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1 (* Title: Complex.thy |
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2 Author: Jacques D. Fleuriot |
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3 Copyright: 2001 University of Edinburgh |
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4 Conversion to Isar and new proofs by Lawrence C Paulson, 2003/4 |
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5 *) |
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6 |
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7 header {* Complex Numbers: Rectangular and Polar Representations *} |
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8 |
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9 theory Complex |
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10 imports Transcendental |
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11 begin |
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12 |
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13 datatype complex = Complex real real |
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14 |
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15 primrec |
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16 Re :: "complex \<Rightarrow> real" |
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17 where |
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18 Re: "Re (Complex x y) = x" |
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19 |
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20 primrec |
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21 Im :: "complex \<Rightarrow> real" |
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22 where |
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23 Im: "Im (Complex x y) = y" |
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24 |
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25 lemma complex_surj [simp]: "Complex (Re z) (Im z) = z" |
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26 by (induct z) simp |
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27 |
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28 lemma complex_equality [intro?]: "\<lbrakk>Re x = Re y; Im x = Im y\<rbrakk> \<Longrightarrow> x = y" |
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29 by (induct x, induct y) simp |
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30 |
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31 lemma expand_complex_eq: "x = y \<longleftrightarrow> Re x = Re y \<and> Im x = Im y" |
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32 by (induct x, induct y) simp |
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33 |
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34 lemmas complex_Re_Im_cancel_iff = expand_complex_eq |
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35 |
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36 |
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37 subsection {* Addition and Subtraction *} |
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38 |
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39 instantiation complex :: ab_group_add |
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40 begin |
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41 |
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42 definition |
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43 complex_zero_def: "0 = Complex 0 0" |
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44 |
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45 definition |
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46 complex_add_def: "x + y = Complex (Re x + Re y) (Im x + Im y)" |
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47 |
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48 definition |
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49 complex_minus_def: "- x = Complex (- Re x) (- Im x)" |
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50 |
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51 definition |
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52 complex_diff_def: "x - (y\<Colon>complex) = x + - y" |
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53 |
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54 lemma Complex_eq_0 [simp]: "Complex a b = 0 \<longleftrightarrow> a = 0 \<and> b = 0" |
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55 by (simp add: complex_zero_def) |
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56 |
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57 lemma complex_Re_zero [simp]: "Re 0 = 0" |
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58 by (simp add: complex_zero_def) |
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59 |
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60 lemma complex_Im_zero [simp]: "Im 0 = 0" |
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61 by (simp add: complex_zero_def) |
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62 |
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63 lemma complex_add [simp]: |
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64 "Complex a b + Complex c d = Complex (a + c) (b + d)" |
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65 by (simp add: complex_add_def) |
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66 |
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67 lemma complex_Re_add [simp]: "Re (x + y) = Re x + Re y" |
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68 by (simp add: complex_add_def) |
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69 |
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70 lemma complex_Im_add [simp]: "Im (x + y) = Im x + Im y" |
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71 by (simp add: complex_add_def) |
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72 |
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73 lemma complex_minus [simp]: |
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74 "- (Complex a b) = Complex (- a) (- b)" |
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75 by (simp add: complex_minus_def) |
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76 |
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77 lemma complex_Re_minus [simp]: "Re (- x) = - Re x" |
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78 by (simp add: complex_minus_def) |
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79 |
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80 lemma complex_Im_minus [simp]: "Im (- x) = - Im x" |
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81 by (simp add: complex_minus_def) |
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82 |
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83 lemma complex_diff [simp]: |
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84 "Complex a b - Complex c d = Complex (a - c) (b - d)" |
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85 by (simp add: complex_diff_def) |
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86 |
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87 lemma complex_Re_diff [simp]: "Re (x - y) = Re x - Re y" |
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88 by (simp add: complex_diff_def) |
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89 |
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90 lemma complex_Im_diff [simp]: "Im (x - y) = Im x - Im y" |
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91 by (simp add: complex_diff_def) |
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92 |
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93 instance |
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94 by intro_classes (simp_all add: complex_add_def complex_diff_def) |
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95 |
