolon}\ semigroup{\isachardot}assoc{\isacharparenright}\isanewline
-\ \ \isacommand{also}\ \isacommand{have}\ {\isachardoublequote}{\isachardot}{\isachardot}{\isachardot}\ {\isacharequal}\ {\isacharparenleft}x{\isasyminv}{\isacharparenright}{\isasyminv}\ {\isasymodot}\ x{\isasyminv}\ {\isasymodot}\ x\ {\isasymodot}\ x{\isasyminv}{\isachardoublequote}\isanewline
-\ \ \ \ \isacommand{by}\ {\isacharparenleft}simp\ only{\isacharcolon}\ group{\isachardot}left{\isacharunderscore}inverse{\isacharparenright}\isanewline
-\ \ \isacommand{also}\ \isacommand{have}\ {\isachardoublequote}{\isachardot}{\isachardot}{\isachardot}\ {\isacharequal}\ {\isacharparenleft}x{\isasyminv}{\isacharparenright}{\isasyminv}\ {\isasymodot}\ {\isacharparenleft}x{\isasyminv}\ {\isasymodot}\ x{\isacharparenright}\ {\isasymodot}\ x{\isasyminv}{\isachardoublequote}\isanewline
-\ \ \ \ \isacommand{by}\ {\isacharparenleft}simp\ only{\isacharcolon}\ semigroup{\isachardot}assoc{\isacharparenright}\isanewline
-\ \ \isacommand{also}\ \isacommand{have}\ {\isachardoublequote}{\isachardot}{\isachardot}{\isachardot}\ {\isacharequal}\ {\isacharparenleft}x{\isasyminv}{\isacharparenright}{\isasyminv}\ {\isasymodot}\ {\isasymunit}\ {\isasymodot}\ x{\isasyminv}{\isachardoublequote}\isanewline
-\ \ \ \ \isacommand{by}\ {\isacharparenleft}simp\ only{\isacharcolon}\ group{\isachardot}left{\isacharunderscore}inverse{\isacharparenright}\isanewline
-\ \ \isacommand{also}\ \isacommand{have}\ {\isachardoublequote}{\isachardot}{\isachardot}{\isachardot}\ {\isacharequal}\ {\isacharparenleft}x{\isasyminv}{\isacharparenright}{\isasyminv}\ {\isasymodot}\ {\isacharparenleft}{\isasymunit}\ {\isasymodot}\ x{\isasyminv}{\isacharparenright}{\isachardoublequote}\isanewline
-\ \ \ \ \isacommand{by}\ {\isacharparenleft}simp\ only{\isacharcolon}\ semigroup{\isachardot}assoc{\isacharparenright}\isanewline
-\ \ \isacommand{also}\ \isacommand{have}\ {\isachardoublequote}{\isachardot}{\isachardot}{\isachardot}\ {\isacharequal}\ {\isacharparenleft}x{\isasyminv}{\isacharparenright}{\isasyminv}\ {\isasymodot}\ x{\isasyminv}{\isachardoublequote}\isanewline
-\ \ \ \ \isacommand{by}\ {\isacharparenleft}simp\ only{\isacharcolon}\ group{\isachardot}left{\isacharunderscore}unit{\isacharparenright}\isanewline
-\ \ \isacommand{also}\ \isacommand{have}\ {\isachardoublequote}{\isachardot}{\isachardot}{\isachardot}\ {\isacharequal}\ {\isasymunit}{\isachardoublequote}\isanewline
-\ \ \ \ \isacommand{by}\ {\isacharparenleft}simp\ only{\isacharcolon}\ group{\isachardot}left{\isacharunderscore}inverse{\isacharparenright}\isanewline
-\ \ \isacommand{finally}\ \isacommand{show}\ {\isacharquery}thesis\ \isacommand{{\isachardot}}\isanewline
-\isacommand{qed}%
+\ \ \isamarkupfalse%
+\isacommand{have}\ {\isachardoublequote}x\ {\isasymodot}\ x{\isasyminv}\ {\isacharequal}\ {\isasymunit}\ {\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}\ {\isasymunit}\ {\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}\ {\isasymunit}\ {\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}{\isasymunit}\ {\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}\ {\isasymunit}{\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}\isamarkupfalse%
+%
 \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}%
+\isamarkuptrue%
 \isacommand{theorem}\ group{\isacharunderscore}right{\isacharunderscore}unit{\isacharcolon}\ {\isachardoublequote}x\ {\isasymodot}\ {\isasymunit}\ {\isacharequal}\ {\isacharparenleft}x{\isasymColon}{\isacharprime}a{\isasymColon}group{\isacharparenright}{\isachardoublequote}\isanewline
+\isamarkupfalse%
 \isacommand{proof}\ {\isacharminus}\isanewline
-\ \ \isacommand{have}\ {\isachardoublequote}x\ {\isasymodot}\ {\isasymunit}\ {\isacharequal}\ x\ {\isasymodot}\ {\isacharparenleft}x{\isasyminv}\ {\isasymodot}\ x{\isacharparenright}{\isachardoublequote}\isanewline
-\ \ \ \ \isacommand{by}\ {\isacharparenleft}simp\ only{\isacharcolon}\ group{\isachardot}left{\isacharunderscore}inverse{\isacharparenright}\isanewline
-\ \ \isacommand{also}\ \isacommand{have}\ {\isachardoublequote}{\isachardot}{\isachardot}{\isachardot}\ {\isacharequal}\ x\ {\isasymodot}\ x{\isasyminv}\ {\isasymodot}\ x{\isachardoublequote}\isanewline
-\ \ \ \ \isacommand{by}\ {\isacharparenleft}simp\ only{\isacharcolon}\ semigroup{\isachardot}assoc{\isacharparenright}\isanewline
-\ \ \isacommand{also}\ \isacommand{have}\ {\isachardoublequote}{\isachardot}{\isachardot}{\isachardot}\ {\isacharequal}\ {\isasymunit}\ {\isasymodot}\ x{\isachardoublequote}\isanewline
-\ \ \ \ \isacommand{by}\ {\isacharparenleft}simp\ only{\isacharcolon}\ group{\isacharunderscore}right{\isacharunderscore}inverse{\isacharparenright}\isanewline
-\ \ \isacommand{also}\ \isacommand{have}\ {\isachardoublequote}{\isachardot}{\isachardot}{\isachardot}\ {\isacharequal}\ x{\isachardoublequote}\isanewline
-\ \ \ \ \isacommand{by}\ {\isacharparenleft}simp\ only{\isacharcolon}\ group{\isachardot}left{\isacharunderscore}unit{\isacharparenright}\isanewline
-\ \ \isacommand{finally}\ \isacommand{show}\ {\isacharquery}thesis\ \isacommand{{\isachardot}}\isanewline
-\isacommand{qed}%
+\ \ \isamarkupfalse%
+\isacommand{have}\ {\isachardoublequote}x\ {\isasymodot}\ {\isasymunit}\ {\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}\ {\isasymunit}\ {\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}\isamarkupfalse%
+%
 \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}

