# HG changeset patch
# User ballarin
# Date 1212489262 -7200
# Node ID d1d35284542fbcb029a0b0ec3eb4b9303868a11e
# Parent  9681f347b6f54cb71fa49c815dba70ba25e6d8a5
New version covering interpretation.

diff -r 9681f347b6f5 -r d1d35284542f doc-src/Locales/IsaMakefile
--- a/doc-src/Locales/IsaMakefile	Tue Jun 03 11:55:35 2008 +0200
+++ b/doc-src/Locales/IsaMakefile	Tue Jun 03 12:34:22 2008 +0200
@@ -23,7 +23,8 @@
 HOL:
 	@cd $(SRC)/HOL; $(ISATOOL) make HOL
 
-$(LOG)/HOL-Locales.gz: $(OUT)/HOL Locales/ROOT.ML Locales/Locales.thy \
+$(LOG)/HOL-Locales.gz: $(OUT)/HOL Locales/ROOT.ML Locales/Examples.thy \
+  Locales/Examples1.thy Locales/Examples2.thy Locales/Examples3.thy \
   Locales/document/root.tex Locales/document/root.bib
 	@$(USEDIR) $(OUT)/HOL Locales
 
diff -r 9681f347b6f5 -r d1d35284542f doc-src/Locales/Locales/Examples.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/Locales/Locales/Examples.thy	Tue Jun 03 12:34:22 2008 +0200
@@ -0,0 +1,683 @@
+(*<*)
+theory Examples
+imports GCD
+begin
+
+hide const Lattices.lattice
+pretty_setmargin 65
+
+(*>*)
+
+(*
+text {* The following presentation will use notation of
+  Isabelle's meta logic, hence a few sentences to explain this.
+  The logical
+  primitives are universal quantification (@{text "\<And>"}), entailment
+  (@{text "\<Longrightarrow>"}) and equality (@{text "\<equiv>"}).  Variables (not bound
+  variables) are sometimes preceded by a question mark.  The logic is
+  typed.  Type variables are denoted by @{text "'a"}, @{text "'b"}
+  etc., and @{text "\<Rightarrow>"} is the function type.  Double brackets @{text
+  "\<lbrakk>"} and @{text "\<rbrakk>"} are used to abbreviate nested entailment.
+*}
+*)
+
+section {* Introduction *}
+
+text {*
+  Locales are based on contexts.  A \emph{context} can be seen as a
+  formula schema
+\[
+  @{text "\<And>x\<^sub>1\<dots>x\<^sub>n. \<lbrakk> A\<^sub>1; \<dots> ;A\<^sub>m \<rbrakk> \<Longrightarrow> \<dots>"}
+\]
+  where variables @{text "x\<^sub>1"}, \ldots, @{text "x\<^sub>n"} are called
+  \emph{parameters} and the premises $@{text "A\<^sub>1"}, \ldots,
+  @{text "A\<^sub>m"}$ \emph{assumptions}.  A formula @{text "C"}
+  is a \emph{theorem} in the context if it is a conclusion
+\[
+%\label{eq-fact-in-context}
+  @{text "\<And>x\<^sub>1\<dots>x\<^sub>n. \<lbrakk> A\<^sub>1; \<dots> ;A\<^sub>m \<rbrakk> \<Longrightarrow> C"}.
+\]
+  Isabelle/Isar's notion of context goes beyond this logical view.
+  Its contexts record, in a consecutive order, proved
+  conclusions along with attributes, which
+  may control proof procedures.  Contexts also contain syntax information
+  for parameters and for terms depending on them.
+  *}
+
+section {* Simple Locales *}
+
+text {*
+  Locales can be seen as persistent contexts.  In its simplest form, a
+  \emph{locale declaration} consists of a sequence of context elements
+  declaring parameters (keyword \isakeyword{fixes}) and assumptions
+  (keyword \isakeyword{assumes}).  The following is the specification of
+  partial orders, as locale @{text partial_order}.
+  *}
+
+  locale partial_order =
+    fixes le :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infixl "\<sqsubseteq>" 50)
+    assumes refl [intro, simp]: "x \<sqsubseteq> x"
+      and anti_sym [intro]: "\<lbrakk> x \<sqsubseteq> y; y \<sqsubseteq> x \<rbrakk> \<Longrightarrow> x = y"
+      and trans [trans]: "\<lbrakk> x \<sqsubseteq> y; y \<sqsubseteq> z \<rbrakk> \<Longrightarrow> x \<sqsubseteq> z"
+
+text {* The parameter of this locale is @{term le}, with infix syntax
+  @{text \<sqsubseteq>}.  There is an implicit type parameter @{typ "'a"}.  It
+  is not necessary to declare parameter types: most general types will
+  be inferred from the context elements for all parameters.
+
+  The above declaration not only introduces the locale, it also
+  defines the \emph{locale predicate} @{term partial_order} with
+  definition @{thm [source] partial_order_def}:
+  @{thm [display, indent=2] partial_order_def}
+
+  The specification of a locale is fixed, but its list of conclusions
+  may be extended through Isar commands that take a \emph{target} argument.
+  In the following, \isakeyword{definition} and 
+  \isakeyword{theorem} are illustrated.
+  Table~\ref{tab:commands-with-target} lists Isar commands that accept
+  a target.  There are various ways of specifying the target.  A
+  target for a single command may be indicated with keyword
+  \isakeyword{in} in the following way:
+
+\begin{table}
+\hrule
+\vspace{2ex}
+\begin{center}
+\begin{tabular}{ll}
+  \isakeyword{definition} & definition through an equation \\
+  \isakeyword{inductive} & inductive definition \\
+  \isakeyword{fun}, \isakeyword{function} & recursive function \\
+  \isakeyword{abbreviation} & syntactic abbreviation \\
+  \isakeyword{theorem}, etc.\ & theorem statement with proof \\
+  \isakeyword{theorems}, etc.\ & redeclaration of theorems
+\end{tabular}
+\end{center}
+\hrule
+\caption{Isar commands that accept a target.}
+\label{tab:commands-with-target}
+\end{table}
+  *}
+
+  definition (in partial_order)
+    less :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infixl "\<sqsubset>" 50)
+    where "(x \<sqsubset> y) = (x \<sqsubseteq> y \<and> x \<noteq> y)"
+
+text {* A definition in a locale depends on the locale parameters.
+  Here, a global constant @{term partial_order.less} is declared, which is lifted over the
+  locale parameter @{term le}.  Its definition is the global theorem
+  @{thm [source] partial_order.less_def}:
+  @{thm [display, indent=2] partial_order.less_def}
+  At the same time, the locale is extended by syntax information
+  hiding this construction in the context of the locale.  That is,
+  @{term "partial_order.less le"} is printed and parsed as infix
+  @{text \<sqsubset>}.  Finally, the conclusion of the definition is added to
+  the locale, @{thm [locale=partial_order, source] less_def}:
+  @{thm [locale=partial_order, display, indent=2] less_def}
+  *}
+
+text {* As an example of a theorem statement in the locale, here is the
+  derivation of a transitivity law. *}
+
+  lemma (in partial_order) less_le_trans [trans]:
+    "\<lbrakk> x \<sqsubset> y; y \<sqsubseteq> z \<rbrakk> \<Longrightarrow> x \<sqsubset> z"
+    unfolding %visible less_def by %visible (blast intro: trans)
+
+text {* In the context of the proof, assumptions and theorems of the
+  locale may be used.  Attributes are effective: @{text anti_sym} was
+  declared as introduction rule, hence it is in the context's set of
+  rules used by the classical reasoner by default.  *}
+
+text {* When working with locales, sequences of commands with the same
+  target are frequent.  A block of commands, delimited by
+  \isakeyword{begin} and \isakeyword{end}, makes a theory-like style
+  of working possible.  All commands inside the block refer to the
+  same target.  A block may immediately follow a locale
+  declaration, which makes that locale the target.  Alternatively the
+  target for a block may be given with the \isakeyword{context}
+  command.
+
+  In the block below, notions of infimum and supremum together with
+  theorems are introduced for partial orders.
+  *}
+
+  context partial_order begin
+
+  definition
+    is_inf where "is_inf x y i =
+      (i \<sqsubseteq> x \<and> i \<sqsubseteq> y \<and> (\<forall>z. z \<sqsubseteq> x \<and> z \<sqsubseteq> y \<longrightarrow> z \<sqsubseteq> i))"
+
+  definition
+    is_sup where "is_sup x y s =
+      (x \<sqsubseteq> s \<and> y \<sqsubseteq> s \<and> (\<forall>z. x \<sqsubseteq> z \<and> y \<sqsubseteq> z \<longrightarrow> s \<sqsubseteq> z))"
+
+  lemma %invisible is_infI [intro?]: "i \<sqsubseteq> x \<Longrightarrow> i \<sqsubseteq> y \<Longrightarrow>
+      (\<And>z. z \<sqsubseteq> x \<Longrightarrow> z \<sqsubseteq> y \<Longrightarrow> z \<sqsubseteq> i) \<Longrightarrow> is_inf x y i"
+    by (unfold is_inf_def) blast
+
+  lemma %invisible is_inf_lower [elim?]:
+    "is_inf x y i \<Longrightarrow> (i \<sqsubseteq> x \<Longrightarrow> i \<sqsubseteq> y \<Longrightarrow> C) \<Longrightarrow> C"
+    by (unfold is_inf_def) blast
+
+  lemma %invisible is_inf_greatest [elim?]:
+      "is_inf x y i \<Longrightarrow> z \<sqsubseteq> x \<Longrightarrow> z \<sqsubseteq> y \<Longrightarrow> z \<sqsubseteq> i"
+    by (unfold is_inf_def) blast
+
+  theorem is_inf_uniq: "\<lbrakk>is_inf x y i; is_inf x y i'\<rbrakk> \<Longrightarrow> i = i'"
+  proof -
+    assume inf: "is_inf x y i"
+    assume inf': "is_inf x y i'"
+    show ?thesis
+    proof (rule anti_sym)
+      from inf' show "i \<sqsubseteq> i'"
+      proof (rule is_inf_greatest)
+	from inf show "i \<sqsubseteq> x" ..
+	from inf show "i \<sqsubseteq> y" ..
+      qed
+      from inf show "i' \<sqsubseteq> i"
+      proof (rule is_inf_greatest)
+	from inf' show "i' \<sqsubseteq> x" ..
+	from inf' show "i' \<sqsubseteq> y" ..
+      qed
+    qed
+  qed
+
+  theorem %invisible is_inf_related [elim?]: "x \<sqsubseteq> y \<Longrightarrow> is_inf x y x"
+  proof -
+    assume "x \<sqsubseteq> y"
+    show ?thesis
+    proof
+      show "x \<sqsubseteq> x" ..
+      show "x \<sqsubseteq> y" by fact
+      fix z assume "z \<sqsubseteq> x" and "z \<sqsubseteq> y" show "z \<sqsubseteq> x" by fact
+    qed
+  qed
+
+  lemma %invisible is_supI [intro?]: "x \<sqsubseteq> s \<Longrightarrow> y \<sqsubseteq> s \<Longrightarrow>
+      (\<And>z. x \<sqsubseteq> z \<Longrightarrow> y \<sqsubseteq> z \<Longrightarrow> s \<sqsubseteq> z) \<Longrightarrow> is_sup x y s"
+    by (unfold is_sup_def) blast
+
+  lemma %invisible is_sup_least [elim?]:
+      "is_sup x y s \<Longrightarrow> x \<sqsubseteq> z \<Longrightarrow> y \<sqsubseteq> z \<Longrightarrow> s \<sqsubseteq> z"
+    by (unfold is_sup_def) blast
+
+  lemma %invisible is_sup_upper [elim?]:
+      "is_sup x y s \<Longrightarrow> (x \<sqsubseteq> s \<Longrightarrow> y \<sqsubseteq> s \<Longrightarrow> C) \<Longrightarrow> C"
+    by (unfold is_sup_def) blast
+
+  theorem is_sup_uniq: "\<lbrakk>is_sup x y s; is_sup x y s'\<rbrakk> \<Longrightarrow> s = s'"
+  proof -
+    assume sup: "is_sup x y s"
+    assume sup': "is_sup x y s'"
+    show ?thesis
+    proof (rule anti_sym)
+      from sup show "s \<sqsubseteq> s'"
+      proof (rule is_sup_least)
+	from sup' show "x \<sqsubseteq> s'" ..
+	from sup' show "y \<sqsubseteq> s'" ..
+      qed
+      from sup' show "s' \<sqsubseteq> s"
+      proof (rule is_sup_least)
+	from sup show "x \<sqsubseteq> s" ..
+	from sup show "y \<sqsubseteq> s" ..
+      qed
+    qed
+  qed
+
+  theorem %invisible is_sup_related [elim?]: "x \<sqsubseteq> y \<Longrightarrow> is_sup x y y"
+  proof -
+    assume "x \<sqsubseteq> y"
+    show ?thesis
+    proof
+      show "x \<sqsubseteq> y" by fact
+      show "y \<sqsubseteq> y" ..
+      fix z assume "x \<sqsubseteq> z" and "y \<sqsubseteq> z"
+      show "y \<sqsubseteq> z" by fact
+    qed
+  qed
+
+  end
+
+text {* In fact, many more theorems need to be shown for a usable
+  theory of partial orders.  The
+  above two serve as illustrative examples. *}
+
+text {*
+  Two commands are provided to inspect locales:
+  \isakeyword{print\_locales} lists the names of all locales of the
+  theory; \isakeyword{print\_locale}~$n$ prints the parameters and
+  assumptions of locale $n$; \isakeyword{print\_locale!}~$n$
+  additionally outputs the conclusions.
+
+  The syntax of the locale commands discussed in this tutorial is
+  shown in Table~\ref{tab:commands}.  See the
+  Isabelle/Isar Reference Manual~\cite{IsarRef}
+  for full documentation.  *}
+
+
+section {* Import *}
+
+text {* 
+\label{sec:import}
+
+  Algebraic structures are commonly defined by adding operations and
+  properties to existing structures.  For example, partial orders
+  are extended to lattices and total orders.  Lattices are extended to
+  distributive lattices.
+
+  With locales, this inheritance is achieved through \emph{import} of a
+  locale.  The import comes before the context elements.
+  *}
+
+  locale lattice = partial_order +
+    assumes ex_inf: "\<exists>inf. partial_order.is_inf le x y inf"
+      and ex_sup: "\<exists>sup. partial_order.is_sup le x y sup"
+  begin
+
+text {* Note that the assumptions above refer to the predicates for infimum
+  and supremum defined in @{text partial_order}.  In the current
+  implementation of locales, syntax from definitions of the imported
+  locale is unavailable in the locale declaration, neither are their
+  names.  Hence we refer to the constants of the theory.  The names
+  and syntax is available below, in the context of the locale.  *}
+
+  definition
+    meet (infixl "\<sqinter>" 70) where "x \<sqinter> y = (THE inf. is_inf x y inf)"
+
+  definition
+    join (infixl "\<squnion>" 65) where "x \<squnion> y = (THE sup. is_sup x y sup)"
+
+  lemma %invisible meet_equality [elim?]: "is_inf x y i \<Longrightarrow> x \<sqinter> y = i"
+  proof (unfold meet_def)
+    assume "is_inf x y i"
+    then show "(THE i. is_inf x y i) = i"
+      by (rule the_equality) (rule is_inf_uniq [OF _ `is_inf x y i`])
+  qed
+
+  lemma %invisible meetI [intro?]:
+      "i \<sqsubseteq> x \<Longrightarrow> i \<sqsubseteq> y \<Longrightarrow> (\<And>z. z \<sqsubseteq> x \<Longrightarrow> z \<sqsubseteq> y \<Longrightarrow> z \<sqsubseteq> i) \<Longrightarrow> x \<sqinter> y = i"
+    by (rule meet_equality, rule is_infI) blast+
+
+  lemma %invisible is_inf_meet [intro?]: "is_inf x y (x \<sqinter> y)"
+  proof (unfold meet_def)
+    from ex_inf obtain i where "is_inf x y i" ..
+    then show "is_inf x y (THE i. is_inf x y i)"
+      by (rule theI) (rule is_inf_uniq [OF _ `is_inf x y i`])
+  qed
+
+  lemma %invisible meet_left [intro?]:
+    "x \<sqinter> y \<sqsubseteq> x"
+    by (rule is_inf_lower) (rule is_inf_meet)
+
+  lemma %invisible meet_right [intro?]:
+    "x \<sqinter> y \<sqsubseteq> y"
+    by (rule is_inf_lower) (rule is_inf_meet)
+
+  lemma %invisible meet_le [intro?]:
+    "\<lbrakk> z \<sqsubseteq> x; z \<sqsubseteq> y \<rbrakk> \<Longrightarrow> z \<sqsubseteq> x \<sqinter> y"
+    by (rule is_inf_greatest) (rule is_inf_meet)
+
+  lemma %invisible join_equality [elim?]: "is_sup x y s \<Longrightarrow> x \<squnion> y = s"
+  proof (unfold join_def)
+    assume "is_sup x y s"
+    then show "(THE s. is_sup x y s) = s"
+      by (rule the_equality) (rule is_sup_uniq [OF _ `is_sup x y s`])
+  qed
+
+  lemma %invisible joinI [intro?]: "x \<sqsubseteq> s \<Longrightarrow> y \<sqsubseteq> s \<Longrightarrow>
+      (\<And>z. x \<sqsubseteq> z \<Longrightarrow> y \<sqsubseteq> z \<Longrightarrow> s \<sqsubseteq> z) \<Longrightarrow> x \<squnion> y = s"
+    by (rule join_equality, rule is_supI) blast+
+
+  lemma %invisible is_sup_join [intro?]: "is_sup x y (x \<squnion> y)"
+  proof (unfold join_def)
+    from ex_sup obtain s where "is_sup x y s" ..
+    then show "is_sup x y (THE s. is_sup x y s)"
+      by (rule theI) (rule is_sup_uniq [OF _ `is_sup x y s`])
+  qed
+
+  lemma %invisible join_left [intro?]:
+    "x \<sqsubseteq> x \<squnion> y"
+    by (rule is_sup_upper) (rule is_sup_join)
+
+  lemma %invisible join_right [intro?]:
+    "y \<sqsubseteq> x \<squnion> y"
+    by (rule is_sup_upper) (rule is_sup_join)
+
+  lemma %invisible join_le [intro?]:
+    "\<lbrakk> x \<sqsubseteq> z; y \<sqsubseteq> z \<rbrakk> \<Longrightarrow> x \<squnion> y \<sqsubseteq> z"
+    by (rule is_sup_least) (rule is_sup_join)
+
+  theorem %invisible meet_assoc: "(x \<sqinter> y) \<sqinter> z = x \<sqinter> (y \<sqinter> z)"
+  proof (rule meetI)
+    show "x \<sqinter> (y \<sqinter> z) \<sqsubseteq> x \<sqinter> y"
+    proof
+      show "x \<sqinter> (y \<sqinter> z) \<sqsubseteq> x" ..
+      show "x \<sqinter> (y \<sqinter> z) \<sqsubseteq> y"
+      proof -
+	have "x \<sqinter> (y \<sqinter> z) \<sqsubseteq> y \<sqinter> z" ..
+	also have "\<dots> \<sqsubseteq> y" ..
+	finally show ?thesis .
+      qed
+    qed
+    show "x \<sqinter> (y \<sqinter> z) \<sqsubseteq> z"
+    proof -
+      have "x \<sqinter> (y \<sqinter> z) \<sqsubseteq> y \<sqinter> z" ..
+      also have "\<dots> \<sqsubseteq> z" ..
+      finally show ?thesis .
+    qed
+    fix w assume "w \<sqsubseteq> x \<sqinter> y" and "w \<sqsubseteq> z"
+    show "w \<sqsubseteq> x \<sqinter> (y \<sqinter> z)"
+    proof
+      show "w \<sqsubseteq> x"
+      proof -
+	have "w \<sqsubseteq> x \<sqinter> y" by fact
+	also have "\<dots> \<sqsubseteq> x" ..
+	finally show ?thesis .
+      qed
+      show "w \<sqsubseteq> y \<sqinter> z"
+      proof
+	show "w \<sqsubseteq> y"
+	proof -
+	  have "w \<sqsubseteq> x \<sqinter> y" by fact
+	  also have "\<dots> \<sqsubseteq> y" ..
+	  finally show ?thesis .
+	qed
+	show "w \<sqsubseteq> z" by fact
+      qed
+    qed
+  qed
+
+  theorem %invisible meet_commute: "x \<sqinter> y = y \<sqinter> x"
+  proof (rule meetI)
+    show "y \<sqinter> x \<sqsubseteq> x" ..
+    show "y \<sqinter> x \<sqsubseteq> y" ..
+    fix z assume "z \<sqsubseteq> y" and "z \<sqsubseteq> x"
+    then show "z \<sqsubseteq> y \<sqinter> x" ..
+  qed
+
+  theorem %invisible meet_join_absorb: "x \<sqinter> (x \<squnion> y) = x"
+  proof (rule meetI)
+    show "x \<sqsubseteq> x" ..
+    show "x \<sqsubseteq> x \<squnion> y" ..
+    fix z assume "z \<sqsubseteq> x" and "z \<sqsubseteq> x \<squnion> y"
+    show "z \<sqsubseteq> x" by fact
+  qed
+
+  theorem %invisible join_assoc: "(x \<squnion> y) \<squnion> z = x \<squnion> (y \<squnion> z)"
+  proof (rule joinI)
+    show "x \<squnion> y \<sqsubseteq> x \<squnion> (y \<squnion> z)"
+    proof
+      show "x \<sqsubseteq> x \<squnion> (y \<squnion> z)" ..
+      show "y \<sqsubseteq> x \<squnion> (y \<squnion> z)"
+      proof -
+	have "y \<sqsubseteq> y \<squnion> z" ..
+	also have "... \<sqsubseteq> x \<squnion> (y \<squnion> z)" ..
+	finally show ?thesis .
+      qed
+    qed
+    show "z \<sqsubseteq> x \<squnion> (y \<squnion> z)"
+    proof -
+      have "z \<sqsubseteq> y \<squnion> z"  ..
+      also have "... \<sqsubseteq> x \<squnion> (y \<squnion> z)" ..
+      finally show ?thesis .
+    qed
+    fix w assume "x \<squnion> y \<sqsubseteq> w" and "z \<sqsubseteq> w"
+    show "x \<squnion> (y \<squnion> z) \<sqsubseteq> w"
+    proof
+      show "x \<sqsubseteq> w"
+      proof -
+	have "x \<sqsubseteq> x \<squnion> y" ..
+	also have "\<dots> \<sqsubseteq> w" by fact
+	finally show ?thesis .
+      qed
+      show "y \<squnion> z \<sqsubseteq> w"
+      proof
+	show "y \<sqsubseteq> w"
+	proof -
+	  have "y \<sqsubseteq> x \<squnion> y" ..
+	  also have "... \<sqsubseteq> w" by fact
+	  finally show ?thesis .
+	qed
+	show "z \<sqsubseteq> w" by fact
+      qed
+    qed
+  qed
+
+  theorem %invisible join_commute: "x \<squnion> y = y \<squnion> x"
+  proof (rule joinI)
+    show "x \<sqsubseteq> y \<squnion> x" ..
+    show "y \<sqsubseteq> y \<squnion> x" ..
+    fix z assume "y \<sqsubseteq> z" and "x \<sqsubseteq> z"
+    then show "y \<squnion> x \<sqsubseteq> z" ..
+  qed
+
+  theorem %invisible join_meet_absorb: "x \<squnion> (x \<sqinter> y) = x"
+  proof (rule joinI)
+    show "x \<sqsubseteq> x" ..
+    show "x \<sqinter> y \<sqsubseteq> x" ..
+    fix z assume "x \<sqsubseteq> z" and "x \<sqinter> y \<sqsubseteq> z"
+    show "x \<sqsubseteq> z" by fact
+  qed
+
+  theorem %invisible meet_idem: "x \<sqinter> x = x"
+  proof -
+    have "x \<sqinter> (x \<squnion> (x \<sqinter> x)) = x" by (rule meet_join_absorb)
+    also have "x \<squnion> (x \<sqinter> x) = x" by (rule join_meet_absorb)
+    finally show ?thesis .
+  qed
+
+  theorem %invisible meet_related [elim?]: "x \<sqsubseteq> y \<Longrightarrow> x \<sqinter> y = x"
+  proof (rule meetI)
+    assume "x \<sqsubseteq> y"
+    show "x \<sqsubseteq> x" ..
+    show "x \<sqsubseteq> y" by fact
+    fix z assume "z \<sqsubseteq> x" and "z \<sqsubseteq> y"
+    show "z \<sqsubseteq> x" by fact
+  qed
+
+  theorem %invisible meet_related2 [elim?]: "y \<sqsubseteq> x \<Longrightarrow> x \<sqinter> y = y"
+    by (drule meet_related) (simp add: meet_commute)
+
+  theorem %invisible join_related [elim?]: "x \<sqsubseteq> y \<Longrightarrow> x \<squnion> y = y"
+  proof (rule joinI)
+    assume "x \<sqsubseteq> y"
+    show "y \<sqsubseteq> y" ..
+    show "x \<sqsubseteq> y" by fact
+    fix z assume "x \<sqsubseteq> z" and "y \<sqsubseteq> z"
+    show "y \<sqsubseteq> z" by fact
+  qed
+
+  theorem %invisible join_related2 [elim?]: "y \<sqsubseteq> x \<Longrightarrow> x \<squnion> y = x"
+    by (drule join_related) (simp add: join_commute)
+
+  theorem %invisible meet_connection: "(x \<sqsubseteq> y) = (x \<sqinter> y = x)"
+  proof
+    assume "x \<sqsubseteq> y"
+    then have "is_inf x y x" ..
+    then show "x \<sqinter> y = x" ..
+  next
+    have "x \<sqinter> y \<sqsubseteq> y" ..
+    also assume "x \<sqinter> y = x"
+    finally show "x \<sqsubseteq> y" .
+  qed
+
+  theorem %invisible join_connection: "(x \<sqsubseteq> y) = (x \<squnion> y = y)"
+  proof
+    assume "x \<sqsubseteq> y"
+    then have "is_sup x y y" ..
+    then show "x \<squnion> y = y" ..
+  next
+    have "x \<sqsubseteq> x \<squnion> y" ..
+    also assume "x \<squnion> y = y"
+    finally show "x \<sqsubseteq> y" .
+  qed
+
+  theorem %invisible meet_connection2: "(x \<sqsubseteq> y) = (y \<sqinter> x = x)"
+    using meet_commute meet_connection by simp
+
+  theorem %invisible join_connection2: "(x \<sqsubseteq> y) = (x \<squnion> y = y)"
+    using join_commute join_connection by simp
+
+  text %invisible {* Naming according to Jacobson I, p.\ 459. *}
+  lemmas %invisible L1 = join_commute meet_commute
+  lemmas %invisible L2 = join_assoc meet_assoc
+  (* lemmas L3 = join_idem meet_idem *)
+  lemmas %invisible L4 = join_meet_absorb meet_join_absorb
+
+  end
+
+text {* Locales for total orders and distributive lattices follow.
+  Each comes with an example theorem. *}
+
+  locale total_order = partial_order +
+    assumes total: "x \<sqsubseteq> y \<or> y \<sqsubseteq> x"
+
+  lemma (in total_order) less_total: "x \<sqsubset> y \<or> x = y \<or> y \<sqsubset> x"
+    using total
+    by (unfold less_def) blast
+
+  locale distrib_lattice = lattice +
+    assumes meet_distr:
+      "lattice.meet le x (lattice.join le y z) =
+      lattice.join le (lattice.meet le x y) (lattice.meet le x z)"
+
+  lemma (in distrib_lattice) join_distr:
+    "x \<squnion> (y \<sqinter> z) = (x \<squnion> y) \<sqinter> (x \<squnion> z)"  (* txt {* Jacobson I, p.\ 462 *} *)
+    proof -
+    have "x \<squnion> (y \<sqinter> z) = (x \<squnion> (x \<sqinter> z)) \<squnion> (y \<sqinter> z)" by (simp add: L4)
+    also have "... = x \<squnion> ((x \<sqinter> z) \<squnion> (y \<sqinter> z))" by (simp add: L2)
+    also have "... = x \<squnion> ((x \<squnion> y) \<sqinter> z)" by (simp add: L1 meet_distr)
+    also have "... = ((x \<squnion> y) \<sqinter> x) \<squnion> ((x \<squnion> y) \<sqinter> z)" by (simp add: L1 L4)
+    also have "... = (x \<squnion> y) \<sqinter> (x \<squnion> z)" by (simp add: meet_distr)
+    finally show ?thesis .
+  qed
+
+text {*
+  The locale hierachy obtained through these declarations is shown in Figure~\ref{fig:lattices}(a).
+
+\begin{figure}
+\hrule \vspace{2ex}
+\begin{center}
+\subfigure[Declared hierachy]{
+\begin{tikzpicture}
+  \node (po) at (0,0) {@{text partial_order}};
+  \node (lat) at (-1.5,-1) {@{text lattice}};
+  \node (dlat) at (-1.5,-2) {@{text distrib_lattice}};
+  \node (to) at (1.5,-1) {@{text total_order}};
+  \draw (po) -- (lat);
+  \draw (lat) -- (dlat);
+  \draw (po) -- (to);
+%  \draw[->, dashed] (lat) -- (to);
+\end{tikzpicture}
+} \\
+\subfigure[Total orders are lattices]{
+\begin{tikzpicture}
+  \node (po) at (0,0) {@{text partial_order}};
+  \node (lat) at (0,-1) {@{text lattice}};
+  \node (dlat) at (-1.5,-2) {@{text distrib_lattice}};
+  \node (to) at (1.5,-2) {@{text total_order}};
+  \draw (po) -- (lat);
+  \draw (lat) -- (dlat);
+  \draw (lat) -- (to);
+%  \draw[->, dashed] (dlat) -- (to);
+\end{tikzpicture}
+} \quad
+\subfigure[Total orders are distributive lattices]{
+\begin{tikzpicture}
+  \node (po) at (0,0) {@{text partial_order}};
+  \node (lat) at (0,-1) {@{text lattice}};
+  \node (dlat) at (0,-2) {@{text distrib_lattice}};
+  \node (to) at (0,-3) {@{text total_order}};
+  \draw (po) -- (lat);
+  \draw (lat) -- (dlat);
+  \draw (dlat) -- (to);
+\end{tikzpicture}
+}
+\end{center}
+\hrule
+\caption{Hierarchy of Lattice Locales.}
+\label{fig:lattices}
+\end{figure}
+  *}
+
+section {* Changing the Locale Hierarchy *}
+
+text {*
+\label{sec:changing-the-hierarchy}
+
+  Total orders are lattices.  Hence, by deriving the lattice
+  axioms for total orders, the hierarchy may be changed
+  and @{text lattice} be placed between @{text partial_order}
+  and @{text total_order}, as shown in Figure~\ref{fig:lattices}(b).
+  Changes to the locale hierarchy may be declared
+  with the \isakeyword{interpretation} command. *}
+
+  interpretation %visible total_order \<subseteq> lattice
+
+txt {* This enters the context of locale @{text total_order}, in
+  which the goal @{subgoals [display]} must be shown.  First, the
+  locale predicate needs to be unfolded --- for example using its
+  definition or by introduction rules
+  provided by the locale package.  The methods @{text intro_locales}
+  and @{text unfold_locales} automate this.  They are aware of the
+  current context and dependencies between locales and automatically
+  discharge goals implied by these.  While @{text unfold_locales}
+  always unfolds locale predicates to assumptions, @{text
+  intro_locales} only unfolds definitions along the locale
+  hierarchy, leaving a goal consisting of predicates defined by the
+  locale package.  Occasionally the latter is of advantage since the goal
+  is smaller.
+
+  For the current goal, we would like to get hold of
+  the assumptions of @{text lattice}, hence @{text unfold_locales}
+  is appropriate. *}
+
+  proof unfold_locales
+
+txt {* Since both @{text lattice} and @{text total_order}
+  inherit @{text partial_order}, the assumptions of the latter are
+  discharged, and the only subgoals that remain are the assumptions
+  introduced in @{text lattice} @{subgoals [display]}
+  The proof for the first subgoal is *}
+
+    fix x y
+    from total have "is_inf x y (if x \<sqsubseteq> y then x else y)"
+      by (auto simp: is_inf_def)
+    then show "\<exists>inf. is_inf x y inf" ..
+txt {* The proof for the second subgoal is analogous and not
+  reproduced here. *}
+  next %invisible
+    fix x y
+    from total have "is_sup x y (if x \<sqsubseteq> y then y else x)"
+      by (auto simp: is_sup_def)
+    then show "\<exists>sup. is_sup x y sup" .. qed %visible
+
+text {* Similarly, total orders are distributive lattices. *}
+
+  interpretation total_order \<subseteq> distrib_lattice
+  proof unfold_locales
+    fix %"proof" x y z
+    show "x \<sqinter> (y \<squnion> z) = x \<sqinter> y \<squnion> x \<sqinter> z" (is "?l = ?r")
+      txt {* Jacobson I, p.\ 462 *}
+    proof -
+      { assume c: "y \<sqsubseteq> x" "z \<sqsubseteq> x"
+	from c have "?l = y \<squnion> z"
+	  by (metis c join_connection2 join_related2 meet_related2 total)
+	also from c have "... = ?r" by (metis meet_related2)
+	finally have "?l = ?r" . }
+      moreover
+      { assume c: "x \<sqsubseteq> y \<or> x \<sqsubseteq> z"
+	from c have "?l = x"
+	  by (metis join_connection2 join_related2 meet_connection total trans)
+	also from c have "... = ?r"
+	  by (metis join_commute join_related2 meet_connection meet_related2 total)
+	finally have "?l = ?r" . }
+      moreover note total
+      ultimately show ?thesis by blast
+    qed
+  qed
+
+text {* The locale hierarchy is now as shown in Figure~\ref{fig:lattices}(c). *}
+
+(*<*)
+end
+(*>*)
diff -r 9681f347b6f5 -r d1d35284542f doc-src/Locales/Locales/Examples1.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/Locales/Locales/Examples1.thy	Tue Jun 03 12:34:22 2008 +0200
@@ -0,0 +1,94 @@
+(*<*)
+theory Examples1
+imports Examples
+begin
+
+(*>*)
+
+section {* Use of Locales in Theories and Proofs *}
+
+text {* Locales enable to prove theorems abstractly, relative to
+  sets of assumptions.  These theorems can then be used in other
+  contexts where the assumptions themselves, or
+  instances of the assumptions, are theorems.  This form of theorem
+  reuse is called \emph{interpretation}.
+
+  The changes of the locale
+  hierarchy from the previous sections are examples of
+  interpretations.  The command \isakeyword{interpretation} $l_1
+  \subseteq l_2$ is said to \emph{interpret} locale $l_2$ in the
+  context of $l_1$.  It causes all theorems of $l_2$ to be made
+  available in $l_1$.  The interpretation is \emph{dynamic}: not only
+  theorems already present in $l_2$ are available in $l_1$.  Theorems
+  that will be added to $l_2$ in future will automatically be
+  propagated to $l_1$.
+
+  Locales can also be interpreted in the contexts of theories and
+  structured proofs.  These interpretations are dynamic, too.
+  Theorems added to locales will be propagated to theories.
+  In this section the interpretation in
+  theories is illustrated; interpretation in proofs is analogous.
+  As an example consider, the type of natural numbers @{typ nat}.  The
+  order relation @{text \<le>} is a total order over @{typ nat},
+  divisibility @{text dvd} is a distributive lattice.
+
+  We start with the
+  interpretation that @{text \<le>} is a partial order.  The facilities of
+  the interpretation command are explored in three versions.
+  *}
+
+
+subsection {* First Version: Replacement of Parameters Only *}
+
+text {*
+\label{sec:po-first}
+
+  In the most basic form, interpretation just replaces the locale
+  parameters by terms.  The following command interprets the locale
+  @{text partial_order} in the global context of the theory.  The
+  parameter @{term le} is replaced by @{term "op \<le> :: nat \<Rightarrow> nat \<Rightarrow> bool"}. *} 
+
+  interpretation %visible nat: partial_order ["op \<le> :: nat \<Rightarrow> nat \<Rightarrow> bool"]
+txt {* The locale name is succeeded by a \emph{parameter
+  instantiation}.  In general, this is a list of terms, which refer to
+  the parameters in the order of declaration in the locale.  The
+  locale name is preceded by an optional \emph{interpretation prefix},
+  which is used to qualify names from the locale in the global
+  context.
+
+  The command creates the goal @{subgoals [display]} which can be shown
+  easily:%
+\footnote{Note that @{text op} binds tighter than functions
+  application: parentheses around @{text "op \<le>"} are not necessary.}
+ *}
+    by unfold_locales auto
+
+text {*  Now theorems from the locale are available in the theory,
+  interpreted for natural numbers, for example @{thm [source]
+  nat.trans}: @{thm [display, indent=2] nat.trans}
+
+  In order to avoid unwanted hiding of theorems, interpretation
+  accepts a qualifier, @{text nat} in the example, which is applied to
+  all names processed by the
+  interpretation.  If present the qualifier is mandatory --- that is,
+  the above theorem cannot be referred to simply as @{text trans}.
+  *}
+
+
+subsection {* Second Version: Replacement of Definitions *}
+
+text {* The above interpretation also creates the theorem
+  @{thm [source] nat.less_le_trans}: @{thm [display, indent=2]
+  nat.less_le_trans}
+  Here, @{term "partial_order.less (op \<le> :: nat \<Rightarrow> nat \<Rightarrow> bool)"}
+  represents the strict order, although @{text "<"} is defined on
+  @{typ nat}.  Interpretation enables to map concepts
+  introduced in locales through definitions to the corresponding
+  concepts of the theory.%
+\footnote{This applies not only to \isakeyword{definition} but also to
+  \isakeyword{inductive}, \isakeyword{fun} and \isakeyword{function}.} 
+  *}
+
+(*<*)
+end
+(*>*)
diff -r 9681f347b6f5 -r d1d35284542f doc-src/Locales/Locales/Examples2.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/Locales/Locales/Examples2.thy	Tue Jun 03 12:34:22 2008 +0200
@@ -0,0 +1,34 @@
+(*<*)
+theory Examples2
+imports Examples
+begin
+
+(*>*)
+
+text {* This is achieved by unfolding suitable equations during
+  interpretation.  These equations are given after the keyword
+  \isakeyword{where} and require proofs.  The revised command,
+  replacing @{text "\<sqsubset>"} by @{text "<"}, is: *}
+
+interpretation %visible nat: partial_order ["op \<le> :: [nat, nat] \<Rightarrow> bool"]
+  where "partial_order.less op \<le> (x::nat) y = (x < y)"
+proof -
+  txt {* The goals are @{subgoals [display]}
+    The proof that @{text \<le>} is a partial order is a above. *}
+  show "partial_order (op \<le> :: nat \<Rightarrow> nat \<Rightarrow> bool)"
+    by unfold_locales auto
+  txt {* The second goal is shown by unfolding the
+    definition of @{term "partial_order.less"}. *}
+  show "partial_order.less op \<le> (x::nat) y = (x < y)"
+    unfolding partial_order.less_def [OF `partial_order op \<le>`]
+    by auto
+qed
+
+text {* Note that the above proof is not in the context of a locale.
+  Hence, the correct interpretation of @{text
+  "partial_order.less_def"} is obtained manually with @{text OF}.
+  *}
+
+(*<*)
+end
+(*>*)
\ No newline at end of file
diff -r 9681f347b6f5 -r d1d35284542f doc-src/Locales/Locales/Examples3.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/Locales/Locales/Examples3.thy	Tue Jun 03 12:34:22 2008 +0200
@@ -0,0 +1,547 @@
+(*<*)
+theory Examples3
+imports Examples
+begin
+
+(*>*)
+
+subsection {* Third Version: Local Interpretation *}
+
+text {* In the above example, the fact that @{text \<le>} is a partial
+  order for the natural numbers was used in the proof of the
+  second goal.  In general, proofs of the equations may involve
+  theorems implied by the fact the assumptions of the instantiated
+  locale hold for the instantiating structure.  If these theorems have
+  been shown abstractly in the locale they can be made available
+  conveniently in the context through an auxiliary local interpretation (keyword
+  \isakeyword{interpret}).  This interpretation is inside the proof of the global
+  interpretation.  The third revision of the example illustrates this.  *}
+
+interpretation %visible nat: partial_order ["op \<le> :: nat \<Rightarrow> nat \<Rightarrow> bool"]
+  where "partial_order.less (op \<le>) (x::nat) y = (x < y)"
+proof -
+  show "partial_order (op \<le> :: nat \<Rightarrow> nat \<Rightarrow> bool)"
+    by unfold_locales auto
+  then interpret nat: partial_order ["op \<le> :: [nat, nat] \<Rightarrow> bool"] .
+  show "partial_order.less (op \<le>) (x::nat) y = (x < y)"
+    unfolding nat.less_def by auto
+qed
+
+text {* The inner interpretation does not require an
+  elaborate new proof, it is immediate from the preceeding fact and
+  proved with ``.''.
+  This interpretation enriches the local proof context by
+  the very theorems also obtained in the interpretation from
+  Section~\ref{sec:po-first}, and @{text nat.less_def} may directly be
+  used to unfold the definition.  Theorems from the local
+  interpretation disappear after leaving the proof context --- that
+  is, after the closing \isakeyword{qed} --- and are
+  then replaced by those with the desired substitutions of the strict
+  order. *}
+
+
+subsection {* Further Interpretations *}
+
+text {* Further interpretations are necessary to reuse theorems from
+  the other locales.  In @{text lattice} the operations @{text \<sqinter>} and
+  @{text \<squnion>} are substituted by @{term "min :: nat \<Rightarrow> nat \<Rightarrow> nat"} and
+  @{term "max :: nat \<Rightarrow> nat \<Rightarrow> nat"}.  The entire proof for the
+  interpretation is reproduced in order to give an example of a more
+  elaborate interpretation proof.  *}
+
+interpretation %visible nat: lattice ["op \<le> :: nat \<Rightarrow> nat \<Rightarrow> bool"]
+  where "lattice.meet op \<le> (x::nat) y = min x y"
+    and "lattice.join op \<le> (x::nat) y = max x y"
+proof -
+  show "lattice (op \<le> :: nat \<Rightarrow> nat \<Rightarrow> bool)"
+    txt {* We have already shown that this is a partial order, *}
+    apply unfold_locales
+    txt {* hence only the lattice axioms remain to be shown: @{subgoals
+      [display]}  After unfolding @{text is_inf} and @{text is_sup}, *}
+    apply (unfold nat.is_inf_def nat.is_sup_def)
+    txt {* the goals become @{subgoals [display]} which can be solved
+      by Presburger arithmetic. *}
+    by arith+
+  txt {* In order to show the equations, we put ourselves in a
+    situation where the lattice theorems can be used in a convenient way. *}
+  then interpret nat: lattice ["op \<le> :: nat \<Rightarrow> nat \<Rightarrow> bool"] .
+  show "lattice.meet op \<le> (x::nat) y = min x y"
+    by (bestsimp simp: nat.meet_def nat.is_inf_def)
+  show "lattice.join op \<le> (x::nat) y = max x y"
+    by (bestsimp simp: nat.join_def nat.is_sup_def)
+qed
+
+text {* That the relation @{text \<le>} is a total order completes this
+  sequence of interpretations. *}
+
+interpretation %visible nat: total_order ["op \<le> :: nat \<Rightarrow> nat \<Rightarrow> bool"]
+  by unfold_locales arith
+
+text {* Theorems that are available in the theory at this point are shown in
+  Table~\ref{tab:nat-lattice}.
+
+\begin{table}
+\hrule
+\vspace{2ex}
+\begin{center}
+\begin{tabular}{l}
+  @{thm [source] nat.less_def} from locale @{text partial_order}: \\
+  \quad @{thm nat.less_def} \\
+  @{thm [source] nat.meet_left} from locale @{text lattice}: \\
+  \quad @{thm nat.meet_left} \\
+  @{thm [source] nat.join_distr} from locale @{text distrib_lattice}: \\
+  \quad @{thm nat.join_distr} \\
+  @{thm [source] nat.less_total} from locale @{text total_order}: \\
+  \quad @{thm nat.less_total}
+\end{tabular}
+\end{center}
+\hrule
+\caption{Interpreted theorems for @{text \<le>} on the natural numbers.}
+\label{tab:nat-lattice}
+\end{table}
+
+  Note that since the locale hierarchy reflects that total orders are
+  distributive lattices, an explicit interpretation of distributive
+  lattices for the order relation on natural numbers is not neccessary.
+
+  Why not push this idea further and just give the last interpretation
+  as a single interpretation instead of the sequence of three?  The
+  reasons for this are twofold:
+\begin{itemize}
+\item
+  Often it is easier to work in an incremental fashion, because later
+  interpretations require theorems provided by earlier
+  interpretations.
+\item
+  Assume that a definition is made in some locale $l_1$, and that $l_2$
+  imports $l_1$.  Let an equation for the definition be
+  proved in an interpretation of $l_2$.  The equation will be unfolded
+  in interpretations of theorems added to $l_2$ or below in the import
+  hierarchy, but not for theorems added above $l_2$.
+  Hence, an equation interpreting a definition should always be given in
+  an interpretation of the locale where the definition is made, not in
+  an interpretation of a locale further down the hierarchy.
+\end{itemize}
+  *}
+
+
+subsection {* Lattice @{text "dvd"} on @{typ nat} *}
+
+text {* Divisibility on the natural numbers is a distributive lattice
+  but not a total order.  Interpretation again proceeds
+  incrementally. *}
+
+interpretation nat_dvd: partial_order ["op dvd :: nat \<Rightarrow> nat \<Rightarrow> bool"]
+  where "partial_order.less op dvd (x::nat) y = (x dvd y \<and> x \<noteq> y)"
+proof -
+  show "partial_order (op dvd :: nat \<Rightarrow> nat \<Rightarrow> bool)"
+    by unfold_locales (auto simp: dvd_def)
+  then interpret nat_dvd: partial_order ["op dvd :: nat \<Rightarrow> nat \<Rightarrow> bool"] .
+  show "partial_order.less op dvd (x::nat) y = (x dvd y \<and> x \<noteq> y)"
+    apply (unfold nat_dvd.less_def)
+    apply auto
+    done
+qed
+
+text {* Note that there is no symbol for strict divisibility.  Instead,
+  interpretation substitutes @{term "x dvd y \<and> x \<noteq> y"}.   *}
+
+interpretation nat_dvd: lattice ["op dvd :: nat \<Rightarrow> nat \<Rightarrow> bool"]
+  where nat_dvd_meet_eq:
+      "lattice.meet op dvd (x::nat) y = gcd (x, y)"
+    and nat_dvd_join_eq:
+      "lattice.join op dvd (x::nat) y = lcm (x, y)"
+proof -
+  show "lattice (op dvd :: nat \<Rightarrow> nat \<Rightarrow> bool)"
+    apply unfold_locales
+    apply (unfold nat_dvd.is_inf_def nat_dvd.is_sup_def)
+    apply (rule_tac x = "gcd (x, y)" in exI)
+    apply auto [1]
+    apply (rule_tac x = "lcm (x, y)" in exI)
+    apply (auto intro: lcm_dvd1 lcm_dvd2 lcm_least)
+    done
+  then interpret nat_dvd: lattice ["op dvd :: nat \<Rightarrow> nat \<Rightarrow> bool"] .
+  show "lattice.meet op dvd (x::nat) y = gcd (x, y)"
+    apply (unfold nat_dvd.meet_def)
+    apply (rule the_equality)
+    apply (unfold nat_dvd.is_inf_def)
+    by auto
+  show "lattice.join op dvd (x::nat) y = lcm (x, y)"
+    apply (unfold nat_dvd.join_def)
+    apply (rule the_equality)
+    apply (unfold nat_dvd.is_sup_def)
+    by (auto intro: lcm_dvd1 lcm_dvd2 lcm_least)
+qed
+
+text {* Equations @{thm [source] nat_dvd_meet_eq} and @{thm [source]
+  nat_dvd_join_eq} are named since they are handy in the proof of
+  the subsequent interpretation. *}
+
+ML %invisible {* set quick_and_dirty *}
+
+(*
+definition
+  is_lcm :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool" where
+  "is_lcm p m n \<longleftrightarrow> m dvd p \<and> n dvd p \<and>
+    (\<forall>d. m dvd d \<longrightarrow> n dvd d \<longrightarrow> p dvd d)"
+
+lemma is_gcd: "is_lcm (lcm (m, n)) m n"
+  by (simp add: is_lcm_def lcm_least)
+
+lemma gcd_lcm_distr_lemma:
+  "[| is_gcd g1 x l1; is_lcm l1 y z; is_gcd g2 x y; is_gcd g3 x z |] ==> is_lcm g1 g2 g3"
+apply (unfold is_gcd_def is_lcm_def dvd_def)
+apply (clarsimp simp: mult_ac)
+apply (blast intro: mult_is_0)
+thm mult_is_0 [THEN iffD1]
+*)
+
+lemma %invisible gcd_lcm_distr:
+  "gcd (x, lcm (y, z)) = lcm (gcd (x, y), gcd (x, z))"
+  sorry
+
+ML %invisible {* reset quick_and_dirty *}
+  
+interpretation %visible nat_dvd:
+  distrib_lattice ["op dvd :: nat \<Rightarrow> nat \<Rightarrow> bool"]
+  apply unfold_locales
+  txt {* @{subgoals [display]} *}
+  apply (unfold nat_dvd_meet_eq nat_dvd_join_eq)
+  txt {* @{subgoals [display]} *}
+  apply (rule gcd_lcm_distr) done
+
+
+text {* Theorems that are available in the theory after these
+  interpretations are shown in Table~\ref{tab:nat-dvd-lattice}.
+
+\begin{table}
+\hrule
+\vspace{2ex}
+\begin{center}
+\begin{tabular}{l}
+  @{thm [source] nat_dvd.less_def} from locale @{text partial_order}: \\
+  \quad @{thm nat_dvd.less_def} \\
+  @{thm [source] nat_dvd.meet_left} from locale @{text lattice}: \\
+  \quad @{thm nat_dvd.meet_left} \\
+  @{thm [source] nat_dvd.join_distr} from locale @{text distrib_lattice}: \\
+  \quad @{thm nat_dvd.join_distr} \\
+\end{tabular}
+\end{center}
+\hrule
+\caption{Interpreted theorems for @{text dvd} on the natural numbers.}
+\label{tab:nat-dvd-lattice}
+\end{table}
+  *}
+
+text {*
+  The full syntax of the interpretation commands is shown in
+  Table~\ref{tab:commands}.  The grammar refers to
+  \textit{expr}, which stands for a \emph{locale} expression.  Locale
+  expressions are discussed in Section~\ref{sec:expressions}.
+  *}
+
+
+section {* Locale Expressions *}
+
+text {*
+\label{sec:expressions}
+
+  A map @{term \<phi>} between partial orders @{text \<sqsubseteq>} and @{text \<preceq>}
+  is called order preserving if @{text "x \<sqsubseteq> y"} implies @{text "\<phi> x \<preceq>
+  \<phi> y"}.  This situation is more complex than those encountered so
+  far: it involves two partial orders, and it is desirable to use the
+  existing locale for both.
+
+  Inspecting the grammar of locale commands in
+  Table~\ref{tab:commands} reveals that the import of a locale can be
+  more than just a single locale.  In general, the import is a
+  \emph{locale expression}.  Locale expressions enable to combine locales
+  and rename parameters.  A locale name is a locale expression.  If
+  $e_1$ and $e_2$ are locale expressions then $e_1 + e_2$ is their
+  \emph{merge}.  If $e$ is an expression, then $e\:q_1 \ldots q_n$ is
+  a \emph{renamed expression} where the parameters in $e$ are renamed
+  to $q_1 \ldots q_n$.  Using a locale expression, a locale for order
+  preserving maps can be declared in the following way.  *}
+
+  locale order_preserving =
+    partial_order + partial_order le' (infixl "\<preceq>" 50) +
+    fixes \<phi> :: "'a \<Rightarrow> 'b"
+    assumes hom_le: "x \<sqsubseteq> y \<Longrightarrow> \<phi> x \<preceq> \<phi> y"
+
+text {* The second line contains the expression, which is the
+  merge of two partial order locales.  The parameter of the second one
+  is @{text le'} with new infix syntax @{text \<preceq>}.  The
+  parameters of the entire locale are @{text le}, @{text le'} and
+  @{text \<phi>}.  This is their \emph{canonical order},
+  which is obtained by a left-to-right traversal of the expression,
+  where only the new parameters are appended to the end of the list.  The
+  parameters introduced in the locale elements of the declaration
+  follow.
+
+  In renamings parameters are referred to by position in the canonical
+  order; an underscore is used to skip a parameter position, which is
+  then not renamed.  Renaming deletes the syntax of a parameter unless
+  a new mixfix annotation is given.
+
+  Parameter renamings are morphisms between locales.  These can be
+  lifted to terms and theorems and thus be applied to assumptions and
+  conclusions.  The assumption of a merge is the conjunction of the
+  assumptions of the merged locale.  The conclusions of a merge are
+  obtained by appending the conclusions of the left locale and of the
+  right locale.  *}
+
+text {* The locale @{text order_preserving} contains theorems for both
+  orders @{text \<sqsubseteq>} and @{text \<preceq>}.  How can one refer to a theorem for
+  a particular order, @{text \<sqsubseteq>} or @{text \<preceq>}?  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: *}
+
+context %invisible order_preserving begin
+
+text {*
+  @{thm [source] le.less_le_trans}: @{thm le.less_le_trans}
+
+  @{thm [source] le_le'_\<phi>.hom_le}: @{thm le_le'_\<phi>.hom_le}
+  *}
+
+text {* When renaming a locale, the morphism is also applied
+  to the qualifiers.  Hence theorems for the partial order @{text \<preceq>}
+  are qualified by @{text le'}.  For example, @{thm [source]
+  le'.less_le_trans}: @{thm [display, indent=2] le'.less_le_trans} *}
+
+end %invisible
+
+text {* This example reveals that there is no infix syntax for the strict
+  version of @{text \<preceq>}!  This can, of course, not be introduced
+  automatically, but it can be declared manually through an abbreviation.
+  *}
+
+  abbreviation (in order_preserving)
+    less' (infixl "\<prec>" 50) where "less' \<equiv> partial_order.less le'"
+
+text {* Now the theorem is displayed nicely as
+  @{thm [locale=order_preserving] le'.less_le_trans}.  *}
+
+text {* Not only names of theorems are qualified.  In fact, all names
+  are qualified, in particular names introduced by definitions and
+  abbreviations.  The name of the strict order of @{text \<sqsubseteq>} is @{text
+  le.less} and therefore @{text le'.less} is the name of the strict
+  order of @{text \<preceq>}.  Hence, the equation in the above abbreviation
+  could have been written as @{text "less' \<equiv> le'.less"}. *}
+
+text {* Two more locales illustrate working with locale expressions.
+  A map @{text \<phi>} is a lattice homomorphism if it preserves meet and join. *}
+
+  locale lattice_hom = lattice + lattice le' (infixl "\<preceq>" 50) +
+    fixes \<phi>
+    assumes hom_meet:
+	"\<phi> (lattice.meet le x y) = lattice.meet le' (\<phi> x) (\<phi> y)"
+      and hom_join:
+	"\<phi> (lattice.join le x y) = lattice.join le' (\<phi> x) (\<phi> y)"
+
+  abbreviation (in lattice_hom)
+    meet' (infixl "\<sqinter>''" 50) where "meet' \<equiv> le'.meet"
+  abbreviation (in lattice_hom)
+    join' (infixl "\<squnion>''" 50) where "join' \<equiv> le'.join"
+
+text {* A homomorphism is an endomorphism if both orders coincide. *}
+
+  locale lattice_end =
+    lattice_hom le (infixl "\<sqsubseteq>" 50) le (infixl "\<sqsubseteq>" 50)
+
+text {* 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
+  imported by @{text lattice_end} identifies the first and second
+  parameter of @{text lattice_hom}.  By looking at the inheritance diagram it would seem
+  that two identical copies of each of the locales @{text
+  partial_order} and @{text lattice} are imported.  This is not the
+  case!  Inheritance paths with identical morphisms are detected and
+  the conclusions of the respecitve locales appear only once.
+
+\begin{figure}
+\hrule \vspace{2ex}
+\begin{center}
+\begin{tikzpicture}
+  \node (o) at (0,0) {@{text partial_order}};
+  \node (oh) at (1.5,-2) {@{text order_preserving}};
+  \node (oh1) at (1.5,-0.7) {$\scriptscriptstyle \sqsubseteq \mapsto \sqsubseteq$};
+  \node (oh2) at (0,-1.3) {$\scriptscriptstyle \sqsubseteq \mapsto \preceq$};
+  \node (l) at (-1.5,-2) {@{text lattice}};
+  \node (lh) at (0,-4) {@{text lattice_hom}};
+  \node (lh1) at (0,-2.7) {$\scriptscriptstyle \sqsubseteq \mapsto \sqsubseteq$};
+  \node (lh2) at (-1.5,-3.3) {$\scriptscriptstyle \sqsubseteq \mapsto \preceq$};
+  \node (le) at (0,-6) {@{text lattice_end}};
+  \node (le1) at (0,-4.8)
+    [anchor=west]{$\scriptscriptstyle \sqsubseteq \mapsto \sqsubseteq$};
+  \node (le2) at (0,-5.2)
+    [anchor=west]{$\scriptscriptstyle \preceq \mapsto \sqsubseteq$};
+  \draw (o) -- (l);
+  \draw[dashed] (oh) -- (lh);
+  \draw (lh) -- (le);
+  \draw (o) .. controls (oh1.south west) .. (oh);
+  \draw (o) .. controls (oh2.north east) .. (oh);
+  \draw (l) .. controls (lh1.south west) .. (lh);
+  \draw (l) .. controls (lh2.north east) .. (lh);
+\end{tikzpicture}
+\end{center}
+\hrule
+\caption{Hierarchy of Homomorphism Locales.}
+\label{fig:hom}
+\end{figure}
+  *}
+
+text {* 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. *}
+
+  interpretation lattice_hom \<subseteq> order_preserving proof unfold_locales
+    fix x y
+    assume "x \<sqsubseteq> y"
+    then have "y = (x \<squnion> y)" by (simp add: le.join_connection)
+    then have "\<phi> y = (\<phi> x \<squnion>' \<phi> y)" by (simp add: hom_join [symmetric])
+    then show "\<phi> x \<preceq> \<phi> y" by (simp add: le'.join_connection)
+  qed
+
+text {* Theorems and other declarations --- syntax, in particular ---
+  from the locale @{text order_preserving} are now active in @{text
+  lattice_hom}, for example
+
+  @{thm [locale=lattice_hom, source] le'.less_le_trans}:
+  @{thm [locale=lattice_hom] le'.less_le_trans}
+  *}
+
+
+
+section {* Further Reading *}
+
+text {* More information on locales and their interpretation is
+  available.  For the locale hierarchy of import and interpretation
+  dependencies see \cite{Ballarin2006a}; interpretations in theories
+  and proofs are covered in \cite{Ballarin2006b}.  In the latter, we
+  show how interpretation in proofs enables to reason about families
+  of algebraic structures, which cannot be expressed with locales
+  directly.
+
+  Haftmann and Wenzel \cite{HaftmannWenzel2007} overcome a restriction
+  of axiomatic type classes through a combination with locale
+  interpretation.  The result is a Haskell-style class system with a
+  facility to generate Haskell code.  Classes are sufficient for
+  simple specifications with a single type parameter.  The locales for
+  orders and lattices presented in this tutorial fall into this
+  category.  Order preserving maps, homomorphisms and vector spaces,
+  on the other hand, do not.
+
+  The original work of Kamm\"uller on locales \cite{KammullerEtAl1999}
+  may be of interest from a historical perspective.  The mathematical
+  background on orders and lattices is taken from Jacobson's textbook
+  on algebra \cite[Chapter~8]{Jacobson1985}.
+  *}
+
+text {*
+\begin{table}
+\hrule
+\vspace{2ex}
+\begin{center}
+\begin{tabular}{l>$c<$l}
+  \multicolumn{3}{l}{Miscellaneous} \\
+
+  \textit{attr-name} & ::=
+  & \textit{name} $|$ \textit{attribute} $|$
+    \textit{name} \textit{attribute} \\[2ex]
+
+  \multicolumn{3}{l}{Context Elements} \\
+
+  \textit{fixes} & ::=
+  & \textit{name} [ ``\textbf{::}'' \textit{type} ]
+    [ ``\textbf{(}'' \textbf{structure} ``\textbf{)}'' $|$
+    \textit{mixfix} ] \\
+\begin{comment}
+  \textit{constrains} & ::=
+  & \textit{name} ``\textbf{::}'' \textit{type} \\
+\end{comment}
+  \textit{assumes} & ::=
+  & [ \textit{attr-name} ``\textbf{:}'' ] \textit{proposition} \\
+\begin{comment}
+  \textit{defines} & ::=
+  & [ \textit{attr-name} ``\textbf{:}'' ] \textit{proposition} \\
+  \textit{notes} & ::=
+  & [ \textit{attr-name} ``\textbf{=}'' ]
+    ( \textit{qualified-name} [ \textit{attribute} ] )$^+$ \\
+\end{comment}
+
+  \textit{element} & ::=
+  & \textbf{fixes} \textit{fixes} ( \textbf{and} \textit{fixes} )$^*$ \\
+\begin{comment}
+  & |
+  & \textbf{constrains} \textit{constrains}
+    ( \textbf{and} \textit{constrains} )$^*$ \\
+\end{comment}
+  & |
+  & \textbf{assumes} \textit{assumes} ( \textbf{and} \textit{assumes} )$^*$ \\[2ex]
+%\begin{comment}
+%  & |
+%  & \textbf{defines} \textit{defines} ( \textbf{and} \textit{defines} )$^*$ \\
+%  & |
+%  & \textbf{notes} \textit{notes} ( \textbf{and} \textit{notes} )$^*$ \\
+%\end{comment}
+
+  \multicolumn{3}{l}{Locale Expressions} \\
+
+  \textit{rename} & ::=
+  & \textit{name} [ \textit{mixfix} ] $|$ ``\textbf{\_}'' \\
+  \textit{expr}  & ::= 
+  & \textit{renamed-expr} ( ``\textbf{+}'' \textit{renamed-expr} )$^*$ \\
+  \textit{renamed-expr} & ::=
+  & ( \textit{qualified-name} $|$
+    ``\textbf{(}'' \textit{expr} ``\textbf{)}'' ) \textit{rename}$^*$ \\[2ex]
+
+  \multicolumn{3}{l}{Declaration of Locales} \\
+
+  \textit{locale} & ::=
+  & \textit{element}$^+$ \\
+  & | & \textit{locale-expr} [ ``\textbf{+}'' \textit{element}$^+$ ] \\
+  \textit{toplevel} & ::=
+  & \textbf{locale} \textit{name} [ ``\textbf{=}''
+    \textit{locale} ] \\[2ex]
+
+  \multicolumn{3}{l}{Interpretation} \\
+
+  \textit{equation} & ::= & [ \textit{attr-name} ``\textbf{:}'' ]
+    \textit{prop} \\
+  \textit{insts} & ::= & [ ``\textbf{[}'' \textit{term}$^+$
+    ``\textbf{]}'' ] \\
+  & & [ \textbf{where} \textit{equation} ( \textbf{and}
+    \textit{equation} )$^*$ ] \\
+  \textit{toplevel} & ::=
+  & \textbf{interpretation} \textit{name} ( ``$<$'' $|$
+    ``$\subseteq$'' ) \textit{expr} \textit{proof} \\
+  & |
+  & \textbf{interpretation} [ \textit{attr-name} ``\textbf{:}'' ]
+    \textit{expr} \textit{insts} \textit{proof} \\
+  & |
+  & \textbf{interpret} [ \textit{attr-name} ``\textbf{:}'' ]
+    \textit{expr} \textit{insts} \textit{proof} \\[2ex]
+
+  \multicolumn{3}{l}{Diagnostics} \\
+
+  \textit{toplevel} & ::=
+  & \textbf{print\_locale} [ ``\textbf{!}'' ] \textit{locale} \\
+  & | & \textbf{print\_locales} 
+\end{tabular}
+\end{center}
+\hrule
+\caption{Syntax of Locale Commands.}
+\label{tab:commands}
+\end{table}
+  *}
+
+text {* \textbf{Acknowledgements.}  Alexander Krauss, Tobias Nipkow,
+  Christian Sternagel and Makarius Wenzel have made useful comments on
+  a draft of this document. *}
+
+(*<*)
+end
+(*>*)
diff -r 9681f347b6f5 -r d1d35284542f doc-src/Locales/Locales/Locales.thy
--- a/doc-src/Locales/Locales/Locales.thy	Tue Jun 03 11:55:35 2008 +0200
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,1172 +0,0 @@
-(*
-  Title:  Locales in Isabelle/Isar
-  Id:	  $Id$
-  Author: Clemens Ballarin, started 31 January 2003
-  Copyright (c) 2003 by Clemens Ballarin
-*)
-
-(*<*)
-theory Locales imports Main begin
-
-ML {* print_mode := "type_brackets" :: !print_mode; *}
-(*>*)
-
-section {* Overview *}
-
-text {*
-  The present text is based on~\cite{Ballarin2004a}.  It was updated
-  for for Isabelle2005, but does not cover locale interpretation.
-
-  Locales are an extension of the Isabelle proof assistant.  They
-  provide support for modular reasoning. Locales were initially
-  developed by Kamm\"uller~\cite{Kammuller2000} to support reasoning
-  in abstract algebra, but are applied also in other domains --- for
-  example, bytecode verification~\cite{Klein2003}.
-
-  Kamm\"uller's original design, implemented in Isabelle99, provides, in
-  addition to
-  means for declaring locales, a set of ML functions that were used
-  along with ML tactics in a proof.  In the meantime, the input format
-  for proof in Isabelle has changed and users write proof
-  scripts in ML only rarely if at all.  Two new proof styles are
-  available, and can
-  be used interchangeably: linear proof scripts that closely resemble ML
-  tactics, and the structured Isar proof language by
-  Wenzel~\cite{Wenzel2002a}.  Subsequently, Wenzel re-implemented
-  locales for
-  the new proof format.  The implementation, available with
-  Isabelle2002 or later, constitutes a complete re-design and exploits that
-  both Isar and locales are based on the notion of context,
-  and thus locales are seen as a natural extension of Isar.
-  Nevertheless, locales can also be used with proof scripts:
-  their use does not require a deep understanding of the structured
-  Isar proof style.
-
-  At the same time, Wenzel considerably extended locales.  The most
-  important addition are locale expressions, which allow to combine
-  locales more freely.  Previously only
-  linear inheritance was possible.  Now locales support multiple
-  inheritance through a normalisation algorithm.  New are also
-  structures, which provide special syntax for locale parameters that
-  represent algebraic structures.
-
-  Unfortunately, Wenzel provided only an implementation but hardly any
-  documentation.  Besides providing documentation, the present paper
-  is a high-level description of locales, and in particular locale
-  expressions.  It is meant as a first step towards the semantics of
-  locales, and also as a base for comparing locales with module concepts
-  in other provers.  It also constitutes the base for future
-  extensions of locales in Isabelle.
-  The description was derived mainly by experimenting
-  with locales and partially also by inspecting the code.
-
-  The main contribution of the author of the present paper is the
-  abstract description of Wenzel's version of locales, and in
-  particular of the normalisation algorithm for locale expressions (see
-  Section~\ref{sec-normal-forms}).  Contributions to the
-  implementation are confined to bug fixes and to provisions that
-  enable the use of locales with linear proof scripts.
-
-  Concepts are introduced along with examples, so that the text can be
-  used as tutorial.  It is assumed that the reader is somewhat
-  familiar with Isabelle proof scripts.  Examples have been phrased as
-  structured
-  Isar proofs.  However, in order to understand the key concepts,
-  including locales expressions and their normalisation, detailed
-  knowledge of Isabelle is not necessary. 
-
-\nocite{Nipkow2003,Wenzel2002b,Wenzel2003}
-*}
-
-section {* Locales: Beyond Proof Contexts *}
-
-text {*
-  In tactic-based provers the application of a sequence of proof
-  tactics leads to a proof state.  This state is usually hard to
-  predict from looking at the tactic script, unless one replays the
-  proof step-by-step.  The structured proof language Isar is
-  different.  It is additionally based on \emph{proof contexts},
-  which are directly visible in Isar scripts, and since tactic
-  sequences tend to be short, this commonly leads to clearer proof
-  scripts.
-
-  Goals are stated with the \textbf{theorem}
-  command.  This is followed by a proof.  When discharging a goal
-  requires an elaborate argument
-  (rather than the application of a single tactic) a new context
-  may be entered (\textbf{proof}).  Inside the context, variables may
-  be fixed (\textbf{fix}), assumptions made (\textbf{assume}) and
-  intermediate goals stated (\textbf{have}) and proved.  The
-  assumptions must be dischargeable by premises of the surrounding
-  goal, and once this goal has been proved (\textbf{show}) the proof context
-  can be closed (\textbf{qed}). Contexts inherit from surrounding
-  contexts, but it is not possible
-  to export from them (with exception of the proved goal);
-  they ``disappear'' after the closing \textbf{qed}.
-  Facts may have attributes --- for example, identifying them as
-  default to the simplifier or classical reasoner.
-
-  Locales extend proof contexts in various ways:
-  \begin{itemize}
-  \item
-    Locales are usually \emph{named}.  This makes them persistent.
-  \item
-    Fixed variables may have \emph{syntax}.
