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1 (* Title: HOL/Group.thy |
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2 ID: $Id$ |
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3 Author: Gertrud Bauer and Markus Wenzel, TU Muenchen |
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4 Lawrence C Paulson, University of Cambridge |
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5 Revised and decoupled from Ring_and_Field.thy |
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6 by Steven Obua, TU Muenchen, in May 2004 |
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7 License: GPL (GNU GENERAL PUBLIC LICENSE) |
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8 *) |
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9 |
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10 header {* Ordered Groups *} |
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11 |
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12 theory OrderedGroup = Inductive + LOrder: |
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13 |
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14 text {* |
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15 The theory of partially ordered groups is taken from the books: |
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16 \begin{itemize} |
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17 \item \emph{Lattice Theory} by Garret Birkhoff, American Mathematical Society 1979 |
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18 \item \emph{Partially Ordered Algebraic Systems}, Pergamon Press 1963 |
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19 \end{itemize} |
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20 Most of the used notions can also be looked up in |
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21 \begin{itemize} |
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22 \item \emph{www.mathworld.com} by Eric Weisstein et. al. |
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23 \item \emph{Algebra I} by van der Waerden, Springer. |
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24 \end{itemize} |
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25 *} |
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26 |
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27 subsection {* Semigroups, Groups *} |
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28 |
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29 axclass semigroup_add \<subseteq> plus |
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30 add_assoc: "(a + b) + c = a + (b + c)" |
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31 |
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32 axclass ab_semigroup_add \<subseteq> semigroup_add |
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33 add_commute: "a + b = b + a" |
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34 |
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35 lemma add_left_commute: "a + (b + c) = b + (a + (c::'a::ab_semigroup_add))" |
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36 by (rule mk_left_commute [of "op +", OF add_assoc add_commute]) |
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37 |
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38 theorems add_ac = add_assoc add_commute add_left_commute |
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39 |
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40 axclass semigroup_mult \<subseteq> times |
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41 mult_assoc: "(a * b) * c = a * (b * c)" |
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42 |
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43 axclass ab_semigroup_mult \<subseteq> semigroup_mult |
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44 mult_commute: "a * b = b * a" |
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45 |
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46 lemma mult_left_commute: "a * (b * c) = b * (a * (c::'a::ab_semigroup_mult))" |
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47 by (rule mk_left_commute [of "op *", OF mult_assoc mult_commute]) |
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48 |
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49 theorems mult_ac = mult_assoc mult_commute mult_left_commute |
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50 |
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51 axclass comm_monoid_add \<subseteq> zero, ab_semigroup_add |
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52 add_0[simp]: "0 + a = a" |
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53 |
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54 axclass monoid_mult \<subseteq> one, semigroup_mult |
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55 mult_1_left[simp]: "1 * a = a" |
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56 mult_1_right[simp]: "a * 1 = a" |
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57 |
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58 axclass comm_monoid_mult \<subseteq> one, ab_semigroup_mult |
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59 mult_1: "1 * a = a" |
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60 |
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61 instance comm_monoid_mult \<subseteq> monoid_mult |
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62 by (intro_classes, insert mult_1, simp_all add: mult_commute, auto) |
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63 |
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64 axclass cancel_semigroup_add \<subseteq> semigroup_add |
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65 add_left_imp_eq: "a + b = a + c \<Longrightarrow> b = c" |
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66 add_right_imp_eq: "b + a = c + a \<Longrightarrow> b = c" |
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67 |
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68 axclass cancel_ab_semigroup_add \<subseteq> ab_semigroup_add |
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69 add_imp_eq: "a + b = a + c \<Longrightarrow> b = c" |
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70 |
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71 instance cancel_ab_semigroup_add \<subseteq> cancel_semigroup_add |
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72 proof |
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73 { |
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74 fix a b c :: 'a |
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75 assume "a + b = a + c" |
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76 thus "b = c" by (rule add_imp_eq) |
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77 } |
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78 note f = this |
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79 fix a b c :: 'a |
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80 assume "b + a = c + a" |
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81 hence "a + b = a + c" by (simp only: add_commute) |
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82 thus "b = c" by (rule f) |
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83 qed |
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84 |
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85 axclass ab_group_add \<subseteq> minus, comm_monoid_add |
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86 left_minus[simp]: " - a + a = 0" |
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87 diff_minus: "a - b = a + (-b)" |
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88 |
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89 instance ab_group_add \<subseteq> cancel_ab_semigroup_add |
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90 proof |
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91 fix a b c :: 'a |
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92 assume "a + b = a + c" |
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93 hence "-a + a + b = -a + a + c" by (simp only: add_assoc) |
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94 thus "b = c" by simp |
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95 qed |
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96 |
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97 lemma add_0_right [simp]: "a + 0 = (a::'a::comm_monoid_add)" |
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98 proof - |
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99 have "a + 0 = 0 + a" by (simp only: add_commute) |
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100 also have "... = a" by simp |
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101 finally show ?thesis . |
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102 qed |
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103 |
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104 lemma add_left_cancel [simp]: |
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105 "(a + b = a + c) = (b = (c::'a::cancel_semigroup_add))" |
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106 by (blast dest: add_left_imp_eq) |
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107 |
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108 lemma add_right_cancel [simp]: |
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109 "(b + a = c + a) = (b = (c::'a::cancel_semigroup_add))" |
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110 by (blast dest: add_right_imp_eq) |
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111 |
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112 lemma right_minus [simp]: "a + -(a::'a::ab_group_add) = 0" |
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113 proof - |
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114 have "a + -a = -a + a" by (simp add: add_ac) |
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115 also have "... = 0" by simp |
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116 finally show ?thesis . |
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117 qed |
