author  wenzelm 
Tue, 01 Sep 2015 22:32:58 +0200  
changeset 61076  bdc1e2f0a86a 
parent 60758  d8d85a8172b5 
child 61238  e3d8a313a649 
permissions  rwrr 
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(* Title: HOL/Fields.thy 
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eliminated hard tabulators, guessing at each author's individual tabwidth;
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2 
Author: Gertrud Bauer 
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eliminated hard tabulators, guessing at each author's individual tabwidth;
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3 
Author: Steven Obua 
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eliminated hard tabulators, guessing at each author's individual tabwidth;
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4 
Author: Tobias Nipkow 
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eliminated hard tabulators, guessing at each author's individual tabwidth;
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5 
Author: Lawrence C Paulson 
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eliminated hard tabulators, guessing at each author's individual tabwidth;
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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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*) 
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HOL: installation of Ring_and_Field as the basis for Naturals and Reals
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9 

60758  10 
section \<open>Fields\<close> 
25152  11 

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theory Fields 
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imports Rings 
25186  14 
begin 
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60758  16 
subsection \<open>Division rings\<close> 
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60758  18 
text \<open> 
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A division ring is like a field, but without the commutativity requirement. 
60758  20 
\<close> 
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21 

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class inverse = divide + 
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fixes inverse :: "'a \<Rightarrow> 'a" 
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begin 
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25 

d46de31a50c4
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26 
abbreviation inverse_divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "'/" 70) 
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27 
where 
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28 
"inverse_divide \<equiv> divide" 
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29 

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30 
end 
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31 

60758  32 
text\<open>Lemmas @{text divide_simps} move division to the outside and eliminates them on (in)equalities.\<close> 
56481  33 

57950  34 
named_theorems divide_simps "rewrite rules to eliminate divisions" 
56481  35 

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class division_ring = ring_1 + inverse + 
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assumes left_inverse [simp]: "a \<noteq> 0 \<Longrightarrow> inverse a * a = 1" 
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38 
assumes right_inverse [simp]: "a \<noteq> 0 \<Longrightarrow> a * inverse a = 1" 
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39 
assumes divide_inverse: "a / b = a * inverse b" 
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40 
assumes inverse_zero [simp]: "inverse 0 = 0" 
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begin 
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42 

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43 
subclass ring_1_no_zero_divisors 
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44 
proof 
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45 
fix a b :: 'a 
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assume a: "a \<noteq> 0" and b: "b \<noteq> 0" 
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47 
show "a * b \<noteq> 0" 
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48 
proof 
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49 
assume ab: "a * b = 0" 
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50 
hence "0 = inverse a * (a * b) * inverse b" by simp 
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also have "\<dots> = (inverse a * a) * (b * inverse b)" 
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52 
by (simp only: mult.assoc) 
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53 
also have "\<dots> = 1" using a b by simp 
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54 
finally show False by simp 
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55 
qed 
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56 
qed 
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57 

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58 
lemma nonzero_imp_inverse_nonzero: 
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59 
"a \<noteq> 0 \<Longrightarrow> inverse a \<noteq> 0" 
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60 
proof 
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61 
assume ianz: "inverse a = 0" 
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62 
assume "a \<noteq> 0" 
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63 
hence "1 = a * inverse a" by simp 
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64 
also have "... = 0" by (simp add: ianz) 
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65 
finally have "1 = 0" . 
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66 
thus False by (simp add: eq_commute) 
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67 
qed 
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68 

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69 
lemma inverse_zero_imp_zero: 
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70 
"inverse a = 0 \<Longrightarrow> a = 0" 
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71 
apply (rule classical) 
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72 
apply (drule nonzero_imp_inverse_nonzero) 
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73 
apply auto 
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74 
done 
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75 

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76 
lemma inverse_unique: 
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77 
assumes ab: "a * b = 1" 
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78 
shows "inverse a = b" 
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79 
proof  
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80 
have "a \<noteq> 0" using ab by (cases "a = 0") simp_all 
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81 
moreover have "inverse a * (a * b) = inverse a" by (simp add: ab) 
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82 
ultimately show ?thesis by (simp add: mult.assoc [symmetric]) 
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83 
qed 
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84 

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85 
lemma nonzero_inverse_minus_eq: 
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86 
"a \<noteq> 0 \<Longrightarrow> inverse ( a) =  inverse a" 
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87 
by (rule inverse_unique) simp 
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88 

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89 
lemma nonzero_inverse_inverse_eq: 
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90 
"a \<noteq> 0 \<Longrightarrow> inverse (inverse a) = a" 
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91 
by (rule inverse_unique) simp 
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92 

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93 
lemma nonzero_inverse_eq_imp_eq: 
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94 
assumes "inverse a = inverse b" and "a \<noteq> 0" and "b \<noteq> 0" 
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95 
shows "a = b" 
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96 
proof  
60758  97 
from \<open>inverse a = inverse b\<close> 
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98 
have "inverse (inverse a) = inverse (inverse b)" by (rule arg_cong) 
60758  99 
with \<open>a \<noteq> 0\<close> and \<open>b \<noteq> 0\<close> show "a = b" 
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100 
by (simp add: nonzero_inverse_inverse_eq) 
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101 
qed 
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102 

5bce8ff0d9ae
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103 
lemma inverse_1 [simp]: "inverse 1 = 1" 
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104 
by (rule inverse_unique) simp 
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105 

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Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
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106 
lemma nonzero_inverse_mult_distrib: 
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107 
assumes "a \<noteq> 0" and "b \<noteq> 0" 
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108 
shows "inverse (a * b) = inverse b * inverse a" 
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109 
proof  
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110 
have "a * (b * inverse b) * inverse a = 1" using assms by simp 
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111 
hence "a * b * (inverse b * inverse a) = 1" by (simp only: mult.assoc) 
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112 
thus ?thesis by (rule inverse_unique) 
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113 
qed 
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114 

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115 
lemma division_ring_inverse_add: 
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116 
"a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> inverse a + inverse b = inverse a * (a + b) * inverse b" 
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117 
by (simp add: algebra_simps) 
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118 

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119 
lemma division_ring_inverse_diff: 
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120 
"a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> inverse a  inverse b = inverse a * (b  a) * inverse b" 
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121 
by (simp add: algebra_simps) 
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122 

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123 
lemma right_inverse_eq: "b \<noteq> 0 \<Longrightarrow> a / b = 1 \<longleftrightarrow> a = b" 
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124 
proof 
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125 
assume neq: "b \<noteq> 0" 
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126 
{ 
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127 
hence "a = (a / b) * b" by (simp add: divide_inverse mult.assoc) 
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128 
also assume "a / b = 1" 
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129 
finally show "a = b" by simp 
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130 
next 
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131 
assume "a = b" 
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132 
with neq show "a / b = 1" by (simp add: divide_inverse) 
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133 
} 
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134 
qed 
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135 

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136 
lemma nonzero_inverse_eq_divide: "a \<noteq> 0 \<Longrightarrow> inverse a = 1 / a" 
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137 
by (simp add: divide_inverse) 
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138 

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139 
lemma divide_self [simp]: "a \<noteq> 0 \<Longrightarrow> a / a = 1" 
5bce8ff0d9ae
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140 
by (simp add: divide_inverse) 
5bce8ff0d9ae
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huffman
parents:
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141 

56481  142 
lemma inverse_eq_divide [field_simps, divide_simps]: "inverse a = 1 / a" 
44064
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parents:
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diff
changeset

143 
by (simp add: divide_inverse) 
5bce8ff0d9ae
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huffman
parents:
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diff
changeset

144 

5bce8ff0d9ae
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parents:
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145 
lemma add_divide_distrib: "(a+b) / c = a/c + b/c" 
5bce8ff0d9ae
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parents:
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diff
changeset

146 
by (simp add: divide_inverse algebra_simps) 
5bce8ff0d9ae
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huffman
parents:
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diff
changeset

147 

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changeset

148 
lemma times_divide_eq_right [simp]: "a * (b / c) = (a * b) / c" 
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149 
by (simp add: divide_inverse mult.assoc) 
44064
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huffman
parents:
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diff
changeset

150 

5bce8ff0d9ae
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parents:
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changeset

151 
lemma minus_divide_left: " (a / b) = (a) / b" 
5bce8ff0d9ae
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changeset

152 
by (simp add: divide_inverse) 
5bce8ff0d9ae
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huffman
parents:
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diff
changeset

153 

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changeset

154 
lemma nonzero_minus_divide_right: "b \<noteq> 0 ==>  (a / b) = a / ( b)" 
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changeset

155 
by (simp add: divide_inverse nonzero_inverse_minus_eq) 
5bce8ff0d9ae
moved division ring stuff from Rings.thy to Fields.thy
huffman
parents:
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diff
changeset

156 

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parents:
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diff
changeset

157 
lemma nonzero_minus_divide_divide: "b \<noteq> 0 ==> (a) / (b) = a / b" 
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parents:
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diff
changeset

158 
by (simp add: divide_inverse nonzero_inverse_minus_eq) 
5bce8ff0d9ae
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huffman
parents:
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diff
changeset

159 

56479
91958d4b30f7
revert c1bbd3e22226, a14831ac3023, and 36489d77c484: divide_minus_left/right are again simp rules
hoelzl
parents:
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diff
changeset

160 
lemma divide_minus_left [simp]: "(a) / b =  (a / b)" 
44064
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huffman
parents:
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diff
changeset

161 
by (simp add: divide_inverse) 
5bce8ff0d9ae
moved division ring stuff from Rings.thy to Fields.thy
huffman
parents:
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diff
changeset

162 

5bce8ff0d9ae
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parents:
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diff
changeset

163 
lemma diff_divide_distrib: "(a  b) / c = a / c  b / c" 
56479
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revert c1bbd3e22226, a14831ac3023, and 36489d77c484: divide_minus_left/right are again simp rules
hoelzl
parents:
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diff
changeset

164 
using add_divide_distrib [of a " b" c] by simp 
44064
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moved division ring stuff from Rings.thy to Fields.thy
huffman
parents:
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diff
changeset

165 

5bce8ff0d9ae
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parents:
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changeset

166 
lemma nonzero_eq_divide_eq [field_simps]: "c \<noteq> 0 \<Longrightarrow> a = b / c \<longleftrightarrow> a * c = b" 
5bce8ff0d9ae
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huffman
parents:
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changeset

167 
proof  
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huffman
parents:
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diff
changeset

168 
assume [simp]: "c \<noteq> 0" 
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moved division ring stuff from Rings.thy to Fields.thy
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parents:
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changeset

169 
have "a = b / c \<longleftrightarrow> a * c = (b / c) * c" by simp 
57512
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diff
changeset

170 
also have "... \<longleftrightarrow> a * c = b" by (simp add: divide_inverse mult.assoc) 
44064
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parents:
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changeset

171 
finally show ?thesis . 
5bce8ff0d9ae
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huffman
parents:
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diff
changeset

172 
qed 
5bce8ff0d9ae
moved division ring stuff from Rings.thy to Fields.thy
huffman
parents:
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diff
changeset

173 

5bce8ff0d9ae
moved division ring stuff from Rings.thy to Fields.thy
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parents:
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changeset

174 
lemma nonzero_divide_eq_eq [field_simps]: "c \<noteq> 0 \<Longrightarrow> b / c = a \<longleftrightarrow> b = a * c" 
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parents:
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changeset

