author | blanchet |
Wed, 26 Sep 2012 10:41:36 +0200 | |
changeset 49596 | c3536db7e938 |
parent 44921 | 58eef4843641 |
child 49962 | a8cc904a6820 |
permissions | -rw-r--r-- |
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(* Title: HOL/Rings.thy |
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Author: Gertrud Bauer |
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Author: Steven Obua |
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Author: Tobias Nipkow |
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Author: Lawrence C Paulson |
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Author: Markus Wenzel |
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Author: Jeremy Avigad |
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*) |
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|
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10 |
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 + |
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assumes left_distrib[algebra_simps, field_simps]: "(a + b) * c = a * c + b * c" |
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assumes right_distrib[algebra_simps, field_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 + |
25062 | 29 |
assumes mult_zero_left [simp]: "0 * a = 0" |
30 |
assumes mult_zero_right [simp]: "a * 0 = 0" |
|
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|
22390 | 32 |
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 |
25186 | 35 |
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) |
|
25152 | 42 |
next |
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fix a :: 'a |
|
29667 | 44 |
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 |
14940 | 47 |
|
25186 | 48 |
end |
25152 | 49 |
|
22390 | 50 |
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 |
14738 | 56 |
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 |
|
25152 | 66 |
class comm_semiring_0 = comm_semiring + comm_monoid_add + mult_zero |
67 |
begin |
|
68 |
||
27516 | 69 |
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 |
14940 | 75 |
|
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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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|
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class zero_neq_one = zero + one + |
25062 | 83 |
assumes zero_neq_one [simp]: "0 \<noteq> 1" |
26193 | 84 |
begin |
85 |
||
86 |
lemma one_neq_zero [simp]: "1 \<noteq> 0" |
|
29667 | 87 |
by (rule not_sym) (rule zero_neq_one) |
26193 | 88 |
|
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end |
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|
22390 | 91 |
class semiring_1 = zero_neq_one + semiring_0 + monoid_mult |
14504 | 92 |
|
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text {* Abstract divisibility *} |
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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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"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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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 |
22390 | 110 |
(*previously almost_semiring*) |
25152 | 111 |
begin |
14738 | 112 |
|
27516 | 113 |
subclass semiring_1 .. |
25152 | 114 |
|
29925 | 115 |
lemma dvd_refl[simp]: "a dvd a" |
28559 | 116 |
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 - |
28559 | 124 |
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) |
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then show ?thesis .. |
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128 |
qed |
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129 |
|
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130 |
lemma dvd_0_left_iff [no_atp, simp]: "0 dvd a \<longleftrightarrow> a = 0" |
29667 | 131 |
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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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 |
|
30042 | 141 |
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" |
29667 | 150 |
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" |
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by (rule dvd_mult2) (rule dvd_refl) |
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154 |
|
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lemma mult_dvd_mono: |
30042 | 156 |
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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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 - |
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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 |
|
22390 | 186 |
class no_zero_divisors = zero + times + |
25062 | 187 |
assumes no_zero_divisors: "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> a * b \<noteq> 0" |
36719 | 188 |
begin |
189 |
||
190 |
lemma divisors_zero: |
|
191 |
assumes "a * b = 0" |
|
192 |
shows "a = 0 \<or> b = 0" |
|
193 |
proof (rule classical) |
|
194 |
assume "\<not> (a = 0 \<or> b = 0)" |
|
195 |
then have "a \<noteq> 0" and "b \<noteq> 0" by auto |
|
196 |
with no_zero_divisors have "a * b \<noteq> 0" by blast |
|
197 |
with assms show ?thesis by simp |
|
198 |
qed |
|
199 |
||
200 |
end |
|
14504 | 201 |
|
29904 | 202 |
class semiring_1_cancel = semiring + cancel_comm_monoid_add |
203 |
+ zero_neq_one + monoid_mult |
|
25267 | 204 |
begin |
14940 | 205 |
|
27516 | 206 |
subclass semiring_0_cancel .. |
25512
4134f7c782e2
using intro_locales instead of unfold_locales if appropriate
haftmann
parents:
25450
diff
changeset
|
207 |
|
27516 | 208 |
subclass semiring_1 .. |
25267 | 209 |
|
210 |
end |
|
21199
2d83f93c3580
* Added annihilation axioms ("x * 0 = 0") to axclass semiring_0.
krauss
parents:
20633
diff
changeset
|
211 |
|
29904 | 212 |
class comm_semiring_1_cancel = comm_semiring + cancel_comm_monoid_add |
213 |
+ zero_neq_one + comm_monoid_mult |
|
25267 | 214 |
begin |
14738 | 215 |
|
27516 | 216 |
subclass semiring_1_cancel .. |
217 |
subclass comm_semiring_0_cancel .. |
|
218 |
subclass comm_semiring_1 .. |
|
25267 | 219 |
|
220 |
end |
|
25152 | 221 |
|
22390 | 222 |
class ring = semiring + ab_group_add |
25267 | 223 |
begin |
25152 | 224 |
|
27516 | 225 |
subclass semiring_0_cancel .. |
25152 | 226 |
|
227 |
text {* Distribution rules *} |
|
228 |
||
229 |
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
|
230 |
by (rule minus_unique) (simp add: left_distrib [symmetric]) |
25152 | 231 |
|
232 |
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
|
233 |
by (rule minus_unique) (simp add: right_distrib [symmetric]) |
25152 | 234 |
|
29407
5ef7e97fd9e4
move lemmas mult_minus{left,right} inside class ring
huffman
parents:
29406
diff
changeset
|
235 |
text{*Extract signs from products*} |
35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35631
diff
changeset
|
236 |
lemmas mult_minus_left [simp, no_atp] = minus_mult_left [symmetric] |
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35631
diff
changeset
|
237 |
lemmas mult_minus_right [simp,no_atp] = minus_mult_right [symmetric] |
29407
5ef7e97fd9e4
move lemmas mult_minus{left,right} inside class ring
huffman
parents:
29406
diff
changeset
|
238 |
|
25152 | 239 |
lemma minus_mult_minus [simp]: "- a * - b = a * b" |
29667 | 240 |
by simp |
25152 | 241 |
|
242 |
lemma minus_mult_commute: "- a * b = a * - b" |
|
29667 | 243 |
by simp |
244 |
||
36348
89c54f51f55a
dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents:
36304
diff
changeset
|
245 |
lemma right_diff_distrib[algebra_simps, field_simps]: "a * (b - c) = a * b - a * c" |
29667 | 246 |
by (simp add: right_distrib diff_minus) |
247 |
||
36348
89c54f51f55a
dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents:
36304
diff
changeset
|
248 |
