1 (* Title: HOL/Real/HahnBanach/Subspace.thy |
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2 Author: Gertrud Bauer, TU Munich |
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3 *) |
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4 |
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5 header {* Subspaces *} |
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6 |
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7 theory Subspace |
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8 imports VectorSpace |
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9 begin |
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10 |
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11 subsection {* Definition *} |
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12 |
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13 text {* |
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14 A non-empty subset @{text U} of a vector space @{text V} is a |
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15 \emph{subspace} of @{text V}, iff @{text U} is closed under addition |
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16 and scalar multiplication. |
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17 *} |
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18 |
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19 locale subspace = |
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20 fixes U :: "'a\<Colon>{minus, plus, zero, uminus} set" and V |
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21 assumes non_empty [iff, intro]: "U \<noteq> {}" |
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22 and subset [iff]: "U \<subseteq> V" |
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23 and add_closed [iff]: "x \<in> U \<Longrightarrow> y \<in> U \<Longrightarrow> x + y \<in> U" |
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24 and mult_closed [iff]: "x \<in> U \<Longrightarrow> a \<cdot> x \<in> U" |
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25 |
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26 notation (symbols) |
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27 subspace (infix "\<unlhd>" 50) |
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28 |
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29 declare vectorspace.intro [intro?] subspace.intro [intro?] |
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30 |
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31 lemma subspace_subset [elim]: "U \<unlhd> V \<Longrightarrow> U \<subseteq> V" |
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32 by (rule subspace.subset) |
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33 |
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34 lemma (in subspace) subsetD [iff]: "x \<in> U \<Longrightarrow> x \<in> V" |
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35 using subset by blast |
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36 |
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37 lemma subspaceD [elim]: "U \<unlhd> V \<Longrightarrow> x \<in> U \<Longrightarrow> x \<in> V" |
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38 by (rule subspace.subsetD) |
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39 |
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40 lemma rev_subspaceD [elim?]: "x \<in> U \<Longrightarrow> U \<unlhd> V \<Longrightarrow> x \<in> V" |
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41 by (rule subspace.subsetD) |
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42 |
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43 lemma (in subspace) diff_closed [iff]: |
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44 assumes "vectorspace V" |
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45 assumes x: "x \<in> U" and y: "y \<in> U" |
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46 shows "x - y \<in> U" |
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47 proof - |
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48 interpret vectorspace V by fact |
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49 from x y show ?thesis by (simp add: diff_eq1 negate_eq1) |
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50 qed |
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51 |
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52 text {* |
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53 \medskip Similar as for linear spaces, the existence of the zero |
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54 element in every subspace follows from the non-emptiness of the |
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55 carrier set and by vector space laws. |
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56 *} |
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57 |
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58 lemma (in subspace) zero [intro]: |
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59 assumes "vectorspace V" |
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60 shows "0 \<in> U" |
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61 proof - |
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62 interpret V: vectorspace V by fact |
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63 have "U \<noteq> {}" by (rule non_empty) |
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64 then obtain x where x: "x \<in> U" by blast |
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65 then have "x \<in> V" .. then have "0 = x - x" by simp |
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66 also from `vectorspace V` x x have "\<dots> \<in> U" by (rule diff_closed) |
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67 finally show ?thesis . |
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68 qed |
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69 |
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70 lemma (in subspace) neg_closed [iff]: |
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71 assumes "vectorspace V" |
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72 assumes x: "x \<in> U" |
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73 shows "- x \<in> U" |
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74 proof - |
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75 interpret vectorspace V by fact |
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76 from x show ?thesis by (simp add: negate_eq1) |
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77 qed |
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78 |
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79 text {* \medskip Further derived laws: every subspace is a vector space. *} |
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80 |
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81 lemma (in subspace) vectorspace [iff]: |
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82 assumes "vectorspace V" |
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83 shows "vectorspace U" |
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84 proof - |
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85 interpret vectorspace V by fact |
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86 show ?thesis |
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87 proof |
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88 show "U \<noteq> {}" .. |
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89 fix x y z assume x: "x \<in> U" and y: "y \<in> U" and z: "z \<in> U" |
