# Theory Completion

theory Completion
imports Cfun
```(*  Title:      HOL/HOLCF/Completion.thy
Author:     Brian Huffman
*)

section ‹Defining algebraic domains by ideal completion›

theory Completion
imports Cfun
begin

subsection ‹Ideals over a preorder›

locale preorder =
fixes r :: "'a::type ⇒ 'a ⇒ bool" (infix "≼" 50)
assumes r_refl: "x ≼ x"
assumes r_trans: "⟦x ≼ y; y ≼ z⟧ ⟹ x ≼ z"
begin

definition
ideal :: "'a set ⇒ bool" where
"ideal A = ((∃x. x ∈ A) ∧ (∀x∈A. ∀y∈A. ∃z∈A. x ≼ z ∧ y ≼ z) ∧
(∀x y. x ≼ y ⟶ y ∈ A ⟶ x ∈ A))"

lemma idealI:
assumes "∃x. x ∈ A"
assumes "⋀x y. ⟦x ∈ A; y ∈ A⟧ ⟹ ∃z∈A. x ≼ z ∧ y ≼ z"
assumes "⋀x y. ⟦x ≼ y; y ∈ A⟧ ⟹ x ∈ A"
shows "ideal A"
unfolding ideal_def using assms by fast

lemma idealD1:
"ideal A ⟹ ∃x. x ∈ A"
unfolding ideal_def by fast

lemma idealD2:
"⟦ideal A; x ∈ A; y ∈ A⟧ ⟹ ∃z∈A. x ≼ z ∧ y ≼ z"
unfolding ideal_def by fast

lemma idealD3:
"⟦ideal A; x ≼ y; y ∈ A⟧ ⟹ x ∈ A"
unfolding ideal_def by fast

lemma ideal_principal: "ideal {x. x ≼ 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: "∃A. A ∈ {A. ideal A}"
by (fast intro: ideal_principal)

text ‹The set of ideals is a cpo›

lemma ideal_UN:
fixes A :: "nat ⇒ 'a set"
assumes ideal_A: "⋀i. ideal (A i)"
assumes chain_A: "⋀i j. i ≤ j ⟹ A i ⊆ A j"
shows "ideal (⋃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 max.cobounded1]])
apply (drule subsetD [OF chain_A [OF max.cobounded2]])
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 ⇒ 'b::below"
assumes type: "type_definition Rep Abs {S. ideal S}"
assumes below: "⋀x y. x ⊑ y ⟷ Rep x ⊆ 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 ⇒ 'b::po"
assumes type: "type_definition Rep Abs {S. ideal S}"
assumes below: "⋀x y. x ⊑ y ⟷ Rep x ⊆ Rep y"
assumes S: "chain S"
shows typedef_ideal_lub: "range S <<| Abs (⋃i. Rep (S i))"
and typedef_ideal_rep_lub: "Rep (⨆i. S i) = (⋃i. Rep (S i))"
proof -
have 1: "ideal (⋃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 (⋃i. Rep (S i))) = (⋃i. Rep (S i))"
by (simp add: type_definition.Abs_inverse [OF type])
show 3: "range S <<| Abs (⋃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: "(⨆i. S i) = Abs (⋃i. Rep (S i))"
by (rule lub_eqI)
show 5: "Rep (⨆i. S i) = (⋃i. Rep (S i))"
qed

lemma typedef_ideal_cpo:
fixes Abs :: "'a set ⇒ 'b::po"
assumes type: "type_definition Rep Abs {S. ideal S}"
assumes below: "⋀x y. x ⊑ y ⟷ Rep x ⊆ Rep y"
shows "OFCLASS('b, cpo_class)"
by standard (rule exI, erule typedef_ideal_lub [OF type below])

end

interpretation below: preorder "below :: 'a::po ⇒ 'a ⇒ 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: "⟦∃u. S <<| u; x ∈ S⟧ ⟹ x ⊑ lub S"
apply (erule exE, drule is_lub_lub)
apply (drule is_lubD1)
apply (erule (1) is_ubD)
done

lemma is_lub_thelub_ex: "⟦∃u. S <<| u; S <| x⟧ ⟹ lub S ⊑ x"
by (erule exE, drule is_lub_lub, erule is_lubD2)

subsection ‹Locale for ideal completion›

hide_const (open) Filter.principal

locale ideal_completion = preorder +
fixes principal :: "'a::type ⇒ 'b::cpo"
fixes rep :: "'b::cpo ⇒ 'a::type set"
assumes ideal_rep: "⋀x. ideal (rep x)"
assumes rep_lub: "⋀Y. chain Y ⟹ rep (⨆i. Y i) = (⋃i. rep (Y i))"
