--- a/src/HOLCF/Pcpodef.thy Tue Jul 26 18:22:55 2005 +0200
+++ b/src/HOLCF/Pcpodef.thy Tue Jul 26 18:24:29 2005 +0200
@@ -24,9 +24,9 @@
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+)
+ apply (rule type_definition.Rep_inject [OF type, THEN iffD1])
+ apply (erule (1) antisym_less)
+ apply (erule (1) trans_less)
done
@@ -40,55 +40,55 @@
admissible predicate.
*}
-lemma chain_Rep:
+lemma monofun_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)
+ shows "monofun Rep"
+by (rule monofunI, unfold less)
-lemma lub_Rep_in_A:
+lemmas ch2ch_Rep = ch2ch_monofun [OF monofun_Rep]
+lemmas ub2ub_Rep = ub2ub_monofun [OF monofun_Rep]
+
+lemma Abs_inverse_lub_Rep:
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])
+ 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 less], rule_format])
apply (rule type_definition.Rep [OF type])
done
-theorem typedef_is_lub:
+theorem typedef_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))"
+ shows "chain S \<Longrightarrow> range S <<| Abs (\<Squnion>i. Rep (S i))"
+ apply (frule ch2ch_Rep [OF less])
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
+ apply (simp only: less Abs_inverse_lub_Rep [OF type less adm])
+ apply (erule is_ub_thelub)
+ apply (simp only: less Abs_inverse_lub_Rep [OF type less adm])
+ apply (erule is_lub_thelub)
+ apply (erule ub2ub_Rep [OF less])
done
+lemmas typedef_thelub = typedef_lub [THEN thelubI, standard]
+
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
+proof
+ fix S::"nat \<Rightarrow> 'b" assume "chain S"
+ hence "range S <<| Abs (\<Squnion>i. Rep (S i))"
+ by (rule typedef_lub [OF type less adm])
+ thus "\<exists>x. range S <<| x" ..
+qed
subsubsection {* Continuity of @{term Rep} and @{term Abs} *}
@@ -102,11 +102,10 @@
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 (simp only: typedef_thelub [OF type less adm])
+ apply (simp only: Abs_inverse_lub_Rep [OF type less adm])
apply (rule thelubE [OF _ refl])
- apply (erule chain_Rep [OF less])
+ apply (erule ch2ch_Rep [OF less])
done
text {*
@@ -115,28 +114,31 @@
composing it with another continuous function.
*}
+theorem typedef_is_lubI:
+ assumes less: "op \<sqsubseteq> \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
+ shows "range (\<lambda>i. Rep (S i)) <<| Rep x \<Longrightarrow> range S <<| x"
+ apply (rule is_lubI)
+ apply (rule ub_rangeI)
+ apply (subst less)
+ apply (erule is_ub_lub)
+ apply (subst less)
+ apply (erule is_lub_lub)
+ apply (erule ub2ub_Rep [OF less])
+done
+
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 adm: "adm (\<lambda>x. x \<in> A)" (* not used *)
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: cont2contlubE [OF cont_f])
- apply (rule is_lub_thelub)
- apply (erule ch2ch_cont [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])
+ apply (rule typedef_is_lubI [OF less])
+ apply (simp only: type_definition.Abs_inverse [OF type f_in_A])
+ apply (erule cont_f [THEN contE])
done
subsection {* Proving a subtype is pointed *}
@@ -146,7 +148,7 @@
the defining subset has a least element.
*}
-theorem typedef_pcpo:
+theorem typedef_pcpo_generic:
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"
@@ -165,13 +167,13 @@
if the defining subset contains @{term \<bottom>}.
*}
-theorem typedef_pcpo_UU:
+theorem typedef_pcpo:
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)
+by (rule typedef_pcpo_generic [OF type less UU_in_A], rule minimal)
subsubsection {* Strictness of @{term Rep} and @{term Abs} *}