src/HOL/Number_Theory/Euclidean_Algorithm.thy
 author haftmann Fri Jun 19 07:53:33 2015 +0200 (2015-06-19) changeset 60516 0826b7025d07 parent 60439 b765e08f8bc0 child 60517 f16e4fb20652 permissions -rw-r--r--
generalized some theorems about integral domains and moved to HOL theories
1 (* Author: Manuel Eberl *)
3 section {* Abstract euclidean algorithm *}
5 theory Euclidean_Algorithm
6 imports Complex_Main
7 begin
9 context semidom_divide
10 begin
12 lemma dvd_div_mult_self [simp]:
13   "a dvd b \<Longrightarrow> b div a * a = b"
14   by (cases "a = 0") (auto elim: dvdE simp add: ac_simps)
16 lemma dvd_mult_div_cancel [simp]:
17   "a dvd b \<Longrightarrow> a * (b div a) = b"
18   using dvd_div_mult_self [of a b] by (simp add: ac_simps)
20 lemma div_mult_swap:
21   assumes "c dvd b"
22   shows "a * (b div c) = (a * b) div c"
23 proof (cases "c = 0")
24   case True then show ?thesis by simp
25 next
26   case False from assms obtain d where "b = c * d" ..
27   moreover from False have "a * divide (d * c) c = ((a * d) * c) div c"
28     by simp
29   ultimately show ?thesis by (simp add: ac_simps)
30 qed
32 lemma dvd_div_mult:
33   assumes "c dvd b"
34   shows "b div c * a = (b * a) div c"
35   using assms div_mult_swap [of c b a] by (simp add: ac_simps)
38 text \<open>Units: invertible elements in a ring\<close>
40 abbreviation is_unit :: "'a \<Rightarrow> bool"
41 where
42   "is_unit a \<equiv> a dvd 1"
44 lemma not_is_unit_0 [simp]:
45   "\<not> is_unit 0"
46   by simp
48 lemma unit_imp_dvd [dest]:
49   "is_unit b \<Longrightarrow> b dvd a"
50   by (rule dvd_trans [of _ 1]) simp_all
52 lemma unit_dvdE:
53   assumes "is_unit a"
54   obtains c where "a \<noteq> 0" and "b = a * c"
55 proof -
56   from assms have "a dvd b" by auto
57   then obtain c where "b = a * c" ..
58   moreover from assms have "a \<noteq> 0" by auto
59   ultimately show thesis using that by blast
60 qed
62 lemma dvd_unit_imp_unit:
63   "a dvd b \<Longrightarrow> is_unit b \<Longrightarrow> is_unit a"
64   by (rule dvd_trans)
66 lemma unit_div_1_unit [simp, intro]:
67   assumes "is_unit a"
68   shows "is_unit (1 div a)"
69 proof -
70   from assms have "1 = 1 div a * a" by simp
71   then show "is_unit (1 div a)" by (rule dvdI)
72 qed
74 lemma is_unitE [elim?]:
75   assumes "is_unit a"
76   obtains b where "a \<noteq> 0" and "b \<noteq> 0"
77     and "is_unit b" and "1 div a = b" and "1 div b = a"
78     and "a * b = 1" and "c div a = c * b"
79 proof (rule that)
80   def b \<equiv> "1 div a"
81   then show "1 div a = b" by simp
82   from b_def is_unit a show "is_unit b" by simp
83   from is_unit a and is_unit b show "a \<noteq> 0" and "b \<noteq> 0" by auto
84   from b_def is_unit a show "a * b = 1" by simp
85   then have "1 = a * b" ..
86   with b_def b \<noteq> 0 show "1 div b = a" by simp
87   from is_unit a have "a dvd c" ..
88   then obtain d where "c = a * d" ..
89   with a \<noteq> 0 a * b = 1 show "c div a = c * b"
90     by (simp add: mult.assoc mult.left_commute [of a])
91 qed
93 lemma unit_prod [intro]:
94   "is_unit a \<Longrightarrow> is_unit b \<Longrightarrow> is_unit (a * b)"
95   by (subst mult_1_left [of 1, symmetric]) (rule mult_dvd_mono)
97 lemma unit_div [intro]:
98   "is_unit a \<Longrightarrow> is_unit b \<Longrightarrow> is_unit (a div b)"
99   by (erule is_unitE [of b a]) (simp add: ac_simps unit_prod)
101 lemma mult_unit_dvd_iff:
102   assumes "is_unit b"
103   shows "a * b dvd c \<longleftrightarrow> a dvd c"
104 proof
105   assume "a * b dvd c"
106   with assms show "a dvd c"
107     by (simp add: dvd_mult_left)
108 next
109   assume "a dvd c"
110   then obtain k where "c = a * k" ..
111   with assms have "c = (a * b) * (1 div b * k)"
112     by (simp add: mult_ac)
113   then show "a * b dvd c" by (rule dvdI)
114 qed
116 lemma dvd_mult_unit_iff:
117   assumes "is_unit b"
118   shows "a dvd c * b \<longleftrightarrow> a dvd c"
119 proof
120   assume "a dvd c * b"
121   with assms have "c * b dvd c * (b * (1 div b))"
122     by (subst mult_assoc [symmetric]) simp
123   also from is_unit b have "b * (1 div b) = 1" by (rule is_unitE) simp
124   finally have "c * b dvd c" by simp
125   with a dvd c * b show "a dvd c" by (rule dvd_trans)
126 next
127   assume "a dvd c"
128   then show "a dvd c * b" by simp
129 qed
131 lemma div_unit_dvd_iff:
132   "is_unit b \<Longrightarrow> a div b dvd c \<longleftrightarrow> a dvd c"
133   by (erule is_unitE [of _ a]) (auto simp add: mult_unit_dvd_iff)
135 lemma dvd_div_unit_iff:
136   "is_unit b \<Longrightarrow> a dvd c div b \<longleftrightarrow> a dvd c"
137   by (erule is_unitE [of _ c]) (simp add: dvd_mult_unit_iff)
139 lemmas unit_dvd_iff = mult_unit_dvd_iff div_unit_dvd_iff
140   dvd_mult_unit_iff dvd_div_unit_iff -- \<open>FIXME consider fact collection\<close>
142 lemma unit_mult_div_div [simp]:
143   "is_unit a \<Longrightarrow> b * (1 div a) = b div a"
144   by (erule is_unitE [of _ b]) simp
146 lemma unit_div_mult_self [simp]:
147   "is_unit a \<Longrightarrow> b div a * a = b"
148   by (rule dvd_div_mult_self) auto
150 lemma unit_div_1_div_1 [simp]:
151   "is_unit a \<Longrightarrow> 1 div (1 div a) = a"
152   by (erule is_unitE) simp
154 lemma unit_div_mult_swap:
155   "is_unit c \<Longrightarrow> a * (b div c) = (a * b) div c"
156   by (erule unit_dvdE [of _ b]) (simp add: mult.left_commute [of _ c])
158 lemma unit_div_commute:
159   "is_unit b \<Longrightarrow> (a div b) * c = (a * c) div b"
160   using unit_div_mult_swap [of b c a] by (simp add: ac_simps)
162 lemma unit_eq_div1:
163   "is_unit b \<Longrightarrow> a div b = c \<longleftrightarrow> a = c * b"
164   by (auto elim: is_unitE)
166 lemma unit_eq_div2:
167   "is_unit b \<Longrightarrow> a = c div b \<longleftrightarrow> a * b = c"
168   using unit_eq_div1 [of b c a] by auto
170 lemma unit_mult_left_cancel:
171   assumes "is_unit a"
172   shows "a * b = a * c \<longleftrightarrow> b = c" (is "?P \<longleftrightarrow> ?Q")
173   using assms mult_cancel_left [of a b c] by auto
175 lemma unit_mult_right_cancel:
176   "is_unit a \<Longrightarrow> b * a = c * a \<longleftrightarrow> b = c"
177   using unit_mult_left_cancel [of a b c] by (auto simp add: ac_simps)
179 lemma unit_div_cancel:
180   assumes "is_unit a"
181   shows "b div a = c div a \<longleftrightarrow> b = c"
182 proof -
183   from assms have "is_unit (1 div a)" by simp
184   then have "b * (1 div a) = c * (1 div a) \<longleftrightarrow> b = c"
185     by (rule unit_mult_right_cancel)
186   with assms show ?thesis by simp
187 qed
190 text \<open>Associated elements in a ring – an equivalence relation induced by the quasi-order divisibility \<close>
192 definition associated :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
193 where
194   "associated a b \<longleftrightarrow> a dvd b \<and> b dvd a"
196 lemma associatedI:
197   "a dvd b \<Longrightarrow> b dvd a \<Longrightarrow> associated a b"
198   by (simp add: associated_def)
200 lemma associatedD1:
201   "associated a b \<Longrightarrow> a dvd b"
202   by (simp add: associated_def)
204 lemma associatedD2:
205   "associated a b \<Longrightarrow> b dvd a"
206   by (simp add: associated_def)
208 lemma associated_refl [simp]:
209   "associated a a"
210   by (auto intro: associatedI)
212 lemma associated_sym:
213   "associated b a \<longleftrightarrow> associated a b"
214   by (auto intro: associatedI dest: associatedD1 associatedD2)
216 lemma associated_trans:
217   "associated a b \<Longrightarrow> associated b c \<Longrightarrow> associated a c"
218   by (auto intro: associatedI dvd_trans dest: associatedD1 associatedD2)
220 lemma equivp_associated:
221   "equivp associated"
222 proof (rule equivpI)
223   show "reflp associated"
224     by (rule reflpI) simp
225   show "symp associated"
226     by (rule sympI) (simp add: associated_sym)
227   show "transp associated"
228     by (rule transpI) (fact associated_trans)
229 qed
231 lemma associated_0 [simp]:
232   "associated 0 b \<longleftrightarrow> b = 0"
233   "associated a 0 \<longleftrightarrow> a = 0"
234   by (auto dest: associatedD1 associatedD2)
236 lemma associated_unit:
237   "associated a b \<Longrightarrow> is_unit a \<Longrightarrow> is_unit b"
238   using dvd_unit_imp_unit by (auto dest!: associatedD1 associatedD2)
240 lemma is_unit_associatedI:
241   assumes "is_unit c" and "a = c * b"
242   shows "associated a b"
243 proof (rule associatedI)
244   from a = c * b show "b dvd a" by auto
245   from is_unit c obtain d where "c * d = 1" by (rule is_unitE)
246   moreover from a = c * b have "d * a = d * (c * b)" by simp
247   ultimately have "b = a * d" by (simp add: ac_simps)
248   then show "a dvd b" ..
249 qed
251 lemma associated_is_unitE:
252   assumes "associated a b"
253   obtains c where "is_unit c" and "a = c * b"
254 proof (cases "b = 0")
255   case True with assms have "is_unit 1" and "a = 1 * b" by simp_all
256   with that show thesis .
257 next
258   case False
259   from assms have "a dvd b" and "b dvd a" by (auto dest: associatedD1 associatedD2)
260   then obtain c d where "b = a * d" and "a = b * c" by (blast elim: dvdE)
261   then have "a = c * b" and "(c * d) * b = 1 * b" by (simp_all add: ac_simps)
262   with False have "c * d = 1" using mult_cancel_right [of "c * d" b 1] by simp
263   then have "is_unit c" by auto
264   with a = c * b that show thesis by blast
265 qed
267 lemmas unit_simps = mult_unit_dvd_iff div_unit_dvd_iff dvd_mult_unit_iff
268   dvd_div_unit_iff unit_div_mult_swap unit_div_commute
269   unit_mult_left_cancel unit_mult_right_cancel unit_div_cancel
270   unit_eq_div1 unit_eq_div2
272 end
274 lemma is_unit_int:
275   "is_unit (k::int) \<longleftrightarrow> k = 1 \<or> k = - 1"
276   by auto
279 text {*
280   A Euclidean semiring is a semiring upon which the Euclidean algorithm can be
281   implemented. It must provide:
282   \begin{itemize}
283   \item division with remainder
284   \item a size function such that @{term "size (a mod b) < size b"}
285         for any @{term "b \<noteq> 0"}
286   \item a normalization factor such that two associated numbers are equal iff
287         they are the same when divd by their normalization factors.