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118 |
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119 lemma right_minus_eq: "(a - b = 0) = (a = (b::'a::ab_group_add))" |
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120 proof |
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121 have "a = a - b + b" by (simp add: diff_minus add_ac) |
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122 also assume "a - b = 0" |
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123 finally show "a = b" by simp |
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124 next |
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125 assume "a = b" |
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126 thus "a - b = 0" by (simp add: diff_minus) |
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127 qed |
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128 |
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129 lemma minus_minus [simp]: "- (- (a::'a::ab_group_add)) = a" |
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130 proof (rule add_left_cancel [of "-a", THEN iffD1]) |
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131 show "(-a + -(-a) = -a + a)" |
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132 by simp |
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133 qed |
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134 |
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135 lemma equals_zero_I: "a+b = 0 ==> -a = (b::'a::ab_group_add)" |
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136 apply (rule right_minus_eq [THEN iffD1, symmetric]) |
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137 apply (simp add: diff_minus add_commute) |
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138 done |
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139 |
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140 lemma minus_zero [simp]: "- 0 = (0::'a::ab_group_add)" |
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141 by (simp add: equals_zero_I) |
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142 |
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143 lemma diff_self [simp]: "a - (a::'a::ab_group_add) = 0" |
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144 by (simp add: diff_minus) |
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145 |
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146 lemma diff_0 [simp]: "(0::'a::ab_group_add) - a = -a" |
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147 by (simp add: diff_minus) |
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148 |
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149 lemma diff_0_right [simp]: "a - (0::'a::ab_group_add) = a" |
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150 by (simp add: diff_minus) |
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151 |
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152 lemma diff_minus_eq_add [simp]: "a - - b = a + (b::'a::ab_group_add)" |
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153 by (simp add: diff_minus) |
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154 |
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155 lemma neg_equal_iff_equal [simp]: "(-a = -b) = (a = (b::'a::ab_group_add))" |
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156 proof |
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157 assume "- a = - b" |
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158 hence "- (- a) = - (- b)" |
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159 by simp |
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160 thus "a=b" by simp |
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161 next |
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162 assume "a=b" |
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163 thus "-a = -b" by simp |
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164 qed |
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165 |
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166 lemma neg_equal_0_iff_equal [simp]: "(-a = 0) = (a = (0::'a::ab_group_add))" |
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167 by (subst neg_equal_iff_equal [symmetric], simp) |
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168 |
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169 lemma neg_0_equal_iff_equal [simp]: "(0 = -a) = (0 = (a::'a::ab_group_add))" |
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170 by (subst neg_equal_iff_equal [symmetric], simp) |
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171 |
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172 text{*The next two equations can make the simplifier loop!*} |
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173 |
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174 lemma equation_minus_iff: "(a = - b) = (b = - (a::'a::ab_group_add))" |
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175 proof - |
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176 have "(- (-a) = - b) = (- a = b)" by (rule neg_equal_iff_equal) |
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177 thus ?thesis by (simp add: eq_commute) |
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178 qed |
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179 |
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180 lemma minus_equation_iff: "(- a = b) = (- (b::'a::ab_group_add) = a)" |
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181 proof - |
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182 have "(- a = - (-b)) = (a = -b)" by (rule neg_equal_iff_equal) |
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183 thus ?thesis by (simp add: eq_commute) |
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184 qed |
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185 |
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186 lemma minus_add_distrib [simp]: "- (a + b) = -a + -(b::'a::ab_group_add)" |
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187 apply (rule equals_zero_I) |
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188 apply (simp add: add_ac) |
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189 done |
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190 |
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191 lemma minus_diff_eq [simp]: "- (a - b) = b - (a::'a::ab_group_add)" |
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192 by (simp add: diff_minus add_commute) |
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193 |
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194 subsection {* (Partially) Ordered Groups *} |
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195 |
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196 axclass pordered_ab_semigroup_add \<subseteq> order, ab_semigroup_add |
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197 add_left_mono: "a \<le> b \<Longrightarrow> c + a \<le> c + b" |
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198 |
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199 axclass pordered_cancel_ab_semigroup_add \<subseteq> pordered_ab_semigroup_add, cancel_ab_semigroup_add |
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200 |
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201 instance pordered_cancel_ab_semigroup_add \<subseteq> pordered_ab_semigroup_add .. |
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202 |
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203 axclass pordered_ab_semigroup_add_imp_le \<subseteq> pordered_cancel_ab_semigroup_add |
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204 add_le_imp_le_left: "c + a \<le> c + b \<Longrightarrow> a \<le> b" |
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205 |
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206 axclass pordered_ab_group_add \<subseteq> ab_group_add, pordered_ab_semigroup_add |
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207 |
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208 instance pordered_ab_group_add \<subseteq> pordered_ab_semigroup_add_imp_le |
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209 proof |
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210 fix a b c :: 'a |
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211 assume "c + a \<le> c + b" |
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212 hence "(-c) + (c + a) \<le> (-c) + (c + b)" by (rule add_left_mono) |
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213 hence "((-c) + c) + a \<le> ((-c) + c) + b" by (simp only: add_assoc) |
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214 thus "a \<le> b" by simp |
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215 qed |
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216 |
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217 axclass ordered_cancel_ab_semigroup_add \<subseteq> pordered_cancel_ab_semigroup_add, linorder |
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218 |
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219 instance ordered_cancel_ab_semigroup_add \<subseteq> pordered_ab_semigroup_add_imp_le |
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220 proof |
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221 fix a b c :: 'a |
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222 assume le: "c + a <= c + b" |
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223 show "a <= b" |
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224 proof (rule ccontr) |
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225 assume w: "~ a \<le> b" |
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226 hence "b <= a" by (simp add: linorder_not_le) |
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227 hence le2: "c+b <= c+a" by (rule add_left_mono) |
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228 have "a = b" |
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229 apply (insert le) |
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230 apply (insert le2) |
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231 apply (drule order_antisym, simp_all) |
