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1 (* Title: HOLCF/CompactBasis.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 {* Compact bases of domains *} |
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7 |
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8 theory CompactBasis |
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9 imports Bifinite SetPcpo |
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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" |
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16 assumes refl: "r x x" |
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17 assumes trans: "\<lbrakk>r x y; r y z\<rbrakk> \<Longrightarrow> r x 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. r x z \<and> r y z) \<and> |
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23 (\<forall>x y. r x 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. r x z \<and> r y z" |
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28 assumes "\<And>x y. \<lbrakk>r x 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. r x z \<and> r y 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; r x 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. r x 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: trans) |
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53 done |
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54 |
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55 lemma ideal_principal: "ideal {x. r x 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: refl) |
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59 apply (rule_tac x=z in bexI, fast) |
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60 apply (fast intro: refl) |
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61 apply (fast intro: trans) |
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62 done |
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63 |
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64 lemma directed_image_ideal: |
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65 assumes A: "ideal A" |
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66 assumes f: "\<And>x y. r x y \<Longrightarrow> f x \<sqsubseteq> f y" |
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67 shows "directed (f ` A)" |
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68 apply (rule directedI) |
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69 apply (cut_tac idealD1 [OF A], fast) |
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70 apply (clarify, rename_tac a b) |
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71 apply (drule (1) idealD2 [OF A]) |
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72 apply (clarify, rename_tac c) |
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73 apply (rule_tac x="f c" in rev_bexI) |
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74 apply (erule imageI) |
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75 apply (simp add: f) |
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76 done |
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77 |
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78 lemma adm_ideal: "adm (\<lambda>A. ideal A)" |
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79 unfolding ideal_def by (intro adm_lemmas adm_set_lemmas) |
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80 |
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81 end |
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82 |
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83 subsection {* Defining functions in terms of basis elements *} |
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84 |
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85 lemma (in preorder) lub_image_principal: |
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86 assumes f: "\<And>x y. r x y \<Longrightarrow> f x \<sqsubseteq> f y" |
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87 shows "(\<Squnion>x\<in>{x. r x y}. f x) = f y" |
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88 apply (rule thelubI) |
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89 apply (rule is_lub_maximal) |
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90 apply (rule ub_imageI) |
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91 apply (simp add: f) |
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92 apply (rule imageI) |
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93 apply (simp add: refl) |
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94 done |
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95 |
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96 lemma finite_directed_has_lub: "\<lbrakk>finite S; directed S\<rbrakk> \<Longrightarrow> \<exists>u. S <<| u" |
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97 apply (drule (1) directed_finiteD, rule subset_refl) |
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98 apply (erule bexE) |
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99 apply (rule_tac x=z in exI) |
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100 apply (erule (1) is_lub_maximal) |
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101 done |
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102 |
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103 lemma is_ub_thelub0: "\<lbrakk>\<exists>u. S <<| u; x \<in> S\<rbrakk> \<Longrightarrow> x \<sqsubseteq> lub S" |
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104 apply (erule exE, drule lubI) |
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105 apply (drule is_lubD1) |
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106 apply (erule (1) is_ubD) |
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107 done |
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108 |
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109 lemma is_lub_thelub0: "\<lbrakk>\<exists>u. S <<| u; S <| x\<rbrakk> \<Longrightarrow> lub S \<sqsubseteq> x" |
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110 by (erule exE, drule lubI, erule is_lub_lub) |
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111 |
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112 locale bifinite_basis = preorder + |
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113 fixes principal :: "'a::type \<Rightarrow> 'b::cpo" |
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114 fixes approxes :: "'b::cpo \<Rightarrow> 'a::type set" |
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115 assumes ideal_approxes: "\<And>x. preorder.ideal r (approxes x)" |
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116 assumes cont_approxes: "cont approxes" |
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117 assumes approxes_principal: "\<And>a. approxes (principal a) = {b. r b a}" |
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118 assumes subset_approxesD: "\<And>x y. approxes x \<subseteq> approxes y \<Longrightarrow> x \<sqsubseteq> y" |
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119 |
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120 fixes take :: "nat \<Rightarrow> 'a::type \<Rightarrow> 'a" |
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121 assumes take_less: "r (take n a) a" |
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122 assumes take_take: "take n (take n a) = take n a" |
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123 assumes take_mono: "r a b \<Longrightarrow> r (take n a) (take n b)" |