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96 end |
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97 |
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98 |
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99 |
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100 subsection {* Multiplication and Division *} |
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101 |
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102 instantiation complex :: "{field, division_by_zero}" |
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103 begin |
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104 |
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105 definition |
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106 complex_one_def: "1 = Complex 1 0" |
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107 |
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108 definition |
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109 complex_mult_def: "x * y = |
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110 Complex (Re x * Re y - Im x * Im y) (Re x * Im y + Im x * Re y)" |
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111 |
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112 definition |
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113 complex_inverse_def: "inverse x = |
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114 Complex (Re x / ((Re x)\<twosuperior> + (Im x)\<twosuperior>)) (- Im x / ((Re x)\<twosuperior> + (Im x)\<twosuperior>))" |
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115 |
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116 definition |
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117 complex_divide_def: "x / (y\<Colon>complex) = x * inverse y" |
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118 |
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119 lemma Complex_eq_1 [simp]: "(Complex a b = 1) = (a = 1 \<and> b = 0)" |
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120 by (simp add: complex_one_def) |
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121 |
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122 lemma complex_Re_one [simp]: "Re 1 = 1" |
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123 by (simp add: complex_one_def) |
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124 |
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125 lemma complex_Im_one [simp]: "Im 1 = 0" |
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126 by (simp add: complex_one_def) |
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127 |
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128 lemma complex_mult [simp]: |
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129 "Complex a b * Complex c d = Complex (a * c - b * d) (a * d + b * c)" |
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130 by (simp add: complex_mult_def) |
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131 |
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132 lemma complex_Re_mult [simp]: "Re (x * y) = Re x * Re y - Im x * Im y" |
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133 by (simp add: complex_mult_def) |
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134 |
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135 lemma complex_Im_mult [simp]: "Im (x * y) = Re x * Im y + Im x * Re y" |
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136 by (simp add: complex_mult_def) |
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137 |
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138 lemma complex_inverse [simp]: |
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139 "inverse (Complex a b) = Complex (a / (a\<twosuperior> + b\<twosuperior>)) (- b / (a\<twosuperior> + b\<twosuperior>))" |
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140 by (simp add: complex_inverse_def) |
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141 |
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142 lemma complex_Re_inverse: |
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143 "Re (inverse x) = Re x / ((Re x)\<twosuperior> + (Im x)\<twosuperior>)" |
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144 by (simp add: complex_inverse_def) |
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145 |
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146 lemma complex_Im_inverse: |
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147 "Im (inverse x) = - Im x / ((Re x)\<twosuperior> + (Im x)\<twosuperior>)" |
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148 by (simp add: complex_inverse_def) |
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149 |
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150 instance |
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151 by intro_classes (simp_all add: complex_mult_def |
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152 right_distrib left_distrib right_diff_distrib left_diff_distrib |
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153 complex_inverse_def complex_divide_def |
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154 power2_eq_square add_divide_distrib [symmetric] |
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155 expand_complex_eq) |
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156 |
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157 end |
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158 |
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159 |
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160 subsection {* Exponentiation *} |
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161 |
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162 instantiation complex :: recpower |
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163 begin |
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164 |
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165 primrec power_complex where |
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166 complexpow_0: "z ^ 0 = (1\<Colon>complex)" |
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167 | complexpow_Suc: "z ^ Suc n = (z\<Colon>complex) * z ^ n" |
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168 |
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169 instance by intro_classes simp_all |
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170 |
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171 end |
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172 |
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173 |
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174 subsection {* Numerals and Arithmetic *} |
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175 |
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176 instantiation complex :: number_ring |
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177 begin |
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178 |
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179 definition number_of_complex where |
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180 complex_number_of_def: "number_of w = (of_int w \<Colon> complex)" |
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181 |
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182 instance |
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183 by intro_classes (simp only: complex_number_of_def) |
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184 |
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185 end |
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186 |
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187 lemma complex_Re_of_nat [simp]: "Re (of_nat n) = of_nat n" |
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188 by (induct n) simp_all |
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189 |
