src/HOL/Number_Theory/Euclidean_Algorithm.thy
 author haftmann Thu Jun 25 15:01:43 2015 +0200 (2015-06-25) changeset 60572 718b1ba06429 parent 60571 c9fdf2080447 child 60582 d694f217ee41 permissions -rw-r--r--
streamlined definitions and primitive lemma of euclidean algorithm, including code generation
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 definition Lcm_eucl :: "'a set \<Rightarrow> 'a" -- \<open>
213   Somewhat complicated definition of Lcm that has the advantage of working
214   for infinite sets as well\<close>
215 where
216   "Lcm_eucl A = (if \<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) then
217      let l = SOME l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l =
218        (LEAST n. \<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n)
219        in l div normalization_factor l
220       else 0)"
222 definition Gcd_eucl :: "'a set \<Rightarrow> 'a"
223 where
224   "Gcd_eucl A = Lcm_eucl {d. \<forall>a\<in>A. d dvd a}"
226 lemma gcd_eucl_0:
227   "gcd_eucl a 0 = a div normalization_factor a"
228   by (simp add: gcd_eucl.simps [of a 0])
230 lemma gcd_eucl_0_left:
231   "gcd_eucl 0 a = a div normalization_factor a"
232   by (simp add: gcd_eucl.simps [of 0 a])
234 lemma gcd_eucl_non_0:
235   "b \<noteq> 0 \<Longrightarrow> gcd_eucl a b = gcd_eucl b (a mod b)"
236   by (cases "b dvd a")
237     (simp_all add: gcd_eucl.simps [of a b] gcd_eucl.simps [of b 0])
239 lemma gcd_eucl_code [code]:
240   "gcd_eucl a b = (if b = 0 then a div normalization_factor a else gcd_eucl b (a mod b))"
241   by (auto simp add: gcd_eucl_non_0 gcd_eucl_0 gcd_eucl_0_left)
243 end
245 class euclidean_semiring_gcd = euclidean_semiring + gcd + Gcd +
246   assumes gcd_gcd_eucl: "gcd = gcd_eucl" and lcm_lcm_eucl: "lcm = lcm_eucl"
247   assumes Gcd_Gcd_eucl: "Gcd = Gcd_eucl" and Lcm_Lcm_eucl: "Lcm = Lcm_eucl"
248 begin
250 lemma gcd_0_left:
251   "gcd 0 a = a div normalization_factor a"
252   unfolding gcd_gcd_eucl by (fact gcd_eucl_0_left)
254 lemma gcd_0:
255   "gcd a 0 = a div normalization_factor a"
256   unfolding gcd_gcd_eucl by (fact gcd_eucl_0)
258 lemma gcd_non_0:
259   "b \<noteq> 0 \<Longrightarrow> gcd a b = gcd b (a mod b)"
260   unfolding gcd_gcd_eucl by (fact gcd_eucl_non_0)
262 lemma gcd_dvd1 [iff]: "gcd a b dvd a"
263   and gcd_dvd2 [iff]: "gcd a b dvd b"
264   by (induct a b rule: gcd_eucl_induct)
265     (simp_all add: gcd_0 gcd_non_0 dvd_mod_iff)
267 lemma dvd_gcd_D1: "k dvd gcd m n \<Longrightarrow> k dvd m"
268   by (rule dvd_trans, assumption, rule gcd_dvd1)
270 lemma dvd_gcd_D2: "k dvd gcd m n \<Longrightarrow> k dvd n"
271   by (rule dvd_trans, assumption, rule gcd_dvd2)
273 lemma gcd_greatest:
274   fixes k a b :: 'a
275   shows "k dvd a \<Longrightarrow> k dvd b \<Longrightarrow> k dvd gcd a b"
276 proof (induct a b rule: gcd_eucl_induct)
277   case (zero a) from zero(1) show ?case by (rule dvd_trans) (simp add: gcd_0)
278 next
279   case (mod a b)
280   then show ?case
281     by (simp add: gcd_non_0 dvd_mod_iff)
282 qed
284 lemma dvd_gcd_iff:
285   "k dvd gcd a b \<longleftrightarrow> k dvd a \<and> k dvd b"
286   by (blast intro!: gcd_greatest intro: dvd_trans)
288 lemmas gcd_greatest_iff = dvd_gcd_iff
290 lemma gcd_zero [simp]:
291   "gcd a b = 0 \<longleftrightarrow> a = 0 \<and> b = 0"
292   by (metis dvd_0_left dvd_refl gcd_dvd1 gcd_dvd2 gcd_greatest)+
294 lemma normalization_factor_gcd [simp]:
295   "normalization_factor (gcd a b) = (if a = 0 \<and> b = 0 then 0 else 1)" (is "?f a b = ?g a b")
296   by (induct a b rule: gcd_eucl_induct)
297     (auto simp add: gcd_0 gcd_non_0)
299 lemma gcdI:
300   "k dvd a \<Longrightarrow> k dvd b \<Longrightarrow> (\<And>l. l dvd a \<Longrightarrow> l dvd b \<Longrightarrow> l dvd k)
301     \<Longrightarrow> normalization_factor k = (if k = 0 then 0 else 1) \<Longrightarrow> k = gcd a b"
302   by (intro normed_associated_imp_eq) (auto simp: associated_def intro: gcd_greatest)
304 sublocale gcd!: abel_semigroup gcd
305 proof
306   fix a b c
307   show "gcd (gcd a b) c = gcd a (gcd b c)"
308   proof (rule gcdI)
309     have "gcd (gcd a b) c dvd gcd a b" "gcd a b dvd a" by simp_all
310     then show "gcd (gcd a b) c dvd a" by (rule dvd_trans)
311     have "gcd (gcd a b) c dvd gcd a b" "gcd a b dvd b" by simp_all
312     hence "gcd (gcd a b) c dvd b" by (rule dvd_trans)
313     moreover have "gcd (gcd a b) c dvd c" by simp
314     ultimately show "gcd (gcd a b) c dvd gcd b c"
315       by (rule gcd_greatest)
316     show "normalization_factor (gcd (gcd a b) c) =  (if gcd (gcd a b) c = 0 then 0 else 1)"
317       by auto
318     fix l assume "l dvd a" and "l dvd gcd b c"
319     with dvd_trans[OF _ gcd_dvd1] and dvd_trans[OF _ gcd_dvd2]
320       have "l dvd b" and "l dvd c" by blast+
321     with \<open>l dvd a\<close> show "l dvd gcd (gcd a b) c"
322       by (intro gcd_greatest)
323   qed
324 next
325   fix a b
326   show "gcd a b = gcd b a"
327     by (rule gcdI) (simp_all add: gcd_greatest)
328 qed
330 lemma gcd_unique: "d dvd a \<and> d dvd b \<and>
331     normalization_factor d = (if d = 0 then 0 else 1) \<and>
332     (\<forall>e. e dvd a \<and> e dvd b \<longrightarrow> e dvd d) \<longleftrightarrow> d = gcd a b"
333   by (rule, auto intro: gcdI simp: gcd_greatest)
335 lemma gcd_dvd_prod: "gcd a b dvd k * b"
336   using mult_dvd_mono [of 1] by auto
338 lemma gcd_1_left [simp]: "gcd 1 a = 1"
339   by (rule sym, rule gcdI, simp_all)
341 lemma gcd_1 [simp]: "gcd a 1 = 1"
342   by (rule sym, rule gcdI, simp_all)
344 lemma gcd_proj2_if_dvd:
345   "b dvd a \<Longrightarrow> gcd a b = b div normalization_factor b"
346   by (cases "b = 0", simp_all add: dvd_eq_mod_eq_0 gcd_non_0 gcd_0)
348 lemma gcd_proj1_if_dvd:
349   "a dvd b \<Longrightarrow> gcd a b = a div normalization_factor a"
350   by (subst gcd.commute, simp add: gcd_proj2_if_dvd)
352 lemma gcd_proj1_iff: "gcd m n = m div normalization_factor m \<longleftrightarrow> m dvd n"
353 proof
354   assume A: "gcd m n = m div normalization_factor m"
355   show "m dvd n"
356   proof (cases "m = 0")
357     assume [simp]: "m \<noteq> 0"
358     from A have B: "m = gcd m n * normalization_factor m"
359       by (simp add: unit_eq_div2)
360     show ?thesis by (subst B, simp add: mult_unit_dvd_iff)
361   qed (insert A, simp)
362 next
363   assume "m dvd n"
364   then show "gcd m n = m div normalization_factor m" by (rule gcd_proj1_if_dvd)
365 qed
367 lemma gcd_proj2_iff: "gcd m n = n div normalization_factor n \<longleftrightarrow> n dvd m"
368   by (subst gcd.commute, simp add: gcd_proj1_iff)
370 lemma gcd_mod1 [simp]:
371   "gcd (a mod b) b = gcd a b"
372   by (rule gcdI, metis dvd_mod_iff gcd_dvd1 gcd_dvd2, simp_all add: gcd_greatest dvd_mod_iff)
374 lemma gcd_mod2 [simp]:
375   "gcd a (b mod a) = gcd a b"
376   by (rule gcdI, simp, metis dvd_mod_iff gcd_dvd1 gcd_dvd2, simp_all add: gcd_greatest dvd_mod_iff)
378 lemma gcd_mult_distrib':
379   "c div normalization_factor c * gcd a b = gcd (c * a) (c * b)"
380 proof (cases "c = 0")
381   case True then show ?thesis by (simp_all add: gcd_0)
382 next
383   case False then have [simp]: "is_unit (normalization_factor c)" by simp
384   show ?thesis
385   proof (induct a b rule: gcd_eucl_induct)
386     case (zero a) show ?case
387     proof (cases "a = 0")
388       case True then show ?thesis by (simp add: gcd_0)
389     next
390       case False then have "is_unit (normalization_factor a)" by simp
391       then show ?thesis
392         by (simp add: gcd_0 unit_div_commute unit_div_mult_swap normalization_factor_mult is_unit_div_mult2_eq)
393     qed
394     case (mod a b)
395     then show ?case by (simp add: mult_mod_right gcd.commute)
396   qed
397 qed
399 lemma gcd_mult_distrib:
400   "k * gcd a b = gcd (k*a) (k*b) * normalization_factor k"
401 proof-
402   let ?nf = "normalization_factor"
403   from gcd_mult_distrib'
404     have "gcd (k*a) (k*b) = k div ?nf k * gcd a b" ..
