(*  Title:      HOLCF/Domain.thy

     ID:         $Id$

     Author:     Brian Huffman

 *)

 header {* Domain package *}

 theory Domain

 imports Ssum Sprod Up One Tr Fixrec

 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_less: "(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_less: "(rep\<cdot>x \<sqsubseteq> rep\<cdot>y) = (x \<sqsubseteq> y)"

   by (rule iso.abs_less [OF swap])

 lemma abs_eq: "(abs\<cdot>x = abs\<cdot>y) = (x = y)"

   by (simp add: po_eq_conv abs_less)

 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 (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)

   then show "adm (\<lambda>y. \<not> x \<sqsubseteq> y)" using abs_less 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

 end