src/HOL/Number_Theory/Euclidean_Algorithm.thy
 author haftmann Wed Feb 17 21:51:58 2016 +0100 (2016-02-17) changeset 62353 7f927120b5a2 parent 62348 9a5f43dac883 child 62422 4aa35fd6c152 permissions -rw-r--r--
dropped various legacy fact bindings and tuned proofs
1 (* Author: Manuel Eberl *)
3 section \<open>Abstract euclidean algorithm\<close>
5 theory Euclidean_Algorithm
6 imports Main "~~/src/HOL/GCD" "~~/src/HOL/Library/Polynomial"
7 begin
9 text \<open>
10   A Euclidean semiring is a semiring upon which the Euclidean algorithm can be
11   implemented. It must provide:
12   \begin{itemize}
13   \item division with remainder
14   \item a size function such that @{term "size (a mod b) < size b"}
15         for any @{term "b \<noteq> 0"}
16   \end{itemize}
17   The existence of these functions makes it possible to derive gcd and lcm functions
18   for any Euclidean semiring.
19 \<close>
20 class euclidean_semiring = semiring_div + normalization_semidom +
21   fixes euclidean_size :: "'a \<Rightarrow> nat"
22   assumes mod_size_less:
23     "b \<noteq> 0 \<Longrightarrow> euclidean_size (a mod b) < euclidean_size b"
24   assumes size_mult_mono:
25     "b \<noteq> 0 \<Longrightarrow> euclidean_size a \<le> euclidean_size (a * b)"
26 begin
28 lemma euclidean_division:
29   fixes a :: 'a and b :: 'a
30   assumes "b \<noteq> 0"
31   obtains s and t where "a = s * b + t"
32     and "euclidean_size t < euclidean_size b"
33 proof -
34   from div_mod_equality [of a b 0]
35      have "a = a div b * b + a mod b" by simp
36   with that and assms show ?thesis by (auto simp add: mod_size_less)
37 qed
39 lemma dvd_euclidean_size_eq_imp_dvd:
40   assumes "a \<noteq> 0" and b_dvd_a: "b dvd a" and size_eq: "euclidean_size a = euclidean_size b"
41   shows "a dvd b"
42 proof (rule ccontr)
43   assume "\<not> a dvd b"
44   then have "b mod a \<noteq> 0" by (simp add: mod_eq_0_iff_dvd)
45   from b_dvd_a have b_dvd_mod: "b dvd b mod a" by (simp add: dvd_mod_iff)
46   from b_dvd_mod obtain c where "b mod a = b * c" unfolding dvd_def by blast
47     with \<open>b mod a \<noteq> 0\<close> have "c \<noteq> 0" by auto
48   with \<open>b mod a = b * c\<close> have "euclidean_size (b mod a) \<ge> euclidean_size b"
49       using size_mult_mono by force
50   moreover from \<open>\<not> a dvd b\<close> and \<open>a \<noteq> 0\<close>
51   have "euclidean_size (b mod a) < euclidean_size a"
52       using mod_size_less by blast
53   ultimately show False using size_eq by simp
54 qed
56 function gcd_eucl :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
57 where
58   "gcd_eucl a b = (if b = 0 then normalize a else gcd_eucl b (a mod b))"
59   by pat_completeness simp
60 termination
61   by (relation "measure (euclidean_size \<circ> snd)") (simp_all add: mod_size_less)
63 declare gcd_eucl.simps [simp del]
65 lemma gcd_eucl_induct [case_names zero mod]:
66   assumes H1: "\<And>b. P b 0"
67   and H2: "\<And>a b. b \<noteq> 0 \<Longrightarrow> P b (a mod b) \<Longrightarrow> P a b"
68   shows "P a b"
69 proof (induct a b rule: gcd_eucl.induct)
70   case ("1" a b)
71   show ?case
72   proof (cases "b = 0")
73     case True then show "P a b" by simp (rule H1)
74   next
75     case False
76     then have "P b (a mod b)"
