author | wenzelm |
Fri, 05 Apr 2019 17:05:32 +0200 | |
changeset 70067 | 9b34dbeb1103 |
parent 69661 | a03a63b81f44 |
child 70094 | a93e6472ac9c |
permissions | -rw-r--r-- |
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renamed OrderedGroup to Groups; split theory Ring_and_Field into Rings Fields
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1 |
(* Title: HOL/Rings.thy |
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2 |
Author: Gertrud Bauer |
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parents:
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3 |
Author: Steven Obua |
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parents:
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4 |
Author: Tobias Nipkow |
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eliminated hard tabulators, guessing at each author's individual tab-width;
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parents:
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5 |
Author: Lawrence C Paulson |
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parents:
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6 |
Author: Markus Wenzel |
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parents:
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7 |
Author: Jeremy Avigad |
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HOL: installation of Ring_and_Field as the basis for Naturals and Reals
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parents:
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8 |
*) |
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HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
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9 |
|
60758 | 10 |
section \<open>Rings\<close> |
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11 |
|
35050
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renamed OrderedGroup to Groups; split theory Ring_and_Field into Rings Fields
haftmann
parents:
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12 |
theory Rings |
69661 | 13 |
imports Groups Set Fun |
15131 | 14 |
begin |
14504 | 15 |
|
22390 | 16 |
class semiring = ab_semigroup_add + semigroup_mult + |
58776
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use NO_MATCH-simproc for distribution rules in field_simps, otherwise field_simps on '(a / (c + d)) * (e + f)' can be non-terminating
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parents:
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17 |
assumes distrib_right[algebra_simps]: "(a + b) * c = a * c + b * c" |
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
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18 |
assumes distrib_left[algebra_simps]: "a * (b + c) = a * b + a * c" |
25152 | 19 |
begin |
20 |
||
63325 | 21 |
text \<open>For the \<open>combine_numerals\<close> simproc\<close> |
22 |
lemma combine_common_factor: "a * e + (b * e + c) = (a + b) * e + c" |
|
23 |
by (simp add: distrib_right ac_simps) |
|
25152 | 24 |
|
25 |
end |
|
14504 | 26 |
|
22390 | 27 |
class mult_zero = times + zero + |
25062 | 28 |
assumes mult_zero_left [simp]: "0 * a = 0" |
29 |
assumes mult_zero_right [simp]: "a * 0 = 0" |
|
58195 | 30 |
begin |
31 |
||
63325 | 32 |
lemma mult_not_zero: "a * b \<noteq> 0 \<Longrightarrow> a \<noteq> 0 \<and> b \<noteq> 0" |
58195 | 33 |
by auto |
34 |
||
35 |
end |
|
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* Added annihilation axioms ("x * 0 = 0") to axclass semiring_0.
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36 |
|
58198 | 37 |
class semiring_0 = semiring + comm_monoid_add + mult_zero |
38 |
||
29904 | 39 |
class semiring_0_cancel = semiring + cancel_comm_monoid_add |
25186 | 40 |
begin |
14504 | 41 |
|
25186 | 42 |
subclass semiring_0 |
28823 | 43 |
proof |
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parents:
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fix a :: 'a |
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have "0 * a + 0 * a = 0 * a + 0" |
46 |
by (simp add: distrib_right [symmetric]) |
|
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then show "0 * a = 0" |
|
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by (simp only: add_left_cancel) |
|
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have "a * 0 + a * 0 = a * 0 + 0" |
|
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by (simp add: distrib_left [symmetric]) |
|
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then show "a * 0 = 0" |
|
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by (simp only: add_left_cancel) |
|
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* Added annihilation axioms ("x * 0 = 0") to axclass semiring_0.
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parents:
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53 |
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 |
63588 | 64 |
show "(a + b) * c = a * c + b * c" |
65 |
by (simp add: distrib) |
|
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have "a * (b + c) = (b + c) * a" |
|
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by (simp add: ac_simps) |
|
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also have "\<dots> = b * a + c * a" |
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by (simp only: distrib) |
|
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also have "\<dots> = a * b + a * c" |
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by (simp add: ac_simps) |
|
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finally show "a * (b + c) = a * b + a * c" |
|
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by blast |
|
14504 | 74 |
qed |
75 |
||
25152 | 76 |
end |
14504 | 77 |
|
25152 | 78 |
class comm_semiring_0 = comm_semiring + comm_monoid_add + mult_zero |
79 |
begin |
|
80 |
||
27516 | 81 |
subclass semiring_0 .. |
25152 | 82 |
|
83 |
end |
|
14504 | 84 |
|
29904 | 85 |
class comm_semiring_0_cancel = comm_semiring + cancel_comm_monoid_add |
25186 | 86 |
begin |
14940 | 87 |
|
27516 | 88 |
subclass semiring_0_cancel .. |
14940 | 89 |
|
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instances comm_semiring_0_cancel < comm_semiring_0, comm_ring < comm_semiring_0_cancel
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parents:
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subclass comm_semiring_0 .. |
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instances comm_semiring_0_cancel < comm_semiring_0, comm_ring < comm_semiring_0_cancel
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parents:
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91 |
|
25186 | 92 |
end |
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* Added annihilation axioms ("x * 0 = 0") to axclass semiring_0.
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93 |
|
22390 | 94 |
class zero_neq_one = zero + one + |
25062 | 95 |
assumes zero_neq_one [simp]: "0 \<noteq> 1" |
26193 | 96 |
begin |
97 |
||
98 |
lemma one_neq_zero [simp]: "1 \<noteq> 0" |
|
63325 | 99 |
by (rule not_sym) (rule zero_neq_one) |
26193 | 100 |
|
54225 | 101 |
definition of_bool :: "bool \<Rightarrow> 'a" |
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where "of_bool p = (if p then 1 else 0)" |
54225 | 103 |
|
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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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||
63325 | 109 |
lemma of_bool_eq_iff: "of_bool p = of_bool q \<longleftrightarrow> p = q" |
54225 | 110 |
by (simp add: of_bool_def) |
111 |
||
63325 | 112 |
lemma split_of_bool [split]: "P (of_bool p) \<longleftrightarrow> (p \<longrightarrow> P 1) \<and> (\<not> p \<longrightarrow> P 0)" |
55187 | 113 |
by (cases p) simp_all |
114 |
||
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lemma split_of_bool_asm: "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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Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents:
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diff
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117 |
|
24af00b010cf
Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents:
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118 |
end |
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|
22390 | 120 |
class semiring_1 = zero_neq_one + semiring_0 + monoid_mult |
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more fundamental definition of div and mod on int
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121 |
begin |
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more fundamental definition of div and mod on int
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122 |
|
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more fundamental definition of div and mod on int
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lemma (in semiring_1) of_bool_conj: |
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124 |
"of_bool (P \<and> Q) = of_bool P * of_bool Q" |
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more fundamental definition of div and mod on int
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125 |
by auto |
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more fundamental definition of div and mod on int
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126 |
|
212a3334e7da
more fundamental definition of div and mod on int
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parents:
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127 |
end |
14504 | 128 |
|
60758 | 129 |
text \<open>Abstract divisibility\<close> |
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130 |
|
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131 |
class dvd = times |
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132 |
begin |
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133 |
|
63325 | 134 |
definition dvd :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infix "dvd" 50) |
135 |
where "b dvd a \<longleftrightarrow> (\<exists>k. a = b * k)" |
|
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136 |
|
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moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
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137 |
lemma dvdI [intro?]: "a = b * k \<Longrightarrow> b dvd a" |
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parents:
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138 |
unfolding dvd_def .. |
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moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
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parents:
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139 |
|
68251 | 140 |
lemma dvdE [elim]: "b dvd a \<Longrightarrow> (\<And>k. a = b * k \<Longrightarrow> P) \<Longrightarrow> P" |
60562
24af00b010cf
Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents:
60529
diff
changeset
|
141 |
unfolding dvd_def by blast |
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moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
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diff
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142 |
|
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
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parents:
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diff
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143 |
end |
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moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
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parents:
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|
144 |
|
59009
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generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
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diff
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145 |
context comm_monoid_mult |
25152 | 146 |
begin |
14738 | 147 |
|
59009
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generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
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diff
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148 |
subclass dvd . |
25152 | 149 |
|
63325 | 150 |
lemma dvd_refl [simp]: "a dvd a" |
28559 | 151 |
proof |
152 |
show "a = a * 1" by simp |
|
27651
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153 |
qed |
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moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
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parents:
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|
154 |
|
62349
7c23469b5118
cleansed junk-producing interpretations for gcd/lcm on nat altogether
haftmann
parents:
62347
diff
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155 |
lemma dvd_trans [trans]: |
27651
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moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
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parents:
27516
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156 |
assumes "a dvd b" and "b dvd c" |
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
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diff
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|
157 |
shows "a dvd c" |
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
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diff
changeset
|
158 |
proof - |
63588 | 159 |
from assms obtain v where "b = a * v" |
160 |
by (auto elim!: dvdE) |
|
161 |
moreover from assms obtain w where "c = b * w" |
|
162 |
by (auto elim!: dvdE) |
|
163 |
ultimately have "c = a * (v * w)" |
|
164 |
by (simp add: mult.assoc) |
|
28559 | 165 |
then show ?thesis .. |
27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
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diff
changeset
|
166 |
qed |
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27516
diff
changeset
|
167 |
|
63325 | 168 |
lemma subset_divisors_dvd: "{c. c dvd a} \<subseteq> {c. c dvd b} \<longleftrightarrow> a dvd b" |
62366 | 169 |
by (auto simp add: subset_iff intro: dvd_trans) |
170 |
||
63325 | 171 |
lemma strict_subset_divisors_dvd: "{c. c dvd a} \<subset> {c. c dvd b} \<longleftrightarrow> a dvd b \<and> \<not> b dvd a" |
62366 | 172 |
by (auto simp add: subset_iff intro: dvd_trans) |
173 |
||
63325 | 174 |
lemma one_dvd [simp]: "1 dvd a" |
59009
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generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
59000
diff
changeset
|
175 |
by (auto intro!: dvdI) |
28559 | 176 |
|
63325 | 177 |
lemma dvd_mult [simp]: "a dvd c \<Longrightarrow> a dvd (b * c)" |
59009
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
59000
diff
changeset
|
178 |
by (auto intro!: mult.left_commute dvdI elim!: dvdE) |
27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27516
diff
changeset
|
179 |
|
63325 | 180 |
lemma dvd_mult2 [simp]: "a dvd b \<Longrightarrow> a dvd (b * c)" |
60562
24af00b010cf
Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents:
60529
diff
changeset
|
181 |
using dvd_mult [of a b c] by (simp add: ac_simps) |
27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27516
diff
changeset
|
182 |
|
63325 | 183 |
lemma dvd_triv_right [simp]: "a dvd b * a" |
59009
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
59000
diff
changeset
|
184 |
by (rule dvd_mult) (rule dvd_refl) |
27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27516
diff
changeset
|
185 |
|
63325 | 186 |
lemma dvd_triv_left [simp]: "a dvd a * b" |
59009
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
59000
diff
changeset
|
187 |
by (rule dvd_mult2) (rule dvd_refl) |
27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27516
diff
changeset
|
188 |
|
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27516
diff
changeset
|
189 |
lemma mult_dvd_mono: |
30042 | 190 |
assumes "a dvd b" |
191 |
and "c dvd d" |
|
27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27516
diff
changeset
|
192 |
shows "a * c dvd b * d" |
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27516
diff
changeset
|
193 |
proof - |
60758 | 194 |
from \<open>a dvd b\<close> obtain b' where "b = a * b'" .. |
195 |
moreover from \<open>c dvd d\<close> obtain d' where "d = c * d'" .. |
|
63588 | 196 |
ultimately have "b * d = (a * c) * (b' * d')" |
197 |
by (simp add: ac_simps) |
|
27651
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moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27516
diff
changeset
|
198 |
then show ?thesis .. |
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27516
diff
changeset
|
199 |
qed |
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27516
diff
changeset
|
200 |
|
63325 | 201 |
lemma dvd_mult_left: "a * b dvd c \<Longrightarrow> a dvd c" |
59009
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
59000
diff
changeset
|
202 |
by (simp add: dvd_def mult.assoc) blast |
27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27516
diff
changeset
|
203 |
|
63325 | 204 |
lemma dvd_mult_right: "a * b dvd c \<Longrightarrow> b dvd c" |
59009
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
59000
diff
changeset
|
205 |
using dvd_mult_left [of b a c] by (simp add: ac_simps) |
60562
24af00b010cf
Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents:
60529
diff
changeset
|
206 |
|
59009
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
59000
diff
changeset
|
207 |
end |
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
59000
diff
changeset
|
208 |
|
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
59000
diff
changeset
|
209 |
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
|
210 |
begin |
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
59000
diff
changeset
|
211 |
|
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
59000
diff
changeset
|
212 |
subclass semiring_1 .. |
27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27516
diff
changeset
|
213 |
|
63325 | 214 |
lemma dvd_0_left_iff [simp]: "0 dvd a \<longleftrightarrow> a = 0" |
59009
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
59000
diff
changeset
|
215 |
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
|
216 |
|
63325 | 217 |
lemma dvd_0_right [iff]: "a dvd 0" |
59009
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
59000
diff
changeset
|
218 |
proof |
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
59000
diff
changeset
|
219 |
show "0 = a * 0" by simp |
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
59000
diff
changeset
|
220 |
qed |
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
59000
diff
changeset
|
221 |
|
63325 | 222 |
lemma dvd_0_left: "0 dvd a \<Longrightarrow> a = 0" |
59009
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
59000
diff
changeset
|
223 |
by simp |
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
59000
diff
changeset
|
224 |
|
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
59000
diff
changeset
|
225 |
lemma dvd_add [simp]: |
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
59000
diff
changeset
|
226 |
assumes "a dvd b" and "a dvd c" |
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
59000
diff
changeset
|
227 |
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
|
228 |
proof - |
60758 | 229 |
from \<open>a dvd b\<close> obtain b' where "b = a * b'" .. |
230 |
moreover from \<open>a dvd c\<close> obtain c' where "c = a * c'" .. |
|
63588 | 231 |
ultimately have "b + c = a * (b' + c')" |
232 |
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
|
233 |
then show ?thesis .. |
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27516
diff
changeset
|
234 |
qed |
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27516
diff
changeset
|
235 |
|
25152 | 236 |
end |
14421
ee97b6463cb4
new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents:
14398
diff
changeset
|
237 |
|
29904 | 238 |
class semiring_1_cancel = semiring + cancel_comm_monoid_add |
239 |
+ zero_neq_one + monoid_mult |
|
25267 | 240 |
begin |
14940 | 241 |
|
27516 | 242 |
subclass semiring_0_cancel .. |
25512
4134f7c782e2
using intro_locales instead of unfold_locales if appropriate
haftmann
parents:
25450
diff
changeset
|
243 |
|
27516 | 244 |
subclass semiring_1 .. |
25267 | 245 |
|
246 |
end |
|
21199
2d83f93c3580
* Added annihilation axioms ("x * 0 = 0") to axclass semiring_0.
krauss
parents:
20633
diff
changeset
|
247 |
|
63325 | 248 |
class comm_semiring_1_cancel = |
249 |
comm_semiring + cancel_comm_monoid_add + zero_neq_one + comm_monoid_mult + |
|
60562
24af00b010cf
Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents:
60529
diff
changeset
|
250 |
assumes right_diff_distrib' [algebra_simps]: "a * (b - c) = a * b - a * c" |
25267 | 251 |
begin |
14738 | 252 |
|
27516 | 253 |
subclass semiring_1_cancel .. |
254 |
subclass comm_semiring_0_cancel .. |
|
255 |
subclass comm_semiring_1 .. |
|
25267 | 256 |
|
63325 | 257 |
lemma left_diff_distrib' [algebra_simps]: "(b - c) * a = b * a - c * a" |
59816
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
258 |
by (simp add: algebra_simps) |
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
259 |
|
63325 | 260 |
lemma dvd_add_times_triv_left_iff [simp]: "a dvd c * a + b \<longleftrightarrow> a dvd b" |
59816
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
261 |
proof - |
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
262 |
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
|
263 |
proof |
63325 | 264 |
assume ?Q |
265 |
then show ?P by simp |
|
59816
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
266 |
next |
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
267 |
assume ?P |
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
268 |
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
|
269 |
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
|
270 |
then have "b = a * d - a * c" by simp |
60562
24af00b010cf
Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents:
60529
diff
changeset
|
271 |
then have "b = a * (d - c)" by (simp add: algebra_simps) |
59816
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
272 |
then show ?Q .. |
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 |
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
|
275 |
qed |
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
276 |
|
63325 | 277 |
lemma dvd_add_times_triv_right_iff [simp]: "a dvd b + c * a \<longleftrightarrow> a dvd b" |
59816
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
278 |
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
|
279 |
|
63325 | 280 |
lemma dvd_add_triv_left_iff [simp]: "a dvd a + b \<longleftrightarrow> a dvd b" |
59816
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 |
|
63325 | 283 |
lemma dvd_add_triv_right_iff [simp]: "a dvd b + a \<longleftrightarrow> a dvd b" |
59816
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
284 |
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
|
285 |
|
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
286 |
lemma dvd_add_right_iff: |
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
287 |
assumes "a dvd b" |
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
288 |
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
|
289 |
proof |
63325 | 290 |
assume ?P |
291 |
then obtain d where "b + c = a * d" .. |
|
60758 | 292 |
moreover from \<open>a dvd b\<close> obtain e where "b = a * e" .. |
59816
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 |
63325 | 299 |
assume ?Q |
300 |
with assms show ?P by simp |
|
59816
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
301 |
qed |
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
302 |
|
63325 | 303 |
lemma dvd_add_left_iff: "a dvd c \<Longrightarrow> a dvd b + c \<longleftrightarrow> a dvd b" |
304 |
using dvd_add_right_iff [of a c b] by (simp add: ac_simps) |
|
59816
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
305 |
|
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
306 |
end |
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
307 |
|
22390 | 308 |
class ring = semiring + ab_group_add |
25267 | 309 |
begin |
25152 | 310 |
|
27516 | 311 |
subclass semiring_0_cancel .. |
25152 | 312 |
|
60758 | 313 |
text \<open>Distribution rules\<close> |
25152 | 314 |
|
315 |
lemma minus_mult_left: "- (a * b) = - a * b" |
|
63325 | 316 |
by (rule minus_unique) (simp add: distrib_right [symmetric]) |
25152 | 317 |
|
318 |
lemma minus_mult_right: "- (a * b) = a * - b" |
|
63325 | 319 |
by (rule minus_unique) (simp add: distrib_left [symmetric]) |
25152 | 320 |
|
63325 | 321 |
text \<open>Extract signs from products\<close> |
54147
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
52435
diff
changeset
|
322 |
lemmas mult_minus_left [simp] = minus_mult_left [symmetric] |
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
52435
diff
changeset
|
323 |
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
|
324 |
|
25152 | 325 |
lemma minus_mult_minus [simp]: "- a * - b = a * b" |
63325 | 326 |
by simp |
25152 | 327 |
|
328 |
lemma minus_mult_commute: "- a * b = a * - b" |
|
63325 | 329 |
by simp |
29667 | 330 |
|
63325 | 331 |
lemma right_diff_distrib [algebra_simps]: "a * (b - c) = a * b - a * c" |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54225
