--- a/src/HOLCF/Domain.thy Wed Jun 13 18:30:16 2007 +0200
+++ b/src/HOLCF/Domain.thy Wed Jun 13 18:30:17 2007 +0200
@@ -11,6 +11,7 @@
defaultsort pcpo
+
subsection {* Continuous isomorphisms *}
text {* A locale for continuous isomorphisms *}
@@ -20,41 +21,42 @@
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 (in iso) swap: "iso rep abs"
-by (rule iso.intro [OF rep_iso abs_iso])
+lemma swap: "iso rep abs"
+ by (rule iso.intro [OF rep_iso abs_iso])
-lemma (in iso) abs_less: "(abs\<cdot>x \<sqsubseteq> abs\<cdot>y) = (x \<sqsubseteq> y)"
+lemma abs_less: "(abs\<cdot>x \<sqsubseteq> abs\<cdot>y) = (x \<sqsubseteq> y)"
proof
assume "abs\<cdot>x \<sqsubseteq> abs\<cdot>y"
- hence "rep\<cdot>(abs\<cdot>x) \<sqsubseteq> rep\<cdot>(abs\<cdot>y)" by (rule monofun_cfun_arg)
- thus "x \<sqsubseteq> y" by simp
+ 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"
- thus "abs\<cdot>x \<sqsubseteq> abs\<cdot>y" by (rule monofun_cfun_arg)
+ then show "abs\<cdot>x \<sqsubseteq> abs\<cdot>y" by (rule monofun_cfun_arg)
qed
-lemma (in iso) rep_less: "(rep\<cdot>x \<sqsubseteq> rep\<cdot>y) = (x \<sqsubseteq> y)"
-by (rule iso.abs_less [OF swap])
+lemma rep_less: "(rep\<cdot>x \<sqsubseteq> rep\<cdot>y) = (x \<sqsubseteq> y)"
+ by (rule iso.abs_less [OF swap])
-lemma (in iso) abs_eq: "(abs\<cdot>x = abs\<cdot>y) = (x = y)"
-by (simp add: po_eq_conv abs_less)
+lemma abs_eq: "(abs\<cdot>x = abs\<cdot>y) = (x = y)"
+ by (simp add: po_eq_conv abs_less)
-lemma (in iso) rep_eq: "(rep\<cdot>x = rep\<cdot>y) = (x = y)"
-by (rule iso.abs_eq [OF swap])
+lemma rep_eq: "(rep\<cdot>x = rep\<cdot>y) = (x = y)"
+ by (rule iso.abs_eq [OF swap])
-lemma (in iso) abs_strict: "abs\<cdot>\<bottom> = \<bottom>"
+lemma abs_strict: "abs\<cdot>\<bottom> = \<bottom>"
proof -
have "\<bottom> \<sqsubseteq> rep\<cdot>\<bottom>" ..
- hence "abs\<cdot>\<bottom> \<sqsubseteq> abs\<cdot>(rep\<cdot>\<bottom>)" by (rule monofun_cfun_arg)
- hence "abs\<cdot>\<bottom> \<sqsubseteq> \<bottom>" by simp
- thus ?thesis by (rule UU_I)
+ 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 (in iso) rep_strict: "rep\<cdot>\<bottom> = \<bottom>"
-by (rule iso.abs_strict [OF swap])
+lemma rep_strict: "rep\<cdot>\<bottom> = \<bottom>"
+ by (rule iso.abs_strict [OF swap])
-lemma (in iso) abs_defin': "abs\<cdot>x = \<bottom> \<Longrightarrow> x = \<bottom>"
+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>"
@@ -62,49 +64,52 @@
finally show "x = \<bottom>" .