lemma left_diff_distrib[algebra_simps, field_simps]: "(a - b) * c = a * c - b * c" |
29667 | 249 |
by (simp add: left_distrib diff_minus) |
25152 | 250 |
|
35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35631
diff
changeset
|
251 |
lemmas ring_distribs[no_atp] = |
25152 | 252 |
right_distrib left_distrib left_diff_distrib right_diff_distrib |
253 |
||
25230 | 254 |
lemma eq_add_iff1: |
255 |
"a * e + c = b * e + d \<longleftrightarrow> (a - b) * e + c = d" |
|
29667 | 256 |
by (simp add: algebra_simps) |
25230 | 257 |
|
258 |
lemma eq_add_iff2: |
|
259 |
"a * e + c = b * e + d \<longleftrightarrow> c = (b - a) * e + d" |
|
29667 | 260 |
by (simp add: algebra_simps) |
25230 | 261 |
|
25152 | 262 |
end |
263 |
||
35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35631
diff
changeset
|
264 |
lemmas ring_distribs[no_atp] = |
25152 | 265 |
right_distrib left_distrib left_diff_distrib right_diff_distrib |
266 |
||
22390 | 267 |
class comm_ring = comm_semiring + ab_group_add |
25267 | 268 |
begin |
14738 | 269 |
|
27516 | 270 |
subclass ring .. |
28141
193c3ea0f63b
instances comm_semiring_0_cancel < comm_semiring_0, comm_ring < comm_semiring_0_cancel
huffman
parents:
27651
diff
changeset
|
271 |
subclass comm_semiring_0_cancel .. |
25267 | 272 |
|
44350
63cddfbc5a09
replace lemma realpow_two_diff with new lemma square_diff_square_factored
huffman
parents:
44346
diff
changeset
|
273 |
lemma square_diff_square_factored: |
63cddfbc5a09
replace lemma realpow_two_diff with new lemma square_diff_square_factored
huffman
parents:
44346
diff
changeset
|
274 |
"x * x - y * y = (x + y) * (x - y)" |
63cddfbc5a09
replace lemma realpow_two_diff with new lemma square_diff_square_factored
huffman
parents:
44346
diff
changeset
|
275 |
by (simp add: algebra_simps) |
63cddfbc5a09
replace lemma realpow_two_diff with new lemma square_diff_square_factored
huffman
parents:
44346
diff
changeset
|
276 |
|
25267 | 277 |
end |
14738 | 278 |
|
22390 | 279 |
class ring_1 = ring + zero_neq_one + monoid_mult |
25267 | 280 |
begin |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
281 |
|
27516 | 282 |
subclass semiring_1_cancel .. |
25267 | 283 |
|
44346
00dd3c4dabe0
rename real_squared_diff_one_factored to square_diff_one_factored and move to Rings.thy
huffman
parents:
44064
diff
changeset
|
284 |
lemma square_diff_one_factored: |
00dd3c4dabe0
rename real_squared_diff_one_factored to square_diff_one_factored and move to Rings.thy
huffman
parents:
44064
diff
changeset
|
285 |
"x * x - 1 = (x + 1) * (x - 1)" |
00dd3c4dabe0
rename real_squared_diff_one_factored to square_diff_one_factored and move to Rings.thy
huffman
parents:
44064
diff
changeset
|
286 |
by (simp add: algebra_simps) |
00dd3c4dabe0
rename real_squared_diff_one_factored to square_diff_one_factored and move to Rings.thy
huffman
parents:
44064
diff
changeset
|
287 |
|
25267 | 288 |
end |
25152 | 289 |
|
22390 | 290 |
class comm_ring_1 = comm_ring + zero_neq_one + comm_monoid_mult |
291 |
(*previously ring*) |
|
25267 | 292 |
begin |
14738 | 293 |
|
27516 | 294 |
subclass ring_1 .. |
295 |
subclass comm_semiring_1_cancel .. |
|
25267 | 296 |
|
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
|
297 |
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
|
298 |
proof |
6d10cf26b5dc
add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents:
29407
diff
changeset
|
299 |
assume "x dvd - y" |
6d10cf26b5dc
add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents:
29407
diff
changeset
|
300 |
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
|
301 |
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
|
302 |
next |
6d10cf26b5dc
add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents:
29407
diff
changeset
|
303 |
assume "x dvd y" |
6d10cf26b5dc
add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents:
29407
diff
changeset
|
304 |
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
|
305 |
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
|
306 |
qed |
6d10cf26b5dc
add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents:
29407
diff
changeset
|
307 |
|
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
|
308 |
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
|
309 |
proof |
6d10cf26b5dc
add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents:
29407
diff
changeset
|
310 |
assume "- x dvd y" |
6d10cf26b5dc
add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents:
29407
diff
changeset
|
311 |
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
|
312 |
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
|
313 |
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
|
314 |
next |
6d10cf26b5dc
add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents:
29407
diff
changeset
|
315 |
assume "x dvd y" |
6d10cf26b5dc
add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents:
29407
diff
changeset
|
316 |
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
|
317 |
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
|
318 |
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
|
319 |
qed |
6d10cf26b5dc
add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents:
29407
diff
changeset
|
320 |
|
30042 | 321 |
lemma dvd_diff[simp]: "x dvd y \<Longrightarrow> x dvd z \<Longrightarrow> x dvd (y - z)" |
35216 | 322 |
by (simp only: diff_minus dvd_add dvd_minus_iff) |
29409 | 323 |
|
25267 | 324 |
end |
25152 | 325 |
|
22990
775e9de3db48
added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents:
22987
diff
changeset
|
326 |
class ring_no_zero_divisors = ring + no_zero_divisors |
25230 | 327 |
begin |
328 |
||
329 |
lemma mult_eq_0_iff [simp]: |
|
330 |
shows "a * b = 0 \<longleftrightarrow> (a = 0 \<or> b = 0)" |
|
331 |
proof (cases "a = 0 \<or> b = 0") |
|
332 |
case False then have "a \<noteq> 0" and "b \<noteq> 0" by auto |
|
333 |
then show ?thesis using no_zero_divisors by simp |
|
334 |
next |
|
335 |
case True then show ?thesis by auto |
|
336 |
qed |
|
337 |
||
26193 | 338 |
text{*Cancellation of equalities with a common factor*} |
35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35631
diff
changeset
|
339 |
lemma mult_cancel_right [simp, no_atp]: |
26193 | 340 |
"a * c = b * c \<longleftrightarrow> c = 0 \<or> a = b" |
341 |
proof - |
|
342 |
have "(a * c = b * c) = ((a - b) * c = 0)" |
|
35216 | 343 |
by (simp add: algebra_simps) |
344 |
thus ?thesis by (simp add: disj_commute) |
|
26193 | 345 |
qed |
346 |
||
35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35631
diff
changeset
|
347 |
lemma mult_cancel_left [simp, no_atp]: |
26193 | 348 |
"c * a = c * b \<longleftrightarrow> c = 0 \<or> a = b" |
349 |
proof - |
|
350 |
have "(c * a = c * b) = (c * (a - b) = 0)" |
|
35216 | 351 |
by (simp add: algebra_simps) |
352 |
thus ?thesis by simp |
|
26193 | 353 |
qed |
354 |
||
25230 | 355 |
end |
22990
775e9de3db48
added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents:
22987
diff
changeset
|
356 |
|
23544 | 357 |
class ring_1_no_zero_divisors = ring_1 + ring_no_zero_divisors |
26274 | 358 |
begin |
359 |
||
36970 | 360 |
lemma square_eq_1_iff: |
36821
9207505d1ee5
move lemma real_mult_is_one to Rings.thy, renamed to square_eq_1_iff
huffman
parents:
36719
diff
changeset
|
361 |
"x * x = 1 \<longleftrightarrow> x = 1 \<or> x = - 1" |
9207505d1ee5
move lemma real_mult_is_one to Rings.thy, renamed to square_eq_1_iff
huffman
parents:
36719
diff
changeset
|
362 |
proof - |
9207505d1ee5
move lemma real_mult_is_one to Rings.thy, renamed to square_eq_1_iff
huffman
parents:
36719
diff
changeset
|
363 |
have "(x - 1) * (x + 1) = x * x - 1" |
9207505d1ee5
move lemma real_mult_is_one to Rings.thy, renamed to square_eq_1_iff
huffman
parents:
36719
diff
changeset
|
364 |
by (simp add: algebra_simps) |
9207505d1ee5
move lemma real_mult_is_one to Rings.thy, renamed to square_eq_1_iff
