1 (* Title: Real/RealOrd.thy |
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2 ID: $Id$ |
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3 Author: Jacques D. Fleuriot and Lawrence C. Paulson |
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4 Copyright: 1998 University of Cambridge |
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5 *) |
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6 |
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7 header{*The Reals Form an Ordered Field, etc.*} |
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8 |
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9 theory RealOrd = RealDef: |
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10 |
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11 defs (overloaded) |
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12 real_abs_def: "abs (r::real) == (if 0 \<le> r then r else -r)" |
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13 |
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14 |
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15 |
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16 subsection{*Properties of Less-Than Or Equals*} |
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17 |
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18 lemma real_leI: "~(w < z) ==> z \<le> (w::real)" |
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19 by (unfold real_le_def, assumption) |
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20 |
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21 lemma real_leD: "z\<le>w ==> ~(w<(z::real))" |
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22 by (unfold real_le_def, assumption) |
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23 |
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24 lemma not_real_leE: "~ z \<le> w ==> w<(z::real)" |
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25 by (unfold real_le_def, blast) |
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26 |
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27 lemma real_le_imp_less_or_eq: "!!(x::real). x \<le> y ==> x < y | x = y" |
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28 apply (unfold real_le_def) |
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29 apply (cut_tac real_linear) |
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30 apply (blast elim: real_less_irrefl real_less_asym) |
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31 done |
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32 |
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33 lemma real_less_or_eq_imp_le: "z<w | z=w ==> z \<le>(w::real)" |
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34 apply (unfold real_le_def) |
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35 apply (cut_tac real_linear) |
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36 apply (fast elim: real_less_irrefl real_less_asym) |
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37 done |
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38 |
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39 lemma real_le_less: "(x \<le> (y::real)) = (x < y | x=y)" |
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40 by (blast intro!: real_less_or_eq_imp_le dest!: real_le_imp_less_or_eq) |
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41 |
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42 lemma real_le_refl: "w \<le> (w::real)" |
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43 by (simp add: real_le_less) |
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44 |
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45 lemma real_le_trans: "[| i \<le> j; j \<le> k |] ==> i \<le> (k::real)" |
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46 apply (drule real_le_imp_less_or_eq) |
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47 apply (drule real_le_imp_less_or_eq) |
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48 apply (rule real_less_or_eq_imp_le) |
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49 apply (blast intro: real_less_trans) |
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50 done |
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51 |
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52 lemma real_le_anti_sym: "[| z \<le> w; w \<le> z |] ==> z = (w::real)" |
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53 apply (drule real_le_imp_less_or_eq) |
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54 apply (drule real_le_imp_less_or_eq) |
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55 apply (fast elim: real_less_irrefl real_less_asym) |
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56 done |
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57 |
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58 (* Axiom 'order_less_le' of class 'order': *) |
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59 lemma real_less_le: "((w::real) < z) = (w \<le> z & w \<noteq> z)" |
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60 apply (simp add: real_le_def real_neq_iff) |
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61 apply (blast elim!: real_less_asym) |
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62 done |
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63 |
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64 instance real :: order |
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65 by (intro_classes, |
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66 (assumption | |
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67 rule real_le_refl real_le_trans real_le_anti_sym real_less_le)+) |
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68 |
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69 (* Axiom 'linorder_linear' of class 'linorder': *) |
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70 lemma real_le_linear: "(z::real) \<le> w | w \<le> z" |
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71 apply (simp add: real_le_less) |
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72 apply (cut_tac real_linear, blast) |
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73 done |
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74 |
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75 instance real :: linorder |
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76 by (intro_classes, rule real_le_linear) |
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77 |
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78 |
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79 subsection{*Theorems About the Ordering*} |
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80 |
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81 lemma real_gt_zero_preal_Ex: "(0 < x) = (\<exists>y. x = real_of_preal y)" |
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82 apply (auto simp add: real_of_preal_zero_less) |
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83 apply (cut_tac x = x in real_of_preal_trichotomy) |
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84 apply (blast elim!: real_less_irrefl real_of_preal_not_minus_gt_zero [THEN notE]) |
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85 done |
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86 |
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87 lemma real_gt_preal_preal_Ex: |
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88 "real_of_preal z < x ==> \<exists>y. x = real_of_preal y" |
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89 by (blast dest!: real_of_preal_zero_less [THEN real_less_trans] |
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90 intro: real_gt_zero_preal_Ex [THEN iffD1]) |
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91 |
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92 lemma real_ge_preal_preal_Ex: |
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93 "real_of_preal z \<le> x ==> \<exists>y. x = real_of_preal y" |
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94 by (blast dest: order_le_imp_less_or_eq real_gt_preal_preal_Ex) |
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95 |
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96 lemma real_less_all_preal: "y \<le> 0 ==> \<forall>x. y < real_of_preal x" |
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97 by (auto elim: order_le_imp_less_or_eq [THEN disjE] |
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98 intro: real_of_preal_zero_less [THEN [2] real_less_trans] |
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99 simp add: real_of_preal_zero_less) |
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100 |
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101 lemma real_less_all_real2: "~ 0 < y ==> \<forall>x. y < real_of_preal x" |
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102 by (blast intro!: real_less_all_preal real_leI) |
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103 |
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104 lemma real_of_preal_le_iff: |
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105 "(real_of_preal m1 \<le> real_of_preal m2) = (m1 \<le> m2)" |