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124 assumes take_chain: "r (take n a) (take (Suc n) a)" |
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125 assumes finite_range_take: "finite (range (take n))" |
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126 assumes take_covers: "\<exists>n. take n a = a" |
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127 begin |
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128 |
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129 lemma finite_take_approxes: "finite (take n ` approxes x)" |
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130 by (rule finite_subset [OF image_mono [OF subset_UNIV] finite_range_take]) |
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131 |
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132 lemma basis_fun_lemma0: |
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133 fixes f :: "'a::type \<Rightarrow> 'c::cpo" |
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134 assumes f_mono: "\<And>a b. r a b \<Longrightarrow> f a \<sqsubseteq> f b" |
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135 shows "\<exists>u. f ` take i ` approxes x <<| u" |
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136 apply (rule finite_directed_has_lub) |
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137 apply (rule finite_imageI) |
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138 apply (rule finite_take_approxes) |
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139 apply (subst image_image) |
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140 apply (rule directed_image_ideal) |
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141 apply (rule ideal_approxes) |
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142 apply (rule f_mono) |
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143 apply (erule take_mono) |
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144 done |
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145 |
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146 lemma basis_fun_lemma1: |
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147 fixes f :: "'a::type \<Rightarrow> 'c::cpo" |
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148 assumes f_mono: "\<And>a b. r a b \<Longrightarrow> f a \<sqsubseteq> f b" |
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149 shows "chain (\<lambda>i. lub (f ` take i ` approxes x))" |
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150 apply (rule chainI) |
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151 apply (rule is_lub_thelub0) |
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152 apply (rule basis_fun_lemma0, erule f_mono) |
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153 apply (rule is_ubI, clarsimp, rename_tac a) |
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154 apply (rule trans_less [OF f_mono [OF take_chain]]) |
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155 apply (rule is_ub_thelub0) |
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156 apply (rule basis_fun_lemma0, erule f_mono) |
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157 apply simp |
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158 done |
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159 |
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160 lemma basis_fun_lemma2: |
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161 fixes f :: "'a::type \<Rightarrow> 'c::cpo" |
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162 assumes f_mono: "\<And>a b. r a b \<Longrightarrow> f a \<sqsubseteq> f b" |
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163 shows "f ` approxes x <<| (\<Squnion>i. lub (f ` take i ` approxes x))" |
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164 apply (rule is_lubI) |
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165 apply (rule ub_imageI, rename_tac a) |
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166 apply (cut_tac a=a in take_covers, erule exE, rename_tac i) |
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167 apply (erule subst) |
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168 apply (rule rev_trans_less) |
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169 apply (rule_tac x=i in is_ub_thelub) |
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170 apply (rule basis_fun_lemma1, erule f_mono) |
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171 apply (rule is_ub_thelub0) |
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172 apply (rule basis_fun_lemma0, erule f_mono) |
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173 apply simp |
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174 apply (rule is_lub_thelub [OF _ ub_rangeI]) |
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175 apply (rule basis_fun_lemma1, erule f_mono) |
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176 apply (rule is_lub_thelub0) |
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177 apply (rule basis_fun_lemma0, erule f_mono) |
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178 apply (rule is_ubI, clarsimp, rename_tac a) |
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179 apply (rule trans_less [OF f_mono [OF take_less]]) |
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180 apply (erule (1) ub_imageD) |
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181 done |
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182 |
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183 lemma basis_fun_lemma: |
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184 fixes f :: "'a::type \<Rightarrow> 'c::cpo" |
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185 assumes f_mono: "\<And>a b. r a b \<Longrightarrow> f a \<sqsubseteq> f b" |
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186 shows "\<exists>u. f ` approxes x <<| u" |
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187 by (rule exI, rule basis_fun_lemma2, erule f_mono) |
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188 |
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189 lemma approxes_mono: "x \<sqsubseteq> y \<Longrightarrow> approxes x \<subseteq> approxes y" |
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190 apply (drule cont_approxes [THEN cont2mono, THEN monofunE]) |
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191 apply (simp add: set_cpo_simps) |
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192 done |
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193 |
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194 lemma approxes_contlub: |
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195 "chain Y \<Longrightarrow> approxes (\<Squnion>i. Y i) = (\<Union>i. approxes (Y i))" |
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196 by (simp add: cont2contlubE [OF cont_approxes] set_cpo_simps) |
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197 |
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198 lemma less_def: "(x \<sqsubseteq> y) = (approxes x \<subseteq> approxes y)" |
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199 by (rule iffI [OF approxes_mono subset_approxesD]) |
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200 |
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201 lemma approxes_eq: "approxes x = {a. principal a \<sqsubseteq> x}" |
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202 unfolding less_def approxes_principal |
