doc-src/Locales/Locales/document/Examples3.tex

 %
 \endisadelimvisible
 %
 \isatagvisible
 \isacommand{interpretation}\isamarkupfalse%
-\ nat{\isacharcolon}\ partial{\isacharunderscore}order\ {\isacharbrackleft}{\isachardoublequoteopen}op\ {\isasymle}\ {\isacharcolon}{\isacharcolon}\ nat\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}{\isacharbrackright}\isanewline
+\ nat{\isacharcolon}\ partial{\isacharunderscore}order\ {\isachardoublequoteopen}op\ {\isasymle}\ {\isacharcolon}{\isacharcolon}\ nat\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\isanewline
 \ \ \isakeyword{where}\ {\isachardoublequoteopen}partial{\isacharunderscore}order{\isachardot}less\ {\isacharparenleft}op\ {\isasymle}{\isacharparenright}\ {\isacharparenleft}x{\isacharcolon}{\isacharcolon}nat{\isacharparenright}\ y\ {\isacharequal}\ {\isacharparenleft}x\ {\isacharless}\ y{\isacharparenright}{\isachardoublequoteclose}\isanewline
 \isacommand{proof}\isamarkupfalse%
 \ {\isacharminus}\isanewline
 \ \ \isacommand{show}\isamarkupfalse%
 \ {\isachardoublequoteopen}partial{\isacharunderscore}order\ {\isacharparenleft}op\ {\isasymle}\ {\isacharcolon}{\isacharcolon}\ nat\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ bool{\isacharparenright}{\isachardoublequoteclose}\isanewline
 \ \ \ \ \isacommand{by}\isamarkupfalse%
 \ unfold{\isacharunderscore}locales\ auto\isanewline
 \ \ \isacommand{then}\isamarkupfalse%
 \ \isacommand{interpret}\isamarkupfalse%
-\ nat{\isacharcolon}\ partial{\isacharunderscore}order\ {\isacharbrackleft}{\isachardoublequoteopen}op\ {\isasymle}\ {\isacharcolon}{\isacharcolon}\ {\isacharbrackleft}nat{\isacharcomma}\ nat{\isacharbrackright}\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}{\isacharbrackright}\ \isacommand{{\isachardot}}\isamarkupfalse%
+\ nat{\isacharcolon}\ partial{\isacharunderscore}order\ {\isachardoublequoteopen}op\ {\isasymle}\ {\isacharcolon}{\isacharcolon}\ {\isacharbrackleft}nat{\isacharcomma}\ nat{\isacharbrackright}\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isacommand{{\isachardot}}\isamarkupfalse%
 \isanewline
 \ \ \isacommand{show}\isamarkupfalse%
 \ {\isachardoublequoteopen}partial{\isacharunderscore}order{\isachardot}less\ {\isacharparenleft}op\ {\isasymle}{\isacharparenright}\ {\isacharparenleft}x{\isacharcolon}{\isacharcolon}nat{\isacharparenright}\ y\ {\isacharequal}\ {\isacharparenleft}x\ {\isacharless}\ y{\isacharparenright}{\isachardoublequoteclose}\isanewline
 \ \ \ \ \isacommand{unfolding}\isamarkupfalse%
 \ nat{\isachardot}less{\isacharunderscore}def\ \isacommand{by}\isamarkupfalse%

 \isamarkuptrue%
 %
 \begin{isamarkuptext}%
 Further interpretations are necessary to reuse theorems from
   the other locales.  In \isa{lattice} the operations \isa{{\isasymsqinter}} and
-  \isa{{\isasymsqunion}} are substituted by \isa{ord{\isacharunderscore}class{\isachardot}min} and
-  \isa{ord{\isacharunderscore}class{\isachardot}max}.  The entire proof for the
+  \isa{{\isasymsqunion}} are substituted by \isa{min} and
+  \isa{max}.  The entire proof for the
   interpretation is reproduced in order to give an example of a more
   elaborate interpretation proof.%
 \end{isamarkuptext}%
 \isamarkuptrue%
 %

