src/HOL/Number_Theory/Euclidean_Algorithm.thy
 author haftmann Mon Nov 17 14:55:33 2014 +0100 (2014-11-17) changeset 59009 348561aa3869 parent 58953 2e19b392d9e3 child 59010 ec2b4270a502 permissions -rw-r--r--
generalized lemmas (particularly concerning dvd) as far as appropriate
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 lemma dvd_setprod [intro]:
13   assumes "finite A" and "x \<in> A"
14   shows "f x dvd setprod f A"
15 proof
16   from finite A have "setprod f (insert x (A - {x})) = f x * setprod f (A - {x})"
17     by (intro setprod.insert) auto
18   also from x \<in> A have "insert x (A - {x}) = A" by blast
19   finally show "setprod f A = f x * setprod f (A - {x})" .
20 qed
22 end
24 context semiring_div
25 begin
27 definition ring_inv :: "'a \<Rightarrow> 'a"
28 where
29   "ring_inv x = 1 div x"
31 definition is_unit :: "'a \<Rightarrow> bool"
32 where
33   "is_unit x \<longleftrightarrow> x dvd 1"
35 definition associated :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
36 where
37   "associated x y \<longleftrightarrow> x dvd y \<and> y dvd x"
39 lemma unit_prod [intro]:
40   "is_unit x \<Longrightarrow> is_unit y \<Longrightarrow> is_unit (x * y)"
41   unfolding is_unit_def by (subst mult_1_left [of 1, symmetric], rule mult_dvd_mono)
43 lemma unit_ring_inv:
44   "is_unit y \<Longrightarrow> x div y = x * ring_inv y"
45   by (simp add: div_mult_swap ring_inv_def is_unit_def)
47 lemma unit_ring_inv_ring_inv [simp]:
48   "is_unit x \<Longrightarrow> ring_inv (ring_inv x) = x"
49   unfolding is_unit_def ring_inv_def
50   by (metis div_mult_mult1_if div_mult_self1_is_id dvd_mult_div_cancel mult_1_right)
52 lemma inv_imp_eq_ring_inv:
53   "a * b = 1 \<Longrightarrow> ring_inv a = b"
54   by (metis dvd_mult_div_cancel dvd_mult_right mult_1_right mult.left_commute one_dvd ring_inv_def)
56 lemma ring_inv_is_inv1 [simp]:
57   "is_unit a \<Longrightarrow> a * ring_inv a = 1"
58   unfolding is_unit_def ring_inv_def by simp
60 lemma ring_inv_is_inv2 [simp]:
61   "is_unit a \<Longrightarrow> ring_inv a * a = 1"
62   by (simp add: ac_simps)
64 lemma unit_ring_inv_unit [simp, intro]:
65   assumes "is_unit x"
66   shows "is_unit (ring_inv x)"
67 proof -
68   from assms have "1 = ring_inv x * x" by simp
69   then show "is_unit (ring_inv x)" unfolding is_unit_def by (rule dvdI)
70 qed
72 lemma mult_unit_dvd_iff:
73   "is_unit y \<Longrightarrow> x * y dvd z \<longleftrightarrow> x dvd z"
74 proof
75   assume "is_unit y" "x * y dvd z"
76   then show "x dvd z" by (simp add: dvd_mult_left)
77 next
78   assume "is_unit y" "x dvd z"
79   then obtain k where "z = x * k" unfolding dvd_def by blast
80   with is_unit y have "z = (x * y) * (ring_inv y * k)"
81       by (simp add: mult_ac)
82   then show "x * y dvd z" by (rule dvdI)
83 qed
85 lemma div_unit_dvd_iff:
86   "is_unit y \<Longrightarrow> x div y dvd z \<longleftrightarrow> x dvd z"
87   by (subst unit_ring_inv) (assumption, simp add: mult_unit_dvd_iff)
89 lemma dvd_mult_unit_iff:
90   "is_unit y \<Longrightarrow> x dvd z * y \<longleftrightarrow> x dvd z"
91 proof
92   assume "is_unit y" and "x dvd z * y"
93   have "z * y dvd z * (y * ring_inv y)" by (subst mult_assoc [symmetric]) simp
94   also from is_unit y have "y * ring_inv y = 1" by simp
95   finally have "z * y dvd z" by simp
96   with x dvd z * y show "x dvd z" by (rule dvd_trans)
97 next
98   assume "x dvd z"
99   then show "x dvd z * y" by simp
100 qed
102 lemma dvd_div_unit_iff:
103   "is_unit y \<Longrightarrow> x dvd z div y \<longleftrightarrow> x dvd z"
104   by (subst unit_ring_inv) (assumption, simp add: dvd_mult_unit_iff)
106 lemmas unit_dvd_iff = mult_unit_dvd_iff div_unit_dvd_iff dvd_mult_unit_iff dvd_div_unit_iff
108 lemma unit_div [intro]:
109   "is_unit x \<Longrightarrow> is_unit y \<Longrightarrow> is_unit (x div y)"
110   by (subst unit_ring_inv) (assumption, rule unit_prod, simp_all)
112 lemma unit_div_mult_swap:
113   "is_unit z \<Longrightarrow> x * (y div z) = x * y div z"
114   by (simp only: unit_ring_inv [of _ y] unit_ring_inv [of _ "x*y"] ac_simps)
116 lemma unit_div_commute:
117   "is_unit y \<Longrightarrow> x div y * z = x * z div y"
118   by (simp only: unit_ring_inv [of _ x] unit_ring_inv [of _ "x*z"] ac_simps)
120 lemma unit_imp_dvd [dest]:
121   "is_unit y \<Longrightarrow> y dvd x"
122   by (rule dvd_trans [of _ 1]) (simp_all add: is_unit_def)
124 lemma dvd_unit_imp_unit:
125   "is_unit y \<Longrightarrow> x dvd y \<Longrightarrow> is_unit x"
126   by (unfold is_unit_def) (rule dvd_trans)
128 lemma ring_inv_0 [simp]:
129   "ring_inv 0 = 0"
130   unfolding ring_inv_def by simp
132 lemma unit_ring_inv'1:
133   assumes "is_unit y"
134   shows "x div (y * z) = x * ring_inv y div z"
135 proof -
136   from assms have "x div (y * z) = x * (ring_inv y * y) div (y * z)"
137     by simp
138   also have "... = y * (x * ring_inv y) div (y * z)"
139     by (simp only: mult_ac)
140   also have "... = x * ring_inv y div z"
141     by (cases "y = 0", simp, rule div_mult_mult1)
142   finally show ?thesis .
143 qed
145 lemma associated_comm:
146   "associated x y \<Longrightarrow> associated y x"
147   by (simp add: associated_def)
149 lemma associated_0 [simp]:
150   "associated 0 b \<longleftrightarrow> b = 0"
151   "associated a 0 \<longleftrightarrow> a = 0"
152   unfolding associated_def by simp_all
154 lemma associated_unit:
155   "is_unit x \<Longrightarrow> associated x y \<Longrightarrow> is_unit y"
156   unfolding associated_def by (fast dest: dvd_unit_imp_unit)
158 lemma is_unit_1 [simp]:
159   "is_unit 1"
160   unfolding is_unit_def by simp
162 lemma not_is_unit_0 [simp]:
163   "\<not> is_unit 0"
164   unfolding is_unit_def by auto
166 lemma unit_mult_left_cancel:
167   assumes "is_unit x"
168   shows "(x * y) = (x * z) \<longleftrightarrow> y = z"
169 proof -
170   from assms have "x \<noteq> 0" by auto
171   then show ?thesis by (metis div_mult_self1_is_id)
172 qed
174 lemma unit_mult_right_cancel:
175   "is_unit x \<Longrightarrow> (y * x) = (z * x) \<longleftrightarrow> y = z"
176   by (simp add: ac_simps unit_mult_left_cancel)
178 lemma unit_div_cancel:
179   "is_unit x \<Longrightarrow> (y div x) = (z div x) \<longleftrightarrow> y = z"
180   apply (subst unit_ring_inv[of _ y], assumption)
181   apply (subst unit_ring_inv[of _ z], assumption)
182   apply (rule unit_mult_right_cancel, erule unit_ring_inv_unit)
183   done
185 lemma unit_eq_div1:
186   "is_unit y \<Longrightarrow> x div y = z \<longleftrightarrow> x = z * y"
187   apply (subst unit_ring_inv, assumption)
188   apply (subst unit_mult_right_cancel[symmetric], assumption)
189   apply (subst mult_assoc, subst ring_inv_is_inv2, assumption, simp)
190   done
192 lemma unit_eq_div2:
193   "is_unit y \<Longrightarrow> x = z div y \<longleftrightarrow> x * y = z"
194   by (subst (1 2) eq_commute, simp add: unit_eq_div1, subst eq_commute, rule refl)
196 lemma associated_iff_div_unit:
197   "associated x y \<longleftrightarrow> (\<exists>z. is_unit z \<and> x = z * y)"
198 proof
199   assume "associated x y"
200   show "\<exists>z. is_unit z \<and> x = z * y"
201   proof (cases "x = 0")
202     assume "x = 0"
203     then show "\<exists>z. is_unit z \<and> x = z * y" using associated x y
204         by (intro exI[of _ 1], simp add: associated_def)
205   next
206     assume [simp]: "x \<noteq> 0"
207     hence [simp]: "x dvd y" "y dvd x" using associated x y
208         unfolding associated_def by simp_all
209     hence "1 = x div y * (y div x)"
210       by (simp add: div_mult_swap)
211     hence "is_unit (x div y)" unfolding is_unit_def by (rule dvdI)
212     moreover have "x = (x div y) * y" by simp
213     ultimately show ?thesis by blast
214   qed
215 next
216   assume "\<exists>z. is_unit z \<and> x = z * y"
217   then obtain z where "is_unit z" and "x = z * y" by blast
218   hence "y = x * ring_inv z" by (simp add: algebra_simps)
219   hence "x dvd y" by simp
220   moreover from x = z * y have "y dvd x" by simp
221   ultimately show "associated x y" unfolding associated_def by simp
222 qed
224 lemmas unit_simps = mult_unit_dvd_iff div_unit_dvd_iff dvd_mult_unit_iff
225   dvd_div_unit_iff unit_div_mult_swap unit_div_commute
226   unit_mult_left_cancel unit_mult_right_cancel unit_div_cancel
227   unit_eq_div1 unit_eq_div2
229 end
231 context ring_div
232 begin
234 lemma is_unit_neg [simp]:
235   "is_unit (- x) \<Longrightarrow> is_unit x"
236   unfolding is_unit_def by simp
238 lemma is_unit_neg_1 [simp]:
239   "is_unit (-1)"
240   unfolding is_unit_def by simp
242 end
244 lemma is_unit_nat [simp]:
245   "is_unit (x::nat) \<longleftrightarrow> x = 1"
246   unfolding is_unit_def by simp
248 lemma is_unit_int:
249   "is_unit (x::int) \<longleftrightarrow> x = 1 \<or> x = -1"
250   unfolding is_unit_def by auto
252 text {*
253   A Euclidean semiring is a semiring upon which the Euclidean algorithm can be
254   implemented. It must provide:
255   \begin{itemize}
256   \item division with remainder
257   \item a size function such that @{term "size (a mod b) < size b"}
258         for any @{term "b \<noteq> 0"}
259   \item a normalisation factor such that two associated numbers are equal iff
260         they are the same when divided by their normalisation factors.
261   \end{itemize}
262   The existence of these functions makes it possible to derive gcd and lcm functions
263   for any Euclidean semiring.
