1 (* Title : HOL/Real/RComplete.thy |
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2 ID : $Id$ |
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3 Author : Jacques D. Fleuriot, University of Edinburgh |
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4 Author : Larry Paulson, University of Cambridge |
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5 Author : Jeremy Avigad, Carnegie Mellon University |
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6 Author : Florian Zuleger, Johannes Hoelzl, and Simon Funke, TU Muenchen |
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7 *) |
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8 |
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9 header {* Completeness of the Reals; Floor and Ceiling Functions *} |
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10 |
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11 theory RComplete |
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12 imports Lubs RealDef |
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13 begin |
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14 |
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15 lemma real_sum_of_halves: "x/2 + x/2 = (x::real)" |
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16 by simp |
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17 |
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18 |
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19 subsection {* Completeness of Positive Reals *} |
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20 |
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21 text {* |
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22 Supremum property for the set of positive reals |
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23 |
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24 Let @{text "P"} be a non-empty set of positive reals, with an upper |
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25 bound @{text "y"}. Then @{text "P"} has a least upper bound |
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26 (written @{text "S"}). |
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27 |
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28 FIXME: Can the premise be weakened to @{text "\<forall>x \<in> P. x\<le> y"}? |
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29 *} |
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30 |
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31 lemma posreal_complete: |
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32 assumes positive_P: "\<forall>x \<in> P. (0::real) < x" |
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33 and not_empty_P: "\<exists>x. x \<in> P" |
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34 and upper_bound_Ex: "\<exists>y. \<forall>x \<in> P. x<y" |
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35 shows "\<exists>S. \<forall>y. (\<exists>x \<in> P. y < x) = (y < S)" |
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36 proof (rule exI, rule allI) |
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37 fix y |
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38 let ?pP = "{w. real_of_preal w \<in> P}" |
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39 |
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40 show "(\<exists>x\<in>P. y < x) = (y < real_of_preal (psup ?pP))" |
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41 proof (cases "0 < y") |
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42 assume neg_y: "\<not> 0 < y" |
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43 show ?thesis |
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44 proof |
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45 assume "\<exists>x\<in>P. y < x" |
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46 have "\<forall>x. y < real_of_preal x" |
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47 using neg_y by (rule real_less_all_real2) |
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48 thus "y < real_of_preal (psup ?pP)" .. |
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49 next |
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50 assume "y < real_of_preal (psup ?pP)" |
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51 obtain "x" where x_in_P: "x \<in> P" using not_empty_P .. |
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52 hence "0 < x" using positive_P by simp |
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53 hence "y < x" using neg_y by simp |
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54 thus "\<exists>x \<in> P. y < x" using x_in_P .. |
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55 qed |
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56 next |
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57 assume pos_y: "0 < y" |
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58 |
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59 then obtain py where y_is_py: "y = real_of_preal py" |
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60 by (auto simp add: real_gt_zero_preal_Ex) |
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61 |
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62 obtain a where "a \<in> P" using not_empty_P .. |
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63 with positive_P have a_pos: "0 < a" .. |
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64 then obtain pa where "a = real_of_preal pa" |
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65 by (auto simp add: real_gt_zero_preal_Ex) |
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66 hence "pa \<in> ?pP" using `a \<in> P` by auto |
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67 hence pP_not_empty: "?pP \<noteq> {}" by auto |
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68 |
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69 obtain sup where sup: "\<forall>x \<in> P. x < sup" |
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70 using upper_bound_Ex .. |
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71 from this and `a \<in> P` have "a < sup" .. |
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72 hence "0 < sup" using a_pos by arith |
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73 then obtain possup where "sup = real_of_preal possup" |
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74 by (auto simp add: real_gt_zero_preal_Ex) |
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75 hence "\<forall>X \<in> ?pP. X \<le> possup" |
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76 using sup by (auto simp add: real_of_preal_lessI) |
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77 with pP_not_empty have psup: "\<And>Z. (\<exists>X \<in> ?pP. Z < X) = (Z < psup ?pP)" |
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78 by (rule preal_complete) |
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79 |
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80 show ?thesis |
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81 proof |
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82 assume "\<exists>x \<in> P. y < x" |
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83 then obtain x where x_in_P: "x \<in> P" and y_less_x: "y < x" .. |
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84 hence "0 < x" using pos_y by arith |
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85 then obtain px where x_is_px: "x = real_of_preal px" |
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86 by (auto simp add: real_gt_zero_preal_Ex) |
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87 |
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88 have py_less_X: "\<exists>X \<in> ?pP. py < X" |
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89 proof |
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90 show "py < px" using y_is_py and x_is_px and y_less_x |
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91 by (simp add: real_of_preal_lessI) |
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92 show "px \<in> ?pP" using x_in_P and x_is_px by simp |
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93 qed |
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94 |
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95 have "(\<exists>X \<in> ?pP. py < X) ==> (py < psup ?pP)" |
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96 using psup by simp |
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97 hence "py < psup ?pP" using py_less_X by simp |
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98 thus "y < real_of_preal (psup {w. real_of_preal w \<in> P})" |
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99 using y_is_py and pos_y by (simp add: real_of_preal_lessI) |
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100 next |
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101 assume y_less_psup: "y < real_of_preal (psup ?pP)" |
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102 |
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103 hence "py < psup ?pP" using y_is_py |
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104 by (simp add: real_of_preal_lessI) |
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105 then obtain "X" where py_less_X: "py < X" and X_in_pP: "X \<in> ?pP" |
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106 using psup by auto |
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107 then obtain x where x_is_X: "x = real_of_preal X" |
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108 by (simp add: real_gt_zero_preal_Ex) |
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109 hence "y < x" using py_less_X and y_is_py |
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110 by (simp add: real_of_preal_lessI) |
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111 |
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112 moreover have "x \<in> P" using x_is_X and X_in_pP by simp |
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113 |
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114 ultimately show "\<exists> x \<in> P. y < x" .. |
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115 qed |
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116 qed |
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117 qed |
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118 |
