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1 (* Title: HOLCF/Universal.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 theory Universal |
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7 imports CompactBasis NatIso |
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8 begin |
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9 |
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10 subsection {* Basis datatype *} |
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11 |
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12 types ubasis = nat |
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13 |
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14 definition |
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15 node :: "nat \<Rightarrow> ubasis \<Rightarrow> ubasis set \<Rightarrow> ubasis" |
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16 where |
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17 "node i x A = Suc (prod2nat (i, prod2nat (x, set2nat A)))" |
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18 |
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19 lemma node_not_0 [simp]: "node i x A \<noteq> 0" |
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20 unfolding node_def by simp |
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21 |
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22 lemma node_gt_0 [simp]: "0 < node i x A" |
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23 unfolding node_def by simp |
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24 |
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25 lemma node_inject [simp]: |
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26 "\<lbrakk>finite A; finite B\<rbrakk> |
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27 \<Longrightarrow> node i x A = node j y B \<longleftrightarrow> i = j \<and> x = y \<and> A = B" |
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28 unfolding node_def by simp |
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29 |
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30 lemma node_gt0: "i < node i x A" |
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31 unfolding node_def less_Suc_eq_le |
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32 by (rule le_prod2nat_1) |
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33 |
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34 lemma node_gt1: "x < node i x A" |
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35 unfolding node_def less_Suc_eq_le |
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36 by (rule order_trans [OF le_prod2nat_1 le_prod2nat_2]) |
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37 |
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38 lemma nat_less_power2: "n < 2^n" |
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39 by (induct n) simp_all |
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40 |
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41 lemma node_gt2: "\<lbrakk>finite A; y \<in> A\<rbrakk> \<Longrightarrow> y < node i x A" |
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42 unfolding node_def less_Suc_eq_le set2nat_def |
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43 apply (rule order_trans [OF _ le_prod2nat_2]) |
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44 apply (rule order_trans [OF _ le_prod2nat_2]) |
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45 apply (rule order_trans [where y="setsum (op ^ 2) {y}"]) |
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46 apply (simp add: nat_less_power2 [THEN order_less_imp_le]) |
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47 apply (erule setsum_mono2, simp, simp) |
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48 done |
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49 |
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50 lemma eq_prod2nat_pairI: |
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51 "\<lbrakk>fst (nat2prod x) = a; snd (nat2prod x) = b\<rbrakk> \<Longrightarrow> x = prod2nat (a, b)" |
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52 by (erule subst, erule subst, simp) |
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53 |
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54 lemma node_cases: |
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55 assumes 1: "x = 0 \<Longrightarrow> P" |
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56 assumes 2: "\<And>i y A. \<lbrakk>finite A; x = node i y A\<rbrakk> \<Longrightarrow> P" |
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57 shows "P" |
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58 apply (cases x) |
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59 apply (erule 1) |
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60 apply (rule 2) |
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61 apply (rule finite_nat2set) |
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62 apply (simp add: node_def) |
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63 apply (rule eq_prod2nat_pairI [OF refl]) |
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64 apply (rule eq_prod2nat_pairI [OF refl refl]) |
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65 done |
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66 |
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67 lemma node_induct: |
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68 assumes 1: "P 0" |
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69 assumes 2: "\<And>i x A. \<lbrakk>P x; finite A; \<forall>y\<in>A. P y\<rbrakk> \<Longrightarrow> P (node i x A)" |
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70 shows "P x" |
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71 apply (induct x rule: nat_less_induct) |
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72 apply (case_tac n rule: node_cases) |
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73 apply (simp add: 1) |
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74 apply (simp add: 2 node_gt1 node_gt2) |
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75 done |
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76 |
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77 subsection {* Basis ordering *} |
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78 |
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79 inductive |
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80 ubasis_le :: "nat \<Rightarrow> nat \<Rightarrow> bool" |
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81 where |
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82 ubasis_le_refl: "ubasis_le x x" |
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83 | ubasis_le_trans: |
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84 "\<lbrakk>ubasis_le x y; ubasis_le y z\<rbrakk> \<Longrightarrow> ubasis_le x z" |
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85 | ubasis_le_lower: |
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86 "finite A \<Longrightarrow> ubasis_le x (node i x A)" |
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87 | ubasis_le_upper: |
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88 "\<lbrakk>finite A; y \<in> A; ubasis_le x y\<rbrakk> \<Longrightarrow> ubasis_le (node i x A) y" |
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89 |
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90 lemma ubasis_le_minimal: "ubasis_le 0 x" |
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91 apply (induct x rule: node_induct) |
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92 apply (rule ubasis_le_refl) |
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93 apply (erule ubasis_le_trans) |
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94 apply (erule ubasis_le_lower) |
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95 done |
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96 |
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97 subsubsection {* Generic take function *} |
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98 |
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99 function |
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100 ubasis_until :: "(ubasis \<Rightarrow> bool) \<Rightarrow> ubasis \<Rightarrow> ubasis" |
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101 where |
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102 "ubasis_until P 0 = 0" |
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103 | "finite A \<Longrightarrow> ubasis_until P (node i x A) = |