405   also have "... = k * gcd a b div ?nf k"
406     by (metis dvd_div_mult dvd_eq_mod_eq_0 mod_0 normalization_factor_dvd)
407   finally show ?thesis
408     by simp
409 qed
411 lemma euclidean_size_gcd_le1 [simp]:
412   assumes "a \<noteq> 0"
413   shows "euclidean_size (gcd a b) \<le> euclidean_size a"
414 proof -
415    have "gcd a b dvd a" by (rule gcd_dvd1)
416    then obtain c where A: "a = gcd a b * c" unfolding dvd_def by blast
417    with \<open>a \<noteq> 0\<close> show ?thesis by (subst (2) A, intro size_mult_mono) auto
418 qed
420 lemma euclidean_size_gcd_le2 [simp]:
421   "b \<noteq> 0 \<Longrightarrow> euclidean_size (gcd a b) \<le> euclidean_size b"
422   by (subst gcd.commute, rule euclidean_size_gcd_le1)
424 lemma euclidean_size_gcd_less1:
425   assumes "a \<noteq> 0" and "\<not>a dvd b"
426   shows "euclidean_size (gcd a b) < euclidean_size a"
427 proof (rule ccontr)
428   assume "\<not>euclidean_size (gcd a b) < euclidean_size a"
429   with \<open>a \<noteq> 0\<close> have "euclidean_size (gcd a b) = euclidean_size a"
430     by (intro le_antisym, simp_all)
431   with assms have "a dvd gcd a b" by (auto intro: dvd_euclidean_size_eq_imp_dvd)
432   hence "a dvd b" using dvd_gcd_D2 by blast
433   with \<open>\<not>a dvd b\<close> show False by contradiction
434 qed
436 lemma euclidean_size_gcd_less2:
437   assumes "b \<noteq> 0" and "\<not>b dvd a"
438   shows "euclidean_size (gcd a b) < euclidean_size b"
439   using assms by (subst gcd.commute, rule euclidean_size_gcd_less1)
441 lemma gcd_mult_unit1: "is_unit a \<Longrightarrow> gcd (b * a) c = gcd b c"
442   apply (rule gcdI)
443   apply (rule dvd_trans, rule gcd_dvd1, simp add: unit_simps)
444   apply (rule gcd_dvd2)
445   apply (rule gcd_greatest, simp add: unit_simps, assumption)
446   apply (subst normalization_factor_gcd, simp add: gcd_0)
447   done
449 lemma gcd_mult_unit2: "is_unit a \<Longrightarrow> gcd b (c * a) = gcd b c"
450   by (subst gcd.commute, subst gcd_mult_unit1, assumption, rule gcd.commute)
452 lemma gcd_div_unit1: "is_unit a \<Longrightarrow> gcd (b div a) c = gcd b c"
453   by (erule is_unitE [of _ b]) (simp add: gcd_mult_unit1)
455 lemma gcd_div_unit2: "is_unit a \<Longrightarrow> gcd b (c div a) = gcd b c"
456   by (erule is_unitE [of _ c]) (simp add: gcd_mult_unit2)
458 lemma gcd_idem: "gcd a a = a div normalization_factor a"
459   by (cases "a = 0") (simp add: gcd_0_left, rule sym, rule gcdI, simp_all)
461 lemma gcd_right_idem: "gcd (gcd a b) b = gcd a b"
462   apply (rule gcdI)
463   apply (simp add: ac_simps)
464   apply (rule gcd_dvd2)
465   apply (rule gcd_greatest, erule (1) gcd_greatest, assumption)
466   apply simp
467   done
469 lemma gcd_left_idem: "gcd a (gcd a b) = gcd a b"
470   apply (rule gcdI)
471   apply simp
472   apply (rule dvd_trans, rule gcd_dvd2, rule gcd_dvd2)
473   apply (rule gcd_greatest, assumption, erule gcd_greatest, assumption)
474   apply simp
475   done
477 lemma comp_fun_idem_gcd: "comp_fun_idem gcd"
478 proof
479   fix a b show "gcd a \<circ> gcd b = gcd b \<circ> gcd a"
480     by (simp add: fun_eq_iff ac_simps)
481 next
482   fix a show "gcd a \<circ> gcd a = gcd a"
483     by (simp add: fun_eq_iff gcd_left_idem)
484 qed
486 lemma coprime_dvd_mult:
487   assumes "gcd c b = 1" and "c dvd a * b"
488   shows "c dvd a"
489 proof -
490   let ?nf = "normalization_factor"
491   from assms gcd_mult_distrib [of a c b]
492     have A: "a = gcd (a * c) (a * b) * ?nf a" by simp
493   from \<open>c dvd a * b\<close> show ?thesis by (subst A, simp_all add: gcd_greatest)
494 qed
496 lemma coprime_dvd_mult_iff:
497   "gcd c b = 1 \<Longrightarrow> (c dvd a * b) = (c dvd a)"
498   by (rule, rule coprime_dvd_mult, simp_all)
500 lemma gcd_dvd_antisym:
501   "gcd a b dvd gcd c d \<Longrightarrow> gcd c d dvd gcd a b \<Longrightarrow> gcd a b = gcd c d"
502 proof (rule gcdI)
503   assume A: "gcd a b dvd gcd c d" and B: "gcd c d dvd gcd a b"
504   have "gcd c d dvd c" by simp
505   with A show "gcd a b dvd c" by (rule dvd_trans)
506   have "gcd c d dvd d" by simp
507   with A show "gcd a b dvd d" by (rule dvd_trans)
508   show "normalization_factor (gcd a b) = (if gcd a b = 0 then 0 else 1)"
509     by simp
510   fix l assume "l dvd c" and "l dvd d"
511   hence "l dvd gcd c d" by (rule gcd_greatest)
512   from this and B show "l dvd gcd a b" by (rule dvd_trans)
513 qed
515 lemma gcd_mult_cancel:
516   assumes "gcd k n = 1"
517   shows "gcd (k * m) n = gcd m n"
518 proof (rule gcd_dvd_antisym)
519   have "gcd (gcd (k * m) n) k = gcd (gcd k n) (k * m)" by (simp add: ac_simps)
520   also note \<open>gcd k n = 1\<close>
521   finally have "gcd (gcd (k * m) n) k = 1" by simp
522   hence "gcd (k * m) n dvd m" by (rule coprime_dvd_mult, simp add: ac_simps)
523   moreover have "gcd (k * m) n dvd n" by simp
524   ultimately show "gcd (k * m) n dvd gcd m n" by (rule gcd_greatest)
525   have "gcd m n dvd (k * m)" and "gcd m n dvd n" by simp_all
526   then show "gcd m n dvd gcd (k * m) n" by (rule gcd_greatest)
527 qed
529 lemma coprime_crossproduct:
530   assumes [simp]: "gcd a d = 1" "gcd b c = 1"
531   shows "associated (a * c) (b * d) \<longleftrightarrow> associated a b \<and> associated c d" (is "?lhs \<longleftrightarrow> ?rhs")
532 proof
533   assume ?rhs then show ?lhs unfolding associated_def by (fast intro: mult_dvd_mono)
534 next
535   assume ?lhs
536   from \<open>?lhs\<close> have "a dvd b * d" unfolding associated_def by (metis dvd_mult_left)
537   hence "a dvd b" by (simp add: coprime_dvd_mult_iff)
538   moreover from \<open>?lhs\<close> have "b dvd a * c" unfolding associated_def by (metis dvd_mult_left)
539   hence "b dvd a" by (simp add: coprime_dvd_mult_iff)
540   moreover from \<open>?lhs\<close> have "c dvd d * b"
541     unfolding associated_def by (auto dest: dvd_mult_right simp add: ac_simps)
542   hence "c dvd d" by (simp add: coprime_dvd_mult_iff gcd.commute)
543   moreover from \<open>?lhs\<close> have "d dvd c * a"
544     unfolding associated_def by (auto dest: dvd_mult_right simp add: ac_simps)
545   hence "d dvd c" by (simp add: coprime_dvd_mult_iff gcd.commute)
546   ultimately show ?rhs unfolding associated_def by simp
547 qed
549 lemma gcd_add1 [simp]:
550   "gcd (m + n) n = gcd m n"
551   by (cases "n = 0", simp_all add: gcd_non_0)
553 lemma gcd_add2 [simp]:
554   "gcd m (m + n) = gcd m n"
555   using gcd_add1 [of n m] by (simp add: ac_simps)
558   "gcd m (k * m + n) = gcd m n"
559 proof -
560   have "gcd m ((k * m + n) mod m) = gcd m (k * m + n)"
561     by (fact gcd_mod2)
562   then show ?thesis by simp
563 qed
565 lemma coprimeI: "(\<And>l. \<lbrakk>l dvd a; l dvd b\<rbrakk> \<Longrightarrow> l dvd 1) \<Longrightarrow> gcd a b = 1"
566   by (rule sym, rule gcdI, simp_all)
568 lemma coprime: "gcd a b = 1 \<longleftrightarrow> (\<forall>d. d dvd a \<and> d dvd b \<longleftrightarrow> is_unit d)"
569   by (auto intro: coprimeI gcd_greatest dvd_gcd_D1 dvd_gcd_D2)
571 lemma div_gcd_coprime:
572   assumes nz: "a \<noteq> 0 \<or> b \<noteq> 0"
573   defines [simp]: "d \<equiv> gcd a b"
574   defines [simp]: "a' \<equiv> a div d" and [simp]: "b' \<equiv> b div d"
575   shows "gcd a' b' = 1"
576 proof (rule coprimeI)
577   fix l assume "l dvd a'" "l dvd b'"
578   then obtain s t where "a' = l * s" "b' = l * t" unfolding dvd_def by blast
579   moreover have "a = a' * d" "b = b' * d" by simp_all
580   ultimately have "a = (l * d) * s" "b = (l * d) * t"
581     by (simp_all only: ac_simps)
582   hence "l*d dvd a" and "l*d dvd b" by (simp_all only: dvd_triv_left)
583   hence "l*d dvd d" by (simp add: gcd_greatest)
584   then obtain u where "d = l * d * u" ..