 %
 \endisadelimvisible
 %
 \isatagvisible
 \isacommand{interpretation}\isamarkupfalse%
-\ nat{\isacharcolon}\ lattice\ {\isacharbrackleft}{\isachardoublequoteopen}op\ {\isasymle}\ {\isacharcolon}{\isacharcolon}\ nat\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}{\isacharbrackright}\isanewline
+\ nat{\isacharcolon}\ lattice\ {\isachardoublequoteopen}op\ {\isasymle}\ {\isacharcolon}{\isacharcolon}\ nat\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\isanewline
 \ \ \isakeyword{where}\ {\isachardoublequoteopen}lattice{\isachardot}meet\ op\ {\isasymle}\ {\isacharparenleft}x{\isacharcolon}{\isacharcolon}nat{\isacharparenright}\ y\ {\isacharequal}\ min\ x\ y{\isachardoublequoteclose}\isanewline
 \ \ \ \ \isakeyword{and}\ {\isachardoublequoteopen}lattice{\isachardot}join\ op\ {\isasymle}\ {\isacharparenleft}x{\isacharcolon}{\isacharcolon}nat{\isacharparenright}\ y\ {\isacharequal}\ max\ x\ y{\isachardoublequoteclose}\isanewline
 \isacommand{proof}\isamarkupfalse%
 \ {\isacharminus}\isanewline
 \ \ \isacommand{show}\isamarkupfalse%

     situation where the lattice theorems can be used in a convenient way.%
 \end{isamarkuptxt}%
 \isamarkuptrue%
 \ \ \isacommand{then}\isamarkupfalse%
 \ \isacommand{interpret}\isamarkupfalse%
-\ nat{\isacharcolon}\ lattice\ {\isacharbrackleft}{\isachardoublequoteopen}op\ {\isasymle}\ {\isacharcolon}{\isacharcolon}\ nat\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}{\isacharbrackright}\ \isacommand{{\isachardot}}\isamarkupfalse%
+\ nat{\isacharcolon}\ lattice\ {\isachardoublequoteopen}op\ {\isasymle}\ {\isacharcolon}{\isacharcolon}\ nat\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isacommand{{\isachardot}}\isamarkupfalse%
 \isanewline
 \ \ \isacommand{show}\isamarkupfalse%
 \ {\isachardoublequoteopen}lattice{\isachardot}meet\ op\ {\isasymle}\ {\isacharparenleft}x{\isacharcolon}{\isacharcolon}nat{\isacharparenright}\ y\ {\isacharequal}\ min\ x\ y{\isachardoublequoteclose}\isanewline
 \ \ \ \ \isacommand{by}\isamarkupfalse%
 \ {\isacharparenleft}bestsimp\ simp{\isacharcolon}\ nat{\isachardot}meet{\isacharunderscore}def\ nat{\isachardot}is{\isacharunderscore}inf{\isacharunderscore}def{\isacharparenright}\isanewline

 %
 \endisadelimvisible
 %
 \isatagvisible
 \isacommand{interpretation}\isamarkupfalse%
-\ nat{\isacharcolon}\ total{\isacharunderscore}order\ {\isacharbrackleft}{\isachardoublequoteopen}op\ {\isasymle}\ {\isacharcolon}{\isacharcolon}\ nat\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}{\isacharbrackright}\isanewline
+\ nat{\isacharcolon}\ total{\isacharunderscore}order\ {\isachardoublequoteopen}op\ {\isasymle}\ {\isacharcolon}{\isacharcolon}\ nat\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\isanewline
 \ \ \isacommand{by}\isamarkupfalse%
 \ unfold{\isacharunderscore}locales\ arith%
 \endisatagvisible
 {\isafoldvisible}%
 %

