src/HOL/Number_Theory/Euclidean_Algorithm.thy
 author haftmann Thu Jun 25 15:01:42 2015 +0200 (2015-06-25) changeset 60571 c9fdf2080447 parent 60569 f2f1f6860959 child 60572 718b1ba06429 permissions -rw-r--r--
euclidean algorithm on polynomials
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"
86 proof (cases "b = 0", simp, cases "a = 0", metis associated_0(1) normalization_0_iff, rule iffI)
87   let ?nf = normalization_factor
88   assume "a \<noteq> 0" "b \<noteq> 0" "a div ?nf a = b div ?nf b"
89   hence "a = b * (?nf a div ?nf b)"
90     apply (subst (asm) unit_eq_div1, blast, subst (asm) unit_div_commute, blast)
91     apply (subst div_mult_swap, simp, simp)
92     done
93   with \<open>a \<noteq> 0\<close> \<open>b \<noteq> 0\<close> have "\<exists>c. is_unit c \<and> a = c * b"
94     by (intro exI[of _ "?nf a div ?nf b"], force simp: mult_ac)
95   then obtain c where "is_unit c" and "a = c * b" by blast
96   then show "associated a b" by (rule is_unit_associatedI)
97 next
98   let ?nf = normalization_factor
99   assume "a \<noteq> 0" "b \<noteq> 0" "associated a b"
100   then obtain c where "is_unit c" and "a = c * b" by (blast elim: associated_is_unitE)
101   then show "a div ?nf a = b div ?nf b"
102     apply (simp only: \<open>a = c * b\<close> normalization_factor_mult normalization_factor_unit)
103     apply (rule div_mult_mult1, force)
104     done
105   qed
107 lemma normed_associated_imp_eq:
108   "associated a b \<Longrightarrow> normalization_factor a \<in> {0, 1} \<Longrightarrow> normalization_factor b \<in> {0, 1} \<Longrightarrow> a = b"
109   by (simp add: associated_iff_normed_eq, elim disjE, simp_all)
111 lemma normed_dvd [iff]:
112   "a div normalization_factor a dvd a"
113 proof (cases "a = 0")
114   case True then show ?thesis by simp
115 next
116   case False
117   then have "a = a div normalization_factor a * normalization_factor a"
118     by (auto intro: unit_div_mult_self)
119   then show ?thesis ..
120 qed
122 lemma dvd_normed [iff]:
123   "a dvd a div normalization_factor a"
124 proof (cases "a = 0")
125   case True then show ?thesis by simp
126 next
127   case False
128   then have "a div normalization_factor a = a * (1 div normalization_factor a)"
129     by (auto intro: unit_mult_div_div)
130   then show ?thesis ..
131 qed
133 lemma associated_normed:
134   "associated (a div normalization_factor a) a"
135   by (rule associatedI) simp_all
137 lemma normalization_factor_dvd' [simp]:
138   "normalization_factor a dvd a"
139   by (cases "a = 0", simp_all)
141 lemmas normalization_factor_dvd_iff [simp] =
142   unit_dvd_iff [OF normalization_factor_is_unit]
144 lemma euclidean_division:
145   fixes a :: 'a and b :: 'a
146   assumes "b \<noteq> 0" and "\<not> b dvd a"
147   obtains s and t where "a = s * b + t"
148     and "euclidean_size t < euclidean_size b"
149 proof -
150   from div_mod_equality [of a b 0]
151      have "a = a div b * b + a mod b" by simp
152   with that and assms show ?thesis by (auto simp add: mod_size_less)
153 qed
155 lemma dvd_euclidean_size_eq_imp_dvd:
156   assumes "a \<noteq> 0" and b_dvd_a: "b dvd a" and size_eq: "euclidean_size a = euclidean_size b"
157   shows "a dvd b"
158 proof (rule ccontr)
159   assume "\<not> a dvd b"
160   then have "b mod a \<noteq> 0" by (simp add: mod_eq_0_iff_dvd)
161   from b_dvd_a have b_dvd_mod: "b dvd b mod a" by (simp add: dvd_mod_iff)
162   from b_dvd_mod obtain c where "b mod a = b * c" unfolding dvd_def by blast
163     with \<open>b mod a \<noteq> 0\<close> have "c \<noteq> 0" by auto
164   with \<open>b mod a = b * c\<close> have "euclidean_size (b mod a) \<ge> euclidean_size b"
165       using size_mult_mono by force
166   moreover from \<open>\<not> a dvd b\<close> and \<open>a \<noteq> 0\<close>
167   have "euclidean_size (b mod a) < euclidean_size a"
168       using mod_size_less by blast
169   ultimately show False using size_eq by simp
170 qed
172 function gcd_eucl :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
173 where
174   "gcd_eucl a b = (if b = 0 then a div normalization_factor a
175     else if b dvd a then b div normalization_factor b
176     else gcd_eucl b (a mod b))"
177   by (pat_completeness, simp)
178 termination
179   by (relation "measure (euclidean_size \<circ> snd)") (simp_all add: mod_size_less)
181 declare gcd_eucl.simps [simp del]
183 lemma gcd_eucl_induct [case_names zero mod]:
184   assumes H1: "\<And>b. P b 0"
185   and H2: "\<And>a b. b \<noteq> 0 \<Longrightarrow> P b (a mod b) \<Longrightarrow> P a b"
186   shows "P a b"
187 proof (induct a b rule: gcd_eucl.induct)
188   case ("1" a b)
189   show ?case
190   proof (cases "b = 0")
191     case True then show "P a b" by simp (rule H1)
192   next
193     case False
194     have "P b (a mod b)"
195     proof (cases "b dvd a")
196       case False with \<open>b \<noteq> 0\<close> show "P b (a mod b)"
197         by (rule "1.hyps")
198     next
199       case True then have "a mod b = 0"
200         by (simp add: mod_eq_0_iff_dvd)
201       then show "P b (a mod b)" by simp (rule H1)
202     qed
203     with \<open>b \<noteq> 0\<close> show "P a b"
204       by (blast intro: H2)
205   qed
206 qed
208 definition lcm_eucl :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
209 where
210   "lcm_eucl a b = a * b div (gcd_eucl a b * normalization_factor (a * b))"
212   (* Somewhat complicated definition of Lcm that has the advantage of working
213      for infinite sets as well *)
215 definition Lcm_eucl :: "'a set \<Rightarrow> 'a"
216 where
217   "Lcm_eucl A = (if \<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) then
218      let l = SOME l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l =
219        (LEAST n. \<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n)
220        in l div normalization_factor l
221       else 0)"
223 definition Gcd_eucl :: "'a set \<Rightarrow> 'a"
224 where
225   "Gcd_eucl A = Lcm_eucl {d. \<forall>a\<in>A. d dvd a}"
227 end
229 class euclidean_semiring_gcd = euclidean_semiring + gcd + Gcd +
230   assumes gcd_gcd_eucl: "gcd = gcd_eucl" and lcm_lcm_eucl: "lcm = lcm_eucl"
231   assumes Gcd_Gcd_eucl: "Gcd = Gcd_eucl" and Lcm_Lcm_eucl: "Lcm = Lcm_eucl"
232 begin
234 lemma gcd_red:
235   "gcd a b = gcd b (a mod b)"
236   by (cases "b dvd a")
237     (auto simp add: gcd_gcd_eucl gcd_eucl.simps [of a b] gcd_eucl.simps [of 0 a] gcd_eucl.simps [of b 0])
239 lemma gcd_non_0:
240   "b \<noteq> 0 \<Longrightarrow> gcd a b = gcd b (a mod b)"
241   by (rule gcd_red)
243 lemma gcd_0_left:
244   "gcd 0 a = a div normalization_factor a"
245    by (simp only: gcd_gcd_eucl, subst gcd_eucl.simps, subst gcd_eucl.simps, simp add: Let_def)
247 lemma gcd_0:
248   "gcd a 0 = a div normalization_factor a"
249   by (simp only: gcd_gcd_eucl, subst gcd_eucl.simps, simp add: Let_def)
251 lemma gcd_dvd1 [iff]: "gcd a b dvd a"
252   and gcd_dvd2 [iff]: "gcd a b dvd b"
253   by (induct a b rule: gcd_eucl_induct)
254     (simp_all add: gcd_0 gcd_non_0 dvd_mod_iff)
256 lemma dvd_gcd_D1: "k dvd gcd m n \<Longrightarrow> k dvd m"
257   by (rule dvd_trans, assumption, rule gcd_dvd1)
259 lemma dvd_gcd_D2: "k dvd gcd m n \<Longrightarrow> k dvd n"
260   by (rule dvd_trans, assumption, rule gcd_dvd2)
262 lemma gcd_greatest:
263   fixes k a b :: 'a