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232 done |
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233 with w show False |
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234 by (simp add: linorder_not_le [symmetric]) |
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235 qed |
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236 qed |
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237 |
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238 lemma add_right_mono: "a \<le> (b::'a::pordered_ab_semigroup_add) ==> a + c \<le> b + c" |
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239 by (simp add: add_commute[of _ c] add_left_mono) |
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240 |
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241 text {* non-strict, in both arguments *} |
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242 lemma add_mono: |
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243 "[|a \<le> b; c \<le> d|] ==> a + c \<le> b + (d::'a::pordered_ab_semigroup_add)" |
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244 apply (erule add_right_mono [THEN order_trans]) |
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245 apply (simp add: add_commute add_left_mono) |
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246 done |
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247 |
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248 lemma add_strict_left_mono: |
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249 "a < b ==> c + a < c + (b::'a::pordered_cancel_ab_semigroup_add)" |
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250 by (simp add: order_less_le add_left_mono) |
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251 |
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252 lemma add_strict_right_mono: |
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253 "a < b ==> a + c < b + (c::'a::pordered_cancel_ab_semigroup_add)" |
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254 by (simp add: add_commute [of _ c] add_strict_left_mono) |
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255 |
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256 text{*Strict monotonicity in both arguments*} |
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257 lemma add_strict_mono: "[|a<b; c<d|] ==> a + c < b + (d::'a::pordered_cancel_ab_semigroup_add)" |
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258 apply (erule add_strict_right_mono [THEN order_less_trans]) |
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259 apply (erule add_strict_left_mono) |
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260 done |
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261 |
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262 lemma add_less_le_mono: |
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263 "[| a<b; c\<le>d |] ==> a + c < b + (d::'a::pordered_cancel_ab_semigroup_add)" |
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264 apply (erule add_strict_right_mono [THEN order_less_le_trans]) |
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265 apply (erule add_left_mono) |
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266 done |
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267 |
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268 lemma add_le_less_mono: |
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269 "[| a\<le>b; c<d |] ==> a + c < b + (d::'a::pordered_cancel_ab_semigroup_add)" |
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270 apply (erule add_right_mono [THEN order_le_less_trans]) |
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271 apply (erule add_strict_left_mono) |
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272 done |
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273 |
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274 lemma add_less_imp_less_left: |
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275 assumes less: "c + a < c + b" shows "a < (b::'a::pordered_ab_semigroup_add_imp_le)" |
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276 proof - |
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277 from less have le: "c + a <= c + b" by (simp add: order_le_less) |
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278 have "a <= b" |
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279 apply (insert le) |
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280 apply (drule add_le_imp_le_left) |
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281 by (insert le, drule add_le_imp_le_left, assumption) |
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282 moreover have "a \<noteq> b" |
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283 proof (rule ccontr) |
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284 assume "~(a \<noteq> b)" |
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285 then have "a = b" by simp |
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286 then have "c + a = c + b" by simp |
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287 with less show "False"by simp |
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288 qed |
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289 ultimately show "a < b" by (simp add: order_le_less) |
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290 qed |
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291 |
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292 lemma add_less_imp_less_right: |
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293 "a + c < b + c ==> a < (b::'a::pordered_ab_semigroup_add_imp_le)" |
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294 apply (rule add_less_imp_less_left [of c]) |
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295 apply (simp add: add_commute) |
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296 done |
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297 |
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298 lemma add_less_cancel_left [simp]: |
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299 "(c+a < c+b) = (a < (b::'a::pordered_ab_semigroup_add_imp_le))" |
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300 by (blast intro: add_less_imp_less_left add_strict_left_mono) |
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301 |
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302 lemma add_less_cancel_right [simp]: |
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303 "(a+c < b+c) = (a < (b::'a::pordered_ab_semigroup_add_imp_le))" |
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304 by (blast intro: add_less_imp_less_right add_strict_right_mono) |
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305 |
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306 lemma add_le_cancel_left [simp]: |
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307 "(c+a \<le> c+b) = (a \<le> (b::'a::pordered_ab_semigroup_add_imp_le))" |
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308 by (auto, drule add_le_imp_le_left, simp_all add: add_left_mono) |
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309 |
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310 lemma add_le_cancel_right [simp]: |
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311 "(a+c \<le> b+c) = (a \<le> (b::'a::pordered_ab_semigroup_add_imp_le))" |
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312 by (simp add: add_commute[of a c] add_commute[of b c]) |
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313 |
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314 lemma add_le_imp_le_right: |
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315 "a + c \<le> b + c ==> a \<le> (b::'a::pordered_ab_semigroup_add_imp_le)" |
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316 by simp |
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317 |
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318 lemma add_increasing: "[|0\<le>a; b\<le>c|] ==> b \<le> a + (c::'a::{pordered_ab_semigroup_add_imp_le, comm_monoid_add})" |
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319 by (insert add_mono [of 0 a b c], simp) |
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320 |
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321 subsection {* Ordering Rules for Unary Minus *} |
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322 |
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323 lemma le_imp_neg_le: |
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324 assumes "a \<le> (b::'a::{pordered_ab_semigroup_add_imp_le, ab_group_add})" shows "-b \<le> -a" |
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325 proof - |
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326 have "-a+a \<le> -a+b" |
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327 by (rule add_left_mono) |
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328 hence "0 \<le> -a+b" |
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329 by simp |
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330 hence "0 + (-b) \<le> (-a + b) + (-b)" |
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331 by (rule add_right_mono) |
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332 thus ?thesis |
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333 by (simp add: add_assoc) |
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334 qed |
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335 |
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336 lemma neg_le_iff_le [simp]: "(-b \<le> -a) = (a \<le> (b::'a::pordered_ab_group_add))" |
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337 proof |
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338 assume "- b \<le> - a" |
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339 hence "- (- a) \<le> - (- b)" |
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340 by (rule le_imp_neg_le) |
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341 thus "a\<le>b" by simp |
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342 next |
