1 (* Title: HOL/Real/RealVector.thy |
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2 Author: Brian Huffman |
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3 *) |
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4 |
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5 header {* Vector Spaces and Algebras over the Reals *} |
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6 |
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7 theory RealVector |
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8 imports "~~/src/HOL/RealPow" |
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9 begin |
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10 |
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11 subsection {* Locale for additive functions *} |
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12 |
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13 locale additive = |
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14 fixes f :: "'a::ab_group_add \<Rightarrow> 'b::ab_group_add" |
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15 assumes add: "f (x + y) = f x + f y" |
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16 begin |
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17 |
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18 lemma zero: "f 0 = 0" |
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19 proof - |
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20 have "f 0 = f (0 + 0)" by simp |
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21 also have "\<dots> = f 0 + f 0" by (rule add) |
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22 finally show "f 0 = 0" by simp |
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23 qed |
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24 |
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25 lemma minus: "f (- x) = - f x" |
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26 proof - |
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27 have "f (- x) + f x = f (- x + x)" by (rule add [symmetric]) |
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28 also have "\<dots> = - f x + f x" by (simp add: zero) |
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29 finally show "f (- x) = - f x" by (rule add_right_imp_eq) |
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30 qed |
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31 |
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32 lemma diff: "f (x - y) = f x - f y" |
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33 by (simp add: diff_def add minus) |
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34 |
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35 lemma setsum: "f (setsum g A) = (\<Sum>x\<in>A. f (g x))" |
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36 apply (cases "finite A") |
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37 apply (induct set: finite) |
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38 apply (simp add: zero) |
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39 apply (simp add: add) |
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40 apply (simp add: zero) |
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41 done |
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42 |
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43 end |
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44 |
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45 subsection {* Vector spaces *} |
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46 |
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47 locale vector_space = |
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48 fixes scale :: "'a::field \<Rightarrow> 'b::ab_group_add \<Rightarrow> 'b" |
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49 assumes scale_right_distrib: "scale a (x + y) = scale a x + scale a y" |
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50 and scale_left_distrib: "scale (a + b) x = scale a x + scale b x" |
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51 and scale_scale [simp]: "scale a (scale b x) = scale (a * b) x" |
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52 and scale_one [simp]: "scale 1 x = x" |
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53 begin |
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54 |
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55 lemma scale_left_commute: |
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56 "scale a (scale b x) = scale b (scale a x)" |
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57 by (simp add: mult_commute) |
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58 |
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59 lemma scale_zero_left [simp]: "scale 0 x = 0" |
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60 and scale_minus_left [simp]: "scale (- a) x = - (scale a x)" |
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61 and scale_left_diff_distrib: "scale (a - b) x = scale a x - scale b x" |
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62 proof - |
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63 interpret s: additive "\<lambda>a. scale a x" |
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64 proof qed (rule scale_left_distrib) |
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65 show "scale 0 x = 0" by (rule s.zero) |
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66 show "scale (- a) x = - (scale a x)" by (rule s.minus) |
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67 show "scale (a - b) x = scale a x - scale b x" by (rule s.diff) |
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68 qed |
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69 |
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70 lemma scale_zero_right [simp]: "scale a 0 = 0" |
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71 and scale_minus_right [simp]: "scale a (- x) = - (scale a x)" |
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72 and scale_right_diff_distrib: "scale a (x - y) = scale a x - scale a y" |
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73 proof - |
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74 interpret s: additive "\<lambda>x. scale a x" |
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75 proof qed (rule scale_right_distrib) |
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76 show "scale a 0 = 0" by (rule s.zero) |
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77 show "scale a (- x) = - (scale a x)" by (rule s.minus) |
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78 show "scale a (x - y) = scale a x - scale a y" by (rule s.diff) |
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79 qed |
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80 |
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81 lemma scale_eq_0_iff [simp]: |
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82 "scale a x = 0 \<longleftrightarrow> a = 0 \<or> x = 0" |
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83 proof cases |
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84 assume "a = 0" thus ?thesis by simp |
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85 next |
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86 assume anz [simp]: "a \<noteq> 0" |
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87 { assume "scale a x = 0" |
