author  paulson 
Tue, 02 Dec 2003 11:48:15 +0100  
changeset 14270  342451d763f9 
parent 14269  502a7c95de73 
child 14272  5efbb548107d 
permissions  rwrr 
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(* Title: HOL/Ring_and_Field.thy 
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ID: $Id$ 
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Author: Gertrud Bauer and Markus Wenzel, TU Muenchen 
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Lawrence C Paulson, University of Cambridge 
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License: GPL (GNU GENERAL PUBLIC LICENSE) 
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*) 
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header {* 
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\title{Ring and field structures} 
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\author{Gertrud Bauer and Markus Wenzel} 
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*} 
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theory Ring_and_Field = Inductive: 
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text{*Lemmas and extension to semirings by L. C. Paulson*} 
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subsection {* Abstract algebraic structures *} 
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axclass semiring \<subseteq> zero, one, plus, times 
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add_assoc: "(a + b) + c = a + (b + c)" 
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add_commute: "a + b = b + a" 
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left_zero [simp]: "0 + a = a" 
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mult_assoc: "(a * b) * c = a * (b * c)" 
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mult_commute: "a * b = b * a" 
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More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
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mult_1 [simp]: "1 * a = a" 
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left_distrib: "(a + b) * c = a * c + b * c" 
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zero_neq_one [simp]: "0 \<noteq> 1" 
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axclass ring \<subseteq> semiring, minus 
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left_minus [simp]: " a + a = 0" 
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diff_minus: "a  b = a + (b)" 
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axclass ordered_semiring \<subseteq> semiring, linorder 
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add_left_mono: "a \<le> b ==> c + a \<le> c + b" 
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mult_strict_left_mono: "a < b ==> 0 < c ==> c * a < c * b" 
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axclass ordered_ring \<subseteq> ordered_semiring, ring 
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abs_if: "\<bar>a\<bar> = (if a < 0 then a else a)" 
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axclass field \<subseteq> ring, inverse 
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left_inverse [simp]: "a \<noteq> 0 ==> inverse a * a = 1" 
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divide_inverse: "b \<noteq> 0 ==> a / b = a * inverse b" 
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axclass ordered_field \<subseteq> ordered_ring, field 
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axclass division_by_zero \<subseteq> zero, inverse 
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inverse_zero [simp]: "inverse 0 = 0" 
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divide_zero [simp]: "a / 0 = 0" 
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14270  53 
subsection {* Derived Rules for Addition *} 
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lemma right_zero [simp]: "a + 0 = (a::'a::semiring)" 
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56 
proof  
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have "a + 0 = 0 + a" by (simp only: add_commute) 
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also have "... = a" by simp 
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finally show ?thesis . 
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qed 
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lemma add_left_commute: "a + (b + c) = b + (a + (c::'a::semiring))" 
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by (rule mk_left_commute [of "op +", OF add_assoc add_commute]) 
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theorems add_ac = add_assoc add_commute add_left_commute 
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lemma right_minus [simp]: "a + (a::'a::ring) = 0" 
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proof  
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have "a + a = a + a" by (simp add: add_ac) 
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also have "... = 0" by simp 
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finally show ?thesis . 
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qed 
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lemma right_minus_eq: "(a  b = 0) = (a = (b::'a::ring))" 
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75 
proof 
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have "a = a  b + b" by (simp add: diff_minus add_ac) 
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also assume "a  b = 0" 
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78 
finally show "a = b" by simp 
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next 
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assume "a = b" 
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thus "a  b = 0" by (simp add: diff_minus) 
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82 
qed 
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83 

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lemma add_left_cancel [simp]: 
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"(a + b = a + c) = (b = (c::'a::ring))" 
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86 
proof 
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87 
assume eq: "a + b = a + c" 
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88 
hence "(a + a) + b = (a + a) + c" 
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89 
by (simp only: eq add_assoc) 
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thus "b = c" by simp 
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91 
next 
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92 
assume eq: "b = c" 
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thus "a + b = a + c" by simp 
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94 
qed 
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95 

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lemma add_right_cancel [simp]: 
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"(b + a = c + a) = (b = (c::'a::ring))" 
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98 
by (simp add: add_commute) 
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99 

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lemma minus_minus [simp]: " ( (a::'a::ring)) = a" 
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101 
proof (rule add_left_cancel [of "a", THEN iffD1]) 
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102 
show "(a + (a) = a + a)" 
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103 
by simp 
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104 
qed 
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105 

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lemma equals_zero_I: "a+b = 0 ==> a = (b::'a::ring)" 
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107 
apply (rule right_minus_eq [THEN iffD1, symmetric]) 
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108 
apply (simp add: diff_minus add_commute) 
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109 
done 
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110 

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lemma minus_zero [simp]: " 0 = (0::'a::ring)" 
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112 
by (simp add: equals_zero_I) 
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113 

14270  114 
lemma diff_self [simp]: "a  (a::'a::ring) = 0" 
115 
by (simp add: diff_minus) 

116 

117 
lemma diff_0 [simp]: "(0::'a::ring)  a = a" 

118 
by (simp add: diff_minus) 

119 

120 
lemma diff_0_right [simp]: "a  (0::'a::ring) = a" 

121 
by (simp add: diff_minus) 

122 

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lemma neg_equal_iff_equal [simp]: "(a = b) = (a = (b::'a::ring))" 
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124 
proof 
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125 
assume " a =  b" 
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126 
hence " ( a) =  ( b)" 
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127 
by simp 
14266  128 
thus "a=b" by simp 
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129 
next 
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130 
assume "a=b" 
14266  131 
thus "a = b" by simp 
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132 
qed 
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133 