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343 assume "a\<le>b" |
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344 thus "-b \<le> -a" by (rule le_imp_neg_le) |
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345 qed |
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346 |
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347 lemma neg_le_0_iff_le [simp]: "(-a \<le> 0) = (0 \<le> (a::'a::pordered_ab_group_add))" |
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348 by (subst neg_le_iff_le [symmetric], simp) |
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349 |
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350 lemma neg_0_le_iff_le [simp]: "(0 \<le> -a) = (a \<le> (0::'a::pordered_ab_group_add))" |
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351 by (subst neg_le_iff_le [symmetric], simp) |
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352 |
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353 lemma neg_less_iff_less [simp]: "(-b < -a) = (a < (b::'a::pordered_ab_group_add))" |
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354 by (force simp add: order_less_le) |
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355 |
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356 lemma neg_less_0_iff_less [simp]: "(-a < 0) = (0 < (a::'a::pordered_ab_group_add))" |
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357 by (subst neg_less_iff_less [symmetric], simp) |
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358 |
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359 lemma neg_0_less_iff_less [simp]: "(0 < -a) = (a < (0::'a::pordered_ab_group_add))" |
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360 by (subst neg_less_iff_less [symmetric], simp) |
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361 |
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362 text{*The next several equations can make the simplifier loop!*} |
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363 |
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364 lemma less_minus_iff: "(a < - b) = (b < - (a::'a::pordered_ab_group_add))" |
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365 proof - |
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366 have "(- (-a) < - b) = (b < - a)" by (rule neg_less_iff_less) |
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367 thus ?thesis by simp |
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368 qed |
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369 |
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370 lemma minus_less_iff: "(- a < b) = (- b < (a::'a::pordered_ab_group_add))" |
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371 proof - |
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372 have "(- a < - (-b)) = (- b < a)" by (rule neg_less_iff_less) |
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373 thus ?thesis by simp |
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374 qed |
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375 |
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376 lemma le_minus_iff: "(a \<le> - b) = (b \<le> - (a::'a::pordered_ab_group_add))" |
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377 proof - |
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378 have mm: "!! a (b::'a). (-(-a)) < -b \<Longrightarrow> -(-b) < -a" by (simp only: minus_less_iff) |
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379 have "(- (- a) <= -b) = (b <= - a)" |
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380 apply (auto simp only: order_le_less) |
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381 apply (drule mm) |
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382 apply (simp_all) |
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383 apply (drule mm[simplified], assumption) |
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384 done |
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385 then show ?thesis by simp |
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386 qed |
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387 |
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388 lemma minus_le_iff: "(- a \<le> b) = (- b \<le> (a::'a::pordered_ab_group_add))" |
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389 by (auto simp add: order_le_less minus_less_iff) |
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390 |
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391 lemma add_diff_eq: "a + (b - c) = (a + b) - (c::'a::ab_group_add)" |
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392 by (simp add: diff_minus add_ac) |
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393 |
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394 lemma diff_add_eq: "(a - b) + c = (a + c) - (b::'a::ab_group_add)" |
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395 by (simp add: diff_minus add_ac) |
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396 |
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397 lemma diff_eq_eq: "(a-b = c) = (a = c + (b::'a::ab_group_add))" |
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398 by (auto simp add: diff_minus add_assoc) |
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399 |
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400 lemma eq_diff_eq: "(a = c-b) = (a + (b::'a::ab_group_add) = c)" |
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401 by (auto simp add: diff_minus add_assoc) |
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402 |
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403 lemma diff_diff_eq: "(a - b) - c = a - (b + (c::'a::ab_group_add))" |
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404 by (simp add: diff_minus add_ac) |
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405 |
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406 lemma diff_diff_eq2: "a - (b - c) = (a + c) - (b::'a::ab_group_add)" |
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407 by (simp add: diff_minus add_ac) |
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408 |
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409 lemma diff_add_cancel: "a - b + b = (a::'a::ab_group_add)" |
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410 by (simp add: diff_minus add_ac) |
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411 |
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412 lemma add_diff_cancel: "a + b - b = (a::'a::ab_group_add)" |
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413 by (simp add: diff_minus add_ac) |
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414 |
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415 text{*Further subtraction laws for ordered rings*} |
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416 |
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417 lemma less_iff_diff_less_0: "(a < b) = (a - b < (0::'a::pordered_ab_group_add))" |
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418 proof - |
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419 have "(a < b) = (a + (- b) < b + (-b))" |
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420 by (simp only: add_less_cancel_right) |
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421 also have "... = (a - b < 0)" by (simp add: diff_minus) |
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422 finally show ?thesis . |
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423 qed |
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424 |
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425 lemma diff_less_eq: "(a-b < c) = (a < c + (b::'a::pordered_ab_group_add))" |
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426 apply (subst less_iff_diff_less_0) |
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427 apply (rule less_iff_diff_less_0 [of _ c, THEN ssubst]) |
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428 apply (simp add: diff_minus add_ac) |
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429 done |
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430 |
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431 lemma less_diff_eq: "(a < c-b) = (a + (b::'a::pordered_ab_group_add) < c)" |
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432 apply (subst less_iff_diff_less_0) |
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433 apply (rule less_iff_diff_less_0 [of _ "c-b", THEN ssubst]) |
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434 apply (simp add: diff_minus add_ac) |
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435 done |
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436 |
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437 lemma diff_le_eq: "(a-b \<le> c) = (a \<le> c + (b::'a::pordered_ab_group_add))" |
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438 by (auto simp add: order_le_less diff_less_eq diff_add_cancel add_diff_cancel) |
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439 |
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440 lemma le_diff_eq: "(a \<le> c-b) = (a + (b::'a::pordered_ab_group_add) \<le> c)" |
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441 by (auto simp add: order_le_less less_diff_eq diff_add_cancel add_diff_cancel) |
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442 |
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443 text{*This list of rewrites simplifies (in)equalities by bringing subtractions |
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444 to the top and then moving negative terms to the other side. |
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445 Use with @{text add_ac}*} |
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446 lemmas compare_rls = |
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447 diff_minus [symmetric] |
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448 add_diff_eq diff_add_eq diff_diff_eq diff_diff_eq2 |
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449 diff_less_eq less_diff_eq diff_le_eq le_diff_eq |
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450 diff_eq_eq eq_diff_eq |
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451 |
