src/HOL/Number_Theory/Euclidean_Algorithm.thy
 author haftmann Fri Jun 12 21:53:05 2015 +0200 (2015-06-12) changeset 60438 e1c345094813 parent 60437 63edc650cf67 child 60439 b765e08f8bc0 permissions -rw-r--r--
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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 mult_cancel_right [simp]:
13   "a * c = b * c \<longleftrightarrow> c = 0 \<or> a = b"
14 proof (cases "c = 0")
15   case True then show ?thesis by simp
16 next
17   case False
18   { assume "a * c = b * c"
19     then have "a * c div c = b * c div c"
20       by simp
21     with False have "a = b"
22       by simp
23   } then show ?thesis by auto
24 qed
26 lemma mult_cancel_left [simp]:
27   "c * a = c * b \<longleftrightarrow> c = 0 \<or> a = b"
28   using mult_cancel_right [of a c b] by (simp add: ac_simps)
30 end
32 context semidom_divide
33 begin
35 lemma div_self [simp]:
36   assumes "a \<noteq> 0"
37   shows "a div a = 1"
38   using assms nonzero_mult_divide_cancel_left [of a 1] by simp
40 lemma dvd_div_mult_self [simp]:
41   "a dvd b \<Longrightarrow> b div a * a = b"
42   by (cases "a = 0") (auto elim: dvdE simp add: ac_simps)
44 lemma dvd_mult_div_cancel [simp]:
45   "a dvd b \<Longrightarrow> a * (b div a) = b"
46   using dvd_div_mult_self [of a b] by (simp add: ac_simps)
48 lemma div_mult_swap:
49   assumes "c dvd b"
50   shows "a * (b div c) = (a * b) div c"
51 proof (cases "c = 0")
52   case True then show ?thesis by simp
53 next
54   case False from assms obtain d where "b = c * d" ..
55   moreover from False have "a * divide (d * c) c = ((a * d) * c) div c"
56     by simp
57   ultimately show ?thesis by (simp add: ac_simps)
58 qed
60 lemma dvd_div_mult:
61   assumes "c dvd b"
62   shows "b div c * a = (b * a) div c"
63   using assms div_mult_swap [of c b a] by (simp add: ac_simps)
66 text \<open>Units: invertible elements in a ring\<close>
68 abbreviation is_unit :: "'a \<Rightarrow> bool"
69 where
70   "is_unit a \<equiv> a dvd 1"
72 lemma not_is_unit_0 [simp]:
73   "\<not> is_unit 0"
74   by simp
76 lemma unit_imp_dvd [dest]:
77   "is_unit b \<Longrightarrow> b dvd a"
78   by (rule dvd_trans [of _ 1]) simp_all
80 lemma unit_dvdE:
81   assumes "is_unit a"
82   obtains c where "a \<noteq> 0" and "b = a * c"
83 proof -
84   from assms have "a dvd b" by auto
85   then obtain c where "b = a * c" ..
86   moreover from assms have "a \<noteq> 0" by auto
87   ultimately show thesis using that by blast
88 qed
90 lemma dvd_unit_imp_unit:
91   "a dvd b \<Longrightarrow> is_unit b \<Longrightarrow> is_unit a"
92   by (rule dvd_trans)
94 lemma unit_div_1_unit [simp, intro]:
95   assumes "is_unit a"
96   shows "is_unit (1 div a)"
97 proof -
98   from assms have "1 = 1 div a * a" by simp
99   then show "is_unit (1 div a)" by (rule dvdI)
100 qed
102 lemma is_unitE [elim?]:
103   assumes "is_unit a"
104   obtains b where "a \<noteq> 0" and "b \<noteq> 0"
105     and "is_unit b" and "1 div a = b" and "1 div b = a"
106     and "a * b = 1" and "c div a = c * b"
107 proof (rule that)
108   def b \<equiv> "1 div a"
109   then show "1 div a = b" by simp
110   from b_def is_unit a show "is_unit b" by simp
111   from is_unit a and is_unit b show "a \<noteq> 0" and "b \<noteq> 0" by auto
112   from b_def is_unit a show "a * b = 1" by simp
113   then have "1 = a * b" ..
114   with b_def b \<noteq> 0 show "1 div b = a" by simp
115   from is_unit a have "a dvd c" ..
116   then obtain d where "c = a * d" ..
117   with a \<noteq> 0 a * b = 1 show "c div a = c * b"
118     by (simp add: mult.assoc mult.left_commute [of a])
119 qed
121 lemma unit_prod [intro]:
122   "is_unit a \<Longrightarrow> is_unit b \<Longrightarrow> is_unit (a * b)"
123   by (subst mult_1_left [of 1, symmetric]) (rule mult_dvd_mono)
125 lemma unit_div [intro]:
126   "is_unit a \<Longrightarrow> is_unit b \<Longrightarrow> is_unit (a div b)"
127   by (erule is_unitE [of b a]) (simp add: ac_simps unit_prod)
129 lemma mult_unit_dvd_iff:
130   assumes "is_unit b"
131   shows "a * b dvd c \<longleftrightarrow> a dvd c"
132 proof
133   assume "a * b dvd c"
134   with assms show "a dvd c"
135     by (simp add: dvd_mult_left)
136 next
137   assume "a dvd c"
138   then obtain k where "c = a * k" ..
139   with assms have "c = (a * b) * (1 div b * k)"
140     by (simp add: mult_ac)
141   then show "a * b dvd c" by (rule dvdI)
142 qed
144 lemma dvd_mult_unit_iff:
145   assumes "is_unit b"
146   shows "a dvd c * b \<longleftrightarrow> a dvd c"
147 proof
148   assume "a dvd c * b"
149   with assms have "c * b dvd c * (b * (1 div b))"
150     by (subst mult_assoc [symmetric]) simp
151   also from is_unit b have "b * (1 div b) = 1" by (rule is_unitE) simp
152   finally have "c * b dvd c" by simp
153   with a dvd c * b show "a dvd c" by (rule dvd_trans)
154 next
155   assume "a dvd c"
156   then show "a dvd c * b" by simp
157 qed
159 lemma div_unit_dvd_iff:
160   "is_unit b \<Longrightarrow> a div b dvd c \<longleftrightarrow> a dvd c"
161   by (erule is_unitE [of _ a]) (auto simp add: mult_unit_dvd_iff)
163 lemma dvd_div_unit_iff:
164   "is_unit b \<Longrightarrow> a dvd c div b \<longleftrightarrow> a dvd c"
165   by (erule is_unitE [of _ c]) (simp add: dvd_mult_unit_iff)
167 lemmas unit_dvd_iff = mult_unit_dvd_iff div_unit_dvd_iff
168   dvd_mult_unit_iff dvd_div_unit_iff -- \<open>FIXME consider fact collection\<close>
170 lemma unit_mult_div_div [simp]:
171   "is_unit a \<Longrightarrow> b * (1 div a) = b div a"
172   by (erule is_unitE [of _ b]) simp
174 lemma unit_div_mult_self [simp]:
175   "is_unit a \<Longrightarrow> b div a * a = b"
176   by (rule dvd_div_mult_self) auto
178 lemma unit_div_1_div_1 [simp]:
179   "is_unit a \<Longrightarrow> 1 div (1 div a) = a"
180   by (erule is_unitE) simp
182 lemma unit_div_mult_swap:
183   "is_unit c \<Longrightarrow> a * (b div c) = (a * b) div c"
184   by (erule unit_dvdE [of _ b]) (simp add: mult.left_commute [of _ c])
186 lemma unit_div_commute:
187   "is_unit b \<Longrightarrow> (a div b) * c = (a * c) div b"
188   using unit_div_mult_swap [of b c a] by (simp add: ac_simps)
190 lemma unit_eq_div1:
191   "is_unit b \<Longrightarrow> a div b = c \<longleftrightarrow> a = c * b"
192   by (auto elim: is_unitE)
194 lemma unit_eq_div2:
195   "is_unit b \<Longrightarrow> a = c div b \<longleftrightarrow> a * b = c"
196   using unit_eq_div1 [of b c a] by auto
198 lemma unit_mult_left_cancel:
199   assumes "is_unit a"
200   shows "a * b = a * c \<longleftrightarrow> b = c" (is "?P \<longleftrightarrow> ?Q")
201   using assms mult_cancel_left [of a b c] by auto
203 lemma unit_mult_right_cancel:
204   "is_unit a \<Longrightarrow> b * a = c * a \<longleftrightarrow> b = c"
205   using unit_mult_left_cancel [of a b c] by (auto simp add: ac_simps)
207 lemma unit_div_cancel:
208   assumes "is_unit a"
209   shows "b div a = c div a \<longleftrightarrow> b = c"
210 proof -
211   from assms have "is_unit (1 div a)" by simp
212   then have "b * (1 div a) = c * (1 div a) \<longleftrightarrow> b = c"
213     by (rule unit_mult_right_cancel)
214   with assms show ?thesis by simp
215 qed
218 text \<open>Associated elements in a ring – an equivalence relation induced by the quasi-order divisibility \<close>
220 definition associated :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
221 where
222   "associated a b \<longleftrightarrow> a dvd b \<and> b dvd a"
224 lemma associatedI:
225   "a dvd b \<Longrightarrow> b dvd a \<Longrightarrow> associated a b"
226   by (simp add: associated_def)
228 lemma associatedD1:
229   "associated a b \<Longrightarrow> a dvd b"
230   by (simp add: associated_def)
232 lemma associatedD2:
233   "associated a b \<Longrightarrow> b dvd a"
234   by (simp add: associated_def)
236 lemma associated_refl [simp]:
237   "associated a a"
238   by (auto intro: associatedI)
240 lemma associated_sym:
241   "associated b a \<longleftrightarrow> associated a b"
242   by (auto intro: associatedI dest: associatedD1 associatedD2)
244 lemma associated_trans:
245   "associated a b \<Longrightarrow> associated b c \<Longrightarrow> associated a c"
246   by (auto intro: associatedI dvd_trans dest: associatedD1 associatedD2)
248 lemma equivp_associated:
249   "equivp associated"
250 proof (rule equivpI)
251   show "reflp associated"
252     by (rule reflpI) simp
253   show "symp associated"
254     by (rule sympI) (simp add: associated_sym)
255   show "transp associated"
256     by (rule transpI) (fact associated_trans)
257 qed
259 lemma associated_0 [simp]:
260   "associated 0 b \<longleftrightarrow> b = 0"
261   "associated a 0 \<longleftrightarrow> a = 0"
262   by (auto dest: associatedD1 associatedD2)
264 lemma associated_unit:
265   "associated a b \<Longrightarrow> is_unit a \<Longrightarrow> is_unit b"
266   using dvd_unit_imp_unit by (auto dest!: associatedD1 associatedD2)
268 lemma is_unit_associatedI:
269   assumes "is_unit c" and "a = c * b"
270   shows "associated a b"
271 proof (rule associatedI)
272   from a = c * b show "b dvd a" by auto
273   from is_unit c obtain d where "c * d = 1" by (rule is_unitE)
274   moreover from a = c * b have "d * a = d * (c * b)" by simp
275   ultimately have "b = a * d" by (simp add: ac_simps)
276   then show "a dvd b" ..
277 qed
279 lemma associated_is_unitE:
280   assumes "associated a b"
281   obtains c where "is_unit c" and "a = c * b"
282 proof (cases "b = 0")
283   case True with assms have "is_unit 1" and "a = 1 * b" by simp_all
284   with that show thesis .