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104 (if P (node i x A) then node i x A else ubasis_until P x)" |
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105 apply clarify |
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106 apply (rule_tac x=b in node_cases) |
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107 apply simp |
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108 apply simp |
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109 apply fast |
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110 apply simp |
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111 apply simp |
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112 apply simp |
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113 done |
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114 |
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115 termination ubasis_until |
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116 apply (relation "measure snd") |
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117 apply (rule wf_measure) |
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118 apply (simp add: node_gt1) |
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119 done |
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120 |
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121 lemma ubasis_until: "P 0 \<Longrightarrow> P (ubasis_until P x)" |
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122 by (induct x rule: node_induct) simp_all |
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123 |
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124 lemma ubasis_until': "0 < ubasis_until P x \<Longrightarrow> P (ubasis_until P x)" |
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125 by (induct x rule: node_induct) auto |
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126 |
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127 lemma ubasis_until_same: "P x \<Longrightarrow> ubasis_until P x = x" |
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128 by (induct x rule: node_induct) simp_all |
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129 |
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130 lemma ubasis_until_idem: |
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131 "P 0 \<Longrightarrow> ubasis_until P (ubasis_until P x) = ubasis_until P x" |
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132 by (rule ubasis_until_same [OF ubasis_until]) |
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133 |
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134 lemma ubasis_until_0: |
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135 "\<forall>x. x \<noteq> 0 \<longrightarrow> \<not> P x \<Longrightarrow> ubasis_until P x = 0" |
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136 by (induct x rule: node_induct) simp_all |
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137 |
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138 lemma ubasis_until_less: "ubasis_le (ubasis_until P x) x" |
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139 apply (induct x rule: node_induct) |
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140 apply (simp add: ubasis_le_refl) |
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141 apply (simp add: ubasis_le_refl) |
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142 apply (rule impI) |
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143 apply (erule ubasis_le_trans) |
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144 apply (erule ubasis_le_lower) |
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145 done |
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146 |
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147 lemma ubasis_until_chain: |
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148 assumes PQ: "\<And>x. P x \<Longrightarrow> Q x" |
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149 shows "ubasis_le (ubasis_until P x) (ubasis_until Q x)" |
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150 apply (induct x rule: node_induct) |
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151 apply (simp add: ubasis_le_refl) |
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152 apply (simp add: ubasis_le_refl) |
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153 apply (simp add: PQ) |
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154 apply clarify |
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155 apply (rule ubasis_le_trans) |
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156 apply (rule ubasis_until_less) |
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157 apply (erule ubasis_le_lower) |
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158 done |
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159 |
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160 lemma ubasis_until_mono: |
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161 assumes "\<And>i x A y. \<lbrakk>finite A; P (node i x A); y \<in> A; ubasis_le x y\<rbrakk> \<Longrightarrow> P y" |
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162 shows "ubasis_le x y \<Longrightarrow> ubasis_le (ubasis_until P x) (ubasis_until P y)" |
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163 apply (induct set: ubasis_le) |
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164 apply (rule ubasis_le_refl) |
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165 apply (erule (1) ubasis_le_trans) |
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166 apply (simp add: ubasis_le_refl) |
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167 apply (rule impI) |
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168 apply (rule ubasis_le_trans) |
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169 apply (rule ubasis_until_less) |
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170 apply (erule ubasis_le_lower) |
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171 apply simp |
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172 apply (rule impI) |
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173 apply (subst ubasis_until_same) |
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174 apply (erule (3) prems) |
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175 apply (erule (2) ubasis_le_upper) |
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176 done |
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177 |
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178 lemma finite_range_ubasis_until: |
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179 "finite {x. P x} \<Longrightarrow> finite (range (ubasis_until P))" |
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180 apply (rule finite_subset [where B="insert 0 {x. P x}"]) |
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181 apply (clarsimp simp add: ubasis_until') |
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182 apply simp |
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183 done |
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184 |
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185 subsubsection {* Take function for @{typ ubasis} *} |
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186 |
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187 definition |
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188 ubasis_take :: "nat \<Rightarrow> ubasis \<Rightarrow> ubasis" |
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189 where |
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190 "ubasis_take n = ubasis_until (\<lambda>x. x \<le> n)" |
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191 |
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192 lemma ubasis_take_le: "ubasis_take n x \<le> n" |
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193 unfolding ubasis_take_def by (rule ubasis_until, rule le0) |
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194 |
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195 lemma ubasis_take_same: "x \<le> n \<Longrightarrow> ubasis_take n x = x" |
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196 unfolding ubasis_take_def by (rule ubasis_until_same) |
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197 |
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198 lemma ubasis_take_idem: "ubasis_take n (ubasis_take n x) = ubasis_take n x" |
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199 by (rule ubasis_take_same [OF ubasis_take_le]) |
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200 |
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201 lemma ubasis_take_0 [simp]: "ubasis_take 0 x = 0" |
