author  haftmann 
Mon, 26 Sep 2016 07:56:54 +0200  
changeset 63947  559f0882d6a6 
parent 63924  f91766530e13 
child 63950  cdc1e59aa513 
permissions  rwrr 
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(* Title: HOL/Rings.thy 
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2 
Author: Gertrud Bauer 
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3 
Author: Steven Obua 
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4 
Author: Tobias Nipkow 
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5 
Author: Lawrence C Paulson 
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6 
Author: Markus Wenzel 
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Author: Jeremy Avigad 
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*) 
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60758  10 
section \<open>Rings\<close> 
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theory Rings 
63588  13 
imports Groups Set 
15131  14 
begin 
14504  15 

22390  16 
class semiring = ab_semigroup_add + semigroup_mult + 
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assumes distrib_right[algebra_simps]: "(a + b) * c = a * c + b * c" 
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assumes distrib_left[algebra_simps]: "a * (b + c) = a * b + a * c" 
25152  19 
begin 
20 

63325  21 
text \<open>For the \<open>combine_numerals\<close> simproc\<close> 
22 
lemma combine_common_factor: "a * e + (b * e + c) = (a + b) * e + c" 

23 
by (simp add: distrib_right ac_simps) 

25152  24 

25 
end 

14504  26 

22390  27 
class mult_zero = times + zero + 
25062  28 
assumes mult_zero_left [simp]: "0 * a = 0" 
29 
assumes mult_zero_right [simp]: "a * 0 = 0" 

58195  30 
begin 
31 

63325  32 
lemma mult_not_zero: "a * b \<noteq> 0 \<Longrightarrow> a \<noteq> 0 \<and> b \<noteq> 0" 
58195  33 
by auto 
34 

35 
end 

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58198  37 
class semiring_0 = semiring + comm_monoid_add + mult_zero 
38 

29904  39 
class semiring_0_cancel = semiring + cancel_comm_monoid_add 
25186  40 
begin 
14504  41 

25186  42 
subclass semiring_0 
28823  43 
proof 
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fix a :: 'a 
63588  45 
have "0 * a + 0 * a = 0 * a + 0" 
46 
by (simp add: distrib_right [symmetric]) 

47 
then show "0 * a = 0" 

48 
by (simp only: add_left_cancel) 

49 
have "a * 0 + a * 0 = a * 0 + 0" 

50 
by (simp add: distrib_left [symmetric]) 

51 
then show "a * 0 = 0" 

52 
by (simp only: add_left_cancel) 

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qed 
14940  54 

25186  55 
end 
25152  56 

22390  57 
class comm_semiring = ab_semigroup_add + ab_semigroup_mult + 
25062  58 
assumes distrib: "(a + b) * c = a * c + b * c" 
25152  59 
begin 
14504  60 

25152  61 
subclass semiring 
28823  62 
proof 
14738  63 
fix a b c :: 'a 
63588  64 
show "(a + b) * c = a * c + b * c" 
65 
by (simp add: distrib) 

66 
have "a * (b + c) = (b + c) * a" 

67 
by (simp add: ac_simps) 

68 
also have "\<dots> = b * a + c * a" 

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by (simp only: distrib) 

70 
also have "\<dots> = a * b + a * c" 

71 
by (simp add: ac_simps) 

72 
finally show "a * (b + c) = a * b + a * c" 

73 
by blast 

14504  74 
qed 
75 

25152  76 
end 
14504  77 

25152  78 
class comm_semiring_0 = comm_semiring + comm_monoid_add + mult_zero 
79 
begin 

80 

27516  81 
subclass semiring_0 .. 
25152  82 

83 
end 

14504  84 

29904  85 
class comm_semiring_0_cancel = comm_semiring + cancel_comm_monoid_add 
25186  86 
begin 
14940  87 

27516  88 
subclass semiring_0_cancel .. 
14940  89 

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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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25186  92 
end 
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22390  94 
class zero_neq_one = zero + one + 
25062  95 
assumes zero_neq_one [simp]: "0 \<noteq> 1" 
26193  96 
begin 
97 

98 
lemma one_neq_zero [simp]: "1 \<noteq> 0" 

63325  99 
by (rule not_sym) (rule zero_neq_one) 
26193  100 

54225  101 
definition of_bool :: "bool \<Rightarrow> 'a" 
63325  102 
where "of_bool p = (if p then 1 else 0)" 
54225  103 

104 
lemma of_bool_eq [simp, code]: 

105 
"of_bool False = 0" 

106 
"of_bool True = 1" 

107 
by (simp_all add: of_bool_def) 

108 

63325  109 
lemma of_bool_eq_iff: "of_bool p = of_bool q \<longleftrightarrow> p = q" 
54225  110 
by (simp add: of_bool_def) 
111 

63325  112 
lemma split_of_bool [split]: "P (of_bool p) \<longleftrightarrow> (p \<longrightarrow> P 1) \<and> (\<not> p \<longrightarrow> P 0)" 
55187  113 
by (cases p) simp_all 
114 

63325  115 
lemma split_of_bool_asm: "P (of_bool p) \<longleftrightarrow> \<not> (p \<and> \<not> P 1 \<or> \<not> p \<and> \<not> P 0)" 
55187  116 
by (cases p) simp_all 
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end 
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22390  120 
class semiring_1 = zero_neq_one + semiring_0 + monoid_mult 
14504  121 

60758  122 
text \<open>Abstract divisibility\<close> 
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class dvd = times 
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begin 
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63325  127 
definition dvd :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infix "dvd" 50) 
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where "b dvd a \<longleftrightarrow> (\<exists>k. a = b * k)" 

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lemma dvdI [intro?]: "a = b * k \<Longrightarrow> b dvd a" 
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unfolding dvd_def .. 
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lemma dvdE [elim?]: "b dvd a \<Longrightarrow> (\<And>k. a = b * k \<Longrightarrow> P) \<Longrightarrow> P" 
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unfolding dvd_def by blast 
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end 
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context comm_monoid_mult 
25152  139 
begin 
14738  140 

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subclass dvd . 
25152  142 

63325  143 
lemma dvd_refl [simp]: "a dvd a" 
28559  144 
proof 
145 
show "a = a * 1" by simp 

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qed 
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lemma dvd_trans [trans]: 
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assumes "a dvd b" and "b dvd c" 
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shows "a dvd c" 
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proof  
63588  152 
from assms obtain v where "b = a * v" 
153 
by (auto elim!: dvdE) 

154 
moreover from assms obtain w where "c = b * w" 

155 
by (auto elim!: dvdE) 

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ultimately have "c = a * (v * w)" 

157 
by (simp add: mult.assoc) 

28559  158 
then show ?thesis .. 
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qed 
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63325  161 
lemma subset_divisors_dvd: "{c. c dvd a} \<subseteq> {c. c dvd b} \<longleftrightarrow> a dvd b" 
62366  162 
by (auto simp add: subset_iff intro: dvd_trans) 
163 

63325  164 
lemma strict_subset_divisors_dvd: "{c. c dvd a} \<subset> {c. c dvd b} \<longleftrightarrow> a dvd b \<and> \<not> b dvd a" 
62366  165 
by (auto simp add: subset_iff intro: dvd_trans) 
166 

63325  167 
lemma one_dvd [simp]: "1 dvd a" 
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by (auto intro!: dvdI) 
28559  169 

63325  170 
lemma dvd_mult [simp]: "a dvd c \<Longrightarrow> a dvd (b * c)" 
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by (auto intro!: mult.left_commute dvdI elim!: dvdE) 
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63325  173 
lemma dvd_mult2 [simp]: "a dvd b \<Longrightarrow> a dvd (b * c)" 
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using dvd_mult [of a b c] by (simp add: ac_simps) 
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63325  176 
lemma dvd_triv_right [simp]: "a dvd b * a" 
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by (rule dvd_mult) (rule dvd_refl) 
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63325  179 
lemma dvd_triv_left [simp]: "a dvd a * b" 
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by (rule dvd_mult2) (rule dvd_refl) 
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lemma mult_dvd_mono: 
30042  183 
assumes "a dvd b" 
184 
and "c dvd d" 

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shows "a * c dvd b * d" 
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proof  
60758  187 
from \<open>a dvd b\<close> obtain b' where "b = a * b'" .. 
188 
moreover from \<open>c dvd d\<close> obtain d' where "d = c * d'" .. 