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88 hence "scale (inverse a) (scale a x) = 0" by simp |
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89 hence "x = 0" by simp } |
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90 thus ?thesis by force |
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91 qed |
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92 |
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93 lemma scale_left_imp_eq: |
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94 "\<lbrakk>a \<noteq> 0; scale a x = scale a y\<rbrakk> \<Longrightarrow> x = y" |
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95 proof - |
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96 assume nonzero: "a \<noteq> 0" |
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97 assume "scale a x = scale a y" |
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98 hence "scale a (x - y) = 0" |
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99 by (simp add: scale_right_diff_distrib) |
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100 hence "x - y = 0" by (simp add: nonzero) |
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101 thus "x = y" by (simp only: right_minus_eq) |
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102 qed |
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103 |
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104 lemma scale_right_imp_eq: |
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105 "\<lbrakk>x \<noteq> 0; scale a x = scale b x\<rbrakk> \<Longrightarrow> a = b" |
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106 proof - |
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107 assume nonzero: "x \<noteq> 0" |
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108 assume "scale a x = scale b x" |
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109 hence "scale (a - b) x = 0" |
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110 by (simp add: scale_left_diff_distrib) |
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111 hence "a - b = 0" by (simp add: nonzero) |
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112 thus "a = b" by (simp only: right_minus_eq) |
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113 qed |
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114 |
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115 lemma scale_cancel_left: |
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116 "scale a x = scale a y \<longleftrightarrow> x = y \<or> a = 0" |
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117 by (auto intro: scale_left_imp_eq) |
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118 |
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119 lemma scale_cancel_right: |
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120 "scale a x = scale b x \<longleftrightarrow> a = b \<or> x = 0" |
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121 by (auto intro: scale_right_imp_eq) |
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122 |
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123 end |
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124 |
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125 subsection {* Real vector spaces *} |
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126 |
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127 class scaleR = type + |
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128 fixes scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a" (infixr "*\<^sub>R" 75) |
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129 begin |
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130 |
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131 abbreviation |
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132 divideR :: "'a \<Rightarrow> real \<Rightarrow> 'a" (infixl "'/\<^sub>R" 70) |
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133 where |
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134 "x /\<^sub>R r == scaleR (inverse r) x" |
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135 |
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136 end |
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137 |
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138 instantiation real :: scaleR |
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139 begin |
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140 |
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141 definition |
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142 real_scaleR_def [simp]: "scaleR a x = a * x" |
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143 |
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144 instance .. |
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145 |
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146 end |
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147 |
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148 class real_vector = scaleR + ab_group_add + |
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149 assumes scaleR_right_distrib: "scaleR a (x + y) = scaleR a x + scaleR a y" |
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150 and scaleR_left_distrib: "scaleR (a + b) x = scaleR a x + scaleR b x" |
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151 and scaleR_scaleR [simp]: "scaleR a (scaleR b x) = scaleR (a * b) x" |
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152 and scaleR_one [simp]: "scaleR 1 x = x" |
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153 |
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154 interpretation real_vector!: |
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155 vector_space "scaleR :: real \<Rightarrow> 'a \<Rightarrow> 'a::real_vector" |
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156 apply unfold_locales |
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157 apply (rule scaleR_right_distrib) |
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158 apply (rule scaleR_left_distrib) |
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159 apply (rule scaleR_scaleR) |
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160 apply (rule scaleR_one) |
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161 done |
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162 |
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163 text {* Recover original theorem names *} |
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164 |
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165 lemmas scaleR_left_commute = real_vector.scale_left_commute |
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166 lemmas scaleR_zero_left = real_vector.scale_zero_left |
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167 lemmas scaleR_minus_left = real_vector.scale_minus_left |
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168 lemmas scaleR_left_diff_distrib = real_vector.scale_left_diff_distrib |
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169 lemmas scaleR_zero_right = real_vector.scale_zero_right |
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170 lemmas scaleR_minus_right = real_vector.scale_minus_right |
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171 lemmas scaleR_right_diff_distrib = real_vector.scale_right_diff_distrib |
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172 lemmas scaleR_eq_0_iff = real_vector.scale_eq_0_iff |