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119 text {* |
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120 \medskip Completeness properties using @{text "isUb"}, @{text "isLub"} etc. |
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121 *} |
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122 |
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123 lemma real_isLub_unique: "[| isLub R S x; isLub R S y |] ==> x = (y::real)" |
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124 apply (frule isLub_isUb) |
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125 apply (frule_tac x = y in isLub_isUb) |
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126 apply (blast intro!: order_antisym dest!: isLub_le_isUb) |
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127 done |
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128 |
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129 |
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130 text {* |
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131 \medskip Completeness theorem for the positive reals (again). |
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132 *} |
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133 |
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134 lemma posreals_complete: |
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135 assumes positive_S: "\<forall>x \<in> S. 0 < x" |
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136 and not_empty_S: "\<exists>x. x \<in> S" |
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137 and upper_bound_Ex: "\<exists>u. isUb (UNIV::real set) S u" |
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138 shows "\<exists>t. isLub (UNIV::real set) S t" |
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139 proof |
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140 let ?pS = "{w. real_of_preal w \<in> S}" |
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141 |
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142 obtain u where "isUb UNIV S u" using upper_bound_Ex .. |
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143 hence sup: "\<forall>x \<in> S. x \<le> u" by (simp add: isUb_def setle_def) |
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144 |
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145 obtain x where x_in_S: "x \<in> S" using not_empty_S .. |
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146 hence x_gt_zero: "0 < x" using positive_S by simp |
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147 have "x \<le> u" using sup and x_in_S .. |
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148 hence "0 < u" using x_gt_zero by arith |
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149 |
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150 then obtain pu where u_is_pu: "u = real_of_preal pu" |
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151 by (auto simp add: real_gt_zero_preal_Ex) |
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152 |
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153 have pS_less_pu: "\<forall>pa \<in> ?pS. pa \<le> pu" |
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154 proof |
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155 fix pa |
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156 assume "pa \<in> ?pS" |
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157 then obtain a where "a \<in> S" and "a = real_of_preal pa" |
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158 by simp |
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159 moreover hence "a \<le> u" using sup by simp |
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160 ultimately show "pa \<le> pu" |
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161 using sup and u_is_pu by (simp add: real_of_preal_le_iff) |
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162 qed |
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163 |
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164 have "\<forall>y \<in> S. y \<le> real_of_preal (psup ?pS)" |
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165 proof |
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166 fix y |
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167 assume y_in_S: "y \<in> S" |
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168 hence "0 < y" using positive_S by simp |
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169 then obtain py where y_is_py: "y = real_of_preal py" |
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170 by (auto simp add: real_gt_zero_preal_Ex) |
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171 hence py_in_pS: "py \<in> ?pS" using y_in_S by simp |
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172 with pS_less_pu have "py \<le> psup ?pS" |
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173 by (rule preal_psup_le) |
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174 thus "y \<le> real_of_preal (psup ?pS)" |
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175 using y_is_py by (simp add: real_of_preal_le_iff) |
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176 qed |
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177 |
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178 moreover { |
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179 fix x |
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180 assume x_ub_S: "\<forall>y\<in>S. y \<le> x" |
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181 have "real_of_preal (psup ?pS) \<le> x" |
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182 proof - |
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183 obtain "s" where s_in_S: "s \<in> S" using not_empty_S .. |
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184 hence s_pos: "0 < s" using positive_S by simp |
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185 |
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186 hence "\<exists> ps. s = real_of_preal ps" by (simp add: real_gt_zero_preal_Ex) |
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187 then obtain "ps" where s_is_ps: "s = real_of_preal ps" .. |
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188 hence ps_in_pS: "ps \<in> {w. real_of_preal w \<in> S}" using s_in_S by simp |
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189 |
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190 from x_ub_S have "s \<le> x" using s_in_S .. |
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191 hence "0 < x" using s_pos by simp |
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192 hence "\<exists> px. x = real_of_preal px" by (simp add: real_gt_zero_preal_Ex) |
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193 then obtain "px" where x_is_px: "x = real_of_preal px" .. |
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194 |
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195 have "\<forall>pe \<in> ?pS. pe \<le> px" |
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196 proof |
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197 fix pe |
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198 assume "pe \<in> ?pS" |
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199 hence "real_of_preal pe \<in> S" by simp |
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200 hence "real_of_preal pe \<le> x" using x_ub_S by simp |
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201 thus "pe \<le> px" using x_is_px by (simp add: real_of_preal_le_iff) |
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202 qed |
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203 |
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204 moreover have "?pS \<noteq> {}" using ps_in_pS by auto |
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205 ultimately have "(psup ?pS) \<le> px" by (simp add: psup_le_ub) |
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206 thus "real_of_preal (psup ?pS) \<le> x" using x_is_px by (simp add: real_of_preal_le_iff) |
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207 qed |
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208 } |
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209 ultimately show "isLub UNIV S (real_of_preal (psup ?pS))" |
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210 by (simp add: isLub_def leastP_def isUb_def setle_def setge_def) |
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211 qed |
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212 |
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213 text {* |
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214 \medskip reals Completeness (again!) |
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215 *} |
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216 |
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217 lemma reals_complete: |
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218 assumes notempty_S: "\<exists>X. X \<in> S" |
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219 and exists_Ub: "\<exists>Y. isUb (UNIV::real set) S Y" |
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220 shows "\<exists>t. isLub (UNIV :: real set) S t" |
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221 proof - |
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222 obtain X where X_in_S: "X \<in> S" using notempty_S .. |
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223 obtain Y where Y_isUb: "isUb (UNIV::real set) S Y" |
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224 using exists_Ub .. |
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225 let ?SHIFT = "{z. \<exists>x \<in>S. z = x + (-X) + 1} \<inter> {x. 0 < x}" |
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226 |
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227 { |
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228 fix x |
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229 assume "isUb (UNIV::real set) S x" |
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230 hence S_le_x: "\<forall> y \<in> S. y <= x" |
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231 by (simp add: isUb_def setle_def) |
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232 { |
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233 fix s |
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234 assume "s \<in> {z. \<exists>x\<in>S. z = x + - X + 1}" |
