1 (* Title: HOLCF/Completion.thy |
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2 Author: Brian Huffman |
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3 *) |
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4 |
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5 header {* Defining algebraic domains by ideal completion *} |
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6 |
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7 theory Completion |
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8 imports Plain_HOLCF |
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9 begin |
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10 |
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11 subsection {* Ideals over a preorder *} |
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12 |
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13 locale preorder = |
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14 fixes r :: "'a::type \<Rightarrow> 'a \<Rightarrow> bool" (infix "\<preceq>" 50) |
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15 assumes r_refl: "x \<preceq> x" |
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16 assumes r_trans: "\<lbrakk>x \<preceq> y; y \<preceq> z\<rbrakk> \<Longrightarrow> x \<preceq> z" |
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17 begin |
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18 |
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19 definition |
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20 ideal :: "'a set \<Rightarrow> bool" where |
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21 "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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22 (\<forall>x y. x \<preceq> y \<longrightarrow> y \<in> A \<longrightarrow> x \<in> A))" |
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23 |
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24 lemma idealI: |
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25 assumes "\<exists>x. x \<in> A" |
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26 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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27 assumes "\<And>x y. \<lbrakk>x \<preceq> y; y \<in> A\<rbrakk> \<Longrightarrow> x \<in> A" |
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28 shows "ideal A" |
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29 unfolding ideal_def using prems by fast |
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30 |
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31 lemma idealD1: |
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32 "ideal A \<Longrightarrow> \<exists>x. x \<in> A" |
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33 unfolding ideal_def by fast |
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34 |
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35 lemma idealD2: |
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36 "\<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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37 unfolding ideal_def by fast |
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38 |
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39 lemma idealD3: |
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40 "\<lbrakk>ideal A; x \<preceq> y; y \<in> A\<rbrakk> \<Longrightarrow> x \<in> A" |
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41 unfolding ideal_def by fast |
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42 |
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43 lemma ideal_principal: "ideal {x. x \<preceq> z}" |
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44 apply (rule idealI) |
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45 apply (rule_tac x=z in exI) |
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46 apply (fast intro: r_refl) |
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47 apply (rule_tac x=z in bexI, fast) |
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48 apply (fast intro: r_refl) |
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49 apply (fast intro: r_trans) |
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50 done |
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51 |
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52 lemma ex_ideal: "\<exists>A. ideal A" |
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53 by (rule exI, rule ideal_principal) |
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54 |
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55 lemma lub_image_principal: |
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56 assumes f: "\<And>x y. x \<preceq> y \<Longrightarrow> f x \<sqsubseteq> f y" |
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57 shows "(\<Squnion>x\<in>{x. x \<preceq> y}. f x) = f y" |
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58 apply (rule lub_eqI) |
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59 apply (rule is_lub_maximal) |
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60 apply (rule ub_imageI) |
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61 apply (simp add: f) |
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62 apply (rule imageI) |
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63 apply (simp add: r_refl) |
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64 done |
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65 |
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66 text {* The set of ideals is a cpo *} |
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67 |
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68 lemma ideal_UN: |
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69 fixes A :: "nat \<Rightarrow> 'a set" |
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70 assumes ideal_A: "\<And>i. ideal (A i)" |
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71 assumes chain_A: "\<And>i j. i \<le> j \<Longrightarrow> A i \<subseteq> A j" |
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72 shows "ideal (\<Union>i. A i)" |
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73 apply (rule idealI) |
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74 apply (cut_tac idealD1 [OF ideal_A], fast) |
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75 apply (clarify, rename_tac i j) |
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76 apply (drule subsetD [OF chain_A [OF le_maxI1]]) |
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77 apply (drule subsetD [OF chain_A [OF le_maxI2]]) |
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78 apply (drule (1) idealD2 [OF ideal_A]) |
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79 apply blast |