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202 unfolding ubasis_take_def by (simp add: ubasis_until_0) |
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203 |
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204 lemma ubasis_take_less: "ubasis_le (ubasis_take n x) x" |
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205 unfolding ubasis_take_def by (rule ubasis_until_less) |
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206 |
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207 lemma ubasis_take_chain: "ubasis_le (ubasis_take n x) (ubasis_take (Suc n) x)" |
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208 unfolding ubasis_take_def by (rule ubasis_until_chain) simp |
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209 |
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210 lemma ubasis_take_mono: |
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211 assumes "ubasis_le x y" |
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212 shows "ubasis_le (ubasis_take n x) (ubasis_take n y)" |
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213 unfolding ubasis_take_def |
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214 apply (rule ubasis_until_mono [OF _ prems]) |
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215 apply (frule (2) order_less_le_trans [OF node_gt2]) |
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216 apply (erule order_less_imp_le) |
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217 done |
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218 |
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219 lemma finite_range_ubasis_take: "finite (range (ubasis_take n))" |
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220 apply (rule finite_subset [where B="{..n}"]) |
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221 apply (simp add: subset_eq ubasis_take_le) |
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222 apply simp |
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223 done |
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224 |
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225 lemma ubasis_take_covers: "\<exists>n. ubasis_take n x = x" |
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226 apply (rule exI [where x=x]) |
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227 apply (simp add: ubasis_take_same) |
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228 done |
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229 |
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230 interpretation udom: preorder [ubasis_le] |
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231 apply default |
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232 apply (rule ubasis_le_refl) |
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233 apply (erule (1) ubasis_le_trans) |
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234 done |
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235 |
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236 interpretation udom: basis_take [ubasis_le ubasis_take] |
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237 apply default |
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238 apply (rule ubasis_take_less) |
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239 apply (rule ubasis_take_idem) |
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240 apply (erule ubasis_take_mono) |
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241 apply (rule ubasis_take_chain) |
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242 apply (rule finite_range_ubasis_take) |
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243 apply (rule ubasis_take_covers) |
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244 done |
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245 |
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246 subsection {* Defining the universal domain by ideal completion *} |
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247 |
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248 typedef (open) udom = "{S. udom.ideal S}" |
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249 by (fast intro: udom.ideal_principal) |
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250 |
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251 instantiation udom :: sq_ord |
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252 begin |
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253 |
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254 definition |
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255 "x \<sqsubseteq> y \<longleftrightarrow> Rep_udom x \<subseteq> Rep_udom y" |
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256 |
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257 instance .. |
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258 end |
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259 |
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260 instance udom :: po |
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261 by (rule udom.typedef_ideal_po |
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262 [OF type_definition_udom sq_le_udom_def]) |
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263 |
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264 instance udom :: cpo |
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265 by (rule udom.typedef_ideal_cpo |
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266 [OF type_definition_udom sq_le_udom_def]) |
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267 |
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268 lemma Rep_udom_lub: |
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269 "chain Y \<Longrightarrow> Rep_udom (\<Squnion>i. Y i) = (\<Union>i. Rep_udom (Y i))" |
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270 by (rule udom.typedef_ideal_rep_contlub |
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271 [OF type_definition_udom sq_le_udom_def]) |
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272 |
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273 lemma ideal_Rep_udom: "udom.ideal (Rep_udom xs)" |
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274 by (rule Rep_udom [unfolded mem_Collect_eq]) |
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275 |
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276 definition |
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277 udom_principal :: "nat \<Rightarrow> udom" where |
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278 "udom_principal t = Abs_udom {u. ubasis_le u t}" |
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279 |
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280 lemma Rep_udom_principal: |
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281 "Rep_udom (udom_principal t) = {u. ubasis_le u t}" |
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282 unfolding udom_principal_def |
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283 by (simp add: Abs_udom_inverse udom.ideal_principal) |
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284 |
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285 interpretation udom: |
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286 ideal_completion [ubasis_le ubasis_take udom_principal Rep_udom] |
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287 apply unfold_locales |
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288 apply (rule ideal_Rep_udom) |
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289 apply (erule Rep_udom_lub) |
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290 apply (rule Rep_udom_principal) |
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291 apply (simp only: sq_le_udom_def) |
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292 done |
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293 |
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294 text {* Universal domain is pointed *} |
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295 |
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296 lemma udom_minimal: "udom_principal 0 \<sqsubseteq> x" |
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297 apply (induct x rule: udom.principal_induct) |
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298 apply (simp, simp add: ubasis_le_minimal) |
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299 done |
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300 |
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301 instance udom :: pcpo |
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302 by intro_classes (fast intro: udom_minimal) |
