src/HOL/HOLCF/Completion.thy
author wenzelm
Sat, 07 Apr 2012 16:41:59 +0200
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child 54863 82acc20ded73
permissions -rw-r--r--
explicit checks stable_finished_theory/stable_command allow parallel asynchronous command transactions; tuned;
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(*  Title:      HOL/HOLCF/Completion.thy
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    Author:     Brian Huffman
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*)
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header {* Defining algebraic domains by ideal completion *}
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theory Completion
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imports Plain_HOLCF
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begin
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subsection {* Ideals over a preorder *}
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locale preorder =
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  fixes r :: "'a::type \<Rightarrow> 'a \<Rightarrow> bool" (infix "\<preceq>" 50)
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  assumes r_refl: "x \<preceq> x"
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  assumes r_trans: "\<lbrakk>x \<preceq> y; y \<preceq> z\<rbrakk> \<Longrightarrow> x \<preceq> z"
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begin
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definition
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  ideal :: "'a set \<Rightarrow> bool" where
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  "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>
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    (\<forall>x y. x \<preceq> y \<longrightarrow> y \<in> A \<longrightarrow> x \<in> A))"
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lemma idealI:
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  assumes "\<exists>x. x \<in> A"
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  assumes "\<And>x y. \<lbrakk>x \<in> A; y \<in> A\<rbrakk> \<Longrightarrow> \<exists>z\<in>A. x \<preceq> z \<and> y \<preceq> z"
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  assumes "\<And>x y. \<lbrakk>x \<preceq> y; y \<in> A\<rbrakk> \<Longrightarrow> x \<in> A"
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  shows "ideal A"
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unfolding ideal_def using assms by fast
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lemma idealD1:
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  "ideal A \<Longrightarrow> \<exists>x. x \<in> A"
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unfolding ideal_def by fast
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lemma idealD2:
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  "\<lbrakk>ideal A; x \<in> A; y \<in> A\<rbrakk> \<Longrightarrow> \<exists>z\<in>A. x \<preceq> z \<and> y \<preceq> z"
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unfolding ideal_def by fast
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lemma idealD3:
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  "\<lbrakk>ideal A; x \<preceq> y; y \<in> A\<rbrakk> \<Longrightarrow> x \<in> A"
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unfolding ideal_def by fast
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lemma ideal_principal: "ideal {x. x \<preceq> z}"
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apply (rule idealI)
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apply (rule_tac x=z in exI)
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apply (fast intro: r_refl)
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apply (rule_tac x=z in bexI, fast)
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apply (fast intro: r_refl)
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apply (fast intro: r_trans)
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done
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lemma ex_ideal: "\<exists>A. A \<in> {A. ideal A}"
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by (fast intro: ideal_principal)
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text {* The set of ideals is a cpo *}
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lemma ideal_UN:
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  fixes A :: "nat \<Rightarrow> 'a set"
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  assumes ideal_A: "\<And>i. ideal (A i)"
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  assumes chain_A: "\<And>i j. i \<le> j \<Longrightarrow> A i \<subseteq> A j"
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  shows "ideal (\<Union>i. A i)"
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 apply (rule idealI)
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   apply (cut_tac idealD1 [OF ideal_A], fast)
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  apply (clarify, rename_tac i j)
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  apply (drule subsetD [OF chain_A [OF le_maxI1]])
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  apply (drule subsetD [OF chain_A [OF le_maxI2]])
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  apply (drule (1) idealD2 [OF ideal_A])
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  apply blast
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 apply clarify
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 apply (drule (1) idealD3 [OF ideal_A])
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 apply fast
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done
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lemma typedef_ideal_po:
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  fixes Abs :: "'a set \<Rightarrow> 'b::below"
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  assumes type: "type_definition Rep Abs {S. ideal S}"
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  assumes below: "\<And>x y. x \<sqsubseteq> y \<longleftrightarrow> Rep x \<subseteq> Rep y"
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  shows "OFCLASS('b, po_class)"
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 apply (intro_classes, unfold below)
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   apply (rule subset_refl)
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  apply (erule (1) subset_trans)
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 apply (rule type_definition.Rep_inject [OF type, THEN iffD1])
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 apply (erule (1) subset_antisym)
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done
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lemma
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  fixes Abs :: "'a set \<Rightarrow> 'b::po"
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  assumes type: "type_definition Rep Abs {S. ideal S}"
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  assumes below: "\<And>x y. x \<sqsubseteq> y \<longleftrightarrow> Rep x \<subseteq> Rep y"
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  assumes S: "chain S"
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  shows typedef_ideal_lub: "range S <<| Abs (\<Union>i. Rep (S i))"
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    and typedef_ideal_rep_lub: "Rep (\<Squnion>i. S i) = (\<Union>i. Rep (S i))"