-  \item
-    It is possible to \emph{add} and \emph{export} facts.
-  \item
-    Locales can be combined and modified with \emph{locale
-    expressions}.
-  \end{itemize}
-  The Locales facility extends the Isar language: it provides new ways
-  of stating and managing facts, but it does not modify the language
-  for proofs.  Its purpose is to support writing modular proofs.
-*}
-
-section {* Simple Locales *}
-
-subsection {* Syntax and Terminology *}
-
-text {*
-  The grammar of Isar is extended by commands for locales as shown in
-  Figure~\ref{fig-grammar}.
-  A key concept, introduced by Wenzel, is that
-  locales are (internally) lists
-  of \emph{context elements}.  There are five kinds, identified
-  by the keywords \textbf{fixes}, \textbf{constrains},
-  \textbf{assumes}, \textbf{defines} and \textbf{notes}.
-
-  \begin{figure}
-  \hrule
-  \vspace{2ex}
-  \begin{small}
-  \begin{tabular}{l>$c<$l}
-  \textit{attr-name} & ::=
-  & \textit{name} $|$ \textit{attribute} $|$
-    \textit{name} \textit{attribute} \\
-
-  \textit{locale-expr}  & ::= 
-  & \textit{locale-expr1} ( ``\textbf{+}'' \textit{locale-expr1} )$^*$ \\
-  \textit{locale-expr1} & ::=
-  & ( \textit{qualified-name} $|$
-    ``\textbf{(}'' \textit{locale-expr} ``\textbf{)}'' ) \\
-  & & ( \textit{name} [ \textit{mixfix} ] $|$ ``\textbf{\_}'' )$^*$ \\
-
-  \textit{fixes} & ::=
-  & \textit{name} [ ``\textbf{::}'' \textit{type} ]
-    [ ``\textbf{(}'' \textbf{structure} ``\textbf{)}'' $|$
-    \textit{mixfix} ] \\
-  \textit{constrains} & ::=
-  & \textit{name} ``\textbf{::}'' \textit{type} \\
-  \textit{assumes} & ::=
-  & [ \textit{attr-name} ``\textbf{:}'' ] \textit{proposition} \\
-  \textit{defines} & ::=
-  & [ \textit{attr-name} ``\textbf{:}'' ] \textit{proposition} \\
-  \textit{notes} & ::=
-  & [ \textit{attr-name} ``\textbf{=}'' ]
-    ( \textit{qualified-name} [ \textit{attribute} ] )$^+$ \\
-
-  \textit{element} & ::=
-  & \textbf{fixes} \textit{fixes} ( \textbf{and} \textit{fixes} )$^*$ \\
-  & |
-  & \textbf{constrains} \textit{constrains}
-    ( \textbf{and} \textit{constrains} )$^*$ \\
-  & |
-  & \textbf{assumes} \textit{assumes} ( \textbf{and} \textit{assumes} )$^*$ \\
-  & |
-  & \textbf{defines} \textit{defines} ( \textbf{and} \textit{defines} )$^*$ \\
-  & |
-  & \textbf{notes} \textit{notes} ( \textbf{and} \textit{notes} )$^*$ \\
-  \textit{element1} & ::=
-  & \textit{element} \\
-  & | & \textbf{includes} \textit{locale-expr} \\
-
-  \textit{locale} & ::=
-  & \textit{element}$^+$ \\
-  & | & \textit{locale-expr} [ ``\textbf{+}'' \textit{element}$^+$ ] \\
-
-  \textit{in-target} & ::=
-  & ``\textbf{(}'' \textbf{in} \textit{qualified-name} ``\textbf{)}'' \\
-
-  \textit{theorem} & ::= & ( \textbf{theorem} $|$ \textbf{lemma} $|$
-    \textbf{corollary} ) [ \textit{in-target} ] [ \textit{attr-name} ] \\
-
-  \textit{theory-level} & ::= & \ldots \\
-  & | & \textbf{locale} \textit{name} [ ``\textbf{=}''
-    \textit{locale} ] \\
-  % note: legacy "locale (open)" omitted.
-  & | & ( \textbf{theorems} $|$ \textbf{lemmas} ) \\
-  & & [ \textit{in-target} ] [ \textit{attr-name} ``\textbf{=}'' ]
-    ( \textit{qualified-name} [ \textit{attribute} ] )$^+$ \\
-  & | & \textbf{declare} [ \textit{in-target} ] ( \textit{qualified-name}
-    [ \textit{attribute} ] )$^+$ \\
-  & | & \textit{theorem} \textit{proposition} \textit{proof} \\
-  & | & \textit{theorem} \textit{element1}$^*$
-    \textbf{shows} \textit{proposition} \textit{proof} \\
-  & | & \textbf{print\_locale} \textit{locale} \\
-  & | & \textbf{print\_locales}
-  \end{tabular}
-  \end{small}
-  \vspace{2ex}
-  \hrule
-  \caption{Locales extend the grammar of Isar.}
-  \label{fig-grammar}
-  \end{figure}
-
-  At the theory level --- that is, at the outer syntactic level of an
-  Isabelle input file --- \textbf{locale} declares a named
-  locale.  Other kinds of locales,
-  locale expressions and unnamed locales, will be introduced later.  When
-  declaring a named locale, it is possible to \emph{import} another
-  named locale, or indeed several ones by importing a locale
-  expression.  The second part of the declaration, also optional,
-  consists of a number of context element declarations.
-
-  A number of Isar commands have an additional, optional \emph{target}
-  argument, which always refers to a named locale.  These commands
-  are \textbf{theorem} (together with \textbf{lemma} and
-  \textbf{corollary}),  \textbf{theorems} (and
-  \textbf{lemmas}), and \textbf{declare}.  The effect of specifying a target is
-  that these commands focus on the specified locale, not the
-  surrounding theory.  Commands that are used to
-  prove new theorems will add them not to the theory, but to the
-  locale.  Similarly, \textbf{declare} modifies attributes of theorems
-  that belong to the specified target.  Additionally, for
-  \textbf{theorem} (and related commands), theorems stored in the target
-  can be used in the associated proof scripts.
-
-  The Locales package provides a \emph{long goals format} for
-  propositions stated with \textbf{theorem} (and friends).  While
-  normally a goal is just a formula, a long goal is a list of context
-  elements, followed by the keyword \textbf{shows}, followed by the
-  formula.  Roughly speaking, the context elements are
-  (additional) premises.  For an example, see
-  Section~\ref{sec-includes}.  The list of context elements in a long goal
-  is also called \emph{unnamed locale}.
-
-  Finally, there are two commands to inspect locales when working in
-  interactive mode: \textbf{print\_locales} prints the names of all
-  targets
-  visible in the current theory, \textbf{print\_locale} outputs the
-  elements of a named locale or locale expression.
-
-  The following presentation will use notation of
-  Isabelle's meta logic, hence a few sentences to explain this.
-  The logical
-  primitives are universal quantification (@{text "\<And>"}), entailment
-  (@{text "\<Longrightarrow>"}) and equality (@{text "\<equiv>"}).  Variables (not bound
-  variables) are sometimes preceded by a question mark.  The logic is
-  typed.  Type variables are denoted by @{text "'a"}, @{text "'b"}
-  etc., and @{text "\<Rightarrow>"} is the function type.  Double brackets @{text
-  "\<lbrakk>"} and @{text "\<rbrakk>"} are used to abbreviate nested entailment.
-*}
-
-subsection {* Parameters, Assumptions and Facts *}
-
-text {*
-  From a logical point of view a \emph{context} is a formula schema of
-  the form
-\[
-  @{text "\<And>x\<^sub>1\<dots>x\<^sub>n. \<lbrakk> C\<^sub>1; \<dots> ;C\<^sub>m \<rbrakk> \<Longrightarrow> \<dots>"}
-\]
-  The variables $@{text "x\<^sub>1"}, \ldots, @{text "x\<^sub>n"}$ are
-  called \emph{parameters}, the premises $@{text "C\<^sub>1"}, \ldots,
-  @{text "C\<^sub>n"}$ \emph{assumptions}.  A formula @{text "F"}
-  holds in this context if
-\begin{equation}
-\label{eq-fact-in-context}
-  @{text "\<And>x\<^sub>1\<dots>x\<^sub>n. \<lbrakk> C\<^sub>1; \<dots> ;C\<^sub>m \<rbrakk> \<Longrightarrow> F"}
-\end{equation}
-  is valid.  The formula is called a \emph{fact} of the context.
-
-  A locale allows fixing the parameters @{text
-  "x\<^sub>1, \<dots>, x\<^sub>n"} and making the assumptions @{text
-  "C\<^sub>1, \<dots>, C\<^sub>m"}.  This implicitly builds the context in
-  which the formula @{text "F"} can be established.
-  Parameters of a locale correspond to the context element
-  \textbf{fixes}, and assumptions may be declared with
-  \textbf{assumes}.  Using these context elements one can define
-  the specification of semigroups.
-*}
-
-locale semi =
-  fixes prod :: "['a, 'a] \<Rightarrow> 'a" (infixl "\<cdot>" 70)
-  assumes assoc: "(x \<cdot> y) \<cdot> z = x \<cdot> (y \<cdot> z)"
-
-text {*
-  The parameter @{term prod} has a
-  syntax annotation enabling the infix ``@{text "\<cdot>"}'' in the
-  assumption of associativity.  Parameters may have arbitrary mixfix
-  syntax, like constants.  In the example, the type of @{term prod} is
-  specified explicitly.  This is not necessary.  If no type is
-  specified, a most general type is inferred simultaneously for all
-  parameters, taking into account all assumptions (and type
-  specifications of parameters, if present).%
-\footnote{Type inference also takes into account type constraints,
-  definitions and import, as introduced later.}
-
-  Free variables in assumptions are implicitly universally quantified,
-  unless they are parameters.  Hence the context defined by the locale
-  @{term "semi"} is
-\[
-  @{text "\<And>prod. \<lbrakk> \<And>x y z. prod (prod x y) z = prod x (prod y z)
-  \<rbrakk> \<Longrightarrow> \<dots>"}
-\]
-  The locale can be extended to commutative semigroups.
-*}
-
-locale comm_semi = semi +
-  assumes comm: "x \<cdot> y = y \<cdot> x"
-
-text {*
-  This locale \emph{imports} all elements of @{term "semi"}.  The
-  latter locale is called the import of @{term "comm_semi"}. The
-  definition adds commutativity, hence its context is
-\begin{align*%
-}
-  @{text "\<And>prod. \<lbrakk> "} & 
-  @{text "\<And>x y z. prod (prod x y) z = prod x (prod y z);"} \\
-  & @{text "\<And>x y. prod x y = prod y x \<rbrakk> \<Longrightarrow> \<dots>"}
-\end{align*%
-}
-  One may now derive facts --- for example, left-commutativity --- in
-  the context of @{term "comm_semi"} by specifying this locale as
-  target, and by referring to the names of the assumptions @{text
-  assoc} and @{text comm} in the proof.
-*}
-
-theorem (in comm_semi) lcomm:
-  "x \<cdot> (y \<cdot> z) = y \<cdot> (x \<cdot> z)"
-proof -
-  have "x \<cdot> (y \<cdot> z) = (x \<cdot> y) \<cdot> z" by (simp add: assoc)
-  also have "\<dots> = (y \<cdot> x) \<cdot> z" by (simp add: comm)
-  also have "\<dots> = y \<cdot> (x \<cdot> z)" by (simp add: assoc)
-  finally show ?thesis .
-qed
-
-text {*
-  In this equational Isar proof, ``@{text "\<dots>"}'' refers to the
-  right hand side of the preceding equation.
-  After the proof is finished, the fact @{text "lcomm"} is added to
-  the locale @{term "comm_semi"}.  This is done by adding a
-  \textbf{notes} element to the internal representation of the locale,
-  as explained the next section.
-*}
-
-subsection {* Locale Predicates and the Internal Representation of
-  Locales *}
-
-text {*
-\label{sec-locale-predicates}
-  In mathematical texts, often arbitrary but fixed objects with
-  certain properties are considered --- for instance, an arbitrary but
-  fixed group $G$ --- with the purpose of establishing facts valid for
-  any group.  These facts are subsequently used on other objects that
-  also have these properties.
-
-  Locales permit the same style of reasoning.  Exporting a fact $F$
-  generalises the fixed parameters and leads to a (valid) formula of the
-  form of equation~(\ref{eq-fact-in-context}).  If a locale has many
-  assumptions
-  (possibly accumulated through a number of imports) this formula can
-  become large and cumbersome.  Therefore, Wenzel introduced 
-  predicates that abbreviate the assumptions of locales.  These
-  predicates are not confined to the locale but are visible in the
-  surrounding theory.
-
-  The definition of the locale @{term semi} generates the \emph{locale
-  predicate} @{term semi} over the type of the parameter @{term prod},
-  hence the predicate's type is @{typ "(['a, 'a] \<Rightarrow> 'a)
-  \<Rightarrow> bool"}.  Its
-  definition is
-\begin{equation}
-  \tag*{@{thm [source] semi_def}:} @{thm semi_def}.
-\end{equation}
-  In the case where the locale has no import, the generated
-  predicate abbreviates all assumptions and is over the parameters
-  that occur in these assumptions.
-
-  The situation is more complicated when a locale extends
-  another locale, as is the case for @{term comm_semi}.  Two
-  predicates are defined.  The predicate
-  @{term comm_semi_axioms} corresponds to the new assumptions and is
-  called \emph{delta predicate}, the locale
-  predicate @{term comm_semi} captures the content of all the locale,
-  including the import.
-  If a locale has neither assumptions nor import, no predicate is
-  defined.  If a locale has import but no assumptions, only the locale
-  predicate is defined.
-*}
-(*<*)
-ML {*
-  val [comm_semi_ax1, comm_semi_ax2] = thms "comm_semi.axioms";
-  bind_thm ("comm_semi_ax1", comm_semi_ax1);
-  bind_thm ("comm_semi_ax2", comm_semi_ax2);
-*}
-(*>*)
-text {*
-  The Locales package generates a number of theorems for locale and
-  delta predicates.  All predicates have a definition and an
-  introduction rule.  Locale predicates that are defined in terms of
-  other predicates (which is the case if and only if the locale has
-  import) also have a number of elimination rules (called
-  \emph{axioms}).  All generated theorems for the predicates of the
-  locales @{text semi} and @{text comm_semi} are shown in
-  Figures~\ref{fig-theorems-semi} and~\ref{fig-theorems-comm-semi},
-  respectively.
-  \begin{figure}
-  \hrule
-  \vspace{2ex}
-  Theorems generated for the predicate @{term semi}.
-  \begin{gather}
-    \tag*{@{thm [source] semi_def}:} @{thm semi_def} \\
-    \tag*{@{thm [source] semi.intro}:} @{thm semi.intro}
-  \end{gather}
-  \hrule
-  \caption{Theorems for the locale predicate @{term "semi"}.}
-  \label{fig-theorems-semi}
-  \end{figure}
-
-  \begin{figure}[h]
-  \hrule
-  \vspace{2ex}
-  Theorems generated for the predicate @{term "comm_semi_axioms"}.
-  \begin{gather}
-    \tag*{@{thm [source] comm_semi_axioms_def}:} @{thm
-    comm_semi_axioms_def} \\                        
-    \tag*{@{thm [source] comm_semi_axioms.intro}:} @{thm
-    comm_semi_axioms.intro}                       
-  \end{gather}
-  Theorems generated for the predicate @{term "comm_semi"}.
-  \begin{gather}
-    \tag*{@{thm [source] comm_semi_def}:} @{thm
-    comm_semi_def} \\                          
-    \tag*{@{thm [source] comm_semi.intro}:} @{thm
-    comm_semi.intro} \\
-    \tag*{@{thm [source] comm_semi.axioms}:} \mbox{} \\
-    \notag @{thm comm_semi_ax1} \\
-    \notag @{thm comm_semi_ax2}               
-  \end{gather} 
-  \hrule
-  \caption{Theorems for the predicates @{term "comm_semi_axioms"} and
-    @{term "comm_semi"}.}
-  \label{fig-theorems-comm-semi}
-  \end{figure}
-  Note that the theorems generated by a locale
-  definition may be inspected immediately after the definition in the
-  Proof General interface \cite{Aspinall2000} of Isabelle through
-  the menu item ``Isabelle/Isar$>$Show me $\ldots>$Theorems''.
-
-  Locale and delta predicates are used also in the internal
-  representation of locales as lists of context elements.  While all
-  \textbf{fixes} in a
-  declaration generate internal \textbf{fixes}, all assumptions
-  of one locale declaration contribute to one internal \textbf{assumes}
-  element.  The internal representation of @{term semi} is
-\[
-\begin{array}{ll}
-  \textbf{fixes} & @{text prod} :: @{typ [quotes] "['a, 'a] \<Rightarrow> 'a"}
-    (\textbf{infixl} @{text [quotes] "\<cdot>"} 70) \\
-  \textbf{assumes} & @{text [quotes] "semi prod"} \\
-  \textbf{notes} & @{text assoc}: @{text [quotes] "?x \<cdot> ?y \<cdot> ?z = ?x \<cdot> (?y \<cdot>
-    ?z)"}
-\end{array}
-\]
-  and the internal representation of @{text [quotes] comm_semi} is
-\begin{equation}
-\label{eq-comm-semi}
-\begin{array}{ll}
-  \textbf{fixes} & @{text prod} :: @{typ [quotes] "['a, 'a] \<Rightarrow> 'a"}
-    ~(\textbf{infixl}~@{text [quotes] "\<cdot>"}~70) \\
-  \textbf{assumes} & @{text [quotes] "semi prod"} \\
-  \textbf{notes} & @{text assoc}: @{text [quotes] "?x \<cdot> ?y \<cdot> ?z = ?x \<cdot> (?y \<cdot>
-    ?z)"} \\
-  \textbf{assumes} & @{text [quotes] "comm_semi_axioms prod"} \\
-  \textbf{notes} & @{text comm}: @{text [quotes] "?x \<cdot> ?y = ?y \<cdot> ?x"} \\
-  \textbf{notes} & @{text lcomm}: @{text [quotes] "?x \<cdot> (?y \<cdot> ?z) = ?y \<cdot> (?x \<cdot>
-    ?z)"}
-\end{array}
-\end{equation}
-  The \textbf{notes} elements store facts of the
-  locales.  The facts @{text assoc} and @{text comm} were added
-  during the declaration of the locales.  They stem from assumptions,
-  which are trivially facts.  The fact @{text lcomm} was
-  added later, after finishing the proof in the respective
-  \textbf{theorem} command above.
-
-  By using \textbf{notes} in a declaration, facts can be added
-  to a locale directly.  Of course, these must be theorems.
-  Typical use of this feature includes adding theorems that are not
-  usually used as a default rewrite rules by the simplifier to the
-  simpset of the locale by a \textbf{notes} element with the attribute
-  @{text "[simp]"}.  This way it is also possible to add specialised
-  versions of
-  theorems to a locale by instantiating locale parameters for unknowns
-  or locale assumptions for premises.
-*}
-
-subsection {* Definitions *}
-
-text {*
-  Definitions were available in Kamm\"uller's version of Locales, and
-  they are in Wenzel's.  
-  The context element \textbf{defines} adds a definition of the form
-  @{text "p x\<^sub>1 \<dots> x\<^sub>n \<equiv> t"} as an assumption, where @{term p} is a
-  parameter of the locale (possibly an imported parameter), and @{text
-  t} a term that may contain the @{text "x\<^sub>i"}.  The parameter may
-  neither occur in a previous \textbf{assumes} or \textbf{defines}
-  element, nor on the right hand side of the definition.  Hence
-  recursion is not allowed.
-  The parameter may, however, occur in subsequent \textbf{assumes} and
-  on the right hand side of subsequent \textbf{defines}.  We call
-  @{term p} \emph{defined parameter}.
-*}
-
-locale semi2 = semi +
-  fixes rprod (infixl "\<odot>" 70)
-  defines rprod_def: "rprod x y \<equiv> y \<cdot> x "
-
-text {*
-  This locale extends @{term semi} by a second binary operation @{text
-  [quotes] \<odot>} that is like @{text [quotes] \<cdot>} but with
-  reversed arguments.  The
-  definition of the locale generates the predicate @{term semi2},
-  which is equivalent to @{text semi}, but no @{term "semi2_axioms"}.
-  The difference between \textbf{assumes} and \textbf{defines} lies in
-  the way parameters are treated on export.
-*}
-
-subsection {* Export *}
-
-text {*
-  A fact is exported out
-  of a locale by generalising over the parameters and adding
-  assumptions as premises.  For brevity of the exported theorems,
-  locale predicates are used.  Exported facts are referenced by
-  writing qualified names consisting of the locale name and the fact name ---
-  for example,
-\begin{equation}
-  \tag*{@{thm [source] semi.assoc}:} @{thm semi.assoc}.
-\end{equation}
-  Defined parameters receive special treatment.  Instead of adding a
-  premise for the definition, the definition is unfolded in the
-  exported theorem.  In order to illustrate this we prove that the
-  reverse operation @{text [quotes] \<odot>} defined in the locale
-  @{text semi2} is also associative.
-*}
-
-theorem (in semi2) r_assoc: "(x \<odot> y) \<odot> z = x \<odot> (y \<odot> z)"
-  by (simp only: rprod_def assoc)                                    
-
-text {*
-  The exported fact is
-\begin{equation}
-  \tag*{@{thm [source] semi2.r_assoc}:} @{thm semi2.r_assoc}.
-\end{equation}
-  The defined parameter is not present but is replaced by its
-  definition.  Note that the definition itself is not exported, hence
-  there is no @{text "semi2.rprod_def"}.%
-\footnote{The definition could alternatively be exported using a
-  let-construct if there was one in Isabelle's meta-logic.  Let is
-  usually defined in object-logics.}
-*}
-
-section {* Locale Expressions *}
-
-text {*
-  Locale expressions provide a simple language for combining
-  locales.  They are an effective means of building complex
-  specifications from simple ones.  Locale expressions are the main
-  innovation of the version of Locales discussed here.  Locale
-  expressions are also reason for introducing locale predicates.
-*}
-
-subsection {* Rename and Merge *}
-
-text {*
-  The grammar of locale expressions is part of the grammar in
-  Figure~\ref{fig-grammar}.  Locale names are locale
-  expressions, and
-  further expressions are obtained by \emph{rename} and \emph{merge}.
-\begin{description}
-\item[Rename.]
-  The locale expression $e\: q_1 \ldots q_n$ denotes
-  the locale of $e$ where pa\-ra\-me\-ters, in the order in
-  which they are fixed, are renamed to $q_1$ to $q_n$.
-  The expression is only well-formed if $n$ does not
-  exceed the number of parameters of $e$.  Underscores denote
-  parameters that are not renamed.
-  Renaming by default removes mixfix syntax, but new syntax may be
-  specified.
-\item[Merge.]
-  The locale expression $e_1 + e_2$ denotes
-  the locale obtained by merging the locales of $e_1$
-  and $e_2$.  This locale contains the context elements
-  of $e_1$, followed by the context elements of $e_2$.
-
-  In actual fact, the semantics of the merge operation
-  is more complicated.  If $e_1$ and $e_2$ are expressions containing
-  the same name, followed by
-  identical parameter lists, then the merge of both will contain
-  the elements of those locales only once.  Details are explained in
-  Section~\ref{sec-normal-forms} below.
-
-  The merge operation is associative but not commutative.  The latter
-  is because parameters of $e_1$ appear
-  before parameters of $e_2$ in the composite expression.
-\end{description}
-
-  Rename can be used if a different parameter name seems more
-  appropriate --- for example, when moving from groups to rings, a
-  parameter @{text G} representing the group could be changed to
-  @{text R}.  Besides of this stylistic use, renaming is important in
-  combination with merge.  Both operations are used in the following
-  specification of semigroup homomorphisms.
-*}
-
-locale semi_hom = comm_semi sum (infixl "\<oplus>" 65) + comm_semi +
-  fixes hom
-  assumes hom: "hom (x \<oplus> y) = hom x \<cdot> hom y"
-
-text {*
-  This locale defines a context with three parameters @{text "sum"},
-  @{text "prod"} and @{text "hom"}.  The first two parameters have
-  mixfix syntax.  They are associative operations,
-  the first of type @{typeof [locale=semi_hom] prod}, the second of
-  type @{typeof [locale=semi_hom] sum}.  
-
-  How are facts that are imported via a locale expression identified?
-  Facts are always introduced in a named locale (either in the
-  locale's declaration, or by using the locale as target in
-  \textbf{theorem}), and their names are
-  qualified by the parameter names of this locale.
-  Hence the full name of associativity in @{text "semi"} is
-  @{text "prod.assoc"}.  Renaming parameters of a target also renames
-  the qualifier of facts.  Hence, associativity of @{text "sum"} is
-  @{text "sum.assoc"}.  Several parameters are separated by
-  underscores in qualifiers.  For example, the full name of the fact
-  @{text "hom"} in the locale @{text "semi_hom"} is @{text
-  "sum_prod_hom.hom"}.
-
-  The following example is quite artificial, it illustrates the use of
-  facts, though.
-*}
-
-theorem (in semi_hom) "hom x \<cdot> (hom y \<cdot> hom z) = hom (x \<oplus> (y \<oplus> z))"
-proof -
-  have "hom x \<cdot> (hom y \<cdot> hom z) = hom y \<cdot> (hom x \<cdot> hom z)"
-    by (simp add: prod.lcomm)
-  also have "\<dots> = hom (y \<oplus> (x \<oplus> z))" by (simp add: hom)
-  also have "\<dots> = hom (x \<oplus> (y \<oplus> z))" by (simp add: sum.lcomm)
-  finally show ?thesis .
-qed
-
-text {*
-  Importing via a locale expression imports all facts of
-  the imported locales, hence both @{text "sum.lcomm"} and @{text
-  "prod.lcomm"} are
-  available in @{text "hom_semi"}.  The import is dynamic --- that is,
-  whenever facts are added to a locale, they automatically
-  become available in subsequent \textbf{theorem} commands that use
-  the locale as a target, or a locale importing the locale.
-*}
-
-subsection {* Normal Forms *}
-
-text_raw {*
-\label{sec-normal-forms}
-\newcommand{\I}{\mathcal{I}}
-\newcommand{\F}{\mathcal{F}}
-\newcommand{\N}{\mathcal{N}}
-\newcommand{\C}{\mathcal{C}}
-\newcommand{\App}{\mathbin{\overline{@}}}
-*}
-
-text {*
-  Locale expressions are interpreted in a two-step process.  First, an
-  expression is normalised, then it is converted to a list of context
-  elements.
-
-  Normal forms are based on \textbf{locale} declarations.  These
-  consist of an import section followed by a list of context
-  elements.  Let $\I(l)$ denote the locale expression imported by
-  locale $l$.  If $l$ has no import then $\I(l) = \varepsilon$.
-  Likewise, let $\F(l)$ denote the list of context elements, also
-  called the \emph{context fragment} of $l$.  Note that $\F(l)$
-  contains only those context elements that are stated in the
-  declaration of $l$, not imported ones.
-
-\paragraph{Example 1.}  Consider the locales @{text semi} and @{text
-  "comm_semi"}.  We have $\I(@{text semi}) = \varepsilon$ and
-  $\I(@{text "comm_semi"}) = @{text semi}$, and the context fragments
-  are
-\begin{align*%
-}
-  \F(@{text semi}) & = \left[
-\begin{array}{ll}
-  \textbf{fixes} & @{text prod} :: @{typ [quotes] "['a, 'a] \<Rightarrow> 'a"}
-    ~(\textbf{infixl}~@{text [quotes] "\<cdot>"}~70) \\
-  \textbf{assumes} & @{text [quotes] "semi prod"} \\
-  \textbf{notes} & @{text assoc}: @{text  [quotes]"?x \<cdot> ?y \<cdot> ?z = ?x \<cdot> (?y \<cdot>
-    ?z)"}
-\end{array} \right], \\
-  \F(@{text "comm_semi"}) & = \left[
-\begin{array}{ll}
-  \textbf{assumes} & @{text [quotes] "comm_semi_axioms prod"} \\
-  \textbf{notes} & @{text comm}: @{text [quotes] "?x \<cdot> ?y = ?y \<cdot> ?x"} \\
-  \textbf{notes} & @{text lcomm}: @{text [quotes] "?x \<cdot> (?y \<cdot> ?z) = ?y \<cdot> (?x \<cdot>
-    ?z)"}
-\end{array} \right].
-\end{align*%
-}
-  Let $\pi_0(\F(l))$ denote the list of parameters defined in the
-  \textbf{fixes} elements of $\F(l)$ in the order of their occurrence.
-  The list of parameters of a locale expression $\pi(e)$ is defined as
-  follows:
-\begin{align*%
-}
-  \pi(l) & = \pi(\I(l)) \App \pi_0(\F(l)) \text{, for named locale $l$.} \\
-  \pi(e\: q_1 \ldots q_n) & = \text{$[q_1, \ldots, q_n, p_{n+1}, \ldots,
-    p_{m}]$, where $\pi(e) = [p_1, \ldots, p_m]$.} \\
-  \pi(e_1 + e_2) & = \pi(e_1) \App \pi(e_2)
-\end{align*%
-}
-  The operation $\App$ concatenates two lists but omits elements from
-  the second list that are also present in the first list.
-  It is not possible to rename more parameters than there
-  are present in an expression --- that is, $n \le m$ --- otherwise
-  the renaming is illegal.  If $q_i
-  = \_$ then the $i$th entry of the resulting list is $p_i$.
-
-  In the normalisation phase, imports of named locales are unfolded, and
-  renames and merges are recursively propagated to the imported locale
-  expressions.  The result is a list of locale names, each with a full
-  list of parameters, where locale names occurring with the same parameter
-  list twice are removed.  Let $\N$ denote normalisation.  It is defined
-  by these equations:
-\begin{align*%
-}
-  \N(l) & = \N(\I(l)) \App [l\:\pi(l)] \text{, for named locale $l$.} \\
-  \N(e\: q_1 \ldots q_n) & = \N(e)\:[q_1 \ldots q_n/\pi(e)] \\
-  \N(e_1 + e_2) & = \N(e_1) \App \N(e_2)
-\end{align*%
-}
-  Normalisation yields a list of \emph{identifiers}.  An identifier
-  consists of a locale name and a (possibly empty) list of parameters.
-
-  In the second phase, the list of identifiers $\N(e)$ is converted to
-  a list of context elements $\C(e)$ by converting each identifier to
-  a list of context elements, and flattening the obtained list.
-  Conversion of the identifier $l\:q_1 \ldots q_n$ yields the list of
-  context elements $\F(l)$, but with the following modifications:
-\begin{itemize}
-\item
-  Rename the parameter in the $i$th \textbf{fixes} element of $\F(l)$
-  to $q_i$, $i = 1, \ldots, n$.  If the parameter name is actually
-  changed then delete the syntax annotation.  Renaming a parameter may
-  also change its type.
-\item
-  Perform the same renamings on all occurrences of parameters (fixed
-  variables) in \textbf{assumes}, \textbf{defines} and \textbf{notes}
-  elements.
-\item
-  Qualify names of facts by $q_1\_\ldots\_q_n$.
-\end{itemize}
-  The locale expression is \emph{well-formed} if it contains no
-  illegal renamings and the following conditions on $\C(e)$ hold,
-  otherwise the expression is rejected:
-\begin{itemize}
-\item Parameters in \textbf{fixes} are distinct;
-\item Free variables in \textbf{assumes} and
-  \textbf{defines} occur in preceding \textbf{fixes};%
-\footnote{This restriction is relaxed for contexts obtained with
-  \textbf{includes}, see Section~\ref{sec-includes}.}
-\item Parameters defined in \textbf{defines} must neither occur in
-  preceding \textbf{assumes} nor \textbf{defines}.
-\end{itemize}
-*}
-
-subsection {* Examples *}
-
-text {*
-\paragraph{Example 2.}
-  We obtain the context fragment $\C(@{text "comm_semi"})$ of the
-  locale @{text "comm_semi"}.  First, the parameters are computed.
-\begin{align*%
-}
-  \pi(@{text "semi"}) & = [@{text prod}] \\
-  \pi(@{text "comm_semi"}) & = \pi(@{text "semi"}) \App []
-     = [@{text prod}]
-\end{align*%
-}
-  Next, the normal form of the locale expression
-  @{text "comm_semi"} is obtained.
-\begin{align*%
-}
-  \N(@{text "semi"}) & = [@{text semi} @{text prod}] \\
-  \N(@{text "comm_semi"}) & = \N(@{text "semi"}) \App
-       [@{text "comm_semi prod"}]
-   = [@{text "semi prod"}, @{text "comm_semi prod"}]
-\end{align*%
-}
-  Converting this to a list of context elements leads to the
-  list~(\ref{eq-comm-semi}) shown in
-  Section~\ref{sec-locale-predicates}, but with fact names qualified
-  by @{text prod} --- for example, @{text "prod.assoc"}.
-  Qualification was omitted to keep the presentation simple.
-  Isabelle's scoping rules identify the most recent fact with
-  qualified name $x.a$ when a fact with name $a$ is requested.
-
-\paragraph{Example 3.}
-  The locale expression @{text "comm_semi sum"} involves
-  renaming.  Computing parameters yields $\pi(@{text "comm_semi sum"})
-  = [@{text sum}]$, the normal form is
-\begin{align*%
-}
-  \N(@{text "comm_semi sum"}) & =
-  \N(@{text "comm_semi"})[@{text sum}/@{text prod}] =
-  [@{text "semi sum"}, @{text "comm_semi sum"}]
-\end{align*%
-}
-  and the list of context elements
-\[
-\begin{array}{ll}
-  \textbf{fixes} & @{text sum} :: @{typ [quotes] "['a, 'a] \<Rightarrow> 'a"}
-    ~(\textbf{infixl}~@{text [quotes] "\<oplus>"}~65) \\
-  \textbf{assumes} & @{text [quotes] "semi sum"} \\
-  \textbf{notes} & @{text sum.assoc}: @{text [quotes] "(?x \<oplus> ?y) \<oplus> ?z
-    = sum ?x (sum ?y ?z)"} \\
-  \textbf{assumes} & @{text [quotes] "comm_semi_axioms sum"} \\
-  \textbf{notes} & @{text sum.comm}: @{text [quotes] "?x \<oplus> ?y = 
-    ?y \<oplus> ?x"} \\
-  \textbf{notes} & @{text sum.lcomm}: @{text [quotes] "?x \<oplus> (?y \<oplus> ?z)
-    = ?y \<oplus> (?x \<oplus> ?z)"}
-\end{array}
-\]
-
-\paragraph{Example 4.}
-  The context defined by the locale @{text "semi_hom"} involves
-  merging two copies of @{text "comm_semi"}.  We obtain the parameter
-  list and normal form:
-\begin{align*%
-}
-  \pi(@{text "semi_hom"}) & = \pi(@{text "comm_semi sum"} +
-       @{text "comm_semi"}) \App [@{text hom}] \\
-     & = (\pi(@{text "comm_semi sum"}) \App \pi(@{text "comm_semi"}))
-       \App [@{text hom}] \\
-     & = ([@{text sum}] \App [@{text prod}]) \App [@{text hom}]
-     = [@{text sum}, @{text prod}, @{text hom}] \\
-  \N(@{text "semi_hom"}) & =
-       \N(@{text "comm_semi sum"} + @{text "comm_semi"}) \App \\
-     & \phantom{==}
-       [@{text "semi_hom sum prod hom"}] \\
-     & = (\N(@{text "comm_semi sum"}) \App \N(@{text "comm_semi"})) \App \\
-     & \phantom{==}
-       [@{text "semi_hom sum prod hom"}] \\
-     & = ([@{text "semi sum"}, @{text "comm_semi sum"}] \App %\\
-%     & \phantom{==}
-       [@{text "semi prod"}, @{text "comm_semi prod"}]) \App \\
-     & \phantom{==}
-       [@{text "semi_hom sum prod hom"}] \\
-     & = [@{text "semi sum"}, @{text "comm_semi sum"},
-       @{text "semi prod"}, @{text "comm_semi prod"}, \\
-     & \phantom{==}
-       @{text "semi_hom sum prod hom"}].
-\end{align*%
-}
-  Hence $\C(@{text "semi_hom"})$, shown below, is again well-formed.
-\[
-\begin{array}{ll}
-  \textbf{fixes} & @{text sum} :: @{typ [quotes] "['a, 'a] \<Rightarrow> 'a"}
-    ~(\textbf{infixl}~@{text [quotes] "\<oplus>"}~65) \\
-  \textbf{assumes} & @{text [quotes] "semi sum"} \\
-  \textbf{notes} & @{text sum.assoc}: @{text [quotes] "(?x \<oplus> ?y) \<oplus> ?z
-    = ?x \<oplus> (?y \<oplus> ?z)"} \\
-  \textbf{assumes} & @{text [quotes] "comm_semi_axioms sum"} \\
-  \textbf{notes} & @{text sum.comm}: @{text [quotes] "?x \<oplus> ?y = ?y \<oplus> ?x"} \\
-  \textbf{notes} & @{text sum.lcomm}: @{text [quotes] "?x \<oplus> (?y \<oplus> ?z)
-    = ?y \<oplus> (?x \<oplus> ?z)"} \\
-  \textbf{fixes} & @{text prod} :: @{typ [quotes] "['b, 'b] \<Rightarrow> 'b"}
-    ~(\textbf{infixl}~@{text [quotes] "\<cdot>"}~70) \\
-  \textbf{assumes} & @{text [quotes] "semi prod"} \\
-  \textbf{notes} & @{text prod.assoc}: @{text [quotes] "?x \<cdot> ?y \<cdot> ?z = ?x \<cdot> (?y \<cdot>
-    ?z)"} \\
-  \textbf{assumes} & @{text [quotes] "comm_semi_axioms prod"} \\
-  \textbf{notes} & @{text prod.comm}: @{text [quotes] "?x \<cdot> ?y = ?y \<cdot> ?x"} \\
-  \textbf{notes} & @{text prod.lcomm}: @{text [quotes] "?x \<cdot> (?y \<cdot> ?z) = ?y \<cdot> (?x \<cdot>
-    ?z)"} \\
-  \textbf{fixes} & @{text hom} :: @{typ [quotes] "'a \<Rightarrow> 'b"} \\
-  \textbf{assumes} & @{text [quotes] "semi_hom_axioms sum"} \\
-  \textbf{notes} & @{text "sum_prod_hom.hom"}:
-    @{text "hom (x \<oplus> y) = hom x \<cdot> hom y"}
-\end{array}
-\]
-
-\paragraph{Example 5.}
-  In this example, a locale expression leading to a list of context
-  elements that is not well-defined is encountered, and it is illustrated
-  how normalisation deals with multiple inheritance.
-  Consider the specification of monads (in the algebraic sense)
-  and monoids.
-*}
-
-locale monad =
-  fixes prod :: "['a, 'a] \<Rightarrow> 'a" (infixl "\<cdot>" 70) and one :: 'a ("\<one>" 100)
-  assumes l_one: "\<one> \<cdot> x = x" and r_one: "x \<cdot> \<one> = x"
-
-text {*
-  Monoids are both semigroups and monads and one would want to
-  specify them as locale expression @{text "semi + monad"}.
-  Unfortunately, this expression is not well-formed.  Its normal form
-\begin{align*%
-}
-  \N(@{text "monad"}) & = [@{text "monad prod"}] \\
-  \N(@{text "semi"}+@{text "monad"}) & =
-       \N(@{text "semi"}) \App \N(@{text "monad"})
-     = [@{text "semi prod"}, @{text "monad prod"}]
-\end{align*%
-}
-  leads to a list containing the context element
-\[
-  \textbf{fixes}~@{text prod} :: @{typ [quotes] "['a, 'a] \<Rightarrow> 'a"}
-    ~(\textbf{infixl}~@{text [quotes] "\<cdot>"}~70)
-\]
-  twice and thus violating the first criterion of well-formedness.  To
-  avoid this problem, one can
-  introduce a new locale @{text magma} with the sole purpose of fixing the
-  parameter and defining its syntax.  The specifications of semigroup
-  and monad are changed so that they import @{text magma}.
-*}
-
-locale magma = fixes prod (infixl "\<cdot>" 70)
-
-locale semi' = magma + assumes assoc: "(x \<cdot> y) \<cdot> z = x \<cdot> (y \<cdot> z)"
-locale monad' = magma + fixes one ("\<one>" 100)
-  assumes l_one: "\<one> \<cdot> x = x" and r_one: "x \<cdot> \<one> = x"
-
-text {*
-  Normalisation now yields
-\begin{align*%
-}
-  \N(@{text "semi' + monad'"}) & =
-       \N(@{text "semi'"}) \App \N(@{text "monad'"}) \\
-     & = (\N(@{text magma}) \App [@{text "semi' prod"}]) \App
-         (\N(@{text magma}) \App [@{text "monad' prod"}]) \\
-     & = [@{text "magma prod"}, @{text "semi' prod"}] \App
-         [@{text "magma prod"}, @{text "monad' prod"}]) \\
-     & = [@{text "magma prod"}, @{text "semi' prod"},
-          @{text "monad' prod"}]
-\end{align*%
-}
-  where the second occurrence of @{text "magma prod"} is eliminated.
-  The reader is encouraged to check, using the \textbf{print\_locale}
-  command, that the list of context elements generated from this is
-  indeed well-formed.
-
-  It follows that importing
-  parameters is more flexible than fixing them using a context element.
-  The Locale package provides the predefined locale @{term var} that
-  can be used to import parameters if no
-  particular mixfix syntax is required.
-  Its definition is
-\begin{center}
-  \textbf{locale} @{text var} = \textbf{fixes} @{text "x_"}
-\end{center}
-  The use of the internal variable @{text "x_"}
-  enforces that the parameter is renamed before being used, because
-  internal variables may not occur in the input syntax.  Using
-  @{text var}, the locale @{text magma} could have been replaced by
-  the locale expression
-\begin{center}
-  @{text var} @{text prod} (\textbf{infixl} @{text [quotes] \<cdot>} 70)
-\end{center}
-  in the above locale declarations.
-*}
-
-subsection {* Includes *}
-
-text {*
-\label{sec-includes}
-  The context element \textbf{includes} takes a locale expression $e$
-  as argument.  It can only occur in long goals, where it
-  adds, like a target, locale context to the proof context.  Unlike
-  with targets, the proved theorem is not stored
-  in the locale.  Instead, it is exported immediately.
-*}
-
-theorem lcomm2:
-  includes comm_semi shows "x \<cdot> (y \<cdot> z) = y \<cdot> (x \<cdot> z)"
-proof -
-  have "x \<cdot> (y \<cdot> z) = (x \<cdot> y) \<cdot> z" by (simp add: assoc)
-  also have "\<dots> = (y \<cdot> x) \<cdot> z" by (simp add: comm)
-  also have "\<dots> = y \<cdot> (x \<cdot> z)" by (simp add: assoc)
-  finally show ?thesis .
-qed
-
-text {*
-  This proof is identical to the proof of @{text lcomm}.  The use of
-  \textbf{includes} provides the same context and facts as when using
-  @{term comm_semi} as target.  On the other hand, @{thm [source]
-  lcomm2} is not added as a fact to the locale @{term comm_semi}, but
-  is directly visible in the theory.  The theorem is
-\[
-  @{thm lcomm2}.
-\]
-  Note that it is possible to
-  combine a target and (several) \textbf{includes} in a goal statement, thus
-  using contexts of several locales but storing the theorem in only
-  one of them.
-*}
-
-section {* Structures *}
-
-text {*
-\label{sec-structures}
-  The specifications of semigroups and monoids that served as examples
-  in previous sections modelled each operation of an algebraic
-  structure as a single parameter.  This is rather inconvenient for
-  structures with many operations, and also unnatural.  In accordance
-  to mathematical texts, one would rather fix two groups instead of
-  two sets of operations.
-
-  The approach taken in Isabelle is to encode algebraic structures
-  with suitable types (in Isabelle/HOL usually records).  An issue to
-  be addressed by
-  locales is syntax for algebraic structures.  This is the purpose of
-  the \textbf{(structure)} annotation in \textbf{fixes}, introduced by
-  Wenzel.  We illustrate this, independently of record types, with a
-  different formalisation of semigroups.
-
-  Let @{text "'a semi_type"} be a not further specified type that
-  represents semigroups over the carrier type @{typ "'a"}.  Let @{text
-  "s_op"} be an operation that maps an object of @{text "'a semi_type"} to
-  a binary operation.
-*}
-
-typedecl 'a semi_type
-consts s_op :: "['a semi_type, 'a, 'a] \<Rightarrow> 'a" (infixl "\<star>\<index>" 70)
-
-text {*
-  Although @{text "s_op"} is a ternary operation, it is declared
-  infix.  The syntax annotation contains the token  @{text \<index>}
-  (\verb.\<index>.), which refers to the first argument.  This syntax is only
-  effective in the context of a locale, and only if the first argument
-  is a
-  \emph{structural} parameter --- that is, a parameter with annotation
-  \textbf{(structure)}.  The token has the effect of subscripting the
-  parameter --- by bracketing it between \verb.\<^bsub>. and  \verb.\<^esub>..
-  Additionally, the subscript of the first structural parameter may be
-  omitted, as in this specification of semigroups with structures:
-*}
-
-locale comm_semi' =
-  fixes G :: "'a semi_type" (structure)
-  assumes assoc: "(x \<star> y) \<star> z = x \<star> (y \<star> z)" and comm: "x \<star> y = y \<star> x"
-
-text {*
-  Here @{text "x \<star> y"} is equivalent to @{text "x \<star>\<^bsub>G\<^esub> y"} and
-  abbreviates @{text "s_op G x y"}.  A specification of homomorphisms
-  requires a second structural parameter.
-*}
-
-locale semi'_hom = comm_semi' + comm_semi' H +
-  fixes hom
-  assumes hom: "hom (x \<star> y) = hom x \<star>\<^bsub>H\<^esub> hom y"
-
-text {*
-  The parameter @{text H} is defined in the second \textbf{fixes}
-  element of $\C(@{term "semi'_comm"})$. Hence @{text "\<star>\<^bsub>H\<^esub>"}
-  abbreviates @{text "s_op H x y"}.  The same construction can be done
-  with records instead of an \textit{ad-hoc} type.  *}
-
-record 'a semi = prod :: "['a, 'a] \<Rightarrow> 'a" (infixl "\<bullet>\<index>" 70)
-
-text {*
-  This declares the types @{typ "'a semi"} and  @{typ "('a, 'b)
-  semi_scheme"}.  The latter is an extensible record, where the second
-  type argument is the type of the extension field.  For details on
-  records, see \cite{NipkowEtAl2002} Chapter~8.3.
-*}
-
-locale semi_w_records = struct G +
-  assumes assoc: "(x \<bullet> y) \<bullet> z = x \<bullet> (y \<bullet> z)"
-
-text {*
-  The type @{typ "('a, 'b) semi_scheme"} is inferred for the parameter @{term
-  G}.  Using subtyping on records, the specification can be extended
-  to groups easily.
-*}
-
-record 'a group = "'a semi" +
-  one :: "'a" ("l\<index>" 100)
-  inv :: "'a \<Rightarrow> 'a" ("inv\<index> _" [81] 80)
-locale group_w_records = semi_w_records +
-  assumes l_one: "l \<bullet> x = x" and l_inv: "inv x \<bullet> x = l"
-
-text {*
- Finally, the predefined locale
-\begin{center}
-  \textbf{locale} \textit{struct} = \textbf{fixes} @{text "S_"}
-    \textbf{(structure)}.
-\end{center}
-  is analogous to @{text "var"}.  
-  More examples on the use of structures, including groups, rings and
-  polynomials can be found in the Isabelle distribution in the
-  session HOL-Algebra.
-*}
-
-section {* Conclusions and Outlook *}
-
-text {*
-  Locales provide a simple means of modular reasoning.  They enable to
-  abbreviate frequently occurring context statements and maintain facts
-  valid in these contexts.  Importantly, using structures, they allow syntax to be
-  effective only in certain contexts, and thus to mimic common
-  practice in mathematics, where notation is chosen very flexibly.
-  This is also known as literate formalisation \cite{Bailey1998}.
-  Locale expressions allow to duplicate and merge
-  specifications.  This is a necessity, for example, when reasoning about
-  homomorphisms.  Normalisation makes it possible to deal with
-  diamond-shaped inheritance structures, and generally with directed
-  acyclic graphs.  The combination of locales with record
-  types in higher-order logic provides an effective means for
-  specifying algebraic structures: locale import and record subtyping
-  provide independent hierarchies for specifications and structure
-  elements.  Rich examples for this can be found in
-  the Isabelle distribution in the session HOL-Algebra.
-
-  The primary reason for writing this report was to provide a better
-  understanding of locales in Isar.  Wenzel provided hardly any
-  documentation, with the exception of \cite{Wenzel2002b}.  The
-  present report should make it easier for users of Isabelle to take
-  advantage of locales.
-
-  The report is also a base for future extensions.  These include
-  improved syntax for structures.  Identifying them by numbers seems
-  unnatural and can be confusing if more than two structures are
-  involved --- for example, when reasoning about universal
-  properties --- and numbering them by order of occurrence seems
-  arbitrary.  Another desirable feature is \emph{instantiation}.  One
-  may, in the course of a theory development, construct objects that
-  fulfil the specification of a locale.  These objects are possibly
-  defined in the context of another locale.  Instantiation should make it
-  simple to specialise abstract facts for the object under
-  consideration and to use the specified facts.
-
-  A detailed comparison of locales with module systems in type theory
-  has not been undertaken yet, but could be beneficial.  For example,
-  a module system for Coq has recently been presented by Chrzaszcz
-  \cite{Chrzaszcz2003,Chrzaszcz2004}.  While the
-  latter usually constitute extensions of the calculus, locales are
-  a rather thin layer that does not change Isabelle's meta logic.
-  Locales mainly manage specifications and facts.  Functors, like
-  the constructor for polynomial rings, remain objects of the
-  logic.
-
-  \textbf{Acknowledgements.}  Lawrence C.\ Paulson and Norbert
-  Schirmer made useful comments on a draft of this paper.
-*}
-                                
-(*<*)
-end
-(*>*)
\ No newline at end of file
diff -r 9681f347b6f5 -r d1d35284542f doc-src/Locales/Locales/ROOT.ML
--- a/doc-src/Locales/Locales/ROOT.ML	Tue Jun 03 11:55:35 2008 +0200
+++ b/doc-src/Locales/Locales/ROOT.ML	Tue Jun 03 12:34:22 2008 +0200
@@ -1,6 +1,4 @@
-(*
-  no_document use_thy "ThisTheory";
-  use_thy "ThatTheory";
-*)
-
-use_thy "Locales";
+no_document use_thy "GCD";
+use_thy "Examples1";
+use_thy "Examples2";
+use_thy "Examples3";
diff -r 9681f347b6f5 -r d1d35284542f doc-src/Locales/Locales/document/Examples.tex
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/Locales/Locales/document/Examples.tex	Tue Jun 03 12:34:22 2008 +0200
@@ -0,0 +1,1403 @@
+%
+\begin{isabellebody}%
+\def\isabellecontext{Examples}%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+%
+\isatagtheory
+%
+\endisatagtheory
+{\isafoldtheory}%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+%
+\isamarkupsection{Introduction%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+Locales are based on contexts.  A \emph{context} can be seen as a
+  formula schema
+\[
+  \isa{{\isasymAnd}x\isactrlsub {\isadigit{1}}{\isasymdots}x\isactrlsub n{\isachardot}\ {\isasymlbrakk}\ A\isactrlsub {\isadigit{1}}{\isacharsemicolon}\ {\isasymdots}\ {\isacharsemicolon}A\isactrlsub m\ {\isasymrbrakk}\ {\isasymLongrightarrow}\ {\isasymdots}}
+\]
+  where variables \isa{x\isactrlsub {\isadigit{1}}}, \ldots, \isa{x\isactrlsub n} are called
+  \emph{parameters} and the premises $\isa{A\isactrlsub {\isadigit{1}}}, \ldots,
+  \isa{A\isactrlsub m}$ \emph{assumptions}.  A formula \isa{C}
+  is a \emph{theorem} in the context if it is a conclusion
+\[
+%\label{eq-fact-in-context}
+  \isa{{\isasymAnd}x\isactrlsub {\isadigit{1}}{\isasymdots}x\isactrlsub n{\isachardot}\ {\isasymlbrakk}\ A\isactrlsub {\isadigit{1}}{\isacharsemicolon}\ {\isasymdots}\ {\isacharsemicolon}A\isactrlsub m\ {\isasymrbrakk}\ {\isasymLongrightarrow}\ C}.
+\]
+  Isabelle/Isar's notion of context goes beyond this logical view.
+  Its contexts record, in a consecutive order, proved
+  conclusions along with attributes, which
+  may control proof procedures.  Contexts also contain syntax information
+  for parameters and for terms depending on them.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsection{Simple Locales%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+Locales can be seen as persistent contexts.  In its simplest form, a
+  \emph{locale declaration} consists of a sequence of context elements
+  declaring parameters (keyword \isakeyword{fixes}) and assumptions
+  (keyword \isakeyword{assumes}).  The following is the specification of
+  partial orders, as locale \isa{partial{\isacharunderscore}order}.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\ \ \isacommand{locale}\isamarkupfalse%
+\ partial{\isacharunderscore}order\ {\isacharequal}\isanewline
+\ \ \ \ \isakeyword{fixes}\ le\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ {\isacharparenleft}\isakeyword{infixl}\ {\isachardoublequoteopen}{\isasymsqsubseteq}{\isachardoublequoteclose}\ {\isadigit{5}}{\isadigit{0}}{\isacharparenright}\isanewline
+\ \ \ \ \isakeyword{assumes}\ refl\ {\isacharbrackleft}intro{\isacharcomma}\ simp{\isacharbrackright}{\isacharcolon}\ {\isachardoublequoteopen}x\ {\isasymsqsubseteq}\ x{\isachardoublequoteclose}\isanewline
+\ \ \ \ \ \ \isakeyword{and}\ anti{\isacharunderscore}sym\ {\isacharbrackleft}intro{\isacharbrackright}{\isacharcolon}\ {\isachardoublequoteopen}{\isasymlbrakk}\ x\ {\isasymsqsubseteq}\ y{\isacharsemicolon}\ y\ {\isasymsqsubseteq}\ x\ {\isasymrbrakk}\ {\isasymLongrightarrow}\ x\ {\isacharequal}\ y{\isachardoublequoteclose}\isanewline
+\ \ \ \ \ \ \isakeyword{and}\ trans\ {\isacharbrackleft}trans{\isacharbrackright}{\isacharcolon}\ {\isachardoublequoteopen}{\isasymlbrakk}\ x\ {\isasymsqsubseteq}\ y{\isacharsemicolon}\ y\ {\isasymsqsubseteq}\ z\ {\isasymrbrakk}\ {\isasymLongrightarrow}\ x\ {\isasymsqsubseteq}\ z{\isachardoublequoteclose}%
+\begin{isamarkuptext}%
+The parameter of this locale is \isa{le}, with infix syntax
+  \isa{{\isasymsqsubseteq}}.  There is an implicit type parameter \isa{{\isacharprime}a}.  It
+  is not necessary to declare parameter types: most general types will
+  be inferred from the context elements for all parameters.
+
+  The above declaration not only introduces the locale, it also
+  defines the \emph{locale predicate} \isa{partial{\isacharunderscore}order} with
+  definition \isa{partial{\isacharunderscore}order{\isacharunderscore}def}:
+  \begin{isabelle}%
+\ \ partial{\isacharunderscore}order\ {\isacharquery}le\ {\isasymequiv}\isanewline
+\isaindent{\ \ }{\isacharparenleft}{\isasymforall}x{\isachardot}\ {\isacharquery}le\ x\ x{\isacharparenright}\ {\isasymand}\isanewline
+\isaindent{\ \ }{\isacharparenleft}{\isasymforall}x\ y{\isachardot}\ {\isacharquery}le\ x\ y\ {\isasymlongrightarrow}\ {\isacharquery}le\ y\ x\ {\isasymlongrightarrow}\ x\ {\isacharequal}\ y{\isacharparenright}\ {\isasymand}\isanewline
+\isaindent{\ \ }{\isacharparenleft}{\isasymforall}x\ y\ z{\isachardot}\ {\isacharquery}le\ x\ y\ {\isasymlongrightarrow}\ {\isacharquery}le\ y\ z\ {\isasymlongrightarrow}\ {\isacharquery}le\ x\ z{\isacharparenright}%
+\end{isabelle}
+
+  The specification of a locale is fixed, but its list of conclusions
+  may be extended through Isar commands that take a \emph{target} argument.
+  In the following, \isakeyword{definition} and 
+  \isakeyword{theorem} are illustrated.
+  Table~\ref{tab:commands-with-target} lists Isar commands that accept
+  a target.  There are various ways of specifying the target.  A
+  target for a single command may be indicated with keyword
+  \isakeyword{in} in the following way:
+
+\begin{table}
+\hrule
+\vspace{2ex}
+\begin{center}
+\begin{tabular}{ll}
+  \isakeyword{definition} & definition through an equation \\
+  \isakeyword{inductive} & inductive definition \\
+  \isakeyword{fun}, \isakeyword{function} & recursive function \\
+  \isakeyword{abbreviation} & syntactic abbreviation \\
+  \isakeyword{theorem}, etc.\ & theorem statement with proof \\
+  \isakeyword{theorems}, etc.\ & redeclaration of theorems
+\end{tabular}
+\end{center}
+\hrule
+\caption{Isar commands that accept a target.}
+\label{tab:commands-with-target}
+\end{table}%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\ \ \isacommand{definition}\isamarkupfalse%
+\ {\isacharparenleft}\isakeyword{in}\ partial{\isacharunderscore}order{\isacharparenright}\isanewline
+\ \ \ \ less\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ {\isacharparenleft}\isakeyword{infixl}\ {\isachardoublequoteopen}{\isasymsqsubset}{\isachardoublequoteclose}\ {\isadigit{5}}{\isadigit{0}}{\isacharparenright}\isanewline
+\ \ \ \ \isakeyword{where}\ {\isachardoublequoteopen}{\isacharparenleft}x\ {\isasymsqsubset}\ y{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}x\ {\isasymsqsubseteq}\ y\ {\isasymand}\ x\ {\isasymnoteq}\ y{\isacharparenright}{\isachardoublequoteclose}%
+\begin{isamarkuptext}%
+A definition in a locale depends on the locale parameters.
+  Here, a global constant \isa{partial{\isacharunderscore}order{\isachardot}less} is declared, which is lifted over the
+  locale parameter \isa{le}.  Its definition is the global theorem
+  \isa{partial{\isacharunderscore}order{\isachardot}less{\isacharunderscore}def}:
+  \begin{isabelle}%
+\ \ partial{\isacharunderscore}order\ {\isacharquery}le\ {\isasymLongrightarrow}\isanewline
+\isaindent{\ \ }partial{\isacharunderscore}order{\isachardot}less\ {\isacharquery}le\ {\isacharquery}x\ {\isacharquery}y\ {\isacharequal}\ {\isacharparenleft}{\isacharquery}le\ {\isacharquery}x\ {\isacharquery}y\ {\isasymand}\ {\isacharquery}x\ {\isasymnoteq}\ {\isacharquery}y{\isacharparenright}%
+\end{isabelle}
+  At the same time, the locale is extended by syntax information
+  hiding this construction in the context of the locale.  That is,
+  \isa{partial{\isacharunderscore}order{\isachardot}less\ le} is printed and parsed as infix
+  \isa{{\isasymsqsubset}}.  Finally, the conclusion of the definition is added to
+  the locale, \isa{less{\isacharunderscore}def}:
+  \begin{isabelle}%
+\ \ {\isacharparenleft}{\isacharquery}x\ {\isasymsqsubset}\ {\isacharquery}y{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}{\isacharquery}x\ {\isasymsqsubseteq}\ {\isacharquery}y\ {\isasymand}\ {\isacharquery}x\ {\isasymnoteq}\ {\isacharquery}y{\isacharparenright}%
+\end{isabelle}%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+As an example of a theorem statement in the locale, here is the
+  derivation of a transitivity law.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\ \ \isacommand{lemma}\isamarkupfalse%
+\ {\isacharparenleft}\isakeyword{in}\ partial{\isacharunderscore}order{\isacharparenright}\ less{\isacharunderscore}le{\isacharunderscore}trans\ {\isacharbrackleft}trans{\isacharbrackright}{\isacharcolon}\isanewline
+\ \ \ \ {\isachardoublequoteopen}{\isasymlbrakk}\ x\ {\isasymsqsubset}\ y{\isacharsemicolon}\ y\ {\isasymsqsubseteq}\ z\ {\isasymrbrakk}\ {\isasymLongrightarrow}\ x\ {\isasymsqsubset}\ z{\isachardoublequoteclose}\isanewline
+%
+\isadelimvisible
+\ \ \ \ %
+\endisadelimvisible
+%
+\isatagvisible
+\isacommand{unfolding}\isamarkupfalse%
+\ less{\isacharunderscore}def\ \isacommand{by}\isamarkupfalse%
+\ {\isacharparenleft}blast\ intro{\isacharcolon}\ trans{\isacharparenright}%
+\endisatagvisible
+{\isafoldvisible}%
+%
+\isadelimvisible
+%
+\endisadelimvisible
+%
+\begin{isamarkuptext}%
+In the context of the proof, assumptions and theorems of the
+  locale may be used.  Attributes are effective: \isa{anti{\isacharunderscore}sym} was
+  declared as introduction rule, hence it is in the context's set of
+  rules used by the classical reasoner by default.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+When working with locales, sequences of commands with the same
+  target are frequent.  A block of commands, delimited by
+  \isakeyword{begin} and \isakeyword{end}, makes a theory-like style
+  of working possible.  All commands inside the block refer to the
+  same target.  A block may immediately follow a locale
+  declaration, which makes that locale the target.  Alternatively the
+  target for a block may be given with the \isakeyword{context}
+  command.
+
+  In the block below, notions of infimum and supremum together with
+  theorems are introduced for partial orders.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\ \ \isacommand{context}\isamarkupfalse%
+\ partial{\isacharunderscore}order\ \isakeyword{begin}\isanewline
+\isanewline
+\ \ \isacommand{definition}\isamarkupfalse%
+\isanewline
+\ \ \ \ is{\isacharunderscore}inf\ \isakeyword{where}\ {\isachardoublequoteopen}is{\isacharunderscore}inf\ x\ y\ i\ {\isacharequal}\isanewline
+\ \ \ \ \ \ {\isacharparenleft}i\ {\isasymsqsubseteq}\ x\ {\isasymand}\ i\ {\isasymsqsubseteq}\ y\ {\isasymand}\ {\isacharparenleft}{\isasymforall}z{\isachardot}\ z\ {\isasymsqsubseteq}\ x\ {\isasymand}\ z\ {\isasymsqsubseteq}\ y\ {\isasymlongrightarrow}\ z\ {\isasymsqsubseteq}\ i{\isacharparenright}{\isacharparenright}{\isachardoublequoteclose}\isanewline
+\isanewline
+\ \ \isacommand{definition}\isamarkupfalse%
+\isanewline
+\ \ \ \ is{\isacharunderscore}sup\ \isakeyword{where}\ {\isachardoublequoteopen}is{\isacharunderscore}sup\ x\ y\ s\ {\isacharequal}\isanewline
+\ \ \ \ \ \ {\isacharparenleft}x\ {\isasymsqsubseteq}\ s\ {\isasymand}\ y\ {\isasymsqsubseteq}\ s\ {\isasymand}\ {\isacharparenleft}{\isasymforall}z{\isachardot}\ x\ {\isasymsqsubseteq}\ z\ {\isasymand}\ y\ {\isasymsqsubseteq}\ z\ {\isasymlongrightarrow}\ s\ {\isasymsqsubseteq}\ z{\isacharparenright}{\isacharparenright}{\isachardoublequoteclose}\isanewline
+%
+\isadeliminvisible
+\isanewline
+\ \ %
+\endisadeliminvisible
+%
+\isataginvisible
+\isacommand{lemma}\isamarkupfalse%
+\ is{\isacharunderscore}infI\ {\isacharbrackleft}intro{\isacharquery}{\isacharbrackright}{\isacharcolon}\ {\isachardoublequoteopen}i\ {\isasymsqsubseteq}\ x\ {\isasymLongrightarrow}\ i\ {\isasymsqsubseteq}\ y\ {\isasymLongrightarrow}\isanewline
+\ \ \ \ \ \ {\isacharparenleft}{\isasymAnd}z{\isachardot}\ z\ {\isasymsqsubseteq}\ x\ {\isasymLongrightarrow}\ z\ {\isasymsqsubseteq}\ y\ {\isasymLongrightarrow}\ z\ {\isasymsqsubseteq}\ i{\isacharparenright}\ {\isasymLongrightarrow}\ is{\isacharunderscore}inf\ x\ y\ i{\isachardoublequoteclose}\isanewline
+\ \ \ \ \isacommand{by}\isamarkupfalse%
+\ {\isacharparenleft}unfold\ is{\isacharunderscore}inf{\isacharunderscore}def{\isacharparenright}\ blast\isanewline
+\isanewline
+\ \ \isacommand{lemma}\isamarkupfalse%
+\ is{\isacharunderscore}inf{\isacharunderscore}lower\ {\isacharbrackleft}elim{\isacharquery}{\isacharbrackright}{\isacharcolon}\isanewline
+\ \ \ \ {\isachardoublequoteopen}is{\isacharunderscore}inf\ x\ y\ i\ {\isasymLongrightarrow}\ {\isacharparenleft}i\ {\isasymsqsubseteq}\ x\ {\isasymLongrightarrow}\ i\ {\isasymsqsubseteq}\ y\ {\isasymLongrightarrow}\ C{\isacharparenright}\ {\isasymLongrightarrow}\ C{\isachardoublequoteclose}\isanewline
+\ \ \ \ \isacommand{by}\isamarkupfalse%
+\ {\isacharparenleft}unfold\ is{\isacharunderscore}inf{\isacharunderscore}def{\isacharparenright}\ blast\isanewline
+\isanewline
+\ \ \isacommand{lemma}\isamarkupfalse%
+\ is{\isacharunderscore}inf{\isacharunderscore}greatest\ {\isacharbrackleft}elim{\isacharquery}{\isacharbrackright}{\isacharcolon}\isanewline
+\ \ \ \ \ \ {\isachardoublequoteopen}is{\isacharunderscore}inf\ x\ y\ i\ {\isasymLongrightarrow}\ z\ {\isasymsqsubseteq}\ x\ {\isasymLongrightarrow}\ z\ {\isasymsqsubseteq}\ y\ {\isasymLongrightarrow}\ z\ {\isasymsqsubseteq}\ i{\isachardoublequoteclose}\isanewline
+\ \ \ \ \isacommand{by}\isamarkupfalse%
+\ {\isacharparenleft}unfold\ is{\isacharunderscore}inf{\isacharunderscore}def{\isacharparenright}\ blast%
+\endisataginvisible
+{\isafoldinvisible}%
+%
+\isadeliminvisible
+\isanewline
+%
+\endisadeliminvisible
+\isanewline
+\ \ \isacommand{theorem}\isamarkupfalse%
+\ is{\isacharunderscore}inf{\isacharunderscore}uniq{\isacharcolon}\ {\isachardoublequoteopen}{\isasymlbrakk}is{\isacharunderscore}inf\ x\ y\ i{\isacharsemicolon}\ is{\isacharunderscore}inf\ x\ y\ i{\isacharprime}{\isasymrbrakk}\ {\isasymLongrightarrow}\ i\ {\isacharequal}\ i{\isacharprime}{\isachardoublequoteclose}\isanewline
+%
+\isadelimproof
+\ \ %
+\endisadelimproof
+%
+\isatagproof
+\isacommand{proof}\isamarkupfalse%
+\ {\isacharminus}\isanewline
+\ \ \ \ \isacommand{assume}\isamarkupfalse%
+\ inf{\isacharcolon}\ {\isachardoublequoteopen}is{\isacharunderscore}inf\ x\ y\ i{\isachardoublequoteclose}\isanewline
+\ \ \ \ \isacommand{assume}\isamarkupfalse%
+\ inf{\isacharprime}{\isacharcolon}\ {\isachardoublequoteopen}is{\isacharunderscore}inf\ x\ y\ i{\isacharprime}{\isachardoublequoteclose}\isanewline
+\ \ \ \ \isacommand{show}\isamarkupfalse%
+\ {\isacharquery}thesis\isanewline
+\ \ \ \ \isacommand{proof}\isamarkupfalse%
+\ {\isacharparenleft}rule\ anti{\isacharunderscore}sym{\isacharparenright}\isanewline
+\ \ \ \ \ \ \isacommand{from}\isamarkupfalse%
+\ inf{\isacharprime}\ \isacommand{show}\isamarkupfalse%
+\ {\isachardoublequoteopen}i\ {\isasymsqsubseteq}\ i{\isacharprime}{\isachardoublequoteclose}\isanewline
+\ \ \ \ \ \ \isacommand{proof}\isamarkupfalse%
+\ {\isacharparenleft}rule\ is{\isacharunderscore}inf{\isacharunderscore}greatest{\isacharparenright}\isanewline
+\ \ \ \ \ \ \ \ \isacommand{from}\isamarkupfalse%
+\ inf\ \isacommand{show}\isamarkupfalse%
+\ {\isachardoublequoteopen}i\ {\isasymsqsubseteq}\ x{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse%
+\isanewline
+\ \ \ \ \ \ \ \ \isacommand{from}\isamarkupfalse%
