1 (* Title : HOL/NSA/HyperDef.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 section\<open>Construction of Hyperreals Using Ultrafilters\<close> |
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8 |
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9 theory HyperDef |
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10 imports Complex_Main HyperNat |
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11 begin |
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12 |
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13 type_synonym hypreal = "real star" |
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14 |
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15 abbreviation |
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16 hypreal_of_real :: "real => real star" where |
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17 "hypreal_of_real == star_of" |
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18 |
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19 abbreviation |
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20 hypreal_of_hypnat :: "hypnat \<Rightarrow> hypreal" where |
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21 "hypreal_of_hypnat \<equiv> of_hypnat" |
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22 |
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23 definition |
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24 omega :: hypreal ("\<omega>") where |
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25 \<comment> \<open>an infinite number \<open>= [<1,2,3,...>]\<close>\<close> |
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26 "\<omega> = star_n (\<lambda>n. real (Suc n))" |
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27 |
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28 definition |
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29 epsilon :: hypreal ("\<epsilon>") where |
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30 \<comment> \<open>an infinitesimal number \<open>= [<1,1/2,1/3,...>]\<close>\<close> |
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31 "\<epsilon> = star_n (\<lambda>n. inverse (real (Suc n)))" |
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32 |
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33 |
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34 subsection \<open>Real vector class instances\<close> |
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35 |
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36 instantiation star :: (scaleR) scaleR |
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37 begin |
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38 |
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39 definition |
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40 star_scaleR_def [transfer_unfold]: "scaleR r \<equiv> *f* (scaleR r)" |
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41 |
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42 instance .. |
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43 |
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44 end |
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45 |
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46 lemma Standard_scaleR [simp]: "x \<in> Standard \<Longrightarrow> scaleR r x \<in> Standard" |
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47 by (simp add: star_scaleR_def) |
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48 |
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49 lemma star_of_scaleR [simp]: "star_of (scaleR r x) = scaleR r (star_of x)" |
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50 by transfer (rule refl) |
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51 |
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52 instance star :: (real_vector) real_vector |
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53 proof |
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54 fix a b :: real |
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55 show "\<And>x y::'a star. scaleR a (x + y) = scaleR a x + scaleR a y" |
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56 by transfer (rule scaleR_right_distrib) |
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57 show "\<And>x::'a star. scaleR (a + b) x = scaleR a x + scaleR b x" |
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58 by transfer (rule scaleR_left_distrib) |
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59 show "\<And>x::'a star. scaleR a (scaleR b x) = scaleR (a * b) x" |
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60 by transfer (rule scaleR_scaleR) |
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61 show "\<And>x::'a star. scaleR 1 x = x" |
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62 by transfer (rule scaleR_one) |
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63 qed |
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64 |
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65 instance star :: (real_algebra) real_algebra |
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66 proof |
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67 fix a :: real |
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68 show "\<And>x y::'a star. scaleR a x * y = scaleR a (x * y)" |
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69 by transfer (rule mult_scaleR_left) |
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70 show "\<And>x y::'a star. x * scaleR a y = scaleR a (x * y)" |
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71 by transfer (rule mult_scaleR_right) |
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72 qed |
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73 |
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74 instance star :: (real_algebra_1) real_algebra_1 .. |
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75 |
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76 instance star :: (real_div_algebra) real_div_algebra .. |
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77 |
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78 instance star :: (field_char_0) field_char_0 .. |
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79 |
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80 instance star :: (real_field) real_field .. |
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81 |
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82 lemma star_of_real_def [transfer_unfold]: "of_real r = star_of (of_real r)" |
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83 by (unfold of_real_def, transfer, rule refl) |
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84 |
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85 lemma Standard_of_real [simp]: "of_real r \<in> Standard" |
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86 by (simp add: star_of_real_def) |