huffman
parents:
36719
diff
changeset
|
365 |
hence "x * x = 1 \<longleftrightarrow> (x - 1) * (x + 1) = 0" |
9207505d1ee5
move lemma real_mult_is_one to Rings.thy, renamed to square_eq_1_iff
huffman
parents:
36719
diff
changeset
|
366 |
by simp |
9207505d1ee5
move lemma real_mult_is_one to Rings.thy, renamed to square_eq_1_iff
huffman
parents:
36719
diff
changeset
|
367 |
thus ?thesis |
9207505d1ee5
move lemma real_mult_is_one to Rings.thy, renamed to square_eq_1_iff
huffman
parents:
36719
diff
changeset
|
368 |
by (simp add: eq_neg_iff_add_eq_0) |
9207505d1ee5
move lemma real_mult_is_one to Rings.thy, renamed to square_eq_1_iff
huffman
parents:
36719
diff
changeset
|
369 |
qed |
9207505d1ee5
move lemma real_mult_is_one to Rings.thy, renamed to square_eq_1_iff
huffman
parents:
36719
diff
changeset
|
370 |
|
26274 | 371 |
lemma mult_cancel_right1 [simp]: |
372 |
"c = b * c \<longleftrightarrow> c = 0 \<or> b = 1" |
|
29667 | 373 |
by (insert mult_cancel_right [of 1 c b], force) |
26274 | 374 |
|
375 |
lemma mult_cancel_right2 [simp]: |
|
376 |
"a * c = c \<longleftrightarrow> c = 0 \<or> a = 1" |
|
29667 | 377 |
by (insert mult_cancel_right [of a c 1], simp) |
26274 | 378 |
|
379 |
lemma mult_cancel_left1 [simp]: |
|
380 |
"c = c * b \<longleftrightarrow> c = 0 \<or> b = 1" |
|
29667 | 381 |
by (insert mult_cancel_left [of c 1 b], force) |
26274 | 382 |
|
383 |
lemma mult_cancel_left2 [simp]: |
|
384 |
"c * a = c \<longleftrightarrow> c = 0 \<or> a = 1" |
|
29667 | 385 |
by (insert mult_cancel_left [of c a 1], simp) |
26274 | 386 |
|
387 |
end |
|
22990
775e9de3db48
added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents:
22987
diff
changeset
|
388 |
|
22390 | 389 |
class idom = comm_ring_1 + no_zero_divisors |
25186 | 390 |
begin |
14421
ee97b6463cb4
new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents:
14398
diff
changeset
|
391 |
|
27516 | 392 |
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
|
393 |
|
29915
2146e512cec9
generalize lemma fps_square_eq_iff, move to Ring_and_Field
huffman
parents:
29904
diff
changeset
|
394 |
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
|
395 |
proof |
2146e512cec9
generalize lemma fps_square_eq_iff, move to Ring_and_Field
huffman
parents:
29904
diff
changeset
|
396 |
assume "a * a = b * b" |
2146e512cec9
generalize lemma fps_square_eq_iff, move to Ring_and_Field
huffman
parents:
29904
diff
changeset
|
397 |
then have "(a - b) * (a + b) = 0" |
2146e512cec9
generalize lemma fps_square_eq_iff, move to Ring_and_Field
huffman
parents:
29904
diff
changeset
|
398 |
by (simp add: algebra_simps) |
2146e512cec9
generalize lemma fps_square_eq_iff, move to Ring_and_Field
huffman
parents:
29904
diff
changeset
|
399 |
then show "a = b \<or> a = - b" |
35216 | 400 |
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
|
401 |
next |
2146e512cec9
generalize lemma fps_square_eq_iff, move to Ring_and_Field
huffman
parents:
29904
diff
changeset
|
402 |
assume "a = b \<or> a = - b" |
2146e512cec9
generalize lemma fps_square_eq_iff, move to Ring_and_Field
huffman
parents:
29904
diff
changeset
|
403 |
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
|
404 |
qed |
2146e512cec9
generalize lemma fps_square_eq_iff, move to Ring_and_Field
huffman
parents:
29904
diff
changeset
|
405 |
|
29981
7d0ed261b712
generalize int_dvd_cancel_factor simproc to idom class
huffman
parents:
29949
diff
changeset
|
406 |
lemma dvd_mult_cancel_right [simp]: |
7d0ed261b712
generalize int_dvd_cancel_factor simproc to idom class
huffman
parents:
29949
diff
changeset
|
407 |
"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
|
408 |
proof - |
7d0ed261b712
generalize int_dvd_cancel_factor simproc to idom class
huffman
parents:
29949
diff
changeset
|
409 |
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
|
410 |
unfolding dvd_def by (simp add: mult_ac) |
7d0ed261b712
generalize int_dvd_cancel_factor simproc to idom class
huffman
parents:
29949
diff
changeset
|
411 |
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
|
412 |
unfolding dvd_def by simp |
7d0ed261b712
generalize int_dvd_cancel_factor simproc to idom class
huffman
parents:
29949
diff
changeset
|
413 |
finally show ?thesis . |
7d0ed261b712
generalize int_dvd_cancel_factor simproc to idom class
huffman
parents:
29949
diff
changeset
|
414 |
qed |
7d0ed261b712
generalize int_dvd_cancel_factor simproc to idom class
huffman
parents:
29949
diff
changeset
|
415 |
|
7d0ed261b712
generalize int_dvd_cancel_factor simproc to idom class
huffman
parents:
29949
diff
changeset
|
416 |
lemma dvd_mult_cancel_left [simp]: |
7d0ed261b712
generalize int_dvd_cancel_factor simproc to idom class
huffman
parents:
29949
diff
changeset
|
417 |
"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
|
418 |
proof - |
7d0ed261b712
generalize int_dvd_cancel_factor simproc to idom class
huffman
parents:
29949
diff
changeset
|
419 |
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
|
420 |
unfolding dvd_def by (simp add: mult_ac) |
7d0ed261b712
generalize int_dvd_cancel_factor simproc to idom class
huffman
parents:
29949
diff
changeset
|
421 |
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
|
422 |
unfolding dvd_def by simp |
7d0ed261b712
generalize int_dvd_cancel_factor simproc to idom class
huffman
parents:
29949
diff
changeset
|
423 |
finally show ?thesis . |
7d0ed261b712
generalize int_dvd_cancel_factor simproc to idom class
huffman
parents:
29949
diff
changeset
|
424 |
qed |
7d0ed261b712
generalize int_dvd_cancel_factor simproc to idom class
huffman
parents:
29949
diff
changeset
|
425 |
|
25186 | 426 |
end |
25152 | 427 |
|
35302 | 428 |
text {* |
429 |
The theory of partially ordered rings is taken from the books: |
|
430 |
\begin{itemize} |
|
431 |
\item \emph{Lattice Theory} by Garret Birkhoff, American Mathematical Society 1979 |
|
432 |
\item \emph{Partially Ordered Algebraic Systems}, Pergamon Press 1963 |
|
433 |
\end{itemize} |
|
434 |
Most of the used notions can also be looked up in |
|
435 |
\begin{itemize} |
|
436 |
\item \url{http://www.mathworld.com} by Eric Weisstein et. al. |
|
437 |
\item \emph{Algebra I} by van der Waerden, Springer. |
|
438 |
\end{itemize} |
|
439 |
*} |
|
440 |
||
38642
8fa437809c67
dropped type classes mult_mono and mult_mono1; tuned names of technical rule duplicates
haftmann
parents:
37767
diff
changeset
|
441 |
class ordered_semiring = semiring + comm_monoid_add + ordered_ab_semigroup_add + |
8fa437809c67
dropped type classes mult_mono and mult_mono1; tuned names of technical rule duplicates
haftmann
parents:
37767
diff
changeset
|
442 |
assumes mult_left_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> c * a \<le> c * b" |
8fa437809c67
dropped type classes mult_mono and mult_mono1; tuned names of technical rule duplicates
haftmann
parents:
37767
diff
changeset
|
443 |
assumes mult_right_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * c" |
25230 | 444 |
begin |
445 |
||
446 |
lemma mult_mono: |
|
38642
8fa437809c67
dropped type classes mult_mono and mult_mono1; tuned names of technical rule duplicates
haftmann
parents:
37767
diff
changeset
|
447 |
"a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * d" |
25230 | 448 |
apply (erule mult_right_mono [THEN order_trans], assumption) |
449 |
apply (erule mult_left_mono, assumption) |
|
450 |
done |
|
451 |
||
452 |
lemma mult_mono': |
|
38642
8fa437809c67
dropped type classes mult_mono and mult_mono1; tuned names of technical rule duplicates
haftmann
parents:
37767
diff
changeset
|
453 |
"a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * d" |
25230 | 454 |
apply (rule mult_mono) |
455 |
apply (fast intro: order_trans)+ |
|
456 |
done |
|
457 |
||
458 |
end |
|
21199
2d83f93c3580
* Added annihilation axioms ("x * 0 = 0") to axclass semiring_0.
