src/HOL/Number_Theory/Euclidean_Algorithm.thy
 author haftmann Sat Jun 27 20:20:36 2015 +0200 (2015-06-27) changeset 60599 f8bb070dc98b parent 60598 78ca5674c66a child 60600 87fbfea0bd0a permissions -rw-r--r--
tuned proof
1 (* Author: Manuel Eberl *)
3 section \<open>Abstract euclidean algorithm\<close>
5 theory Euclidean_Algorithm
6 imports Complex_Main "~~/src/HOL/Library/Polynomial"
7 begin
9 text \<open>
10   A Euclidean semiring is a semiring upon which the Euclidean algorithm can be
11   implemented. It must provide:
12   \begin{itemize}
13   \item division with remainder
14   \item a size function such that @{term "size (a mod b) < size b"}
15         for any @{term "b \<noteq> 0"}
16   \item a normalization factor such that two associated numbers are equal iff
17         they are the same when divd by their normalization factors.
18   \end{itemize}
19   The existence of these functions makes it possible to derive gcd and lcm functions
20   for any Euclidean semiring.
21 \<close>
22 class euclidean_semiring = semiring_div +
23   fixes euclidean_size :: "'a \<Rightarrow> nat"
24   fixes normalization_factor :: "'a \<Rightarrow> 'a"
25   assumes mod_size_less:
26     "b \<noteq> 0 \<Longrightarrow> \<not> b dvd a \<Longrightarrow> euclidean_size (a mod b) < euclidean_size b"
27   assumes size_mult_mono:
28     "b \<noteq> 0 \<Longrightarrow> euclidean_size (a * b) \<ge> euclidean_size a"
29   assumes normalization_factor_is_unit [intro,simp]:
30     "a \<noteq> 0 \<Longrightarrow> is_unit (normalization_factor a)"
31   assumes normalization_factor_mult: "normalization_factor (a * b) =
32     normalization_factor a * normalization_factor b"
33   assumes normalization_factor_unit: "is_unit a \<Longrightarrow> normalization_factor a = a"
34   assumes normalization_factor_0 [simp]: "normalization_factor 0 = 0"
35 begin
37 lemma normalization_factor_dvd [simp]:
38   "a \<noteq> 0 \<Longrightarrow> normalization_factor a dvd b"
39   by (rule unit_imp_dvd, simp)
41 lemma normalization_factor_1 [simp]:
42   "normalization_factor 1 = 1"
43   by (simp add: normalization_factor_unit)
45 lemma normalization_factor_0_iff [simp]:
46   "normalization_factor a = 0 \<longleftrightarrow> a = 0"
47 proof
48   assume "normalization_factor a = 0"
49   hence "\<not> is_unit (normalization_factor a)"
50     by simp
51   then show "a = 0" by auto
52 qed simp
54 lemma normalization_factor_pow:
55   "normalization_factor (a ^ n) = normalization_factor a ^ n"
56   by (induct n) (simp_all add: normalization_factor_mult power_Suc2)
58 lemma normalization_correct [simp]:
59   "normalization_factor (a div normalization_factor a) = (if a = 0 then 0 else 1)"
60 proof (cases "a = 0", simp)
61   assume "a \<noteq> 0"
62   let ?nf = "normalization_factor"
63   from normalization_factor_is_unit[OF \<open>a \<noteq> 0\<close>] have "?nf a \<noteq> 0"
64     by auto
65   have "?nf (a div ?nf a) * ?nf (?nf a) = ?nf (a div ?nf a * ?nf a)"
66     by (simp add: normalization_factor_mult)
67   also have "a div ?nf a * ?nf a = a" using \<open>a \<noteq> 0\<close>
68     by simp
69   also have "?nf (?nf a) = ?nf a" using \<open>a \<noteq> 0\<close>
70     normalization_factor_is_unit normalization_factor_unit by simp
71   finally have "normalization_factor (a div normalization_factor a) = 1"
72     using \<open>?nf a \<noteq> 0\<close> by (metis div_mult_self2_is_id div_self)
73   with \<open>a \<noteq> 0\<close> show ?thesis by simp
74 qed
76 lemma normalization_0_iff [simp]:
77   "a div normalization_factor a = 0 \<longleftrightarrow> a = 0"
78   by (cases "a = 0", simp, subst unit_eq_div1, blast, simp)
80 lemma mult_div_normalization [simp]:
81   "b * (1 div normalization_factor a) = b div normalization_factor a"
82   by (cases "a = 0") simp_all
84 lemma associated_iff_normed_eq:
85   "associated a b \<longleftrightarrow> a div normalization_factor a = b div normalization_factor b" (is "?P \<longleftrightarrow> ?Q")
86 proof (cases "a = 0 \<or> b = 0")
87   case True then show ?thesis by (auto dest: sym)
88 next
89   case False then have "a \<noteq> 0" and "b \<noteq> 0" by auto
90   show ?thesis
91   proof
92     assume ?Q
93     from \<open>a \<noteq> 0\<close> \<open>b \<noteq> 0\<close>
94     have "is_unit (normalization_factor a div normalization_factor b)"
95       by auto
96     moreover from \<open>?Q\<close> \<open>a \<noteq> 0\<close> \<open>b \<noteq> 0\<close>
97     have "a = (normalization_factor a div normalization_factor b) * b"
98       by (simp add: ac_simps div_mult_swap unit_eq_div1)
99     ultimately show "associated a b" by (rule is_unit_associatedI)
100   next
101     assume ?P
102     then obtain c where "is_unit c" and "a = c * b"
103       by (blast elim: associated_is_unitE)
104     then show ?Q
105       by (auto simp add: normalization_factor_mult normalization_factor_unit)
106   qed
107 qed
109 lemma normed_associated_imp_eq:
110   "associated a b \<Longrightarrow> normalization_factor a \<in> {0, 1} \<Longrightarrow> normalization_factor b \<in> {0, 1} \<Longrightarrow> a = b"
111   by (simp add: associated_iff_normed_eq, elim disjE, simp_all)
113 lemma normed_dvd [iff]:
114   "a div normalization_factor a dvd a"
115 proof (cases "a = 0")
116   case True then show ?thesis by simp
117 next
118   case False
119   then have "a = a div normalization_factor a * normalization_factor a"
120     by (auto intro: unit_div_mult_self)
121   then show ?thesis ..
122 qed
124 lemma dvd_normed [iff]:
125   "a dvd a div normalization_factor a"
126 proof (cases "a = 0")
127   case True then show ?thesis by simp
128 next
129   case False
130   then have "a div normalization_factor a = a * (1 div normalization_factor a)"
131     by (auto intro: unit_mult_div_div)
132   then show ?thesis ..
133 qed
135 lemma associated_normed:
136   "associated (a div normalization_factor a) a"
137   by (rule associatedI) simp_all
139 lemma normalization_factor_dvd' [simp]:
140   "normalization_factor a dvd a"
141   by (cases "a = 0", simp_all)
143 lemmas normalization_factor_dvd_iff [simp] =
144   unit_dvd_iff [OF normalization_factor_is_unit]
146 lemma euclidean_division:
147   fixes a :: 'a and b :: 'a
148   assumes "b \<noteq> 0" and "\<not> b dvd a"
149   obtains s and t where "a = s * b + t"
150     and "euclidean_size t < euclidean_size b"
151 proof -
152   from div_mod_equality [of a b 0]
153      have "a = a div b * b + a mod b" by simp
154   with that and assms show ?thesis by (auto simp add: mod_size_less)
155 qed
157 lemma dvd_euclidean_size_eq_imp_dvd:
158   assumes "a \<noteq> 0" and b_dvd_a: "b dvd a" and size_eq: "euclidean_size a = euclidean_size b"
159   shows "a dvd b"
160 proof (rule ccontr)
161   assume "\<not> a dvd b"
162   then have "b mod a \<noteq> 0" by (simp add: mod_eq_0_iff_dvd)
163   from b_dvd_a have b_dvd_mod: "b dvd b mod a" by (simp add: dvd_mod_iff)
164   from b_dvd_mod obtain c where "b mod a = b * c" unfolding dvd_def by blast
165     with \<open>b mod a \<noteq> 0\<close> have "c \<noteq> 0" by auto
166   with \<open>b mod a = b * c\<close> have "euclidean_size (b mod a) \<ge> euclidean_size b"
167       using size_mult_mono by force
168   moreover from \<open>\<not> a dvd b\<close> and \<open>a \<noteq> 0\<close>
169   have "euclidean_size (b mod a) < euclidean_size a"
170       using mod_size_less by blast
171   ultimately show False using size_eq by simp
172 qed
174 function gcd_eucl :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
175 where
176   "gcd_eucl a b = (if b = 0 then a div normalization_factor a
177     else if b dvd a then b div normalization_factor b
178     else gcd_eucl b (a mod b))"
179   by pat_completeness simp
180 termination
181   by (relation "measure (euclidean_size \<circ> snd)") (simp_all add: mod_size_less)
183 declare gcd_eucl.simps [simp del]
185 lemma gcd_eucl_induct [case_names zero mod]:
186   assumes H1: "\<And>b. P b 0"
187   and H2: "\<And>a b. b \<noteq> 0 \<Longrightarrow> P b (a mod b) \<Longrightarrow> P a b"
188   shows "P a b"
189 proof (induct a b rule: gcd_eucl.induct)
190   case ("1" a b)
191   show ?case
192   proof (cases "b = 0")
193     case True then show "P a b" by simp (rule H1)
194   next
195     case False
196     have "P b (a mod b)"
197     proof (cases "b dvd a")
198       case False with \<open>b \<noteq> 0\<close> show "P b (a mod b)"
199         by (rule "1.hyps")
200     next
201       case True then have "a mod b = 0"
202         by (simp add: mod_eq_0_iff_dvd)
203       then show "P b (a mod b)" by simp (rule H1)
204     qed
205     with \<open>b \<noteq> 0\<close> show "P a b"
206       by (blast intro: H2)
207   qed
208 qed
210 definition lcm_eucl :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
211 where
212   "lcm_eucl a b = a * b div (gcd_eucl a b * normalization_factor (a * b))"
214 definition Lcm_eucl :: "'a set \<Rightarrow> 'a" -- \<open>
215   Somewhat complicated definition of Lcm that has the advantage of working
216   for infinite sets as well\<close>
217 where
218   "Lcm_eucl A = (if \<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) then
219      let l = SOME l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l =
220        (LEAST n. \<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n)
221        in l div normalization_factor l
222       else 0)"
224 definition Gcd_eucl :: "'a set \<Rightarrow> 'a"
225 where
226   "Gcd_eucl A = Lcm_eucl {d. \<forall>a\<in>A. d dvd a}"
228 lemma gcd_eucl_0:
229   "gcd_eucl a 0 = a div normalization_factor a"
230   by (simp add: gcd_eucl.simps [of a 0])
232 lemma gcd_eucl_0_left:
233   "gcd_eucl 0 a = a div normalization_factor a"
234   by (simp add: gcd_eucl.simps [of 0 a])