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87 |
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88 lemma star_of_of_real [simp]: "star_of (of_real r) = of_real r" |
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89 by transfer (rule refl) |
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90 |
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91 lemma of_real_eq_star_of [simp]: "of_real = star_of" |
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92 proof |
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93 fix r :: real |
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94 show "of_real r = star_of r" |
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95 by transfer simp |
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96 qed |
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97 |
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98 lemma Reals_eq_Standard: "(\<real> :: hypreal set) = Standard" |
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99 by (simp add: Reals_def Standard_def) |
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100 |
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101 |
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102 subsection \<open>Injection from @{typ hypreal}\<close> |
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103 |
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104 definition |
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105 of_hypreal :: "hypreal \<Rightarrow> 'a::real_algebra_1 star" where |
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106 [transfer_unfold]: "of_hypreal = *f* of_real" |
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107 |
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108 lemma Standard_of_hypreal [simp]: |
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109 "r \<in> Standard \<Longrightarrow> of_hypreal r \<in> Standard" |
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110 by (simp add: of_hypreal_def) |
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111 |
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112 lemma of_hypreal_0 [simp]: "of_hypreal 0 = 0" |
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113 by transfer (rule of_real_0) |
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114 |
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115 lemma of_hypreal_1 [simp]: "of_hypreal 1 = 1" |
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116 by transfer (rule of_real_1) |
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117 |
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118 lemma of_hypreal_add [simp]: |
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119 "\<And>x y. of_hypreal (x + y) = of_hypreal x + of_hypreal y" |
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120 by transfer (rule of_real_add) |
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121 |
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122 lemma of_hypreal_minus [simp]: "\<And>x. of_hypreal (- x) = - of_hypreal x" |
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123 by transfer (rule of_real_minus) |
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124 |
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125 lemma of_hypreal_diff [simp]: |
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126 "\<And>x y. of_hypreal (x - y) = of_hypreal x - of_hypreal y" |
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127 by transfer (rule of_real_diff) |
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128 |
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129 lemma of_hypreal_mult [simp]: |
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130 "\<And>x y. of_hypreal (x * y) = of_hypreal x * of_hypreal y" |
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131 by transfer (rule of_real_mult) |
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132 |
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133 lemma of_hypreal_inverse [simp]: |
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134 "\<And>x. of_hypreal (inverse x) = |
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135 inverse (of_hypreal x :: 'a::{real_div_algebra, division_ring} star)" |
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136 by transfer (rule of_real_inverse) |
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137 |
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138 lemma of_hypreal_divide [simp]: |
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139 "\<And>x y. of_hypreal (x / y) = |
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140 (of_hypreal x / of_hypreal y :: 'a::{real_field, field} star)" |
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141 by transfer (rule of_real_divide) |
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142 |
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143 lemma of_hypreal_eq_iff [simp]: |
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144 "\<And>x y. (of_hypreal x = of_hypreal y) = (x = y)" |
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145 by transfer (rule of_real_eq_iff) |
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146 |
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147 lemma of_hypreal_eq_0_iff [simp]: |
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148 "\<And>x. (of_hypreal x = 0) = (x = 0)" |
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149 by transfer (rule of_real_eq_0_iff) |
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150 |
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151 |
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152 subsection\<open>Properties of @{term starrel}\<close> |
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153 |
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154 lemma lemma_starrel_refl [simp]: "x \<in> starrel `` {x}" |
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155 by (simp add: starrel_def) |
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156 |
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157 lemma starrel_in_hypreal [simp]: "starrel``{x}:star" |
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158 by (simp add: star_def starrel_def quotient_def, blast) |
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159 |
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160 declare Abs_star_inject [simp] Abs_star_inverse [simp] |
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161 declare equiv_starrel [THEN eq_equiv_class_iff, simp] |
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162 |
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163 subsection\<open>@{term hypreal_of_real}: |
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164 the Injection from @{typ real} to @{typ hypreal}\<close> |
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165 |
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166 lemma inj_star_of: "inj star_of" |