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452 |
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453 subsection{*Lemmas for the @{text cancel_numerals} simproc*} |
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454 |
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455 lemma eq_iff_diff_eq_0: "(a = b) = (a-b = (0::'a::ab_group_add))" |
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456 by (simp add: compare_rls) |
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457 |
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458 lemma le_iff_diff_le_0: "(a \<le> b) = (a-b \<le> (0::'a::pordered_ab_group_add))" |
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459 by (simp add: compare_rls) |
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460 |
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461 subsection {* Lattice Ordered (Abelian) Groups *} |
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462 |
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463 axclass lordered_ab_group_meet < pordered_ab_group_add, meet_semilorder |
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464 |
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465 axclass lordered_ab_group_join < pordered_ab_group_add, join_semilorder |
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466 |
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467 lemma add_meet_distrib_left: "a + (meet b c) = meet (a + b) (a + (c::'a::{pordered_ab_group_add, meet_semilorder}))" |
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468 apply (rule order_antisym) |
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469 apply (rule meet_imp_le, simp_all add: meet_join_le) |
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470 apply (rule add_le_imp_le_left [of "-a"]) |
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471 apply (simp only: add_assoc[symmetric], simp) |
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472 apply (rule meet_imp_le) |
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473 apply (rule add_le_imp_le_left[of "a"], simp only: add_assoc[symmetric], simp add: meet_join_le)+ |
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474 done |
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475 |
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476 lemma add_join_distrib_left: "a + (join b c) = join (a + b) (a+ (c::'a::{pordered_ab_group_add, join_semilorder}))" |
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477 apply (rule order_antisym) |
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478 apply (rule add_le_imp_le_left [of "-a"]) |
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479 apply (simp only: add_assoc[symmetric], simp) |
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480 apply (rule join_imp_le) |
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481 apply (rule add_le_imp_le_left [of "a"], simp only: add_assoc[symmetric], simp add: meet_join_le)+ |
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482 apply (rule join_imp_le) |
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483 apply (simp_all add: meet_join_le) |
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484 done |
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485 |
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486 lemma is_join_neg_meet: "is_join (% (a::'a::{pordered_ab_group_add, meet_semilorder}) b. - (meet (-a) (-b)))" |
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487 apply (auto simp add: is_join_def) |
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488 apply (rule_tac c="meet (-a) (-b)" in add_le_imp_le_right, simp, simp add: add_meet_distrib_left meet_join_le) |
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489 apply (rule_tac c="meet (-a) (-b)" in add_le_imp_le_right, simp, simp add: add_meet_distrib_left meet_join_le) |
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490 apply (subst neg_le_iff_le[symmetric]) |
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491 apply (simp add: meet_imp_le) |
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492 done |
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493 |
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494 lemma is_meet_neg_join: "is_meet (% (a::'a::{pordered_ab_group_add, join_semilorder}) b. - (join (-a) (-b)))" |
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495 apply (auto simp add: is_meet_def) |
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496 apply (rule_tac c="join (-a) (-b)" in add_le_imp_le_right, simp, simp add: add_join_distrib_left meet_join_le) |
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497 apply (rule_tac c="join (-a) (-b)" in add_le_imp_le_right, simp, simp add: add_join_distrib_left meet_join_le) |
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498 apply (subst neg_le_iff_le[symmetric]) |
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499 apply (simp add: join_imp_le) |
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500 done |
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501 |
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502 axclass lordered_ab_group \<subseteq> pordered_ab_group_add, lorder |
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503 |
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504 instance lordered_ab_group_meet \<subseteq> lordered_ab_group |
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505 proof |
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506 show "? j. is_join (j::'a\<Rightarrow>'a\<Rightarrow>('a::lordered_ab_group_meet))" by (blast intro: is_join_neg_meet) |
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507 qed |
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508 |
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509 instance lordered_ab_group_join \<subseteq> lordered_ab_group |
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510 proof |
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511 show "? m. is_meet (m::'a\<Rightarrow>'a\<Rightarrow>('a::lordered_ab_group_join))" by (blast intro: is_meet_neg_join) |
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512 qed |
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513 |
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514 lemma add_join_distrib_right: "(join a b) + (c::'a::lordered_ab_group) = join (a+c) (b+c)" |
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515 proof - |
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516 have "c + (join a b) = join (c+a) (c+b)" by (simp add: add_join_distrib_left) |
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517 thus ?thesis by (simp add: add_commute) |
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518 qed |
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519 |
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520 lemma add_meet_distrib_right: "(meet a b) + (c::'a::lordered_ab_group) = meet (a+c) (b+c)" |
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521 proof - |
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522 have "c + (meet a b) = meet (c+a) (c+b)" by (simp add: add_meet_distrib_left) |
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523 thus ?thesis by (simp add: add_commute) |
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524 qed |
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525 |
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526 lemmas add_meet_join_distribs = add_meet_distrib_right add_meet_distrib_left add_join_distrib_right add_join_distrib_left |
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527 |
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528 lemma join_eq_neg_meet: "join a (b::'a::lordered_ab_group) = - meet (-a) (-b)" |
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529 by (simp add: is_join_unique[OF is_join_join is_join_neg_meet]) |
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530 |
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531 lemma meet_eq_neg_join: "meet a (b::'a::lordered_ab_group) = - join (-a) (-b)" |
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532 by (simp add: is_meet_unique[OF is_meet_meet is_meet_neg_join]) |
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533 |
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534 lemma add_eq_meet_join: "a + b = (join a b) + (meet a (b::'a::lordered_ab_group))" |
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535 proof - |
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536 have "0 = - meet 0 (a-b) + meet (a-b) 0" by (simp add: meet_comm) |
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537 hence "0 = join 0 (b-a) + meet (a-b) 0" by (simp add: meet_eq_neg_join) |
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538 hence "0 = (-a + join a b) + (meet a b + (-b))" |
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539 apply (simp add: add_join_distrib_left add_meet_distrib_right) |
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540 by (simp add: diff_minus add_commute) |
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541 thus ?thesis |
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542 apply (simp add: compare_rls) |
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543 apply (subst add_left_cancel[symmetric, of "a+b" "join a b + meet a b" "-a"]) |
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544 apply (simp only: add_assoc, simp add: add_assoc[symmetric]) |
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545 done |
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546 qed |
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547 |
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548 subsection {* Positive Part, Negative Part, Absolute Value *} |
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549 |
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550 constdefs |
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551 pprt :: "'a \<Rightarrow> ('a::lordered_ab_group)" |
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552 "pprt x == join x 0" |
