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1 (* Title: HOLCF/Completion.thy |
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2 ID: $Id$ |
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3 Author: Brian Huffman |
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4 *) |
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5 |
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6 header {* Defining bifinite domains by ideal completion *} |
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7 |
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8 theory Completion |
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9 imports Bifinite |
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10 begin |
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11 |
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12 subsection {* Ideals over a preorder *} |
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13 |
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14 locale preorder = |
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15 fixes r :: "'a::type \<Rightarrow> 'a \<Rightarrow> bool" (infix "\<preceq>" 50) |
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16 assumes r_refl: "x \<preceq> x" |
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17 assumes r_trans: "\<lbrakk>x \<preceq> y; y \<preceq> z\<rbrakk> \<Longrightarrow> x \<preceq> z" |
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18 begin |
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19 |
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20 definition |
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21 ideal :: "'a set \<Rightarrow> bool" where |
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22 "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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23 (\<forall>x y. x \<preceq> y \<longrightarrow> y \<in> A \<longrightarrow> x \<in> A))" |
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24 |
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25 lemma idealI: |
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26 assumes "\<exists>x. x \<in> A" |
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27 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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28 assumes "\<And>x y. \<lbrakk>x \<preceq> y; y \<in> A\<rbrakk> \<Longrightarrow> x \<in> A" |
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29 shows "ideal A" |
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30 unfolding ideal_def using prems by fast |
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31 |
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32 lemma idealD1: |
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33 "ideal A \<Longrightarrow> \<exists>x. x \<in> A" |
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34 unfolding ideal_def by fast |
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35 |
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36 lemma idealD2: |
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37 "\<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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38 unfolding ideal_def by fast |
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39 |
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40 lemma idealD3: |
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41 "\<lbrakk>ideal A; x \<preceq> y; y \<in> A\<rbrakk> \<Longrightarrow> x \<in> A" |
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42 unfolding ideal_def by fast |
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43 |
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44 lemma ideal_directed_finite: |
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45 assumes A: "ideal A" |
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46 shows "\<lbrakk>finite U; U \<subseteq> A\<rbrakk> \<Longrightarrow> \<exists>z\<in>A. \<forall>x\<in>U. x \<preceq> z" |
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47 apply (induct U set: finite) |
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48 apply (simp add: idealD1 [OF A]) |
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49 apply (simp, clarify, rename_tac y) |
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50 apply (drule (1) idealD2 [OF A]) |
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51 apply (clarify, erule_tac x=z in rev_bexI) |
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52 apply (fast intro: r_trans) |
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53 done |
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54 |
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55 lemma ideal_principal: "ideal {x. x \<preceq> z}" |
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56 apply (rule idealI) |
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57 apply (rule_tac x=z in exI) |
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58 apply (fast intro: r_refl) |
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59 apply (rule_tac x=z in bexI, fast) |
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60 apply (fast intro: r_refl) |
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61 apply (fast intro: r_trans) |
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62 done |
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63 |
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64 lemma ex_ideal: "\<exists>A. ideal A" |
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65 by (rule exI, rule ideal_principal) |
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66 |
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67 lemma directed_image_ideal: |
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68 assumes A: "ideal A" |
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69 assumes f: "\<And>x y. x \<preceq> y \<Longrightarrow> f x \<sqsubseteq> f y" |
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70 shows "directed (f ` A)" |
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71 apply (rule directedI) |
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72 apply (cut_tac idealD1 [OF A], fast) |
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73 apply (clarify, rename_tac a b) |
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74 apply (drule (1) idealD2 [OF A]) |
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75 apply (clarify, rename_tac c) |
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76 apply (rule_tac x="f c" in rev_bexI) |
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77 apply (erule imageI) |
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78 apply (simp add: f) |
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79 done |
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80 |
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81 lemma lub_image_principal: |
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82 assumes f: "\<And>x y. x \<preceq> y \<Longrightarrow> f x \<sqsubseteq> f y" |
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83 shows "(\<Squnion>x\<in>{x. x \<preceq> y}. f x) = f y" |
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84 apply (rule thelubI) |
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85 apply (rule is_lub_maximal) |
