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1 (* Title: HOLCF/Bifinite.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 {* Bifinite domains and approximation *} |
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7 |
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8 theory Bifinite |
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9 imports Cfun |
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10 begin |
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11 |
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12 subsection {* Bifinite domains *} |
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13 |
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14 axclass approx < pcpo |
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15 |
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16 consts approx :: "nat \<Rightarrow> 'a::approx \<rightarrow> 'a" |
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17 |
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18 axclass bifinite < approx |
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19 chain_approx_app: "chain (\<lambda>i. approx i\<cdot>x)" |
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20 lub_approx_app [simp]: "(\<Squnion>i. approx i\<cdot>x) = x" |
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21 approx_idem: "approx i\<cdot>(approx i\<cdot>x) = approx i\<cdot>x" |
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22 finite_fixes_approx: "finite {x. approx i\<cdot>x = x}" |
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23 |
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24 lemma finite_range_imp_finite_fixes: |
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25 "finite {x. \<exists>y. x = f y} \<Longrightarrow> finite {x. f x = x}" |
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26 apply (subgoal_tac "{x. f x = x} \<subseteq> {x. \<exists>y. x = f y}") |
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27 apply (erule (1) finite_subset) |
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28 apply (clarify, erule subst, rule exI, rule refl) |
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29 done |
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30 |
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31 lemma chain_approx [simp]: |
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32 "chain (approx :: nat \<Rightarrow> 'a::bifinite \<rightarrow> 'a)" |
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33 apply (rule chainI) |
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34 apply (rule less_cfun_ext) |
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35 apply (rule chainE) |
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36 apply (rule chain_approx_app) |
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37 done |
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38 |
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39 lemma lub_approx [simp]: "(\<Squnion>i. approx i) = (\<Lambda>(x::'a::bifinite). x)" |
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40 by (rule ext_cfun, simp add: contlub_cfun_fun) |
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41 |
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42 lemma approx_less: "approx i\<cdot>x \<sqsubseteq> (x::'a::bifinite)" |
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43 apply (subgoal_tac "approx i\<cdot>x \<sqsubseteq> (\<Squnion>i. approx i\<cdot>x)", simp) |
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44 apply (rule is_ub_thelub, simp) |
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45 done |
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46 |
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47 lemma approx_strict [simp]: "approx i\<cdot>(\<bottom>::'a::bifinite) = \<bottom>" |
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48 by (rule UU_I, rule approx_less) |
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49 |
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50 lemma approx_approx1: |
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51 "i \<le> j \<Longrightarrow> approx i\<cdot>(approx j\<cdot>x) = approx i\<cdot>(x::'a::bifinite)" |
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52 apply (rule antisym_less) |
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53 apply (rule monofun_cfun_arg [OF approx_less]) |
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54 apply (rule sq_ord_eq_less_trans [OF approx_idem [symmetric]]) |
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55 apply (rule monofun_cfun_arg) |
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56 apply (rule monofun_cfun_fun) |
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57 apply (erule chain_mono3 [OF chain_approx]) |
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58 done |
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59 |
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60 lemma approx_approx2: |
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61 "j \<le> i \<Longrightarrow> approx i\<cdot>(approx j\<cdot>x) = approx j\<cdot>(x::'a::bifinite)" |
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62 apply (rule antisym_less) |
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63 apply (rule approx_less) |
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64 apply (rule sq_ord_eq_less_trans [OF approx_idem [symmetric]]) |
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65 apply (rule monofun_cfun_fun) |
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66 apply (erule chain_mono3 [OF chain_approx]) |
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67 done |
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68 |
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69 lemma approx_approx [simp]: |
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70 "approx i\<cdot>(approx j\<cdot>x) = approx (min i j)\<cdot>(x::'a::bifinite)" |
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71 apply (rule_tac x=i and y=j in linorder_le_cases) |
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72 apply (simp add: approx_approx1 min_def) |
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73 apply (simp add: approx_approx2 min_def) |
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74 done |
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75 |
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76 lemma idem_fixes_eq_range: |
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77 "\<forall>x. f (f x) = f x \<Longrightarrow> {x. f x = x} = {y. \<exists>x. y = f x}" |
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78 by (auto simp add: eq_sym_conv) |
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79 |
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80 lemma finite_approx: "finite {y::'a::bifinite. \<exists>x. y = approx n\<cdot>x}" |
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81 using finite_fixes_approx by (simp add: idem_fixes_eq_range) |
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82 |
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83 lemma finite_range_approx: |
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84 "finite (range (\<lambda>x::'a::bifinite. approx n\<cdot>x))" |
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85 by (simp add: image_def finite_approx) |
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86 |
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87 lemma compact_approx [simp]: |
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88 fixes x :: "'a::bifinite" |
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89 shows "compact (approx n\<cdot>x)" |
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90 proof (rule compactI2) |
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91 fix Y::"nat \<Rightarrow> 'a" |
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92 assume Y: "chain Y" |
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93 have "finite_chain (\<lambda>i. approx n\<cdot>(Y i))" |
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94 proof (rule finite_range_imp_finch) |
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95 show "chain (\<lambda>i. approx n\<cdot>(Y i))" |
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96 using Y by simp |
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97 have "range (\<lambda>i. approx n\<cdot>(Y i)) \<subseteq> {x. approx n\<cdot>x = x}" |
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98 by clarsimp |
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99 thus "finite (range (\<lambda>i. approx n\<cdot>(Y i)))" |
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100 using finite_fixes_approx by (rule finite_subset) |
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101 qed |
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102 hence "\<exists>j. (\<Squnion>i. approx n\<cdot>(Y i)) = approx n\<cdot>(Y j)" |
