author  wenzelm 
Sat, 07 Apr 2012 16:41:59 +0200  
changeset 47389  e8552cba702d 
parent 44921  58eef4843641 
child 52435  6646bb548c6b 
permissions  rwrr 
35050
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(* Title: HOL/Fields.thy 
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2 
Author: Gertrud Bauer 
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3 
Author: Steven Obua 
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4 
Author: Tobias Nipkow 
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5 
Author: Lawrence C Paulson 
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6 
Author: Markus Wenzel 
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7 
Author: Jeremy Avigad 
14265
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HOL: installation of Ring_and_Field as the basis for Naturals and Reals
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8 
*) 
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HOL: installation of Ring_and_Field as the basis for Naturals and Reals
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9 

35050
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10 
header {* Fields *} 
25152  11 

35050
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12 
theory Fields 
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imports Rings 
25186  14 
begin 
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15 

44064
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subsection {* Division rings *} 
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17 

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text {* 
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A division ring is like a field, but without the commutativity requirement. 
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*} 
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21 

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class inverse = 
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23 
fixes inverse :: "'a \<Rightarrow> 'a" 
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24 
and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "'/" 70) 
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25 

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class division_ring = ring_1 + inverse + 
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27 
assumes left_inverse [simp]: "a \<noteq> 0 \<Longrightarrow> inverse a * a = 1" 
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28 
assumes right_inverse [simp]: "a \<noteq> 0 \<Longrightarrow> a * inverse a = 1" 
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29 
assumes divide_inverse: "a / b = a * inverse b" 
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30 
begin 
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31 

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32 
subclass ring_1_no_zero_divisors 
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33 
proof 
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34 
fix a b :: 'a 
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35 
assume a: "a \<noteq> 0" and b: "b \<noteq> 0" 
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36 
show "a * b \<noteq> 0" 
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37 
proof 
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38 
assume ab: "a * b = 0" 
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39 
hence "0 = inverse a * (a * b) * inverse b" by simp 
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40 
also have "\<dots> = (inverse a * a) * (b * inverse b)" 
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41 
by (simp only: mult_assoc) 
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42 
also have "\<dots> = 1" using a b by simp 
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43 
finally show False by simp 
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44 
qed 
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45 
qed 
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46 

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47 
lemma nonzero_imp_inverse_nonzero: 
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48 
"a \<noteq> 0 \<Longrightarrow> inverse a \<noteq> 0" 
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49 
proof 
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50 
assume ianz: "inverse a = 0" 
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51 
assume "a \<noteq> 0" 
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52 
hence "1 = a * inverse a" by simp 
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53 
also have "... = 0" by (simp add: ianz) 
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54 
finally have "1 = 0" . 
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55 
thus False by (simp add: eq_commute) 
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56 
qed 
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57 

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58 
lemma inverse_zero_imp_zero: 
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59 
"inverse a = 0 \<Longrightarrow> a = 0" 
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60 
apply (rule classical) 
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61 
apply (drule nonzero_imp_inverse_nonzero) 
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62 
apply auto 
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63 
done 
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64 

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65 
lemma inverse_unique: 
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66 
assumes ab: "a * b = 1" 
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67 
shows "inverse a = b" 
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68 
proof  
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69 
have "a \<noteq> 0" using ab by (cases "a = 0") simp_all 
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70 
moreover have "inverse a * (a * b) = inverse a" by (simp add: ab) 
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71 
ultimately show ?thesis by (simp add: mult_assoc [symmetric]) 
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72 
qed 
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73 

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74 
lemma nonzero_inverse_minus_eq: 
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75 
"a \<noteq> 0 \<Longrightarrow> inverse ( a) =  inverse a" 
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76 
by (rule inverse_unique) simp 
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77 

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78 
lemma nonzero_inverse_inverse_eq: 
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79 
"a \<noteq> 0 \<Longrightarrow> inverse (inverse a) = a" 
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80 
by (rule inverse_unique) simp 
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81 

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82 
lemma nonzero_inverse_eq_imp_eq: 
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83 
assumes "inverse a = inverse b" and "a \<noteq> 0" and "b \<noteq> 0" 
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84 
shows "a = b" 
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85 
proof  
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86 
from `inverse a = inverse b` 
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87 
have "inverse (inverse a) = inverse (inverse b)" by (rule arg_cong) 
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88 
with `a \<noteq> 0` and `b \<noteq> 0` show "a = b" 
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89 
by (simp add: nonzero_inverse_inverse_eq) 
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90 
qed 
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91 

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92 
lemma inverse_1 [simp]: "inverse 1 = 1" 
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93 
by (rule inverse_unique) simp 
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94 

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95 
lemma nonzero_inverse_mult_distrib: 
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96 
assumes "a \<noteq> 0" and "b \<noteq> 0" 
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97 
shows "inverse (a * b) = inverse b * inverse a" 
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98 
proof  
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99 
have "a * (b * inverse b) * inverse a = 1" using assms by simp 
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100 
hence "a * b * (inverse b * inverse a) = 1" by (simp only: mult_assoc) 
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101 
thus ?thesis by (rule inverse_unique) 
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102 
qed 
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103 

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104 
lemma division_ring_inverse_add: 
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105 
"a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> inverse a + inverse b = inverse a * (a + b) * inverse b" 
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moved division ring stuff from Rings.thy to Fields.thy
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106 
by (simp add: algebra_simps) 
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moved division ring stuff from Rings.thy to Fields.thy
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107 

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108 
lemma division_ring_inverse_diff: 
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109 
"a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> inverse a  inverse b = inverse a * (b  a) * inverse b" 
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moved division ring stuff from Rings.thy to Fields.thy
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110 
by (simp add: algebra_simps) 
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moved division ring stuff from Rings.thy to Fields.thy
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111 

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112 
lemma right_inverse_eq: "b \<noteq> 0 \<Longrightarrow> a / b = 1 \<longleftrightarrow> a = b" 
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113 
proof 
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114 
assume neq: "b \<noteq> 0" 
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115 
{ 
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116 
hence "a = (a / b) * b" by (simp add: divide_inverse mult_assoc) 
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117 
also assume "a / b = 1" 
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118 
finally show "a = b" by simp 
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119 
next 
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120 
assume "a = b" 
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121 
with neq show "a / b = 1" by (simp add: divide_inverse) 
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122 
} 
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123 
qed 
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moved division ring stuff from Rings.thy to Fields.thy
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124 

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125 
lemma nonzero_inverse_eq_divide: "a \<noteq> 0 \<Longrightarrow> inverse a = 1 / a" 
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moved division ring stuff from Rings.thy to Fields.thy
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126 
by (simp add: divide_inverse) 
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moved division ring stuff from Rings.thy to Fields.thy
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127 

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128 
lemma divide_self [simp]: "a \<noteq> 0 \<Longrightarrow> a / a = 1" 
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moved division ring stuff from Rings.thy to Fields.thy
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129 
by (simp add: divide_inverse) 
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moved division ring stuff from Rings.thy to Fields.thy
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130 

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131 
lemma divide_zero_left [simp]: "0 / a = 0" 
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132 
by (simp add: divide_inverse) 
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moved division ring stuff from Rings.thy to Fields.thy
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133 

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134 
lemma inverse_eq_divide: "inverse a = 1 / a" 
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moved division ring stuff from Rings.thy to Fields.thy
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135 
by (simp add: divide_inverse) 
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moved division ring stuff from Rings.thy to Fields.thy
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136 

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137 
lemma add_divide_distrib: "(a+b) / c = a/c + b/c" 
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138 
by (simp add: divide_inverse algebra_simps) 
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139 

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140 
lemma divide_1 [simp]: "a / 1 = a" 
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141 
by (simp add: divide_inverse) 
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142 

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143 
lemma times_divide_eq_right [simp]: "a * (b / c) = (a * b) / c" 
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144 
by (simp add: divide_inverse mult_assoc) 
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145 

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146 
lemma minus_divide_left: " (a / b) = (a) / b" 
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147 
by (simp add: divide_inverse) 
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148 

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149 
lemma nonzero_minus_divide_right: "b \<noteq> 0 ==>  (a / b) = a / ( b)" 
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150 
by (simp add: divide_inverse nonzero_inverse_minus_eq) 
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151 

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152 
lemma nonzero_minus_divide_divide: "b \<noteq> 0 ==> (a) / (b) = a / b" 
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153 
by (simp add: divide_inverse nonzero_inverse_minus_eq) 
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154 

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155 
lemma divide_minus_left [simp, no_atp]: "(a) / b =  (a / b)" 
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156 
by (simp add: divide_inverse) 
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157 