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86 apply (rule ub_imageI) |
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87 apply (simp add: f) |
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88 apply (rule imageI) |
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89 apply (simp add: r_refl) |
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90 done |
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91 |
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92 text {* The set of ideals is a cpo *} |
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93 |
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94 lemma ideal_UN: |
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95 fixes A :: "nat \<Rightarrow> 'a set" |
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96 assumes ideal_A: "\<And>i. ideal (A i)" |
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97 assumes chain_A: "\<And>i j. i \<le> j \<Longrightarrow> A i \<subseteq> A j" |
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98 shows "ideal (\<Union>i. A i)" |
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99 apply (rule idealI) |
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100 apply (cut_tac idealD1 [OF ideal_A], fast) |
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101 apply (clarify, rename_tac i j) |
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102 apply (drule subsetD [OF chain_A [OF le_maxI1]]) |
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103 apply (drule subsetD [OF chain_A [OF le_maxI2]]) |
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104 apply (drule (1) idealD2 [OF ideal_A]) |
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105 apply blast |
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106 apply clarify |
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107 apply (drule (1) idealD3 [OF ideal_A]) |
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108 apply fast |
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109 done |
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110 |
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111 lemma typedef_ideal_po: |
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112 fixes Abs :: "'a set \<Rightarrow> 'b::sq_ord" |
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113 assumes type: "type_definition Rep Abs {S. ideal S}" |
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114 assumes less: "\<And>x y. x \<sqsubseteq> y \<longleftrightarrow> Rep x \<subseteq> Rep y" |
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115 shows "OFCLASS('b, po_class)" |
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116 apply (intro_classes, unfold less) |
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117 apply (rule subset_refl) |
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118 apply (erule (1) subset_trans) |
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119 apply (rule type_definition.Rep_inject [OF type, THEN iffD1]) |
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120 apply (erule (1) subset_antisym) |
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121 done |
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122 |
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123 lemma |
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124 fixes Abs :: "'a set \<Rightarrow> 'b::po" |
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125 assumes type: "type_definition Rep Abs {S. ideal S}" |
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126 assumes less: "\<And>x y. x \<sqsubseteq> y \<longleftrightarrow> Rep x \<subseteq> Rep y" |
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127 assumes S: "chain S" |
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128 shows typedef_ideal_lub: "range S <<| Abs (\<Union>i. Rep (S i))" |
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129 and typedef_ideal_rep_contlub: "Rep (\<Squnion>i. S i) = (\<Union>i. Rep (S i))" |
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130 proof - |
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131 have 1: "ideal (\<Union>i. Rep (S i))" |
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132 apply (rule ideal_UN) |
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133 apply (rule type_definition.Rep [OF type, unfolded mem_Collect_eq]) |
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134 apply (subst less [symmetric]) |
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135 apply (erule chain_mono [OF S]) |
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136 done |
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137 hence 2: "Rep (Abs (\<Union>i. Rep (S i))) = (\<Union>i. Rep (S i))" |
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138 by (simp add: type_definition.Abs_inverse [OF type]) |
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139 show 3: "range S <<| Abs (\<Union>i. Rep (S i))" |
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140 apply (rule is_lubI) |
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141 apply (rule is_ubI) |
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142 apply (simp add: less 2, fast) |
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143 apply (simp add: less 2 is_ub_def, fast) |
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144 done |
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145 hence 4: "(\<Squnion>i. S i) = Abs (\<Union>i. Rep (S i))" |
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146 by (rule thelubI) |
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147 show 5: "Rep (\<Squnion>i. S i) = (\<Union>i. Rep (S i))" |
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148 by (simp add: 4 2) |
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149 qed |
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150 |
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151 lemma typedef_ideal_cpo: |
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152 fixes Abs :: "'a set \<Rightarrow> 'b::po" |
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153 assumes type: "type_definition Rep Abs {S. ideal S}" |
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154 assumes less: "\<And>x y. x \<sqsubseteq> y \<longleftrightarrow> Rep x \<subseteq> Rep y" |
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155 shows "OFCLASS('b, cpo_class)" |
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156 by (default, rule exI, erule typedef_ideal_lub [OF type less]) |
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157 |
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158 end |
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159 |
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160 interpretation sq_le: preorder ["sq_le :: 'a::po \<Rightarrow> 'a \<Rightarrow> bool"] |
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161 apply unfold_locales |
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162 apply (rule refl_less) |
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163 apply (erule (1) trans_less) |
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164 done |
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165 |
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166 subsection {* Defining functions in terms of basis elements *} |
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167 |