 \begin{center}
 \begin{tabular}{l}
   \isa{nat{\isachardot}less{\isacharunderscore}def} from locale \isa{partial{\isacharunderscore}order}: \\
   \quad \isa{{\isacharparenleft}{\isacharquery}x\ {\isacharless}\ {\isacharquery}y{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}{\isacharquery}x\ {\isasymle}\ {\isacharquery}y\ {\isasymand}\ {\isacharquery}x\ {\isasymnoteq}\ {\isacharquery}y{\isacharparenright}} \\
   \isa{nat{\isachardot}meet{\isacharunderscore}left} from locale \isa{lattice}: \\
-  \quad \isa{ord{\isacharunderscore}class{\isachardot}min\ {\isacharquery}x\ {\isacharquery}y\ {\isasymle}\ {\isacharquery}x} \\
+  \quad \isa{min\ {\isacharquery}x\ {\isacharquery}y\ {\isasymle}\ {\isacharquery}x} \\
   \isa{nat{\isachardot}join{\isacharunderscore}distr} from locale \isa{distrib{\isacharunderscore}lattice}: \\
-  \quad \isa{ord{\isacharunderscore}class{\isachardot}max\ {\isacharquery}x\ {\isacharparenleft}ord{\isacharunderscore}class{\isachardot}min\ {\isacharquery}y\ {\isacharquery}z{\isacharparenright}\ {\isacharequal}\ ord{\isacharunderscore}class{\isachardot}min\ {\isacharparenleft}ord{\isacharunderscore}class{\isachardot}max\ {\isacharquery}x\ {\isacharquery}y{\isacharparenright}\ {\isacharparenleft}ord{\isacharunderscore}class{\isachardot}max\ {\isacharquery}x\ {\isacharquery}z{\isacharparenright}} \\
+  \quad \isa{lattice{\isachardot}join\ op\ {\isasymle}\ {\isacharquery}x\ {\isacharparenleft}lattice{\isachardot}meet\ op\ {\isasymle}\ {\isacharquery}y\ {\isacharquery}z{\isacharparenright}\ {\isacharequal}\ lattice{\isachardot}meet\ op\ {\isasymle}\ {\isacharparenleft}lattice{\isachardot}join\ op\ {\isasymle}\ {\isacharquery}x\ {\isacharquery}y{\isacharparenright}\ {\isacharparenleft}lattice{\isachardot}join\ op\ {\isasymle}\ {\isacharquery}x\ {\isacharquery}z{\isacharparenright}} \\
   \isa{nat{\isachardot}less{\isacharunderscore}total} from locale \isa{total{\isacharunderscore}order}: \\
-  \quad \isa{{\isacharquery}x\ {\isacharless}\ {\isacharquery}y\ {\isasymor}\ {\isacharquery}x\ {\isacharequal}\ {\isacharquery}y\ {\isasymor}\ {\isacharquery}y\ {\isacharless}\ {\isacharquery}x}
+  \quad \isa{partial{\isacharunderscore}order{\isachardot}less\ op\ {\isasymle}\ {\isacharquery}x\ {\isacharquery}y\ {\isasymor}\ {\isacharquery}x\ {\isacharequal}\ {\isacharquery}y\ {\isasymor}\ partial{\isacharunderscore}order{\isachardot}less\ op\ {\isasymle}\ {\isacharquery}y\ {\isacharquery}x}
 \end{tabular}
 \end{center}
 \hrule
 \caption{Interpreted theorems for \isa{{\isasymle}} on the natural numbers.}
 \label{tab:nat-lattice}

   but not a total order.  Interpretation again proceeds
   incrementally.%
 \end{isamarkuptext}%
 \isamarkuptrue%
 \isacommand{interpretation}\isamarkupfalse%
-\ nat{\isacharunderscore}dvd{\isacharcolon}\ partial{\isacharunderscore}order\ {\isacharbrackleft}{\isachardoublequoteopen}op\ dvd\ {\isacharcolon}{\isacharcolon}\ nat\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}{\isacharbrackright}\isanewline
+\ nat{\isacharunderscore}dvd{\isacharcolon}\ partial{\isacharunderscore}order\ {\isachardoublequoteopen}op\ dvd\ {\isacharcolon}{\isacharcolon}\ nat\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\isanewline
 \ \ \isakeyword{where}\ {\isachardoublequoteopen}partial{\isacharunderscore}order{\isachardot}less\ op\ dvd\ {\isacharparenleft}x{\isacharcolon}{\isacharcolon}nat{\isacharparenright}\ y\ {\isacharequal}\ {\isacharparenleft}x\ dvd\ y\ {\isasymand}\ x\ {\isasymnoteq}\ y{\isacharparenright}{\isachardoublequoteclose}\isanewline
 %
 \isadelimproof
 %
 \endisadelimproof

 \ {\isachardoublequoteopen}partial{\isacharunderscore}order\ {\isacharparenleft}op\ dvd\ {\isacharcolon}{\isacharcolon}\ nat\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ bool{\isacharparenright}{\isachardoublequoteclose}\isanewline
 \ \ \ \ \isacommand{by}\isamarkupfalse%
 \ unfold{\isacharunderscore}locales\ {\isacharparenleft}auto\ simp{\isacharcolon}\ dvd{\isacharunderscore}def{\isacharparenright}\isanewline
 \ \ \isacommand{then}\isamarkupfalse%
 \ \isacommand{interpret}\isamarkupfalse%
-\ nat{\isacharunderscore}dvd{\isacharcolon}\ partial{\isacharunderscore}order\ {\isacharbrackleft}{\isachardoublequoteopen}op\ dvd\ {\isacharcolon}{\isacharcolon}\ nat\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}{\isacharbrackright}\ \isacommand{{\isachardot}}\isamarkupfalse%
+\ nat{\isacharunderscore}dvd{\isacharcolon}\ partial{\isacharunderscore}order\ {\isachardoublequoteopen}op\ dvd\ {\isacharcolon}{\isacharcolon}\ nat\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isacommand{{\isachardot}}\isamarkupfalse%
 \isanewline
 \ \ \isacommand{show}\isamarkupfalse%
 \ {\isachardoublequoteopen}partial{\isacharunderscore}order{\isachardot}less\ op\ dvd\ {\isacharparenleft}x{\isacharcolon}{\isacharcolon}nat{\isacharparenright}\ y\ {\isacharequal}\ {\isacharparenleft}x\ dvd\ y\ {\isasymand}\ x\ {\isasymnoteq}\ y{\isacharparenright}{\isachardoublequoteclose}\isanewline
 \ \ \ \ \isacommand{apply}\isamarkupfalse%
 \ {\isacharparenleft}unfold\ nat{\isacharunderscore}dvd{\isachardot}less{\isacharunderscore}def{\isacharparenright}\isanewline