285 next
286   case False
287   from assms have "a dvd b" and "b dvd a" by (auto dest: associatedD1 associatedD2)
288   then obtain c d where "b = a * d" and "a = b * c" by (blast elim: dvdE)
289   then have "a = c * b" and "(c * d) * b = 1 * b" by (simp_all add: ac_simps)
290   with False have "c * d = 1" using mult_cancel_right [of "c * d" b 1] by simp
291   then have "is_unit c" by auto
292   with a = c * b that show thesis by blast
293 qed
295 lemmas unit_simps = mult_unit_dvd_iff div_unit_dvd_iff dvd_mult_unit_iff
296   dvd_div_unit_iff unit_div_mult_swap unit_div_commute
297   unit_mult_left_cancel unit_mult_right_cancel unit_div_cancel
298   unit_eq_div1 unit_eq_div2
300 end
302 lemma is_unit_int:
303   "is_unit (k::int) \<longleftrightarrow> k = 1 \<or> k = - 1"
304   by auto
307 text {*
308   A Euclidean semiring is a semiring upon which the Euclidean algorithm can be
309   implemented. It must provide:
310   \begin{itemize}
311   \item division with remainder
312   \item a size function such that @{term "size (a mod b) < size b"}
313         for any @{term "b \<noteq> 0"}
314   \item a normalization factor such that two associated numbers are equal iff
315         they are the same when divd by their normalization factors.
316   \end{itemize}
317   The existence of these functions makes it possible to derive gcd and lcm functions
318   for any Euclidean semiring.
319 *}
320 class euclidean_semiring = semiring_div +
321   fixes euclidean_size :: "'a \<Rightarrow> nat"
322   fixes normalization_factor :: "'a \<Rightarrow> 'a"
323   assumes mod_size_less [simp]:
324     "b \<noteq> 0 \<Longrightarrow> euclidean_size (a mod b) < euclidean_size b"
325   assumes size_mult_mono:
326     "b \<noteq> 0 \<Longrightarrow> euclidean_size (a * b) \<ge> euclidean_size a"
327   assumes normalization_factor_is_unit [intro,simp]:
328     "a \<noteq> 0 \<Longrightarrow> is_unit (normalization_factor a)"
329   assumes normalization_factor_mult: "normalization_factor (a * b) =
330     normalization_factor a * normalization_factor b"
331   assumes normalization_factor_unit: "is_unit a \<Longrightarrow> normalization_factor a = a"
332   assumes normalization_factor_0 [simp]: "normalization_factor 0 = 0"
333 begin
335 lemma normalization_factor_dvd [simp]:
336   "a \<noteq> 0 \<Longrightarrow> normalization_factor a dvd b"
337   by (rule unit_imp_dvd, simp)
339 lemma normalization_factor_1 [simp]:
340   "normalization_factor 1 = 1"
341   by (simp add: normalization_factor_unit)
343 lemma normalization_factor_0_iff [simp]:
344   "normalization_factor a = 0 \<longleftrightarrow> a = 0"
345 proof
346   assume "normalization_factor a = 0"
347   hence "\<not> is_unit (normalization_factor a)"
348     by simp
349   then show "a = 0" by auto
350 qed simp
352 lemma normalization_factor_pow:
353   "normalization_factor (a ^ n) = normalization_factor a ^ n"
354   by (induct n) (simp_all add: normalization_factor_mult power_Suc2)
356 lemma normalization_correct [simp]:
357   "normalization_factor (a div normalization_factor a) = (if a = 0 then 0 else 1)"
358 proof (cases "a = 0", simp)
359   assume "a \<noteq> 0"
360   let ?nf = "normalization_factor"
361   from normalization_factor_is_unit[OF a \<noteq> 0] have "?nf a \<noteq> 0"
362     by auto
363   have "?nf (a div ?nf a) * ?nf (?nf a) = ?nf (a div ?nf a * ?nf a)"
364     by (simp add: normalization_factor_mult)
365   also have "a div ?nf a * ?nf a = a" using a \<noteq> 0
366     by simp
367   also have "?nf (?nf a) = ?nf a" using a \<noteq> 0
368     normalization_factor_is_unit normalization_factor_unit by simp
369   finally have "normalization_factor (a div normalization_factor a) = 1"
370     using ?nf a \<noteq> 0 by (metis div_mult_self2_is_id div_self)
371   with a \<noteq> 0 show ?thesis by simp
372 qed
374 lemma normalization_0_iff [simp]:
375   "a div normalization_factor a = 0 \<longleftrightarrow> a = 0"
376   by (cases "a = 0", simp, subst unit_eq_div1, blast, simp)
378 lemma mult_div_normalization [simp]:
379   "b * (1 div normalization_factor a) = b div normalization_factor a"
380   by (cases "a = 0") simp_all
382 lemma associated_iff_normed_eq:
383   "associated a b \<longleftrightarrow> a div normalization_factor a = b div normalization_factor b"
384 proof (cases "b = 0", simp, cases "a = 0", metis associated_0(1) normalization_0_iff, rule iffI)
385   let ?nf = normalization_factor
386   assume "a \<noteq> 0" "b \<noteq> 0" "a div ?nf a = b div ?nf b"
387   hence "a = b * (?nf a div ?nf b)"
388     apply (subst (asm) unit_eq_div1, blast, subst (asm) unit_div_commute, blast)
389     apply (subst div_mult_swap, simp, simp)
390     done
391   with a \<noteq> 0 b \<noteq> 0 have "\<exists>c. is_unit c \<and> a = c * b"
392     by (intro exI[of _ "?nf a div ?nf b"], force simp: mult_ac)
393   then obtain c where "is_unit c" and "a = c * b" by blast
394   then show "associated a b" by (rule is_unit_associatedI)
395 next
396   let ?nf = normalization_factor
397   assume "a \<noteq> 0" "b \<noteq> 0" "associated a b"
398   then obtain c where "is_unit c" and "a = c * b" by (blast elim: associated_is_unitE)
399   then show "a div ?nf a = b div ?nf b"
400     apply (simp only: a = c * b normalization_factor_mult normalization_factor_unit)
401     apply (rule div_mult_mult1, force)
402     done
403   qed
405 lemma normed_associated_imp_eq:
406   "associated a b \<Longrightarrow> normalization_factor a \<in> {0, 1} \<Longrightarrow> normalization_factor b \<in> {0, 1} \<Longrightarrow> a = b"
407   by (simp add: associated_iff_normed_eq, elim disjE, simp_all)
409 lemmas normalization_factor_dvd_iff [simp] =
410   unit_dvd_iff [OF normalization_factor_is_unit]
412 lemma euclidean_division:
413   fixes a :: 'a and b :: 'a
414   assumes "b \<noteq> 0"
415   obtains s and t where "a = s * b + t"
416     and "euclidean_size t < euclidean_size b"
417 proof -
418   from div_mod_equality[of a b 0]
419      have "a = a div b * b + a mod b" by simp
420   with that and assms show ?thesis by force
421 qed
423 lemma dvd_euclidean_size_eq_imp_dvd:
424   assumes "a \<noteq> 0" and b_dvd_a: "b dvd a" and size_eq: "euclidean_size a = euclidean_size b"
425   shows "a dvd b"
426 proof (subst dvd_eq_mod_eq_0, rule ccontr)
427   assume "b mod a \<noteq> 0"
428   from b_dvd_a have b_dvd_mod: "b dvd b mod a" by (simp add: dvd_mod_iff)
429   from b_dvd_mod obtain c where "b mod a = b * c" unfolding dvd_def by blast
430     with b mod a \<noteq> 0 have "c \<noteq> 0" by auto
431   with b mod a = b * c have "euclidean_size (b mod a) \<ge> euclidean_size b"
432       using size_mult_mono by force
433   moreover from a \<noteq> 0 have "euclidean_size (b mod a) < euclidean_size a"
434       using mod_size_less by blast
435   ultimately show False using size_eq by simp
436 qed
438 function gcd_eucl :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
439 where
440   "gcd_eucl a b = (if b = 0 then a div normalization_factor a else gcd_eucl b (a mod b))"
441   by (pat_completeness, simp)
442 termination by (relation "measure (euclidean_size \<circ> snd)", simp_all)
444 declare gcd_eucl.simps [simp del]
446 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"
447 proof (induct a b rule: gcd_eucl.induct)
448   case ("1" m n)
449     then show ?case by (cases "n = 0") auto
450 qed
452 definition lcm_eucl :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
453 where
454   "lcm_eucl a b = a * b div (gcd_eucl a b * normalization_factor (a * b))"
456   (* Somewhat complicated definition of Lcm that has the advantage of working
457      for infinite sets as well *)
459 definition Lcm_eucl :: "'a set \<Rightarrow> 'a"
460 where
461   "Lcm_eucl A = (if \<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) then
462      let l = SOME l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l =
463        (LEAST n. \<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n)
464        in l div normalization_factor l
465       else 0)"
467 definition Gcd_eucl :: "'a set \<Rightarrow> 'a"
468 where
469   "Gcd_eucl A = Lcm_eucl {d. \<forall>a\<in>A. d dvd a}"
471 end
473 class euclidean_semiring_gcd = euclidean_semiring + gcd + Gcd +
474   assumes gcd_gcd_eucl: "gcd = gcd_eucl" and lcm_lcm_eucl: "lcm = lcm_eucl"
475   assumes Gcd_Gcd_eucl: "Gcd = Gcd_eucl" and Lcm_Lcm_eucl: "Lcm = Lcm_eucl"
476 begin
478 lemma gcd_red:
479   "gcd a b = gcd b (a mod b)"
480   by (metis gcd_eucl.simps mod_0 mod_by_0 gcd_gcd_eucl)
482 lemma gcd_non_0:
483   "b \<noteq> 0 \<Longrightarrow> gcd a b = gcd b (a mod b)"
484   by (rule gcd_red)
486 lemma gcd_0_left:
487   "gcd 0 a = a div normalization_factor a"
488    by (simp only: gcd_gcd_eucl, subst gcd_eucl.simps, subst gcd_eucl.simps, simp add: Let_def)
490 lemma gcd_0:
491   "gcd a 0 = a div normalization_factor a"
492   by (simp only: gcd_gcd_eucl, subst gcd_eucl.simps, simp add: Let_def)
494 lemma gcd_dvd1 [iff]: "gcd a b dvd a"
495   and gcd_dvd2 [iff]: "gcd a b dvd b"
