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1 (* Title : NthRoot.thy |
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2 Author : Jacques D. Fleuriot |
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3 Copyright : 1998 University of Cambridge |
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4 Conversion to Isar and new proofs by Lawrence C Paulson, 2004 |
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5 *) |
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6 |
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7 header {* Nth Roots of Real Numbers *} |
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8 |
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9 theory NthRoot |
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10 imports Parity Deriv |
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11 begin |
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12 |
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13 subsection {* Existence of Nth Root *} |
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14 |
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15 text {* Existence follows from the Intermediate Value Theorem *} |
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16 |
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17 lemma realpow_pos_nth: |
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18 assumes n: "0 < n" |
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19 assumes a: "0 < a" |
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20 shows "\<exists>r>0. r ^ n = (a::real)" |
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21 proof - |
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22 have "\<exists>r\<ge>0. r \<le> (max 1 a) \<and> r ^ n = a" |
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23 proof (rule IVT) |
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24 show "0 ^ n \<le> a" using n a by (simp add: power_0_left) |
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25 show "0 \<le> max 1 a" by simp |
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26 from n have n1: "1 \<le> n" by simp |
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27 have "a \<le> max 1 a ^ 1" by simp |
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28 also have "max 1 a ^ 1 \<le> max 1 a ^ n" |
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29 using n1 by (rule power_increasing, simp) |
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30 finally show "a \<le> max 1 a ^ n" . |
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31 show "\<forall>r. 0 \<le> r \<and> r \<le> max 1 a \<longrightarrow> isCont (\<lambda>x. x ^ n) r" |
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32 by (simp add: isCont_power) |
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33 qed |
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34 then obtain r where r: "0 \<le> r \<and> r ^ n = a" by fast |
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35 with n a have "r \<noteq> 0" by (auto simp add: power_0_left) |
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36 with r have "0 < r \<and> r ^ n = a" by simp |
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37 thus ?thesis .. |
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38 qed |
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39 |
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40 (* Used by Integration/RealRandVar.thy in AFP *) |
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41 lemma realpow_pos_nth2: "(0::real) < a \<Longrightarrow> \<exists>r>0. r ^ Suc n = a" |
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42 by (blast intro: realpow_pos_nth) |
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43 |
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44 text {* Uniqueness of nth positive root *} |
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45 |
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46 lemma realpow_pos_nth_unique: |
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47 "\<lbrakk>0 < n; 0 < a\<rbrakk> \<Longrightarrow> \<exists>!r. 0 < r \<and> r ^ n = (a::real)" |
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48 apply (auto intro!: realpow_pos_nth) |
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49 apply (rule_tac n=n in power_eq_imp_eq_base, simp_all) |
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50 done |
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51 |
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52 subsection {* Nth Root *} |
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53 |
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54 text {* We define roots of negative reals such that |
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55 @{term "root n (- x) = - root n x"}. This allows |
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56 us to omit side conditions from many theorems. *} |
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57 |
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58 definition |
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59 root :: "[nat, real] \<Rightarrow> real" where |
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60 "root n x = (if 0 < x then (THE u. 0 < u \<and> u ^ n = x) else |
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61 if x < 0 then - (THE u. 0 < u \<and> u ^ n = - x) else 0)" |
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62 |
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63 lemma real_root_zero [simp]: "root n 0 = 0" |
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64 unfolding root_def by simp |
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65 |
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66 lemma real_root_minus: "0 < n \<Longrightarrow> root n (- x) = - root n x" |
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67 unfolding root_def by simp |
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68 |
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69 lemma real_root_gt_zero: "\<lbrakk>0 < n; 0 < x\<rbrakk> \<Longrightarrow> 0 < root n x" |
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70 apply (simp add: root_def) |
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71 apply (drule (1) realpow_pos_nth_unique) |
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72 apply (erule theI' [THEN conjunct1]) |
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73 done |
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74 |
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75 lemma real_root_pow_pos: (* TODO: rename *) |
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76 "\<lbrakk>0 < n; 0 < x\<rbrakk> \<Longrightarrow> root n x ^ n = x" |
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77 apply (simp add: root_def) |
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78 apply (drule (1) realpow_pos_nth_unique) |
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79 apply (erule theI' [THEN conjunct2]) |
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80 done |
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81 |
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82 lemma real_root_pow_pos2 [simp]: (* TODO: rename *) |
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83 "\<lbrakk>0 < n; 0 \<le> x\<rbrakk> \<Longrightarrow> root n x ^ n = x" |
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84 by (auto simp add: order_le_less real_root_pow_pos) |