 Note that there is no symbol for strict divisibility.  Instead,
   interpretation substitutes \isa{x\ dvd\ y\ {\isasymand}\ x\ {\isasymnoteq}\ y}.%
 \end{isamarkuptext}%
 \isamarkuptrue%
 \isacommand{interpretation}\isamarkupfalse%
-\ nat{\isacharunderscore}dvd{\isacharcolon}\ lattice\ {\isacharbrackleft}{\isachardoublequoteopen}op\ dvd\ {\isacharcolon}{\isacharcolon}\ nat\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}{\isacharbrackright}\isanewline
+\ nat{\isacharunderscore}dvd{\isacharcolon}\ lattice\ {\isachardoublequoteopen}op\ dvd\ {\isacharcolon}{\isacharcolon}\ nat\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\isanewline
 \ \ \isakeyword{where}\ nat{\isacharunderscore}dvd{\isacharunderscore}meet{\isacharunderscore}eq{\isacharcolon}\isanewline
 \ \ \ \ \ \ {\isachardoublequoteopen}lattice{\isachardot}meet\ op\ dvd\ {\isacharequal}\ gcd{\isachardoublequoteclose}\isanewline
 \ \ \ \ \isakeyword{and}\ nat{\isacharunderscore}dvd{\isacharunderscore}join{\isacharunderscore}eq{\isacharcolon}\isanewline
 \ \ \ \ \ \ {\isachardoublequoteopen}lattice{\isachardot}join\ op\ dvd\ {\isacharequal}\ lcm{\isachardoublequoteclose}\isanewline
 %

 \ {\isacharparenleft}auto\ intro{\isacharcolon}\ lcm{\isacharunderscore}dvd{\isadigit{1}}\ lcm{\isacharunderscore}dvd{\isadigit{2}}\ lcm{\isacharunderscore}least{\isacharparenright}\isanewline
 \ \ \ \ \isacommand{done}\isamarkupfalse%
 \isanewline
 \ \ \isacommand{then}\isamarkupfalse%
 \ \isacommand{interpret}\isamarkupfalse%
-\ nat{\isacharunderscore}dvd{\isacharcolon}\ lattice\ {\isacharbrackleft}{\isachardoublequoteopen}op\ dvd\ {\isacharcolon}{\isacharcolon}\ nat\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}{\isacharbrackright}\ \isacommand{{\isachardot}}\isamarkupfalse%
+\ nat{\isacharunderscore}dvd{\isacharcolon}\ lattice\ {\isachardoublequoteopen}op\ dvd\ {\isacharcolon}{\isacharcolon}\ nat\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isacommand{{\isachardot}}\isamarkupfalse%
 \isanewline
 \ \ \isacommand{show}\isamarkupfalse%
 \ {\isachardoublequoteopen}lattice{\isachardot}meet\ op\ dvd\ {\isacharequal}\ gcd{\isachardoublequoteclose}\isanewline
 \ \ \ \ \isacommand{apply}\isamarkupfalse%
 \ {\isacharparenleft}auto\ simp\ add{\isacharcolon}\ expand{\isacharunderscore}fun{\isacharunderscore}eq{\isacharparenright}\isanewline

 \endisadelimvisible
 %
 \isatagvisible
 \isacommand{interpretation}\isamarkupfalse%
 \ nat{\isacharunderscore}dvd{\isacharcolon}\isanewline
-\ \ distrib{\isacharunderscore}lattice\ {\isacharbrackleft}{\isachardoublequoteopen}op\ dvd\ {\isacharcolon}{\isacharcolon}\ nat\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}{\isacharbrackright}\isanewline
+\ \ distrib{\isacharunderscore}lattice\ {\isachardoublequoteopen}op\ dvd\ {\isacharcolon}{\isacharcolon}\ nat\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\isanewline
 \ \ \isacommand{apply}\isamarkupfalse%
 \ unfold{\isacharunderscore}locales%
 \begin{isamarkuptxt}%
 \begin{isabelle}%
 \ {\isadigit{1}}{\isachardot}\ {\isasymAnd}x\ y\ z{\isachardot}\isanewline