236 lemma gcd_eucl_non_0:
237   "b \<noteq> 0 \<Longrightarrow> gcd_eucl a b = gcd_eucl b (a mod b)"
238   by (cases "b dvd a")
239     (simp_all add: gcd_eucl.simps [of a b] gcd_eucl.simps [of b 0])
241 lemma gcd_eucl_code [code]:
242   "gcd_eucl a b = (if b = 0 then a div normalization_factor a else gcd_eucl b (a mod b))"
243   by (auto simp add: gcd_eucl_non_0 gcd_eucl_0 gcd_eucl_0_left)
245 end
247 class euclidean_ring = euclidean_semiring + idom
248 begin
250 function euclid_ext :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a" where
251   "euclid_ext a b =
252      (if b = 0 then
253         let c = 1 div normalization_factor a in (c, 0, a * c)
254       else if b dvd a then
255         let c = 1 div normalization_factor b in (0, c, b * c)
256       else
257         case euclid_ext b (a mod b) of
258             (s, t, c) \<Rightarrow> (t, s - t * (a div b), c))"
259   by pat_completeness simp
260 termination
261   by (relation "measure (euclidean_size \<circ> snd)") (simp_all add: mod_size_less)
263 declare euclid_ext.simps [simp del]
265 lemma euclid_ext_0:
266   "euclid_ext a 0 = (1 div normalization_factor a, 0, a div normalization_factor a)"
267   by (simp add: euclid_ext.simps [of a 0])
269 lemma euclid_ext_left_0:
270   "euclid_ext 0 a = (0, 1 div normalization_factor a, a div normalization_factor a)"
271   by (simp add: euclid_ext.simps [of 0 a])
273 lemma euclid_ext_non_0:
274   "b \<noteq> 0 \<Longrightarrow> euclid_ext a b = (case euclid_ext b (a mod b) of
275     (s, t, c) \<Rightarrow> (t, s - t * (a div b), c))"
276   by (cases "b dvd a")
277     (simp_all add: euclid_ext.simps [of a b] euclid_ext.simps [of b 0])
279 lemma euclid_ext_code [code]:
280   "euclid_ext a b = (if b = 0 then (1 div normalization_factor a, 0, a div normalization_factor a)
281     else let (s, t, c) = euclid_ext b (a mod b) in  (t, s - t * (a div b), c))"
282   by (simp add: euclid_ext.simps [of a b] euclid_ext.simps [of b 0])
284 lemma euclid_ext_correct:
285   "case euclid_ext a b of (s, t, c) \<Rightarrow> s * a + t * b = c"
286 proof (induct a b rule: gcd_eucl_induct)
287   case (zero a) then show ?case
288     by (simp add: euclid_ext_0 ac_simps)
289 next
290   case (mod a b)
291   obtain s t c where stc: "euclid_ext b (a mod b) = (s,t,c)"
292     by (cases "euclid_ext b (a mod b)") blast
293   with mod have "c = s * b + t * (a mod b)" by simp
294   also have "... = t * ((a div b) * b + a mod b) + (s - t * (a div b)) * b"
295     by (simp add: algebra_simps)
296   also have "(a div b) * b + a mod b = a" using mod_div_equality .
297   finally show ?case
298     by (subst euclid_ext.simps) (simp add: stc mod ac_simps)
299 qed
301 definition euclid_ext' :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a"
302 where
303   "euclid_ext' a b = (case euclid_ext a b of (s, t, _) \<Rightarrow> (s, t))"
305 lemma euclid_ext'_0: "euclid_ext' a 0 = (1 div normalization_factor a, 0)"
306   by (simp add: euclid_ext'_def euclid_ext_0)
308 lemma euclid_ext'_left_0: "euclid_ext' 0 a = (0, 1 div normalization_factor a)"
309   by (simp add: euclid_ext'_def euclid_ext_left_0)
311 lemma euclid_ext'_non_0: "b \<noteq> 0 \<Longrightarrow> euclid_ext' a b = (snd (euclid_ext' b (a mod b)),
312   fst (euclid_ext' b (a mod b)) - snd (euclid_ext' b (a mod b)) * (a div b))"
313   by (simp add: euclid_ext'_def euclid_ext_non_0 split_def)
315 end
317 class euclidean_semiring_gcd = euclidean_semiring + gcd + Gcd +
318   assumes gcd_gcd_eucl: "gcd = gcd_eucl" and lcm_lcm_eucl: "lcm = lcm_eucl"
319   assumes Gcd_Gcd_eucl: "Gcd = Gcd_eucl" and Lcm_Lcm_eucl: "Lcm = Lcm_eucl"
320 begin
322 lemma gcd_0_left:
323   "gcd 0 a = a div normalization_factor a"
324   unfolding gcd_gcd_eucl by (fact gcd_eucl_0_left)
326 lemma gcd_0:
327   "gcd a 0 = a div normalization_factor a"
328   unfolding gcd_gcd_eucl by (fact gcd_eucl_0)
330 lemma gcd_non_0:
331   "b \<noteq> 0 \<Longrightarrow> gcd a b = gcd b (a mod b)"
332   unfolding gcd_gcd_eucl by (fact gcd_eucl_non_0)
334 lemma gcd_dvd1 [iff]: "gcd a b dvd a"
335   and gcd_dvd2 [iff]: "gcd a b dvd b"
336   by (induct a b rule: gcd_eucl_induct)
337     (simp_all add: gcd_0 gcd_non_0 dvd_mod_iff)
339 lemma dvd_gcd_D1: "k dvd gcd m n \<Longrightarrow> k dvd m"
340   by (rule dvd_trans, assumption, rule gcd_dvd1)
342 lemma dvd_gcd_D2: "k dvd gcd m n \<Longrightarrow> k dvd n"
343   by (rule dvd_trans, assumption, rule gcd_dvd2)
345 lemma gcd_greatest:
346   fixes k a b :: 'a
347   shows "k dvd a \<Longrightarrow> k dvd b \<Longrightarrow> k dvd gcd a b"
348 proof (induct a b rule: gcd_eucl_induct)
349   case (zero a) from zero(1) show ?case by (rule dvd_trans) (simp add: gcd_0)
350 next
351   case (mod a b)
352   then show ?case
353     by (simp add: gcd_non_0 dvd_mod_iff)
354 qed
356 lemma dvd_gcd_iff:
357   "k dvd gcd a b \<longleftrightarrow> k dvd a \<and> k dvd b"
358   by (blast intro!: gcd_greatest intro: dvd_trans)
360 lemmas gcd_greatest_iff = dvd_gcd_iff
362 lemma gcd_zero [simp]:
363   "gcd a b = 0 \<longleftrightarrow> a = 0 \<and> b = 0"
364   by (metis dvd_0_left dvd_refl gcd_dvd1 gcd_dvd2 gcd_greatest)+
366 lemma normalization_factor_gcd [simp]:
367   "normalization_factor (gcd a b) = (if a = 0 \<and> b = 0 then 0 else 1)" (is "?f a b = ?g a b")
368   by (induct a b rule: gcd_eucl_induct)
369     (auto simp add: gcd_0 gcd_non_0)
371 lemma gcdI:
372   "k dvd a \<Longrightarrow> k dvd b \<Longrightarrow> (\<And>l. l dvd a \<Longrightarrow> l dvd b \<Longrightarrow> l dvd k)
373     \<Longrightarrow> normalization_factor k = (if k = 0 then 0 else 1) \<Longrightarrow> k = gcd a b"
374   by (intro normed_associated_imp_eq) (auto simp: associated_def intro: gcd_greatest)
376 sublocale gcd!: abel_semigroup gcd
377 proof
378   fix a b c
379   show "gcd (gcd a b) c = gcd a (gcd b c)"
380   proof (rule gcdI)
381     have "gcd (gcd a b) c dvd gcd a b" "gcd a b dvd a" by simp_all
382     then show "gcd (gcd a b) c dvd a" by (rule dvd_trans)
383     have "gcd (gcd a b) c dvd gcd a b" "gcd a b dvd b" by simp_all
384     hence "gcd (gcd a b) c dvd b" by (rule dvd_trans)
385     moreover have "gcd (gcd a b) c dvd c" by simp
386     ultimately show "gcd (gcd a b) c dvd gcd b c"
387       by (rule gcd_greatest)
388     show "normalization_factor (gcd (gcd a b) c) =  (if gcd (gcd a b) c = 0 then 0 else 1)"
389       by auto
390     fix l assume "l dvd a" and "l dvd gcd b c"
391     with dvd_trans[OF _ gcd_dvd1] and dvd_trans[OF _ gcd_dvd2]
392       have "l dvd b" and "l dvd c" by blast+
393     with \<open>l dvd a\<close> show "l dvd gcd (gcd a b) c"
394       by (intro gcd_greatest)
395   qed
396 next
397   fix a b
398   show "gcd a b = gcd b a"
399     by (rule gcdI) (simp_all add: gcd_greatest)
400 qed
402 lemma gcd_unique: "d dvd a \<and> d dvd b \<and>
403     normalization_factor d = (if d = 0 then 0 else 1) \<and>
404     (\<forall>e. e dvd a \<and> e dvd b \<longrightarrow> e dvd d) \<longleftrightarrow> d = gcd a b"
405   by (rule, auto intro: gcdI simp: gcd_greatest)
407 lemma gcd_dvd_prod: "gcd a b dvd k * b"
408   using mult_dvd_mono [of 1] by auto
410 lemma gcd_1_left [simp]: "gcd 1 a = 1"
411   by (rule sym, rule gcdI, simp_all)
413 lemma gcd_1 [simp]: "gcd a 1 = 1"
414   by (rule sym, rule gcdI, simp_all)
416 lemma gcd_proj2_if_dvd:
417   "b dvd a \<Longrightarrow> gcd a b = b div normalization_factor b"
418   by (cases "b = 0", simp_all add: dvd_eq_mod_eq_0 gcd_non_0 gcd_0)
420 lemma gcd_proj1_if_dvd:
421   "a dvd b \<Longrightarrow> gcd a b = a div normalization_factor a"
422   by (subst gcd.commute, simp add: gcd_proj2_if_dvd)
424 lemma gcd_proj1_iff: "gcd m n = m div normalization_factor m \<longleftrightarrow> m dvd n"
425 proof
426   assume A: "gcd m n = m div normalization_factor m"
427   show "m dvd n"
428   proof (cases "m = 0")
429     assume [simp]: "m \<noteq> 0"
430     from A have B: "m = gcd m n * normalization_factor m"
431       by (simp add: unit_eq_div2)
432     show ?thesis by (subst B, simp add: mult_unit_dvd_iff)
433   qed (insert A, simp)
434 next
435   assume "m dvd n"
436   then show "gcd m n = m div normalization_factor m" by (rule gcd_proj1_if_dvd)
437 qed
439 lemma gcd_proj2_iff: "gcd m n = n div normalization_factor n \<longleftrightarrow> n dvd m"
440   by (subst gcd.commute, simp add: gcd_proj1_iff)
442 lemma gcd_mod1 [simp]:
443   "gcd (a mod b) b = gcd a b"
444   by (rule gcdI, metis dvd_mod_iff gcd_dvd1 gcd_dvd2, simp_all add: gcd_greatest dvd_mod_iff)
446 lemma gcd_mod2 [simp]:
447   "gcd a (b mod a) = gcd a b"
448   by (rule gcdI, simp, metis dvd_mod_iff gcd_dvd1 gcd_dvd2, simp_all add: gcd_greatest dvd_mod_iff)
450 lemma gcd_mult_distrib':
451   "c div normalization_factor c * gcd a b = gcd (c * a) (c * b)"
452 proof (cases "c = 0")
453   case True then show ?thesis by (simp_all add: gcd_0)
454 next
455   case False then have [simp]: "is_unit (normalization_factor c)" by simp
456   show ?thesis
457   proof (induct a b rule: gcd_eucl_induct)
458     case (zero a) show ?case
459     proof (cases "a = 0")
460       case True then show ?thesis by (simp add: gcd_0)
461     next
462       case False then have "is_unit (normalization_factor a)" by simp
463       then show ?thesis
464         by (simp add: gcd_0 unit_div_commute unit_div_mult_swap normalization_factor_mult is_unit_div_mult2_eq)
465     qed
466     case (mod a b)
467     then show ?case by (simp add: mult_mod_right gcd.commute)
468   qed
469 qed
471 lemma gcd_mult_distrib:
472   "k * gcd a b = gcd (k*a) (k*b) * normalization_factor k"
473 proof-
474   let ?nf = "normalization_factor"
475   from gcd_mult_distrib'
476     have "gcd (k*a) (k*b) = k div ?nf k * gcd a b" ..