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85 |
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86 lemma odd_real_root_pow: "odd n \<Longrightarrow> root n x ^ n = x" |
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87 apply (rule_tac x=0 and y=x in linorder_le_cases) |
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88 apply (erule (1) real_root_pow_pos2 [OF odd_pos]) |
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89 apply (subgoal_tac "root n (- x) ^ n = - x") |
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90 apply (simp add: real_root_minus odd_pos) |
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91 apply (simp add: odd_pos) |
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92 done |
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93 |
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94 lemma real_root_ge_zero: "\<lbrakk>0 < n; 0 \<le> x\<rbrakk> \<Longrightarrow> 0 \<le> root n x" |
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95 by (auto simp add: order_le_less real_root_gt_zero) |
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96 |
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97 lemma real_root_power_cancel: "\<lbrakk>0 < n; 0 \<le> x\<rbrakk> \<Longrightarrow> root n (x ^ n) = x" |
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98 apply (subgoal_tac "0 \<le> x ^ n") |
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99 apply (subgoal_tac "0 \<le> root n (x ^ n)") |
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100 apply (subgoal_tac "root n (x ^ n) ^ n = x ^ n") |
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101 apply (erule (3) power_eq_imp_eq_base) |
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102 apply (erule (1) real_root_pow_pos2) |
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103 apply (erule (1) real_root_ge_zero) |
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104 apply (erule zero_le_power) |
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105 done |
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106 |
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107 lemma odd_real_root_power_cancel: "odd n \<Longrightarrow> root n (x ^ n) = x" |
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108 apply (rule_tac x=0 and y=x in linorder_le_cases) |
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109 apply (erule (1) real_root_power_cancel [OF odd_pos]) |
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110 apply (subgoal_tac "root n ((- x) ^ n) = - x") |
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111 apply (simp add: real_root_minus odd_pos) |
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112 apply (erule real_root_power_cancel [OF odd_pos], simp) |
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113 done |
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114 |
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115 lemma real_root_pos_unique: |
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116 "\<lbrakk>0 < n; 0 \<le> y; y ^ n = x\<rbrakk> \<Longrightarrow> root n x = y" |
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117 by (erule subst, rule real_root_power_cancel) |
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118 |
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119 lemma odd_real_root_unique: |
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120 "\<lbrakk>odd n; y ^ n = x\<rbrakk> \<Longrightarrow> root n x = y" |
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121 by (erule subst, rule odd_real_root_power_cancel) |
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122 |
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123 lemma real_root_one [simp]: "0 < n \<Longrightarrow> root n 1 = 1" |
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124 by (simp add: real_root_pos_unique) |
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125 |
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126 text {* Root function is strictly monotonic, hence injective *} |
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127 |
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128 lemma real_root_less_mono_lemma: |
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129 "\<lbrakk>0 < n; 0 \<le> x; x < y\<rbrakk> \<Longrightarrow> root n x < root n y" |
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130 apply (subgoal_tac "0 \<le> y") |
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131 apply (subgoal_tac "root n x ^ n < root n y ^ n") |
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132 apply (erule power_less_imp_less_base) |
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133 apply (erule (1) real_root_ge_zero) |
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134 apply simp |
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135 apply simp |
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136 done |
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137 |
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138 lemma real_root_less_mono: "\<lbrakk>0 < n; x < y\<rbrakk> \<Longrightarrow> root n x < root n y" |
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139 apply (cases "0 \<le> x") |
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140 apply (erule (2) real_root_less_mono_lemma) |
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141 apply (cases "0 \<le> y") |
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142 apply (rule_tac y=0 in order_less_le_trans) |
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143 apply (subgoal_tac "0 < root n (- x)") |
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144 apply (simp add: real_root_minus) |
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145 apply (simp add: real_root_gt_zero) |
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146 apply (simp add: real_root_ge_zero) |
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147 apply (subgoal_tac "root n (- y) < root n (- x)") |
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148 apply (simp add: real_root_minus) |
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149 apply (simp add: real_root_less_mono_lemma) |
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150 done |
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151 |
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152 lemma real_root_le_mono: "\<lbrakk>0 < n; x \<le> y\<rbrakk> \<Longrightarrow> root n x \<le> root n y" |
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153 by (auto simp add: order_le_less real_root_less_mono) |
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154 |
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155 lemma real_root_less_iff [simp]: |
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156 "0 < n \<Longrightarrow> (root n x < root n y) = (x < y)" |
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157 apply (cases "x < y") |
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158 apply (simp add: real_root_less_mono) |
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159 apply (simp add: linorder_not_less real_root_le_mono) |
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160 done |
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161 |
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162 lemma real_root_le_iff [simp]: |
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163 "0 < n \<Longrightarrow> (root n x \<le> root n y) = (x \<le> y)" |