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235 hence "\<exists> x \<in> S. s = x + -X + 1" .. |
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236 then obtain x1 where "x1 \<in> S" and "s = x1 + (-X) + 1" .. |
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237 moreover hence "x1 \<le> x" using S_le_x by simp |
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238 ultimately have "s \<le> x + - X + 1" by arith |
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239 } |
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240 then have "isUb (UNIV::real set) ?SHIFT (x + (-X) + 1)" |
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241 by (auto simp add: isUb_def setle_def) |
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242 } note S_Ub_is_SHIFT_Ub = this |
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243 |
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244 hence "isUb UNIV ?SHIFT (Y + (-X) + 1)" using Y_isUb by simp |
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245 hence "\<exists>Z. isUb UNIV ?SHIFT Z" .. |
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246 moreover have "\<forall>y \<in> ?SHIFT. 0 < y" by auto |
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247 moreover have shifted_not_empty: "\<exists>u. u \<in> ?SHIFT" |
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248 using X_in_S and Y_isUb by auto |
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249 ultimately obtain t where t_is_Lub: "isLub UNIV ?SHIFT t" |
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250 using posreals_complete [of ?SHIFT] by blast |
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251 |
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252 show ?thesis |
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253 proof |
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254 show "isLub UNIV S (t + X + (-1))" |
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255 proof (rule isLubI2) |
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256 { |
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257 fix x |
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258 assume "isUb (UNIV::real set) S x" |
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259 hence "isUb (UNIV::real set) (?SHIFT) (x + (-X) + 1)" |
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260 using S_Ub_is_SHIFT_Ub by simp |
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261 hence "t \<le> (x + (-X) + 1)" |
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262 using t_is_Lub by (simp add: isLub_le_isUb) |
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263 hence "t + X + -1 \<le> x" by arith |
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264 } |
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265 then show "(t + X + -1) <=* Collect (isUb UNIV S)" |
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266 by (simp add: setgeI) |
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267 next |
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268 show "isUb UNIV S (t + X + -1)" |
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269 proof - |
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270 { |
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271 fix y |
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272 assume y_in_S: "y \<in> S" |
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273 have "y \<le> t + X + -1" |
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274 proof - |
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275 obtain "u" where u_in_shift: "u \<in> ?SHIFT" using shifted_not_empty .. |
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276 hence "\<exists> x \<in> S. u = x + - X + 1" by simp |
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277 then obtain "x" where x_and_u: "u = x + - X + 1" .. |
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278 have u_le_t: "u \<le> t" using u_in_shift and t_is_Lub by (simp add: isLubD2) |
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279 |
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280 show ?thesis |
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281 proof cases |
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282 assume "y \<le> x" |
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283 moreover have "x = u + X + - 1" using x_and_u by arith |
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284 moreover have "u + X + - 1 \<le> t + X + -1" using u_le_t by arith |
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285 ultimately show "y \<le> t + X + -1" by arith |
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286 next |
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287 assume "~(y \<le> x)" |
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288 hence x_less_y: "x < y" by arith |
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289 |
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290 have "x + (-X) + 1 \<in> ?SHIFT" using x_and_u and u_in_shift by simp |
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291 hence "0 < x + (-X) + 1" by simp |
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292 hence "0 < y + (-X) + 1" using x_less_y by arith |
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293 hence "y + (-X) + 1 \<in> ?SHIFT" using y_in_S by simp |
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294 hence "y + (-X) + 1 \<le> t" using t_is_Lub by (simp add: isLubD2) |
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295 thus ?thesis by simp |
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296 qed |
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297 qed |
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298 } |
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299 then show ?thesis by (simp add: isUb_def setle_def) |
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300 qed |
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301 qed |
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302 qed |
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303 qed |
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304 |
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305 |
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306 subsection {* The Archimedean Property of the Reals *} |
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307 |
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308 theorem reals_Archimedean: |
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309 assumes x_pos: "0 < x" |
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310 shows "\<exists>n. inverse (real (Suc n)) < x" |
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311 proof (rule ccontr) |
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312 assume contr: "\<not> ?thesis" |
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313 have "\<forall>n. x * real (Suc n) <= 1" |
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314 proof |
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315 fix n |
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316 from contr have "x \<le> inverse (real (Suc n))" |
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317 by (simp add: linorder_not_less) |
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318 hence "x \<le> (1 / (real (Suc n)))" |
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319 by (simp add: inverse_eq_divide) |
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320 moreover have "0 \<le> real (Suc n)" |
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321 by (rule real_of_nat_ge_zero) |
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322 ultimately have "x * real (Suc n) \<le> (1 / real (Suc n)) * real (Suc n)" |
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323 by (rule mult_right_mono) |
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324 thus "x * real (Suc n) \<le> 1" by simp |
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325 qed |
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326 hence "{z. \<exists>n. z = x * (real (Suc n))} *<= 1" |
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327 by (simp add: setle_def, safe, rule spec) |
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328 hence "isUb (UNIV::real set) {z. \<exists>n. z = x * (real (Suc n))} 1" |
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329 by (simp add: isUbI) |
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330 hence "\<exists>Y. isUb (UNIV::real set) {z. \<exists>n. z = x* (real (Suc n))} Y" .. |
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331 moreover have "\<exists>X. X \<in> {z. \<exists>n. z = x* (real (Suc n))}" by auto |
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332 ultimately have "\<exists>t. isLub UNIV {z. \<exists>n. z = x * real (Suc n)} t" |
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333 by (simp add: reals_complete) |
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334 then obtain "t" where |
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335 t_is_Lub: "isLub UNIV {z. \<exists>n. z = x * real (Suc n)} t" .. |
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336 |
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337 have "\<forall>n::nat. x * real n \<le> t + - x" |
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338 proof |
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339 fix n |
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340 from t_is_Lub have "x * real (Suc n) \<le> t" |
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341 by (simp add: isLubD2) |
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342 hence "x * (real n) + x \<le> t" |
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343 by (simp add: right_distrib real_of_nat_Suc) |
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344 thus "x * (real n) \<le> t + - x" by arith |
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345 qed |
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346 |
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347 hence "\<forall>m. x * real (Suc m) \<le> t + - x" by simp |
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348 hence "{z. \<exists>n. z = x * (real (Suc n))} *<= (t + - x)" |