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190 lemma complex_Im_of_nat [simp]: "Im (of_nat n) = 0" |
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191 by (induct n) simp_all |
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192 |
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193 lemma complex_Re_of_int [simp]: "Re (of_int z) = of_int z" |
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194 by (cases z rule: int_diff_cases) simp |
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195 |
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196 lemma complex_Im_of_int [simp]: "Im (of_int z) = 0" |
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197 by (cases z rule: int_diff_cases) simp |
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198 |
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199 lemma complex_Re_number_of [simp]: "Re (number_of v) = number_of v" |
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200 unfolding number_of_eq by (rule complex_Re_of_int) |
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201 |
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202 lemma complex_Im_number_of [simp]: "Im (number_of v) = 0" |
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203 unfolding number_of_eq by (rule complex_Im_of_int) |
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204 |
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205 lemma Complex_eq_number_of [simp]: |
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206 "(Complex a b = number_of w) = (a = number_of w \<and> b = 0)" |
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207 by (simp add: expand_complex_eq) |
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208 |
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209 |
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210 subsection {* Scalar Multiplication *} |
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211 |
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212 instantiation complex :: real_field |
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213 begin |
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214 |
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215 definition |
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216 complex_scaleR_def: "scaleR r x = Complex (r * Re x) (r * Im x)" |
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217 |
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218 lemma complex_scaleR [simp]: |
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219 "scaleR r (Complex a b) = Complex (r * a) (r * b)" |
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220 unfolding complex_scaleR_def by simp |
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221 |
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222 lemma complex_Re_scaleR [simp]: "Re (scaleR r x) = r * Re x" |
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223 unfolding complex_scaleR_def by simp |
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224 |
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225 lemma complex_Im_scaleR [simp]: "Im (scaleR r x) = r * Im x" |
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226 unfolding complex_scaleR_def by simp |
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227 |
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228 instance |
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229 proof |
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230 fix a b :: real and x y :: complex |
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231 show "scaleR a (x + y) = scaleR a x + scaleR a y" |
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232 by (simp add: expand_complex_eq right_distrib) |
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233 show "scaleR (a + b) x = scaleR a x + scaleR b x" |
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234 by (simp add: expand_complex_eq left_distrib) |
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235 show "scaleR a (scaleR b x) = scaleR (a * b) x" |
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236 by (simp add: expand_complex_eq mult_assoc) |
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237 show "scaleR 1 x = x" |
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238 by (simp add: expand_complex_eq) |
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239 show "scaleR a x * y = scaleR a (x * y)" |
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240 by (simp add: expand_complex_eq ring_simps) |
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241 show "x * scaleR a y = scaleR a (x * y)" |
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242 by (simp add: expand_complex_eq ring_simps) |
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243 qed |
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244 |
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245 end |
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246 |
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247 |
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248 subsection{* Properties of Embedding from Reals *} |
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249 |
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250 abbreviation |
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251 complex_of_real :: "real \<Rightarrow> complex" where |
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252 "complex_of_real \<equiv> of_real" |
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253 |
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254 lemma complex_of_real_def: "complex_of_real r = Complex r 0" |
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255 by (simp add: of_real_def complex_scaleR_def) |
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256 |
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257 lemma Re_complex_of_real [simp]: "Re (complex_of_real z) = z" |
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258 by (simp add: complex_of_real_def) |
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259 |
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260 lemma Im_complex_of_real [simp]: "Im (complex_of_real z) = 0" |
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261 by (simp add: complex_of_real_def) |
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262 |
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263 lemma Complex_add_complex_of_real [simp]: |
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264 "Complex x y + complex_of_real r = Complex (x+r) y" |
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265 by (simp add: complex_of_real_def) |
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266 |
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267 lemma complex_of_real_add_Complex [simp]: |
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268 "complex_of_real r + Complex x y = Complex (r+x) y" |
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269 by (simp add: complex_of_real_def) |
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270 |
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271 lemma Complex_mult_complex_of_real: |
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272 "Complex x y * complex_of_real r = Complex (x*r) (y*r)" |
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273 by (simp add: complex_of_real_def) |
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274 |
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275 lemma complex_of_real_mult_Complex: |
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276 "complex_of_real r * Complex x y = Complex (r*x) (r*y)" |
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277 by (simp add: complex_of_real_def) |
