author | wenzelm |
Sat, 13 Mar 2010 14:44:47 +0100 | |
changeset 35743 | c506c029a082 |
parent 35631 | 0b8a5fd339ab |
child 35828 | 46cfc4b8112e |
permissions | -rw-r--r-- |
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(* Title: HOL/Rings.thy |
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Author: Gertrud Bauer |
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3 |
Author: Steven Obua |
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Author: Tobias Nipkow |
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Author: Lawrence C Paulson |
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6 |
Author: Markus Wenzel |
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7 |
Author: Jeremy Avigad |
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*) |
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9 |
|
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header {* Rings *} |
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|
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theory Rings |
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13 |
imports Groups |
15131 | 14 |
begin |
14504 | 15 |
|
22390 | 16 |
class semiring = ab_semigroup_add + semigroup_mult + |
29667 | 17 |
assumes left_distrib[algebra_simps]: "(a + b) * c = a * c + b * c" |
18 |
assumes right_distrib[algebra_simps]: "a * (b + c) = a * b + a * c" |
|
25152 | 19 |
begin |
20 |
||
21 |
text{*For the @{text combine_numerals} simproc*} |
|
22 |
lemma combine_common_factor: |
|
23 |
"a * e + (b * e + c) = (a + b) * e + c" |
|
29667 | 24 |
by (simp add: left_distrib add_ac) |
25152 | 25 |
|
26 |
end |
|
14504 | 27 |
|
22390 | 28 |
class mult_zero = times + zero + |
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assumes mult_zero_left [simp]: "0 * a = 0" |
30 |
assumes mult_zero_right [simp]: "a * 0 = 0" |
|
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|
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class semiring_0 = semiring + comm_monoid_add + mult_zero |
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|
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class semiring_0_cancel = semiring + cancel_comm_monoid_add |
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begin |
14504 | 36 |
|
25186 | 37 |
subclass semiring_0 |
28823 | 38 |
proof |
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fix a :: 'a |
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have "0 * a + 0 * a = 0 * a + 0" by (simp add: left_distrib [symmetric]) |
41 |
thus "0 * a = 0" by (simp only: add_left_cancel) |
|
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next |
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fix a :: 'a |
|
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have "a * 0 + a * 0 = a * 0 + 0" by (simp add: right_distrib [symmetric]) |
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thus "a * 0 = 0" by (simp only: add_left_cancel) |
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qed |
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|
25186 | 48 |
end |
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|
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class comm_semiring = ab_semigroup_add + ab_semigroup_mult + |
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assumes distrib: "(a + b) * c = a * c + b * c" |
25152 | 52 |
begin |
14504 | 53 |
|
25152 | 54 |
subclass semiring |
28823 | 55 |
proof |
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fix a b c :: 'a |
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show "(a + b) * c = a * c + b * c" by (simp add: distrib) |
|
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have "a * (b + c) = (b + c) * a" by (simp add: mult_ac) |
|
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also have "... = b * a + c * a" by (simp only: distrib) |
|
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also have "... = a * b + a * c" by (simp add: mult_ac) |
|
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finally show "a * (b + c) = a * b + a * c" by blast |
|
14504 | 62 |
qed |
63 |
||
25152 | 64 |
end |
14504 | 65 |
|
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class comm_semiring_0 = comm_semiring + comm_monoid_add + mult_zero |
67 |
begin |
|
68 |
||
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subclass semiring_0 .. |
25152 | 70 |
|
71 |
end |
|
14504 | 72 |
|
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class comm_semiring_0_cancel = comm_semiring + cancel_comm_monoid_add |
25186 | 74 |
begin |
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|
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subclass semiring_0_cancel .. |
14940 | 77 |
|
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subclass comm_semiring_0 .. |
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instances comm_semiring_0_cancel < comm_semiring_0, comm_ring < comm_semiring_0_cancel
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79 |
|
25186 | 80 |
end |
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81 |
|
22390 | 82 |
class zero_neq_one = zero + one + |
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assumes zero_neq_one [simp]: "0 \<noteq> 1" |
26193 | 84 |
begin |
85 |
||
86 |
lemma one_neq_zero [simp]: "1 \<noteq> 0" |
|
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by (rule not_sym) (rule zero_neq_one) |
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|
89 |
end |
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|
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class semiring_1 = zero_neq_one + semiring_0 + monoid_mult |
14504 | 92 |
|
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text {* Abstract divisibility *} |
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94 |
|
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class dvd = times |
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begin |
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97 |
|
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definition dvd :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infixl "dvd" 50) where |
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[code del]: "b dvd a \<longleftrightarrow> (\<exists>k. a = b * k)" |
|
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100 |
|
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lemma dvdI [intro?]: "a = b * k \<Longrightarrow> b dvd a" |
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102 |
unfolding dvd_def .. |
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103 |
|
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lemma dvdE [elim?]: "b dvd a \<Longrightarrow> (\<And>k. a = b * k \<Longrightarrow> P) \<Longrightarrow> P" |
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unfolding dvd_def by blast |
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106 |
|
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107 |
end |
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108 |
|
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109 |
class comm_semiring_1 = zero_neq_one + comm_semiring_0 + comm_monoid_mult + dvd |
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(*previously almost_semiring*) |
25152 | 111 |
begin |
14738 | 112 |
|
27516 | 113 |
subclass semiring_1 .. |
25152 | 114 |
|
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lemma dvd_refl[simp]: "a dvd a" |
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proof |
117 |
show "a = a * 1" by simp |
|
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118 |
qed |
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119 |
|
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lemma dvd_trans: |
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assumes "a dvd b" and "b dvd c" |
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shows "a dvd c" |
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123 |
proof - |
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from assms obtain v where "b = a * v" by (auto elim!: dvdE) |
125 |
moreover from assms obtain w where "c = b * w" by (auto elim!: dvdE) |
|
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ultimately have "c = a * (v * w)" by (simp add: mult_assoc) |
28559 | 127 |
then show ?thesis .. |
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128 |
qed |
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129 |
|
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lemma dvd_0_left_iff [noatp, simp]: "0 dvd a \<longleftrightarrow> a = 0" |
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by (auto intro: dvd_refl elim!: dvdE) |
28559 | 132 |
|
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lemma dvd_0_right [iff]: "a dvd 0" |
|
134 |
proof |
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show "0 = a * 0" by simp |
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136 |
qed |
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137 |
|
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lemma one_dvd [simp]: "1 dvd a" |
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by (auto intro!: dvdI) |
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140 |
|
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lemma dvd_mult[simp]: "a dvd c \<Longrightarrow> a dvd (b * c)" |
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by (auto intro!: mult_left_commute dvdI elim!: dvdE) |
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143 |
|
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lemma dvd_mult2[simp]: "a dvd b \<Longrightarrow> a dvd (b * c)" |
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apply (subst mult_commute) |
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apply (erule dvd_mult) |
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done |
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148 |
|
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lemma dvd_triv_right [simp]: "a dvd b * a" |