assumes rep_principal: "⋀a. rep (principal a) = {b. b ≼ a}"
assumes belowI: "⋀x y. rep x ⊆ rep y ⟹ x ⊑ y"
assumes countable: "∃f::'a ⇒ nat. inj f"
begin

lemma rep_mono: "x ⊑ y ⟹ rep x ⊆ rep y"
apply (frule bin_chain)
apply (drule rep_lub)
apply (simp only: lub_eqI [OF is_lub_bin_chain])
apply (rule subsetI, rule UN_I [where a=0], simp_all)
done

lemma below_def: "x ⊑ y ⟷ rep x ⊆ rep y"
by (rule iffI [OF rep_mono belowI])

lemma principal_below_iff_mem_rep: "principal a ⊑ x ⟷ a ∈ rep x"
unfolding below_def rep_principal
by (auto intro: r_refl elim: idealD3 [OF ideal_rep])

lemma principal_below_iff [simp]: "principal a ⊑ principal b ⟷ a ≼ b"

lemma principal_eq_iff: "principal a = principal b ⟷ a ≼ b ∧ b ≼ a"
unfolding po_eq_conv [where 'a='b] principal_below_iff ..

lemma eq_iff: "x = y ⟷ rep x = rep y"
unfolding po_eq_conv below_def by auto

lemma principal_mono: "a ≼ b ⟹ principal a ⊑ principal b"
by (simp only: principal_below_iff)

lemma ch2ch_principal [simp]:
"∀i. Y i ≼ Y (Suc i) ⟹ chain (λi. principal (Y i))"

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 "∀i. Y i ≼ Y (Suc i)" and "x = (⨆i. principal (Y i))"
proof -
obtain count :: "'a ⇒ nat" where inj: "inj count"
using countable ..
define enum where "enum i = (THE a. count a = i)" for i
have enum_count [simp]: "⋀x. enum (count x) = x"
unfolding enum_def by (simp add: inj_eq [OF inj])
define a where "a = (LEAST i. enum i ∈ rep x)"
define b where "b i = (LEAST j. enum j ∈ rep x ∧ ¬ enum j ≼ enum i)" for i
define c where "c i j = (LEAST k. enum k ∈ rep x ∧ enum i ≼ enum k ∧ enum j ≼ enum k)" for i j
define P where "P i ⟷ (∃j. enum j ∈ rep x ∧ ¬ enum j ≼ enum i)" for i
define X where "X = rec_nat a (λn 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: "⋀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 ∈ 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: "⋀i. P i ⟹ enum i ∈ rep x
⟹ enum (b i) ∈ rep x ∧ ¬ enum (b i) ≼ enum i"
unfolding P_def b_def by (erule LeastI2_ex, simp)
have c: "⋀i j. enum i ∈ rep x ⟹ enum j ∈ rep x
⟹ enum (c i j) ∈ rep x ∧ enum i ≼ enum (c i j) ∧ enum j ≼ 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: "⋀n. enum (X n) ∈ rep x"
apply (induct_tac n)
apply (clarsimp simp add: X_Suc, rename_tac n)
done
have X_chain: "⋀n. enum (X n) ≼ enum (X (Suc n))"
apply (clarsimp simp add: X_Suc r_refl)
apply (simp add: b c X_mem)
done
have less_b: "⋀n i. n < b i ⟹ enum n ∈ rep x ⟹ enum n ≼ enum i"
unfolding b_def by (drule not_less_Least, simp)
have X_covers: "⋀n. ∀k≤n. enum k ∈ rep x ⟶ enum k ≼ 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 (drule (1) less_b)
apply (erule r_trans)
apply (simp add: b c X_mem)
done
have 1: "∀i. enum (X i) ≼ enum (X (Suc i))"
have 2: "x = (⨆n. principal (enum (X n)))"
apply (simp add: eq_iff rep_lub 1 rep_principal)
apply (auto, rename_tac a)
apply (subgoal_tac "∃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 P: "⋀a. P (principal a)"
shows "P x"
apply (rule obtain_principal_chain [of x])
done

lemma compact_imp_principal: "compact x ⟹ ∃a. x = principal a"
apply (rule obtain_principal_chain [of x])
apply (drule adm_compact_neq [OF _ cont_id])
