(* 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