      \caption{Monoids and groups: according to definition, and by proof}
      \label{fig:monoid-group}
    \end{center}
  \end{figure}%
 \end{isamarkuptext}%
+\isamarkuptrue%
 \isacommand{instance}\ monoid\ {\isasymsubseteq}\ semigroup\isanewline
+\isamarkupfalse%
 \isacommand{proof}\isanewline
-\ \ \isacommand{fix}\ x\ y\ z\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharprime}a{\isasymColon}monoid{\isachardoublequote}\isanewline
-\ \ \isacommand{show}\ {\isachardoublequote}x\ {\isasymodot}\ y\ {\isasymodot}\ z\ {\isacharequal}\ x\ {\isasymodot}\ {\isacharparenleft}y\ {\isasymodot}\ z{\isacharparenright}{\isachardoublequote}\isanewline
-\ \ \ \ \isacommand{by}\ {\isacharparenleft}rule\ monoid{\isachardot}assoc{\isacharparenright}\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}\isanewline
 \isanewline
+\isamarkupfalse%
 \isacommand{instance}\ group\ {\isasymsubseteq}\ monoid\isanewline
+\isamarkupfalse%
 \isacommand{proof}\isanewline
-\ \ \isacommand{fix}\ x\ y\ z\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharprime}a{\isasymColon}group{\isachardoublequote}\isanewline
-\ \ \isacommand{show}\ {\isachardoublequote}x\ {\isasymodot}\ y\ {\isasymodot}\ z\ {\isacharequal}\ x\ {\isasymodot}\ {\isacharparenleft}y\ {\isasymodot}\ z{\isacharparenright}{\isachardoublequote}\isanewline
-\ \ \ \ \isacommand{by}\ {\isacharparenleft}rule\ semigroup{\isachardot}assoc{\isacharparenright}\isanewline
-\ \ \isacommand{show}\ {\isachardoublequote}{\isasymunit}\ {\isasymodot}\ x\ {\isacharequal}\ x{\isachardoublequote}\isanewline
-\ \ \ \ \isacommand{by}\ {\isacharparenleft}rule\ group{\isachardot}left{\isacharunderscore}unit{\isacharparenright}\isanewline
-\ \ \isacommand{show}\ {\isachardoublequote}x\ {\isasymodot}\ {\isasymunit}\ {\isacharequal}\ x{\isachardoublequote}\isanewline
-\ \ \ \ \isacommand{by}\ {\isacharparenleft}rule\ group{\isacharunderscore}right{\isacharunderscore}unit{\isacharparenright}\isanewline
-\isacommand{qed}%
+\ \ \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}{\isasymunit}\ {\isasymodot}\ x\ {\isacharequal}\ x{\isachardoublequote}\isanewline
+\ \ \ \ \isamarkupfalse%
+\isacommand{by}\ {\isacharparenleft}rule\ group{\isachardot}left{\isacharunderscore}unit{\isacharparenright}\isanewline
+\ \ \isamarkupfalse%
+\isacommand{show}\ {\isachardoublequote}x\ {\isasymodot}\ {\isasymunit}\ {\isacharequal}\ x{\isachardoublequote}\isanewline
+\ \ \ \ \isamarkupfalse%
+\isacommand{by}\ {\isacharparenleft}rule\ group{\isacharunderscore}right{\isacharunderscore}unit{\isacharparenright}\isanewline
+\isamarkupfalse%
+\isacommand{qed}\isamarkupfalse%
+%
 \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