175 
proof  
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huffman
parents:
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diff
changeset

176 
assume [simp]: "c \<noteq> 0" 
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parents:
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changeset

177 
have "b / c = a \<longleftrightarrow> (b / c) * c = a * c" by simp 
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diff
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178 
also have "... \<longleftrightarrow> b = a * c" by (simp add: divide_inverse mult.assoc) 
44064
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huffman
parents:
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diff
changeset

179 
finally show ?thesis . 
5bce8ff0d9ae
moved division ring stuff from Rings.thy to Fields.thy
huffman
parents:
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diff
changeset

180 
qed 
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huffman
parents:
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diff
changeset

181 

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182 
lemma nonzero_neg_divide_eq_eq [field_simps]: "b \<noteq> 0 \<Longrightarrow>  (a / b) = c \<longleftrightarrow>  a = c * b" 
59535  183 
using nonzero_divide_eq_eq[of b "a" c] by simp 
56441  184 

56480
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changeset

185 
lemma nonzero_neg_divide_eq_eq2 [field_simps]: "b \<noteq> 0 \<Longrightarrow> c =  (a / b) \<longleftrightarrow> c * b =  a" 
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changeset

186 
using nonzero_neg_divide_eq_eq[of b a c] by auto 
56441  187 

44064
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188 
lemma divide_eq_imp: "c \<noteq> 0 \<Longrightarrow> b = a * c \<Longrightarrow> b / c = a" 
57512
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parents:
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diff
changeset

189 
by (simp add: divide_inverse mult.assoc) 
44064
5bce8ff0d9ae
moved division ring stuff from Rings.thy to Fields.thy
huffman
parents:
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diff
changeset

190 

5bce8ff0d9ae
moved division ring stuff from Rings.thy to Fields.thy
huffman
parents:
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changeset

191 
lemma eq_divide_imp: "c \<noteq> 0 \<Longrightarrow> a * c = b \<Longrightarrow> a = b / c" 
57512
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haftmann
parents:
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diff
changeset

192 
by (drule sym) (simp add: divide_inverse mult.assoc) 
44064
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huffman
parents:
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diff
changeset

193 

56445  194 
lemma add_divide_eq_iff [field_simps]: 
195 
"z \<noteq> 0 \<Longrightarrow> x + y / z = (x * z + y) / z" 

196 
by (simp add: add_divide_distrib nonzero_eq_divide_eq) 

197 

198 
lemma divide_add_eq_iff [field_simps]: 

199 
"z \<noteq> 0 \<Longrightarrow> x / z + y = (x + y * z) / z" 

200 
by (simp add: add_divide_distrib nonzero_eq_divide_eq) 

201 

202 
lemma diff_divide_eq_iff [field_simps]: 

203 
"z \<noteq> 0 \<Longrightarrow> x  y / z = (x * z  y) / z" 

204 
by (simp add: diff_divide_distrib nonzero_eq_divide_eq eq_diff_eq) 

205 

56480
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parents:
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changeset

206 
lemma minus_divide_add_eq_iff [field_simps]: 
093ea91498e6
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parents:
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diff
changeset

207 
"z \<noteq> 0 \<Longrightarrow>  (x / z) + y = ( x + y * z) / z" 
59535  208 
by (simp add: add_divide_distrib diff_divide_eq_iff) 
56480
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hoelzl
parents:
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diff
changeset

209 

56445  210 
lemma divide_diff_eq_iff [field_simps]: 
211 
"z \<noteq> 0 \<Longrightarrow> x / z  y = (x  y * z) / z" 

212 
by (simp add: field_simps) 

213 

56480
093ea91498e6
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hoelzl
parents:
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diff
changeset

214 
lemma minus_divide_diff_eq_iff [field_simps]: 
093ea91498e6
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hoelzl
parents:
56479
diff
changeset

215 
"z \<noteq> 0 \<Longrightarrow>  (x / z)  y = ( x  y * z) / z" 
59535  216 
by (simp add: divide_diff_eq_iff[symmetric]) 
56480
093ea91498e6
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hoelzl
parents:
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diff
changeset

217 

60353
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset

218 
lemma division_ring_divide_zero [simp]: 
44064
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huffman
parents:
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diff
changeset

219 
"a / 0 = 0" 
5bce8ff0d9ae
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huffman
parents:
42904
diff
changeset

220 
by (simp add: divide_inverse) 
5bce8ff0d9ae
moved division ring stuff from Rings.thy to Fields.thy
huffman
parents:
42904
diff
changeset

221 

5bce8ff0d9ae
moved division ring stuff from Rings.thy to Fields.thy
huffman
parents:
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diff
changeset

222 
lemma divide_self_if [simp]: 
5bce8ff0d9ae
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huffman
parents:
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diff
changeset

223 
"a / a = (if a = 0 then 0 else 1)" 
5bce8ff0d9ae
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huffman
parents:
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diff
changeset

224 
by simp 
5bce8ff0d9ae
moved division ring stuff from Rings.thy to Fields.thy
huffman
parents:
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diff
changeset

225 

5bce8ff0d9ae
moved division ring stuff from Rings.thy to Fields.thy
huffman
parents:
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diff
changeset

226 
lemma inverse_nonzero_iff_nonzero [simp]: 
5bce8ff0d9ae
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huffman
parents:
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diff
changeset

227 
"inverse a = 0 \<longleftrightarrow> a = 0" 
5bce8ff0d9ae
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huffman
parents:
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diff
changeset

228 
by rule (fact inverse_zero_imp_zero, simp) 
5bce8ff0d9ae
moved division ring stuff from Rings.thy to Fields.thy
huffman
parents:
42904
diff
changeset

229 

5bce8ff0d9ae
moved division ring stuff from Rings.thy to Fields.thy
huffman
parents:
42904
diff
changeset

230 
lemma inverse_minus_eq [simp]: 
5bce8ff0d9ae
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huffman
parents:
42904
diff
changeset

231 
"inverse ( a) =  inverse a" 
5bce8ff0d9ae
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huffman
parents:
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diff
changeset

232 
proof cases 
5bce8ff0d9ae
moved division ring stuff from Rings.thy to Fields.thy
huffman
parents:
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diff
changeset

233 
assume "a=0" thus ?thesis by simp 
5bce8ff0d9ae
moved division ring stuff from Rings.thy to Fields.thy
huffman
parents:
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diff
changeset

234 
next 
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59557
diff
changeset

235 
assume "a\<noteq>0" 
44064
5bce8ff0d9ae
moved division ring stuff from Rings.thy to Fields.thy
huffman
parents:
42904
diff
changeset

236 
thus ?thesis by (simp add: nonzero_inverse_minus_eq) 
5bce8ff0d9ae
moved division ring stuff from Rings.thy to Fields.thy
huffman
parents:
42904
diff
changeset

237 
qed 
5bce8ff0d9ae
moved division ring stuff from Rings.thy to Fields.thy
huffman
parents:
42904
diff
changeset

238 

5bce8ff0d9ae
moved division ring stuff from Rings.thy to Fields.thy
huffman
parents:
42904
diff
changeset

239 
lemma inverse_inverse_eq [simp]: 
5bce8ff0d9ae
moved division ring stuff from Rings.thy to Fields.thy
huffman
parents:
42904
diff
changeset

240 
"inverse (inverse a) = a" 
5bce8ff0d9ae
moved division ring stuff from Rings.thy to Fields.thy
huffman
parents:
42904
diff
changeset

241 
proof cases 
5bce8ff0d9ae
moved division ring stuff from Rings.thy to Fields.thy
huffman
parents:
42904
diff
changeset

242 
assume "a=0" thus ?thesis by simp 
5bce8ff0d9ae
moved division ring stuff from Rings.thy to Fields.thy
huffman
parents:
42904
diff
changeset

243 
next 
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59557
diff
changeset

244 
assume "a\<noteq>0" 
44064
5bce8ff0d9ae
moved division ring stuff from Rings.thy to Fields.thy
huffman
parents:
42904
diff
changeset

245 
thus ?thesis by (simp add: nonzero_inverse_inverse_eq) 
5bce8ff0d9ae
moved division ring stuff from Rings.thy to Fields.thy
huffman
parents:
42904
diff
changeset

246 
qed 
5bce8ff0d9ae
moved division ring stuff from Rings.thy to Fields.thy
huffman
parents:
42904
diff
changeset

247 

44680  248 
lemma inverse_eq_imp_eq: 
249 
"inverse a = inverse b \<Longrightarrow> a = b" 

250 
by (drule arg_cong [where f="inverse"], simp) 

251 

252 
lemma inverse_eq_iff_eq [simp]: 

253 
"inverse a = inverse b \<longleftrightarrow> a = b" 

254 
by (force dest!: inverse_eq_imp_eq) 

255 

56481  256 
lemma add_divide_eq_if_simps [divide_simps]: 
257 
"a + b / z = (if z = 0 then a else (a * z + b) / z)" 

258 
"a / z + b = (if z = 0 then b else (a + b * z) / z)" 

259 
" (a / z) + b = (if z = 0 then b else (a + b * z) / z)" 

260 
"a  b / z = (if z = 0 then a else (a * z  b) / z)" 

261 
"a / z  b = (if z = 0 then b else (a  b * z) / z)" 

262 
" (a / z)  b = (if z = 0 then b else ( a  b * z) / z)" 

263 
by (simp_all add: add_divide_eq_iff divide_add_eq_iff diff_divide_eq_iff divide_diff_eq_iff 

264 
minus_divide_diff_eq_iff) 

265 

266 
lemma [divide_simps]: 

267 
shows divide_eq_eq: "b / c = a \<longleftrightarrow> (if c \<noteq> 0 then b = a * c else a = 0)" 

268 
and eq_divide_eq: "a = b / c \<longleftrightarrow> (if c \<noteq> 0 then a * c = b else a = 0)" 

269 
and minus_divide_eq_eq: " (b / c) = a \<longleftrightarrow> (if c \<noteq> 0 then  b = a * c else a = 0)" 

270 
and eq_minus_divide_eq: "a =  (b / c) \<longleftrightarrow> (if c \<noteq> 0 then a * c =  b else a = 0)" 

271 
by (auto simp add: field_simps) 

272 

44064
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huffman
parents:
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diff
changeset

273 
end 
5bce8ff0d9ae
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huffman
parents:
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diff
changeset

274 

60758  275 
subsection \<open>Fields\<close> 
44064
5bce8ff0d9ae
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parents:
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diff
changeset

276 

22987
550709aa8e66
instance division_ring < no_zero_divisors; clean up field instance proofs
huffman
parents:
22842
diff
changeset

277 
class field = comm_ring_1 + inverse + 
35084  278 
assumes field_inverse: "a \<noteq> 0 \<Longrightarrow> inverse a * a = 1" 
279 
assumes field_divide_inverse: "a / b = a * inverse b" 

59867
58043346ca64
given up separate type classes demanding `inverse 0 = 0`
haftmann
parents:
59779
diff
changeset

280 
assumes field_inverse_zero: "inverse 0 = 0" 
25267  281 
begin 
20496
23eb6034c06d
added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents:
19404
diff
changeset

