(*  Title:      HOLCF/Completion.thy

     Author:     Brian Huffman

 *)

 header {* Defining algebraic domains by ideal completion *}

 theory Completion

 imports Plain_HOLCF

 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_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 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_lub: "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 is_ub_thelub_ex: "\<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_thelub_ex: "\<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 ideal_completion = preorder +

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

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

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

   assumes rep_lub: "\<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"

   assumes countable: "\<exists>f::'a \<Rightarrow> nat. inj f"

 begin

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

 apply (frule bin_chain)

 apply (drule rep_lub)

 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 eq_iff: "x = y \<longleftrightarrow> rep x = rep y"

 unfolding po_eq_conv below_def by auto

 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 ch2ch_principal [simp]:

   "\<forall>i. Y i \<preceq> Y (Suc i) \<Longrightarrow> chain (\<lambda>i. principal (Y i))"

 by (simp add: chainI principal_mono)

 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

 subsubsection {* Principal ideals approximate all elements *}

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

 by (rule compactI2, simp add: principal_below_iff_mem_rep rep_lub)

 text {* Construct a chain whose lub is the same as a given ideal *}

 lemma obtain_principal_chain:

   obtains Y where "\<forall>i. Y i \<preceq> Y (Suc i)" and "x = (\<Squnion>i. principal (Y i))"

 proof -

   obtain count :: "'a \<Rightarrow> nat" where inj: "inj count"

     using countable ..

   def enum \<equiv> "\<lambda>i. THE a. count a = i"

   have enum_count [simp]: "\<And>x. enum (count x) = x"

     unfolding enum_def by (simp add: inj_eq [OF inj])

   def a \<equiv> "LEAST i. enum i \<in> rep x"

   def b \<equiv> "\<lambda>i. LEAST j. enum j \<in> rep x \<and> \<not> enum j \<preceq> enum i"

   def c \<equiv> "\<lambda>i j. LEAST k. enum k \<in> rep x \<and> enum i \<preceq> enum k \<and> enum j \<preceq> enum k"

   def P \<equiv> "\<lambda>i. \<exists>j. enum j \<in> rep x \<and> \<not> enum j \<preceq> enum i"

   def X \<equiv> "nat_rec a (\<lambda>n i. if P i then c i (b i) else i)"

   have X_0: "X 0 = a" unfolding X_def by simp

   have X_Suc: "\<And>n. X (Suc n) = (if P (X n) then c (X n) (b (X n)) else X n)"

     unfolding X_def by simp

   have a_mem: "enum a \<in> rep x"

     unfolding a_def

     apply (rule LeastI_ex)

     apply (cut_tac ideal_rep [of x])

     apply (drule idealD1)

     apply (clarify, rename_tac a)

     apply (rule_tac x="count a" in exI, simp)

     done

   have b: "\<And>i. P i \<Longrightarrow> enum i \<in> rep x

     \<Longrightarrow> enum (b i) \<in> rep x \<and> \<not> enum (b i) \<preceq> enum i"

     unfolding P_def b_def by (erule LeastI2_ex, simp)

   have c: "\<And>i j. enum i \<in> rep x \<Longrightarrow> enum j \<in> rep x

     \<Longrightarrow> enum (c i j) \<in> rep x \<and> enum i \<preceq> enum (c i j) \<and> enum j \<preceq> enum (c i j)"

     unfolding c_def

     apply (drule (1) idealD2 [OF ideal_rep], clarify)

     apply (rule_tac a="count z" in LeastI2, simp, simp)

     done

   have X_mem: "\<And>n. enum (X n) \<in> rep x"

     apply (induct_tac n)

     apply (simp add: X_0 a_mem)

     apply (clarsimp simp add: X_Suc, rename_tac n)

     apply (simp add: b c)

     done

   have X_chain: "\<And>n. enum (X n) \<preceq> enum (X (Suc n))"

     apply (clarsimp simp add: X_Suc r_refl)

     apply (simp add: b c X_mem)

     done

   have less_b: "\<And>n i. n < b i \<Longrightarrow> enum n \<in> rep x \<Longrightarrow> enum n \<preceq> enum i"

     unfolding b_def by (drule not_less_Least, simp)

   have X_covers: "\<And>n. \<forall>k\<le>n. enum k \<in> rep x \<longrightarrow> enum k \<preceq> enum (X n)"

     apply (induct_tac n)

     apply (clarsimp simp add: X_0 a_def)

     apply (drule_tac k=0 in Least_le, simp add: r_refl)

     apply (clarsimp, rename_tac n k)

     apply (erule le_SucE)

     apply (rule r_trans [OF _ X_chain], simp)