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167 by (rule inj_onI, simp) |
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168 |
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169 lemma mem_Rep_star_iff: "(X \<in> Rep_star x) = (x = star_n X)" |
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170 by (cases x, simp add: star_n_def) |
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171 |
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172 lemma Rep_star_star_n_iff [simp]: |
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173 "(X \<in> Rep_star (star_n Y)) = (eventually (\<lambda>n. Y n = X n) \<U>)" |
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174 by (simp add: star_n_def) |
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175 |
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176 lemma Rep_star_star_n: "X \<in> Rep_star (star_n X)" |
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177 by simp |
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178 |
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179 subsection\<open>Properties of @{term star_n}\<close> |
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180 |
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181 lemma star_n_add: |
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182 "star_n X + star_n Y = star_n (%n. X n + Y n)" |
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183 by (simp only: star_add_def starfun2_star_n) |
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184 |
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185 lemma star_n_minus: |
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186 "- star_n X = star_n (%n. -(X n))" |
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187 by (simp only: star_minus_def starfun_star_n) |
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188 |
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189 lemma star_n_diff: |
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190 "star_n X - star_n Y = star_n (%n. X n - Y n)" |
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191 by (simp only: star_diff_def starfun2_star_n) |
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192 |
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193 lemma star_n_mult: |
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194 "star_n X * star_n Y = star_n (%n. X n * Y n)" |
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195 by (simp only: star_mult_def starfun2_star_n) |
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196 |
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197 lemma star_n_inverse: |
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198 "inverse (star_n X) = star_n (%n. inverse(X n))" |
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199 by (simp only: star_inverse_def starfun_star_n) |
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200 |
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201 lemma star_n_le: |
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202 "star_n X \<le> star_n Y = (eventually (\<lambda>n. X n \<le> Y n) FreeUltrafilterNat)" |
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203 by (simp only: star_le_def starP2_star_n) |
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204 |
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205 lemma star_n_less: |
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206 "star_n X < star_n Y = (eventually (\<lambda>n. X n < Y n) FreeUltrafilterNat)" |
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207 by (simp only: star_less_def starP2_star_n) |
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208 |
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209 lemma star_n_zero_num: "0 = star_n (%n. 0)" |
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210 by (simp only: star_zero_def star_of_def) |
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211 |
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212 lemma star_n_one_num: "1 = star_n (%n. 1)" |
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213 by (simp only: star_one_def star_of_def) |
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214 |
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215 lemma star_n_abs: "\<bar>star_n X\<bar> = star_n (%n. \<bar>X n\<bar>)" |
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216 by (simp only: star_abs_def starfun_star_n) |
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217 |
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218 lemma hypreal_omega_gt_zero [simp]: "0 < \<omega>" |
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219 by (simp add: omega_def star_n_zero_num star_n_less) |
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220 |
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221 subsection\<open>Existence of Infinite Hyperreal Number\<close> |
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222 |
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223 text\<open>Existence of infinite number not corresponding to any real number. |
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224 Use assumption that member @{term FreeUltrafilterNat} is not finite.\<close> |
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225 |
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226 |
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227 text\<open>A few lemmas first\<close> |
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228 |
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229 lemma lemma_omega_empty_singleton_disj: |
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230 "{n::nat. x = real n} = {} \<or> (\<exists>y. {n::nat. x = real n} = {y})" |
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231 by force |
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232 |
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233 lemma lemma_finite_omega_set: "finite {n::nat. x = real n}" |
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234 using lemma_omega_empty_singleton_disj [of x] by auto |
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235 |
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236 lemma not_ex_hypreal_of_real_eq_omega: |
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237 "~ (\<exists>x. hypreal_of_real x = \<omega>)" |
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238 apply (simp add: omega_def star_of_def star_n_eq_iff) |
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239 apply clarify |
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240 apply (rule_tac x2="x-1" in lemma_finite_omega_set [THEN FreeUltrafilterNat.finite, THEN notE]) |
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241 apply (erule eventually_mono) |
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242 apply auto |
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243 done |