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by (rule dvd_mult) (rule dvd_refl) |
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151 |
|
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lemma dvd_triv_left [simp]: "a dvd a * b" |
29667 | 153 |
by (rule dvd_mult2) (rule dvd_refl) |
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154 |
|
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lemma mult_dvd_mono: |
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assumes "a dvd b" |
157 |
and "c dvd d" |
|
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shows "a * c dvd b * d" |
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159 |
proof - |
30042 | 160 |
from `a dvd b` obtain b' where "b = a * b'" .. |
161 |
moreover from `c dvd d` obtain d' where "d = c * d'" .. |
|
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ultimately have "b * d = (a * c) * (b' * d')" by (simp add: mult_ac) |
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163 |
then show ?thesis .. |
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164 |
qed |
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165 |
|
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lemma dvd_mult_left: "a * b dvd c \<Longrightarrow> a dvd c" |
29667 | 167 |
by (simp add: dvd_def mult_assoc, blast) |
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168 |
|
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lemma dvd_mult_right: "a * b dvd c \<Longrightarrow> b dvd c" |
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unfolding mult_ac [of a] by (rule dvd_mult_left) |
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171 |
|
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172 |
lemma dvd_0_left: "0 dvd a \<Longrightarrow> a = 0" |
29667 | 173 |
by simp |
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174 |
|
29925 | 175 |
lemma dvd_add[simp]: |
176 |
assumes "a dvd b" and "a dvd c" shows "a dvd (b + c)" |
|
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177 |
proof - |
29925 | 178 |
from `a dvd b` obtain b' where "b = a * b'" .. |
179 |
moreover from `a dvd c` obtain c' where "c = a * c'" .. |
|
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ultimately have "b + c = a * (b' + c')" by (simp add: right_distrib) |
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181 |
then show ?thesis .. |
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182 |
qed |
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183 |
|
25152 | 184 |
end |
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185 |
|
29925 | 186 |
|
22390 | 187 |
class no_zero_divisors = zero + times + |
25062 | 188 |
assumes no_zero_divisors: "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> a * b \<noteq> 0" |
14504 | 189 |
|
29904 | 190 |
class semiring_1_cancel = semiring + cancel_comm_monoid_add |
191 |
+ zero_neq_one + monoid_mult |
|
25267 | 192 |
begin |
14940 | 193 |
|
27516 | 194 |
subclass semiring_0_cancel .. |
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195 |
|
27516 | 196 |
subclass semiring_1 .. |
25267 | 197 |
|
198 |
end |
|
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|
199 |
|
29904 | 200 |
class comm_semiring_1_cancel = comm_semiring + cancel_comm_monoid_add |
201 |
+ zero_neq_one + comm_monoid_mult |
|
25267 | 202 |
begin |
14738 | 203 |
|
27516 | 204 |
subclass semiring_1_cancel .. |
205 |
subclass comm_semiring_0_cancel .. |
|
206 |
subclass comm_semiring_1 .. |
|
25267 | 207 |
|
208 |
end |
|
25152 | 209 |
|
22390 | 210 |
class ring = semiring + ab_group_add |
25267 | 211 |
begin |
25152 | 212 |
|
27516 | 213 |
subclass semiring_0_cancel .. |
25152 | 214 |
|
215 |
text {* Distribution rules *} |
|
216 |
||
217 |
lemma minus_mult_left: "- (a * b) = - a * b" |
|
34146
14595e0c27e8
rename equals_zero_I to minus_unique (keep old name too)
huffman
parents:
33676
diff
changeset
|
218 |
by (rule minus_unique) (simp add: left_distrib [symmetric]) |
25152 | 219 |
|
220 |
lemma minus_mult_right: "- (a * b) = a * - b" |
|
34146
14595e0c27e8
rename equals_zero_I to minus_unique (keep old name too)
huffman
parents:
33676
diff
changeset
|
221 |
by (rule minus_unique) (simp add: right_distrib [symmetric]) |
25152 | 222 |
|
29407
5ef7e97fd9e4
move lemmas mult_minus{left,right} inside class ring
huffman
parents:
29406
diff
changeset
|
223 |
text{*Extract signs from products*} |
29833 | 224 |
lemmas mult_minus_left [simp, noatp] = minus_mult_left [symmetric] |
225 |
lemmas mult_minus_right [simp,noatp] = minus_mult_right [symmetric] |
|
29407
5ef7e97fd9e4
move lemmas mult_minus{left,right} inside class ring
huffman
parents:
29406
diff
changeset
|
226 |
|
25152 | 227 |
lemma minus_mult_minus [simp]: "- a * - b = a * b" |
29667 | 228 |
by simp |
25152 | 229 |
|
230 |
lemma minus_mult_commute: "- a * b = a * - b" |
|
29667 | 231 |
by simp |
232 |
||
233 |
lemma right_diff_distrib[algebra_simps]: "a * (b - c) = a * b - a * c" |
|
234 |
by (simp add: right_distrib diff_minus) |
|
235 |
||
236 |
lemma left_diff_distrib[algebra_simps]: "(a - b) * c = a * c - b * c" |
|
237 |
by (simp add: left_distrib diff_minus) |
|
25152 | 238 |
|
29833 | 239 |
lemmas ring_distribs[noatp] = |
25152 | 240 |
right_distrib left_distrib left_diff_distrib right_diff_distrib |
241 |
||
29667 | 242 |
text{*Legacy - use @{text algebra_simps} *} |
29833 | 243 |
lemmas ring_simps[noatp] = algebra_simps |
25230 | 244 |
|
245 |
lemma eq_add_iff1: |
|
246 |
"a * e + c = b * e + d \<longleftrightarrow> (a - b) * e + c = d" |
|
29667 | 247 |
by (simp add: algebra_simps) |
25230 | 248 |
|
249 |
lemma eq_add_iff2: |
|
250 |
"a * e + c = b * e + d \<longleftrightarrow> c = (b - a) * e + d" |
|
29667 | 251 |
by (simp add: algebra_simps) |
25230 | 252 |
|
25152 | 253 |
end |
254 |
||
29833 | 255 |
lemmas ring_distribs[noatp] = |
25152 | 256 |
right_distrib left_distrib left_diff_distrib right_diff_distrib |
257 |
||
22390 | 258 |
class comm_ring = comm_semiring + ab_group_add |
25267 | 259 |
begin |
14738 | 260 |
|
27516 | 261 |
subclass ring .. |
28141
193c3ea0f63b
instances comm_semiring_0_cancel < comm_semiring_0, comm_ring < comm_semiring_0_cancel
huffman
parents:
27651
diff
changeset
|
262 |
subclass comm_semiring_0_cancel .. |
25267 | 263 |
|
264 |
end |
|
14738 | 265 |
|
22390 | 266 |
class ring_1 = ring + zero_neq_one + monoid_mult |
25267 | 267 |
begin |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
268 |
|
27516 | 269 |
subclass semiring_1_cancel .. |
25267 | 270 |
|
271 |
end |
|
25152 | 272 |
|
22390 | 273 |
class comm_ring_1 = comm_ring + zero_neq_one + comm_monoid_mult |
274 |
(*previously ring*) |
|
25267 | 275 |
begin |
14738 | 276 |
|
27516 | 277 |
subclass ring_1 .. |
278 |
subclass comm_semiring_1_cancel .. |
|
25267 | 279 |
|
29465
b2cfb5d0a59e
change dvd_minus_iff, minus_dvd_iff from [iff] to [simp] (due to problems with Library/Primes.thy)
huffman
parents:
29461
diff
changeset
|
280 |
lemma dvd_minus_iff [simp]: "x dvd - y \<longleftrightarrow> x dvd y" |
29408
6d10cf26b5dc
add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents:
29407
diff
changeset
|
281 |
proof |
6d10cf26b5dc
add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents:
29407
diff
changeset
|
282 |
assume "x dvd - y" |
6d10cf26b5dc
add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents:
29407
diff
changeset
|
283 |
then have "x dvd - 1 * - y" by (rule dvd_mult) |
6d10cf26b5dc
add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents:
29407
diff
changeset
|
284 |
then show "x dvd y" by simp |
6d10cf26b5dc
add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents:
29407
diff
changeset
|
285 |
next |
6d10cf26b5dc
add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents:
29407
diff
changeset
|
286 |
assume "x dvd y" |
6d10cf26b5dc
add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents:
29407
diff
changeset
|
287 |
then have "x dvd - 1 * y" by (rule dvd_mult) |
6d10cf26b5dc
add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents:
29407
diff
changeset
|
288 |
then show "x dvd - y" by simp |
6d10cf26b5dc
add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents:
29407
diff
changeset
|
289 |
qed |
6d10cf26b5dc
add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents:
29407
diff
changeset
|
290 |
|
29465
b2cfb5d0a59e
change dvd_minus_iff, minus_dvd_iff from [iff] to [simp] (due to problems with Library/Primes.thy)
huffman
parents:
29461
diff
changeset
|
291 |
lemma minus_dvd_iff [simp]: "- x dvd y \<longleftrightarrow> x dvd y" |
29408
6d10cf26b5dc
add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents:
29407
diff
changeset
|
292 |
proof |
6d10cf26b5dc
add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents:
29407
diff
changeset
|
293 |
assume "- x dvd y" |
6d10cf26b5dc
add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents:
29407
diff
changeset
|
294 |
then obtain k where "y = - x * k" .. |
6d10cf26b5dc
add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents:
29407
diff
changeset
|
295 |
then have "y = x * - k" by simp |
6d10cf26b5dc
add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents:
29407
diff
changeset
|
296 |
then show "x dvd y" .. |
6d10cf26b5dc
add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents:
29407
diff
changeset
|
297 |
next |
6d10cf26b5dc
add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents:
29407
diff
changeset
|
298 |
assume "x dvd y" |
6d10cf26b5dc
add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents:
29407
diff
changeset
|
299 |
then obtain k where "y = x * k" .. |
6d10cf26b5dc
add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents:
29407
diff
changeset
|
300 |
then have "y = - x * - k" by simp |
6d10cf26b5dc
add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents:
29407
diff
changeset
|
301 |
then show "- x dvd y" .. |
6d10cf26b5dc
add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents:
29407
diff