585   then have "d * (l * u) = d" by (simp add: ac_simps)
586   moreover from nz have "d \<noteq> 0" by simp
587   with div_mult_self1_is_id have "d * (l * u) div d = l * u" .
588   ultimately have "1 = l * u"
589     using \<open>d \<noteq> 0\<close> by simp
590   then show "l dvd 1" ..
591 qed
593 lemma coprime_mult:
594   assumes da: "gcd d a = 1" and db: "gcd d b = 1"
595   shows "gcd d (a * b) = 1"
596   apply (subst gcd.commute)
597   using da apply (subst gcd_mult_cancel)
598   apply (subst gcd.commute, assumption)
599   apply (subst gcd.commute, rule db)
600   done
602 lemma coprime_lmult:
603   assumes dab: "gcd d (a * b) = 1"
604   shows "gcd d a = 1"
605 proof (rule coprimeI)
606   fix l assume "l dvd d" and "l dvd a"
607   hence "l dvd a * b" by simp
608   with \<open>l dvd d\<close> and dab show "l dvd 1" by (auto intro: gcd_greatest)
609 qed
611 lemma coprime_rmult:
612   assumes dab: "gcd d (a * b) = 1"
613   shows "gcd d b = 1"
614 proof (rule coprimeI)
615   fix l assume "l dvd d" and "l dvd b"
616   hence "l dvd a * b" by simp
617   with \<open>l dvd d\<close> and dab show "l dvd 1" by (auto intro: gcd_greatest)
618 qed
620 lemma coprime_mul_eq: "gcd d (a * b) = 1 \<longleftrightarrow> gcd d a = 1 \<and> gcd d b = 1"
621   using coprime_rmult[of d a b] coprime_lmult[of d a b] coprime_mult[of d a b] by blast
623 lemma gcd_coprime:
624   assumes c: "gcd a b \<noteq> 0" and a: "a = a' * gcd a b" and b: "b = b' * gcd a b"
625   shows "gcd a' b' = 1"
626 proof -
627   from c have "a \<noteq> 0 \<or> b \<noteq> 0" by simp
628   with div_gcd_coprime have "gcd (a div gcd a b) (b div gcd a b) = 1" .
629   also from assms have "a div gcd a b = a'" by (metis div_mult_self2_is_id)+
630   also from assms have "b div gcd a b = b'" by (metis div_mult_self2_is_id)+
631   finally show ?thesis .
632 qed
634 lemma coprime_power:
635   assumes "0 < n"
636   shows "gcd a (b ^ n) = 1 \<longleftrightarrow> gcd a b = 1"
637 using assms proof (induct n)
638   case (Suc n) then show ?case
639     by (cases n) (simp_all add: coprime_mul_eq)
640 qed simp
642 lemma gcd_coprime_exists:
643   assumes nz: "gcd a b \<noteq> 0"
644   shows "\<exists>a' b'. a = a' * gcd a b \<and> b = b' * gcd a b \<and> gcd a' b' = 1"
645   apply (rule_tac x = "a div gcd a b" in exI)
646   apply (rule_tac x = "b div gcd a b" in exI)
647   apply (insert nz, auto intro: div_gcd_coprime)
648   done
650 lemma coprime_exp:
651   "gcd d a = 1 \<Longrightarrow> gcd d (a^n) = 1"
652   by (induct n, simp_all add: coprime_mult)
654 lemma coprime_exp2 [intro]:
655   "gcd a b = 1 \<Longrightarrow> gcd (a^n) (b^m) = 1"
656   apply (rule coprime_exp)
657   apply (subst gcd.commute)
658   apply (rule coprime_exp)
659   apply (subst gcd.commute)
660   apply assumption
661   done
663 lemma gcd_exp:
664   "gcd (a^n) (b^n) = (gcd a b) ^ n"
665 proof (cases "a = 0 \<and> b = 0")
666   assume "a = 0 \<and> b = 0"
667   then show ?thesis by (cases n, simp_all add: gcd_0_left)
668 next
669   assume A: "\<not>(a = 0 \<and> b = 0)"
670   hence "1 = gcd ((a div gcd a b)^n) ((b div gcd a b)^n)"
671     using div_gcd_coprime by (subst sym, auto simp: div_gcd_coprime)
672   hence "(gcd a b) ^ n = (gcd a b) ^ n * ..." by simp
673   also note gcd_mult_distrib
674   also have "normalization_factor ((gcd a b)^n) = 1"
675     by (simp add: normalization_factor_pow A)
676   also have "(gcd a b)^n * (a div gcd a b)^n = a^n"
677     by (subst ac_simps, subst div_power, simp, rule dvd_div_mult_self, rule dvd_power_same, simp)
678   also have "(gcd a b)^n * (b div gcd a b)^n = b^n"
679     by (subst ac_simps, subst div_power, simp, rule dvd_div_mult_self, rule dvd_power_same, simp)
680   finally show ?thesis by simp
681 qed
683 lemma coprime_common_divisor:
684   "gcd a b = 1 \<Longrightarrow> a dvd a \<Longrightarrow> a dvd b \<Longrightarrow> is_unit a"
685   apply (subgoal_tac "a dvd gcd a b")
686   apply simp
687   apply (erule (1) gcd_greatest)
688   done
690 lemma division_decomp:
691   assumes dc: "a dvd b * c"
692   shows "\<exists>b' c'. a = b' * c' \<and> b' dvd b \<and> c' dvd c"
693 proof (cases "gcd a b = 0")
694   assume "gcd a b = 0"
695   hence "a = 0 \<and> b = 0" by simp
696   hence "a = 0 * c \<and> 0 dvd b \<and> c dvd c" by simp
697   then show ?thesis by blast
698 next
699   let ?d = "gcd a b"
700   assume "?d \<noteq> 0"
701   from gcd_coprime_exists[OF this]
702     obtain a' b' where ab': "a = a' * ?d" "b = b' * ?d" "gcd a' b' = 1"
703     by blast
704   from ab'(1) have "a' dvd a" unfolding dvd_def by blast
705   with dc have "a' dvd b*c" using dvd_trans[of a' a "b*c"] by simp
706   from dc ab'(1,2) have "a'*?d dvd (b'*?d) * c" by simp
707   hence "?d * a' dvd ?d * (b' * c)" by (simp add: mult_ac)
708   with \<open>?d \<noteq> 0\<close> have "a' dvd b' * c" by simp
709   with coprime_dvd_mult[OF ab'(3)]
710     have "a' dvd c" by (subst (asm) ac_simps, blast)
711   with ab'(1) have "a = ?d * a' \<and> ?d dvd b \<and> a' dvd c" by (simp add: mult_ac)
712   then show ?thesis by blast
713 qed
715 lemma pow_divs_pow:
716   assumes ab: "a ^ n dvd b ^ n" and n: "n \<noteq> 0"
717   shows "a dvd b"
718 proof (cases "gcd a b = 0")
719   assume "gcd a b = 0"
720   then show ?thesis by simp
721 next
722   let ?d = "gcd a b"
723   assume "?d \<noteq> 0"
724   from n obtain m where m: "n = Suc m" by (cases n, simp_all)
725   from \<open>?d \<noteq> 0\<close> have zn: "?d ^ n \<noteq> 0" by (rule power_not_zero)
726   from gcd_coprime_exists[OF \<open>?d \<noteq> 0\<close>]
727     obtain a' b' where ab': "a = a' * ?d" "b = b' * ?d" "gcd a' b' = 1"
728     by blast
729   from ab have "(a' * ?d) ^ n dvd (b' * ?d) ^ n"
730     by (simp add: ab'(1,2)[symmetric])
731   hence "?d^n * a'^n dvd ?d^n * b'^n"
732     by (simp only: power_mult_distrib ac_simps)
733   with zn have "a'^n dvd b'^n" by simp
734   hence "a' dvd b'^n" using dvd_trans[of a' "a'^n" "b'^n"] by (simp add: m)
735   hence "a' dvd b'^m * b'" by (simp add: m ac_simps)
736   with coprime_dvd_mult[OF coprime_exp[OF ab'(3), of m]]
737     have "a' dvd b'" by (subst (asm) ac_simps, blast)
738   hence "a'*?d dvd b'*?d" by (rule mult_dvd_mono, simp)
739   with ab'(1,2) show ?thesis by simp
740 qed
742 lemma pow_divs_eq [simp]:
743   "n \<noteq> 0 \<Longrightarrow> a ^ n dvd b ^ n \<longleftrightarrow> a dvd b"
744   by (auto intro: pow_divs_pow dvd_power_same)
746 lemma divs_mult:
747   assumes mr: "m dvd r" and nr: "n dvd r" and mn: "gcd m n = 1"
748   shows "m * n dvd r"
749 proof -
750   from mr nr obtain m' n' where m': "r = m*m'" and n': "r = n*n'"
751     unfolding dvd_def by blast
752   from mr n' have "m dvd n'*n" by (simp add: ac_simps)
753   hence "m dvd n'" using coprime_dvd_mult_iff[OF mn] by simp
754   then obtain k where k: "n' = m*k" unfolding dvd_def by blast