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proof -
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  have 1: "ideal (\<Union>i. Rep (S i))"
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    apply (rule ideal_UN)
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     apply (rule type_definition.Rep [OF type, unfolded mem_Collect_eq])
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    apply (subst below [symmetric])
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    apply (erule chain_mono [OF S])
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    done
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  hence 2: "Rep (Abs (\<Union>i. Rep (S i))) = (\<Union>i. Rep (S i))"
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    by (simp add: type_definition.Abs_inverse [OF type])
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  show 3: "range S <<| Abs (\<Union>i. Rep (S i))"
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    apply (rule is_lubI)
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     apply (rule is_ubI)
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     apply (simp add: below 2, fast)
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    apply (simp add: below 2 is_ub_def, fast)
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    done
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  hence 4: "(\<Squnion>i. S i) = Abs (\<Union>i. Rep (S i))"
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    by (rule lub_eqI)
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  show 5: "Rep (\<Squnion>i. S i) = (\<Union>i. Rep (S i))"
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    by (simp add: 4 2)
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qed
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lemma typedef_ideal_cpo:
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  fixes Abs :: "'a set \<Rightarrow> 'b::po"
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  assumes type: "type_definition Rep Abs {S. ideal S}"
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  assumes below: "\<And>x y. x \<sqsubseteq> y \<longleftrightarrow> Rep x \<subseteq> Rep y"
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  shows "OFCLASS('b, cpo_class)"
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by (default, rule exI, erule typedef_ideal_lub [OF type below])
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end
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interpretation below: preorder "below :: 'a::po \<Rightarrow> 'a \<Rightarrow> bool"
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apply unfold_locales
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apply (rule below_refl)
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apply (erule (1) below_trans)
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done
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subsection {* Lemmas about least upper bounds *}
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lemma is_ub_thelub_ex: "\<lbrakk>\<exists>u. S <<| u; x \<in> S\<rbrakk> \<Longrightarrow> x \<sqsubseteq> lub S"
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apply (erule exE, drule is_lub_lub)
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apply (drule is_lubD1)
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apply (erule (1) is_ubD)
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done
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lemma is_lub_thelub_ex: "\<lbrakk>\<exists>u. S <<| u; S <| x\<rbrakk> \<Longrightarrow> lub S \<sqsubseteq> x"
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by (erule exE, drule is_lub_lub, erule is_lubD2)
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subsection {* Locale for ideal completion *}
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locale ideal_completion = preorder +
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  fixes principal :: "'a::type \<Rightarrow> 'b::cpo"
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  fixes rep :: "'b::cpo \<Rightarrow> 'a::type set"
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  assumes ideal_rep: "\<And>x. ideal (rep x)"
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   146
  assumes rep_lub: "\<And>Y. chain Y \<Longrightarrow> rep (\<Squnion>i. Y i) = (\<Union>i. rep (Y i))"
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   147
  assumes rep_principal: "\<And>a. rep (principal a) = {b. b \<preceq> a}"
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diff changeset
   148
  assumes belowI: "\<And>x y. rep x \<subseteq> rep y \<Longrightarrow> x \<sqsubseteq> y"
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   149
  assumes countable: "\<exists>f::'a \<Rightarrow> nat. inj f"
27404
62171da527d6 split Completion.thy from CompactBasis.thy
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   150
begin
62171da527d6 split Completion.thy from CompactBasis.thy
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   151
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   152
lemma rep_mono: "x \<sqsubseteq> y \<Longrightarrow> rep x \<subseteq> rep y"
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   153
apply (frule bin_chain)
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diff changeset
   154
apply (drule rep_lub)
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   155
apply (simp only: lub_eqI [OF is_lub_bin_chain])
28133
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   156
apply (rule subsetI, rule UN_I [where a=0], simp_all)
218252dfd81e reorganize subsections
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   157
done
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diff changeset
   158
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   159
lemma below_def: "x \<sqsubseteq> y \<longleftrightarrow> rep x \<subseteq> rep y"
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   160
by (rule iffI [OF rep_mono belowI])
28133
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diff changeset
   161
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   162
lemma principal_below_iff_mem_rep: "principal a \<sqsubseteq> x \<longleftrightarrow> a \<in> rep x"
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diff changeset
   163
unfolding below_def rep_principal
7a67a8832da8 simplify ideal completion proofs
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   164
by (auto intro: r_refl elim: idealD3 [OF ideal_rep])
28133
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diff changeset
   165
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   166
lemma principal_below_iff [simp]: "principal a \<sqsubseteq> principal b \<longleftrightarrow> a \<preceq> b"
99fe356cbbc2 rename constant sq_le to below; rename class sq_ord to below; less->below in many lemma names
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   167
by (simp add: principal_below_iff_mem_rep rep_principal)
28133
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diff changeset
   168
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   169
lemma principal_eq_iff: "principal a = principal b \<longleftrightarrow> a \<preceq> b \<and> b \<preceq> a"
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   170
unfolding po_eq_conv [where 'a='b] principal_below_iff ..
