# HG changeset patch
# User huffman
# Date 1120601273 -7200
# Node ID 92925e30ff59656a612ade7b2ee8f692840fc9e1
# Parent  24b494ff8f0fd83b21e089a07e15d221bc1aaf07
renamed to Pcpodef.thy

diff -r 24b494ff8f0f -r 92925e30ff59 src/HOLCF/TypedefPcpo.thy
--- a/src/HOLCF/TypedefPcpo.thy	Wed Jul 06 00:07:34 2005 +0200
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,202 +0,0 @@
-(*  Title:      HOLCF/TypedefPcpo.thy
-    ID:         $Id$
-    Author:     Brian Huffman
-*)
-
-header {* Subtypes of pcpos *}
-
-theory TypedefPcpo
-imports Adm
-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 typedef 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
-
-end