--- a/src/HOLCF/Domain_Aux.thy Mon Mar 08 08:12:48 2010 -0800
+++ b/src/HOLCF/Domain_Aux.thy Mon Mar 08 09:33:05 2010 -0800
@@ -10,6 +10,107 @@
("Tools/Domain/domain_take_proofs.ML")
begin
+subsection {* Continuous isomorphisms *}
+
+text {* A locale for continuous isomorphisms *}
+
+locale iso =
+ fixes abs :: "'a \<rightarrow> 'b"
+ fixes rep :: "'b \<rightarrow> 'a"
+ assumes abs_iso [simp]: "rep\<cdot>(abs\<cdot>x) = x"
+ assumes rep_iso [simp]: "abs\<cdot>(rep\<cdot>y) = y"
+begin
+
+lemma swap: "iso rep abs"
+ by (rule iso.intro [OF rep_iso abs_iso])
+
+lemma abs_below: "(abs\<cdot>x \<sqsubseteq> abs\<cdot>y) = (x \<sqsubseteq> y)"
+proof
+ assume "abs\<cdot>x \<sqsubseteq> abs\<cdot>y"
+ then have "rep\<cdot>(abs\<cdot>x) \<sqsubseteq> rep\<cdot>(abs\<cdot>y)" by (rule monofun_cfun_arg)
+ then show "x \<sqsubseteq> y" by simp
+next
+ assume "x \<sqsubseteq> y"
+ then show "abs\<cdot>x \<sqsubseteq> abs\<cdot>y" by (rule monofun_cfun_arg)
+qed
+
+lemma rep_below: "(rep\<cdot>x \<sqsubseteq> rep\<cdot>y) = (x \<sqsubseteq> y)"
+ by (rule iso.abs_below [OF swap])
+
+lemma abs_eq: "(abs\<cdot>x = abs\<cdot>y) = (x = y)"
+ by (simp add: po_eq_conv abs_below)
+
+lemma rep_eq: "(rep\<cdot>x = rep\<cdot>y) = (x = y)"
+ by (rule iso.abs_eq [OF swap])
+
+lemma abs_strict: "abs\<cdot>\<bottom> = \<bottom>"
+proof -
+ have "\<bottom> \<sqsubseteq> rep\<cdot>\<bottom>" ..
+ then have "abs\<cdot>\<bottom> \<sqsubseteq> abs\<cdot>(rep\<cdot>\<bottom>)" by (rule monofun_cfun_arg)
+ then have "abs\<cdot>\<bottom> \<sqsubseteq> \<bottom>" by simp
+ then show ?thesis by (rule UU_I)
+qed
+
+lemma rep_strict: "rep\<cdot>\<bottom> = \<bottom>"
+ by (rule iso.abs_strict [OF swap])
+
+lemma abs_defin': "abs\<cdot>x = \<bottom> \<Longrightarrow> x = \<bottom>"
+proof -
+ have "x = rep\<cdot>(abs\<cdot>x)" by simp
+ also assume "abs\<cdot>x = \<bottom>"
+ also note rep_strict
+ finally show "x = \<bottom>" .
+qed
+
+lemma rep_defin': "rep\<cdot>z = \<bottom> \<Longrightarrow> z = \<bottom>"
+ by (rule iso.abs_defin' [OF swap])
+
+lemma abs_defined: "z \<noteq> \<bottom> \<Longrightarrow> abs\<cdot>z \<noteq> \<bottom>"
+ by (erule contrapos_nn, erule abs_defin')
+
+lemma rep_defined: "z \<noteq> \<bottom> \<Longrightarrow> rep\<cdot>z \<noteq> \<bottom>"
+ by (rule iso.abs_defined [OF iso.swap]) (rule iso_axioms)
+
+lemma abs_defined_iff: "(abs\<cdot>x = \<bottom>) = (x = \<bottom>)"
+ by (auto elim: abs_defin' intro: abs_strict)
+
+lemma rep_defined_iff: "(rep\<cdot>x = \<bottom>) = (x = \<bottom>)"
+ by (rule iso.abs_defined_iff [OF iso.swap]) (rule iso_axioms)
+
+lemma casedist_rule: "rep\<cdot>x = \<bottom> \<or> P \<Longrightarrow> x = \<bottom> \<or> P"
+ by (simp add: rep_defined_iff)
+
+lemma compact_abs_rev: "compact (abs\<cdot>x) \<Longrightarrow> compact x"
+proof (unfold compact_def)
+ assume "adm (\<lambda>y. \<not> abs\<cdot>x \<sqsubseteq> y)"
+ with cont_Rep_CFun2
+ have "adm (\<lambda>y. \<not> abs\<cdot>x \<sqsubseteq> abs\<cdot>y)" by (rule adm_subst)
+ then show "adm (\<lambda>y. \<not> x \<sqsubseteq> y)" using abs_below by simp
+qed
+
+lemma compact_rep_rev: "compact (rep\<cdot>x) \<Longrightarrow> compact x"
+ by (rule iso.compact_abs_rev [OF iso.swap]) (rule iso_axioms)
+
+lemma compact_abs: "compact x \<Longrightarrow> compact (abs\<cdot>x)"
+ by (rule compact_rep_rev) simp
+
+lemma compact_rep: "compact x \<Longrightarrow> compact (rep\<cdot>x)"
+ by (rule iso.compact_abs [OF iso.swap]) (rule iso_axioms)
+
+lemma iso_swap: "(x = abs\<cdot>y) = (rep\<cdot>x = y)"
+proof
+ assume "x = abs\<cdot>y"
+ then have "rep\<cdot>x = rep\<cdot>(abs\<cdot>y)" by simp
