(*  Title:      HOLCF/Completion.thy

     ID:         $Id$

     Author:     Brian Huffman

 *)

 header {* Defining bifinite domains by ideal completion *}

 theory Completion

 imports Bifinite

 begin

 subsection {* Ideals over a preorder *}

 locale preorder =

   fixes r :: "'a::type \<Rightarrow> 'a \<Rightarrow> bool" (infix "\<preceq>" 50)

   assumes r_refl: "x \<preceq> x"

   assumes r_trans: "\<lbrakk>x \<preceq> y; y \<preceq> z\<rbrakk> \<Longrightarrow> x \<preceq> z"

 begin

 definition

   ideal :: "'a set \<Rightarrow> bool" where

   "ideal A = ((\<exists>x. x \<in> A) \<and> (\<forall>x\<in>A. \<forall>y\<in>A. \<exists>z\<in>A. x \<preceq> z \<and> y \<preceq> z) \<and>

     (\<forall>x y. x \<preceq> y \<longrightarrow> y \<in> A \<longrightarrow> x \<in> A))"

 lemma idealI:

   assumes "\<exists>x. x \<in> A"

   assumes "\<And>x y. \<lbrakk>x \<in> A; y \<in> A\<rbrakk> \<Longrightarrow> \<exists>z\<in>A. x \<preceq> z \<and> y \<preceq> z"

   assumes "\<And>x y. \<lbrakk>x \<preceq> y; y \<in> A\<rbrakk> \<Longrightarrow> x \<in> A"

   shows "ideal A"

 unfolding ideal_def using prems by fast

 lemma idealD1:

   "ideal A \<Longrightarrow> \<exists>x. x \<in> A"

 unfolding ideal_def by fast

 lemma idealD2:

   "\<lbrakk>ideal A; x \<in> A; y \<in> A\<rbrakk> \<Longrightarrow> \<exists>z\<in>A. x \<preceq> z \<and> y \<preceq> z"

 unfolding ideal_def by fast

 lemma idealD3:

   "\<lbrakk>ideal A; x \<preceq> y; y \<in> A\<rbrakk> \<Longrightarrow> x \<in> A"

 unfolding ideal_def by fast

 lemma ideal_directed_finite:

   assumes A: "ideal A"

   shows "\<lbrakk>finite U; U \<subseteq> A\<rbrakk> \<Longrightarrow> \<exists>z\<in>A. \<forall>x\<in>U. x \<preceq> z"

 apply (induct U set: finite)

 apply (simp add: idealD1 [OF A])

 apply (simp, clarify, rename_tac y)

 apply (drule (1) idealD2 [OF A])

 apply (clarify, erule_tac x=z in rev_bexI)

 apply (fast intro: r_trans)

 done

 lemma ideal_principal: "ideal {x. x \<preceq> z}"

 apply (rule idealI)

 apply (rule_tac x=z in exI)

 apply (fast intro: r_refl)

 apply (rule_tac x=z in bexI, fast)

 apply (fast intro: r_refl)

 apply (fast intro: r_trans)

 done

 lemma ex_ideal: "\<exists>A. ideal A"

 by (rule exI, rule ideal_principal)

 lemma directed_image_ideal:

   assumes A: "ideal A"

   assumes f: "\<And>x y. x \<preceq> y \<Longrightarrow> f x \<sqsubseteq> f y"

   shows "directed (f ` A)"

 apply (rule directedI)

 apply (cut_tac idealD1 [OF A], fast)

 apply (clarify, rename_tac a b)

 apply (drule (1) idealD2 [OF A])

 apply (clarify, rename_tac c)

 apply (rule_tac x="f c" in rev_bexI)

 apply (erule imageI)

 apply (simp add: f)

 done

 lemma lub_image_principal:

   assumes f: "\<And>x y. x \<preceq> y \<Longrightarrow> f x \<sqsubseteq> f y"