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106 apply (auto intro!: preal_leI simp add: linorder_not_less [symmetric]) |
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107 done |
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108 |
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109 |
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110 subsection{*Monotonicity of Addition*} |
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111 |
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112 lemma real_add_left_cancel: "((x::real) + y = x + z) = (y = z)" |
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113 apply safe |
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114 apply (drule_tac f = "%t. (-x) + t" in arg_cong) |
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115 apply (simp add: real_add_assoc [symmetric]) |
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116 done |
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117 |
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118 |
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119 lemma real_mult_order: "[| 0 < x; 0 < y |] ==> (0::real) < x * y" |
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120 apply (auto simp add: real_gt_zero_preal_Ex) |
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121 apply (rule_tac x = "y*ya" in exI) |
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122 apply (simp (no_asm_use) add: real_of_preal_mult) |
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123 done |
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124 |
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125 lemma real_minus_add_distrib [simp]: "-(x + y) = (-x) + (- y :: real)" |
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126 apply (rule_tac z = x in eq_Abs_REAL) |
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127 apply (rule_tac z = y in eq_Abs_REAL) |
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128 apply (auto simp add: real_minus real_add) |
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129 done |
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130 |
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131 (*Alternative definition for real_less*) |
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132 lemma real_less_add_positive_left_Ex: "R < S ==> \<exists>T::real. 0 < T & R + T = S" |
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133 apply (rule_tac x = R in real_of_preal_trichotomyE) |
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134 apply (rule_tac [!] x = S in real_of_preal_trichotomyE) |
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135 apply (auto dest!: preal_less_add_left_Ex simp add: real_of_preal_not_minus_gt_all real_of_preal_add real_of_preal_not_less_zero real_less_not_refl real_of_preal_not_minus_gt_zero) |
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136 apply (rule_tac x = "real_of_preal D" in exI) |
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137 apply (rule_tac [2] x = "real_of_preal m+real_of_preal ma" in exI) |
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138 apply (rule_tac [3] x = "real_of_preal D" in exI) |
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139 apply (auto simp add: real_of_preal_zero_less real_of_preal_sum_zero_less real_add_assoc) |
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140 apply (simp add: real_add_assoc [symmetric]) |
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141 done |
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142 |
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143 lemma real_less_sum_gt_zero: "(W < S) ==> (0 < S + (-W::real))" |
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144 apply (drule real_less_add_positive_left_Ex) |
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145 apply (auto simp add: real_add_minus real_add_zero_right real_add_ac) |
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146 done |
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147 |
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148 lemma real_lemma_change_eq_subj: "!!S::real. T = S + W ==> S = T + (-W)" |
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149 by (simp add: real_add_ac) |
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150 |
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151 (* FIXME: long! *) |
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152 lemma real_sum_gt_zero_less: "(0 < S + (-W::real)) ==> (W < S)" |
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153 apply (rule ccontr) |
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154 apply (drule real_leI [THEN real_le_imp_less_or_eq]) |
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155 apply (auto simp add: real_less_not_refl) |
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156 apply (drule real_less_add_positive_left_Ex, clarify, simp) |
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157 apply (drule real_lemma_change_eq_subj, auto) |
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158 apply (drule real_less_sum_gt_zero) |
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159 apply (auto elim: real_less_asym simp add: real_add_left_commute [of W] real_add_ac) |
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160 done |
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161 |
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162 lemma real_mult_less_mono2: "[| (0::real) < z; x < y |] ==> z * x < z * y" |
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163 apply (rule real_sum_gt_zero_less) |
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164 apply (drule real_less_sum_gt_zero [of x y]) |
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165 apply (drule real_mult_order, assumption) |
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166 apply (simp add: real_add_mult_distrib2) |
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167 done |
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168 |
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169 lemma real_less_sum_gt_0_iff: "(0 < S + (-W::real)) = (W < S)" |
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170 by (blast intro: real_less_sum_gt_zero real_sum_gt_zero_less) |
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171 |
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172 lemma real_less_eq_diff: "(x<y) = (x-y < (0::real))" |
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173 apply (unfold real_diff_def) |
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174 apply (subst real_minus_zero_less_iff [symmetric]) |
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175 apply (simp add: real_add_commute real_less_sum_gt_0_iff) |
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176 done |
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177 |
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178 lemma real_less_eqI: "(x::real) - y = x' - y' ==> (x<y) = (x'<y')" |
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179 apply (subst real_less_eq_diff) |
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180 apply (rule_tac y1 = y in real_less_eq_diff [THEN ssubst], simp) |
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181 done |
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182 |
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183 lemma real_le_eqI: "(x::real) - y = x' - y' ==> (y\<le>x) = (y'\<le>x')" |
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184 apply (drule real_less_eqI) |
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185 apply (simp add: real_le_def) |
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186 done |
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187 |
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188 lemma real_add_left_mono: "x \<le> y ==> z + x \<le> z + (y::real)" |
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189 apply (rule real_le_eqI [THEN iffD1]) |
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190 prefer 2 apply assumption |
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191 apply (simp add: real_diff_def real_add_ac) |
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192 done |
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193 |
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194 |
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195 subsection{*The Reals Form an Ordered Field*} |
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196 |
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197 instance real :: inverse .. |
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198 |
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199 instance real :: ordered_field |
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200 proof |
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201 fix x y z :: real |
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202 show "(x + y) + z = x + (y + z)" by (rule real_add_assoc) |
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203 show "x + y = y + x" by (rule real_add_commute) |
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204 show "0 + x = x" by simp |