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164 apply (cases "x \<le> y") |
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165 apply (simp add: real_root_le_mono) |
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166 apply (simp add: linorder_not_le real_root_less_mono) |
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167 done |
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168 |
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169 lemma real_root_eq_iff [simp]: |
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170 "0 < n \<Longrightarrow> (root n x = root n y) = (x = y)" |
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171 by (simp add: order_eq_iff) |
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172 |
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173 lemmas real_root_gt_0_iff [simp] = real_root_less_iff [where x=0, simplified] |
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174 lemmas real_root_lt_0_iff [simp] = real_root_less_iff [where y=0, simplified] |
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175 lemmas real_root_ge_0_iff [simp] = real_root_le_iff [where x=0, simplified] |
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176 lemmas real_root_le_0_iff [simp] = real_root_le_iff [where y=0, simplified] |
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177 lemmas real_root_eq_0_iff [simp] = real_root_eq_iff [where y=0, simplified] |
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178 |
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179 lemma real_root_gt_1_iff [simp]: "0 < n \<Longrightarrow> (1 < root n y) = (1 < y)" |
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180 by (insert real_root_less_iff [where x=1], simp) |
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181 |
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182 lemma real_root_lt_1_iff [simp]: "0 < n \<Longrightarrow> (root n x < 1) = (x < 1)" |
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183 by (insert real_root_less_iff [where y=1], simp) |
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184 |
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185 lemma real_root_ge_1_iff [simp]: "0 < n \<Longrightarrow> (1 \<le> root n y) = (1 \<le> y)" |
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186 by (insert real_root_le_iff [where x=1], simp) |
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187 |
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188 lemma real_root_le_1_iff [simp]: "0 < n \<Longrightarrow> (root n x \<le> 1) = (x \<le> 1)" |
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189 by (insert real_root_le_iff [where y=1], simp) |
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190 |
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191 lemma real_root_eq_1_iff [simp]: "0 < n \<Longrightarrow> (root n x = 1) = (x = 1)" |
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192 by (insert real_root_eq_iff [where y=1], simp) |
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193 |
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194 text {* Roots of roots *} |
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195 |
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196 lemma real_root_Suc_0 [simp]: "root (Suc 0) x = x" |
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197 by (simp add: odd_real_root_unique) |
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198 |
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199 lemma real_root_pos_mult_exp: |
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200 "\<lbrakk>0 < m; 0 < n; 0 < x\<rbrakk> \<Longrightarrow> root (m * n) x = root m (root n x)" |
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201 by (rule real_root_pos_unique, simp_all add: power_mult) |
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202 |
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203 lemma real_root_mult_exp: |
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204 "\<lbrakk>0 < m; 0 < n\<rbrakk> \<Longrightarrow> root (m * n) x = root m (root n x)" |
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205 apply (rule linorder_cases [where x=x and y=0]) |
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206 apply (subgoal_tac "root (m * n) (- x) = root m (root n (- x))") |
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207 apply (simp add: real_root_minus) |
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208 apply (simp_all add: real_root_pos_mult_exp) |
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209 done |
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210 |
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211 lemma real_root_commute: |
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212 "\<lbrakk>0 < m; 0 < n\<rbrakk> \<Longrightarrow> root m (root n x) = root n (root m x)" |
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213 by (simp add: real_root_mult_exp [symmetric] mult_commute) |
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214 |
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215 text {* Monotonicity in first argument *} |
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216 |
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217 lemma real_root_strict_decreasing: |
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218 "\<lbrakk>0 < n; n < N; 1 < x\<rbrakk> \<Longrightarrow> root N x < root n x" |
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219 apply (subgoal_tac "root n (root N x) ^ n < root N (root n x) ^ N", simp) |
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220 apply (simp add: real_root_commute power_strict_increasing |
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221 del: real_root_pow_pos2) |
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222 done |
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223 |
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224 lemma real_root_strict_increasing: |
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225 "\<lbrakk>0 < n; n < N; 0 < x; x < 1\<rbrakk> \<Longrightarrow> root n x < root N x" |
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226 apply (subgoal_tac "root N (root n x) ^ N < root n (root N x) ^ n", simp) |
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227 apply (simp add: real_root_commute power_strict_decreasing |
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228 del: real_root_pow_pos2) |
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229 done |
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230 |
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231 lemma real_root_decreasing: |
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232 "\<lbrakk>0 < n; n < N; 1 \<le> x\<rbrakk> \<Longrightarrow> root N x \<le> root n x" |
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233 by (auto simp add: order_le_less real_root_strict_decreasing) |
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234 |
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235 lemma real_root_increasing: |
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236 "\<lbrakk>0 < n; n < N; 0 \<le> x; x \<le> 1\<rbrakk> \<Longrightarrow> root n x \<le> root N x" |
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237 by (auto simp add: order_le_less real_root_strict_increasing) |
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238 |