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diff changeset
   171
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   172
lemma eq_iff: "x = y \<longleftrightarrow> rep x = rep y"
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   173
unfolding po_eq_conv below_def by auto
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diff changeset
   174
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   175
lemma principal_mono: "a \<preceq> b \<Longrightarrow> principal a \<sqsubseteq> principal b"
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99fe356cbbc2 rename constant sq_le to below; rename class sq_ord to below; less->below in many lemma names
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   176
by (simp only: principal_below_iff)
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diff changeset
   177
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   178
lemma ch2ch_principal [simp]:
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   179
  "\<forall>i. Y i \<preceq> Y (Suc i) \<Longrightarrow> chain (\<lambda>i. principal (Y i))"
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   180
by (simp add: chainI principal_mono)
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   181
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   182
subsubsection {* Principal ideals approximate all elements *}
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   183
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   184
lemma compact_principal [simp]: "compact (principal a)"
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3af9b0df3521 rename rep_contlub lemmas to rep_lub
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parents: 40502
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   185
by (rule compactI2, simp add: principal_below_iff_mem_rep rep_lub)
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   186
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   187
text {* Construct a chain whose lub is the same as a given ideal *}
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   188
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   189
lemma obtain_principal_chain:
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   190
  obtains Y where "\<forall>i. Y i \<preceq> Y (Suc i)" and "x = (\<Squnion>i. principal (Y i))"
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   191
proof -
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   192
  obtain count :: "'a \<Rightarrow> nat" where inj: "inj count"
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   193
    using countable ..
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   194
  def enum \<equiv> "\<lambda>i. THE a. count a = i"
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   195
  have enum_count [simp]: "\<And>x. enum (count x) = x"
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diff changeset
   196
    unfolding enum_def by (simp add: inj_eq [OF inj])
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diff changeset
   197
  def a \<equiv> "LEAST i. enum i \<in> rep x"
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diff changeset
   198
  def b \<equiv> "\<lambda>i. LEAST j. enum j \<in> rep x \<and> \<not> enum j \<preceq> enum i"
b525988432e9 major reorganization/simplification of HOLCF type classes:
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diff changeset
   199
  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"
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diff changeset
   200
  def P \<equiv> "\<lambda>i. \<exists>j. enum j \<in> rep x \<and> \<not> enum j \<preceq> enum i"
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   201
  def X \<equiv> "nat_rec a (\<lambda>n i. if P i then c i (b i) else i)"
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   202
  have X_0: "X 0 = a" unfolding X_def by simp
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diff changeset
   203
  have X_Suc: "\<And>n. X (Suc n) = (if P (X n) then c (X n) (b (X n)) else X n)"
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diff changeset
   204
    unfolding X_def by simp
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   205
  have a_mem: "enum a \<in> rep x"
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diff changeset
   206
    unfolding a_def
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   207
    apply (rule LeastI_ex)
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diff changeset
   208
    apply (cut_tac ideal_rep [of x])
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diff changeset
   209
    apply (drule idealD1)
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diff changeset
   210
    apply (clarify, rename_tac a)
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diff changeset
   211
    apply (rule_tac x="count a" in exI, simp)
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diff changeset
   212
    done
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diff changeset
   213
  have b: "\<And>i. P i \<Longrightarrow> enum i \<in> rep x
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   214
    \<Longrightarrow> enum (b i) \<in> rep x \<and> \<not> enum (b i) \<preceq> enum i"
b525988432e9 major reorganization/simplification of HOLCF type classes:
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diff changeset
   215
    unfolding P_def b_def by (erule LeastI2_ex, simp)
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parents: 39967
diff changeset
   216
  have c: "\<And>i j. enum i \<in> rep x \<Longrightarrow> enum j \<in> rep x
b525988432e9 major reorganization/simplification of HOLCF type classes:
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diff changeset
   217
    \<Longrightarrow> enum (c i j) \<in> rep x \<and> enum i \<preceq> enum (c i j) \<and> enum j \<preceq> enum (c i j)"
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diff changeset
   218
    unfolding c_def
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diff changeset
   219
    apply (drule (1) idealD2 [OF ideal_rep], clarify)
b525988432e9 major reorganization/simplification of HOLCF type classes:
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diff changeset
   220
    apply (rule_tac a="count z" in LeastI2, simp, simp)
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diff changeset
   221
    done
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diff changeset
   222
  have X_mem: "\<And>n. enum (X n) \<in> rep x"
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diff changeset