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90 fix a b :: real |
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91 from x y show "x + y \<in> U" by simp |
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92 from x show "a \<cdot> x \<in> U" by simp |
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93 from x y z show "(x + y) + z = x + (y + z)" by (simp add: add_ac) |
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94 from x y show "x + y = y + x" by (simp add: add_ac) |
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95 from x show "x - x = 0" by simp |
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96 from x show "0 + x = x" by simp |
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97 from x y show "a \<cdot> (x + y) = a \<cdot> x + a \<cdot> y" by (simp add: distrib) |
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98 from x show "(a + b) \<cdot> x = a \<cdot> x + b \<cdot> x" by (simp add: distrib) |
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99 from x show "(a * b) \<cdot> x = a \<cdot> b \<cdot> x" by (simp add: mult_assoc) |
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100 from x show "1 \<cdot> x = x" by simp |
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101 from x show "- x = - 1 \<cdot> x" by (simp add: negate_eq1) |
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102 from x y show "x - y = x + - y" by (simp add: diff_eq1) |
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103 qed |
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104 qed |
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105 |
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106 |
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107 text {* The subspace relation is reflexive. *} |
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108 |
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109 lemma (in vectorspace) subspace_refl [intro]: "V \<unlhd> V" |
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110 proof |
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111 show "V \<noteq> {}" .. |
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112 show "V \<subseteq> V" .. |
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113 fix x y assume x: "x \<in> V" and y: "y \<in> V" |
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114 fix a :: real |
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115 from x y show "x + y \<in> V" by simp |
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116 from x show "a \<cdot> x \<in> V" by simp |
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117 qed |
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118 |
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119 text {* The subspace relation is transitive. *} |
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120 |
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121 lemma (in vectorspace) subspace_trans [trans]: |
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122 "U \<unlhd> V \<Longrightarrow> V \<unlhd> W \<Longrightarrow> U \<unlhd> W" |
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123 proof |
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124 assume uv: "U \<unlhd> V" and vw: "V \<unlhd> W" |
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125 from uv show "U \<noteq> {}" by (rule subspace.non_empty) |
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126 show "U \<subseteq> W" |
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127 proof - |
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128 from uv have "U \<subseteq> V" by (rule subspace.subset) |
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129 also from vw have "V \<subseteq> W" by (rule subspace.subset) |
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130 finally show ?thesis . |
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131 qed |
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132 fix x y assume x: "x \<in> U" and y: "y \<in> U" |
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133 from uv and x y show "x + y \<in> U" by (rule subspace.add_closed) |
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134 from uv and x show "\<And>a. a \<cdot> x \<in> U" by (rule subspace.mult_closed) |
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135 qed |
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136 |
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137 |
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138 subsection {* Linear closure *} |
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139 |
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140 text {* |
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141 The \emph{linear closure} of a vector @{text x} is the set of all |
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142 scalar multiples of @{text x}. |
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143 *} |
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144 |
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145 definition |
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146 lin :: "('a::{minus, plus, zero}) \<Rightarrow> 'a set" where |
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147 "lin x = {a \<cdot> x | a. True}" |
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148 |
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149 lemma linI [intro]: "y = a \<cdot> x \<Longrightarrow> y \<in> lin x" |
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150 unfolding lin_def by blast |
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151 |
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152 lemma linI' [iff]: "a \<cdot> x \<in> lin x" |
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153 unfolding lin_def by blast |
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154 |
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155 lemma linE [elim]: "x \<in> lin v \<Longrightarrow> (\<And>a::real. x = a \<cdot> v \<Longrightarrow> C) \<Longrightarrow> C" |
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156 unfolding lin_def by blast |
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157 |
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158 |
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159 text {* Every vector is contained in its linear closure. *} |
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160 |
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161 lemma (in vectorspace) x_lin_x [iff]: "x \<in> V \<Longrightarrow> x \<in> lin x" |
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162 proof - |
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163 assume "x \<in> V" |
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164 then have "x = 1 \<cdot> x" by simp |
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165 also have "\<dots> \<in> lin x" .. |
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166 finally show ?thesis . |
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167 qed |
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168 |
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169 lemma (in vectorspace) "0_lin_x" [iff]: "x \<in> V \<Longrightarrow> 0 \<in> lin x" |
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170 proof |
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171 assume "x \<in> V" |
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172 then show "0 = 0 \<cdot> x" by simp |