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lemma neg_equal_0_iff_equal [simp]: "(a = 0) = (a = (0::'a::ring))" 
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135 
by (subst neg_equal_iff_equal [symmetric], simp) 
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136 

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137 
lemma neg_0_equal_iff_equal [simp]: "(0 = a) = (0 = (a::'a::ring))" 
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138 
by (subst neg_equal_iff_equal [symmetric], simp) 
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139 

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140 

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141 
subsection {* Derived rules for multiplication *} 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

142 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

143 
lemma mult_1_right [simp]: "a * (1::'a::semiring) = a" 
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

144 
proof  
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

145 
have "a * 1 = 1 * a" by (simp add: mult_commute) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

146 
also have "... = a" by simp 
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

147 
finally show ?thesis . 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

148 
qed 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

149 

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

150 
lemma mult_left_commute: "a * (b * c) = b * (a * (c::'a::semiring))" 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

151 
by (rule mk_left_commute [of "op *", OF mult_assoc mult_commute]) 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

152 

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

153 
theorems mult_ac = mult_assoc mult_commute mult_left_commute 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

154 

14269  155 
lemma right_inverse [simp]: 
14270  156 
assumes not0: "a \<noteq> 0" shows "a * inverse (a::'a::field) = 1" 
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

157 
proof  
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

158 
have "a * inverse a = inverse a * a" by (simp add: mult_ac) 
14269  159 
also have "... = 1" using not0 by simp 
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

160 
finally show ?thesis . 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

161 
qed 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

162 

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

163 
lemma right_inverse_eq: "b \<noteq> 0 ==> (a / b = 1) = (a = (b::'a::field))" 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

164 
proof 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

165 
assume neq: "b \<noteq> 0" 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

166 
{ 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

167 
hence "a = (a / b) * b" by (simp add: divide_inverse mult_ac) 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

168 
also assume "a / b = 1" 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

169 
finally show "a = b" by simp 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

170 
next 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

171 
assume "a = b" 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

172 
with neq show "a / b = 1" by (simp add: divide_inverse) 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

173 
} 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

174 
qed 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

175 

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

176 
lemma divide_self [simp]: "a \<noteq> 0 ==> a / (a::'a::field) = 1" 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

177 
by (simp add: divide_inverse) 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

178 

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

179 
lemma mult_left_zero [simp]: "0 * a = (0::'a::ring)" 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

180 
proof  
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

181 
have "0*a + 0*a = 0*a + 0" 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

182 
by (simp add: left_distrib [symmetric]) 
14266  183 
thus ?thesis by (simp only: add_left_cancel) 
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

184 
qed 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

185 

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

186 
lemma mult_right_zero [simp]: "a * 0 = (0::'a::ring)" 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

187 
by (simp add: mult_commute) 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

188 

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

189 

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

190 
subsection {* Distribution rules *} 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

191 

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

192 
lemma right_distrib: "a * (b + c) = a * b + a * (c::'a::semiring)" 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

193 
proof  
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

194 
have "a * (b + c) = (b + c) * a" by (simp add: mult_ac) 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

195 
also have "... = b * a + c * a" by (simp only: left_distrib) 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

196 
also have "... = a * b + a * c" by (simp add: mult_ac) 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

197 
finally show ?thesis . 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

198 
qed 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

199 

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

200 
theorems ring_distrib = right_distrib left_distrib 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

201 

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

202 
lemma minus_add_distrib [simp]: " (a + b) = a + (b::'a::ring)" 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

203 
apply (rule equals_zero_I) 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

204 
apply (simp add: add_ac) 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

205 
done 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

206 

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

207 
lemma minus_mult_left: " (a * b) = (a) * (b::'a::ring)" 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

208 
apply (rule equals_zero_I) 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

209 
apply (simp add: left_distrib [symmetric]) 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

210 
done 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

211 

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

212 
lemma minus_mult_right: " (a * b) = a * (b::'a::ring)" 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

213 
apply (rule equals_zero_I) 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

214 
apply (simp add: right_distrib [symmetric]) 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

215 
done 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

216 

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

217 
lemma minus_mult_minus [simp]: "( a) * ( b) = a * (b::'a::ring)" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

218 
by (simp add: minus_mult_left [symmetric] minus_mult_right [symmetric]) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

219 

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

220 
lemma right_diff_distrib: "a * (b  c) = a * b  a * (c::'a::ring)" 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

221 
by (simp add: right_distrib diff_minus 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

222 
minus_mult_left [symmetric] minus_mult_right [symmetric]) 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

223 

14270  224 
lemma minus_diff_eq [simp]: " (a  b) = b  (a::'a::ring)" 
225 
by (simp add: diff_minus add_commute) 

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

226 

14270  227 

228 
subsection {* Ordering Rules for Addition *} 

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

229 

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

230 
lemma add_right_mono: "a \<le> (b::'a::ordered_semiring) ==> a + c \<le> b + c" 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

231 
by (simp add: add_commute [of _ c] add_left_mono) 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

232 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

233 
text {* nonstrict, in both arguments *} 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

234 
lemma add_mono: "[a \<le> b; c \<le> d] ==> a + c \<le> b + (d::'a::ordered_semiring)" 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

235 
apply (erule add_right_mono [THEN order_trans]) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

236 
apply (simp add: add_commute add_left_mono) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

237 
done 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

238 

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

239 
lemma add_strict_left_mono: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

240 
"a < b ==> c + a < c + (b::'a::ordered_ring)" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

241 
by (simp add: order_less_le add_left_mono) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

242 

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

243 
lemma add_strict_right_mono: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

244 
"a < b ==> a + c < b + (c::'a::ordered_ring)" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