63588  189 
ultimately have "b * d = (a * c) * (b' * d')" 
190 
by (simp add: ac_simps) 

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then show ?thesis .. 
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qed 
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63325  194 
lemma dvd_mult_left: "a * b dvd c \<Longrightarrow> a dvd c" 
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by (simp add: dvd_def mult.assoc) blast 
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63325  197 
lemma dvd_mult_right: "a * b dvd c \<Longrightarrow> b dvd c" 
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using dvd_mult_left [of b a c] by (simp add: ac_simps) 
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end 
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class comm_semiring_1 = zero_neq_one + comm_semiring_0 + comm_monoid_mult 
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begin 
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204 

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subclass semiring_1 .. 
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63325  207 
lemma dvd_0_left_iff [simp]: "0 dvd a \<longleftrightarrow> a = 0" 
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by (auto intro: dvd_refl elim!: dvdE) 
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63325  210 
lemma dvd_0_right [iff]: "a dvd 0" 
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211 
proof 
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show "0 = a * 0" by simp 
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qed 
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63325  215 
lemma dvd_0_left: "0 dvd a \<Longrightarrow> a = 0" 
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by simp 
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lemma dvd_add [simp]: 
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assumes "a dvd b" and "a dvd c" 
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shows "a dvd (b + c)" 
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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'" .. 

63588  224 
ultimately have "b + c = a * (b' + c')" 
225 
by (simp add: distrib_left) 

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then show ?thesis .. 
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qed 
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25152  229 
end 
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29904  231 
class semiring_1_cancel = semiring + cancel_comm_monoid_add 
232 
+ zero_neq_one + monoid_mult 

25267  233 
begin 
14940  234 

27516  235 
subclass semiring_0_cancel .. 
25512
4134f7c782e2
using intro_locales instead of unfold_locales if appropriate
haftmann
parents:
25450
diff
changeset

236 

27516  237 
subclass semiring_1 .. 
25267  238 

239 
end 

21199
2d83f93c3580
* Added annihilation axioms ("x * 0 = 0") to axclass semiring_0.
krauss
parents:
20633
diff
changeset

240 

63325  241 
class comm_semiring_1_cancel = 
242 
comm_semiring + cancel_comm_monoid_add + zero_neq_one + comm_monoid_mult + 

60562
24af00b010cf
Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents:
60529
diff
changeset

243 
assumes right_diff_distrib' [algebra_simps]: "a * (b  c) = a * b  a * c" 
25267  244 
begin 
14738  245 

27516  246 
subclass semiring_1_cancel .. 
247 
subclass comm_semiring_0_cancel .. 

248 
subclass comm_semiring_1 .. 

25267  249 

63325  250 
lemma left_diff_distrib' [algebra_simps]: "(b  c) * a = b * a  c * a" 
59816
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset

251 
by (simp add: algebra_simps) 
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset

252 

63325  253 
lemma dvd_add_times_triv_left_iff [simp]: "a dvd c * a + b \<longleftrightarrow> a dvd b" 
59816
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset

254 
proof  
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset

255 
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

256 
proof 
63325  257 
assume ?Q 
258 
then show ?P by simp 

59816
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 

63325  270 
lemma dvd_add_times_triv_right_iff [simp]: "a dvd b + c * a \<longleftrightarrow> a dvd b" 
59816
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset

271 
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

272 

63325  273 
lemma dvd_add_triv_left_iff [simp]: "a dvd a + b \<longleftrightarrow> a dvd b" 
59816
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset

274 
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

275 

63325  276 
lemma dvd_add_triv_right_iff [simp]: "a dvd b + a \<longleftrightarrow> a dvd b" 
59816
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset

277 
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

278 

034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset

279 
lemma dvd_add_right_iff: 
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset

280 
assumes "a dvd b" 
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset

281 
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

282 
proof 
63325  283 
assume ?P 
284 
then obtain d where "b + c = a * d" .. 

60758  285 
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

286 
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

287 
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

288 
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

289 
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

290 
then show ?Q .. 
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset

291 
next 
63325  292 
assume ?Q 
293 
with assms show ?P by simp 

59816
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset

294 
qed 
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset

295 

63325  296 
lemma dvd_add_left_iff: "a dvd c \<Longrightarrow> a dvd b + c \<longleftrightarrow> a dvd b" 
297 
using dvd_add_right_iff [of a c b] by (simp add: ac_simps) 

59816
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset

298 

034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset

299 
end 
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset

300 

22390  301 
class ring = semiring + ab_group_add 
25267  302 
begin 
25152  303 

27516  304 
subclass semiring_0_cancel .. 
25152  305 

60758  306 
text \<open>Distribution rules\<close> 
25152  307 

308 
lemma minus_mult_left: " (a * b) =  a * b" 

63325  309 
by (rule minus_unique) (simp add: distrib_right [symmetric]) 
25152  310 

311 
lemma minus_mult_right: " (a * b) = a *  b" 

63325  312 
by (rule minus_unique) (simp add: distrib_left [symmetric]) 
25152  313 

63325  314 
text \<open>Extract signs from products\<close> 
54147
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
52435
diff
changeset

315 
lemmas mult_minus_left [simp] = minus_mult_left [symmetric] 
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
52435
diff
changeset

316 
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

317 

25152  318 
lemma minus_mult_minus [simp]: " a *  b = a * b" 
63325  319 
by simp 
25152  320 

321 
lemma minus_mult_commute: " a * b = a *  b" 

63325  322 
by simp 
29667  323 

63325  324 
lemma right_diff_distrib [algebra_simps]: "a * (b  c) = a * b  a * c" 
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54225
diff
changeset

325 
using distrib_left [of a b "c "] by simp 
29667  326 

63325  327 
lemma left_diff_distrib [algebra_simps]: "(a  b) * c = a * c  b * c" 
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54225
diff
changeset

328 
using distrib_right [of a " b" c] by simp 
25152  329 

63325  330 
lemmas ring_distribs = distrib_left distrib_right left_diff_distrib right_diff_distrib 
25152  331 

63325  332 
lemma eq_add_iff1: "a * e + c = b * e + d \<longleftrightarrow> (a  b) * e + c = d" 
333 
by (simp add: algebra_simps) 

25230  334 

63325  335 
lemma eq_add_iff2: "a * e + c = b * e + d \<longleftrightarrow> c = (b  a) * e + d" 
336 
by (simp add: algebra_simps) 

25230  337 

25152  338 
end 
339 

63325  340 
lemmas ring_distribs = distrib_left distrib_right left_diff_distrib right_diff_distrib 
25152  341 

22390  342 
class comm_ring = comm_semiring + ab_group_add 
25267  343 
begin 
14738  344 

27516  345 
subclass ring .. 
28141
193c3ea0f63b
instances comm_semiring_0_cancel < comm_semiring_0, comm_ring < comm_semiring_0_cancel
huffman
parents:
27651
diff
changeset

346 
subclass comm_semiring_0_cancel .. 
25267  347 

63325  348 
lemma square_diff_square_factored: "x * x  y * y = (x + y) * (x  y)" 
44350
63cddfbc5a09
replace lemma realpow_two_diff with new lemma square_diff_square_factored
huffman
parents:
44346
diff
changeset

349 
by (simp add: algebra_simps) 
63cddfbc5a09
replace lemma realpow_two_diff with new lemma square_diff_square_factored
huffman
parents:
44346
diff
changeset

350 

25267  351 
end 
14738  352 

22390  353 
class ring_1 = ring + zero_neq_one + monoid_mult 
25267  354 
begin 
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

355 

27516  356 
subclass semiring_1_cancel .. 
25267  357 

63325  358 
lemma square_diff_one_factored: "x * x  1 = (x + 1) * (x  1)" 
44346
00dd3c4dabe0
rename real_squared_diff_one_factored to square_diff_one_factored and move to Rings.thy
huffman
parents:
44064
diff
changeset

