author | haftmann |
Sat, 28 Mar 2015 21:32:48 +0100 | |
changeset 59833 | ab828c2c5d67 |
parent 59832 | d5ccdca16cca |
child 59865 | 8a20dd967385 |
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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|
58889 | 10 |
section {* Rings *} |
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11 |
|
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theory Rings |
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imports Groups |
15131 | 14 |
begin |
14504 | 15 |
|
22390 | 16 |
class semiring = ab_semigroup_add + semigroup_mult + |
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assumes distrib_right[algebra_simps]: "(a + b) * c = a * c + b * c" |
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assumes distrib_left[algebra_simps]: "a * (b + c) = a * b + a * c" |
25152 | 19 |
begin |
20 |
||
21 |
text{*For the @{text combine_numerals} simproc*} |
|
22 |
lemma combine_common_factor: |
|
23 |
"a * e + (b * e + c) = (a + b) * e + c" |
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by (simp add: distrib_right ac_simps) |
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" |
|
58195 | 31 |
begin |
32 |
||
33 |
lemma mult_not_zero: |
|
34 |
"a * b \<noteq> 0 \<Longrightarrow> a \<noteq> 0 \<and> b \<noteq> 0" |
|
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by auto |
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||
37 |
end |
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|
58198 | 39 |
class semiring_0 = semiring + comm_monoid_add + mult_zero |
40 |
||
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class semiring_0_cancel = semiring + cancel_comm_monoid_add |
25186 | 42 |
begin |
14504 | 43 |
|
25186 | 44 |
subclass semiring_0 |
28823 | 45 |
proof |
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fix a :: 'a |
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have "0 * a + 0 * a = 0 * a + 0" by (simp add: distrib_right [symmetric]) |
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thus "0 * a = 0" by (simp only: add_left_cancel) |
25152 | 49 |
next |
50 |
fix a :: 'a |
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have "a * 0 + a * 0 = a * 0 + 0" by (simp add: distrib_left [symmetric]) |
29667 | 52 |
thus "a * 0 = 0" by (simp only: add_left_cancel) |
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qed |
14940 | 54 |
|
25186 | 55 |
end |
25152 | 56 |
|
22390 | 57 |
class comm_semiring = ab_semigroup_add + ab_semigroup_mult + |
25062 | 58 |
assumes distrib: "(a + b) * c = a * c + b * c" |
25152 | 59 |
begin |
14504 | 60 |
|
25152 | 61 |
subclass semiring |
28823 | 62 |
proof |
14738 | 63 |
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: ac_simps) |
14738 | 66 |
also have "... = b * a + c * a" by (simp only: distrib) |
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also have "... = a * b + a * c" by (simp add: ac_simps) |
14738 | 68 |
finally show "a * (b + c) = a * b + a * c" by blast |
14504 | 69 |
qed |
70 |
||
25152 | 71 |
end |
14504 | 72 |
|
25152 | 73 |
class comm_semiring_0 = comm_semiring + comm_monoid_add + mult_zero |
74 |
begin |
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75 |
||
27516 | 76 |
subclass semiring_0 .. |
25152 | 77 |
|
78 |
end |
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14504 | 79 |
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29904 | 80 |
class comm_semiring_0_cancel = comm_semiring + cancel_comm_monoid_add |
25186 | 81 |
begin |
14940 | 82 |
|
27516 | 83 |
subclass semiring_0_cancel .. |
14940 | 84 |
|
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subclass comm_semiring_0 .. |
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86 |
|
25186 | 87 |
end |
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|
22390 | 89 |
class zero_neq_one = zero + one + |
25062 | 90 |
assumes zero_neq_one [simp]: "0 \<noteq> 1" |
26193 | 91 |
begin |
92 |
||
93 |
lemma one_neq_zero [simp]: "1 \<noteq> 0" |
|
29667 | 94 |
by (rule not_sym) (rule zero_neq_one) |
26193 | 95 |
|
54225 | 96 |
definition of_bool :: "bool \<Rightarrow> 'a" |
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where |
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"of_bool p = (if p then 1 else 0)" |
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lemma of_bool_eq [simp, code]: |
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"of_bool False = 0" |
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"of_bool True = 1" |
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by (simp_all add: of_bool_def) |
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lemma of_bool_eq_iff: |
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"of_bool p = of_bool q \<longleftrightarrow> p = q" |
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by (simp add: of_bool_def) |
|
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||
55187 | 109 |
lemma split_of_bool [split]: |
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"P (of_bool p) \<longleftrightarrow> (p \<longrightarrow> P 1) \<and> (\<not> p \<longrightarrow> P 0)" |
|
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by (cases p) simp_all |
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||
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lemma split_of_bool_asm: |
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"P (of_bool p) \<longleftrightarrow> \<not> (p \<and> \<not> P 1 \<or> \<not> p \<and> \<not> P 0)" |
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by (cases p) simp_all |
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||
54225 | 117 |
end |
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|
22390 | 119 |
class semiring_1 = zero_neq_one + semiring_0 + monoid_mult |
14504 | 120 |
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text {* Abstract divisibility *} |
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class dvd = times |
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begin |
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|
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definition dvd :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infix "dvd" 50) where |
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"b dvd a \<longleftrightarrow> (\<exists>k. a = b * k)" |
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128 |
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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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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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134 |
|
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end |
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136 |
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context comm_monoid_mult |
25152 | 138 |
begin |
14738 | 139 |
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subclass dvd . |
25152 | 141 |
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lemma dvd_refl [simp]: |
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"a dvd a" |
28559 | 144 |
proof |
145 |
show "a = a * 1" by simp |
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qed |
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147 |
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lemma dvd_trans: |
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149 |
assumes "a dvd b" and "b dvd c" |
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150 |
shows "a dvd c" |
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151 |
proof - |
28559 | 152 |
from assms obtain v where "b = a * v" by (auto elim!: dvdE) |
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moreover from assms obtain w where "c = b * w" by (auto elim!: dvdE) |
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ultimately have "c = a * (v * w)" by (simp add: mult.assoc) |
28559 | 155 |
then show ?thesis .. |
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156 |
qed |
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157 |
|
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lemma one_dvd [simp]: |
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"1 dvd a" |
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by (auto intro!: dvdI) |
28559 | 161 |
|
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lemma dvd_mult [simp]: |
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163 |
"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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165 |
|
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lemma dvd_mult2 [simp]: |
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"a dvd b \<Longrightarrow> a dvd (b * c)" |
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using dvd_mult [of a b c] by (simp add: ac_simps) |
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169 |
|
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lemma dvd_triv_right [simp]: |
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"a dvd b * a" |
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by (rule dvd_mult) (rule dvd_refl) |
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173 |
|
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lemma dvd_triv_left [simp]: |
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"a dvd a * b" |
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by (rule dvd_mult2) (rule dvd_refl) |
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177 |
|
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178 |
lemma mult_dvd_mono: |
30042 | 179 |
assumes "a dvd b" |
180 |
and "c dvd d" |
|
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181 |
shows "a * c dvd b * d" |
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182 |
proof - |
30042 | 183 |
from `a dvd b` obtain b' where "b = a * b'" .. |
184 |
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: ac_simps) |
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then show ?thesis .. |
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187 |
qed |
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188 |
|
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189 |
lemma dvd_mult_left: |
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190 |
"a * b dvd c \<Longrightarrow> a dvd c" |
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by (simp add: dvd_def mult.assoc) blast |
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192 |
|
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lemma dvd_mult_right: |
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194 |
"a * b dvd c \<Longrightarrow> b dvd c" |
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using dvd_mult_left [of b a c] by (simp add: ac_simps) |
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196 |
|
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197 |
end |
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|
198 |
|
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
59000
diff
changeset
|
199 |
class comm_semiring_1 = zero_neq_one + comm_semiring_0 + comm_monoid_mult |
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
59000
diff
changeset
|
200 |
begin |
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
59000
diff
changeset
|
201 |
|
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
59000
diff
changeset
|
202 |
subclass semiring_1 .. |
27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27516
diff
changeset
|
203 |
|
59009
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
59000
diff
changeset
|
204 |
lemma dvd_0_left_iff [simp]: |
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
59000
diff
changeset
|
205 |
"0 dvd a \<longleftrightarrow> a = 0" |
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
59000
diff
changeset
|
206 |
by (auto intro: dvd_refl elim!: dvdE) |
27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27516
diff
changeset
|
207 |
|
59009
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
59000
diff
changeset
|
208 |
lemma dvd_0_right [iff]: |
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
59000
diff
changeset
|
209 |
"a dvd 0" |
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
59000
diff
changeset
|
210 |
proof |
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
59000
diff
changeset
|
211 |
show "0 = a * 0" by simp |
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
59000
diff
changeset
|
212 |
qed |
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
59000
diff
changeset
|
213 |
|
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
59000
diff
changeset
|
214 |
lemma dvd_0_left: |
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
59000
diff
changeset
|
215 |
"0 dvd a \<Longrightarrow> a = 0" |
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
59000
diff
changeset
|
216 |
by simp |
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
59000
diff
changeset
|
217 |
|
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
59000
diff
changeset
|
218 |
lemma dvd_add [simp]: |
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
59000
diff
changeset
|
219 |
assumes "a dvd b" and "a dvd c" |
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
59000
diff
changeset
|
220 |
shows "a dvd (b + c)" |
27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27516
diff
changeset
|
221 |
proof - |
29925 | 222 |
from `a dvd b` obtain b' where "b = a * b'" .. |
223 |
moreover from `a dvd c` obtain c' where "c = a * c'" .. |
|
49962
a8cc904a6820
Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents:
44921
diff
changeset
|
224 |
ultimately have "b + c = a * (b' + c')" by (simp add: distrib_left) |
27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27516
diff
changeset
|
225 |
then show ?thesis .. |
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27516
diff
changeset
|
226 |
qed |
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27516
diff
changeset
|
227 |
|
25152 | 228 |
end |
14421
ee97b6463cb4
new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents:
14398
diff
changeset
|
229 |
|
29904 | 230 |
class semiring_1_cancel = semiring + cancel_comm_monoid_add |
231 |
+ zero_neq_one + monoid_mult |
|
25267 | 232 |
begin |
14940 | 233 |
|
27516 | 234 |
subclass semiring_0_cancel .. |
25512
4134f7c782e2
using intro_locales instead of unfold_locales if appropriate
haftmann
parents:
25450
diff
changeset
|
235 |
|
27516 | 236 |
subclass semiring_1 .. |
25267 | 237 |
|
238 |
end |
|
21199
2d83f93c3580
* Added annihilation axioms ("x * 0 = 0") to axclass semiring_0.