496 proof (induct a b rule: gcd_eucl.induct)
497   fix a b :: 'a
498   assume IH1: "b \<noteq> 0 \<Longrightarrow> gcd b (a mod b) dvd b"
499   assume IH2: "b \<noteq> 0 \<Longrightarrow> gcd b (a mod b) dvd (a mod b)"
501   have "gcd a b dvd a \<and> gcd a b dvd b"
502   proof (cases "b = 0")
503     case True
504       then show ?thesis by (cases "a = 0", simp_all add: gcd_0)
505   next
506     case False
507       with IH1 and IH2 show ?thesis by (simp add: gcd_non_0 dvd_mod_iff)
508   qed
509   then show "gcd a b dvd a" "gcd a b dvd b" by simp_all
510 qed
512 lemma dvd_gcd_D1: "k dvd gcd m n \<Longrightarrow> k dvd m"
513   by (rule dvd_trans, assumption, rule gcd_dvd1)
515 lemma dvd_gcd_D2: "k dvd gcd m n \<Longrightarrow> k dvd n"
516   by (rule dvd_trans, assumption, rule gcd_dvd2)
518 lemma gcd_greatest:
519   fixes k a b :: 'a
520   shows "k dvd a \<Longrightarrow> k dvd b \<Longrightarrow> k dvd gcd a b"
521 proof (induct a b rule: gcd_eucl.induct)
522   case (1 a b)
523   show ?case
524     proof (cases "b = 0")
525       assume "b = 0"
526       with 1 show ?thesis by (cases "a = 0", simp_all add: gcd_0)
527     next
528       assume "b \<noteq> 0"
529       with 1 show ?thesis by (simp add: gcd_non_0 dvd_mod_iff)
530     qed
531 qed
533 lemma dvd_gcd_iff:
534   "k dvd gcd a b \<longleftrightarrow> k dvd a \<and> k dvd b"
535   by (blast intro!: gcd_greatest intro: dvd_trans)
537 lemmas gcd_greatest_iff = dvd_gcd_iff
539 lemma gcd_zero [simp]:
540   "gcd a b = 0 \<longleftrightarrow> a = 0 \<and> b = 0"
541   by (metis dvd_0_left dvd_refl gcd_dvd1 gcd_dvd2 gcd_greatest)+
543 lemma normalization_factor_gcd [simp]:
544   "normalization_factor (gcd a b) = (if a = 0 \<and> b = 0 then 0 else 1)" (is "?f a b = ?g a b")
545 proof (induct a b rule: gcd_eucl.induct)
546   fix a b :: 'a
547   assume IH: "b \<noteq> 0 \<Longrightarrow> ?f b (a mod b) = ?g b (a mod b)"
548   then show "?f a b = ?g a b" by (cases "b = 0", auto simp: gcd_non_0 gcd_0)
549 qed
551 lemma gcdI:
552   "k dvd a \<Longrightarrow> k dvd b \<Longrightarrow> (\<And>l. l dvd a \<Longrightarrow> l dvd b \<Longrightarrow> l dvd k)
553     \<Longrightarrow> normalization_factor k = (if k = 0 then 0 else 1) \<Longrightarrow> k = gcd a b"
554   by (intro normed_associated_imp_eq) (auto simp: associated_def intro: gcd_greatest)
556 sublocale gcd!: abel_semigroup gcd
557 proof
558   fix a b c
559   show "gcd (gcd a b) c = gcd a (gcd b c)"
560   proof (rule gcdI)
561     have "gcd (gcd a b) c dvd gcd a b" "gcd a b dvd a" by simp_all
562     then show "gcd (gcd a b) c dvd a" by (rule dvd_trans)
563     have "gcd (gcd a b) c dvd gcd a b" "gcd a b dvd b" by simp_all
564     hence "gcd (gcd a b) c dvd b" by (rule dvd_trans)
565     moreover have "gcd (gcd a b) c dvd c" by simp
566     ultimately show "gcd (gcd a b) c dvd gcd b c"
567       by (rule gcd_greatest)
568     show "normalization_factor (gcd (gcd a b) c) =  (if gcd (gcd a b) c = 0 then 0 else 1)"
569       by auto
570     fix l assume "l dvd a" and "l dvd gcd b c"
571     with dvd_trans[OF _ gcd_dvd1] and dvd_trans[OF _ gcd_dvd2]
572       have "l dvd b" and "l dvd c" by blast+
573     with l dvd a show "l dvd gcd (gcd a b) c"
574       by (intro gcd_greatest)
575   qed
576 next
577   fix a b
578   show "gcd a b = gcd b a"
579     by (rule gcdI) (simp_all add: gcd_greatest)
580 qed
582 lemma gcd_unique: "d dvd a \<and> d dvd b \<and>
583     normalization_factor d = (if d = 0 then 0 else 1) \<and>
584     (\<forall>e. e dvd a \<and> e dvd b \<longrightarrow> e dvd d) \<longleftrightarrow> d = gcd a b"
585   by (rule, auto intro: gcdI simp: gcd_greatest)
587 lemma gcd_dvd_prod: "gcd a b dvd k * b"
588   using mult_dvd_mono [of 1] by auto
590 lemma gcd_1_left [simp]: "gcd 1 a = 1"
591   by (rule sym, rule gcdI, simp_all)
593 lemma gcd_1 [simp]: "gcd a 1 = 1"
594   by (rule sym, rule gcdI, simp_all)
596 lemma gcd_proj2_if_dvd:
597   "b dvd a \<Longrightarrow> gcd a b = b div normalization_factor b"
598   by (cases "b = 0", simp_all add: dvd_eq_mod_eq_0 gcd_non_0 gcd_0)
600 lemma gcd_proj1_if_dvd:
601   "a dvd b \<Longrightarrow> gcd a b = a div normalization_factor a"
602   by (subst gcd.commute, simp add: gcd_proj2_if_dvd)
604 lemma gcd_proj1_iff: "gcd m n = m div normalization_factor m \<longleftrightarrow> m dvd n"
605 proof
606   assume A: "gcd m n = m div normalization_factor m"
607   show "m dvd n"
608   proof (cases "m = 0")
609     assume [simp]: "m \<noteq> 0"
610     from A have B: "m = gcd m n * normalization_factor m"
611       by (simp add: unit_eq_div2)
612     show ?thesis by (subst B, simp add: mult_unit_dvd_iff)
613   qed (insert A, simp)
614 next
615   assume "m dvd n"
616   then show "gcd m n = m div normalization_factor m" by (rule gcd_proj1_if_dvd)
617 qed
619 lemma gcd_proj2_iff: "gcd m n = n div normalization_factor n \<longleftrightarrow> n dvd m"
620   by (subst gcd.commute, simp add: gcd_proj1_iff)
622 lemma gcd_mod1 [simp]:
623   "gcd (a mod b) b = gcd a b"
624   by (rule gcdI, metis dvd_mod_iff gcd_dvd1 gcd_dvd2, simp_all add: gcd_greatest dvd_mod_iff)
626 lemma gcd_mod2 [simp]:
627   "gcd a (b mod a) = gcd a b"
628   by (rule gcdI, simp, metis dvd_mod_iff gcd_dvd1 gcd_dvd2, simp_all add: gcd_greatest dvd_mod_iff)
630 lemma normalization_factor_dvd' [simp]:
631   "normalization_factor a dvd a"
632   by (cases "a = 0", simp_all)
634 lemma gcd_mult_distrib':
635   "k div normalization_factor k * gcd a b = gcd (k*a) (k*b)"
636 proof (induct a b rule: gcd_eucl.induct)
637   case (1 a b)
638   show ?case
639   proof (cases "b = 0")
640     case True
641     then show ?thesis by (simp add: normalization_factor_mult gcd_0 algebra_simps div_mult_div_if_dvd)
642   next
643     case False
644     hence "k div normalization_factor k * gcd a b =  gcd (k * b) (k * (a mod b))"
645       using 1 by (subst gcd_red, simp)
646     also have "... = gcd (k * a) (k * b)"
647       by (simp add: mult_mod_right gcd.commute)
648     finally show ?thesis .
649   qed
650 qed
652 lemma gcd_mult_distrib:
653   "k * gcd a b = gcd (k*a) (k*b) * normalization_factor k"
654 proof-
655   let ?nf = "normalization_factor"
656   from gcd_mult_distrib'
657     have "gcd (k*a) (k*b) = k div ?nf k * gcd a b" ..
658   also have "... = k * gcd a b div ?nf k"
659     by (metis dvd_div_mult dvd_eq_mod_eq_0 mod_0 normalization_factor_dvd)
660   finally show ?thesis
661     by simp
662 qed
664 lemma euclidean_size_gcd_le1 [simp]:
665   assumes "a \<noteq> 0"
666   shows "euclidean_size (gcd a b) \<le> euclidean_size a"
667 proof -
668    have "gcd a b dvd a" by (rule gcd_dvd1)
669    then obtain c where A: "a = gcd a b * c" unfolding dvd_def by blast
670    with a \<noteq> 0 show ?thesis by (subst (2) A, intro size_mult_mono) auto
671 qed
673 lemma euclidean_size_gcd_le2 [simp]:
674   "b \<noteq> 0 \<Longrightarrow> euclidean_size (gcd a b) \<le> euclidean_size b"
675   by (subst gcd.commute, rule euclidean_size_gcd_le1)
677 lemma euclidean_size_gcd_less1:
678   assumes "a \<noteq> 0" and "\<not>a dvd b"
679   shows "euclidean_size (gcd a b) < euclidean_size a"
680 proof (rule ccontr)
681   assume "\<not>euclidean_size (gcd a b) < euclidean_size a"
682   with a \<noteq> 0 have "euclidean_size (gcd a b) = euclidean_size a"
683     by (intro le_antisym, simp_all)
684   with assms have "a dvd gcd a b" by (auto intro: dvd_euclidean_size_eq_imp_dvd)
685   hence "a dvd b" using dvd_gcd_D2 by blast
686   with \<not>a dvd b show False by contradiction
687 qed
689 lemma euclidean_size_gcd_less2:
690   assumes "b \<noteq> 0" and "\<not>b dvd a"
691   shows "euclidean_size (gcd a b) < euclidean_size b"
692   using assms by (subst gcd.commute, rule euclidean_size_gcd_less1)
694 lemma gcd_mult_unit1: "is_unit a \<Longrightarrow> gcd (b * a) c = gcd b c"
695   apply (rule gcdI)
696   apply (rule dvd_trans, rule gcd_dvd1, simp add: unit_simps)
697   apply (rule gcd_dvd2)
698   apply (rule gcd_greatest, simp add: unit_simps, assumption)
699   apply (subst normalization_factor_gcd, simp add: gcd_0)
700   done
702 lemma gcd_mult_unit2: "is_unit a \<Longrightarrow> gcd b (c * a) = gcd b c"
703   by (subst gcd.commute, subst gcd_mult_unit1, assumption, rule gcd.commute)
705 lemma gcd_div_unit1: "is_unit a \<Longrightarrow> gcd (b div a) c = gcd b c"
706   by (erule is_unitE [of _ b]) (simp add: gcd_mult_unit1)
708 lemma gcd_div_unit2: "is_unit a \<Longrightarrow> gcd b (c div a) = gcd b c"
709   by (erule is_unitE [of _ c]) (simp add: gcd_mult_unit2)
711 lemma gcd_idem: "gcd a a = a div normalization_factor a"
712   by (cases "a = 0") (simp add: gcd_0_left, rule sym, rule gcdI, simp_all)
714 lemma gcd_right_idem: "gcd (gcd a b) b = gcd a b"
715   apply (rule gcdI)
716   apply (simp add: ac_simps)
717   apply (rule gcd_dvd2)
718   apply (rule gcd_greatest, erule (1) gcd_greatest, assumption)
719   apply simp
720   done
722 lemma gcd_left_idem: "gcd a (gcd a b) = gcd a b"
723   apply (rule gcdI)
724   apply simp
725   apply (rule dvd_trans, rule gcd_dvd2, rule gcd_dvd2)