245 
by (simp add: add_commute [of _ c] add_strict_left_mono) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

246 

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

247 
text{*Strict monotonicity in both arguments*} 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

248 
lemma add_strict_mono: "[a<b; c<d] ==> a + c < b + (d::'a::ordered_ring)" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

249 
apply (erule add_strict_right_mono [THEN order_less_trans]) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

250 
apply (erule add_strict_left_mono) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

251 
done 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

252 

14270  253 
lemma add_less_imp_less_left: 
254 
assumes less: "c + a < c + b" shows "a < (b::'a::ordered_ring)" 

255 
proof  

256 
have "c + (c + a) < c + (c + b)" 

257 
by (rule add_strict_left_mono [OF less]) 

258 
thus "a < b" by (simp add: add_assoc [symmetric]) 

259 
qed 

260 

261 
lemma add_less_imp_less_right: 

262 
"a + c < b + c ==> a < (b::'a::ordered_ring)" 

263 
apply (rule add_less_imp_less_left [of c]) 

264 
apply (simp add: add_commute) 

265 
done 

266 

267 
lemma add_less_cancel_left [simp]: 

268 
"(c+a < c+b) = (a < (b::'a::ordered_ring))" 

269 
by (blast intro: add_less_imp_less_left add_strict_left_mono) 

270 

271 
lemma add_less_cancel_right [simp]: 

272 
"(a+c < b+c) = (a < (b::'a::ordered_ring))" 

273 
by (blast intro: add_less_imp_less_right add_strict_right_mono) 

274 

275 
lemma add_le_cancel_left [simp]: 

276 
"(c+a \<le> c+b) = (a \<le> (b::'a::ordered_ring))" 

277 
by (simp add: linorder_not_less [symmetric]) 

278 

279 
lemma add_le_cancel_right [simp]: 

280 
"(a+c \<le> b+c) = (a \<le> (b::'a::ordered_ring))" 

281 
by (simp add: linorder_not_less [symmetric]) 

282 

283 
lemma add_le_imp_le_left: 

284 
"c + a \<le> c + b ==> a \<le> (b::'a::ordered_ring)" 

285 
by simp 

286 

287 
lemma add_le_imp_le_right: 

288 
"a + c \<le> b + c ==> a \<le> (b::'a::ordered_ring)" 

289 
by simp 

290 

291 

292 
subsection {* Ordering Rules for Unary Minus *} 

293 

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

294 
lemma le_imp_neg_le: 
14269  295 
assumes "a \<le> (b::'a::ordered_ring)" shows "b \<le> a" 
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

296 
proof  
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

297 
have "a+a \<le> a+b" 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

298 
by (rule add_left_mono) 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

299 
hence "0 \<le> a+b" 
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

300 
by simp 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

301 
hence "0 + (b) \<le> (a + b) + (b)" 
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

302 
by (rule add_right_mono) 
14266  303 
thus ?thesis 
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

304 
by (simp add: add_assoc) 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

305 
qed 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

306 

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

307 
lemma neg_le_iff_le [simp]: "(b \<le> a) = (a \<le> (b::'a::ordered_ring))" 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

308 
proof 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

309 
assume " b \<le>  a" 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

310 
hence " ( a) \<le>  ( b)" 
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

311 
by (rule le_imp_neg_le) 
14266  312 
thus "a\<le>b" by simp 
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

313 
next 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

314 
assume "a\<le>b" 
14266  315 
thus "b \<le> a" by (rule le_imp_neg_le) 
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

316 
qed 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

317 

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

318 
lemma neg_le_0_iff_le [simp]: "(a \<le> 0) = (0 \<le> (a::'a::ordered_ring))" 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

319 
by (subst neg_le_iff_le [symmetric], simp) 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

320 

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

321 
lemma neg_0_le_iff_le [simp]: "(0 \<le> a) = (a \<le> (0::'a::ordered_ring))" 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

322 
by (subst neg_le_iff_le [symmetric], simp) 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

323 

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

324 
lemma neg_less_iff_less [simp]: "(b < a) = (a < (b::'a::ordered_ring))" 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

325 
by (force simp add: order_less_le) 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

326 

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

327 
lemma neg_less_0_iff_less [simp]: "(a < 0) = (0 < (a::'a::ordered_ring))" 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

328 
by (subst neg_less_iff_less [symmetric], simp) 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

329 

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

330 
lemma neg_0_less_iff_less [simp]: "(0 < a) = (a < (0::'a::ordered_ring))" 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

331 
by (subst neg_less_iff_less [symmetric], simp) 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

332 

14270  333 

334 
subsection{*Subtraction Laws*} 

335 

336 
lemma add_diff_eq: "a + (b  c) = (a + b)  (c::'a::ring)" 

337 
by (simp add: diff_minus add_ac) 

338 

339 
lemma diff_add_eq: "(a  b) + c = (a + c)  (b::'a::ring)" 

340 
by (simp add: diff_minus add_ac) 

341 

342 
lemma diff_eq_eq: "(ab = c) = (a = c + (b::'a::ring))" 

343 
by (auto simp add: diff_minus add_assoc) 

344 

345 
lemma eq_diff_eq: "(a = cb) = (a + (b::'a::ring) = c)" 

346 
by (auto simp add: diff_minus add_assoc) 

347 

348 
lemma diff_diff_eq: "(a  b)  c = a  (b + (c::'a::ring))" 

349 
by (simp add: diff_minus add_ac) 

350 

351 
lemma diff_diff_eq2: "a  (b  c) = (a + c)  (b::'a::ring)" 

352 
by (simp add: diff_minus add_ac) 

353 

354 
text{*Further subtraction laws for ordered rings*} 

355 

356 
lemma less_eq_diff: "(a < b) = (a  b < (0::'a::ordered_ring))" 

357 
proof  

358 
have "(a < b) = (a + ( b) < b + (b))" 

359 
by (simp only: add_less_cancel_right) 

360 
also have "... = (a  b < 0)" by (simp add: diff_minus) 

361 
finally show ?thesis . 