77       by (rule "1.hyps")
78     with \<open>b \<noteq> 0\<close> show "P a b"
79       by (blast intro: H2)
80   qed
81 qed
83 definition lcm_eucl :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
84 where
85   "lcm_eucl a b = normalize (a * b) div gcd_eucl a b"
87 definition Lcm_eucl :: "'a set \<Rightarrow> 'a" -- \<open>
88   Somewhat complicated definition of Lcm that has the advantage of working
89   for infinite sets as well\<close>
90 where
91   "Lcm_eucl A = (if \<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) then
92      let l = SOME l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l =
93        (LEAST n. \<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n)
94        in normalize l
95       else 0)"
97 definition Gcd_eucl :: "'a set \<Rightarrow> 'a"
98 where
99   "Gcd_eucl A = Lcm_eucl {d. \<forall>a\<in>A. d dvd a}"
101 lemma gcd_eucl_0:
102   "gcd_eucl a 0 = normalize a"
103   by (simp add: gcd_eucl.simps [of a 0])
105 lemma gcd_eucl_0_left:
106   "gcd_eucl 0 a = normalize a"
107   by (simp_all add: gcd_eucl_0 gcd_eucl.simps [of 0 a])
109 lemma gcd_eucl_non_0:
110   "b \<noteq> 0 \<Longrightarrow> gcd_eucl a b = gcd_eucl b (a mod b)"
111   by (simp add: gcd_eucl.simps [of a b] gcd_eucl.simps [of b 0])
113 end
115 class euclidean_ring = euclidean_semiring + idom
116 begin
118 function euclid_ext :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a" where
119   "euclid_ext a b =
120      (if b = 0 then
121         (1 div unit_factor a, 0, normalize a)
122       else
123         case euclid_ext b (a mod b) of
124             (s, t, c) \<Rightarrow> (t, s - t * (a div b), c))"
125   by pat_completeness simp
126 termination
127   by (relation "measure (euclidean_size \<circ> snd)") (simp_all add: mod_size_less)
129 declare euclid_ext.simps [simp del]
131 lemma euclid_ext_0:
132   "euclid_ext a 0 = (1 div unit_factor a, 0, normalize a)"
133   by (simp add: euclid_ext.simps [of a 0])
135 lemma euclid_ext_left_0:
136   "euclid_ext 0 a = (0, 1 div unit_factor a, normalize a)"
137   by (simp add: euclid_ext_0 euclid_ext.simps [of 0 a])
139 lemma euclid_ext_non_0:
140   "b \<noteq> 0 \<Longrightarrow> euclid_ext a b = (case euclid_ext b (a mod b) of
141     (s, t, c) \<Rightarrow> (t, s - t * (a div b), c))"
142   by (simp add: euclid_ext.simps [of a b] euclid_ext.simps [of b 0])
144 lemma euclid_ext_code [code]:
145   "euclid_ext a b = (if b = 0 then (1 div unit_factor a, 0, normalize a)
146     else let (s, t, c) = euclid_ext b (a mod b) in  (t, s - t * (a div b), c))"
147   by (simp add: euclid_ext.simps [of a b] euclid_ext.simps [of b 0])
149 lemma euclid_ext_correct:
150   "case euclid_ext a b of (s, t, c) \<Rightarrow> s * a + t * b = c"
151 proof (induct a b rule: gcd_eucl_induct)
152   case (zero a) then show ?case
153     by (simp add: euclid_ext_0 ac_simps)
154 next
155   case (mod a b)
156   obtain s t c where stc: "euclid_ext b (a mod b) = (s,t,c)"
157     by (cases "euclid_ext b (a mod b)") blast
158   with mod have "c = s * b + t * (a mod b)" by simp
159   also have "... = t * ((a div b) * b + a mod b) + (s - t * (a div b)) * b"
160     by (simp add: algebra_simps)
161   also have "(a div b) * b + a mod b = a" using mod_div_equality .
