more standard document preparation within session context;

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/Locales/Examples.thy	Mon Aug 27 21:19:16 2012 +0200
@@ -0,0 +1,770 @@
+theory Examples
+imports Main
+begin
+
+pretty_setmargin %invisible 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 the 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
+\[
+  @{text "\<And>x\<^sub>1\<dots>x\<^sub>n. \<lbrakk> A\<^sub>1; \<dots> ;A\<^sub>m \<rbrakk> \<Longrightarrow> C"}.
+\]
+  Isabelle/Isar's notion of context goes beyond this logical view.
+  Its contexts record, in a consecutive order, proved
+  conclusions along with \emph{attributes}, which can provide context
+  specific configuration information for proof procedures and concrete
+  syntax.  From a logical perspective, locales are just contexts that
+  have been made persistent.  To the user, though, they provide
+  powerful means for declaring and combining contexts, and for the
+  reuse of theorems proved in these contexts.
+  *}
+
+section {* Simple Locales *}
+
+text {*
+  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 (in partial_order) {* The parameter of this locale is~@{text le},
+  which is a binary predicate with infix syntax~@{text \<sqsubseteq>}.  The
+  parameter syntax is available in the subsequent
+  assumptions, which are the familiar partial order axioms.
+
+  Isabelle recognises unbound names as free variables.  In locale
+  assumptions, these are implicitly universally quantified.  That is,
+  @{term "\<lbrakk> x \<sqsubseteq> y; y \<sqsubseteq> z \<rbrakk> \<Longrightarrow> x \<sqsubseteq> z"} in fact means
+  @{term "\<And>x y z. \<lbrakk> x \<sqsubseteq> y; y \<sqsubseteq> z \<rbrakk> \<Longrightarrow> x \<sqsubseteq> z"}.
+
+  Two commands are provided to inspect locales:
+  \isakeyword{print\_locales} lists the names of all locales of the
+  current theory; \isakeyword{print\_locale}~$n$ prints the parameters
+  and assumptions of locale $n$; the variation \isakeyword{print\_locale!}~$n$
+  additionally outputs the conclusions that are stored in the locale.
+  We may inspect the new locale
+  by issuing \isakeyword{print\_locale!} @{term partial_order}.  The output
+  is the following list of context elements.
+\begin{small}
+\begin{alltt}
+  \isakeyword{fixes} le :: "'a \(\Rightarrow\) 'a \(\Rightarrow\)  bool" (\isakeyword{infixl} "\(\sqsubseteq\)" 50)
+  \isakeyword{assumes} "partial_order op \(\sqsubseteq\)"
+  \isakeyword{notes} assumption
+    refl [intro, simp] = `?x \(\sqsubseteq\) ?x`
+    \isakeyword{and}
+    anti_sym [intro] = `\(\isasymlbrakk\)?x \(\sqsubseteq\) ?y; ?y \(\sqsubseteq\) ?x\(\isasymrbrakk\) \(\Longrightarrow\) ?x = ?y`
+    \isakeyword{and}
+    trans [trans] = `\(\isasymlbrakk\)?x \(\sqsubseteq\) ?y; ?y \(\sqsubseteq\) ?z\(\isasymrbrakk\) \(\Longrightarrow\) ?x \(\sqsubseteq\) ?z`
+\end{alltt}
+\end{small}
+  The keyword \isakeyword{notes} denotes a conclusion element.  There
+  is one conclusion, which was added automatically.  Instead, there is
+  only one assumption, namely @{term "partial_order le"}.  The locale
+  declaration has introduced the predicate @{term
+  partial_order} to the theory.  This predicate is the
+  \emph{locale predicate}.  Its definition may be inspected by
+  issuing \isakeyword{thm} @{thm [source] partial_order_def}.
+  @{thm [display, indent=2] partial_order_def}
+  In our example, this is a unary predicate over the parameter of the
+  locale.  It is equivalent to the original assumptions, which have
+  been turned into conclusions and are
+  available as theorems in the context of the locale.  The names and
+  attributes from the locale declaration are associated to these
+  theorems and are effective in the context of the locale.
+
+  Each conclusion has a \emph{foundational theorem} as counterpart
+  in the theory.  Technically, this is simply the theorem composed
+  of context and conclusion.  For the transitivity theorem, this is
+  @{thm [source] partial_order.trans}:
+  @{thm [display, indent=2] partial_order.trans}
+*}
+
+subsection {* Targets: Extending Locales *}
+
+text {*
+  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.  Isar provides 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{primrec} & primitive recursion \\
+  \isakeyword{fun}, \isakeyword{function} & general recursion \\
+  \isakeyword{abbreviation} & syntactic abbreviation \\
+  \isakeyword{theorem}, etc.\ & theorem statement with proof \\
+  \isakeyword{theorems}, etc.\ & redeclaration of theorems \\
+  \isakeyword{text}, etc.\ & document markup
+\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 (in partial_order) {* The strict order @{text less} with infix
+  syntax~@{text \<sqsubset>} is
+  defined in terms of the locale parameter~@{text le} and the general
+  equality of the object logic we work in.  The definition generates a
+  \emph{foundational constant}
+  @{term partial_order.less} with definition @{thm [source]
+  partial_order.less_def}:
+  @{thm [display, indent=2] partial_order.less_def}
+  At the same time, the locale is extended by syntax transformations
+  hiding this construction in the context of the locale.  Here, the
+  abbreviation @{text less} is available for
+  @{text "partial_order.less le"}, and it is printed
+  and parsed as infix~@{text \<sqsubset>}.  Finally, the conclusion @{thm [source]
+  less_def} is added to the locale:
+  @{thm [display, indent=2] less_def}
+*}
+
+text {* The treatment of theorem statements is more straightforward.
+  As an example, here is the derivation of a transitivity law for the
+  strict order relation. *}
+
+  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, conclusions of the
+  locale may be used like theorems.  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.  *}
+
+subsection {* Context Blocks *}
+
+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.
+
+  This style of working is illustrated in the block below, where
+  notions of infimum and supremum for partial orders are introduced,
+  together with theorems about their uniqueness.  *}
+
+  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 {* The syntax of the locale commands discussed in this tutorial is
+  shown in Table~\ref{tab:commands}.  The grammar is complete with the
+  exception of the context elements \isakeyword{constrains} and
+  \isakeyword{defines}, which are provided for backward
+  compatibility.  See the Isabelle/Isar Reference
+  Manual~\cite{IsarRef} for full documentation.  *}
+
+
+section {* Import \label{sec:import} *}
+
+text {* 
+  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. *}
+
+text {*
+  With locales, this kind of inheritance is achieved through
+  \emph{import} of locales.  The import part of a locale declaration,
+  if present, precedes the context elements.  Here is an example,
+  where partial orders are extended to lattices.
+  *}
+
+  locale lattice = partial_order +
+    assumes ex_inf: "\<exists>inf. is_inf x y inf"
+      and ex_sup: "\<exists>sup. is_sup x y sup"
+  begin
+
+text {* These assumptions refer to the predicates for infimum
+  and supremum defined for @{text partial_order} in the previous
+  section.  We now introduce the notions of meet and join.  *}
+
+  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 to
+  establish a sufficiently rich landscape of locales for
+  further examples in this tutorial.  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: "x \<sqinter> (y \<squnion> z) = x \<sqinter> y \<squnion> x \<sqinter> 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 hierarchy obtained through these declarations is shown in
+  Figure~\ref{fig:lattices}(a).
+
+\begin{figure}
+\hrule \vspace{2ex}
+\begin{center}
+\subfigure[Declared hierarchy]{
+\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
+  \label{sec:changing-the-hierarchy} *}
+
+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}.  Locales generalise
+  interpretation from theorems to conclusions, enabling the reuse of
+  definitions and other constructs that are not part of the
+  specifications of the locales.
+
+  The first form of interpretation we will consider in this tutorial
+  is provided by the \isakeyword{sublocale} command.  It enables to
+  modify the import hierarchy to reflect the \emph{logical} relation
+  between locales.
+
+  Consider the locale hierarchy from Figure~\ref{fig:lattices}(a).
+  Total orders are lattices, although this is not reflected here, and
+  definitions, theorems and other conclusions
+  from @{term lattice} are not available in @{term total_order}.  To
+  obtain the situation in Figure~\ref{fig:lattices}(b), it is
+  sufficient to add the conclusions of the latter locale to the former.
+  The \isakeyword{sublocale} command does exactly this.
+  The declaration \isakeyword{sublocale} $l_1
+  \subseteq l_2$ causes locale $l_2$ to be \emph{interpreted} in the
+  context of $l_1$.  This means that all conclusions of $l_2$ are made
+  available in $l_1$.
+
+  Of course, the change of hierarchy must be supported by a theorem
+  that reflects, in our example, that total orders are indeed
+  lattices.  Therefore the \isakeyword{sublocale} command generates a
+  goal, which must be discharged by the user.  This is illustrated in
+  the following paragraphs.  First the sublocale relation is stated.
+*}
+
+  sublocale %visible total_order \<subseteq> lattice
+
+txt {* \normalsize
+  This enters the context of locale @{text total_order}, in
+  which the goal @{subgoals [display]} must be shown.
+  Now the
+  locale predicate needs to be unfolded --- for example, using its
+  definition or by introduction rules
+  provided by the locale package.  For automation, the locale package
+  provides the methods @{text intro_locales} and @{text
+  unfold_locales}.  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}, which need to be shown, hence
+  @{text unfold_locales} is appropriate. *}
+
+  proof unfold_locales
+
+txt {* \normalsize
+  Since the fact that both lattices and total orders are partial
+  orders is already reflected in the locale hierarchy, the assumptions
+  of @{text partial_order} are discharged automatically, and only the
+  assumptions introduced in @{text lattice} remain as subgoals
+  @{subgoals [display]}
+  The proof for the first subgoal is obtained by constructing an
+  infimum, whose existence is implied by totality. *}
+
+    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 {* \normalsize
+   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, we may establish that total orders are distributive
+  lattices with a second \isakeyword{sublocale} statement. *}
+
+  sublocale 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). *}
+
+text {*
+  Locale interpretation is \emph{dynamic}.  The statement
+  \isakeyword{sublocale} $l_1 \subseteq l_2$ will not just add the
+  current conclusions of $l_2$ to $l_1$.  Rather the dependency is
+  stored, and conclusions that will be
+  added to $l_2$ in future are automatically propagated to $l_1$.
+  The sublocale relation is transitive --- that is, propagation takes
+  effect along chains of sublocales.  Even cycles in the sublocale relation are
+  supported, as long as these cycles do not lead to infinite chains.
+  Details are discussed in the technical report \cite{Ballarin2006a}.
+  See also Section~\ref{sec:infinite-chains} of this tutorial.  *}
+
+end

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/Locales/Examples1.thy	Mon Aug 27 21:19:16 2012 +0200
@@ -0,0 +1,89 @@
+theory Examples1
+imports Examples
+begin
+text {* \vspace{-5ex} *}
+section {* Use of Locales in Theories and Proofs
+  \label{sec:interpretation} *}
+
+text {*
+  Locales can be interpreted in the contexts of theories and
+  structured proofs.  These interpretations are dynamic, too.
+  Conclusions of locales will be propagated to the current theory or
+  the current proof context.%
+\footnote{Strictly speaking, only interpretation in theories is
+  dynamic since it is not possible to change locales or the locale
+  hierarchy from within a proof.}
+  The focus of this section is on
+  interpretation in theories, but we will also encounter
+  interpretations in proofs, in
+  Section~\ref{sec:local-interpretation}.
+
+  As an example, consider the type of integers @{typ int}.  The
+  relation @{term "op \<le>"} is a total order over @{typ int}.  We start
+  with the interpretation that @{term "op \<le>"} is a partial order.  The
+  facilities of the interpretation command are explored gradually in
+  three versions.
+  *}
+
+
+subsection {* First Version: Replacement of Parameters Only
+  \label{sec:po-first} *}
+
+text {*
+  The command \isakeyword{interpretation} is for the interpretation of
+  locale in theories.  In the following example, the parameter of locale
+  @{text partial_order} is replaced by @{term "op \<le> :: int \<Rightarrow> int \<Rightarrow>
+  bool"} and the locale instance is interpreted in the current
+  theory. *}
+
+  interpretation %visible int: partial_order "op \<le> :: int \<Rightarrow> int \<Rightarrow> bool"
+txt {* \normalsize
+  The argument of the command is a simple \emph{locale expression}
+  consisting of the name of the interpreted locale, which is
+  preceded by the qualifier @{text "int:"} and succeeded by a
+  white-space-separated list of terms, which provide a full
+  instantiation of the locale parameters.  The parameters are referred
+  to by order of declaration, which is also the order in which
+  \isakeyword{print\_locale} outputs them.  The locale has only a
+  single parameter, hence the list of instantiation terms is a
+  singleton.
+
+  The command creates the goal
+  @{subgoals [display]} which can be shown easily:
+ *}
+    by unfold_locales auto
+
+text {*  The effect of the command is that instances of all
+  conclusions of the locale are available in the theory, where names
+  are prefixed by the qualifier.  For example, transitivity for @{typ
+  int} is named @{thm [source] int.trans} and is the following
+  theorem:
+  @{thm [display, indent=2] int.trans}
+  It is not possible to reference this theorem simply as @{text
+  trans}.  This prevents unwanted hiding of existing theorems of the
+  theory by an interpretation. *}
+
+
+subsection {* Second Version: Replacement of Definitions *}
+
+text {* Not only does the above interpretation qualify theorem names.
+  The prefix @{text int} is applied to all names introduced in locale
+  conclusions including names introduced in definitions.  The
+  qualified name @{text int.less} is short for
+  the interpretation of the definition, which is @{term int.less}.
+  Qualified name and expanded form may be used almost
+  interchangeably.%
+\footnote{Since @{term "op \<le>"} is polymorphic, for @{term int.less} a
+  more general type will be inferred than for @{text int.less} which
+  is over type @{typ int}.}
+  The latter is preferred on output, as for example in the theorem
+  @{thm [source] int.less_le_trans}: @{thm [display, indent=2]
+  int.less_le_trans}
+  Both notations for the strict order are not satisfactory.  The
+  constant @{term "op <"} is the strict order for @{typ int}.
+  In order to allow for the desired replacement, interpretation
+  accepts \emph{equations} in addition to the parameter instantiation.
+  These follow the locale expression and are indicated with the
+  keyword \isakeyword{where}.  This is the revised interpretation:
+  *}
+end

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/Locales/Examples2.thy	Mon Aug 27 21:19:16 2012 +0200
@@ -0,0 +1,24 @@
+theory Examples2
+imports Examples
+begin
+text {* \vspace{-5ex} *}
+  interpretation %visible int: partial_order "op \<le> :: [int, int] \<Rightarrow> bool"
+    where "int.less x y = (x < y)"
+  proof -
+    txt {* \normalsize The goals are now:
+      @{subgoals [display]}
+      The proof that~@{text \<le>} is a partial order is as above. *}
+    show "partial_order (op \<le> :: int \<Rightarrow> int \<Rightarrow> bool)"
+      by unfold_locales auto
+    txt {* \normalsize The second goal is shown by unfolding the
+      definition of @{term "partial_order.less"}. *}
+    show "partial_order.less op \<le> x 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 the
+  interpreted locale.  Hence, the premise of @{text
+  "partial_order.less_def"} is discharged manually with @{text OF}.
+  *}
+end

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/Locales/Examples3.thy	Mon Aug 27 21:19:16 2012 +0200
@@ -0,0 +1,659 @@
+theory Examples3
+imports Examples
+begin
+text {* \vspace{-5ex} *}
+subsection {* Third Version: Local Interpretation
+  \label{sec:local-interpretation} *}
+
+text {* In the above example, the fact that @{term "op \<le>"} is a partial
+  order for the integers was used in the second goal to
+  discharge the premise in the definition of @{text "op \<sqsubset>"}.  In
+  general, proofs of the equations not only may involve definitions
+  from the interpreted locale but arbitrarily complex arguments in the
+  context of the locale.  Therefore it would be convenient to have the
+  interpreted locale conclusions temporarily available in the proof.
+  This can be achieved by a locale interpretation in the proof body.
+  The command for local interpretations is \isakeyword{interpret}.  We
+  repeat the example from the previous section to illustrate this. *}
+
+  interpretation %visible int: partial_order "op \<le> :: int \<Rightarrow> int \<Rightarrow> bool"
+    where "int.less x y = (x < y)"
+  proof -
+    show "partial_order (op \<le> :: int \<Rightarrow> int \<Rightarrow> bool)"
+      by unfold_locales auto
+    then interpret int: partial_order "op \<le> :: [int, int] \<Rightarrow> bool" .
+    show "int.less x y = (x < y)"
+      unfolding int.less_def by auto
+  qed
+
+text {* The inner interpretation is immediate from the preceding fact
+  and proved by assumption (Isar short hand ``.'').  It enriches the
+  local proof context by the theorems
+  also obtained in the interpretation from Section~\ref{sec:po-first},
+  and @{text int.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 succeeding
+  \isakeyword{next} or \isakeyword{qed} statement. *}
+
+
+subsection {* Further Interpretations *}
+
+text {* Further interpretations are necessary for
+  the other locales.  In @{text lattice} the operations~@{text \<sqinter>}
+  and~@{text \<squnion>} are substituted by @{term "min :: int \<Rightarrow> int \<Rightarrow> int"}
+  and @{term "max :: int \<Rightarrow> int \<Rightarrow> int"}.  The entire proof for the
+  interpretation is reproduced to give an example of a more
+  elaborate interpretation proof.  Note that the equations are named
+  so they can be used in a later example.  *}
+
+  interpretation %visible int: lattice "op \<le> :: int \<Rightarrow> int \<Rightarrow> bool"
+    where int_min_eq: "int.meet x y = min x y"
+      and int_max_eq: "int.join x y = max x y"
+  proof -
+    show "lattice (op \<le> :: int \<Rightarrow> int \<Rightarrow> bool)"
+      txt {* \normalsize We have already shown that this is a partial
+        order, *}
+      apply unfold_locales
+      txt {* \normalsize hence only the lattice axioms remain to be
+        shown.
+        @{subgoals [display]}
+        By @{text is_inf} and @{text is_sup}, *}
+      apply (unfold int.is_inf_def int.is_sup_def)
+      txt {* \normalsize the goals are transformed to these
+        statements:
+        @{subgoals [display]}
+        This is Presburger arithmetic, which can be solved by the
+        method @{text arith}. *}
+      by arith+
+    txt {* \normalsize 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 int: lattice "op \<le> :: int \<Rightarrow> int \<Rightarrow> bool" .
+    show "int.meet x y = min x y"
+      by (bestsimp simp: int.meet_def int.is_inf_def)
+    show "int.join x y = max x y"
+      by (bestsimp simp: int.join_def int.is_sup_def)
+  qed
+
+text {* Next follows that @{text "op \<le>"} is a total order, again for
+  the integers. *}
+
+  interpretation %visible int: total_order "op \<le> :: int \<Rightarrow> int \<Rightarrow> bool"
+    by unfold_locales arith
+
+text {* Theorems that are available in the theory at this point are shown in
+  Table~\ref{tab:int-lattice}.  Two points are worth noting:
+
+\begin{table}
+\hrule
+\vspace{2ex}
+\begin{center}
+\begin{tabular}{l}
+  @{thm [source] int.less_def} from locale @{text partial_order}: \\
+  \quad @{thm int.less_def} \\
+  @{thm [source] int.meet_left} from locale @{text lattice}: \\
+  \quad @{thm int.meet_left} \\
+  @{thm [source] int.join_distr} from locale @{text distrib_lattice}: \\
+  \quad @{thm int.join_distr} \\
+  @{thm [source] int.less_total} from locale @{text total_order}: \\
+  \quad @{thm int.less_total}
+\end{tabular}
+\end{center}
+\hrule
+\caption{Interpreted theorems for~@{text \<le>} on the integers.}
+\label{tab:int-lattice}
+\end{table}
+
+\begin{itemize}
+\item
+  Locale @{text distrib_lattice} was also interpreted.  Since the
+  locale hierarchy reflects that total orders are distributive
+  lattices, the interpretation of the latter was inserted
+  automatically with the interpretation of the former.  In general,
+  interpretation traverses the locale hierarchy upwards and interprets
+  all encountered locales, regardless whether imported or proved via
+  the \isakeyword{sublocale} command.  Existing interpretations are
+  skipped avoiding duplicate work.
+\item
+  The predicate @{term "op <"} appears in theorem @{thm [source]
+  int.less_total}
+  although an equation for the replacement of @{text "op \<sqsubset>"} was only
+  given in the interpretation of @{text partial_order}.  The
+  interpretation equations are pushed downwards the hierarchy for
+  related interpretations --- that is, for interpretations that share
+  the instances of parameters they have in common.
+\end{itemize}
+  *}
+
+text {* The interpretations for a locale $n$ within the current
+  theory may be inspected with \isakeyword{print\_interps}~$n$.  This
+  prints the list of instances of $n$, for which interpretations exist.
+  For example, \isakeyword{print\_interps} @{term partial_order}
+  outputs the following:
+\begin{small}
+\begin{alltt}
+  int! : partial_order "op \(\le\)"
+\end{alltt}
+\end{small}
+  Of course, there is only one interpretation.
+  The interpretation qualifier on the left is decorated with an
+  exclamation point.  This means that it is mandatory.  Qualifiers
+  can either be \emph{mandatory} or \emph{optional}, designated by
+  ``!'' or ``?'' respectively.  Mandatory qualifiers must occur in a
+  name reference while optional ones need not.  Mandatory qualifiers
+  prevent accidental hiding of names, while optional qualifiers can be
+  more convenient to use.  For \isakeyword{interpretation}, the
+  default is ``!''.
+*}
+
+
+section {* Locale Expressions \label{sec:expressions} *}
+
+text {*
+  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.
+
+  A locale for order preserving maps requires three parameters: @{text
+  le}~(\isakeyword{infixl}~@{text \<sqsubseteq>}) and @{text
+  le'}~(\isakeyword{infixl}~@{text \<preceq>}) for the orders and~@{text \<phi>}
+  for the map.
+
+  In order to reuse the existing locale for partial orders, which has
+  the single parameter~@{text le}, it must be imported twice, once
+  mapping its parameter to~@{text le} from the new locale and once
+  to~@{text le'}.  This can be achieved with a compound locale
+  expression.
+
+  In general, a locale expression is a sequence of \emph{locale instances}
+  separated by~``$\textbf{+}$'' and followed by a \isakeyword{for}
+  clause.
+  An instance has the following format:
+\begin{quote}
+  \textit{qualifier} \textbf{:} \textit{locale-name}
+  \textit{parameter-instantiation}
+\end{quote}
+  We have already seen locale instances as arguments to
+  \isakeyword{interpretation} in Section~\ref{sec:interpretation}.
+  As before, the qualifier serves to disambiguate names from
+  different instances of the same locale.  While in
+  \isakeyword{interpretation} qualifiers default to mandatory, in
+  import and in the \isakeyword{sublocale} command, they default to
+  optional.
+
+  Since the parameters~@{text le} and~@{text le'} are to be partial
+  orders, our locale for order preserving maps will import the these
+  instances:
+\begin{small}
+\begin{alltt}
+  le: partial_order le
+  le': partial_order le'
+\end{alltt}
+\end{small}
+  For matter of convenience we choose to name parameter names and
+  qualifiers alike.  This is an arbitrary decision.  Technically, qualifiers
+  and parameters are unrelated.
+
+  Having determined the instances, let us turn to the \isakeyword{for}
+  clause.  It serves to declare locale parameters in the same way as
+  the context element \isakeyword{fixes} does.  Context elements can
+  only occur after the import section, and therefore the parameters
+  referred to in the instances must be declared in the \isakeyword{for}
+  clause.  The \isakeyword{for} clause is also where the syntax of these
+  parameters is declared.
+
+  Two context elements for the map parameter~@{text \<phi>} and the
+  assumptions that it is order preserving complete the locale
+  declaration. *}
+
+  locale order_preserving =
+    le: partial_order le + le': partial_order le'
+      for le (infixl "\<sqsubseteq>" 50) and le' (infixl "\<preceq>" 50) +
+    fixes \<phi>
+    assumes hom_le: "x \<sqsubseteq> y \<Longrightarrow> \<phi> x \<preceq> \<phi> y"
+
+text (in order_preserving) {* Here are examples of theorems that are
+  available in the locale:
+
+  \hspace*{1em}@{thm [source] hom_le}: @{thm hom_le}
+
+  \hspace*{1em}@{thm [source] le.less_le_trans}: @{thm le.less_le_trans}
+
+  \hspace*{1em}@{thm [source] le'.less_le_trans}:
+  @{thm [display, indent=4] le'.less_le_trans}
+  While there is infix syntax for the strict operation associated to
+  @{term "op \<sqsubseteq>"}, there is none for the strict version of @{term
+  "op \<preceq>"}.  The abbreviation @{text less} with its infix syntax is only
+  available for the original instance it was declared for.  We may
+  introduce the abbreviation @{text less'} with infix syntax~@{text \<prec>}
+  with the following declaration: *}
+
+  abbreviation (in order_preserving)
+    less' (infixl "\<prec>" 50) where "less' \<equiv> partial_order.less le'"
+
+text (in order_preserving) {* Now the theorem is displayed nicely as
+  @{thm [source]  le'.less_le_trans}:
+  @{thm [display, indent=2] le'.less_le_trans} *}
+
+text {* There are short notations for locale expressions.  These are
+  discussed in the following. *}
+
+
+subsection {* Default Instantiations *}
+
+text {* 
+  It is possible to omit parameter instantiations.  The
+  instantiation then defaults to the name of
+  the parameter itself.  For example, the locale expression @{text
+  partial_order} is short for @{text "partial_order le"}, since the
+  locale's single parameter is~@{text le}.  We took advantage of this
+  in the \isakeyword{sublocale} declarations of
+  Section~\ref{sec:changing-the-hierarchy}. *}
+
+
+subsection {* Implicit Parameters \label{sec:implicit-parameters} *}
+
+text {* In a locale expression that occurs within a locale
+  declaration, omitted parameters additionally extend the (possibly
+  empty) \isakeyword{for} clause.
+
+  The \isakeyword{for} clause is a general construct of Isabelle/Isar
+  to mark names occurring in the preceding declaration as ``arbitrary
+  but fixed''.  This is necessary for example, if the name is already
+  bound in a surrounding context.  In a locale expression, names
+  occurring in parameter instantiations should be bound by a
+  \isakeyword{for} clause whenever these names are not introduced
+  elsewhere in the context --- for example, on the left hand side of a
+  \isakeyword{sublocale} declaration.
+
+  There is an exception to this rule in locale declarations, where the
+  \isakeyword{for} clause serves to declare locale parameters.  Here,
+  locale parameters for which no parameter instantiation is given are
+  implicitly added, with their mixfix syntax, at the beginning of the
+  \isakeyword{for} clause.  For example, in a locale declaration, the
+  expression @{text partial_order} is short for
+\begin{small}
+\begin{alltt}
+  partial_order le \isakeyword{for} le (\isakeyword{infixl} "\(\sqsubseteq\)" 50)\textrm{.}
+\end{alltt}
+\end{small}
+  This short hand was used in the locale declarations throughout
+  Section~\ref{sec:import}.
+ *}
+
+text{*
+  The following locale declarations provide more examples.  A
+  map~@{text \<phi>} is a lattice homomorphism if it preserves meet and
+  join. *}
+
+  locale lattice_hom =
+    le: lattice + le': lattice le' for le' (infixl "\<preceq>" 50) +
+    fixes \<phi>
+    assumes hom_meet: "\<phi> (x \<sqinter> y) = le'.meet (\<phi> x) (\<phi> y)"
+      and hom_join: "\<phi> (x \<squnion> y) = le'.join (\<phi> x) (\<phi> y)"
+
+text {* The parameter instantiation in the first instance of @{term
+  lattice} is omitted.  This causes the parameter~@{text le} to be
+  added to the \isakeyword{for} clause, and the locale has
+  parameters~@{text le},~@{text le'} and, of course,~@{text \<phi>}.
+
+  Before turning to the second example, we complete the locale by
+  providing infix syntax for the meet and join operations of the
+  second lattice.
+*}
+
+  context lattice_hom
+  begin
+  abbreviation meet' (infixl "\<sqinter>''" 50) where "meet' \<equiv> le'.meet"
+  abbreviation join' (infixl "\<squnion>''" 50) where "join' \<equiv> le'.join"
+  end
+
+text {* The next example makes radical use of the short hand
+  facilities.  A homomorphism is an endomorphism if both orders
+  coincide. *}
+
+  locale lattice_end = lattice_hom _ le
+
+text {* The notation~@{text _} enables to omit a parameter in a
+  positional instantiation.  The omitted parameter,~@{text le} becomes
+  the parameter of the declared locale and is, in the following
+  position, used to instantiate the second parameter of @{text
+  lattice_hom}.  The effect is that of identifying the first in second
+  parameter of the homomorphism locale. *}
+
+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.  Parameter instantiations
+  are indicated by $\sqsubseteq \mapsto \preceq$ etc.  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 by @{text
+  lattice_end}.  This is not the case!  Inheritance paths with
+  identical morphisms are automatically detected and
+  the conclusions of the respective 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: *}
+
+  sublocale 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 (in lattice_hom) {*
+  Theorems and other declarations --- syntax, in particular --- from
+  the locale @{text order_preserving} are now active in @{text
+  lattice_hom}, for example
+  @{thm [source] hom_le}:
+  @{thm [display, indent=2] hom_le}
+  This theorem will be useful in the following section.
+  *}
+
+
+section {* Conditional Interpretation *}
+
+text {* There are situations where an interpretation is not possible
+  in the general case since the desired property is only valid if
+  certain conditions are fulfilled.  Take, for example, the function
+  @{text "\<lambda>i. n * i"} that scales its argument by a constant factor.
+  This function is order preserving (and even a lattice endomorphism)
+  with respect to @{term "op \<le>"} provided @{text "n \<ge> 0"}.
+
+  It is not possible to express this using a global interpretation,
+  because it is in general unspecified whether~@{term n} is
+  non-negative, but one may make an interpretation in an inner context
+  of a proof where full information is available.
+  This is not fully satisfactory either, since potentially
+  interpretations may be required to make interpretations in many
+  contexts.  What is
+  required is an interpretation that depends on the condition --- and
+  this can be done with the \isakeyword{sublocale} command.  For this
+  purpose, we introduce a locale for the condition. *}
+
+  locale non_negative =
+    fixes n :: int
+    assumes non_neg: "0 \<le> n"
+
+text {* It is again convenient to make the interpretation in an
+  incremental fashion, first for order preserving maps, the for
+  lattice endomorphisms. *}
+
+  sublocale non_negative \<subseteq>
+      order_preserving "op \<le>" "op \<le>" "\<lambda>i. n * i"
+    using non_neg by unfold_locales (rule mult_left_mono)
+
+text {* While the proof of the previous interpretation
+  is straightforward from monotonicity lemmas for~@{term "op *"}, the
+  second proof follows a useful pattern. *}
+
+  sublocale %visible non_negative \<subseteq> lattice_end "op \<le>" "\<lambda>i. n * i"
+  proof (unfold_locales, unfold int_min_eq int_max_eq)
+    txt {* \normalsize Unfolding the locale predicates \emph{and} the
+      interpretation equations immediately yields two subgoals that
+      reflect the core conjecture.
+      @{subgoals [display]}
+      It is now necessary to show, in the context of @{term
+      non_negative}, that multiplication by~@{term n} commutes with
+      @{term min} and @{term max}. *}
+  qed (auto simp: hom_le)
+
+text (in order_preserving) {* The lemma @{thm [source] hom_le}
+  simplifies a proof that would have otherwise been lengthy and we may
+  consider making it a default rule for the simplifier: *}
+
+  lemmas (in order_preserving) hom_le [simp]
+
+
+subsection {* Avoiding Infinite Chains of Interpretations
+  \label{sec:infinite-chains} *}
+
+text {* Similar situations arise frequently in formalisations of
+  abstract algebra where it is desirable to express that certain
+  constructions preserve certain properties.  For example, polynomials
+  over rings are rings, or --- an example from the domain where the
+  illustrations of this tutorial are taken from --- a partial order
+  may be obtained for a function space by point-wise lifting of the
+  partial order of the co-domain.  This corresponds to the following
+  interpretation: *}
+
+  sublocale %visible partial_order \<subseteq> f: partial_order "\<lambda>f g. \<forall>x. f x \<sqsubseteq> g x"
+    oops
+
+text {* Unfortunately this is a cyclic interpretation that leads to an
+  infinite chain, namely
+  @{text [display, indent=2] "partial_order \<subseteq> partial_order (\<lambda>f g. \<forall>x. f x \<sqsubseteq> g x) \<subseteq>
+  partial_order (\<lambda>f g. \<forall>x y. f x y \<sqsubseteq> g x y) \<subseteq>  \<dots>"}
+  and the interpretation is rejected.
+
+  Instead it is necessary to declare a locale that is logically
+  equivalent to @{term partial_order} but serves to collect facts
+  about functions spaces where the co-domain is a partial order, and
+  to make the interpretation in its context: *}
+
+  locale fun_partial_order = partial_order
+
+  sublocale fun_partial_order \<subseteq>
+      f: partial_order "\<lambda>f g. \<forall>x. f x \<sqsubseteq> g x"
+    by unfold_locales (fast,rule,fast,blast intro: trans)
+
+text {* It is quite common in abstract algebra that such a construction
+  maps a hierarchy of algebraic structures (or specifications) to a
+  related hierarchy.  By means of the same lifting, a function space
+  is a lattice if its co-domain is a lattice: *}
+
+  locale fun_lattice = fun_partial_order + lattice
+
+  sublocale fun_lattice \<subseteq> f: lattice "\<lambda>f g. \<forall>x. f x \<sqsubseteq> g x"
+  proof unfold_locales
+    fix f g
+    have "partial_order.is_inf (\<lambda>f g. \<forall>x. f x \<sqsubseteq> g x) f g (\<lambda>x. f x \<sqinter> g x)"
+      apply (rule is_infI) apply rule+ apply (drule spec, assumption)+ done
+    then show "\<exists>inf. partial_order.is_inf (\<lambda>f g. \<forall>x. f x \<sqsubseteq> g x) f g inf"
+      by fast
+  next
+    fix f g
+    have "partial_order.is_sup (\<lambda>f g. \<forall>x. f x \<sqsubseteq> g x) f g (\<lambda>x. f x \<squnion> g x)"
+      apply (rule is_supI) apply rule+ apply (drule spec, assumption)+ done
+    then show "\<exists>sup. partial_order.is_sup (\<lambda>f g. \<forall>x. f x \<sqsubseteq> g x) f g sup"
+      by fast
+  qed
+
+
+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, I
+  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 ML and 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 locales reimplementation for Isabelle 2009 provides, among other
+  improvements, a clean integration with Isabelle/Isar's local theory
+  mechanisms, which are described in another paper by Haftmann and
+  Wenzel~\cite{HaftmannWenzel2009}.
+
+  The original work of Kamm\"uller on locales~\cite{KammullerEtAl1999}
+  may be of interest from a historical perspective.  My previous
+  report on locales and locale expressions~\cite{Ballarin2004a}
+  describes a simpler form of expressions than available now and is
+  outdated. The mathematical background on orders and lattices is
+  taken from Jacobson's textbook on algebra~\cite[Chapter~8]{Jacobson1985}.
+
+  The sources of this tutorial, which include all proofs, are
+  available with the Isabelle distribution at
+  \url{http://isabelle.in.tum.de}.
+  *}
+
+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} \\
+  \textit{qualifier} & ::=
+  & \textit{name} [``\textbf{?}'' $|$ ``\textbf{!}''] \\[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{pos-insts} & ::=
+  & ( \textit{term} $|$ ``\textbf{\_}'' )$^*$ \\
+  \textit{named-insts} & ::=
+  & \textbf{where} \textit{name} ``\textbf{=}'' \textit{term}
+  ( \textbf{and} \textit{name} ``\textbf{=}'' \textit{term} )$^*$ \\
+  \textit{instance} & ::=
+  & [ \textit{qualifier} ``\textbf{:}'' ]
+    \textit{name} ( \textit{pos-insts} $|$ \textit{named-inst} ) \\
+  \textit{expression}  & ::= 
+  & \textit{instance} ( ``\textbf{+}'' \textit{instance} )$^*$
+    [ \textbf{for} \textit{fixes} ( \textbf{and} \textit{fixes} )$^*$ ] \\[2ex]
+
+  \multicolumn{3}{l}{Declaration of Locales} \\
+
+  \textit{locale} & ::=
+  & \textit{element}$^+$ \\
+  & | & \textit{expression} [ ``\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{equations} & ::= &  \textbf{where} \textit{equation} ( \textbf{and}
+    \textit{equation} )$^*$  \\
+  \textit{toplevel} & ::=
+  & \textbf{sublocale} \textit{name} ( ``$<$'' $|$
+    ``$\subseteq$'' ) \textit{expression} \textit{proof} \\
+  & |
+  & \textbf{interpretation}
+    \textit{expression} [ \textit{equations} ] \textit{proof} \\
+  & |
+  & \textbf{interpret}
+    \textit{expression} \textit{proof} \\[2ex]
+
+  \multicolumn{3}{l}{Diagnostics} \\
+
+  \textit{toplevel} & ::=
+  & \textbf{print\_locales} \\
+  & | & \textbf{print\_locale} [ ``\textbf{!}'' ] \textit{name} \\
+  & | & \textbf{print\_interps} \textit{name}
+\end{tabular}
+\end{center}
+\hrule
+\caption{Syntax of Locale Commands.}
+\label{tab:commands}
+\end{table}
+  *}
+
+text {* \textbf{Revision History.}  For the present third revision of
+  the tutorial, much of the explanatory text
+  was rewritten.  Inheritance of interpretation equations is
+  available with the forthcoming release of Isabelle, which at the
+  time of editing these notes is expected for the end of 2009.
+  The second revision accommodates changes introduced by the locales
+  reimplementation for Isabelle 2009.  Most notably locale expressions
+  have been generalised from renaming to instantiation.  *}
+
+text {* \textbf{Acknowledgements.}  Alexander Krauss, Tobias Nipkow,
+  Randy Pollack, Andreas Schropp, Christian Sternagel and Makarius Wenzel
+  have made
+  useful comments on earlier versions of this document.  The section
+  on conditional interpretation was inspired by a number of e-mail
+  enquiries the author received from locale users, and which suggested
+  that this use case is important enough to deserve explicit
+  explanation.  The term \emph{conditional interpretation} is due to
+  Larry Paulson. *}
+
+end