264 *}
265 class euclidean_semiring = semiring_div +
266   fixes euclidean_size :: "'a \<Rightarrow> nat"
267   fixes normalisation_factor :: "'a \<Rightarrow> 'a"
268   assumes mod_size_less [simp]:
269     "b \<noteq> 0 \<Longrightarrow> euclidean_size (a mod b) < euclidean_size b"
270   assumes size_mult_mono:
271     "b \<noteq> 0 \<Longrightarrow> euclidean_size (a * b) \<ge> euclidean_size a"
272   assumes normalisation_factor_is_unit [intro,simp]:
273     "a \<noteq> 0 \<Longrightarrow> is_unit (normalisation_factor a)"
274   assumes normalisation_factor_mult: "normalisation_factor (a * b) =
275     normalisation_factor a * normalisation_factor b"
276   assumes normalisation_factor_unit: "is_unit x \<Longrightarrow> normalisation_factor x = x"
277   assumes normalisation_factor_0 [simp]: "normalisation_factor 0 = 0"
278 begin
280 lemma normalisation_factor_dvd [simp]:
281   "a \<noteq> 0 \<Longrightarrow> normalisation_factor a dvd b"
282   by (rule unit_imp_dvd, simp)
284 lemma normalisation_factor_1 [simp]:
285   "normalisation_factor 1 = 1"
286   by (simp add: normalisation_factor_unit)
288 lemma normalisation_factor_0_iff [simp]:
289   "normalisation_factor x = 0 \<longleftrightarrow> x = 0"
290 proof
291   assume "normalisation_factor x = 0"
292   hence "\<not> is_unit (normalisation_factor x)"
293     by (metis not_is_unit_0)
294   then show "x = 0" by force
295 next
296   assume "x = 0"
297   then show "normalisation_factor x = 0" by simp
298 qed
300 lemma normalisation_factor_pow:
301   "normalisation_factor (x ^ n) = normalisation_factor x ^ n"
302   by (induct n) (simp_all add: normalisation_factor_mult power_Suc2)
304 lemma normalisation_correct [simp]:
305   "normalisation_factor (x div normalisation_factor x) = (if x = 0 then 0 else 1)"
306 proof (cases "x = 0", simp)
307   assume "x \<noteq> 0"
308   let ?nf = "normalisation_factor"
309   from normalisation_factor_is_unit[OF x \<noteq> 0] have "?nf x \<noteq> 0"
310     by (metis not_is_unit_0)
311   have "?nf (x div ?nf x) * ?nf (?nf x) = ?nf (x div ?nf x * ?nf x)"
312     by (simp add: normalisation_factor_mult)
313   also have "x div ?nf x * ?nf x = x" using x \<noteq> 0
314     by simp
315   also have "?nf (?nf x) = ?nf x" using x \<noteq> 0
316     normalisation_factor_is_unit normalisation_factor_unit by simp
317   finally show ?thesis using x \<noteq> 0 and ?nf x \<noteq> 0
318     by (metis div_mult_self2_is_id div_self)
319 qed
321 lemma normalisation_0_iff [simp]:
322   "x div normalisation_factor x = 0 \<longleftrightarrow> x = 0"
323   by (cases "x = 0", simp, subst unit_eq_div1, blast, simp)
325 lemma associated_iff_normed_eq:
326   "associated a b \<longleftrightarrow> a div normalisation_factor a = b div normalisation_factor b"
327 proof (cases "b = 0", simp, cases "a = 0", metis associated_0(1) normalisation_0_iff, rule iffI)
328   let ?nf = normalisation_factor
329   assume "a \<noteq> 0" "b \<noteq> 0" "a div ?nf a = b div ?nf b"
330   hence "a = b * (?nf a div ?nf b)"
331     apply (subst (asm) unit_eq_div1, blast, subst (asm) unit_div_commute, blast)
332     apply (subst div_mult_swap, simp, simp)
333     done
334   with a \<noteq> 0 b \<noteq> 0 have "\<exists>z. is_unit z \<and> a = z * b"
335     by (intro exI[of _ "?nf a div ?nf b"], force simp: mult_ac)
336   with associated_iff_div_unit show "associated a b" by simp
337 next
338   let ?nf = normalisation_factor
339   assume "a \<noteq> 0" "b \<noteq> 0" "associated a b"
340   with associated_iff_div_unit obtain z where "is_unit z" and "a = z * b" by blast
341   then show "a div ?nf a = b div ?nf b"
342     apply (simp only: a = z * b normalisation_factor_mult normalisation_factor_unit)
343     apply (rule div_mult_mult1, force)
344     done
345   qed
347 lemma normed_associated_imp_eq:
348   "associated a b \<Longrightarrow> normalisation_factor a \<in> {0, 1} \<Longrightarrow> normalisation_factor b \<in> {0, 1} \<Longrightarrow> a = b"
349   by (simp add: associated_iff_normed_eq, elim disjE, simp_all)
351 lemmas normalisation_factor_dvd_iff [simp] =
352   unit_dvd_iff [OF normalisation_factor_is_unit]
354 lemma euclidean_division:
355   fixes a :: 'a and b :: 'a
356   assumes "b \<noteq> 0"
357   obtains s and t where "a = s * b + t"
358     and "euclidean_size t < euclidean_size b"
359 proof -
360   from div_mod_equality[of a b 0]
361      have "a = a div b * b + a mod b" by simp
362   with that and assms show ?thesis by force
363 qed
365 lemma dvd_euclidean_size_eq_imp_dvd:
366   assumes "a \<noteq> 0" and b_dvd_a: "b dvd a" and size_eq: "euclidean_size a = euclidean_size b"
367   shows "a dvd b"
368 proof (subst dvd_eq_mod_eq_0, rule ccontr)
369   assume "b mod a \<noteq> 0"
370   from b_dvd_a have b_dvd_mod: "b dvd b mod a" by (simp add: dvd_mod_iff)
371   from b_dvd_mod obtain c where "b mod a = b * c" unfolding dvd_def by blast
372     with b mod a \<noteq> 0 have "c \<noteq> 0" by auto
373   with b mod a = b * c have "euclidean_size (b mod a) \<ge> euclidean_size b"
374       using size_mult_mono by force
375   moreover from a \<noteq> 0 have "euclidean_size (b mod a) < euclidean_size a"
376       using mod_size_less by blast
377   ultimately show False using size_eq by simp
378 qed
380 function gcd_eucl :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
381 where
382   "gcd_eucl a b = (if b = 0 then a div normalisation_factor a else gcd_eucl b (a mod b))"
383   by (pat_completeness, simp)
384 termination by (relation "measure (euclidean_size \<circ> snd)", simp_all)
386 declare gcd_eucl.simps [simp del]
388 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"
389 proof (induct a b rule: gcd_eucl.induct)
390   case ("1" m n)
391     then show ?case by (cases "n = 0") auto
392 qed
394 definition lcm_eucl :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
395 where
396   "lcm_eucl a b = a * b div (gcd_eucl a b * normalisation_factor (a * b))"
398   (* Somewhat complicated definition of Lcm that has the advantage of working
399      for infinite sets as well *)
401 definition Lcm_eucl :: "'a set \<Rightarrow> 'a"
402 where
403   "Lcm_eucl A = (if \<exists>l. l \<noteq> 0 \<and> (\<forall>x\<in>A. x dvd l) then
404      let l = SOME l. l \<noteq> 0 \<and> (\<forall>x\<in>A. x dvd l) \<and> euclidean_size l =
405        (LEAST n. \<exists>l. l \<noteq> 0 \<and> (\<forall>x\<in>A. x dvd l) \<and> euclidean_size l = n)
406        in l div normalisation_factor l
407       else 0)"
409 definition Gcd_eucl :: "'a set \<Rightarrow> 'a"
410 where
411   "Gcd_eucl A = Lcm_eucl {d. \<forall>a\<in>A. d dvd a}"
413 end
415 class euclidean_semiring_gcd = euclidean_semiring + gcd + Gcd +
416   assumes gcd_gcd_eucl: "gcd = gcd_eucl" and lcm_lcm_eucl: "lcm = lcm_eucl"
417   assumes Gcd_Gcd_eucl: "Gcd = Gcd_eucl" and Lcm_Lcm_eucl: "Lcm = Lcm_eucl"
418 begin
420 lemma gcd_red:
421   "gcd x y = gcd y (x mod y)"
422   by (metis gcd_eucl.simps mod_0 mod_by_0 gcd_gcd_eucl)
424 lemma gcd_non_0:
425   "y \<noteq> 0 \<Longrightarrow> gcd x y = gcd y (x mod y)"
426   by (rule gcd_red)
428 lemma gcd_0_left:
429   "gcd 0 x = x div normalisation_factor x"
430    by (simp only: gcd_gcd_eucl, subst gcd_eucl.simps, subst gcd_eucl.simps, simp add: Let_def)
432 lemma gcd_0:
433   "gcd x 0 = x div normalisation_factor x"
434   by (simp only: gcd_gcd_eucl, subst gcd_eucl.simps, simp add: Let_def)
436 lemma gcd_dvd1 [iff]: "gcd x y dvd x"
437   and gcd_dvd2 [iff]: "gcd x y dvd y"
438 proof (induct x y rule: gcd_eucl.induct)
439   fix x y :: 'a
440   assume IH1: "y \<noteq> 0 \<Longrightarrow> gcd y (x mod y) dvd y"
441   assume IH2: "y \<noteq> 0 \<Longrightarrow> gcd y (x mod y) dvd (x mod y)"
443   have "gcd x y dvd x \<and> gcd x y dvd y"
444   proof (cases "y = 0")
445     case True
446       then show ?thesis by (cases "x = 0", simp_all add: gcd_0)
447   next
448     case False
449       with IH1 and IH2 show ?thesis by (simp add: gcd_non_0 dvd_mod_iff)
450   qed
451   then show "gcd x y dvd x" "gcd x y dvd y" by simp_all
452 qed
454 lemma dvd_gcd_D1: "k dvd gcd m n \<Longrightarrow> k dvd m"
455   by (rule dvd_trans, assumption, rule gcd_dvd1)
457 lemma dvd_gcd_D2: "k dvd gcd m n \<Longrightarrow> k dvd n"
458   by (rule dvd_trans, assumption, rule gcd_dvd2)
460 lemma gcd_greatest:
461   fixes k x y :: 'a
462   shows "k dvd x \<Longrightarrow> k dvd y \<Longrightarrow> k dvd gcd x y"
463 proof (induct x y rule: gcd_eucl.induct)
464   case (1 x y)
465   show ?case
466     proof (cases "y = 0")
467       assume "y = 0"
468       with 1 show ?thesis by (cases "x = 0", simp_all add: gcd_0)
469     next
470       assume "y \<noteq> 0"
471       with 1 show ?thesis by (simp add: gcd_non_0 dvd_mod_iff)
472     qed
473 qed
475 lemma dvd_gcd_iff:
476   "k dvd gcd x y \<longleftrightarrow> k dvd x \<and> k dvd y"
477   by (blast intro!: gcd_greatest intro: dvd_trans)
479 lemmas gcd_greatest_iff = dvd_gcd_iff
481 lemma gcd_zero [simp]:
482   "gcd x y = 0 \<longleftrightarrow> x = 0 \<and> y = 0"
483   by (metis dvd_0_left dvd_refl gcd_dvd1 gcd_dvd2 gcd_greatest)+
485 lemma normalisation_factor_gcd [simp]:
486   "normalisation_factor (gcd x y) = (if x = 0 \<and> y = 0 then 0 else 1)" (is "?f x y = ?g x y")
487 proof (induct x y rule: gcd_eucl.induct)
488   fix x y :: 'a
489   assume IH: "y \<noteq> 0 \<Longrightarrow> ?f y (x mod y) = ?g y (x mod y)"
490   then show "?f x y = ?g x y" by (cases "y = 0", auto simp: gcd_non_0 gcd_0)
491 qed
493 lemma gcdI:
494   "k dvd x \<Longrightarrow> k dvd y \<Longrightarrow> (\<And>l. l dvd x \<Longrightarrow> l dvd y \<Longrightarrow> l dvd k)
495     \<Longrightarrow> normalisation_factor k = (if k = 0 then 0 else 1) \<Longrightarrow> k = gcd x y"
496   by (intro normed_associated_imp_eq) (auto simp: associated_def intro: gcd_greatest)
498 sublocale gcd!: abel_semigroup gcd
499 proof
500   fix x y z
501   show "gcd (gcd x y) z = gcd x (gcd y z)"
502   proof (rule gcdI)
503     have "gcd (gcd x y) z dvd gcd x y" "gcd x y dvd x" by simp_all
504     then show "gcd (gcd x y) z dvd x" by (rule dvd_trans)
505     have "gcd (gcd x y) z dvd gcd x y" "gcd x y dvd y" by simp_all
506     hence "gcd (gcd x y) z dvd y" by (rule dvd_trans)
507     moreover have "gcd (gcd x y) z dvd z" by simp
508     ultimately show "gcd (gcd x y) z dvd gcd y z"
509       by (rule gcd_greatest)
510     show "normalisation_factor (gcd (gcd x y) z) =  (if gcd (gcd x y) z = 0 then 0 else 1)"
511       by auto
512     fix l assume "l dvd x" and "l dvd gcd y z"
513     with dvd_trans[OF _ gcd_dvd1] and dvd_trans[OF _ gcd_dvd2]