+\ inf\ \isacommand{show}\isamarkupfalse%
+\ {\isachardoublequoteopen}i\ {\isasymsqsubseteq}\ y{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse%
+\isanewline
+\ \ \ \ \ \ \isacommand{qed}\isamarkupfalse%
+\isanewline
+\ \ \ \ \ \ \isacommand{from}\isamarkupfalse%
+\ inf\ \isacommand{show}\isamarkupfalse%
+\ {\isachardoublequoteopen}i{\isacharprime}\ {\isasymsqsubseteq}\ i{\isachardoublequoteclose}\isanewline
+\ \ \ \ \ \ \isacommand{proof}\isamarkupfalse%
+\ {\isacharparenleft}rule\ is{\isacharunderscore}inf{\isacharunderscore}greatest{\isacharparenright}\isanewline
+\ \ \ \ \ \ \ \ \isacommand{from}\isamarkupfalse%
+\ inf{\isacharprime}\ \isacommand{show}\isamarkupfalse%
+\ {\isachardoublequoteopen}i{\isacharprime}\ {\isasymsqsubseteq}\ x{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse%
+\isanewline
+\ \ \ \ \ \ \ \ \isacommand{from}\isamarkupfalse%
+\ inf{\isacharprime}\ \isacommand{show}\isamarkupfalse%
+\ {\isachardoublequoteopen}i{\isacharprime}\ {\isasymsqsubseteq}\ y{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse%
+\isanewline
+\ \ \ \ \ \ \isacommand{qed}\isamarkupfalse%
+\isanewline
+\ \ \ \ \isacommand{qed}\isamarkupfalse%
+\isanewline
+\ \ \isacommand{qed}\isamarkupfalse%
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+\isanewline
+%
+\endisadelimproof
+%
+\isadeliminvisible
+\isanewline
+\ \ %
+\endisadeliminvisible
+%
+\isataginvisible
+\isacommand{theorem}\isamarkupfalse%
+\ is{\isacharunderscore}inf{\isacharunderscore}related\ {\isacharbrackleft}elim{\isacharquery}{\isacharbrackright}{\isacharcolon}\ {\isachardoublequoteopen}x\ {\isasymsqsubseteq}\ y\ {\isasymLongrightarrow}\ is{\isacharunderscore}inf\ x\ y\ x{\isachardoublequoteclose}\isanewline
+\ \ \isacommand{proof}\isamarkupfalse%
+\ {\isacharminus}\isanewline
+\ \ \ \ \isacommand{assume}\isamarkupfalse%
+\ {\isachardoublequoteopen}x\ {\isasymsqsubseteq}\ y{\isachardoublequoteclose}\isanewline
+\ \ \ \ \isacommand{show}\isamarkupfalse%
+\ {\isacharquery}thesis\isanewline
+\ \ \ \ \isacommand{proof}\isamarkupfalse%
+\isanewline
+\ \ \ \ \ \ \isacommand{show}\isamarkupfalse%
+\ {\isachardoublequoteopen}x\ {\isasymsqsubseteq}\ x{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse%
+\isanewline
+\ \ \ \ \ \ \isacommand{show}\isamarkupfalse%
+\ {\isachardoublequoteopen}x\ {\isasymsqsubseteq}\ y{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
+\ fact\isanewline
+\ \ \ \ \ \ \isacommand{fix}\isamarkupfalse%
+\ z\ \isacommand{assume}\isamarkupfalse%
+\ {\isachardoublequoteopen}z\ {\isasymsqsubseteq}\ x{\isachardoublequoteclose}\ \isakeyword{and}\ {\isachardoublequoteopen}z\ {\isasymsqsubseteq}\ y{\isachardoublequoteclose}\ \isacommand{show}\isamarkupfalse%
+\ {\isachardoublequoteopen}z\ {\isasymsqsubseteq}\ x{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
+\ fact\isanewline
+\ \ \ \ \isacommand{qed}\isamarkupfalse%
+\isanewline
+\ \ \isacommand{qed}\isamarkupfalse%
+\isanewline
+\isanewline
+\ \ \isacommand{lemma}\isamarkupfalse%
+\ is{\isacharunderscore}supI\ {\isacharbrackleft}intro{\isacharquery}{\isacharbrackright}{\isacharcolon}\ {\isachardoublequoteopen}x\ {\isasymsqsubseteq}\ s\ {\isasymLongrightarrow}\ y\ {\isasymsqsubseteq}\ s\ {\isasymLongrightarrow}\isanewline
+\ \ \ \ \ \ {\isacharparenleft}{\isasymAnd}z{\isachardot}\ x\ {\isasymsqsubseteq}\ z\ {\isasymLongrightarrow}\ y\ {\isasymsqsubseteq}\ z\ {\isasymLongrightarrow}\ s\ {\isasymsqsubseteq}\ z{\isacharparenright}\ {\isasymLongrightarrow}\ is{\isacharunderscore}sup\ x\ y\ s{\isachardoublequoteclose}\isanewline
+\ \ \ \ \isacommand{by}\isamarkupfalse%
+\ {\isacharparenleft}unfold\ is{\isacharunderscore}sup{\isacharunderscore}def{\isacharparenright}\ blast\isanewline
+\isanewline
+\ \ \isacommand{lemma}\isamarkupfalse%
+\ is{\isacharunderscore}sup{\isacharunderscore}least\ {\isacharbrackleft}elim{\isacharquery}{\isacharbrackright}{\isacharcolon}\isanewline
+\ \ \ \ \ \ {\isachardoublequoteopen}is{\isacharunderscore}sup\ x\ y\ s\ {\isasymLongrightarrow}\ x\ {\isasymsqsubseteq}\ z\ {\isasymLongrightarrow}\ y\ {\isasymsqsubseteq}\ z\ {\isasymLongrightarrow}\ s\ {\isasymsqsubseteq}\ z{\isachardoublequoteclose}\isanewline
+\ \ \ \ \isacommand{by}\isamarkupfalse%
+\ {\isacharparenleft}unfold\ is{\isacharunderscore}sup{\isacharunderscore}def{\isacharparenright}\ blast\isanewline
+\isanewline
+\ \ \isacommand{lemma}\isamarkupfalse%
+\ is{\isacharunderscore}sup{\isacharunderscore}upper\ {\isacharbrackleft}elim{\isacharquery}{\isacharbrackright}{\isacharcolon}\isanewline
+\ \ \ \ \ \ {\isachardoublequoteopen}is{\isacharunderscore}sup\ x\ y\ s\ {\isasymLongrightarrow}\ {\isacharparenleft}x\ {\isasymsqsubseteq}\ s\ {\isasymLongrightarrow}\ y\ {\isasymsqsubseteq}\ s\ {\isasymLongrightarrow}\ C{\isacharparenright}\ {\isasymLongrightarrow}\ C{\isachardoublequoteclose}\isanewline
+\ \ \ \ \isacommand{by}\isamarkupfalse%
+\ {\isacharparenleft}unfold\ is{\isacharunderscore}sup{\isacharunderscore}def{\isacharparenright}\ blast%
+\endisataginvisible
+{\isafoldinvisible}%
+%
+\isadeliminvisible
+\isanewline
+%
+\endisadeliminvisible
+\isanewline
+\ \ \isacommand{theorem}\isamarkupfalse%
+\ is{\isacharunderscore}sup{\isacharunderscore}uniq{\isacharcolon}\ {\isachardoublequoteopen}{\isasymlbrakk}is{\isacharunderscore}sup\ x\ y\ s{\isacharsemicolon}\ is{\isacharunderscore}sup\ x\ y\ s{\isacharprime}{\isasymrbrakk}\ {\isasymLongrightarrow}\ s\ {\isacharequal}\ s{\isacharprime}{\isachardoublequoteclose}\isanewline
+%
+\isadelimproof
+\ \ %
+\endisadelimproof
+%
+\isatagproof
+\isacommand{proof}\isamarkupfalse%
+\ {\isacharminus}\isanewline
+\ \ \ \ \isacommand{assume}\isamarkupfalse%
+\ sup{\isacharcolon}\ {\isachardoublequoteopen}is{\isacharunderscore}sup\ x\ y\ s{\isachardoublequoteclose}\isanewline
+\ \ \ \ \isacommand{assume}\isamarkupfalse%
+\ sup{\isacharprime}{\isacharcolon}\ {\isachardoublequoteopen}is{\isacharunderscore}sup\ x\ y\ s{\isacharprime}{\isachardoublequoteclose}\isanewline
+\ \ \ \ \isacommand{show}\isamarkupfalse%
+\ {\isacharquery}thesis\isanewline
+\ \ \ \ \isacommand{proof}\isamarkupfalse%
+\ {\isacharparenleft}rule\ anti{\isacharunderscore}sym{\isacharparenright}\isanewline
+\ \ \ \ \ \ \isacommand{from}\isamarkupfalse%
+\ sup\ \isacommand{show}\isamarkupfalse%
+\ {\isachardoublequoteopen}s\ {\isasymsqsubseteq}\ s{\isacharprime}{\isachardoublequoteclose}\isanewline
+\ \ \ \ \ \ \isacommand{proof}\isamarkupfalse%
+\ {\isacharparenleft}rule\ is{\isacharunderscore}sup{\isacharunderscore}least{\isacharparenright}\isanewline
+\ \ \ \ \ \ \ \ \isacommand{from}\isamarkupfalse%
+\ sup{\isacharprime}\ \isacommand{show}\isamarkupfalse%
+\ {\isachardoublequoteopen}x\ {\isasymsqsubseteq}\ s{\isacharprime}{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse%
+\isanewline
+\ \ \ \ \ \ \ \ \isacommand{from}\isamarkupfalse%
+\ sup{\isacharprime}\ \isacommand{show}\isamarkupfalse%
+\ {\isachardoublequoteopen}y\ {\isasymsqsubseteq}\ s{\isacharprime}{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse%
+\isanewline
+\ \ \ \ \ \ \isacommand{qed}\isamarkupfalse%
+\isanewline
+\ \ \ \ \ \ \isacommand{from}\isamarkupfalse%
+\ sup{\isacharprime}\ \isacommand{show}\isamarkupfalse%
+\ {\isachardoublequoteopen}s{\isacharprime}\ {\isasymsqsubseteq}\ s{\isachardoublequoteclose}\isanewline
+\ \ \ \ \ \ \isacommand{proof}\isamarkupfalse%
+\ {\isacharparenleft}rule\ is{\isacharunderscore}sup{\isacharunderscore}least{\isacharparenright}\isanewline
+\ \ \ \ \ \ \ \ \isacommand{from}\isamarkupfalse%
+\ sup\ \isacommand{show}\isamarkupfalse%
+\ {\isachardoublequoteopen}x\ {\isasymsqsubseteq}\ s{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse%
+\isanewline
+\ \ \ \ \ \ \ \ \isacommand{from}\isamarkupfalse%
+\ sup\ \isacommand{show}\isamarkupfalse%
+\ {\isachardoublequoteopen}y\ {\isasymsqsubseteq}\ s{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse%
+\isanewline
+\ \ \ \ \ \ \isacommand{qed}\isamarkupfalse%
+\isanewline
+\ \ \ \ \isacommand{qed}\isamarkupfalse%
+\isanewline
+\ \ \isacommand{qed}\isamarkupfalse%
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+\isanewline
+%
+\endisadelimproof
+%
+\isadeliminvisible
+\isanewline
+\ \ %
+\endisadeliminvisible
+%
+\isataginvisible
+\isacommand{theorem}\isamarkupfalse%
+\ is{\isacharunderscore}sup{\isacharunderscore}related\ {\isacharbrackleft}elim{\isacharquery}{\isacharbrackright}{\isacharcolon}\ {\isachardoublequoteopen}x\ {\isasymsqsubseteq}\ y\ {\isasymLongrightarrow}\ is{\isacharunderscore}sup\ x\ y\ y{\isachardoublequoteclose}\isanewline
+\ \ \isacommand{proof}\isamarkupfalse%
+\ {\isacharminus}\isanewline
+\ \ \ \ \isacommand{assume}\isamarkupfalse%
+\ {\isachardoublequoteopen}x\ {\isasymsqsubseteq}\ y{\isachardoublequoteclose}\isanewline
+\ \ \ \ \isacommand{show}\isamarkupfalse%
+\ {\isacharquery}thesis\isanewline
+\ \ \ \ \isacommand{proof}\isamarkupfalse%
+\isanewline
+\ \ \ \ \ \ \isacommand{show}\isamarkupfalse%
+\ {\isachardoublequoteopen}x\ {\isasymsqsubseteq}\ y{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
+\ fact\isanewline
+\ \ \ \ \ \ \isacommand{show}\isamarkupfalse%
+\ {\isachardoublequoteopen}y\ {\isasymsqsubseteq}\ y{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse%
+\isanewline
+\ \ \ \ \ \ \isacommand{fix}\isamarkupfalse%
+\ z\ \isacommand{assume}\isamarkupfalse%
+\ {\isachardoublequoteopen}x\ {\isasymsqsubseteq}\ z{\isachardoublequoteclose}\ \isakeyword{and}\ {\isachardoublequoteopen}y\ {\isasymsqsubseteq}\ z{\isachardoublequoteclose}\isanewline
+\ \ \ \ \ \ \isacommand{show}\isamarkupfalse%
+\ {\isachardoublequoteopen}y\ {\isasymsqsubseteq}\ z{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
+\ fact\isanewline
+\ \ \ \ \isacommand{qed}\isamarkupfalse%
+\isanewline
+\ \ \isacommand{qed}\isamarkupfalse%
+%
+\endisataginvisible
+{\isafoldinvisible}%
+%
+\isadeliminvisible
+\isanewline
+%
+\endisadeliminvisible
+\isanewline
+\ \ \isacommand{end}\isamarkupfalse%
+%
+\begin{isamarkuptext}%
+In fact, many more theorems need to be shown for a usable
+  theory of partial orders.  The
+  above two serve as illustrative examples.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+Two commands are provided to inspect locales:
+  \isakeyword{print\_locales} lists the names of all locales of the
+  theory; \isakeyword{print\_locale}~$n$ prints the parameters and
+  assumptions of locale $n$; \isakeyword{print\_locale!}~$n$
+  additionally outputs the conclusions.
+
+  The syntax of the locale commands discussed in this tutorial is
+  shown in Table~\ref{tab:commands}.  See the
+  Isabelle/Isar Reference Manual~\cite{IsarRef}
+  for full documentation.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsection{Import%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+\label{sec:import}
+
+  Algebraic structures are commonly defined by adding operations and
+  properties to existing structures.  For example, partial orders
+  are extended to lattices and total orders.  Lattices are extended to
+  distributive lattices.
+
+  With locales, this inheritance is achieved through \emph{import} of a
+  locale.  The import comes before the context elements.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\ \ \isacommand{locale}\isamarkupfalse%
+\ lattice\ {\isacharequal}\ partial{\isacharunderscore}order\ {\isacharplus}\isanewline
+\ \ \ \ \isakeyword{assumes}\ ex{\isacharunderscore}inf{\isacharcolon}\ {\isachardoublequoteopen}{\isasymexists}inf{\isachardot}\ partial{\isacharunderscore}order{\isachardot}is{\isacharunderscore}inf\ le\ x\ y\ inf{\isachardoublequoteclose}\isanewline
+\ \ \ \ \ \ \isakeyword{and}\ ex{\isacharunderscore}sup{\isacharcolon}\ {\isachardoublequoteopen}{\isasymexists}sup{\isachardot}\ partial{\isacharunderscore}order{\isachardot}is{\isacharunderscore}sup\ le\ x\ y\ sup{\isachardoublequoteclose}\isanewline
+\ \ \isakeyword{begin}%
+\begin{isamarkuptext}%
+Note that the assumptions above refer to the predicates for infimum
+  and supremum defined in \isa{partial{\isacharunderscore}order}.  In the current
+  implementation of locales, syntax from definitions of the imported
+  locale is unavailable in the locale declaration, neither are their
+  names.  Hence we refer to the constants of the theory.  The names
+  and syntax is available below, in the context of the locale.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\ \ \isacommand{definition}\isamarkupfalse%
+\isanewline
+\ \ \ \ meet\ {\isacharparenleft}\isakeyword{infixl}\ {\isachardoublequoteopen}{\isasymsqinter}{\isachardoublequoteclose}\ {\isadigit{7}}{\isadigit{0}}{\isacharparenright}\ \isakeyword{where}\ {\isachardoublequoteopen}x\ {\isasymsqinter}\ y\ {\isacharequal}\ {\isacharparenleft}THE\ inf{\isachardot}\ is{\isacharunderscore}inf\ x\ y\ inf{\isacharparenright}{\isachardoublequoteclose}\isanewline
+\isanewline
+\ \ \isacommand{definition}\isamarkupfalse%
+\isanewline
+\ \ \ \ join\ {\isacharparenleft}\isakeyword{infixl}\ {\isachardoublequoteopen}{\isasymsqunion}{\isachardoublequoteclose}\ {\isadigit{6}}{\isadigit{5}}{\isacharparenright}\ \isakeyword{where}\ {\isachardoublequoteopen}x\ {\isasymsqunion}\ y\ {\isacharequal}\ {\isacharparenleft}THE\ sup{\isachardot}\ is{\isacharunderscore}sup\ x\ y\ sup{\isacharparenright}{\isachardoublequoteclose}\isanewline
+%
+\isadeliminvisible
+\isanewline
+\ \ %
+\endisadeliminvisible
+%
+\isataginvisible
+\isacommand{lemma}\isamarkupfalse%
+\ meet{\isacharunderscore}equality\ {\isacharbrackleft}elim{\isacharquery}{\isacharbrackright}{\isacharcolon}\ {\isachardoublequoteopen}is{\isacharunderscore}inf\ x\ y\ i\ {\isasymLongrightarrow}\ x\ {\isasymsqinter}\ y\ {\isacharequal}\ i{\isachardoublequoteclose}\isanewline
+\ \ \isacommand{proof}\isamarkupfalse%
+\ {\isacharparenleft}unfold\ meet{\isacharunderscore}def{\isacharparenright}\isanewline
+\ \ \ \ \isacommand{assume}\isamarkupfalse%
+\ {\isachardoublequoteopen}is{\isacharunderscore}inf\ x\ y\ i{\isachardoublequoteclose}\isanewline
+\ \ \ \ \isacommand{then}\isamarkupfalse%
+\ \isacommand{show}\isamarkupfalse%
+\ {\isachardoublequoteopen}{\isacharparenleft}THE\ i{\isachardot}\ is{\isacharunderscore}inf\ x\ y\ i{\isacharparenright}\ {\isacharequal}\ i{\isachardoublequoteclose}\isanewline
+\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
+\ {\isacharparenleft}rule\ the{\isacharunderscore}equality{\isacharparenright}\ {\isacharparenleft}rule\ is{\isacharunderscore}inf{\isacharunderscore}uniq\ {\isacharbrackleft}OF\ {\isacharunderscore}\ {\isacharbackquoteopen}is{\isacharunderscore}inf\ x\ y\ i{\isacharbackquoteclose}{\isacharbrackright}{\isacharparenright}\isanewline
+\ \ \isacommand{qed}\isamarkupfalse%
+\isanewline
+\isanewline
+\ \ \isacommand{lemma}\isamarkupfalse%
+\ meetI\ {\isacharbrackleft}intro{\isacharquery}{\isacharbrackright}{\isacharcolon}\isanewline
+\ \ \ \ \ \ {\isachardoublequoteopen}i\ {\isasymsqsubseteq}\ x\ {\isasymLongrightarrow}\ i\ {\isasymsqsubseteq}\ y\ {\isasymLongrightarrow}\ {\isacharparenleft}{\isasymAnd}z{\isachardot}\ z\ {\isasymsqsubseteq}\ x\ {\isasymLongrightarrow}\ z\ {\isasymsqsubseteq}\ y\ {\isasymLongrightarrow}\ z\ {\isasymsqsubseteq}\ i{\isacharparenright}\ {\isasymLongrightarrow}\ x\ {\isasymsqinter}\ y\ {\isacharequal}\ i{\isachardoublequoteclose}\isanewline
+\ \ \ \ \isacommand{by}\isamarkupfalse%
+\ {\isacharparenleft}rule\ meet{\isacharunderscore}equality{\isacharcomma}\ rule\ is{\isacharunderscore}infI{\isacharparenright}\ blast{\isacharplus}\isanewline
+\isanewline
+\ \ \isacommand{lemma}\isamarkupfalse%
+\ is{\isacharunderscore}inf{\isacharunderscore}meet\ {\isacharbrackleft}intro{\isacharquery}{\isacharbrackright}{\isacharcolon}\ {\isachardoublequoteopen}is{\isacharunderscore}inf\ x\ y\ {\isacharparenleft}x\ {\isasymsqinter}\ y{\isacharparenright}{\isachardoublequoteclose}\isanewline
+\ \ \isacommand{proof}\isamarkupfalse%
+\ {\isacharparenleft}unfold\ meet{\isacharunderscore}def{\isacharparenright}\isanewline
+\ \ \ \ \isacommand{from}\isamarkupfalse%
+\ ex{\isacharunderscore}inf\ \isacommand{obtain}\isamarkupfalse%
+\ i\ \isakeyword{where}\ {\isachardoublequoteopen}is{\isacharunderscore}inf\ x\ y\ i{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse%
+\isanewline
+\ \ \ \ \isacommand{then}\isamarkupfalse%
+\ \isacommand{show}\isamarkupfalse%
+\ {\isachardoublequoteopen}is{\isacharunderscore}inf\ x\ y\ {\isacharparenleft}THE\ i{\isachardot}\ is{\isacharunderscore}inf\ x\ y\ i{\isacharparenright}{\isachardoublequoteclose}\isanewline
+\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
+\ {\isacharparenleft}rule\ theI{\isacharparenright}\ {\isacharparenleft}rule\ is{\isacharunderscore}inf{\isacharunderscore}uniq\ {\isacharbrackleft}OF\ {\isacharunderscore}\ {\isacharbackquoteopen}is{\isacharunderscore}inf\ x\ y\ i{\isacharbackquoteclose}{\isacharbrackright}{\isacharparenright}\isanewline
+\ \ \isacommand{qed}\isamarkupfalse%
+\isanewline
+\isanewline
+\ \ \isacommand{lemma}\isamarkupfalse%
+\ meet{\isacharunderscore}left\ {\isacharbrackleft}intro{\isacharquery}{\isacharbrackright}{\isacharcolon}\isanewline
+\ \ \ \ {\isachardoublequoteopen}x\ {\isasymsqinter}\ y\ {\isasymsqsubseteq}\ x{\isachardoublequoteclose}\isanewline
+\ \ \ \ \isacommand{by}\isamarkupfalse%
+\ {\isacharparenleft}rule\ is{\isacharunderscore}inf{\isacharunderscore}lower{\isacharparenright}\ {\isacharparenleft}rule\ is{\isacharunderscore}inf{\isacharunderscore}meet{\isacharparenright}\isanewline
+\isanewline
+\ \ \isacommand{lemma}\isamarkupfalse%
+\ meet{\isacharunderscore}right\ {\isacharbrackleft}intro{\isacharquery}{\isacharbrackright}{\isacharcolon}\isanewline
+\ \ \ \ {\isachardoublequoteopen}x\ {\isasymsqinter}\ y\ {\isasymsqsubseteq}\ y{\isachardoublequoteclose}\isanewline
+\ \ \ \ \isacommand{by}\isamarkupfalse%
+\ {\isacharparenleft}rule\ is{\isacharunderscore}inf{\isacharunderscore}lower{\isacharparenright}\ {\isacharparenleft}rule\ is{\isacharunderscore}inf{\isacharunderscore}meet{\isacharparenright}\isanewline
+\isanewline
+\ \ \isacommand{lemma}\isamarkupfalse%
+\ meet{\isacharunderscore}le\ {\isacharbrackleft}intro{\isacharquery}{\isacharbrackright}{\isacharcolon}\isanewline
+\ \ \ \ {\isachardoublequoteopen}{\isasymlbrakk}\ z\ {\isasymsqsubseteq}\ x{\isacharsemicolon}\ z\ {\isasymsqsubseteq}\ y\ {\isasymrbrakk}\ {\isasymLongrightarrow}\ z\ {\isasymsqsubseteq}\ x\ {\isasymsqinter}\ y{\isachardoublequoteclose}\isanewline
+\ \ \ \ \isacommand{by}\isamarkupfalse%
+\ {\isacharparenleft}rule\ is{\isacharunderscore}inf{\isacharunderscore}greatest{\isacharparenright}\ {\isacharparenleft}rule\ is{\isacharunderscore}inf{\isacharunderscore}meet{\isacharparenright}\isanewline
+\isanewline
+\ \ \isacommand{lemma}\isamarkupfalse%
+\ join{\isacharunderscore}equality\ {\isacharbrackleft}elim{\isacharquery}{\isacharbrackright}{\isacharcolon}\ {\isachardoublequoteopen}is{\isacharunderscore}sup\ x\ y\ s\ {\isasymLongrightarrow}\ x\ {\isasymsqunion}\ y\ {\isacharequal}\ s{\isachardoublequoteclose}\isanewline
+\ \ \isacommand{proof}\isamarkupfalse%
+\ {\isacharparenleft}unfold\ join{\isacharunderscore}def{\isacharparenright}\isanewline
+\ \ \ \ \isacommand{assume}\isamarkupfalse%
+\ {\isachardoublequoteopen}is{\isacharunderscore}sup\ x\ y\ s{\isachardoublequoteclose}\isanewline
+\ \ \ \ \isacommand{then}\isamarkupfalse%
+\ \isacommand{show}\isamarkupfalse%
+\ {\isachardoublequoteopen}{\isacharparenleft}THE\ s{\isachardot}\ is{\isacharunderscore}sup\ x\ y\ s{\isacharparenright}\ {\isacharequal}\ s{\isachardoublequoteclose}\isanewline
+\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
+\ {\isacharparenleft}rule\ the{\isacharunderscore}equality{\isacharparenright}\ {\isacharparenleft}rule\ is{\isacharunderscore}sup{\isacharunderscore}uniq\ {\isacharbrackleft}OF\ {\isacharunderscore}\ {\isacharbackquoteopen}is{\isacharunderscore}sup\ x\ y\ s{\isacharbackquoteclose}{\isacharbrackright}{\isacharparenright}\isanewline
+\ \ \isacommand{qed}\isamarkupfalse%
+\isanewline
+\isanewline
+\ \ \isacommand{lemma}\isamarkupfalse%
+\ joinI\ {\isacharbrackleft}intro{\isacharquery}{\isacharbrackright}{\isacharcolon}\ {\isachardoublequoteopen}x\ {\isasymsqsubseteq}\ s\ {\isasymLongrightarrow}\ y\ {\isasymsqsubseteq}\ s\ {\isasymLongrightarrow}\isanewline
+\ \ \ \ \ \ {\isacharparenleft}{\isasymAnd}z{\isachardot}\ x\ {\isasymsqsubseteq}\ z\ {\isasymLongrightarrow}\ y\ {\isasymsqsubseteq}\ z\ {\isasymLongrightarrow}\ s\ {\isasymsqsubseteq}\ z{\isacharparenright}\ {\isasymLongrightarrow}\ x\ {\isasymsqunion}\ y\ {\isacharequal}\ s{\isachardoublequoteclose}\isanewline
+\ \ \ \ \isacommand{by}\isamarkupfalse%
+\ {\isacharparenleft}rule\ join{\isacharunderscore}equality{\isacharcomma}\ rule\ is{\isacharunderscore}supI{\isacharparenright}\ blast{\isacharplus}\isanewline
+\isanewline
+\ \ \isacommand{lemma}\isamarkupfalse%
+\ is{\isacharunderscore}sup{\isacharunderscore}join\ {\isacharbrackleft}intro{\isacharquery}{\isacharbrackright}{\isacharcolon}\ {\isachardoublequoteopen}is{\isacharunderscore}sup\ x\ y\ {\isacharparenleft}x\ {\isasymsqunion}\ y{\isacharparenright}{\isachardoublequoteclose}\isanewline
+\ \ \isacommand{proof}\isamarkupfalse%
+\ {\isacharparenleft}unfold\ join{\isacharunderscore}def{\isacharparenright}\isanewline
+\ \ \ \ \isacommand{from}\isamarkupfalse%
+\ ex{\isacharunderscore}sup\ \isacommand{obtain}\isamarkupfalse%
+\ s\ \isakeyword{where}\ {\isachardoublequoteopen}is{\isacharunderscore}sup\ x\ y\ s{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse%
+\isanewline
+\ \ \ \ \isacommand{then}\isamarkupfalse%
+\ \isacommand{show}\isamarkupfalse%
+\ {\isachardoublequoteopen}is{\isacharunderscore}sup\ x\ y\ {\isacharparenleft}THE\ s{\isachardot}\ is{\isacharunderscore}sup\ x\ y\ s{\isacharparenright}{\isachardoublequoteclose}\isanewline
+\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
+\ {\isacharparenleft}rule\ theI{\isacharparenright}\ {\isacharparenleft}rule\ is{\isacharunderscore}sup{\isacharunderscore}uniq\ {\isacharbrackleft}OF\ {\isacharunderscore}\ {\isacharbackquoteopen}is{\isacharunderscore}sup\ x\ y\ s{\isacharbackquoteclose}{\isacharbrackright}{\isacharparenright}\isanewline
+\ \ \isacommand{qed}\isamarkupfalse%
+\isanewline
+\isanewline
+\ \ \isacommand{lemma}\isamarkupfalse%
+\ join{\isacharunderscore}left\ {\isacharbrackleft}intro{\isacharquery}{\isacharbrackright}{\isacharcolon}\isanewline
+\ \ \ \ {\isachardoublequoteopen}x\ {\isasymsqsubseteq}\ x\ {\isasymsqunion}\ y{\isachardoublequoteclose}\isanewline
+\ \ \ \ \isacommand{by}\isamarkupfalse%
+\ {\isacharparenleft}rule\ is{\isacharunderscore}sup{\isacharunderscore}upper{\isacharparenright}\ {\isacharparenleft}rule\ is{\isacharunderscore}sup{\isacharunderscore}join{\isacharparenright}\isanewline
+\isanewline
+\ \ \isacommand{lemma}\isamarkupfalse%
+\ join{\isacharunderscore}right\ {\isacharbrackleft}intro{\isacharquery}{\isacharbrackright}{\isacharcolon}\isanewline
+\ \ \ \ {\isachardoublequoteopen}y\ {\isasymsqsubseteq}\ x\ {\isasymsqunion}\ y{\isachardoublequoteclose}\isanewline
+\ \ \ \ \isacommand{by}\isamarkupfalse%
+\ {\isacharparenleft}rule\ is{\isacharunderscore}sup{\isacharunderscore}upper{\isacharparenright}\ {\isacharparenleft}rule\ is{\isacharunderscore}sup{\isacharunderscore}join{\isacharparenright}\isanewline
+\isanewline
+\ \ \isacommand{lemma}\isamarkupfalse%
+\ join{\isacharunderscore}le\ {\isacharbrackleft}intro{\isacharquery}{\isacharbrackright}{\isacharcolon}\isanewline
+\ \ \ \ {\isachardoublequoteopen}{\isasymlbrakk}\ x\ {\isasymsqsubseteq}\ z{\isacharsemicolon}\ y\ {\isasymsqsubseteq}\ z\ {\isasymrbrakk}\ {\isasymLongrightarrow}\ x\ {\isasymsqunion}\ y\ {\isasymsqsubseteq}\ z{\isachardoublequoteclose}\isanewline
+\ \ \ \ \isacommand{by}\isamarkupfalse%
+\ {\isacharparenleft}rule\ is{\isacharunderscore}sup{\isacharunderscore}least{\isacharparenright}\ {\isacharparenleft}rule\ is{\isacharunderscore}sup{\isacharunderscore}join{\isacharparenright}\isanewline
+\isanewline
+\ \ \isacommand{theorem}\isamarkupfalse%
+\ meet{\isacharunderscore}assoc{\isacharcolon}\ {\isachardoublequoteopen}{\isacharparenleft}x\ {\isasymsqinter}\ y{\isacharparenright}\ {\isasymsqinter}\ z\ {\isacharequal}\ x\ {\isasymsqinter}\ {\isacharparenleft}y\ {\isasymsqinter}\ z{\isacharparenright}{\isachardoublequoteclose}\isanewline
+\ \ \isacommand{proof}\isamarkupfalse%
+\ {\isacharparenleft}rule\ meetI{\isacharparenright}\isanewline
+\ \ \ \ \isacommand{show}\isamarkupfalse%
+\ {\isachardoublequoteopen}x\ {\isasymsqinter}\ {\isacharparenleft}y\ {\isasymsqinter}\ z{\isacharparenright}\ {\isasymsqsubseteq}\ x\ {\isasymsqinter}\ y{\isachardoublequoteclose}\isanewline
+\ \ \ \ \isacommand{proof}\isamarkupfalse%
+\isanewline
+\ \ \ \ \ \ \isacommand{show}\isamarkupfalse%
+\ {\isachardoublequoteopen}x\ {\isasymsqinter}\ {\isacharparenleft}y\ {\isasymsqinter}\ z{\isacharparenright}\ {\isasymsqsubseteq}\ x{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse%
+\isanewline
+\ \ \ \ \ \ \isacommand{show}\isamarkupfalse%
+\ {\isachardoublequoteopen}x\ {\isasymsqinter}\ {\isacharparenleft}y\ {\isasymsqinter}\ z{\isacharparenright}\ {\isasymsqsubseteq}\ y{\isachardoublequoteclose}\isanewline
+\ \ \ \ \ \ \isacommand{proof}\isamarkupfalse%
+\ {\isacharminus}\isanewline
+\ \ \ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
+\ {\isachardoublequoteopen}x\ {\isasymsqinter}\ {\isacharparenleft}y\ {\isasymsqinter}\ z{\isacharparenright}\ {\isasymsqsubseteq}\ y\ {\isasymsqinter}\ z{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse%
+\isanewline
+\ \ \ \ \ \ \ \ \isacommand{also}\isamarkupfalse%
+\ \isacommand{have}\isamarkupfalse%
+\ {\isachardoublequoteopen}{\isasymdots}\ {\isasymsqsubseteq}\ y{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse%
+\isanewline
+\ \ \ \ \ \ \ \ \isacommand{finally}\isamarkupfalse%
+\ \isacommand{show}\isamarkupfalse%
+\ {\isacharquery}thesis\ \isacommand{{\isachardot}}\isamarkupfalse%
+\isanewline
+\ \ \ \ \ \ \isacommand{qed}\isamarkupfalse%
+\isanewline
+\ \ \ \ \isacommand{qed}\isamarkupfalse%
+\isanewline
+\ \ \ \ \isacommand{show}\isamarkupfalse%
+\ {\isachardoublequoteopen}x\ {\isasymsqinter}\ {\isacharparenleft}y\ {\isasymsqinter}\ z{\isacharparenright}\ {\isasymsqsubseteq}\ z{\isachardoublequoteclose}\isanewline
+\ \ \ \ \isacommand{proof}\isamarkupfalse%
+\ {\isacharminus}\isanewline
+\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
+\ {\isachardoublequoteopen}x\ {\isasymsqinter}\ {\isacharparenleft}y\ {\isasymsqinter}\ z{\isacharparenright}\ {\isasymsqsubseteq}\ y\ {\isasymsqinter}\ z{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse%
+\isanewline
+\ \ \ \ \ \ \isacommand{also}\isamarkupfalse%
+\ \isacommand{have}\isamarkupfalse%
+\ {\isachardoublequoteopen}{\isasymdots}\ {\isasymsqsubseteq}\ z{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse%
+\isanewline
+\ \ \ \ \ \ \isacommand{finally}\isamarkupfalse%
+\ \isacommand{show}\isamarkupfalse%
+\ {\isacharquery}thesis\ \isacommand{{\isachardot}}\isamarkupfalse%
+\isanewline
+\ \ \ \ \isacommand{qed}\isamarkupfalse%
+\isanewline
+\ \ \ \ \isacommand{fix}\isamarkupfalse%
+\ w\ \isacommand{assume}\isamarkupfalse%
+\ {\isachardoublequoteopen}w\ {\isasymsqsubseteq}\ x\ {\isasymsqinter}\ y{\isachardoublequoteclose}\ \isakeyword{and}\ {\isachardoublequoteopen}w\ {\isasymsqsubseteq}\ z{\isachardoublequoteclose}\isanewline
+\ \ \ \ \isacommand{show}\isamarkupfalse%
+\ {\isachardoublequoteopen}w\ {\isasymsqsubseteq}\ x\ {\isasymsqinter}\ {\isacharparenleft}y\ {\isasymsqinter}\ z{\isacharparenright}{\isachardoublequoteclose}\isanewline
+\ \ \ \ \isacommand{proof}\isamarkupfalse%
+\isanewline
+\ \ \ \ \ \ \isacommand{show}\isamarkupfalse%
+\ {\isachardoublequoteopen}w\ {\isasymsqsubseteq}\ x{\isachardoublequoteclose}\isanewline
+\ \ \ \ \ \ \isacommand{proof}\isamarkupfalse%
+\ {\isacharminus}\isanewline
+\ \ \ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
+\ {\isachardoublequoteopen}w\ {\isasymsqsubseteq}\ x\ {\isasymsqinter}\ y{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
+\ fact\isanewline
+\ \ \ \ \ \ \ \ \isacommand{also}\isamarkupfalse%
+\ \isacommand{have}\isamarkupfalse%
+\ {\isachardoublequoteopen}{\isasymdots}\ {\isasymsqsubseteq}\ x{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse%
+\isanewline
+\ \ \ \ \ \ \ \ \isacommand{finally}\isamarkupfalse%
+\ \isacommand{show}\isamarkupfalse%
+\ {\isacharquery}thesis\ \isacommand{{\isachardot}}\isamarkupfalse%
+\isanewline
+\ \ \ \ \ \ \isacommand{qed}\isamarkupfalse%
+\isanewline
+\ \ \ \ \ \ \isacommand{show}\isamarkupfalse%
+\ {\isachardoublequoteopen}w\ {\isasymsqsubseteq}\ y\ {\isasymsqinter}\ z{\isachardoublequoteclose}\isanewline
+\ \ \ \ \ \ \isacommand{proof}\isamarkupfalse%
+\isanewline
+\ \ \ \ \ \ \ \ \isacommand{show}\isamarkupfalse%
+\ {\isachardoublequoteopen}w\ {\isasymsqsubseteq}\ y{\isachardoublequoteclose}\isanewline
+\ \ \ \ \ \ \ \ \isacommand{proof}\isamarkupfalse%
+\ {\isacharminus}\isanewline
+\ \ \ \ \ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
+\ {\isachardoublequoteopen}w\ {\isasymsqsubseteq}\ x\ {\isasymsqinter}\ y{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
+\ fact\isanewline
+\ \ \ \ \ \ \ \ \ \ \isacommand{also}\isamarkupfalse%
+\ \isacommand{have}\isamarkupfalse%
+\ {\isachardoublequoteopen}{\isasymdots}\ {\isasymsqsubseteq}\ y{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse%
+\isanewline
+\ \ \ \ \ \ \ \ \ \ \isacommand{finally}\isamarkupfalse%
+\ \isacommand{show}\isamarkupfalse%
+\ {\isacharquery}thesis\ \isacommand{{\isachardot}}\isamarkupfalse%
+\isanewline
+\ \ \ \ \ \ \ \ \isacommand{qed}\isamarkupfalse%
+\isanewline
+\ \ \ \ \ \ \ \ \isacommand{show}\isamarkupfalse%
+\ {\isachardoublequoteopen}w\ {\isasymsqsubseteq}\ z{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
+\ fact\isanewline
+\ \ \ \ \ \ \isacommand{qed}\isamarkupfalse%
+\isanewline
+\ \ \ \ \isacommand{qed}\isamarkupfalse%
+\isanewline
+\ \ \isacommand{qed}\isamarkupfalse%
+\isanewline
+\isanewline
+\ \ \isacommand{theorem}\isamarkupfalse%
+\ meet{\isacharunderscore}commute{\isacharcolon}\ {\isachardoublequoteopen}x\ {\isasymsqinter}\ y\ {\isacharequal}\ y\ {\isasymsqinter}\ x{\isachardoublequoteclose}\isanewline
+\ \ \isacommand{proof}\isamarkupfalse%
+\ {\isacharparenleft}rule\ meetI{\isacharparenright}\isanewline
+\ \ \ \ \isacommand{show}\isamarkupfalse%
+\ {\isachardoublequoteopen}y\ {\isasymsqinter}\ x\ {\isasymsqsubseteq}\ x{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse%
+\isanewline
+\ \ \ \ \isacommand{show}\isamarkupfalse%
+\ {\isachardoublequoteopen}y\ {\isasymsqinter}\ x\ {\isasymsqsubseteq}\ y{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse%
+\isanewline
+\ \ \ \ \isacommand{fix}\isamarkupfalse%
+\ z\ \isacommand{assume}\isamarkupfalse%
+\ {\isachardoublequoteopen}z\ {\isasymsqsubseteq}\ y{\isachardoublequoteclose}\ \isakeyword{and}\ {\isachardoublequoteopen}z\ {\isasymsqsubseteq}\ x{\isachardoublequoteclose}\isanewline
+\ \ \ \ \isacommand{then}\isamarkupfalse%
+\ \isacommand{show}\isamarkupfalse%
+\ {\isachardoublequoteopen}z\ {\isasymsqsubseteq}\ y\ {\isasymsqinter}\ x{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse%
+\isanewline
+\ \ \isacommand{qed}\isamarkupfalse%
+\isanewline
+\isanewline
+\ \ \isacommand{theorem}\isamarkupfalse%
+\ meet{\isacharunderscore}join{\isacharunderscore}absorb{\isacharcolon}\ {\isachardoublequoteopen}x\ {\isasymsqinter}\ {\isacharparenleft}x\ {\isasymsqunion}\ y{\isacharparenright}\ {\isacharequal}\ x{\isachardoublequoteclose}\isanewline
+\ \ \isacommand{proof}\isamarkupfalse%
+\ {\isacharparenleft}rule\ meetI{\isacharparenright}\isanewline
+\ \ \ \ \isacommand{show}\isamarkupfalse%
+\ {\isachardoublequoteopen}x\ {\isasymsqsubseteq}\ x{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse%
+\isanewline
+\ \ \ \ \isacommand{show}\isamarkupfalse%
+\ {\isachardoublequoteopen}x\ {\isasymsqsubseteq}\ x\ {\isasymsqunion}\ y{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse%
+\isanewline
+\ \ \ \ \isacommand{fix}\isamarkupfalse%
+\ z\ \isacommand{assume}\isamarkupfalse%
+\ {\isachardoublequoteopen}z\ {\isasymsqsubseteq}\ x{\isachardoublequoteclose}\ \isakeyword{and}\ {\isachardoublequoteopen}z\ {\isasymsqsubseteq}\ x\ {\isasymsqunion}\ y{\isachardoublequoteclose}\isanewline
+\ \ \ \ \isacommand{show}\isamarkupfalse%
+\ {\isachardoublequoteopen}z\ {\isasymsqsubseteq}\ x{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
+\ fact\isanewline
+\ \ \isacommand{qed}\isamarkupfalse%
+\isanewline
+\isanewline
+\ \ \isacommand{theorem}\isamarkupfalse%
+\ join{\isacharunderscore}assoc{\isacharcolon}\ {\isachardoublequoteopen}{\isacharparenleft}x\ {\isasymsqunion}\ y{\isacharparenright}\ {\isasymsqunion}\ z\ {\isacharequal}\ x\ {\isasymsqunion}\ {\isacharparenleft}y\ {\isasymsqunion}\ z{\isacharparenright}{\isachardoublequoteclose}\isanewline
+\ \ \isacommand{proof}\isamarkupfalse%
+\ {\isacharparenleft}rule\ joinI{\isacharparenright}\isanewline
+\ \ \ \ \isacommand{show}\isamarkupfalse%
+\ {\isachardoublequoteopen}x\ {\isasymsqunion}\ y\ {\isasymsqsubseteq}\ x\ {\isasymsqunion}\ {\isacharparenleft}y\ {\isasymsqunion}\ z{\isacharparenright}{\isachardoublequoteclose}\isanewline
+\ \ \ \ \isacommand{proof}\isamarkupfalse%
+\isanewline
+\ \ \ \ \ \ \isacommand{show}\isamarkupfalse%
+\ {\isachardoublequoteopen}x\ {\isasymsqsubseteq}\ x\ {\isasymsqunion}\ {\isacharparenleft}y\ {\isasymsqunion}\ z{\isacharparenright}{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse%
+\isanewline
+\ \ \ \ \ \ \isacommand{show}\isamarkupfalse%
+\ {\isachardoublequoteopen}y\ {\isasymsqsubseteq}\ x\ {\isasymsqunion}\ {\isacharparenleft}y\ {\isasymsqunion}\ z{\isacharparenright}{\isachardoublequoteclose}\isanewline
+\ \ \ \ \ \ \isacommand{proof}\isamarkupfalse%
+\ {\isacharminus}\isanewline
+\ \ \ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
+\ {\isachardoublequoteopen}y\ {\isasymsqsubseteq}\ y\ {\isasymsqunion}\ z{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse%
+\isanewline
+\ \ \ \ \ \ \ \ \isacommand{also}\isamarkupfalse%
+\ \isacommand{have}\isamarkupfalse%
+\ {\isachardoublequoteopen}{\isachardot}{\isachardot}{\isachardot}\ {\isasymsqsubseteq}\ x\ {\isasymsqunion}\ {\isacharparenleft}y\ {\isasymsqunion}\ z{\isacharparenright}{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse%
+\isanewline
+\ \ \ \ \ \ \ \ \isacommand{finally}\isamarkupfalse%
+\ \isacommand{show}\isamarkupfalse%
+\ {\isacharquery}thesis\ \isacommand{{\isachardot}}\isamarkupfalse%
+\isanewline
+\ \ \ \ \ \ \isacommand{qed}\isamarkupfalse%
+\isanewline
+\ \ \ \ \isacommand{qed}\isamarkupfalse%
+\isanewline
+\ \ \ \ \isacommand{show}\isamarkupfalse%
+\ {\isachardoublequoteopen}z\ {\isasymsqsubseteq}\ x\ {\isasymsqunion}\ {\isacharparenleft}y\ {\isasymsqunion}\ z{\isacharparenright}{\isachardoublequoteclose}\isanewline
+\ \ \ \ \isacommand{proof}\isamarkupfalse%
+\ {\isacharminus}\isanewline
+\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
+\ {\isachardoublequoteopen}z\ {\isasymsqsubseteq}\ y\ {\isasymsqunion}\ z{\isachardoublequoteclose}\ \ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse%
+\isanewline
+\ \ \ \ \ \ \isacommand{also}\isamarkupfalse%
+\ \isacommand{have}\isamarkupfalse%
+\ {\isachardoublequoteopen}{\isachardot}{\isachardot}{\isachardot}\ {\isasymsqsubseteq}\ x\ {\isasymsqunion}\ {\isacharparenleft}y\ {\isasymsqunion}\ z{\isacharparenright}{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse%
+\isanewline
+\ \ \ \ \ \ \isacommand{finally}\isamarkupfalse%
+\ \isacommand{show}\isamarkupfalse%
+\ {\isacharquery}thesis\ \isacommand{{\isachardot}}\isamarkupfalse%
+\isanewline
+\ \ \ \ \isacommand{qed}\isamarkupfalse%
+\isanewline
+\ \ \ \ \isacommand{fix}\isamarkupfalse%
+\ w\ \isacommand{assume}\isamarkupfalse%
+\ {\isachardoublequoteopen}x\ {\isasymsqunion}\ y\ {\isasymsqsubseteq}\ w{\isachardoublequoteclose}\ \isakeyword{and}\ {\isachardoublequoteopen}z\ {\isasymsqsubseteq}\ w{\isachardoublequoteclose}\isanewline
+\ \ \ \ \isacommand{show}\isamarkupfalse%
+\ {\isachardoublequoteopen}x\ {\isasymsqunion}\ {\isacharparenleft}y\ {\isasymsqunion}\ z{\isacharparenright}\ {\isasymsqsubseteq}\ w{\isachardoublequoteclose}\isanewline
+\ \ \ \ \isacommand{proof}\isamarkupfalse%
+\isanewline
+\ \ \ \ \ \ \isacommand{show}\isamarkupfalse%
+\ {\isachardoublequoteopen}x\ {\isasymsqsubseteq}\ w{\isachardoublequoteclose}\isanewline
+\ \ \ \ \ \ \isacommand{proof}\isamarkupfalse%
+\ {\isacharminus}\isanewline
+\ \ \ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
+\ {\isachardoublequoteopen}x\ {\isasymsqsubseteq}\ x\ {\isasymsqunion}\ y{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse%
+\isanewline
+\ \ \ \ \ \ \ \ \isacommand{also}\isamarkupfalse%
+\ \isacommand{have}\isamarkupfalse%
+\ {\isachardoublequoteopen}{\isasymdots}\ {\isasymsqsubseteq}\ w{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
+\ fact\isanewline
+\ \ \ \ \ \ \ \ \isacommand{finally}\isamarkupfalse%
+\ \isacommand{show}\isamarkupfalse%
+\ {\isacharquery}thesis\ \isacommand{{\isachardot}}\isamarkupfalse%
+\isanewline
+\ \ \ \ \ \ \isacommand{qed}\isamarkupfalse%
+\isanewline
+\ \ \ \ \ \ \isacommand{show}\isamarkupfalse%
+\ {\isachardoublequoteopen}y\ {\isasymsqunion}\ z\ {\isasymsqsubseteq}\ w{\isachardoublequoteclose}\isanewline
+\ \ \ \ \ \ \isacommand{proof}\isamarkupfalse%
+\isanewline
+\ \ \ \ \ \ \ \ \isacommand{show}\isamarkupfalse%
+\ {\isachardoublequoteopen}y\ {\isasymsqsubseteq}\ w{\isachardoublequoteclose}\isanewline
+\ \ \ \ \ \ \ \ \isacommand{proof}\isamarkupfalse%
+\ {\isacharminus}\isanewline
+\ \ \ \ \ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
+\ {\isachardoublequoteopen}y\ {\isasymsqsubseteq}\ x\ {\isasymsqunion}\ y{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse%
+\isanewline
+\ \ \ \ \ \ \ \ \ \ \isacommand{also}\isamarkupfalse%
+\ \isacommand{have}\isamarkupfalse%
+\ {\isachardoublequoteopen}{\isachardot}{\isachardot}{\isachardot}\ {\isasymsqsubseteq}\ w{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
+\ fact\isanewline
+\ \ \ \ \ \ \ \ \ \ \isacommand{finally}\isamarkupfalse%
+\ \isacommand{show}\isamarkupfalse%
+\ {\isacharquery}thesis\ \isacommand{{\isachardot}}\isamarkupfalse%
+\isanewline
+\ \ \ \ \ \ \ \ \isacommand{qed}\isamarkupfalse%
+\isanewline
+\ \ \ \ \ \ \ \ \isacommand{show}\isamarkupfalse%
+\ {\isachardoublequoteopen}z\ {\isasymsqsubseteq}\ w{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
+\ fact\isanewline
+\ \ \ \ \ \ \isacommand{qed}\isamarkupfalse%
+\isanewline
+\ \ \ \ \isacommand{qed}\isamarkupfalse%
+\isanewline
+\ \ \isacommand{qed}\isamarkupfalse%
+\isanewline
+\isanewline
+\ \ \isacommand{theorem}\isamarkupfalse%
+\ join{\isacharunderscore}commute{\isacharcolon}\ {\isachardoublequoteopen}x\ {\isasymsqunion}\ y\ {\isacharequal}\ y\ {\isasymsqunion}\ x{\isachardoublequoteclose}\isanewline
+\ \ \isacommand{proof}\isamarkupfalse%
+\ {\isacharparenleft}rule\ joinI{\isacharparenright}\isanewline
+\ \ \ \ \isacommand{show}\isamarkupfalse%
+\ {\isachardoublequoteopen}x\ {\isasymsqsubseteq}\ y\ {\isasymsqunion}\ x{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse%
+\isanewline
+\ \ \ \ \isacommand{show}\isamarkupfalse%
+\ {\isachardoublequoteopen}y\ {\isasymsqsubseteq}\ y\ {\isasymsqunion}\ x{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse%
+\isanewline
+\ \ \ \ \isacommand{fix}\isamarkupfalse%
+\ z\ \isacommand{assume}\isamarkupfalse%
+\ {\isachardoublequoteopen}y\ {\isasymsqsubseteq}\ z{\isachardoublequoteclose}\ \isakeyword{and}\ {\isachardoublequoteopen}x\ {\isasymsqsubseteq}\ z{\isachardoublequoteclose}\isanewline
+\ \ \ \ \isacommand{then}\isamarkupfalse%
+\ \isacommand{show}\isamarkupfalse%
+\ {\isachardoublequoteopen}y\ {\isasymsqunion}\ x\ {\isasymsqsubseteq}\ z{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse%
+\isanewline
+\ \ \isacommand{qed}\isamarkupfalse%
+\isanewline
+\isanewline
+\ \ \isacommand{theorem}\isamarkupfalse%
+\ join{\isacharunderscore}meet{\isacharunderscore}absorb{\isacharcolon}\ {\isachardoublequoteopen}x\ {\isasymsqunion}\ {\isacharparenleft}x\ {\isasymsqinter}\ y{\isacharparenright}\ {\isacharequal}\ x{\isachardoublequoteclose}\isanewline
+\ \ \isacommand{proof}\isamarkupfalse%
+\ {\isacharparenleft}rule\ joinI{\isacharparenright}\isanewline
+\ \ \ \ \isacommand{show}\isamarkupfalse%
+\ {\isachardoublequoteopen}x\ {\isasymsqsubseteq}\ x{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse%
+\isanewline
+\ \ \ \ \isacommand{show}\isamarkupfalse%
+\ {\isachardoublequoteopen}x\ {\isasymsqinter}\ y\ {\isasymsqsubseteq}\ x{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse%
+\isanewline
+\ \ \ \ \isacommand{fix}\isamarkupfalse%
+\ z\ \isacommand{assume}\isamarkupfalse%
+\ {\isachardoublequoteopen}x\ {\isasymsqsubseteq}\ z{\isachardoublequoteclose}\ \isakeyword{and}\ {\isachardoublequoteopen}x\ {\isasymsqinter}\ y\ {\isasymsqsubseteq}\ z{\isachardoublequoteclose}\isanewline
+\ \ \ \ \isacommand{show}\isamarkupfalse%
+\ {\isachardoublequoteopen}x\ {\isasymsqsubseteq}\ z{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
+\ fact\isanewline
+\ \ \isacommand{qed}\isamarkupfalse%
+\isanewline
+\isanewline
+\ \ \isacommand{theorem}\isamarkupfalse%
+\ meet{\isacharunderscore}idem{\isacharcolon}\ {\isachardoublequoteopen}x\ {\isasymsqinter}\ x\ {\isacharequal}\ x{\isachardoublequoteclose}\isanewline
+\ \ \isacommand{proof}\isamarkupfalse%
+\ {\isacharminus}\isanewline
+\ \ \ \ \isacommand{have}\isamarkupfalse%
+\ {\isachardoublequoteopen}x\ {\isasymsqinter}\ {\isacharparenleft}x\ {\isasymsqunion}\ {\isacharparenleft}x\ {\isasymsqinter}\ x{\isacharparenright}{\isacharparenright}\ {\isacharequal}\ x{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
+\ {\isacharparenleft}rule\ meet{\isacharunderscore}join{\isacharunderscore}absorb{\isacharparenright}\isanewline
+\ \ \ \ \isacommand{also}\isamarkupfalse%
+\ \isacommand{have}\isamarkupfalse%
+\ {\isachardoublequoteopen}x\ {\isasymsqunion}\ {\isacharparenleft}x\ {\isasymsqinter}\ x{\isacharparenright}\ {\isacharequal}\ x{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
+\ {\isacharparenleft}rule\ join{\isacharunderscore}meet{\isacharunderscore}absorb{\isacharparenright}\isanewline
+\ \ \ \ \isacommand{finally}\isamarkupfalse%
+\ \isacommand{show}\isamarkupfalse%
+\ {\isacharquery}thesis\ \isacommand{{\isachardot}}\isamarkupfalse%
+\isanewline
+\ \ \isacommand{qed}\isamarkupfalse%
+\isanewline
+\isanewline
+\ \ \isacommand{theorem}\isamarkupfalse%
+\ meet{\isacharunderscore}related\ {\isacharbrackleft}elim{\isacharquery}{\isacharbrackright}{\isacharcolon}\ {\isachardoublequoteopen}x\ {\isasymsqsubseteq}\ y\ {\isasymLongrightarrow}\ x\ {\isasymsqinter}\ y\ {\isacharequal}\ x{\isachardoublequoteclose}\isanewline
+\ \ \isacommand{proof}\isamarkupfalse%
+\ {\isacharparenleft}rule\ meetI{\isacharparenright}\isanewline
+\ \ \ \ \isacommand{assume}\isamarkupfalse%
+\ {\isachardoublequoteopen}x\ {\isasymsqsubseteq}\ y{\isachardoublequoteclose}\isanewline
+\ \ \ \ \isacommand{show}\isamarkupfalse%
+\ {\isachardoublequoteopen}x\ {\isasymsqsubseteq}\ x{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse%
+\isanewline
+\ \ \ \ \isacommand{show}\isamarkupfalse%
+\ {\isachardoublequoteopen}x\ {\isasymsqsubseteq}\ y{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
+\ fact\isanewline
+\ \ \ \ \isacommand{fix}\isamarkupfalse%
+\ z\ \isacommand{assume}\isamarkupfalse%
+\ {\isachardoublequoteopen}z\ {\isasymsqsubseteq}\ x{\isachardoublequoteclose}\ \isakeyword{and}\ {\isachardoublequoteopen}z\ {\isasymsqsubseteq}\ y{\isachardoublequoteclose}\isanewline
+\ \ \ \ \isacommand{show}\isamarkupfalse%
+\ {\isachardoublequoteopen}z\ {\isasymsqsubseteq}\ x{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
+\ fact\isanewline
+\ \ \isacommand{qed}\isamarkupfalse%
+\isanewline
+\isanewline
+\ \ \isacommand{theorem}\isamarkupfalse%
+\ meet{\isacharunderscore}related{\isadigit{2}}\ {\isacharbrackleft}elim{\isacharquery}{\isacharbrackright}{\isacharcolon}\ {\isachardoublequoteopen}y\ {\isasymsqsubseteq}\ x\ {\isasymLongrightarrow}\ x\ {\isasymsqinter}\ y\ {\isacharequal}\ y{\isachardoublequoteclose}\isanewline
+\ \ \ \ \isacommand{by}\isamarkupfalse%
+\ {\isacharparenleft}drule\ meet{\isacharunderscore}related{\isacharparenright}\ {\isacharparenleft}simp\ add{\isacharcolon}\ meet{\isacharunderscore}commute{\isacharparenright}\isanewline
+\isanewline
+\ \ \isacommand{theorem}\isamarkupfalse%
+\ join{\isacharunderscore}related\ {\isacharbrackleft}elim{\isacharquery}{\isacharbrackright}{\isacharcolon}\ {\isachardoublequoteopen}x\ {\isasymsqsubseteq}\ y\ {\isasymLongrightarrow}\ x\ {\isasymsqunion}\ y\ {\isacharequal}\ y{\isachardoublequoteclose}\isanewline
+\ \ \isacommand{proof}\isamarkupfalse%
+\ {\isacharparenleft}rule\ joinI{\isacharparenright}\isanewline
+\ \ \ \ \isacommand{assume}\isamarkupfalse%
+\ {\isachardoublequoteopen}x\ {\isasymsqsubseteq}\ y{\isachardoublequoteclose}\isanewline
+\ \ \ \ \isacommand{show}\isamarkupfalse%
+\ {\isachardoublequoteopen}y\ {\isasymsqsubseteq}\ y{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse%
+\isanewline
+\ \ \ \ \isacommand{show}\isamarkupfalse%
+\ {\isachardoublequoteopen}x\ {\isasymsqsubseteq}\ y{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
+\ fact\isanewline
+\ \ \ \ \isacommand{fix}\isamarkupfalse%
+\ z\ \isacommand{assume}\isamarkupfalse%
+\ {\isachardoublequoteopen}x\ {\isasymsqsubseteq}\ z{\isachardoublequoteclose}\ \isakeyword{and}\ {\isachardoublequoteopen}y\ {\isasymsqsubseteq}\ z{\isachardoublequoteclose}\isanewline
+\ \ \ \ \isacommand{show}\isamarkupfalse%
+\ {\isachardoublequoteopen}y\ {\isasymsqsubseteq}\ z{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
+\ fact\isanewline
+\ \ \isacommand{qed}\isamarkupfalse%
+\isanewline
+\isanewline
+\ \ \isacommand{theorem}\isamarkupfalse%
+\ join{\isacharunderscore}related{\isadigit{2}}\ {\isacharbrackleft}elim{\isacharquery}{\isacharbrackright}{\isacharcolon}\ {\isachardoublequoteopen}y\ {\isasymsqsubseteq}\ x\ {\isasymLongrightarrow}\ x\ {\isasymsqunion}\ y\ {\isacharequal}\ x{\isachardoublequoteclose}\isanewline
+\ \ \ \ \isacommand{by}\isamarkupfalse%
+\ {\isacharparenleft}drule\ join{\isacharunderscore}related{\isacharparenright}\ {\isacharparenleft}simp\ add{\isacharcolon}\ join{\isacharunderscore}commute{\isacharparenright}\isanewline
+\isanewline
+\ \ \isacommand{theorem}\isamarkupfalse%
+\ meet{\isacharunderscore}connection{\isacharcolon}\ {\isachardoublequoteopen}{\isacharparenleft}x\ {\isasymsqsubseteq}\ y{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}x\ {\isasymsqinter}\ y\ {\isacharequal}\ x{\isacharparenright}{\isachardoublequoteclose}\isanewline
+\ \ \isacommand{proof}\isamarkupfalse%
+\isanewline
+\ \ \ \ \isacommand{assume}\isamarkupfalse%
+\ {\isachardoublequoteopen}x\ {\isasymsqsubseteq}\ y{\isachardoublequoteclose}\isanewline
+\ \ \ \ \isacommand{then}\isamarkupfalse%
+\ \isacommand{have}\isamarkupfalse%
+\ {\isachardoublequoteopen}is{\isacharunderscore}inf\ x\ y\ x{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse%
+\isanewline
+\ \ \ \ \isacommand{then}\isamarkupfalse%
+\ \isacommand{show}\isamarkupfalse%
+\ {\isachardoublequoteopen}x\ {\isasymsqinter}\ y\ {\isacharequal}\ x{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse%
+\isanewline
+\ \ \isacommand{next}\isamarkupfalse%
+\isanewline
+\ \ \ \ \isacommand{have}\isamarkupfalse%
+\ {\isachardoublequoteopen}x\ {\isasymsqinter}\ y\ {\isasymsqsubseteq}\ y{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse%
+\isanewline
+\ \ \ \ \isacommand{also}\isamarkupfalse%
+\ \isacommand{assume}\isamarkupfalse%
+\ {\isachardoublequoteopen}x\ {\isasymsqinter}\ y\ {\isacharequal}\ x{\isachardoublequoteclose}\isanewline
+\ \ \ \ \isacommand{finally}\isamarkupfalse%
+\ \isacommand{show}\isamarkupfalse%
+\ {\isachardoublequoteopen}x\ {\isasymsqsubseteq}\ y{\isachardoublequoteclose}\ \isacommand{{\isachardot}}\isamarkupfalse%
+\isanewline
+\ \ \isacommand{qed}\isamarkupfalse%
+\isanewline
+\isanewline
+\ \ \isacommand{theorem}\isamarkupfalse%
+\ join{\isacharunderscore}connection{\isacharcolon}\ {\isachardoublequoteopen}{\isacharparenleft}x\ {\isasymsqsubseteq}\ y{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}x\ {\isasymsqunion}\ y\ {\isacharequal}\ y{\isacharparenright}{\isachardoublequoteclose}\isanewline
+\ \ \isacommand{proof}\isamarkupfalse%
+\isanewline
+\ \ \ \ \isacommand{assume}\isamarkupfalse%
+\ {\isachardoublequoteopen}x\ {\isasymsqsubseteq}\ y{\isachardoublequoteclose}\isanewline
+\ \ \ \ \isacommand{then}\isamarkupfalse%
+\ \isacommand{have}\isamarkupfalse%
+\ {\isachardoublequoteopen}is{\isacharunderscore}sup\ x\ y\ y{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse%
+\isanewline
+\ \ \ \ \isacommand{then}\isamarkupfalse%
+\ \isacommand{show}\isamarkupfalse%
+\ {\isachardoublequoteopen}x\ {\isasymsqunion}\ y\ {\isacharequal}\ y{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse%
+\isanewline
+\ \ \isacommand{next}\isamarkupfalse%
+\isanewline
+\ \ \ \ \isacommand{have}\isamarkupfalse%
+\ {\isachardoublequoteopen}x\ {\isasymsqsubseteq}\ x\ {\isasymsqunion}\ y{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse%
+\isanewline
+\ \ \ \ \isacommand{also}\isamarkupfalse%
+\ \isacommand{assume}\isamarkupfalse%
+\ {\isachardoublequoteopen}x\ {\isasymsqunion}\ y\ {\isacharequal}\ y{\isachardoublequoteclose}\isanewline
+\ \ \ \ \isacommand{finally}\isamarkupfalse%
+\ \isacommand{show}\isamarkupfalse%
+\ {\isachardoublequoteopen}x\ {\isasymsqsubseteq}\ y{\isachardoublequoteclose}\ \isacommand{{\isachardot}}\isamarkupfalse%
+\isanewline
+\ \ \isacommand{qed}\isamarkupfalse%
+\isanewline
+\isanewline
+\ \ \isacommand{theorem}\isamarkupfalse%
+\ meet{\isacharunderscore}connection{\isadigit{2}}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharparenleft}x\ {\isasymsqsubseteq}\ y{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}y\ {\isasymsqinter}\ x\ {\isacharequal}\ x{\isacharparenright}{\isachardoublequoteclose}\isanewline
+\ \ \ \ \isacommand{using}\isamarkupfalse%
+\ meet{\isacharunderscore}commute\ meet{\isacharunderscore}connection\ \isacommand{by}\isamarkupfalse%
+\ simp\isanewline
+\isanewline
+\ \ \isacommand{theorem}\isamarkupfalse%
+\ join{\isacharunderscore}connection{\isadigit{2}}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharparenleft}x\ {\isasymsqsubseteq}\ y{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}x\ {\isasymsqunion}\ y\ {\isacharequal}\ y{\isacharparenright}{\isachardoublequoteclose}\isanewline
+\ \ \ \ \isacommand{using}\isamarkupfalse%
+\ join{\isacharunderscore}commute\ join{\isacharunderscore}connection\ \isacommand{by}\isamarkupfalse%
+\ simp%
+\begin{isamarkuptext}%
+Naming according to Jacobson I, p.\ 459.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\ \ \isacommand{lemmas}\isamarkupfalse%
+\ L{\isadigit{1}}\ {\isacharequal}\ join{\isacharunderscore}commute\ meet{\isacharunderscore}commute\isanewline
+\ \ \isacommand{lemmas}\isamarkupfalse%
+\ L{\isadigit{2}}\ {\isacharequal}\ join{\isacharunderscore}assoc\ meet{\isacharunderscore}assoc\isanewline
+\ \ \isanewline
+\ \ \isacommand{lemmas}\isamarkupfalse%
+\ L{\isadigit{4}}\ {\isacharequal}\ join{\isacharunderscore}meet{\isacharunderscore}absorb\ meet{\isacharunderscore}join{\isacharunderscore}absorb%
+\endisataginvisible
+{\isafoldinvisible}%
+%
+\isadeliminvisible
+\isanewline
+%
+\endisadeliminvisible
+\isanewline
+\ \ \isacommand{end}\isamarkupfalse%
+%
+\begin{isamarkuptext}%
+Locales for total orders and distributive lattices follow.
+  Each comes with an example theorem.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\ \ \isacommand{locale}\isamarkupfalse%
+\ total{\isacharunderscore}order\ {\isacharequal}\ partial{\isacharunderscore}order\ {\isacharplus}\isanewline
+\ \ \ \ \isakeyword{assumes}\ total{\isacharcolon}\ {\isachardoublequoteopen}x\ {\isasymsqsubseteq}\ y\ {\isasymor}\ y\ {\isasymsqsubseteq}\ x{\isachardoublequoteclose}\isanewline
+\isanewline
+\ \ \isacommand{lemma}\isamarkupfalse%
+\ {\isacharparenleft}\isakeyword{in}\ total{\isacharunderscore}order{\isacharparenright}\ less{\isacharunderscore}total{\isacharcolon}\ {\isachardoublequoteopen}x\ {\isasymsqsubset}\ y\ {\isasymor}\ x\ {\isacharequal}\ y\ {\isasymor}\ y\ {\isasymsqsubset}\ x{\isachardoublequoteclose}\isanewline
+%
+\isadelimproof
+\ \ \ \ %
+\endisadelimproof
+%
+\isatagproof
+\isacommand{using}\isamarkupfalse%
+\ total\isanewline
+\ \ \ \ \isacommand{by}\isamarkupfalse%
+\ {\isacharparenleft}unfold\ less{\isacharunderscore}def{\isacharparenright}\ blast%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+\isanewline
+%
+\endisadelimproof
+\isanewline
+\ \ \isacommand{locale}\isamarkupfalse%
+\ distrib{\isacharunderscore}lattice\ {\isacharequal}\ lattice\ {\isacharplus}\isanewline
+\ \ \ \ \isakeyword{assumes}\ meet{\isacharunderscore}distr{\isacharcolon}\isanewline
+\ \ \ \ \ \ {\isachardoublequoteopen}lattice{\isachardot}meet\ le\ x\ {\isacharparenleft}lattice{\isachardot}join\ le\ y\ z{\isacharparenright}\ {\isacharequal}\isanewline
+\ \ \ \ \ \ lattice{\isachardot}join\ le\ {\isacharparenleft}lattice{\isachardot}meet\ le\ x\ y{\isacharparenright}\ {\isacharparenleft}lattice{\isachardot}meet\ le\ x\ z{\isacharparenright}{\isachardoublequoteclose}\isanewline
+\isanewline
+\ \ \isacommand{lemma}\isamarkupfalse%
+\ {\isacharparenleft}\isakeyword{in}\ distrib{\isacharunderscore}lattice{\isacharparenright}\ join{\isacharunderscore}distr{\isacharcolon}\isanewline
+\ \ \ \ {\isachardoublequoteopen}x\ {\isasymsqunion}\ {\isacharparenleft}y\ {\isasymsqinter}\ z{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}x\ {\isasymsqunion}\ y{\isacharparenright}\ {\isasymsqinter}\ {\isacharparenleft}x\ {\isasymsqunion}\ z{\isacharparenright}{\isachardoublequoteclose}\ \ \isanewline
+%
+\isadelimproof
+\ \ \ \ %
+\endisadelimproof
+%
+\isatagproof
+\isacommand{proof}\isamarkupfalse%
+\ {\isacharminus}\isanewline
+\ \ \ \ \isacommand{have}\isamarkupfalse%
+\ {\isachardoublequoteopen}x\ {\isasymsqunion}\ {\isacharparenleft}y\ {\isasymsqinter}\ z{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}x\ {\isasymsqunion}\ {\isacharparenleft}x\ {\isasymsqinter}\ z{\isacharparenright}{\isacharparenright}\ {\isasymsqunion}\ {\isacharparenleft}y\ {\isasymsqinter}\ z{\isacharparenright}{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
+\ {\isacharparenleft}simp\ add{\isacharcolon}\ L{\isadigit{4}}{\isacharparenright}\isanewline
+\ \ \ \ \isacommand{also}\isamarkupfalse%
+\ \isacommand{have}\isamarkupfalse%
+\ {\isachardoublequoteopen}{\isachardot}{\isachardot}{\isachardot}\ {\isacharequal}\ x\ {\isasymsqunion}\ {\isacharparenleft}{\isacharparenleft}x\ {\isasymsqinter}\ z{\isacharparenright}\ {\isasymsqunion}\ {\isacharparenleft}y\ {\isasymsqinter}\ z{\isacharparenright}{\isacharparenright}{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
+\ {\isacharparenleft}simp\ add{\isacharcolon}\ L{\isadigit{2}}{\isacharparenright}\isanewline
+\ \ \ \ \isacommand{also}\isamarkupfalse%
+\ \isacommand{have}\isamarkupfalse%
+\ {\isachardoublequoteopen}{\isachardot}{\isachardot}{\isachardot}\ {\isacharequal}\ x\ {\isasymsqunion}\ {\isacharparenleft}{\isacharparenleft}x\ {\isasymsqunion}\ y{\isacharparenright}\ {\isasymsqinter}\ z{\isacharparenright}{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
+\ {\isacharparenleft}simp\ add{\isacharcolon}\ L{\isadigit{1}}\ meet{\isacharunderscore}distr{\isacharparenright}\isanewline
+\ \ \ \ \isacommand{also}\isamarkupfalse%
+\ \isacommand{have}\isamarkupfalse%
+\ {\isachardoublequoteopen}{\isachardot}{\isachardot}{\isachardot}\ {\isacharequal}\ {\isacharparenleft}{\isacharparenleft}x\ {\isasymsqunion}\ y{\isacharparenright}\ {\isasymsqinter}\ x{\isacharparenright}\ {\isasymsqunion}\ {\isacharparenleft}{\isacharparenleft}x\ {\isasymsqunion}\ y{\isacharparenright}\ {\isasymsqinter}\ z{\isacharparenright}{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
+\ {\isacharparenleft}simp\ add{\isacharcolon}\ L{\isadigit{1}}\ L{\isadigit{4}}{\isacharparenright}\isanewline
+\ \ \ \ \isacommand{also}\isamarkupfalse%
+\ \isacommand{have}\isamarkupfalse%
+\ {\isachardoublequoteopen}{\isachardot}{\isachardot}{\isachardot}\ {\isacharequal}\ {\isacharparenleft}x\ {\isasymsqunion}\ y{\isacharparenright}\ {\isasymsqinter}\ {\isacharparenleft}x\ {\isasymsqunion}\ z{\isacharparenright}{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
+\ {\isacharparenleft}simp\ add{\isacharcolon}\ meet{\isacharunderscore}distr{\isacharparenright}\isanewline
+\ \ \ \ \isacommand{finally}\isamarkupfalse%
+\ \isacommand{show}\isamarkupfalse%
+\ {\isacharquery}thesis\ \isacommand{{\isachardot}}\isamarkupfalse%
+\isanewline
+\ \ \isacommand{qed}\isamarkupfalse%
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\begin{isamarkuptext}%
+The locale hierachy obtained through these declarations is shown in Figure~\ref{fig:lattices}(a).
+
+\begin{figure}
+\hrule \vspace{2ex}
+\begin{center}
+\subfigure[Declared hierachy]{
+\begin{tikzpicture}
+  \node (po) at (0,0) {\isa{partial{\isacharunderscore}order}};
+  \node (lat) at (-1.5,-1) {\isa{lattice}};
+  \node (dlat) at (-1.5,-2) {\isa{distrib{\isacharunderscore}lattice}};
+  \node (to) at (1.5,-1) {\isa{total{\isacharunderscore}order}};
+  \draw (po) -- (lat);
+  \draw (lat) -- (dlat);
+  \draw (po) -- (to);
+%  \draw[->, dashed] (lat) -- (to);
+\end{tikzpicture}
+} \\
+\subfigure[Total orders are lattices]{
+\begin{tikzpicture}
+  \node (po) at (0,0) {\isa{partial{\isacharunderscore}order}};
+  \node (lat) at (0,-1) {\isa{lattice}};
+  \node (dlat) at (-1.5,-2) {\isa{distrib{\isacharunderscore}lattice}};
+  \node (to) at (1.5,-2) {\isa{total{\isacharunderscore}order}};
+  \draw (po) -- (lat);
+  \draw (lat) -- (dlat);
+  \draw (lat) -- (to);
+%  \draw[->, dashed] (dlat) -- (to);
+\end{tikzpicture}
+} \quad
+\subfigure[Total orders are distributive lattices]{
+\begin{tikzpicture}
+  \node (po) at (0,0) {\isa{partial{\isacharunderscore}order}};
+  \node (lat) at (0,-1) {\isa{lattice}};
+  \node (dlat) at (0,-2) {\isa{distrib{\isacharunderscore}lattice}};
+  \node (to) at (0,-3) {\isa{total{\isacharunderscore}order}};
+  \draw (po) -- (lat);
+  \draw (lat) -- (dlat);
+  \draw (dlat) -- (to);
+\end{tikzpicture}
+}
+\end{center}
+\hrule
+\caption{Hierarchy of Lattice Locales.}
+\label{fig:lattices}
+\end{figure}%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsection{Changing the Locale Hierarchy%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+\label{sec:changing-the-hierarchy}
+
+  Total orders are lattices.  Hence, by deriving the lattice
+  axioms for total orders, the hierarchy may be changed
+  and \isa{lattice} be placed between \isa{partial{\isacharunderscore}order}
+  and \isa{total{\isacharunderscore}order}, as shown in Figure~\ref{fig:lattices}(b).
+  Changes to the locale hierarchy may be declared
+  with the \isakeyword{interpretation} command.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimvisible
+\ \ %
+\endisadelimvisible
+%
+\isatagvisible
+\isacommand{interpretation}\isamarkupfalse%
+\ total{\isacharunderscore}order\ {\isasymsubseteq}\ lattice%
+\begin{isamarkuptxt}%
+This enters the context of locale \isa{total{\isacharunderscore}order}, in
+  which the goal \begin{isabelle}%
+\ {\isadigit{1}}{\isachardot}\ lattice\ op\ {\isasymsqsubseteq}%
+\end{isabelle} must be shown.  First, the
+  locale predicate needs to be unfolded --- for example using its
+  definition or by introduction rules
+  provided by the locale package.  The methods \isa{intro{\isacharunderscore}locales}
+  and \isa{unfold{\isacharunderscore}locales} automate this.  They are aware of the
+  current context and dependencies between locales and automatically
+  discharge goals implied by these.  While \isa{unfold{\isacharunderscore}locales}
+  always unfolds locale predicates to assumptions, \isa{intro{\isacharunderscore}locales} only unfolds definitions along the locale
+  hierarchy, leaving a goal consisting of predicates defined by the
+  locale package.  Occasionally the latter is of advantage since the goal
+  is smaller.
+
+  For the current goal, we would like to get hold of
+  the assumptions of \isa{lattice}, hence \isa{unfold{\isacharunderscore}locales}
+  is appropriate.%
+\end{isamarkuptxt}%
+\isamarkuptrue%
+\ \ \isacommand{proof}\isamarkupfalse%
+\ unfold{\isacharunderscore}locales%
+\begin{isamarkuptxt}%
+Since both \isa{lattice} and \isa{total{\isacharunderscore}order}
+  inherit \isa{partial{\isacharunderscore}order}, the assumptions of the latter are
+  discharged, and the only subgoals that remain are the assumptions
+  introduced in \isa{lattice} \begin{isabelle}%