755   with n' have "r = m * n * k" by (simp add: mult_ac)
756   then show ?thesis unfolding dvd_def by blast
757 qed
759 lemma coprime_plus_one [simp]: "gcd (n + 1) n = 1"
760   by (subst add_commute, simp)
762 lemma setprod_coprime [rule_format]:
763   "(\<forall>i\<in>A. gcd (f i) a = 1) \<longrightarrow> gcd (\<Prod>i\<in>A. f i) a = 1"
764   apply (cases "finite A")
765   apply (induct set: finite)
766   apply (auto simp add: gcd_mult_cancel)
767   done
769 lemma coprime_divisors:
770   assumes "d dvd a" "e dvd b" "gcd a b = 1"
771   shows "gcd d e = 1"
772 proof -
773   from assms obtain k l where "a = d * k" "b = e * l"
774     unfolding dvd_def by blast
775   with assms have "gcd (d * k) (e * l) = 1" by simp
776   hence "gcd (d * k) e = 1" by (rule coprime_lmult)
777   also have "gcd (d * k) e = gcd e (d * k)" by (simp add: ac_simps)
778   finally have "gcd e d = 1" by (rule coprime_lmult)
779   then show ?thesis by (simp add: ac_simps)
780 qed
782 lemma invertible_coprime:
783   assumes "a * b mod m = 1"
784   shows "coprime a m"
785 proof -
786   from assms have "coprime m (a * b mod m)"
787     by simp
788   then have "coprime m (a * b)"
789     by simp
790   then have "coprime m a"
791     by (rule coprime_lmult)
792   then show ?thesis
793     by (simp add: ac_simps)
794 qed
796 lemma lcm_gcd:
797   "lcm a b = a * b div (gcd a b * normalization_factor (a*b))"
798   by (simp only: lcm_lcm_eucl gcd_gcd_eucl lcm_eucl_def)
800 lemma lcm_gcd_prod:
801   "lcm a b * gcd a b = a * b div normalization_factor (a*b)"
802 proof (cases "a * b = 0")
803   let ?nf = normalization_factor
804   assume "a * b \<noteq> 0"
805   hence "gcd a b \<noteq> 0" by simp
806   from lcm_gcd have "lcm a b * gcd a b = gcd a b * (a * b div (?nf (a*b) * gcd a b))"
807     by (simp add: mult_ac)
808   also from \<open>a * b \<noteq> 0\<close> have "... = a * b div ?nf (a*b)"
809     by (simp add: div_mult_swap mult.commute)
810   finally show ?thesis .
811 qed (auto simp add: lcm_gcd)
813 lemma lcm_dvd1 [iff]:
814   "a dvd lcm a b"
815 proof (cases "a*b = 0")
816   assume "a * b \<noteq> 0"
817   hence "gcd a b \<noteq> 0" by simp
818   let ?c = "1 div normalization_factor (a * b)"
819   from \<open>a * b \<noteq> 0\<close> have [simp]: "is_unit (normalization_factor (a * b))" by simp
820   from lcm_gcd_prod[of a b] have "lcm a b * gcd a b = a * ?c * b"
821     by (simp add: div_mult_swap unit_div_commute)
822   hence "lcm a b * gcd a b div gcd a b = a * ?c * b div gcd a b" by simp
823   with \<open>gcd a b \<noteq> 0\<close> have "lcm a b = a * ?c * b div gcd a b"
824     by (subst (asm) div_mult_self2_is_id, simp_all)
825   also have "... = a * (?c * b div gcd a b)"
826     by (metis div_mult_swap gcd_dvd2 mult_assoc)
827   finally show ?thesis by (rule dvdI)
828 qed (auto simp add: lcm_gcd)
830 lemma lcm_least:
831   "\<lbrakk>a dvd k; b dvd k\<rbrakk> \<Longrightarrow> lcm a b dvd k"
832 proof (cases "k = 0")
833   let ?nf = normalization_factor
834   assume "k \<noteq> 0"
835   hence "is_unit (?nf k)" by simp
836   hence "?nf k \<noteq> 0" by (metis not_is_unit_0)
837   assume A: "a dvd k" "b dvd k"
838   hence "gcd a b \<noteq> 0" using \<open>k \<noteq> 0\<close> by auto
839   from A obtain r s where ar: "k = a * r" and bs: "k = b * s"
840     unfolding dvd_def by blast
841   with \<open>k \<noteq> 0\<close> have "r * s \<noteq> 0"
842     by auto (drule sym [of 0], simp)
843   hence "is_unit (?nf (r * s))" by simp
844   let ?c = "?nf k div ?nf (r*s)"
845   from \<open>is_unit (?nf k)\<close> and \<open>is_unit (?nf (r * s))\<close> have "is_unit ?c" by (rule unit_div)
846   hence "?c \<noteq> 0" using not_is_unit_0 by fast
847   from ar bs have "k * k * gcd s r = ?nf k * k * gcd (k * s) (k * r)"
848     by (subst mult_assoc, subst gcd_mult_distrib[of k s r], simp only: ac_simps)
849   also have "... = ?nf k * k * gcd ((r*s) * a) ((r*s) * b)"
850     by (subst (3) \<open>k = a * r\<close>, subst (3) \<open>k = b * s\<close>, simp add: algebra_simps)
851   also have "... = ?c * r*s * k * gcd a b" using \<open>r * s \<noteq> 0\<close>
852     by (subst gcd_mult_distrib'[symmetric], simp add: algebra_simps unit_simps)
853   finally have "(a*r) * (b*s) * gcd s r = ?c * k * r * s * gcd a b"
854     by (subst ar[symmetric], subst bs[symmetric], simp add: mult_ac)
855   hence "a * b * gcd s r * (r * s) = ?c * k * gcd a b * (r * s)"
856     by (simp add: algebra_simps)
857   hence "?c * k * gcd a b = a * b * gcd s r" using \<open>r * s \<noteq> 0\<close>
858     by (metis div_mult_self2_is_id)
859   also have "... = lcm a b * gcd a b * gcd s r * ?nf (a*b)"
860     by (subst lcm_gcd_prod[of a b], metis gcd_mult_distrib gcd_mult_distrib')
861   also have "... = lcm a b * gcd s r * ?nf (a*b) * gcd a b"
862     by (simp add: algebra_simps)
863   finally have "k * ?c = lcm a b * gcd s r * ?nf (a*b)" using \<open>gcd a b \<noteq> 0\<close>
864     by (metis mult.commute div_mult_self2_is_id)
865   hence "k = lcm a b * (gcd s r * ?nf (a*b)) div ?c" using \<open>?c \<noteq> 0\<close>
866     by (metis div_mult_self2_is_id mult_assoc)
867   also have "... = lcm a b * (gcd s r * ?nf (a*b) div ?c)" using \<open>is_unit ?c\<close>
868     by (simp add: unit_simps)
869   finally show ?thesis by (rule dvdI)
870 qed simp
872 lemma lcm_zero:
873   "lcm a b = 0 \<longleftrightarrow> a = 0 \<or> b = 0"
874 proof -
875   let ?nf = normalization_factor
876   {
877     assume "a \<noteq> 0" "b \<noteq> 0"
878     hence "a * b div ?nf (a * b) \<noteq> 0" by (simp add: no_zero_divisors)
879     moreover from \<open>a \<noteq> 0\<close> and \<open>b \<noteq> 0\<close> have "gcd a b \<noteq> 0" by simp
880     ultimately have "lcm a b \<noteq> 0" using lcm_gcd_prod[of a b] by (intro notI, simp)
881   } moreover {
882     assume "a = 0 \<or> b = 0"
883     hence "lcm a b = 0" by (elim disjE, simp_all add: lcm_gcd)
884   }
885   ultimately show ?thesis by blast
886 qed
888 lemmas lcm_0_iff = lcm_zero
890 lemma gcd_lcm:
891   assumes "lcm a b \<noteq> 0"
892   shows "gcd a b = a * b div (lcm a b * normalization_factor (a * b))"
893 proof-
894   from assms have "gcd a b \<noteq> 0" by (simp add: lcm_zero)
895   let ?c = "normalization_factor (a * b)"
896   from \<open>lcm a b \<noteq> 0\<close> have "?c \<noteq> 0" by (intro notI, simp add: lcm_zero no_zero_divisors)
897   hence "is_unit ?c" by simp
898   from lcm_gcd_prod [of a b] have "gcd a b = a * b div ?c div lcm a b"
899     by (subst (2) div_mult_self2_is_id[OF \<open>lcm a b \<noteq> 0\<close>, symmetric], simp add: mult_ac)
900   also from \<open>is_unit ?c\<close> have "... = a * b div (lcm a b * ?c)"
901     by (metis \<open>?c \<noteq> 0\<close> div_mult_mult1 dvd_mult_div_cancel mult_commute normalization_factor_dvd')
902   finally show ?thesis .