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158 
lemma diff_divide_distrib: "(a  b) / c = a / c  b / c" 
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159 
by (simp add: diff_minus add_divide_distrib) 
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160 

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161 
lemma nonzero_eq_divide_eq [field_simps]: "c \<noteq> 0 \<Longrightarrow> a = b / c \<longleftrightarrow> a * c = b" 
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162 
proof  
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163 
assume [simp]: "c \<noteq> 0" 
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164 
have "a = b / c \<longleftrightarrow> a * c = (b / c) * c" by simp 
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165 
also have "... \<longleftrightarrow> a * c = b" by (simp add: divide_inverse mult_assoc) 
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166 
finally show ?thesis . 
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167 
qed 
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168 

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169 
lemma nonzero_divide_eq_eq [field_simps]: "c \<noteq> 0 \<Longrightarrow> b / c = a \<longleftrightarrow> b = a * c" 
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170 
proof  
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171 
assume [simp]: "c \<noteq> 0" 
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172 
have "b / c = a \<longleftrightarrow> (b / c) * c = a * c" by simp 
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173 
also have "... \<longleftrightarrow> b = a * c" by (simp add: divide_inverse mult_assoc) 
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174 
finally show ?thesis . 
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175 
qed 
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176 

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177 
lemma divide_eq_imp: "c \<noteq> 0 \<Longrightarrow> b = a * c \<Longrightarrow> b / c = a" 
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178 
by (simp add: divide_inverse mult_assoc) 
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179 

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180 
lemma eq_divide_imp: "c \<noteq> 0 \<Longrightarrow> a * c = b \<Longrightarrow> a = b / c" 
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181 
by (drule sym) (simp add: divide_inverse mult_assoc) 
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182 

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183 
end 
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184 

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185 
class division_ring_inverse_zero = division_ring + 
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186 
assumes inverse_zero [simp]: "inverse 0 = 0" 
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187 
begin 
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188 

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189 
lemma divide_zero [simp]: 
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190 
"a / 0 = 0" 
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191 
by (simp add: divide_inverse) 
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192 

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193 
lemma divide_self_if [simp]: 
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194 
"a / a = (if a = 0 then 0 else 1)" 
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195 
by simp 
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196 

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197 
lemma inverse_nonzero_iff_nonzero [simp]: 
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198 
"inverse a = 0 \<longleftrightarrow> a = 0" 
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199 
by rule (fact inverse_zero_imp_zero, simp) 
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200 

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201 
lemma inverse_minus_eq [simp]: 
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202 
"inverse ( a) =  inverse a" 
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203 
proof cases 
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204 
assume "a=0" thus ?thesis by simp 
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205 
next 
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206 
assume "a\<noteq>0" 
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207 
thus ?thesis by (simp add: nonzero_inverse_minus_eq) 
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208 
qed 
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209 

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210 
lemma inverse_inverse_eq [simp]: 
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211 
"inverse (inverse a) = a" 
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212 
proof cases 
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213 
assume "a=0" thus ?thesis by simp 
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214 
next 
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215 
assume "a\<noteq>0" 
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216 
thus ?thesis by (simp add: nonzero_inverse_inverse_eq) 
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217 
qed 
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218 

44680  219 
lemma inverse_eq_imp_eq: 
220 
"inverse a = inverse b \<Longrightarrow> a = b" 

221 
by (drule arg_cong [where f="inverse"], simp) 

222 

223 
lemma inverse_eq_iff_eq [simp]: 

224 
"inverse a = inverse b \<longleftrightarrow> a = b" 

225 
by (force dest!: inverse_eq_imp_eq) 

226 

44064
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227 
end 
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228 

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229 
subsection {* Fields *} 
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230 

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231 
class field = comm_ring_1 + inverse + 
35084  232 
assumes field_inverse: "a \<noteq> 0 \<Longrightarrow> inverse a * a = 1" 
233 
assumes field_divide_inverse: "a / b = a * inverse b" 

25267  234 
begin 
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235 

25267  236 
subclass division_ring 
28823  237 
proof 
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238 
fix a :: 'a 
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239 
assume "a \<noteq> 0" 
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240 
thus "inverse a * a = 1" by (rule field_inverse) 
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241 
thus "a * inverse a = 1" by (simp only: mult_commute) 
35084  242 
next 
243 
fix a b :: 'a 

244 
show "a / b = a * inverse b" by (rule field_divide_inverse) 

14738  245 
qed 
25230  246 

27516  247 
subclass idom .. 
25230  248 

30630  249 
text{*There is no slick version using division by zero.*} 
250 
lemma inverse_add: 

251 
"[ a \<noteq> 0; b \<noteq> 0 ] 

252 
==> inverse a + inverse b = (a + b) * inverse a * inverse b" 

253 
by (simp add: division_ring_inverse_add mult_ac) 

254 

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255 
lemma nonzero_mult_divide_mult_cancel_left [simp, no_atp]: 
30630  256 
assumes [simp]: "b\<noteq>0" and [simp]: "c\<noteq>0" shows "(c*a)/(c*b) = a/b" 
257 
proof  

258 
have "(c*a)/(c*b) = c * a * (inverse b * inverse c)" 

259 
by (simp add: divide_inverse nonzero_inverse_mult_distrib) 

260 
also have "... = a * inverse b * (inverse c * c)" 

261 
by (simp only: mult_ac) 

262 
also have "... = a * inverse b" by simp 

263 
finally show ?thesis by (simp add: divide_inverse) 

264 
qed 

265 

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266 
lemma nonzero_mult_divide_mult_cancel_right [simp, no_atp]: 
30630  267 
"\<lbrakk>b \<noteq> 0; c \<noteq> 0\<rbrakk> \<Longrightarrow> (a * c) / (b * c) = a / b" 
268 
by (simp add: mult_commute [of _ c]) 

269 

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270 
lemma times_divide_eq_left [simp]: "(b / c) * a = (b * a) / c" 
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271 
by (simp add: divide_inverse mult_ac) 
30630  272 

44921  273 
text{*It's not obvious whether @{text times_divide_eq} should be 
274 
simprules or not. Their effect is to gather terms into one big 

275 
fraction, like a*b*c / x*y*z. The rationale for that is unclear, but 

276 
many proofs seem to need them.*} 

277 

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278 
lemmas times_divide_eq [no_atp] = times_divide_eq_right times_divide_eq_left 
30630  279 

280 
lemma add_frac_eq: 

281 
assumes "y \<noteq> 0" and "z \<noteq> 0" 

282 
shows "x / y + w / z = (x * z + w * y) / (y * z)" 

283 
proof  

284 
have "x / y + w / z = (x * z) / (y * z) + (y * w) / (y * z)" 

285 
using assms by simp 

286 
also have "\<dots> = (x * z + y * w) / (y * z)" 

287 
by (simp only: add_divide_distrib) 

288 
finally show ?thesis 

289 
by (simp only: mult_commute) 

290 
qed 

291 

292 
text{*Special Cancellation Simprules for Division*} 

293 

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294 
lemma nonzero_mult_divide_cancel_right [simp, no_atp]: 
30630  295 
"b \<noteq> 0 \<Longrightarrow> a * b / b = a" 
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296 
using nonzero_mult_divide_mult_cancel_right [of 1 b a] by simp 
30630  297 

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298 
lemma nonzero_mult_divide_cancel_left [simp, no_atp]: 
30630  299 
"a \<noteq> 0 \<Longrightarrow> a * b / a = b" 
300 
using nonzero_mult_divide_mult_cancel_left [of 1 a b] by simp 

301 

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302 
lemma nonzero_divide_mult_cancel_right [simp, no_atp]: 
30630  303 
"\<lbrakk>a \<noteq> 0; b \<noteq> 0\<rbrakk> \<Longrightarrow> b / (a * b) = 1 / a" 
304 
using nonzero_mult_divide_mult_cancel_right [of a b 1] by simp 

305 

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306 
lemma nonzero_divide_mult_cancel_left [simp, no_atp]: 
30630  307 
"\<lbrakk>a \<noteq> 0; b \<noteq> 0\<rbrakk> \<Longrightarrow> a / (a * b) = 1 / b" 
308 
using nonzero_mult_divide_mult_cancel_left [of b a 1] by simp 

309 

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310 
lemma nonzero_mult_divide_mult_cancel_left2 [simp, no_atp]: 
30630  311 
"\<lbrakk>b \<noteq> 0; c \<noteq> 0\<rbrakk> \<Longrightarrow> (c * a) / (b * c) = a / b" 
312 
using nonzero_mult_divide_mult_cancel_left [of b c a] by (simp add: mult_ac) 