diff
changeset
|
332 |
using distrib_left [of a b "-c "] by simp |
29667 | 333 |
|
63325 | 334 |
lemma left_diff_distrib [algebra_simps]: "(a - b) * c = a * c - b * c" |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54225
diff
changeset
|
335 |
using distrib_right [of a "- b" c] by simp |
25152 | 336 |
|
63325 | 337 |
lemmas ring_distribs = distrib_left distrib_right left_diff_distrib right_diff_distrib |
25152 | 338 |
|
63325 | 339 |
lemma eq_add_iff1: "a * e + c = b * e + d \<longleftrightarrow> (a - b) * e + c = d" |
340 |
by (simp add: algebra_simps) |
|
25230 | 341 |
|
63325 | 342 |
lemma eq_add_iff2: "a * e + c = b * e + d \<longleftrightarrow> c = (b - a) * e + d" |
343 |
by (simp add: algebra_simps) |
|
25230 | 344 |
|
25152 | 345 |
end |
346 |
||
63325 | 347 |
lemmas ring_distribs = distrib_left distrib_right left_diff_distrib right_diff_distrib |
25152 | 348 |
|
22390 | 349 |
class comm_ring = comm_semiring + ab_group_add |
25267 | 350 |
begin |
14738 | 351 |
|
27516 | 352 |
subclass ring .. |
28141
193c3ea0f63b
instances comm_semiring_0_cancel < comm_semiring_0, comm_ring < comm_semiring_0_cancel
huffman
parents:
27651
diff
changeset
|
353 |
subclass comm_semiring_0_cancel .. |
25267 | 354 |
|
63325 | 355 |
lemma square_diff_square_factored: "x * x - y * y = (x + y) * (x - y)" |
44350
63cddfbc5a09
replace lemma realpow_two_diff with new lemma square_diff_square_factored
huffman
parents:
44346
diff
changeset
|
356 |
by (simp add: algebra_simps) |
63cddfbc5a09
replace lemma realpow_two_diff with new lemma square_diff_square_factored
huffman
parents:
44346
diff
changeset
|
357 |
|
25267 | 358 |
end |
14738 | 359 |
|
22390 | 360 |
class ring_1 = ring + zero_neq_one + monoid_mult |
25267 | 361 |
begin |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
362 |
|
27516 | 363 |
subclass semiring_1_cancel .. |
25267 | 364 |
|
63325 | 365 |
lemma square_diff_one_factored: "x * x - 1 = (x + 1) * (x - 1)" |
44346
00dd3c4dabe0
rename real_squared_diff_one_factored to square_diff_one_factored and move to Rings.thy
huffman
parents:
44064
diff
changeset
|
366 |
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
|
367 |
|
25267 | 368 |
end |
25152 | 369 |
|
22390 | 370 |
class comm_ring_1 = comm_ring + zero_neq_one + comm_monoid_mult |
25267 | 371 |
begin |
14738 | 372 |
|
27516 | 373 |
subclass ring_1 .. |
60562
24af00b010cf
Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents:
60529
diff
changeset
|
374 |
subclass comm_semiring_1_cancel |
59816
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
375 |
by unfold_locales (simp add: algebra_simps) |
58647 | 376 |
|
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
|
377 |
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
|
378 |
proof |
6d10cf26b5dc
add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents:
29407
diff
changeset
|
379 |
assume "x dvd - y" |
6d10cf26b5dc
add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents:
29407
diff
changeset
|
380 |
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
|
381 |
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
|
382 |
next |
6d10cf26b5dc
add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents:
29407
diff
changeset
|
383 |
assume "x dvd y" |
6d10cf26b5dc
add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents:
29407
diff
changeset
|
384 |
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
|
385 |
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
|
386 |
qed |
6d10cf26b5dc
add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents:
29407
diff
changeset
|
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 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
|
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 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
|
392 |
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
|
393 |
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
|
394 |
next |
6d10cf26b5dc
add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents:
29407
diff
changeset
|
395 |
assume "x dvd y" |
6d10cf26b5dc
add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents:
29407
diff
changeset
|
396 |
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
|
397 |
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
|
398 |
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
|
399 |
qed |
6d10cf26b5dc
add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents:
29407
diff
changeset
|
400 |
|
63325 | 401 |
lemma dvd_diff [simp]: "x dvd y \<Longrightarrow> x dvd z \<Longrightarrow> x dvd (y - z)" |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54225
diff
changeset
|
402 |
using dvd_add [of x y "- z"] by simp |
29409 | 403 |
|
25267 | 404 |
end |
25152 | 405 |
|
59833
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset
|
406 |
class semiring_no_zero_divisors = semiring_0 + |
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset
|
407 |
assumes no_zero_divisors: "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> a * b \<noteq> 0" |
25230 | 408 |
begin |
409 |
||
59833
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset
|
410 |
lemma divisors_zero: |
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset
|
411 |
assumes "a * b = 0" |
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset
|
412 |
shows "a = 0 \<or> b = 0" |
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset
|
413 |
proof (rule classical) |
63325 | 414 |
assume "\<not> ?thesis" |
59833
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset
|
415 |
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
|
416 |
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
|
417 |
with assms show ?thesis by simp |
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset
|
418 |
qed |
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset
|
419 |
|
63325 | 420 |
lemma mult_eq_0_iff [simp]: "a * b = 0 \<longleftrightarrow> a = 0 \<or> b = 0" |
25230 | 421 |
proof (cases "a = 0 \<or> b = 0") |
63325 | 422 |
case False |
423 |
then have "a \<noteq> 0" and "b \<noteq> 0" by auto |
|
25230 | 424 |
then show ?thesis using no_zero_divisors by simp |
425 |
next |
|
63325 | 426 |
case True |
427 |
then show ?thesis by auto |
|
25230 | 428 |
qed |
429 |
||
58952
5d82cdef6c1b
equivalence rules for structures without zero divisors
haftmann
parents:
58889
diff
changeset
|
430 |
end |
5d82cdef6c1b
equivalence rules for structures without zero divisors
haftmann
parents:
58889
diff
changeset
|
431 |
|
62481
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62390
diff
changeset
|
432 |
class semiring_1_no_zero_divisors = semiring_1 + semiring_no_zero_divisors |
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62390
diff
changeset
|
433 |
|
60516
0826b7025d07
generalized some theorems about integral domains and moved to HOL theories
haftmann
parents:
60429
diff
changeset
|
434 |
class semiring_no_zero_divisors_cancel = semiring_no_zero_divisors + |
0826b7025d07
generalized some theorems about integral domains and moved to HOL theories
haftmann
parents:
60429
diff
changeset
|
435 |
assumes mult_cancel_right [simp]: "a * c = b * c \<longleftrightarrow> c = 0 \<or> a = b" |
0826b7025d07
generalized some theorems about integral domains and moved to HOL theories
haftmann
parents:
60429
diff
changeset
|
436 |
and mult_cancel_left [simp]: "c * a = c * b \<longleftrightarrow> c = 0 \<or> a = b" |
58952
5d82cdef6c1b
equivalence rules for structures without zero divisors
haftmann
parents:
58889
diff
changeset
|
437 |
begin |
5d82cdef6c1b
equivalence rules for structures without zero divisors
haftmann
parents:
58889
diff
changeset
|
438 |
|
63325 | 439 |
lemma mult_left_cancel: "c \<noteq> 0 \<Longrightarrow> c * a = c * b \<longleftrightarrow> a = b" |
60562
24af00b010cf
Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents:
60529
diff
changeset
|
440 |
by simp |
56217
dc429a5b13c4
Some rationalisation of basic lemmas
paulson <lp15@cam.ac.uk>
parents:
55912
diff
changeset
|
441 |
|
63325 | 442 |
lemma mult_right_cancel: "c \<noteq> 0 \<Longrightarrow> a * c = b * c \<longleftrightarrow> a = b" |
60562
24af00b010cf
Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents:
60529
diff
changeset
|
443 |
by simp |
56217
dc429a5b13c4
Some rationalisation of basic lemmas
paulson <lp15@cam.ac.uk>
parents:
55912
diff
changeset
|
444 |
|
25230 | 445 |
end |
22990
775e9de3db48
added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents:
22987
diff
changeset
|
446 |
|
60516
0826b7025d07
generalized some theorems about integral domains and moved to HOL theories
haftmann
parents:
60429
diff
changeset
|
447 |
class ring_no_zero_divisors = ring + semiring_no_zero_divisors |
0826b7025d07
generalized some theorems about integral domains and moved to HOL theories
haftmann
parents:
60429
diff
changeset
|
448 |
begin |
0826b7025d07
generalized some theorems about integral domains and moved to HOL theories
haftmann
parents:
60429
diff
changeset
|
449 |
|
0826b7025d07
generalized some theorems about integral domains and moved to HOL theories
haftmann
parents:
60429
diff
changeset
|
450 |
subclass semiring_no_zero_divisors_cancel |
0826b7025d07
generalized some theorems about integral domains and moved to HOL theories
haftmann
parents:
60429
diff
changeset
|
451 |
proof |
0826b7025d07
generalized some theorems about integral domains and moved to HOL theories
haftmann
parents:
60429
diff
changeset
|
452 |
fix a b c |
0826b7025d07
generalized some theorems about integral domains and moved to HOL theories
haftmann
parents:
60429
diff
changeset
|
453 |
have "a * c = b * c \<longleftrightarrow> (a - b) * c = 0" |
0826b7025d07
generalized some theorems about integral domains and moved to HOL theories
haftmann
parents:
60429
diff
changeset
|
454 |
by (simp add: algebra_simps) |
0826b7025d07
generalized some theorems about integral domains and moved to HOL theories
haftmann
parents:
60429
diff
changeset
|
455 |
also have "\<dots> \<longleftrightarrow> c = 0 \<or> a = b" |
0826b7025d07
generalized some theorems about integral domains and moved to HOL theories
haftmann
parents:
60429
diff
changeset
|
456 |
by auto |
0826b7025d07
generalized some theorems about integral domains and moved to HOL theories
haftmann
parents:
60429
diff
changeset
|
457 |
finally show "a * c = b * c \<longleftrightarrow> c = 0 \<or> a = b" . |
0826b7025d07
generalized some theorems about integral domains and moved to HOL theories
haftmann
parents:
60429
diff
changeset
|
458 |
have "c * a = c * b \<longleftrightarrow> c * (a - b) = 0" |
0826b7025d07
generalized some theorems about integral domains and moved to HOL theories
haftmann
parents:
60429
diff
changeset
|
459 |
by (simp add: algebra_simps) |
0826b7025d07
generalized some theorems about integral domains and moved to HOL theories
haftmann
parents:
60429
diff
changeset
|
460 |
also have "\<dots> \<longleftrightarrow> c = 0 \<or> a = b" |
0826b7025d07
generalized some theorems about integral domains and moved to HOL theories
haftmann
parents:
60429
diff
changeset
|
461 |
by auto |
0826b7025d07
generalized some theorems about integral domains and moved to HOL theories
haftmann
parents:
60429
diff
changeset
|
462 |
finally show "c * a = c * b \<longleftrightarrow> c = 0 \<or> a = b" . |
0826b7025d07
generalized some theorems about integral domains and moved to HOL theories
haftmann
parents:
60429
diff
changeset
|
463 |
qed |
0826b7025d07
generalized some theorems about integral domains and moved to HOL theories
haftmann
parents:
60429
diff
changeset
|
464 |
|
0826b7025d07
generalized some theorems about integral domains and moved to HOL theories
haftmann
parents:
60429
diff
changeset
|
465 |
end |
0826b7025d07
generalized some theorems about integral domains and moved to HOL theories
haftmann
parents:
60429
diff
changeset
|
466 |
|
23544 | 467 |
class ring_1_no_zero_divisors = ring_1 + ring_no_zero_divisors |
26274 | 468 |
begin |
469 |
||
62481
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62390
diff
changeset
|
470 |
subclass semiring_1_no_zero_divisors .. |
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62390
diff
changeset
|
471 |
|
63325 | 472 |
lemma square_eq_1_iff: "x * x = 1 \<longleftrightarrow> x = 1 \<or> x = - 1" |
36821
9207505d1ee5
move lemma real_mult_is_one to Rings.thy, renamed to square_eq_1_iff
huffman
parents:
36719
diff
changeset
|
473 |
proof - |
9207505d1ee5
move lemma real_mult_is_one to Rings.thy, renamed to square_eq_1_iff
huffman
parents:
36719
diff
changeset
|
474 |
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
|
475 |
by (simp add: algebra_simps) |
63325 | 476 |
then have "x * x = 1 \<longleftrightarrow> (x - 1) * (x + 1) = 0" |
36821
9207505d1ee5
move lemma real_mult_is_one to Rings.thy, renamed to square_eq_1_iff
huffman
parents:
36719
diff
changeset
|
477 |
by simp |
63325 | 478 |
then show ?thesis |
36821
9207505d1ee5
move lemma real_mult_is_one to Rings.thy, renamed to square_eq_1_iff
huffman
parents:
36719
diff
changeset
|
479 |
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
|
480 |
qed |
9207505d1ee5
move lemma real_mult_is_one to Rings.thy, renamed to square_eq_1_iff
huffman
parents:
36719
diff
changeset
|
481 |
|
63325 | 482 |
lemma mult_cancel_right1 [simp]: "c = b * c \<longleftrightarrow> c = 0 \<or> b = 1" |
483 |
using mult_cancel_right [of 1 c b] by auto |
|
26274 | 484 |
|
63325 | 485 |
lemma mult_cancel_right2 [simp]: "a * c = c \<longleftrightarrow> c = 0 \<or> a = 1" |
486 |
using mult_cancel_right [of a c 1] by simp |
|
60562
24af00b010cf
Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents:
60529
diff
changeset
|
487 |
|
63325 | 488 |
lemma mult_cancel_left1 [simp]: "c = c * b \<longleftrightarrow> c = 0 \<or> b = 1" |
489 |
using mult_cancel_left [of c 1 b] by force |
|
26274 | 490 |
|
63325 | 491 |
lemma mult_cancel_left2 [simp]: "c * a = c \<longleftrightarrow> c = 0 \<or> a = 1" |
492 |
using mult_cancel_left [of c a 1] by simp |
|
26274 | 493 |
|
494 |
end |
|
22990
775e9de3db48
added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents:
22987
diff
changeset
|
495 |
|
60562
24af00b010cf
Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents:
60529
diff
changeset
|
496 |
class semidom = comm_semiring_1_cancel + semiring_no_zero_divisors |
62481
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62390
diff
changeset
|
497 |
begin |
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62390
diff
changeset
|
498 |
|
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62390
diff
changeset
|
499 |
subclass semiring_1_no_zero_divisors .. |
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62390
diff
changeset
|
500 |
|
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62390
diff
changeset
|
501 |
end |
59833
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset
|
502 |
|
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset
|
503 |
class idom = comm_ring_1 + semiring_no_zero_divisors |
25186 | 504 |
begin |
14421
ee97b6463cb4
new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents:
14398
diff
changeset
|
505 |
|
59833
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset
|
506 |
subclass semidom .. |
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset
|
507 |
|
27516 | 508 |
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
|
509 |
|
63325 | 510 |
lemma dvd_mult_cancel_right [simp]: "a * c dvd b * c \<longleftrightarrow> c = 0 \<or> a dvd b" |
29981
7d0ed261b712
generalize int_dvd_cancel_factor simproc to idom class
huffman
parents:
29949
diff
changeset
|
511 |
proof - |
7d0ed261b712
generalize int_dvd_cancel_factor simproc to idom class
huffman
parents:
29949
diff
changeset
|
512 |
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
|
513 |
unfolding dvd_def by (simp add: ac_simps) |
29981
7d0ed261b712
generalize int_dvd_cancel_factor simproc to idom class
huffman
parents:
29949
diff
changeset
|
514 |
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
|
515 |
unfolding dvd_def by simp |
7d0ed261b712
generalize int_dvd_cancel_factor simproc to idom class
huffman
parents:
29949
diff
changeset
|
516 |
finally show ?thesis . |
7d0ed261b712
generalize int_dvd_cancel_factor simproc to idom class
huffman
parents:
29949
diff
changeset
|
517 |
qed |
7d0ed261b712
generalize int_dvd_cancel_factor simproc to idom class
huffman
parents:
29949
diff
changeset
|
518 |
|
63325 | 519 |
lemma dvd_mult_cancel_left [simp]: "c * a dvd c * b \<longleftrightarrow> c = 0 \<or> a dvd b" |
29981
7d0ed261b712
generalize int_dvd_cancel_factor simproc to idom class
huffman
parents:
29949
diff
changeset
|
520 |
proof - |
7d0ed261b712
generalize int_dvd_cancel_factor simproc to idom class
huffman
parents:
29949
diff
changeset
|
521 |
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
|
522 |
unfolding dvd_def by (simp add: ac_simps) |
29981
7d0ed261b712
generalize int_dvd_cancel_factor simproc to idom class
huffman
parents:
29949
diff
changeset
|
523 |
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
|
524 |
unfolding dvd_def by simp |
7d0ed261b712
generalize int_dvd_cancel_factor simproc to idom class
huffman
parents:
29949
diff
changeset
|
525 |
finally show ?thesis . |
7d0ed261b712
generalize int_dvd_cancel_factor simproc to idom class
huffman
parents:
29949
diff
changeset
|
526 |
qed |
7d0ed261b712
generalize int_dvd_cancel_factor simproc to idom class
huffman
parents:
29949
diff
changeset
|
527 |
|
60516
0826b7025d07
generalized some theorems about integral domains and moved to HOL theories
haftmann
parents:
60429
diff
changeset
|
528 |
lemma square_eq_iff: "a * a = b * b \<longleftrightarrow> a = b \<or> a = - b" |
59833
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset
|
529 |
proof |
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset
|
530 |
assume "a * a = b * b" |
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset
|
531 |
then have "(a - b) * (a + b) = 0" |
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset
|
532 |
by (simp add: algebra_simps) |
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset
|
533 |
then show "a = b \<or> a = - b" |
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset
|
534 |
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
|
535 |
next |
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset
|
536 |
assume "a = b \<or> a = - b" |
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset
|
537 |
then show "a * a = b * b" by auto |
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset
|
538 |
qed |
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset
|
539 |
|
25186 | 540 |
end |
25152 | 541 |
|
64290 | 542 |
class idom_abs_sgn = idom + abs + sgn + |
543 |
assumes sgn_mult_abs: "sgn a * \<bar>a\<bar> = a" |
|
544 |
and sgn_sgn [simp]: "sgn (sgn a) = sgn a" |
|
545 |
and abs_abs [simp]: "\<bar>\<bar>a\<bar>\<bar> = \<bar>a\<bar>" |
|
546 |
and abs_0 [simp]: "\<bar>0\<bar> = 0" |
|
547 |
and sgn_0 [simp]: "sgn 0 = 0" |
|
548 |
and sgn_1 [simp]: "sgn 1 = 1" |
|
549 |
and sgn_minus_1: "sgn (- 1) = - 1" |
|
550 |
and sgn_mult: "sgn (a * b) = sgn a * sgn b" |
|
551 |
begin |
|
552 |
||
553 |
lemma sgn_eq_0_iff: |
|
554 |
"sgn a = 0 \<longleftrightarrow> a = 0" |
|
555 |
proof - |
|
556 |
{ assume "sgn a = 0" |
|
557 |
then have "sgn a * \<bar>a\<bar> = 0" |
|
558 |
by simp |
|
559 |
then have "a = 0" |
|
560 |
by (simp add: sgn_mult_abs) |
|
561 |
} then show ?thesis |
|
562 |
by auto |
|
563 |
qed |
|
564 |
||
565 |
lemma abs_eq_0_iff: |
|
566 |
"\<bar>a\<bar> = 0 \<longleftrightarrow> a = 0" |
|
567 |
proof - |
|
568 |
{ assume "\<bar>a\<bar> = 0" |
|
569 |
then have "sgn a * \<bar>a\<bar> = 0" |
|
570 |
by simp |
|
571 |
then have "a = 0" |
|
572 |
by (simp add: sgn_mult_abs) |
|
573 |
} then show ?thesis |
|
574 |
by auto |
|
575 |
qed |
|
576 |
||
577 |
lemma abs_mult_sgn: |
|
578 |
"\<bar>a\<bar> * sgn a = a" |
|
579 |
using sgn_mult_abs [of a] by (simp add: ac_simps) |
|
580 |
||
581 |
lemma abs_1 [simp]: |
|
582 |
"\<bar>1\<bar> = 1" |
|
583 |
using sgn_mult_abs [of 1] by simp |
|
584 |
||
585 |
lemma sgn_abs [simp]: |
|
586 |
"\<bar>sgn a\<bar> = of_bool (a \<noteq> 0)" |
|
587 |
using sgn_mult_abs [of "sgn a"] mult_cancel_left [of "sgn a" "\<bar>sgn a\<bar>" 1] |
|
588 |
by (auto simp add: sgn_eq_0_iff) |
|
589 |
||
590 |
lemma abs_sgn [simp]: |
|
591 |
"sgn \<bar>a\<bar> = of_bool (a \<noteq> 0)" |
|
592 |
using sgn_mult_abs [of "\<bar>a\<bar>"] mult_cancel_right [of "sgn \<bar>a\<bar>" "\<bar>a\<bar>" 1] |
|
593 |
by (auto simp add: abs_eq_0_iff) |
|
594 |
||
595 |
lemma abs_mult: |
|
596 |
"\<bar>a * b\<bar> = \<bar>a\<bar> * \<bar>b\<bar>" |
|
597 |
proof (cases "a = 0 \<or> b = 0") |
|
598 |
case True |
|
599 |
then show ?thesis |
|
600 |
by auto |
|
601 |
next |
|
602 |
case False |
|
603 |
then have *: "sgn (a * b) \<noteq> 0" |
|
604 |
by (simp add: sgn_eq_0_iff) |
|
605 |
from abs_mult_sgn [of "a * b"] abs_mult_sgn [of a] abs_mult_sgn [of b] |
|
606 |
have "\<bar>a * b\<bar> * sgn (a * b) = \<bar>a\<bar> * sgn a * \<bar>b\<bar> * sgn b" |
|
607 |
by (simp add: ac_simps) |
|
608 |
then have "\<bar>a * b\<bar> * sgn (a * b) = \<bar>a\<bar> * \<bar>b\<bar> * sgn (a * b)" |
|
609 |
by (simp add: sgn_mult ac_simps) |
|
610 |
with * show ?thesis |
|
611 |
by simp |
|
612 |
qed |
|
613 |
||
614 |
lemma sgn_minus [simp]: |
|
615 |
"sgn (- a) = - sgn a" |
|
616 |
proof - |
|
617 |
from sgn_minus_1 have "sgn (- 1 * a) = - 1 * sgn a" |
|
618 |
by (simp only: sgn_mult) |
|
619 |
then show ?thesis |
|
620 |
by simp |
|
621 |
qed |
|
622 |
||
623 |
lemma abs_minus [simp]: |
|
624 |
"\<bar>- a\<bar> = \<bar>a\<bar>" |
|
625 |
proof - |
|
626 |
have [simp]: "\<bar>- 1\<bar> = 1" |
|
627 |
using sgn_mult_abs [of "- 1"] by simp |
|
628 |
then have "\<bar>- 1 * a\<bar> = 1 * \<bar>a\<bar>" |
|
629 |
by (simp only: abs_mult) |
|
630 |
then show ?thesis |
|
631 |
by simp |
|
632 |
qed |
|
633 |
||
634 |
end |
|
635 |
||
60758 | 636 |
text \<open> |
35302 | 637 |
The theory of partially ordered rings is taken from the books: |
63325 | 638 |
\<^item> \<^emph>\<open>Lattice Theory\<close> by Garret Birkhoff, American Mathematical Society, 1979 |
639 |
\<^item> \<^emph>\<open>Partially Ordered Algebraic Systems\<close>, Pergamon Press, 1963 |
|
640 |
||
60562
24af00b010cf
Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents:
60529
diff
changeset
|
641 |
Most of the used notions can also be looked up in |
63680 | 642 |
\<^item> \<^url>\<open>http://www.mathworld.com\<close> by Eric Weisstein et. al. |
63325 | 643 |
\<^item> \<^emph>\<open>Algebra I\<close> by van der Waerden, Springer |
60758 | 644 |
\<close> |
35302 | 645 |
|
63950
cdc1e59aa513
syntactic type class for operation mod named after mod;
haftmann
parents:
63947
diff
changeset
|
646 |
text \<open>Syntactic division operator\<close> |
cdc1e59aa513
syntactic type class for operation mod named after mod;
haftmann
parents:
63947
diff
changeset
|
647 |
|
60353
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset
|
648 |
class divide = |
60429
d3d1e185cd63
uniform _ div _ as infix syntax for ring division
haftmann
parents:
60353
diff
changeset
|
649 |
fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "div" 70) |
60353
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset
|
650 |
|
69593 | 651 |
setup \<open>Sign.add_const_constraint (\<^const_name>\<open>divide\<close>, SOME \<^typ>\<open>'a \<Rightarrow> 'a \<Rightarrow> 'a\<close>)\<close> |
60353
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset
|
652 |
|
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset
|
653 |
context semiring |
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset
|
654 |
begin |
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset
|
655 |
|
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset
|
656 |
lemma [field_simps]: |
60429
d3d1e185cd63
uniform _ div _ as infix syntax for ring division
haftmann
parents:
60353
diff
changeset
|
657 |
shows distrib_left_NO_MATCH: "NO_MATCH (x div y) a \<Longrightarrow> a * (b + c) = a * b + a * c" |
d3d1e185cd63
uniform _ div _ as infix syntax for ring division
haftmann
parents:
60353
diff
changeset
|
658 |
and distrib_right_NO_MATCH: "NO_MATCH (x div y) c \<Longrightarrow> (a + b) * c = a * c + b * c" |
60353
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset
|
659 |
by (rule distrib_left distrib_right)+ |
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset
|
660 |
|
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset
|
661 |
end |
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset
|
662 |
|
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset
|
663 |
context ring |
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset
|
664 |
begin |
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset
|
665 |
|
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset
|
666 |
lemma [field_simps]: |
60429
d3d1e185cd63
uniform _ div _ as infix syntax for ring division
haftmann
parents:
60353
diff
changeset
|
667 |
shows left_diff_distrib_NO_MATCH: "NO_MATCH (x div y) c \<Longrightarrow> (a - b) * c = a * c - b * c" |
d3d1e185cd63
uniform _ div _ as infix syntax for ring division
haftmann
parents:
60353
diff
changeset
|
668 |
and right_diff_distrib_NO_MATCH: "NO_MATCH (x div y) a \<Longrightarrow> a * (b - c) = a * b - a * c" |