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303 |
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304 lemma inst_udom_pcpo: "\<bottom> = udom_principal 0" |
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305 by (rule udom_minimal [THEN UU_I, symmetric]) |
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306 |
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307 text {* Universal domain is bifinite *} |
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308 |
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309 instantiation udom :: bifinite |
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310 begin |
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311 |
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312 definition |
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313 approx_udom_def: "approx = udom.completion_approx" |
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314 |
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315 instance |
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316 apply (intro_classes, unfold approx_udom_def) |
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317 apply (rule udom.chain_completion_approx) |
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318 apply (rule udom.lub_completion_approx) |
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319 apply (rule udom.completion_approx_idem) |
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320 apply (rule udom.finite_fixes_completion_approx) |
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321 done |
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322 |
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323 end |
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324 |
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325 lemma approx_udom_principal [simp]: |
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326 "approx n\<cdot>(udom_principal x) = udom_principal (ubasis_take n x)" |
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327 unfolding approx_udom_def |
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328 by (rule udom.completion_approx_principal) |
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329 |
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330 lemma approx_eq_udom_principal: |
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331 "\<exists>a\<in>Rep_udom x. approx n\<cdot>x = udom_principal (ubasis_take n a)" |
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332 unfolding approx_udom_def |
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333 by (rule udom.completion_approx_eq_principal) |
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334 |
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335 |
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336 subsection {* Universality of @{typ udom} *} |
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337 |
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338 defaultsort bifinite |
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339 |
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340 subsubsection {* Choosing a maximal element from a finite set *} |
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341 |
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342 lemma finite_has_maximal: |
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343 fixes A :: "'a::po set" |
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344 shows "\<lbrakk>finite A; A \<noteq> {}\<rbrakk> \<Longrightarrow> \<exists>x\<in>A. \<forall>y\<in>A. x \<sqsubseteq> y \<longrightarrow> x = y" |
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345 proof (induct rule: finite_ne_induct) |
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346 case (singleton x) |
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347 show ?case by simp |
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348 next |
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349 case (insert a A) |
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350 from `\<exists>x\<in>A. \<forall>y\<in>A. x \<sqsubseteq> y \<longrightarrow> x = y` |
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351 obtain x where x: "x \<in> A" |
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352 and x_eq: "\<And>y. \<lbrakk>y \<in> A; x \<sqsubseteq> y\<rbrakk> \<Longrightarrow> x = y" by fast |
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353 show ?case |
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354 proof (intro bexI ballI impI) |
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355 fix y |
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356 assume "y \<in> insert a A" and "(if x \<sqsubseteq> a then a else x) \<sqsubseteq> y" |
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357 thus "(if x \<sqsubseteq> a then a else x) = y" |
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358 apply auto |
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359 apply (frule (1) trans_less) |
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360 apply (frule (1) x_eq) |
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361 apply (rule antisym_less, assumption) |
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362 apply simp |
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363 apply (erule (1) x_eq) |
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364 done |
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365 next |
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366 show "(if x \<sqsubseteq> a then a else x) \<in> insert a A" |
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367 by (simp add: x) |
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368 qed |
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369 qed |
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370 |
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371 definition |
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372 choose :: "'a compact_basis set \<Rightarrow> 'a compact_basis" |
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373 where |
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374 "choose A = (SOME x. x \<in> {x\<in>A. \<forall>y\<in>A. x \<sqsubseteq> y \<longrightarrow> x = y})" |
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375 |
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376 lemma choose_lemma: |
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377 "\<lbrakk>finite A; A \<noteq> {}\<rbrakk> \<Longrightarrow> choose A \<in> {x\<in>A. \<forall>y\<in>A. x \<sqsubseteq> y \<longrightarrow> x = y}" |
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378 unfolding choose_def |
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379 apply (rule someI_ex) |
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380 apply (frule (1) finite_has_maximal, fast) |
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381 done |
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382 |
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383 lemma maximal_choose: |
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384 "\<lbrakk>finite A; y \<in> A; choose A \<sqsubseteq> y\<rbrakk> \<Longrightarrow> choose A = y" |
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385 apply (cases "A = {}", simp) |
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386 apply (frule (1) choose_lemma, simp) |
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387 done |
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388 |
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389 lemma choose_in: "\<lbrakk>finite A; A \<noteq> {}\<rbrakk> \<Longrightarrow> choose A \<in> A" |
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390 by (frule (1) choose_lemma, simp) |
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391 |
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392 function |
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393 choose_pos :: "'a compact_basis set \<Rightarrow> 'a compact_basis \<Rightarrow> nat" |
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394 where |
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395 "choose_pos A x = |
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396 (if finite A \<and> x \<in> A \<and> x \<noteq> choose A |
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397 then Suc (choose_pos (A - {choose A}) x) else 0)" |
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398 by auto |
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399 |
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400 termination choose_pos |