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80 apply clarify |
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81 apply (drule (1) idealD3 [OF ideal_A]) |
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82 apply fast |
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83 done |
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84 |
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85 lemma typedef_ideal_po: |
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86 fixes Abs :: "'a set \<Rightarrow> 'b::below" |
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87 assumes type: "type_definition Rep Abs {S. ideal S}" |
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88 assumes below: "\<And>x y. x \<sqsubseteq> y \<longleftrightarrow> Rep x \<subseteq> Rep y" |
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89 shows "OFCLASS('b, po_class)" |
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90 apply (intro_classes, unfold below) |
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91 apply (rule subset_refl) |
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92 apply (erule (1) subset_trans) |
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93 apply (rule type_definition.Rep_inject [OF type, THEN iffD1]) |
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94 apply (erule (1) subset_antisym) |
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95 done |
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96 |
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97 lemma |
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98 fixes Abs :: "'a set \<Rightarrow> 'b::po" |
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99 assumes type: "type_definition Rep Abs {S. ideal S}" |
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100 assumes below: "\<And>x y. x \<sqsubseteq> y \<longleftrightarrow> Rep x \<subseteq> Rep y" |
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101 assumes S: "chain S" |
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102 shows typedef_ideal_lub: "range S <<| Abs (\<Union>i. Rep (S i))" |
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103 and typedef_ideal_rep_lub: "Rep (\<Squnion>i. S i) = (\<Union>i. Rep (S i))" |
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104 proof - |
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105 have 1: "ideal (\<Union>i. Rep (S i))" |
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106 apply (rule ideal_UN) |
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107 apply (rule type_definition.Rep [OF type, unfolded mem_Collect_eq]) |
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108 apply (subst below [symmetric]) |
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109 apply (erule chain_mono [OF S]) |
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110 done |
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111 hence 2: "Rep (Abs (\<Union>i. Rep (S i))) = (\<Union>i. Rep (S i))" |
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112 by (simp add: type_definition.Abs_inverse [OF type]) |
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113 show 3: "range S <<| Abs (\<Union>i. Rep (S i))" |
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114 apply (rule is_lubI) |
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115 apply (rule is_ubI) |
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116 apply (simp add: below 2, fast) |
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117 apply (simp add: below 2 is_ub_def, fast) |
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118 done |
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119 hence 4: "(\<Squnion>i. S i) = Abs (\<Union>i. Rep (S i))" |
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120 by (rule lub_eqI) |
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121 show 5: "Rep (\<Squnion>i. S i) = (\<Union>i. Rep (S i))" |
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122 by (simp add: 4 2) |
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123 qed |
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124 |
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125 lemma typedef_ideal_cpo: |
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126 fixes Abs :: "'a set \<Rightarrow> 'b::po" |
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127 assumes type: "type_definition Rep Abs {S. ideal S}" |
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128 assumes below: "\<And>x y. x \<sqsubseteq> y \<longleftrightarrow> Rep x \<subseteq> Rep y" |
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129 shows "OFCLASS('b, cpo_class)" |
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130 by (default, rule exI, erule typedef_ideal_lub [OF type below]) |
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131 |
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132 end |
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133 |
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134 interpretation below: preorder "below :: 'a::po \<Rightarrow> 'a \<Rightarrow> bool" |
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135 apply unfold_locales |
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136 apply (rule below_refl) |
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137 apply (erule (1) below_trans) |
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138 done |
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139 |
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140 subsection {* Lemmas about least upper bounds *} |
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141 |
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142 lemma is_ub_thelub_ex: "\<lbrakk>\<exists>u. S <<| u; x \<in> S\<rbrakk> \<Longrightarrow> x \<sqsubseteq> lub S" |
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143 apply (erule exE, drule is_lub_lub) |
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144 apply (drule is_lubD1) |
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145 apply (erule (1) is_ubD) |
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146 done |
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147 |
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148 lemma is_lub_thelub_ex: "\<lbrakk>\<exists>u. S <<| u; S <| x\<rbrakk> \<Longrightarrow> lub S \<sqsubseteq> x" |
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149 by (erule exE, drule is_lub_lub, erule is_lubD2) |
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150 |
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151 subsection {* Locale for ideal completion *} |
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152 |