362 
qed 

363 

364 
lemma diff_less_eq: "(ab < c) = (a < c + (b::'a::ordered_ring))" 

365 
apply (subst less_eq_diff) 

366 
apply (rule less_eq_diff [of _ c, THEN ssubst]) 

367 
apply (simp add: diff_minus add_ac) 

368 
done 

369 

370 
lemma less_diff_eq: "(a < cb) = (a + (b::'a::ordered_ring) < c)" 

371 
apply (subst less_eq_diff) 

372 
apply (rule less_eq_diff [of _ "cb", THEN ssubst]) 

373 
apply (simp add: diff_minus add_ac) 

374 
done 

375 

376 
lemma diff_le_eq: "(ab \<le> c) = (a \<le> c + (b::'a::ordered_ring))" 

377 
by (simp add: linorder_not_less [symmetric] less_diff_eq) 

378 

379 
lemma le_diff_eq: "(a \<le> cb) = (a + (b::'a::ordered_ring) \<le> c)" 

380 
by (simp add: linorder_not_less [symmetric] diff_less_eq) 

381 

382 
text{*This list of rewrites simplifies (in)equalities by bringing subtractions 

383 
to the top and then moving negative terms to the other side. 

384 
Use with @{text add_ac}*} 

385 
lemmas compare_rls = 

386 
diff_minus [symmetric] 

387 
add_diff_eq diff_add_eq diff_diff_eq diff_diff_eq2 

388 
diff_less_eq less_diff_eq diff_le_eq le_diff_eq 

389 
diff_eq_eq eq_diff_eq 

390 

391 

392 
subsection {* Ordering Rules for Multiplication *} 

393 

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

394 
lemma mult_strict_right_mono: 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

395 
"[a < b; 0 < c] ==> a * c < b * (c::'a::ordered_semiring)" 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

396 
by (simp add: mult_commute [of _ c] mult_strict_left_mono) 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

397 

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

398 
lemma mult_left_mono: 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

399 
"[a \<le> b; 0 \<le> c] ==> c * a \<le> c * (b::'a::ordered_ring)" 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

400 
apply (case_tac "c=0", simp) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

401 
apply (force simp add: mult_strict_left_mono order_le_less) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

402 
done 
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

403 

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

404 
lemma mult_right_mono: 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

405 
"[a \<le> b; 0 \<le> c] ==> a*c \<le> b * (c::'a::ordered_ring)" 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

406 
by (simp add: mult_left_mono mult_commute [of _ c]) 
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

407 

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

408 
lemma mult_strict_left_mono_neg: 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

409 
"[b < a; c < 0] ==> c * a < c * (b::'a::ordered_ring)" 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

410 
apply (drule mult_strict_left_mono [of _ _ "c"]) 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

411 
apply (simp_all add: minus_mult_left [symmetric]) 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

412 
done 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

413 

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

414 
lemma mult_strict_right_mono_neg: 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

415 
"[b < a; c < 0] ==> a * c < b * (c::'a::ordered_ring)" 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

416 
apply (drule mult_strict_right_mono [of _ _ "c"]) 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

417 
apply (simp_all add: minus_mult_right [symmetric]) 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

418 
done 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

419 

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

420 

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

421 
subsection{* Products of Signs *} 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

422 

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

423 
lemma mult_pos: "[ (0::'a::ordered_ring) < a; 0 < b ] ==> 0 < a*b" 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

424 
by (drule mult_strict_left_mono [of 0 b], auto) 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

425 

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

426 
lemma mult_pos_neg: "[ (0::'a::ordered_ring) < a; b < 0 ] ==> a*b < 0" 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

427 
by (drule mult_strict_left_mono [of b 0], auto) 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

428 

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

429 
lemma mult_neg: "[ a < (0::'a::ordered_ring); b < 0 ] ==> 0 < a*b" 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

430 
by (drule mult_strict_right_mono_neg, auto) 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

431 

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

432 
lemma zero_less_mult_pos: "[ 0 < a*b; 0 < a] ==> 0 < (b::'a::ordered_ring)" 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

433 
apply (case_tac "b\<le>0") 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

434 
apply (auto simp add: order_le_less linorder_not_less) 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

435 
apply (drule_tac mult_pos_neg [of a b]) 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

436 
apply (auto dest: order_less_not_sym) 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

437 
done 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

438 

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

439 
lemma zero_less_mult_iff: 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

440 
"((0::'a::ordered_ring) < a*b) = (0 < a & 0 < b  a < 0 & b < 0)" 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

441 
apply (auto simp add: order_le_less linorder_not_less mult_pos mult_neg) 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

442 
apply (blast dest: zero_less_mult_pos) 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

443 
apply (simp add: mult_commute [of a b]) 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

444 
apply (blast dest: zero_less_mult_pos) 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

445 
done 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

446 

14266  447 
lemma mult_eq_0_iff [simp]: "(a*b = (0::'a::ordered_ring)) = (a = 0  b = 0)" 
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

448 
apply (case_tac "a < 0") 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

449 
apply (auto simp add: linorder_not_less order_le_less linorder_neq_iff) 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

450 
apply (force dest: mult_strict_right_mono_neg mult_strict_right_mono)+ 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

451 
done 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

452 

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

453 
lemma zero_le_mult_iff: 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