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173 qed |
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174 |
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175 text {* Any linear closure is a subspace. *} |
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176 |
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177 lemma (in vectorspace) lin_subspace [intro]: |
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178 "x \<in> V \<Longrightarrow> lin x \<unlhd> V" |
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179 proof |
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180 assume x: "x \<in> V" |
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181 then show "lin x \<noteq> {}" by (auto simp add: x_lin_x) |
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182 show "lin x \<subseteq> V" |
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183 proof |
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184 fix x' assume "x' \<in> lin x" |
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185 then obtain a where "x' = a \<cdot> x" .. |
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186 with x show "x' \<in> V" by simp |
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187 qed |
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188 fix x' x'' assume x': "x' \<in> lin x" and x'': "x'' \<in> lin x" |
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189 show "x' + x'' \<in> lin x" |
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190 proof - |
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191 from x' obtain a' where "x' = a' \<cdot> x" .. |
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192 moreover from x'' obtain a'' where "x'' = a'' \<cdot> x" .. |
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193 ultimately have "x' + x'' = (a' + a'') \<cdot> x" |
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194 using x by (simp add: distrib) |
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195 also have "\<dots> \<in> lin x" .. |
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196 finally show ?thesis . |
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197 qed |
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198 fix a :: real |
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199 show "a \<cdot> x' \<in> lin x" |
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200 proof - |
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201 from x' obtain a' where "x' = a' \<cdot> x" .. |
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202 with x have "a \<cdot> x' = (a * a') \<cdot> x" by (simp add: mult_assoc) |
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203 also have "\<dots> \<in> lin x" .. |
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204 finally show ?thesis . |
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205 qed |
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206 qed |
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207 |
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208 |
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209 text {* Any linear closure is a vector space. *} |
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210 |
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211 lemma (in vectorspace) lin_vectorspace [intro]: |
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212 assumes "x \<in> V" |
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213 shows "vectorspace (lin x)" |
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214 proof - |
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215 from `x \<in> V` have "subspace (lin x) V" |
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216 by (rule lin_subspace) |
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217 from this and vectorspace_axioms show ?thesis |
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218 by (rule subspace.vectorspace) |
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219 qed |
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220 |
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221 |
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222 subsection {* Sum of two vectorspaces *} |
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223 |
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224 text {* |
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225 The \emph{sum} of two vectorspaces @{text U} and @{text V} is the |
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226 set of all sums of elements from @{text U} and @{text V}. |
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227 *} |
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228 |
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229 instantiation "fun" :: (type, type) plus |
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230 begin |
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231 |
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232 definition |
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233 sum_def: "plus_fun U V = {u + v | u v. u \<in> U \<and> v \<in> V}" (* FIXME not fully general!? *) |
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234 |
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235 instance .. |
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236 |
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237 end |
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238 |
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239 lemma sumE [elim]: |
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240 "x \<in> U + V \<Longrightarrow> (\<And>u v. x = u + v \<Longrightarrow> u \<in> U \<Longrightarrow> v \<in> V \<Longrightarrow> C) \<Longrightarrow> C" |
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241 unfolding sum_def by blast |
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242 |
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243 lemma sumI [intro]: |
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244 "u \<in> U \<Longrightarrow> v \<in> V \<Longrightarrow> x = u + v \<Longrightarrow> x \<in> U + V" |
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245 unfolding sum_def by blast |
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246 |
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247 lemma sumI' [intro]: |
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248 "u \<in> U \<Longrightarrow> v \<in> V \<Longrightarrow> u + v \<in> U + V" |
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249 unfolding sum_def by blast |
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250 |
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251 text {* @{text U} is a subspace of @{text "U + V"}. *} |
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252 |
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253 lemma subspace_sum1 [iff]: |
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254 assumes "vectorspace U" "vectorspace V" |
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255 shows "U \<unlhd> U + V" |
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256 proof - |
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257 interpret vectorspace U by fact |
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258 interpret vectorspace V by fact |
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259 show ?thesis |
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260 proof |
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261 show "U \<noteq> {}" .. |