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205 show "- x + x = 0" by simp |
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206 show "x - y = x + (-y)" by (simp add: real_diff_def) |
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207 show "(x * y) * z = x * (y * z)" by (rule real_mult_assoc) |
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208 show "x * y = y * x" by (rule real_mult_commute) |
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209 show "1 * x = x" by simp |
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210 show "(x + y) * z = x * z + y * z" by (simp add: real_add_mult_distrib) |
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211 show "0 \<noteq> (1::real)" by (rule real_zero_not_eq_one) |
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212 show "x \<le> y ==> z + x \<le> z + y" by (rule real_add_left_mono) |
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213 show "x < y ==> 0 < z ==> z * x < z * y" by (simp add: real_mult_less_mono2) |
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214 show "\<bar>x\<bar> = (if x < 0 then -x else x)" |
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215 by (auto dest: order_le_less_trans simp add: real_abs_def linorder_not_le) |
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216 show "x \<noteq> 0 ==> inverse x * x = 1" by simp |
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217 show "y \<noteq> 0 ==> x / y = x * inverse y" by (simp add: real_divide_def) |
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218 qed |
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219 |
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220 |
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221 lemma real_zero_less_one: "0 < (1::real)" |
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222 by (rule Ring_and_Field.zero_less_one) |
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223 |
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224 lemma real_add_less_mono: "[| R1 < S1; R2 < S2 |] ==> R1+R2 < S1+(S2::real)" |
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225 by (rule Ring_and_Field.add_strict_mono) |
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226 |
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227 lemma real_add_le_mono: "[|i\<le>j; k\<le>l |] ==> i + k \<le> j + (l::real)" |
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228 by (rule Ring_and_Field.add_mono) |
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229 |
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230 lemma real_le_minus_iff: "(-s \<le> -r) = ((r::real) \<le> s)" |
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231 by (rule Ring_and_Field.neg_le_iff_le) |
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232 |
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233 lemma real_le_square [simp]: "(0::real) \<le> x*x" |
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234 by (rule Ring_and_Field.zero_le_square) |
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235 |
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236 |
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237 subsection{*Division Lemmas*} |
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238 |
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239 (** Inverse of zero! Useful to simplify certain equations **) |
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240 |
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241 lemma INVERSE_ZERO: "inverse 0 = (0::real)" |
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242 apply (unfold real_inverse_def) |
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243 apply (rule someI2) |
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244 apply (auto simp add: real_zero_not_eq_one) |
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245 done |
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246 |
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247 lemma DIVISION_BY_ZERO: "a / (0::real) = 0" |
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248 by (simp add: real_divide_def INVERSE_ZERO) |
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249 |
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250 instance real :: division_by_zero |
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251 proof |
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252 fix x :: real |
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253 show "inverse 0 = (0::real)" by (rule INVERSE_ZERO) |
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254 show "x/0 = 0" by (rule DIVISION_BY_ZERO) |
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255 qed |
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256 |
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257 lemma real_mult_left_cancel: "(c::real) \<noteq> 0 ==> (c*a=c*b) = (a=b)" |
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258 by auto |
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259 |
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260 lemma real_mult_right_cancel: "(c::real) \<noteq> 0 ==> (a*c=b*c) = (a=b)" |
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261 by auto |
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262 |
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263 lemma real_mult_left_cancel_ccontr: "c*a \<noteq> c*b ==> a \<noteq> b" |
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264 by auto |
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265 |
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266 lemma real_mult_right_cancel_ccontr: "a*c \<noteq> b*c ==> a \<noteq> b" |
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267 by auto |
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268 |
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269 lemma real_inverse_not_zero: "x \<noteq> 0 ==> inverse(x::real) \<noteq> 0" |
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270 by (rule Ring_and_Field.nonzero_imp_inverse_nonzero) |
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271 |
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272 lemma real_mult_not_zero: "[| x \<noteq> 0; y \<noteq> 0 |] ==> x * y \<noteq> (0::real)" |
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273 by simp |
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274 |
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275 lemma real_inverse_1: "inverse((1::real)) = (1::real)" |
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276 by (rule Ring_and_Field.inverse_1) |
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277 |
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278 lemma real_minus_inverse: "inverse(-x) = -inverse(x::real)" |
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279 by (rule Ring_and_Field.inverse_minus_eq) |
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280 |
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281 lemma real_inverse_distrib: "inverse(x*y) = inverse(x)*inverse(y::real)" |
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282 by (rule Ring_and_Field.inverse_mult_distrib) |
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283 |
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284 lemma real_add_divide_distrib: "(x+y)/(z::real) = x/z + y/z" |
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285 by (rule Ring_and_Field.add_divide_distrib) |
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286 |
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287 |
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288 subsection{*More Lemmas*} |
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289 |
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290 lemma real_add_right_cancel: "(y + (x::real)= z + x) = (y = z)" |
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291 by (rule Ring_and_Field.add_right_cancel) |
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292 |
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293 lemma real_add_less_mono1: "v < (w::real) ==> v + z < w + z" |
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294 by (rule Ring_and_Field.add_strict_right_mono) |
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295 |
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296 lemma real_add_le_mono1: "v \<le> (w::real) ==> v + z \<le> w + z" |
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297 by (rule Ring_and_Field.add_right_mono) |
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298 |
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299 lemma real_add_less_le_mono: "[| w'<w; z'\<le>z |] ==> w' + z' < w + (z::real)" |
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300 apply (erule add_strict_right_mono [THEN order_less_le_trans]) |
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301 apply (erule add_left_mono) |
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302 done |
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303 |
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304 lemma real_add_le_less_mono: |
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305 "!!z z'::real. [| w'\<le>w; z'<z |] ==> w' + z' < w + z" |