krauss
parents:
20633
diff
changeset
|
239 |
|
29904 | 240 |
class comm_semiring_1_cancel = comm_semiring + cancel_comm_monoid_add |
241 |
+ zero_neq_one + comm_monoid_mult |
|
25267 | 242 |
begin |
14738 | 243 |
|
27516 | 244 |
subclass semiring_1_cancel .. |
245 |
subclass comm_semiring_0_cancel .. |
|
246 |
subclass comm_semiring_1 .. |
|
25267 | 247 |
|
248 |
end |
|
25152 | 249 |
|
59816
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
250 |
class comm_semiring_1_diff_distrib = comm_semiring_1_cancel + |
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
251 |
assumes right_diff_distrib' [algebra_simps]: "a * (b - c) = a * b - a * c" |
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
252 |
begin |
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
253 |
|
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
254 |
lemma left_diff_distrib' [algebra_simps]: |
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
255 |
"(b - c) * a = b * a - c * a" |
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
256 |
by (simp add: algebra_simps) |
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
257 |
|
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
258 |
lemma dvd_add_times_triv_left_iff [simp]: |
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
259 |
"a dvd c * a + b \<longleftrightarrow> a dvd b" |
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
260 |
proof - |
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
261 |
have "a dvd a * c + b \<longleftrightarrow> a dvd b" (is "?P \<longleftrightarrow> ?Q") |
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
262 |
proof |
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
263 |
assume ?Q then show ?P by simp |
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
264 |
next |
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
265 |
assume ?P |
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
266 |
then obtain d where "a * c + b = a * d" .. |
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
267 |
then have "a * c + b - a * c = a * d - a * c" by simp |
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
268 |
then have "b = a * d - a * c" by simp |
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
269 |
then have "b = a * (d - c)" by (simp add: algebra_simps) |
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
270 |
then show ?Q .. |
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
271 |
qed |
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
272 |
then show "a dvd c * a + b \<longleftrightarrow> a dvd b" by (simp add: ac_simps) |
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
273 |
qed |
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
274 |
|
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
275 |
lemma dvd_add_times_triv_right_iff [simp]: |
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
276 |
"a dvd b + c * a \<longleftrightarrow> a dvd b" |
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
277 |
using dvd_add_times_triv_left_iff [of a c b] by (simp add: ac_simps) |
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
278 |
|
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
279 |
lemma dvd_add_triv_left_iff [simp]: |
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
280 |
"a dvd a + b \<longleftrightarrow> a dvd b" |
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
281 |
using dvd_add_times_triv_left_iff [of a 1 b] by simp |
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
282 |
|
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
283 |
lemma dvd_add_triv_right_iff [simp]: |
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
284 |
"a dvd b + a \<longleftrightarrow> a dvd b" |
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
285 |
using dvd_add_times_triv_right_iff [of a b 1] by simp |
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
286 |
|
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
287 |
lemma dvd_add_right_iff: |
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
288 |
assumes "a dvd b" |
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
289 |
shows "a dvd b + c \<longleftrightarrow> a dvd c" (is "?P \<longleftrightarrow> ?Q") |
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
290 |
proof |
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
291 |
assume ?P then obtain d where "b + c = a * d" .. |
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
292 |
moreover from `a dvd b` obtain e where "b = a * e" .. |
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
293 |
ultimately have "a * e + c = a * d" by simp |
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
294 |
then have "a * e + c - a * e = a * d - a * e" by simp |
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
295 |
then have "c = a * d - a * e" by simp |
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
296 |
then have "c = a * (d - e)" by (simp add: algebra_simps) |
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
297 |
then show ?Q .. |
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
298 |
next |
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
299 |
assume ?Q with assms show ?P by simp |
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
300 |
qed |
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
301 |
|
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
302 |
lemma dvd_add_left_iff: |
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
303 |
assumes "a dvd c" |
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
304 |
shows "a dvd b + c \<longleftrightarrow> a dvd b" |
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
305 |
using assms dvd_add_right_iff [of a c b] by (simp add: ac_simps) |
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
306 |
|
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
307 |
end |
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
308 |
|
22390 | 309 |
class ring = semiring + ab_group_add |
25267 | 310 |
begin |
25152 | 311 |
|
27516 | 312 |
subclass semiring_0_cancel .. |
25152 | 313 |
|
314 |
text {* Distribution rules *} |
|
315 |
||
316 |
lemma minus_mult_left: "- (a * b) = - a * b" |
|
49962
a8cc904a6820
Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents:
44921
diff
changeset
|
317 |
by (rule minus_unique) (simp add: distrib_right [symmetric]) |
25152 | 318 |
|
319 |
lemma minus_mult_right: "- (a * b) = a * - b" |
|
49962
a8cc904a6820
Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents:
44921
diff
changeset
|
320 |
by (rule minus_unique) (simp add: distrib_left [symmetric]) |
25152 | 321 |
|
29407
5ef7e97fd9e4
move lemmas mult_minus{left,right} inside class ring
huffman
parents:
29406
diff
changeset
|
322 |
text{*Extract signs from products*} |
54147
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
52435
diff
changeset
|
323 |
lemmas mult_minus_left [simp] = minus_mult_left [symmetric] |
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
52435
diff
changeset
|
324 |
lemmas mult_minus_right [simp] = minus_mult_right [symmetric] |
29407
5ef7e97fd9e4
move lemmas mult_minus{left,right} inside class ring
huffman
parents:
29406
diff
changeset
|
325 |
|
25152 | 326 |
lemma minus_mult_minus [simp]: "- a * - b = a * b" |
29667 | 327 |
by simp |
25152 | 328 |
|
329 |
lemma minus_mult_commute: "- a * b = a * - b" |
|
29667 | 330 |
by simp |
331 |
||
58776
95e58e04e534
use NO_MATCH-simproc for distribution rules in field_simps, otherwise field_simps on '(a / (c + d)) * (e + f)' can be non-terminating
hoelzl
parents:
58649
diff
changeset
|
332 |
lemma right_diff_distrib [algebra_simps]: |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54225
diff
changeset
|
333 |
"a * (b - c) = a * b - a * c" |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54225
diff
changeset
|
334 |
using distrib_left [of a b "-c "] by simp |
29667 | 335 |
|
58776
95e58e04e534
use NO_MATCH-simproc for distribution rules in field_simps, otherwise field_simps on '(a / (c + d)) * (e + f)' can be non-terminating
hoelzl
parents:
58649
diff
changeset
|
336 |
lemma left_diff_distrib [algebra_simps]: |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54225
diff
changeset
|
337 |
"(a - b) * c = a * c - b * c" |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54225
diff
changeset
|
338 |
using distrib_right [of a "- b" c] by simp |
25152 | 339 |
|
54147
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
52435
diff
changeset
|
340 |
lemmas ring_distribs = |
49962
a8cc904a6820
Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents:
44921
diff
changeset
|
341 |
distrib_left distrib_right left_diff_distrib right_diff_distrib |
25152 | 342 |
|
25230 | 343 |
lemma eq_add_iff1: |
344 |
"a * e + c = b * e + d \<longleftrightarrow> (a - b) * e + c = d" |
|
29667 | 345 |
by (simp add: algebra_simps) |
25230 | 346 |
|
347 |
lemma eq_add_iff2: |
|
348 |
"a * e + c = b * e + d \<longleftrightarrow> c = (b - a) * e + d" |
|
29667 | 349 |
by (simp add: algebra_simps) |
25230 | 350 |
|
25152 | 351 |
end |
352 |
||
54147
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
52435
diff
changeset
|
353 |
lemmas ring_distribs = |
49962
a8cc904a6820
Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents:
44921
diff
changeset
|
354 |
distrib_left distrib_right left_diff_distrib right_diff_distrib |
25152 | 355 |
|
22390 | 356 |
class comm_ring = comm_semiring + ab_group_add |
25267 | 357 |
begin |
14738 | 358 |
|
27516 | 359 |
subclass ring .. |
28141
193c3ea0f63b
instances comm_semiring_0_cancel < comm_semiring_0, comm_ring < comm_semiring_0_cancel
huffman
parents:
27651
diff
changeset
|
360 |
subclass comm_semiring_0_cancel .. |
25267 | 361 |
|
44350
63cddfbc5a09
replace lemma realpow_two_diff with new lemma square_diff_square_factored
huffman
parents:
44346
diff
changeset
|
362 |
lemma square_diff_square_factored: |
63cddfbc5a09
replace lemma realpow_two_diff with new lemma square_diff_square_factored
huffman
parents:
44346
diff
changeset
|
363 |
"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
|
364 |
by (simp add: algebra_simps) |
63cddfbc5a09