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262 show "U \<subseteq> U + V" |
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263 proof |
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264 fix x assume x: "x \<in> U" |
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265 moreover have "0 \<in> V" .. |
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266 ultimately have "x + 0 \<in> U + V" .. |
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267 with x show "x \<in> U + V" by simp |
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268 qed |
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269 fix x y assume x: "x \<in> U" and "y \<in> U" |
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270 then show "x + y \<in> U" by simp |
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271 from x show "\<And>a. a \<cdot> x \<in> U" by simp |
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272 qed |
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273 qed |
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274 |
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275 text {* The sum of two subspaces is again a subspace. *} |
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276 |
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277 lemma sum_subspace [intro?]: |
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278 assumes "subspace U E" "vectorspace E" "subspace V E" |
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279 shows "U + V \<unlhd> E" |
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280 proof - |
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281 interpret subspace U E by fact |
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282 interpret vectorspace E by fact |
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283 interpret subspace V E by fact |
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284 show ?thesis |
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285 proof |
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286 have "0 \<in> U + V" |
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287 proof |
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288 show "0 \<in> U" using `vectorspace E` .. |
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289 show "0 \<in> V" using `vectorspace E` .. |
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290 show "(0::'a) = 0 + 0" by simp |
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291 qed |
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292 then show "U + V \<noteq> {}" by blast |
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293 show "U + V \<subseteq> E" |
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294 proof |
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295 fix x assume "x \<in> U + V" |
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296 then obtain u v where "x = u + v" and |
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297 "u \<in> U" and "v \<in> V" .. |
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298 then show "x \<in> E" by simp |
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299 qed |
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300 fix x y assume x: "x \<in> U + V" and y: "y \<in> U + V" |
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301 show "x + y \<in> U + V" |
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302 proof - |
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303 from x obtain ux vx where "x = ux + vx" and "ux \<in> U" and "vx \<in> V" .. |
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304 moreover |
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305 from y obtain uy vy where "y = uy + vy" and "uy \<in> U" and "vy \<in> V" .. |
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306 ultimately |
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307 have "ux + uy \<in> U" |
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308 and "vx + vy \<in> V" |
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309 and "x + y = (ux + uy) + (vx + vy)" |
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310 using x y by (simp_all add: add_ac) |
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311 then show ?thesis .. |
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312 qed |
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313 fix a show "a \<cdot> x \<in> U + V" |
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314 proof - |
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315 from x obtain u v where "x = u + v" and "u \<in> U" and "v \<in> V" .. |
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316 then have "a \<cdot> u \<in> U" and "a \<cdot> v \<in> V" |
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317 and "a \<cdot> x = (a \<cdot> u) + (a \<cdot> v)" by (simp_all add: distrib) |
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318 then show ?thesis .. |
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319 qed |
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320 qed |
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321 qed |
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322 |
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323 text{* The sum of two subspaces is a vectorspace. *} |
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324 |
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325 lemma sum_vs [intro?]: |
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326 "U \<unlhd> E \<Longrightarrow> V \<unlhd> E \<Longrightarrow> vectorspace E \<Longrightarrow> vectorspace (U + V)" |
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327 by (rule subspace.vectorspace) (rule sum_subspace) |
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328 |
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329 |
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330 subsection {* Direct sums *} |
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331 |
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332 text {* |
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333 The sum of @{text U} and @{text V} is called \emph{direct}, iff the |
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334 zero element is the only common element of @{text U} and @{text |
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335 V}. For every element @{text x} of the direct sum of @{text U} and |
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336 @{text V} the decomposition in @{text "x = u + v"} with |
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337 @{text "u \<in> U"} and @{text "v \<in> V"} is unique. |
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338 *} |
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339 |
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340 lemma decomp: |
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341 assumes "vectorspace E" "subspace U E" "subspace V E" |
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342 assumes direct: "U \<inter> V = {0}" |
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343 and u1: "u1 \<in> U" and u2: "u2 \<in> U" |
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344 and v1: "v1 \<in> V" and v2: "v2 \<in> V" |
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345 and sum: "u1 + v1 = u2 + v2" |
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346 shows "u1 = u2 \<and> v1 = v2" |
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347 proof - |