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349 by (auto simp add: setle_def) |
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350 hence "isUb (UNIV::real set) {z. \<exists>n. z = x * (real (Suc n))} (t + (-x))" |
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351 by (simp add: isUbI) |
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352 hence "t \<le> t + - x" |
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353 using t_is_Lub by (simp add: isLub_le_isUb) |
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354 thus False using x_pos by arith |
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355 qed |
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356 |
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357 text {* |
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358 There must be other proofs, e.g. @{text "Suc"} of the largest |
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359 integer in the cut representing @{text "x"}. |
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360 *} |
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361 |
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362 lemma reals_Archimedean2: "\<exists>n. (x::real) < real (n::nat)" |
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363 proof cases |
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364 assume "x \<le> 0" |
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365 hence "x < real (1::nat)" by simp |
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366 thus ?thesis .. |
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367 next |
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368 assume "\<not> x \<le> 0" |
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369 hence x_greater_zero: "0 < x" by simp |
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370 hence "0 < inverse x" by simp |
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371 then obtain n where "inverse (real (Suc n)) < inverse x" |
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372 using reals_Archimedean by blast |
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373 hence "inverse (real (Suc n)) * x < inverse x * x" |
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374 using x_greater_zero by (rule mult_strict_right_mono) |
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375 hence "inverse (real (Suc n)) * x < 1" |
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376 using x_greater_zero by simp |
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377 hence "real (Suc n) * (inverse (real (Suc n)) * x) < real (Suc n) * 1" |
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378 by (rule mult_strict_left_mono) simp |
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379 hence "x < real (Suc n)" |
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380 by (simp add: ring_simps) |
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381 thus "\<exists>(n::nat). x < real n" .. |
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382 qed |
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383 |
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384 lemma reals_Archimedean3: |
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385 assumes x_greater_zero: "0 < x" |
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386 shows "\<forall>(y::real). \<exists>(n::nat). y < real n * x" |
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387 proof |
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388 fix y |
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389 have x_not_zero: "x \<noteq> 0" using x_greater_zero by simp |
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390 obtain n where "y * inverse x < real (n::nat)" |
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391 using reals_Archimedean2 .. |
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392 hence "y * inverse x * x < real n * x" |
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393 using x_greater_zero by (simp add: mult_strict_right_mono) |
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394 hence "x * inverse x * y < x * real n" |
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395 by (simp add: ring_simps) |
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396 hence "y < real (n::nat) * x" |
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397 using x_not_zero by (simp add: ring_simps) |
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398 thus "\<exists>(n::nat). y < real n * x" .. |
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399 qed |
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400 |
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401 lemma reals_Archimedean6: |
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402 "0 \<le> r ==> \<exists>(n::nat). real (n - 1) \<le> r & r < real (n)" |
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403 apply (insert reals_Archimedean2 [of r], safe) |
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404 apply (subgoal_tac "\<exists>x::nat. r < real x \<and> (\<forall>y. r < real y \<longrightarrow> x \<le> y)", auto) |
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405 apply (rule_tac x = x in exI) |
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406 apply (case_tac x, simp) |
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407 apply (rename_tac x') |
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408 apply (drule_tac x = x' in spec, simp) |
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409 apply (rule_tac x="LEAST n. r < real n" in exI, safe) |
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410 apply (erule LeastI, erule Least_le) |
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411 done |
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412 |
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413 lemma reals_Archimedean6a: "0 \<le> r ==> \<exists>n. real (n) \<le> r & r < real (Suc n)" |
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414 by (drule reals_Archimedean6) auto |
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415 |
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416 lemma reals_Archimedean_6b_int: |
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417 "0 \<le> r ==> \<exists>n::int. real n \<le> r & r < real (n+1)" |
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418 apply (drule reals_Archimedean6a, auto) |
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419 apply (rule_tac x = "int n" in exI) |
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420 apply (simp add: real_of_int_real_of_nat real_of_nat_Suc) |
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421 done |
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422 |
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423 lemma reals_Archimedean_6c_int: |
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424 "r < 0 ==> \<exists>n::int. real n \<le> r & r < real (n+1)" |
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425 apply (rule reals_Archimedean_6b_int [of "-r", THEN exE], simp, auto) |
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426 apply (rename_tac n) |
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427 apply (drule order_le_imp_less_or_eq, auto) |
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428 apply (rule_tac x = "- n - 1" in exI) |
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429 apply (rule_tac [2] x = "- n" in exI, auto) |
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430 done |
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431 |
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432 |
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433 subsection{*Density of the Rational Reals in the Reals*} |
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434 |
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435 text{* This density proof is due to Stefan Richter and was ported by TN. The |
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436 original source is \emph{Real Analysis} by H.L. Royden. |
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437 It employs the Archimedean property of the reals. *} |
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438 |
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439 lemma Rats_dense_in_nn_real: fixes x::real |
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440 assumes "0\<le>x" and "x<y" shows "\<exists>r \<in> \<rat>. x<r \<and> r<y" |
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441 proof - |
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442 from `x<y` have "0 < y-x" by simp |
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443 with reals_Archimedean obtain q::nat |
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444 where q: "inverse (real q) < y-x" and "0 < real q" by auto |
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445 def p \<equiv> "LEAST n. y \<le> real (Suc n)/real q" |
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446 from reals_Archimedean2 obtain n::nat where "y * real q < real n" by auto |
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447 with `0 < real q` have ex: "y \<le> real n/real q" (is "?P n") |
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448 by (simp add: pos_less_divide_eq[THEN sym]) |
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449 also from assms have "\<not> y \<le> real (0::nat) / real q" by simp |
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450 ultimately have main: "(LEAST n. y \<le> real n/real q) = Suc p" |
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451 by (unfold p_def) (rule Least_Suc) |
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452 also from ex have "?P (LEAST x. ?P x)" by (rule LeastI) |
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453 ultimately have suc: "y \<le> real (Suc p) / real q" by simp |
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454 def r \<equiv> "real p/real q" |
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455 have "x = y-(y-x)" by simp |
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456 also from suc q have "\<dots> < real (Suc p)/real q - inverse (real q)" by arith |
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457 also have "\<dots> = real p / real q" |
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458 by (simp only: inverse_eq_divide real_diff_def real_of_nat_Suc |