krauss
parents:
20633
diff
changeset
|
459 |
|
38642
8fa437809c67
dropped type classes mult_mono and mult_mono1; tuned names of technical rule duplicates
haftmann
parents:
37767
diff
changeset
|
460 |
class ordered_cancel_semiring = ordered_semiring + cancel_comm_monoid_add |
25267 | 461 |
begin |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
462 |
|
27516 | 463 |
subclass semiring_0_cancel .. |
23521 | 464 |
|
25230 | 465 |
lemma mult_nonneg_nonneg: "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> a * b" |
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
466 |
using mult_left_mono [of 0 b a] by simp |
25230 | 467 |
|
468 |
lemma mult_nonneg_nonpos: "0 \<le> a \<Longrightarrow> b \<le> 0 \<Longrightarrow> a * b \<le> 0" |
|
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
469 |
using mult_left_mono [of b 0 a] by simp |
30692
44ea10bc07a7
clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents:
30650
diff
changeset
|
470 |
|
44ea10bc07a7
clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents:
30650
diff
changeset
|
471 |
lemma mult_nonpos_nonneg: "a \<le> 0 \<Longrightarrow> 0 \<le> b \<Longrightarrow> a * b \<le> 0" |
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
472 |
using mult_right_mono [of a 0 b] by simp |
30692
44ea10bc07a7
clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents:
30650
diff
changeset
|
473 |
|
44ea10bc07a7
clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents:
30650
diff
changeset
|
474 |
text {* Legacy - use @{text mult_nonpos_nonneg} *} |
25230 | 475 |
lemma mult_nonneg_nonpos2: "0 \<le> a \<Longrightarrow> b \<le> 0 \<Longrightarrow> b * a \<le> 0" |
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
476 |
by (drule mult_right_mono [of b 0], auto) |
25230 | 477 |
|
26234 | 478 |
lemma split_mult_neg_le: "(0 \<le> a & b \<le> 0) | (a \<le> 0 & 0 \<le> b) \<Longrightarrow> a * b \<le> 0" |
29667 | 479 |
by (auto simp add: mult_nonneg_nonpos mult_nonneg_nonpos2) |
25230 | 480 |
|
481 |
end |
|
482 |
||
38642
8fa437809c67
dropped type classes mult_mono and mult_mono1; tuned names of technical rule duplicates
haftmann
parents:
37767
diff
changeset
|
483 |
class linordered_semiring = ordered_semiring + linordered_cancel_ab_semigroup_add |
25267 | 484 |
begin |
25230 | 485 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
486 |
subclass ordered_cancel_semiring .. |
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
487 |
|
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
488 |
subclass ordered_comm_monoid_add .. |
25304
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset
|
489 |
|
25230 | 490 |
lemma mult_left_less_imp_less: |
491 |
"c * a < c * b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a < b" |
|
29667 | 492 |
by (force simp add: mult_left_mono not_le [symmetric]) |
25230 | 493 |
|
494 |
lemma mult_right_less_imp_less: |
|
495 |
"a * c < b * c \<Longrightarrow> 0 \<le> c \<Longrightarrow> a < b" |
|
29667 | 496 |
by (force simp add: mult_right_mono not_le [symmetric]) |
23521 | 497 |
|
25186 | 498 |
end |
25152 | 499 |
|
35043
07dbdf60d5ad
dropped accidental duplication of "lin" prefix from cs. 108662d50512
haftmann
parents:
35032
diff
changeset
|
500 |
class linordered_semiring_1 = linordered_semiring + semiring_1 |
36622
e393a91f86df
Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents:
36348
diff
changeset
|
501 |
begin |
e393a91f86df
Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents:
36348
diff
changeset
|
502 |
|
e393a91f86df
Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents:
36348
diff
changeset
|
503 |
lemma convex_bound_le: |
e393a91f86df
Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents:
36348
diff
changeset
|
504 |
assumes "x \<le> a" "y \<le> a" "0 \<le> u" "0 \<le> v" "u + v = 1" |
e393a91f86df
Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents:
36348
diff
changeset
|
505 |
shows "u * x + v * y \<le> a" |
e393a91f86df
Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents:
36348
diff
changeset
|
506 |
proof- |
e393a91f86df
Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents:
36348
diff
changeset
|
507 |
from assms have "u * x + v * y \<le> u * a + v * a" |
e393a91f86df
Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents:
36348
diff
changeset
|
508 |
by (simp add: add_mono mult_left_mono) |
e393a91f86df
Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents:
36348
diff
changeset
|
509 |
thus ?thesis using assms unfolding left_distrib[symmetric] by simp |
e393a91f86df
Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents:
36348
diff
changeset
|
510 |
qed |
e393a91f86df
Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents:
36348
diff
changeset
|
511 |
|
e393a91f86df
Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents:
36348
diff
changeset
|
512 |
end |
35043
07dbdf60d5ad
dropped accidental duplication of "lin" prefix from cs. 108662d50512
haftmann
parents:
35032
diff
changeset
|
513 |
|
07dbdf60d5ad
dropped accidental duplication of "lin" prefix from cs. 108662d50512
haftmann
parents:
35032
diff
changeset
|
514 |
class linordered_semiring_strict = semiring + comm_monoid_add + linordered_cancel_ab_semigroup_add + |
25062 | 515 |
assumes mult_strict_left_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b" |
516 |
assumes mult_strict_right_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> a * c < b * c" |
|
25267 | 517 |
begin |
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset
|
518 |
|
27516 | 519 |
subclass semiring_0_cancel .. |
14940 | 520 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
521 |
subclass linordered_semiring |
28823 | 522 |
proof |
23550 | 523 |
fix a b c :: 'a |
524 |
assume A: "a \<le> b" "0 \<le> c" |
|
525 |
from A show "c * a \<le> c * b" |
|
25186 | 526 |
unfolding le_less |
527 |
using mult_strict_left_mono by (cases "c = 0") auto |
|
23550 | 528 |
from A show "a * c \<le> b * c" |
25152 | 529 |
unfolding le_less |
25186 | 530 |
using mult_strict_right_mono by (cases "c = 0") auto |
25152 | 531 |
qed |
532 |
||
25230 | 533 |
lemma mult_left_le_imp_le: |
534 |
"c * a \<le> c * b \<Longrightarrow> 0 < c \<Longrightarrow> a \<le> b" |
|
29667 | 535 |
by (force simp add: mult_strict_left_mono _not_less [symmetric]) |
25230 | 536 |
|
537 |
lemma mult_right_le_imp_le: |
|
538 |
"a * c \<le> b * c \<Longrightarrow> 0 < c \<Longrightarrow> a \<le> b" |
|
29667 | 539 |
by (force simp add: mult_strict_right_mono not_less [symmetric]) |
25230 | 540 |
|
30692
44ea10bc07a7
clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents:
30650
diff
changeset
|
541 |
lemma mult_pos_pos: "0 < a \<Longrightarrow> 0 < b \<Longrightarrow> 0 < a * b" |
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
542 |
using mult_strict_left_mono [of 0 b a] by simp |
30692
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 |
lemma mult_pos_neg: "0 < a \<Longrightarrow> b < 0 \<Longrightarrow> a * b < 0" |
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
545 |
using mult_strict_left_mono [of b 0 a] by simp |
30692
44ea10bc07a7
clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents:
30650
diff
changeset
|
546 |
|
44ea10bc07a7
clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents:
30650
diff
changeset
|
547 |
lemma mult_neg_pos: "a < 0 \<Longrightarrow> 0 < b \<Longrightarrow> a * b < 0" |
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
548 |
using mult_strict_right_mono [of a 0 b] by simp |
30692
44ea10bc07a7
clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents:
30650
diff
changeset
|
549 |
|
44ea10bc07a7
clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents:
30650
diff
changeset
|
550 |
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
|
551 |
lemma mult_pos_neg2: "0 < a \<Longrightarrow> b < 0 \<Longrightarrow> b * a < 0" |
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
552 |
by (drule mult_strict_right_mono [of b 0], auto) |
25230 | 553 |
|
554 |
lemma zero_less_mult_pos: |
|
555 |
"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
|
556 |
apply (cases "b\<le>0") |
25230 | 557 |
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
|
558 |
apply (drule_tac mult_pos_neg [of a b]) |
25230 | 559 |
apply (auto dest: less_not_sym) |
560 |
done |
|
561 |
||
562 |
lemma zero_less_mult_pos2: |
|
563 |
"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
|
564 |
apply (cases "b\<le>0") |
25230 | 565 |
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
|
566 |
apply (drule_tac mult_pos_neg2 [of a b]) |
25230 | 567 |
apply (auto dest: less_not_sym) |
568 |
done |
|
569 |
||
26193 | 570 |
text{*Strict monotonicity in both arguments*} |
571 |
lemma mult_strict_mono: |
|
572 |
assumes "a < b" and "c < d" and "0 < b" and "0 \<le> c" |
|
573 |
shows "a * c < b * d" |
|
574 |
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
|
575 |
apply (simp add: mult_pos_pos) |
26193 | 576 |
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
|
577 |
apply (force simp add: le_less) |
26193 | 578 |
apply (erule mult_strict_left_mono, assumption) |
579 |
done |
|
580 |
||
581 |
text{*This weaker variant has more natural premises*} |
|
582 |
lemma mult_strict_mono': |
|
583 |