  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

 
  \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{{\isasymunit}} forms an Abelian group.%
 \end{isamarkuptext}%
+\isamarkuptrue%
 \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}{\isasymunit}\ {\isasymequiv}\ False{\isachardoublequote}%
+\ \ unit{\isacharunderscore}bool{\isacharunderscore}def{\isacharcolon}\ {\isachardoublequote}{\isasymunit}\ {\isasymequiv}\ False{\isachardoublequote}\isamarkupfalse%
+%
 \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

 
  \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}%
+\isamarkuptrue%
 \isacommand{instance}\ bool\ {\isacharcolon}{\isacharcolon}\ agroup\isanewline
+\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
-\ \ \isacommand{fix}\ x\ y\ z\isanewline
-\ \ \isacommand{show}\ {\isachardoublequote}{\isacharparenleft}{\isacharparenleft}x\ {\isasymnoteq}\ y{\isacharparenright}\ {\isasymnoteq}\ z{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}x\ {\isasymnoteq}\ {\isacharparenleft}y\ {\isasymnoteq}\ z{\isacharparenright}{\isacharparenright}{\isachardoublequote}\ \isacommand{by}\ blast\isanewline
-\ \ \isacommand{show}\ {\isachardoublequote}{\isacharparenleft}False\ {\isasymnoteq}\ x{\isacharparenright}\ {\isacharequal}\ x{\isachardoublequote}\ \isacommand{by}\ blast\isanewline
-\ \ \isacommand{show}\ {\isachardoublequote}{\isacharparenleft}x\ {\isasymnoteq}\ x{\isacharparenright}\ {\isacharequal}\ False{\isachardoublequote}\ \isacommand{by}\ blast\isanewline
-\ \ \isacommand{show}\ {\isachardoublequote}{\isacharparenleft}x\ {\isasymnoteq}\ y{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}y\ {\isasymnoteq}\ x{\isacharparenright}{\isachardoublequote}\ \isacommand{by}\ blast\isanewline
-\isacommand{qed}%
+\ \ \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}\isamarkupfalse%
+%
 \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.

  (e.g.\ if \isa{{\isasymodot}} is defined as list append).  Thus, the
  characteristic constants \isa{{\isasymodot}}, \isa{{\isasyminv}}, \isa{{\isasymunit}}
  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

 
  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}%
+\isamarkuptrue%
 \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}%
+\ \ 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}\isamarkupfalse%
+%
 \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}%
+\isamarkuptrue%
 \isacommand{instance}\ {\isacharasterisk}\ {\isacharcolon}{\isacharcolon}\ {\isacharparenleft}semigroup{\isacharcomma}\ semigroup{\isacharparenright}\ semigroup\isanewline
+\isamarkupfalse%
 \isacommand{proof}\ {\isacharparenleft}intro{\isacharunderscore}classes{\isacharcomma}\ unfold\ times{\isacharunderscore}prod{\isacharunderscore}def{\isacharparenright}\isanewline
-\ \ \isacommand{fix}\ p\ q\ r\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharprime}a{\isasymColon}semigroup\ {\isasymtimes}\ {\isacharprime}b{\isasymColon}semigroup{\isachardoublequote}\isanewline
-\ \ \isacommand{show}\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
-\ \ \ \ \isacommand{by}\ {\isacharparenleft}simp\ add{\isacharcolon}\ semigroup{\isachardot}assoc{\isacharparenright}\isanewline
-\isacommand{qed}%
+\ \ \ \ \isamarkupfalse%
+\isacommand{by}\ {\isacharparenleft}simp\ add{\isacharcolon}\ semigroup{\isachardot}assoc{\isacharparenright}\isanewline
+\isamarkupfalse%
+\isacommand{qed}\isamarkupfalse%
+%
 \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}%
-\isacommand{end}\end{isabellebody}%
+\isamarkuptrue%
+\isacommand{end}\isamarkupfalse%
+\end{isabellebody}%
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