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306 apply (erule add_right_mono [THEN order_le_less_trans]) |
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307 apply (erule add_strict_left_mono) |
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308 done |
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309 |
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310 lemma real_less_add_right_cancel: "!!(A::real). A + C < B + C ==> A < B" |
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311 by (rule Ring_and_Field.add_less_imp_less_right) |
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312 |
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313 lemma real_less_add_left_cancel: "!!(A::real). C + A < C + B ==> A < B" |
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314 by (rule Ring_and_Field.add_less_imp_less_left) |
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315 |
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316 lemma real_le_add_right_cancel: "!!(A::real). A + C \<le> B + C ==> A \<le> B" |
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317 by (rule Ring_and_Field.add_le_imp_le_right) |
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318 |
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319 lemma real_le_add_left_cancel: "!!(A::real). C + A \<le> C + B ==> A \<le> B" |
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320 by (rule Ring_and_Field.add_le_imp_le_left) |
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321 |
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322 lemma real_add_right_cancel_less: "(v+z < w+z) = (v < (w::real))" |
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323 by (rule Ring_and_Field.add_less_cancel_right) |
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324 |
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325 lemma real_add_left_cancel_less: "(z+v < z+w) = (v < (w::real))" |
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326 by (rule Ring_and_Field.add_less_cancel_left) |
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327 |
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328 lemma real_add_right_cancel_le: "(v+z \<le> w+z) = (v \<le> (w::real))" |
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329 by (rule Ring_and_Field.add_le_cancel_right) |
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330 |
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331 lemma real_add_left_cancel_le: "(z+v \<le> z+w) = (v \<le> (w::real))" |
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332 by (rule Ring_and_Field.add_le_cancel_left) |
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333 |
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334 |
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335 subsection{*Inverse and Division*} |
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336 |
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337 lemma real_inverse_gt_0: "0 < x ==> 0 < inverse (x::real)" |
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338 by (rule Ring_and_Field.positive_imp_inverse_positive) |
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339 |
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340 lemma real_inverse_less_0: "x < 0 ==> inverse (x::real) < 0" |
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341 by (rule Ring_and_Field.negative_imp_inverse_negative) |
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342 |
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343 lemma real_mult_less_mono1: "[| (0::real) < z; x < y |] ==> x*z < y*z" |
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344 by (rule Ring_and_Field.mult_strict_right_mono) |
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345 |
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346 text{*The precondition could be weakened to @{term "0\<le>x"}*} |
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347 lemma real_mult_less_mono: |
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348 "[| u<v; x<y; (0::real) < v; 0 < x |] ==> u*x < v* y" |
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349 by (simp add: Ring_and_Field.mult_strict_mono order_less_imp_le) |
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350 |
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351 lemma real_mult_less_iff1 [simp]: "(0::real) < z ==> (x*z < y*z) = (x < y)" |
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352 by (force elim: order_less_asym |
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353 simp add: Ring_and_Field.mult_less_cancel_right) |
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354 |
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355 lemma real_mult_le_cancel_iff1 [simp]: "(0::real) < z ==> (x*z \<le> y*z) = (x\<le>y)" |
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356 by (auto simp add: real_le_def) |
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357 |
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358 lemma real_mult_le_cancel_iff2 [simp]: "(0::real) < z ==> (z*x \<le> z*y) = (x\<le>y)" |
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359 by (force elim: order_less_asym |
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360 simp add: Ring_and_Field.mult_le_cancel_left) |
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361 |
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362 text{*Only two uses?*} |
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363 lemma real_mult_less_mono': |
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364 "[| x < y; r1 < r2; (0::real) \<le> r1; 0 \<le> x|] ==> r1 * x < r2 * y" |
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365 by (rule Ring_and_Field.mult_strict_mono') |
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366 |
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367 lemma real_inverse_less_swap: |
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368 "[| 0 < r; r < x |] ==> inverse x < inverse (r::real)" |
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369 by (rule Ring_and_Field.less_imp_inverse_less) |
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370 |
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371 (*FIXME: remove the [iff], since the general theorem is already [simp]*) |
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372 lemma real_mult_is_0 [iff]: "(x*y = 0) = (x = 0 | y = (0::real))" |
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373 by (rule Ring_and_Field.mult_eq_0_iff) |
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374 |
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375 lemma real_inverse_add: |
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376 "[| x \<noteq> 0; y \<noteq> 0 |] |
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377 ==> inverse x + inverse y = (x + y) * inverse (x * (y::real))" |
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378 by (simp add: Ring_and_Field.inverse_add mult_assoc) |
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379 |
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380 text{*FIXME: delete or at least combine the next two lemmas*} |
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381 lemma real_sum_squares_cancel: "x * x + y * y = 0 ==> x = (0::real)" |
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382 apply (drule Ring_and_Field.equals_zero_I [THEN sym]) |
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383 apply (cut_tac x = y in real_le_square) |
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384 apply (auto, drule real_le_anti_sym, auto) |
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385 done |
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386 |
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387 lemma real_sum_squares_cancel2: "x * x + y * y = 0 ==> y = (0::real)" |
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388 apply (rule_tac y = x in real_sum_squares_cancel) |
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389 apply (simp add: real_add_commute) |
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390 done |
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391 |
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392 |
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393 subsection{*Convenient Biconditionals for Products of Signs*} |
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394 |
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395 lemma real_0_less_mult_iff: "((0::real) < x*y) = (0<x & 0<y | x<0 & y<0)" |
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396 by (rule Ring_and_Field.zero_less_mult_iff) |
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397 |
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398 lemma real_0_le_mult_iff: "((0::real)\<le>x*y) = (0\<le>x & 0\<le>y | x\<le>0 & y\<le>0)" |
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399 by (rule Ring_and_Field.zero_le_mult_iff) |
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400 |
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401 lemma real_mult_less_0_iff: "(x*y < (0::real)) = (0<x & y<0 | x<0 & 0<y)" |
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402 by (rule Ring_and_Field.mult_less_0_iff) |
