author  paulson 
Tue, 27 Jan 2004 15:39:51 +0100  
changeset 14365  3d4df8c166ae 
parent 14353  79f9fbef9106 
child 14368  2763da611ad9 
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 
14269  4 
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, L. C. Paulson and Markus Wenzel} 
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*} 
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theory Ring_and_Field = Inductive: 
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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" 
14288  20 
add_0 [simp]: "0 + a = a" 
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add_left_imp_eq: "a + b = a + c ==> b=c" 
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{*This axiom is needed for semirings only: for rings, etc., it is 
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redundant. Including it allows many more of the following results 
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to be proved for semirings too. The drawback is that this redundant 
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axiom must be proved for instances of rings.*} 
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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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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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zero_less_one: "0 < 1" {*This axiom too is needed for semirings only.*} 
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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  57 
subsection {* Derived Rules for Addition *} 
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14288  59 
lemma add_0_right [simp]: "a + 0 = (a::'a::semiring)" 
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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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79 
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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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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86 
qed 
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87 

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lemma add_left_cancel [simp]: 
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"(a + b = a + c) = (b = (c::'a::semiring))" 
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by (blast dest: add_left_imp_eq) 
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lemma add_right_cancel [simp]: 
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"(b + a = c + a) = (b = (c::'a::semiring))" 
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94 
by (simp add: add_commute) 
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lemma minus_minus [simp]: " ( (a::'a::ring)) = a" 
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proof (rule add_left_cancel [of "a", THEN iffD1]) 
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show "(a + (a) = a + a)" 
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99 
by simp 
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qed 
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lemma equals_zero_I: "a+b = 0 ==> a = (b::'a::ring)" 
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103 
apply (rule right_minus_eq [THEN iffD1, symmetric]) 
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104 
apply (simp add: diff_minus add_commute) 
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done 
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lemma minus_zero [simp]: " 0 = (0::'a::ring)" 
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by (simp add: equals_zero_I) 
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14270  110 
lemma diff_self [simp]: "a  (a::'a::ring) = 0" 
111 
by (simp add: diff_minus) 

112 

113 
lemma diff_0 [simp]: "(0::'a::ring)  a = a" 

114 
by (simp add: diff_minus) 

115 

116 
lemma diff_0_right [simp]: "a  (0::'a::ring) = a" 

117 
by (simp add: diff_minus) 

118 

14288  119 
lemma diff_minus_eq_add [simp]: "a   b = a + (b::'a::ring)" 
120 
by (simp add: diff_minus) 

121 

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lemma neg_equal_iff_equal [simp]: "(a = b) = (a = (b::'a::ring))" 
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123 
proof 
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124 
assume " a =  b" 
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125 
hence " ( a) =  ( b)" 
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126 
by simp 
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thus "a=b" by simp 
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128 
next 
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assume "a=b" 
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thus "a = b" by simp 
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131 
qed 
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132 

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lemma neg_equal_0_iff_equal [simp]: "(a = 0) = (a = (0::'a::ring))" 
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134 
by (subst neg_equal_iff_equal [symmetric], simp) 
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135 

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

14272
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139 
text{*The next two equations can make the simplifier loop!*} 
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140 

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141 
lemma equation_minus_iff: "(a =  b) = (b =  (a::'a::ring))" 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

142 
proof  
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

143 
have "( (a) =  b) = ( a = b)" by (rule neg_equal_iff_equal) 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

144 
thus ?thesis by (simp add: eq_commute) 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

145 
qed 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

146 

5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

147 
lemma minus_equation_iff: "( a = b) = ( (b::'a::ring) = a)" 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

148 
proof  
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

149 
have "( a =  (b)) = (a = b)" by (rule neg_equal_iff_equal) 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

150 
thus ?thesis by (simp add: eq_commute) 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

151 
qed 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

152 

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

153 

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

154 
subsection {* Derived rules for multiplication *} 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

155 

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

156 
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

157 
proof  
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

158 
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

159 
also have "... = a" 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 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

164 
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

165 

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

166 
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

167 

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

168 
lemma mult_zero_left [simp]: "0 * a = (0::'a::semiring)" 
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

169 
proof  
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

170 
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

171 
by (simp add: left_distrib [symmetric]) 
14266  172 
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

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

174 

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

175 
lemma mult_zero_right [simp]: "a * 0 = (0::'a::semiring)" 
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

176 
by (simp add: mult_commute) 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

177 

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

180 

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

181 
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

182 
proof  
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

183 
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

184 
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

185 
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

186 
finally show ?thesis . 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

187 
qed 
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 
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

190 

14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

191 
text{*For the @{text combine_numerals} simproc*} 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

192 
lemma combine_common_factor: "a*e + (b*e + c) = (a+b)*e + (c::'a::semiring)" 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

193 
by (simp add: left_distrib add_ac) 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

194 

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

195 
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

196 
apply (rule equals_zero_I) 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

197 
apply (simp add: add_ac) 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

198 
done 
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 
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

201 
apply (rule equals_zero_I) 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

202 
apply (simp add: left_distrib [symmetric]) 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

203 
done 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

204 

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

205 
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

206 
apply (rule equals_zero_I) 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

207 
apply (simp add: right_distrib [symmetric]) 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

208 
done 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

209 

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

210 
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

211 
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

212 

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

213 
lemma minus_mult_commute: "( a) * b = a * ( b::'a::ring)" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

214 
by (simp add: minus_mult_left [symmetric] minus_mult_right [symmetric]) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

215 

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

216 
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

217 
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

218 
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

219 

14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

220 
lemma left_diff_distrib: "(a  b) * c = a * c  b * (c::'a::ring)" 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

221 
by (simp add: mult_commute [of _ c] right_diff_distrib) 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

222 

14270  223 
lemma minus_diff_eq [simp]: " (a  b) = b  (a::'a::ring)" 
224 
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

225 

14270  226 

227 
subsection {* Ordering Rules for Addition *} 

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

228 

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

229 
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

230 
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

231 

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

232 
text {* nonstrict, in both arguments *} 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

233 
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

234 
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

235 
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

236 
done 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

237 

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

238 
lemma add_strict_left_mono: 
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset

239 
"a < b ==> c + a < c + (b::'a::ordered_semiring)" 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

240 
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

241 

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

242 
lemma add_strict_right_mono: 
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset

243 
"a < b ==> a + c < b + (c::'a::ordered_semiring)" 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

244 
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

245 

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

246 
text{*Strict monotonicity in both arguments*} 
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset

247 
lemma add_strict_mono: "[a<b; c<d] ==> a + c < b + (d::'a::ordered_semiring)" 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

248 
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

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

250 
done 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

251 

14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset

252 
lemma add_less_le_mono: "[ a<b; c\<le>d ] ==> a + c < b + (d::'a::ordered_semiring)" 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset

253 
apply (erule add_strict_right_mono [THEN order_less_le_trans]) 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset

254 
apply (erule add_left_mono) 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset

255 
done 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset

256 

a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset

257 
lemma add_le_less_mono: 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset

258 
"[ a\<le>b; c<d ] ==> a + c < b + (d::'a::ordered_semiring)" 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset

259 
apply (erule add_right_mono [THEN order_le_less_trans]) 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset

260 
apply (erule add_strict_left_mono) 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset

261 
done 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset

262 

14270  263 
lemma add_less_imp_less_left: 
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset

264 
assumes less: "c + a < c + b" shows "a < (b::'a::ordered_semiring)" 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset

265 
proof (rule ccontr) 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset

266 
assume "~ a < b" 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset

267 
hence "b \<le> a" by (simp add: linorder_not_less) 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset

268 
hence "c+b \<le> c+a" by (rule add_left_mono) 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset

269 
with this and less show False 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset

270 
by (simp add: linorder_not_less [symmetric]) 
14270  271 
qed 
272 

273 
lemma add_less_imp_less_right: 

14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset

274 
"a + c < b + c ==> a < (b::'a::ordered_semiring)" 
14270  275 
apply (rule add_less_imp_less_left [of c]) 
276 
apply (simp add: add_commute) 

277 
done 

278 

279 
lemma add_less_cancel_left [simp]: 

14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset

280 
"(c+a < c+b) = (a < (b::'a::ordered_semiring))" 
14270  281 
by (blast intro: add_less_imp_less_left add_strict_left_mono) 
282 

283 
lemma add_less_cancel_right [simp]: 

14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset

284 
"(a+c < b+c) = (a < (b::'a::ordered_semiring))" 
14270  285 
by (blast intro: add_less_imp_less_right add_strict_right_mono) 
286 

287 
lemma add_le_cancel_left [simp]: 

14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset

288 
"(c+a \<le> c+b) = (a \<le> (b::'a::ordered_semiring))" 
14270  289 
by (simp add: linorder_not_less [symmetric]) 
290 

291 
lemma add_le_cancel_right [simp]: 

14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset

292 
"(a+c \<le> b+c) = (a \<le> (b::'a::ordered_semiring))" 
14270  293 
by (simp add: linorder_not_less [symmetric]) 
294 

295 
lemma add_le_imp_le_left: 

14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset

296 
"c + a \<le> c + b ==> a \<le> (b::'a::ordered_semiring)" 
14270  297 
by simp 
298 

299 
lemma add_le_imp_le_right: 

14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset

300 
"a + c \<le> b + c ==> a \<le> (b::'a::ordered_semiring)" 
14270  301 
by simp 
302 

303 

304 
subsection {* Ordering Rules for Unary Minus *} 

305 

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

306 
lemma le_imp_neg_le: 
14269  307 
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

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

309 
have "a+a \<le> a+b" 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

310 
by (rule add_left_mono) 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

311 
hence "0 \<le> a+b" 
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

312 
by simp 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

313 
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

314 
by (rule add_right_mono) 
14266  315 
thus ?thesis 
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

316 
by (simp add: add_assoc) 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

317 
qed 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

318 

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

319 
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

320 
proof 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

321 
assume " b \<le>  a" 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

322 
hence " ( a) \<le>  ( b)" 
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

323 
by (rule le_imp_neg_le) 
14266  324 
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

325 
next 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

326 
assume "a\<le>b" 
14266  327 
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

328 
qed 
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_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

331 
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

332 

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

333 
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

334 
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

335 

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

336 
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

337 
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

338 

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

339 
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

340 
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

341 

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

342 
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

343 
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

344 

14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

345 
text{*The next several equations can make the simplifier loop!*} 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

346 

5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

347 
lemma less_minus_iff: "(a <  b) = (b <  (a::'a::ordered_ring))" 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

348 
proof  
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

349 
have "( (a) <  b) = (b <  a)" by (rule neg_less_iff_less) 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

350 
thus ?thesis by simp 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

351 
qed 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

352 

5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

353 
lemma minus_less_iff: "( a < b) = ( b < (a::'a::ordered_ring))" 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

354 
proof  
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

355 
have "( a <  (b)) = ( b < a)" by (rule neg_less_iff_less) 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

356 
thus ?thesis by simp 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

357 
qed 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

358 

5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

359 
lemma le_minus_iff: "(a \<le>  b) = (b \<le>  (a::'a::ordered_ring))" 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

360 
apply (simp add: linorder_not_less [symmetric]) 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

361 
apply (rule minus_less_iff) 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

362 
done 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

363 

5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

364 
lemma minus_le_iff: "( a \<le> b) = ( b \<le> (a::'a::ordered_ring))" 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

365 
apply (simp add: linorder_not_less [symmetric]) 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

366 
apply (rule less_minus_iff) 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

367 
done 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

368 

14270  369 

370 
subsection{*Subtraction Laws*} 

371 

372 
lemma add_diff_eq: "a + (b  c) = (a + b)  (c::'a::ring)" 

373 
by (simp add: diff_minus add_ac) 

374 

375 
lemma diff_add_eq: "(a  b) + c = (a + c)  (b::'a::ring)" 

376 
by (simp add: diff_minus add_ac) 

377 

378 
lemma diff_eq_eq: "(ab = c) = (a = c + (b::'a::ring))" 

379 
by (auto simp add: diff_minus add_assoc) 

380 

381 
lemma eq_diff_eq: "(a = cb) = (a + (b::'a::ring) = c)" 

382 
by (auto simp add: diff_minus add_assoc) 

383 

384 
lemma diff_diff_eq: "(a  b)  c = a  (b + (c::'a::ring))" 

385 
by (simp add: diff_minus add_ac) 

386 

387 
lemma diff_diff_eq2: "a  (b  c) = (a + c)  (b::'a::ring)" 

388 
by (simp add: diff_minus add_ac) 

389 

390 
text{*Further subtraction laws for ordered rings*} 

391 

14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

392 
lemma less_iff_diff_less_0: "(a < b) = (a  b < (0::'a::ordered_ring))" 
14270  393 
proof  
394 
have "(a < b) = (a + ( b) < b + (b))" 

395 
by (simp only: add_less_cancel_right) 

396 
also have "... = (a  b < 0)" by (simp add: diff_minus) 

397 
finally show ?thesis . 