60353
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset
|
669 |
by (rule left_diff_distrib right_diff_distrib)+ |
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset
|
670 |
|
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset
|
671 |
end |
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset
|
672 |
|
69593 | 673 |
setup \<open>Sign.add_const_constraint (\<^const_name>\<open>divide\<close>, SOME \<^typ>\<open>'a::divide \<Rightarrow> 'a \<Rightarrow> 'a\<close>)\<close> |
60353
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset
|
674 |
|
63950
cdc1e59aa513
syntactic type class for operation mod named after mod;
haftmann
parents:
63947
diff
changeset
|
675 |
text \<open>Algebraic classes with division\<close> |
cdc1e59aa513
syntactic type class for operation mod named after mod;
haftmann
parents:
63947
diff
changeset
|
676 |
|
60353
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset
|
677 |
class semidom_divide = semidom + divide + |
64240 | 678 |
assumes nonzero_mult_div_cancel_right [simp]: "b \<noteq> 0 \<Longrightarrow> (a * b) div b = a" |
679 |
assumes div_by_0 [simp]: "a div 0 = 0" |
|
60353
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset
|
680 |
begin |
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset
|
681 |
|
64240 | 682 |
lemma nonzero_mult_div_cancel_left [simp]: "a \<noteq> 0 \<Longrightarrow> (a * b) div a = b" |
683 |
using nonzero_mult_div_cancel_right [of a b] by (simp add: ac_simps) |
|
60353
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset
|
684 |
|
60516
0826b7025d07
generalized some theorems about integral domains and moved to HOL theories
haftmann
parents:
60429
diff
changeset
|
685 |
subclass semiring_no_zero_divisors_cancel |
0826b7025d07
generalized some theorems about integral domains and moved to HOL theories
haftmann
parents:
60429
diff
changeset
|
686 |
proof |
63325 | 687 |
show *: "a * c = b * c \<longleftrightarrow> c = 0 \<or> a = b" for a b c |
688 |
proof (cases "c = 0") |
|
689 |
case True |
|
690 |
then show ?thesis by simp |
|
691 |
next |
|
692 |
case False |
|
63588 | 693 |
have "a = b" if "a * c = b * c" |
694 |
proof - |
|
695 |
from that have "a * c div c = b * c div c" |
|
63325 | 696 |
by simp |
63588 | 697 |
with False show ?thesis |
63325 | 698 |
by simp |
63588 | 699 |
qed |
63325 | 700 |
then show ?thesis by auto |
701 |
qed |
|
702 |
show "c * a = c * b \<longleftrightarrow> c = 0 \<or> a = b" for a b c |
|
703 |
using * [of a c b] by (simp add: ac_simps) |
|
60516
0826b7025d07
generalized some theorems about integral domains and moved to HOL theories
haftmann
parents:
60429
diff
changeset
|
704 |
qed |
0826b7025d07
generalized some theorems about integral domains and moved to HOL theories
haftmann
parents:
60429
diff
changeset
|
705 |
|
63325 | 706 |
lemma div_self [simp]: "a \<noteq> 0 \<Longrightarrow> a div a = 1" |
64240 | 707 |
using nonzero_mult_div_cancel_left [of a 1] by simp |
60516
0826b7025d07
generalized some theorems about integral domains and moved to HOL theories
haftmann
parents:
60429
diff
changeset
|
708 |
|
64240 | 709 |
lemma div_0 [simp]: "0 div a = 0" |
60570 | 710 |
proof (cases "a = 0") |
63325 | 711 |
case True |
712 |
then show ?thesis by simp |
|
60570 | 713 |
next |
63325 | 714 |
case False |
715 |
then have "a * 0 div a = 0" |
|
64240 | 716 |
by (rule nonzero_mult_div_cancel_left) |
60570 | 717 |
then show ?thesis by simp |
62376
85f38d5f8807
Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
hoelzl
parents:
62366
diff
changeset
|
718 |
qed |
60570 | 719 |
|
64240 | 720 |
lemma div_by_1 [simp]: "a div 1 = a" |
721 |
using nonzero_mult_div_cancel_left [of 1 a] by simp |
|
60690 | 722 |
|
64591
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64290
diff
changeset
|
723 |
lemma dvd_div_eq_0_iff: |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64290
diff
changeset
|
724 |
assumes "b dvd a" |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64290
diff
changeset
|
725 |
shows "a div b = 0 \<longleftrightarrow> a = 0" |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64290
diff
changeset
|
726 |
using assms by (elim dvdE, cases "b = 0") simp_all |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64290
diff
changeset
|
727 |
|
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64290
diff
changeset
|
728 |
lemma dvd_div_eq_cancel: |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64290
diff
changeset
|
729 |
"a div c = b div c \<Longrightarrow> c dvd a \<Longrightarrow> c dvd b \<Longrightarrow> a = b" |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64290
diff
changeset
|
730 |
by (elim dvdE, cases "c = 0") simp_all |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64290
diff
changeset
|
731 |
|
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64290
diff
changeset
|
732 |
lemma dvd_div_eq_iff: |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64290
diff
changeset
|
733 |
"c dvd a \<Longrightarrow> c dvd b \<Longrightarrow> a div c = b div c \<longleftrightarrow> a = b" |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64290
diff
changeset
|
734 |
by (elim dvdE, cases "c = 0") simp_all |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64290
diff
changeset
|
735 |
|
69661 | 736 |
lemma inj_on_mult: |
737 |
"inj_on ((*) a) A" if "a \<noteq> 0" |
|
738 |
proof (rule inj_onI) |
|
739 |
fix b c |
|
740 |
assume "a * b = a * c" |
|
741 |
then have "a * b div a = a * c div a" |
|
742 |
by (simp only:) |
|
743 |
with that show "b = c" |
|
744 |
by simp |
|
745 |
qed |
|
746 |
||
60867 | 747 |
end |
748 |
||
749 |
class idom_divide = idom + semidom_divide |
|
64591
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64290
diff
changeset
|
750 |
begin |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64290
diff
changeset
|
751 |
|
64592
7759f1766189
more fine-grained type class hierarchy for div and mod
haftmann
parents:
64591
diff
changeset
|
752 |
lemma dvd_neg_div: |
64591
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64290
diff
changeset
|
753 |
assumes "b dvd a" |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64290
diff
changeset
|
754 |
shows "- a div b = - (a div b)" |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64290
diff
changeset
|
755 |
proof (cases "b = 0") |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64290
diff
changeset
|
756 |
case True |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64290
diff
changeset
|
757 |
then show ?thesis by simp |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64290
diff
changeset
|
758 |
next |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64290
diff
changeset
|
759 |
case False |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64290
diff
changeset
|
760 |
from assms obtain c where "a = b * c" .. |
64592
7759f1766189
more fine-grained type class hierarchy for div and mod
haftmann
parents:
64591
diff
changeset
|
761 |
then have "- a div b = (b * - c) div b" |
7759f1766189
more fine-grained type class hierarchy for div and mod
haftmann
parents:
64591
diff
changeset
|
762 |
by simp |
7759f1766189
more fine-grained type class hierarchy for div and mod
haftmann
parents:
64591
diff
changeset
|
763 |
from False also have "\<dots> = - c" |
7759f1766189
more fine-grained type class hierarchy for div and mod
haftmann
parents:
64591
diff
changeset
|
764 |
by (rule nonzero_mult_div_cancel_left) |
7759f1766189
more fine-grained type class hierarchy for div and mod
haftmann
parents:
64591
diff
changeset
|
765 |
with False \<open>a = b * c\<close> show ?thesis |
64591
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64290
diff
changeset
|
766 |
by simp |
64592
7759f1766189
more fine-grained type class hierarchy for div and mod
haftmann
parents:
64591
diff
changeset
|
767 |
qed |
7759f1766189
more fine-grained type class hierarchy for div and mod
haftmann
parents:
64591
diff
changeset
|
768 |
|
7759f1766189
more fine-grained type class hierarchy for div and mod
haftmann
parents:
64591
diff
changeset
|
769 |
lemma dvd_div_neg: |
7759f1766189
more fine-grained type class hierarchy for div and mod
haftmann
parents:
64591
diff
changeset
|
770 |
assumes "b dvd a" |
7759f1766189
more fine-grained type class hierarchy for div and mod
haftmann
parents:
64591
diff
changeset
|
771 |
shows "a div - b = - (a div b)" |
7759f1766189
more fine-grained type class hierarchy for div and mod
haftmann
parents:
64591
diff
changeset
|
772 |
proof (cases "b = 0") |
7759f1766189
more fine-grained type class hierarchy for div and mod
haftmann
parents:
64591
diff
changeset
|
773 |
case True |
7759f1766189
more fine-grained type class hierarchy for div and mod
haftmann
parents:
64591
diff
changeset
|
774 |
then show ?thesis by simp |
7759f1766189
more fine-grained type class hierarchy for div and mod
haftmann
parents:
64591
diff
changeset
|
775 |
next |
7759f1766189
more fine-grained type class hierarchy for div and mod
haftmann
parents:
64591
diff
changeset
|
776 |
case False |
7759f1766189
more fine-grained type class hierarchy for div and mod
haftmann
parents:
64591
diff
changeset
|
777 |
then have "- b \<noteq> 0" |
7759f1766189
more fine-grained type class hierarchy for div and mod
haftmann
parents:
64591
diff
changeset
|
778 |
by simp |
7759f1766189
more fine-grained type class hierarchy for div and mod
haftmann
parents:
64591
diff
changeset
|
779 |
from assms obtain c where "a = b * c" .. |
7759f1766189
more fine-grained type class hierarchy for div and mod
haftmann
parents:
64591
diff
changeset
|
780 |
then have "a div - b = (- b * - c) div - b" |
7759f1766189
more fine-grained type class hierarchy for div and mod
haftmann
parents:
64591
diff
changeset
|
781 |
by simp |
7759f1766189
more fine-grained type class hierarchy for div and mod
haftmann
parents:
64591
diff
changeset
|
782 |
from \<open>- b \<noteq> 0\<close> also have "\<dots> = - c" |
7759f1766189
more fine-grained type class hierarchy for div and mod
haftmann
parents:
64591
diff
changeset
|
783 |
by (rule nonzero_mult_div_cancel_left) |
7759f1766189
more fine-grained type class hierarchy for div and mod
haftmann
parents:
64591
diff
changeset
|
784 |
with False \<open>a = b * c\<close> show ?thesis |
64591
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64290
diff
changeset
|
785 |
by simp |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64290
diff
changeset
|
786 |
qed |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64290
diff
changeset
|
787 |
|
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64290
diff
changeset
|
788 |
end |
60867 | 789 |
|
790 |
class algebraic_semidom = semidom_divide |
|
791 |
begin |
|
792 |
||
793 |
text \<open> |
|
69593 | 794 |
Class \<^class>\<open>algebraic_semidom\<close> enriches a integral domain |
60867 | 795 |
by notions from algebra, like units in a ring. |
796 |
It is a separate class to avoid spoiling fields with notions |
|
797 |
which are degenerated there. |
|
798 |
\<close> |
|
799 |
||
60690 | 800 |
lemma dvd_times_left_cancel_iff [simp]: |
801 |
assumes "a \<noteq> 0" |
|
63588 | 802 |
shows "a * b dvd a * c \<longleftrightarrow> b dvd c" |
803 |
(is "?lhs \<longleftrightarrow> ?rhs") |
|
60690 | 804 |
proof |
63588 | 805 |
assume ?lhs |
63325 | 806 |
then obtain d where "a * c = a * b * d" .. |
60690 | 807 |
with assms have "c = b * d" by (simp add: ac_simps) |
63588 | 808 |
then show ?rhs .. |
60690 | 809 |
next |
63588 | 810 |
assume ?rhs |
63325 | 811 |
then obtain d where "c = b * d" .. |
60690 | 812 |
then have "a * c = a * b * d" by (simp add: ac_simps) |
63588 | 813 |
then show ?lhs .. |
60690 | 814 |
qed |
62376
85f38d5f8807
Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
hoelzl
parents:
62366
diff
changeset
|
815 |
|
60690 | 816 |
lemma dvd_times_right_cancel_iff [simp]: |
817 |
assumes "a \<noteq> 0" |
|
63588 | 818 |
shows "b * a dvd c * a \<longleftrightarrow> b dvd c" |
63325 | 819 |
using dvd_times_left_cancel_iff [of a b c] assms by (simp add: ac_simps) |
62376
85f38d5f8807
Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
hoelzl
parents:
62366
diff
changeset
|
820 |
|
60690 | 821 |
lemma div_dvd_iff_mult: |
822 |
assumes "b \<noteq> 0" and "b dvd a" |
|
823 |
shows "a div b dvd c \<longleftrightarrow> a dvd c * b" |
|
824 |
proof - |
|
825 |
from \<open>b dvd a\<close> obtain d where "a = b * d" .. |
|
826 |
with \<open>b \<noteq> 0\<close> show ?thesis by (simp add: ac_simps) |
|
827 |
qed |
|
828 |
||
829 |
lemma dvd_div_iff_mult: |
|
830 |
assumes "c \<noteq> 0" and "c dvd b" |
|
831 |
shows "a dvd b div c \<longleftrightarrow> a * c dvd b" |
|
832 |
proof - |
|
833 |
from \<open>c dvd b\<close> obtain d where "b = c * d" .. |
|
834 |
with \<open>c \<noteq> 0\<close> show ?thesis by (simp add: mult.commute [of a]) |
|
835 |
qed |
|
836 |
||
60867 | 837 |
lemma div_dvd_div [simp]: |
838 |
assumes "a dvd b" and "a dvd c" |
|
839 |
shows "b div a dvd c div a \<longleftrightarrow> b dvd c" |
|
840 |
proof (cases "a = 0") |
|
63325 | 841 |
case True |
842 |
with assms show ?thesis by simp |
|
60867 | 843 |
next |
844 |
case False |
|
845 |
moreover from assms obtain k l where "b = a * k" and "c = a * l" |
|
846 |
by (auto elim!: dvdE) |
|
847 |
ultimately show ?thesis by simp |
|
848 |
qed |
|
60353
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset
|
849 |
|
60867 | 850 |
lemma div_add [simp]: |
851 |
assumes "c dvd a" and "c dvd b" |
|
852 |
shows "(a + b) div c = a div c + b div c" |
|
853 |
proof (cases "c = 0") |
|
63325 | 854 |
case True |
855 |
then show ?thesis by simp |
|
60867 | 856 |
next |
857 |
case False |
|
858 |
moreover from assms obtain k l where "a = c * k" and "b = c * l" |
|
859 |
by (auto elim!: dvdE) |
|
860 |
moreover have "c * k + c * l = c * (k + l)" |
|
861 |
by (simp add: algebra_simps) |
|
862 |
ultimately show ?thesis |
|
863 |
by simp |
|
864 |
qed |
|
60517
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
865 |
|
60867 | 866 |
lemma div_mult_div_if_dvd: |
867 |
assumes "b dvd a" and "d dvd c" |
|
868 |
shows "(a div b) * (c div d) = (a * c) div (b * d)" |
|
869 |
proof (cases "b = 0 \<or> c = 0") |
|
63325 | 870 |
case True |
871 |
with assms show ?thesis by auto |
|
60867 | 872 |
next |
873 |
case False |
|
874 |
moreover from assms obtain k l where "a = b * k" and "c = d * l" |
|
875 |
by (auto elim!: dvdE) |
|
876 |
moreover have "b * k * (d * l) div (b * d) = (b * d) * (k * l) div (b * d)" |
|
877 |
by (simp add: ac_simps) |
|
878 |
ultimately show ?thesis by simp |
|
879 |
qed |
|
880 |
||
881 |
lemma dvd_div_eq_mult: |
|
882 |
assumes "a \<noteq> 0" and "a dvd b" |
|
883 |
shows "b div a = c \<longleftrightarrow> b = c * a" |
|
63588 | 884 |
(is "?lhs \<longleftrightarrow> ?rhs") |
60867 | 885 |
proof |
63588 | 886 |
assume ?rhs |
887 |
then show ?lhs by (simp add: assms) |
|
60867 | 888 |
next |
63588 | 889 |
assume ?lhs |
60867 | 890 |
then have "b div a * a = c * a" by simp |
63325 | 891 |
moreover from assms have "b div a * a = b" |
60867 | 892 |
by (auto elim!: dvdE simp add: ac_simps) |
63588 | 893 |
ultimately show ?rhs by simp |
60867 | 894 |
qed |
60688
01488b559910
avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset
|
895 |
|
63325 | 896 |
lemma dvd_div_mult_self [simp]: "a dvd b \<Longrightarrow> b div a * a = b" |
60517
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
897 |
by (cases "a = 0") (auto elim: dvdE simp add: ac_simps) |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
898 |
|
63325 | 899 |
lemma dvd_mult_div_cancel [simp]: "a dvd b \<Longrightarrow> a * (b div a) = b" |
60517
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
900 |
using dvd_div_mult_self [of a b] by (simp add: ac_simps) |
60562
24af00b010cf
Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents:
60529
diff
changeset
|
901 |
|
60517
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
902 |
lemma div_mult_swap: |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
903 |
assumes "c dvd b" |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
904 |
shows "a * (b div c) = (a * b) div c" |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
905 |
proof (cases "c = 0") |
63325 | 906 |
case True |
907 |
then show ?thesis by simp |
|
60517
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
908 |
next |
63325 | 909 |
case False |
910 |
from assms obtain d where "b = c * d" .. |
|
60517
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
911 |
moreover from False have "a * divide (d * c) c = ((a * d) * c) div c" |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
912 |
by simp |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
913 |
ultimately show ?thesis by (simp add: ac_simps) |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
914 |
qed |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
915 |
|
63325 | 916 |
lemma dvd_div_mult: "c dvd b \<Longrightarrow> b div c * a = (b * a) div c" |
917 |
using div_mult_swap [of c b a] by (simp add: ac_simps) |
|
60517
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
918 |
|
60570 | 919 |
lemma dvd_div_mult2_eq: |
920 |
assumes "b * c dvd a" |
|
921 |
shows "a div (b * c) = a div b div c" |
|
63325 | 922 |
proof - |
923 |
from assms obtain k where "a = b * c * k" .. |
|
60570 | 924 |
then show ?thesis |
925 |
by (cases "b = 0 \<or> c = 0") (auto, simp add: ac_simps) |
|
926 |
qed |
|
927 |
||
60867 | 928 |
lemma dvd_div_div_eq_mult: |
929 |
assumes "a \<noteq> 0" "c \<noteq> 0" and "a dvd b" "c dvd d" |
|
63588 | 930 |
shows "b div a = d div c \<longleftrightarrow> b * c = a * d" |
931 |
(is "?lhs \<longleftrightarrow> ?rhs") |
|
60867 | 932 |
proof - |
933 |
from assms have "a * c \<noteq> 0" by simp |
|
63588 | 934 |
then have "?lhs \<longleftrightarrow> b div a * (a * c) = d div c * (a * c)" |
60867 | 935 |
by simp |
936 |
also have "\<dots> \<longleftrightarrow> (a * (b div a)) * c = (c * (d div c)) * a" |
|
937 |
by (simp add: ac_simps) |
|
938 |
also have "\<dots> \<longleftrightarrow> (a * b div a) * c = (c * d div c) * a" |
|
939 |
using assms by (simp add: div_mult_swap) |
|
63588 | 940 |
also have "\<dots> \<longleftrightarrow> ?rhs" |
60867 | 941 |
using assms by (simp add: ac_simps) |
942 |
finally show ?thesis . |
|
943 |
qed |
|
944 |
||
63359 | 945 |
lemma dvd_mult_imp_div: |
946 |
assumes "a * c dvd b" |
|
947 |
shows "a dvd b div c" |
|
948 |
proof (cases "c = 0") |
|
949 |
case True then show ?thesis by simp |
|
950 |
next |
|
951 |
case False |
|
952 |
from \<open>a * c dvd b\<close> obtain d where "b = a * c * d" .. |
|
63588 | 953 |
with False show ?thesis |
954 |
by (simp add: mult.commute [of a] mult.assoc) |
|
63359 | 955 |
qed |
956 |
||
64592
7759f1766189
more fine-grained type class hierarchy for div and mod
haftmann
parents:
64591
diff
changeset
|
957 |
lemma div_div_eq_right: |
7759f1766189
more fine-grained type class hierarchy for div and mod
haftmann
parents:
64591
diff
changeset
|
958 |
assumes "c dvd b" "b dvd a" |
7759f1766189
more fine-grained type class hierarchy for div and mod
haftmann
parents:
64591
diff
changeset
|
959 |
shows "a div (b div c) = a div b * c" |
7759f1766189
more fine-grained type class hierarchy for div and mod
haftmann
parents:
64591
diff
changeset
|
960 |
proof (cases "c = 0 \<or> b = 0") |
7759f1766189
more fine-grained type class hierarchy for div and mod
haftmann
parents:
64591
diff
changeset
|
961 |
case True |
7759f1766189
more fine-grained type class hierarchy for div and mod
haftmann
parents:
64591
diff
changeset
|
962 |
then show ?thesis |
7759f1766189
more fine-grained type class hierarchy for div and mod
haftmann
parents:
64591
diff
changeset
|
963 |
by auto |
7759f1766189
more fine-grained type class hierarchy for div and mod
haftmann
parents:
64591
diff
changeset
|
964 |
next |
7759f1766189
more fine-grained type class hierarchy for div and mod
haftmann
parents:
64591
diff
changeset
|
965 |
case False |
7759f1766189
more fine-grained type class hierarchy for div and mod
haftmann
parents:
64591
diff
changeset
|
966 |
from assms obtain r s where "b = c * r" and "a = c * r * s" |
7759f1766189
more fine-grained type class hierarchy for div and mod
haftmann
parents:
64591
diff
changeset
|
967 |
by (blast elim: dvdE) |
7759f1766189
more fine-grained type class hierarchy for div and mod
haftmann
parents:
64591
diff
changeset
|
968 |
moreover with False have "r \<noteq> 0" |
7759f1766189
more fine-grained type class hierarchy for div and mod
haftmann
parents:
64591
diff
changeset
|
969 |
by auto |
7759f1766189
more fine-grained type class hierarchy for div and mod
haftmann
parents:
64591
diff
changeset
|
970 |
ultimately show ?thesis using False |
7759f1766189
more fine-grained type class hierarchy for div and mod
haftmann
parents:
64591
diff
changeset
|
971 |
by simp (simp add: mult.commute [of _ r] mult.assoc mult.commute [of c]) |
7759f1766189
more fine-grained type class hierarchy for div and mod
haftmann
parents:
64591
diff
changeset
|
972 |
qed |
7759f1766189
more fine-grained type class hierarchy for div and mod
haftmann
parents:
64591
diff
changeset
|
973 |
|
7759f1766189
more fine-grained type class hierarchy for div and mod
haftmann
parents:
64591
diff
changeset
|
974 |
lemma div_div_div_same: |
7759f1766189
more fine-grained type class hierarchy for div and mod
haftmann
parents:
64591
diff
changeset
|
975 |
assumes "d dvd b" "b dvd a" |
7759f1766189
more fine-grained type class hierarchy for div and mod
haftmann
parents:
64591
diff
changeset
|
976 |
shows "(a div d) div (b div d) = a div b" |
7759f1766189
more fine-grained type class hierarchy for div and mod
haftmann
parents:
64591
diff
changeset
|
977 |
proof (cases "b = 0 \<or> d = 0") |
7759f1766189
more fine-grained type class hierarchy for div and mod
haftmann
parents:
64591
diff
changeset
|
978 |
case True |
7759f1766189
more fine-grained type class hierarchy for div and mod
haftmann
parents:
64591
diff
changeset
|
979 |
with assms show ?thesis |
7759f1766189
more fine-grained type class hierarchy for div and mod
haftmann
parents:
64591
diff
changeset
|
980 |
by auto |
7759f1766189
more fine-grained type class hierarchy for div and mod
haftmann
parents:
64591
diff
changeset
|
981 |
next |
7759f1766189
more fine-grained type class hierarchy for div and mod
haftmann
parents:
64591
diff
changeset
|
982 |
case False |
7759f1766189
more fine-grained type class hierarchy for div and mod
haftmann
parents:
64591
diff
changeset
|
983 |
from assms obtain r s |
7759f1766189
more fine-grained type class hierarchy for div and mod
haftmann
parents:
64591
diff
changeset
|
984 |
where "a = d * r * s" and "b = d * r" |
7759f1766189
more fine-grained type class hierarchy for div and mod
haftmann
parents:
64591
diff
changeset
|
985 |
by (blast elim: dvdE) |
7759f1766189
more fine-grained type class hierarchy for div and mod
haftmann
parents:
64591
diff
changeset
|
986 |
with False show ?thesis |
7759f1766189
more fine-grained type class hierarchy for div and mod
haftmann
parents:
64591
diff
changeset
|
987 |
by simp (simp add: ac_simps) |
7759f1766189
more fine-grained type class hierarchy for div and mod
haftmann
parents:
64591
diff
changeset
|
988 |
qed |
7759f1766189
more fine-grained type class hierarchy for div and mod
haftmann
parents:
64591
diff
changeset
|
989 |
|
60562
24af00b010cf
Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents:
60529
diff
changeset
|
990 |
|
60517
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
991 |
text \<open>Units: invertible elements in a ring\<close> |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
992 |
|
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
993 |
abbreviation is_unit :: "'a \<Rightarrow> bool" |
63325 | 994 |
where "is_unit a \<equiv> a dvd 1" |
60517
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
995 |
|
63325 | 996 |
lemma not_is_unit_0 [simp]: "\<not> is_unit 0" |
60517
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
997 |
by simp |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
998 |
|
63325 | 999 |
lemma unit_imp_dvd [dest]: "is_unit b \<Longrightarrow> b dvd a" |
60517
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
1000 |
by (rule dvd_trans [of _ 1]) simp_all |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
1001 |
|
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
1002 |
lemma unit_dvdE: |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
1003 |
assumes "is_unit a" |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
1004 |
obtains c where "a \<noteq> 0" and "b = a * c" |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
1005 |
proof - |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
1006 |
from assms have "a dvd b" by auto |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
1007 |
then obtain c where "b = a * c" .. |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
1008 |
moreover from assms have "a \<noteq> 0" by auto |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
1009 |
ultimately show thesis using that by blast |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
1010 |
qed |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
1011 |
|
63325 | 1012 |
lemma dvd_unit_imp_unit: "a dvd b \<Longrightarrow> is_unit b \<Longrightarrow> is_unit a" |
60517
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
