src/HOL/HOLCF/Cpodef.thy

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(*  Title:      HOL/HOLCF/Cpodef.thy
    Author:     Brian Huffman
*)

section \<open>Subtypes of pcpos\<close>

theory Cpodef
  imports Adm
  keywords "pcpodef" "cpodef" :: thy_goal_defn
begin

subsection \<open>Proving a subtype is a partial order\<close>

text \<open>
  A subtype of a partial order is itself a partial order,
  if the ordering is defined in the standard way.
\<close>

setup \<open>Sign.add_const_constraint (\<^const_name>\<open>Porder.below\<close>, NONE)\<close>

theorem typedef_po:
  fixes Abs :: "'a::po \<Rightarrow> 'b::type"
  assumes type: "type_definition Rep Abs A"
    and below: "(\<sqsubseteq>) \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
  shows "OFCLASS('b, po_class)"
  apply (intro_classes, unfold below)
    apply (rule below_refl)
   apply (erule (1) below_trans)
  apply (rule type_definition.Rep_inject [OF type, THEN iffD1])
  apply (erule (1) below_antisym)
  done

setup \<open>Sign.add_const_constraint (\<^const_name>\<open>Porder.below\<close>, SOME \<^typ>\<open>'a::below \<Rightarrow> 'a::below \<Rightarrow> bool\<close>)\<close>


subsection \<open>Proving a subtype is finite\<close>

lemma typedef_finite_UNIV:
  fixes Abs :: "'a::type \<Rightarrow> 'b::type"
  assumes type: "type_definition Rep Abs A"
  shows "finite A \<Longrightarrow> finite (UNIV :: 'b set)"
proof -
  assume "finite A"
  then have "finite (Abs ` A)"
    by (rule finite_imageI)
  then show "finite (UNIV :: 'b set)"
    by (simp only: type_definition.Abs_image [OF type])
qed


subsection \<open>Proving a subtype is chain-finite\<close>

lemma ch2ch_Rep:
  assumes below: "(\<sqsubseteq>) \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
  shows "chain S \<Longrightarrow> chain (\<lambda>i. Rep (S i))"
  unfolding chain_def below .

theorem typedef_chfin:
  fixes Abs :: "'a::chfin \<Rightarrow> 'b::po"
  assumes type: "type_definition Rep Abs A"
    and below: "(\<sqsubseteq>) \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
  shows "OFCLASS('b, chfin_class)"
  apply intro_classes
  apply (drule ch2ch_Rep [OF below])
  apply (drule chfin)
  apply (unfold max_in_chain_def)
  apply (simp add: type_definition.Rep_inject [OF type])
  done


subsection \<open>Proving a subtype is complete\<close>

text \<open>
  A subtype of a cpo is itself a cpo if the ordering is
  defined in the standard way, and the defining subset
  is closed with respect to limits of chains.  A set is
  closed if and only if membership in the set is an
  admissible predicate.
\<close>

lemma typedef_is_lubI:
  assumes below: "(\<sqsubseteq>) \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
  shows "range (\<lambda>i. Rep (S i)) <<| Rep x \<Longrightarrow> range S <<| x"
  by (simp add: is_lub_def is_ub_def below)

lemma Abs_inverse_lub_Rep:
  fixes Abs :: "'a::cpo \<Rightarrow> 'b::po"
  assumes type: "type_definition Rep Abs A"
    and below: "(\<sqsubseteq>) \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
    and adm:  "adm (\<lambda>x. x \<in> A)"
  shows "chain S \<Longrightarrow> Rep (Abs (\<Squnion>i. Rep (S i))) = (\<Squnion>i. Rep (S i))"
  apply (rule type_definition.Abs_inverse [OF type])
  apply (erule admD [OF adm ch2ch_Rep [OF below]])
  apply (rule type_definition.Rep [OF type])
  done