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103 by (simp add: finite_chain_def maxinch_is_thelub Y) |
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104 then obtain j where j: "(\<Squnion>i. approx n\<cdot>(Y i)) = approx n\<cdot>(Y j)" .. |
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105 |
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106 assume "approx n\<cdot>x \<sqsubseteq> (\<Squnion>i. Y i)" |
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107 hence "approx n\<cdot>(approx n\<cdot>x) \<sqsubseteq> approx n\<cdot>(\<Squnion>i. Y i)" |
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108 by (rule monofun_cfun_arg) |
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109 hence "approx n\<cdot>x \<sqsubseteq> (\<Squnion>i. approx n\<cdot>(Y i))" |
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110 by (simp add: contlub_cfun_arg Y) |
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111 hence "approx n\<cdot>x \<sqsubseteq> approx n\<cdot>(Y j)" |
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112 using j by simp |
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113 hence "approx n\<cdot>x \<sqsubseteq> Y j" |
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114 using approx_less by (rule trans_less) |
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115 thus "\<exists>j. approx n\<cdot>x \<sqsubseteq> Y j" .. |
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116 qed |
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117 |
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118 lemma bifinite_compact_eq_approx: |
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119 fixes x :: "'a::bifinite" |
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120 assumes x: "compact x" |
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121 shows "\<exists>i. approx i\<cdot>x = x" |
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122 proof - |
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123 have chain: "chain (\<lambda>i. approx i\<cdot>x)" by simp |
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124 have less: "x \<sqsubseteq> (\<Squnion>i. approx i\<cdot>x)" by simp |
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125 obtain i where i: "x \<sqsubseteq> approx i\<cdot>x" |
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126 using compactD2 [OF x chain less] .. |
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127 with approx_less have "approx i\<cdot>x = x" |
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128 by (rule antisym_less) |
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129 thus "\<exists>i. approx i\<cdot>x = x" .. |
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130 qed |
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131 |
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132 lemma bifinite_compact_iff: |
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133 "compact (x::'a::bifinite) = (\<exists>n. approx n\<cdot>x = x)" |
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134 apply (rule iffI) |
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135 apply (erule bifinite_compact_eq_approx) |
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136 apply (erule exE) |
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137 apply (erule subst) |
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138 apply (rule compact_approx) |
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139 done |
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140 |
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141 lemma approx_induct: |
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142 assumes adm: "adm P" and P: "\<And>n x. P (approx n\<cdot>x)" |
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143 shows "P (x::'a::bifinite)" |
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144 proof - |
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145 have "P (\<Squnion>n. approx n\<cdot>x)" |
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146 by (rule admD [OF adm], simp, simp add: P) |
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147 thus "P x" by simp |
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148 qed |
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149 |
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150 lemma bifinite_less_ext: |
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151 fixes x y :: "'a::bifinite" |
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152 shows "(\<And>i. approx i\<cdot>x \<sqsubseteq> approx i\<cdot>y) \<Longrightarrow> x \<sqsubseteq> y" |
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153 apply (subgoal_tac "(\<Squnion>i. approx i\<cdot>x) \<sqsubseteq> (\<Squnion>i. approx i\<cdot>y)", simp) |
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154 apply (rule lub_mono [rule_format], simp, simp, simp) |
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155 done |
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156 |
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157 subsection {* Instance for continuous function space *} |
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158 |
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159 lemma finite_range_lemma: |
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160 fixes h :: "'a::cpo \<rightarrow> 'b::cpo" |
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161 fixes k :: "'c::cpo \<rightarrow> 'd::cpo" |
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162 shows "\<lbrakk>finite {y. \<exists>x. y = h\<cdot>x}; finite {y. \<exists>x. y = k\<cdot>x}\<rbrakk> |
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163 \<Longrightarrow> finite {g. \<exists>f. g = (\<Lambda> x. k\<cdot>(f\<cdot>(h\<cdot>x)))}" |
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164 apply (rule_tac f="\<lambda>g. {(h\<cdot>x, y) |x y. y = g\<cdot>x}" in finite_imageD) |
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165 apply (rule_tac B="Pow ({y. \<exists>x. y = h\<cdot>x} \<times> {y. \<exists>x. y = k\<cdot>x})" |
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166 in finite_subset) |
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167 apply (rule image_subsetI) |
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168 apply (clarsimp, fast) |
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169 apply simp |
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170 apply (rule inj_onI) |
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171 apply (clarsimp simp add: expand_set_eq) |
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172 apply (rule ext_cfun, simp) |
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173 apply (drule_tac x="h\<cdot>x" in spec) |
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174 apply (drule_tac x="k\<cdot>(f\<cdot>(h\<cdot>x))" in spec) |
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175 apply (drule iffD1, fast) |
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176 apply clarsimp |
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177 done |
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178 |
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179 instance "->" :: (bifinite, bifinite) approx .. |
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180 |
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181 defs (overloaded) |
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182 approx_cfun_def: |
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183 "approx \<equiv> \<lambda>n. \<Lambda> f x. approx n\<cdot>(f\<cdot>(approx n\<cdot>x))" |
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184 |
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185 instance "->" :: (bifinite, bifinite) bifinite |
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186 apply (intro_classes, unfold approx_cfun_def) |
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187 apply simp |
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188 apply (simp add: lub_distribs eta_cfun) |
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189 apply simp |
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190 apply simp |
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191 apply (rule finite_range_imp_finite_fixes) |
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192 apply (intro finite_range_lemma finite_approx) |
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193 done |
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194 |
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195 lemma approx_cfun: "approx n\<cdot>f\<cdot>x = approx n\<cdot>(f\<cdot>(approx n\<cdot>x))" |
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196 by (simp add: approx_cfun_def) |
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197 |
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198 end |