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168 lemma finite_directed_contains_lub: |
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169 "\<lbrakk>finite S; directed S\<rbrakk> \<Longrightarrow> \<exists>u\<in>S. S <<| u" |
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170 apply (drule (1) directed_finiteD, rule subset_refl) |
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171 apply (erule bexE) |
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172 apply (rule rev_bexI, assumption) |
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173 apply (erule (1) is_lub_maximal) |
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174 done |
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175 |
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176 lemma lub_finite_directed_in_self: |
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177 "\<lbrakk>finite S; directed S\<rbrakk> \<Longrightarrow> lub S \<in> S" |
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178 apply (drule (1) finite_directed_contains_lub, clarify) |
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179 apply (drule thelubI, simp) |
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180 done |
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181 |
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182 lemma finite_directed_has_lub: "\<lbrakk>finite S; directed S\<rbrakk> \<Longrightarrow> \<exists>u. S <<| u" |
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183 by (drule (1) finite_directed_contains_lub, fast) |
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184 |
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185 lemma is_ub_thelub0: "\<lbrakk>\<exists>u. S <<| u; x \<in> S\<rbrakk> \<Longrightarrow> x \<sqsubseteq> lub S" |
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186 apply (erule exE, drule lubI) |
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187 apply (drule is_lubD1) |
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188 apply (erule (1) is_ubD) |
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189 done |
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190 |
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191 lemma is_lub_thelub0: "\<lbrakk>\<exists>u. S <<| u; S <| x\<rbrakk> \<Longrightarrow> lub S \<sqsubseteq> x" |
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192 by (erule exE, drule lubI, erule is_lub_lub) |
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193 |
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194 locale basis_take = preorder + |
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195 fixes take :: "nat \<Rightarrow> 'a::type \<Rightarrow> 'a" |
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196 assumes take_less: "take n a \<preceq> a" |
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197 assumes take_take: "take n (take n a) = take n a" |
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198 assumes take_mono: "a \<preceq> b \<Longrightarrow> take n a \<preceq> take n b" |
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199 assumes take_chain: "take n a \<preceq> take (Suc n) a" |
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200 assumes finite_range_take: "finite (range (take n))" |
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201 assumes take_covers: "\<exists>n. take n a = a" |
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202 begin |
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203 |
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204 lemma take_chain_less: "m < n \<Longrightarrow> take m a \<preceq> take n a" |
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205 by (erule less_Suc_induct, rule take_chain, erule (1) r_trans) |
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206 |
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207 lemma take_chain_le: "m \<le> n \<Longrightarrow> take m a \<preceq> take n a" |
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208 by (cases "m = n", simp add: r_refl, simp add: take_chain_less) |
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209 |
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210 end |
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211 |
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212 locale ideal_completion = basis_take + |
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213 fixes principal :: "'a::type \<Rightarrow> 'b::cpo" |
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214 fixes rep :: "'b::cpo \<Rightarrow> 'a::type set" |
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215 assumes ideal_rep: "\<And>x. preorder.ideal r (rep x)" |
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216 assumes rep_contlub: "\<And>Y. chain Y \<Longrightarrow> rep (\<Squnion>i. Y i) = (\<Union>i. rep (Y i))" |
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217 assumes rep_principal: "\<And>a. rep (principal a) = {b. b \<preceq> a}" |
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218 assumes subset_repD: "\<And>x y. rep x \<subseteq> rep y \<Longrightarrow> x \<sqsubseteq> y" |
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219 begin |
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220 |
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221 lemma finite_take_rep: "finite (take n ` rep x)" |
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222 by (rule finite_subset [OF image_mono [OF subset_UNIV] finite_range_take]) |
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223 |
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224 lemma basis_fun_lemma0: |
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225 fixes f :: "'a::type \<Rightarrow> 'c::cpo" |
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226 assumes f_mono: "\<And>a b. a \<preceq> b \<Longrightarrow> f a \<sqsubseteq> f b" |
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227 shows "\<exists>u. f ` take i ` rep x <<| u" |
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228 apply (rule finite_directed_has_lub) |
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229 apply (rule finite_imageI) |
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230 apply (rule finite_take_rep) |
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231 apply (subst image_image) |
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232 apply (rule directed_image_ideal) |
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233 apply (rule ideal_rep) |
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234 apply (rule f_mono) |
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235 apply (erule take_mono) |
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236 done |
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237 |
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238 lemma basis_fun_lemma1: |
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239 fixes f :: "'a::type \<Rightarrow> 'c::cpo" |
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240 assumes f_mono: "\<And>a b. a \<preceq> b \<Longrightarrow> f a \<sqsubseteq> f b" |
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241 shows "chain (\<lambda>i. lub (f ` take i ` rep x))" |
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242 apply (rule chainI) |