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203 apply safe |
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204 apply (erule (1) idealD3 [OF ideal_approxes]) |
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205 apply (erule subsetD, simp add: refl) |
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206 done |
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207 |
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208 lemma mem_approxes_iff: "(a \<in> approxes x) = (principal a \<sqsubseteq> x)" |
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209 by (simp add: approxes_eq) |
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210 |
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211 lemma principal_less_iff: "(principal a \<sqsubseteq> x) = (a \<in> approxes x)" |
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212 by (simp add: approxes_eq) |
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213 |
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214 lemma approxesD: "a \<in> approxes x \<Longrightarrow> principal a \<sqsubseteq> x" |
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215 by (simp add: approxes_eq) |
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216 |
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217 lemma principal_mono: "r a b \<Longrightarrow> principal a \<sqsubseteq> principal b" |
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218 by (rule approxesD, simp add: approxes_principal) |
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219 |
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220 lemma lessI: "(\<And>a. principal a \<sqsubseteq> x \<Longrightarrow> principal a \<sqsubseteq> u) \<Longrightarrow> x \<sqsubseteq> u" |
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221 unfolding principal_less_iff |
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222 by (simp add: less_def subset_def) |
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223 |
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224 lemma lub_principal_approxes: "principal ` approxes x <<| x" |
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225 apply (rule is_lubI) |
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226 apply (rule ub_imageI) |
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227 apply (erule approxesD) |
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228 apply (subst less_def) |
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229 apply (rule subsetI) |
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230 apply (drule (1) ub_imageD) |
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231 apply (simp add: approxes_eq) |
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232 done |
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233 |
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234 definition |
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235 basis_fun :: "('a::type \<Rightarrow> 'c::cpo) \<Rightarrow> 'b \<rightarrow> 'c" where |
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236 "basis_fun = (\<lambda>f. (\<Lambda> x. lub (f ` approxes x)))" |
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237 |
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238 lemma basis_fun_beta: |
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239 fixes f :: "'a::type \<Rightarrow> 'c::cpo" |
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240 assumes f_mono: "\<And>a b. r a b \<Longrightarrow> f a \<sqsubseteq> f b" |
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241 shows "basis_fun f\<cdot>x = lub (f ` approxes x)" |
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242 unfolding basis_fun_def |
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243 proof (rule beta_cfun) |
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244 have lub: "\<And>x. \<exists>u. f ` approxes x <<| u" |
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245 using f_mono by (rule basis_fun_lemma) |
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246 show cont: "cont (\<lambda>x. lub (f ` approxes x))" |
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247 apply (rule contI2) |
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248 apply (rule monofunI) |
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249 apply (rule is_lub_thelub0 [OF lub ub_imageI]) |
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250 apply (rule is_ub_thelub0 [OF lub imageI]) |
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251 apply (erule (1) subsetD [OF approxes_mono]) |
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252 apply (rule is_lub_thelub0 [OF lub ub_imageI]) |
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253 apply (simp add: approxes_contlub, clarify) |
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254 apply (erule rev_trans_less [OF is_ub_thelub]) |
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255 apply (erule is_ub_thelub0 [OF lub imageI]) |
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256 done |
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257 qed |
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258 |
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259 lemma basis_fun_principal: |
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260 fixes f :: "'a::type \<Rightarrow> 'c::cpo" |
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261 assumes f_mono: "\<And>a b. r a b \<Longrightarrow> f a \<sqsubseteq> f b" |
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262 shows "basis_fun f\<cdot>(principal a) = f a" |
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263 apply (subst basis_fun_beta, erule f_mono) |
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264 apply (subst approxes_principal) |
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265 apply (rule lub_image_principal, erule f_mono) |
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266 done |
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267 |
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268 lemma basis_fun_mono: |
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269 assumes f_mono: "\<And>a b. r a b \<Longrightarrow> f a \<sqsubseteq> f b" |
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270 assumes g_mono: "\<And>a b. r a b \<Longrightarrow> g a \<sqsubseteq> g b" |
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271 assumes less: "\<And>a. f a \<sqsubseteq> g a" |
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272 shows "basis_fun f \<sqsubseteq> basis_fun g" |
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273 apply (rule less_cfun_ext) |
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274 apply (simp only: basis_fun_beta f_mono g_mono) |
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275 apply (rule is_lub_thelub0) |
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276 apply (rule basis_fun_lemma, erule f_mono) |
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277 apply (rule ub_imageI, rename_tac a) |
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278 apply (rule trans_less [OF less]) |
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279 apply (rule is_ub_thelub0) |
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280 apply (rule basis_fun_lemma, erule g_mono) |
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281 apply (erule imageI) |
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282 done |