replace lemma realpow_two_diff with new lemma square_diff_square_factored
huffman
parents:
44346
diff
changeset
|
365 |
|
25267 | 366 |
end |
14738 | 367 |
|
22390 | 368 |
class ring_1 = ring + zero_neq_one + monoid_mult |
25267 | 369 |
begin |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
370 |
|
27516 | 371 |
subclass semiring_1_cancel .. |
25267 | 372 |
|
44346
00dd3c4dabe0
rename real_squared_diff_one_factored to square_diff_one_factored and move to Rings.thy
huffman
parents:
44064
diff
changeset
|
373 |
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
|
374 |
"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
|
375 |
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
|
376 |
|
25267 | 377 |
end |
25152 | 378 |
|
22390 | 379 |
class comm_ring_1 = comm_ring + zero_neq_one + comm_monoid_mult |
25267 | 380 |
begin |
14738 | 381 |
|
27516 | 382 |
subclass ring_1 .. |
383 |
subclass comm_semiring_1_cancel .. |
|
25267 | 384 |
|
59816
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
385 |
subclass comm_semiring_1_diff_distrib |
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
386 |
by unfold_locales (simp add: algebra_simps) |
58647 | 387 |
|
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
|
388 |
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
|
389 |
proof |
6d10cf26b5dc
add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents:
29407
diff
changeset
|
390 |
assume "x dvd - y" |
6d10cf26b5dc
add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents:
29407
diff
changeset
|
391 |
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
|
392 |
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
|
393 |
next |
6d10cf26b5dc
add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents:
29407
diff
changeset
|
394 |
assume "x dvd y" |
6d10cf26b5dc
add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents:
29407
diff
changeset
|
395 |
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
|
396 |
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
|
397 |
qed |
6d10cf26b5dc
add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents:
29407
diff
changeset
|
398 |
|
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
|
399 |
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
|
400 |
proof |
6d10cf26b5dc
add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents:
29407
diff
changeset
|
401 |
assume "- x dvd y" |
6d10cf26b5dc
add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents:
29407
diff
changeset
|
402 |
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
|
403 |
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
|
404 |
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
|
405 |
next |
6d10cf26b5dc
add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents:
29407
diff
changeset
|
406 |
assume "x dvd y" |
6d10cf26b5dc
add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents:
29407
diff
changeset
|
407 |
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
|
408 |
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
|
409 |
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
|
410 |
qed |
6d10cf26b5dc
add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents:
29407
diff
changeset
|
411 |
|
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54225
diff
changeset
|
412 |
lemma dvd_diff [simp]: |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54225
diff
changeset
|
413 |
"x dvd y \<Longrightarrow> x dvd z \<Longrightarrow> x dvd (y - z)" |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54225
diff
changeset
|
414 |
using dvd_add [of x y "- z"] by simp |
29409 | 415 |
|
25267 | 416 |
end |
25152 | 417 |
|
59833
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset
|
418 |
class semiring_no_zero_divisors = semiring_0 + |
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset
|
419 |
assumes no_zero_divisors: "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> a * b \<noteq> 0" |
25230 | 420 |
begin |
421 |
||
59833
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset
|
422 |
lemma divisors_zero: |
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset
|
423 |
assumes "a * b = 0" |
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset
|
424 |
shows "a = 0 \<or> b = 0" |
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset
|
425 |
proof (rule classical) |
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset
|
426 |
assume "\<not> (a = 0 \<or> b = 0)" |
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset
|
427 |
then have "a \<noteq> 0" and "b \<noteq> 0" by auto |
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset
|
428 |
with no_zero_divisors have "a * b \<noteq> 0" by blast |
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset
|
429 |
with assms show ?thesis by simp |
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset
|
430 |
qed |
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset
|
431 |
|
25230 | 432 |
lemma mult_eq_0_iff [simp]: |
58952
5d82cdef6c1b
equivalence rules for structures without zero divisors
haftmann
parents:
58889
diff
changeset
|
433 |
shows "a * b = 0 \<longleftrightarrow> a = 0 \<or> b = 0" |
25230 | 434 |
proof (cases "a = 0 \<or> b = 0") |
435 |
case False then have "a \<noteq> 0" and "b \<noteq> 0" by auto |
|
436 |
then show ?thesis using no_zero_divisors by simp |
|
437 |
next |
|
438 |
case True then show ?thesis by auto |
|
439 |
qed |
|
440 |
||
58952
5d82cdef6c1b
equivalence rules for structures without zero divisors
haftmann
parents:
58889
diff
changeset
|
441 |
end |
5d82cdef6c1b
equivalence rules for structures without zero divisors
haftmann
parents:
58889
diff
changeset
|
442 |
|
5d82cdef6c1b
equivalence rules for structures without zero divisors
haftmann
parents:
58889
diff
changeset
|
443 |
class ring_no_zero_divisors = ring + semiring_no_zero_divisors |
5d82cdef6c1b
equivalence rules for structures without zero divisors
haftmann
parents:
58889
diff
changeset
|
444 |
begin |
5d82cdef6c1b
equivalence rules for structures without zero divisors
haftmann
parents:
58889
diff
changeset
|
445 |
|
26193 | 446 |
text{*Cancellation of equalities with a common factor*} |
54147
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
52435
diff
changeset
|
447 |
lemma mult_cancel_right [simp]: |
26193 | 448 |
"a * c = b * c \<longleftrightarrow> c = 0 \<or> a = b" |
449 |
proof - |
|
450 |
have "(a * c = b * c) = ((a - b) * c = 0)" |
|
35216 | 451 |
by (simp add: algebra_simps) |
452 |
thus ?thesis by (simp add: disj_commute) |
|
26193 | 453 |
qed |
454 |
||
54147
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
52435
diff
changeset
|
455 |
lemma mult_cancel_left [simp]: |
26193 | 456 |
"c * a = c * b \<longleftrightarrow> c = 0 \<or> a = b" |
457 |
proof - |
|
458 |
have "(c * a = c * b) = (c * (a - b) = 0)" |
|
35216 | 459 |
by (simp add: algebra_simps) |
460 |
thus ?thesis by simp |
|
26193 | 461 |
qed |
462 |
||
58952
5d82cdef6c1b
equivalence rules for structures without zero divisors
haftmann
parents:
58889
diff
changeset
|
463 |
lemma mult_left_cancel: |
5d82cdef6c1b
equivalence rules for structures without zero divisors
haftmann
parents:
58889
diff
changeset
|
464 |
"c \<noteq> 0 \<Longrightarrow> c * a = c * b \<longleftrightarrow> a = b" |
5d82cdef6c1b
equivalence rules for structures without zero divisors
haftmann
parents:
58889
diff
changeset
|
465 |
by simp |
56217
dc429a5b13c4
Some rationalisation of basic lemmas
paulson <lp15@cam.ac.uk>
parents:
55912
diff
changeset
|
466 |
|
58952
5d82cdef6c1b
equivalence rules for structures without zero divisors
haftmann
parents:
58889
diff
changeset
|
467 |
lemma mult_right_cancel: |
5d82cdef6c1b
equivalence rules for structures without zero divisors
haftmann
parents:
58889
diff
changeset
|
468 |
"c \<noteq> 0 \<Longrightarrow> a * c = b * c \<longleftrightarrow> a = b" |
5d82cdef6c1b
equivalence rules for structures without zero divisors
haftmann
parents:
58889
diff
changeset
|
469 |
by simp |
56217
dc429a5b13c4
Some rationalisation of basic lemmas
paulson <lp15@cam.ac.uk>
parents:
55912
diff
changeset
|
470 |
|
25230 | 471 |
end |
22990
775e9de3db48
added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents:
22987
diff
changeset
|
472 |
|
23544 | 473 |
class ring_1_no_zero_divisors = ring_1 + ring_no_zero_divisors |
26274 | 474 |
begin |
475 |
||
36970 | 476 |
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
|
477 |
"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
|
478 |
proof - |
9207505d1ee5
move lemma real_mult_is_one to Rings.thy, renamed to square_eq_1_iff
huffman
parents:
36719
diff
changeset
|
479 |
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
|
480 |
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
|
481 |
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
|
482 |
by simp |
9207505d1ee5
move lemma real_mult_is_one to Rings.thy, renamed to square_eq_1_iff
huffman
parents:
36719
diff
changeset
|
483 |
thus ?thesis |
9207505d1ee5
move lemma real_mult_is_one to Rings.thy, renamed to square_eq_1_iff
huffman
parents:
36719
diff
changeset
|
484 |
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
|
485 |
qed |
9207505d1ee5
move lemma real_mult_is_one to Rings.thy, renamed to square_eq_1_iff
huffman
parents:
36719
diff
changeset
|
486 |
|
26274 | 487 |
lemma mult_cancel_right1 [simp]: |
488 |
"c = b * c \<longleftrightarrow> c = 0 \<or> b = 1" |
|
29667 | 489 |
by (insert mult_cancel_right [of 1 c b], force) |
26274 | 490 |
|
491 |
lemma mult_cancel_right2 [simp]: |
|
492 |
"a * c = c \<longleftrightarrow> c = 0 \<or> a = 1" |
|
29667 | 493 |
by (insert mult_cancel_right [of a c 1], simp) |
26274 | 494 |
|
495 |
lemma mult_cancel_left1 [simp]: |
|
496 |
"c = c * b \<longleftrightarrow> c = 0 \<or> b = 1" |
|
29667 | 497 |
by (insert mult_cancel_left [of c 1 b], force) |
26274 | 498 |
|
499 |
lemma mult_cancel_left2 [simp]: |
|
500 |
"c * a = c \<longleftrightarrow> c = 0 \<or> a = 1" |
|
29667 | 501 |
by (insert mult_cancel_left [of c a 1], simp) |
26274 | 502 |
|
503 |
end |
|
22990
775e9de3db48
added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents:
22987
diff
changeset
|
504 |
|
59833
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset