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401 apply (relation "measure (card \<circ> fst)", simp) |
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402 apply clarsimp |
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403 apply (rule card_Diff1_less) |
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404 apply assumption |
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405 apply (erule choose_in) |
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406 apply clarsimp |
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407 done |
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408 |
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409 declare choose_pos.simps [simp del] |
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410 |
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411 lemma choose_pos_choose: "finite A \<Longrightarrow> choose_pos A (choose A) = 0" |
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412 by (simp add: choose_pos.simps) |
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413 |
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414 lemma inj_on_choose_pos [OF refl]: |
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415 "\<lbrakk>card A = n; finite A\<rbrakk> \<Longrightarrow> inj_on (choose_pos A) A" |
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416 apply (induct n arbitrary: A) |
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417 apply simp |
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418 apply (case_tac "A = {}", simp) |
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419 apply (frule (1) choose_in) |
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420 apply (rule inj_onI) |
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421 apply (drule_tac x="A - {choose A}" in meta_spec, simp) |
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422 apply (simp add: choose_pos.simps) |
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423 apply (simp split: split_if_asm) |
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424 apply (erule (1) inj_onD, simp, simp) |
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425 done |
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426 |
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427 lemma choose_pos_bounded [OF refl]: |
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428 "\<lbrakk>card A = n; finite A; x \<in> A\<rbrakk> \<Longrightarrow> choose_pos A x < n" |
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429 apply (induct n arbitrary: A) |
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430 apply simp |
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431 apply (case_tac "A = {}", simp) |
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432 apply (frule (1) choose_in) |
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433 apply (subst choose_pos.simps) |
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434 apply simp |
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435 done |
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436 |
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437 lemma choose_pos_lessD: |
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438 "\<lbrakk>choose_pos A x < choose_pos A y; finite A; x \<in> A; y \<in> A\<rbrakk> \<Longrightarrow> \<not> x \<sqsubseteq> y" |
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439 apply (induct A x arbitrary: y rule: choose_pos.induct) |
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440 apply simp |
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441 apply (case_tac "x = choose A") |
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442 apply simp |
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443 apply (rule notI) |
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444 apply (frule (2) maximal_choose) |
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445 apply simp |
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446 apply (case_tac "y = choose A") |
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447 apply (simp add: choose_pos_choose) |
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448 apply (drule_tac x=y in meta_spec) |
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449 apply simp |
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450 apply (erule meta_mp) |
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451 apply (simp add: choose_pos.simps) |
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452 done |
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453 |
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454 subsubsection {* Rank of basis elements *} |
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455 |
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456 primrec |
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457 cb_take :: "nat \<Rightarrow> 'a compact_basis \<Rightarrow> 'a compact_basis" |
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458 where |
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459 "cb_take 0 = (\<lambda>x. compact_bot)" |
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460 | "cb_take (Suc n) = compact_take n" |
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461 |
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462 lemma cb_take_covers: "\<exists>n. cb_take n x = x" |
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463 apply (rule exE [OF compact_basis.take_covers [where a=x]]) |
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464 apply (rename_tac n, rule_tac x="Suc n" in exI, simp) |
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465 done |
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466 |
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467 lemma cb_take_less: "cb_take n x \<sqsubseteq> x" |
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468 by (cases n, simp, simp add: compact_basis.take_less) |
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469 |
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470 lemma cb_take_idem: "cb_take n (cb_take n x) = cb_take n x" |
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471 by (cases n, simp, simp add: compact_basis.take_take) |
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472 |
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473 lemma cb_take_mono: "x \<sqsubseteq> y \<Longrightarrow> cb_take n x \<sqsubseteq> cb_take n y" |
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474 by (cases n, simp, simp add: compact_basis.take_mono) |
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475 |
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476 lemma cb_take_chain_le: "m \<le> n \<Longrightarrow> cb_take m x \<sqsubseteq> cb_take n x" |
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477 apply (cases m, simp) |
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478 apply (cases n, simp) |
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479 apply (simp add: compact_basis.take_chain_le) |
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480 done |
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481 |
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482 lemma range_const: "range (\<lambda>x. c) = {c}" |
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483 by auto |
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484 |
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485 lemma finite_range_cb_take: "finite (range (cb_take n))" |
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486 apply (cases n) |
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487 apply (simp add: range_const) |
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488 apply (simp add: compact_basis.finite_range_take) |
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489 done |
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490 |
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491 definition |
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492 rank :: "'a compact_basis \<Rightarrow> nat" |
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493 where |
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494 "rank x = (LEAST n. cb_take n x = x)" |
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495 |
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496 lemma compact_approx_rank: "cb_take (rank x) x = x" |
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497 unfolding rank_def |
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498 apply (rule LeastI_ex) |
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499 apply (rule cb_take_covers) |