726   apply (rule gcd_greatest, assumption, erule gcd_greatest, assumption)
727   apply simp
728   done
730 lemma comp_fun_idem_gcd: "comp_fun_idem gcd"
731 proof
732   fix a b show "gcd a \<circ> gcd b = gcd b \<circ> gcd a"
733     by (simp add: fun_eq_iff ac_simps)
734 next
735   fix a show "gcd a \<circ> gcd a = gcd a"
736     by (simp add: fun_eq_iff gcd_left_idem)
737 qed
739 lemma coprime_dvd_mult:
740   assumes "gcd c b = 1" and "c dvd a * b"
741   shows "c dvd a"
742 proof -
743   let ?nf = "normalization_factor"
744   from assms gcd_mult_distrib [of a c b]
745     have A: "a = gcd (a * c) (a * b) * ?nf a" by simp
746   from c dvd a * b show ?thesis by (subst A, simp_all add: gcd_greatest)
747 qed
749 lemma coprime_dvd_mult_iff:
750   "gcd c b = 1 \<Longrightarrow> (c dvd a * b) = (c dvd a)"
751   by (rule, rule coprime_dvd_mult, simp_all)
753 lemma gcd_dvd_antisym:
754   "gcd a b dvd gcd c d \<Longrightarrow> gcd c d dvd gcd a b \<Longrightarrow> gcd a b = gcd c d"
755 proof (rule gcdI)
756   assume A: "gcd a b dvd gcd c d" and B: "gcd c d dvd gcd a b"
757   have "gcd c d dvd c" by simp
758   with A show "gcd a b dvd c" by (rule dvd_trans)
759   have "gcd c d dvd d" by simp
760   with A show "gcd a b dvd d" by (rule dvd_trans)
761   show "normalization_factor (gcd a b) = (if gcd a b = 0 then 0 else 1)"
762     by simp
763   fix l assume "l dvd c" and "l dvd d"
764   hence "l dvd gcd c d" by (rule gcd_greatest)
765   from this and B show "l dvd gcd a b" by (rule dvd_trans)
766 qed
768 lemma gcd_mult_cancel:
769   assumes "gcd k n = 1"
770   shows "gcd (k * m) n = gcd m n"
771 proof (rule gcd_dvd_antisym)
772   have "gcd (gcd (k * m) n) k = gcd (gcd k n) (k * m)" by (simp add: ac_simps)
773   also note gcd k n = 1
774   finally have "gcd (gcd (k * m) n) k = 1" by simp
775   hence "gcd (k * m) n dvd m" by (rule coprime_dvd_mult, simp add: ac_simps)
776   moreover have "gcd (k * m) n dvd n" by simp
777   ultimately show "gcd (k * m) n dvd gcd m n" by (rule gcd_greatest)
778   have "gcd m n dvd (k * m)" and "gcd m n dvd n" by simp_all
779   then show "gcd m n dvd gcd (k * m) n" by (rule gcd_greatest)
780 qed
782 lemma coprime_crossproduct:
783   assumes [simp]: "gcd a d = 1" "gcd b c = 1"
784   shows "associated (a * c) (b * d) \<longleftrightarrow> associated a b \<and> associated c d" (is "?lhs \<longleftrightarrow> ?rhs")
785 proof
786   assume ?rhs then show ?lhs unfolding associated_def by (fast intro: mult_dvd_mono)
787 next
788   assume ?lhs
789   from ?lhs have "a dvd b * d" unfolding associated_def by (metis dvd_mult_left)
790   hence "a dvd b" by (simp add: coprime_dvd_mult_iff)
791   moreover from ?lhs have "b dvd a * c" unfolding associated_def by (metis dvd_mult_left)
792   hence "b dvd a" by (simp add: coprime_dvd_mult_iff)
793   moreover from ?lhs have "c dvd d * b"
794     unfolding associated_def by (auto dest: dvd_mult_right simp add: ac_simps)
795   hence "c dvd d" by (simp add: coprime_dvd_mult_iff gcd.commute)
796   moreover from ?lhs have "d dvd c * a"
797     unfolding associated_def by (auto dest: dvd_mult_right simp add: ac_simps)
798   hence "d dvd c" by (simp add: coprime_dvd_mult_iff gcd.commute)
799   ultimately show ?rhs unfolding associated_def by simp
800 qed
802 lemma gcd_add1 [simp]:
803   "gcd (m + n) n = gcd m n"
804   by (cases "n = 0", simp_all add: gcd_non_0)
806 lemma gcd_add2 [simp]:
807   "gcd m (m + n) = gcd m n"
808   using gcd_add1 [of n m] by (simp add: ac_simps)
810 lemma gcd_add_mult: "gcd m (k * m + n) = gcd m n"
811   by (subst gcd.commute, subst gcd_red, simp)
813 lemma coprimeI: "(\<And>l. \<lbrakk>l dvd a; l dvd b\<rbrakk> \<Longrightarrow> l dvd 1) \<Longrightarrow> gcd a b = 1"
814   by (rule sym, rule gcdI, simp_all)
816 lemma coprime: "gcd a b = 1 \<longleftrightarrow> (\<forall>d. d dvd a \<and> d dvd b \<longleftrightarrow> is_unit d)"
817   by (auto intro: coprimeI gcd_greatest dvd_gcd_D1 dvd_gcd_D2)
819 lemma div_gcd_coprime:
820   assumes nz: "a \<noteq> 0 \<or> b \<noteq> 0"
821   defines [simp]: "d \<equiv> gcd a b"
822   defines [simp]: "a' \<equiv> a div d" and [simp]: "b' \<equiv> b div d"
823   shows "gcd a' b' = 1"
824 proof (rule coprimeI)
825   fix l assume "l dvd a'" "l dvd b'"
826   then obtain s t where "a' = l * s" "b' = l * t" unfolding dvd_def by blast
827   moreover have "a = a' * d" "b = b' * d" by simp_all
828   ultimately have "a = (l * d) * s" "b = (l * d) * t"
829     by (simp_all only: ac_simps)
830   hence "l*d dvd a" and "l*d dvd b" by (simp_all only: dvd_triv_left)
831   hence "l*d dvd d" by (simp add: gcd_greatest)
832   then obtain u where "d = l * d * u" ..
833   then have "d * (l * u) = d" by (simp add: ac_simps)
834   moreover from nz have "d \<noteq> 0" by simp
835   with div_mult_self1_is_id have "d * (l * u) div d = l * u" .
836   ultimately have "1 = l * u"
837     using d \<noteq> 0 by simp
838   then show "l dvd 1" ..
839 qed
841 lemma coprime_mult:
842   assumes da: "gcd d a = 1" and db: "gcd d b = 1"
843   shows "gcd d (a * b) = 1"
844   apply (subst gcd.commute)
845   using da apply (subst gcd_mult_cancel)
846   apply (subst gcd.commute, assumption)
847   apply (subst gcd.commute, rule db)
848   done
850 lemma coprime_lmult:
851   assumes dab: "gcd d (a * b) = 1"
852   shows "gcd d a = 1"
853 proof (rule coprimeI)
854   fix l assume "l dvd d" and "l dvd a"
855   hence "l dvd a * b" by simp
856   with l dvd d and dab show "l dvd 1" by (auto intro: gcd_greatest)
857 qed
859 lemma coprime_rmult:
860   assumes dab: "gcd d (a * b) = 1"
861   shows "gcd d b = 1"
862 proof (rule coprimeI)
863   fix l assume "l dvd d" and "l dvd b"
864   hence "l dvd a * b" by simp
865   with l dvd d and dab show "l dvd 1" by (auto intro: gcd_greatest)
866 qed
868 lemma coprime_mul_eq: "gcd d (a * b) = 1 \<longleftrightarrow> gcd d a = 1 \<and> gcd d b = 1"
869   using coprime_rmult[of d a b] coprime_lmult[of d a b] coprime_mult[of d a b] by blast
871 lemma gcd_coprime:
872   assumes c: "gcd a b \<noteq> 0" and a: "a = a' * gcd a b" and b: "b = b' * gcd a b"
873   shows "gcd a' b' = 1"
874 proof -
875   from c have "a \<noteq> 0 \<or> b \<noteq> 0" by simp
876   with div_gcd_coprime have "gcd (a div gcd a b) (b div gcd a b) = 1" .
877   also from assms have "a div gcd a b = a'" by (metis div_mult_self2_is_id)+
878   also from assms have "b div gcd a b = b'" by (metis div_mult_self2_is_id)+
879   finally show ?thesis .
880 qed
882 lemma coprime_power:
883   assumes "0 < n"
884   shows "gcd a (b ^ n) = 1 \<longleftrightarrow> gcd a b = 1"
885 using assms proof (induct n)
886   case (Suc n) then show ?case
887     by (cases n) (simp_all add: coprime_mul_eq)
888 qed simp
890 lemma gcd_coprime_exists:
891   assumes nz: "gcd a b \<noteq> 0"
892   shows "\<exists>a' b'. a = a' * gcd a b \<and> b = b' * gcd a b \<and> gcd a' b' = 1"
893   apply (rule_tac x = "a div gcd a b" in exI)
894   apply (rule_tac x = "b div gcd a b" in exI)
895   apply (insert nz, auto intro: div_gcd_coprime)
896   done
898 lemma coprime_exp:
899   "gcd d a = 1 \<Longrightarrow> gcd d (a^n) = 1"
900   by (induct n, simp_all add: coprime_mult)
902 lemma coprime_exp2 [intro]:
903   "gcd a b = 1 \<Longrightarrow> gcd (a^n) (b^m) = 1"
904   apply (rule coprime_exp)
905   apply (subst gcd.commute)
906   apply (rule coprime_exp)
907   apply (subst gcd.commute)
908   apply assumption
909   done
911 lemma gcd_exp:
912   "gcd (a^n) (b^n) = (gcd a b) ^ n"
913 proof (cases "a = 0 \<and> b = 0")
914   assume "a = 0 \<and> b = 0"
915   then show ?thesis by (cases n, simp_all add: gcd_0_left)
916 next
917   assume A: "\<not>(a = 0 \<and> b = 0)"
918   hence "1 = gcd ((a div gcd a b)^n) ((b div gcd a b)^n)"
919     using div_gcd_coprime by (subst sym, auto simp: div_gcd_coprime)
920   hence "(gcd a b) ^ n = (gcd a b) ^ n * ..." by simp
921   also note gcd_mult_distrib
922   also have "normalization_factor ((gcd a b)^n) = 1"
923     by (simp add: normalization_factor_pow A)
924   also have "(gcd a b)^n * (a div gcd a b)^n = a^n"
925     by (subst ac_simps, subst div_power, simp, rule dvd_div_mult_self, rule dvd_power_same, simp)
926   also have "(gcd a b)^n * (b div gcd a b)^n = b^n"
927     by (subst ac_simps, subst div_power, simp, rule dvd_div_mult_self, rule dvd_power_same, simp)
928   finally show ?thesis by simp
929 qed
931 lemma coprime_common_divisor:
932   "gcd a b = 1 \<Longrightarrow> a dvd a \<Longrightarrow> a dvd b \<Longrightarrow> is_unit a"
933   apply (subgoal_tac "a dvd gcd a b")
934   apply simp
935   apply (erule (1) gcd_greatest)
936   done
938 lemma division_decomp:
939   assumes dc: "a dvd b * c"
940   shows "\<exists>b' c'. a = b' * c' \<and> b' dvd b \<and> c' dvd c"
941 proof (cases "gcd a b = 0")
942   assume "gcd a b = 0"
943   hence "a = 0 \<and> b = 0" by simp
944   hence "a = 0 * c \<and> 0 dvd b \<and> c dvd c" by simp
945   then show ?thesis by blast
946 next
947   let ?d = "gcd a b"
948   assume "?d \<noteq> 0"
949   from gcd_coprime_exists[OF this]