477   also have "... = k * gcd a b div ?nf k"
478     by (metis dvd_div_mult dvd_eq_mod_eq_0 mod_0 normalization_factor_dvd)
479   finally show ?thesis
480     by simp
481 qed
483 lemma euclidean_size_gcd_le1 [simp]:
484   assumes "a \<noteq> 0"
485   shows "euclidean_size (gcd a b) \<le> euclidean_size a"
486 proof -
487    have "gcd a b dvd a" by (rule gcd_dvd1)
488    then obtain c where A: "a = gcd a b * c" unfolding dvd_def by blast
489    with \<open>a \<noteq> 0\<close> show ?thesis by (subst (2) A, intro size_mult_mono) auto
490 qed
492 lemma euclidean_size_gcd_le2 [simp]:
493   "b \<noteq> 0 \<Longrightarrow> euclidean_size (gcd a b) \<le> euclidean_size b"
494   by (subst gcd.commute, rule euclidean_size_gcd_le1)
496 lemma euclidean_size_gcd_less1:
497   assumes "a \<noteq> 0" and "\<not>a dvd b"
498   shows "euclidean_size (gcd a b) < euclidean_size a"
499 proof (rule ccontr)
500   assume "\<not>euclidean_size (gcd a b) < euclidean_size a"
501   with \<open>a \<noteq> 0\<close> have "euclidean_size (gcd a b) = euclidean_size a"
502     by (intro le_antisym, simp_all)
503   with assms have "a dvd gcd a b" by (auto intro: dvd_euclidean_size_eq_imp_dvd)
504   hence "a dvd b" using dvd_gcd_D2 by blast
505   with \<open>\<not>a dvd b\<close> show False by contradiction
506 qed
508 lemma euclidean_size_gcd_less2:
509   assumes "b \<noteq> 0" and "\<not>b dvd a"
510   shows "euclidean_size (gcd a b) < euclidean_size b"
511   using assms by (subst gcd.commute, rule euclidean_size_gcd_less1)
513 lemma gcd_mult_unit1: "is_unit a \<Longrightarrow> gcd (b * a) c = gcd b c"
514   apply (rule gcdI)
515   apply (rule dvd_trans, rule gcd_dvd1, simp add: unit_simps)
516   apply (rule gcd_dvd2)
517   apply (rule gcd_greatest, simp add: unit_simps, assumption)
518   apply (subst normalization_factor_gcd, simp add: gcd_0)
519   done
521 lemma gcd_mult_unit2: "is_unit a \<Longrightarrow> gcd b (c * a) = gcd b c"
522   by (subst gcd.commute, subst gcd_mult_unit1, assumption, rule gcd.commute)
524 lemma gcd_div_unit1: "is_unit a \<Longrightarrow> gcd (b div a) c = gcd b c"
525   by (erule is_unitE [of _ b]) (simp add: gcd_mult_unit1)
527 lemma gcd_div_unit2: "is_unit a \<Longrightarrow> gcd b (c div a) = gcd b c"
528   by (erule is_unitE [of _ c]) (simp add: gcd_mult_unit2)
530 lemma gcd_idem: "gcd a a = a div normalization_factor a"
531   by (cases "a = 0") (simp add: gcd_0_left, rule sym, rule gcdI, simp_all)
533 lemma gcd_right_idem: "gcd (gcd a b) b = gcd a b"
534   apply (rule gcdI)
535   apply (simp add: ac_simps)
536   apply (rule gcd_dvd2)
537   apply (rule gcd_greatest, erule (1) gcd_greatest, assumption)
538   apply simp
539   done
541 lemma gcd_left_idem: "gcd a (gcd a b) = gcd a b"
542   apply (rule gcdI)
543   apply simp
544   apply (rule dvd_trans, rule gcd_dvd2, rule gcd_dvd2)
545   apply (rule gcd_greatest, assumption, erule gcd_greatest, assumption)
546   apply simp
547   done
549 lemma comp_fun_idem_gcd: "comp_fun_idem gcd"
550 proof
551   fix a b show "gcd a \<circ> gcd b = gcd b \<circ> gcd a"
552     by (simp add: fun_eq_iff ac_simps)
553 next
554   fix a show "gcd a \<circ> gcd a = gcd a"
555     by (simp add: fun_eq_iff gcd_left_idem)
556 qed
558 lemma coprime_dvd_mult:
559   assumes "gcd c b = 1" and "c dvd a * b"
560   shows "c dvd a"
561 proof -
562   let ?nf = "normalization_factor"
563   from assms gcd_mult_distrib [of a c b]
564     have A: "a = gcd (a * c) (a * b) * ?nf a" by simp
565   from \<open>c dvd a * b\<close> show ?thesis by (subst A, simp_all add: gcd_greatest)
566 qed
568 lemma coprime_dvd_mult_iff:
569   "gcd c b = 1 \<Longrightarrow> (c dvd a * b) = (c dvd a)"
570   by (rule, rule coprime_dvd_mult, simp_all)
572 lemma gcd_dvd_antisym:
573   "gcd a b dvd gcd c d \<Longrightarrow> gcd c d dvd gcd a b \<Longrightarrow> gcd a b = gcd c d"
574 proof (rule gcdI)
575   assume A: "gcd a b dvd gcd c d" and B: "gcd c d dvd gcd a b"
576   have "gcd c d dvd c" by simp
577   with A show "gcd a b dvd c" by (rule dvd_trans)
578   have "gcd c d dvd d" by simp
579   with A show "gcd a b dvd d" by (rule dvd_trans)
580   show "normalization_factor (gcd a b) = (if gcd a b = 0 then 0 else 1)"
581     by simp
582   fix l assume "l dvd c" and "l dvd d"
583   hence "l dvd gcd c d" by (rule gcd_greatest)
584   from this and B show "l dvd gcd a b" by (rule dvd_trans)
585 qed
587 lemma gcd_mult_cancel:
588   assumes "gcd k n = 1"
589   shows "gcd (k * m) n = gcd m n"
590 proof (rule gcd_dvd_antisym)
591   have "gcd (gcd (k * m) n) k = gcd (gcd k n) (k * m)" by (simp add: ac_simps)
592   also note \<open>gcd k n = 1\<close>
593   finally have "gcd (gcd (k * m) n) k = 1" by simp
594   hence "gcd (k * m) n dvd m" by (rule coprime_dvd_mult, simp add: ac_simps)
595   moreover have "gcd (k * m) n dvd n" by simp
596   ultimately show "gcd (k * m) n dvd gcd m n" by (rule gcd_greatest)
597   have "gcd m n dvd (k * m)" and "gcd m n dvd n" by simp_all
598   then show "gcd m n dvd gcd (k * m) n" by (rule gcd_greatest)
599 qed
601 lemma coprime_crossproduct:
602   assumes [simp]: "gcd a d = 1" "gcd b c = 1"
603   shows "associated (a * c) (b * d) \<longleftrightarrow> associated a b \<and> associated c d" (is "?lhs \<longleftrightarrow> ?rhs")
604 proof
605   assume ?rhs then show ?lhs unfolding associated_def by (fast intro: mult_dvd_mono)
606 next
607   assume ?lhs
608   from \<open>?lhs\<close> have "a dvd b * d" unfolding associated_def by (metis dvd_mult_left)
609   hence "a dvd b" by (simp add: coprime_dvd_mult_iff)
610   moreover from \<open>?lhs\<close> have "b dvd a * c" unfolding associated_def by (metis dvd_mult_left)
611   hence "b dvd a" by (simp add: coprime_dvd_mult_iff)
612   moreover from \<open>?lhs\<close> have "c dvd d * b"
613     unfolding associated_def by (auto dest: dvd_mult_right simp add: ac_simps)
614   hence "c dvd d" by (simp add: coprime_dvd_mult_iff gcd.commute)
615   moreover from \<open>?lhs\<close> have "d dvd c * a"
616     unfolding associated_def by (auto dest: dvd_mult_right simp add: ac_simps)
617   hence "d dvd c" by (simp add: coprime_dvd_mult_iff gcd.commute)
618   ultimately show ?rhs unfolding associated_def by simp
619 qed
621 lemma gcd_add1 [simp]:
622   "gcd (m + n) n = gcd m n"
623   by (cases "n = 0", simp_all add: gcd_non_0)
625 lemma gcd_add2 [simp]:
626   "gcd m (m + n) = gcd m n"
627   using gcd_add1 [of n m] by (simp add: ac_simps)
630   "gcd m (k * m + n) = gcd m n"
631 proof -
632   have "gcd m ((k * m + n) mod m) = gcd m (k * m + n)"
633     by (fact gcd_mod2)
634   then show ?thesis by simp
635 qed
637 lemma coprimeI: "(\<And>l. \<lbrakk>l dvd a; l dvd b\<rbrakk> \<Longrightarrow> l dvd 1) \<Longrightarrow> gcd a b = 1"
638   by (rule sym, rule gcdI, simp_all)
640 lemma coprime: "gcd a b = 1 \<longleftrightarrow> (\<forall>d. d dvd a \<and> d dvd b \<longleftrightarrow> is_unit d)"
641   by (auto intro: coprimeI gcd_greatest dvd_gcd_D1 dvd_gcd_D2)
643 lemma div_gcd_coprime:
644   assumes nz: "a \<noteq> 0 \<or> b \<noteq> 0"
645   defines [simp]: "d \<equiv> gcd a b"
646   defines [simp]: "a' \<equiv> a div d" and [simp]: "b' \<equiv> b div d"
647   shows "gcd a' b' = 1"
648 proof (rule coprimeI)
649   fix l assume "l dvd a'" "l dvd b'"
650   then obtain s t where "a' = l * s" "b' = l * t" unfolding dvd_def by blast
651   moreover have "a = a' * d" "b = b' * d" by simp_all
652   ultimately have "a = (l * d) * s" "b = (l * d) * t"
653     by (simp_all only: ac_simps)
654   hence "l*d dvd a" and "l*d dvd b" by (simp_all only: dvd_triv_left)
655   hence "l*d dvd d" by (simp add: gcd_greatest)
656   then obtain u where "d = l * d * u" ..