+\ {\isadigit{1}}{\isachardot}\ {\isasymAnd}x\ y{\isachardot}\ {\isasymexists}inf{\isachardot}\ is{\isacharunderscore}inf\ x\ y\ inf\isanewline
+\ {\isadigit{2}}{\isachardot}\ {\isasymAnd}x\ y{\isachardot}\ {\isasymexists}sup{\isachardot}\ is{\isacharunderscore}sup\ x\ y\ sup%
+\end{isabelle}
+  The proof for the first subgoal is%
+\end{isamarkuptxt}%
+\isamarkuptrue%
+\ \ \ \ \isacommand{fix}\isamarkupfalse%
+\ x\ y\isanewline
+\ \ \ \ \isacommand{from}\isamarkupfalse%
+\ total\ \isacommand{have}\isamarkupfalse%
+\ {\isachardoublequoteopen}is{\isacharunderscore}inf\ x\ y\ {\isacharparenleft}if\ x\ {\isasymsqsubseteq}\ y\ then\ x\ else\ y{\isacharparenright}{\isachardoublequoteclose}\isanewline
+\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
+\ {\isacharparenleft}auto\ simp{\isacharcolon}\ is{\isacharunderscore}inf{\isacharunderscore}def{\isacharparenright}\isanewline
+\ \ \ \ \isacommand{then}\isamarkupfalse%
+\ \isacommand{show}\isamarkupfalse%
+\ {\isachardoublequoteopen}{\isasymexists}inf{\isachardot}\ is{\isacharunderscore}inf\ x\ y\ inf{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse%
+%
+\begin{isamarkuptxt}%
+The proof for the second subgoal is analogous and not
+  reproduced here.%
+\end{isamarkuptxt}%
+\isamarkuptrue%
+%
+\endisatagvisible
+{\isafoldvisible}%
+%
+\isadelimvisible
+%
+\endisadelimvisible
+%
+\isadeliminvisible
+\ \ %
+\endisadeliminvisible
+%
+\isataginvisible
+\isacommand{next}\isamarkupfalse%
+\isanewline
+\ \ \ \ \isacommand{fix}\isamarkupfalse%
+\ x\ y\isanewline
+\ \ \ \ \isacommand{from}\isamarkupfalse%
+\ total\ \isacommand{have}\isamarkupfalse%
+\ {\isachardoublequoteopen}is{\isacharunderscore}sup\ x\ y\ {\isacharparenleft}if\ x\ {\isasymsqsubseteq}\ y\ then\ y\ else\ x{\isacharparenright}{\isachardoublequoteclose}\isanewline
+\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
+\ {\isacharparenleft}auto\ simp{\isacharcolon}\ is{\isacharunderscore}sup{\isacharunderscore}def{\isacharparenright}\isanewline
+\ \ \ \ \isacommand{then}\isamarkupfalse%
+\ \isacommand{show}\isamarkupfalse%
+\ {\isachardoublequoteopen}{\isasymexists}sup{\isachardot}\ is{\isacharunderscore}sup\ x\ y\ sup{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse%
+%
+\endisataginvisible
+{\isafoldinvisible}%
+%
+\isadeliminvisible
+%
+\endisadeliminvisible
+%
+\isadelimvisible
+\ %
+\endisadelimvisible
+%
+\isatagvisible
+\isacommand{qed}\isamarkupfalse%
+%
+\endisatagvisible
+{\isafoldvisible}%
+%
+\isadelimvisible
+%
+\endisadelimvisible
+%
+\begin{isamarkuptext}%
+Similarly, total orders are distributive lattices.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\ \ \isacommand{interpretation}\isamarkupfalse%
+\ total{\isacharunderscore}order\ {\isasymsubseteq}\ distrib{\isacharunderscore}lattice\isanewline
+%
+\isadelimproof
+\ \ %
+\endisadelimproof
+%
+\isatagproof
+\isacommand{proof}\isamarkupfalse%
+\ unfold{\isacharunderscore}locales\isanewline
+\ \ \ \ \isacommand{fix}\isamarkupfalse%
+\ x\ y\ z\isanewline
+\ \ \ \ \isacommand{show}\isamarkupfalse%
+\ {\isachardoublequoteopen}x\ {\isasymsqinter}\ {\isacharparenleft}y\ {\isasymsqunion}\ z{\isacharparenright}\ {\isacharequal}\ x\ {\isasymsqinter}\ y\ {\isasymsqunion}\ x\ {\isasymsqinter}\ z{\isachardoublequoteclose}\ {\isacharparenleft}\isakeyword{is}\ {\isachardoublequoteopen}{\isacharquery}l\ {\isacharequal}\ {\isacharquery}r{\isachardoublequoteclose}{\isacharparenright}%
+\begin{isamarkuptxt}%
+Jacobson I, p.\ 462%
+\end{isamarkuptxt}%
+\isamarkuptrue%
+\ \ \ \ \isacommand{proof}\isamarkupfalse%
+\ {\isacharminus}\isanewline
+\ \ \ \ \ \ \isacommand{{\isacharbraceleft}}\isamarkupfalse%
+\ \isacommand{assume}\isamarkupfalse%
+\ c{\isacharcolon}\ {\isachardoublequoteopen}y\ {\isasymsqsubseteq}\ x{\isachardoublequoteclose}\ {\isachardoublequoteopen}z\ {\isasymsqsubseteq}\ x{\isachardoublequoteclose}\isanewline
+\ \ \ \ \ \ \ \ \isacommand{from}\isamarkupfalse%
+\ c\ \isacommand{have}\isamarkupfalse%
+\ {\isachardoublequoteopen}{\isacharquery}l\ {\isacharequal}\ y\ {\isasymsqunion}\ z{\isachardoublequoteclose}\isanewline
+\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
+\ {\isacharparenleft}metis\ c\ join{\isacharunderscore}connection{\isadigit{2}}\ join{\isacharunderscore}related{\isadigit{2}}\ meet{\isacharunderscore}related{\isadigit{2}}\ total{\isacharparenright}\isanewline
+\ \ \ \ \ \ \ \ \isacommand{also}\isamarkupfalse%
+\ \isacommand{from}\isamarkupfalse%
+\ c\ \isacommand{have}\isamarkupfalse%
+\ {\isachardoublequoteopen}{\isachardot}{\isachardot}{\isachardot}\ {\isacharequal}\ {\isacharquery}r{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
+\ {\isacharparenleft}metis\ meet{\isacharunderscore}related{\isadigit{2}}{\isacharparenright}\isanewline
+\ \ \ \ \ \ \ \ \isacommand{finally}\isamarkupfalse%
+\ \isacommand{have}\isamarkupfalse%
+\ {\isachardoublequoteopen}{\isacharquery}l\ {\isacharequal}\ {\isacharquery}r{\isachardoublequoteclose}\ \isacommand{{\isachardot}}\isamarkupfalse%
+\ \isacommand{{\isacharbraceright}}\isamarkupfalse%
+\isanewline
+\ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
+\isanewline
+\ \ \ \ \ \ \isacommand{{\isacharbraceleft}}\isamarkupfalse%
+\ \isacommand{assume}\isamarkupfalse%
+\ c{\isacharcolon}\ {\isachardoublequoteopen}x\ {\isasymsqsubseteq}\ y\ {\isasymor}\ x\ {\isasymsqsubseteq}\ z{\isachardoublequoteclose}\isanewline
+\ \ \ \ \ \ \ \ \isacommand{from}\isamarkupfalse%
+\ c\ \isacommand{have}\isamarkupfalse%
+\ {\isachardoublequoteopen}{\isacharquery}l\ {\isacharequal}\ x{\isachardoublequoteclose}\isanewline
+\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
+\ {\isacharparenleft}metis\ join{\isacharunderscore}connection{\isadigit{2}}\ join{\isacharunderscore}related{\isadigit{2}}\ meet{\isacharunderscore}connection\ total\ trans{\isacharparenright}\isanewline
+\ \ \ \ \ \ \ \ \isacommand{also}\isamarkupfalse%
+\ \isacommand{from}\isamarkupfalse%
+\ c\ \isacommand{have}\isamarkupfalse%
+\ {\isachardoublequoteopen}{\isachardot}{\isachardot}{\isachardot}\ {\isacharequal}\ {\isacharquery}r{\isachardoublequoteclose}\isanewline
+\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
+\ {\isacharparenleft}metis\ join{\isacharunderscore}commute\ join{\isacharunderscore}related{\isadigit{2}}\ meet{\isacharunderscore}connection\ meet{\isacharunderscore}related{\isadigit{2}}\ total{\isacharparenright}\isanewline
+\ \ \ \ \ \ \ \ \isacommand{finally}\isamarkupfalse%
+\ \isacommand{have}\isamarkupfalse%
+\ {\isachardoublequoteopen}{\isacharquery}l\ {\isacharequal}\ {\isacharquery}r{\isachardoublequoteclose}\ \isacommand{{\isachardot}}\isamarkupfalse%
+\ \isacommand{{\isacharbraceright}}\isamarkupfalse%
+\isanewline
+\ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
+\ \isacommand{note}\isamarkupfalse%
+\ total\isanewline
+\ \ \ \ \ \ \isacommand{ultimately}\isamarkupfalse%
+\ \isacommand{show}\isamarkupfalse%
+\ {\isacharquery}thesis\ \isacommand{by}\isamarkupfalse%
+\ blast\isanewline
+\ \ \ \ \isacommand{qed}\isamarkupfalse%
+\isanewline
+\ \ \isacommand{qed}\isamarkupfalse%
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\begin{isamarkuptext}%
+The locale hierarchy is now as shown in Figure~\ref{fig:lattices}(c).%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+%
+\isatagtheory
+%
+\endisatagtheory
+{\isafoldtheory}%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+\end{isabellebody}%
+%%% Local Variables:
+%%% mode: latex
+%%% TeX-master: "root"
+%%% End:
diff -r 9681f347b6f5 -r d1d35284542f doc-src/Locales/Locales/document/Examples1.tex
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/Locales/Locales/document/Examples1.tex	Tue Jun 03 12:34:22 2008 +0200
@@ -0,0 +1,150 @@
+%
+\begin{isabellebody}%
+\def\isabellecontext{Examples{\isadigit{1}}}%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+%
+\isatagtheory
+%
+\endisatagtheory
+{\isafoldtheory}%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+%
+\isamarkupsection{Use of Locales in Theories and Proofs%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+Locales enable to prove theorems abstractly, relative to
+  sets of assumptions.  These theorems can then be used in other
+  contexts where the assumptions themselves, or
+  instances of the assumptions, are theorems.  This form of theorem
+  reuse is called \emph{interpretation}.
+
+  The changes of the locale
+  hierarchy from the previous sections are examples of
+  interpretations.  The command \isakeyword{interpretation} $l_1
+  \subseteq l_2$ is said to \emph{interpret} locale $l_2$ in the
+  context of $l_1$.  It causes all theorems of $l_2$ to be made
+  available in $l_1$.  The interpretation is \emph{dynamic}: not only
+  theorems already present in $l_2$ are available in $l_1$.  Theorems
+  that will be added to $l_2$ in future will automatically be
+  propagated to $l_1$.
+
+  Locales can also be interpreted in the contexts of theories and
+  structured proofs.  These interpretations are dynamic, too.
+  Theorems added to locales will be propagated to theories.
+  In this section the interpretation in
+  theories is illustrated; interpretation in proofs is analogous.
+  As an example consider, the type of natural numbers \isa{nat}.  The
+  order relation \isa{{\isasymle}} is a total order over \isa{nat},
+  divisibility \isa{dvd} is a distributive lattice.
+
+  We start with the
+  interpretation that \isa{{\isasymle}} is a partial order.  The facilities of
+  the interpretation command are explored in three versions.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsubsection{First Version: Replacement of Parameters Only%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+\label{sec:po-first}
+
+  In the most basic form, interpretation just replaces the locale
+  parameters by terms.  The following command interprets the locale
+  \isa{partial{\isacharunderscore}order} in the global context of the theory.  The
+  parameter \isa{le} is replaced by \isa{op\ {\isasymle}}.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimvisible
+\ \ %
+\endisadelimvisible
+%
+\isatagvisible
+\isacommand{interpretation}\isamarkupfalse%
+\ nat{\isacharcolon}\ partial{\isacharunderscore}order\ {\isacharbrackleft}{\isachardoublequoteopen}op\ {\isasymle}\ {\isacharcolon}{\isacharcolon}\ nat\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}{\isacharbrackright}%
+\begin{isamarkuptxt}%
+The locale name is succeeded by a \emph{parameter
+  instantiation}.  In general, this is a list of terms, which refer to
+  the parameters in the order of declaration in the locale.  The
+  locale name is preceded by an optional \emph{interpretation prefix},
+  which is used to qualify names from the locale in the global
+  context.
+
+  The command creates the goal \begin{isabelle}%
+\ {\isadigit{1}}{\isachardot}\ partial{\isacharunderscore}order\ op\ {\isasymle}%
+\end{isabelle} which can be shown
+  easily:%
+\footnote{Note that \isa{op} binds tighter than functions
+  application: parentheses around \isa{op\ {\isasymle}} are not necessary.}%
+\end{isamarkuptxt}%
+\isamarkuptrue%
+\ \ \ \ \isacommand{by}\isamarkupfalse%
+\ unfold{\isacharunderscore}locales\ auto%
+\endisatagvisible
+{\isafoldvisible}%
+%
+\isadelimvisible
+%
+\endisadelimvisible
+%
+\begin{isamarkuptext}%
+Now theorems from the locale are available in the theory,
+  interpreted for natural numbers, for example \isa{nat{\isachardot}trans}: \begin{isabelle}%
+\ \ {\isasymlbrakk}{\isacharquery}x\ {\isasymle}\ {\isacharquery}y{\isacharsemicolon}\ {\isacharquery}y\ {\isasymle}\ {\isacharquery}z{\isasymrbrakk}\ {\isasymLongrightarrow}\ {\isacharquery}x\ {\isasymle}\ {\isacharquery}z%
+\end{isabelle}
+
+  In order to avoid unwanted hiding of theorems, interpretation
+  accepts a qualifier, \isa{nat} in the example, which is applied to
+  all names processed by the
+  interpretation.  If present the qualifier is mandatory --- that is,
+  the above theorem cannot be referred to simply as \isa{trans}.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsubsection{Second Version: Replacement of Definitions%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+The above interpretation also creates the theorem
+  \isa{nat{\isachardot}less{\isacharunderscore}le{\isacharunderscore}trans}: \begin{isabelle}%
+\ \ {\isasymlbrakk}partial{\isacharunderscore}order{\isachardot}less\ op\ {\isasymle}\ {\isacharquery}x\ {\isacharquery}y{\isacharsemicolon}\ {\isacharquery}y\ {\isasymle}\ {\isacharquery}z{\isasymrbrakk}\isanewline
+\isaindent{\ \ }{\isasymLongrightarrow}\ partial{\isacharunderscore}order{\isachardot}less\ op\ {\isasymle}\ {\isacharquery}x\ {\isacharquery}z%
+\end{isabelle}
+  Here, \isa{partial{\isacharunderscore}order{\isachardot}less\ op\ {\isasymle}}
+  represents the strict order, although \isa{{\isacharless}} is defined on
+  \isa{nat}.  Interpretation enables to map concepts
+  introduced in locales through definitions to the corresponding
+  concepts of the theory.%
+\footnote{This applies not only to \isakeyword{definition} but also to
+  \isakeyword{inductive}, \isakeyword{fun} and \isakeyword{function}.}%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+%
+\isatagtheory
+%
+\endisatagtheory
+{\isafoldtheory}%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+\end{isabellebody}%
+%%% Local Variables:
+%%% mode: latex
+%%% TeX-master: "root"
+%%% End:
diff -r 9681f347b6f5 -r d1d35284542f doc-src/Locales/Locales/document/Examples2.tex
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/Locales/Locales/document/Examples2.tex	Tue Jun 03 12:34:22 2008 +0200
@@ -0,0 +1,90 @@
+%
+\begin{isabellebody}%
+\def\isabellecontext{Examples{\isadigit{2}}}%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+%
+\isatagtheory
+%
+\endisatagtheory
+{\isafoldtheory}%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+%
+\begin{isamarkuptext}%
+This is achieved by unfolding suitable equations during
+  interpretation.  These equations are given after the keyword
+  \isakeyword{where} and require proofs.  The revised command,
+  replacing \isa{{\isasymsqsubset}} by \isa{{\isacharless}}, is:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimvisible
+%
+\endisadelimvisible
+%
+\isatagvisible
+\isacommand{interpretation}\isamarkupfalse%
+\ nat{\isacharcolon}\ partial{\isacharunderscore}order\ {\isacharbrackleft}{\isachardoublequoteopen}op\ {\isasymle}\ {\isacharcolon}{\isacharcolon}\ {\isacharbrackleft}nat{\isacharcomma}\ nat{\isacharbrackright}\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}{\isacharbrackright}\isanewline
+\ \ \isakeyword{where}\ {\isachardoublequoteopen}partial{\isacharunderscore}order{\isachardot}less\ op\ {\isasymle}\ {\isacharparenleft}x{\isacharcolon}{\isacharcolon}nat{\isacharparenright}\ y\ {\isacharequal}\ {\isacharparenleft}x\ {\isacharless}\ y{\isacharparenright}{\isachardoublequoteclose}\isanewline
+\isacommand{proof}\isamarkupfalse%
+\ {\isacharminus}%
+\begin{isamarkuptxt}%
+The goals are \begin{isabelle}%
+\ {\isadigit{1}}{\isachardot}\ partial{\isacharunderscore}order\ op\ {\isasymle}\isanewline
+\ {\isadigit{2}}{\isachardot}\ partial{\isacharunderscore}order{\isachardot}less\ op\ {\isasymle}\ x\ y\ {\isacharequal}\ {\isacharparenleft}x\ {\isacharless}\ y{\isacharparenright}%
+\end{isabelle}
+    The proof that \isa{{\isasymle}} is a partial order is a above.%
+\end{isamarkuptxt}%
+\isamarkuptrue%
+\ \ \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%
+\begin{isamarkuptxt}%
+The second goal is shown by unfolding the
+    definition of \isa{partial{\isacharunderscore}order{\isachardot}less}.%
+\end{isamarkuptxt}%
+\isamarkuptrue%
+\ \ \isacommand{show}\isamarkupfalse%
+\ {\isachardoublequoteopen}partial{\isacharunderscore}order{\isachardot}less\ op\ {\isasymle}\ {\isacharparenleft}x{\isacharcolon}{\isacharcolon}nat{\isacharparenright}\ y\ {\isacharequal}\ {\isacharparenleft}x\ {\isacharless}\ y{\isacharparenright}{\isachardoublequoteclose}\isanewline
+\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
+\ partial{\isacharunderscore}order{\isachardot}less{\isacharunderscore}def\ {\isacharbrackleft}OF\ {\isacharbackquoteopen}partial{\isacharunderscore}order\ op\ {\isasymle}{\isacharbackquoteclose}{\isacharbrackright}\isanewline
+\ \ \ \ \isacommand{by}\isamarkupfalse%
+\ auto\isanewline
+\isacommand{qed}\isamarkupfalse%
+%
+\endisatagvisible
+{\isafoldvisible}%
+%
+\isadelimvisible
+%
+\endisadelimvisible
+%
+\begin{isamarkuptext}%
+Note that the above proof is not in the context of a locale.
+  Hence, the correct interpretation of \isa{partial{\isacharunderscore}order{\isachardot}less{\isacharunderscore}def} is obtained manually with \isa{OF}.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+%
+\isatagtheory
+%
+\endisatagtheory
+{\isafoldtheory}%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+\end{isabellebody}%
+%%% Local Variables:
+%%% mode: latex
+%%% TeX-master: "root"
+%%% End:
diff -r 9681f347b6f5 -r d1d35284542f doc-src/Locales/Locales/document/Examples3.tex
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/Locales/Locales/document/Examples3.tex	Tue Jun 03 12:34:22 2008 +0200
@@ -0,0 +1,850 @@
+%
+\begin{isabellebody}%
+\def\isabellecontext{Examples{\isadigit{3}}}%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+%
+\isatagtheory
+%
+\endisatagtheory
+{\isafoldtheory}%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+%
+\isamarkupsubsection{Third Version: Local Interpretation%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+In the above example, the fact that \isa{{\isasymle}} is a partial
+  order for the natural numbers was used in the proof of the
+  second goal.  In general, proofs of the equations may involve
+  theorems implied by the fact the assumptions of the instantiated
+  locale hold for the instantiating structure.  If these theorems have
+  been shown abstractly in the locale they can be made available
+  conveniently in the context through an auxiliary local interpretation (keyword
+  \isakeyword{interpret}).  This interpretation is inside the proof of the global
+  interpretation.  The third revision of the example illustrates this.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimvisible
+%
+\endisadelimvisible
+%
+\isatagvisible
+\isacommand{interpretation}\isamarkupfalse%
+\ nat{\isacharcolon}\ partial{\isacharunderscore}order\ {\isacharbrackleft}{\isachardoublequoteopen}op\ {\isasymle}\ {\isacharcolon}{\isacharcolon}\ nat\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}{\isacharbrackright}\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%
+\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%
+\ auto\isanewline
+\isacommand{qed}\isamarkupfalse%
+%
+\endisatagvisible
+{\isafoldvisible}%
+%
+\isadelimvisible
+%
+\endisadelimvisible
+%
+\begin{isamarkuptext}%
+The inner interpretation does not require an
+  elaborate new proof, it is immediate from the preceeding fact and
+  proved with ``.''.
+  This interpretation enriches the local proof context by
+  the very theorems also obtained in the interpretation from
+  Section~\ref{sec:po-first}, and \isa{nat{\isachardot}less{\isacharunderscore}def} may directly be
+  used to unfold the definition.  Theorems from the local
+  interpretation disappear after leaving the proof context --- that
+  is, after the closing \isakeyword{qed} --- and are
+  then replaced by those with the desired substitutions of the strict
+  order.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsubsection{Further Interpretations%
+}
+\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{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%
+%
+\isadelimvisible
+%
+\endisadelimvisible
+%
+\isatagvisible
+\isacommand{interpretation}\isamarkupfalse%
+\ nat{\isacharcolon}\ lattice\ {\isacharbrackleft}{\isachardoublequoteopen}op\ {\isasymle}\ {\isacharcolon}{\isacharcolon}\ nat\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}{\isacharbrackright}\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%
+\ {\isachardoublequoteopen}lattice\ {\isacharparenleft}op\ {\isasymle}\ {\isacharcolon}{\isacharcolon}\ nat\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ bool{\isacharparenright}{\isachardoublequoteclose}%
+\begin{isamarkuptxt}%
+We have already shown that this is a partial order,%
+\end{isamarkuptxt}%
+\isamarkuptrue%
+\ \ \ \ \isacommand{apply}\isamarkupfalse%
+\ unfold{\isacharunderscore}locales%
+\begin{isamarkuptxt}%
+hence only the lattice axioms remain to be shown: \begin{isabelle}%
+\ {\isadigit{1}}{\isachardot}\ {\isasymAnd}x\ y{\isachardot}\ {\isasymexists}inf{\isachardot}\ partial{\isacharunderscore}order{\isachardot}is{\isacharunderscore}inf\ op\ {\isasymle}\ x\ y\ inf\isanewline
+\ {\isadigit{2}}{\isachardot}\ {\isasymAnd}x\ y{\isachardot}\ {\isasymexists}sup{\isachardot}\ partial{\isacharunderscore}order{\isachardot}is{\isacharunderscore}sup\ op\ {\isasymle}\ x\ y\ sup%
+\end{isabelle}  After unfolding \isa{is{\isacharunderscore}inf} and \isa{is{\isacharunderscore}sup},%
+\end{isamarkuptxt}%
+\isamarkuptrue%
+\ \ \ \ \isacommand{apply}\isamarkupfalse%
+\ {\isacharparenleft}unfold\ nat{\isachardot}is{\isacharunderscore}inf{\isacharunderscore}def\ nat{\isachardot}is{\isacharunderscore}sup{\isacharunderscore}def{\isacharparenright}%
+\begin{isamarkuptxt}%
+the goals become \begin{isabelle}%
+\ {\isadigit{1}}{\isachardot}\ {\isasymAnd}x\ y{\isachardot}\ {\isasymexists}inf{\isasymle}x{\isachardot}\ inf\ {\isasymle}\ y\ {\isasymand}\ {\isacharparenleft}{\isasymforall}z{\isachardot}\ z\ {\isasymle}\ x\ {\isasymand}\ z\ {\isasymle}\ y\ {\isasymlongrightarrow}\ z\ {\isasymle}\ inf{\isacharparenright}\isanewline
+\ {\isadigit{2}}{\isachardot}\ {\isasymAnd}x\ y{\isachardot}\ {\isasymexists}sup{\isasymge}x{\isachardot}\ y\ {\isasymle}\ sup\ {\isasymand}\ {\isacharparenleft}{\isasymforall}z{\isachardot}\ x\ {\isasymle}\ z\ {\isasymand}\ y\ {\isasymle}\ z\ {\isasymlongrightarrow}\ sup\ {\isasymle}\ z{\isacharparenright}%
+\end{isabelle} which can be solved
+      by Presburger arithmetic.%
+\end{isamarkuptxt}%
+\isamarkuptrue%
+\ \ \ \ \isacommand{by}\isamarkupfalse%
+\ arith{\isacharplus}%
+\begin{isamarkuptxt}%
+In order to show the equations, we put ourselves in a
+    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%
+\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
+\ \ \isacommand{show}\isamarkupfalse%
+\ {\isachardoublequoteopen}lattice{\isachardot}join\ op\ {\isasymle}\ {\isacharparenleft}x{\isacharcolon}{\isacharcolon}nat{\isacharparenright}\ y\ {\isacharequal}\ max\ x\ y{\isachardoublequoteclose}\isanewline
+\ \ \ \ \isacommand{by}\isamarkupfalse%
+\ {\isacharparenleft}bestsimp\ simp{\isacharcolon}\ nat{\isachardot}join{\isacharunderscore}def\ nat{\isachardot}is{\isacharunderscore}sup{\isacharunderscore}def{\isacharparenright}\isanewline
+\isacommand{qed}\isamarkupfalse%
+%
+\endisatagvisible
+{\isafoldvisible}%
+%
+\isadelimvisible
+%
+\endisadelimvisible
+%
+\begin{isamarkuptext}%
+That the relation \isa{{\isasymle}} is a total order completes this
+  sequence of interpretations.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimvisible
+%
+\endisadelimvisible
+%
+\isatagvisible
+\isacommand{interpretation}\isamarkupfalse%
+\ nat{\isacharcolon}\ total{\isacharunderscore}order\ {\isacharbrackleft}{\isachardoublequoteopen}op\ {\isasymle}\ {\isacharcolon}{\isacharcolon}\ nat\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}{\isacharbrackright}\isanewline
+\ \ \isacommand{by}\isamarkupfalse%
+\ unfold{\isacharunderscore}locales\ arith%
+\endisatagvisible
+{\isafoldvisible}%
+%
+\isadelimvisible
+%
+\endisadelimvisible
+%
+\begin{isamarkuptext}%
+Theorems that are available in the theory at this point are shown in
+  Table~\ref{tab:nat-lattice}.
+
+\begin{table}
+\hrule
+\vspace{2ex}
+\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{min\ {\isacharquery}x\ {\isacharquery}y\ {\isasymle}\ {\isacharquery}x} \\
+  \isa{nat{\isachardot}join{\isacharunderscore}distr} from locale \isa{distrib{\isacharunderscore}lattice}: \\
+  \quad \isa{max\ {\isacharquery}x\ {\isacharparenleft}min\ {\isacharquery}y\ {\isacharquery}z{\isacharparenright}\ {\isacharequal}\ min\ {\isacharparenleft}max\ {\isacharquery}x\ {\isacharquery}y{\isacharparenright}\ {\isacharparenleft}max\ {\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}
+\end{tabular}
+\end{center}
+\hrule
+\caption{Interpreted theorems for \isa{{\isasymle}} on the natural numbers.}
+\label{tab:nat-lattice}
+\end{table}
+
+  Note that since the locale hierarchy reflects that total orders are
+  distributive lattices, an explicit interpretation of distributive
+  lattices for the order relation on natural numbers is not neccessary.
+
+  Why not push this idea further and just give the last interpretation
+  as a single interpretation instead of the sequence of three?  The
+  reasons for this are twofold:
+\begin{itemize}
+\item
+  Often it is easier to work in an incremental fashion, because later
+  interpretations require theorems provided by earlier
+  interpretations.
+\item
+  Assume that a definition is made in some locale $l_1$, and that $l_2$
+  imports $l_1$.  Let an equation for the definition be
+  proved in an interpretation of $l_2$.  The equation will be unfolded
+  in interpretations of theorems added to $l_2$ or below in the import
+  hierarchy, but not for theorems added above $l_2$.
+  Hence, an equation interpreting a definition should always be given in
+  an interpretation of the locale where the definition is made, not in
+  an interpretation of a locale further down the hierarchy.
+\end{itemize}%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsubsection{Lattice \isa{dvd} on \isa{nat}%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+Divisibility on the natural numbers is a distributive 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
+\ \ \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
+%
+\isatagproof
+\isacommand{proof}\isamarkupfalse%
+\ {\isacharminus}\isanewline
+\ \ \isacommand{show}\isamarkupfalse%
+\ {\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%
+\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
+\ \ \ \ \isacommand{apply}\isamarkupfalse%
+\ auto\isanewline
+\ \ \ \ \isacommand{done}\isamarkupfalse%
+\isanewline
+\isacommand{qed}\isamarkupfalse%
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\begin{isamarkuptext}%
+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
+\ \ \isakeyword{where}\ nat{\isacharunderscore}dvd{\isacharunderscore}meet{\isacharunderscore}eq{\isacharcolon}\isanewline
+\ \ \ \ \ \ {\isachardoublequoteopen}lattice{\isachardot}meet\ op\ dvd\ {\isacharparenleft}x{\isacharcolon}{\isacharcolon}nat{\isacharparenright}\ y\ {\isacharequal}\ gcd\ {\isacharparenleft}x{\isacharcomma}\ y{\isacharparenright}{\isachardoublequoteclose}\isanewline
+\ \ \ \ \isakeyword{and}\ nat{\isacharunderscore}dvd{\isacharunderscore}join{\isacharunderscore}eq{\isacharcolon}\isanewline
+\ \ \ \ \ \ {\isachardoublequoteopen}lattice{\isachardot}join\ op\ dvd\ {\isacharparenleft}x{\isacharcolon}{\isacharcolon}nat{\isacharparenright}\ y\ {\isacharequal}\ lcm\ {\isacharparenleft}x{\isacharcomma}\ y{\isacharparenright}{\isachardoublequoteclose}\isanewline
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+\isacommand{proof}\isamarkupfalse%
+\ {\isacharminus}\isanewline
+\ \ \isacommand{show}\isamarkupfalse%
+\ {\isachardoublequoteopen}lattice\ {\isacharparenleft}op\ dvd\ {\isacharcolon}{\isacharcolon}\ nat\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ bool{\isacharparenright}{\isachardoublequoteclose}\isanewline
+\ \ \ \ \isacommand{apply}\isamarkupfalse%
+\ unfold{\isacharunderscore}locales\isanewline
+\ \ \ \ \isacommand{apply}\isamarkupfalse%
+\ {\isacharparenleft}unfold\ nat{\isacharunderscore}dvd{\isachardot}is{\isacharunderscore}inf{\isacharunderscore}def\ nat{\isacharunderscore}dvd{\isachardot}is{\isacharunderscore}sup{\isacharunderscore}def{\isacharparenright}\isanewline
+\ \ \ \ \isacommand{apply}\isamarkupfalse%
+\ {\isacharparenleft}rule{\isacharunderscore}tac\ x\ {\isacharequal}\ {\isachardoublequoteopen}gcd\ {\isacharparenleft}x{\isacharcomma}\ y{\isacharparenright}{\isachardoublequoteclose}\ \isakeyword{in}\ exI{\isacharparenright}\isanewline
+\ \ \ \ \isacommand{apply}\isamarkupfalse%
+\ auto\ {\isacharbrackleft}{\isadigit{1}}{\isacharbrackright}\isanewline
+\ \ \ \ \isacommand{apply}\isamarkupfalse%
+\ {\isacharparenleft}rule{\isacharunderscore}tac\ x\ {\isacharequal}\ {\isachardoublequoteopen}lcm\ {\isacharparenleft}x{\isacharcomma}\ y{\isacharparenright}{\isachardoublequoteclose}\ \isakeyword{in}\ exI{\isacharparenright}\isanewline
+\ \ \ \ \isacommand{apply}\isamarkupfalse%
+\ {\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%
+\isanewline
+\ \ \isacommand{show}\isamarkupfalse%
+\ {\isachardoublequoteopen}lattice{\isachardot}meet\ op\ dvd\ {\isacharparenleft}x{\isacharcolon}{\isacharcolon}nat{\isacharparenright}\ y\ {\isacharequal}\ gcd\ {\isacharparenleft}x{\isacharcomma}\ y{\isacharparenright}{\isachardoublequoteclose}\isanewline
+\ \ \ \ \isacommand{apply}\isamarkupfalse%
+\ {\isacharparenleft}unfold\ nat{\isacharunderscore}dvd{\isachardot}meet{\isacharunderscore}def{\isacharparenright}\isanewline
+\ \ \ \ \isacommand{apply}\isamarkupfalse%
+\ {\isacharparenleft}rule\ the{\isacharunderscore}equality{\isacharparenright}\isanewline
+\ \ \ \ \isacommand{apply}\isamarkupfalse%
+\ {\isacharparenleft}unfold\ nat{\isacharunderscore}dvd{\isachardot}is{\isacharunderscore}inf{\isacharunderscore}def{\isacharparenright}\isanewline
+\ \ \ \ \isacommand{by}\isamarkupfalse%
+\ auto\isanewline
+\ \ \isacommand{show}\isamarkupfalse%
+\ {\isachardoublequoteopen}lattice{\isachardot}join\ op\ dvd\ {\isacharparenleft}x{\isacharcolon}{\isacharcolon}nat{\isacharparenright}\ y\ {\isacharequal}\ lcm\ {\isacharparenleft}x{\isacharcomma}\ y{\isacharparenright}{\isachardoublequoteclose}\isanewline
+\ \ \ \ \isacommand{apply}\isamarkupfalse%
+\ {\isacharparenleft}unfold\ nat{\isacharunderscore}dvd{\isachardot}join{\isacharunderscore}def{\isacharparenright}\isanewline
+\ \ \ \ \isacommand{apply}\isamarkupfalse%
+\ {\isacharparenleft}rule\ the{\isacharunderscore}equality{\isacharparenright}\isanewline
+\ \ \ \ \isacommand{apply}\isamarkupfalse%
+\ {\isacharparenleft}unfold\ nat{\isacharunderscore}dvd{\isachardot}is{\isacharunderscore}sup{\isacharunderscore}def{\isacharparenright}\isanewline
+\ \ \ \ \isacommand{by}\isamarkupfalse%
+\ {\isacharparenleft}auto\ intro{\isacharcolon}\ lcm{\isacharunderscore}dvd{\isadigit{1}}\ lcm{\isacharunderscore}dvd{\isadigit{2}}\ lcm{\isacharunderscore}least{\isacharparenright}\isanewline
+\isacommand{qed}\isamarkupfalse%
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\begin{isamarkuptext}%
+Equations \isa{nat{\isacharunderscore}dvd{\isacharunderscore}meet{\isacharunderscore}eq} and \isa{nat{\isacharunderscore}dvd{\isacharunderscore}join{\isacharunderscore}eq} are named since they are handy in the proof of
+  the subsequent interpretation.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadeliminvisible
+%
+\endisadeliminvisible
+%
+\isataginvisible
+\isacommand{ML}\isamarkupfalse%
+\ {\isacharverbatimopen}\ set\ quick{\isacharunderscore}and{\isacharunderscore}dirty\ {\isacharverbatimclose}\isanewline
+\isanewline
+\isanewline
+\isanewline
+\isacommand{lemma}\isamarkupfalse%
+\ gcd{\isacharunderscore}lcm{\isacharunderscore}distr{\isacharcolon}\isanewline
+\ \ {\isachardoublequoteopen}gcd\ {\isacharparenleft}x{\isacharcomma}\ lcm\ {\isacharparenleft}y{\isacharcomma}\ z{\isacharparenright}{\isacharparenright}\ {\isacharequal}\ lcm\ {\isacharparenleft}gcd\ {\isacharparenleft}x{\isacharcomma}\ y{\isacharparenright}{\isacharcomma}\ gcd\ {\isacharparenleft}x{\isacharcomma}\ z{\isacharparenright}{\isacharparenright}{\isachardoublequoteclose}\isanewline
+\ \ \isacommand{sorry}\isamarkupfalse%
+\isanewline
+\isanewline
+\isacommand{ML}\isamarkupfalse%
+\ {\isacharverbatimopen}\ reset\ quick{\isacharunderscore}and{\isacharunderscore}dirty\ {\isacharverbatimclose}%
+\endisataginvisible
+{\isafoldinvisible}%
+%
+\isadeliminvisible
+%
+\endisadeliminvisible
+\isanewline
+%
+\isadelimvisible
+\ \ \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
+\ \ \isacommand{apply}\isamarkupfalse%
+\ unfold{\isacharunderscore}locales%
+\begin{isamarkuptxt}%
+\begin{isabelle}%
+\ {\isadigit{1}}{\isachardot}\ {\isasymAnd}x\ y\ z{\isachardot}\isanewline
+\isaindent{\ {\isadigit{1}}{\isachardot}\ \ \ \ }lattice{\isachardot}meet\ op\ dvd\ x\ {\isacharparenleft}lattice{\isachardot}join\ op\ dvd\ y\ z{\isacharparenright}\ {\isacharequal}\isanewline
+\isaindent{\ {\isadigit{1}}{\isachardot}\ \ \ \ }lattice{\isachardot}join\ op\ dvd\ {\isacharparenleft}lattice{\isachardot}meet\ op\ dvd\ x\ y{\isacharparenright}\isanewline
+\isaindent{\ {\isadigit{1}}{\isachardot}\ \ \ \ \ }{\isacharparenleft}lattice{\isachardot}meet\ op\ dvd\ x\ z{\isacharparenright}%
+\end{isabelle}%
+\end{isamarkuptxt}%
+\isamarkuptrue%
+\ \ \isacommand{apply}\isamarkupfalse%
+\ {\isacharparenleft}unfold\ nat{\isacharunderscore}dvd{\isacharunderscore}meet{\isacharunderscore}eq\ nat{\isacharunderscore}dvd{\isacharunderscore}join{\isacharunderscore}eq{\isacharparenright}%
+\begin{isamarkuptxt}%
+\begin{isabelle}%
+\ {\isadigit{1}}{\isachardot}\ {\isasymAnd}x\ y\ z{\isachardot}\ gcd\ {\isacharparenleft}x{\isacharcomma}\ lcm\ {\isacharparenleft}y{\isacharcomma}\ z{\isacharparenright}{\isacharparenright}\ {\isacharequal}\ lcm\ {\isacharparenleft}gcd\ {\isacharparenleft}x{\isacharcomma}\ y{\isacharparenright}{\isacharcomma}\ gcd\ {\isacharparenleft}x{\isacharcomma}\ z{\isacharparenright}{\isacharparenright}%
+\end{isabelle}%
+\end{isamarkuptxt}%
+\isamarkuptrue%
+\ \ \isacommand{apply}\isamarkupfalse%
+\ {\isacharparenleft}rule\ gcd{\isacharunderscore}lcm{\isacharunderscore}distr{\isacharparenright}\ \isacommand{done}\isamarkupfalse%
+%
+\endisatagvisible
+{\isafoldvisible}%
+%
+\isadelimvisible
+%
+\endisadelimvisible
+%
+\begin{isamarkuptext}%
+Theorems that are available in the theory after these
+  interpretations are shown in Table~\ref{tab:nat-dvd-lattice}.
+
+\begin{table}
+\hrule
+\vspace{2ex}
+\begin{center}
+\begin{tabular}{l}
+  \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\ {\isacharparenleft}{\isacharquery}x{\isacharcomma}\ {\isacharquery}y{\isacharparenright}\ dvd\ {\isacharquery}x} \\
+  \isa{nat{\isacharunderscore}dvd{\isachardot}join{\isacharunderscore}distr} from locale \isa{distrib{\isacharunderscore}lattice}: \\
+  \quad \isa{lcm\ {\isacharparenleft}{\isacharquery}x{\isacharcomma}\ gcd\ {\isacharparenleft}{\isacharquery}y{\isacharcomma}\ {\isacharquery}z{\isacharparenright}{\isacharparenright}\ {\isacharequal}\ gcd\ {\isacharparenleft}lcm\ {\isacharparenleft}{\isacharquery}x{\isacharcomma}\ {\isacharquery}y{\isacharparenright}{\isacharcomma}\ lcm\ {\isacharparenleft}{\isacharquery}x{\isacharcomma}\ {\isacharquery}z{\isacharparenright}{\isacharparenright}} \\
+\end{tabular}
+\end{center}
+\hrule
+\caption{Interpreted theorems for \isa{dvd} on the natural numbers.}
+\label{tab:nat-dvd-lattice}
+\end{table}%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+The full syntax of the interpretation commands is shown in
+  Table~\ref{tab:commands}.  The grammar refers to
+  \textit{expr}, which stands for a \emph{locale} expression.  Locale
+  expressions are discussed in Section~\ref{sec:expressions}.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsection{Locale Expressions%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+\label{sec:expressions}
+
+  A map \isa{{\isasymphi}} between partial orders \isa{{\isasymsqsubseteq}} and \isa{{\isasympreceq}}
+  is called order preserving if \isa{x\ {\isasymsqsubseteq}\ y} implies \isa{{\isasymphi}\ x\ {\isasympreceq}\ {\isasymphi}\ y}.  This situation is more complex than those encountered so
+  far: it involves two partial orders, and it is desirable to use the
+  existing locale for both.
+
+  Inspecting the grammar of locale commands in
+  Table~\ref{tab:commands} reveals that the import of a locale can be
+  more than just a single locale.  In general, the import is a
+  \emph{locale expression}.  Locale expressions enable to combine locales
+  and rename parameters.  A locale name is a locale expression.  If
+  $e_1$ and $e_2$ are locale expressions then $e_1 + e_2$ is their
+  \emph{merge}.  If $e$ is an expression, then $e\:q_1 \ldots q_n$ is
+  a \emph{renamed expression} where the parameters in $e$ are renamed
+  to $q_1 \ldots q_n$.  Using a locale expression, a locale for order
+  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
+\ \ \ \ \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
+  is \isa{le{\isacharprime}} with new infix syntax \isa{{\isasympreceq}}.  The
+  parameters of the entire locale are \isa{le}, \isa{le{\isacharprime}} and
+  \isa{{\isasymphi}}.  This is their \emph{canonical order},
+  which is obtained by a left-to-right traversal of the expression,
+  where only the new parameters are appended to the end of the list.  The
+  parameters introduced in the locale elements of the declaration
+  follow.
+
+  In renamings parameters are referred to by position in the canonical
+  order; an underscore is used to skip a parameter position, which is
+  then not renamed.  Renaming deletes the syntax of a parameter unless
+  a new mixfix annotation is given.
+
+  Parameter renamings are morphisms between locales.  These can be
+  lifted to terms and theorems and thus be applied to assumptions and
+  conclusions.  The assumption of a merge is the conjunction of the
+  assumptions of the merged locale.  The conclusions of a merge are
+  obtained by appending the conclusions of the left locale and of the
+  right locale.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+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:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadeliminvisible
+%
+\endisadeliminvisible
+%
+\isataginvisible
+\isacommand{context}\isamarkupfalse%
+\ order{\isacharunderscore}preserving\ \isakeyword{begin}%
+\endisataginvisible
+{\isafoldinvisible}%
+%
+\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}%
+\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}%
+\ \ {\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
+%
+\isataginvisible
+\isacommand{end}\isamarkupfalse%
+%
+\endisataginvisible
+{\isafoldinvisible}%
+%
+\isadeliminvisible
+%
+\endisadeliminvisible
+%
+\begin{isamarkuptext}%
+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%
+\ {\isacharparenleft}\isakeyword{in}\ order{\isacharunderscore}preserving{\isacharparenright}\isanewline
+\ \ \ \ less{\isacharprime}\ {\isacharparenleft}\isakeyword{infixl}\ {\isachardoublequoteopen}{\isasymprec}{\isachardoublequoteclose}\ {\isadigit{5}}{\isadigit{0}}{\isacharparenright}\ \isakeyword{where}\ {\isachardoublequoteopen}less{\isacharprime}\ {\isasymequiv}\ partial{\isacharunderscore}order{\isachardot}less\ le{\isacharprime}{\isachardoublequoteclose}%
+\begin{isamarkuptext}%
+Now the theorem is displayed nicely as
+  \isa{{\isasymlbrakk}{\isacharquery}x\ {\isasymprec}\ {\isacharquery}y{\isacharsemicolon}\ {\isacharquery}y\ {\isasympreceq}\ {\isacharquery}z{\isasymrbrakk}\ {\isasymLongrightarrow}\ {\isacharquery}x\ {\isasymprec}\ {\isacharquery}z}.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+Not only names of theorems are qualified.  In fact, all names
+  are qualified, in particular names introduced by definitions and
+  abbreviations.  The name of the strict order of \isa{{\isasymsqsubseteq}} is \isa{le{\isachardot}less} and therefore \isa{le{\isacharprime}{\isachardot}less} is the name of the strict
+  order of \isa{{\isasympreceq}}.  Hence, the equation in the above abbreviation
+  could have been written as \isa{less{\isacharprime}\ {\isasymequiv}\ le{\isacharprime}{\isachardot}less}.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+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
+\ \ \ \ \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
+\ \ \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}%
+\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}%
+\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
+  imported by \isa{lattice{\isacharunderscore}end} identifies the first and second
+  parameter of \isa{lattice{\isacharunderscore}hom}.  By looking at the inheritance diagram it would seem
+  that two identical copies of each of the locales \isa{partial{\isacharunderscore}order} and \isa{lattice} are imported.  This is not the
+  case!  Inheritance paths with identical morphisms are detected and
+  the conclusions of the respecitve locales appear only once.
+
+\begin{figure}
+\hrule \vspace{2ex}
+\begin{center}
+\begin{tikzpicture}
+  \node (o) at (0,0) {\isa{partial{\isacharunderscore}order}};
+  \node (oh) at (1.5,-2) {\isa{order{\isacharunderscore}preserving}};
+  \node (oh1) at (1.5,-0.7) {$\scriptscriptstyle \sqsubseteq \mapsto \sqsubseteq$};
+  \node (oh2) at (0,-1.3) {$\scriptscriptstyle \sqsubseteq \mapsto \preceq$};
+  \node (l) at (-1.5,-2) {\isa{lattice}};
+  \node (lh) at (0,-4) {\isa{lattice{\isacharunderscore}hom}};
+  \node (lh1) at (0,-2.7) {$\scriptscriptstyle \sqsubseteq \mapsto \sqsubseteq$};
+  \node (lh2) at (-1.5,-3.3) {$\scriptscriptstyle \sqsubseteq \mapsto \preceq$};
+  \node (le) at (0,-6) {\isa{lattice{\isacharunderscore}end}};
+  \node (le1) at (0,-4.8)
+    [anchor=west]{$\scriptscriptstyle \sqsubseteq \mapsto \sqsubseteq$};
+  \node (le2) at (0,-5.2)
+    [anchor=west]{$\scriptscriptstyle \preceq \mapsto \sqsubseteq$};
+  \draw (o) -- (l);
+  \draw[dashed] (oh) -- (lh);
+  \draw (lh) -- (le);
+  \draw (o) .. controls (oh1.south west) .. (oh);
+  \draw (o) .. controls (oh2.north east) .. (oh);
+  \draw (l) .. controls (lh1.south west) .. (lh);
+  \draw (l) .. controls (lh2.north east) .. (lh);
+\end{tikzpicture}
+\end{center}
+\hrule
+\caption{Hierarchy of Homomorphism Locales.}
+\label{fig:hom}
+\end{figure}%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+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%
+\ lattice{\isacharunderscore}hom\ {\isasymsubseteq}\ order{\isacharunderscore}preserving%
+\isadelimproof
+\ %
+\endisadelimproof
+%
+\isatagproof
+\isacommand{proof}\isamarkupfalse%
+\ unfold{\isacharunderscore}locales\isanewline
+\ \ \ \ \isacommand{fix}\isamarkupfalse%
+\ x\ y\isanewline
+\ \ \ \ \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
+\ \ \ \ \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
+\ \ \isacommand{qed}\isamarkupfalse%
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\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{{\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%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+More information on locales and their interpretation is
+  available.  For the locale hierarchy of import and interpretation
+  dependencies see \cite{Ballarin2006a}; interpretations in theories
+  and proofs are covered in \cite{Ballarin2006b}.  In the latter, we
+  show how interpretation in proofs enables to reason about families
+  of algebraic structures, which cannot be expressed with locales
+  directly.
+
+  Haftmann and Wenzel \cite{HaftmannWenzel2007} overcome a restriction
+  of axiomatic type classes through a combination with locale
+  interpretation.  The result is a Haskell-style class system with a
+  facility to generate Haskell code.  Classes are sufficient for
+  simple specifications with a single type parameter.  The locales for
+  orders and lattices presented in this tutorial fall into this
+  category.  Order preserving maps, homomorphisms and vector spaces,
+  on the other hand, do not.
+
+  The original work of Kamm\"uller on locales \cite{KammullerEtAl1999}
+  may be of interest from a historical perspective.  The mathematical
+  background on orders and lattices is taken from Jacobson's textbook
+  on algebra \cite[Chapter~8]{Jacobson1985}.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+\begin{table}
+\hrule
+\vspace{2ex}
+\begin{center}
+\begin{tabular}{l>$c<$l}
+  \multicolumn{3}{l}{Miscellaneous} \\
+
+  \textit{attr-name} & ::=
+  & \textit{name} $|$ \textit{attribute} $|$
+    \textit{name} \textit{attribute} \\[2ex]
+
+  \multicolumn{3}{l}{Context Elements} \\
+
+  \textit{fixes} & ::=
+  & \textit{name} [ ``\textbf{::}'' \textit{type} ]
+    [ ``\textbf{(}'' \textbf{structure} ``\textbf{)}'' $|$
+    \textit{mixfix} ] \\
+\begin{comment}
+  \textit{constrains} & ::=
+  & \textit{name} ``\textbf{::}'' \textit{type} \\
+\end{comment}
+  \textit{assumes} & ::=
+  & [ \textit{attr-name} ``\textbf{:}'' ] \textit{proposition} \\
+\begin{comment}
+  \textit{defines} & ::=
+  & [ \textit{attr-name} ``\textbf{:}'' ] \textit{proposition} \\
+  \textit{notes} & ::=
+  & [ \textit{attr-name} ``\textbf{=}'' ]
+    ( \textit{qualified-name} [ \textit{attribute} ] )$^+$ \\
+\end{comment}
+
+  \textit{element} & ::=
+  & \textbf{fixes} \textit{fixes} ( \textbf{and} \textit{fixes} )$^*$ \\
+\begin{comment}
+  & |
+  & \textbf{constrains} \textit{constrains}
+    ( \textbf{and} \textit{constrains} )$^*$ \\
+\end{comment}
+  & |
+  & \textbf{assumes} \textit{assumes} ( \textbf{and} \textit{assumes} )$^*$ \\[2ex]
+%\begin{comment}
+%  & |
+%  & \textbf{defines} \textit{defines} ( \textbf{and} \textit{defines} )$^*$ \\
+%  & |
+%  & \textbf{notes} \textit{notes} ( \textbf{and} \textit{notes} )$^*$ \\
+%\end{comment}
+
+  \multicolumn{3}{l}{Locale Expressions} \\
+
+  \textit{rename} & ::=
+  & \textit{name} [ \textit{mixfix} ] $|$ ``\textbf{\_}'' \\
+  \textit{expr}  & ::= 
+  & \textit{renamed-expr} ( ``\textbf{+}'' \textit{renamed-expr} )$^*$ \\
+  \textit{renamed-expr} & ::=
+  & ( \textit{qualified-name} $|$
+    ``\textbf{(}'' \textit{expr} ``\textbf{)}'' ) \textit{rename}$^*$ \\[2ex]
+
+  \multicolumn{3}{l}{Declaration of Locales} \\
+
+  \textit{locale} & ::=
+  & \textit{element}$^+$ \\
+  & | & \textit{locale-expr} [ ``\textbf{+}'' \textit{element}$^+$ ] \\
+  \textit{toplevel} & ::=
+  & \textbf{locale} \textit{name} [ ``\textbf{=}''
+    \textit{locale} ] \\[2ex]
+
+  \multicolumn{3}{l}{Interpretation} \\
+
+  \textit{equation} & ::= & [ \textit{attr-name} ``\textbf{:}'' ]
+    \textit{prop} \\
+  \textit{insts} & ::= & [ ``\textbf{[}'' \textit{term}$^+$
+    ``\textbf{]}'' ] \\
+  & & [ \textbf{where} \textit{equation} ( \textbf{and}
+    \textit{equation} )$^*$ ] \\
+  \textit{toplevel} & ::=
+  & \textbf{interpretation} \textit{name} ( ``$<$'' $|$
+    ``$\subseteq$'' ) \textit{expr} \textit{proof} \\
+  & |
+  & \textbf{interpretation} [ \textit{attr-name} ``\textbf{:}'' ]
+    \textit{expr} \textit{insts} \textit{proof} \\
+  & |
+  & \textbf{interpret} [ \textit{attr-name} ``\textbf{:}'' ]
+    \textit{expr} \textit{insts} \textit{proof} \\[2ex]
+
+  \multicolumn{3}{l}{Diagnostics} \\
+
+  \textit{toplevel} & ::=
+  & \textbf{print\_locale} [ ``\textbf{!}'' ] \textit{locale} \\
+  & | & \textbf{print\_locales} 
+\end{tabular}
+\end{center}
+\hrule
+\caption{Syntax of Locale Commands.}
+\label{tab:commands}
+\end{table}%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+\textbf{Acknowledgements.}  Alexander Krauss, Tobias Nipkow,
+  Christian Sternagel and Makarius Wenzel have made useful comments on
+  a draft of this document.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+%
+\isatagtheory
+%
+\endisatagtheory
+{\isafoldtheory}%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+\end{isabellebody}%
+%%% Local Variables:
+%%% mode: latex
+%%% TeX-master: "root"
+%%% End:
diff -r 9681f347b6f5 -r d1d35284542f doc-src/Locales/Locales/document/GCD.tex
diff -r 9681f347b6f5 -r d1d35284542f doc-src/Locales/Locales/document/Locales.tex
--- a/doc-src/Locales/Locales/document/Locales.tex	Tue Jun 03 11:55:35 2008 +0200
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,1320 +0,0 @@
-%
-\begin{isabellebody}%
-\def\isabellecontext{Locales}%
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isadelimML
-%
-\endisadelimML
-%
-\isatagML
-%
-\endisatagML
-{\isafoldML}%
-%
-\isadelimML
-%
-\endisadelimML
-%
-\isamarkupsection{Overview%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-The present text is based on~\cite{Ballarin2004a}.  It was updated
-  for for Isabelle2005, but does not cover locale interpretation.
-
-  Locales are an extension of the Isabelle proof assistant.  They
-  provide support for modular reasoning. Locales were initially
-  developed by Kamm\"uller~\cite{Kammuller2000} to support reasoning
-  in abstract algebra, but are applied also in other domains --- for
-  example, bytecode verification~\cite{Klein2003}.
-
-  Kamm\"uller's original design, implemented in Isabelle99, provides, in
-  addition to
-  means for declaring locales, a set of ML functions that were used
-  along with ML tactics in a proof.  In the meantime, the input format
-  for proof in Isabelle has changed and users write proof
-  scripts in ML only rarely if at all.  Two new proof styles are
-  available, and can
-  be used interchangeably: linear proof scripts that closely resemble ML
-  tactics, and the structured Isar proof language by
-  Wenzel~\cite{Wenzel2002a}.  Subsequently, Wenzel re-implemented
-  locales for
-  the new proof format.  The implementation, available with
-  Isabelle2002 or later, constitutes a complete re-design and exploits that
-  both Isar and locales are based on the notion of context,
-  and thus locales are seen as a natural extension of Isar.
-  Nevertheless, locales can also be used with proof scripts:
-  their use does not require a deep understanding of the structured
-  Isar proof style.
-
-  At the same time, Wenzel considerably extended locales.  The most
-  important addition are locale expressions, which allow to combine
-  locales more freely.  Previously only
-  linear inheritance was possible.  Now locales support multiple
-  inheritance through a normalisation algorithm.  New are also
-  structures, which provide special syntax for locale parameters that
-  represent algebraic structures.
-
-  Unfortunately, Wenzel provided only an implementation but hardly any
-  documentation.  Besides providing documentation, the present paper
-  is a high-level description of locales, and in particular locale
-  expressions.  It is meant as a first step towards the semantics of
-  locales, and also as a base for comparing locales with module concepts
-  in other provers.  It also constitutes the base for future
-  extensions of locales in Isabelle.
-  The description was derived mainly by experimenting
-  with locales and partially also by inspecting the code.
-
-  The main contribution of the author of the present paper is the
-  abstract description of Wenzel's version of locales, and in
-  particular of the normalisation algorithm for locale expressions (see
-  Section~\ref{sec-normal-forms}).  Contributions to the
-  implementation are confined to bug fixes and to provisions that
-  enable the use of locales with linear proof scripts.
-
-  Concepts are introduced along with examples, so that the text can be
-  used as tutorial.  It is assumed that the reader is somewhat
-  familiar with Isabelle proof scripts.  Examples have been phrased as
-  structured
-  Isar proofs.  However, in order to understand the key concepts,
-  including locales expressions and their normalisation, detailed
-  knowledge of Isabelle is not necessary. 
-
-\nocite{Nipkow2003,Wenzel2002b,Wenzel2003}%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsection{Locales: Beyond Proof Contexts%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-In tactic-based provers the application of a sequence of proof
-  tactics leads to a proof state.  This state is usually hard to
-  predict from looking at the tactic script, unless one replays the
-  proof step-by-step.  The structured proof language Isar is
-  different.  It is additionally based on \emph{proof contexts},
-  which are directly visible in Isar scripts, and since tactic
-  sequences tend to be short, this commonly leads to clearer proof
-  scripts.
-
-  Goals are stated with the \textbf{theorem}
-  command.  This is followed by a proof.  When discharging a goal
-  requires an elaborate argument
-  (rather than the application of a single tactic) a new context
-  may be entered (\textbf{proof}).  Inside the context, variables may
-  be fixed (\textbf{fix}), assumptions made (\textbf{assume}) and
-  intermediate goals stated (\textbf{have}) and proved.  The
-  assumptions must be dischargeable by premises of the surrounding
-  goal, and once this goal has been proved (\textbf{show}) the proof context
-  can be closed (\textbf{qed}). Contexts inherit from surrounding
-  contexts, but it is not possible
-  to export from them (with exception of the proved goal);
-  they ``disappear'' after the closing \textbf{qed}.
-  Facts may have attributes --- for example, identifying them as
-  default to the simplifier or classical reasoner.
-
-  Locales extend proof contexts in various ways:
-  \begin{itemize}
-  \item
-    Locales are usually \emph{named}.  This makes them persistent.
-  \item
-    Fixed variables may have \emph{syntax}.
-  \item
-    It is possible to \emph{add} and \emph{export} facts.
-  \item
-    Locales can be combined and modified with \emph{locale
-    expressions}.
-  \end{itemize}
-  The Locales facility extends the Isar language: it provides new ways
-  of stating and managing facts, but it does not modify the language
-  for proofs.  Its purpose is to support writing modular proofs.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsection{Simple Locales%
-}
-\isamarkuptrue%
-%
-\isamarkupsubsection{Syntax and Terminology%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-The grammar of Isar is extended by commands for locales as shown in
-  Figure~\ref{fig-grammar}.
-  A key concept, introduced by Wenzel, is that
-  locales are (internally) lists
-  of \emph{context elements}.  There are five kinds, identified
-  by the keywords \textbf{fixes}, \textbf{constrains},
-  \textbf{assumes}, \textbf{defines} and \textbf{notes}.
-
-  \begin{figure}
-  \hrule
-  \vspace{2ex}
-  \begin{small}
-  \begin{tabular}{l>$c<$l}
-  \textit{attr-name} & ::=
-  & \textit{name} $|$ \textit{attribute} $|$
-    \textit{name} \textit{attribute} \\
-
-  \textit{locale-expr}  & ::= 
-  & \textit{locale-expr1} ( ``\textbf{+}'' \textit{locale-expr1} )$^*$ \\
-  \textit{locale-expr1} & ::=
-  & ( \textit{qualified-name} $|$
-    ``\textbf{(}'' \textit{locale-expr} ``\textbf{)}'' ) \\
-  & & ( \textit{name} [ \textit{mixfix} ] $|$ ``\textbf{\_}'' )$^*$ \\
-
-  \textit{fixes} & ::=
-  & \textit{name} [ ``\textbf{::}'' \textit{type} ]
-    [ ``\textbf{(}'' \textbf{structure} ``\textbf{)}'' $|$
-    \textit{mixfix} ] \\
-  \textit{constrains} & ::=
-  & \textit{name} ``\textbf{::}'' \textit{type} \\
-  \textit{assumes} & ::=
-  & [ \textit{attr-name} ``\textbf{:}'' ] \textit{proposition} \\
-  \textit{defines} & ::=
-  & [ \textit{attr-name} ``\textbf{:}'' ] \textit{proposition} \\
-  \textit{notes} & ::=
-  & [ \textit{attr-name} ``\textbf{=}'' ]
-    ( \textit{qualified-name} [ \textit{attribute} ] )$^+$ \\
-
-  \textit{element} & ::=
-  & \textbf{fixes} \textit{fixes} ( \textbf{and} \textit{fixes} )$^*$ \\
-  & |
-  & \textbf{constrains} \textit{constrains}
-    ( \textbf{and} \textit{constrains} )$^*$ \\
-  & |
-  & \textbf{assumes} \textit{assumes} ( \textbf{and} \textit{assumes} )$^*$ \\
-  & |
-  & \textbf{defines} \textit{defines} ( \textbf{and} \textit{defines} )$^*$ \\
-  & |
-  & \textbf{notes} \textit{notes} ( \textbf{and} \textit{notes} )$^*$ \\
-  \textit{element1} & ::=
-  & \textit{element} \\
-  & | & \textbf{includes} \textit{locale-expr} \\
-
-  \textit{locale} & ::=
-  & \textit{element}$^+$ \\
-  & | & \textit{locale-expr} [ ``\textbf{+}'' \textit{element}$^+$ ] \\
-
-  \textit{in-target} & ::=
-  & ``\textbf{(}'' \textbf{in} \textit{qualified-name} ``\textbf{)}'' \\
-
-  \textit{theorem} & ::= & ( \textbf{theorem} $|$ \textbf{lemma} $|$
-    \textbf{corollary} ) [ \textit{in-target} ] [ \textit{attr-name} ] \\
-
-  \textit{theory-level} & ::= & \ldots \\
-  & | & \textbf{locale} \textit{name} [ ``\textbf{=}''
-    \textit{locale} ] \\
-  % note: legacy "locale (open)" omitted.
-  & | & ( \textbf{theorems} $|$ \textbf{lemmas} ) \\
-  & & [ \textit{in-target} ] [ \textit{attr-name} ``\textbf{=}'' ]
-    ( \textit{qualified-name} [ \textit{attribute} ] )$^+$ \\
-  & | & \textbf{declare} [ \textit{in-target} ] ( \textit{qualified-name}
-    [ \textit{attribute} ] )$^+$ \\
-  & | & \textit{theorem} \textit{proposition} \textit{proof} \\
-  & | & \textit{theorem} \textit{element1}$^*$
-    \textbf{shows} \textit{proposition} \textit{proof} \\
-  & | & \textbf{print\_locale} \textit{locale} \\
-  & | & \textbf{print\_locales}
-  \end{tabular}
-  \end{small}
-  \vspace{2ex}
-  \hrule
-  \caption{Locales extend the grammar of Isar.}
-  \label{fig-grammar}
-  \end{figure}
-
-  At the theory level --- that is, at the outer syntactic level of an
-  Isabelle input file --- \textbf{locale} declares a named
-  locale.  Other kinds of locales,
-  locale expressions and unnamed locales, will be introduced later.  When
-  declaring a named locale, it is possible to \emph{import} another
-  named locale, or indeed several ones by importing a locale
-  expression.  The second part of the declaration, also optional,
-  consists of a number of context element declarations.
-
-  A number of Isar commands have an additional, optional \emph{target}
-  argument, which always refers to a named locale.  These commands
-  are \textbf{theorem} (together with \textbf{lemma} and
-  \textbf{corollary}),  \textbf{theorems} (and
-  \textbf{lemmas}), and \textbf{declare}.  The effect of specifying a target is
-  that these commands focus on the specified locale, not the
-  surrounding theory.  Commands that are used to
-  prove new theorems will add them not to the theory, but to the
-  locale.  Similarly, \textbf{declare} modifies attributes of theorems
-  that belong to the specified target.  Additionally, for
-  \textbf{theorem} (and related commands), theorems stored in the target
-  can be used in the associated proof scripts.
-
-  The Locales package provides a \emph{long goals format} for
-  propositions stated with \textbf{theorem} (and friends).  While
-  normally a goal is just a formula, a long goal is a list of context
-  elements, followed by the keyword \textbf{shows}, followed by the
-  formula.  Roughly speaking, the context elements are
-  (additional) premises.  For an example, see
-  Section~\ref{sec-includes}.  The list of context elements in a long goal
-  is also called \emph{unnamed locale}.
-
-  Finally, there are two commands to inspect locales when working in
-  interactive mode: \textbf{print\_locales} prints the names of all
-  targets
-  visible in the current theory, \textbf{print\_locale} outputs the
-  elements of a named locale or locale expression.
-
-  The following presentation will use notation of
-  Isabelle's meta logic, hence a few sentences to explain this.
-  The logical
-  primitives are universal quantification (\isa{{\isasymAnd}}), entailment
-  (\isa{{\isasymLongrightarrow}}) and equality (\isa{{\isasymequiv}}).  Variables (not bound
-  variables) are sometimes preceded by a question mark.  The logic is
-  typed.  Type variables are denoted by \isa{{\isacharprime}a}, \isa{{\isacharprime}b}
-  etc., and \isa{{\isasymRightarrow}} is the function type.  Double brackets \isa{{\isasymlbrakk}} and \isa{{\isasymrbrakk}} are used to abbreviate nested entailment.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsubsection{Parameters, Assumptions and Facts%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-From a logical point of view a \emph{context} is a formula schema of
-  the form
-\[
-  \isa{{\isasymAnd}x\isactrlsub {\isadigit{1}}{\isasymdots}x\isactrlsub n{\isachardot}\ {\isasymlbrakk}\ C\isactrlsub {\isadigit{1}}{\isacharsemicolon}\ {\isasymdots}\ {\isacharsemicolon}C\isactrlsub m\ {\isasymrbrakk}\ {\isasymLongrightarrow}\ {\isasymdots}}
-\]
-  The variables $\isa{x\isactrlsub {\isadigit{1}}}, \ldots, \isa{x\isactrlsub n}$ are
-  called \emph{parameters}, the premises $\isa{C\isactrlsub {\isadigit{1}}}, \ldots,
-  \isa{C\isactrlsub n}$ \emph{assumptions}.  A formula \isa{F}
-  holds in this context if
-\begin{equation}
-\label{eq-fact-in-context}
-  \isa{{\isasymAnd}x\isactrlsub {\isadigit{1}}{\isasymdots}x\isactrlsub n{\isachardot}\ {\isasymlbrakk}\ C\isactrlsub {\isadigit{1}}{\isacharsemicolon}\ {\isasymdots}\ {\isacharsemicolon}C\isactrlsub m\ {\isasymrbrakk}\ {\isasymLongrightarrow}\ F}
-\end{equation}