288   \end{itemize}
289   The existence of these functions makes it possible to derive gcd and lcm functions
290   for any Euclidean semiring.
291 *}
292 class euclidean_semiring = semiring_div +
293   fixes euclidean_size :: "'a \<Rightarrow> nat"
294   fixes normalization_factor :: "'a \<Rightarrow> 'a"
295   assumes mod_size_less [simp]:
296     "b \<noteq> 0 \<Longrightarrow> euclidean_size (a mod b) < euclidean_size b"
297   assumes size_mult_mono:
298     "b \<noteq> 0 \<Longrightarrow> euclidean_size (a * b) \<ge> euclidean_size a"
299   assumes normalization_factor_is_unit [intro,simp]:
300     "a \<noteq> 0 \<Longrightarrow> is_unit (normalization_factor a)"
301   assumes normalization_factor_mult: "normalization_factor (a * b) =
302     normalization_factor a * normalization_factor b"
303   assumes normalization_factor_unit: "is_unit a \<Longrightarrow> normalization_factor a = a"
304   assumes normalization_factor_0 [simp]: "normalization_factor 0 = 0"
305 begin
307 lemma normalization_factor_dvd [simp]:
308   "a \<noteq> 0 \<Longrightarrow> normalization_factor a dvd b"
309   by (rule unit_imp_dvd, simp)
311 lemma normalization_factor_1 [simp]:
312   "normalization_factor 1 = 1"
313   by (simp add: normalization_factor_unit)
315 lemma normalization_factor_0_iff [simp]:
316   "normalization_factor a = 0 \<longleftrightarrow> a = 0"
317 proof
318   assume "normalization_factor a = 0"
319   hence "\<not> is_unit (normalization_factor a)"
320     by simp
321   then show "a = 0" by auto
322 qed simp
324 lemma normalization_factor_pow:
325   "normalization_factor (a ^ n) = normalization_factor a ^ n"
326   by (induct n) (simp_all add: normalization_factor_mult power_Suc2)
328 lemma normalization_correct [simp]:
329   "normalization_factor (a div normalization_factor a) = (if a = 0 then 0 else 1)"
330 proof (cases "a = 0", simp)
331   assume "a \<noteq> 0"
332   let ?nf = "normalization_factor"
333   from normalization_factor_is_unit[OF a \<noteq> 0] have "?nf a \<noteq> 0"
334     by auto
335   have "?nf (a div ?nf a) * ?nf (?nf a) = ?nf (a div ?nf a * ?nf a)"
336     by (simp add: normalization_factor_mult)
337   also have "a div ?nf a * ?nf a = a" using a \<noteq> 0
338     by simp
339   also have "?nf (?nf a) = ?nf a" using a \<noteq> 0
340     normalization_factor_is_unit normalization_factor_unit by simp
341   finally have "normalization_factor (a div normalization_factor a) = 1"
342     using ?nf a \<noteq> 0 by (metis div_mult_self2_is_id div_self)
343   with a \<noteq> 0 show ?thesis by simp
344 qed
346 lemma normalization_0_iff [simp]:
347   "a div normalization_factor a = 0 \<longleftrightarrow> a = 0"
348   by (cases "a = 0", simp, subst unit_eq_div1, blast, simp)
350 lemma mult_div_normalization [simp]:
351   "b * (1 div normalization_factor a) = b div normalization_factor a"
352   by (cases "a = 0") simp_all
354 lemma associated_iff_normed_eq:
355   "associated a b \<longleftrightarrow> a div normalization_factor a = b div normalization_factor b"
356 proof (cases "b = 0", simp, cases "a = 0", metis associated_0(1) normalization_0_iff, rule iffI)
357   let ?nf = normalization_factor
358   assume "a \<noteq> 0" "b \<noteq> 0" "a div ?nf a = b div ?nf b"
359   hence "a = b * (?nf a div ?nf b)"
360     apply (subst (asm) unit_eq_div1, blast, subst (asm) unit_div_commute, blast)
361     apply (subst div_mult_swap, simp, simp)
362     done
363   with a \<noteq> 0 b \<noteq> 0 have "\<exists>c. is_unit c \<and> a = c * b"
364     by (intro exI[of _ "?nf a div ?nf b"], force simp: mult_ac)
365   then obtain c where "is_unit c" and "a = c * b" by blast
366   then show "associated a b" by (rule is_unit_associatedI)
367 next
368   let ?nf = normalization_factor
369   assume "a \<noteq> 0" "b \<noteq> 0" "associated a b"
370   then obtain c where "is_unit c" and "a = c * b" by (blast elim: associated_is_unitE)
371   then show "a div ?nf a = b div ?nf b"
372     apply (simp only: a = c * b normalization_factor_mult normalization_factor_unit)
373     apply (rule div_mult_mult1, force)
374     done
375   qed
377 lemma normed_associated_imp_eq:
378   "associated a b \<Longrightarrow> normalization_factor a \<in> {0, 1} \<Longrightarrow> normalization_factor b \<in> {0, 1} \<Longrightarrow> a = b"
379   by (simp add: associated_iff_normed_eq, elim disjE, simp_all)
381 lemmas normalization_factor_dvd_iff [simp] =
382   unit_dvd_iff [OF normalization_factor_is_unit]
384 lemma euclidean_division:
385   fixes a :: 'a and b :: 'a
386   assumes "b \<noteq> 0"
387   obtains s and t where "a = s * b + t"
388     and "euclidean_size t < euclidean_size b"
389 proof -
390   from div_mod_equality[of a b 0]
391      have "a = a div b * b + a mod b" by simp
392   with that and assms show ?thesis by force
393 qed
395 lemma dvd_euclidean_size_eq_imp_dvd:
396   assumes "a \<noteq> 0" and b_dvd_a: "b dvd a" and size_eq: "euclidean_size a = euclidean_size b"
397   shows "a dvd b"
398 proof (subst dvd_eq_mod_eq_0, rule ccontr)
399   assume "b mod a \<noteq> 0"
400   from b_dvd_a have b_dvd_mod: "b dvd b mod a" by (simp add: dvd_mod_iff)
401   from b_dvd_mod obtain c where "b mod a = b * c" unfolding dvd_def by blast
402     with b mod a \<noteq> 0 have "c \<noteq> 0" by auto
403   with b mod a = b * c have "euclidean_size (b mod a) \<ge> euclidean_size b"
404       using size_mult_mono by force
405   moreover from a \<noteq> 0 have "euclidean_size (b mod a) < euclidean_size a"
406       using mod_size_less by blast
407   ultimately show False using size_eq by simp
408 qed
410 function gcd_eucl :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
411 where
412   "gcd_eucl a b = (if b = 0 then a div normalization_factor a else gcd_eucl b (a mod b))"
413   by (pat_completeness, simp)
414 termination by (relation "measure (euclidean_size \<circ> snd)", simp_all)
416 declare gcd_eucl.simps [simp del]
418 lemma gcd_induct: "\<lbrakk>\<And>b. P b 0; \<And>a b. 0 \<noteq> b \<Longrightarrow> P b (a mod b) \<Longrightarrow> P a b\<rbrakk> \<Longrightarrow> P a b"
419 proof (induct a b rule: gcd_eucl.induct)
420   case ("1" m n)
421     then show ?case by (cases "n = 0") auto
422 qed
424 definition lcm_eucl :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
425 where
426   "lcm_eucl a b = a * b div (gcd_eucl a b * normalization_factor (a * b))"
428   (* Somewhat complicated definition of Lcm that has the advantage of working
429      for infinite sets as well *)
431 definition Lcm_eucl :: "'a set \<Rightarrow> 'a"
432 where
433   "Lcm_eucl A = (if \<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) then
434      let l = SOME l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l =
435        (LEAST n. \<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n)
436        in l div normalization_factor l
437       else 0)"
439 definition Gcd_eucl :: "'a set \<Rightarrow> 'a"
440 where
441   "Gcd_eucl A = Lcm_eucl {d. \<forall>a\<in>A. d dvd a}"
443 end
445 class euclidean_semiring_gcd = euclidean_semiring + gcd + Gcd +
446   assumes gcd_gcd_eucl: "gcd = gcd_eucl" and lcm_lcm_eucl: "lcm = lcm_eucl"
447   assumes Gcd_Gcd_eucl: "Gcd = Gcd_eucl" and Lcm_Lcm_eucl: "Lcm = Lcm_eucl"
448 begin
450 lemma gcd_red:
451   "gcd a b = gcd b (a mod b)"
452   by (metis gcd_eucl.simps mod_0 mod_by_0 gcd_gcd_eucl)
454 lemma gcd_non_0:
455   "b \<noteq> 0 \<Longrightarrow> gcd a b = gcd b (a mod b)"
456   by (rule gcd_red)
458 lemma gcd_0_left:
459   "gcd 0 a = a div normalization_factor a"
460    by (simp only: gcd_gcd_eucl, subst gcd_eucl.simps, subst gcd_eucl.simps, simp add: Let_def)
462 lemma gcd_0:
463   "gcd a 0 = a div normalization_factor a"
464   by (simp only: gcd_gcd_eucl, subst gcd_eucl.simps, simp add: Let_def)
466 lemma gcd_dvd1 [iff]: "gcd a b dvd a"
467   and gcd_dvd2 [iff]: "gcd a b dvd b"
468 proof (induct a b rule: gcd_eucl.induct)
469   fix a b :: 'a
470   assume IH1: "b \<noteq> 0 \<Longrightarrow> gcd b (a mod b) dvd b"
471   assume IH2: "b \<noteq> 0 \<Longrightarrow> gcd b (a mod b) dvd (a mod b)"
473   have "gcd a b dvd a \<and> gcd a b dvd b"
474   proof (cases "b = 0")
475     case True
476       then show ?thesis by (cases "a = 0", simp_all add: gcd_0)
477   next
478     case False
479       with IH1 and IH2 show ?thesis by (simp add: gcd_non_0 dvd_mod_iff)
480   qed
481   then show "gcd a b dvd a" "gcd a b dvd b" by simp_all
482 qed
484 lemma dvd_gcd_D1: "k dvd gcd m n \<Longrightarrow> k dvd m"
485   by (rule dvd_trans, assumption, rule gcd_dvd1)
487 lemma dvd_gcd_D2: "k dvd gcd m n \<Longrightarrow> k dvd n"
488   by (rule dvd_trans, assumption, rule gcd_dvd2)
490 lemma gcd_greatest:
491   fixes k a b :: 'a