398 
qed 

399 

400 
lemma diff_less_eq: "(ab < c) = (a < c + (b::'a::ordered_ring))" 

14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

401 
apply (subst less_iff_diff_less_0) 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

402 
apply (rule less_iff_diff_less_0 [of _ c, THEN ssubst]) 
14270  403 
apply (simp add: diff_minus add_ac) 
404 
done 

405 

406 
lemma less_diff_eq: "(a < cb) = (a + (b::'a::ordered_ring) < c)" 

14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

407 
apply (subst less_iff_diff_less_0) 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

408 
apply (rule less_iff_diff_less_0 [of _ "cb", THEN ssubst]) 
14270  409 
apply (simp add: diff_minus add_ac) 
410 
done 

411 

412 
lemma diff_le_eq: "(ab \<le> c) = (a \<le> c + (b::'a::ordered_ring))" 

413 
by (simp add: linorder_not_less [symmetric] less_diff_eq) 

414 

415 
lemma le_diff_eq: "(a \<le> cb) = (a + (b::'a::ordered_ring) \<le> c)" 

416 
by (simp add: linorder_not_less [symmetric] diff_less_eq) 

417 

418 
text{*This list of rewrites simplifies (in)equalities by bringing subtractions 

419 
to the top and then moving negative terms to the other side. 

420 
Use with @{text add_ac}*} 

421 
lemmas compare_rls = 

422 
diff_minus [symmetric] 

423 
add_diff_eq diff_add_eq diff_diff_eq diff_diff_eq2 

424 
diff_less_eq less_diff_eq diff_le_eq le_diff_eq 

425 
diff_eq_eq eq_diff_eq 

426 

427 

14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

428 
subsection{*Lemmas for the @{text cancel_numerals} simproc*} 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

429 

5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

430 
lemma eq_iff_diff_eq_0: "(a = b) = (ab = (0::'a::ring))" 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

431 
by (simp add: compare_rls) 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

432 

5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

433 
lemma le_iff_diff_le_0: "(a \<le> b) = (ab \<le> (0::'a::ordered_ring))" 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

434 
by (simp add: compare_rls) 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

435 

5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

436 
lemma eq_add_iff1: 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

437 
"(a*e + c = b*e + d) = ((ab)*e + c = (d::'a::ring))" 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

438 
apply (simp add: diff_minus left_distrib add_ac) 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

439 
apply (simp add: compare_rls minus_mult_left [symmetric]) 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

440 
done 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

441 

5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

442 
lemma eq_add_iff2: 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

443 
"(a*e + c = b*e + d) = (c = (ba)*e + (d::'a::ring))" 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

444 
apply (simp add: diff_minus left_distrib add_ac) 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

445 
apply (simp add: compare_rls minus_mult_left [symmetric]) 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

446 
done 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

447 

5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

448 
lemma less_add_iff1: 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

449 
"(a*e + c < b*e + d) = ((ab)*e + c < (d::'a::ordered_ring))" 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

450 
apply (simp add: diff_minus left_distrib add_ac) 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

451 
apply (simp add: compare_rls minus_mult_left [symmetric]) 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

452 
done 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

453 

5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

454 
lemma less_add_iff2: 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

455 
"(a*e + c < b*e + d) = (c < (ba)*e + (d::'a::ordered_ring))" 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

456 
apply (simp add: diff_minus left_distrib add_ac) 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

457 
apply (simp add: compare_rls minus_mult_left [symmetric]) 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

458 
done 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

459 

5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

460 
lemma le_add_iff1: 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

461 
"(a*e + c \<le> b*e + d) = ((ab)*e + c \<le> (d::'a::ordered_ring))" 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

462 
apply (simp add: diff_minus left_distrib add_ac) 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

463 
apply (simp add: compare_rls minus_mult_left [symmetric]) 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

464 
done 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

465 

5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

466 
lemma le_add_iff2: 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

467 
"(a*e + c \<le> b*e + d) = (c \<le> (ba)*e + (d::'a::ordered_ring))" 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

468 
apply (simp add: diff_minus left_distrib add_ac) 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

469 
apply (simp add: compare_rls minus_mult_left [symmetric]) 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

470 
done 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

471 

5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

472 

14270  473 
subsection {* Ordering Rules for Multiplication *} 
474 

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

475 
lemma mult_strict_right_mono: 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

476 
"[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

477 
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

478 

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

479 
lemma mult_left_mono: 
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset

480 
"[a \<le> b; 0 \<le> c] ==> c * a \<le> c * (b::'a::ordered_semiring)" 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

481 
apply (case_tac "c=0", simp) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

482 
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

483 
done 
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

484 

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

485 
lemma mult_right_mono: 
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset

486 
"[a \<le> b; 0 \<le> c] ==> a*c \<le> b * (c::'a::ordered_semiring)" 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

487 
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

488 

14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset

489 
lemma mult_left_le_imp_le: 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset

490 
"[c*a \<le> c*b; 0 < c] ==> a \<le> (b::'a::ordered_semiring)" 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset

491 
by (force simp add: mult_strict_left_mono linorder_not_less [symmetric]) 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset

492 

744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset

493 
lemma mult_right_le_imp_le: 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset

494 
"[a*c \<le> b*c; 0 < c] ==> a \<le> (b::'a::ordered_semiring)" 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset

495 
by (force simp add: mult_strict_right_mono linorder_not_less [symmetric]) 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset

496 

744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset

497 
lemma mult_left_less_imp_less: 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset

498 
"[c*a < c*b; 0 \<le> c] ==> a < (b::'a::ordered_semiring)" 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset

499 
by (force simp add: mult_left_mono linorder_not_le [symmetric]) 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset

500 

744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset

501 
lemma mult_right_less_imp_less: 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset

502 
"[a*c < b*c; 0 \<le> c] ==> a < (b::'a::ordered_semiring)" 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset

503 
by (force simp add: mult_right_mono linorder_not_le [symmetric]) 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset

504 

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

505 
lemma mult_strict_left_mono_neg: 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

506 
"[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

507 
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

508 
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

509 
done 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

510 

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

511 
lemma mult_strict_right_mono_neg: 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

512 
"[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

513 
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

514 
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

515 
done 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

516 

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

517 

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

518 
subsection{* Products of Signs *} 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

519 

14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset

520 
lemma mult_pos: "[ (0::'a::ordered_semiring) < a; 0 < b ] ==> 0 < a*b" 
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