313 

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314 
lemma nonzero_mult_divide_mult_cancel_right2 [simp, no_atp]: 
30630  315 
"\<lbrakk>b \<noteq> 0; c \<noteq> 0\<rbrakk> \<Longrightarrow> (a * c) / (c * b) = a / b" 
316 
using nonzero_mult_divide_mult_cancel_right [of b c a] by (simp add: mult_ac) 

317 

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318 
lemma add_divide_eq_iff [field_simps]: 
30630  319 
"z \<noteq> 0 \<Longrightarrow> x + y / z = (z * x + y) / z" 
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320 
by (simp add: add_divide_distrib) 
30630  321 

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322 
lemma divide_add_eq_iff [field_simps]: 
30630  323 
"z \<noteq> 0 \<Longrightarrow> x / z + y = (x + z * y) / z" 
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324 
by (simp add: add_divide_distrib) 
30630  325 

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326 
lemma diff_divide_eq_iff [field_simps]: 
30630  327 
"z \<noteq> 0 \<Longrightarrow> x  y / z = (z * x  y) / z" 
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328 
by (simp add: diff_divide_distrib) 
30630  329 

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330 
lemma divide_diff_eq_iff [field_simps]: 
30630  331 
"z \<noteq> 0 \<Longrightarrow> x / z  y = (x  z * y) / z" 
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332 
by (simp add: diff_divide_distrib) 
30630  333 

334 
lemma diff_frac_eq: 

335 
"y \<noteq> 0 \<Longrightarrow> z \<noteq> 0 \<Longrightarrow> x / y  w / z = (x * z  w * y) / (y * z)" 

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336 
by (simp add: field_simps) 
30630  337 

338 
lemma frac_eq_eq: 

339 
"y \<noteq> 0 \<Longrightarrow> z \<noteq> 0 \<Longrightarrow> (x / y = w / z) = (x * z = w * y)" 

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340 
by (simp add: field_simps) 
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341 

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342 
end 
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343 

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344 
class field_inverse_zero = field + 
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345 
assumes field_inverse_zero: "inverse 0 = 0" 
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346 
begin 
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347 

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348 
subclass division_ring_inverse_zero proof 
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349 
qed (fact field_inverse_zero) 
25230  350 

14270  351 
text{*This version builds in division by zero while also reorienting 
352 
the righthand side.*} 

353 
lemma inverse_mult_distrib [simp]: 

36409  354 
"inverse (a * b) = inverse a * inverse b" 
355 
proof cases 

356 
assume "a \<noteq> 0 & b \<noteq> 0" 

357 
thus ?thesis by (simp add: nonzero_inverse_mult_distrib mult_ac) 

358 
next 

359 
assume "~ (a \<noteq> 0 & b \<noteq> 0)" 

360 
thus ?thesis by force 

361 
qed 

14270  362 

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363 
lemma inverse_divide [simp]: 
36409  364 
"inverse (a / b) = b / a" 
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365 
by (simp add: divide_inverse mult_commute) 
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366 

23389  367 

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368 
text {* Calculations with fractions *} 
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avigad
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369 

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370 
text{* There is a whole bunch of simprules just for class @{text 
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371 
field} but none for class @{text field} and @{text nonzero_divides} 
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372 
because the latter are covered by a simproc. *} 
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373 

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374 
lemma mult_divide_mult_cancel_left: 
36409  375 
"c \<noteq> 0 \<Longrightarrow> (c * a) / (c * b) = a / b" 
21328  376 
apply (cases "b = 0") 
35216  377 
apply simp_all 
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378 
done 
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379 

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380 
lemma mult_divide_mult_cancel_right: 
36409  381 
"c \<noteq> 0 \<Longrightarrow> (a * c) / (b * c) = a / b" 
21328  382 
apply (cases "b = 0") 
35216  383 
apply simp_all 
14321  384 
done 
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385 

36409  386 
lemma divide_divide_eq_right [simp, no_atp]: 
387 
"a / (b / c) = (a * c) / b" 

388 
by (simp add: divide_inverse mult_ac) 

14288  389 

36409  390 
lemma divide_divide_eq_left [simp, no_atp]: 
391 
"(a / b) / c = a / (b * c)" 

392 
by (simp add: divide_inverse mult_assoc) 

14288  393 

23389  394 

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395 
text {*Special Cancellation Simprules for Division*} 
15234
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396 

36409  397 
lemma mult_divide_mult_cancel_left_if [simp,no_atp]: 
398 
shows "(c * a) / (c * b) = (if c = 0 then 0 else a / b)" 

399 
by (simp add: mult_divide_mult_cancel_left) 

23413
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changeset

400 

15234
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changeset

401 

36301
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402 
text {* Division and Unary Minus *} 
14293  403 

36409  404 
lemma minus_divide_right: 
405 
" (a / b) = a /  b" 

406 
by (simp add: divide_inverse) 

14430
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changeset

407 

35828
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blanchet
parents:
35579
diff
changeset

408 
lemma divide_minus_right [simp, no_atp]: 
36409  409 
"a /  b =  (a / b)" 
410 
by (simp add: divide_inverse) 

30630  411 

412 
lemma minus_divide_divide: 

36409  413 
"( a) / ( b) = a / b" 
21328  414 
apply (cases "b=0", simp) 
14293  415 
apply (simp add: nonzero_minus_divide_divide) 
416 
done 

417 

23482  418 
lemma eq_divide_eq: 
36409  419 
"a = b / c \<longleftrightarrow> (if c \<noteq> 0 then a * c = b else a = 0)" 
420 
by (simp add: nonzero_eq_divide_eq) 

23482  421 

422 
lemma divide_eq_eq: 

36409  423 
"b / c = a \<longleftrightarrow> (if c \<noteq> 0 then b = a * c else a = 0)" 
424 
by (force simp add: nonzero_divide_eq_eq) 

14293  425 

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426 
lemma inverse_eq_1_iff [simp]: 
36409  427 
"inverse x = 1 \<longleftrightarrow> x = 1" 
428 
by (insert inverse_eq_iff_eq [of x 1], simp) 

23389  429 

36409  430 
lemma divide_eq_0_iff [simp, no_atp]: 
431 
"a / b = 0 \<longleftrightarrow> a = 0 \<or> b = 0" 

432 
by (simp add: divide_inverse) 

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433 

36409  434 
lemma divide_cancel_right [simp, no_atp]: 
435 
"a / c = b / c \<longleftrightarrow> c = 0 \<or> a = b" 

436 
apply (cases "c=0", simp) 

437 
apply (simp add: divide_inverse) 

438 
done 

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439 

36409  440 
lemma divide_cancel_left [simp, no_atp]: 
441 
"c / a = c / b \<longleftrightarrow> c = 0 \<or> a = b" 

442 
apply (cases "c=0", simp) 

443 
apply (simp add: divide_inverse) 

444 
done 

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445 

36409  446 
lemma divide_eq_1_iff [simp, no_atp]: 
447 
"a / b = 1 \<longleftrightarrow> b \<noteq> 0 \<and> a = b" 

448 
apply (cases "b=0", simp) 

449 
apply (simp add: right_inverse_eq) 

450 
done 

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451 

36409  452 
lemma one_eq_divide_iff [simp, no_atp]: 
453 
"1 = a / b \<longleftrightarrow> b \<noteq> 0 \<and> a = b" 

454 
by (simp add: eq_commute [of 1]) 

455 

36719  456 
lemma times_divide_times_eq: 
457 
"(x / y) * (z / w) = (x * z) / (y * w)" 

458 
by simp 

459 

460 
lemma add_frac_num: 

461 
"y \<noteq> 0 \<Longrightarrow> x / y + z = (x + z * y) / y" 

462 
by (simp add: add_divide_distrib) 

463 

464 
lemma add_num_frac: 

465 
"y \<noteq> 0 \<Longrightarrow> z + x / y = (x + z * y) / y" 

466 
by (simp add: add_divide_distrib add.commute) 

467 

36409  468 
end 
36301
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469 

72f4d079ebf8
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changeset

470 

44064
5bce8ff0d9ae
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huffman
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42904
diff
changeset

471 
subsection {* Ordered fields *} 
36301
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472 

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473 
class linordered_field = field + linordered_idom 
72f4d079ebf8
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474 
begin 
14268
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paulson
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changeset

475 

14277
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changeset

476 
lemma positive_imp_inverse_positive: 
36301
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haftmann
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changeset

477 
assumes a_gt_0: "0 < a" 
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changeset

478 
shows "0 < inverse a" 
23482  479 
proof  
14268
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paulson
parents:
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diff
changeset

480 
have "0 < a * inverse a" 
36301
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changeset

481 
by (simp add: a_gt_0 [THEN less_imp_not_eq2]) 
14268
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paulson
parents:
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diff
changeset