162   finally show ?case
163     by (subst euclid_ext.simps) (simp add: stc mod ac_simps)
164 qed
166 definition euclid_ext' :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a"
167 where
168   "euclid_ext' a b = (case euclid_ext a b of (s, t, _) \<Rightarrow> (s, t))"
170 lemma euclid_ext'_0: "euclid_ext' a 0 = (1 div unit_factor a, 0)"
171   by (simp add: euclid_ext'_def euclid_ext_0)
173 lemma euclid_ext'_left_0: "euclid_ext' 0 a = (0, 1 div unit_factor a)"
174   by (simp add: euclid_ext'_def euclid_ext_left_0)
176 lemma euclid_ext'_non_0: "b \<noteq> 0 \<Longrightarrow> euclid_ext' a b = (snd (euclid_ext' b (a mod b)),
177   fst (euclid_ext' b (a mod b)) - snd (euclid_ext' b (a mod b)) * (a div b))"
178   by (simp add: euclid_ext'_def euclid_ext_non_0 split_def)
180 end
182 class euclidean_semiring_gcd = euclidean_semiring + gcd + Gcd +
183   assumes gcd_gcd_eucl: "gcd = gcd_eucl" and lcm_lcm_eucl: "lcm = lcm_eucl"
184   assumes Gcd_Gcd_eucl: "Gcd = Gcd_eucl" and Lcm_Lcm_eucl: "Lcm = Lcm_eucl"
185 begin
187 lemma gcd_0_left:
188   "gcd 0 a = normalize a"
189   unfolding gcd_gcd_eucl by (fact gcd_eucl_0_left)
191 lemma gcd_0:
192   "gcd a 0 = normalize a"
193   unfolding gcd_gcd_eucl by (fact gcd_eucl_0)
195 lemma gcd_non_0:
196   "b \<noteq> 0 \<Longrightarrow> gcd a b = gcd b (a mod b)"
197   unfolding gcd_gcd_eucl by (fact gcd_eucl_non_0)
199 lemma gcd_dvd1 [iff]: "gcd a b dvd a"
200   and gcd_dvd2 [iff]: "gcd a b dvd b"
201   by (induct a b rule: gcd_eucl_induct)
202     (simp_all add: gcd_0 gcd_non_0 dvd_mod_iff)
204 lemma dvd_gcd_D1: "k dvd gcd m n \<Longrightarrow> k dvd m"
205   by (rule dvd_trans, assumption, rule gcd_dvd1)
207 lemma dvd_gcd_D2: "k dvd gcd m n \<Longrightarrow> k dvd n"
208   by (rule dvd_trans, assumption, rule gcd_dvd2)
210 lemma gcd_greatest:
211   fixes k a b :: 'a
212   shows "k dvd a \<Longrightarrow> k dvd b \<Longrightarrow> k dvd gcd a b"
213 proof (induct a b rule: gcd_eucl_induct)
214   case (zero a) from zero(1) show ?case by (rule dvd_trans) (simp add: gcd_0)
215 next
216   case (mod a b)
217   then show ?case
218     by (simp add: gcd_non_0 dvd_mod_iff)
219 qed
221 lemma dvd_gcd_iff:
222   "k dvd gcd a b \<longleftrightarrow> k dvd a \<and> k dvd b"
223   by (blast intro!: gcd_greatest intro: dvd_trans)
225 lemmas gcd_greatest_iff = dvd_gcd_iff
227 lemma gcd_zero [simp]:
228   "gcd a b = 0 \<longleftrightarrow> a = 0 \<and> b = 0"
229   by (metis dvd_0_left dvd_refl gcd_dvd1 gcd_dvd2 gcd_greatest)+
231 lemma normalize_gcd [simp]:
232   "normalize (gcd a b) = gcd a b"
233   by (induct a b rule: gcd_eucl_induct) (simp_all add: gcd_0 gcd_non_0)
235 lemma gcdI:
236   assumes "c dvd a" and "c dvd b" and greatest: "\<And>d. d dvd a \<Longrightarrow> d dvd b \<Longrightarrow> d dvd c"
237     and "normalize c = c"
238   shows "c = gcd a b"
239   by (rule associated_eqI) (auto simp: assms intro: gcd_greatest)
241 sublocale gcd: abel_semigroup gcd
242 proof
243   fix a b c
244   show "gcd (gcd a b) c = gcd a (gcd b c)"
245   proof (rule gcdI)
246     have "gcd (gcd a b) c dvd gcd a b" "gcd a b dvd a" by simp_all