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153 locale ideal_completion = preorder + |
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154 fixes principal :: "'a::type \<Rightarrow> 'b::cpo" |
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155 fixes rep :: "'b::cpo \<Rightarrow> 'a::type set" |
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156 assumes ideal_rep: "\<And>x. ideal (rep x)" |
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157 assumes rep_lub: "\<And>Y. chain Y \<Longrightarrow> rep (\<Squnion>i. Y i) = (\<Union>i. rep (Y i))" |
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158 assumes rep_principal: "\<And>a. rep (principal a) = {b. b \<preceq> a}" |
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159 assumes subset_repD: "\<And>x y. rep x \<subseteq> rep y \<Longrightarrow> x \<sqsubseteq> y" |
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160 assumes countable: "\<exists>f::'a \<Rightarrow> nat. inj f" |
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161 begin |
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162 |
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163 lemma rep_mono: "x \<sqsubseteq> y \<Longrightarrow> rep x \<subseteq> rep y" |
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164 apply (frule bin_chain) |
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165 apply (drule rep_lub) |
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166 apply (simp only: lub_eqI [OF is_lub_bin_chain]) |
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167 apply (rule subsetI, rule UN_I [where a=0], simp_all) |
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168 done |
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169 |
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170 lemma below_def: "x \<sqsubseteq> y \<longleftrightarrow> rep x \<subseteq> rep y" |
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171 by (rule iffI [OF rep_mono subset_repD]) |
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172 |
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173 lemma rep_eq: "rep x = {a. principal a \<sqsubseteq> x}" |
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174 unfolding below_def rep_principal |
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175 apply safe |
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176 apply (erule (1) idealD3 [OF ideal_rep]) |
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177 apply (erule subsetD, simp add: r_refl) |
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178 done |
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179 |
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180 lemma mem_rep_iff_principal_below: "a \<in> rep x \<longleftrightarrow> principal a \<sqsubseteq> x" |
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181 by (simp add: rep_eq) |
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182 |
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183 lemma principal_below_iff_mem_rep: "principal a \<sqsubseteq> x \<longleftrightarrow> a \<in> rep x" |
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184 by (simp add: rep_eq) |
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185 |
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186 lemma principal_below_iff [simp]: "principal a \<sqsubseteq> principal b \<longleftrightarrow> a \<preceq> b" |
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187 by (simp add: principal_below_iff_mem_rep rep_principal) |
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188 |
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189 lemma principal_eq_iff: "principal a = principal b \<longleftrightarrow> a \<preceq> b \<and> b \<preceq> a" |
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190 unfolding po_eq_conv [where 'a='b] principal_below_iff .. |
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191 |
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192 lemma eq_iff: "x = y \<longleftrightarrow> rep x = rep y" |
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193 unfolding po_eq_conv below_def by auto |
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194 |
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195 lemma repD: "a \<in> rep x \<Longrightarrow> principal a \<sqsubseteq> x" |
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196 by (simp add: rep_eq) |
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197 |
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198 lemma principal_mono: "a \<preceq> b \<Longrightarrow> principal a \<sqsubseteq> principal b" |
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199 by (simp only: principal_below_iff) |
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200 |
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201 lemma ch2ch_principal [simp]: |
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202 "\<forall>i. Y i \<preceq> Y (Suc i) \<Longrightarrow> chain (\<lambda>i. principal (Y i))" |
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203 by (simp add: chainI principal_mono) |
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204 |
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205 lemma lub_principal_rep: "principal ` rep x <<| x" |
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206 apply (rule is_lubI) |
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207 apply (rule ub_imageI) |
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208 apply (erule repD) |
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209 apply (subst below_def) |
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210 apply (rule subsetI) |
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211 apply (drule (1) ub_imageD) |
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212 apply (simp add: rep_eq) |
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213 done |
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214 |
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215 subsubsection {* Principal ideals approximate all elements *} |
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216 |
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217 lemma compact_principal [simp]: "compact (principal a)" |
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218 by (rule compactI2, simp add: principal_below_iff_mem_rep rep_lub) |
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219 |
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220 text {* Construct a chain whose lub is the same as a given ideal *} |
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221 |
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222 lemma obtain_principal_chain: |