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278 |
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279 |
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280 subsection {* Vector Norm *} |
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281 |
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282 instantiation complex :: real_normed_field |
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283 begin |
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284 |
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285 definition |
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286 complex_norm_def: "norm z = sqrt ((Re z)\<twosuperior> + (Im z)\<twosuperior>)" |
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287 |
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288 abbreviation |
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289 cmod :: "complex \<Rightarrow> real" where |
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290 "cmod \<equiv> norm" |
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291 |
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292 definition |
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293 complex_sgn_def: "sgn x = x /\<^sub>R cmod x" |
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294 |
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295 lemmas cmod_def = complex_norm_def |
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296 |
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297 lemma complex_norm [simp]: "cmod (Complex x y) = sqrt (x\<twosuperior> + y\<twosuperior>)" |
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298 by (simp add: complex_norm_def) |
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299 |
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300 instance |
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301 proof |
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302 fix r :: real and x y :: complex |
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303 show "0 \<le> norm x" |
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304 by (induct x) simp |
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305 show "(norm x = 0) = (x = 0)" |
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306 by (induct x) simp |
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307 show "norm (x + y) \<le> norm x + norm y" |
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308 by (induct x, induct y) |
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309 (simp add: real_sqrt_sum_squares_triangle_ineq) |
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310 show "norm (scaleR r x) = \<bar>r\<bar> * norm x" |
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311 by (induct x) |
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312 (simp add: power_mult_distrib right_distrib [symmetric] real_sqrt_mult) |
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313 show "norm (x * y) = norm x * norm y" |
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314 by (induct x, induct y) |
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315 (simp add: real_sqrt_mult [symmetric] power2_eq_square ring_simps) |
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316 show "sgn x = x /\<^sub>R cmod x" by(simp add: complex_sgn_def) |
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317 qed |
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318 |
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319 end |
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320 |
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321 lemma cmod_unit_one [simp]: "cmod (Complex (cos a) (sin a)) = 1" |
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322 by simp |
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323 |
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324 lemma cmod_complex_polar [simp]: |
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325 "cmod (complex_of_real r * Complex (cos a) (sin a)) = abs r" |
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326 by (simp add: norm_mult) |
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327 |
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328 lemma complex_Re_le_cmod: "Re x \<le> cmod x" |
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329 unfolding complex_norm_def |
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330 by (rule real_sqrt_sum_squares_ge1) |
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331 |
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332 lemma complex_mod_minus_le_complex_mod [simp]: "- cmod x \<le> cmod x" |
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333 by (rule order_trans [OF _ norm_ge_zero], simp) |
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334 |
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335 lemma complex_mod_triangle_ineq2 [simp]: "cmod(b + a) - cmod b \<le> cmod a" |
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336 by (rule ord_le_eq_trans [OF norm_triangle_ineq2], simp) |
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337 |
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338 lemmas real_sum_squared_expand = power2_sum [where 'a=real] |
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339 |
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340 lemma abs_Re_le_cmod: "\<bar>Re x\<bar> \<le> cmod x" |
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341 by (cases x) simp |
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342 |
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343 lemma abs_Im_le_cmod: "\<bar>Im x\<bar> \<le> cmod x" |
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344 by (cases x) simp |
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345 |
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346 subsection {* Completeness of the Complexes *} |
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347 |
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348 interpretation Re: bounded_linear ["Re"] |
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349 apply (unfold_locales, simp, simp) |
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350 apply (rule_tac x=1 in exI) |
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351 apply (simp add: complex_norm_def) |
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352 done |
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353 |
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354 interpretation Im: bounded_linear ["Im"] |
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355 apply (unfold_locales, simp, simp) |
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356 apply (rule_tac x=1 in exI) |
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357 apply (simp add: complex_norm_def) |
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358 done |
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359 |
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360 lemma LIMSEQ_Complex: |
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361 "\<lbrakk>X ----> a; Y ----> b\<rbrakk> \<Longrightarrow> (\<lambda>n. Complex (X n) (Y n)) ----> Complex a b" |
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362 apply (rule LIMSEQ_I) |
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363 apply (subgoal_tac "0 < r / sqrt 2") |
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364 apply (drule_tac r="r / sqrt 2" in LIMSEQ_D, safe) |
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365 apply (drule_tac r="r / sqrt 2" in LIMSEQ_D, safe) |
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366 apply (rename_tac M N, rule_tac x="max M N" in exI, safe) |
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367 apply (simp add: real_sqrt_sum_squares_less) |
