src/HOLCF/Completion.thy

Hilbert_Classical: sequential loading due to @{prf}, which joins within a critical section (via options);

(*  Title:      HOLCF/Completion.thy
    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::below"
  assumes type: "type_definition Rep Abs {S. ideal S}"
  assumes below: "\<And>x y. x \<sqsubseteq> y \<longleftrightarrow> Rep x \<subseteq> Rep y"
  shows "OFCLASS('b, po_class)"
 apply (intro_classes, unfold below)
   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 below: "\<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 below [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: below 2, fast)
    apply (simp add: below 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 below: "\<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 below])

end

interpretation below: preorder "below :: 'a::po \<Rightarrow> 'a \<Rightarrow> bool"
apply unfold_locales
apply (rule below_refl)
apply (erule (1) below_trans)
done

subsection {* Lemmas about least upper bounds *}

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)

subsection {* Locale for ideal completion *}

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 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 below_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 below_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_below: "a \<in> rep x \<longleftrightarrow> principal a \<sqsubseteq> x"
by (simp add: rep_eq)

lemma principal_below_iff_mem_rep: "principal a \<sqsubseteq> x \<longleftrightarrow> a \<in> rep x"
by (simp add: rep_eq)

lemma principal_below_iff [simp]: "principal a \<sqsubseteq> principal b \<longleftrightarrow> a \<preceq> b"
by (simp add: principal_below_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_below_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_below_iff)

lemma belowI: "(\<And>a. principal a \<sqsubseteq> x \<Longrightarrow> principal a \<sqsubseteq> u) \<Longrightarrow> x \<sqsubseteq> u"
unfolding principal_below_iff_mem_rep
by (simp add: below_def subset_eq)

lemma lub_principal_rep: "principal ` rep x <<| x"
apply (rule is_lubI)
apply (rule ub_imageI)
apply (erule repD)
apply (subst below_def)
apply (rule subsetI)
apply (drule (1) ub_imageD)
apply (simp add: rep_eq)
done

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

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_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 below_trans [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_below_trans)
   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 below_trans [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 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_below_trans [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 below: "\<And>a. f a \<sqsubseteq> g a"
  shows "basis_fun f \<sqsubseteq> basis_fun g"
 apply (rule below_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 below_trans [OF below])
 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_below_iff_mem_rep rep_contlub)

subsection {* Bifiniteness of ideal completions *}

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