assumes "a < b" and "c < d" and "0 \<le> a" and "0 \<le> c" |
|
584 |
shows "a * c < b * d" |
|
29667 | 585 |
by (rule mult_strict_mono) (insert assms, auto) |
26193 | 586 |
|
587 |
lemma mult_less_le_imp_less: |
|
588 |
assumes "a < b" and "c \<le> d" and "0 \<le> a" and "0 < c" |
|
589 |
shows "a * c < b * d" |
|
590 |
using assms apply (subgoal_tac "a * c < b * c") |
|
591 |
apply (erule less_le_trans) |
|
592 |
apply (erule mult_left_mono) |
|
593 |
apply simp |
|
594 |
apply (erule mult_strict_right_mono) |
|
595 |
apply assumption |
|
596 |
done |
|
597 |
||
598 |
lemma mult_le_less_imp_less: |
|
599 |
assumes "a \<le> b" and "c < d" and "0 < a" and "0 \<le> c" |
|
600 |
shows "a * c < b * d" |
|
601 |
using assms apply (subgoal_tac "a * c \<le> b * c") |
|
602 |
apply (erule le_less_trans) |
|
603 |
apply (erule mult_strict_left_mono) |
|
604 |
apply simp |
|
605 |
apply (erule mult_right_mono) |
|
606 |
apply simp |
|
607 |
done |
|
608 |
||
609 |
lemma mult_less_imp_less_left: |
|
610 |
assumes less: "c * a < c * b" and nonneg: "0 \<le> c" |
|
611 |
shows "a < b" |
|
612 |
proof (rule ccontr) |
|
613 |
assume "\<not> a < b" |
|
614 |
hence "b \<le> a" by (simp add: linorder_not_less) |
|
615 |
hence "c * b \<le> c * a" using nonneg by (rule mult_left_mono) |
|
29667 | 616 |
with this and less show False by (simp add: not_less [symmetric]) |
26193 | 617 |
qed |
618 |
||
619 |
lemma mult_less_imp_less_right: |
|
620 |
assumes less: "a * c < b * c" and nonneg: "0 \<le> c" |
|
621 |
shows "a < b" |
|
622 |
proof (rule ccontr) |
|
623 |
assume "\<not> a < b" |
|
624 |
hence "b \<le> a" by (simp add: linorder_not_less) |
|
625 |
hence "b * c \<le> a * c" using nonneg by (rule mult_right_mono) |
|
29667 | 626 |
with this and less show False by (simp add: not_less [symmetric]) |
26193 | 627 |
qed |
628 |
||
25230 | 629 |
end |
630 |
||
35097
4554bb2abfa3
dropped last occurence of the linlinordered accident
haftmann
parents:
35092
diff
changeset
|
631 |
class linordered_semiring_1_strict = linordered_semiring_strict + semiring_1 |
36622
e393a91f86df
Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents:
36348
diff
changeset
|
632 |
begin |
e393a91f86df
Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents:
36348
diff
changeset
|
633 |
|
e393a91f86df
Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents:
36348
diff
changeset
|
634 |
subclass linordered_semiring_1 .. |
e393a91f86df
Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents:
36348
diff
changeset
|
635 |
|
e393a91f86df
Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents:
36348
diff
changeset
|
636 |
lemma convex_bound_lt: |
e393a91f86df
Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents:
36348
diff
changeset
|
637 |
assumes "x < a" "y < a" "0 \<le> u" "0 \<le> v" "u + v = 1" |
e393a91f86df
Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents:
36348
diff
changeset
|
638 |
shows "u * x + v * y < a" |
e393a91f86df
Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents:
36348
diff
changeset
|
639 |
proof - |
e393a91f86df
Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents:
36348
diff
changeset
|
640 |
from assms have "u * x + v * y < u * a + v * a" |
e393a91f86df
Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents:
36348
diff
changeset
|
641 |
by (cases "u = 0") |
e393a91f86df
Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents:
36348
diff
changeset
|
642 |
(auto intro!: add_less_le_mono mult_strict_left_mono mult_left_mono) |
e393a91f86df
Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents:
36348
diff
changeset
|
643 |
thus ?thesis using assms unfolding left_distrib[symmetric] by simp |
e393a91f86df
Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents:
36348
diff
changeset
|
644 |
qed |
e393a91f86df
Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents:
36348
diff
changeset
|
645 |
|
e393a91f86df
Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents:
36348
diff
changeset
|
646 |
end |
33319 | 647 |
|
38642
8fa437809c67
dropped type classes mult_mono and mult_mono1; tuned names of technical rule duplicates
haftmann
parents:
37767
diff
changeset
|
648 |
class ordered_comm_semiring = comm_semiring_0 + ordered_ab_semigroup_add + |
8fa437809c67
dropped type classes mult_mono and mult_mono1; tuned names of technical rule duplicates
haftmann
parents:
37767
diff
changeset
|
649 |
assumes comm_mult_left_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> c * a \<le> c * b" |
25186 | 650 |
begin |
25152 | 651 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
652 |
subclass ordered_semiring |
28823 | 653 |
proof |
21199
2d83f93c3580
* Added annihilation axioms ("x * 0 = 0") to axclass semiring_0.
krauss
parents:
20633
diff
changeset
|
654 |
fix a b c :: 'a |
23550 | 655 |
assume "a \<le> b" "0 \<le> c" |
38642
8fa437809c67
dropped type classes mult_mono and mult_mono1; tuned names of technical rule duplicates
haftmann
parents:
37767
diff
changeset
|
656 |
thus "c * a \<le> c * b" by (rule comm_mult_left_mono) |
23550 | 657 |
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
|
658 |
qed |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
659 |
|
25267 | 660 |
end |
661 |
||
38642
8fa437809c67
dropped type classes mult_mono and mult_mono1; tuned names of technical rule duplicates
haftmann
parents:
37767
diff
changeset
|
662 |
class ordered_cancel_comm_semiring = ordered_comm_semiring + cancel_comm_monoid_add |
25267 | 663 |
begin |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
664 |
|
38642
8fa437809c67
dropped type classes mult_mono and mult_mono1; tuned names of technical rule duplicates
haftmann
parents:
37767
diff
changeset
|
665 |
subclass comm_semiring_0_cancel .. |
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
666 |
subclass ordered_comm_semiring .. |
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
667 |
subclass ordered_cancel_semiring .. |
25267 | 668 |
|
669 |
end |
|
670 |
||
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
671 |
class linordered_comm_semiring_strict = comm_semiring_0 + linordered_cancel_ab_semigroup_add + |
38642
8fa437809c67
dropped type classes mult_mono and mult_mono1; tuned names of technical rule duplicates
haftmann
parents:
37767
diff
changeset
|
672 |
assumes comm_mult_strict_left_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b" |
25267 | 673 |
begin |
674 |
||
35043
07dbdf60d5ad
dropped accidental duplication of "lin" prefix from cs. 108662d50512
haftmann
parents:
35032
diff
changeset
|
675 |
subclass linordered_semiring_strict |
28823 | 676 |
proof |
23550 | 677 |
fix a b c :: 'a |
678 |
assume "a < b" "0 < c" |
|
38642
8fa437809c67
dropped type classes mult_mono and mult_mono1; tuned names of technical rule duplicates
haftmann
parents:
37767
diff
changeset
|
679 |
thus "c * a < c * b" by (rule comm_mult_strict_left_mono) |
23550 | 680 |
thus "a * c < b * c" by (simp only: mult_commute) |
681 |
qed |
|
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
682 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
683 |
subclass ordered_cancel_comm_semiring |
28823 | 684 |
proof |
23550 | 685 |
fix a b c :: 'a |
686 |
assume "a \<le> b" "0 \<le> c" |
|
687 |
thus "c * a \<le> c * b" |
|
25186 | 688 |
unfolding le_less |
26193 | 689 |
using mult_strict_left_mono by (cases "c = 0") auto |
23550 | 690 |
qed |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
691 |
|
25267 | 692 |
end |
25230 | 693 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
694 |
class ordered_ring = ring + ordered_cancel_semiring |
25267 | 695 |
begin |
25230 | 696 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
697 |
subclass ordered_ab_group_add .. |
14270 | 698 |
|
25230 | 699 |
lemma less_add_iff1: |
700 |
"a * e + c < b * e + d \<longleftrightarrow> (a - b) * e + c < d" |
|
29667 | 701 |
by (simp add: algebra_simps) |
25230 | 702 |
|
703 |
lemma less_add_iff2: |
|
704 |
"a * e + c < b * e + d \<longleftrightarrow> c < (b - a) * e + d" |
|
29667 | 705 |
by (simp add: algebra_simps) |
25230 | 706 |
|
707 |
lemma le_add_iff1: |
|
708 |
"a * e + c \<le> b * e + d \<longleftrightarrow> (a - b) * e + c \<le> d" |
|
29667 | 709 |
by (simp add: algebra_simps) |
25230 | 710 |
|
711 |
lemma le_add_iff2: |
|
712 |
"a * e + c \<le> b * e + d \<longleftrightarrow> c \<le> (b - a) * e + d" |
|
29667 | 713 |
by (simp add: algebra_simps) |
25230 | 714 |
|
715 |
lemma mult_left_mono_neg: |
|
716 |
"b \<le> a \<Longrightarrow> c \<le> 0 \<Longrightarrow> c * a \<le> c * b" |
|
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
717 |
apply (drule mult_left_mono [of _ _ "- c"]) |
35216 | 718 |
apply simp_all |
25230 | 719 |
done |
720 |
||
721 |
lemma mult_right_mono_neg: |
|
722 |
"b \<le> a \<Longrightarrow> c \<le> 0 \<Longrightarrow> a * c \<le> b * c" |
|
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
723 |
apply (drule mult_right_mono [of _ _ "- c"]) |
35216 | 724 |
apply simp_all |
25230 | 725 |
done |
726 |
||
30692
44ea10bc07a7
clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents:
30650
diff
changeset
|
727 |
lemma mult_nonpos_nonpos: "a \<le> 0 \<Longrightarrow> b \<le> 0 \<Longrightarrow> 0 \<le> a * b" |
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