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243 apply (rule is_lub_thelub0) |
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244 apply (rule basis_fun_lemma0, erule f_mono) |
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245 apply (rule is_ubI, clarsimp, rename_tac a) |
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246 apply (rule sq_le.trans_less [OF f_mono [OF take_chain]]) |
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247 apply (rule is_ub_thelub0) |
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248 apply (rule basis_fun_lemma0, erule f_mono) |
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249 apply simp |
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250 done |
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251 |
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252 lemma basis_fun_lemma2: |
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253 fixes f :: "'a::type \<Rightarrow> 'c::cpo" |
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254 assumes f_mono: "\<And>a b. a \<preceq> b \<Longrightarrow> f a \<sqsubseteq> f b" |
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255 shows "f ` rep x <<| (\<Squnion>i. lub (f ` take i ` rep x))" |
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256 apply (rule is_lubI) |
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257 apply (rule ub_imageI, rename_tac a) |
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258 apply (cut_tac a=a in take_covers, erule exE, rename_tac i) |
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259 apply (erule subst) |
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260 apply (rule rev_trans_less) |
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261 apply (rule_tac x=i in is_ub_thelub) |
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262 apply (rule basis_fun_lemma1, erule f_mono) |
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263 apply (rule is_ub_thelub0) |
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264 apply (rule basis_fun_lemma0, erule f_mono) |
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265 apply simp |
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266 apply (rule is_lub_thelub [OF _ ub_rangeI]) |
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267 apply (rule basis_fun_lemma1, erule f_mono) |
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268 apply (rule is_lub_thelub0) |
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269 apply (rule basis_fun_lemma0, erule f_mono) |
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270 apply (rule is_ubI, clarsimp, rename_tac a) |
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271 apply (rule sq_le.trans_less [OF f_mono [OF take_less]]) |
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272 apply (erule (1) ub_imageD) |
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273 done |
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274 |
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275 lemma basis_fun_lemma: |
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276 fixes f :: "'a::type \<Rightarrow> 'c::cpo" |
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277 assumes f_mono: "\<And>a b. a \<preceq> b \<Longrightarrow> f a \<sqsubseteq> f b" |
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278 shows "\<exists>u. f ` rep x <<| u" |
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279 by (rule exI, rule basis_fun_lemma2, erule f_mono) |
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280 |
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281 lemma rep_mono: "x \<sqsubseteq> y \<Longrightarrow> rep x \<subseteq> rep y" |
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282 apply (frule bin_chain) |
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283 apply (drule rep_contlub) |
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284 apply (simp only: thelubI [OF lub_bin_chain]) |
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285 apply (rule subsetI, rule UN_I [where a=0], simp_all) |
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286 done |
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287 |
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288 lemma less_def: "x \<sqsubseteq> y \<longleftrightarrow> rep x \<subseteq> rep y" |
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289 by (rule iffI [OF rep_mono subset_repD]) |
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290 |
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291 lemma rep_eq: "rep x = {a. principal a \<sqsubseteq> x}" |
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292 unfolding less_def rep_principal |
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293 apply safe |
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294 apply (erule (1) idealD3 [OF ideal_rep]) |
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295 apply (erule subsetD, simp add: r_refl) |
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296 done |
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297 |
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298 lemma mem_rep_iff_principal_less: "a \<in> rep x \<longleftrightarrow> principal a \<sqsubseteq> x" |
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299 by (simp add: rep_eq) |
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300 |
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301 lemma principal_less_iff_mem_rep: "principal a \<sqsubseteq> x \<longleftrightarrow> a \<in> rep x" |
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302 by (simp add: rep_eq) |
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303 |
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304 lemma principal_less_iff [simp]: "principal a \<sqsubseteq> principal b \<longleftrightarrow> a \<preceq> b" |
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305 by (simp add: principal_less_iff_mem_rep rep_principal) |
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306 |
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307 lemma principal_eq_iff: "principal a = principal b \<longleftrightarrow> a \<preceq> b \<and> b \<preceq> a" |
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308 unfolding po_eq_conv [where 'a='b] principal_less_iff .. |
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309 |
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310 lemma repD: "a \<in> rep x \<Longrightarrow> principal a \<sqsubseteq> x" |
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311 by (simp add: rep_eq) |
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312 |
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313 lemma principal_mono: "a \<preceq> b \<Longrightarrow> principal a \<sqsubseteq> principal b" |
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314 by (simp only: principal_less_iff) |
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315 |
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316 lemma lessI: "(\<And>a. principal a \<sqsubseteq> x \<Longrightarrow> principal a \<sqsubseteq> u) \<Longrightarrow> x \<sqsubseteq> u" |
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317 unfolding principal_less_iff_mem_rep |
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318 by (simp add: less_def subset_eq) |
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319 |