482 
thus "0 < inverse a" 
36301
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changeset

483 
by (simp add: a_gt_0 [THEN less_not_sym] zero_less_mult_iff) 
23482  484 
qed 
14268
5cf13e80be0e
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paulson
parents:
14267
diff
changeset

485 

14277
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changeset

486 
lemma negative_imp_inverse_negative: 
36301
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changeset

487 
"a < 0 \<Longrightarrow> inverse a < 0" 
72f4d079ebf8
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haftmann
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diff
changeset

488 
by (insert positive_imp_inverse_positive [of "a"], 
72f4d079ebf8
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haftmann
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diff
changeset

489 
simp add: nonzero_inverse_minus_eq less_imp_not_eq) 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

490 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
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diff
changeset

491 
lemma inverse_le_imp_le: 
36301
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changeset

492 
assumes invle: "inverse a \<le> inverse b" and apos: "0 < a" 
72f4d079ebf8
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haftmann
parents:
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diff
changeset

493 
shows "b \<le> a" 
23482  494 
proof (rule classical) 
14268
5cf13e80be0e
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paulson
parents:
14267
diff
changeset

495 
assume "~ b \<le> a" 
23482  496 
hence "a < b" by (simp add: linorder_not_le) 
36301
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haftmann
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changeset

497 
hence bpos: "0 < b" by (blast intro: apos less_trans) 
14268
5cf13e80be0e
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paulson
parents:
14267
diff
changeset

498 
hence "a * inverse a \<le> a * inverse b" 
36301
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haftmann
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changeset

499 
by (simp add: apos invle less_imp_le mult_left_mono) 
14268
5cf13e80be0e
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paulson
parents:
14267
diff
changeset

500 
hence "(a * inverse a) * b \<le> (a * inverse b) * b" 
36301
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changeset

501 
by (simp add: bpos less_imp_le mult_right_mono) 
72f4d079ebf8
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haftmann
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changeset

502 
thus "b \<le> a" by (simp add: mult_assoc apos bpos less_imp_not_eq2) 
23482  503 
qed 
14268
5cf13e80be0e
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paulson
parents:
14267
diff
changeset

504 

14277
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changeset

505 
lemma inverse_positive_imp_positive: 
36301
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changeset

506 
assumes inv_gt_0: "0 < inverse a" and nz: "a \<noteq> 0" 
72f4d079ebf8
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haftmann
parents:
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changeset

507 
shows "0 < a" 
23389  508 
proof  
14277
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more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

509 
have "0 < inverse (inverse a)" 
23389  510 
using inv_gt_0 by (rule positive_imp_inverse_positive) 
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

511 
thus "0 < a" 
23389  512 
using nz by (simp add: nonzero_inverse_inverse_eq) 
513 
qed 

14277
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parents:
14272
diff
changeset

514 

36301
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changeset

515 
lemma inverse_negative_imp_negative: 
72f4d079ebf8
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haftmann
parents:
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diff
changeset

516 
assumes inv_less_0: "inverse a < 0" and nz: "a \<noteq> 0" 
72f4d079ebf8
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haftmann
parents:
35828
diff
changeset

517 
shows "a < 0" 
72f4d079ebf8
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haftmann
parents:
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diff
changeset

518 
proof  
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

519 
have "inverse (inverse a) < 0" 
72f4d079ebf8
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haftmann
parents:
35828
diff
changeset

520 
using inv_less_0 by (rule negative_imp_inverse_negative) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

521 
thus "a < 0" using nz by (simp add: nonzero_inverse_inverse_eq) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
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diff
changeset

522 
qed 
72f4d079ebf8
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haftmann
parents:
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diff
changeset

523 

72f4d079ebf8
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haftmann
parents:
35828
diff
changeset

524 
lemma linordered_field_no_lb: 
72f4d079ebf8
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haftmann
parents:
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diff
changeset

525 
"\<forall>x. \<exists>y. y < x" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

526 
proof 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

527 
fix x::'a 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

528 
have m1: " (1::'a) < 0" by simp 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

529 
from add_strict_right_mono[OF m1, where c=x] 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

530 
have "( 1) + x < x" by simp 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

531 
thus "\<exists>y. y < x" by blast 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

532 
qed 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

533 

72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

534 
lemma linordered_field_no_ub: 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

535 
"\<forall> x. \<exists>y. y > x" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

536 
proof 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

537 
fix x::'a 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

538 
have m1: " (1::'a) > 0" by simp 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

539 
from add_strict_right_mono[OF m1, where c=x] 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

540 
have "1 + x > x" by simp 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

541 
thus "\<exists>y. y > x" by blast 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

542 
qed 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

543 

72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

544 
lemma less_imp_inverse_less: 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

545 
assumes less: "a < b" and apos: "0 < a" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

546 
shows "inverse b < inverse a" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

547 
proof (rule ccontr) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

548 
assume "~ inverse b < inverse a" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

549 
hence "inverse a \<le> inverse b" by simp 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

550 
hence "~ (a < b)" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

551 
by (simp add: not_less inverse_le_imp_le [OF _ apos]) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

552 
thus False by (rule notE [OF _ less]) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

553 
qed 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

554 

72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

555 
lemma inverse_less_imp_less: 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

556 
"inverse a < inverse b \<Longrightarrow> 0 < a \<Longrightarrow> b < a" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

557 
apply (simp add: less_le [of "inverse a"] less_le [of "b"]) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

558 
apply (force dest!: inverse_le_imp_le nonzero_inverse_eq_imp_eq) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

559 
done 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

560 

72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

561 
text{*Both premises are essential. Consider 1 and 1.*} 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

562 
lemma inverse_less_iff_less [simp,no_atp]: 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

563 
"0 < a \<Longrightarrow> 0 < b \<Longrightarrow> inverse a < inverse b \<longleftrightarrow> b < a" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

564 
by (blast intro: less_imp_inverse_less dest: inverse_less_imp_less) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

565 

72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

566 
lemma le_imp_inverse_le: 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

567 
"a \<le> b \<Longrightarrow> 0 < a \<Longrightarrow> inverse b \<le> inverse a" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

568 
by (force simp add: le_less less_imp_inverse_less) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

569 

72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

570 
lemma inverse_le_iff_le [simp,no_atp]: 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

571 
"0 < a \<Longrightarrow> 0 < b \<Longrightarrow> inverse a \<le> inverse b \<longleftrightarrow> b \<le> a" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

572 
by (blast intro: le_imp_inverse_le dest: inverse_le_imp_le) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

573 

72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

574 

72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

575 
text{*These results refer to both operands being negative. The oppositesign 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

576 
case is trivial, since inverse preserves signs.*} 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

577 
lemma inverse_le_imp_le_neg: 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

578 
"inverse a \<le> inverse b \<Longrightarrow> b < 0 \<Longrightarrow> b \<le> a" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

579 
apply (rule classical) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

580 
apply (subgoal_tac "a < 0") 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

581 
prefer 2 apply force 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

582 
apply (insert inverse_le_imp_le [of "b" "a"]) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

583 
apply (simp add: nonzero_inverse_minus_eq) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

584 
done 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

585 

72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

586 
lemma less_imp_inverse_less_neg: 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

587 
"a < b \<Longrightarrow> b < 0 \<Longrightarrow> inverse b < inverse a" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

588 
apply (subgoal_tac "a < 0") 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

589 
prefer 2 apply (blast intro: less_trans) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

590 
apply (insert less_imp_inverse_less [of "b" "a"]) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

591 
apply (simp add: nonzero_inverse_minus_eq) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

592 
done 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

593 

72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

594 
lemma inverse_less_imp_less_neg: 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

595 
"inverse a < inverse b \<Longrightarrow> b < 0 \<Longrightarrow> b < a" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

596 
apply (rule classical) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

597 
apply (subgoal_tac "a < 0") 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

598 
prefer 2 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

599 
apply force 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

600 
apply (insert inverse_less_imp_less [of "b" "a"]) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

601 
apply (simp add: nonzero_inverse_minus_eq) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

602 
done 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

603 

72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

604 
lemma inverse_less_iff_less_neg [simp,no_atp]: 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

605 
"a < 0 \<Longrightarrow> b < 0 \<Longrightarrow> inverse a < inverse b \<longleftrightarrow> b < a" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

606 
apply (insert inverse_less_iff_less [of "b" "a"]) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

607 
apply (simp del: inverse_less_iff_less 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

608 
add: nonzero_inverse_minus_eq) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

609 
done 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

610 

72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

611 
lemma le_imp_inverse_le_neg: 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

612 
"a \<le> b \<Longrightarrow> b < 0 ==> inverse b \<le> inverse a" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

613 
by (force simp add: le_less less_imp_inverse_less_neg) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

614 

72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

615 
lemma inverse_le_iff_le_neg [simp,no_atp]: 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

616 
"a < 0 \<Longrightarrow> b < 0 \<Longrightarrow> inverse a \<le> inverse b \<longleftrightarrow> b \<le> a" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

617 
by (blast intro: le_imp_inverse_le_neg dest: inverse_le_imp_le_neg) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

618 

36774  619 
lemma one_less_inverse: 
620 
"0 < a \<Longrightarrow> a < 1 \<Longrightarrow> 1 < inverse a" 

621 
using less_imp_inverse_less [of a 1, unfolded inverse_1] . 

