author | wenzelm |
Mon, 07 Dec 2015 10:38:04 +0100 | |
changeset 61799 | 4cf66f21b764 |
parent 61762 | d50b993b4fb9 |
child 61944 | 5d06ecfdb472 |
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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3 |
Author: Steven Obua |
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Author: Tobias Nipkow |
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Author: Lawrence C Paulson |
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parents:
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6 |
Author: Markus Wenzel |
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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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8 |
*) |
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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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9 |
|
60758 | 10 |
section \<open>Rings\<close> |
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parents:
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11 |
|
35050
9f841f20dca6
renamed OrderedGroup to Groups; split theory Ring_and_Field into Rings Fields
haftmann
parents:
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12 |
theory Rings |
9f841f20dca6
renamed OrderedGroup to Groups; split theory Ring_and_Field into Rings Fields
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parents:
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13 |
imports Groups |
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
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parents:
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18 |
assumes distrib_left[algebra_simps]: "a * (b + c) = a * b + a * c" |
25152 | 19 |
begin |
20 |
||
61799 | 21 |
text\<open>For the \<open>combine_numerals\<close> simproc\<close> |
25152 | 22 |
lemma combine_common_factor: |
23 |
"a * e + (b * e + c) = (a + b) * e + c" |
|
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24 |
by (simp add: distrib_right ac_simps) |
25152 | 25 |
|
26 |
end |
|
14504 | 27 |
|
22390 | 28 |
class mult_zero = times + zero + |
25062 | 29 |
assumes mult_zero_left [simp]: "0 * a = 0" |
30 |
assumes mult_zero_right [simp]: "a * 0 = 0" |
|
58195 | 31 |
begin |
32 |
||
33 |
lemma mult_not_zero: |
|
34 |
"a * b \<noteq> 0 \<Longrightarrow> a \<noteq> 0 \<and> b \<noteq> 0" |
|
35 |
by auto |
|
36 |
||
37 |
end |
|
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38 |
|
58198 | 39 |
class semiring_0 = semiring + comm_monoid_add + mult_zero |
40 |
||
29904 | 41 |
class semiring_0_cancel = semiring + cancel_comm_monoid_add |
25186 | 42 |
begin |
14504 | 43 |
|
25186 | 44 |
subclass semiring_0 |
28823 | 45 |
proof |
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parents:
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|
46 |
fix a :: 'a |
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Renamed {left,right}_distrib to distrib_{right,left}.
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parents:
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|
47 |
have "0 * a + 0 * a = 0 * a + 0" by (simp add: distrib_right [symmetric]) |
29667 | 48 |
thus "0 * a = 0" by (simp only: add_left_cancel) |
25152 | 49 |
next |
50 |
fix a :: 'a |
|
49962
a8cc904a6820
Renamed {left,right}_distrib to distrib_{right,left}.
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parents:
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51 |
have "a * 0 + a * 0 = a * 0 + 0" by (simp add: distrib_left [symmetric]) |
29667 | 52 |
thus "a * 0 = 0" by (simp only: add_left_cancel) |
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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 |
64 |
show "(a + b) * c = a * c + b * c" by (simp add: distrib) |
|
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65 |
have "a * (b + c) = (b + c) * a" by (simp add: ac_simps) |
14738 | 66 |
also have "... = b * a + c * a" by (simp only: distrib) |
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67 |
also have "... = a * b + a * c" by (simp add: ac_simps) |
14738 | 68 |
finally show "a * (b + c) = a * b + a * c" by blast |
14504 | 69 |
qed |
70 |
||
25152 | 71 |
end |
14504 | 72 |
|
25152 | 73 |
class comm_semiring_0 = comm_semiring + comm_monoid_add + mult_zero |
74 |
begin |
|
75 |
||
27516 | 76 |
subclass semiring_0 .. |
25152 | 77 |
|
78 |
end |
|
14504 | 79 |
|
29904 | 80 |
class comm_semiring_0_cancel = comm_semiring + cancel_comm_monoid_add |
25186 | 81 |
begin |
14940 | 82 |
|
27516 | 83 |
subclass semiring_0_cancel .. |
14940 | 84 |
|
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instances comm_semiring_0_cancel < comm_semiring_0, comm_ring < comm_semiring_0_cancel
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85 |
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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86 |
|
25186 | 87 |
end |
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88 |
|
22390 | 89 |
class zero_neq_one = zero + one + |
25062 | 90 |
assumes zero_neq_one [simp]: "0 \<noteq> 1" |
26193 | 91 |
begin |
92 |
||
93 |
lemma one_neq_zero [simp]: "1 \<noteq> 0" |
|
29667 | 94 |
by (rule not_sym) (rule zero_neq_one) |
26193 | 95 |
|
54225 | 96 |
definition of_bool :: "bool \<Rightarrow> 'a" |
97 |
where |
|
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98 |
"of_bool p = (if p then 1 else 0)" |
54225 | 99 |
|
100 |
lemma of_bool_eq [simp, code]: |
|
101 |
"of_bool False = 0" |
|
102 |
"of_bool True = 1" |
|
103 |
by (simp_all add: of_bool_def) |
|
104 |
||
105 |
lemma of_bool_eq_iff: |
|
106 |
"of_bool p = of_bool q \<longleftrightarrow> p = q" |
|
107 |
by (simp add: of_bool_def) |
|
108 |
||
55187 | 109 |
lemma split_of_bool [split]: |
110 |
"P (of_bool p) \<longleftrightarrow> (p \<longrightarrow> P 1) \<and> (\<not> p \<longrightarrow> P 0)" |
|
111 |
by (cases p) simp_all |
|
112 |
||
113 |
lemma split_of_bool_asm: |
|
114 |
"P (of_bool p) \<longleftrightarrow> \<not> (p \<and> \<not> P 1 \<or> \<not> p \<and> \<not> P 0)" |
|
115 |
by (cases p) simp_all |
|
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
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116 |
|
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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diff
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|
117 |
end |
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parents:
diff
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|
118 |
|
22390 | 119 |
class semiring_1 = zero_neq_one + semiring_0 + monoid_mult |
14504 | 120 |
|
60758 | 121 |
text \<open>Abstract divisibility\<close> |
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122 |
|
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123 |
class dvd = times |
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124 |
begin |
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125 |
|
50420 | 126 |
definition dvd :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infix "dvd" 50) where |
37767 | 127 |
"b dvd a \<longleftrightarrow> (\<exists>k. a = b * k)" |
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128 |
|
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129 |
lemma dvdI [intro?]: "a = b * k \<Longrightarrow> b dvd a" |
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parents:
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130 |
unfolding dvd_def .. |
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131 |
|
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132 |
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
|
133 |
unfolding dvd_def by blast |
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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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|
134 |
|
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
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parents:
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|
135 |
end |
16a26996c30e
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parents:
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|
136 |
|
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parents:
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diff
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|
137 |
context comm_monoid_mult |
25152 | 138 |
begin |
14738 | 139 |
|
59009
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generalized lemmas (particularly concerning dvd) as far as appropriate
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parents:
59000
diff
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|
140 |
subclass dvd . |
25152 | 141 |
|
59009
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generalized lemmas (particularly concerning dvd) as far as appropriate
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parents:
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142 |
lemma dvd_refl [simp]: |
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parents:
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143 |
"a dvd a" |
28559 | 144 |
proof |
145 |
show "a = a * 1" by simp |
|
27651
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parents:
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|
146 |
qed |
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27516
diff
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|
147 |
|
16a26996c30e
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parents:
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diff
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|
148 |
lemma dvd_trans: |
16a26996c30e
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parents:
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diff
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|
149 |
assumes "a dvd b" and "b dvd c" |
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27516
diff
changeset
|
150 |
shows "a dvd c" |
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27516
diff
changeset
|
151 |
proof - |
28559 | 152 |
from assms obtain v where "b = a * v" by (auto elim!: dvdE) |
153 |
moreover from assms obtain w where "c = b * w" by (auto elim!: dvdE) |
|
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haftmann
parents:
56544
diff
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|
154 |
ultimately have "c = a * (v * w)" by (simp add: mult.assoc) |
28559 | 155 |
then show ?thesis .. |
27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27516
diff
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|
156 |
qed |
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27516
diff
changeset
|
157 |
|
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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|
158 |
lemma one_dvd [simp]: |
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haftmann
parents:
59000
diff
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|
159 |
"1 dvd a" |
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
59000
diff
changeset
|
160 |
by (auto intro!: dvdI) |
28559 | 161 |
|
59009
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
59000
diff
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|
162 |
lemma dvd_mult [simp]: |
348561aa3869
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haftmann
parents:
59000
diff
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|
163 |
"a dvd c \<Longrightarrow> a dvd (b * c)" |
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
59000
diff
changeset
|
164 |
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
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|
165 |
|
59009
348561aa3869
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haftmann
parents:
59000
diff
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|
166 |
lemma dvd_mult2 [simp]: |
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
59000
diff
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|
167 |
"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
|
168 |
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
|
169 |
|
59009
348561aa3869
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haftmann
parents:
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diff
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|
170 |
lemma dvd_triv_right [simp]: |
348561aa3869
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parents:
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diff
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|
171 |
"a dvd b * a" |
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parents:
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diff
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|
172 |
by (rule dvd_mult) (rule dvd_refl) |
27651
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haftmann
parents:
27516
diff
changeset
|
173 |
|
59009
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haftmann
parents:
59000
diff
changeset
|
174 |
lemma dvd_triv_left [simp]: |
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haftmann
parents:
59000
diff
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|
175 |
"a dvd a * b" |
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generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
59000
diff
changeset
|
176 |
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
|
177 |
|
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27516
diff
changeset
|
178 |
lemma mult_dvd_mono: |
30042 | 179 |
assumes "a dvd b" |
180 |
and "c dvd d" |
|
27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27516
diff
changeset
|
181 |
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
|
182 |
proof - |
60758 | 183 |
from \<open>a dvd b\<close> obtain b' where "b = a * b'" .. |
184 |
moreover from \<open>c dvd d\<close> obtain d' where "d = c * d'" .. |
|
57514
bdc2c6b40bf2
prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents:
57512
diff
changeset
|
185 |
ultimately have "b * d = (a * c) * (b' * d')" 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
|
186 |
then show ?thesis .. |
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27516
diff
changeset
|
187 |
qed |
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27516
diff
changeset
|
188 |
|
59009
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
59000
diff
changeset
|
189 |
lemma dvd_mult_left: |
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
59000
diff
changeset
|
190 |
"a * b dvd c \<Longrightarrow> a dvd c" |
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
59000
diff
changeset
|
191 |
by (simp add: dvd_def mult.assoc) blast |
27651
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moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27516
diff
changeset
|
192 |
|
59009
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
59000
diff
changeset
|
193 |
lemma dvd_mult_right: |
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
59000
diff
changeset
|
194 |
"a * b dvd c \<Longrightarrow> b dvd c" |
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
59000
diff
changeset
|
195 |
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
|
196 |
|
59009
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
59000
diff
changeset
|
197 |
end |
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
59000
diff
changeset
|
198 |
|
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
59000
diff
changeset
|
199 |
class comm_semiring_1 = zero_neq_one + comm_semiring_0 + comm_monoid_mult |
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
59000
diff
changeset
|
200 |
begin |
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
59000
diff
changeset
|
201 |
|
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
59000
diff
changeset
|
202 |
subclass semiring_1 .. |
27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27516
diff
changeset
|
203 |
|
59009
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
59000
diff
changeset
|
204 |
lemma dvd_0_left_iff [simp]: |
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
59000
diff
changeset
|
205 |
"0 dvd a \<longleftrightarrow> a = 0" |
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
59000
diff
changeset
|
206 |
by (auto intro: dvd_refl elim!: dvdE) |
27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27516
diff
changeset
|
207 |
|
59009
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
59000
diff
changeset
|
208 |
lemma dvd_0_right [iff]: |
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
59000
diff
changeset
|
209 |
"a dvd 0" |
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
59000
diff
changeset
|
210 |
proof |
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
59000
diff
changeset
|
211 |
show "0 = a * 0" by simp |
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
59000
diff
changeset
|
212 |
qed |
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
59000
diff
changeset
|
213 |
|
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
59000
diff
changeset
|
214 |
lemma dvd_0_left: |
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
59000
diff
changeset
|
215 |
"0 dvd a \<Longrightarrow> a = 0" |
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
59000
diff
changeset
|
216 |
by simp |
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
59000
diff
changeset
|
217 |
|
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
59000
diff
changeset
|
218 |
lemma dvd_add [simp]: |
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
59000
diff
changeset
|
219 |
assumes "a dvd b" and "a dvd c" |
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
59000
diff
changeset
|
220 |
shows "a dvd (b + c)" |
27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27516
diff
changeset
|
221 |
proof - |
60758 | 222 |
from \<open>a dvd b\<close> obtain b' where "b = a * b'" .. |
223 |
moreover from \<open>a dvd c\<close> obtain c' where "c = a * c'" .. |
|
49962
a8cc904a6820
Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents:
44921
diff
changeset
|
224 |
ultimately have "b + c = a * (b' + c')" by (simp add: distrib_left) |
27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27516
diff
changeset
|
225 |
then show ?thesis .. |
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27516
diff
changeset
|
226 |
qed |
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27516
diff
changeset
|
227 |
|
25152 | 228 |
end |
14421
ee97b6463cb4
new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents:
14398
diff
changeset
|
229 |
|
29904 | 230 |
class semiring_1_cancel = semiring + cancel_comm_monoid_add |
231 |
+ zero_neq_one + monoid_mult |
|
25267 | 232 |
begin |
14940 | 233 |
|
27516 | 234 |
subclass semiring_0_cancel .. |
25512
4134f7c782e2
using intro_locales instead of unfold_locales if appropriate
haftmann
parents:
25450
diff
changeset
|
235 |
|
27516 | 236 |
subclass semiring_1 .. |
25267 | 237 |
|
238 |
end |
|
21199
2d83f93c3580
* Added annihilation axioms ("x * 0 = 0") to axclass semiring_0.