454 
"((0::'a::ordered_ring) \<le> a*b) = (0 \<le> a & 0 \<le> b  a \<le> 0 & b \<le> 0)" 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

455 
by (auto simp add: eq_commute [of 0] order_le_less linorder_not_less 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

456 
zero_less_mult_iff) 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

457 

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

458 
lemma mult_less_0_iff: 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

459 
"(a*b < (0::'a::ordered_ring)) = (0 < a & b < 0  a < 0 & 0 < b)" 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

460 
apply (insert zero_less_mult_iff [of "a" b]) 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

461 
apply (force simp add: minus_mult_left[symmetric]) 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

462 
done 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

463 

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

464 
lemma mult_le_0_iff: 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

465 
"(a*b \<le> (0::'a::ordered_ring)) = (0 \<le> a & b \<le> 0  a \<le> 0 & 0 \<le> b)" 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

466 
apply (insert zero_le_mult_iff [of "a" b]) 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

467 
apply (force simp add: minus_mult_left[symmetric]) 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

468 
done 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

469 

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

470 
lemma zero_le_square: "(0::'a::ordered_ring) \<le> a*a" 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

471 
by (simp add: zero_le_mult_iff linorder_linear) 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

472 

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

473 
lemma zero_less_one: "(0::'a::ordered_ring) < 1" 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

474 
apply (insert zero_le_square [of 1]) 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

475 
apply (simp add: order_less_le) 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

476 
done 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

477 

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

478 
lemma zero_le_one: "(0::'a::ordered_ring) \<le> 1" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

479 
by (rule zero_less_one [THEN order_less_imp_le]) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

480 

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

481 

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

482 
subsection{*More Monotonicity*} 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

483 

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

484 
lemma mult_left_mono_neg: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

485 
"[b \<le> a; c \<le> 0] ==> c * a \<le> c * (b::'a::ordered_ring)" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

486 
apply (drule mult_left_mono [of _ _ "c"]) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

487 
apply (simp_all add: minus_mult_left [symmetric]) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

488 
done 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

489 

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

490 
lemma mult_right_mono_neg: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

491 
"[b \<le> a; c \<le> 0] ==> a * c \<le> b * (c::'a::ordered_ring)" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

492 
by (simp add: mult_left_mono_neg mult_commute [of _ c]) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

493 

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

494 
text{*Strict monotonicity in both arguments*} 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

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

496 
"[a<b; c<d; 0<b; 0\<le>c] ==> a * c < b * (d::'a::ordered_ring)" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

497 
apply (case_tac "c=0") 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

498 
apply (simp add: mult_pos) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

499 
apply (erule mult_strict_right_mono [THEN order_less_trans]) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

500 
apply (force simp add: order_le_less) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

501 
apply (erule mult_strict_left_mono, assumption) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

502 
done 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

503 

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

504 
text{*This weaker variant has more natural premises*} 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

505 
lemma mult_strict_mono': 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

506 
"[ a<b; c<d; 0 \<le> a; 0 \<le> c] ==> a * c < b * (d::'a::ordered_ring)" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

507 
apply (rule mult_strict_mono) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

508 
apply (blast intro: order_le_less_trans)+ 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

509 
done 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

510 

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

511 
lemma mult_mono: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

512 
"[a \<le> b; c \<le> d; 0 \<le> b; 0 \<le> c] 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

513 
==> a * c \<le> b * (d::'a::ordered_ring)" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

514 
apply (erule mult_right_mono [THEN order_trans], assumption) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

515 
apply (erule mult_left_mono, assumption) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

516 
done 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

517 

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

518 

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

519 
subsection{*Cancellation Laws for Relationships With a Common Factor*} 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

520 

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

521 
text{*Cancellation laws for @{term "c*a < c*b"} and @{term "a*c < b*c"}, 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

522 
also with the relations @{text "\<le>"} and equality.*} 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

523 

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

524 
lemma mult_less_cancel_right: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

525 
"(a*c < b*c) = ((0 < c & a < b)  (c < 0 & b < (a::'a::ordered_ring)))" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

526 
apply (case_tac "c = 0") 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

527 
apply (auto simp add: linorder_neq_iff mult_strict_right_mono 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

528 
mult_strict_right_mono_neg) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

529 
apply (auto simp add: linorder_not_less 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

530 
linorder_not_le [symmetric, of "a*c"] 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

531 
linorder_not_le [symmetric, of a]) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

532 
apply (erule_tac [!] notE) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

533 
apply (auto simp add: order_less_imp_le mult_right_mono 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

534 
mult_right_mono_neg) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

535 
done 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

536 

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

537 
lemma mult_less_cancel_left: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

538 
"(c*a < c*b) = ((0 < c & a < b)  (c < 0 & b < (a::'a::ordered_ring)))" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

539 
by (simp add: mult_commute [of c] mult_less_cancel_right) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

540 

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

541 
lemma mult_le_cancel_right: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

542 
"(a*c \<le> b*c) = ((0<c > a\<le>b) & (c<0 > b \<le> (a::'a::ordered_ring)))" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

543 
by (simp add: linorder_not_less [symmetric] mult_less_cancel_right) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

544 

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

545 
lemma mult_le_cancel_left: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

546 
"(c*a \<le> c*b) = ((0<c > a\<le>b) & (c<0 > b \<le> (a::'a::ordered_ring)))" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

547 
by (simp add: mult_commute [of c] mult_le_cancel_right) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

548 

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

549 
lemma mult_less_imp_less_left: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

550 
"[c*a < c*b; 0 < c] ==> a < (b::'a::ordered_ring)" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

551 
by (force elim: order_less_asym simp add: mult_less_cancel_left) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

552 

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

553 
lemma mult_less_imp_less_right: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