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244 |
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245 lemma hypreal_of_real_not_eq_omega: "hypreal_of_real x \<noteq> \<omega>" |
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246 by (insert not_ex_hypreal_of_real_eq_omega, auto) |
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247 |
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248 text\<open>Existence of infinitesimal number also not corresponding to any |
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249 real number\<close> |
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250 |
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251 lemma lemma_epsilon_empty_singleton_disj: |
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252 "{n::nat. x = inverse(real(Suc n))} = {} | |
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253 (\<exists>y. {n::nat. x = inverse(real(Suc n))} = {y})" |
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254 by auto |
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255 |
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256 lemma lemma_finite_epsilon_set: "finite {n. x = inverse(real(Suc n))}" |
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257 by (cut_tac x = x in lemma_epsilon_empty_singleton_disj, auto) |
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258 |
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259 lemma not_ex_hypreal_of_real_eq_epsilon: "~ (\<exists>x. hypreal_of_real x = \<epsilon>)" |
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260 by (auto simp add: epsilon_def star_of_def star_n_eq_iff |
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261 lemma_finite_epsilon_set [THEN FreeUltrafilterNat.finite] simp del: of_nat_Suc) |
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262 |
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263 lemma hypreal_of_real_not_eq_epsilon: "hypreal_of_real x \<noteq> \<epsilon>" |
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264 by (insert not_ex_hypreal_of_real_eq_epsilon, auto) |
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265 |
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266 lemma hypreal_epsilon_not_zero: "\<epsilon> \<noteq> 0" |
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267 by (simp add: epsilon_def star_zero_def star_of_def star_n_eq_iff FreeUltrafilterNat.proper |
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268 del: star_of_zero) |
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269 |
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270 lemma hypreal_epsilon_inverse_omega: "\<epsilon> = inverse \<omega>" |
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271 by (simp add: epsilon_def omega_def star_n_inverse) |
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272 |
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273 lemma hypreal_epsilon_gt_zero: "0 < \<epsilon>" |
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274 by (simp add: hypreal_epsilon_inverse_omega) |
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275 |
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276 subsection\<open>Absolute Value Function for the Hyperreals\<close> |
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277 |
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278 lemma hrabs_add_less: "[| \<bar>x\<bar> < r; \<bar>y\<bar> < s |] ==> \<bar>x + y\<bar> < r + (s::hypreal)" |
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279 by (simp add: abs_if split: if_split_asm) |
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280 |
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281 lemma hrabs_less_gt_zero: "\<bar>x\<bar> < r ==> (0::hypreal) < r" |
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282 by (blast intro!: order_le_less_trans abs_ge_zero) |
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283 |
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284 lemma hrabs_disj: "\<bar>x\<bar> = (x::'a::abs_if) \<or> \<bar>x\<bar> = -x" |
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285 by (simp add: abs_if) |
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286 |
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287 lemma hrabs_add_lemma_disj: "(y::hypreal) + - x + (y + - z) = \<bar>x + - z\<bar> ==> y = z | x = y" |
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288 by (simp add: abs_if split add: if_split_asm) |
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289 |
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290 |
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291 subsection\<open>Embedding the Naturals into the Hyperreals\<close> |
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292 |
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293 abbreviation |
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294 hypreal_of_nat :: "nat => hypreal" where |
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295 "hypreal_of_nat == of_nat" |
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296 |
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297 lemma SNat_eq: "Nats = {n. \<exists>N. n = hypreal_of_nat N}" |
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298 by (simp add: Nats_def image_def) |
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299 |
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300 (*------------------------------------------------------------*) |
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301 (* naturals embedded in hyperreals *) |
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302 (* is a hyperreal c.f. NS extension *) |
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303 (*------------------------------------------------------------*) |
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304 |
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305 lemma hypreal_of_nat: "hypreal_of_nat m = star_n (%n. real m)" |
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306 by (simp add: star_of_def [symmetric]) |
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307 |
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308 declaration \<open> |
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309 K (Lin_Arith.add_inj_thms [@{thm star_of_le} RS iffD2, |
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310 @{thm star_of_less} RS iffD2, @{thm star_of_eq} RS iffD2] |
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311 #> Lin_Arith.add_simps [@{thm star_of_zero}, @{thm star_of_one}, |
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312 @{thm star_of_numeral}, @{thm star_of_add}, |
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313 @{thm star_of_minus}, @{thm star_of_diff}, @{thm star_of_mult}] |
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314 #> Lin_Arith.add_inj_const (@{const_name "StarDef.star_of"}, @{typ "real \<Rightarrow> hypreal"})) |