krauss
parents:
20633
diff
changeset
|
239 |
|
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
|
240 |
class comm_semiring_1_cancel = comm_semiring + cancel_comm_monoid_add + |
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
|
241 |
zero_neq_one + comm_monoid_mult + |
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
|
242 |
assumes right_diff_distrib' [algebra_simps]: "a * (b - c) = a * b - a * c" |
25267 | 243 |
begin |
14738 | 244 |
|
27516 | 245 |
subclass semiring_1_cancel .. |
246 |
subclass comm_semiring_0_cancel .. |
|
247 |
subclass comm_semiring_1 .. |
|
25267 | 248 |
|
59816
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
249 |
lemma left_diff_distrib' [algebra_simps]: |
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
250 |
"(b - c) * a = b * a - c * a" |
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
251 |
by (simp add: algebra_simps) |
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
252 |
|
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
253 |
lemma dvd_add_times_triv_left_iff [simp]: |
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
254 |
"a dvd c * a + b \<longleftrightarrow> a dvd b" |
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
255 |
proof - |
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
256 |
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
|
257 |
proof |
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
258 |
assume ?Q then show ?P by simp |
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
259 |
next |
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
260 |
assume ?P |
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
261 |
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
|
262 |
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
|
263 |
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
|
264 |
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
|
265 |
then show ?Q .. |
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
266 |
qed |
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
267 |
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
|
268 |
qed |
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
269 |
|
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
270 |
lemma dvd_add_times_triv_right_iff [simp]: |
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
271 |
"a dvd b + c * a \<longleftrightarrow> a dvd b" |
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
272 |
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
|
273 |
|
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
274 |
lemma dvd_add_triv_left_iff [simp]: |
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
275 |
"a dvd a + b \<longleftrightarrow> a dvd b" |
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
276 |
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
|
277 |
|
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
278 |
lemma dvd_add_triv_right_iff [simp]: |
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
279 |
"a dvd b + a \<longleftrightarrow> a dvd b" |
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
280 |
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
|
281 |
|
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
282 |
lemma dvd_add_right_iff: |
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
283 |
assumes "a dvd b" |
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
284 |
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
|
285 |
proof |
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
286 |
assume ?P then obtain d where "b + c = a * d" .. |
60758 | 287 |
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
|
288 |
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
|
289 |
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
|
290 |
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
|
291 |
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
|
292 |
then show ?Q .. |
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
293 |
next |
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
294 |
assume ?Q with assms show ?P by simp |
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
295 |
qed |
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
296 |
|
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
297 |
lemma dvd_add_left_iff: |
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
298 |
assumes "a dvd c" |
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
299 |
shows "a dvd b + c \<longleftrightarrow> a dvd b" |
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
300 |
using assms dvd_add_right_iff [of a c b] by (simp add: ac_simps) |
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
301 |
|
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
302 |
end |
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
303 |
|
22390 | 304 |
class ring = semiring + ab_group_add |
25267 | 305 |
begin |
25152 | 306 |
|
27516 | 307 |
subclass semiring_0_cancel .. |
25152 | 308 |
|
60758 | 309 |
text \<open>Distribution rules\<close> |
25152 | 310 |
|
311 |
lemma minus_mult_left: "- (a * b) = - 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
|
312 |
by (rule minus_unique) (simp add: distrib_right [symmetric]) |
25152 | 313 |
|
314 |
lemma minus_mult_right: "- (a * b) = 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
|
315 |
by (rule minus_unique) (simp add: distrib_left [symmetric]) |
25152 | 316 |
|
60758 | 317 |
text\<open>Extract signs from products\<close> |
54147
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
52435
diff
changeset
|
318 |
lemmas mult_minus_left [simp] = minus_mult_left [symmetric] |
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
52435
diff
changeset
|
319 |
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
|
320 |
|
25152 | 321 |
lemma minus_mult_minus [simp]: "- a * - b = a * b" |
29667 | 322 |
by simp |
25152 | 323 |
|
324 |
lemma minus_mult_commute: "- a * b = a * - b" |
|
29667 | 325 |
by simp |
326 |
||
58776
95e58e04e534
use NO_MATCH-simproc for distribution rules in field_simps, otherwise field_simps on '(a / (c + d)) * (e + f)' can be non-terminating
hoelzl
parents:
58649
diff
changeset
|
327 |
lemma right_diff_distrib [algebra_simps]: |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54225
diff
changeset
|
328 |
"a * (b - c) = a * b - a * c" |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54225
diff
changeset
|
329 |
using distrib_left [of a b "-c "] by simp |
29667 | 330 |
|
58776
95e58e04e534
use NO_MATCH-simproc for distribution rules in field_simps, otherwise field_simps on '(a / (c + d)) * (e + f)' can be non-terminating
hoelzl
parents:
58649
diff
changeset
|
331 |
lemma left_diff_distrib [algebra_simps]: |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54225
diff
changeset
|
332 |
"(a - b) * c = a * c - b * c" |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54225
diff
changeset
|
333 |
using distrib_right [of a "- b" c] by simp |
25152 | 334 |
|
54147
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
52435
diff
changeset
|
335 |
lemmas ring_distribs = |
49962
a8cc904a6820
Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents:
44921
diff
changeset
|
336 |
distrib_left distrib_right left_diff_distrib right_diff_distrib |
25152 | 337 |
|
25230 | 338 |
lemma eq_add_iff1: |
339 |
"a * e + c = b * e + d \<longleftrightarrow> (a - b) * e + c = d" |
|
29667 | 340 |
by (simp add: algebra_simps) |
25230 | 341 |
|
342 |
lemma eq_add_iff2: |
|
343 |
"a * e + c = b * e + d \<longleftrightarrow> c = (b - a) * e + d" |
|
29667 | 344 |
by (simp add: algebra_simps) |
25230 | 345 |
|
25152 | 346 |
end |
347 |
||
54147
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
52435
diff
changeset
|
348 |
lemmas ring_distribs = |
49962
a8cc904a6820
Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents:
44921
diff
changeset
|
349 |
distrib_left distrib_right left_diff_distrib right_diff_distrib |
25152 | 350 |
|
22390 | 351 |
class comm_ring = comm_semiring + ab_group_add |
25267 | 352 |
begin |
14738 | 353 |
|
27516 | 354 |
subclass ring .. |
28141
193c3ea0f63b
instances comm_semiring_0_cancel < comm_semiring_0, comm_ring < comm_semiring_0_cancel
huffman
parents:
27651
diff
changeset
|
355 |
subclass comm_semiring_0_cancel .. |
25267 | 356 |
|
44350
63cddfbc5a09
replace lemma realpow_two_diff with new lemma square_diff_square_factored
huffman
parents:
44346
diff
changeset
|
357 |
lemma square_diff_square_factored: |
63cddfbc5a09
replace lemma realpow_two_diff with new lemma square_diff_square_factored
huffman
parents:
44346
diff
changeset
|
358 |
"x * x - y * y = (x + y) * (x - y)" |
63cddfbc5a09
replace lemma realpow_two_diff with new lemma square_diff_square_factored
huffman
parents:
44346
diff
changeset
|
359 |
by (simp add: algebra_simps) |
63cddfbc5a09
replace lemma realpow_two_diff with new lemma square_diff_square_factored
huffman
parents:
44346
diff
changeset
|
360 |
|
25267 | 361 |
end |
14738 | 362 |
|
22390 | 363 |
class ring_1 = ring + zero_neq_one + monoid_mult |
25267 | 364 |
begin |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
365 |
|
27516 | 366 |
subclass semiring_1_cancel .. |
25267 | 367 |
|
44346
00dd3c4dabe0
rename real_squared_diff_one_factored to square_diff_one_factored and move to Rings.thy
huffman
parents:
44064
diff
changeset
|
368 |
lemma square_diff_one_factored: |
00dd3c4dabe0
rename real_squared_diff_one_factored to square_diff_one_factored and move to Rings.thy
huffman
parents:
44064
diff
changeset
|
369 |
"x * x - 1 = (x + 1) * (x - 1)" |
00dd3c4dabe0
rename real_squared_diff_one_factored to square_diff_one_factored and move to Rings.thy
huffman
parents:
44064
diff
changeset
|
370 |
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
|
371 |
|
25267 | 372 |
end |
25152 | 373 |
|
22390 | 374 |
class comm_ring_1 = comm_ring + zero_neq_one + comm_monoid_mult |
25267 | 375 |
begin |
14738 | 376 |
|
27516 | 377 |
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
|
378 |
subclass comm_semiring_1_cancel |
59816
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset
|
379 |
by unfold_locales (simp add: algebra_simps) |
58647 | 380 |
|
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
|
381 |
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
|
382 |
proof |
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 |
next |
6d10cf26b5dc
add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents:
29407
diff
changeset
|
387 |
assume "x dvd y" |
6d10cf26b5dc
add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents:
29407
diff
changeset
|
388 |
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
|
389 |
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
|
390 |
qed |
6d10cf26b5dc
add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents:
29407
diff
changeset
|
391 |
|
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
|
392 |
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
|
393 |
proof |
6d10cf26b5dc
add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents:
29407
diff
changeset
|
394 |
assume "- x dvd y" |
6d10cf26b5dc
add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents:
29407
diff
changeset
|
395 |
then 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
|
396 |
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
|
397 |
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
|
398 |
next |
6d10cf26b5dc
add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents:
29407
diff
changeset
|
399 |
assume "x dvd y" |
6d10cf26b5dc
add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents:
29407
diff
changeset
|
400 |
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
|
401 |
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
|
402 |
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
|
403 |
qed |
6d10cf26b5dc
add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents:
29407
diff
changeset
|
404 |
|
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54225
diff
changeset
|
405 |
lemma dvd_diff [simp]: |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54225
diff
changeset
|
406 |
"x dvd y \<Longrightarrow> x dvd z \<Longrightarrow> x dvd (y - z)" |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54225
diff
changeset
|
407 |
using dvd_add [of x y "- z"] by simp |
29409 | 408 |
|
25267 | 409 |
end |
25152 | 410 |
|
59833
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset
|
411 |
class semiring_no_zero_divisors = semiring_0 + |
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset
|
412 |
assumes no_zero_divisors: "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> a * b \<noteq> 0" |
25230 | 413 |
begin |
414 |
||
59833
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset
|
415 |
lemma divisors_zero: |
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset
|
416 |
assumes "a * b = 0" |
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset
|
417 |
shows "a = 0 \<or> b = 0" |
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset
|
418 |
proof (rule classical) |
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset
|
419 |
assume "\<not> (a = 0 \<or> b = 0)" |
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset
|
420 |
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
|
421 |
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
|
422 |
with assms show ?thesis by simp |
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset
|
423 |
qed |
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset
|
424 |
|
25230 | 425 |
lemma mult_eq_0_iff [simp]: |
58952
5d82cdef6c1b
equivalence rules for structures without zero divisors
haftmann
parents:
58889
diff
changeset
|
426 |
shows "a * b = 0 \<longleftrightarrow> a = 0 \<or> b = 0" |
25230 | 427 |
proof (cases "a = 0 \<or> b = 0") |
428 |
case False then have "a \<noteq> 0" and "b \<noteq> 0" by auto |
|
429 |
then show ?thesis using no_zero_divisors by simp |
|
430 |
next |
|
431 |
case True then show ?thesis by auto |
|
432 |
qed |
|
433 |
||
58952
5d82cdef6c1b
equivalence rules for structures without zero divisors
haftmann
parents:
58889
diff
changeset
|
434 |
end |
5d82cdef6c1b
equivalence rules for structures without zero divisors
haftmann
parents:
58889
diff
changeset
|
435 |
|
60516
0826b7025d07
generalized some theorems about integral domains and moved to HOL theories
haftmann
parents:
60429
diff
changeset
|
436 |
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
|
437 |
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
|
438 |
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
|
439 |
begin |
5d82cdef6c1b
equivalence rules for structures without zero divisors
haftmann
parents:
58889
diff
changeset
|
440 |
|
5d82cdef6c1b
equivalence rules for structures without zero divisors
haftmann
parents:
58889
diff
changeset
|
441 |
lemma mult_left_cancel: |
5d82cdef6c1b
equivalence rules for structures without zero divisors
haftmann
parents:
58889
diff
changeset
|
442 |
"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
|
443 |
by simp |
56217
dc429a5b13c4
Some rationalisation of basic lemmas
paulson <lp15@cam.ac.uk>
parents:
55912
diff
changeset
|
444 |
|
58952
5d82cdef6c1b
equivalence rules for structures without zero divisors
haftmann
parents:
58889
diff
changeset
|
445 |
lemma mult_right_cancel: |
5d82cdef6c1b
equivalence rules for structures without zero divisors
haftmann
parents:
58889
diff
changeset
|
446 |
"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
|
447 |
by simp |
56217
dc429a5b13c4
Some rationalisation of basic lemmas
paulson <lp15@cam.ac.uk>
parents:
55912
diff
changeset
|
448 |
|
25230 | 449 |
end |
22990
775e9de3db48
added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents:
22987
diff
changeset
|
450 |
|
60516
0826b7025d07
generalized some theorems about integral domains and moved to HOL theories
haftmann
parents:
60429
diff
changeset
|
451 |
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
|
452 |
begin |
0826b7025d07
generalized some theorems about integral domains and moved to HOL theories
haftmann
parents:
60429
diff
changeset
|
453 |
|
0826b7025d07
generalized some theorems about integral domains and moved to HOL theories
haftmann
parents:
60429
diff
changeset
|
454 |
subclass semiring_no_zero_divisors_cancel |
0826b7025d07
generalized some theorems about integral domains and moved to HOL theories
haftmann
parents:
60429
diff
changeset
|
455 |
proof |
0826b7025d07
generalized some theorems about integral domains and moved to HOL theories
haftmann
parents:
60429
diff
changeset
|
456 |
fix a b c |
0826b7025d07
generalized some theorems about integral domains and moved to HOL theories
haftmann
parents:
60429
diff
changeset
|
457 |
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
|
458 |
by (simp add: algebra_simps) |
0826b7025d07
generalized some theorems about integral domains and moved to HOL theories
haftmann
parents:
60429
diff
changeset
|
459 |
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
|
460 |
by auto |
0826b7025d07
generalized some theorems about integral domains and moved to HOL theories
haftmann
parents:
60429
diff
changeset
|
461 |
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
|
462 |
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
|
463 |
by (simp add: algebra_simps) |
0826b7025d07
generalized some theorems about integral domains and moved to HOL theories
haftmann
parents:
60429
diff
changeset
|
464 |
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
|
465 |
by auto |
0826b7025d07
generalized some theorems about integral domains and moved to HOL theories
haftmann
parents:
60429
diff
changeset
|
466 |
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
|
467 |
qed |
0826b7025d07
generalized some theorems about integral domains and moved to HOL theories
haftmann
parents:
60429
diff
changeset
|
468 |
|
0826b7025d07
generalized some theorems about integral domains and moved to HOL theories
haftmann
parents:
60429
diff
changeset
|
469 |
end |
0826b7025d07
generalized some theorems about integral domains and moved to HOL theories
haftmann
parents:
60429
diff
changeset
|
470 |
|
23544 | 471 |
class ring_1_no_zero_divisors = ring_1 + ring_no_zero_divisors |
26274 | 472 |
begin |
473 |
||
36970 | 474 |
lemma square_eq_1_iff: |
36821
9207505d1ee5
move lemma real_mult_is_one to Rings.thy, renamed to square_eq_1_iff
huffman
parents:
36719
diff
changeset
|
475 |
"x * x = 1 \<longleftrightarrow> x = 1 \<or> x = - 1" |
9207505d1ee5
move lemma real_mult_is_one to Rings.thy, renamed to square_eq_1_iff
huffman
parents:
36719
diff
changeset
|
476 |
proof - |
9207505d1ee5
move lemma real_mult_is_one to Rings.thy, renamed to square_eq_1_iff
huffman
parents:
36719
diff
changeset
|
477 |
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
|
478 |
by (simp add: algebra_simps) |
9207505d1ee5
move lemma real_mult_is_one to Rings.thy, renamed to square_eq_1_iff
huffman
parents:
36719
diff
changeset
|
479 |
hence "x * x = 1 \<longleftrightarrow> (x - 1) * (x + 1) = 0" |
9207505d1ee5
move lemma real_mult_is_one to Rings.thy, renamed to square_eq_1_iff
huffman
parents:
36719
diff
changeset
|
480 |
by simp |
9207505d1ee5
move lemma real_mult_is_one to Rings.thy, renamed to square_eq_1_iff
huffman
parents:
36719
diff
changeset
|
481 |
thus ?thesis |
9207505d1ee5
move lemma real_mult_is_one to Rings.thy, renamed to square_eq_1_iff
huffman
parents:
36719
diff
changeset
|
482 |
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
|
483 |
qed |
9207505d1ee5
move lemma real_mult_is_one to Rings.thy, renamed to square_eq_1_iff
huffman
parents:
36719
diff
changeset
|
484 |
|
26274 | 485 |
lemma mult_cancel_right1 [simp]: |
486 |
"c = b * c \<longleftrightarrow> c = 0 \<or> b = 1" |
|
29667 | 487 |
by (insert mult_cancel_right [of 1 c b], force) |
26274 | 488 |
|
489 |
lemma mult_cancel_right2 [simp]: |
|
490 |
"a * c = c \<longleftrightarrow> c = 0 \<or> a = 1" |
|
29667 | 491 |
by (insert mult_cancel_right [of a c 1], 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
|
492 |
|
26274 | 493 |
lemma mult_cancel_left1 [simp]: |
494 |
"c = c * b \<longleftrightarrow> c = 0 \<or> b = 1" |
|
29667 | 495 |
by (insert mult_cancel_left [of c 1 b], force) |
26274 | 496 |
|
497 |
lemma mult_cancel_left2 [simp]: |
|
498 |
"c * a = c \<longleftrightarrow> c = 0 \<or> a = 1" |
|
29667 | 499 |
by (insert mult_cancel_left [of c a 1], simp) |
26274 | 500 |
|
501 |
end |
|
22990
775e9de3db48
added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents:
22987
diff
changeset
|
502 |