492   shows "k dvd a \<Longrightarrow> k dvd b \<Longrightarrow> k dvd gcd a b"
493 proof (induct a b rule: gcd_eucl.induct)
494   case (1 a b)
495   show ?case
496     proof (cases "b = 0")
497       assume "b = 0"
498       with 1 show ?thesis by (cases "a = 0", simp_all add: gcd_0)
499     next
500       assume "b \<noteq> 0"
501       with 1 show ?thesis by (simp add: gcd_non_0 dvd_mod_iff)
502     qed
503 qed
505 lemma dvd_gcd_iff:
506   "k dvd gcd a b \<longleftrightarrow> k dvd a \<and> k dvd b"
507   by (blast intro!: gcd_greatest intro: dvd_trans)
509 lemmas gcd_greatest_iff = dvd_gcd_iff
511 lemma gcd_zero [simp]:
512   "gcd a b = 0 \<longleftrightarrow> a = 0 \<and> b = 0"
513   by (metis dvd_0_left dvd_refl gcd_dvd1 gcd_dvd2 gcd_greatest)+
515 lemma normalization_factor_gcd [simp]:
516   "normalization_factor (gcd a b) = (if a = 0 \<and> b = 0 then 0 else 1)" (is "?f a b = ?g a b")
517 proof (induct a b rule: gcd_eucl.induct)
518   fix a b :: 'a
519   assume IH: "b \<noteq> 0 \<Longrightarrow> ?f b (a mod b) = ?g b (a mod b)"
520   then show "?f a b = ?g a b" by (cases "b = 0", auto simp: gcd_non_0 gcd_0)
521 qed
523 lemma gcdI:
524   "k dvd a \<Longrightarrow> k dvd b \<Longrightarrow> (\<And>l. l dvd a \<Longrightarrow> l dvd b \<Longrightarrow> l dvd k)
525     \<Longrightarrow> normalization_factor k = (if k = 0 then 0 else 1) \<Longrightarrow> k = gcd a b"
526   by (intro normed_associated_imp_eq) (auto simp: associated_def intro: gcd_greatest)
528 sublocale gcd!: abel_semigroup gcd
529 proof
530   fix a b c
531   show "gcd (gcd a b) c = gcd a (gcd b c)"
532   proof (rule gcdI)
533     have "gcd (gcd a b) c dvd gcd a b" "gcd a b dvd a" by simp_all
534     then show "gcd (gcd a b) c dvd a" by (rule dvd_trans)
535     have "gcd (gcd a b) c dvd gcd a b" "gcd a b dvd b" by simp_all
536     hence "gcd (gcd a b) c dvd b" by (rule dvd_trans)
537     moreover have "gcd (gcd a b) c dvd c" by simp
538     ultimately show "gcd (gcd a b) c dvd gcd b c"
539       by (rule gcd_greatest)
540     show "normalization_factor (gcd (gcd a b) c) =  (if gcd (gcd a b) c = 0 then 0 else 1)"
541       by auto
542     fix l assume "l dvd a" and "l dvd gcd b c"
543     with dvd_trans[OF _ gcd_dvd1] and dvd_trans[OF _ gcd_dvd2]
544       have "l dvd b" and "l dvd c" by blast+
545     with l dvd a show "l dvd gcd (gcd a b) c"
546       by (intro gcd_greatest)
547   qed
548 next
549   fix a b
550   show "gcd a b = gcd b a"
551     by (rule gcdI) (simp_all add: gcd_greatest)
552 qed
554 lemma gcd_unique: "d dvd a \<and> d dvd b \<and>
555     normalization_factor d = (if d = 0 then 0 else 1) \<and>
556     (\<forall>e. e dvd a \<and> e dvd b \<longrightarrow> e dvd d) \<longleftrightarrow> d = gcd a b"
557   by (rule, auto intro: gcdI simp: gcd_greatest)
559 lemma gcd_dvd_prod: "gcd a b dvd k * b"
560   using mult_dvd_mono [of 1] by auto
562 lemma gcd_1_left [simp]: "gcd 1 a = 1"
563   by (rule sym, rule gcdI, simp_all)
565 lemma gcd_1 [simp]: "gcd a 1 = 1"
566   by (rule sym, rule gcdI, simp_all)
568 lemma gcd_proj2_if_dvd:
569   "b dvd a \<Longrightarrow> gcd a b = b div normalization_factor b"
570   by (cases "b = 0", simp_all add: dvd_eq_mod_eq_0 gcd_non_0 gcd_0)
572 lemma gcd_proj1_if_dvd:
573   "a dvd b \<Longrightarrow> gcd a b = a div normalization_factor a"
574   by (subst gcd.commute, simp add: gcd_proj2_if_dvd)
576 lemma gcd_proj1_iff: "gcd m n = m div normalization_factor m \<longleftrightarrow> m dvd n"
577 proof
578   assume A: "gcd m n = m div normalization_factor m"
579   show "m dvd n"
580   proof (cases "m = 0")
581     assume [simp]: "m \<noteq> 0"
582     from A have B: "m = gcd m n * normalization_factor m"
583       by (simp add: unit_eq_div2)
584     show ?thesis by (subst B, simp add: mult_unit_dvd_iff)
585   qed (insert A, simp)
586 next
587   assume "m dvd n"
588   then show "gcd m n = m div normalization_factor m" by (rule gcd_proj1_if_dvd)
589 qed
591 lemma gcd_proj2_iff: "gcd m n = n div normalization_factor n \<longleftrightarrow> n dvd m"
592   by (subst gcd.commute, simp add: gcd_proj1_iff)
594 lemma gcd_mod1 [simp]:
595   "gcd (a mod b) b = gcd a b"
596   by (rule gcdI, metis dvd_mod_iff gcd_dvd1 gcd_dvd2, simp_all add: gcd_greatest dvd_mod_iff)
598 lemma gcd_mod2 [simp]:
599   "gcd a (b mod a) = gcd a b"
600   by (rule gcdI, simp, metis dvd_mod_iff gcd_dvd1 gcd_dvd2, simp_all add: gcd_greatest dvd_mod_iff)
602 lemma normalization_factor_dvd' [simp]:
603   "normalization_factor a dvd a"
604   by (cases "a = 0", simp_all)
606 lemma gcd_mult_distrib':
607   "k div normalization_factor k * gcd a b = gcd (k*a) (k*b)"
608 proof (induct a b rule: gcd_eucl.induct)
609   case (1 a b)
610   show ?case
611   proof (cases "b = 0")
612     case True
613     then show ?thesis by (simp add: normalization_factor_mult gcd_0 algebra_simps div_mult_div_if_dvd)
614   next
615     case False
616     hence "k div normalization_factor k * gcd a b =  gcd (k * b) (k * (a mod b))"
617       using 1 by (subst gcd_red, simp)
618     also have "... = gcd (k * a) (k * b)"
619       by (simp add: mult_mod_right gcd.commute)
620     finally show ?thesis .
621   qed
622 qed
624 lemma gcd_mult_distrib:
625   "k * gcd a b = gcd (k*a) (k*b) * normalization_factor k"
626 proof-
627   let ?nf = "normalization_factor"
628   from gcd_mult_distrib'
629     have "gcd (k*a) (k*b) = k div ?nf k * gcd a b" ..
630   also have "... = k * gcd a b div ?nf k"
631     by (metis dvd_div_mult dvd_eq_mod_eq_0 mod_0 normalization_factor_dvd)
632   finally show ?thesis
633     by simp
634 qed
636 lemma euclidean_size_gcd_le1 [simp]:
637   assumes "a \<noteq> 0"
638   shows "euclidean_size (gcd a b) \<le> euclidean_size a"
639 proof -
640    have "gcd a b dvd a" by (rule gcd_dvd1)
641    then obtain c where A: "a = gcd a b * c" unfolding dvd_def by blast
642    with a \<noteq> 0 show ?thesis by (subst (2) A, intro size_mult_mono) auto
643 qed
645 lemma euclidean_size_gcd_le2 [simp]:
646   "b \<noteq> 0 \<Longrightarrow> euclidean_size (gcd a b) \<le> euclidean_size b"
647   by (subst gcd.commute, rule euclidean_size_gcd_le1)
649 lemma euclidean_size_gcd_less1:
650   assumes "a \<noteq> 0" and "\<not>a dvd b"
651   shows "euclidean_size (gcd a b) < euclidean_size a"
652 proof (rule ccontr)
653   assume "\<not>euclidean_size (gcd a b) < euclidean_size a"
654   with a \<noteq> 0 have "euclidean_size (gcd a b) = euclidean_size a"
655     by (intro le_antisym, simp_all)
656   with assms have "a dvd gcd a b" by (auto intro: dvd_euclidean_size_eq_imp_dvd)
657   hence "a dvd b" using dvd_gcd_D2 by blast
658   with \<not>a dvd b show False by contradiction
659 qed
661 lemma euclidean_size_gcd_less2:
662   assumes "b \<noteq> 0" and "\<not>b dvd a"
663   shows "euclidean_size (gcd a b) < euclidean_size b"
664   using assms by (subst gcd.commute, rule euclidean_size_gcd_less1)
666 lemma gcd_mult_unit1: "is_unit a \<Longrightarrow> gcd (b * a) c = gcd b c"
667   apply (rule gcdI)
668   apply (rule dvd_trans, rule gcd_dvd1, simp add: unit_simps)
669   apply (rule gcd_dvd2)
670   apply (rule gcd_greatest, simp add: unit_simps, assumption)
671   apply (subst normalization_factor_gcd, simp add: gcd_0)
672   done
674 lemma gcd_mult_unit2: "is_unit a \<Longrightarrow> gcd b (c * a) = gcd b c"
675   by (subst gcd.commute, subst gcd_mult_unit1, assumption, rule gcd.commute)
677 lemma gcd_div_unit1: "is_unit a \<Longrightarrow> gcd (b div a) c = gcd b c"
678   by (erule is_unitE [of _ b]) (simp add: gcd_mult_unit1)
680 lemma gcd_div_unit2: "is_unit a \<Longrightarrow> gcd b (c div a) = gcd b c"
681   by (erule is_unitE [of _ c]) (simp add: gcd_mult_unit2)
683 lemma gcd_idem: "gcd a a = a div normalization_factor a"
684   by (cases "a = 0") (simp add: gcd_0_left, rule sym, rule gcdI, simp_all)
686 lemma gcd_right_idem: "gcd (gcd a b) b = gcd a b"
687   apply (rule gcdI)
688   apply (simp add: ac_simps)
689   apply (rule gcd_dvd2)
690   apply (rule gcd_greatest, erule (1) gcd_greatest, assumption)
691   apply simp
692   done
694 lemma gcd_left_idem: "gcd a (gcd a b) = gcd a b"
695   apply (rule gcdI)
696   apply simp
697   apply (rule dvd_trans, rule gcd_dvd2, rule gcd_dvd2)
698   apply (rule gcd_greatest, assumption, erule gcd_greatest, assumption)
699   apply simp
700   done
702 lemma comp_fun_idem_gcd: "comp_fun_idem gcd"
703 proof
704   fix a b show "gcd a \<circ> gcd b = gcd b \<circ> gcd a"
705     by (simp add: fun_eq_iff ac_simps)
706 next
707   fix a show "gcd a \<circ> gcd a = gcd a"
708     by (simp add: fun_eq_iff gcd_left_idem)
709 qed
711 lemma coprime_dvd_mult:
712   assumes "gcd c b = 1" and "c dvd a * b"
713   shows "c dvd a"
714 proof -
715   let ?nf = "normalization_factor"
716   from assms gcd_mult_distrib [of a c b]