657   then have "d * (l * u) = d" by (simp add: ac_simps)
658   moreover from nz have "d \<noteq> 0" by simp
659   with div_mult_self1_is_id have "d * (l * u) div d = l * u" .
660   ultimately have "1 = l * u"
661     using \<open>d \<noteq> 0\<close> by simp
662   then show "l dvd 1" ..
663 qed
665 lemma coprime_mult:
666   assumes da: "gcd d a = 1" and db: "gcd d b = 1"
667   shows "gcd d (a * b) = 1"
668   apply (subst gcd.commute)
669   using da apply (subst gcd_mult_cancel)
670   apply (subst gcd.commute, assumption)
671   apply (subst gcd.commute, rule db)
672   done
674 lemma coprime_lmult:
675   assumes dab: "gcd d (a * b) = 1"
676   shows "gcd d a = 1"
677 proof (rule coprimeI)
678   fix l assume "l dvd d" and "l dvd a"
679   hence "l dvd a * b" by simp
680   with \<open>l dvd d\<close> and dab show "l dvd 1" by (auto intro: gcd_greatest)
681 qed
683 lemma coprime_rmult:
684   assumes dab: "gcd d (a * b) = 1"
685   shows "gcd d b = 1"
686 proof (rule coprimeI)
687   fix l assume "l dvd d" and "l dvd b"
688   hence "l dvd a * b" by simp
689   with \<open>l dvd d\<close> and dab show "l dvd 1" by (auto intro: gcd_greatest)
690 qed
692 lemma coprime_mul_eq: "gcd d (a * b) = 1 \<longleftrightarrow> gcd d a = 1 \<and> gcd d b = 1"
693   using coprime_rmult[of d a b] coprime_lmult[of d a b] coprime_mult[of d a b] by blast
695 lemma gcd_coprime:
696   assumes c: "gcd a b \<noteq> 0" and a: "a = a' * gcd a b" and b: "b = b' * gcd a b"
697   shows "gcd a' b' = 1"
698 proof -
699   from c have "a \<noteq> 0 \<or> b \<noteq> 0" by simp
700   with div_gcd_coprime have "gcd (a div gcd a b) (b div gcd a b) = 1" .
701   also from assms have "a div gcd a b = a'" by (metis div_mult_self2_is_id)+
702   also from assms have "b div gcd a b = b'" by (metis div_mult_self2_is_id)+
703   finally show ?thesis .
704 qed
706 lemma coprime_power:
707   assumes "0 < n"
708   shows "gcd a (b ^ n) = 1 \<longleftrightarrow> gcd a b = 1"
709 using assms proof (induct n)
710   case (Suc n) then show ?case
711     by (cases n) (simp_all add: coprime_mul_eq)
712 qed simp
714 lemma gcd_coprime_exists:
715   assumes nz: "gcd a b \<noteq> 0"
716   shows "\<exists>a' b'. a = a' * gcd a b \<and> b = b' * gcd a b \<and> gcd a' b' = 1"
717   apply (rule_tac x = "a div gcd a b" in exI)
718   apply (rule_tac x = "b div gcd a b" in exI)
719   apply (insert nz, auto intro: div_gcd_coprime)
720   done
722 lemma coprime_exp:
723   "gcd d a = 1 \<Longrightarrow> gcd d (a^n) = 1"
724   by (induct n, simp_all add: coprime_mult)
726 lemma coprime_exp2 [intro]:
727   "gcd a b = 1 \<Longrightarrow> gcd (a^n) (b^m) = 1"
728   apply (rule coprime_exp)
729   apply (subst gcd.commute)
730   apply (rule coprime_exp)
731   apply (subst gcd.commute)
732   apply assumption
733   done
735 lemma gcd_exp:
736   "gcd (a^n) (b^n) = (gcd a b) ^ n"
737 proof (cases "a = 0 \<and> b = 0")
738   assume "a = 0 \<and> b = 0"
739   then show ?thesis by (cases n, simp_all add: gcd_0_left)
740 next
741   assume A: "\<not>(a = 0 \<and> b = 0)"
742   hence "1 = gcd ((a div gcd a b)^n) ((b div gcd a b)^n)"
743     using div_gcd_coprime by (subst sym, auto simp: div_gcd_coprime)
744   hence "(gcd a b) ^ n = (gcd a b) ^ n * ..." by simp
745   also note gcd_mult_distrib
746   also have "normalization_factor ((gcd a b)^n) = 1"
747     by (simp add: normalization_factor_pow A)
748   also have "(gcd a b)^n * (a div gcd a b)^n = a^n"
749     by (subst ac_simps, subst div_power, simp, rule dvd_div_mult_self, rule dvd_power_same, simp)
750   also have "(gcd a b)^n * (b div gcd a b)^n = b^n"
751     by (subst ac_simps, subst div_power, simp, rule dvd_div_mult_self, rule dvd_power_same, simp)
752   finally show ?thesis by simp
753 qed
755 lemma coprime_common_divisor:
756   "gcd a b = 1 \<Longrightarrow> a dvd a \<Longrightarrow> a dvd b \<Longrightarrow> is_unit a"
757   apply (subgoal_tac "a dvd gcd a b")
758   apply simp
759   apply (erule (1) gcd_greatest)
760   done
762 lemma division_decomp:
763   assumes dc: "a dvd b * c"
764   shows "\<exists>b' c'. a = b' * c' \<and> b' dvd b \<and> c' dvd c"
765 proof (cases "gcd a b = 0")
766   assume "gcd a b = 0"
767   hence "a = 0 \<and> b = 0" by simp
768   hence "a = 0 * c \<and> 0 dvd b \<and> c dvd c" by simp
769   then show ?thesis by blast
770 next
771   let ?d = "gcd a b"
772   assume "?d \<noteq> 0"
773   from gcd_coprime_exists[OF this]
774     obtain a' b' where ab': "a = a' * ?d" "b = b' * ?d" "gcd a' b' = 1"
775     by blast
776   from ab'(1) have "a' dvd a" unfolding dvd_def by blast
777   with dc have "a' dvd b*c" using dvd_trans[of a' a "b*c"] by simp
778   from dc ab'(1,2) have "a'*?d dvd (b'*?d) * c" by simp
779   hence "?d * a' dvd ?d * (b' * c)" by (simp add: mult_ac)
780   with \<open>?d \<noteq> 0\<close> have "a' dvd b' * c" by simp
781   with coprime_dvd_mult[OF ab'(3)]
782     have "a' dvd c" by (subst (asm) ac_simps, blast)
783   with ab'(1) have "a = ?d * a' \<and> ?d dvd b \<and> a' dvd c" by (simp add: mult_ac)
784   then show ?thesis by blast
785 qed
787 lemma pow_divs_pow:
788   assumes ab: "a ^ n dvd b ^ n" and n: "n \<noteq> 0"
789   shows "a dvd b"
790 proof (cases "gcd a b = 0")
791   assume "gcd a b = 0"
792   then show ?thesis by simp
793 next
794   let ?d = "gcd a b"
795   assume "?d \<noteq> 0"
796   from n obtain m where m: "n = Suc m" by (cases n, simp_all)
797   from \<open>?d \<noteq> 0\<close> have zn: "?d ^ n \<noteq> 0" by (rule power_not_zero)
798   from gcd_coprime_exists[OF \<open>?d \<noteq> 0\<close>]
799     obtain a' b' where ab': "a = a' * ?d" "b = b' * ?d" "gcd a' b' = 1"
800     by blast
801   from ab have "(a' * ?d) ^ n dvd (b' * ?d) ^ n"
802     by (simp add: ab'(1,2)[symmetric])
803   hence "?d^n * a'^n dvd ?d^n * b'^n"
804     by (simp only: power_mult_distrib ac_simps)
805   with zn have "a'^n dvd b'^n" by simp
806   hence "a' dvd b'^n" using dvd_trans[of a' "a'^n" "b'^n"] by (simp add: m)
807   hence "a' dvd b'^m * b'" by (simp add: m ac_simps)
808   with coprime_dvd_mult[OF coprime_exp[OF ab'(3), of m]]
809     have "a' dvd b'" by (subst (asm) ac_simps, blast)
810   hence "a'*?d dvd b'*?d" by (rule mult_dvd_mono, simp)
811   with ab'(1,2) show ?thesis by simp
812 qed
814 lemma pow_divs_eq [simp]:
815   "n \<noteq> 0 \<Longrightarrow> a ^ n dvd b ^ n \<longleftrightarrow> a dvd b"
816   by (auto intro: pow_divs_pow dvd_power_same)
818 lemma divs_mult:
819   assumes mr: "m dvd r" and nr: "n dvd r" and mn: "gcd m n = 1"
820   shows "m * n dvd r"
821 proof -
822   from mr nr obtain m' n' where m': "r = m*m'" and n': "r = n*n'"
823     unfolding dvd_def by blast
824   from mr n' have "m dvd n'*n" by (simp add: ac_simps)
825   hence "m dvd n'" using coprime_dvd_mult_iff[OF mn] by simp
826   then obtain k where k: "n' = m*k" unfolding dvd_def by blast
827   with n' have "r = m * n * k" by (simp add: mult_ac)
828   then show ?thesis unfolding dvd_def by blast
829 qed
831 lemma coprime_plus_one [simp]: "gcd (n + 1) n = 1"
832   by (subst add_commute, simp)
834 lemma setprod_coprime [rule_format]:
835   "(\<forall>i\<in>A. gcd (f i) a = 1) \<longrightarrow> gcd (\<Prod>i\<in>A. f i) a = 1"
836   apply (cases "finite A")
837   apply (induct set: finite)
838   apply (auto simp add: gcd_mult_cancel)
839   done
841 lemma coprime_divisors:
842   assumes "d dvd a" "e dvd b" "gcd a b = 1"
843   shows "gcd d e = 1"
844 proof -
845   from assms obtain k l where "a = d * k" "b = e * l"
846     unfolding dvd_def by blast
847   with assms have "gcd (d * k) (e * l) = 1" by simp
848   hence "gcd (d * k) e = 1" by (rule coprime_lmult)
849   also have "gcd (d * k) e = gcd e (d * k)" by (simp add: ac_simps)
850   finally have "gcd e d = 1" by (rule coprime_lmult)
851   then show ?thesis by (simp add: ac_simps)
852 qed
854 lemma invertible_coprime:
855   assumes "a * b mod m = 1"
856   shows "coprime a m"
857 proof -
858   from assms have "coprime m (a * b mod m)"
859     by simp
860   then have "coprime m (a * b)"
861     by simp
862   then have "coprime m a"
863     by (rule coprime_lmult)
864   then show ?thesis
865     by (simp add: ac_simps)
866 qed
868 lemma lcm_gcd:
869   "lcm a b = a * b div (gcd a b * normalization_factor (a*b))"
870   by (simp only: lcm_lcm_eucl gcd_gcd_eucl lcm_eucl_def)
872 lemma lcm_gcd_prod:
873   "lcm a b * gcd a b = a * b div normalization_factor (a*b)"
874 proof (cases "a * b = 0")
875   let ?nf = normalization_factor
876   assume "a * b \<noteq> 0"
877   hence "gcd a b \<noteq> 0" by simp
878   from lcm_gcd have "lcm a b * gcd a b = gcd a b * (a * b div (?nf (a*b) * gcd a b))"
879     by (simp add: mult_ac)
880   also from \<open>a * b \<noteq> 0\<close> have "... = a * b div ?nf (a*b)"
881     by (simp add: div_mult_swap mult.commute)
882   finally show ?thesis .
