(* Title: HOL/HOLCF/Completion.thy
Author: Brian Huffman
*)
section \<open>Defining algebraic domains by ideal completion\<close>
theory Completion
imports Cfun
begin
subsection \<open>Ideals over a preorder\<close>
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) \
(\<forall>x y. x \<preceq> y \<longrightarrow> y \<in> A \<longrightarrow> x \<in> 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 exI [where x = z])
apply (fast intro: r_refl)
apply (rule bexI [where x = z], 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 \<open>The set of ideals is a cpo\<close>
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)
using idealD1 [OF ideal_A] apply fast
apply (clarify)
subgoal for 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
done
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))"
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 \ 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 \<open>Lemmas about least upper bounds\<close>
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 \<open>Locale for ideal completion\<close>
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"
by (simp add: principal_below_iff_mem_rep rep_principal)
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 (\i. principal (Y i))"
by (simp add: chainI principal_mono)
subsubsection \<open>Principal ideals approximate all elements\<close>
lemma compact_principal [simp]: "compact (principal a)"
by (rule compactI2, simp add: principal_below_iff_mem_rep rep_lub)
text \<open>Construct a chain whose lub is the same as a given ideal\<close>
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 (\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: "\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 (insert ideal_rep [of x])
apply (drule idealD1)
apply (clarify)
subgoal for a by (rule exI [where x="count a"]) simp
done
have b: "\i. P i \ enum i \ 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: "\i j. enum i \ rep x \ enum j \ 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)
subgoal for \<dots> z by (rule LeastI2 [where a="count z"], simp, simp)
done
have X_mem: "enum (X n) \ rep x" for n
proof (induct n)
case 0
then show ?case by (simp add: X_0 a_mem)
next
case (Suc n)
with b c show ?case by (auto simp: X_Suc)
qed
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: "\k\n. enum k \ rep x \ enum k \ enum (X n)" for n
proof (induct n)
case 0
then show ?case
apply (clarsimp simp add: X_0 a_def)
apply (drule Least_le [where k=0], simp add: r_refl)
done
next
case (Suc n)
then show ?case
apply clarsimp
apply (erule le_SucE)
apply (rule r_trans [OF _ X_chain], simp)
apply (cases "P (X n)", simp add: X_Suc)
apply (rule linorder_cases [where x="b (X n)" and y="Suc n"])
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
qed
have 1: "\i. enum (X i) \ enum (X (Suc i))"
by (simp add: X_chain)
have "x = (\n. principal (enum (X n)))"
apply (simp add: eq_iff rep_lub 1 rep_principal)
apply auto
subgoal for 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)
done
subgoal
apply (erule idealD3 [OF ideal_rep])
apply (rule X_mem)
done
done
with 1 show ?thesis ..
qed
lemma principal_induct:
assumes adm: "adm P"
assumes P: "\a. P (principal a)"
shows "P x"
apply (rule obtain_principal_chain [of x])
apply (simp add: admD [OF adm] P)
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 (\i. principal (Y i))")
apply (drule (2) admD2, fast, simp)
done
subsection \<open>Defining functions in terms of basis elements\<close>
definition
extension :: "('a::type \ 'c::cpo) \ 'b \ 'c" where
"extension = (\f. (\ 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 (\i. 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)
subgoal for a
apply (clarsimp simp add: rep_x)
apply (drule f_mono)
apply (erule below_lub [OF chain])
done
apply (rule lub_below [OF chain])
subgoal for \<dots> n
apply (drule ub_imageD [where x="Y n"])
apply (simp add: rep_x, fast intro: r_refl)
apply assumption
done
done
then show ?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 (\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 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 (simp add: f_mono)
apply (rule imageI)
apply (simp add: r_refl)
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)
subgoal for x a
apply (rule below_trans [OF below])
apply (rule is_ub_thelub_ex)
apply (rule extension_lemma, erule g_mono)
apply (erule imageI)
done
done
lemma cont_extension:
assumes f_mono: "\a b x. a \ b \ f x a \ f x b"
assumes f_cont: "\a. cont (\x. f x a)"
shows "cont (\x. extension (\a. 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
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