359 
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

360 

25267  361 
end 
25152  362 

22390  363 
class comm_ring_1 = comm_ring + zero_neq_one + comm_monoid_mult 
25267  364 
begin 
14738  365 

27516  366 
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

367 
subclass comm_semiring_1_cancel 
59816
034b13f4efae
distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents:
59557
diff
changeset

368 
by unfold_locales (simp add: algebra_simps) 
58647  369 

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

370 
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

371 
proof 
6d10cf26b5dc
add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents:
29407
diff
changeset

372 
assume "x dvd  y" 
6d10cf26b5dc
add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents:
29407
diff
changeset

373 
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

374 
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

375 
next 
6d10cf26b5dc
add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents:
29407
diff
changeset

376 
assume "x dvd y" 
6d10cf26b5dc
add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents:
29407
diff
changeset

377 
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

378 
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

379 
qed 
6d10cf26b5dc
add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents:
29407
diff
changeset

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 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

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 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

385 
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

386 
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

387 
next 
6d10cf26b5dc
add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents:
29407
diff
changeset

388 
assume "x dvd y" 
6d10cf26b5dc
add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents:
29407
diff
changeset

389 
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

390 
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

391 
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

392 
qed 
6d10cf26b5dc
add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents:
29407
diff
changeset

393 

63325  394 
lemma dvd_diff [simp]: "x dvd y \<Longrightarrow> x dvd z \<Longrightarrow> x dvd (y  z)" 
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54225
diff
changeset

395 
using dvd_add [of x y " z"] by simp 
29409  396 

25267  397 
end 
25152  398 

59833
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset

399 
class semiring_no_zero_divisors = semiring_0 + 
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset

400 
assumes no_zero_divisors: "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> a * b \<noteq> 0" 
25230  401 
begin 
402 

59833
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset

403 
lemma divisors_zero: 
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset

404 
assumes "a * b = 0" 
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset

405 
shows "a = 0 \<or> b = 0" 
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset

406 
proof (rule classical) 
63325  407 
assume "\<not> ?thesis" 
59833
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset

408 
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

409 
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

410 
with assms show ?thesis by simp 
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset

411 
qed 
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset

412 

63325  413 
lemma mult_eq_0_iff [simp]: "a * b = 0 \<longleftrightarrow> a = 0 \<or> b = 0" 
25230  414 
proof (cases "a = 0 \<or> b = 0") 
63325  415 
case False 
416 
then have "a \<noteq> 0" and "b \<noteq> 0" by auto 

25230  417 
then show ?thesis using no_zero_divisors by simp 
418 
next 

63325  419 
case True 
420 
then show ?thesis by auto 

25230  421 
qed 
422 

58952
5d82cdef6c1b
equivalence rules for structures without zero divisors
haftmann
parents:
58889
diff
changeset

423 
end 
5d82cdef6c1b
equivalence rules for structures without zero divisors
haftmann
parents:
58889
diff
changeset

424 

62481
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62390
diff
changeset

425 
class semiring_1_no_zero_divisors = semiring_1 + semiring_no_zero_divisors 
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62390
diff
changeset

426 

60516
0826b7025d07
generalized some theorems about integral domains and moved to HOL theories
haftmann
parents:
60429
diff
changeset

427 
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

428 
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

429 
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

430 
begin 
5d82cdef6c1b
equivalence rules for structures without zero divisors
haftmann
parents:
58889
diff
changeset

431 

63325  432 
lemma mult_left_cancel: "c \<noteq> 0 \<Longrightarrow> c * a = c * b \<longleftrightarrow> a = b" 
60562
24af00b010cf
Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents:
60529
diff
changeset

433 
by simp 
56217
dc429a5b13c4
Some rationalisation of basic lemmas
paulson <lp15@cam.ac.uk>
parents:
55912
diff
changeset

434 

63325  435 
lemma mult_right_cancel: "c \<noteq> 0 \<Longrightarrow> a * c = b * c \<longleftrightarrow> a = b" 
60562
24af00b010cf
Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents:
60529
diff
changeset

436 
by simp 
56217
dc429a5b13c4
Some rationalisation of basic lemmas
paulson <lp15@cam.ac.uk>
parents:
55912
diff
changeset

437 

25230  438 
end 
22990
775e9de3db48
added classes ring_no_zero_divisors and dom (noncommutative version of idom);
huffman
parents:
22987
diff
changeset

439 

60516
0826b7025d07
generalized some theorems about integral domains and moved to HOL theories
haftmann
parents:
60429
diff
changeset

440 
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

441 
begin 
0826b7025d07
generalized some theorems about integral domains and moved to HOL theories
haftmann
parents:
60429
diff
changeset

442 

0826b7025d07
generalized some theorems about integral domains and moved to HOL theories
haftmann
parents:
60429
diff
changeset

443 
subclass semiring_no_zero_divisors_cancel 
0826b7025d07
generalized some theorems about integral domains and moved to HOL theories
haftmann
parents:
60429
diff
changeset

444 
proof 
0826b7025d07
generalized some theorems about integral domains and moved to HOL theories
haftmann
parents:
60429
diff
changeset

445 
fix a b c 
0826b7025d07
generalized some theorems about integral domains and moved to HOL theories
haftmann
parents:
60429
diff
changeset

446 
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

447 
by (simp add: algebra_simps) 
0826b7025d07
generalized some theorems about integral domains and moved to HOL theories
haftmann
parents:
60429
diff
changeset

448 
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

449 
by auto 
0826b7025d07
generalized some theorems about integral domains and moved to HOL theories
haftmann
parents:
60429
diff
changeset

450 
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

451 
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

452 
by (simp add: algebra_simps) 
0826b7025d07
generalized some theorems about integral domains and moved to HOL theories
haftmann
parents:
60429
diff
changeset

453 
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

454 
by auto 
0826b7025d07
generalized some theorems about integral domains and moved to HOL theories
haftmann
parents:
60429
diff
changeset

455 
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

456 
qed 
0826b7025d07
generalized some theorems about integral domains and moved to HOL theories
haftmann
parents:
60429
diff
changeset

457 

0826b7025d07
generalized some theorems about integral domains and moved to HOL theories
haftmann
parents:
60429
diff
changeset

458 
end 
0826b7025d07
generalized some theorems about integral domains and moved to HOL theories
haftmann
parents:
60429
diff
changeset

459 

23544  460 
class ring_1_no_zero_divisors = ring_1 + ring_no_zero_divisors 
26274  461 
begin 
462 

62481
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62390
diff
changeset

463 
subclass semiring_1_no_zero_divisors .. 
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62390
diff
changeset

464 

63325  465 
lemma square_eq_1_iff: "x * x = 1 \<longleftrightarrow> x = 1 \<or> x =  1" 
36821
9207505d1ee5
move lemma real_mult_is_one to Rings.thy, renamed to square_eq_1_iff
huffman
parents:
36719
diff
changeset

466 
proof  
9207505d1ee5
move lemma real_mult_is_one to Rings.thy, renamed to square_eq_1_iff
huffman
parents:
36719
diff
changeset

467 
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

468 
by (simp add: algebra_simps) 
63325  469 
then have "x * x = 1 \<longleftrightarrow> (x  1) * (x + 1) = 0" 
36821
9207505d1ee5
move lemma real_mult_is_one to Rings.thy, renamed to square_eq_1_iff
huffman
parents:
36719
diff
changeset

470 
by simp 
63325  471 
then show ?thesis 
36821
9207505d1ee5
move lemma real_mult_is_one to Rings.thy, renamed to square_eq_1_iff
huffman
parents:
36719
diff
changeset

472 
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

473 
qed 
9207505d1ee5
move lemma real_mult_is_one to Rings.thy, renamed to square_eq_1_iff
huffman
parents:
36719
diff
changeset

474 

63325  475 
lemma mult_cancel_right1 [simp]: "c = b * c \<longleftrightarrow> c = 0 \<or> b = 1" 
476 
using mult_cancel_right [of 1 c b] by auto 