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368 apply (simp add: divide_pos_pos) |
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369 done |
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370 |
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371 instance complex :: banach |
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372 proof |
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373 fix X :: "nat \<Rightarrow> complex" |
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374 assume X: "Cauchy X" |
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375 from Re.Cauchy [OF X] have 1: "(\<lambda>n. Re (X n)) ----> lim (\<lambda>n. Re (X n))" |
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376 by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff) |
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377 from Im.Cauchy [OF X] have 2: "(\<lambda>n. Im (X n)) ----> lim (\<lambda>n. Im (X n))" |
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378 by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff) |
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379 have "X ----> Complex (lim (\<lambda>n. Re (X n))) (lim (\<lambda>n. Im (X n)))" |
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380 using LIMSEQ_Complex [OF 1 2] by simp |
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381 thus "convergent X" |
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382 by (rule convergentI) |
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383 qed |
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384 |
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385 |
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386 subsection {* The Complex Number @{term "\<i>"} *} |
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387 |
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388 definition |
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389 "ii" :: complex ("\<i>") where |
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390 i_def: "ii \<equiv> Complex 0 1" |
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391 |
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392 lemma complex_Re_i [simp]: "Re ii = 0" |
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393 by (simp add: i_def) |
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394 |
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395 lemma complex_Im_i [simp]: "Im ii = 1" |
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396 by (simp add: i_def) |
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397 |
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398 lemma Complex_eq_i [simp]: "(Complex x y = ii) = (x = 0 \<and> y = 1)" |
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399 by (simp add: i_def) |
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400 |
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401 lemma complex_i_not_zero [simp]: "ii \<noteq> 0" |
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402 by (simp add: expand_complex_eq) |
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403 |
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404 lemma complex_i_not_one [simp]: "ii \<noteq> 1" |
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405 by (simp add: expand_complex_eq) |
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406 |
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407 lemma complex_i_not_number_of [simp]: "ii \<noteq> number_of w" |
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408 by (simp add: expand_complex_eq) |
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409 |
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410 lemma i_mult_Complex [simp]: "ii * Complex a b = Complex (- b) a" |
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411 by (simp add: expand_complex_eq) |
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412 |
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413 lemma Complex_mult_i [simp]: "Complex a b * ii = Complex (- b) a" |
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414 by (simp add: expand_complex_eq) |
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415 |
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416 lemma i_complex_of_real [simp]: "ii * complex_of_real r = Complex 0 r" |
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417 by (simp add: i_def complex_of_real_def) |
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418 |
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419 lemma complex_of_real_i [simp]: "complex_of_real r * ii = Complex 0 r" |
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420 by (simp add: i_def complex_of_real_def) |
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421 |
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422 lemma i_squared [simp]: "ii * ii = -1" |
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423 by (simp add: i_def) |
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424 |
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425 lemma power2_i [simp]: "ii\<twosuperior> = -1" |
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426 by (simp add: power2_eq_square) |
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427 |
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428 lemma inverse_i [simp]: "inverse ii = - ii" |
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429 by (rule inverse_unique, simp) |
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430 |
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431 |
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432 subsection {* Complex Conjugation *} |
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433 |
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434 definition |
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435 cnj :: "complex \<Rightarrow> complex" where |
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436 "cnj z = Complex (Re z) (- Im z)" |
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437 |
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438 lemma complex_cnj [simp]: "cnj (Complex a b) = Complex a (- b)" |
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439 by (simp add: cnj_def) |
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440 |
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441 lemma complex_Re_cnj [simp]: "Re (cnj x) = Re x" |
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442 by (simp add: cnj_def) |
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443 |
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444 lemma complex_Im_cnj [simp]: "Im (cnj x) = - Im x" |
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445 by (simp add: cnj_def) |
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446 |
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447 lemma complex_cnj_cancel_iff [simp]: "(cnj x = cnj y) = (x = y)" |
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448 by (simp add: expand_complex_eq) |
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449 |
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450 lemma complex_cnj_cnj [simp]: "cnj (cnj z) = z" |
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451 by (simp add: cnj_def) |
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452 |
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453 lemma complex_cnj_zero [simp]: "cnj 0 = 0" |
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454 by (simp add: expand_complex_eq) |
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455 |
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456 lemma complex_cnj_zero_iff [iff]: "(cnj z = 0) = (z = 0)" |
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457 by (simp add: expand_complex_eq) |
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458 |