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283 |
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284 lemma compact_principal: "compact (principal a)" |
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285 by (rule compactI2, simp add: principal_less_iff approxes_contlub) |
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286 |
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287 lemma chain_basis_fun_take: |
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288 "chain (\<lambda>i. basis_fun (\<lambda>a. principal (take i a)))" |
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289 apply (rule chainI) |
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290 apply (rule basis_fun_mono) |
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291 apply (erule principal_mono [OF take_mono]) |
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292 apply (erule principal_mono [OF take_mono]) |
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293 apply (rule principal_mono [OF take_chain]) |
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294 done |
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295 |
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296 lemma lub_basis_fun_take: |
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297 "(\<Squnion>i. basis_fun (\<lambda>a. principal (take i a))\<cdot>x) = x" |
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298 apply (simp add: basis_fun_beta principal_mono take_mono) |
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299 apply (subst image_image [where f=principal, symmetric]) |
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300 apply (rule unique_lub [OF _ lub_principal_approxes]) |
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301 apply (rule basis_fun_lemma2, erule principal_mono) |
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302 done |
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303 |
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304 lemma finite_directed_contains_lub: |
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305 "\<lbrakk>finite S; directed S\<rbrakk> \<Longrightarrow> \<exists>u\<in>S. S <<| u" |
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306 apply (drule (1) directed_finiteD, rule subset_refl) |
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307 apply (erule bexE) |
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308 apply (rule rev_bexI, assumption) |
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309 apply (erule (1) is_lub_maximal) |
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310 done |
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311 |
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312 lemma lub_finite_directed_in_self: |
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313 "\<lbrakk>finite S; directed S\<rbrakk> \<Longrightarrow> lub S \<in> S" |
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314 apply (drule (1) directed_finiteD, rule subset_refl) |
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315 apply (erule bexE) |
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316 apply (drule (1) is_lub_maximal) |
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317 apply (drule thelubI) |
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318 apply simp |
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319 done |
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320 |
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321 lemma basis_fun_take_eq_principal: |
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322 "\<exists>a\<in>approxes x. |
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323 basis_fun (\<lambda>a. principal (take i a))\<cdot>x = principal (take i a)" |
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324 apply (simp add: basis_fun_beta principal_mono take_mono) |
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325 apply (subst image_image [where f=principal, symmetric]) |
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326 apply (subgoal_tac "finite (principal ` take i ` approxes x)") |
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327 apply (subgoal_tac "directed (principal ` take i ` approxes x)") |
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328 apply (drule (1) lub_finite_directed_in_self, fast) |
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329 apply (subst image_image) |
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330 apply (rule directed_image_ideal) |
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331 apply (rule ideal_approxes) |
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332 apply (erule principal_mono [OF take_mono]) |
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333 apply (rule finite_imageI) |
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334 apply (rule finite_take_approxes) |
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335 done |
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336 |
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337 lemma principal_induct: |
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338 assumes adm: "adm P" |
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339 assumes P: "\<And>a. P (principal a)" |
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340 shows "P x" |
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341 apply (subgoal_tac "P (\<Squnion>i. basis_fun (\<lambda>a. principal (take i a))\<cdot>x)") |
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342 apply (simp add: lub_basis_fun_take) |
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343 apply (rule admD [rule_format, OF adm]) |
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344 apply (simp add: chain_basis_fun_take) |
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345 apply (cut_tac x=x and i=i in basis_fun_take_eq_principal) |
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346 apply (clarify, simp add: P) |
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347 done |
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348 |
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349 lemma finite_fixes_basis_fun_take: |
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350 "finite {x. basis_fun (\<lambda>a. principal (take i a))\<cdot>x = x}" (is "finite ?S") |
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351 apply (subgoal_tac "?S \<subseteq> principal ` range (take i)") |
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352 apply (erule finite_subset) |
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353 apply (rule finite_imageI) |
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354 apply (rule finite_range_take) |
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355 apply (clarify, erule subst) |
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356 apply (cut_tac x=x and i=i in basis_fun_take_eq_principal) |
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357 apply fast |
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358 done |
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359 |
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360 end |
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361 |
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362 |
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363 subsection {* Compact bases of bifinite domains *} |