changeset
|
302 |
qed |
6d10cf26b5dc
add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents:
29407
diff
changeset
|
303 |
|
30042 | 304 |
lemma dvd_diff[simp]: "x dvd y \<Longrightarrow> x dvd z \<Longrightarrow> x dvd (y - z)" |
35216 | 305 |
by (simp only: diff_minus dvd_add dvd_minus_iff) |
29409 | 306 |
|
25267 | 307 |
end |
25152 | 308 |
|
22990
775e9de3db48
added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents:
22987
diff
changeset
|
309 |
class ring_no_zero_divisors = ring + no_zero_divisors |
25230 | 310 |
begin |
311 |
||
312 |
lemma mult_eq_0_iff [simp]: |
|
313 |
shows "a * b = 0 \<longleftrightarrow> (a = 0 \<or> b = 0)" |
|
314 |
proof (cases "a = 0 \<or> b = 0") |
|
315 |
case False then have "a \<noteq> 0" and "b \<noteq> 0" by auto |
|
316 |
then show ?thesis using no_zero_divisors by simp |
|
317 |
next |
|
318 |
case True then show ?thesis by auto |
|
319 |
qed |
|
320 |
||
26193 | 321 |
text{*Cancellation of equalities with a common factor*} |
322 |
lemma mult_cancel_right [simp, noatp]: |
|
323 |
"a * c = b * c \<longleftrightarrow> c = 0 \<or> a = b" |
|
324 |
proof - |
|
325 |
have "(a * c = b * c) = ((a - b) * c = 0)" |
|
35216 | 326 |
by (simp add: algebra_simps) |
327 |
thus ?thesis by (simp add: disj_commute) |
|
26193 | 328 |
qed |
329 |
||
330 |
lemma mult_cancel_left [simp, noatp]: |
|
331 |
"c * a = c * b \<longleftrightarrow> c = 0 \<or> a = b" |
|
332 |
proof - |
|
333 |
have "(c * a = c * b) = (c * (a - b) = 0)" |
|
35216 | 334 |
by (simp add: algebra_simps) |
335 |
thus ?thesis by simp |
|
26193 | 336 |
qed |
337 |
||
25230 | 338 |
end |
22990
775e9de3db48
added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents:
22987
diff
changeset
|
339 |
|
23544 | 340 |
class ring_1_no_zero_divisors = ring_1 + ring_no_zero_divisors |
26274 | 341 |
begin |
342 |
||
343 |
lemma mult_cancel_right1 [simp]: |
|
344 |
"c = b * c \<longleftrightarrow> c = 0 \<or> b = 1" |
|
29667 | 345 |
by (insert mult_cancel_right [of 1 c b], force) |
26274 | 346 |
|
347 |
lemma mult_cancel_right2 [simp]: |
|
348 |
"a * c = c \<longleftrightarrow> c = 0 \<or> a = 1" |
|
29667 | 349 |
by (insert mult_cancel_right [of a c 1], simp) |
26274 | 350 |
|
351 |
lemma mult_cancel_left1 [simp]: |
|
352 |
"c = c * b \<longleftrightarrow> c = 0 \<or> b = 1" |
|
29667 | 353 |
by (insert mult_cancel_left [of c 1 b], force) |
26274 | 354 |
|
355 |
lemma mult_cancel_left2 [simp]: |
|
356 |
"c * a = c \<longleftrightarrow> c = 0 \<or> a = 1" |
|
29667 | 357 |
by (insert mult_cancel_left [of c a 1], simp) |
26274 | 358 |
|
359 |
end |
|
22990
775e9de3db48
added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents:
22987
diff
changeset
|
360 |
|
22390 | 361 |
class idom = comm_ring_1 + no_zero_divisors |
25186 | 362 |
begin |
14421
ee97b6463cb4
new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents:
14398
diff
changeset
|
363 |
|
27516 | 364 |
subclass ring_1_no_zero_divisors .. |
22990
775e9de3db48
added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents:
22987
diff
changeset
|
365 |
|
29915
2146e512cec9
generalize lemma fps_square_eq_iff, move to Ring_and_Field
huffman
parents:
29904
diff
changeset
|
366 |
lemma square_eq_iff: "a * a = b * b \<longleftrightarrow> (a = b \<or> a = - b)" |
2146e512cec9
generalize lemma fps_square_eq_iff, move to Ring_and_Field
huffman
parents:
29904
diff
changeset
|
367 |
proof |
2146e512cec9
generalize lemma fps_square_eq_iff, move to Ring_and_Field
huffman
parents:
29904
diff
changeset
|
368 |
assume "a * a = b * b" |
2146e512cec9
generalize lemma fps_square_eq_iff, move to Ring_and_Field
huffman
parents:
29904
diff
changeset
|
369 |
then have "(a - b) * (a + b) = 0" |
2146e512cec9
generalize lemma fps_square_eq_iff, move to Ring_and_Field
huffman
parents:
29904
diff
changeset
|
370 |
by (simp add: algebra_simps) |
2146e512cec9
generalize lemma fps_square_eq_iff, move to Ring_and_Field
huffman
parents:
29904
diff
changeset
|
371 |
then show "a = b \<or> a = - b" |
35216 | 372 |
by (simp add: eq_neg_iff_add_eq_0) |
29915
2146e512cec9
generalize lemma fps_square_eq_iff, move to Ring_and_Field
huffman
parents:
29904
diff
changeset
|
373 |
next |
2146e512cec9
generalize lemma fps_square_eq_iff, move to Ring_and_Field
huffman
parents:
29904
diff
changeset
|
374 |
assume "a = b \<or> a = - b" |
2146e512cec9
generalize lemma fps_square_eq_iff, move to Ring_and_Field
huffman
parents:
29904
diff
changeset
|
375 |
then show "a * a = b * b" by auto |
2146e512cec9
generalize lemma fps_square_eq_iff, move to Ring_and_Field
huffman
parents:
29904
diff
changeset
|
376 |
qed |
2146e512cec9
generalize lemma fps_square_eq_iff, move to Ring_and_Field
huffman
parents:
29904
diff
changeset
|
377 |
|
29981
7d0ed261b712
generalize int_dvd_cancel_factor simproc to idom class
huffman
parents:
29949
diff
changeset
|
378 |
lemma dvd_mult_cancel_right [simp]: |
7d0ed261b712
generalize int_dvd_cancel_factor simproc to idom class
huffman
parents:
29949
diff
changeset
|
379 |
"a * c dvd b * c \<longleftrightarrow> c = 0 \<or> a dvd b" |
7d0ed261b712
generalize int_dvd_cancel_factor simproc to idom class
huffman
parents:
29949
diff
changeset
|
380 |
proof - |
7d0ed261b712
generalize int_dvd_cancel_factor simproc to idom class
huffman
parents:
29949
diff
changeset
|
381 |
have "a * c dvd b * c \<longleftrightarrow> (\<exists>k. b * c = (a * k) * c)" |
7d0ed261b712
generalize int_dvd_cancel_factor simproc to idom class
huffman
parents:
29949
diff
changeset
|
382 |
unfolding dvd_def by (simp add: mult_ac) |
7d0ed261b712
generalize int_dvd_cancel_factor simproc to idom class
huffman
parents:
29949
diff
changeset
|
383 |
also have "(\<exists>k. b * c = (a * k) * c) \<longleftrightarrow> c = 0 \<or> a dvd b" |
7d0ed261b712
generalize int_dvd_cancel_factor simproc to idom class
huffman
parents:
29949
diff
changeset
|
384 |
unfolding dvd_def by simp |
7d0ed261b712
generalize int_dvd_cancel_factor simproc to idom class
huffman
parents:
29949
diff
changeset
|
385 |
finally show ?thesis . |
7d0ed261b712
generalize int_dvd_cancel_factor simproc to idom class
huffman
parents:
29949
diff
changeset
|
386 |
qed |
7d0ed261b712
generalize int_dvd_cancel_factor simproc to idom class
huffman
parents:
29949
diff
changeset
|
387 |
|
7d0ed261b712
generalize int_dvd_cancel_factor simproc to idom class
huffman
parents:
29949
diff
changeset
|
388 |
lemma dvd_mult_cancel_left [simp]: |
7d0ed261b712
generalize int_dvd_cancel_factor simproc to idom class
huffman
parents:
29949
diff
changeset
|
389 |
"c * a dvd c * b \<longleftrightarrow> c = 0 \<or> a dvd b" |
7d0ed261b712
generalize int_dvd_cancel_factor simproc to idom class
huffman
parents:
29949
diff
changeset
|
390 |
proof - |
7d0ed261b712
generalize int_dvd_cancel_factor simproc to idom class
huffman
parents:
29949
diff
changeset
|
391 |
have "c * a dvd c * b \<longleftrightarrow> (\<exists>k. b * c = (a * k) * c)" |
7d0ed261b712
generalize int_dvd_cancel_factor simproc to idom class
huffman
parents:
29949
diff
changeset
|
392 |
unfolding dvd_def by (simp add: mult_ac) |
7d0ed261b712
generalize int_dvd_cancel_factor simproc to idom class
huffman
parents:
29949
diff
changeset
|
393 |
also have "(\<exists>k. b * c = (a * k) * c) \<longleftrightarrow> c = 0 \<or> a dvd b" |
7d0ed261b712
generalize int_dvd_cancel_factor simproc to idom class
huffman
parents:
29949
diff
changeset
|
394 |
unfolding dvd_def by simp |
7d0ed261b712
generalize int_dvd_cancel_factor simproc to idom class
huffman
parents:
29949
diff
changeset
|
395 |
finally show ?thesis . |
7d0ed261b712
generalize int_dvd_cancel_factor simproc to idom class
huffman
parents:
29949
diff
changeset
|
396 |
qed |
7d0ed261b712
generalize int_dvd_cancel_factor simproc to idom class
huffman
parents:
29949
diff
changeset
|
397 |
|
25186 | 398 |
end |
25152 | 399 |
|
35083 | 400 |
class inverse = |
401 |
fixes inverse :: "'a \<Rightarrow> 'a" |
|
402 |
and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "'/" 70) |
|
403 |
||
22390 | 404 |
class division_ring = ring_1 + inverse + |
25062 | 405 |
assumes left_inverse [simp]: "a \<noteq> 0 \<Longrightarrow> inverse a * a = 1" |
406 |
assumes right_inverse [simp]: "a \<noteq> 0 \<Longrightarrow> a * inverse a = 1" |
|
35083 | 407 |
assumes divide_inverse: "a / b = a * inverse b" |
25186 | 408 |
begin |
20496
23eb6034c06d
added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents:
19404
diff
changeset
|
409 |
|
25186 | 410 |
subclass ring_1_no_zero_divisors |
28823 | 411 |
proof |
22987
550709aa8e66
instance division_ring < no_zero_divisors; clean up field instance proofs
huffman
parents:
22842
diff
changeset
|
412 |
fix a b :: 'a |
550709aa8e66
instance division_ring < no_zero_divisors; clean up field instance proofs
huffman
parents:
22842
diff
changeset
|
413 |
assume a: "a \<noteq> 0" and b: "b \<noteq> 0" |
550709aa8e66
instance division_ring < no_zero_divisors; clean up field instance proofs
huffman
parents:
22842
diff
changeset
|
414 |
show "a * b \<noteq> 0" |
550709aa8e66
instance division_ring < no_zero_divisors; clean up field instance proofs
huffman
parents:
22842
diff
changeset
|
415 |
proof |
550709aa8e66
instance division_ring < no_zero_divisors; clean up field instance proofs
huffman
parents:
22842
diff
changeset
|
416 |
assume ab: "a * b = 0" |
29667 | 417 |
hence "0 = inverse a * (a * b) * inverse b" by simp |
22987
550709aa8e66
instance division_ring < no_zero_divisors; clean up field instance proofs
huffman
parents:
22842
diff
changeset
|
418 |
also have "\<dots> = (inverse a * a) * (b * inverse b)" |
550709aa8e66
instance division_ring < no_zero_divisors; clean up field instance proofs
huffman
parents:
22842
diff
changeset
|
419 |
by (simp only: mult_assoc) |
29667 | 420 |
also have "\<dots> = 1" using a b by simp |
421 |
finally show False by simp |
|
22987
550709aa8e66
instance division_ring < no_zero_divisors; clean up field instance proofs
huffman
parents:
22842
diff
changeset
|
422 |
qed |
550709aa8e66
instance division_ring < no_zero_divisors; clean up field instance proofs
huffman
parents:
22842
diff
changeset
|
423 |
qed |
20496
23eb6034c06d
added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents:
19404
diff
changeset
|
424 |
|
26274 | 425 |
lemma nonzero_imp_inverse_nonzero: |
426 |
"a \<noteq> 0 \<Longrightarrow> inverse a \<noteq> 0" |
|
427 |
proof |
|
428 |
assume ianz: "inverse a = 0" |
|
429 |
assume "a \<noteq> 0" |
|
430 |
hence "1 = a * inverse a" by simp |
|
431 |
also have "... = 0" by (simp add: ianz) |
|
432 |
finally have "1 = 0" . |
|
433 |
thus False by (simp add: eq_commute) |
|
434 |
qed |
|
435 |
||
436 |
lemma inverse_zero_imp_zero: |
|
437 |
"inverse a = 0 \<Longrightarrow> a = 0" |
|
438 |
apply (rule classical) |
|
439 |
apply (drule nonzero_imp_inverse_nonzero) |
|
440 |
apply auto |
|
441 |
done |
|
442 |
||
443 |
lemma inverse_unique: |
|
444 |
assumes ab: "a * b = 1" |
|
445 |
shows "inverse a = b" |
|
446 |
proof - |
|
447 |
have "a \<noteq> 0" using ab by (cases "a = 0") simp_all |
|
29406 | 448 |
moreover have "inverse a * (a * b) = inverse a" by (simp add: ab) |
449 |
ultimately show ?thesis by (simp add: mult_assoc [symmetric]) |
|
26274 | 450 |
qed |
451 |
||
29406 | 452 |
lemma nonzero_inverse_minus_eq: |
453 |
"a \<noteq> 0 \<Longrightarrow> inverse (- a) = - inverse a" |
|
29667 | 454 |
by (rule inverse_unique) simp |
29406 | 455 |
|
456 |
lemma nonzero_inverse_inverse_eq: |
|
457 |
"a \<noteq> 0 \<Longrightarrow> inverse (inverse a) = a" |
|
29667 | 458 |
by (rule inverse_unique) simp |
29406 | 459 |
|
460 |
lemma nonzero_inverse_eq_imp_eq: |
|
461 |
assumes "inverse a = inverse b" and "a \<noteq> 0" and "b \<noteq> 0" |
|
462 |
shows "a = b" |
|
463 |
proof - |
|
464 |
from `inverse a = inverse b` |
|
29667 | 465 |
have "inverse (inverse a) = inverse (inverse b)" by (rule arg_cong) |
29406 | 466 |
with `a \<noteq> 0` and `b \<noteq> 0` show "a = b" |
467 |
by (simp add: nonzero_inverse_inverse_eq) |
|
468 |
qed |
|
469 |
||
470 |
lemma inverse_1 [simp]: "inverse 1 = 1" |
|
29667 | 471 |
by (rule inverse_unique) simp |
29406 | 472 |
|
26274 | 473 |
lemma nonzero_inverse_mult_distrib: |
29406 | 474 |
assumes "a \<noteq> 0" and "b \<noteq> 0" |
26274 | 475 |
shows "inverse (a * b) = inverse b * inverse a" |
476 |
proof - |
|
29667 | 477 |
have "a * (b * inverse b) * inverse a = 1" using assms by simp |
478 |
hence "a * b * (inverse b * inverse a) = 1" by (simp only: mult_assoc) |
|
479 |
thus ?thesis by (rule inverse_unique) |
|
26274 | 480 |
qed |
481 |
||
482 |
lemma division_ring_inverse_add: |
|
483 |
"a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> inverse a + inverse b = inverse a * (a + b) * inverse b" |
|
29667 | 484 |
by (simp add: algebra_simps) |
26274 | 485 |
|
486 |
lemma division_ring_inverse_diff: |
|
487 |
"a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> inverse a - inverse b = inverse a * (b - a) * inverse b" |
|
29667 | 488 |
by (simp add: algebra_simps) |
26274 | 489 |
|
25186 | 490 |
end |
25152 | 491 |
|
22390 | 492 |
class mult_mono = times + zero + ord + |
25062 | 493 |
assumes mult_left_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> c * a \<le> c * b" |
494 |
assumes mult_right_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * c" |
|
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
495 |
|
35302 | 496 |
text {* |
497 |
The theory of partially ordered rings is taken from the books: |
|
498 |
\begin{itemize} |
|
499 |
\item \emph{Lattice Theory} by Garret Birkhoff, American Mathematical Society 1979 |
|
500 |
\item \emph{Partially Ordered Algebraic Systems}, Pergamon Press 1963 |
|
501 |
\end{itemize} |
|
502 |
Most of the used notions can also be looked up in |
|
503 |
\begin{itemize} |
|
504 |
\item \url{http://www.mathworld.com} by Eric Weisstein et. al. |
|
505 |
\item \emph{Algebra I} by van der Waerden, Springer. |
|
506 |
\end{itemize} |
|
507 |
*} |
|
508 |
||
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
509 |
class ordered_semiring = mult_mono + semiring_0 + ordered_ab_semigroup_add |
25230 | 510 |
begin |
511 |
||
512 |
lemma mult_mono: |
|
513 |
"a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> c |
|
514 |
\<Longrightarrow> a * c \<le> b * d" |
|
515 |
apply (erule mult_right_mono [THEN order_trans], assumption) |
|
516 |
apply (erule mult_left_mono, assumption) |
|
517 |
done |
|
518 |
||
519 |
lemma mult_mono': |
|
520 |
"a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> c |
|
521 |
\<Longrightarrow> a * c \<le> b * d" |
|
522 |
apply (rule mult_mono) |
|
523 |
apply (fast intro: order_trans)+ |
|
524 |
done |
|
525 |
||
526 |
end |
|
21199
2d83f93c3580
* Added annihilation axioms ("x * 0 = 0") to axclass semiring_0.
krauss
parents:
20633
diff
changeset
|
527 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
528 |
class ordered_cancel_semiring = mult_mono + ordered_ab_semigroup_add |
29904 | 529 |
+ semiring + cancel_comm_monoid_add |
25267 | 530 |
begin |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
531 |
|
27516 | 532 |
subclass semiring_0_cancel .. |
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
533 |
subclass ordered_semiring .. |
23521 | 534 |
|
25230 | 535 |
lemma mult_nonneg_nonneg: "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> a * b" |
30692
44ea10bc07a7
clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents:
30650
diff
changeset
|
536 |
using mult_left_mono [of zero b a] by simp |
25230 | 537 |
|
538 |
lemma mult_nonneg_nonpos: "0 \<le> a \<Longrightarrow> b \<le> 0 \<Longrightarrow> a * b \<le> 0" |
|
30692
44ea10bc07a7
clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents:
30650
diff
changeset
|
539 |
using mult_left_mono [of b zero a] by simp |
44ea10bc07a7
clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents:
30650
diff
changeset
|
540 |
|
44ea10bc07a7
clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents:
30650
diff
changeset
|
541 |
lemma mult_nonpos_nonneg: "a \<le> 0 \<Longrightarrow> 0 \<le> b \<Longrightarrow> a * b \<le> 0" |
44ea10bc07a7
clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents:
30650
diff
changeset
|
542 |
using mult_right_mono [of a zero b] by simp |
44ea10bc07a7
clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents:
30650
diff
changeset
|
543 |
|
44ea10bc07a7
clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents:
30650
diff
changeset
|
544 |
text {* Legacy - use @{text mult_nonpos_nonneg} *} |
25230 | 545 |
lemma mult_nonneg_nonpos2: "0 \<le> a \<Longrightarrow> b \<le> 0 \<Longrightarrow> b * a \<le> 0" |
29667 | 546 |
by (drule mult_right_mono [of b zero], auto) |
25230 | 547 |
|
26234 | 548 |
lemma split_mult_neg_le: "(0 \<le> a & b \<le> 0) | (a \<le> 0 & 0 \<le> b) \<Longrightarrow> a * b \<le> 0" |
29667 | 549 |
by (auto simp add: mult_nonneg_nonpos mult_nonneg_nonpos2) |
25230 | 550 |
|
551 |
end |
|
552 |
||
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
553 |
class linordered_semiring = semiring + comm_monoid_add + linordered_cancel_ab_semigroup_add + mult_mono |
25267 | 554 |
begin |
25230 | 555 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
556 |
subclass ordered_cancel_semiring .. |
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
557 |
|
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
558 |
subclass ordered_comm_monoid_add .. |
25304
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset
|
559 |
|
25230 | 560 |
lemma mult_left_less_imp_less: |
561 |
"c * a < c * b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a < b" |
|
29667 | 562 |
by (force simp add: mult_left_mono not_le [symmetric]) |
25230 | 563 |
|
564 |
lemma mult_right_less_imp_less: |
|
565 |
"a * c < b * c \<Longrightarrow> 0 \<le> c \<Longrightarrow> a < b" |
|
29667 | 566 |
by (force simp add: mult_right_mono not_le [symmetric]) |
23521 | 567 |
|
25186 | 568 |
end |
25152 | 569 |
|
35043
07dbdf60d5ad
dropped accidental duplication of "lin" prefix from cs. 108662d50512
haftmann
parents:
35032
diff
changeset
|
570 |
class linordered_semiring_1 = linordered_semiring + semiring_1 |
07dbdf60d5ad
dropped accidental duplication of "lin" prefix from cs. 108662d50512
haftmann
parents:
35032
diff
changeset
|
571 |
|
07dbdf60d5ad
dropped accidental duplication of "lin" prefix from cs. 108662d50512
haftmann
parents:
35032
diff
changeset
|
572 |
class linordered_semiring_strict = semiring + comm_monoid_add + linordered_cancel_ab_semigroup_add + |
25062 | 573 |
assumes mult_strict_left_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b" |
574 |
assumes mult_strict_right_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> a * c < b * c" |
|
25267 | 575 |
begin |
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset
|
576 |
|
27516 | 577 |
subclass semiring_0_cancel .. |
14940 | 578 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
579 |
subclass linordered_semiring |
28823 | 580 |
proof |
23550 | 581 |
fix a b c :: 'a |
582 |
assume A: "a \<le> b" "0 \<le> c" |
|
583 |
from A show "c * a \<le> c * b" |
|
25186 | 584 |
unfolding le_less |
585 |
using mult_strict_left_mono by (cases "c = 0") auto |
|
23550 | 586 |
from A show "a * c \<le> b * c" |
25152 | 587 |
unfolding le_less |
25186 | 588 |
using mult_strict_right_mono by (cases "c = 0") auto |
25152 | 589 |
qed |
590 |
||
25230 | 591 |
lemma mult_left_le_imp_le: |
592 |
"c * a \<le> c * b \<Longrightarrow> 0 < c \<Longrightarrow> a \<le> b" |
|
29667 | 593 |
by (force simp add: mult_strict_left_mono _not_less [symmetric]) |
25230 | 594 |
|
595 |
lemma mult_right_le_imp_le: |
|
596 |
"a * c \<le> b * c \<Longrightarrow> 0 < c \<Longrightarrow> a \<le> b" |
|
29667 | 597 |
by (force simp add: mult_strict_right_mono not_less [symmetric]) |
25230 | 598 |
|
30692
44ea10bc07a7
clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents:
30650
diff
changeset
|
599 |