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239 text {* Roots of multiplication and division *} |
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240 |
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241 lemma real_root_mult_lemma: |
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242 "\<lbrakk>0 < n; 0 \<le> x; 0 \<le> y\<rbrakk> \<Longrightarrow> root n (x * y) = root n x * root n y" |
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243 by (simp add: real_root_pos_unique mult_nonneg_nonneg power_mult_distrib) |
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244 |
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245 lemma real_root_inverse_lemma: |
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246 "\<lbrakk>0 < n; 0 \<le> x\<rbrakk> \<Longrightarrow> root n (inverse x) = inverse (root n x)" |
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247 by (simp add: real_root_pos_unique power_inverse [symmetric]) |
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248 |
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249 lemma real_root_mult: |
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250 assumes n: "0 < n" |
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251 shows "root n (x * y) = root n x * root n y" |
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252 proof (rule linorder_le_cases, rule_tac [!] linorder_le_cases) |
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253 assume "0 \<le> x" and "0 \<le> y" |
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254 thus ?thesis by (rule real_root_mult_lemma [OF n]) |
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255 next |
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256 assume "0 \<le> x" and "y \<le> 0" |
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257 hence "0 \<le> x" and "0 \<le> - y" by simp_all |
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258 hence "root n (x * - y) = root n x * root n (- y)" |
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259 by (rule real_root_mult_lemma [OF n]) |
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260 thus ?thesis by (simp add: real_root_minus [OF n]) |
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261 next |
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262 assume "x \<le> 0" and "0 \<le> y" |
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263 hence "0 \<le> - x" and "0 \<le> y" by simp_all |
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264 hence "root n (- x * y) = root n (- x) * root n y" |
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265 by (rule real_root_mult_lemma [OF n]) |
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266 thus ?thesis by (simp add: real_root_minus [OF n]) |
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267 next |
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268 assume "x \<le> 0" and "y \<le> 0" |
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269 hence "0 \<le> - x" and "0 \<le> - y" by simp_all |
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270 hence "root n (- x * - y) = root n (- x) * root n (- y)" |
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271 by (rule real_root_mult_lemma [OF n]) |
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272 thus ?thesis by (simp add: real_root_minus [OF n]) |
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273 qed |
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274 |
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275 lemma real_root_inverse: |
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276 assumes n: "0 < n" |
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277 shows "root n (inverse x) = inverse (root n x)" |
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278 proof (rule linorder_le_cases) |
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279 assume "0 \<le> x" |
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280 thus ?thesis by (rule real_root_inverse_lemma [OF n]) |
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281 next |
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282 assume "x \<le> 0" |
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283 hence "0 \<le> - x" by simp |
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284 hence "root n (inverse (- x)) = inverse (root n (- x))" |
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285 by (rule real_root_inverse_lemma [OF n]) |
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286 thus ?thesis by (simp add: real_root_minus [OF n]) |
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287 qed |
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288 |
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289 lemma real_root_divide: |
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290 "0 < n \<Longrightarrow> root n (x / y) = root n x / root n y" |
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291 by (simp add: divide_inverse real_root_mult real_root_inverse) |
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292 |
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293 lemma real_root_power: |
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294 "0 < n \<Longrightarrow> root n (x ^ k) = root n x ^ k" |
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295 by (induct k, simp_all add: real_root_mult) |
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296 |
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297 lemma real_root_abs: "0 < n \<Longrightarrow> root n \<bar>x\<bar> = \<bar>root n x\<bar>" |
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298 by (simp add: abs_if real_root_minus) |
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299 |
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300 text {* Continuity and derivatives *} |
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301 |
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302 lemma isCont_root_pos: |
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303 assumes n: "0 < n" |
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304 assumes x: "0 < x" |
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305 shows "isCont (root n) x" |
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306 proof - |
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307 have "isCont (root n) (root n x ^ n)" |
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308 proof (rule isCont_inverse_function [where f="\<lambda>a. a ^ n"]) |
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309 show "0 < root n x" using n x by simp |
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310 show "\<forall>z. \<bar>z - root n x\<bar> \<le> root n x \<longrightarrow> root n (z ^ n) = z" |
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311 by (simp add: abs_le_iff real_root_power_cancel n) |
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312 show "\<forall>z. \<bar>z - root n x\<bar> \<le> root n x \<longrightarrow> isCont (\<lambda>a. a ^ n) z" |
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313 by (simp add: isCont_power) |
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314 qed |
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315 thus ?thesis using n x by simp |
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316 qed |
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317 |
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318 lemma isCont_root_neg: |