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459 minus_divide_left add_divide_distrib[THEN sym]) simp |
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460 finally have "x<r" by (unfold r_def) |
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461 have "p<Suc p" .. also note main[THEN sym] |
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462 finally have "\<not> ?P p" by (rule not_less_Least) |
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463 hence "r<y" by (simp add: r_def) |
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464 from r_def have "r \<in> \<rat>" by simp |
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465 with `x<r` `r<y` show ?thesis by fast |
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466 qed |
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467 |
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468 theorem Rats_dense_in_real: fixes x y :: real |
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469 assumes "x<y" shows "\<exists>r \<in> \<rat>. x<r \<and> r<y" |
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470 proof - |
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471 from reals_Archimedean2 obtain n::nat where "-x < real n" by auto |
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472 hence "0 \<le> x + real n" by arith |
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473 also from `x<y` have "x + real n < y + real n" by arith |
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474 ultimately have "\<exists>r \<in> \<rat>. x + real n < r \<and> r < y + real n" |
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475 by(rule Rats_dense_in_nn_real) |
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476 then obtain r where "r \<in> \<rat>" and r2: "x + real n < r" |
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477 and r3: "r < y + real n" |
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478 by blast |
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479 have "r - real n = r + real (int n)/real (-1::int)" by simp |
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480 also from `r\<in>\<rat>` have "r + real (int n)/real (-1::int) \<in> \<rat>" by simp |
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481 also from r2 have "x < r - real n" by arith |
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482 moreover from r3 have "r - real n < y" by arith |
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483 ultimately show ?thesis by fast |
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484 qed |
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485 |
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486 |
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487 subsection{*Floor and Ceiling Functions from the Reals to the Integers*} |
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488 |
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489 definition |
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490 floor :: "real => int" where |
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491 [code del]: "floor r = (LEAST n::int. r < real (n+1))" |
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492 |
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493 definition |
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494 ceiling :: "real => int" where |
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495 "ceiling r = - floor (- r)" |
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496 |
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497 notation (xsymbols) |
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498 floor ("\<lfloor>_\<rfloor>") and |
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499 ceiling ("\<lceil>_\<rceil>") |
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500 |
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501 notation (HTML output) |
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502 floor ("\<lfloor>_\<rfloor>") and |
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503 ceiling ("\<lceil>_\<rceil>") |
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504 |
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505 |
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506 lemma number_of_less_real_of_int_iff [simp]: |
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507 "((number_of n) < real (m::int)) = (number_of n < m)" |
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508 apply auto |
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509 apply (rule real_of_int_less_iff [THEN iffD1]) |
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510 apply (drule_tac [2] real_of_int_less_iff [THEN iffD2], auto) |
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511 done |
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512 |
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513 lemma number_of_less_real_of_int_iff2 [simp]: |
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514 "(real (m::int) < (number_of n)) = (m < number_of n)" |
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515 apply auto |
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516 apply (rule real_of_int_less_iff [THEN iffD1]) |
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517 apply (drule_tac [2] real_of_int_less_iff [THEN iffD2], auto) |
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518 done |
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519 |
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520 lemma number_of_le_real_of_int_iff [simp]: |
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521 "((number_of n) \<le> real (m::int)) = (number_of n \<le> m)" |
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522 by (simp add: linorder_not_less [symmetric]) |
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523 |
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524 lemma number_of_le_real_of_int_iff2 [simp]: |
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525 "(real (m::int) \<le> (number_of n)) = (m \<le> number_of n)" |
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526 by (simp add: linorder_not_less [symmetric]) |
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527 |
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528 lemma floor_zero [simp]: "floor 0 = 0" |
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529 apply (simp add: floor_def del: real_of_int_add) |
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530 apply (rule Least_equality) |
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531 apply simp_all |
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532 done |
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533 |
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534 lemma floor_real_of_nat_zero [simp]: "floor (real (0::nat)) = 0" |
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535 by auto |
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536 |
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537 lemma floor_real_of_nat [simp]: "floor (real (n::nat)) = int n" |
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538 apply (simp only: floor_def) |
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539 apply (rule Least_equality) |
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540 apply (drule_tac [2] real_of_int_of_nat_eq [THEN ssubst]) |
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541 apply (drule_tac [2] real_of_int_less_iff [THEN iffD1]) |
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542 apply simp_all |
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543 done |
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544 |
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545 lemma floor_minus_real_of_nat [simp]: "floor (- real (n::nat)) = - int n" |
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546 apply (simp only: floor_def) |
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547 apply (rule Least_equality) |
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548 apply (drule_tac [2] real_of_int_of_nat_eq [THEN ssubst]) |
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549 apply (drule_tac [2] real_of_int_minus [THEN sym, THEN subst]) |
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550 apply (drule_tac [2] real_of_int_less_iff [THEN iffD1]) |
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551 apply simp_all |
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552 done |
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553 |
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554 lemma floor_real_of_int [simp]: "floor (real (n::int)) = n" |
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555 apply (simp only: floor_def) |
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556 apply (rule Least_equality) |
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557 apply auto |
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558 done |
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559 |
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560 lemma floor_minus_real_of_int [simp]: "floor (- real (n::int)) = - n" |
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561 apply (simp only: floor_def) |
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562 apply (rule Least_equality) |
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563 apply (drule_tac [2] real_of_int_minus [THEN sym, THEN subst]) |
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564 apply auto |
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565 done |
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566 |
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567 lemma real_lb_ub_int: " \<exists>n::int. real n \<le> r & r < real (n+1)" |
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568 apply (case_tac "r < 0") |
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569 apply (blast intro: reals_Archimedean_6c_int) |
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570 apply (simp only: linorder_not_less) |
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571 apply (blast intro: reals_Archimedean_6b_int reals_Archimedean_6c_int) |
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572 done |
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573 |