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403 |
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404 lemma real_mult_le_0_iff: "(x*y \<le> (0::real)) = (0\<le>x & y\<le>0 | x\<le>0 & 0\<le>y)" |
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405 by (rule Ring_and_Field.mult_le_0_iff) |
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406 |
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407 subsection{*Hardly Used Theorems to be Deleted*} |
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408 |
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409 lemma real_add_less_mono2: "!!(A::real). A < B ==> C + A < C + B" |
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410 by simp |
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411 |
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412 lemma real_add_order: "[| 0 < x; 0 < y |] ==> (0::real) < x + y" |
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413 apply (erule order_less_trans) |
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414 apply (drule real_add_less_mono2, simp) |
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415 done |
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416 |
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417 lemma real_le_add_order: "[| 0 \<le> x; 0 \<le> y |] ==> (0::real) \<le> x + y" |
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418 apply (drule order_le_imp_less_or_eq)+ |
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419 apply (auto intro: real_add_order order_less_imp_le) |
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420 done |
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421 |
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422 |
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423 subsection{*An Embedding of the Naturals in the Reals*} |
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424 |
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425 lemma real_of_posnat_one: "real_of_posnat 0 = (1::real)" |
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426 by (simp add: real_of_posnat_def pnat_one_iff [symmetric] |
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427 real_of_preal_def symmetric real_one_def) |
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428 |
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429 lemma real_of_posnat_two: "real_of_posnat (Suc 0) = (1::real) + (1::real)" |
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430 by (simp add: real_of_posnat_def real_of_preal_def real_one_def pnat_two_eq |
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431 real_add |
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432 prat_of_pnat_add [symmetric] preal_of_prat_add [symmetric] |
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433 pnat_add_ac) |
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434 |
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435 lemma real_of_posnat_add: |
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436 "real_of_posnat n1 + real_of_posnat n2 = real_of_posnat (n1 + n2) + (1::real)" |
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437 apply (unfold real_of_posnat_def) |
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438 apply (simp (no_asm_use) add: real_of_posnat_one [symmetric] real_of_posnat_def real_of_preal_add [symmetric] preal_of_prat_add [symmetric] prat_of_pnat_add [symmetric] pnat_of_nat_add) |
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439 done |
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440 |
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441 lemma real_of_posnat_add_one: |
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442 "real_of_posnat (n + 1) = real_of_posnat n + (1::real)" |
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443 apply (rule_tac x1 = " (1::real) " in real_add_right_cancel [THEN iffD1]) |
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444 apply (rule real_of_posnat_add [THEN subst]) |
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445 apply (simp (no_asm_use) add: real_of_posnat_two real_add_assoc) |
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446 done |
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447 |
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448 lemma real_of_posnat_Suc: |
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449 "real_of_posnat (Suc n) = real_of_posnat n + (1::real)" |
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450 by (subst real_of_posnat_add_one [symmetric], simp) |
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451 |
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452 lemma inj_real_of_posnat: "inj(real_of_posnat)" |
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453 apply (rule inj_onI) |
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454 apply (unfold real_of_posnat_def) |
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455 apply (drule inj_real_of_preal [THEN injD]) |
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456 apply (drule inj_preal_of_prat [THEN injD]) |
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457 apply (drule inj_prat_of_pnat [THEN injD]) |
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458 apply (erule inj_pnat_of_nat [THEN injD]) |
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459 done |
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460 |
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461 lemma real_of_nat_zero [simp]: "real (0::nat) = 0" |
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462 by (simp add: real_of_nat_def real_of_posnat_one) |
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463 |
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464 lemma real_of_nat_one [simp]: "real (Suc 0) = (1::real)" |
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465 by (simp add: real_of_nat_def real_of_posnat_two real_add_assoc) |
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466 |
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467 lemma real_of_nat_add [simp]: |
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468 "real (m + n) = real (m::nat) + real n" |
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469 apply (simp add: real_of_nat_def add_ac) |
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470 apply (simp add: real_of_posnat_add add_assoc [symmetric]) |
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471 apply (simp add: add_commute) |
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472 apply (simp add: add_assoc [symmetric]) |
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473 done |
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474 |
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475 (*Not for addsimps: often the LHS is used to represent a positive natural*) |
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476 lemma real_of_nat_Suc: "real (Suc n) = real n + (1::real)" |
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477 by (simp add: real_of_nat_def real_of_posnat_Suc real_add_ac) |
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478 |
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479 lemma real_of_nat_less_iff [iff]: |
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480 "(real (n::nat) < real m) = (n < m)" |
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481 by (auto simp add: real_of_nat_def real_of_posnat_def) |
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482 |
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483 lemma real_of_nat_le_iff [iff]: "(real (n::nat) \<le> real m) = (n \<le> m)" |
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484 by (simp add: linorder_not_less [symmetric]) |
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485 |
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486 lemma inj_real_of_nat: "inj (real :: nat => real)" |
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487 apply (rule inj_onI) |
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488 apply (auto intro!: inj_real_of_posnat [THEN injD] |
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489 simp add: real_of_nat_def real_add_right_cancel) |
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490 done |
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491 |
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492 lemma real_of_nat_ge_zero [iff]: "0 \<le> real (n::nat)" |
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493 apply (induct_tac "n") |
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494 apply (auto simp add: real_of_nat_Suc) |
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495 apply (drule real_add_le_less_mono) |
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496 apply (rule real_zero_less_one) |
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497 apply (simp add: order_less_imp_le) |
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498 done |
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499 |
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500 lemma real_of_nat_mult [simp]: "real (m * n) = real (m::nat) * real n" |
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501 apply (induct_tac "m") |