+ then show "rep\<cdot>x = y" by simp
+next
+ assume "rep\<cdot>x = y"
+ then have "abs\<cdot>(rep\<cdot>x) = abs\<cdot>y" by simp
+ then show "x = abs\<cdot>y" by simp
+qed
+
+end
+
+
subsection {* Proofs about take functions *}
text {*
@@ -45,6 +146,104 @@
ultimately show ?thesis by simp
qed
+lemma lub_ID_take_lemma:
+ assumes "chain t" and "(\<Squnion>n. t n) = ID"
+ assumes "\<And>n. t n\<cdot>x = t n\<cdot>y" shows "x = y"
+proof -
+ have "(\<Squnion>n. t n\<cdot>x) = (\<Squnion>n. t n\<cdot>y)"
+ using assms(3) by simp
+ then have "(\<Squnion>n. t n)\<cdot>x = (\<Squnion>n. t n)\<cdot>y"
+ using assms(1) by (simp add: lub_distribs)
+ then show "x = y"
+ using assms(2) by simp
+qed
+
+lemma lub_ID_reach:
+ assumes "chain t" and "(\<Squnion>n. t n) = ID"
+ shows "(\<Squnion>n. t n\<cdot>x) = x"
+using assms by (simp add: lub_distribs)
+
+
+subsection {* Finiteness *}
+
+text {*
+ Let a ``decisive'' function be a deflation that maps every input to
+ either itself or bottom. Then if a domain's take functions are all
+ decisive, then all values in the domain are finite.
+*}
+
+definition
+ decisive :: "('a::pcpo \<rightarrow> 'a) \<Rightarrow> bool"
+where
+ "decisive d \<longleftrightarrow> (\<forall>x. d\<cdot>x = x \<or> d\<cdot>x = \<bottom>)"
+
+lemma decisiveI: "(\<And>x. d\<cdot>x = x \<or> d\<cdot>x = \<bottom>) \<Longrightarrow> decisive d"
+ unfolding decisive_def by simp
+
+lemma decisive_cases:
+ assumes "decisive d" obtains "d\<cdot>x = x" | "d\<cdot>x = \<bottom>"
+using assms unfolding decisive_def by auto
+
+lemma decisive_bottom: "decisive \<bottom>"
+ unfolding decisive_def by simp
+
+lemma decisive_ID: "decisive ID"
+ unfolding decisive_def by simp
+
+lemma decisive_ssum_map:
+ assumes f: "decisive f"
+ assumes g: "decisive g"
+ shows "decisive (ssum_map\<cdot>f\<cdot>g)"
+apply (rule decisiveI, rename_tac s)
+apply (case_tac s, simp_all)
+apply (rule_tac x=x in decisive_cases [OF f], simp_all)
+apply (rule_tac x=y in decisive_cases [OF g], simp_all)
+done
+
+lemma decisive_sprod_map:
+ assumes f: "decisive f"
+ assumes g: "decisive g"
+ shows "decisive (sprod_map\<cdot>f\<cdot>g)"
+apply (rule decisiveI, rename_tac s)
+apply (case_tac s, simp_all)
+apply (rule_tac x=x in decisive_cases [OF f], simp_all)
+apply (rule_tac x=y in decisive_cases [OF g], simp_all)
+done
+
+lemma decisive_abs_rep:
+ fixes abs rep
+ assumes iso: "iso abs rep"
+ assumes d: "decisive d"
+ shows "decisive (abs oo d oo rep)"
+apply (rule decisiveI)
+apply (rule_tac x="rep\<cdot>x" in decisive_cases [OF d])
+apply (simp add: iso.rep_iso [OF iso])
+apply (simp add: iso.abs_strict [OF iso])
+done
+
+lemma lub_ID_finite:
+ assumes chain: "chain d"
+ assumes lub: "(\<Squnion>n. d n) = ID"
+ assumes decisive: "\<And>n. decisive (d n)"
+ shows "\<exists>n. d n\<cdot>x = x"
+proof -
+ have 1: "chain (\<lambda>n. d n\<cdot>x)" using chain by simp
+ have 2: "(\<Squnion>n. d n\<cdot>x) = x" using chain lub by (rule lub_ID_reach)
+ have "\<forall>n. d n\<cdot>x = x \<or> d n\<cdot>x = \<bottom>"
+ using decisive unfolding decisive_def by simp
+ hence "range (\<lambda>n. d n\<cdot>x) \<subseteq> {x, \<bottom>}"
+ by auto
+ hence "finite (range (\<lambda>n. d n\<cdot>x))"
+ by (rule finite_subset, simp)
+ with 1 have "finite_chain (\<lambda>n. d n\<cdot>x)"
+ by (rule finite_range_imp_finch)
+ then have "\<exists>n. (\<Squnion>n. d n\<cdot>x) = d n\<cdot>x"
+ unfolding finite_chain_def by (auto simp add: maxinch_is_thelub)
+ with 2 show "\<exists>n. d n\<cdot>x = x" by (auto elim: sym)
+qed
+
+subsection {* ML setup *}
+
use "Tools/Domain/domain_take_proofs.ML"
end