282 

25267  283 
subclass division_ring 
28823  284 
proof 
22987
550709aa8e66
instance division_ring < no_zero_divisors; clean up field instance proofs
huffman
parents:
22842
diff
changeset

285 
fix a :: 'a 
550709aa8e66
instance division_ring < no_zero_divisors; clean up field instance proofs
huffman
parents:
22842
diff
changeset

286 
assume "a \<noteq> 0" 
550709aa8e66
instance division_ring < no_zero_divisors; clean up field instance proofs
huffman
parents:
22842
diff
changeset

287 
thus "inverse a * a = 1" by (rule field_inverse) 
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
56571
diff
changeset

288 
thus "a * inverse a = 1" by (simp only: mult.commute) 
35084  289 
next 
290 
fix a b :: 'a 

291 
show "a / b = a * inverse b" by (rule field_divide_inverse) 

59867
58043346ca64
given up separate type classes demanding `inverse 0 = 0`
haftmann
parents:
59779
diff
changeset

292 
next 
58043346ca64
given up separate type classes demanding `inverse 0 = 0`
haftmann
parents:
59779
diff
changeset

293 
show "inverse 0 = 0" 
58043346ca64
given up separate type classes demanding `inverse 0 = 0`
haftmann
parents:
59779
diff
changeset

294 
by (fact field_inverse_zero) 
14738  295 
qed 
25230  296 

60353
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset

297 
subclass idom_divide 
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset

298 
proof 
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset

299 
fix b a 
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset

300 
assume "b \<noteq> 0" 
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset

301 
then show "a * b / b = a" 
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset

302 
by (simp add: divide_inverse ac_simps) 
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset

303 
next 
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset

304 
fix a 
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset

305 
show "a / 0 = 0" 
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset

306 
by (simp add: divide_inverse) 
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset

307 
qed 
25230  308 

60758  309 
text\<open>There is no slick version using division by zero.\<close> 
30630  310 
lemma inverse_add: 
60353
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset

311 
"a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> inverse a + inverse b = (a + b) * inverse a * inverse b" 
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset

312 
by (simp add: division_ring_inverse_add ac_simps) 
30630  313 

54147
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killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
53374
diff
changeset

314 
lemma nonzero_mult_divide_mult_cancel_left [simp]: 
60353
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset

315 
assumes [simp]: "c \<noteq> 0" 
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset

316 
shows "(c * a) / (c * b) = a / b" 
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset

317 
proof (cases "b = 0") 
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset

318 
case True then show ?thesis by simp 
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset

319 
next 
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset

320 
case False 
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset

321 
then have "(c*a)/(c*b) = c * a * (inverse b * inverse c)" 
30630  322 
by (simp add: divide_inverse nonzero_inverse_mult_distrib) 
323 
also have "... = a * inverse b * (inverse c * c)" 

57514
bdc2c6b40bf2
prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents:
57512
diff
changeset

324 
by (simp only: ac_simps) 
30630  325 
also have "... = a * inverse b" by simp 
326 
finally show ?thesis by (simp add: divide_inverse) 

327 
qed 

328 

54147
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killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
53374
diff
changeset

329 
lemma nonzero_mult_divide_mult_cancel_right [simp]: 
60353
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset

330 
"c \<noteq> 0 \<Longrightarrow> (a * c) / (b * c) = a / b" 
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset

331 
using nonzero_mult_divide_mult_cancel_left [of c a b] by (simp add: ac_simps) 
30630  332 

36304
6984744e6b34
less special treatment of times_divide_eq [simp]
haftmann
parents:
36301
diff
changeset

333 
lemma times_divide_eq_left [simp]: "(b / c) * a = (b * a) / c" 
57514
bdc2c6b40bf2
prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents:
57512
diff
changeset

334 
by (simp add: divide_inverse ac_simps) 
30630  335 

336 
lemma add_frac_eq: 

337 
assumes "y \<noteq> 0" and "z \<noteq> 0" 

338 
shows "x / y + w / z = (x * z + w * y) / (y * z)" 

339 
proof  

340 
have "x / y + w / z = (x * z) / (y * z) + (y * w) / (y * z)" 

341 
using assms by simp 

342 
also have "\<dots> = (x * z + y * w) / (y * z)" 

343 
by (simp only: add_divide_distrib) 

344 
finally show ?thesis 

57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
56571
diff
changeset

345 
by (simp only: mult.commute) 
30630  346 
qed 
347 

60758  348 
text\<open>Special Cancellation Simprules for Division\<close> 
30630  349 

54147
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killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
53374
diff
changeset

350 
lemma nonzero_divide_mult_cancel_right [simp]: 
60353
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset

351 
"b \<noteq> 0 \<Longrightarrow> b / (a * b) = 1 / a" 
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset

352 
using nonzero_mult_divide_mult_cancel_right [of b 1 a] by simp 
30630  353 

54147
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
53374
diff
changeset

354 
lemma nonzero_divide_mult_cancel_left [simp]: 
60353
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset

355 
"a \<noteq> 0 \<Longrightarrow> a / (a * b) = 1 / b" 
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset

356 
using nonzero_mult_divide_mult_cancel_left [of a 1 b] by simp 
30630  357 

54147
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
53374
diff
changeset

358 
lemma nonzero_mult_divide_mult_cancel_left2 [simp]: 
60353
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset

359 
"c \<noteq> 0 \<Longrightarrow> (c * a) / (b * c) = a / b" 
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset

360 
using nonzero_mult_divide_mult_cancel_left [of c a b] by (simp add: ac_simps) 
30630  361 

54147
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
53374
diff
changeset

362 
lemma nonzero_mult_divide_mult_cancel_right2 [simp]: 
60353
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset

363 
"c \<noteq> 0 \<Longrightarrow> (a * c) / (c * b) = a / b" 
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset

364 
using nonzero_mult_divide_mult_cancel_right [of b c a] by (simp add: ac_simps) 
30630  365 

366 
lemma diff_frac_eq: 

367 
"y \<noteq> 0 \<Longrightarrow> z \<noteq> 0 \<Longrightarrow> x / y  w / z = (x * z  w * y) / (y * z)" 

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

368 
by (simp add: field_simps) 
30630  369 

370 
lemma frac_eq_eq: 

371 
"y \<noteq> 0 \<Longrightarrow> z \<noteq> 0 \<Longrightarrow> (x / y = w / z) = (x * z = w * y)" 

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

372 
by (simp add: field_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

373 

58512
dc4d76dfa8f0
moved lemmas out of Int.thy which have nothing to do with int
haftmann
parents:
57950
diff
changeset

374 
lemma divide_minus1 [simp]: "x /  1 =  x" 
dc4d76dfa8f0
moved lemmas out of Int.thy which have nothing to do with int
haftmann
parents:
57950
diff
changeset

375 
using nonzero_minus_divide_right [of "1" x] by simp 
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59557
diff
changeset

376 

60758  377 
text\<open>This version builds in division by zero while also reorienting 
378 
the righthand side.\<close> 

14270  379 
lemma inverse_mult_distrib [simp]: 
36409  380 
"inverse (a * b) = inverse a * inverse b" 
381 
proof cases 

59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59557
diff
changeset

382 
assume "a \<noteq> 0 & b \<noteq> 0" 
57514
bdc2c6b40bf2
prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents:
57512
diff
changeset

383 
thus ?thesis by (simp add: nonzero_inverse_mult_distrib ac_simps) 
36409  384 
next 
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59557
diff
changeset

385 
assume "~ (a \<noteq> 0 & b \<noteq> 0)" 
36409  386 
thus ?thesis by force 
387 
qed 

14270  388 

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

389 
lemma inverse_divide [simp]: 
36409  390 
"inverse (a / b) = b / a" 
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
56571
diff
changeset

391 
by (simp add: divide_inverse mult.commute) 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

392 

23389  393 

60758  394 
text \<open>Calculations with fractions\<close> 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

395 

60758  396 
text\<open>There is a whole bunch of simprules just for class @{text 
23413
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset

397 
field} but none for class @{text field} and @{text nonzero_divides} 
60758  398 
because the latter are covered by a simproc.\<close> 
23413
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset

399 

5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset

400 
lemma mult_divide_mult_cancel_left: 
36409  401 
"c \<noteq> 0 \<Longrightarrow> (c * a) / (c * b) = a / b" 
21328  402 
apply (cases "b = 0") 
35216  403 
apply simp_all 
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

404 
done 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

405 

23413
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset

406 
lemma mult_divide_mult_cancel_right: 
36409  407 
"c \<noteq> 0 \<Longrightarrow> (a * c) / (b * c) = a / b" 
21328  408 
apply (cases "b = 0") 
35216  409 
apply simp_all 
14321  410 
done 
23413
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset

411 

54147
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
53374
diff
changeset

412 
lemma divide_divide_eq_right [simp]: 
36409  413 
"a / (b / c) = (a * c) / b" 
57514
bdc2c6b40bf2
prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents:
57512
diff
changeset

414 
by (simp add: divide_inverse ac_simps) 
14288  415 

54147
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
53374
diff
changeset

416 
lemma divide_divide_eq_left [simp]: 
36409  417 
"(a / b) / c = a / (b * c)" 
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
56571
diff
changeset

418 
by (simp add: divide_inverse mult.assoc) 
14288  419 

56365
713f9b9a7e51
New theorems for extracting quotients
paulson <lp15@cam.ac.uk>
parents:
55718
diff
changeset

420 
lemma divide_divide_times_eq: 
713f9b9a7e51
New theorems for extracting quotients
paulson <lp15@cam.ac.uk>
parents:
55718
diff
changeset

421 
"(x / y) / (z / w) = (x * w) / (y * z)" 
713f9b9a7e51
New theorems for extracting quotients
paulson <lp15@cam.ac.uk>
parents:
55718
diff
changeset

422 
by simp 
23389  423 

60758  424 
text \<open>Special Cancellation Simprules for Division\<close> 
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

425 

54147
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
53374
diff
changeset

426 
lemma mult_divide_mult_cancel_left_if [simp]: 
36409  427 
shows "(c * a) / (c * b) = (if c = 0 then 0 else a / b)" 
60353
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset

428 
by simp 
23413
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset

429 

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

430 

60758  431 
text \<open>Division and Unary Minus\<close> 
14293  432 

36409  433 
lemma minus_divide_right: 
434 
" (a / b) = a /  b" 

435 
by (simp add: divide_inverse) 

14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset

436 

56479
91958d4b30f7
revert c1bbd3e22226, a14831ac3023, and 36489d77c484: divide_minus_left/right are again simp rules
hoelzl
parents:
56445
diff
changeset

437 
lemma divide_minus_right [simp]: 
36409  438 
"a /  b =  (a / b)" 
439 
by (simp add: divide_inverse) 

30630  440 

56479
91958d4b30f7
revert c1bbd3e22226, a14831ac3023, and 36489d77c484: divide_minus_left/right are again simp rules
hoelzl
parents:
56445
diff
changeset

441 
lemma minus_divide_divide: 
36409  442 
"( a) / ( b) = a / b" 
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59557
diff
changeset

443 
apply (cases "b=0", simp) 
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59557
diff
changeset

444 
apply (simp add: nonzero_minus_divide_divide) 
14293  445 
done 
446 

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

447 
lemma inverse_eq_1_iff [simp]: 
36409  448 
"inverse x = 1 \<longleftrightarrow> x = 1" 
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59557
diff
changeset