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500 done |
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501 |
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502 lemma rank_leD: "rank x \<le> n \<Longrightarrow> cb_take n x = x" |
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503 apply (rule antisym_less [OF cb_take_less]) |
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504 apply (subst compact_approx_rank [symmetric]) |
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505 apply (erule cb_take_chain_le) |
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506 done |
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507 |
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508 lemma rank_leI: "cb_take n x = x \<Longrightarrow> rank x \<le> n" |
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509 unfolding rank_def by (rule Least_le) |
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510 |
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511 lemma rank_le_iff: "rank x \<le> n \<longleftrightarrow> cb_take n x = x" |
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512 by (rule iffI [OF rank_leD rank_leI]) |
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513 |
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514 definition |
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515 rank_le :: "'a compact_basis \<Rightarrow> 'a compact_basis set" |
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516 where |
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517 "rank_le x = {y. rank y \<le> rank x}" |
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518 |
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519 definition |
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520 rank_lt :: "'a compact_basis \<Rightarrow> 'a compact_basis set" |
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521 where |
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522 "rank_lt x = {y. rank y < rank x}" |
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523 |
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524 definition |
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525 rank_eq :: "'a compact_basis \<Rightarrow> 'a compact_basis set" |
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526 where |
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527 "rank_eq x = {y. rank y = rank x}" |
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528 |
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529 lemma rank_eq_cong: "rank x = rank y \<Longrightarrow> rank_eq x = rank_eq y" |
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530 unfolding rank_eq_def by simp |
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531 |
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532 lemma rank_lt_cong: "rank x = rank y \<Longrightarrow> rank_lt x = rank_lt y" |
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533 unfolding rank_lt_def by simp |
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534 |
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535 lemma rank_eq_subset: "rank_eq x \<subseteq> rank_le x" |
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536 unfolding rank_eq_def rank_le_def by auto |
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537 |
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538 lemma rank_lt_subset: "rank_lt x \<subseteq> rank_le x" |
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539 unfolding rank_lt_def rank_le_def by auto |
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540 |
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541 lemma finite_rank_le: "finite (rank_le x)" |
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542 unfolding rank_le_def |
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543 apply (rule finite_subset [where B="range (cb_take (rank x))"]) |
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544 apply clarify |
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545 apply (rule range_eqI) |
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546 apply (erule rank_leD [symmetric]) |
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547 apply (rule finite_range_cb_take) |
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548 done |
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549 |
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550 lemma finite_rank_eq: "finite (rank_eq x)" |
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551 by (rule finite_subset [OF rank_eq_subset finite_rank_le]) |
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552 |
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553 lemma finite_rank_lt: "finite (rank_lt x)" |
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554 by (rule finite_subset [OF rank_lt_subset finite_rank_le]) |
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555 |
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556 lemma rank_lt_Int_rank_eq: "rank_lt x \<inter> rank_eq x = {}" |
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557 unfolding rank_lt_def rank_eq_def rank_le_def by auto |
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558 |
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559 lemma rank_lt_Un_rank_eq: "rank_lt x \<union> rank_eq x = rank_le x" |
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560 unfolding rank_lt_def rank_eq_def rank_le_def by auto |
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561 |
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562 subsubsection {* Reordering of basis elements *} |
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563 |
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564 definition |
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565 reorder :: "'a compact_basis \<Rightarrow> nat" |
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566 where |
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567 "reorder x = card (rank_lt x) + choose_pos (rank_eq x) x" |
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568 |
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569 lemma reorder_bounded: "reorder x < card (rank_le x)" |
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570 unfolding reorder_def |
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571 apply (rule ord_less_eq_trans) |
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572 apply (rule add_strict_left_mono) |
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573 apply (rule choose_pos_bounded) |
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574 apply (rule finite_rank_eq) |
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575 apply (simp add: rank_eq_def) |
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576 apply (subst card_Un_disjoint [symmetric]) |
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577 apply (rule finite_rank_lt) |
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578 apply (rule finite_rank_eq) |
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579 apply (rule rank_lt_Int_rank_eq) |
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580 apply (simp add: rank_lt_Un_rank_eq) |
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581 done |
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582 |
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583 lemma reorder_ge: "card (rank_lt x) \<le> reorder x" |
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584 unfolding reorder_def by simp |
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585 |
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586 lemma reorder_rank_mono: |
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587 fixes x y :: "'a compact_basis" |
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588 shows "rank x < rank y \<Longrightarrow> reorder x < reorder y" |
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589 apply (rule less_le_trans [OF reorder_bounded]) |
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590 apply (rule order_trans [OF _ reorder_ge]) |
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591 apply (rule card_mono) |
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592 apply (rule finite_rank_lt) |
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593 apply (simp add: rank_le_def rank_lt_def subset_eq) |
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594 done |
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595 |
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596 lemma reorder_eqD: "reorder x = reorder y \<Longrightarrow> x = y" |
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597 apply (rule linorder_cases [where x="rank x" and y="rank y"]) |