|
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
|
503 |
class semidom = comm_semiring_1_cancel + semiring_no_zero_divisors |
59833
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset
|
504 |
|
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset
|
505 |
class idom = comm_ring_1 + semiring_no_zero_divisors |
25186 | 506 |
begin |
14421
ee97b6463cb4
new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents:
14398
diff
changeset
|
507 |
|
59833
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset
|
508 |
subclass semidom .. |
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset
|
509 |
|
27516 | 510 |
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
|
511 |
|
29981
7d0ed261b712
generalize int_dvd_cancel_factor simproc to idom class
huffman
parents:
29949
diff
changeset
|
512 |
lemma dvd_mult_cancel_right [simp]: |
7d0ed261b712
generalize int_dvd_cancel_factor simproc to idom class
huffman
parents:
29949
diff
changeset
|
513 |
"a * c dvd b * c \<longleftrightarrow> c = 0 \<or> a dvd b" |
7d0ed261b712
generalize int_dvd_cancel_factor simproc to idom class
huffman
parents:
29949
diff
changeset
|
514 |
proof - |
7d0ed261b712
generalize int_dvd_cancel_factor simproc to idom class
huffman
parents:
29949
diff
changeset
|
515 |
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
|
516 |
unfolding dvd_def by (simp add: ac_simps) |
29981
7d0ed261b712
generalize int_dvd_cancel_factor simproc to idom class
huffman
parents:
29949
diff
changeset
|
517 |
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
|
518 |
unfolding dvd_def by simp |
7d0ed261b712
generalize int_dvd_cancel_factor simproc to idom class
huffman
parents:
29949
diff
changeset
|
519 |
finally show ?thesis . |
7d0ed261b712
generalize int_dvd_cancel_factor simproc to idom class
huffman
parents:
29949
diff
changeset
|
520 |
qed |
7d0ed261b712
generalize int_dvd_cancel_factor simproc to idom class
huffman
parents:
29949
diff
changeset
|
521 |
|
7d0ed261b712
generalize int_dvd_cancel_factor simproc to idom class
huffman
parents:
29949
diff
changeset
|
522 |
lemma dvd_mult_cancel_left [simp]: |
7d0ed261b712
generalize int_dvd_cancel_factor simproc to idom class
huffman
parents:
29949
diff
changeset
|
523 |
"c * a dvd c * b \<longleftrightarrow> c = 0 \<or> a dvd b" |
7d0ed261b712
generalize int_dvd_cancel_factor simproc to idom class
huffman
parents:
29949
diff
changeset
|
524 |
proof - |
7d0ed261b712
generalize int_dvd_cancel_factor simproc to idom class
huffman
parents:
29949
diff
changeset
|
525 |
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
|
526 |
unfolding dvd_def by (simp add: ac_simps) |
29981
7d0ed261b712
generalize int_dvd_cancel_factor simproc to idom class
huffman
parents:
29949
diff
changeset
|
527 |
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
|
528 |
unfolding dvd_def by simp |
7d0ed261b712
generalize int_dvd_cancel_factor simproc to idom class
huffman
parents:
29949
diff
changeset
|
529 |
finally show ?thesis . |
7d0ed261b712
generalize int_dvd_cancel_factor simproc to idom class
huffman
parents:
29949
diff
changeset
|
530 |
qed |
7d0ed261b712
generalize int_dvd_cancel_factor simproc to idom class
huffman
parents:
29949
diff
changeset
|
531 |
|
60516
0826b7025d07
generalized some theorems about integral domains and moved to HOL theories
haftmann
parents:
60429
diff
changeset
|
532 |
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
|
533 |
proof |
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset
|
534 |
assume "a * a = b * b" |
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset
|
535 |
then have "(a - b) * (a + b) = 0" |
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset
|
536 |
by (simp add: algebra_simps) |
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset
|
537 |
then show "a = b \<or> a = - b" |
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset
|
538 |
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
|
539 |
next |
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset
|
540 |
assume "a = b \<or> a = - b" |
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset
|
541 |
then show "a * a = b * b" by auto |
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset
|
542 |
qed |
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset
|
543 |
|
25186 | 544 |
end |
25152 | 545 |
|
60758 | 546 |
text \<open> |
35302 | 547 |
The theory of partially ordered rings is taken from the books: |
548 |
\begin{itemize} |
|
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
|
549 |
\item \emph{Lattice Theory} by Garret Birkhoff, American Mathematical Society 1979 |
35302 | 550 |
\item \emph{Partially Ordered Algebraic Systems}, Pergamon Press 1963 |
551 |
\end{itemize} |
|
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
|
552 |
Most of the used notions can also be looked up in |
35302 | 553 |
\begin{itemize} |
54703 | 554 |
\item @{url "http://www.mathworld.com"} by Eric Weisstein et. al. |
35302 | 555 |
\item \emph{Algebra I} by van der Waerden, Springer. |
556 |
\end{itemize} |
|
60758 | 557 |
\<close> |
35302 | 558 |
|
60353
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset
|
559 |
class divide = |
60429
d3d1e185cd63
uniform _ div _ as infix syntax for ring division
haftmann
parents:
60353
diff
changeset
|
560 |
fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "div" 70) |
60353
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset
|
561 |
|
60758 | 562 |
setup \<open>Sign.add_const_constraint (@{const_name "divide"}, SOME @{typ "'a \<Rightarrow> 'a \<Rightarrow> 'a"})\<close> |
60353
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset
|
563 |
|
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset
|
564 |
context semiring |
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset
|
565 |
begin |
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset
|
566 |
|
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset
|
567 |
lemma [field_simps]: |
60429
d3d1e185cd63
uniform _ div _ as infix syntax for ring division
haftmann
parents:
60353
diff
changeset
|
568 |
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
|
569 |
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
|
570 |
by (rule distrib_left distrib_right)+ |
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset
|
571 |
|
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset
|
572 |
end |
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset
|
573 |
|
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset
|
574 |
context ring |
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset
|
575 |
begin |
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset
|
576 |
|
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset
|
577 |
lemma [field_simps]: |
60429
d3d1e185cd63
uniform _ div _ as infix syntax for ring division
haftmann
parents:
60353
diff
changeset
|
578 |
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
|
579 |
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
|
580 |
by (rule left_diff_distrib right_diff_distrib)+ |
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset
|
581 |
|
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset
|
582 |
end |
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset
|
583 |
|
60758 | 584 |
setup \<open>Sign.add_const_constraint (@{const_name "divide"}, SOME @{typ "'a::divide \<Rightarrow> 'a \<Rightarrow> 'a"})\<close> |
60353
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset
|
585 |
|
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset
|
586 |
class semidom_divide = semidom + divide + |
60429
d3d1e185cd63
uniform _ div _ as infix syntax for ring division
haftmann
parents:
60353
diff
changeset
|
587 |
assumes nonzero_mult_divide_cancel_right [simp]: "b \<noteq> 0 \<Longrightarrow> (a * b) div b = a" |
d3d1e185cd63
uniform _ div _ as infix syntax for ring division
haftmann
parents:
60353
diff
changeset
|
588 |
assumes divide_zero [simp]: "a div 0 = 0" |
60353
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset
|
589 |
begin |
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset
|
590 |
|
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset
|
591 |
lemma nonzero_mult_divide_cancel_left [simp]: |
60429
d3d1e185cd63
uniform _ div _ as infix syntax for ring division
haftmann
parents:
60353
diff
changeset
|
592 |
"a \<noteq> 0 \<Longrightarrow> (a * b) div a = b" |
60353
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset
|
593 |
using nonzero_mult_divide_cancel_right [of a b] by (simp add: ac_simps) |
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset
|
594 |
|
60516
0826b7025d07
generalized some theorems about integral domains and moved to HOL theories
haftmann
parents:
60429
diff
changeset
|
595 |
subclass semiring_no_zero_divisors_cancel |
0826b7025d07
generalized some theorems about integral domains and moved to HOL theories
haftmann
parents:
60429
diff
changeset
|
596 |
proof |
0826b7025d07
generalized some theorems about integral domains and moved to HOL theories
haftmann
parents:
60429
diff
changeset
|
597 |
fix a b c |
0826b7025d07
generalized some theorems about integral domains and moved to HOL theories
haftmann
parents:
60429
diff
changeset
|
598 |
{ fix a b c |
0826b7025d07
generalized some theorems about integral domains and moved to HOL theories
haftmann
parents:
60429
diff
changeset
|
599 |
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
|
600 |
proof (cases "c = 0") |
0826b7025d07
generalized some theorems about integral domains and moved to HOL theories
haftmann
parents:
60429
diff
changeset
|
601 |
case True then show ?thesis by simp |
0826b7025d07
generalized some theorems about integral domains and moved to HOL theories
haftmann
parents:
60429
diff
changeset
|
602 |
next |
0826b7025d07
generalized some theorems about integral domains and moved to HOL theories
haftmann
parents:
60429
diff
changeset
|
603 |
case False |
0826b7025d07
generalized some theorems about integral domains and moved to HOL theories
haftmann
parents:
60429
diff
changeset
|
604 |
{ assume "a * c = b * c" |
0826b7025d07
generalized some theorems about integral domains and moved to HOL theories
haftmann
parents:
60429
diff
changeset
|
605 |
then have "a * c div c = b * c div c" |
0826b7025d07
generalized some theorems about integral domains and moved to HOL theories
haftmann
parents:
60429
diff
changeset
|
606 |
by simp |
0826b7025d07
generalized some theorems about integral domains and moved to HOL theories
haftmann
parents:
60429
diff
changeset
|
607 |
with False have "a = b" |
0826b7025d07
generalized some theorems about integral domains and moved to HOL theories
haftmann
parents:
60429
diff
changeset
|
608 |
by simp |
0826b7025d07
generalized some theorems about integral domains and moved to HOL theories
haftmann
parents:
60429
diff
changeset
|
609 |
} then show ?thesis by auto |
0826b7025d07
generalized some theorems about integral domains and moved to HOL theories
haftmann
parents:
60429
diff
changeset
|
610 |
qed |
0826b7025d07
generalized some theorems about integral domains and moved to HOL theories
haftmann
parents:
60429
diff
changeset
|
611 |
} |
0826b7025d07
generalized some theorems about integral domains and moved to HOL theories
haftmann
parents:
60429
diff
changeset
|
612 |
from this [of a c b] |
0826b7025d07
generalized some theorems about integral domains and moved to HOL theories
haftmann
parents:
60429
diff
changeset
|
613 |
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
|
614 |
by (simp add: ac_simps) |
0826b7025d07
generalized some theorems about integral domains and moved to HOL theories
haftmann
parents:
60429
diff
changeset
|
615 |
qed |
0826b7025d07
generalized some theorems about integral domains and moved to HOL theories
haftmann
parents:
60429
diff
changeset
|
616 |
|
0826b7025d07
generalized some theorems about integral domains and moved to HOL theories
haftmann
parents:
60429
diff
changeset
|
617 |
lemma div_self [simp]: |
0826b7025d07
generalized some theorems about integral domains and moved to HOL theories
haftmann
parents:
60429
diff
changeset
|
618 |
assumes "a \<noteq> 0" |
0826b7025d07
generalized some theorems about integral domains and moved to HOL theories
haftmann
parents:
60429
diff
changeset
|
619 |
shows "a div a = 1" |
0826b7025d07
generalized some theorems about integral domains and moved to HOL theories
haftmann
parents:
60429
diff
changeset
|
620 |
using assms nonzero_mult_divide_cancel_left [of a 1] by simp |
0826b7025d07
generalized some theorems about integral domains and moved to HOL theories
haftmann
parents:
60429
diff
changeset
|
621 |
|
60570 | 622 |
lemma divide_zero_left [simp]: |
623 |
"0 div a = 0" |
|
624 |
proof (cases "a = 0") |
|
625 |
case True then show ?thesis by simp |
|
626 |
next |
|
627 |
case False then have "a * 0 div a = 0" |
|
628 |
by (rule nonzero_mult_divide_cancel_left) |
|
629 |
then show ?thesis by simp |
|
630 |
qed |
|
631 |
||
60690 | 632 |
lemma divide_1 [simp]: |
633 |
"a div 1 = a" |
|
634 |
using nonzero_mult_divide_cancel_left [of 1 a] by simp |
|
635 |
||
60867 | 636 |
end |
637 |
||
638 |
class idom_divide = idom + semidom_divide |
|
639 |
||
640 |
class algebraic_semidom = semidom_divide |
|
641 |
begin |
|
642 |
||
643 |
text \<open> |
|
644 |
Class @{class algebraic_semidom} enriches a integral domain |
|
645 |
by notions from algebra, like units in a ring. |
|
646 |
It is a separate class to avoid spoiling fields with notions |
|
647 |
which are degenerated there. |
|
648 |
\<close> |
|
649 |
||
60690 | 650 |
lemma dvd_times_left_cancel_iff [simp]: |
651 |
assumes "a \<noteq> 0" |
|
652 |
shows "a * b dvd a * c \<longleftrightarrow> b dvd c" (is "?P \<longleftrightarrow> ?Q") |
|
653 |
proof |
|
654 |
assume ?P then obtain d where "a * c = a * b * d" .. |
|
655 |
with assms have "c = b * d" by (simp add: ac_simps) |
|
656 |
then show ?Q .. |
|
657 |
next |
|
658 |
assume ?Q then obtain d where "c = b * d" .. |
|
659 |
then have "a * c = a * b * d" by (simp add: ac_simps) |
|
660 |
then show ?P .. |
|
661 |
qed |
|
662 |
||
663 |
lemma dvd_times_right_cancel_iff [simp]: |
|
664 |
assumes "a \<noteq> 0" |
|
665 |
shows "b * a dvd c * a \<longleftrightarrow> b dvd c" (is "?P \<longleftrightarrow> ?Q") |
|
666 |
using dvd_times_left_cancel_iff [of a b c] assms by (simp add: ac_simps) |
|
667 |
||
668 |
lemma div_dvd_iff_mult: |
|
669 |
assumes "b \<noteq> 0" and "b dvd a" |
|
670 |
shows "a div b dvd c \<longleftrightarrow> a dvd c * b" |
|
671 |
proof - |
|
672 |
from \<open>b dvd a\<close> obtain d where "a = b * d" .. |
|
673 |
with \<open>b \<noteq> 0\<close> show ?thesis by (simp add: ac_simps) |
|
674 |
qed |
|
675 |
||
676 |
lemma dvd_div_iff_mult: |
|
677 |
assumes "c \<noteq> 0" and "c dvd b" |
|
678 |
shows "a dvd b div c \<longleftrightarrow> a * c dvd b" |
|
679 |
proof - |
|
680 |
from \<open>c dvd b\<close> obtain d where "b = c * d" .. |
|
681 |
with \<open>c \<noteq> 0\<close> show ?thesis by (simp add: mult.commute [of a]) |
|
682 |
qed |
|
683 |
||
60867 | 684 |
lemma div_dvd_div [simp]: |
685 |
assumes "a dvd b" and "a dvd c" |
|
686 |
shows "b div a dvd c div a \<longleftrightarrow> b dvd c" |
|
687 |
proof (cases "a = 0") |
|
688 |
case True with assms show ?thesis by simp |
|
689 |
next |
|
690 |
case False |
|
691 |
moreover from assms obtain k l where "b = a * k" and "c = a * l" |
|
692 |
by (auto elim!: dvdE) |
|
693 |
ultimately show ?thesis by simp |
|
694 |
qed |
|
60353
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset
|
695 |
|
60867 | 696 |
lemma div_add [simp]: |
697 |
assumes "c dvd a" and "c dvd b" |
|
698 |
shows "(a + b) div c = a div c + b div c" |
|
699 |
proof (cases "c = 0") |
|
700 |
case True then show ?thesis by simp |
|
701 |
next |
|
702 |
case False |
|
703 |
moreover from assms obtain k l where "a = c * k" and "b = c * l" |
|
704 |
by (auto elim!: dvdE) |
|
705 |
moreover have "c * k + c * l = c * (k + l)" |
|
706 |
by (simp add: algebra_simps) |
|
707 |
ultimately show ?thesis |
|
708 |
by simp |
|
709 |
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
|
710 |
|
60867 | 711 |
lemma div_mult_div_if_dvd: |
712 |
assumes "b dvd a" and "d dvd c" |
|
713 |
shows "(a div b) * (c div d) = (a * c) div (b * d)" |
|
714 |
proof (cases "b = 0 \<or> c = 0") |
|
715 |
case True with assms show ?thesis by auto |
|
716 |
next |
|
717 |
case False |
|
718 |
moreover from assms obtain k l where "a = b * k" and "c = d * l" |
|
719 |
by (auto elim!: dvdE) |
|
720 |
moreover have "b * k * (d * l) div (b * d) = (b * d) * (k * l) div (b * d)" |
|
721 |
by (simp add: ac_simps) |
|
722 |
ultimately show ?thesis by simp |
|
723 |
qed |
|
724 |
||
725 |
lemma dvd_div_eq_mult: |
|
726 |
assumes "a \<noteq> 0" and "a dvd b" |
|
727 |
shows "b div a = c \<longleftrightarrow> b = c * a" |
|
728 |
proof |
|
729 |
assume "b = c * a" |
|
730 |
then show "b div a = c" by (simp add: assms) |
|
731 |
next |
|
732 |
assume "b div a = c" |
|
733 |
then have "b div a * a = c * a" by simp |
|
734 |
moreover from \<open>a \<noteq> 0\<close> \<open>a dvd b\<close> have "b div a * a = b" |
|
735 |
by (auto elim!: dvdE simp add: ac_simps) |
|
736 |
ultimately show "b = c * a" by simp |
|
737 |
qed |
|
60688
01488b559910
avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset
|
738 |
|
60517
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
739 |
lemma dvd_div_mult_self [simp]: |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
740 |
"a dvd b \<Longrightarrow> b div a * a = b" |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
741 |
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
|
742 |
|
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
743 |
lemma dvd_mult_div_cancel [simp]: |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
744 |
"a dvd b \<Longrightarrow> a * (b div a) = b" |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
745 |
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
|
746 |
|
60517
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
747 |
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
|
748 |
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
|
749 |
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
|
750 |
proof (cases "c = 0") |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
751 |
case True then 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
|
752 |
next |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
753 |
case False from assms obtain d where "b = c * d" .. |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
754 |
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
|
755 |
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
|
756 |
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
|
757 |
qed |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
758 |
|
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
759 |
lemma dvd_div_mult: |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
760 |
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
|
761 |
shows "b div c * a = (b * a) 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
|
762 |
using assms div_mult_swap [of c b 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
|
763 |
|
60570 | 764 |
lemma dvd_div_mult2_eq: |
765 |
assumes "b * c dvd a" |
|
766 |
shows "a div (b * c) = a div b div c" |
|
767 |
using assms proof |
|
768 |
fix k |
|
769 |
assume "a = b * c * k" |
|
770 |
then show ?thesis |
|
771 |
by (cases "b = 0 \<or> c = 0") (auto, simp add: ac_simps) |
|
772 |
qed |
|
773 |
||
60867 | 774 |
lemma dvd_div_div_eq_mult: |
775 |
assumes "a \<noteq> 0" "c \<noteq> 0" and "a dvd b" "c dvd d" |
|
776 |
shows "b div a = d div c \<longleftrightarrow> b * c = a * d" (is "?P \<longleftrightarrow> ?Q") |
|
777 |
proof - |
|
778 |
from assms have "a * c \<noteq> 0" by simp |
|
779 |
then have "?P \<longleftrightarrow> b div a * (a * c) = d div c * (a * c)" |
|
780 |
by simp |
|
781 |
also have "\<dots> \<longleftrightarrow> (a * (b div a)) * c = (c * (d div c)) * a" |
|
782 |
by (simp add: ac_simps) |
|
783 |
also have "\<dots> \<longleftrightarrow> (a * b div a) * c = (c * d div c) * a" |
|
784 |
using assms by (simp add: div_mult_swap) |
|
785 |
also have "\<dots> \<longleftrightarrow> ?Q" |
|
786 |
using assms by (simp add: ac_simps) |
|
787 |
finally show ?thesis . |
|
788 |
qed |
|
789 |
||
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
|
790 |
|
60517
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
791 |
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