449 
by (insert inverse_eq_iff_eq [of x 1], simp) 
23389  450 

54147
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
53374
diff
changeset

451 
lemma divide_eq_0_iff [simp]: 
36409  452 
"a / b = 0 \<longleftrightarrow> a = 0 \<or> b = 0" 
453 
by (simp add: divide_inverse) 

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

454 

54147
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
53374
diff
changeset

455 
lemma divide_cancel_right [simp]: 
36409  456 
"a / c = b / c \<longleftrightarrow> c = 0 \<or> a = b" 
457 
apply (cases "c=0", simp) 

458 
apply (simp add: divide_inverse) 

459 
done 

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

460 

54147
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
53374
diff
changeset

461 
lemma divide_cancel_left [simp]: 
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59557
diff
changeset

462 
"c / a = c / b \<longleftrightarrow> c = 0 \<or> a = b" 
36409  463 
apply (cases "c=0", simp) 
464 
apply (simp add: divide_inverse) 

465 
done 

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

466 

54147
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
53374
diff
changeset

467 
lemma divide_eq_1_iff [simp]: 
36409  468 
"a / b = 1 \<longleftrightarrow> b \<noteq> 0 \<and> a = b" 
469 
apply (cases "b=0", simp) 

470 
apply (simp add: right_inverse_eq) 

471 
done 

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

472 

54147
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
53374
diff
changeset

473 
lemma one_eq_divide_iff [simp]: 
36409  474 
"1 = a / b \<longleftrightarrow> b \<noteq> 0 \<and> a = b" 
475 
by (simp add: eq_commute [of 1]) 

476 

36719  477 
lemma times_divide_times_eq: 
478 
"(x / y) * (z / w) = (x * z) / (y * w)" 

479 
by simp 

480 

481 
lemma add_frac_num: 

482 
"y \<noteq> 0 \<Longrightarrow> x / y + z = (x + z * y) / y" 

483 
by (simp add: add_divide_distrib) 

484 

485 
lemma add_num_frac: 

486 
"y \<noteq> 0 \<Longrightarrow> z + x / y = (x + z * y) / y" 

487 
by (simp add: add_divide_distrib add.commute) 

488 

36409  489 
end 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

490 

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

491 

60758  492 
subsection \<open>Ordered fields\<close> 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

493 

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

494 
class linordered_field = field + linordered_idom 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

495 
begin 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

496 

59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59557
diff
changeset

497 
lemma positive_imp_inverse_positive: 
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59557
diff
changeset

498 
assumes a_gt_0: "0 < a" 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

499 
shows "0 < inverse a" 
23482  500 
proof  
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59557
diff
changeset

501 
have "0 < a * inverse a" 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

502 
by (simp add: a_gt_0 [THEN less_imp_not_eq2]) 
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59557
diff
changeset

503 
thus "0 < inverse a" 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

504 
by (simp add: a_gt_0 [THEN less_not_sym] zero_less_mult_iff) 
23482  505 
qed 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

506 

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

507 
lemma negative_imp_inverse_negative: 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

508 
"a < 0 \<Longrightarrow> inverse a < 0" 
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59557
diff
changeset

509 
by (insert positive_imp_inverse_positive [of "a"], 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

510 
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

511 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

512 
lemma inverse_le_imp_le: 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

513 
assumes invle: "inverse a \<le> inverse b" and apos: "0 < a" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

514 
shows "b \<le> a" 
23482  515 
proof (rule classical) 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

516 
assume "~ b \<le> a" 
23482  517 
hence "a < b" by (simp add: linorder_not_le) 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

518 
hence bpos: "0 < b" by (blast intro: apos less_trans) 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

519 
hence "a * inverse a \<le> a * inverse b" 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

520 
by (simp add: apos invle less_imp_le mult_left_mono) 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

521 
hence "(a * inverse a) * b \<le> (a * inverse b) * b" 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

522 
by (simp add: bpos less_imp_le mult_right_mono) 
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
56571
diff
changeset

523 
thus "b \<le> a" by (simp add: mult.assoc apos bpos less_imp_not_eq2) 
23482  524 
qed 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

525 

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

526 
lemma inverse_positive_imp_positive: 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

527 
assumes inv_gt_0: "0 < inverse a" and nz: "a \<noteq> 0" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

528 
shows "0 < a" 
23389  529 
proof  
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

530 
have "0 < inverse (inverse a)" 
23389  531 
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

532 
thus "0 < a" 
23389  533 
using nz by (simp add: nonzero_inverse_inverse_eq) 
534 
qed 

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

535 

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

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

537 
assumes inv_less_0: "inverse a < 0" and nz: "a \<noteq> 0" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

538 
shows "a < 0" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

539 
proof  
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

540 
have "inverse (inverse a) < 0" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

541 
using inv_less_0 by (rule negative_imp_inverse_negative) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

542 
thus "a < 0" using nz by (simp add: nonzero_inverse_inverse_eq) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

543 
qed 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

544 

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

545 
lemma linordered_field_no_lb: 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

546 
"\<forall>x. \<exists>y. y < x" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

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

548 
fix x::'a 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

549 
have m1: " (1::'a) < 0" by simp 
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59557
diff
changeset

550 
from add_strict_right_mono[OF m1, where c=x] 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

551 
have "( 1) + x < x" by simp 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

552 
thus "\<exists>y. y < x" by blast 
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 linordered_field_no_ub: 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

556 
"\<forall> x. \<exists>y. y > x" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

557 
proof 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

558 
fix x::'a 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

559 
have m1: " (1::'a) > 0" by simp 
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59557
diff
changeset

560 
from add_strict_right_mono[OF m1, where c=x] 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

561 
have "1 + x > x" by simp 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

562 
thus "\<exists>y. y > x" by blast 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

563 
qed 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

564 

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

565 
lemma less_imp_inverse_less: 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

566 
assumes less: "a < b" and apos: "0 < a" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

567 
shows "inverse b < inverse a" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

568 
proof (rule ccontr) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

569 
assume "~ inverse b < inverse a" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

570 
hence "inverse a \<le> inverse b" by simp 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

571 
hence "~ (a < b)" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

572 
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

573 
thus False by (rule notE [OF _ less]) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

574 
qed 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

575 

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

576 
lemma inverse_less_imp_less: 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

577 
"inverse a < inverse b \<Longrightarrow> 0 < a \<Longrightarrow> b < a" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

578 
apply (simp add: less_le [of "inverse a"] less_le [of "b"]) 
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59557
diff
changeset

579 
apply (force dest!: inverse_le_imp_le nonzero_inverse_eq_imp_eq) 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

580 
done 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

581 

60758  582 
text\<open>Both premises are essential. Consider 1 and 1.\<close> 
54147
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
53374
diff
changeset

583 
lemma inverse_less_iff_less [simp]: 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

584 
"0 < a \<Longrightarrow> 0 < b \<Longrightarrow> inverse a < inverse b \<longleftrightarrow> b < a" 
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59557
diff
changeset

585 
by (blast intro: less_imp_inverse_less dest: inverse_less_imp_less) 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

586 

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

587 
lemma le_imp_inverse_le: 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

588 
"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

589 
by (force simp add: le_less less_imp_inverse_less) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

590 

54147
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
53374
diff
changeset

591 
lemma inverse_le_iff_le [simp]: 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

592 
"0 < a \<Longrightarrow> 0 < b \<Longrightarrow> inverse a \<le> inverse b \<longleftrightarrow> b \<le> a" 
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59557
diff
changeset

593 
by (blast intro: le_imp_inverse_le dest: inverse_le_imp_le) 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

594 

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

595 

60758  596 
text\<open>These results refer to both operands being negative. The oppositesign 
597 
case is trivial, since inverse preserves signs.\<close> 

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

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

599 
"inverse a \<le> inverse b \<Longrightarrow> b < 0 \<Longrightarrow> b \<le> a" 
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59557
diff
changeset

600 
apply (rule classical) 
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59557
diff
changeset

601 
apply (subgoal_tac "a < 0") 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

602 
prefer 2 apply force 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

603 
apply (insert inverse_le_imp_le [of "b" "a"]) 
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59557
diff
changeset

604 
apply (simp add: nonzero_inverse_minus_eq) 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

605 
done 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

606 

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

607 
lemma less_imp_inverse_less_neg: 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

608 
"a < b \<Longrightarrow> b < 0 \<Longrightarrow> inverse b < inverse a" 
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59557
diff
changeset

609 
apply (subgoal_tac "a < 0") 
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59557
diff
changeset

610 
prefer 2 apply (blast intro: less_trans) 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

611 
apply (insert less_imp_inverse_less [of "b" "a"]) 
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59557
diff
changeset

612 
apply (simp add: nonzero_inverse_minus_eq) 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

613 
done 
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_less_imp_less_neg: 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

616 
"inverse a < inverse b \<Longrightarrow> b < 0 \<Longrightarrow> b < a" 
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59557
diff
changeset

617 
apply (rule classical) 
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59557
diff
changeset

618 
apply (subgoal_tac "a < 0") 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

619 
prefer 2 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

620 
apply force 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

621 
apply (insert inverse_less_imp_less [of "b" "a"]) 
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59557
diff
changeset

622 
apply (simp add: nonzero_inverse_minus_eq) 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

623 
done 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

624 

54147
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
53374
diff
changeset

625 
lemma inverse_less_iff_less_neg [simp]: 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

626 
"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

627 
apply (insert inverse_less_iff_less [of "b" "a"]) 
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59557
diff
changeset

628 
apply (simp del: inverse_less_iff_less 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

629 
add: nonzero_inverse_minus_eq) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

630 
done 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

631 

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

632 
lemma le_imp_inverse_le_neg: 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

633 
"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

634 
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

635 

54147
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
53374
diff
changeset

636 
lemma inverse_le_iff_le_neg [simp]: 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

637 
"a < 0 \<Longrightarrow> b < 0 \<Longrightarrow> inverse a \<le> inverse b \<longleftrightarrow> b \<le> a" 
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59557
diff
changeset

638 
by (blast intro: le_imp_inverse_le_neg dest: inverse_le_imp_le_neg) 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

639 

36774  640 
lemma one_less_inverse: 
641 
"0 < a \<Longrightarrow> a < 1 \<Longrightarrow> 1 < inverse a" 

642 
using less_imp_inverse_less [of a 1, unfolded inverse_1] . 

643 

644 
lemma one_le_inverse: 

645 
"0 < a \<Longrightarrow> a \<le> 1 \<Longrightarrow> 1 \<le> inverse a" 

646 
using le_imp_inverse_le [of a 1, unfolded inverse_1] . 