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574 lemma lemma_floor: |
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575 assumes a1: "real m \<le> r" and a2: "r < real n + 1" |
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576 shows "m \<le> (n::int)" |
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577 proof - |
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578 have "real m < real n + 1" using a1 a2 by (rule order_le_less_trans) |
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579 also have "... = real (n + 1)" by simp |
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580 finally have "m < n + 1" by (simp only: real_of_int_less_iff) |
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581 thus ?thesis by arith |
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582 qed |
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583 |
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584 lemma real_of_int_floor_le [simp]: "real (floor r) \<le> r" |
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585 apply (simp add: floor_def Least_def) |
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586 apply (insert real_lb_ub_int [of r], safe) |
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587 apply (rule theI2) |
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588 apply auto |
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589 done |
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590 |
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591 lemma floor_mono: "x < y ==> floor x \<le> floor y" |
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592 apply (simp add: floor_def Least_def) |
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593 apply (insert real_lb_ub_int [of x]) |
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594 apply (insert real_lb_ub_int [of y], safe) |
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595 apply (rule theI2) |
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596 apply (rule_tac [3] theI2) |
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597 apply simp |
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598 apply (erule conjI) |
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599 apply (auto simp add: order_eq_iff int_le_real_less) |
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600 done |
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601 |
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602 lemma floor_mono2: "x \<le> y ==> floor x \<le> floor y" |
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603 by (auto dest: order_le_imp_less_or_eq simp add: floor_mono) |
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604 |
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605 lemma lemma_floor2: "real n < real (x::int) + 1 ==> n \<le> x" |
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606 by (auto intro: lemma_floor) |
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607 |
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608 lemma real_of_int_floor_cancel [simp]: |
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609 "(real (floor x) = x) = (\<exists>n::int. x = real n)" |
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610 apply (simp add: floor_def Least_def) |
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611 apply (insert real_lb_ub_int [of x], erule exE) |
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612 apply (rule theI2) |
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613 apply (auto intro: lemma_floor) |
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614 done |
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615 |
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616 lemma floor_eq: "[| real n < x; x < real n + 1 |] ==> floor x = n" |
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617 apply (simp add: floor_def) |
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618 apply (rule Least_equality) |
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619 apply (auto intro: lemma_floor) |
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620 done |
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621 |
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622 lemma floor_eq2: "[| real n \<le> x; x < real n + 1 |] ==> floor x = n" |
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623 apply (simp add: floor_def) |
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624 apply (rule Least_equality) |
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625 apply (auto intro: lemma_floor) |
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626 done |
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627 |
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628 lemma floor_eq3: "[| real n < x; x < real (Suc n) |] ==> nat(floor x) = n" |
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629 apply (rule inj_int [THEN injD]) |
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630 apply (simp add: real_of_nat_Suc) |
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631 apply (simp add: real_of_nat_Suc floor_eq floor_eq [where n = "int n"]) |
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632 done |
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633 |
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634 lemma floor_eq4: "[| real n \<le> x; x < real (Suc n) |] ==> nat(floor x) = n" |
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635 apply (drule order_le_imp_less_or_eq) |
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636 apply (auto intro: floor_eq3) |
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637 done |
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638 |
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639 lemma floor_number_of_eq [simp]: |
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640 "floor(number_of n :: real) = (number_of n :: int)" |
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641 apply (subst real_number_of [symmetric]) |
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642 apply (rule floor_real_of_int) |
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643 done |
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644 |
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645 lemma floor_one [simp]: "floor 1 = 1" |
|
646 apply (rule trans) |
|
647 prefer 2 |
|
648 apply (rule floor_real_of_int) |
|
649 apply simp |
|
650 done |
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651 |
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652 lemma real_of_int_floor_ge_diff_one [simp]: "r - 1 \<le> real(floor r)" |
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653 apply (simp add: floor_def Least_def) |
|
654 apply (insert real_lb_ub_int [of r], safe) |
|
655 apply (rule theI2) |
|
656 apply (auto intro: lemma_floor) |
|
657 done |
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658 |
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659 lemma real_of_int_floor_gt_diff_one [simp]: "r - 1 < real(floor r)" |
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660 apply (simp add: floor_def Least_def) |
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661 apply (insert real_lb_ub_int [of r], safe) |
|
662 apply (rule theI2) |
|
663 apply (auto intro: lemma_floor) |
|
664 done |
|
665 |
|
666 lemma real_of_int_floor_add_one_ge [simp]: "r \<le> real(floor r) + 1" |
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667 apply (insert real_of_int_floor_ge_diff_one [of r]) |
|
668 apply (auto simp del: real_of_int_floor_ge_diff_one) |
|
669 done |
|
670 |
|
671 lemma real_of_int_floor_add_one_gt [simp]: "r < real(floor r) + 1" |
|
672 apply (insert real_of_int_floor_gt_diff_one [of r]) |
|
673 apply (auto simp del: real_of_int_floor_gt_diff_one) |
|
674 done |
|
675 |
|
676 lemma le_floor: "real a <= x ==> a <= floor x" |
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677 apply (subgoal_tac "a < floor x + 1") |
|
678 apply arith |
|
679 apply (subst real_of_int_less_iff [THEN sym]) |
|
680 apply simp |
|
681 apply (insert real_of_int_floor_add_one_gt [of x]) |
|
682 apply arith |
|
683 done |
|
684 |
|
685 lemma real_le_floor: "a <= floor x ==> real a <= x" |
|
686 apply (rule order_trans) |
|
687 prefer 2 |
|
688 apply (rule real_of_int_floor_le) |
|
689 apply (subst real_of_int_le_iff) |
|
690 apply assumption |
|
691 done |
|
692 |
|
693 lemma le_floor_eq: "(a <= floor x) = (real a <= x)" |
|
694 apply (rule iffI) |
|
695 apply (erule real_le_floor) |
|
696 apply (erule le_floor) |
|
697 done |
|
698 |
|
699 lemma le_floor_eq_number_of [simp]: |
|
700 "(number_of n <= floor x) = (number_of n <= x)" |
|
701 by (simp add: le_floor_eq) |
|
702 |
|
703 lemma le_floor_eq_zero [simp]: "(0 <= floor x) = (0 <= x)" |
|
704 by (simp add: le_floor_eq) |
|
705 |
|
706 lemma le_floor_eq_one [simp]: "(1 <= floor x) = (1 <= x)" |
|
707 by (simp add: le_floor_eq) |
|
708 |
|
709 lemma floor_less_eq: "(floor x < a) = (x < real a)" |
|
710 apply (subst linorder_not_le [THEN sym])+ |
|
711 apply simp |
|
712 apply (rule le_floor_eq) |
|
713 done |
|
714 |
|
715 lemma floor_less_eq_number_of [simp]: |
|
716 "(floor x < number_of n) = (x < number_of n)" |
|
717 by (simp add: floor_less_eq) |
|
718 |
|
719 lemma floor_less_eq_zero [simp]: "(floor x < 0) = (x < 0)" |
|
720 by (simp add: floor_less_eq) |
|
721 |
|
722 lemma floor_less_eq_one [simp]: "(floor x < 1) = (x < 1)" |
|
723 by (simp add: floor_less_eq) |
|
724 |
|
725 lemma less_floor_eq: "(a < floor x) = (real a + 1 <= x)" |
|
726 apply (insert le_floor_eq [of "a + 1" x]) |
|
727 apply auto |
|
728 done |
|
729 |
|
730 lemma less_floor_eq_number_of [simp]: |
|
731 "(number_of n < floor x) = (number_of n + 1 <= x)" |
|
732 by (simp add: less_floor_eq) |
|
733 |
|
734 lemma less_floor_eq_zero [simp]: "(0 < floor x) = (1 <= x)" |
|
735 by (simp add: less_floor_eq) |
|
736 |
|
737 lemma less_floor_eq_one [simp]: "(1 < floor x) = (2 <= x)" |
|
738 by (simp add: less_floor_eq) |
|
739 |
|
740 lemma floor_le_eq: "(floor x <= a) = (x < real a + 1)" |
|
741 apply (insert floor_less_eq [of x "a + 1"]) |
|
742 apply auto |
|
743 done |
|
744 |
|
745 lemma floor_le_eq_number_of [simp]: |
|
746 "(floor x <= number_of n) = (x < number_of n + 1)" |
|
747 by (simp add: floor_le_eq) |
|
748 |
|
749 lemma floor_le_eq_zero [simp]: "(floor x <= 0) = (x < 1)" |
|