950     obtain a' b' where ab': "a = a' * ?d" "b = b' * ?d" "gcd a' b' = 1"
951     by blast
952   from ab'(1) have "a' dvd a" unfolding dvd_def by blast
953   with dc have "a' dvd b*c" using dvd_trans[of a' a "b*c"] by simp
954   from dc ab'(1,2) have "a'*?d dvd (b'*?d) * c" by simp
955   hence "?d * a' dvd ?d * (b' * c)" by (simp add: mult_ac)
956   with ?d \<noteq> 0 have "a' dvd b' * c" by simp
957   with coprime_dvd_mult[OF ab'(3)]
958     have "a' dvd c" by (subst (asm) ac_simps, blast)
959   with ab'(1) have "a = ?d * a' \<and> ?d dvd b \<and> a' dvd c" by (simp add: mult_ac)
960   then show ?thesis by blast
961 qed
963 lemma pow_divs_pow:
964   assumes ab: "a ^ n dvd b ^ n" and n: "n \<noteq> 0"
965   shows "a dvd b"
966 proof (cases "gcd a b = 0")
967   assume "gcd a b = 0"
968   then show ?thesis by simp
969 next
970   let ?d = "gcd a b"
971   assume "?d \<noteq> 0"
972   from n obtain m where m: "n = Suc m" by (cases n, simp_all)
973   from ?d \<noteq> 0 have zn: "?d ^ n \<noteq> 0" by (rule power_not_zero)
974   from gcd_coprime_exists[OF ?d \<noteq> 0]
975     obtain a' b' where ab': "a = a' * ?d" "b = b' * ?d" "gcd a' b' = 1"
976     by blast
977   from ab have "(a' * ?d) ^ n dvd (b' * ?d) ^ n"
978     by (simp add: ab'(1,2)[symmetric])
979   hence "?d^n * a'^n dvd ?d^n * b'^n"
980     by (simp only: power_mult_distrib ac_simps)
981   with zn have "a'^n dvd b'^n" by simp
982   hence "a' dvd b'^n" using dvd_trans[of a' "a'^n" "b'^n"] by (simp add: m)
983   hence "a' dvd b'^m * b'" by (simp add: m ac_simps)
984   with coprime_dvd_mult[OF coprime_exp[OF ab'(3), of m]]
985     have "a' dvd b'" by (subst (asm) ac_simps, blast)
986   hence "a'*?d dvd b'*?d" by (rule mult_dvd_mono, simp)
987   with ab'(1,2) show ?thesis by simp
988 qed
990 lemma pow_divs_eq [simp]:
991   "n \<noteq> 0 \<Longrightarrow> a ^ n dvd b ^ n \<longleftrightarrow> a dvd b"
992   by (auto intro: pow_divs_pow dvd_power_same)
994 lemma divs_mult:
995   assumes mr: "m dvd r" and nr: "n dvd r" and mn: "gcd m n = 1"
996   shows "m * n dvd r"
997 proof -
998   from mr nr obtain m' n' where m': "r = m*m'" and n': "r = n*n'"
999     unfolding dvd_def by blast
1000   from mr n' have "m dvd n'*n" by (simp add: ac_simps)
1001   hence "m dvd n'" using coprime_dvd_mult_iff[OF mn] by simp
1002   then obtain k where k: "n' = m*k" unfolding dvd_def by blast
1003   with n' have "r = m * n * k" by (simp add: mult_ac)
1004   then show ?thesis unfolding dvd_def by blast
1005 qed
1007 lemma coprime_plus_one [simp]: "gcd (n + 1) n = 1"
1008   by (subst add_commute, simp)
1010 lemma setprod_coprime [rule_format]:
1011   "(\<forall>i\<in>A. gcd (f i) a = 1) \<longrightarrow> gcd (\<Prod>i\<in>A. f i) a = 1"
1012   apply (cases "finite A")
1013   apply (induct set: finite)
1014   apply (auto simp add: gcd_mult_cancel)
1015   done
1017 lemma coprime_divisors:
1018   assumes "d dvd a" "e dvd b" "gcd a b = 1"
1019   shows "gcd d e = 1"
1020 proof -
1021   from assms obtain k l where "a = d * k" "b = e * l"
1022     unfolding dvd_def by blast
1023   with assms have "gcd (d * k) (e * l) = 1" by simp
1024   hence "gcd (d * k) e = 1" by (rule coprime_lmult)
1025   also have "gcd (d * k) e = gcd e (d * k)" by (simp add: ac_simps)
1026   finally have "gcd e d = 1" by (rule coprime_lmult)
1027   then show ?thesis by (simp add: ac_simps)
1028 qed
1030 lemma invertible_coprime:
1031   assumes "a * b mod m = 1"
1032   shows "coprime a m"
1033 proof -
1034   from assms have "coprime m (a * b mod m)"
1035     by simp
1036   then have "coprime m (a * b)"
1037     by simp
1038   then have "coprime m a"
1039     by (rule coprime_lmult)
1040   then show ?thesis
1041     by (simp add: ac_simps)
1042 qed
1044 lemma lcm_gcd:
1045   "lcm a b = a * b div (gcd a b * normalization_factor (a*b))"
1046   by (simp only: lcm_lcm_eucl gcd_gcd_eucl lcm_eucl_def)
1048 lemma lcm_gcd_prod:
1049   "lcm a b * gcd a b = a * b div normalization_factor (a*b)"
1050 proof (cases "a * b = 0")
1051   let ?nf = normalization_factor
1052   assume "a * b \<noteq> 0"
1053   hence "gcd a b \<noteq> 0" by simp
1054   from lcm_gcd have "lcm a b * gcd a b = gcd a b * (a * b div (?nf (a*b) * gcd a b))"
1055     by (simp add: mult_ac)
1056   also from a * b \<noteq> 0 have "... = a * b div ?nf (a*b)"
1057     by (simp add: div_mult_swap mult.commute)
1058   finally show ?thesis .
1059 qed (auto simp add: lcm_gcd)
1061 lemma lcm_dvd1 [iff]:
1062   "a dvd lcm a b"
1063 proof (cases "a*b = 0")
1064   assume "a * b \<noteq> 0"
1065   hence "gcd a b \<noteq> 0" by simp
1066   let ?c = "1 div normalization_factor (a * b)"
1067   from a * b \<noteq> 0 have [simp]: "is_unit (normalization_factor (a * b))" by simp
1068   from lcm_gcd_prod[of a b] have "lcm a b * gcd a b = a * ?c * b"
1069     by (simp add: div_mult_swap unit_div_commute)
1070   hence "lcm a b * gcd a b div gcd a b = a * ?c * b div gcd a b" by simp
1071   with gcd a b \<noteq> 0 have "lcm a b = a * ?c * b div gcd a b"
1072     by (subst (asm) div_mult_self2_is_id, simp_all)
1073   also have "... = a * (?c * b div gcd a b)"
1074     by (metis div_mult_swap gcd_dvd2 mult_assoc)
1075   finally show ?thesis by (rule dvdI)
1076 qed (auto simp add: lcm_gcd)
1078 lemma lcm_least:
1079   "\<lbrakk>a dvd k; b dvd k\<rbrakk> \<Longrightarrow> lcm a b dvd k"
1080 proof (cases "k = 0")
1081   let ?nf = normalization_factor
1082   assume "k \<noteq> 0"
1083   hence "is_unit (?nf k)" by simp
1084   hence "?nf k \<noteq> 0" by (metis not_is_unit_0)
1085   assume A: "a dvd k" "b dvd k"
1086   hence "gcd a b \<noteq> 0" using k \<noteq> 0 by auto
1087   from A obtain r s where ar: "k = a * r" and bs: "k = b * s"
1088     unfolding dvd_def by blast
1089   with k \<noteq> 0 have "r * s \<noteq> 0"
1090     by auto (drule sym [of 0], simp)
1091   hence "is_unit (?nf (r * s))" by simp
1092   let ?c = "?nf k div ?nf (r*s)"
1093   from is_unit (?nf k) and is_unit (?nf (r * s)) have "is_unit ?c" by (rule unit_div)
1094   hence "?c \<noteq> 0" using not_is_unit_0 by fast
1095   from ar bs have "k * k * gcd s r = ?nf k * k * gcd (k * s) (k * r)"
1096     by (subst mult_assoc, subst gcd_mult_distrib[of k s r], simp only: ac_simps)
1097   also have "... = ?nf k * k * gcd ((r*s) * a) ((r*s) * b)"
1098     by (subst (3) k = a * r, subst (3) k = b * s, simp add: algebra_simps)
1099   also have "... = ?c * r*s * k * gcd a b" using r * s \<noteq> 0
1100     by (subst gcd_mult_distrib'[symmetric], simp add: algebra_simps unit_simps)
1101   finally have "(a*r) * (b*s) * gcd s r = ?c * k * r * s * gcd a b"
1102     by (subst ar[symmetric], subst bs[symmetric], simp add: mult_ac)
1103   hence "a * b * gcd s r * (r * s) = ?c * k * gcd a b * (r * s)"
1104     by (simp add: algebra_simps)
1105   hence "?c * k * gcd a b = a * b * gcd s r" using r * s \<noteq> 0
1106     by (metis div_mult_self2_is_id)
1107   also have "... = lcm a b * gcd a b * gcd s r * ?nf (a*b)"
1108     by (subst lcm_gcd_prod[of a b], metis gcd_mult_distrib gcd_mult_distrib')
1109   also have "... = lcm a b * gcd s r * ?nf (a*b) * gcd a b"
1110     by (simp add: algebra_simps)
1111   finally have "k * ?c = lcm a b * gcd s r * ?nf (a*b)" using gcd a b \<noteq> 0
1112     by (metis mult.commute div_mult_self2_is_id)
1113   hence "k = lcm a b * (gcd s r * ?nf (a*b)) div ?c" using ?c \<noteq> 0
1114     by (metis div_mult_self2_is_id mult_assoc)
1115   also have "... = lcm a b * (gcd s r * ?nf (a*b) div ?c)" using is_unit ?c
1116     by (simp add: unit_simps)
1117   finally show ?thesis by (rule dvdI)
1118 qed simp
1120 lemma lcm_zero:
1121   "lcm a b = 0 \<longleftrightarrow> a = 0 \<or> b = 0"
1122 proof -
1123   let ?nf = normalization_factor
1124   {
1125     assume "a \<noteq> 0" "b \<noteq> 0"
1126     hence "a * b div ?nf (a * b) \<noteq> 0" by (simp add: no_zero_divisors)
1127     moreover from a \<noteq> 0 and b \<noteq> 0 have "gcd a b \<noteq> 0" by simp
1128     ultimately have "lcm a b \<noteq> 0" using lcm_gcd_prod[of a b] by (intro notI, simp)
1129   } moreover {
1130     assume "a = 0 \<or> b = 0"
1131     hence "lcm a b = 0" by (elim disjE, simp_all add: lcm_gcd)
1132   }
1133   ultimately show ?thesis by blast
1134 qed
1136 lemmas lcm_0_iff = lcm_zero
1138 lemma gcd_lcm:
1139   assumes "lcm a b \<noteq> 0"
1140   shows "gcd a b = a * b div (lcm a b * normalization_factor (a * b))"
1141 proof-
1142   from assms have "gcd a b \<noteq> 0" by (simp add: lcm_zero)
1143   let ?c = "normalization_factor (a * b)"
1144   from lcm a b \<noteq> 0 have "?c \<noteq> 0" by (intro notI, simp add: lcm_zero no_zero_divisors)
1145   hence "is_unit ?c" by simp
1146   from lcm_gcd_prod [of a b] have "gcd a b = a * b div ?c div lcm a b"
1147     by (subst (2) div_mult_self2_is_id[OF lcm a b \<noteq> 0, symmetric], simp add: mult_ac)
1148   also from is_unit ?c have "... = a * b div (lcm a b * ?c)"
1149     by (metis ?c \<noteq> 0 div_mult_mult1 dvd_mult_div_cancel mult_commute normalization_factor_dvd')
1150   finally show ?thesis .