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319 "\<lbrakk>0 < n; x < 0\<rbrakk> \<Longrightarrow> isCont (root n) x" |
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320 apply (subgoal_tac "isCont (\<lambda>x. - root n (- x)) x") |
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321 apply (simp add: real_root_minus) |
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322 apply (rule isCont_o2 [OF isCont_minus [OF isCont_ident]]) |
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323 apply (simp add: isCont_minus isCont_root_pos) |
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324 done |
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325 |
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326 lemma isCont_root_zero: |
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327 "0 < n \<Longrightarrow> isCont (root n) 0" |
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328 unfolding isCont_def |
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329 apply (rule LIM_I) |
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330 apply (rule_tac x="r ^ n" in exI, safe) |
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331 apply (simp) |
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332 apply (simp add: real_root_abs [symmetric]) |
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333 apply (rule_tac n="n" in power_less_imp_less_base, simp_all) |
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334 done |
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335 |
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336 lemma isCont_real_root: "0 < n \<Longrightarrow> isCont (root n) x" |
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337 apply (rule_tac x=x and y=0 in linorder_cases) |
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338 apply (simp_all add: isCont_root_pos isCont_root_neg isCont_root_zero) |
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339 done |
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340 |
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341 lemma DERIV_real_root: |
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342 assumes n: "0 < n" |
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343 assumes x: "0 < x" |
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344 shows "DERIV (root n) x :> inverse (real n * root n x ^ (n - Suc 0))" |
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345 proof (rule DERIV_inverse_function) |
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346 show "0 < x" using x . |
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347 show "x < x + 1" by simp |
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348 show "\<forall>y. 0 < y \<and> y < x + 1 \<longrightarrow> root n y ^ n = y" |
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349 using n by simp |
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350 show "DERIV (\<lambda>x. x ^ n) (root n x) :> real n * root n x ^ (n - Suc 0)" |
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351 by (rule DERIV_pow) |
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352 show "real n * root n x ^ (n - Suc 0) \<noteq> 0" |
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353 using n x by simp |
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354 show "isCont (root n) x" |
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355 using n by (rule isCont_real_root) |
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356 qed |
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357 |
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358 lemma DERIV_odd_real_root: |
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359 assumes n: "odd n" |
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360 assumes x: "x \<noteq> 0" |
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361 shows "DERIV (root n) x :> inverse (real n * root n x ^ (n - Suc 0))" |
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362 proof (rule DERIV_inverse_function) |
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363 show "x - 1 < x" by simp |
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364 show "x < x + 1" by simp |
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365 show "\<forall>y. x - 1 < y \<and> y < x + 1 \<longrightarrow> root n y ^ n = y" |
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366 using n by (simp add: odd_real_root_pow) |
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367 show "DERIV (\<lambda>x. x ^ n) (root n x) :> real n * root n x ^ (n - Suc 0)" |
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368 by (rule DERIV_pow) |
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369 show "real n * root n x ^ (n - Suc 0) \<noteq> 0" |
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370 using odd_pos [OF n] x by simp |
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371 show "isCont (root n) x" |
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372 using odd_pos [OF n] by (rule isCont_real_root) |
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373 qed |
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374 |
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375 subsection {* Square Root *} |
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376 |
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377 definition |
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378 sqrt :: "real \<Rightarrow> real" where |
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379 "sqrt = root 2" |
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380 |
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381 lemma pos2: "0 < (2::nat)" by simp |
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382 |
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383 lemma real_sqrt_unique: "\<lbrakk>y\<twosuperior> = x; 0 \<le> y\<rbrakk> \<Longrightarrow> sqrt x = y" |
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384 unfolding sqrt_def by (rule real_root_pos_unique [OF pos2]) |
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385 |
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386 lemma real_sqrt_abs [simp]: "sqrt (x\<twosuperior>) = \<bar>x\<bar>" |
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387 apply (rule real_sqrt_unique) |
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388 apply (rule power2_abs) |
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389 apply (rule abs_ge_zero) |
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390 done |
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391 |
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392 lemma real_sqrt_pow2 [simp]: "0 \<le> x \<Longrightarrow> (sqrt x)\<twosuperior> = x" |
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393 unfolding sqrt_def by (rule real_root_pow_pos2 [OF pos2]) |
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394 |
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395 lemma real_sqrt_pow2_iff [simp]: "((sqrt x)\<twosuperior> = x) = (0 \<le> x)" |
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396 apply (rule iffI) |
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397 apply (erule subst) |
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398 apply (rule zero_le_power2) |
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399 apply (erule real_sqrt_pow2) |
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400 done |
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401 |