lemma mult_pos_pos: "0 < a \<Longrightarrow> 0 < b \<Longrightarrow> 0 < a * b" |
44ea10bc07a7
clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents:
30650
diff
changeset
|
600 |
using mult_strict_left_mono [of zero b a] by simp |
44ea10bc07a7
clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents:
30650
diff
changeset
|
601 |
|
44ea10bc07a7
clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents:
30650
diff
changeset
|
602 |
lemma mult_pos_neg: "0 < a \<Longrightarrow> b < 0 \<Longrightarrow> a * b < 0" |
44ea10bc07a7
clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents:
30650
diff
changeset
|
603 |
using mult_strict_left_mono [of b zero a] by simp |
44ea10bc07a7
clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents:
30650
diff
changeset
|
604 |
|
44ea10bc07a7
clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents:
30650
diff
changeset
|
605 |
lemma mult_neg_pos: "a < 0 \<Longrightarrow> 0 < b \<Longrightarrow> a * b < 0" |
44ea10bc07a7
clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents:
30650
diff
changeset
|
606 |
using mult_strict_right_mono [of a zero b] by simp |
44ea10bc07a7
clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents:
30650
diff
changeset
|
607 |
|
44ea10bc07a7
clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents:
30650
diff
changeset
|
608 |
text {* Legacy - use @{text mult_neg_pos} *} |
44ea10bc07a7
clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents:
30650
diff
changeset
|
609 |
lemma mult_pos_neg2: "0 < a \<Longrightarrow> b < 0 \<Longrightarrow> b * a < 0" |
29667 | 610 |
by (drule mult_strict_right_mono [of b zero], auto) |
25230 | 611 |
|
612 |
lemma zero_less_mult_pos: |
|
613 |
"0 < a * b \<Longrightarrow> 0 < a \<Longrightarrow> 0 < b" |
|
30692
44ea10bc07a7
clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents:
30650
diff
changeset
|
614 |
apply (cases "b\<le>0") |
25230 | 615 |
apply (auto simp add: le_less not_less) |
30692
44ea10bc07a7
clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents:
30650
diff
changeset
|
616 |
apply (drule_tac mult_pos_neg [of a b]) |
25230 | 617 |
apply (auto dest: less_not_sym) |
618 |
done |
|
619 |
||
620 |
lemma zero_less_mult_pos2: |
|
621 |
"0 < b * a \<Longrightarrow> 0 < a \<Longrightarrow> 0 < b" |
|
30692
44ea10bc07a7
clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents:
30650
diff
changeset
|
622 |
apply (cases "b\<le>0") |
25230 | 623 |
apply (auto simp add: le_less not_less) |
30692
44ea10bc07a7
clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents:
30650
diff
changeset
|
624 |
apply (drule_tac mult_pos_neg2 [of a b]) |
25230 | 625 |
apply (auto dest: less_not_sym) |
626 |
done |
|
627 |
||
26193 | 628 |
text{*Strict monotonicity in both arguments*} |
629 |
lemma mult_strict_mono: |
|
630 |
assumes "a < b" and "c < d" and "0 < b" and "0 \<le> c" |
|
631 |
shows "a * c < b * d" |
|
632 |
using assms apply (cases "c=0") |
|
30692
44ea10bc07a7
clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents:
30650
diff
changeset
|
633 |
apply (simp add: mult_pos_pos) |
26193 | 634 |
apply (erule mult_strict_right_mono [THEN less_trans]) |
30692
44ea10bc07a7
clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents:
30650
diff
changeset
|
635 |
apply (force simp add: le_less) |
26193 | 636 |
apply (erule mult_strict_left_mono, assumption) |
637 |
done |
|
638 |
||
639 |
text{*This weaker variant has more natural premises*} |
|
640 |
lemma mult_strict_mono': |
|
641 |
assumes "a < b" and "c < d" and "0 \<le> a" and "0 \<le> c" |
|
642 |
shows "a * c < b * d" |
|
29667 | 643 |
by (rule mult_strict_mono) (insert assms, auto) |
26193 | 644 |
|
645 |
lemma mult_less_le_imp_less: |
|
646 |
assumes "a < b" and "c \<le> d" and "0 \<le> a" and "0 < c" |
|
647 |
shows "a * c < b * d" |
|
648 |
using assms apply (subgoal_tac "a * c < b * c") |
|
649 |
apply (erule less_le_trans) |
|
650 |
apply (erule mult_left_mono) |
|
651 |
apply simp |
|
652 |
apply (erule mult_strict_right_mono) |
|
653 |
apply assumption |
|
654 |
done |
|
655 |
||
656 |
lemma mult_le_less_imp_less: |
|
657 |
assumes "a \<le> b" and "c < d" and "0 < a" and "0 \<le> c" |
|
658 |
shows "a * c < b * d" |
|
659 |
using assms apply (subgoal_tac "a * c \<le> b * c") |
|
660 |
apply (erule le_less_trans) |
|
661 |
apply (erule mult_strict_left_mono) |
|
662 |
apply simp |
|
663 |
apply (erule mult_right_mono) |
|
664 |
apply simp |
|
665 |
done |
|
666 |
||
667 |
lemma mult_less_imp_less_left: |
|
668 |
assumes less: "c * a < c * b" and nonneg: "0 \<le> c" |
|
669 |
shows "a < b" |
|
670 |
proof (rule ccontr) |
|
671 |
assume "\<not> a < b" |
|
672 |
hence "b \<le> a" by (simp add: linorder_not_less) |
|
673 |
hence "c * b \<le> c * a" using nonneg by (rule mult_left_mono) |
|
29667 | 674 |
with this and less show False by (simp add: not_less [symmetric]) |
26193 | 675 |
qed |
676 |
||
677 |
lemma mult_less_imp_less_right: |
|
678 |
assumes less: "a * c < b * c" and nonneg: "0 \<le> c" |
|
679 |
shows "a < b" |
|
680 |
proof (rule ccontr) |
|
681 |
assume "\<not> a < b" |
|
682 |
hence "b \<le> a" by (simp add: linorder_not_less) |
|
683 |
hence "b * c \<le> a * c" using nonneg by (rule mult_right_mono) |
|
29667 | 684 |
with this and less show False by (simp add: not_less [symmetric]) |
26193 | 685 |
qed |
686 |
||
25230 | 687 |
end |
688 |
||
35097
4554bb2abfa3
dropped last occurence of the linlinordered accident
haftmann
parents:
35092
diff
changeset
|
689 |
class linordered_semiring_1_strict = linordered_semiring_strict + semiring_1 |
33319 | 690 |
|
22390 | 691 |
class mult_mono1 = times + zero + ord + |
25230 | 692 |
assumes mult_mono1: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> c * a \<le> c * b" |
14270 | 693 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
694 |
class ordered_comm_semiring = comm_semiring_0 |
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
695 |
+ ordered_ab_semigroup_add + mult_mono1 |
25186 | 696 |
begin |
25152 | 697 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
698 |
subclass ordered_semiring |
28823 | 699 |
proof |
21199
2d83f93c3580
* Added annihilation axioms ("x * 0 = 0") to axclass semiring_0.
krauss
parents:
20633
diff
changeset
|
700 |
fix a b c :: 'a |
23550 | 701 |
assume "a \<le> b" "0 \<le> c" |
25230 | 702 |
thus "c * a \<le> c * b" by (rule mult_mono1) |
23550 | 703 |
thus "a * c \<le> b * c" by (simp only: mult_commute) |
21199
2d83f93c3580
* Added annihilation axioms ("x * 0 = 0") to axclass semiring_0.
krauss
parents:
20633
diff
changeset
|
704 |
qed |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
705 |
|
25267 | 706 |
end |
707 |
||
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
708 |
class ordered_cancel_comm_semiring = comm_semiring_0_cancel |
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
709 |
+ ordered_ab_semigroup_add + mult_mono1 |
25267 | 710 |
begin |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
711 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
712 |
subclass ordered_comm_semiring .. |
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
713 |
subclass ordered_cancel_semiring .. |
25267 | 714 |
|
715 |
end |
|
716 |
||
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
717 |
class linordered_comm_semiring_strict = comm_semiring_0 + linordered_cancel_ab_semigroup_add + |
26193 | 718 |
assumes mult_strict_left_mono_comm: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b" |
25267 | 719 |
begin |
720 |
||
35043
07dbdf60d5ad
dropped accidental duplication of "lin" prefix from cs. 108662d50512
haftmann
parents:
35032
diff
changeset
|
721 |
subclass linordered_semiring_strict |
28823 | 722 |
proof |
23550 | 723 |
fix a b c :: 'a |
724 |
assume "a < b" "0 < c" |
|
26193 | 725 |
thus "c * a < c * b" by (rule mult_strict_left_mono_comm) |
23550 | 726 |
thus "a * c < b * c" by (simp only: mult_commute) |
727 |
qed |
|
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
728 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
729 |
subclass ordered_cancel_comm_semiring |
28823 | 730 |
proof |
23550 | 731 |
fix a b c :: 'a |
732 |
assume "a \<le> b" "0 \<le> c" |
|
733 |
thus "c * a \<le> c * b" |
|
25186 | 734 |
unfolding le_less |
26193 | 735 |
using mult_strict_left_mono by (cases "c = 0") auto |
23550 | 736 |
qed |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
737 |
|
25267 | 738 |
end |
25230 | 739 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
740 |
class ordered_ring = ring + ordered_cancel_semiring |
25267 | 741 |
begin |
25230 | 742 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
743 |
subclass ordered_ab_group_add .. |
14270 | 744 |
|
29667 | 745 |
text{*Legacy - use @{text algebra_simps} *} |
29833 | 746 |
lemmas ring_simps[noatp] = algebra_simps |
25230 | 747 |
|
748 |
lemma less_add_iff1: |
|
749 |
"a * e + c < b * e + d \<longleftrightarrow> (a - b) * e + c < d" |
|
29667 | 750 |
by (simp add: algebra_simps) |
25230 | 751 |
|
752 |
lemma less_add_iff2: |
|
753 |
"a * e + c < b * e + d \<longleftrightarrow> c < (b - a) * e + d" |
|
29667 | 754 |
by (simp add: algebra_simps) |
25230 | 755 |
|
756 |
lemma le_add_iff1: |
|
757 |
"a * e + c \<le> b * e + d \<longleftrightarrow> (a - b) * e + c \<le> d" |
|
29667 | 758 |
by (simp add: algebra_simps) |
25230 | 759 |
|
760 |
lemma le_add_iff2: |
|
761 |
"a * e + c \<le> b * e + d \<longleftrightarrow> c \<le> (b - a) * e + d" |
|
29667 | 762 |
by (simp add: algebra_simps) |
25230 | 763 |
|
764 |
lemma mult_left_mono_neg: |
|
765 |
"b \<le> a \<Longrightarrow> c \<le> 0 \<Longrightarrow> c * a \<le> c * b" |
|
766 |
apply (drule mult_left_mono [of _ _ "uminus c"]) |
|
35216 | 767 |
apply simp_all |
25230 | 768 |
done |
769 |
||
770 |
lemma mult_right_mono_neg: |
|
771 |
"b \<le> a \<Longrightarrow> c \<le> 0 \<Longrightarrow> a * c \<le> b * c" |
|
772 |
apply (drule mult_right_mono [of _ _ "uminus c"]) |
|
35216 | 773 |
apply simp_all |