247     then show "gcd (gcd a b) c dvd a" by (rule dvd_trans)
248     have "gcd (gcd a b) c dvd gcd a b" "gcd a b dvd b" by simp_all
249     hence "gcd (gcd a b) c dvd b" by (rule dvd_trans)
250     moreover have "gcd (gcd a b) c dvd c" by simp
251     ultimately show "gcd (gcd a b) c dvd gcd b c"
252       by (rule gcd_greatest)
253     show "normalize (gcd (gcd a b) c) = gcd (gcd a b) c"
254       by auto
255     fix l assume "l dvd a" and "l dvd gcd b c"
256     with dvd_trans [OF _ gcd_dvd1] and dvd_trans [OF _ gcd_dvd2]
257       have "l dvd b" and "l dvd c" by blast+
258     with \<open>l dvd a\<close> show "l dvd gcd (gcd a b) c"
259       by (intro gcd_greatest)
260   qed
261 next
262   fix a b
263   show "gcd a b = gcd b a"
264     by (rule gcdI) (simp_all add: gcd_greatest)
265 qed
267 lemma gcd_unique: "d dvd a \<and> d dvd b \<and>
268     normalize d = d \<and>
269     (\<forall>e. e dvd a \<and> e dvd b \<longrightarrow> e dvd d) \<longleftrightarrow> d = gcd a b"
270   by rule (auto intro: gcdI simp: gcd_greatest)
272 lemma gcd_dvd_prod: "gcd a b dvd k * b"
273   using mult_dvd_mono [of 1] by auto
275 lemma gcd_1_left [simp]: "gcd 1 a = 1"
276   by (rule sym, rule gcdI, simp_all)
278 lemma gcd_1 [simp]: "gcd a 1 = 1"
279   by (rule sym, rule gcdI, simp_all)
281 lemma gcd_proj2_if_dvd:
282   "b dvd a \<Longrightarrow> gcd a b = normalize b"
283   by (cases "b = 0", simp_all add: dvd_eq_mod_eq_0 gcd_non_0 gcd_0)
285 lemma gcd_proj1_if_dvd:
286   "a dvd b \<Longrightarrow> gcd a b = normalize a"
287   by (subst gcd.commute, simp add: gcd_proj2_if_dvd)
289 lemma gcd_proj1_iff: "gcd m n = normalize m \<longleftrightarrow> m dvd n"
290 proof
291   assume A: "gcd m n = normalize m"
292   show "m dvd n"
293   proof (cases "m = 0")
294     assume [simp]: "m \<noteq> 0"
295     from A have B: "m = gcd m n * unit_factor m"
296       by (simp add: unit_eq_div2)
297     show ?thesis by (subst B, simp add: mult_unit_dvd_iff)
298   qed (insert A, simp)
299 next
300   assume "m dvd n"
301   then show "gcd m n = normalize m" by (rule gcd_proj1_if_dvd)
302 qed
304 lemma gcd_proj2_iff: "gcd m n = normalize n \<longleftrightarrow> n dvd m"
305   using gcd_proj1_iff [of n m] by (simp add: ac_simps)
307 lemma gcd_mod1 [simp]:
308   "gcd (a mod b) b = gcd a b"
309   by (rule gcdI, metis dvd_mod_iff gcd_dvd1 gcd_dvd2, simp_all add: gcd_greatest dvd_mod_iff)
311 lemma gcd_mod2 [simp]:
312   "gcd a (b mod a) = gcd a b"
313   by (rule gcdI, simp, metis dvd_mod_iff gcd_dvd1 gcd_dvd2, simp_all add: gcd_greatest dvd_mod_iff)
315 lemma gcd_mult_distrib':
316   "normalize c * gcd a b = gcd (c * a) (c * b)"
317 proof (cases "c = 0")
318   case True then show ?thesis by (simp_all add: gcd_0)
319 next
320   case False then have [simp]: "is_unit (unit_factor c)" by simp
321   show ?thesis
322   proof (induct a b rule: gcd_eucl_induct)
323     case (zero a) show ?case
324     proof (cases "a = 0")
325       case True then show ?thesis by (simp add: gcd_0)
326     next
327       case False
328       then show ?thesis by (simp add: gcd_0 normalize_mult)