728 |
using mult_right_mono_neg [of a 0 b] by simp |
25230 | 729 |
|
730 |
lemma split_mult_pos_le: |
|
731 |
"(0 \<le> a \<and> 0 \<le> b) \<or> (a \<le> 0 \<and> b \<le> 0) \<Longrightarrow> 0 \<le> a * b" |
|
29667 | 732 |
by (auto simp add: mult_nonneg_nonneg mult_nonpos_nonpos) |
25186 | 733 |
|
734 |
end |
|
14270 | 735 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
736 |
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
|
737 |
begin |
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset
|
738 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
739 |
subclass ordered_ring .. |
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
740 |
|
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
741 |
subclass ordered_ab_group_add_abs |
28823 | 742 |
proof |
25304
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset
|
743 |
fix a b |
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset
|
744 |
show "\<bar>a + b\<bar> \<le> \<bar>a\<bar> + \<bar>b\<bar>" |
35216 | 745 |
by (auto simp add: abs_if not_less) |
746 |
(auto simp del: minus_add_distrib simp add: minus_add_distrib [symmetric], |
|
36977
71c8973a604b
declare add_nonneg_nonneg [simp]; remove now-redundant lemmas realpow_two_le_order(2)
huffman
parents:
36970
diff
changeset
|
747 |
auto intro!: less_imp_le add_neg_neg) |
35216 | 748 |
qed (auto simp add: abs_if) |
25304
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset
|
749 |
|
35631
0b8a5fd339ab
generalize some lemmas from class linordered_ring_strict to linordered_ring
huffman
parents:
35302
diff
changeset
|
750 |
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
|
751 |
using linear [of 0 a] |
0b8a5fd339ab
generalize some lemmas from class linordered_ring_strict to linordered_ring
huffman
parents:
35302
diff
changeset
|
752 |
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
|
753 |
|
0b8a5fd339ab
generalize some lemmas from class linordered_ring_strict to linordered_ring
huffman
parents:
35302
diff
changeset
|
754 |
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
|
755 |
by (simp add: not_less) |
0b8a5fd339ab
generalize some lemmas from class linordered_ring_strict to linordered_ring
huffman
parents:
35302
diff
changeset
|
756 |
|
25304
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset
|
757 |
end |
23521 | 758 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
759 |
(* 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
|
760 |
Basically, linordered_ring + no_zero_divisors = linordered_ring_strict. |
25230 | 761 |
*) |
35043
07dbdf60d5ad
dropped accidental duplication of "lin" prefix from cs. 108662d50512
haftmann
parents:
35032
diff
changeset
|
762 |
class linordered_ring_strict = ring + linordered_semiring_strict |
25304
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset
|
763 |
+ ordered_ab_group_add + abs_if |
25230 | 764 |
begin |
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset
|
765 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
766 |
subclass linordered_ring .. |
25304
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset
|
767 |
|
30692
44ea10bc07a7
clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents:
30650
diff
changeset
|
768 |
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
|
769 |
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
|
770 |
|
44ea10bc07a7
clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents:
30650
diff
changeset
|
771 |
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
|
772 |
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
|
773 |
|
44ea10bc07a7
clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents:
30650
diff
changeset
|
774 |
lemma mult_neg_neg: "a < 0 \<Longrightarrow> b < 0 \<Longrightarrow> 0 < a * b" |
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
775 |
using mult_strict_right_mono_neg [of a 0 b] by simp |
14738 | 776 |
|
25917 | 777 |
subclass ring_no_zero_divisors |
28823 | 778 |
proof |
25917 | 779 |
fix a b |
780 |
assume "a \<noteq> 0" then have A: "a < 0 \<or> 0 < a" by (simp add: neq_iff) |
|
781 |
assume "b \<noteq> 0" then have B: "b < 0 \<or> 0 < b" by (simp add: neq_iff) |
|
782 |
have "a * b < 0 \<or> 0 < a * b" |
|
783 |
proof (cases "a < 0") |
|
784 |
case True note A' = this |
|
785 |
show ?thesis proof (cases "b < 0") |
|
786 |
case True with A' |
|
787 |
show ?thesis by (auto dest: mult_neg_neg) |
|
788 |
next |
|
789 |
case False with B have "0 < b" by auto |
|
790 |
with A' show ?thesis by (auto dest: mult_strict_right_mono) |
|
791 |
qed |
|
792 |
next |
|
793 |
case False with A have A': "0 < a" by auto |
|
794 |
show ?thesis proof (cases "b < 0") |
|
795 |
case True with A' |
|
796 |
show ?thesis by (auto dest: mult_strict_right_mono_neg) |
|
797 |
next |
|
798 |
case False with B have "0 < b" by auto |
|
799 |
with A' show ?thesis by (auto dest: mult_pos_pos) |
|
800 |
qed |
|
801 |
qed |
|
802 |
then show "a * b \<noteq> 0" by (simp add: neq_iff) |
|
803 |
qed |
|
25304
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset
|
804 |
|
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
805 |
lemma zero_less_mult_iff: |
25917 | 806 |
"0 < a * b \<longleftrightarrow> 0 < a \<and> 0 < b \<or> a < 0 \<and> b < 0" |
807 |
apply (auto simp add: mult_pos_pos mult_neg_neg) |
|
808 |
apply (simp_all add: not_less le_less) |
|
809 |
apply (erule disjE) apply assumption defer |
|
810 |
apply (erule disjE) defer apply (drule sym) apply simp |
|
811 |
apply (erule disjE) defer apply (drule sym) apply simp |
|
812 |
apply (erule disjE) apply assumption apply (drule sym) apply simp |
|
813 |
apply (drule sym) apply simp |
|
814 |
apply (blast dest: zero_less_mult_pos) |
|
25230 | 815 |
apply (blast dest: zero_less_mult_pos2) |
816 |
done |
|
22990
775e9de3db48
added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents:
22987
diff
changeset
|
817 |
|
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
818 |
lemma zero_le_mult_iff: |
25917 | 819 |
"0 \<le> a * b \<longleftrightarrow> 0 \<le> a \<and> 0 \<le> b \<or> a \<le> 0 \<and> b \<le> 0" |
29667 | 820 |
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
|
821 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
822 |
lemma mult_less_0_iff: |
25917 | 823 |
"a * b < 0 \<longleftrightarrow> 0 < a \<and> b < 0 \<or> a < 0 \<and> 0 < b" |
35216 | 824 |
apply (insert zero_less_mult_iff [of "-a" b]) |
825 |
apply force |
|
25917 | 826 |
done |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
827 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
828 |
lemma mult_le_0_iff: |
25917 | 829 |
"a * b \<le> 0 \<longleftrightarrow> 0 \<le> a \<and> b \<le> 0 \<or> a \<le> 0 \<and> 0 \<le> b" |
830 |
apply (insert zero_le_mult_iff [of "-a" b]) |
|
35216 | 831 |
apply force |
25917 | 832 |
done |
833 |
||
26193 | 834 |
text{*Cancellation laws for @{term "c*a < c*b"} and @{term "a*c < b*c"}, |
835 |
also with the relations @{text "\<le>"} and equality.*} |
|
836 |
||
837 |
text{*These ``disjunction'' versions produce two cases when the comparison is |
|
838 |
an assumption, but effectively four when the comparison is a goal.*} |
|
839 |
||
840 |
lemma mult_less_cancel_right_disj: |
|
841 |
"a * c < b * c \<longleftrightarrow> 0 < c \<and> a < b \<or> c < 0 \<and> b < a" |
|
842 |
apply (cases "c = 0") |
|
843 |
apply (auto simp add: neq_iff mult_strict_right_mono |
|
844 |
mult_strict_right_mono_neg) |
|
845 |
apply (auto simp add: not_less |
|
846 |
not_le [symmetric, of "a*c"] |
|
847 |
not_le [symmetric, of a]) |
|
848 |
apply (erule_tac [!] notE) |
|
849 |
apply (auto simp add: less_imp_le mult_right_mono |
|
850 |
mult_right_mono_neg) |
|
851 |
done |
|
852 |
||
853 |
lemma mult_less_cancel_left_disj: |
|
854 |
"c * a < c * b \<longleftrightarrow> 0 < c \<and> a < b \<or> c < 0 \<and> b < a" |
|
855 |
apply (cases "c = 0") |
|
856 |
apply (auto simp add: neq_iff mult_strict_left_mono |
|
857 |
mult_strict_left_mono_neg) |
|
858 |
apply (auto simp add: not_less |
|
859 |
not_le [symmetric, of "c*a"] |
|
860 |
not_le [symmetric, of a]) |
|
861 |
apply (erule_tac [!] notE) |
|
862 |
apply (auto simp add: less_imp_le mult_left_mono |
|
863 |
mult_left_mono_neg) |
|
864 |
done |
|
865 |
||
866 |
text{*The ``conjunction of implication'' lemmas produce two cases when the |
|
867 |
comparison is a goal, but give four when the comparison is an assumption.*} |
|
868 |
||
869 |
lemma mult_less_cancel_right: |
|
870 |
"a * c < b * c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < b) \<and> (c \<le> 0 \<longrightarrow> b < a)" |
|
871 |
using mult_less_cancel_right_disj [of a c b] by auto |
|
872 |
||
873 |
lemma mult_less_cancel_left: |
|
874 |
"c * a < c * b \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < b) \<and> (c \<le> 0 \<longrightarrow> b < a)" |
|
875 |
using mult_less_cancel_left_disj [of c a b] by auto |
|
876 |
||
877 |
lemma mult_le_cancel_right: |
|
878 |
"a * c \<le> b * c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)" |
|
29667 | 879 |
by (simp add: not_less [symmetric] mult_less_cancel_right_disj) |
26193 | 880 |
|
881 |
lemma mult_le_cancel_left: |
|
882 |
"c * a \<le> c * b \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)" |