26274  477 

63325  478 
lemma mult_cancel_right2 [simp]: "a * c = c \<longleftrightarrow> c = 0 \<or> a = 1" 
479 
using mult_cancel_right [of a c 1] by simp 

60562
24af00b010cf
Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents:
60529
diff
changeset

480 

63325  481 
lemma mult_cancel_left1 [simp]: "c = c * b \<longleftrightarrow> c = 0 \<or> b = 1" 
482 
using mult_cancel_left [of c 1 b] by force 

26274  483 

63325  484 
lemma mult_cancel_left2 [simp]: "c * a = c \<longleftrightarrow> c = 0 \<or> a = 1" 
485 
using mult_cancel_left [of c a 1] by simp 

26274  486 

487 
end 

22990
775e9de3db48
added classes ring_no_zero_divisors and dom (noncommutative version of idom);
huffman
parents:
22987
diff
changeset

488 

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

489 
class semidom = comm_semiring_1_cancel + semiring_no_zero_divisors 
62481
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62390
diff
changeset

490 
begin 
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62390
diff
changeset

491 

b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62390
diff
changeset

492 
subclass semiring_1_no_zero_divisors .. 
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62390
diff
changeset

493 

b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62390
diff
changeset

494 
end 
59833
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset

495 

ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset

496 
class idom = comm_ring_1 + semiring_no_zero_divisors 
25186  497 
begin 
14421
ee97b6463cb4
new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents:
14398
diff
changeset

498 

59833
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset

499 
subclass semidom .. 
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset

500 

27516  501 
subclass ring_1_no_zero_divisors .. 
22990
775e9de3db48
added classes ring_no_zero_divisors and dom (noncommutative version of idom);
huffman
parents:
22987
diff
changeset

502 

63325  503 
lemma dvd_mult_cancel_right [simp]: "a * c dvd b * c \<longleftrightarrow> c = 0 \<or> a dvd b" 
29981
7d0ed261b712
generalize int_dvd_cancel_factor simproc to idom class
huffman
parents:
29949
diff
changeset

504 
proof  
7d0ed261b712
generalize int_dvd_cancel_factor simproc to idom class
huffman
parents:
29949
diff
changeset

505 
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

506 
unfolding dvd_def by (simp add: ac_simps) 
29981
7d0ed261b712
generalize int_dvd_cancel_factor simproc to idom class
huffman
parents:
29949
diff
changeset

507 
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

508 
unfolding dvd_def by simp 
7d0ed261b712
generalize int_dvd_cancel_factor simproc to idom class
huffman
parents:
29949
diff
changeset

509 
finally show ?thesis . 
7d0ed261b712
generalize int_dvd_cancel_factor simproc to idom class
huffman
parents:
29949
diff
changeset

510 
qed 
7d0ed261b712
generalize int_dvd_cancel_factor simproc to idom class
huffman
parents:
29949
diff
changeset

511 

63325  512 
lemma dvd_mult_cancel_left [simp]: "c * a dvd c * b \<longleftrightarrow> c = 0 \<or> a dvd b" 
29981
7d0ed261b712
generalize int_dvd_cancel_factor simproc to idom class
huffman
parents:
29949
diff
changeset

513 
proof  
7d0ed261b712
generalize int_dvd_cancel_factor simproc to idom class
huffman
parents:
29949
diff
changeset

514 
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

515 
unfolding dvd_def by (simp add: ac_simps) 
29981
7d0ed261b712
generalize int_dvd_cancel_factor simproc to idom class
huffman
parents:
29949
diff
changeset

516 
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

517 
unfolding dvd_def by simp 
7d0ed261b712
generalize int_dvd_cancel_factor simproc to idom class
huffman
parents:
29949
diff
changeset

518 
finally show ?thesis . 
7d0ed261b712
generalize int_dvd_cancel_factor simproc to idom class
huffman
parents:
29949
diff
changeset

519 
qed 
7d0ed261b712
generalize int_dvd_cancel_factor simproc to idom class
huffman
parents:
29949
diff
changeset

520 

60516
0826b7025d07
generalized some theorems about integral domains and moved to HOL theories
haftmann
parents:
60429
diff
changeset

521 
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

522 
proof 
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset

523 
assume "a * a = b * b" 
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset

524 
then have "(a  b) * (a + b) = 0" 
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset

525 
by (simp add: algebra_simps) 
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset

526 
then show "a = b \<or> a =  b" 
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset

527 
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

528 
next 
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset

529 
assume "a = b \<or> a =  b" 
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset

530 
then show "a * a = b * b" by auto 
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset

531 
qed 
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59832
diff
changeset

532 

25186  533 
end 
25152  534 

60758  535 
text \<open> 
35302  536 
The theory of partially ordered rings is taken from the books: 
63325  537 
\<^item> \<^emph>\<open>Lattice Theory\<close> by Garret Birkhoff, American Mathematical Society, 1979 
538 
\<^item> \<^emph>\<open>Partially Ordered Algebraic Systems\<close>, Pergamon Press, 1963 

539 

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

540 
Most of the used notions can also be looked up in 
63680  541 
\<^item> \<^url>\<open>http://www.mathworld.com\<close> by Eric Weisstein et. al. 
63325  542 
\<^item> \<^emph>\<open>Algebra I\<close> by van der Waerden, Springer 
60758  543 
\<close> 
35302  544 

60353
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset

545 
class divide = 
60429
d3d1e185cd63
uniform _ div _ as infix syntax for ring division
haftmann
parents:
60353
diff
changeset

546 
fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "div" 70) 
60353
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset

547 

60758  548 
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

549 

838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset

550 
context semiring 
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset

551 
begin 
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset

552 

838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset

553 
lemma [field_simps]: 
60429
d3d1e185cd63
uniform _ div _ as infix syntax for ring division
haftmann
parents:
60353
diff
changeset

554 
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

555 
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

556 
by (rule distrib_left distrib_right)+ 
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset

557 

838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset

558 
end 
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset

559 

838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset

560 
context ring 
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset

561 
begin 
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset

562 

838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset

563 
lemma [field_simps]: 
60429
d3d1e185cd63
uniform _ div _ as infix syntax for ring division
haftmann
parents:
60353
diff
changeset

564 
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

565 
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

566 
by (rule left_diff_distrib right_diff_distrib)+ 
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset

567 

838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset

568 
end 
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset

569 

60758  570 
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

571 

838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset

572 
class semidom_divide = semidom + divide + 
60429
d3d1e185cd63
uniform _ div _ as infix syntax for ring division
haftmann
parents:
60353
diff
changeset

573 
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

574 
assumes divide_zero [simp]: "a div 0 = 0" 
60353
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 

63325  577 
lemma nonzero_mult_divide_cancel_left [simp]: "a \<noteq> 0 \<Longrightarrow> (a * b) div a = b" 
60353
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset

578 
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

579 

60516
0826b7025d07
generalized some theorems about integral domains and moved to HOL theories
haftmann
parents:
60429
diff
changeset

580 
subclass semiring_no_zero_divisors_cancel 
0826b7025d07
generalized some theorems about integral domains and moved to HOL theories
haftmann
parents:
60429
diff
changeset

581 
proof 
63325  582 
show *: "a * c = b * c \<longleftrightarrow> c = 0 \<or> a = b" for a b c 
583 
proof (cases "c = 0") 

584 
case True 

585 
then show ?thesis by simp 

586 
next 

587 
case False 

63588  588 
have "a = b" if "a * c = b * c" 
589 
proof  

590 
from that have "a * c div c = b * c div c" 

63325  591 
by simp 
63588  592 
with False show ?thesis 
63325  593 
by simp 
63588  594 
qed 
63325  595 
then show ?thesis by auto 
596 
qed 

597 
show "c * a = c * b \<longleftrightarrow> c = 0 \<or> a = b" for a b c 

598 
using * [of a c b] by (simp add: ac_simps) 

60516
0826b7025d07
generalized some theorems about integral domains and moved to HOL theories
haftmann
parents:
60429
diff
changeset

599 
qed 
0826b7025d07
generalized some theorems about integral domains and moved to HOL theories
haftmann
parents:
60429
diff
changeset

