src/HOL/Number_Theory/Euclidean_Algorithm.thy
 author haftmann Fri Jun 12 08:53:23 2015 +0200 (2015-06-12) changeset 60432 68d75cff8809 parent 60431 db9c67b760f1 child 60433 720f210c5b1d permissions -rw-r--r--
given up trivial definition
1 (* Author: Manuel Eberl *)
3 section {* Abstract euclidean algorithm *}
5 theory Euclidean_Algorithm
6 imports Complex_Main
7 begin
9 context semiring_div
10 begin
12 abbreviation is_unit :: "'a \<Rightarrow> bool"
13 where
14   "is_unit a \<equiv> a dvd 1"
16 definition associated :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
17 where
18   "associated a b \<longleftrightarrow> a dvd b \<and> b dvd a"
20 lemma unit_prod [intro]:
21   "is_unit a \<Longrightarrow> is_unit b \<Longrightarrow> is_unit (a * b)"
22   by (subst mult_1_left [of 1, symmetric], rule mult_dvd_mono)
24 lemma unit_divide_1:
25   "is_unit b \<Longrightarrow> a div b = a * divide 1 b"
26   by (simp add: div_mult_swap)
28 lemma unit_divide_1_divide_1 [simp]:
29   "is_unit a \<Longrightarrow> divide 1 (divide 1 a) = a"
30   by (metis div_mult_mult1_if div_mult_self1_is_id dvd_mult_div_cancel mult_1_right)
32 lemma inv_imp_eq_divide_1:
33   "a * b = 1 \<Longrightarrow> divide 1 a = b"
34   by (metis dvd_mult_div_cancel dvd_mult_right mult_1_right mult.left_commute one_dvd)
36 lemma unit_divide_1_unit [simp, intro]:
37   assumes "is_unit a"
38   shows "is_unit (divide 1 a)"
39 proof -
40   from assms have "1 = divide 1 a * a" by simp
41   then show "is_unit (divide 1 a)" by (rule dvdI)
42 qed
44 lemma mult_unit_dvd_iff:
45   "is_unit b \<Longrightarrow> a * b dvd c \<longleftrightarrow> a dvd c"
46 proof
47   assume "is_unit b" "a * b dvd c"
48   then show "a dvd c" by (simp add: dvd_mult_left)
49 next
50   assume "is_unit b" "a dvd c"
51   then obtain k where "c = a * k" unfolding dvd_def by blast
52   with is_unit b have "c = (a * b) * (divide 1 b * k)"
53       by (simp add: mult_ac)
54   then show "a * b dvd c" by (rule dvdI)
55 qed
57 lemma div_unit_dvd_iff:
58   "is_unit b \<Longrightarrow> a div b dvd c \<longleftrightarrow> a dvd c"
59   by (subst unit_divide_1) (assumption, simp add: mult_unit_dvd_iff)
61 lemma dvd_mult_unit_iff:
62   "is_unit b \<Longrightarrow> a dvd c * b \<longleftrightarrow> a dvd c"
63 proof
64   assume "is_unit b" and "a dvd c * b"
65   have "c * b dvd c * (b * divide 1 b)" by (subst mult_assoc [symmetric]) simp
66   also from is_unit b have "b * divide 1 b = 1" by simp
67   finally have "c * b dvd c" by simp
68   with a dvd c * b show "a dvd c" by (rule dvd_trans)
69 next
70   assume "a dvd c"
71   then show "a dvd c * b" by simp
72 qed
74 lemma dvd_div_unit_iff:
75   "is_unit b \<Longrightarrow> a dvd c div b \<longleftrightarrow> a dvd c"
76   by (subst unit_divide_1) (assumption, simp add: dvd_mult_unit_iff)
78 lemmas unit_dvd_iff = mult_unit_dvd_iff div_unit_dvd_iff dvd_mult_unit_iff dvd_div_unit_iff
80 lemma unit_div [intro]:
81   "is_unit a \<Longrightarrow> is_unit b \<Longrightarrow> is_unit (a div b)"
82   by (subst unit_divide_1) (assumption, rule unit_prod, simp_all)
84 lemma unit_div_mult_swap:
85   "is_unit c \<Longrightarrow> a * (b div c) = a * b div c"
86   by (simp only: unit_divide_1 [of _ b] unit_divide_1 [of _ "a*b"] ac_simps)
88 lemma unit_div_commute:
89   "is_unit b \<Longrightarrow> a div b * c = a * c div b"
90   by (simp only: unit_divide_1 [of _ a] unit_divide_1 [of _ "a*c"] ac_simps)
92 lemma unit_imp_dvd [dest]:
93   "is_unit b \<Longrightarrow> b dvd a"
94   by (rule dvd_trans [of _ 1]) simp_all
96 lemma dvd_unit_imp_unit:
97   "is_unit b \<Longrightarrow> a dvd b \<Longrightarrow> is_unit a"
98   by (rule dvd_trans)
100 lemma unit_divide_1'1:
101   assumes "is_unit b"
102   shows "a div (b * c) = a * divide 1 b div c"
103 proof -
104   from assms have "a div (b * c) = a * (divide 1 b * b) div (b * c)"
105     by simp
106   also have "... = b * (a * divide 1 b) div (b * c)"
107     by (simp only: mult_ac)
108   also have "... = a * divide 1 b div c"
109     by (cases "b = 0", simp, rule div_mult_mult1)
110   finally show ?thesis .
111 qed
113 lemma associated_comm:
114   "associated a b \<Longrightarrow> associated b a"
115   by (simp add: associated_def)
117 lemma associated_0 [simp]:
118   "associated 0 b \<longleftrightarrow> b = 0"
119   "associated a 0 \<longleftrightarrow> a = 0"
120   unfolding associated_def by simp_all
122 lemma associated_unit:
123   "is_unit a \<Longrightarrow> associated a b \<Longrightarrow> is_unit b"
124   unfolding associated_def using dvd_unit_imp_unit by auto
126 lemma is_unit_1 [simp]:
127   "is_unit 1"
128   by simp
130 lemma not_is_unit_0 [simp]:
131   "\<not> is_unit 0"
132   by auto
134 lemma unit_mult_left_cancel:
135   assumes "is_unit a"
136   shows "(a * b) = (a * c) \<longleftrightarrow> b = c"
137 proof -
138   from assms have "a \<noteq> 0" by auto
139   then show ?thesis by (metis div_mult_self1_is_id)
140 qed
142 lemma unit_mult_right_cancel:
143   "is_unit a \<Longrightarrow> (b * a) = (c * a) \<longleftrightarrow> b = c"
144   by (simp add: ac_simps unit_mult_left_cancel)
146 lemma unit_div_cancel:
147   "is_unit a \<Longrightarrow> (b div a) = (c div a) \<longleftrightarrow> b = c"
148   apply (subst unit_divide_1[of _ b], assumption)
149   apply (subst unit_divide_1[of _ c], assumption)
150   apply (rule unit_mult_right_cancel, erule unit_divide_1_unit)
151   done
153 lemma unit_eq_div1:
154   "is_unit b \<Longrightarrow> a div b = c \<longleftrightarrow> a = c * b"
155   apply (subst unit_divide_1, assumption)
156   apply (subst unit_mult_right_cancel[symmetric], assumption)
157   apply (subst mult_assoc, subst dvd_div_mult_self, assumption, simp)
158   done
160 lemma unit_eq_div2:
161   "is_unit b \<Longrightarrow> a = c div b \<longleftrightarrow> a * b = c"
162   by (subst (1 2) eq_commute, simp add: unit_eq_div1, subst eq_commute, rule refl)
164 lemma associated_iff_div_unit:
165   "associated a b \<longleftrightarrow> (\<exists>c. is_unit c \<and> a = c * b)"
166 proof
167   assume "associated a b"
168   show "\<exists>c. is_unit c \<and> a = c * b"
169   proof (cases "a = 0")
170     assume "a = 0"
171     then show "\<exists>c. is_unit c \<and> a = c * b" using associated a b
172         by (intro exI[of _ 1], simp add: associated_def)
173   next
174     assume [simp]: "a \<noteq> 0"
175     hence [simp]: "a dvd b" "b dvd a" using associated a b
176         unfolding associated_def by simp_all
177     hence "1 = a div b * (b div a)"
178       by (simp add: div_mult_swap)
179     hence "is_unit (a div b)" ..
180     moreover have "a = (a div b) * b" by simp
181     ultimately show ?thesis by blast
182   qed
183 next
184   assume "\<exists>c. is_unit c \<and> a = c * b"
185   then obtain c where "is_unit c" and "a = c * b" by blast
186   hence "b = a * divide 1 c" by (simp add: algebra_simps)
187   hence "a dvd b" by simp
188   moreover from a = c * b have "b dvd a" by simp
189   ultimately show "associated a b" unfolding associated_def by simp
190 qed
192 lemmas unit_simps = mult_unit_dvd_iff div_unit_dvd_iff dvd_mult_unit_iff
193   dvd_div_unit_iff unit_div_mult_swap unit_div_commute
194   unit_mult_left_cancel unit_mult_right_cancel unit_div_cancel
195   unit_eq_div1 unit_eq_div2
197 end
199 context ring_div
200 begin
202 lemma is_unit_neg [simp]:
203   "is_unit (- a) \<Longrightarrow> is_unit a"
204   by simp
206 lemma is_unit_neg_1 [simp]:
207   "is_unit (-1)"
208   by simp
210 end
212 lemma is_unit_nat [simp]:
213   "is_unit (a::nat) \<longleftrightarrow> a = 1"
214   by simp
216 lemma is_unit_int:
217   "is_unit (a::int) \<longleftrightarrow> a = 1 \<or> a = -1"
218   by auto
220 text {*
221   A Euclidean semiring is a semiring upon which the Euclidean algorithm can be
222   implemented. It must provide:
223   \begin{itemize}
224   \item division with remainder
225   \item a size function such that @{term "size (a mod b) < size b"}
226         for any @{term "b \<noteq> 0"}
227   \item a normalisation factor such that two associated numbers are equal iff
228         they are the same when divided by their normalisation factors.
229   \end{itemize}
230   The existence of these functions makes it possible to derive gcd and lcm functions
231   for any Euclidean semiring.