|
792 |
|
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
793 |
abbreviation is_unit :: "'a \<Rightarrow> bool" |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
794 |
where |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
795 |
"is_unit a \<equiv> a dvd 1" |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
796 |
|
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
797 |
lemma not_is_unit_0 [simp]: |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
798 |
"\<not> is_unit 0" |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
799 |
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
|
800 |
|
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
|
801 |
lemma unit_imp_dvd [dest]: |
60517
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
802 |
"is_unit b \<Longrightarrow> b dvd a" |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
803 |
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
|
804 |
|
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
805 |
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
|
806 |
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
|
807 |
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
|
808 |
proof - |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
809 |
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
|
810 |
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
|
811 |
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
|
812 |
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
|
813 |
qed |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
814 |
|
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
815 |
lemma dvd_unit_imp_unit: |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
816 |
"a dvd b \<Longrightarrow> is_unit b \<Longrightarrow> 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
|
817 |
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
|
818 |
|
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
819 |
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
|
820 |
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
|
821 |
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
|
822 |
proof - |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
823 |
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
|
824 |
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
|
825 |
qed |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
826 |
|
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
827 |
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
|
828 |
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
|
829 |
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
|
830 |
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
|
831 |
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
|
832 |
proof (rule that) |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
833 |
def b \<equiv> "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
|
834 |
then show "1 div a = b" by simp |
60758 | 835 |
from b_def \<open>is_unit a\<close> show "is_unit b" by simp |
836 |
from \<open>is_unit a\<close> and \<open>is_unit b\<close> show "a \<noteq> 0" and "b \<noteq> 0" by auto |
|
837 |
from b_def \<open>is_unit a\<close> 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
|
838 |
then have "1 = a * b" .. |
60758 | 839 |
with b_def \<open>b \<noteq> 0\<close> show "1 div b = a" by simp |
840 |
from \<open>is_unit a\<close> 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
|
841 |
then obtain d where "c = a * d" .. |
60758 | 842 |
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
|
843 |
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
|
844 |
qed |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
845 |
|
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
846 |
lemma unit_prod [intro]: |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
847 |
"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
|
848 |
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
|
849 |
|
60517
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
850 |
lemma unit_div [intro]: |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
851 |
"is_unit a \<Longrightarrow> is_unit b \<Longrightarrow> is_unit (a 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
|
852 |
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
|
853 |
|
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
854 |
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
|
855 |
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
|
856 |
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
|
857 |
proof |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
858 |
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
|
859 |
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
|
860 |
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
|
861 |
next |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
862 |
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
|
863 |
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
|
864 |
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
|
865 |
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
|
866 |
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
|
867 |
qed |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
868 |
|
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
869 |
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
|
870 |
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
|
871 |
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
|
872 |
proof |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
873 |
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
|
874 |
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
|
875 |
by (subst mult_assoc [symmetric]) simp |
60758 | 876 |
also from \<open>is_unit b\<close> have "b * (1 div b) = 1" 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
|
877 |
finally have "c * b dvd c" by simp |
60758 | 878 |
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
|
879 |
next |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
880 |
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
|
881 |
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
|
882 |
qed |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
883 |
|
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
884 |
lemma div_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
|
885 |
"is_unit b \<Longrightarrow> a div 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
|
886 |
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
|
887 |
|
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
888 |
lemma dvd_div_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
|
889 |
"is_unit b \<Longrightarrow> a dvd c div 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
|
890 |
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
|
891 |
|
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
892 |
lemmas unit_dvd_iff = mult_unit_dvd_iff div_unit_dvd_iff |
61799 | 893 |
dvd_mult_unit_iff dvd_div_unit_iff \<comment> \<open>FIXME consider fact collection\<close> |
60517
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
894 |
|
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
895 |
lemma unit_mult_div_div [simp]: |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
896 |
"is_unit a \<Longrightarrow> b * (1 div a) = b 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
|
897 |
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
|
898 |
|
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
899 |
lemma unit_div_mult_self [simp]: |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
900 |
"is_unit a \<Longrightarrow> b div a * a = b" |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
901 |
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
|
902 |
|
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
903 |
lemma unit_div_1_div_1 [simp]: |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
904 |
"is_unit a \<Longrightarrow> 1 div (1 div a) = a" |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
905 |
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
|
906 |
|
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
907 |
lemma unit_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
|
908 |
"is_unit c \<Longrightarrow> 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
|
909 |
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
|
910 |
|
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
911 |
lemma unit_div_commute: |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
912 |
"is_unit b \<Longrightarrow> (a div b) * c = (a * c) 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
|
913 |
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
|
914 |
|
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
915 |
lemma unit_eq_div1: |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
916 |
"is_unit b \<Longrightarrow> a div b = c \<longleftrightarrow> 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
|
917 |
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
|
918 |
|
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
919 |
lemma 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
|
920 |
"is_unit b \<Longrightarrow> a = c div b \<longleftrightarrow> a * 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
|
921 |
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
|
922 |
|
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
923 |
lemma unit_mult_left_cancel: |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
924 |
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
|
925 |
shows "a * b = a * c \<longleftrightarrow> b = c" (is "?P \<longleftrightarrow> ?Q") |
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
|
926 |
using assms 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
|
927 |
|
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
928 |
lemma 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
|
929 |
"is_unit a \<Longrightarrow> b * a = c * 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
|
930 |
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
|
931 |
|
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
932 |
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
|
933 |
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
|
934 |
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
|
935 |
proof - |
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset
|
936 |
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
|
937 |
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
|
938 |
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
|
939 |
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
|
940 |
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
|
941 |
|
60570 | 942 |
lemma is_unit_div_mult2_eq: |
943 |
assumes "is_unit b" and "is_unit c" |
|
944 |
shows "a div (b * c) = a div b div c" |
|
945 |
proof - |
|
946 |
from assms have "is_unit (b * c)" by (simp add: unit_prod) |
|
947 |
then have "b * c dvd a" |
|
948 |
by (rule unit_imp_dvd) |
|
949 |
then show ?thesis |
|
950 |
by (rule dvd_div_mult2_eq) |
|
951 |
qed |
|
952 |
||
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
|
953 |
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
|
954 |
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
|
955 |
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
|
956 |
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
|
957 |
|
60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
958 |
lemma is_unit_divide_mult_cancel_left: |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
959 |
assumes "a \<noteq> 0" and "is_unit b" |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
960 |
shows "a div (a * b) = 1 div b" |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
961 |
proof - |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
962 |
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
|
963 |
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
|
964 |
with assms show ?thesis by simp |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
965 |
qed |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
966 |
|
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
967 |
lemma is_unit_divide_mult_cancel_right: |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
968 |
assumes "a \<noteq> 0" and "is_unit b" |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
969 |
shows "a div (b * a) = 1 div b" |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
970 |
using assms is_unit_divide_mult_cancel_left [of a b] by (simp add: ac_simps) |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
971 |
|
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
972 |
end |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
973 |
|
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
974 |
class normalization_semidom = algebraic_semidom + |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
975 |
fixes normalize :: "'a \<Rightarrow> 'a" |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
976 |
and unit_factor :: "'a \<Rightarrow> 'a" |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
977 |
assumes unit_factor_mult_normalize [simp]: "unit_factor a * normalize a = a" |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
978 |
assumes normalize_0 [simp]: "normalize 0 = 0" |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
979 |
and unit_factor_0 [simp]: "unit_factor 0 = 0" |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
980 |
assumes is_unit_normalize: |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
981 |
"is_unit a \<Longrightarrow> normalize a = 1" |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
982 |
assumes unit_factor_is_unit [iff]: |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
983 |
"a \<noteq> 0 \<Longrightarrow> is_unit (unit_factor a)" |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
984 |
assumes unit_factor_mult: "unit_factor (a * b) = unit_factor a * unit_factor b" |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
985 |
begin |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
986 |
|
60688
01488b559910
avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset
|
987 |
text \<open> |
01488b559910
avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset
|
988 |
Class @{class normalization_semidom} cultivates the idea that |
01488b559910
avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset
|
989 |
each integral domain can be split into equivalence classes |
01488b559910
avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset
|
990 |
whose representants are associated, i.e. divide each other. |
01488b559910
avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset
|
991 |
@{const normalize} specifies a canonical representant for each equivalence |
01488b559910
avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset
|
992 |
class. The rationale behind this is that it is easier to reason about equality |
01488b559910
avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset
|
993 |
than equivalences, hence we prefer to think about equality of normalized |
01488b559910
avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset
|
994 |
values rather than associated elements. |
01488b559910
avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset
|
995 |
\<close> |
01488b559910
avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset
|
996 |
|
60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
997 |
lemma unit_factor_dvd [simp]: |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
998 |
"a \<noteq> 0 \<Longrightarrow> unit_factor a dvd b" |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
999 |
by (rule unit_imp_dvd) simp |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1000 |
|
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1001 |
lemma unit_factor_self [simp]: |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1002 |
"unit_factor a dvd a" |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1003 |
by (cases "a = 0") simp_all |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1004 |
|
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1005 |
lemma normalize_mult_unit_factor [simp]: |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1006 |
"normalize a * unit_factor a = a" |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1007 |
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
|
1008 |
|
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1009 |
lemma normalize_eq_0_iff [simp]: |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1010 |
"normalize a = 0 \<longleftrightarrow> a = 0" (is "?P \<longleftrightarrow> ?Q") |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1011 |
proof |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1012 |
assume ?P |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1013 |
moreover have "unit_factor a * normalize a = a" by simp |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1014 |
ultimately show ?Q by simp |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1015 |
next |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1016 |
assume ?Q then show ?P by simp |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1017 |
qed |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1018 |
|
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1019 |
lemma unit_factor_eq_0_iff [simp]: |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1020 |
"unit_factor a = 0 \<longleftrightarrow> a = 0" (is "?P \<longleftrightarrow> ?Q") |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1021 |
proof |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1022 |
assume ?P |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1023 |
moreover have "unit_factor a * normalize a = a" by simp |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1024 |
ultimately show ?Q by simp |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1025 |
next |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1026 |
assume ?Q then show ?P by simp |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1027 |
qed |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1028 |
|
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1029 |
lemma is_unit_unit_factor: |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1030 |
assumes "is_unit a" shows "unit_factor a = a" |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1031 |
proof - |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1032 |
from assms have "normalize a = 1" by (rule is_unit_normalize) |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1033 |
moreover from unit_factor_mult_normalize have "unit_factor a * normalize a = a" . |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1034 |
ultimately show ?thesis by simp |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1035 |
qed |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1036 |
|
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1037 |
lemma unit_factor_1 [simp]: |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1038 |
"unit_factor 1 = 1" |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1039 |
by (rule is_unit_unit_factor) simp |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1040 |
|
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1041 |
lemma normalize_1 [simp]: |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1042 |
"normalize 1 = 1" |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1043 |
by (rule is_unit_normalize) simp |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1044 |
|
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1045 |
lemma normalize_1_iff: |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1046 |
"normalize a = 1 \<longleftrightarrow> is_unit a" (is "?P \<longleftrightarrow> ?Q") |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1047 |
proof |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1048 |
assume ?Q then show ?P by (rule is_unit_normalize) |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1049 |
next |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1050 |
assume ?P |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1051 |
then have "a \<noteq> 0" by auto |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1052 |
from \<open>?P\<close> have "unit_factor a * normalize a = unit_factor a * 1" |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1053 |
by simp |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1054 |