600 

63325  601 
lemma div_self [simp]: "a \<noteq> 0 \<Longrightarrow> a div a = 1" 
602 
using nonzero_mult_divide_cancel_left [of a 1] by simp 

60516
0826b7025d07
generalized some theorems about integral domains and moved to HOL theories
haftmann
parents:
60429
diff
changeset

603 

63325  604 
lemma divide_zero_left [simp]: "0 div a = 0" 
60570  605 
proof (cases "a = 0") 
63325  606 
case True 
607 
then show ?thesis by simp 

60570  608 
next 
63325  609 
case False 
610 
then have "a * 0 div a = 0" 

60570  611 
by (rule nonzero_mult_divide_cancel_left) 
612 
then show ?thesis by simp 

62376
85f38d5f8807
Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
hoelzl
parents:
62366
diff
changeset

613 
qed 
60570  614 

63325  615 
lemma divide_1 [simp]: "a div 1 = a" 
60690  616 
using nonzero_mult_divide_cancel_left [of 1 a] by simp 
617 

60867  618 
end 
619 

620 
class idom_divide = idom + semidom_divide 

621 

622 
class algebraic_semidom = semidom_divide 

623 
begin 

624 

625 
text \<open> 

626 
Class @{class algebraic_semidom} enriches a integral domain 

627 
by notions from algebra, like units in a ring. 

628 
It is a separate class to avoid spoiling fields with notions 

629 
which are degenerated there. 

630 
\<close> 

631 

60690  632 
lemma dvd_times_left_cancel_iff [simp]: 
633 
assumes "a \<noteq> 0" 

63588  634 
shows "a * b dvd a * c \<longleftrightarrow> b dvd c" 
635 
(is "?lhs \<longleftrightarrow> ?rhs") 

60690  636 
proof 
63588  637 
assume ?lhs 
63325  638 
then obtain d where "a * c = a * b * d" .. 
60690  639 
with assms have "c = b * d" by (simp add: ac_simps) 
63588  640 
then show ?rhs .. 
60690  641 
next 
63588  642 
assume ?rhs 
63325  643 
then obtain d where "c = b * d" .. 
60690  644 
then have "a * c = a * b * d" by (simp add: ac_simps) 
63588  645 
then show ?lhs .. 
60690  646 
qed 
62376
85f38d5f8807
Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
hoelzl
parents:
62366
diff
changeset

647 

60690  648 
lemma dvd_times_right_cancel_iff [simp]: 
649 
assumes "a \<noteq> 0" 

63588  650 
shows "b * a dvd c * a \<longleftrightarrow> b dvd c" 
63325  651 
using dvd_times_left_cancel_iff [of a b c] assms by (simp add: ac_simps) 
62376
85f38d5f8807
Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
hoelzl
parents:
62366
diff
changeset

652 

60690  653 
lemma div_dvd_iff_mult: 
654 
assumes "b \<noteq> 0" and "b dvd a" 

655 
shows "a div b dvd c \<longleftrightarrow> a dvd c * b" 

656 
proof  

657 
from \<open>b dvd a\<close> obtain d where "a = b * d" .. 

658 
with \<open>b \<noteq> 0\<close> show ?thesis by (simp add: ac_simps) 

659 
qed 

660 

661 
lemma dvd_div_iff_mult: 

662 
assumes "c \<noteq> 0" and "c dvd b" 

663 
shows "a dvd b div c \<longleftrightarrow> a * c dvd b" 

664 
proof  

665 
from \<open>c dvd b\<close> obtain d where "b = c * d" .. 

666 
with \<open>c \<noteq> 0\<close> show ?thesis by (simp add: mult.commute [of a]) 

667 
qed 

668 

60867  669 
lemma div_dvd_div [simp]: 
670 
assumes "a dvd b" and "a dvd c" 

671 
shows "b div a dvd c div a \<longleftrightarrow> b dvd c" 

672 
proof (cases "a = 0") 

63325  673 
case True 
674 
with assms show ?thesis by simp 

60867  675 
next 
676 
case False 

677 
moreover from assms obtain k l where "b = a * k" and "c = a * l" 

678 
by (auto elim!: dvdE) 

679 
ultimately show ?thesis by simp 

680 
qed 

60353
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60352
diff
changeset

681 

60867  682 
lemma div_add [simp]: 
683 
assumes "c dvd a" and "c dvd b" 

684 
shows "(a + b) div c = a div c + b div c" 

685 
proof (cases "c = 0") 

63325  686 
case True 
687 
then show ?thesis by simp 

60867  688 
next 
689 
case False 

690 
moreover from assms obtain k l where "a = c * k" and "b = c * l" 

691 
by (auto elim!: dvdE) 

692 
moreover have "c * k + c * l = c * (k + l)" 

693 
by (simp add: algebra_simps) 

694 
ultimately show ?thesis 

695 
by simp 

696 
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

697 

60867  698 
lemma div_mult_div_if_dvd: 
699 
assumes "b dvd a" and "d dvd c" 

700 
shows "(a div b) * (c div d) = (a * c) div (b * d)" 

701 
proof (cases "b = 0 \<or> c = 0") 

63325  702 
case True 
703 
with assms show ?thesis by auto 

60867  704 
next 
705 
case False 

706 
moreover from assms obtain k l where "a = b * k" and "c = d * l" 

707 
by (auto elim!: dvdE) 

708 
moreover have "b * k * (d * l) div (b * d) = (b * d) * (k * l) div (b * d)" 

709 
by (simp add: ac_simps) 

710 
ultimately show ?thesis by simp 

711 
qed 

712 

713 
lemma dvd_div_eq_mult: 

714 
assumes "a \<noteq> 0" and "a dvd b" 

715 
shows "b div a = c \<longleftrightarrow> b = c * a" 

63588  716 
(is "?lhs \<longleftrightarrow> ?rhs") 
60867  717 
proof 
63588  718 
assume ?rhs 
719 
then show ?lhs by (simp add: assms) 

60867  720 
next 
63588  721 
assume ?lhs 
60867  722 
then have "b div a * a = c * a" by simp 
63325  723 
moreover from assms have "b div a * a = b" 
60867  724 
by (auto elim!: dvdE simp add: ac_simps) 
63588  725 
ultimately show ?rhs by simp 
60867  726 
qed 
60688
01488b559910
avoid explicit definition of the relation of associated elements in a ring  prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset

727 

63325  728 
lemma dvd_div_mult_self [simp]: "a dvd b \<Longrightarrow> b div a * a = b" 
60517
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset

729 
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

730 

63325  731 
lemma dvd_mult_div_cancel [simp]: "a dvd b \<Longrightarrow> a * (b div a) = b" 
60517
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset

732 
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

733 

60517
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset

734 
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

735 
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

736 
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

737 
proof (cases "c = 0") 
63325  738 
case True 
739 
then show ?thesis by simp 

60517
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset

740 
next 
63325  741 
case False 
742 
from assms obtain d where "b = c * d" .. 

60517
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset

743 
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

744 
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

745 
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

746 
qed 
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset

747 

63325  748 
lemma dvd_div_mult: "c dvd b \<Longrightarrow> b div c * a = (b * a) div c" 
749 
using div_mult_swap [of c b a] by (simp add: ac_simps) 

60517
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset

750 

60570  751 
lemma dvd_div_mult2_eq: 
752 
assumes "b * c dvd a" 

753 
shows "a div (b * c) = a div b div c" 

63325  754 
proof  
755 
from assms obtain k where "a = b * c * k" .. 