232 *}
233 class euclidean_semiring = semiring_div +
234   fixes euclidean_size :: "'a \<Rightarrow> nat"
235   fixes normalisation_factor :: "'a \<Rightarrow> 'a"
236   assumes mod_size_less [simp]:
237     "b \<noteq> 0 \<Longrightarrow> euclidean_size (a mod b) < euclidean_size b"
238   assumes size_mult_mono:
239     "b \<noteq> 0 \<Longrightarrow> euclidean_size (a * b) \<ge> euclidean_size a"
240   assumes normalisation_factor_is_unit [intro,simp]:
241     "a \<noteq> 0 \<Longrightarrow> is_unit (normalisation_factor a)"
242   assumes normalisation_factor_mult: "normalisation_factor (a * b) =
243     normalisation_factor a * normalisation_factor b"
244   assumes normalisation_factor_unit: "is_unit a \<Longrightarrow> normalisation_factor a = a"
245   assumes normalisation_factor_0 [simp]: "normalisation_factor 0 = 0"
246 begin
248 lemma normalisation_factor_dvd [simp]:
249   "a \<noteq> 0 \<Longrightarrow> normalisation_factor a dvd b"
250   by (rule unit_imp_dvd, simp)
252 lemma normalisation_factor_1 [simp]:
253   "normalisation_factor 1 = 1"
254   by (simp add: normalisation_factor_unit)
256 lemma normalisation_factor_0_iff [simp]:
257   "normalisation_factor a = 0 \<longleftrightarrow> a = 0"
258 proof
259   assume "normalisation_factor a = 0"
260   hence "\<not> is_unit (normalisation_factor a)"
261     by (metis not_is_unit_0)
262   then show "a = 0" by force
263 next
264   assume "a = 0"
265   then show "normalisation_factor a = 0" by simp
266 qed
268 lemma normalisation_factor_pow:
269   "normalisation_factor (a ^ n) = normalisation_factor a ^ n"
270   by (induct n) (simp_all add: normalisation_factor_mult power_Suc2)
272 lemma normalisation_correct [simp]:
273   "normalisation_factor (a div normalisation_factor a) = (if a = 0 then 0 else 1)"
274 proof (cases "a = 0", simp)
275   assume "a \<noteq> 0"
276   let ?nf = "normalisation_factor"
277   from normalisation_factor_is_unit[OF a \<noteq> 0] have "?nf a \<noteq> 0"
278     by (metis not_is_unit_0)
279   have "?nf (a div ?nf a) * ?nf (?nf a) = ?nf (a div ?nf a * ?nf a)"
280     by (simp add: normalisation_factor_mult)
281   also have "a div ?nf a * ?nf a = a" using a \<noteq> 0
282     by simp
283   also have "?nf (?nf a) = ?nf a" using a \<noteq> 0
284     normalisation_factor_is_unit normalisation_factor_unit by simp
285   finally show ?thesis using a \<noteq> 0 and ?nf a \<noteq> 0
286     by (metis div_mult_self2_is_id div_self)
287 qed
289 lemma normalisation_0_iff [simp]:
290   "a div normalisation_factor a = 0 \<longleftrightarrow> a = 0"
291   by (cases "a = 0", simp, subst unit_eq_div1, blast, simp)
293 lemma associated_iff_normed_eq:
294   "associated a b \<longleftrightarrow> a div normalisation_factor a = b div normalisation_factor b"
295 proof (cases "b = 0", simp, cases "a = 0", metis associated_0(1) normalisation_0_iff, rule iffI)
296   let ?nf = normalisation_factor
297   assume "a \<noteq> 0" "b \<noteq> 0" "a div ?nf a = b div ?nf b"
298   hence "a = b * (?nf a div ?nf b)"
299     apply (subst (asm) unit_eq_div1, blast, subst (asm) unit_div_commute, blast)
300     apply (subst div_mult_swap, simp, simp)
301     done
302   with a \<noteq> 0 b \<noteq> 0 have "\<exists>c. is_unit c \<and> a = c * b"
303     by (intro exI[of _ "?nf a div ?nf b"], force simp: mult_ac)
304   with associated_iff_div_unit show "associated a b" by simp
305 next
306   let ?nf = normalisation_factor
307   assume "a \<noteq> 0" "b \<noteq> 0" "associated a b"
308   with associated_iff_div_unit obtain c where "is_unit c" and "a = c * b" by blast
309   then show "a div ?nf a = b div ?nf b"
310     apply (simp only: a = c * b normalisation_factor_mult normalisation_factor_unit)
311     apply (rule div_mult_mult1, force)
312     done
313   qed
315 lemma normed_associated_imp_eq:
316   "associated a b \<Longrightarrow> normalisation_factor a \<in> {0, 1} \<Longrightarrow> normalisation_factor b \<in> {0, 1} \<Longrightarrow> a = b"
317   by (simp add: associated_iff_normed_eq, elim disjE, simp_all)
319 lemmas normalisation_factor_dvd_iff [simp] =
320   unit_dvd_iff [OF normalisation_factor_is_unit]
322 lemma euclidean_division:
323   fixes a :: 'a and b :: 'a
324   assumes "b \<noteq> 0"
325   obtains s and t where "a = s * b + t"
326     and "euclidean_size t < euclidean_size b"
327 proof -
328   from div_mod_equality[of a b 0]
329      have "a = a div b * b + a mod b" by simp
330   with that and assms show ?thesis by force
331 qed
333 lemma dvd_euclidean_size_eq_imp_dvd:
334   assumes "a \<noteq> 0" and b_dvd_a: "b dvd a" and size_eq: "euclidean_size a = euclidean_size b"
335   shows "a dvd b"
336 proof (subst dvd_eq_mod_eq_0, rule ccontr)
337   assume "b mod a \<noteq> 0"
338   from b_dvd_a have b_dvd_mod: "b dvd b mod a" by (simp add: dvd_mod_iff)
339   from b_dvd_mod obtain c where "b mod a = b * c" unfolding dvd_def by blast
340     with b mod a \<noteq> 0 have "c \<noteq> 0" by auto
341   with b mod a = b * c have "euclidean_size (b mod a) \<ge> euclidean_size b"
342       using size_mult_mono by force
343   moreover from a \<noteq> 0 have "euclidean_size (b mod a) < euclidean_size a"
344       using mod_size_less by blast
345   ultimately show False using size_eq by simp
346 qed
348 function gcd_eucl :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
349 where
350   "gcd_eucl a b = (if b = 0 then a div normalisation_factor a else gcd_eucl b (a mod b))"
351   by (pat_completeness, simp)
352 termination by (relation "measure (euclidean_size \<circ> snd)", simp_all)
354 declare gcd_eucl.simps [simp del]
356 lemma gcd_induct: "\<lbrakk>\<And>b. P b 0; \<And>a b. 0 \<noteq> b \<Longrightarrow> P b (a mod b) \<Longrightarrow> P a b\<rbrakk> \<Longrightarrow> P a b"
357 proof (induct a b rule: gcd_eucl.induct)
358   case ("1" m n)
359     then show ?case by (cases "n = 0") auto
360 qed
362 definition lcm_eucl :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
363 where
364   "lcm_eucl a b = a * b div (gcd_eucl a b * normalisation_factor (a * b))"
366   (* Somewhat complicated definition of Lcm that has the advantage of working
367      for infinite sets as well *)
369 definition Lcm_eucl :: "'a set \<Rightarrow> 'a"
370 where
371   "Lcm_eucl A = (if \<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) then
372      let l = SOME l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l =
373        (LEAST n. \<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n)
374        in l div normalisation_factor l
375       else 0)"
377 definition Gcd_eucl :: "'a set \<Rightarrow> 'a"
378 where
379   "Gcd_eucl A = Lcm_eucl {d. \<forall>a\<in>A. d dvd a}"
381 end
383 class euclidean_semiring_gcd = euclidean_semiring + gcd + Gcd +
384   assumes gcd_gcd_eucl: "gcd = gcd_eucl" and lcm_lcm_eucl: "lcm = lcm_eucl"
385   assumes Gcd_Gcd_eucl: "Gcd = Gcd_eucl" and Lcm_Lcm_eucl: "Lcm = Lcm_eucl"
386 begin
388 lemma gcd_red:
389   "gcd a b = gcd b (a mod b)"
390   by (metis gcd_eucl.simps mod_0 mod_by_0 gcd_gcd_eucl)
392 lemma gcd_non_0:
393   "b \<noteq> 0 \<Longrightarrow> gcd a b = gcd b (a mod b)"
394   by (rule gcd_red)
396 lemma gcd_0_left:
397   "gcd 0 a = a div normalisation_factor a"
398    by (simp only: gcd_gcd_eucl, subst gcd_eucl.simps, subst gcd_eucl.simps, simp add: Let_def)
400 lemma gcd_0:
401   "gcd a 0 = a div normalisation_factor a"
402   by (simp only: gcd_gcd_eucl, subst gcd_eucl.simps, simp add: Let_def)
404 lemma gcd_dvd1 [iff]: "gcd a b dvd a"
405   and gcd_dvd2 [iff]: "gcd a b dvd b"
406 proof (induct a b rule: gcd_eucl.induct)
407   fix a b :: 'a
408   assume IH1: "b \<noteq> 0 \<Longrightarrow> gcd b (a mod b) dvd b"
409   assume IH2: "b \<noteq> 0 \<Longrightarrow> gcd b (a mod b) dvd (a mod b)"
411   have "gcd a b dvd a \<and> gcd a b dvd b"
412   proof (cases "b = 0")
413     case True
414       then show ?thesis by (cases "a = 0", simp_all add: gcd_0)
415   next
416     case False
417       with IH1 and IH2 show ?thesis by (simp add: gcd_non_0 dvd_mod_iff)
418   qed
419   then show "gcd a b dvd a" "gcd a b dvd b" by simp_all
420 qed
422 lemma dvd_gcd_D1: "k dvd gcd m n \<Longrightarrow> k dvd m"
423   by (rule dvd_trans, assumption, rule gcd_dvd1)
425 lemma dvd_gcd_D2: "k dvd gcd m n \<Longrightarrow> k dvd n"
426   by (rule dvd_trans, assumption, rule gcd_dvd2)
428 lemma gcd_greatest:
429   fixes k a b :: 'a
430   shows "k dvd a \<Longrightarrow> k dvd b \<Longrightarrow> k dvd gcd a b"
431 proof (induct a b rule: gcd_eucl.induct)
432   case (1 a b)
433   show ?case
434     proof (cases "b = 0")
435       assume "b = 0"
436       with 1 show ?thesis by (cases "a = 0", simp_all add: gcd_0)
437     next
438       assume "b \<noteq> 0"
439       with 1 show ?thesis by (simp add: gcd_non_0 dvd_mod_iff)
440     qed
441 qed
443 lemma dvd_gcd_iff:
444   "k dvd gcd a b \<longleftrightarrow> k dvd a \<and> k dvd b"
445   by (blast intro!: gcd_greatest intro: dvd_trans)
447 lemmas gcd_greatest_iff = dvd_gcd_iff
449 lemma gcd_zero [simp]:
450   "gcd a b = 0 \<longleftrightarrow> a = 0 \<and> b = 0"
451   by (metis dvd_0_left dvd_refl gcd_dvd1 gcd_dvd2 gcd_greatest)+
453 lemma normalisation_factor_gcd [simp]:
454   "normalisation_factor (gcd a b) = (if a = 0 \<and> b = 0 then 0 else 1)" (is "?f a b = ?g a b")
455 proof (induct a b rule: gcd_eucl.induct)
456   fix a b :: 'a
457   assume IH: "b \<noteq> 0 \<Longrightarrow> ?f b (a mod b) = ?g b (a mod b)"
458   then show "?f a b = ?g a b" by (cases "b = 0", auto simp: gcd_non_0 gcd_0)
459 qed
461 lemma gcdI:
462   "k dvd a \<Longrightarrow> k dvd b \<Longrightarrow> (\<And>l. l dvd a \<Longrightarrow> l dvd b \<Longrightarrow> l dvd k)
463     \<Longrightarrow> normalisation_factor k = (if k = 0 then 0 else 1) \<Longrightarrow> k = gcd a b"
464   by (intro normed_associated_imp_eq) (auto simp: associated_def intro: gcd_greatest)
466 sublocale gcd!: abel_semigroup gcd
467 proof
468   fix a b c
469   show "gcd (gcd a b) c = gcd a (gcd b c)"
470   proof (rule gcdI)
471     have "gcd (gcd a b) c dvd gcd a b" "gcd a b dvd a" by simp_all
472     then show "gcd (gcd a b) c dvd a" by (rule dvd_trans)
473     have "gcd (gcd a b) c dvd gcd a b" "gcd a b dvd b" by simp_all
474     hence "gcd (gcd a b) c dvd b" by (rule dvd_trans)
475     moreover have "gcd (gcd a b) c dvd c" by simp
476     ultimately show "gcd (gcd a b) c dvd gcd b c"
477       by (rule gcd_greatest)
478     show "normalisation_factor (gcd (gcd a b) c) =  (if gcd (gcd a b) c = 0 then 0 else 1)"
479       by auto
480     fix l assume "l dvd a" and "l dvd gcd b c"
481     with dvd_trans[OF _ gcd_dvd1] and dvd_trans[OF _ gcd_dvd2]