521 
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

522 

14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset

523 
lemma mult_pos_neg: "[ (0::'a::ordered_semiring) < a; b < 0 ] ==> a*b < 0" 
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

524 
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

525 

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

526 
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

527 
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

528 

14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset

529 
lemma zero_less_mult_pos: 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset

530 
"[ 0 < a*b; 0 < a] ==> 0 < (b::'a::ordered_semiring)" 
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

531 
apply (case_tac "b\<le>0") 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

532 
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

533 
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

534 
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

535 
done 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

536 

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

537 
lemma zero_less_mult_iff: 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

538 
"((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

539 
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

540 
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

541 
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

542 
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

543 
done 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

544 

14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset

545 
text{*A field has no "zero divisors", and this theorem holds without the 
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

546 
assumption of an ordering. See @{text field_mult_eq_0_iff} below.*} 
14266  547 
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

548 
apply (case_tac "a < 0") 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

549 
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

550 
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

551 
done 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

552 

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

553 
lemma zero_le_mult_iff: 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

554 
"((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

555 
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

556 
zero_less_mult_iff) 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

557 

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

558 
lemma mult_less_0_iff: 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

559 
"(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

560 
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

561 
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

562 
done 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

563 

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

564 
lemma mult_le_0_iff: 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

565 
"(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

566 
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

567 
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

568 
done 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

569 

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

570 
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

571 
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

572 

14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset

573 
lemma zero_le_one: "(0::'a::ordered_semiring) \<le> 1" 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

574 
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

575 

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

576 

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

577 
subsection{*More Monotonicity*} 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

578 

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

579 
lemma mult_left_mono_neg: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

580 
"[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

581 
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

582 
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

583 
done 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

584 

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

585 
lemma mult_right_mono_neg: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

586 
"[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

587 
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

588 

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

589 
text{*Strict monotonicity in both arguments*} 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

590 
lemma mult_strict_mono: 
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset

591 
"[a<b; c<d; 0<b; 0\<le>c] ==> a * c < b * (d::'a::ordered_semiring)" 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

592 
apply (case_tac "c=0") 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

593 
apply (simp add: mult_pos) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

594 
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

595 
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

596 
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

597 
done 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

598 

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

599 
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

600 
lemma mult_strict_mono': 
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset

601 
"[ a<b; c<d; 0 \<le> a; 0 \<le> c] ==> a * c < b * (d::'a::ordered_semiring)" 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

602 
apply (rule mult_strict_mono) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

603 
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

604 
done 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

605 

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

606 
lemma mult_mono: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

607 
"[a \<le> b; c \<le> d; 0 \<le> b; 0 \<le> c] 
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset

608 
==> a * c \<le> b * (d::'a::ordered_semiring)" 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

609 
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

610 
apply (erule mult_left_mono, assumption) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

611 
done 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

612 

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

613 

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

614 
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

615 

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

616 
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

617 
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

618 

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

619 
lemma mult_less_cancel_right: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

620 
"(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

621 
apply (case_tac "c = 0") 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

622 
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

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

624 
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

625 
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

626 
linorder_not_le [symmetric, of a]) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

627 
apply (erule_tac [!] notE) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

628 
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

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

630 
done 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

631 

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

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

633 
"(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

634 
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

635 

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

636 
lemma mult_le_cancel_right: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

637 
"(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

638 
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

639 

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

640 
lemma mult_le_cancel_left: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

641 
"(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

642 
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

643 

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

644 
lemma mult_less_imp_less_left: 
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset

645 
assumes less: "c*a < c*b" and nonneg: "0 \<le> c" 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset

646 
shows "a < (b::'a::ordered_semiring)" 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset

647 
proof (rule ccontr) 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset

648 
assume "~ a < b" 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset

649 
hence "b \<le> a" by (simp add: linorder_not_less) 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset

650 
hence "c*b \<le> c*a" by (rule mult_left_mono) 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset

651 
with this and less show False 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset

652 
by (simp add: linorder_not_less [symmetric]) 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset

653 
qed 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

654 

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

655 
lemma mult_less_imp_less_right: 
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset

656 
"[a*c < b*c; 0 \<le> c] ==> a < (b::'a::ordered_semiring)" 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset

657 
by (rule mult_less_imp_less_left, simp add: mult_commute) 
14268
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 
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

660 
lemma mult_cancel_right [simp]: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

661 
"(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

662 
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

663 
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

664 
simp add: linorder_neq_iff) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

665 
done 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

666 

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

667 
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

668 
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

669 
lemma mult_cancel_left [simp]: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

670 
"(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

671 
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

672 

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

673 

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

674 
subsection {* Fields *} 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

675 

14288  676 
lemma right_inverse [simp]: 
677 
assumes not0: "a \<noteq> 0" shows "a * inverse (a::'a::field) = 1" 

678 
proof  

679 
have "a * inverse a = inverse a * a" by (simp add: mult_ac) 

680 
also have "... = 1" using not0 by simp 

681 
finally show ?thesis . 

682 
qed 

683 

684 
lemma right_inverse_eq: "b \<noteq> 0 ==> (a / b = 1) = (a = (b::'a::field))" 

685 
proof 

686 
assume neq: "b \<noteq> 0" 

687 
{ 

688 
hence "a = (a / b) * b" by (simp add: divide_inverse mult_ac) 

689 
also assume "a / b = 1" 

690 
finally show "a = b" by simp 

691 
next 

692 
assume "a = b" 

693 
with neq show "a / b = 1" by (simp add: divide_inverse) 

694 
} 

695 
qed 

696 

697 
lemma nonzero_inverse_eq_divide: "a \<noteq> 0 ==> inverse (a::'a::field) = 1/a" 

698 
by (simp add: divide_inverse) 

699 

700 
lemma divide_self [simp]: "a \<noteq> 0 ==> a / (a::'a::field) = 1" 

701 
by (simp add: divide_inverse) 

702 

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

703 
lemma divide_inverse_zero: "a/b = a * inverse(b::'a::{field,division_by_zero})" 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

704 
apply (case_tac "b = 0") 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

705 
apply (simp_all add: divide_inverse) 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

706 
done 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

707 

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

708 
lemma divide_zero_left [simp]: "0/a = (0::'a::{field,division_by_zero})" 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

709 
by (simp add: divide_inverse_zero) 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

710 

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

711 
lemma inverse_eq_divide: "inverse (a::'a::{field,division_by_zero}) = 1/a" 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

712 
by (simp add: divide_inverse_zero) 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