514       have "l dvd y" and "l dvd z" by blast+
515     with l dvd x show "l dvd gcd (gcd x y) z"
516       by (intro gcd_greatest)
517   qed
518 next
519   fix x y
520   show "gcd x y = gcd y x"
521     by (rule gcdI) (simp_all add: gcd_greatest)
522 qed
524 lemma gcd_unique: "d dvd a \<and> d dvd b \<and>
525     normalisation_factor d = (if d = 0 then 0 else 1) \<and>
526     (\<forall>e. e dvd a \<and> e dvd b \<longrightarrow> e dvd d) \<longleftrightarrow> d = gcd a b"
527   by (rule, auto intro: gcdI simp: gcd_greatest)
529 lemma gcd_dvd_prod: "gcd a b dvd k * b"
530   using mult_dvd_mono [of 1] by auto
532 lemma gcd_1_left [simp]: "gcd 1 x = 1"
533   by (rule sym, rule gcdI, simp_all)
535 lemma gcd_1 [simp]: "gcd x 1 = 1"
536   by (rule sym, rule gcdI, simp_all)
538 lemma gcd_proj2_if_dvd:
539   "y dvd x \<Longrightarrow> gcd x y = y div normalisation_factor y"
540   by (cases "y = 0", simp_all add: dvd_eq_mod_eq_0 gcd_non_0 gcd_0)
542 lemma gcd_proj1_if_dvd:
543   "x dvd y \<Longrightarrow> gcd x y = x div normalisation_factor x"
544   by (subst gcd.commute, simp add: gcd_proj2_if_dvd)
546 lemma gcd_proj1_iff: "gcd m n = m div normalisation_factor m \<longleftrightarrow> m dvd n"
547 proof
548   assume A: "gcd m n = m div normalisation_factor m"
549   show "m dvd n"
550   proof (cases "m = 0")
551     assume [simp]: "m \<noteq> 0"
552     from A have B: "m = gcd m n * normalisation_factor m"
553       by (simp add: unit_eq_div2)
554     show ?thesis by (subst B, simp add: mult_unit_dvd_iff)
555   qed (insert A, simp)
556 next
557   assume "m dvd n"
558   then show "gcd m n = m div normalisation_factor m" by (rule gcd_proj1_if_dvd)
559 qed
561 lemma gcd_proj2_iff: "gcd m n = n div normalisation_factor n \<longleftrightarrow> n dvd m"
562   by (subst gcd.commute, simp add: gcd_proj1_iff)
564 lemma gcd_mod1 [simp]:
565   "gcd (x mod y) y = gcd x y"
566   by (rule gcdI, metis dvd_mod_iff gcd_dvd1 gcd_dvd2, simp_all add: gcd_greatest dvd_mod_iff)
568 lemma gcd_mod2 [simp]:
569   "gcd x (y mod x) = gcd x y"
570   by (rule gcdI, simp, metis dvd_mod_iff gcd_dvd1 gcd_dvd2, simp_all add: gcd_greatest dvd_mod_iff)
572 lemma normalisation_factor_dvd' [simp]:
573   "normalisation_factor x dvd x"
574   by (cases "x = 0", simp_all)
576 lemma gcd_mult_distrib':
577   "k div normalisation_factor k * gcd x y = gcd (k*x) (k*y)"
578 proof (induct x y rule: gcd_eucl.induct)
579   case (1 x y)
580   show ?case
581   proof (cases "y = 0")
582     case True
583     then show ?thesis by (simp add: normalisation_factor_mult gcd_0 algebra_simps div_mult_div_if_dvd)
584   next
585     case False
586     hence "k div normalisation_factor k * gcd x y =  gcd (k * y) (k * (x mod y))"
587       using 1 by (subst gcd_red, simp)
588     also have "... = gcd (k * x) (k * y)"
589       by (simp add: mult_mod_right gcd.commute)
590     finally show ?thesis .
591   qed
592 qed
594 lemma gcd_mult_distrib:
595   "k * gcd x y = gcd (k*x) (k*y) * normalisation_factor k"
596 proof-
597   let ?nf = "normalisation_factor"
598   from gcd_mult_distrib'
599     have "gcd (k*x) (k*y) = k div ?nf k * gcd x y" ..
600   also have "... = k * gcd x y div ?nf k"
601     by (metis dvd_div_mult dvd_eq_mod_eq_0 mod_0 normalisation_factor_dvd)
602   finally show ?thesis
603     by simp
604 qed
606 lemma euclidean_size_gcd_le1 [simp]:
607   assumes "a \<noteq> 0"
608   shows "euclidean_size (gcd a b) \<le> euclidean_size a"
609 proof -
610    have "gcd a b dvd a" by (rule gcd_dvd1)
611    then obtain c where A: "a = gcd a b * c" unfolding dvd_def by blast
612    with a \<noteq> 0 show ?thesis by (subst (2) A, intro size_mult_mono) auto
613 qed
615 lemma euclidean_size_gcd_le2 [simp]:
616   "b \<noteq> 0 \<Longrightarrow> euclidean_size (gcd a b) \<le> euclidean_size b"
617   by (subst gcd.commute, rule euclidean_size_gcd_le1)
619 lemma euclidean_size_gcd_less1:
620   assumes "a \<noteq> 0" and "\<not>a dvd b"
621   shows "euclidean_size (gcd a b) < euclidean_size a"
622 proof (rule ccontr)
623   assume "\<not>euclidean_size (gcd a b) < euclidean_size a"
624   with a \<noteq> 0 have "euclidean_size (gcd a b) = euclidean_size a"
625     by (intro le_antisym, simp_all)
626   with assms have "a dvd gcd a b" by (auto intro: dvd_euclidean_size_eq_imp_dvd)
627   hence "a dvd b" using dvd_gcd_D2 by blast
628   with \<not>a dvd b show False by contradiction
629 qed
631 lemma euclidean_size_gcd_less2:
632   assumes "b \<noteq> 0" and "\<not>b dvd a"
633   shows "euclidean_size (gcd a b) < euclidean_size b"
634   using assms by (subst gcd.commute, rule euclidean_size_gcd_less1)
636 lemma gcd_mult_unit1: "is_unit a \<Longrightarrow> gcd (x*a) y = gcd x y"
637   apply (rule gcdI)
638   apply (rule dvd_trans, rule gcd_dvd1, simp add: unit_simps)
639   apply (rule gcd_dvd2)
640   apply (rule gcd_greatest, simp add: unit_simps, assumption)
641   apply (subst normalisation_factor_gcd, simp add: gcd_0)
642   done
644 lemma gcd_mult_unit2: "is_unit a \<Longrightarrow> gcd x (y*a) = gcd x y"
645   by (subst gcd.commute, subst gcd_mult_unit1, assumption, rule gcd.commute)
647 lemma gcd_div_unit1: "is_unit a \<Longrightarrow> gcd (x div a) y = gcd x y"
648   by (simp add: unit_ring_inv gcd_mult_unit1)
650 lemma gcd_div_unit2: "is_unit a \<Longrightarrow> gcd x (y div a) = gcd x y"
651   by (simp add: unit_ring_inv gcd_mult_unit2)
653 lemma gcd_idem: "gcd x x = x div normalisation_factor x"
654   by (cases "x = 0") (simp add: gcd_0_left, rule sym, rule gcdI, simp_all)
656 lemma gcd_right_idem: "gcd (gcd p q) q = gcd p q"
657   apply (rule gcdI)
658   apply (simp add: ac_simps)
659   apply (rule gcd_dvd2)
660   apply (rule gcd_greatest, erule (1) gcd_greatest, assumption)
661   apply simp
662   done
664 lemma gcd_left_idem: "gcd p (gcd p q) = gcd p q"
665   apply (rule gcdI)
666   apply simp
667   apply (rule dvd_trans, rule gcd_dvd2, rule gcd_dvd2)
668   apply (rule gcd_greatest, assumption, erule gcd_greatest, assumption)
669   apply simp
670   done
672 lemma comp_fun_idem_gcd: "comp_fun_idem gcd"
673 proof
674   fix a b show "gcd a \<circ> gcd b = gcd b \<circ> gcd a"
675     by (simp add: fun_eq_iff ac_simps)
676 next
677   fix a show "gcd a \<circ> gcd a = gcd a"
678     by (simp add: fun_eq_iff gcd_left_idem)
679 qed
681 lemma coprime_dvd_mult:
682   assumes "gcd k n = 1" and "k dvd m * n"
683   shows "k dvd m"
684 proof -
685   let ?nf = "normalisation_factor"
686   from assms gcd_mult_distrib [of m k n]
687     have A: "m = gcd (m * k) (m * n) * ?nf m" by simp
688   from k dvd m * n show ?thesis by (subst A, simp_all add: gcd_greatest)
689 qed
691 lemma coprime_dvd_mult_iff:
692   "gcd k n = 1 \<Longrightarrow> (k dvd m * n) = (k dvd m)"
693   by (rule, rule coprime_dvd_mult, simp_all)
695 lemma gcd_dvd_antisym:
696   "gcd a b dvd gcd c d \<Longrightarrow> gcd c d dvd gcd a b \<Longrightarrow> gcd a b = gcd c d"
697 proof (rule gcdI)
698   assume A: "gcd a b dvd gcd c d" and B: "gcd c d dvd gcd a b"
699   have "gcd c d dvd c" by simp
700   with A show "gcd a b dvd c" by (rule dvd_trans)
701   have "gcd c d dvd d" by simp
702   with A show "gcd a b dvd d" by (rule dvd_trans)
703   show "normalisation_factor (gcd a b) = (if gcd a b = 0 then 0 else 1)"
704     by simp
705   fix l assume "l dvd c" and "l dvd d"
706   hence "l dvd gcd c d" by (rule gcd_greatest)
707   from this and B show "l dvd gcd a b" by (rule dvd_trans)
708 qed
710 lemma gcd_mult_cancel:
711   assumes "gcd k n = 1"
712   shows "gcd (k * m) n = gcd m n"
713 proof (rule gcd_dvd_antisym)
714   have "gcd (gcd (k * m) n) k = gcd (gcd k n) (k * m)" by (simp add: ac_simps)
715   also note gcd k n = 1
716   finally have "gcd (gcd (k * m) n) k = 1" by simp
717   hence "gcd (k * m) n dvd m" by (rule coprime_dvd_mult, simp add: ac_simps)
718   moreover have "gcd (k * m) n dvd n" by simp
719   ultimately show "gcd (k * m) n dvd gcd m n" by (rule gcd_greatest)
720   have "gcd m n dvd (k * m)" and "gcd m n dvd n" by simp_all
721   then show "gcd m n dvd gcd (k * m) n" by (rule gcd_greatest)
722 qed
724 lemma coprime_crossproduct:
725   assumes [simp]: "gcd a d = 1" "gcd b c = 1"
726   shows "associated (a * c) (b * d) \<longleftrightarrow> associated a b \<and> associated c d" (is "?lhs \<longleftrightarrow> ?rhs")
727 proof
728   assume ?rhs then show ?lhs unfolding associated_def by (fast intro: mult_dvd_mono)
729 next
730   assume ?lhs
731   from ?lhs have "a dvd b * d" unfolding associated_def by (metis dvd_mult_left)
732   hence "a dvd b" by (simp add: coprime_dvd_mult_iff)
733   moreover from ?lhs have "b dvd a * c" unfolding associated_def by (metis dvd_mult_left)
734   hence "b dvd a" by (simp add: coprime_dvd_mult_iff)
735   moreover from ?lhs have "c dvd d * b"
736     unfolding associated_def by (auto dest: dvd_mult_right simp add: ac_simps)
737   hence "c dvd d" by (simp add: coprime_dvd_mult_iff gcd.commute)
738   moreover from ?lhs have "d dvd c * a"
739     unfolding associated_def by (auto dest: dvd_mult_right simp add: ac_simps)
740   hence "d dvd c" by (simp add: coprime_dvd_mult_iff gcd.commute)
741   ultimately show ?rhs unfolding associated_def by simp
742 qed
744 lemma gcd_add1 [simp]:
745   "gcd (m + n) n = gcd m n"
746   by (cases "n = 0", simp_all add: gcd_non_0)
748 lemma gcd_add2 [simp]:
749   "gcd m (m + n) = gcd m n"
750   using gcd_add1 [of n m] by (simp add: ac_simps)
752 lemma gcd_add_mult: "gcd m (k * m + n) = gcd m n"
753   by (subst gcd.commute, subst gcd_red, simp)
755 lemma coprimeI: "(\<And>l. \<lbrakk>l dvd x; l dvd y\<rbrakk> \<Longrightarrow> l dvd 1) \<Longrightarrow> gcd x y = 1"
756   by (rule sym, rule gcdI, simp_all)
758 lemma coprime: "gcd a b = 1 \<longleftrightarrow> (\<forall>d. d dvd a \<and> d dvd b \<longleftrightarrow> is_unit d)"
759   by (auto simp: is_unit_def intro: coprimeI gcd_greatest dvd_gcd_D1 dvd_gcd_D2)
761 lemma div_gcd_coprime:
762   assumes nz: "a \<noteq> 0 \<or> b \<noteq> 0"
763   defines [simp]: "d \<equiv> gcd a b"
764   defines [simp]: "a' \<equiv> a div d" and [simp]: "b' \<equiv> b div d"
765   shows "gcd a' b' = 1"
766 proof (rule coprimeI)
767   fix l assume "l dvd a'" "l dvd b'"
768   then obtain s t where "a' = l * s" "b' = l * t" unfolding dvd_def by blast
769   moreover have "a = a' * d" "b = b' * d" by simp_all
770   ultimately have "a = (l * d) * s" "b = (l * d) * t"
771     by (simp_all only: ac_simps)
772   hence "l*d dvd a" and "l*d dvd b" by (simp_all only: dvd_triv_left)
773   hence "l*d dvd d" by (simp add: gcd_greatest)
774   then obtain u where "d = l * d * u" ..