then have "unit_factor a = a" |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1055 |
by simp |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1056 |
moreover have "is_unit (unit_factor a)" |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1057 |
using \<open>a \<noteq> 0\<close> by simp |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1058 |
ultimately show ?Q by simp |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1059 |
qed |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1060 |
|
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1061 |
lemma div_normalize [simp]: |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1062 |
"a div normalize a = unit_factor a" |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1063 |
proof (cases "a = 0") |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1064 |
case True then show ?thesis by simp |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1065 |
next |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1066 |
case False then have "normalize a \<noteq> 0" by simp |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1067 |
with nonzero_mult_divide_cancel_right |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1068 |
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
|
1069 |
then show ?thesis by simp |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1070 |
qed |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1071 |
|
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1072 |
lemma div_unit_factor [simp]: |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1073 |
"a div unit_factor a = normalize a" |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1074 |
proof (cases "a = 0") |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1075 |
case True then show ?thesis by simp |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1076 |
next |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1077 |
case False then have "unit_factor a \<noteq> 0" by simp |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1078 |
with nonzero_mult_divide_cancel_left |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1079 |
have "unit_factor a * normalize a div unit_factor a = normalize a" by blast |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1080 |
then show ?thesis by simp |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1081 |
qed |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1082 |
|
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1083 |
lemma normalize_div [simp]: |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1084 |
"normalize a div a = 1 div unit_factor a" |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1085 |
proof (cases "a = 0") |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1086 |
case True then show ?thesis by simp |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1087 |
next |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1088 |
case False |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1089 |
have "normalize a div a = normalize a div (unit_factor a * normalize a)" |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1090 |
by simp |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1091 |
also have "\<dots> = 1 div unit_factor a" |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1092 |
using False by (subst is_unit_divide_mult_cancel_right) simp_all |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1093 |
finally show ?thesis . |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1094 |
qed |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1095 |
|
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1096 |
lemma mult_one_div_unit_factor [simp]: |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1097 |
"a * (1 div unit_factor b) = a div unit_factor b" |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1098 |
by (cases "b = 0") simp_all |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1099 |
|
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1100 |
lemma normalize_mult: |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1101 |
"normalize (a * b) = normalize a * normalize b" |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1102 |
proof (cases "a = 0 \<or> b = 0") |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1103 |
case True then show ?thesis by auto |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1104 |
next |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1105 |
case False |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1106 |
from unit_factor_mult_normalize have "unit_factor (a * b) * normalize (a * b) = a * b" . |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1107 |
then have "normalize (a * b) = a * b div unit_factor (a * b)" by simp |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1108 |
also have "\<dots> = a * b div unit_factor (b * a)" by (simp add: ac_simps) |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1109 |
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
|
1110 |
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
|
1111 |
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
|
1112 |
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
|
1113 |
also have "\<dots> = normalize a * normalize b" |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1114 |
using False by (simp add: mult.commute [of a] mult.commute [of "normalize a"] unit_div_mult_swap [symmetric]) |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1115 |
finally show ?thesis . |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1116 |
qed |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1117 |
|
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1118 |
lemma unit_factor_idem [simp]: |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1119 |
"unit_factor (unit_factor a) = unit_factor a" |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1120 |
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
|
1121 |
|
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1122 |
lemma normalize_unit_factor [simp]: |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1123 |
"a \<noteq> 0 \<Longrightarrow> normalize (unit_factor a) = 1" |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1124 |
by (rule is_unit_normalize) simp |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1125 |
|
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1126 |
lemma normalize_idem [simp]: |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1127 |
"normalize (normalize a) = normalize a" |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1128 |
proof (cases "a = 0") |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1129 |
case True then show ?thesis by simp |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1130 |
next |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1131 |
case False |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1132 |
have "normalize a = normalize (unit_factor a * normalize a)" by simp |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1133 |
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
|
1134 |
by (simp only: normalize_mult) |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1135 |
finally show ?thesis using False by simp_all |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1136 |
qed |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1137 |
|
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1138 |
lemma unit_factor_normalize [simp]: |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1139 |
assumes "a \<noteq> 0" |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1140 |
shows "unit_factor (normalize a) = 1" |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1141 |
proof - |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1142 |
from assms have "normalize a \<noteq> 0" by simp |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1143 |
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
|
1144 |
by (simp only: unit_factor_mult_normalize) |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1145 |
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
|
1146 |
by simp |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1147 |
with \<open>normalize a \<noteq> 0\<close> |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1148 |
have "unit_factor (normalize a) * normalize a div normalize a = normalize a div normalize a" |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1149 |
by simp |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1150 |
with \<open>normalize a \<noteq> 0\<close> |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1151 |
show ?thesis by simp |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1152 |
qed |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1153 |
|
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1154 |
lemma dvd_unit_factor_div: |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1155 |
assumes "b dvd a" |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1156 |
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
|
1157 |
proof - |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1158 |
from assms have "a = a div b * b" |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1159 |
by simp |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1160 |
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
|
1161 |
by simp |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1162 |
then show ?thesis |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1163 |
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
|
1164 |
qed |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1165 |
|
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1166 |
lemma dvd_normalize_div: |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1167 |
assumes "b dvd a" |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1168 |
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
|
1169 |
proof - |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1170 |
from assms have "a = a div b * b" |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1171 |
by simp |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1172 |
then have "normalize a = normalize (a div b * b)" |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1173 |
by simp |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1174 |
then show ?thesis |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1175 |
by (cases "b = 0") (simp_all add: normalize_mult) |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1176 |
qed |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1177 |
|
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1178 |
lemma normalize_dvd_iff [simp]: |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1179 |
"normalize a dvd b \<longleftrightarrow> a dvd b" |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1180 |
proof - |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1181 |
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
|
1182 |
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
|
1183 |
by (cases "a = 0") simp_all |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1184 |
then show ?thesis by simp |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1185 |
qed |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1186 |
|
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1187 |
lemma dvd_normalize_iff [simp]: |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1188 |
"a dvd normalize b \<longleftrightarrow> a dvd b" |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1189 |
proof - |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1190 |
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
|
1191 |
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
|
1192 |
by (cases "b = 0") simp_all |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1193 |
then show ?thesis by simp |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1194 |
qed |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1195 |
|
60688
01488b559910
avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset
|
1196 |
text \<open> |
01488b559910
avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset
|
1197 |
We avoid an explicit definition of associated elements but prefer |
01488b559910
avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset
|
1198 |
explicit normalisation instead. In theory we could define an abbreviation |
01488b559910
avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset
|
1199 |
like @{prop "associated a b \<longleftrightarrow> normalize a = normalize b"} but this is |
01488b559910
avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset
|
1200 |
counterproductive without suggestive infix syntax, which we do not want |
01488b559910
avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset
|
1201 |
to sacrifice for this purpose here. |
01488b559910
avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset
|
1202 |
\<close> |
60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1203 |
|
60688
01488b559910
avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset
|
1204 |
lemma associatedI: |
01488b559910
avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset
|
1205 |
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
|
1206 |
shows "normalize a = normalize b" |
60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1207 |
proof (cases "a = 0 \<or> b = 0") |
60688
01488b559910
avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset
|
1208 |
case True with assms show ?thesis by auto |
60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1209 |
next |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1210 |
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
|
1211 |
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
|
1212 |
moreover from \<open>b dvd a\<close> obtain d where a: "a = b * d" .. |
01488b559910
avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset
|
1213 |
ultimately have "b * 1 = b * (c * d)" by (simp add: ac_simps) |
01488b559910
avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset
|
1214 |
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
|
1215 |
unfolding mult_cancel_left by simp |
01488b559910
avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset
|
1216 |
then have "is_unit c" and "is_unit d" by auto |
01488b559910
avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset
|
1217 |
with a b show ?thesis by (simp add: normalize_mult is_unit_normalize) |
01488b559910
avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset
|
1218 |
qed |
01488b559910
avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset
|
1219 |
|
01488b559910
avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset
|
1220 |
lemma associatedD1: |
01488b559910
avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset
|
1221 |
"normalize a = normalize b \<Longrightarrow> 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
|
1222 |
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
|
1223 |
by simp |
01488b559910
avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset
|
1224 |
|
01488b559910
avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset
|
1225 |
lemma associatedD2: |
01488b559910
avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset
|
1226 |
"normalize a = normalize b \<Longrightarrow> 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
|
1227 |
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
|
1228 |
by simp |
01488b559910
avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset
|
1229 |
|
01488b559910
avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset
|
1230 |
lemma associated_unit: |
01488b559910
avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset
|
1231 |
"normalize a = normalize b \<Longrightarrow> is_unit a \<Longrightarrow> is_unit b" |
01488b559910
avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset
|
1232 |
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
|
1233 |
|
01488b559910
avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset
|
1234 |
lemma associated_iff_dvd: |
01488b559910
avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset
|
1235 |
"normalize a = normalize b \<longleftrightarrow> a dvd b \<and> b dvd a" (is "?P \<longleftrightarrow> ?Q") |
01488b559910
avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset
|
1236 |
proof |
01488b559910
avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset
|
1237 |
assume ?Q then show ?P by (auto intro!: associatedI) |
01488b559910
avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset
|
1238 |
next |
01488b559910
avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset
|
1239 |
assume ?P |
01488b559910
avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset
|
1240 |
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
|
1241 |
by simp |
01488b559910
avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset
|
1242 |
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
|
1243 |
by (simp add: ac_simps) |
01488b559910
avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset
|
1244 |
show ?Q |
01488b559910
avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset
|
1245 |
proof (cases "a = 0 \<or> b = 0") |
01488b559910
avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset
|
1246 |
case True with \<open>?P\<close> show ?thesis by auto |
60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1247 |
next |
60688
01488b559910
avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset
|
1248 |
case False |
01488b559910
avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset
|
1249 |
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
|
1250 |
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
|
1251 |
with * show ?thesis by simp |
60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1252 |
qed |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1253 |
qed |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1254 |
|
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1255 |
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
|
1256 |
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
|
1257 |
assumes "normalize a = a" and "normalize b = b" |
60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1258 |
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
|
1259 |
proof - |
01488b559910
avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset
|
1260 |
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
|
1261 |
unfolding associated_iff_dvd by simp |
01488b559910
avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset
|
1262 |
with \<open>normalize a = a\<close> have "a = normalize b" by simp |
01488b559910
avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset
|
1263 |
with \<open>normalize b = b\<close> show "a = b" by simp |
60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1264 |
qed |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1265 |
|
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1266 |
end |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset
|
1267 |
|
38642
8fa437809c67
dropped type classes mult_mono and mult_mono1; tuned names of technical rule duplicates
haftmann
parents:
37767
diff
changeset
|
1268 |
class ordered_semiring = semiring + comm_monoid_add + ordered_ab_semigroup_add + |
8fa437809c67
dropped type classes mult_mono and mult_mono1; tuned names of technical rule duplicates
haftmann
parents:
37767
diff
changeset
|
1269 |
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
|
1270 |
assumes mult_right_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * c" |
25230 | 1271 |
begin |
1272 |
||
1273 |
lemma mult_mono: |
|
38642
8fa437809c67
dropped type classes mult_mono and mult_mono1; tuned names of technical rule duplicates
haftmann
parents:
37767
diff
changeset
|
1274 |
"a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * d" |
25230 | 1275 |
apply (erule mult_right_mono [THEN order_trans], assumption) |
1276 |
apply (erule mult_left_mono, assumption) |
|
1277 |
done |
|
1278 |
||
1279 |
lemma mult_mono': |
|
38642
8fa437809c67
dropped type classes mult_mono and mult_mono1; tuned names of technical rule duplicates
haftmann
parents:
37767
diff
changeset
|
1280 |
"a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * d" |
25230 | 1281 |
apply (rule mult_mono) |
1282 |
apply (fast intro: order_trans)+ |
|
1283 |
done |
|
1284 |
||
1285 |
end |
|
21199
2d83f93c3580
* Added annihilation axioms ("x * 0 = 0") to axclass semiring_0.