-  is valid.  The formula is called a \emph{fact} of the context.
-
-  A locale allows fixing the parameters \isa{x\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ x\isactrlsub n} and making the assumptions \isa{C\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ C\isactrlsub m}.  This implicitly builds the context in
-  which the formula \isa{F} can be established.
-  Parameters of a locale correspond to the context element
-  \textbf{fixes}, and assumptions may be declared with
-  \textbf{assumes}.  Using these context elements one can define
-  the specification of semigroups.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{locale}\isamarkupfalse%
-\ semi\ {\isacharequal}\isanewline
-\ \ \isakeyword{fixes}\ prod\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharbrackleft}{\isacharprime}a{\isacharcomma}\ {\isacharprime}a{\isacharbrackright}\ {\isasymRightarrow}\ {\isacharprime}a{\isachardoublequoteclose}\ {\isacharparenleft}\isakeyword{infixl}\ {\isachardoublequoteopen}{\isasymcdot}{\isachardoublequoteclose}\ {\isadigit{7}}{\isadigit{0}}{\isacharparenright}\isanewline
-\ \ \isakeyword{assumes}\ assoc{\isacharcolon}\ {\isachardoublequoteopen}{\isacharparenleft}x\ {\isasymcdot}\ y{\isacharparenright}\ {\isasymcdot}\ z\ {\isacharequal}\ x\ {\isasymcdot}\ {\isacharparenleft}y\ {\isasymcdot}\ z{\isacharparenright}{\isachardoublequoteclose}%
-\begin{isamarkuptext}%
-The parameter \isa{prod} has a
-  syntax annotation enabling the infix ``\isa{{\isasymcdot}}'' in the
-  assumption of associativity.  Parameters may have arbitrary mixfix
-  syntax, like constants.  In the example, the type of \isa{prod} is
-  specified explicitly.  This is not necessary.  If no type is
-  specified, a most general type is inferred simultaneously for all
-  parameters, taking into account all assumptions (and type
-  specifications of parameters, if present).%
-\footnote{Type inference also takes into account type constraints,
-  definitions and import, as introduced later.}
-
-  Free variables in assumptions are implicitly universally quantified,
-  unless they are parameters.  Hence the context defined by the locale
-  \isa{semi} is
-\[
-  \isa{{\isasymAnd}prod{\isachardot}\ {\isasymlbrakk}\ {\isasymAnd}x\ y\ z{\isachardot}\ prod\ {\isacharparenleft}prod\ x\ y{\isacharparenright}\ z\ {\isacharequal}\ prod\ x\ {\isacharparenleft}prod\ y\ z{\isacharparenright}\ {\isasymrbrakk}\ {\isasymLongrightarrow}\ {\isasymdots}}
-\]
-  The locale can be extended to commutative semigroups.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{locale}\isamarkupfalse%
-\ comm{\isacharunderscore}semi\ {\isacharequal}\ semi\ {\isacharplus}\isanewline
-\ \ \isakeyword{assumes}\ comm{\isacharcolon}\ {\isachardoublequoteopen}x\ {\isasymcdot}\ y\ {\isacharequal}\ y\ {\isasymcdot}\ x{\isachardoublequoteclose}%
-\begin{isamarkuptext}%
-This locale \emph{imports} all elements of \isa{semi}.  The
-  latter locale is called the import of \isa{comm{\isacharunderscore}semi}. The
-  definition adds commutativity, hence its context is
-\begin{align*%
-}
-  \isa{{\isasymAnd}prod{\isachardot}\ {\isasymlbrakk}} & 
-  \isa{{\isasymAnd}x\ y\ z{\isachardot}\ prod\ {\isacharparenleft}prod\ x\ y{\isacharparenright}\ z\ {\isacharequal}\ prod\ x\ {\isacharparenleft}prod\ y\ z{\isacharparenright}{\isacharsemicolon}} \\
-  & \isa{{\isasymAnd}x\ y{\isachardot}\ prod\ x\ y\ {\isacharequal}\ prod\ y\ x\ {\isasymrbrakk}\ {\isasymLongrightarrow}\ {\isasymdots}}
-\end{align*%
-}
-  One may now derive facts --- for example, left-commutativity --- in
-  the context of \isa{comm{\isacharunderscore}semi} by specifying this locale as
-  target, and by referring to the names of the assumptions \isa{assoc} and \isa{comm} in the proof.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{theorem}\isamarkupfalse%
-\ {\isacharparenleft}\isakeyword{in}\ comm{\isacharunderscore}semi{\isacharparenright}\ lcomm{\isacharcolon}\isanewline
-\ \ {\isachardoublequoteopen}x\ {\isasymcdot}\ {\isacharparenleft}y\ {\isasymcdot}\ z{\isacharparenright}\ {\isacharequal}\ y\ {\isasymcdot}\ {\isacharparenleft}x\ {\isasymcdot}\ z{\isacharparenright}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymcdot}\ {\isacharparenleft}y\ {\isasymcdot}\ z{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}x\ {\isasymcdot}\ y{\isacharparenright}\ {\isasymcdot}\ z{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}simp\ add{\isacharcolon}\ assoc{\isacharparenright}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}\ {\isacharparenleft}y\ {\isasymcdot}\ x{\isacharparenright}\ {\isasymcdot}\ z{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}simp\ add{\isacharcolon}\ comm{\isacharparenright}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}\ y\ {\isasymcdot}\ {\isacharparenleft}x\ {\isasymcdot}\ z{\isacharparenright}{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}simp\ add{\isacharcolon}\ assoc{\isacharparenright}\isanewline
-\ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}thesis\ \isacommand{{\isachardot}}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-In this equational Isar proof, ``\isa{{\isasymdots}}'' refers to the
-  right hand side of the preceding equation.
-  After the proof is finished, the fact \isa{lcomm} is added to
-  the locale \isa{comm{\isacharunderscore}semi}.  This is done by adding a
-  \textbf{notes} element to the internal representation of the locale,
-  as explained the next section.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsubsection{Locale Predicates and the Internal Representation of
-  Locales%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-\label{sec-locale-predicates}
-  In mathematical texts, often arbitrary but fixed objects with
-  certain properties are considered --- for instance, an arbitrary but
-  fixed group $G$ --- with the purpose of establishing facts valid for
-  any group.  These facts are subsequently used on other objects that
-  also have these properties.
-
-  Locales permit the same style of reasoning.  Exporting a fact $F$
-  generalises the fixed parameters and leads to a (valid) formula of the
-  form of equation~(\ref{eq-fact-in-context}).  If a locale has many
-  assumptions
-  (possibly accumulated through a number of imports) this formula can
-  become large and cumbersome.  Therefore, Wenzel introduced 
-  predicates that abbreviate the assumptions of locales.  These
-  predicates are not confined to the locale but are visible in the
-  surrounding theory.
-
-  The definition of the locale \isa{semi} generates the \emph{locale
-  predicate} \isa{semi} over the type of the parameter \isa{prod},
-  hence the predicate's type is \isa{{\isacharparenleft}{\isacharbrackleft}{\isacharprime}a{\isacharcomma}\ {\isacharprime}a{\isacharbrackright}\ {\isasymRightarrow}\ {\isacharprime}a{\isacharparenright}\ {\isasymRightarrow}\ bool}.  Its
-  definition is
-\begin{equation}
-  \tag*{\isa{semi{\isacharunderscore}def}:} \isa{semi\ {\isacharquery}prod\ {\isasymequiv}\ {\isasymforall}x\ y\ z{\isachardot}\ {\isacharquery}prod\ {\isacharparenleft}{\isacharquery}prod\ x\ y{\isacharparenright}\ z\ {\isacharequal}\ {\isacharquery}prod\ x\ {\isacharparenleft}{\isacharquery}prod\ y\ z{\isacharparenright}}.
-\end{equation}
-  In the case where the locale has no import, the generated
-  predicate abbreviates all assumptions and is over the parameters
-  that occur in these assumptions.
-
-  The situation is more complicated when a locale extends
-  another locale, as is the case for \isa{comm{\isacharunderscore}semi}.  Two
-  predicates are defined.  The predicate
-  \isa{comm{\isacharunderscore}semi{\isacharunderscore}axioms} corresponds to the new assumptions and is
-  called \emph{delta predicate}, the locale
-  predicate \isa{comm{\isacharunderscore}semi} captures the content of all the locale,
-  including the import.
-  If a locale has neither assumptions nor import, no predicate is
-  defined.  If a locale has import but no assumptions, only the locale
-  predicate is defined.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimML
-%
-\endisadelimML
-%
-\isatagML
-%
-\endisatagML
-{\isafoldML}%
-%
-\isadelimML
-%
-\endisadelimML
-%
-\begin{isamarkuptext}%
-The Locales package generates a number of theorems for locale and
-  delta predicates.  All predicates have a definition and an
-  introduction rule.  Locale predicates that are defined in terms of
-  other predicates (which is the case if and only if the locale has
-  import) also have a number of elimination rules (called
-  \emph{axioms}).  All generated theorems for the predicates of the
-  locales \isa{semi} and \isa{comm{\isacharunderscore}semi} are shown in
-  Figures~\ref{fig-theorems-semi} and~\ref{fig-theorems-comm-semi},
-  respectively.
-  \begin{figure}
-  \hrule
-  \vspace{2ex}
-  Theorems generated for the predicate \isa{semi}.
-  \begin{gather}
-    \tag*{\isa{semi{\isacharunderscore}def}:} \isa{semi\ {\isacharquery}prod\ {\isasymequiv}\ {\isasymforall}x\ y\ z{\isachardot}\ {\isacharquery}prod\ {\isacharparenleft}{\isacharquery}prod\ x\ y{\isacharparenright}\ z\ {\isacharequal}\ {\isacharquery}prod\ x\ {\isacharparenleft}{\isacharquery}prod\ y\ z{\isacharparenright}} \\
-    \tag*{\isa{semi{\isachardot}intro}:} \isa{{\isacharparenleft}{\isasymAnd}x\ y\ z{\isachardot}\ {\isacharquery}prod\ {\isacharparenleft}{\isacharquery}prod\ x\ y{\isacharparenright}\ z\ {\isacharequal}\ {\isacharquery}prod\ x\ {\isacharparenleft}{\isacharquery}prod\ y\ z{\isacharparenright}{\isacharparenright}\ {\isasymLongrightarrow}\ semi\ {\isacharquery}prod}
-  \end{gather}
-  \hrule
-  \caption{Theorems for the locale predicate \isa{semi}.}
-  \label{fig-theorems-semi}
-  \end{figure}
-
-  \begin{figure}[h]
-  \hrule
-  \vspace{2ex}
-  Theorems generated for the predicate \isa{comm{\isacharunderscore}semi{\isacharunderscore}axioms}.
-  \begin{gather}
-    \tag*{\isa{comm{\isacharunderscore}semi{\isacharunderscore}axioms{\isacharunderscore}def}:} \isa{comm{\isacharunderscore}semi{\isacharunderscore}axioms\ {\isacharquery}prod\ {\isasymequiv}\ {\isasymforall}x\ y{\isachardot}\ {\isacharquery}prod\ x\ y\ {\isacharequal}\ {\isacharquery}prod\ y\ x} \\                        
-    \tag*{\isa{comm{\isacharunderscore}semi{\isacharunderscore}axioms{\isachardot}intro}:} \isa{{\isacharparenleft}{\isasymAnd}x\ y{\isachardot}\ {\isacharquery}prod\ x\ y\ {\isacharequal}\ {\isacharquery}prod\ y\ x{\isacharparenright}\ {\isasymLongrightarrow}\ comm{\isacharunderscore}semi{\isacharunderscore}axioms\ {\isacharquery}prod}                       
-  \end{gather}
-  Theorems generated for the predicate \isa{comm{\isacharunderscore}semi}.
-  \begin{gather}
-    \tag*{\isa{comm{\isacharunderscore}semi{\isacharunderscore}def}:} \isa{comm{\isacharunderscore}semi\ {\isacharquery}prod\ {\isasymequiv}\ semi\ {\isacharquery}prod\ {\isasymand}\ comm{\isacharunderscore}semi{\isacharunderscore}axioms\ {\isacharquery}prod} \\                          
-    \tag*{\isa{comm{\isacharunderscore}semi{\isachardot}intro}:} \isa{{\isasymlbrakk}semi\ {\isacharquery}prod{\isacharsemicolon}\ comm{\isacharunderscore}semi{\isacharunderscore}axioms\ {\isacharquery}prod{\isasymrbrakk}\ {\isasymLongrightarrow}\ comm{\isacharunderscore}semi\ {\isacharquery}prod} \\
-    \tag*{\isa{comm{\isacharunderscore}semi{\isachardot}axioms}:} \mbox{} \\
-    \notag \isa{comm{\isacharunderscore}semi\ {\isacharquery}prod\ {\isasymLongrightarrow}\ semi\ {\isacharquery}prod} \\
-    \notag \isa{comm{\isacharunderscore}semi\ {\isacharquery}prod\ {\isasymLongrightarrow}\ comm{\isacharunderscore}semi{\isacharunderscore}axioms\ {\isacharquery}prod}               
-  \end{gather} 
-  \hrule
-  \caption{Theorems for the predicates \isa{comm{\isacharunderscore}semi{\isacharunderscore}axioms} and
-    \isa{comm{\isacharunderscore}semi}.}
-  \label{fig-theorems-comm-semi}
-  \end{figure}
-  Note that the theorems generated by a locale
-  definition may be inspected immediately after the definition in the
-  Proof General interface \cite{Aspinall2000} of Isabelle through
-  the menu item ``Isabelle/Isar$>$Show me $\ldots>$Theorems''.
-
-  Locale and delta predicates are used also in the internal
-  representation of locales as lists of context elements.  While all
-  \textbf{fixes} in a
-  declaration generate internal \textbf{fixes}, all assumptions
-  of one locale declaration contribute to one internal \textbf{assumes}
-  element.  The internal representation of \isa{semi} is
-\[
-\begin{array}{ll}
-  \textbf{fixes} & \isa{prod} :: \isa{{\isachardoublequote}{\isacharbrackleft}{\isacharprime}a{\isacharcomma}\ {\isacharprime}a{\isacharbrackright}\ {\isasymRightarrow}\ {\isacharprime}a{\isachardoublequote}}
-    (\textbf{infixl} \isa{{\isachardoublequote}{\isasymcdot}{\isachardoublequote}} 70) \\
-  \textbf{assumes} & \isa{{\isachardoublequote}semi\ prod{\isachardoublequote}} \\
-  \textbf{notes} & \isa{assoc}: \isa{{\isachardoublequote}{\isacharquery}x\ {\isasymcdot}\ {\isacharquery}y\ {\isasymcdot}\ {\isacharquery}z\ {\isacharequal}\ {\isacharquery}x\ {\isasymcdot}\ {\isacharparenleft}{\isacharquery}y\ {\isasymcdot}\ {\isacharquery}z{\isacharparenright}{\isachardoublequote}}
-\end{array}
-\]
-  and the internal representation of \isa{{\isachardoublequote}comm{\isacharunderscore}semi{\isachardoublequote}} is
-\begin{equation}
-\label{eq-comm-semi}
-\begin{array}{ll}
-  \textbf{fixes} & \isa{prod} :: \isa{{\isachardoublequote}{\isacharbrackleft}{\isacharprime}a{\isacharcomma}\ {\isacharprime}a{\isacharbrackright}\ {\isasymRightarrow}\ {\isacharprime}a{\isachardoublequote}}
-    ~(\textbf{infixl}~\isa{{\isachardoublequote}{\isasymcdot}{\isachardoublequote}}~70) \\
-  \textbf{assumes} & \isa{{\isachardoublequote}semi\ prod{\isachardoublequote}} \\
-  \textbf{notes} & \isa{assoc}: \isa{{\isachardoublequote}{\isacharquery}x\ {\isasymcdot}\ {\isacharquery}y\ {\isasymcdot}\ {\isacharquery}z\ {\isacharequal}\ {\isacharquery}x\ {\isasymcdot}\ {\isacharparenleft}{\isacharquery}y\ {\isasymcdot}\ {\isacharquery}z{\isacharparenright}{\isachardoublequote}} \\
-  \textbf{assumes} & \isa{{\isachardoublequote}comm{\isacharunderscore}semi{\isacharunderscore}axioms\ prod{\isachardoublequote}} \\
-  \textbf{notes} & \isa{comm}: \isa{{\isachardoublequote}{\isacharquery}x\ {\isasymcdot}\ {\isacharquery}y\ {\isacharequal}\ {\isacharquery}y\ {\isasymcdot}\ {\isacharquery}x{\isachardoublequote}} \\
-  \textbf{notes} & \isa{lcomm}: \isa{{\isachardoublequote}{\isacharquery}x\ {\isasymcdot}\ {\isacharparenleft}{\isacharquery}y\ {\isasymcdot}\ {\isacharquery}z{\isacharparenright}\ {\isacharequal}\ {\isacharquery}y\ {\isasymcdot}\ {\isacharparenleft}{\isacharquery}x\ {\isasymcdot}\ {\isacharquery}z{\isacharparenright}{\isachardoublequote}}
-\end{array}
-\end{equation}
-  The \textbf{notes} elements store facts of the
-  locales.  The facts \isa{assoc} and \isa{comm} were added
-  during the declaration of the locales.  They stem from assumptions,
-  which are trivially facts.  The fact \isa{lcomm} was
-  added later, after finishing the proof in the respective
-  \textbf{theorem} command above.
-
-  By using \textbf{notes} in a declaration, facts can be added
-  to a locale directly.  Of course, these must be theorems.
-  Typical use of this feature includes adding theorems that are not
-  usually used as a default rewrite rules by the simplifier to the
-  simpset of the locale by a \textbf{notes} element with the attribute
-  \isa{{\isacharbrackleft}simp{\isacharbrackright}}.  This way it is also possible to add specialised
-  versions of
-  theorems to a locale by instantiating locale parameters for unknowns
-  or locale assumptions for premises.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsubsection{Definitions%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-Definitions were available in Kamm\"uller's version of Locales, and
-  they are in Wenzel's.  
-  The context element \textbf{defines} adds a definition of the form
-  \isa{p\ x\isactrlsub {\isadigit{1}}\ {\isasymdots}\ x\isactrlsub n\ {\isasymequiv}\ t} as an assumption, where \isa{p} is a
-  parameter of the locale (possibly an imported parameter), and \isa{t} a term that may contain the \isa{x\isactrlsub i}.  The parameter may
-  neither occur in a previous \textbf{assumes} or \textbf{defines}
-  element, nor on the right hand side of the definition.  Hence
-  recursion is not allowed.
-  The parameter may, however, occur in subsequent \textbf{assumes} and
-  on the right hand side of subsequent \textbf{defines}.  We call
-  \isa{p} \emph{defined parameter}.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{locale}\isamarkupfalse%
-\ semi{\isadigit{2}}\ {\isacharequal}\ semi\ {\isacharplus}\isanewline
-\ \ \isakeyword{fixes}\ rprod\ {\isacharparenleft}\isakeyword{infixl}\ {\isachardoublequoteopen}{\isasymodot}{\isachardoublequoteclose}\ {\isadigit{7}}{\isadigit{0}}{\isacharparenright}\isanewline
-\ \ \isakeyword{defines}\ rprod{\isacharunderscore}def{\isacharcolon}\ {\isachardoublequoteopen}rprod\ x\ y\ {\isasymequiv}\ y\ {\isasymcdot}\ x\ {\isachardoublequoteclose}%
-\begin{isamarkuptext}%
-This locale extends \isa{semi} by a second binary operation \isa{{\isachardoublequote}{\isasymodot}{\isachardoublequote}} that is like \isa{{\isachardoublequote}{\isasymcdot}{\isachardoublequote}} but with
-  reversed arguments.  The
-  definition of the locale generates the predicate \isa{semi{\isadigit{2}}},
-  which is equivalent to \isa{semi}, but no \isa{semi{\isadigit{2}}{\isacharunderscore}axioms}.
-  The difference between \textbf{assumes} and \textbf{defines} lies in
-  the way parameters are treated on export.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsubsection{Export%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-A fact is exported out
-  of a locale by generalising over the parameters and adding
-  assumptions as premises.  For brevity of the exported theorems,
-  locale predicates are used.  Exported facts are referenced by
-  writing qualified names consisting of the locale name and the fact name ---
-  for example,
-\begin{equation}
-  \tag*{\isa{semi{\isachardot}assoc}:} \isa{semi\ {\isacharquery}prod\ {\isasymLongrightarrow}\ {\isacharquery}prod\ {\isacharparenleft}{\isacharquery}prod\ {\isacharquery}x\ {\isacharquery}y{\isacharparenright}\ {\isacharquery}z\ {\isacharequal}\ {\isacharquery}prod\ {\isacharquery}x\ {\isacharparenleft}{\isacharquery}prod\ {\isacharquery}y\ {\isacharquery}z{\isacharparenright}}.
-\end{equation}
-  Defined parameters receive special treatment.  Instead of adding a
-  premise for the definition, the definition is unfolded in the
-  exported theorem.  In order to illustrate this we prove that the
-  reverse operation \isa{{\isachardoublequote}{\isasymodot}{\isachardoublequote}} defined in the locale
-  \isa{semi{\isadigit{2}}} is also associative.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{theorem}\isamarkupfalse%
-\ {\isacharparenleft}\isakeyword{in}\ semi{\isadigit{2}}{\isacharparenright}\ r{\isacharunderscore}assoc{\isacharcolon}\ {\isachardoublequoteopen}{\isacharparenleft}x\ {\isasymodot}\ y{\isacharparenright}\ {\isasymodot}\ z\ {\isacharequal}\ x\ {\isasymodot}\ {\isacharparenleft}y\ {\isasymodot}\ z{\isacharparenright}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}simp\ only{\isacharcolon}\ rprod{\isacharunderscore}def\ assoc{\isacharparenright}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-The exported fact is
-\begin{equation}
-  \tag*{\isa{semi{\isadigit{2}}{\isachardot}r{\isacharunderscore}assoc}:} \isa{semi{\isadigit{2}}\ {\isacharquery}prod\ {\isasymLongrightarrow}\ {\isacharquery}prod\ {\isacharquery}z\ {\isacharparenleft}{\isacharquery}prod\ {\isacharquery}y\ {\isacharquery}x{\isacharparenright}\ {\isacharequal}\ {\isacharquery}prod\ {\isacharparenleft}{\isacharquery}prod\ {\isacharquery}z\ {\isacharquery}y{\isacharparenright}\ {\isacharquery}x}.
-\end{equation}
-  The defined parameter is not present but is replaced by its
-  definition.  Note that the definition itself is not exported, hence
-  there is no \isa{semi{\isadigit{2}}{\isachardot}rprod{\isacharunderscore}def}.%
-\footnote{The definition could alternatively be exported using a
-  let-construct if there was one in Isabelle's meta-logic.  Let is
-  usually defined in object-logics.}%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsection{Locale Expressions%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-Locale expressions provide a simple language for combining
-  locales.  They are an effective means of building complex
-  specifications from simple ones.  Locale expressions are the main
-  innovation of the version of Locales discussed here.  Locale
-  expressions are also reason for introducing locale predicates.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsubsection{Rename and Merge%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-The grammar of locale expressions is part of the grammar in
-  Figure~\ref{fig-grammar}.  Locale names are locale
-  expressions, and
-  further expressions are obtained by \emph{rename} and \emph{merge}.
-\begin{description}
-\item[Rename.]
-  The locale expression $e\: q_1 \ldots q_n$ denotes
-  the locale of $e$ where pa\-ra\-me\-ters, in the order in
-  which they are fixed, are renamed to $q_1$ to $q_n$.
-  The expression is only well-formed if $n$ does not
-  exceed the number of parameters of $e$.  Underscores denote
-  parameters that are not renamed.
-  Renaming by default removes mixfix syntax, but new syntax may be
-  specified.
-\item[Merge.]
-  The locale expression $e_1 + e_2$ denotes
-  the locale obtained by merging the locales of $e_1$
-  and $e_2$.  This locale contains the context elements
-  of $e_1$, followed by the context elements of $e_2$.
-
-  In actual fact, the semantics of the merge operation
-  is more complicated.  If $e_1$ and $e_2$ are expressions containing
-  the same name, followed by
-  identical parameter lists, then the merge of both will contain
-  the elements of those locales only once.  Details are explained in
-  Section~\ref{sec-normal-forms} below.
-
-  The merge operation is associative but not commutative.  The latter
-  is because parameters of $e_1$ appear
-  before parameters of $e_2$ in the composite expression.
-\end{description}
-
-  Rename can be used if a different parameter name seems more
-  appropriate --- for example, when moving from groups to rings, a
-  parameter \isa{G} representing the group could be changed to
-  \isa{R}.  Besides of this stylistic use, renaming is important in
-  combination with merge.  Both operations are used in the following
-  specification of semigroup homomorphisms.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{locale}\isamarkupfalse%
-\ semi{\isacharunderscore}hom\ {\isacharequal}\ comm{\isacharunderscore}semi\ sum\ {\isacharparenleft}\isakeyword{infixl}\ {\isachardoublequoteopen}{\isasymoplus}{\isachardoublequoteclose}\ {\isadigit{6}}{\isadigit{5}}{\isacharparenright}\ {\isacharplus}\ comm{\isacharunderscore}semi\ {\isacharplus}\isanewline
-\ \ \isakeyword{fixes}\ hom\isanewline
-\ \ \isakeyword{assumes}\ hom{\isacharcolon}\ {\isachardoublequoteopen}hom\ {\isacharparenleft}x\ {\isasymoplus}\ y{\isacharparenright}\ {\isacharequal}\ hom\ x\ {\isasymcdot}\ hom\ y{\isachardoublequoteclose}%
-\begin{isamarkuptext}%
-This locale defines a context with three parameters \isa{sum},
-  \isa{prod} and \isa{hom}.  The first two parameters have
-  mixfix syntax.  They are associative operations,
-  the first of type \isa{{\isacharbrackleft}{\isacharprime}a{\isacharcomma}\ {\isacharprime}a{\isacharbrackright}\ {\isasymRightarrow}\ {\isacharprime}a}, the second of
-  type \isa{{\isacharbrackleft}{\isacharprime}b{\isacharcomma}\ {\isacharprime}b{\isacharbrackright}\ {\isasymRightarrow}\ {\isacharprime}b}.  
-
-  How are facts that are imported via a locale expression identified?
-  Facts are always introduced in a named locale (either in the
-  locale's declaration, or by using the locale as target in
-  \textbf{theorem}), and their names are
-  qualified by the parameter names of this locale.
-  Hence the full name of associativity in \isa{semi} is
-  \isa{prod{\isachardot}assoc}.  Renaming parameters of a target also renames
-  the qualifier of facts.  Hence, associativity of \isa{sum} is
-  \isa{sum{\isachardot}assoc}.  Several parameters are separated by
-  underscores in qualifiers.  For example, the full name of the fact
-  \isa{hom} in the locale \isa{semi{\isacharunderscore}hom} is \isa{sum{\isacharunderscore}prod{\isacharunderscore}hom{\isachardot}hom}.
-
-  The following example is quite artificial, it illustrates the use of
-  facts, though.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{theorem}\isamarkupfalse%
-\ {\isacharparenleft}\isakeyword{in}\ semi{\isacharunderscore}hom{\isacharparenright}\ {\isachardoublequoteopen}hom\ x\ {\isasymcdot}\ {\isacharparenleft}hom\ y\ {\isasymcdot}\ hom\ z{\isacharparenright}\ {\isacharequal}\ hom\ {\isacharparenleft}x\ {\isasymoplus}\ {\isacharparenleft}y\ {\isasymoplus}\ z{\isacharparenright}{\isacharparenright}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}hom\ x\ {\isasymcdot}\ {\isacharparenleft}hom\ y\ {\isasymcdot}\ hom\ z{\isacharparenright}\ {\isacharequal}\ hom\ y\ {\isasymcdot}\ {\isacharparenleft}hom\ x\ {\isasymcdot}\ hom\ z{\isacharparenright}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}simp\ add{\isacharcolon}\ prod{\isachardot}lcomm{\isacharparenright}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}\ hom\ {\isacharparenleft}y\ {\isasymoplus}\ {\isacharparenleft}x\ {\isasymoplus}\ z{\isacharparenright}{\isacharparenright}{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}simp\ add{\isacharcolon}\ hom{\isacharparenright}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}\ hom\ {\isacharparenleft}x\ {\isasymoplus}\ {\isacharparenleft}y\ {\isasymoplus}\ z{\isacharparenright}{\isacharparenright}{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}simp\ add{\isacharcolon}\ sum{\isachardot}lcomm{\isacharparenright}\isanewline
-\ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}thesis\ \isacommand{{\isachardot}}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-Importing via a locale expression imports all facts of
-  the imported locales, hence both \isa{sum{\isachardot}lcomm} and \isa{prod{\isachardot}lcomm} are
-  available in \isa{hom{\isacharunderscore}semi}.  The import is dynamic --- that is,
-  whenever facts are added to a locale, they automatically
-  become available in subsequent \textbf{theorem} commands that use
-  the locale as a target, or a locale importing the locale.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsubsection{Normal Forms%
-}
-\isamarkuptrue%
-%
-\label{sec-normal-forms}
-\newcommand{\I}{\mathcal{I}}
-\newcommand{\F}{\mathcal{F}}
-\newcommand{\N}{\mathcal{N}}
-\newcommand{\C}{\mathcal{C}}
-\newcommand{\App}{\mathbin{\overline{@}}}
-%
-\begin{isamarkuptext}%
-Locale expressions are interpreted in a two-step process.  First, an
-  expression is normalised, then it is converted to a list of context
-  elements.
-
-  Normal forms are based on \textbf{locale} declarations.  These
-  consist of an import section followed by a list of context
-  elements.  Let $\I(l)$ denote the locale expression imported by
-  locale $l$.  If $l$ has no import then $\I(l) = \varepsilon$.
-  Likewise, let $\F(l)$ denote the list of context elements, also
-  called the \emph{context fragment} of $l$.  Note that $\F(l)$
-  contains only those context elements that are stated in the
-  declaration of $l$, not imported ones.
-
-\paragraph{Example 1.}  Consider the locales \isa{semi} and \isa{comm{\isacharunderscore}semi}.  We have $\I(\isa{semi}) = \varepsilon$ and
-  $\I(\isa{comm{\isacharunderscore}semi}) = \isa{semi}$, and the context fragments
-  are
-\begin{align*%
-}
-  \F(\isa{semi}) & = \left[
-\begin{array}{ll}
-  \textbf{fixes} & \isa{prod} :: \isa{{\isachardoublequote}{\isacharbrackleft}{\isacharprime}a{\isacharcomma}\ {\isacharprime}a{\isacharbrackright}\ {\isasymRightarrow}\ {\isacharprime}a{\isachardoublequote}}
-    ~(\textbf{infixl}~\isa{{\isachardoublequote}{\isasymcdot}{\isachardoublequote}}~70) \\
-  \textbf{assumes} & \isa{{\isachardoublequote}semi\ prod{\isachardoublequote}} \\
-  \textbf{notes} & \isa{assoc}: \isa{{\isachardoublequote}{\isacharquery}x\ {\isasymcdot}\ {\isacharquery}y\ {\isasymcdot}\ {\isacharquery}z\ {\isacharequal}\ {\isacharquery}x\ {\isasymcdot}\ {\isacharparenleft}{\isacharquery}y\ {\isasymcdot}\ {\isacharquery}z{\isacharparenright}{\isachardoublequote}}
-\end{array} \right], \\
-  \F(\isa{comm{\isacharunderscore}semi}) & = \left[
-\begin{array}{ll}
-  \textbf{assumes} & \isa{{\isachardoublequote}comm{\isacharunderscore}semi{\isacharunderscore}axioms\ prod{\isachardoublequote}} \\
-  \textbf{notes} & \isa{comm}: \isa{{\isachardoublequote}{\isacharquery}x\ {\isasymcdot}\ {\isacharquery}y\ {\isacharequal}\ {\isacharquery}y\ {\isasymcdot}\ {\isacharquery}x{\isachardoublequote}} \\
-  \textbf{notes} & \isa{lcomm}: \isa{{\isachardoublequote}{\isacharquery}x\ {\isasymcdot}\ {\isacharparenleft}{\isacharquery}y\ {\isasymcdot}\ {\isacharquery}z{\isacharparenright}\ {\isacharequal}\ {\isacharquery}y\ {\isasymcdot}\ {\isacharparenleft}{\isacharquery}x\ {\isasymcdot}\ {\isacharquery}z{\isacharparenright}{\isachardoublequote}}
-\end{array} \right].
-\end{align*%
-}
-  Let $\pi_0(\F(l))$ denote the list of parameters defined in the
-  \textbf{fixes} elements of $\F(l)$ in the order of their occurrence.
-  The list of parameters of a locale expression $\pi(e)$ is defined as
-  follows:
-\begin{align*%
-}
-  \pi(l) & = \pi(\I(l)) \App \pi_0(\F(l)) \text{, for named locale $l$.} \\
-  \pi(e\: q_1 \ldots q_n) & = \text{$[q_1, \ldots, q_n, p_{n+1}, \ldots,
-    p_{m}]$, where $\pi(e) = [p_1, \ldots, p_m]$.} \\
-  \pi(e_1 + e_2) & = \pi(e_1) \App \pi(e_2)
-\end{align*%
-}
-  The operation $\App$ concatenates two lists but omits elements from
-  the second list that are also present in the first list.
-  It is not possible to rename more parameters than there
-  are present in an expression --- that is, $n \le m$ --- otherwise
-  the renaming is illegal.  If $q_i
-  = \_$ then the $i$th entry of the resulting list is $p_i$.
-
-  In the normalisation phase, imports of named locales are unfolded, and
-  renames and merges are recursively propagated to the imported locale
-  expressions.  The result is a list of locale names, each with a full
-  list of parameters, where locale names occurring with the same parameter
-  list twice are removed.  Let $\N$ denote normalisation.  It is defined
-  by these equations:
-\begin{align*%
-}
-  \N(l) & = \N(\I(l)) \App [l\:\pi(l)] \text{, for named locale $l$.} \\
-  \N(e\: q_1 \ldots q_n) & = \N(e)\:[q_1 \ldots q_n/\pi(e)] \\
-  \N(e_1 + e_2) & = \N(e_1) \App \N(e_2)
-\end{align*%
-}
-  Normalisation yields a list of \emph{identifiers}.  An identifier
-  consists of a locale name and a (possibly empty) list of parameters.
-
-  In the second phase, the list of identifiers $\N(e)$ is converted to
-  a list of context elements $\C(e)$ by converting each identifier to
-  a list of context elements, and flattening the obtained list.
-  Conversion of the identifier $l\:q_1 \ldots q_n$ yields the list of
-  context elements $\F(l)$, but with the following modifications:
-\begin{itemize}
-\item
-  Rename the parameter in the $i$th \textbf{fixes} element of $\F(l)$
-  to $q_i$, $i = 1, \ldots, n$.  If the parameter name is actually
-  changed then delete the syntax annotation.  Renaming a parameter may
-  also change its type.
-\item
-  Perform the same renamings on all occurrences of parameters (fixed
-  variables) in \textbf{assumes}, \textbf{defines} and \textbf{notes}
-  elements.
-\item
-  Qualify names of facts by $q_1\_\ldots\_q_n$.
-\end{itemize}
-  The locale expression is \emph{well-formed} if it contains no
-  illegal renamings and the following conditions on $\C(e)$ hold,
-  otherwise the expression is rejected:
-\begin{itemize}
-\item Parameters in \textbf{fixes} are distinct;
-\item Free variables in \textbf{assumes} and
-  \textbf{defines} occur in preceding \textbf{fixes};%
-\footnote{This restriction is relaxed for contexts obtained with
-  \textbf{includes}, see Section~\ref{sec-includes}.}
-\item Parameters defined in \textbf{defines} must neither occur in
-  preceding \textbf{assumes} nor \textbf{defines}.
-\end{itemize}%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsubsection{Examples%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-\paragraph{Example 2.}
-  We obtain the context fragment $\C(\isa{comm{\isacharunderscore}semi})$ of the
-  locale \isa{comm{\isacharunderscore}semi}.  First, the parameters are computed.
-\begin{align*%
-}
-  \pi(\isa{semi}) & = [\isa{prod}] \\
-  \pi(\isa{comm{\isacharunderscore}semi}) & = \pi(\isa{semi}) \App []
-     = [\isa{prod}]
-\end{align*%
-}
-  Next, the normal form of the locale expression
-  \isa{comm{\isacharunderscore}semi} is obtained.
-\begin{align*%
-}
-  \N(\isa{semi}) & = [\isa{semi} \isa{prod}] \\
-  \N(\isa{comm{\isacharunderscore}semi}) & = \N(\isa{semi}) \App
-       [\isa{comm{\isacharunderscore}semi\ prod}]
-   = [\isa{semi\ prod}, \isa{comm{\isacharunderscore}semi\ prod}]
-\end{align*%
-}
-  Converting this to a list of context elements leads to the
-  list~(\ref{eq-comm-semi}) shown in
-  Section~\ref{sec-locale-predicates}, but with fact names qualified
-  by \isa{prod} --- for example, \isa{prod{\isachardot}assoc}.
-  Qualification was omitted to keep the presentation simple.
-  Isabelle's scoping rules identify the most recent fact with
-  qualified name $x.a$ when a fact with name $a$ is requested.
-
-\paragraph{Example 3.}
-  The locale expression \isa{comm{\isacharunderscore}semi\ sum} involves
-  renaming.  Computing parameters yields $\pi(\isa{comm{\isacharunderscore}semi\ sum})
-  = [\isa{sum}]$, the normal form is
-\begin{align*%
-}
-  \N(\isa{comm{\isacharunderscore}semi\ sum}) & =
-  \N(\isa{comm{\isacharunderscore}semi})[\isa{sum}/\isa{prod}] =
-  [\isa{semi\ sum}, \isa{comm{\isacharunderscore}semi\ sum}]
-\end{align*%
-}
-  and the list of context elements
-\[
-\begin{array}{ll}
-  \textbf{fixes} & \isa{sum} :: \isa{{\isachardoublequote}{\isacharbrackleft}{\isacharprime}a{\isacharcomma}\ {\isacharprime}a{\isacharbrackright}\ {\isasymRightarrow}\ {\isacharprime}a{\isachardoublequote}}
-    ~(\textbf{infixl}~\isa{{\isachardoublequote}{\isasymoplus}{\isachardoublequote}}~65) \\
-  \textbf{assumes} & \isa{{\isachardoublequote}semi\ sum{\isachardoublequote}} \\
-  \textbf{notes} & \isa{sum{\isachardot}assoc}: \isa{{\isachardoublequote}{\isacharparenleft}{\isacharquery}x\ {\isasymoplus}\ {\isacharquery}y{\isacharparenright}\ {\isasymoplus}\ {\isacharquery}z\ {\isacharequal}\ sum\ {\isacharquery}x\ {\isacharparenleft}sum\ {\isacharquery}y\ {\isacharquery}z{\isacharparenright}{\isachardoublequote}} \\
-  \textbf{assumes} & \isa{{\isachardoublequote}comm{\isacharunderscore}semi{\isacharunderscore}axioms\ sum{\isachardoublequote}} \\
-  \textbf{notes} & \isa{sum{\isachardot}comm}: \isa{{\isachardoublequote}{\isacharquery}x\ {\isasymoplus}\ {\isacharquery}y\ {\isacharequal}\ {\isacharquery}y\ {\isasymoplus}\ {\isacharquery}x{\isachardoublequote}} \\
-  \textbf{notes} & \isa{sum{\isachardot}lcomm}: \isa{{\isachardoublequote}{\isacharquery}x\ {\isasymoplus}\ {\isacharparenleft}{\isacharquery}y\ {\isasymoplus}\ {\isacharquery}z{\isacharparenright}\ {\isacharequal}\ {\isacharquery}y\ {\isasymoplus}\ {\isacharparenleft}{\isacharquery}x\ {\isasymoplus}\ {\isacharquery}z{\isacharparenright}{\isachardoublequote}}
-\end{array}
-\]
-
-\paragraph{Example 4.}
-  The context defined by the locale \isa{semi{\isacharunderscore}hom} involves
-  merging two copies of \isa{comm{\isacharunderscore}semi}.  We obtain the parameter
-  list and normal form:
-\begin{align*%
-}
-  \pi(\isa{semi{\isacharunderscore}hom}) & = \pi(\isa{comm{\isacharunderscore}semi\ sum} +
-       \isa{comm{\isacharunderscore}semi}) \App [\isa{hom}] \\
-     & = (\pi(\isa{comm{\isacharunderscore}semi\ sum}) \App \pi(\isa{comm{\isacharunderscore}semi}))
-       \App [\isa{hom}] \\
-     & = ([\isa{sum}] \App [\isa{prod}]) \App [\isa{hom}]
-     = [\isa{sum}, \isa{prod}, \isa{hom}] \\
-  \N(\isa{semi{\isacharunderscore}hom}) & =
-       \N(\isa{comm{\isacharunderscore}semi\ sum} + \isa{comm{\isacharunderscore}semi}) \App \\
-     & \phantom{==}
-       [\isa{semi{\isacharunderscore}hom\ sum\ prod\ hom}] \\
-     & = (\N(\isa{comm{\isacharunderscore}semi\ sum}) \App \N(\isa{comm{\isacharunderscore}semi})) \App \\
-     & \phantom{==}
-       [\isa{semi{\isacharunderscore}hom\ sum\ prod\ hom}] \\
-     & = ([\isa{semi\ sum}, \isa{comm{\isacharunderscore}semi\ sum}] \App %\\
-%     & \phantom{==}
-       [\isa{semi\ prod}, \isa{comm{\isacharunderscore}semi\ prod}]) \App \\
-     & \phantom{==}
-       [\isa{semi{\isacharunderscore}hom\ sum\ prod\ hom}] \\
-     & = [\isa{semi\ sum}, \isa{comm{\isacharunderscore}semi\ sum},
-       \isa{semi\ prod}, \isa{comm{\isacharunderscore}semi\ prod}, \\
-     & \phantom{==}
-       \isa{semi{\isacharunderscore}hom\ sum\ prod\ hom}].
-\end{align*%
-}
-  Hence $\C(\isa{semi{\isacharunderscore}hom})$, shown below, is again well-formed.
-\[
-\begin{array}{ll}
-  \textbf{fixes} & \isa{sum} :: \isa{{\isachardoublequote}{\isacharbrackleft}{\isacharprime}a{\isacharcomma}\ {\isacharprime}a{\isacharbrackright}\ {\isasymRightarrow}\ {\isacharprime}a{\isachardoublequote}}
-    ~(\textbf{infixl}~\isa{{\isachardoublequote}{\isasymoplus}{\isachardoublequote}}~65) \\
-  \textbf{assumes} & \isa{{\isachardoublequote}semi\ sum{\isachardoublequote}} \\
-  \textbf{notes} & \isa{sum{\isachardot}assoc}: \isa{{\isachardoublequote}{\isacharparenleft}{\isacharquery}x\ {\isasymoplus}\ {\isacharquery}y{\isacharparenright}\ {\isasymoplus}\ {\isacharquery}z\ {\isacharequal}\ {\isacharquery}x\ {\isasymoplus}\ {\isacharparenleft}{\isacharquery}y\ {\isasymoplus}\ {\isacharquery}z{\isacharparenright}{\isachardoublequote}} \\
-  \textbf{assumes} & \isa{{\isachardoublequote}comm{\isacharunderscore}semi{\isacharunderscore}axioms\ sum{\isachardoublequote}} \\
-  \textbf{notes} & \isa{sum{\isachardot}comm}: \isa{{\isachardoublequote}{\isacharquery}x\ {\isasymoplus}\ {\isacharquery}y\ {\isacharequal}\ {\isacharquery}y\ {\isasymoplus}\ {\isacharquery}x{\isachardoublequote}} \\
-  \textbf{notes} & \isa{sum{\isachardot}lcomm}: \isa{{\isachardoublequote}{\isacharquery}x\ {\isasymoplus}\ {\isacharparenleft}{\isacharquery}y\ {\isasymoplus}\ {\isacharquery}z{\isacharparenright}\ {\isacharequal}\ {\isacharquery}y\ {\isasymoplus}\ {\isacharparenleft}{\isacharquery}x\ {\isasymoplus}\ {\isacharquery}z{\isacharparenright}{\isachardoublequote}} \\
-  \textbf{fixes} & \isa{prod} :: \isa{{\isachardoublequote}{\isacharbrackleft}{\isacharprime}b{\isacharcomma}\ {\isacharprime}b{\isacharbrackright}\ {\isasymRightarrow}\ {\isacharprime}b{\isachardoublequote}}
-    ~(\textbf{infixl}~\isa{{\isachardoublequote}{\isasymcdot}{\isachardoublequote}}~70) \\
-  \textbf{assumes} & \isa{{\isachardoublequote}semi\ prod{\isachardoublequote}} \\
-  \textbf{notes} & \isa{prod{\isachardot}assoc}: \isa{{\isachardoublequote}{\isacharquery}x\ {\isasymcdot}\ {\isacharquery}y\ {\isasymcdot}\ {\isacharquery}z\ {\isacharequal}\ {\isacharquery}x\ {\isasymcdot}\ {\isacharparenleft}{\isacharquery}y\ {\isasymcdot}\ {\isacharquery}z{\isacharparenright}{\isachardoublequote}} \\
-  \textbf{assumes} & \isa{{\isachardoublequote}comm{\isacharunderscore}semi{\isacharunderscore}axioms\ prod{\isachardoublequote}} \\
-  \textbf{notes} & \isa{prod{\isachardot}comm}: \isa{{\isachardoublequote}{\isacharquery}x\ {\isasymcdot}\ {\isacharquery}y\ {\isacharequal}\ {\isacharquery}y\ {\isasymcdot}\ {\isacharquery}x{\isachardoublequote}} \\
-  \textbf{notes} & \isa{prod{\isachardot}lcomm}: \isa{{\isachardoublequote}{\isacharquery}x\ {\isasymcdot}\ {\isacharparenleft}{\isacharquery}y\ {\isasymcdot}\ {\isacharquery}z{\isacharparenright}\ {\isacharequal}\ {\isacharquery}y\ {\isasymcdot}\ {\isacharparenleft}{\isacharquery}x\ {\isasymcdot}\ {\isacharquery}z{\isacharparenright}{\isachardoublequote}} \\
-  \textbf{fixes} & \isa{hom} :: \isa{{\isachardoublequote}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}b{\isachardoublequote}} \\
-  \textbf{assumes} & \isa{{\isachardoublequote}semi{\isacharunderscore}hom{\isacharunderscore}axioms\ sum{\isachardoublequote}} \\
-  \textbf{notes} & \isa{sum{\isacharunderscore}prod{\isacharunderscore}hom{\isachardot}hom}:
-    \isa{hom\ {\isacharparenleft}x\ {\isasymoplus}\ y{\isacharparenright}\ {\isacharequal}\ hom\ x\ {\isasymcdot}\ hom\ y}
-\end{array}
-\]
-
-\paragraph{Example 5.}
-  In this example, a locale expression leading to a list of context
-  elements that is not well-defined is encountered, and it is illustrated
-  how normalisation deals with multiple inheritance.
-  Consider the specification of monads (in the algebraic sense)
-  and monoids.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{locale}\isamarkupfalse%
-\ monad\ {\isacharequal}\isanewline
-\ \ \isakeyword{fixes}\ prod\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharbrackleft}{\isacharprime}a{\isacharcomma}\ {\isacharprime}a{\isacharbrackright}\ {\isasymRightarrow}\ {\isacharprime}a{\isachardoublequoteclose}\ {\isacharparenleft}\isakeyword{infixl}\ {\isachardoublequoteopen}{\isasymcdot}{\isachardoublequoteclose}\ {\isadigit{7}}{\isadigit{0}}{\isacharparenright}\ \isakeyword{and}\ one\ {\isacharcolon}{\isacharcolon}\ {\isacharprime}a\ {\isacharparenleft}{\isachardoublequoteopen}{\isasymone}{\isachardoublequoteclose}\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}{\isacharparenright}\isanewline
-\ \ \isakeyword{assumes}\ l{\isacharunderscore}one{\isacharcolon}\ {\isachardoublequoteopen}{\isasymone}\ {\isasymcdot}\ x\ {\isacharequal}\ x{\isachardoublequoteclose}\ \isakeyword{and}\ r{\isacharunderscore}one{\isacharcolon}\ {\isachardoublequoteopen}x\ {\isasymcdot}\ {\isasymone}\ {\isacharequal}\ x{\isachardoublequoteclose}%
-\begin{isamarkuptext}%
-Monoids are both semigroups and monads and one would want to
-  specify them as locale expression \isa{semi\ {\isacharplus}\ monad}.
-  Unfortunately, this expression is not well-formed.  Its normal form
-\begin{align*%
-}
-  \N(\isa{monad}) & = [\isa{monad\ prod}] \\
-  \N(\isa{semi}+\isa{monad}) & =
-       \N(\isa{semi}) \App \N(\isa{monad})
-     = [\isa{semi\ prod}, \isa{monad\ prod}]
-\end{align*%
-}
-  leads to a list containing the context element
-\[
-  \textbf{fixes}~\isa{prod} :: \isa{{\isachardoublequote}{\isacharbrackleft}{\isacharprime}a{\isacharcomma}\ {\isacharprime}a{\isacharbrackright}\ {\isasymRightarrow}\ {\isacharprime}a{\isachardoublequote}}
-    ~(\textbf{infixl}~\isa{{\isachardoublequote}{\isasymcdot}{\isachardoublequote}}~70)
-\]
-  twice and thus violating the first criterion of well-formedness.  To
-  avoid this problem, one can
-  introduce a new locale \isa{magma} with the sole purpose of fixing the
-  parameter and defining its syntax.  The specifications of semigroup
-  and monad are changed so that they import \isa{magma}.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{locale}\isamarkupfalse%
-\ magma\ {\isacharequal}\ \isakeyword{fixes}\ prod\ {\isacharparenleft}\isakeyword{infixl}\ {\isachardoublequoteopen}{\isasymcdot}{\isachardoublequoteclose}\ {\isadigit{7}}{\isadigit{0}}{\isacharparenright}\isanewline
-\isanewline
-\isacommand{locale}\isamarkupfalse%
-\ semi{\isacharprime}\ {\isacharequal}\ magma\ {\isacharplus}\ \isakeyword{assumes}\ assoc{\isacharcolon}\ {\isachardoublequoteopen}{\isacharparenleft}x\ {\isasymcdot}\ y{\isacharparenright}\ {\isasymcdot}\ z\ {\isacharequal}\ x\ {\isasymcdot}\ {\isacharparenleft}y\ {\isasymcdot}\ z{\isacharparenright}{\isachardoublequoteclose}\isanewline
-\isacommand{locale}\isamarkupfalse%
-\ monad{\isacharprime}\ {\isacharequal}\ magma\ {\isacharplus}\ \isakeyword{fixes}\ one\ {\isacharparenleft}{\isachardoublequoteopen}{\isasymone}{\isachardoublequoteclose}\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}{\isacharparenright}\isanewline
-\ \ \isakeyword{assumes}\ l{\isacharunderscore}one{\isacharcolon}\ {\isachardoublequoteopen}{\isasymone}\ {\isasymcdot}\ x\ {\isacharequal}\ x{\isachardoublequoteclose}\ \isakeyword{and}\ r{\isacharunderscore}one{\isacharcolon}\ {\isachardoublequoteopen}x\ {\isasymcdot}\ {\isasymone}\ {\isacharequal}\ x{\isachardoublequoteclose}%
-\begin{isamarkuptext}%
-Normalisation now yields
-\begin{align*%
-}
-  \N(\isa{semi{\isacharprime}\ {\isacharplus}\ monad{\isacharprime}}) & =
-       \N(\isa{semi{\isacharprime}}) \App \N(\isa{monad{\isacharprime}}) \\
-     & = (\N(\isa{magma}) \App [\isa{semi{\isacharprime}\ prod}]) \App
-         (\N(\isa{magma}) \App [\isa{monad{\isacharprime}\ prod}]) \\
-     & = [\isa{magma\ prod}, \isa{semi{\isacharprime}\ prod}] \App
-         [\isa{magma\ prod}, \isa{monad{\isacharprime}\ prod}]) \\
-     & = [\isa{magma\ prod}, \isa{semi{\isacharprime}\ prod},
-          \isa{monad{\isacharprime}\ prod}]
-\end{align*%
-}
-  where the second occurrence of \isa{magma\ prod} is eliminated.
-  The reader is encouraged to check, using the \textbf{print\_locale}
-  command, that the list of context elements generated from this is
-  indeed well-formed.
-
-  It follows that importing
-  parameters is more flexible than fixing them using a context element.
-  The Locale package provides the predefined locale \isa{var} that
-  can be used to import parameters if no
-  particular mixfix syntax is required.
-  Its definition is
-\begin{center}
-  \textbf{locale} \isa{var} = \textbf{fixes} \isa{x{\isacharunderscore}}
-\end{center}
-  The use of the internal variable \isa{x{\isacharunderscore}}
-  enforces that the parameter is renamed before being used, because
-  internal variables may not occur in the input syntax.  Using
-  \isa{var}, the locale \isa{magma} could have been replaced by
-  the locale expression
-\begin{center}
-  \isa{var} \isa{prod} (\textbf{infixl} \isa{{\isachardoublequote}{\isasymcdot}{\isachardoublequote}} 70)
-\end{center}
-  in the above locale declarations.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsubsection{Includes%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-\label{sec-includes}
-  The context element \textbf{includes} takes a locale expression $e$
-  as argument.  It can only occur in long goals, where it
-  adds, like a target, locale context to the proof context.  Unlike
-  with targets, the proved theorem is not stored
-  in the locale.  Instead, it is exported immediately.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{theorem}\isamarkupfalse%
-\ lcomm{\isadigit{2}}{\isacharcolon}\isanewline
-\ \ \isakeyword{includes}\ comm{\isacharunderscore}semi\ \isakeyword{shows}\ {\isachardoublequoteopen}x\ {\isasymcdot}\ {\isacharparenleft}y\ {\isasymcdot}\ z{\isacharparenright}\ {\isacharequal}\ y\ {\isasymcdot}\ {\isacharparenleft}x\ {\isasymcdot}\ z{\isacharparenright}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymcdot}\ {\isacharparenleft}y\ {\isasymcdot}\ z{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}x\ {\isasymcdot}\ y{\isacharparenright}\ {\isasymcdot}\ z{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}simp\ add{\isacharcolon}\ assoc{\isacharparenright}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}\ {\isacharparenleft}y\ {\isasymcdot}\ x{\isacharparenright}\ {\isasymcdot}\ z{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}simp\ add{\isacharcolon}\ comm{\isacharparenright}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}\ y\ {\isasymcdot}\ {\isacharparenleft}x\ {\isasymcdot}\ z{\isacharparenright}{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}simp\ add{\isacharcolon}\ assoc{\isacharparenright}\isanewline
-\ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}thesis\ \isacommand{{\isachardot}}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-This proof is identical to the proof of \isa{lcomm}.  The use of
-  \textbf{includes} provides the same context and facts as when using
-  \isa{comm{\isacharunderscore}semi} as target.  On the other hand, \isa{lcomm{\isadigit{2}}} is not added as a fact to the locale \isa{comm{\isacharunderscore}semi}, but
-  is directly visible in the theory.  The theorem is
-\[
-  \isa{comm{\isacharunderscore}semi\ {\isacharquery}prod\ {\isasymLongrightarrow}\ {\isacharquery}prod\ {\isacharquery}x\ {\isacharparenleft}{\isacharquery}prod\ {\isacharquery}y\ {\isacharquery}z{\isacharparenright}\ {\isacharequal}\ {\isacharquery}prod\ {\isacharquery}y\ {\isacharparenleft}{\isacharquery}prod\ {\isacharquery}x\ {\isacharquery}z{\isacharparenright}}.
-\]
-  Note that it is possible to
-  combine a target and (several) \textbf{includes} in a goal statement, thus
-  using contexts of several locales but storing the theorem in only
-  one of them.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsection{Structures%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-\label{sec-structures}
-  The specifications of semigroups and monoids that served as examples
-  in previous sections modelled each operation of an algebraic
-  structure as a single parameter.  This is rather inconvenient for
-  structures with many operations, and also unnatural.  In accordance
-  to mathematical texts, one would rather fix two groups instead of
-  two sets of operations.
-
-  The approach taken in Isabelle is to encode algebraic structures
-  with suitable types (in Isabelle/HOL usually records).  An issue to
-  be addressed by
-  locales is syntax for algebraic structures.  This is the purpose of
-  the \textbf{(structure)} annotation in \textbf{fixes}, introduced by
-  Wenzel.  We illustrate this, independently of record types, with a
-  different formalisation of semigroups.
-
-  Let \isa{{\isacharprime}a\ semi{\isacharunderscore}type} be a not further specified type that
-  represents semigroups over the carrier type \isa{{\isacharprime}a}.  Let \isa{s{\isacharunderscore}op} be an operation that maps an object of \isa{{\isacharprime}a\ semi{\isacharunderscore}type} to
-  a binary operation.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{typedecl}\isamarkupfalse%
-\ {\isacharprime}a\ semi{\isacharunderscore}type\isanewline
-\isacommand{consts}\isamarkupfalse%
-\ s{\isacharunderscore}op\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharbrackleft}{\isacharprime}a\ semi{\isacharunderscore}type{\isacharcomma}\ {\isacharprime}a{\isacharcomma}\ {\isacharprime}a{\isacharbrackright}\ {\isasymRightarrow}\ {\isacharprime}a{\isachardoublequoteclose}\ {\isacharparenleft}\isakeyword{infixl}\ {\isachardoublequoteopen}{\isasymstar}{\isasymindex}{\isachardoublequoteclose}\ {\isadigit{7}}{\isadigit{0}}{\isacharparenright}%
-\begin{isamarkuptext}%
-Although \isa{s{\isacharunderscore}op} is a ternary operation, it is declared
-  infix.  The syntax annotation contains the token  \isa{{\isasymindex}}
-  (\verb.\<index>.), which refers to the first argument.  This syntax is only
-  effective in the context of a locale, and only if the first argument
-  is a
-  \emph{structural} parameter --- that is, a parameter with annotation
-  \textbf{(structure)}.  The token has the effect of subscripting the
-  parameter --- by bracketing it between \verb.\<^bsub>. and  \verb.\<^esub>..
-  Additionally, the subscript of the first structural parameter may be
-  omitted, as in this specification of semigroups with structures:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{locale}\isamarkupfalse%
-\ comm{\isacharunderscore}semi{\isacharprime}\ {\isacharequal}\isanewline
-\ \ \isakeyword{fixes}\ G\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharprime}a\ semi{\isacharunderscore}type{\isachardoublequoteclose}\ {\isacharparenleft}\isakeyword{structure}{\isacharparenright}\isanewline
-\ \ \isakeyword{assumes}\ assoc{\isacharcolon}\ {\isachardoublequoteopen}{\isacharparenleft}x\ {\isasymstar}\ y{\isacharparenright}\ {\isasymstar}\ z\ {\isacharequal}\ x\ {\isasymstar}\ {\isacharparenleft}y\ {\isasymstar}\ z{\isacharparenright}{\isachardoublequoteclose}\ \isakeyword{and}\ comm{\isacharcolon}\ {\isachardoublequoteopen}x\ {\isasymstar}\ y\ {\isacharequal}\ y\ {\isasymstar}\ x{\isachardoublequoteclose}%
-\begin{isamarkuptext}%
-Here \isa{x\ {\isasymstar}\ y} is equivalent to \isa{x\ {\isasymstar}\isactrlbsub G\isactrlesub \ y} and
-  abbreviates \isa{s{\isacharunderscore}op\ G\ x\ y}.  A specification of homomorphisms
-  requires a second structural parameter.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{locale}\isamarkupfalse%
-\ semi{\isacharprime}{\isacharunderscore}hom\ {\isacharequal}\ comm{\isacharunderscore}semi{\isacharprime}\ {\isacharplus}\ comm{\isacharunderscore}semi{\isacharprime}\ H\ {\isacharplus}\isanewline
-\ \ \isakeyword{fixes}\ hom\isanewline
-\ \ \isakeyword{assumes}\ hom{\isacharcolon}\ {\isachardoublequoteopen}hom\ {\isacharparenleft}x\ {\isasymstar}\ y{\isacharparenright}\ {\isacharequal}\ hom\ x\ {\isasymstar}\isactrlbsub H\isactrlesub \ hom\ y{\isachardoublequoteclose}%
-\begin{isamarkuptext}%
-The parameter \isa{H} is defined in the second \textbf{fixes}
-  element of $\C(\isa{semi{\isacharprime}{\isacharunderscore}comm})$. Hence \isa{{\isasymstar}\isactrlbsub H\isactrlesub }
-  abbreviates \isa{s{\isacharunderscore}op\ H\ x\ y}.  The same construction can be done
-  with records instead of an \textit{ad-hoc} type.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{record}\isamarkupfalse%
-\ {\isacharprime}a\ semi\ {\isacharequal}\ prod\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharbrackleft}{\isacharprime}a{\isacharcomma}\ {\isacharprime}a{\isacharbrackright}\ {\isasymRightarrow}\ {\isacharprime}a{\isachardoublequoteclose}\ {\isacharparenleft}\isakeyword{infixl}\ {\isachardoublequoteopen}{\isasymbullet}{\isasymindex}{\isachardoublequoteclose}\ {\isadigit{7}}{\isadigit{0}}{\isacharparenright}%
-\begin{isamarkuptext}%
-This declares the types \isa{{\isacharprime}a\ semi} and  \isa{{\isacharparenleft}{\isacharprime}a{\isacharcomma}\ {\isacharprime}b{\isacharparenright}\ semi{\isacharunderscore}scheme}.  The latter is an extensible record, where the second
-  type argument is the type of the extension field.  For details on
-  records, see \cite{NipkowEtAl2002} Chapter~8.3.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{locale}\isamarkupfalse%
-\ semi{\isacharunderscore}w{\isacharunderscore}records\ {\isacharequal}\ struct\ G\ {\isacharplus}\isanewline
-\ \ \isakeyword{assumes}\ assoc{\isacharcolon}\ {\isachardoublequoteopen}{\isacharparenleft}x\ {\isasymbullet}\ y{\isacharparenright}\ {\isasymbullet}\ z\ {\isacharequal}\ x\ {\isasymbullet}\ {\isacharparenleft}y\ {\isasymbullet}\ z{\isacharparenright}{\isachardoublequoteclose}%
-\begin{isamarkuptext}%
-The type \isa{{\isacharparenleft}{\isacharprime}a{\isacharcomma}\ {\isacharprime}b{\isacharparenright}\ semi{\isacharunderscore}scheme} is inferred for the parameter \isa{G}.  Using subtyping on records, the specification can be extended
-  to groups easily.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{record}\isamarkupfalse%
-\ {\isacharprime}a\ group\ {\isacharequal}\ {\isachardoublequoteopen}{\isacharprime}a\ semi{\isachardoublequoteclose}\ {\isacharplus}\isanewline
-\ \ one\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharprime}a{\isachardoublequoteclose}\ {\isacharparenleft}{\isachardoublequoteopen}l{\isasymindex}{\isachardoublequoteclose}\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}{\isacharparenright}\isanewline
-\ \ inv\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a{\isachardoublequoteclose}\ {\isacharparenleft}{\isachardoublequoteopen}inv{\isasymindex}\ {\isacharunderscore}{\isachardoublequoteclose}\ {\isacharbrackleft}{\isadigit{8}}{\isadigit{1}}{\isacharbrackright}\ {\isadigit{8}}{\isadigit{0}}{\isacharparenright}\isanewline
-\isacommand{locale}\isamarkupfalse%
-\ group{\isacharunderscore}w{\isacharunderscore}records\ {\isacharequal}\ semi{\isacharunderscore}w{\isacharunderscore}records\ {\isacharplus}\isanewline
-\ \ \isakeyword{assumes}\ l{\isacharunderscore}one{\isacharcolon}\ {\isachardoublequoteopen}l\ {\isasymbullet}\ x\ {\isacharequal}\ x{\isachardoublequoteclose}\ \isakeyword{and}\ l{\isacharunderscore}inv{\isacharcolon}\ {\isachardoublequoteopen}inv\ x\ {\isasymbullet}\ x\ {\isacharequal}\ l{\isachardoublequoteclose}%
-\begin{isamarkuptext}%
-Finally, the predefined locale
-\begin{center}
-  \textbf{locale} \textit{struct} = \textbf{fixes} \isa{S{\isacharunderscore}}
-    \textbf{(structure)}.
-\end{center}
-  is analogous to \isa{var}.  
-  More examples on the use of structures, including groups, rings and
-  polynomials can be found in the Isabelle distribution in the
-  session HOL-Algebra.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsection{Conclusions and Outlook%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-Locales provide a simple means of modular reasoning.  They enable to
-  abbreviate frequently occurring context statements and maintain facts
-  valid in these contexts.  Importantly, using structures, they allow syntax to be
-  effective only in certain contexts, and thus to mimic common
-  practice in mathematics, where notation is chosen very flexibly.
-  This is also known as literate formalisation \cite{Bailey1998}.
-  Locale expressions allow to duplicate and merge
-  specifications.  This is a necessity, for example, when reasoning about
-  homomorphisms.  Normalisation makes it possible to deal with
-  diamond-shaped inheritance structures, and generally with directed
-  acyclic graphs.  The combination of locales with record
-  types in higher-order logic provides an effective means for
-  specifying algebraic structures: locale import and record subtyping
-  provide independent hierarchies for specifications and structure
-  elements.  Rich examples for this can be found in
-  the Isabelle distribution in the session HOL-Algebra.
-
-  The primary reason for writing this report was to provide a better
-  understanding of locales in Isar.  Wenzel provided hardly any
-  documentation, with the exception of \cite{Wenzel2002b}.  The
-  present report should make it easier for users of Isabelle to take
-  advantage of locales.
-
-  The report is also a base for future extensions.  These include
-  improved syntax for structures.  Identifying them by numbers seems
-  unnatural and can be confusing if more than two structures are
-  involved --- for example, when reasoning about universal
-  properties --- and numbering them by order of occurrence seems
-  arbitrary.  Another desirable feature is \emph{instantiation}.  One
-  may, in the course of a theory development, construct objects that
-  fulfil the specification of a locale.  These objects are possibly
-  defined in the context of another locale.  Instantiation should make it
-  simple to specialise abstract facts for the object under
-  consideration and to use the specified facts.
-
-  A detailed comparison of locales with module systems in type theory
-  has not been undertaken yet, but could be beneficial.  For example,
-  a module system for Coq has recently been presented by Chrzaszcz
-  \cite{Chrzaszcz2003,Chrzaszcz2004}.  While the
-  latter usually constitute extensions of the calculus, locales are
-  a rather thin layer that does not change Isabelle's meta logic.
-  Locales mainly manage specifications and facts.  Functors, like
-  the constructor for polynomial rings, remain objects of the
-  logic.
-
-  \textbf{Acknowledgements.}  Lawrence C.\ Paulson and Norbert
-  Schirmer made useful comments on a draft of this paper.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-\end{isabellebody}%
-%%% Local Variables:
-%%% mode: latex
-%%% TeX-master: "root"
-%%% End:
diff -r 9681f347b6f5 -r d1d35284542f doc-src/Locales/Locales/document/root.bib
--- a/doc-src/Locales/Locales/document/root.bib	Tue Jun 03 11:55:35 2008 +0200
+++ b/doc-src/Locales/Locales/document/root.bib	Tue Jun 03 12:34:22 2008 +0200
@@ -1,173 +1,81 @@
-@phdthesis{Bailey1998,
-  author = "Anthony Bailey",
-  title = "The machine-checked literate formalisation of algebra in type theory",
-  school = "University of Manchester",
-  month = jan,
-  year = 1998,
-  available = { CB }
+@unpublished{IsarRef,
+  author = "Markus Wenzel",
+  title = "The {Isabelle/Isar} Reference Manual",
+  note = "Part of the Isabelle distribution, \url{http://isabelle.in.tum.de/doc/isar-ref.pdf}."
 }
 