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173 lemmas scaleR_left_imp_eq = real_vector.scale_left_imp_eq |
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174 lemmas scaleR_right_imp_eq = real_vector.scale_right_imp_eq |
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175 lemmas scaleR_cancel_left = real_vector.scale_cancel_left |
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176 lemmas scaleR_cancel_right = real_vector.scale_cancel_right |
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177 |
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178 class real_algebra = real_vector + ring + |
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179 assumes mult_scaleR_left [simp]: "scaleR a x * y = scaleR a (x * y)" |
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180 and mult_scaleR_right [simp]: "x * scaleR a y = scaleR a (x * y)" |
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181 |
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182 class real_algebra_1 = real_algebra + ring_1 |
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183 |
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184 class real_div_algebra = real_algebra_1 + division_ring |
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185 |
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186 class real_field = real_div_algebra + field |
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187 |
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188 instance real :: real_field |
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189 apply (intro_classes, unfold real_scaleR_def) |
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190 apply (rule right_distrib) |
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191 apply (rule left_distrib) |
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192 apply (rule mult_assoc [symmetric]) |
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193 apply (rule mult_1_left) |
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194 apply (rule mult_assoc) |
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195 apply (rule mult_left_commute) |
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196 done |
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197 |
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198 interpretation scaleR_left!: additive "(\<lambda>a. scaleR a x::'a::real_vector)" |
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199 proof qed (rule scaleR_left_distrib) |
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200 |
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201 interpretation scaleR_right!: additive "(\<lambda>x. scaleR a x::'a::real_vector)" |
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202 proof qed (rule scaleR_right_distrib) |
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203 |
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204 lemma nonzero_inverse_scaleR_distrib: |
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205 fixes x :: "'a::real_div_algebra" shows |
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206 "\<lbrakk>a \<noteq> 0; x \<noteq> 0\<rbrakk> \<Longrightarrow> inverse (scaleR a x) = scaleR (inverse a) (inverse x)" |
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207 by (rule inverse_unique, simp) |
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208 |
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209 lemma inverse_scaleR_distrib: |
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210 fixes x :: "'a::{real_div_algebra,division_by_zero}" |
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211 shows "inverse (scaleR a x) = scaleR (inverse a) (inverse x)" |
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212 apply (case_tac "a = 0", simp) |
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213 apply (case_tac "x = 0", simp) |
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214 apply (erule (1) nonzero_inverse_scaleR_distrib) |
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215 done |
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216 |
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217 |
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218 subsection {* Embedding of the Reals into any @{text real_algebra_1}: |
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219 @{term of_real} *} |
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220 |
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221 definition |
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222 of_real :: "real \<Rightarrow> 'a::real_algebra_1" where |
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223 "of_real r = scaleR r 1" |
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224 |
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225 lemma scaleR_conv_of_real: "scaleR r x = of_real r * x" |
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226 by (simp add: of_real_def) |
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227 |
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228 lemma of_real_0 [simp]: "of_real 0 = 0" |
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229 by (simp add: of_real_def) |
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230 |
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231 lemma of_real_1 [simp]: "of_real 1 = 1" |
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232 by (simp add: of_real_def) |
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233 |
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234 lemma of_real_add [simp]: "of_real (x + y) = of_real x + of_real y" |
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235 by (simp add: of_real_def scaleR_left_distrib) |
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236 |
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237 lemma of_real_minus [simp]: "of_real (- x) = - of_real x" |
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238 by (simp add: of_real_def) |
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239 |
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240 lemma of_real_diff [simp]: "of_real (x - y) = of_real x - of_real y" |
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241 by (simp add: of_real_def scaleR_left_diff_distrib) |
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242 |
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243 lemma of_real_mult [simp]: "of_real (x * y) = of_real x * of_real y" |
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244 by (simp add: of_real_def mult_commute) |
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245 |
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246 lemma nonzero_of_real_inverse: |
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247 "x \<noteq> 0 \<Longrightarrow> of_real (inverse x) = |
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248 inverse (of_real x :: 'a::real_div_algebra)" |
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249 by (simp add: of_real_def nonzero_inverse_scaleR_distrib) |
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250 |
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251 lemma of_real_inverse [simp]: |
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252 "of_real (inverse x) = |
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253 inverse (of_real x :: 'a::{real_div_algebra,division_by_zero})" |
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254 by (simp add: of_real_def inverse_scaleR_distrib) |