883 qed (auto simp add: lcm_gcd)
885 lemma lcm_dvd1 [iff]:
886   "a dvd lcm a b"
887 proof (cases "a*b = 0")
888   assume "a * b \<noteq> 0"
889   hence "gcd a b \<noteq> 0" by simp
890   let ?c = "1 div normalization_factor (a * b)"
891   from \<open>a * b \<noteq> 0\<close> have [simp]: "is_unit (normalization_factor (a * b))" by simp
892   from lcm_gcd_prod[of a b] have "lcm a b * gcd a b = a * ?c * b"
893     by (simp add: div_mult_swap unit_div_commute)
894   hence "lcm a b * gcd a b div gcd a b = a * ?c * b div gcd a b" by simp
895   with \<open>gcd a b \<noteq> 0\<close> have "lcm a b = a * ?c * b div gcd a b"
896     by (subst (asm) div_mult_self2_is_id, simp_all)
897   also have "... = a * (?c * b div gcd a b)"
898     by (metis div_mult_swap gcd_dvd2 mult_assoc)
899   finally show ?thesis by (rule dvdI)
900 qed (auto simp add: lcm_gcd)
902 lemma lcm_least:
903   "\<lbrakk>a dvd k; b dvd k\<rbrakk> \<Longrightarrow> lcm a b dvd k"
904 proof (cases "k = 0")
905   let ?nf = normalization_factor
906   assume "k \<noteq> 0"
907   hence "is_unit (?nf k)" by simp
908   hence "?nf k \<noteq> 0" by (metis not_is_unit_0)
909   assume A: "a dvd k" "b dvd k"
910   hence "gcd a b \<noteq> 0" using \<open>k \<noteq> 0\<close> by auto
911   from A obtain r s where ar: "k = a * r" and bs: "k = b * s"
912     unfolding dvd_def by blast
913   with \<open>k \<noteq> 0\<close> have "r * s \<noteq> 0"
914     by auto (drule sym [of 0], simp)
915   hence "is_unit (?nf (r * s))" by simp
916   let ?c = "?nf k div ?nf (r*s)"
917   from \<open>is_unit (?nf k)\<close> and \<open>is_unit (?nf (r * s))\<close> have "is_unit ?c" by (rule unit_div)
918   hence "?c \<noteq> 0" using not_is_unit_0 by fast
919   from ar bs have "k * k * gcd s r = ?nf k * k * gcd (k * s) (k * r)"
920     by (subst mult_assoc, subst gcd_mult_distrib[of k s r], simp only: ac_simps)
921   also have "... = ?nf k * k * gcd ((r*s) * a) ((r*s) * b)"
922     by (subst (3) \<open>k = a * r\<close>, subst (3) \<open>k = b * s\<close>, simp add: algebra_simps)
923   also have "... = ?c * r*s * k * gcd a b" using \<open>r * s \<noteq> 0\<close>
924     by (subst gcd_mult_distrib'[symmetric], simp add: algebra_simps unit_simps)
925   finally have "(a*r) * (b*s) * gcd s r = ?c * k * r * s * gcd a b"
926     by (subst ar[symmetric], subst bs[symmetric], simp add: mult_ac)
927   hence "a * b * gcd s r * (r * s) = ?c * k * gcd a b * (r * s)"
928     by (simp add: algebra_simps)
929   hence "?c * k * gcd a b = a * b * gcd s r" using \<open>r * s \<noteq> 0\<close>
930     by (metis div_mult_self2_is_id)
931   also have "... = lcm a b * gcd a b * gcd s r * ?nf (a*b)"
932     by (subst lcm_gcd_prod[of a b], metis gcd_mult_distrib gcd_mult_distrib')
933   also have "... = lcm a b * gcd s r * ?nf (a*b) * gcd a b"
934     by (simp add: algebra_simps)
935   finally have "k * ?c = lcm a b * gcd s r * ?nf (a*b)" using \<open>gcd a b \<noteq> 0\<close>
936     by (metis mult.commute div_mult_self2_is_id)
937   hence "k = lcm a b * (gcd s r * ?nf (a*b)) div ?c" using \<open>?c \<noteq> 0\<close>
938     by (metis div_mult_self2_is_id mult_assoc)
939   also have "... = lcm a b * (gcd s r * ?nf (a*b) div ?c)" using \<open>is_unit ?c\<close>
940     by (simp add: unit_simps)
941   finally show ?thesis by (rule dvdI)
942 qed simp
944 lemma lcm_zero:
945   "lcm a b = 0 \<longleftrightarrow> a = 0 \<or> b = 0"
946 proof -
947   let ?nf = normalization_factor
948   {
949     assume "a \<noteq> 0" "b \<noteq> 0"
950     hence "a * b div ?nf (a * b) \<noteq> 0" by (simp add: no_zero_divisors)
951     moreover from \<open>a \<noteq> 0\<close> and \<open>b \<noteq> 0\<close> have "gcd a b \<noteq> 0" by simp
952     ultimately have "lcm a b \<noteq> 0" using lcm_gcd_prod[of a b] by (intro notI, simp)
953   } moreover {
954     assume "a = 0 \<or> b = 0"
955     hence "lcm a b = 0" by (elim disjE, simp_all add: lcm_gcd)
956   }
957   ultimately show ?thesis by blast
958 qed
960 lemmas lcm_0_iff = lcm_zero
962 lemma gcd_lcm:
963   assumes "lcm a b \<noteq> 0"
964   shows "gcd a b = a * b div (lcm a b * normalization_factor (a * b))"
965 proof-
966   from assms have "gcd a b \<noteq> 0" by (simp add: lcm_zero)
967   let ?c = "normalization_factor (a * b)"
968   from \<open>lcm a b \<noteq> 0\<close> have "?c \<noteq> 0" by (intro notI, simp add: lcm_zero no_zero_divisors)
969   hence "is_unit ?c" by simp
970   from lcm_gcd_prod [of a b] have "gcd a b = a * b div ?c div lcm a b"
971     by (subst (2) div_mult_self2_is_id[OF \<open>lcm a b \<noteq> 0\<close>, symmetric], simp add: mult_ac)
972   also from \<open>is_unit ?c\<close> have "... = a * b div (lcm a b * ?c)"
973     by (metis \<open>?c \<noteq> 0\<close> div_mult_mult1 dvd_mult_div_cancel mult_commute normalization_factor_dvd')
974   finally show ?thesis .
975 qed
977 lemma normalization_factor_lcm [simp]:
978   "normalization_factor (lcm a b) = (if a = 0 \<or> b = 0 then 0 else 1)"
979 proof (cases "a = 0 \<or> b = 0")
980   case True then show ?thesis
981     by (auto simp add: lcm_gcd)
982 next
983   case False
984   let ?nf = normalization_factor
985   from lcm_gcd_prod[of a b]
986     have "?nf (lcm a b) * ?nf (gcd a b) = ?nf (a*b) div ?nf (a*b)"
987     by (metis div_by_0 div_self normalization_correct normalization_factor_0 normalization_factor_mult)
988   also have "... = (if a*b = 0 then 0 else 1)"
989     by simp
990   finally show ?thesis using False by simp
991 qed
993 lemma lcm_dvd2 [iff]: "b dvd lcm a b"
994   using lcm_dvd1 [of b a] by (simp add: lcm_gcd ac_simps)
996 lemma lcmI:
997   "\<lbrakk>a dvd k; b dvd k; \<And>l. a dvd l \<Longrightarrow> b dvd l \<Longrightarrow> k dvd l;
998     normalization_factor k = (if k = 0 then 0 else 1)\<rbrakk> \<Longrightarrow> k = lcm a b"
999   by (intro normed_associated_imp_eq) (auto simp: associated_def intro: lcm_least)
1001 sublocale lcm!: abel_semigroup lcm
1002 proof
1003   fix a b c
1004   show "lcm (lcm a b) c = lcm a (lcm b c)"
1005   proof (rule lcmI)
1006     have "a dvd lcm a b" and "lcm a b dvd lcm (lcm a b) c" by simp_all
1007     then show "a dvd lcm (lcm a b) c" by (rule dvd_trans)
1009     have "b dvd lcm a b" and "lcm a b dvd lcm (lcm a b) c" by simp_all
1010     hence "b dvd lcm (lcm a b) c" by (rule dvd_trans)
1011     moreover have "c dvd lcm (lcm a b) c" by simp
1012     ultimately show "lcm b c dvd lcm (lcm a b) c" by (rule lcm_least)
1014     fix l assume "a dvd l" and "lcm b c dvd l"
1015     have "b dvd lcm b c" by simp
1016     from this and \<open>lcm b c dvd l\<close> have "b dvd l" by (rule dvd_trans)
1017     have "c dvd lcm b c" by simp
1018     from this and \<open>lcm b c dvd l\<close> have "c dvd l" by (rule dvd_trans)
1019     from \<open>a dvd l\<close> and \<open>b dvd l\<close> have "lcm a b dvd l" by (rule lcm_least)
1020     from this and \<open>c dvd l\<close> show "lcm (lcm a b) c dvd l" by (rule lcm_least)
1021   qed (simp add: lcm_zero)
1022 next
1023   fix a b
1024   show "lcm a b = lcm b a"
1025     by (simp add: lcm_gcd ac_simps)
1026 qed
1028 lemma dvd_lcm_D1:
1029   "lcm m n dvd k \<Longrightarrow> m dvd k"
1030   by (rule dvd_trans, rule lcm_dvd1, assumption)
1032 lemma dvd_lcm_D2:
1033   "lcm m n dvd k \<Longrightarrow> n dvd k"
1034   by (rule dvd_trans, rule lcm_dvd2, assumption)
1036 lemma gcd_dvd_lcm [simp]:
1037   "gcd a b dvd lcm a b"
1038   by (metis dvd_trans gcd_dvd2 lcm_dvd2)
1040 lemma lcm_1_iff:
1041   "lcm a b = 1 \<longleftrightarrow> is_unit a \<and> is_unit b"
1042 proof
1043   assume "lcm a b = 1"
1044   then show "is_unit a \<and> is_unit b" by auto
1045 next
1046   assume "is_unit a \<and> is_unit b"
1047   hence "a dvd 1" and "b dvd 1" by simp_all
1048   hence "is_unit (lcm a b)" by (rule lcm_least)
1049   hence "lcm a b = normalization_factor (lcm a b)"
1050     by (subst normalization_factor_unit, simp_all)
1051   also have "\<dots> = 1" using \<open>is_unit a \<and> is_unit b\<close>
1052     by auto
1053   finally show "lcm a b = 1" .