25230 | 774 |
done |
775 |
||
30692
44ea10bc07a7
clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents:
30650
diff
changeset
|
776 |
lemma mult_nonpos_nonpos: "a \<le> 0 \<Longrightarrow> b \<le> 0 \<Longrightarrow> 0 \<le> a * b" |
44ea10bc07a7
clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents:
30650
diff
changeset
|
777 |
using mult_right_mono_neg [of a zero b] by simp |
25230 | 778 |
|
779 |
lemma split_mult_pos_le: |
|
780 |
"(0 \<le> a \<and> 0 \<le> b) \<or> (a \<le> 0 \<and> b \<le> 0) \<Longrightarrow> 0 \<le> a * b" |
|
29667 | 781 |
by (auto simp add: mult_nonneg_nonneg mult_nonpos_nonpos) |
25186 | 782 |
|
783 |
end |
|
14270 | 784 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
785 |
class linordered_ring = ring + linordered_semiring + linordered_ab_group_add + abs_if |
25304
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset
|
786 |
begin |
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset
|
787 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
788 |
subclass ordered_ring .. |
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
789 |
|
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
790 |
subclass ordered_ab_group_add_abs |
28823 | 791 |
proof |
25304
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset
|
792 |
fix a b |
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset
|
793 |
show "\<bar>a + b\<bar> \<le> \<bar>a\<bar> + \<bar>b\<bar>" |
35216 | 794 |
by (auto simp add: abs_if not_less) |
795 |
(auto simp del: minus_add_distrib simp add: minus_add_distrib [symmetric], |
|
796 |
auto intro: add_nonneg_nonneg, auto intro!: less_imp_le add_neg_neg) |
|
797 |
qed (auto simp add: abs_if) |
|
25304
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset
|
798 |
|
35631
0b8a5fd339ab
generalize some lemmas from class linordered_ring_strict to linordered_ring
huffman
parents:
35302
diff
changeset
|
799 |
lemma zero_le_square [simp]: "0 \<le> a * a" |
0b8a5fd339ab
generalize some lemmas from class linordered_ring_strict to linordered_ring
huffman
parents:
35302
diff
changeset
|
800 |
using linear [of 0 a] |
0b8a5fd339ab
generalize some lemmas from class linordered_ring_strict to linordered_ring
huffman
parents:
35302
diff
changeset
|
801 |
by (auto simp add: mult_nonneg_nonneg mult_nonpos_nonpos) |
0b8a5fd339ab
generalize some lemmas from class linordered_ring_strict to linordered_ring
huffman
parents:
35302
diff
changeset
|
802 |
|
0b8a5fd339ab
generalize some lemmas from class linordered_ring_strict to linordered_ring
huffman
parents:
35302
diff
changeset
|
803 |
lemma not_square_less_zero [simp]: "\<not> (a * a < 0)" |
0b8a5fd339ab
generalize some lemmas from class linordered_ring_strict to linordered_ring
huffman
parents:
35302
diff
changeset
|
804 |
by (simp add: not_less) |
0b8a5fd339ab
generalize some lemmas from class linordered_ring_strict to linordered_ring
huffman
parents:
35302
diff
changeset
|
805 |
|
25304
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset
|
806 |
end |
23521 | 807 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
808 |
(* The "strict" suffix can be seen as describing the combination of linordered_ring and no_zero_divisors. |
35043
07dbdf60d5ad
dropped accidental duplication of "lin" prefix from cs. 108662d50512
haftmann
parents:
35032
diff
changeset
|
809 |
Basically, linordered_ring + no_zero_divisors = linordered_ring_strict. |
25230 | 810 |
*) |
35043
07dbdf60d5ad
dropped accidental duplication of "lin" prefix from cs. 108662d50512
haftmann
parents:
35032
diff
changeset
|
811 |
class linordered_ring_strict = ring + linordered_semiring_strict |
25304
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset
|
812 |
+ ordered_ab_group_add + abs_if |
25230 | 813 |
begin |
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset
|
814 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
815 |
subclass linordered_ring .. |
25304
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset
|
816 |
|
30692
44ea10bc07a7
clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents:
30650
diff
changeset
|
817 |
lemma mult_strict_left_mono_neg: "b < a \<Longrightarrow> c < 0 \<Longrightarrow> c * a < c * b" |
44ea10bc07a7
clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents:
30650
diff
changeset
|
818 |
using mult_strict_left_mono [of b a "- c"] by simp |
44ea10bc07a7
clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents:
30650
diff
changeset
|
819 |
|
44ea10bc07a7
clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents:
30650
diff
changeset
|
820 |
lemma mult_strict_right_mono_neg: "b < a \<Longrightarrow> c < 0 \<Longrightarrow> a * c < b * c" |
44ea10bc07a7
clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents:
30650
diff
changeset
|
821 |
using mult_strict_right_mono [of b a "- c"] by simp |
44ea10bc07a7
clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents:
30650
diff
changeset
|
822 |
|
44ea10bc07a7
clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents:
30650
diff
changeset
|
823 |
lemma mult_neg_neg: "a < 0 \<Longrightarrow> b < 0 \<Longrightarrow> 0 < a * b" |
44ea10bc07a7
clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents:
30650
diff
changeset
|
824 |
using mult_strict_right_mono_neg [of a zero b] by simp |
14738 | 825 |
|
25917 | 826 |
subclass ring_no_zero_divisors |
28823 | 827 |
proof |
25917 | 828 |
fix a b |
829 |
assume "a \<noteq> 0" then have A: "a < 0 \<or> 0 < a" by (simp add: neq_iff) |
|
830 |
assume "b \<noteq> 0" then have B: "b < 0 \<or> 0 < b" by (simp add: neq_iff) |
|
831 |
have "a * b < 0 \<or> 0 < a * b" |
|
832 |
proof (cases "a < 0") |
|
833 |
case True note A' = this |
|
834 |
show ?thesis proof (cases "b < 0") |
|
835 |
case True with A' |
|
836 |
show ?thesis by (auto dest: mult_neg_neg) |
|
837 |
next |
|
838 |
case False with B have "0 < b" by auto |
|
839 |
with A' show ?thesis by (auto dest: mult_strict_right_mono) |
|
840 |
qed |
|
841 |
next |
|
842 |
case False with A have A': "0 < a" by auto |
|
843 |
show ?thesis proof (cases "b < 0") |
|
844 |
case True with A' |
|
845 |
show ?thesis by (auto dest: mult_strict_right_mono_neg) |
|
846 |
next |
|
847 |
case False with B have "0 < b" by auto |
|
848 |
with A' show ?thesis by (auto dest: mult_pos_pos) |
|
849 |
qed |
|
850 |
qed |
|
851 |
then show "a * b \<noteq> 0" by (simp add: neq_iff) |
|
852 |
qed |
|
25304
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset
|
853 |
|
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
854 |
lemma zero_less_mult_iff: |
25917 | 855 |
"0 < a * b \<longleftrightarrow> 0 < a \<and> 0 < b \<or> a < 0 \<and> b < 0" |
856 |
apply (auto simp add: mult_pos_pos mult_neg_neg) |
|
857 |
apply (simp_all add: not_less le_less) |
|
858 |
apply (erule disjE) apply assumption defer |
|
859 |
apply (erule disjE) defer apply (drule sym) apply simp |
|
860 |
apply (erule disjE) defer apply (drule sym) apply simp |
|
861 |
apply (erule disjE) apply assumption apply (drule sym) apply simp |
|
862 |
apply (drule sym) apply simp |
|
863 |
apply (blast dest: zero_less_mult_pos) |
|
25230 | 864 |
apply (blast dest: zero_less_mult_pos2) |
865 |
done |
|
22990
775e9de3db48
added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents:
22987
diff
changeset
|
866 |
|
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
867 |
lemma zero_le_mult_iff: |
25917 | 868 |
"0 \<le> a * b \<longleftrightarrow> 0 \<le> a \<and> 0 \<le> b \<or> a \<le> 0 \<and> b \<le> 0" |
29667 | 869 |
by (auto simp add: eq_commute [of 0] le_less not_less zero_less_mult_iff) |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
870 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
871 |
lemma mult_less_0_iff: |
25917 | 872 |
"a * b < 0 \<longleftrightarrow> 0 < a \<and> b < 0 \<or> a < 0 \<and> 0 < b" |
35216 | 873 |
apply (insert zero_less_mult_iff [of "-a" b]) |
874 |
apply force |
|
25917 | 875 |
done |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
876 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
877 |
lemma mult_le_0_iff: |
25917 | 878 |
"a * b \<le> 0 \<longleftrightarrow> 0 \<le> a \<and> b \<le> 0 \<or> a \<le> 0 \<and> 0 \<le> b" |
879 |
apply (insert zero_le_mult_iff [of "-a" b]) |
|
35216 | 880 |
apply force |
25917 | 881 |
done |
882 |
||
26193 | 883 |
text{*Cancellation laws for @{term "c*a < c*b"} and @{term "a*c < b*c"}, |
884 |
also with the relations @{text "\<le>"} and equality.*} |
|
885 |
||
886 |
text{*These ``disjunction'' versions produce two cases when the comparison is |
|
887 |
an assumption, but effectively four when the comparison is a goal.*} |
|
888 |
||
889 |
lemma mult_less_cancel_right_disj: |
|
890 |
"a * c < b * c \<longleftrightarrow> 0 < c \<and> a < b \<or> c < 0 \<and> b < a" |
|
891 |
apply (cases "c = 0") |
|
892 |
apply (auto simp add: neq_iff mult_strict_right_mono |
|
893 |
mult_strict_right_mono_neg) |
|
894 |
apply (auto simp add: not_less |
|
895 |
not_le [symmetric, of "a*c"] |
|
896 |
not_le [symmetric, of a]) |
|
897 |
apply (erule_tac [!] notE) |
|
898 |
apply (auto simp add: less_imp_le mult_right_mono |
|
899 |
mult_right_mono_neg) |
|
900 |
done |
|
901 |
||
902 |
lemma mult_less_cancel_left_disj: |
|
903 |
"c * a < c * b \<longleftrightarrow> 0 < c \<and> a < b \<or> c < 0 \<and> b < a" |
|
904 |
apply (cases "c = 0") |
|
905 |
apply (auto simp add: neq_iff mult_strict_left_mono |
|
906 |
mult_strict_left_mono_neg) |
|
907 |
apply (auto simp add: not_less |
|
908 |
not_le [symmetric, of "c*a"] |
|
909 |
not_le [symmetric, of a]) |
|
910 |
apply (erule_tac [!] notE) |
|
911 |
apply (auto simp add: less_imp_le mult_left_mono |
|
912 |
mult_left_mono_neg) |
|
913 |
done |
|
914 |
||
915 |
text{*The ``conjunction of implication'' lemmas produce two cases when the |
|
916 |
comparison is a goal, but give four when the comparison is an assumption.*} |
|
917 |
||
918 |
lemma mult_less_cancel_right: |
|
919 |