482       have "l dvd b" and "l dvd c" by blast+
483     with l dvd a show "l dvd gcd (gcd a b) c"
484       by (intro gcd_greatest)
485   qed
486 next
487   fix a b
488   show "gcd a b = gcd b a"
489     by (rule gcdI) (simp_all add: gcd_greatest)
490 qed
492 lemma gcd_unique: "d dvd a \<and> d dvd b \<and>
493     normalisation_factor d = (if d = 0 then 0 else 1) \<and>
494     (\<forall>e. e dvd a \<and> e dvd b \<longrightarrow> e dvd d) \<longleftrightarrow> d = gcd a b"
495   by (rule, auto intro: gcdI simp: gcd_greatest)
497 lemma gcd_dvd_prod: "gcd a b dvd k * b"
498   using mult_dvd_mono [of 1] by auto
500 lemma gcd_1_left [simp]: "gcd 1 a = 1"
501   by (rule sym, rule gcdI, simp_all)
503 lemma gcd_1 [simp]: "gcd a 1 = 1"
504   by (rule sym, rule gcdI, simp_all)
506 lemma gcd_proj2_if_dvd:
507   "b dvd a \<Longrightarrow> gcd a b = b div normalisation_factor b"
508   by (cases "b = 0", simp_all add: dvd_eq_mod_eq_0 gcd_non_0 gcd_0)
510 lemma gcd_proj1_if_dvd:
511   "a dvd b \<Longrightarrow> gcd a b = a div normalisation_factor a"
512   by (subst gcd.commute, simp add: gcd_proj2_if_dvd)
514 lemma gcd_proj1_iff: "gcd m n = m div normalisation_factor m \<longleftrightarrow> m dvd n"
515 proof
516   assume A: "gcd m n = m div normalisation_factor m"
517   show "m dvd n"
518   proof (cases "m = 0")
519     assume [simp]: "m \<noteq> 0"
520     from A have B: "m = gcd m n * normalisation_factor m"
521       by (simp add: unit_eq_div2)
522     show ?thesis by (subst B, simp add: mult_unit_dvd_iff)
523   qed (insert A, simp)
524 next
525   assume "m dvd n"
526   then show "gcd m n = m div normalisation_factor m" by (rule gcd_proj1_if_dvd)
527 qed
529 lemma gcd_proj2_iff: "gcd m n = n div normalisation_factor n \<longleftrightarrow> n dvd m"
530   by (subst gcd.commute, simp add: gcd_proj1_iff)
532 lemma gcd_mod1 [simp]:
533   "gcd (a mod b) b = gcd a b"
534   by (rule gcdI, metis dvd_mod_iff gcd_dvd1 gcd_dvd2, simp_all add: gcd_greatest dvd_mod_iff)
536 lemma gcd_mod2 [simp]:
537   "gcd a (b mod a) = gcd a b"
538   by (rule gcdI, simp, metis dvd_mod_iff gcd_dvd1 gcd_dvd2, simp_all add: gcd_greatest dvd_mod_iff)
540 lemma normalisation_factor_dvd' [simp]:
541   "normalisation_factor a dvd a"
542   by (cases "a = 0", simp_all)
544 lemma gcd_mult_distrib':
545   "k div normalisation_factor k * gcd a b = gcd (k*a) (k*b)"
546 proof (induct a b rule: gcd_eucl.induct)
547   case (1 a b)
548   show ?case
549   proof (cases "b = 0")
550     case True
551     then show ?thesis by (simp add: normalisation_factor_mult gcd_0 algebra_simps div_mult_div_if_dvd)
552   next
553     case False
554     hence "k div normalisation_factor k * gcd a b =  gcd (k * b) (k * (a mod b))"
555       using 1 by (subst gcd_red, simp)
556     also have "... = gcd (k * a) (k * b)"
557       by (simp add: mult_mod_right gcd.commute)
558     finally show ?thesis .
559   qed
560 qed
562 lemma gcd_mult_distrib:
563   "k * gcd a b = gcd (k*a) (k*b) * normalisation_factor k"
564 proof-
565   let ?nf = "normalisation_factor"
566   from gcd_mult_distrib'
567     have "gcd (k*a) (k*b) = k div ?nf k * gcd a b" ..
568   also have "... = k * gcd a b div ?nf k"
569     by (metis dvd_div_mult dvd_eq_mod_eq_0 mod_0 normalisation_factor_dvd)
570   finally show ?thesis
571     by simp
572 qed
574 lemma euclidean_size_gcd_le1 [simp]:
575   assumes "a \<noteq> 0"
576   shows "euclidean_size (gcd a b) \<le> euclidean_size a"
577 proof -
578    have "gcd a b dvd a" by (rule gcd_dvd1)
579    then obtain c where A: "a = gcd a b * c" unfolding dvd_def by blast
580    with a \<noteq> 0 show ?thesis by (subst (2) A, intro size_mult_mono) auto
581 qed
583 lemma euclidean_size_gcd_le2 [simp]:
584   "b \<noteq> 0 \<Longrightarrow> euclidean_size (gcd a b) \<le> euclidean_size b"
585   by (subst gcd.commute, rule euclidean_size_gcd_le1)
587 lemma euclidean_size_gcd_less1:
588   assumes "a \<noteq> 0" and "\<not>a dvd b"
589   shows "euclidean_size (gcd a b) < euclidean_size a"
590 proof (rule ccontr)
591   assume "\<not>euclidean_size (gcd a b) < euclidean_size a"
592   with a \<noteq> 0 have "euclidean_size (gcd a b) = euclidean_size a"
593     by (intro le_antisym, simp_all)
594   with assms have "a dvd gcd a b" by (auto intro: dvd_euclidean_size_eq_imp_dvd)
595   hence "a dvd b" using dvd_gcd_D2 by blast
596   with \<not>a dvd b show False by contradiction
597 qed
599 lemma euclidean_size_gcd_less2:
600   assumes "b \<noteq> 0" and "\<not>b dvd a"
601   shows "euclidean_size (gcd a b) < euclidean_size b"
602   using assms by (subst gcd.commute, rule euclidean_size_gcd_less1)
604 lemma gcd_mult_unit1: "is_unit a \<Longrightarrow> gcd (b * a) c = gcd b c"
605   apply (rule gcdI)
606   apply (rule dvd_trans, rule gcd_dvd1, simp add: unit_simps)
607   apply (rule gcd_dvd2)
608   apply (rule gcd_greatest, simp add: unit_simps, assumption)
609   apply (subst normalisation_factor_gcd, simp add: gcd_0)
610   done
612 lemma gcd_mult_unit2: "is_unit a \<Longrightarrow> gcd b (c * a) = gcd b c"
613   by (subst gcd.commute, subst gcd_mult_unit1, assumption, rule gcd.commute)
615 lemma gcd_div_unit1: "is_unit a \<Longrightarrow> gcd (b div a) c = gcd b c"
616   by (subst unit_divide_1) (simp_all add: gcd_mult_unit1)
618 lemma gcd_div_unit2: "is_unit a \<Longrightarrow> gcd b (c div a) = gcd b c"
619   by (subst unit_divide_1) (simp_all add: gcd_mult_unit2)
621 lemma gcd_idem: "gcd a a = a div normalisation_factor a"
622   by (cases "a = 0") (simp add: gcd_0_left, rule sym, rule gcdI, simp_all)
624 lemma gcd_right_idem: "gcd (gcd a b) b = gcd a b"
625   apply (rule gcdI)
626   apply (simp add: ac_simps)
627   apply (rule gcd_dvd2)
628   apply (rule gcd_greatest, erule (1) gcd_greatest, assumption)
629   apply simp
630   done
632 lemma gcd_left_idem: "gcd a (gcd a b) = gcd a b"
633   apply (rule gcdI)
634   apply simp
635   apply (rule dvd_trans, rule gcd_dvd2, rule gcd_dvd2)
636   apply (rule gcd_greatest, assumption, erule gcd_greatest, assumption)
637   apply simp
638   done
640 lemma comp_fun_idem_gcd: "comp_fun_idem gcd"
641 proof
642   fix a b show "gcd a \<circ> gcd b = gcd b \<circ> gcd a"
643     by (simp add: fun_eq_iff ac_simps)
644 next
645   fix a show "gcd a \<circ> gcd a = gcd a"
646     by (simp add: fun_eq_iff gcd_left_idem)
647 qed
649 lemma coprime_dvd_mult:
650   assumes "gcd c b = 1" and "c dvd a * b"
651   shows "c dvd a"
652 proof -
653   let ?nf = "normalisation_factor"
654   from assms gcd_mult_distrib [of a c b]
655     have A: "a = gcd (a * c) (a * b) * ?nf a" by simp
656   from c dvd a * b show ?thesis by (subst A, simp_all add: gcd_greatest)
657 qed
659 lemma coprime_dvd_mult_iff:
660   "gcd c b = 1 \<Longrightarrow> (c dvd a * b) = (c dvd a)"
661   by (rule, rule coprime_dvd_mult, simp_all)
663 lemma gcd_dvd_antisym:
664   "gcd a b dvd gcd c d \<Longrightarrow> gcd c d dvd gcd a b \<Longrightarrow> gcd a b = gcd c d"
665 proof (rule gcdI)
666   assume A: "gcd a b dvd gcd c d" and B: "gcd c d dvd gcd a b"
667   have "gcd c d dvd c" by simp
668   with A show "gcd a b dvd c" by (rule dvd_trans)
669   have "gcd c d dvd d" by simp
670   with A show "gcd a b dvd d" by (rule dvd_trans)
671   show "normalisation_factor (gcd a b) = (if gcd a b = 0 then 0 else 1)"
672     by simp
673   fix l assume "l dvd c" and "l dvd d"
674   hence "l dvd gcd c d" by (rule gcd_greatest)
675   from this and B show "l dvd gcd a b" by (rule dvd_trans)
676 qed
678 lemma gcd_mult_cancel:
679   assumes "gcd k n = 1"
680   shows "gcd (k * m) n = gcd m n"
681 proof (rule gcd_dvd_antisym)
682   have "gcd (gcd (k * m) n) k = gcd (gcd k n) (k * m)" by (simp add: ac_simps)
683   also note gcd k n = 1
684   finally have "gcd (gcd (k * m) n) k = 1" by simp
685   hence "gcd (k * m) n dvd m" by (rule coprime_dvd_mult, simp add: ac_simps)
686   moreover have "gcd (k * m) n dvd n" by simp
687   ultimately show "gcd (k * m) n dvd gcd m n" by (rule gcd_greatest)
688   have "gcd m n dvd (k * m)" and "gcd m n dvd n" by simp_all
689   then show "gcd m n dvd gcd (k * m) n" by (rule gcd_greatest)
690 qed
692 lemma coprime_crossproduct:
693   assumes [simp]: "gcd a d = 1" "gcd b c = 1"
694   shows "associated (a * c) (b * d) \<longleftrightarrow> associated a b \<and> associated c d" (is "?lhs \<longleftrightarrow> ?rhs")
695 proof
696   assume ?rhs then show ?lhs unfolding associated_def by (fast intro: mult_dvd_mono)
697 next
698   assume ?lhs
699   from ?lhs have "a dvd b * d" unfolding associated_def by (metis dvd_mult_left)
700   hence "a dvd b" by (simp add: coprime_dvd_mult_iff)
701   moreover from ?lhs have "b dvd a * c" unfolding associated_def by (metis dvd_mult_left)
702   hence "b dvd a" by (simp add: coprime_dvd_mult_iff)
703   moreover from ?lhs have "c dvd d * b"
704     unfolding associated_def by (auto dest: dvd_mult_right simp add: ac_simps)
705   hence "c dvd d" by (simp add: coprime_dvd_mult_iff gcd.commute)
706   moreover from ?lhs have "d dvd c * a"
707     unfolding associated_def by (auto dest: dvd_mult_right simp add: ac_simps)
708   hence "d dvd c" by (simp add: coprime_dvd_mult_iff gcd.commute)
709   ultimately show ?rhs unfolding associated_def by simp
710 qed
712 lemma gcd_add1 [simp]:
713   "gcd (m + n) n = gcd m n"
714   by (cases "n = 0", simp_all add: gcd_non_0)
716 lemma gcd_add2 [simp]:
717   "gcd m (m + n) = gcd m n"
718   using gcd_add1 [of n m] by (simp add: ac_simps)
720 lemma gcd_add_mult: "gcd m (k * m + n) = gcd m n"
721   by (subst gcd.commute, subst gcd_red, simp)
723 lemma coprimeI: "(\<And>l. \<lbrakk>l dvd a; l dvd b\<rbrakk> \<Longrightarrow> l dvd 1) \<Longrightarrow> gcd a b = 1"
724   by (rule sym, rule gcdI, simp_all)
726 lemma coprime: "gcd a b = 1 \<longleftrightarrow> (\<forall>d. d dvd a \<and> d dvd b \<longleftrightarrow> is_unit d)"
727   by (auto intro: coprimeI gcd_greatest dvd_gcd_D1 dvd_gcd_D2)
729 lemma div_gcd_coprime:
730   assumes nz: "a \<noteq> 0 \<or> b \<noteq> 0"
731   defines [simp]: "d \<equiv> gcd a b"
732   defines [simp]: "a' \<equiv> a div d" and [simp]: "b' \<equiv> b div d"
733   shows "gcd a' b' = 1"
734 proof (rule coprimeI)
735   fix l assume "l dvd a'" "l dvd b'"
736   then obtain s t where "a' = l * s" "b' = l * t" unfolding dvd_def by blast
737   moreover have "a = a' * d" "b = b' * d" by simp_all
738   ultimately have "a = (l * d) * s" "b = (l * d) * t"
739     by (simp_all only: ac_simps)
740   hence "l*d dvd a" and "l*d dvd b" by (simp_all only: dvd_triv_left)
741   hence "l*d dvd d" by (simp add: gcd_greatest)
742   then obtain u where "d = l * d * u" ..