713 

14293  714 
lemma nonzero_add_divide_distrib: "c \<noteq> 0 ==> (a+b)/(c::'a::field) = a/c + b/c" 
715 
by (simp add: divide_inverse left_distrib) 

716 

717 
lemma add_divide_distrib: "(a+b)/(c::'a::{field,division_by_zero}) = a/c + b/c" 

718 
apply (case_tac "c=0", simp) 

719 
apply (simp add: nonzero_add_divide_distrib) 

720 
done 

721 

14270  722 
text{*Compared with @{text mult_eq_0_iff}, this version removes the requirement 
723 
of an ordering.*} 

14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset

724 
lemma field_mult_eq_0_iff [simp]: "(a*b = (0::'a::field)) = (a = 0  b = 0)" 
14270  725 
proof cases 
726 
assume "a=0" thus ?thesis by simp 

727 
next 

728 
assume anz [simp]: "a\<noteq>0" 

729 
thus ?thesis 

730 
proof auto 

731 
assume "a * b = 0" 

732 
hence "inverse a * (a * b) = 0" by simp 

733 
thus "b = 0" by (simp (no_asm_use) add: mult_assoc [symmetric]) 

734 
qed 

735 
qed 

736 

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

737 
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

738 
lemma field_mult_cancel_right_lemma: 
14269  739 
assumes cnz: "c \<noteq> (0::'a::field)" 
740 
and eq: "a*c = b*c" 

741 
shows "a=b" 

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

742 
proof  
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

743 
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

744 
by (simp add: eq) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

745 
thus "a=b" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

746 
by (simp add: mult_assoc cnz) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

747 
qed 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

748 

14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset

749 
lemma field_mult_cancel_right [simp]: 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

750 
"(a*c = b*c) = (c = (0::'a::field)  a=b)" 
14269  751 
proof cases 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

752 
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

753 
next 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

754 
assume "c\<noteq>0" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

755 
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

756 
qed 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

757 

14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset

758 
lemma field_mult_cancel_left [simp]: 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

759 
"(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

760 
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

761 

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

762 
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

763 
proof 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

764 
assume ianz: "inverse a = 0" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

765 
assume "a \<noteq> 0" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

766 
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

767 
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

768 
finally have "1 = (0::'a::field)" . 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

769 
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

770 
qed 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

771 

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

772 

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

773 
subsection{*Basic Properties of @{term inverse}*} 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

774 

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

775 
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

776 
apply (rule ccontr) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

777 
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

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

781 
"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

782 
apply (rule ccontr) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

783 
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

784 
done 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

785 

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

786 
lemma inverse_nonzero_iff_nonzero [simp]: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

787 
"(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

788 
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

789 

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

790 
lemma nonzero_inverse_minus_eq: 
14269  791 
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

792 
proof  
14269  793 
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

794 
by simp 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

795 
thus ?thesis 
14269  796 
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

797 
qed 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

798 

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

799 
lemma inverse_minus_eq [simp]: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

800 
"inverse(a) = inverse(a::'a::{field,division_by_zero})"; 
14269  801 
proof cases 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

802 
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

803 
next 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

804 
assume "a\<noteq>0" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

805 
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

806 
qed 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

807 

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

808 
lemma nonzero_inverse_eq_imp_eq: 
14269  809 
assumes inveq: "inverse a = inverse b" 
810 
and anz: "a \<noteq> 0" 

811 
and bnz: "b \<noteq> 0" 

812 
shows "a = (b::'a::field)" 

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

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

814 
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

815 
by (simp add: inveq) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

816 
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

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

818 
thus "a = b" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

819 
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

820 
qed 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

821 

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

822 
lemma inverse_eq_imp_eq: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

823 
"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

824 
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

825 
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

826 
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

827 
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

828 
done 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

829 

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

830 
lemma inverse_eq_iff_eq [simp]: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

831 
"(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

832 
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

833 

14270  834 
lemma nonzero_inverse_inverse_eq: 
835 
assumes [simp]: "a \<noteq> 0" shows "inverse(inverse (a::'a::field)) = a" 

836 
proof  

837 
have "(inverse (inverse a) * inverse a) * a = a" 

838 
by (simp add: nonzero_imp_inverse_nonzero) 

839 
thus ?thesis 

840 
by (simp add: mult_assoc) 

841 
qed 

842 

843 
lemma inverse_inverse_eq [simp]: 

844 
"inverse(inverse (a::'a::{field,division_by_zero})) = a" 

845 
proof cases 

846 
assume "a=0" thus ?thesis by simp 

847 
next 

848 
assume "a\<noteq>0" 

849 
thus ?thesis by (simp add: nonzero_inverse_inverse_eq) 

850 
qed 

851 

852 
lemma inverse_1 [simp]: "inverse 1 = (1::'a::field)" 

853 
proof  

854 
have "inverse 1 * 1 = (1::'a::field)" 

855 
by (rule left_inverse [OF zero_neq_one [symmetric]]) 

856 
thus ?thesis by simp 

857 
qed 

858 

859 
lemma nonzero_inverse_mult_distrib: 

860 
assumes anz: "a \<noteq> 0" 

861 
and bnz: "b \<noteq> 0" 

862 
shows "inverse(a*b) = inverse(b) * inverse(a::'a::field)" 

863 
proof  

864 
have "inverse(a*b) * (a * b) * inverse(b) = inverse(b)" 

865 
by (simp add: field_mult_eq_0_iff anz bnz) 

866 
hence "inverse(a*b) * a = inverse(b)" 

867 
by (simp add: mult_assoc bnz) 

868 
hence "inverse(a*b) * a * inverse(a) = inverse(b) * inverse(a)" 

869 
by simp 

870 
thus ?thesis 

871 
by (simp add: mult_assoc anz) 

872 
qed 

873 

874 
text{*This version builds in division by zero while also reorienting 

875 
the righthand side.*} 

876 
lemma inverse_mult_distrib [simp]: 

877 
"inverse(a*b) = inverse(a) * inverse(b::'a::{field,division_by_zero})" 

878 
proof cases 

879 
assume "a \<noteq> 0 & b \<noteq> 0" 

880 
thus ?thesis by (simp add: nonzero_inverse_mult_distrib mult_commute) 

881 
next 

882 
assume "~ (a \<noteq> 0 & b \<noteq> 0)" 

883 
thus ?thesis by force 

884 
qed 

885 

886 
text{*There is no slick version using division by zero.*} 

887 
lemma inverse_add: 

888 
"[a \<noteq> 0; b \<noteq> 0] 

889 
==> inverse a + inverse b = (a+b) * inverse a * inverse (b::'a::field)" 