1151 qed
1153 lemma normalization_factor_lcm [simp]:
1154   "normalization_factor (lcm a b) = (if a = 0 \<or> b = 0 then 0 else 1)"
1155 proof (cases "a = 0 \<or> b = 0")
1156   case True then show ?thesis
1157     by (auto simp add: lcm_gcd)
1158 next
1159   case False
1160   let ?nf = normalization_factor
1161   from lcm_gcd_prod[of a b]
1162     have "?nf (lcm a b) * ?nf (gcd a b) = ?nf (a*b) div ?nf (a*b)"
1163     by (metis div_by_0 div_self normalization_correct normalization_factor_0 normalization_factor_mult)
1164   also have "... = (if a*b = 0 then 0 else 1)"
1165     by simp
1166   finally show ?thesis using False by simp
1167 qed
1169 lemma lcm_dvd2 [iff]: "b dvd lcm a b"
1170   using lcm_dvd1 [of b a] by (simp add: lcm_gcd ac_simps)
1172 lemma lcmI:
1173   "\<lbrakk>a dvd k; b dvd k; \<And>l. a dvd l \<Longrightarrow> b dvd l \<Longrightarrow> k dvd l;
1174     normalization_factor k = (if k = 0 then 0 else 1)\<rbrakk> \<Longrightarrow> k = lcm a b"
1175   by (intro normed_associated_imp_eq) (auto simp: associated_def intro: lcm_least)
1177 sublocale lcm!: abel_semigroup lcm
1178 proof
1179   fix a b c
1180   show "lcm (lcm a b) c = lcm a (lcm b c)"
1181   proof (rule lcmI)
1182     have "a dvd lcm a b" and "lcm a b dvd lcm (lcm a b) c" by simp_all
1183     then show "a dvd lcm (lcm a b) c" by (rule dvd_trans)
1185     have "b dvd lcm a b" and "lcm a b dvd lcm (lcm a b) c" by simp_all
1186     hence "b dvd lcm (lcm a b) c" by (rule dvd_trans)
1187     moreover have "c dvd lcm (lcm a b) c" by simp
1188     ultimately show "lcm b c dvd lcm (lcm a b) c" by (rule lcm_least)
1190     fix l assume "a dvd l" and "lcm b c dvd l"
1191     have "b dvd lcm b c" by simp
1192     from this and lcm b c dvd l have "b dvd l" by (rule dvd_trans)
1193     have "c dvd lcm b c" by simp
1194     from this and lcm b c dvd l have "c dvd l" by (rule dvd_trans)
1195     from a dvd l and b dvd l have "lcm a b dvd l" by (rule lcm_least)
1196     from this and c dvd l show "lcm (lcm a b) c dvd l" by (rule lcm_least)
1197   qed (simp add: lcm_zero)
1198 next
1199   fix a b
1200   show "lcm a b = lcm b a"
1201     by (simp add: lcm_gcd ac_simps)
1202 qed
1204 lemma dvd_lcm_D1:
1205   "lcm m n dvd k \<Longrightarrow> m dvd k"
1206   by (rule dvd_trans, rule lcm_dvd1, assumption)
1208 lemma dvd_lcm_D2:
1209   "lcm m n dvd k \<Longrightarrow> n dvd k"
1210   by (rule dvd_trans, rule lcm_dvd2, assumption)
1212 lemma gcd_dvd_lcm [simp]:
1213   "gcd a b dvd lcm a b"
1214   by (metis dvd_trans gcd_dvd2 lcm_dvd2)
1216 lemma lcm_1_iff:
1217   "lcm a b = 1 \<longleftrightarrow> is_unit a \<and> is_unit b"
1218 proof
1219   assume "lcm a b = 1"
1220   then show "is_unit a \<and> is_unit b" by auto
1221 next
1222   assume "is_unit a \<and> is_unit b"
1223   hence "a dvd 1" and "b dvd 1" by simp_all
1224   hence "is_unit (lcm a b)" by (rule lcm_least)
1225   hence "lcm a b = normalization_factor (lcm a b)"
1226     by (subst normalization_factor_unit, simp_all)
1227   also have "\<dots> = 1" using is_unit a \<and> is_unit b
1228     by auto
1229   finally show "lcm a b = 1" .
1230 qed
1232 lemma lcm_0_left [simp]:
1233   "lcm 0 a = 0"
1234   by (rule sym, rule lcmI, simp_all)
1236 lemma lcm_0 [simp]:
1237   "lcm a 0 = 0"
1238   by (rule sym, rule lcmI, simp_all)
1240 lemma lcm_unique:
1241   "a dvd d \<and> b dvd d \<and>
1242   normalization_factor d = (if d = 0 then 0 else 1) \<and>
1243   (\<forall>e. a dvd e \<and> b dvd e \<longrightarrow> d dvd e) \<longleftrightarrow> d = lcm a b"
1244   by (rule, auto intro: lcmI simp: lcm_least lcm_zero)
1246 lemma dvd_lcm_I1 [simp]:
1247   "k dvd m \<Longrightarrow> k dvd lcm m n"
1248   by (metis lcm_dvd1 dvd_trans)
1250 lemma dvd_lcm_I2 [simp]:
1251   "k dvd n \<Longrightarrow> k dvd lcm m n"
1252   by (metis lcm_dvd2 dvd_trans)
1254 lemma lcm_1_left [simp]:
1255   "lcm 1 a = a div normalization_factor a"
1256   by (cases "a = 0") (simp, rule sym, rule lcmI, simp_all)
1258 lemma lcm_1_right [simp]:
1259   "lcm a 1 = a div normalization_factor a"
1260   using lcm_1_left [of a] by (simp add: ac_simps)
1262 lemma lcm_coprime:
1263   "gcd a b = 1 \<Longrightarrow> lcm a b = a * b div normalization_factor (a*b)"
1264   by (subst lcm_gcd) simp
1266 lemma lcm_proj1_if_dvd:
1267   "b dvd a \<Longrightarrow> lcm a b = a div normalization_factor a"
1268   by (cases "a = 0") (simp, rule sym, rule lcmI, simp_all)
1270 lemma lcm_proj2_if_dvd:
1271   "a dvd b \<Longrightarrow> lcm a b = b div normalization_factor b"
1272   using lcm_proj1_if_dvd [of a b] by (simp add: ac_simps)
1274 lemma lcm_proj1_iff:
1275   "lcm m n = m div normalization_factor m \<longleftrightarrow> n dvd m"
1276 proof
1277   assume A: "lcm m n = m div normalization_factor m"
1278   show "n dvd m"
1279   proof (cases "m = 0")
1280     assume [simp]: "m \<noteq> 0"
1281     from A have B: "m = lcm m n * normalization_factor m"
1282       by (simp add: unit_eq_div2)
1283     show ?thesis by (subst B, simp)
1284   qed simp
1285 next
1286   assume "n dvd m"
1287   then show "lcm m n = m div normalization_factor m" by (rule lcm_proj1_if_dvd)
1288 qed
1290 lemma lcm_proj2_iff:
1291   "lcm m n = n div normalization_factor n \<longleftrightarrow> m dvd n"
1292   using lcm_proj1_iff [of n m] by (simp add: ac_simps)
1294 lemma euclidean_size_lcm_le1:
1295   assumes "a \<noteq> 0" and "b \<noteq> 0"
1296   shows "euclidean_size a \<le> euclidean_size (lcm a b)"
1297 proof -
1298   have "a dvd lcm a b" by (rule lcm_dvd1)
1299   then obtain c where A: "lcm a b = a * c" unfolding dvd_def by blast
1300   with a \<noteq> 0 and b \<noteq> 0 have "c \<noteq> 0" by (auto simp: lcm_zero)
1301   then show ?thesis by (subst A, intro size_mult_mono)
1302 qed
1304 lemma euclidean_size_lcm_le2:
1305   "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> euclidean_size b \<le> euclidean_size (lcm a b)"
1306   using euclidean_size_lcm_le1 [of b a] by (simp add: ac_simps)
1308 lemma euclidean_size_lcm_less1:
1309   assumes "b \<noteq> 0" and "\<not>b dvd a"
1310   shows "euclidean_size a < euclidean_size (lcm a b)"
1311 proof (rule ccontr)
1312   from assms have "a \<noteq> 0" by auto
1313   assume "\<not>euclidean_size a < euclidean_size (lcm a b)"
1314   with a \<noteq> 0 and b \<noteq> 0 have "euclidean_size (lcm a b) = euclidean_size a"
1315     by (intro le_antisym, simp, intro euclidean_size_lcm_le1)
1316   with assms have "lcm a b dvd a"
1317     by (rule_tac dvd_euclidean_size_eq_imp_dvd) (auto simp: lcm_zero)
1318   hence "b dvd a" by (rule dvd_lcm_D2)
1319   with \<not>b dvd a show False by contradiction
1320 qed
1322 lemma euclidean_size_lcm_less2:
1323   assumes "a \<noteq> 0" and "\<not>a dvd b"
1324   shows "euclidean_size b < euclidean_size (lcm a b)"
1325   using assms euclidean_size_lcm_less1 [of a b] by (simp add: ac_simps)
1327 lemma lcm_mult_unit1:
1328   "is_unit a \<Longrightarrow> lcm (b * a) c = lcm b c"
1329   apply (rule lcmI)
1330   apply (rule dvd_trans[of _ "b * a"], simp, rule lcm_dvd1)
1331   apply (rule lcm_dvd2)
1332   apply (rule lcm_least, simp add: unit_simps, assumption)
1333   apply (subst normalization_factor_lcm, simp add: lcm_zero)
1334   done
1336 lemma lcm_mult_unit2:
1337   "is_unit a \<Longrightarrow> lcm b (c * a) = lcm b c"
1338   using lcm_mult_unit1 [of a c b] by (simp add: ac_simps)
1340 lemma lcm_div_unit1:
1341   "is_unit a \<Longrightarrow> lcm (b div a) c = lcm b c"
1342   by (erule is_unitE [of _ b]) (simp add: lcm_mult_unit1)
1344 lemma lcm_div_unit2:
1345   "is_unit a \<Longrightarrow> lcm b (c div a) = lcm b c"
1346   by (erule is_unitE [of _ c]) (simp add: lcm_mult_unit2)
1348 lemma lcm_left_idem:
1349   "lcm a (lcm a b) = lcm a b"
1350   apply (rule lcmI)
1351   apply simp
1352   apply (subst lcm.assoc [symmetric], rule lcm_dvd2)
1353   apply (rule lcm_least, assumption)
1354   apply (erule (1) lcm_least)
1355   apply (auto simp: lcm_zero)
1356   done
1358 lemma lcm_right_idem:
1359   "lcm (lcm a b) b = lcm a b"
1360   apply (rule lcmI)
1361   apply (subst lcm.assoc, rule lcm_dvd1)
1362   apply (rule lcm_dvd2)
1363   apply (rule lcm_least, erule (1) lcm_least, assumption)
1364   apply (auto simp: lcm_zero)
1365   done
1367 lemma comp_fun_idem_lcm: "comp_fun_idem lcm"
1368 proof
1369   fix a b show "lcm a \<circ> lcm b = lcm b \<circ> lcm a"
1370     by (simp add: fun_eq_iff ac_simps)
1371 next
1372   fix a show "lcm a \<circ> lcm a = lcm a" unfolding o_def
1373     by (intro ext, simp add: lcm_left_idem)
1374 qed
1376 lemma dvd_Lcm [simp]: "a \<in> A \<Longrightarrow> a dvd Lcm A"
1377   and Lcm_dvd [simp]: "(\<forall>a\<in>A. a dvd l') \<Longrightarrow> Lcm A dvd l'"
1378   and normalization_factor_Lcm [simp]:
1379           "normalization_factor (Lcm A) = (if Lcm A = 0 then 0 else 1)"
1380 proof -
1381   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>
1382     normalization_factor (Lcm A) = (if Lcm A = 0 then 0 else 1)" (is ?thesis)
1383   proof (cases "\<exists>l. l \<noteq>  0 \<and> (\<forall>a\<in>A. a dvd l)")
1384     case False
1385     hence "Lcm A = 0" by (auto simp: Lcm_Lcm_eucl Lcm_eucl_def)
1386     with False show ?thesis by auto
1387   next
1388     case True
1389     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
1390     def n \<equiv> "LEAST n. \<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"
1391     def l \<equiv> "SOME l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"
1392     have "\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"
1393       apply (subst n_def)
1394       apply (rule LeastI[of _ "euclidean_size l\<^sub>0"])
1395       apply (rule exI[of _ l\<^sub>0])
1396       apply (simp add: l\<^sub>0_props)
1397       done
1398     from someI_ex[OF this] have "l \<noteq> 0" and "\<forall>a\<in>A. a dvd l" and "euclidean_size l = n"
1399       unfolding l_def by simp_all
1400     {
1401       fix l' assume "\<forall>a\<in>A. a dvd l'"
1402       with \<forall>a\<in>A. a dvd l have "\<forall>a\<in>A. a dvd gcd l l'" by (auto intro: gcd_greatest)
1403       moreover from l \<noteq> 0 have "gcd l l' \<noteq> 0" by simp
1404       ultimately have "\<exists>b. b \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd b) \<and> euclidean_size b = euclidean_size (gcd l l')"
1405         by (intro exI[of _ "gcd l l'"], auto)
1406       hence "euclidean_size (gcd l l') \<ge> n" by (subst n_def) (rule Least_le)
1407       moreover have "euclidean_size (gcd l l') \<le> n"
1408       proof -
1409         have "gcd l l' dvd l" by simp
1410         then obtain a where "l = gcd l l' * a" unfolding dvd_def by blast
1411         with l \<noteq> 0 have "a \<noteq> 0" by auto
1412         hence "euclidean_size (gcd l l') \<le> euclidean_size (gcd l l' * a)"
1413           by (rule size_mult_mono)
1414         also have "gcd l l' * a = l" using l = gcd l l' * a ..