554 
"[a*c < b*c; 0 < c] ==> a < (b::'a::ordered_ring)" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

555 
by (force elim: order_less_asym simp add: mult_less_cancel_right) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

556 

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

557 
text{*Cancellation of equalities with a common factor*} 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

558 
lemma mult_cancel_right [simp]: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

559 
"(a*c = b*c) = (c = (0::'a::ordered_ring)  a=b)" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

560 
apply (cut_tac linorder_less_linear [of 0 c]) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

561 
apply (force dest: mult_strict_right_mono_neg mult_strict_right_mono 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

562 
simp add: linorder_neq_iff) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

563 
done 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

564 

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

565 
text{*These cancellation theorems require an ordering. Versions are proved 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

566 
below that work for fields without an ordering.*} 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

567 
lemma mult_cancel_left [simp]: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

568 
"(c*a = c*b) = (c = (0::'a::ordered_ring)  a=b)" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

569 
by (simp add: mult_commute [of c] mult_cancel_right) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

570 

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

571 

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

572 
subsection {* Absolute Value *} 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

573 

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

574 
text{*But is it really better than just rewriting with @{text abs_if}?*} 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

575 
lemma abs_split: 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

576 
"P(abs(a::'a::ordered_ring)) = ((0 \<le> a > P a) & (a < 0 > P(a)))" 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

577 
by (force dest: order_less_le_trans simp add: abs_if linorder_not_less) 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

578 

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

579 
lemma abs_zero [simp]: "abs 0 = (0::'a::ordered_ring)" 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

580 
by (simp add: abs_if) 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

581 

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

582 
lemma abs_mult: "abs (x * y) = abs x * abs (y::'a::ordered_ring)" 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

583 
apply (case_tac "x=0  y=0", force) 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

584 
apply (auto elim: order_less_asym 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

585 
simp add: abs_if mult_less_0_iff linorder_neq_iff 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

586 
minus_mult_left [symmetric] minus_mult_right [symmetric]) 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

587 
done 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

588 

14266  589 
lemma abs_eq_0 [simp]: "(abs x = 0) = (x = (0::'a::ordered_ring))" 
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

590 
by (simp add: abs_if) 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

591 

14266  592 
lemma zero_less_abs_iff [simp]: "(0 < abs x) = (x ~= (0::'a::ordered_ring))" 
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

593 
by (simp add: abs_if linorder_neq_iff) 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

594 

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

595 

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

596 
subsection {* Fields *} 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

597 

14270  598 
text{*Compared with @{text mult_eq_0_iff}, this version removes the requirement 
599 
of an ordering.*} 

600 
lemma field_mult_eq_0_iff: "(a*b = (0::'a::field)) = (a = 0  b = 0)" 

601 
proof cases 

602 
assume "a=0" thus ?thesis by simp 

603 
next 

604 
assume anz [simp]: "a\<noteq>0" 

605 
thus ?thesis 

606 
proof auto 

607 
assume "a * b = 0" 

608 
hence "inverse a * (a * b) = 0" by simp 

609 
thus "b = 0" by (simp (no_asm_use) add: mult_assoc [symmetric]) 

610 
qed 

611 
qed 

612 

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

613 
text{*Cancellation of equalities with a common factor*} 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

614 
lemma field_mult_cancel_right_lemma: 
14269  615 
assumes cnz: "c \<noteq> (0::'a::field)" 
616 
and eq: "a*c = b*c" 

617 
shows "a=b" 

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

618 
proof  
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

619 
have "(a * c) * inverse c = (b * c) * inverse c" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

620 
by (simp add: eq) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

621 
thus "a=b" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

622 
by (simp add: mult_assoc cnz) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

623 
qed 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

624 

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

625 
lemma field_mult_cancel_right: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

626 
"(a*c = b*c) = (c = (0::'a::field)  a=b)" 
14269  627 
proof cases 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

628 
assume "c=0" thus ?thesis by simp 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

629 
next 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

630 
assume "c\<noteq>0" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

631 
thus ?thesis by (force dest: field_mult_cancel_right_lemma) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

632 
qed 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

633 

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

634 
lemma field_mult_cancel_left: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

635 
"(c*a = c*b) = (c = (0::'a::field)  a=b)" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

636 
by (simp add: mult_commute [of c] field_mult_cancel_right) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

637 

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

638 
lemma nonzero_imp_inverse_nonzero: "a \<noteq> 0 ==> inverse a \<noteq> (0::'a::field)" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

639 
proof 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

640 
assume ianz: "inverse a = 0" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

641 
assume "a \<noteq> 0" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

642 
hence "1 = a * inverse a" by simp 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

643 
also have "... = 0" by (simp add: ianz) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

644 
finally have "1 = (0::'a::field)" . 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

645 
thus False by (simp add: eq_commute) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

646 
qed 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

647 

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

648 
lemma inverse_zero_imp_zero: "inverse a = 0 ==> a = (0::'a::field)" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

649 
apply (rule ccontr) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

650 
apply (blast dest: nonzero_imp_inverse_nonzero) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

651 
done 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

652 

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

653 
lemma inverse_nonzero_imp_nonzero: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

654 
"inverse a = 0 ==> a = (0::'a::field)" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

655 
apply (rule ccontr) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

656 
apply (blast dest: nonzero_imp_inverse_nonzero) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

657 
done 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

658 

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

659 
lemma inverse_nonzero_iff_nonzero [simp]: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

660 
"(inverse a = 0) = (a = (0::'a::{field,division_by_zero}))" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

661 
by (force dest: inverse_nonzero_imp_nonzero) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

