src/HOLCF/Pcpodef.thy
author huffman
Tue Jul 26 18:24:29 2005 +0200 (2005-07-26)
changeset 16918 d0fdc7b9a33f
parent 16738 b70bac29b11d
child 17812 703005988cfe
permissions -rw-r--r--
cleaned up; renamed some theorems
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(*  Title:      HOLCF/Pcpodef.thy
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    ID:         $Id$
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    Author:     Brian Huffman
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*)
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header {* Subtypes of pcpos *}
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theory Pcpodef
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imports Adm
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uses ("pcpodef_package.ML")
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begin
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subsection {* Proving a subtype is a partial order *}
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text {*
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  A subtype of a partial order is itself a partial order,
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  if the ordering is defined in the standard way.
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*}
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theorem typedef_po:
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  fixes Abs :: "'a::po \<Rightarrow> 'b::sq_ord"
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  assumes type: "type_definition Rep Abs A"
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    and less: "op \<sqsubseteq> \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
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  shows "OFCLASS('b, po_class)"
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 apply (intro_classes, unfold less)
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   apply (rule refl_less)
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  apply (rule type_definition.Rep_inject [OF type, THEN iffD1])
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  apply (erule (1) antisym_less)
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 apply (erule (1) trans_less)
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done
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subsection {* Proving a subtype is complete *}
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text {*
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  A subtype of a cpo is itself a cpo if the ordering is
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  defined in the standard way, and the defining subset
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  is closed with respect to limits of chains.  A set is
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  closed if and only if membership in the set is an
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  admissible predicate.
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*}
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lemma monofun_Rep:
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  assumes less: "op \<sqsubseteq> \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
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  shows "monofun Rep"
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by (rule monofunI, unfold less)
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lemmas ch2ch_Rep = ch2ch_monofun [OF monofun_Rep]
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lemmas ub2ub_Rep = ub2ub_monofun [OF monofun_Rep]
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lemma Abs_inverse_lub_Rep:
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  fixes Abs :: "'a::cpo \<Rightarrow> 'b::po"
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  assumes type: "type_definition Rep Abs A"
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    and less: "op \<sqsubseteq> \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
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    and adm:  "adm (\<lambda>x. x \<in> A)"
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  shows "chain S \<Longrightarrow> Rep (Abs (\<Squnion>i. Rep (S i))) = (\<Squnion>i. Rep (S i))"
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 apply (rule type_definition.Abs_inverse [OF type])
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 apply (erule admD [OF adm ch2ch_Rep [OF less], rule_format])
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 apply (rule type_definition.Rep [OF type])
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done
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theorem typedef_lub:
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  fixes Abs :: "'a::cpo \<Rightarrow> 'b::po"
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  assumes type: "type_definition Rep Abs A"
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    and less: "op \<sqsubseteq> \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
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    and adm: "adm (\<lambda>x. x \<in> A)"
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  shows "chain S \<Longrightarrow> range S <<| Abs (\<Squnion>i. Rep (S i))"
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 apply (frule ch2ch_Rep [OF less])
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 apply (rule is_lubI)
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  apply (rule ub_rangeI)
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  apply (simp only: less Abs_inverse_lub_Rep [OF type less adm])
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  apply (erule is_ub_thelub)
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 apply (simp only: less Abs_inverse_lub_Rep [OF type less adm])
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 apply (erule is_lub_thelub)
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 apply (erule ub2ub_Rep [OF less])
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done
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lemmas typedef_thelub = typedef_lub [THEN thelubI, standard]
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theorem typedef_cpo:
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  fixes Abs :: "'a::cpo \<Rightarrow> 'b::po"
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  assumes type: "type_definition Rep Abs A"
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    and less: "op \<sqsubseteq> \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
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    and adm: "adm (\<lambda>x. x \<in> A)"
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  shows "OFCLASS('b, cpo_class)"
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proof
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  fix S::"nat \<Rightarrow> 'b" assume "chain S"
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  hence "range S <<| Abs (\<Squnion>i. Rep (S i))"
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    by (rule typedef_lub [OF type less adm])
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  thus "\<exists>x. range S <<| x" ..
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qed
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subsubsection {* Continuity of @{term Rep} and @{term Abs} *}
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text {* For any sub-cpo, the @{term Rep} function is continuous. *}
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theorem typedef_cont_Rep:
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  fixes Abs :: "'a::cpo \<Rightarrow> 'b::cpo"
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  assumes type: "type_definition Rep Abs A"
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    and less: "op \<sqsubseteq> \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
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    and adm: "adm (\<lambda>x. x \<in> A)"
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  shows "cont Rep"
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 apply (rule contI)
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 apply (simp only: typedef_thelub [OF type less adm])
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 apply (simp only: Abs_inverse_lub_Rep [OF type less adm])
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 apply (rule thelubE [OF _ refl])
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 apply (erule ch2ch_Rep [OF less])
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done
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text {*
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  For a sub-cpo, we can make the @{term Abs} function continuous
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  only if we restrict its domain to the defining subset by
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  composing it with another continuous function.