|
505 |
class semidom = comm_semiring_1_cancel + semiring_no_zero_divisors |
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset
|
506 |
|
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset
|
507 |
class idom = comm_ring_1 + semiring_no_zero_divisors |
25186 | 508 |
begin |
14421
ee97b6463cb4
new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents:
14398
diff
changeset
|
509 |
|
59833
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset
|
510 |
subclass semidom .. |
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset
|
511 |
|
27516 | 512 |
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
|
513 |
|
29981
7d0ed261b712
generalize int_dvd_cancel_factor simproc to idom class
huffman
parents:
29949
diff
changeset
|
514 |
lemma dvd_mult_cancel_right [simp]: |
7d0ed261b712
generalize int_dvd_cancel_factor simproc to idom class
huffman
parents:
29949
diff
changeset
|
515 |
"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
|
516 |
proof - |
7d0ed261b712
generalize int_dvd_cancel_factor simproc to idom class
huffman
parents:
29949
diff
changeset
|
517 |
have "a * c dvd b * c \<longleftrightarrow> (\<exists>k. b * c = (a * k) * c)" |
57514
bdc2c6b40bf2
prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents:
57512
diff
changeset
|
518 |
unfolding dvd_def by (simp add: ac_simps) |
29981
7d0ed261b712
generalize int_dvd_cancel_factor simproc to idom class
huffman
parents:
29949
diff
changeset
|
519 |
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
|
520 |
unfolding dvd_def by simp |
7d0ed261b712
generalize int_dvd_cancel_factor simproc to idom class
huffman
parents:
29949
diff
changeset
|
521 |
finally show ?thesis . |
7d0ed261b712
generalize int_dvd_cancel_factor simproc to idom class
huffman
parents:
29949
diff
changeset
|
522 |
qed |
7d0ed261b712
generalize int_dvd_cancel_factor simproc to idom class
huffman
parents:
29949
diff
changeset
|
523 |
|
7d0ed261b712
generalize int_dvd_cancel_factor simproc to idom class
huffman
parents:
29949
diff
changeset
|
524 |
lemma dvd_mult_cancel_left [simp]: |
7d0ed261b712
generalize int_dvd_cancel_factor simproc to idom class
huffman
parents:
29949
diff
changeset
|
525 |
"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
|
526 |
proof - |
7d0ed261b712
generalize int_dvd_cancel_factor simproc to idom class
huffman
parents:
29949
diff
changeset
|
527 |
have "c * a dvd c * b \<longleftrightarrow> (\<exists>k. b * c = (a * k) * c)" |
57514
bdc2c6b40bf2
prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents:
57512
diff
changeset
|
528 |
unfolding dvd_def by (simp add: ac_simps) |
29981
7d0ed261b712
generalize int_dvd_cancel_factor simproc to idom class
huffman
parents:
29949
diff
changeset
|
529 |
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
|
530 |
unfolding dvd_def by simp |
7d0ed261b712
generalize int_dvd_cancel_factor simproc to idom class
huffman
parents:
29949
diff
changeset
|
531 |
finally show ?thesis . |
7d0ed261b712
generalize int_dvd_cancel_factor simproc to idom class
huffman
parents:
29949
diff
changeset
|
532 |
qed |
7d0ed261b712
generalize int_dvd_cancel_factor simproc to idom class
huffman
parents:
29949
diff
changeset
|
533 |
|
59833
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset
|
534 |
lemma square_eq_iff: "a * a = b * b \<longleftrightarrow> (a = b \<or> a = - b)" |
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset
|
535 |
proof |
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset
|
536 |
assume "a * a = b * b" |
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset
|
537 |
then have "(a - b) * (a + b) = 0" |
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset
|
538 |
by (simp add: algebra_simps) |
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset
|
539 |
then show "a = b \<or> a = - b" |
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset
|
540 |
by (simp add: eq_neg_iff_add_eq_0) |
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset
|
541 |
next |
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset
|
542 |
assume "a = b \<or> a = - b" |
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset
|
543 |
then show "a * a = b * b" by auto |
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset
|
544 |
qed |
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset
|
545 |
|
25186 | 546 |
end |
25152 | 547 |
|
35302 | 548 |
text {* |
549 |
The theory of partially ordered rings is taken from the books: |
|
550 |
\begin{itemize} |
|
551 |
\item \emph{Lattice Theory} by Garret Birkhoff, American Mathematical Society 1979 |
|
552 |
\item \emph{Partially Ordered Algebraic Systems}, Pergamon Press 1963 |
|
553 |
\end{itemize} |
|
554 |
Most of the used notions can also be looked up in |
|
555 |
\begin{itemize} |
|
54703 | 556 |
\item @{url "http://www.mathworld.com"} by Eric Weisstein et. al. |
35302 | 557 |
\item \emph{Algebra I} by van der Waerden, Springer. |
558 |
\end{itemize} |
|
559 |
*} |
|
560 |
||
38642
8fa437809c67
dropped type classes mult_mono and mult_mono1; tuned names of technical rule duplicates
haftmann
parents:
37767
diff
changeset
|
561 |
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
|
562 |
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
|
563 |
assumes mult_right_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * c" |
25230 | 564 |
begin |
565 |
||
566 |
lemma mult_mono: |
|
38642
8fa437809c67
dropped type classes mult_mono and mult_mono1; tuned names of technical rule duplicates
haftmann
parents:
37767
diff
changeset
|
567 |
"a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * d" |
25230 | 568 |
apply (erule mult_right_mono [THEN order_trans], assumption) |
569 |
apply (erule mult_left_mono, assumption) |
|
570 |
done |
|
571 |
||
572 |
lemma mult_mono': |
|
38642
8fa437809c67
dropped type classes mult_mono and mult_mono1; tuned names of technical rule duplicates
haftmann
parents:
37767
diff
changeset
|
573 |
"a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * d" |
25230 | 574 |
apply (rule mult_mono) |
575 |
apply (fast intro: order_trans)+ |
|
576 |
done |
|
577 |
||
578 |
end |
|
21199
2d83f93c3580
* Added annihilation axioms ("x * 0 = 0") to axclass semiring_0.
krauss
parents:
20633
diff
changeset
|
579 |
|
38642
8fa437809c67
dropped type classes mult_mono and mult_mono1; tuned names of technical rule duplicates
haftmann
parents:
37767
diff
changeset
|
580 |
class ordered_cancel_semiring = ordered_semiring + cancel_comm_monoid_add |
25267 | 581 |
begin |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
582 |
|
27516 | 583 |
subclass semiring_0_cancel .. |
23521 | 584 |
|
56536 | 585 |
lemma mult_nonneg_nonneg[simp]: "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
|
586 |
using mult_left_mono [of 0 b a] by simp |
25230 | 587 |
|
588 |
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
|
589 |
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
|
590 |
|
44ea10bc07a7
clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents:
30650
diff
changeset
|
591 |
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
|
592 |
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
|
593 |
|
44ea10bc07a7
clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents:
30650
diff
changeset
|
594 |
text {* Legacy - use @{text mult_nonpos_nonneg} *} |
25230 | 595 |
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
|
596 |
by (drule mult_right_mono [of b 0], auto) |
25230 | 597 |
|
26234 | 598 |
lemma split_mult_neg_le: "(0 \<le> a & b \<le> 0) | (a \<le> 0 & 0 \<le> b) \<Longrightarrow> a * b \<le> 0" |
29667 | 599 |
by (auto simp add: mult_nonneg_nonpos mult_nonneg_nonpos2) |
25230 | 600 |
|
601 |
end |
|
602 |
||
38642
8fa437809c67
dropped type classes mult_mono and mult_mono1; tuned names of technical rule duplicates
haftmann
parents:
37767
diff
changeset
|
603 |
class linordered_semiring = ordered_semiring + linordered_cancel_ab_semigroup_add |
25267 | 604 |
begin |
25230 | 605 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
606 |
subclass ordered_cancel_semiring .. |
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
607 |
|
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
608 |
subclass ordered_comm_monoid_add .. |
25304
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset
|
609 |
|
25230 | 610 |
lemma mult_left_less_imp_less: |
611 |
"c * a < c * b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a < b" |
|
29667 | 612 |
by (force simp add: mult_left_mono not_le [symmetric]) |
25230 | 613 |
|
614 |
lemma mult_right_less_imp_less: |
|
615 |
"a * c < b * c \<Longrightarrow> 0 \<le> c \<Longrightarrow> a < b" |
|
29667 | 616 |
by (force simp add: mult_right_mono not_le [symmetric]) |
23521 | 617 |
|
25186 | 618 |
end |
25152 | 619 |
|
35043
07dbdf60d5ad
dropped accidental duplication of "lin" prefix from cs. 108662d50512
haftmann
parents:
35032
diff
changeset
|
620 |
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
|
621 |
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
|
622 |
|
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
|
623 |
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
|
624 |
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
|
625 |
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
|
626 |
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
|
627 |
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
|
628 |
by (simp add: add_mono mult_left_mono) |
49962
a8cc904a6820
Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents:
44921
diff
changeset
|
629 |
thus ?thesis using assms unfolding distrib_right[symmetric] by simp |
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
|
630 |
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
|
631 |
|
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 |
end |
35043
07dbdf60d5ad
dropped accidental duplication of "lin" prefix from cs. 108662d50512
haftmann
parents:
35032
diff
changeset
|
633 |
|
07dbdf60d5ad
dropped accidental duplication of "lin" prefix from cs. 108662d50512
haftmann
parents:
35032
diff
changeset
|
634 |
class linordered_semiring_strict = semiring + comm_monoid_add + linordered_cancel_ab_semigroup_add + |