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223 obtains Y where "\<forall>i. Y i \<preceq> Y (Suc i)" and "x = (\<Squnion>i. principal (Y i))" |
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224 proof - |
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225 obtain count :: "'a \<Rightarrow> nat" where inj: "inj count" |
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226 using countable .. |
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227 def enum \<equiv> "\<lambda>i. THE a. count a = i" |
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228 have enum_count [simp]: "\<And>x. enum (count x) = x" |
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229 unfolding enum_def by (simp add: inj_eq [OF inj]) |
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230 def a \<equiv> "LEAST i. enum i \<in> rep x" |
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231 def b \<equiv> "\<lambda>i. LEAST j. enum j \<in> rep x \<and> \<not> enum j \<preceq> enum i" |
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232 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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233 def P \<equiv> "\<lambda>i. \<exists>j. enum j \<in> rep x \<and> \<not> enum j \<preceq> enum i" |
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234 def X \<equiv> "nat_rec a (\<lambda>n i. if P i then c i (b i) else i)" |
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235 have X_0: "X 0 = a" unfolding X_def by simp |
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236 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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237 unfolding X_def by simp |
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238 have a_mem: "enum a \<in> rep x" |
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239 unfolding a_def |
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240 apply (rule LeastI_ex) |
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241 apply (cut_tac ideal_rep [of x]) |
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242 apply (drule idealD1) |
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243 apply (clarify, rename_tac a) |
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244 apply (rule_tac x="count a" in exI, simp) |
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245 done |
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246 have b: "\<And>i. P i \<Longrightarrow> enum i \<in> rep x |
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247 \<Longrightarrow> enum (b i) \<in> rep x \<and> \<not> enum (b i) \<preceq> enum i" |
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248 unfolding P_def b_def by (erule LeastI2_ex, simp) |
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249 have c: "\<And>i j. enum i \<in> rep x \<Longrightarrow> enum j \<in> rep x |
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250 \<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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251 unfolding c_def |
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252 apply (drule (1) idealD2 [OF ideal_rep], clarify) |
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253 apply (rule_tac a="count z" in LeastI2, simp, simp) |
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254 done |
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255 have X_mem: "\<And>n. enum (X n) \<in> rep x" |
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256 apply (induct_tac n) |
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257 apply (simp add: X_0 a_mem) |
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258 apply (clarsimp simp add: X_Suc, rename_tac n) |
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259 apply (simp add: b c) |
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260 done |
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261 have X_chain: "\<And>n. enum (X n) \<preceq> enum (X (Suc n))" |
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262 apply (clarsimp simp add: X_Suc r_refl) |
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263 apply (simp add: b c X_mem) |
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264 done |
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265 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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266 unfolding b_def by (drule not_less_Least, simp) |
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267 have X_covers: "\<And>n. \<forall>k\<le>n. enum k \<in> rep x \<longrightarrow> enum k \<preceq> enum (X n)" |
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268 apply (induct_tac n) |
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269 apply (clarsimp simp add: X_0 a_def) |
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270 apply (drule_tac k=0 in Least_le, simp add: r_refl) |
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271 apply (clarsimp, rename_tac n k) |
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272 apply (erule le_SucE) |
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273 apply (rule r_trans [OF _ X_chain], simp) |
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274 apply (case_tac "P (X n)", simp add: X_Suc) |
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275 apply (rule_tac x="b (X n)" and y="Suc n" in linorder_cases) |
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276 apply (simp only: less_Suc_eq_le) |
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277 apply (drule spec, drule (1) mp, simp add: b X_mem) |
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278 apply (simp add: c X_mem) |
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279 apply (drule (1) less_b) |
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280 apply (erule r_trans) |
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281 apply (simp add: b c X_mem) |
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282 apply (simp add: X_Suc) |
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283 apply (simp add: P_def) |
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284 done |
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285 have 1: "\<forall>i. enum (X i) \<preceq> enum (X (Suc i))" |
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286 by (simp add: X_chain) |
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287 have 2: "x = (\<Squnion>n. principal (enum (X n)))" |
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288 apply (simp add: eq_iff rep_lub 1 rep_principal) |
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289 apply (auto, rename_tac a) |