--- a/doc-src/Locales/Locales/Examples.thy	Mon Aug 27 21:04:37 2012 +0200
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,770 +0,0 @@
-theory Examples
-imports Main
-begin
-
-pretty_setmargin %invisible 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 the 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
-\[
-  @{text "\<And>x\<^sub>1\<dots>x\<^sub>n. \<lbrakk> A\<^sub>1; \<dots> ;A\<^sub>m \<rbrakk> \<Longrightarrow> C"}.
-\]
-  Isabelle/Isar's notion of context goes beyond this logical view.
-  Its contexts record, in a consecutive order, proved
-  conclusions along with \emph{attributes}, which can provide context
-  specific configuration information for proof procedures and concrete
-  syntax.  From a logical perspective, locales are just contexts that
-  have been made persistent.  To the user, though, they provide
-  powerful means for declaring and combining contexts, and for the
-  reuse of theorems proved in these contexts.
-  *}
-
-section {* Simple Locales *}
-
-text {*
-  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 (in partial_order) {* The parameter of this locale is~@{text le},
-  which is a binary predicate with infix syntax~@{text \<sqsubseteq>}.  The
-  parameter syntax is available in the subsequent
-  assumptions, which are the familiar partial order axioms.
-
-  Isabelle recognises unbound names as free variables.  In locale
-  assumptions, these are implicitly universally quantified.  That is,
-  @{term "\<lbrakk> x \<sqsubseteq> y; y \<sqsubseteq> z \<rbrakk> \<Longrightarrow> x \<sqsubseteq> z"} in fact means
-  @{term "\<And>x y z. \<lbrakk> x \<sqsubseteq> y; y \<sqsubseteq> z \<rbrakk> \<Longrightarrow> x \<sqsubseteq> z"}.
-
-  Two commands are provided to inspect locales:
-  \isakeyword{print\_locales} lists the names of all locales of the
-  current theory; \isakeyword{print\_locale}~$n$ prints the parameters
-  and assumptions of locale $n$; the variation \isakeyword{print\_locale!}~$n$
-  additionally outputs the conclusions that are stored in the locale.
-  We may inspect the new locale
-  by issuing \isakeyword{print\_locale!} @{term partial_order}.  The output
-  is the following list of context elements.
-\begin{small}
-\begin{alltt}
-  \isakeyword{fixes} le :: "'a \(\Rightarrow\) 'a \(\Rightarrow\)  bool" (\isakeyword{infixl} "\(\sqsubseteq\)" 50)
-  \isakeyword{assumes} "partial_order op \(\sqsubseteq\)"
-  \isakeyword{notes} assumption
-    refl [intro, simp] = `?x \(\sqsubseteq\) ?x`
-    \isakeyword{and}
-    anti_sym [intro] = `\(\isasymlbrakk\)?x \(\sqsubseteq\) ?y; ?y \(\sqsubseteq\) ?x\(\isasymrbrakk\) \(\Longrightarrow\) ?x = ?y`
-    \isakeyword{and}
-    trans [trans] = `\(\isasymlbrakk\)?x \(\sqsubseteq\) ?y; ?y \(\sqsubseteq\) ?z\(\isasymrbrakk\) \(\Longrightarrow\) ?x \(\sqsubseteq\) ?z`
-\end{alltt}
-\end{small}
-  The keyword \isakeyword{notes} denotes a conclusion element.  There
-  is one conclusion, which was added automatically.  Instead, there is
-  only one assumption, namely @{term "partial_order le"}.  The locale
-  declaration has introduced the predicate @{term
-  partial_order} to the theory.  This predicate is the
-  \emph{locale predicate}.  Its definition may be inspected by
-  issuing \isakeyword{thm} @{thm [source] partial_order_def}.
-  @{thm [display, indent=2] partial_order_def}
-  In our example, this is a unary predicate over the parameter of the
-  locale.  It is equivalent to the original assumptions, which have
-  been turned into conclusions and are
-  available as theorems in the context of the locale.  The names and
-  attributes from the locale declaration are associated to these
-  theorems and are effective in the context of the locale.
-
-  Each conclusion has a \emph{foundational theorem} as counterpart
-  in the theory.  Technically, this is simply the theorem composed
-  of context and conclusion.  For the transitivity theorem, this is
-  @{thm [source] partial_order.trans}:
-  @{thm [display, indent=2] partial_order.trans}
-*}
-
-subsection {* Targets: Extending Locales *}
-
-text {*
-  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.  Isar provides 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{primrec} & primitive recursion \\
-  \isakeyword{fun}, \isakeyword{function} & general recursion \\
-  \isakeyword{abbreviation} & syntactic abbreviation \\
-  \isakeyword{theorem}, etc.\ & theorem statement with proof \\
-  \isakeyword{theorems}, etc.\ & redeclaration of theorems \\
-  \isakeyword{text}, etc.\ & document markup
-\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 (in partial_order) {* The strict order @{text less} with infix
-  syntax~@{text \<sqsubset>} is
-  defined in terms of the locale parameter~@{text le} and the general
-  equality of the object logic we work in.  The definition generates a
-  \emph{foundational constant}
-  @{term partial_order.less} with definition @{thm [source]
-  partial_order.less_def}:
-  @{thm [display, indent=2] partial_order.less_def}
-  At the same time, the locale is extended by syntax transformations
-  hiding this construction in the context of the locale.  Here, the
-  abbreviation @{text less} is available for
-  @{text "partial_order.less le"}, and it is printed
-  and parsed as infix~@{text \<sqsubset>}.  Finally, the conclusion @{thm [source]
-  less_def} is added to the locale:
-  @{thm [display, indent=2] less_def}
-*}
-
-text {* The treatment of theorem statements is more straightforward.
-  As an example, here is the derivation of a transitivity law for the
-  strict order relation. *}
-
-  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, conclusions of the
-  locale may be used like theorems.  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.  *}
-
-subsection {* Context Blocks *}
-
-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.
-
-  This style of working is illustrated in the block below, where
-  notions of infimum and supremum for partial orders are introduced,
-  together with theorems about their uniqueness.  *}
-
-  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 {* The syntax of the locale commands discussed in this tutorial is
-  shown in Table~\ref{tab:commands}.  The grammar is complete with the
-  exception of the context elements \isakeyword{constrains} and
-  \isakeyword{defines}, which are provided for backward
-  compatibility.  See the Isabelle/Isar Reference
-  Manual~\cite{IsarRef} for full documentation.  *}
-
-
-section {* Import \label{sec:import} *}
-
-text {* 
-  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. *}
-
-text {*
-  With locales, this kind of inheritance is achieved through
-  \emph{import} of locales.  The import part of a locale declaration,
-  if present, precedes the context elements.  Here is an example,
-  where partial orders are extended to lattices.
-  *}
-
-  locale lattice = partial_order +
-    assumes ex_inf: "\<exists>inf. is_inf x y inf"
-      and ex_sup: "\<exists>sup. is_sup x y sup"
-  begin
-
-text {* These assumptions refer to the predicates for infimum
-  and supremum defined for @{text partial_order} in the previous
-  section.  We now introduce the notions of meet and join.  *}
-
-  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 to
-  establish a sufficiently rich landscape of locales for
-  further examples in this tutorial.  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: "x \<sqinter> (y \<squnion> z) = x \<sqinter> y \<squnion> x \<sqinter> 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 hierarchy obtained through these declarations is shown in
-  Figure~\ref{fig:lattices}(a).
-
-\begin{figure}
-\hrule \vspace{2ex}
-\begin{center}
-\subfigure[Declared hierarchy]{
-\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
-  \label{sec:changing-the-hierarchy} *}
-
-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}.  Locales generalise
-  interpretation from theorems to conclusions, enabling the reuse of
-  definitions and other constructs that are not part of the
-  specifications of the locales.
-
-  The first form of interpretation we will consider in this tutorial
-  is provided by the \isakeyword{sublocale} command.  It enables to
-  modify the import hierarchy to reflect the \emph{logical} relation
-  between locales.
-
-  Consider the locale hierarchy from Figure~\ref{fig:lattices}(a).
-  Total orders are lattices, although this is not reflected here, and
-  definitions, theorems and other conclusions
-  from @{term lattice} are not available in @{term total_order}.  To
-  obtain the situation in Figure~\ref{fig:lattices}(b), it is
-  sufficient to add the conclusions of the latter locale to the former.
-  The \isakeyword{sublocale} command does exactly this.
-  The declaration \isakeyword{sublocale} $l_1
-  \subseteq l_2$ causes locale $l_2$ to be \emph{interpreted} in the
-  context of $l_1$.  This means that all conclusions of $l_2$ are made
-  available in $l_1$.
-
-  Of course, the change of hierarchy must be supported by a theorem
-  that reflects, in our example, that total orders are indeed
-  lattices.  Therefore the \isakeyword{sublocale} command generates a
-  goal, which must be discharged by the user.  This is illustrated in
-  the following paragraphs.  First the sublocale relation is stated.
-*}
-
-  sublocale %visible total_order \<subseteq> lattice
-
-txt {* \normalsize
-  This enters the context of locale @{text total_order}, in
-  which the goal @{subgoals [display]} must be shown.
-  Now the
-  locale predicate needs to be unfolded --- for example, using its
-  definition or by introduction rules
-  provided by the locale package.  For automation, the locale package
-  provides the methods @{text intro_locales} and @{text
-  unfold_locales}.  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}, which need to be shown, hence
-  @{text unfold_locales} is appropriate. *}
-
-  proof unfold_locales
-
-txt {* \normalsize
-  Since the fact that both lattices and total orders are partial
-  orders is already reflected in the locale hierarchy, the assumptions
-  of @{text partial_order} are discharged automatically, and only the
-  assumptions introduced in @{text lattice} remain as subgoals
-  @{subgoals [display]}
-  The proof for the first subgoal is obtained by constructing an
-  infimum, whose existence is implied by totality. *}
-
-    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 {* \normalsize
-   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, we may establish that total orders are distributive
-  lattices with a second \isakeyword{sublocale} statement. *}
-
-  sublocale 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). *}
-
-text {*
-  Locale interpretation is \emph{dynamic}.  The statement
-  \isakeyword{sublocale} $l_1 \subseteq l_2$ will not just add the
-  current conclusions of $l_2$ to $l_1$.  Rather the dependency is
-  stored, and conclusions that will be
-  added to $l_2$ in future are automatically propagated to $l_1$.
-  The sublocale relation is transitive --- that is, propagation takes
-  effect along chains of sublocales.  Even cycles in the sublocale relation are
-  supported, as long as these cycles do not lead to infinite chains.
-  Details are discussed in the technical report \cite{Ballarin2006a}.
-  See also Section~\ref{sec:infinite-chains} of this tutorial.  *}
-
-end

--- a/doc-src/Locales/Locales/Examples1.thy	Mon Aug 27 21:04:37 2012 +0200
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,89 +0,0 @@
-theory Examples1
-imports Examples
-begin
-text {* \vspace{-5ex} *}
-section {* Use of Locales in Theories and Proofs
-  \label{sec:interpretation} *}
-
-text {*
-  Locales can be interpreted in the contexts of theories and
-  structured proofs.  These interpretations are dynamic, too.
-  Conclusions of locales will be propagated to the current theory or
-  the current proof context.%
-\footnote{Strictly speaking, only interpretation in theories is
-  dynamic since it is not possible to change locales or the locale
-  hierarchy from within a proof.}
-  The focus of this section is on
-  interpretation in theories, but we will also encounter
-  interpretations in proofs, in
-  Section~\ref{sec:local-interpretation}.
-
-  As an example, consider the type of integers @{typ int}.  The
-  relation @{term "op \<le>"} is a total order over @{typ int}.  We start
-  with the interpretation that @{term "op \<le>"} is a partial order.  The
-  facilities of the interpretation command are explored gradually in
-  three versions.
-  *}
-
-
-subsection {* First Version: Replacement of Parameters Only
-  \label{sec:po-first} *}
-
-text {*
-  The command \isakeyword{interpretation} is for the interpretation of
-  locale in theories.  In the following example, the parameter of locale
-  @{text partial_order} is replaced by @{term "op \<le> :: int \<Rightarrow> int \<Rightarrow>
-  bool"} and the locale instance is interpreted in the current
-  theory. *}
-
-  interpretation %visible int: partial_order "op \<le> :: int \<Rightarrow> int \<Rightarrow> bool"
-txt {* \normalsize
-  The argument of the command is a simple \emph{locale expression}
-  consisting of the name of the interpreted locale, which is
-  preceded by the qualifier @{text "int:"} and succeeded by a
-  white-space-separated list of terms, which provide a full
-  instantiation of the locale parameters.  The parameters are referred
-  to by order of declaration, which is also the order in which
-  \isakeyword{print\_locale} outputs them.  The locale has only a
-  single parameter, hence the list of instantiation terms is a
-  singleton.
-
-  The command creates the goal
-  @{subgoals [display]} which can be shown easily:
- *}
-    by unfold_locales auto
-
-text {*  The effect of the command is that instances of all
-  conclusions of the locale are available in the theory, where names
-  are prefixed by the qualifier.  For example, transitivity for @{typ
-  int} is named @{thm [source] int.trans} and is the following
-  theorem:
-  @{thm [display, indent=2] int.trans}
-  It is not possible to reference this theorem simply as @{text
-  trans}.  This prevents unwanted hiding of existing theorems of the
-  theory by an interpretation. *}
-
-
-subsection {* Second Version: Replacement of Definitions *}
-
-text {* Not only does the above interpretation qualify theorem names.
-  The prefix @{text int} is applied to all names introduced in locale
-  conclusions including names introduced in definitions.  The
-  qualified name @{text int.less} is short for
-  the interpretation of the definition, which is @{term int.less}.
-  Qualified name and expanded form may be used almost
-  interchangeably.%
-\footnote{Since @{term "op \<le>"} is polymorphic, for @{term int.less} a
-  more general type will be inferred than for @{text int.less} which
-  is over type @{typ int}.}
-  The latter is preferred on output, as for example in the theorem
-  @{thm [source] int.less_le_trans}: @{thm [display, indent=2]
-  int.less_le_trans}
-  Both notations for the strict order are not satisfactory.  The
-  constant @{term "op <"} is the strict order for @{typ int}.
-  In order to allow for the desired replacement, interpretation
-  accepts \emph{equations} in addition to the parameter instantiation.
-  These follow the locale expression and are indicated with the
-  keyword \isakeyword{where}.  This is the revised interpretation:
-  *}
-end

--- a/doc-src/Locales/Locales/Examples2.thy	Mon Aug 27 21:04:37 2012 +0200
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,24 +0,0 @@
-theory Examples2
-imports Examples
-begin
-text {* \vspace{-5ex} *}
-  interpretation %visible int: partial_order "op \<le> :: [int, int] \<Rightarrow> bool"
-    where "int.less x y = (x < y)"
-  proof -
-    txt {* \normalsize The goals are now:
-      @{subgoals [display]}
-      The proof that~@{text \<le>} is a partial order is as above. *}
-    show "partial_order (op \<le> :: int \<Rightarrow> int \<Rightarrow> bool)"
-      by unfold_locales auto
-    txt {* \normalsize The second goal is shown by unfolding the
-      definition of @{term "partial_order.less"}. *}
-    show "partial_order.less op \<le> x 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 the
-  interpreted locale.  Hence, the premise of @{text
-  "partial_order.less_def"} is discharged manually with @{text OF}.
-  *}
-end