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502 apply (auto simp add: real_of_nat_Suc real_add_mult_distrib real_add_commute) |
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503 done |
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504 |
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505 lemma real_of_nat_inject [iff]: "(real (n::nat) = real m) = (n = m)" |
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506 by (auto dest: inj_real_of_nat [THEN injD]) |
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507 |
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508 lemma real_of_nat_diff [rule_format]: |
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509 "n \<le> m --> real (m - n) = real (m::nat) - real n" |
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510 apply (induct_tac "m", simp) |
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511 apply (simp add: real_diff_def Suc_diff_le le_Suc_eq real_of_nat_Suc add_ac) |
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512 apply (simp add: add_left_commute [of _ "- 1"]) |
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513 done |
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514 |
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515 lemma real_of_nat_zero_iff: "(real (n::nat) = 0) = (n = 0)" |
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516 proof |
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517 assume "real n = 0" |
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518 have "real n = real (0::nat)" by simp |
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519 then show "n = 0" by (simp only: real_of_nat_inject) |
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520 next |
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521 show "n = 0 \<Longrightarrow> real n = 0" by simp |
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522 qed |
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523 |
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524 lemma real_of_nat_neg_int [simp]: "neg z ==> real (nat z) = 0" |
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525 by (simp add: neg_nat real_of_nat_zero) |
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526 |
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527 |
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528 lemma real_mult_le_le_mono1: "[| (0::real) <=z; x<=y |] ==> z*x<=z*y" |
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529 by (rule Ring_and_Field.mult_left_mono) |
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530 |
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531 lemma real_mult_le_le_mono2: "[| (0::real)<=z; x<=y |] ==> x*z<=y*z" |
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532 by (rule Ring_and_Field.mult_right_mono) |
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533 |
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534 (*Used just below and in HahnBanach/Aux.thy*) |
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535 lemma real_mult_le_less_mono1: "[| (0::real) \<le> z; x < y |] ==> x*z \<le> y*z" |
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536 apply (rule real_less_or_eq_imp_le) |
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537 apply (drule order_le_imp_less_or_eq) |
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538 apply (auto intro: real_mult_less_mono1) |
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539 done |
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540 |
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541 lemma real_inverse_unique: "x*y = (1::real) ==> y = inverse x" |
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542 apply (case_tac "x \<noteq> 0") |
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543 apply (rule_tac c1 = x in real_mult_left_cancel [THEN iffD1], auto) |
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544 done |
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545 |
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546 lemma real_inverse_gt_one: "[| (0::real) < x; x < 1 |] ==> 1 < inverse x" |
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547 by (auto dest: real_inverse_less_swap) |
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548 |
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549 lemma real_of_nat_gt_zero_cancel_iff: "(0 < real (n::nat)) = (0 < n)" |
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550 by (rule real_of_nat_less_iff [THEN subst], auto) |
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551 declare real_of_nat_gt_zero_cancel_iff [simp] |
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552 |
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553 lemma real_of_nat_le_zero_cancel_iff: "(real (n::nat) <= 0) = (n = 0)" |
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554 apply (rule real_of_nat_zero [THEN subst]) |
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555 apply (subst real_of_nat_le_iff, auto) |
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556 done |
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557 declare real_of_nat_le_zero_cancel_iff [simp] |
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558 |
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559 lemma not_real_of_nat_less_zero: "~ real (n::nat) < 0" |
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560 apply (simp (no_asm) add: real_le_def [symmetric] real_of_nat_ge_zero) |
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561 done |
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562 declare not_real_of_nat_less_zero [simp] |
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563 |
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564 lemma real_of_nat_ge_zero_cancel_iff: |
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565 "(0 <= real (n::nat)) = (0 <= n)" |
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566 apply (unfold real_le_def le_def) |
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567 apply (simp (no_asm)) |
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568 done |
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569 declare real_of_nat_ge_zero_cancel_iff [simp] |
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570 |
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571 lemma real_of_nat_num_if: |
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572 "real n = (if n=0 then 0 else 1 + real ((n::nat) - 1))" |
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573 apply (case_tac "n", simp) |
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574 apply (simp add: real_of_nat_Suc add_commute) |
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575 done |
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576 |
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577 lemma real_mult_self_sum_ge_zero: "(0::real) \<le> x*x + y*y" |
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578 proof - |
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579 have "0 + 0 \<le> x*x + y*y" by (blast intro: add_mono zero_le_square) |
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580 thus ?thesis by simp |
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581 qed |
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582 |
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583 declare real_mult_self_sum_ge_zero [simp] |
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584 |
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585 ML |
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586 {* |
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587 val real_abs_def = thm "real_abs_def"; |
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588 |
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589 val real_less_eq_diff = thm "real_less_eq_diff"; |
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590 |
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591 val real_add_right_cancel = thm"real_add_right_cancel"; |
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592 val real_mult_congruent2_lemma = thm"real_mult_congruent2_lemma"; |
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593 val real_mult_congruent2 = thm"real_mult_congruent2"; |
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594 val real_mult = thm"real_mult"; |
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595 val real_mult_commute = thm"real_mult_commute"; |
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596 val real_mult_assoc = thm"real_mult_assoc"; |
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597 val real_mult_left_commute = thm"real_mult_left_commute"; |
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598 val real_mult_1 = thm"real_mult_1"; |
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599 val real_mult_1_right = thm"real_mult_1_right"; |
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600 val real_mult_0 = thm"real_mult_0"; |
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601 val real_mult_0_right = thm"real_mult_0_right"; |
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602 val real_mult_minus_eq1 = thm"real_mult_minus_eq1"; |