775   then have "d * (l * u) = d" by (simp add: ac_simps)
776   moreover from nz have "d \<noteq> 0" by simp
777   with div_mult_self1_is_id have "d * (l * u) div d = l * u" .
778   ultimately have "1 = l * u"
779     using d \<noteq> 0 by simp
780   then show "l dvd 1" ..
781 qed
783 lemma coprime_mult:
784   assumes da: "gcd d a = 1" and db: "gcd d b = 1"
785   shows "gcd d (a * b) = 1"
786   apply (subst gcd.commute)
787   using da apply (subst gcd_mult_cancel)
788   apply (subst gcd.commute, assumption)
789   apply (subst gcd.commute, rule db)
790   done
792 lemma coprime_lmult:
793   assumes dab: "gcd d (a * b) = 1"
794   shows "gcd d a = 1"
795 proof (rule coprimeI)
796   fix l assume "l dvd d" and "l dvd a"
797   hence "l dvd a * b" by simp
798   with l dvd d and dab show "l dvd 1" by (auto intro: gcd_greatest)
799 qed
801 lemma coprime_rmult:
802   assumes dab: "gcd d (a * b) = 1"
803   shows "gcd d b = 1"
804 proof (rule coprimeI)
805   fix l assume "l dvd d" and "l dvd b"
806   hence "l dvd a * b" by simp
807   with l dvd d and dab show "l dvd 1" by (auto intro: gcd_greatest)
808 qed
810 lemma coprime_mul_eq: "gcd d (a * b) = 1 \<longleftrightarrow> gcd d a = 1 \<and> gcd d b = 1"
811   using coprime_rmult[of d a b] coprime_lmult[of d a b] coprime_mult[of d a b] by blast
813 lemma gcd_coprime:
814   assumes z: "gcd a b \<noteq> 0" and a: "a = a' * gcd a b" and b: "b = b' * gcd a b"
815   shows "gcd a' b' = 1"
816 proof -
817   from z have "a \<noteq> 0 \<or> b \<noteq> 0" by simp
818   with div_gcd_coprime have "gcd (a div gcd a b) (b div gcd a b) = 1" .
819   also from assms have "a div gcd a b = a'" by (metis div_mult_self2_is_id)+
820   also from assms have "b div gcd a b = b'" by (metis div_mult_self2_is_id)+
821   finally show ?thesis .
822 qed
824 lemma coprime_power:
825   assumes "0 < n"
826   shows "gcd a (b ^ n) = 1 \<longleftrightarrow> gcd a b = 1"
827 using assms proof (induct n)
828   case (Suc n) then show ?case
829     by (cases n) (simp_all add: coprime_mul_eq)
830 qed simp
832 lemma gcd_coprime_exists:
833   assumes nz: "gcd a b \<noteq> 0"
834   shows "\<exists>a' b'. a = a' * gcd a b \<and> b = b' * gcd a b \<and> gcd a' b' = 1"
835   apply (rule_tac x = "a div gcd a b" in exI)
836   apply (rule_tac x = "b div gcd a b" in exI)
837   apply (insert nz, auto intro: div_gcd_coprime)
838   done
840 lemma coprime_exp:
841   "gcd d a = 1 \<Longrightarrow> gcd d (a^n) = 1"
842   by (induct n, simp_all add: coprime_mult)
844 lemma coprime_exp2 [intro]:
845   "gcd a b = 1 \<Longrightarrow> gcd (a^n) (b^m) = 1"
846   apply (rule coprime_exp)
847   apply (subst gcd.commute)
848   apply (rule coprime_exp)
849   apply (subst gcd.commute)
850   apply assumption
851   done
853 lemma gcd_exp:
854   "gcd (a^n) (b^n) = (gcd a b) ^ n"
855 proof (cases "a = 0 \<and> b = 0")
856   assume "a = 0 \<and> b = 0"
857   then show ?thesis by (cases n, simp_all add: gcd_0_left)
858 next
859   assume A: "\<not>(a = 0 \<and> b = 0)"
860   hence "1 = gcd ((a div gcd a b)^n) ((b div gcd a b)^n)"
861     using div_gcd_coprime by (subst sym, auto simp: div_gcd_coprime)
862   hence "(gcd a b) ^ n = (gcd a b) ^ n * ..." by simp
863   also note gcd_mult_distrib
864   also have "normalisation_factor ((gcd a b)^n) = 1"
865     by (simp add: normalisation_factor_pow A)
866   also have "(gcd a b)^n * (a div gcd a b)^n = a^n"
867     by (subst ac_simps, subst div_power, simp, rule dvd_div_mult_self, rule dvd_power_same, simp)
868   also have "(gcd a b)^n * (b div gcd a b)^n = b^n"
869     by (subst ac_simps, subst div_power, simp, rule dvd_div_mult_self, rule dvd_power_same, simp)
870   finally show ?thesis by simp
871 qed
873 lemma coprime_common_divisor:
874   "gcd a b = 1 \<Longrightarrow> x dvd a \<Longrightarrow> x dvd b \<Longrightarrow> is_unit x"
875   apply (subgoal_tac "x dvd gcd a b")
876   apply (simp add: is_unit_def)
877   apply (erule (1) gcd_greatest)
878   done
880 lemma division_decomp:
881   assumes dc: "a dvd b * c"
882   shows "\<exists>b' c'. a = b' * c' \<and> b' dvd b \<and> c' dvd c"
883 proof (cases "gcd a b = 0")
884   assume "gcd a b = 0"
885   hence "a = 0 \<and> b = 0" by simp
886   hence "a = 0 * c \<and> 0 dvd b \<and> c dvd c" by simp
887   then show ?thesis by blast
888 next
889   let ?d = "gcd a b"
890   assume "?d \<noteq> 0"
891   from gcd_coprime_exists[OF this]
892     obtain a' b' where ab': "a = a' * ?d" "b = b' * ?d" "gcd a' b' = 1"
893     by blast
894   from ab'(1) have "a' dvd a" unfolding dvd_def by blast
895   with dc have "a' dvd b*c" using dvd_trans[of a' a "b*c"] by simp
896   from dc ab'(1,2) have "a'*?d dvd (b'*?d) * c" by simp
897   hence "?d * a' dvd ?d * (b' * c)" by (simp add: mult_ac)
898   with ?d \<noteq> 0 have "a' dvd b' * c" by simp
899   with coprime_dvd_mult[OF ab'(3)]
900     have "a' dvd c" by (subst (asm) ac_simps, blast)
901   with ab'(1) have "a = ?d * a' \<and> ?d dvd b \<and> a' dvd c" by (simp add: mult_ac)
902   then show ?thesis by blast
903 qed
905 lemma pow_divides_pow:
906   assumes ab: "a ^ n dvd b ^ n" and n: "n \<noteq> 0"
907   shows "a dvd b"
908 proof (cases "gcd a b = 0")
909   assume "gcd a b = 0"
910   then show ?thesis by simp
911 next
912   let ?d = "gcd a b"
913   assume "?d \<noteq> 0"
914   from n obtain m where m: "n = Suc m" by (cases n, simp_all)
915   from ?d \<noteq> 0 have zn: "?d ^ n \<noteq> 0" by (rule power_not_zero)
916   from gcd_coprime_exists[OF ?d \<noteq> 0]
917     obtain a' b' where ab': "a = a' * ?d" "b = b' * ?d" "gcd a' b' = 1"
918     by blast
919   from ab have "(a' * ?d) ^ n dvd (b' * ?d) ^ n"
920     by (simp add: ab'(1,2)[symmetric])
921   hence "?d^n * a'^n dvd ?d^n * b'^n"
922     by (simp only: power_mult_distrib ac_simps)
923   with zn have "a'^n dvd b'^n" by simp
924   hence "a' dvd b'^n" using dvd_trans[of a' "a'^n" "b'^n"] by (simp add: m)
925   hence "a' dvd b'^m * b'" by (simp add: m ac_simps)
926   with coprime_dvd_mult[OF coprime_exp[OF ab'(3), of m]]
927     have "a' dvd b'" by (subst (asm) ac_simps, blast)
928   hence "a'*?d dvd b'*?d" by (rule mult_dvd_mono, simp)
929   with ab'(1,2) show ?thesis by simp
930 qed
932 lemma pow_divides_eq [simp]:
933   "n \<noteq> 0 \<Longrightarrow> a ^ n dvd b ^ n \<longleftrightarrow> a dvd b"
934   by (auto intro: pow_divides_pow dvd_power_same)
936 lemma divides_mult:
937   assumes mr: "m dvd r" and nr: "n dvd r" and mn: "gcd m n = 1"
938   shows "m * n dvd r"
939 proof -
940   from mr nr obtain m' n' where m': "r = m*m'" and n': "r = n*n'"
941     unfolding dvd_def by blast
942   from mr n' have "m dvd n'*n" by (simp add: ac_simps)
943   hence "m dvd n'" using coprime_dvd_mult_iff[OF mn] by simp
944   then obtain k where k: "n' = m*k" unfolding dvd_def by blast
945   with n' have "r = m * n * k" by (simp add: mult_ac)
946   then show ?thesis unfolding dvd_def by blast
947 qed
949 lemma coprime_plus_one [simp]: "gcd (n + 1) n = 1"
950   by (subst add_commute, simp)
952 lemma setprod_coprime [rule_format]:
953   "(\<forall>i\<in>A. gcd (f i) x = 1) \<longrightarrow> gcd (\<Prod>i\<in>A. f i) x = 1"
954   apply (cases "finite A")
955   apply (induct set: finite)
956   apply (auto simp add: gcd_mult_cancel)
957   done
959 lemma coprime_divisors:
960   assumes "d dvd a" "e dvd b" "gcd a b = 1"
961   shows "gcd d e = 1"
962 proof -
963   from assms obtain k l where "a = d * k" "b = e * l"
964     unfolding dvd_def by blast
965   with assms have "gcd (d * k) (e * l) = 1" by simp
966   hence "gcd (d * k) e = 1" by (rule coprime_lmult)
967   also have "gcd (d * k) e = gcd e (d * k)" by (simp add: ac_simps)
968   finally have "gcd e d = 1" by (rule coprime_lmult)
969   then show ?thesis by (simp add: ac_simps)
970 qed
972 lemma invertible_coprime:
973   assumes "x * y mod m = 1"
974   shows "coprime x m"
975 proof -
976   from assms have "coprime m (x * y mod m)"
977     by simp
978   then have "coprime m (x * y)"
979     by simp
980   then have "coprime m x"
981     by (rule coprime_lmult)
982   then show ?thesis
983     by (simp add: ac_simps)
984 qed
986 lemma lcm_gcd:
987   "lcm a b = a * b div (gcd a b * normalisation_factor (a*b))"
988   by (simp only: lcm_lcm_eucl gcd_gcd_eucl lcm_eucl_def)
990 lemma lcm_gcd_prod:
991   "lcm a b * gcd a b = a * b div normalisation_factor (a*b)"
992 proof (cases "a * b = 0")
993   let ?nf = normalisation_factor
994   assume "a * b \<noteq> 0"
995   hence "gcd a b \<noteq> 0" by simp
996   from lcm_gcd have "lcm a b * gcd a b = gcd a b * (a * b div (?nf (a*b) * gcd a b))"
997     by (simp add: mult_ac)
998   also from a * b \<noteq> 0 have "... = a * b div ?nf (a*b)"
999     by (simp_all add: unit_ring_inv'1 unit_ring_inv)
1000   finally show ?thesis .