1415         also note euclidean_size l = n
1416         finally show "euclidean_size (gcd l l') \<le> n" .
1417       qed
1418       ultimately have "euclidean_size l = euclidean_size (gcd l l')"
1419         by (intro le_antisym, simp_all add: euclidean_size l = n)
1420       with l \<noteq> 0 have "l dvd gcd l l'" by (blast intro: dvd_euclidean_size_eq_imp_dvd)
1421       hence "l dvd l'" by (blast dest: dvd_gcd_D2)
1422     }
1424     with (\<forall>a\<in>A. a dvd l) and normalization_factor_is_unit[OF l \<noteq> 0] and l \<noteq> 0
1425       have "(\<forall>a\<in>A. a dvd l div normalization_factor l) \<and>
1426         (\<forall>l'. (\<forall>a\<in>A. a dvd l') \<longrightarrow> l div normalization_factor l dvd l') \<and>
1427         normalization_factor (l div normalization_factor l) =
1428         (if l div normalization_factor l = 0 then 0 else 1)"
1429       by (auto simp: unit_simps)
1430     also from True have "l div normalization_factor l = Lcm A"
1431       by (simp add: Lcm_Lcm_eucl Lcm_eucl_def Let_def n_def l_def)
1432     finally show ?thesis .
1433   qed
1434   note A = this
1436   {fix a assume "a \<in> A" then show "a dvd Lcm A" using A by blast}
1437   {fix l' assume "\<forall>a\<in>A. a dvd l'" then show "Lcm A dvd l'" using A by blast}
1438   from A show "normalization_factor (Lcm A) = (if Lcm A = 0 then 0 else 1)" by blast
1439 qed
1441 lemma LcmI:
1442   "(\<And>a. a\<in>A \<Longrightarrow> a dvd l) \<Longrightarrow> (\<And>l'. (\<forall>a\<in>A. a dvd l') \<Longrightarrow> l dvd l') \<Longrightarrow>
1443       normalization_factor l = (if l = 0 then 0 else 1) \<Longrightarrow> l = Lcm A"
1444   by (intro normed_associated_imp_eq)
1445     (auto intro: Lcm_dvd dvd_Lcm simp: associated_def)
1447 lemma Lcm_subset:
1448   "A \<subseteq> B \<Longrightarrow> Lcm A dvd Lcm B"
1449   by (blast intro: Lcm_dvd dvd_Lcm)
1451 lemma Lcm_Un:
1452   "Lcm (A \<union> B) = lcm (Lcm A) (Lcm B)"
1453   apply (rule lcmI)
1454   apply (blast intro: Lcm_subset)
1455   apply (blast intro: Lcm_subset)
1456   apply (intro Lcm_dvd ballI, elim UnE)
1457   apply (rule dvd_trans, erule dvd_Lcm, assumption)
1458   apply (rule dvd_trans, erule dvd_Lcm, assumption)
1459   apply simp
1460   done
1462 lemma Lcm_1_iff:
1463   "Lcm A = 1 \<longleftrightarrow> (\<forall>a\<in>A. is_unit a)"
1464 proof
1465   assume "Lcm A = 1"
1466   then show "\<forall>a\<in>A. is_unit a" by auto
1467 qed (rule LcmI [symmetric], auto)
1469 lemma Lcm_no_units:
1470   "Lcm A = Lcm (A - {a. is_unit a})"
1471 proof -
1472   have "(A - {a. is_unit a}) \<union> {a\<in>A. is_unit a} = A" by blast
1473   hence "Lcm A = lcm (Lcm (A - {a. is_unit a})) (Lcm {a\<in>A. is_unit a})"
1474     by (simp add: Lcm_Un[symmetric])
1475   also have "Lcm {a\<in>A. is_unit a} = 1" by (simp add: Lcm_1_iff)
1476   finally show ?thesis by simp
1477 qed
1479 lemma Lcm_empty [simp]:
1480   "Lcm {} = 1"
1481   by (simp add: Lcm_1_iff)
1483 lemma Lcm_eq_0 [simp]:
1484   "0 \<in> A \<Longrightarrow> Lcm A = 0"
1485   by (drule dvd_Lcm) simp
1487 lemma Lcm0_iff':
1488   "Lcm A = 0 \<longleftrightarrow> \<not>(\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l))"
1489 proof
1490   assume "Lcm A = 0"
1491   show "\<not>(\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l))"
1492   proof
1493     assume ex: "\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l)"
1494     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
1495     def n \<equiv> "LEAST n. \<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"
1496     def l \<equiv> "SOME l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"
1497     have "\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"
1498       apply (subst n_def)
1499       apply (rule LeastI[of _ "euclidean_size l\<^sub>0"])
1500       apply (rule exI[of _ l\<^sub>0])
1501       apply (simp add: l\<^sub>0_props)
1502       done
1503     from someI_ex[OF this] have "l \<noteq> 0" unfolding l_def by simp_all
1504     hence "l div normalization_factor l \<noteq> 0" by simp
1505     also from ex have "l div normalization_factor l = Lcm A"
1506        by (simp only: Lcm_Lcm_eucl Lcm_eucl_def n_def l_def if_True Let_def)
1507     finally show False using Lcm A = 0 by contradiction
1508   qed
1509 qed (simp only: Lcm_Lcm_eucl Lcm_eucl_def if_False)
1511 lemma Lcm0_iff [simp]:
1512   "finite A \<Longrightarrow> Lcm A = 0 \<longleftrightarrow> 0 \<in> A"
1513 proof -
1514   assume "finite A"
1515   have "0 \<in> A \<Longrightarrow> Lcm A = 0"  by (intro dvd_0_left dvd_Lcm)
1516   moreover {
1517     assume "0 \<notin> A"
1518     hence "\<Prod>A \<noteq> 0"
1519       apply (induct rule: finite_induct[OF finite A])
1520       apply simp
1521       apply (subst setprod.insert, assumption, assumption)
1522       apply (rule no_zero_divisors)
1523       apply blast+
1524       done
1525     moreover from finite A have "\<forall>a\<in>A. a dvd \<Prod>A" by blast
1526     ultimately have "\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l)" by blast
1527     with Lcm0_iff' have "Lcm A \<noteq> 0" by simp
1528   }
1529   ultimately show "Lcm A = 0 \<longleftrightarrow> 0 \<in> A" by blast
1530 qed
1532 lemma Lcm_no_multiple:
1533   "(\<forall>m. m \<noteq> 0 \<longrightarrow> (\<exists>a\<in>A. \<not>a dvd m)) \<Longrightarrow> Lcm A = 0"
1534 proof -
1535   assume "\<forall>m. m \<noteq> 0 \<longrightarrow> (\<exists>a\<in>A. \<not>a dvd m)"
1536   hence "\<not>(\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l))" by blast
1537   then show "Lcm A = 0" by (simp only: Lcm_Lcm_eucl Lcm_eucl_def if_False)
1538 qed
1540 lemma Lcm_insert [simp]:
1541   "Lcm (insert a A) = lcm a (Lcm A)"
1542 proof (rule lcmI)
1543   fix l assume "a dvd l" and "Lcm A dvd l"
1544   hence "\<forall>a\<in>A. a dvd l" by (blast intro: dvd_trans dvd_Lcm)
1545   with a dvd l show "Lcm (insert a A) dvd l" by (force intro: Lcm_dvd)
1546 qed (auto intro: Lcm_dvd dvd_Lcm)
1548 lemma Lcm_finite:
1549   assumes "finite A"
1550   shows "Lcm A = Finite_Set.fold lcm 1 A"
1551   by (induct rule: finite.induct[OF finite A])
1552     (simp_all add: comp_fun_idem.fold_insert_idem[OF comp_fun_idem_lcm])
1554 lemma Lcm_set [code_unfold]:
1555   "Lcm (set xs) = fold lcm xs 1"
1556   using comp_fun_idem.fold_set_fold[OF comp_fun_idem_lcm] Lcm_finite by (simp add: ac_simps)
1558 lemma Lcm_singleton [simp]:
1559   "Lcm {a} = a div normalization_factor a"
1560   by simp
1562 lemma Lcm_2 [simp]:
1563   "Lcm {a,b} = lcm a b"
1564   by (simp only: Lcm_insert Lcm_empty lcm_1_right)
1565     (cases "b = 0", simp, rule lcm_div_unit2, simp)
1567 lemma Lcm_coprime:
1568   assumes "finite A" and "A \<noteq> {}"
1569   assumes "\<And>a b. a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<noteq> b \<Longrightarrow> gcd a b = 1"
1570   shows "Lcm A = \<Prod>A div normalization_factor (\<Prod>A)"
1571 using assms proof (induct rule: finite_ne_induct)
1572   case (insert a A)
1573   have "Lcm (insert a A) = lcm a (Lcm A)" by simp
1574   also from insert have "Lcm A = \<Prod>A div normalization_factor (\<Prod>A)" by blast
1575   also have "lcm a \<dots> = lcm a (\<Prod>A)" by (cases "\<Prod>A = 0") (simp_all add: lcm_div_unit2)
1576   also from insert have "gcd a (\<Prod>A) = 1" by (subst gcd.commute, intro setprod_coprime) auto
1577   with insert have "lcm a (\<Prod>A) = \<Prod>(insert a A) div normalization_factor (\<Prod>(insert a A))"
1578     by (simp add: lcm_coprime)
1579   finally show ?case .