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320 lemma lub_principal_rep: "principal ` rep x <<| x" |
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321 apply (rule is_lubI) |
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322 apply (rule ub_imageI) |
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323 apply (erule repD) |
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324 apply (subst less_def) |
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325 apply (rule subsetI) |
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326 apply (drule (1) ub_imageD) |
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327 apply (simp add: rep_eq) |
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328 done |
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329 |
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330 definition |
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331 basis_fun :: "('a::type \<Rightarrow> 'c::cpo) \<Rightarrow> 'b \<rightarrow> 'c" where |
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332 "basis_fun = (\<lambda>f. (\<Lambda> x. lub (f ` rep x)))" |
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333 |
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334 lemma basis_fun_beta: |
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335 fixes f :: "'a::type \<Rightarrow> 'c::cpo" |
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336 assumes f_mono: "\<And>a b. a \<preceq> b \<Longrightarrow> f a \<sqsubseteq> f b" |
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337 shows "basis_fun f\<cdot>x = lub (f ` rep x)" |
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338 unfolding basis_fun_def |
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339 proof (rule beta_cfun) |
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340 have lub: "\<And>x. \<exists>u. f ` rep x <<| u" |
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341 using f_mono by (rule basis_fun_lemma) |
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342 show cont: "cont (\<lambda>x. lub (f ` rep x))" |
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343 apply (rule contI2) |
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344 apply (rule monofunI) |
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345 apply (rule is_lub_thelub0 [OF lub ub_imageI]) |
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346 apply (rule is_ub_thelub0 [OF lub imageI]) |
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347 apply (erule (1) subsetD [OF rep_mono]) |
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348 apply (rule is_lub_thelub0 [OF lub ub_imageI]) |
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349 apply (simp add: rep_contlub, clarify) |
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350 apply (erule rev_trans_less [OF is_ub_thelub]) |
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351 apply (erule is_ub_thelub0 [OF lub imageI]) |
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352 done |
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353 qed |
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354 |
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355 lemma basis_fun_principal: |
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356 fixes f :: "'a::type \<Rightarrow> 'c::cpo" |
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357 assumes f_mono: "\<And>a b. a \<preceq> b \<Longrightarrow> f a \<sqsubseteq> f b" |
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358 shows "basis_fun f\<cdot>(principal a) = f a" |
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359 apply (subst basis_fun_beta, erule f_mono) |
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360 apply (subst rep_principal) |
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361 apply (rule lub_image_principal, erule f_mono) |
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362 done |
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363 |
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364 lemma basis_fun_mono: |
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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 assumes g_mono: "\<And>a b. a \<preceq> b \<Longrightarrow> g a \<sqsubseteq> g b" |
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367 assumes less: "\<And>a. f a \<sqsubseteq> g a" |
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368 shows "basis_fun f \<sqsubseteq> basis_fun g" |
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369 apply (rule less_cfun_ext) |
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370 apply (simp only: basis_fun_beta f_mono g_mono) |
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371 apply (rule is_lub_thelub0) |
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372 apply (rule basis_fun_lemma, erule f_mono) |
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373 apply (rule ub_imageI, rename_tac a) |
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374 apply (rule sq_le.trans_less [OF less]) |
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375 apply (rule is_ub_thelub0) |
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376 apply (rule basis_fun_lemma, erule g_mono) |
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377 apply (erule imageI) |
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378 done |
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379 |
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380 lemma compact_principal [simp]: "compact (principal a)" |
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381 by (rule compactI2, simp add: principal_less_iff_mem_rep rep_contlub) |
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382 |
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383 definition |
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384 completion_approx :: "nat \<Rightarrow> 'b \<rightarrow> 'b" where |
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385 "completion_approx = (\<lambda>i. basis_fun (\<lambda>a. principal (take i a)))" |
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386 |
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387 lemma completion_approx_beta: |
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388 "completion_approx i\<cdot>x = (\<Squnion>a\<in>rep x. principal (take i a))" |
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389 unfolding completion_approx_def |
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390 by (simp add: basis_fun_beta principal_mono take_mono) |
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391 |
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392 lemma completion_approx_principal: |
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393 "completion_approx i\<cdot>(principal a) = principal (take i a)" |
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394 unfolding completion_approx_def |
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395 by (simp add: basis_fun_principal principal_mono take_mono) |
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396 |
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397 lemma chain_completion_approx: "chain completion_approx" |
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398 unfolding completion_approx_def |
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399 apply (rule chainI) |