717     have A: "a = gcd (a * c) (a * b) * ?nf a" by simp
718   from c dvd a * b show ?thesis by (subst A, simp_all add: gcd_greatest)
719 qed
721 lemma coprime_dvd_mult_iff:
722   "gcd c b = 1 \<Longrightarrow> (c dvd a * b) = (c dvd a)"
723   by (rule, rule coprime_dvd_mult, simp_all)
725 lemma gcd_dvd_antisym:
726   "gcd a b dvd gcd c d \<Longrightarrow> gcd c d dvd gcd a b \<Longrightarrow> gcd a b = gcd c d"
727 proof (rule gcdI)
728   assume A: "gcd a b dvd gcd c d" and B: "gcd c d dvd gcd a b"
729   have "gcd c d dvd c" by simp
730   with A show "gcd a b dvd c" by (rule dvd_trans)
731   have "gcd c d dvd d" by simp
732   with A show "gcd a b dvd d" by (rule dvd_trans)
733   show "normalization_factor (gcd a b) = (if gcd a b = 0 then 0 else 1)"
734     by simp
735   fix l assume "l dvd c" and "l dvd d"
736   hence "l dvd gcd c d" by (rule gcd_greatest)
737   from this and B show "l dvd gcd a b" by (rule dvd_trans)
738 qed
740 lemma gcd_mult_cancel:
741   assumes "gcd k n = 1"
742   shows "gcd (k * m) n = gcd m n"
743 proof (rule gcd_dvd_antisym)
744   have "gcd (gcd (k * m) n) k = gcd (gcd k n) (k * m)" by (simp add: ac_simps)
745   also note gcd k n = 1
746   finally have "gcd (gcd (k * m) n) k = 1" by simp
747   hence "gcd (k * m) n dvd m" by (rule coprime_dvd_mult, simp add: ac_simps)
748   moreover have "gcd (k * m) n dvd n" by simp
749   ultimately show "gcd (k * m) n dvd gcd m n" by (rule gcd_greatest)
750   have "gcd m n dvd (k * m)" and "gcd m n dvd n" by simp_all
751   then show "gcd m n dvd gcd (k * m) n" by (rule gcd_greatest)
752 qed
754 lemma coprime_crossproduct:
755   assumes [simp]: "gcd a d = 1" "gcd b c = 1"
756   shows "associated (a * c) (b * d) \<longleftrightarrow> associated a b \<and> associated c d" (is "?lhs \<longleftrightarrow> ?rhs")
757 proof
758   assume ?rhs then show ?lhs unfolding associated_def by (fast intro: mult_dvd_mono)
759 next
760   assume ?lhs
761   from ?lhs have "a dvd b * d" unfolding associated_def by (metis dvd_mult_left)
762   hence "a dvd b" by (simp add: coprime_dvd_mult_iff)
763   moreover from ?lhs have "b dvd a * c" unfolding associated_def by (metis dvd_mult_left)
764   hence "b dvd a" by (simp add: coprime_dvd_mult_iff)
765   moreover from ?lhs have "c dvd d * b"
766     unfolding associated_def by (auto dest: dvd_mult_right simp add: ac_simps)
767   hence "c dvd d" by (simp add: coprime_dvd_mult_iff gcd.commute)
768   moreover from ?lhs have "d dvd c * a"
769     unfolding associated_def by (auto dest: dvd_mult_right simp add: ac_simps)
770   hence "d dvd c" by (simp add: coprime_dvd_mult_iff gcd.commute)
771   ultimately show ?rhs unfolding associated_def by simp
772 qed
774 lemma gcd_add1 [simp]:
775   "gcd (m + n) n = gcd m n"
776   by (cases "n = 0", simp_all add: gcd_non_0)
778 lemma gcd_add2 [simp]:
779   "gcd m (m + n) = gcd m n"
780   using gcd_add1 [of n m] by (simp add: ac_simps)
782 lemma gcd_add_mult: "gcd m (k * m + n) = gcd m n"
783   by (subst gcd.commute, subst gcd_red, simp)
785 lemma coprimeI: "(\<And>l. \<lbrakk>l dvd a; l dvd b\<rbrakk> \<Longrightarrow> l dvd 1) \<Longrightarrow> gcd a b = 1"
786   by (rule sym, rule gcdI, simp_all)
788 lemma coprime: "gcd a b = 1 \<longleftrightarrow> (\<forall>d. d dvd a \<and> d dvd b \<longleftrightarrow> is_unit d)"
789   by (auto intro: coprimeI gcd_greatest dvd_gcd_D1 dvd_gcd_D2)
791 lemma div_gcd_coprime:
792   assumes nz: "a \<noteq> 0 \<or> b \<noteq> 0"
793   defines [simp]: "d \<equiv> gcd a b"
794   defines [simp]: "a' \<equiv> a div d" and [simp]: "b' \<equiv> b div d"
795   shows "gcd a' b' = 1"
796 proof (rule coprimeI)
797   fix l assume "l dvd a'" "l dvd b'"
798   then obtain s t where "a' = l * s" "b' = l * t" unfolding dvd_def by blast
799   moreover have "a = a' * d" "b = b' * d" by simp_all
800   ultimately have "a = (l * d) * s" "b = (l * d) * t"
801     by (simp_all only: ac_simps)
802   hence "l*d dvd a" and "l*d dvd b" by (simp_all only: dvd_triv_left)
803   hence "l*d dvd d" by (simp add: gcd_greatest)
804   then obtain u where "d = l * d * u" ..
805   then have "d * (l * u) = d" by (simp add: ac_simps)
806   moreover from nz have "d \<noteq> 0" by simp
807   with div_mult_self1_is_id have "d * (l * u) div d = l * u" .
808   ultimately have "1 = l * u"
809     using d \<noteq> 0 by simp
810   then show "l dvd 1" ..
811 qed
813 lemma coprime_mult:
814   assumes da: "gcd d a = 1" and db: "gcd d b = 1"
815   shows "gcd d (a * b) = 1"
816   apply (subst gcd.commute)
817   using da apply (subst gcd_mult_cancel)
818   apply (subst gcd.commute, assumption)
819   apply (subst gcd.commute, rule db)
820   done
822 lemma coprime_lmult:
823   assumes dab: "gcd d (a * b) = 1"
824   shows "gcd d a = 1"
825 proof (rule coprimeI)
826   fix l assume "l dvd d" and "l dvd a"
827   hence "l dvd a * b" by simp
828   with l dvd d and dab show "l dvd 1" by (auto intro: gcd_greatest)
829 qed
831 lemma coprime_rmult:
832   assumes dab: "gcd d (a * b) = 1"
833   shows "gcd d b = 1"
834 proof (rule coprimeI)
835   fix l assume "l dvd d" and "l dvd b"
836   hence "l dvd a * b" by simp
837   with l dvd d and dab show "l dvd 1" by (auto intro: gcd_greatest)
838 qed
840 lemma coprime_mul_eq: "gcd d (a * b) = 1 \<longleftrightarrow> gcd d a = 1 \<and> gcd d b = 1"
841   using coprime_rmult[of d a b] coprime_lmult[of d a b] coprime_mult[of d a b] by blast
843 lemma gcd_coprime:
844   assumes c: "gcd a b \<noteq> 0" and a: "a = a' * gcd a b" and b: "b = b' * gcd a b"
845   shows "gcd a' b' = 1"
846 proof -
847   from c have "a \<noteq> 0 \<or> b \<noteq> 0" by simp
848   with div_gcd_coprime have "gcd (a div gcd a b) (b div gcd a b) = 1" .
849   also from assms have "a div gcd a b = a'" by (metis div_mult_self2_is_id)+
850   also from assms have "b div gcd a b = b'" by (metis div_mult_self2_is_id)+
851   finally show ?thesis .
852 qed
854 lemma coprime_power:
855   assumes "0 < n"
856   shows "gcd a (b ^ n) = 1 \<longleftrightarrow> gcd a b = 1"
857 using assms proof (induct n)
858   case (Suc n) then show ?case
859     by (cases n) (simp_all add: coprime_mul_eq)
860 qed simp
862 lemma gcd_coprime_exists:
863   assumes nz: "gcd a b \<noteq> 0"
864   shows "\<exists>a' b'. a = a' * gcd a b \<and> b = b' * gcd a b \<and> gcd a' b' = 1"
865   apply (rule_tac x = "a div gcd a b" in exI)
866   apply (rule_tac x = "b div gcd a b" in exI)
867   apply (insert nz, auto intro: div_gcd_coprime)
868   done
870 lemma coprime_exp:
871   "gcd d a = 1 \<Longrightarrow> gcd d (a^n) = 1"
872   by (induct n, simp_all add: coprime_mult)
874 lemma coprime_exp2 [intro]:
875   "gcd a b = 1 \<Longrightarrow> gcd (a^n) (b^m) = 1"
876   apply (rule coprime_exp)
877   apply (subst gcd.commute)
878   apply (rule coprime_exp)
879   apply (subst gcd.commute)
880   apply assumption
881   done
883 lemma gcd_exp:
884   "gcd (a^n) (b^n) = (gcd a b) ^ n"
885 proof (cases "a = 0 \<and> b = 0")
886   assume "a = 0 \<and> b = 0"
887   then show ?thesis by (cases n, simp_all add: gcd_0_left)
888 next
889   assume A: "\<not>(a = 0 \<and> b = 0)"
890   hence "1 = gcd ((a div gcd a b)^n) ((b div gcd a b)^n)"
891     using div_gcd_coprime by (subst sym, auto simp: div_gcd_coprime)
892   hence "(gcd a b) ^ n = (gcd a b) ^ n * ..." by simp
893   also note gcd_mult_distrib
894   also have "normalization_factor ((gcd a b)^n) = 1"
895     by (simp add: normalization_factor_pow A)
896   also have "(gcd a b)^n * (a div gcd a b)^n = a^n"
897     by (subst ac_simps, subst div_power, simp, rule dvd_div_mult_self, rule dvd_power_same, simp)
898   also have "(gcd a b)^n * (b div gcd a b)^n = b^n"
899     by (subst ac_simps, subst div_power, simp, rule dvd_div_mult_self, rule dvd_power_same, simp)
900   finally show ?thesis by simp
901 qed
903 lemma coprime_common_divisor:
904   "gcd a b = 1 \<Longrightarrow> a dvd a \<Longrightarrow> a dvd b \<Longrightarrow> is_unit a"
905   apply (subgoal_tac "a dvd gcd a b")
906   apply simp
907   apply (erule (1) gcd_greatest)
908   done
910 lemma division_decomp:
911   assumes dc: "a dvd b * c"
912   shows "\<exists>b' c'. a = b' * c' \<and> b' dvd b \<and> c' dvd c"
913 proof (cases "gcd a b = 0")
914   assume "gcd a b = 0"
915   hence "a = 0 \<and> b = 0" by simp
916   hence "a = 0 * c \<and> 0 dvd b \<and> c dvd c" by simp
917   then show ?thesis by blast
918 next
919   let ?d = "gcd a b"
920   assume "?d \<noteq> 0"
921   from gcd_coprime_exists[OF this]
922     obtain a' b' where ab': "a = a' * ?d" "b = b' * ?d" "gcd a' b' = 1"
923     by blast
924   from ab'(1) have "a' dvd a" unfolding dvd_def by blast
925   with dc have "a' dvd b*c" using dvd_trans[of a' a "b*c"] by simp
926   from dc ab'(1,2) have "a'*?d dvd (b'*?d) * c" by simp