743   then have "d * (l * u) = d" by (simp add: ac_simps)
744   moreover from nz have "d \<noteq> 0" by simp
745   with div_mult_self1_is_id have "d * (l * u) div d = l * u" .
746   ultimately have "1 = l * u"
747     using d \<noteq> 0 by simp
748   then show "l dvd 1" ..
749 qed
751 lemma coprime_mult:
752   assumes da: "gcd d a = 1" and db: "gcd d b = 1"
753   shows "gcd d (a * b) = 1"
754   apply (subst gcd.commute)
755   using da apply (subst gcd_mult_cancel)
756   apply (subst gcd.commute, assumption)
757   apply (subst gcd.commute, rule db)
758   done
760 lemma coprime_lmult:
761   assumes dab: "gcd d (a * b) = 1"
762   shows "gcd d a = 1"
763 proof (rule coprimeI)
764   fix l assume "l dvd d" and "l dvd a"
765   hence "l dvd a * b" by simp
766   with l dvd d and dab show "l dvd 1" by (auto intro: gcd_greatest)
767 qed
769 lemma coprime_rmult:
770   assumes dab: "gcd d (a * b) = 1"
771   shows "gcd d b = 1"
772 proof (rule coprimeI)
773   fix l assume "l dvd d" and "l dvd b"
774   hence "l dvd a * b" by simp
775   with l dvd d and dab show "l dvd 1" by (auto intro: gcd_greatest)
776 qed
778 lemma coprime_mul_eq: "gcd d (a * b) = 1 \<longleftrightarrow> gcd d a = 1 \<and> gcd d b = 1"
779   using coprime_rmult[of d a b] coprime_lmult[of d a b] coprime_mult[of d a b] by blast
781 lemma gcd_coprime:
782   assumes c: "gcd a b \<noteq> 0" and a: "a = a' * gcd a b" and b: "b = b' * gcd a b"
783   shows "gcd a' b' = 1"
784 proof -
785   from c have "a \<noteq> 0 \<or> b \<noteq> 0" by simp
786   with div_gcd_coprime have "gcd (a div gcd a b) (b div gcd a b) = 1" .
787   also from assms have "a div gcd a b = a'" by (metis div_mult_self2_is_id)+
788   also from assms have "b div gcd a b = b'" by (metis div_mult_self2_is_id)+
789   finally show ?thesis .
790 qed
792 lemma coprime_power:
793   assumes "0 < n"
794   shows "gcd a (b ^ n) = 1 \<longleftrightarrow> gcd a b = 1"
795 using assms proof (induct n)
796   case (Suc n) then show ?case
797     by (cases n) (simp_all add: coprime_mul_eq)
798 qed simp
800 lemma gcd_coprime_exists:
801   assumes nz: "gcd a b \<noteq> 0"
802   shows "\<exists>a' b'. a = a' * gcd a b \<and> b = b' * gcd a b \<and> gcd a' b' = 1"
803   apply (rule_tac x = "a div gcd a b" in exI)
804   apply (rule_tac x = "b div gcd a b" in exI)
805   apply (insert nz, auto intro: div_gcd_coprime)
806   done
808 lemma coprime_exp:
809   "gcd d a = 1 \<Longrightarrow> gcd d (a^n) = 1"
810   by (induct n, simp_all add: coprime_mult)
812 lemma coprime_exp2 [intro]:
813   "gcd a b = 1 \<Longrightarrow> gcd (a^n) (b^m) = 1"
814   apply (rule coprime_exp)
815   apply (subst gcd.commute)
816   apply (rule coprime_exp)
817   apply (subst gcd.commute)
818   apply assumption
819   done
821 lemma gcd_exp:
822   "gcd (a^n) (b^n) = (gcd a b) ^ n"
823 proof (cases "a = 0 \<and> b = 0")
824   assume "a = 0 \<and> b = 0"
825   then show ?thesis by (cases n, simp_all add: gcd_0_left)
826 next
827   assume A: "\<not>(a = 0 \<and> b = 0)"
828   hence "1 = gcd ((a div gcd a b)^n) ((b div gcd a b)^n)"
829     using div_gcd_coprime by (subst sym, auto simp: div_gcd_coprime)
830   hence "(gcd a b) ^ n = (gcd a b) ^ n * ..." by simp
831   also note gcd_mult_distrib
832   also have "normalisation_factor ((gcd a b)^n) = 1"
833     by (simp add: normalisation_factor_pow A)
834   also have "(gcd a b)^n * (a div gcd a b)^n = a^n"
835     by (subst ac_simps, subst div_power, simp, rule dvd_div_mult_self, rule dvd_power_same, simp)
836   also have "(gcd a b)^n * (b div gcd a b)^n = b^n"
837     by (subst ac_simps, subst div_power, simp, rule dvd_div_mult_self, rule dvd_power_same, simp)
838   finally show ?thesis by simp
839 qed
841 lemma coprime_common_divisor:
842   "gcd a b = 1 \<Longrightarrow> a dvd a \<Longrightarrow> a dvd b \<Longrightarrow> is_unit a"
843   apply (subgoal_tac "a dvd gcd a b")
844   apply simp
845   apply (erule (1) gcd_greatest)
846   done
848 lemma division_decomp:
849   assumes dc: "a dvd b * c"
850   shows "\<exists>b' c'. a = b' * c' \<and> b' dvd b \<and> c' dvd c"
851 proof (cases "gcd a b = 0")
852   assume "gcd a b = 0"
853   hence "a = 0 \<and> b = 0" by simp
854   hence "a = 0 * c \<and> 0 dvd b \<and> c dvd c" by simp
855   then show ?thesis by blast
856 next
857   let ?d = "gcd a b"
858   assume "?d \<noteq> 0"
859   from gcd_coprime_exists[OF this]
860     obtain a' b' where ab': "a = a' * ?d" "b = b' * ?d" "gcd a' b' = 1"
861     by blast
862   from ab'(1) have "a' dvd a" unfolding dvd_def by blast
863   with dc have "a' dvd b*c" using dvd_trans[of a' a "b*c"] by simp
864   from dc ab'(1,2) have "a'*?d dvd (b'*?d) * c" by simp
865   hence "?d * a' dvd ?d * (b' * c)" by (simp add: mult_ac)
866   with ?d \<noteq> 0 have "a' dvd b' * c" by simp
867   with coprime_dvd_mult[OF ab'(3)]
868     have "a' dvd c" by (subst (asm) ac_simps, blast)
869   with ab'(1) have "a = ?d * a' \<and> ?d dvd b \<and> a' dvd c" by (simp add: mult_ac)
870   then show ?thesis by blast
871 qed
873 lemma pow_divides_pow:
874   assumes ab: "a ^ n dvd b ^ n" and n: "n \<noteq> 0"
875   shows "a dvd b"
876 proof (cases "gcd a b = 0")
877   assume "gcd a b = 0"
878   then show ?thesis by simp
879 next
880   let ?d = "gcd a b"
881   assume "?d \<noteq> 0"
882   from n obtain m where m: "n = Suc m" by (cases n, simp_all)
883   from ?d \<noteq> 0 have zn: "?d ^ n \<noteq> 0" by (rule power_not_zero)
884   from gcd_coprime_exists[OF ?d \<noteq> 0]
885     obtain a' b' where ab': "a = a' * ?d" "b = b' * ?d" "gcd a' b' = 1"
886     by blast
887   from ab have "(a' * ?d) ^ n dvd (b' * ?d) ^ n"
888     by (simp add: ab'(1,2)[symmetric])
889   hence "?d^n * a'^n dvd ?d^n * b'^n"
890     by (simp only: power_mult_distrib ac_simps)
891   with zn have "a'^n dvd b'^n" by simp
892   hence "a' dvd b'^n" using dvd_trans[of a' "a'^n" "b'^n"] by (simp add: m)
893   hence "a' dvd b'^m * b'" by (simp add: m ac_simps)
894   with coprime_dvd_mult[OF coprime_exp[OF ab'(3), of m]]
895     have "a' dvd b'" by (subst (asm) ac_simps, blast)
896   hence "a'*?d dvd b'*?d" by (rule mult_dvd_mono, simp)
897   with ab'(1,2) show ?thesis by simp
898 qed
900 lemma pow_divides_eq [simp]:
901   "n \<noteq> 0 \<Longrightarrow> a ^ n dvd b ^ n \<longleftrightarrow> a dvd b"
902   by (auto intro: pow_divides_pow dvd_power_same)
904 lemma divides_mult:
905   assumes mr: "m dvd r" and nr: "n dvd r" and mn: "gcd m n = 1"
906   shows "m * n dvd r"
907 proof -
908   from mr nr obtain m' n' where m': "r = m*m'" and n': "r = n*n'"
909     unfolding dvd_def by blast
910   from mr n' have "m dvd n'*n" by (simp add: ac_simps)
911   hence "m dvd n'" using coprime_dvd_mult_iff[OF mn] by simp
912   then obtain k where k: "n' = m*k" unfolding dvd_def by blast
913   with n' have "r = m * n * k" by (simp add: mult_ac)
914   then show ?thesis unfolding dvd_def by blast
915 qed
917 lemma coprime_plus_one [simp]: "gcd (n + 1) n = 1"
918   by (subst add_commute, simp)
920 lemma setprod_coprime [rule_format]:
921   "(\<forall>i\<in>A. gcd (f i) a = 1) \<longrightarrow> gcd (\<Prod>i\<in>A. f i) a = 1"
922   apply (cases "finite A")
923   apply (induct set: finite)
924   apply (auto simp add: gcd_mult_cancel)
925   done
927 lemma coprime_divisors:
928   assumes "d dvd a" "e dvd b" "gcd a b = 1"
929   shows "gcd d e = 1"
930 proof -
931   from assms obtain k l where "a = d * k" "b = e * l"
932     unfolding dvd_def by blast
933   with assms have "gcd (d * k) (e * l) = 1" by simp
934   hence "gcd (d * k) e = 1" by (rule coprime_lmult)
935   also have "gcd (d * k) e = gcd e (d * k)" by (simp add: ac_simps)
936   finally have "gcd e d = 1" by (rule coprime_lmult)
937   then show ?thesis by (simp add: ac_simps)
938 qed
940 lemma invertible_coprime:
941   assumes "a * b mod m = 1"
942   shows "coprime a m"
943 proof -
944   from assms have "coprime m (a * b mod m)"
945     by simp
946   then have "coprime m (a * b)"
947     by simp
948   then have "coprime m a"
949     by (rule coprime_lmult)
950   then show ?thesis
951     by (simp add: ac_simps)
952 qed
954 lemma lcm_gcd:
955   "lcm a b = a * b div (gcd a b * normalisation_factor (a*b))"
956   by (simp only: lcm_lcm_eucl gcd_gcd_eucl lcm_eucl_def)
958 lemma lcm_gcd_prod:
959   "lcm a b * gcd a b = a * b div normalisation_factor (a*b)"
960 proof (cases "a * b = 0")
961   let ?nf = normalisation_factor
962   assume "a * b \<noteq> 0"
963   hence "gcd a b \<noteq> 0" by simp
964   from lcm_gcd have "lcm a b * gcd a b = gcd a b * (a * b div (?nf (a*b) * gcd a b))"
965     by (simp add: mult_ac)
966   also from a * b \<noteq> 0 have "... = a * b div ?nf (a*b)"
967     by (simp add: div_mult_swap mult.commute)
968   finally show ?thesis .