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290 apply (subgoal_tac "\<exists>i. a = enum i", erule exE) |
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291 apply (rule_tac x=i in exI, simp add: X_covers) |
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292 apply (rule_tac x="count a" in exI, simp) |
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293 apply (erule idealD3 [OF ideal_rep]) |
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294 apply (rule X_mem) |
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295 done |
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296 from 1 2 show ?thesis .. |
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297 qed |
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298 |
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299 lemma principal_induct: |
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300 assumes adm: "adm P" |
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301 assumes P: "\<And>a. P (principal a)" |
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302 shows "P x" |
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303 apply (rule obtain_principal_chain [of x]) |
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304 apply (simp add: admD [OF adm] P) |
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305 done |
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306 |
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307 lemma principal_induct2: |
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308 "\<lbrakk>\<And>y. adm (\<lambda>x. P x y); \<And>x. adm (\<lambda>y. P x y); |
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309 \<And>a b. P (principal a) (principal b)\<rbrakk> \<Longrightarrow> P x y" |
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310 apply (rule_tac x=y in spec) |
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311 apply (rule_tac x=x in principal_induct, simp) |
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312 apply (rule allI, rename_tac y) |
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313 apply (rule_tac x=y in principal_induct, simp) |
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314 apply simp |
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315 done |
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316 |
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317 lemma compact_imp_principal: "compact x \<Longrightarrow> \<exists>a. x = principal a" |
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318 apply (rule obtain_principal_chain [of x]) |
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319 apply (drule adm_compact_neq [OF _ cont_id]) |
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320 apply (subgoal_tac "chain (\<lambda>i. principal (Y i))") |
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321 apply (drule (2) admD2, fast, simp) |
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322 done |
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323 |
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324 lemma obtain_compact_chain: |
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325 obtains Y :: "nat \<Rightarrow> 'b" |
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326 where "chain Y" and "\<forall>i. compact (Y i)" and "x = (\<Squnion>i. Y i)" |
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327 apply (rule obtain_principal_chain [of x]) |
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328 apply (rule_tac Y="\<lambda>i. principal (Y i)" in that, simp_all) |
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329 done |
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330 |
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331 subsection {* Defining functions in terms of basis elements *} |
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332 |
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333 definition |
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334 basis_fun :: "('a::type \<Rightarrow> 'c::cpo) \<Rightarrow> 'b \<rightarrow> 'c" where |
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335 "basis_fun = (\<lambda>f. (\<Lambda> x. lub (f ` rep x)))" |
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336 |
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337 lemma basis_fun_lemma: |
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338 fixes f :: "'a::type \<Rightarrow> 'c::cpo" |
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339 assumes f_mono: "\<And>a b. a \<preceq> b \<Longrightarrow> f a \<sqsubseteq> f b" |
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340 shows "\<exists>u. f ` rep x <<| u" |
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341 proof - |
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342 obtain Y where Y: "\<forall>i. Y i \<preceq> Y (Suc i)" |
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343 and x: "x = (\<Squnion>i. principal (Y i))" |
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344 by (rule obtain_principal_chain [of x]) |
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345 have chain: "chain (\<lambda>i. f (Y i))" |
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346 by (rule chainI, simp add: f_mono Y) |
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347 have rep_x: "rep x = (\<Union>n. {a. a \<preceq> Y n})" |
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348 by (simp add: x rep_lub Y rep_principal) |
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349 have "f ` rep x <<| (\<Squnion>n. f (Y n))" |
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350 apply (rule is_lubI) |
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351 apply (rule ub_imageI, rename_tac a) |
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352 apply (clarsimp simp add: rep_x) |
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353 apply (drule f_mono) |
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354 apply (erule below_lub [OF chain]) |
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355 apply (rule lub_below [OF chain]) |
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356 apply (drule_tac x="Y n" in ub_imageD) |
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357 apply (simp add: rep_x, fast intro: r_refl) |
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358 apply assumption |
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359 done |
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360 thus ?thesis .. |
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361 qed |
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362 |
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363 lemma basis_fun_beta: |