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603 val real_minus_mult_eq1 = thm"real_minus_mult_eq1"; |
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604 val real_mult_minus_eq2 = thm"real_mult_minus_eq2"; |
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605 val real_minus_mult_eq2 = thm"real_minus_mult_eq2"; |
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606 val real_mult_minus_1 = thm"real_mult_minus_1"; |
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607 val real_mult_minus_1_right = thm"real_mult_minus_1_right"; |
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608 val real_minus_mult_cancel = thm"real_minus_mult_cancel"; |
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609 val real_minus_mult_commute = thm"real_minus_mult_commute"; |
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610 val real_add_assoc_cong = thm"real_add_assoc_cong"; |
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611 val real_add_mult_distrib = thm"real_add_mult_distrib"; |
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612 val real_add_mult_distrib2 = thm"real_add_mult_distrib2"; |
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613 val real_diff_mult_distrib = thm"real_diff_mult_distrib"; |
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614 val real_diff_mult_distrib2 = thm"real_diff_mult_distrib2"; |
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615 val real_zero_not_eq_one = thm"real_zero_not_eq_one"; |
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616 val real_zero_iff = thm"real_zero_iff"; |
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617 val preal_le_linear = thm"preal_le_linear"; |
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618 val real_mult_inv_right_ex = thm"real_mult_inv_right_ex"; |
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619 val real_mult_inv_left_ex = thm"real_mult_inv_left_ex"; |
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620 val real_mult_inv_left = thm"real_mult_inv_left"; |
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621 val real_mult_inv_right = thm"real_mult_inv_right"; |
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622 val preal_lemma_eq_rev_sum = thm"preal_lemma_eq_rev_sum"; |
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623 val preal_add_left_commute_cancel = thm"preal_add_left_commute_cancel"; |
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624 val preal_lemma_for_not_refl = thm"preal_lemma_for_not_refl"; |
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625 val real_less_not_refl = thm"real_less_not_refl"; |
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626 val real_less_irrefl = thm"real_less_irrefl"; |
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627 val real_not_refl2 = thm"real_not_refl2"; |
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628 val preal_lemma_trans = thm"preal_lemma_trans"; |
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629 val real_less_trans = thm"real_less_trans"; |
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630 val real_less_not_sym = thm"real_less_not_sym"; |
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631 val real_less_asym = thm"real_less_asym"; |
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632 val real_of_preal_add = thm"real_of_preal_add"; |
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633 val real_of_preal_mult = thm"real_of_preal_mult"; |
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634 val real_of_preal_ExI = thm"real_of_preal_ExI"; |
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635 val real_of_preal_ExD = thm"real_of_preal_ExD"; |
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636 val real_of_preal_iff = thm"real_of_preal_iff"; |
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637 val real_of_preal_trichotomy = thm"real_of_preal_trichotomy"; |
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638 val real_of_preal_trichotomyE = thm"real_of_preal_trichotomyE"; |
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639 val real_of_preal_lessD = thm"real_of_preal_lessD"; |
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640 val real_of_preal_lessI = thm"real_of_preal_lessI"; |
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641 val real_of_preal_less_iff1 = thm"real_of_preal_less_iff1"; |
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642 val real_of_preal_minus_less_self = thm"real_of_preal_minus_less_self"; |
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643 val real_of_preal_minus_less_zero = thm"real_of_preal_minus_less_zero"; |
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644 val real_of_preal_not_minus_gt_zero = thm"real_of_preal_not_minus_gt_zero"; |
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645 val real_of_preal_zero_less = thm"real_of_preal_zero_less"; |
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646 val real_of_preal_not_less_zero = thm"real_of_preal_not_less_zero"; |
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647 val real_minus_minus_zero_less = thm"real_minus_minus_zero_less"; |
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648 val real_of_preal_sum_zero_less = thm"real_of_preal_sum_zero_less"; |
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649 val real_of_preal_minus_less_all = thm"real_of_preal_minus_less_all"; |
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650 val real_of_preal_not_minus_gt_all = thm"real_of_preal_not_minus_gt_all"; |
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651 val real_of_preal_minus_less_rev1 = thm"real_of_preal_minus_less_rev1"; |
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652 val real_of_preal_minus_less_rev2 = thm"real_of_preal_minus_less_rev2"; |
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653 val real_of_preal_minus_less_rev_iff = thm"real_of_preal_minus_less_rev_iff"; |
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654 val real_linear = thm"real_linear"; |
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655 val real_neq_iff = thm"real_neq_iff"; |
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656 val real_linear_less2 = thm"real_linear_less2"; |
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657 val real_leI = thm"real_leI"; |
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658 val real_leD = thm"real_leD"; |
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659 val not_real_leE = thm"not_real_leE"; |
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660 val real_le_imp_less_or_eq = thm"real_le_imp_less_or_eq"; |
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661 val real_less_or_eq_imp_le = thm"real_less_or_eq_imp_le"; |
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662 val real_le_less = thm"real_le_less"; |
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663 val real_le_refl = thm"real_le_refl"; |
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664 val real_le_linear = thm"real_le_linear"; |
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665 val real_le_trans = thm"real_le_trans"; |
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666 val real_le_anti_sym = thm"real_le_anti_sym"; |
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667 val real_less_le = thm"real_less_le"; |
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668 val real_minus_zero_less_iff = thm"real_minus_zero_less_iff"; |
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669 val real_minus_zero_less_iff2 = thm"real_minus_zero_less_iff2"; |
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670 val real_less_add_positive_left_Ex = thm"real_less_add_positive_left_Ex"; |
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671 val real_less_sum_gt_zero = thm"real_less_sum_gt_zero"; |
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672 val real_sum_gt_zero_less = thm"real_sum_gt_zero_less"; |
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673 |
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674 val real_gt_zero_preal_Ex = thm "real_gt_zero_preal_Ex"; |
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675 val real_gt_preal_preal_Ex = thm "real_gt_preal_preal_Ex"; |
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676 val real_ge_preal_preal_Ex = thm "real_ge_preal_preal_Ex"; |
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677 val real_less_all_preal = thm "real_less_all_preal"; |
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678 val real_less_all_real2 = thm "real_less_all_real2"; |
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679 val real_of_preal_le_iff = thm "real_of_preal_le_iff"; |
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680 val real_mult_order = thm "real_mult_order"; |
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681 val real_zero_less_one = thm "real_zero_less_one"; |
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682 val real_add_right_cancel_less = thm "real_add_right_cancel_less"; |
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683 val real_add_left_cancel_less = thm "real_add_left_cancel_less"; |
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684 val real_add_right_cancel_le = thm "real_add_right_cancel_le"; |