1054 qed
1056 lemma lcm_0_left [simp]:
1057   "lcm 0 a = 0"
1058   by (rule sym, rule lcmI, simp_all)
1060 lemma lcm_0 [simp]:
1061   "lcm a 0 = 0"
1062   by (rule sym, rule lcmI, simp_all)
1064 lemma lcm_unique:
1065   "a dvd d \<and> b dvd d \<and>
1066   normalization_factor d = (if d = 0 then 0 else 1) \<and>
1067   (\<forall>e. a dvd e \<and> b dvd e \<longrightarrow> d dvd e) \<longleftrightarrow> d = lcm a b"
1068   by (rule, auto intro: lcmI simp: lcm_least lcm_zero)
1070 lemma dvd_lcm_I1 [simp]:
1071   "k dvd m \<Longrightarrow> k dvd lcm m n"
1072   by (metis lcm_dvd1 dvd_trans)
1074 lemma dvd_lcm_I2 [simp]:
1075   "k dvd n \<Longrightarrow> k dvd lcm m n"
1076   by (metis lcm_dvd2 dvd_trans)
1078 lemma lcm_1_left [simp]:
1079   "lcm 1 a = a div normalization_factor a"
1080   by (cases "a = 0") (simp, rule sym, rule lcmI, simp_all)
1082 lemma lcm_1_right [simp]:
1083   "lcm a 1 = a div normalization_factor a"
1084   using lcm_1_left [of a] by (simp add: ac_simps)
1086 lemma lcm_coprime:
1087   "gcd a b = 1 \<Longrightarrow> lcm a b = a * b div normalization_factor (a*b)"
1088   by (subst lcm_gcd) simp
1090 lemma lcm_proj1_if_dvd:
1091   "b dvd a \<Longrightarrow> lcm a b = a div normalization_factor a"
1092   by (cases "a = 0") (simp, rule sym, rule lcmI, simp_all)
1094 lemma lcm_proj2_if_dvd:
1095   "a dvd b \<Longrightarrow> lcm a b = b div normalization_factor b"
1096   using lcm_proj1_if_dvd [of a b] by (simp add: ac_simps)
1098 lemma lcm_proj1_iff:
1099   "lcm m n = m div normalization_factor m \<longleftrightarrow> n dvd m"
1100 proof
1101   assume A: "lcm m n = m div normalization_factor m"
1102   show "n dvd m"
1103   proof (cases "m = 0")
1104     assume [simp]: "m \<noteq> 0"
1105     from A have B: "m = lcm m n * normalization_factor m"
1106       by (simp add: unit_eq_div2)
1107     show ?thesis by (subst B, simp)
1108   qed simp
1109 next
1110   assume "n dvd m"
1111   then show "lcm m n = m div normalization_factor m" by (rule lcm_proj1_if_dvd)
1112 qed
1114 lemma lcm_proj2_iff:
1115   "lcm m n = n div normalization_factor n \<longleftrightarrow> m dvd n"
1116   using lcm_proj1_iff [of n m] by (simp add: ac_simps)
1118 lemma euclidean_size_lcm_le1:
1119   assumes "a \<noteq> 0" and "b \<noteq> 0"
1120   shows "euclidean_size a \<le> euclidean_size (lcm a b)"
1121 proof -
1122   have "a dvd lcm a b" by (rule lcm_dvd1)
1123   then obtain c where A: "lcm a b = a * c" unfolding dvd_def by blast
1124   with \<open>a \<noteq> 0\<close> and \<open>b \<noteq> 0\<close> have "c \<noteq> 0" by (auto simp: lcm_zero)
1125   then show ?thesis by (subst A, intro size_mult_mono)
1126 qed
1128 lemma euclidean_size_lcm_le2:
1129   "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> euclidean_size b \<le> euclidean_size (lcm a b)"
1130   using euclidean_size_lcm_le1 [of b a] by (simp add: ac_simps)
1132 lemma euclidean_size_lcm_less1:
1133   assumes "b \<noteq> 0" and "\<not>b dvd a"
1134   shows "euclidean_size a < euclidean_size (lcm a b)"
1135 proof (rule ccontr)
1136   from assms have "a \<noteq> 0" by auto
1137   assume "\<not>euclidean_size a < euclidean_size (lcm a b)"
1138   with \<open>a \<noteq> 0\<close> and \<open>b \<noteq> 0\<close> have "euclidean_size (lcm a b) = euclidean_size a"
1139     by (intro le_antisym, simp, intro euclidean_size_lcm_le1)
1140   with assms have "lcm a b dvd a"
1141     by (rule_tac dvd_euclidean_size_eq_imp_dvd) (auto simp: lcm_zero)
1142   hence "b dvd a" by (rule dvd_lcm_D2)
1143   with \<open>\<not>b dvd a\<close> show False by contradiction
1144 qed
1146 lemma euclidean_size_lcm_less2:
1147   assumes "a \<noteq> 0" and "\<not>a dvd b"
1148   shows "euclidean_size b < euclidean_size (lcm a b)"
1149   using assms euclidean_size_lcm_less1 [of a b] by (simp add: ac_simps)
1151 lemma lcm_mult_unit1:
1152   "is_unit a \<Longrightarrow> lcm (b * a) c = lcm b c"
1153   apply (rule lcmI)
1154   apply (rule dvd_trans[of _ "b * a"], simp, rule lcm_dvd1)
1155   apply (rule lcm_dvd2)
1156   apply (rule lcm_least, simp add: unit_simps, assumption)
1157   apply (subst normalization_factor_lcm, simp add: lcm_zero)
1158   done
1160 lemma lcm_mult_unit2:
1161   "is_unit a \<Longrightarrow> lcm b (c * a) = lcm b c"
1162   using lcm_mult_unit1 [of a c b] by (simp add: ac_simps)
1164 lemma lcm_div_unit1:
1165   "is_unit a \<Longrightarrow> lcm (b div a) c = lcm b c"
1166   by (erule is_unitE [of _ b]) (simp add: lcm_mult_unit1)
1168 lemma lcm_div_unit2:
1169   "is_unit a \<Longrightarrow> lcm b (c div a) = lcm b c"
1170   by (erule is_unitE [of _ c]) (simp add: lcm_mult_unit2)
1172 lemma lcm_left_idem:
1173   "lcm a (lcm a b) = lcm a b"
1174   apply (rule lcmI)
1175   apply simp
1176   apply (subst lcm.assoc [symmetric], rule lcm_dvd2)
1177   apply (rule lcm_least, assumption)
1178   apply (erule (1) lcm_least)
1179   apply (auto simp: lcm_zero)
1180   done
1182 lemma lcm_right_idem:
1183   "lcm (lcm a b) b = lcm a b"
1184   apply (rule lcmI)
1185   apply (subst lcm.assoc, rule lcm_dvd1)
1186   apply (rule lcm_dvd2)
1187   apply (rule lcm_least, erule (1) lcm_least, assumption)
1188   apply (auto simp: lcm_zero)
1189   done
1191 lemma comp_fun_idem_lcm: "comp_fun_idem lcm"
1192 proof
1193   fix a b show "lcm a \<circ> lcm b = lcm b \<circ> lcm a"
1194     by (simp add: fun_eq_iff ac_simps)
1195 next
1196   fix a show "lcm a \<circ> lcm a = lcm a" unfolding o_def
1197     by (intro ext, simp add: lcm_left_idem)
1198 qed
1200 lemma dvd_Lcm [simp]: "a \<in> A \<Longrightarrow> a dvd Lcm A"
1201   and Lcm_dvd [simp]: "(\<forall>a\<in>A. a dvd l') \<Longrightarrow> Lcm A dvd l'"
1202   and normalization_factor_Lcm [simp]:
1203           "normalization_factor (Lcm A) = (if Lcm A = 0 then 0 else 1)"
1204 proof -
1205   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>
1206     normalization_factor (Lcm A) = (if Lcm A = 0 then 0 else 1)" (is ?thesis)
1207   proof (cases "\<exists>l. l \<noteq>  0 \<and> (\<forall>a\<in>A. a dvd l)")
1208     case False
1209     hence "Lcm A = 0" by (auto simp: Lcm_Lcm_eucl Lcm_eucl_def)
1210     with False show ?thesis by auto
1211   next
1212     case True
1213     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
1214     def n \<equiv> "LEAST n. \<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"
1215     def l \<equiv> "SOME l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"
1216     have "\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"
1217       apply (subst n_def)
1218       apply (rule LeastI[of _ "euclidean_size l\<^sub>0"])
1219       apply (rule exI[of _ l\<^sub>0])
1220       apply (simp add: l\<^sub>0_props)
1221       done
1222     from someI_ex[OF this] have "l \<noteq> 0" and "\<forall>a\<in>A. a dvd l" and "euclidean_size l = n"
1223       unfolding l_def by simp_all
1224     {
1225       fix l' assume "\<forall>a\<in>A. a dvd l'"
1226       with \<open>\<forall>a\<in>A. a dvd l\<close> have "\<forall>a\<in>A. a dvd gcd l l'" by (auto intro: gcd_greatest)
1227       moreover from \<open>l \<noteq> 0\<close> have "gcd l l' \<noteq> 0" by simp
1228       ultimately have "\<exists>b. b \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd b) \<and> euclidean_size b = euclidean_size (gcd l l')"
1229         by (intro exI[of _ "gcd l l'"], auto)
1230       hence "euclidean_size (gcd l l') \<ge> n" by (subst n_def) (rule Least_le)
1231       moreover have "euclidean_size (gcd l l') \<le> n"
1232       proof -
1233         have "gcd l l' dvd l" by simp
1234         then obtain a where "l = gcd l l' * a" unfolding dvd_def by blast
1235         with \<open>l \<noteq> 0\<close> have "a \<noteq> 0" by auto
1236         hence "euclidean_size (gcd l l') \<le> euclidean_size (gcd l l' * a)"
1237           by (rule size_mult_mono)
1238         also have "gcd l l' * a = l" using \<open>l = gcd l l' * a\<close> ..
