Theory Examples

theory Examples
imports Main
begin

(*
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 "⋀"}), entailment
  (@{text "⟹"}) and equality (@{text "≡"}).  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 "⇒"} is the function type.  Double brackets~@{text
  "⟦"} and~@{text "⟧"} are used to abbreviate nested entailment.
*}
*)

section ‹Introduction›

text ‹
  Locales are based on contexts.  A \emph{context} can be seen as a
  formula schema
\[
  ‹⋀x⇩1…x⇩n. ⟦ A⇩1; … ;A⇩m ⟧ ⟹ …›
\]
  where the variables~‹x⇩1›, \ldots,~‹x⇩n› are called
  \emph{parameters} and the premises $‹A⇩1›, \ldots,~‹A⇩m›$ \emph{assumptions}.  A formula~‹C›
  is a \emph{theorem} in the context if it is a conclusion
\[
  ‹⋀x⇩1…x⇩n. ⟦ A⇩1; … ;A⇩m ⟧ ⟹ C›.
\]
  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 ‹partial_order›.
›

  locale partial_order =
    fixes le :: "'a ⇒ 'a ⇒ bool" (infixl "⊑" 50)
    assumes refl [intro, simp]: "x ⊑ x"
      and anti_sym [intro]: "⟦ x ⊑ y; y ⊑ x ⟧ ⟹ x = y"
      and trans [trans]: "⟦ x ⊑ y; y ⊑ z ⟧ ⟹ x ⊑ z"

text (in partial_order) ‹The parameter of this locale is~‹le›,
  which is a binary predicate with infix syntax~‹⊑›.  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>‹⟦ x ⊑ y; y ⊑ z ⟧ ⟹ x ⊑ z› in fact means
  \<^term>‹⋀x y z. ⟦ x ⊑ y; y ⊑ z ⟧ ⟹ x ⊑ 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 (\(\sqsubseteq\))"
  \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}

  This differs from the declaration.  The assumptions have turned into
  conclusions, denoted by the keyword \isakeyword{notes}.  Also,
  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 ⇒ 'a ⇒ bool" (infixl "⊏" 50)
    where "(x ⊏ y) = (x ⊑ y ∧ x ≠ y)"

text (in partial_order) ‹The strict order ‹less› with infix
  syntax~‹⊏› is
  defined in terms of the locale parameter~‹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 ‹less› is available for
  ‹partial_order.less le›, and it is printed
  and parsed as infix~‹⊏›.  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]:
    "⟦ x ⊏ y; y ⊑ z ⟧ ⟹ x ⊏ 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: ‹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 ⊑ x ∧ i ⊑ y ∧ (∀z. z ⊑ x ∧ z ⊑ y ⟶ z ⊑ i))"

  definition
    is_sup where "is_sup x y s =
      (x ⊑ s ∧ y ⊑ s ∧ (∀z. x ⊑ z ∧ y ⊑ z ⟶ s ⊑ z))"

  lemma %invisible is_infI [intro?]: "i ⊑ x ⟹ i ⊑ y ⟹
      (⋀z. z ⊑ x ⟹ z ⊑ y ⟹ z ⊑ i) ⟹ is_inf x y i"
    by (unfold is_inf_def) blast

  lemma %invisible is_inf_lower [elim?]:
    "is_inf x y i ⟹ (i ⊑ x ⟹ i ⊑ y ⟹ C) ⟹ C"
    by (unfold is_inf_def) blast

  lemma %invisible is_inf_greatest [elim?]:
      "is_inf x y i ⟹ z ⊑ x ⟹ z ⊑ y ⟹ z ⊑ i"
    by (unfold is_inf_def) blast

  theorem is_inf_uniq: "⟦is_inf x y i; is_inf x y i'⟧ ⟹ 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 ⊑ i'"
      proof (rule is_inf_greatest)
        from inf show "i ⊑ x" ..
        from inf show "i ⊑ y" ..
      qed
      from inf show "i' ⊑ i"
      proof (rule is_inf_greatest)
        from inf' show "i' ⊑ x" ..
        from inf' show "i' ⊑ y" ..
      qed
    qed
  qed

  theorem %invisible is_inf_related [elim?]: "x ⊑ y ⟹ is_inf x y x"
  proof -
    assume "x ⊑ y"
    show ?thesis
    proof
      show "x ⊑ x" ..
      show "x ⊑ y" by fact
      fix z assume "z ⊑ x" and "z ⊑ y" show "z ⊑ x" by fact
    qed
  qed