25062 | 635 |
assumes mult_strict_left_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b" |
636 |
assumes mult_strict_right_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> a * c < b * c" |
|
25267 | 637 |
begin |
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset
|
638 |
|
27516 | 639 |
subclass semiring_0_cancel .. |
14940 | 640 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
641 |
subclass linordered_semiring |
28823 | 642 |
proof |
23550 | 643 |
fix a b c :: 'a |
644 |
assume A: "a \<le> b" "0 \<le> c" |
|
645 |
from A show "c * a \<le> c * b" |
|
25186 | 646 |
unfolding le_less |
647 |
using mult_strict_left_mono by (cases "c = 0") auto |
|
23550 | 648 |
from A show "a * c \<le> b * c" |
25152 | 649 |
unfolding le_less |
25186 | 650 |
using mult_strict_right_mono by (cases "c = 0") auto |
25152 | 651 |
qed |
652 |
||
25230 | 653 |
lemma mult_left_le_imp_le: |
654 |
"c * a \<le> c * b \<Longrightarrow> 0 < c \<Longrightarrow> a \<le> b" |
|
29667 | 655 |
by (force simp add: mult_strict_left_mono _not_less [symmetric]) |
25230 | 656 |
|
657 |
lemma mult_right_le_imp_le: |
|
658 |
"a * c \<le> b * c \<Longrightarrow> 0 < c \<Longrightarrow> a \<le> b" |
|
29667 | 659 |
by (force simp add: mult_strict_right_mono not_less [symmetric]) |
25230 | 660 |
|
56544 | 661 |
lemma mult_pos_pos[simp]: "0 < a \<Longrightarrow> 0 < b \<Longrightarrow> 0 < a * b" |
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
662 |
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
|
663 |
|
44ea10bc07a7
clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents:
30650
diff
changeset
|
664 |
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
|
665 |
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
|
666 |
|
44ea10bc07a7
clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents:
30650
diff
changeset
|
667 |
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
|
668 |
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
|
669 |
|
44ea10bc07a7
clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents:
30650
diff
changeset
|
670 |
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
|
671 |
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
|
672 |
by (drule mult_strict_right_mono [of b 0], auto) |
25230 | 673 |
|
674 |
lemma zero_less_mult_pos: |
|
675 |
"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
|
676 |
apply (cases "b\<le>0") |
25230 | 677 |
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
|
678 |
apply (drule_tac mult_pos_neg [of a b]) |
25230 | 679 |
apply (auto dest: less_not_sym) |
680 |
done |
|
681 |
||
682 |
lemma zero_less_mult_pos2: |
|
683 |
"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
|
684 |
apply (cases "b\<le>0") |
25230 | 685 |
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
|
686 |
apply (drule_tac mult_pos_neg2 [of a b]) |
25230 | 687 |
apply (auto dest: less_not_sym) |
688 |
done |
|
689 |
||
26193 | 690 |
text{*Strict monotonicity in both arguments*} |
691 |
lemma mult_strict_mono: |
|
692 |
assumes "a < b" and "c < d" and "0 < b" and "0 \<le> c" |
|
693 |
shows "a * c < b * d" |
|
694 |
using assms apply (cases "c=0") |
|
56544 | 695 |
apply (simp) |
26193 | 696 |
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
|
697 |
apply (force simp add: le_less) |
26193 | 698 |
apply (erule mult_strict_left_mono, assumption) |
699 |
done |
|
700 |
||
701 |
text{*This weaker variant has more natural premises*} |
|
702 |
lemma mult_strict_mono': |
|
703 |
assumes "a < b" and "c < d" and "0 \<le> a" and "0 \<le> c" |
|
704 |
shows "a * c < b * d" |
|
29667 | 705 |
by (rule mult_strict_mono) (insert assms, auto) |
26193 | 706 |
|
707 |
lemma mult_less_le_imp_less: |
|
708 |
assumes "a < b" and "c \<le> d" and "0 \<le> a" and "0 < c" |
|
709 |
shows "a * c < b * d" |
|
710 |
using assms apply (subgoal_tac "a * c < b * c") |
|
711 |
apply (erule less_le_trans) |
|
712 |
apply (erule mult_left_mono) |
|
713 |
apply simp |
|
714 |
apply (erule mult_strict_right_mono) |
|
715 |
apply assumption |
|
716 |
done |
|
717 |
||
718 |
lemma mult_le_less_imp_less: |
|
719 |
assumes "a \<le> b" and "c < d" and "0 < a" and "0 \<le> c" |
|
720 |
shows "a * c < b * d" |
|
721 |
using assms apply (subgoal_tac "a * c \<le> b * c") |
|
722 |
apply (erule le_less_trans) |
|
723 |
apply (erule mult_strict_left_mono) |
|
724 |
apply simp |
|
725 |
apply (erule mult_right_mono) |
|
726 |
apply simp |
|
727 |
done |
|
728 |
||
25230 | 729 |
end |
730 |
||
35097
4554bb2abfa3
dropped last occurence of the linlinordered accident
haftmann
parents:
35092
diff
changeset
|
731 |
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
|
732 |
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
|
733 |
|
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
|
734 |
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
|
735 |
|
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
|
736 |
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
|
737 |
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
|
738 |
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
|
739 |
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
|
740 |
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
|
741 |
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
|
742 |
(auto intro!: add_less_le_mono mult_strict_left_mono mult_left_mono) |
49962
a8cc904a6820
Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents:
44921
diff
changeset
|
743 |
thus ?thesis using assms unfolding distrib_right[symmetric] by simp |
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
|
744 |
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
|
745 |
|
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
|
746 |
end |
33319 | 747 |
|
38642
8fa437809c67
dropped type classes mult_mono and mult_mono1; tuned names of technical rule duplicates
haftmann
parents:
37767
diff
changeset
|
748 |
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
|
749 |
assumes comm_mult_left_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> c * a \<le> c * b" |
25186 | 750 |
begin |
25152 | 751 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
752 |
subclass ordered_semiring |
28823 | 753 |
proof |
21199
2d83f93c3580
* Added annihilation axioms ("x * 0 = 0") to axclass semiring_0.
krauss
parents:
20633
diff
changeset
|
754 |
fix a b c :: 'a |
23550 | 755 |
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
|
756 |
thus "c * a \<le> c * b" by (rule comm_mult_left_mono) |
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
56544
diff
changeset
|
757 |
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
|
758 |
qed |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
759 |
|
25267 | 760 |
end |
761 |
||
38642
8fa437809c67
dropped type classes mult_mono and mult_mono1; tuned names of technical rule duplicates
haftmann
parents:
37767
diff
changeset
|
762 |
class ordered_cancel_comm_semiring = ordered_comm_semiring + cancel_comm_monoid_add |
25267 | 763 |
begin |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
764 |
|
38642
8fa437809c67
dropped type classes mult_mono and mult_mono1; tuned names of technical rule duplicates
haftmann
parents:
37767
diff
changeset
|
765 |
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
|
766 |
subclass ordered_comm_semiring .. |
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
767 |
subclass ordered_cancel_semiring .. |
25267 | 768 |
|
769 |
end |
|
770 |
||
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
771 |
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
|
772 |
assumes comm_mult_strict_left_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b" |
25267 | 773 |
begin |
774 |
||
35043
07dbdf60d5ad
dropped accidental duplication of "lin" prefix from cs. 108662d50512
haftmann
parents:
35032
diff
changeset
|
775 |
subclass linordered_semiring_strict |
28823 | 776 |
proof |
23550 | 777 |
fix a b c :: 'a |
778 |
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
|
779 |
thus "c * a < c * b" by (rule comm_mult_strict_left_mono) |
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
56544
diff
changeset
|
780 |
thus "a * c < b * c" by (simp only: mult.commute) |
23550 | 781 |
qed |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
782 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
783 |
subclass ordered_cancel_comm_semiring |
28823 | 784 |
proof |
23550 | 785 |
fix a b c :: 'a |
786 |
assume "a \<le> b" "0 \<le> c" |
|
787 |
thus "c * a \<le> c * b" |
|
25186 | 788 |
unfolding le_less |
26193 | 789 |
using mult_strict_left_mono by (cases "c = 0") auto |
23550 | 790 |
qed |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
791 |
|
25267 | 792 |
end |
25230 | 793 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
794 |
class ordered_ring = ring + ordered_cancel_semiring |
25267 | 795 |
begin |
25230 | 796 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
797 |
subclass ordered_ab_group_add .. |
14270 | 798 |
|
25230 | 799 |
lemma less_add_iff1: |
800 |
"a * e + c < b * e + d \<longleftrightarrow> (a - b) * e + c < d" |
|
29667 | 801 |
by (simp add: algebra_simps) |
25230 | 802 |
|
803 |
lemma less_add_iff2: |
|
804 |
"a * e + c < b * e + d \<longleftrightarrow> c < (b - a) * e + d" |
|
29667 | 805 |
by (simp add: algebra_simps) |
25230 | 806 |
|
807 |
lemma le_add_iff1: |
|
808 |
"a * e + c \<le> b * e + d \<longleftrightarrow> (a - b) * e + c \<le> d" |
|
29667 | 809 |
by (simp add: algebra_simps) |
25230 | 810 |
|
811 |
lemma le_add_iff2: |
|
812 |
"a * e + c \<le> b * e + d \<longleftrightarrow> c \<le> (b - a) * e + d" |
|
29667 | 813 |
by (simp add: algebra_simps) |
25230 | 814 |
|
815 |
lemma mult_left_mono_neg: |
|
816 |
"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
|
817 |
apply (drule mult_left_mono [of _ _ "- c"]) |
35216 | 818 |
apply simp_all |
25230 | 819 |
done |
820 |
||
821 |
lemma mult_right_mono_neg: |
|
822 |
"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
|
823 |
apply (drule mult_right_mono [of _ _ "- c"]) |
35216 | 824 |
apply simp_all |
25230 | 825 |
done |
826 |
||
30692
44ea10bc07a7
clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents:
30650
diff
changeset
|
827 |
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
|
828 |
using mult_right_mono_neg [of a 0 b] by simp |
25230 | 829 |
|
830 |
lemma split_mult_pos_le: |
|
831 |
"(0 \<le> a \<and> 0 \<le> b) \<or> (a \<le> 0 \<and> b \<le> 0) \<Longrightarrow> 0 \<le> a * b" |
|
56536 | 832 |
by (auto simp add: mult_nonpos_nonpos) |
25186 | 833 |
|
834 |
end |
|
14270 | 835 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
836 |
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
|
837 |
begin |
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset
|
838 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
839 |
subclass ordered_ring .. |
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
840 |
|
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
841 |
subclass ordered_ab_group_add_abs |
28823 | 842 |
proof |
25304
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset
|
843 |
fix a b |
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset
|
844 |
show "\<bar>a + b\<bar> \<le> \<bar>a\<bar> + \<bar>b\<bar>" |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54225
diff
changeset
|
845 |
by (auto simp add: abs_if not_le not_less algebra_simps simp del: add.commute dest: add_neg_neg add_nonneg_nonneg) |
35216 | 846 |
qed (auto simp add: abs_if) |
25304
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset
|
847 |
|
35631
0b8a5fd339ab
generalize some lemmas from class linordered_ring_strict to linordered_ring
huffman
parents:
35302
diff
changeset
|
848 |
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
|
849 |
using linear [of 0 a] |
56536 | 850 |
by (auto simp add: mult_nonpos_nonpos) |
35631
0b8a5fd339ab
generalize some lemmas from class linordered_ring_strict to linordered_ring
huffman
parents:
35302
diff
changeset
|
851 |
|
0b8a5fd339ab
generalize some lemmas from class linordered_ring_strict to linordered_ring
huffman
parents:
35302
diff
changeset
|
852 |
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
|
853 |
by (simp add: not_less) |
0b8a5fd339ab
generalize some lemmas from class linordered_ring_strict to linordered_ring
huffman
parents:
35302
diff
changeset
|
854 |
|
25304
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset
|
855 |
end |
23521 | 856 |
|
35043
07dbdf60d5ad
dropped accidental duplication of "lin" prefix from cs. 108662d50512
haftmann
parents:
35032
diff
changeset
|
857 |
class linordered_ring_strict = ring + linordered_semiring_strict |
25304
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset
|
858 |
+ ordered_ab_group_add + abs_if |
25230 | 859 |
begin |
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset
|
860 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
861 |
subclass linordered_ring .. |
25304
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset
|
862 |
|
30692
44ea10bc07a7
clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents:
30650
diff
changeset
|
863 |
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
|
864 |
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
|
865 |
|
44ea10bc07a7
clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents:
30650
diff
changeset
|
866 |
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
|
867 |
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
|
868 |
|
44ea10bc07a7
clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents:
30650
diff
changeset
|
869 |
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
|
870 |
using mult_strict_right_mono_neg [of a 0 b] by simp |
14738 | 871 |
|
25917 | 872 |
subclass ring_no_zero_divisors |
28823 | 873 |
proof |
25917 | 874 |
fix a b |
875 |
assume "a \<noteq> 0" then have A: "a < 0 \<or> 0 < a" by (simp add: neq_iff) |
|
876 |
assume "b \<noteq> 0" then have B: "b < 0 \<or> 0 < b" by (simp add: neq_iff) |
|
877 |
have "a * b < 0 \<or> 0 < a * b" |
|
878 |
proof (cases "a < 0") |
|
879 |
case True note A' = this |
|
880 |
show ?thesis proof (cases "b < 0") |
|
881 |
case True with A' |
|
882 |
show ?thesis by (auto dest: mult_neg_neg) |
|
883 |
next |
|
884 |
case False with B have "0 < b" by auto |
|
885 |
with A' show ?thesis by (auto dest: mult_strict_right_mono) |
|
886 |
qed |
|
887 |
next |
|
888 |
case False with A have A': "0 < a" by auto |
|
889 |
show ?thesis proof (cases "b < 0") |
|
890 |
case True with A' |
|
891 |
show ?thesis by (auto dest: mult_strict_right_mono_neg) |
|
892 |
next |
|
893 |
case False with B have "0 < b" by auto |
|
56544 | 894 |
with A' show ?thesis by auto |
25917 | 895 |
qed |
896 |
qed |
|
897 |
then show "a * b \<noteq> 0" by (simp add: neq_iff) |
|
898 |
qed |
|
25304
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset
|
899 |
|
56480
093ea91498e6
field_simps: better support for negation and division, and power
hoelzl
parents:
56217
diff
changeset
|
900 |
lemma zero_less_mult_iff: "0 < a * b \<longleftrightarrow> 0 < a \<and> 0 < b \<or> a < 0 \<and> b < 0" |
093ea91498e6
field_simps: better support for negation and division, and power
hoelzl
parents:
56217
diff
changeset
|
901 |
by (cases a 0 b 0 rule: linorder_cases[case_product linorder_cases]) |
56544 | 902 |
(auto simp add: mult_neg_neg not_less le_less dest: zero_less_mult_pos zero_less_mult_pos2) |
22990
775e9de3db48
added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents:
22987
diff
changeset
|
903 |
|
56480
093ea91498e6
field_simps: better support for negation and division, and power
hoelzl
parents:
56217
diff
changeset
|
904 |
lemma zero_le_mult_iff: "0 \<le> a * b \<longleftrightarrow> 0 \<le> a \<and> 0 \<le> b \<or> a \<le> 0 \<and> b \<le> 0" |
093ea91498e6
field_simps: better support for negation and division, and power
hoelzl
parents:
56217
diff
changeset
|
905 |
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
|
906 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
907 |
lemma mult_less_0_iff: |
25917 | 908 |
"a * b < 0 \<longleftrightarrow> 0 < a \<and> b < 0 \<or> a < 0 \<and> 0 < b" |
35216 | 909 |
apply (insert zero_less_mult_iff [of "-a" b]) |
910 |
apply force |
|
25917 | 911 |
done |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
912 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
913 |
lemma mult_le_0_iff: |
25917 | 914 |
"a * b \<le> 0 \<longleftrightarrow> 0 \<le> a \<and> b \<le> 0 \<or> a \<le> 0 \<and> 0 \<le> b" |
915 |
apply (insert zero_le_mult_iff [of "-a" b]) |
|
35216 | 916 |
apply force |
25917 | 917 |
done |
918 |
||
26193 | 919 |
text{*Cancellation laws for @{term "c*a < c*b"} and @{term "a*c < b*c"}, |
920 |
also with the relations @{text "\<le>"} and equality.*} |
|
921 |
||
922 |
text{*These ``disjunction'' versions produce two cases when the comparison is |
|
923 |
an assumption, but effectively four when the comparison is a goal.*} |
|
924 |
||
925 |
lemma mult_less_cancel_right_disj: |
|
926 |
"a * c < b * c \<longleftrightarrow> 0 < c \<and> a < b \<or> c < 0 \<and> b < a" |
|
927 |
apply (cases "c = 0") |
|
928 |
apply (auto simp add: neq_iff mult_strict_right_mono |
|
929 |
mult_strict_right_mono_neg) |
|
930 |
apply (auto simp add: not_less |
|
931 |
not_le [symmetric, of "a*c"] |
|
932 |
not_le [symmetric, of a]) |
|
933 |
apply (erule_tac [!] notE) |
|
934 |
apply (auto simp add: less_imp_le mult_right_mono |
|
935 |
mult_right_mono_neg) |
|
936 |
done |
|
937 |
||
938 |
lemma mult_less_cancel_left_disj: |
|
939 |
"c * a < c * b \<longleftrightarrow> 0 < c \<and> a < b \<or> c < 0 \<and> b < a" |
|
940 |
apply (cases "c = 0") |
|
941 |
apply (auto simp add: neq_iff mult_strict_left_mono |
|
942 |
mult_strict_left_mono_neg) |
|
943 |
apply (auto simp add: not_less |
|
944 |
not_le [symmetric, of "c*a"] |
|
945 |
not_le [symmetric, of a]) |
|
946 |
apply (erule_tac [!] notE) |
|
947 |
apply (auto simp add: less_imp_le mult_left_mono |
|
948 |
mult_left_mono_neg) |
|
949 |
done |
|
950 |
||
951 |
text{*The ``conjunction of implication'' lemmas produce two cases when the |
|
952 |
comparison is a goal, but give four when the comparison is an assumption.*} |
|
953 |
||
954 |
lemma mult_less_cancel_right: |
|
955 |
"a * c < b * c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < b) \<and> (c \<le> 0 \<longrightarrow> b < a)" |
|
956 |
using mult_less_cancel_right_disj [of a c b] by auto |
|
957 |
||
958 |
lemma mult_less_cancel_left: |
|
959 |
"c * a < c * b \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < b) \<and> (c \<le> 0 \<longrightarrow> b < a)" |
|
960 |
using mult_less_cancel_left_disj [of c a b] by auto |
|
961 |
||
962 |
lemma mult_le_cancel_right: |
|
963 |
"a * c \<le> b * c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)" |
|
29667 | 964 |
by (simp add: not_less [symmetric] mult_less_cancel_right_disj) |
26193 | 965 |
|
966 |
lemma mult_le_cancel_left: |
|
967 |
"c * a \<le> c * b \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)" |
|
29667 | 968 |
by (simp add: not_less [symmetric] mult_less_cancel_left_disj) |
26193 | 969 |
|
30649
57753e0ec1d4
1. New cancellation simprocs for common factors in inequations
nipkow
parents:
30242
diff
changeset
|
970 |
lemma mult_le_cancel_left_pos: |
57753e0ec1d4
1. New cancellation simprocs for common factors in inequations
nipkow
parents:
30242
diff
changeset
|
971 |
"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
|
972 |
by (auto simp: mult_le_cancel_left) |
57753e0ec1d4
1. New cancellation simprocs for common factors in inequations
nipkow
parents:
30242
diff
changeset
|
973 |
|
57753e0ec1d4
1. New cancellation simprocs for common factors in inequations
nipkow
parents:
30242
diff
changeset
|
974 |
lemma mult_le_cancel_left_neg: |
57753e0ec1d4
1. New cancellation simprocs for common factors in inequations
nipkow
parents:
30242
diff
changeset
|
975 |
"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
|
976 |
by (auto simp: mult_le_cancel_left) |
57753e0ec1d4
1. New cancellation simprocs for common factors in inequations
nipkow
parents:
30242
diff
changeset
|
977 |
|
57753e0ec1d4
1. New cancellation simprocs for common factors in inequations
nipkow
parents:
30242
diff
changeset
|
978 |
lemma mult_less_cancel_left_pos: |
57753e0ec1d4