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402 lemma real_sqrt_zero [simp]: "sqrt 0 = 0" |
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403 unfolding sqrt_def by (rule real_root_zero) |
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404 |
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405 lemma real_sqrt_one [simp]: "sqrt 1 = 1" |
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406 unfolding sqrt_def by (rule real_root_one [OF pos2]) |
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407 |
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408 lemma real_sqrt_minus: "sqrt (- x) = - sqrt x" |
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409 unfolding sqrt_def by (rule real_root_minus [OF pos2]) |
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410 |
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411 lemma real_sqrt_mult: "sqrt (x * y) = sqrt x * sqrt y" |
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412 unfolding sqrt_def by (rule real_root_mult [OF pos2]) |
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413 |
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414 lemma real_sqrt_inverse: "sqrt (inverse x) = inverse (sqrt x)" |
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415 unfolding sqrt_def by (rule real_root_inverse [OF pos2]) |
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416 |
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417 lemma real_sqrt_divide: "sqrt (x / y) = sqrt x / sqrt y" |
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418 unfolding sqrt_def by (rule real_root_divide [OF pos2]) |
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419 |
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420 lemma real_sqrt_power: "sqrt (x ^ k) = sqrt x ^ k" |
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421 unfolding sqrt_def by (rule real_root_power [OF pos2]) |
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422 |
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423 lemma real_sqrt_gt_zero: "0 < x \<Longrightarrow> 0 < sqrt x" |
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424 unfolding sqrt_def by (rule real_root_gt_zero [OF pos2]) |
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425 |
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426 lemma real_sqrt_ge_zero: "0 \<le> x \<Longrightarrow> 0 \<le> sqrt x" |
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427 unfolding sqrt_def by (rule real_root_ge_zero [OF pos2]) |
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428 |
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429 lemma real_sqrt_less_mono: "x < y \<Longrightarrow> sqrt x < sqrt y" |
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430 unfolding sqrt_def by (rule real_root_less_mono [OF pos2]) |
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431 |
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432 lemma real_sqrt_le_mono: "x \<le> y \<Longrightarrow> sqrt x \<le> sqrt y" |
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433 unfolding sqrt_def by (rule real_root_le_mono [OF pos2]) |
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434 |
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435 lemma real_sqrt_less_iff [simp]: "(sqrt x < sqrt y) = (x < y)" |
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436 unfolding sqrt_def by (rule real_root_less_iff [OF pos2]) |
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437 |
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438 lemma real_sqrt_le_iff [simp]: "(sqrt x \<le> sqrt y) = (x \<le> y)" |
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439 unfolding sqrt_def by (rule real_root_le_iff [OF pos2]) |
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440 |
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441 lemma real_sqrt_eq_iff [simp]: "(sqrt x = sqrt y) = (x = y)" |
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442 unfolding sqrt_def by (rule real_root_eq_iff [OF pos2]) |
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443 |
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444 lemmas real_sqrt_gt_0_iff [simp] = real_sqrt_less_iff [where x=0, simplified] |
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445 lemmas real_sqrt_lt_0_iff [simp] = real_sqrt_less_iff [where y=0, simplified] |
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446 lemmas real_sqrt_ge_0_iff [simp] = real_sqrt_le_iff [where x=0, simplified] |
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447 lemmas real_sqrt_le_0_iff [simp] = real_sqrt_le_iff [where y=0, simplified] |
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448 lemmas real_sqrt_eq_0_iff [simp] = real_sqrt_eq_iff [where y=0, simplified] |
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449 |
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450 lemmas real_sqrt_gt_1_iff [simp] = real_sqrt_less_iff [where x=1, simplified] |
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451 lemmas real_sqrt_lt_1_iff [simp] = real_sqrt_less_iff [where y=1, simplified] |
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452 lemmas real_sqrt_ge_1_iff [simp] = real_sqrt_le_iff [where x=1, simplified] |
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453 lemmas real_sqrt_le_1_iff [simp] = real_sqrt_le_iff [where y=1, simplified] |
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454 lemmas real_sqrt_eq_1_iff [simp] = real_sqrt_eq_iff [where y=1, simplified] |
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455 |
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456 lemma isCont_real_sqrt: "isCont sqrt x" |
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457 unfolding sqrt_def by (rule isCont_real_root [OF pos2]) |
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458 |
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459 lemma DERIV_real_sqrt: |
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460 "0 < x \<Longrightarrow> DERIV sqrt x :> inverse (sqrt x) / 2" |
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461 unfolding sqrt_def by (rule DERIV_real_root [OF pos2, simplified]) |
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462 |
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463 lemma not_real_square_gt_zero [simp]: "(~ (0::real) < x*x) = (x = 0)" |
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464 apply auto |
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465 apply (cut_tac x = x and y = 0 in linorder_less_linear) |
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466 apply (simp add: zero_less_mult_iff) |
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467 done |
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468 |
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469 lemma real_sqrt_abs2 [simp]: "sqrt(x*x) = \<bar>x\<bar>" |
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470 apply (subst power2_eq_square [symmetric]) |
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471 apply (rule real_sqrt_abs) |
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472 done |
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473 |
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474 lemma real_sqrt_pow2_gt_zero: "0 < x ==> 0 < (sqrt x)\<twosuperior>" |
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475 by simp (* TODO: delete *) |
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476 |