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255 |
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256 lemma nonzero_of_real_divide: |
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257 "y \<noteq> 0 \<Longrightarrow> of_real (x / y) = |
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258 (of_real x / of_real y :: 'a::real_field)" |
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259 by (simp add: divide_inverse nonzero_of_real_inverse) |
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260 |
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261 lemma of_real_divide [simp]: |
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262 "of_real (x / y) = |
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263 (of_real x / of_real y :: 'a::{real_field,division_by_zero})" |
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264 by (simp add: divide_inverse) |
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265 |
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266 lemma of_real_power [simp]: |
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267 "of_real (x ^ n) = (of_real x :: 'a::{real_algebra_1,recpower}) ^ n" |
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268 by (induct n) (simp_all add: power_Suc) |
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269 |
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270 lemma of_real_eq_iff [simp]: "(of_real x = of_real y) = (x = y)" |
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271 by (simp add: of_real_def scaleR_cancel_right) |
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272 |
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273 lemmas of_real_eq_0_iff [simp] = of_real_eq_iff [of _ 0, simplified] |
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274 |
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275 lemma of_real_eq_id [simp]: "of_real = (id :: real \<Rightarrow> real)" |
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276 proof |
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277 fix r |
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278 show "of_real r = id r" |
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279 by (simp add: of_real_def) |
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280 qed |
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281 |
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282 text{*Collapse nested embeddings*} |
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283 lemma of_real_of_nat_eq [simp]: "of_real (of_nat n) = of_nat n" |
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284 by (induct n) auto |
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285 |
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286 lemma of_real_of_int_eq [simp]: "of_real (of_int z) = of_int z" |
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287 by (cases z rule: int_diff_cases, simp) |
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288 |
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289 lemma of_real_number_of_eq: |
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290 "of_real (number_of w) = (number_of w :: 'a::{number_ring,real_algebra_1})" |
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291 by (simp add: number_of_eq) |
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292 |
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293 text{*Every real algebra has characteristic zero*} |
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294 instance real_algebra_1 < ring_char_0 |
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295 proof |
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296 fix m n :: nat |
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297 have "(of_real (of_nat m) = (of_real (of_nat n)::'a)) = (m = n)" |
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298 by (simp only: of_real_eq_iff of_nat_eq_iff) |
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299 thus "(of_nat m = (of_nat n::'a)) = (m = n)" |
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300 by (simp only: of_real_of_nat_eq) |
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301 qed |
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302 |
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303 instance real_field < field_char_0 .. |
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304 |
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305 |
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306 subsection {* The Set of Real Numbers *} |
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307 |
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308 definition |
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309 Reals :: "'a::real_algebra_1 set" where |
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310 [code del]: "Reals \<equiv> range of_real" |
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311 |
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312 notation (xsymbols) |
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313 Reals ("\<real>") |
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314 |
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315 lemma Reals_of_real [simp]: "of_real r \<in> Reals" |
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316 by (simp add: Reals_def) |
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317 |
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318 lemma Reals_of_int [simp]: "of_int z \<in> Reals" |
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319 by (subst of_real_of_int_eq [symmetric], rule Reals_of_real) |
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320 |
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321 lemma Reals_of_nat [simp]: "of_nat n \<in> Reals" |
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322 by (subst of_real_of_nat_eq [symmetric], rule Reals_of_real) |
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323 |
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324 lemma Reals_number_of [simp]: |
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325 "(number_of w::'a::{number_ring,real_algebra_1}) \<in> Reals" |
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326 by (subst of_real_number_of_eq [symmetric], rule Reals_of_real) |
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327 |
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328 lemma Reals_0 [simp]: "0 \<in> Reals" |
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329 apply (unfold Reals_def) |
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330 apply (rule range_eqI) |
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331 apply (rule of_real_0 [symmetric]) |
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332 done |
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333 |
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334 lemma Reals_1 [simp]: "1 \<in> Reals" |
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335 apply (unfold Reals_def) |
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336 apply (rule range_eqI) |
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337 apply (rule of_real_1 [symmetric]) |
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338 done |
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339 |
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340 lemma Reals_add [simp]: "\<lbrakk>a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> a + b \<in> Reals" |