1239         also note \<open>euclidean_size l = n\<close>
1240         finally show "euclidean_size (gcd l l') \<le> n" .
1241       qed
1242       ultimately have "euclidean_size l = euclidean_size (gcd l l')"
1243         by (intro le_antisym, simp_all add: \<open>euclidean_size l = n\<close>)
1244       with \<open>l \<noteq> 0\<close> have "l dvd gcd l l'" by (blast intro: dvd_euclidean_size_eq_imp_dvd)
1245       hence "l dvd l'" by (blast dest: dvd_gcd_D2)
1246     }
1248     with \<open>(\<forall>a\<in>A. a dvd l)\<close> and normalization_factor_is_unit[OF \<open>l \<noteq> 0\<close>] and \<open>l \<noteq> 0\<close>
1249       have "(\<forall>a\<in>A. a dvd l div normalization_factor l) \<and>
1250         (\<forall>l'. (\<forall>a\<in>A. a dvd l') \<longrightarrow> l div normalization_factor l dvd l') \<and>
1251         normalization_factor (l div normalization_factor l) =
1252         (if l div normalization_factor l = 0 then 0 else 1)"
1253       by (auto simp: unit_simps)
1254     also from True have "l div normalization_factor l = Lcm A"
1255       by (simp add: Lcm_Lcm_eucl Lcm_eucl_def Let_def n_def l_def)
1256     finally show ?thesis .
1257   qed
1258   note A = this
1260   {fix a assume "a \<in> A" then show "a dvd Lcm A" using A by blast}
1261   {fix l' assume "\<forall>a\<in>A. a dvd l'" then show "Lcm A dvd l'" using A by blast}
1262   from A show "normalization_factor (Lcm A) = (if Lcm A = 0 then 0 else 1)" by blast
1263 qed
1265 lemma LcmI:
1266   "(\<And>a. a\<in>A \<Longrightarrow> a dvd l) \<Longrightarrow> (\<And>l'. (\<forall>a\<in>A. a dvd l') \<Longrightarrow> l dvd l') \<Longrightarrow>
1267       normalization_factor l = (if l = 0 then 0 else 1) \<Longrightarrow> l = Lcm A"
1268   by (intro normed_associated_imp_eq)
1269     (auto intro: Lcm_dvd dvd_Lcm simp: associated_def)
1271 lemma Lcm_subset:
1272   "A \<subseteq> B \<Longrightarrow> Lcm A dvd Lcm B"
1273   by (blast intro: Lcm_dvd dvd_Lcm)
1275 lemma Lcm_Un:
1276   "Lcm (A \<union> B) = lcm (Lcm A) (Lcm B)"
1277   apply (rule lcmI)
1278   apply (blast intro: Lcm_subset)
1279   apply (blast intro: Lcm_subset)
1280   apply (intro Lcm_dvd ballI, elim UnE)
1281   apply (rule dvd_trans, erule dvd_Lcm, assumption)
1282   apply (rule dvd_trans, erule dvd_Lcm, assumption)
1283   apply simp
1284   done
1286 lemma Lcm_1_iff:
1287   "Lcm A = 1 \<longleftrightarrow> (\<forall>a\<in>A. is_unit a)"
1288 proof
1289   assume "Lcm A = 1"
1290   then show "\<forall>a\<in>A. is_unit a" by auto
1291 qed (rule LcmI [symmetric], auto)
1293 lemma Lcm_no_units:
1294   "Lcm A = Lcm (A - {a. is_unit a})"
1295 proof -
1296   have "(A - {a. is_unit a}) \<union> {a\<in>A. is_unit a} = A" by blast
1297   hence "Lcm A = lcm (Lcm (A - {a. is_unit a})) (Lcm {a\<in>A. is_unit a})"
1298     by (simp add: Lcm_Un[symmetric])
1299   also have "Lcm {a\<in>A. is_unit a} = 1" by (simp add: Lcm_1_iff)
1300   finally show ?thesis by simp
1301 qed
1303 lemma Lcm_empty [simp]:
1304   "Lcm {} = 1"
1305   by (simp add: Lcm_1_iff)
1307 lemma Lcm_eq_0 [simp]:
1308   "0 \<in> A \<Longrightarrow> Lcm A = 0"
1309   by (drule dvd_Lcm) simp
1311 lemma Lcm0_iff':
1312   "Lcm A = 0 \<longleftrightarrow> \<not>(\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l))"
1313 proof
1314   assume "Lcm A = 0"
1315   show "\<not>(\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l))"
1316   proof
1317     assume ex: "\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l)"
1318     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
1319     def n \<equiv> "LEAST n. \<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"
1320     def l \<equiv> "SOME l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"
1321     have "\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"
1322       apply (subst n_def)
1323       apply (rule LeastI[of _ "euclidean_size l\<^sub>0"])
1324       apply (rule exI[of _ l\<^sub>0])
1325       apply (simp add: l\<^sub>0_props)
1326       done
1327     from someI_ex[OF this] have "l \<noteq> 0" unfolding l_def by simp_all
1328     hence "l div normalization_factor l \<noteq> 0" by simp
1329     also from ex have "l div normalization_factor l = Lcm A"
1330        by (simp only: Lcm_Lcm_eucl Lcm_eucl_def n_def l_def if_True Let_def)
1331     finally show False using \<open>Lcm A = 0\<close> by contradiction
1332   qed
1333 qed (simp only: Lcm_Lcm_eucl Lcm_eucl_def if_False)
1335 lemma Lcm0_iff [simp]:
1336   "finite A \<Longrightarrow> Lcm A = 0 \<longleftrightarrow> 0 \<in> A"
1337 proof -
1338   assume "finite A"
1339   have "0 \<in> A \<Longrightarrow> Lcm A = 0"  by (intro dvd_0_left dvd_Lcm)
1340   moreover {
1341     assume "0 \<notin> A"
1342     hence "\<Prod>A \<noteq> 0"
1343       apply (induct rule: finite_induct[OF \<open>finite A\<close>])
1344       apply simp
1345       apply (subst setprod.insert, assumption, assumption)
1346       apply (rule no_zero_divisors)
1347       apply blast+
1348       done
1349     moreover from \<open>finite A\<close> have "\<forall>a\<in>A. a dvd \<Prod>A" by blast
1350     ultimately have "\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l)" by blast
1351     with Lcm0_iff' have "Lcm A \<noteq> 0" by simp
1352   }
1353   ultimately show "Lcm A = 0 \<longleftrightarrow> 0 \<in> A" by blast
1354 qed
1356 lemma Lcm_no_multiple:
1357   "(\<forall>m. m \<noteq> 0 \<longrightarrow> (\<exists>a\<in>A. \<not>a dvd m)) \<Longrightarrow> Lcm A = 0"
1358 proof -
1359   assume "\<forall>m. m \<noteq> 0 \<longrightarrow> (\<exists>a\<in>A. \<not>a dvd m)"
1360   hence "\<not>(\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l))" by blast
1361   then show "Lcm A = 0" by (simp only: Lcm_Lcm_eucl Lcm_eucl_def if_False)
1362 qed
1364 lemma Lcm_insert [simp]:
1365   "Lcm (insert a A) = lcm a (Lcm A)"
1366 proof (rule lcmI)
1367   fix l assume "a dvd l" and "Lcm A dvd l"
1368   hence "\<forall>a\<in>A. a dvd l" by (blast intro: dvd_trans dvd_Lcm)
1369   with \<open>a dvd l\<close> show "Lcm (insert a A) dvd l" by (force intro: Lcm_dvd)
1370 qed (auto intro: Lcm_dvd dvd_Lcm)
1372 lemma Lcm_finite:
1373   assumes "finite A"
1374   shows "Lcm A = Finite_Set.fold lcm 1 A"
1375   by (induct rule: finite.induct[OF \<open>finite A\<close>])
1376     (simp_all add: comp_fun_idem.fold_insert_idem[OF comp_fun_idem_lcm])
1378 lemma Lcm_set [code_unfold]:
1379   "Lcm (set xs) = fold lcm xs 1"
1380   using comp_fun_idem.fold_set_fold[OF comp_fun_idem_lcm] Lcm_finite by (simp add: ac_simps)
1382 lemma Lcm_singleton [simp]:
1383   "Lcm {a} = a div normalization_factor a"
1384   by simp
1386 lemma Lcm_2 [simp]:
1387   "Lcm {a,b} = lcm a b"
1388   by (simp only: Lcm_insert Lcm_empty lcm_1_right)
1389     (cases "b = 0", simp, rule lcm_div_unit2, simp)
1391 lemma Lcm_coprime:
1392   assumes "finite A" and "A \<noteq> {}"
1393   assumes "\<And>a b. a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<noteq> b \<Longrightarrow> gcd a b = 1"
1394   shows "Lcm A = \<Prod>A div normalization_factor (\<Prod>A)"
1395 using assms proof (induct rule: finite_ne_induct)
1396   case (insert a A)
1397   have "Lcm (insert a A) = lcm a (Lcm A)" by simp
1398   also from insert have "Lcm A = \<Prod>A div normalization_factor (\<Prod>A)" by blast
1399   also have "lcm a \<dots> = lcm a (\<Prod>A)" by (cases "\<Prod>A = 0") (simp_all add: lcm_div_unit2)
1400   also from insert have "gcd a (\<Prod>A) = 1" by (subst gcd.commute, intro setprod_coprime) auto
1401   with insert have "lcm a (\<Prod>A) = \<Prod>(insert a A) div normalization_factor (\<Prod>(insert a A))"
1402     by (simp add: lcm_coprime)
1403   finally show ?case .