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364 |
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365 defaultsort bifinite |
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366 |
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367 typedef (open) 'a compact_basis = "{x::'a::bifinite. compact x}" |
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368 by (fast intro: compact_approx) |
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369 |
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370 lemma compact_Rep_compact_basis [simp]: "compact (Rep_compact_basis a)" |
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371 by (rule Rep_compact_basis [simplified]) |
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372 |
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373 lemma Rep_Abs_compact_basis_approx [simp]: |
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374 "Rep_compact_basis (Abs_compact_basis (approx n\<cdot>x)) = approx n\<cdot>x" |
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375 by (rule Abs_compact_basis_inverse, simp) |
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376 |
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377 lemma compact_imp_Rep_compact_basis: |
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378 "compact x \<Longrightarrow> \<exists>y. x = Rep_compact_basis y" |
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379 by (rule exI, rule Abs_compact_basis_inverse [symmetric], simp) |
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380 |
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381 definition |
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382 compact_le :: "'a compact_basis \<Rightarrow> 'a compact_basis \<Rightarrow> bool" where |
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383 "compact_le = (\<lambda>x y. Rep_compact_basis x \<sqsubseteq> Rep_compact_basis y)" |
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384 |
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385 lemma compact_le_refl: "compact_le x x" |
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386 unfolding compact_le_def by (rule refl_less) |
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387 |
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388 lemma compact_le_trans: "\<lbrakk>compact_le x y; compact_le y z\<rbrakk> \<Longrightarrow> compact_le x z" |
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389 unfolding compact_le_def by (rule trans_less) |
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390 |
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391 lemma compact_le_antisym: "\<lbrakk>compact_le x y; compact_le y x\<rbrakk> \<Longrightarrow> x = y" |
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392 unfolding compact_le_def |
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393 apply (rule Rep_compact_basis_inject [THEN iffD1]) |
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394 apply (erule (1) antisym_less) |
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395 done |
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396 |
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397 interpretation compact_le: preorder [compact_le] |
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398 by (rule preorder.intro, rule compact_le_refl, rule compact_le_trans) |
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399 |
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400 text {* minimal compact element *} |
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401 |
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402 definition |
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403 compact_bot :: "'a compact_basis" where |
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404 "compact_bot = Abs_compact_basis \<bottom>" |
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405 |
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406 lemma Rep_compact_bot: "Rep_compact_basis compact_bot = \<bottom>" |
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407 unfolding compact_bot_def by (simp add: Abs_compact_basis_inverse) |
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408 |
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409 lemma compact_minimal [simp]: "compact_le compact_bot a" |
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410 unfolding compact_le_def Rep_compact_bot by simp |
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411 |
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412 text {* compacts *} |
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413 |
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414 definition |
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415 compacts :: "'a \<Rightarrow> 'a compact_basis set" where |
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416 "compacts = (\<lambda>x. {a. Rep_compact_basis a \<sqsubseteq> x})" |
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417 |
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418 lemma ideal_compacts: "compact_le.ideal (compacts w)" |
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419 unfolding compacts_def |
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420 apply (rule compact_le.idealI) |
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421 apply (rule_tac x="Abs_compact_basis (approx 0\<cdot>w)" in exI) |
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422 apply (simp add: approx_less) |
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423 apply simp |
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424 apply (cut_tac a=x in compact_Rep_compact_basis) |
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425 apply (cut_tac a=y in compact_Rep_compact_basis) |
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426 apply (drule bifinite_compact_eq_approx) |
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427 apply (drule bifinite_compact_eq_approx) |
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428 apply (clarify, rename_tac i j) |
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429 apply (rule_tac x="Abs_compact_basis (approx (max i j)\<cdot>w)" in exI) |
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430 apply (simp add: approx_less compact_le_def) |
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431 apply (erule subst, erule subst) |
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432 apply (simp add: monofun_cfun chain_mono3 [OF chain_approx]) |
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433 apply (simp add: compact_le_def) |
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434 apply (erule (1) trans_less) |
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435 done |
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436 |
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437 lemma compacts_Rep_compact_basis: |
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438 "compacts (Rep_compact_basis b) = {a. compact_le a b}" |
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439 unfolding compacts_def compact_le_def .. |
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440 |
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441 lemma cont_compacts: "cont compacts" |
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442 unfolding compacts_def |
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443 apply (rule contI2) |
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444 apply (rule monofunI) |