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315 \<close> |
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316 |
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317 simproc_setup fast_arith_hypreal ("(m::hypreal) < n" | "(m::hypreal) <= n" | "(m::hypreal) = n") = |
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318 \<open>K Lin_Arith.simproc\<close> |
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319 |
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320 |
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321 subsection \<open>Exponentials on the Hyperreals\<close> |
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322 |
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323 lemma hpowr_0 [simp]: "r ^ 0 = (1::hypreal)" |
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324 by (rule power_0) |
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325 |
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326 lemma hpowr_Suc [simp]: "r ^ (Suc n) = (r::hypreal) * (r ^ n)" |
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327 by (rule power_Suc) |
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328 |
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329 lemma hrealpow_two: "(r::hypreal) ^ Suc (Suc 0) = r * r" |
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330 by simp |
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331 |
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332 lemma hrealpow_two_le [simp]: "(0::hypreal) \<le> r ^ Suc (Suc 0)" |
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333 by (auto simp add: zero_le_mult_iff) |
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334 |
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335 lemma hrealpow_two_le_add_order [simp]: |
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336 "(0::hypreal) \<le> u ^ Suc (Suc 0) + v ^ Suc (Suc 0)" |
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337 by (simp only: hrealpow_two_le add_nonneg_nonneg) |
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338 |
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339 lemma hrealpow_two_le_add_order2 [simp]: |
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340 "(0::hypreal) \<le> u ^ Suc (Suc 0) + v ^ Suc (Suc 0) + w ^ Suc (Suc 0)" |
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341 by (simp only: hrealpow_two_le add_nonneg_nonneg) |
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342 |
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343 lemma hypreal_add_nonneg_eq_0_iff: |
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344 "[| 0 \<le> x; 0 \<le> y |] ==> (x+y = 0) = (x = 0 & y = (0::hypreal))" |
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345 by arith |
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346 |
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347 |
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348 text\<open>FIXME: DELETE THESE\<close> |
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349 lemma hypreal_three_squares_add_zero_iff: |
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350 "(x*x + y*y + z*z = 0) = (x = 0 & y = 0 & z = (0::hypreal))" |
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351 apply (simp only: zero_le_square add_nonneg_nonneg hypreal_add_nonneg_eq_0_iff, auto) |
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352 done |
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353 |
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354 lemma hrealpow_three_squares_add_zero_iff [simp]: |
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355 "(x ^ Suc (Suc 0) + y ^ Suc (Suc 0) + z ^ Suc (Suc 0) = (0::hypreal)) = |
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356 (x = 0 & y = 0 & z = 0)" |
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357 by (simp only: hypreal_three_squares_add_zero_iff hrealpow_two) |
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358 |
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359 (*FIXME: This and RealPow.abs_realpow_two should be replaced by an abstract |
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360 result proved in Rings or Fields*) |
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361 lemma hrabs_hrealpow_two [simp]: "\<bar>x ^ Suc (Suc 0)\<bar> = (x::hypreal) ^ Suc (Suc 0)" |
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362 by (simp add: abs_mult) |
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363 |
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364 lemma two_hrealpow_ge_one [simp]: "(1::hypreal) \<le> 2 ^ n" |
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365 by (insert power_increasing [of 0 n "2::hypreal"], simp) |
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366 |
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367 lemma hrealpow: |
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368 "star_n X ^ m = star_n (%n. (X n::real) ^ m)" |
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369 apply (induct_tac "m") |
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370 apply (auto simp add: star_n_one_num star_n_mult power_0) |
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371 done |
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372 |
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373 lemma hrealpow_sum_square_expand: |
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374 "(x + (y::hypreal)) ^ Suc (Suc 0) = |
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375 x ^ Suc (Suc 0) + y ^ Suc (Suc 0) + (hypreal_of_nat (Suc (Suc 0)))*x*y" |
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376 by (simp add: distrib_left distrib_right) |
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377 |
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378 lemma power_hypreal_of_real_numeral: |
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379 "(numeral v :: hypreal) ^ n = hypreal_of_real ((numeral v) ^ n)" |
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380 by simp |
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381 declare power_hypreal_of_real_numeral [of _ "numeral w", simp] for w |
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382 |
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383 lemma power_hypreal_of_real_neg_numeral: |
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384 "(- numeral v :: hypreal) ^ n = hypreal_of_real ((- numeral v) ^ n)" |
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385 by simp |
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386 declare power_hypreal_of_real_neg_numeral [of _ "numeral w", simp] for w |
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387 (* |
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388 lemma hrealpow_HFinite: |
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389 fixes x :: "'a::{real_normed_algebra,power} star" |