622 

623 
lemma one_le_inverse: 

624 
"0 < a \<Longrightarrow> a \<le> 1 \<Longrightarrow> 1 \<le> inverse a" 

625 
using le_imp_inverse_le [of a 1, unfolded inverse_1] . 

626 

36348
89c54f51f55a
dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents:
36343
diff
changeset

627 
lemma pos_le_divide_eq [field_simps]: "0 < c ==> (a \<le> b/c) = (a*c \<le> b)" 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

628 
proof  
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

629 
assume less: "0<c" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

630 
hence "(a \<le> b/c) = (a*c \<le> (b/c)*c)" 
36304
6984744e6b34
less special treatment of times_divide_eq [simp]
haftmann
parents:
36301
diff
changeset

631 
by (simp add: mult_le_cancel_right less_not_sym [OF less] del: times_divide_eq) 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

632 
also have "... = (a*c \<le> b)" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

633 
by (simp add: less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

634 
finally show ?thesis . 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

635 
qed 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

636 

36348
89c54f51f55a
dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents:
36343
diff
changeset

637 
lemma neg_le_divide_eq [field_simps]: "c < 0 ==> (a \<le> b/c) = (b \<le> a*c)" 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

638 
proof  
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

639 
assume less: "c<0" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

640 
hence "(a \<le> b/c) = ((b/c)*c \<le> a*c)" 
36304
6984744e6b34
less special treatment of times_divide_eq [simp]
haftmann
parents:
36301
diff
changeset

641 
by (simp add: mult_le_cancel_right less_not_sym [OF less] del: times_divide_eq) 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

642 
also have "... = (b \<le> a*c)" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

643 
by (simp add: less_imp_not_eq [OF less] divide_inverse mult_assoc) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

644 
finally show ?thesis . 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

645 
qed 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

646 

36348
89c54f51f55a
dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents:
36343
diff
changeset

647 
lemma pos_less_divide_eq [field_simps]: 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

648 
"0 < c ==> (a < b/c) = (a*c < b)" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

649 
proof  
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

650 
assume less: "0<c" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

651 
hence "(a < b/c) = (a*c < (b/c)*c)" 
36304
6984744e6b34
less special treatment of times_divide_eq [simp]
haftmann
parents:
36301
diff
changeset

652 
by (simp add: mult_less_cancel_right_disj less_not_sym [OF less] del: times_divide_eq) 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

653 
also have "... = (a*c < b)" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

654 
by (simp add: less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

655 
finally show ?thesis . 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

656 
qed 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

657 

36348
89c54f51f55a
dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents:
36343
diff
changeset

658 
lemma neg_less_divide_eq [field_simps]: 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

659 
"c < 0 ==> (a < b/c) = (b < a*c)" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

660 
proof  
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

661 
assume less: "c<0" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

662 
hence "(a < b/c) = ((b/c)*c < a*c)" 
36304
6984744e6b34
less special treatment of times_divide_eq [simp]
haftmann
parents:
36301
diff
changeset

663 
by (simp add: mult_less_cancel_right_disj less_not_sym [OF less] del: times_divide_eq) 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

664 
also have "... = (b < a*c)" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

665 
by (simp add: less_imp_not_eq [OF less] divide_inverse mult_assoc) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

666 
finally show ?thesis . 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

667 
qed 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

668 

36348
89c54f51f55a
dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents:
36343
diff
changeset

669 
lemma pos_divide_less_eq [field_simps]: 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

670 
"0 < c ==> (b/c < a) = (b < a*c)" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

671 
proof  
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

672 
assume less: "0<c" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

673 
hence "(b/c < a) = ((b/c)*c < a*c)" 
36304
6984744e6b34
less special treatment of times_divide_eq [simp]
haftmann
parents:
36301
diff
changeset

674 
by (simp add: mult_less_cancel_right_disj less_not_sym [OF less] del: times_divide_eq) 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

675 
also have "... = (b < a*c)" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

676 
by (simp add: less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

677 
finally show ?thesis . 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

678 
qed 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

679 

36348
89c54f51f55a
dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents:
36343
diff
changeset

680 
lemma neg_divide_less_eq [field_simps]: 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

681 
"c < 0 ==> (b/c < a) = (a*c < b)" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

682 
proof  
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

683 
assume less: "c<0" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

684 
hence "(b/c < a) = (a*c < (b/c)*c)" 
36304
6984744e6b34
less special treatment of times_divide_eq [simp]
haftmann
parents:
36301
diff
changeset

685 
by (simp add: mult_less_cancel_right_disj less_not_sym [OF less] del: times_divide_eq) 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

686 
also have "... = (a*c < b)" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

687 
by (simp add: less_imp_not_eq [OF less] divide_inverse mult_assoc) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

688 
finally show ?thesis . 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

689 
qed 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

690 

36348
89c54f51f55a
dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents:
36343
diff
changeset

691 
lemma pos_divide_le_eq [field_simps]: "0 < c ==> (b/c \<le> a) = (b \<le> a*c)" 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

692 
proof  
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

693 
assume less: "0<c" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

694 
hence "(b/c \<le> a) = ((b/c)*c \<le> a*c)" 
36304
6984744e6b34
less special treatment of times_divide_eq [simp]
haftmann
parents:
36301
diff
changeset

695 
by (simp add: mult_le_cancel_right less_not_sym [OF less] del: times_divide_eq) 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

696 
also have "... = (b \<le> a*c)" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

697 
by (simp add: less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

698 
finally show ?thesis . 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

699 
qed 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

700 

36348
89c54f51f55a
dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents:
36343
diff
changeset

701 
lemma neg_divide_le_eq [field_simps]: "c < 0 ==> (b/c \<le> a) = (a*c \<le> b)" 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

702 
proof  
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

703 
assume less: "c<0" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

704 
hence "(b/c \<le> a) = (a*c \<le> (b/c)*c)" 
36304
6984744e6b34
less special treatment of times_divide_eq [simp]
haftmann
parents:
36301
diff
changeset

705 
by (simp add: mult_le_cancel_right less_not_sym [OF less] del: times_divide_eq) 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

706 
also have "... = (a*c \<le> b)" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

707 
by (simp add: less_imp_not_eq [OF less] divide_inverse mult_assoc) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

708 
finally show ?thesis . 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

709 
qed 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

710 

72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

711 
text{* Lemmas @{text sign_simps} is a first attempt to automate proofs 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

712 
of positivity/negativity needed for @{text field_simps}. Have not added @{text 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

713 
sign_simps} to @{text field_simps} because the former can lead to case 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

714 
explosions. *} 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

715 

36348
89c54f51f55a
dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents:
36343
diff
changeset

716 
lemmas sign_simps [no_atp] = algebra_simps 
89c54f51f55a
dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents:
36343
diff
changeset

717 
zero_less_mult_iff mult_less_0_iff 
89c54f51f55a
dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents:
36343
diff
changeset

718 

89c54f51f55a
dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents:
36343
diff
changeset

719 
lemmas (in ) sign_simps [no_atp] = algebra_simps 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

720 
zero_less_mult_iff mult_less_0_iff 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

721 

72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

722 
(* Only works once linear arithmetic is installed: 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

723 
text{*An example:*} 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

724 
lemma fixes a b c d e f :: "'a::linordered_field" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

725 
shows "\<lbrakk>a>b; c<d; e<f; 0 < u \<rbrakk> \<Longrightarrow> 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

726 
((ab)*(cd)*(ef))/((cd)*(ef)*(ab)) < 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

727 
((ef)*(ab)*(cd))/((ef)*(ab)*(cd)) + u" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

728 
apply(subgoal_tac "(cd)*(ef)*(ab) > 0") 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

729 
prefer 2 apply(simp add:sign_simps) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

730 
apply(subgoal_tac "(cd)*(ef)*(ab)*u > 0") 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

731 
prefer 2 apply(simp add:sign_simps) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

732 
apply(simp add:field_simps) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