890 
apply (simp add: left_distrib mult_assoc) 

891 
apply (simp add: mult_commute [of "inverse a"]) 

892 
apply (simp add: mult_assoc [symmetric] add_commute) 

893 
done 

894 

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

895 
lemma inverse_divide [simp]: 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

896 
"inverse (a/b) = b / (a::'a::{field,division_by_zero})" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

897 
by (simp add: divide_inverse_zero mult_commute) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

898 

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

899 
lemma nonzero_mult_divide_cancel_left: 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

900 
assumes [simp]: "b\<noteq>0" and [simp]: "c\<noteq>0" 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

901 
shows "(c*a)/(c*b) = a/(b::'a::field)" 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

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

903 
have "(c*a)/(c*b) = c * a * (inverse b * inverse c)" 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

904 
by (simp add: field_mult_eq_0_iff divide_inverse 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

905 
nonzero_inverse_mult_distrib) 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

906 
also have "... = a * inverse b * (inverse c * c)" 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

907 
by (simp only: mult_ac) 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

908 
also have "... = a * inverse b" 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

909 
by simp 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

910 
finally show ?thesis 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

911 
by (simp add: divide_inverse) 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

912 
qed 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

913 

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

914 
lemma mult_divide_cancel_left: 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

915 
"c\<noteq>0 ==> (c*a) / (c*b) = a / (b::'a::{field,division_by_zero})" 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

916 
apply (case_tac "b = 0") 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

917 
apply (simp_all add: nonzero_mult_divide_cancel_left) 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

918 
done 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

919 

14321  920 
lemma nonzero_mult_divide_cancel_right: 
921 
"[b\<noteq>0; c\<noteq>0] ==> (a*c) / (b*c) = a/(b::'a::field)" 

922 
by (simp add: mult_commute [of _ c] nonzero_mult_divide_cancel_left) 

923 

924 
lemma mult_divide_cancel_right: 

925 
"c\<noteq>0 ==> (a*c) / (b*c) = a / (b::'a::{field,division_by_zero})" 

926 
apply (case_tac "b = 0") 

927 
apply (simp_all add: nonzero_mult_divide_cancel_right) 

928 
done 

929 

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

930 
(*For ExtractCommonTerm*) 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

931 
lemma mult_divide_cancel_eq_if: 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

932 
"(c*a) / (c*b) = 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

933 
(if c=0 then 0 else a / (b::'a::{field,division_by_zero}))" 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

934 
by (simp add: mult_divide_cancel_left) 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

935 

14284
f1abe67c448a
reorganisation of Real/RealArith0.ML; more `Isar scripts
paulson
parents:
14277
diff
changeset

936 
lemma divide_1 [simp]: "a/1 = (a::'a::field)" 
f1abe67c448a
reorganisation of Real/RealArith0.ML; more `Isar scripts
paulson
parents:
14277
diff
changeset

937 
by (simp add: divide_inverse [OF not_sym]) 
f1abe67c448a
reorganisation of Real/RealArith0.ML; more `Isar scripts
paulson
parents:
14277
diff
changeset

938 

14288  939 
lemma times_divide_eq_right [simp]: 
940 
"a * (b/c) = (a*b) / (c::'a::{field,division_by_zero})" 

941 
by (simp add: divide_inverse_zero mult_assoc) 

942 

943 
lemma times_divide_eq_left [simp]: 

944 
"(b/c) * a = (b*a) / (c::'a::{field,division_by_zero})" 

945 
by (simp add: divide_inverse_zero mult_ac) 

946 

947 
lemma divide_divide_eq_right [simp]: 

948 
"a / (b/c) = (a*c) / (b::'a::{field,division_by_zero})" 

949 
by (simp add: divide_inverse_zero mult_ac) 

950 

951 
lemma divide_divide_eq_left [simp]: 

952 
"(a / b) / (c::'a::{field,division_by_zero}) = a / (b*c)" 

953 
by (simp add: divide_inverse_zero mult_assoc) 

954 

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

955 

14293  956 
subsection {* Division and Unary Minus *} 
957 

958 
lemma nonzero_minus_divide_left: "b \<noteq> 0 ==>  (a/b) = (a) / (b::'a::field)" 

959 
by (simp add: divide_inverse minus_mult_left) 

960 

961 
lemma nonzero_minus_divide_right: "b \<noteq> 0 ==>  (a/b) = a / (b::'a::field)" 

962 
by (simp add: divide_inverse nonzero_inverse_minus_eq minus_mult_right) 

963 

964 
lemma nonzero_minus_divide_divide: "b \<noteq> 0 ==> (a)/(b) = a / (b::'a::field)" 

965 
by (simp add: divide_inverse nonzero_inverse_minus_eq) 

966 

967 
lemma minus_divide_left: " (a/b) = (a) / (b::'a::{field,division_by_zero})" 

968 
apply (case_tac "b=0", simp) 

969 
apply (simp add: nonzero_minus_divide_left) 

970 
done 

971 

972 
lemma minus_divide_right: " (a/b) = a / (b::'a::{field,division_by_zero})" 

973 
apply (case_tac "b=0", simp) 

974 
by (rule nonzero_minus_divide_right) 

975 

976 
text{*The effect is to extract signs from divisions*} 

977 
declare minus_divide_left [symmetric, simp] 

978 
declare minus_divide_right [symmetric, simp] 

979 

980 
lemma minus_divide_divide [simp]: 

981 
"(a)/(b) = a / (b::'a::{field,division_by_zero})" 

982 
apply (case_tac "b=0", simp) 

983 
apply (simp add: nonzero_minus_divide_divide) 

984 
done 

985 

986 

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

987 
subsection {* Ordered Fields *} 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

988 

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

989 
lemma positive_imp_inverse_positive: 
14269  990 
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

991 
proof  
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

992 
have "0 < a * inverse a" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

993 
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

994 
thus "0 < inverse a" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

995 
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

996 
qed 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

997 

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

998 
lemma negative_imp_inverse_negative: 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

999 
"a < 0 ==> inverse a < (0::'a::ordered_field)" 
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1000 
by (insert positive_imp_inverse_positive [of "a"], 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1001 
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

1002 

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

1003 
lemma inverse_le_imp_le: 
14269  1004 
assumes invle: "inverse a \<le> inverse b" 
1005 
and apos: "0 < a" 

1006 
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

1007 
proof (rule classical) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1008 
assume "~ b \<le> a" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1009 
hence "a < b" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1010 
by (simp add: linorder_not_le) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1011 
hence bpos: "0 < b" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1012 
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