theorem typedef_is_lub:
  fixes Abs :: "'a::cpo \<Rightarrow> 'b::po"
  assumes type: "type_definition Rep Abs A"
    and below: "(\<sqsubseteq>) \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
    and adm: "adm (\<lambda>x. x \<in> A)"
  assumes S: "chain S"
  shows "range S <<| Abs (\<Squnion>i. Rep (S i))"
proof -
  from S have "chain (\<lambda>i. Rep (S i))"
    by (rule ch2ch_Rep [OF below])
  then have "range (\<lambda>i. Rep (S i)) <<| (\<Squnion>i. Rep (S i))"
    by (rule cpo_lubI)
  then have "range (\<lambda>i. Rep (S i)) <<| Rep (Abs (\<Squnion>i. Rep (S i)))"
    by (simp only: Abs_inverse_lub_Rep [OF type below adm S])
  then show "range S <<| Abs (\<Squnion>i. Rep (S i))"
    by (rule typedef_is_lubI [OF below])
qed

lemmas typedef_lub = typedef_is_lub [THEN lub_eqI]

theorem typedef_cpo:
  fixes Abs :: "'a::cpo \<Rightarrow> 'b::po"
  assumes type: "type_definition Rep Abs A"
    and below: "(\<sqsubseteq>) \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
    and adm: "adm (\<lambda>x. x \<in> A)"
  shows "OFCLASS('b, cpo_class)"
proof
  fix S :: "nat \<Rightarrow> 'b"
  assume "chain S"
  then have "range S <<| Abs (\<Squnion>i. Rep (S i))"
    by (rule typedef_is_lub [OF type below adm])
  then show "\<exists>x. range S <<| x" ..
qed


subsubsection \<open>Continuity of \emph{Rep} and \emph{Abs}\<close>

text \<open>For any sub-cpo, the \<^term>\<open>Rep\<close> function is continuous.\<close>

theorem typedef_cont_Rep:
  fixes Abs :: "'a::cpo \<Rightarrow> 'b::cpo"
  assumes type: "type_definition Rep Abs A"
    and below: "(\<sqsubseteq>) \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
    and adm: "adm (\<lambda>x. x \<in> A)"
  shows "cont (\<lambda>x. f x) \<Longrightarrow> cont (\<lambda>x. Rep (f x))"
  apply (erule cont_apply [OF _ _ cont_const])
  apply (rule contI)
  apply (simp only: typedef_lub [OF type below adm])
  apply (simp only: Abs_inverse_lub_Rep [OF type below adm])
  apply (rule cpo_lubI)
  apply (erule ch2ch_Rep [OF below])
  done

text \<open>
  For a sub-cpo, we can make the \<^term>\<open>Abs\<close> function continuous
  only if we restrict its domain to the defining subset by
  composing it with another continuous function.
\<close>

theorem typedef_cont_Abs:
  fixes Abs :: "'a::cpo \<Rightarrow> 'b::cpo"
  fixes f :: "'c::cpo \<Rightarrow> 'a::cpo"
  assumes type: "type_definition Rep Abs A"
    and below: "(\<sqsubseteq>) \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
    and adm: "adm (\<lambda>x. x \<in> A)" (* not used *)
    and f_in_A: "\<And>x. f x \<in> A"
  shows "cont f \<Longrightarrow> cont (\<lambda>x. Abs (f x))"
  unfolding cont_def is_lub_def is_ub_def ball_simps below
  by (simp add: type_definition.Abs_inverse [OF type f_in_A])


subsection \<open>Proving subtype elements are compact\<close>

theorem typedef_compact:
  fixes Abs :: "'a::cpo \<Rightarrow> 'b::cpo"
  assumes type: "type_definition Rep Abs A"
    and below: "(\<sqsubseteq>) \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
    and adm: "adm (\<lambda>x. x \<in> A)"
  shows "compact (Rep k) \<Longrightarrow> compact k"
proof (unfold compact_def)
  have cont_Rep: "cont Rep"
    by (rule typedef_cont_Rep [OF type below adm cont_id])
  assume "adm (\<lambda>x. Rep k \<notsqsubseteq> x)"
  with cont_Rep have "adm (\<lambda>x. Rep k \<notsqsubseteq> Rep x)" by (rule adm_subst)
  then show "adm (\<lambda>x. k \<notsqsubseteq> x)" by (unfold below)
qed


subsection \<open>Proving a subtype is pointed\<close>

text \<open>
  A subtype of a cpo has a least element if and only if
  the defining subset has a least element.
\<close>