1404 qed simp
1406 lemma Lcm_coprime':
1407   "card A \<noteq> 0 \<Longrightarrow> (\<And>a b. a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<noteq> b \<Longrightarrow> gcd a b = 1)
1408     \<Longrightarrow> Lcm A = \<Prod>A div normalization_factor (\<Prod>A)"
1409   by (rule Lcm_coprime) (simp_all add: card_eq_0_iff)
1411 lemma Gcd_Lcm:
1412   "Gcd A = Lcm {d. \<forall>a\<in>A. d dvd a}"
1413   by (simp add: Gcd_Gcd_eucl Lcm_Lcm_eucl Gcd_eucl_def)
1415 lemma Gcd_dvd [simp]: "a \<in> A \<Longrightarrow> Gcd A dvd a"
1416   and dvd_Gcd [simp]: "(\<forall>a\<in>A. g' dvd a) \<Longrightarrow> g' dvd Gcd A"
1417   and normalization_factor_Gcd [simp]:
1418     "normalization_factor (Gcd A) = (if Gcd A = 0 then 0 else 1)"
1419 proof -
1420   fix a assume "a \<in> A"
1421   hence "Lcm {d. \<forall>a\<in>A. d dvd a} dvd a" by (intro Lcm_dvd) blast
1422   then show "Gcd A dvd a" by (simp add: Gcd_Lcm)
1423 next
1424   fix g' assume "\<forall>a\<in>A. g' dvd a"
1425   hence "g' dvd Lcm {d. \<forall>a\<in>A. d dvd a}" by (intro dvd_Lcm) blast
1426   then show "g' dvd Gcd A" by (simp add: Gcd_Lcm)
1427 next
1428   show "normalization_factor (Gcd A) = (if Gcd A = 0 then 0 else 1)"
1429     by (simp add: Gcd_Lcm)
1430 qed
1432 lemma GcdI:
1433   "(\<And>a. a\<in>A \<Longrightarrow> l dvd a) \<Longrightarrow> (\<And>l'. (\<forall>a\<in>A. l' dvd a) \<Longrightarrow> l' dvd l) \<Longrightarrow>
1434     normalization_factor l = (if l = 0 then 0 else 1) \<Longrightarrow> l = Gcd A"
1435   by (intro normed_associated_imp_eq)
1436     (auto intro: Gcd_dvd dvd_Gcd simp: associated_def)
1438 lemma Lcm_Gcd:
1439   "Lcm A = Gcd {m. \<forall>a\<in>A. a dvd m}"
1440   by (rule LcmI[symmetric]) (auto intro: dvd_Gcd Gcd_dvd)
1442 lemma Gcd_0_iff:
1443   "Gcd A = 0 \<longleftrightarrow> A \<subseteq> {0}"
1444   apply (rule iffI)
1445   apply (rule subsetI, drule Gcd_dvd, simp)
1446   apply (auto intro: GcdI[symmetric])
1447   done
1449 lemma Gcd_empty [simp]:
1450   "Gcd {} = 0"
1451   by (simp add: Gcd_0_iff)
1453 lemma Gcd_1:
1454   "1 \<in> A \<Longrightarrow> Gcd A = 1"
1455   by (intro GcdI[symmetric]) (auto intro: Gcd_dvd dvd_Gcd)
1457 lemma Gcd_insert [simp]:
1458   "Gcd (insert a A) = gcd a (Gcd A)"
1459 proof (rule gcdI)
1460   fix l assume "l dvd a" and "l dvd Gcd A"
1461   hence "\<forall>a\<in>A. l dvd a" by (blast intro: dvd_trans Gcd_dvd)
1462   with \<open>l dvd a\<close> show "l dvd Gcd (insert a A)" by (force intro: Gcd_dvd)
1463 qed auto
1465 lemma Gcd_finite:
1466   assumes "finite A"
1467   shows "Gcd A = Finite_Set.fold gcd 0 A"
1468   by (induct rule: finite.induct[OF \<open>finite A\<close>])
1469     (simp_all add: comp_fun_idem.fold_insert_idem[OF comp_fun_idem_gcd])
1471 lemma Gcd_set [code_unfold]:
1472   "Gcd (set xs) = fold gcd xs 0"
1473   using comp_fun_idem.fold_set_fold[OF comp_fun_idem_gcd] Gcd_finite by (simp add: ac_simps)
1475 lemma Gcd_singleton [simp]: "Gcd {a} = a div normalization_factor a"
1476   by (simp add: gcd_0)
1478 lemma Gcd_2 [simp]: "Gcd {a,b} = gcd a b"
1479   by (simp only: Gcd_insert Gcd_empty gcd_0) (cases "b = 0", simp, rule gcd_div_unit2, simp)
1481 subclass semiring_gcd
1482   by unfold_locales (simp_all add: gcd_greatest_iff)
1484 end
1486 text \<open>
1487   A Euclidean ring is a Euclidean semiring with additive inverses. It provides a
1488   few more lemmas; in particular, Bezout's lemma holds for any Euclidean ring.
1489 \<close>
1491 class euclidean_ring_gcd = euclidean_semiring_gcd + idom
1492 begin
1494 subclass euclidean_ring ..
1496 subclass ring_gcd ..
1498 lemma euclid_ext_gcd [simp]:
1499   "(case euclid_ext a b of (_, _ , t) \<Rightarrow> t) = gcd a b"
1500   by (induct a b rule: gcd_eucl_induct)
1501     (simp_all add: euclid_ext_0 gcd_0 euclid_ext_non_0 ac_simps split: prod.split prod.split_asm)
1503 lemma euclid_ext_gcd' [simp]:
1504   "euclid_ext a b = (r, s, t) \<Longrightarrow> t = gcd a b"
1505   by (insert euclid_ext_gcd[of a b], drule (1) subst, simp)
1507 lemma euclid_ext'_correct:
1508   "fst (euclid_ext' a b) * a + snd (euclid_ext' a b) * b = gcd a b"
1509 proof-
1510   obtain s t c where "euclid_ext a b = (s,t,c)"
1511     by (cases "euclid_ext a b", blast)
1512   with euclid_ext_correct[of a b] euclid_ext_gcd[of a b]
1513     show ?thesis unfolding euclid_ext'_def by simp
1514 qed
1516 lemma bezout: "\<exists>s t. s * a + t * b = gcd a b"
1517   using euclid_ext'_correct by blast
1519 lemma gcd_neg1 [simp]:
1520   "gcd (-a) b = gcd a b"
1521   by (rule sym, rule gcdI, simp_all add: gcd_greatest)
1523 lemma gcd_neg2 [simp]:
1524   "gcd a (-b) = gcd a b"
1525   by (rule sym, rule gcdI, simp_all add: gcd_greatest)
1527 lemma gcd_neg_numeral_1 [simp]:
1528   "gcd (- numeral n) a = gcd (numeral n) a"
1529   by (fact gcd_neg1)
1531 lemma gcd_neg_numeral_2 [simp]:
1532   "gcd a (- numeral n) = gcd a (numeral n)"
1533   by (fact gcd_neg2)
1535 lemma gcd_diff1: "gcd (m - n) n = gcd m n"
1536   by (subst diff_conv_add_uminus, subst gcd_neg2[symmetric],  subst gcd_add1, simp)
1538 lemma gcd_diff2: "gcd (n - m) n = gcd m n"
1539   by (subst gcd_neg1[symmetric], simp only: minus_diff_eq gcd_diff1)
1541 lemma coprime_minus_one [simp]: "gcd (n - 1) n = 1"
1542 proof -
1543   have "gcd (n - 1) n = gcd n (n - 1)" by (fact gcd.commute)
1544   also have "\<dots> = gcd ((n - 1) + 1) (n - 1)" by simp
1545   also have "\<dots> = 1" by (rule coprime_plus_one)
1546   finally show ?thesis .
1547 qed
1549 lemma lcm_neg1 [simp]: "lcm (-a) b = lcm a b"
1550   by (rule sym, rule lcmI, simp_all add: lcm_least lcm_zero)
1552 lemma lcm_neg2 [simp]: "lcm a (-b) = lcm a b"
1553   by (rule sym, rule lcmI, simp_all add: lcm_least lcm_zero)
1555 lemma lcm_neg_numeral_1 [simp]: "lcm (- numeral n) a = lcm (numeral n) a"
1556   by (fact lcm_neg1)
1558 lemma lcm_neg_numeral_2 [simp]: "lcm a (- numeral n) = lcm a (numeral n)"
1559   by (fact lcm_neg2)
1561 end
1564 subsection \<open>Typical instances\<close>
1566 instantiation nat :: euclidean_semiring
1567 begin
1569 definition [simp]:
1570   "euclidean_size_nat = (id :: nat \<Rightarrow> nat)"
1572 definition [simp]:
1573   "normalization_factor_nat (n::nat) = (if n = 0 then 0 else 1 :: nat)"
1575 instance proof
1576 qed simp_all
1578 end
1580 instantiation int :: euclidean_ring
1581 begin
1583 definition [simp]:
1584   "euclidean_size_int = (nat \<circ> abs :: int \<Rightarrow> nat)"
1586 definition [simp]:
1587   "normalization_factor_int = (sgn :: int \<Rightarrow> int)"
1589 instance
1590 proof (default, goals)
1591   case 2
1592   then show ?case by (auto simp add: abs_mult nat_mult_distrib)
1593 next
1594   case 3
1595   then show ?case by (simp add: zsgn_def)
1596 next
1597   case 5
1598   then show ?case by (auto simp: zsgn_def)
1599 next
1600   case 6
1601   then show ?case by (auto split: abs_split simp: zsgn_def)
1602 qed (auto simp: sgn_times split: abs_split)
1604 end
1606 instantiation poly :: (field) euclidean_ring
1607 begin
1609 definition euclidean_size_poly :: "'a poly \<Rightarrow> nat"
1610   where "euclidean_size = (degree :: 'a poly \<Rightarrow> nat)"
1612 definition normalization_factor_poly :: "'a poly \<Rightarrow> 'a poly"
1613   where "normalization_factor p = monom (coeff p (degree p)) 0"
1615 instance
1616 proof (default, unfold euclidean_size_poly_def normalization_factor_poly_def)
1617   fix p q :: "'a poly"
1618   assume "q \<noteq> 0" and "\<not> q dvd p"
1619   then show "degree (p mod q) < degree q"
1620     using degree_mod_less [of q p] by (simp add: mod_eq_0_iff_dvd)
1621 next
1622   fix p q :: "'a poly"
1623   assume "q \<noteq> 0"
1624   from \<open>q \<noteq> 0\<close> show "degree p \<le> degree (p * q)"
1625     by (rule degree_mult_right_le)
1626   from \<open>q \<noteq> 0\<close> show "is_unit (monom (coeff q (degree q)) 0)"
1627     by (auto intro: is_unit_monom_0)
1628 next
1629   fix p :: "'a poly"
1630   show "monom (coeff p (degree p)) 0 = p" if "is_unit p"
1631     using that by (fact is_unit_monom_trival)
1632 next
1633   fix p q :: "'a poly"
1634   show "monom (coeff (p * q) (degree (p * q))) 0 =
1635     monom (coeff p (degree p)) 0 * monom (coeff q (degree q)) 0"
1636     by (simp add: monom_0 coeff_degree_mult)
1637 next
1638   show "monom (coeff 0 (degree 0)) 0 = 0"
1639     by simp
1640 qed
1642 end
1644 end