1013 |
by (rule dvd_trans) |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
1014 |
|
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
1015 |
lemma unit_div_1_unit [simp, intro]: |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
1016 |
assumes "is_unit a" |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
1017 |
shows "is_unit (1 div a)" |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
1018 |
proof - |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
1019 |
from assms have "1 = 1 div a * a" by simp |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
1020 |
then show "is_unit (1 div a)" by (rule dvdI) |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
1021 |
qed |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
1022 |
|
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
1023 |
lemma is_unitE [elim?]: |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
1024 |
assumes "is_unit a" |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
1025 |
obtains b where "a \<noteq> 0" and "b \<noteq> 0" |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
1026 |
and "is_unit b" and "1 div a = b" and "1 div b = a" |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
1027 |
and "a * b = 1" and "c div a = c * b" |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
1028 |
proof (rule that) |
63040 | 1029 |
define b where "b = 1 div a" |
60517
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
1030 |
then show "1 div a = b" by simp |
63325 | 1031 |
from assms b_def show "is_unit b" by simp |
1032 |
with assms show "a \<noteq> 0" and "b \<noteq> 0" by auto |
|
1033 |
from assms b_def show "a * b = 1" by simp |
|
60517
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
1034 |
then have "1 = a * b" .. |
60758 | 1035 |
with b_def \<open>b \<noteq> 0\<close> show "1 div b = a" by simp |
63325 | 1036 |
from assms have "a dvd c" .. |
60517
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
1037 |
then obtain d where "c = a * d" .. |
60758 | 1038 |
with \<open>a \<noteq> 0\<close> \<open>a * b = 1\<close> show "c div a = c * b" |
60517
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
1039 |
by (simp add: mult.assoc mult.left_commute [of a]) |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
1040 |
qed |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
1041 |
|
63325 | 1042 |
lemma unit_prod [intro]: "is_unit a \<Longrightarrow> is_unit b \<Longrightarrow> is_unit (a * b)" |
60562
24af00b010cf
Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents:
60529
diff
changeset
|
1043 |
by (subst mult_1_left [of 1, symmetric]) (rule mult_dvd_mono) |
24af00b010cf
Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents:
60529
diff
changeset
|
1044 |
|
63325 | 1045 |
lemma is_unit_mult_iff: "is_unit (a * b) \<longleftrightarrow> is_unit a \<and> is_unit b" |
62366 | 1046 |
by (auto dest: dvd_mult_left dvd_mult_right) |
1047 |
||
63325 | 1048 |
lemma unit_div [intro]: "is_unit a \<Longrightarrow> is_unit b \<Longrightarrow> is_unit (a div b)" |
60517
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
1049 |
by (erule is_unitE [of b a]) (simp add: ac_simps unit_prod) |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
1050 |
|
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
1051 |
lemma mult_unit_dvd_iff: |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
1052 |
assumes "is_unit b" |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
1053 |
shows "a * b dvd c \<longleftrightarrow> a dvd c" |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
1054 |
proof |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
1055 |
assume "a * b dvd c" |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
1056 |
with assms show "a dvd c" |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
1057 |
by (simp add: dvd_mult_left) |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
1058 |
next |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
1059 |
assume "a dvd c" |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
1060 |
then obtain k where "c = a * k" .. |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
1061 |
with assms have "c = (a * b) * (1 div b * k)" |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
1062 |
by (simp add: mult_ac) |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
1063 |
then show "a * b dvd c" by (rule dvdI) |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
1064 |
qed |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
1065 |
|
63924 | 1066 |
lemma mult_unit_dvd_iff': "is_unit a \<Longrightarrow> (a * b) dvd c \<longleftrightarrow> b dvd c" |
1067 |
using mult_unit_dvd_iff [of a b c] by (simp add: ac_simps) |
|
1068 |
||
60517
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
1069 |
lemma dvd_mult_unit_iff: |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
1070 |
assumes "is_unit b" |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
1071 |
shows "a dvd c * b \<longleftrightarrow> a dvd c" |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
1072 |
proof |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
1073 |
assume "a dvd c * b" |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
1074 |
with assms have "c * b dvd c * (b * (1 div b))" |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
1075 |
by (subst mult_assoc [symmetric]) simp |
63325 | 1076 |
also from assms have "b * (1 div b) = 1" |
1077 |
by (rule is_unitE) simp |
|
60517
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
1078 |
finally have "c * b dvd c" by simp |
60758 | 1079 |
with \<open>a dvd c * b\<close> show "a dvd c" by (rule dvd_trans) |
60517
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
1080 |
next |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
1081 |
assume "a dvd c" |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
1082 |
then show "a dvd c * b" by simp |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
1083 |
qed |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
1084 |
|
63924 | 1085 |
lemma dvd_mult_unit_iff': "is_unit b \<Longrightarrow> a dvd b * c \<longleftrightarrow> a dvd c" |
1086 |
using dvd_mult_unit_iff [of b a c] by (simp add: ac_simps) |
|
1087 |
||
63325 | 1088 |
lemma div_unit_dvd_iff: "is_unit b \<Longrightarrow> a div b dvd c \<longleftrightarrow> a dvd c" |
60517
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
1089 |
by (erule is_unitE [of _ a]) (auto simp add: mult_unit_dvd_iff) |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
1090 |
|
63325 | 1091 |
lemma dvd_div_unit_iff: "is_unit b \<Longrightarrow> a dvd c div b \<longleftrightarrow> a dvd c" |
60517
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
1092 |
by (erule is_unitE [of _ c]) (simp add: dvd_mult_unit_iff) |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
1093 |
|
63924 | 1094 |
lemmas unit_dvd_iff = mult_unit_dvd_iff mult_unit_dvd_iff' |
1095 |
dvd_mult_unit_iff dvd_mult_unit_iff' |
|
1096 |
div_unit_dvd_iff dvd_div_unit_iff (* FIXME consider named_theorems *) |
|
60517
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
1097 |
|
63325 | 1098 |
lemma unit_mult_div_div [simp]: "is_unit a \<Longrightarrow> b * (1 div a) = b div a" |
60517
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
1099 |
by (erule is_unitE [of _ b]) simp |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
1100 |
|
63325 | 1101 |
lemma unit_div_mult_self [simp]: "is_unit a \<Longrightarrow> b div a * a = b" |
60517
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
1102 |
by (rule dvd_div_mult_self) auto |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
1103 |
|
63325 | 1104 |
lemma unit_div_1_div_1 [simp]: "is_unit a \<Longrightarrow> 1 div (1 div a) = a" |
60517
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
1105 |
by (erule is_unitE) simp |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
1106 |
|
63325 | 1107 |
lemma unit_div_mult_swap: "is_unit c \<Longrightarrow> a * (b div c) = (a * b) div c" |
60517
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
1108 |
by (erule unit_dvdE [of _ b]) (simp add: mult.left_commute [of _ c]) |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
1109 |
|
63325 | 1110 |
lemma unit_div_commute: "is_unit b \<Longrightarrow> (a div b) * c = (a * c) div b" |
60517
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
1111 |
using unit_div_mult_swap [of b c a] by (simp add: ac_simps) |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
1112 |
|
63325 | 1113 |
lemma unit_eq_div1: "is_unit b \<Longrightarrow> a div b = c \<longleftrightarrow> a = c * b" |
60517
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
1114 |
by (auto elim: is_unitE) |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
1115 |
|
63325 | 1116 |
lemma unit_eq_div2: "is_unit b \<Longrightarrow> a = c div b \<longleftrightarrow> a * b = c" |
60517
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
1117 |
using unit_eq_div1 [of b c a] by auto |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
1118 |
|
63325 | 1119 |
lemma unit_mult_left_cancel: "is_unit a \<Longrightarrow> a * b = a * c \<longleftrightarrow> b = c" |
1120 |
using mult_cancel_left [of a b c] by auto |
|
60517
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
1121 |
|
63325 | 1122 |
lemma unit_mult_right_cancel: "is_unit a \<Longrightarrow> b * a = c * a \<longleftrightarrow> b = c" |
60517
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
1123 |
using unit_mult_left_cancel [of a b c] by (auto simp add: ac_simps) |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
1124 |
|
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
1125 |
lemma unit_div_cancel: |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
1126 |
assumes "is_unit a" |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
1127 |
shows "b div a = c div a \<longleftrightarrow> b = c" |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
1128 |
proof - |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
1129 |
from assms have "is_unit (1 div a)" by simp |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
1130 |
then have "b * (1 div a) = c * (1 div a) \<longleftrightarrow> b = c" |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
1131 |
by (rule unit_mult_right_cancel) |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
1132 |
with assms show ?thesis by simp |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
1133 |
qed |
60562
24af00b010cf
Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents:
60529
diff
changeset
|
1134 |
|
60570 | 1135 |
lemma is_unit_div_mult2_eq: |
1136 |
assumes "is_unit b" and "is_unit c" |
|
1137 |
shows "a div (b * c) = a div b div c" |
|
1138 |
proof - |
|
63325 | 1139 |
from assms have "is_unit (b * c)" |
1140 |
by (simp add: unit_prod) |
|
60570 | 1141 |
then have "b * c dvd a" |
1142 |
by (rule unit_imp_dvd) |
|
1143 |
then show ?thesis |
|
1144 |
by (rule dvd_div_mult2_eq) |
|
1145 |
qed |
|
1146 |
||
60562
24af00b010cf
Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents:
60529
diff
changeset
|
1147 |
lemmas unit_simps = mult_unit_dvd_iff div_unit_dvd_iff dvd_mult_unit_iff |
60517
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
1148 |
dvd_div_unit_iff unit_div_mult_swap unit_div_commute |
60562
24af00b010cf
Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents:
60529
diff
changeset
|
1149 |
unit_mult_left_cancel unit_mult_right_cancel unit_div_cancel |
60517
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
1150 |
unit_eq_div1 unit_eq_div2 |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
1151 |
|
64240 | 1152 |
lemma is_unit_div_mult_cancel_left: |
60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1153 |
assumes "a \<noteq> 0" and "is_unit b" |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1154 |
shows "a div (a * b) = 1 div b" |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1155 |
proof - |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1156 |
from assms have "a div (a * b) = a div a div b" |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1157 |
by (simp add: mult_unit_dvd_iff dvd_div_mult2_eq) |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1158 |
with assms show ?thesis by simp |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1159 |
qed |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1160 |
|
64240 | 1161 |
lemma is_unit_div_mult_cancel_right: |
60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1162 |
assumes "a \<noteq> 0" and "is_unit b" |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1163 |
shows "a div (b * a) = 1 div b" |
64240 | 1164 |
using assms is_unit_div_mult_cancel_left [of a b] by (simp add: ac_simps) |
60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1165 |
|
64591
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64290
diff
changeset
|
1166 |
lemma unit_div_eq_0_iff: |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64290
diff
changeset
|
1167 |
assumes "is_unit b" |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64290
diff
changeset
|
1168 |
shows "a div b = 0 \<longleftrightarrow> a = 0" |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64290
diff
changeset
|
1169 |
by (rule dvd_div_eq_0_iff) (insert assms, auto) |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64290
diff
changeset
|
1170 |
|
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64290
diff
changeset
|
1171 |
lemma div_mult_unit2: |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64290
diff
changeset
|
1172 |
"is_unit c \<Longrightarrow> b dvd a \<Longrightarrow> a div (b * c) = a div b div c" |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64290
diff
changeset
|
1173 |
by (rule dvd_div_mult2_eq) (simp_all add: mult_unit_dvd_iff) |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64290
diff
changeset
|
1174 |
|
67051 | 1175 |
|
1176 |
text \<open>Coprimality\<close> |
|
1177 |
||
1178 |
definition coprime :: "'a \<Rightarrow> 'a \<Rightarrow> bool" |
|
1179 |
where "coprime a b \<longleftrightarrow> (\<forall>c. c dvd a \<longrightarrow> c dvd b \<longrightarrow> is_unit c)" |
|
1180 |
||
1181 |
lemma coprimeI: |
|
1182 |
assumes "\<And>c. c dvd a \<Longrightarrow> c dvd b \<Longrightarrow> is_unit c" |
|
1183 |
shows "coprime a b" |
|
1184 |
using assms by (auto simp: coprime_def) |
|
1185 |
||
1186 |
lemma not_coprimeI: |
|
1187 |
assumes "c dvd a" and "c dvd b" and "\<not> is_unit c" |
|
1188 |
shows "\<not> coprime a b" |
|
1189 |
using assms by (auto simp: coprime_def) |
|
1190 |
||
1191 |
lemma coprime_common_divisor: |
|
1192 |
"is_unit c" if "coprime a b" and "c dvd a" and "c dvd b" |
|
1193 |
using that by (auto simp: coprime_def) |
|
1194 |
||
1195 |
lemma not_coprimeE: |
|
1196 |
assumes "\<not> coprime a b" |
|
1197 |
obtains c where "c dvd a" and "c dvd b" and "\<not> is_unit c" |
|
1198 |
using assms by (auto simp: coprime_def) |
|
1199 |
||
1200 |
lemma coprime_imp_coprime: |
|
1201 |
"coprime a b" if "coprime c d" |
|
1202 |
and "\<And>e. \<not> is_unit e \<Longrightarrow> e dvd a \<Longrightarrow> e dvd b \<Longrightarrow> e dvd c" |
|
1203 |
and "\<And>e. \<not> is_unit e \<Longrightarrow> e dvd a \<Longrightarrow> e dvd b \<Longrightarrow> e dvd d" |
|
1204 |
proof (rule coprimeI) |
|
1205 |
fix e |
|
1206 |
assume "e dvd a" and "e dvd b" |
|
1207 |
with that have "e dvd c" and "e dvd d" |
|
1208 |
by (auto intro: dvd_trans) |
|
1209 |
with \<open>coprime c d\<close> show "is_unit e" |
|
1210 |
by (rule coprime_common_divisor) |
|
1211 |
qed |
|
1212 |
||
1213 |
lemma coprime_divisors: |
|
1214 |
"coprime a b" if "a dvd c" "b dvd d" and "coprime c d" |
|
1215 |
using \<open>coprime c d\<close> proof (rule coprime_imp_coprime) |
|
1216 |
fix e |
|
1217 |
assume "e dvd a" then show "e dvd c" |
|
1218 |
using \<open>a dvd c\<close> by (rule dvd_trans) |
|
1219 |
assume "e dvd b" then show "e dvd d" |
|
1220 |
using \<open>b dvd d\<close> by (rule dvd_trans) |
|
1221 |
qed |
|
1222 |
||
1223 |
lemma coprime_self [simp]: |
|
1224 |
"coprime a a \<longleftrightarrow> is_unit a" (is "?P \<longleftrightarrow> ?Q") |
|
1225 |
proof |
|
1226 |
assume ?P |
|
1227 |
then show ?Q |
|
1228 |
by (rule coprime_common_divisor) simp_all |
|
1229 |
next |
|
1230 |
assume ?Q |
|
1231 |
show ?P |
|
1232 |
by (rule coprimeI) (erule dvd_unit_imp_unit, rule \<open>?Q\<close>) |
|
1233 |
qed |
|
1234 |
||
1235 |
lemma coprime_commute [ac_simps]: |
|
1236 |
"coprime b a \<longleftrightarrow> coprime a b" |
|
1237 |
unfolding coprime_def by auto |
|
1238 |
||
1239 |
lemma is_unit_left_imp_coprime: |
|
1240 |
"coprime a b" if "is_unit a" |
|
1241 |
proof (rule coprimeI) |
|
1242 |
fix c |
|
1243 |
assume "c dvd a" |
|
1244 |
with that show "is_unit c" |
|
1245 |
by (auto intro: dvd_unit_imp_unit) |
|
1246 |
qed |
|
1247 |
||
1248 |
lemma is_unit_right_imp_coprime: |
|
1249 |
"coprime a b" if "is_unit b" |
|
1250 |
using that is_unit_left_imp_coprime [of b a] by (simp add: ac_simps) |
|
1251 |
||
1252 |
lemma coprime_1_left [simp]: |
|
1253 |
"coprime 1 a" |
|
1254 |
by (rule coprimeI) |
|
1255 |
||
1256 |
lemma coprime_1_right [simp]: |
|
1257 |
"coprime a 1" |
|
1258 |
by (rule coprimeI) |
|
1259 |
||
1260 |
lemma coprime_0_left_iff [simp]: |
|
1261 |
"coprime 0 a \<longleftrightarrow> is_unit a" |
|
1262 |
by (auto intro: coprimeI dvd_unit_imp_unit coprime_common_divisor [of 0 a a]) |
|
1263 |
||
1264 |
lemma coprime_0_right_iff [simp]: |
|
1265 |
"coprime a 0 \<longleftrightarrow> is_unit a" |
|
1266 |
using coprime_0_left_iff [of a] by (simp add: ac_simps) |
|
1267 |
||
1268 |
lemma coprime_mult_self_left_iff [simp]: |
|
1269 |
"coprime (c * a) (c * b) \<longleftrightarrow> is_unit c \<and> coprime a b" |
|
1270 |
by (auto intro: coprime_common_divisor) |
|
1271 |
(rule coprimeI, auto intro: coprime_common_divisor simp add: dvd_mult_unit_iff')+ |
|
1272 |
||
1273 |
lemma coprime_mult_self_right_iff [simp]: |
|
1274 |
"coprime (a * c) (b * c) \<longleftrightarrow> is_unit c \<and> coprime a b" |
|
1275 |
using coprime_mult_self_left_iff [of c a b] by (simp add: ac_simps) |
|
1276 |
||
67234
ab10ea1d6fd0
Some lemmas on complex numbers and coprimality
eberlm <eberlm@in.tum.de>
parents:
67226
diff
changeset
|
1277 |
lemma coprime_absorb_left: |
ab10ea1d6fd0
Some lemmas on complex numbers and coprimality
eberlm <eberlm@in.tum.de>
parents:
67226
diff
changeset
|
1278 |
assumes "x dvd y" |
ab10ea1d6fd0
Some lemmas on complex numbers and coprimality
eberlm <eberlm@in.tum.de>
parents:
67226
diff
changeset
|
1279 |
shows "coprime x y \<longleftrightarrow> is_unit x" |
ab10ea1d6fd0
Some lemmas on complex numbers and coprimality
eberlm <eberlm@in.tum.de>
parents:
67226
diff
changeset
|
1280 |
using assms coprime_common_divisor is_unit_left_imp_coprime by auto |
ab10ea1d6fd0
Some lemmas on complex numbers and coprimality
eberlm <eberlm@in.tum.de>
parents:
67226
diff
changeset
|
1281 |
|
ab10ea1d6fd0
Some lemmas on complex numbers and coprimality
eberlm <eberlm@in.tum.de>
parents:
67226
diff
changeset
|
1282 |
lemma coprime_absorb_right: |
ab10ea1d6fd0
Some lemmas on complex numbers and coprimality
eberlm <eberlm@in.tum.de>
parents:
67226
diff
changeset
|
1283 |
assumes "y dvd x" |
ab10ea1d6fd0
Some lemmas on complex numbers and coprimality
eberlm <eberlm@in.tum.de>
parents:
67226
diff
changeset
|
1284 |
shows "coprime x y \<longleftrightarrow> is_unit y" |
ab10ea1d6fd0
Some lemmas on complex numbers and coprimality
eberlm <eberlm@in.tum.de>
parents:
67226
diff
changeset
|
1285 |
using assms coprime_common_divisor is_unit_right_imp_coprime by auto |
ab10ea1d6fd0
Some lemmas on complex numbers and coprimality
eberlm <eberlm@in.tum.de>
parents:
67226
diff
changeset
|
1286 |
|
60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1287 |
end |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1288 |
|
64848
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64713
diff
changeset
|
1289 |
class unit_factor = |
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64713
diff
changeset
|
1290 |
fixes unit_factor :: "'a \<Rightarrow> 'a" |
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64713
diff
changeset
|
1291 |
|
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64713
diff
changeset
|
1292 |
class semidom_divide_unit_factor = semidom_divide + unit_factor + |
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64713
diff
changeset
|
1293 |
assumes unit_factor_0 [simp]: "unit_factor 0 = 0" |
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64713
diff
changeset
|
1294 |
and is_unit_unit_factor: "a dvd 1 \<Longrightarrow> unit_factor a = a" |
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64713
diff
changeset
|
1295 |
and unit_factor_is_unit: "a \<noteq> 0 \<Longrightarrow> unit_factor a dvd 1" |
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64713
diff
changeset
|
1296 |
and unit_factor_mult: "unit_factor (a * b) = unit_factor a * unit_factor b" |
67226 | 1297 |
\<comment> \<open>This fine-grained hierarchy will later on allow lean normalization of polynomials\<close> |
64848
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64713
diff
changeset
|
1298 |
|
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64713
diff
changeset
|
1299 |
class normalization_semidom = algebraic_semidom + semidom_divide_unit_factor + |
60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1300 |
fixes normalize :: "'a \<Rightarrow> 'a" |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1301 |
assumes unit_factor_mult_normalize [simp]: "unit_factor a * normalize a = a" |
63588 | 1302 |
and normalize_0 [simp]: "normalize 0 = 0" |
60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1303 |
begin |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1304 |
|
60688
01488b559910
avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset
|
1305 |
text \<open> |
69593 | 1306 |
Class \<^class>\<open>normalization_semidom\<close> cultivates the idea that each integral |
63588 | 1307 |
domain can be split into equivalence classes whose representants are |
69593 | 1308 |
associated, i.e. divide each other. \<^const>\<open>normalize\<close> specifies a canonical |
63588 | 1309 |
representant for each equivalence class. The rationale behind this is that |
1310 |
it is easier to reason about equality than equivalences, hence we prefer to |
|
1311 |
think about equality of normalized values rather than associated elements. |
|
60688
01488b559910
avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset
|
1312 |
\<close> |
01488b559910
avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset
|
1313 |
|
64848
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64713
diff
changeset
|
1314 |
declare unit_factor_is_unit [iff] |
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64713
diff
changeset
|
1315 |
|
63325 | 1316 |
lemma unit_factor_dvd [simp]: "a \<noteq> 0 \<Longrightarrow> unit_factor a dvd b" |
60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1317 |
by (rule unit_imp_dvd) simp |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1318 |
|
63325 | 1319 |
lemma unit_factor_self [simp]: "unit_factor a dvd a" |
62376
85f38d5f8807
Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
hoelzl
parents:
62366
diff
changeset
|
1320 |
by (cases "a = 0") simp_all |
85f38d5f8807
Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
hoelzl
parents:
62366
diff
changeset
|
1321 |
|
63325 | 1322 |
lemma normalize_mult_unit_factor [simp]: "normalize a * unit_factor a = a" |
60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1323 |
using unit_factor_mult_normalize [of a] by (simp add: ac_simps) |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1324 |
|
63325 | 1325 |
lemma normalize_eq_0_iff [simp]: "normalize a = 0 \<longleftrightarrow> a = 0" |
63588 | 1326 |
(is "?lhs \<longleftrightarrow> ?rhs") |
60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1327 |
proof |
63588 | 1328 |
assume ?lhs |
60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1329 |
moreover have "unit_factor a * normalize a = a" by simp |
63588 | 1330 |
ultimately show ?rhs by simp |
60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1331 |
next |
63588 | 1332 |
assume ?rhs |
1333 |
then show ?lhs by simp |
|
60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1334 |
qed |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1335 |
|
63325 | 1336 |
lemma unit_factor_eq_0_iff [simp]: "unit_factor a = 0 \<longleftrightarrow> a = 0" |
63588 | 1337 |
(is "?lhs \<longleftrightarrow> ?rhs") |
60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1338 |
proof |
63588 | 1339 |
assume ?lhs |
60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1340 |
moreover have "unit_factor a * normalize a = a" by simp |
63588 | 1341 |
ultimately show ?rhs by simp |
60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1342 |
next |
63588 | 1343 |
assume ?rhs |
1344 |
then show ?lhs by simp |
|
60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1345 |
qed |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1346 |
|
64848
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64713
diff
changeset
|
1347 |
lemma div_unit_factor [simp]: "a div unit_factor a = normalize a" |
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64713
diff
changeset
|
1348 |
proof (cases "a = 0") |