903 qed
905 lemma normalization_factor_lcm [simp]:
906   "normalization_factor (lcm a b) = (if a = 0 \<or> b = 0 then 0 else 1)"
907 proof (cases "a = 0 \<or> b = 0")
908   case True then show ?thesis
909     by (auto simp add: lcm_gcd)
910 next
911   case False
912   let ?nf = normalization_factor
913   from lcm_gcd_prod[of a b]
914     have "?nf (lcm a b) * ?nf (gcd a b) = ?nf (a*b) div ?nf (a*b)"
915     by (metis div_by_0 div_self normalization_correct normalization_factor_0 normalization_factor_mult)
916   also have "... = (if a*b = 0 then 0 else 1)"
917     by simp
918   finally show ?thesis using False by simp
919 qed
921 lemma lcm_dvd2 [iff]: "b dvd lcm a b"
922   using lcm_dvd1 [of b a] by (simp add: lcm_gcd ac_simps)
924 lemma lcmI:
925   "\<lbrakk>a dvd k; b dvd k; \<And>l. a dvd l \<Longrightarrow> b dvd l \<Longrightarrow> k dvd l;
926     normalization_factor k = (if k = 0 then 0 else 1)\<rbrakk> \<Longrightarrow> k = lcm a b"
927   by (intro normed_associated_imp_eq) (auto simp: associated_def intro: lcm_least)
929 sublocale lcm!: abel_semigroup lcm
930 proof
931   fix a b c
932   show "lcm (lcm a b) c = lcm a (lcm b c)"
933   proof (rule lcmI)
934     have "a dvd lcm a b" and "lcm a b dvd lcm (lcm a b) c" by simp_all
935     then show "a dvd lcm (lcm a b) c" by (rule dvd_trans)
937     have "b dvd lcm a b" and "lcm a b dvd lcm (lcm a b) c" by simp_all
938     hence "b dvd lcm (lcm a b) c" by (rule dvd_trans)
939     moreover have "c dvd lcm (lcm a b) c" by simp
940     ultimately show "lcm b c dvd lcm (lcm a b) c" by (rule lcm_least)
942     fix l assume "a dvd l" and "lcm b c dvd l"
943     have "b dvd lcm b c" by simp
944     from this and \<open>lcm b c dvd l\<close> have "b dvd l" by (rule dvd_trans)
945     have "c dvd lcm b c" by simp
946     from this and \<open>lcm b c dvd l\<close> have "c dvd l" by (rule dvd_trans)
947     from \<open>a dvd l\<close> and \<open>b dvd l\<close> have "lcm a b dvd l" by (rule lcm_least)
948     from this and \<open>c dvd l\<close> show "lcm (lcm a b) c dvd l" by (rule lcm_least)
949   qed (simp add: lcm_zero)
950 next
951   fix a b
952   show "lcm a b = lcm b a"
953     by (simp add: lcm_gcd ac_simps)
954 qed
956 lemma dvd_lcm_D1:
957   "lcm m n dvd k \<Longrightarrow> m dvd k"
958   by (rule dvd_trans, rule lcm_dvd1, assumption)
960 lemma dvd_lcm_D2:
961   "lcm m n dvd k \<Longrightarrow> n dvd k"
962   by (rule dvd_trans, rule lcm_dvd2, assumption)
964 lemma gcd_dvd_lcm [simp]:
965   "gcd a b dvd lcm a b"
966   by (metis dvd_trans gcd_dvd2 lcm_dvd2)
968 lemma lcm_1_iff:
969   "lcm a b = 1 \<longleftrightarrow> is_unit a \<and> is_unit b"
970 proof
971   assume "lcm a b = 1"
972   then show "is_unit a \<and> is_unit b" by auto
973 next
974   assume "is_unit a \<and> is_unit b"
975   hence "a dvd 1" and "b dvd 1" by simp_all
976   hence "is_unit (lcm a b)" by (rule lcm_least)
977   hence "lcm a b = normalization_factor (lcm a b)"
978     by (subst normalization_factor_unit, simp_all)
979   also have "\<dots> = 1" using \<open>is_unit a \<and> is_unit b\<close>
980     by auto
981   finally show "lcm a b = 1" .
982 qed
984 lemma lcm_0_left [simp]:
985   "lcm 0 a = 0"
986   by (rule sym, rule lcmI, simp_all)
988 lemma lcm_0 [simp]:
989   "lcm a 0 = 0"
990   by (rule sym, rule lcmI, simp_all)
992 lemma lcm_unique:
993   "a dvd d \<and> b dvd d \<and>
994   normalization_factor d = (if d = 0 then 0 else 1) \<and>
995   (\<forall>e. a dvd e \<and> b dvd e \<longrightarrow> d dvd e) \<longleftrightarrow> d = lcm a b"
996   by (rule, auto intro: lcmI simp: lcm_least lcm_zero)
998 lemma dvd_lcm_I1 [simp]:
999   "k dvd m \<Longrightarrow> k dvd lcm m n"
1000   by (metis lcm_dvd1 dvd_trans)
1002 lemma dvd_lcm_I2 [simp]:
1003   "k dvd n \<Longrightarrow> k dvd lcm m n"
1004   by (metis lcm_dvd2 dvd_trans)
1006 lemma lcm_1_left [simp]:
1007   "lcm 1 a = a div normalization_factor a"
1008   by (cases "a = 0") (simp, rule sym, rule lcmI, simp_all)
1010 lemma lcm_1_right [simp]:
1011   "lcm a 1 = a div normalization_factor a"
1012   using lcm_1_left [of a] by (simp add: ac_simps)
1014 lemma lcm_coprime:
1015   "gcd a b = 1 \<Longrightarrow> lcm a b = a * b div normalization_factor (a*b)"
1016   by (subst lcm_gcd) simp
1018 lemma lcm_proj1_if_dvd:
1019   "b dvd a \<Longrightarrow> lcm a b = a div normalization_factor a"
1020   by (cases "a = 0") (simp, rule sym, rule lcmI, simp_all)
1022 lemma lcm_proj2_if_dvd:
1023   "a dvd b \<Longrightarrow> lcm a b = b div normalization_factor b"
1024   using lcm_proj1_if_dvd [of a b] by (simp add: ac_simps)
1026 lemma lcm_proj1_iff:
1027   "lcm m n = m div normalization_factor m \<longleftrightarrow> n dvd m"
1028 proof
1029   assume A: "lcm m n = m div normalization_factor m"
1030   show "n dvd m"
1031   proof (cases "m = 0")
1032     assume [simp]: "m \<noteq> 0"
1033     from A have B: "m = lcm m n * normalization_factor m"
1034       by (simp add: unit_eq_div2)
1035     show ?thesis by (subst B, simp)
1036   qed simp
1037 next
1038   assume "n dvd m"
1039   then show "lcm m n = m div normalization_factor m" by (rule lcm_proj1_if_dvd)
1040 qed
1042 lemma lcm_proj2_iff:
1043   "lcm m n = n div normalization_factor n \<longleftrightarrow> m dvd n"
1044   using lcm_proj1_iff [of n m] by (simp add: ac_simps)
1046 lemma euclidean_size_lcm_le1:
1047   assumes "a \<noteq> 0" and "b \<noteq> 0"
1048   shows "euclidean_size a \<le> euclidean_size (lcm a b)"
1049 proof -
1050   have "a dvd lcm a b" by (rule lcm_dvd1)
1051   then obtain c where A: "lcm a b = a * c" unfolding dvd_def by blast
1052   with \<open>a \<noteq> 0\<close> and \<open>b \<noteq> 0\<close> have "c \<noteq> 0" by (auto simp: lcm_zero)
1053   then show ?thesis by (subst A, intro size_mult_mono)
1054 qed
1056 lemma euclidean_size_lcm_le2:
1057   "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> euclidean_size b \<le> euclidean_size (lcm a b)"
1058   using euclidean_size_lcm_le1 [of b a] by (simp add: ac_simps)
1060 lemma euclidean_size_lcm_less1:
1061   assumes "b \<noteq> 0" and "\<not>b dvd a"
1062   shows "euclidean_size a < euclidean_size (lcm a b)"
1063 proof (rule ccontr)
1064   from assms have "a \<noteq> 0" by auto
1065   assume "\<not>euclidean_size a < euclidean_size (lcm a b)"
1066   with \<open>a \<noteq> 0\<close> and \<open>b \<noteq> 0\<close> have "euclidean_size (lcm a b) = euclidean_size a"
1067     by (intro le_antisym, simp, intro euclidean_size_lcm_le1)
1068   with assms have "lcm a b dvd a"
1069     by (rule_tac dvd_euclidean_size_eq_imp_dvd) (auto simp: lcm_zero)
1070   hence "b dvd a" by (rule dvd_lcm_D2)
1071   with \<open>\<not>b dvd a\<close> show False by contradiction
1072 qed
1074 lemma euclidean_size_lcm_less2:
1075   assumes "a \<noteq> 0" and "\<not>a dvd b"
1076   shows "euclidean_size b < euclidean_size (lcm a b)"
1077   using assms euclidean_size_lcm_less1 [of a b] by (simp add: ac_simps)
1079 lemma lcm_mult_unit1:
1080   "is_unit a \<Longrightarrow> lcm (b * a) c = lcm b c"
1081   apply (rule lcmI)
1082   apply (rule dvd_trans[of _ "b * a"], simp, rule lcm_dvd1)
1083   apply (rule lcm_dvd2)
1084   apply (rule lcm_least, simp add: unit_simps, assumption)
1085   apply (subst normalization_factor_lcm, simp add: lcm_zero)
1086   done
1088 lemma lcm_mult_unit2:
1089   "is_unit a \<Longrightarrow> lcm b (c * a) = lcm b c"
1090   using lcm_mult_unit1 [of a c b] by (simp add: ac_simps)
1092 lemma lcm_div_unit1:
1093   "is_unit a \<Longrightarrow> lcm (b div a) c = lcm b c"
1094   by (erule is_unitE [of _ b]) (simp add: lcm_mult_unit1)
1096 lemma lcm_div_unit2:
1097   "is_unit a \<Longrightarrow> lcm b (c div a) = lcm b c"
1098   by (erule is_unitE [of _ c]) (simp add: lcm_mult_unit2)
1100 lemma lcm_left_idem:
1101   "lcm a (lcm a b) = lcm a b"
1102   apply (rule lcmI)
1103   apply simp
1104   apply (subst lcm.assoc [symmetric], rule lcm_dvd2)
1105   apply (rule lcm_least, assumption)
1106   apply (erule (1) lcm_least)
1107   apply (auto simp: lcm_zero)
1108   done
1110 lemma lcm_right_idem:
1111   "lcm (lcm a b) b = lcm a b"
1112   apply (rule lcmI)
1113   apply (subst lcm.assoc, rule lcm_dvd1)
1114   apply (rule lcm_dvd2)
1115   apply (rule lcm_least, erule (1) lcm_least, assumption)
1116   apply (auto simp: lcm_zero)
1117   done
1119 lemma comp_fun_idem_lcm: "comp_fun_idem lcm"
1120 proof
1121   fix a b show "lcm a \<circ> lcm b = lcm b \<circ> lcm a"
1122     by (simp add: fun_eq_iff ac_simps)
1123 next
1124   fix a show "lcm a \<circ> lcm a = lcm a" unfolding o_def
1125     by (intro ext, simp add: lcm_left_idem)
1126 qed
1128 lemma dvd_Lcm [simp]: "a \<in> A \<Longrightarrow> a dvd Lcm A"
1129   and Lcm_dvd [simp]: "(\<forall>a\<in>A. a dvd l') \<Longrightarrow> Lcm A dvd l'"
1130   and normalization_factor_Lcm [simp]:
1131           "normalization_factor (Lcm A) = (if Lcm A = 0 then 0 else 1)"
1132 proof -
1133   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>
1134     normalization_factor (Lcm A) = (if Lcm A = 0 then 0 else 1)" (is ?thesis)
1135   proof (cases "\<exists>l. l \<noteq>  0 \<and> (\<forall>a\<in>A. a dvd l)")
1136     case False
1137     hence "Lcm A = 0" by (auto simp: Lcm_Lcm_eucl Lcm_eucl_def)
1138     with False show ?thesis by auto
1139   next
1140     case True
1141     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
1142     def n \<equiv> "LEAST n. \<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"
1143     def l \<equiv> "SOME l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"
1144     have "\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"
1145       apply (subst n_def)
1146       apply (rule LeastI[of _ "euclidean_size l\<^sub>0"])
1147       apply (rule exI[of _ l\<^sub>0])
1148       apply (simp add: l\<^sub>0_props)
1149       done
1150     from someI_ex[OF this] have "l \<noteq> 0" and "\<forall>a\<in>A. a dvd l" and "euclidean_size l = n"
1151       unfolding l_def by simp_all
1152     {
1153       fix l' assume "\<forall>a\<in>A. a dvd l'"
1154       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)
1155       moreover from \<open>l \<noteq> 0\<close> have "gcd l l' \<noteq> 0" by simp
1156       ultimately have "\<exists>b. b \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd b) \<and> euclidean_size b = euclidean_size (gcd l l')"
1157         by (intro exI[of _ "gcd l l'"], auto)
1158       hence "euclidean_size (gcd l l') \<ge> n" by (subst n_def) (rule Least_le)
1159       moreover have "euclidean_size (gcd l l') \<le> n"
1160       proof -
1161         have "gcd l l' dvd l" by simp
1162         then obtain a where "l = gcd l l' * a" unfolding dvd_def by blast
1163         with \<open>l \<noteq> 0\<close> have "a \<noteq> 0" by auto
1164         hence "euclidean_size (gcd l l') \<le> euclidean_size (gcd l l' * a)"
1165           by (rule size_mult_mono)
1166         also have "gcd l l' * a = l" using \<open>l = gcd l l' * a\<close> ..
1167         also note \<open>euclidean_size l = n\<close>
1168         finally show "euclidean_size (gcd l l') \<le> n" .
1169       qed
1170       ultimately have "euclidean_size l = euclidean_size (gcd l l')"
1171         by (intro le_antisym, simp_all add: \<open>euclidean_size l = n\<close>)
1172       with \<open>l \<noteq> 0\<close> have "l dvd gcd l l'" by (blast intro: dvd_euclidean_size_eq_imp_dvd)
1173       hence "l dvd l'" by (blast dest: dvd_gcd_D2)
1174     }
1176     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>
1177       have "(\<forall>a\<in>A. a dvd l div normalization_factor l) \<and>
1178         (\<forall>l'. (\<forall>a\<in>A. a dvd l') \<longrightarrow> l div normalization_factor l dvd l') \<and>
1179         normalization_factor (l div normalization_factor l) =
1180         (if l div normalization_factor l = 0 then 0 else 1)"
1181       by (auto simp: unit_simps)
1182     also from True have "l div normalization_factor l = Lcm A"
1183       by (simp add: Lcm_Lcm_eucl Lcm_eucl_def Let_def n_def l_def)
1184     finally show ?thesis .
1185   qed
1186   note A = this
1188   {fix a assume "a \<in> A" then show "a dvd Lcm A" using A by blast}
1189   {fix l' assume "\<forall>a\<in>A. a dvd l'" then show "Lcm A dvd l'" using A by blast}
1190   from A show "normalization_factor (Lcm A) = (if Lcm A = 0 then 0 else 1)" by blast
1191 qed
1193 lemma LcmI:
1194   "(\<And>a. a\<in>A \<Longrightarrow> a dvd l) \<Longrightarrow> (\<And>l'. (\<forall>a\<in>A. a dvd l') \<Longrightarrow> l dvd l') \<Longrightarrow>
1195       normalization_factor l = (if l = 0 then 0 else 1) \<Longrightarrow> l = Lcm A"
1196   by (intro normed_associated_imp_eq)
1197     (auto intro: Lcm_dvd dvd_Lcm simp: associated_def)
1199 lemma Lcm_subset:
1200   "A \<subseteq> B \<Longrightarrow> Lcm A dvd Lcm B"
1201   by (blast intro: Lcm_dvd dvd_Lcm)
1203 lemma Lcm_Un:
1204   "Lcm (A \<union> B) = lcm (Lcm A) (Lcm B)"
1205   apply (rule lcmI)
1206   apply (blast intro: Lcm_subset)
1207   apply (blast intro: Lcm_subset)
1208   apply (intro Lcm_dvd ballI, elim UnE)
1209   apply (rule dvd_trans, erule dvd_Lcm, assumption)
1210   apply (rule dvd_trans, erule dvd_Lcm, assumption)
1211   apply simp
1212   done
1214 lemma Lcm_1_iff:
1215   "Lcm A = 1 \<longleftrightarrow> (\<forall>a\<in>A. is_unit a)"
1216 proof
1217   assume "Lcm A = 1"
1218   then show "\<forall>a\<in>A. is_unit a" by auto
1219 qed (rule LcmI [symmetric], auto)
1221 lemma Lcm_no_units:
1222   "Lcm A = Lcm (A - {a. is_unit a})"
1223 proof -
1224   have "(A - {a. is_unit a}) \<union> {a\<in>A. is_unit a} = A" by blast
1225   hence "Lcm A = lcm (Lcm (A - {a. is_unit a})) (Lcm {a\<in>A. is_unit a})"
1226     by (simp add: Lcm_Un[symmetric])
1227   also have "Lcm {a\<in>A. is_unit a} = 1" by (simp add: Lcm_1_iff)
1228   finally show ?thesis by simp
1229 qed
1231 lemma Lcm_empty [simp]:
1232   "Lcm {} = 1"
1233   by (simp add: Lcm_1_iff)
1235 lemma Lcm_eq_0 [simp]:
1236   "0 \<in> A \<Longrightarrow> Lcm A = 0"
1237   by (drule dvd_Lcm) simp
1239 lemma Lcm0_iff':
1240   "Lcm A = 0 \<longleftrightarrow> \<not>(\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l))"
1241 proof
1242   assume "Lcm A = 0"
1243   show "\<not>(\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l))"
1244   proof
1245     assume ex: "\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l)"
1246     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
1247     def n \<equiv> "LEAST n. \<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"
1248     def l \<equiv> "SOME l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"
1249     have "\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"