   \isa{nat{\isacharunderscore}dvd{\isachardot}less{\isacharunderscore}def} from locale \isa{partial{\isacharunderscore}order}: \\
   \quad \isa{{\isacharparenleft}{\isacharquery}x\ dvd\ {\isacharquery}y\ {\isasymand}\ {\isacharquery}x\ {\isasymnoteq}\ {\isacharquery}y{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}{\isacharquery}x\ dvd\ {\isacharquery}y\ {\isasymand}\ {\isacharquery}x\ {\isasymnoteq}\ {\isacharquery}y{\isacharparenright}} \\
   \isa{nat{\isacharunderscore}dvd{\isachardot}meet{\isacharunderscore}left} from locale \isa{lattice}: \\
   \quad \isa{gcd\ {\isacharquery}x\ {\isacharquery}y\ dvd\ {\isacharquery}x} \\
   \isa{nat{\isacharunderscore}dvd{\isachardot}join{\isacharunderscore}distr} from locale \isa{distrib{\isacharunderscore}lattice}: \\
-  \quad \isa{lcm\ {\isacharquery}x\ {\isacharparenleft}gcd\ {\isacharquery}y\ {\isacharquery}z{\isacharparenright}\ {\isacharequal}\ gcd\ {\isacharparenleft}lcm\ {\isacharquery}x\ {\isacharquery}y{\isacharparenright}\ {\isacharparenleft}lcm\ {\isacharquery}x\ {\isacharquery}z{\isacharparenright}} \\
+  \quad \isa{lattice{\isachardot}join\ op\ dvd\ {\isacharquery}x\ {\isacharparenleft}lattice{\isachardot}meet\ op\ dvd\ {\isacharquery}y\ {\isacharquery}z{\isacharparenright}\ {\isacharequal}\ lattice{\isachardot}meet\ op\ dvd\ {\isacharparenleft}lattice{\isachardot}join\ op\ dvd\ {\isacharquery}x\ {\isacharquery}y{\isacharparenright}\ {\isacharparenleft}lattice{\isachardot}join\ op\ dvd\ {\isacharquery}x\ {\isacharquery}z{\isacharparenright}} \\
 \end{tabular}
 \end{center}
 \hrule
 \caption{Interpreted theorems for \isa{dvd} on the natural numbers.}
 \label{tab:nat-dvd-lattice}

   preserving maps can be declared in the following way.%
 \end{isamarkuptext}%
 \isamarkuptrue%
 \ \ \isacommand{locale}\isamarkupfalse%
 \ order{\isacharunderscore}preserving\ {\isacharequal}\isanewline
-\ \ \ \ partial{\isacharunderscore}order\ {\isacharplus}\ partial{\isacharunderscore}order\ le{\isacharprime}\ {\isacharparenleft}\isakeyword{infixl}\ {\isachardoublequoteopen}{\isasympreceq}{\isachardoublequoteclose}\ {\isadigit{5}}{\isadigit{0}}{\isacharparenright}\ {\isacharplus}\isanewline
+\ \ \ \ partial{\isacharunderscore}order\ {\isacharplus}\ po{\isacharprime}{\isacharcolon}\ partial{\isacharunderscore}order\ le{\isacharprime}\ \isakeyword{for}\ le{\isacharprime}\ {\isacharparenleft}\isakeyword{infixl}\ {\isachardoublequoteopen}{\isasympreceq}{\isachardoublequoteclose}\ {\isadigit{5}}{\isadigit{0}}{\isacharparenright}\ {\isacharplus}\isanewline
 \ \ \ \ \isakeyword{fixes}\ {\isasymphi}\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}b{\isachardoublequoteclose}\isanewline
 \ \ \ \ \isakeyword{assumes}\ hom{\isacharunderscore}le{\isacharcolon}\ {\isachardoublequoteopen}x\ {\isasymsqsubseteq}\ y\ {\isasymLongrightarrow}\ {\isasymphi}\ x\ {\isasympreceq}\ {\isasymphi}\ y{\isachardoublequoteclose}%
 \begin{isamarkuptext}%
 The second line contains the expression, which is the
   merge of two partial order locales.  The parameter of the second one

   right locale.%
 \end{isamarkuptext}%
 \isamarkuptrue%
 %
 \begin{isamarkuptext}%
-The locale \isa{order{\isacharunderscore}preserving} contains theorems for both
+% FIXME needs update
+  The locale \isa{order{\isacharunderscore}preserving} contains theorems for both
   orders \isa{{\isasymsqsubseteq}} and \isa{{\isasympreceq}}.  How can one refer to a theorem for
   a particular order, \isa{{\isasymsqsubseteq}} or \isa{{\isasympreceq}}?  Names in locales are
   qualified by the locale parameters.  More precisely, a name is
   qualified by the parameters of the locale in which its declaration
   occurs.  Here are examples:%