1. New cancellation simprocs for common factors in inequations
nipkow
parents:
30242
diff
changeset
|
979 |
"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
|
980 |
by (auto simp: mult_less_cancel_left) |
57753e0ec1d4
1. New cancellation simprocs for common factors in inequations
nipkow
parents:
30242
diff
changeset
|
981 |
|
57753e0ec1d4
1. New cancellation simprocs for common factors in inequations
nipkow
parents:
30242
diff
changeset
|
982 |
lemma mult_less_cancel_left_neg: |
57753e0ec1d4
1. New cancellation simprocs for common factors in inequations
nipkow
parents:
30242
diff
changeset
|
983 |
"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
|
984 |
by (auto simp: mult_less_cancel_left) |
57753e0ec1d4
1. New cancellation simprocs for common factors in inequations
nipkow
parents:
30242
diff
changeset
|
985 |
|
25917 | 986 |
end |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
987 |
|
30692
44ea10bc07a7
clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents:
30650
diff
changeset
|
988 |
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
|
989 |
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
|
990 |
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
|
991 |
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
|
992 |
mult_neg_pos mult_neg_neg |
25230 | 993 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
994 |
class ordered_comm_ring = comm_ring + ordered_comm_semiring |
25267 | 995 |
begin |
25230 | 996 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
997 |
subclass ordered_ring .. |
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
998 |
subclass ordered_cancel_comm_semiring .. |
25230 | 999 |
|
25267 | 1000 |
end |
25230 | 1001 |
|
59833
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset
|
1002 |
class linordered_semidom = semidom + linordered_comm_semiring_strict + |
25230 | 1003 |
assumes zero_less_one [simp]: "0 < 1" |
1004 |
begin |
|
1005 |
||
1006 |
lemma pos_add_strict: |
|
1007 |
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
|
1008 |
using add_strict_mono [of 0 a b c] by simp |
25230 | 1009 |
|
26193 | 1010 |
lemma zero_le_one [simp]: "0 \<le> 1" |
29667 | 1011 |
by (rule zero_less_one [THEN less_imp_le]) |
26193 | 1012 |
|
1013 |
lemma not_one_le_zero [simp]: "\<not> 1 \<le> 0" |
|
29667 | 1014 |
by (simp add: not_le) |
26193 | 1015 |
|
1016 |
lemma not_one_less_zero [simp]: "\<not> 1 < 0" |
|
29667 | 1017 |
by (simp add: not_less) |
26193 | 1018 |
|
1019 |
lemma less_1_mult: |
|
1020 |
assumes "1 < m" and "1 < n" |
|
1021 |
shows "1 < m * n" |
|
1022 |
using assms mult_strict_mono [of 1 m 1 n] |
|
1023 |
by (simp add: less_trans [OF zero_less_one]) |
|
1024 |
||
59000 | 1025 |
lemma mult_left_le: "c \<le> 1 \<Longrightarrow> 0 \<le> a \<Longrightarrow> a * c \<le> a" |
1026 |
using mult_left_mono[of c 1 a] by simp |
|
1027 |
||
1028 |
lemma mult_le_one: "a \<le> 1 \<Longrightarrow> 0 \<le> b \<Longrightarrow> b \<le> 1 \<Longrightarrow> a * b \<le> 1" |
|
1029 |
using mult_mono[of a 1 b 1] by simp |
|
1030 |
||
25230 | 1031 |
end |
1032 |
||
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
1033 |
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
|
1034 |
linordered_comm_semiring_strict + ordered_ab_group_add + |
25230 | 1035 |
abs_if + sgn_if |
25917 | 1036 |
begin |
1037 |
||
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
|
1038 |
subclass linordered_semiring_1_strict .. |
35043
07dbdf60d5ad
dropped accidental duplication of "lin" prefix from cs. 108662d50512
haftmann
parents:
35032
diff
changeset
|
1039 |
subclass linordered_ring_strict .. |
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
1040 |
subclass ordered_comm_ring .. |
27516 | 1041 |
subclass idom .. |
25917 | 1042 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
1043 |
subclass linordered_semidom |
28823 | 1044 |
proof |
26193 | 1045 |
have "0 \<le> 1 * 1" by (rule zero_le_square) |
1046 |
thus "0 < 1" by (simp add: le_less) |
|
25917 | 1047 |
qed |
1048 |
||
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
1049 |
lemma linorder_neqE_linordered_idom: |
26193 | 1050 |
assumes "x \<noteq> y" obtains "x < y" | "y < x" |
1051 |
using assms by (rule neqE) |
|
1052 |
||
26274 | 1053 |
text {* These cancellation simprules also produce two cases when the comparison is a goal. *} |
1054 |
||
1055 |
lemma mult_le_cancel_right1: |
|
1056 |
"c \<le> b * c \<longleftrightarrow> (0 < c \<longrightarrow> 1 \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> 1)" |
|
29667 | 1057 |
by (insert mult_le_cancel_right [of 1 c b], simp) |
26274 | 1058 |
|
1059 |
lemma mult_le_cancel_right2: |
|
1060 |
"a * c \<le> c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> 1) \<and> (c < 0 \<longrightarrow> 1 \<le> a)" |
|
29667 | 1061 |
by (insert mult_le_cancel_right [of a c 1], simp) |
26274 | 1062 |
|
1063 |
lemma mult_le_cancel_left1: |
|
1064 |
"c \<le> c * b \<longleftrightarrow> (0 < c \<longrightarrow> 1 \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> 1)" |
|
29667 | 1065 |
by (insert mult_le_cancel_left [of c 1 b], simp) |
26274 | 1066 |
|
1067 |
lemma mult_le_cancel_left2: |
|
1068 |
"c * a \<le> c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> 1) \<and> (c < 0 \<longrightarrow> 1 \<le> a)" |
|
29667 | 1069 |
by (insert mult_le_cancel_left [of c a 1], simp) |
26274 | 1070 |
|
1071 |
lemma mult_less_cancel_right1: |
|
1072 |
"c < b * c \<longleftrightarrow> (0 \<le> c \<longrightarrow> 1 < b) \<and> (c \<le> 0 \<longrightarrow> b < 1)" |
|
29667 | 1073 |
by (insert mult_less_cancel_right [of 1 c b], simp) |
26274 | 1074 |
|
1075 |
lemma mult_less_cancel_right2: |
|
1076 |
"a * c < c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < 1) \<and> (c \<le> 0 \<longrightarrow> 1 < a)" |
|
29667 | 1077 |
by (insert mult_less_cancel_right [of a c 1], simp) |
26274 | 1078 |
|
1079 |
lemma mult_less_cancel_left1: |
|
1080 |
"c < c * b \<longleftrightarrow> (0 \<le> c \<longrightarrow> 1 < b) \<and> (c \<le> 0 \<longrightarrow> b < 1)" |
|
29667 | 1081 |
by (insert mult_less_cancel_left [of c 1 b], simp) |
26274 | 1082 |
|
1083 |
lemma mult_less_cancel_left2: |
|
1084 |
"c * a < c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < 1) \<and> (c \<le> 0 \<longrightarrow> 1 < a)" |
|
29667 | 1085 |
by (insert mult_less_cancel_left [of c a 1], simp) |
26274 | 1086 |
|
27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27516
diff
changeset
|
1087 |
lemma sgn_sgn [simp]: |
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27516
diff
changeset
|
1088 |
"sgn (sgn a) = sgn a" |
29700 | 1089 |
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
|
1090 |
|
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27516
diff
changeset
|
1091 |
lemma sgn_0_0: |
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27516
diff
changeset
|
1092 |
"sgn a = 0 \<longleftrightarrow> a = 0" |
29700 | 1093 |
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
|
1094 |
|
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27516
diff
changeset
|
1095 |
lemma sgn_1_pos: |
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27516
diff
changeset
|
1096 |
"sgn a = 1 \<longleftrightarrow> a > 0" |
35216 | 1097 |
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
|
1098 |
|
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27516
diff
changeset
|
1099 |
lemma sgn_1_neg: |
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27516
diff
changeset
|
1100 |
"sgn a = - 1 \<longleftrightarrow> a < 0" |
35216 | 1101 |
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
|
1102 |
|
29940 | 1103 |
lemma sgn_pos [simp]: |
1104 |
"0 < a \<Longrightarrow> sgn a = 1" |
|
1105 |
unfolding sgn_1_pos . |
|
1106 |
||
1107 |
lemma sgn_neg [simp]: |
|
1108 |
"a < 0 \<Longrightarrow> sgn a = - 1" |
|
1109 |
unfolding sgn_1_neg . |
|
1110 |
||
27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27516
diff
changeset
|
1111 |
lemma sgn_times: |
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27516
diff
changeset
|
1112 |
"sgn (a * b) = sgn a * sgn b" |
29667 | 1113 |
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
|
1114 |
|
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1115 |
lemma abs_sgn: "\<bar>k\<bar> = k * sgn k" |
29700 | 1116 |
unfolding sgn_if abs_if by auto |
1117 |
||
29940 | 1118 |
lemma sgn_greater [simp]: |
1119 |
"0 < sgn a \<longleftrightarrow> 0 < a" |
|
1120 |
unfolding sgn_if by auto |
|
1121 |
||
1122 |
lemma sgn_less [simp]: |
|
1123 |
"sgn a < 0 \<longleftrightarrow> a < 0" |
|
1124 |
unfolding sgn_if by auto |
|
1125 |
||
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1126 |
lemma abs_dvd_iff [simp]: "\<bar>m\<bar> dvd k \<longleftrightarrow> m dvd k" |
29949 | 1127 |
by (simp add: abs_if) |
1128 |
||
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1129 |
lemma dvd_abs_iff [simp]: "m dvd \<bar>k\<bar> \<longleftrightarrow> m dvd k" |
29949 | 1130 |
by (simp add: abs_if) |
29653 | 1131 |
|
33676
802f5e233e48
moved lemma from Algebra/IntRing to Ring_and_Field
nipkow
parents:
33364
diff
changeset
|
1132 |
lemma dvd_if_abs_eq: |
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1133 |
"\<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
|
1134 |
by(subst abs_dvd_iff[symmetric]) simp |
802f5e233e48
moved lemma from Algebra/IntRing to Ring_and_Field
nipkow
parents:
33364
diff
changeset
|
1135 |
|
55912 | 1136 |
text {* The following lemmas can be proven in more general structures, but |
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54250
diff
changeset
|
1137 |
are dangerous as simp rules in absence of @{thm neg_equal_zero}, |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54250
diff
changeset
|
1138 |
@{thm neg_less_pos}, @{thm neg_less_eq_nonneg}. *} |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54250
diff
changeset
|
1139 |
|
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54250
diff
changeset
|
1140 |
lemma equation_minus_iff_1 [simp, no_atp]: |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54250
diff
changeset
|
1141 |
"1 = - a \<longleftrightarrow> a = - 1" |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54250
