(*  Title:      HOLCF/Domain.thy

    ID:         $Id$

    Author:     Brian Huffman

*)

header {* Domain package *}

theory Domain

imports Ssum Sprod Up One Tr Fixrec

(*

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_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

next

  assume "x \<sqsubseteq> y"

  thus "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 (in iso) 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 (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>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 (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 (rule iso.abs_defined [OF iso.swap])

lemma (in iso) 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 (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

qed

lemma (in iso) compact_rep_rev: "compact (rep\<cdot>x) \<Longrightarrow> compact x"

by (rule iso.compact_abs_rev [OF iso.swap])

lemma (in iso) 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 (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 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 simp: norm_hhf_eq)

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 {*

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_compact_abs = thm "iso.compact_abs";

val iso_compact_rep = thm "iso.compact_rep";

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