60570  756 
then show ?thesis 
757 
by (cases "b = 0 \<or> c = 0") (auto, simp add: ac_simps) 

758 
qed 

759 

60867  760 
lemma dvd_div_div_eq_mult: 
761 
assumes "a \<noteq> 0" "c \<noteq> 0" and "a dvd b" "c dvd d" 

63588  762 
shows "b div a = d div c \<longleftrightarrow> b * c = a * d" 
763 
(is "?lhs \<longleftrightarrow> ?rhs") 

60867  764 
proof  
765 
from assms have "a * c \<noteq> 0" by simp 

63588  766 
then have "?lhs \<longleftrightarrow> b div a * (a * c) = d div c * (a * c)" 
60867  767 
by simp 
768 
also have "\<dots> \<longleftrightarrow> (a * (b div a)) * c = (c * (d div c)) * a" 

769 
by (simp add: ac_simps) 

770 
also have "\<dots> \<longleftrightarrow> (a * b div a) * c = (c * d div c) * a" 

771 
using assms by (simp add: div_mult_swap) 

63588  772 
also have "\<dots> \<longleftrightarrow> ?rhs" 
60867  773 
using assms by (simp add: ac_simps) 
774 
finally show ?thesis . 

775 
qed 

776 

63359  777 
lemma dvd_mult_imp_div: 
778 
assumes "a * c dvd b" 

779 
shows "a dvd b div c" 

780 
proof (cases "c = 0") 

781 
case True then show ?thesis by simp 

782 
next 

783 
case False 

784 
from \<open>a * c dvd b\<close> obtain d where "b = a * c * d" .. 

63588  785 
with False show ?thesis 
786 
by (simp add: mult.commute [of a] mult.assoc) 

63359  787 
qed 
788 

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

789 

60517
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset

790 
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

791 

f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset

792 
abbreviation is_unit :: "'a \<Rightarrow> bool" 
63325  793 
where "is_unit a \<equiv> a dvd 1" 
60517
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset

794 

63325  795 
lemma not_is_unit_0 [simp]: "\<not> is_unit 0" 
60517
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset

796 
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

797 

63325  798 
lemma unit_imp_dvd [dest]: "is_unit b \<Longrightarrow> b dvd a" 
60517
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset

799 
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

800 

f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset

801 
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

802 
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

803 
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

804 
proof  
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset

805 
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

806 
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

807 
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

808 
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

809 
qed 
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset

810 

63325  811 
lemma dvd_unit_imp_unit: "a dvd b \<Longrightarrow> is_unit b \<Longrightarrow> is_unit a" 
60517
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset

812 
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

813 

f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset

814 
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

815 
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

816 
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

817 
proof  
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset

818 
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

819 
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

820 
qed 
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset

821 

f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset

822 
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

823 
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

824 
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

825 
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

826 
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

827 
proof (rule that) 
63040  828 
define b where "b = 1 div a" 
60517
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset

829 
then show "1 div a = b" by simp 
63325  830 
from assms b_def show "is_unit b" by simp 
831 
with assms show "a \<noteq> 0" and "b \<noteq> 0" by auto 

832 
from assms b_def show "a * b = 1" by simp 

60517
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset

833 
then have "1 = a * b" .. 
60758  834 
with b_def \<open>b \<noteq> 0\<close> show "1 div b = a" by simp 
63325  835 
from assms have "a dvd c" .. 
60517
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset

836 
then obtain d where "c = a * d" .. 
60758  837 
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

838 
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

839 
qed 
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset

840 

63325  841 
lemma unit_prod [intro]: "is_unit a \<Longrightarrow> is_unit b \<Longrightarrow> is_unit (a * b)" 
60562
24af00b010cf
Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents:
60529
diff
changeset

842 
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

843 

63325  844 
lemma is_unit_mult_iff: "is_unit (a * b) \<longleftrightarrow> is_unit a \<and> is_unit b" 
62366  845 
by (auto dest: dvd_mult_left dvd_mult_right) 
846 

63325  847 
lemma unit_div [intro]: "is_unit a \<Longrightarrow> is_unit b \<Longrightarrow> is_unit (a div b)" 
60517
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset

848 
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

849 

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 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

851 
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

852 
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

853 
proof 
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset

854 
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

855 
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

856 
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

857 
next 
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 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 
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

860 
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

861 
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

862 
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

863 
qed 
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset

864 

63924  865 
lemma mult_unit_dvd_iff': "is_unit a \<Longrightarrow> (a * b) dvd c \<longleftrightarrow> b dvd c" 
866 
using mult_unit_dvd_iff [of a b c] by (simp add: ac_simps) 

867 

60517
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset

868 
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

869 
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

870 
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

871 
proof 
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset

872 
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

873 
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

874 
by (subst mult_assoc [symmetric]) simp 
63325  875 
also from assms have "b * (1 div b) = 1" 
876 
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 

63924  884 
lemma dvd_mult_unit_iff': "is_unit b \<Longrightarrow> a dvd b * c \<longleftrightarrow> a dvd c" 
885 
using dvd_mult_unit_iff [of b a c] by (simp add: ac_simps) 

886 

63325  887 
lemma div_unit_dvd_iff: "is_unit b \<Longrightarrow> a div b dvd c \<longleftrightarrow> a dvd c" 
60517
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset

888 
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

889 

63325  890 
lemma dvd_div_unit_iff: "is_unit b \<Longrightarrow> a dvd c div b \<longleftrightarrow> a dvd c" 
60517
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset

891 
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

892 

63924  893 
lemmas unit_dvd_iff = mult_unit_dvd_iff mult_unit_dvd_iff' 
894 
dvd_mult_unit_iff dvd_mult_unit_iff' 

895 
div_unit_dvd_iff dvd_div_unit_iff (* FIXME consider named_theorems *) 

60517
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset

896 

63325  897 
lemma unit_mult_div_div [simp]: "is_unit a \<Longrightarrow> b * (1 div a) = b div a" 
60517
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset

898 
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

899 

63325  900 
lemma unit_div_mult_self [simp]: "is_unit a \<Longrightarrow> b div a * a = b" 
60517
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset

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 

63325  903 
lemma unit_div_1_div_1 [simp]: "is_unit a \<Longrightarrow> 1 div (1 div a) = a" 
60517
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset

904 
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

905 

63325  906 
lemma unit_div_mult_swap: "is_unit c \<Longrightarrow> a * (b div c) = (a * b) div c" 
60517
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset

907 
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

908 

63325  909 
lemma unit_div_commute: "is_unit b \<Longrightarrow> (a div b) * c = (a * c) div b" 
60517
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset

910 
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

911 

63325  912 
lemma unit_eq_div1: "is_unit b \<Longrightarrow> a div b = c \<longleftrightarrow> a = c * b" 
60517
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset

913 
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

914 

63325  915 
lemma unit_eq_div2: "is_unit b \<Longrightarrow> a = c div b \<longleftrightarrow> a * b = c" 
60517
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset

916 
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

917 

63325  918 
lemma unit_mult_left_cancel: "is_unit a \<Longrightarrow> a * b = a * c \<longleftrightarrow> b = c" 
919 
using mult_cancel_left [of a b c] by auto 

60517
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset

920 

63325  921 
lemma unit_mult_right_cancel: "is_unit a \<Longrightarrow> b * a = c * a \<longleftrightarrow> b = c" 
60517
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset

922 
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

923 

f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset

924 
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

925 
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

926 
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

927 
proof  
f16e4fb20652
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents:
60516
diff
changeset

928 
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

929 
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

930 
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

931 
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

932 
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

933 

60570  934 
lemma is_unit_div_mult2_eq: 
935 
assumes "is_unit b" and "is_unit c" 

936 
shows "a div (b * c) = a div b div c" 

937 
proof  

63325  938 
from assms have "is_unit (b * c)" 
939 
by (simp add: unit_prod) 

60570  940 
then have "b * c dvd a" 
941 
by (rule unit_imp_dvd) 

942 
then show ?thesis 

943 
by (rule dvd_div_mult2_eq) 

944 
qed 

945 

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

946 
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

947 
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

948 
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

949 
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

950 

60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

951 
lemma is_unit_divide_mult_cancel_left: 
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

952 
assumes "a \<noteq> 0" and "is_unit b" 
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

953 
shows "a div (a * b) = 1 div b" 
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

954 
proof  
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

955 
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

956 
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

957 
with assms show ?thesis by simp 
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

958 
qed 
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

959 

cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

960 
lemma is_unit_divide_mult_cancel_right: 
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

961 
assumes "a \<noteq> 0" and "is_unit b" 
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

962 
shows "a div (b * a) = 1 div b" 
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