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459 lemma complex_cnj_add: "cnj (x + y) = cnj x + cnj y" |
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460 by (simp add: expand_complex_eq) |
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461 |
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462 lemma complex_cnj_diff: "cnj (x - y) = cnj x - cnj y" |
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463 by (simp add: expand_complex_eq) |
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464 |
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465 lemma complex_cnj_minus: "cnj (- x) = - cnj x" |
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466 by (simp add: expand_complex_eq) |
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467 |
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468 lemma complex_cnj_one [simp]: "cnj 1 = 1" |
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469 by (simp add: expand_complex_eq) |
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470 |
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471 lemma complex_cnj_mult: "cnj (x * y) = cnj x * cnj y" |
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472 by (simp add: expand_complex_eq) |
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473 |
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474 lemma complex_cnj_inverse: "cnj (inverse x) = inverse (cnj x)" |
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475 by (simp add: complex_inverse_def) |
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476 |
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477 lemma complex_cnj_divide: "cnj (x / y) = cnj x / cnj y" |
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478 by (simp add: complex_divide_def complex_cnj_mult complex_cnj_inverse) |
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479 |
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480 lemma complex_cnj_power: "cnj (x ^ n) = cnj x ^ n" |
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481 by (induct n, simp_all add: complex_cnj_mult) |
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482 |
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483 lemma complex_cnj_of_nat [simp]: "cnj (of_nat n) = of_nat n" |
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484 by (simp add: expand_complex_eq) |
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485 |
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486 lemma complex_cnj_of_int [simp]: "cnj (of_int z) = of_int z" |
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487 by (simp add: expand_complex_eq) |
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488 |
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489 lemma complex_cnj_number_of [simp]: "cnj (number_of w) = number_of w" |
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490 by (simp add: expand_complex_eq) |
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491 |
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492 lemma complex_cnj_scaleR: "cnj (scaleR r x) = scaleR r (cnj x)" |
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493 by (simp add: expand_complex_eq) |
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494 |
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495 lemma complex_mod_cnj [simp]: "cmod (cnj z) = cmod z" |
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496 by (simp add: complex_norm_def) |
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497 |
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498 lemma complex_cnj_complex_of_real [simp]: "cnj (of_real x) = of_real x" |
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499 by (simp add: expand_complex_eq) |
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500 |
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501 lemma complex_cnj_i [simp]: "cnj ii = - ii" |
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502 by (simp add: expand_complex_eq) |
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503 |
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504 lemma complex_add_cnj: "z + cnj z = complex_of_real (2 * Re z)" |
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505 by (simp add: expand_complex_eq) |
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506 |
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507 lemma complex_diff_cnj: "z - cnj z = complex_of_real (2 * Im z) * ii" |
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508 by (simp add: expand_complex_eq) |
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509 |
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510 lemma complex_mult_cnj: "z * cnj z = complex_of_real ((Re z)\<twosuperior> + (Im z)\<twosuperior>)" |
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511 by (simp add: expand_complex_eq power2_eq_square) |
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512 |
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513 lemma complex_mod_mult_cnj: "cmod (z * cnj z) = (cmod z)\<twosuperior>" |
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514 by (simp add: norm_mult power2_eq_square) |
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515 |
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516 interpretation cnj: bounded_linear ["cnj"] |
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517 apply (unfold_locales) |
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518 apply (rule complex_cnj_add) |
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519 apply (rule complex_cnj_scaleR) |
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520 apply (rule_tac x=1 in exI, simp) |
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521 done |
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522 |
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523 |
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524 subsection{*The Functions @{term sgn} and @{term arg}*} |
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525 |
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526 text {*------------ Argand -------------*} |
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527 |
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528 definition |
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529 arg :: "complex => real" where |
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530 "arg z = (SOME a. Re(sgn z) = cos a & Im(sgn z) = sin a & -pi < a & a \<le> pi)" |
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531 |
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532 lemma sgn_eq: "sgn z = z / complex_of_real (cmod z)" |
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533 by (simp add: complex_sgn_def divide_inverse scaleR_conv_of_real mult_commute) |
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534 |
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535 lemma i_mult_eq: "ii * ii = complex_of_real (-1)" |
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536 by (simp add: i_def complex_of_real_def) |
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537 |
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538 lemma i_mult_eq2 [simp]: "ii * ii = -(1::complex)" |
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539 by (simp add: i_def complex_one_def) |
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540 |
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541 lemma complex_eq_cancel_iff2 [simp]: |
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542 "(Complex x y = complex_of_real xa) = (x = xa & y = 0)" |
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543 by (simp add: complex_of_real_def) |
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544 |
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545 lemma Re_sgn [simp]: "Re(sgn z) = Re(z)/cmod z" |