264   shows "k dvd a \<Longrightarrow> k dvd b \<Longrightarrow> k dvd gcd a b"
265 proof (induct a b rule: gcd_eucl_induct)
266   case (zero a) from zero(1) show ?case by (rule dvd_trans) (simp add: gcd_0)
267 next
268   case (mod a b)
269   then show ?case
270     by (simp add: gcd_non_0 dvd_mod_iff)
271 qed
273 lemma dvd_gcd_iff:
274   "k dvd gcd a b \<longleftrightarrow> k dvd a \<and> k dvd b"
275   by (blast intro!: gcd_greatest intro: dvd_trans)
277 lemmas gcd_greatest_iff = dvd_gcd_iff
279 lemma gcd_zero [simp]:
280   "gcd a b = 0 \<longleftrightarrow> a = 0 \<and> b = 0"
281   by (metis dvd_0_left dvd_refl gcd_dvd1 gcd_dvd2 gcd_greatest)+
283 lemma normalization_factor_gcd [simp]:
284   "normalization_factor (gcd a b) = (if a = 0 \<and> b = 0 then 0 else 1)" (is "?f a b = ?g a b")
285   by (induct a b rule: gcd_eucl_induct)
286     (auto simp add: gcd_0 gcd_non_0)
288 lemma gcdI:
289   "k dvd a \<Longrightarrow> k dvd b \<Longrightarrow> (\<And>l. l dvd a \<Longrightarrow> l dvd b \<Longrightarrow> l dvd k)
290     \<Longrightarrow> normalization_factor k = (if k = 0 then 0 else 1) \<Longrightarrow> k = gcd a b"
291   by (intro normed_associated_imp_eq) (auto simp: associated_def intro: gcd_greatest)
293 sublocale gcd!: abel_semigroup gcd
294 proof
295   fix a b c
296   show "gcd (gcd a b) c = gcd a (gcd b c)"
297   proof (rule gcdI)
298     have "gcd (gcd a b) c dvd gcd a b" "gcd a b dvd a" by simp_all
299     then show "gcd (gcd a b) c dvd a" by (rule dvd_trans)
300     have "gcd (gcd a b) c dvd gcd a b" "gcd a b dvd b" by simp_all
301     hence "gcd (gcd a b) c dvd b" by (rule dvd_trans)
302     moreover have "gcd (gcd a b) c dvd c" by simp
303     ultimately show "gcd (gcd a b) c dvd gcd b c"
304       by (rule gcd_greatest)
305     show "normalization_factor (gcd (gcd a b) c) =  (if gcd (gcd a b) c = 0 then 0 else 1)"
306       by auto
307     fix l assume "l dvd a" and "l dvd gcd b c"
308     with dvd_trans[OF _ gcd_dvd1] and dvd_trans[OF _ gcd_dvd2]
309       have "l dvd b" and "l dvd c" by blast+
310     with \<open>l dvd a\<close> show "l dvd gcd (gcd a b) c"
311       by (intro gcd_greatest)
312   qed
313 next
314   fix a b
315   show "gcd a b = gcd b a"
316     by (rule gcdI) (simp_all add: gcd_greatest)
317 qed
319 lemma gcd_unique: "d dvd a \<and> d dvd b \<and>
320     normalization_factor d = (if d = 0 then 0 else 1) \<and>
321     (\<forall>e. e dvd a \<and> e dvd b \<longrightarrow> e dvd d) \<longleftrightarrow> d = gcd a b"
322   by (rule, auto intro: gcdI simp: gcd_greatest)
324 lemma gcd_dvd_prod: "gcd a b dvd k * b"
325   using mult_dvd_mono [of 1] by auto
327 lemma gcd_1_left [simp]: "gcd 1 a = 1"
328   by (rule sym, rule gcdI, simp_all)
330 lemma gcd_1 [simp]: "gcd a 1 = 1"
331   by (rule sym, rule gcdI, simp_all)
333 lemma gcd_proj2_if_dvd:
334   "b dvd a \<Longrightarrow> gcd a b = b div normalization_factor b"
335   by (cases "b = 0", simp_all add: dvd_eq_mod_eq_0 gcd_non_0 gcd_0)
337 lemma gcd_proj1_if_dvd:
338   "a dvd b \<Longrightarrow> gcd a b = a div normalization_factor a"
339   by (subst gcd.commute, simp add: gcd_proj2_if_dvd)
341 lemma gcd_proj1_iff: "gcd m n = m div normalization_factor m \<longleftrightarrow> m dvd n"
342 proof
343   assume A: "gcd m n = m div normalization_factor m"
344   show "m dvd n"
345   proof (cases "m = 0")
346     assume [simp]: "m \<noteq> 0"
347     from A have B: "m = gcd m n * normalization_factor m"
348       by (simp add: unit_eq_div2)
349     show ?thesis by (subst B, simp add: mult_unit_dvd_iff)
350   qed (insert A, simp)
351 next
352   assume "m dvd n"
353   then show "gcd m n = m div normalization_factor m" by (rule gcd_proj1_if_dvd)
354 qed
356 lemma gcd_proj2_iff: "gcd m n = n div normalization_factor n \<longleftrightarrow> n dvd m"
357   by (subst gcd.commute, simp add: gcd_proj1_iff)
359 lemma gcd_mod1 [simp]:
360   "gcd (a mod b) b = gcd a b"
361   by (rule gcdI, metis dvd_mod_iff gcd_dvd1 gcd_dvd2, simp_all add: gcd_greatest dvd_mod_iff)
363 lemma gcd_mod2 [simp]:
364   "gcd a (b mod a) = gcd a b"
365   by (rule gcdI, simp, metis dvd_mod_iff gcd_dvd1 gcd_dvd2, simp_all add: gcd_greatest dvd_mod_iff)
367 lemma gcd_mult_distrib':
368   "c div normalization_factor c * gcd a b = gcd (c * a) (c * b)"
369 proof (cases "c = 0")
370   case True then show ?thesis by (simp_all add: gcd_0)
371 next
372   case False then have [simp]: "is_unit (normalization_factor c)" by simp
373   show ?thesis
374   proof (induct a b rule: gcd_eucl_induct)
375     case (zero a) show ?case
376     proof (cases "a = 0")
377       case True then show ?thesis by (simp add: gcd_0)
378     next
379       case False then have "is_unit (normalization_factor a)" by simp
380       then show ?thesis
381         by (simp add: gcd_0 unit_div_commute unit_div_mult_swap normalization_factor_mult is_unit_div_mult2_eq)
382     qed
383     case (mod a b)
384     then show ?case by (simp add: mult_mod_right gcd.commute)
385   qed
386 qed
388 lemma gcd_mult_distrib:
389   "k * gcd a b = gcd (k*a) (k*b) * normalization_factor k"
390 proof-
391   let ?nf = "normalization_factor"
392   from gcd_mult_distrib'
393     have "gcd (k*a) (k*b) = k div ?nf k * gcd a b" ..
394   also have "... = k * gcd a b div ?nf k"
395     by (metis dvd_div_mult dvd_eq_mod_eq_0 mod_0 normalization_factor_dvd)
396   finally show ?thesis
397     by simp
398 qed
400 lemma euclidean_size_gcd_le1 [simp]:
401   assumes "a \<noteq> 0"
402   shows "euclidean_size (gcd a b) \<le> euclidean_size a"
403 proof -
404    have "gcd a b dvd a" by (rule gcd_dvd1)
405    then obtain c where A: "a = gcd a b * c" unfolding dvd_def by blast
406    with \<open>a \<noteq> 0\<close> show ?thesis by (subst (2) A, intro size_mult_mono) auto
407 qed
409 lemma euclidean_size_gcd_le2 [simp]:
410   "b \<noteq> 0 \<Longrightarrow> euclidean_size (gcd a b) \<le> euclidean_size b"
411   by (subst gcd.commute, rule euclidean_size_gcd_le1)
413 lemma euclidean_size_gcd_less1:
414   assumes "a \<noteq> 0" and "\<not>a dvd b"
415   shows "euclidean_size (gcd a b) < euclidean_size a"
416 proof (rule ccontr)
417   assume "\<not>euclidean_size (gcd a b) < euclidean_size a"
418   with \<open>a \<noteq> 0\<close> have "euclidean_size (gcd a b) = euclidean_size a"
419     by (intro le_antisym, simp_all)
420   with assms have "a dvd gcd a b" by (auto intro: dvd_euclidean_size_eq_imp_dvd)
421   hence "a dvd b" using dvd_gcd_D2 by blast
422   with \<open>\<not>a dvd b\<close> show False by contradiction
423 qed
425 lemma euclidean_size_gcd_less2:
426   assumes "b \<noteq> 0" and "\<not>b dvd a"
427   shows "euclidean_size (gcd a b) < euclidean_size b"
428   using assms by (subst gcd.commute, rule euclidean_size_gcd_less1)
430 lemma gcd_mult_unit1: "is_unit a \<Longrightarrow> gcd (b * a) c = gcd b c"
431   apply (rule gcdI)
432   apply (rule dvd_trans, rule gcd_dvd1, simp add: unit_simps)
433   apply (rule gcd_dvd2)
434   apply (rule gcd_greatest, simp add: unit_simps, assumption)
435   apply (subst normalization_factor_gcd, simp add: gcd_0)
436   done
438 lemma gcd_mult_unit2: "is_unit a \<Longrightarrow> gcd b (c * a) = gcd b c"
439   by (subst gcd.commute, subst gcd_mult_unit1, assumption, rule gcd.commute)
441 lemma gcd_div_unit1: "is_unit a \<Longrightarrow> gcd (b div a) c = gcd b c"