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64713
diff
changeset
|
1349 |
case True |
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64713
diff
changeset
|
1350 |
then show ?thesis by simp |
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64713
diff
changeset
|
1351 |
next |
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64713
diff
changeset
|
1352 |
case False |
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64713
diff
changeset
|
1353 |
then have "unit_factor a \<noteq> 0" |
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64713
diff
changeset
|
1354 |
by simp |
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64713
diff
changeset
|
1355 |
with nonzero_mult_div_cancel_left |
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64713
diff
changeset
|
1356 |
have "unit_factor a * normalize a div unit_factor a = normalize a" |
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64713
diff
changeset
|
1357 |
by blast |
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64713
diff
changeset
|
1358 |
then show ?thesis by simp |
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64713
diff
changeset
|
1359 |
qed |
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64713
diff
changeset
|
1360 |
|
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64713
diff
changeset
|
1361 |
lemma normalize_div [simp]: "normalize a div a = 1 div unit_factor a" |
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64713
diff
changeset
|
1362 |
proof (cases "a = 0") |
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64713
diff
changeset
|
1363 |
case True |
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64713
diff
changeset
|
1364 |
then show ?thesis by simp |
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64713
diff
changeset
|
1365 |
next |
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64713
diff
changeset
|
1366 |
case False |
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64713
diff
changeset
|
1367 |
have "normalize a div a = normalize a div (unit_factor a * normalize a)" |
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64713
diff
changeset
|
1368 |
by simp |
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64713
diff
changeset
|
1369 |
also have "\<dots> = 1 div unit_factor a" |
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64713
diff
changeset
|
1370 |
using False by (subst is_unit_div_mult_cancel_right) simp_all |
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64713
diff
changeset
|
1371 |
finally show ?thesis . |
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64713
diff
changeset
|
1372 |
qed |
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64713
diff
changeset
|
1373 |
|
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64713
diff
changeset
|
1374 |
lemma is_unit_normalize: |
63325 | 1375 |
assumes "is_unit a" |
64848
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64713
diff
changeset
|
1376 |
shows "normalize a = 1" |
62376
85f38d5f8807
Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
hoelzl
parents:
62366
diff
changeset
|
1377 |
proof - |
64848
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64713
diff
changeset
|
1378 |
from assms have "unit_factor a = a" |
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64713
diff
changeset
|
1379 |
by (rule is_unit_unit_factor) |
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64713
diff
changeset
|
1380 |
moreover from assms have "a \<noteq> 0" |
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64713
diff
changeset
|
1381 |
by auto |
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64713
diff
changeset
|
1382 |
moreover have "normalize a = a div unit_factor a" |
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64713
diff
changeset
|
1383 |
by simp |
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64713
diff
changeset
|
1384 |
ultimately show ?thesis |
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64713
diff
changeset
|
1385 |
by simp |
60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1386 |
qed |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1387 |
|
63325 | 1388 |
lemma unit_factor_1 [simp]: "unit_factor 1 = 1" |
60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1389 |
by (rule is_unit_unit_factor) simp |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1390 |
|
63325 | 1391 |
lemma normalize_1 [simp]: "normalize 1 = 1" |
60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1392 |
by (rule is_unit_normalize) simp |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1393 |
|
63325 | 1394 |
lemma normalize_1_iff: "normalize a = 1 \<longleftrightarrow> is_unit a" |
63588 | 1395 |
(is "?lhs \<longleftrightarrow> ?rhs") |
60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1396 |
proof |
63588 | 1397 |
assume ?rhs |
1398 |
then show ?lhs by (rule is_unit_normalize) |
|
60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1399 |
next |
63588 | 1400 |
assume ?lhs |
1401 |
then have "unit_factor a * normalize a = unit_factor a * 1" |
|
60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1402 |
by simp |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1403 |
then have "unit_factor a = a" |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1404 |
by simp |
63588 | 1405 |
moreover |
1406 |
from \<open>?lhs\<close> have "a \<noteq> 0" by auto |
|
1407 |
then have "is_unit (unit_factor a)" by simp |
|
1408 |
ultimately show ?rhs by simp |
|
60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1409 |
qed |
62376
85f38d5f8807
Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
hoelzl
parents:
62366
diff
changeset
|
1410 |
|
63325 | 1411 |
lemma div_normalize [simp]: "a div normalize a = unit_factor a" |
60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1412 |
proof (cases "a = 0") |
63325 | 1413 |
case True |
1414 |
then show ?thesis by simp |
|
60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1415 |
next |
63325 | 1416 |
case False |
1417 |
then have "normalize a \<noteq> 0" by simp |
|
64240 | 1418 |
with nonzero_mult_div_cancel_right |
60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1419 |
have "unit_factor a * normalize a div normalize a = unit_factor a" by blast |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1420 |
then show ?thesis by simp |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1421 |
qed |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1422 |
|
63325 | 1423 |
lemma mult_one_div_unit_factor [simp]: "a * (1 div unit_factor b) = a div unit_factor b" |
60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1424 |
by (cases "b = 0") simp_all |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1425 |
|
63947 | 1426 |
lemma inv_unit_factor_eq_0_iff [simp]: |
1427 |
"1 div unit_factor a = 0 \<longleftrightarrow> a = 0" |
|
1428 |
(is "?lhs \<longleftrightarrow> ?rhs") |
|
1429 |
proof |
|
1430 |
assume ?lhs |
|
1431 |
then have "a * (1 div unit_factor a) = a * 0" |
|
1432 |
by simp |
|
1433 |
then show ?rhs |
|
1434 |
by simp |
|
1435 |
next |
|
1436 |
assume ?rhs |
|
1437 |
then show ?lhs by simp |
|
1438 |
qed |
|
1439 |
||
63325 | 1440 |
lemma normalize_mult: "normalize (a * b) = normalize a * normalize b" |
60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1441 |
proof (cases "a = 0 \<or> b = 0") |
63325 | 1442 |
case True |
1443 |
then show ?thesis by auto |
|
60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1444 |
next |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1445 |
case False |
63588 | 1446 |
have "unit_factor (a * b) * normalize (a * b) = a * b" |
1447 |
by (rule unit_factor_mult_normalize) |
|
63325 | 1448 |
then have "normalize (a * b) = a * b div unit_factor (a * b)" |
1449 |
by simp |
|
1450 |
also have "\<dots> = a * b div unit_factor (b * a)" |
|
1451 |
by (simp add: ac_simps) |
|
60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1452 |
also have "\<dots> = a * b div unit_factor b div unit_factor a" |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1453 |
using False by (simp add: unit_factor_mult is_unit_div_mult2_eq [symmetric]) |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1454 |
also have "\<dots> = a * (b div unit_factor b) div unit_factor a" |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1455 |
using False by (subst unit_div_mult_swap) simp_all |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1456 |
also have "\<dots> = normalize a * normalize b" |
63325 | 1457 |
using False |
1458 |
by (simp add: mult.commute [of a] mult.commute [of "normalize a"] unit_div_mult_swap [symmetric]) |
|
60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1459 |
finally show ?thesis . |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1460 |
qed |
62376
85f38d5f8807
Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
hoelzl
parents:
62366
diff
changeset
|
1461 |
|
63325 | 1462 |
lemma unit_factor_idem [simp]: "unit_factor (unit_factor a) = unit_factor a" |
60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1463 |
by (cases "a = 0") (auto intro: is_unit_unit_factor) |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1464 |
|
63325 | 1465 |
lemma normalize_unit_factor [simp]: "a \<noteq> 0 \<Longrightarrow> normalize (unit_factor a) = 1" |
60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1466 |
by (rule is_unit_normalize) simp |
62376
85f38d5f8807
Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
hoelzl
parents:
62366
diff
changeset
|
1467 |
|
63325 | 1468 |
lemma normalize_idem [simp]: "normalize (normalize a) = normalize a" |
60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1469 |
proof (cases "a = 0") |
63325 | 1470 |
case True |
1471 |
then show ?thesis by simp |
|
60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1472 |
next |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1473 |
case False |
63325 | 1474 |
have "normalize a = normalize (unit_factor a * normalize a)" |
1475 |
by simp |
|
60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1476 |
also have "\<dots> = normalize (unit_factor a) * normalize (normalize a)" |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1477 |
by (simp only: normalize_mult) |
63325 | 1478 |
finally show ?thesis |
1479 |
using False by simp_all |
|
60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1480 |
qed |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1481 |
|
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1482 |
lemma unit_factor_normalize [simp]: |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1483 |
assumes "a \<noteq> 0" |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1484 |
shows "unit_factor (normalize a) = 1" |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1485 |
proof - |
63325 | 1486 |
from assms have *: "normalize a \<noteq> 0" |
1487 |
by simp |
|
60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1488 |
have "unit_factor (normalize a) * normalize (normalize a) = normalize a" |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1489 |
by (simp only: unit_factor_mult_normalize) |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1490 |
then have "unit_factor (normalize a) * normalize a = normalize a" |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1491 |
by simp |
63325 | 1492 |
with * have "unit_factor (normalize a) * normalize a div normalize a = normalize a div normalize a" |
60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1493 |
by simp |
63325 | 1494 |
with * show ?thesis |
1495 |
by simp |
|
60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1496 |
qed |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1497 |
|
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1498 |
lemma dvd_unit_factor_div: |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1499 |
assumes "b dvd a" |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1500 |
shows "unit_factor (a div b) = unit_factor a div unit_factor b" |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1501 |
proof - |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1502 |
from assms have "a = a div b * b" |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1503 |
by simp |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1504 |
then have "unit_factor a = unit_factor (a div b * b)" |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1505 |
by simp |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1506 |
then show ?thesis |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1507 |
by (cases "b = 0") (simp_all add: unit_factor_mult) |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1508 |
qed |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1509 |
|
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1510 |
lemma dvd_normalize_div: |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1511 |
assumes "b dvd a" |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1512 |
shows "normalize (a div b) = normalize a div normalize b" |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1513 |
proof - |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1514 |
from assms have "a = a div b * b" |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1515 |
by simp |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1516 |
then have "normalize a = normalize (a div b * b)" |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1517 |
by simp |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1518 |
then show ?thesis |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1519 |
by (cases "b = 0") (simp_all add: normalize_mult) |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1520 |
qed |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1521 |
|
63325 | 1522 |
lemma normalize_dvd_iff [simp]: "normalize a dvd b \<longleftrightarrow> a dvd b" |
60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1523 |
proof - |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1524 |
have "normalize a dvd b \<longleftrightarrow> unit_factor a * normalize a dvd b" |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1525 |
using mult_unit_dvd_iff [of "unit_factor a" "normalize a" b] |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1526 |
by (cases "a = 0") simp_all |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1527 |
then show ?thesis by simp |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1528 |
qed |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1529 |
|
63325 | 1530 |
lemma dvd_normalize_iff [simp]: "a dvd normalize b \<longleftrightarrow> a dvd b" |
60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1531 |
proof - |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1532 |
have "a dvd normalize b \<longleftrightarrow> a dvd normalize b * unit_factor b" |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1533 |
using dvd_mult_unit_iff [of "unit_factor b" a "normalize b"] |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1534 |
by (cases "b = 0") simp_all |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1535 |
then show ?thesis by simp |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1536 |
qed |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1537 |
|
65811 | 1538 |
lemma normalize_idem_imp_unit_factor_eq: |
1539 |
assumes "normalize a = a" |
|
1540 |
shows "unit_factor a = of_bool (a \<noteq> 0)" |
|
1541 |
proof (cases "a = 0") |
|
1542 |
case True |
|
1543 |
then show ?thesis |
|
1544 |
by simp |
|
1545 |
next |
|
1546 |
case False |
|
1547 |
then show ?thesis |
|
1548 |
using assms unit_factor_normalize [of a] by simp |
|
1549 |
qed |
|
1550 |
||
1551 |
lemma normalize_idem_imp_is_unit_iff: |
|
1552 |
assumes "normalize a = a" |
|
1553 |
shows "is_unit a \<longleftrightarrow> a = 1" |
|
1554 |
using assms by (cases "a = 0") (auto dest: is_unit_normalize) |
|
1555 |
||
67051 | 1556 |
lemma coprime_normalize_left_iff [simp]: |
1557 |
"coprime (normalize a) b \<longleftrightarrow> coprime a b" |
|
1558 |
by (rule; rule coprimeI) (auto intro: coprime_common_divisor) |
|
1559 |
||
1560 |
lemma coprime_normalize_right_iff [simp]: |
|
1561 |
"coprime a (normalize b) \<longleftrightarrow> coprime a b" |
|
1562 |
using coprime_normalize_left_iff [of b a] by (simp add: ac_simps) |
|
1563 |
||
60688
01488b559910
avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset
|
1564 |
text \<open> |
63588 | 1565 |
We avoid an explicit definition of associated elements but prefer explicit |
69593 | 1566 |
normalisation instead. In theory we could define an abbreviation like \<^prop>\<open>associated a b \<longleftrightarrow> normalize a = normalize b\<close> but this is counterproductive |
63588 | 1567 |
without suggestive infix syntax, which we do not want to sacrifice for this |
1568 |
purpose here. |
|
60688
01488b559910
avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset
|
1569 |
\<close> |
60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1570 |
|
60688
01488b559910
avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset
|
1571 |
lemma associatedI: |
01488b559910
avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset
|
1572 |
assumes "a dvd b" and "b dvd a" |
01488b559910
avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset
|
1573 |
shows "normalize a = normalize b" |
60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1574 |
proof (cases "a = 0 \<or> b = 0") |
63325 | 1575 |
case True |
1576 |
with assms show ?thesis by auto |
|
60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1577 |
next |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1578 |
case False |
60688
01488b559910
avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset
|
1579 |
from \<open>a dvd b\<close> obtain c where b: "b = a * c" .. |
01488b559910
avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset
|
1580 |
moreover from \<open>b dvd a\<close> obtain d where a: "a = b * d" .. |
63325 | 1581 |
ultimately have "b * 1 = b * (c * d)" |
1582 |
by (simp add: ac_simps) |
|
60688
01488b559910
avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset
|
1583 |
with False have "1 = c * d" |
01488b559910
avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset
|
1584 |
unfolding mult_cancel_left by simp |
63325 | 1585 |
then have "is_unit c" and "is_unit d" |
1586 |
by auto |
|
1587 |
with a b show ?thesis |
|
1588 |
by (simp add: normalize_mult is_unit_normalize) |
|
60688
01488b559910
avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset
|
1589 |
qed |
01488b559910
avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset
|
1590 |
|
63325 | 1591 |
lemma associatedD1: "normalize a = normalize b \<Longrightarrow> a dvd b" |
60688
01488b559910
avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset
|
1592 |
using dvd_normalize_iff [of _ b, symmetric] normalize_dvd_iff [of a _, symmetric] |
01488b559910
avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset
|
1593 |
by simp |
01488b559910
avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset
|
1594 |
|
63325 | 1595 |
lemma associatedD2: "normalize a = normalize b \<Longrightarrow> b dvd a" |
60688
01488b559910
avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset
|
1596 |
using dvd_normalize_iff [of _ a, symmetric] normalize_dvd_iff [of b _, symmetric] |
01488b559910
avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset
|
1597 |
by simp |
01488b559910
avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset
|
1598 |
|
63325 | 1599 |
lemma associated_unit: "normalize a = normalize b \<Longrightarrow> is_unit a \<Longrightarrow> is_unit b" |
60688
01488b559910
avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset
|
1600 |
using dvd_unit_imp_unit by (auto dest!: associatedD1 associatedD2) |
01488b559910
avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset
|
1601 |
|
63325 | 1602 |
lemma associated_iff_dvd: "normalize a = normalize b \<longleftrightarrow> a dvd b \<and> b dvd a" |
63588 | 1603 |
(is "?lhs \<longleftrightarrow> ?rhs") |
60688
01488b559910
avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset
|
1604 |
proof |
63588 | 1605 |
assume ?rhs |
1606 |
then show ?lhs by (auto intro!: associatedI) |
|
60688
01488b559910
avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset
|
1607 |
next |
63588 | 1608 |
assume ?lhs |
60688
01488b559910
avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset
|
1609 |
then have "unit_factor a * normalize a = unit_factor a * normalize b" |
01488b559910
avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset
|
1610 |
by simp |
01488b559910
avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset
|
1611 |
then have *: "normalize b * unit_factor a = a" |
01488b559910
avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset
|
1612 |
by (simp add: ac_simps) |
63588 | 1613 |
show ?rhs |
60688
01488b559910
avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset
|
1614 |
proof (cases "a = 0 \<or> b = 0") |
63325 | 1615 |
case True |
63588 | 1616 |
with \<open>?lhs\<close> show ?thesis by auto |
60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1617 |
next |
62376
85f38d5f8807
Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
hoelzl
parents:
62366
diff
changeset
|
1618 |
case False |
60688
01488b559910
avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset
|
1619 |
then have "b dvd normalize b * unit_factor a" and "normalize b * unit_factor a dvd b" |
01488b559910
avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset
|
1620 |
by (simp_all add: mult_unit_dvd_iff dvd_mult_unit_iff) |
01488b559910
avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset
|
1621 |
with * show ?thesis by simp |
60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1622 |
qed |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1623 |
qed |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1624 |
|
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1625 |
lemma associated_eqI: |
60688
01488b559910
avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset
|
1626 |
assumes "a dvd b" and "b dvd a" |
01488b559910
avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset
|
1627 |
assumes "normalize a = a" and "normalize b = b" |
60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1628 |
shows "a = b" |
60688
01488b559910
avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset
|
1629 |
proof - |
01488b559910
avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset
|
1630 |
from assms have "normalize a = normalize b" |
01488b559910
avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset
|
1631 |
unfolding associated_iff_dvd by simp |
63588 | 1632 |
with \<open>normalize a = a\<close> have "a = normalize b" |
1633 |
by simp |
|
1634 |
with \<open>normalize b = b\<close> show "a = b" |
|
1635 |
by simp |
|
60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1636 |
qed |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1637 |
|
64591
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64290
diff
changeset
|
1638 |
lemma normalize_unit_factor_eqI: |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64290
diff
changeset
|
1639 |
assumes "normalize a = normalize b" |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64290
diff
changeset
|
1640 |
and "unit_factor a = unit_factor b" |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64290
diff
changeset
|
1641 |
shows "a = b" |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64290
diff
changeset
|
1642 |
proof - |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64290
diff
changeset
|
1643 |
from assms have "unit_factor a * normalize a = unit_factor b * normalize b" |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64290
diff
changeset
|
1644 |
by simp |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64290
diff
changeset
|
1645 |
then show ?thesis |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64290
diff
changeset
|
1646 |
by simp |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64290
diff
changeset
|
1647 |
qed |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64290
diff
changeset
|
1648 |
|
60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1649 |
end |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1650 |
|
64164
38c407446400
separate type class for arbitrary quotient and remainder partitions
haftmann
parents:
63950
diff
changeset
|
1651 |
|
38c407446400
separate type class for arbitrary quotient and remainder partitions
haftmann
parents:
63950
diff
changeset
|
1652 |
text \<open>Syntactic division remainder operator\<close> |
38c407446400
separate type class for arbitrary quotient and remainder partitions
haftmann
parents:
63950
diff
changeset
|
1653 |
|
38c407446400
separate type class for arbitrary quotient and remainder partitions
haftmann
parents:
63950
diff
changeset
|
1654 |
class modulo = dvd + divide + |
38c407446400
separate type class for arbitrary quotient and remainder partitions
haftmann
parents:
63950
diff
changeset
|
1655 |
fixes modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "mod" 70) |
38c407446400
separate type class for arbitrary quotient and remainder partitions
haftmann
parents:
63950
diff
changeset
|
1656 |
|
38c407446400
separate type class for arbitrary quotient and remainder partitions
haftmann
parents:
63950
diff
changeset
|
1657 |
text \<open>Arbitrary quotient and remainder partitions\<close> |
38c407446400