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477 lemma real_sqrt_not_eq_zero: "0 < x ==> sqrt x \<noteq> 0" |
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478 by simp (* TODO: delete *) |
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479 |
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480 lemma real_inv_sqrt_pow2: "0 < x ==> inverse (sqrt(x)) ^ 2 = inverse x" |
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481 by (simp add: power_inverse [symmetric]) |
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482 |
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483 lemma real_sqrt_eq_zero_cancel: "[| 0 \<le> x; sqrt(x) = 0|] ==> x = 0" |
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484 by simp |
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485 |
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486 lemma real_sqrt_ge_one: "1 \<le> x ==> 1 \<le> sqrt x" |
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487 by simp |
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488 |
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489 lemma real_sqrt_two_gt_zero [simp]: "0 < sqrt 2" |
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490 by simp |
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491 |
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492 lemma real_sqrt_two_ge_zero [simp]: "0 \<le> sqrt 2" |
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493 by simp |
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494 |
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495 lemma real_sqrt_two_gt_one [simp]: "1 < sqrt 2" |
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496 by simp |
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497 |
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498 lemma sqrt_divide_self_eq: |
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499 assumes nneg: "0 \<le> x" |
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500 shows "sqrt x / x = inverse (sqrt x)" |
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501 proof cases |
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502 assume "x=0" thus ?thesis by simp |
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503 next |
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504 assume nz: "x\<noteq>0" |
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505 hence pos: "0<x" using nneg by arith |
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506 show ?thesis |
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507 proof (rule right_inverse_eq [THEN iffD1, THEN sym]) |
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508 show "sqrt x / x \<noteq> 0" by (simp add: divide_inverse nneg nz) |
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509 show "inverse (sqrt x) / (sqrt x / x) = 1" |
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510 by (simp add: divide_inverse mult_assoc [symmetric] |
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511 power2_eq_square [symmetric] real_inv_sqrt_pow2 pos nz) |
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512 qed |
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513 qed |
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514 |
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515 lemma real_divide_square_eq [simp]: "(((r::real) * a) / (r * r)) = a / r" |
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516 apply (simp add: divide_inverse) |
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517 apply (case_tac "r=0") |
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518 apply (auto simp add: mult_ac) |
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519 done |
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520 |
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521 lemma lemma_real_divide_sqrt_less: "0 < u ==> u / sqrt 2 < u" |
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522 by (simp add: divide_less_eq mult_compare_simps) |
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523 |
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524 lemma four_x_squared: |
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525 fixes x::real |
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526 shows "4 * x\<twosuperior> = (2 * x)\<twosuperior>" |
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527 by (simp add: power2_eq_square) |
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528 |
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529 subsection {* Square Root of Sum of Squares *} |
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530 |
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531 lemma real_sqrt_mult_self_sum_ge_zero [simp]: "0 \<le> sqrt(x*x + y*y)" |
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532 by (rule real_sqrt_ge_zero [OF sum_squares_ge_zero]) |
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533 |
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534 lemma real_sqrt_sum_squares_ge_zero [simp]: "0 \<le> sqrt (x\<twosuperior> + y\<twosuperior>)" |
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535 by simp |
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536 |
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537 declare real_sqrt_sum_squares_ge_zero [THEN abs_of_nonneg, simp] |
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538 |
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539 lemma real_sqrt_sum_squares_mult_ge_zero [simp]: |
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540 "0 \<le> sqrt ((x\<twosuperior> + y\<twosuperior>)*(xa\<twosuperior> + ya\<twosuperior>))" |
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541 by (auto intro!: real_sqrt_ge_zero simp add: zero_le_mult_iff) |
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542 |
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543 lemma real_sqrt_sum_squares_mult_squared_eq [simp]: |
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544 "sqrt ((x\<twosuperior> + y\<twosuperior>) * (xa\<twosuperior> + ya\<twosuperior>)) ^ 2 = (x\<twosuperior> + y\<twosuperior>) * (xa\<twosuperior> + ya\<twosuperior>)" |
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545 by (auto simp add: zero_le_mult_iff) |
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546 |
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547 lemma real_sqrt_sum_squares_eq_cancel: "sqrt (x\<twosuperior> + y\<twosuperior>) = x \<Longrightarrow> y = 0" |
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548 by (drule_tac f = "%x. x\<twosuperior>" in arg_cong, simp) |
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549 |
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550 lemma real_sqrt_sum_squares_eq_cancel2: "sqrt (x\<twosuperior> + y\<twosuperior>) = y \<Longrightarrow> x = 0" |
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551 by (drule_tac f = "%x. x\<twosuperior>" in arg_cong, simp) |
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552 |
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553 lemma real_sqrt_sum_squares_ge1 [simp]: "x \<le> sqrt (x\<twosuperior> + y\<twosuperior>)" |
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554 by (rule power2_le_imp_le, simp_all) |
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555 |
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556 lemma real_sqrt_sum_squares_ge2 [simp]: "y \<le> sqrt (x\<twosuperior> + y\<twosuperior>)" |
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557 by (rule power2_le_imp_le, simp_all) |