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553 nprt :: "'a \<Rightarrow> ('a::lordered_ab_group)" |
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554 "nprt x == meet x 0" |
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555 |
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556 lemma prts: "a = pprt a + nprt a" |
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557 by (simp add: pprt_def nprt_def add_eq_meet_join[symmetric]) |
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558 |
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559 lemma zero_le_pprt[simp]: "0 \<le> pprt a" |
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560 by (simp add: pprt_def meet_join_le) |
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561 |
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562 lemma nprt_le_zero[simp]: "nprt a \<le> 0" |
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563 by (simp add: nprt_def meet_join_le) |
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564 |
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565 lemma le_eq_neg: "(a \<le> -b) = (a + b \<le> (0::_::lordered_ab_group))" (is "?l = ?r") |
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566 proof - |
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567 have a: "?l \<longrightarrow> ?r" |
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568 apply (auto) |
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569 apply (rule add_le_imp_le_right[of _ "-b" _]) |
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570 apply (simp add: add_assoc) |
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571 done |
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572 have b: "?r \<longrightarrow> ?l" |
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573 apply (auto) |
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574 apply (rule add_le_imp_le_right[of _ "b" _]) |
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575 apply (simp) |
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576 done |
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577 from a b show ?thesis by blast |
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578 qed |
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579 |
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580 lemma join_0_imp_0: "join a (-a) = 0 \<Longrightarrow> a = (0::'a::lordered_ab_group)" |
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581 proof - |
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582 { |
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583 fix a::'a |
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584 assume hyp: "join a (-a) = 0" |
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585 hence "join a (-a) + a = a" by (simp) |
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586 hence "join (a+a) 0 = a" by (simp add: add_join_distrib_right) |
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587 hence "join (a+a) 0 <= a" by (simp) |
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588 hence "0 <= a" by (blast intro: order_trans meet_join_le) |
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589 } |
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590 note p = this |
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591 thm p |
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592 assume hyp:"join a (-a) = 0" |
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593 hence hyp2:"join (-a) (-(-a)) = 0" by (simp add: join_comm) |
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594 from p[OF hyp] p[OF hyp2] show "a = 0" by simp |
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595 qed |
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596 |
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597 lemma meet_0_imp_0: "meet a (-a) = 0 \<Longrightarrow> a = (0::'a::lordered_ab_group)" |
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598 apply (simp add: meet_eq_neg_join) |
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599 apply (simp add: join_comm) |
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600 apply (subst join_0_imp_0) |
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601 by auto |
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602 |
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603 lemma join_0_eq_0[simp]: "(join a (-a) = 0) = (a = (0::'a::lordered_ab_group))" |
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604 by (auto, erule join_0_imp_0) |
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605 |
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606 lemma meet_0_eq_0[simp]: "(meet a (-a) = 0) = (a = (0::'a::lordered_ab_group))" |
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607 by (auto, erule meet_0_imp_0) |
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608 |
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609 lemma zero_le_double_add_iff_zero_le_single_add[simp]: "(0 \<le> a + a) = (0 \<le> (a::'a::lordered_ab_group))" |
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610 proof |
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611 assume "0 <= a + a" |
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612 hence a:"meet (a+a) 0 = 0" by (simp add: le_def_meet meet_comm) |
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613 have "(meet a 0)+(meet a 0) = meet (meet (a+a) 0) a" (is "?l=_") by (simp add: add_meet_join_distribs meet_aci) |
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614 hence "?l = 0 + meet a 0" by (simp add: a, simp add: meet_comm) |
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615 hence "meet a 0 = 0" by (simp only: add_right_cancel) |
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616 then show "0 <= a" by (simp add: le_def_meet meet_comm) |
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617 next |
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618 assume a: "0 <= a" |
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619 show "0 <= a + a" by (simp add: add_mono[OF a a, simplified]) |
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620 qed |
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621 |
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622 lemma double_add_le_zero_iff_single_add_le_zero[simp]: "(a + a <= 0) = ((a::'a::lordered_ab_group) <= 0)" |
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623 proof - |
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624 have "(a + a <= 0) = (0 <= -(a+a))" by (subst le_minus_iff, simp) |
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625 moreover have "\<dots> = (a <= 0)" by (simp add: zero_le_double_add_iff_zero_le_single_add) |
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626 ultimately show ?thesis by blast |
|
627 qed |
|
628 |
|
629 lemma double_add_less_zero_iff_single_less_zero[simp]: "(a+a<0) = ((a::'a::{pordered_ab_group_add,linorder}) < 0)" (is ?s) |
|
630 proof cases |
|
631 assume a: "a < 0" |
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632 thus ?s by (simp add: add_strict_mono[OF a a, simplified]) |
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633 next |
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634 assume "~(a < 0)" |
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635 hence a:"0 <= a" by (simp) |
|
636 hence "0 <= a+a" by (simp add: add_mono[OF a a, simplified]) |
|
637 hence "~(a+a < 0)" by simp |
|
638 with a show ?thesis by simp |
|
639 qed |
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640 |
|
641 axclass lordered_ab_group_abs \<subseteq> lordered_ab_group |
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642 abs_lattice: "abs x = join x (-x)" |
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643 |
|
644 lemma abs_zero[simp]: "abs 0 = (0::'a::lordered_ab_group_abs)" |
|
645 by (simp add: abs_lattice) |
|
646 |
|
647 lemma abs_eq_0[simp]: "(abs a = 0) = (a = (0::'a::lordered_ab_group_abs))" |
|
648 by (simp add: abs_lattice) |
|
649 |
|
650 lemma abs_0_eq[simp]: "(0 = abs a) = (a = (0::'a::lordered_ab_group_abs))" |
|
651 proof - |
|
652 have "(0 = abs a) = (abs a = 0)" by (simp only: eq_ac) |
|
653 thus ?thesis by simp |
|
654 qed |
|
655 |
|
656 lemma neg_meet_eq_join[simp]: "- meet a (b::_::lordered_ab_group) = join (-a) (-b)" |
|
657 by (simp add: meet_eq_neg_join) |
|
658 |
|
659 lemma neg_join_eq_meet[simp]: "- join a (b::_::lordered_ab_group) = meet (-a) (-b)" |
|
660 by (simp del: neg_meet_eq_join add: join_eq_neg_meet) |
|
661 |
|
662 lemma join_eq_if: "join a (-a) = (if a < 0 then -a else (a::'a::{lordered_ab_group, linorder}))" |
|
663 proof - |
|
664 note b = add_le_cancel_right[of a a "-a",symmetric,simplified] |
|
665 have c: "a + a = 0 \<Longrightarrow> -a = a" by (rule add_right_imp_eq[of _ a], simp) |
|
666 show ?thesis |
|
667 apply (auto simp add: join_max max_def b linorder_not_less) |
|
668 apply (drule order_antisym, auto) |
|
669 done |
|
670 qed |
|
671 |
|
672 lemma abs_if_lattice: "\<bar>a\<bar> = (if a < 0 then -a else (a::'a::{lordered_ab_group_abs, linorder}))" |
|
673 proof - |
|
674 show ?thesis by (simp add: abs_lattice join_eq_if) |
|
675 qed |
|
676 |
|
677 lemma abs_ge_zero[simp]: "0 \<le> abs (a::'a::lordered_ab_group_abs)" |
|
678 proof - |
|
679 have a:"a <= abs a" and b:"-a <= abs a" by (auto simp add: abs_lattice meet_join_le) |
|
680 show ?thesis by (rule add_mono[OF a b, simplified]) |
|
681 qed |
|
682 |
|
683 lemma abs_le_zero_iff [simp]: "(abs a \<le> (0::'a::lordered_ab_group_abs)) = (a = 0)" |
|
684 proof |
|
685 assume "abs a <= 0" |
|
686 hence "abs a = 0" by (auto dest: order_antisym) |
|
687 thus "a = 0" by simp |
|
688 next |
|
689 assume "a = 0" |
|
690 thus "abs a <= 0" by simp |
|
691 qed |
|
692 |
|
693 lemma zero_less_abs_iff [simp]: "(0 < abs a) = (a \<noteq> (0::'a::lordered_ab_group_abs))" |
|
694 by (simp add: order_less_le) |
|
695 |
|
696 lemma abs_not_less_zero [simp]: "~ abs a < (0::'a::lordered_ab_group_abs)" |
|
697 proof - |
|
698 have a:"!! x (y::_::order). x <= y \<Longrightarrow> ~(y < x)" by auto |
|
699 show ?thesis by (simp add: a) |
|
700 qed |
|