   223
    apply (induct_tac n)
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diff changeset
   224
    apply (simp add: X_0 a_mem)
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diff changeset
   225
    apply (clarsimp simp add: X_Suc, rename_tac n)
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diff changeset
   226
    apply (simp add: b c)
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diff changeset
   227
    done
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diff changeset
   228
  have X_chain: "\<And>n. enum (X n) \<preceq> enum (X (Suc n))"
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diff changeset
   229
    apply (clarsimp simp add: X_Suc r_refl)
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diff changeset
   230
    apply (simp add: b c X_mem)
b525988432e9 major reorganization/simplification of HOLCF type classes:
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diff changeset
   231
    done
b525988432e9 major reorganization/simplification of HOLCF type classes:
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diff changeset
   232
  have less_b: "\<And>n i. n < b i \<Longrightarrow> enum n \<in> rep x \<Longrightarrow> enum n \<preceq> enum i"
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diff changeset
   233
    unfolding b_def by (drule not_less_Least, simp)
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diff changeset
   234
  have X_covers: "\<And>n. \<forall>k\<le>n. enum k \<in> rep x \<longrightarrow> enum k \<preceq> enum (X n)"
b525988432e9 major reorganization/simplification of HOLCF type classes:
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diff changeset
   235
    apply (induct_tac n)
b525988432e9 major reorganization/simplification of HOLCF type classes:
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parents: 39967
diff changeset
   236
    apply (clarsimp simp add: X_0 a_def)
b525988432e9 major reorganization/simplification of HOLCF type classes:
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parents: 39967
diff changeset
   237
    apply (drule_tac k=0 in Least_le, simp add: r_refl)
b525988432e9 major reorganization/simplification of HOLCF type classes:
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parents: 39967
diff changeset
   238
    apply (clarsimp, rename_tac n k)
b525988432e9 major reorganization/simplification of HOLCF type classes:
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parents: 39967
diff changeset
   239
    apply (erule le_SucE)
b525988432e9 major reorganization/simplification of HOLCF type classes:
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parents: 39967
diff changeset
   240
    apply (rule r_trans [OF _ X_chain], simp)
b525988432e9 major reorganization/simplification of HOLCF type classes:
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parents: 39967
diff changeset
   241
    apply (case_tac "P (X n)", simp add: X_Suc)
b525988432e9 major reorganization/simplification of HOLCF type classes:
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parents: 39967
diff changeset
   242
    apply (rule_tac x="b (X n)" and y="Suc n" in linorder_cases)
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39967
diff changeset
   243
    apply (simp only: less_Suc_eq_le)
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39967
diff changeset
   244
    apply (drule spec, drule (1) mp, simp add: b X_mem)
b525988432e9 major reorganization/simplification of HOLCF type classes:
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parents: 39967
diff changeset
   245
    apply (simp add: c X_mem)
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39967
diff changeset
   246
    apply (drule (1) less_b)
b525988432e9 major reorganization/simplification of HOLCF type classes:
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parents: 39967
diff changeset
   247
    apply (erule r_trans)
b525988432e9 major reorganization/simplification of HOLCF type classes:
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parents: 39967
diff changeset
   248
    apply (simp add: b c X_mem)
b525988432e9 major reorganization/simplification of HOLCF type classes:
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parents: 39967
diff changeset
   249
    apply (simp add: X_Suc)
b525988432e9 major reorganization/simplification of HOLCF type classes:
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diff changeset
   250
    apply (simp add: P_def)
b525988432e9 major reorganization/simplification of HOLCF type classes:
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diff changeset
   251
    done
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diff changeset
   252
  have 1: "\<forall>i. enum (X i) \<preceq> enum (X (Suc i))"
b525988432e9 major reorganization/simplification of HOLCF type classes:
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diff changeset
   253
    by (simp add: X_chain)
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parents: 39967
diff changeset
   254
  have 2: "x = (\<Squnion>n. principal (enum (X n)))"
40769
3af9b0df3521 rename rep_contlub lemmas to rep_lub
huffman
parents: 40502
diff changeset
   255
    apply (simp add: eq_iff rep_lub 1 rep_principal)
39974
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parents: 39967
diff changeset
   256
    apply (auto, rename_tac a)
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39967
diff changeset
   257
    apply (subgoal_tac "\<exists>i. a = enum i", erule exE)
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39967
diff changeset
   258
    apply (rule_tac x=i in exI, simp add: X_covers)
b525988432e9 major reorganization/simplification of HOLCF type classes:
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parents: 39967
diff changeset
   259
    apply (rule_tac x="count a" in exI, simp)
b525988432e9 major reorganization/simplification of HOLCF type classes:
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parents: 39967
diff changeset
   260
    apply (erule idealD3 [OF ideal_rep])
b525988432e9 major reorganization/simplification of HOLCF type classes:
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parents: 39967
diff changeset
   261
    apply (rule X_mem)
b525988432e9 major reorganization/simplification of HOLCF type classes:
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parents: 39967
diff changeset
   262
    done
b525988432e9 major reorganization/simplification of HOLCF type classes:
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diff changeset
   263
  from 1 2 show ?thesis ..