647 

59546
3850a2d20f19
times_divide_eq rules are already [simp] despite of comment
haftmann
parents:
59535
diff
changeset

648 
lemma pos_le_divide_eq [field_simps]: 
3850a2d20f19
times_divide_eq rules are already [simp] despite of comment
haftmann
parents:
59535
diff
changeset

649 
assumes "0 < c" 
3850a2d20f19
times_divide_eq rules are already [simp] despite of comment
haftmann
parents:
59535
diff
changeset

650 
shows "a \<le> b / c \<longleftrightarrow> a * c \<le> b" 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

651 
proof  
59546
3850a2d20f19
times_divide_eq rules are already [simp] despite of comment
haftmann
parents:
59535
diff
changeset

652 
from assms have "a \<le> b / c \<longleftrightarrow> a * c \<le> (b / c) * c" 
3850a2d20f19
times_divide_eq rules are already [simp] despite of comment
haftmann
parents:
59535
diff
changeset

653 
using mult_le_cancel_right [of a c "b * inverse c"] by (auto simp add: field_simps) 
3850a2d20f19
times_divide_eq rules are already [simp] despite of comment
haftmann
parents:
59535
diff
changeset

654 
also have "... \<longleftrightarrow> a * c \<le> b" 
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59557
diff
changeset

655 
by (simp add: less_imp_not_eq2 [OF assms] divide_inverse mult.assoc) 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

656 
finally show ?thesis . 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

657 
qed 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

658 

59546
3850a2d20f19
times_divide_eq rules are already [simp] despite of comment
haftmann
parents:
59535
diff
changeset

659 
lemma pos_less_divide_eq [field_simps]: 
3850a2d20f19
times_divide_eq rules are already [simp] despite of comment
haftmann
parents:
59535
diff
changeset

660 
assumes "0 < c" 
3850a2d20f19
times_divide_eq rules are already [simp] despite of comment
haftmann
parents:
59535
diff
changeset

661 
shows "a < b / c \<longleftrightarrow> a * c < b" 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

662 
proof  
59546
3850a2d20f19
times_divide_eq rules are already [simp] despite of comment
haftmann
parents:
59535
diff
changeset

663 
from assms have "a < b / c \<longleftrightarrow> a * c < (b / c) * c" 
3850a2d20f19
times_divide_eq rules are already [simp] despite of comment
haftmann
parents:
59535
diff
changeset

664 
using mult_less_cancel_right [of a c "b / c"] by auto 
3850a2d20f19
times_divide_eq rules are already [simp] despite of comment
haftmann
parents:
59535
diff
changeset

665 
also have "... = (a*c < b)" 
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59557
diff
changeset

666 
by (simp add: less_imp_not_eq2 [OF assms] divide_inverse mult.assoc) 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

667 
finally show ?thesis . 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

668 
qed 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

669 

59546
3850a2d20f19
times_divide_eq rules are already [simp] despite of comment
haftmann
parents:
59535
diff
changeset

670 
lemma neg_less_divide_eq [field_simps]: 
3850a2d20f19
times_divide_eq rules are already [simp] despite of comment
haftmann
parents:
59535
diff
changeset

671 
assumes "c < 0" 
3850a2d20f19
times_divide_eq rules are already [simp] despite of comment
haftmann
parents:
59535
diff
changeset

672 
shows "a < b / c \<longleftrightarrow> b < a * c" 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

673 
proof  
59546
3850a2d20f19
times_divide_eq rules are already [simp] despite of comment
haftmann
parents:
59535
diff
changeset

674 
from assms have "a < b / c \<longleftrightarrow> (b / c) * c < a * c" 
3850a2d20f19
times_divide_eq rules are already [simp] despite of comment
haftmann
parents:
59535
diff
changeset

675 
using mult_less_cancel_right [of "b / c" c a] by auto 
3850a2d20f19
times_divide_eq rules are already [simp] despite of comment
haftmann
parents:
59535
diff
changeset

676 
also have "... \<longleftrightarrow> b < a * c" 
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59557
diff
changeset

677 
by (simp add: less_imp_not_eq [OF assms] divide_inverse mult.assoc) 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

678 
finally show ?thesis . 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

679 
qed 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

680 

59546
3850a2d20f19
times_divide_eq rules are already [simp] despite of comment
haftmann
parents:
59535
diff
changeset

681 
lemma neg_le_divide_eq [field_simps]: 
3850a2d20f19
times_divide_eq rules are already [simp] despite of comment
haftmann
parents:
59535
diff
changeset

682 
assumes "c < 0" 
3850a2d20f19
times_divide_eq rules are already [simp] despite of comment
haftmann
parents:
59535
diff
changeset

683 
shows "a \<le> b / c \<longleftrightarrow> b \<le> a * c" 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

684 
proof  
59546
3850a2d20f19
times_divide_eq rules are already [simp] despite of comment
haftmann
parents:
59535
diff
changeset

685 
from assms have "a \<le> b / c \<longleftrightarrow> (b / c) * c \<le> a * c" 
3850a2d20f19
times_divide_eq rules are already [simp] despite of comment
haftmann
parents:
59535
diff
changeset

686 
using mult_le_cancel_right [of "b * inverse c" c a] by (auto simp add: field_simps) 
3850a2d20f19
times_divide_eq rules are already [simp] despite of comment
haftmann
parents:
59535
diff
changeset

687 
also have "... \<longleftrightarrow> b \<le> a * c" 
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59557
diff
changeset

688 
by (simp add: less_imp_not_eq [OF assms] divide_inverse mult.assoc) 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

689 
finally show ?thesis . 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

690 
qed 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

691 

59546
3850a2d20f19
times_divide_eq rules are already [simp] despite of comment
haftmann
parents:
59535
diff
changeset

692 
lemma pos_divide_le_eq [field_simps]: 
3850a2d20f19
times_divide_eq rules are already [simp] despite of comment
haftmann
parents:
59535
diff
changeset

693 
assumes "0 < c" 
3850a2d20f19
times_divide_eq rules are already [simp] despite of comment
haftmann
parents:
59535
diff
changeset

694 
shows "b / c \<le> a \<longleftrightarrow> b \<le> a * c" 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

695 
proof  
59546
3850a2d20f19
times_divide_eq rules are already [simp] despite of comment
haftmann
parents:
59535
diff
changeset

696 
from assms have "b / c \<le> a \<longleftrightarrow> (b / c) * c \<le> a * c" 
3850a2d20f19
times_divide_eq rules are already [simp] despite of comment
haftmann
parents:
59535
diff
changeset

697 
using mult_le_cancel_right [of "b / c" c a] by auto 
3850a2d20f19
times_divide_eq rules are already [simp] despite of comment
haftmann
parents:
59535
diff
changeset

698 
also have "... \<longleftrightarrow> b \<le> a * c" 
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59557
diff
changeset

699 
by (simp add: less_imp_not_eq2 [OF assms] divide_inverse mult.assoc) 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

700 
finally show ?thesis . 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

701 
qed 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

702 

59546
3850a2d20f19
times_divide_eq rules are already [simp] despite of comment
haftmann
parents:
59535
diff
changeset

703 
lemma pos_divide_less_eq [field_simps]: 
3850a2d20f19
times_divide_eq rules are already [simp] despite of comment
haftmann
parents:
59535
diff
changeset

704 
assumes "0 < c" 
3850a2d20f19
times_divide_eq rules are already [simp] despite of comment
haftmann
parents:
59535
diff
changeset

705 
shows "b / c < a \<longleftrightarrow> b < a * c" 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

706 
proof  
59546
3850a2d20f19
times_divide_eq rules are already [simp] despite of comment
haftmann
parents:
59535
diff
changeset

707 
from assms have "b / c < a \<longleftrightarrow> (b / c) * c < a * c" 
3850a2d20f19
times_divide_eq rules are already [simp] despite of comment
haftmann
parents:
59535
diff
changeset

708 
using mult_less_cancel_right [of "b / c" c a] by auto 
3850a2d20f19
times_divide_eq rules are already [simp] despite of comment
haftmann
parents:
59535
diff
changeset

709 
also have "... \<longleftrightarrow> b < a * c" 
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59557
diff
changeset

710 
by (simp add: less_imp_not_eq2 [OF assms] divide_inverse mult.assoc) 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

711 
finally show ?thesis . 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

712 
qed 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

713 

59546
3850a2d20f19
times_divide_eq rules are already [simp] despite of comment
haftmann
parents:
59535
diff
changeset

714 
lemma neg_divide_le_eq [field_simps]: 
3850a2d20f19
times_divide_eq rules are already [simp] despite of comment
haftmann
parents:
59535
diff
changeset

715 
assumes "c < 0" 
3850a2d20f19
times_divide_eq rules are already [simp] despite of comment
haftmann
parents:
59535
diff
changeset

716 
shows "b / c \<le> a \<longleftrightarrow> a * c \<le> b" 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

717 
proof  
59546
3850a2d20f19
times_divide_eq rules are already [simp] despite of comment
haftmann
parents:
59535
diff
changeset

718 
from assms have "b / c \<le> a \<longleftrightarrow> a * c \<le> (b / c) * c" 
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59557
diff
changeset

719 
using mult_le_cancel_right [of a c "b / c"] by auto 
59546
3850a2d20f19
times_divide_eq rules are already [simp] despite of comment
haftmann
parents:
59535
diff
changeset

720 
also have "... \<longleftrightarrow> a * c \<le> b" 
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59557
diff
changeset

721 
by (simp add: less_imp_not_eq [OF assms] divide_inverse mult.assoc) 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

722 
finally show ?thesis . 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

723 
qed 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

724 

59546
3850a2d20f19
times_divide_eq rules are already [simp] despite of comment
haftmann
parents:
59535
diff
changeset

725 
lemma neg_divide_less_eq [field_simps]: 
3850a2d20f19
times_divide_eq rules are already [simp] despite of comment
haftmann
parents:
59535
diff
changeset

726 
assumes "c < 0" 
3850a2d20f19
times_divide_eq rules are already [simp] despite of comment
haftmann
parents:
59535
diff
changeset

727 
shows "b / c < a \<longleftrightarrow> a * c < b" 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

728 
proof  
59546
3850a2d20f19
times_divide_eq rules are already [simp] despite of comment
haftmann
parents:
59535
diff
changeset

729 
from assms have "b / c < a \<longleftrightarrow> a * c < b / c * c" 
3850a2d20f19
times_divide_eq rules are already [simp] despite of comment
haftmann
parents:
59535
diff
changeset

730 
using mult_less_cancel_right [of a c "b / c"] by auto 
3850a2d20f19
times_divide_eq rules are already [simp] despite of comment
haftmann
parents:
59535
diff
changeset

731 
also have "... \<longleftrightarrow> a * c < b" 
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59557
diff
changeset

732 
by (simp add: less_imp_not_eq [OF assms] divide_inverse mult.assoc) 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

733 
finally show ?thesis . 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

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

735 

60758  736 
text\<open>The following @{text field_simps} rules are necessary, as minus is always moved atop of 
737 
division but we want to get rid of division.\<close> 

56480
093ea91498e6
field_simps: better support for negation and division, and power
hoelzl
parents:
56479
diff
changeset

738 

093ea91498e6
field_simps: better support for negation and division, and power
hoelzl
parents:
56479
diff
changeset

739 
lemma pos_le_minus_divide_eq [field_simps]: "0 < c \<Longrightarrow> a \<le>  (b / c) \<longleftrightarrow> a * c \<le>  b" 
093ea91498e6
field_simps: better support for negation and division, and power
hoelzl
parents:
56479
diff
changeset

740 
unfolding minus_divide_left by (rule pos_le_divide_eq) 
093ea91498e6
field_simps: better support for negation and division, and power
hoelzl
parents:
56479
diff
changeset

741 

093ea91498e6
field_simps: better support for negation and division, and power
hoelzl
parents:
56479
diff
changeset