|
29667 | 883 |
by (simp add: not_less [symmetric] mult_less_cancel_left_disj) |
26193 | 884 |
|
30649
57753e0ec1d4
1. New cancellation simprocs for common factors in inequations
nipkow
parents:
30242
diff
changeset
|
885 |
lemma mult_le_cancel_left_pos: |
57753e0ec1d4
1. New cancellation simprocs for common factors in inequations
nipkow
parents:
30242
diff
changeset
|
886 |
"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
|
887 |
by (auto simp: mult_le_cancel_left) |
57753e0ec1d4
1. New cancellation simprocs for common factors in inequations
nipkow
parents:
30242
diff
changeset
|
888 |
|
57753e0ec1d4
1. New cancellation simprocs for common factors in inequations
nipkow
parents:
30242
diff
changeset
|
889 |
lemma mult_le_cancel_left_neg: |
57753e0ec1d4
1. New cancellation simprocs for common factors in inequations
nipkow
parents:
30242
diff
changeset
|
890 |
"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
|
891 |
by (auto simp: mult_le_cancel_left) |
57753e0ec1d4
1. New cancellation simprocs for common factors in inequations
nipkow
parents:
30242
diff
changeset
|
892 |
|
57753e0ec1d4
1. New cancellation simprocs for common factors in inequations
nipkow
parents:
30242
diff
changeset
|
893 |
lemma mult_less_cancel_left_pos: |
57753e0ec1d4
1. New cancellation simprocs for common factors in inequations
nipkow
parents:
30242
diff
changeset
|
894 |
"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
|
895 |
by (auto simp: mult_less_cancel_left) |
57753e0ec1d4
1. New cancellation simprocs for common factors in inequations
nipkow
parents:
30242
diff
changeset
|
896 |
|
57753e0ec1d4
1. New cancellation simprocs for common factors in inequations
nipkow
parents:
30242
diff
changeset
|
897 |
lemma mult_less_cancel_left_neg: |
57753e0ec1d4
1. New cancellation simprocs for common factors in inequations
nipkow
parents:
30242
diff
changeset
|
898 |
"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
|
899 |
by (auto simp: mult_less_cancel_left) |
57753e0ec1d4
1. New cancellation simprocs for common factors in inequations
nipkow
parents:
30242
diff
changeset
|
900 |
|
25917 | 901 |
end |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
902 |
|
30692
44ea10bc07a7
clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents:
30650
diff
changeset
|
903 |
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
|
904 |
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
|
905 |
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
|
906 |
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
|
907 |
mult_neg_pos mult_neg_neg |
25230 | 908 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
909 |
class ordered_comm_ring = comm_ring + ordered_comm_semiring |
25267 | 910 |
begin |
25230 | 911 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
912 |
subclass ordered_ring .. |
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
913 |
subclass ordered_cancel_comm_semiring .. |
25230 | 914 |
|
25267 | 915 |
end |
25230 | 916 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
917 |
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
|
918 |
(*previously linordered_semiring*) |
25230 | 919 |
assumes zero_less_one [simp]: "0 < 1" |
920 |
begin |
|
921 |
||
922 |
lemma pos_add_strict: |
|
923 |
shows "0 < a \<Longrightarrow> b < c \<Longrightarrow> b < a + c" |
|
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
924 |
using add_strict_mono [of 0 a b c] by simp |
25230 | 925 |
|
26193 | 926 |
lemma zero_le_one [simp]: "0 \<le> 1" |
29667 | 927 |
by (rule zero_less_one [THEN less_imp_le]) |
26193 | 928 |
|
929 |
lemma not_one_le_zero [simp]: "\<not> 1 \<le> 0" |
|
29667 | 930 |
by (simp add: not_le) |
26193 | 931 |
|
932 |
lemma not_one_less_zero [simp]: "\<not> 1 < 0" |
|
29667 | 933 |
by (simp add: not_less) |
26193 | 934 |
|
935 |
lemma less_1_mult: |
|
936 |
assumes "1 < m" and "1 < n" |
|
937 |
shows "1 < m * n" |
|
938 |
using assms mult_strict_mono [of 1 m 1 n] |
|
939 |
by (simp add: less_trans [OF zero_less_one]) |
|
940 |
||
25230 | 941 |
end |
942 |
||
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
943 |
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
|
944 |
linordered_comm_semiring_strict + ordered_ab_group_add + |
25230 | 945 |
abs_if + sgn_if |
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
946 |
(*previously linordered_ring*) |
25917 | 947 |
begin |
948 |
||
36622
e393a91f86df
Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents:
36348
diff
changeset
|
949 |
subclass linordered_semiring_1_strict .. |
35043
07dbdf60d5ad
dropped accidental duplication of "lin" prefix from cs. 108662d50512
haftmann
parents:
35032
diff
changeset
|
950 |
subclass linordered_ring_strict .. |
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
951 |
subclass ordered_comm_ring .. |
27516 | 952 |
subclass idom .. |
25917 | 953 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
954 |
subclass linordered_semidom |
28823 | 955 |
proof |
26193 | 956 |
have "0 \<le> 1 * 1" by (rule zero_le_square) |
957 |
thus "0 < 1" by (simp add: le_less) |
|
25917 | 958 |
qed |
959 |
||
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
960 |
lemma linorder_neqE_linordered_idom: |
26193 | 961 |
assumes "x \<noteq> y" obtains "x < y" | "y < x" |
962 |
using assms by (rule neqE) |
|
963 |
||
26274 | 964 |
text {* These cancellation simprules also produce two cases when the comparison is a goal. *} |
965 |
||
966 |
lemma mult_le_cancel_right1: |
|
967 |
"c \<le> b * c \<longleftrightarrow> (0 < c \<longrightarrow> 1 \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> 1)" |
|
29667 | 968 |
by (insert mult_le_cancel_right [of 1 c b], simp) |
26274 | 969 |
|
970 |
lemma mult_le_cancel_right2: |
|
971 |
"a * c \<le> c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> 1) \<and> (c < 0 \<longrightarrow> 1 \<le> a)" |
|
29667 | 972 |
by (insert mult_le_cancel_right [of a c 1], simp) |
26274 | 973 |
|
974 |
lemma mult_le_cancel_left1: |
|
975 |
"c \<le> c * b \<longleftrightarrow> (0 < c \<longrightarrow> 1 \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> 1)" |
|
29667 | 976 |
by (insert mult_le_cancel_left [of c 1 b], simp) |
26274 | 977 |
|
978 |
lemma mult_le_cancel_left2: |
|
979 |
"c * a \<le> c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> 1) \<and> (c < 0 \<longrightarrow> 1 \<le> a)" |
|
29667 | 980 |
by (insert mult_le_cancel_left [of c a 1], simp) |
26274 | 981 |
|
982 |
lemma mult_less_cancel_right1: |
|
983 |
"c < b * c \<longleftrightarrow> (0 \<le> c \<longrightarrow> 1 < b) \<and> (c \<le> 0 \<longrightarrow> b < 1)" |
|
29667 | 984 |
by (insert mult_less_cancel_right [of 1 c b], simp) |
26274 | 985 |
|
986 |
lemma mult_less_cancel_right2: |
|
987 |
"a * c < c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < 1) \<and> (c \<le> 0 \<longrightarrow> 1 < a)" |
|
29667 | 988 |
by (insert mult_less_cancel_right [of a c 1], simp) |
26274 | 989 |
|
990 |
lemma mult_less_cancel_left1: |
|
991 |
"c < c * b \<longleftrightarrow> (0 \<le> c \<longrightarrow> 1 < b) \<and> (c \<le> 0 \<longrightarrow> b < 1)" |
|
29667 | 992 |
by (insert mult_less_cancel_left [of c 1 b], simp) |
26274 | 993 |
|
994 |
lemma mult_less_cancel_left2: |
|
995 |
"c * a < c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < 1) \<and> (c \<le> 0 \<longrightarrow> 1 < a)" |
|
29667 | 996 |
by (insert mult_less_cancel_left [of c a 1], simp) |
26274 | 997 |
|
27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27516
diff
changeset
|
998 |
lemma sgn_sgn [simp]: |
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27516
diff
changeset
|
999 |
"sgn (sgn a) = sgn a" |
29700 | 1000 |
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
|
1001 |
|
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27516
diff
changeset
|
1002 |
lemma sgn_0_0: |
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27516
diff
changeset
|
1003 |
"sgn a = 0 \<longleftrightarrow> a = 0" |
29700 | 1004 |
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
|
1005 |
|
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27516
diff
changeset
|
1006 |
lemma sgn_1_pos: |
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27516
diff
changeset
|
1007 |
"sgn a = 1 \<longleftrightarrow> a > 0" |
35216 | 1008 |
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
|
1009 |
|
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27516
diff
changeset
|
1010 |
lemma sgn_1_neg: |
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27516
diff
changeset
|
1011 |
"sgn a = - 1 \<longleftrightarrow> a < 0" |
35216 | 1012 |
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
|
1013 |
|
29940 | 1014 |
lemma sgn_pos [simp]: |
1015 |
"0 < a \<Longrightarrow> sgn a = 1" |
|
1016 |
unfolding sgn_1_pos . |
|
1017 |
||
1018 |
lemma sgn_neg [simp]: |
|
1019 |
"a < 0 \<Longrightarrow> sgn a = - 1" |
|
1020 |
unfolding sgn_1_neg . |
|
1021 |
||
27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27516
diff
changeset
|
1022 |
lemma sgn_times: |
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27516
diff
changeset
|
1023 |
"sgn (a * b) = sgn a * sgn b" |
29667 | 1024 |
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
|
1025 |