b525988432e9 major reorganization/simplification of HOLCF type classes:
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diff changeset
   264
qed
b525988432e9 major reorganization/simplification of HOLCF type classes:
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diff changeset
   265
b525988432e9 major reorganization/simplification of HOLCF type classes:
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diff changeset
   266
lemma principal_induct:
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diff changeset
   267
  assumes adm: "adm P"
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diff changeset
   268
  assumes P: "\<And>a. P (principal a)"
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diff changeset
   269
  shows "P x"
b525988432e9 major reorganization/simplification of HOLCF type classes:
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parents: 39967
diff changeset
   270
apply (rule obtain_principal_chain [of x])
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diff changeset
   271
apply (simp add: admD [OF adm] P)
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diff changeset
   272
done
b525988432e9 major reorganization/simplification of HOLCF type classes:
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diff changeset
   273
b525988432e9 major reorganization/simplification of HOLCF type classes:
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diff changeset
   274
lemma compact_imp_principal: "compact x \<Longrightarrow> \<exists>a. x = principal a"
b525988432e9 major reorganization/simplification of HOLCF type classes:
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parents: 39967
diff changeset
   275
apply (rule obtain_principal_chain [of x])
b525988432e9 major reorganization/simplification of HOLCF type classes:
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parents: 39967
diff changeset
   276
apply (drule adm_compact_neq [OF _ cont_id])
b525988432e9 major reorganization/simplification of HOLCF type classes:
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parents: 39967
diff changeset
   277
apply (subgoal_tac "chain (\<lambda>i. principal (Y i))")
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39967
diff changeset
   278
apply (drule (2) admD2, fast, simp)
b525988432e9 major reorganization/simplification of HOLCF type classes:
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parents: 39967
diff changeset
   279
done
b525988432e9 major reorganization/simplification of HOLCF type classes:
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diff changeset
   280
28133
218252dfd81e reorganize subsections
huffman
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diff changeset
   281
subsection {* Defining functions in terms of basis elements *}
218252dfd81e reorganize subsections
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diff changeset
   282
218252dfd81e reorganize subsections
huffman
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diff changeset
   283
definition
41394
51c866d1b53b rename function ideal_completion.basis_fun to ideal_completion.extension
huffman
parents: 41182
diff changeset
   284
  extension :: "('a::type \<Rightarrow> 'c::cpo) \<Rightarrow> 'b \<rightarrow> 'c" where
51c866d1b53b rename function ideal_completion.basis_fun to ideal_completion.extension
huffman
parents: 41182
diff changeset
   285
  "extension = (\<lambda>f. (\<Lambda> x. lub (f ` rep x)))"
28133
218252dfd81e reorganize subsections
huffman
parents: 28053
diff changeset
   286
41394
51c866d1b53b rename function ideal_completion.basis_fun to ideal_completion.extension
huffman
parents: 41182
diff changeset
   287
lemma extension_lemma:
27404
62171da527d6 split Completion.thy from CompactBasis.thy
huffman
parents:
diff changeset
   288
  fixes f :: "'a::type \<Rightarrow> 'c::cpo"
62171da527d6 split Completion.thy from CompactBasis.thy
huffman
parents:
diff changeset
   289
  assumes f_mono: "\<And>a b. a \<preceq> b \<Longrightarrow> f a \<sqsubseteq> f b"
62171da527d6 split Completion.thy from CompactBasis.thy
huffman
parents:
diff changeset
   290
  shows "\<exists>u. f ` rep x <<| u"
39974
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39967
diff changeset
   291
proof -
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39967
diff changeset
   292
  obtain Y where Y: "\<forall>i. Y i \<preceq> Y (Suc i)"
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39967
diff changeset
   293
  and x: "x = (\<Squnion>i. principal (Y i))"
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39967
diff changeset
   294
    by (rule obtain_principal_chain [of x])
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39967
diff changeset
   295
  have chain: "chain (\<lambda>i. f (Y i))"
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39967
diff changeset
   296
    by (rule chainI, simp add: f_mono Y)
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39967
diff changeset
   297
  have rep_x: "rep x = (\<Union>n. {a. a \<preceq> Y n})"
40769
3af9b0df3521 rename rep_contlub lemmas to rep_lub
huffman
parents: 40502
diff changeset
   298
    by (simp add: x rep_lub Y rep_principal)
39974
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39967
diff changeset
   299
  have "f ` rep x <<| (\<Squnion>n. f (Y n))"
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39967
diff changeset
   300
    apply (rule is_lubI)
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39967
diff changeset
   301
    apply (rule ub_imageI, rename_tac a)
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39967
diff changeset
   302
    apply (clarsimp simp add: rep_x)
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39967
diff changeset
   303
    apply (drule f_mono)
40500
ee9c8d36318e add lemmas lub_below, below_lub; simplify some proofs; remove some unused lemmas
huffman
parents: 40002
diff changeset
   304
    apply (erule below_lub [OF chain])
ee9c8d36318e add lemmas lub_below, below_lub; simplify some proofs; remove some unused lemmas
huffman
parents: 40002
diff changeset
   305
    apply (rule lub_below [OF chain])
ee9c8d36318e add lemmas lub_below, below_lub; simplify some proofs; remove some unused lemmas
huffman
parents: 40002
diff changeset
   306
    apply (drule_tac x="Y n" in ub_imageD)
39974
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39967
diff changeset
   307
    apply (simp add: rep_x, fast intro: r_refl)
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39967
diff changeset
   308
    apply assumption
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39967
diff changeset
   309
    done
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39967
diff changeset
   310
  thus ?thesis ..