733 
done 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

734 
*) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

735 

72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

736 
lemma divide_pos_pos: 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

737 
"0 < x ==> 0 < y ==> 0 < x / y" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

738 
by(simp add:field_simps) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

739 

72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

740 
lemma divide_nonneg_pos: 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

741 
"0 <= x ==> 0 < y ==> 0 <= x / y" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

742 
by(simp add:field_simps) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

743 

72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

744 
lemma divide_neg_pos: 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

745 
"x < 0 ==> 0 < y ==> x / y < 0" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

746 
by(simp add:field_simps) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

747 

72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

748 
lemma divide_nonpos_pos: 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

749 
"x <= 0 ==> 0 < y ==> x / y <= 0" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

750 
by(simp add:field_simps) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

751 

72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

752 
lemma divide_pos_neg: 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

753 
"0 < x ==> y < 0 ==> x / y < 0" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

754 
by(simp add:field_simps) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

755 

72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

756 
lemma divide_nonneg_neg: 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

757 
"0 <= x ==> y < 0 ==> x / y <= 0" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

758 
by(simp add:field_simps) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

759 

72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

760 
lemma divide_neg_neg: 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

761 
"x < 0 ==> y < 0 ==> 0 < x / y" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

762 
by(simp add:field_simps) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

763 

72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

764 
lemma divide_nonpos_neg: 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

765 
"x <= 0 ==> y < 0 ==> 0 <= x / y" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

766 
by(simp add:field_simps) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

767 

72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

768 
lemma divide_strict_right_mono: 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

769 
"[a < b; 0 < c] ==> a / c < b / c" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

770 
by (simp add: less_imp_not_eq2 divide_inverse mult_strict_right_mono 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

771 
positive_imp_inverse_positive) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

772 

72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

773 

72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

774 
lemma divide_strict_right_mono_neg: 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

775 
"[b < a; c < 0] ==> a / c < b / c" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

776 
apply (drule divide_strict_right_mono [of _ _ "c"], simp) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

777 
apply (simp add: less_imp_not_eq nonzero_minus_divide_right [symmetric]) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

778 
done 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

779 

72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

780 
text{*The last premise ensures that @{term a} and @{term b} 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

781 
have the same sign*} 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

782 
lemma divide_strict_left_mono: 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

783 
"[b < a; 0 < c; 0 < a*b] ==> c / a < c / b" 
44921  784 
by (auto simp: field_simps zero_less_mult_iff mult_strict_right_mono) 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

785 

72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

786 
lemma divide_left_mono: 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

787 
"[b \<le> a; 0 \<le> c; 0 < a*b] ==> c / a \<le> c / b" 
44921  788 
by (auto simp: field_simps zero_less_mult_iff mult_right_mono) 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

789 

72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

790 
lemma divide_strict_left_mono_neg: 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

791 
"[a < b; c < 0; 0 < a*b] ==> c / a < c / b" 
44921  792 
by (auto simp: field_simps zero_less_mult_iff mult_strict_right_mono_neg) 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

793 

72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

794 
lemma mult_imp_div_pos_le: "0 < y ==> x <= z * y ==> 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

795 
x / y <= z" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

796 
by (subst pos_divide_le_eq, assumption+) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

797 

72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

798 
lemma mult_imp_le_div_pos: "0 < y ==> z * y <= x ==> 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

799 
z <= x / y" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

800 
by(simp add:field_simps) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

801 

72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

802 
lemma mult_imp_div_pos_less: "0 < y ==> x < z * y ==> 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

803 
x / y < z" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

804 
by(simp add:field_simps) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

805 

72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

806 
lemma mult_imp_less_div_pos: "0 < y ==> z * y < x ==> 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

807 
z < x / y" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

808 
by(simp add:field_simps) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

809 

72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

810 
lemma frac_le: "0 <= x ==> 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

811 
x <= y ==> 0 < w ==> w <= z ==> x / z <= y / w" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

812 
apply (rule mult_imp_div_pos_le) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

813 
apply simp 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

814 
apply (subst times_divide_eq_left) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

815 
apply (rule mult_imp_le_div_pos, assumption) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

816 
apply (rule mult_mono) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

817 
apply simp_all 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

818 
done 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

819 

72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

820 
lemma frac_less: "0 <= x ==> 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

821 
x < y ==> 0 < w ==> w <= z ==> x / z < y / w" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

822 
apply (rule mult_imp_div_pos_less) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

823 
apply simp 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

824 
apply (subst times_divide_eq_left) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

825 
apply (rule mult_imp_less_div_pos, assumption) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

826 
apply (erule mult_less_le_imp_less) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

827 
apply simp_all 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

828 
done 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

829 

72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

830 
lemma frac_less2: "0 < x ==> 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

831 
x <= y ==> 0 < w ==> w < z ==> x / z < y / w" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

832 
apply (rule mult_imp_div_pos_less) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

833 
apply simp_all 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

834 
apply (rule mult_imp_less_div_pos, assumption) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

835 
apply (erule mult_le_less_imp_less) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

836 
apply simp_all 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

837 
done 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

838 

72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

839 
lemma less_half_sum: "a < b ==> a < (a+b) / (1+1)" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

840 
by (simp add: field_simps zero_less_two) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

841 

72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

842 
lemma gt_half_sum: "a < b ==> (a+b)/(1+1) < b" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

843 
by (simp add: field_simps zero_less_two) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

844 

72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

845 
subclass dense_linorder 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

846 
proof 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

847 
fix x y :: 'a 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

848 
from less_add_one show "\<exists>y. x < y" .. 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

849 
from less_add_one have "x + ( 1) < (x + 1) + ( 1)" by (rule add_strict_right_mono) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

850 
then have "x  1 < x + 1  1" by (simp only: diff_minus [symmetric]) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

851 
then have "x  1 < x" by (simp add: algebra_simps) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

852 
then show "\<exists>y. y < x" .. 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

853 
show "x < y \<Longrightarrow> \<exists>z>x. z < y" by (blast intro!: less_half_sum gt_half_sum) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

854 
qed 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

855 

72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

856 
lemma nonzero_abs_inverse: 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

857 
"a \<noteq> 0 ==> \<bar>inverse a\<bar> = inverse \<bar>a\<bar>" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

858 
apply (auto simp add: neq_iff abs_if nonzero_inverse_minus_eq 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

859 
negative_imp_inverse_negative) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

860 
apply (blast intro: positive_imp_inverse_positive elim: less_asym) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

861 
done 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

862 

72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

863 
lemma nonzero_abs_divide: 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

864 
"b \<noteq> 0 ==> \<bar>a / b\<bar> = \<bar>a\<bar> / \<bar>b\<bar>" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

865 
by (simp add: divide_inverse abs_mult nonzero_abs_inverse) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

866 

72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

867 
lemma field_le_epsilon: 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

868 
assumes e: "\<And>e. 0 < e \<Longrightarrow> x \<le> y + e" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

869 
shows "x \<le> y" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

870 
proof (rule dense_le) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

871 
fix t assume "t < x" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

872 
hence "0 < x  t" by (simp add: less_diff_eq) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

873 
from e [OF this] have "x + 0 \<le> x + (y  t)" by (simp add: algebra_simps) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

874 
then have "0 \<le> y  t" by (simp only: add_le_cancel_left) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

875 
then show "t \<le> y" by (simp add: algebra_simps) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

876 
qed 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

877 

72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

878 
end 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

879 

36414  880 
class linordered_field_inverse_zero = linordered_field + field_inverse_zero 
36348
89c54f51f55a
dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents:
36343
diff
changeset

881 
begin 
89c54f51f55a
dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents:
36343
diff
changeset

882 

36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

883 
lemma le_divide_eq: 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

884 
"(a \<le> b/c) = 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

885 
(if 0 < c then a*c \<le> b 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

886 
else if c < 0 then b \<le> a*c 
36409  887 
else a \<le> 0)" 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

888 
apply (cases "c=0", simp) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

889 
apply (force simp add: pos_le_divide_eq neg_le_divide_eq linorder_neq_iff) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

890 
done 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

891 

14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

892 
lemma inverse_positive_iff_positive [simp]: 
36409  893 
"(0 < inverse a) = (0 < a)" 
21328  894 
apply (cases "a = 0", simp) 
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

895 
apply (blast intro: inverse_positive_imp_positive positive_imp_inverse_positive) 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

896 
done 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

897 

ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

898 
lemma inverse_negative_iff_negative [simp]: 
36409  899 
"(inverse a < 0) = (a < 0)" 
21328  900 
apply (cases "a = 0", simp) 
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

901 
apply (blast intro: inverse_negative_imp_negative negative_imp_inverse_negative) 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