1580 qed simp
1582 lemma Lcm_coprime':
1583   "card A \<noteq> 0 \<Longrightarrow> (\<And>a b. a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<noteq> b \<Longrightarrow> gcd a b = 1)
1584     \<Longrightarrow> Lcm A = \<Prod>A div normalization_factor (\<Prod>A)"
1585   by (rule Lcm_coprime) (simp_all add: card_eq_0_iff)
1587 lemma Gcd_Lcm:
1588   "Gcd A = Lcm {d. \<forall>a\<in>A. d dvd a}"
1589   by (simp add: Gcd_Gcd_eucl Lcm_Lcm_eucl Gcd_eucl_def)
1591 lemma Gcd_dvd [simp]: "a \<in> A \<Longrightarrow> Gcd A dvd a"
1592   and dvd_Gcd [simp]: "(\<forall>a\<in>A. g' dvd a) \<Longrightarrow> g' dvd Gcd A"
1593   and normalization_factor_Gcd [simp]:
1594     "normalization_factor (Gcd A) = (if Gcd A = 0 then 0 else 1)"
1595 proof -
1596   fix a assume "a \<in> A"
1597   hence "Lcm {d. \<forall>a\<in>A. d dvd a} dvd a" by (intro Lcm_dvd) blast
1598   then show "Gcd A dvd a" by (simp add: Gcd_Lcm)
1599 next
1600   fix g' assume "\<forall>a\<in>A. g' dvd a"
1601   hence "g' dvd Lcm {d. \<forall>a\<in>A. d dvd a}" by (intro dvd_Lcm) blast
1602   then show "g' dvd Gcd A" by (simp add: Gcd_Lcm)
1603 next
1604   show "normalization_factor (Gcd A) = (if Gcd A = 0 then 0 else 1)"
1605     by (simp add: Gcd_Lcm)
1606 qed
1608 lemma GcdI:
1609   "(\<And>a. a\<in>A \<Longrightarrow> l dvd a) \<Longrightarrow> (\<And>l'. (\<forall>a\<in>A. l' dvd a) \<Longrightarrow> l' dvd l) \<Longrightarrow>
1610     normalization_factor l = (if l = 0 then 0 else 1) \<Longrightarrow> l = Gcd A"
1611   by (intro normed_associated_imp_eq)
1612     (auto intro: Gcd_dvd dvd_Gcd simp: associated_def)
1614 lemma Lcm_Gcd:
1615   "Lcm A = Gcd {m. \<forall>a\<in>A. a dvd m}"
1616   by (rule LcmI[symmetric]) (auto intro: dvd_Gcd Gcd_dvd)
1618 lemma Gcd_0_iff:
1619   "Gcd A = 0 \<longleftrightarrow> A \<subseteq> {0}"
1620   apply (rule iffI)
1621   apply (rule subsetI, drule Gcd_dvd, simp)
1622   apply (auto intro: GcdI[symmetric])
1623   done
1625 lemma Gcd_empty [simp]:
1626   "Gcd {} = 0"
1627   by (simp add: Gcd_0_iff)
1629 lemma Gcd_1:
1630   "1 \<in> A \<Longrightarrow> Gcd A = 1"
1631   by (intro GcdI[symmetric]) (auto intro: Gcd_dvd dvd_Gcd)
1633 lemma Gcd_insert [simp]:
1634   "Gcd (insert a A) = gcd a (Gcd A)"
1635 proof (rule gcdI)
1636   fix l assume "l dvd a" and "l dvd Gcd A"
1637   hence "\<forall>a\<in>A. l dvd a" by (blast intro: dvd_trans Gcd_dvd)
1638   with l dvd a show "l dvd Gcd (insert a A)" by (force intro: Gcd_dvd)
1639 qed auto
1641 lemma Gcd_finite:
1642   assumes "finite A"
1643   shows "Gcd A = Finite_Set.fold gcd 0 A"
1644   by (induct rule: finite.induct[OF finite A])
1645     (simp_all add: comp_fun_idem.fold_insert_idem[OF comp_fun_idem_gcd])
1647 lemma Gcd_set [code_unfold]:
1648   "Gcd (set xs) = fold gcd xs 0"
1649   using comp_fun_idem.fold_set_fold[OF comp_fun_idem_gcd] Gcd_finite by (simp add: ac_simps)
1651 lemma Gcd_singleton [simp]: "Gcd {a} = a div normalization_factor a"
1652   by (simp add: gcd_0)
1654 lemma Gcd_2 [simp]: "Gcd {a,b} = gcd a b"
1655   by (simp only: Gcd_insert Gcd_empty gcd_0) (cases "b = 0", simp, rule gcd_div_unit2, simp)
1657 end
1659 text {*
1660   A Euclidean ring is a Euclidean semiring with additive inverses. It provides a
1661   few more lemmas; in particular, Bezout's lemma holds for any Euclidean ring.
1662 *}
1664 class euclidean_ring = euclidean_semiring + idom
1666 class euclidean_ring_gcd = euclidean_semiring_gcd + idom
1667 begin
1669 subclass euclidean_ring ..
1671 lemma gcd_neg1 [simp]:
1672   "gcd (-a) b = gcd a b"
1673   by (rule sym, rule gcdI, simp_all add: gcd_greatest)
1675 lemma gcd_neg2 [simp]:
1676   "gcd a (-b) = gcd a b"
1677   by (rule sym, rule gcdI, simp_all add: gcd_greatest)
1679 lemma gcd_neg_numeral_1 [simp]:
1680   "gcd (- numeral n) a = gcd (numeral n) a"
1681   by (fact gcd_neg1)
1683 lemma gcd_neg_numeral_2 [simp]:
1684   "gcd a (- numeral n) = gcd a (numeral n)"
1685   by (fact gcd_neg2)
1687 lemma gcd_diff1: "gcd (m - n) n = gcd m n"
1688   by (subst diff_conv_add_uminus, subst gcd_neg2[symmetric],  subst gcd_add1, simp)
1690 lemma gcd_diff2: "gcd (n - m) n = gcd m n"
1691   by (subst gcd_neg1[symmetric], simp only: minus_diff_eq gcd_diff1)
1693 lemma coprime_minus_one [simp]: "gcd (n - 1) n = 1"
1694 proof -
1695   have "gcd (n - 1) n = gcd n (n - 1)" by (fact gcd.commute)
1696   also have "\<dots> = gcd ((n - 1) + 1) (n - 1)" by simp
1697   also have "\<dots> = 1" by (rule coprime_plus_one)
1698   finally show ?thesis .
1699 qed
1701 lemma lcm_neg1 [simp]: "lcm (-a) b = lcm a b"
1702   by (rule sym, rule lcmI, simp_all add: lcm_least lcm_zero)
1704 lemma lcm_neg2 [simp]: "lcm a (-b) = lcm a b"
1705   by (rule sym, rule lcmI, simp_all add: lcm_least lcm_zero)
1707 lemma lcm_neg_numeral_1 [simp]: "lcm (- numeral n) a = lcm (numeral n) a"
1708   by (fact lcm_neg1)
1710 lemma lcm_neg_numeral_2 [simp]: "lcm a (- numeral n) = lcm a (numeral n)"
1711   by (fact lcm_neg2)
1713 function euclid_ext :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a" where
1714   "euclid_ext a b =
1715      (if b = 0 then
1716         let c = 1 div normalization_factor a in (c, 0, a * c)
1717       else
1718         case euclid_ext b (a mod b) of
1719             (s,t,c) \<Rightarrow> (t, s - t * (a div b), c))"
1720   by (pat_completeness, simp)
1721   termination by (relation "measure (euclidean_size \<circ> snd)", simp_all)
1723 declare euclid_ext.simps [simp del]
1725 lemma euclid_ext_0:
1726   "euclid_ext a 0 = (1 div normalization_factor a, 0, a div normalization_factor a)"
1727   by (subst euclid_ext.simps) (simp add: Let_def)
1729 lemma euclid_ext_non_0:
1730   "b \<noteq> 0 \<Longrightarrow> euclid_ext a b = (case euclid_ext b (a mod b) of
1731     (s,t,c) \<Rightarrow> (t, s - t * (a div b), c))"
1732   by (subst euclid_ext.simps) simp
1734 definition euclid_ext' :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a"
1735 where
1736   "euclid_ext' a b = (case euclid_ext a b of (s, t, _) \<Rightarrow> (s, t))"
1738 lemma euclid_ext_gcd [simp]:
1739   "(case euclid_ext a b of (_,_,t) \<Rightarrow> t) = gcd a b"
1740 proof (induct a b rule: euclid_ext.induct)
1741   case (1 a b)
1742   then show ?case
1743   proof (cases "b = 0")
1744     case True
1745       then show ?thesis by
1746         (simp add: euclid_ext_0 unit_div mult_ac unit_simps gcd_0)
1747     next
1748     case False with 1 show ?thesis
1749       by (simp add: euclid_ext_non_0 ac_simps split: prod.split prod.split_asm)
1750     qed
1751 qed
1753 lemma euclid_ext_gcd' [simp]:
1754   "euclid_ext a b = (r, s, t) \<Longrightarrow> t = gcd a b"
1755   by (insert euclid_ext_gcd[of a b], drule (1) subst, simp)
1757 lemma euclid_ext_correct:
1758   "case euclid_ext a b of (s,t,c) \<Rightarrow> s*a + t*b = c"
1759 proof (induct a b rule: euclid_ext.induct)
1760   case (1 a b)
1761   show ?case
1762   proof (cases "b = 0")
1763     case True
1764     then show ?thesis by (simp add: euclid_ext_0 mult_ac)
1765   next
1766     case False
1767     obtain s t c where stc: "euclid_ext b (a mod b) = (s,t,c)"
1768       by (cases "euclid_ext b (a mod b)", blast)
1769     from 1 have "c = s * b + t * (a mod b)" by (simp add: stc False)
1770     also have "... = t*((a div b)*b + a mod b) + (s - t * (a div b))*b"
1771       by (simp add: algebra_simps)
1772     also have "(a div b)*b + a mod b = a" using mod_div_equality .
1773     finally show ?thesis
1774       by (subst euclid_ext.simps, simp add: False stc)
1775     qed
1776 qed
1778 lemma euclid_ext'_correct:
1779   "fst (euclid_ext' a b) * a + snd (euclid_ext' a b) * b = gcd a b"
1780 proof-
1781   obtain s t c where "euclid_ext a b = (s,t,c)"
1782     by (cases "euclid_ext a b", blast)
1783   with euclid_ext_correct[of a b] euclid_ext_gcd[of a b]
1784     show ?thesis unfolding euclid_ext'_def by simp
1785 qed
1787 lemma bezout: "\<exists>s t. s * a + t * b = gcd a b"
1788   using euclid_ext'_correct by blast
1790 lemma euclid_ext'_0 [simp]: "euclid_ext' a 0 = (1 div normalization_factor a, 0)"
1791   by (simp add: bezw_def euclid_ext'_def euclid_ext_0)
1793 lemma euclid_ext'_non_0: "b \<noteq> 0 \<Longrightarrow> euclid_ext' a b = (snd (euclid_ext' b (a mod b)),
1794   fst (euclid_ext' b (a mod b)) - snd (euclid_ext' b (a mod b)) * (a div b))"
1795   by (cases "euclid_ext b (a mod b)")
1796     (simp add: euclid_ext'_def euclid_ext_non_0)
1798 end
1800 instantiation nat :: euclidean_semiring
1801 begin
1803 definition [simp]:
1804   "euclidean_size_nat = (id :: nat \<Rightarrow> nat)"
1806 definition [simp]:
1807   "normalization_factor_nat (n::nat) = (if n = 0 then 0 else 1 :: nat)"
1809 instance proof
1810 qed simp_all
1812 end
1814 instantiation int :: euclidean_ring
1815 begin
1817 definition [simp]:
1818   "euclidean_size_int = (nat \<circ> abs :: int \<Rightarrow> nat)"
1820 definition [simp]:
1821   "normalization_factor_int = (sgn :: int \<Rightarrow> int)"
1823 instance proof
1824   case goal2 then show ?case by (auto simp add: abs_mult nat_mult_distrib)
1825 next
1826   case goal3 then show ?case by (simp add: zsgn_def)
1827 next
1828   case goal5 then show ?case by (auto simp: zsgn_def)
1829 next
1830   case goal6 then show ?case by (auto split: abs_split simp: zsgn_def)
1831 qed (auto simp: sgn_times split: abs_split)
1833 end
1835 end