separate type class for arbitrary quotient and remainder partitions
haftmann
parents:
63950
diff
changeset
|
1658 |
|
38c407446400
separate type class for arbitrary quotient and remainder partitions
haftmann
parents:
63950
diff
changeset
|
1659 |
class semiring_modulo = comm_semiring_1_cancel + divide + modulo + |
64242
93c6f0da5c70
more standardized theorem names for facts involving the div and mod identity
haftmann
parents:
64240
diff
changeset
|
1660 |
assumes div_mult_mod_eq: "a div b * b + a mod b = a" |
64164
38c407446400
separate type class for arbitrary quotient and remainder partitions
haftmann
parents:
63950
diff
changeset
|
1661 |
begin |
38c407446400
separate type class for arbitrary quotient and remainder partitions
haftmann
parents:
63950
diff
changeset
|
1662 |
|
38c407446400
separate type class for arbitrary quotient and remainder partitions
haftmann
parents:
63950
diff
changeset
|
1663 |
lemma mod_div_decomp: |
38c407446400
separate type class for arbitrary quotient and remainder partitions
haftmann
parents:
63950
diff
changeset
|
1664 |
fixes a b |
38c407446400
separate type class for arbitrary quotient and remainder partitions
haftmann
parents:
63950
diff
changeset
|
1665 |
obtains q r where "q = a div b" and "r = a mod b" |
38c407446400
separate type class for arbitrary quotient and remainder partitions
haftmann
parents:
63950
diff
changeset
|
1666 |
and "a = q * b + r" |
38c407446400
separate type class for arbitrary quotient and remainder partitions
haftmann
parents:
63950
diff
changeset
|
1667 |
proof - |
64242
93c6f0da5c70
more standardized theorem names for facts involving the div and mod identity
haftmann
parents:
64240
diff
changeset
|
1668 |
from div_mult_mod_eq have "a = a div b * b + a mod b" by simp |
64164
38c407446400
separate type class for arbitrary quotient and remainder partitions
haftmann
parents:
63950
diff
changeset
|
1669 |
moreover have "a div b = a div b" .. |
38c407446400
separate type class for arbitrary quotient and remainder partitions
haftmann
parents:
63950
diff
changeset
|
1670 |
moreover have "a mod b = a mod b" .. |
38c407446400
separate type class for arbitrary quotient and remainder partitions
haftmann
parents:
63950
diff
changeset
|
1671 |
note that ultimately show thesis by blast |
38c407446400
separate type class for arbitrary quotient and remainder partitions
haftmann
parents:
63950
diff
changeset
|
1672 |
qed |
38c407446400
separate type class for arbitrary quotient and remainder partitions
haftmann
parents:
63950
diff
changeset
|
1673 |
|
64242
93c6f0da5c70
more standardized theorem names for facts involving the div and mod identity
haftmann
parents:
64240
diff
changeset
|
1674 |
lemma mult_div_mod_eq: "b * (a div b) + a mod b = a" |
93c6f0da5c70
more standardized theorem names for facts involving the div and mod identity
haftmann
parents:
64240
diff
changeset
|
1675 |
using div_mult_mod_eq [of a b] by (simp add: ac_simps) |
64164
38c407446400
separate type class for arbitrary quotient and remainder partitions
haftmann
parents:
63950
diff
changeset
|
1676 |
|
64242
93c6f0da5c70
more standardized theorem names for facts involving the div and mod identity
haftmann
parents:
64240
diff
changeset
|
1677 |
lemma mod_div_mult_eq: "a mod b + a div b * b = a" |
93c6f0da5c70
more standardized theorem names for facts involving the div and mod identity
haftmann
parents:
64240
diff
changeset
|
1678 |
using div_mult_mod_eq [of a b] by (simp add: ac_simps) |
64164
38c407446400
separate type class for arbitrary quotient and remainder partitions
haftmann
parents:
63950
diff
changeset
|
1679 |
|
64242
93c6f0da5c70
more standardized theorem names for facts involving the div and mod identity
haftmann
parents:
64240
diff
changeset
|
1680 |
lemma mod_mult_div_eq: "a mod b + b * (a div b) = a" |
93c6f0da5c70
more standardized theorem names for facts involving the div and mod identity
haftmann
parents:
64240
diff
changeset
|
1681 |
using div_mult_mod_eq [of a b] by (simp add: ac_simps) |
64164
38c407446400
separate type class for arbitrary quotient and remainder partitions
haftmann
parents:
63950
diff
changeset
|
1682 |
|
64242
93c6f0da5c70
more standardized theorem names for facts involving the div and mod identity
haftmann
parents:
64240
diff
changeset
|
1683 |
lemma minus_div_mult_eq_mod: "a - a div b * b = a mod b" |
93c6f0da5c70
more standardized theorem names for facts involving the div and mod identity
haftmann
parents:
64240
diff
changeset
|
1684 |
by (rule add_implies_diff [symmetric]) (fact mod_div_mult_eq) |
64164
38c407446400
separate type class for arbitrary quotient and remainder partitions
haftmann
parents:
63950
diff
changeset
|
1685 |
|
64242
93c6f0da5c70
more standardized theorem names for facts involving the div and mod identity
haftmann
parents:
64240
diff
changeset
|
1686 |
lemma minus_mult_div_eq_mod: "a - b * (a div b) = a mod b" |
93c6f0da5c70
more standardized theorem names for facts involving the div and mod identity
haftmann
parents:
64240
diff
changeset
|
1687 |
by (rule add_implies_diff [symmetric]) (fact mod_mult_div_eq) |
64164
38c407446400
separate type class for arbitrary quotient and remainder partitions
haftmann
parents:
63950
diff
changeset
|
1688 |
|
64242
93c6f0da5c70
more standardized theorem names for facts involving the div and mod identity
haftmann
parents:
64240
diff
changeset
|
1689 |
lemma minus_mod_eq_div_mult: "a - a mod b = a div b * b" |
93c6f0da5c70
more standardized theorem names for facts involving the div and mod identity
haftmann
parents:
64240
diff
changeset
|
1690 |
by (rule add_implies_diff [symmetric]) (fact div_mult_mod_eq) |
64164
38c407446400
separate type class for arbitrary quotient and remainder partitions
haftmann
parents:
63950
diff
changeset
|
1691 |
|
64242
93c6f0da5c70
more standardized theorem names for facts involving the div and mod identity
haftmann
parents:
64240
diff
changeset
|
1692 |
lemma minus_mod_eq_mult_div: "a - a mod b = b * (a div b)" |
93c6f0da5c70
more standardized theorem names for facts involving the div and mod identity
haftmann
parents:
64240
diff
changeset
|
1693 |
by (rule add_implies_diff [symmetric]) (fact mult_div_mod_eq) |
64164
38c407446400
separate type class for arbitrary quotient and remainder partitions
haftmann
parents:
63950
diff
changeset
|
1694 |
|
68253 | 1695 |
lemma [nitpick_unfold]: |
1696 |
"a mod b = a - a div b * b" |
|
1697 |
by (fact minus_div_mult_eq_mod [symmetric]) |
|
1698 |
||
64164
38c407446400
separate type class for arbitrary quotient and remainder partitions
haftmann
parents:
63950
diff
changeset
|
1699 |
end |
64242
93c6f0da5c70
more standardized theorem names for facts involving the div and mod identity
haftmann
parents:
64240
diff
changeset
|
1700 |
|
64164
38c407446400
separate type class for arbitrary quotient and remainder partitions
haftmann
parents:
63950
diff
changeset
|
1701 |
|
66807 | 1702 |
text \<open>Quotient and remainder in integral domains\<close> |
1703 |
||
1704 |
class semidom_modulo = algebraic_semidom + semiring_modulo |
|
1705 |
begin |
|
1706 |
||
1707 |
lemma mod_0 [simp]: "0 mod a = 0" |
|
1708 |
using div_mult_mod_eq [of 0 a] by simp |
|
1709 |
||
1710 |
lemma mod_by_0 [simp]: "a mod 0 = a" |
|
1711 |
using div_mult_mod_eq [of a 0] by simp |
|
1712 |
||
1713 |
lemma mod_by_1 [simp]: |
|
1714 |
"a mod 1 = 0" |
|
1715 |
proof - |
|
1716 |
from div_mult_mod_eq [of a one] div_by_1 have "a + a mod 1 = a" by simp |
|
1717 |
then have "a + a mod 1 = a + 0" by simp |
|
1718 |
then show ?thesis by (rule add_left_imp_eq) |
|
1719 |
qed |
|
1720 |
||
1721 |
lemma mod_self [simp]: |
|
1722 |
"a mod a = 0" |
|
1723 |
using div_mult_mod_eq [of a a] by simp |
|
1724 |
||
1725 |
lemma dvd_imp_mod_0 [simp]: |
|
67084 | 1726 |
"b mod a = 0" if "a dvd b" |
1727 |
using that minus_div_mult_eq_mod [of b a] by simp |
|
66807 | 1728 |
|
68252
8b284d24f434
automatic classical rule to derive a dvd b from b mod a = 0
haftmann
parents:
68251
diff
changeset
|
1729 |
lemma mod_0_imp_dvd [dest!]: |
67084 | 1730 |
"b dvd a" if "a mod b = 0" |
66807 | 1731 |
proof - |
67084 | 1732 |
have "b dvd (a div b) * b" by simp |
66807 | 1733 |
also have "(a div b) * b = a" |
67084 | 1734 |
using div_mult_mod_eq [of a b] by (simp add: that) |
66807 | 1735 |
finally show ?thesis . |
1736 |
qed |
|
1737 |
||
1738 |
lemma mod_eq_0_iff_dvd: |
|
1739 |
"a mod b = 0 \<longleftrightarrow> b dvd a" |
|
1740 |
by (auto intro: mod_0_imp_dvd) |
|
1741 |
||
1742 |
lemma dvd_eq_mod_eq_0 [nitpick_unfold, code]: |
|
1743 |
"a dvd b \<longleftrightarrow> b mod a = 0" |
|
1744 |
by (simp add: mod_eq_0_iff_dvd) |
|
1745 |
||
1746 |
lemma dvd_mod_iff: |
|
1747 |
assumes "c dvd b" |
|
1748 |
shows "c dvd a mod b \<longleftrightarrow> c dvd a" |
|
1749 |
proof - |
|
1750 |
from assms have "(c dvd a mod b) \<longleftrightarrow> (c dvd ((a div b) * b + a mod b))" |
|
1751 |
by (simp add: dvd_add_right_iff) |
|
1752 |
also have "(a div b) * b + a mod b = a" |
|
1753 |
using div_mult_mod_eq [of a b] by simp |
|
1754 |
finally show ?thesis . |
|
1755 |
qed |
|
1756 |
||
1757 |
lemma dvd_mod_imp_dvd: |
|
1758 |
assumes "c dvd a mod b" and "c dvd b" |
|
1759 |
shows "c dvd a" |
|
1760 |
using assms dvd_mod_iff [of c b a] by simp |
|
1761 |
||
66808
1907167b6038
elementary definition of division on natural numbers
haftmann
parents:
66807
diff
changeset
|
1762 |
lemma dvd_minus_mod [simp]: |
1907167b6038
elementary definition of division on natural numbers
haftmann
parents:
66807
diff
changeset
|
1763 |
"b dvd a - a mod b" |
1907167b6038
elementary definition of division on natural numbers
haftmann
parents:
66807
diff
changeset
|
1764 |
by (simp add: minus_mod_eq_div_mult) |
1907167b6038
elementary definition of division on natural numbers
haftmann
parents:
66807
diff
changeset
|
1765 |
|
66810 | 1766 |
lemma cancel_div_mod_rules: |
1767 |
"((a div b) * b + a mod b) + c = a + c" |
|
1768 |
"(b * (a div b) + a mod b) + c = a + c" |
|
1769 |
by (simp_all add: div_mult_mod_eq mult_div_mod_eq) |
|
1770 |
||
66807 | 1771 |
end |
1772 |
||
66810 | 1773 |
text \<open>Interlude: basic tool support for algebraic and arithmetic calculations\<close> |
1774 |
||
1775 |
named_theorems arith "arith facts -- only ground formulas" |
|
69605 | 1776 |
ML_file \<open>Tools/arith_data.ML\<close> |
1777 |
||
1778 |
ML_file \<open>~~/src/Provers/Arith/cancel_div_mod.ML\<close> |
|
66810 | 1779 |
|
1780 |
ML \<open> |
|
1781 |
structure Cancel_Div_Mod_Ring = Cancel_Div_Mod |
|
1782 |
( |
|
69593 | 1783 |
val div_name = \<^const_name>\<open>divide\<close>; |
1784 |
val mod_name = \<^const_name>\<open>modulo\<close>; |
|
66810 | 1785 |
val mk_binop = HOLogic.mk_binop; |
1786 |
val mk_sum = Arith_Data.mk_sum; |
|
1787 |
val dest_sum = Arith_Data.dest_sum; |
|
1788 |
||
1789 |
val div_mod_eqs = map mk_meta_eq @{thms cancel_div_mod_rules}; |
|
1790 |
||
1791 |
val prove_eq_sums = Arith_Data.prove_conv2 all_tac (Arith_Data.simp_all_tac |
|
1792 |
@{thms diff_conv_add_uminus add_0_left add_0_right ac_simps}) |
|
1793 |
) |
|
1794 |
\<close> |
|
1795 |
||
1796 |
simproc_setup cancel_div_mod_int ("(a::'a::semidom_modulo) + b") = |
|
1797 |
\<open>K Cancel_Div_Mod_Ring.proc\<close> |
|
1798 |
||
66807 | 1799 |
class idom_modulo = idom + semidom_modulo |
1800 |
begin |
|
1801 |
||
1802 |
subclass idom_divide .. |
|
1803 |
||
1804 |
lemma div_diff [simp]: |
|
1805 |
"c dvd a \<Longrightarrow> c dvd b \<Longrightarrow> (a - b) div c = a div c - b div c" |
|
1806 |
using div_add [of _ _ "- b"] by (simp add: dvd_neg_div) |
|
1807 |
||
1808 |
end |
|
1809 |
||
1810 |
||
62376
85f38d5f8807
Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
hoelzl
parents:
62366
diff
changeset
|
1811 |
class ordered_semiring = semiring + ordered_comm_monoid_add + |
38642
8fa437809c67
dropped type classes mult_mono and mult_mono1; tuned names of technical rule duplicates
haftmann
parents:
37767
diff
changeset
|
1812 |
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
|
1813 |
assumes mult_right_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * c" |
25230 | 1814 |
begin |
1815 |
||
63325 | 1816 |
lemma mult_mono: "a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * d" |
1817 |
apply (erule (1) mult_right_mono [THEN order_trans]) |
|
1818 |
apply (erule (1) mult_left_mono) |
|
1819 |
done |
|
25230 | 1820 |
|
63325 | 1821 |
lemma mult_mono': "a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * d" |
63588 | 1822 |
by (rule mult_mono) (fast intro: order_trans)+ |
25230 | 1823 |
|
1824 |
end |
|
21199
2d83f93c3580
* Added annihilation axioms ("x * 0 = 0") to axclass semiring_0.
krauss
parents:
20633
diff
changeset
|
1825 |
|
62377
ace69956d018
moved more proofs to ordered_comm_monoid_add; introduced strict_ordered_ab_semigroup/comm_monoid_add
hoelzl
parents:
62376
diff
changeset
|
1826 |
class ordered_semiring_0 = semiring_0 + ordered_semiring |
25267 | 1827 |
begin |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1828 |
|
63325 | 1829 |
lemma mult_nonneg_nonneg [simp]: "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> a * b" |
1830 |
using mult_left_mono [of 0 b a] by simp |
|
25230 | 1831 |
|
1832 |
lemma mult_nonneg_nonpos: "0 \<le> a \<Longrightarrow> b \<le> 0 \<Longrightarrow> a * b \<le> 0" |
|
63325 | 1833 |
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
|
1834 |
|
44ea10bc07a7
clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents:
30650
diff
changeset
|
1835 |
lemma mult_nonpos_nonneg: "a \<le> 0 \<Longrightarrow> 0 \<le> b \<Longrightarrow> a * b \<le> 0" |
63325 | 1836 |
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
|
1837 |
|
63588 | 1838 |
text \<open>Legacy -- use @{thm [source] mult_nonpos_nonneg}.\<close> |
60562
24af00b010cf
Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents:
60529
diff
changeset
|
1839 |
lemma mult_nonneg_nonpos2: "0 \<le> a \<Longrightarrow> b \<le> 0 \<Longrightarrow> b * a \<le> 0" |
63588 | 1840 |
by (drule mult_right_mono [of b 0]) auto |
25230 | 1841 |
|
62378
85ed00c1fe7c
generalize more theorems to support enat and ennreal
hoelzl
parents:
62377
diff
changeset
|
1842 |
lemma split_mult_neg_le: "(0 \<le> a \<and> b \<le> 0) \<or> (a \<le> 0 \<and> 0 \<le> b) \<Longrightarrow> a * b \<le> 0" |
63325 | 1843 |
by (auto simp add: mult_nonneg_nonpos mult_nonneg_nonpos2) |
25230 | 1844 |
|
1845 |
end |
|
1846 |
||
62377
ace69956d018
moved more proofs to ordered_comm_monoid_add; introduced strict_ordered_ab_semigroup/comm_monoid_add
hoelzl
parents:
62376
diff
changeset
|
1847 |
class ordered_cancel_semiring = ordered_semiring + cancel_comm_monoid_add |
ace69956d018
moved more proofs to ordered_comm_monoid_add; introduced strict_ordered_ab_semigroup/comm_monoid_add
hoelzl
parents:
62376
diff
changeset
|
1848 |
begin |
ace69956d018
moved more proofs to ordered_comm_monoid_add; introduced strict_ordered_ab_semigroup/comm_monoid_add
hoelzl
parents:
62376
diff
changeset
|
1849 |
|
ace69956d018
moved more proofs to ordered_comm_monoid_add; introduced strict_ordered_ab_semigroup/comm_monoid_add
hoelzl
parents:
62376
diff
changeset
|
1850 |
subclass semiring_0_cancel .. |
63588 | 1851 |
|
62377
ace69956d018
moved more proofs to ordered_comm_monoid_add; introduced strict_ordered_ab_semigroup/comm_monoid_add
hoelzl
parents:
62376
diff
changeset
|
1852 |
subclass ordered_semiring_0 .. |
ace69956d018
moved more proofs to ordered_comm_monoid_add; introduced strict_ordered_ab_semigroup/comm_monoid_add
hoelzl
parents:
62376
diff
changeset
|
1853 |
|
ace69956d018
moved more proofs to ordered_comm_monoid_add; introduced strict_ordered_ab_semigroup/comm_monoid_add
hoelzl
parents:
62376
diff
changeset
|
1854 |
end |
ace69956d018
moved more proofs to ordered_comm_monoid_add; introduced strict_ordered_ab_semigroup/comm_monoid_add
hoelzl
parents:
62376
diff
changeset
|
1855 |
|
38642
8fa437809c67
dropped type classes mult_mono and mult_mono1; tuned names of technical rule duplicates
haftmann
parents:
37767
diff
changeset
|
1856 |
class linordered_semiring = ordered_semiring + linordered_cancel_ab_semigroup_add |
25267 | 1857 |
begin |
25230 | 1858 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
1859 |
subclass ordered_cancel_semiring .. |
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
1860 |
|
62376
85f38d5f8807
Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
hoelzl
parents:
62366
diff
changeset
|
1861 |
subclass ordered_cancel_comm_monoid_add .. |
25304
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset
|
1862 |
|
63456
3365c8ec67bd
sharing simp rules between ordered monoids and rings
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63359
diff
changeset
|
1863 |
subclass ordered_ab_semigroup_monoid_add_imp_le .. |
3365c8ec67bd
sharing simp rules between ordered monoids and rings
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63359
diff
changeset
|
1864 |
|
63325 | 1865 |
lemma mult_left_less_imp_less: "c * a < c * b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a < b" |
1866 |
by (force simp add: mult_left_mono not_le [symmetric]) |
|
60562
24af00b010cf
Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents:
60529
diff
changeset
|
1867 |
|
63325 | 1868 |
lemma mult_right_less_imp_less: "a * c < b * c \<Longrightarrow> 0 \<le> c \<Longrightarrow> a < b" |
1869 |
by (force simp add: mult_right_mono not_le [symmetric]) |
|
23521 | 1870 |
|
25186 | 1871 |
end |
25152 | 1872 |
|
66937 | 1873 |
class zero_less_one = order + zero + one + |
1874 |
assumes zero_less_one [simp]: "0 < 1" |
|
1875 |
||
1876 |
class linordered_semiring_1 = linordered_semiring + semiring_1 + zero_less_one |
|
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
|
1877 |
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
|
1878 |
|
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
|
1879 |
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
|
1880 |
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
|
1881 |
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
|
1882 |
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
|
1883 |
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
|
1884 |
by (simp add: add_mono mult_left_mono) |
63325 | 1885 |
with assms show ?thesis |
1886 |
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
|
1887 |
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
|
1888 |
|
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
|
1889 |
end |
35043
07dbdf60d5ad
dropped accidental duplication of "lin" prefix from cs. 108662d50512
haftmann
parents:
35032
diff
changeset
|
1890 |
|
07dbdf60d5ad
dropped accidental duplication of "lin" prefix from cs. 108662d50512
haftmann
parents:
35032
diff
changeset
|
1891 |
class linordered_semiring_strict = semiring + comm_monoid_add + linordered_cancel_ab_semigroup_add + |
25062 | 1892 |
assumes mult_strict_left_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b" |
1893 |
assumes mult_strict_right_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> a * c < b * c" |
|
25267 | 1894 |
begin |
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset
|
1895 |
|
27516 | 1896 |
subclass semiring_0_cancel .. |
14940 | 1897 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
1898 |
subclass linordered_semiring |
28823 | 1899 |
proof |
23550 | 1900 |
fix a b c :: 'a |
63588 | 1901 |
assume *: "a \<le> b" "0 \<le> c" |
1902 |
then show "c * a \<le> c * b" |
|
25186 | 1903 |
unfolding le_less |
1904 |
using mult_strict_left_mono by (cases "c = 0") auto |
|
63588 | 1905 |
from * show "a * c \<le> b * c" |
25152 | 1906 |
unfolding le_less |
25186 | 1907 |
using mult_strict_right_mono by (cases "c = 0") auto |
25152 | 1908 |
qed |
1909 |
||
63325 | 1910 |
lemma mult_left_le_imp_le: "c * a \<le> c * b \<Longrightarrow> 0 < c \<Longrightarrow> a \<le> b" |
1911 |
by (auto simp add: mult_strict_left_mono _not_less [symmetric]) |
|
60562
24af00b010cf
Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents:
60529
diff
changeset
|
1912 |
|
63325 | 1913 |
lemma mult_right_le_imp_le: "a * c \<le> b * c \<Longrightarrow> 0 < c \<Longrightarrow> a \<le> b" |
1914 |
by (auto simp add: mult_strict_right_mono not_less [symmetric]) |
|
25230 | 1915 |
|
56544 | 1916 |
lemma mult_pos_pos[simp]: "0 < a \<Longrightarrow> 0 < b \<Longrightarrow> 0 < a * b" |
63325 | 1917 |
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
|
1918 |
|
44ea10bc07a7
clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents:
30650
diff
changeset
|
1919 |
lemma mult_pos_neg: "0 < a \<Longrightarrow> b < 0 \<Longrightarrow> a * b < 0" |
63325 | 1920 |
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
|
1921 |
|
44ea10bc07a7
clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents:
30650
diff
changeset
|
1922 |
lemma mult_neg_pos: "a < 0 \<Longrightarrow> 0 < b \<Longrightarrow> a * b < 0" |
63325 | 1923 |
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
|
1924 |
|
63588 | 1925 |
text \<open>Legacy -- use @{thm [source] mult_neg_pos}.\<close> |
60562
24af00b010cf
Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents:
60529
diff
changeset
|
1926 |
lemma mult_pos_neg2: "0 < a \<Longrightarrow> b < 0 \<Longrightarrow> b * a < 0" |
63588 | 1927 |
by (drule mult_strict_right_mono [of b 0]) auto |
25230 | 1928 |
|
63325 | 1929 |
lemma zero_less_mult_pos: "0 < a * b \<Longrightarrow> 0 < a \<Longrightarrow> 0 < b" |
1930 |
apply (cases "b \<le> 0") |
|
1931 |
apply (auto simp add: le_less not_less) |
|
1932 |
apply (drule_tac mult_pos_neg [of a b]) |
|
1933 |
apply (auto dest: less_not_sym) |
|
1934 |
done |
|
25230 | 1935 |
|
63325 | 1936 |
lemma zero_less_mult_pos2: "0 < b * a \<Longrightarrow> 0 < a \<Longrightarrow> 0 < b" |
1937 |
apply (cases "b \<le> 0") |
|
1938 |
apply (auto simp add: le_less not_less) |
|
1939 |
apply (drule_tac mult_pos_neg2 [of a b]) |
|
1940 |
apply (auto dest: less_not_sym) |
|
1941 |
done |
|
1942 |
||
1943 |
text \<open>Strict monotonicity in both arguments\<close> |
|
26193 | 1944 |
lemma mult_strict_mono: |
1945 |
assumes "a < b" and "c < d" and "0 < b" and "0 \<le> c" |
|
1946 |
shows "a * c < b * d" |
|
63325 | 1947 |
using assms |
1948 |
apply (cases "c = 0") |
|
63588 | 1949 |
apply simp |
26193 | 1950 |
apply (erule mult_strict_right_mono [THEN less_trans]) |
63588 | 1951 |
apply (auto simp add: le_less) |
63325 | 1952 |
apply (erule (1) mult_strict_left_mono) |
26193 | 1953 |
done |
1954 |
||
63325 | 1955 |
text \<open>This weaker variant has more natural premises\<close> |
26193 | 1956 |
lemma mult_strict_mono': |
1957 |
assumes "a < b" and "c < d" and "0 \<le> a" and "0 \<le> c" |
|
1958 |
shows "a * c < b * d" |
|
63325 | 1959 |
by (rule mult_strict_mono) (insert assms, auto) |
26193 | 1960 |
|
1961 |
lemma mult_less_le_imp_less: |
|
1962 |
assumes "a < b" and "c \<le> d" and "0 \<le> a" and "0 < c" |
|
1963 |
shows "a * c < b * d" |
|
63325 | 1964 |
using assms |
1965 |
apply (subgoal_tac "a * c < b * c") |
|
63588 | 1966 |
apply (erule less_le_trans) |
1967 |
apply (erule mult_left_mono) |
|
1968 |
apply simp |
|
63325 | 1969 |
apply (erule (1) mult_strict_right_mono) |
26193 | 1970 |
done |
1971 |
||
1972 |
lemma mult_le_less_imp_less: |
|
1973 |
assumes "a \<le> b" and "c < d" and "0 < a" and "0 \<le> c" |
|
1974 |
shows "a * c < b * d" |
|
63325 | 1975 |
using assms |
1976 |
apply (subgoal_tac "a * c \<le> b * c") |
|
63588 | 1977 |
apply (erule le_less_trans) |
1978 |
apply (erule mult_strict_left_mono) |
|
1979 |
apply simp |
|
63325 | 1980 |
apply (erule (1) mult_right_mono) |
26193 | 1981 |
done |
1982 |
||
25230 | 1983 |
end |
1984 |
||
66937 | 1985 |
class linordered_semiring_1_strict = linordered_semiring_strict + semiring_1 + zero_less_one |
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
|
1986 |
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
|
1987 |
|
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
|
1988 |
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
|
1989 |
|
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
|
1990 |
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
|
1991 |
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
|
1992 |
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
|
1993 |
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
|
1994 |
from assms have "u * x + v * y < u * a + v * a" |
63325 | 1995 |
by (cases "u = 0") (auto intro!: add_less_le_mono mult_strict_left_mono mult_left_mono) |
1996 |
with assms show ?thesis |
|
1997 |
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
|
1998 |
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
|
1999 |
|
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
|
2000 |
end |
33319 | 2001 |
|
60562
24af00b010cf
Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents:
60529
diff
changeset
|
2002 |
class ordered_comm_semiring = comm_semiring_0 + ordered_ab_semigroup_add + |
38642
8fa437809c67
dropped type classes mult_mono and mult_mono1; tuned names of technical rule duplicates
haftmann
parents:
37767
diff
changeset
|
2003 |
assumes comm_mult_left_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> c * a \<le> c * b" |
25186 | 2004 |
begin |
25152 | 2005 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
2006 |
subclass ordered_semiring |
28823 | 2007 |
proof |
21199
2d83f93c3580
* Added annihilation axioms ("x * 0 = 0") to axclass semiring_0.