902 
done 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

903 

ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

904 
lemma inverse_nonnegative_iff_nonnegative [simp]: 
36409  905 
"0 \<le> inverse a \<longleftrightarrow> 0 \<le> a" 
906 
by (simp add: not_less [symmetric]) 

14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

907 

ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

908 
lemma inverse_nonpositive_iff_nonpositive [simp]: 
36409  909 
"inverse a \<le> 0 \<longleftrightarrow> a \<le> 0" 
910 
by (simp add: not_less [symmetric]) 

14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

911 

14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

912 
lemma one_less_inverse_iff: 
36409  913 
"1 < inverse x \<longleftrightarrow> 0 < x \<and> x < 1" 
23482  914 
proof cases 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

915 
assume "0 < x" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

916 
with inverse_less_iff_less [OF zero_less_one, of x] 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

917 
show ?thesis by simp 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

918 
next 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

919 
assume notless: "~ (0 < x)" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

920 
have "~ (1 < inverse x)" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

921 
proof 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

922 
assume "1 < inverse x" 
36409  923 
also with notless have "... \<le> 0" by simp 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

924 
also have "... < 1" by (rule zero_less_one) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

925 
finally show False by auto 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

926 
qed 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

927 
with notless show ?thesis by simp 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

928 
qed 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

929 

3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

930 
lemma one_le_inverse_iff: 
36409  931 
"1 \<le> inverse x \<longleftrightarrow> 0 < x \<and> x \<le> 1" 
932 
proof (cases "x = 1") 

933 
case True then show ?thesis by simp 

934 
next 

935 
case False then have "inverse x \<noteq> 1" by simp 

936 
then have "1 \<noteq> inverse x" by blast 

937 
then have "1 \<le> inverse x \<longleftrightarrow> 1 < inverse x" by (simp add: le_less) 

938 
with False show ?thesis by (auto simp add: one_less_inverse_iff) 

939 
qed 

14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

940 

3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

941 
lemma inverse_less_1_iff: 
36409  942 
"inverse x < 1 \<longleftrightarrow> x \<le> 0 \<or> 1 < x" 
943 
by (simp add: not_le [symmetric] one_le_inverse_iff) 

14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

944 

3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

945 
lemma inverse_le_1_iff: 
36409  946 
"inverse x \<le> 1 \<longleftrightarrow> x \<le> 0 \<or> 1 \<le> x" 
947 
by (simp add: not_less [symmetric] one_less_inverse_iff) 

14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

948 

14288  949 
lemma divide_le_eq: 
950 
"(b/c \<le> a) = 

951 
(if 0 < c then b \<le> a*c 

952 
else if c < 0 then a*c \<le> b 

36409  953 
else 0 \<le> a)" 
21328  954 
apply (cases "c=0", simp) 
36409  955 
apply (force simp add: pos_divide_le_eq neg_divide_le_eq) 
14288  956 
done 
957 

958 
lemma less_divide_eq: 

959 
"(a < b/c) = 

960 
(if 0 < c then a*c < b 

961 
else if c < 0 then b < a*c 

36409  962 
else a < 0)" 
21328  963 
apply (cases "c=0", simp) 
36409  964 
apply (force simp add: pos_less_divide_eq neg_less_divide_eq) 
14288  965 
done 
966 

967 
lemma divide_less_eq: 

968 
"(b/c < a) = 

969 
(if 0 < c then b < a*c 

970 
else if c < 0 then a*c < b 

36409  971 
else 0 < a)" 
21328  972 
apply (cases "c=0", simp) 
36409  973 
apply (force simp add: pos_divide_less_eq neg_divide_less_eq) 
14288  974 
done 
975 

36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

976 
text {*Division and Signs*} 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

977 

c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

978 
lemma zero_less_divide_iff: 
36409  979 
"(0 < a/b) = (0 < a & 0 < b  a < 0 & b < 0)" 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

980 
by (simp add: divide_inverse zero_less_mult_iff) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

981 

c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

982 
lemma divide_less_0_iff: 
36409  983 
"(a/b < 0) = 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

984 
(0 < a & b < 0  a < 0 & 0 < b)" 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

985 
by (simp add: divide_inverse mult_less_0_iff) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

986 

c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

987 
lemma zero_le_divide_iff: 
36409  988 
"(0 \<le> a/b) = 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

989 
(0 \<le> a & 0 \<le> b  a \<le> 0 & b \<le> 0)" 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

990 
by (simp add: divide_inverse zero_le_mult_iff) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

991 

c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

992 
lemma divide_le_0_iff: 
36409  993 
"(a/b \<le> 0) = 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

994 
(0 \<le> a & b \<le> 0  a \<le> 0 & 0 \<le> b)" 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

995 
by (simp add: divide_inverse mult_le_0_iff) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

996 

36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

997 
text {* Division and the Number One *} 
14353
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset

998 

79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset

999 
text{*Simplify expressions equated with 1*} 
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset

1000 

35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35579
diff
changeset

1001 
lemma zero_eq_1_divide_iff [simp,no_atp]: 
36409  1002 
"(0 = 1/a) = (a = 0)" 
23482  1003 
apply (cases "a=0", simp) 
1004 
apply (auto simp add: nonzero_eq_divide_eq) 

14353
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset

1005 
done 
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset

1006 

35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35579
diff
changeset

1007 
lemma one_divide_eq_0_iff [simp,no_atp]: 
36409  1008 
"(1/a = 0) = (a = 0)" 
23482  1009 
apply (cases "a=0", simp) 
1010 
apply (insert zero_neq_one [THEN not_sym]) 

1011 
apply (auto simp add: nonzero_divide_eq_eq) 

14353
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset

1012 
done 
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset

1013 

79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset

1014 
text{*Simplify expressions such as @{text "0 < 1/x"} to @{text "0 < x"}*} 
36423  1015 

1016 
lemma zero_le_divide_1_iff [simp, no_atp]: 

1017 
"0 \<le> 1 / a \<longleftrightarrow> 0 \<le> a" 

1018 
by (simp add: zero_le_divide_iff) 

17085  1019 

36423  1020 
lemma zero_less_divide_1_iff [simp, no_atp]: 
1021 
"0 < 1 / a \<longleftrightarrow> 0 < a" 

1022 
by (simp add: zero_less_divide_iff) 

1023 

1024 
lemma divide_le_0_1_iff [simp, no_atp]: 

1025 
"1 / a \<le> 0 \<longleftrightarrow> a \<le> 0" 

1026 
by (simp add: divide_le_0_iff) 

1027 

1028 
lemma divide_less_0_1_iff [simp, no_atp]: 

1029 
"1 / a < 0 \<longleftrightarrow> a < 0" 

1030 
by (simp add: divide_less_0_iff) 

14353
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset

1031 

14293  1032 
lemma divide_right_mono: 
36409  1033 
"[a \<le> b; 0 \<le> c] ==> a/c \<le> b/c" 
1034 
by (force simp add: divide_strict_right_mono le_less) 

14293  1035 

36409  1036 
lemma divide_right_mono_neg: "a <= b 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

1037 
==> c <= 0 ==> b / c <= a / c" 
23482  1038 
apply (drule divide_right_mono [of _ _ " c"]) 
1039 
apply auto 

16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

1040 
done 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

1041 

36409  1042 
lemma divide_left_mono_neg: "a <= b 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

1043 
==> c <= 0 ==> 0 < a * b ==> c / a <= c / b" 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

1044 
apply (drule divide_left_mono [of _ _ " c"]) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

1045 
apply (auto simp add: mult_commute) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

1046 
done 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

1047 

42904  1048 
lemma inverse_le_iff: 
1049 
"inverse a \<le> inverse b \<longleftrightarrow> (0 < a * b \<longrightarrow> b \<le> a) \<and> (a * b \<le> 0 \<longrightarrow> a \<le> b)" 

1050 
proof  

1051 
{ assume "a < 0" 

1052 
then have "inverse a < 0" by simp 

1053 
moreover assume "0 < b" 

1054 
then have "0 < inverse b" by simp 

1055 
ultimately have "inverse a < inverse b" by (rule less_trans) 

1056 
then have "inverse a \<le> inverse b" by simp } 

1057 
moreover 

1058 
{ assume "b < 0" 

1059 
then have "inverse b < 0" by simp 

1060 
moreover assume "0 < a" 

1061 
then have "0 < inverse a" by simp 

1062 
ultimately have "inverse b < inverse a" by (rule less_trans) 

1063 
then have "\<not> inverse a \<le> inverse b" by simp } 

1064 
ultimately show ?thesis 

1065 
by (cases a 0 b 0 rule: linorder_cases[case_product linorder_cases]) 