742 
lemma neg_le_minus_divide_eq [field_simps]: "c < 0 \<Longrightarrow> a \<le>  (b / c) \<longleftrightarrow>  b \<le> a * c" 
093ea91498e6
field_simps: better support for negation and division, and power
hoelzl
parents:
56479
diff
changeset

743 
unfolding minus_divide_left by (rule neg_le_divide_eq) 
093ea91498e6
field_simps: better support for negation and division, and power
hoelzl
parents:
56479
diff
changeset

744 

093ea91498e6
field_simps: better support for negation and division, and power
hoelzl
parents:
56479
diff
changeset

745 
lemma pos_less_minus_divide_eq [field_simps]: "0 < c \<Longrightarrow> a <  (b / c) \<longleftrightarrow> a * c <  b" 
093ea91498e6
field_simps: better support for negation and division, and power
hoelzl
parents:
56479
diff
changeset

746 
unfolding minus_divide_left by (rule pos_less_divide_eq) 
093ea91498e6
field_simps: better support for negation and division, and power
hoelzl
parents:
56479
diff
changeset

747 

093ea91498e6
field_simps: better support for negation and division, and power
hoelzl
parents:
56479
diff
changeset

748 
lemma neg_less_minus_divide_eq [field_simps]: "c < 0 \<Longrightarrow> a <  (b / c) \<longleftrightarrow>  b < a * c" 
093ea91498e6
field_simps: better support for negation and division, and power
hoelzl
parents:
56479
diff
changeset

749 
unfolding minus_divide_left by (rule neg_less_divide_eq) 
093ea91498e6
field_simps: better support for negation and division, and power
hoelzl
parents:
56479
diff
changeset

750 

093ea91498e6
field_simps: better support for negation and division, and power
hoelzl
parents:
56479
diff
changeset

751 
lemma pos_minus_divide_less_eq [field_simps]: "0 < c \<Longrightarrow>  (b / c) < a \<longleftrightarrow>  b < a * c" 
093ea91498e6
field_simps: better support for negation and division, and power
hoelzl
parents:
56479
diff
changeset

752 
unfolding minus_divide_left by (rule pos_divide_less_eq) 
093ea91498e6
field_simps: better support for negation and division, and power
hoelzl
parents:
56479
diff
changeset

753 

093ea91498e6
field_simps: better support for negation and division, and power
hoelzl
parents:
56479
diff
changeset

754 
lemma neg_minus_divide_less_eq [field_simps]: "c < 0 \<Longrightarrow>  (b / c) < a \<longleftrightarrow> a * c <  b" 
093ea91498e6
field_simps: better support for negation and division, and power
hoelzl
parents:
56479
diff
changeset

755 
unfolding minus_divide_left by (rule neg_divide_less_eq) 
093ea91498e6
field_simps: better support for negation and division, and power
hoelzl
parents:
56479
diff
changeset

756 

093ea91498e6
field_simps: better support for negation and division, and power
hoelzl
parents:
56479
diff
changeset

757 
lemma pos_minus_divide_le_eq [field_simps]: "0 < c \<Longrightarrow>  (b / c) \<le> a \<longleftrightarrow>  b \<le> a * c" 
093ea91498e6
field_simps: better support for negation and division, and power
hoelzl
parents:
56479
diff
changeset

758 
unfolding minus_divide_left by (rule pos_divide_le_eq) 
093ea91498e6
field_simps: better support for negation and division, and power
hoelzl
parents:
56479
diff
changeset

759 

093ea91498e6
field_simps: better support for negation and division, and power
hoelzl
parents:
56479
diff
changeset

760 
lemma neg_minus_divide_le_eq [field_simps]: "c < 0 \<Longrightarrow>  (b / c) \<le> a \<longleftrightarrow> a * c \<le>  b" 
093ea91498e6
field_simps: better support for negation and division, and power
hoelzl
parents:
56479
diff
changeset

761 
unfolding minus_divide_left by (rule neg_divide_le_eq) 
093ea91498e6
field_simps: better support for negation and division, and power
hoelzl
parents:
56479
diff
changeset

762 

56365
713f9b9a7e51
New theorems for extracting quotients
paulson <lp15@cam.ac.uk>
parents:
55718
diff
changeset

763 
lemma frac_less_eq: 
713f9b9a7e51
New theorems for extracting quotients
paulson <lp15@cam.ac.uk>
parents:
55718
diff
changeset

764 
"y \<noteq> 0 \<Longrightarrow> z \<noteq> 0 \<Longrightarrow> x / y < w / z \<longleftrightarrow> (x * z  w * y) / (y * z) < 0" 
713f9b9a7e51
New theorems for extracting quotients
paulson <lp15@cam.ac.uk>
parents:
55718
diff
changeset

765 
by (subst less_iff_diff_less_0) (simp add: diff_frac_eq ) 
713f9b9a7e51
New theorems for extracting quotients
paulson <lp15@cam.ac.uk>
parents:
55718
diff
changeset

766 

713f9b9a7e51
New theorems for extracting quotients
paulson <lp15@cam.ac.uk>
parents:
55718
diff
changeset

767 
lemma frac_le_eq: 
713f9b9a7e51
New theorems for extracting quotients
paulson <lp15@cam.ac.uk>
parents:
55718
diff
changeset

768 
"y \<noteq> 0 \<Longrightarrow> z \<noteq> 0 \<Longrightarrow> x / y \<le> w / z \<longleftrightarrow> (x * z  w * y) / (y * z) \<le> 0" 
713f9b9a7e51
New theorems for extracting quotients
paulson <lp15@cam.ac.uk>
parents:
55718
diff
changeset

769 
by (subst le_iff_diff_le_0) (simp add: diff_frac_eq ) 
713f9b9a7e51
New theorems for extracting quotients
paulson <lp15@cam.ac.uk>
parents:
55718
diff
changeset

770 

60758  771 
text\<open>Lemmas @{text sign_simps} is a first attempt to automate proofs 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

772 
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

773 
sign_simps} to @{text field_simps} because the former can lead to case 
60758  774 
explosions.\<close> 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

775 

54147
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
53374
diff
changeset

776 
lemmas sign_simps = algebra_simps zero_less_mult_iff mult_less_0_iff 
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

777 

54147
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
53374
diff
changeset

778 
lemmas (in ) sign_simps = algebra_simps zero_less_mult_iff mult_less_0_iff 
36301
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 
(* Only works once linear arithmetic is installed: 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

781 
text{*An example:*} 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

782 
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

783 
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

784 
((ab)*(cd)*(ef))/((cd)*(ef)*(ab)) < 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

785 
((ef)*(ab)*(cd))/((ef)*(ab)*(cd)) + u" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

786 
apply(subgoal_tac "(cd)*(ef)*(ab) > 0") 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

787 
prefer 2 apply(simp add:sign_simps) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

788 
apply(subgoal_tac "(cd)*(ef)*(ab)*u > 0") 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

789 
prefer 2 apply(simp add:sign_simps) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

790 
apply(simp add:field_simps) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

791 
done 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

792 
*) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

793 

56541  794 
lemma divide_pos_pos[simp]: 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

795 
"0 < x ==> 0 < y ==> 0 < x / y" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

796 
by(simp add:field_simps) 
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 divide_nonneg_pos: 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

799 
"0 <= x ==> 0 < y ==> 0 <= 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 divide_neg_pos: 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

803 
"x < 0 ==> 0 < y ==> x / y < 0" 
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 divide_nonpos_pos: 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

807 
"x <= 0 ==> 0 < y ==> x / y <= 0" 
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 divide_pos_neg: 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

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

812 
by(simp add:field_simps) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

813 

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

814 
lemma divide_nonneg_neg: 
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59557
diff
changeset

815 
"0 <= x ==> y < 0 ==> x / y <= 0" 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

816 
by(simp add:field_simps) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

817 

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

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

819 
"x < 0 ==> y < 0 ==> 0 < x / y" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

820 
by(simp add:field_simps) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

821 

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

822 
lemma divide_nonpos_neg: 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

823 
"x <= 0 ==> y < 0 ==> 0 <= x / y" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

824 
by(simp add:field_simps) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

825 

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

826 
lemma divide_strict_right_mono: 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

827 
"[a < b; 0 < c] ==> a / c < b / c" 
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59557
diff
changeset

828 
by (simp add: less_imp_not_eq2 divide_inverse mult_strict_right_mono 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

829 
positive_imp_inverse_positive) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

830 

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

831 

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

832 
lemma divide_strict_right_mono_neg: 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

833 
"[b < a; c < 0] ==> a / c < b / c" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

834 
apply (drule divide_strict_right_mono [of _ _ "c"], simp) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

835 
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

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

837 

60758  838 
text\<open>The last premise ensures that @{term a} and @{term b} 
839 
have the same sign\<close> 

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

840 
lemma divide_strict_left_mono: 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

841 
"[b < a; 0 < c; 0 < a*b] ==> c / a < c / b" 
44921  842 
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

843 

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

844 
lemma divide_left_mono: 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

845 
"[b \<le> a; 0 \<le> c; 0 < a*b] ==> c / a \<le> c / b" 
44921  846 
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

847 

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

848 
lemma divide_strict_left_mono_neg: 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

849 
"[a < b; c < 0; 0 < a*b] ==> c / a < c / b" 
44921  850 
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

851 

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

852 
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

853 
x / y <= z" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

854 
by (subst pos_divide_le_eq, assumption+) 
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 mult_imp_le_div_pos: "0 < y ==> z * y <= x ==> 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

857 
z <= x / y" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

858 
by(simp add:field_simps) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

859 

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

860 
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

861 
x / y < z" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

862 
by(simp add:field_simps) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

863 

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

864 
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

865 
z < x / y" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

866 
by(simp add:field_simps) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

867 

59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59557
diff
changeset

868 
lemma frac_le: "0 <= x ==> 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

869 
x <= y ==> 0 < w ==> w <= z ==> x / z <= y / w" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

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

871 
apply simp 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

872 
apply (subst times_divide_eq_left) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

873 
apply (rule mult_imp_le_div_pos, assumption) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

874 
apply (rule mult_mono) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

875 
apply simp_all 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

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

877 

59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59557
diff
changeset

878 
lemma frac_less: "0 <= x ==> 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

879 
x < y ==> 0 < w ==> w <= z ==> x / z < y / w" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

880 
apply (rule mult_imp_div_pos_less) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

881 
apply simp 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

882 
apply (subst times_divide_eq_left) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

883 
apply (rule mult_imp_less_div_pos, assumption) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

884 
apply (erule mult_less_le_imp_less) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

885 
apply simp_all 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

886 
done 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

887 

59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59557
diff
changeset

888 
lemma frac_less2: "0 < x ==> 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

889 
x <= y ==> 0 < w ==> w < z ==> x / z < y / w" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

890 
apply (rule mult_imp_div_pos_less) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

891 
apply simp_all 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

892 
apply (rule mult_imp_less_div_pos, assumption) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

893 
apply (erule mult_le_less_imp_less) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

894 
apply simp_all 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

895 
done 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

896 

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

897 
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

898 
by (simp add: field_simps zero_less_two) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

899 

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

900 
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

901 
by (simp add: field_simps zero_less_two) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

902 

53215
5e47c31c6f7c
renamed typeclass dense_linorder to unbounded_dense_linorder
hoelzl
parents:
52435
diff
changeset