-@phdthesis{Kammuller1999a,
-  author = "Florian Kamm{\"u}ller",
-  title = "Modular Reasoning in {Isabelle}",
-  school = "University of Cambridge, Computer Laboratory",
-  note = "Available as Technical Report No. 470.",
-  month = aug,
-  year = 1999,
-  available = { CB }
-}
-
-% TYPES 98
-
-@inproceedings{Kammuller1999b,
-  author = "Florian Kamm{\"u}ller",
-  title = "Modular Structures as Dependent Types in {Isabelle}",
-  pages = "121--132",
-  crossref = "AltenkirchEtAl1999",
+@book {Jacobson1985,
+  author = "Nathan Jacobson",
+  title = "Basic Algebra",
+  volume = "I",
+  publisher = "Freeman",
+  edition = "2nd",
+  year = 1985,
   available = { CB }
 }
 
-@proceedings{AltenkirchEtAl1999,
-  editor = "Thorsten Altenkirch and Wolfgang Naraschewski and Bernhard Reus",
-  title = "Types for Proofs and Programs: International Workshop, {TYPES} '98, {Kloster Irsee, Germany, March 27--31, 1998}, Selected Papers",
-  booktitle = "TYPES '98",
-  publisher = "Springer",
-  series = "LNCS 1657",
-  year = 1999
-}
-% CADE-17
+% TYPES 2006
 