--- a/doc-src/Locales/Locales/Examples3.thy	Mon Aug 27 21:04:37 2012 +0200
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,659 +0,0 @@
-theory Examples3
-imports Examples
-begin
-text {* \vspace{-5ex} *}
-subsection {* Third Version: Local Interpretation
-  \label{sec:local-interpretation} *}
-
-text {* In the above example, the fact that @{term "op \<le>"} is a partial
-  order for the integers was used in the second goal to
-  discharge the premise in the definition of @{text "op \<sqsubset>"}.  In
-  general, proofs of the equations not only may involve definitions
-  from the interpreted locale but arbitrarily complex arguments in the
-  context of the locale.  Therefore it would be convenient to have the
-  interpreted locale conclusions temporarily available in the proof.
-  This can be achieved by a locale interpretation in the proof body.
-  The command for local interpretations is \isakeyword{interpret}.  We
-  repeat the example from the previous section to illustrate this. *}
-
-  interpretation %visible int: partial_order "op \<le> :: int \<Rightarrow> int \<Rightarrow> bool"
-    where "int.less x y = (x < y)"
-  proof -
-    show "partial_order (op \<le> :: int \<Rightarrow> int \<Rightarrow> bool)"
-      by unfold_locales auto
-    then interpret int: partial_order "op \<le> :: [int, int] \<Rightarrow> bool" .
-    show "int.less x y = (x < y)"
-      unfolding int.less_def by auto
-  qed
-
-text {* The inner interpretation is immediate from the preceding fact
-  and proved by assumption (Isar short hand ``.'').  It enriches the
-  local proof context by the theorems
-  also obtained in the interpretation from Section~\ref{sec:po-first},
-  and @{text int.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 succeeding
-  \isakeyword{next} or \isakeyword{qed} statement. *}
-
-
-subsection {* Further Interpretations *}
-
-text {* Further interpretations are necessary for
-  the other locales.  In @{text lattice} the operations~@{text \<sqinter>}
-  and~@{text \<squnion>} are substituted by @{term "min :: int \<Rightarrow> int \<Rightarrow> int"}
-  and @{term "max :: int \<Rightarrow> int \<Rightarrow> int"}.  The entire proof for the
-  interpretation is reproduced to give an example of a more
-  elaborate interpretation proof.  Note that the equations are named
-  so they can be used in a later example.  *}
-
-  interpretation %visible int: lattice "op \<le> :: int \<Rightarrow> int \<Rightarrow> bool"
-    where int_min_eq: "int.meet x y = min x y"
-      and int_max_eq: "int.join x y = max x y"
-  proof -
-    show "lattice (op \<le> :: int \<Rightarrow> int \<Rightarrow> bool)"
-      txt {* \normalsize We have already shown that this is a partial
-        order, *}
-      apply unfold_locales
-      txt {* \normalsize hence only the lattice axioms remain to be
-        shown.
-        @{subgoals [display]}
-        By @{text is_inf} and @{text is_sup}, *}
-      apply (unfold int.is_inf_def int.is_sup_def)
-      txt {* \normalsize the goals are transformed to these
-        statements:
-        @{subgoals [display]}
-        This is Presburger arithmetic, which can be solved by the
-        method @{text arith}. *}
-      by arith+
-    txt {* \normalsize 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 int: lattice "op \<le> :: int \<Rightarrow> int \<Rightarrow> bool" .
-    show "int.meet x y = min x y"
-      by (bestsimp simp: int.meet_def int.is_inf_def)
-    show "int.join x y = max x y"
-      by (bestsimp simp: int.join_def int.is_sup_def)
-  qed
-
-text {* Next follows that @{text "op \<le>"} is a total order, again for
-  the integers. *}
-
-  interpretation %visible int: total_order "op \<le> :: int \<Rightarrow> int \<Rightarrow> bool"
-    by unfold_locales arith
-
-text {* Theorems that are available in the theory at this point are shown in
-  Table~\ref{tab:int-lattice}.  Two points are worth noting:
-
-\begin{table}
-\hrule
-\vspace{2ex}
-\begin{center}
-\begin{tabular}{l}
-  @{thm [source] int.less_def} from locale @{text partial_order}: \\
-  \quad @{thm int.less_def} \\
-  @{thm [source] int.meet_left} from locale @{text lattice}: \\
-  \quad @{thm int.meet_left} \\
-  @{thm [source] int.join_distr} from locale @{text distrib_lattice}: \\
-  \quad @{thm int.join_distr} \\
-  @{thm [source] int.less_total} from locale @{text total_order}: \\
-  \quad @{thm int.less_total}
-\end{tabular}
-\end{center}
-\hrule
-\caption{Interpreted theorems for~@{text \<le>} on the integers.}
-\label{tab:int-lattice}
-\end{table}
-
-\begin{itemize}
-\item
-  Locale @{text distrib_lattice} was also interpreted.  Since the
-  locale hierarchy reflects that total orders are distributive
-  lattices, the interpretation of the latter was inserted
-  automatically with the interpretation of the former.  In general,
-  interpretation traverses the locale hierarchy upwards and interprets
-  all encountered locales, regardless whether imported or proved via
-  the \isakeyword{sublocale} command.  Existing interpretations are
-  skipped avoiding duplicate work.
-\item
-  The predicate @{term "op <"} appears in theorem @{thm [source]
-  int.less_total}
-  although an equation for the replacement of @{text "op \<sqsubset>"} was only
-  given in the interpretation of @{text partial_order}.  The
-  interpretation equations are pushed downwards the hierarchy for
-  related interpretations --- that is, for interpretations that share
-  the instances of parameters they have in common.
-\end{itemize}
-  *}
-
-text {* The interpretations for a locale $n$ within the current
-  theory may be inspected with \isakeyword{print\_interps}~$n$.  This
-  prints the list of instances of $n$, for which interpretations exist.
-  For example, \isakeyword{print\_interps} @{term partial_order}
-  outputs the following:
-\begin{small}
-\begin{alltt}
-  int! : partial_order "op \(\le\)"
-\end{alltt}
-\end{small}
-  Of course, there is only one interpretation.
-  The interpretation qualifier on the left is decorated with an
-  exclamation point.  This means that it is mandatory.  Qualifiers
-  can either be \emph{mandatory} or \emph{optional}, designated by
-  ``!'' or ``?'' respectively.  Mandatory qualifiers must occur in a
-  name reference while optional ones need not.  Mandatory qualifiers
-  prevent accidental hiding of names, while optional qualifiers can be
-  more convenient to use.  For \isakeyword{interpretation}, the
-  default is ``!''.
-*}
-
-
-section {* Locale Expressions \label{sec:expressions} *}
-
-text {*
-  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.
-
-  A locale for order preserving maps requires three parameters: @{text
-  le}~(\isakeyword{infixl}~@{text \<sqsubseteq>}) and @{text
-  le'}~(\isakeyword{infixl}~@{text \<preceq>}) for the orders and~@{text \<phi>}
-  for the map.
-
-  In order to reuse the existing locale for partial orders, which has
-  the single parameter~@{text le}, it must be imported twice, once
-  mapping its parameter to~@{text le} from the new locale and once
-  to~@{text le'}.  This can be achieved with a compound locale
-  expression.
-
-  In general, a locale expression is a sequence of \emph{locale instances}
-  separated by~``$\textbf{+}$'' and followed by a \isakeyword{for}
-  clause.
-  An instance has the following format:
-\begin{quote}
-  \textit{qualifier} \textbf{:} \textit{locale-name}
-  \textit{parameter-instantiation}
-\end{quote}
-  We have already seen locale instances as arguments to
-  \isakeyword{interpretation} in Section~\ref{sec:interpretation}.
-  As before, the qualifier serves to disambiguate names from
-  different instances of the same locale.  While in
-  \isakeyword{interpretation} qualifiers default to mandatory, in
-  import and in the \isakeyword{sublocale} command, they default to
-  optional.
-
-  Since the parameters~@{text le} and~@{text le'} are to be partial
-  orders, our locale for order preserving maps will import the these
-  instances:
-\begin{small}
-\begin{alltt}
-  le: partial_order le
-  le': partial_order le'
-\end{alltt}
-\end{small}
-  For matter of convenience we choose to name parameter names and
-  qualifiers alike.  This is an arbitrary decision.  Technically, qualifiers
-  and parameters are unrelated.
-
-  Having determined the instances, let us turn to the \isakeyword{for}
-  clause.  It serves to declare locale parameters in the same way as
-  the context element \isakeyword{fixes} does.  Context elements can
-  only occur after the import section, and therefore the parameters
-  referred to in the instances must be declared in the \isakeyword{for}
-  clause.  The \isakeyword{for} clause is also where the syntax of these
-  parameters is declared.
-
-  Two context elements for the map parameter~@{text \<phi>} and the
-  assumptions that it is order preserving complete the locale
-  declaration. *}
-
-  locale order_preserving =
-    le: partial_order le + le': partial_order le'
-      for le (infixl "\<sqsubseteq>" 50) and le' (infixl "\<preceq>" 50) +
-    fixes \<phi>
-    assumes hom_le: "x \<sqsubseteq> y \<Longrightarrow> \<phi> x \<preceq> \<phi> y"
-
-text (in order_preserving) {* Here are examples of theorems that are
-  available in the locale:
-
-  \hspace*{1em}@{thm [source] hom_le}: @{thm hom_le}
-
-  \hspace*{1em}@{thm [source] le.less_le_trans}: @{thm le.less_le_trans}
-
-  \hspace*{1em}@{thm [source] le'.less_le_trans}:
-  @{thm [display, indent=4] le'.less_le_trans}
-  While there is infix syntax for the strict operation associated to
-  @{term "op \<sqsubseteq>"}, there is none for the strict version of @{term
-  "op \<preceq>"}.  The abbreviation @{text less} with its infix syntax is only
-  available for the original instance it was declared for.  We may
-  introduce the abbreviation @{text less'} with infix syntax~@{text \<prec>}
-  with the following declaration: *}
-
-  abbreviation (in order_preserving)
-    less' (infixl "\<prec>" 50) where "less' \<equiv> partial_order.less le'"
-
-text (in order_preserving) {* Now the theorem is displayed nicely as
-  @{thm [source]  le'.less_le_trans}:
-  @{thm [display, indent=2] le'.less_le_trans} *}
-
-text {* There are short notations for locale expressions.  These are
-  discussed in the following. *}
-
-
-subsection {* Default Instantiations *}
-
-text {* 
-  It is possible to omit parameter instantiations.  The
-  instantiation then defaults to the name of
-  the parameter itself.  For example, the locale expression @{text
-  partial_order} is short for @{text "partial_order le"}, since the
-  locale's single parameter is~@{text le}.  We took advantage of this
-  in the \isakeyword{sublocale} declarations of
-  Section~\ref{sec:changing-the-hierarchy}. *}
-
-
-subsection {* Implicit Parameters \label{sec:implicit-parameters} *}
-
-text {* In a locale expression that occurs within a locale
-  declaration, omitted parameters additionally extend the (possibly
-  empty) \isakeyword{for} clause.
-
-  The \isakeyword{for} clause is a general construct of Isabelle/Isar
-  to mark names occurring in the preceding declaration as ``arbitrary
-  but fixed''.  This is necessary for example, if the name is already
-  bound in a surrounding context.  In a locale expression, names
-  occurring in parameter instantiations should be bound by a
-  \isakeyword{for} clause whenever these names are not introduced
-  elsewhere in the context --- for example, on the left hand side of a
-  \isakeyword{sublocale} declaration.
-
-  There is an exception to this rule in locale declarations, where the
-  \isakeyword{for} clause serves to declare locale parameters.  Here,
-  locale parameters for which no parameter instantiation is given are
-  implicitly added, with their mixfix syntax, at the beginning of the
-  \isakeyword{for} clause.  For example, in a locale declaration, the
-  expression @{text partial_order} is short for
-\begin{small}
-\begin{alltt}
-  partial_order le \isakeyword{for} le (\isakeyword{infixl} "\(\sqsubseteq\)" 50)\textrm{.}
-\end{alltt}
-\end{small}
-  This short hand was used in the locale declarations throughout
-  Section~\ref{sec:import}.
- *}
-
-text{*
-  The following locale declarations provide more examples.  A
-  map~@{text \<phi>} is a lattice homomorphism if it preserves meet and
-  join. *}
-
-  locale lattice_hom =
-    le: lattice + le': lattice le' for le' (infixl "\<preceq>" 50) +
-    fixes \<phi>
-    assumes hom_meet: "\<phi> (x \<sqinter> y) = le'.meet (\<phi> x) (\<phi> y)"
-      and hom_join: "\<phi> (x \<squnion> y) = le'.join (\<phi> x) (\<phi> y)"
-
-text {* The parameter instantiation in the first instance of @{term
-  lattice} is omitted.  This causes the parameter~@{text le} to be
-  added to the \isakeyword{for} clause, and the locale has
-  parameters~@{text le},~@{text le'} and, of course,~@{text \<phi>}.
-
-  Before turning to the second example, we complete the locale by
-  providing infix syntax for the meet and join operations of the
-  second lattice.
-*}
-
-  context lattice_hom
-  begin
-  abbreviation meet' (infixl "\<sqinter>''" 50) where "meet' \<equiv> le'.meet"
-  abbreviation join' (infixl "\<squnion>''" 50) where "join' \<equiv> le'.join"
-  end
-
-text {* The next example makes radical use of the short hand
-  facilities.  A homomorphism is an endomorphism if both orders
-  coincide. *}
-
-  locale lattice_end = lattice_hom _ le
-
-text {* The notation~@{text _} enables to omit a parameter in a
-  positional instantiation.  The omitted parameter,~@{text le} becomes
-  the parameter of the declared locale and is, in the following
-  position, used to instantiate the second parameter of @{text
-  lattice_hom}.  The effect is that of identifying the first in second
-  parameter of the homomorphism locale. *}
-
-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.  Parameter instantiations
-  are indicated by $\sqsubseteq \mapsto \preceq$ etc.  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 by @{text
-  lattice_end}.  This is not the case!  Inheritance paths with
-  identical morphisms are automatically detected and
-  the conclusions of the respective 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: *}
-
-  sublocale 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 (in lattice_hom) {*
-  Theorems and other declarations --- syntax, in particular --- from
-  the locale @{text order_preserving} are now active in @{text
-  lattice_hom}, for example
-  @{thm [source] hom_le}:
-  @{thm [display, indent=2] hom_le}
-  This theorem will be useful in the following section.
-  *}
-
-
-section {* Conditional Interpretation *}
-
-text {* There are situations where an interpretation is not possible
-  in the general case since the desired property is only valid if
-  certain conditions are fulfilled.  Take, for example, the function
-  @{text "\<lambda>i. n * i"} that scales its argument by a constant factor.
-  This function is order preserving (and even a lattice endomorphism)
-  with respect to @{term "op \<le>"} provided @{text "n \<ge> 0"}.
-
-  It is not possible to express this using a global interpretation,
-  because it is in general unspecified whether~@{term n} is
-  non-negative, but one may make an interpretation in an inner context
-  of a proof where full information is available.
-  This is not fully satisfactory either, since potentially
-  interpretations may be required to make interpretations in many
-  contexts.  What is
-  required is an interpretation that depends on the condition --- and
-  this can be done with the \isakeyword{sublocale} command.  For this
-  purpose, we introduce a locale for the condition. *}
-
-  locale non_negative =
-    fixes n :: int
-    assumes non_neg: "0 \<le> n"
-
-text {* It is again convenient to make the interpretation in an
-  incremental fashion, first for order preserving maps, the for
-  lattice endomorphisms. *}
-
-  sublocale non_negative \<subseteq>
-      order_preserving "op \<le>" "op \<le>" "\<lambda>i. n * i"
-    using non_neg by unfold_locales (rule mult_left_mono)
-
-text {* While the proof of the previous interpretation
-  is straightforward from monotonicity lemmas for~@{term "op *"}, the
-  second proof follows a useful pattern. *}
-
-  sublocale %visible non_negative \<subseteq> lattice_end "op \<le>" "\<lambda>i. n * i"
-  proof (unfold_locales, unfold int_min_eq int_max_eq)
-    txt {* \normalsize Unfolding the locale predicates \emph{and} the
-      interpretation equations immediately yields two subgoals that
-      reflect the core conjecture.
-      @{subgoals [display]}
-      It is now necessary to show, in the context of @{term
-      non_negative}, that multiplication by~@{term n} commutes with
-      @{term min} and @{term max}. *}
-  qed (auto simp: hom_le)
-
-text (in order_preserving) {* The lemma @{thm [source] hom_le}
-  simplifies a proof that would have otherwise been lengthy and we may
-  consider making it a default rule for the simplifier: *}
-
-  lemmas (in order_preserving) hom_le [simp]
-
-
-subsection {* Avoiding Infinite Chains of Interpretations
-  \label{sec:infinite-chains} *}
-
-text {* Similar situations arise frequently in formalisations of
-  abstract algebra where it is desirable to express that certain
-  constructions preserve certain properties.  For example, polynomials
-  over rings are rings, or --- an example from the domain where the
-  illustrations of this tutorial are taken from --- a partial order
-  may be obtained for a function space by point-wise lifting of the
-  partial order of the co-domain.  This corresponds to the following
-  interpretation: *}
-
-  sublocale %visible partial_order \<subseteq> f: partial_order "\<lambda>f g. \<forall>x. f x \<sqsubseteq> g x"
-    oops
-
-text {* Unfortunately this is a cyclic interpretation that leads to an
-  infinite chain, namely
-  @{text [display, indent=2] "partial_order \<subseteq> partial_order (\<lambda>f g. \<forall>x. f x \<sqsubseteq> g x) \<subseteq>
-  partial_order (\<lambda>f g. \<forall>x y. f x y \<sqsubseteq> g x y) \<subseteq>  \<dots>"}
-  and the interpretation is rejected.
-
-  Instead it is necessary to declare a locale that is logically
-  equivalent to @{term partial_order} but serves to collect facts
-  about functions spaces where the co-domain is a partial order, and
-  to make the interpretation in its context: *}
-
-  locale fun_partial_order = partial_order
-
-  sublocale fun_partial_order \<subseteq>
-      f: partial_order "\<lambda>f g. \<forall>x. f x \<sqsubseteq> g x"
-    by unfold_locales (fast,rule,fast,blast intro: trans)
-
-text {* It is quite common in abstract algebra that such a construction
-  maps a hierarchy of algebraic structures (or specifications) to a
-  related hierarchy.  By means of the same lifting, a function space
-  is a lattice if its co-domain is a lattice: *}
-
-  locale fun_lattice = fun_partial_order + lattice
-
-  sublocale fun_lattice \<subseteq> f: lattice "\<lambda>f g. \<forall>x. f x \<sqsubseteq> g x"
-  proof unfold_locales
-    fix f g
-    have "partial_order.is_inf (\<lambda>f g. \<forall>x. f x \<sqsubseteq> g x) f g (\<lambda>x. f x \<sqinter> g x)"
-      apply (rule is_infI) apply rule+ apply (drule spec, assumption)+ done
-    then show "\<exists>inf. partial_order.is_inf (\<lambda>f g. \<forall>x. f x \<sqsubseteq> g x) f g inf"
-      by fast
-  next
-    fix f g
-    have "partial_order.is_sup (\<lambda>f g. \<forall>x. f x \<sqsubseteq> g x) f g (\<lambda>x. f x \<squnion> g x)"
-      apply (rule is_supI) apply rule+ apply (drule spec, assumption)+ done
-    then show "\<exists>sup. partial_order.is_sup (\<lambda>f g. \<forall>x. f x \<sqsubseteq> g x) f g sup"
-      by fast
-  qed
-
-
-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, I
-  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 ML and 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 locales reimplementation for Isabelle 2009 provides, among other
-  improvements, a clean integration with Isabelle/Isar's local theory
-  mechanisms, which are described in another paper by Haftmann and
-  Wenzel~\cite{HaftmannWenzel2009}.
-
-  The original work of Kamm\"uller on locales~\cite{KammullerEtAl1999}
-  may be of interest from a historical perspective.  My previous
-  report on locales and locale expressions~\cite{Ballarin2004a}
-  describes a simpler form of expressions than available now and is
-  outdated. The mathematical background on orders and lattices is
-  taken from Jacobson's textbook on algebra~\cite[Chapter~8]{Jacobson1985}.
-
-  The sources of this tutorial, which include all proofs, are
-  available with the Isabelle distribution at
-  \url{http://isabelle.in.tum.de}.
-  *}
-
-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} \\
-  \textit{qualifier} & ::=
-  & \textit{name} [``\textbf{?}'' $|$ ``\textbf{!}''] \\[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{pos-insts} & ::=
-  & ( \textit{term} $|$ ``\textbf{\_}'' )$^*$ \\
-  \textit{named-insts} & ::=
-  & \textbf{where} \textit{name} ``\textbf{=}'' \textit{term}
-  ( \textbf{and} \textit{name} ``\textbf{=}'' \textit{term} )$^*$ \\
-  \textit{instance} & ::=
-  & [ \textit{qualifier} ``\textbf{:}'' ]
-    \textit{name} ( \textit{pos-insts} $|$ \textit{named-inst} ) \\
-  \textit{expression}  & ::= 
-  & \textit{instance} ( ``\textbf{+}'' \textit{instance} )$^*$
-    [ \textbf{for} \textit{fixes} ( \textbf{and} \textit{fixes} )$^*$ ] \\[2ex]
-
-  \multicolumn{3}{l}{Declaration of Locales} \\
-
-  \textit{locale} & ::=
-  & \textit{element}$^+$ \\
-  & | & \textit{expression} [ ``\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{equations} & ::= &  \textbf{where} \textit{equation} ( \textbf{and}
-    \textit{equation} )$^*$  \\
-  \textit{toplevel} & ::=
-  & \textbf{sublocale} \textit{name} ( ``$<$'' $|$
-    ``$\subseteq$'' ) \textit{expression} \textit{proof} \\
-  & |
-  & \textbf{interpretation}
-    \textit{expression} [ \textit{equations} ] \textit{proof} \\
-  & |
-  & \textbf{interpret}
-    \textit{expression} \textit{proof} \\[2ex]
-
-  \multicolumn{3}{l}{Diagnostics} \\
-
-  \textit{toplevel} & ::=
-  & \textbf{print\_locales} \\
-  & | & \textbf{print\_locale} [ ``\textbf{!}'' ] \textit{name} \\
-  & | & \textbf{print\_interps} \textit{name}
-\end{tabular}
-\end{center}
-\hrule
-\caption{Syntax of Locale Commands.}
-\label{tab:commands}
-\end{table}
-  *}
-
-text {* \textbf{Revision History.}  For the present third revision of
-  the tutorial, much of the explanatory text
-  was rewritten.  Inheritance of interpretation equations is
-  available with the forthcoming release of Isabelle, which at the
-  time of editing these notes is expected for the end of 2009.
-  The second revision accommodates changes introduced by the locales
-  reimplementation for Isabelle 2009.  Most notably locale expressions
-  have been generalised from renaming to instantiation.  *}
-
-text {* \textbf{Acknowledgements.}  Alexander Krauss, Tobias Nipkow,
-  Randy Pollack, Andreas Schropp, Christian Sternagel and Makarius Wenzel
-  have made
-  useful comments on earlier versions of this document.  The section
-  on conditional interpretation was inspired by a number of e-mail
-  enquiries the author received from locale users, and which suggested
-  that this use case is important enough to deserve explicit
-  explanation.  The term \emph{conditional interpretation} is due to
-  Larry Paulson. *}
-
-end