   shows "(\<Squnion>x\<in>{x. x \<preceq> y}. f x) = f y"

 apply (rule thelubI)

 apply (rule is_lub_maximal)

 apply (rule ub_imageI)

 apply (simp add: f)

 apply (rule imageI)

 apply (simp add: r_refl)

 done

 text {* The set of ideals is a cpo *}

 lemma ideal_UN:

   fixes A :: "nat \<Rightarrow> 'a set"

   assumes ideal_A: "\<And>i. ideal (A i)"

   assumes chain_A: "\<And>i j. i \<le> j \<Longrightarrow> A i \<subseteq> A j"

   shows "ideal (\<Union>i. A i)"

  apply (rule idealI)

    apply (cut_tac idealD1 [OF ideal_A], fast)

   apply (clarify, rename_tac i j)

   apply (drule subsetD [OF chain_A [OF le_maxI1]])

   apply (drule subsetD [OF chain_A [OF le_maxI2]])

   apply (drule (1) idealD2 [OF ideal_A])

   apply blast

  apply clarify

  apply (drule (1) idealD3 [OF ideal_A])

  apply fast

 done

 lemma typedef_ideal_po:

   fixes Abs :: "'a set \<Rightarrow> 'b::sq_ord"

   assumes type: "type_definition Rep Abs {S. ideal S}"

   assumes less: "\<And>x y. x \<sqsubseteq> y \<longleftrightarrow> Rep x \<subseteq> Rep y"

   shows "OFCLASS('b, po_class)"

  apply (intro_classes, unfold less)

    apply (rule subset_refl)

   apply (erule (1) subset_trans)

  apply (rule type_definition.Rep_inject [OF type, THEN iffD1])

  apply (erule (1) subset_antisym)

 done

 lemma

   fixes Abs :: "'a set \<Rightarrow> 'b::po"

   assumes type: "type_definition Rep Abs {S. ideal S}"

   assumes less: "\<And>x y. x \<sqsubseteq> y \<longleftrightarrow> Rep x \<subseteq> Rep y"

   assumes S: "chain S"

   shows typedef_ideal_lub: "range S <<| Abs (\<Union>i. Rep (S i))"

     and typedef_ideal_rep_contlub: "Rep (\<Squnion>i. S i) = (\<Union>i. Rep (S i))"

 proof -

   have 1: "ideal (\<Union>i. Rep (S i))"

     apply (rule ideal_UN)

      apply (rule type_definition.Rep [OF type, unfolded mem_Collect_eq])

     apply (subst less [symmetric])

     apply (erule chain_mono [OF S])

     done

   hence 2: "Rep (Abs (\<Union>i. Rep (S i))) = (\<Union>i. Rep (S i))"

     by (simp add: type_definition.Abs_inverse [OF type])

   show 3: "range S <<| Abs (\<Union>i. Rep (S i))"

     apply (rule is_lubI)

      apply (rule is_ubI)

      apply (simp add: less 2, fast)

     apply (simp add: less 2 is_ub_def, fast)

     done

   hence 4: "(\<Squnion>i. S i) = Abs (\<Union>i. Rep (S i))"

     by (rule thelubI)

   show 5: "Rep (\<Squnion>i. S i) = (\<Union>i. Rep (S i))"

     by (simp add: 4 2)

 qed

 lemma typedef_ideal_cpo:

   fixes Abs :: "'a set \<Rightarrow> 'b::po"

   assumes type: "type_definition Rep Abs {S. ideal S}"

   assumes less: "\<And>x y. x \<sqsubseteq> y \<longleftrightarrow> Rep x \<subseteq> Rep y"

   shows "OFCLASS('b, cpo_class)"

 by (default, rule exI, erule typedef_ideal_lub [OF type less])

 end

 interpretation sq_le: preorder ["sq_le :: 'a::po \<Rightarrow> 'a \<Rightarrow> bool"]

 apply unfold_locales

 apply (rule refl_less)

 apply (erule (1) trans_less)

 done

 subsection {* Defining functions in terms of basis elements *}

 lemma finite_directed_contains_lub:

   "\<lbrakk>finite S; directed S\<rbrakk> \<Longrightarrow> \<exists>u\<in>S. S <<| u"

 apply (drule (1) directed_finiteD, rule subset_refl)

 apply (erule bexE)

 apply (rule rev_bexI, assumption)

 apply (erule (1) is_lub_maximal)

 done

 lemma lub_finite_directed_in_self:

   "\<lbrakk>finite S; directed S\<rbrakk> \<Longrightarrow> lub S \<in> S"

 apply (drule (1) finite_directed_contains_lub, clarify)

 apply (drule thelubI, simp)

 done

 lemma finite_directed_has_lub: "\<lbrakk>finite S; directed S\<rbrakk> \<Longrightarrow> \<exists>u. S <<| u"

 by (drule (1) finite_directed_contains_lub, fast)

 lemma is_ub_thelub0: "\<lbrakk>\<exists>u. S <<| u; x \<in> S\<rbrakk> \<Longrightarrow> x \<sqsubseteq> lub S"

 apply (erule exE, drule lubI)

 apply (drule is_lubD1)

 apply (erule (1) is_ubD)

 done

 lemma is_lub_thelub0: "\<lbrakk>\<exists>u. S <<| u; S <| x\<rbrakk> \<Longrightarrow> lub S \<sqsubseteq> x"

 by (erule exE, drule lubI, erule is_lub_lub)

 locale basis_take = preorder +

   fixes take :: "nat \<Rightarrow> 'a::type \<Rightarrow> 'a"

   assumes take_less: "take n a \<preceq> a"

   assumes take_take: "take n (take n a) = take n a"

   assumes take_mono: "a \<preceq> b \<Longrightarrow> take n a \<preceq> take n b"

   assumes take_chain: "take n a \<preceq> take (Suc n) a"

   assumes finite_range_take: "finite (range (take n))"

   assumes take_covers: "\<exists>n. take n a = a"

 begin

 lemma take_chain_less: "m < n \<Longrightarrow> take m a \<preceq> take n a"

 by (erule less_Suc_induct, rule take_chain, erule (1) r_trans)

 lemma take_chain_le: "m \<le> n \<Longrightarrow> take m a \<preceq> take n a"

 by (cases "m = n", simp add: r_refl, simp add: take_chain_less)

 end

 locale ideal_completion = basis_take +

   fixes principal :: "'a::type \<Rightarrow> 'b::cpo"

   fixes rep :: "'b::cpo \<Rightarrow> 'a::type set"

   assumes ideal_rep: "\<And>x. preorder.ideal r (rep x)"

   assumes rep_contlub: "\<And>Y. chain Y \<Longrightarrow> rep (\<Squnion>i. Y i) = (\<Union>i. rep (Y i))"

   assumes rep_principal: "\<And>a. rep (principal a) = {b. b \<preceq> a}"

   assumes subset_repD: "\<And>x y. rep x \<subseteq> rep y \<Longrightarrow> x \<sqsubseteq> y"

 begin

 lemma finite_take_rep: "finite (take n ` rep x)"

 by (rule finite_subset [OF image_mono [OF subset_UNIV] finite_range_take])

 lemma basis_fun_lemma0:

   fixes f :: "'a::type \<Rightarrow> 'c::cpo"

   assumes f_mono: "\<And>a b. a \<preceq> b \<Longrightarrow> f a \<sqsubseteq> f b"

   shows "\<exists>u. f ` take i ` rep x <<| u"

 apply (rule finite_directed_has_lub)

 apply (rule finite_imageI)

 apply (rule finite_take_rep)

 apply (subst image_image)

 apply (rule directed_image_ideal)

 apply (rule ideal_rep)

 apply (rule f_mono)

 apply (erule take_mono)

 done

 lemma basis_fun_lemma1:

   fixes f :: "'a::type \<Rightarrow> 'c::cpo"

   assumes f_mono: "\<And>a b. a \<preceq> b \<Longrightarrow> f a \<sqsubseteq> f b"

   shows "chain (\<lambda>i. lub (f ` take i ` rep x))"

  apply (rule chainI)

  apply (rule is_lub_thelub0)

   apply (rule basis_fun_lemma0, erule f_mono)

  apply (rule is_ubI, clarsimp, rename_tac a)

  apply (rule trans_less [OF f_mono [OF take_chain]])

  apply (rule is_ub_thelub0)

   apply (rule basis_fun_lemma0, erule f_mono)

  apply simp

 done

 lemma basis_fun_lemma2:

   fixes f :: "'a::type \<Rightarrow> 'c::cpo"

   assumes f_mono: "\<And>a b. a \<preceq> b \<Longrightarrow> f a \<sqsubseteq> f b"

   shows "f ` rep x <<| (\<Squnion>i. lub (f ` take i ` rep x))"

  apply (rule is_lubI)

  apply (rule ub_imageI, rename_tac a)

   apply (cut_tac a=a in take_covers, erule exE, rename_tac i)

   apply (erule subst)

   apply (rule rev_trans_less)

    apply (rule_tac x=i in is_ub_thelub)

    apply (rule basis_fun_lemma1, erule f_mono)

   apply (rule is_ub_thelub0)

    apply (rule basis_fun_lemma0, erule f_mono)

   apply simp

  apply (rule is_lub_thelub [OF _ ub_rangeI])

   apply (rule basis_fun_lemma1, erule f_mono)

  apply (rule is_lub_thelub0)

   apply (rule basis_fun_lemma0, erule f_mono)

  apply (rule is_ubI, clarsimp, rename_tac a)

  apply (rule trans_less [OF f_mono [OF take_less]])

  apply (erule (1) ub_imageD)

 done

 lemma basis_fun_lemma:

   fixes f :: "'a::type \<Rightarrow> 'c::cpo"

   assumes f_mono: "\<And>a b. a \<preceq> b \<Longrightarrow> f a \<sqsubseteq> f b"

   shows "\<exists>u. f ` rep x <<| u"

 by (rule exI, rule basis_fun_lemma2, erule f_mono)

 lemma rep_mono: "x \<sqsubseteq> y \<Longrightarrow> rep x \<subseteq> rep y"

 apply (frule bin_chain)

 apply (drule rep_contlub)

 apply (simp only: thelubI [OF lub_bin_chain])

 apply (rule subsetI, rule UN_I [where a=0], simp_all)

 done

 lemma less_def: "x \<sqsubseteq> y \<longleftrightarrow> rep x \<subseteq> rep y"

 by (rule iffI [OF rep_mono subset_repD])

 lemma rep_eq: "rep x = {a. principal a \<sqsubseteq> x}"

 unfolding less_def rep_principal

 apply safe

 apply (erule (1) idealD3 [OF ideal_rep])

 apply (erule subsetD, simp add: r_refl)

 done

 lemma mem_rep_iff_principal_less: "a \<in> rep x \<longleftrightarrow> principal a \<sqsubseteq> x"

 by (simp add: rep_eq)

 lemma principal_less_iff_mem_rep: "principal a \<sqsubseteq> x \<longleftrightarrow> a \<in> rep x"

 by (simp add: rep_eq)

 lemma principal_less_iff [simp]: "principal a \<sqsubseteq> principal b \<longleftrightarrow> a \<preceq> b"

 by (simp add: principal_less_iff_mem_rep rep_principal)

 lemma principal_eq_iff: "principal a = principal b \<longleftrightarrow> a \<preceq> b \<and> b \<preceq> a"

 unfolding po_eq_conv [where 'a='b] principal_less_iff ..