750 by (simp add: floor_le_eq) |
|
751 |
|
752 lemma floor_le_eq_one [simp]: "(floor x <= 1) = (x < 2)" |
|
753 by (simp add: floor_le_eq) |
|
754 |
|
755 lemma floor_add [simp]: "floor (x + real a) = floor x + a" |
|
756 apply (subst order_eq_iff) |
|
757 apply (rule conjI) |
|
758 prefer 2 |
|
759 apply (subgoal_tac "floor x + a < floor (x + real a) + 1") |
|
760 apply arith |
|
761 apply (subst real_of_int_less_iff [THEN sym]) |
|
762 apply simp |
|
763 apply (subgoal_tac "x + real a < real(floor(x + real a)) + 1") |
|
764 apply (subgoal_tac "real (floor x) <= x") |
|
765 apply arith |
|
766 apply (rule real_of_int_floor_le) |
|
767 apply (rule real_of_int_floor_add_one_gt) |
|
768 apply (subgoal_tac "floor (x + real a) < floor x + a + 1") |
|
769 apply arith |
|
770 apply (subst real_of_int_less_iff [THEN sym]) |
|
771 apply simp |
|
772 apply (subgoal_tac "real(floor(x + real a)) <= x + real a") |
|
773 apply (subgoal_tac "x < real(floor x) + 1") |
|
774 apply arith |
|
775 apply (rule real_of_int_floor_add_one_gt) |
|
776 apply (rule real_of_int_floor_le) |
|
777 done |
|
778 |
|
779 lemma floor_add_number_of [simp]: |
|
780 "floor (x + number_of n) = floor x + number_of n" |
|
781 apply (subst floor_add [THEN sym]) |
|
782 apply simp |
|
783 done |
|
784 |
|
785 lemma floor_add_one [simp]: "floor (x + 1) = floor x + 1" |
|
786 apply (subst floor_add [THEN sym]) |
|
787 apply simp |
|
788 done |
|
789 |
|
790 lemma floor_subtract [simp]: "floor (x - real a) = floor x - a" |
|
791 apply (subst diff_minus)+ |
|
792 apply (subst real_of_int_minus [THEN sym]) |
|
793 apply (rule floor_add) |
|
794 done |
|
795 |
|
796 lemma floor_subtract_number_of [simp]: "floor (x - number_of n) = |
|
797 floor x - number_of n" |
|
798 apply (subst floor_subtract [THEN sym]) |
|
799 apply simp |
|
800 done |
|
801 |
|
802 lemma floor_subtract_one [simp]: "floor (x - 1) = floor x - 1" |
|
803 apply (subst floor_subtract [THEN sym]) |
|
804 apply simp |
|
805 done |
|
806 |
|
807 lemma ceiling_zero [simp]: "ceiling 0 = 0" |
|
808 by (simp add: ceiling_def) |
|
809 |
|
810 lemma ceiling_real_of_nat [simp]: "ceiling (real (n::nat)) = int n" |
|
811 by (simp add: ceiling_def) |
|
812 |
|
813 lemma ceiling_real_of_nat_zero [simp]: "ceiling (real (0::nat)) = 0" |
|
814 by auto |
|
815 |
|
816 lemma ceiling_floor [simp]: "ceiling (real (floor r)) = floor r" |
|
817 by (simp add: ceiling_def) |
|
818 |
|
819 lemma floor_ceiling [simp]: "floor (real (ceiling r)) = ceiling r" |
|
820 by (simp add: ceiling_def) |
|
821 |
|
822 lemma real_of_int_ceiling_ge [simp]: "r \<le> real (ceiling r)" |
|
823 apply (simp add: ceiling_def) |
|
824 apply (subst le_minus_iff, simp) |
|
825 done |
|
826 |
|
827 lemma ceiling_mono: "x < y ==> ceiling x \<le> ceiling y" |
|
828 by (simp add: floor_mono ceiling_def) |
|
829 |
|
830 lemma ceiling_mono2: "x \<le> y ==> ceiling x \<le> ceiling y" |
|
831 by (simp add: floor_mono2 ceiling_def) |
|
832 |
|
833 lemma real_of_int_ceiling_cancel [simp]: |
|
834 "(real (ceiling x) = x) = (\<exists>n::int. x = real n)" |
|
835 apply (auto simp add: ceiling_def) |
|
836 apply (drule arg_cong [where f = uminus], auto) |
|
837 apply (rule_tac x = "-n" in exI, auto) |
|
838 done |
|
839 |
|
840 lemma ceiling_eq: "[| real n < x; x < real n + 1 |] ==> ceiling x = n + 1" |
|
841 apply (simp add: ceiling_def) |
|
842 apply (rule minus_equation_iff [THEN iffD1]) |
|
843 apply (simp add: floor_eq [where n = "-(n+1)"]) |
|
844 done |
|
845 |
|
846 lemma ceiling_eq2: "[| real n < x; x \<le> real n + 1 |] ==> ceiling x = n + 1" |
|
847 by (simp add: ceiling_def floor_eq2 [where n = "-(n+1)"]) |
|
848 |
|
849 lemma ceiling_eq3: "[| real n - 1 < x; x \<le> real n |] ==> ceiling x = n" |
|
850 by (simp add: ceiling_def floor_eq2 [where n = "-n"]) |
|
851 |
|
852 lemma ceiling_real_of_int [simp]: "ceiling (real (n::int)) = n" |
|
853 by (simp add: ceiling_def) |
|
854 |
|
855 lemma ceiling_number_of_eq [simp]: |
|
856 "ceiling (number_of n :: real) = (number_of n)" |
|
857 apply (subst real_number_of [symmetric]) |
|
858 apply (rule ceiling_real_of_int) |
|
859 done |
|
860 |
|
861 lemma ceiling_one [simp]: "ceiling 1 = 1" |
|
862 by (unfold ceiling_def, simp) |
|
863 |
|
864 lemma real_of_int_ceiling_diff_one_le [simp]: "real (ceiling r) - 1 \<le> r" |
|
865 apply (rule neg_le_iff_le [THEN iffD1]) |
|
866 apply (simp add: ceiling_def diff_minus) |
|
867 done |
|
868 |
|
869 lemma real_of_int_ceiling_le_add_one [simp]: "real (ceiling r) \<le> r + 1" |
|
870 apply (insert real_of_int_ceiling_diff_one_le [of r]) |
|
871 apply (simp del: real_of_int_ceiling_diff_one_le) |
|
872 done |
|
873 |
|
874 lemma ceiling_le: "x <= real a ==> ceiling x <= a" |
|
875 apply (unfold ceiling_def) |
|
876 apply (subgoal_tac "-a <= floor(- x)") |
|
877 apply simp |
|
878 apply (rule le_floor) |
|
879 apply simp |
|
880 done |
|
881 |
|
882 lemma ceiling_le_real: "ceiling x <= a ==> x <= real a" |
|
883 apply (unfold ceiling_def) |
|
884 apply (subgoal_tac "real(- a) <= - x") |
|
885 apply simp |
|
886 apply (rule real_le_floor) |
|
887 apply simp |
|
888 done |
|
889 |
|
890 lemma ceiling_le_eq: "(ceiling x <= a) = (x <= real a)" |
|
891 apply (rule iffI) |
|
892 apply (erule ceiling_le_real) |
|
893 apply (erule ceiling_le) |
|
894 done |
|
895 |
|
896 lemma ceiling_le_eq_number_of [simp]: |
|
897 "(ceiling x <= number_of n) = (x <= number_of n)" |
|
898 by (simp add: ceiling_le_eq) |
|
899 |
|
900 lemma ceiling_le_zero_eq [simp]: "(ceiling x <= 0) = (x <= 0)" |
|
901 by (simp add: ceiling_le_eq) |
|
902 |
|
903 lemma ceiling_le_eq_one [simp]: "(ceiling x <= 1) = (x <= 1)" |
|
904 by (simp add: ceiling_le_eq) |
|
905 |
|
906 lemma less_ceiling_eq: "(a < ceiling x) = (real a < x)" |
|
907 apply (subst linorder_not_le [THEN sym])+ |
|
908 apply simp |
|
909 apply (rule ceiling_le_eq) |
|
910 done |
|
911 |
|
912 lemma less_ceiling_eq_number_of [simp]: |
|
913 "(number_of n < ceiling x) = (number_of n < x)" |
|
914 by (simp add: less_ceiling_eq) |
|
915 |
|
916 lemma less_ceiling_eq_zero [simp]: "(0 < ceiling x) = (0 < x)" |
|
917 by (simp add: less_ceiling_eq) |
|
918 |
|
919 lemma less_ceiling_eq_one [simp]: "(1 < ceiling x) = (1 < x)" |
|
920 by (simp add: less_ceiling_eq) |
|
921 |
|
922 lemma ceiling_less_eq: "(ceiling x < a) = (x <= real a - 1)" |
|
923 apply (insert ceiling_le_eq [of x "a - 1"]) |
|
924 apply auto |
|
925 done |
|
926 |
|
927 lemma ceiling_less_eq_number_of [simp]: |
|
928 "(ceiling x < number_of n) = (x <= number_of n - 1)" |
|
929 by (simp add: ceiling_less_eq) |
|
930 |
|
931 lemma ceiling_less_eq_zero [simp]: "(ceiling x < 0) = (x <= -1)" |
|
932 by (simp add: ceiling_less_eq) |
|
933 |
|
934 lemma ceiling_less_eq_one [simp]: "(ceiling x < 1) = (x <= 0)" |
|
935 by (simp add: ceiling_less_eq) |
|
936 |
|
937 lemma le_ceiling_eq: "(a <= ceiling x) = (real a - 1 < x)" |
|
938 apply (insert less_ceiling_eq [of "a - 1" x]) |
|
939 apply auto |
|
940 done |
|
941 |
|
942 lemma le_ceiling_eq_number_of [simp]: |
|
943 "(number_of n <= ceiling x) = (number_of n - 1 < x)" |
|
944 by (simp add: le_ceiling_eq) |
|
945 |
|
946 lemma le_ceiling_eq_zero [simp]: "(0 <= ceiling x) = (-1 < x)" |
|
947 by (simp add: le_ceiling_eq) |
|
948 |
|
949 lemma le_ceiling_eq_one [simp]: "(1 <= ceiling x) = (0 < x)" |
|
950 by (simp add: le_ceiling_eq) |
|
951 |
|
952 lemma ceiling_add [simp]: "ceiling (x + real a) = ceiling x + a" |
|
953 apply (unfold ceiling_def, simp) |
|
954 apply (subst real_of_int_minus [THEN sym]) |
|
955 apply (subst floor_add) |
|
956 apply simp |
|
957 done |
|
958 |
|
959 lemma ceiling_add_number_of [simp]: "ceiling (x + number_of n) = |
|
960 ceiling x + number_of n" |
|
961 apply (subst ceiling_add [THEN sym]) |
|
962 apply simp |
|
963 done |
|
964 |
|
965 lemma ceiling_add_one [simp]: "ceiling (x + 1) = ceiling x + 1" |
|
966 apply (subst ceiling_add [THEN sym]) |
|
967 apply simp |
|
968 done |
|
969 |
|
970 lemma ceiling_subtract [simp]: "ceiling (x - real a) = ceiling x - a" |
|
971 apply (subst diff_minus)+ |
|
972 apply (subst real_of_int_minus [THEN sym]) |
|
973 apply (rule ceiling_add) |
|
974 done |
|
975 |
|
976 lemma ceiling_subtract_number_of [simp]: "ceiling (x - number_of n) = |
|
977 ceiling x - number_of n" |
|
978 apply (subst ceiling_subtract [THEN sym]) |
|
979 apply simp |
|
980 done |
|
981 |
|
982 lemma ceiling_subtract_one [simp]: "ceiling (x - 1) = ceiling x - 1" |
|
983 apply (subst ceiling_subtract [THEN sym]) |
|
984 apply simp |
|
985 done |
|
986 |
|
987 subsection {* Versions for the natural numbers *} |
|
988 |
|
989 definition |
|
990 natfloor :: "real => nat" where |
|
991 "natfloor x = nat(floor x)" |
|
992 |
|
993 definition |
|
994 natceiling :: "real => nat" where |
|
995 "natceiling x = nat(ceiling x)" |
|
996 |
|
997 lemma natfloor_zero [simp]: "natfloor 0 = 0" |
|
998 by (unfold natfloor_def, simp) |
|
999 |
|
1000 lemma natfloor_one [simp]: "natfloor 1 = 1" |
|
1001 by (unfold natfloor_def, simp) |
|
1002 |
|
1003 lemma zero_le_natfloor [simp]: "0 <= natfloor x" |
|
1004 by (unfold natfloor_def, simp) |
|
1005 |
|
1006 lemma natfloor_number_of_eq [simp]: "natfloor (number_of n) = number_of n" |
|
1007 by (unfold natfloor_def, simp) |
|
1008 |
|
1009 lemma natfloor_real_of_nat [simp]: "natfloor(real n) = n" |
|
1010 by (unfold natfloor_def, simp) |
|
1011 |
|
1012 lemma real_natfloor_le: "0 <= x ==> real(natfloor x) <= x" |
|
1013 by (unfold natfloor_def, simp) |
|
1014 |
|
1015 lemma natfloor_neg: "x <= 0 ==> natfloor x = 0" |
|
1016 apply (unfold natfloor_def) |
|
1017 apply (subgoal_tac "floor x <= floor 0") |
|
1018 apply simp |
|
1019 apply (erule floor_mono2) |
|
1020 done |
|
1021 |
|
1022 lemma natfloor_mono: "x <= y ==> natfloor x <= natfloor y" |
|
1023 apply (case_tac "0 <= x") |
|
1024 apply (subst natfloor_def)+ |
|
1025 apply (subst nat_le_eq_zle) |
|
1026 apply force |
|
1027 apply (erule floor_mono2) |
|
1028 apply (subst natfloor_neg) |
|
1029 apply simp |
|
1030 apply simp |
|
1031 done |
|
1032 |
|
1033 lemma le_natfloor: "real x <= a ==> x <= natfloor a" |
|
1034 apply (unfold natfloor_def) |
|
1035 apply (subst nat_int [THEN sym]) |
|
1036 apply (subst nat_le_eq_zle) |
|
1037 apply simp |
|
1038 apply (rule le_floor) |
|
1039 apply simp |
|
1040 done |
|
1041 |
|
1042 lemma le_natfloor_eq: "0 <= x ==> (a <= natfloor x) = (real a <= x)" |
|
1043 apply (rule iffI) |
|
1044 apply (rule order_trans) |
|