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39967
diff changeset
   311
qed
27404
62171da527d6 split Completion.thy from CompactBasis.thy
huffman
parents:
diff changeset
   312
41394
51c866d1b53b rename function ideal_completion.basis_fun to ideal_completion.extension
huffman
parents: 41182
diff changeset
   313
lemma extension_beta:
27404
62171da527d6 split Completion.thy from CompactBasis.thy
huffman
parents:
diff changeset
   314
  fixes f :: "'a::type \<Rightarrow> 'c::cpo"
62171da527d6 split Completion.thy from CompactBasis.thy
huffman
parents:
diff changeset
   315
  assumes f_mono: "\<And>a b. a \<preceq> b \<Longrightarrow> f a \<sqsubseteq> f b"
41394
51c866d1b53b rename function ideal_completion.basis_fun to ideal_completion.extension
huffman
parents: 41182
diff changeset
   316
  shows "extension f\<cdot>x = lub (f ` rep x)"
51c866d1b53b rename function ideal_completion.basis_fun to ideal_completion.extension
huffman
parents: 41182
diff changeset
   317
unfolding extension_def
27404
62171da527d6 split Completion.thy from CompactBasis.thy
huffman
parents:
diff changeset
   318
proof (rule beta_cfun)
62171da527d6 split Completion.thy from CompactBasis.thy
huffman
parents:
diff changeset
   319
  have lub: "\<And>x. \<exists>u. f ` rep x <<| u"
41394
51c866d1b53b rename function ideal_completion.basis_fun to ideal_completion.extension
huffman
parents: 41182
diff changeset
   320
    using f_mono by (rule extension_lemma)
27404
62171da527d6 split Completion.thy from CompactBasis.thy
huffman
parents:
diff changeset
   321
  show cont: "cont (\<lambda>x. lub (f ` rep x))"
62171da527d6 split Completion.thy from CompactBasis.thy
huffman
parents:
diff changeset
   322
    apply (rule contI2)
62171da527d6 split Completion.thy from CompactBasis.thy
huffman
parents:
diff changeset
   323
     apply (rule monofunI)
39974
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39967
diff changeset
   324
     apply (rule is_lub_thelub_ex [OF lub ub_imageI])
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39967
diff changeset
   325
     apply (rule is_ub_thelub_ex [OF lub imageI])
27404
62171da527d6 split Completion.thy from CompactBasis.thy
huffman
parents:
diff changeset
   326
     apply (erule (1) subsetD [OF rep_mono])
39974
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39967
diff changeset
   327
    apply (rule is_lub_thelub_ex [OF lub ub_imageI])
40769
3af9b0df3521 rename rep_contlub lemmas to rep_lub
huffman
parents: 40502
diff changeset
   328
    apply (simp add: rep_lub, clarify)
31076
99fe356cbbc2 rename constant sq_le to below; rename class sq_ord to below; less->below in many lemma names
huffman
parents: 30729
diff changeset
   329
    apply (erule rev_below_trans [OF is_ub_thelub])
39974
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39967
diff changeset
   330
    apply (erule is_ub_thelub_ex [OF lub imageI])
27404
62171da527d6 split Completion.thy from CompactBasis.thy
huffman
parents:
diff changeset
   331
    done
62171da527d6 split Completion.thy from CompactBasis.thy
huffman
parents:
diff changeset
   332
qed
62171da527d6 split Completion.thy from CompactBasis.thy
huffman
parents:
diff changeset
   333
41394
51c866d1b53b rename function ideal_completion.basis_fun to ideal_completion.extension
huffman
parents: 41182
diff changeset
   334
lemma extension_principal:
27404
62171da527d6 split Completion.thy from CompactBasis.thy
huffman
parents:
diff changeset
   335
  fixes f :: "'a::type \<Rightarrow> 'c::cpo"
62171da527d6 split Completion.thy from CompactBasis.thy
huffman
parents:
diff changeset
   336
  assumes f_mono: "\<And>a b. a \<preceq> b \<Longrightarrow> f a \<sqsubseteq> f b"
41394
51c866d1b53b rename function ideal_completion.basis_fun to ideal_completion.extension
huffman
parents: 41182
diff changeset
   337
  shows "extension f\<cdot>(principal a) = f a"
51c866d1b53b rename function ideal_completion.basis_fun to ideal_completion.extension
huffman
parents: 41182
diff changeset
   338
apply (subst extension_beta, erule f_mono)
27404
62171da527d6 split Completion.thy from CompactBasis.thy
huffman
parents:
diff changeset
   339
apply (subst rep_principal)
41033
7a67a8832da8 simplify ideal completion proofs
huffman
parents: 40888
diff changeset
   340
apply (rule lub_eqI)
7a67a8832da8 simplify ideal completion proofs