 lemma repD: "a \<in> rep x \<Longrightarrow> principal a \<sqsubseteq> x"

 by (simp add: rep_eq)

 lemma principal_mono: "a \<preceq> b \<Longrightarrow> principal a \<sqsubseteq> principal b"

 by (simp only: principal_less_iff)

 lemma lessI: "(\<And>a. principal a \<sqsubseteq> x \<Longrightarrow> principal a \<sqsubseteq> u) \<Longrightarrow> x \<sqsubseteq> u"

 unfolding principal_less_iff_mem_rep

 by (simp add: less_def subset_eq)

 lemma lub_principal_rep: "principal ` rep x <<| x"

 apply (rule is_lubI)

 apply (rule ub_imageI)

 apply (erule repD)

 apply (subst less_def)

 apply (rule subsetI)

 apply (drule (1) ub_imageD)

 apply (simp add: rep_eq)

 done

 definition

   basis_fun :: "('a::type \<Rightarrow> 'c::cpo) \<Rightarrow> 'b \<rightarrow> 'c" where

   "basis_fun = (\<lambda>f. (\<Lambda> x. lub (f ` rep x)))"

 lemma basis_fun_beta:

   fixes f :: "'a::type \<Rightarrow> 'c::cpo"

   assumes f_mono: "\<And>a b. a \<preceq> b \<Longrightarrow> f a \<sqsubseteq> f b"

   shows "basis_fun f\<cdot>x = lub (f ` rep x)"

 unfolding basis_fun_def

 proof (rule beta_cfun)

   have lub: "\<And>x. \<exists>u. f ` rep x <<| u"

     using f_mono by (rule basis_fun_lemma)

   show cont: "cont (\<lambda>x. lub (f ` rep x))"

     apply (rule contI2)

      apply (rule monofunI)

      apply (rule is_lub_thelub0 [OF lub ub_imageI])

      apply (rule is_ub_thelub0 [OF lub imageI])

      apply (erule (1) subsetD [OF rep_mono])

     apply (rule is_lub_thelub0 [OF lub ub_imageI])

     apply (simp add: rep_contlub, clarify)

     apply (erule rev_trans_less [OF is_ub_thelub])

     apply (erule is_ub_thelub0 [OF lub imageI])

     done

 qed

 lemma basis_fun_principal:

   fixes f :: "'a::type \<Rightarrow> 'c::cpo"

   assumes f_mono: "\<And>a b. a \<preceq> b \<Longrightarrow> f a \<sqsubseteq> f b"

   shows "basis_fun f\<cdot>(principal a) = f a"

 apply (subst basis_fun_beta, erule f_mono)

 apply (subst rep_principal)

 apply (rule lub_image_principal, erule f_mono)

 done

 lemma basis_fun_mono:

   assumes f_mono: "\<And>a b. a \<preceq> b \<Longrightarrow> f a \<sqsubseteq> f b"

   assumes g_mono: "\<And>a b. a \<preceq> b \<Longrightarrow> g a \<sqsubseteq> g b"

   assumes less: "\<And>a. f a \<sqsubseteq> g a"

   shows "basis_fun f \<sqsubseteq> basis_fun g"

  apply (rule less_cfun_ext)

  apply (simp only: basis_fun_beta f_mono g_mono)

  apply (rule is_lub_thelub0)

   apply (rule basis_fun_lemma, erule f_mono)

  apply (rule ub_imageI, rename_tac a)

  apply (rule trans_less [OF less])

  apply (rule is_ub_thelub0)

   apply (rule basis_fun_lemma, erule g_mono)

  apply (erule imageI)

 done

 lemma compact_principal [simp]: "compact (principal a)"

 by (rule compactI2, simp add: principal_less_iff_mem_rep rep_contlub)

 definition

   completion_approx :: "nat \<Rightarrow> 'b \<rightarrow> 'b" where

   "completion_approx = (\<lambda>i. basis_fun (\<lambda>a. principal (take i a)))"

 lemma completion_approx_beta:

   "completion_approx i\<cdot>x = (\<Squnion>a\<in>rep x. principal (take i a))"

 unfolding completion_approx_def

 by (simp add: basis_fun_beta principal_mono take_mono)

 lemma completion_approx_principal:

   "completion_approx i\<cdot>(principal a) = principal (take i a)"

 unfolding completion_approx_def

 by (simp add: basis_fun_principal principal_mono take_mono)

 lemma chain_completion_approx: "chain completion_approx"

 unfolding completion_approx_def

 apply (rule chainI)

 apply (rule basis_fun_mono)

 apply (erule principal_mono [OF take_mono])

 apply (erule principal_mono [OF take_mono])

 apply (rule principal_mono [OF take_chain])

 done

 lemma lub_completion_approx: "(\<Squnion>i. completion_approx i\<cdot>x) = x"

 unfolding completion_approx_beta

  apply (subst image_image [where f=principal, symmetric])

  apply (rule unique_lub [OF _ lub_principal_rep])

  apply (rule basis_fun_lemma2, erule principal_mono)

 done

 lemma completion_approx_eq_principal:

   "\<exists>a\<in>rep x. completion_approx i\<cdot>x = principal (take i a)"

 unfolding completion_approx_beta

  apply (subst image_image [where f=principal, symmetric])

  apply (subgoal_tac "finite (principal ` take i ` rep x)")

   apply (subgoal_tac "directed (principal ` take i ` rep x)")

    apply (drule (1) lub_finite_directed_in_self, fast)

   apply (subst image_image)

   apply (rule directed_image_ideal)

    apply (rule ideal_rep)

   apply (erule principal_mono [OF take_mono])

  apply (rule finite_imageI)

  apply (rule finite_take_rep)

 done

 lemma completion_approx_idem:

   "completion_approx i\<cdot>(completion_approx i\<cdot>x) = completion_approx i\<cdot>x"

 using completion_approx_eq_principal [where i=i and x=x]

 by (auto simp add: completion_approx_principal take_take)

 lemma finite_fixes_completion_approx:

   "finite {x. completion_approx i\<cdot>x = x}" (is "finite ?S")

 apply (subgoal_tac "?S \<subseteq> principal ` range (take i)")

 apply (erule finite_subset)

 apply (rule finite_imageI)

 apply (rule finite_range_take)

 apply (clarify, erule subst)

 apply (cut_tac x=x and i=i in completion_approx_eq_principal)

 apply fast

 done

 lemma principal_induct:

   assumes adm: "adm P"

   assumes P: "\<And>a. P (principal a)"

   shows "P x"

  apply (subgoal_tac "P (\<Squnion>i. completion_approx i\<cdot>x)")

  apply (simp add: lub_completion_approx)

  apply (rule admD [OF adm])

   apply (simp add: chain_completion_approx)

  apply (cut_tac x=x and i=i in completion_approx_eq_principal)

  apply (clarify, simp add: P)

 done

 lemma principal_induct2:

   "\<lbrakk>\<And>y. adm (\<lambda>x. P x y); \<And>x. adm (\<lambda>y. P x y);

     \<And>a b. P (principal a) (principal b)\<rbrakk> \<Longrightarrow> P x y"

 apply (rule_tac x=y in spec)

 apply (rule_tac x=x in principal_induct, simp)

 apply (rule allI, rename_tac y)

 apply (rule_tac x=y in principal_induct, simp)

 apply simp

 done

 lemma compact_imp_principal: "compact x \<Longrightarrow> \<exists>a. x = principal a"

 apply (drule adm_compact_neq [OF _ cont_id])

 apply (drule admD2 [where Y="\<lambda>n. completion_approx n\<cdot>x"])

 apply (simp add: chain_completion_approx)

 apply (simp add: lub_completion_approx)

 apply (erule exE, erule ssubst)

 apply (cut_tac i=i and x=x in completion_approx_eq_principal)

 apply (clarify, erule exI)

 done

 end

 end