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445 apply (simp add: set_cpo_simps) |
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446 apply (fast intro: trans_less) |
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447 apply (simp add: set_cpo_simps) |
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448 apply clarify |
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449 apply simp |
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450 apply (erule (1) compactD2 [OF compact_Rep_compact_basis]) |
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451 done |
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452 |
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453 lemma compacts_lessD: "compacts x \<subseteq> compacts y \<Longrightarrow> x \<sqsubseteq> y" |
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454 apply (subgoal_tac "(\<Squnion>i. approx i\<cdot>x) \<sqsubseteq> y", simp) |
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455 apply (rule admD [rule_format], simp, simp) |
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456 apply (drule_tac c="Abs_compact_basis (approx i\<cdot>x)" in subsetD) |
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457 apply (simp add: compacts_def Abs_compact_basis_inverse approx_less) |
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458 apply (simp add: compacts_def Abs_compact_basis_inverse) |
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459 done |
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460 |
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461 lemma compacts_mono: "x \<sqsubseteq> y \<Longrightarrow> compacts x \<subseteq> compacts y" |
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462 unfolding compacts_def by (fast intro: trans_less) |
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463 |
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464 lemma less_compact_basis_iff: "(x \<sqsubseteq> y) = (compacts x \<subseteq> compacts y)" |
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465 by (rule iffI [OF compacts_mono compacts_lessD]) |
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466 |
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467 lemma compact_basis_induct: |
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468 "\<lbrakk>adm P; \<And>a. P (Rep_compact_basis a)\<rbrakk> \<Longrightarrow> P x" |
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469 apply (erule approx_induct) |
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470 apply (drule_tac x="Abs_compact_basis (approx n\<cdot>x)" in meta_spec) |
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471 apply (simp add: Abs_compact_basis_inverse) |
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472 done |
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473 |
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474 text {* approximation on compact bases *} |
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475 |
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476 definition |
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477 compact_approx :: "nat \<Rightarrow> 'a compact_basis \<Rightarrow> 'a compact_basis" where |
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478 "compact_approx = (\<lambda>n a. Abs_compact_basis (approx n\<cdot>(Rep_compact_basis a)))" |
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479 |
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480 lemma Rep_compact_approx: |
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481 "Rep_compact_basis (compact_approx n a) = approx n\<cdot>(Rep_compact_basis a)" |
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482 unfolding compact_approx_def |
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483 by (simp add: Abs_compact_basis_inverse) |
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484 |
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485 lemmas approx_Rep_compact_basis = Rep_compact_approx [symmetric] |
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486 |
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487 lemma compact_approx_le: |
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488 "compact_le (compact_approx n a) a" |
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489 unfolding compact_le_def |
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490 by (simp add: Rep_compact_approx approx_less) |
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491 |
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492 lemma compact_approx_mono1: |
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493 "i \<le> j \<Longrightarrow> compact_le (compact_approx i a) (compact_approx j a)" |
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494 unfolding compact_le_def |
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495 apply (simp add: Rep_compact_approx) |
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496 apply (rule chain_mono3, simp, assumption) |
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497 done |
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498 |
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499 lemma compact_approx_mono: |
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500 "compact_le a b \<Longrightarrow> compact_le (compact_approx n a) (compact_approx n b)" |
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501 unfolding compact_le_def |
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502 apply (simp add: Rep_compact_approx) |
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503 apply (erule monofun_cfun_arg) |
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504 done |
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505 |
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506 lemma ex_compact_approx_eq: "\<exists>n. compact_approx n a = a" |
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507 apply (simp add: Rep_compact_basis_inject [symmetric]) |
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508 apply (simp add: Rep_compact_approx) |
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509 apply (rule bifinite_compact_eq_approx) |
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510 apply (rule compact_Rep_compact_basis) |
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511 done |
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512 |
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513 lemma compact_approx_idem: |
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514 "compact_approx n (compact_approx n a) = compact_approx n a" |
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515 apply (rule Rep_compact_basis_inject [THEN iffD1]) |
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516 apply (simp add: Rep_compact_approx) |
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517 done |
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518 |
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519 lemma finite_fixes_compact_approx: "finite {a. compact_approx n a = a}" |
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520 apply (subgoal_tac "finite (Rep_compact_basis ` {a. compact_approx n a = a})") |
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521 apply (erule finite_imageD) |
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522 apply (rule inj_onI, simp add: Rep_compact_basis_inject) |
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523 apply (rule finite_subset [OF _ finite_fixes_approx [where i=n]]) |