927   hence "?d * a' dvd ?d * (b' * c)" by (simp add: mult_ac)
928   with ?d \<noteq> 0 have "a' dvd b' * c" by simp
929   with coprime_dvd_mult[OF ab'(3)]
930     have "a' dvd c" by (subst (asm) ac_simps, blast)
931   with ab'(1) have "a = ?d * a' \<and> ?d dvd b \<and> a' dvd c" by (simp add: mult_ac)
932   then show ?thesis by blast
933 qed
935 lemma pow_divs_pow:
936   assumes ab: "a ^ n dvd b ^ n" and n: "n \<noteq> 0"
937   shows "a dvd b"
938 proof (cases "gcd a b = 0")
939   assume "gcd a b = 0"
940   then show ?thesis by simp
941 next
942   let ?d = "gcd a b"
943   assume "?d \<noteq> 0"
944   from n obtain m where m: "n = Suc m" by (cases n, simp_all)
945   from ?d \<noteq> 0 have zn: "?d ^ n \<noteq> 0" by (rule power_not_zero)
946   from gcd_coprime_exists[OF ?d \<noteq> 0]
947     obtain a' b' where ab': "a = a' * ?d" "b = b' * ?d" "gcd a' b' = 1"
948     by blast
949   from ab have "(a' * ?d) ^ n dvd (b' * ?d) ^ n"
950     by (simp add: ab'(1,2)[symmetric])
951   hence "?d^n * a'^n dvd ?d^n * b'^n"
952     by (simp only: power_mult_distrib ac_simps)
953   with zn have "a'^n dvd b'^n" by simp
954   hence "a' dvd b'^n" using dvd_trans[of a' "a'^n" "b'^n"] by (simp add: m)
955   hence "a' dvd b'^m * b'" by (simp add: m ac_simps)
956   with coprime_dvd_mult[OF coprime_exp[OF ab'(3), of m]]
957     have "a' dvd b'" by (subst (asm) ac_simps, blast)
958   hence "a'*?d dvd b'*?d" by (rule mult_dvd_mono, simp)
959   with ab'(1,2) show ?thesis by simp
960 qed
962 lemma pow_divs_eq [simp]:
963   "n \<noteq> 0 \<Longrightarrow> a ^ n dvd b ^ n \<longleftrightarrow> a dvd b"
964   by (auto intro: pow_divs_pow dvd_power_same)
966 lemma divs_mult:
967   assumes mr: "m dvd r" and nr: "n dvd r" and mn: "gcd m n = 1"
968   shows "m * n dvd r"
969 proof -
970   from mr nr obtain m' n' where m': "r = m*m'" and n': "r = n*n'"
971     unfolding dvd_def by blast
972   from mr n' have "m dvd n'*n" by (simp add: ac_simps)
973   hence "m dvd n'" using coprime_dvd_mult_iff[OF mn] by simp
974   then obtain k where k: "n' = m*k" unfolding dvd_def by blast
975   with n' have "r = m * n * k" by (simp add: mult_ac)
976   then show ?thesis unfolding dvd_def by blast
977 qed
979 lemma coprime_plus_one [simp]: "gcd (n + 1) n = 1"
980   by (subst add_commute, simp)
982 lemma setprod_coprime [rule_format]:
983   "(\<forall>i\<in>A. gcd (f i) a = 1) \<longrightarrow> gcd (\<Prod>i\<in>A. f i) a = 1"
984   apply (cases "finite A")
985   apply (induct set: finite)
986   apply (auto simp add: gcd_mult_cancel)
987   done
989 lemma coprime_divisors:
990   assumes "d dvd a" "e dvd b" "gcd a b = 1"
991   shows "gcd d e = 1"
992 proof -
993   from assms obtain k l where "a = d * k" "b = e * l"
994     unfolding dvd_def by blast
995   with assms have "gcd (d * k) (e * l) = 1" by simp
996   hence "gcd (d * k) e = 1" by (rule coprime_lmult)
997   also have "gcd (d * k) e = gcd e (d * k)" by (simp add: ac_simps)
998   finally have "gcd e d = 1" by (rule coprime_lmult)
999   then show ?thesis by (simp add: ac_simps)
1000 qed
1002 lemma invertible_coprime:
1003   assumes "a * b mod m = 1"
1004   shows "coprime a m"
1005 proof -
1006   from assms have "coprime m (a * b mod m)"
1007     by simp
1008   then have "coprime m (a * b)"
1009     by simp
1010   then have "coprime m a"
1011     by (rule coprime_lmult)
1012   then show ?thesis
1013     by (simp add: ac_simps)
1014 qed
1016 lemma lcm_gcd:
1017   "lcm a b = a * b div (gcd a b * normalization_factor (a*b))"
1018   by (simp only: lcm_lcm_eucl gcd_gcd_eucl lcm_eucl_def)
1020 lemma lcm_gcd_prod:
1021   "lcm a b * gcd a b = a * b div normalization_factor (a*b)"
1022 proof (cases "a * b = 0")
1023   let ?nf = normalization_factor
1024   assume "a * b \<noteq> 0"
1025   hence "gcd a b \<noteq> 0" by simp
1026   from lcm_gcd have "lcm a b * gcd a b = gcd a b * (a * b div (?nf (a*b) * gcd a b))"
1027     by (simp add: mult_ac)
1028   also from a * b \<noteq> 0 have "... = a * b div ?nf (a*b)"
1029     by (simp add: div_mult_swap mult.commute)
1030   finally show ?thesis .
1031 qed (auto simp add: lcm_gcd)
1033 lemma lcm_dvd1 [iff]:
1034   "a dvd lcm a b"
1035 proof (cases "a*b = 0")
1036   assume "a * b \<noteq> 0"
1037   hence "gcd a b \<noteq> 0" by simp
1038   let ?c = "1 div normalization_factor (a * b)"
1039   from a * b \<noteq> 0 have [simp]: "is_unit (normalization_factor (a * b))" by simp
1040   from lcm_gcd_prod[of a b] have "lcm a b * gcd a b = a * ?c * b"
1041     by (simp add: div_mult_swap unit_div_commute)
1042   hence "lcm a b * gcd a b div gcd a b = a * ?c * b div gcd a b" by simp
1043   with gcd a b \<noteq> 0 have "lcm a b = a * ?c * b div gcd a b"
1044     by (subst (asm) div_mult_self2_is_id, simp_all)
1045   also have "... = a * (?c * b div gcd a b)"
1046     by (metis div_mult_swap gcd_dvd2 mult_assoc)
1047   finally show ?thesis by (rule dvdI)
1048 qed (auto simp add: lcm_gcd)
1050 lemma lcm_least:
1051   "\<lbrakk>a dvd k; b dvd k\<rbrakk> \<Longrightarrow> lcm a b dvd k"
1052 proof (cases "k = 0")
1053   let ?nf = normalization_factor
1054   assume "k \<noteq> 0"
1055   hence "is_unit (?nf k)" by simp
1056   hence "?nf k \<noteq> 0" by (metis not_is_unit_0)
1057   assume A: "a dvd k" "b dvd k"
1058   hence "gcd a b \<noteq> 0" using k \<noteq> 0 by auto
1059   from A obtain r s where ar: "k = a * r" and bs: "k = b * s"
1060     unfolding dvd_def by blast
1061   with k \<noteq> 0 have "r * s \<noteq> 0"
1062     by auto (drule sym [of 0], simp)
1063   hence "is_unit (?nf (r * s))" by simp
1064   let ?c = "?nf k div ?nf (r*s)"
1065   from is_unit (?nf k) and is_unit (?nf (r * s)) have "is_unit ?c" by (rule unit_div)
1066   hence "?c \<noteq> 0" using not_is_unit_0 by fast
1067   from ar bs have "k * k * gcd s r = ?nf k * k * gcd (k * s) (k * r)"
1068     by (subst mult_assoc, subst gcd_mult_distrib[of k s r], simp only: ac_simps)
1069   also have "... = ?nf k * k * gcd ((r*s) * a) ((r*s) * b)"
1070     by (subst (3) k = a * r, subst (3) k = b * s, simp add: algebra_simps)
1071   also have "... = ?c * r*s * k * gcd a b" using r * s \<noteq> 0
1072     by (subst gcd_mult_distrib'[symmetric], simp add: algebra_simps unit_simps)
1073   finally have "(a*r) * (b*s) * gcd s r = ?c * k * r * s * gcd a b"
1074     by (subst ar[symmetric], subst bs[symmetric], simp add: mult_ac)
1075   hence "a * b * gcd s r * (r * s) = ?c * k * gcd a b * (r * s)"
1076     by (simp add: algebra_simps)
1077   hence "?c * k * gcd a b = a * b * gcd s r" using r * s \<noteq> 0
1078     by (metis div_mult_self2_is_id)
1079   also have "... = lcm a b * gcd a b * gcd s r * ?nf (a*b)"
1080     by (subst lcm_gcd_prod[of a b], metis gcd_mult_distrib gcd_mult_distrib')
1081   also have "... = lcm a b * gcd s r * ?nf (a*b) * gcd a b"
1082     by (simp add: algebra_simps)
1083   finally have "k * ?c = lcm a b * gcd s r * ?nf (a*b)" using gcd a b \<noteq> 0
1084     by (metis mult.commute div_mult_self2_is_id)
1085   hence "k = lcm a b * (gcd s r * ?nf (a*b)) div ?c" using ?c \<noteq> 0
1086     by (metis div_mult_self2_is_id mult_assoc)
1087   also have "... = lcm a b * (gcd s r * ?nf (a*b) div ?c)" using is_unit ?c
1088     by (simp add: unit_simps)
1089   finally show ?thesis by (rule dvdI)
1090 qed simp
1092 lemma lcm_zero:
1093   "lcm a b = 0 \<longleftrightarrow> a = 0 \<or> b = 0"
1094 proof -
1095   let ?nf = normalization_factor
1096   {
1097     assume "a \<noteq> 0" "b \<noteq> 0"
1098     hence "a * b div ?nf (a * b) \<noteq> 0" by (simp add: no_zero_divisors)
1099     moreover from a \<noteq> 0 and b \<noteq> 0 have "gcd a b \<noteq> 0" by simp
1100     ultimately have "lcm a b \<noteq> 0" using lcm_gcd_prod[of a b] by (intro notI, simp)
1101   } moreover {
1102     assume "a = 0 \<or> b = 0"
1103     hence "lcm a b = 0" by (elim disjE, simp_all add: lcm_gcd)
1104   }
1105   ultimately show ?thesis by blast
1106 qed
1108 lemmas lcm_0_iff = lcm_zero
1110 lemma gcd_lcm:
1111   assumes "lcm a b \<noteq> 0"
1112   shows "gcd a b = a * b div (lcm a b * normalization_factor (a * b))"
1113 proof-
1114   from assms have "gcd a b \<noteq> 0" by (simp add: lcm_zero)
1115   let ?c = "normalization_factor (a * b)"
1116   from lcm a b \<noteq> 0 have "?c \<noteq> 0" by (intro notI, simp add: lcm_zero no_zero_divisors)
1117   hence "is_unit ?c" by simp
1118   from lcm_gcd_prod [of a b] have "gcd a b = a * b div ?c div lcm a b"
1119     by (subst (2) div_mult_self2_is_id[OF lcm a b \<noteq> 0, symmetric], simp add: mult_ac)
1120   also from is_unit ?c have "... = a * b div (lcm a b * ?c)"
1121     by (metis ?c \<noteq> 0 div_mult_mult1 dvd_mult_div_cancel mult_commute normalization_factor_dvd')
1122   finally show ?thesis .