963 
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

964 

cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

965 
end 
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 
class normalization_semidom = algebraic_semidom + 
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

968 
fixes normalize :: "'a \<Rightarrow> 'a" 
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

969 
and unit_factor :: "'a \<Rightarrow> 'a" 
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

970 
assumes unit_factor_mult_normalize [simp]: "unit_factor a * normalize a = a" 
63588  971 
and normalize_0 [simp]: "normalize 0 = 0" 
60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

972 
and unit_factor_0 [simp]: "unit_factor 0 = 0" 
63588  973 
and is_unit_normalize: "is_unit a \<Longrightarrow> normalize a = 1" 
974 
and unit_factor_is_unit [iff]: "a \<noteq> 0 \<Longrightarrow> is_unit (unit_factor a)" 

975 
and unit_factor_mult: "unit_factor (a * b) = unit_factor a * unit_factor b" 

60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

976 
begin 
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

977 

60688
01488b559910
avoid explicit definition of the relation of associated elements in a ring  prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset

978 
text \<open> 
63588  979 
Class @{class normalization_semidom} cultivates the idea that each integral 
980 
domain can be split into equivalence classes whose representants are 

981 
associated, i.e. divide each other. @{const normalize} specifies a canonical 

982 
representant for each equivalence class. The rationale behind this is that 

983 
it is easier to reason about equality than equivalences, hence we prefer to 

984 
think about equality of normalized values rather than associated elements. 

60688
01488b559910
avoid explicit definition of the relation of associated elements in a ring  prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset

985 
\<close> 
01488b559910
avoid explicit definition of the relation of associated elements in a ring  prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset

986 

63325  987 
lemma unit_factor_dvd [simp]: "a \<noteq> 0 \<Longrightarrow> unit_factor a dvd b" 
60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

988 
by (rule unit_imp_dvd) simp 
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

989 

63325  990 
lemma unit_factor_self [simp]: "unit_factor a dvd a" 
62376
85f38d5f8807
Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
hoelzl
parents:
62366
diff
changeset

991 
by (cases "a = 0") simp_all 
85f38d5f8807
Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
hoelzl
parents:
62366
diff
changeset

992 

63325  993 
lemma normalize_mult_unit_factor [simp]: "normalize a * unit_factor a = a" 
60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

994 
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

995 

63325  996 
lemma normalize_eq_0_iff [simp]: "normalize a = 0 \<longleftrightarrow> a = 0" 
63588  997 
(is "?lhs \<longleftrightarrow> ?rhs") 
60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

998 
proof 
63588  999 
assume ?lhs 
60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1000 
moreover have "unit_factor a * normalize a = a" by simp 
63588  1001 
ultimately show ?rhs by simp 
60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1002 
next 
63588  1003 
assume ?rhs 
1004 
then show ?lhs by simp 

60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1005 
qed 
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1006 

63325  1007 
lemma unit_factor_eq_0_iff [simp]: "unit_factor a = 0 \<longleftrightarrow> a = 0" 
63588  1008 
(is "?lhs \<longleftrightarrow> ?rhs") 
60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1009 
proof 
63588  1010 
assume ?lhs 
60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1011 
moreover have "unit_factor a * normalize a = a" by simp 
63588  1012 
ultimately show ?rhs by simp 
60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1013 
next 
63588  1014 
assume ?rhs 
1015 
then show ?lhs by simp 

60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1016 
qed 
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1017 

cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1018 
lemma is_unit_unit_factor: 
63325  1019 
assumes "is_unit a" 
1020 
shows "unit_factor a = a" 

62376
85f38d5f8807
Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
hoelzl
parents:
62366
diff
changeset

1021 
proof  
60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1022 
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

1023 
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

1024 
ultimately show ?thesis by simp 
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1025 
qed 
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1026 

63325  1027 
lemma unit_factor_1 [simp]: "unit_factor 1 = 1" 
60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1028 
by (rule is_unit_unit_factor) simp 
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1029 

63325  1030 
lemma normalize_1 [simp]: "normalize 1 = 1" 
60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1031 
by (rule is_unit_normalize) simp 
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1032 

63325  1033 
lemma normalize_1_iff: "normalize a = 1 \<longleftrightarrow> is_unit a" 
63588  1034 
(is "?lhs \<longleftrightarrow> ?rhs") 
60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1035 
proof 
63588  1036 
assume ?rhs 
1037 
then show ?lhs by (rule is_unit_normalize) 

60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1038 
next 
63588  1039 
assume ?lhs 
1040 
then have "unit_factor a * normalize a = unit_factor a * 1" 

60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1041 
by simp 
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1042 
then have "unit_factor a = a" 
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1043 
by simp 
63588  1044 
moreover 
1045 
from \<open>?lhs\<close> have "a \<noteq> 0" by auto 

1046 
then have "is_unit (unit_factor a)" by simp 

1047 
ultimately show ?rhs by simp 

60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1048 
qed 
62376
85f38d5f8807
Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
hoelzl
parents:
62366
diff
changeset

1049 

63325  1050 
lemma div_normalize [simp]: "a div normalize a = unit_factor a" 
60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1051 
proof (cases "a = 0") 
63325  1052 
case True 
1053 
then show ?thesis by simp 

60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1054 
next 
63325  1055 
case False 
1056 
then have "normalize a \<noteq> 0" by simp 

60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1057 
with nonzero_mult_divide_cancel_right 
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1058 
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

1059 
then show ?thesis by simp 
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1060 
qed 
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1061 

63325  1062 
lemma div_unit_factor [simp]: "a div unit_factor a = normalize a" 
60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1063 
proof (cases "a = 0") 
63325  1064 
case True 
1065 
then show ?thesis by simp 

60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1066 
next 
63325  1067 
case False 
1068 
then have "unit_factor a \<noteq> 0" by simp 

60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1069 
with nonzero_mult_divide_cancel_left 
63588  1070 
have "unit_factor a * normalize a div unit_factor a = normalize a" 
1071 
by blast 

60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1072 
then show ?thesis by simp 
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1073 
qed 
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1074 

63325  1075 
lemma normalize_div [simp]: "normalize a div a = 1 div unit_factor a" 
60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1076 
proof (cases "a = 0") 
63325  1077 
case True 
1078 
then show ?thesis by simp 

60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1079 
next 
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1080 
case False 
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1081 
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

1082 
by simp 
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1083 
also have "\<dots> = 1 div unit_factor a" 
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1084 
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

1085 
finally show ?thesis . 
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1086 
qed 
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1087 

63325  1088 
lemma mult_one_div_unit_factor [simp]: "a * (1 div unit_factor b) = a div unit_factor b" 
60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1089 
by (cases "b = 0") simp_all 
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1090 

63947  1091 
lemma inv_unit_factor_eq_0_iff [simp]: 
1092 
"1 div unit_factor a = 0 \<longleftrightarrow> a = 0" 

1093 
(is "?lhs \<longleftrightarrow> ?rhs") 

1094 
proof 

1095 
assume ?lhs 

1096 
then have "a * (1 div unit_factor a) = a * 0" 

1097 
by simp 

1098 
then show ?rhs 

1099 
by simp 

1100 
next 

1101 
assume ?rhs 

1102 
then show ?lhs by simp 

1103 
qed 

1104 

63325  1105 
lemma normalize_mult: "normalize (a * b) = normalize a * normalize b" 
60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1106 
proof (cases "a = 0 \<or> b = 0") 
63325  1107 
case True 
1108 
then show ?thesis by auto 

60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1109 
next 
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1110 
case False 
63588  1111 
have "unit_factor (a * b) * normalize (a * b) = a * b" 
1112 
by (rule unit_factor_mult_normalize) 

63325  1113 
then have "normalize (a * b) = a * b div unit_factor (a * b)" 
1114 
by simp 

1115 
also have "\<dots> = a * b div unit_factor (b * a)" 

1116 
by (simp add: ac_simps) 

60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1117 
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

1118 
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

1119 
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

1120 
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

1121 
also have "\<dots> = normalize a * normalize b" 
63325  1122 
using False 
1123 
by (simp add: mult.commute [of a] mult.commute [of "normalize a"] unit_div_mult_swap [symmetric]) 