1250       apply (subst n_def)
1251       apply (rule LeastI[of _ "euclidean_size l\<^sub>0"])
1252       apply (rule exI[of _ l\<^sub>0])
1253       apply (simp add: l\<^sub>0_props)
1254       done
1255     from someI_ex[OF this] have "l \<noteq> 0" unfolding l_def by simp_all
1256     hence "l div normalization_factor l \<noteq> 0" by simp
1257     also from ex have "l div normalization_factor l = Lcm A"
1258        by (simp only: Lcm_Lcm_eucl Lcm_eucl_def n_def l_def if_True Let_def)
1259     finally show False using \<open>Lcm A = 0\<close> by contradiction
1260   qed
1261 qed (simp only: Lcm_Lcm_eucl Lcm_eucl_def if_False)
1263 lemma Lcm0_iff [simp]:
1264   "finite A \<Longrightarrow> Lcm A = 0 \<longleftrightarrow> 0 \<in> A"
1265 proof -
1266   assume "finite A"
1267   have "0 \<in> A \<Longrightarrow> Lcm A = 0"  by (intro dvd_0_left dvd_Lcm)
1268   moreover {
1269     assume "0 \<notin> A"
1270     hence "\<Prod>A \<noteq> 0"
1271       apply (induct rule: finite_induct[OF \<open>finite A\<close>])
1272       apply simp
1273       apply (subst setprod.insert, assumption, assumption)
1274       apply (rule no_zero_divisors)
1275       apply blast+
1276       done
1277     moreover from \<open>finite A\<close> have "\<forall>a\<in>A. a dvd \<Prod>A" by blast
1278     ultimately have "\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l)" by blast
1279     with Lcm0_iff' have "Lcm A \<noteq> 0" by simp
1280   }
1281   ultimately show "Lcm A = 0 \<longleftrightarrow> 0 \<in> A" by blast
1282 qed
1284 lemma Lcm_no_multiple:
1285   "(\<forall>m. m \<noteq> 0 \<longrightarrow> (\<exists>a\<in>A. \<not>a dvd m)) \<Longrightarrow> Lcm A = 0"
1286 proof -
1287   assume "\<forall>m. m \<noteq> 0 \<longrightarrow> (\<exists>a\<in>A. \<not>a dvd m)"
1288   hence "\<not>(\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l))" by blast
1289   then show "Lcm A = 0" by (simp only: Lcm_Lcm_eucl Lcm_eucl_def if_False)
1290 qed
1292 lemma Lcm_insert [simp]:
1293   "Lcm (insert a A) = lcm a (Lcm A)"
1294 proof (rule lcmI)
1295   fix l assume "a dvd l" and "Lcm A dvd l"
1296   hence "\<forall>a\<in>A. a dvd l" by (blast intro: dvd_trans dvd_Lcm)
1297   with \<open>a dvd l\<close> show "Lcm (insert a A) dvd l" by (force intro: Lcm_dvd)
1298 qed (auto intro: Lcm_dvd dvd_Lcm)
1300 lemma Lcm_finite:
1301   assumes "finite A"
1302   shows "Lcm A = Finite_Set.fold lcm 1 A"
1303   by (induct rule: finite.induct[OF \<open>finite A\<close>])
1304     (simp_all add: comp_fun_idem.fold_insert_idem[OF comp_fun_idem_lcm])
1306 lemma Lcm_set [code_unfold]:
1307   "Lcm (set xs) = fold lcm xs 1"
1308   using comp_fun_idem.fold_set_fold[OF comp_fun_idem_lcm] Lcm_finite by (simp add: ac_simps)
1310 lemma Lcm_singleton [simp]:
1311   "Lcm {a} = a div normalization_factor a"
1312   by simp
1314 lemma Lcm_2 [simp]:
1315   "Lcm {a,b} = lcm a b"
1316   by (simp only: Lcm_insert Lcm_empty lcm_1_right)
1317     (cases "b = 0", simp, rule lcm_div_unit2, simp)
1319 lemma Lcm_coprime:
1320   assumes "finite A" and "A \<noteq> {}"
1321   assumes "\<And>a b. a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<noteq> b \<Longrightarrow> gcd a b = 1"
1322   shows "Lcm A = \<Prod>A div normalization_factor (\<Prod>A)"
1323 using assms proof (induct rule: finite_ne_induct)
1324   case (insert a A)
1325   have "Lcm (insert a A) = lcm a (Lcm A)" by simp
1326   also from insert have "Lcm A = \<Prod>A div normalization_factor (\<Prod>A)" by blast
1327   also have "lcm a \<dots> = lcm a (\<Prod>A)" by (cases "\<Prod>A = 0") (simp_all add: lcm_div_unit2)
1328   also from insert have "gcd a (\<Prod>A) = 1" by (subst gcd.commute, intro setprod_coprime) auto
1329   with insert have "lcm a (\<Prod>A) = \<Prod>(insert a A) div normalization_factor (\<Prod>(insert a A))"
1330     by (simp add: lcm_coprime)
1331   finally show ?case .
1332 qed simp
1334 lemma Lcm_coprime':
1335   "card A \<noteq> 0 \<Longrightarrow> (\<And>a b. a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<noteq> b \<Longrightarrow> gcd a b = 1)
1336     \<Longrightarrow> Lcm A = \<Prod>A div normalization_factor (\<Prod>A)"
1337   by (rule Lcm_coprime) (simp_all add: card_eq_0_iff)
1339 lemma Gcd_Lcm:
1340   "Gcd A = Lcm {d. \<forall>a\<in>A. d dvd a}"
1341   by (simp add: Gcd_Gcd_eucl Lcm_Lcm_eucl Gcd_eucl_def)
1343 lemma Gcd_dvd [simp]: "a \<in> A \<Longrightarrow> Gcd A dvd a"
1344   and dvd_Gcd [simp]: "(\<forall>a\<in>A. g' dvd a) \<Longrightarrow> g' dvd Gcd A"
1345   and normalization_factor_Gcd [simp]:
1346     "normalization_factor (Gcd A) = (if Gcd A = 0 then 0 else 1)"
1347 proof -
1348   fix a assume "a \<in> A"
1349   hence "Lcm {d. \<forall>a\<in>A. d dvd a} dvd a" by (intro Lcm_dvd) blast
1350   then show "Gcd A dvd a" by (simp add: Gcd_Lcm)
1351 next
1352   fix g' assume "\<forall>a\<in>A. g' dvd a"
1353   hence "g' dvd Lcm {d. \<forall>a\<in>A. d dvd a}" by (intro dvd_Lcm) blast
1354   then show "g' dvd Gcd A" by (simp add: Gcd_Lcm)
1355 next
1356   show "normalization_factor (Gcd A) = (if Gcd A = 0 then 0 else 1)"
1357     by (simp add: Gcd_Lcm)
1358 qed
1360 lemma GcdI:
1361   "(\<And>a. a\<in>A \<Longrightarrow> l dvd a) \<Longrightarrow> (\<And>l'. (\<forall>a\<in>A. l' dvd a) \<Longrightarrow> l' dvd l) \<Longrightarrow>
1362     normalization_factor l = (if l = 0 then 0 else 1) \<Longrightarrow> l = Gcd A"
1363   by (intro normed_associated_imp_eq)
1364     (auto intro: Gcd_dvd dvd_Gcd simp: associated_def)
1366 lemma Lcm_Gcd:
1367   "Lcm A = Gcd {m. \<forall>a\<in>A. a dvd m}"
1368   by (rule LcmI[symmetric]) (auto intro: dvd_Gcd Gcd_dvd)
1370 lemma Gcd_0_iff:
1371   "Gcd A = 0 \<longleftrightarrow> A \<subseteq> {0}"
1372   apply (rule iffI)
1373   apply (rule subsetI, drule Gcd_dvd, simp)
1374   apply (auto intro: GcdI[symmetric])
1375   done
1377 lemma Gcd_empty [simp]:
1378   "Gcd {} = 0"
1379   by (simp add: Gcd_0_iff)
1381 lemma Gcd_1:
1382   "1 \<in> A \<Longrightarrow> Gcd A = 1"
1383   by (intro GcdI[symmetric]) (auto intro: Gcd_dvd dvd_Gcd)
1385 lemma Gcd_insert [simp]:
1386   "Gcd (insert a A) = gcd a (Gcd A)"
1387 proof (rule gcdI)
1388   fix l assume "l dvd a" and "l dvd Gcd A"
1389   hence "\<forall>a\<in>A. l dvd a" by (blast intro: dvd_trans Gcd_dvd)
1390   with \<open>l dvd a\<close> show "l dvd Gcd (insert a A)" by (force intro: Gcd_dvd)
1391 qed auto
1393 lemma Gcd_finite:
1394   assumes "finite A"
1395   shows "Gcd A = Finite_Set.fold gcd 0 A"
1396   by (induct rule: finite.induct[OF \<open>finite A\<close>])
1397     (simp_all add: comp_fun_idem.fold_insert_idem[OF comp_fun_idem_gcd])
1399 lemma Gcd_set [code_unfold]:
1400   "Gcd (set xs) = fold gcd xs 0"
1401   using comp_fun_idem.fold_set_fold[OF comp_fun_idem_gcd] Gcd_finite by (simp add: ac_simps)
1403 lemma Gcd_singleton [simp]: "Gcd {a} = a div normalization_factor a"
1404   by (simp add: gcd_0)
1406 lemma Gcd_2 [simp]: "Gcd {a,b} = gcd a b"
1407   by (simp only: Gcd_insert Gcd_empty gcd_0) (cases "b = 0", simp, rule gcd_div_unit2, simp)
1409 subclass semiring_gcd
1410   by unfold_locales (simp_all add: gcd_greatest_iff)
1412 end
1414 text \<open>
1415   A Euclidean ring is a Euclidean semiring with additive inverses. It provides a
1416   few more lemmas; in particular, Bezout's lemma holds for any Euclidean ring.