diff
changeset
|
1142 |
by (fact equation_minus_iff) |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54250
diff
changeset
|
1143 |
|
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54250
diff
changeset
|
1144 |
lemma minus_equation_iff_1 [simp, no_atp]: |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54250
diff
changeset
|
1145 |
"- a = 1 \<longleftrightarrow> a = - 1" |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54250
diff
changeset
|
1146 |
by (subst minus_equation_iff, auto) |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54250
diff
changeset
|
1147 |
|
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54250
diff
changeset
|
1148 |
lemma le_minus_iff_1 [simp, no_atp]: |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54250
diff
changeset
|
1149 |
"1 \<le> - b \<longleftrightarrow> b \<le> - 1" |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54250
diff
changeset
|
1150 |
by (fact le_minus_iff) |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54250
diff
changeset
|
1151 |
|
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54250
diff
changeset
|
1152 |
lemma minus_le_iff_1 [simp, no_atp]: |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54250
diff
changeset
|
1153 |
"- a \<le> 1 \<longleftrightarrow> - 1 \<le> a" |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54250
diff
changeset
|
1154 |
by (fact minus_le_iff) |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54250
diff
changeset
|
1155 |
|
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54250
diff
changeset
|
1156 |
lemma less_minus_iff_1 [simp, no_atp]: |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54250
diff
changeset
|
1157 |
"1 < - b \<longleftrightarrow> b < - 1" |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54250
diff
changeset
|
1158 |
by (fact less_minus_iff) |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54250
diff
changeset
|
1159 |
|
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54250
diff
changeset
|
1160 |
lemma minus_less_iff_1 [simp, no_atp]: |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54250
diff
changeset
|
1161 |
"- a < 1 \<longleftrightarrow> - 1 < a" |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54250
diff
changeset
|
1162 |
by (fact minus_less_iff) |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54250
diff
changeset
|
1163 |
|
25917 | 1164 |
end |
25230 | 1165 |
|
26274 | 1166 |
text {* Simprules for comparisons where common factors can be cancelled. *} |
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
1167 |
|
54147
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
52435
diff
changeset
|
1168 |
lemmas mult_compare_simps = |
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
1169 |
mult_le_cancel_right mult_le_cancel_left |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
1170 |
mult_le_cancel_right1 mult_le_cancel_right2 |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
1171 |
mult_le_cancel_left1 mult_le_cancel_left2 |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
1172 |
mult_less_cancel_right mult_less_cancel_left |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
1173 |
mult_less_cancel_right1 mult_less_cancel_right2 |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
1174 |
mult_less_cancel_left1 mult_less_cancel_left2 |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
1175 |
mult_cancel_right mult_cancel_left |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
1176 |
mult_cancel_right1 mult_cancel_right2 |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
1177 |
mult_cancel_left1 mult_cancel_left2 |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
1178 |
|
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1179 |
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
|
1180 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
1181 |
context linordered_semidom |
25193 | 1182 |
begin |
1183 |
||
1184 |
lemma less_add_one: "a < a + 1" |
|
14293 | 1185 |
proof - |
25193 | 1186 |
have "a + 0 < a + 1" |
23482 | 1187 |
by (blast intro: zero_less_one add_strict_left_mono) |
14293 | 1188 |
thus ?thesis by simp |
1189 |
qed |
|
1190 |
||
25193 | 1191 |
lemma zero_less_two: "0 < 1 + 1" |
29667 | 1192 |
by (blast intro: less_trans zero_less_one less_add_one) |
25193 | 1193 |
|
1194 |
end |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1195 |
|
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1196 |
context linordered_idom |
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1197 |
begin |
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
1198 |
|
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1199 |
lemma mult_right_le_one_le: |
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1200 |
"0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> x * y \<le> x" |
59833
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset
|
1201 |
by (rule mult_left_le) |
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1202 |
|
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1203 |
lemma mult_left_le_one_le: |
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1204 |
"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
|
1205 |
by (auto simp add: mult_le_cancel_right2) |
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1206 |
|
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1207 |
end |
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1208 |
|
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1209 |
text {* Absolute Value *} |
14293 | 1210 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
1211 |
context linordered_idom |
25304
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset
|
1212 |
begin |
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset
|
1213 |
|
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1214 |
lemma mult_sgn_abs: |
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1215 |
"sgn x * \<bar>x\<bar> = x" |
25304
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset
|
1216 |
unfolding abs_if sgn_if by auto |
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset
|
1217 |
|
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1218 |
lemma abs_one [simp]: |
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1219 |
"\<bar>1\<bar> = 1" |
44921 | 1220 |
by (simp add: abs_if) |
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1221 |
|
25304
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset
|
1222 |
end |
24491 | 1223 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
1224 |
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
|
1225 |
assumes abs_eq_mult: |
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset
|
1226 |
"(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
|
1227 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
1228 |
context linordered_idom |
30961 | 1229 |
begin |
1230 |
||
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
1231 |
subclass ordered_ring_abs proof |
35216 | 1232 |
qed (auto simp add: abs_if not_less mult_less_0_iff) |
30961 | 1233 |
|
1234 |
lemma abs_mult: |
|
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1235 |
"\<bar>a * b\<bar> = \<bar>a\<bar> * \<bar>b\<bar>" |
30961 | 1236 |
by (rule abs_eq_mult) auto |
1237 |
||
1238 |
lemma abs_mult_self: |
|
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1239 |
"\<bar>a\<bar> * \<bar>a\<bar> = a * a" |
30961 | 1240 |
by (simp add: abs_if) |
1241 |
||
14294
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1242 |
lemma abs_mult_less: |
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1243 |
"\<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
|
1244 |
proof - |
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1245 |
assume ac: "\<bar>a\<bar> < c" |
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1246 |
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
|
1247 |
assume "\<bar>b\<bar> < d" |
14294
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1248 |
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
|
1249 |
qed |
14293 | 1250 |
|
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1251 |
lemma abs_less_iff: |
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1252 |
"\<bar>a\<bar> < b \<longleftrightarrow> a < b \<and> - a < b" |
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1253 |
by (simp add: less_le abs_le_iff) (auto simp add: abs_if) |
14738 | 1254 |
|
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1255 |
lemma abs_mult_pos: |
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1256 |
"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
|
1257 |
by (simp add: abs_mult) |
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1258 |
|
51520
e9b361845809
move real_isLub_unique to isLub_unique in Lubs; real_sum_of_halves to RealDef; abs_diff_less_iff to Rings
hoelzl
parents:
50420
diff
changeset
|
1259 |
lemma abs_diff_less_iff: |
e9b361845809
move real_isLub_unique to isLub_unique in Lubs; real_sum_of_halves to RealDef; abs_diff_less_iff to Rings
hoelzl
parents:
50420
diff
changeset
|
1260 |
"\<bar>x - a\<bar> < r \<longleftrightarrow> a - r < x \<and> x < a + r" |
e9b361845809
move real_isLub_unique to isLub_unique in Lubs; real_sum_of_halves to RealDef; abs_diff_less_iff to Rings
hoelzl
parents:
50420
diff
changeset
|
1261 |
by (auto simp add: diff_less_eq ac_simps abs_less_iff) |
e9b361845809
move real_isLub_unique to isLub_unique in Lubs; real_sum_of_halves to RealDef; abs_diff_less_iff to Rings
hoelzl
parents:
50420
diff
changeset
|
1262 |
|
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1263 |
end |
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1264 |
|
59557 | 1265 |
hide_fact (open) comm_mult_left_mono comm_mult_strict_left_mono distrib |
1266 |
||
52435
6646bb548c6b
migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents:
51520
diff
changeset
|
1267 |
code_identifier |
6646bb548c6b
migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents:
51520
diff
changeset
|
1268 |
code_module Rings \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith |
33364 | 1269 |
|
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
1270 |
end |