329     qed
330     case (mod a b)
331     then show ?case by (simp add: mult_mod_right gcd.commute)
332   qed
333 qed
335 lemma gcd_mult_distrib:
336   "k * gcd a b = gcd (k * a) (k * b) * unit_factor k"
337 proof-
338   have "normalize k * gcd a b = gcd (k * a) (k * b)"
339     by (simp add: gcd_mult_distrib')
340   then have "normalize k * gcd a b * unit_factor k = gcd (k * a) (k * b) * unit_factor k"
341     by simp
342   then have "normalize k * unit_factor k * gcd a b  = gcd (k * a) (k * b) * unit_factor k"
343     by (simp only: ac_simps)
344   then show ?thesis
345     by simp
346 qed
348 lemma euclidean_size_gcd_le1 [simp]:
349   assumes "a \<noteq> 0"
350   shows "euclidean_size (gcd a b) \<le> euclidean_size a"
351 proof -
352    have "gcd a b dvd a" by (rule gcd_dvd1)
353    then obtain c where A: "a = gcd a b * c" unfolding dvd_def by blast
354    with \<open>a \<noteq> 0\<close> show ?thesis by (subst (2) A, intro size_mult_mono) auto
355 qed
357 lemma euclidean_size_gcd_le2 [simp]:
358   "b \<noteq> 0 \<Longrightarrow> euclidean_size (gcd a b) \<le> euclidean_size b"
359   by (subst gcd.commute, rule euclidean_size_gcd_le1)
361 lemma euclidean_size_gcd_less1:
362   assumes "a \<noteq> 0" and "\<not>a dvd b"
363   shows "euclidean_size (gcd a b) < euclidean_size a"
364 proof (rule ccontr)
365   assume "\<not>euclidean_size (gcd a b) < euclidean_size a"
366   with \<open>a \<noteq> 0\<close> have "euclidean_size (gcd a b) = euclidean_size a"
367     by (intro le_antisym, simp_all)
368   with assms have "a dvd gcd a b" by (auto intro: dvd_euclidean_size_eq_imp_dvd)
369   hence "a dvd b" using dvd_gcd_D2 by blast
370   with \<open>\<not>a dvd b\<close> show False by contradiction
371 qed
373 lemma euclidean_size_gcd_less2:
374   assumes "b \<noteq> 0" and "\<not>b dvd a"
375   shows "euclidean_size (gcd a b) < euclidean_size b"
376   using assms by (subst gcd.commute, rule euclidean_size_gcd_less1)
378 lemma gcd_mult_unit1: "is_unit a \<Longrightarrow> gcd (b * a) c = gcd b c"
379   apply (rule gcdI)
380   apply simp_all
381   apply (rule dvd_trans, rule gcd_dvd1, simp add: unit_simps)
382   apply (rule gcd_greatest, simp add: unit_simps, assumption)
383   done
385 lemma gcd_mult_unit2: "is_unit a \<Longrightarrow> gcd b (c * a) = gcd b c"
386   by (subst gcd.commute, subst gcd_mult_unit1, assumption, rule gcd.commute)
388 lemma gcd_div_unit1: "is_unit a \<Longrightarrow> gcd (b div a) c = gcd b c"
389   by (erule is_unitE [of _ b]) (simp add: gcd_mult_unit1)
391 lemma gcd_div_unit2: "is_unit a \<Longrightarrow> gcd b (c div a) = gcd b c"
392   by (erule is_unitE [of _ c]) (simp add: gcd_mult_unit2)
394 lemma normalize_gcd_left [simp]:
395   "gcd (normalize a) b = gcd a b"
396 proof (cases "a = 0")
397   case True then show ?thesis
398     by simp
399 next
400   case False then have "is_unit (unit_factor a)"
401     by simp
402   moreover have "normalize a = a div unit_factor a"
403     by simp
404   ultimately show ?thesis
405     by (simp only: gcd_div_unit1)
406 qed
408 lemma normalize_gcd_right [simp]:
409   "gcd a (normalize b) = gcd a b"