theorem typedef_pcpo_generic:
  fixes Abs :: "'a::cpo \<Rightarrow> 'b::cpo"
  assumes type: "type_definition Rep Abs A"
    and below: "(\<sqsubseteq>) \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
    and z_in_A: "z \<in> A"
    and z_least: "\<And>x. x \<in> A \<Longrightarrow> z \<sqsubseteq> x"
  shows "OFCLASS('b, pcpo_class)"
  apply (intro_classes)
  apply (rule_tac x="Abs z" in exI, rule allI)
  apply (unfold below)
  apply (subst type_definition.Abs_inverse [OF type z_in_A])
  apply (rule z_least [OF type_definition.Rep [OF type]])
  done

text \<open>
  As a special case, a subtype of a pcpo has a least element
  if the defining subset contains \<^term>\<open>\<bottom>\<close>.
\<close>

theorem typedef_pcpo:
  fixes Abs :: "'a::pcpo \<Rightarrow> 'b::cpo"
  assumes type: "type_definition Rep Abs A"
    and below: "(\<sqsubseteq>) \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
    and bottom_in_A: "\<bottom> \<in> A"
  shows "OFCLASS('b, pcpo_class)"
  by (rule typedef_pcpo_generic [OF type below bottom_in_A], rule minimal)


subsubsection \<open>Strictness of \emph{Rep} and \emph{Abs}\<close>

text \<open>
  For a sub-pcpo where \<^term>\<open>\<bottom>\<close> is a member of the defining
  subset, \<^term>\<open>Rep\<close> and \<^term>\<open>Abs\<close> are both strict.
\<close>

theorem typedef_Abs_strict:
  assumes type: "type_definition Rep Abs A"
    and below: "(\<sqsubseteq>) \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
    and bottom_in_A: "\<bottom> \<in> A"
  shows "Abs \<bottom> = \<bottom>"
  apply (rule bottomI, unfold below)
  apply (simp add: type_definition.Abs_inverse [OF type bottom_in_A])
  done

theorem typedef_Rep_strict:
  assumes type: "type_definition Rep Abs A"
    and below: "(\<sqsubseteq>) \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
    and bottom_in_A: "\<bottom> \<in> A"
  shows "Rep \<bottom> = \<bottom>"
  apply (rule typedef_Abs_strict [OF type below bottom_in_A, THEN subst])
  apply (rule type_definition.Abs_inverse [OF type bottom_in_A])
  done

theorem typedef_Abs_bottom_iff:
  assumes type: "type_definition Rep Abs A"
    and below: "(\<sqsubseteq>) \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
    and bottom_in_A: "\<bottom> \<in> A"
  shows "x \<in> A \<Longrightarrow> (Abs x = \<bottom>) = (x = \<bottom>)"
  apply (rule typedef_Abs_strict [OF type below bottom_in_A, THEN subst])
  apply (simp add: type_definition.Abs_inject [OF type] bottom_in_A)
  done

theorem typedef_Rep_bottom_iff:
  assumes type: "type_definition Rep Abs A"
    and below: "(\<sqsubseteq>) \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
    and bottom_in_A: "\<bottom> \<in> A"
  shows "(Rep x = \<bottom>) = (x = \<bottom>)"
  apply (rule typedef_Rep_strict [OF type below bottom_in_A, THEN subst])
  apply (simp add: type_definition.Rep_inject [OF type])
  done


subsection \<open>Proving a subtype is flat\<close>

theorem typedef_flat:
  fixes Abs :: "'a::flat \<Rightarrow> 'b::pcpo"
  assumes type: "type_definition Rep Abs A"
    and below: "(\<sqsubseteq>) \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
    and bottom_in_A: "\<bottom> \<in> A"
  shows "OFCLASS('b, flat_class)"
  apply (intro_classes)
  apply (unfold below)
  apply (simp add: type_definition.Rep_inject [OF type, symmetric])
  apply (simp add: typedef_Rep_strict [OF type below bottom_in_A])
  apply (simp add: ax_flat)
  done


subsection \<open>HOLCF type definition package\<close>

ML_file \<open>Tools/cpodef.ML\<close>

end