1417 \<close>
1419 class euclidean_ring = euclidean_semiring + idom
1420 begin
1422 function euclid_ext :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a" where
1423   "euclid_ext a b =
1424      (if b = 0 then
1425         let c = 1 div normalization_factor a in (c, 0, a * c)
1426       else if b dvd a then
1427         let c = 1 div normalization_factor b in (0, c, b * c)
1428       else
1429         case euclid_ext b (a mod b) of
1430             (s, t, c) \<Rightarrow> (t, s - t * (a div b), c))"
1431   by pat_completeness simp
1432 termination
1433   by (relation "measure (euclidean_size \<circ> snd)") (simp_all add: mod_size_less)
1435 declare euclid_ext.simps [simp del]
1437 lemma euclid_ext_0:
1438   "euclid_ext a 0 = (1 div normalization_factor a, 0, a div normalization_factor a)"
1439   by (simp add: euclid_ext.simps [of a 0])
1441 lemma euclid_ext_left_0:
1442   "euclid_ext 0 a = (0, 1 div normalization_factor a, a div normalization_factor a)"
1443   by (simp add: euclid_ext.simps [of 0 a])
1445 lemma euclid_ext_non_0:
1446   "b \<noteq> 0 \<Longrightarrow> euclid_ext a b = (case euclid_ext b (a mod b) of
1447     (s, t, c) \<Rightarrow> (t, s - t * (a div b), c))"
1448   by (cases "b dvd a")
1449     (simp_all add: euclid_ext.simps [of a b] euclid_ext.simps [of b 0])
1451 lemma euclid_ext_code [code]:
1452   "euclid_ext a b = (if b = 0 then (1 div normalization_factor a, 0, a div normalization_factor a)
1453     else let (s, t, c) = euclid_ext b (a mod b) in  (t, s - t * (a div b), c))"
1454   by (simp add: euclid_ext.simps [of a b] euclid_ext.simps [of b 0])
1456 lemma euclid_ext_correct:
1457   "case euclid_ext a b of (s, t, c) \<Rightarrow> s * a + t * b = c"
1458 proof (induct a b rule: gcd_eucl_induct)
1459   case (zero a) then show ?case
1460     by (simp add: euclid_ext_0 ac_simps)
1461 next
1462   case (mod a b)
1463   obtain s t c where stc: "euclid_ext b (a mod b) = (s,t,c)"
1464     by (cases "euclid_ext b (a mod b)") blast
1465   with mod have "c = s * b + t * (a mod b)" by simp
1466   also have "... = t * ((a div b) * b + a mod b) + (s - t * (a div b)) * b"
1467     by (simp add: algebra_simps)
1468   also have "(a div b) * b + a mod b = a" using mod_div_equality .
1469   finally show ?case
1470     by (subst euclid_ext.simps) (simp add: stc mod ac_simps)
1471 qed
1473 definition euclid_ext' :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a"
1474 where
1475   "euclid_ext' a b = (case euclid_ext a b of (s, t, _) \<Rightarrow> (s, t))"
1477 lemma euclid_ext'_0: "euclid_ext' a 0 = (1 div normalization_factor a, 0)"
1478   by (simp add: euclid_ext'_def euclid_ext_0)
1480 lemma euclid_ext'_left_0: "euclid_ext' 0 a = (0, 1 div normalization_factor a)"
1481   by (simp add: euclid_ext'_def euclid_ext_left_0)
1483 lemma euclid_ext'_non_0: "b \<noteq> 0 \<Longrightarrow> euclid_ext' a b = (snd (euclid_ext' b (a mod b)),
1484   fst (euclid_ext' b (a mod b)) - snd (euclid_ext' b (a mod b)) * (a div b))"
1485   by (simp add: euclid_ext'_def euclid_ext_non_0 split_def)
1487 end
1489 class euclidean_ring_gcd = euclidean_semiring_gcd + idom
1490 begin
1492 subclass euclidean_ring ..
1494 subclass ring_gcd ..
1496 lemma euclid_ext_gcd [simp]:
1497   "(case euclid_ext a b of (_, _ , t) \<Rightarrow> t) = gcd a b"
1498   by (induct a b rule: gcd_eucl_induct)
1499     (simp_all add: euclid_ext_0 gcd_0 euclid_ext_non_0 ac_simps split: prod.split prod.split_asm)
1501 lemma euclid_ext_gcd' [simp]:
1502   "euclid_ext a b = (r, s, t) \<Longrightarrow> t = gcd a b"
1503   by (insert euclid_ext_gcd[of a b], drule (1) subst, simp)
1505 lemma euclid_ext'_correct:
1506   "fst (euclid_ext' a b) * a + snd (euclid_ext' a b) * b = gcd a b"
1507 proof-
1508   obtain s t c where "euclid_ext a b = (s,t,c)"
1509     by (cases "euclid_ext a b", blast)
1510   with euclid_ext_correct[of a b] euclid_ext_gcd[of a b]
1511     show ?thesis unfolding euclid_ext'_def by simp
1512 qed
1514 lemma bezout: "\<exists>s t. s * a + t * b = gcd a b"
1515   using euclid_ext'_correct by blast
1517 lemma gcd_neg1 [simp]:
1518   "gcd (-a) b = gcd a b"
1519   by (rule sym, rule gcdI, simp_all add: gcd_greatest)
1521 lemma gcd_neg2 [simp]:
1522   "gcd a (-b) = gcd a b"
1523   by (rule sym, rule gcdI, simp_all add: gcd_greatest)
1525 lemma gcd_neg_numeral_1 [simp]:
1526   "gcd (- numeral n) a = gcd (numeral n) a"
1527   by (fact gcd_neg1)
1529 lemma gcd_neg_numeral_2 [simp]:
1530   "gcd a (- numeral n) = gcd a (numeral n)"
1531   by (fact gcd_neg2)
1533 lemma gcd_diff1: "gcd (m - n) n = gcd m n"
1534   by (subst diff_conv_add_uminus, subst gcd_neg2[symmetric],  subst gcd_add1, simp)
1536 lemma gcd_diff2: "gcd (n - m) n = gcd m n"
1537   by (subst gcd_neg1[symmetric], simp only: minus_diff_eq gcd_diff1)
1539 lemma coprime_minus_one [simp]: "gcd (n - 1) n = 1"
1540 proof -
1541   have "gcd (n - 1) n = gcd n (n - 1)" by (fact gcd.commute)
1542   also have "\<dots> = gcd ((n - 1) + 1) (n - 1)" by simp
1543   also have "\<dots> = 1" by (rule coprime_plus_one)
1544   finally show ?thesis .
1545 qed
1547 lemma lcm_neg1 [simp]: "lcm (-a) b = lcm a b"
1548   by (rule sym, rule lcmI, simp_all add: lcm_least lcm_zero)
1550 lemma lcm_neg2 [simp]: "lcm a (-b) = lcm a b"
1551   by (rule sym, rule lcmI, simp_all add: lcm_least lcm_zero)
1553 lemma lcm_neg_numeral_1 [simp]: "lcm (- numeral n) a = lcm (numeral n) a"
1554   by (fact lcm_neg1)
1556 lemma lcm_neg_numeral_2 [simp]: "lcm a (- numeral n) = lcm a (numeral n)"
1557   by (fact lcm_neg2)
1559 end
1562 subsection \<open>Typical instances\<close>
1564 instantiation nat :: euclidean_semiring
1565 begin
1567 definition [simp]:
1568   "euclidean_size_nat = (id :: nat \<Rightarrow> nat)"
1570 definition [simp]:
1571   "normalization_factor_nat (n::nat) = (if n = 0 then 0 else 1 :: nat)"
1573 instance proof
1574 qed simp_all
1576 end
1578 instantiation int :: euclidean_ring
1579 begin
1581 definition [simp]:
1582   "euclidean_size_int = (nat \<circ> abs :: int \<Rightarrow> nat)"
1584 definition [simp]:
1585   "normalization_factor_int = (sgn :: int \<Rightarrow> int)"
1587 instance proof
1588   case goal2 then show ?case by (auto simp add: abs_mult nat_mult_distrib)
1589 next
1590   case goal3 then show ?case by (simp add: zsgn_def)
1591 next
1592   case goal5 then show ?case by (auto simp: zsgn_def)
1593 next
1594   case goal6 then show ?case by (auto split: abs_split simp: zsgn_def)
1595 qed (auto simp: sgn_times split: abs_split)
1597 end
1599 instantiation poly :: (field) euclidean_ring
1600 begin
1602 definition euclidean_size_poly :: "'a poly \<Rightarrow> nat"
1603   where "euclidean_size = (degree :: 'a poly \<Rightarrow> nat)"
1605 definition normalization_factor_poly :: "'a poly \<Rightarrow> 'a poly"
1606   where "normalization_factor p = monom (coeff p (degree p)) 0"
1608 instance
1609 proof (default, unfold euclidean_size_poly_def normalization_factor_poly_def)
1610   fix p q :: "'a poly"
1611   assume "q \<noteq> 0" and "\<not> q dvd p"
1612   then show "degree (p mod q) < degree q"
1613     using degree_mod_less [of q p] by (simp add: mod_eq_0_iff_dvd)
1614 next
1615   fix p q :: "'a poly"
1616   assume "q \<noteq> 0"
1617   from \<open>q \<noteq> 0\<close> show "degree p \<le> degree (p * q)"
1618     by (rule degree_mult_right_le)
1619   from \<open>q \<noteq> 0\<close> show "is_unit (monom (coeff q (degree q)) 0)"
1620     by (auto intro: is_unit_monom_0)
1621 next
1622   fix p :: "'a poly"
1623   show "monom (coeff p (degree p)) 0 = p" if "is_unit p"
1624     using that by (fact is_unit_monom_trival)
1625 next
1626   fix p q :: "'a poly"
1627   show "monom (coeff (p * q) (degree (p * q))) 0 =
1628     monom (coeff p (degree p)) 0 * monom (coeff q (degree q)) 0"
1629     by (simp add: monom_0 coeff_degree_mult)
1630 next
1631   show "monom (coeff 0 (degree 0)) 0 = 0"
1632     by simp
1633 qed
1635 end
1637 end