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558 |
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559 lemma real_sqrt_ge_abs1 [simp]: "\<bar>x\<bar> \<le> sqrt (x\<twosuperior> + y\<twosuperior>)" |
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560 by (rule power2_le_imp_le, simp_all) |
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561 |
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562 lemma real_sqrt_ge_abs2 [simp]: "\<bar>y\<bar> \<le> sqrt (x\<twosuperior> + y\<twosuperior>)" |
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563 by (rule power2_le_imp_le, simp_all) |
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564 |
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565 lemma le_real_sqrt_sumsq [simp]: "x \<le> sqrt (x * x + y * y)" |
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566 by (simp add: power2_eq_square [symmetric]) |
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567 |
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568 lemma power2_sum: |
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569 fixes x y :: "'a::{number_ring,recpower}" |
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570 shows "(x + y)\<twosuperior> = x\<twosuperior> + y\<twosuperior> + 2 * x * y" |
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571 by (simp add: ring_distribs power2_eq_square) |
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572 |
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573 lemma power2_diff: |
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574 fixes x y :: "'a::{number_ring,recpower}" |
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575 shows "(x - y)\<twosuperior> = x\<twosuperior> + y\<twosuperior> - 2 * x * y" |
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576 by (simp add: ring_distribs power2_eq_square) |
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577 |
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578 lemma real_sqrt_sum_squares_triangle_ineq: |
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579 "sqrt ((a + c)\<twosuperior> + (b + d)\<twosuperior>) \<le> sqrt (a\<twosuperior> + b\<twosuperior>) + sqrt (c\<twosuperior> + d\<twosuperior>)" |
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580 apply (rule power2_le_imp_le, simp) |
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581 apply (simp add: power2_sum) |
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582 apply (simp only: mult_assoc right_distrib [symmetric]) |
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583 apply (rule mult_left_mono) |
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584 apply (rule power2_le_imp_le) |
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585 apply (simp add: power2_sum power_mult_distrib) |
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586 apply (simp add: ring_distribs) |
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587 apply (subgoal_tac "0 \<le> b\<twosuperior> * c\<twosuperior> + a\<twosuperior> * d\<twosuperior> - 2 * (a * c) * (b * d)", simp) |
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588 apply (rule_tac b="(a * d - b * c)\<twosuperior>" in ord_le_eq_trans) |
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589 apply (rule zero_le_power2) |
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590 apply (simp add: power2_diff power_mult_distrib) |
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591 apply (simp add: mult_nonneg_nonneg) |
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592 apply simp |
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593 apply (simp add: add_increasing) |
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594 done |
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595 |
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596 lemma real_sqrt_sum_squares_less: |
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597 "\<lbrakk>\<bar>x\<bar> < u / sqrt 2; \<bar>y\<bar> < u / sqrt 2\<rbrakk> \<Longrightarrow> sqrt (x\<twosuperior> + y\<twosuperior>) < u" |
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598 apply (rule power2_less_imp_less, simp) |
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599 apply (drule power_strict_mono [OF _ abs_ge_zero pos2]) |
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600 apply (drule power_strict_mono [OF _ abs_ge_zero pos2]) |
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601 apply (simp add: power_divide) |
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602 apply (drule order_le_less_trans [OF abs_ge_zero]) |
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603 apply (simp add: zero_less_divide_iff) |
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604 done |
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605 |
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606 text{*Needed for the infinitely close relation over the nonstandard |
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607 complex numbers*} |
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608 lemma lemma_sqrt_hcomplex_capprox: |
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609 "[| 0 < u; x < u/2; y < u/2; 0 \<le> x; 0 \<le> y |] ==> sqrt (x\<twosuperior> + y\<twosuperior>) < u" |
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610 apply (rule_tac y = "u/sqrt 2" in order_le_less_trans) |
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611 apply (erule_tac [2] lemma_real_divide_sqrt_less) |
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612 apply (rule power2_le_imp_le) |
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613 apply (auto simp add: real_0_le_divide_iff power_divide) |
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614 apply (rule_tac t = "u\<twosuperior>" in real_sum_of_halves [THEN subst]) |
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615 apply (rule add_mono) |
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616 apply (auto simp add: four_x_squared simp del: realpow_Suc intro: power_mono) |
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617 done |
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618 |
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619 text "Legacy theorem names:" |
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620 lemmas real_root_pos2 = real_root_power_cancel |
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621 lemmas real_root_pos_pos = real_root_gt_zero [THEN order_less_imp_le] |
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622 lemmas real_root_pos_pos_le = real_root_ge_zero |
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623 lemmas real_sqrt_mult_distrib = real_sqrt_mult |
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624 lemmas real_sqrt_mult_distrib2 = real_sqrt_mult |
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625 lemmas real_sqrt_eq_zero_cancel_iff = real_sqrt_eq_0_iff |
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626 |
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627 (* needed for CauchysMeanTheorem.het_base from AFP *) |
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628 lemma real_root_pos: "0 < x \<Longrightarrow> root (Suc n) (x ^ (Suc n)) = x" |
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629 by (rule real_root_power_cancel [OF zero_less_Suc order_less_imp_le]) |
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630 |
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631 end |