1123 qed
1125 lemma normalization_factor_lcm [simp]:
1126   "normalization_factor (lcm a b) = (if a = 0 \<or> b = 0 then 0 else 1)"
1127 proof (cases "a = 0 \<or> b = 0")
1128   case True then show ?thesis
1129     by (auto simp add: lcm_gcd)
1130 next
1131   case False
1132   let ?nf = normalization_factor
1133   from lcm_gcd_prod[of a b]
1134     have "?nf (lcm a b) * ?nf (gcd a b) = ?nf (a*b) div ?nf (a*b)"
1135     by (metis div_by_0 div_self normalization_correct normalization_factor_0 normalization_factor_mult)
1136   also have "... = (if a*b = 0 then 0 else 1)"
1137     by simp
1138   finally show ?thesis using False by simp
1139 qed
1141 lemma lcm_dvd2 [iff]: "b dvd lcm a b"
1142   using lcm_dvd1 [of b a] by (simp add: lcm_gcd ac_simps)
1144 lemma lcmI:
1145   "\<lbrakk>a dvd k; b dvd k; \<And>l. a dvd l \<Longrightarrow> b dvd l \<Longrightarrow> k dvd l;
1146     normalization_factor k = (if k = 0 then 0 else 1)\<rbrakk> \<Longrightarrow> k = lcm a b"
1147   by (intro normed_associated_imp_eq) (auto simp: associated_def intro: lcm_least)
1149 sublocale lcm!: abel_semigroup lcm
1150 proof
1151   fix a b c
1152   show "lcm (lcm a b) c = lcm a (lcm b c)"
1153   proof (rule lcmI)
1154     have "a dvd lcm a b" and "lcm a b dvd lcm (lcm a b) c" by simp_all
1155     then show "a dvd lcm (lcm a b) c" by (rule dvd_trans)
1157     have "b dvd lcm a b" and "lcm a b dvd lcm (lcm a b) c" by simp_all
1158     hence "b dvd lcm (lcm a b) c" by (rule dvd_trans)
1159     moreover have "c dvd lcm (lcm a b) c" by simp
1160     ultimately show "lcm b c dvd lcm (lcm a b) c" by (rule lcm_least)
1162     fix l assume "a dvd l" and "lcm b c dvd l"
1163     have "b dvd lcm b c" by simp
1164     from this and lcm b c dvd l have "b dvd l" by (rule dvd_trans)
1165     have "c dvd lcm b c" by simp
1166     from this and lcm b c dvd l have "c dvd l" by (rule dvd_trans)
1167     from a dvd l and b dvd l have "lcm a b dvd l" by (rule lcm_least)
1168     from this and c dvd l show "lcm (lcm a b) c dvd l" by (rule lcm_least)
1169   qed (simp add: lcm_zero)
1170 next
1171   fix a b
1172   show "lcm a b = lcm b a"
1173     by (simp add: lcm_gcd ac_simps)
1174 qed
1176 lemma dvd_lcm_D1:
1177   "lcm m n dvd k \<Longrightarrow> m dvd k"
1178   by (rule dvd_trans, rule lcm_dvd1, assumption)
1180 lemma dvd_lcm_D2:
1181   "lcm m n dvd k \<Longrightarrow> n dvd k"
1182   by (rule dvd_trans, rule lcm_dvd2, assumption)
1184 lemma gcd_dvd_lcm [simp]:
1185   "gcd a b dvd lcm a b"
1186   by (metis dvd_trans gcd_dvd2 lcm_dvd2)
1188 lemma lcm_1_iff:
1189   "lcm a b = 1 \<longleftrightarrow> is_unit a \<and> is_unit b"
1190 proof
1191   assume "lcm a b = 1"
1192   then show "is_unit a \<and> is_unit b" by auto
1193 next
1194   assume "is_unit a \<and> is_unit b"
1195   hence "a dvd 1" and "b dvd 1" by simp_all
1196   hence "is_unit (lcm a b)" by (rule lcm_least)
1197   hence "lcm a b = normalization_factor (lcm a b)"
1198     by (subst normalization_factor_unit, simp_all)
1199   also have "\<dots> = 1" using is_unit a \<and> is_unit b
1200     by auto
1201   finally show "lcm a b = 1" .
1202 qed
1204 lemma lcm_0_left [simp]:
1205   "lcm 0 a = 0"
1206   by (rule sym, rule lcmI, simp_all)
1208 lemma lcm_0 [simp]:
1209   "lcm a 0 = 0"
1210   by (rule sym, rule lcmI, simp_all)
1212 lemma lcm_unique:
1213   "a dvd d \<and> b dvd d \<and>
1214   normalization_factor d = (if d = 0 then 0 else 1) \<and>
1215   (\<forall>e. a dvd e \<and> b dvd e \<longrightarrow> d dvd e) \<longleftrightarrow> d = lcm a b"
1216   by (rule, auto intro: lcmI simp: lcm_least lcm_zero)
1218 lemma dvd_lcm_I1 [simp]:
1219   "k dvd m \<Longrightarrow> k dvd lcm m n"
1220   by (metis lcm_dvd1 dvd_trans)
1222 lemma dvd_lcm_I2 [simp]:
1223   "k dvd n \<Longrightarrow> k dvd lcm m n"
1224   by (metis lcm_dvd2 dvd_trans)
1226 lemma lcm_1_left [simp]:
1227   "lcm 1 a = a div normalization_factor a"
1228   by (cases "a = 0") (simp, rule sym, rule lcmI, simp_all)
1230 lemma lcm_1_right [simp]:
1231   "lcm a 1 = a div normalization_factor a"
1232   using lcm_1_left [of a] by (simp add: ac_simps)
1234 lemma lcm_coprime:
1235   "gcd a b = 1 \<Longrightarrow> lcm a b = a * b div normalization_factor (a*b)"
1236   by (subst lcm_gcd) simp
1238 lemma lcm_proj1_if_dvd:
1239   "b dvd a \<Longrightarrow> lcm a b = a div normalization_factor a"
1240   by (cases "a = 0") (simp, rule sym, rule lcmI, simp_all)
1242 lemma lcm_proj2_if_dvd:
1243   "a dvd b \<Longrightarrow> lcm a b = b div normalization_factor b"
1244   using lcm_proj1_if_dvd [of a b] by (simp add: ac_simps)
1246 lemma lcm_proj1_iff:
1247   "lcm m n = m div normalization_factor m \<longleftrightarrow> n dvd m"
1248 proof
1249   assume A: "lcm m n = m div normalization_factor m"
1250   show "n dvd m"
1251   proof (cases "m = 0")
1252     assume [simp]: "m \<noteq> 0"
1253     from A have B: "m = lcm m n * normalization_factor m"
1254       by (simp add: unit_eq_div2)
1255     show ?thesis by (subst B, simp)
1256   qed simp
1257 next
1258   assume "n dvd m"
1259   then show "lcm m n = m div normalization_factor m" by (rule lcm_proj1_if_dvd)
1260 qed
1262 lemma lcm_proj2_iff:
1263   "lcm m n = n div normalization_factor n \<longleftrightarrow> m dvd n"
1264   using lcm_proj1_iff [of n m] by (simp add: ac_simps)
1266 lemma euclidean_size_lcm_le1:
1267   assumes "a \<noteq> 0" and "b \<noteq> 0"
1268   shows "euclidean_size a \<le> euclidean_size (lcm a b)"
1269 proof -
1270   have "a dvd lcm a b" by (rule lcm_dvd1)
1271   then obtain c where A: "lcm a b = a * c" unfolding dvd_def by blast
1272   with a \<noteq> 0 and b \<noteq> 0 have "c \<noteq> 0" by (auto simp: lcm_zero)
1273   then show ?thesis by (subst A, intro size_mult_mono)
1274 qed
1276 lemma euclidean_size_lcm_le2:
1277   "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> euclidean_size b \<le> euclidean_size (lcm a b)"
1278   using euclidean_size_lcm_le1 [of b a] by (simp add: ac_simps)
1280 lemma euclidean_size_lcm_less1:
1281   assumes "b \<noteq> 0" and "\<not>b dvd a"
1282   shows "euclidean_size a < euclidean_size (lcm a b)"
1283 proof (rule ccontr)
1284   from assms have "a \<noteq> 0" by auto
1285   assume "\<not>euclidean_size a < euclidean_size (lcm a b)"
1286   with a \<noteq> 0 and b \<noteq> 0 have "euclidean_size (lcm a b) = euclidean_size a"
1287     by (intro le_antisym, simp, intro euclidean_size_lcm_le1)
1288   with assms have "lcm a b dvd a"
1289     by (rule_tac dvd_euclidean_size_eq_imp_dvd) (auto simp: lcm_zero)
1290   hence "b dvd a" by (rule dvd_lcm_D2)
1291   with \<not>b dvd a show False by contradiction
1292 qed
1294 lemma euclidean_size_lcm_less2:
1295   assumes "a \<noteq> 0" and "\<not>a dvd b"
1296   shows "euclidean_size b < euclidean_size (lcm a b)"
1297   using assms euclidean_size_lcm_less1 [of a b] by (simp add: ac_simps)
1299 lemma lcm_mult_unit1:
1300   "is_unit a \<Longrightarrow> lcm (b * a) c = lcm b c"
1301   apply (rule lcmI)
1302   apply (rule dvd_trans[of _ "b * a"], simp, rule lcm_dvd1)
1303   apply (rule lcm_dvd2)
1304   apply (rule lcm_least, simp add: unit_simps, assumption)
1305   apply (subst normalization_factor_lcm, simp add: lcm_zero)
1306   done
1308 lemma lcm_mult_unit2:
1309   "is_unit a \<Longrightarrow> lcm b (c * a) = lcm b c"
1310   using lcm_mult_unit1 [of a c b] by (simp add: ac_simps)
1312 lemma lcm_div_unit1:
1313   "is_unit a \<Longrightarrow> lcm (b div a) c = lcm b c"
1314   by (erule is_unitE [of _ b]) (simp add: lcm_mult_unit1)
1316 lemma lcm_div_unit2:
1317   "is_unit a \<Longrightarrow> lcm b (c div a) = lcm b c"
1318   by (erule is_unitE [of _ c]) (simp add: lcm_mult_unit2)
1320 lemma lcm_left_idem:
1321   "lcm a (lcm a b) = lcm a b"
1322   apply (rule lcmI)
1323   apply simp
1324   apply (subst lcm.assoc [symmetric], rule lcm_dvd2)
1325   apply (rule lcm_least, assumption)
1326   apply (erule (1) lcm_least)
1327   apply (auto simp: lcm_zero)
1328   done
1330 lemma lcm_right_idem:
1331   "lcm (lcm a b) b = lcm a b"
1332   apply (rule lcmI)
1333   apply (subst lcm.assoc, rule lcm_dvd1)
1334   apply (rule lcm_dvd2)
1335   apply (rule lcm_least, erule (1) lcm_least, assumption)
1336   apply (auto simp: lcm_zero)
1337   done
1339 lemma comp_fun_idem_lcm: "comp_fun_idem lcm"
1340 proof
1341   fix a b show "lcm a \<circ> lcm b = lcm b \<circ> lcm a"
1342     by (simp add: fun_eq_iff ac_simps)
1343 next
1344   fix a show "lcm a \<circ> lcm a = lcm a" unfolding o_def
1345     by (intro ext, simp add: lcm_left_idem)
1346 qed
1348 lemma dvd_Lcm [simp]: "a \<in> A \<Longrightarrow> a dvd Lcm A"
1349   and Lcm_dvd [simp]: "(\<forall>a\<in>A. a dvd l') \<Longrightarrow> Lcm A dvd l'"
1350   and normalization_factor_Lcm [simp]:
1351           "normalization_factor (Lcm A) = (if Lcm A = 0 then 0 else 1)"
1352 proof -
1353   have "(\<forall>a\<in>A. a dvd Lcm A) \<and> (\<forall>l'. (\<forall>a\<in>A. a dvd l') \<longrightarrow> Lcm A dvd l') \<and>
1354     normalization_factor (Lcm A) = (if Lcm A = 0 then 0 else 1)" (is ?thesis)
1355   proof (cases "\<exists>l. l \<noteq>  0 \<and> (\<forall>a\<in>A. a dvd l)")
1356     case False
1357     hence "Lcm A = 0" by (auto simp: Lcm_Lcm_eucl Lcm_eucl_def)
1358     with False show ?thesis by auto
1359   next
1360     case True
1361     then obtain l\<^sub>0 where l\<^sub>0_props: "l\<^sub>0 \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l\<^sub>0)" by blast
1362     def n \<equiv> "LEAST n. \<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"
1363     def l \<equiv> "SOME l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"
1364     have "\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"
1365       apply (subst n_def)
1366       apply (rule LeastI[of _ "euclidean_size l\<^sub>0"])
1367       apply (rule exI[of _ l\<^sub>0])
1368       apply (simp add: l\<^sub>0_props)
1369       done
1370     from someI_ex[OF this] have "l \<noteq> 0" and "\<forall>a\<in>A. a dvd l" and "euclidean_size l = n"
1371       unfolding l_def by simp_all
1372     {
1373       fix l' assume "\<forall>a\<in>A. a dvd l'"
1374       with \<forall>a\<in>A. a dvd l have "\<forall>a\<in>A. a dvd gcd l l'" by (auto intro: gcd_greatest)
1375       moreover from l \<noteq> 0 have "gcd l l' \<noteq> 0" by simp
1376       ultimately have "\<exists>b. b \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd b) \<and> euclidean_size b = euclidean_size (gcd l l')"
1377         by (intro exI[of _ "gcd l l'"], auto)
1378       hence "euclidean_size (gcd l l') \<ge> n" by (subst n_def) (rule Least_le)
1379       moreover have "euclidean_size (gcd l l') \<le> n"
1380       proof -
1381         have "gcd l l' dvd l" by simp
1382         then obtain a where "l = gcd l l' * a" unfolding dvd_def by blast
1383         with l \<noteq> 0 have "a \<noteq> 0" by auto
1384         hence "euclidean_size (gcd l l') \<le> euclidean_size (gcd l l' * a)"
1385           by (rule size_mult_mono)
1386         also have "gcd l l' * a = l" using l = gcd l l' * a ..