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390 shows "x \<in> HFinite ==> x ^ n \<in> HFinite" |
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391 apply (induct_tac "n") |
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392 apply (auto simp add: power_Suc intro: HFinite_mult) |
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393 done |
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394 *) |
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395 |
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396 subsection\<open>Powers with Hypernatural Exponents\<close> |
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397 |
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398 definition pow :: "['a::power star, nat star] \<Rightarrow> 'a star" (infixr "pow" 80) where |
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399 hyperpow_def [transfer_unfold]: "R pow N = ( *f2* op ^) R N" |
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400 (* hypernatural powers of hyperreals *) |
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401 |
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402 lemma Standard_hyperpow [simp]: |
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403 "\<lbrakk>r \<in> Standard; n \<in> Standard\<rbrakk> \<Longrightarrow> r pow n \<in> Standard" |
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404 unfolding hyperpow_def by simp |
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405 |
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406 lemma hyperpow: "star_n X pow star_n Y = star_n (%n. X n ^ Y n)" |
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407 by (simp add: hyperpow_def starfun2_star_n) |
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408 |
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409 lemma hyperpow_zero [simp]: |
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410 "\<And>n. (0::'a::{power,semiring_0} star) pow (n + (1::hypnat)) = 0" |
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411 by transfer simp |
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412 |
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413 lemma hyperpow_not_zero: |
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414 "\<And>r n. r \<noteq> (0::'a::{field} star) ==> r pow n \<noteq> 0" |
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415 by transfer (rule power_not_zero) |
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416 |
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417 lemma hyperpow_inverse: |
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418 "\<And>r n. r \<noteq> (0::'a::field star) |
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419 \<Longrightarrow> inverse (r pow n) = (inverse r) pow n" |
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420 by transfer (rule power_inverse [symmetric]) |
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421 |
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422 lemma hyperpow_hrabs: |
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423 "\<And>r n. \<bar>r::'a::{linordered_idom} star\<bar> pow n = \<bar>r pow n\<bar>" |
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424 by transfer (rule power_abs [symmetric]) |
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425 |
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426 lemma hyperpow_add: |
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427 "\<And>r n m. (r::'a::monoid_mult star) pow (n + m) = (r pow n) * (r pow m)" |
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428 by transfer (rule power_add) |
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429 |
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430 lemma hyperpow_one [simp]: |
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431 "\<And>r. (r::'a::monoid_mult star) pow (1::hypnat) = r" |
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432 by transfer (rule power_one_right) |
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433 |
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434 lemma hyperpow_two: |
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435 "\<And>r. (r::'a::monoid_mult star) pow (2::hypnat) = r * r" |
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436 by transfer (rule power2_eq_square) |
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437 |
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438 lemma hyperpow_gt_zero: |
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439 "\<And>r n. (0::'a::{linordered_semidom} star) < r \<Longrightarrow> 0 < r pow n" |
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440 by transfer (rule zero_less_power) |
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441 |
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442 lemma hyperpow_ge_zero: |
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443 "\<And>r n. (0::'a::{linordered_semidom} star) \<le> r \<Longrightarrow> 0 \<le> r pow n" |
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444 by transfer (rule zero_le_power) |
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445 |
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446 lemma hyperpow_le: |
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447 "\<And>x y n. \<lbrakk>(0::'a::{linordered_semidom} star) < x; x \<le> y\<rbrakk> |
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448 \<Longrightarrow> x pow n \<le> y pow n" |
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449 by transfer (rule power_mono [OF _ order_less_imp_le]) |
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450 |
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451 lemma hyperpow_eq_one [simp]: |
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452 "\<And>n. 1 pow n = (1::'a::monoid_mult star)" |
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453 by transfer (rule power_one) |
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454 |
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455 lemma hrabs_hyperpow_minus [simp]: |
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456 "\<And>(a::'a::{linordered_idom} star) n. \<bar>(-a) pow n\<bar> = \<bar>a pow n\<bar>" |
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457 by transfer (rule abs_power_minus) |
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458 |
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459 lemma hyperpow_mult: |
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460 "\<And>r s n. (r * s::'a::{comm_monoid_mult} star) pow n |
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461 = (r pow n) * (s pow n)" |
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462 by transfer (rule power_mult_distrib) |
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463 |
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464 lemma hyperpow_two_le [simp]: |