  lemma %invisible is_supI [intro?]: "x ⊑ s ⟹ y ⊑ s ⟹
      (⋀z. x ⊑ z ⟹ y ⊑ z ⟹ s ⊑ z) ⟹ is_sup x y s"
    by (unfold is_sup_def) blast

  lemma %invisible is_sup_least [elim?]:
      "is_sup x y s ⟹ x ⊑ z ⟹ y ⊑ z ⟹ s ⊑ z"
    by (unfold is_sup_def) blast

  lemma %invisible is_sup_upper [elim?]:
      "is_sup x y s ⟹ (x ⊑ s ⟹ y ⊑ s ⟹ C) ⟹ C"
    by (unfold is_sup_def) blast

  theorem is_sup_uniq: "⟦is_sup x y s; is_sup x y s'⟧ ⟹ 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 ⊑ s'"
      proof (rule is_sup_least)
        from sup' show "x ⊑ s'" ..
        from sup' show "y ⊑ s'" ..
      qed
      from sup' show "s' ⊑ s"
      proof (rule is_sup_least)
        from sup show "x ⊑ s" ..
        from sup show "y ⊑ s" ..
      qed
    qed
  qed

  theorem %invisible is_sup_related [elim?]: "x ⊑ y ⟹ is_sup x y y"
  proof -
    assume "x ⊑ y"
    show ?thesis
    proof
      show "x ⊑ y" by fact
      show "y ⊑ y" ..
      fix z assume "x ⊑ z" and "y ⊑ z"
      show "y ⊑ 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: "∃inf. is_inf x y inf"
      and ex_sup: "∃sup. is_sup x y sup"
  begin

text ‹These assumptions refer to the predicates for infimum
  and supremum defined for ‹partial_order› in the previous
  section.  We now introduce the notions of meet and join,
  together with an example theorem.›

  definition
    meet (infixl "⊓" 70) where "x ⊓ y = (THE inf. is_inf x y inf)"
  definition
    join (infixl "⊔" 65) where "x ⊔ y = (THE sup. is_sup x y sup)"

  lemma %invisible meet_equality [elim?]: "is_inf x y i ⟹ x ⊓ 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 ⊑ x ⟹ i ⊑ y ⟹ (⋀z. z ⊑ x ⟹ z ⊑ y ⟹ z ⊑ i) ⟹ x ⊓ y = i"
    by (rule meet_equality, rule is_infI) blast+

  lemma %invisible is_inf_meet [intro?]: "is_inf x y (x ⊓ 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 meet_left(*<*)[intro?](*>*): "x ⊓ y ⊑ x"
    by (rule is_inf_lower) (rule is_inf_meet)

  lemma %invisible meet_right [intro?]:
    "x ⊓ y ⊑ y"
    by (rule is_inf_lower) (rule is_inf_meet)

  lemma %invisible meet_le [intro?]:
    "⟦ z ⊑ x; z ⊑ y ⟧ ⟹ z ⊑ x ⊓ y"
    by (rule is_inf_greatest) (rule is_inf_meet)

  lemma %invisible join_equality [elim?]: "is_sup x y s ⟹ x ⊔ 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 ⊑ s ⟹ y ⊑ s ⟹
      (⋀z. x ⊑ z ⟹ y ⊑ z ⟹ s ⊑ z) ⟹ x ⊔ y = s"
    by (rule join_equality, rule is_supI) blast+

  lemma %invisible is_sup_join [intro?]: "is_sup x y (x ⊔ 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 ⊑ x ⊔ y"
    by (rule is_sup_upper) (rule is_sup_join)

  lemma %invisible join_right [intro?]:
    "y ⊑ x ⊔ y"
    by (rule is_sup_upper) (rule is_sup_join)

  lemma %invisible join_le [intro?]:
    "⟦ x ⊑ z; y ⊑ z ⟧ ⟹ x ⊔ y ⊑ z"
    by (rule is_sup_least) (rule is_sup_join)