     apply (case_tac "P (X n)", simp add: X_Suc)

     apply (rule_tac x="b (X n)" and y="Suc n" in linorder_cases)

     apply (simp only: less_Suc_eq_le)

     apply (drule spec, drule (1) mp, simp add: b X_mem)

     apply (simp add: c X_mem)

     apply (drule (1) less_b)

     apply (erule r_trans)

     apply (simp add: b c X_mem)

     apply (simp add: X_Suc)

     apply (simp add: P_def)

     done

   have 1: "\<forall>i. enum (X i) \<preceq> enum (X (Suc i))"

     by (simp add: X_chain)

   have 2: "x = (\<Squnion>n. principal (enum (X n)))"

     apply (simp add: eq_iff rep_lub 1 rep_principal)

     apply (auto, rename_tac a)

     apply (subgoal_tac "\<exists>i. a = enum i", erule exE)

     apply (rule_tac x=i in exI, simp add: X_covers)

     apply (rule_tac x="count a" in exI, simp)

     apply (erule idealD3 [OF ideal_rep])

     apply (rule X_mem)

     done

   from 1 2 show ?thesis ..

 qed

 lemma principal_induct:

   assumes adm: "adm P"

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

   shows "P x"

 apply (rule obtain_principal_chain [of x])

 apply (simp add: admD [OF adm] 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 (rule obtain_principal_chain [of x])

 apply (drule adm_compact_neq [OF _ cont_id])

 apply (subgoal_tac "chain (\<lambda>i. principal (Y i))")

 apply (drule (2) admD2, fast, simp)

 done

 lemma obtain_compact_chain:

   obtains Y :: "nat \<Rightarrow> 'b"

   where "chain Y" and "\<forall>i. compact (Y i)" and "x = (\<Squnion>i. Y i)"

 apply (rule obtain_principal_chain [of x])

 apply (rule_tac Y="\<lambda>i. principal (Y i)" in that, simp_all)

 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_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"

 proof -

   obtain Y where Y: "\<forall>i. Y i \<preceq> Y (Suc i)"

   and x: "x = (\<Squnion>i. principal (Y i))"

     by (rule obtain_principal_chain [of x])

   have chain: "chain (\<lambda>i. f (Y i))"

     by (rule chainI, simp add: f_mono Y)

   have rep_x: "rep x = (\<Union>n. {a. a \<preceq> Y n})"

     by (simp add: x rep_lub Y rep_principal)

   have "f ` rep x <<| (\<Squnion>n. f (Y n))"

     apply (rule is_lubI)

     apply (rule ub_imageI, rename_tac a)

     apply (clarsimp simp add: rep_x)

     apply (drule f_mono)

     apply (erule below_lub [OF chain])

     apply (rule lub_below [OF chain])

     apply (drule_tac x="Y n" in ub_imageD)

     apply (simp add: rep_x, fast intro: r_refl)

     apply assumption

     done

   thus ?thesis ..

 qed

 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_thelub_ex [OF lub ub_imageI])

      apply (rule is_ub_thelub_ex [OF lub imageI])

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

     apply (rule is_lub_thelub_ex [OF lub ub_imageI])

     apply (simp add: rep_lub, clarify)

     apply (erule rev_below_trans [OF is_ub_thelub])

     apply (erule is_ub_thelub_ex [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 cfun_belowI)

  apply (simp only: basis_fun_beta f_mono g_mono)

  apply (rule is_lub_thelub_ex)

   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_thelub_ex)

   apply (rule basis_fun_lemma, erule g_mono)

  apply (erule imageI)

 done

 end

 lemma (in preorder) typedef_ideal_completion:

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

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

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

   assumes principal: "\<And>a. principal a = Abs {b. b \<preceq> a}"

   assumes countable: "\<exists>f::'a \<Rightarrow> nat. inj f"

   shows "ideal_completion r principal Rep"

 proof

   interpret type_definition Rep Abs "{S. ideal S}" by fact

   fix a b :: 'a and x y :: 'b and Y :: "nat \<Rightarrow> 'b"

   show "ideal (Rep x)"

     using Rep [of x] by simp

   show "chain Y \<Longrightarrow> Rep (\<Squnion>i. Y i) = (\<Union>i. Rep (Y i))"

     using type below by (rule typedef_ideal_rep_lub)

   show "Rep (principal a) = {b. b \<preceq> a}"

     by (simp add: principal Abs_inverse ideal_principal)

   show "Rep x \<subseteq> Rep y \<Longrightarrow> x \<sqsubseteq> y"

     by (simp only: below)

   show "\<exists>f::'a \<Rightarrow> nat. inj f"

     by (rule countable)

 qed

 end