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*}
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theorem typedef_is_lubI:
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  assumes less: "op \<sqsubseteq> \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
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  shows "range (\<lambda>i. Rep (S i)) <<| Rep x \<Longrightarrow> range S <<| x"
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 apply (rule is_lubI)
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  apply (rule ub_rangeI)
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  apply (subst less)
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  apply (erule is_ub_lub)
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 apply (subst less)
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 apply (erule is_lub_lub)
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 apply (erule ub2ub_Rep [OF less])
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done
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theorem typedef_cont_Abs:
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  fixes Abs :: "'a::cpo \<Rightarrow> 'b::cpo"
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  fixes f :: "'c::cpo \<Rightarrow> 'a::cpo"
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  assumes type: "type_definition Rep Abs A"
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    and less: "op \<sqsubseteq> \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
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    and adm: "adm (\<lambda>x. x \<in> A)" (* not used *)
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    and f_in_A: "\<And>x. f x \<in> A"
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    and cont_f: "cont f"
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  shows "cont (\<lambda>x. Abs (f x))"
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 apply (rule contI)
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 apply (rule typedef_is_lubI [OF less])
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 apply (simp only: type_definition.Abs_inverse [OF type f_in_A])
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 apply (erule cont_f [THEN contE])
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done
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subsection {* Proving a subtype is pointed *}
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text {*
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  A subtype of a cpo has a least element if and only if
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  the defining subset has a least element.
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*}
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theorem typedef_pcpo_generic:
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  fixes Abs :: "'a::cpo \<Rightarrow> 'b::cpo"
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  assumes type: "type_definition Rep Abs A"
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    and less: "op \<sqsubseteq> \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
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    and z_in_A: "z \<in> A"
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    and z_least: "\<And>x. x \<in> A \<Longrightarrow> z \<sqsubseteq> x"
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  shows "OFCLASS('b, pcpo_class)"
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 apply (intro_classes)
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 apply (rule_tac x="Abs z" in exI, rule allI)
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 apply (unfold less)
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 apply (subst type_definition.Abs_inverse [OF type z_in_A])
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 apply (rule z_least [OF type_definition.Rep [OF type]])
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done
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text {*
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  As a special case, a subtype of a pcpo has a least element
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  if the defining subset contains @{term \<bottom>}.
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*}
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theorem typedef_pcpo:
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  fixes Abs :: "'a::pcpo \<Rightarrow> 'b::cpo"
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  assumes type: "type_definition Rep Abs A"
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    and less: "op \<sqsubseteq> \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
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    and UU_in_A: "\<bottom> \<in> A"
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  shows "OFCLASS('b, pcpo_class)"
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by (rule typedef_pcpo_generic [OF type less UU_in_A], rule minimal)
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subsubsection {* Strictness of @{term Rep} and @{term Abs} *}
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text {*
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  For a sub-pcpo where @{term \<bottom>} is a member of the defining
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  subset, @{term Rep} and @{term Abs} are both strict.
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*}
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theorem typedef_Abs_strict:
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  assumes type: "type_definition Rep Abs A"
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    and less: "op \<sqsubseteq> \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
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    and UU_in_A: "\<bottom> \<in> A"
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  shows "Abs \<bottom> = \<bottom>"
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 apply (rule UU_I, unfold less)
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 apply (simp add: type_definition.Abs_inverse [OF type UU_in_A])
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done
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theorem typedef_Rep_strict:
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  assumes type: "type_definition Rep Abs A"
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    and less: "op \<sqsubseteq> \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
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    and UU_in_A: "\<bottom> \<in> A"
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  shows "Rep \<bottom> = \<bottom>"
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 apply (rule typedef_Abs_strict [OF type less UU_in_A, THEN subst])
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 apply (rule type_definition.Abs_inverse [OF type UU_in_A])
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done
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theorem typedef_Abs_defined:
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  assumes type: "type_definition Rep Abs A"
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    and less: "op \<sqsubseteq> \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
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    and UU_in_A: "\<bottom> \<in> A"
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  shows "\<lbrakk>x \<noteq> \<bottom>; x \<in> A\<rbrakk> \<Longrightarrow> Abs x \<noteq> \<bottom>"
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 apply (rule typedef_Abs_strict [OF type less UU_in_A, THEN subst])
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 apply (simp add: type_definition.Abs_inject [OF type] UU_in_A)
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done
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theorem typedef_Rep_defined:
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  assumes type: "type_definition Rep Abs A"
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    and less: "op \<sqsubseteq> \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
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    and UU_in_A: "\<bottom> \<in> A"
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  shows "x \<noteq> \<bottom> \<Longrightarrow> Rep x \<noteq> \<bottom>"
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 apply (rule typedef_Rep_strict [OF type less UU_in_A, THEN subst])
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 apply (simp add: type_definition.Rep_inject [OF type])
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done
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subsection {* HOLCF type definition package *}
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use "pcpodef_package.ML"
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end