krauss
parents:
20633
diff
changeset
|
1286 |
|
38642
8fa437809c67
dropped type classes mult_mono and mult_mono1; tuned names of technical rule duplicates
haftmann
parents:
37767
diff
changeset
|
1287 |
class ordered_cancel_semiring = ordered_semiring + cancel_comm_monoid_add |
25267 | 1288 |
begin |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1289 |
|
27516 | 1290 |
subclass semiring_0_cancel .. |
23521 | 1291 |
|
56536 | 1292 |
lemma mult_nonneg_nonneg[simp]: "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> a * b" |
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1293 |
using mult_left_mono [of 0 b a] by simp |
25230 | 1294 |
|
1295 |
lemma mult_nonneg_nonpos: "0 \<le> a \<Longrightarrow> b \<le> 0 \<Longrightarrow> a * b \<le> 0" |
|
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1296 |
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
|
1297 |
|
44ea10bc07a7
clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents:
30650
diff
changeset
|
1298 |
lemma mult_nonpos_nonneg: "a \<le> 0 \<Longrightarrow> 0 \<le> b \<Longrightarrow> a * b \<le> 0" |
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1299 |
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
|
1300 |
|
61799 | 1301 |
text \<open>Legacy - use \<open>mult_nonpos_nonneg\<close>\<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
|
1302 |
lemma mult_nonneg_nonpos2: "0 \<le> a \<Longrightarrow> b \<le> 0 \<Longrightarrow> b * a \<le> 0" |
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1303 |
by (drule mult_right_mono [of b 0], auto) |
25230 | 1304 |
|
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
|
1305 |
lemma split_mult_neg_le: "(0 \<le> a & b \<le> 0) | (a \<le> 0 & 0 \<le> b) \<Longrightarrow> a * b \<le> 0" |
29667 | 1306 |
by (auto simp add: mult_nonneg_nonpos mult_nonneg_nonpos2) |
25230 | 1307 |
|
1308 |
end |
|
1309 |
||
38642
8fa437809c67
dropped type classes mult_mono and mult_mono1; tuned names of technical rule duplicates
haftmann
parents:
37767
diff
changeset
|
1310 |
class linordered_semiring = ordered_semiring + linordered_cancel_ab_semigroup_add |
25267 | 1311 |
begin |
25230 | 1312 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
1313 |
subclass ordered_cancel_semiring .. |
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
1314 |
|
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
1315 |
subclass ordered_comm_monoid_add .. |
25304
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset
|
1316 |
|
25230 | 1317 |
lemma mult_left_less_imp_less: |
1318 |
"c * a < c * b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a < b" |
|
29667 | 1319 |
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
|
1320 |
|
25230 | 1321 |
lemma mult_right_less_imp_less: |
1322 |
"a * c < b * c \<Longrightarrow> 0 \<le> c \<Longrightarrow> a < b" |
|
29667 | 1323 |
by (force simp add: mult_right_mono not_le [symmetric]) |
23521 | 1324 |
|
25186 | 1325 |
end |
25152 | 1326 |
|
35043
07dbdf60d5ad
dropped accidental duplication of "lin" prefix from cs. 108662d50512
haftmann
parents:
35032
diff
changeset
|
1327 |
class linordered_semiring_1 = linordered_semiring + semiring_1 |
36622
e393a91f86df
Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents:
36348
diff
changeset
|
1328 |
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
|
1329 |
|
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
|
1330 |
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
|
1331 |
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
|
1332 |
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
|
1333 |
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
|
1334 |
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
|
1335 |
by (simp add: add_mono mult_left_mono) |
49962
a8cc904a6820
Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents:
44921
diff
changeset
|
1336 |
thus ?thesis using assms unfolding distrib_right[symmetric] by simp |
36622
e393a91f86df
Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents:
36348
diff
changeset
|
1337 |
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
|
1338 |
|
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
|
1339 |
end |
35043
07dbdf60d5ad
dropped accidental duplication of "lin" prefix from cs. 108662d50512
haftmann
parents:
35032
diff
changeset
|
1340 |
|
07dbdf60d5ad
dropped accidental duplication of "lin" prefix from cs. 108662d50512
haftmann
parents:
35032
diff
changeset
|
1341 |
class linordered_semiring_strict = semiring + comm_monoid_add + linordered_cancel_ab_semigroup_add + |
25062 | 1342 |
assumes mult_strict_left_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b" |
1343 |
assumes mult_strict_right_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> a * c < b * c" |
|
25267 | 1344 |
begin |
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset
|
1345 |
|
27516 | 1346 |
subclass semiring_0_cancel .. |
14940 | 1347 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
1348 |
subclass linordered_semiring |
28823 | 1349 |
proof |
23550 | 1350 |
fix a b c :: 'a |
1351 |
assume A: "a \<le> b" "0 \<le> c" |
|
1352 |
from A show "c * a \<le> c * b" |
|
25186 | 1353 |
unfolding le_less |
1354 |
using mult_strict_left_mono by (cases "c = 0") auto |
|
23550 | 1355 |
from A show "a * c \<le> b * c" |
25152 | 1356 |
unfolding le_less |
25186 | 1357 |
using mult_strict_right_mono by (cases "c = 0") auto |
25152 | 1358 |
qed |
1359 |
||
25230 | 1360 |
lemma mult_left_le_imp_le: |
1361 |
"c * a \<le> c * b \<Longrightarrow> 0 < c \<Longrightarrow> a \<le> b" |
|
29667 | 1362 |
by (force 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
|
1363 |
|
25230 | 1364 |
lemma mult_right_le_imp_le: |
1365 |
"a * c \<le> b * c \<Longrightarrow> 0 < c \<Longrightarrow> a \<le> b" |
|
29667 | 1366 |
by (force simp add: mult_strict_right_mono not_less [symmetric]) |
25230 | 1367 |
|
56544 | 1368 |
lemma mult_pos_pos[simp]: "0 < a \<Longrightarrow> 0 < b \<Longrightarrow> 0 < a * b" |
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1369 |
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
|
1370 |
|
44ea10bc07a7
clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents:
30650
diff
changeset
|
1371 |
lemma mult_pos_neg: "0 < a \<Longrightarrow> b < 0 \<Longrightarrow> a * b < 0" |
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1372 |
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
|
1373 |
|
44ea10bc07a7
clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents:
30650
diff
changeset
|
1374 |
lemma mult_neg_pos: "a < 0 \<Longrightarrow> 0 < b \<Longrightarrow> a * b < 0" |
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1375 |
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
|
1376 |
|
61799 | 1377 |
text \<open>Legacy - use \<open>mult_neg_pos\<close>\<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
|
1378 |
lemma mult_pos_neg2: "0 < a \<Longrightarrow> b < 0 \<Longrightarrow> b * a < 0" |
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1379 |
by (drule mult_strict_right_mono [of b 0], auto) |
25230 | 1380 |
|
1381 |
lemma zero_less_mult_pos: |
|
1382 |
"0 < a * b \<Longrightarrow> 0 < a \<Longrightarrow> 0 < b" |
|
30692
44ea10bc07a7
clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents:
30650
diff
changeset
|
1383 |
apply (cases "b\<le>0") |
25230 | 1384 |
apply (auto simp add: le_less not_less) |
30692
44ea10bc07a7
clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents:
30650
diff
changeset
|
1385 |
apply (drule_tac mult_pos_neg [of a b]) |
25230 | 1386 |
apply (auto dest: less_not_sym) |
1387 |
done |
|
1388 |
||
1389 |
lemma zero_less_mult_pos2: |
|
1390 |
"0 < b * a \<Longrightarrow> 0 < a \<Longrightarrow> 0 < b" |
|
30692
44ea10bc07a7
clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents:
30650
diff
changeset
|
1391 |
apply (cases "b\<le>0") |
25230 | 1392 |
apply (auto simp add: le_less not_less) |
30692
44ea10bc07a7
clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents:
30650
diff
changeset
|
1393 |
apply (drule_tac mult_pos_neg2 [of a b]) |
25230 | 1394 |
apply (auto dest: less_not_sym) |
1395 |
done |
|
1396 |
||
60758 | 1397 |
text\<open>Strict monotonicity in both arguments\<close> |
26193 | 1398 |
lemma mult_strict_mono: |
1399 |
assumes "a < b" and "c < d" and "0 < b" and "0 \<le> c" |
|
1400 |
shows "a * c < b * d" |
|
1401 |
using assms apply (cases "c=0") |
|
56544 | 1402 |
apply (simp) |
26193 | 1403 |
apply (erule mult_strict_right_mono [THEN less_trans]) |
30692
44ea10bc07a7
clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents:
30650
diff
changeset
|
1404 |
apply (force simp add: le_less) |
26193 | 1405 |
apply (erule mult_strict_left_mono, assumption) |
1406 |
done |
|
1407 |
||
60758 | 1408 |
text\<open>This weaker variant has more natural premises\<close> |
26193 | 1409 |
lemma mult_strict_mono': |
1410 |
assumes "a < b" and "c < d" and "0 \<le> a" and "0 \<le> c" |
|
1411 |
shows "a * c < b * d" |
|
29667 | 1412 |
by (rule mult_strict_mono) (insert assms, auto) |
26193 | 1413 |
|
1414 |
lemma mult_less_le_imp_less: |
|
1415 |
assumes "a < b" and "c \<le> d" and "0 \<le> a" and "0 < c" |
|
1416 |
shows "a * c < b * d" |
|
1417 |
using assms apply (subgoal_tac "a * c < b * c") |
|
1418 |
apply (erule less_le_trans) |
|
1419 |
apply (erule mult_left_mono) |
|
1420 |
apply simp |
|
1421 |
apply (erule mult_strict_right_mono) |
|
1422 |
apply assumption |
|
1423 |
done |
|
1424 |
||
1425 |
lemma mult_le_less_imp_less: |
|
1426 |
assumes "a \<le> b" and "c < d" and "0 < a" and "0 \<le> c" |
|
1427 |
shows "a * c < b * d" |
|
1428 |
using assms apply (subgoal_tac "a * c \<le> b * c") |
|
1429 |
apply (erule le_less_trans) |
|
1430 |
apply (erule mult_strict_left_mono) |
|
1431 |
apply simp |
|
1432 |
apply (erule mult_right_mono) |
|
1433 |
apply simp |
|
1434 |
done |
|
1435 |
||
25230 | 1436 |
end |
1437 |
||
35097
4554bb2abfa3
dropped last occurence of the linlinordered accident
haftmann
parents:
35092
diff
changeset
|
1438 |
class linordered_semiring_1_strict = linordered_semiring_strict + semiring_1 |
36622
e393a91f86df
Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents:
36348
diff
changeset
|
1439 |
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
|
1440 |
|
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
|
1441 |
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
|
1442 |
|
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
|
1443 |
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
|
1444 |
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
|
1445 |
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
|
1446 |
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
|
1447 |
from assms have "u * x + v * y < u * a + v * a" |
e393a91f86df
Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents:
36348
diff
changeset
|
1448 |
by (cases "u = 0") |
e393a91f86df
Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents:
36348
diff
changeset
|
1449 |
(auto intro!: add_less_le_mono mult_strict_left_mono mult_left_mono) |
49962
a8cc904a6820
Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents:
44921
diff
changeset
|
1450 |
thus ?thesis using assms unfolding distrib_right[symmetric] by simp |
36622
e393a91f86df
Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents:
36348
diff
changeset
|
1451 |
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
|
1452 |
|
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
|
1453 |
end |
33319 | 1454 |
|
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
|
1455 |
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
|
1456 |
assumes comm_mult_left_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> c * a \<le> c * b" |
25186 | 1457 |
begin |
25152 | 1458 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
1459 |
subclass ordered_semiring |
28823 | 1460 |
proof |
21199
2d83f93c3580
* Added annihilation axioms ("x * 0 = 0") to axclass semiring_0.
krauss
parents:
20633
diff
changeset
|
1461 |
fix a b c :: 'a |
23550 | 1462 |
assume "a \<le> b" "0 \<le> c" |
38642
8fa437809c67
dropped type classes mult_mono and mult_mono1; tuned names of technical rule duplicates
haftmann
parents:
37767
diff
changeset
|
1463 |
thus "c * a \<le> c * b" by (rule comm_mult_left_mono) |
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
56544
diff
changeset
|
1464 |
thus "a * c \<le> b * c" by (simp only: mult.commute) |
21199
2d83f93c3580
* Added annihilation axioms ("x * 0 = 0") to axclass semiring_0.
krauss
parents:
20633
diff
changeset
|
1465 |
qed |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
1466 |
|
25267 | 1467 |
end |
1468 |
||
38642
8fa437809c67
dropped type classes mult_mono and mult_mono1; tuned names of technical rule duplicates
haftmann
parents:
37767
diff
changeset
|
1469 |
class ordered_cancel_comm_semiring = ordered_comm_semiring + cancel_comm_monoid_add |
25267 | 1470 |
begin |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
1471 |
|
38642
8fa437809c67
dropped type classes mult_mono and mult_mono1; tuned names of technical rule duplicates
haftmann
parents:
37767
diff
changeset
|
1472 |
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
|
1473 |
subclass ordered_comm_semiring .. |
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
1474 |
subclass ordered_cancel_semiring .. |
25267 | 1475 |
|
1476 |
end |
|
1477 |
||
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
1478 |
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
|
1479 |
assumes comm_mult_strict_left_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b" |
25267 | 1480 |
begin |
1481 |
||
35043
07dbdf60d5ad
dropped accidental duplication of "lin" prefix from cs. 108662d50512
haftmann
parents:
35032
diff
changeset
|
1482 |
subclass linordered_semiring_strict |
28823 | 1483 |
proof |
23550 | 1484 |
fix a b c :: 'a |
1485 |
assume "a < b" "0 < c" |
|
38642
8fa437809c67
dropped type classes mult_mono and mult_mono1; tuned names of technical rule duplicates
haftmann
parents:
37767
diff
changeset
|
1486 |
thus "c * a < c * b" by (rule comm_mult_strict_left_mono) |
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
56544
diff
changeset
|
1487 |
thus "a * c < b * c" by (simp only: mult.commute) |
23550 | 1488 |
qed |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
1489 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
1490 |
subclass ordered_cancel_comm_semiring |
28823 | 1491 |
proof |
23550 | 1492 |
fix a b c :: 'a |
1493 |
assume "a \<le> b" "0 \<le> c" |
|
1494 |
thus "c * a \<le> c * b" |
|
25186 | 1495 |
unfolding le_less |
26193 | 1496 |
using mult_strict_left_mono by (cases "c = 0") auto |
23550 | 1497 |
qed |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
1498 |
|
25267 | 1499 |
end |
25230 | 1500 |
|
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
|
1501 |
class ordered_ring = ring + ordered_cancel_semiring |
25267 | 1502 |
begin |
25230 | 1503 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
1504 |
subclass ordered_ab_group_add .. |
14270 | 1505 |
|
25230 | 1506 |
lemma less_add_iff1: |
1507 |
"a * e + c < b * e + d \<longleftrightarrow> (a - b) * e + c < d" |
|
29667 | 1508 |
by (simp add: algebra_simps) |
25230 | 1509 |
|
1510 |
lemma less_add_iff2: |
|
1511 |
"a * e + c < b * e + d \<longleftrightarrow> c < (b - a) * e + d" |
|
29667 | 1512 |
by (simp add: algebra_simps) |
25230 | 1513 |
|
1514 |
lemma le_add_iff1: |
|
1515 |
"a * e + c \<le> b * e + d \<longleftrightarrow> (a - b) * e + c \<le> d" |
|
29667 | 1516 |
by (simp add: algebra_simps) |
25230 | 1517 |
|
1518 |
lemma le_add_iff2: |
|
1519 |
"a * e + c \<le> b * e + d \<longleftrightarrow> c \<le> (b - a) * e + d" |
|
29667 | 1520 |
by (simp add: algebra_simps) |
25230 | 1521 |
|
1522 |
lemma mult_left_mono_neg: |
|
1523 |
"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
|
1524 |
apply (drule mult_left_mono [of _ _ "- c"]) |
35216 | 1525 |
apply simp_all |
25230 | 1526 |
done |
1527 |
||
1528 |
lemma mult_right_mono_neg: |
|
1529 |
"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
|
1530 |
apply (drule mult_right_mono [of _ _ "- c"]) |
35216 | 1531 |
apply simp_all |
25230 | 1532 |
done |
1533 |
||
30692
44ea10bc07a7
clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents:
30650
diff
changeset
|
1534 |
lemma mult_nonpos_nonpos: "a \<le> 0 \<Longrightarrow> b \<le> 0 \<Longrightarrow> 0 \<le> a * b" |
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1535 |
using mult_right_mono_neg [of a 0 b] by simp |
25230 | 1536 |
|
1537 |
lemma split_mult_pos_le: |
|
1538 |
"(0 \<le> a \<and> 0 \<le> b) \<or> (a \<le> 0 \<and> b \<le> 0) \<Longrightarrow> 0 \<le> a * b" |
|
56536 | 1539 |
by (auto simp add: mult_nonpos_nonpos) |
25186 | 1540 |
|
1541 |
end |
|
14270 | 1542 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
1543 |
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
|
1544 |
begin |
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset
|
1545 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
1546 |
subclass ordered_ring .. |
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
1547 |
|
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
1548 |
subclass ordered_ab_group_add_abs |
28823 | 1549 |
proof |
25304
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset
|
1550 |
fix a b |
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset
|
1551 |
show "\<bar>a + b\<bar> \<le> \<bar>a\<bar> + \<bar>b\<bar>" |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54225
diff
changeset
|
1552 |
by (auto simp add: abs_if not_le not_less algebra_simps simp del: add.commute dest: add_neg_neg add_nonneg_nonneg) |
35216 | 1553 |
qed (auto simp add: abs_if) |
25304
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset
|
1554 |
|
35631
0b8a5fd339ab
generalize some lemmas from class linordered_ring_strict to linordered_ring
huffman
parents:
35302
diff
changeset
|
1555 |
lemma zero_le_square [simp]: "0 \<le> a * a" |
0b8a5fd339ab
generalize some lemmas from class linordered_ring_strict to linordered_ring
huffman
parents:
35302
diff
changeset
|
1556 |
using linear [of 0 a] |
56536 | 1557 |
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
|
1558 |
|
0b8a5fd339ab
generalize some lemmas from class linordered_ring_strict to linordered_ring
huffman
parents:
35302
diff
changeset
|
1559 |
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
|
1560 |
by (simp add: not_less) |
0b8a5fd339ab
generalize some lemmas from class linordered_ring_strict to linordered_ring
huffman
parents:
35302
diff
changeset
|
1561 |
|
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
|
1562 |
proposition abs_eq_iff: "abs x = abs y \<longleftrightarrow> x = y \<or> x = -y" |
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
|
1563 |
by (auto simp add: abs_if split: split_if_asm) |
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
|
1564 |
|
25304
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset
|
1565 |
end |
23521 | 1566 |
|
35043
07dbdf60d5ad
dropped accidental duplication of "lin" prefix from cs. 108662d50512
haftmann
parents:
35032
diff
changeset
|
1567 |
class linordered_ring_strict = ring + linordered_semiring_strict |
25304
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset
|
1568 |
+ ordered_ab_group_add + abs_if |
25230 | 1569 |
begin |
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset
|
1570 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
1571 |
subclass linordered_ring .. |
25304
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset
|
1572 |
|
30692
44ea10bc07a7
clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents:
30650
diff
changeset
|
1573 |
lemma mult_strict_left_mono_neg: "b < a \<Longrightarrow> c < 0 \<Longrightarrow> c * a < c * b" |
44ea10bc07a7
clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents:
30650
diff
changeset
|
1574 |
using mult_strict_left_mono [of b a "- c"] by simp |
44ea10bc07a7
clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents:
30650
diff
changeset
|
1575 |
|
44ea10bc07a7
clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents:
30650
diff
changeset
|
1576 |
lemma mult_strict_right_mono_neg: "b < a \<Longrightarrow> c < 0 \<Longrightarrow> a * c < b * c" |
44ea10bc07a7
clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents:
30650
diff
changeset
|
1577 |
using mult_strict_right_mono [of b a "- c"] by simp |
44ea10bc07a7
clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents:
30650
diff
changeset
|
1578 |
|
44ea10bc07a7
clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents:
30650
diff
changeset
|
1579 |
lemma mult_neg_neg: "a < 0 \<Longrightarrow> b < 0 \<Longrightarrow> 0 < a * b" |
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1580 |
using mult_strict_right_mono_neg [of a 0 b] by simp |
14738 | 1581 |
|
25917 | 1582 |
subclass ring_no_zero_divisors |
28823 | 1583 |
proof |
25917 | 1584 |
fix a b |
1585 |
assume "a \<noteq> 0" then have A: "a < 0 \<or> 0 < a" by (simp add: neq_iff) |
|
1586 |
assume "b \<noteq> 0" then have B: "b < 0 \<or> 0 < b" by (simp add: neq_iff) |
|
1587 |
have "a * b < 0 \<or> 0 < a * b" |
|
1588 |
proof (cases "a < 0") |
|
1589 |
case True note A' = this |
|
1590 |
show ?thesis proof (cases "b < 0") |
|
1591 |
case True with A' |
|
1592 |
show ?thesis by (auto dest: mult_neg_neg) |
|
1593 |
next |
|
1594 |
case False with B have "0 < b" by auto |
|
1595 |
with A' show ?thesis by (auto dest: mult_strict_right_mono) |
|
1596 |
qed |
|
1597 |
next |
|
1598 |
case False with A have A': "0 < a" by auto |
|
1599 |
show ?thesis proof (cases "b < 0") |
|
1600 |
case True with A' |
|
1601 |
show ?thesis by (auto dest: mult_strict_right_mono_neg) |
|
1602 |
next |
|
1603 |
case False with B have "0 < b" by auto |
|
56544 | 1604 |
with A' show ?thesis by auto |
25917 | 1605 |
qed |
1606 |
qed |
|
1607 |
then show "a * b \<noteq> 0" by (simp add: neq_iff) |
|
1608 |
qed |
|
25304
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset
|
1609 |
|
56480
093ea91498e6
field_simps: better support for negation and division, and power
hoelzl
parents:
56217
diff
changeset
|
1610 |
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
|
1611 |
by (cases a 0 b 0 rule: linorder_cases[case_product linorder_cases]) |
56544 | 1612 |
(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
|
1613 |
|
56480
093ea91498e6
field_simps: better support for negation and division, and power
hoelzl
parents:
56217
diff
changeset
|
1614 |
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
|
1615 |
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
|
1616 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
1617 |
lemma mult_less_0_iff: |
25917 | 1618 |
"a * b < 0 \<longleftrightarrow> 0 < a \<and> b < 0 \<or> a < 0 \<and> 0 < b" |
35216 | 1619 |
apply (insert zero_less_mult_iff [of "-a" b]) |
1620 |
apply force |
|
25917 | 1621 |
done |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
1622 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
1623 |
lemma mult_le_0_iff: |
25917 | 1624 |
"a * b \<le> 0 \<longleftrightarrow> 0 \<le> a \<and> b \<le> 0 \<or> a \<le> 0 \<and> 0 \<le> 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
|
1625 |
apply (insert zero_le_mult_iff [of "-a" b]) |
35216 | 1626 |
apply force |
25917 | 1627 |
done |
1628 |
||
60758 | 1629 |
text\<open>Cancellation laws for @{term "c*a < c*b"} and @{term "a*c < b*c"}, |
61799 | 1630 |
also with the relations \<open>\<le>\<close> and equality.\<close> |
26193 | 1631 |
|
60758 | 1632 |
text\<open>These ``disjunction'' versions produce two cases when the comparison is |
1633 |
an assumption, but effectively four when the comparison is a goal.\<close> |
|
26193 | 1634 |
|
1635 |
lemma mult_less_cancel_right_disj: |
|
1636 |
"a * c < b * c \<longleftrightarrow> 0 < c \<and> a < b \<or> c < 0 \<and> b < a" |
|
1637 |
apply (cases "c = 0") |
|
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
|
1638 |
apply (auto simp add: neq_iff mult_strict_right_mono |
26193 | 1639 |
mult_strict_right_mono_neg) |
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
|
1640 |
apply (auto simp add: not_less |
26193 | 1641 |
not_le [symmetric, of "a*c"] |
1642 |
not_le [symmetric, of a]) |
|
1643 |
apply (erule_tac [!] notE) |
|
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
|
1644 |
apply (auto simp add: less_imp_le mult_right_mono |
26193 | 1645 |
mult_right_mono_neg) |
1646 |
done |
|
1647 |
||
1648 |
lemma mult_less_cancel_left_disj: |
|
1649 |
"c * a < c * b \<longleftrightarrow> 0 < c \<and> a < b \<or> c < 0 \<and> b < a" |
|
1650 |
apply (cases "c = 0") |
|
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
|
1651 |
apply (auto simp add: neq_iff mult_strict_left_mono |
26193 | 1652 |
mult_strict_left_mono_neg) |
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
|
1653 |
apply (auto simp add: not_less |
26193 | 1654 |
not_le [symmetric, of "c*a"] |
1655 |
not_le [symmetric, of a]) |
|
1656 |
apply (erule_tac [!] notE) |
|
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
|
1657 |
apply (auto simp add: less_imp_le mult_left_mono |
26193 | 1658 |
mult_left_mono_neg) |
1659 |
done |
|
1660 |
||
60758 | 1661 |
text\<open>The ``conjunction of implication'' lemmas produce two cases when the |
1662 |
comparison is a goal, but give four when the comparison is an assumption.\<close> |
|
26193 | 1663 |
|
1664 |
lemma mult_less_cancel_right: |
|
1665 |
"a * c < b * c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < b) \<and> (c \<le> 0 \<longrightarrow> b < a)" |
|
1666 |
using mult_less_cancel_right_disj [of a c b] by auto |
|
1667 |
||
1668 |
lemma mult_less_cancel_left: |
|
1669 |
"c * a < c * b \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < b) \<and> (c \<le> 0 \<longrightarrow> b < a)" |
|
1670 |
using mult_less_cancel_left_disj [of c a b] by auto |
|
1671 |
||
1672 |
lemma mult_le_cancel_right: |
|
1673 |
"a * c \<le> b * c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)" |
|
29667 | 1674 |
by (simp add: not_less [symmetric] mult_less_cancel_right_disj) |
26193 | 1675 |
|
1676 |
lemma mult_le_cancel_left: |
|
1677 |
"c * a \<le> c * b \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)" |
|
29667 | 1678 |
by (simp add: not_less [symmetric] mult_less_cancel_left_disj) |
26193 | 1679 |
|
30649
57753e0ec1d4
1. New cancellation simprocs for common factors in inequations
nipkow
parents:
30242
diff
changeset
|
1680 |
lemma mult_le_cancel_left_pos: |
57753e0ec1d4
1. New cancellation simprocs for common factors in inequations
nipkow
parents:
30242
diff
changeset
|
1681 |
"0 < c \<Longrightarrow> c * a \<le> c * b \<longleftrightarrow> a \<le> b" |
57753e0ec1d4
1. New cancellation simprocs for common factors in inequations
nipkow
parents:
30242
diff
changeset
|
1682 |
by (auto simp: mult_le_cancel_left) |
57753e0ec1d4
1. New cancellation simprocs for common factors in inequations
nipkow
parents:
30242
diff
changeset
|
1683 |
|
57753e0ec1d4
1. New cancellation simprocs for common factors in inequations
nipkow
parents:
30242
diff
changeset
|
1684 |
lemma mult_le_cancel_left_neg: |
57753e0ec1d4
1. New cancellation simprocs for common factors in inequations
nipkow
parents:
30242
diff
changeset
|
1685 |
"c < 0 \<Longrightarrow> c * a \<le> c * b \<longleftrightarrow> b \<le> a" |
57753e0ec1d4
1. New cancellation simprocs for common factors in inequations
nipkow
parents:
30242
diff
changeset
|
1686 |
by (auto simp: mult_le_cancel_left) |
57753e0ec1d4
1. New cancellation simprocs for common factors in inequations
nipkow
parents:
30242
diff
changeset
|
1687 |
|
57753e0ec1d4
1. New cancellation simprocs for common factors in inequations
nipkow
parents:
30242
diff
changeset
|
1688 |
lemma mult_less_cancel_left_pos: |
57753e0ec1d4
1. New cancellation simprocs for common factors in inequations
nipkow
parents:
30242
diff
changeset
|
1689 |
"0 < c \<Longrightarrow> c * a < c * b \<longleftrightarrow> a < b" |
57753e0ec1d4
1. New cancellation simprocs for common factors in inequations
nipkow
parents:
30242
diff
changeset
|
1690 |
by (auto simp: mult_less_cancel_left) |
57753e0ec1d4
1. New cancellation simprocs for common factors in inequations
nipkow
parents:
30242
diff
changeset
|
1691 |
|
57753e0ec1d4
1. New cancellation simprocs for common factors in inequations
nipkow
parents:
30242
diff
changeset
|
1692 |
lemma mult_less_cancel_left_neg: |
57753e0ec1d4
1. New cancellation simprocs for common factors in inequations
nipkow
parents:
30242
diff
changeset
|
1693 |
"c < 0 \<Longrightarrow> c * a < c * b \<longleftrightarrow> b < a" |
57753e0ec1d4
1. New cancellation simprocs for common factors in inequations
nipkow
parents:
30242
diff
changeset
|
1694 |
by (auto simp: mult_less_cancel_left) |
57753e0ec1d4
1. New cancellation simprocs for common factors in inequations
nipkow
parents:
30242
diff
changeset
|
1695 |
|
25917 | 1696 |
end |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
1697 |
|
30692
44ea10bc07a7
clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents:
30650
diff
changeset
|
1698 |
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
|
1699 |
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
|
1700 |
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
|
1701 |
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
|
1702 |
mult_neg_pos mult_neg_neg |
25230 | 1703 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
1704 |
class ordered_comm_ring = comm_ring + ordered_comm_semiring |
25267 | 1705 |
begin |
25230 | 1706 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
1707 |
subclass ordered_ring .. |
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
1708 |
subclass ordered_cancel_comm_semiring .. |
25230 | 1709 |
|
25267 | 1710 |
end |
25230 | 1711 |
|
59833
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset
|
1712 |
class linordered_semidom = semidom + linordered_comm_semiring_strict + |
25230 | 1713 |
assumes zero_less_one [simp]: "0 < 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
|
1714 |
assumes le_add_diff_inverse2 [simp]: "b \<le> a \<Longrightarrow> a - b + b = a" |
25230 | 1715 |
begin |
1716 |
||
60758 | 1717 |
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
|
1718 |
|
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
|
1719 |
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
|
1720 |
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
|
1721 |
|
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
|
1722 |
lemma add_diff_inverse: "~ a<b \<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
|
1723 |
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
|
1724 |
|
e5fa1d5d3952
Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents:
60570
diff
changeset
|
1725 |
lemma add_le_imp_le_diff: |
e5fa1d5d3952
Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents:
60570
diff
changeset
|
1726 |
shows "i + k \<le> n \<Longrightarrow> i \<le> n - k" |
e5fa1d5d3952
Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents:
60570
diff
changeset
|
1727 |
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
|
1728 |
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
|
1729 |
apply (simp only: add.assoc [symmetric]) |
e5fa1d5d3952
Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents:
60570
diff
changeset
|
1730 |
using add_implies_diff by fastforce |
e5fa1d5d3952
Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents:
60570
diff
changeset
|
1731 |
|
e5fa1d5d3952
Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents:
60570
diff
changeset
|
1732 |
lemma add_le_add_imp_diff_le: |
e5fa1d5d3952
Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents:
60570
diff
changeset
|
1733 |
assumes a1: "i + k \<le> n" |
e5fa1d5d3952
Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents:
60570
diff
changeset
|
1734 |
and a2: "n \<le> j + k" |
e5fa1d5d3952
Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents:
60570
diff
changeset
|
1735 |
shows "\<lbrakk>i + k \<le> n; n \<le> j + k\<rbrakk> \<Longrightarrow> n - k \<le> j" |
e5fa1d5d3952
Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents:
60570
diff
changeset
|
1736 |
proof - |
e5fa1d5d3952
Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents:
60570
diff
changeset
|
1737 |
have "n - (i + k) + (i + k) = n" |
e5fa1d5d3952
Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents:
60570
diff
changeset
|
1738 |
using a1 by simp |
e5fa1d5d3952
Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents:
60570
diff
changeset
|
1739 |
moreover have "n - k = n - k - i + i" |
e5fa1d5d3952
Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents:
60570
diff
changeset
|
1740 |
using a1 by (simp add: add_le_imp_le_diff) |
e5fa1d5d3952
Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents:
60570
diff
changeset
|
1741 |
ultimately show ?thesis |
e5fa1d5d3952
Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents:
60570
diff
changeset
|
1742 |
using a2 |
e5fa1d5d3952
Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents:
60570
diff
changeset
|
1743 |
apply (simp add: add.assoc [symmetric]) |
e5fa1d5d3952
Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents:
60570
diff
changeset
|
1744 |
apply (rule add_le_imp_le_diff [of _ k "j+k", simplified add_diff_cancel_right']) |
e5fa1d5d3952
Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents:
60570
diff
changeset
|
1745 |
by (simp add: add.commute diff_diff_add) |
e5fa1d5d3952
Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents:
60570
diff
changeset
|
1746 |
qed |
e5fa1d5d3952
Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents:
60570
diff
changeset
|
1747 |
|
25230 | 1748 |
lemma pos_add_strict: |
1749 |
shows "0 < a \<Longrightarrow> b < c \<Longrightarrow> b < a + c" |
|
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1750 |
using add_strict_mono [of 0 a b c] by simp |
25230 | 1751 |
|
26193 | 1752 |
lemma zero_le_one [simp]: "0 \<le> 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
|
1753 |
by (rule zero_less_one [THEN less_imp_le]) |
26193 | 1754 |
|
1755 |
lemma not_one_le_zero [simp]: "\<not> 1 \<le> 0" |
|
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
|
1756 |
by (simp add: not_le) |
26193 | 1757 |
|
1758 |
lemma not_one_less_zero [simp]: "\<not> 1 < 0" |
|
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
|
1759 |
by (simp add: not_less) |
26193 | 1760 |
|
1761 |
lemma less_1_mult: |
|
1762 |
assumes "1 < m" and "1 < n" |
|
1763 |
shows "1 < m * n" |
|
1764 |
using assms mult_strict_mono [of 1 m 1 n] |
|
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
|
1765 |
by (simp add: less_trans [OF zero_less_one]) |
26193 | 1766 |
|
59000 | 1767 |
lemma mult_left_le: "c \<le> 1 \<Longrightarrow> 0 \<le> a \<Longrightarrow> a * c \<le> a" |
1768 |
using mult_left_mono[of c 1 a] by simp |
|
1769 |
||
1770 |
lemma mult_le_one: "a \<le> 1 \<Longrightarrow> 0 \<le> b \<Longrightarrow> b \<le> 1 \<Longrightarrow> a * b \<le> 1" |
|
1771 |
using mult_mono[of a 1 b 1] by simp |
|
1772 |
||
25230 | 1773 |
end |
1774 |
||
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
1775 |
class linordered_idom = comm_ring_1 + |
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
1776 |
linordered_comm_semiring_strict + ordered_ab_group_add + |
25230 | 1777 |
abs_if + sgn_if |
25917 | 1778 |
begin |
1779 |
||
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
|
1780 |
subclass linordered_semiring_1_strict .. |
35043
07dbdf60d5ad