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341 apply (auto simp add: Reals_def) |
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342 apply (rule range_eqI) |
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343 apply (rule of_real_add [symmetric]) |
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344 done |
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345 |
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346 lemma Reals_minus [simp]: "a \<in> Reals \<Longrightarrow> - a \<in> Reals" |
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347 apply (auto simp add: Reals_def) |
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348 apply (rule range_eqI) |
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349 apply (rule of_real_minus [symmetric]) |
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350 done |
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351 |
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352 lemma Reals_diff [simp]: "\<lbrakk>a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> a - b \<in> Reals" |
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353 apply (auto simp add: Reals_def) |
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354 apply (rule range_eqI) |
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355 apply (rule of_real_diff [symmetric]) |
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356 done |
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357 |
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358 lemma Reals_mult [simp]: "\<lbrakk>a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> a * b \<in> Reals" |
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359 apply (auto simp add: Reals_def) |
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360 apply (rule range_eqI) |
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361 apply (rule of_real_mult [symmetric]) |
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362 done |
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363 |
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364 lemma nonzero_Reals_inverse: |
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365 fixes a :: "'a::real_div_algebra" |
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366 shows "\<lbrakk>a \<in> Reals; a \<noteq> 0\<rbrakk> \<Longrightarrow> inverse a \<in> Reals" |
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367 apply (auto simp add: Reals_def) |
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368 apply (rule range_eqI) |
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369 apply (erule nonzero_of_real_inverse [symmetric]) |
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370 done |
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371 |
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372 lemma Reals_inverse [simp]: |
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373 fixes a :: "'a::{real_div_algebra,division_by_zero}" |
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374 shows "a \<in> Reals \<Longrightarrow> inverse a \<in> Reals" |
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375 apply (auto simp add: Reals_def) |
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376 apply (rule range_eqI) |
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377 apply (rule of_real_inverse [symmetric]) |
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378 done |
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379 |
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380 lemma nonzero_Reals_divide: |
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381 fixes a b :: "'a::real_field" |
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382 shows "\<lbrakk>a \<in> Reals; b \<in> Reals; b \<noteq> 0\<rbrakk> \<Longrightarrow> a / b \<in> Reals" |
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383 apply (auto simp add: Reals_def) |
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384 apply (rule range_eqI) |
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385 apply (erule nonzero_of_real_divide [symmetric]) |
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386 done |
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387 |
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388 lemma Reals_divide [simp]: |
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389 fixes a b :: "'a::{real_field,division_by_zero}" |
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390 shows "\<lbrakk>a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> a / b \<in> Reals" |
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391 apply (auto simp add: Reals_def) |
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392 apply (rule range_eqI) |
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393 apply (rule of_real_divide [symmetric]) |
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394 done |
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395 |
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396 lemma Reals_power [simp]: |
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397 fixes a :: "'a::{real_algebra_1,recpower}" |
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398 shows "a \<in> Reals \<Longrightarrow> a ^ n \<in> Reals" |
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399 apply (auto simp add: Reals_def) |
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400 apply (rule range_eqI) |
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401 apply (rule of_real_power [symmetric]) |
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402 done |
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403 |
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404 lemma Reals_cases [cases set: Reals]: |
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405 assumes "q \<in> \<real>" |
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406 obtains (of_real) r where "q = of_real r" |
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407 unfolding Reals_def |
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408 proof - |
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409 from `q \<in> \<real>` have "q \<in> range of_real" unfolding Reals_def . |
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410 then obtain r where "q = of_real r" .. |
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411 then show thesis .. |
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412 qed |
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413 |
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414 lemma Reals_induct [case_names of_real, induct set: Reals]: |
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415 "q \<in> \<real> \<Longrightarrow> (\<And>r. P (of_real r)) \<Longrightarrow> P q" |
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416 by (rule Reals_cases) auto |
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417 |
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418 |
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419 subsection {* Real normed vector spaces *} |
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420 |
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421 class norm = type + |
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422 fixes norm :: "'a \<Rightarrow> real" |
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423 |