huffman
parents: 40888
diff changeset
   341
apply (rule is_lub_maximal)
7a67a8832da8 simplify ideal completion proofs
huffman
parents: 40888
diff changeset
   342
apply (rule ub_imageI)
7a67a8832da8 simplify ideal completion proofs
huffman
parents: 40888
diff changeset
   343
apply (simp add: f_mono)
7a67a8832da8 simplify ideal completion proofs
huffman
parents: 40888
diff changeset
   344
apply (rule imageI)
7a67a8832da8 simplify ideal completion proofs
huffman
parents: 40888
diff changeset
   345
apply (simp add: r_refl)
27404
62171da527d6 split Completion.thy from CompactBasis.thy
huffman
parents:
diff changeset
   346
done
62171da527d6 split Completion.thy from CompactBasis.thy
huffman
parents:
diff changeset
   347
41394
51c866d1b53b rename function ideal_completion.basis_fun to ideal_completion.extension
huffman
parents: 41182
diff changeset
   348
lemma extension_mono:
27404
62171da527d6 split Completion.thy from CompactBasis.thy
huffman
parents:
diff changeset
   349
  assumes f_mono: "\<And>a b. a \<preceq> b \<Longrightarrow> f a \<sqsubseteq> f b"
62171da527d6 split Completion.thy from CompactBasis.thy
huffman
parents:
diff changeset
   350
  assumes g_mono: "\<And>a b. a \<preceq> b \<Longrightarrow> g a \<sqsubseteq> g b"
31076
99fe356cbbc2 rename constant sq_le to below; rename class sq_ord to below; less->below in many lemma names
huffman
parents: 30729
diff changeset
   351
  assumes below: "\<And>a. f a \<sqsubseteq> g a"
41394
51c866d1b53b rename function ideal_completion.basis_fun to ideal_completion.extension
huffman
parents: 41182
diff changeset
   352
  shows "extension f \<sqsubseteq> extension g"
40002
c5b5f7a3a3b1 new theorem names: fun_below_iff, fun_belowI, cfun_eq_iff, cfun_eqI, cfun_below_iff, cfun_belowI
huffman
parents: 39984
diff changeset
   353
 apply (rule cfun_belowI)
41394
51c866d1b53b rename function ideal_completion.basis_fun to ideal_completion.extension
huffman
parents: 41182
diff changeset
   354
 apply (simp only: extension_beta f_mono g_mono)
39974
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39967
diff changeset
   355
 apply (rule is_lub_thelub_ex)
41394
51c866d1b53b rename function ideal_completion.basis_fun to ideal_completion.extension
huffman
parents: 41182
diff changeset
   356
  apply (rule extension_lemma, erule f_mono)
27404
62171da527d6 split Completion.thy from CompactBasis.thy
huffman
parents:
diff changeset
   357
 apply (rule ub_imageI, rename_tac a)
31076
99fe356cbbc2 rename constant sq_le to below; rename class sq_ord to below; less->below in many lemma names
huffman
parents: 30729
diff changeset
   358
 apply (rule below_trans [OF below])
39974
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39967
diff changeset
   359
 apply (rule is_ub_thelub_ex)
41394
51c866d1b53b rename function ideal_completion.basis_fun to ideal_completion.extension
huffman
parents: 41182
diff changeset
   360
  apply (rule extension_lemma, erule g_mono)
27404
62171da527d6 split Completion.thy from CompactBasis.thy
huffman
parents:
diff changeset
   361
 apply (erule imageI)
62171da527d6 split Completion.thy from CompactBasis.thy
huffman
parents:
diff changeset
   362
done
62171da527d6 split Completion.thy from CompactBasis.thy
huffman
parents:
diff changeset
   363
41394
51c866d1b53b rename function ideal_completion.basis_fun to ideal_completion.extension
huffman
parents: 41182
diff changeset
   364
lemma cont_extension:
41182
717404c7d59a add notsqsubseteq syntax
huffman
parents: 41033
diff changeset
   365
  assumes f_mono: "\<And>a b x. a \<preceq> b \<Longrightarrow> f x a \<sqsubseteq> f x b"
717404c7d59a add notsqsubseteq syntax
huffman
parents: 41033
diff changeset
   366
  assumes f_cont: "\<And>a. cont (\<lambda>x. f x a)"
41394
51c866d1b53b rename function ideal_completion.basis_fun to ideal_completion.extension
huffman
parents: 41182
diff changeset
   367
  shows "cont (\<lambda>x. extension (\<lambda>a. f x a))"
41182
717404c7d59a add notsqsubseteq syntax
huffman
parents: 41033
diff changeset
   368
 apply (rule contI2)
717404c7d59a add notsqsubseteq syntax
huffman
parents: 41033
diff changeset
   369
  apply (rule monofunI)
41394
51c866d1b53b rename function ideal_completion.basis_fun to ideal_completion.extension
huffman
parents: 41182
diff changeset
   370
  apply (rule extension_mono, erule f_mono, erule f_mono)