1387         also note euclidean_size l = n
1388         finally show "euclidean_size (gcd l l') \<le> n" .
1389       qed
1390       ultimately have "euclidean_size l = euclidean_size (gcd l l')"
1391         by (intro le_antisym, simp_all add: euclidean_size l = n)
1392       with l \<noteq> 0 have "l dvd gcd l l'" by (blast intro: dvd_euclidean_size_eq_imp_dvd)
1393       hence "l dvd l'" by (blast dest: dvd_gcd_D2)
1394     }
1396     with (\<forall>a\<in>A. a dvd l) and normalization_factor_is_unit[OF l \<noteq> 0] and l \<noteq> 0
1397       have "(\<forall>a\<in>A. a dvd l div normalization_factor l) \<and>
1398         (\<forall>l'. (\<forall>a\<in>A. a dvd l') \<longrightarrow> l div normalization_factor l dvd l') \<and>
1399         normalization_factor (l div normalization_factor l) =
1400         (if l div normalization_factor l = 0 then 0 else 1)"
1401       by (auto simp: unit_simps)
1402     also from True have "l div normalization_factor l = Lcm A"
1403       by (simp add: Lcm_Lcm_eucl Lcm_eucl_def Let_def n_def l_def)
1404     finally show ?thesis .
1405   qed
1406   note A = this
1408   {fix a assume "a \<in> A" then show "a dvd Lcm A" using A by blast}
1409   {fix l' assume "\<forall>a\<in>A. a dvd l'" then show "Lcm A dvd l'" using A by blast}
1410   from A show "normalization_factor (Lcm A) = (if Lcm A = 0 then 0 else 1)" by blast
1411 qed
1413 lemma LcmI:
1414   "(\<And>a. a\<in>A \<Longrightarrow> a dvd l) \<Longrightarrow> (\<And>l'. (\<forall>a\<in>A. a dvd l') \<Longrightarrow> l dvd l') \<Longrightarrow>
1415       normalization_factor l = (if l = 0 then 0 else 1) \<Longrightarrow> l = Lcm A"
1416   by (intro normed_associated_imp_eq)
1417     (auto intro: Lcm_dvd dvd_Lcm simp: associated_def)
1419 lemma Lcm_subset:
1420   "A \<subseteq> B \<Longrightarrow> Lcm A dvd Lcm B"
1421   by (blast intro: Lcm_dvd dvd_Lcm)
1423 lemma Lcm_Un:
1424   "Lcm (A \<union> B) = lcm (Lcm A) (Lcm B)"
1425   apply (rule lcmI)
1426   apply (blast intro: Lcm_subset)
1427   apply (blast intro: Lcm_subset)
1428   apply (intro Lcm_dvd ballI, elim UnE)
1429   apply (rule dvd_trans, erule dvd_Lcm, assumption)
1430   apply (rule dvd_trans, erule dvd_Lcm, assumption)
1431   apply simp
1432   done
1434 lemma Lcm_1_iff:
1435   "Lcm A = 1 \<longleftrightarrow> (\<forall>a\<in>A. is_unit a)"
1436 proof
1437   assume "Lcm A = 1"
1438   then show "\<forall>a\<in>A. is_unit a" by auto
1439 qed (rule LcmI [symmetric], auto)
1441 lemma Lcm_no_units:
1442   "Lcm A = Lcm (A - {a. is_unit a})"
1443 proof -
1444   have "(A - {a. is_unit a}) \<union> {a\<in>A. is_unit a} = A" by blast
1445   hence "Lcm A = lcm (Lcm (A - {a. is_unit a})) (Lcm {a\<in>A. is_unit a})"
1446     by (simp add: Lcm_Un[symmetric])
1447   also have "Lcm {a\<in>A. is_unit a} = 1" by (simp add: Lcm_1_iff)
1448   finally show ?thesis by simp
1449 qed
1451 lemma Lcm_empty [simp]:
1452   "Lcm {} = 1"
1453   by (simp add: Lcm_1_iff)
1455 lemma Lcm_eq_0 [simp]:
1456   "0 \<in> A \<Longrightarrow> Lcm A = 0"
1457   by (drule dvd_Lcm) simp
1459 lemma Lcm0_iff':
1460   "Lcm A = 0 \<longleftrightarrow> \<not>(\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l))"
1461 proof
1462   assume "Lcm A = 0"
1463   show "\<not>(\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l))"
1464   proof
1465     assume ex: "\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l)"
1466     then obtain l\<^sub>0 where l\<^sub>0_props: "l\<^sub>0 \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l\<^sub>0)" by blast
1467     def n \<equiv> "LEAST n. \<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"
1468     def l \<equiv> "SOME l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"
1469     have "\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"
1470       apply (subst n_def)
1471       apply (rule LeastI[of _ "euclidean_size l\<^sub>0"])
1472       apply (rule exI[of _ l\<^sub>0])
1473       apply (simp add: l\<^sub>0_props)
1474       done
1475     from someI_ex[OF this] have "l \<noteq> 0" unfolding l_def by simp_all
1476     hence "l div normalization_factor l \<noteq> 0" by simp
1477     also from ex have "l div normalization_factor l = Lcm A"
1478        by (simp only: Lcm_Lcm_eucl Lcm_eucl_def n_def l_def if_True Let_def)
1479     finally show False using Lcm A = 0 by contradiction
1480   qed
1481 qed (simp only: Lcm_Lcm_eucl Lcm_eucl_def if_False)
1483 lemma Lcm0_iff [simp]:
1484   "finite A \<Longrightarrow> Lcm A = 0 \<longleftrightarrow> 0 \<in> A"
1485 proof -
1486   assume "finite A"
1487   have "0 \<in> A \<Longrightarrow> Lcm A = 0"  by (intro dvd_0_left dvd_Lcm)
1488   moreover {
1489     assume "0 \<notin> A"
1490     hence "\<Prod>A \<noteq> 0"
1491       apply (induct rule: finite_induct[OF finite A])
1492       apply simp
1493       apply (subst setprod.insert, assumption, assumption)
1494       apply (rule no_zero_divisors)
1495       apply blast+
1496       done
1497     moreover from finite A have "\<forall>a\<in>A. a dvd \<Prod>A" by blast
1498     ultimately have "\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l)" by blast
1499     with Lcm0_iff' have "Lcm A \<noteq> 0" by simp
1500   }
1501   ultimately show "Lcm A = 0 \<longleftrightarrow> 0 \<in> A" by blast
1502 qed
1504 lemma Lcm_no_multiple:
1505   "(\<forall>m. m \<noteq> 0 \<longrightarrow> (\<exists>a\<in>A. \<not>a dvd m)) \<Longrightarrow> Lcm A = 0"
1506 proof -
1507   assume "\<forall>m. m \<noteq> 0 \<longrightarrow> (\<exists>a\<in>A. \<not>a dvd m)"
1508   hence "\<not>(\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l))" by blast
1509   then show "Lcm A = 0" by (simp only: Lcm_Lcm_eucl Lcm_eucl_def if_False)
1510 qed
1512 lemma Lcm_insert [simp]:
1513   "Lcm (insert a A) = lcm a (Lcm A)"
1514 proof (rule lcmI)
1515   fix l assume "a dvd l" and "Lcm A dvd l"
1516   hence "\<forall>a\<in>A. a dvd l" by (blast intro: dvd_trans dvd_Lcm)
1517   with a dvd l show "Lcm (insert a A) dvd l" by (force intro: Lcm_dvd)
1518 qed (auto intro: Lcm_dvd dvd_Lcm)
1520 lemma Lcm_finite:
1521   assumes "finite A"
1522   shows "Lcm A = Finite_Set.fold lcm 1 A"
1523   by (induct rule: finite.induct[OF finite A])
1524     (simp_all add: comp_fun_idem.fold_insert_idem[OF comp_fun_idem_lcm])
1526 lemma Lcm_set [code_unfold]:
1527   "Lcm (set xs) = fold lcm xs 1"
1528   using comp_fun_idem.fold_set_fold[OF comp_fun_idem_lcm] Lcm_finite by (simp add: ac_simps)
1530 lemma Lcm_singleton [simp]:
1531   "Lcm {a} = a div normalization_factor a"
1532   by simp
1534 lemma Lcm_2 [simp]:
1535   "Lcm {a,b} = lcm a b"
1536   by (simp only: Lcm_insert Lcm_empty lcm_1_right)
1537     (cases "b = 0", simp, rule lcm_div_unit2, simp)
1539 lemma Lcm_coprime:
1540   assumes "finite A" and "A \<noteq> {}"
1541   assumes "\<And>a b. a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<noteq> b \<Longrightarrow> gcd a b = 1"
1542   shows "Lcm A = \<Prod>A div normalization_factor (\<Prod>A)"
1543 using assms proof (induct rule: finite_ne_induct)
1544   case (insert a A)
1545   have "Lcm (insert a A) = lcm a (Lcm A)" by simp
1546   also from insert have "Lcm A = \<Prod>A div normalization_factor (\<Prod>A)" by blast
1547   also have "lcm a \<dots> = lcm a (\<Prod>A)" by (cases "\<Prod>A = 0") (simp_all add: lcm_div_unit2)
1548   also from insert have "gcd a (\<Prod>A) = 1" by (subst gcd.commute, intro setprod_coprime) auto
1549   with insert have "lcm a (\<Prod>A) = \<Prod>(insert a A) div normalization_factor (\<Prod>(insert a A))"
1550     by (simp add: lcm_coprime)
1551   finally show ?case .