 \isadeliminvisible
 %
 \endisadeliminvisible
 %
 \begin{isamarkuptext}%
-\isa{le{\isachardot}less{\isacharunderscore}le{\isacharunderscore}trans}: \isa{{\isasymlbrakk}{\isacharquery}x\ {\isasymsqsubset}\ {\isacharquery}y{\isacharsemicolon}\ {\isacharquery}y\ {\isasymsqsubseteq}\ {\isacharquery}z{\isasymrbrakk}\ {\isasymLongrightarrow}\ {\isacharquery}x\ {\isasymsqsubset}\ {\isacharquery}z}
-
-  \isa{le{\isacharunderscore}le{\isacharprime}{\isacharunderscore}{\isasymphi}{\isachardot}hom{\isacharunderscore}le}: \isa{{\isacharquery}x\ {\isasymsqsubseteq}\ {\isacharquery}y\ {\isasymLongrightarrow}\ {\isasymphi}\ {\isacharquery}x\ {\isasympreceq}\ {\isasymphi}\ {\isacharquery}y}%
+% FIXME needs update?
+  \isa{less{\isacharunderscore}le{\isacharunderscore}trans}: \isa{{\isasymlbrakk}partial{\isacharunderscore}order{\isachardot}less\ op\ {\isasympreceq}\ {\isacharquery}x\ {\isacharquery}y{\isacharsemicolon}\ {\isacharquery}y\ {\isasympreceq}\ {\isacharquery}z{\isasymrbrakk}\ {\isasymLongrightarrow}\ partial{\isacharunderscore}order{\isachardot}less\ op\ {\isasympreceq}\ {\isacharquery}x\ {\isacharquery}z}
+
+  \isa{hom{\isacharunderscore}le}: \isa{{\isacharquery}x\ {\isasymsqsubseteq}\ {\isacharquery}y\ {\isasymLongrightarrow}\ {\isasymphi}\ {\isacharquery}x\ {\isasympreceq}\ {\isasymphi}\ {\isacharquery}y}%
 \end{isamarkuptext}%
 \isamarkuptrue%
 %
 \begin{isamarkuptext}%
 When renaming a locale, the morphism is also applied
   to the qualifiers.  Hence theorems for the partial order \isa{{\isasympreceq}}
-  are qualified by \isa{le{\isacharprime}}.  For example, \isa{le{\isacharprime}{\isachardot}less{\isacharunderscore}le{\isacharunderscore}trans}: \begin{isabelle}%
+  are qualified by \isa{le{\isacharprime}}.  For example, \isa{po{\isacharprime}{\isachardot}less{\isacharunderscore}le{\isacharunderscore}trans}: \begin{isabelle}%
 \ \ {\isasymlbrakk}partial{\isacharunderscore}order{\isachardot}less\ op\ {\isasympreceq}\ {\isacharquery}x\ {\isacharquery}y{\isacharsemicolon}\ {\isacharquery}y\ {\isasympreceq}\ {\isacharquery}z{\isasymrbrakk}\isanewline
 \isaindent{\ \ }{\isasymLongrightarrow}\ partial{\isacharunderscore}order{\isachardot}less\ op\ {\isasympreceq}\ {\isacharquery}x\ {\isacharquery}z%
 \end{isabelle}%
 \end{isamarkuptext}%
 \isamarkuptrue%

 \isadeliminvisible
 %
 \endisadeliminvisible
 %
 \begin{isamarkuptext}%
-This example reveals that there is no infix syntax for the strict
+% FIXME needs update?
+  This example reveals that there is no infix syntax for the strict
   version of \isa{{\isasympreceq}}!  This can, of course, not be introduced
   automatically, but it can be declared manually through an abbreviation.%
 \end{isamarkuptext}%
 \isamarkuptrue%
 \ \ \isacommand{abbreviation}\isamarkupfalse%