dropped accidental duplication of "lin" prefix from cs. 108662d50512
haftmann
parents:
35032
diff
changeset
|
1781 |
subclass linordered_ring_strict .. |
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
1782 |
subclass ordered_comm_ring .. |
27516 | 1783 |
subclass idom .. |
25917 | 1784 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
1785 |
subclass linordered_semidom |
28823 | 1786 |
proof |
26193 | 1787 |
have "0 \<le> 1 * 1" by (rule zero_le_square) |
1788 |
thus "0 < 1" by (simp add: le_less) |
|
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
|
1789 |
show "\<And>b a. b \<le> a \<Longrightarrow> a - b + 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
|
1790 |
by simp |
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
|
1791 |
qed |
25917 | 1792 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
1793 |
lemma linorder_neqE_linordered_idom: |
26193 | 1794 |
assumes "x \<noteq> y" obtains "x < y" | "y < x" |
1795 |
using assms by (rule neqE) |
|
1796 |
||
60758 | 1797 |
text \<open>These cancellation simprules also produce two cases when the comparison is a goal.\<close> |
26274 | 1798 |
|
1799 |
lemma mult_le_cancel_right1: |
|
1800 |
"c \<le> b * c \<longleftrightarrow> (0 < c \<longrightarrow> 1 \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> 1)" |
|
29667 | 1801 |
by (insert mult_le_cancel_right [of 1 c b], simp) |
26274 | 1802 |
|
1803 |
lemma mult_le_cancel_right2: |
|
1804 |
"a * c \<le> c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> 1) \<and> (c < 0 \<longrightarrow> 1 \<le> a)" |
|
29667 | 1805 |
by (insert mult_le_cancel_right [of a c 1], simp) |
26274 | 1806 |
|
1807 |
lemma mult_le_cancel_left1: |
|
1808 |
"c \<le> c * b \<longleftrightarrow> (0 < c \<longrightarrow> 1 \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> 1)" |
|
29667 | 1809 |
by (insert mult_le_cancel_left [of c 1 b], simp) |
26274 | 1810 |
|
1811 |
lemma mult_le_cancel_left2: |
|
1812 |
"c * a \<le> c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> 1) \<and> (c < 0 \<longrightarrow> 1 \<le> a)" |
|
29667 | 1813 |
by (insert mult_le_cancel_left [of c a 1], simp) |
26274 | 1814 |
|
1815 |
lemma mult_less_cancel_right1: |
|
1816 |
"c < b * c \<longleftrightarrow> (0 \<le> c \<longrightarrow> 1 < b) \<and> (c \<le> 0 \<longrightarrow> b < 1)" |
|
29667 | 1817 |
by (insert mult_less_cancel_right [of 1 c b], simp) |
26274 | 1818 |
|
1819 |
lemma mult_less_cancel_right2: |
|
1820 |
"a * c < c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < 1) \<and> (c \<le> 0 \<longrightarrow> 1 < a)" |
|
29667 | 1821 |
by (insert mult_less_cancel_right [of a c 1], simp) |
26274 | 1822 |
|
1823 |
lemma mult_less_cancel_left1: |
|
1824 |
"c < c * b \<longleftrightarrow> (0 \<le> c \<longrightarrow> 1 < b) \<and> (c \<le> 0 \<longrightarrow> b < 1)" |
|
29667 | 1825 |
by (insert mult_less_cancel_left [of c 1 b], simp) |
26274 | 1826 |
|
1827 |
lemma mult_less_cancel_left2: |
|
1828 |
"c * a < c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < 1) \<and> (c \<le> 0 \<longrightarrow> 1 < a)" |
|
29667 | 1829 |
by (insert mult_less_cancel_left [of c a 1], simp) |
26274 | 1830 |
|
27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27516
diff
changeset
|
1831 |
lemma sgn_sgn [simp]: |
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27516
diff
changeset
|
1832 |
"sgn (sgn a) = sgn a" |
29700 | 1833 |
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
|
1834 |
|
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27516
diff
changeset
|
1835 |
lemma sgn_0_0: |
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27516
diff
changeset
|
1836 |
"sgn a = 0 \<longleftrightarrow> a = 0" |
29700 | 1837 |
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
|
1838 |
|
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27516
diff
changeset
|
1839 |
lemma sgn_1_pos: |
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27516
diff
changeset
|
1840 |
"sgn a = 1 \<longleftrightarrow> a > 0" |
35216 | 1841 |
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
|
1842 |
|
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27516
diff
changeset
|
1843 |
lemma sgn_1_neg: |
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27516
diff
changeset
|
1844 |
"sgn a = - 1 \<longleftrightarrow> a < 0" |
35216 | 1845 |
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
|
1846 |
|
29940 | 1847 |
lemma sgn_pos [simp]: |
1848 |
"0 < a \<Longrightarrow> sgn a = 1" |
|
1849 |
unfolding sgn_1_pos . |
|
1850 |
||
1851 |
lemma sgn_neg [simp]: |
|
1852 |
"a < 0 \<Longrightarrow> sgn a = - 1" |
|
1853 |
unfolding sgn_1_neg . |
|
1854 |
||
27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27516
diff
changeset
|
1855 |
lemma sgn_times: |
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27516
diff
changeset
|
1856 |
"sgn (a * b) = sgn a * sgn b" |
29667 | 1857 |
by (auto simp add: sgn_if zero_less_mult_iff) |
27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27516
diff
changeset
|
1858 |
|
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1859 |
lemma abs_sgn: "\<bar>k\<bar> = k * sgn k" |
29700 | 1860 |
unfolding sgn_if abs_if by auto |
1861 |
||
29940 | 1862 |
lemma sgn_greater [simp]: |
1863 |
"0 < sgn a \<longleftrightarrow> 0 < a" |
|
1864 |
unfolding sgn_if by auto |
|
1865 |
||
1866 |
lemma sgn_less [simp]: |
|
1867 |
"sgn a < 0 \<longleftrightarrow> a < 0" |
|
1868 |
unfolding sgn_if by auto |
|
1869 |
||
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1870 |
lemma abs_dvd_iff [simp]: "\<bar>m\<bar> dvd k \<longleftrightarrow> m dvd k" |
29949 | 1871 |
by (simp add: abs_if) |
1872 |
||
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1873 |
lemma dvd_abs_iff [simp]: "m dvd \<bar>k\<bar> \<longleftrightarrow> m dvd k" |
29949 | 1874 |
by (simp add: abs_if) |
29653 | 1875 |
|
33676
802f5e233e48
moved lemma from Algebra/IntRing to Ring_and_Field
nipkow
parents:
33364
diff
changeset
|
1876 |
lemma dvd_if_abs_eq: |
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1877 |
"\<bar>l\<bar> = \<bar>k\<bar> \<Longrightarrow> l dvd k" |
33676
802f5e233e48
moved lemma from Algebra/IntRing to Ring_and_Field
nipkow
parents:
33364
diff
changeset
|
1878 |
by(subst abs_dvd_iff[symmetric]) simp |
802f5e233e48
moved lemma from Algebra/IntRing to Ring_and_Field
nipkow
parents:
33364
diff
changeset
|
1879 |
|
60758 | 1880 |
text \<open>The following lemmas can be proven in more general structures, but |
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
|
1881 |
are dangerous as simp rules in absence of @{thm neg_equal_zero}, |
60758 | 1882 |
@{thm neg_less_pos}, @{thm neg_less_eq_nonneg}.\<close> |
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54250
diff
changeset
|
1883 |
|
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54250
diff
changeset
|
1884 |
lemma equation_minus_iff_1 [simp, no_atp]: |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54250
diff
changeset
|
1885 |
"1 = - a \<longleftrightarrow> a = - 1" |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54250
diff
changeset
|
1886 |
by (fact equation_minus_iff) |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54250
diff
changeset
|
1887 |
|
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54250
diff
changeset
|
1888 |
lemma minus_equation_iff_1 [simp, no_atp]: |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54250
diff
changeset
|
1889 |
"- a = 1 \<longleftrightarrow> a = - 1" |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54250
diff
changeset
|
1890 |
by (subst minus_equation_iff, auto) |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54250
diff
changeset
|
1891 |
|
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54250
diff
changeset
|
1892 |
lemma le_minus_iff_1 [simp, no_atp]: |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54250
diff
changeset
|
1893 |
"1 \<le> - b \<longleftrightarrow> b \<le> - 1" |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54250
diff
changeset
|
1894 |
by (fact le_minus_iff) |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54250
diff
changeset
|
1895 |
|
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54250
diff
changeset
|
1896 |
lemma minus_le_iff_1 [simp, no_atp]: |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54250
diff
changeset
|
1897 |
"- a \<le> 1 \<longleftrightarrow> - 1 \<le> a" |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54250
diff
changeset
|
1898 |
by (fact minus_le_iff) |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54250
diff
changeset
|
1899 |
|
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54250
diff
changeset
|
1900 |
lemma less_minus_iff_1 [simp, no_atp]: |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54250
diff
changeset
|
1901 |
"1 < - b \<longleftrightarrow> b < - 1" |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54250
diff
changeset
|
1902 |
by (fact less_minus_iff) |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54250
diff
changeset
|
1903 |
|
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54250
diff
changeset
|
1904 |
lemma minus_less_iff_1 [simp, no_atp]: |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54250
diff
changeset
|
1905 |
"- a < 1 \<longleftrightarrow> - 1 < a" |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54250
diff
changeset
|
1906 |
by (fact minus_less_iff) |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54250
diff
changeset
|
1907 |
|
25917 | 1908 |
end |
25230 | 1909 |
|
60758 | 1910 |
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
|
1911 |
|
54147
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
52435
diff
changeset
|
1912 |
lemmas mult_compare_simps = |
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
1913 |
mult_le_cancel_right mult_le_cancel_left |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
1914 |
mult_le_cancel_right1 mult_le_cancel_right2 |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
1915 |
mult_le_cancel_left1 mult_le_cancel_left2 |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
1916 |
mult_less_cancel_right mult_less_cancel_left |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
1917 |
mult_less_cancel_right1 mult_less_cancel_right2 |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
1918 |
mult_less_cancel_left1 mult_less_cancel_left2 |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
1919 |
mult_cancel_right mult_cancel_left |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
1920 |
mult_cancel_right1 mult_cancel_right2 |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
1921 |
mult_cancel_left1 mult_cancel_left2 |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
1922 |
|
60758 | 1923 |
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
|
1924 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
1925 |
context linordered_semidom |
25193 | 1926 |
begin |
1927 |
||
1928 |
lemma less_add_one: "a < a + 1" |
|
14293 | 1929 |
proof - |
25193 | 1930 |
have "a + 0 < a + 1" |
23482 | 1931 |
by (blast intro: zero_less_one add_strict_left_mono) |
14293 | 1932 |
thus ?thesis by simp |
1933 |
qed |
|
1934 |
||
25193 | 1935 |
lemma zero_less_two: "0 < 1 + 1" |
29667 | 1936 |
by (blast intro: less_trans zero_less_one less_add_one) |
25193 | 1937 |
|
1938 |
end |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1939 |
|
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1940 |
context linordered_idom |
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1941 |
begin |
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
1942 |
|
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1943 |
lemma mult_right_le_one_le: |
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1944 |
"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
|
1945 |
by (rule mult_left_le) |
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1946 |
|
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1947 |
lemma mult_left_le_one_le: |
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1948 |
"0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> y * x \<le> x" |
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1949 |
by (auto simp add: mult_le_cancel_right2) |
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1950 |
|
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1951 |
end |
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1952 |
|
60758 | 1953 |
text \<open>Absolute Value\<close> |
14293 | 1954 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
1955 |
context linordered_idom |
25304
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset
|
1956 |
begin |
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset
|
1957 |
|
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1958 |
lemma mult_sgn_abs: |
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1959 |
"sgn x * \<bar>x\<bar> = x" |
25304
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset
|
1960 |
unfolding abs_if sgn_if by auto |
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset
|
1961 |
|
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1962 |
lemma abs_one [simp]: |
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1963 |
"\<bar>1\<bar> = 1" |
44921 | 1964 |
by (simp add: abs_if) |
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1965 |
|
25304
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset
|
1966 |
end |
24491 | 1967 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
1968 |
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
|
1969 |
assumes abs_eq_mult: |
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset
|
1970 |
"(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
|
1971 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
1972 |
context linordered_idom |
30961 | 1973 |
begin |
1974 |
||
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34146
diff
changeset
|
1975 |
subclass ordered_ring_abs proof |
35216 | 1976 |
qed (auto simp add: abs_if not_less mult_less_0_iff) |
30961 | 1977 |
|
1978 |
lemma abs_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
|
1979 |
"\<bar>a * b\<bar> = \<bar>a\<bar> * \<bar>b\<bar>" |
30961 | 1980 |
by (rule abs_eq_mult) auto |
1981 |
||
61649
268d88ec9087
Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents:
60867
diff
changeset
|
1982 |
lemma abs_mult_self [simp]: |
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1983 |
"\<bar>a\<bar> * \<bar>a\<bar> = a * 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
|
1984 |
by (simp add: abs_if) |
30961 | 1985 |
|
14294
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1986 |
lemma abs_mult_less: |
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1987 |
"\<bar>a\<bar> < c \<Longrightarrow> \<bar>b\<bar> < d \<Longrightarrow> \<bar>a\<bar> * \<bar>b\<bar> < c * d" |
14294
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1988 |
proof - |
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1989 |
assume ac: "\<bar>a\<bar> < c" |
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1990 |
hence cpos: "0<c" by (blast intro: le_less_trans abs_ge_zero) |
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1991 |
assume "\<bar>b\<bar> < d" |
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
|
1992 |
thus ?thesis by (simp add: ac cpos mult_strict_mono) |
14294
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1993 |
qed |
14293 | 1994 |
|
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1995 |
lemma abs_less_iff: |
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
|
1996 |
"\<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
|
1997 |
by (simp add: less_le abs_le_iff) (auto simp add: abs_if) |
14738 | 1998 |
|
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
1999 |
lemma abs_mult_pos: |
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
2000 |
"0 \<le> x \<Longrightarrow> \<bar>y\<bar> * x = \<bar>y * x\<bar>" |
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
2001 |
by (simp add: abs_mult) |
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
2002 |
|
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
|
2003 |
lemma abs_diff_less_iff: |
e9b361845809
move real_isLub_unique to isLub_unique in Lubs; real_sum_of_halves to RealDef; abs_diff_less_iff to Rings
hoelzl
parents:
50420
diff
changeset
|
2004 |
"\<bar>x - a\<bar> < r \<longleftrightarrow> a - r < x \<and> x < a + r" |
e9b361845809
move real_isLub_unique to isLub_unique in Lubs; real_sum_of_halves to RealDef; abs_diff_less_iff to Rings
hoelzl
parents:
50420
diff
changeset
|
2005 |
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
|
2006 |
|
59865
8a20dd967385
rationalised and generalised some theorems concerning abs and x^2.
paulson <lp15@cam.ac.uk>
parents:
59833
diff
changeset
|
2007 |
lemma abs_diff_le_iff: |
8a20dd967385
rationalised and generalised some theorems concerning abs and x^2.
paulson <lp15@cam.ac.uk>
parents:
59833
diff
changeset
|
2008 |
"\<bar>x - a\<bar> \<le> r \<longleftrightarrow> a - r \<le> x \<and> x \<le> a + r" |
8a20dd967385
rationalised and generalised some theorems concerning abs and x^2.
paulson <lp15@cam.ac.uk>
parents:
59833
diff
changeset
|
2009 |
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
|
2010 |
|
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset
|
2011 |
end |
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
2012 |
|
59557 | 2013 |
hide_fact (open) comm_mult_left_mono comm_mult_strict_left_mono distrib |
2014 |
||
52435
6646bb548c6b
migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents:
51520
diff
changeset
|
2015 |
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
|
2016 |
code_module Rings \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith |
33364 | 2017 |
|
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
2018 |
end |