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546 by (simp add: complex_sgn_def divide_inverse) |
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547 |
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548 lemma Im_sgn [simp]: "Im(sgn z) = Im(z)/cmod z" |
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549 by (simp add: complex_sgn_def divide_inverse) |
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550 |
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551 lemma complex_inverse_complex_split: |
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552 "inverse(complex_of_real x + ii * complex_of_real y) = |
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553 complex_of_real(x/(x ^ 2 + y ^ 2)) - |
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554 ii * complex_of_real(y/(x ^ 2 + y ^ 2))" |
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555 by (simp add: complex_of_real_def i_def diff_minus divide_inverse) |
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556 |
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557 (*----------------------------------------------------------------------------*) |
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558 (* Many of the theorems below need to be moved elsewhere e.g. Transc. Also *) |
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559 (* many of the theorems are not used - so should they be kept? *) |
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560 (*----------------------------------------------------------------------------*) |
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561 |
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562 lemma cos_arg_i_mult_zero_pos: |
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563 "0 < y ==> cos (arg(Complex 0 y)) = 0" |
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564 apply (simp add: arg_def abs_if) |
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565 apply (rule_tac a = "pi/2" in someI2, auto) |
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566 apply (rule order_less_trans [of _ 0], auto) |
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567 done |
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568 |
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569 lemma cos_arg_i_mult_zero_neg: |
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570 "y < 0 ==> cos (arg(Complex 0 y)) = 0" |
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571 apply (simp add: arg_def abs_if) |
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572 apply (rule_tac a = "- pi/2" in someI2, auto) |
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573 apply (rule order_trans [of _ 0], auto) |
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574 done |
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575 |
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576 lemma cos_arg_i_mult_zero [simp]: |
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577 "y \<noteq> 0 ==> cos (arg(Complex 0 y)) = 0" |
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578 by (auto simp add: linorder_neq_iff cos_arg_i_mult_zero_pos cos_arg_i_mult_zero_neg) |
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579 |
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580 |
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581 subsection{*Finally! Polar Form for Complex Numbers*} |
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582 |
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583 definition |
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584 |
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585 (* abbreviation for (cos a + i sin a) *) |
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586 cis :: "real => complex" where |
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587 "cis a = Complex (cos a) (sin a)" |
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588 |
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589 definition |
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590 (* abbreviation for r*(cos a + i sin a) *) |
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591 rcis :: "[real, real] => complex" where |
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592 "rcis r a = complex_of_real r * cis a" |
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593 |
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594 definition |
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595 (* e ^ (x + iy) *) |
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596 expi :: "complex => complex" where |
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597 "expi z = complex_of_real(exp (Re z)) * cis (Im z)" |
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598 |
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599 lemma complex_split_polar: |
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600 "\<exists>r a. z = complex_of_real r * (Complex (cos a) (sin a))" |
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601 apply (induct z) |
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602 apply (auto simp add: polar_Ex complex_of_real_mult_Complex) |
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603 done |
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604 |
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605 lemma rcis_Ex: "\<exists>r a. z = rcis r a" |
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606 apply (induct z) |
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607 apply (simp add: rcis_def cis_def polar_Ex complex_of_real_mult_Complex) |
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608 done |
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609 |
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610 lemma Re_rcis [simp]: "Re(rcis r a) = r * cos a" |
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611 by (simp add: rcis_def cis_def) |
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612 |
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613 lemma Im_rcis [simp]: "Im(rcis r a) = r * sin a" |
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614 by (simp add: rcis_def cis_def) |
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615 |
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616 lemma sin_cos_squared_add2_mult: "(r * cos a)\<twosuperior> + (r * sin a)\<twosuperior> = r\<twosuperior>" |
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617 proof - |
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618 have "(r * cos a)\<twosuperior> + (r * sin a)\<twosuperior> = r\<twosuperior> * ((cos a)\<twosuperior> + (sin a)\<twosuperior>)" |
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619 by (simp only: power_mult_distrib right_distrib) |
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620 thus ?thesis by simp |
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621 qed |
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622 |
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623 lemma complex_mod_rcis [simp]: "cmod(rcis r a) = abs r" |
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624 by (simp add: rcis_def cis_def sin_cos_squared_add2_mult) |
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625 |
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626 lemma complex_mod_sqrt_Re_mult_cnj: "cmod z = sqrt (Re (z * cnj z))" |
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627 by (simp add: cmod_def power2_eq_square) |
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628 |
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629 lemma complex_In_mult_cnj_zero [simp]: "Im (z * cnj z) = 0" |
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630 by simp |
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631 |