442   by (erule is_unitE [of _ b]) (simp add: gcd_mult_unit1)
444 lemma gcd_div_unit2: "is_unit a \<Longrightarrow> gcd b (c div a) = gcd b c"
445   by (erule is_unitE [of _ c]) (simp add: gcd_mult_unit2)
447 lemma gcd_idem: "gcd a a = a div normalization_factor a"
448   by (cases "a = 0") (simp add: gcd_0_left, rule sym, rule gcdI, simp_all)
450 lemma gcd_right_idem: "gcd (gcd a b) b = gcd a b"
451   apply (rule gcdI)
452   apply (simp add: ac_simps)
453   apply (rule gcd_dvd2)
454   apply (rule gcd_greatest, erule (1) gcd_greatest, assumption)
455   apply simp
456   done
458 lemma gcd_left_idem: "gcd a (gcd a b) = gcd a b"
459   apply (rule gcdI)
460   apply simp
461   apply (rule dvd_trans, rule gcd_dvd2, rule gcd_dvd2)
462   apply (rule gcd_greatest, assumption, erule gcd_greatest, assumption)
463   apply simp
464   done
466 lemma comp_fun_idem_gcd: "comp_fun_idem gcd"
467 proof
468   fix a b show "gcd a \<circ> gcd b = gcd b \<circ> gcd a"
469     by (simp add: fun_eq_iff ac_simps)
470 next
471   fix a show "gcd a \<circ> gcd a = gcd a"
472     by (simp add: fun_eq_iff gcd_left_idem)
473 qed
475 lemma coprime_dvd_mult:
476   assumes "gcd c b = 1" and "c dvd a * b"
477   shows "c dvd a"
478 proof -
479   let ?nf = "normalization_factor"
480   from assms gcd_mult_distrib [of a c b]
481     have A: "a = gcd (a * c) (a * b) * ?nf a" by simp
482   from \<open>c dvd a * b\<close> show ?thesis by (subst A, simp_all add: gcd_greatest)
483 qed
485 lemma coprime_dvd_mult_iff:
486   "gcd c b = 1 \<Longrightarrow> (c dvd a * b) = (c dvd a)"
487   by (rule, rule coprime_dvd_mult, simp_all)
489 lemma gcd_dvd_antisym:
490   "gcd a b dvd gcd c d \<Longrightarrow> gcd c d dvd gcd a b \<Longrightarrow> gcd a b = gcd c d"
491 proof (rule gcdI)
492   assume A: "gcd a b dvd gcd c d" and B: "gcd c d dvd gcd a b"
493   have "gcd c d dvd c" by simp
494   with A show "gcd a b dvd c" by (rule dvd_trans)
495   have "gcd c d dvd d" by simp
496   with A show "gcd a b dvd d" by (rule dvd_trans)
497   show "normalization_factor (gcd a b) = (if gcd a b = 0 then 0 else 1)"
498     by simp
499   fix l assume "l dvd c" and "l dvd d"
500   hence "l dvd gcd c d" by (rule gcd_greatest)
501   from this and B show "l dvd gcd a b" by (rule dvd_trans)
502 qed
504 lemma gcd_mult_cancel:
505   assumes "gcd k n = 1"
506   shows "gcd (k * m) n = gcd m n"
507 proof (rule gcd_dvd_antisym)
508   have "gcd (gcd (k * m) n) k = gcd (gcd k n) (k * m)" by (simp add: ac_simps)
509   also note \<open>gcd k n = 1\<close>
510   finally have "gcd (gcd (k * m) n) k = 1" by simp
511   hence "gcd (k * m) n dvd m" by (rule coprime_dvd_mult, simp add: ac_simps)
512   moreover have "gcd (k * m) n dvd n" by simp
513   ultimately show "gcd (k * m) n dvd gcd m n" by (rule gcd_greatest)
514   have "gcd m n dvd (k * m)" and "gcd m n dvd n" by simp_all
515   then show "gcd m n dvd gcd (k * m) n" by (rule gcd_greatest)
516 qed
518 lemma coprime_crossproduct:
519   assumes [simp]: "gcd a d = 1" "gcd b c = 1"
520   shows "associated (a * c) (b * d) \<longleftrightarrow> associated a b \<and> associated c d" (is "?lhs \<longleftrightarrow> ?rhs")
521 proof
522   assume ?rhs then show ?lhs unfolding associated_def by (fast intro: mult_dvd_mono)
523 next
524   assume ?lhs
525   from \<open>?lhs\<close> have "a dvd b * d" unfolding associated_def by (metis dvd_mult_left)
526   hence "a dvd b" by (simp add: coprime_dvd_mult_iff)
527   moreover from \<open>?lhs\<close> have "b dvd a * c" unfolding associated_def by (metis dvd_mult_left)
528   hence "b dvd a" by (simp add: coprime_dvd_mult_iff)
529   moreover from \<open>?lhs\<close> have "c dvd d * b"
530     unfolding associated_def by (auto dest: dvd_mult_right simp add: ac_simps)
531   hence "c dvd d" by (simp add: coprime_dvd_mult_iff gcd.commute)
532   moreover from \<open>?lhs\<close> have "d dvd c * a"
533     unfolding associated_def by (auto dest: dvd_mult_right simp add: ac_simps)
534   hence "d dvd c" by (simp add: coprime_dvd_mult_iff gcd.commute)
535   ultimately show ?rhs unfolding associated_def by simp
536 qed
538 lemma gcd_add1 [simp]:
539   "gcd (m + n) n = gcd m n"
540   by (cases "n = 0", simp_all add: gcd_non_0)
542 lemma gcd_add2 [simp]:
543   "gcd m (m + n) = gcd m n"
544   using gcd_add1 [of n m] by (simp add: ac_simps)
546 lemma gcd_add_mult: "gcd m (k * m + n) = gcd m n"
547   by (subst gcd.commute, subst gcd_red, simp)
549 lemma coprimeI: "(\<And>l. \<lbrakk>l dvd a; l dvd b\<rbrakk> \<Longrightarrow> l dvd 1) \<Longrightarrow> gcd a b = 1"
550   by (rule sym, rule gcdI, simp_all)
552 lemma coprime: "gcd a b = 1 \<longleftrightarrow> (\<forall>d. d dvd a \<and> d dvd b \<longleftrightarrow> is_unit d)"
553   by (auto intro: coprimeI gcd_greatest dvd_gcd_D1 dvd_gcd_D2)
555 lemma div_gcd_coprime:
556   assumes nz: "a \<noteq> 0 \<or> b \<noteq> 0"
557   defines [simp]: "d \<equiv> gcd a b"
558   defines [simp]: "a' \<equiv> a div d" and [simp]: "b' \<equiv> b div d"
559   shows "gcd a' b' = 1"
560 proof (rule coprimeI)
561   fix l assume "l dvd a'" "l dvd b'"
562   then obtain s t where "a' = l * s" "b' = l * t" unfolding dvd_def by blast
563   moreover have "a = a' * d" "b = b' * d" by simp_all
564   ultimately have "a = (l * d) * s" "b = (l * d) * t"
565     by (simp_all only: ac_simps)
566   hence "l*d dvd a" and "l*d dvd b" by (simp_all only: dvd_triv_left)
567   hence "l*d dvd d" by (simp add: gcd_greatest)
568   then obtain u where "d = l * d * u" ..
569   then have "d * (l * u) = d" by (simp add: ac_simps)
570   moreover from nz have "d \<noteq> 0" by simp
571   with div_mult_self1_is_id have "d * (l * u) div d = l * u" .
572   ultimately have "1 = l * u"
573     using \<open>d \<noteq> 0\<close> by simp
574   then show "l dvd 1" ..
575 qed
577 lemma coprime_mult:
578   assumes da: "gcd d a = 1" and db: "gcd d b = 1"
579   shows "gcd d (a * b) = 1"
580   apply (subst gcd.commute)
581   using da apply (subst gcd_mult_cancel)
582   apply (subst gcd.commute, assumption)
583   apply (subst gcd.commute, rule db)
584   done
586 lemma coprime_lmult:
587   assumes dab: "gcd d (a * b) = 1"
588   shows "gcd d a = 1"
589 proof (rule coprimeI)
590   fix l assume "l dvd d" and "l dvd a"
591   hence "l dvd a * b" by simp
592   with \<open>l dvd d\<close> and dab show "l dvd 1" by (auto intro: gcd_greatest)
593 qed
595 lemma coprime_rmult:
596   assumes dab: "gcd d (a * b) = 1"
597   shows "gcd d b = 1"
598 proof (rule coprimeI)
599   fix l assume "l dvd d" and "l dvd b"
600   hence "l dvd a * b" by simp
601   with \<open>l dvd d\<close> and dab show "l dvd 1" by (auto intro: gcd_greatest)
602 qed
604 lemma coprime_mul_eq: "gcd d (a * b) = 1 \<longleftrightarrow> gcd d a = 1 \<and> gcd d b = 1"
605   using coprime_rmult[of d a b] coprime_lmult[of d a b] coprime_mult[of d a b] by blast
607 lemma gcd_coprime:
608   assumes c: "gcd a b \<noteq> 0" and a: "a = a' * gcd a b" and b: "b = b' * gcd a b"
609   shows "gcd a' b' = 1"
610 proof -
611   from c have "a \<noteq> 0 \<or> b \<noteq> 0" by simp
612   with div_gcd_coprime have "gcd (a div gcd a b) (b div gcd a b) = 1" .
613   also from assms have "a div gcd a b = a'" by (metis div_mult_self2_is_id)+
614   also from assms have "b div gcd a b = b'" by (metis div_mult_self2_is_id)+
615   finally show ?thesis .