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400 apply (rule basis_fun_mono) |
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401 apply (erule principal_mono [OF take_mono]) |
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402 apply (erule principal_mono [OF take_mono]) |
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403 apply (rule principal_mono [OF take_chain]) |
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404 done |
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405 |
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406 lemma lub_completion_approx: "(\<Squnion>i. completion_approx i\<cdot>x) = x" |
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407 unfolding completion_approx_beta |
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408 apply (subst image_image [where f=principal, symmetric]) |
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409 apply (rule unique_lub [OF _ lub_principal_rep]) |
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410 apply (rule basis_fun_lemma2, erule principal_mono) |
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411 done |
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412 |
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413 lemma completion_approx_eq_principal: |
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414 "\<exists>a\<in>rep x. completion_approx i\<cdot>x = principal (take i a)" |
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415 unfolding completion_approx_beta |
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416 apply (subst image_image [where f=principal, symmetric]) |
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417 apply (subgoal_tac "finite (principal ` take i ` rep x)") |
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418 apply (subgoal_tac "directed (principal ` take i ` rep x)") |
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419 apply (drule (1) lub_finite_directed_in_self, fast) |
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420 apply (subst image_image) |
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421 apply (rule directed_image_ideal) |
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422 apply (rule ideal_rep) |
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423 apply (erule principal_mono [OF take_mono]) |
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424 apply (rule finite_imageI) |
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425 apply (rule finite_take_rep) |
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426 done |
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427 |
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428 lemma completion_approx_idem: |
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429 "completion_approx i\<cdot>(completion_approx i\<cdot>x) = completion_approx i\<cdot>x" |
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430 using completion_approx_eq_principal [where i=i and x=x] |
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431 by (auto simp add: completion_approx_principal take_take) |
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432 |
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433 lemma finite_fixes_completion_approx: |
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434 "finite {x. completion_approx i\<cdot>x = x}" (is "finite ?S") |
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435 apply (subgoal_tac "?S \<subseteq> principal ` range (take i)") |
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436 apply (erule finite_subset) |
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437 apply (rule finite_imageI) |
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438 apply (rule finite_range_take) |
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439 apply (clarify, erule subst) |
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440 apply (cut_tac x=x and i=i in completion_approx_eq_principal) |
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441 apply fast |
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442 done |
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443 |
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444 lemma principal_induct: |
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445 assumes adm: "adm P" |
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446 assumes P: "\<And>a. P (principal a)" |
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447 shows "P x" |
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448 apply (subgoal_tac "P (\<Squnion>i. completion_approx i\<cdot>x)") |
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449 apply (simp add: lub_completion_approx) |
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450 apply (rule admD [OF adm]) |
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451 apply (simp add: chain_completion_approx) |
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452 apply (cut_tac x=x and i=i in completion_approx_eq_principal) |
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453 apply (clarify, simp add: P) |
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454 done |
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455 |
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456 lemma principal_induct2: |
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457 "\<lbrakk>\<And>y. adm (\<lambda>x. P x y); \<And>x. adm (\<lambda>y. P x y); |
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458 \<And>a b. P (principal a) (principal b)\<rbrakk> \<Longrightarrow> P x y" |
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459 apply (rule_tac x=y in spec) |
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460 apply (rule_tac x=x in principal_induct, simp) |
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461 apply (rule allI, rename_tac y) |
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462 apply (rule_tac x=y in principal_induct, simp) |
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463 apply simp |
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464 done |
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465 |
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466 lemma compact_imp_principal: "compact x \<Longrightarrow> \<exists>a. x = principal a" |
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467 apply (drule adm_compact_neq [OF _ cont_id]) |
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468 apply (drule admD2 [where Y="\<lambda>n. completion_approx n\<cdot>x"]) |
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469 apply (simp add: chain_completion_approx) |
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470 apply (simp add: lub_completion_approx) |
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471 apply (erule exE, erule ssubst) |
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472 apply (cut_tac i=i and x=x in completion_approx_eq_principal) |
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473 apply (clarify, erule exI) |
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474 done |
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475 |
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476 end |
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477 |
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478 end |