903 
subclass unbounded_dense_linorder 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

904 
proof 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

905 
fix x y :: 'a 
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59557
diff
changeset

906 
from less_add_one show "\<exists>y. x < y" .. 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

907 
from less_add_one have "x + ( 1) < (x + 1) + ( 1)" by (rule add_strict_right_mono) 
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54147
diff
changeset

908 
then have "x  1 < x + 1  1" by simp 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

909 
then have "x  1 < x" by (simp add: algebra_simps) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

910 
then show "\<exists>y. y < x" .. 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

911 
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

912 
qed 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

913 

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

914 
lemma nonzero_abs_inverse: 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

915 
"a \<noteq> 0 ==> \<bar>inverse a\<bar> = inverse \<bar>a\<bar>" 
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59557
diff
changeset

916 
apply (auto simp add: neq_iff abs_if nonzero_inverse_minus_eq 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

917 
negative_imp_inverse_negative) 
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59557
diff
changeset

918 
apply (blast intro: positive_imp_inverse_positive elim: less_asym) 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

919 
done 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

920 

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

921 
lemma nonzero_abs_divide: 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

922 
"b \<noteq> 0 ==> \<bar>a / b\<bar> = \<bar>a\<bar> / \<bar>b\<bar>" 
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59557
diff
changeset

923 
by (simp add: divide_inverse abs_mult nonzero_abs_inverse) 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

924 

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

925 
lemma field_le_epsilon: 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

926 
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

927 
shows "x \<le> y" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

928 
proof (rule dense_le) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

929 
fix t assume "t < x" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

930 
hence "0 < x  t" by (simp add: less_diff_eq) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

931 
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

932 
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

933 
then show "t \<le> y" by (simp add: algebra_simps) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

934 
qed 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

935 

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

936 
lemma inverse_positive_iff_positive [simp]: 
36409  937 
"(0 < inverse a) = (0 < a)" 
21328  938 
apply (cases "a = 0", simp) 
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

939 
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

940 
done 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

941 

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

942 
lemma inverse_negative_iff_negative [simp]: 
36409  943 
"(inverse a < 0) = (a < 0)" 
21328  944 
apply (cases "a = 0", simp) 
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

945 
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

946 
done 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

947 

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

948 
lemma inverse_nonnegative_iff_nonnegative [simp]: 
36409  949 
"0 \<le> inverse a \<longleftrightarrow> 0 \<le> a" 
950 
by (simp add: not_less [symmetric]) 

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

951 

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

952 
lemma inverse_nonpositive_iff_nonpositive [simp]: 
36409  953 
"inverse a \<le> 0 \<longleftrightarrow> a \<le> 0" 
954 
by (simp add: not_less [symmetric]) 

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

955 

56480
093ea91498e6
field_simps: better support for negation and division, and power
hoelzl
parents:
56479
diff
changeset

956 
lemma one_less_inverse_iff: "1 < inverse x \<longleftrightarrow> 0 < x \<and> x < 1" 
093ea91498e6
field_simps: better support for negation and division, and power
hoelzl
parents:
56479
diff
changeset

957 
using less_trans[of 1 x 0 for x] 
093ea91498e6
field_simps: better support for negation and division, and power
hoelzl
parents:
56479
diff
changeset

958 
by (cases x 0 rule: linorder_cases) (auto simp add: field_simps) 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

959 

56480
093ea91498e6
field_simps: better support for negation and division, and power
hoelzl
parents:
56479
diff
changeset

960 
lemma one_le_inverse_iff: "1 \<le> inverse x \<longleftrightarrow> 0 < x \<and> x \<le> 1" 
36409  961 
proof (cases "x = 1") 
962 
case True then show ?thesis by simp 

963 
next 

964 
case False then have "inverse x \<noteq> 1" by simp 

965 
then have "1 \<noteq> inverse x" by blast 

966 
then have "1 \<le> inverse x \<longleftrightarrow> 1 < inverse x" by (simp add: le_less) 

967 
with False show ?thesis by (auto simp add: one_less_inverse_iff) 

968 
qed 

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

969 

56480
093ea91498e6
field_simps: better support for negation and division, and power
hoelzl
parents:
56479
diff
changeset

970 
lemma inverse_less_1_iff: "inverse x < 1 \<longleftrightarrow> x \<le> 0 \<or> 1 < x" 
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59557
diff
changeset

971 
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

972 

56480
093ea91498e6
field_simps: better support for negation and division, and power
hoelzl
parents:
56479
diff
changeset

973 
lemma inverse_le_1_iff: "inverse x \<le> 1 \<longleftrightarrow> x \<le> 0 \<or> 1 \<le> x" 
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59557
diff
changeset

974 
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

975 

56481  976 
lemma [divide_simps]: 
56480
093ea91498e6
field_simps: better support for negation and division, and power
hoelzl
parents:
56479
diff
changeset

977 
shows le_divide_eq: "a \<le> b / c \<longleftrightarrow> (if 0 < c then a * c \<le> b else if c < 0 then b \<le> a * c else a \<le> 0)" 
093ea91498e6
field_simps: better support for negation and division, and power
hoelzl
parents:
56479
diff
changeset

978 
and divide_le_eq: "b / c \<le> a \<longleftrightarrow> (if 0 < c then b \<le> a * c else if c < 0 then a * c \<le> b else 0 \<le> a)" 
093ea91498e6
field_simps: better support for negation and division, and power
hoelzl
parents:
56479
diff
changeset

979 
and less_divide_eq: "a < b / c \<longleftrightarrow> (if 0 < c then a * c < b else if c < 0 then b < a * c else a < 0)" 
093ea91498e6
field_simps: better support for negation and division, and power
hoelzl
parents:
56479
diff
changeset

980 
and divide_less_eq: "b / c < a \<longleftrightarrow> (if 0 < c then b < a * c else if c < 0 then a * c < b else 0 < a)" 
56481  981 
and le_minus_divide_eq: "a \<le>  (b / c) \<longleftrightarrow> (if 0 < c then a * c \<le>  b else if c < 0 then  b \<le> a * c else a \<le> 0)" 
982 
and minus_divide_le_eq: " (b / c) \<le> a \<longleftrightarrow> (if 0 < c then  b \<le> a * c else if c < 0 then a * c \<le>  b else 0 \<le> a)" 

983 
and less_minus_divide_eq: "a <  (b / c) \<longleftrightarrow> (if 0 < c then a * c <  b else if c < 0 then  b < a * c else a < 0)" 

984 
and minus_divide_less_eq: " (b / c) < a \<longleftrightarrow> (if 0 < c then  b < a * c else if c < 0 then a * c <  b else 0 < a)" 

56480
093ea91498e6
field_simps: better support for negation and division, and power
hoelzl
parents:
56479
diff
changeset

985 
by (auto simp: field_simps not_less dest: antisym) 
14288  986 

60758  987 
text \<open>Division and Signs\<close> 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

988 

56480
093ea91498e6
field_simps: better support for negation and division, and power
hoelzl
parents:
56479
diff
changeset

989 
lemma 
093ea91498e6
field_simps: better support for negation and division, and power
hoelzl
parents:
56479
diff
changeset

990 
shows zero_less_divide_iff: "0 < a / b \<longleftrightarrow> 0 < a \<and> 0 < b \<or> a < 0 \<and> b < 0" 
093ea91498e6
field_simps: better support for negation and division, and power
hoelzl
parents:
56479
diff
changeset

991 
and divide_less_0_iff: "a / b < 0 \<longleftrightarrow> 0 < a \<and> b < 0 \<or> a < 0 \<and> 0 < b" 
093ea91498e6
field_simps: better support for negation and division, and power
hoelzl
parents:
56479
diff
changeset

992 
and zero_le_divide_iff: "0 \<le> a / b \<longleftrightarrow> 0 \<le> a \<and> 0 \<le> b \<or> a \<le> 0 \<and> b \<le> 0" 
093ea91498e6
field_simps: better support for negation and division, and power
hoelzl
parents:
56479
diff
changeset

993 
and divide_le_0_iff: "a / b \<le> 0 \<longleftrightarrow> 0 \<le> a \<and> b \<le> 0 \<or> a \<le> 0 \<and> 0 \<le> b" 
56481  994 
by (auto simp add: divide_simps) 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

995 

60758  996 
text \<open>Division and the Number One\<close> 
14353
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset

997 

60758  998 
text\<open>Simplify expressions equated with 1\<close> 
14353
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset

999 

56480
093ea91498e6
field_simps: better support for negation and division, and power
hoelzl
parents:
56479
diff
changeset

1000 
lemma zero_eq_1_divide_iff [simp]: "0 = 1 / a \<longleftrightarrow> a = 0" 
093ea91498e6
field_simps: better support for negation and division, and power
hoelzl
parents:
56479
diff
changeset

1001 
by (cases "a = 0") (auto simp: field_simps) 
14353
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset

1002 

56480
093ea91498e6
field_simps: better support for negation and division, and power
hoelzl
parents:
56479
diff
changeset

1003 
lemma one_divide_eq_0_iff [simp]: "1 / a = 0 \<longleftrightarrow> a = 0" 
093ea91498e6
field_simps: better support for negation and division, and power
hoelzl
parents:
56479
diff
changeset

1004 
using zero_eq_1_divide_iff[of a] by simp 
14353
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset

1005 

60758  1006 
text\<open>Simplify expressions such as @{text "0 < 1/x"} to @{text "0 < x"}\<close> 
36423  1007 

54147
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
53374
diff
changeset

1008 
lemma zero_le_divide_1_iff [simp]: 
36423  1009 
"0 \<le> 1 / a \<longleftrightarrow> 0 \<le> a" 
1010 
by (simp add: zero_le_divide_iff) 

17085  1011 

54147
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
53374
diff
changeset

1012 
lemma zero_less_divide_1_iff [simp]: 
36423  1013 
"0 < 1 / a \<longleftrightarrow> 0 < a" 
1014 
by (simp add: zero_less_divide_iff) 

1015 

54147
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
53374
diff
changeset

1016 
lemma divide_le_0_1_iff [simp]: 
36423  1017 
"1 / a \<le> 0 \<longleftrightarrow> a \<le> 0" 
1018 
by (simp add: divide_le_0_iff) 

1019 

54147
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
53374
diff
changeset

1020 
lemma divide_less_0_1_iff [simp]: 
36423  1021 
"1 / a < 0 \<longleftrightarrow> a < 0" 
1022 
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

1023 

14293  1024 
lemma divide_right_mono: 
36409  1025 
"[a \<le> b; 0 \<le> c] ==> a/c \<le> b/c" 
1026 
by (force simp add: divide_strict_right_mono le_less) 

14293  1027 

59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59557
diff
changeset

1028 
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

1029 
==> c <= 0 ==> b / c <= a / c" 
23482  1030 
apply (drule divide_right_mono [of _ _ " c"]) 
56479
91958d4b30f7
revert c1bbd3e22226, a14831ac3023, and 36489d77c484: divide_minus_left/right are again simp rules
hoelzl
parents:
56445
diff
changeset

1031 
apply auto 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

1032 
done 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

1033 

59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59557
diff
changeset

1034 
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

1035 
==> 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

1036 
apply (drule divide_left_mono [of _ _ " c"]) 
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
56571
diff
changeset