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465 "\<And>r. (0::'a::{monoid_mult,linordered_ring_strict} star) \<le> r pow 2" |
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466 by (auto simp add: hyperpow_two zero_le_mult_iff) |
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467 |
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468 lemma hrabs_hyperpow_two [simp]: |
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469 "\<bar>x pow 2\<bar> = |
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470 (x::'a::{monoid_mult,linordered_ring_strict} star) pow 2" |
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471 by (simp only: abs_of_nonneg hyperpow_two_le) |
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472 |
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473 lemma hyperpow_two_hrabs [simp]: |
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474 "\<bar>x::'a::{linordered_idom} star\<bar> pow 2 = x pow 2" |
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475 by (simp add: hyperpow_hrabs) |
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476 |
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477 text\<open>The precondition could be weakened to @{term "0\<le>x"}\<close> |
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478 lemma hypreal_mult_less_mono: |
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479 "[| u<v; x<y; (0::hypreal) < v; 0 < x |] ==> u*x < v* y" |
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480 by (simp add: mult_strict_mono order_less_imp_le) |
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481 |
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482 lemma hyperpow_two_gt_one: |
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483 "\<And>r::'a::{linordered_semidom} star. 1 < r \<Longrightarrow> 1 < r pow 2" |
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484 by transfer simp |
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485 |
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486 lemma hyperpow_two_ge_one: |
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487 "\<And>r::'a::{linordered_semidom} star. 1 \<le> r \<Longrightarrow> 1 \<le> r pow 2" |
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488 by transfer (rule one_le_power) |
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489 |
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490 lemma two_hyperpow_ge_one [simp]: "(1::hypreal) \<le> 2 pow n" |
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491 apply (rule_tac y = "1 pow n" in order_trans) |
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492 apply (rule_tac [2] hyperpow_le, auto) |
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493 done |
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494 |
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495 lemma hyperpow_minus_one2 [simp]: |
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496 "\<And>n. (- 1) pow (2*n) = (1::hypreal)" |
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497 by transfer (rule power_minus1_even) |
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498 |
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499 lemma hyperpow_less_le: |
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500 "!!r n N. [|(0::hypreal) \<le> r; r \<le> 1; n < N|] ==> r pow N \<le> r pow n" |
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501 by transfer (rule power_decreasing [OF order_less_imp_le]) |
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502 |
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503 lemma hyperpow_SHNat_le: |
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504 "[| 0 \<le> r; r \<le> (1::hypreal); N \<in> HNatInfinite |] |
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505 ==> ALL n: Nats. r pow N \<le> r pow n" |
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506 by (auto intro!: hyperpow_less_le simp add: HNatInfinite_iff) |
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507 |
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508 lemma hyperpow_realpow: |
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509 "(hypreal_of_real r) pow (hypnat_of_nat n) = hypreal_of_real (r ^ n)" |
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510 by transfer (rule refl) |
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511 |
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512 lemma hyperpow_SReal [simp]: |
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513 "(hypreal_of_real r) pow (hypnat_of_nat n) \<in> \<real>" |
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514 by (simp add: Reals_eq_Standard) |
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515 |
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516 lemma hyperpow_zero_HNatInfinite [simp]: |
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517 "N \<in> HNatInfinite ==> (0::hypreal) pow N = 0" |
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518 by (drule HNatInfinite_is_Suc, auto) |
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519 |
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520 lemma hyperpow_le_le: |
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521 "[| (0::hypreal) \<le> r; r \<le> 1; n \<le> N |] ==> r pow N \<le> r pow n" |
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522 apply (drule order_le_less [of n, THEN iffD1]) |
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523 apply (auto intro: hyperpow_less_le) |
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524 done |
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525 |
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526 lemma hyperpow_Suc_le_self2: |
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527 "[| (0::hypreal) \<le> r; r < 1 |] ==> r pow (n + (1::hypnat)) \<le> r" |
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528 apply (drule_tac n = " (1::hypnat) " in hyperpow_le_le) |
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529 apply auto |
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530 done |
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531 |
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532 lemma hyperpow_hypnat_of_nat: "\<And>x. x pow hypnat_of_nat n = x ^ n" |
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533 by transfer (rule refl) |
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534 |
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535 lemma of_hypreal_hyperpow: |
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536 "\<And>x n. of_hypreal (x pow n) = |
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537 (of_hypreal x::'a::{real_algebra_1} star) pow n" |
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538 by transfer (rule of_real_power) |
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539 |
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540 end |
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