616 qed
618 lemma coprime_power:
619   assumes "0 < n"
620   shows "gcd a (b ^ n) = 1 \<longleftrightarrow> gcd a b = 1"
621 using assms proof (induct n)
622   case (Suc n) then show ?case
623     by (cases n) (simp_all add: coprime_mul_eq)
624 qed simp
626 lemma gcd_coprime_exists:
627   assumes nz: "gcd a b \<noteq> 0"
628   shows "\<exists>a' b'. a = a' * gcd a b \<and> b = b' * gcd a b \<and> gcd a' b' = 1"
629   apply (rule_tac x = "a div gcd a b" in exI)
630   apply (rule_tac x = "b div gcd a b" in exI)
631   apply (insert nz, auto intro: div_gcd_coprime)
632   done
634 lemma coprime_exp:
635   "gcd d a = 1 \<Longrightarrow> gcd d (a^n) = 1"
636   by (induct n, simp_all add: coprime_mult)
638 lemma coprime_exp2 [intro]:
639   "gcd a b = 1 \<Longrightarrow> gcd (a^n) (b^m) = 1"
640   apply (rule coprime_exp)
641   apply (subst gcd.commute)
642   apply (rule coprime_exp)
643   apply (subst gcd.commute)
644   apply assumption
645   done
647 lemma gcd_exp:
648   "gcd (a^n) (b^n) = (gcd a b) ^ n"
649 proof (cases "a = 0 \<and> b = 0")
650   assume "a = 0 \<and> b = 0"
651   then show ?thesis by (cases n, simp_all add: gcd_0_left)
652 next
653   assume A: "\<not>(a = 0 \<and> b = 0)"
654   hence "1 = gcd ((a div gcd a b)^n) ((b div gcd a b)^n)"
655     using div_gcd_coprime by (subst sym, auto simp: div_gcd_coprime)
656   hence "(gcd a b) ^ n = (gcd a b) ^ n * ..." by simp
657   also note gcd_mult_distrib
658   also have "normalization_factor ((gcd a b)^n) = 1"
659     by (simp add: normalization_factor_pow A)
660   also have "(gcd a b)^n * (a div gcd a b)^n = a^n"
661     by (subst ac_simps, subst div_power, simp, rule dvd_div_mult_self, rule dvd_power_same, simp)
662   also have "(gcd a b)^n * (b div gcd a b)^n = b^n"
663     by (subst ac_simps, subst div_power, simp, rule dvd_div_mult_self, rule dvd_power_same, simp)
664   finally show ?thesis by simp
665 qed
667 lemma coprime_common_divisor:
668   "gcd a b = 1 \<Longrightarrow> a dvd a \<Longrightarrow> a dvd b \<Longrightarrow> is_unit a"
669   apply (subgoal_tac "a dvd gcd a b")
670   apply simp
671   apply (erule (1) gcd_greatest)
672   done
674 lemma division_decomp:
675   assumes dc: "a dvd b * c"
676   shows "\<exists>b' c'. a = b' * c' \<and> b' dvd b \<and> c' dvd c"
677 proof (cases "gcd a b = 0")
678   assume "gcd a b = 0"
679   hence "a = 0 \<and> b = 0" by simp
680   hence "a = 0 * c \<and> 0 dvd b \<and> c dvd c" by simp
681   then show ?thesis by blast
682 next
683   let ?d = "gcd a b"
684   assume "?d \<noteq> 0"
685   from gcd_coprime_exists[OF this]
686     obtain a' b' where ab': "a = a' * ?d" "b = b' * ?d" "gcd a' b' = 1"
687     by blast
688   from ab'(1) have "a' dvd a" unfolding dvd_def by blast
689   with dc have "a' dvd b*c" using dvd_trans[of a' a "b*c"] by simp
690   from dc ab'(1,2) have "a'*?d dvd (b'*?d) * c" by simp
691   hence "?d * a' dvd ?d * (b' * c)" by (simp add: mult_ac)
692   with \<open>?d \<noteq> 0\<close> have "a' dvd b' * c" by simp
693   with coprime_dvd_mult[OF ab'(3)]
694     have "a' dvd c" by (subst (asm) ac_simps, blast)
695   with ab'(1) have "a = ?d * a' \<and> ?d dvd b \<and> a' dvd c" by (simp add: mult_ac)
696   then show ?thesis by blast
697 qed
699 lemma pow_divs_pow:
700   assumes ab: "a ^ n dvd b ^ n" and n: "n \<noteq> 0"
701   shows "a dvd b"
702 proof (cases "gcd a b = 0")
703   assume "gcd a b = 0"
704   then show ?thesis by simp
705 next
706   let ?d = "gcd a b"
707   assume "?d \<noteq> 0"
708   from n obtain m where m: "n = Suc m" by (cases n, simp_all)
709   from \<open>?d \<noteq> 0\<close> have zn: "?d ^ n \<noteq> 0" by (rule power_not_zero)
710   from gcd_coprime_exists[OF \<open>?d \<noteq> 0\<close>]
711     obtain a' b' where ab': "a = a' * ?d" "b = b' * ?d" "gcd a' b' = 1"
712     by blast
713   from ab have "(a' * ?d) ^ n dvd (b' * ?d) ^ n"
714     by (simp add: ab'(1,2)[symmetric])
715   hence "?d^n * a'^n dvd ?d^n * b'^n"
716     by (simp only: power_mult_distrib ac_simps)
717   with zn have "a'^n dvd b'^n" by simp
718   hence "a' dvd b'^n" using dvd_trans[of a' "a'^n" "b'^n"] by (simp add: m)
719   hence "a' dvd b'^m * b'" by (simp add: m ac_simps)
720   with coprime_dvd_mult[OF coprime_exp[OF ab'(3), of m]]
721     have "a' dvd b'" by (subst (asm) ac_simps, blast)
722   hence "a'*?d dvd b'*?d" by (rule mult_dvd_mono, simp)
723   with ab'(1,2) show ?thesis by simp
724 qed
726 lemma pow_divs_eq [simp]:
727   "n \<noteq> 0 \<Longrightarrow> a ^ n dvd b ^ n \<longleftrightarrow> a dvd b"
728   by (auto intro: pow_divs_pow dvd_power_same)
730 lemma divs_mult:
731   assumes mr: "m dvd r" and nr: "n dvd r" and mn: "gcd m n = 1"
732   shows "m * n dvd r"
733 proof -
734   from mr nr obtain m' n' where m': "r = m*m'" and n': "r = n*n'"
735     unfolding dvd_def by blast
736   from mr n' have "m dvd n'*n" by (simp add: ac_simps)
737   hence "m dvd n'" using coprime_dvd_mult_iff[OF mn] by simp
738   then obtain k where k: "n' = m*k" unfolding dvd_def by blast
739   with n' have "r = m * n * k" by (simp add: mult_ac)
740   then show ?thesis unfolding dvd_def by blast
741 qed
743 lemma coprime_plus_one [simp]: "gcd (n + 1) n = 1"
744   by (subst add_commute, simp)
746 lemma setprod_coprime [rule_format]:
747   "(\<forall>i\<in>A. gcd (f i) a = 1) \<longrightarrow> gcd (\<Prod>i\<in>A. f i) a = 1"
748   apply (cases "finite A")
749   apply (induct set: finite)
750   apply (auto simp add: gcd_mult_cancel)
751   done
753 lemma coprime_divisors:
754   assumes "d dvd a" "e dvd b" "gcd a b = 1"
755   shows "gcd d e = 1"
756 proof -
757   from assms obtain k l where "a = d * k" "b = e * l"
758     unfolding dvd_def by blast
759   with assms have "gcd (d * k) (e * l) = 1" by simp
760   hence "gcd (d * k) e = 1" by (rule coprime_lmult)
761   also have "gcd (d * k) e = gcd e (d * k)" by (simp add: ac_simps)
762   finally have "gcd e d = 1" by (rule coprime_lmult)