969 qed (auto simp add: lcm_gcd)
971 lemma lcm_dvd1 [iff]:
972   "a dvd lcm a b"
973 proof (cases "a*b = 0")
974   assume "a * b \<noteq> 0"
975   hence "gcd a b \<noteq> 0" by simp
976   let ?c = "divide 1 (normalisation_factor (a*b))"
977   from a * b \<noteq> 0 have [simp]: "is_unit (normalisation_factor (a*b))" by simp
978   from lcm_gcd_prod[of a b] have "lcm a b * gcd a b = a * ?c * b"
979     by (simp add: div_mult_swap unit_div_commute)
980   hence "lcm a b * gcd a b div gcd a b = a * ?c * b div gcd a b" by simp
981   with gcd a b \<noteq> 0 have "lcm a b = a * ?c * b div gcd a b"
982     by (subst (asm) div_mult_self2_is_id, simp_all)
983   also have "... = a * (?c * b div gcd a b)"
984     by (metis div_mult_swap gcd_dvd2 mult_assoc)
985   finally show ?thesis by (rule dvdI)
986 qed (auto simp add: lcm_gcd)
988 lemma lcm_least:
989   "\<lbrakk>a dvd k; b dvd k\<rbrakk> \<Longrightarrow> lcm a b dvd k"
990 proof (cases "k = 0")
991   let ?nf = normalisation_factor
992   assume "k \<noteq> 0"
993   hence "is_unit (?nf k)" by simp
994   hence "?nf k \<noteq> 0" by (metis not_is_unit_0)
995   assume A: "a dvd k" "b dvd k"
996   hence "gcd a b \<noteq> 0" using k \<noteq> 0 by auto
997   from A obtain r s where ar: "k = a * r" and bs: "k = b * s"
998     unfolding dvd_def by blast
999   with k \<noteq> 0 have "r * s \<noteq> 0"
1000     by auto (drule sym [of 0], simp)
1001   hence "is_unit (?nf (r * s))" by simp
1002   let ?c = "?nf k div ?nf (r*s)"
1003   from is_unit (?nf k) and is_unit (?nf (r * s)) have "is_unit ?c" by (rule unit_div)
1004   hence "?c \<noteq> 0" using not_is_unit_0 by fast
1005   from ar bs have "k * k * gcd s r = ?nf k * k * gcd (k * s) (k * r)"
1006     by (subst mult_assoc, subst gcd_mult_distrib[of k s r], simp only: ac_simps)
1007   also have "... = ?nf k * k * gcd ((r*s) * a) ((r*s) * b)"
1008     by (subst (3) k = a * r, subst (3) k = b * s, simp add: algebra_simps)
1009   also have "... = ?c * r*s * k * gcd a b" using r * s \<noteq> 0
1010     by (subst gcd_mult_distrib'[symmetric], simp add: algebra_simps unit_simps)
1011   finally have "(a*r) * (b*s) * gcd s r = ?c * k * r * s * gcd a b"
1012     by (subst ar[symmetric], subst bs[symmetric], simp add: mult_ac)
1013   hence "a * b * gcd s r * (r * s) = ?c * k * gcd a b * (r * s)"
1014     by (simp add: algebra_simps)
1015   hence "?c * k * gcd a b = a * b * gcd s r" using r * s \<noteq> 0
1016     by (metis div_mult_self2_is_id)
1017   also have "... = lcm a b * gcd a b * gcd s r * ?nf (a*b)"
1018     by (subst lcm_gcd_prod[of a b], metis gcd_mult_distrib gcd_mult_distrib')
1019   also have "... = lcm a b * gcd s r * ?nf (a*b) * gcd a b"
1020     by (simp add: algebra_simps)
1021   finally have "k * ?c = lcm a b * gcd s r * ?nf (a*b)" using gcd a b \<noteq> 0
1022     by (metis mult.commute div_mult_self2_is_id)
1023   hence "k = lcm a b * (gcd s r * ?nf (a*b)) div ?c" using ?c \<noteq> 0
1024     by (metis div_mult_self2_is_id mult_assoc)
1025   also have "... = lcm a b * (gcd s r * ?nf (a*b) div ?c)" using is_unit ?c
1026     by (simp add: unit_simps)
1027   finally show ?thesis by (rule dvdI)
1028 qed simp
1030 lemma lcm_zero:
1031   "lcm a b = 0 \<longleftrightarrow> a = 0 \<or> b = 0"
1032 proof -
1033   let ?nf = normalisation_factor
1034   {
1035     assume "a \<noteq> 0" "b \<noteq> 0"
1036     hence "a * b div ?nf (a * b) \<noteq> 0" by (simp add: no_zero_divisors)
1037     moreover from a \<noteq> 0 and b \<noteq> 0 have "gcd a b \<noteq> 0" by simp
1038     ultimately have "lcm a b \<noteq> 0" using lcm_gcd_prod[of a b] by (intro notI, simp)
1039   } moreover {
1040     assume "a = 0 \<or> b = 0"
1041     hence "lcm a b = 0" by (elim disjE, simp_all add: lcm_gcd)
1042   }
1043   ultimately show ?thesis by blast
1044 qed
1046 lemmas lcm_0_iff = lcm_zero
1048 lemma gcd_lcm:
1049   assumes "lcm a b \<noteq> 0"
1050   shows "gcd a b = a * b div (lcm a b * normalisation_factor (a * b))"
1051 proof-
1052   from assms have "gcd a b \<noteq> 0" by (simp add: lcm_zero)
1053   let ?c = "normalisation_factor (a*b)"
1054   from lcm a b \<noteq> 0 have "?c \<noteq> 0" by (intro notI, simp add: lcm_zero no_zero_divisors)
1055   hence "is_unit ?c" by simp
1056   from lcm_gcd_prod [of a b] have "gcd a b = a * b div ?c div lcm a b"
1057     by (subst (2) div_mult_self2_is_id[OF lcm a b \<noteq> 0, symmetric], simp add: mult_ac)
1058   also from is_unit ?c have "... = a * b div (?c * lcm a b)"
1059     by (metis local.unit_divide_1 local.unit_divide_1'1)
1060   finally show ?thesis by (simp only: ac_simps)
1061 qed
1063 lemma normalisation_factor_lcm [simp]:
1064   "normalisation_factor (lcm a b) = (if a = 0 \<or> b = 0 then 0 else 1)"
1065 proof (cases "a = 0 \<or> b = 0")
1066   case True then show ?thesis
1067     by (auto simp add: lcm_gcd)
1068 next
1069   case False
1070   let ?nf = normalisation_factor
1071   from lcm_gcd_prod[of a b]
1072     have "?nf (lcm a b) * ?nf (gcd a b) = ?nf (a*b) div ?nf (a*b)"
1073     by (metis div_by_0 div_self normalisation_correct normalisation_factor_0 normalisation_factor_mult)
1074   also have "... = (if a*b = 0 then 0 else 1)"
1075     by simp
1076   finally show ?thesis using False by simp
1077 qed
1079 lemma lcm_dvd2 [iff]: "b dvd lcm a b"
1080   using lcm_dvd1 [of b a] by (simp add: lcm_gcd ac_simps)
1082 lemma lcmI:
1083   "\<lbrakk>a dvd k; b dvd k; \<And>l. a dvd l \<Longrightarrow> b dvd l \<Longrightarrow> k dvd l;
1084     normalisation_factor k = (if k = 0 then 0 else 1)\<rbrakk> \<Longrightarrow> k = lcm a b"
1085   by (intro normed_associated_imp_eq) (auto simp: associated_def intro: lcm_least)
1087 sublocale lcm!: abel_semigroup lcm
1088 proof
1089   fix a b c
1090   show "lcm (lcm a b) c = lcm a (lcm b c)"
1091   proof (rule lcmI)
1092     have "a dvd lcm a b" and "lcm a b dvd lcm (lcm a b) c" by simp_all
1093     then show "a dvd lcm (lcm a b) c" by (rule dvd_trans)
1095     have "b dvd lcm a b" and "lcm a b dvd lcm (lcm a b) c" by simp_all
1096     hence "b dvd lcm (lcm a b) c" by (rule dvd_trans)
1097     moreover have "c dvd lcm (lcm a b) c" by simp
1098     ultimately show "lcm b c dvd lcm (lcm a b) c" by (rule lcm_least)
1100     fix l assume "a dvd l" and "lcm b c dvd l"
1101     have "b dvd lcm b c" by simp
1102     from this and lcm b c dvd l have "b dvd l" by (rule dvd_trans)
1103     have "c dvd lcm b c" by simp
1104     from this and lcm b c dvd l have "c dvd l" by (rule dvd_trans)
1105     from a dvd l and b dvd l have "lcm a b dvd l" by (rule lcm_least)
1106     from this and c dvd l show "lcm (lcm a b) c dvd l" by (rule lcm_least)
1107   qed (simp add: lcm_zero)
1108 next
1109   fix a b
1110   show "lcm a b = lcm b a"
1111     by (simp add: lcm_gcd ac_simps)
1112 qed
1114 lemma dvd_lcm_D1:
1115   "lcm m n dvd k \<Longrightarrow> m dvd k"
1116   by (rule dvd_trans, rule lcm_dvd1, assumption)
1118 lemma dvd_lcm_D2:
1119   "lcm m n dvd k \<Longrightarrow> n dvd k"
1120   by (rule dvd_trans, rule lcm_dvd2, assumption)
1122 lemma gcd_dvd_lcm [simp]:
1123   "gcd a b dvd lcm a b"
1124   by (metis dvd_trans gcd_dvd2 lcm_dvd2)
1126 lemma lcm_1_iff:
1127   "lcm a b = 1 \<longleftrightarrow> is_unit a \<and> is_unit b"
1128 proof
1129   assume "lcm a b = 1"
1130   then show "is_unit a \<and> is_unit b" by auto
1131 next
1132   assume "is_unit a \<and> is_unit b"
1133   hence "a dvd 1" and "b dvd 1" by simp_all
1134   hence "is_unit (lcm a b)" by (rule lcm_least)
1135   hence "lcm a b = normalisation_factor (lcm a b)"
1136     by (subst normalisation_factor_unit, simp_all)
1137   also have "\<dots> = 1" using is_unit a \<and> is_unit b
1138     by auto
1139   finally show "lcm a b = 1" .