"a * c < b * c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < b) \<and> (c \<le> 0 \<longrightarrow> b < a)" |
|
920 |
using mult_less_cancel_right_disj [of a c b] by auto |
|
921 |
||
922 |
lemma mult_less_cancel_left: |
|
923 |
"c * a < c * b \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < b) \<and> (c \<le> 0 \<longrightarrow> b < a)" |
|
924 |
using mult_less_cancel_left_disj [of c a b] by auto |
|
925 |
||
926 |
lemma mult_le_cancel_right: |
|
927 |
"a * c \<le> b * c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)" |
|
29667 | 928 |
by (simp add: not_less [symmetric] mult_less_cancel_right_disj) |
26193 | 929 |
|
930 |
lemma mult_le_cancel_left: |
|
931 |
"c * a \<le> c * b \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)" |
|
29667 | 932 |
by (simp add: not_less [symmetric] mult_less_cancel_left_disj) |
26193 | 933 |
|
30649
57753e0ec1d4
1. New cancellation simprocs for common factors in inequations
nipkow
parents:
30242
diff
changeset
|
934 |
lemma mult_le_cancel_left_pos: |
57753e0ec1d4
1. New cancellation simprocs for common factors in inequations
nipkow
parents:
30242
diff
changeset
|
935 |
"0 < c \<Longrightarrow> c * a \<le> c * b \<longleftrightarrow> a \<le> b" |
57753e0ec1d4
1. New cancellation simprocs for common factors in inequations
nipkow
parents:
30242
diff
changeset
|
936 |
by (auto simp: mult_le_cancel_left) |
57753e0ec1d4
1. New cancellation simprocs for common factors in inequations
nipkow
parents:
30242
diff
changeset
|
937 |
|
57753e0ec1d4
1. New cancellation simprocs for common factors in inequations
nipkow
parents:
30242
diff
changeset
|
938 |
lemma mult_le_cancel_left_neg: |
57753e0ec1d4
1. New cancellation simprocs for common factors in inequations
nipkow
parents:
30242
diff
changeset
|
939 |
"c < 0 \<Longrightarrow> c * a \<le> c * b \<longleftrightarrow> b \<le> a" |
57753e0ec1d4
1. New cancellation simprocs for common factors in inequations
nipkow
parents:
30242
diff
changeset
|
940 |
by (auto simp: mult_le_cancel_left) |
57753e0ec1d4
1. New cancellation simprocs for common factors in inequations
nipkow
parents:
30242
diff
changeset
|
941 |
|
57753e0ec1d4
1. New cancellation simprocs for common factors in inequations
nipkow
parents:
30242
diff
changeset
|
942 |
lemma mult_less_cancel_left_pos: |
57753e0ec1d4
1. New cancellation simprocs for common factors in inequations
nipkow
parents:
30242
diff
changeset
|
943 |
"0 < c \<Longrightarrow> c * a < c * b \<longleftrightarrow> a < b" |
57753e0ec1d4
1. New cancellation simprocs for common factors in inequations
nipkow
parents:
30242
diff
changeset
|
944 |
by (auto simp: mult_less_cancel_left) |
57753e0ec1d4
1. New cancellation simprocs for common factors in inequations
nipkow
parents:
30242
diff
changeset
|
945 |
|
57753e0ec1d4
1. New cancellation simprocs for common factors in inequations
nipkow
parents:
30242
diff
changeset
|
946 |
lemma mult_less_cancel_left_neg: |
57753e0ec1d4
1. New cancellation simprocs for common factors in inequations
nipkow
parents:
30242
diff
changeset
|
947 |
"c < 0 \<Longrightarrow> c * a < c * b \<longleftrightarrow> b < a" |
57753e0ec1d4
1. New cancellation simprocs for common factors in inequations
nipkow
parents:
30242
diff
changeset
|
948 |
by (auto simp: mult_less_cancel_left) |
57753e0ec1d4
1. New cancellation simprocs for common factors in inequations
nipkow
parents:
30242
diff
changeset
|
949 |
|
25917 | 950 |
end |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
951 |
|
29667 | 952 |
text{*Legacy - use @{text algebra_simps} *} |
29833 | 953 |
lemmas ring_simps[noatp] = algebra_simps |
25230 | 954 |
|
30692
44ea10bc07a7
clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents:
30650
diff
changeset
|
955 |
lemmas mult_sign_intros = |
44ea10bc07a7
clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents:
30650
diff
changeset
|
956 |
mult_nonneg_nonneg mult_nonneg_nonpos |
44ea10bc07a7
clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents:
30650
diff
changeset
|
957 |
mult_nonpos_nonneg mult_nonpos_nonpos |
44ea10bc07a7
clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents:
30650
diff
changeset
|
958 |
mult_pos_pos mult_pos_neg |
44ea10bc07a7
clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents:
30650
diff
changeset
|
959 |
mult_neg_pos mult_neg_neg |
25230 | 960 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
961 |
class ordered_comm_ring = comm_ring + ordered_comm_semiring |
25267 | 962 |
begin |
25230 | 963 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
964 |
subclass ordered_ring .. |
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
965 |
subclass ordered_cancel_comm_semiring .. |
25230 | 966 |
|
25267 | 967 |
end |
25230 | 968 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
969 |
class linordered_semidom = comm_semiring_1_cancel + linordered_comm_semiring_strict + |
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
970 |
(*previously linordered_semiring*) |
25230 | 971 |
assumes zero_less_one [simp]: "0 < 1" |
972 |
begin |
|
973 |
||
974 |
lemma pos_add_strict: |
|
975 |
shows "0 < a \<Longrightarrow> b < c \<Longrightarrow> b < a + c" |
|
976 |
using add_strict_mono [of zero a b c] by simp |
|
977 |
||
26193 | 978 |
lemma zero_le_one [simp]: "0 \<le> 1" |
29667 | 979 |
by (rule zero_less_one [THEN less_imp_le]) |
26193 | 980 |
|
981 |
lemma not_one_le_zero [simp]: "\<not> 1 \<le> 0" |
|
29667 | 982 |
by (simp add: not_le) |
26193 | 983 |
|
984 |
lemma not_one_less_zero [simp]: "\<not> 1 < 0" |
|
29667 | 985 |
by (simp add: not_less) |
26193 | 986 |
|
987 |
lemma less_1_mult: |
|
988 |
assumes "1 < m" and "1 < n" |
|
989 |
shows "1 < m * n" |
|
990 |
using assms mult_strict_mono [of 1 m 1 n] |
|
991 |
by (simp add: less_trans [OF zero_less_one]) |
|
992 |
||
25230 | 993 |
end |
994 |
||
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
995 |
class linordered_idom = comm_ring_1 + |
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
996 |
linordered_comm_semiring_strict + ordered_ab_group_add + |
25230 | 997 |
abs_if + sgn_if |
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
998 |
(*previously linordered_ring*) |
25917 | 999 |
begin |
1000 |
||
35043
07dbdf60d5ad
dropped accidental duplication of "lin" prefix from cs. 108662d50512
haftmann
parents:
35032
diff
changeset
|
1001 |
subclass linordered_ring_strict .. |
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
1002 |
subclass ordered_comm_ring .. |
27516 | 1003 |
subclass idom .. |
25917 | 1004 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
1005 |
subclass linordered_semidom |
28823 | 1006 |
proof |
26193 | 1007 |
have "0 \<le> 1 * 1" by (rule zero_le_square) |
1008 |
thus "0 < 1" by (simp add: le_less) |
|
25917 | 1009 |
qed |
1010 |
||
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
1011 |
lemma linorder_neqE_linordered_idom: |
26193 | 1012 |
assumes "x \<noteq> y" obtains "x < y" | "y < x" |
1013 |
using assms by (rule neqE) |
|
1014 |
||
26274 | 1015 |
text {* These cancellation simprules also produce two cases when the comparison is a goal. *} |
1016 |
||
1017 |
lemma mult_le_cancel_right1: |
|
1018 |
"c \<le> b * c \<longleftrightarrow> (0 < c \<longrightarrow> 1 \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> 1)" |
|
29667 | 1019 |
by (insert mult_le_cancel_right [of 1 c b], simp) |
26274 | 1020 |
|
1021 |
lemma mult_le_cancel_right2: |
|
1022 |
"a * c \<le> c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> 1) \<and> (c < 0 \<longrightarrow> 1 \<le> a)" |
|
29667 | 1023 |
by (insert mult_le_cancel_right [of a c 1], simp) |
26274 | 1024 |
|
1025 |
lemma mult_le_cancel_left1: |
|
1026 |
"c \<le> c * b \<longleftrightarrow> (0 < c \<longrightarrow> 1 \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> 1)" |
|
29667 | 1027 |
by (insert mult_le_cancel_left [of c 1 b], simp) |
26274 | 1028 |
|
1029 |
lemma mult_le_cancel_left2: |
|
1030 |
"c * a \<le> c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> 1) \<and> (c < 0 \<longrightarrow> 1 \<le> a)" |
|
29667 | 1031 |
by (insert mult_le_cancel_left [of c a 1], simp) |
26274 | 1032 |
|
1033 |
lemma mult_less_cancel_right1: |
|
1034 |
"c < b * c \<longleftrightarrow> (0 \<le> c \<longrightarrow> 1 < b) \<and> (c \<le> 0 \<longrightarrow> b < 1)" |
|
29667 | 1035 |
by (insert mult_less_cancel_right [of 1 c b], simp) |
26274 | 1036 |
|
1037 |
lemma mult_less_cancel_right2: |
|
1038 |
"a * c < c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < 1) \<and> (c \<le> 0 \<longrightarrow> 1 < a)" |
|
29667 | 1039 |
by (insert mult_less_cancel_right [of a c 1], simp) |
26274 | 1040 |
|
1041 |
lemma mult_less_cancel_left1: |
|
1042 |
"c < c * b \<longleftrightarrow> (0 \<le> c \<longrightarrow> 1 < b) \<and> (c \<le> 0 \<longrightarrow> b < 1)" |
|
29667 | 1043 |
by (insert mult_less_cancel_left [of c 1 b], simp) |
26274 | 1044 |
|
1045 |
lemma mult_less_cancel_left2: |
|
1046 |
"c * a < c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < 1) \<and> (c \<le> 0 \<longrightarrow> 1 < a)" |
|
29667 | 1047 |
by (insert mult_less_cancel_left [of c a 1], simp) |
26274 | 1048 |
|
27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27516
diff
changeset
|
1049 |
lemma sgn_sgn [simp]: |
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27516
diff
changeset
|
1050 |
"sgn (sgn a) = sgn a" |
29700 | 1051 |
unfolding sgn_if by simp |
27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27516
diff
changeset
|
1052 |
|
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27516
diff
changeset
|
1053 |
lemma sgn_0_0: |
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27516
diff
changeset
|
1054 |
"sgn a = 0 \<longleftrightarrow> a = 0" |
29700 | 1055 |
unfolding sgn_if by simp |
27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27516
diff
changeset
|
1056 |
|
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27516
diff
changeset
|
1057 |
lemma sgn_1_pos: |
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27516