662 

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

663 
lemma nonzero_inverse_minus_eq: 
14269  664 
assumes [simp]: "a\<noteq>0" shows "inverse(a) = inverse(a::'a::field)" 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

665 
proof  
14269  666 
have "a * inverse ( a) = a *  inverse a" 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

667 
by simp 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

668 
thus ?thesis 
14269  669 
by (simp only: field_mult_cancel_left, simp) 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

670 
qed 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

671 

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

672 
lemma inverse_minus_eq [simp]: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

673 
"inverse(a) = inverse(a::'a::{field,division_by_zero})"; 
14269  674 
proof cases 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

675 
assume "a=0" thus ?thesis by (simp add: inverse_zero) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

676 
next 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

677 
assume "a\<noteq>0" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

678 
thus ?thesis by (simp add: nonzero_inverse_minus_eq) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

679 
qed 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

680 

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

681 
lemma nonzero_inverse_eq_imp_eq: 
14269  682 
assumes inveq: "inverse a = inverse b" 
683 
and anz: "a \<noteq> 0" 

684 
and bnz: "b \<noteq> 0" 

685 
shows "a = (b::'a::field)" 

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

686 
proof  
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

687 
have "a * inverse b = a * inverse a" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

688 
by (simp add: inveq) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

689 
hence "(a * inverse b) * b = (a * inverse a) * b" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

690 
by simp 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

691 
thus "a = b" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

692 
by (simp add: mult_assoc anz bnz) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

693 
qed 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

694 

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

695 
lemma inverse_eq_imp_eq: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

696 
"inverse a = inverse b ==> a = (b::'a::{field,division_by_zero})" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

697 
apply (case_tac "a=0  b=0") 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

698 
apply (force dest!: inverse_zero_imp_zero 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

699 
simp add: eq_commute [of "0::'a"]) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

700 
apply (force dest!: nonzero_inverse_eq_imp_eq) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

701 
done 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

702 

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

703 
lemma inverse_eq_iff_eq [simp]: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

704 
"(inverse a = inverse b) = (a = (b::'a::{field,division_by_zero}))" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

705 
by (force dest!: inverse_eq_imp_eq) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

706 

14270  707 
lemma nonzero_inverse_inverse_eq: 
708 
assumes [simp]: "a \<noteq> 0" shows "inverse(inverse (a::'a::field)) = a" 

709 
proof  

710 
have "(inverse (inverse a) * inverse a) * a = a" 

711 
by (simp add: nonzero_imp_inverse_nonzero) 

712 
thus ?thesis 

713 
by (simp add: mult_assoc) 

714 
qed 

715 

716 
lemma inverse_inverse_eq [simp]: 

717 
"inverse(inverse (a::'a::{field,division_by_zero})) = a" 

718 
proof cases 

719 
assume "a=0" thus ?thesis by simp 

720 
next 

721 
assume "a\<noteq>0" 

722 
thus ?thesis by (simp add: nonzero_inverse_inverse_eq) 

723 
qed 

724 

725 
lemma inverse_1 [simp]: "inverse 1 = (1::'a::field)" 

726 
proof  

727 
have "inverse 1 * 1 = (1::'a::field)" 

728 
by (rule left_inverse [OF zero_neq_one [symmetric]]) 

729 
thus ?thesis by simp 

730 
qed 

731 

732 
lemma nonzero_inverse_mult_distrib: 

733 
assumes anz: "a \<noteq> 0" 

734 
and bnz: "b \<noteq> 0" 

735 
shows "inverse(a*b) = inverse(b) * inverse(a::'a::field)" 

736 
proof  

737 
have "inverse(a*b) * (a * b) * inverse(b) = inverse(b)" 

738 
by (simp add: field_mult_eq_0_iff anz bnz) 

739 
hence "inverse(a*b) * a = inverse(b)" 

740 
by (simp add: mult_assoc bnz) 

741 
hence "inverse(a*b) * a * inverse(a) = inverse(b) * inverse(a)" 

742 
by simp 

743 
thus ?thesis 

744 
by (simp add: mult_assoc anz) 

745 
qed 

746 

747 
text{*This version builds in division by zero while also reorienting 

748 
the righthand side.*} 

749 
lemma inverse_mult_distrib [simp]: 

750 
"inverse(a*b) = inverse(a) * inverse(b::'a::{field,division_by_zero})" 

751 
proof cases 

752 
assume "a \<noteq> 0 & b \<noteq> 0" 

753 
thus ?thesis by (simp add: nonzero_inverse_mult_distrib mult_commute) 

754 
next 

755 
assume "~ (a \<noteq> 0 & b \<noteq> 0)" 

756 
thus ?thesis by force 

757 
qed 

758 

759 
text{*There is no slick version using division by zero.*} 

760 
lemma inverse_add: 

761 
"[a \<noteq> 0; b \<noteq> 0] 

762 
==> inverse a + inverse b = (a+b) * inverse a * inverse (b::'a::field)" 

763 
apply (simp add: left_distrib mult_assoc) 

764 
apply (simp add: mult_commute [of "inverse a"]) 

765 
apply (simp add: mult_assoc [symmetric] add_commute) 

766 
done 

767 

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

768 

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

769 
subsection {* Ordered Fields *} 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

770 

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

771 
lemma inverse_gt_0: 
14269  772 
assumes a_gt_0: "0 < a" shows "0 < inverse (a::'a::ordered_field)" 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

773 
proof  
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

774 
have "0 < a * inverse a" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

775 
by (simp add: a_gt_0 [THEN order_less_imp_not_eq2] zero_less_one) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

776 
thus "0 < inverse a" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

777 
by (simp add: a_gt_0 [THEN order_less_not_sym] zero_less_mult_iff) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