  theorem %invisible meet_assoc: "(x ⊓ y) ⊓ z = x ⊓ (y ⊓ z)"
  proof (rule meetI)
    show "x ⊓ (y ⊓ z) ⊑ x ⊓ y"
    proof
      show "x ⊓ (y ⊓ z) ⊑ x" ..
      show "x ⊓ (y ⊓ z) ⊑ y"
      proof -
        have "x ⊓ (y ⊓ z) ⊑ y ⊓ z" ..
        also have "… ⊑ y" ..
        finally show ?thesis .
      qed
    qed
    show "x ⊓ (y ⊓ z) ⊑ z"
    proof -
      have "x ⊓ (y ⊓ z) ⊑ y ⊓ z" ..
      also have "… ⊑ z" ..
      finally show ?thesis .
    qed
    fix w assume "w ⊑ x ⊓ y" and "w ⊑ z"
    show "w ⊑ x ⊓ (y ⊓ z)"
    proof
      show "w ⊑ x"
      proof -
        have "w ⊑ x ⊓ y" by fact
        also have "… ⊑ x" ..
        finally show ?thesis .
      qed
      show "w ⊑ y ⊓ z"
      proof
        show "w ⊑ y"
        proof -
          have "w ⊑ x ⊓ y" by fact
          also have "… ⊑ y" ..
          finally show ?thesis .
        qed
        show "w ⊑ z" by fact
      qed
    qed
  qed

  theorem %invisible meet_commute: "x ⊓ y = y ⊓ x"
  proof (rule meetI)
    show "y ⊓ x ⊑ x" ..
    show "y ⊓ x ⊑ y" ..
    fix z assume "z ⊑ y" and "z ⊑ x"
    then show "z ⊑ y ⊓ x" ..
  qed

  theorem %invisible meet_join_absorb: "x ⊓ (x ⊔ y) = x"
  proof (rule meetI)
    show "x ⊑ x" ..
    show "x ⊑ x ⊔ y" ..
    fix z assume "z ⊑ x" and "z ⊑ x ⊔ y"
    show "z ⊑ x" by fact
  qed

  theorem %invisible join_assoc: "(x ⊔ y) ⊔ z = x ⊔ (y ⊔ z)"
  proof (rule joinI)
    show "x ⊔ y ⊑ x ⊔ (y ⊔ z)"
    proof
      show "x ⊑ x ⊔ (y ⊔ z)" ..
      show "y ⊑ x ⊔ (y ⊔ z)"
      proof -
        have "y ⊑ y ⊔ z" ..
        also have "... ⊑ x ⊔ (y ⊔ z)" ..
        finally show ?thesis .
      qed
    qed
    show "z ⊑ x ⊔ (y ⊔ z)"
    proof -
      have "z ⊑ y ⊔ z"  ..
      also have "... ⊑ x ⊔ (y ⊔ z)" ..
      finally show ?thesis .
    qed
    fix w assume "x ⊔ y ⊑ w" and "z ⊑ w"
    show "x ⊔ (y ⊔ z) ⊑ w"
    proof
      show "x ⊑ w"
      proof -
        have "x ⊑ x ⊔ y" ..
        also have "… ⊑ w" by fact
        finally show ?thesis .
      qed
      show "y ⊔ z ⊑ w"
      proof
        show "y ⊑ w"
        proof -
          have "y ⊑ x ⊔ y" ..
          also have "... ⊑ w" by fact
          finally show ?thesis .
        qed
        show "z ⊑ w" by fact
      qed
    qed
  qed

  theorem %invisible join_commute: "x ⊔ y = y ⊔ x"
  proof (rule joinI)
    show "x ⊑ y ⊔ x" ..
    show "y ⊑ y ⊔ x" ..
    fix z assume "y ⊑ z" and "x ⊑ z"
    then show "y ⊔ x ⊑ z" ..
  qed

  theorem %invisible join_meet_absorb: "x ⊔ (x ⊓ y) = x"
  proof (rule joinI)
    show "x ⊑ x" ..
    show "x ⊓ y ⊑ x" ..
    fix z assume "x ⊑ z" and "x ⊓ y ⊑ z"
    show "x ⊑ z" by fact
  qed

  theorem %invisible meet_idem: "x ⊓ x = x"
  proof -
    have "x ⊓ (x ⊔ (x ⊓ x)) = x" by (rule meet_join_absorb)
    also have "x ⊔ (x ⊓ x) = x" by (rule join_meet_absorb)
    finally show ?thesis .
  qed

  theorem %invisible meet_related [elim?]: "x ⊑ y ⟹ x ⊓ y = x"
  proof (rule meetI)
    assume "x ⊑ y"
    show "x ⊑ x" ..
    show "x ⊑ y" by fact
    fix z assume "z ⊑ x" and "z ⊑ y"
    show "z ⊑ x" by fact
  qed