|
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1026 |
lemma abs_sgn: "\<bar>k\<bar> = k * sgn k" |
29700 | 1027 |
unfolding sgn_if abs_if by auto |
1028 |
||
29940 | 1029 |
lemma sgn_greater [simp]: |
1030 |
"0 < sgn a \<longleftrightarrow> 0 < a" |
|
1031 |
unfolding sgn_if by auto |
|
1032 |
||
1033 |
lemma sgn_less [simp]: |
|
1034 |
"sgn a < 0 \<longleftrightarrow> a < 0" |
|
1035 |
unfolding sgn_if by auto |
|
1036 |
||
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1037 |
lemma abs_dvd_iff [simp]: "\<bar>m\<bar> dvd k \<longleftrightarrow> m dvd k" |
29949 | 1038 |
by (simp add: abs_if) |
1039 |
||
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1040 |
lemma dvd_abs_iff [simp]: "m dvd \<bar>k\<bar> \<longleftrightarrow> m dvd k" |
29949 | 1041 |
by (simp add: abs_if) |
29653 | 1042 |
|
33676
802f5e233e48
moved lemma from Algebra/IntRing to Ring_and_Field
nipkow
parents:
33364
diff
changeset
|
1043 |
lemma dvd_if_abs_eq: |
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1044 |
"\<bar>l\<bar> = \<bar>k\<bar> \<Longrightarrow> l dvd k" |
33676
802f5e233e48
moved lemma from Algebra/IntRing to Ring_and_Field
nipkow
parents:
33364
diff
changeset
|
1045 |
by(subst abs_dvd_iff[symmetric]) simp |
802f5e233e48
moved lemma from Algebra/IntRing to Ring_and_Field
nipkow
parents:
33364
diff
changeset
|
1046 |
|
25917 | 1047 |
end |
25230 | 1048 |
|
26274 | 1049 |
text {* Simprules for comparisons where common factors can be cancelled. *} |
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
1050 |
|
35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35631
diff
changeset
|
1051 |
lemmas mult_compare_simps[no_atp] = |
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
1052 |
mult_le_cancel_right mult_le_cancel_left |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
1053 |
mult_le_cancel_right1 mult_le_cancel_right2 |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
1054 |
mult_le_cancel_left1 mult_le_cancel_left2 |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
1055 |
mult_less_cancel_right mult_less_cancel_left |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
1056 |
mult_less_cancel_right1 mult_less_cancel_right2 |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
1057 |
mult_less_cancel_left1 mult_less_cancel_left2 |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
1058 |
mult_cancel_right mult_cancel_left |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
1059 |
mult_cancel_right1 mult_cancel_right2 |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
1060 |
mult_cancel_left1 mult_cancel_left2 |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
1061 |
|
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1062 |
text {* Reasoning about inequalities with division *} |
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1063 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
1064 |
context linordered_semidom |
25193 | 1065 |
begin |
1066 |
||
1067 |
lemma less_add_one: "a < a + 1" |
|
14293 | 1068 |
proof - |
25193 | 1069 |
have "a + 0 < a + 1" |
23482 | 1070 |
by (blast intro: zero_less_one add_strict_left_mono) |
14293 | 1071 |
thus ?thesis by simp |
1072 |
qed |
|
1073 |
||
25193 | 1074 |
lemma zero_less_two: "0 < 1 + 1" |
29667 | 1075 |
by (blast intro: less_trans zero_less_one less_add_one) |
25193 | 1076 |
|
1077 |
end |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1078 |
|
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1079 |
context linordered_idom |
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1080 |
begin |
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
1081 |
|
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1082 |
lemma mult_right_le_one_le: |
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1083 |
"0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> x * y \<le> x" |
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1084 |
by (auto simp add: mult_le_cancel_left2) |
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1085 |
|
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1086 |
lemma mult_left_le_one_le: |
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1087 |
"0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> y * x \<le> x" |
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1088 |
by (auto simp add: mult_le_cancel_right2) |
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1089 |
|
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1090 |
end |
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1091 |
|
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1092 |
text {* Absolute Value *} |
14293 | 1093 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
1094 |
context linordered_idom |
25304
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset
|
1095 |
begin |
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset
|
1096 |
|
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1097 |
lemma mult_sgn_abs: |
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1098 |
"sgn x * \<bar>x\<bar> = x" |
25304
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset
|
1099 |
unfolding abs_if sgn_if by auto |
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset
|
1100 |
|
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1101 |
lemma abs_one [simp]: |
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1102 |
"\<bar>1\<bar> = 1" |
44921 | 1103 |
by (simp add: abs_if) |
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1104 |
|
25304
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset
|
1105 |
end |
24491 | 1106 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
1107 |
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
|
1108 |
assumes abs_eq_mult: |
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset
|
1109 |
"(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
|
1110 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
1111 |
context linordered_idom |
30961 | 1112 |
begin |
1113 |
||
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
1114 |
subclass ordered_ring_abs proof |
35216 | 1115 |
qed (auto simp add: abs_if not_less mult_less_0_iff) |
30961 | 1116 |
|
1117 |
lemma abs_mult: |
|
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1118 |
"\<bar>a * b\<bar> = \<bar>a\<bar> * \<bar>b\<bar>" |
30961 | 1119 |
by (rule abs_eq_mult) auto |
1120 |
||
1121 |
lemma abs_mult_self: |
|
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1122 |
"\<bar>a\<bar> * \<bar>a\<bar> = a * a" |
30961 | 1123 |
by (simp add: abs_if) |
1124 |
||
14294
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1125 |
lemma abs_mult_less: |
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1126 |
"\<bar>a\<bar> < c \<Longrightarrow> \<bar>b\<bar> < d \<Longrightarrow> \<bar>a\<bar> * \<bar>b\<bar> < c * d" |
14294
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1127 |
proof - |
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1128 |
assume ac: "\<bar>a\<bar> < c" |
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1129 |
hence cpos: "0<c" by (blast intro: le_less_trans abs_ge_zero) |
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1130 |
assume "\<bar>b\<bar> < d" |
14294
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1131 |
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
|
1132 |
qed |
14293 | 1133 |
|
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1134 |
lemma less_minus_self_iff: |
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1135 |
"a < - a \<longleftrightarrow> a < 0" |
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1136 |
by (simp only: less_le less_eq_neg_nonpos equal_neg_zero) |
14738 | 1137 |
|
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1138 |
lemma abs_less_iff: |
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1139 |
"\<bar>a\<bar> < b \<longleftrightarrow> a < b \<and> - a < b" |
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1140 |
by (simp add: less_le abs_le_iff) (auto simp add: abs_if) |
14738 | 1141 |
|
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1142 |
lemma abs_mult_pos: |
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1143 |
"0 \<le> x \<Longrightarrow> \<bar>y\<bar> * x = \<bar>y * x\<bar>" |
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1144 |
by (simp add: abs_mult) |
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1145 |
|
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1146 |
end |
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1147 |
|
33364 | 1148 |
code_modulename SML |
35050
9f841f20dca6
renamed OrderedGroup to Groups; split theory Ring_and_Field into Rings Fields
haftmann
parents:
35043
diff
changeset
|
1149 |
Rings Arith |
33364 | 1150 |
|
1151 |
code_modulename OCaml |
|
35050
9f841f20dca6
renamed OrderedGroup to Groups; split theory Ring_and_Field into Rings Fields
haftmann
parents:
35043
diff
changeset
|
1152 |
Rings Arith |
33364 | 1153 |
|
1154 |
code_modulename Haskell |
|
35050
9f841f20dca6
renamed OrderedGroup to Groups; split theory Ring_and_Field into Rings Fields
haftmann
parents:
35043
diff
changeset
|
1155 |
Rings Arith |
33364 | 1156 |
|
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
1157 |
end |