701 |
|
702 lemma abs_ge_self: "a \<le> abs (a::'a::lordered_ab_group_abs)" |
|
703 by (simp add: abs_lattice meet_join_le) |
|
704 |
|
705 lemma abs_ge_minus_self: "-a \<le> abs (a::'a::lordered_ab_group_abs)" |
|
706 by (simp add: abs_lattice meet_join_le) |
|
707 |
|
708 lemma le_imp_join_eq: "a \<le> b \<Longrightarrow> join a b = b" |
|
709 by (simp add: le_def_join) |
|
710 |
|
711 lemma ge_imp_join_eq: "b \<le> a \<Longrightarrow> join a b = a" |
|
712 by (simp add: le_def_join join_aci) |
|
713 |
|
714 lemma le_imp_meet_eq: "a \<le> b \<Longrightarrow> meet a b = a" |
|
715 by (simp add: le_def_meet) |
|
716 |
|
717 lemma ge_imp_meet_eq: "b \<le> a \<Longrightarrow> meet a b = b" |
|
718 by (simp add: le_def_meet meet_aci) |
|
719 |
|
720 lemma abs_prts: "abs (a::_::lordered_ab_group_abs) = pprt a - nprt a" |
|
721 apply (simp add: pprt_def nprt_def diff_minus) |
|
722 apply (simp add: add_meet_join_distribs join_aci abs_lattice[symmetric]) |
|
723 apply (subst le_imp_join_eq, auto) |
|
724 done |
|
725 |
|
726 lemma abs_minus_cancel [simp]: "abs (-a) = abs(a::'a::lordered_ab_group_abs)" |
|
727 by (simp add: abs_lattice join_comm) |
|
728 |
|
729 lemma abs_idempotent [simp]: "abs (abs a) = abs (a::'a::lordered_ab_group_abs)" |
|
730 apply (simp add: abs_lattice[of "abs a"]) |
|
731 apply (subst ge_imp_join_eq) |
|
732 apply (rule order_trans[of _ 0]) |
|
733 by auto |
|
734 |
|
735 lemma zero_le_iff_zero_nprt: "(0 \<le> a) = (nprt a = 0)" |
|
736 by (simp add: le_def_meet nprt_def meet_comm) |
|
737 |
|
738 lemma le_zero_iff_zero_pprt: "(a \<le> 0) = (pprt a = 0)" |
|
739 by (simp add: le_def_join pprt_def join_comm) |
|
740 |
|
741 lemma le_zero_iff_pprt_id: "(0 \<le> a) = (pprt a = a)" |
|
742 by (simp add: le_def_join pprt_def join_comm) |
|
743 |
|
744 lemma zero_le_iff_nprt_id: "(a \<le> 0) = (nprt a = a)" |
|
745 by (simp add: le_def_meet nprt_def meet_comm) |
|
746 |
|
747 lemma iff2imp: "(A=B) \<Longrightarrow> (A \<Longrightarrow> B)" |
|
748 by (simp) |
|
749 |
|
750 lemma imp_abs_id: "0 \<le> a \<Longrightarrow> abs a = (a::'a::lordered_ab_group_abs)" |
|
751 by (simp add: iff2imp[OF zero_le_iff_zero_nprt] iff2imp[OF le_zero_iff_pprt_id] abs_prts) |
|
752 |
|
753 lemma imp_abs_neg_id: "a \<le> 0 \<Longrightarrow> abs a = -(a::'a::lordered_ab_group_abs)" |
|
754 by (simp add: iff2imp[OF le_zero_iff_zero_pprt] iff2imp[OF zero_le_iff_nprt_id] abs_prts) |
|
755 |
|
756 lemma abs_leI: "[|a \<le> b; -a \<le> b|] ==> abs a \<le> (b::'a::lordered_ab_group_abs)" |
|
757 by (simp add: abs_lattice join_imp_le) |
|
758 |
|
759 lemma le_minus_self_iff: "(a \<le> -a) = (a \<le> (0::'a::lordered_ab_group))" |
|
760 proof - |
|
761 from add_le_cancel_left[of "-a" "a+a" "0"] have "(a <= -a) = (a+a <= 0)" |
|
762 by (simp add: add_assoc[symmetric]) |
|
763 thus ?thesis by simp |
|
764 qed |
|
765 |
|
766 lemma minus_le_self_iff: "(-a \<le> a) = (0 \<le> (a::'a::lordered_ab_group))" |
|
767 proof - |
|
768 from add_le_cancel_left[of "-a" "0" "a+a"] have "(-a <= a) = (0 <= a+a)" |
|
769 by (simp add: add_assoc[symmetric]) |
|
770 thus ?thesis by simp |
|
771 qed |
|
772 |
|
773 lemma abs_le_D1: "abs a \<le> b ==> a \<le> (b::'a::lordered_ab_group_abs)" |
|
774 by (insert abs_ge_self, blast intro: order_trans) |
|
775 |
|
776 lemma abs_le_D2: "abs a \<le> b ==> -a \<le> (b::'a::lordered_ab_group_abs)" |
|
777 by (insert abs_le_D1 [of "-a"], simp) |
|
778 |
|
779 lemma abs_le_iff: "(abs a \<le> b) = (a \<le> b & -a \<le> (b::'a::lordered_ab_group_abs))" |
|
780 by (blast intro: abs_leI dest: abs_le_D1 abs_le_D2) |
|
781 |
|
782 lemma abs_triangle_ineq: "abs (a+b) \<le> abs a + abs (b::'a::lordered_ab_group_abs)" |
|
783 proof - |
|
784 have g:"abs a + abs b = join (a+b) (join (-a-b) (join (-a+b) (a + (-b))))" (is "_=join ?m ?n") |
|
785 apply (simp add: abs_lattice add_meet_join_distribs join_aci) |
|
786 by (simp only: diff_minus) |
|
787 have a:"a+b <= join ?m ?n" by (simp add: meet_join_le) |
|
788 have b:"-a-b <= ?n" by (simp add: meet_join_le) |
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789 have c:"?n <= join ?m ?n" by (simp add: meet_join_le) |
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790 from b c have d: "-a-b <= join ?m ?n" by simp |
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791 have e:"-a-b = -(a+b)" by (simp add: diff_minus) |
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792 from a d e have "abs(a+b) <= join ?m ?n" |
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793 by (drule_tac abs_leI, auto) |
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794 with g[symmetric] show ?thesis by simp |
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795 qed |
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796 |
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797 lemma abs_diff_triangle_ineq: |
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798 "\<bar>(a::'a::lordered_ab_group_abs) + b - (c+d)\<bar> \<le> \<bar>a-c\<bar> + \<bar>b-d\<bar>" |
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799 proof - |
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800 have "\<bar>a + b - (c+d)\<bar> = \<bar>(a-c) + (b-d)\<bar>" by (simp add: diff_minus add_ac) |
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801 also have "... \<le> \<bar>a-c\<bar> + \<bar>b-d\<bar>" by (rule abs_triangle_ineq) |
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802 finally show ?thesis . |
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803 qed |
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804 |
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805 ML {* |
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806 val add_zero_left = thm"add_0"; |
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807 val add_zero_right = thm"add_0_right"; |
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808 *} |
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809 |
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810 ML {* |
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811 val add_assoc = thm "add_assoc"; |
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812 val add_commute = thm "add_commute"; |
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813 val add_left_commute = thm "add_left_commute"; |
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814 val add_ac = thms "add_ac"; |
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815 val mult_assoc = thm "mult_assoc"; |
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816 val mult_commute = thm "mult_commute"; |
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817 val mult_left_commute = thm "mult_left_commute"; |
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818 val mult_ac = thms "mult_ac"; |
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819 val add_0 = thm "add_0"; |
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820 val mult_1_left = thm "mult_1_left"; |
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821 val mult_1_right = thm "mult_1_right"; |
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822 val mult_1 = thm "mult_1"; |
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823 val add_left_imp_eq = thm "add_left_imp_eq"; |
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824 val add_right_imp_eq = thm "add_right_imp_eq"; |
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825 val add_imp_eq = thm "add_imp_eq"; |
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826 val left_minus = thm "left_minus"; |
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827 val diff_minus = thm "diff_minus"; |
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828 val add_0_right = thm "add_0_right"; |
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829 val add_left_cancel = thm "add_left_cancel"; |
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830 val add_right_cancel = thm "add_right_cancel"; |
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831 val right_minus = thm "right_minus"; |
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832 val right_minus_eq = thm "right_minus_eq"; |
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833 val minus_minus = thm "minus_minus"; |
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834 val equals_zero_I = thm "equals_zero_I"; |
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835 val minus_zero = thm "minus_zero"; |
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836 val diff_self = thm "diff_self"; |
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837 val diff_0 = thm "diff_0"; |
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838 val diff_0_right = thm "diff_0_right"; |
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839 val diff_minus_eq_add = thm "diff_minus_eq_add"; |
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840 val neg_equal_iff_equal = thm "neg_equal_iff_equal"; |
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841 val neg_equal_0_iff_equal = thm "neg_equal_0_iff_equal"; |
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842 val neg_0_equal_iff_equal = thm "neg_0_equal_iff_equal"; |
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843 val equation_minus_iff = thm "equation_minus_iff"; |
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844 val minus_equation_iff = thm "minus_equation_iff"; |
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845 val minus_add_distrib = thm "minus_add_distrib"; |
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846 val minus_diff_eq = thm "minus_diff_eq"; |
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847 val add_left_mono = thm "add_left_mono"; |
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848 val add_le_imp_le_left = thm "add_le_imp_le_left"; |
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849 val add_right_mono = thm "add_right_mono"; |
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850 val add_mono = thm "add_mono"; |
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851 val add_strict_left_mono = thm "add_strict_left_mono"; |
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852 val add_strict_right_mono = thm "add_strict_right_mono"; |