1140 qed
1142 lemma lcm_0_left [simp]:
1143   "lcm 0 a = 0"
1144   by (rule sym, rule lcmI, simp_all)
1146 lemma lcm_0 [simp]:
1147   "lcm a 0 = 0"
1148   by (rule sym, rule lcmI, simp_all)
1150 lemma lcm_unique:
1151   "a dvd d \<and> b dvd d \<and>
1152   normalisation_factor d = (if d = 0 then 0 else 1) \<and>
1153   (\<forall>e. a dvd e \<and> b dvd e \<longrightarrow> d dvd e) \<longleftrightarrow> d = lcm a b"
1154   by (rule, auto intro: lcmI simp: lcm_least lcm_zero)
1156 lemma dvd_lcm_I1 [simp]:
1157   "k dvd m \<Longrightarrow> k dvd lcm m n"
1158   by (metis lcm_dvd1 dvd_trans)
1160 lemma dvd_lcm_I2 [simp]:
1161   "k dvd n \<Longrightarrow> k dvd lcm m n"
1162   by (metis lcm_dvd2 dvd_trans)
1164 lemma lcm_1_left [simp]:
1165   "lcm 1 a = a div normalisation_factor a"
1166   by (cases "a = 0") (simp, rule sym, rule lcmI, simp_all)
1168 lemma lcm_1_right [simp]:
1169   "lcm a 1 = a div normalisation_factor a"
1170   using lcm_1_left [of a] by (simp add: ac_simps)
1172 lemma lcm_coprime:
1173   "gcd a b = 1 \<Longrightarrow> lcm a b = a * b div normalisation_factor (a*b)"
1174   by (subst lcm_gcd) simp
1176 lemma lcm_proj1_if_dvd:
1177   "b dvd a \<Longrightarrow> lcm a b = a div normalisation_factor a"
1178   by (cases "a = 0") (simp, rule sym, rule lcmI, simp_all)
1180 lemma lcm_proj2_if_dvd:
1181   "a dvd b \<Longrightarrow> lcm a b = b div normalisation_factor b"
1182   using lcm_proj1_if_dvd [of a b] by (simp add: ac_simps)
1184 lemma lcm_proj1_iff:
1185   "lcm m n = m div normalisation_factor m \<longleftrightarrow> n dvd m"
1186 proof
1187   assume A: "lcm m n = m div normalisation_factor m"
1188   show "n dvd m"
1189   proof (cases "m = 0")
1190     assume [simp]: "m \<noteq> 0"
1191     from A have B: "m = lcm m n * normalisation_factor m"
1192       by (simp add: unit_eq_div2)
1193     show ?thesis by (subst B, simp)
1194   qed simp
1195 next
1196   assume "n dvd m"
1197   then show "lcm m n = m div normalisation_factor m" by (rule lcm_proj1_if_dvd)
1198 qed
1200 lemma lcm_proj2_iff:
1201   "lcm m n = n div normalisation_factor n \<longleftrightarrow> m dvd n"
1202   using lcm_proj1_iff [of n m] by (simp add: ac_simps)
1204 lemma euclidean_size_lcm_le1:
1205   assumes "a \<noteq> 0" and "b \<noteq> 0"
1206   shows "euclidean_size a \<le> euclidean_size (lcm a b)"
1207 proof -
1208   have "a dvd lcm a b" by (rule lcm_dvd1)
1209   then obtain c where A: "lcm a b = a * c" unfolding dvd_def by blast
1210   with a \<noteq> 0 and b \<noteq> 0 have "c \<noteq> 0" by (auto simp: lcm_zero)
1211   then show ?thesis by (subst A, intro size_mult_mono)
1212 qed
1214 lemma euclidean_size_lcm_le2:
1215   "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> euclidean_size b \<le> euclidean_size (lcm a b)"
1216   using euclidean_size_lcm_le1 [of b a] by (simp add: ac_simps)
1218 lemma euclidean_size_lcm_less1:
1219   assumes "b \<noteq> 0" and "\<not>b dvd a"
1220   shows "euclidean_size a < euclidean_size (lcm a b)"
1221 proof (rule ccontr)
1222   from assms have "a \<noteq> 0" by auto
1223   assume "\<not>euclidean_size a < euclidean_size (lcm a b)"
1224   with a \<noteq> 0 and b \<noteq> 0 have "euclidean_size (lcm a b) = euclidean_size a"
1225     by (intro le_antisym, simp, intro euclidean_size_lcm_le1)
1226   with assms have "lcm a b dvd a"
1227     by (rule_tac dvd_euclidean_size_eq_imp_dvd) (auto simp: lcm_zero)
1228   hence "b dvd a" by (rule dvd_lcm_D2)
1229   with \<not>b dvd a show False by contradiction
1230 qed
1232 lemma euclidean_size_lcm_less2:
1233   assumes "a \<noteq> 0" and "\<not>a dvd b"
1234   shows "euclidean_size b < euclidean_size (lcm a b)"
1235   using assms euclidean_size_lcm_less1 [of a b] by (simp add: ac_simps)
1237 lemma lcm_mult_unit1:
1238   "is_unit a \<Longrightarrow> lcm (b * a) c = lcm b c"
1239   apply (rule lcmI)
1240   apply (rule dvd_trans[of _ "b * a"], simp, rule lcm_dvd1)
1241   apply (rule lcm_dvd2)
1242   apply (rule lcm_least, simp add: unit_simps, assumption)
1243   apply (subst normalisation_factor_lcm, simp add: lcm_zero)
1244   done
1246 lemma lcm_mult_unit2:
1247   "is_unit a \<Longrightarrow> lcm b (c * a) = lcm b c"
1248   using lcm_mult_unit1 [of a c b] by (simp add: ac_simps)
1250 lemma lcm_div_unit1:
1251   "is_unit a \<Longrightarrow> lcm (b div a) c = lcm b c"
1252   by (metis lcm_mult_unit1 local.unit_divide_1 local.unit_divide_1_unit)
1254 lemma lcm_div_unit2:
1255   "is_unit a \<Longrightarrow> lcm b (c div a) = lcm b c"
1256   by (metis lcm_mult_unit2 local.unit_divide_1 local.unit_divide_1_unit)
1258 lemma lcm_left_idem:
1259   "lcm a (lcm a b) = lcm a b"
1260   apply (rule lcmI)
1261   apply simp
1262   apply (subst lcm.assoc [symmetric], rule lcm_dvd2)
1263   apply (rule lcm_least, assumption)
1264   apply (erule (1) lcm_least)
1265   apply (auto simp: lcm_zero)
1266   done
1268 lemma lcm_right_idem:
1269   "lcm (lcm a b) b = lcm a b"
1270   apply (rule lcmI)
1271   apply (subst lcm.assoc, rule lcm_dvd1)
1272   apply (rule lcm_dvd2)
1273   apply (rule lcm_least, erule (1) lcm_least, assumption)
1274   apply (auto simp: lcm_zero)
1275   done
1277 lemma comp_fun_idem_lcm: "comp_fun_idem lcm"
1278 proof
1279   fix a b show "lcm a \<circ> lcm b = lcm b \<circ> lcm a"
1280     by (simp add: fun_eq_iff ac_simps)
1281 next
1282   fix a show "lcm a \<circ> lcm a = lcm a" unfolding o_def
1283     by (intro ext, simp add: lcm_left_idem)
1284 qed
1286 lemma dvd_Lcm [simp]: "a \<in> A \<Longrightarrow> a dvd Lcm A"
1287   and Lcm_dvd [simp]: "(\<forall>a\<in>A. a dvd l') \<Longrightarrow> Lcm A dvd l'"
1288   and normalisation_factor_Lcm [simp]:
1289           "normalisation_factor (Lcm A) = (if Lcm A = 0 then 0 else 1)"
1290 proof -
1291   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>
1292     normalisation_factor (Lcm A) = (if Lcm A = 0 then 0 else 1)" (is ?thesis)
1293   proof (cases "\<exists>l. l \<noteq>  0 \<and> (\<forall>a\<in>A. a dvd l)")
1294     case False
1295     hence "Lcm A = 0" by (auto simp: Lcm_Lcm_eucl Lcm_eucl_def)
1296     with False show ?thesis by auto
1297   next
1298     case True
1299     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
1300     def n \<equiv> "LEAST n. \<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"
1301     def l \<equiv> "SOME l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"
1302     have "\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"
1303       apply (subst n_def)
1304       apply (rule LeastI[of _ "euclidean_size l\<^sub>0"])
1305       apply (rule exI[of _ l\<^sub>0])
1306       apply (simp add: l\<^sub>0_props)
1307       done
1308     from someI_ex[OF this] have "l \<noteq> 0" and "\<forall>a\<in>A. a dvd l" and "euclidean_size l = n"
1309       unfolding l_def by simp_all
1310     {
1311       fix l' assume "\<forall>a\<in>A. a dvd l'"
1312       with \<forall>a\<in>A. a dvd l have "\<forall>a\<in>A. a dvd gcd l l'" by (auto intro: gcd_greatest)
1313       moreover from l \<noteq> 0 have "gcd l l' \<noteq> 0" by simp
1314       ultimately have "\<exists>b. b \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd b) \<and> euclidean_size b = euclidean_size (gcd l l')"
1315         by (intro exI[of _ "gcd l l'"], auto)
1316       hence "euclidean_size (gcd l l') \<ge> n" by (subst n_def) (rule Least_le)
1317       moreover have "euclidean_size (gcd l l') \<le> n"
1318       proof -
1319         have "gcd l l' dvd l" by simp
1320         then obtain a where "l = gcd l l' * a" unfolding dvd_def by blast
1321         with l \<noteq> 0 have "a \<noteq> 0" by auto
1322         hence "euclidean_size (gcd l l') \<le> euclidean_size (gcd l l' * a)"
1323           by (rule size_mult_mono)
1324         also have "gcd l l' * a = l" using l = gcd l l' * a ..