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598 apply (drule reorder_rank_mono, simp) |
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599 apply (simp add: reorder_def) |
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600 apply (rule inj_on_choose_pos [where A="rank_eq x", THEN inj_onD]) |
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601 apply (rule finite_rank_eq) |
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602 apply (simp cong: rank_lt_cong rank_eq_cong) |
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603 apply (simp add: rank_eq_def) |
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604 apply (simp add: rank_eq_def) |
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605 apply (drule reorder_rank_mono, simp) |
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606 done |
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607 |
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608 lemma inj_reorder: "inj reorder" |
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609 by (rule inj_onI, erule reorder_eqD) |
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610 |
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611 subsubsection {* Embedding and projection on basis elements *} |
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612 |
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613 function |
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614 basis_emb :: "'a compact_basis \<Rightarrow> ubasis" |
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615 where |
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616 "basis_emb x = (if x = compact_bot then 0 else |
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617 node |
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618 (reorder x) |
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619 (case rank x of 0 \<Rightarrow> 0 | Suc k \<Rightarrow> basis_emb (cb_take k x)) |
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620 (basis_emb ` {y. reorder y < reorder x \<and> x \<sqsubseteq> y}))" |
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621 by auto |
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622 |
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623 termination basis_emb |
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624 apply (relation "measure reorder", simp) |
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625 apply simp |
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626 apply (rule reorder_rank_mono) |
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627 apply (simp add: less_Suc_eq_le) |
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628 apply (rule rank_leI) |
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629 apply (rule cb_take_idem) |
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630 apply simp |
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631 done |
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632 |
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633 declare basis_emb.simps [simp del] |
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634 |
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635 lemma basis_emb_compact_bot [simp]: "basis_emb compact_bot = 0" |
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636 by (simp add: basis_emb.simps) |
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637 |
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638 lemma fin1: "finite {y. reorder y < reorder x \<and> x \<sqsubseteq> y}" |
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639 apply (subst Collect_conj_eq) |
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640 apply (rule finite_Int) |
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641 apply (rule disjI1) |
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642 apply (subgoal_tac "finite (reorder -` {n. n < reorder x})", simp) |
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643 apply (rule finite_vimageI [OF _ inj_reorder]) |
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644 apply (simp add: lessThan_def [symmetric]) |
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645 done |
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646 |
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647 lemma fin2: "finite (basis_emb ` {y. reorder y < reorder x \<and> x \<sqsubseteq> y})" |
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648 by (rule finite_imageI [OF fin1]) |
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649 |
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650 lemma basis_emb_mono [OF refl]: |
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651 "\<lbrakk>n = max (reorder x) (reorder y); x \<sqsubseteq> y\<rbrakk> |
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652 \<Longrightarrow> ubasis_le (basis_emb x) (basis_emb y)" |
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653 proof (induct n arbitrary: x y rule: less_induct) |
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654 case (less n) |
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655 assume IH: |
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656 "\<And>(m::nat) (x::'a compact_basis) y. |
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657 \<lbrakk>m < n; m = max (reorder x) (reorder y); x \<sqsubseteq> y\<rbrakk> |
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658 \<Longrightarrow> ubasis_le (basis_emb x) (basis_emb y)" |
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659 assume n: "n = max (reorder x) (reorder y)" |
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660 assume less: "x \<sqsubseteq> y" |
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661 show ?case |
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662 proof (cases) |
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663 assume "x = compact_bot" |
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664 thus ?case by (simp add: ubasis_le_minimal) |
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665 next |
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666 assume x_neq [simp]: "x \<noteq> compact_bot" |
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667 with less have y_neq [simp]: "y \<noteq> compact_bot" |
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668 apply clarify |
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669 apply (drule antisym_less [OF compact_minimal]) |
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670 apply simp |
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671 done |
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672 show ?case |
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673 proof (rule linorder_cases) |
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674 assume 1: "reorder x < reorder y" |
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675 show ?case |
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676 proof (rule linorder_cases) |
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677 assume "rank x < rank y" |
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678 with 1 show ?case |
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679 apply (case_tac "rank y", simp) |
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680 apply (subst basis_emb.simps [where x=y]) |
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681 apply simp |
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682 apply (rule ubasis_le_trans [OF _ ubasis_le_lower [OF fin2]]) |
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683 apply (rule IH [OF _ refl, unfolded n]) |
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684 apply (simp add: less_max_iff_disj) |
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685 apply (rule reorder_rank_mono) |
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686 apply (simp add: less_Suc_eq_le) |
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687 apply (rule rank_leI) |
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688 apply (rule cb_take_idem) |
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689 apply (simp add: less_Suc_eq_le) |
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690 apply (subgoal_tac "cb_take nat x \<sqsubseteq> cb_take nat y") |
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691 apply (simp add: rank_leD) |
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692 apply (rule cb_take_mono [OF less]) |
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693 done |
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694 next |