--- a/doc-src/Locales/Locales/document/Examples.tex	Mon Aug 27 21:04:37 2012 +0200
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,1520 +0,0 @@
-%
-\begin{isabellebody}%
-\def\isabellecontext{Examples}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Examples\isanewline
-\isakeyword{imports}\ Main\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isadeliminvisible
-\isanewline
-%
-\endisadeliminvisible
-%
-\isataginvisible
-\isacommand{pretty{\isaliteral{5F}{\isacharunderscore}}setmargin}\isamarkupfalse%
-\ {\isadigit{6}}{\isadigit{5}}%
-\endisataginvisible
-{\isafoldinvisible}%
-%
-\isadeliminvisible
-%
-\endisadeliminvisible
-%
-\isamarkupsection{Introduction%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-Locales are based on contexts.  A \emph{context} can be seen as a
-  formula schema
-\[
-  \isa{{\isaliteral{5C3C416E643E}{\isasymAnd}}x\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}{\isaliteral{5C3C646F74733E}{\isasymdots}}x\isaliteral{5C3C5E7375623E}{}\isactrlsub n{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}\ A\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}{\isaliteral{3B}{\isacharsemicolon}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{3B}{\isacharsemicolon}}A\isaliteral{5C3C5E7375623E}{}\isactrlsub m\ {\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}}
-\]
-  where the variables~\isa{x\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}}, \ldots,~\isa{x\isaliteral{5C3C5E7375623E}{}\isactrlsub n} are called
-  \emph{parameters} and the premises $\isa{A\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}}, \ldots,~\isa{A\isaliteral{5C3C5E7375623E}{}\isactrlsub m}$ \emph{assumptions}.  A formula~\isa{C}
-  is a \emph{theorem} in the context if it is a conclusion
-\[
-  \isa{{\isaliteral{5C3C416E643E}{\isasymAnd}}x\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}{\isaliteral{5C3C646F74733E}{\isasymdots}}x\isaliteral{5C3C5E7375623E}{}\isactrlsub n{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}\ A\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}{\isaliteral{3B}{\isacharsemicolon}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{3B}{\isacharsemicolon}}A\isaliteral{5C3C5E7375623E}{}\isactrlsub m\ {\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ C}.
-\]
-  Isabelle/Isar's notion of context goes beyond this logical view.
-  Its contexts record, in a consecutive order, proved
-  conclusions along with \emph{attributes}, which can provide context
-  specific configuration information for proof procedures and concrete
-  syntax.  From a logical perspective, locales are just contexts that
-  have been made persistent.  To the user, though, they provide
-  powerful means for declaring and combining contexts, and for the
-  reuse of theorems proved in these contexts.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsection{Simple Locales%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-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{\isaliteral{5F}{\isacharunderscore}}order}.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\ \ \isacommand{locale}\isamarkupfalse%
-\ partial{\isaliteral{5F}{\isacharunderscore}}order\ {\isaliteral{3D}{\isacharequal}}\isanewline
-\ \ \ \ \isakeyword{fixes}\ le\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{27}{\isacharprime}}a\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{27}{\isacharprime}}a\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ bool{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{28}{\isacharparenleft}}\isakeyword{infixl}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}{\isaliteral{22}{\isachardoublequoteclose}}\ {\isadigit{5}}{\isadigit{0}}{\isaliteral{29}{\isacharparenright}}\isanewline
-\ \ \ \ \isakeyword{assumes}\ refl\ {\isaliteral{5B}{\isacharbrackleft}}intro{\isaliteral{2C}{\isacharcomma}}\ simp{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ x{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \ \ \isakeyword{and}\ anti{\isaliteral{5F}{\isacharunderscore}}sym\ {\isaliteral{5B}{\isacharbrackleft}}intro{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}\ x\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ y{\isaliteral{3B}{\isacharsemicolon}}\ y\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ x\ {\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ x\ {\isaliteral{3D}{\isacharequal}}\ y{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \ \ \isakeyword{and}\ trans\ {\isaliteral{5B}{\isacharbrackleft}}trans{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}\ x\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ y{\isaliteral{3B}{\isacharsemicolon}}\ y\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ z\ {\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ x\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ z{\isaliteral{22}{\isachardoublequoteclose}}%
-\begin{isamarkuptext}%
-The parameter of this locale is~\isa{le},
-  which is a binary predicate with infix syntax~\isa{{\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}}.  The
-  parameter syntax is available in the subsequent
-  assumptions, which are the familiar partial order axioms.
-
-  Isabelle recognises unbound names as free variables.  In locale
-  assumptions, these are implicitly universally quantified.  That is,
-  \isa{{\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}x\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ y{\isaliteral{3B}{\isacharsemicolon}}\ y\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ z{\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ x\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ z} in fact means
-  \isa{{\isaliteral{5C3C416E643E}{\isasymAnd}}x\ y\ z{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}x\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ y{\isaliteral{3B}{\isacharsemicolon}}\ y\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ z{\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ x\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ z}.
-
-  Two commands are provided to inspect locales:
-  \isakeyword{print\_locales} lists the names of all locales of the
-  current theory; \isakeyword{print\_locale}~$n$ prints the parameters
-  and assumptions of locale $n$; the variation \isakeyword{print\_locale!}~$n$
-  additionally outputs the conclusions that are stored in the locale.
-  We may inspect the new locale
-  by issuing \isakeyword{print\_locale!} \isa{partial{\isaliteral{5F}{\isacharunderscore}}order}.  The output
-  is the following list of context elements.
-\begin{small}
-\begin{alltt}
-  \isakeyword{fixes} le :: "'a \(\Rightarrow\) 'a \(\Rightarrow\)  bool" (\isakeyword{infixl} "\(\sqsubseteq\)" 50)
-  \isakeyword{assumes} "partial_order op \(\sqsubseteq\)"
-  \isakeyword{notes} assumption
-    refl [intro, simp] = `?x \(\sqsubseteq\) ?x`
-    \isakeyword{and}
-    anti_sym [intro] = `\(\isasymlbrakk\)?x \(\sqsubseteq\) ?y; ?y \(\sqsubseteq\) ?x\(\isasymrbrakk\) \(\Longrightarrow\) ?x = ?y`
-    \isakeyword{and}
-    trans [trans] = `\(\isasymlbrakk\)?x \(\sqsubseteq\) ?y; ?y \(\sqsubseteq\) ?z\(\isasymrbrakk\) \(\Longrightarrow\) ?x \(\sqsubseteq\) ?z`
-\end{alltt}
-\end{small}
-  The keyword \isakeyword{notes} denotes a conclusion element.  There
-  is one conclusion, which was added automatically.  Instead, there is
-  only one assumption, namely \isa{partial{\isaliteral{5F}{\isacharunderscore}}order\ op\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}}.  The locale
-  declaration has introduced the predicate \isa{partial{\isaliteral{5F}{\isacharunderscore}}order} to the theory.  This predicate is the
-  \emph{locale predicate}.  Its definition may be inspected by
-  issuing \isakeyword{thm} \isa{partial{\isaliteral{5F}{\isacharunderscore}}order{\isaliteral{5F}{\isacharunderscore}}def}.
-  \begin{isabelle}%
-\ \ partial{\isaliteral{5F}{\isacharunderscore}}order\ {\isaliteral{3F}{\isacharquery}}le\ {\isaliteral{5C3C65717569763E}{\isasymequiv}}\isanewline
-\isaindent{\ \ }{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}x{\isaliteral{2E}{\isachardot}}\ {\isaliteral{3F}{\isacharquery}}le\ x\ x{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C616E643E}{\isasymand}}\isanewline
-\isaindent{\ \ }{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}x\ y{\isaliteral{2E}{\isachardot}}\ {\isaliteral{3F}{\isacharquery}}le\ x\ y\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ {\isaliteral{3F}{\isacharquery}}le\ y\ x\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ x\ {\isaliteral{3D}{\isacharequal}}\ y{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C616E643E}{\isasymand}}\isanewline
-\isaindent{\ \ }{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}x\ y\ z{\isaliteral{2E}{\isachardot}}\ {\isaliteral{3F}{\isacharquery}}le\ x\ y\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ {\isaliteral{3F}{\isacharquery}}le\ y\ z\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ {\isaliteral{3F}{\isacharquery}}le\ x\ z{\isaliteral{29}{\isacharparenright}}%
-\end{isabelle}
-  In our example, this is a unary predicate over the parameter of the
-  locale.  It is equivalent to the original assumptions, which have
-  been turned into conclusions and are
-  available as theorems in the context of the locale.  The names and
-  attributes from the locale declaration are associated to these
-  theorems and are effective in the context of the locale.
-
-  Each conclusion has a \emph{foundational theorem} as counterpart
-  in the theory.  Technically, this is simply the theorem composed
-  of context and conclusion.  For the transitivity theorem, this is
-  \isa{partial{\isaliteral{5F}{\isacharunderscore}}order{\isaliteral{2E}{\isachardot}}trans}:
-  \begin{isabelle}%
-\ \ {\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}partial{\isaliteral{5F}{\isacharunderscore}}order\ {\isaliteral{3F}{\isacharquery}}le{\isaliteral{3B}{\isacharsemicolon}}\ {\isaliteral{3F}{\isacharquery}}le\ {\isaliteral{3F}{\isacharquery}}x\ {\isaliteral{3F}{\isacharquery}}y{\isaliteral{3B}{\isacharsemicolon}}\ {\isaliteral{3F}{\isacharquery}}le\ {\isaliteral{3F}{\isacharquery}}y\ {\isaliteral{3F}{\isacharquery}}z{\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{3F}{\isacharquery}}le\ {\isaliteral{3F}{\isacharquery}}x\ {\isaliteral{3F}{\isacharquery}}z%
-\end{isabelle}%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsubsection{Targets: Extending Locales%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-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.  Isar provides 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{primrec} & primitive recursion \\
-  \isakeyword{fun}, \isakeyword{function} & general recursion \\
-  \isakeyword{abbreviation} & syntactic abbreviation \\
-  \isakeyword{theorem}, etc.\ & theorem statement with proof \\
-  \isakeyword{theorems}, etc.\ & redeclaration of theorems \\
-  \isakeyword{text}, etc.\ & document markup
-\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%
-\ {\isaliteral{28}{\isacharparenleft}}\isakeyword{in}\ partial{\isaliteral{5F}{\isacharunderscore}}order{\isaliteral{29}{\isacharparenright}}\isanewline
-\ \ \ \ less\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{27}{\isacharprime}}a\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{27}{\isacharprime}}a\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ bool{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{28}{\isacharparenleft}}\isakeyword{infixl}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C73717375627365743E}{\isasymsqsubset}}{\isaliteral{22}{\isachardoublequoteclose}}\ {\isadigit{5}}{\isadigit{0}}{\isaliteral{29}{\isacharparenright}}\isanewline
-\ \ \ \ \isakeyword{where}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}x\ {\isaliteral{5C3C73717375627365743E}{\isasymsqsubset}}\ y{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}x\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ y\ {\isaliteral{5C3C616E643E}{\isasymand}}\ x\ {\isaliteral{5C3C6E6F7465713E}{\isasymnoteq}}\ y{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}%
-\begin{isamarkuptext}%
-The strict order \isa{less} with infix
-  syntax~\isa{{\isaliteral{5C3C73717375627365743E}{\isasymsqsubset}}} is
-  defined in terms of the locale parameter~\isa{le} and the general
-  equality of the object logic we work in.  The definition generates a
-  \emph{foundational constant}
-  \isa{partial{\isaliteral{5F}{\isacharunderscore}}order{\isaliteral{2E}{\isachardot}}less} with definition \isa{partial{\isaliteral{5F}{\isacharunderscore}}order{\isaliteral{2E}{\isachardot}}less{\isaliteral{5F}{\isacharunderscore}}def}:
-  \begin{isabelle}%
-\ \ partial{\isaliteral{5F}{\isacharunderscore}}order\ {\isaliteral{3F}{\isacharquery}}le\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\isanewline
-\isaindent{\ \ }partial{\isaliteral{5F}{\isacharunderscore}}order{\isaliteral{2E}{\isachardot}}less\ {\isaliteral{3F}{\isacharquery}}le\ {\isaliteral{3F}{\isacharquery}}x\ {\isaliteral{3F}{\isacharquery}}y\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{3F}{\isacharquery}}le\ {\isaliteral{3F}{\isacharquery}}x\ {\isaliteral{3F}{\isacharquery}}y\ {\isaliteral{5C3C616E643E}{\isasymand}}\ {\isaliteral{3F}{\isacharquery}}x\ {\isaliteral{5C3C6E6F7465713E}{\isasymnoteq}}\ {\isaliteral{3F}{\isacharquery}}y{\isaliteral{29}{\isacharparenright}}%
-\end{isabelle}
-  At the same time, the locale is extended by syntax transformations
-  hiding this construction in the context of the locale.  Here, the
-  abbreviation \isa{less} is available for
-  \isa{partial{\isaliteral{5F}{\isacharunderscore}}order{\isaliteral{2E}{\isachardot}}less\ le}, and it is printed
-  and parsed as infix~\isa{{\isaliteral{5C3C73717375627365743E}{\isasymsqsubset}}}.  Finally, the conclusion \isa{less{\isaliteral{5F}{\isacharunderscore}}def} is added to the locale:
-  \begin{isabelle}%
-\ \ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{3F}{\isacharquery}}x\ {\isaliteral{5C3C73717375627365743E}{\isasymsqsubset}}\ {\isaliteral{3F}{\isacharquery}}y{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{3F}{\isacharquery}}x\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ {\isaliteral{3F}{\isacharquery}}y\ {\isaliteral{5C3C616E643E}{\isasymand}}\ {\isaliteral{3F}{\isacharquery}}x\ {\isaliteral{5C3C6E6F7465713E}{\isasymnoteq}}\ {\isaliteral{3F}{\isacharquery}}y{\isaliteral{29}{\isacharparenright}}%
-\end{isabelle}%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-The treatment of theorem statements is more straightforward.
-  As an example, here is the derivation of a transitivity law for the
-  strict order relation.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\ \ \isacommand{lemma}\isamarkupfalse%
-\ {\isaliteral{28}{\isacharparenleft}}\isakeyword{in}\ partial{\isaliteral{5F}{\isacharunderscore}}order{\isaliteral{29}{\isacharparenright}}\ less{\isaliteral{5F}{\isacharunderscore}}le{\isaliteral{5F}{\isacharunderscore}}trans\ {\isaliteral{5B}{\isacharbrackleft}}trans{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{3A}{\isacharcolon}}\isanewline
-\ \ \ \ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}\ x\ {\isaliteral{5C3C73717375627365743E}{\isasymsqsubset}}\ y{\isaliteral{3B}{\isacharsemicolon}}\ y\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ z\ {\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ x\ {\isaliteral{5C3C73717375627365743E}{\isasymsqsubset}}\ z{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-%
-\isadelimvisible
-\ \ \ \ %
-\endisadelimvisible
-%
-\isatagvisible
-\isacommand{unfolding}\isamarkupfalse%
-\ less{\isaliteral{5F}{\isacharunderscore}}def\ \isacommand{by}\isamarkupfalse%
-\ {\isaliteral{28}{\isacharparenleft}}blast\ intro{\isaliteral{3A}{\isacharcolon}}\ trans{\isaliteral{29}{\isacharparenright}}%
-\endisatagvisible
-{\isafoldvisible}%
-%
-\isadelimvisible
-%
-\endisadelimvisible
-%
-\begin{isamarkuptext}%
-In the context of the proof, conclusions of the
-  locale may be used like theorems.  Attributes are effective: \isa{anti{\isaliteral{5F}{\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%
-%
-\isamarkupsubsection{Context Blocks%
-}
-\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.
-
-  This style of working is illustrated in the block below, where
-  notions of infimum and supremum for partial orders are introduced,
-  together with theorems about their uniqueness.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\ \ \isacommand{context}\isamarkupfalse%
-\ partial{\isaliteral{5F}{\isacharunderscore}}order\isanewline
-\ \ \isakeyword{begin}\isanewline
-\isanewline
-\ \ \isacommand{definition}\isamarkupfalse%
-\isanewline
-\ \ \ \ is{\isaliteral{5F}{\isacharunderscore}}inf\ \isakeyword{where}\ {\isaliteral{22}{\isachardoublequoteopen}}is{\isaliteral{5F}{\isacharunderscore}}inf\ x\ y\ i\ {\isaliteral{3D}{\isacharequal}}\isanewline
-\ \ \ \ \ \ {\isaliteral{28}{\isacharparenleft}}i\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ x\ {\isaliteral{5C3C616E643E}{\isasymand}}\ i\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ y\ {\isaliteral{5C3C616E643E}{\isasymand}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}z{\isaliteral{2E}{\isachardot}}\ z\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ x\ {\isaliteral{5C3C616E643E}{\isasymand}}\ z\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ y\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ z\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ i{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\isanewline
-\ \ \isacommand{definition}\isamarkupfalse%
-\isanewline
-\ \ \ \ is{\isaliteral{5F}{\isacharunderscore}}sup\ \isakeyword{where}\ {\isaliteral{22}{\isachardoublequoteopen}}is{\isaliteral{5F}{\isacharunderscore}}sup\ x\ y\ s\ {\isaliteral{3D}{\isacharequal}}\isanewline
-\ \ \ \ \ \ {\isaliteral{28}{\isacharparenleft}}x\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ s\ {\isaliteral{5C3C616E643E}{\isasymand}}\ y\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ s\ {\isaliteral{5C3C616E643E}{\isasymand}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}z{\isaliteral{2E}{\isachardot}}\ x\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ z\ {\isaliteral{5C3C616E643E}{\isasymand}}\ y\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ z\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ s\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ z{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-%
-\isadeliminvisible
-\isanewline
-\ \ %
-\endisadeliminvisible
-%
-\isataginvisible
-\isacommand{lemma}\isamarkupfalse%
-\ is{\isaliteral{5F}{\isacharunderscore}}infI\ {\isaliteral{5B}{\isacharbrackleft}}intro{\isaliteral{3F}{\isacharquery}}{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}i\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ x\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ i\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ y\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\isanewline
-\ \ \ \ \ \ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C416E643E}{\isasymAnd}}z{\isaliteral{2E}{\isachardot}}\ z\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ x\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ z\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ y\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ z\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ i{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ is{\isaliteral{5F}{\isacharunderscore}}inf\ x\ y\ i{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isaliteral{28}{\isacharparenleft}}unfold\ is{\isaliteral{5F}{\isacharunderscore}}inf{\isaliteral{5F}{\isacharunderscore}}def{\isaliteral{29}{\isacharparenright}}\ blast\isanewline
-\isanewline
-\ \ \isacommand{lemma}\isamarkupfalse%
-\ is{\isaliteral{5F}{\isacharunderscore}}inf{\isaliteral{5F}{\isacharunderscore}}lower\ {\isaliteral{5B}{\isacharbrackleft}}elim{\isaliteral{3F}{\isacharquery}}{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{3A}{\isacharcolon}}\isanewline
-\ \ \ \ {\isaliteral{22}{\isachardoublequoteopen}}is{\isaliteral{5F}{\isacharunderscore}}inf\ x\ y\ i\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{28}{\isacharparenleft}}i\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ x\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ i\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ y\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ C{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ C{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isaliteral{28}{\isacharparenleft}}unfold\ is{\isaliteral{5F}{\isacharunderscore}}inf{\isaliteral{5F}{\isacharunderscore}}def{\isaliteral{29}{\isacharparenright}}\ blast\isanewline
-\isanewline
-\ \ \isacommand{lemma}\isamarkupfalse%
-\ is{\isaliteral{5F}{\isacharunderscore}}inf{\isaliteral{5F}{\isacharunderscore}}greatest\ {\isaliteral{5B}{\isacharbrackleft}}elim{\isaliteral{3F}{\isacharquery}}{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{3A}{\isacharcolon}}\isanewline
-\ \ \ \ \ \ {\isaliteral{22}{\isachardoublequoteopen}}is{\isaliteral{5F}{\isacharunderscore}}inf\ x\ y\ i\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ z\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ x\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ z\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ y\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ z\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ i{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isaliteral{28}{\isacharparenleft}}unfold\ is{\isaliteral{5F}{\isacharunderscore}}inf{\isaliteral{5F}{\isacharunderscore}}def{\isaliteral{29}{\isacharparenright}}\ blast%
-\endisataginvisible
-{\isafoldinvisible}%
-%
-\isadeliminvisible
-\isanewline
-%
-\endisadeliminvisible
-\isanewline
-\ \ \isacommand{theorem}\isamarkupfalse%
-\ is{\isaliteral{5F}{\isacharunderscore}}inf{\isaliteral{5F}{\isacharunderscore}}uniq{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}is{\isaliteral{5F}{\isacharunderscore}}inf\ x\ y\ i{\isaliteral{3B}{\isacharsemicolon}}\ is{\isaliteral{5F}{\isacharunderscore}}inf\ x\ y\ i{\isaliteral{27}{\isacharprime}}{\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ i\ {\isaliteral{3D}{\isacharequal}}\ i{\isaliteral{27}{\isacharprime}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-%
-\isadelimproof
-\ \ \ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isaliteral{2D}{\isacharminus}}\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ inf{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}is{\isaliteral{5F}{\isacharunderscore}}inf\ x\ y\ i{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ inf{\isaliteral{27}{\isacharprime}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}is{\isaliteral{5F}{\isacharunderscore}}inf\ x\ y\ i{\isaliteral{27}{\isacharprime}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{3F}{\isacharquery}}thesis\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ {\isaliteral{28}{\isacharparenleft}}rule\ anti{\isaliteral{5F}{\isacharunderscore}}sym{\isaliteral{29}{\isacharparenright}}\isanewline
-\ \ \ \ \ \ \isacommand{from}\isamarkupfalse%
-\ inf{\isaliteral{27}{\isacharprime}}\ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}i\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ i{\isaliteral{27}{\isacharprime}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ {\isaliteral{28}{\isacharparenleft}}rule\ is{\isaliteral{5F}{\isacharunderscore}}inf{\isaliteral{5F}{\isacharunderscore}}greatest{\isaliteral{29}{\isacharparenright}}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{from}\isamarkupfalse%
-\ inf\ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}i\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ x{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \isacommand{from}\isamarkupfalse%
-\ inf\ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}i\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ y{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{from}\isamarkupfalse%
-\ inf\ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}i{\isaliteral{27}{\isacharprime}}\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ i{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ {\isaliteral{28}{\isacharparenleft}}rule\ is{\isaliteral{5F}{\isacharunderscore}}inf{\isaliteral{5F}{\isacharunderscore}}greatest{\isaliteral{29}{\isacharparenright}}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{from}\isamarkupfalse%
-\ inf{\isaliteral{27}{\isacharprime}}\ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}i{\isaliteral{27}{\isacharprime}}\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ x{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \isacommand{from}\isamarkupfalse%
-\ inf{\isaliteral{27}{\isacharprime}}\ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}i{\isaliteral{27}{\isacharprime}}\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ y{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\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{\isaliteral{5F}{\isacharunderscore}}inf{\isaliteral{5F}{\isacharunderscore}}related\ {\isaliteral{5B}{\isacharbrackleft}}elim{\isaliteral{3F}{\isacharquery}}{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ y\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ is{\isaliteral{5F}{\isacharunderscore}}inf\ x\ y\ x{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isaliteral{2D}{\isacharminus}}\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ y{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{3F}{\isacharquery}}thesis\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ x{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ y{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{by}\isamarkupfalse%
-\ fact\isanewline
-\ \ \ \ \ \ \isacommand{fix}\isamarkupfalse%
-\ z\ \isacommand{assume}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}z\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ x{\isaliteral{22}{\isachardoublequoteclose}}\ \isakeyword{and}\ {\isaliteral{22}{\isachardoublequoteopen}}z\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ y{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}z\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ x{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{by}\isamarkupfalse%
-\ fact\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isanewline
-\ \ \isacommand{lemma}\isamarkupfalse%
-\ is{\isaliteral{5F}{\isacharunderscore}}supI\ {\isaliteral{5B}{\isacharbrackleft}}intro{\isaliteral{3F}{\isacharquery}}{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ s\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ y\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ s\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\isanewline
-\ \ \ \ \ \ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C416E643E}{\isasymAnd}}z{\isaliteral{2E}{\isachardot}}\ x\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ z\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ y\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ z\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ s\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ z{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ is{\isaliteral{5F}{\isacharunderscore}}sup\ x\ y\ s{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isaliteral{28}{\isacharparenleft}}unfold\ is{\isaliteral{5F}{\isacharunderscore}}sup{\isaliteral{5F}{\isacharunderscore}}def{\isaliteral{29}{\isacharparenright}}\ blast\isanewline
-\isanewline
-\ \ \isacommand{lemma}\isamarkupfalse%
-\ is{\isaliteral{5F}{\isacharunderscore}}sup{\isaliteral{5F}{\isacharunderscore}}least\ {\isaliteral{5B}{\isacharbrackleft}}elim{\isaliteral{3F}{\isacharquery}}{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{3A}{\isacharcolon}}\isanewline
-\ \ \ \ \ \ {\isaliteral{22}{\isachardoublequoteopen}}is{\isaliteral{5F}{\isacharunderscore}}sup\ x\ y\ s\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ x\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ z\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ y\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ z\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ s\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ z{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isaliteral{28}{\isacharparenleft}}unfold\ is{\isaliteral{5F}{\isacharunderscore}}sup{\isaliteral{5F}{\isacharunderscore}}def{\isaliteral{29}{\isacharparenright}}\ blast\isanewline
-\isanewline
-\ \ \isacommand{lemma}\isamarkupfalse%
-\ is{\isaliteral{5F}{\isacharunderscore}}sup{\isaliteral{5F}{\isacharunderscore}}upper\ {\isaliteral{5B}{\isacharbrackleft}}elim{\isaliteral{3F}{\isacharquery}}{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{3A}{\isacharcolon}}\isanewline
-\ \ \ \ \ \ {\isaliteral{22}{\isachardoublequoteopen}}is{\isaliteral{5F}{\isacharunderscore}}sup\ x\ y\ s\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{28}{\isacharparenleft}}x\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ s\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ y\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ s\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ C{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ C{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isaliteral{28}{\isacharparenleft}}unfold\ is{\isaliteral{5F}{\isacharunderscore}}sup{\isaliteral{5F}{\isacharunderscore}}def{\isaliteral{29}{\isacharparenright}}\ blast%
-\endisataginvisible
-{\isafoldinvisible}%
-%
-\isadeliminvisible
-\isanewline
-%
-\endisadeliminvisible
-\isanewline
-\ \ \isacommand{theorem}\isamarkupfalse%
-\ is{\isaliteral{5F}{\isacharunderscore}}sup{\isaliteral{5F}{\isacharunderscore}}uniq{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}is{\isaliteral{5F}{\isacharunderscore}}sup\ x\ y\ s{\isaliteral{3B}{\isacharsemicolon}}\ is{\isaliteral{5F}{\isacharunderscore}}sup\ x\ y\ s{\isaliteral{27}{\isacharprime}}{\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ s\ {\isaliteral{3D}{\isacharequal}}\ s{\isaliteral{27}{\isacharprime}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-%
-\isadelimproof
-\ \ \ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isaliteral{2D}{\isacharminus}}\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ sup{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}is{\isaliteral{5F}{\isacharunderscore}}sup\ x\ y\ s{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ sup{\isaliteral{27}{\isacharprime}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}is{\isaliteral{5F}{\isacharunderscore}}sup\ x\ y\ s{\isaliteral{27}{\isacharprime}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{3F}{\isacharquery}}thesis\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ {\isaliteral{28}{\isacharparenleft}}rule\ anti{\isaliteral{5F}{\isacharunderscore}}sym{\isaliteral{29}{\isacharparenright}}\isanewline
-\ \ \ \ \ \ \isacommand{from}\isamarkupfalse%
-\ sup\ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}s\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ s{\isaliteral{27}{\isacharprime}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ {\isaliteral{28}{\isacharparenleft}}rule\ is{\isaliteral{5F}{\isacharunderscore}}sup{\isaliteral{5F}{\isacharunderscore}}least{\isaliteral{29}{\isacharparenright}}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{from}\isamarkupfalse%
-\ sup{\isaliteral{27}{\isacharprime}}\ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ s{\isaliteral{27}{\isacharprime}}{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \isacommand{from}\isamarkupfalse%
-\ sup{\isaliteral{27}{\isacharprime}}\ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}y\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ s{\isaliteral{27}{\isacharprime}}{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{from}\isamarkupfalse%
-\ sup{\isaliteral{27}{\isacharprime}}\ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}s{\isaliteral{27}{\isacharprime}}\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ s{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ {\isaliteral{28}{\isacharparenleft}}rule\ is{\isaliteral{5F}{\isacharunderscore}}sup{\isaliteral{5F}{\isacharunderscore}}least{\isaliteral{29}{\isacharparenright}}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{from}\isamarkupfalse%
-\ sup\ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ s{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \isacommand{from}\isamarkupfalse%
-\ sup\ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}y\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ s{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\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{\isaliteral{5F}{\isacharunderscore}}sup{\isaliteral{5F}{\isacharunderscore}}related\ {\isaliteral{5B}{\isacharbrackleft}}elim{\isaliteral{3F}{\isacharquery}}{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ y\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ is{\isaliteral{5F}{\isacharunderscore}}sup\ x\ y\ y{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isaliteral{2D}{\isacharminus}}\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ y{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{3F}{\isacharquery}}thesis\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ y{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{by}\isamarkupfalse%
-\ fact\isanewline
-\ \ \ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}y\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ y{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{fix}\isamarkupfalse%
-\ z\ \isacommand{assume}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ z{\isaliteral{22}{\isachardoublequoteclose}}\ \isakeyword{and}\ {\isaliteral{22}{\isachardoublequoteopen}}y\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ z{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}y\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ z{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{by}\isamarkupfalse%
-\ fact\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-%
-\endisataginvisible
-{\isafoldinvisible}%
-%
-\isadeliminvisible
-\isanewline
-%
-\endisadeliminvisible
-\isanewline
-\ \ \isacommand{end}\isamarkupfalse%
-%
-\begin{isamarkuptext}%
-The syntax of the locale commands discussed in this tutorial is
-  shown in Table~\ref{tab:commands}.  The grammar is complete with the
-  exception of the context elements \isakeyword{constrains} and
-  \isakeyword{defines}, which are provided for backward
-  compatibility.  See the Isabelle/Isar Reference
-  Manual~\cite{IsarRef} for full documentation.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsection{Import \label{sec:import}%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-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.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-With locales, this kind of inheritance is achieved through
-  \emph{import} of locales.  The import part of a locale declaration,
-  if present, precedes the context elements.  Here is an example,
-  where partial orders are extended to lattices.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\ \ \isacommand{locale}\isamarkupfalse%
-\ lattice\ {\isaliteral{3D}{\isacharequal}}\ partial{\isaliteral{5F}{\isacharunderscore}}order\ {\isaliteral{2B}{\isacharplus}}\isanewline
-\ \ \ \ \isakeyword{assumes}\ ex{\isaliteral{5F}{\isacharunderscore}}inf{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6578697374733E}{\isasymexists}}inf{\isaliteral{2E}{\isachardot}}\ is{\isaliteral{5F}{\isacharunderscore}}inf\ x\ y\ inf{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \ \ \isakeyword{and}\ ex{\isaliteral{5F}{\isacharunderscore}}sup{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6578697374733E}{\isasymexists}}sup{\isaliteral{2E}{\isachardot}}\ is{\isaliteral{5F}{\isacharunderscore}}sup\ x\ y\ sup{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \isakeyword{begin}%
-\begin{isamarkuptext}%
-These assumptions refer to the predicates for infimum
-  and supremum defined for \isa{partial{\isaliteral{5F}{\isacharunderscore}}order} in the previous
-  section.  We now introduce the notions of meet and join.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\ \ \isacommand{definition}\isamarkupfalse%
-\isanewline
-\ \ \ \ meet\ {\isaliteral{28}{\isacharparenleft}}\isakeyword{infixl}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C7371696E7465723E}{\isasymsqinter}}{\isaliteral{22}{\isachardoublequoteclose}}\ {\isadigit{7}}{\isadigit{0}}{\isaliteral{29}{\isacharparenright}}\ \isakeyword{where}\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C7371696E7465723E}{\isasymsqinter}}\ y\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}THE\ inf{\isaliteral{2E}{\isachardot}}\ is{\isaliteral{5F}{\isacharunderscore}}inf\ x\ y\ inf{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \isacommand{definition}\isamarkupfalse%
-\isanewline
-\ \ \ \ join\ {\isaliteral{28}{\isacharparenleft}}\isakeyword{infixl}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C7371756E696F6E3E}{\isasymsqunion}}{\isaliteral{22}{\isachardoublequoteclose}}\ {\isadigit{6}}{\isadigit{5}}{\isaliteral{29}{\isacharparenright}}\ \isakeyword{where}\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C7371756E696F6E3E}{\isasymsqunion}}\ y\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}THE\ sup{\isaliteral{2E}{\isachardot}}\ is{\isaliteral{5F}{\isacharunderscore}}sup\ x\ y\ sup{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-%
-\isadeliminvisible
-\isanewline
-\ \ %
-\endisadeliminvisible
-%
-\isataginvisible
-\isacommand{lemma}\isamarkupfalse%
-\ meet{\isaliteral{5F}{\isacharunderscore}}equality\ {\isaliteral{5B}{\isacharbrackleft}}elim{\isaliteral{3F}{\isacharquery}}{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}is{\isaliteral{5F}{\isacharunderscore}}inf\ x\ y\ i\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ x\ {\isaliteral{5C3C7371696E7465723E}{\isasymsqinter}}\ y\ {\isaliteral{3D}{\isacharequal}}\ i{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isaliteral{28}{\isacharparenleft}}unfold\ meet{\isaliteral{5F}{\isacharunderscore}}def{\isaliteral{29}{\isacharparenright}}\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}is{\isaliteral{5F}{\isacharunderscore}}inf\ x\ y\ i{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}THE\ i{\isaliteral{2E}{\isachardot}}\ is{\isaliteral{5F}{\isacharunderscore}}inf\ x\ y\ i{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ i{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isaliteral{28}{\isacharparenleft}}rule\ the{\isaliteral{5F}{\isacharunderscore}}equality{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{28}{\isacharparenleft}}rule\ is{\isaliteral{5F}{\isacharunderscore}}inf{\isaliteral{5F}{\isacharunderscore}}uniq\ {\isaliteral{5B}{\isacharbrackleft}}OF\ {\isaliteral{5F}{\isacharunderscore}}\ {\isaliteral{60}{\isacharbackquoteopen}}is{\isaliteral{5F}{\isacharunderscore}}inf\ x\ y\ i{\isaliteral{60}{\isacharbackquoteclose}}{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{29}{\isacharparenright}}\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isanewline
-\ \ \isacommand{lemma}\isamarkupfalse%
-\ meetI\ {\isaliteral{5B}{\isacharbrackleft}}intro{\isaliteral{3F}{\isacharquery}}{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{3A}{\isacharcolon}}\isanewline
-\ \ \ \ \ \ {\isaliteral{22}{\isachardoublequoteopen}}i\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ x\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ i\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ y\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C416E643E}{\isasymAnd}}z{\isaliteral{2E}{\isachardot}}\ z\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ x\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ z\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ y\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ z\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ i{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ x\ {\isaliteral{5C3C7371696E7465723E}{\isasymsqinter}}\ y\ {\isaliteral{3D}{\isacharequal}}\ i{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isaliteral{28}{\isacharparenleft}}rule\ meet{\isaliteral{5F}{\isacharunderscore}}equality{\isaliteral{2C}{\isacharcomma}}\ rule\ is{\isaliteral{5F}{\isacharunderscore}}infI{\isaliteral{29}{\isacharparenright}}\ blast{\isaliteral{2B}{\isacharplus}}\isanewline
-\isanewline
-\ \ \isacommand{lemma}\isamarkupfalse%
-\ is{\isaliteral{5F}{\isacharunderscore}}inf{\isaliteral{5F}{\isacharunderscore}}meet\ {\isaliteral{5B}{\isacharbrackleft}}intro{\isaliteral{3F}{\isacharquery}}{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}is{\isaliteral{5F}{\isacharunderscore}}inf\ x\ y\ {\isaliteral{28}{\isacharparenleft}}x\ {\isaliteral{5C3C7371696E7465723E}{\isasymsqinter}}\ y{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isaliteral{28}{\isacharparenleft}}unfold\ meet{\isaliteral{5F}{\isacharunderscore}}def{\isaliteral{29}{\isacharparenright}}\isanewline
-\ \ \ \ \isacommand{from}\isamarkupfalse%
-\ ex{\isaliteral{5F}{\isacharunderscore}}inf\ \isacommand{obtain}\isamarkupfalse%
-\ i\ \isakeyword{where}\ {\isaliteral{22}{\isachardoublequoteopen}}is{\isaliteral{5F}{\isacharunderscore}}inf\ x\ y\ i{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}is{\isaliteral{5F}{\isacharunderscore}}inf\ x\ y\ {\isaliteral{28}{\isacharparenleft}}THE\ i{\isaliteral{2E}{\isachardot}}\ is{\isaliteral{5F}{\isacharunderscore}}inf\ x\ y\ i{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isaliteral{28}{\isacharparenleft}}rule\ theI{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{28}{\isacharparenleft}}rule\ is{\isaliteral{5F}{\isacharunderscore}}inf{\isaliteral{5F}{\isacharunderscore}}uniq\ {\isaliteral{5B}{\isacharbrackleft}}OF\ {\isaliteral{5F}{\isacharunderscore}}\ {\isaliteral{60}{\isacharbackquoteopen}}is{\isaliteral{5F}{\isacharunderscore}}inf\ x\ y\ i{\isaliteral{60}{\isacharbackquoteclose}}{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{29}{\isacharparenright}}\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isanewline
-\ \ \isacommand{lemma}\isamarkupfalse%
-\ meet{\isaliteral{5F}{\isacharunderscore}}left\ {\isaliteral{5B}{\isacharbrackleft}}intro{\isaliteral{3F}{\isacharquery}}{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{3A}{\isacharcolon}}\isanewline
-\ \ \ \ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C7371696E7465723E}{\isasymsqinter}}\ y\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ x{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isaliteral{28}{\isacharparenleft}}rule\ is{\isaliteral{5F}{\isacharunderscore}}inf{\isaliteral{5F}{\isacharunderscore}}lower{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{28}{\isacharparenleft}}rule\ is{\isaliteral{5F}{\isacharunderscore}}inf{\isaliteral{5F}{\isacharunderscore}}meet{\isaliteral{29}{\isacharparenright}}\isanewline
-\isanewline
-\ \ \isacommand{lemma}\isamarkupfalse%
-\ meet{\isaliteral{5F}{\isacharunderscore}}right\ {\isaliteral{5B}{\isacharbrackleft}}intro{\isaliteral{3F}{\isacharquery}}{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{3A}{\isacharcolon}}\isanewline
-\ \ \ \ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C7371696E7465723E}{\isasymsqinter}}\ y\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ y{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isaliteral{28}{\isacharparenleft}}rule\ is{\isaliteral{5F}{\isacharunderscore}}inf{\isaliteral{5F}{\isacharunderscore}}lower{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{28}{\isacharparenleft}}rule\ is{\isaliteral{5F}{\isacharunderscore}}inf{\isaliteral{5F}{\isacharunderscore}}meet{\isaliteral{29}{\isacharparenright}}\isanewline
-\isanewline
-\ \ \isacommand{lemma}\isamarkupfalse%
-\ meet{\isaliteral{5F}{\isacharunderscore}}le\ {\isaliteral{5B}{\isacharbrackleft}}intro{\isaliteral{3F}{\isacharquery}}{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{3A}{\isacharcolon}}\isanewline
-\ \ \ \ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}\ z\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ x{\isaliteral{3B}{\isacharsemicolon}}\ z\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ y\ {\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ z\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ x\ {\isaliteral{5C3C7371696E7465723E}{\isasymsqinter}}\ y{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isaliteral{28}{\isacharparenleft}}rule\ is{\isaliteral{5F}{\isacharunderscore}}inf{\isaliteral{5F}{\isacharunderscore}}greatest{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{28}{\isacharparenleft}}rule\ is{\isaliteral{5F}{\isacharunderscore}}inf{\isaliteral{5F}{\isacharunderscore}}meet{\isaliteral{29}{\isacharparenright}}\isanewline
-\isanewline
-\ \ \isacommand{lemma}\isamarkupfalse%
-\ join{\isaliteral{5F}{\isacharunderscore}}equality\ {\isaliteral{5B}{\isacharbrackleft}}elim{\isaliteral{3F}{\isacharquery}}{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}is{\isaliteral{5F}{\isacharunderscore}}sup\ x\ y\ s\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ x\ {\isaliteral{5C3C7371756E696F6E3E}{\isasymsqunion}}\ y\ {\isaliteral{3D}{\isacharequal}}\ s{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isaliteral{28}{\isacharparenleft}}unfold\ join{\isaliteral{5F}{\isacharunderscore}}def{\isaliteral{29}{\isacharparenright}}\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}is{\isaliteral{5F}{\isacharunderscore}}sup\ x\ y\ s{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}THE\ s{\isaliteral{2E}{\isachardot}}\ is{\isaliteral{5F}{\isacharunderscore}}sup\ x\ y\ s{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ s{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isaliteral{28}{\isacharparenleft}}rule\ the{\isaliteral{5F}{\isacharunderscore}}equality{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{28}{\isacharparenleft}}rule\ is{\isaliteral{5F}{\isacharunderscore}}sup{\isaliteral{5F}{\isacharunderscore}}uniq\ {\isaliteral{5B}{\isacharbrackleft}}OF\ {\isaliteral{5F}{\isacharunderscore}}\ {\isaliteral{60}{\isacharbackquoteopen}}is{\isaliteral{5F}{\isacharunderscore}}sup\ x\ y\ s{\isaliteral{60}{\isacharbackquoteclose}}{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{29}{\isacharparenright}}\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isanewline
-\ \ \isacommand{lemma}\isamarkupfalse%
-\ joinI\ {\isaliteral{5B}{\isacharbrackleft}}intro{\isaliteral{3F}{\isacharquery}}{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ s\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ y\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ s\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\isanewline
-\ \ \ \ \ \ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C416E643E}{\isasymAnd}}z{\isaliteral{2E}{\isachardot}}\ x\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ z\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ y\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ z\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ s\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ z{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ x\ {\isaliteral{5C3C7371756E696F6E3E}{\isasymsqunion}}\ y\ {\isaliteral{3D}{\isacharequal}}\ s{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isaliteral{28}{\isacharparenleft}}rule\ join{\isaliteral{5F}{\isacharunderscore}}equality{\isaliteral{2C}{\isacharcomma}}\ rule\ is{\isaliteral{5F}{\isacharunderscore}}supI{\isaliteral{29}{\isacharparenright}}\ blast{\isaliteral{2B}{\isacharplus}}\isanewline
-\isanewline
-\ \ \isacommand{lemma}\isamarkupfalse%
-\ is{\isaliteral{5F}{\isacharunderscore}}sup{\isaliteral{5F}{\isacharunderscore}}join\ {\isaliteral{5B}{\isacharbrackleft}}intro{\isaliteral{3F}{\isacharquery}}{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}is{\isaliteral{5F}{\isacharunderscore}}sup\ x\ y\ {\isaliteral{28}{\isacharparenleft}}x\ {\isaliteral{5C3C7371756E696F6E3E}{\isasymsqunion}}\ y{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isaliteral{28}{\isacharparenleft}}unfold\ join{\isaliteral{5F}{\isacharunderscore}}def{\isaliteral{29}{\isacharparenright}}\isanewline
-\ \ \ \ \isacommand{from}\isamarkupfalse%
-\ ex{\isaliteral{5F}{\isacharunderscore}}sup\ \isacommand{obtain}\isamarkupfalse%
-\ s\ \isakeyword{where}\ {\isaliteral{22}{\isachardoublequoteopen}}is{\isaliteral{5F}{\isacharunderscore}}sup\ x\ y\ s{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}is{\isaliteral{5F}{\isacharunderscore}}sup\ x\ y\ {\isaliteral{28}{\isacharparenleft}}THE\ s{\isaliteral{2E}{\isachardot}}\ is{\isaliteral{5F}{\isacharunderscore}}sup\ x\ y\ s{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isaliteral{28}{\isacharparenleft}}rule\ theI{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{28}{\isacharparenleft}}rule\ is{\isaliteral{5F}{\isacharunderscore}}sup{\isaliteral{5F}{\isacharunderscore}}uniq\ {\isaliteral{5B}{\isacharbrackleft}}OF\ {\isaliteral{5F}{\isacharunderscore}}\ {\isaliteral{60}{\isacharbackquoteopen}}is{\isaliteral{5F}{\isacharunderscore}}sup\ x\ y\ s{\isaliteral{60}{\isacharbackquoteclose}}{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{29}{\isacharparenright}}\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isanewline
-\ \ \isacommand{lemma}\isamarkupfalse%
-\ join{\isaliteral{5F}{\isacharunderscore}}left\ {\isaliteral{5B}{\isacharbrackleft}}intro{\isaliteral{3F}{\isacharquery}}{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{3A}{\isacharcolon}}\isanewline
-\ \ \ \ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ x\ {\isaliteral{5C3C7371756E696F6E3E}{\isasymsqunion}}\ y{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isaliteral{28}{\isacharparenleft}}rule\ is{\isaliteral{5F}{\isacharunderscore}}sup{\isaliteral{5F}{\isacharunderscore}}upper{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{28}{\isacharparenleft}}rule\ is{\isaliteral{5F}{\isacharunderscore}}sup{\isaliteral{5F}{\isacharunderscore}}join{\isaliteral{29}{\isacharparenright}}\isanewline
-\isanewline
-\ \ \isacommand{lemma}\isamarkupfalse%
-\ join{\isaliteral{5F}{\isacharunderscore}}right\ {\isaliteral{5B}{\isacharbrackleft}}intro{\isaliteral{3F}{\isacharquery}}{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{3A}{\isacharcolon}}\isanewline
-\ \ \ \ {\isaliteral{22}{\isachardoublequoteopen}}y\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ x\ {\isaliteral{5C3C7371756E696F6E3E}{\isasymsqunion}}\ y{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isaliteral{28}{\isacharparenleft}}rule\ is{\isaliteral{5F}{\isacharunderscore}}sup{\isaliteral{5F}{\isacharunderscore}}upper{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{28}{\isacharparenleft}}rule\ is{\isaliteral{5F}{\isacharunderscore}}sup{\isaliteral{5F}{\isacharunderscore}}join{\isaliteral{29}{\isacharparenright}}\isanewline
-\isanewline
-\ \ \isacommand{lemma}\isamarkupfalse%
-\ join{\isaliteral{5F}{\isacharunderscore}}le\ {\isaliteral{5B}{\isacharbrackleft}}intro{\isaliteral{3F}{\isacharquery}}{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{3A}{\isacharcolon}}\isanewline
-\ \ \ \ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}\ x\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ z{\isaliteral{3B}{\isacharsemicolon}}\ y\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ z\ {\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ x\ {\isaliteral{5C3C7371756E696F6E3E}{\isasymsqunion}}\ y\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ z{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isaliteral{28}{\isacharparenleft}}rule\ is{\isaliteral{5F}{\isacharunderscore}}sup{\isaliteral{5F}{\isacharunderscore}}least{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{28}{\isacharparenleft}}rule\ is{\isaliteral{5F}{\isacharunderscore}}sup{\isaliteral{5F}{\isacharunderscore}}join{\isaliteral{29}{\isacharparenright}}\isanewline
-\isanewline
-\ \ \isacommand{theorem}\isamarkupfalse%
-\ meet{\isaliteral{5F}{\isacharunderscore}}assoc{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}x\ {\isaliteral{5C3C7371696E7465723E}{\isasymsqinter}}\ y{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C7371696E7465723E}{\isasymsqinter}}\ z\ {\isaliteral{3D}{\isacharequal}}\ x\ {\isaliteral{5C3C7371696E7465723E}{\isasymsqinter}}\ {\isaliteral{28}{\isacharparenleft}}y\ {\isaliteral{5C3C7371696E7465723E}{\isasymsqinter}}\ z{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isaliteral{28}{\isacharparenleft}}rule\ meetI{\isaliteral{29}{\isacharparenright}}\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C7371696E7465723E}{\isasymsqinter}}\ {\isaliteral{28}{\isacharparenleft}}y\ {\isaliteral{5C3C7371696E7465723E}{\isasymsqinter}}\ z{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ x\ {\isaliteral{5C3C7371696E7465723E}{\isasymsqinter}}\ y{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C7371696E7465723E}{\isasymsqinter}}\ {\isaliteral{28}{\isacharparenleft}}y\ {\isaliteral{5C3C7371696E7465723E}{\isasymsqinter}}\ z{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ x{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C7371696E7465723E}{\isasymsqinter}}\ {\isaliteral{28}{\isacharparenleft}}y\ {\isaliteral{5C3C7371696E7465723E}{\isasymsqinter}}\ z{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ y{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ {\isaliteral{2D}{\isacharminus}}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C7371696E7465723E}{\isasymsqinter}}\ {\isaliteral{28}{\isacharparenleft}}y\ {\isaliteral{5C3C7371696E7465723E}{\isasymsqinter}}\ z{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ y\ {\isaliteral{5C3C7371696E7465723E}{\isasymsqinter}}\ z{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ y{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{3F}{\isacharquery}}thesis\ \isacommand{{\isaliteral{2E}{\isachardot}}}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C7371696E7465723E}{\isasymsqinter}}\ {\isaliteral{28}{\isacharparenleft}}y\ {\isaliteral{5C3C7371696E7465723E}{\isasymsqinter}}\ z{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ z{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ {\isaliteral{2D}{\isacharminus}}\isanewline
-\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C7371696E7465723E}{\isasymsqinter}}\ {\isaliteral{28}{\isacharparenleft}}y\ {\isaliteral{5C3C7371696E7465723E}{\isasymsqinter}}\ z{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ y\ {\isaliteral{5C3C7371696E7465723E}{\isasymsqinter}}\ z{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ z{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{3F}{\isacharquery}}thesis\ \isacommand{{\isaliteral{2E}{\isachardot}}}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{fix}\isamarkupfalse%
-\ w\ \isacommand{assume}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}w\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ x\ {\isaliteral{5C3C7371696E7465723E}{\isasymsqinter}}\ y{\isaliteral{22}{\isachardoublequoteclose}}\ \isakeyword{and}\ {\isaliteral{22}{\isachardoublequoteopen}}w\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ z{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}w\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ x\ {\isaliteral{5C3C7371696E7465723E}{\isasymsqinter}}\ {\isaliteral{28}{\isacharparenleft}}y\ {\isaliteral{5C3C7371696E7465723E}{\isasymsqinter}}\ z{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}w\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ x{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ {\isaliteral{2D}{\isacharminus}}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}w\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ x\ {\isaliteral{5C3C7371696E7465723E}{\isasymsqinter}}\ y{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{by}\isamarkupfalse%
-\ fact\isanewline
-\ \ \ \ \ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ x{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{3F}{\isacharquery}}thesis\ \isacommand{{\isaliteral{2E}{\isachardot}}}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}w\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ y\ {\isaliteral{5C3C7371696E7465723E}{\isasymsqinter}}\ z{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \ \ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}w\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ y{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ {\isaliteral{2D}{\isacharminus}}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}w\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ x\ {\isaliteral{5C3C7371696E7465723E}{\isasymsqinter}}\ y{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{by}\isamarkupfalse%
-\ fact\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ y{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{3F}{\isacharquery}}thesis\ \isacommand{{\isaliteral{2E}{\isachardot}}}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}w\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ z{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{by}\isamarkupfalse%
-\ fact\isanewline
-\ \ \ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isanewline
-\ \ \isacommand{theorem}\isamarkupfalse%
-\ meet{\isaliteral{5F}{\isacharunderscore}}commute{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C7371696E7465723E}{\isasymsqinter}}\ y\ {\isaliteral{3D}{\isacharequal}}\ y\ {\isaliteral{5C3C7371696E7465723E}{\isasymsqinter}}\ x{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isaliteral{28}{\isacharparenleft}}rule\ meetI{\isaliteral{29}{\isacharparenright}}\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}y\ {\isaliteral{5C3C7371696E7465723E}{\isasymsqinter}}\ x\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ x{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}y\ {\isaliteral{5C3C7371696E7465723E}{\isasymsqinter}}\ x\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ y{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{fix}\isamarkupfalse%
-\ z\ \isacommand{assume}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}z\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ y{\isaliteral{22}{\isachardoublequoteclose}}\ \isakeyword{and}\ {\isaliteral{22}{\isachardoublequoteopen}}z\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ x{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}z\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ y\ {\isaliteral{5C3C7371696E7465723E}{\isasymsqinter}}\ x{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isanewline
-\ \ \isacommand{theorem}\isamarkupfalse%
-\ meet{\isaliteral{5F}{\isacharunderscore}}join{\isaliteral{5F}{\isacharunderscore}}absorb{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C7371696E7465723E}{\isasymsqinter}}\ {\isaliteral{28}{\isacharparenleft}}x\ {\isaliteral{5C3C7371756E696F6E3E}{\isasymsqunion}}\ y{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ x{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isaliteral{28}{\isacharparenleft}}rule\ meetI{\isaliteral{29}{\isacharparenright}}\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ x{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ x\ {\isaliteral{5C3C7371756E696F6E3E}{\isasymsqunion}}\ y{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{fix}\isamarkupfalse%
-\ z\ \isacommand{assume}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}z\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ x{\isaliteral{22}{\isachardoublequoteclose}}\ \isakeyword{and}\ {\isaliteral{22}{\isachardoublequoteopen}}z\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ x\ {\isaliteral{5C3C7371756E696F6E3E}{\isasymsqunion}}\ y{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}z\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ x{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{by}\isamarkupfalse%
-\ fact\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isanewline
-\ \ \isacommand{theorem}\isamarkupfalse%
-\ join{\isaliteral{5F}{\isacharunderscore}}assoc{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}x\ {\isaliteral{5C3C7371756E696F6E3E}{\isasymsqunion}}\ y{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C7371756E696F6E3E}{\isasymsqunion}}\ z\ {\isaliteral{3D}{\isacharequal}}\ x\ {\isaliteral{5C3C7371756E696F6E3E}{\isasymsqunion}}\ {\isaliteral{28}{\isacharparenleft}}y\ {\isaliteral{5C3C7371756E696F6E3E}{\isasymsqunion}}\ z{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isaliteral{28}{\isacharparenleft}}rule\ joinI{\isaliteral{29}{\isacharparenright}}\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C7371756E696F6E3E}{\isasymsqunion}}\ y\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ x\ {\isaliteral{5C3C7371756E696F6E3E}{\isasymsqunion}}\ {\isaliteral{28}{\isacharparenleft}}y\ {\isaliteral{5C3C7371756E696F6E3E}{\isasymsqunion}}\ z{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ x\ {\isaliteral{5C3C7371756E696F6E3E}{\isasymsqunion}}\ {\isaliteral{28}{\isacharparenleft}}y\ {\isaliteral{5C3C7371756E696F6E3E}{\isasymsqunion}}\ z{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}y\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ x\ {\isaliteral{5C3C7371756E696F6E3E}{\isasymsqunion}}\ {\isaliteral{28}{\isacharparenleft}}y\ {\isaliteral{5C3C7371756E696F6E3E}{\isasymsqunion}}\ z{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ {\isaliteral{2D}{\isacharminus}}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}y\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ y\ {\isaliteral{5C3C7371756E696F6E3E}{\isasymsqunion}}\ z{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ x\ {\isaliteral{5C3C7371756E696F6E3E}{\isasymsqunion}}\ {\isaliteral{28}{\isacharparenleft}}y\ {\isaliteral{5C3C7371756E696F6E3E}{\isasymsqunion}}\ z{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{3F}{\isacharquery}}thesis\ \isacommand{{\isaliteral{2E}{\isachardot}}}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}z\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ x\ {\isaliteral{5C3C7371756E696F6E3E}{\isasymsqunion}}\ {\isaliteral{28}{\isacharparenleft}}y\ {\isaliteral{5C3C7371756E696F6E3E}{\isasymsqunion}}\ z{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ {\isaliteral{2D}{\isacharminus}}\isanewline
-\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}z\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ y\ {\isaliteral{5C3C7371756E696F6E3E}{\isasymsqunion}}\ z{\isaliteral{22}{\isachardoublequoteclose}}\ \ \isacommand{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ x\ {\isaliteral{5C3C7371756E696F6E3E}{\isasymsqunion}}\ {\isaliteral{28}{\isacharparenleft}}y\ {\isaliteral{5C3C7371756E696F6E3E}{\isasymsqunion}}\ z{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{3F}{\isacharquery}}thesis\ \isacommand{{\isaliteral{2E}{\isachardot}}}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{fix}\isamarkupfalse%
-\ w\ \isacommand{assume}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C7371756E696F6E3E}{\isasymsqunion}}\ y\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ w{\isaliteral{22}{\isachardoublequoteclose}}\ \isakeyword{and}\ {\isaliteral{22}{\isachardoublequoteopen}}z\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ w{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C7371756E696F6E3E}{\isasymsqunion}}\ {\isaliteral{28}{\isacharparenleft}}y\ {\isaliteral{5C3C7371756E696F6E3E}{\isasymsqunion}}\ z{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ w{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ w{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ {\isaliteral{2D}{\isacharminus}}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ x\ {\isaliteral{5C3C7371756E696F6E3E}{\isasymsqunion}}\ y{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ w{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{by}\isamarkupfalse%
-\ fact\isanewline
-\ \ \ \ \ \ \ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{3F}{\isacharquery}}thesis\ \isacommand{{\isaliteral{2E}{\isachardot}}}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}y\ {\isaliteral{5C3C7371756E696F6E3E}{\isasymsqunion}}\ z\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ w{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \ \ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}y\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ w{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ {\isaliteral{2D}{\isacharminus}}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}y\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ x\ {\isaliteral{5C3C7371756E696F6E3E}{\isasymsqunion}}\ y{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ w{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{by}\isamarkupfalse%
-\ fact\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{3F}{\isacharquery}}thesis\ \isacommand{{\isaliteral{2E}{\isachardot}}}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}z\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ w{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{by}\isamarkupfalse%
-\ fact\isanewline
-\ \ \ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isanewline
-\ \ \isacommand{theorem}\isamarkupfalse%
-\ join{\isaliteral{5F}{\isacharunderscore}}commute{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C7371756E696F6E3E}{\isasymsqunion}}\ y\ {\isaliteral{3D}{\isacharequal}}\ y\ {\isaliteral{5C3C7371756E696F6E3E}{\isasymsqunion}}\ x{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isaliteral{28}{\isacharparenleft}}rule\ joinI{\isaliteral{29}{\isacharparenright}}\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ y\ {\isaliteral{5C3C7371756E696F6E3E}{\isasymsqunion}}\ x{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}y\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ y\ {\isaliteral{5C3C7371756E696F6E3E}{\isasymsqunion}}\ x{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{fix}\isamarkupfalse%
-\ z\ \isacommand{assume}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}y\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ z{\isaliteral{22}{\isachardoublequoteclose}}\ \isakeyword{and}\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ z{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}y\ {\isaliteral{5C3C7371756E696F6E3E}{\isasymsqunion}}\ x\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ z{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isanewline
-\ \ \isacommand{theorem}\isamarkupfalse%
-\ join{\isaliteral{5F}{\isacharunderscore}}meet{\isaliteral{5F}{\isacharunderscore}}absorb{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C7371756E696F6E3E}{\isasymsqunion}}\ {\isaliteral{28}{\isacharparenleft}}x\ {\isaliteral{5C3C7371696E7465723E}{\isasymsqinter}}\ y{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ x{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isaliteral{28}{\isacharparenleft}}rule\ joinI{\isaliteral{29}{\isacharparenright}}\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ x{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C7371696E7465723E}{\isasymsqinter}}\ y\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ x{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{fix}\isamarkupfalse%
-\ z\ \isacommand{assume}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ z{\isaliteral{22}{\isachardoublequoteclose}}\ \isakeyword{and}\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C7371696E7465723E}{\isasymsqinter}}\ y\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ z{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ z{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{by}\isamarkupfalse%
-\ fact\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isanewline
-\ \ \isacommand{theorem}\isamarkupfalse%
-\ meet{\isaliteral{5F}{\isacharunderscore}}idem{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C7371696E7465723E}{\isasymsqinter}}\ x\ {\isaliteral{3D}{\isacharequal}}\ x{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isaliteral{2D}{\isacharminus}}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C7371696E7465723E}{\isasymsqinter}}\ {\isaliteral{28}{\isacharparenleft}}x\ {\isaliteral{5C3C7371756E696F6E3E}{\isasymsqunion}}\ {\isaliteral{28}{\isacharparenleft}}x\ {\isaliteral{5C3C7371696E7465723E}{\isasymsqinter}}\ x{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ x{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{by}\isamarkupfalse%
-\ {\isaliteral{28}{\isacharparenleft}}rule\ meet{\isaliteral{5F}{\isacharunderscore}}join{\isaliteral{5F}{\isacharunderscore}}absorb{\isaliteral{29}{\isacharparenright}}\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C7371756E696F6E3E}{\isasymsqunion}}\ {\isaliteral{28}{\isacharparenleft}}x\ {\isaliteral{5C3C7371696E7465723E}{\isasymsqinter}}\ x{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ x{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{by}\isamarkupfalse%
-\ {\isaliteral{28}{\isacharparenleft}}rule\ join{\isaliteral{5F}{\isacharunderscore}}meet{\isaliteral{5F}{\isacharunderscore}}absorb{\isaliteral{29}{\isacharparenright}}\isanewline
-\ \ \ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{3F}{\isacharquery}}thesis\ \isacommand{{\isaliteral{2E}{\isachardot}}}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isanewline
-\ \ \isacommand{theorem}\isamarkupfalse%
-\ meet{\isaliteral{5F}{\isacharunderscore}}related\ {\isaliteral{5B}{\isacharbrackleft}}elim{\isaliteral{3F}{\isacharquery}}{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ y\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ x\ {\isaliteral{5C3C7371696E7465723E}{\isasymsqinter}}\ y\ {\isaliteral{3D}{\isacharequal}}\ x{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isaliteral{28}{\isacharparenleft}}rule\ meetI{\isaliteral{29}{\isacharparenright}}\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ y{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ x{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ y{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{by}\isamarkupfalse%
-\ fact\isanewline
-\ \ \ \ \isacommand{fix}\isamarkupfalse%
-\ z\ \isacommand{assume}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}z\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ x{\isaliteral{22}{\isachardoublequoteclose}}\ \isakeyword{and}\ {\isaliteral{22}{\isachardoublequoteopen}}z\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ y{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}z\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ x{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{by}\isamarkupfalse%
-\ fact\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isanewline
-\ \ \isacommand{theorem}\isamarkupfalse%
-\ meet{\isaliteral{5F}{\isacharunderscore}}related{\isadigit{2}}\ {\isaliteral{5B}{\isacharbrackleft}}elim{\isaliteral{3F}{\isacharquery}}{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}y\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ x\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ x\ {\isaliteral{5C3C7371696E7465723E}{\isasymsqinter}}\ y\ {\isaliteral{3D}{\isacharequal}}\ y{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isaliteral{28}{\isacharparenleft}}drule\ meet{\isaliteral{5F}{\isacharunderscore}}related{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{28}{\isacharparenleft}}simp\ add{\isaliteral{3A}{\isacharcolon}}\ meet{\isaliteral{5F}{\isacharunderscore}}commute{\isaliteral{29}{\isacharparenright}}\isanewline
-\isanewline
-\ \ \isacommand{theorem}\isamarkupfalse%
-\ join{\isaliteral{5F}{\isacharunderscore}}related\ {\isaliteral{5B}{\isacharbrackleft}}elim{\isaliteral{3F}{\isacharquery}}{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ y\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ x\ {\isaliteral{5C3C7371756E696F6E3E}{\isasymsqunion}}\ y\ {\isaliteral{3D}{\isacharequal}}\ y{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isaliteral{28}{\isacharparenleft}}rule\ joinI{\isaliteral{29}{\isacharparenright}}\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ y{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}y\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ y{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ y{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{by}\isamarkupfalse%
-\ fact\isanewline
-\ \ \ \ \isacommand{fix}\isamarkupfalse%
-\ z\ \isacommand{assume}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ z{\isaliteral{22}{\isachardoublequoteclose}}\ \isakeyword{and}\ {\isaliteral{22}{\isachardoublequoteopen}}y\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ z{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}y\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ z{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{by}\isamarkupfalse%
-\ fact\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isanewline
-\ \ \isacommand{theorem}\isamarkupfalse%
-\ join{\isaliteral{5F}{\isacharunderscore}}related{\isadigit{2}}\ {\isaliteral{5B}{\isacharbrackleft}}elim{\isaliteral{3F}{\isacharquery}}{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}y\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ x\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ x\ {\isaliteral{5C3C7371756E696F6E3E}{\isasymsqunion}}\ y\ {\isaliteral{3D}{\isacharequal}}\ x{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isaliteral{28}{\isacharparenleft}}drule\ join{\isaliteral{5F}{\isacharunderscore}}related{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{28}{\isacharparenleft}}simp\ add{\isaliteral{3A}{\isacharcolon}}\ join{\isaliteral{5F}{\isacharunderscore}}commute{\isaliteral{29}{\isacharparenright}}\isanewline
-\isanewline
-\ \ \isacommand{theorem}\isamarkupfalse%
-\ meet{\isaliteral{5F}{\isacharunderscore}}connection{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}x\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ y{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}x\ {\isaliteral{5C3C7371696E7465723E}{\isasymsqinter}}\ y\ {\isaliteral{3D}{\isacharequal}}\ x{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ y{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}is{\isaliteral{5F}{\isacharunderscore}}inf\ x\ y\ x{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C7371696E7465723E}{\isasymsqinter}}\ y\ {\isaliteral{3D}{\isacharequal}}\ x{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C7371696E7465723E}{\isasymsqinter}}\ y\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ y{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{assume}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C7371696E7465723E}{\isasymsqinter}}\ y\ {\isaliteral{3D}{\isacharequal}}\ x{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ y{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{{\isaliteral{2E}{\isachardot}}}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isanewline
-\ \ \isacommand{theorem}\isamarkupfalse%
-\ join{\isaliteral{5F}{\isacharunderscore}}connection{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}x\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ y{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}x\ {\isaliteral{5C3C7371756E696F6E3E}{\isasymsqunion}}\ y\ {\isaliteral{3D}{\isacharequal}}\ y{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ y{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}is{\isaliteral{5F}{\isacharunderscore}}sup\ x\ y\ y{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C7371756E696F6E3E}{\isasymsqunion}}\ y\ {\isaliteral{3D}{\isacharequal}}\ y{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ x\ {\isaliteral{5C3C7371756E696F6E3E}{\isasymsqunion}}\ y{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{assume}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C7371756E696F6E3E}{\isasymsqunion}}\ y\ {\isaliteral{3D}{\isacharequal}}\ y{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ y{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{{\isaliteral{2E}{\isachardot}}}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isanewline
-\ \ \isacommand{theorem}\isamarkupfalse%
-\ meet{\isaliteral{5F}{\isacharunderscore}}connection{\isadigit{2}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}x\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ y{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}y\ {\isaliteral{5C3C7371696E7465723E}{\isasymsqinter}}\ x\ {\isaliteral{3D}{\isacharequal}}\ x{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ meet{\isaliteral{5F}{\isacharunderscore}}commute\ meet{\isaliteral{5F}{\isacharunderscore}}connection\ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isanewline
-\ \ \isacommand{theorem}\isamarkupfalse%
-\ join{\isaliteral{5F}{\isacharunderscore}}connection{\isadigit{2}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}x\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ y{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}x\ {\isaliteral{5C3C7371756E696F6E3E}{\isasymsqunion}}\ y\ {\isaliteral{3D}{\isacharequal}}\ y{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ join{\isaliteral{5F}{\isacharunderscore}}commute\ join{\isaliteral{5F}{\isacharunderscore}}connection\ \isacommand{by}\isamarkupfalse%
-\ simp%
-\begin{isamarkuptext}%
-Naming according to Jacobson I, p.\ 459.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\ \ \isacommand{lemmas}\isamarkupfalse%
-\ L{\isadigit{1}}\ {\isaliteral{3D}{\isacharequal}}\ join{\isaliteral{5F}{\isacharunderscore}}commute\ meet{\isaliteral{5F}{\isacharunderscore}}commute\isanewline
-\ \ \isacommand{lemmas}\isamarkupfalse%
-\ L{\isadigit{2}}\ {\isaliteral{3D}{\isacharequal}}\ join{\isaliteral{5F}{\isacharunderscore}}assoc\ meet{\isaliteral{5F}{\isacharunderscore}}assoc\isanewline
-\ \ \isanewline
-\ \ \isacommand{lemmas}\isamarkupfalse%
-\ L{\isadigit{4}}\ {\isaliteral{3D}{\isacharequal}}\ join{\isaliteral{5F}{\isacharunderscore}}meet{\isaliteral{5F}{\isacharunderscore}}absorb\ meet{\isaliteral{5F}{\isacharunderscore}}join{\isaliteral{5F}{\isacharunderscore}}absorb%
-\endisataginvisible
-{\isafoldinvisible}%
-%
-\isadeliminvisible
-\isanewline
-%
-\endisadeliminvisible
-\isanewline
-\ \ \isacommand{end}\isamarkupfalse%
-%
-\begin{isamarkuptext}%
-Locales for total orders and distributive lattices follow to
-  establish a sufficiently rich landscape of locales for
-  further examples in this tutorial.  Each comes with an example
-  theorem.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\ \ \isacommand{locale}\isamarkupfalse%
-\ total{\isaliteral{5F}{\isacharunderscore}}order\ {\isaliteral{3D}{\isacharequal}}\ partial{\isaliteral{5F}{\isacharunderscore}}order\ {\isaliteral{2B}{\isacharplus}}\isanewline
-\ \ \ \ \isakeyword{assumes}\ total{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ y\ {\isaliteral{5C3C6F723E}{\isasymor}}\ y\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ x{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\isanewline
-\ \ \isacommand{lemma}\isamarkupfalse%
-\ {\isaliteral{28}{\isacharparenleft}}\isakeyword{in}\ total{\isaliteral{5F}{\isacharunderscore}}order{\isaliteral{29}{\isacharparenright}}\ less{\isaliteral{5F}{\isacharunderscore}}total{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C73717375627365743E}{\isasymsqsubset}}\ y\ {\isaliteral{5C3C6F723E}{\isasymor}}\ x\ {\isaliteral{3D}{\isacharequal}}\ y\ {\isaliteral{5C3C6F723E}{\isasymor}}\ y\ {\isaliteral{5C3C73717375627365743E}{\isasymsqsubset}}\ x{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-%
-\isadelimproof
-\ \ \ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ total\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isaliteral{28}{\isacharparenleft}}unfold\ less{\isaliteral{5F}{\isacharunderscore}}def{\isaliteral{29}{\isacharparenright}}\ blast%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\ \ \isacommand{locale}\isamarkupfalse%
-\ distrib{\isaliteral{5F}{\isacharunderscore}}lattice\ {\isaliteral{3D}{\isacharequal}}\ lattice\ {\isaliteral{2B}{\isacharplus}}\isanewline
-\ \ \ \ \isakeyword{assumes}\ meet{\isaliteral{5F}{\isacharunderscore}}distr{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C7371696E7465723E}{\isasymsqinter}}\ {\isaliteral{28}{\isacharparenleft}}y\ {\isaliteral{5C3C7371756E696F6E3E}{\isasymsqunion}}\ z{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ x\ {\isaliteral{5C3C7371696E7465723E}{\isasymsqinter}}\ y\ {\isaliteral{5C3C7371756E696F6E3E}{\isasymsqunion}}\ x\ {\isaliteral{5C3C7371696E7465723E}{\isasymsqinter}}\ z{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\isanewline
-\ \ \isacommand{lemma}\isamarkupfalse%
-\ {\isaliteral{28}{\isacharparenleft}}\isakeyword{in}\ distrib{\isaliteral{5F}{\isacharunderscore}}lattice{\isaliteral{29}{\isacharparenright}}\ join{\isaliteral{5F}{\isacharunderscore}}distr{\isaliteral{3A}{\isacharcolon}}\isanewline
-\ \ \ \ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C7371756E696F6E3E}{\isasymsqunion}}\ {\isaliteral{28}{\isacharparenleft}}y\ {\isaliteral{5C3C7371696E7465723E}{\isasymsqinter}}\ z{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}x\ {\isaliteral{5C3C7371756E696F6E3E}{\isasymsqunion}}\ y{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C7371696E7465723E}{\isasymsqinter}}\ {\isaliteral{28}{\isacharparenleft}}x\ {\isaliteral{5C3C7371756E696F6E3E}{\isasymsqunion}}\ z{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\ \ \isanewline
-%
-\isadelimproof
-\ \ \ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isaliteral{2D}{\isacharminus}}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C7371756E696F6E3E}{\isasymsqunion}}\ {\isaliteral{28}{\isacharparenleft}}y\ {\isaliteral{5C3C7371696E7465723E}{\isasymsqinter}}\ z{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}x\ {\isaliteral{5C3C7371756E696F6E3E}{\isasymsqunion}}\ {\isaliteral{28}{\isacharparenleft}}x\ {\isaliteral{5C3C7371696E7465723E}{\isasymsqinter}}\ z{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C7371756E696F6E3E}{\isasymsqunion}}\ {\isaliteral{28}{\isacharparenleft}}y\ {\isaliteral{5C3C7371696E7465723E}{\isasymsqinter}}\ z{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{by}\isamarkupfalse%
-\ {\isaliteral{28}{\isacharparenleft}}simp\ add{\isaliteral{3A}{\isacharcolon}}\ L{\isadigit{4}}{\isaliteral{29}{\isacharparenright}}\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}\ {\isaliteral{3D}{\isacharequal}}\ x\ {\isaliteral{5C3C7371756E696F6E3E}{\isasymsqunion}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{28}{\isacharparenleft}}x\ {\isaliteral{5C3C7371696E7465723E}{\isasymsqinter}}\ z{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C7371756E696F6E3E}{\isasymsqunion}}\ {\isaliteral{28}{\isacharparenleft}}y\ {\isaliteral{5C3C7371696E7465723E}{\isasymsqinter}}\ z{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{by}\isamarkupfalse%
-\ {\isaliteral{28}{\isacharparenleft}}simp\ add{\isaliteral{3A}{\isacharcolon}}\ L{\isadigit{2}}{\isaliteral{29}{\isacharparenright}}\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}\ {\isaliteral{3D}{\isacharequal}}\ x\ {\isaliteral{5C3C7371756E696F6E3E}{\isasymsqunion}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{28}{\isacharparenleft}}x\ {\isaliteral{5C3C7371756E696F6E3E}{\isasymsqunion}}\ y{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C7371696E7465723E}{\isasymsqinter}}\ z{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{by}\isamarkupfalse%
-\ {\isaliteral{28}{\isacharparenleft}}simp\ add{\isaliteral{3A}{\isacharcolon}}\ L{\isadigit{1}}\ meet{\isaliteral{5F}{\isacharunderscore}}distr{\isaliteral{29}{\isacharparenright}}\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{28}{\isacharparenleft}}x\ {\isaliteral{5C3C7371756E696F6E3E}{\isasymsqunion}}\ y{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C7371696E7465723E}{\isasymsqinter}}\ x{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C7371756E696F6E3E}{\isasymsqunion}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{28}{\isacharparenleft}}x\ {\isaliteral{5C3C7371756E696F6E3E}{\isasymsqunion}}\ y{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C7371696E7465723E}{\isasymsqinter}}\ z{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{by}\isamarkupfalse%
-\ {\isaliteral{28}{\isacharparenleft}}simp\ add{\isaliteral{3A}{\isacharcolon}}\ L{\isadigit{1}}\ L{\isadigit{4}}{\isaliteral{29}{\isacharparenright}}\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}x\ {\isaliteral{5C3C7371756E696F6E3E}{\isasymsqunion}}\ y{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C7371696E7465723E}{\isasymsqinter}}\ {\isaliteral{28}{\isacharparenleft}}x\ {\isaliteral{5C3C7371756E696F6E3E}{\isasymsqunion}}\ z{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{by}\isamarkupfalse%
-\ {\isaliteral{28}{\isacharparenleft}}simp\ add{\isaliteral{3A}{\isacharcolon}}\ meet{\isaliteral{5F}{\isacharunderscore}}distr{\isaliteral{29}{\isacharparenright}}\isanewline
-\ \ \ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{3F}{\isacharquery}}thesis\ \isacommand{{\isaliteral{2E}{\isachardot}}}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-The locale hierarchy obtained through these declarations is shown in
-  Figure~\ref{fig:lattices}(a).
-
-\begin{figure}
-\hrule \vspace{2ex}
-\begin{center}
-\subfigure[Declared hierarchy]{
-\begin{tikzpicture}
-  \node (po) at (0,0) {\isa{partial{\isaliteral{5F}{\isacharunderscore}}order}};
-  \node (lat) at (-1.5,-1) {\isa{lattice}};
-  \node (dlat) at (-1.5,-2) {\isa{distrib{\isaliteral{5F}{\isacharunderscore}}lattice}};
-  \node (to) at (1.5,-1) {\isa{total{\isaliteral{5F}{\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{\isaliteral{5F}{\isacharunderscore}}order}};
-  \node (lat) at (0,-1) {\isa{lattice}};
-  \node (dlat) at (-1.5,-2) {\isa{distrib{\isaliteral{5F}{\isacharunderscore}}lattice}};
-  \node (to) at (1.5,-2) {\isa{total{\isaliteral{5F}{\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{\isaliteral{5F}{\isacharunderscore}}order}};
-  \node (lat) at (0,-1) {\isa{lattice}};
-  \node (dlat) at (0,-2) {\isa{distrib{\isaliteral{5F}{\isacharunderscore}}lattice}};
-  \node (to) at (0,-3) {\isa{total{\isaliteral{5F}{\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
-  \label{sec:changing-the-hierarchy}%
-}
-\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}.  Locales generalise
-  interpretation from theorems to conclusions, enabling the reuse of
-  definitions and other constructs that are not part of the
-  specifications of the locales.
-
-  The first form of interpretation we will consider in this tutorial
-  is provided by the \isakeyword{sublocale} command.  It enables to
-  modify the import hierarchy to reflect the \emph{logical} relation
-  between locales.
-
-  Consider the locale hierarchy from Figure~\ref{fig:lattices}(a).
-  Total orders are lattices, although this is not reflected here, and
-  definitions, theorems and other conclusions
-  from \isa{lattice} are not available in \isa{total{\isaliteral{5F}{\isacharunderscore}}order}.  To
-  obtain the situation in Figure~\ref{fig:lattices}(b), it is
-  sufficient to add the conclusions of the latter locale to the former.
-  The \isakeyword{sublocale} command does exactly this.
-  The declaration \isakeyword{sublocale} $l_1
-  \subseteq l_2$ causes locale $l_2$ to be \emph{interpreted} in the
-  context of $l_1$.  This means that all conclusions of $l_2$ are made
-  available in $l_1$.
-
-  Of course, the change of hierarchy must be supported by a theorem
-  that reflects, in our example, that total orders are indeed
-  lattices.  Therefore the \isakeyword{sublocale} command generates a
-  goal, which must be discharged by the user.  This is illustrated in
-  the following paragraphs.  First the sublocale relation is stated.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimvisible
-\ \ %
-\endisadelimvisible
-%
-\isatagvisible
-\isacommand{sublocale}\isamarkupfalse%
-\ total{\isaliteral{5F}{\isacharunderscore}}order\ {\isaliteral{5C3C73756273657465713E}{\isasymsubseteq}}\ lattice%
-\begin{isamarkuptxt}%
-\normalsize
-  This enters the context of locale \isa{total{\isaliteral{5F}{\isacharunderscore}}order}, in
-  which the goal \begin{isabelle}%
-\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ lattice\ op\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}%
-\end{isabelle} must be shown.
-  Now the
-  locale predicate needs to be unfolded --- for example, using its
-  definition or by introduction rules
-  provided by the locale package.  For automation, the locale package
-  provides the methods \isa{intro{\isaliteral{5F}{\isacharunderscore}}locales} and \isa{unfold{\isaliteral{5F}{\isacharunderscore}}locales}.  They are aware of the
-  current context and dependencies between locales and automatically
-  discharge goals implied by these.  While \isa{unfold{\isaliteral{5F}{\isacharunderscore}}locales}
-  always unfolds locale predicates to assumptions, \isa{intro{\isaliteral{5F}{\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}, which need to be shown, hence
-  \isa{unfold{\isaliteral{5F}{\isacharunderscore}}locales} is appropriate.%
-\end{isamarkuptxt}%
-\isamarkuptrue%
-\ \ \isacommand{proof}\isamarkupfalse%
-\ unfold{\isaliteral{5F}{\isacharunderscore}}locales%
-\begin{isamarkuptxt}%
-\normalsize
-  Since the fact that both lattices and total orders are partial
-  orders is already reflected in the locale hierarchy, the assumptions
-  of \isa{partial{\isaliteral{5F}{\isacharunderscore}}order} are discharged automatically, and only the
-  assumptions introduced in \isa{lattice} remain as subgoals
-  \begin{isabelle}%
-\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}x\ y{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C6578697374733E}{\isasymexists}}inf{\isaliteral{2E}{\isachardot}}\ is{\isaliteral{5F}{\isacharunderscore}}inf\ x\ y\ inf\isanewline
-\ {\isadigit{2}}{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}x\ y{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C6578697374733E}{\isasymexists}}sup{\isaliteral{2E}{\isachardot}}\ is{\isaliteral{5F}{\isacharunderscore}}sup\ x\ y\ sup%
-\end{isabelle}
-  The proof for the first subgoal is obtained by constructing an
-  infimum, whose existence is implied by totality.%
-\end{isamarkuptxt}%
-\isamarkuptrue%
-\ \ \ \ \isacommand{fix}\isamarkupfalse%
-\ x\ y\isanewline
-\ \ \ \ \isacommand{from}\isamarkupfalse%
-\ total\ \isacommand{have}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}is{\isaliteral{5F}{\isacharunderscore}}inf\ x\ y\ {\isaliteral{28}{\isacharparenleft}}if\ x\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ y\ then\ x\ else\ y{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isaliteral{28}{\isacharparenleft}}auto\ simp{\isaliteral{3A}{\isacharcolon}}\ is{\isaliteral{5F}{\isacharunderscore}}inf{\isaliteral{5F}{\isacharunderscore}}def{\isaliteral{29}{\isacharparenright}}\isanewline
-\ \ \ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6578697374733E}{\isasymexists}}inf{\isaliteral{2E}{\isachardot}}\ is{\isaliteral{5F}{\isacharunderscore}}inf\ x\ y\ inf{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}}\isamarkupfalse%
-%
-\begin{isamarkuptxt}%
-\normalsize
-   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%
-\ {\isaliteral{22}{\isachardoublequoteopen}}is{\isaliteral{5F}{\isacharunderscore}}sup\ x\ y\ {\isaliteral{28}{\isacharparenleft}}if\ x\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ y\ then\ y\ else\ x{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isaliteral{28}{\isacharparenleft}}auto\ simp{\isaliteral{3A}{\isacharcolon}}\ is{\isaliteral{5F}{\isacharunderscore}}sup{\isaliteral{5F}{\isacharunderscore}}def{\isaliteral{29}{\isacharparenright}}\isanewline
-\ \ \ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6578697374733E}{\isasymexists}}sup{\isaliteral{2E}{\isachardot}}\ is{\isaliteral{5F}{\isacharunderscore}}sup\ x\ y\ sup{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}}\isamarkupfalse%
-%
-\endisataginvisible
-{\isafoldinvisible}%
-%
-\isadeliminvisible
-%
-\endisadeliminvisible
-%
-\isadelimvisible
-\ %
-\endisadelimvisible
-%
-\isatagvisible
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagvisible
-{\isafoldvisible}%
-%
-\isadelimvisible
-%
-\endisadelimvisible
-%
-\begin{isamarkuptext}%
-Similarly, we may establish that total orders are distributive
-  lattices with a second \isakeyword{sublocale} statement.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\ \ \isacommand{sublocale}\isamarkupfalse%
-\ total{\isaliteral{5F}{\isacharunderscore}}order\ {\isaliteral{5C3C73756273657465713E}{\isasymsubseteq}}\ distrib{\isaliteral{5F}{\isacharunderscore}}lattice\isanewline
-%
-\isadelimproof
-\ \ \ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ unfold{\isaliteral{5F}{\isacharunderscore}}locales\isanewline
-\ \ \ \ \isacommand{fix}\isamarkupfalse%
-\ x\ y\ z\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C7371696E7465723E}{\isasymsqinter}}\ {\isaliteral{28}{\isacharparenleft}}y\ {\isaliteral{5C3C7371756E696F6E3E}{\isasymsqunion}}\ z{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ x\ {\isaliteral{5C3C7371696E7465723E}{\isasymsqinter}}\ y\ {\isaliteral{5C3C7371756E696F6E3E}{\isasymsqunion}}\ x\ {\isaliteral{5C3C7371696E7465723E}{\isasymsqinter}}\ z{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{28}{\isacharparenleft}}\isakeyword{is}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{3F}{\isacharquery}}l\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{3F}{\isacharquery}}r{\isaliteral{22}{\isachardoublequoteclose}}{\isaliteral{29}{\isacharparenright}}%
-\begin{isamarkuptxt}%
-Jacobson I, p.\ 462%
-\end{isamarkuptxt}%
-\isamarkuptrue%
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ {\isaliteral{2D}{\isacharminus}}\isanewline
-\ \ \ \ \ \ \isacommand{{\isaliteral{7B}{\isacharbraceleft}}}\isamarkupfalse%
-\ \isacommand{assume}\isamarkupfalse%
-\ c{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}y\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ x{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{22}{\isachardoublequoteopen}}z\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ x{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{from}\isamarkupfalse%
-\ c\ \isacommand{have}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{3F}{\isacharquery}}l\ {\isaliteral{3D}{\isacharequal}}\ y\ {\isaliteral{5C3C7371756E696F6E3E}{\isasymsqunion}}\ z{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isaliteral{28}{\isacharparenleft}}metis\ c\ join{\isaliteral{5F}{\isacharunderscore}}connection{\isadigit{2}}\ join{\isaliteral{5F}{\isacharunderscore}}related{\isadigit{2}}\ meet{\isaliteral{5F}{\isacharunderscore}}related{\isadigit{2}}\ total{\isaliteral{29}{\isacharparenright}}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{from}\isamarkupfalse%
-\ c\ \isacommand{have}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{3F}{\isacharquery}}r{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{by}\isamarkupfalse%
-\ {\isaliteral{28}{\isacharparenleft}}metis\ meet{\isaliteral{5F}{\isacharunderscore}}related{\isadigit{2}}{\isaliteral{29}{\isacharparenright}}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{3F}{\isacharquery}}l\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{3F}{\isacharquery}}r{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{{\isaliteral{2E}{\isachardot}}}\isamarkupfalse%
-\ \isacommand{{\isaliteral{7D}{\isacharbraceright}}}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{{\isaliteral{7B}{\isacharbraceleft}}}\isamarkupfalse%
-\ \isacommand{assume}\isamarkupfalse%
-\ c{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ y\ {\isaliteral{5C3C6F723E}{\isasymor}}\ x\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ z{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{from}\isamarkupfalse%
-\ c\ \isacommand{have}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{3F}{\isacharquery}}l\ {\isaliteral{3D}{\isacharequal}}\ x{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isaliteral{28}{\isacharparenleft}}metis\ join{\isaliteral{5F}{\isacharunderscore}}connection{\isadigit{2}}\ join{\isaliteral{5F}{\isacharunderscore}}related{\isadigit{2}}\ meet{\isaliteral{5F}{\isacharunderscore}}connection\ total\ trans{\isaliteral{29}{\isacharparenright}}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{from}\isamarkupfalse%
-\ c\ \isacommand{have}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{3F}{\isacharquery}}r{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isaliteral{28}{\isacharparenleft}}metis\ join{\isaliteral{5F}{\isacharunderscore}}commute\ join{\isaliteral{5F}{\isacharunderscore}}related{\isadigit{2}}\ meet{\isaliteral{5F}{\isacharunderscore}}connection\ meet{\isaliteral{5F}{\isacharunderscore}}related{\isadigit{2}}\ total{\isaliteral{29}{\isacharparenright}}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{3F}{\isacharquery}}l\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{3F}{\isacharquery}}r{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{{\isaliteral{2E}{\isachardot}}}\isamarkupfalse%
-\ \isacommand{{\isaliteral{7D}{\isacharbraceright}}}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{note}\isamarkupfalse%
-\ total\isanewline
-\ \ \ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{3F}{\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%
-%
-\begin{isamarkuptext}%
-Locale interpretation is \emph{dynamic}.  The statement
-  \isakeyword{sublocale} $l_1 \subseteq l_2$ will not just add the
-  current conclusions of $l_2$ to $l_1$.  Rather the dependency is
-  stored, and conclusions that will be
-  added to $l_2$ in future are automatically propagated to $l_1$.
-  The sublocale relation is transitive --- that is, propagation takes
-  effect along chains of sublocales.  Even cycles in the sublocale relation are
-  supported, as long as these cycles do not lead to infinite chains.
-  Details are discussed in the technical report \cite{Ballarin2006a}.
-  See also Section~\ref{sec:infinite-chains} of this tutorial.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-\isanewline
-\end{isabellebody}%
-%%% Local Variables:
-%%% mode: latex
-%%% TeX-master: "root"
-%%% End:

--- a/doc-src/Locales/Locales/document/Examples1.tex	Mon Aug 27 21:04:37 2012 +0200
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,159 +0,0 @@
-%
-\begin{isabellebody}%
-\def\isabellecontext{Examples{\isadigit{1}}}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Examples{\isadigit{1}}\isanewline
-\isakeyword{imports}\ Examples\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-\vspace{-5ex}%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsection{Use of Locales in Theories and Proofs
-  \label{sec:interpretation}%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-Locales can be interpreted in the contexts of theories and
-  structured proofs.  These interpretations are dynamic, too.
-  Conclusions of locales will be propagated to the current theory or
-  the current proof context.%
-\footnote{Strictly speaking, only interpretation in theories is
-  dynamic since it is not possible to change locales or the locale
-  hierarchy from within a proof.}
-  The focus of this section is on
-  interpretation in theories, but we will also encounter
-  interpretations in proofs, in
-  Section~\ref{sec:local-interpretation}.
-
-  As an example, consider the type of integers \isa{int}.  The
-  relation \isa{op\ {\isaliteral{5C3C6C653E}{\isasymle}}} is a total order over \isa{int}.  We start
-  with the interpretation that \isa{op\ {\isaliteral{5C3C6C653E}{\isasymle}}} is a partial order.  The
-  facilities of the interpretation command are explored gradually in
-  three versions.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsubsection{First Version: Replacement of Parameters Only
-  \label{sec:po-first}%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-The command \isakeyword{interpretation} is for the interpretation of
-  locale in theories.  In the following example, the parameter of locale
-  \isa{partial{\isaliteral{5F}{\isacharunderscore}}order} is replaced by \isa{op\ {\isaliteral{5C3C6C653E}{\isasymle}}} and the locale instance is interpreted in the current
-  theory.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimvisible
-\ \ %
-\endisadelimvisible
-%
-\isatagvisible
-\isacommand{interpretation}\isamarkupfalse%
-\ int{\isaliteral{3A}{\isacharcolon}}\ partial{\isaliteral{5F}{\isacharunderscore}}order\ {\isaliteral{22}{\isachardoublequoteopen}}op\ {\isaliteral{5C3C6C653E}{\isasymle}}\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ int\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ int\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ bool{\isaliteral{22}{\isachardoublequoteclose}}%
-\begin{isamarkuptxt}%
-\normalsize
-  The argument of the command is a simple \emph{locale expression}
-  consisting of the name of the interpreted locale, which is
-  preceded by the qualifier \isa{int{\isaliteral{3A}{\isacharcolon}}} and succeeded by a
-  white-space-separated list of terms, which provide a full
-  instantiation of the locale parameters.  The parameters are referred
-  to by order of declaration, which is also the order in which
-  \isakeyword{print\_locale} outputs them.  The locale has only a
-  single parameter, hence the list of instantiation terms is a
-  singleton.
-
-  The command creates the goal
-  \begin{isabelle}%
-\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ partial{\isaliteral{5F}{\isacharunderscore}}order\ op\ {\isaliteral{5C3C6C653E}{\isasymle}}%
-\end{isabelle} which can be shown easily:%
-\end{isamarkuptxt}%
-\isamarkuptrue%
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ unfold{\isaliteral{5F}{\isacharunderscore}}locales\ auto%
-\endisatagvisible
-{\isafoldvisible}%
-%
-\isadelimvisible
-%
-\endisadelimvisible
-%
-\begin{isamarkuptext}%
-The effect of the command is that instances of all
-  conclusions of the locale are available in the theory, where names
-  are prefixed by the qualifier.  For example, transitivity for \isa{int} is named \isa{int{\isaliteral{2E}{\isachardot}}trans} and is the following
-  theorem:
-  \begin{isabelle}%
-\ \ {\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}{\isaliteral{3F}{\isacharquery}}x\ {\isaliteral{5C3C6C653E}{\isasymle}}\ {\isaliteral{3F}{\isacharquery}}y{\isaliteral{3B}{\isacharsemicolon}}\ {\isaliteral{3F}{\isacharquery}}y\ {\isaliteral{5C3C6C653E}{\isasymle}}\ {\isaliteral{3F}{\isacharquery}}z{\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{3F}{\isacharquery}}x\ {\isaliteral{5C3C6C653E}{\isasymle}}\ {\isaliteral{3F}{\isacharquery}}z%
-\end{isabelle}
-  It is not possible to reference this theorem simply as \isa{trans}.  This prevents unwanted hiding of existing theorems of the
-  theory by an interpretation.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsubsection{Second Version: Replacement of Definitions%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-Not only does the above interpretation qualify theorem names.
-  The prefix \isa{int} is applied to all names introduced in locale
-  conclusions including names introduced in definitions.  The
-  qualified name \isa{int{\isaliteral{2E}{\isachardot}}less} is short for
-  the interpretation of the definition, which is \isa{partial{\isaliteral{5F}{\isacharunderscore}}order{\isaliteral{2E}{\isachardot}}less\ op\ {\isaliteral{5C3C6C653E}{\isasymle}}}.
-  Qualified name and expanded form may be used almost
-  interchangeably.%
-\footnote{Since \isa{op\ {\isaliteral{5C3C6C653E}{\isasymle}}} is polymorphic, for \isa{partial{\isaliteral{5F}{\isacharunderscore}}order{\isaliteral{2E}{\isachardot}}less\ op\ {\isaliteral{5C3C6C653E}{\isasymle}}} a
-  more general type will be inferred than for \isa{int{\isaliteral{2E}{\isachardot}}less} which
-  is over type \isa{int}.}
-  The latter is preferred on output, as for example in the theorem
-  \isa{int{\isaliteral{2E}{\isachardot}}less{\isaliteral{5F}{\isacharunderscore}}le{\isaliteral{5F}{\isacharunderscore}}trans}: \begin{isabelle}%
-\ \ {\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}partial{\isaliteral{5F}{\isacharunderscore}}order{\isaliteral{2E}{\isachardot}}less\ op\ {\isaliteral{5C3C6C653E}{\isasymle}}\ {\isaliteral{3F}{\isacharquery}}x\ {\isaliteral{3F}{\isacharquery}}y{\isaliteral{3B}{\isacharsemicolon}}\ {\isaliteral{3F}{\isacharquery}}y\ {\isaliteral{5C3C6C653E}{\isasymle}}\ {\isaliteral{3F}{\isacharquery}}z{\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\isanewline
-\isaindent{\ \ }{\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ partial{\isaliteral{5F}{\isacharunderscore}}order{\isaliteral{2E}{\isachardot}}less\ op\ {\isaliteral{5C3C6C653E}{\isasymle}}\ {\isaliteral{3F}{\isacharquery}}x\ {\isaliteral{3F}{\isacharquery}}z%
-\end{isabelle}
-  Both notations for the strict order are not satisfactory.  The
-  constant \isa{op\ {\isaliteral{3C}{\isacharless}}} is the strict order for \isa{int}.
-  In order to allow for the desired replacement, interpretation
-  accepts \emph{equations} in addition to the parameter instantiation.
-  These follow the locale expression and are indicated with the
-  keyword \isakeyword{where}.  This is the revised interpretation:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-\isanewline
-\end{isabellebody}%
-%%% Local Variables:
-%%% mode: latex
-%%% TeX-master: "root"
-%%% End:

--- a/doc-src/Locales/Locales/document/Examples2.tex	Mon Aug 27 21:04:37 2012 +0200
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,93 +0,0 @@
-%
-\begin{isabellebody}%
-\def\isabellecontext{Examples{\isadigit{2}}}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Examples{\isadigit{2}}\isanewline
-\isakeyword{imports}\ Examples\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-\vspace{-5ex}%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimvisible
-\ \ %
-\endisadelimvisible
-%
-\isatagvisible
-\isacommand{interpretation}\isamarkupfalse%
-\ int{\isaliteral{3A}{\isacharcolon}}\ partial{\isaliteral{5F}{\isacharunderscore}}order\ {\isaliteral{22}{\isachardoublequoteopen}}op\ {\isaliteral{5C3C6C653E}{\isasymle}}\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{5B}{\isacharbrackleft}}int{\isaliteral{2C}{\isacharcomma}}\ int{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ bool{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \isakeyword{where}\ {\isaliteral{22}{\isachardoublequoteopen}}int{\isaliteral{2E}{\isachardot}}less\ x\ y\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}x\ {\isaliteral{3C}{\isacharless}}\ y{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isaliteral{2D}{\isacharminus}}%
-\begin{isamarkuptxt}%
-\normalsize The goals are now:
-      \begin{isabelle}%
-\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ partial{\isaliteral{5F}{\isacharunderscore}}order\ op\ {\isaliteral{5C3C6C653E}{\isasymle}}\isanewline
-\ {\isadigit{2}}{\isaliteral{2E}{\isachardot}}\ partial{\isaliteral{5F}{\isacharunderscore}}order{\isaliteral{2E}{\isachardot}}less\ op\ {\isaliteral{5C3C6C653E}{\isasymle}}\ x\ y\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}x\ {\isaliteral{3C}{\isacharless}}\ y{\isaliteral{29}{\isacharparenright}}%
-\end{isabelle}
-      The proof that~\isa{{\isaliteral{5C3C6C653E}{\isasymle}}} is a partial order is as above.%
-\end{isamarkuptxt}%
-\isamarkuptrue%
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}partial{\isaliteral{5F}{\isacharunderscore}}order\ {\isaliteral{28}{\isacharparenleft}}op\ {\isaliteral{5C3C6C653E}{\isasymle}}\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ int\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ int\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ bool{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ unfold{\isaliteral{5F}{\isacharunderscore}}locales\ auto%
-\begin{isamarkuptxt}%
-\normalsize The second goal is shown by unfolding the
-      definition of \isa{partial{\isaliteral{5F}{\isacharunderscore}}order{\isaliteral{2E}{\isachardot}}less}.%
-\end{isamarkuptxt}%
-\isamarkuptrue%
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}partial{\isaliteral{5F}{\isacharunderscore}}order{\isaliteral{2E}{\isachardot}}less\ op\ {\isaliteral{5C3C6C653E}{\isasymle}}\ x\ y\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}x\ {\isaliteral{3C}{\isacharless}}\ y{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ partial{\isaliteral{5F}{\isacharunderscore}}order{\isaliteral{2E}{\isachardot}}less{\isaliteral{5F}{\isacharunderscore}}def\ {\isaliteral{5B}{\isacharbrackleft}}OF\ {\isaliteral{60}{\isacharbackquoteopen}}partial{\isaliteral{5F}{\isacharunderscore}}order\ op\ {\isaliteral{5C3C6C653E}{\isasymle}}{\isaliteral{60}{\isacharbackquoteclose}}{\isaliteral{5D}{\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 the
-  interpreted locale.  Hence, the premise of \isa{partial{\isaliteral{5F}{\isacharunderscore}}order{\isaliteral{2E}{\isachardot}}less{\isaliteral{5F}{\isacharunderscore}}def} is discharged manually with \isa{OF}.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-\isanewline
-\end{isabellebody}%
-%%% Local Variables:
-%%% mode: latex
-%%% TeX-master: "root"
-%%% End:

--- a/doc-src/Locales/Locales/document/Examples3.tex	Mon Aug 27 21:04:37 2012 +0200
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,975 +0,0 @@
-%
-\begin{isabellebody}%
-\def\isabellecontext{Examples{\isadigit{3}}}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Examples{\isadigit{3}}\isanewline
-\isakeyword{imports}\ Examples\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-\vspace{-5ex}%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsubsection{Third Version: Local Interpretation
-  \label{sec:local-interpretation}%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-In the above example, the fact that \isa{op\ {\isaliteral{5C3C6C653E}{\isasymle}}} is a partial
-  order for the integers was used in the second goal to
-  discharge the premise in the definition of \isa{op\ {\isaliteral{5C3C73717375627365743E}{\isasymsqsubset}}}.  In
-  general, proofs of the equations not only may involve definitions
-  from the interpreted locale but arbitrarily complex arguments in the
-  context of the locale.  Therefore it would be convenient to have the
-  interpreted locale conclusions temporarily available in the proof.
-  This can be achieved by a locale interpretation in the proof body.
-  The command for local interpretations is \isakeyword{interpret}.  We
-  repeat the example from the previous section to illustrate this.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimvisible
-\ \ %
-\endisadelimvisible
-%
-\isatagvisible
-\isacommand{interpretation}\isamarkupfalse%
-\ int{\isaliteral{3A}{\isacharcolon}}\ partial{\isaliteral{5F}{\isacharunderscore}}order\ {\isaliteral{22}{\isachardoublequoteopen}}op\ {\isaliteral{5C3C6C653E}{\isasymle}}\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ int\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ int\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ bool{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \isakeyword{where}\ {\isaliteral{22}{\isachardoublequoteopen}}int{\isaliteral{2E}{\isachardot}}less\ x\ y\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}x\ {\isaliteral{3C}{\isacharless}}\ y{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isaliteral{2D}{\isacharminus}}\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}partial{\isaliteral{5F}{\isacharunderscore}}order\ {\isaliteral{28}{\isacharparenleft}}op\ {\isaliteral{5C3C6C653E}{\isasymle}}\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ int\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ int\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ bool{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ unfold{\isaliteral{5F}{\isacharunderscore}}locales\ auto\isanewline
-\ \ \ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{interpret}\isamarkupfalse%
-\ int{\isaliteral{3A}{\isacharcolon}}\ partial{\isaliteral{5F}{\isacharunderscore}}order\ {\isaliteral{22}{\isachardoublequoteopen}}op\ {\isaliteral{5C3C6C653E}{\isasymle}}\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{5B}{\isacharbrackleft}}int{\isaliteral{2C}{\isacharcomma}}\ int{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ bool{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{{\isaliteral{2E}{\isachardot}}}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}int{\isaliteral{2E}{\isachardot}}less\ x\ y\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}x\ {\isaliteral{3C}{\isacharless}}\ y{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ int{\isaliteral{2E}{\isachardot}}less{\isaliteral{5F}{\isacharunderscore}}def\ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-%
-\endisatagvisible
-{\isafoldvisible}%
-%
-\isadelimvisible
-%
-\endisadelimvisible
-%
-\begin{isamarkuptext}%
-The inner interpretation is immediate from the preceding fact
-  and proved by assumption (Isar short hand ``.'').  It enriches the
-  local proof context by the theorems
-  also obtained in the interpretation from Section~\ref{sec:po-first},
-  and \isa{int{\isaliteral{2E}{\isachardot}}less{\isaliteral{5F}{\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 succeeding
-  \isakeyword{next} or \isakeyword{qed} statement.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsubsection{Further Interpretations%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-Further interpretations are necessary for
-  the other locales.  In \isa{lattice} the operations~\isa{{\isaliteral{5C3C7371696E7465723E}{\isasymsqinter}}}
-  and~\isa{{\isaliteral{5C3C7371756E696F6E3E}{\isasymsqunion}}} are substituted by \isa{min}
-  and \isa{max}.  The entire proof for the
-  interpretation is reproduced to give an example of a more
-  elaborate interpretation proof.  Note that the equations are named
-  so they can be used in a later example.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimvisible
-\ \ %
-\endisadelimvisible
-%
-\isatagvisible
-\isacommand{interpretation}\isamarkupfalse%
-\ int{\isaliteral{3A}{\isacharcolon}}\ lattice\ {\isaliteral{22}{\isachardoublequoteopen}}op\ {\isaliteral{5C3C6C653E}{\isasymle}}\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ int\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ int\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ bool{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \isakeyword{where}\ int{\isaliteral{5F}{\isacharunderscore}}min{\isaliteral{5F}{\isacharunderscore}}eq{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}int{\isaliteral{2E}{\isachardot}}meet\ x\ y\ {\isaliteral{3D}{\isacharequal}}\ min\ x\ y{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \ \ \isakeyword{and}\ int{\isaliteral{5F}{\isacharunderscore}}max{\isaliteral{5F}{\isacharunderscore}}eq{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}int{\isaliteral{2E}{\isachardot}}join\ x\ y\ {\isaliteral{3D}{\isacharequal}}\ max\ x\ y{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isaliteral{2D}{\isacharminus}}\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}lattice\ {\isaliteral{28}{\isacharparenleft}}op\ {\isaliteral{5C3C6C653E}{\isasymle}}\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ int\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ int\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ bool{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}%
-\begin{isamarkuptxt}%
-\normalsize We have already shown that this is a partial
-        order,%
-\end{isamarkuptxt}%
-\isamarkuptrue%
-\ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ unfold{\isaliteral{5F}{\isacharunderscore}}locales%
-\begin{isamarkuptxt}%
-\normalsize hence only the lattice axioms remain to be
-        shown.
-        \begin{isabelle}%
-\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}x\ y{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C6578697374733E}{\isasymexists}}inf{\isaliteral{2E}{\isachardot}}\ partial{\isaliteral{5F}{\isacharunderscore}}order{\isaliteral{2E}{\isachardot}}is{\isaliteral{5F}{\isacharunderscore}}inf\ op\ {\isaliteral{5C3C6C653E}{\isasymle}}\ x\ y\ inf\isanewline
-\ {\isadigit{2}}{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}x\ y{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C6578697374733E}{\isasymexists}}sup{\isaliteral{2E}{\isachardot}}\ partial{\isaliteral{5F}{\isacharunderscore}}order{\isaliteral{2E}{\isachardot}}is{\isaliteral{5F}{\isacharunderscore}}sup\ op\ {\isaliteral{5C3C6C653E}{\isasymle}}\ x\ y\ sup%
-\end{isabelle}
-        By \isa{is{\isaliteral{5F}{\isacharunderscore}}inf} and \isa{is{\isaliteral{5F}{\isacharunderscore}}sup},%
-\end{isamarkuptxt}%
-\isamarkuptrue%
-\ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isaliteral{28}{\isacharparenleft}}unfold\ int{\isaliteral{2E}{\isachardot}}is{\isaliteral{5F}{\isacharunderscore}}inf{\isaliteral{5F}{\isacharunderscore}}def\ int{\isaliteral{2E}{\isachardot}}is{\isaliteral{5F}{\isacharunderscore}}sup{\isaliteral{5F}{\isacharunderscore}}def{\isaliteral{29}{\isacharparenright}}%
-\begin{isamarkuptxt}%
-\normalsize the goals are transformed to these
-        statements:
-        \begin{isabelle}%
-\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}x\ y{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C6578697374733E}{\isasymexists}}inf{\isaliteral{5C3C6C653E}{\isasymle}}x{\isaliteral{2E}{\isachardot}}\ inf\ {\isaliteral{5C3C6C653E}{\isasymle}}\ y\ {\isaliteral{5C3C616E643E}{\isasymand}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}z{\isaliteral{2E}{\isachardot}}\ z\ {\isaliteral{5C3C6C653E}{\isasymle}}\ x\ {\isaliteral{5C3C616E643E}{\isasymand}}\ z\ {\isaliteral{5C3C6C653E}{\isasymle}}\ y\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ z\ {\isaliteral{5C3C6C653E}{\isasymle}}\ inf{\isaliteral{29}{\isacharparenright}}\isanewline
-\ {\isadigit{2}}{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}x\ y{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C6578697374733E}{\isasymexists}}sup{\isaliteral{5C3C67653E}{\isasymge}}x{\isaliteral{2E}{\isachardot}}\ y\ {\isaliteral{5C3C6C653E}{\isasymle}}\ sup\ {\isaliteral{5C3C616E643E}{\isasymand}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}z{\isaliteral{2E}{\isachardot}}\ x\ {\isaliteral{5C3C6C653E}{\isasymle}}\ z\ {\isaliteral{5C3C616E643E}{\isasymand}}\ y\ {\isaliteral{5C3C6C653E}{\isasymle}}\ z\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ sup\ {\isaliteral{5C3C6C653E}{\isasymle}}\ z{\isaliteral{29}{\isacharparenright}}%
-\end{isabelle}
-        This is Presburger arithmetic, which can be solved by the
-        method \isa{arith}.%
-\end{isamarkuptxt}%
-\isamarkuptrue%
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ arith{\isaliteral{2B}{\isacharplus}}%
-\begin{isamarkuptxt}%
-\normalsize 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%
-\ int{\isaliteral{3A}{\isacharcolon}}\ lattice\ {\isaliteral{22}{\isachardoublequoteopen}}op\ {\isaliteral{5C3C6C653E}{\isasymle}}\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ int\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ int\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ bool{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{{\isaliteral{2E}{\isachardot}}}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}int{\isaliteral{2E}{\isachardot}}meet\ x\ y\ {\isaliteral{3D}{\isacharequal}}\ min\ x\ y{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isaliteral{28}{\isacharparenleft}}bestsimp\ simp{\isaliteral{3A}{\isacharcolon}}\ int{\isaliteral{2E}{\isachardot}}meet{\isaliteral{5F}{\isacharunderscore}}def\ int{\isaliteral{2E}{\isachardot}}is{\isaliteral{5F}{\isacharunderscore}}inf{\isaliteral{5F}{\isacharunderscore}}def{\isaliteral{29}{\isacharparenright}}\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}int{\isaliteral{2E}{\isachardot}}join\ x\ y\ {\isaliteral{3D}{\isacharequal}}\ max\ x\ y{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isaliteral{28}{\isacharparenleft}}bestsimp\ simp{\isaliteral{3A}{\isacharcolon}}\ int{\isaliteral{2E}{\isachardot}}join{\isaliteral{5F}{\isacharunderscore}}def\ int{\isaliteral{2E}{\isachardot}}is{\isaliteral{5F}{\isacharunderscore}}sup{\isaliteral{5F}{\isacharunderscore}}def{\isaliteral{29}{\isacharparenright}}\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-%
-\endisatagvisible
-{\isafoldvisible}%
-%
-\isadelimvisible
-%
-\endisadelimvisible
-%
-\begin{isamarkuptext}%
-Next follows that \isa{op\ {\isaliteral{5C3C6C653E}{\isasymle}}} is a total order, again for
-  the integers.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimvisible
-\ \ %
-\endisadelimvisible
-%
-\isatagvisible
-\isacommand{interpretation}\isamarkupfalse%
-\ int{\isaliteral{3A}{\isacharcolon}}\ total{\isaliteral{5F}{\isacharunderscore}}order\ {\isaliteral{22}{\isachardoublequoteopen}}op\ {\isaliteral{5C3C6C653E}{\isasymle}}\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ int\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ int\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ bool{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ unfold{\isaliteral{5F}{\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:int-lattice}.  Two points are worth noting:
-
-\begin{table}
-\hrule
-\vspace{2ex}
-\begin{center}
-\begin{tabular}{l}
-  \isa{int{\isaliteral{2E}{\isachardot}}less{\isaliteral{5F}{\isacharunderscore}}def} from locale \isa{partial{\isaliteral{5F}{\isacharunderscore}}order}: \\
-  \quad \isa{{\isaliteral{28}{\isacharparenleft}}{\isaliteral{3F}{\isacharquery}}x\ {\isaliteral{3C}{\isacharless}}\ {\isaliteral{3F}{\isacharquery}}y{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{3F}{\isacharquery}}x\ {\isaliteral{5C3C6C653E}{\isasymle}}\ {\isaliteral{3F}{\isacharquery}}y\ {\isaliteral{5C3C616E643E}{\isasymand}}\ {\isaliteral{3F}{\isacharquery}}x\ {\isaliteral{5C3C6E6F7465713E}{\isasymnoteq}}\ {\isaliteral{3F}{\isacharquery}}y{\isaliteral{29}{\isacharparenright}}} \\
-  \isa{int{\isaliteral{2E}{\isachardot}}meet{\isaliteral{5F}{\isacharunderscore}}left} from locale \isa{lattice}: \\
-  \quad \isa{min\ {\isaliteral{3F}{\isacharquery}}x\ {\isaliteral{3F}{\isacharquery}}y\ {\isaliteral{5C3C6C653E}{\isasymle}}\ {\isaliteral{3F}{\isacharquery}}x} \\
-  \isa{int{\isaliteral{2E}{\isachardot}}join{\isaliteral{5F}{\isacharunderscore}}distr} from locale \isa{distrib{\isaliteral{5F}{\isacharunderscore}}lattice}: \\
-  \quad \isa{max\ {\isaliteral{3F}{\isacharquery}}x\ {\isaliteral{28}{\isacharparenleft}}min\ {\isaliteral{3F}{\isacharquery}}y\ {\isaliteral{3F}{\isacharquery}}z{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ min\ {\isaliteral{28}{\isacharparenleft}}max\ {\isaliteral{3F}{\isacharquery}}x\ {\isaliteral{3F}{\isacharquery}}y{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{28}{\isacharparenleft}}max\ {\isaliteral{3F}{\isacharquery}}x\ {\isaliteral{3F}{\isacharquery}}z{\isaliteral{29}{\isacharparenright}}} \\
-  \isa{int{\isaliteral{2E}{\isachardot}}less{\isaliteral{5F}{\isacharunderscore}}total} from locale \isa{total{\isaliteral{5F}{\isacharunderscore}}order}: \\
-  \quad \isa{{\isaliteral{3F}{\isacharquery}}x\ {\isaliteral{3C}{\isacharless}}\ {\isaliteral{3F}{\isacharquery}}y\ {\isaliteral{5C3C6F723E}{\isasymor}}\ {\isaliteral{3F}{\isacharquery}}x\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{3F}{\isacharquery}}y\ {\isaliteral{5C3C6F723E}{\isasymor}}\ {\isaliteral{3F}{\isacharquery}}y\ {\isaliteral{3C}{\isacharless}}\ {\isaliteral{3F}{\isacharquery}}x}
-\end{tabular}
-\end{center}
-\hrule
-\caption{Interpreted theorems for~\isa{{\isaliteral{5C3C6C653E}{\isasymle}}} on the integers.}
-\label{tab:int-lattice}
-\end{table}
-
-\begin{itemize}
-\item
-  Locale \isa{distrib{\isaliteral{5F}{\isacharunderscore}}lattice} was also interpreted.  Since the
-  locale hierarchy reflects that total orders are distributive
-  lattices, the interpretation of the latter was inserted
-  automatically with the interpretation of the former.  In general,
-  interpretation traverses the locale hierarchy upwards and interprets
-  all encountered locales, regardless whether imported or proved via
-  the \isakeyword{sublocale} command.  Existing interpretations are
-  skipped avoiding duplicate work.
-\item
-  The predicate \isa{op\ {\isaliteral{3C}{\isacharless}}} appears in theorem \isa{int{\isaliteral{2E}{\isachardot}}less{\isaliteral{5F}{\isacharunderscore}}total}
-  although an equation for the replacement of \isa{op\ {\isaliteral{5C3C73717375627365743E}{\isasymsqsubset}}} was only
-  given in the interpretation of \isa{partial{\isaliteral{5F}{\isacharunderscore}}order}.  The
-  interpretation equations are pushed downwards the hierarchy for
-  related interpretations --- that is, for interpretations that share
-  the instances of parameters they have in common.
-\end{itemize}%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-The interpretations for a locale $n$ within the current
-  theory may be inspected with \isakeyword{print\_interps}~$n$.  This
-  prints the list of instances of $n$, for which interpretations exist.
-  For example, \isakeyword{print\_interps} \isa{partial{\isaliteral{5F}{\isacharunderscore}}order}
-  outputs the following:
-\begin{small}
-\begin{alltt}
-  int! : partial_order "op \(\le\)"
-\end{alltt}
-\end{small}
-  Of course, there is only one interpretation.
-  The interpretation qualifier on the left is decorated with an
-  exclamation point.  This means that it is mandatory.  Qualifiers
-  can either be \emph{mandatory} or \emph{optional}, designated by
-  ``!'' or ``?'' respectively.  Mandatory qualifiers must occur in a
-  name reference while optional ones need not.  Mandatory qualifiers
-  prevent accidental hiding of names, while optional qualifiers can be
-  more convenient to use.  For \isakeyword{interpretation}, the
-  default is ``!''.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsection{Locale Expressions \label{sec:expressions}%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-A map~\isa{{\isaliteral{5C3C7068693E}{\isasymphi}}} between partial orders~\isa{{\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}} and~\isa{{\isaliteral{5C3C7072656365713E}{\isasympreceq}}}
-  is called order preserving if \isa{x\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ y} implies \isa{{\isaliteral{5C3C7068693E}{\isasymphi}}\ x\ {\isaliteral{5C3C7072656365713E}{\isasympreceq}}\ {\isaliteral{5C3C7068693E}{\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.
-
-  A locale for order preserving maps requires three parameters: \isa{le}~(\isakeyword{infixl}~\isa{{\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}}) and \isa{le{\isaliteral{27}{\isacharprime}}}~(\isakeyword{infixl}~\isa{{\isaliteral{5C3C7072656365713E}{\isasympreceq}}}) for the orders and~\isa{{\isaliteral{5C3C7068693E}{\isasymphi}}}
-  for the map.
-
-  In order to reuse the existing locale for partial orders, which has
-  the single parameter~\isa{le}, it must be imported twice, once
-  mapping its parameter to~\isa{le} from the new locale and once
-  to~\isa{le{\isaliteral{27}{\isacharprime}}}.  This can be achieved with a compound locale
-  expression.
-
-  In general, a locale expression is a sequence of \emph{locale instances}
-  separated by~``$\textbf{+}$'' and followed by a \isakeyword{for}
-  clause.
-  An instance has the following format:
-\begin{quote}
-  \textit{qualifier} \textbf{:} \textit{locale-name}
-  \textit{parameter-instantiation}
-\end{quote}
-  We have already seen locale instances as arguments to
-  \isakeyword{interpretation} in Section~\ref{sec:interpretation}.
-  As before, the qualifier serves to disambiguate names from
-  different instances of the same locale.  While in
-  \isakeyword{interpretation} qualifiers default to mandatory, in
-  import and in the \isakeyword{sublocale} command, they default to
-  optional.
-
-  Since the parameters~\isa{le} and~\isa{le{\isaliteral{27}{\isacharprime}}} are to be partial
-  orders, our locale for order preserving maps will import the these
-  instances:
-\begin{small}
-\begin{alltt}
-  le: partial_order le
-  le': partial_order le'
-\end{alltt}
-\end{small}
-  For matter of convenience we choose to name parameter names and
-  qualifiers alike.  This is an arbitrary decision.  Technically, qualifiers
-  and parameters are unrelated.
-
-  Having determined the instances, let us turn to the \isakeyword{for}
-  clause.  It serves to declare locale parameters in the same way as
-  the context element \isakeyword{fixes} does.  Context elements can
-  only occur after the import section, and therefore the parameters
-  referred to in the instances must be declared in the \isakeyword{for}
-  clause.  The \isakeyword{for} clause is also where the syntax of these
-  parameters is declared.
-
-  Two context elements for the map parameter~\isa{{\isaliteral{5C3C7068693E}{\isasymphi}}} and the
-  assumptions that it is order preserving complete the locale
-  declaration.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\ \ \isacommand{locale}\isamarkupfalse%
-\ order{\isaliteral{5F}{\isacharunderscore}}preserving\ {\isaliteral{3D}{\isacharequal}}\isanewline
-\ \ \ \ le{\isaliteral{3A}{\isacharcolon}}\ partial{\isaliteral{5F}{\isacharunderscore}}order\ le\ {\isaliteral{2B}{\isacharplus}}\ le{\isaliteral{27}{\isacharprime}}{\isaliteral{3A}{\isacharcolon}}\ partial{\isaliteral{5F}{\isacharunderscore}}order\ le{\isaliteral{27}{\isacharprime}}\isanewline
-\ \ \ \ \ \ \isakeyword{for}\ le\ {\isaliteral{28}{\isacharparenleft}}\isakeyword{infixl}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}{\isaliteral{22}{\isachardoublequoteclose}}\ {\isadigit{5}}{\isadigit{0}}{\isaliteral{29}{\isacharparenright}}\ \isakeyword{and}\ le{\isaliteral{27}{\isacharprime}}\ {\isaliteral{28}{\isacharparenleft}}\isakeyword{infixl}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C7072656365713E}{\isasympreceq}}{\isaliteral{22}{\isachardoublequoteclose}}\ {\isadigit{5}}{\isadigit{0}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{2B}{\isacharplus}}\isanewline
-\ \ \ \ \isakeyword{fixes}\ {\isaliteral{5C3C7068693E}{\isasymphi}}\isanewline
-\ \ \ \ \isakeyword{assumes}\ hom{\isaliteral{5F}{\isacharunderscore}}le{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ y\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{5C3C7068693E}{\isasymphi}}\ x\ {\isaliteral{5C3C7072656365713E}{\isasympreceq}}\ {\isaliteral{5C3C7068693E}{\isasymphi}}\ y{\isaliteral{22}{\isachardoublequoteclose}}%
-\begin{isamarkuptext}%
-Here are examples of theorems that are
-  available in the locale:
-
-  \hspace*{1em}\isa{hom{\isaliteral{5F}{\isacharunderscore}}le}: \isa{{\isaliteral{3F}{\isacharquery}}x\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ {\isaliteral{3F}{\isacharquery}}y\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{5C3C7068693E}{\isasymphi}}\ {\isaliteral{3F}{\isacharquery}}x\ {\isaliteral{5C3C7072656365713E}{\isasympreceq}}\ {\isaliteral{5C3C7068693E}{\isasymphi}}\ {\isaliteral{3F}{\isacharquery}}y}
-
-  \hspace*{1em}\isa{le{\isaliteral{2E}{\isachardot}}less{\isaliteral{5F}{\isacharunderscore}}le{\isaliteral{5F}{\isacharunderscore}}trans}: \isa{{\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}{\isaliteral{3F}{\isacharquery}}x\ {\isaliteral{5C3C73717375627365743E}{\isasymsqsubset}}\ {\isaliteral{3F}{\isacharquery}}y{\isaliteral{3B}{\isacharsemicolon}}\ {\isaliteral{3F}{\isacharquery}}y\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ {\isaliteral{3F}{\isacharquery}}z{\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{3F}{\isacharquery}}x\ {\isaliteral{5C3C73717375627365743E}{\isasymsqsubset}}\ {\isaliteral{3F}{\isacharquery}}z}
-
-  \hspace*{1em}\isa{le{\isaliteral{27}{\isacharprime}}{\isaliteral{2E}{\isachardot}}less{\isaliteral{5F}{\isacharunderscore}}le{\isaliteral{5F}{\isacharunderscore}}trans}:
-  \begin{isabelle}%
-\ \ \ \ {\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}partial{\isaliteral{5F}{\isacharunderscore}}order{\isaliteral{2E}{\isachardot}}less\ op\ {\isaliteral{5C3C7072656365713E}{\isasympreceq}}\ {\isaliteral{3F}{\isacharquery}}x\ {\isaliteral{3F}{\isacharquery}}y{\isaliteral{3B}{\isacharsemicolon}}\ {\isaliteral{3F}{\isacharquery}}y\ {\isaliteral{5C3C7072656365713E}{\isasympreceq}}\ {\isaliteral{3F}{\isacharquery}}z{\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\isanewline
-\isaindent{\ \ \ \ }{\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ partial{\isaliteral{5F}{\isacharunderscore}}order{\isaliteral{2E}{\isachardot}}less\ op\ {\isaliteral{5C3C7072656365713E}{\isasympreceq}}\ {\isaliteral{3F}{\isacharquery}}x\ {\isaliteral{3F}{\isacharquery}}z%
-\end{isabelle}
-  While there is infix syntax for the strict operation associated to
-  \isa{op\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}}, there is none for the strict version of \isa{op\ {\isaliteral{5C3C7072656365713E}{\isasympreceq}}}.  The abbreviation \isa{less} with its infix syntax is only
-  available for the original instance it was declared for.  We may
-  introduce the abbreviation \isa{less{\isaliteral{27}{\isacharprime}}} with infix syntax~\isa{{\isaliteral{5C3C707265633E}{\isasymprec}}}
-  with the following declaration:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\ \ \isacommand{abbreviation}\isamarkupfalse%
-\ {\isaliteral{28}{\isacharparenleft}}\isakeyword{in}\ order{\isaliteral{5F}{\isacharunderscore}}preserving{\isaliteral{29}{\isacharparenright}}\isanewline
-\ \ \ \ less{\isaliteral{27}{\isacharprime}}\ {\isaliteral{28}{\isacharparenleft}}\isakeyword{infixl}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C707265633E}{\isasymprec}}{\isaliteral{22}{\isachardoublequoteclose}}\ {\isadigit{5}}{\isadigit{0}}{\isaliteral{29}{\isacharparenright}}\ \isakeyword{where}\ {\isaliteral{22}{\isachardoublequoteopen}}less{\isaliteral{27}{\isacharprime}}\ {\isaliteral{5C3C65717569763E}{\isasymequiv}}\ partial{\isaliteral{5F}{\isacharunderscore}}order{\isaliteral{2E}{\isachardot}}less\ le{\isaliteral{27}{\isacharprime}}{\isaliteral{22}{\isachardoublequoteclose}}%
-\begin{isamarkuptext}%
-Now the theorem is displayed nicely as
-  \isa{le{\isaliteral{27}{\isacharprime}}{\isaliteral{2E}{\isachardot}}less{\isaliteral{5F}{\isacharunderscore}}le{\isaliteral{5F}{\isacharunderscore}}trans}:
-  \begin{isabelle}%
-\ \ {\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}{\isaliteral{3F}{\isacharquery}}x\ {\isaliteral{5C3C707265633E}{\isasymprec}}\ {\isaliteral{3F}{\isacharquery}}y{\isaliteral{3B}{\isacharsemicolon}}\ {\isaliteral{3F}{\isacharquery}}y\ {\isaliteral{5C3C7072656365713E}{\isasympreceq}}\ {\isaliteral{3F}{\isacharquery}}z{\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{3F}{\isacharquery}}x\ {\isaliteral{5C3C707265633E}{\isasymprec}}\ {\isaliteral{3F}{\isacharquery}}z%
-\end{isabelle}%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-There are short notations for locale expressions.  These are
-  discussed in the following.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsubsection{Default Instantiations%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-It is possible to omit parameter instantiations.  The
-  instantiation then defaults to the name of
-  the parameter itself.  For example, the locale expression \isa{partial{\isaliteral{5F}{\isacharunderscore}}order} is short for \isa{partial{\isaliteral{5F}{\isacharunderscore}}order\ le}, since the
-  locale's single parameter is~\isa{le}.  We took advantage of this
-  in the \isakeyword{sublocale} declarations of
-  Section~\ref{sec:changing-the-hierarchy}.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsubsection{Implicit Parameters \label{sec:implicit-parameters}%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-In a locale expression that occurs within a locale
-  declaration, omitted parameters additionally extend the (possibly
-  empty) \isakeyword{for} clause.
-
-  The \isakeyword{for} clause is a general construct of Isabelle/Isar
-  to mark names occurring in the preceding declaration as ``arbitrary
-  but fixed''.  This is necessary for example, if the name is already
-  bound in a surrounding context.  In a locale expression, names
-  occurring in parameter instantiations should be bound by a
-  \isakeyword{for} clause whenever these names are not introduced
-  elsewhere in the context --- for example, on the left hand side of a
-  \isakeyword{sublocale} declaration.
-
-  There is an exception to this rule in locale declarations, where the
-  \isakeyword{for} clause serves to declare locale parameters.  Here,
-  locale parameters for which no parameter instantiation is given are
-  implicitly added, with their mixfix syntax, at the beginning of the
-  \isakeyword{for} clause.  For example, in a locale declaration, the
-  expression \isa{partial{\isaliteral{5F}{\isacharunderscore}}order} is short for
-\begin{small}
-\begin{alltt}
-  partial_order le \isakeyword{for} le (\isakeyword{infixl} "\(\sqsubseteq\)" 50)\textrm{.}
-\end{alltt}
-\end{small}
-  This short hand was used in the locale declarations throughout
-  Section~\ref{sec:import}.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-The following locale declarations provide more examples.  A
-  map~\isa{{\isaliteral{5C3C7068693E}{\isasymphi}}} is a lattice homomorphism if it preserves meet and
-  join.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\ \ \isacommand{locale}\isamarkupfalse%
-\ lattice{\isaliteral{5F}{\isacharunderscore}}hom\ {\isaliteral{3D}{\isacharequal}}\isanewline
-\ \ \ \ le{\isaliteral{3A}{\isacharcolon}}\ lattice\ {\isaliteral{2B}{\isacharplus}}\ le{\isaliteral{27}{\isacharprime}}{\isaliteral{3A}{\isacharcolon}}\ lattice\ le{\isaliteral{27}{\isacharprime}}\ \isakeyword{for}\ le{\isaliteral{27}{\isacharprime}}\ {\isaliteral{28}{\isacharparenleft}}\isakeyword{infixl}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C7072656365713E}{\isasympreceq}}{\isaliteral{22}{\isachardoublequoteclose}}\ {\isadigit{5}}{\isadigit{0}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{2B}{\isacharplus}}\isanewline
-\ \ \ \ \isakeyword{fixes}\ {\isaliteral{5C3C7068693E}{\isasymphi}}\isanewline
-\ \ \ \ \isakeyword{assumes}\ hom{\isaliteral{5F}{\isacharunderscore}}meet{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C7068693E}{\isasymphi}}\ {\isaliteral{28}{\isacharparenleft}}x\ {\isaliteral{5C3C7371696E7465723E}{\isasymsqinter}}\ y{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ le{\isaliteral{27}{\isacharprime}}{\isaliteral{2E}{\isachardot}}meet\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C7068693E}{\isasymphi}}\ x{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C7068693E}{\isasymphi}}\ y{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \ \ \isakeyword{and}\ hom{\isaliteral{5F}{\isacharunderscore}}join{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C7068693E}{\isasymphi}}\ {\isaliteral{28}{\isacharparenleft}}x\ {\isaliteral{5C3C7371756E696F6E3E}{\isasymsqunion}}\ y{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ le{\isaliteral{27}{\isacharprime}}{\isaliteral{2E}{\isachardot}}join\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C7068693E}{\isasymphi}}\ x{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C7068693E}{\isasymphi}}\ y{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}%
-\begin{isamarkuptext}%
-The parameter instantiation in the first instance of \isa{lattice} is omitted.  This causes the parameter~\isa{le} to be
-  added to the \isakeyword{for} clause, and the locale has
-  parameters~\isa{le},~\isa{le{\isaliteral{27}{\isacharprime}}} and, of course,~\isa{{\isaliteral{5C3C7068693E}{\isasymphi}}}.
-
-  Before turning to the second example, we complete the locale by
-  providing infix syntax for the meet and join operations of the
-  second lattice.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\ \ \isacommand{context}\isamarkupfalse%
-\ lattice{\isaliteral{5F}{\isacharunderscore}}hom\isanewline
-\ \ \isakeyword{begin}\isanewline
-\ \ \isacommand{abbreviation}\isamarkupfalse%
-\ meet{\isaliteral{27}{\isacharprime}}\ {\isaliteral{28}{\isacharparenleft}}\isakeyword{infixl}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C7371696E7465723E}{\isasymsqinter}}{\isaliteral{27}{\isacharprime}}{\isaliteral{27}{\isacharprime}}{\isaliteral{22}{\isachardoublequoteclose}}\ {\isadigit{5}}{\isadigit{0}}{\isaliteral{29}{\isacharparenright}}\ \isakeyword{where}\ {\isaliteral{22}{\isachardoublequoteopen}}meet{\isaliteral{27}{\isacharprime}}\ {\isaliteral{5C3C65717569763E}{\isasymequiv}}\ le{\isaliteral{27}{\isacharprime}}{\isaliteral{2E}{\isachardot}}meet{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \isacommand{abbreviation}\isamarkupfalse%
-\ join{\isaliteral{27}{\isacharprime}}\ {\isaliteral{28}{\isacharparenleft}}\isakeyword{infixl}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C7371756E696F6E3E}{\isasymsqunion}}{\isaliteral{27}{\isacharprime}}{\isaliteral{27}{\isacharprime}}{\isaliteral{22}{\isachardoublequoteclose}}\ {\isadigit{5}}{\isadigit{0}}{\isaliteral{29}{\isacharparenright}}\ \isakeyword{where}\ {\isaliteral{22}{\isachardoublequoteopen}}join{\isaliteral{27}{\isacharprime}}\ {\isaliteral{5C3C65717569763E}{\isasymequiv}}\ le{\isaliteral{27}{\isacharprime}}{\isaliteral{2E}{\isachardot}}join{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \isacommand{end}\isamarkupfalse%
-%
-\begin{isamarkuptext}%
-The next example makes radical use of the short hand
-  facilities.  A homomorphism is an endomorphism if both orders
-  coincide.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\ \ \isacommand{locale}\isamarkupfalse%
-\ lattice{\isaliteral{5F}{\isacharunderscore}}end\ {\isaliteral{3D}{\isacharequal}}\ lattice{\isaliteral{5F}{\isacharunderscore}}hom\ {\isaliteral{5F}{\isacharunderscore}}\ le%
-\begin{isamarkuptext}%
-The notation~\isa{{\isaliteral{5F}{\isacharunderscore}}} enables to omit a parameter in a
-  positional instantiation.  The omitted parameter,~\isa{le} becomes
-  the parameter of the declared locale and is, in the following
-  position, used to instantiate the second parameter of \isa{lattice{\isaliteral{5F}{\isacharunderscore}}hom}.  The effect is that of identifying the first in second
-  parameter of the homomorphism locale.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\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.  Parameter instantiations
-  are indicated by $\sqsubseteq \mapsto \preceq$ etc.  By looking at
-  the inheritance diagram it would seem
-  that two identical copies of each of the locales \isa{partial{\isaliteral{5F}{\isacharunderscore}}order} and \isa{lattice} are imported by \isa{lattice{\isaliteral{5F}{\isacharunderscore}}end}.  This is not the case!  Inheritance paths with
-  identical morphisms are automatically detected and
-  the conclusions of the respective locales appear only once.
-
-\begin{figure}
-\hrule \vspace{2ex}
-\begin{center}
-\begin{tikzpicture}
-  \node (o) at (0,0) {\isa{partial{\isaliteral{5F}{\isacharunderscore}}order}};
-  \node (oh) at (1.5,-2) {\isa{order{\isaliteral{5F}{\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{\isaliteral{5F}{\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{\isaliteral{5F}{\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{sublocale}\isamarkupfalse%
-\ lattice{\isaliteral{5F}{\isacharunderscore}}hom\ {\isaliteral{5C3C73756273657465713E}{\isasymsubseteq}}\ order{\isaliteral{5F}{\isacharunderscore}}preserving\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ unfold{\isaliteral{5F}{\isacharunderscore}}locales\isanewline
-\ \ \ \ \isacommand{fix}\isamarkupfalse%
-\ x\ y\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ y{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}y\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}x\ {\isaliteral{5C3C7371756E696F6E3E}{\isasymsqunion}}\ y{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{by}\isamarkupfalse%
-\ {\isaliteral{28}{\isacharparenleft}}simp\ add{\isaliteral{3A}{\isacharcolon}}\ le{\isaliteral{2E}{\isachardot}}join{\isaliteral{5F}{\isacharunderscore}}connection{\isaliteral{29}{\isacharparenright}}\isanewline
-\ \ \ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C7068693E}{\isasymphi}}\ y\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C7068693E}{\isasymphi}}\ x\ {\isaliteral{5C3C7371756E696F6E3E}{\isasymsqunion}}{\isaliteral{27}{\isacharprime}}\ {\isaliteral{5C3C7068693E}{\isasymphi}}\ y{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{by}\isamarkupfalse%
-\ {\isaliteral{28}{\isacharparenleft}}simp\ add{\isaliteral{3A}{\isacharcolon}}\ hom{\isaliteral{5F}{\isacharunderscore}}join\ {\isaliteral{5B}{\isacharbrackleft}}symmetric{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{29}{\isacharparenright}}\isanewline
-\ \ \ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C7068693E}{\isasymphi}}\ x\ {\isaliteral{5C3C7072656365713E}{\isasympreceq}}\ {\isaliteral{5C3C7068693E}{\isasymphi}}\ y{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{by}\isamarkupfalse%
-\ {\isaliteral{28}{\isacharparenleft}}simp\ add{\isaliteral{3A}{\isacharcolon}}\ le{\isaliteral{27}{\isacharprime}}{\isaliteral{2E}{\isachardot}}join{\isaliteral{5F}{\isacharunderscore}}connection{\isaliteral{29}{\isacharparenright}}\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-Theorems and other declarations --- syntax, in particular --- from
-  the locale \isa{order{\isaliteral{5F}{\isacharunderscore}}preserving} are now active in \isa{lattice{\isaliteral{5F}{\isacharunderscore}}hom}, for example
-  \isa{hom{\isaliteral{5F}{\isacharunderscore}}le}:
-  \begin{isabelle}%
-\ \ {\isaliteral{3F}{\isacharquery}}x\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ {\isaliteral{3F}{\isacharquery}}y\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{5C3C7068693E}{\isasymphi}}\ {\isaliteral{3F}{\isacharquery}}x\ {\isaliteral{5C3C7072656365713E}{\isasympreceq}}\ {\isaliteral{5C3C7068693E}{\isasymphi}}\ {\isaliteral{3F}{\isacharquery}}y%
-\end{isabelle}
-  This theorem will be useful in the following section.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsection{Conditional Interpretation%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-There are situations where an interpretation is not possible
-  in the general case since the desired property is only valid if
-  certain conditions are fulfilled.  Take, for example, the function
-  \isa{{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}i{\isaliteral{2E}{\isachardot}}\ n\ {\isaliteral{2A}{\isacharasterisk}}\ i} that scales its argument by a constant factor.
-  This function is order preserving (and even a lattice endomorphism)
-  with respect to \isa{op\ {\isaliteral{5C3C6C653E}{\isasymle}}} provided \isa{n\ {\isaliteral{5C3C67653E}{\isasymge}}\ {\isadigit{0}}}.
-
-  It is not possible to express this using a global interpretation,
-  because it is in general unspecified whether~\isa{n} is
-  non-negative, but one may make an interpretation in an inner context
-  of a proof where full information is available.
-  This is not fully satisfactory either, since potentially
-  interpretations may be required to make interpretations in many
-  contexts.  What is
-  required is an interpretation that depends on the condition --- and
-  this can be done with the \isakeyword{sublocale} command.  For this
-  purpose, we introduce a locale for the condition.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\ \ \isacommand{locale}\isamarkupfalse%
-\ non{\isaliteral{5F}{\isacharunderscore}}negative\ {\isaliteral{3D}{\isacharequal}}\isanewline
-\ \ \ \ \isakeyword{fixes}\ n\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ int\isanewline
-\ \ \ \ \isakeyword{assumes}\ non{\isaliteral{5F}{\isacharunderscore}}neg{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isadigit{0}}\ {\isaliteral{5C3C6C653E}{\isasymle}}\ n{\isaliteral{22}{\isachardoublequoteclose}}%
-\begin{isamarkuptext}%
-It is again convenient to make the interpretation in an
-  incremental fashion, first for order preserving maps, the for
-  lattice endomorphisms.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\ \ \isacommand{sublocale}\isamarkupfalse%
-\ non{\isaliteral{5F}{\isacharunderscore}}negative\ {\isaliteral{5C3C73756273657465713E}{\isasymsubseteq}}\isanewline
-\ \ \ \ \ \ order{\isaliteral{5F}{\isacharunderscore}}preserving\ {\isaliteral{22}{\isachardoublequoteopen}}op\ {\isaliteral{5C3C6C653E}{\isasymle}}{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{22}{\isachardoublequoteopen}}op\ {\isaliteral{5C3C6C653E}{\isasymle}}{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}i{\isaliteral{2E}{\isachardot}}\ n\ {\isaliteral{2A}{\isacharasterisk}}\ i{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-%
-\isadelimproof
-\ \ \ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ non{\isaliteral{5F}{\isacharunderscore}}neg\ \isacommand{by}\isamarkupfalse%
-\ unfold{\isaliteral{5F}{\isacharunderscore}}locales\ {\isaliteral{28}{\isacharparenleft}}rule\ mult{\isaliteral{5F}{\isacharunderscore}}left{\isaliteral{5F}{\isacharunderscore}}mono{\isaliteral{29}{\isacharparenright}}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-While the proof of the previous interpretation
-  is straightforward from monotonicity lemmas for~\isa{op\ {\isaliteral{2A}{\isacharasterisk}}}, the
-  second proof follows a useful pattern.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimvisible
-\ \ %
-\endisadelimvisible
-%
-\isatagvisible
-\isacommand{sublocale}\isamarkupfalse%
-\ non{\isaliteral{5F}{\isacharunderscore}}negative\ {\isaliteral{5C3C73756273657465713E}{\isasymsubseteq}}\ lattice{\isaliteral{5F}{\isacharunderscore}}end\ {\isaliteral{22}{\isachardoublequoteopen}}op\ {\isaliteral{5C3C6C653E}{\isasymle}}{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}i{\isaliteral{2E}{\isachardot}}\ n\ {\isaliteral{2A}{\isacharasterisk}}\ i{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isaliteral{28}{\isacharparenleft}}unfold{\isaliteral{5F}{\isacharunderscore}}locales{\isaliteral{2C}{\isacharcomma}}\ unfold\ int{\isaliteral{5F}{\isacharunderscore}}min{\isaliteral{5F}{\isacharunderscore}}eq\ int{\isaliteral{5F}{\isacharunderscore}}max{\isaliteral{5F}{\isacharunderscore}}eq{\isaliteral{29}{\isacharparenright}}%
-\begin{isamarkuptxt}%
-\normalsize Unfolding the locale predicates \emph{and} the
-      interpretation equations immediately yields two subgoals that
-      reflect the core conjecture.
-      \begin{isabelle}%
-\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}x\ y{\isaliteral{2E}{\isachardot}}\ n\ {\isaliteral{2A}{\isacharasterisk}}\ min\ x\ y\ {\isaliteral{3D}{\isacharequal}}\ min\ {\isaliteral{28}{\isacharparenleft}}n\ {\isaliteral{2A}{\isacharasterisk}}\ x{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{28}{\isacharparenleft}}n\ {\isaliteral{2A}{\isacharasterisk}}\ y{\isaliteral{29}{\isacharparenright}}\isanewline
-\ {\isadigit{2}}{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}x\ y{\isaliteral{2E}{\isachardot}}\ n\ {\isaliteral{2A}{\isacharasterisk}}\ max\ x\ y\ {\isaliteral{3D}{\isacharequal}}\ max\ {\isaliteral{28}{\isacharparenleft}}n\ {\isaliteral{2A}{\isacharasterisk}}\ x{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{28}{\isacharparenleft}}n\ {\isaliteral{2A}{\isacharasterisk}}\ y{\isaliteral{29}{\isacharparenright}}%
-\end{isabelle}
-      It is now necessary to show, in the context of \isa{non{\isaliteral{5F}{\isacharunderscore}}negative}, that multiplication by~\isa{n} commutes with
-      \isa{min} and \isa{max}.%
-\end{isamarkuptxt}%
-\isamarkuptrue%
-\ \ \isacommand{qed}\isamarkupfalse%
-\ {\isaliteral{28}{\isacharparenleft}}auto\ simp{\isaliteral{3A}{\isacharcolon}}\ hom{\isaliteral{5F}{\isacharunderscore}}le{\isaliteral{29}{\isacharparenright}}%
-\endisatagvisible
-{\isafoldvisible}%
-%
-\isadelimvisible
-%
-\endisadelimvisible
-%
-\begin{isamarkuptext}%
-The lemma \isa{hom{\isaliteral{5F}{\isacharunderscore}}le}
-  simplifies a proof that would have otherwise been lengthy and we may
-  consider making it a default rule for the simplifier:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\ \ \isacommand{lemmas}\isamarkupfalse%
-\ {\isaliteral{28}{\isacharparenleft}}\isakeyword{in}\ order{\isaliteral{5F}{\isacharunderscore}}preserving{\isaliteral{29}{\isacharparenright}}\ hom{\isaliteral{5F}{\isacharunderscore}}le\ {\isaliteral{5B}{\isacharbrackleft}}simp{\isaliteral{5D}{\isacharbrackright}}%
-\isamarkupsubsection{Avoiding Infinite Chains of Interpretations
-  \label{sec:infinite-chains}%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-Similar situations arise frequently in formalisations of
-  abstract algebra where it is desirable to express that certain
-  constructions preserve certain properties.  For example, polynomials
-  over rings are rings, or --- an example from the domain where the
-  illustrations of this tutorial are taken from --- a partial order
-  may be obtained for a function space by point-wise lifting of the
-  partial order of the co-domain.  This corresponds to the following
-  interpretation:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimvisible
-\ \ %
-\endisadelimvisible
-%
-\isatagvisible
-\isacommand{sublocale}\isamarkupfalse%
-\ partial{\isaliteral{5F}{\isacharunderscore}}order\ {\isaliteral{5C3C73756273657465713E}{\isasymsubseteq}}\ f{\isaliteral{3A}{\isacharcolon}}\ partial{\isaliteral{5F}{\isacharunderscore}}order\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}f\ g{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}x{\isaliteral{2E}{\isachardot}}\ f\ x\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ g\ x{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \isacommand{oops}\isamarkupfalse%
-%
-\endisatagvisible
-{\isafoldvisible}%
-%
-\isadelimvisible
-%
-\endisadelimvisible
-%
-\begin{isamarkuptext}%
-Unfortunately this is a cyclic interpretation that leads to an
-  infinite chain, namely
-  \begin{isabelle}%
-\ \ partial{\isaliteral{5F}{\isacharunderscore}}order\ {\isaliteral{5C3C73756273657465713E}{\isasymsubseteq}}\ partial{\isaliteral{5F}{\isacharunderscore}}order\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}f\ g{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}x{\isaliteral{2E}{\isachardot}}\ f\ x\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ g\ x{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C73756273657465713E}{\isasymsubseteq}}\isanewline
-\isaindent{\ \ }\ \ partial{\isaliteral{5F}{\isacharunderscore}}order\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}f\ g{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}x\ y{\isaliteral{2E}{\isachardot}}\ f\ x\ y\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ g\ x\ y{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C73756273657465713E}{\isasymsubseteq}}\ \ {\isaliteral{5C3C646F74733E}{\isasymdots}}%
-\end{isabelle}
-  and the interpretation is rejected.
-
-  Instead it is necessary to declare a locale that is logically
-  equivalent to \isa{partial{\isaliteral{5F}{\isacharunderscore}}order} but serves to collect facts
-  about functions spaces where the co-domain is a partial order, and
-  to make the interpretation in its context:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\ \ \isacommand{locale}\isamarkupfalse%
-\ fun{\isaliteral{5F}{\isacharunderscore}}partial{\isaliteral{5F}{\isacharunderscore}}order\ {\isaliteral{3D}{\isacharequal}}\ partial{\isaliteral{5F}{\isacharunderscore}}order\isanewline
-\isanewline
-\ \ \isacommand{sublocale}\isamarkupfalse%
-\ fun{\isaliteral{5F}{\isacharunderscore}}partial{\isaliteral{5F}{\isacharunderscore}}order\ {\isaliteral{5C3C73756273657465713E}{\isasymsubseteq}}\isanewline
-\ \ \ \ \ \ f{\isaliteral{3A}{\isacharcolon}}\ partial{\isaliteral{5F}{\isacharunderscore}}order\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}f\ g{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}x{\isaliteral{2E}{\isachardot}}\ f\ x\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ g\ x{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-%
-\isadelimproof
-\ \ \ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ unfold{\isaliteral{5F}{\isacharunderscore}}locales\ {\isaliteral{28}{\isacharparenleft}}fast{\isaliteral{2C}{\isacharcomma}}rule{\isaliteral{2C}{\isacharcomma}}fast{\isaliteral{2C}{\isacharcomma}}blast\ intro{\isaliteral{3A}{\isacharcolon}}\ trans{\isaliteral{29}{\isacharparenright}}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-It is quite common in abstract algebra that such a construction
-  maps a hierarchy of algebraic structures (or specifications) to a
-  related hierarchy.  By means of the same lifting, a function space
-  is a lattice if its co-domain is a lattice:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\ \ \isacommand{locale}\isamarkupfalse%
-\ fun{\isaliteral{5F}{\isacharunderscore}}lattice\ {\isaliteral{3D}{\isacharequal}}\ fun{\isaliteral{5F}{\isacharunderscore}}partial{\isaliteral{5F}{\isacharunderscore}}order\ {\isaliteral{2B}{\isacharplus}}\ lattice\isanewline
-\isanewline
-\ \ \isacommand{sublocale}\isamarkupfalse%
-\ fun{\isaliteral{5F}{\isacharunderscore}}lattice\ {\isaliteral{5C3C73756273657465713E}{\isasymsubseteq}}\ f{\isaliteral{3A}{\isacharcolon}}\ lattice\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}f\ g{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}x{\isaliteral{2E}{\isachardot}}\ f\ x\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ g\ x{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ unfold{\isaliteral{5F}{\isacharunderscore}}locales\isanewline
-\ \ \ \ \isacommand{fix}\isamarkupfalse%
-\ f\ g\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}partial{\isaliteral{5F}{\isacharunderscore}}order{\isaliteral{2E}{\isachardot}}is{\isaliteral{5F}{\isacharunderscore}}inf\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}f\ g{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}x{\isaliteral{2E}{\isachardot}}\ f\ x\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ g\ x{\isaliteral{29}{\isacharparenright}}\ f\ g\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}x{\isaliteral{2E}{\isachardot}}\ f\ x\ {\isaliteral{5C3C7371696E7465723E}{\isasymsqinter}}\ g\ x{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isaliteral{28}{\isacharparenleft}}rule\ is{\isaliteral{5F}{\isacharunderscore}}infI{\isaliteral{29}{\isacharparenright}}\ \isacommand{apply}\isamarkupfalse%
-\ rule{\isaliteral{2B}{\isacharplus}}\ \isacommand{apply}\isamarkupfalse%
-\ {\isaliteral{28}{\isacharparenleft}}drule\ spec{\isaliteral{2C}{\isacharcomma}}\ assumption{\isaliteral{29}{\isacharparenright}}{\isaliteral{2B}{\isacharplus}}\ \isacommand{done}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6578697374733E}{\isasymexists}}inf{\isaliteral{2E}{\isachardot}}\ partial{\isaliteral{5F}{\isacharunderscore}}order{\isaliteral{2E}{\isachardot}}is{\isaliteral{5F}{\isacharunderscore}}inf\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}f\ g{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}x{\isaliteral{2E}{\isachardot}}\ f\ x\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ g\ x{\isaliteral{29}{\isacharparenright}}\ f\ g\ inf{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fast\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{fix}\isamarkupfalse%
-\ f\ g\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}partial{\isaliteral{5F}{\isacharunderscore}}order{\isaliteral{2E}{\isachardot}}is{\isaliteral{5F}{\isacharunderscore}}sup\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}f\ g{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}x{\isaliteral{2E}{\isachardot}}\ f\ x\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ g\ x{\isaliteral{29}{\isacharparenright}}\ f\ g\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}x{\isaliteral{2E}{\isachardot}}\ f\ x\ {\isaliteral{5C3C7371756E696F6E3E}{\isasymsqunion}}\ g\ x{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isaliteral{28}{\isacharparenleft}}rule\ is{\isaliteral{5F}{\isacharunderscore}}supI{\isaliteral{29}{\isacharparenright}}\ \isacommand{apply}\isamarkupfalse%
-\ rule{\isaliteral{2B}{\isacharplus}}\ \isacommand{apply}\isamarkupfalse%
-\ {\isaliteral{28}{\isacharparenleft}}drule\ spec{\isaliteral{2C}{\isacharcomma}}\ assumption{\isaliteral{29}{\isacharparenright}}{\isaliteral{2B}{\isacharplus}}\ \isacommand{done}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6578697374733E}{\isasymexists}}sup{\isaliteral{2E}{\isachardot}}\ partial{\isaliteral{5F}{\isacharunderscore}}order{\isaliteral{2E}{\isachardot}}is{\isaliteral{5F}{\isacharunderscore}}sup\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}f\ g{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}x{\isaliteral{2E}{\isachardot}}\ f\ x\ {\isaliteral{5C3C737173756273657465713E}{\isasymsqsubseteq}}\ g\ x{\isaliteral{29}{\isacharparenright}}\ f\ g\ sup{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fast\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\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, I
-  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 ML and 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 locales reimplementation for Isabelle 2009 provides, among other
-  improvements, a clean integration with Isabelle/Isar's local theory
-  mechanisms, which are described in another paper by Haftmann and
-  Wenzel~\cite{HaftmannWenzel2009}.
-
-  The original work of Kamm\"uller on locales~\cite{KammullerEtAl1999}
-  may be of interest from a historical perspective.  My previous
-  report on locales and locale expressions~\cite{Ballarin2004a}
-  describes a simpler form of expressions than available now and is
-  outdated. The mathematical background on orders and lattices is
-  taken from Jacobson's textbook on algebra~\cite[Chapter~8]{Jacobson1985}.
-
-  The sources of this tutorial, which include all proofs, are
-  available with the Isabelle distribution at
-  \url{http://isabelle.in.tum.de}.%
-\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} \\
-  \textit{qualifier} & ::=
-  & \textit{name} [``\textbf{?}'' $|$ ``\textbf{!}''] \\[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{pos-insts} & ::=
-  & ( \textit{term} $|$ ``\textbf{\_}'' )$^*$ \\
-  \textit{named-insts} & ::=
-  & \textbf{where} \textit{name} ``\textbf{=}'' \textit{term}
-  ( \textbf{and} \textit{name} ``\textbf{=}'' \textit{term} )$^*$ \\
-  \textit{instance} & ::=
-  & [ \textit{qualifier} ``\textbf{:}'' ]
-    \textit{name} ( \textit{pos-insts} $|$ \textit{named-inst} ) \\
-  \textit{expression}  & ::= 
-  & \textit{instance} ( ``\textbf{+}'' \textit{instance} )$^*$
-    [ \textbf{for} \textit{fixes} ( \textbf{and} \textit{fixes} )$^*$ ] \\[2ex]
-
-  \multicolumn{3}{l}{Declaration of Locales} \\
-
-  \textit{locale} & ::=
-  & \textit{element}$^+$ \\
-  & | & \textit{expression} [ ``\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{equations} & ::= &  \textbf{where} \textit{equation} ( \textbf{and}
-    \textit{equation} )$^*$  \\
-  \textit{toplevel} & ::=
-  & \textbf{sublocale} \textit{name} ( ``$<$'' $|$
-    ``$\subseteq$'' ) \textit{expression} \textit{proof} \\
-  & |
-  & \textbf{interpretation}
-    \textit{expression} [ \textit{equations} ] \textit{proof} \\
-  & |
-  & \textbf{interpret}
-    \textit{expression} \textit{proof} \\[2ex]
-
-  \multicolumn{3}{l}{Diagnostics} \\
-
-  \textit{toplevel} & ::=
-  & \textbf{print\_locales} \\
-  & | & \textbf{print\_locale} [ ``\textbf{!}'' ] \textit{name} \\
-  & | & \textbf{print\_interps} \textit{name}
-\end{tabular}
-\end{center}
-\hrule
-\caption{Syntax of Locale Commands.}
-\label{tab:commands}
-\end{table}%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-\textbf{Revision History.}  For the present third revision of
-  the tutorial, much of the explanatory text
-  was rewritten.  Inheritance of interpretation equations is
-  available with the forthcoming release of Isabelle, which at the
-  time of editing these notes is expected for the end of 2009.
-  The second revision accommodates changes introduced by the locales
-  reimplementation for Isabelle 2009.  Most notably locale expressions
-  have been generalised from renaming to instantiation.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-\textbf{Acknowledgements.}  Alexander Krauss, Tobias Nipkow,
-  Randy Pollack, Andreas Schropp, Christian Sternagel and Makarius Wenzel
-  have made
-  useful comments on earlier versions of this document.  The section
-  on conditional interpretation was inspired by a number of e-mail
-  enquiries the author received from locale users, and which suggested
-  that this use case is important enough to deserve explicit
-  explanation.  The term \emph{conditional interpretation} is due to
-  Larry Paulson.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-\isanewline
-\end{isabellebody}%
-%%% Local Variables:
-%%% mode: latex
-%%% TeX-master: "root"
-%%% End:

--- a/doc-src/Locales/Locales/document/root.bib	Mon Aug 27 21:04:37 2012 +0200
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,117 +0,0 @@
-@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}."
-}
-
-@book {Jacobson1985,
-  author = "Nathan Jacobson",
-  title = "Basic Algebra",
-  volume = "I",
-  publisher = "Freeman",
-  edition = "2nd",
-  year = 1985,
-  available = { CB }
-}
-
-% TYPES 2006
-
-@inproceedings{HaftmannWenzel2007,
-  author = "Florian Haftmann and Makarius Wenzel",
-  title = "Constructive Type Classes in {Isabelle}",
-  pages = "160--174",
-  crossref = "AltenkirchMcBride2007",
-  available = { CB }
-}
-
-@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 4502",
-  year = 2007
-}
-
-
-@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
-}
-
-% TYPES 2003
-
-@inproceedings{Ballarin2004a,
-  author = "Clemens Ballarin",
-  title = "Locales and Locale Expressions in {Isabelle/Isar}",
-  pages = "34--50",
-  crossref = "BerardiEtAl2004"
-}
-
-@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
-}
-
-% TYPES 2008
-
-@inproceedings{HaftmannWenzel2009,
-  author = "Florian Haftmann and Makarius Wenzel",
-  title = "Local theory specifications in {Isabelle}/{Isar}",
-  pages = "153--168",
-  crossref = "BerardiEtAl2009"
-}
-
-@proceedings{BerardiEtAl2009,
-  editor = "Stefano Berardi and Ferruccio Damiani and Ugo de Liguoro",
-  title = "Types for Proofs and Programs, TYPES 2008, Torino, Italy",
-  booktitle = "Types for Proofs and Programs, TYPES 2008, Torino, Italy",
-  series = "LNCS 5497",
-  publisher = "Springer",
-  year = 2009
-}
-
-% MKM 2006
-
-@inproceedings{Ballarin2006b,
-  author = "Clemens Ballarin",
-  title = "Interpretation of Locales in {Isabelle}: Theories and Proof Contexts",
-  pages = "31--43",
-  crossref = "BorweinFarmer2006"
-}
-
-@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 }
-}
-
-% TPHOLs 1999
-
-@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 }
-}
-
-@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
-}

--- a/doc-src/Locales/Locales/document/root.tex	Mon Aug 27 21:04:37 2012 +0200
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,62 +0,0 @@
-\documentclass[11pt,a4paper]{article}
-\usepackage{amsmath}
-\usepackage{../../../../lib/texinputs/isabelle,../../../../lib/texinputs/isabellesym}
-\usepackage{verbatim}
-\usepackage{alltt}
-\usepackage{array}
-
-\usepackage{amssymb}
-
-\usepackage{../../../pdfsetup}
-
-\usepackage{ifpdf}
-\ifpdf\relax\else\def\pgfsysdriver{pgfsys-dvi.def}\fi
-\usepackage{tikz}
-\usepackage{subfigure}
-
-\isadroptag{theory}
-\isafoldtag{proof}
-
-% urls in roman style, theory text in typewriter
-\urlstyle{rm}
-\isabellestyle{tt}
-
-
-\begin{document}
-
-\title{Tutorial to Locales and Locale Interpretation%
-\thanks{Published in L.~Lamb\'an, A.~Romero, J.~Rubio, editors, {\em Contribuciones Cient\'{\i}ficas en honor de Mirian Andr\'es.}  Servicio de Publicaciones de la Universidad de La Rioja, Logro\~no, Spain, 2010.  Reproduced by permission.}}
-\author{Clemens Ballarin}
-\date{}
-
-\maketitle
-
-\begin{abstract}
-  Locales are Isabelle's approach for dealing with parametric
-  theories.  They have been designed as a module system for a
-  theorem prover that can adequately represent the complex
-  inter-dependencies between structures found in abstract algebra, but
-  have proven fruitful also in other applications --- for example,
-  software verification.
-
-  Both design and implementation of locales have evolved considerably
-  since Kamm\"uller did his initial experiments.  Today, locales
-  are a simple yet powerful extension of the Isar proof language.
-  The present tutorial covers all major facilities of locales.  It is
-  intended for locale novices; familiarity with Isabelle and Isar is
-  presumed.
-\end{abstract}
-
-\parindent 0pt\parskip 0.5ex
-
-\input{session}
-
-\bibliographystyle{abbrv}
-\bibliography{root}
-
-\end{document}
-
-%%% Local Variables:
-%%% mode: latex
-%%% TeX-master: t
-%%% End:

--- a/doc-src/Locales/Locales/document/session.tex	Mon Aug 27 21:04:37 2012 +0200
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,12 +0,0 @@
-\input{Examples.tex}
-
-\input{Examples1.tex}
-
-\input{Examples2.tex}
-
-\input{Examples3.tex}
-
-%%% Local Variables:
-%%% mode: latex
-%%% TeX-master: "root"
-%%% End:

--- a/doc-src/Locales/Makefile	Mon Aug 27 21:04:37 2012 +0200
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,39 +0,0 @@
-
-## targets
-
-default: dvi
-
-
-## dependencies
-
-include ../Makefile.in
-
-NAME = locales
-
-FILES = Locales/document/root.tex Locales/document/root.bib \
-  Locales/document/session.tex Locales/document/Examples.tex \
-  Locales/document/Examples1.tex Locales/document/Examples2.tex \
-  Locales/document/Examples3.tex \
-  ../../lib/texinputs/isabelle.sty ../../lib/texinputs/isabellesym.sty ../pdfsetup.sty 
-
-dvi: $(NAME).dvi
-
-$(NAME).dvi: $(FILES)
-	cd Locales/document && \
-	$(LATEX) root && \
-	$(BIBTEX) root && \
-	$(LATEX) root && \
-	$(LATEX) root && \
-	$(LATEX) root
-	mv Locales/document/root.dvi $(NAME).dvi
-
-pdf: $(NAME).pdf
-
-$(NAME).pdf: $(FILES)
-	cd Locales/document && \
-	$(PDFLATEX) root && \
-	$(BIBTEX) root && \
-	$(PDFLATEX) root && \
-	$(PDFLATEX) root && \
-	$(PDFLATEX) root
-	mv Locales/document/root.pdf $(NAME).pdf

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/Locales/document/build	Mon Aug 27 21:19:16 2012 +0200
@@ -0,0 +1,16 @@
+#!/bin/bash
+
+set -e
+
+FORMAT="$1"
+VARIANT="$2"
+
+"$ISABELLE_TOOL" latex -o sty
+"$ISABELLE_TOOL" latex -o "$FORMAT"
+"$ISABELLE_TOOL" latex -o bbl
+"$ISABELLE_TOOL" latex -o "$FORMAT"
+"$ISABELLE_TOOL" latex -o "$FORMAT"
+"$ISABELLE_HOME/doc-src/sedindex" root
+[ -f root.out ] && "$ISABELLE_HOME/doc-src/fixbookmarks" root.out
+"$ISABELLE_TOOL" latex -o "$FORMAT"
+"$ISABELLE_TOOL" latex -o "$FORMAT"

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/Locales/document/root.bib	Mon Aug 27 21:19:16 2012 +0200
@@ -0,0 +1,117 @@
+@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}."
+}
+
+@book {Jacobson1985,
+  author = "Nathan Jacobson",
+  title = "Basic Algebra",
+  volume = "I",
+  publisher = "Freeman",
+  edition = "2nd",
+  year = 1985,
+  available = { CB }
+}
+
+% TYPES 2006
+
+@inproceedings{HaftmannWenzel2007,
+  author = "Florian Haftmann and Makarius Wenzel",
+  title = "Constructive Type Classes in {Isabelle}",
+  pages = "160--174",
+  crossref = "AltenkirchMcBride2007",
+  available = { CB }
+}
+
+@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 4502",
+  year = 2007
+}
+
+
+@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
+}
+
+% TYPES 2003
+
+@inproceedings{Ballarin2004a,
+  author = "Clemens Ballarin",
+  title = "Locales and Locale Expressions in {Isabelle/Isar}",
+  pages = "34--50",
+  crossref = "BerardiEtAl2004"
+}
+
+@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
+}
+
+% TYPES 2008
+
+@inproceedings{HaftmannWenzel2009,
+  author = "Florian Haftmann and Makarius Wenzel",
+  title = "Local theory specifications in {Isabelle}/{Isar}",
+  pages = "153--168",
+  crossref = "BerardiEtAl2009"
+}
+
+@proceedings{BerardiEtAl2009,
+  editor = "Stefano Berardi and Ferruccio Damiani and Ugo de Liguoro",
+  title = "Types for Proofs and Programs, TYPES 2008, Torino, Italy",
+  booktitle = "Types for Proofs and Programs, TYPES 2008, Torino, Italy",
+  series = "LNCS 5497",
+  publisher = "Springer",
+  year = 2009
+}
+
+% MKM 2006
+
+@inproceedings{Ballarin2006b,
+  author = "Clemens Ballarin",
+  title = "Interpretation of Locales in {Isabelle}: Theories and Proof Contexts",
+  pages = "31--43",
+  crossref = "BorweinFarmer2006"
+}
+
+@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 }
+}
+
+% TPHOLs 1999
+
+@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 }
+}
+
+@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
+}

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/Locales/document/root.tex	Mon Aug 27 21:19:16 2012 +0200
@@ -0,0 +1,62 @@
+\documentclass[11pt,a4paper]{article}
+\usepackage{amsmath}
+\usepackage{isabelle,isabellesym}
+\usepackage{verbatim}
+\usepackage{alltt}
+\usepackage{array}
+
+\usepackage{amssymb}
+
+\usepackage{pdfsetup}
+
+\usepackage{ifpdf}
+\ifpdf\relax\else\def\pgfsysdriver{pgfsys-dvi.def}\fi
+\usepackage{tikz}
+\usepackage{subfigure}
+
+\isadroptag{theory}
+\isafoldtag{proof}
+
+% urls in roman style, theory text in typewriter
+\urlstyle{rm}
+\isabellestyle{tt}
+
+
+\begin{document}
+
+\title{Tutorial to Locales and Locale Interpretation%
+\thanks{Published in L.~Lamb\'an, A.~Romero, J.~Rubio, editors, {\em Contribuciones Cient\'{\i}ficas en honor de Mirian Andr\'es.}  Servicio de Publicaciones de la Universidad de La Rioja, Logro\~no, Spain, 2010.  Reproduced by permission.}}
+\author{Clemens Ballarin}
+\date{}
+
+\maketitle
+
+\begin{abstract}
+  Locales are Isabelle's approach for dealing with parametric
+  theories.  They have been designed as a module system for a
+  theorem prover that can adequately represent the complex
+  inter-dependencies between structures found in abstract algebra, but
+  have proven fruitful also in other applications --- for example,
+  software verification.
+
+  Both design and implementation of locales have evolved considerably
+  since Kamm\"uller did his initial experiments.  Today, locales
+  are a simple yet powerful extension of the Isar proof language.
+  The present tutorial covers all major facilities of locales.  It is
+  intended for locale novices; familiarity with Isabelle and Isar is
+  presumed.
+\end{abstract}
+
+\parindent 0pt\parskip 0.5ex
+
+\input{session}
+
+\bibliographystyle{abbrv}
+\bibliography{root}
+
+\end{document}
+
+%%% Local Variables:
+%%% mode: latex
+%%% TeX-master: t
+%%% End:

--- a/doc-src/ROOT	Mon Aug 27 21:04:37 2012 +0200
+++ b/doc-src/ROOT	Mon Aug 27 21:19:16 2012 +0200
@@ -101,13 +101,15 @@
     "~~/src/HOL/Library/OptionalSugar"
   theories Sugar
 
-session Locales (doc) in "Locales/Locales" = HOL +
-  options [browser_info = false, document = false,
-    document_dump = document, document_dump_mode = "tex"]
+session Locales (doc) in "Locales" = HOL +
+  options [document_variants = "locales"]
   theories
     Examples1
     Examples2
     Examples3
+  files
+    "document/build"
+    "document/root.tex"
 
 session Logics (doc) in "Logics" = Pure +
   options [document_variants = "logics"]