1325         also note euclidean_size l = n
1326         finally show "euclidean_size (gcd l l') \<le> n" .
1327       qed
1328       ultimately have "euclidean_size l = euclidean_size (gcd l l')"
1329         by (intro le_antisym, simp_all add: euclidean_size l = n)
1330       with l \<noteq> 0 have "l dvd gcd l l'" by (blast intro: dvd_euclidean_size_eq_imp_dvd)
1331       hence "l dvd l'" by (blast dest: dvd_gcd_D2)
1332     }
1334     with (\<forall>a\<in>A. a dvd l) and normalisation_factor_is_unit[OF l \<noteq> 0] and l \<noteq> 0
1335       have "(\<forall>a\<in>A. a dvd l div normalisation_factor l) \<and>
1336         (\<forall>l'. (\<forall>a\<in>A. a dvd l') \<longrightarrow> l div normalisation_factor l dvd l') \<and>
1337         normalisation_factor (l div normalisation_factor l) =
1338         (if l div normalisation_factor l = 0 then 0 else 1)"
1339       by (auto simp: unit_simps)
1340     also from True have "l div normalisation_factor l = Lcm A"
1341       by (simp add: Lcm_Lcm_eucl Lcm_eucl_def Let_def n_def l_def)
1342     finally show ?thesis .
1343   qed
1344   note A = this
1346   {fix a assume "a \<in> A" then show "a dvd Lcm A" using A by blast}
1347   {fix l' assume "\<forall>a\<in>A. a dvd l'" then show "Lcm A dvd l'" using A by blast}
1348   from A show "normalisation_factor (Lcm A) = (if Lcm A = 0 then 0 else 1)" by blast
1349 qed
1351 lemma LcmI:
1352   "(\<And>a. a\<in>A \<Longrightarrow> a dvd l) \<Longrightarrow> (\<And>l'. (\<forall>a\<in>A. a dvd l') \<Longrightarrow> l dvd l') \<Longrightarrow>
1353       normalisation_factor l = (if l = 0 then 0 else 1) \<Longrightarrow> l = Lcm A"
1354   by (intro normed_associated_imp_eq)
1355     (auto intro: Lcm_dvd dvd_Lcm simp: associated_def)
1357 lemma Lcm_subset:
1358   "A \<subseteq> B \<Longrightarrow> Lcm A dvd Lcm B"
1359   by (blast intro: Lcm_dvd dvd_Lcm)
1361 lemma Lcm_Un:
1362   "Lcm (A \<union> B) = lcm (Lcm A) (Lcm B)"
1363   apply (rule lcmI)
1364   apply (blast intro: Lcm_subset)
1365   apply (blast intro: Lcm_subset)
1366   apply (intro Lcm_dvd ballI, elim UnE)
1367   apply (rule dvd_trans, erule dvd_Lcm, assumption)
1368   apply (rule dvd_trans, erule dvd_Lcm, assumption)
1369   apply simp
1370   done
1372 lemma Lcm_1_iff:
1373   "Lcm A = 1 \<longleftrightarrow> (\<forall>a\<in>A. is_unit a)"
1374 proof
1375   assume "Lcm A = 1"
1376   then show "\<forall>a\<in>A. is_unit a" by auto
1377 qed (rule LcmI [symmetric], auto)
1379 lemma Lcm_no_units:
1380   "Lcm A = Lcm (A - {a. is_unit a})"
1381 proof -
1382   have "(A - {a. is_unit a}) \<union> {a\<in>A. is_unit a} = A" by blast
1383   hence "Lcm A = lcm (Lcm (A - {a. is_unit a})) (Lcm {a\<in>A. is_unit a})"
1384     by (simp add: Lcm_Un[symmetric])
1385   also have "Lcm {a\<in>A. is_unit a} = 1" by (simp add: Lcm_1_iff)
1386   finally show ?thesis by simp
1387 qed
1389 lemma Lcm_empty [simp]:
1390   "Lcm {} = 1"
1391   by (simp add: Lcm_1_iff)
1393 lemma Lcm_eq_0 [simp]:
1394   "0 \<in> A \<Longrightarrow> Lcm A = 0"
1395   by (drule dvd_Lcm) simp
1397 lemma Lcm0_iff':
1398   "Lcm A = 0 \<longleftrightarrow> \<not>(\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l))"
1399 proof
1400   assume "Lcm A = 0"
1401   show "\<not>(\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l))"
1402   proof
1403     assume ex: "\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l)"
1404     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
1405     def n \<equiv> "LEAST n. \<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"
1406     def l \<equiv> "SOME l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"
1407     have "\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"
1408       apply (subst n_def)
1409       apply (rule LeastI[of _ "euclidean_size l\<^sub>0"])
1410       apply (rule exI[of _ l\<^sub>0])
1411       apply (simp add: l\<^sub>0_props)
1412       done
1413     from someI_ex[OF this] have "l \<noteq> 0" unfolding l_def by simp_all
1414     hence "l div normalisation_factor l \<noteq> 0" by simp
1415     also from ex have "l div normalisation_factor l = Lcm A"
1416        by (simp only: Lcm_Lcm_eucl Lcm_eucl_def n_def l_def if_True Let_def)
1417     finally show False using Lcm A = 0 by contradiction
1418   qed
1419 qed (simp only: Lcm_Lcm_eucl Lcm_eucl_def if_False)
1421 lemma Lcm0_iff [simp]:
1422   "finite A \<Longrightarrow> Lcm A = 0 \<longleftrightarrow> 0 \<in> A"
1423 proof -
1424   assume "finite A"
1425   have "0 \<in> A \<Longrightarrow> Lcm A = 0"  by (intro dvd_0_left dvd_Lcm)
1426   moreover {
1427     assume "0 \<notin> A"
1428     hence "\<Prod>A \<noteq> 0"
1429       apply (induct rule: finite_induct[OF finite A])
1430       apply simp
1431       apply (subst setprod.insert, assumption, assumption)
1432       apply (rule no_zero_divisors)
1433       apply blast+
1434       done
1435     moreover from finite A have "\<forall>a\<in>A. a dvd \<Prod>A" by blast
1436     ultimately have "\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l)" by blast
1437     with Lcm0_iff' have "Lcm A \<noteq> 0" by simp
1438   }
1439   ultimately show "Lcm A = 0 \<longleftrightarrow> 0 \<in> A" by blast
1440 qed
1442 lemma Lcm_no_multiple:
1443   "(\<forall>m. m \<noteq> 0 \<longrightarrow> (\<exists>a\<in>A. \<not>a dvd m)) \<Longrightarrow> Lcm A = 0"
1444 proof -
1445   assume "\<forall>m. m \<noteq> 0 \<longrightarrow> (\<exists>a\<in>A. \<not>a dvd m)"
1446   hence "\<not>(\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l))" by blast
1447   then show "Lcm A = 0" by (simp only: Lcm_Lcm_eucl Lcm_eucl_def if_False)
1448 qed
1450 lemma Lcm_insert [simp]:
1451   "Lcm (insert a A) = lcm a (Lcm A)"
1452 proof (rule lcmI)
1453   fix l assume "a dvd l" and "Lcm A dvd l"
1454   hence "\<forall>a\<in>A. a dvd l" by (blast intro: dvd_trans dvd_Lcm)
1455   with a dvd l show "Lcm (insert a A) dvd l" by (force intro: Lcm_dvd)
1456 qed (auto intro: Lcm_dvd dvd_Lcm)
1458 lemma Lcm_finite:
1459   assumes "finite A"
1460   shows "Lcm A = Finite_Set.fold lcm 1 A"
1461   by (induct rule: finite.induct[OF finite A])
1462     (simp_all add: comp_fun_idem.fold_insert_idem[OF comp_fun_idem_lcm])
1464 lemma Lcm_set [code_unfold]:
1465   "Lcm (set xs) = fold lcm xs 1"
1466   using comp_fun_idem.fold_set_fold[OF comp_fun_idem_lcm] Lcm_finite by (simp add: ac_simps)
1468 lemma Lcm_singleton [simp]:
1469   "Lcm {a} = a div normalisation_factor a"
1470   by simp
1472 lemma Lcm_2 [simp]:
1473   "Lcm {a,b} = lcm a b"
1474   by (simp only: Lcm_insert Lcm_empty lcm_1_right)
1475     (cases "b = 0", simp, rule lcm_div_unit2, simp)
1477 lemma Lcm_coprime:
1478   assumes "finite A" and "A \<noteq> {}"
1479   assumes "\<And>a b. a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<noteq> b \<Longrightarrow> gcd a b = 1"
1480   shows "Lcm A = \<Prod>A div normalisation_factor (\<Prod>A)"
1481 using assms proof (induct rule: finite_ne_induct)
1482   case (insert a A)
1483   have "Lcm (insert a A) = lcm a (Lcm A)" by simp
1484   also from insert have "Lcm A = \<Prod>A div normalisation_factor (\<Prod>A)" by blast
1485   also have "lcm a \<dots> = lcm a (\<Prod>A)" by (cases "\<Prod>A = 0") (simp_all add: lcm_div_unit2)
1486   also from insert have "gcd a (\<Prod>A) = 1" by (subst gcd.commute, intro setprod_coprime) auto
1487   with insert have "lcm a (\<Prod>A) = \<Prod>(insert a A) div normalisation_factor (\<Prod>(insert a A))"
1488     by (simp add: lcm_coprime)
1489   finally show ?case .
