renamed from TypedefPcpo.thy;
added theorems Rep_defined, Abs_defined;
uses pcpodef_package.ML

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOLCF/Pcpodef.thy	Wed Jul 06 00:04:31 2005 +0200
@@ -0,0 +1,223 @@
+(*  Title:      HOLCF/Pcpodef.thy
+    ID:         $Id$
+    Author:     Brian Huffman
+*)
+
+header {* Subtypes of pcpos *}
+
+theory Pcpodef
+imports Adm
+uses ("pcpodef_package.ML")
+begin
+
+subsection {* Proving a subtype is a partial order *}
+
+text {*
+  A subtype of a partial order is itself a partial order,
+  if the ordering is defined in the standard way.
+*}
+
+theorem typedef_po:
+  fixes Abs :: "'a::po \<Rightarrow> 'b::sq_ord"
+  assumes type: "type_definition Rep Abs A"
+    and less: "op \<sqsubseteq> \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
+  shows "OFCLASS('b, po_class)"
+ apply (intro_classes, unfold less)
+   apply (rule refl_less)
+  apply (subst type_definition.Rep_inject [OF type, symmetric])
+  apply (rule antisym_less, assumption+)
+ apply (rule trans_less, assumption+)
+done
+
+
+subsection {* Proving a subtype is complete *}
+
+text {*
+  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.
+*}
+
+lemma chain_Rep:
+  assumes less: "op \<sqsubseteq> \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
+  shows "chain S \<Longrightarrow> chain (\<lambda>n. Rep (S n))"
+by (rule chainI, drule chainE, unfold less)
+
+lemma lub_Rep_in_A:
+  fixes Abs :: "'a::cpo \<Rightarrow> 'b::po"
+  assumes type: "type_definition Rep Abs A"
+    and less: "op \<sqsubseteq> \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
+    and adm:  "adm (\<lambda>x. x \<in> A)"
+  shows "chain S \<Longrightarrow> (LUB n. Rep (S n)) \<in> A"
+ apply (erule admD [OF adm chain_Rep [OF less], rule_format])
+ 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 less: "op \<sqsubseteq> \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
+    and adm: "adm (\<lambda>x. x \<in> A)"
+  shows "chain S \<Longrightarrow> range S <<| Abs (LUB n. Rep (S n))"
+ apply (rule is_lubI)
+  apply (rule ub_rangeI)
+  apply (subst less)
+  apply (subst type_definition.Abs_inverse [OF type])
+   apply (erule lub_Rep_in_A [OF type less adm])
+  apply (rule is_ub_thelub)
+  apply (erule chain_Rep [OF less])
+ apply (subst less)
+ apply (subst type_definition.Abs_inverse [OF type])
+  apply (erule lub_Rep_in_A [OF type less adm])
+ apply (rule is_lub_thelub)
+  apply (erule chain_Rep [OF less])
+ apply (rule ub_rangeI)
+ apply (drule ub_rangeD)
+ apply (unfold less)
+ apply assumption
+done
+
+theorem typedef_cpo:
+  fixes Abs :: "'a::cpo \<Rightarrow> 'b::po"
+  assumes type: "type_definition Rep Abs A"
+    and less: "op \<sqsubseteq> \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
+    and adm: "adm (\<lambda>x. x \<in> A)"
+  shows "OFCLASS('b, cpo_class)"
+ apply (intro_classes)
+ apply (rule_tac x="Abs (LUB n. Rep (S n))" in exI)
+ apply (erule typedef_is_lub [OF type less adm])
+done
+
+
+subsubsection {* Continuity of @{term Rep} and @{term Abs} *}
+
+text {* For any sub-cpo, the @{term Rep} function is continuous. *}
+
+theorem typedef_cont_Rep:
+  fixes Abs :: "'a::cpo \<Rightarrow> 'b::cpo"
+  assumes type: "type_definition Rep Abs A"
+    and less: "op \<sqsubseteq> \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
+    and adm: "adm (\<lambda>x. x \<in> A)"
+  shows "cont Rep"
+ apply (rule contI)
+ apply (simp only: typedef_is_lub [OF type less adm, THEN thelubI])
+ apply (subst type_definition.Abs_inverse [OF type])
+  apply (erule lub_Rep_in_A [OF type less adm])
+ apply (rule thelubE [OF _ refl])
+ apply (erule chain_Rep [OF less])
+done
+
+text {*
+  For a sub-cpo, we can make the @{term Abs} function continuous
+  only if we restrict its domain to the defining subset by
+  composing it with another continuous function.
