(* Title: HOLCF/Domain.thy
Author: Brian Huffman
*)
header {* Domain package *}
theory Domain
imports Ssum Sprod Up One Tr Fixrec Representable
uses
("Tools/cont_consts.ML")
("Tools/cont_proc.ML")
("Tools/Domain/domain_constructors.ML")
("Tools/Domain/domain_library.ML")
("Tools/Domain/domain_axioms.ML")
("Tools/Domain/domain_theorems.ML")
("Tools/Domain/domain_extender.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"
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 {* 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 upE)
apply simp
apply fast
apply (force intro!: up_defined)
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!: spair_defined)
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 upE)
apply simp
apply fast
apply (force intro!: spair_defined)
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
apply force
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 {* Combinators for building copy functions *}
lemmas domain_map_stricts =
ssum_map_strict sprod_map_strict u_map_strict
lemmas domain_map_simps =
ssum_map_sinl ssum_map_sinr sprod_map_spair u_map_up
subsection {* Installing the domain package *}
lemmas con_strict_rules =
sinl_strict sinr_strict spair_strict1 spair_strict2
lemmas con_defin_rules =
sinl_defined sinr_defined spair_defined up_defined ONE_defined
lemmas con_defined_iff_rules =
sinl_defined_iff sinr_defined_iff spair_strict_iff up_defined ONE_defined
lemmas con_below_iff_rules =
sinl_below sinr_below sinl_below_sinr sinr_below_sinl con_defined_iff_rules
lemmas con_eq_iff_rules =
sinl_eq sinr_eq sinl_eq_sinr sinr_eq_sinl con_defined_iff_rules
lemmas sel_strict_rules =
cfcomp2 sscase1 sfst_strict ssnd_strict fup1
lemma sel_app_extra_rules:
"sscase\<cdot>ID\<cdot>\<bottom>\<cdot>(sinr\<cdot>x) = \<bottom>"
"sscase\<cdot>ID\<cdot>\<bottom>\<cdot>(sinl\<cdot>x) = x"
"sscase\<cdot>\<bottom>\<cdot>ID\<cdot>(sinl\<cdot>x) = \<bottom>"
"sscase\<cdot>\<bottom>\<cdot>ID\<cdot>(sinr\<cdot>x) = x"
"fup\<cdot>ID\<cdot>(up\<cdot>x) = x"
by (cases "x = \<bottom>", simp, simp)+
lemmas sel_app_rules =
sel_strict_rules sel_app_extra_rules
ssnd_spair sfst_spair up_defined spair_defined
lemmas sel_defined_iff_rules =
cfcomp2 sfst_defined_iff ssnd_defined_iff
lemmas take_con_rules =
ID1 ssum_map_sinl' ssum_map_sinr' ssum_map_strict
sprod_map_spair' sprod_map_strict u_map_up u_map_strict
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)
use "Tools/cont_consts.ML"
use "Tools/cont_proc.ML"
use "Tools/Domain/domain_library.ML"
use "Tools/Domain/domain_axioms.ML"
use "Tools/Domain/domain_constructors.ML"
use "Tools/Domain/domain_theorems.ML"
use "Tools/Domain/domain_extender.ML"
end