1001 qed (auto simp add: lcm_gcd)
1003 lemma lcm_dvd1 [iff]:
1004   "x dvd lcm x y"
1005 proof (cases "x*y = 0")
1006   assume "x * y \<noteq> 0"
1007   hence "gcd x y \<noteq> 0" by simp
1008   let ?c = "ring_inv (normalisation_factor (x*y))"
1009   from x * y \<noteq> 0 have [simp]: "is_unit (normalisation_factor (x*y))" by simp
1010   from lcm_gcd_prod[of x y] have "lcm x y * gcd x y = x * ?c * y"
1011     by (simp add: mult_ac unit_ring_inv)
1012   hence "lcm x y * gcd x y div gcd x y = x * ?c * y div gcd x y" by simp
1013   with gcd x y \<noteq> 0 have "lcm x y = x * ?c * y div gcd x y"
1014     by (subst (asm) div_mult_self2_is_id, simp_all)
1015   also have "... = x * (?c * y div gcd x y)"
1016     by (metis div_mult_swap gcd_dvd2 mult_assoc)
1017   finally show ?thesis by (rule dvdI)
1018 qed (auto simp add: lcm_gcd)
1020 lemma lcm_least:
1021   "\<lbrakk>a dvd k; b dvd k\<rbrakk> \<Longrightarrow> lcm a b dvd k"
1022 proof (cases "k = 0")
1023   let ?nf = normalisation_factor
1024   assume "k \<noteq> 0"
1025   hence "is_unit (?nf k)" by simp
1026   hence "?nf k \<noteq> 0" by (metis not_is_unit_0)
1027   assume A: "a dvd k" "b dvd k"
1028   hence "gcd a b \<noteq> 0" using k \<noteq> 0 by auto
1029   from A obtain r s where ar: "k = a * r" and bs: "k = b * s"
1030     unfolding dvd_def by blast
1031   with k \<noteq> 0 have "r * s \<noteq> 0"
1032     by auto (drule sym [of 0], simp)
1033   hence "is_unit (?nf (r * s))" by simp
1034   let ?c = "?nf k div ?nf (r*s)"
1035   from is_unit (?nf k) and is_unit (?nf (r * s)) have "is_unit ?c" by (rule unit_div)
1036   hence "?c \<noteq> 0" using not_is_unit_0 by fast
1037   from ar bs have "k * k * gcd s r = ?nf k * k * gcd (k * s) (k * r)"
1038     by (subst mult_assoc, subst gcd_mult_distrib[of k s r], simp only: ac_simps)
1039   also have "... = ?nf k * k * gcd ((r*s) * a) ((r*s) * b)"
1040     by (subst (3) k = a * r, subst (3) k = b * s, simp add: algebra_simps)
1041   also have "... = ?c * r*s * k * gcd a b" using r * s \<noteq> 0
1042     by (subst gcd_mult_distrib'[symmetric], simp add: algebra_simps unit_simps)
1043   finally have "(a*r) * (b*s) * gcd s r = ?c * k * r * s * gcd a b"
1044     by (subst ar[symmetric], subst bs[symmetric], simp add: mult_ac)
1045   hence "a * b * gcd s r * (r * s) = ?c * k * gcd a b * (r * s)"
1046     by (simp add: algebra_simps)
1047   hence "?c * k * gcd a b = a * b * gcd s r" using r * s \<noteq> 0
1048     by (metis div_mult_self2_is_id)
1049   also have "... = lcm a b * gcd a b * gcd s r * ?nf (a*b)"
1050     by (subst lcm_gcd_prod[of a b], metis gcd_mult_distrib gcd_mult_distrib')
1051   also have "... = lcm a b * gcd s r * ?nf (a*b) * gcd a b"
1052     by (simp add: algebra_simps)
1053   finally have "k * ?c = lcm a b * gcd s r * ?nf (a*b)" using gcd a b \<noteq> 0
1054     by (metis mult.commute div_mult_self2_is_id)
1055   hence "k = lcm a b * (gcd s r * ?nf (a*b)) div ?c" using ?c \<noteq> 0
1056     by (metis div_mult_self2_is_id mult_assoc)
1057   also have "... = lcm a b * (gcd s r * ?nf (a*b) div ?c)" using is_unit ?c
1058     by (simp add: unit_simps)
1059   finally show ?thesis by (rule dvdI)
1060 qed simp
1062 lemma lcm_zero:
1063   "lcm a b = 0 \<longleftrightarrow> a = 0 \<or> b = 0"
1064 proof -
1065   let ?nf = normalisation_factor
1066   {
1067     assume "a \<noteq> 0" "b \<noteq> 0"
1068     hence "a * b div ?nf (a * b) \<noteq> 0" by (simp add: no_zero_divisors)
1069     moreover from a \<noteq> 0 and b \<noteq> 0 have "gcd a b \<noteq> 0" by simp
1070     ultimately have "lcm a b \<noteq> 0" using lcm_gcd_prod[of a b] by (intro notI, simp)
1071   } moreover {
1072     assume "a = 0 \<or> b = 0"
1073     hence "lcm a b = 0" by (elim disjE, simp_all add: lcm_gcd)
1074   }
1075   ultimately show ?thesis by blast
1076 qed
1078 lemmas lcm_0_iff = lcm_zero
1080 lemma gcd_lcm:
1081   assumes "lcm a b \<noteq> 0"
1082   shows "gcd a b = a * b div (lcm a b * normalisation_factor (a * b))"
1083 proof-
1084   from assms have "gcd a b \<noteq> 0" by (simp add: lcm_zero)
1085   let ?c = "normalisation_factor (a*b)"
1086   from lcm a b \<noteq> 0 have "?c \<noteq> 0" by (intro notI, simp add: lcm_zero no_zero_divisors)
1087   hence "is_unit ?c" by simp
1088   from lcm_gcd_prod [of a b] have "gcd a b = a * b div ?c div lcm a b"
1089     by (subst (2) div_mult_self2_is_id[OF lcm a b \<noteq> 0, symmetric], simp add: mult_ac)
1090   also from is_unit ?c have "... = a * b div (?c * lcm a b)"
1091     by (simp only: unit_ring_inv'1 unit_ring_inv)
1092   finally show ?thesis by (simp only: ac_simps)
1093 qed
1095 lemma normalisation_factor_lcm [simp]:
1096   "normalisation_factor (lcm a b) = (if a = 0 \<or> b = 0 then 0 else 1)"
1097 proof (cases "a = 0 \<or> b = 0")
1098   case True then show ?thesis
1099     by (auto simp add: lcm_gcd)
1100 next
1101   case False
1102   let ?nf = normalisation_factor
1103   from lcm_gcd_prod[of a b]
1104     have "?nf (lcm a b) * ?nf (gcd a b) = ?nf (a*b) div ?nf (a*b)"
1105     by (metis div_by_0 div_self normalisation_correct normalisation_factor_0 normalisation_factor_mult)
1106   also have "... = (if a*b = 0 then 0 else 1)"
1107     by simp
1108   finally show ?thesis using False by simp
1109 qed
1111 lemma lcm_dvd2 [iff]: "y dvd lcm x y"
1112   using lcm_dvd1 [of y x] by (simp add: lcm_gcd ac_simps)
1114 lemma lcmI:
1115   "\<lbrakk>x dvd k; y dvd k; \<And>l. x dvd l \<Longrightarrow> y dvd l \<Longrightarrow> k dvd l;
1116     normalisation_factor k = (if k = 0 then 0 else 1)\<rbrakk> \<Longrightarrow> k = lcm x y"
1117   by (intro normed_associated_imp_eq) (auto simp: associated_def intro: lcm_least)
1119 sublocale lcm!: abel_semigroup lcm
1120 proof
1121   fix x y z
1122   show "lcm (lcm x y) z = lcm x (lcm y z)"
1123   proof (rule lcmI)
1124     have "x dvd lcm x y" and "lcm x y dvd lcm (lcm x y) z" by simp_all
1125     then show "x dvd lcm (lcm x y) z" by (rule dvd_trans)
1127     have "y dvd lcm x y" and "lcm x y dvd lcm (lcm x y) z" by simp_all
1128     hence "y dvd lcm (lcm x y) z" by (rule dvd_trans)
1129     moreover have "z dvd lcm (lcm x y) z" by simp
1130     ultimately show "lcm y z dvd lcm (lcm x y) z" by (rule lcm_least)
1132     fix l assume "x dvd l" and "lcm y z dvd l"
1133     have "y dvd lcm y z" by simp
1134     from this and lcm y z dvd l have "y dvd l" by (rule dvd_trans)
1135     have "z dvd lcm y z" by simp
1136     from this and lcm y z dvd l have "z dvd l" by (rule dvd_trans)
1137     from x dvd l and y dvd l have "lcm x y dvd l" by (rule lcm_least)
1138     from this and z dvd l show "lcm (lcm x y) z dvd l" by (rule lcm_least)
1139   qed (simp add: lcm_zero)
1140 next
1141   fix x y
1142   show "lcm x y = lcm y x"
1143     by (simp add: lcm_gcd ac_simps)
1144 qed
1146 lemma dvd_lcm_D1:
1147   "lcm m n dvd k \<Longrightarrow> m dvd k"
1148   by (rule dvd_trans, rule lcm_dvd1, assumption)
1150 lemma dvd_lcm_D2:
1151   "lcm m n dvd k \<Longrightarrow> n dvd k"
1152   by (rule dvd_trans, rule lcm_dvd2, assumption)
1154 lemma gcd_dvd_lcm [simp]:
1155   "gcd a b dvd lcm a b"
1156   by (metis dvd_trans gcd_dvd2 lcm_dvd2)
1158 lemma lcm_1_iff:
1159   "lcm a b = 1 \<longleftrightarrow> is_unit a \<and> is_unit b"
1160 proof
1161   assume "lcm a b = 1"
1162   then show "is_unit a \<and> is_unit b" unfolding is_unit_def by auto
1163 next
1164   assume "is_unit a \<and> is_unit b"
1165   hence "a dvd 1" and "b dvd 1" unfolding is_unit_def by simp_all
1166   hence "is_unit (lcm a b)" unfolding is_unit_def by (rule lcm_least)
1167   hence "lcm a b = normalisation_factor (lcm a b)"
1168     by (subst normalisation_factor_unit, simp_all)
1169   also have "\<dots> = 1" using is_unit a \<and> is_unit b by (auto simp add: is_unit_def)
1170   finally show "lcm a b = 1" .