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424 instantiation real :: norm |
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425 begin |
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426 |
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427 definition |
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428 real_norm_def [simp]: "norm r \<equiv> \<bar>r\<bar>" |
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429 |
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430 instance .. |
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431 |
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432 end |
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433 |
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434 class sgn_div_norm = scaleR + norm + sgn + |
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435 assumes sgn_div_norm: "sgn x = x /\<^sub>R norm x" |
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436 |
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437 class real_normed_vector = real_vector + sgn_div_norm + |
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438 assumes norm_ge_zero [simp]: "0 \<le> norm x" |
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439 and norm_eq_zero [simp]: "norm x = 0 \<longleftrightarrow> x = 0" |
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440 and norm_triangle_ineq: "norm (x + y) \<le> norm x + norm y" |
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441 and norm_scaleR: "norm (scaleR a x) = \<bar>a\<bar> * norm x" |
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442 |
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443 class real_normed_algebra = real_algebra + real_normed_vector + |
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444 assumes norm_mult_ineq: "norm (x * y) \<le> norm x * norm y" |
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445 |
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446 class real_normed_algebra_1 = real_algebra_1 + real_normed_algebra + |
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447 assumes norm_one [simp]: "norm 1 = 1" |
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448 |
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449 class real_normed_div_algebra = real_div_algebra + real_normed_vector + |
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450 assumes norm_mult: "norm (x * y) = norm x * norm y" |
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451 |
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452 class real_normed_field = real_field + real_normed_div_algebra |
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453 |
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454 instance real_normed_div_algebra < real_normed_algebra_1 |
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455 proof |
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456 fix x y :: 'a |
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457 show "norm (x * y) \<le> norm x * norm y" |
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458 by (simp add: norm_mult) |
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459 next |
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460 have "norm (1 * 1::'a) = norm (1::'a) * norm (1::'a)" |
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461 by (rule norm_mult) |
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462 thus "norm (1::'a) = 1" by simp |
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463 qed |
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464 |
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465 instance real :: real_normed_field |
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466 apply (intro_classes, unfold real_norm_def real_scaleR_def) |
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467 apply (simp add: real_sgn_def) |
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468 apply (rule abs_ge_zero) |
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469 apply (rule abs_eq_0) |
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470 apply (rule abs_triangle_ineq) |
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471 apply (rule abs_mult) |
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472 apply (rule abs_mult) |
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473 done |
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474 |
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475 lemma norm_zero [simp]: "norm (0::'a::real_normed_vector) = 0" |
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476 by simp |
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477 |
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478 lemma zero_less_norm_iff [simp]: |
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479 fixes x :: "'a::real_normed_vector" |
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480 shows "(0 < norm x) = (x \<noteq> 0)" |
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481 by (simp add: order_less_le) |
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482 |
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483 lemma norm_not_less_zero [simp]: |
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484 fixes x :: "'a::real_normed_vector" |
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485 shows "\<not> norm x < 0" |
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486 by (simp add: linorder_not_less) |
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487 |
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488 lemma norm_le_zero_iff [simp]: |
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489 fixes x :: "'a::real_normed_vector" |
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490 shows "(norm x \<le> 0) = (x = 0)" |
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491 by (simp add: order_le_less) |
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492 |
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493 lemma norm_minus_cancel [simp]: |
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494 fixes x :: "'a::real_normed_vector" |
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495 shows "norm (- x) = norm x" |
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496 proof - |
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497 have "norm (- x) = norm (scaleR (- 1) x)" |
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498 by (simp only: scaleR_minus_left scaleR_one) |
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499 also have "\<dots> = \<bar>- 1\<bar> * norm x" |
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500 by (rule norm_scaleR) |
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501 finally show ?thesis by simp |
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502 qed |
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503 |
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504 lemma norm_minus_commute: |
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505 fixes a b :: "'a::real_normed_vector" |
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506 shows "norm (a - b) = norm (b - a)" |
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507 proof - |
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508 have "norm (- (b - a)) = norm (b - a)" |
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509 by (rule norm_minus_cancel) |
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510 thus ?thesis by simp |