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853 val add_strict_mono = thm "add_strict_mono"; |
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854 val add_less_le_mono = thm "add_less_le_mono"; |
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855 val add_le_less_mono = thm "add_le_less_mono"; |
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856 val add_less_imp_less_left = thm "add_less_imp_less_left"; |
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857 val add_less_imp_less_right = thm "add_less_imp_less_right"; |
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858 val add_less_cancel_left = thm "add_less_cancel_left"; |
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859 val add_less_cancel_right = thm "add_less_cancel_right"; |
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860 val add_le_cancel_left = thm "add_le_cancel_left"; |
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861 val add_le_cancel_right = thm "add_le_cancel_right"; |
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862 val add_le_imp_le_right = thm "add_le_imp_le_right"; |
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863 val add_increasing = thm "add_increasing"; |
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864 val le_imp_neg_le = thm "le_imp_neg_le"; |
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865 val neg_le_iff_le = thm "neg_le_iff_le"; |
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866 val neg_le_0_iff_le = thm "neg_le_0_iff_le"; |
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867 val neg_0_le_iff_le = thm "neg_0_le_iff_le"; |
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868 val neg_less_iff_less = thm "neg_less_iff_less"; |
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869 val neg_less_0_iff_less = thm "neg_less_0_iff_less"; |
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870 val neg_0_less_iff_less = thm "neg_0_less_iff_less"; |
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871 val less_minus_iff = thm "less_minus_iff"; |
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872 val minus_less_iff = thm "minus_less_iff"; |
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873 val le_minus_iff = thm "le_minus_iff"; |
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874 val minus_le_iff = thm "minus_le_iff"; |
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875 val add_diff_eq = thm "add_diff_eq"; |
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876 val diff_add_eq = thm "diff_add_eq"; |
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877 val diff_eq_eq = thm "diff_eq_eq"; |
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878 val eq_diff_eq = thm "eq_diff_eq"; |
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879 val diff_diff_eq = thm "diff_diff_eq"; |
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880 val diff_diff_eq2 = thm "diff_diff_eq2"; |
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881 val diff_add_cancel = thm "diff_add_cancel"; |
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882 val add_diff_cancel = thm "add_diff_cancel"; |
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883 val less_iff_diff_less_0 = thm "less_iff_diff_less_0"; |
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884 val diff_less_eq = thm "diff_less_eq"; |
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885 val less_diff_eq = thm "less_diff_eq"; |
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886 val diff_le_eq = thm "diff_le_eq"; |
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887 val le_diff_eq = thm "le_diff_eq"; |
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888 val compare_rls = thms "compare_rls"; |
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889 val eq_iff_diff_eq_0 = thm "eq_iff_diff_eq_0"; |
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890 val le_iff_diff_le_0 = thm "le_iff_diff_le_0"; |
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891 val add_meet_distrib_left = thm "add_meet_distrib_left"; |
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892 val add_join_distrib_left = thm "add_join_distrib_left"; |
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893 val is_join_neg_meet = thm "is_join_neg_meet"; |
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894 val is_meet_neg_join = thm "is_meet_neg_join"; |
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895 val add_join_distrib_right = thm "add_join_distrib_right"; |
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896 val add_meet_distrib_right = thm "add_meet_distrib_right"; |
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897 val add_meet_join_distribs = thms "add_meet_join_distribs"; |
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898 val join_eq_neg_meet = thm "join_eq_neg_meet"; |
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899 val meet_eq_neg_join = thm "meet_eq_neg_join"; |
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900 val add_eq_meet_join = thm "add_eq_meet_join"; |
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901 val prts = thm "prts"; |
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902 val zero_le_pprt = thm "zero_le_pprt"; |
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903 val nprt_le_zero = thm "nprt_le_zero"; |
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904 val le_eq_neg = thm "le_eq_neg"; |
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905 val join_0_imp_0 = thm "join_0_imp_0"; |
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906 val meet_0_imp_0 = thm "meet_0_imp_0"; |
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907 val join_0_eq_0 = thm "join_0_eq_0"; |
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908 val meet_0_eq_0 = thm "meet_0_eq_0"; |
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909 val zero_le_double_add_iff_zero_le_single_add = thm "zero_le_double_add_iff_zero_le_single_add"; |
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910 val double_add_le_zero_iff_single_add_le_zero = thm "double_add_le_zero_iff_single_add_le_zero"; |
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911 val double_add_less_zero_iff_single_less_zero = thm "double_add_less_zero_iff_single_less_zero"; |
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912 val abs_lattice = thm "abs_lattice"; |
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913 val abs_zero = thm "abs_zero"; |
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914 val abs_eq_0 = thm "abs_eq_0"; |
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915 val abs_0_eq = thm "abs_0_eq"; |
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916 val neg_meet_eq_join = thm "neg_meet_eq_join"; |
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917 val neg_join_eq_meet = thm "neg_join_eq_meet"; |
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918 val join_eq_if = thm "join_eq_if"; |
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919 val abs_if_lattice = thm "abs_if_lattice"; |
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920 val abs_ge_zero = thm "abs_ge_zero"; |
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921 val abs_le_zero_iff = thm "abs_le_zero_iff"; |
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922 val zero_less_abs_iff = thm "zero_less_abs_iff"; |
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923 val abs_not_less_zero = thm "abs_not_less_zero"; |
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924 val abs_ge_self = thm "abs_ge_self"; |
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925 val abs_ge_minus_self = thm "abs_ge_minus_self"; |
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926 val le_imp_join_eq = thm "le_imp_join_eq"; |
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927 val ge_imp_join_eq = thm "ge_imp_join_eq"; |
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928 val le_imp_meet_eq = thm "le_imp_meet_eq"; |
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929 val ge_imp_meet_eq = thm "ge_imp_meet_eq"; |
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930 val abs_prts = thm "abs_prts"; |
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931 val abs_minus_cancel = thm "abs_minus_cancel"; |
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932 val abs_idempotent = thm "abs_idempotent"; |
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933 val zero_le_iff_zero_nprt = thm "zero_le_iff_zero_nprt"; |
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934 val le_zero_iff_zero_pprt = thm "le_zero_iff_zero_pprt"; |
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935 val le_zero_iff_pprt_id = thm "le_zero_iff_pprt_id"; |
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936 val zero_le_iff_nprt_id = thm "zero_le_iff_nprt_id"; |
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937 val iff2imp = thm "iff2imp"; |
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938 val imp_abs_id = thm "imp_abs_id"; |
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939 val imp_abs_neg_id = thm "imp_abs_neg_id"; |
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940 val abs_leI = thm "abs_leI"; |
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941 val le_minus_self_iff = thm "le_minus_self_iff"; |
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942 val minus_le_self_iff = thm "minus_le_self_iff"; |
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943 val abs_le_D1 = thm "abs_le_D1"; |
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944 val abs_le_D2 = thm "abs_le_D2"; |
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945 val abs_le_iff = thm "abs_le_iff"; |
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946 val abs_triangle_ineq = thm "abs_triangle_ineq"; |
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947 val abs_diff_triangle_ineq = thm "abs_diff_triangle_ineq"; |
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948 *} |
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949 |
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950 end |