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632 |
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633 (*---------------------------------------------------------------------------*) |
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634 (* (r1 * cis a) * (r2 * cis b) = r1 * r2 * cis (a + b) *) |
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635 (*---------------------------------------------------------------------------*) |
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636 |
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637 lemma cis_rcis_eq: "cis a = rcis 1 a" |
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638 by (simp add: rcis_def) |
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639 |
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640 lemma rcis_mult: "rcis r1 a * rcis r2 b = rcis (r1*r2) (a + b)" |
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641 by (simp add: rcis_def cis_def cos_add sin_add right_distrib right_diff_distrib |
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642 complex_of_real_def) |
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643 |
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644 lemma cis_mult: "cis a * cis b = cis (a + b)" |
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645 by (simp add: cis_rcis_eq rcis_mult) |
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646 |
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647 lemma cis_zero [simp]: "cis 0 = 1" |
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648 by (simp add: cis_def complex_one_def) |
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649 |
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650 lemma rcis_zero_mod [simp]: "rcis 0 a = 0" |
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651 by (simp add: rcis_def) |
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652 |
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653 lemma rcis_zero_arg [simp]: "rcis r 0 = complex_of_real r" |
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654 by (simp add: rcis_def) |
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655 |
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656 lemma complex_of_real_minus_one: |
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657 "complex_of_real (-(1::real)) = -(1::complex)" |
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658 by (simp add: complex_of_real_def complex_one_def) |
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659 |
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660 lemma complex_i_mult_minus [simp]: "ii * (ii * x) = - x" |
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661 by (simp add: mult_assoc [symmetric]) |
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662 |
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663 |
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664 lemma cis_real_of_nat_Suc_mult: |
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665 "cis (real (Suc n) * a) = cis a * cis (real n * a)" |
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666 by (simp add: cis_def real_of_nat_Suc left_distrib cos_add sin_add right_distrib) |
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667 |
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668 lemma DeMoivre: "(cis a) ^ n = cis (real n * a)" |
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669 apply (induct_tac "n") |
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670 apply (auto simp add: cis_real_of_nat_Suc_mult) |
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671 done |
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672 |
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673 lemma DeMoivre2: "(rcis r a) ^ n = rcis (r ^ n) (real n * a)" |
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674 by (simp add: rcis_def power_mult_distrib DeMoivre) |
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675 |
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676 lemma cis_inverse [simp]: "inverse(cis a) = cis (-a)" |
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677 by (simp add: cis_def complex_inverse_complex_split diff_minus) |
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678 |
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679 lemma rcis_inverse: "inverse(rcis r a) = rcis (1/r) (-a)" |
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680 by (simp add: divide_inverse rcis_def) |
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681 |
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682 lemma cis_divide: "cis a / cis b = cis (a - b)" |
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683 by (simp add: complex_divide_def cis_mult real_diff_def) |
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684 |
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685 lemma rcis_divide: "rcis r1 a / rcis r2 b = rcis (r1/r2) (a - b)" |
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686 apply (simp add: complex_divide_def) |
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687 apply (case_tac "r2=0", simp) |
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688 apply (simp add: rcis_inverse rcis_mult real_diff_def) |
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689 done |
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690 |
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691 lemma Re_cis [simp]: "Re(cis a) = cos a" |
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692 by (simp add: cis_def) |
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693 |
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694 lemma Im_cis [simp]: "Im(cis a) = sin a" |
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695 by (simp add: cis_def) |
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696 |
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697 lemma cos_n_Re_cis_pow_n: "cos (real n * a) = Re(cis a ^ n)" |
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698 by (auto simp add: DeMoivre) |
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699 |
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700 lemma sin_n_Im_cis_pow_n: "sin (real n * a) = Im(cis a ^ n)" |
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701 by (auto simp add: DeMoivre) |
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702 |
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703 lemma expi_add: "expi(a + b) = expi(a) * expi(b)" |
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704 by (simp add: expi_def exp_add cis_mult [symmetric] mult_ac) |
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705 |
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706 lemma expi_zero [simp]: "expi (0::complex) = 1" |
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707 by (simp add: expi_def) |
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708 |
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709 lemma complex_expi_Ex: "\<exists>a r. z = complex_of_real r * expi a" |
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710 apply (insert rcis_Ex [of z]) |
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711 apply (auto simp add: expi_def rcis_def mult_assoc [symmetric]) |
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712 apply (rule_tac x = "ii * complex_of_real a" in exI, auto) |
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713 done |
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714 |
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715 lemma expi_two_pi_i [simp]: "expi((2::complex) * complex_of_real pi * ii) = 1" |
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716 by (simp add: expi_def cis_def) |
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717 |
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718 end |