krauss
parents:
20633
diff
changeset
|
2008 |
fix a b c :: 'a |
23550 | 2009 |
assume "a \<le> b" "0 \<le> c" |
63325 | 2010 |
then show "c * a \<le> c * b" by (rule comm_mult_left_mono) |
2011 |
then show "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
|
2012 |
qed |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
2013 |
|
25267 | 2014 |
end |
2015 |
||
38642
8fa437809c67
dropped type classes mult_mono and mult_mono1; tuned names of technical rule duplicates
haftmann
parents:
37767
diff
changeset
|
2016 |
class ordered_cancel_comm_semiring = ordered_comm_semiring + cancel_comm_monoid_add |
25267 | 2017 |
begin |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
2018 |
|
38642
8fa437809c67
dropped type classes mult_mono and mult_mono1; tuned names of technical rule duplicates
haftmann
parents:
37767
diff
changeset
|
2019 |
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
|
2020 |
subclass ordered_comm_semiring .. |
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
2021 |
subclass ordered_cancel_semiring .. |
25267 | 2022 |
|
2023 |
end |
|
2024 |
||
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
2025 |
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
|
2026 |
assumes comm_mult_strict_left_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b" |
25267 | 2027 |
begin |
2028 |
||
35043
07dbdf60d5ad
dropped accidental duplication of "lin" prefix from cs. 108662d50512
haftmann
parents:
35032
diff
changeset
|
2029 |
subclass linordered_semiring_strict |
28823 | 2030 |
proof |
23550 | 2031 |
fix a b c :: 'a |
2032 |
assume "a < b" "0 < c" |
|
63588 | 2033 |
then show "c * a < c * b" |
2034 |
by (rule comm_mult_strict_left_mono) |
|
2035 |
then show "a * c < b * c" |
|
2036 |
by (simp only: mult.commute) |
|
23550 | 2037 |
qed |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
2038 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
2039 |
subclass ordered_cancel_comm_semiring |
28823 | 2040 |
proof |
23550 | 2041 |
fix a b c :: 'a |
2042 |
assume "a \<le> b" "0 \<le> c" |
|
63325 | 2043 |
then show "c * a \<le> c * b" |
25186 | 2044 |
unfolding le_less |
26193 | 2045 |
using mult_strict_left_mono by (cases "c = 0") auto |
23550 | 2046 |
qed |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
2047 |
|
25267 | 2048 |
end |
25230 | 2049 |
|
60562
24af00b010cf
Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents:
60529
diff
changeset
|
2050 |
class ordered_ring = ring + ordered_cancel_semiring |
25267 | 2051 |
begin |
25230 | 2052 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
2053 |
subclass ordered_ab_group_add .. |
14270 | 2054 |
|
63325 | 2055 |
lemma less_add_iff1: "a * e + c < b * e + d \<longleftrightarrow> (a - b) * e + c < d" |
2056 |
by (simp add: algebra_simps) |
|
25230 | 2057 |
|
63325 | 2058 |
lemma less_add_iff2: "a * e + c < b * e + d \<longleftrightarrow> c < (b - a) * e + d" |
2059 |
by (simp add: algebra_simps) |
|
25230 | 2060 |
|
63325 | 2061 |
lemma le_add_iff1: "a * e + c \<le> b * e + d \<longleftrightarrow> (a - b) * e + c \<le> d" |
2062 |
by (simp add: algebra_simps) |
|
25230 | 2063 |
|
63325 | 2064 |
lemma le_add_iff2: "a * e + c \<le> b * e + d \<longleftrightarrow> c \<le> (b - a) * e + d" |
2065 |
by (simp add: algebra_simps) |
|
25230 | 2066 |
|
63325 | 2067 |
lemma mult_left_mono_neg: "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
|
2068 |
apply (drule mult_left_mono [of _ _ "- c"]) |
35216 | 2069 |
apply simp_all |
25230 | 2070 |
done |
2071 |
||
63325 | 2072 |
lemma mult_right_mono_neg: "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
|
2073 |
apply (drule mult_right_mono [of _ _ "- c"]) |
35216 | 2074 |
apply simp_all |
25230 | 2075 |
done |
2076 |
||
30692
44ea10bc07a7
clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents:
30650
diff
changeset
|
2077 |
lemma mult_nonpos_nonpos: "a \<le> 0 \<Longrightarrow> b \<le> 0 \<Longrightarrow> 0 \<le> a * b" |
63325 | 2078 |
using mult_right_mono_neg [of a 0 b] by simp |
25230 | 2079 |
|
63325 | 2080 |
lemma split_mult_pos_le: "(0 \<le> a \<and> 0 \<le> b) \<or> (a \<le> 0 \<and> b \<le> 0) \<Longrightarrow> 0 \<le> a * b" |
2081 |
by (auto simp add: mult_nonpos_nonpos) |
|
25186 | 2082 |
|
2083 |
end |
|
14270 | 2084 |
|
64290 | 2085 |
class abs_if = minus + uminus + ord + zero + abs + |
2086 |
assumes abs_if: "\<bar>a\<bar> = (if a < 0 then - a else a)" |
|
2087 |
||
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
2088 |
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
|
2089 |
begin |
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset
|
2090 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
2091 |
subclass ordered_ring .. |
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
2092 |
|
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
2093 |
subclass ordered_ab_group_add_abs |
28823 | 2094 |
proof |
25304
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset
|
2095 |
fix a b |
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset
|
2096 |
show "\<bar>a + b\<bar> \<le> \<bar>a\<bar> + \<bar>b\<bar>" |
63325 | 2097 |
by (auto simp add: abs_if not_le not_less algebra_simps |
2098 |
simp del: add.commute dest: add_neg_neg add_nonneg_nonneg) |
|
63588 | 2099 |
qed (auto simp: abs_if) |
25304
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset
|
2100 |
|
35631
0b8a5fd339ab
generalize some lemmas from class linordered_ring_strict to linordered_ring
huffman
parents:
35302
diff
changeset
|
2101 |
lemma zero_le_square [simp]: "0 \<le> a * a" |
63325 | 2102 |
using linear [of 0 a] 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
|
2103 |
|
0b8a5fd339ab
generalize some lemmas from class linordered_ring_strict to linordered_ring
huffman
parents:
35302
diff
changeset
|
2104 |
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
|
2105 |
by (simp add: not_less) |
0b8a5fd339ab
generalize some lemmas from class linordered_ring_strict to linordered_ring
huffman
parents:
35302
diff
changeset
|
2106 |
|
61944 | 2107 |
proposition abs_eq_iff: "\<bar>x\<bar> = \<bar>y\<bar> \<longleftrightarrow> x = y \<or> x = -y" |
62390 | 2108 |
by (auto simp add: abs_if split: if_split_asm) |
61762
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61649
diff
changeset
|
2109 |
|
64848
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64713
diff
changeset
|
2110 |
lemma abs_eq_iff': |
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64713
diff
changeset
|
2111 |
"\<bar>a\<bar> = b \<longleftrightarrow> b \<ge> 0 \<and> (a = b \<or> a = - b)" |
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64713
diff
changeset
|
2112 |
by (cases "a \<ge> 0") auto |
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64713
diff
changeset
|
2113 |
|
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64713
diff
changeset
|
2114 |
lemma eq_abs_iff': |
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64713
diff
changeset
|
2115 |
"a = \<bar>b\<bar> \<longleftrightarrow> a \<ge> 0 \<and> (b = a \<or> b = - a)" |
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64713
diff
changeset
|
2116 |
using abs_eq_iff' [of b a] by auto |
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64713
diff
changeset
|
2117 |
|
63325 | 2118 |
lemma sum_squares_ge_zero: "0 \<le> x * x + y * y" |
62347 | 2119 |
by (intro add_nonneg_nonneg zero_le_square) |
2120 |
||
63325 | 2121 |
lemma not_sum_squares_lt_zero: "\<not> x * x + y * y < 0" |
62347 | 2122 |
by (simp add: not_less sum_squares_ge_zero) |
2123 |
||
25304
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset
|
2124 |
end |
23521 | 2125 |
|
35043
07dbdf60d5ad
dropped accidental duplication of "lin" prefix from cs. 108662d50512
haftmann
parents:
35032
diff
changeset
|
2126 |
class linordered_ring_strict = ring + linordered_semiring_strict |
25304
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset
|
2127 |
+ ordered_ab_group_add + abs_if |
25230 | 2128 |
begin |
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset
|
2129 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
2130 |
subclass linordered_ring .. |
25304
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset
|
2131 |
|
30692
44ea10bc07a7
clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents:
30650
diff
changeset
|
2132 |
lemma mult_strict_left_mono_neg: "b < a \<Longrightarrow> c < 0 \<Longrightarrow> c * a < c * b" |
63325 | 2133 |
using mult_strict_left_mono [of b a "- c"] by simp |
30692
44ea10bc07a7
clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents:
30650
diff
changeset
|
2134 |
|
44ea10bc07a7
clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents:
30650
diff
changeset
|
2135 |
lemma mult_strict_right_mono_neg: "b < a \<Longrightarrow> c < 0 \<Longrightarrow> a * c < b * c" |
63325 | 2136 |
using mult_strict_right_mono [of b a "- c"] by simp |
30692
44ea10bc07a7
clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents:
30650
diff
changeset
|
2137 |
|
44ea10bc07a7
clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents:
30650
diff
changeset
|
2138 |
lemma mult_neg_neg: "a < 0 \<Longrightarrow> b < 0 \<Longrightarrow> 0 < a * b" |
63325 | 2139 |
using mult_strict_right_mono_neg [of a 0 b] by simp |
14738 | 2140 |
|
25917 | 2141 |
subclass ring_no_zero_divisors |
28823 | 2142 |
proof |
25917 | 2143 |
fix a b |
63325 | 2144 |
assume "a \<noteq> 0" |
63588 | 2145 |
then have a: "a < 0 \<or> 0 < a" by (simp add: neq_iff) |
63325 | 2146 |
assume "b \<noteq> 0" |
63588 | 2147 |
then have b: "b < 0 \<or> 0 < b" by (simp add: neq_iff) |
25917 | 2148 |
have "a * b < 0 \<or> 0 < a * b" |
2149 |
proof (cases "a < 0") |
|
63588 | 2150 |
case True |
63325 | 2151 |
show ?thesis |
2152 |
proof (cases "b < 0") |
|
2153 |
case True |
|
63588 | 2154 |
with \<open>a < 0\<close> show ?thesis by (auto dest: mult_neg_neg) |
25917 | 2155 |
next |
63325 | 2156 |
case False |
63588 | 2157 |
with b have "0 < b" by auto |
2158 |
with \<open>a < 0\<close> show ?thesis by (auto dest: mult_strict_right_mono) |
|
25917 | 2159 |
qed |
2160 |
next |
|
63325 | 2161 |
case False |
63588 | 2162 |
with a have "0 < a" by auto |
63325 | 2163 |
show ?thesis |
2164 |
proof (cases "b < 0") |
|
2165 |
case True |
|
63588 | 2166 |
with \<open>0 < a\<close> show ?thesis |
63325 | 2167 |
by (auto dest: mult_strict_right_mono_neg) |
25917 | 2168 |
next |
63325 | 2169 |
case False |
63588 | 2170 |
with b have "0 < b" by auto |
2171 |
with \<open>0 < a\<close> show ?thesis by auto |
|
25917 | 2172 |
qed |
2173 |
qed |
|
63325 | 2174 |
then show "a * b \<noteq> 0" |
2175 |
by (simp add: neq_iff) |
|
25917 | 2176 |
qed |
25304
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset
|
2177 |
|
56480
093ea91498e6
field_simps: better support for negation and division, and power
hoelzl
parents:
56217
diff
changeset
|
2178 |
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
|
2179 |
by (cases a 0 b 0 rule: linorder_cases[case_product linorder_cases]) |
56544 | 2180 |
(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
|
2181 |
|
56480
093ea91498e6
field_simps: better support for negation and division, and power
hoelzl
parents:
56217
diff
changeset
|
2182 |
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
|
2183 |
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
|
2184 |
|
63325 | 2185 |
lemma mult_less_0_iff: "a * b < 0 \<longleftrightarrow> 0 < a \<and> b < 0 \<or> a < 0 \<and> 0 < b" |
2186 |
using zero_less_mult_iff [of "- a" b] by auto |
|
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
2187 |
|
63325 | 2188 |
lemma mult_le_0_iff: "a * b \<le> 0 \<longleftrightarrow> 0 \<le> a \<and> b \<le> 0 \<or> a \<le> 0 \<and> 0 \<le> b" |
2189 |
using zero_le_mult_iff [of "- a" b] by auto |
|
25917 | 2190 |
|
63325 | 2191 |
text \<open> |
69593 | 2192 |
Cancellation laws for \<^term>\<open>c * a < c * b\<close> and \<^term>\<open>a * c < b * c\<close>, |
63325 | 2193 |
also with the relations \<open>\<le>\<close> and equality. |
2194 |
\<close> |
|
26193 | 2195 |
|
63325 | 2196 |
text \<open> |
2197 |
These ``disjunction'' versions produce two cases when the comparison is |
|
2198 |
an assumption, but effectively four when the comparison is a goal. |
|
2199 |
\<close> |
|
26193 | 2200 |
|
63325 | 2201 |
lemma mult_less_cancel_right_disj: "a * c < b * c \<longleftrightarrow> 0 < c \<and> a < b \<or> c < 0 \<and> b < a" |
26193 | 2202 |
apply (cases "c = 0") |
63588 | 2203 |
apply (auto simp add: neq_iff mult_strict_right_mono mult_strict_right_mono_neg) |
2204 |
apply (auto simp add: not_less not_le [symmetric, of "a*c"] not_le [symmetric, of a]) |
|
2205 |
apply (erule_tac [!] notE) |
|
2206 |
apply (auto simp add: less_imp_le mult_right_mono mult_right_mono_neg) |
|
26193 | 2207 |
done |
2208 |
||
63325 | 2209 |
lemma mult_less_cancel_left_disj: "c * a < c * b \<longleftrightarrow> 0 < c \<and> a < b \<or> c < 0 \<and> b < a" |
26193 | 2210 |
apply (cases "c = 0") |
63588 | 2211 |
apply (auto simp add: neq_iff mult_strict_left_mono mult_strict_left_mono_neg) |
2212 |
apply (auto simp add: not_less not_le [symmetric, of "c * a"] not_le [symmetric, of a]) |
|
2213 |
apply (erule_tac [!] notE) |
|
2214 |
apply (auto simp add: less_imp_le mult_left_mono mult_left_mono_neg) |
|
26193 | 2215 |
done |
2216 |
||
63325 | 2217 |
text \<open> |
2218 |
The ``conjunction of implication'' lemmas produce two cases when the |
|
2219 |
comparison is a goal, but give four when the comparison is an assumption. |
|
2220 |
\<close> |
|
26193 | 2221 |
|
63325 | 2222 |
lemma mult_less_cancel_right: "a * c < b * c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < b) \<and> (c \<le> 0 \<longrightarrow> b < a)" |
26193 | 2223 |
using mult_less_cancel_right_disj [of a c b] by auto |
2224 |
||
63325 | 2225 |
lemma mult_less_cancel_left: "c * a < c * b \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < b) \<and> (c \<le> 0 \<longrightarrow> b < a)" |
26193 | 2226 |
using mult_less_cancel_left_disj [of c a b] by auto |
2227 |
||
63325 | 2228 |
lemma mult_le_cancel_right: "a * c \<le> b * c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)" |
2229 |
by (simp add: not_less [symmetric] mult_less_cancel_right_disj) |
|
26193 | 2230 |
|
63325 | 2231 |
lemma mult_le_cancel_left: "c * a \<le> c * b \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)" |
2232 |
by (simp add: not_less [symmetric] mult_less_cancel_left_disj) |
|
26193 | 2233 |
|
63325 | 2234 |
lemma mult_le_cancel_left_pos: "0 < c \<Longrightarrow> c * a \<le> c * b \<longleftrightarrow> a \<le> b" |
2235 |
by (auto simp: mult_le_cancel_left) |
|
30649
57753e0ec1d4
1. New cancellation simprocs for common factors in inequations
nipkow
parents:
30242
diff
changeset
|
2236 |
|
63325 | 2237 |
lemma mult_le_cancel_left_neg: "c < 0 \<Longrightarrow> c * a \<le> c * b \<longleftrightarrow> b \<le> a" |
2238 |
by (auto simp: mult_le_cancel_left) |
|
30649
57753e0ec1d4
1. New cancellation simprocs for common factors in inequations
nipkow
parents:
30242
diff
changeset
|
2239 |
|
63325 | 2240 |
lemma mult_less_cancel_left_pos: "0 < c \<Longrightarrow> c * a < c * b \<longleftrightarrow> a < b" |
2241 |
by (auto simp: mult_less_cancel_left) |
|
30649
57753e0ec1d4
1. New cancellation simprocs for common factors in inequations
nipkow
parents:
30242
diff
changeset
|
2242 |
|
63325 | 2243 |
lemma mult_less_cancel_left_neg: "c < 0 \<Longrightarrow> c * a < c * b \<longleftrightarrow> b < a" |
2244 |
by (auto simp: mult_less_cancel_left) |
|
30649
57753e0ec1d4
1. New cancellation simprocs for common factors in inequations
nipkow
parents:
30242
diff
changeset
|
2245 |
|
25917 | 2246 |
end |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
2247 |
|
30692
44ea10bc07a7
clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents:
30650
diff
changeset
|
2248 |
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
|
2249 |
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
|
2250 |
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
|
2251 |
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
|
2252 |
mult_neg_pos mult_neg_neg |
25230 | 2253 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
2254 |
class ordered_comm_ring = comm_ring + ordered_comm_semiring |
25267 | 2255 |
begin |
25230 | 2256 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
2257 |
subclass ordered_ring .. |
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
2258 |
subclass ordered_cancel_comm_semiring .. |
25230 | 2259 |
|
25267 | 2260 |
end |
25230 | 2261 |
|
67689
2c38ffd6ec71
type class linordered_nonzero_semiring has new axiom to guarantee characteristic 0
paulson <lp15@cam.ac.uk>
parents:
67234
diff
changeset
|
2262 |
class linordered_nonzero_semiring = ordered_comm_semiring + monoid_mult + linorder + zero_less_one + |
2c38ffd6ec71
type class linordered_nonzero_semiring has new axiom to guarantee characteristic 0
paulson <lp15@cam.ac.uk>
parents:
67234
diff
changeset
|
2263 |
assumes add_mono1: "a < b \<Longrightarrow> a + 1 < b + 1" |
62378
85ed00c1fe7c
generalize more theorems to support enat and ennreal
hoelzl
parents:
62377
diff
changeset
|
2264 |
begin |
85ed00c1fe7c
generalize more theorems to support enat and ennreal
hoelzl
parents:
62377
diff
changeset
|
2265 |
|
85ed00c1fe7c
generalize more theorems to support enat and ennreal
hoelzl
parents:
62377
diff
changeset
|
2266 |
subclass zero_neq_one |
63325 | 2267 |
by standard (insert zero_less_one, blast) |
62378
85ed00c1fe7c
generalize more theorems to support enat and ennreal
hoelzl
parents:
62377
diff
changeset
|
2268 |
|
85ed00c1fe7c
generalize more theorems to support enat and ennreal
hoelzl
parents:
62377
diff
changeset
|
2269 |
subclass comm_semiring_1 |
63325 | 2270 |
by standard (rule mult_1_left) |
62378
85ed00c1fe7c
generalize more theorems to support enat and ennreal
hoelzl
parents:
62377
diff
changeset
|
2271 |
|
85ed00c1fe7c
generalize more theorems to support enat and ennreal
hoelzl
parents:
62377
diff
changeset
|
2272 |
lemma zero_le_one [simp]: "0 \<le> 1" |
63325 | 2273 |
by (rule zero_less_one [THEN less_imp_le]) |
62378
85ed00c1fe7c
generalize more theorems to support enat and ennreal
hoelzl
parents:
62377
diff
changeset
|
2274 |
|
85ed00c1fe7c
generalize more theorems to support enat and ennreal
hoelzl
parents:
62377
diff
changeset
|
2275 |
lemma not_one_le_zero [simp]: "\<not> 1 \<le> 0" |
63325 | 2276 |
by (simp add: not_le) |
62378
85ed00c1fe7c
generalize more theorems to support enat and ennreal
hoelzl
parents:
62377
diff
changeset
|
2277 |
|
85ed00c1fe7c
generalize more theorems to support enat and ennreal
hoelzl
parents:
62377
diff
changeset
|
2278 |
lemma not_one_less_zero [simp]: "\<not> 1 < 0" |
63325 | 2279 |
by (simp add: not_less) |
62378
85ed00c1fe7c
generalize more theorems to support enat and ennreal
hoelzl
parents:
62377
diff
changeset
|
2280 |
|
85ed00c1fe7c
generalize more theorems to support enat and ennreal
hoelzl
parents:
62377
diff
changeset
|
2281 |
lemma mult_left_le: "c \<le> 1 \<Longrightarrow> 0 \<le> a \<Longrightarrow> a * c \<le> a" |
85ed00c1fe7c
generalize more theorems to support enat and ennreal
hoelzl
parents:
62377
diff
changeset
|
2282 |
using mult_left_mono[of c 1 a] by simp |
85ed00c1fe7c
generalize more theorems to support enat and ennreal
hoelzl
parents:
62377
diff
changeset
|
2283 |
|
85ed00c1fe7c
generalize more theorems to support enat and ennreal
hoelzl
parents:
62377
diff
changeset
|
2284 |
lemma mult_le_one: "a \<le> 1 \<Longrightarrow> 0 \<le> b \<Longrightarrow> b \<le> 1 \<Longrightarrow> a * b \<le> 1" |
85ed00c1fe7c
generalize more theorems to support enat and ennreal
hoelzl
parents:
62377
diff
changeset
|
2285 |
using mult_mono[of a 1 b 1] by simp |
85ed00c1fe7c
generalize more theorems to support enat and ennreal
hoelzl
parents:
62377
diff
changeset
|
2286 |
|
85ed00c1fe7c
generalize more theorems to support enat and ennreal
hoelzl
parents:
62377
diff
changeset
|
2287 |
lemma zero_less_two: "0 < 1 + 1" |
85ed00c1fe7c
generalize more theorems to support enat and ennreal
hoelzl
parents:
62377
diff
changeset
|
2288 |
using add_pos_pos[OF zero_less_one zero_less_one] . |
85ed00c1fe7c
generalize more theorems to support enat and ennreal
hoelzl
parents:
62377
diff
changeset
|
2289 |
|
85ed00c1fe7c
generalize more theorems to support enat and ennreal
hoelzl
parents:
62377
diff
changeset
|
2290 |
end |
85ed00c1fe7c
generalize more theorems to support enat and ennreal
hoelzl
parents:
62377
diff
changeset
|
2291 |
|
85ed00c1fe7c
generalize more theorems to support enat and ennreal
hoelzl
parents:
62377
diff
changeset
|
2292 |
class linordered_semidom = semidom + linordered_comm_semiring_strict + zero_less_one + |
60562
24af00b010cf
Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents:
60529
diff
changeset
|
2293 |
assumes le_add_diff_inverse2 [simp]: "b \<le> a \<Longrightarrow> a - b + b = a" |
25230 | 2294 |
begin |
2295 |
||
67689
2c38ffd6ec71
type class linordered_nonzero_semiring has new axiom to guarantee characteristic 0
paulson <lp15@cam.ac.uk>
parents:
67234
diff
changeset
|
2296 |
subclass linordered_nonzero_semiring |
2c38ffd6ec71
type class linordered_nonzero_semiring has new axiom to guarantee characteristic 0
paulson <lp15@cam.ac.uk>
parents:
67234
diff
changeset
|
2297 |
proof |
2c38ffd6ec71
type class linordered_nonzero_semiring has new axiom to guarantee characteristic 0
paulson <lp15@cam.ac.uk>
parents:
67234
diff
changeset
|
2298 |
show "a + 1 < b + 1" if "a < b" for a b |
2c38ffd6ec71
type class linordered_nonzero_semiring has new axiom to guarantee characteristic 0
paulson <lp15@cam.ac.uk>
parents:
67234
diff
changeset
|
2299 |
proof (rule ccontr, simp add: not_less) |
2c38ffd6ec71
type class linordered_nonzero_semiring has new axiom to guarantee characteristic 0
paulson <lp15@cam.ac.uk>
parents:
67234
diff
changeset
|
2300 |
assume "b \<le> a" |
2c38ffd6ec71
type class linordered_nonzero_semiring has new axiom to guarantee characteristic 0
paulson <lp15@cam.ac.uk>
parents:
67234
diff
changeset
|
2301 |
with that show False |
2c38ffd6ec71
type class linordered_nonzero_semiring has new axiom to guarantee characteristic 0
paulson <lp15@cam.ac.uk>
parents:
67234
diff
changeset
|
2302 |
by (simp add: ) |
2c38ffd6ec71
type class linordered_nonzero_semiring has new axiom to guarantee characteristic 0
paulson <lp15@cam.ac.uk>