1171 qed
1173 lemma lcm_0_left [simp]:
1174   "lcm 0 x = 0"
1175   by (rule sym, rule lcmI, simp_all)
1177 lemma lcm_0 [simp]:
1178   "lcm x 0 = 0"
1179   by (rule sym, rule lcmI, simp_all)
1181 lemma lcm_unique:
1182   "a dvd d \<and> b dvd d \<and>
1183   normalisation_factor d = (if d = 0 then 0 else 1) \<and>
1184   (\<forall>e. a dvd e \<and> b dvd e \<longrightarrow> d dvd e) \<longleftrightarrow> d = lcm a b"
1185   by (rule, auto intro: lcmI simp: lcm_least lcm_zero)
1187 lemma dvd_lcm_I1 [simp]:
1188   "k dvd m \<Longrightarrow> k dvd lcm m n"
1189   by (metis lcm_dvd1 dvd_trans)
1191 lemma dvd_lcm_I2 [simp]:
1192   "k dvd n \<Longrightarrow> k dvd lcm m n"
1193   by (metis lcm_dvd2 dvd_trans)
1195 lemma lcm_1_left [simp]:
1196   "lcm 1 x = x div normalisation_factor x"
1197   by (cases "x = 0") (simp, rule sym, rule lcmI, simp_all)
1199 lemma lcm_1_right [simp]:
1200   "lcm x 1 = x div normalisation_factor x"
1201   by (simp add: ac_simps)
1203 lemma lcm_coprime:
1204   "gcd a b = 1 \<Longrightarrow> lcm a b = a * b div normalisation_factor (a*b)"
1205   by (subst lcm_gcd) simp
1207 lemma lcm_proj1_if_dvd:
1208   "y dvd x \<Longrightarrow> lcm x y = x div normalisation_factor x"
1209   by (cases "x = 0") (simp, rule sym, rule lcmI, simp_all)
1211 lemma lcm_proj2_if_dvd:
1212   "x dvd y \<Longrightarrow> lcm x y = y div normalisation_factor y"
1213   using lcm_proj1_if_dvd [of x y] by (simp add: ac_simps)
1215 lemma lcm_proj1_iff:
1216   "lcm m n = m div normalisation_factor m \<longleftrightarrow> n dvd m"
1217 proof
1218   assume A: "lcm m n = m div normalisation_factor m"
1219   show "n dvd m"
1220   proof (cases "m = 0")
1221     assume [simp]: "m \<noteq> 0"
1222     from A have B: "m = lcm m n * normalisation_factor m"
1223       by (simp add: unit_eq_div2)
1224     show ?thesis by (subst B, simp)
1225   qed simp
1226 next
1227   assume "n dvd m"
1228   then show "lcm m n = m div normalisation_factor m" by (rule lcm_proj1_if_dvd)
1229 qed
1231 lemma lcm_proj2_iff:
1232   "lcm m n = n div normalisation_factor n \<longleftrightarrow> m dvd n"
1233   using lcm_proj1_iff [of n m] by (simp add: ac_simps)
1235 lemma euclidean_size_lcm_le1:
1236   assumes "a \<noteq> 0" and "b \<noteq> 0"
1237   shows "euclidean_size a \<le> euclidean_size (lcm a b)"
1238 proof -
1239   have "a dvd lcm a b" by (rule lcm_dvd1)
1240   then obtain c where A: "lcm a b = a * c" unfolding dvd_def by blast
1241   with a \<noteq> 0 and b \<noteq> 0 have "c \<noteq> 0" by (auto simp: lcm_zero)
1242   then show ?thesis by (subst A, intro size_mult_mono)
1243 qed
1245 lemma euclidean_size_lcm_le2:
1246   "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> euclidean_size b \<le> euclidean_size (lcm a b)"
1247   using euclidean_size_lcm_le1 [of b a] by (simp add: ac_simps)
1249 lemma euclidean_size_lcm_less1:
1250   assumes "b \<noteq> 0" and "\<not>b dvd a"
1251   shows "euclidean_size a < euclidean_size (lcm a b)"
1252 proof (rule ccontr)
1253   from assms have "a \<noteq> 0" by auto
1254   assume "\<not>euclidean_size a < euclidean_size (lcm a b)"
1255   with a \<noteq> 0 and b \<noteq> 0 have "euclidean_size (lcm a b) = euclidean_size a"
1256     by (intro le_antisym, simp, intro euclidean_size_lcm_le1)
1257   with assms have "lcm a b dvd a"
1258     by (rule_tac dvd_euclidean_size_eq_imp_dvd) (auto simp: lcm_zero)
1259   hence "b dvd a" by (rule dvd_lcm_D2)
1260   with \<not>b dvd a show False by contradiction
1261 qed
1263 lemma euclidean_size_lcm_less2:
1264   assumes "a \<noteq> 0" and "\<not>a dvd b"
1265   shows "euclidean_size b < euclidean_size (lcm a b)"
1266   using assms euclidean_size_lcm_less1 [of a b] by (simp add: ac_simps)
1268 lemma lcm_mult_unit1:
1269   "is_unit a \<Longrightarrow> lcm (x*a) y = lcm x y"
1270   apply (rule lcmI)
1271   apply (rule dvd_trans[of _ "x*a"], simp, rule lcm_dvd1)
1272   apply (rule lcm_dvd2)
1273   apply (rule lcm_least, simp add: unit_simps, assumption)
1274   apply (subst normalisation_factor_lcm, simp add: lcm_zero)
1275   done
1277 lemma lcm_mult_unit2:
1278   "is_unit a \<Longrightarrow> lcm x (y*a) = lcm x y"
1279   using lcm_mult_unit1 [of a y x] by (simp add: ac_simps)
1281 lemma lcm_div_unit1:
1282   "is_unit a \<Longrightarrow> lcm (x div a) y = lcm x y"
1283   by (simp add: unit_ring_inv lcm_mult_unit1)
1285 lemma lcm_div_unit2:
1286   "is_unit a \<Longrightarrow> lcm x (y div a) = lcm x y"
1287   by (simp add: unit_ring_inv lcm_mult_unit2)
1289 lemma lcm_left_idem:
1290   "lcm p (lcm p q) = lcm p q"
1291   apply (rule lcmI)
1292   apply simp
1293   apply (subst lcm.assoc [symmetric], rule lcm_dvd2)
1294   apply (rule lcm_least, assumption)
1295   apply (erule (1) lcm_least)
1296   apply (auto simp: lcm_zero)
1297   done
1299 lemma lcm_right_idem:
1300   "lcm (lcm p q) q = lcm p q"
1301   apply (rule lcmI)
1302   apply (subst lcm.assoc, rule lcm_dvd1)
1303   apply (rule lcm_dvd2)
1304   apply (rule lcm_least, erule (1) lcm_least, assumption)
1305   apply (auto simp: lcm_zero)
1306   done
1308 lemma comp_fun_idem_lcm: "comp_fun_idem lcm"
1309 proof
1310   fix a b show "lcm a \<circ> lcm b = lcm b \<circ> lcm a"
1311     by (simp add: fun_eq_iff ac_simps)
1312 next
1313   fix a show "lcm a \<circ> lcm a = lcm a" unfolding o_def
1314     by (intro ext, simp add: lcm_left_idem)
1315 qed
1317 lemma dvd_Lcm [simp]: "x \<in> A \<Longrightarrow> x dvd Lcm A"
1318   and Lcm_dvd [simp]: "(\<forall>x\<in>A. x dvd l') \<Longrightarrow> Lcm A dvd l'"
1319   and normalisation_factor_Lcm [simp]:
1320           "normalisation_factor (Lcm A) = (if Lcm A = 0 then 0 else 1)"
1321 proof -
1322   have "(\<forall>x\<in>A. x dvd Lcm A) \<and> (\<forall>l'. (\<forall>x\<in>A. x dvd l') \<longrightarrow> Lcm A dvd l') \<and>
1323     normalisation_factor (Lcm A) = (if Lcm A = 0 then 0 else 1)" (is ?thesis)
1324   proof (cases "\<exists>l. l \<noteq>  0 \<and> (\<forall>x\<in>A. x dvd l)")
1325     case False
1326     hence "Lcm A = 0" by (auto simp: Lcm_Lcm_eucl Lcm_eucl_def)
1327     with False show ?thesis by auto
1328   next
1329     case True
1330     then obtain l\<^sub>0 where l\<^sub>0_props: "l\<^sub>0 \<noteq> 0 \<and> (\<forall>x\<in>A. x dvd l\<^sub>0)" by blast
1331     def n \<equiv> "LEAST n. \<exists>l. l \<noteq> 0 \<and> (\<forall>x\<in>A. x dvd l) \<and> euclidean_size l = n"
1332     def l \<equiv> "SOME l. l \<noteq> 0 \<and> (\<forall>x\<in>A. x dvd l) \<and> euclidean_size l = n"
1333     have "\<exists>l. l \<noteq> 0 \<and> (\<forall>x\<in>A. x dvd l) \<and> euclidean_size l = n"
1334       apply (subst n_def)
1335       apply (rule LeastI[of _ "euclidean_size l\<^sub>0"])
1336       apply (rule exI[of _ l\<^sub>0])
1337       apply (simp add: l\<^sub>0_props)
1338       done
1339     from someI_ex[OF this] have "l \<noteq> 0" and "\<forall>x\<in>A. x dvd l" and "euclidean_size l = n"
1340       unfolding l_def by simp_all
1341     {
1342       fix l' assume "\<forall>x\<in>A. x dvd l'"
1343       with \<forall>x\<in>A. x dvd l have "\<forall>x\<in>A. x dvd gcd l l'" by (auto intro: gcd_greatest)
1344       moreover from l \<noteq> 0 have "gcd l l' \<noteq> 0" by simp
1345       ultimately have "\<exists>b. b \<noteq> 0 \<and> (\<forall>x\<in>A. x dvd b) \<and> euclidean_size b = euclidean_size (gcd l l')"
1346         by (intro exI[of _ "gcd l l'"], auto)
1347       hence "euclidean_size (gcd l l') \<ge> n" by (subst n_def) (rule Least_le)
1348       moreover have "euclidean_size (gcd l l') \<le> n"
1349       proof -
1350         have "gcd l l' dvd l" by simp
1351         then obtain a where "l = gcd l l' * a" unfolding dvd_def by blast
1352         with l \<noteq> 0 have "a \<noteq> 0" by auto
1353         hence "euclidean_size (gcd l l') \<le> euclidean_size (gcd l l' * a)"
1354           by (rule size_mult_mono)
1355         also have "gcd l l' * a = l" using l = gcd l l' * a ..