qed
-lemma (in iso) rep_defin': "rep\<cdot>z = \<bottom> \<Longrightarrow> z = \<bottom>"
-by (rule iso.abs_defin' [OF swap])
+lemma rep_defin': "rep\<cdot>z = \<bottom> \<Longrightarrow> z = \<bottom>"
+ by (rule iso.abs_defin' [OF swap])
-lemma (in iso) abs_defined: "z \<noteq> \<bottom> \<Longrightarrow> abs\<cdot>z \<noteq> \<bottom>"
-by (erule contrapos_nn, erule abs_defin')
+lemma abs_defined: "z \<noteq> \<bottom> \<Longrightarrow> abs\<cdot>z \<noteq> \<bottom>"
+ by (erule contrapos_nn, erule abs_defin')
-lemma (in iso) rep_defined: "z \<noteq> \<bottom> \<Longrightarrow> rep\<cdot>z \<noteq> \<bottom>"
-by (rule iso.abs_defined [OF iso.swap])
+lemma rep_defined: "z \<noteq> \<bottom> \<Longrightarrow> rep\<cdot>z \<noteq> \<bottom>"
+ by (rule iso.abs_defined [OF iso.swap]) (rule iso_axioms)
-lemma (in iso) abs_defined_iff: "(abs\<cdot>x = \<bottom>) = (x = \<bottom>)"
-by (auto elim: abs_defin' intro: abs_strict)
+lemma abs_defined_iff: "(abs\<cdot>x = \<bottom>) = (x = \<bottom>)"
+ by (auto elim: abs_defin' intro: abs_strict)
-lemma (in iso) rep_defined_iff: "(rep\<cdot>x = \<bottom>) = (x = \<bottom>)"
-by (rule iso.abs_defined_iff [OF iso.swap])
+lemma rep_defined_iff: "(rep\<cdot>x = \<bottom>) = (x = \<bottom>)"
+ by (rule iso.abs_defined_iff [OF iso.swap]) (rule iso_axioms)
lemma (in iso) 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)
- thus "adm (\<lambda>y. \<not> x \<sqsubseteq> y)" using abs_less by simp
+ then show "adm (\<lambda>y. \<not> x \<sqsubseteq> y)" using abs_less by simp
qed
-lemma (in iso) compact_rep_rev: "compact (rep\<cdot>x) \<Longrightarrow> compact x"
-by (rule iso.compact_abs_rev [OF iso.swap])
+lemma compact_rep_rev: "compact (rep\<cdot>x) \<Longrightarrow> compact x"
+ by (rule iso.compact_abs_rev [OF iso.swap]) (rule iso_axioms)
-lemma (in iso) compact_abs: "compact x \<Longrightarrow> compact (abs\<cdot>x)"
-by (rule compact_rep_rev, simp)
+lemma compact_abs: "compact x \<Longrightarrow> compact (abs\<cdot>x)"
+ by (rule compact_rep_rev) simp
-lemma (in iso) compact_rep: "compact x \<Longrightarrow> compact (rep\<cdot>x)"
-by (rule iso.compact_abs [OF iso.swap])
+lemma compact_rep: "compact x \<Longrightarrow> compact (rep\<cdot>x)"
+ by (rule iso.compact_abs [OF iso.swap]) (rule iso_axioms)
-lemma (in iso) iso_swap: "(x = abs\<cdot>y) = (rep\<cdot>x = y)"
+lemma iso_swap: "(x = abs\<cdot>y) = (rep\<cdot>x = y)"
proof
assume "x = abs\<cdot>y"
- hence "rep\<cdot>x = rep\<cdot>(abs\<cdot>y)" by simp
- thus "rep\<cdot>x = y" by simp
+ 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"
- hence "abs\<cdot>(rep\<cdot>x) = abs\<cdot>y" by simp
- thus "x = abs\<cdot>y" by simp
+ then have "abs\<cdot>(rep\<cdot>x) = abs\<cdot>y" by simp
+ then show "x = abs\<cdot>y" by simp
qed
+end
+
+
subsection {* Casedist *}
lemma ex_one_defined_iff:
@@ -114,7 +119,7 @@
apply simp
apply simp
apply force
-done
+ done
lemma ex_up_defined_iff:
"(\<exists>x. P x \<and> x \<noteq> \<bottom>) = (\<exists>x. P (up\<cdot>x))"
@@ -123,7 +128,7 @@
apply simp
apply fast
apply (force intro!: up_defined)
-done
+ done
lemma ex_sprod_defined_iff:
"(\<exists>y. P y \<and> y \<noteq> \<bottom>) =
@@ -133,7 +138,7 @@
apply simp
apply fast
apply (force intro!: spair_defined)
-done
+ done
lemma ex_sprod_up_defined_iff:
"(\<exists>y. P y \<and> y \<noteq> \<bottom>) =
@@ -145,7 +150,7 @@
apply simp
apply fast
apply (force intro!: spair_defined)
-done
+ done
lemma ex_ssum_defined_iff:
"(\<exists>x. P x \<and> x \<noteq> \<bottom>) =
@@ -161,10 +166,10 @@
apply (erule disjE)
apply force
apply force
-done
+ done
lemma exh_start: "p = \<bottom> \<or> (\<exists>x. p = x \<and> x \<noteq> \<bottom>)"
-by auto
+ by auto
lemmas ex_defined_iffs =
ex_ssum_defined_iff
@@ -176,16 +181,16 @@
text {* Rules for turning exh into casedist *}
lemma exh_casedist0: "\<lbrakk>R; R \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P" (* like make_elim *)
-by auto
+ by auto
lemma exh_casedist1: "((P \<or> Q \<Longrightarrow> R) \<Longrightarrow> S) \<equiv> (\<lbrakk>P \<Longrightarrow> R; Q \<Longrightarrow> R\<rbrakk> \<Longrightarrow> S)"
-by rule auto
+ by rule auto
lemma exh_casedist2: "(\<exists>x. P x \<Longrightarrow> Q) \<equiv> (\<And>x. P x \<Longrightarrow> Q)"
-by rule auto
+ by rule auto
lemma exh_casedist3: "(P \<and> Q \<Longrightarrow> R) \<equiv> (P \<Longrightarrow> Q \<Longrightarrow> R)"
-by rule auto
+ by rule auto
lemmas exh_casedists = exh_casedist1 exh_casedist2 exh_casedist3