1045 prefer 2 |
|
1046 apply (erule real_natfloor_le) |
|
1047 apply (subst real_of_nat_le_iff) |
|
1048 apply assumption |
|
1049 apply (erule le_natfloor) |
|
1050 done |
|
1051 |
|
1052 lemma le_natfloor_eq_number_of [simp]: |
|
1053 "~ neg((number_of n)::int) ==> 0 <= x ==> |
|
1054 (number_of n <= natfloor x) = (number_of n <= x)" |
|
1055 apply (subst le_natfloor_eq, assumption) |
|
1056 apply simp |
|
1057 done |
|
1058 |
|
1059 lemma le_natfloor_eq_one [simp]: "(1 <= natfloor x) = (1 <= x)" |
|
1060 apply (case_tac "0 <= x") |
|
1061 apply (subst le_natfloor_eq, assumption, simp) |
|
1062 apply (rule iffI) |
|
1063 apply (subgoal_tac "natfloor x <= natfloor 0") |
|
1064 apply simp |
|
1065 apply (rule natfloor_mono) |
|
1066 apply simp |
|
1067 apply simp |
|
1068 done |
|
1069 |
|
1070 lemma natfloor_eq: "real n <= x ==> x < real n + 1 ==> natfloor x = n" |
|
1071 apply (unfold natfloor_def) |
|
1072 apply (subst nat_int [THEN sym]);back; |
|
1073 apply (subst eq_nat_nat_iff) |
|
1074 apply simp |
|
1075 apply simp |
|
1076 apply (rule floor_eq2) |
|
1077 apply auto |
|
1078 done |
|
1079 |
|
1080 lemma real_natfloor_add_one_gt: "x < real(natfloor x) + 1" |
|
1081 apply (case_tac "0 <= x") |
|
1082 apply (unfold natfloor_def) |
|
1083 apply simp |
|
1084 apply simp_all |
|
1085 done |
|
1086 |
|
1087 lemma real_natfloor_gt_diff_one: "x - 1 < real(natfloor x)" |
|
1088 apply (simp add: compare_rls) |
|
1089 apply (rule real_natfloor_add_one_gt) |
|
1090 done |
|
1091 |
|
1092 lemma ge_natfloor_plus_one_imp_gt: "natfloor z + 1 <= n ==> z < real n" |
|
1093 apply (subgoal_tac "z < real(natfloor z) + 1") |
|
1094 apply arith |
|
1095 apply (rule real_natfloor_add_one_gt) |
|
1096 done |
|
1097 |
|
1098 lemma natfloor_add [simp]: "0 <= x ==> natfloor (x + real a) = natfloor x + a" |
|
1099 apply (unfold natfloor_def) |
|
1100 apply (subgoal_tac "real a = real (int a)") |
|
1101 apply (erule ssubst) |
|
1102 apply (simp add: nat_add_distrib del: real_of_int_of_nat_eq) |
|
1103 apply simp |
|
1104 done |
|
1105 |
|
1106 lemma natfloor_add_number_of [simp]: |
|
1107 "~neg ((number_of n)::int) ==> 0 <= x ==> |
|
1108 natfloor (x + number_of n) = natfloor x + number_of n" |
|
1109 apply (subst natfloor_add [THEN sym]) |
|
1110 apply simp_all |
|
1111 done |
|
1112 |
|
1113 lemma natfloor_add_one: "0 <= x ==> natfloor(x + 1) = natfloor x + 1" |
|
1114 apply (subst natfloor_add [THEN sym]) |
|
1115 apply assumption |
|
1116 apply simp |
|
1117 done |
|
1118 |
|
1119 lemma natfloor_subtract [simp]: "real a <= x ==> |
|
1120 natfloor(x - real a) = natfloor x - a" |
|
1121 apply (unfold natfloor_def) |
|
1122 apply (subgoal_tac "real a = real (int a)") |
|
1123 apply (erule ssubst) |
|
1124 apply (simp del: real_of_int_of_nat_eq) |
|
1125 apply simp |
|
1126 done |
|
1127 |
|
1128 lemma natceiling_zero [simp]: "natceiling 0 = 0" |
|
1129 by (unfold natceiling_def, simp) |
|
1130 |
|
1131 lemma natceiling_one [simp]: "natceiling 1 = 1" |
|
1132 by (unfold natceiling_def, simp) |
|
1133 |
|
1134 lemma zero_le_natceiling [simp]: "0 <= natceiling x" |
|
1135 by (unfold natceiling_def, simp) |
|
1136 |
|
1137 lemma natceiling_number_of_eq [simp]: "natceiling (number_of n) = number_of n" |
|
1138 by (unfold natceiling_def, simp) |
|
1139 |
|
1140 lemma natceiling_real_of_nat [simp]: "natceiling(real n) = n" |
|
1141 by (unfold natceiling_def, simp) |
|
1142 |
|
1143 lemma real_natceiling_ge: "x <= real(natceiling x)" |
|
1144 apply (unfold natceiling_def) |
|
1145 apply (case_tac "x < 0") |
|
1146 apply simp |
|
1147 apply (subst real_nat_eq_real) |
|
1148 apply (subgoal_tac "ceiling 0 <= ceiling x") |
|
1149 apply simp |
|
1150 apply (rule ceiling_mono2) |
|
1151 apply simp |
|
1152 apply simp |
|
1153 done |
|
1154 |
|
1155 lemma natceiling_neg: "x <= 0 ==> natceiling x = 0" |
|
1156 apply (unfold natceiling_def) |
|
1157 apply simp |
|
1158 done |
|
1159 |
|
1160 lemma natceiling_mono: "x <= y ==> natceiling x <= natceiling y" |
|
1161 apply (case_tac "0 <= x") |
|
1162 apply (subst natceiling_def)+ |
|
1163 apply (subst nat_le_eq_zle) |
|
1164 apply (rule disjI2) |
|
1165 apply (subgoal_tac "real (0::int) <= real(ceiling y)") |
|
1166 apply simp |
|
1167 apply (rule order_trans) |
|
1168 apply simp |
|
1169 apply (erule order_trans) |
|
1170 apply simp |
|
1171 apply (erule ceiling_mono2) |
|
1172 apply (subst natceiling_neg) |
|
1173 apply simp_all |
|
1174 done |
|
1175 |
|
1176 lemma natceiling_le: "x <= real a ==> natceiling x <= a" |
|
1177 apply (unfold natceiling_def) |
|
1178 apply (case_tac "x < 0") |
|
1179 apply simp |
|
1180 apply (subst nat_int [THEN sym]);back; |
|
1181 apply (subst nat_le_eq_zle) |
|
1182 apply simp |
|
1183 apply (rule ceiling_le) |
|
1184 apply simp |
|
1185 done |
|
1186 |
|
1187 lemma natceiling_le_eq: "0 <= x ==> (natceiling x <= a) = (x <= real a)" |
|
1188 apply (rule iffI) |
|
1189 apply (rule order_trans) |
|
1190 apply (rule real_natceiling_ge) |
|
1191 apply (subst real_of_nat_le_iff) |
|
1192 apply assumption |
|
1193 apply (erule natceiling_le) |
|
1194 done |
|
1195 |
|
1196 lemma natceiling_le_eq_number_of [simp]: |
|
1197 "~ neg((number_of n)::int) ==> 0 <= x ==> |
|
1198 (natceiling x <= number_of n) = (x <= number_of n)" |
|
1199 apply (subst natceiling_le_eq, assumption) |
|
1200 apply simp |
|
1201 done |
|
1202 |
|
1203 lemma natceiling_le_eq_one: "(natceiling x <= 1) = (x <= 1)" |
|
1204 apply (case_tac "0 <= x") |
|
1205 apply (subst natceiling_le_eq) |
|
1206 apply assumption |
|
1207 apply simp |
|
1208 apply (subst natceiling_neg) |
|
1209 apply simp |
|
1210 apply simp |
|
1211 done |
|
1212 |
|
1213 lemma natceiling_eq: "real n < x ==> x <= real n + 1 ==> natceiling x = n + 1" |
|
1214 apply (unfold natceiling_def) |
|
1215 apply (simplesubst nat_int [THEN sym]) back back |
|
1216 apply (subgoal_tac "nat(int n) + 1 = nat(int n + 1)") |
|
1217 apply (erule ssubst) |
|
1218 apply (subst eq_nat_nat_iff) |
|
1219 apply (subgoal_tac "ceiling 0 <= ceiling x") |
|
1220 apply simp |
|
1221 apply (rule ceiling_mono2) |
|
1222 apply force |
|
1223 apply force |
|
1224 apply (rule ceiling_eq2) |
|
1225 apply (simp, simp) |
|
1226 apply (subst nat_add_distrib) |
|
1227 apply auto |
|
1228 done |
|
1229 |
|
1230 lemma natceiling_add [simp]: "0 <= x ==> |
|
1231 natceiling (x + real a) = natceiling x + a" |
|
1232 apply (unfold natceiling_def) |
|
1233 apply (subgoal_tac "real a = real (int a)") |
|
1234 apply (erule ssubst) |
|
1235 apply (simp del: real_of_int_of_nat_eq) |
|
1236 apply (subst nat_add_distrib) |
|
1237 apply (subgoal_tac "0 = ceiling 0") |
|
1238 apply (erule ssubst) |
|
1239 apply (erule ceiling_mono2) |
|
1240 apply simp_all |
|
1241 done |
|
1242 |
|
1243 lemma natceiling_add_number_of [simp]: |
|
1244 "~ neg ((number_of n)::int) ==> 0 <= x ==> |
|
1245 natceiling (x + number_of n) = natceiling x + number_of n" |
|
1246 apply (subst natceiling_add [THEN sym]) |
|
1247 apply simp_all |
|
1248 done |
|
1249 |
|
1250 lemma natceiling_add_one: "0 <= x ==> natceiling(x + 1) = natceiling x + 1" |
|
1251 apply (subst natceiling_add [THEN sym]) |
|
1252 apply assumption |
|
1253 apply simp |
|
1254 done |
|
1255 |
|
1256 lemma natceiling_subtract [simp]: "real a <= x ==> |
|
1257 natceiling(x - real a) = natceiling x - a" |
|
1258 apply (unfold natceiling_def) |
|
1259 apply (subgoal_tac "real a = real (int a)") |
|
1260 apply (erule ssubst) |
|
1261 apply (simp del: real_of_int_of_nat_eq) |
|
1262 apply simp |
|
1263 done |
|
1264 |
|
1265 lemma natfloor_div_nat: "1 <= x ==> y > 0 ==> |
|
1266 natfloor (x / real y) = natfloor x div y" |
|
1267 proof - |
|
1268 assume "1 <= (x::real)" and "(y::nat) > 0" |
|
1269 have "natfloor x = (natfloor x) div y * y + (natfloor x) mod y" |
|
1270 by simp |
|
1271 then have a: "real(natfloor x) = real ((natfloor x) div y) * real y + |
|
1272 real((natfloor x) mod y)" |
|
1273 by (simp only: real_of_nat_add [THEN sym] real_of_nat_mult [THEN sym]) |
|
1274 have "x = real(natfloor x) + (x - real(natfloor x))" |
|
1275 by simp |
|
1276 then have "x = real ((natfloor x) div y) * real y + |
|
1277 real((natfloor x) mod y) + (x - real(natfloor x))" |
|
1278 by (simp add: a) |
|
1279 then have "x / real y = ... / real y" |
|
1280 by simp |
|
1281 also have "... = real((natfloor x) div y) + real((natfloor x) mod y) / |
|
1282 real y + (x - real(natfloor x)) / real y" |
|
1283 by (auto simp add: ring_simps add_divide_distrib |
|
1284 diff_divide_distrib prems) |
|
1285 finally have "natfloor (x / real y) = natfloor(...)" by simp |
|
1286 also have "... = natfloor(real((natfloor x) mod y) / |
|
1287 real y + (x - real(natfloor x)) / real y + real((natfloor x) div y))" |
|
1288 by (simp add: add_ac) |
|
1289 also have "... = natfloor(real((natfloor x) mod y) / |
|
1290 real y + (x - real(natfloor x)) / real y) + (natfloor x) div y" |
|
1291 apply (rule natfloor_add) |
|
1292 apply (rule add_nonneg_nonneg) |
|
1293 apply (rule divide_nonneg_pos) |
|
1294 apply simp |
|
1295 apply (simp add: prems) |
|
1296 apply (rule divide_nonneg_pos) |
|
1297 apply (simp add: compare_rls) |
|
1298 apply (rule real_natfloor_le) |
|
1299 apply (insert prems, auto) |
|
1300 done |
|
1301 also have "natfloor(real((natfloor x) mod y) / |
|
1302 real y + (x - real(natfloor x)) / real y) = 0" |
|
1303 apply (rule natfloor_eq) |
|
1304 apply simp |
|
1305 apply (rule add_nonneg_nonneg) |
|
1306 apply (rule divide_nonneg_pos) |
|
1307 apply force |
|
1308 apply (force simp add: prems) |
|
1309 apply (rule divide_nonneg_pos) |
|
1310 apply (simp add: compare_rls) |
|
1311 apply (rule real_natfloor_le) |
|
1312 apply (auto simp add: prems) |
|
1313 apply (insert prems, arith) |
|
1314 apply (simp add: add_divide_distrib [THEN sym]) |
|
1315 apply (subgoal_tac "real y = real y - 1 + 1") |
|
1316 apply (erule ssubst) |
|
1317 apply (rule add_le_less_mono) |
|
1318 apply (simp add: compare_rls) |
|
1319 apply (subgoal_tac "real(natfloor x mod y) + 1 = |
|
1320 real(natfloor x mod y + 1)") |
|
1321 apply (erule ssubst) |
|
1322 apply (subst real_of_nat_le_iff) |
|
1323 apply (subgoal_tac "natfloor x mod y < y") |
|
1324 apply arith |
|
1325 apply (rule mod_less_divisor) |
|
1326 apply auto |
|
1327 apply (simp add: compare_rls) |
|
1328 apply (subst add_commute) |
|
1329 apply (rule real_natfloor_add_one_gt) |
|
1330 done |
|
1331 finally show ?thesis by simp |
|
1332 qed |
|
1333 |
|
1334 end |
|