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511 qed |
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512 |
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513 lemma norm_triangle_ineq2: |
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514 fixes a b :: "'a::real_normed_vector" |
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515 shows "norm a - norm b \<le> norm (a - b)" |
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516 proof - |
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517 have "norm (a - b + b) \<le> norm (a - b) + norm b" |
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518 by (rule norm_triangle_ineq) |
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519 thus ?thesis by simp |
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520 qed |
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521 |
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522 lemma norm_triangle_ineq3: |
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523 fixes a b :: "'a::real_normed_vector" |
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524 shows "\<bar>norm a - norm b\<bar> \<le> norm (a - b)" |
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525 apply (subst abs_le_iff) |
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526 apply auto |
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527 apply (rule norm_triangle_ineq2) |
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528 apply (subst norm_minus_commute) |
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529 apply (rule norm_triangle_ineq2) |
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530 done |
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531 |
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532 lemma norm_triangle_ineq4: |
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533 fixes a b :: "'a::real_normed_vector" |
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534 shows "norm (a - b) \<le> norm a + norm b" |
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535 proof - |
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536 have "norm (a + - b) \<le> norm a + norm (- b)" |
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537 by (rule norm_triangle_ineq) |
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538 thus ?thesis |
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539 by (simp only: diff_minus norm_minus_cancel) |
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540 qed |
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541 |
|
542 lemma norm_diff_ineq: |
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543 fixes a b :: "'a::real_normed_vector" |
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544 shows "norm a - norm b \<le> norm (a + b)" |
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545 proof - |
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546 have "norm a - norm (- b) \<le> norm (a - - b)" |
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547 by (rule norm_triangle_ineq2) |
|
548 thus ?thesis by simp |
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549 qed |
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550 |
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551 lemma norm_diff_triangle_ineq: |
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552 fixes a b c d :: "'a::real_normed_vector" |
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553 shows "norm ((a + b) - (c + d)) \<le> norm (a - c) + norm (b - d)" |
|
554 proof - |
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555 have "norm ((a + b) - (c + d)) = norm ((a - c) + (b - d))" |
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556 by (simp add: diff_minus add_ac) |
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557 also have "\<dots> \<le> norm (a - c) + norm (b - d)" |
|
558 by (rule norm_triangle_ineq) |
|
559 finally show ?thesis . |
|
560 qed |
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561 |
|
562 lemma abs_norm_cancel [simp]: |
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563 fixes a :: "'a::real_normed_vector" |
|
564 shows "\<bar>norm a\<bar> = norm a" |
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565 by (rule abs_of_nonneg [OF norm_ge_zero]) |
|
566 |
|
567 lemma norm_add_less: |
|
568 fixes x y :: "'a::real_normed_vector" |
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569 shows "\<lbrakk>norm x < r; norm y < s\<rbrakk> \<Longrightarrow> norm (x + y) < r + s" |
|
570 by (rule order_le_less_trans [OF norm_triangle_ineq add_strict_mono]) |
|
571 |
|
572 lemma norm_mult_less: |
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573 fixes x y :: "'a::real_normed_algebra" |
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574 shows "\<lbrakk>norm x < r; norm y < s\<rbrakk> \<Longrightarrow> norm (x * y) < r * s" |
|
575 apply (rule order_le_less_trans [OF norm_mult_ineq]) |
|
576 apply (simp add: mult_strict_mono') |
|
577 done |
|
578 |
|
579 lemma norm_of_real [simp]: |
|
580 "norm (of_real r :: 'a::real_normed_algebra_1) = \<bar>r\<bar>" |
|
581 unfolding of_real_def by (simp add: norm_scaleR) |
|
582 |
|
583 lemma norm_number_of [simp]: |
|
584 "norm (number_of w::'a::{number_ring,real_normed_algebra_1}) |
|
585 = \<bar>number_of w\<bar>" |
|
586 by (subst of_real_number_of_eq [symmetric], rule norm_of_real) |
|
587 |
|
588 lemma norm_of_int [simp]: |
|
589 "norm (of_int z::'a::real_normed_algebra_1) = \<bar>of_int z\<bar>" |
|
590 by (subst of_real_of_int_eq [symmetric], rule norm_of_real) |
|
591 |
|
592 lemma norm_of_nat [simp]: |
|
593 "norm (of_nat n::'a::real_normed_algebra_1) = of_nat n" |
|
594 apply (subst of_real_of_nat_eq [symmetric]) |
|
595 apply (subst norm_of_real, simp) |
|
596 done |
|
597 |
|
598 lemma nonzero_norm_inverse: |
|
599 fixes a :: "'a::real_normed_div_algebra" |
|
600 shows "a \<noteq> 0 \<Longrightarrow> norm (inverse a) = inverse (norm a)" |
|
601 apply (rule inverse_unique [symmetric]) |
|
602 apply (simp add: norm_mult [symmetric]) |
|
603 done |
|
604 |
|
605 lemma norm_inverse: |
|
606 fixes a :: "'a::{real_normed_div_algebra,division_by_zero}" |
|
607 shows "norm (inverse a) = inverse (norm a)" |
|
608 apply (case_tac "a = 0", simp) |
|
609 apply (erule nonzero_norm_inverse) |
|
610 done |
|
611 |
|
612 lemma nonzero_norm_divide: |
|
613 fixes a b :: "'a::real_normed_field" |
|
614 shows "b \<noteq> 0 \<Longrightarrow> norm (a / b) = norm a / norm b" |
|
615 by (simp add: divide_inverse norm_mult nonzero_norm_inverse) |
|
616 |
|
617 lemma norm_divide: |
|
618 fixes a b :: "'a::{real_normed_field,division_by_zero}" |
|
619 shows "norm (a / b) = norm a / norm b" |
|
620 by (simp add: divide_inverse norm_mult norm_inverse) |
|
621 |
|
622 lemma norm_power_ineq: |
|
623 fixes x :: "'a::{real_normed_algebra_1,recpower}" |
|
624 shows "norm (x ^ n) \<le> norm x ^ n" |
|
625 proof (induct n) |
|
626 case 0 show "norm (x ^ 0) \<le> norm x ^ 0" by simp |
|
627 next |
|
628 case (Suc n) |
|
629 have "norm (x * x ^ n) \<le> norm x * norm (x ^ n)" |
|
630 by (rule norm_mult_ineq) |
|
631 also from Suc have "\<dots> \<le> norm x * norm x ^ n" |
|
632 using norm_ge_zero by (rule mult_left_mono) |
|
633 finally show "norm (x ^ Suc n) \<le> norm x ^ Suc n" |
|
634 by (simp add: power_Suc) |
|
635 qed |
|
636 |
|
637 lemma norm_power: |
|
638 fixes x :: "'a::{real_normed_div_algebra,recpower}" |
|
639 shows "norm (x ^ n) = norm x ^ n" |
|
640 by (induct n) (simp_all add: power_Suc norm_mult) |