763   then show ?thesis by (simp add: ac_simps)
764 qed
766 lemma invertible_coprime:
767   assumes "a * b mod m = 1"
768   shows "coprime a m"
769 proof -
770   from assms have "coprime m (a * b mod m)"
771     by simp
772   then have "coprime m (a * b)"
773     by simp
774   then have "coprime m a"
775     by (rule coprime_lmult)
776   then show ?thesis
777     by (simp add: ac_simps)
778 qed
780 lemma lcm_gcd:
781   "lcm a b = a * b div (gcd a b * normalization_factor (a*b))"
782   by (simp only: lcm_lcm_eucl gcd_gcd_eucl lcm_eucl_def)
784 lemma lcm_gcd_prod:
785   "lcm a b * gcd a b = a * b div normalization_factor (a*b)"
786 proof (cases "a * b = 0")
787   let ?nf = normalization_factor
788   assume "a * b \<noteq> 0"
789   hence "gcd a b \<noteq> 0" by simp
790   from lcm_gcd have "lcm a b * gcd a b = gcd a b * (a * b div (?nf (a*b) * gcd a b))"
791     by (simp add: mult_ac)
792   also from \<open>a * b \<noteq> 0\<close> have "... = a * b div ?nf (a*b)"
793     by (simp add: div_mult_swap mult.commute)
794   finally show ?thesis .
795 qed (auto simp add: lcm_gcd)
797 lemma lcm_dvd1 [iff]:
798   "a dvd lcm a b"
799 proof (cases "a*b = 0")
800   assume "a * b \<noteq> 0"
801   hence "gcd a b \<noteq> 0" by simp
802   let ?c = "1 div normalization_factor (a * b)"
803   from \<open>a * b \<noteq> 0\<close> have [simp]: "is_unit (normalization_factor (a * b))" by simp
804   from lcm_gcd_prod[of a b] have "lcm a b * gcd a b = a * ?c * b"
805     by (simp add: div_mult_swap unit_div_commute)
806   hence "lcm a b * gcd a b div gcd a b = a * ?c * b div gcd a b" by simp
807   with \<open>gcd a b \<noteq> 0\<close> have "lcm a b = a * ?c * b div gcd a b"
808     by (subst (asm) div_mult_self2_is_id, simp_all)
809   also have "... = a * (?c * b div gcd a b)"
810     by (metis div_mult_swap gcd_dvd2 mult_assoc)
811   finally show ?thesis by (rule dvdI)
812 qed (auto simp add: lcm_gcd)
814 lemma lcm_least:
815   "\<lbrakk>a dvd k; b dvd k\<rbrakk> \<Longrightarrow> lcm a b dvd k"
816 proof (cases "k = 0")
817   let ?nf = normalization_factor
818   assume "k \<noteq> 0"
819   hence "is_unit (?nf k)" by simp
820   hence "?nf k \<noteq> 0" by (metis not_is_unit_0)
821   assume A: "a dvd k" "b dvd k"
822   hence "gcd a b \<noteq> 0" using \<open>k \<noteq> 0\<close> by auto
823   from A obtain r s where ar: "k = a * r" and bs: "k = b * s"
824     unfolding dvd_def by blast
825   with \<open>k \<noteq> 0\<close> have "r * s \<noteq> 0"
826     by auto (drule sym [of 0], simp)
827   hence "is_unit (?nf (r * s))" by simp
828   let ?c = "?nf k div ?nf (r*s)"
829   from \<open>is_unit (?nf k)\<close> and \<open>is_unit (?nf (r * s))\<close> have "is_unit ?c" by (rule unit_div)
830   hence "?c \<noteq> 0" using not_is_unit_0 by fast
831   from ar bs have "k * k * gcd s r = ?nf k * k * gcd (k * s) (k * r)"
832     by (subst mult_assoc, subst gcd_mult_distrib[of k s r], simp only: ac_simps)
833   also have "... = ?nf k * k * gcd ((r*s) * a) ((r*s) * b)"
834     by (subst (3) \<open>k = a * r\<close>, subst (3) \<open>k = b * s\<close>, simp add: algebra_simps)
835   also have "... = ?c * r*s * k * gcd a b" using \<open>r * s \<noteq> 0\<close>
836     by (subst gcd_mult_distrib'[symmetric], simp add: algebra_simps unit_simps)
837   finally have "(a*r) * (b*s) * gcd s r = ?c * k * r * s * gcd a b"
838     by (subst ar[symmetric], subst bs[symmetric], simp add: mult_ac)
839   hence "a * b * gcd s r * (r * s) = ?c * k * gcd a b * (r * s)"
840     by (simp add: algebra_simps)
841   hence "?c * k * gcd a b = a * b * gcd s r" using \<open>r * s \<noteq> 0\<close>
842     by (metis div_mult_self2_is_id)
843   also have "... = lcm a b * gcd a b * gcd s r * ?nf (a*b)"
844     by (subst lcm_gcd_prod[of a b], metis gcd_mult_distrib gcd_mult_distrib')
845   also have "... = lcm a b * gcd s r * ?nf (a*b) * gcd a b"
846     by (simp add: algebra_simps)
847   finally have "k * ?c = lcm a b * gcd s r * ?nf (a*b)" using \<open>gcd a b \<noteq> 0\<close>
848     by (metis mult.commute div_mult_self2_is_id)
849   hence "k = lcm a b * (gcd s r * ?nf (a*b)) div ?c" using \<open>?c \<noteq> 0\<close>
850     by (metis div_mult_self2_is_id mult_assoc)
851   also have "... = lcm a b * (gcd s r * ?nf (a*b) div ?c)" using \<open>is_unit ?c\<close>
852     by (simp add: unit_simps)
853   finally show ?thesis by (rule dvdI)
854 qed simp
856 lemma lcm_zero:
857   "lcm a b = 0 \<longleftrightarrow> a = 0 \<or> b = 0"
858 proof -
859   let ?nf = normalization_factor
860   {
861     assume "a \<noteq> 0" "b \<noteq> 0"
862     hence "a * b div ?nf (a * b) \<noteq> 0" by (simp add: no_zero_divisors)
863     moreover from \<open>a \<noteq> 0\<close> and \<open>b \<noteq> 0\<close> have "gcd a b \<noteq> 0" by simp
864     ultimately have "lcm a b \<noteq> 0" using lcm_gcd_prod[of a b] by (intro notI, simp)
865   } moreover {
866     assume "a = 0 \<or> b = 0"
867     hence "lcm a b = 0" by (elim disjE, simp_all add: lcm_gcd)
868   }
869   ultimately show ?thesis by blast
870 qed
872 lemmas lcm_0_iff = lcm_zero
874 lemma gcd_lcm:
875   assumes "lcm a b \<noteq> 0"
876   shows "gcd a b = a * b div (lcm a b * normalization_factor (a * b))"
877 proof-
878   from assms have "gcd a b \<noteq> 0" by (simp add: lcm_zero)
879   let ?c = "normalization_factor (a * b)"
880   from \<open>lcm a b \<noteq> 0\<close> have "?c \<noteq> 0" by (intro notI, simp add: lcm_zero no_zero_divisors)
881   hence "is_unit ?c" by simp
882   from lcm_gcd_prod [of a b] have "gcd a b = a * b div ?c div lcm a b"
883     by (subst (2) div_mult_self2_is_id[OF \<open>lcm a b \<noteq> 0\<close>, symmetric], simp add: mult_ac)
884   also from \<open>is_unit ?c\<close> have "... = a * b div (lcm a b * ?c)"
885     by (metis \<open>?c \<noteq> 0\<close> div_mult_mult1 dvd_mult_div_cancel mult_commute normalization_factor_dvd')
886   finally show ?thesis .