 Two more locales illustrate working with locale expressions.
   A map \isa{{\isasymphi}} is a lattice homomorphism if it preserves meet and join.%
 \end{isamarkuptext}%
 \isamarkuptrue%
 \ \ \isacommand{locale}\isamarkupfalse%
-\ lattice{\isacharunderscore}hom\ {\isacharequal}\ lattice\ {\isacharplus}\ lattice\ le{\isacharprime}\ {\isacharparenleft}\isakeyword{infixl}\ {\isachardoublequoteopen}{\isasympreceq}{\isachardoublequoteclose}\ {\isadigit{5}}{\isadigit{0}}{\isacharparenright}\ {\isacharplus}\isanewline
+\ lattice{\isacharunderscore}hom\ {\isacharequal}\ lattice\ {\isacharplus}\ lat{\isacharprime}{\isacharbang}{\isacharcolon}\ lattice\ le{\isacharprime}\ \isakeyword{for}\ le{\isacharprime}\ {\isacharparenleft}\isakeyword{infixl}\ {\isachardoublequoteopen}{\isasympreceq}{\isachardoublequoteclose}\ {\isadigit{5}}{\isadigit{0}}{\isacharparenright}\ {\isacharplus}\isanewline
 \ \ \ \ \isakeyword{fixes}\ {\isasymphi}\isanewline
 \ \ \ \ \isakeyword{assumes}\ hom{\isacharunderscore}meet{\isacharcolon}\isanewline
 \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymphi}\ {\isacharparenleft}lattice{\isachardot}meet\ le\ x\ y{\isacharparenright}\ {\isacharequal}\ lattice{\isachardot}meet\ le{\isacharprime}\ {\isacharparenleft}{\isasymphi}\ x{\isacharparenright}\ {\isacharparenleft}{\isasymphi}\ y{\isacharparenright}{\isachardoublequoteclose}\isanewline
 \ \ \ \ \ \ \isakeyword{and}\ hom{\isacharunderscore}join{\isacharcolon}\isanewline
 \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymphi}\ {\isacharparenleft}lattice{\isachardot}join\ le\ x\ y{\isacharparenright}\ {\isacharequal}\ lattice{\isachardot}join\ le{\isacharprime}\ {\isacharparenleft}{\isasymphi}\ x{\isacharparenright}\ {\isacharparenleft}{\isasymphi}\ y{\isacharparenright}{\isachardoublequoteclose}\isanewline
 \isanewline
 \ \ \isacommand{abbreviation}\isamarkupfalse%
 \ {\isacharparenleft}\isakeyword{in}\ lattice{\isacharunderscore}hom{\isacharparenright}\isanewline
-\ \ \ \ meet{\isacharprime}\ {\isacharparenleft}\isakeyword{infixl}\ {\isachardoublequoteopen}{\isasymsqinter}{\isacharprime}{\isacharprime}{\isachardoublequoteclose}\ {\isadigit{5}}{\isadigit{0}}{\isacharparenright}\ \isakeyword{where}\ {\isachardoublequoteopen}meet{\isacharprime}\ {\isasymequiv}\ le{\isacharprime}{\isachardot}meet{\isachardoublequoteclose}\isanewline
+\ \ \ \ meet{\isacharprime}\ {\isacharparenleft}\isakeyword{infixl}\ {\isachardoublequoteopen}{\isasymsqinter}{\isacharprime}{\isacharprime}{\isachardoublequoteclose}\ {\isadigit{5}}{\isadigit{0}}{\isacharparenright}\ \isakeyword{where}\ {\isachardoublequoteopen}meet{\isacharprime}\ {\isasymequiv}\ lat{\isacharprime}{\isachardot}meet{\isachardoublequoteclose}\isanewline
 \ \ \isacommand{abbreviation}\isamarkupfalse%
 \ {\isacharparenleft}\isakeyword{in}\ lattice{\isacharunderscore}hom{\isacharparenright}\isanewline
-\ \ \ \ join{\isacharprime}\ {\isacharparenleft}\isakeyword{infixl}\ {\isachardoublequoteopen}{\isasymsqunion}{\isacharprime}{\isacharprime}{\isachardoublequoteclose}\ {\isadigit{5}}{\isadigit{0}}{\isacharparenright}\ \isakeyword{where}\ {\isachardoublequoteopen}join{\isacharprime}\ {\isasymequiv}\ le{\isacharprime}{\isachardot}join{\isachardoublequoteclose}%
+\ \ \ \ join{\isacharprime}\ {\isacharparenleft}\isakeyword{infixl}\ {\isachardoublequoteopen}{\isasymsqunion}{\isacharprime}{\isacharprime}{\isachardoublequoteclose}\ {\isadigit{5}}{\isadigit{0}}{\isacharparenright}\ \isakeyword{where}\ {\isachardoublequoteopen}join{\isacharprime}\ {\isasymequiv}\ lat{\isacharprime}{\isachardot}join{\isachardoublequoteclose}%
 \begin{isamarkuptext}%
 A homomorphism is an endomorphism if both orders coincide.%
 \end{isamarkuptext}%
 \isamarkuptrue%
 \ \ \isacommand{locale}\isamarkupfalse%
 \ lattice{\isacharunderscore}end\ {\isacharequal}\isanewline
-\ \ \ \ lattice{\isacharunderscore}hom\ le\ {\isacharparenleft}\isakeyword{infixl}\ {\isachardoublequoteopen}{\isasymsqsubseteq}{\isachardoublequoteclose}\ {\isadigit{5}}{\isadigit{0}}{\isacharparenright}\ le\ {\isacharparenleft}\isakeyword{infixl}\ {\isachardoublequoteopen}{\isasymsqsubseteq}{\isachardoublequoteclose}\ {\isadigit{5}}{\isadigit{0}}{\isacharparenright}%
+\ \ \ \ lattice{\isacharunderscore}hom\ le\ le\ \isakeyword{for}\ le\ {\isacharparenleft}\isakeyword{infixl}\ {\isachardoublequoteopen}{\isasymsqsubseteq}{\isachardoublequoteclose}\ {\isadigit{5}}{\isadigit{0}}{\isacharparenright}%
 \begin{isamarkuptext}%
 The inheritance diagram of the situation we have now is shown
   in Figure~\ref{fig:hom}, where the dashed line depicts an
   interpretation which is introduced below.  Renamings are
   indicated by $\sqsubseteq \mapsto \preceq$ etc.  The expression