778 
qed 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

779 

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

780 
lemma inverse_less_0: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

781 
"a < 0 ==> inverse a < (0::'a::ordered_field)" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

782 
by (insert inverse_gt_0 [of "a"], 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

783 
simp add: nonzero_inverse_minus_eq order_less_imp_not_eq) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

784 

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

785 
lemma inverse_le_imp_le: 
14269  786 
assumes invle: "inverse a \<le> inverse b" 
787 
and apos: "0 < a" 

788 
shows "b \<le> (a::'a::ordered_field)" 

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

789 
proof (rule classical) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

790 
assume "~ b \<le> a" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

791 
hence "a < b" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

792 
by (simp add: linorder_not_le) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

793 
hence bpos: "0 < b" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

794 
by (blast intro: apos order_less_trans) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

795 
hence "a * inverse a \<le> a * inverse b" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

796 
by (simp add: apos invle order_less_imp_le mult_left_mono) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

797 
hence "(a * inverse a) * b \<le> (a * inverse b) * b" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

798 
by (simp add: bpos order_less_imp_le mult_right_mono) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

799 
thus "b \<le> a" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

800 
by (simp add: mult_assoc apos bpos order_less_imp_not_eq2) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

801 
qed 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

802 

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

803 
lemma less_imp_inverse_less: 
14269  804 
assumes less: "a < b" 
805 
and apos: "0 < a" 

806 
shows "inverse b < inverse (a::'a::ordered_field)" 

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

807 
proof (rule ccontr) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

808 
assume "~ inverse b < inverse a" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

809 
hence "inverse a \<le> inverse b" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

810 
by (simp add: linorder_not_less) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

811 
hence "~ (a < b)" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

812 
by (simp add: linorder_not_less inverse_le_imp_le [OF _ apos]) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

813 
thus False 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

814 
by (rule notE [OF _ less]) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

815 
qed 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

816 

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

817 
lemma inverse_less_imp_less: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

818 
"[inverse a < inverse b; 0 < a] ==> b < (a::'a::ordered_field)" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

819 
apply (simp add: order_less_le [of "inverse a"] order_less_le [of "b"]) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

820 
apply (force dest!: inverse_le_imp_le nonzero_inverse_eq_imp_eq) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

821 
done 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

822 

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

823 
text{*Both premises are essential. Consider 1 and 1.*} 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

824 
lemma inverse_less_iff_less [simp]: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

825 
"[0 < a; 0 < b] 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

826 
==> (inverse a < inverse b) = (b < (a::'a::ordered_field))" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

827 
by (blast intro: less_imp_inverse_less dest: inverse_less_imp_less) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

828 

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

829 
lemma le_imp_inverse_le: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

830 
"[a \<le> b; 0 < a] ==> inverse b \<le> inverse (a::'a::ordered_field)" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

831 
by (force simp add: order_le_less less_imp_inverse_less) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

832 

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

833 
lemma inverse_le_iff_le [simp]: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

834 
"[0 < a; 0 < b] 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

835 
==> (inverse a \<le> inverse b) = (b \<le> (a::'a::ordered_field))" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

836 
by (blast intro: le_imp_inverse_le dest: inverse_le_imp_le) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

837 

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

838 

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

839 
text{*These results refer to both operands being negative. The oppositesign 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

840 
case is trivial, since inverse preserves signs.*} 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

841 
lemma inverse_le_imp_le_neg: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

842 
"[inverse a \<le> inverse b; b < 0] ==> b \<le> (a::'a::ordered_field)" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

843 
apply (rule classical) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

844 
apply (subgoal_tac "a < 0") 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

845 
prefer 2 apply (force simp add: linorder_not_le intro: order_less_trans) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

846 
apply (insert inverse_le_imp_le [of "b" "a"]) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

847 
apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

848 
done 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

849 

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

850 
lemma less_imp_inverse_less_neg: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

851 
"[a < b; b < 0] ==> inverse b < inverse (a::'a::ordered_field)" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

852 
apply (subgoal_tac "a < 0") 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

853 
prefer 2 apply (blast intro: order_less_trans) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

854 
apply (insert less_imp_inverse_less [of "b" "a"]) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

855 
apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

856 
done 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

857 

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

858 
lemma inverse_less_imp_less_neg: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

859 
"[inverse a < inverse b; b < 0] ==> b < (a::'a::ordered_field)" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

860 
apply (rule classical) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

861 
apply (subgoal_tac "a < 0") 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

862 
prefer 2 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

863 
apply (force simp add: linorder_not_less intro: order_le_less_trans) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

864 
apply (insert inverse_less_imp_less [of "b" "a"]) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

865 
apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

866 
done 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

867 

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

868 
lemma inverse_less_iff_less_neg [simp]: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

869 
"[a < 0; b < 0] 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

870 
==> (inverse a < inverse b) = (b < (a::'a::ordered_field))" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

871 
apply (insert inverse_less_iff_less [of "b" "a"]) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

872 
apply (simp del: inverse_less_iff_less 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

873 
add: order_less_imp_not_eq nonzero_inverse_minus_eq) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

874 
done 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

875 

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

876 
lemma le_imp_inverse_le_neg: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

877 
"[a \<le> b; b < 0] ==> inverse b \<le> inverse (a::'a::ordered_field)" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

878 
by (force simp add: order_le_less less_imp_inverse_less_neg) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

879 

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

880 
lemma inverse_le_iff_le_neg [simp]: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

881 
"[a < 0; b < 0] 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

882 
==> (inverse a \<le> inverse b) = (b \<le> (a::'a::ordered_field))" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

883 
by (blast intro: le_imp_inverse_le_neg dest: inverse_le_imp_le_neg) 
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

884 

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

885 
end 