1037 
apply (auto simp add: mult.commute) 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

1038 
done 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

1039 

56480
093ea91498e6
field_simps: better support for negation and division, and power
hoelzl
parents:
56479
diff
changeset

1040 
lemma inverse_le_iff: "inverse a \<le> inverse b \<longleftrightarrow> (0 < a * b \<longrightarrow> b \<le> a) \<and> (a * b \<le> 0 \<longrightarrow> a \<le> b)" 
093ea91498e6
field_simps: better support for negation and division, and power
hoelzl
parents:
56479
diff
changeset

1041 
by (cases a 0 b 0 rule: linorder_cases[case_product linorder_cases]) 
093ea91498e6
field_simps: better support for negation and division, and power
hoelzl
parents:
56479
diff
changeset

1042 
(auto simp add: field_simps zero_less_mult_iff mult_le_0_iff) 
42904  1043 

56480
093ea91498e6
field_simps: better support for negation and division, and power
hoelzl
parents:
56479
diff
changeset

1044 
lemma inverse_less_iff: "inverse a < inverse b \<longleftrightarrow> (0 < a * b \<longrightarrow> b < a) \<and> (a * b \<le> 0 \<longrightarrow> a < b)" 
42904  1045 
by (subst less_le) (auto simp: inverse_le_iff) 
1046 

56480
093ea91498e6
field_simps: better support for negation and division, and power
hoelzl
parents:
56479
diff
changeset

1047 
lemma divide_le_cancel: "a / c \<le> b / c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)" 
42904  1048 
by (simp add: divide_inverse mult_le_cancel_right) 
1049 

56480
093ea91498e6
field_simps: better support for negation and division, and power
hoelzl
parents:
56479
diff
changeset

1050 
lemma divide_less_cancel: "a / c < b / c \<longleftrightarrow> (0 < c \<longrightarrow> a < b) \<and> (c < 0 \<longrightarrow> b < a) \<and> c \<noteq> 0" 
42904  1051 
by (auto simp add: divide_inverse mult_less_cancel_right) 
1052 

60758  1053 
text\<open>Simplify quotients that are compared with the value 1.\<close> 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

1054 

54147
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
53374
diff
changeset

1055 
lemma le_divide_eq_1: 
36409  1056 
"(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

1057 
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

1058 

54147
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
53374
diff
changeset

1059 
lemma divide_le_eq_1: 
36409  1060 
"(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

1061 
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

1062 

54147
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
53374
diff
changeset

1063 
lemma less_divide_eq_1: 
36409  1064 
"(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

1065 
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

1066 

54147
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
53374
diff
changeset

1067 
lemma divide_less_eq_1: 
36409  1068 
"(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

1069 
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

1070 

56571
f4635657d66f
added divide_nonneg_nonneg and co; made it a simp rule
hoelzl
parents:
56541
diff
changeset

1071 
lemma divide_nonneg_nonneg [simp]: 
f4635657d66f
added divide_nonneg_nonneg and co; made it a simp rule
hoelzl
parents:
56541
diff
changeset

1072 
"0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> 0 \<le> x / y" 
f4635657d66f
added divide_nonneg_nonneg and co; made it a simp rule
hoelzl
parents:
56541
diff
changeset

1073 
by (auto simp add: divide_simps) 
f4635657d66f
added divide_nonneg_nonneg and co; made it a simp rule
hoelzl
parents:
56541
diff
changeset

1074 

f4635657d66f
added divide_nonneg_nonneg and co; made it a simp rule
hoelzl
parents:
56541
diff
changeset

1075 
lemma divide_nonpos_nonpos: 
f4635657d66f
added divide_nonneg_nonneg and co; made it a simp rule
hoelzl
parents:
56541
diff
changeset

1076 
"x \<le> 0 \<Longrightarrow> y \<le> 0 \<Longrightarrow> 0 \<le> x / y" 
f4635657d66f
added divide_nonneg_nonneg and co; made it a simp rule
hoelzl
parents:
56541
diff
changeset

1077 
by (auto simp add: divide_simps) 
f4635657d66f
added divide_nonneg_nonneg and co; made it a simp rule
hoelzl
parents:
56541
diff
changeset

1078 

f4635657d66f
added divide_nonneg_nonneg and co; made it a simp rule
hoelzl
parents:
56541
diff
changeset

1079 
lemma divide_nonneg_nonpos: 
f4635657d66f
added divide_nonneg_nonneg and co; made it a simp rule
hoelzl
parents:
56541
diff
changeset

1080 
"0 \<le> x \<Longrightarrow> y \<le> 0 \<Longrightarrow> x / y \<le> 0" 
f4635657d66f
added divide_nonneg_nonneg and co; made it a simp rule
hoelzl
parents:
56541
diff
changeset

1081 
by (auto simp add: divide_simps) 
f4635657d66f
added divide_nonneg_nonneg and co; made it a simp rule
hoelzl
parents:
56541
diff
changeset

1082 

f4635657d66f
added divide_nonneg_nonneg and co; made it a simp rule
hoelzl
parents:
56541
diff
changeset

1083 
lemma divide_nonpos_nonneg: 
f4635657d66f
added divide_nonneg_nonneg and co; made it a simp rule
hoelzl
parents:
56541
diff
changeset

1084 
"x \<le> 0 \<Longrightarrow> 0 \<le> y \<Longrightarrow> x / y \<le> 0" 
f4635657d66f
added divide_nonneg_nonneg and co; made it a simp rule
hoelzl
parents:
56541
diff
changeset

1085 
by (auto simp add: divide_simps) 
23389  1086 

60758  1087 
text \<open>Conditional Simplification Rules: No Case Splits\<close> 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

1088 

54147
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
53374
diff
changeset

1089 
lemma le_divide_eq_1_pos [simp]: 
36409  1090 
"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

1091 
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

1092 

54147
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
53374
diff
changeset

1093 
lemma le_divide_eq_1_neg [simp]: 
36409  1094 
"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

1095 
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

1096 

54147
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
53374
diff
changeset

1097 
lemma divide_le_eq_1_pos [simp]: 
36409  1098 
"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

1099 
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

1100 

54147
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
53374
diff
changeset

1101 
lemma divide_le_eq_1_neg [simp]: 
36409  1102 
"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

1103 
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

1104 

54147
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
53374
diff
changeset

1105 
lemma less_divide_eq_1_pos [simp]: 
36409  1106 
"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

1107 
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

1108 

54147
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
53374
diff
changeset

1109 
lemma less_divide_eq_1_neg [simp]: 
36409  1110 
"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

1111 
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

1112 

54147
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
53374
diff
changeset

1113 
lemma divide_less_eq_1_pos [simp]: 
36409  1114 
"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

1115 
by (auto simp add: divide_less_eq) 
bb99c2e705ca
tidied, and added missing thm divide_less_eq_1_neg
paulson
parents:
18623
diff
changeset

1116 

54147
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
53374
diff
changeset

1117 
lemma divide_less_eq_1_neg [simp]: 
36409  1118 
"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

1119 
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

1120 

54147
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
53374
diff
changeset

1121 
lemma eq_divide_eq_1 [simp]: 
36409  1122 
"(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

1123 
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

1124 

54147
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
53374
diff
changeset

1125 
lemma divide_eq_eq_1 [simp]: 
36409  1126 
"(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

1127 
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

1128 

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

1129 
lemma abs_inverse [simp]: 
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59557
diff
changeset

1130 
"\<bar>inverse a\<bar> = 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

1131 
inverse \<bar>a\<bar>" 
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59557
diff
changeset

1132 
apply (cases "a=0", simp) 
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59557
diff
changeset

1133 
apply (simp add: nonzero_abs_inverse) 
14294
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset

1134 
done 
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset

1135 

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

1136 
lemma abs_divide [simp]: 
36409  1137 
"\<bar>a / b\<bar> = \<bar>a\<bar> / \<bar>b\<bar>" 
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59557
diff
changeset

1138 
apply (cases "b=0", simp) 
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59557
diff
changeset

1139 
apply (simp add: nonzero_abs_divide) 
14294
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset

1140 
done 
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset

1141 

59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59557
diff
changeset

1142 
lemma abs_div_pos: "0 < y ==> 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

1143 
\<bar>x\<bar> / y = \<bar>x / y\<bar>" 
25304
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset

1144 
apply (subst abs_divide) 
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset

1145 
apply (simp add: order_less_imp_le) 
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset

1146 
done 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

1147 

59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59557
diff
changeset

1148 
lemma zero_le_divide_abs_iff [simp]: "(0 \<le> a / abs b) = (0 \<le> a  b = 0)" 
55718
34618f031ba9
A few lemmas about summations, etc.
paulson <lp15@cam.ac.uk>
parents:
54230
diff
changeset

1149 
by (auto simp: zero_le_divide_iff) 
34618f031ba9
A few lemmas about summations, etc.
paulson <lp15@cam.ac.uk>
parents:
54230
diff
changeset

1150 

59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59557
diff
changeset

1151 
lemma divide_le_0_abs_iff [simp]: "(a / abs b \<le> 0) = (a \<le> 0  b = 0)" 
55718
34618f031ba9
A few lemmas about summations, etc.
paulson <lp15@cam.ac.uk>
parents:
54230
diff
changeset

1152 
by (auto simp: divide_le_0_iff) 
34618f031ba9
A few lemmas about summations, etc.
paulson <lp15@cam.ac.uk>
parents:
54230
diff
changeset

1153 

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

1154 
lemma field_le_mult_one_interval: 
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35216
diff
changeset

1155 
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

1156 
shows "x \<le> y" 
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35216
diff
changeset

1157 
proof (cases "0 < x") 
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35216
diff
changeset

1158 
assume "0 < x" 
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35216
diff
changeset

1159 
thus ?thesis 
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35216
diff
changeset

1160 
using dense_le_bounded[of 0 1 "y/x"] * 
60758  1161 
unfolding le_divide_eq if_P[OF \<open>0 < x\<close>] by simp 
35579
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35216
diff
changeset

1162 
next 
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35216
diff
changeset

1163 
assume "\<not>0 < x" hence "x \<le> 0" by simp 
61076  1164 
obtain s::'a where s: "0 < s" "s < 1" using dense[of 0 "1::'a"] by auto 
60758  1165 
hence "x \<le> s * x" using mult_le_cancel_right[of 1 x s] \<open>x \<le> 0\<close> by auto 
35579
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35216
diff
changeset

1166 
also note *[OF s] 
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35216
diff
changeset

1167 
finally show ?thesis . 
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35216
diff
changeset

1168 
qed 
35090
88cc65ae046e
moved lemma field_le_epsilon from Real.thy to Fields.thy
haftmann
parents:
35084
diff
changeset

1169 

36409  1170 
end 
1171 

59557  1172 
hide_fact (open) field_inverse field_divide_inverse field_inverse_zero 
1173 

52435
6646bb548c6b
migration from code_(consttypeclassinstance) to code_printing and from code_module to code_identifier
haftmann
parents:
44921
diff
changeset

1174 
code_identifier 
6646bb548c6b
migration from code_(consttypeclassinstance) to code_printing and from code_module to code_identifier
haftmann
parents:
44921
diff
changeset

1175 
code_module Fields \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith 
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59557
diff
changeset

1176 

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

1177 
end 