887 qed
889 lemma normalization_factor_lcm [simp]:
890   "normalization_factor (lcm a b) = (if a = 0 \<or> b = 0 then 0 else 1)"
891 proof (cases "a = 0 \<or> b = 0")
892   case True then show ?thesis
893     by (auto simp add: lcm_gcd)
894 next
895   case False
896   let ?nf = normalization_factor
897   from lcm_gcd_prod[of a b]
898     have "?nf (lcm a b) * ?nf (gcd a b) = ?nf (a*b) div ?nf (a*b)"
899     by (metis div_by_0 div_self normalization_correct normalization_factor_0 normalization_factor_mult)
900   also have "... = (if a*b = 0 then 0 else 1)"
901     by simp
902   finally show ?thesis using False by simp
903 qed
905 lemma lcm_dvd2 [iff]: "b dvd lcm a b"
906   using lcm_dvd1 [of b a] by (simp add: lcm_gcd ac_simps)
908 lemma lcmI:
909   "\<lbrakk>a dvd k; b dvd k; \<And>l. a dvd l \<Longrightarrow> b dvd l \<Longrightarrow> k dvd l;
910     normalization_factor k = (if k = 0 then 0 else 1)\<rbrakk> \<Longrightarrow> k = lcm a b"
911   by (intro normed_associated_imp_eq) (auto simp: associated_def intro: lcm_least)
913 sublocale lcm!: abel_semigroup lcm
914 proof
915   fix a b c
916   show "lcm (lcm a b) c = lcm a (lcm b c)"
917   proof (rule lcmI)
918     have "a dvd lcm a b" and "lcm a b dvd lcm (lcm a b) c" by simp_all
919     then show "a dvd lcm (lcm a b) c" by (rule dvd_trans)
921     have "b dvd lcm a b" and "lcm a b dvd lcm (lcm a b) c" by simp_all
922     hence "b dvd lcm (lcm a b) c" by (rule dvd_trans)
923     moreover have "c dvd lcm (lcm a b) c" by simp
924     ultimately show "lcm b c dvd lcm (lcm a b) c" by (rule lcm_least)
926     fix l assume "a dvd l" and "lcm b c dvd l"
927     have "b dvd lcm b c" by simp
928     from this and \<open>lcm b c dvd l\<close> have "b dvd l" by (rule dvd_trans)
929     have "c dvd lcm b c" by simp
930     from this and \<open>lcm b c dvd l\<close> have "c dvd l" by (rule dvd_trans)
931     from \<open>a dvd l\<close> and \<open>b dvd l\<close> have "lcm a b dvd l" by (rule lcm_least)
932     from this and \<open>c dvd l\<close> show "lcm (lcm a b) c dvd l" by (rule lcm_least)
933   qed (simp add: lcm_zero)
934 next
935   fix a b
936   show "lcm a b = lcm b a"
937     by (simp add: lcm_gcd ac_simps)
938 qed
940 lemma dvd_lcm_D1:
941   "lcm m n dvd k \<Longrightarrow> m dvd k"
942   by (rule dvd_trans, rule lcm_dvd1, assumption)
944 lemma dvd_lcm_D2:
945   "lcm m n dvd k \<Longrightarrow> n dvd k"
946   by (rule dvd_trans, rule lcm_dvd2, assumption)
948 lemma gcd_dvd_lcm [simp]:
949   "gcd a b dvd lcm a b"
950   by (metis dvd_trans gcd_dvd2 lcm_dvd2)
952 lemma lcm_1_iff:
953   "lcm a b = 1 \<longleftrightarrow> is_unit a \<and> is_unit b"
954 proof
955   assume "lcm a b = 1"
956   then show "is_unit a \<and> is_unit b" by auto
957 next
958   assume "is_unit a \<and> is_unit b"
959   hence "a dvd 1" and "b dvd 1" by simp_all
960   hence "is_unit (lcm a b)" by (rule lcm_least)
961   hence "lcm a b = normalization_factor (lcm a b)"
962     by (subst normalization_factor_unit, simp_all)
963   also have "\<dots> = 1" using \<open>is_unit a \<and> is_unit b\<close>
964     by auto
965   finally show "lcm a b = 1" .
966 qed
968 lemma lcm_0_left [simp]:
969   "lcm 0 a = 0"
970   by (rule sym, rule lcmI, simp_all)
972 lemma lcm_0 [simp]:
973   "lcm a 0 = 0"
974   by (rule sym, rule lcmI, simp_all)
976 lemma lcm_unique:
977   "a dvd d \<and> b dvd d \<and>
978   normalization_factor d = (if d = 0 then 0 else 1) \<and>
979   (\<forall>e. a dvd e \<and> b dvd e \<longrightarrow> d dvd e) \<longleftrightarrow> d = lcm a b"
980   by (rule, auto intro: lcmI simp: lcm_least lcm_zero)
982 lemma dvd_lcm_I1 [simp]:
983   "k dvd m \<Longrightarrow> k dvd lcm m n"
984   by (metis lcm_dvd1 dvd_trans)
986 lemma dvd_lcm_I2 [simp]:
987   "k dvd n \<Longrightarrow> k dvd lcm m n"
988   by (metis lcm_dvd2 dvd_trans)
990 lemma lcm_1_left [simp]:
991   "lcm 1 a = a div normalization_factor a"
992   by (cases "a = 0") (simp, rule sym, rule lcmI, simp_all)
994 lemma lcm_1_right [simp]:
995   "lcm a 1 = a div normalization_factor a"
996   using lcm_1_left [of a] by (simp add: ac_simps)
998 lemma lcm_coprime:
999   "gcd a b = 1 \<Longrightarrow> lcm a b = a * b div normalization_factor (a*b)"
1000   by (subst lcm_gcd) simp
1002 lemma lcm_proj1_if_dvd:
1003   "b dvd a \<Longrightarrow> lcm a b = a div normalization_factor a"
1004   by (cases "a = 0") (simp, rule sym, rule lcmI, simp_all)
1006 lemma lcm_proj2_if_dvd:
1007   "a dvd b \<Longrightarrow> lcm a b = b div normalization_factor b"
1008   using lcm_proj1_if_dvd [of a b] by (simp add: ac_simps)
1010 lemma lcm_proj1_iff:
1011   "lcm m n = m div normalization_factor m \<longleftrightarrow> n dvd m"
1012 proof
1013   assume A: "lcm m n = m div normalization_factor m"
1014   show "n dvd m"
1015   proof (cases "m = 0")
1016     assume [simp]: "m \<noteq> 0"
1017     from A have B: "m = lcm m n * normalization_factor m"
1018       by (simp add: unit_eq_div2)
1019     show ?thesis by (subst B, simp)
1020   qed simp
1021 next
1022   assume "n dvd m"
1023   then show "lcm m n = m div normalization_factor m" by (rule lcm_proj1_if_dvd)
1024 qed
1026 lemma lcm_proj2_iff:
1027   "lcm m n = n div normalization_factor n \<longleftrightarrow> m dvd n"
1028   using lcm_proj1_iff [of n m] by (simp add: ac_simps)
1030 lemma euclidean_size_lcm_le1:
1031   assumes "a \<noteq> 0" and "b \<noteq> 0"
1032   shows "euclidean_size a \<le> euclidean_size (lcm a b)"
1033 proof -
1034   have "a dvd lcm a b" by (rule lcm_dvd1)
1035   then obtain c where A: "lcm a b = a * c" unfolding dvd_def by blast
1036   with \<open>a \<noteq> 0\<close> and \<open>b \<noteq> 0\<close> have "c \<noteq> 0" by (auto simp: lcm_zero)
1037   then show ?thesis by (subst A, intro size_mult_mono)
1038 qed
1040 lemma euclidean_size_lcm_le2:
1041   "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> euclidean_size b \<le> euclidean_size (lcm a b)"
1042   using euclidean_size_lcm_le1 [of b a] by (simp add: ac_simps)
1044 lemma euclidean_size_lcm_less1:
1045   assumes "b \<noteq> 0" and "\<not>b dvd a"
1046   shows "euclidean_size a < euclidean_size (lcm a b)"
1047 proof (rule ccontr)
1048   from assms have "a \<noteq> 0" by auto
1049   assume "\<not>euclidean_size a < euclidean_size (lcm a b)"
1050   with \<open>a \<noteq> 0\<close> and \<open>b \<noteq> 0\<close> have "euclidean_size (lcm a b) = euclidean_size a"
1051     by (intro le_antisym, simp, intro euclidean_size_lcm_le1)
1052   with assms have "lcm a b dvd a"
1053     by (rule_tac dvd_euclidean_size_eq_imp_dvd) (auto simp: lcm_zero)
1054   hence "b dvd a" by (rule dvd_lcm_D2)
1055   with \<open>\<not>b dvd a\<close> show False by contradiction
1056 qed
1058 lemma euclidean_size_lcm_less2:
1059   assumes "a \<noteq> 0" and "\<not>a dvd b"
1060   shows "euclidean_size b < euclidean_size (lcm a b)"
1061   using assms euclidean_size_lcm_less1 [of a b] by (simp add: ac_simps)
1063 lemma lcm_mult_unit1:
1064   "is_unit a \<Longrightarrow> lcm (b * a) c = lcm b c"
1065   apply (rule lcmI)
1066   apply (rule dvd_trans[of _ "b * a"], simp, rule lcm_dvd1)
1067   apply (rule lcm_dvd2)
1068   apply (rule lcm_least, simp add: unit_simps, assumption)
1069   apply (subst normalization_factor_lcm, simp add: lcm_zero)
1070   done
1072 lemma lcm_mult_unit2:
1073   "is_unit a \<Longrightarrow> lcm b (c * a) = lcm b c"
1074   using lcm_mult_unit1 [of a c b] by (simp add: ac_simps)
1076 lemma lcm_div_unit1:
1077   "is_unit a \<Longrightarrow> lcm (b div a) c = lcm b c"
1078   by (erule is_unitE [of _ b]) (simp add: lcm_mult_unit1)
1080 lemma lcm_div_unit2:
1081   "is_unit a \<Longrightarrow> lcm b (c div a) = lcm b c"
1082   by (erule is_unitE [of _ c]) (simp add: lcm_mult_unit2)
1084 lemma lcm_left_idem:
1085   "lcm a (lcm a b) = lcm a b"
1086   apply (rule lcmI)
1087   apply simp
1088   apply (subst lcm.assoc [symmetric], rule lcm_dvd2)
1089   apply (rule lcm_least, assumption)
1090   apply (erule (1) lcm_least)
1091   apply (auto simp: lcm_zero)
1092   done
1094 lemma lcm_right_idem:
1095   "lcm (lcm a b) b = lcm a b"
1096   apply (rule lcmI)
1097   apply (subst lcm.assoc, rule lcm_dvd1)
1098   apply (rule lcm_dvd2)
1099   apply (rule lcm_least, erule (1) lcm_least, assumption)
1100   apply (auto simp: lcm_zero)
1101   done
1103 lemma comp_fun_idem_lcm: "comp_fun_idem lcm"
1104 proof
1105   fix a b show "lcm a \<circ> lcm b = lcm b \<circ> lcm a"
1106     by (simp add: fun_eq_iff ac_simps)
1107 next
1108   fix a show "lcm a \<circ> lcm a = lcm a" unfolding o_def
1109     by (intro ext, simp add: lcm_left_idem)
1110 qed
1112 lemma dvd_Lcm [simp]: "a \<in> A \<Longrightarrow> a dvd Lcm A"
1113   and Lcm_dvd [simp]: "(\<forall>a\<in>A. a dvd l') \<Longrightarrow> Lcm A dvd l'"
1114   and normalization_factor_Lcm [simp]:
1115           "normalization_factor (Lcm A) = (if Lcm A = 0 then 0 else 1)"
1116 proof -
1117   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>
1118     normalization_factor (Lcm A) = (if Lcm A = 0 then 0 else 1)" (is ?thesis)
1119   proof (cases "\<exists>l. l \<noteq>  0 \<and> (\<forall>a\<in>A. a dvd l)")
1120     case False
1121     hence "Lcm A = 0" by (auto simp: Lcm_Lcm_eucl Lcm_eucl_def)
1122     with False show ?thesis by auto
1123   next
1124     case True
1125     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
1126     def n \<equiv> "LEAST n. \<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"
1127     def l \<equiv> "SOME l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"
1128     have "\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"
1129       apply (subst n_def)
1130       apply (rule LeastI[of _ "euclidean_size l\<^sub>0"])
1131       apply (rule exI[of _ l\<^sub>0])
1132       apply (simp add: l\<^sub>0_props)
1133       done
1134     from someI_ex[OF this] have "l \<noteq> 0" and "\<forall>a\<in>A. a dvd l" and "euclidean_size l = n"
1135       unfolding l_def by simp_all
1136     {
1137       fix l' assume "\<forall>a\<in>A. a dvd l'"
1138       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)
1139       moreover from \<open>l \<noteq> 0\<close> have "gcd l l' \<noteq> 0" by simp
1140       ultimately have "\<exists>b. b \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd b) \<and> euclidean_size b = euclidean_size (gcd l l')"
1141         by (intro exI[of _ "gcd l l'"], auto)
1142       hence "euclidean_size (gcd l l') \<ge> n" by (subst n_def) (rule Least_le)
1143       moreover have "euclidean_size (gcd l l') \<le> n"
1144       proof -
1145         have "gcd l l' dvd l" by simp
1146         then obtain a where "l = gcd l l' * a" unfolding dvd_def by blast
1147         with \<open>l \<noteq> 0\<close> have "a \<noteq> 0" by auto
1148         hence "euclidean_size (gcd l l') \<le> euclidean_size (gcd l l' * a)"
1149           by (rule size_mult_mono)
1150         also have "gcd l l' * a = l" using \<open>l = gcd l l' * a\<close> ..
1151         also note \<open>euclidean_size l = n\<close>
1152         finally show "euclidean_size (gcd l l') \<le> n" .
1153       qed
1154       ultimately have "euclidean_size l = euclidean_size (gcd l l')"
1155         by (intro le_antisym, simp_all add: \<open>euclidean_size l = n\<close>)
1156       with \<open>l \<noteq> 0\<close> have "l dvd gcd l l'" by (blast intro: dvd_euclidean_size_eq_imp_dvd)
1157       hence "l dvd l'" by (blast dest: dvd_gcd_D2)
1158     }
1160     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>
1161       have "(\<forall>a\<in>A. a dvd l div normalization_factor l) \<and>
1162         (\<forall>l'. (\<forall>a\<in>A. a dvd l') \<longrightarrow> l div normalization_factor l dvd l') \<and>
1163         normalization_factor (l div normalization_factor l) =
1164         (if l div normalization_factor l = 0 then 0 else 1)"
1165       by (auto simp: unit_simps)
1166     also from True have "l div normalization_factor l = Lcm A"
1167       by (simp add: Lcm_Lcm_eucl Lcm_eucl_def Let_def n_def l_def)
1168     finally show ?thesis .