+*}
+
+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 less: "op \<sqsubseteq> \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
+    and adm: "adm (\<lambda>x. x \<in> A)"
+    and f_in_A: "\<And>x. f x \<in> A"
+    and cont_f: "cont f"
+  shows "cont (\<lambda>x. Abs (f x))"
+ apply (rule contI)
+ apply (rule is_lubI)
+  apply (rule ub_rangeI)
+  apply (simp only: less type_definition.Abs_inverse [OF type f_in_A])
+  apply (rule monofun_fun_arg [OF cont2mono [OF cont_f]])
+  apply (erule is_ub_thelub)
+ apply (simp only: less type_definition.Abs_inverse [OF type f_in_A])
+ apply (simp only: contlubE [OF cont2contlub [OF cont_f]])
+ apply (rule is_lub_thelub)
+  apply (erule ch2ch_monofun [OF cont2mono [OF cont_f]])
+ apply (rule ub_rangeI)
+ apply (drule_tac i=i in ub_rangeD)
+ apply (simp only: less type_definition.Abs_inverse [OF type f_in_A])
+done
+
+subsection {* Proving a subtype is pointed *}
+
+text {*
+  A subtype of a cpo has a least element if and only if
+  the defining subset has a least element.
+*}
+
+theorem typedef_pcpo:
+  fixes Abs :: "'a::cpo \<Rightarrow> 'b::cpo"
+  assumes type: "type_definition Rep Abs A"
+    and less: "op \<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 less)
+ apply (subst type_definition.Abs_inverse [OF type z_in_A])
+ apply (rule z_least [OF type_definition.Rep [OF type]])
+done
+
+text {*
+  As a special case, a subtype of a pcpo has a least element
+  if the defining subset contains @{term \<bottom>}.
+*}
+
+theorem typedef_pcpo_UU:
+  fixes Abs :: "'a::pcpo \<Rightarrow> 'b::cpo"
+  assumes type: "type_definition Rep Abs A"
+    and less: "op \<sqsubseteq> \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
+    and UU_in_A: "\<bottom> \<in> A"
+  shows "OFCLASS('b, pcpo_class)"
+by (rule typedef_pcpo [OF type less UU_in_A], rule minimal)
+
+subsubsection {* Strictness of @{term Rep} and @{term Abs} *}
+
+text {*
+  For a sub-pcpo where @{term \<bottom>} is a member of the defining
+  subset, @{term Rep} and @{term Abs} are both strict.
+*}
+
+theorem typedef_Abs_strict:
+  assumes type: "type_definition Rep Abs A"
+    and less: "op \<sqsubseteq> \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
+    and UU_in_A: "\<bottom> \<in> A"
+  shows "Abs \<bottom> = \<bottom>"
+ apply (rule UU_I, unfold less)
+ apply (simp add: type_definition.Abs_inverse [OF type UU_in_A])
+done
+
+theorem typedef_Rep_strict:
+  assumes type: "type_definition Rep Abs A"
+    and less: "op \<sqsubseteq> \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
+    and UU_in_A: "\<bottom> \<in> A"
+  shows "Rep \<bottom> = \<bottom>"
+ apply (rule typedef_Abs_strict [OF type less UU_in_A, THEN subst])
+ apply (rule type_definition.Abs_inverse [OF type UU_in_A])
+done
+
+theorem typedef_Abs_defined:
+  assumes type: "type_definition Rep Abs A"
+    and less: "op \<sqsubseteq> \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
+    and UU_in_A: "\<bottom> \<in> A"
+  shows "\<lbrakk>x \<noteq> \<bottom>; x \<in> A\<rbrakk> \<Longrightarrow> Abs x \<noteq> \<bottom>"
+ apply (rule typedef_Abs_strict [OF type less UU_in_A, THEN subst])
+ apply (simp add: type_definition.Abs_inject [OF type] UU_in_A)
+done
+
+theorem typedef_Rep_defined:
+  assumes type: "type_definition Rep Abs A"
+    and less: "op \<sqsubseteq> \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
+    and UU_in_A: "\<bottom> \<in> A"
+  shows "x \<noteq> \<bottom> \<Longrightarrow> Rep x \<noteq> \<bottom>"
+ apply (rule typedef_Rep_strict [OF type less UU_in_A, THEN subst])
+ apply (simp add: type_definition.Rep_inject [OF type])
+done
+
+subsection {* HOLCF type definition package *}
+
+use "pcpodef_package.ML"
+
+end