 It can be shown easily that a lattice homomorphism is order
   preserving.  As the final example of this section, a locale
   interpretation is used to assert this.%
 \end{isamarkuptext}%
 \isamarkuptrue%
-\ \ \isacommand{interpretation}\isamarkupfalse%
+\ \ \isacommand{sublocale}\isamarkupfalse%
 \ lattice{\isacharunderscore}hom\ {\isasymsubseteq}\ order{\isacharunderscore}preserving%
 \isadelimproof
 \ %
 \endisadelimproof
 %

 \ \ \ \ \isacommand{assume}\isamarkupfalse%
 \ {\isachardoublequoteopen}x\ {\isasymsqsubseteq}\ y{\isachardoublequoteclose}\isanewline
 \ \ \ \ \isacommand{then}\isamarkupfalse%
 \ \isacommand{have}\isamarkupfalse%
 \ {\isachardoublequoteopen}y\ {\isacharequal}\ {\isacharparenleft}x\ {\isasymsqunion}\ y{\isacharparenright}{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}simp\ add{\isacharcolon}\ le{\isachardot}join{\isacharunderscore}connection{\isacharparenright}\isanewline
+\ {\isacharparenleft}simp\ add{\isacharcolon}\ join{\isacharunderscore}connection{\isacharparenright}\isanewline
 \ \ \ \ \isacommand{then}\isamarkupfalse%
 \ \isacommand{have}\isamarkupfalse%
 \ {\isachardoublequoteopen}{\isasymphi}\ y\ {\isacharequal}\ {\isacharparenleft}{\isasymphi}\ x\ {\isasymsqunion}{\isacharprime}\ {\isasymphi}\ y{\isacharparenright}{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
 \ {\isacharparenleft}simp\ add{\isacharcolon}\ hom{\isacharunderscore}join\ {\isacharbrackleft}symmetric{\isacharbrackright}{\isacharparenright}\isanewline
 \ \ \ \ \isacommand{then}\isamarkupfalse%
 \ \isacommand{show}\isamarkupfalse%
 \ {\isachardoublequoteopen}{\isasymphi}\ x\ {\isasympreceq}\ {\isasymphi}\ y{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}simp\ add{\isacharcolon}\ le{\isacharprime}{\isachardot}join{\isacharunderscore}connection{\isacharparenright}\isanewline
+\ {\isacharparenleft}simp\ add{\isacharcolon}\ lat{\isacharprime}{\isachardot}join{\isacharunderscore}connection{\isacharparenright}\isanewline
 \ \ \isacommand{qed}\isamarkupfalse%
 %
 \endisatagproof
 {\isafoldproof}%
 %

 %
 \begin{isamarkuptext}%
 Theorems and other declarations --- syntax, in particular ---
   from the locale \isa{order{\isacharunderscore}preserving} are now active in \isa{lattice{\isacharunderscore}hom}, for example
 
-  \isa{le{\isacharprime}{\isachardot}less{\isacharunderscore}le{\isacharunderscore}trans}:
+  \isa{lat{\isacharprime}{\isachardot}less{\isacharunderscore}le{\isacharunderscore}trans}:
   \isa{{\isasymlbrakk}{\isacharquery}x\ {\isasymprec}\ {\isacharquery}y{\isacharsemicolon}\ {\isacharquery}y\ {\isasympreceq}\ {\isacharquery}z{\isasymrbrakk}\ {\isasymLongrightarrow}\ {\isacharquery}x\ {\isasymprec}\ {\isacharquery}z}%
 \end{isamarkuptext}%
 \isamarkuptrue%
 %
 \isamarkupsection{Further Reading%