diff
changeset
|
1058 |
"sgn a = 1 \<longleftrightarrow> a > 0" |
35216 | 1059 |
unfolding sgn_if by simp |
27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27516
diff
changeset
|
1060 |
|
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27516
diff
changeset
|
1061 |
lemma sgn_1_neg: |
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27516
diff
changeset
|
1062 |
"sgn a = - 1 \<longleftrightarrow> a < 0" |
35216 | 1063 |
unfolding sgn_if by auto |
27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27516
diff
changeset
|
1064 |
|
29940 | 1065 |
lemma sgn_pos [simp]: |
1066 |
"0 < a \<Longrightarrow> sgn a = 1" |
|
1067 |
unfolding sgn_1_pos . |
|
1068 |
||
1069 |
lemma sgn_neg [simp]: |
|
1070 |
"a < 0 \<Longrightarrow> sgn a = - 1" |
|
1071 |
unfolding sgn_1_neg . |
|
1072 |
||
27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27516
diff
changeset
|
1073 |
lemma sgn_times: |
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27516
diff
changeset
|
1074 |
"sgn (a * b) = sgn a * sgn b" |
29667 | 1075 |
by (auto simp add: sgn_if zero_less_mult_iff) |
27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27516
diff
changeset
|
1076 |
|
29653 | 1077 |
lemma abs_sgn: "abs k = k * sgn k" |
29700 | 1078 |
unfolding sgn_if abs_if by auto |
1079 |
||
29940 | 1080 |
lemma sgn_greater [simp]: |
1081 |
"0 < sgn a \<longleftrightarrow> 0 < a" |
|
1082 |
unfolding sgn_if by auto |
|
1083 |
||
1084 |
lemma sgn_less [simp]: |
|
1085 |
"sgn a < 0 \<longleftrightarrow> a < 0" |
|
1086 |
unfolding sgn_if by auto |
|
1087 |
||
29949 | 1088 |
lemma abs_dvd_iff [simp]: "(abs m) dvd k \<longleftrightarrow> m dvd k" |
1089 |
by (simp add: abs_if) |
|
1090 |
||
1091 |
lemma dvd_abs_iff [simp]: "m dvd (abs k) \<longleftrightarrow> m dvd k" |
|
1092 |
by (simp add: abs_if) |
|
29653 | 1093 |
|
33676
802f5e233e48
moved lemma from Algebra/IntRing to Ring_and_Field
nipkow
parents:
33364
diff
changeset
|
1094 |
lemma dvd_if_abs_eq: |
802f5e233e48
moved lemma from Algebra/IntRing to Ring_and_Field
nipkow
parents:
33364
diff
changeset
|
1095 |
"abs l = abs (k) \<Longrightarrow> l dvd k" |
802f5e233e48
moved lemma from Algebra/IntRing to Ring_and_Field
nipkow
parents:
33364
diff
changeset
|
1096 |
by(subst abs_dvd_iff[symmetric]) simp |
802f5e233e48
moved lemma from Algebra/IntRing to Ring_and_Field
nipkow
parents:
33364
diff
changeset
|
1097 |
|
25917 | 1098 |
end |
25230 | 1099 |
|
26274 | 1100 |
text {* Simprules for comparisons where common factors can be cancelled. *} |
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
1101 |
|
29833 | 1102 |
lemmas mult_compare_simps[noatp] = |
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
1103 |
mult_le_cancel_right mult_le_cancel_left |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
1104 |
mult_le_cancel_right1 mult_le_cancel_right2 |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
1105 |
mult_le_cancel_left1 mult_le_cancel_left2 |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
1106 |
mult_less_cancel_right mult_less_cancel_left |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
1107 |
mult_less_cancel_right1 mult_less_cancel_right2 |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
1108 |
mult_less_cancel_left1 mult_less_cancel_left2 |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
1109 |
mult_cancel_right mult_cancel_left |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
1110 |
mult_cancel_right1 mult_cancel_right2 |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
1111 |
mult_cancel_left1 mult_cancel_left2 |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
1112 |
|
26274 | 1113 |
-- {* FIXME continue localization here *} |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1114 |
|
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1115 |
subsection {* Reasoning about inequalities with division *} |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1116 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
1117 |
lemma mult_right_le_one_le: "0 <= (x::'a::linordered_idom) ==> 0 <= y ==> y <= 1 |
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1118 |
==> x * y <= x" |
35216 | 1119 |
by (auto simp add: mult_le_cancel_left2) |
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1120 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
1121 |
lemma mult_left_le_one_le: "0 <= (x::'a::linordered_idom) ==> 0 <= y ==> y <= 1 |
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1122 |
==> y * x <= x" |
35216 | 1123 |
by (auto simp add: mult_le_cancel_right2) |
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1124 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
1125 |
context linordered_semidom |
25193 | 1126 |
begin |
1127 |
||
1128 |
lemma less_add_one: "a < a + 1" |
|
14293 | 1129 |
proof - |
25193 | 1130 |
have "a + 0 < a + 1" |
23482 | 1131 |
by (blast intro: zero_less_one add_strict_left_mono) |
14293 | 1132 |
thus ?thesis by simp |
1133 |
qed |
|
1134 |
||
25193 | 1135 |
lemma zero_less_two: "0 < 1 + 1" |
29667 | 1136 |
by (blast intro: less_trans zero_less_one less_add_one) |
25193 | 1137 |
|
1138 |
end |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1139 |
|
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
1140 |
|
14293 | 1141 |
subsection {* Absolute Value *} |
1142 |
||
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
1143 |
context linordered_idom |
25304
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset
|
1144 |
begin |
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset
|
1145 |
|
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset
|
1146 |
lemma mult_sgn_abs: "sgn x * abs x = x" |
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset
|
1147 |
unfolding abs_if sgn_if by auto |
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset
|
1148 |
|
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset
|
1149 |
end |
24491 | 1150 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
1151 |
lemma abs_one [simp]: "abs 1 = (1::'a::linordered_idom)" |
29667 | 1152 |
by (simp add: abs_if zero_less_one [THEN order_less_not_sym]) |
25304
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset
|
1153 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
1154 |
class ordered_ring_abs = ordered_ring + ordered_ab_group_add_abs + |
25304
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset
|
1155 |
assumes abs_eq_mult: |
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset
|
1156 |
"(0 \<le> a \<or> a \<le> 0) \<and> (0 \<le> b \<or> b \<le> 0) \<Longrightarrow> \<bar>a * b\<bar> = \<bar>a\<bar> * \<bar>b\<bar>" |
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset
|
1157 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
1158 |
context linordered_idom |
30961 | 1159 |
begin |
1160 |
||
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
1161 |
subclass ordered_ring_abs proof |
35216 | 1162 |
qed (auto simp add: abs_if not_less mult_less_0_iff) |
30961 | 1163 |
|
1164 |
lemma abs_mult: |
|
1165 |
"abs (a * b) = abs a * abs b" |
|
1166 |
by (rule abs_eq_mult) auto |
|
1167 |
||
1168 |
lemma abs_mult_self: |
|
1169 |
"abs a * abs a = a * a" |
|
1170 |
by (simp add: abs_if) |
|
1171 |
||
1172 |
end |
|
14294
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1173 |
|
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1174 |
lemma abs_mult_less: |
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
1175 |
"[| abs a < c; abs b < d |] ==> abs a * abs b < c*(d::'a::linordered_idom)" |
14294
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1176 |
proof - |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1177 |
assume ac: "abs a < c" |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1178 |
hence cpos: "0<c" by (blast intro: order_le_less_trans abs_ge_zero) |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1179 |
assume "abs b < d" |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1180 |
thus ?thesis by (simp add: ac cpos mult_strict_mono) |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1181 |
qed |
14293 | 1182 |
|
29833 | 1183 |
lemmas eq_minus_self_iff[noatp] = equal_neg_zero |
14738 | 1184 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
1185 |
lemma less_minus_self_iff: "(a < -a) = (a < (0::'a::linordered_idom))" |
25304
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset
|
1186 |
unfolding order_less_le less_eq_neg_nonpos equal_neg_zero .. |
14738 | 1187 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
1188 |
lemma abs_less_iff: "(abs a < b) = (a < b & -a < (b::'a::linordered_idom))" |
14738 | 1189 |
apply (simp add: order_less_le abs_le_iff) |
35216 | 1190 |
apply (auto simp add: abs_if) |
14738 | 1191 |
done |
1192 |
||
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
1193 |
lemma abs_mult_pos: "(0::'a::linordered_idom) <= x ==> |
25304
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset
|
1194 |
(abs y) * x = abs (y * x)" |
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset
|
1195 |
apply (subst abs_mult) |
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset
|
1196 |
apply simp |
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset
|
1197 |
done |
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1198 |
|
33364 | 1199 |
code_modulename SML |
35050
9f841f20dca6
renamed OrderedGroup to Groups; split theory Ring_and_Field into Rings Fields
haftmann
parents:
35043
diff
changeset
|
1200 |
Rings Arith |
33364 | 1201 |
|
1202 |
code_modulename OCaml |
|
35050
9f841f20dca6
renamed OrderedGroup to Groups; split theory Ring_and_Field into Rings Fields
haftmann
parents:
35043
diff
changeset
|
1203 |
Rings Arith |
33364 | 1204 |
|
1205 |
code_modulename Haskell |
|
35050
9f841f20dca6
renamed OrderedGroup to Groups; split theory Ring_and_Field into Rings Fields
haftmann
parents:
35043
diff
changeset
|
1206 |
Rings Arith |
33364 | 1207 |
|
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
1208 |
end |