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524 apply (rule subsetI, clarify, rename_tac a) |
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525 apply (simp add: Rep_compact_basis_inject [symmetric]) |
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526 apply (simp add: Rep_compact_approx) |
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527 done |
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528 |
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529 lemma finite_range_compact_approx: "finite (range (compact_approx n))" |
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530 apply (cut_tac n=n in finite_fixes_compact_approx) |
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531 apply (simp add: idem_fixes_eq_range compact_approx_idem) |
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532 apply (simp add: image_def) |
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533 done |
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534 |
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535 interpretation compact_basis: |
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536 bifinite_basis [compact_le Rep_compact_basis compacts compact_approx] |
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537 apply unfold_locales |
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538 apply (rule ideal_compacts) |
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539 apply (rule cont_compacts) |
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540 apply (rule compacts_Rep_compact_basis) |
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541 apply (erule compacts_lessD) |
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542 apply (rule compact_approx_le) |
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543 apply (rule compact_approx_idem) |
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544 apply (erule compact_approx_mono) |
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545 apply (rule compact_approx_mono1, simp) |
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546 apply (rule finite_range_compact_approx) |
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547 apply (rule ex_compact_approx_eq) |
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548 done |
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549 |
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550 |
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551 subsection {* A compact basis for powerdomains *} |
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552 |
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553 typedef 'a pd_basis = |
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554 "{S::'a::bifinite compact_basis set. finite S \<and> S \<noteq> {}}" |
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555 by (rule_tac x="{arbitrary}" in exI, simp) |
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556 |
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557 lemma finite_Rep_pd_basis [simp]: "finite (Rep_pd_basis u)" |
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558 by (insert Rep_pd_basis [of u, unfolded pd_basis_def]) simp |
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559 |
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560 lemma Rep_pd_basis_nonempty [simp]: "Rep_pd_basis u \<noteq> {}" |
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561 by (insert Rep_pd_basis [of u, unfolded pd_basis_def]) simp |
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562 |
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563 text {* unit and plus *} |
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564 |
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565 definition |
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566 PDUnit :: "'a compact_basis \<Rightarrow> 'a pd_basis" where |
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567 "PDUnit = (\<lambda>x. Abs_pd_basis {x})" |
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568 |
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569 definition |
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570 PDPlus :: "'a pd_basis \<Rightarrow> 'a pd_basis \<Rightarrow> 'a pd_basis" where |
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571 "PDPlus t u = Abs_pd_basis (Rep_pd_basis t \<union> Rep_pd_basis u)" |
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572 |
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573 lemma Rep_PDUnit: |
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574 "Rep_pd_basis (PDUnit x) = {x}" |
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575 unfolding PDUnit_def by (rule Abs_pd_basis_inverse) (simp add: pd_basis_def) |
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576 |
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577 lemma Rep_PDPlus: |
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578 "Rep_pd_basis (PDPlus u v) = Rep_pd_basis u \<union> Rep_pd_basis v" |
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579 unfolding PDPlus_def by (rule Abs_pd_basis_inverse) (simp add: pd_basis_def) |
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580 |
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581 lemma PDUnit_inject [simp]: "(PDUnit a = PDUnit b) = (a = b)" |
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582 unfolding Rep_pd_basis_inject [symmetric] Rep_PDUnit by simp |
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583 |
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584 lemma PDPlus_assoc: "PDPlus (PDPlus t u) v = PDPlus t (PDPlus u v)" |
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585 unfolding Rep_pd_basis_inject [symmetric] Rep_PDPlus by (rule Un_assoc) |
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586 |
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587 lemma PDPlus_commute: "PDPlus t u = PDPlus u t" |
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588 unfolding Rep_pd_basis_inject [symmetric] Rep_PDPlus by (rule Un_commute) |
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589 |
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590 lemma PDPlus_absorb: "PDPlus t t = t" |
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591 unfolding Rep_pd_basis_inject [symmetric] Rep_PDPlus by (rule Un_absorb) |
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592 |
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593 lemma pd_basis_induct1: |
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594 assumes PDUnit: "\<And>a. P (PDUnit a)" |
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595 assumes PDPlus: "\<And>a t. P t \<Longrightarrow> P (PDPlus (PDUnit a) t)" |
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596 shows "P x" |
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597 apply (induct x, unfold pd_basis_def, clarify) |
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598 apply (erule (1) finite_ne_induct) |
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599 apply (cut_tac a=x in PDUnit) |
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600 apply (simp add: PDUnit_def) |
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601 apply (drule_tac a=x in PDPlus) |
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602 apply (simp add: PDUnit_def PDPlus_def Abs_pd_basis_inverse [unfolded pd_basis_def]) |
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603 done |