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685 val real_add_left_cancel_le = thm "real_add_left_cancel_le"; |
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686 val real_add_less_mono1 = thm "real_add_less_mono1"; |
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687 val real_add_le_mono1 = thm "real_add_le_mono1"; |
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688 val real_add_less_le_mono = thm "real_add_less_le_mono"; |
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689 val real_add_le_less_mono = thm "real_add_le_less_mono"; |
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690 val real_add_less_mono2 = thm "real_add_less_mono2"; |
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691 val real_less_add_right_cancel = thm "real_less_add_right_cancel"; |
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692 val real_less_add_left_cancel = thm "real_less_add_left_cancel"; |
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693 val real_le_add_right_cancel = thm "real_le_add_right_cancel"; |
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694 val real_le_add_left_cancel = thm "real_le_add_left_cancel"; |
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695 val real_add_order = thm "real_add_order"; |
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696 val real_le_add_order = thm "real_le_add_order"; |
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697 val real_add_less_mono = thm "real_add_less_mono"; |
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698 val real_add_le_mono = thm "real_add_le_mono"; |
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699 val real_le_minus_iff = thm "real_le_minus_iff"; |
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700 val real_le_square = thm "real_le_square"; |
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701 val real_mult_less_mono1 = thm "real_mult_less_mono1"; |
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702 val real_mult_less_mono2 = thm "real_mult_less_mono2"; |
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703 |
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704 val real_inverse_gt_0 = thm "real_inverse_gt_0"; |
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705 val real_inverse_less_0 = thm "real_inverse_less_0"; |
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706 val real_mult_less_iff1 = thm "real_mult_less_iff1"; |
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707 val real_mult_le_cancel_iff1 = thm "real_mult_le_cancel_iff1"; |
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708 val real_mult_le_cancel_iff2 = thm "real_mult_le_cancel_iff2"; |
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709 val real_mult_less_mono = thm "real_mult_less_mono"; |
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710 val real_mult_less_mono' = thm "real_mult_less_mono'"; |
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711 val real_inverse_less_swap = thm "real_inverse_less_swap"; |
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712 val real_mult_is_0 = thm "real_mult_is_0"; |
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713 val real_inverse_add = thm "real_inverse_add"; |
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714 val real_sum_squares_cancel = thm "real_sum_squares_cancel"; |
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715 val real_sum_squares_cancel2 = thm "real_sum_squares_cancel2"; |
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716 val real_0_less_mult_iff = thm "real_0_less_mult_iff"; |
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717 val real_0_le_mult_iff = thm "real_0_le_mult_iff"; |
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718 val real_mult_less_0_iff = thm "real_mult_less_0_iff"; |
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719 val real_mult_le_0_iff = thm "real_mult_le_0_iff"; |
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720 |
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721 val INVERSE_ZERO = thm"INVERSE_ZERO"; |
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722 val DIVISION_BY_ZERO = thm"DIVISION_BY_ZERO"; |
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723 val real_mult_left_cancel = thm"real_mult_left_cancel"; |
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724 val real_mult_right_cancel = thm"real_mult_right_cancel"; |
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725 val real_mult_left_cancel_ccontr = thm"real_mult_left_cancel_ccontr"; |
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726 val real_mult_right_cancel_ccontr = thm"real_mult_right_cancel_ccontr"; |
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727 val real_inverse_not_zero = thm"real_inverse_not_zero"; |
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728 val real_mult_not_zero = thm"real_mult_not_zero"; |
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729 val real_inverse_1 = thm"real_inverse_1"; |
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730 val real_minus_inverse = thm"real_minus_inverse"; |
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731 val real_inverse_distrib = thm"real_inverse_distrib"; |
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732 val real_add_divide_distrib = thm"real_add_divide_distrib"; |
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733 |
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734 val real_of_posnat_one = thm "real_of_posnat_one"; |
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735 val real_of_posnat_two = thm "real_of_posnat_two"; |
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736 val real_of_posnat_add = thm "real_of_posnat_add"; |
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737 val real_of_posnat_add_one = thm "real_of_posnat_add_one"; |
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738 val real_of_posnat_Suc = thm "real_of_posnat_Suc"; |
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739 val inj_real_of_posnat = thm "inj_real_of_posnat"; |
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740 val real_of_nat_zero = thm "real_of_nat_zero"; |
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741 val real_of_nat_one = thm "real_of_nat_one"; |
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742 val real_of_nat_add = thm "real_of_nat_add"; |
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743 val real_of_nat_Suc = thm "real_of_nat_Suc"; |
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744 val real_of_nat_less_iff = thm "real_of_nat_less_iff"; |
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745 val real_of_nat_le_iff = thm "real_of_nat_le_iff"; |
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746 val inj_real_of_nat = thm "inj_real_of_nat"; |
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747 val real_of_nat_ge_zero = thm "real_of_nat_ge_zero"; |
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748 val real_of_nat_mult = thm "real_of_nat_mult"; |
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749 val real_of_nat_inject = thm "real_of_nat_inject"; |
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750 val real_of_nat_diff = thm "real_of_nat_diff"; |
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751 val real_of_nat_zero_iff = thm "real_of_nat_zero_iff"; |
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752 val real_of_nat_neg_int = thm "real_of_nat_neg_int"; |
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753 |
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754 val real_mult_le_le_mono1 = thm "real_mult_le_le_mono1"; |
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755 val real_mult_le_le_mono2 = thm "real_mult_le_le_mono2"; |
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756 val real_inverse_unique = thm "real_inverse_unique"; |
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757 val real_inverse_gt_one = thm "real_inverse_gt_one"; |
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758 val real_of_nat_gt_zero_cancel_iff = thm "real_of_nat_gt_zero_cancel_iff"; |
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759 val real_of_nat_le_zero_cancel_iff = thm "real_of_nat_le_zero_cancel_iff"; |
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760 val not_real_of_nat_less_zero = thm "not_real_of_nat_less_zero"; |
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761 val real_of_nat_ge_zero_cancel_iff = thm "real_of_nat_ge_zero_cancel_iff"; |
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762 val real_of_nat_num_if = thm "real_of_nat_num_if"; |
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763 |
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764 val real_minus_add_distrib = thm"real_minus_add_distrib"; |
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765 val real_add_left_cancel = thm"real_add_left_cancel"; |
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766 val real_mult_self_sum_ge_zero = thm "real_mult_self_sum_ge_zero"; |
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767 *} |
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768 |
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769 end |
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