41182
717404c7d59a add notsqsubseteq syntax
huffman
parents: 41033
diff changeset
   371
  apply (erule cont2monofunE [OF f_cont])
717404c7d59a add notsqsubseteq syntax
huffman
parents: 41033
diff changeset
   372
 apply (rule cfun_belowI)
717404c7d59a add notsqsubseteq syntax
huffman
parents: 41033
diff changeset
   373
 apply (rule principal_induct, simp)
717404c7d59a add notsqsubseteq syntax
huffman
parents: 41033
diff changeset
   374
 apply (simp only: contlub_cfun_fun)
41394
51c866d1b53b rename function ideal_completion.basis_fun to ideal_completion.extension
huffman
parents: 41182
diff changeset
   375
 apply (simp only: extension_principal f_mono)
41182
717404c7d59a add notsqsubseteq syntax
huffman
parents: 41033
diff changeset
   376
 apply (simp add: cont2contlubE [OF f_cont])
717404c7d59a add notsqsubseteq syntax
huffman
parents: 41033
diff changeset
   377
done
717404c7d59a add notsqsubseteq syntax
huffman
parents: 41033
diff changeset
   378
27404
62171da527d6 split Completion.thy from CompactBasis.thy
huffman
parents:
diff changeset
   379
end
62171da527d6 split Completion.thy from CompactBasis.thy
huffman
parents:
diff changeset
   380
39984
0300d5170622 add lemma typedef_ideal_completion
huffman
parents: 39983
diff changeset
   381
lemma (in preorder) typedef_ideal_completion:
0300d5170622 add lemma typedef_ideal_completion
huffman
parents: 39983
diff changeset
   382
  fixes Abs :: "'a set \<Rightarrow> 'b::cpo"
0300d5170622 add lemma typedef_ideal_completion
huffman
parents: 39983
diff changeset
   383
  assumes type: "type_definition Rep Abs {S. ideal S}"
0300d5170622 add lemma typedef_ideal_completion
huffman
parents: 39983
diff changeset
   384
  assumes below: "\<And>x y. x \<sqsubseteq> y \<longleftrightarrow> Rep x \<subseteq> Rep y"
0300d5170622 add lemma typedef_ideal_completion
huffman
parents: 39983
diff changeset
   385
  assumes principal: "\<And>a. principal a = Abs {b. b \<preceq> a}"
0300d5170622 add lemma typedef_ideal_completion
huffman
parents: 39983
diff changeset
   386
  assumes countable: "\<exists>f::'a \<Rightarrow> nat. inj f"
0300d5170622 add lemma typedef_ideal_completion
huffman
parents: 39983
diff changeset
   387
  shows "ideal_completion r principal Rep"
0300d5170622 add lemma typedef_ideal_completion
huffman
parents: 39983
diff changeset
   388
proof
0300d5170622 add lemma typedef_ideal_completion
huffman
parents: 39983
diff changeset
   389
  interpret type_definition Rep Abs "{S. ideal S}" by fact
0300d5170622 add lemma typedef_ideal_completion
huffman
parents: 39983
diff changeset
   390
  fix a b :: 'a and x y :: 'b and Y :: "nat \<Rightarrow> 'b"
0300d5170622 add lemma typedef_ideal_completion
huffman
parents: 39983
diff changeset
   391
  show "ideal (Rep x)"
0300d5170622 add lemma typedef_ideal_completion
huffman
parents: 39983
diff changeset
   392
    using Rep [of x] by simp
0300d5170622 add lemma typedef_ideal_completion
huffman
parents: 39983
diff changeset
   393
  show "chain Y \<Longrightarrow> Rep (\<Squnion>i. Y i) = (\<Union>i. Rep (Y i))"
40769
3af9b0df3521 rename rep_contlub lemmas to rep_lub
huffman
parents: 40502
diff changeset
   394
    using type below by (rule typedef_ideal_rep_lub)
39984
0300d5170622 add lemma typedef_ideal_completion
huffman
parents: 39983
diff changeset
   395
  show "Rep (principal a) = {b. b \<preceq> a}"
0300d5170622 add lemma typedef_ideal_completion
huffman
parents: 39983
diff changeset
   396
    by (simp add: principal Abs_inverse ideal_principal)
0300d5170622 add lemma typedef_ideal_completion
huffman
parents: 39983
diff changeset
   397
  show "Rep x \<subseteq> Rep y \<Longrightarrow> x \<sqsubseteq> y"
0300d5170622 add lemma typedef_ideal_completion
huffman
parents: 39983
diff changeset
   398
    by (simp only: below)
0300d5170622 add lemma typedef_ideal_completion
huffman
parents: 39983
diff changeset
   399
  show "\<exists>f::'a \<Rightarrow> nat. inj f"
0300d5170622 add lemma typedef_ideal_completion
huffman
parents: 39983
diff changeset
   400
    by (rule countable)
0300d5170622 add lemma typedef_ideal_completion
huffman
parents: 39983
diff changeset
   401
qed
0300d5170622 add lemma typedef_ideal_completion
huffman
parents: 39983
diff changeset
   402
27404
62171da527d6 split Completion.thy from CompactBasis.thy
huffman
parents:
diff changeset
   403
end