1013 
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

1014 
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

1015 
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

1016 
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

1017 
thus "b \<le> a" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1018 
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

1019 
qed 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1020 

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

1021 
lemma inverse_positive_imp_positive: 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1022 
assumes inv_gt_0: "0 < inverse a" 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1023 
and [simp]: "a \<noteq> 0" 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1024 
shows "0 < (a::'a::ordered_field)" 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1025 
proof  
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1026 
have "0 < inverse (inverse a)" 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1027 
by (rule positive_imp_inverse_positive) 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1028 
thus "0 < a" 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1029 
by (simp add: nonzero_inverse_inverse_eq) 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1030 
qed 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1031 

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

1032 
lemma inverse_positive_iff_positive [simp]: 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1033 
"(0 < inverse a) = (0 < (a::'a::{ordered_field,division_by_zero}))" 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1034 
apply (case_tac "a = 0", simp) 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1035 
apply (blast intro: inverse_positive_imp_positive positive_imp_inverse_positive) 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1036 
done 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1037 

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

1038 
lemma inverse_negative_imp_negative: 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1039 
assumes inv_less_0: "inverse a < 0" 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1040 
and [simp]: "a \<noteq> 0" 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1041 
shows "a < (0::'a::ordered_field)" 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1042 
proof  
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1043 
have "inverse (inverse a) < 0" 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1044 
by (rule negative_imp_inverse_negative) 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1045 
thus "a < 0" 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1046 
by (simp add: nonzero_inverse_inverse_eq) 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1047 
qed 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1048 

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

1049 
lemma inverse_negative_iff_negative [simp]: 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1050 
"(inverse a < 0) = (a < (0::'a::{ordered_field,division_by_zero}))" 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1051 
apply (case_tac "a = 0", simp) 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1052 
apply (blast intro: inverse_negative_imp_negative negative_imp_inverse_negative) 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1053 
done 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1054 

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

1055 
lemma inverse_nonnegative_iff_nonnegative [simp]: 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1056 
"(0 \<le> inverse a) = (0 \<le> (a::'a::{ordered_field,division_by_zero}))" 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1057 
by (simp add: linorder_not_less [symmetric]) 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1058 

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

1059 
lemma inverse_nonpositive_iff_nonpositive [simp]: 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1060 
"(inverse a \<le> 0) = (a \<le> (0::'a::{ordered_field,division_by_zero}))" 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1061 
by (simp add: linorder_not_less [symmetric]) 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1062 

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

1063 

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

1064 
subsection{*AntiMonotonicity of @{term inverse}*} 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1065 

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

1066 
lemma less_imp_inverse_less: 
14269  1067 
assumes less: "a < b" 
1068 
and apos: "0 < a" 

1069 
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

1070 
proof (rule ccontr) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1071 
assume "~ inverse b < inverse a" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1072 
hence "inverse a \<le> inverse b" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1073 
by (simp add: linorder_not_less) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1074 
hence "~ (a < b)" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1075 
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

1076 
thus False 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1077 
by (rule notE [OF _ less]) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1078 
qed 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1079 

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

1080 
lemma inverse_less_imp_less: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1081 
"[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

1082 
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

1083 
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

1084 
done 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1085 

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

1086 
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

1087 
lemma inverse_less_iff_less [simp]: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1088 
"[0 < a; 0 < b] 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1089 
==> (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

1090 
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

1091 

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

1092 
lemma le_imp_inverse_le: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1093 
"[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

1094 
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

1095 

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

1096 
lemma inverse_le_iff_le [simp]: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1097 
"[0 < a; 0 < b] 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1098 
==> (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

1099 
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

1100 

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

1101 

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

1102 
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

1103 
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

1104 
lemma inverse_le_imp_le_neg: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1105 
"[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

1106 
apply (rule classical) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1107 
apply (subgoal_tac "a < 0") 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1108 
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

1109 
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

1110 
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

1111 
done 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1112 

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

1113 
lemma less_imp_inverse_less_neg: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1114 
"[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

1115 
apply (subgoal_tac "a < 0") 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1116 
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

1117 
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

1118 
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

1119 
done 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1120 

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

1121 
lemma inverse_less_imp_less_neg: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1122 
"[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

1123 
apply (rule classical) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1124 
apply (subgoal_tac "a < 0") 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1125 
prefer 2 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1126 
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

1127 
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

1128 
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

1129 
done 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1130 

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

1131 
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

1132 
"[a < 0; b < 0] 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1133 
==> (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

1134 
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

1135 
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

1136 
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

1137 
done 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1138 

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

1139 
lemma le_imp_inverse_le_neg: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1140 
"[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

1141 
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

1142 

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

1143 
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

1144 
"[a < 0; b < 0] 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1145 
==> (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

1146 
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

1147 

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

1148 

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

1149 
subsection{*Inverses and the Number One*} 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1150 

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

1151 
lemma one_less_inverse_iff: 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1152 
"(1 < inverse x) = (0 < x & x < (1::'a::{ordered_field,division_by_zero}))"proof cases 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1153 
assume "0 < x" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1154 
with inverse_less_iff_less [OF zero_less_one, of x] 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1155 
show ?thesis by simp 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1156 
next 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1157 
assume notless: "~ (0 < x)" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1158 
have "~ (1 < inverse x)" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1159 
proof 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1160 
assume "1 < inverse x" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1161 
also with notless have "... \<le> 0" by (simp add: linorder_not_less) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1162 
also have "... < 1" by (rule zero_less_one) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1163 
finally show False by auto 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1164 
qed 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1165 
with notless show ?thesis by simp 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1166 
qed 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1167 

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

1168 
lemma inverse_eq_1_iff [simp]: 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1169 
"(inverse x = 1) = (x = (1::'a::{field,division_by_zero}))" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1170 
by (insert inverse_eq_iff_eq [of x 1], simp) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1171 

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

1172 
lemma one_le_inverse_iff: 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1173 
"(1 \<le> inverse x) = (0 < x & x \<le> (1::'a::{ordered_field,division_by_zero}))" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1174 
by (force simp add: order_le_less one_less_inverse_iff zero_less_one 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1175 
eq_commute [of 1]) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1176 

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

1177 
lemma inverse_less_1_iff: 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1178 
"(inverse x < 1) = (x \<le> 0  1 < (x::'a::{ordered_field,division_by_zero}))" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1179 
by (simp add: linorder_not_le [symmetric] one_le_inverse_iff) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1180 