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348 interpret vectorspace E by fact |
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349 interpret subspace U E by fact |
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350 interpret subspace V E by fact |
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351 show ?thesis |
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352 proof |
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353 have U: "vectorspace U" (* FIXME: use interpret *) |
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354 using `subspace U E` `vectorspace E` by (rule subspace.vectorspace) |
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355 have V: "vectorspace V" |
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356 using `subspace V E` `vectorspace E` by (rule subspace.vectorspace) |
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357 from u1 u2 v1 v2 and sum have eq: "u1 - u2 = v2 - v1" |
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358 by (simp add: add_diff_swap) |
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359 from u1 u2 have u: "u1 - u2 \<in> U" |
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360 by (rule vectorspace.diff_closed [OF U]) |
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361 with eq have v': "v2 - v1 \<in> U" by (simp only:) |
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362 from v2 v1 have v: "v2 - v1 \<in> V" |
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363 by (rule vectorspace.diff_closed [OF V]) |
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364 with eq have u': " u1 - u2 \<in> V" by (simp only:) |
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365 |
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366 show "u1 = u2" |
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367 proof (rule add_minus_eq) |
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368 from u1 show "u1 \<in> E" .. |
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369 from u2 show "u2 \<in> E" .. |
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370 from u u' and direct show "u1 - u2 = 0" by blast |
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371 qed |
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372 show "v1 = v2" |
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373 proof (rule add_minus_eq [symmetric]) |
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374 from v1 show "v1 \<in> E" .. |
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375 from v2 show "v2 \<in> E" .. |
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376 from v v' and direct show "v2 - v1 = 0" by blast |
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377 qed |
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378 qed |
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379 qed |
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380 |
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381 text {* |
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382 An application of the previous lemma will be used in the proof of |
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383 the Hahn-Banach Theorem (see page \pageref{decomp-H-use}): for any |
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384 element @{text "y + a \<cdot> x\<^sub>0"} of the direct sum of a |
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385 vectorspace @{text H} and the linear closure of @{text "x\<^sub>0"} |
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386 the components @{text "y \<in> H"} and @{text a} are uniquely |
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387 determined. |
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388 *} |
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389 |
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390 lemma decomp_H': |
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391 assumes "vectorspace E" "subspace H E" |
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392 assumes y1: "y1 \<in> H" and y2: "y2 \<in> H" |
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393 and x': "x' \<notin> H" "x' \<in> E" "x' \<noteq> 0" |
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394 and eq: "y1 + a1 \<cdot> x' = y2 + a2 \<cdot> x'" |
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395 shows "y1 = y2 \<and> a1 = a2" |
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396 proof - |
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397 interpret vectorspace E by fact |
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398 interpret subspace H E by fact |
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399 show ?thesis |
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400 proof |
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401 have c: "y1 = y2 \<and> a1 \<cdot> x' = a2 \<cdot> x'" |
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402 proof (rule decomp) |
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403 show "a1 \<cdot> x' \<in> lin x'" .. |
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404 show "a2 \<cdot> x' \<in> lin x'" .. |
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405 show "H \<inter> lin x' = {0}" |
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406 proof |
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407 show "H \<inter> lin x' \<subseteq> {0}" |
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408 proof |
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409 fix x assume x: "x \<in> H \<inter> lin x'" |
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410 then obtain a where xx': "x = a \<cdot> x'" |
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411 by blast |
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412 have "x = 0" |
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413 proof cases |
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414 assume "a = 0" |
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415 with xx' and x' show ?thesis by simp |
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416 next |
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417 assume a: "a \<noteq> 0" |
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418 from x have "x \<in> H" .. |
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419 with xx' have "inverse a \<cdot> a \<cdot> x' \<in> H" by simp |
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420 with a and x' have "x' \<in> H" by (simp add: mult_assoc2) |
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421 with `x' \<notin> H` show ?thesis by contradiction |
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422 qed |
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423 then show "x \<in> {0}" .. |
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424 qed |
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425 show "{0} \<subseteq> H \<inter> lin x'" |
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426 proof - |
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427 have "0 \<in> H" using `vectorspace E` .. |
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428 moreover have "0 \<in> lin x'" using `x' \<in> E` .. |
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429 ultimately show ?thesis by blast |
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430 qed |
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431 qed |
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432 show "lin x' \<unlhd> E" using `x' \<in> E` .. |