|
641 |
|
642 |
|
643 subsection {* Sign function *} |
|
644 |
|
645 lemma norm_sgn: |
|
646 "norm (sgn(x::'a::real_normed_vector)) = (if x = 0 then 0 else 1)" |
|
647 by (simp add: sgn_div_norm norm_scaleR) |
|
648 |
|
649 lemma sgn_zero [simp]: "sgn(0::'a::real_normed_vector) = 0" |
|
650 by (simp add: sgn_div_norm) |
|
651 |
|
652 lemma sgn_zero_iff: "(sgn(x::'a::real_normed_vector) = 0) = (x = 0)" |
|
653 by (simp add: sgn_div_norm) |
|
654 |
|
655 lemma sgn_minus: "sgn (- x) = - sgn(x::'a::real_normed_vector)" |
|
656 by (simp add: sgn_div_norm) |
|
657 |
|
658 lemma sgn_scaleR: |
|
659 "sgn (scaleR r x) = scaleR (sgn r) (sgn(x::'a::real_normed_vector))" |
|
660 by (simp add: sgn_div_norm norm_scaleR mult_ac) |
|
661 |
|
662 lemma sgn_one [simp]: "sgn (1::'a::real_normed_algebra_1) = 1" |
|
663 by (simp add: sgn_div_norm) |
|
664 |
|
665 lemma sgn_of_real: |
|
666 "sgn (of_real r::'a::real_normed_algebra_1) = of_real (sgn r)" |
|
667 unfolding of_real_def by (simp only: sgn_scaleR sgn_one) |
|
668 |
|
669 lemma sgn_mult: |
|
670 fixes x y :: "'a::real_normed_div_algebra" |
|
671 shows "sgn (x * y) = sgn x * sgn y" |
|
672 by (simp add: sgn_div_norm norm_mult mult_commute) |
|
673 |
|
674 lemma real_sgn_eq: "sgn (x::real) = x / \<bar>x\<bar>" |
|
675 by (simp add: sgn_div_norm divide_inverse) |
|
676 |
|
677 lemma real_sgn_pos: "0 < (x::real) \<Longrightarrow> sgn x = 1" |
|
678 unfolding real_sgn_eq by simp |
|
679 |
|
680 lemma real_sgn_neg: "(x::real) < 0 \<Longrightarrow> sgn x = -1" |
|
681 unfolding real_sgn_eq by simp |
|
682 |
|
683 |
|
684 subsection {* Bounded Linear and Bilinear Operators *} |
|
685 |
|
686 locale bounded_linear = additive + |
|
687 constrains f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector" |
|
688 assumes scaleR: "f (scaleR r x) = scaleR r (f x)" |
|
689 assumes bounded: "\<exists>K. \<forall>x. norm (f x) \<le> norm x * K" |
|
690 begin |
|
691 |
|
692 lemma pos_bounded: |
|
693 "\<exists>K>0. \<forall>x. norm (f x) \<le> norm x * K" |
|
694 proof - |
|
695 obtain K where K: "\<And>x. norm (f x) \<le> norm x * K" |
|
696 using bounded by fast |
|
697 show ?thesis |
|
698 proof (intro exI impI conjI allI) |
|
699 show "0 < max 1 K" |
|
700 by (rule order_less_le_trans [OF zero_less_one le_maxI1]) |
|
701 next |
|
702 fix x |
|
703 have "norm (f x) \<le> norm x * K" using K . |
|
704 also have "\<dots> \<le> norm x * max 1 K" |
|
705 by (rule mult_left_mono [OF le_maxI2 norm_ge_zero]) |
|
706 finally show "norm (f x) \<le> norm x * max 1 K" . |
|
707 qed |
|
708 qed |
|
709 |
|
710 lemma nonneg_bounded: |
|
711 "\<exists>K\<ge>0. \<forall>x. norm (f x) \<le> norm x * K" |
|
712 proof - |
|
713 from pos_bounded |
|
714 show ?thesis by (auto intro: order_less_imp_le) |
|
715 qed |
|
716 |
|
717 end |
|
718 |
|
719 locale bounded_bilinear = |
|
720 fixes prod :: "['a::real_normed_vector, 'b::real_normed_vector] |
|
721 \<Rightarrow> 'c::real_normed_vector" |
|
722 (infixl "**" 70) |
|
723 assumes add_left: "prod (a + a') b = prod a b + prod a' b" |
|
724 assumes add_right: "prod a (b + b') = prod a b + prod a b'" |
|
725 assumes scaleR_left: "prod (scaleR r a) b = scaleR r (prod a b)" |
|
726 assumes scaleR_right: "prod a (scaleR r b) = scaleR r (prod a b)" |
|
727 assumes bounded: "\<exists>K. \<forall>a b. norm (prod a b) \<le> norm a * norm b * K" |
|
728 begin |
|
729 |
|
730 lemma pos_bounded: |
|
731 "\<exists>K>0. \<forall>a b. norm (a ** b) \<le> norm a * norm b * K" |
|
732 apply (cut_tac bounded, erule exE) |
|
733 apply (rule_tac x="max 1 K" in exI, safe) |
|
734 apply (rule order_less_le_trans [OF zero_less_one le_maxI1]) |
|
735 apply (drule spec, drule spec, erule order_trans) |
|
736 apply (rule mult_left_mono [OF le_maxI2]) |
|
737 apply (intro mult_nonneg_nonneg norm_ge_zero) |
|
738 done |
|
739 |
|
740 lemma nonneg_bounded: |
|
741 "\<exists>K\<ge>0. \<forall>a b. norm (a ** b) \<le> norm a * norm b * K" |
|
742 proof - |
|
743 from pos_bounded |
|
744 show ?thesis by (auto intro: order_less_imp_le) |
|
745 qed |
|
746 |
|
747 lemma additive_right: "additive (\<lambda>b. prod a b)" |
|
748 by (rule additive.intro, rule add_right) |
|
749 |
|
750 lemma additive_left: "additive (\<lambda>a. prod a b)" |
|
751 by (rule additive.intro, rule add_left) |
|
752 |
|
753 lemma zero_left: "prod 0 b = 0" |
|
754 by (rule additive.zero [OF additive_left]) |
|
755 |
|
756 lemma zero_right: "prod a 0 = 0" |
|
757 by (rule additive.zero [OF additive_right]) |
|
758 |
|
759 lemma minus_left: "prod (- a) b = - prod a b" |
|
760 by (rule additive.minus [OF additive_left]) |
|
761 |
|
762 lemma minus_right: "prod a (- b) = - prod a b" |
|
763 by (rule additive.minus [OF additive_right]) |
|
764 |
|
765 lemma diff_left: |
|
766 "prod (a - a') b = prod a b - prod a' b" |
|
767 by (rule additive.diff [OF additive_left]) |
|
768 |
|
769 lemma diff_right: |
|
770 "prod a (b - b') = prod a b - prod a b'" |
|
771 by (rule additive.diff [OF additive_right]) |
|
772 |
|
773 lemma bounded_linear_left: |
|
774 "bounded_linear (\<lambda>a. a ** b)" |
|
775 apply (unfold_locales) |
|
776 apply (rule add_left) |
|
777 apply (rule scaleR_left) |
|
778 apply (cut_tac bounded, safe) |
|
779 apply (rule_tac x="norm b * K" in exI) |
|
780 apply (simp add: mult_ac) |
|
781 done |
|
782 |
|
783 lemma bounded_linear_right: |
|
784 "bounded_linear (\<lambda>b. a ** b)" |
|
785 apply (unfold_locales) |
|
786 apply (rule add_right) |
|
787 apply (rule scaleR_right) |
|
788 apply (cut_tac bounded, safe) |
|
789 apply (rule_tac x="norm a * K" in exI) |
|
790 apply (simp add: mult_ac) |
|
791 done |
|
792 |
|
793 lemma prod_diff_prod: |
|
794 "(x ** y - a ** b) = (x - a) ** (y - b) + (x - a) ** b + a ** (y - b)" |
|
795 by (simp add: diff_left diff_right) |
|
796 |
|
797 end |
|
798 |
|
799 interpretation mult!: |
|
800 bounded_bilinear "op * :: 'a \<Rightarrow> 'a \<Rightarrow> 'a::real_normed_algebra" |
|
801 apply (rule bounded_bilinear.intro) |
|
802 apply (rule left_distrib) |
|
803 apply (rule right_distrib) |
|
804 apply (rule mult_scaleR_left) |
|
805 apply (rule mult_scaleR_right) |
|
806 apply (rule_tac x="1" in exI) |
|
807 apply (simp add: norm_mult_ineq) |
|
808 done |
|
809 |
|
810 interpretation mult_left!: |
|
811 bounded_linear "(\<lambda>x::'a::real_normed_algebra. x * y)" |
|
812 by (rule mult.bounded_linear_left) |
|
813 |
|
814 interpretation mult_right!: |
|
815 bounded_linear "(\<lambda>y::'a::real_normed_algebra. x * y)" |
|
816 by (rule mult.bounded_linear_right) |
|
817 |
|
818 interpretation divide!: |
|
819 bounded_linear "(\<lambda>x::'a::real_normed_field. x / y)" |
|
820 unfolding divide_inverse by (rule mult.bounded_linear_left) |
|
821 |
|
822 interpretation scaleR!: bounded_bilinear "scaleR" |
|
823 apply (rule bounded_bilinear.intro) |
|
824 apply (rule scaleR_left_distrib) |
|
825 apply (rule scaleR_right_distrib) |
|
826 apply simp |
|
827 apply (rule scaleR_left_commute) |
|
828 apply (rule_tac x="1" in exI) |
|
829 apply (simp add: norm_scaleR) |
|
830 done |
|
831 |
|
832 interpretation scaleR_left!: bounded_linear "\<lambda>r. scaleR r x" |
|
833 by (rule scaleR.bounded_linear_left) |
|
834 |
|
835 interpretation scaleR_right!: bounded_linear "\<lambda>x. scaleR r x" |
|
836 by (rule scaleR.bounded_linear_right) |
|
837 |
|
838 interpretation of_real!: bounded_linear "\<lambda>r. of_real r" |
|
839 unfolding of_real_def by (rule scaleR.bounded_linear_left) |
|
840 |
|
841 end |
|