1169   qed
1170   note A = this
1172   {fix a assume "a \<in> A" then show "a dvd Lcm A" using A by blast}
1173   {fix l' assume "\<forall>a\<in>A. a dvd l'" then show "Lcm A dvd l'" using A by blast}
1174   from A show "normalization_factor (Lcm A) = (if Lcm A = 0 then 0 else 1)" by blast
1175 qed
1177 lemma LcmI:
1178   "(\<And>a. a\<in>A \<Longrightarrow> a dvd l) \<Longrightarrow> (\<And>l'. (\<forall>a\<in>A. a dvd l') \<Longrightarrow> l dvd l') \<Longrightarrow>
1179       normalization_factor l = (if l = 0 then 0 else 1) \<Longrightarrow> l = Lcm A"
1180   by (intro normed_associated_imp_eq)
1181     (auto intro: Lcm_dvd dvd_Lcm simp: associated_def)
1183 lemma Lcm_subset:
1184   "A \<subseteq> B \<Longrightarrow> Lcm A dvd Lcm B"
1185   by (blast intro: Lcm_dvd dvd_Lcm)
1187 lemma Lcm_Un:
1188   "Lcm (A \<union> B) = lcm (Lcm A) (Lcm B)"
1189   apply (rule lcmI)
1190   apply (blast intro: Lcm_subset)
1191   apply (blast intro: Lcm_subset)
1192   apply (intro Lcm_dvd ballI, elim UnE)
1193   apply (rule dvd_trans, erule dvd_Lcm, assumption)
1194   apply (rule dvd_trans, erule dvd_Lcm, assumption)
1195   apply simp
1196   done
1198 lemma Lcm_1_iff:
1199   "Lcm A = 1 \<longleftrightarrow> (\<forall>a\<in>A. is_unit a)"
1200 proof
1201   assume "Lcm A = 1"
1202   then show "\<forall>a\<in>A. is_unit a" by auto
1203 qed (rule LcmI [symmetric], auto)
1205 lemma Lcm_no_units:
1206   "Lcm A = Lcm (A - {a. is_unit a})"
1207 proof -
1208   have "(A - {a. is_unit a}) \<union> {a\<in>A. is_unit a} = A" by blast
1209   hence "Lcm A = lcm (Lcm (A - {a. is_unit a})) (Lcm {a\<in>A. is_unit a})"
1210     by (simp add: Lcm_Un[symmetric])
1211   also have "Lcm {a\<in>A. is_unit a} = 1" by (simp add: Lcm_1_iff)
1212   finally show ?thesis by simp
1213 qed
1215 lemma Lcm_empty [simp]:
1216   "Lcm {} = 1"
1217   by (simp add: Lcm_1_iff)
1219 lemma Lcm_eq_0 [simp]:
1220   "0 \<in> A \<Longrightarrow> Lcm A = 0"
1221   by (drule dvd_Lcm) simp
1223 lemma Lcm0_iff':
1224   "Lcm A = 0 \<longleftrightarrow> \<not>(\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l))"
1225 proof
1226   assume "Lcm A = 0"
1227   show "\<not>(\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l))"
1228   proof
1229     assume ex: "\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l)"
1230     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
1231     def n \<equiv> "LEAST n. \<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"
1232     def l \<equiv> "SOME l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"
1233     have "\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"
1234       apply (subst n_def)
1235       apply (rule LeastI[of _ "euclidean_size l\<^sub>0"])
1236       apply (rule exI[of _ l\<^sub>0])
1237       apply (simp add: l\<^sub>0_props)
1238       done
1239     from someI_ex[OF this] have "l \<noteq> 0" unfolding l_def by simp_all
1240     hence "l div normalization_factor l \<noteq> 0" by simp
1241     also from ex have "l div normalization_factor l = Lcm A"
1242        by (simp only: Lcm_Lcm_eucl Lcm_eucl_def n_def l_def if_True Let_def)
1243     finally show False using \<open>Lcm A = 0\<close> by contradiction
1244   qed
1245 qed (simp only: Lcm_Lcm_eucl Lcm_eucl_def if_False)
1247 lemma Lcm0_iff [simp]:
1248   "finite A \<Longrightarrow> Lcm A = 0 \<longleftrightarrow> 0 \<in> A"
1249 proof -
1250   assume "finite A"
1251   have "0 \<in> A \<Longrightarrow> Lcm A = 0"  by (intro dvd_0_left dvd_Lcm)
1252   moreover {
1253     assume "0 \<notin> A"
1254     hence "\<Prod>A \<noteq> 0"
1255       apply (induct rule: finite_induct[OF \<open>finite A\<close>])
1256       apply simp
1257       apply (subst setprod.insert, assumption, assumption)
1258       apply (rule no_zero_divisors)
1259       apply blast+
1260       done
1261     moreover from \<open>finite A\<close> have "\<forall>a\<in>A. a dvd \<Prod>A" by blast
1262     ultimately have "\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l)" by blast
1263     with Lcm0_iff' have "Lcm A \<noteq> 0" by simp
1264   }
1265   ultimately show "Lcm A = 0 \<longleftrightarrow> 0 \<in> A" by blast
1266 qed
1268 lemma Lcm_no_multiple:
1269   "(\<forall>m. m \<noteq> 0 \<longrightarrow> (\<exists>a\<in>A. \<not>a dvd m)) \<Longrightarrow> Lcm A = 0"
1270 proof -
1271   assume "\<forall>m. m \<noteq> 0 \<longrightarrow> (\<exists>a\<in>A. \<not>a dvd m)"
1272   hence "\<not>(\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l))" by blast
1273   then show "Lcm A = 0" by (simp only: Lcm_Lcm_eucl Lcm_eucl_def if_False)
1274 qed
1276 lemma Lcm_insert [simp]:
1277   "Lcm (insert a A) = lcm a (Lcm A)"
1278 proof (rule lcmI)
1279   fix l assume "a dvd l" and "Lcm A dvd l"
1280   hence "\<forall>a\<in>A. a dvd l" by (blast intro: dvd_trans dvd_Lcm)
1281   with \<open>a dvd l\<close> show "Lcm (insert a A) dvd l" by (force intro: Lcm_dvd)
1282 qed (auto intro: Lcm_dvd dvd_Lcm)
1284 lemma Lcm_finite:
1285   assumes "finite A"
1286   shows "Lcm A = Finite_Set.fold lcm 1 A"
1287   by (induct rule: finite.induct[OF \<open>finite A\<close>])
1288     (simp_all add: comp_fun_idem.fold_insert_idem[OF comp_fun_idem_lcm])
1290 lemma Lcm_set [code_unfold]:
1291   "Lcm (set xs) = fold lcm xs 1"
1292   using comp_fun_idem.fold_set_fold[OF comp_fun_idem_lcm] Lcm_finite by (simp add: ac_simps)
1294 lemma Lcm_singleton [simp]:
1295   "Lcm {a} = a div normalization_factor a"
1296   by simp
1298 lemma Lcm_2 [simp]:
1299   "Lcm {a,b} = lcm a b"
1300   by (simp only: Lcm_insert Lcm_empty lcm_1_right)
1301     (cases "b = 0", simp, rule lcm_div_unit2, simp)
1303 lemma Lcm_coprime:
1304   assumes "finite A" and "A \<noteq> {}"
1305   assumes "\<And>a b. a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<noteq> b \<Longrightarrow> gcd a b = 1"
1306   shows "Lcm A = \<Prod>A div normalization_factor (\<Prod>A)"
1307 using assms proof (induct rule: finite_ne_induct)
1308   case (insert a A)
1309   have "Lcm (insert a A) = lcm a (Lcm A)" by simp
1310   also from insert have "Lcm A = \<Prod>A div normalization_factor (\<Prod>A)" by blast
1311   also have "lcm a \<dots> = lcm a (\<Prod>A)" by (cases "\<Prod>A = 0") (simp_all add: lcm_div_unit2)
1312   also from insert have "gcd a (\<Prod>A) = 1" by (subst gcd.commute, intro setprod_coprime) auto
1313   with insert have "lcm a (\<Prod>A) = \<Prod>(insert a A) div normalization_factor (\<Prod>(insert a A))"
1314     by (simp add: lcm_coprime)
1315   finally show ?case .
1316 qed simp
1318 lemma Lcm_coprime':
1319   "card A \<noteq> 0 \<Longrightarrow> (\<And>a b. a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<noteq> b \<Longrightarrow> gcd a b = 1)
1320     \<Longrightarrow> Lcm A = \<Prod>A div normalization_factor (\<Prod>A)"
1321   by (rule Lcm_coprime) (simp_all add: card_eq_0_iff)
1323 lemma Gcd_Lcm:
1324   "Gcd A = Lcm {d. \<forall>a\<in>A. d dvd a}"
1325   by (simp add: Gcd_Gcd_eucl Lcm_Lcm_eucl Gcd_eucl_def)
1327 lemma Gcd_dvd [simp]: "a \<in> A \<Longrightarrow> Gcd A dvd a"
1328   and dvd_Gcd [simp]: "(\<forall>a\<in>A. g' dvd a) \<Longrightarrow> g' dvd Gcd A"
1329   and normalization_factor_Gcd [simp]:
1330     "normalization_factor (Gcd A) = (if Gcd A = 0 then 0 else 1)"
1331 proof -
1332   fix a assume "a \<in> A"
1333   hence "Lcm {d. \<forall>a\<in>A. d dvd a} dvd a" by (intro Lcm_dvd) blast
1334   then show "Gcd A dvd a" by (simp add: Gcd_Lcm)
1335 next
1336   fix g' assume "\<forall>a\<in>A. g' dvd a"
1337   hence "g' dvd Lcm {d. \<forall>a\<in>A. d dvd a}" by (intro dvd_Lcm) blast
1338   then show "g' dvd Gcd A" by (simp add: Gcd_Lcm)
1339 next
1340   show "normalization_factor (Gcd A) = (if Gcd A = 0 then 0 else 1)"
1341     by (simp add: Gcd_Lcm)
1342 qed
1344 lemma GcdI:
1345   "(\<And>a. a\<in>A \<Longrightarrow> l dvd a) \<Longrightarrow> (\<And>l'. (\<forall>a\<in>A. l' dvd a) \<Longrightarrow> l' dvd l) \<Longrightarrow>
1346     normalization_factor l = (if l = 0 then 0 else 1) \<Longrightarrow> l = Gcd A"
1347   by (intro normed_associated_imp_eq)
1348     (auto intro: Gcd_dvd dvd_Gcd simp: associated_def)
1350 lemma Lcm_Gcd:
1351   "Lcm A = Gcd {m. \<forall>a\<in>A. a dvd m}"
1352   by (rule LcmI[symmetric]) (auto intro: dvd_Gcd Gcd_dvd)
1354 lemma Gcd_0_iff:
1355   "Gcd A = 0 \<longleftrightarrow> A \<subseteq> {0}"
1356   apply (rule iffI)
1357   apply (rule subsetI, drule Gcd_dvd, simp)
1358   apply (auto intro: GcdI[symmetric])
1359   done
1361 lemma Gcd_empty [simp]:
1362   "Gcd {} = 0"
1363   by (simp add: Gcd_0_iff)
1365 lemma Gcd_1:
1366   "1 \<in> A \<Longrightarrow> Gcd A = 1"
1367   by (intro GcdI[symmetric]) (auto intro: Gcd_dvd dvd_Gcd)
1369 lemma Gcd_insert [simp]:
1370   "Gcd (insert a A) = gcd a (Gcd A)"
1371 proof (rule gcdI)
1372   fix l assume "l dvd a" and "l dvd Gcd A"
1373   hence "\<forall>a\<in>A. l dvd a" by (blast intro: dvd_trans Gcd_dvd)
1374   with \<open>l dvd a\<close> show "l dvd Gcd (insert a A)" by (force intro: Gcd_dvd)
1375 qed auto
1377 lemma Gcd_finite:
1378   assumes "finite A"
1379   shows "Gcd A = Finite_Set.fold gcd 0 A"
1380   by (induct rule: finite.induct[OF \<open>finite A\<close>])
1381     (simp_all add: comp_fun_idem.fold_insert_idem[OF comp_fun_idem_gcd])
1383 lemma Gcd_set [code_unfold]:
1384   "Gcd (set xs) = fold gcd xs 0"
1385   using comp_fun_idem.fold_set_fold[OF comp_fun_idem_gcd] Gcd_finite by (simp add: ac_simps)
1387 lemma Gcd_singleton [simp]: "Gcd {a} = a div normalization_factor a"
1388   by (simp add: gcd_0)
1390 lemma Gcd_2 [simp]: "Gcd {a,b} = gcd a b"
1391   by (simp only: Gcd_insert Gcd_empty gcd_0) (cases "b = 0", simp, rule gcd_div_unit2, simp)
1393 subclass semiring_gcd
1394   by unfold_locales (simp_all add: gcd_greatest_iff)
1396 end
1398 text \<open>
1399   A Euclidean ring is a Euclidean semiring with additive inverses. It provides a
1400   few more lemmas; in particular, Bezout's lemma holds for any Euclidean ring.