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695 assume "rank x = rank y" |
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696 with 1 show ?case |
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697 apply (simp add: reorder_def) |
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698 apply (simp cong: rank_lt_cong rank_eq_cong) |
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699 apply (drule choose_pos_lessD) |
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700 apply (rule finite_rank_eq) |
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701 apply (simp add: rank_eq_def) |
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702 apply (simp add: rank_eq_def) |
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703 apply (simp add: less) |
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704 done |
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705 next |
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706 assume "rank x > rank y" |
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707 hence "reorder x > reorder y" |
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708 by (rule reorder_rank_mono) |
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709 with 1 show ?case by simp |
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710 qed |
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711 next |
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712 assume "reorder x = reorder y" |
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713 hence "x = y" by (rule reorder_eqD) |
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714 thus ?case by (simp add: ubasis_le_refl) |
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715 next |
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716 assume "reorder x > reorder y" |
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717 with less show ?case |
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718 apply (simp add: basis_emb.simps [where x=x]) |
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719 apply (rule ubasis_le_upper [OF fin2], simp) |
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720 apply (cases "rank x") |
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721 apply (simp add: ubasis_le_minimal) |
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722 apply simp |
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723 apply (rule IH [OF _ refl, unfolded n]) |
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724 apply (simp add: less_max_iff_disj) |
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725 apply (rule reorder_rank_mono) |
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726 apply (simp add: less_Suc_eq_le) |
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727 apply (rule rank_leI) |
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728 apply (rule cb_take_idem) |
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729 apply (erule rev_trans_less) |
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730 apply (rule cb_take_less) |
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731 done |
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732 qed |
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733 qed |
|
734 qed |
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735 |
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736 lemma inj_basis_emb: "inj basis_emb" |
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737 apply (rule inj_onI) |
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738 apply (case_tac "x = compact_bot") |
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739 apply (case_tac [!] "y = compact_bot") |
|
740 apply simp |
|
741 apply (simp add: basis_emb.simps) |
|
742 apply (simp add: basis_emb.simps) |
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743 apply (simp add: basis_emb.simps) |
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744 apply (simp add: fin2 inj_eq [OF inj_reorder]) |
|
745 done |
|
746 |
|
747 definition |
|
748 basis_prj :: "nat \<Rightarrow> 'a compact_basis" |
|
749 where |
|
750 "basis_prj x = inv basis_emb |
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751 (ubasis_until (\<lambda>x. x \<in> range (basis_emb :: 'a compact_basis \<Rightarrow> nat)) x)" |
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752 |
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753 lemma basis_prj_basis_emb: "\<And>x. basis_prj (basis_emb x) = x" |
|
754 unfolding basis_prj_def |
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755 apply (subst ubasis_until_same) |
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756 apply (rule rangeI) |
|
757 apply (rule inv_f_f) |
|
758 apply (rule inj_basis_emb) |
|
759 done |
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760 |
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761 lemma basis_prj_node: |
|
762 "\<lbrakk>finite A; node i x A \<notin> range (basis_emb :: 'a compact_basis \<Rightarrow> nat)\<rbrakk> |
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763 \<Longrightarrow> basis_prj (node i x A) = (basis_prj x :: 'a compact_basis)" |
|
764 unfolding basis_prj_def by simp |
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765 |
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766 lemma basis_prj_0: "basis_prj 0 = compact_bot" |
|
767 apply (subst basis_emb_compact_bot [symmetric]) |
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768 apply (rule basis_prj_basis_emb) |
|
769 done |
|
770 |
|
771 lemma basis_prj_mono: "ubasis_le x y \<Longrightarrow> basis_prj x \<sqsubseteq> basis_prj y" |
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772 apply (erule ubasis_le.induct) |
|
773 apply (rule refl_less) |
|
774 apply (erule (1) trans_less) |
|
775 apply (case_tac "node i x A \<in> range (basis_emb :: 'a compact_basis \<Rightarrow> nat)") |
|
776 apply (erule rangeE, rename_tac a) |
|
777 apply (case_tac "a = compact_bot", simp) |
|
778 apply (simp add: basis_prj_basis_emb) |
|
779 apply (simp add: basis_emb.simps) |
|
780 apply (clarsimp simp add: fin2) |
|
781 apply (case_tac "rank a", simp) |
|
782 apply (simp add: basis_prj_0) |
|
783 apply (simp add: basis_prj_basis_emb) |
|
784 apply (rule cb_take_less) |
|
785 apply (simp add: basis_prj_node) |
|
786 apply (case_tac "node i x A \<in> range (basis_emb :: 'a compact_basis \<Rightarrow> nat)") |
|
787 apply (erule rangeE, rename_tac a) |
|
788 apply (case_tac "a = compact_bot", simp) |
|
789 apply (simp add: basis_prj_basis_emb) |
|
790 apply (simp add: basis_emb.simps) |
|
791 apply (clarsimp simp add: fin2) |
|
792 apply (case_tac "rank a", simp add: basis_prj_basis_emb) |
|
793 apply (simp add: basis_prj_basis_emb) |
|
794 apply (simp add: basis_prj_node) |
|
795 done |
|
796 |
|
797 lemma basis_emb_prj_less: "ubasis_le (basis_emb (basis_prj x)) x" |
|
798 unfolding basis_prj_def |
|
799 apply (subst f_inv_f [where f=basis_emb]) |
|
800 apply (rule ubasis_until) |
|
801 apply (rule range_eqI [where x=compact_bot]) |
|
802 apply simp |
|
803 apply (rule ubasis_until_less) |
|
804 done |
|
805 |
|
806 hide (open) const |
|
807 node |
|
808 choose |
|
809 choose_pos |
|
810 reorder |
|
811 |
|
812 subsubsection {* EP-pair from any bifinite domain into @{typ udom} *} |
|
813 |
|
814 definition |
|
815 udom_emb :: "'a::bifinite \<rightarrow> udom" |
|
816 where |
|
817 "udom_emb = compact_basis.basis_fun (\<lambda>x. udom_principal (basis_emb x))" |
|
818 |
|
819 definition |
|
820 udom_prj :: "udom \<rightarrow> 'a::bifinite" |
|
821 where |
|
822 "udom_prj = udom.basis_fun (\<lambda>x. Rep_compact_basis (basis_prj x))" |
|
823 |
|
824 lemma udom_emb_principal: |
|
825 "udom_emb\<cdot>(Rep_compact_basis x) = udom_principal (basis_emb x)" |
|
826 unfolding udom_emb_def |
|
827 apply (rule compact_basis.basis_fun_principal) |
|
828 apply (rule udom.principal_mono) |
|
829 apply (erule basis_emb_mono) |
|
830 done |
|
831 |
|
832 lemma udom_prj_principal: |
|
833 "udom_prj\<cdot>(udom_principal x) = Rep_compact_basis (basis_prj x)" |
|
834 unfolding udom_prj_def |
|
835 apply (rule udom.basis_fun_principal) |
|
836 apply (rule compact_basis.principal_mono) |
|
837 apply (erule basis_prj_mono) |
|
838 done |
|
839 |
|
840 lemma ep_pair_udom: "ep_pair udom_emb udom_prj" |
|
841 apply default |
|
842 apply (rule compact_basis.principal_induct, simp) |
|
843 apply (simp add: udom_emb_principal udom_prj_principal) |
|
844 apply (simp add: basis_prj_basis_emb) |
|
845 apply (rule udom.principal_induct, simp) |
|
846 apply (simp add: udom_emb_principal udom_prj_principal) |
|
847 apply (rule basis_emb_prj_less) |
|
848 done |
|
849 |
|
850 end |