-@inproceedings{Kammuller2000,
-  author = "Florian Kamm{\"u}ller",
-  title = "Modular Reasoning in {Isabelle}",
-  pages = "99--114",
-  crossref = "McAllester2000",
-  available = { CB }
-}
-
-@proceedings{McAllester2000,
-  editor = "David McAllester",
-  title = "17th International Conference on Automated Deduction, Pittsburgh, PA, USA, June 17--20, 2000: Proceedings",
-  booktitle = "CADE 17",
-  publisher = "Springer",
-  series = "LNCS 1831",
-  year = 2000
-}
-
-@book{NipkowEtAl2002,
-  author = "Tobias Nipkow and Lawrence C. Paulson and Markus Wenzel",
-  title = "{Isabelle/HOL}: A Proof Assistant for Higher-Order Logic",
-  publisher = "Springer",
-  series = "LNCS 2283",
-  year = 2002,
+@inproceedings{HaftmannWenzel2007,
+  author = "Florian Haftmann and Makarius Wenzel",
+  title = "Constructive Type Classes in {Isabelle}",
+  pages = "160--174",
+  crossref = "AltenkirchMcBride2007",
   available = { CB }
 }
 
-% TYPES 2002
-
-@inproceedings{Nipkow2003,
-  author = "Tobias Nipkow",
-  title = "Structured Proofs in {Isar/HOL}",
-  pages = "259--278",
-  crossref = "GeuversWiedijk2003"
-}
-
-@proceedings{GeuversWiedijk2003,
-  editor = "H. Geuvers and F. Wiedijk",
-  title = "Types for Proofs and Programs (TYPES 2002)",
-  booktitle = "TYPES 2002",
+@proceedings{AltenkirchMcBride2007,
+  editor = "Thorsten Altenkirch and Connor McBride",
+  title = "Types for Proofs and Programs, TYPES 2006, Nottingham, UK",
+  booktitle = "Types for Proofs and Programs, TYPES 2006, Nottingham, UK",
   publisher = "Springer",
-  series = "LNCS 2646",
-  year = 2003
+  series = "LNCS 4502",
+  year = 2007
 }
 