1066 
(auto simp: not_less zero_less_mult_iff mult_le_0_iff) 

1067 
qed 

1068 

1069 
lemma inverse_less_iff: 

1070 
"inverse a < inverse b \<longleftrightarrow> (0 < a * b \<longrightarrow> b < a) \<and> (a * b \<le> 0 \<longrightarrow> a < b)" 

1071 
by (subst less_le) (auto simp: inverse_le_iff) 

1072 

1073 
lemma divide_le_cancel: 

1074 
"a / c \<le> b / c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)" 

1075 
by (simp add: divide_inverse mult_le_cancel_right) 

1076 

1077 
lemma divide_less_cancel: 

1078 
"a / c < b / c \<longleftrightarrow> (0 < c \<longrightarrow> a < b) \<and> (c < 0 \<longrightarrow> b < a) \<and> c \<noteq> 0" 

1079 
by (auto simp add: divide_inverse mult_less_cancel_right) 

1080 

16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

1081 
text{*Simplify quotients that are compared with the value 1.*} 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

1082 

35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35579
diff
changeset

1083 
lemma le_divide_eq_1 [no_atp]: 
36409  1084 
"(1 \<le> b / a) = ((0 < a & a \<le> b)  (a < 0 & b \<le> a))" 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

1085 
by (auto simp add: le_divide_eq) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

1086 

35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35579
diff
changeset

1087 
lemma divide_le_eq_1 [no_atp]: 
36409  1088 
"(b / a \<le> 1) = ((0 < a & b \<le> a)  (a < 0 & a \<le> b)  a=0)" 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

1089 
by (auto simp add: divide_le_eq) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

1090 

35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35579
diff
changeset

1091 
lemma less_divide_eq_1 [no_atp]: 
36409  1092 
"(1 < b / a) = ((0 < a & a < b)  (a < 0 & b < a))" 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

1093 
by (auto simp add: less_divide_eq) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

1094 

35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35579
diff
changeset

1095 
lemma divide_less_eq_1 [no_atp]: 
36409  1096 
"(b / a < 1) = ((0 < a & b < a)  (a < 0 & a < b)  a=0)" 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

1097 
by (auto simp add: divide_less_eq) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

1098 

23389  1099 

36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

1100 
text {*Conditional Simplification Rules: No Case Splits*} 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

1101 

35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35579
diff
changeset

1102 
lemma le_divide_eq_1_pos [simp,no_atp]: 
36409  1103 
"0 < a \<Longrightarrow> (1 \<le> b/a) = (a \<le> b)" 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

1104 
by (auto simp add: le_divide_eq) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

1105 

35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35579
diff
changeset

1106 
lemma le_divide_eq_1_neg [simp,no_atp]: 
36409  1107 
"a < 0 \<Longrightarrow> (1 \<le> b/a) = (b \<le> a)" 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

1108 
by (auto simp add: le_divide_eq) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

1109 

35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35579
diff
changeset

1110 
lemma divide_le_eq_1_pos [simp,no_atp]: 
36409  1111 
"0 < a \<Longrightarrow> (b/a \<le> 1) = (b \<le> a)" 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

1112 
by (auto simp add: divide_le_eq) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

1113 

35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35579
diff
changeset

1114 
lemma divide_le_eq_1_neg [simp,no_atp]: 
36409  1115 
"a < 0 \<Longrightarrow> (b/a \<le> 1) = (a \<le> b)" 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

1116 
by (auto simp add: divide_le_eq) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

1117 

35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35579
diff
changeset

1118 
lemma less_divide_eq_1_pos [simp,no_atp]: 
36409  1119 
"0 < a \<Longrightarrow> (1 < b/a) = (a < b)" 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

1120 
by (auto simp add: less_divide_eq) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

1121 

35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35579
diff
changeset

1122 
lemma less_divide_eq_1_neg [simp,no_atp]: 
36409  1123 
"a < 0 \<Longrightarrow> (1 < b/a) = (b < a)" 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

1124 
by (auto simp add: less_divide_eq) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

1125 

35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35579
diff
changeset

1126 
lemma divide_less_eq_1_pos [simp,no_atp]: 
36409  1127 
"0 < a \<Longrightarrow> (b/a < 1) = (b < a)" 
18649
bb99c2e705ca
tidied, and added missing thm divide_less_eq_1_neg
paulson
parents:
18623
diff
changeset

1128 
by (auto simp add: divide_less_eq) 
bb99c2e705ca
tidied, and added missing thm divide_less_eq_1_neg
paulson
parents:
18623
diff
changeset

1129 

35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35579
diff
changeset

1130 
lemma divide_less_eq_1_neg [simp,no_atp]: 
36409  1131 
"a < 0 \<Longrightarrow> b/a < 1 <> a < b" 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

1132 
by (auto simp add: divide_less_eq) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

1133 

35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35579
diff
changeset

1134 
lemma eq_divide_eq_1 [simp,no_atp]: 
36409  1135 
"(1 = b/a) = ((a \<noteq> 0 & a = b))" 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

1136 
by (auto simp add: eq_divide_eq) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

1137 

35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35579
diff
changeset

1138 
lemma divide_eq_eq_1 [simp,no_atp]: 
36409  1139 
"(b/a = 1) = ((a \<noteq> 0 & a = b))" 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

1140 
by (auto simp add: divide_eq_eq) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

1141 

14294
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset

1142 
lemma abs_inverse [simp]: 
36409  1143 
"\<bar>inverse a\<bar> = 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

1144 
inverse \<bar>a\<bar>" 
21328  1145 
apply (cases "a=0", simp) 
14294
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset

1146 
apply (simp add: nonzero_abs_inverse) 
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset

1147 
done 
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset

1148 

15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

1149 
lemma abs_divide [simp]: 
36409  1150 
"\<bar>a / b\<bar> = \<bar>a\<bar> / \<bar>b\<bar>" 
21328  1151 
apply (cases "b=0", simp) 
14294
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset

1152 
apply (simp add: nonzero_abs_divide) 
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset

1153 
done 
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset

1154 

36409  1155 
lemma abs_div_pos: "0 < y ==> 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

1156 
\<bar>x\<bar> / y = \<bar>x / y\<bar>" 
25304
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset

1157 
apply (subst abs_divide) 
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset

1158 
apply (simp add: order_less_imp_le) 
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset

1159 
done 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

1160 

35579
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35216
diff
changeset

1161 
lemma field_le_mult_one_interval: 
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35216
diff
changeset

1162 
assumes *: "\<And>z. \<lbrakk> 0 < z ; z < 1 \<rbrakk> \<Longrightarrow> z * x \<le> y" 
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35216
diff
changeset

1163 
shows "x \<le> y" 
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35216
diff
changeset

1164 
proof (cases "0 < x") 
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35216
diff
changeset

1165 
assume "0 < x" 
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35216
diff
changeset

1166 
thus ?thesis 
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35216
diff
changeset

1167 
using dense_le_bounded[of 0 1 "y/x"] * 
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35216
diff
changeset

1168 
unfolding le_divide_eq if_P[OF `0 < x`] by simp 
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35216
diff
changeset

1169 
next 
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35216
diff
changeset

1170 
assume "\<not>0 < x" hence "x \<le> 0" by simp 
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35216
diff
changeset

1171 
obtain s::'a where s: "0 < s" "s < 1" using dense[of 0 "1\<Colon>'a"] by auto 
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35216
diff
changeset

1172 
hence "x \<le> s * x" using mult_le_cancel_right[of 1 x s] `x \<le> 0` by auto 
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35216
diff
changeset

1173 
also note *[OF s] 
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35216
diff
changeset

1174 
finally show ?thesis . 
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35216
diff
changeset

1175 
qed 
35090
88cc65ae046e
moved lemma field_le_epsilon from Real.thy to Fields.thy
haftmann
parents:
35084
diff
changeset

1176 

36409  1177 
end 
1178 

33364  1179 
code_modulename SML 
35050
9f841f20dca6
renamed OrderedGroup to Groups; split theory Ring_and_Field into Rings Fields
haftmann
parents:
35043
diff
changeset

1180 
Fields Arith 
33364  1181 

1182 
code_modulename OCaml 

35050
9f841f20dca6
renamed OrderedGroup to Groups; split theory Ring_and_Field into Rings Fields
haftmann
parents:
35043
diff
changeset

1183 
Fields Arith 
33364  1184 

1185 
code_modulename Haskell 

35050
9f841f20dca6
renamed OrderedGroup to Groups; split theory Ring_and_Field into Rings Fields
haftmann
parents:
35043
diff
changeset

1186 
Fields Arith 
33364  1187 

14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

1188 
end 