1490 qed simp
1492 lemma Lcm_coprime':
1493   "card A \<noteq> 0 \<Longrightarrow> (\<And>a b. a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<noteq> b \<Longrightarrow> gcd a b = 1)
1494     \<Longrightarrow> Lcm A = \<Prod>A div normalisation_factor (\<Prod>A)"
1495   by (rule Lcm_coprime) (simp_all add: card_eq_0_iff)
1497 lemma Gcd_Lcm:
1498   "Gcd A = Lcm {d. \<forall>a\<in>A. d dvd a}"
1499   by (simp add: Gcd_Gcd_eucl Lcm_Lcm_eucl Gcd_eucl_def)
1501 lemma Gcd_dvd [simp]: "a \<in> A \<Longrightarrow> Gcd A dvd a"
1502   and dvd_Gcd [simp]: "(\<forall>a\<in>A. g' dvd a) \<Longrightarrow> g' dvd Gcd A"
1503   and normalisation_factor_Gcd [simp]:
1504     "normalisation_factor (Gcd A) = (if Gcd A = 0 then 0 else 1)"
1505 proof -
1506   fix a assume "a \<in> A"
1507   hence "Lcm {d. \<forall>a\<in>A. d dvd a} dvd a" by (intro Lcm_dvd) blast
1508   then show "Gcd A dvd a" by (simp add: Gcd_Lcm)
1509 next
1510   fix g' assume "\<forall>a\<in>A. g' dvd a"
1511   hence "g' dvd Lcm {d. \<forall>a\<in>A. d dvd a}" by (intro dvd_Lcm) blast
1512   then show "g' dvd Gcd A" by (simp add: Gcd_Lcm)
1513 next
1514   show "normalisation_factor (Gcd A) = (if Gcd A = 0 then 0 else 1)"
1515     by (simp add: Gcd_Lcm)
1516 qed
1518 lemma GcdI:
1519   "(\<And>a. a\<in>A \<Longrightarrow> l dvd a) \<Longrightarrow> (\<And>l'. (\<forall>a\<in>A. l' dvd a) \<Longrightarrow> l' dvd l) \<Longrightarrow>
1520     normalisation_factor l = (if l = 0 then 0 else 1) \<Longrightarrow> l = Gcd A"
1521   by (intro normed_associated_imp_eq)
1522     (auto intro: Gcd_dvd dvd_Gcd simp: associated_def)
1524 lemma Lcm_Gcd:
1525   "Lcm A = Gcd {m. \<forall>a\<in>A. a dvd m}"
1526   by (rule LcmI[symmetric]) (auto intro: dvd_Gcd Gcd_dvd)
1528 lemma Gcd_0_iff:
1529   "Gcd A = 0 \<longleftrightarrow> A \<subseteq> {0}"
1530   apply (rule iffI)
1531   apply (rule subsetI, drule Gcd_dvd, simp)
1532   apply (auto intro: GcdI[symmetric])
1533   done
1535 lemma Gcd_empty [simp]:
1536   "Gcd {} = 0"
1537   by (simp add: Gcd_0_iff)
1539 lemma Gcd_1:
1540   "1 \<in> A \<Longrightarrow> Gcd A = 1"
1541   by (intro GcdI[symmetric]) (auto intro: Gcd_dvd dvd_Gcd)
1543 lemma Gcd_insert [simp]:
1544   "Gcd (insert a A) = gcd a (Gcd A)"
1545 proof (rule gcdI)
1546   fix l assume "l dvd a" and "l dvd Gcd A"
1547   hence "\<forall>a\<in>A. l dvd a" by (blast intro: dvd_trans Gcd_dvd)
1548   with l dvd a show "l dvd Gcd (insert a A)" by (force intro: Gcd_dvd)
1549 qed auto
1551 lemma Gcd_finite:
1552   assumes "finite A"
1553   shows "Gcd A = Finite_Set.fold gcd 0 A"
1554   by (induct rule: finite.induct[OF finite A])
1555     (simp_all add: comp_fun_idem.fold_insert_idem[OF comp_fun_idem_gcd])
1557 lemma Gcd_set [code_unfold]:
1558   "Gcd (set xs) = fold gcd xs 0"
1559   using comp_fun_idem.fold_set_fold[OF comp_fun_idem_gcd] Gcd_finite by (simp add: ac_simps)
1561 lemma Gcd_singleton [simp]: "Gcd {a} = a div normalisation_factor a"
1562   by (simp add: gcd_0)
1564 lemma Gcd_2 [simp]: "Gcd {a,b} = gcd a b"
1565   by (simp only: Gcd_insert Gcd_empty gcd_0) (cases "b = 0", simp, rule gcd_div_unit2, simp)
1567 end
1569 text {*
1570   A Euclidean ring is a Euclidean semiring with additive inverses. It provides a
1571   few more lemmas; in particular, Bezout's lemma holds for any Euclidean ring.
1572 *}
1574 class euclidean_ring = euclidean_semiring + idom
1576 class euclidean_ring_gcd = euclidean_semiring_gcd + idom
1577 begin
1579 subclass euclidean_ring ..
1581 lemma gcd_neg1 [simp]:
1582   "gcd (-a) b = gcd a b"
1583   by (rule sym, rule gcdI, simp_all add: gcd_greatest)
1585 lemma gcd_neg2 [simp]:
1586   "gcd a (-b) = gcd a b"
1587   by (rule sym, rule gcdI, simp_all add: gcd_greatest)
1589 lemma gcd_neg_numeral_1 [simp]:
1590   "gcd (- numeral n) a = gcd (numeral n) a"
1591   by (fact gcd_neg1)
1593 lemma gcd_neg_numeral_2 [simp]:
1594   "gcd a (- numeral n) = gcd a (numeral n)"
1595   by (fact gcd_neg2)
1597 lemma gcd_diff1: "gcd (m - n) n = gcd m n"
1598   by (subst diff_conv_add_uminus, subst gcd_neg2[symmetric],  subst gcd_add1, simp)
1600 lemma gcd_diff2: "gcd (n - m) n = gcd m n"
1601   by (subst gcd_neg1[symmetric], simp only: minus_diff_eq gcd_diff1)
1603 lemma coprime_minus_one [simp]: "gcd (n - 1) n = 1"
1604 proof -
1605   have "gcd (n - 1) n = gcd n (n - 1)" by (fact gcd.commute)
1606   also have "\<dots> = gcd ((n - 1) + 1) (n - 1)" by simp
1607   also have "\<dots> = 1" by (rule coprime_plus_one)
1608   finally show ?thesis .
1609 qed
1611 lemma lcm_neg1 [simp]: "lcm (-a) b = lcm a b"
1612   by (rule sym, rule lcmI, simp_all add: lcm_least lcm_zero)
1614 lemma lcm_neg2 [simp]: "lcm a (-b) = lcm a b"
1615   by (rule sym, rule lcmI, simp_all add: lcm_least lcm_zero)
1617 lemma lcm_neg_numeral_1 [simp]: "lcm (- numeral n) a = lcm (numeral n) a"
1618   by (fact lcm_neg1)
1620 lemma lcm_neg_numeral_2 [simp]: "lcm a (- numeral n) = lcm a (numeral n)"
1621   by (fact lcm_neg2)
1623 function euclid_ext :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a" where
1624   "euclid_ext a b =
1625      (if b = 0 then
1626         let c = divide 1 (normalisation_factor a) in (c, 0, a * c)
1627       else
1628         case euclid_ext b (a mod b) of
1629             (s,t,c) \<Rightarrow> (t, s - t * (a div b), c))"
1630   by (pat_completeness, simp)
1631   termination by (relation "measure (euclidean_size \<circ> snd)", simp_all)
1633 declare euclid_ext.simps [simp del]
1635 lemma euclid_ext_0:
1636   "euclid_ext a 0 = (divide 1 (normalisation_factor a), 0, a * divide 1 (normalisation_factor a))"
1637   by (subst euclid_ext.simps, simp add: Let_def)
1639 lemma euclid_ext_non_0:
1640   "b \<noteq> 0 \<Longrightarrow> euclid_ext a b = (case euclid_ext b (a mod b) of
1641     (s,t,c) \<Rightarrow> (t, s - t * (a div b), c))"
1642   by (subst euclid_ext.simps, simp)
1644 definition euclid_ext' :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a"
1645 where
1646   "euclid_ext' a b = (case euclid_ext a b of (s, t, _) \<Rightarrow> (s, t))"
1648 lemma euclid_ext_gcd [simp]:
1649   "(case euclid_ext a b of (_,_,t) \<Rightarrow> t) = gcd a b"
1650 proof (induct a b rule: euclid_ext.induct)
1651   case (1 a b)
1652   then show ?case
1653   proof (cases "b = 0")
1654     case True
1655       then show ?thesis by (cases "a = 0")
1656         (simp_all add: euclid_ext_0 unit_div mult_ac unit_simps gcd_0)
1657     next
1658     case False with 1 show ?thesis
1659       by (simp add: euclid_ext_non_0 ac_simps split: prod.split prod.split_asm)
1660     qed
1661 qed
1663 lemma euclid_ext_gcd' [simp]:
1664   "euclid_ext a b = (r, s, t) \<Longrightarrow> t = gcd a b"
1665   by (insert euclid_ext_gcd[of a b], drule (1) subst, simp)
1667 lemma euclid_ext_correct:
1668   "case euclid_ext a b of (s,t,c) \<Rightarrow> s*a + t*b = c"
1669 proof (induct a b rule: euclid_ext.induct)
1670   case (1 a b)
1671   show ?case
1672   proof (cases "b = 0")
1673     case True
1674     then show ?thesis by (simp add: euclid_ext_0 mult_ac)
1675   next
1676     case False
1677     obtain s t c where stc: "euclid_ext b (a mod b) = (s,t,c)"
1678       by (cases "euclid_ext b (a mod b)", blast)
1679     from 1 have "c = s * b + t * (a mod b)" by (simp add: stc False)
1680     also have "... = t*((a div b)*b + a mod b) + (s - t * (a div b))*b"
1681       by (simp add: algebra_simps)
1682     also have "(a div b)*b + a mod b = a" using mod_div_equality .
1683     finally show ?thesis
1684       by (subst euclid_ext.simps, simp add: False stc)
1685     qed
1686 qed
1688 lemma euclid_ext'_correct:
1689   "fst (euclid_ext' a b) * a + snd (euclid_ext' a b) * b = gcd a b"
1690 proof-
1691   obtain s t c where "euclid_ext a b = (s,t,c)"
1692     by (cases "euclid_ext a b", blast)
1693   with euclid_ext_correct[of a b] euclid_ext_gcd[of a b]
1694     show ?thesis unfolding euclid_ext'_def by simp
1695 qed
1697 lemma bezout: "\<exists>s t. s * a + t * b = gcd a b"
1698   using euclid_ext'_correct by blast
1700 lemma euclid_ext'_0 [simp]: "euclid_ext' a 0 = (divide 1 (normalisation_factor a), 0)"
1701   by (simp add: bezw_def euclid_ext'_def euclid_ext_0)
1703 lemma euclid_ext'_non_0: "b \<noteq> 0 \<Longrightarrow> euclid_ext' a b = (snd (euclid_ext' b (a mod b)),
1704   fst (euclid_ext' b (a mod b)) - snd (euclid_ext' b (a mod b)) * (a div b))"
1705   by (cases "euclid_ext b (a mod b)")
1706     (simp add: euclid_ext'_def euclid_ext_non_0)
1708 end
1710 instantiation nat :: euclidean_semiring
1711 begin
1713 definition [simp]:
1714   "euclidean_size_nat = (id :: nat \<Rightarrow> nat)"
1716 definition [simp]:
1717   "normalisation_factor_nat (n::nat) = (if n = 0 then 0 else 1 :: nat)"
1719 instance proof
1720 qed simp_all
1722 end
1724 instantiation int :: euclidean_ring
1725 begin
1727 definition [simp]:
1728   "euclidean_size_int = (nat \<circ> abs :: int \<Rightarrow> nat)"
1730 definition [simp]:
1731   "normalisation_factor_int = (sgn :: int \<Rightarrow> int)"
1733 instance proof
1734   case goal2 then show ?case by (auto simp add: abs_mult nat_mult_distrib)
1735 next
1736   case goal3 then show ?case by (simp add: zsgn_def)
1737 next
1738   case goal5 then show ?case by (auto simp: zsgn_def)
1739 next
1740   case goal6 then show ?case by (auto split: abs_split simp: zsgn_def)
1741 qed (auto simp: sgn_times split: abs_split)
1743 end
1745 end