parents:
67234
diff
changeset
|
2303 |
qed |
2c38ffd6ec71
type class linordered_nonzero_semiring has new axiom to guarantee characteristic 0
paulson <lp15@cam.ac.uk>
parents:
67234
diff
changeset
|
2304 |
qed |
62378
85ed00c1fe7c
generalize more theorems to support enat and ennreal
hoelzl
parents:
62377
diff
changeset
|
2305 |
|
60758 | 2306 |
text \<open>Addition is the inverse of subtraction.\<close> |
60562
24af00b010cf
Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents:
60529
diff
changeset
|
2307 |
|
24af00b010cf
Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents:
60529
diff
changeset
|
2308 |
lemma le_add_diff_inverse [simp]: "b \<le> a \<Longrightarrow> b + (a - b) = a" |
24af00b010cf
Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents:
60529
diff
changeset
|
2309 |
by (frule le_add_diff_inverse2) (simp add: add.commute) |
24af00b010cf
Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents:
60529
diff
changeset
|
2310 |
|
62378
85ed00c1fe7c
generalize more theorems to support enat and ennreal
hoelzl
parents:
62377
diff
changeset
|
2311 |
lemma add_diff_inverse: "\<not> a < b \<Longrightarrow> b + (a - b) = a" |
60562
24af00b010cf
Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents:
60529
diff
changeset
|
2312 |
by simp |
60615
e5fa1d5d3952
Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents:
60570
diff
changeset
|
2313 |
|
63325 | 2314 |
lemma add_le_imp_le_diff: "i + k \<le> n \<Longrightarrow> i \<le> n - k" |
60615
e5fa1d5d3952
Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents:
60570
diff
changeset
|
2315 |
apply (subst add_le_cancel_right [where c=k, symmetric]) |
e5fa1d5d3952
Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents:
60570
diff
changeset
|
2316 |
apply (frule le_add_diff_inverse2) |
e5fa1d5d3952
Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents:
60570
diff
changeset
|
2317 |
apply (simp only: add.assoc [symmetric]) |
63588 | 2318 |
using add_implies_diff |
2319 |
apply fastforce |
|
63325 | 2320 |
done |
60615
e5fa1d5d3952
Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents:
60570
diff
changeset
|
2321 |
|
62376
85f38d5f8807
Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
hoelzl
parents:
62366
diff
changeset
|
2322 |
lemma add_le_add_imp_diff_le: |
63325 | 2323 |
assumes 1: "i + k \<le> n" |
2324 |
and 2: "n \<le> j + k" |
|
2325 |
shows "i + k \<le> n \<Longrightarrow> n \<le> j + k \<Longrightarrow> n - k \<le> j" |
|
60615
e5fa1d5d3952
Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents:
60570
diff
changeset
|
2326 |
proof - |
e5fa1d5d3952
Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents:
60570
diff
changeset
|
2327 |
have "n - (i + k) + (i + k) = n" |
63325 | 2328 |
using 1 by simp |
60615
e5fa1d5d3952
Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents:
60570
diff
changeset
|
2329 |
moreover have "n - k = n - k - i + i" |
63325 | 2330 |
using 1 by (simp add: add_le_imp_le_diff) |
60615
e5fa1d5d3952
Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents:
60570
diff
changeset
|
2331 |
ultimately show ?thesis |
63325 | 2332 |
using 2 |
60615
e5fa1d5d3952
Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents:
60570
diff
changeset
|
2333 |
apply (simp add: add.assoc [symmetric]) |
63325 | 2334 |
apply (rule add_le_imp_le_diff [of _ k "j + k", simplified add_diff_cancel_right']) |
2335 |
apply (simp add: add.commute diff_diff_add) |
|
2336 |
done |
|
60615
e5fa1d5d3952
Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents:
60570
diff
changeset
|
2337 |
qed |
e5fa1d5d3952
Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents:
60570
diff
changeset
|
2338 |
|
63325 | 2339 |
lemma less_1_mult: "1 < m \<Longrightarrow> 1 < n \<Longrightarrow> 1 < m * n" |
62378
85ed00c1fe7c
generalize more theorems to support enat and ennreal
hoelzl
parents:
62377
diff
changeset
|
2340 |
using mult_strict_mono [of 1 m 1 n] by (simp add: less_trans [OF zero_less_one]) |
59000 | 2341 |
|
25230 | 2342 |
end |
2343 |
||
66937 | 2344 |
class linordered_idom = comm_ring_1 + linordered_comm_semiring_strict + |
2345 |
ordered_ab_group_add + abs_if + sgn + |
|
64290 | 2346 |
assumes sgn_if: "sgn x = (if x = 0 then 0 else if 0 < x then 1 else - 1)" |
25917 | 2347 |
begin |
2348 |
||
35043
07dbdf60d5ad
dropped accidental duplication of "lin" prefix from cs. 108662d50512
haftmann
parents:
35032
diff
changeset
|
2349 |
subclass linordered_ring_strict .. |
66937 | 2350 |
|
2351 |
subclass linordered_semiring_1_strict |
|
2352 |
proof |
|
2353 |
have "0 \<le> 1 * 1" |
|
2354 |
by (fact zero_le_square) |
|
2355 |
then show "0 < 1" |
|
2356 |
by (simp add: le_less) |
|
2357 |
qed |
|
2358 |
||
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
2359 |
subclass ordered_comm_ring .. |
27516 | 2360 |
subclass idom .. |
25917 | 2361 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
2362 |
subclass linordered_semidom |
66937 | 2363 |
by standard simp |
25917 | 2364 |
|
64290 | 2365 |
subclass idom_abs_sgn |
2366 |
by standard |
|
2367 |
(auto simp add: sgn_if abs_if zero_less_mult_iff) |
|
2368 |
||
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
2369 |
lemma linorder_neqE_linordered_idom: |
63325 | 2370 |
assumes "x \<noteq> y" |
2371 |
obtains "x < y" | "y < x" |
|
26193 | 2372 |
using assms by (rule neqE) |
2373 |
||
63588 | 2374 |
text \<open>These cancellation simp rules also produce two cases when the comparison is a goal.\<close> |
26274 | 2375 |
|
63325 | 2376 |
lemma mult_le_cancel_right1: "c \<le> b * c \<longleftrightarrow> (0 < c \<longrightarrow> 1 \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> 1)" |
2377 |
using mult_le_cancel_right [of 1 c b] by simp |
|
26274 | 2378 |
|
63325 | 2379 |
lemma mult_le_cancel_right2: "a * c \<le> c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> 1) \<and> (c < 0 \<longrightarrow> 1 \<le> a)" |
2380 |
using mult_le_cancel_right [of a c 1] by simp |
|
26274 | 2381 |
|
63325 | 2382 |
lemma mult_le_cancel_left1: "c \<le> c * b \<longleftrightarrow> (0 < c \<longrightarrow> 1 \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> 1)" |
2383 |
using mult_le_cancel_left [of c 1 b] by simp |
|
26274 | 2384 |
|
63325 | 2385 |
lemma mult_le_cancel_left2: "c * a \<le> c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> 1) \<and> (c < 0 \<longrightarrow> 1 \<le> a)" |
2386 |
using mult_le_cancel_left [of c a 1] by simp |
|
26274 | 2387 |
|
63325 | 2388 |
lemma mult_less_cancel_right1: "c < b * c \<longleftrightarrow> (0 \<le> c \<longrightarrow> 1 < b) \<and> (c \<le> 0 \<longrightarrow> b < 1)" |
2389 |
using mult_less_cancel_right [of 1 c b] by simp |
|
26274 | 2390 |
|
63325 | 2391 |
lemma mult_less_cancel_right2: "a * c < c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < 1) \<and> (c \<le> 0 \<longrightarrow> 1 < a)" |
2392 |
using mult_less_cancel_right [of a c 1] by simp |
|
26274 | 2393 |
|
63325 | 2394 |
lemma mult_less_cancel_left1: "c < c * b \<longleftrightarrow> (0 \<le> c \<longrightarrow> 1 < b) \<and> (c \<le> 0 \<longrightarrow> b < 1)" |
2395 |
using mult_less_cancel_left [of c 1 b] by simp |
|
26274 | 2396 |
|
63325 | 2397 |
lemma mult_less_cancel_left2: "c * a < c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < 1) \<and> (c \<le> 0 \<longrightarrow> 1 < a)" |
2398 |
using mult_less_cancel_left [of c a 1] by simp |
|
26274 | 2399 |
|
63325 | 2400 |
lemma sgn_0_0: "sgn a = 0 \<longleftrightarrow> a = 0" |
64290 | 2401 |
by (fact sgn_eq_0_iff) |
27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27516
diff
changeset
|
2402 |
|
63325 | 2403 |
lemma sgn_1_pos: "sgn a = 1 \<longleftrightarrow> a > 0" |
2404 |
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
|
2405 |
|
63325 | 2406 |
lemma sgn_1_neg: "sgn a = - 1 \<longleftrightarrow> a < 0" |
2407 |
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
|
2408 |
|
63325 | 2409 |
lemma sgn_pos [simp]: "0 < a \<Longrightarrow> sgn a = 1" |
2410 |
by (simp only: sgn_1_pos) |
|
29940 | 2411 |
|
63325 | 2412 |
lemma sgn_neg [simp]: "a < 0 \<Longrightarrow> sgn a = - 1" |
2413 |
by (simp only: sgn_1_neg) |
|
29940 | 2414 |
|
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
2415 |
lemma abs_sgn: "\<bar>k\<bar> = k * sgn k" |
63325 | 2416 |
unfolding sgn_if abs_if by auto |
29700 | 2417 |
|
63325 | 2418 |
lemma sgn_greater [simp]: "0 < sgn a \<longleftrightarrow> 0 < a" |
29940 | 2419 |
unfolding sgn_if by auto |
2420 |
||
63325 | 2421 |
lemma sgn_less [simp]: "sgn a < 0 \<longleftrightarrow> a < 0" |
29940 | 2422 |
unfolding sgn_if by auto |
2423 |
||
64239 | 2424 |
lemma abs_sgn_eq_1 [simp]: |
2425 |
"a \<noteq> 0 \<Longrightarrow> \<bar>sgn a\<bar> = 1" |
|
64290 | 2426 |
by simp |
64239 | 2427 |
|
63325 | 2428 |
lemma abs_sgn_eq: "\<bar>sgn a\<bar> = (if a = 0 then 0 else 1)" |
62347 | 2429 |
by (simp add: sgn_if) |
2430 |
||
64713 | 2431 |
lemma sgn_mult_self_eq [simp]: |
2432 |
"sgn a * sgn a = of_bool (a \<noteq> 0)" |
|
2433 |
by (cases "a > 0") simp_all |
|
2434 |
||
2435 |
lemma abs_mult_self_eq [simp]: |
|
2436 |
"\<bar>a\<bar> * \<bar>a\<bar> = a * a" |
|
2437 |
by (cases "a > 0") simp_all |
|
2438 |
||
2439 |
lemma same_sgn_sgn_add: |
|
2440 |
"sgn (a + b) = sgn a" if "sgn b = sgn a" |
|
2441 |
proof (cases a 0 rule: linorder_cases) |
|
2442 |
case equal |
|
2443 |
with that show ?thesis |
|
2444 |
by simp |
|
2445 |
next |
|
2446 |
case less |
|
2447 |
with that have "b < 0" |
|
2448 |
by (simp add: sgn_1_neg) |
|
2449 |
with \<open>a < 0\<close> have "a + b < 0" |
|
2450 |
by (rule add_neg_neg) |
|
2451 |
with \<open>a < 0\<close> show ?thesis |
|
2452 |
by simp |
|
2453 |
next |
|
2454 |
case greater |
|
2455 |
with that have "b > 0" |
|
2456 |
by (simp add: sgn_1_pos) |
|
2457 |
with \<open>a > 0\<close> have "a + b > 0" |
|
2458 |
by (rule add_pos_pos) |
|
2459 |
with \<open>a > 0\<close> show ?thesis |
|
2460 |
by simp |
|
2461 |
qed |
|
2462 |
||
2463 |
lemma same_sgn_abs_add: |
|
2464 |
"\<bar>a + b\<bar> = \<bar>a\<bar> + \<bar>b\<bar>" if "sgn b = sgn a" |
|
2465 |
proof - |
|
2466 |
have "a + b = sgn a * \<bar>a\<bar> + sgn b * \<bar>b\<bar>" |
|
2467 |
by (simp add: sgn_mult_abs) |
|
2468 |
also have "\<dots> = sgn a * (\<bar>a\<bar> + \<bar>b\<bar>)" |
|
2469 |
using that by (simp add: algebra_simps) |
|
2470 |
finally show ?thesis |
|
2471 |
by (auto simp add: abs_mult) |
|
2472 |
qed |
|
2473 |
||
66816
212a3334e7da
more fundamental definition of div and mod on int
haftmann
parents:
66810
diff
changeset
|
2474 |
lemma sgn_not_eq_imp: |
212a3334e7da
more fundamental definition of div and mod on int
haftmann
parents:
66810
diff
changeset
|
2475 |
"sgn a = - sgn b" if "sgn b \<noteq> sgn a" and "sgn a \<noteq> 0" and "sgn b \<noteq> 0" |
212a3334e7da
more fundamental definition of div and mod on int
haftmann
parents:
66810
diff
changeset
|
2476 |
using that by (cases "a < 0") (auto simp add: sgn_0_0 sgn_1_pos sgn_1_neg) |
212a3334e7da
more fundamental definition of div and mod on int
haftmann
parents:
66810
diff
changeset
|
2477 |
|
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
2478 |
lemma abs_dvd_iff [simp]: "\<bar>m\<bar> dvd k \<longleftrightarrow> m dvd k" |
29949 | 2479 |
by (simp add: abs_if) |
2480 |
||
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
2481 |
lemma dvd_abs_iff [simp]: "m dvd \<bar>k\<bar> \<longleftrightarrow> m dvd k" |
29949 | 2482 |
by (simp add: abs_if) |
29653 | 2483 |
|
63325 | 2484 |
lemma dvd_if_abs_eq: "\<bar>l\<bar> = \<bar>k\<bar> \<Longrightarrow> l dvd k" |
2485 |
by (subst abs_dvd_iff [symmetric]) simp |
|
33676
802f5e233e48
moved lemma from Algebra/IntRing to Ring_and_Field
nipkow
parents:
33364
diff
changeset
|
2486 |
|
63325 | 2487 |
text \<open> |
2488 |
The following lemmas can be proven in more general structures, but |
|
2489 |
are dangerous as simp rules in absence of @{thm neg_equal_zero}, |
|
2490 |
@{thm neg_less_pos}, @{thm neg_less_eq_nonneg}. |
|
2491 |
\<close> |
|
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54250
diff
changeset
|
2492 |
|
63325 | 2493 |
lemma equation_minus_iff_1 [simp, no_atp]: "1 = - a \<longleftrightarrow> a = - 1" |
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54250
diff
changeset
|
2494 |
by (fact equation_minus_iff) |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54250
diff
changeset
|
2495 |
|
63325 | 2496 |
lemma minus_equation_iff_1 [simp, no_atp]: "- a = 1 \<longleftrightarrow> a = - 1" |
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54250
diff
changeset
|
2497 |
by (subst minus_equation_iff, auto) |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54250
diff
changeset
|
2498 |
|
63325 | 2499 |
lemma le_minus_iff_1 [simp, no_atp]: "1 \<le> - b \<longleftrightarrow> b \<le> - 1" |
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54250
diff
changeset
|
2500 |
by (fact le_minus_iff) |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54250
diff
changeset
|
2501 |
|
63325 | 2502 |
lemma minus_le_iff_1 [simp, no_atp]: "- a \<le> 1 \<longleftrightarrow> - 1 \<le> a" |
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54250
diff
changeset
|
2503 |
by (fact minus_le_iff) |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54250
diff
changeset
|
2504 |
|
63325 | 2505 |
lemma less_minus_iff_1 [simp, no_atp]: "1 < - b \<longleftrightarrow> b < - 1" |
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54250
diff
changeset
|
2506 |
by (fact less_minus_iff) |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54250
diff
changeset
|
2507 |
|
63325 | 2508 |
lemma minus_less_iff_1 [simp, no_atp]: "- a < 1 \<longleftrightarrow> - 1 < a" |
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54250
diff
changeset
|
2509 |
by (fact minus_less_iff) |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54250
diff
changeset
|
2510 |
|
66793
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
65811
diff
changeset
|
2511 |
lemma add_less_zeroD: |
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
65811
diff
changeset
|
2512 |
shows "x+y < 0 \<Longrightarrow> x<0 \<or> y<0" |
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
65811
diff
changeset
|
2513 |
by (auto simp: not_less intro: le_less_trans [of _ "x+y"]) |
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
65811
diff
changeset
|
2514 |
|
25917 | 2515 |
end |
25230 | 2516 |
|
60758 | 2517 |
text \<open>Simprules for comparisons where common factors can be cancelled.\<close> |
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
2518 |
|
54147
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
52435
diff
changeset
|
2519 |
lemmas mult_compare_simps = |
63325 | 2520 |
mult_le_cancel_right mult_le_cancel_left |
2521 |
mult_le_cancel_right1 mult_le_cancel_right2 |
|
2522 |
mult_le_cancel_left1 mult_le_cancel_left2 |
|
2523 |
mult_less_cancel_right mult_less_cancel_left |
|
2524 |
mult_less_cancel_right1 mult_less_cancel_right2 |
|
2525 |
mult_less_cancel_left1 mult_less_cancel_left2 |
|
2526 |
mult_cancel_right mult_cancel_left |
|
2527 |
mult_cancel_right1 mult_cancel_right2 |
|
2528 |
mult_cancel_left1 mult_cancel_left2 |
|
2529 |
||
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
2530 |
|
60758 | 2531 |
text \<open>Reasoning about inequalities with division\<close> |
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
2532 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
2533 |
context linordered_semidom |
25193 | 2534 |
begin |
2535 |
||
2536 |
lemma less_add_one: "a < a + 1" |
|
14293 | 2537 |
proof - |
25193 | 2538 |
have "a + 0 < a + 1" |
23482 | 2539 |
by (blast intro: zero_less_one add_strict_left_mono) |
63325 | 2540 |
then show ?thesis by simp |
14293 | 2541 |
qed |
2542 |
||
25193 | 2543 |
end |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
2544 |
|
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
2545 |
context linordered_idom |
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
2546 |
begin |
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
2547 |
|
63325 | 2548 |
lemma mult_right_le_one_le: "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
|
2549 |
by (rule mult_left_le) |
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
2550 |
|
63325 | 2551 |
lemma mult_left_le_one_le: "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> y * x \<le> x" |
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
2552 |
by (auto simp add: mult_le_cancel_right2) |
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
2553 |
|
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
2554 |
end |
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
2555 |
|
60758 | 2556 |
text \<open>Absolute Value\<close> |
14293 | 2557 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
2558 |
context linordered_idom |
25304
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset
|
2559 |
begin |
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset
|
2560 |
|
63325 | 2561 |
lemma mult_sgn_abs: "sgn x * \<bar>x\<bar> = x" |
64290 | 2562 |
by (fact sgn_mult_abs) |
25304
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset
|
2563 |
|
64290 | 2564 |
lemma abs_one: "\<bar>1\<bar> = 1" |
2565 |
by (fact abs_1) |
|
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
2566 |
|
25304
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset
|
2567 |
end |
24491 | 2568 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
2569 |
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
|
2570 |
assumes abs_eq_mult: |
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset
|
2571 |
"(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
|
2572 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
2573 |
context linordered_idom |
30961 | 2574 |
begin |
2575 |
||
63325 | 2576 |
subclass ordered_ring_abs |
63588 | 2577 |
by standard (auto simp: abs_if not_less mult_less_0_iff) |
30961 | 2578 |
|
67051 | 2579 |
lemma abs_mult_self: "\<bar>a\<bar> * \<bar>a\<bar> = a * a" |
2580 |
by (fact abs_mult_self_eq) |
|
30961 | 2581 |
|
14294
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
2582 |
lemma abs_mult_less: |
63325 | 2583 |
assumes ac: "\<bar>a\<bar> < c" |
2584 |
and bd: "\<bar>b\<bar> < d" |
|
2585 |
shows "\<bar>a\<bar> * \<bar>b\<bar> < c * d" |
|
14294
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
2586 |
proof - |
63325 | 2587 |
from ac have "0 < c" |
2588 |
by (blast intro: le_less_trans abs_ge_zero) |
|
2589 |
with bd show ?thesis by (simp add: ac mult_strict_mono) |
|
14294
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
2590 |
qed |
14293 | 2591 |
|
63325 | 2592 |
lemma abs_less_iff: "\<bar>a\<bar> < b \<longleftrightarrow> a < b \<and> - a < b" |
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
2593 |
by (simp add: less_le abs_le_iff) (auto simp add: abs_if) |
14738 | 2594 |
|
63325 | 2595 |
lemma abs_mult_pos: "0 \<le> x \<Longrightarrow> \<bar>y\<bar> * x = \<bar>y * x\<bar>" |
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
2596 |
by (simp add: abs_mult) |
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
2597 |
|
63325 | 2598 |
lemma abs_diff_less_iff: "\<bar>x - a\<bar> < r \<longleftrightarrow> a - r < x \<and> x < a + r" |
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
|
2599 |
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
|
2600 |
|
63325 | 2601 |
lemma abs_diff_le_iff: "\<bar>x - a\<bar> \<le> r \<longleftrightarrow> a - r \<le> x \<and> x \<le> a + r" |
59865
8a20dd967385
rationalised and generalised some theorems concerning abs and x^2.
paulson <lp15@cam.ac.uk>
parents:
59833
diff
changeset
|
2602 |
by (auto simp add: diff_le_eq ac_simps abs_le_iff) |
8a20dd967385
rationalised and generalised some theorems concerning abs and x^2.
paulson <lp15@cam.ac.uk>
parents:
59833
diff
changeset
|
2603 |
|
62626
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62608
diff
changeset
|
2604 |
lemma abs_add_one_gt_zero: "0 < 1 + \<bar>x\<bar>" |
63325 | 2605 |
by (auto simp: abs_if not_less intro: zero_less_one add_strict_increasing less_trans) |
62626
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62608
diff
changeset
|
2606 |
|
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
2607 |
end |
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
2608 |
|
62376
85f38d5f8807
Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
hoelzl
parents:
62366
diff
changeset
|
2609 |
subsection \<open>Dioids\<close> |
85f38d5f8807
Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
hoelzl
parents:
62366
diff
changeset
|
2610 |
|
63325 | 2611 |
text \<open> |
2612 |
Dioids are the alternative extensions of semirings, a semiring can |
|
2613 |
either be a ring or a dioid but never both. |
|
2614 |
\<close> |
|
62376
85f38d5f8807
Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
hoelzl
parents:
62366
diff
changeset
|
2615 |
|
85f38d5f8807
Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
hoelzl
parents:
62366
diff
changeset
|
2616 |
class dioid = semiring_1 + canonically_ordered_monoid_add |
85f38d5f8807
Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
hoelzl
parents:
62366
diff
changeset
|
2617 |
begin |
85f38d5f8807
Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
hoelzl
parents:
62366
diff
changeset
|
2618 |
|
85f38d5f8807
Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
hoelzl
parents:
62366
diff
changeset
|
2619 |
subclass ordered_semiring |
63325 | 2620 |
by standard (auto simp: le_iff_add distrib_left distrib_right) |
62376
85f38d5f8807
Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
hoelzl
parents:
62366
diff
changeset
|
2621 |
|
85f38d5f8807
Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
hoelzl
parents:
62366
diff
changeset
|
2622 |
end |
85f38d5f8807
Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
hoelzl
parents:
62366
diff
changeset
|
2623 |
|
85f38d5f8807
Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
hoelzl
parents:
62366
diff
changeset
|
2624 |
|
59557 | 2625 |
hide_fact (open) comm_mult_left_mono comm_mult_strict_left_mono distrib |
2626 |
||
52435
6646bb548c6b
migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents:
51520
diff
changeset
|
2627 |
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
|
2628 |
code_module Rings \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith |
33364 | 2629 |
|
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
2630 |
end |