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

1181 
lemma inverse_le_1_iff: 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1182 
"(inverse x \<le> 1) = (x \<le> 0  1 \<le> (x::'a::{ordered_field,division_by_zero}))" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1183 
by (simp add: linorder_not_less [symmetric] one_less_inverse_iff) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1184 

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

1185 

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

1186 
subsection{*Division and Signs*} 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1187 

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

1188 
lemma zero_less_divide_iff: 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1189 
"((0::'a::{ordered_field,division_by_zero}) < a/b) = (0 < a & 0 < b  a < 0 & b < 0)" 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1190 
by (simp add: divide_inverse_zero zero_less_mult_iff) 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1191 

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

1192 
lemma divide_less_0_iff: 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1193 
"(a/b < (0::'a::{ordered_field,division_by_zero})) = 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1194 
(0 < a & b < 0  a < 0 & 0 < b)" 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1195 
by (simp add: divide_inverse_zero mult_less_0_iff) 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1196 

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

1197 
lemma zero_le_divide_iff: 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1198 
"((0::'a::{ordered_field,division_by_zero}) \<le> a/b) = 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1199 
(0 \<le> a & 0 \<le> b  a \<le> 0 & b \<le> 0)" 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1200 
by (simp add: divide_inverse_zero zero_le_mult_iff) 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1201 

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

1202 
lemma divide_le_0_iff: 
14288  1203 
"(a/b \<le> (0::'a::{ordered_field,division_by_zero})) = 
1204 
(0 \<le> a & b \<le> 0  a \<le> 0 & 0 \<le> b)" 

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

1205 
by (simp add: divide_inverse_zero mult_le_0_iff) 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1206 

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

1207 
lemma divide_eq_0_iff [simp]: 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1208 
"(a/b = 0) = (a=0  b=(0::'a::{field,division_by_zero}))" 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1209 
by (simp add: divide_inverse_zero field_mult_eq_0_iff) 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1210 

14288  1211 

1212 
subsection{*Simplification of Inequalities Involving Literal Divisors*} 

1213 

1214 
lemma pos_le_divide_eq: "0 < (c::'a::ordered_field) ==> (a \<le> b/c) = (a*c \<le> b)" 

1215 
proof  

1216 
assume less: "0<c" 

1217 
hence "(a \<le> b/c) = (a*c \<le> (b/c)*c)" 

1218 
by (simp add: mult_le_cancel_right order_less_not_sym [OF less]) 

1219 
also have "... = (a*c \<le> b)" 

1220 
by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 

1221 
finally show ?thesis . 

1222 
qed 

1223 

1224 

1225 
lemma neg_le_divide_eq: "c < (0::'a::ordered_field) ==> (a \<le> b/c) = (b \<le> a*c)" 

1226 
proof  

1227 
assume less: "c<0" 

1228 
hence "(a \<le> b/c) = ((b/c)*c \<le> a*c)" 

1229 
by (simp add: mult_le_cancel_right order_less_not_sym [OF less]) 

1230 
also have "... = (b \<le> a*c)" 

1231 
by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 

1232 
finally show ?thesis . 

1233 
qed 

1234 

1235 
lemma le_divide_eq: 

1236 
"(a \<le> b/c) = 

1237 
(if 0 < c then a*c \<le> b 

1238 
else if c < 0 then b \<le> a*c 

1239 
else a \<le> (0::'a::{ordered_field,division_by_zero}))" 

1240 
apply (case_tac "c=0", simp) 

1241 
apply (force simp add: pos_le_divide_eq neg_le_divide_eq linorder_neq_iff) 

1242 
done 

1243 

1244 
lemma pos_divide_le_eq: "0 < (c::'a::ordered_field) ==> (b/c \<le> a) = (b \<le> a*c)" 

1245 
proof  

1246 
assume less: "0<c" 

1247 
hence "(b/c \<le> a) = ((b/c)*c \<le> a*c)" 

1248 
by (simp add: mult_le_cancel_right order_less_not_sym [OF less]) 

1249 
also have "... = (b \<le> a*c)" 

1250 
by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 

1251 
finally show ?thesis . 

1252 
qed 

1253 

1254 
lemma neg_divide_le_eq: "c < (0::'a::ordered_field) ==> (b/c \<le> a) = (a*c \<le> b)" 

1255 
proof  

1256 
assume less: "c<0" 

1257 
hence "(b/c \<le> a) = (a*c \<le> (b/c)*c)" 

1258 
by (simp add: mult_le_cancel_right order_less_not_sym [OF less]) 

1259 
also have "... = (a*c \<le> b)" 

1260 
by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 

1261 
finally show ?thesis . 

1262 
qed 

1263 

1264 
lemma divide_le_eq: 

1265 
"(b/c \<le> a) = 

1266 
(if 0 < c then b \<le> a*c 

1267 
else if c < 0 then a*c \<le> b 

1268 
else 0 \<le> (a::'a::{ordered_field,division_by_zero}))" 

1269 
apply (case_tac "c=0", simp) 

1270 
apply (force simp add: pos_divide_le_eq neg_divide_le_eq linorder_neq_iff) 

1271 
done 

1272 

1273 

1274 
lemma pos_less_divide_eq: 

1275 
"0 < (c::'a::ordered_field) ==> (a < b/c) = (a*c < b)" 

1276 
proof  

1277 
assume less: "0<c" 

1278 
hence "(a < b/c) = (a*c < (b/c)*c)" 

1279 
by (simp add: mult_less_cancel_right order_less_not_sym [OF less]) 

1280 
also have "... = (a*c < b)" 

1281 
by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 

1282 
finally show ?thesis . 

1283 
qed 

1284 

1285 
lemma neg_less_divide_eq: 

1286 
"c < (0::'a::ordered_field) ==> (a < b/c) = (b < a*c)" 

1287 
proof  

1288 
assume less: "c<0" 

1289 
hence "(a < b/c) = ((b/c)*c < a*c)" 

1290 
by (simp add: mult_less_cancel_right order_less_not_sym [OF less]) 

1291 
also have "... = (b < a*c)" 

1292 
by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 

1293 
finally show ?thesis . 

1294 
qed 

1295 

1296 
lemma less_divide_eq: 

1297 
"(a < b/c) = 

1298 
(if 0 < c then a*c < b 

1299 
else if c < 0 then b < a*c 

1300 
else a < (0::'a::{ordered_field,division_by_zero}))" 

1301 
apply (case_tac "c=0", simp) 

1302 
apply (force simp add: pos_less_divide_eq neg_less_divide_eq linorder_neq_iff) 

1303 
done 

1304 