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364 fixes f :: "'a::type \<Rightarrow> 'c::cpo" |
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365 assumes f_mono: "\<And>a b. a \<preceq> b \<Longrightarrow> f a \<sqsubseteq> f b" |
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366 shows "basis_fun f\<cdot>x = lub (f ` rep x)" |
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367 unfolding basis_fun_def |
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368 proof (rule beta_cfun) |
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369 have lub: "\<And>x. \<exists>u. f ` rep x <<| u" |
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370 using f_mono by (rule basis_fun_lemma) |
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371 show cont: "cont (\<lambda>x. lub (f ` rep x))" |
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372 apply (rule contI2) |
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373 apply (rule monofunI) |
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374 apply (rule is_lub_thelub_ex [OF lub ub_imageI]) |
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375 apply (rule is_ub_thelub_ex [OF lub imageI]) |
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376 apply (erule (1) subsetD [OF rep_mono]) |
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377 apply (rule is_lub_thelub_ex [OF lub ub_imageI]) |
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378 apply (simp add: rep_lub, clarify) |
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379 apply (erule rev_below_trans [OF is_ub_thelub]) |
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380 apply (erule is_ub_thelub_ex [OF lub imageI]) |
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381 done |
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382 qed |
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383 |
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384 lemma basis_fun_principal: |
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385 fixes f :: "'a::type \<Rightarrow> 'c::cpo" |
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386 assumes f_mono: "\<And>a b. a \<preceq> b \<Longrightarrow> f a \<sqsubseteq> f b" |
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387 shows "basis_fun f\<cdot>(principal a) = f a" |
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388 apply (subst basis_fun_beta, erule f_mono) |
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389 apply (subst rep_principal) |
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390 apply (rule lub_image_principal, erule f_mono) |
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391 done |
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392 |
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393 lemma basis_fun_mono: |
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394 assumes f_mono: "\<And>a b. a \<preceq> b \<Longrightarrow> f a \<sqsubseteq> f b" |
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395 assumes g_mono: "\<And>a b. a \<preceq> b \<Longrightarrow> g a \<sqsubseteq> g b" |
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396 assumes below: "\<And>a. f a \<sqsubseteq> g a" |
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397 shows "basis_fun f \<sqsubseteq> basis_fun g" |
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398 apply (rule cfun_belowI) |
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399 apply (simp only: basis_fun_beta f_mono g_mono) |
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400 apply (rule is_lub_thelub_ex) |
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401 apply (rule basis_fun_lemma, erule f_mono) |
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402 apply (rule ub_imageI, rename_tac a) |
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403 apply (rule below_trans [OF below]) |
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404 apply (rule is_ub_thelub_ex) |
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405 apply (rule basis_fun_lemma, erule g_mono) |
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406 apply (erule imageI) |
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407 done |
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408 |
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409 end |
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410 |
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411 lemma (in preorder) typedef_ideal_completion: |
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412 fixes Abs :: "'a set \<Rightarrow> 'b::cpo" |
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413 assumes type: "type_definition Rep Abs {S. ideal S}" |
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414 assumes below: "\<And>x y. x \<sqsubseteq> y \<longleftrightarrow> Rep x \<subseteq> Rep y" |
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415 assumes principal: "\<And>a. principal a = Abs {b. b \<preceq> a}" |
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416 assumes countable: "\<exists>f::'a \<Rightarrow> nat. inj f" |
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417 shows "ideal_completion r principal Rep" |
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418 proof |
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419 interpret type_definition Rep Abs "{S. ideal S}" by fact |
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420 fix a b :: 'a and x y :: 'b and Y :: "nat \<Rightarrow> 'b" |
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421 show "ideal (Rep x)" |
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422 using Rep [of x] by simp |
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423 show "chain Y \<Longrightarrow> Rep (\<Squnion>i. Y i) = (\<Union>i. Rep (Y i))" |
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424 using type below by (rule typedef_ideal_rep_lub) |
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425 show "Rep (principal a) = {b. b \<preceq> a}" |
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426 by (simp add: principal Abs_inverse ideal_principal) |
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427 show "Rep x \<subseteq> Rep y \<Longrightarrow> x \<sqsubseteq> y" |
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428 by (simp only: below) |
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429 show "\<exists>f::'a \<Rightarrow> nat. inj f" |
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430 by (rule countable) |
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431 qed |
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432 |
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433 end |
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