1356         also note euclidean_size l = n
1357         finally show "euclidean_size (gcd l l') \<le> n" .
1358       qed
1359       ultimately have "euclidean_size l = euclidean_size (gcd l l')"
1360         by (intro le_antisym, simp_all add: euclidean_size l = n)
1361       with l \<noteq> 0 have "l dvd gcd l l'" by (blast intro: dvd_euclidean_size_eq_imp_dvd)
1362       hence "l dvd l'" by (blast dest: dvd_gcd_D2)
1363     }
1365     with (\<forall>x\<in>A. x dvd l) and normalisation_factor_is_unit[OF l \<noteq> 0] and l \<noteq> 0
1366       have "(\<forall>x\<in>A. x dvd l div normalisation_factor l) \<and>
1367         (\<forall>l'. (\<forall>x\<in>A. x dvd l') \<longrightarrow> l div normalisation_factor l dvd l') \<and>
1368         normalisation_factor (l div normalisation_factor l) =
1369         (if l div normalisation_factor l = 0 then 0 else 1)"
1370       by (auto simp: unit_simps)
1371     also from True have "l div normalisation_factor l = Lcm A"
1372       by (simp add: Lcm_Lcm_eucl Lcm_eucl_def Let_def n_def l_def)
1373     finally show ?thesis .
1374   qed
1375   note A = this
1377   {fix x assume "x \<in> A" then show "x dvd Lcm A" using A by blast}
1378   {fix l' assume "\<forall>x\<in>A. x dvd l'" then show "Lcm A dvd l'" using A by blast}
1379   from A show "normalisation_factor (Lcm A) = (if Lcm A = 0 then 0 else 1)" by blast
1380 qed
1382 lemma LcmI:
1383   "(\<And>x. x\<in>A \<Longrightarrow> x dvd l) \<Longrightarrow> (\<And>l'. (\<forall>x\<in>A. x dvd l') \<Longrightarrow> l dvd l') \<Longrightarrow>
1384       normalisation_factor l = (if l = 0 then 0 else 1) \<Longrightarrow> l = Lcm A"
1385   by (intro normed_associated_imp_eq)
1386     (auto intro: Lcm_dvd dvd_Lcm simp: associated_def)
1388 lemma Lcm_subset:
1389   "A \<subseteq> B \<Longrightarrow> Lcm A dvd Lcm B"
1390   by (blast intro: Lcm_dvd dvd_Lcm)
1392 lemma Lcm_Un:
1393   "Lcm (A \<union> B) = lcm (Lcm A) (Lcm B)"
1394   apply (rule lcmI)
1395   apply (blast intro: Lcm_subset)
1396   apply (blast intro: Lcm_subset)
1397   apply (intro Lcm_dvd ballI, elim UnE)
1398   apply (rule dvd_trans, erule dvd_Lcm, assumption)
1399   apply (rule dvd_trans, erule dvd_Lcm, assumption)
1400   apply simp
1401   done
1403 lemma Lcm_1_iff:
1404   "Lcm A = 1 \<longleftrightarrow> (\<forall>x\<in>A. is_unit x)"
1405 proof
1406   assume "Lcm A = 1"
1407   then show "\<forall>x\<in>A. is_unit x" unfolding is_unit_def by auto
1408 qed (rule LcmI [symmetric], auto)
1410 lemma Lcm_no_units:
1411   "Lcm A = Lcm (A - {x. is_unit x})"
1412 proof -
1413   have "(A - {x. is_unit x}) \<union> {x\<in>A. is_unit x} = A" by blast
1414   hence "Lcm A = lcm (Lcm (A - {x. is_unit x})) (Lcm {x\<in>A. is_unit x})"
1415     by (simp add: Lcm_Un[symmetric])
1416   also have "Lcm {x\<in>A. is_unit x} = 1" by (simp add: Lcm_1_iff)
1417   finally show ?thesis by simp
1418 qed
1420 lemma Lcm_empty [simp]:
1421   "Lcm {} = 1"
1422   by (simp add: Lcm_1_iff)
1424 lemma Lcm_eq_0 [simp]:
1425   "0 \<in> A \<Longrightarrow> Lcm A = 0"
1426   by (drule dvd_Lcm) simp
1428 lemma Lcm0_iff':
1429   "Lcm A = 0 \<longleftrightarrow> \<not>(\<exists>l. l \<noteq> 0 \<and> (\<forall>x\<in>A. x dvd l))"
1430 proof
1431   assume "Lcm A = 0"
1432   show "\<not>(\<exists>l. l \<noteq> 0 \<and> (\<forall>x\<in>A. x dvd l))"
1433   proof
1434     assume ex: "\<exists>l. l \<noteq> 0 \<and> (\<forall>x\<in>A. x dvd l)"
1435     then obtain l\<^sub>0 where l\<^sub>0_props: "l\<^sub>0 \<noteq> 0 \<and> (\<forall>x\<in>A. x dvd l\<^sub>0)" by blast
1436     def n \<equiv> "LEAST n. \<exists>l. l \<noteq> 0 \<and> (\<forall>x\<in>A. x dvd l) \<and> euclidean_size l = n"
1437     def l \<equiv> "SOME l. l \<noteq> 0 \<and> (\<forall>x\<in>A. x dvd l) \<and> euclidean_size l = n"
1438     have "\<exists>l. l \<noteq> 0 \<and> (\<forall>x\<in>A. x dvd l) \<and> euclidean_size l = n"
1439       apply (subst n_def)
1440       apply (rule LeastI[of _ "euclidean_size l\<^sub>0"])
1441       apply (rule exI[of _ l\<^sub>0])
1442       apply (simp add: l\<^sub>0_props)
1443       done
1444     from someI_ex[OF this] have "l \<noteq> 0" unfolding l_def by simp_all
1445     hence "l div normalisation_factor l \<noteq> 0" by simp
1446     also from ex have "l div normalisation_factor l = Lcm A"
1447        by (simp only: Lcm_Lcm_eucl Lcm_eucl_def n_def l_def if_True Let_def)
1448     finally show False using Lcm A = 0 by contradiction
1449   qed
1450 qed (simp only: Lcm_Lcm_eucl Lcm_eucl_def if_False)
1452 lemma Lcm0_iff [simp]:
1453   "finite A \<Longrightarrow> Lcm A = 0 \<longleftrightarrow> 0 \<in> A"
1454 proof -
1455   assume "finite A"
1456   have "0 \<in> A \<Longrightarrow> Lcm A = 0"  by (intro dvd_0_left dvd_Lcm)
1457   moreover {
1458     assume "0 \<notin> A"
1459     hence "\<Prod>A \<noteq> 0"
1460       apply (induct rule: finite_induct[OF finite A])
1461       apply simp
1462       apply (subst setprod.insert, assumption, assumption)
1463       apply (rule no_zero_divisors)
1464       apply blast+
1465       done
1466     moreover from finite A have "\<forall>x\<in>A. x dvd \<Prod>A" by (intro ballI dvd_setprod)
1467     ultimately have "\<exists>l. l \<noteq> 0 \<and> (\<forall>x\<in>A. x dvd l)" by blast
1468     with Lcm0_iff' have "Lcm A \<noteq> 0" by simp
1469   }
1470   ultimately show "Lcm A = 0 \<longleftrightarrow> 0 \<in> A" by blast
1471 qed
1473 lemma Lcm_no_multiple:
1474   "(\<forall>m. m \<noteq> 0 \<longrightarrow> (\<exists>x\<in>A. \<not>x dvd m)) \<Longrightarrow> Lcm A = 0"
1475 proof -
1476   assume "\<forall>m. m \<noteq> 0 \<longrightarrow> (\<exists>x\<in>A. \<not>x dvd m)"
1477   hence "\<not>(\<exists>l. l \<noteq> 0 \<and> (\<forall>x\<in>A. x dvd l))" by blast
1478   then show "Lcm A = 0" by (simp only: Lcm_Lcm_eucl Lcm_eucl_def if_False)
1479 qed
1481 lemma Lcm_insert [simp]:
1482   "Lcm (insert a A) = lcm a (Lcm A)"
1483 proof (rule lcmI)
1484   fix l assume "a dvd l" and "Lcm A dvd l"
1485   hence "\<forall>x\<in>A. x dvd l" by (blast intro: dvd_trans dvd_Lcm)
1486   with a dvd l show "Lcm (insert a A) dvd l" by (force intro: Lcm_dvd)
1487 qed (auto intro: Lcm_dvd dvd_Lcm)
1489 lemma Lcm_finite:
1490   assumes "finite A"
1491   shows "Lcm A = Finite_Set.fold lcm 1 A"
1492   by (induct rule: finite.induct[OF finite A])
1493     (simp_all add: comp_fun_idem.fold_insert_idem[OF comp_fun_idem_lcm])
1495 lemma Lcm_set [code, code_unfold]:
1496   "Lcm (set xs) = fold lcm xs 1"
1497   using comp_fun_idem.fold_set_fold[OF comp_fun_idem_lcm] Lcm_finite by (simp add: ac_simps)
1499 lemma Lcm_singleton [simp]:
1500   "Lcm {a} = a div normalisation_factor a"
1501   by simp
1503 lemma Lcm_2 [simp]:
1504   "Lcm {a,b} = lcm a b"
1505   by (simp only: Lcm_insert Lcm_empty lcm_1_right)
1506     (cases "b = 0", simp, rule lcm_div_unit2, simp)
1508 lemma Lcm_coprime:
1509   assumes "finite A" and "A \<noteq> {}"
1510   assumes "\<And>a b. a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<noteq> b \<Longrightarrow> gcd a b = 1"
1511   shows "Lcm A = \<Prod>A div normalisation_factor (\<Prod>A)"
1512 using assms proof (induct rule: finite_ne_induct)
1513   case (insert a A)
1514   have "Lcm (insert a A) = lcm a (Lcm A)" by simp
1515   also from insert have "Lcm A = \<Prod>A div normalisation_factor (\<Prod>A)" by blast
1516   also have "lcm a \<dots> = lcm a (\<Prod>A)" by (cases "\<Prod>A = 0") (simp_all add: lcm_div_unit2)
1517   also from insert have "gcd a (\<Prod>A) = 1" by (subst gcd.commute, intro setprod_coprime) auto
1518   with insert have "lcm a (\<Prod>A) = \<Prod>(insert a A) div normalisation_factor (\<Prod>(insert a A))"
1519     by (simp add: lcm_coprime)
1520   finally show ?case .
