--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOLCF/Domain.thy Sat Apr 16 00:16:44 2005 +0200
@@ -0,0 +1,180 @@
+(* Title: HOLCF/Domain.thy
+ ID: $Id$
+ Author: Brian Huffman
+ License: GPL (GNU GENERAL PUBLIC LICENSE)
+*)
+
+header {* Domain package *}
+
+theory Domain
+imports Ssum Sprod One Up
+files
+ ("domain/library.ML")
+ ("domain/syntax.ML")
+ ("domain/axioms.ML")
+ ("domain/theorems.ML")
+ ("domain/extender.ML")
+ ("domain/interface.ML")
+begin
+
+defaultsort pcpo
+
+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"
+
+lemma (in iso) swap: "iso rep abs"
+by (rule iso.intro [OF rep_iso abs_iso])
+
+lemma (in iso) 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)
+qed
+
+lemma (in iso) rep_strict: "rep\<cdot>\<bottom> = \<bottom>"
+by (rule iso.abs_strict [OF swap])
+
+lemma (in iso) abs_defin': "abs\<cdot>z = \<bottom> \<Longrightarrow> z = \<bottom>"
+proof -
+ assume A: "abs\<cdot>z = \<bottom>"
+ have "z = rep\<cdot>(abs\<cdot>z)" by simp
+ also have "\<dots> = rep\<cdot>\<bottom>" by (simp only: A)
+ also note rep_strict
+ finally show "z = \<bottom>" .
+qed
+
+lemma (in iso) 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 (in iso) rep_defined: "z \<noteq> \<bottom> \<Longrightarrow> rep\<cdot>z \<noteq> \<bottom>"
+by (erule contrapos_nn, erule rep_defin')
+
+lemma (in iso) 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
+next
+ assume "rep\<cdot>x = y"
+ hence "abs\<cdot>(rep\<cdot>x) = abs\<cdot>y" by simp
+ thus "x = abs\<cdot>y" by simp
+qed
+
+subsection {* Casedist *}
+
+lemma ex_one_defined_iff:
+ "(\<exists>x. P x \<and> x \<noteq> \<bottom>) = P ONE"
+ apply safe
+ apply (rule_tac p=x in oneE)
+ apply simp
+ apply simp
+ apply force
+done
+
+lemma ex_up_defined_iff:
+ "(\<exists>x. P x \<and> x \<noteq> \<bottom>) = (\<exists>x. P (up\<cdot>x))"
+ apply safe
+ apply (rule_tac p=x in upE1)
+ apply simp
+ apply fast
+ apply (force intro!: defined_up)
+done
+
+lemma ex_sprod_defined_iff:
+ "(\<exists>y. P y \<and> y \<noteq> \<bottom>) =
+ (\<exists>x y. (P (:x, y:) \<and> x \<noteq> \<bottom>) \<and> y \<noteq> \<bottom>)"
+ apply safe
+ apply (rule_tac p=y in sprodE)
+ apply simp
+ apply fast
+ apply (force intro!: defined_spair)
+done
+
+lemma ex_sprod_up_defined_iff:
+ "(\<exists>y. P y \<and> y \<noteq> \<bottom>) =
+ (\<exists>x y. P (:up\<cdot>x, y:) \<and> y \<noteq> \<bottom>)"
+ apply safe
+ apply (rule_tac p=y in sprodE)
+ apply simp
+ apply (rule_tac p=x in upE1)
+ apply simp
+ apply fast
+ apply (force intro!: defined_spair)
+done
+
+lemma ex_ssum_defined_iff:
+ "(\<exists>x. P x \<and> x \<noteq> \<bottom>) =
+ ((\<exists>x. P (sinl\<cdot>x) \<and> x \<noteq> \<bottom>) \<or>
+ (\<exists>x. P (sinr\<cdot>x) \<and> x \<noteq> \<bottom>))"
+ apply (rule iffI)
+ apply (erule exE)
+ apply (erule conjE)
+ apply (rule_tac p=x in ssumE)
+ apply simp
+ apply (rule disjI1, fast)
+ apply (rule disjI2, fast)
+ apply (erule disjE)
+ apply (force intro: defined_sinl)
+ apply (force intro: defined_sinr)
+done
+
+lemma exh_start: "p = \<bottom> \<or> (\<exists>x. p = x \<and> x \<noteq> \<bottom>)"
+by auto
+
+lemmas ex_defined_iffs =
+ ex_ssum_defined_iff
+ ex_sprod_up_defined_iff
+ ex_sprod_defined_iff
+ ex_up_defined_iff
+ ex_one_defined_iff
+
+text {* Rules for turning exh into casedist *}
+
+lemma exh_casedist0: "\<lbrakk>R; R \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P" (* like make_elim *)
+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
+
+lemma exh_casedist2: "(\<exists>x. P x \<Longrightarrow> Q) \<equiv> (\<And>x. P x \<Longrightarrow> Q)"
+by rule auto
+
+lemma exh_casedist3: "(P \<and> Q \<Longrightarrow> R) \<equiv> (P \<Longrightarrow> Q \<Longrightarrow> R)"
+by rule auto
+
+lemmas exh_casedists = exh_casedist1 exh_casedist2 exh_casedist3
+
+
+subsection {* Setting up the package *}
+
+ML_setup {*
+val iso_intro = thm "iso.intro";
+val iso_abs_iso = thm "iso.abs_iso";
+val iso_rep_iso = thm "iso.rep_iso";
+val iso_abs_strict = thm "iso.abs_strict";
+val iso_rep_strict = thm "iso.rep_strict";
+val iso_abs_defin' = thm "iso.abs_defin'";
+val iso_rep_defin' = thm "iso.rep_defin'";
+val iso_abs_defined = thm "iso.abs_defined";
+val iso_rep_defined = thm "iso.rep_defined";
+val iso_iso_swap = thm "iso.iso_swap";
+
+val exh_start = thm "exh_start";
+val ex_defined_iffs = thms "ex_defined_iffs";
+val exh_casedist0 = thm "exh_casedist0";
+val exh_casedists = thms "exh_casedists";
+*}
+
+end