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604 |
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605 lemma pd_basis_induct: |
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606 assumes PDUnit: "\<And>a. P (PDUnit a)" |
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607 assumes PDPlus: "\<And>t u. \<lbrakk>P t; P u\<rbrakk> \<Longrightarrow> P (PDPlus t u)" |
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608 shows "P x" |
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609 apply (induct x rule: pd_basis_induct1) |
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610 apply (rule PDUnit, erule PDPlus [OF PDUnit]) |
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611 done |
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612 |
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613 text {* fold-pd *} |
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614 |
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615 definition |
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616 fold_pd :: |
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617 "('a compact_basis \<Rightarrow> 'b::type) \<Rightarrow> ('b \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a pd_basis \<Rightarrow> 'b" |
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618 where "fold_pd g f t = fold1 f (g ` Rep_pd_basis t)" |
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619 |
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620 lemma (in ACIf) fold_pd_PDUnit: |
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621 "fold_pd g f (PDUnit x) = g x" |
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622 unfolding fold_pd_def Rep_PDUnit by simp |
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623 |
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624 lemma (in ACIf) fold_pd_PDPlus: |
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625 "fold_pd g f (PDPlus t u) = f (fold_pd g f t) (fold_pd g f u)" |
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626 unfolding fold_pd_def Rep_PDPlus by (simp add: image_Un fold1_Un2) |
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627 |
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628 text {* approx-pd *} |
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629 |
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630 definition |
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631 approx_pd :: "nat \<Rightarrow> 'a pd_basis \<Rightarrow> 'a pd_basis" where |
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632 "approx_pd n = (\<lambda>t. Abs_pd_basis (compact_approx n ` Rep_pd_basis t))" |
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633 |
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634 lemma Rep_approx_pd: |
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635 "Rep_pd_basis (approx_pd n t) = compact_approx n ` Rep_pd_basis t" |
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636 unfolding approx_pd_def |
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637 apply (rule Abs_pd_basis_inverse) |
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638 apply (simp add: pd_basis_def) |
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639 done |
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640 |
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641 lemma approx_pd_simps [simp]: |
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642 "approx_pd n (PDUnit a) = PDUnit (compact_approx n a)" |
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643 "approx_pd n (PDPlus t u) = PDPlus (approx_pd n t) (approx_pd n u)" |
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644 apply (simp_all add: Rep_pd_basis_inject [symmetric]) |
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645 apply (simp_all add: Rep_approx_pd Rep_PDUnit Rep_PDPlus image_Un) |
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646 done |
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647 |
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648 lemma approx_pd_idem: "approx_pd n (approx_pd n t) = approx_pd n t" |
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649 apply (induct t rule: pd_basis_induct) |
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650 apply (simp add: compact_approx_idem) |
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651 apply simp |
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652 done |
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653 |
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654 lemma range_image_f: "range (image f) = Pow (range f)" |
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655 apply (safe, fast) |
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656 apply (rule_tac x="f -` x" in range_eqI) |
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657 apply (simp, fast) |
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658 done |
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659 |
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660 lemma finite_range_approx_pd: "finite (range (approx_pd n))" |
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661 apply (subgoal_tac "finite (Rep_pd_basis ` range (approx_pd n))") |
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662 apply (erule finite_imageD) |
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663 apply (rule inj_onI, simp add: Rep_pd_basis_inject) |
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664 apply (subst image_image) |
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665 apply (subst Rep_approx_pd) |
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666 apply (simp only: range_composition) |
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667 apply (rule finite_subset [OF image_mono [OF subset_UNIV]]) |
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668 apply (simp add: range_image_f) |
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669 apply (rule finite_range_compact_approx) |
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670 done |
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671 |
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672 lemma ex_approx_pd_eq: "\<exists>n. approx_pd n t = t" |
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673 apply (subgoal_tac "\<exists>n. \<forall>m\<ge>n. approx_pd m t = t", fast) |
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674 apply (induct t rule: pd_basis_induct) |
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675 apply (cut_tac a=a in ex_compact_approx_eq) |
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676 apply (clarify, rule_tac x=n in exI) |
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677 apply (clarify, simp) |
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678 apply (rule compact_le_antisym) |
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679 apply (rule compact_approx_le) |
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680 apply (drule_tac a=a in compact_approx_mono1) |
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681 apply simp |
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682 apply (clarify, rename_tac i j) |
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683 apply (rule_tac x="max i j" in exI, simp) |
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684 done |
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685 |
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686 end |