1521 qed simp
1523 lemma Lcm_coprime':
1524   "card A \<noteq> 0 \<Longrightarrow> (\<And>a b. a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<noteq> b \<Longrightarrow> gcd a b = 1)
1525     \<Longrightarrow> Lcm A = \<Prod>A div normalisation_factor (\<Prod>A)"
1526   by (rule Lcm_coprime) (simp_all add: card_eq_0_iff)
1528 lemma Gcd_Lcm:
1529   "Gcd A = Lcm {d. \<forall>x\<in>A. d dvd x}"
1530   by (simp add: Gcd_Gcd_eucl Lcm_Lcm_eucl Gcd_eucl_def)
1532 lemma Gcd_dvd [simp]: "x \<in> A \<Longrightarrow> Gcd A dvd x"
1533   and dvd_Gcd [simp]: "(\<forall>x\<in>A. g' dvd x) \<Longrightarrow> g' dvd Gcd A"
1534   and normalisation_factor_Gcd [simp]:
1535     "normalisation_factor (Gcd A) = (if Gcd A = 0 then 0 else 1)"
1536 proof -
1537   fix x assume "x \<in> A"
1538   hence "Lcm {d. \<forall>x\<in>A. d dvd x} dvd x" by (intro Lcm_dvd) blast
1539   then show "Gcd A dvd x" by (simp add: Gcd_Lcm)
1540 next
1541   fix g' assume "\<forall>x\<in>A. g' dvd x"
1542   hence "g' dvd Lcm {d. \<forall>x\<in>A. d dvd x}" by (intro dvd_Lcm) blast
1543   then show "g' dvd Gcd A" by (simp add: Gcd_Lcm)
1544 next
1545   show "normalisation_factor (Gcd A) = (if Gcd A = 0 then 0 else 1)"
1546     by (simp add: Gcd_Lcm)
1547 qed
1549 lemma GcdI:
1550   "(\<And>x. x\<in>A \<Longrightarrow> l dvd x) \<Longrightarrow> (\<And>l'. (\<forall>x\<in>A. l' dvd x) \<Longrightarrow> l' dvd l) \<Longrightarrow>
1551     normalisation_factor l = (if l = 0 then 0 else 1) \<Longrightarrow> l = Gcd A"
1552   by (intro normed_associated_imp_eq)
1553     (auto intro: Gcd_dvd dvd_Gcd simp: associated_def)
1555 lemma Lcm_Gcd:
1556   "Lcm A = Gcd {m. \<forall>x\<in>A. x dvd m}"
1557   by (rule LcmI[symmetric]) (auto intro: dvd_Gcd Gcd_dvd)
1559 lemma Gcd_0_iff:
1560   "Gcd A = 0 \<longleftrightarrow> A \<subseteq> {0}"
1561   apply (rule iffI)
1562   apply (rule subsetI, drule Gcd_dvd, simp)
1563   apply (auto intro: GcdI[symmetric])
1564   done
1566 lemma Gcd_empty [simp]:
1567   "Gcd {} = 0"
1568   by (simp add: Gcd_0_iff)
1570 lemma Gcd_1:
1571   "1 \<in> A \<Longrightarrow> Gcd A = 1"
1572   by (intro GcdI[symmetric]) (auto intro: Gcd_dvd dvd_Gcd)
1574 lemma Gcd_insert [simp]:
1575   "Gcd (insert a A) = gcd a (Gcd A)"
1576 proof (rule gcdI)
1577   fix l assume "l dvd a" and "l dvd Gcd A"
1578   hence "\<forall>x\<in>A. l dvd x" by (blast intro: dvd_trans Gcd_dvd)
1579   with l dvd a show "l dvd Gcd (insert a A)" by (force intro: Gcd_dvd)
1580 qed auto
1582 lemma Gcd_finite:
1583   assumes "finite A"
1584   shows "Gcd A = Finite_Set.fold gcd 0 A"
1585   by (induct rule: finite.induct[OF finite A])
1586     (simp_all add: comp_fun_idem.fold_insert_idem[OF comp_fun_idem_gcd])
1588 lemma Gcd_set [code, code_unfold]:
1589   "Gcd (set xs) = fold gcd xs 0"
1590   using comp_fun_idem.fold_set_fold[OF comp_fun_idem_gcd] Gcd_finite by (simp add: ac_simps)
1592 lemma Gcd_singleton [simp]: "Gcd {a} = a div normalisation_factor a"
1593   by (simp add: gcd_0)
1595 lemma Gcd_2 [simp]: "Gcd {a,b} = gcd a b"
1596   by (simp only: Gcd_insert Gcd_empty gcd_0) (cases "b = 0", simp, rule gcd_div_unit2, simp)
1598 end
1600 text {*
1601   A Euclidean ring is a Euclidean semiring with additive inverses. It provides a
1602   few more lemmas; in particular, Bezout's lemma holds for any Euclidean ring.
1603 *}
1605 class euclidean_ring = euclidean_semiring + idom
1607 class euclidean_ring_gcd = euclidean_semiring_gcd + idom
1608 begin
1610 subclass euclidean_ring ..
1612 lemma gcd_neg1 [simp]:
1613   "gcd (-x) y = gcd x y"
1614   by (rule sym, rule gcdI, simp_all add: gcd_greatest)
1616 lemma gcd_neg2 [simp]:
1617   "gcd x (-y) = gcd x y"
1618   by (rule sym, rule gcdI, simp_all add: gcd_greatest)
1620 lemma gcd_neg_numeral_1 [simp]:
1621   "gcd (- numeral n) x = gcd (numeral n) x"
1622   by (fact gcd_neg1)
1624 lemma gcd_neg_numeral_2 [simp]:
1625   "gcd x (- numeral n) = gcd x (numeral n)"
1626   by (fact gcd_neg2)
1628 lemma gcd_diff1: "gcd (m - n) n = gcd m n"
1629   by (subst diff_conv_add_uminus, subst gcd_neg2[symmetric],  subst gcd_add1, simp)
1631 lemma gcd_diff2: "gcd (n - m) n = gcd m n"
1632   by (subst gcd_neg1[symmetric], simp only: minus_diff_eq gcd_diff1)
1634 lemma coprime_minus_one [simp]: "gcd (n - 1) n = 1"
1635 proof -
1636   have "gcd (n - 1) n = gcd n (n - 1)" by (fact gcd.commute)
1637   also have "\<dots> = gcd ((n - 1) + 1) (n - 1)" by simp
1638   also have "\<dots> = 1" by (rule coprime_plus_one)
1639   finally show ?thesis .
1640 qed
1642 lemma lcm_neg1 [simp]: "lcm (-x) y = lcm x y"
1643   by (rule sym, rule lcmI, simp_all add: lcm_least lcm_zero)
1645 lemma lcm_neg2 [simp]: "lcm x (-y) = lcm x y"
1646   by (rule sym, rule lcmI, simp_all add: lcm_least lcm_zero)
1648 lemma lcm_neg_numeral_1 [simp]: "lcm (- numeral n) x = lcm (numeral n) x"
1649   by (fact lcm_neg1)
1651 lemma lcm_neg_numeral_2 [simp]: "lcm x (- numeral n) = lcm x (numeral n)"
1652   by (fact lcm_neg2)
1654 function euclid_ext :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a" where
1655   "euclid_ext a b =
1656      (if b = 0 then
1657         let x = ring_inv (normalisation_factor a) in (x, 0, a * x)
1658       else
1659         case euclid_ext b (a mod b) of
1660             (s,t,c) \<Rightarrow> (t, s - t * (a div b), c))"
1661   by (pat_completeness, simp)
1662   termination by (relation "measure (euclidean_size \<circ> snd)", simp_all)
1664 declare euclid_ext.simps [simp del]
1666 lemma euclid_ext_0:
1667   "euclid_ext a 0 = (ring_inv (normalisation_factor a), 0, a * ring_inv (normalisation_factor a))"
1668   by (subst euclid_ext.simps, simp add: Let_def)
1670 lemma euclid_ext_non_0:
1671   "b \<noteq> 0 \<Longrightarrow> euclid_ext a b = (case euclid_ext b (a mod b) of
1672     (s,t,c) \<Rightarrow> (t, s - t * (a div b), c))"
1673   by (subst euclid_ext.simps, simp)
1675 definition euclid_ext' :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a"
1676 where
1677   "euclid_ext' a b = (case euclid_ext a b of (s, t, _) \<Rightarrow> (s, t))"
1679 lemma euclid_ext_gcd [simp]:
1680   "(case euclid_ext a b of (_,_,t) \<Rightarrow> t) = gcd a b"
1681 proof (induct a b rule: euclid_ext.induct)
1682   case (1 a b)
1683   then show ?case
1684   proof (cases "b = 0")
1685     case True
1686       then show ?thesis by (cases "a = 0")
1687         (simp_all add: euclid_ext_0 unit_div mult_ac unit_simps gcd_0)
1688     next
1689     case False with 1 show ?thesis
1690       by (simp add: euclid_ext_non_0 ac_simps split: prod.split prod.split_asm)
1691     qed
1692 qed
1694 lemma euclid_ext_gcd' [simp]:
1695   "euclid_ext a b = (r, s, t) \<Longrightarrow> t = gcd a b"
1696   by (insert euclid_ext_gcd[of a b], drule (1) subst, simp)
1698 lemma euclid_ext_correct:
1699   "case euclid_ext x y of (s,t,c) \<Rightarrow> s*x + t*y = c"
1700 proof (induct x y rule: euclid_ext.induct)
1701   case (1 x y)
1702   show ?case
1703   proof (cases "y = 0")
1704     case True
1705     then show ?thesis by (simp add: euclid_ext_0 mult_ac)
1706   next
1707     case False
1708     obtain s t c where stc: "euclid_ext y (x mod y) = (s,t,c)"
1709       by (cases "euclid_ext y (x mod y)", blast)
1710     from 1 have "c = s * y + t * (x mod y)" by (simp add: stc False)
1711     also have "... = t*((x div y)*y + x mod y) + (s - t * (x div y))*y"
1712       by (simp add: algebra_simps)
1713     also have "(x div y)*y + x mod y = x" using mod_div_equality .
1714     finally show ?thesis
1715       by (subst euclid_ext.simps, simp add: False stc)
1716     qed
1717 qed
1719 lemma euclid_ext'_correct:
1720   "fst (euclid_ext' a b) * a + snd (euclid_ext' a b) * b = gcd a b"
1721 proof-
1722   obtain s t c where "euclid_ext a b = (s,t,c)"
1723     by (cases "euclid_ext a b", blast)
1724   with euclid_ext_correct[of a b] euclid_ext_gcd[of a b]
1725     show ?thesis unfolding euclid_ext'_def by simp
1726 qed
1728 lemma bezout: "\<exists>s t. s * x + t * y = gcd x y"
1729   using euclid_ext'_correct by blast
1731 lemma euclid_ext'_0 [simp]: "euclid_ext' x 0 = (ring_inv (normalisation_factor x), 0)"
1732   by (simp add: bezw_def euclid_ext'_def euclid_ext_0)
1734 lemma euclid_ext'_non_0: "y \<noteq> 0 \<Longrightarrow> euclid_ext' x y = (snd (euclid_ext' y (x mod y)),
1735   fst (euclid_ext' y (x mod y)) - snd (euclid_ext' y (x mod y)) * (x div y))"
1736   by (cases "euclid_ext y (x mod y)")
1737     (simp add: euclid_ext'_def euclid_ext_non_0)
1739 end
1741 instantiation nat :: euclidean_semiring
1742 begin
1744 definition [simp]:
1745   "euclidean_size_nat = (id :: nat \<Rightarrow> nat)"
1747 definition [simp]:
1748   "normalisation_factor_nat (n::nat) = (if n = 0 then 0 else 1 :: nat)"
1750 instance proof
1751 qed (simp_all add: is_unit_def)
1753 end
1755 instantiation int :: euclidean_ring
1756 begin
1758 definition [simp]:
1759   "euclidean_size_int = (nat \<circ> abs :: int \<Rightarrow> nat)"
1761 definition [simp]:
1762   "normalisation_factor_int = (sgn :: int \<Rightarrow> int)"
1764 instance proof
1765   case goal2 then show ?case by (auto simp add: abs_mult nat_mult_distrib)
1766 next
1767   case goal3 then show ?case by (simp add: zsgn_def is_unit_def)
1768 next
1769   case goal5 then show ?case by (auto simp: zsgn_def is_unit_def)
1770 next
1771   case goal6 then show ?case by (auto split: abs_split simp: zsgn_def is_unit_def)
1772 qed (auto simp: sgn_times split: abs_split)
1774 end
1776 end