-@phdthesis{Wenzel2002a,
-  author = "Markus Wenzel",
-  title = "{Isabelle/Isar} --- a versatile environment for human-readable formal proof documents",
-  school = "Technische Universit{\"a}t M{\"u}nchen",
-  note = "Electronically published as http://tumb1.biblio.tu-muenchen.de/publ/diss/in/2002/wenzel.html.",
-  year = 2002
-}
 
-@unpublished{Wenzel2002b,
-  author = "Markus Wenzel",
-  title = "Using locales in {Isabelle/Isar}",
-  note = "Part of the Isabelle2003 distribution, file src/HOL/ex/Locales.thy.  Distribution of Isabelle available at http://isabelle.in.tum.de",
-  year = 2002
+@techreport{Ballarin2006a,
+  author = "Clemens Ballarin",
+  title = "Interpretation of Locales in {Isabelle}: Managing Dependencies between Locales",
+  institution = "Technische Universit{\"a}t M{\"u}nchen",
+  number = "TUM-I0607",
+  year = 2006
 }
 
-@unpublished{Wenzel2003,
-  author = "Markus Wenzel",
-  title = "The {Isabelle/Isar} Reference Manual",
-  note = "Part of the Isabelle2003 distribution, available at http://isabelle.in.tum.de",
-  year = 2003
+% MKM 2006
+
+@inproceedings{Ballarin2006b,
+  author = "Clemens Ballarin",
+  title = "Interpretation of Locales in {Isabelle}: Theories and Proof Contexts",
+  pages = "31--43",
+  crossref = "BorweinFarmer2006"
 }
 
-% TPHOLs 2003
-
-@inproceedings{Chrzaszcz2003,
-  author = "Jacek Chrz{\c{a}}szcz",
-  title = "Implementing Modules in the {Coq} System",
-  pages = "270--286",
-  crossref = "BasinWolff2003",
+@proceedings{BorweinFarmer2006,
+  editor = "Jonathan M. Borwein and William M. Farmer",
+  title = "Mathematical knowledge management, MKM 2006, Wokingham, UK",
+  booktitle = "Mathematical knowledge management, MKM 2006, Wokingham, UK",
+  series = "LNCS 4108",
+  publisher = "Springer",
+  year = 2006,
   available = { CB }
 }
 
-@proceedings{BasinWolff2003,
-  editor = "David Basin and Burkhart Wolff",
-  title = "Theorem proving in higher order logics: 16th International Conference, TPHOLs 2003, Rome, Italy, September 2003: proceedings",
-  booktitle = "TPHOLs 2003",
-  publisher = "Springer",
-  series = "LNCS 2758",
-  year = 2003
-}
+% TPHOLs 1999
 
-@PhdThesis{Klein2003,
-  author = "Gerwin Klein",
-  title = "Verified Java Bytecode Verification",
-  school = "Institut f{\"u}r Informatik, Technische Universit{\"a}t M{\"u}nchen",
-  year = 2003
-}
-
-% TACAS 2000
-
-@inproceedings{Aspinall2000,
-  author = "David Aspinall",
-  title = "Proof General: A Generic Tool for Proof Development",
-  pages = "38--42",
-  crossref = "GrafSchwartzbach2000"
+@inproceedings{KammullerEtAl1999,
+  author = "Florian Kamm{\"u}ller and Markus Wenzel and Lawrence C. Paulson",
+  title = "Locales: A Sectioning Concept for {Isabelle}",
+  pages = "149--165",
+  crossref = "BertotEtAl1999",
+  available = { CB }
 }
 
-@proceedings{GrafSchwartzbach2000,
-  editor    = {Susanne Graf and Michael I. Schwartzbach},
-  title     = {Tools and Algorithms for Construction and Analysis of Systems, 6th International Conference, TACAS 2000, Berlin, Germany, March 25 -- April 2, 2000, Proceedings},
-  booktitle     = "TACAS 2000",
-  publisher = {Springer},
-  series    = {LNCS 1785},
-  year      = {2000},
-}
-
-% TYPES 2004
-
-@inproceedings{Ballarin2004a,
-  author = "Clemens Ballarin",
-  title = "Locales and Locale Expressions in {Isabelle/Isar}",
-  pages = "34--50",
-  crossref = "BerardiEtAl2004"
+@book{BertotEtAl1999,
+  editor = "Y. Bertot and G. Dowek and A. Hirschowitz and C. Paulin and L. Th{\'e}ry",
+  title = "Theorem Proving in Higher Order Logics: TPHOLs'99, Nice, France",
+  booktitle = "Theorem Proving in Higher Order Logics: TPHOLs'99, Nice, France",
+  publisher = "Springer",
+  series = "LNCS 1690",
+  year = 1999
 }
-
-@inproceedings{Chrzaszcz2004,
-  author = "Jacek Chrz{\c{a}}szcz",
-  title = "Modules in {Coq} Are and Will Be Correct",
-  pages = "130--136",
-  crossref = "BerardiEtAl2004",
-  available = { CB }
-}
-@proceedings{BerardiEtAl2004,
-  editor = "Stefano Berardi and Mario Coppo and Ferruccio Damiani",
-  title = "Types for Proofs and Programs, TYPES 2003, Torino, Italy",
-  booktitle = "Types for Proofs and Programs, TYPES 2003, Torino, Italy",
-  publisher = "Springer",
-  series = "LNCS 3085",
-  year = 2004
-}
diff -r 9681f347b6f5 -r d1d35284542f doc-src/Locales/Locales/document/root.tex
--- a/doc-src/Locales/Locales/document/root.tex	Tue Jun 03 11:55:35 2008 +0200
+++ b/doc-src/Locales/Locales/document/root.tex	Tue Jun 03 12:34:22 2008 +0200
@@ -1,48 +1,81 @@
-\documentclass[leqno]{article}
-\usepackage{../isabelle,../isabellesym}
-
+\documentclass[11pt,a4paper]{article}
 \usepackage{amsmath}
-\usepackage{amssymb}
+\usepackage{tikz}
+\usepackage{subfigure}
+\usepackage{isabelle,isabellesym}
+\usepackage{verbatim}
 \usepackage{array}
 
+% further packages required for unusual symbols (see also
+% isabellesym.sty), use only when needed
+
+\usepackage{amssymb}
+  %for \<leadsto>, \<box>, \<diamond>, \<sqsupset>, \<mho>, \<Join>,
+  %\<lhd>, \<lesssim>, \<greatersim>, \<lessapprox>, \<greaterapprox>,
+  %\<triangleq>, \<yen>, \<lozenge>
+
+%\usepackage[greek,english]{babel}
+  %option greek for \<euro>
+  %option english (default language) for \<guillemotleft>, \<guillemotright>
+
+%\usepackage[latin1]{inputenc}
+  %for \<onesuperior>, \<onequarter>, \<twosuperior>, \<onehalf>,
+  %\<threesuperior>, \<threequarters>, \<degree>
+
+%\usepackage[only,bigsqcap]{stmaryrd}
+  %for \<Sqinter>
+
+%\usepackage{eufrak}
+  %for \<AA> ... \<ZZ>, \<aa> ... \<zz> (also included in amssymb)
+
+%\usepackage{textcomp}
+  %for \<cent>, \<currency>
+
 % this should be the last package used
-\usepackage{../pdfsetup}
+\usepackage{pdfsetup}
 
+\isafoldtag{proof}
+
+% urls in roman style, theory text in typewriter
 \urlstyle{rm}
 \isabellestyle{tt}
-%\renewcommand{\isacharunderscore}{\_}%
-% the latter is not necessary with \isabellestyle{tt}
+
 
 \begin{document}
 
-\title{Locales and Locale Expressions in Isabelle/Isar}
-\author{Clemens Ballarin \\
-  Fakult\"at f\"ur Informatik \\ Technische Universit\"at M\"unchen \\
-  85748 Garching, Germany \\
-  ballarin@in.tum.de}
+\title{Tutorial to Locales and Locale Interpretation}
+\author{Clemens Ballarin}
 \date{}
 
 \maketitle
 
+\thispagestyle{myheadings}
+\markright{Technical Report TUM-I0723, Technische Universit\"at M\"unchen, 2007}
+
 \begin{abstract}
-  Locales provide a module system for the Isabelle proof assistant.
-  Recently, locales have been ported to the new Isar format for
-  structured proofs.  At the same time, they have been extended by
-  locale expressions, a language for composing locale specifications,
-  and by structures, which provide syntax for algebraic structures.
-  The present paper presents both and is suitable as a tutorial to
-  locales in Isar, because it covers both basics and recent
-  extensions, and contains many examples.
+  Locales are Isabelle's mechanism to deal with parametric theories.
+  We present typical examples of locale specifications,
+  along with interpretations between locales to change their
+  hierarchic dependencies and interpretations to reuse locales in
+  theory contexts and proofs.
+
+  This tutorial is intended for locale novices; familiarity with
+  Isabelle and Isar is presumed.
 \end{abstract}
 
-%\tableofcontents
-
 \parindent 0pt\parskip 0.5ex
 
-% include generated text of all theories
+% generated text of all theories
 \input{session}
 
-\bibliographystyle{plain}
+% optional bibliography
+\newpage
+\bibliographystyle{abbrv}
 \bibliography{root}
 
 \end{document}
+
+%%% Local Variables:
+%%% mode: latex
+%%% TeX-master: t
+%%% End:
diff -r 9681f347b6f5 -r d1d35284542f doc-src/Locales/Locales/document/session.tex
--- a/doc-src/Locales/Locales/document/session.tex	Tue Jun 03 11:55:35 2008 +0200
+++ b/doc-src/Locales/Locales/document/session.tex	Tue Jun 03 12:34:22 2008 +0200
@@ -1,4 +1,12 @@
-\input{Locales.tex}
+\input{GCD.tex}
+
+\input{Examples.tex}
+
+\input{Examples1.tex}
+
+\input{Examples2.tex}
+
+\input{Examples3.tex}
 
 %%% Local Variables:
 %%% mode: latex
diff -r 9681f347b6f5 -r d1d35284542f doc-src/Locales/Makefile
--- a/doc-src/Locales/Makefile	Tue Jun 03 11:55:35 2008 +0200
+++ b/doc-src/Locales/Makefile	Tue Jun 03 12:34:22 2008 +0200
@@ -17,7 +17,9 @@
 NAME = locales
 
 FILES = Locales/document/root.tex Locales/document/root.bib \
-  Locales/document/session.tex Locales/document/Locales.tex \
+  Locales/document/session.tex Locales/document/Examples.tex \
+  Locales/document/Examples1.tex Locales/document/Examples2.tex \
+  Locales/document/Examples3.tex \
   ../isabelle.sty ../isabellesym.sty ../pdfsetup.sty 
 
 dvi: $(NAME).dvi