  theorem %invisible meet_related2 [elim?]: "y ⊑ x ⟹ x ⊓ y = y"
    by (drule meet_related) (simp add: meet_commute)

  theorem %invisible join_related [elim?]: "x ⊑ y ⟹ x ⊔ y = y"
  proof (rule joinI)
    assume "x ⊑ y"
    show "y ⊑ y" ..
    show "x ⊑ y" by fact
    fix z assume "x ⊑ z" and "y ⊑ z"
    show "y ⊑ z" by fact
  qed

  theorem %invisible join_related2 [elim?]: "y ⊑ x ⟹ x ⊔ y = x"
    by (drule join_related) (simp add: join_commute)

  theorem %invisible meet_connection: "(x ⊑ y) = (x ⊓ y = x)"
  proof
    assume "x ⊑ y"
    then have "is_inf x y x" ..
    then show "x ⊓ y = x" ..
  next
    have "x ⊓ y ⊑ y" ..
    also assume "x ⊓ y = x"
    finally show "x ⊑ y" .
  qed

  theorem %invisible join_connection: "(x ⊑ y) = (x ⊔ y = y)"
  proof
    assume "x ⊑ y"
    then have "is_sup x y y" ..
    then show "x ⊔ y = y" ..
  next
    have "x ⊑ x ⊔ y" ..
    also assume "x ⊔ y = y"
    finally show "x ⊑ y" .
  qed

  theorem %invisible meet_connection2: "(x ⊑ y) = (y ⊓ x = x)"
    using meet_commute meet_connection by simp

  theorem %invisible join_connection2: "(x ⊑ y) = (x ⊔ 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.›

  locale total_order = partial_order +
    assumes total: "x ⊑ y ∨ y ⊑ x"

  lemma (in total_order) less_total: "x ⊏ y ∨ x = y ∨ y ⊏ x"
    using total
    by (unfold less_def) blast

  locale distrib_lattice = lattice +
    assumes meet_distr: "x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z"

  lemma (in distrib_lattice) join_distr:
    "x ⊔ (y ⊓ z) = (x ⊔ y) ⊓ (x ⊔ z)"  (* txt {* Jacobson I, p.\ 462 *} *)
    proof -
    have "x ⊔ (y ⊓ z) = (x ⊔ (x ⊓ z)) ⊔ (y ⊓ z)" by (simp add: L4)
    also have "... = x ⊔ ((x ⊓ z) ⊔ (y ⊓ z))" by (simp add: L2)
    also have "... = x ⊔ ((x ⊔ y) ⊓ z)" by (simp add: L1 meet_distr)
    also have "... = ((x ⊔ y) ⊓ x) ⊔ ((x ⊔ y) ⊓ z)" by (simp add: L1 L4)
    also have "... = (x ⊔ y) ⊓ (x ⊔ 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) {‹partial_order›};
  \node (lat) at (-1.5,-1) {‹lattice›};
  \node (dlat) at (-1.5,-2) {‹distrib_lattice›};
  \node (to) at (1.5,-1) {‹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) {‹partial_order›};
  \node (lat) at (0,-1) {‹lattice›};
  \node (dlat) at (-1.5,-2) {‹distrib_lattice›};
  \node (to) at (1.5,-2) {‹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) {‹partial_order›};
  \node (lat) at (0,-1) {‹lattice›};
  \node (dlat) at (0,-2) {‹distrib_lattice›};
  \node (to) at (0,-3) {‹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 ⊆ lattice

txt ‹\normalsize
  This enters the context of locale ‹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 ‹intro_locales› and ‹unfold_locales›.  They are aware of the
  current context and dependencies between locales and automatically
  discharge goals implied by these.  While ‹unfold_locales›
  always unfolds locale predicates to assumptions, ‹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 ‹lattice›, which need to be shown, hence
  ‹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 ‹partial_order› are discharged automatically, and only the
  assumptions introduced in ‹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 ⊑ y then x else y)"
      by (auto simp: is_inf_def)
    then show "∃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 ⊑ y then y else x)"
      by (auto simp: is_sup_def)
    then show "∃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 ⊆ distrib_lattice
    proof unfold_locales
    fix %"proof" x y z
    show "x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z" (is "?l = ?r")
      txt ‹Jacobson I, p.\ 462›
    proof -
      { assume c: "y ⊑ x" "z ⊑ x"
        from c have "?l = y ⊔ 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 ⊑ y ∨ x ⊑ 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