1401 \<close>
1403 class euclidean_ring = euclidean_semiring + idom
1405 class euclidean_ring_gcd = euclidean_semiring_gcd + idom
1406 begin
1408 subclass euclidean_ring ..
1410 subclass ring_gcd ..
1412 lemma gcd_neg1 [simp]:
1413   "gcd (-a) b = gcd a b"
1414   by (rule sym, rule gcdI, simp_all add: gcd_greatest)
1416 lemma gcd_neg2 [simp]:
1417   "gcd a (-b) = gcd a b"
1418   by (rule sym, rule gcdI, simp_all add: gcd_greatest)
1420 lemma gcd_neg_numeral_1 [simp]:
1421   "gcd (- numeral n) a = gcd (numeral n) a"
1422   by (fact gcd_neg1)
1424 lemma gcd_neg_numeral_2 [simp]:
1425   "gcd a (- numeral n) = gcd a (numeral n)"
1426   by (fact gcd_neg2)
1428 lemma gcd_diff1: "gcd (m - n) n = gcd m n"
1429   by (subst diff_conv_add_uminus, subst gcd_neg2[symmetric],  subst gcd_add1, simp)
1431 lemma gcd_diff2: "gcd (n - m) n = gcd m n"
1432   by (subst gcd_neg1[symmetric], simp only: minus_diff_eq gcd_diff1)
1434 lemma coprime_minus_one [simp]: "gcd (n - 1) n = 1"
1435 proof -
1436   have "gcd (n - 1) n = gcd n (n - 1)" by (fact gcd.commute)
1437   also have "\<dots> = gcd ((n - 1) + 1) (n - 1)" by simp
1438   also have "\<dots> = 1" by (rule coprime_plus_one)
1439   finally show ?thesis .
1440 qed
1442 lemma lcm_neg1 [simp]: "lcm (-a) b = lcm a b"
1443   by (rule sym, rule lcmI, simp_all add: lcm_least lcm_zero)
1445 lemma lcm_neg2 [simp]: "lcm a (-b) = lcm a b"
1446   by (rule sym, rule lcmI, simp_all add: lcm_least lcm_zero)
1448 lemma lcm_neg_numeral_1 [simp]: "lcm (- numeral n) a = lcm (numeral n) a"
1449   by (fact lcm_neg1)
1451 lemma lcm_neg_numeral_2 [simp]: "lcm a (- numeral n) = lcm a (numeral n)"
1452   by (fact lcm_neg2)
1454 function euclid_ext :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a" where
1455   "euclid_ext a b =
1456      (if b = 0 then
1457         let c = 1 div normalization_factor a in (c, 0, a * c)
1458       else if b dvd a then
1459         let c = 1 div normalization_factor b in (0, c, b * c)
1460       else
1461         case euclid_ext b (a mod b) of
1462             (s, t, c) \<Rightarrow> (t, s - t * (a div b), c))"
1463   by (pat_completeness, simp)
1464   termination by (relation "measure (euclidean_size \<circ> snd)")
1465     (simp_all add: mod_size_less)
1467 declare euclid_ext.simps [simp del]
1469 lemma euclid_ext_0:
1470   "euclid_ext a 0 = (1 div normalization_factor a, 0, a div normalization_factor a)"
1471   by (subst euclid_ext.simps) (simp add: Let_def)
1473 lemma euclid_ext_non_0:
1474   "b \<noteq> 0 \<Longrightarrow> euclid_ext a b = (case euclid_ext b (a mod b) of
1475     (s, t, c) \<Rightarrow> (t, s - t * (a div b), c))"
1476   apply (subst euclid_ext.simps)
1477   apply (auto simp add: split: if_splits)
1478   apply (subst euclid_ext.simps)
1479   apply (auto simp add: split: if_splits)
1480   done
1482 definition euclid_ext' :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a"
1483 where
1484   "euclid_ext' a b = (case euclid_ext a b of (s, t, _) \<Rightarrow> (s, t))"
1486 lemma euclid_ext_gcd [simp]:
1487   "(case euclid_ext a b of (_, _ , t) \<Rightarrow> t) = gcd a b"
1488   by (induct a b rule: gcd_eucl_induct)
1489     (simp_all add: euclid_ext_0 gcd_0 euclid_ext_non_0 ac_simps split: prod.split prod.split_asm)
1491 lemma euclid_ext_gcd' [simp]:
1492   "euclid_ext a b = (r, s, t) \<Longrightarrow> t = gcd a b"
1493   by (insert euclid_ext_gcd[of a b], drule (1) subst, simp)
1495 lemma euclid_ext_correct:
1496   "case euclid_ext a b of (s, t, c) \<Rightarrow> s * a + t * b = c"
1497 proof (induct a b rule: gcd_eucl_induct)
1498   case (zero a) then show ?case
1499     by (simp add: euclid_ext_0 ac_simps)
1500 next
1501   case (mod a b)
1502   obtain s t c where stc: "euclid_ext b (a mod b) = (s,t,c)"
1503     by (cases "euclid_ext b (a mod b)") blast
1504   with mod have "c = s * b + t * (a mod b)" by simp
1505   also have "... = t * ((a div b) * b + a mod b) + (s - t * (a div b)) * b"
1506     by (simp add: algebra_simps)
1507   also have "(a div b) * b + a mod b = a" using mod_div_equality .
1508   finally show ?case
1509     by (subst euclid_ext.simps) (simp add: stc mod ac_simps)
1510 qed
1512 lemma euclid_ext'_correct:
1513   "fst (euclid_ext' a b) * a + snd (euclid_ext' a b) * b = gcd a b"
1514 proof-
1515   obtain s t c where "euclid_ext a b = (s,t,c)"
1516     by (cases "euclid_ext a b", blast)
1517   with euclid_ext_correct[of a b] euclid_ext_gcd[of a b]
1518     show ?thesis unfolding euclid_ext'_def by simp
1519 qed
1521 lemma bezout: "\<exists>s t. s * a + t * b = gcd a b"
1522   using euclid_ext'_correct by blast
1524 lemma euclid_ext'_0 [simp]: "euclid_ext' a 0 = (1 div normalization_factor a, 0)"
1525   by (simp add: bezw_def euclid_ext'_def euclid_ext_0)
1527 lemma euclid_ext'_non_0: "b \<noteq> 0 \<Longrightarrow> euclid_ext' a b = (snd (euclid_ext' b (a mod b)),
1528   fst (euclid_ext' b (a mod b)) - snd (euclid_ext' b (a mod b)) * (a div b))"
1529   by (cases "euclid_ext b (a mod b)")
1530     (simp add: euclid_ext'_def euclid_ext_non_0)
1532 end
1534 instantiation nat :: euclidean_semiring
1535 begin
1537 definition [simp]:
1538   "euclidean_size_nat = (id :: nat \<Rightarrow> nat)"
1540 definition [simp]:
1541   "normalization_factor_nat (n::nat) = (if n = 0 then 0 else 1 :: nat)"
1543 instance proof
1544 qed simp_all
1546 end
1548 instantiation int :: euclidean_ring
1549 begin
1551 definition [simp]:
1552   "euclidean_size_int = (nat \<circ> abs :: int \<Rightarrow> nat)"
1554 definition [simp]:
1555   "normalization_factor_int = (sgn :: int \<Rightarrow> int)"
1557 instance proof
1558   case goal2 then show ?case by (auto simp add: abs_mult nat_mult_distrib)
1559 next
1560   case goal3 then show ?case by (simp add: zsgn_def)
1561 next
1562   case goal5 then show ?case by (auto simp: zsgn_def)
1563 next
1564   case goal6 then show ?case by (auto split: abs_split simp: zsgn_def)
1565 qed (auto simp: sgn_times split: abs_split)
1567 end
1569 instantiation poly :: (field) euclidean_semiring
1570 begin
1572 definition euclidean_size_poly :: "'a poly \<Rightarrow> nat"
1573   where "euclidean_size = (degree :: 'a poly \<Rightarrow> nat)"
1575 definition normalization_factor_poly :: "'a poly \<Rightarrow> 'a poly"
1576   where "normalization_factor p = monom (coeff p (degree p)) 0"
1578 instance
1579 proof (default, unfold euclidean_size_poly_def normalization_factor_poly_def)
1580   fix p q :: "'a poly"
1581   assume "q \<noteq> 0" and "\<not> q dvd p"
1582   then show "degree (p mod q) < degree q"
1583     using degree_mod_less [of q p] by (simp add: mod_eq_0_iff_dvd)
1584 next
1585   fix p q :: "'a poly"
1586   assume "q \<noteq> 0"
1587   from \<open>q \<noteq> 0\<close> show "degree p \<le> degree (p * q)"
1588     by (rule degree_mult_right_le)
1589   from \<open>q \<noteq> 0\<close> show "is_unit (monom (coeff q (degree q)) 0)"
1590     by (auto intro: is_unit_monom_0)
1591 next
1592   fix p :: "'a poly"
1593   show "monom (coeff p (degree p)) 0 = p" if "is_unit p"
1594     using that by (fact is_unit_monom_trival)
1595 next
1596   fix p q :: "'a poly"
1597   show "monom (coeff (p * q) (degree (p * q))) 0 =
1598     monom (coeff p (degree p)) 0 * monom (coeff q (degree q)) 0"
1599     by (simp add: monom_0 coeff_degree_mult)
1600 next
1601   show "monom (coeff 0 (degree 0)) 0 = 0"
1602     by simp
1603 qed
1605 end
1607 end