apply (subgoal_tac "chain (λi. principal (Y i))")
apply (drule (2) admD2, fast, simp)
done

subsection ‹Defining functions in terms of basis elements›

definition
extension :: "('a::type ⇒ 'c::cpo) ⇒ 'b → 'c" where
"extension = (λf. (Λ x. lub (f ` rep x)))"

lemma extension_lemma:
fixes f :: "'a::type ⇒ 'c::cpo"
assumes f_mono: "⋀a b. a ≼ b ⟹ f a ⊑ f b"
shows "∃u. f ` rep x <<| u"
proof -
obtain Y where Y: "∀i. Y i ≼ Y (Suc i)"
and x: "x = (⨆i. principal (Y i))"
by (rule obtain_principal_chain [of x])
have chain: "chain (λi. f (Y i))"
by (rule chainI, simp add: f_mono Y)
have rep_x: "rep x = (⋃n. {a. a ≼ Y n})"
by (simp add: x rep_lub Y rep_principal)
have "f ` rep x <<| (⨆n. f (Y n))"
apply (rule is_lubI)
apply (rule ub_imageI, rename_tac a)
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 extension_beta:
fixes f :: "'a::type ⇒ 'c::cpo"
assumes f_mono: "⋀a b. a ≼ b ⟹ f a ⊑ f b"
shows "extension f⋅x = lub (f ` rep x)"
unfolding extension_def
proof (rule beta_cfun)
have lub: "⋀x. ∃u. f ` rep x <<| u"
using f_mono by (rule extension_lemma)
show cont: "cont (λx. 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 (erule rev_below_trans [OF is_ub_thelub])
apply (erule is_ub_thelub_ex [OF lub imageI])
done
qed

lemma extension_principal:
fixes f :: "'a::type ⇒ 'c::cpo"
assumes f_mono: "⋀a b. a ≼ b ⟹ f a ⊑ f b"
shows "extension f⋅(principal a) = f a"
apply (subst extension_beta, erule f_mono)
apply (subst rep_principal)
apply (rule lub_eqI)
apply (rule is_lub_maximal)
apply (rule ub_imageI)
apply (rule imageI)
done

lemma extension_mono:
assumes f_mono: "⋀a b. a ≼ b ⟹ f a ⊑ f b"
assumes g_mono: "⋀a b. a ≼ b ⟹ g a ⊑ g b"
assumes below: "⋀a. f a ⊑ g a"
shows "extension f ⊑ extension g"
apply (rule cfun_belowI)
apply (simp only: extension_beta f_mono g_mono)
apply (rule is_lub_thelub_ex)
apply (rule extension_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 extension_lemma, erule g_mono)
apply (erule imageI)
done

lemma cont_extension:
assumes f_mono: "⋀a b x. a ≼ b ⟹ f x a ⊑ f x b"
assumes f_cont: "⋀a. cont (λx. f x a)"
shows "cont (λx. extension (λa. f x a))"
apply (rule contI2)
apply (rule monofunI)
apply (rule extension_mono, erule f_mono, erule f_mono)
apply (erule cont2monofunE [OF f_cont])
apply (rule cfun_belowI)
apply (rule principal_induct, simp)
apply (simp only: contlub_cfun_fun)
apply (simp only: extension_principal f_mono)
apply (simp add: cont2contlubE [OF f_cont])
done

end

lemma (in preorder) typedef_ideal_completion:
fixes Abs :: "'a set ⇒ 'b::cpo"
assumes type: "type_definition Rep Abs {S. ideal S}"
assumes below: "⋀x y. x ⊑ y ⟷ Rep x ⊆ Rep y"
assumes principal: "⋀a. principal a = Abs {b. b ≼ a}"
assumes countable: "∃f::'a ⇒ 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 ⇒ 'b"
show "ideal (Rep x)"
using Rep [of x] by simp
show "chain Y ⟹ Rep (⨆i. Y i) = (⋃i. Rep (Y i))"
using type below by (rule typedef_ideal_rep_lub)
show "Rep (principal a) = {b. b ≼ a}"
by (simp add: principal Abs_inverse ideal_principal)
show "Rep x ⊆ Rep y ⟹ x ⊑ y"
by (simp only: below)
show "∃f::'a ⇒ nat. inj f"
by (rule countable)
qed

end
```