60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1124 
finally show ?thesis . 
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1125 
qed 
62376
85f38d5f8807
Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
hoelzl
parents:
62366
diff
changeset

1126 

63325  1127 
lemma unit_factor_idem [simp]: "unit_factor (unit_factor a) = unit_factor a" 
60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1128 
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

1129 

63325  1130 
lemma normalize_unit_factor [simp]: "a \<noteq> 0 \<Longrightarrow> normalize (unit_factor a) = 1" 
60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1131 
by (rule is_unit_normalize) simp 
62376
85f38d5f8807
Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
hoelzl
parents:
62366
diff
changeset

1132 

63325  1133 
lemma normalize_idem [simp]: "normalize (normalize a) = normalize a" 
60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1134 
proof (cases "a = 0") 
63325  1135 
case True 
1136 
then show ?thesis by simp 

60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1137 
next 
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1138 
case False 
63325  1139 
have "normalize a = normalize (unit_factor a * normalize a)" 
1140 
by simp 

60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1141 
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

1142 
by (simp only: normalize_mult) 
63325  1143 
finally show ?thesis 
1144 
using False by simp_all 

60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1145 
qed 
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1146 

cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1147 
lemma unit_factor_normalize [simp]: 
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1148 
assumes "a \<noteq> 0" 
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1149 
shows "unit_factor (normalize a) = 1" 
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1150 
proof  
63325  1151 
from assms have *: "normalize a \<noteq> 0" 
1152 
by simp 

60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1153 
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

1154 
by (simp only: unit_factor_mult_normalize) 
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1155 
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

1156 
by simp 
63325  1157 
with * have "unit_factor (normalize a) * normalize a div normalize a = normalize a div normalize a" 
60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1158 
by simp 
63325  1159 
with * show ?thesis 
1160 
by simp 

60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1161 
qed 
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1162 

cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1163 
lemma dvd_unit_factor_div: 
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1164 
assumes "b dvd a" 
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1165 
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

1166 
proof  
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1167 
from assms have "a = a div b * b" 
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1168 
by simp 
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1169 
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

1170 
by simp 
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1171 
then show ?thesis 
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1172 
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

1173 
qed 
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1174 

cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1175 
lemma dvd_normalize_div: 
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1176 
assumes "b dvd a" 
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1177 
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

1178 
proof  
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1179 
from assms have "a = a div b * b" 
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1180 
by simp 
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1181 
then have "normalize a = normalize (a div b * b)" 
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1182 
by simp 
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1183 
then show ?thesis 
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1184 
by (cases "b = 0") (simp_all add: normalize_mult) 
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 

63325  1187 
lemma normalize_dvd_iff [simp]: "normalize a dvd b \<longleftrightarrow> a dvd b" 
60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1188 
proof  
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1189 
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

1190 
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

1191 
by (cases "a = 0") simp_all 
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1192 
then show ?thesis by simp 
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1193 
qed 
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1194 

63325  1195 
lemma dvd_normalize_iff [simp]: "a dvd normalize b \<longleftrightarrow> a dvd b" 
60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1196 
proof  
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1197 
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

1198 
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

1199 
by (cases "b = 0") simp_all 
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1200 
then show ?thesis by simp 
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1201 
qed 
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1202 

60688
01488b559910
avoid explicit definition of the relation of associated elements in a ring  prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset

1203 
text \<open> 
63588  1204 
We avoid an explicit definition of associated elements but prefer explicit 
1205 
normalisation instead. In theory we could define an abbreviation like @{prop 

1206 
"associated a b \<longleftrightarrow> normalize a = normalize b"} but this is counterproductive 

1207 
without suggestive infix syntax, which we do not want to sacrifice for this 

1208 
purpose here. 

60688
01488b559910
avoid explicit definition of the relation of associated elements in a ring  prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset

1209 
\<close> 
60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1210 

60688
01488b559910
avoid explicit definition of the relation of associated elements in a ring  prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset

1211 
lemma associatedI: 
01488b559910
avoid explicit definition of the relation of associated elements in a ring  prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset

1212 
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

1213 
shows "normalize a = normalize b" 
60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1214 
proof (cases "a = 0 \<or> b = 0") 
63325  1215 
case True 
1216 
with assms show ?thesis by auto 

60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1217 
next 
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1218 
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

1219 
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

1220 
moreover from \<open>b dvd a\<close> obtain d where a: "a = b * d" .. 
63325  1221 
ultimately have "b * 1 = b * (c * d)" 
1222 
by (simp add: ac_simps) 

60688
01488b559910
avoid explicit definition of the relation of associated elements in a ring  prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset

1223 
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

1224 
unfolding mult_cancel_left by simp 
63325  1225 
then have "is_unit c" and "is_unit d" 
1226 
by auto 

1227 
with a b show ?thesis 

1228 
by (simp add: normalize_mult is_unit_normalize) 

60688
01488b559910
avoid explicit definition of the relation of associated elements in a ring  prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset

1229 
qed 
01488b559910
avoid explicit definition of the relation of associated elements in a ring  prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset

1230 

63325  1231 
lemma associatedD1: "normalize a = normalize b \<Longrightarrow> a dvd b" 
60688
01488b559910
avoid explicit definition of the relation of associated elements in a ring  prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset

1232 
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

1233 
by simp 
01488b559910
avoid explicit definition of the relation of associated elements in a ring  prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset

1234 

63325  1235 
lemma associatedD2: "normalize a = normalize b \<Longrightarrow> b dvd a" 
60688
01488b559910
avoid explicit definition of the relation of associated elements in a ring  prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset

1236 
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

1237 
by simp 
01488b559910
avoid explicit definition of the relation of associated elements in a ring  prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset

1238 

63325  1239 
lemma associated_unit: "normalize a = normalize b \<Longrightarrow> is_unit a \<Longrightarrow> is_unit b" 
60688
01488b559910
avoid explicit definition of the relation of associated elements in a ring  prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset

1240 
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

1241 

63325  1242 
lemma associated_iff_dvd: "normalize a = normalize b \<longleftrightarrow> a dvd b \<and> b dvd a" 
63588  1243 
(is "?lhs \<longleftrightarrow> ?rhs") 
60688
01488b559910
avoid explicit definition of the relation of associated elements in a ring  prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset

1244 
proof 
63588  1245 
assume ?rhs 
1246 
then show ?lhs by (auto intro!: associatedI) 

60688
01488b559910
avoid explicit definition of the relation of associated elements in a ring  prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset

1247 
next 
63588  1248 
assume ?lhs 
60688
01488b559910
avoid explicit definition of the relation of associated elements in a ring  prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset

1249 
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

1250 
by simp 
01488b559910
avoid explicit definition of the relation of associated elements in a ring  prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset

1251 
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

1252 
by (simp add: ac_simps) 
63588  1253 
show ?rhs 
60688
01488b559910
avoid explicit definition of the relation of associated elements in a ring  prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset

1254 
proof (cases "a = 0 \<or> b = 0") 
63325  1255 
case True 
63588  1256 
with \<open>?lhs\<close> show ?thesis by auto 
60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1257 
next 
62376
85f38d5f8807
Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
hoelzl
parents:
62366
diff
changeset

1258 
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

1259 
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

1260 
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

1261 
with * show ?thesis by simp 
60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1262 
qed 
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1263 
qed 
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1264 

cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1265 
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

1266 
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

1267 
assumes "normalize a = a" and "normalize b = b" 
60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1268 
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

1269 
proof  
01488b559910
avoid explicit definition of the relation of associated elements in a ring  prefer explicit normalization instead
haftmann
parents:
60685
diff
changeset

1270 
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

1271 
unfolding associated_iff_dvd by simp 
63588  1272 
with \<open>normalize a = a\<close> have "a = normalize b" 
1273 
by simp 

1274 
with \<open>normalize b = b\<close> show "a = b" 

1275 
by simp 

60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1276 
qed 
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60615
diff
changeset

1277 

cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
ha