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433 qed (rule `vectorspace E`, rule `subspace H E`, rule y1, rule y2, rule eq) |
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434 then show "y1 = y2" .. |
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435 from c have "a1 \<cdot> x' = a2 \<cdot> x'" .. |
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436 with x' show "a1 = a2" by (simp add: mult_right_cancel) |
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437 qed |
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438 qed |
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439 |
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440 text {* |
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441 Since for any element @{text "y + a \<cdot> x'"} of the direct sum of a |
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442 vectorspace @{text H} and the linear closure of @{text x'} the |
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443 components @{text "y \<in> H"} and @{text a} are unique, it follows from |
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444 @{text "y \<in> H"} that @{text "a = 0"}. |
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445 *} |
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446 |
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447 lemma decomp_H'_H: |
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448 assumes "vectorspace E" "subspace H E" |
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449 assumes t: "t \<in> H" |
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450 and x': "x' \<notin> H" "x' \<in> E" "x' \<noteq> 0" |
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451 shows "(SOME (y, a). t = y + a \<cdot> x' \<and> y \<in> H) = (t, 0)" |
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452 proof - |
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453 interpret vectorspace E by fact |
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454 interpret subspace H E by fact |
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455 show ?thesis |
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456 proof (rule, simp_all only: split_paired_all split_conv) |
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457 from t x' show "t = t + 0 \<cdot> x' \<and> t \<in> H" by simp |
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458 fix y and a assume ya: "t = y + a \<cdot> x' \<and> y \<in> H" |
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459 have "y = t \<and> a = 0" |
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460 proof (rule decomp_H') |
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461 from ya x' show "y + a \<cdot> x' = t + 0 \<cdot> x'" by simp |
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462 from ya show "y \<in> H" .. |
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463 qed (rule `vectorspace E`, rule `subspace H E`, rule t, (rule x')+) |
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464 with t x' show "(y, a) = (y + a \<cdot> x', 0)" by simp |
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465 qed |
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466 qed |
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467 |
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468 text {* |
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469 The components @{text "y \<in> H"} and @{text a} in @{text "y + a \<cdot> x'"} |
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470 are unique, so the function @{text h'} defined by |
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471 @{text "h' (y + a \<cdot> x') = h y + a \<cdot> \<xi>"} is definite. |
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472 *} |
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473 |
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474 lemma h'_definite: |
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475 fixes H |
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476 assumes h'_def: |
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477 "h' \<equiv> (\<lambda>x. let (y, a) = SOME (y, a). (x = y + a \<cdot> x' \<and> y \<in> H) |
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478 in (h y) + a * xi)" |
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479 and x: "x = y + a \<cdot> x'" |
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480 assumes "vectorspace E" "subspace H E" |
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481 assumes y: "y \<in> H" |
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482 and x': "x' \<notin> H" "x' \<in> E" "x' \<noteq> 0" |
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483 shows "h' x = h y + a * xi" |
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484 proof - |
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485 interpret vectorspace E by fact |
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486 interpret subspace H E by fact |
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487 from x y x' have "x \<in> H + lin x'" by auto |
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488 have "\<exists>!p. (\<lambda>(y, a). x = y + a \<cdot> x' \<and> y \<in> H) p" (is "\<exists>!p. ?P p") |
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489 proof (rule ex_ex1I) |
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490 from x y show "\<exists>p. ?P p" by blast |
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491 fix p q assume p: "?P p" and q: "?P q" |
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492 show "p = q" |
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493 proof - |
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494 from p have xp: "x = fst p + snd p \<cdot> x' \<and> fst p \<in> H" |
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495 by (cases p) simp |
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496 from q have xq: "x = fst q + snd q \<cdot> x' \<and> fst q \<in> H" |
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497 by (cases q) simp |
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498 have "fst p = fst q \<and> snd p = snd q" |
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499 proof (rule decomp_H') |
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500 from xp show "fst p \<in> H" .. |
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501 from xq show "fst q \<in> H" .. |
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502 from xp and xq show "fst p + snd p \<cdot> x' = fst q + snd q \<cdot> x'" |
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503 by simp |
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504 qed (rule `vectorspace E`, rule `subspace H E`, (rule x')+) |
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505 then show ?thesis by (cases p, cases q) simp |
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506 qed |
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507 qed |
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508 then have eq: "(SOME (y, a). x = y + a \<cdot> x' \<and> y \<in> H) = (y, a)" |
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509 by (rule some1_equality) (simp add: x y) |
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510 with h'_def show "h' x = h y + a * xi" by (simp add: Let_def) |
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511 qed |
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512 |
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513 end |
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