1552 qed simp
1554 lemma Lcm_coprime':
1555   "card A \<noteq> 0 \<Longrightarrow> (\<And>a b. a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<noteq> b \<Longrightarrow> gcd a b = 1)
1556     \<Longrightarrow> Lcm A = \<Prod>A div normalization_factor (\<Prod>A)"
1557   by (rule Lcm_coprime) (simp_all add: card_eq_0_iff)
1559 lemma Gcd_Lcm:
1560   "Gcd A = Lcm {d. \<forall>a\<in>A. d dvd a}"
1561   by (simp add: Gcd_Gcd_eucl Lcm_Lcm_eucl Gcd_eucl_def)
1563 lemma Gcd_dvd [simp]: "a \<in> A \<Longrightarrow> Gcd A dvd a"
1564   and dvd_Gcd [simp]: "(\<forall>a\<in>A. g' dvd a) \<Longrightarrow> g' dvd Gcd A"
1565   and normalization_factor_Gcd [simp]:
1566     "normalization_factor (Gcd A) = (if Gcd A = 0 then 0 else 1)"
1567 proof -
1568   fix a assume "a \<in> A"
1569   hence "Lcm {d. \<forall>a\<in>A. d dvd a} dvd a" by (intro Lcm_dvd) blast
1570   then show "Gcd A dvd a" by (simp add: Gcd_Lcm)
1571 next
1572   fix g' assume "\<forall>a\<in>A. g' dvd a"
1573   hence "g' dvd Lcm {d. \<forall>a\<in>A. d dvd a}" by (intro dvd_Lcm) blast
1574   then show "g' dvd Gcd A" by (simp add: Gcd_Lcm)
1575 next
1576   show "normalization_factor (Gcd A) = (if Gcd A = 0 then 0 else 1)"
1577     by (simp add: Gcd_Lcm)
1578 qed
1580 lemma GcdI:
1581   "(\<And>a. a\<in>A \<Longrightarrow> l dvd a) \<Longrightarrow> (\<And>l'. (\<forall>a\<in>A. l' dvd a) \<Longrightarrow> l' dvd l) \<Longrightarrow>
1582     normalization_factor l = (if l = 0 then 0 else 1) \<Longrightarrow> l = Gcd A"
1583   by (intro normed_associated_imp_eq)
1584     (auto intro: Gcd_dvd dvd_Gcd simp: associated_def)
1586 lemma Lcm_Gcd:
1587   "Lcm A = Gcd {m. \<forall>a\<in>A. a dvd m}"
1588   by (rule LcmI[symmetric]) (auto intro: dvd_Gcd Gcd_dvd)
1590 lemma Gcd_0_iff:
1591   "Gcd A = 0 \<longleftrightarrow> A \<subseteq> {0}"
1592   apply (rule iffI)
1593   apply (rule subsetI, drule Gcd_dvd, simp)
1594   apply (auto intro: GcdI[symmetric])
1595   done
1597 lemma Gcd_empty [simp]:
1598   "Gcd {} = 0"
1599   by (simp add: Gcd_0_iff)
1601 lemma Gcd_1:
1602   "1 \<in> A \<Longrightarrow> Gcd A = 1"
1603   by (intro GcdI[symmetric]) (auto intro: Gcd_dvd dvd_Gcd)
1605 lemma Gcd_insert [simp]:
1606   "Gcd (insert a A) = gcd a (Gcd A)"
1607 proof (rule gcdI)
1608   fix l assume "l dvd a" and "l dvd Gcd A"
1609   hence "\<forall>a\<in>A. l dvd a" by (blast intro: dvd_trans Gcd_dvd)
1610   with l dvd a show "l dvd Gcd (insert a A)" by (force intro: Gcd_dvd)
1611 qed auto
1613 lemma Gcd_finite:
1614   assumes "finite A"
1615   shows "Gcd A = Finite_Set.fold gcd 0 A"
1616   by (induct rule: finite.induct[OF finite A])
1617     (simp_all add: comp_fun_idem.fold_insert_idem[OF comp_fun_idem_gcd])
1619 lemma Gcd_set [code_unfold]:
1620   "Gcd (set xs) = fold gcd xs 0"
1621   using comp_fun_idem.fold_set_fold[OF comp_fun_idem_gcd] Gcd_finite by (simp add: ac_simps)
1623 lemma Gcd_singleton [simp]: "Gcd {a} = a div normalization_factor a"
1624   by (simp add: gcd_0)
1626 lemma Gcd_2 [simp]: "Gcd {a,b} = gcd a b"
1627   by (simp only: Gcd_insert Gcd_empty gcd_0) (cases "b = 0", simp, rule gcd_div_unit2, simp)
1629 subclass semiring_gcd
1630   by unfold_locales (simp_all add: gcd_greatest_iff)
1632 end
1634 text {*
1635   A Euclidean ring is a Euclidean semiring with additive inverses. It provides a
1636   few more lemmas; in particular, Bezout's lemma holds for any Euclidean ring.
1637 *}
1639 class euclidean_ring = euclidean_semiring + idom
1641 class euclidean_ring_gcd = euclidean_semiring_gcd + idom
1642 begin
1644 subclass euclidean_ring ..
1646 subclass ring_gcd ..
1648 lemma gcd_neg1 [simp]:
1649   "gcd (-a) b = gcd a b"
1650   by (rule sym, rule gcdI, simp_all add: gcd_greatest)
1652 lemma gcd_neg2 [simp]:
1653   "gcd a (-b) = gcd a b"
1654   by (rule sym, rule gcdI, simp_all add: gcd_greatest)
1656 lemma gcd_neg_numeral_1 [simp]:
1657   "gcd (- numeral n) a = gcd (numeral n) a"
1658   by (fact gcd_neg1)
1660 lemma gcd_neg_numeral_2 [simp]:
1661   "gcd a (- numeral n) = gcd a (numeral n)"
1662   by (fact gcd_neg2)
1664 lemma gcd_diff1: "gcd (m - n) n = gcd m n"
1665   by (subst diff_conv_add_uminus, subst gcd_neg2[symmetric],  subst gcd_add1, simp)
1667 lemma gcd_diff2: "gcd (n - m) n = gcd m n"
1668   by (subst gcd_neg1[symmetric], simp only: minus_diff_eq gcd_diff1)
1670 lemma coprime_minus_one [simp]: "gcd (n - 1) n = 1"
1671 proof -
1672   have "gcd (n - 1) n = gcd n (n - 1)" by (fact gcd.commute)
1673   also have "\<dots> = gcd ((n - 1) + 1) (n - 1)" by simp
1674   also have "\<dots> = 1" by (rule coprime_plus_one)
1675   finally show ?thesis .
1676 qed
1678 lemma lcm_neg1 [simp]: "lcm (-a) b = lcm a b"
1679   by (rule sym, rule lcmI, simp_all add: lcm_least lcm_zero)
1681 lemma lcm_neg2 [simp]: "lcm a (-b) = lcm a b"
1682   by (rule sym, rule lcmI, simp_all add: lcm_least lcm_zero)
1684 lemma lcm_neg_numeral_1 [simp]: "lcm (- numeral n) a = lcm (numeral n) a"
1685   by (fact lcm_neg1)
1687 lemma lcm_neg_numeral_2 [simp]: "lcm a (- numeral n) = lcm a (numeral n)"
1688   by (fact lcm_neg2)
1690 function euclid_ext :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a" where
1691   "euclid_ext a b =
1692      (if b = 0 then
1693         let c = 1 div normalization_factor a in (c, 0, a * c)
1694       else
1695         case euclid_ext b (a mod b) of
1696             (s,t,c) \<Rightarrow> (t, s - t * (a div b), c))"
1697   by (pat_completeness, simp)
1698   termination by (relation "measure (euclidean_size \<circ> snd)", simp_all)
1700 declare euclid_ext.simps [simp del]
1702 lemma euclid_ext_0:
1703   "euclid_ext a 0 = (1 div normalization_factor a, 0, a div normalization_factor a)"
1704   by (subst euclid_ext.simps) (simp add: Let_def)
1706 lemma euclid_ext_non_0:
1707   "b \<noteq> 0 \<Longrightarrow> euclid_ext a b = (case euclid_ext b (a mod b) of
1708     (s,t,c) \<Rightarrow> (t, s - t * (a div b), c))"
1709   by (subst euclid_ext.simps) simp
1711 definition euclid_ext' :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a"
1712 where
1713   "euclid_ext' a b = (case euclid_ext a b of (s, t, _) \<Rightarrow> (s, t))"
1715 lemma euclid_ext_gcd [simp]:
1716   "(case euclid_ext a b of (_,_,t) \<Rightarrow> t) = gcd a b"
1717 proof (induct a b rule: euclid_ext.induct)
1718   case (1 a b)
1719   then show ?case
1720   proof (cases "b = 0")
1721     case True
1722       then show ?thesis by
1723         (simp add: euclid_ext_0 unit_div mult_ac unit_simps gcd_0)
1724     next
1725     case False with 1 show ?thesis
1726       by (simp add: euclid_ext_non_0 ac_simps split: prod.split prod.split_asm)
1727     qed
1728 qed
1730 lemma euclid_ext_gcd' [simp]:
1731   "euclid_ext a b = (r, s, t) \<Longrightarrow> t = gcd a b"
1732   by (insert euclid_ext_gcd[of a b], drule (1) subst, simp)
1734 lemma euclid_ext_correct:
1735   "case euclid_ext a b of (s,t,c) \<Rightarrow> s*a + t*b = c"
1736 proof (induct a b rule: euclid_ext.induct)
1737   case (1 a b)
1738   show ?case
1739   proof (cases "b = 0")
1740     case True
1741     then show ?thesis by (simp add: euclid_ext_0 mult_ac)
1742   next
1743     case False
1744     obtain s t c where stc: "euclid_ext b (a mod b) = (s,t,c)"
1745       by (cases "euclid_ext b (a mod b)", blast)
1746     from 1 have "c = s * b + t * (a mod b)" by (simp add: stc False)
1747     also have "... = t*((a div b)*b + a mod b) + (s - t * (a div b))*b"
1748       by (simp add: algebra_simps)
1749     also have "(a div b)*b + a mod b = a" using mod_div_equality .
1750     finally show ?thesis
1751       by (subst euclid_ext.simps, simp add: False stc)
1752     qed
1753 qed
1755 lemma euclid_ext'_correct:
1756   "fst (euclid_ext' a b) * a + snd (euclid_ext' a b) * b = gcd a b"
1757 proof-
1758   obtain s t c where "euclid_ext a b = (s,t,c)"
1759     by (cases "euclid_ext a b", blast)
1760   with euclid_ext_correct[of a b] euclid_ext_gcd[of a b]
1761     show ?thesis unfolding euclid_ext'_def by simp
1762 qed
1764 lemma bezout: "\<exists>s t. s * a + t * b = gcd a b"
1765   using euclid_ext'_correct by blast
1767 lemma euclid_ext'_0 [simp]: "euclid_ext' a 0 = (1 div normalization_factor a, 0)"
1768   by (simp add: bezw_def euclid_ext'_def euclid_ext_0)
1770 lemma euclid_ext'_non_0: "b \<noteq> 0 \<Longrightarrow> euclid_ext' a b = (snd (euclid_ext' b (a mod b)),
1771   fst (euclid_ext' b (a mod b)) - snd (euclid_ext' b (a mod b)) * (a div b))"
1772   by (cases "euclid_ext b (a mod b)")
1773     (simp add: euclid_ext'_def euclid_ext_non_0)
1775 end
1777 instantiation nat :: euclidean_semiring
1778 begin
1780 definition [simp]:
1781   "euclidean_size_nat = (id :: nat \<Rightarrow> nat)"
1783 definition [simp]:
1784   "normalization_factor_nat (n::nat) = (if n = 0 then 0 else 1 :: nat)"
1786 instance proof
1787 qed simp_all
1789 end
1791 instantiation int :: euclidean_ring
1792 begin
1794 definition [simp]:
1795   "euclidean_size_int = (nat \<circ> abs :: int \<Rightarrow> nat)"
1797 definition [simp]:
1798   "normalization_factor_int = (sgn :: int \<Rightarrow> int)"
1800 instance proof
1801   case goal2 then show ?case by (auto simp add: abs_mult nat_mult_distrib)
1802 next
1803   case goal3 then show ?case by (simp add: zsgn_def)
1804 next
1805   case goal5 then show ?case by (auto simp: zsgn_def)
1806 next
1807   case goal6 then show ?case by (auto split: abs_split simp: zsgn_def)
1808 qed (auto simp: sgn_times split: abs_split)
1810 end
1812 end