Theory Completion

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theory Completion
imports Bifinite

(*  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 => 'a => bool" (infix "\<preceq>" 50)
assumes r_refl: "x \<preceq> x"
assumes r_trans: "[|x \<preceq> y; y \<preceq> z|] ==> x \<preceq> z"
begin


definition
ideal :: "'a set => bool" where
"ideal A = ((∃x. x ∈ A) ∧ (∀x∈A. ∀y∈A. ∃z∈A. x \<preceq> z ∧ y \<preceq> z) ∧
(∀x y. x \<preceq> y --> y ∈ A --> x ∈ A))"


lemma idealI:
assumes "∃x. x ∈ A"
assumes "!!x y. [|x ∈ A; y ∈ A|] ==> ∃z∈A. x \<preceq> z ∧ y \<preceq> z"
assumes "!!x y. [|x \<preceq> y; y ∈ A|] ==> x ∈ A"
shows "ideal A"

unfolding ideal_def using prems 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 \<preceq> z ∧ y \<preceq> z"

unfolding ideal_def by fast

lemma idealD3:
"[|ideal A; x \<preceq> y; y ∈ A|] ==> x ∈ A"

unfolding ideal_def by fast

lemma ideal_directed_finite:
assumes A: "ideal A"
shows "[|finite U; U ⊆ A|] ==> ∃z∈A. ∀x∈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: "∃A. ideal A"
by (rule exI, rule ideal_principal)

lemma directed_image_ideal:
assumes A: "ideal A"
assumes f: "!!x y. x \<preceq> y ==> 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: "!!x y. x \<preceq> y ==> f x \<sqsubseteq> f y"
shows "(\<Squnion>x∈{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 => 'a set"
assumes ideal_A: "!!i. ideal (A i)"
assumes chain_A: "!!i j. i ≤ j ==> A i ⊆ 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 => 'b::below"
assumes type: "type_definition Rep Abs {S. ideal S}"
assumes below: "!!x y. x \<sqsubseteq> 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 \<sqsubseteq> y <-> Rep x ⊆ 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 => 'b::po"
assumes type: "type_definition Rep Abs {S. ideal S}"
assumes below: "!!x y. x \<sqsubseteq> y <-> Rep x ⊆ 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 => 'a => 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:
"[|finite S; directed S|] ==> ∃u∈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:
"[|finite S; directed S|] ==> lub S ∈ S"

apply (drule (1) finite_directed_contains_lub, clarify)
apply (drule thelubI, simp)
done

lemma finite_directed_has_lub: "[|finite S; directed S|] ==> ∃u. S <<| u"
by (drule (1) finite_directed_contains_lub, fast)

lemma is_ub_thelub0: "[|∃u. S <<| u; x ∈ S|] ==> x \<sqsubseteq> lub S"
apply (erule exE, drule lubI)
apply (drule is_lubD1)
apply (erule (1) is_ubD)
done

lemma is_lub_thelub0: "[|∃u. S <<| u; S <| x|] ==> 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 => 'a::type => '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 ==> 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: "∃n. take n a = a"
begin


lemma take_chain_less: "m < n ==> take m a \<preceq> take n a"
by (erule less_Suc_induct, rule take_chain, erule (1) r_trans)

lemma take_chain_le: "m ≤ n ==> 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 => 'b::cpo"
fixes rep :: "'b::cpo => 'a::type set"
assumes ideal_rep: "!!x. preorder.ideal r (rep x)"
assumes rep_contlub: "!!Y. chain Y ==> rep (\<Squnion>i. Y i) = (\<Union>i. rep (Y i))"
assumes rep_principal: "!!a. rep (principal a) = {b. b \<preceq> a}"
assumes subset_repD: "!!x y. rep x ⊆ rep y ==> 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 ==> rep x ⊆ 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 <-> rep x ⊆ 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 ∈ rep x <-> principal a \<sqsubseteq> x"
by (simp add: rep_eq)

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

lemma principal_below_iff [simp]: "principal a \<sqsubseteq> principal b <-> a \<preceq> b"
by (simp add: principal_below_iff_mem_rep rep_principal)

lemma principal_eq_iff: "principal a = principal b <-> a \<preceq> b ∧ b \<preceq> a"
unfolding po_eq_conv [where 'a='b] principal_below_iff ..

lemma repD: "a ∈ rep x ==> principal a \<sqsubseteq> x"
by (simp add: rep_eq)

lemma principal_mono: "a \<preceq> b ==> principal a \<sqsubseteq> principal b"
by (simp only: principal_below_iff)

lemma belowI: "(!!a. principal a \<sqsubseteq> x ==> principal a \<sqsubseteq> u) ==> 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 => 'c::cpo) => 'b -> 'c" where
"basis_fun = (λf. (Λ x. lub (f ` rep x)))"


lemma basis_fun_lemma0:
fixes f :: "'a::type => 'c::cpo"
assumes f_mono: "!!a b. a \<preceq> b ==> f a \<sqsubseteq> f b"
shows "∃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 => 'c::cpo"
assumes f_mono: "!!a b. a \<preceq> b ==> f a \<sqsubseteq> f b"
shows "chain (λi. 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 => 'c::cpo"
assumes f_mono: "!!a b. a \<preceq> b ==> 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 => 'c::cpo"
assumes f_mono: "!!a b. a \<preceq> b ==> f a \<sqsubseteq> f b"
shows "∃u. f ` rep x <<| u"

by (rule exI, rule basis_fun_lemma2, erule f_mono)

lemma basis_fun_beta:
fixes f :: "'a::type => 'c::cpo"
assumes f_mono: "!!a b. a \<preceq> b ==> f a \<sqsubseteq> f b"
shows "basis_fun f·x = lub (f ` rep x)"

unfolding basis_fun_def
proof (rule beta_cfun)
have lub: "!!x. ∃u. f ` rep x <<| u"
using f_mono by (rule basis_fun_lemma)
show cont: "cont (λx. 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 => 'c::cpo"
assumes f_mono: "!!a b. a \<preceq> b ==> f a \<sqsubseteq> f b"
shows "basis_fun f·(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: "!!a b. a \<preceq> b ==> f a \<sqsubseteq> f b"
assumes g_mono: "!!a b. a \<preceq> b ==> g a \<sqsubseteq> g b"
assumes below: "!!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 => 'b -> 'b" where
"completion_approx = (λi. basis_fun (λa. principal (take i a)))"


lemma completion_approx_beta:
"completion_approx i·x = (\<Squnion>a∈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·(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·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:
"∃a∈rep x. completion_approx i·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·(completion_approx i·x) = completion_approx i·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·x = x}" (is "finite ?S")

apply (subgoal_tac "?S ⊆ 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: "!!a. P (principal a)"
shows "P x"

apply (subgoal_tac "P (\<Squnion>i. completion_approx i·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:
"[|!!y. adm (λx. P x y); !!x. adm (λy. P x y);
!!a b. P (principal a) (principal b)|] ==> 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 ==> ∃a. x = principal a"
apply (drule adm_compact_neq [OF _ cont_id])
apply (drule admD2 [where Y="λn. completion_approx n·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