410   using normalize_gcd_left [of b a] by (simp add: ac_simps)
412 lemma gcd_idem: "gcd a a = normalize a"
413   by (cases "a = 0") (simp add: gcd_0_left, rule sym, rule gcdI, simp_all)
415 lemma gcd_right_idem: "gcd (gcd a b) b = gcd a b"
416   apply (rule gcdI)
417   apply (simp add: ac_simps)
418   apply (rule gcd_dvd2)
419   apply (rule gcd_greatest, erule (1) gcd_greatest, assumption)
420   apply simp
421   done
423 lemma gcd_left_idem: "gcd a (gcd a b) = gcd a b"
424   apply (rule gcdI)
425   apply simp
426   apply (rule dvd_trans, rule gcd_dvd2, rule gcd_dvd2)
427   apply (rule gcd_greatest, assumption, erule gcd_greatest, assumption)
428   apply simp
429   done
431 lemma comp_fun_idem_gcd: "comp_fun_idem gcd"
432 proof
433   fix a b show "gcd a \<circ> gcd b = gcd b \<circ> gcd a"
434     by (simp add: fun_eq_iff ac_simps)
435 next
436   fix a show "gcd a \<circ> gcd a = gcd a"
437     by (simp add: fun_eq_iff gcd_left_idem)
438 qed
440 lemma coprime_dvd_mult:
441   assumes "gcd c b = 1" and "c dvd a * b"
442   shows "c dvd a"
443 proof -
444   let ?nf = "unit_factor"
445   from assms gcd_mult_distrib [of a c b]
446     have A: "a = gcd (a * c) (a * b) * ?nf a" by simp
447   from \<open>c dvd a * b\<close> show ?thesis by (subst A, simp_all add: gcd_greatest)
448 qed
450 lemma coprime_dvd_mult_iff:
451   "gcd c b = 1 \<Longrightarrow> (c dvd a * b) = (c dvd a)"
452   by (rule, rule coprime_dvd_mult, simp_all)
454 lemma gcd_dvd_antisym:
455   "gcd a b dvd gcd c d \<Longrightarrow> gcd c d dvd gcd a b \<Longrightarrow> gcd a b = gcd c d"
456 proof (rule gcdI)
457   assume A: "gcd a b dvd gcd c d" and B: "gcd c d dvd gcd a b"
458   have "gcd c d dvd c" by simp
459   with A show "gcd a b dvd c" by (rule dvd_trans)
460   have "gcd c d dvd d" by simp
461   with A show "gcd a b dvd d" by (rule dvd_trans)
462   show "normalize (gcd a b) = gcd a b"
463     by simp
464   fix l assume "l dvd c" and "l dvd d"
465   hence "l dvd gcd c d" by (rule gcd_greatest)
466   from this and B show "l dvd gcd a b" by (rule dvd_trans)
467 qed
469 lemma gcd_mult_cancel:
470   assumes "gcd k n = 1"
471   shows "gcd (k * m) n = gcd m n"
472 proof (rule gcd_dvd_antisym)
473   have "gcd (gcd (k * m) n) k = gcd (gcd k n) (k * m)" by (simp add: ac_simps)
474   also note \<open>gcd k n = 1\<close>
475   finally have "gcd (gcd (k * m) n) k = 1" by simp
476   hence "gcd (k * m) n dvd m" by (rule coprime_dvd_mult, simp add: ac_simps)
477   moreover have "gcd (k * m) n dvd n" by simp
478   ultimately show "gcd (k * m) n dvd gcd m n" by (rule gcd_greatest)
479   have "gcd m n dvd (k * m)" and "gcd m n dvd n" by simp_all
480   then show "gcd m n dvd gcd (k * m) n" by (rule gcd_greatest)
481 qed
483 lemma coprime_crossproduct:
484   assumes [simp]: "gcd a d = 1" "gcd b c = 1"
485   shows "normalize (a * c) = normalize (b * d) \<longleftrightarrow> normalize a  = normalize b \<and> normalize c = normalize d"
486     (is "?lhs \<longleftrightarrow> ?rhs")
487 proof
488   assume ?rhs
489   then have "a dvd b" "b dvd a" "c dvd d" "d dvd c" by (simp_all add: associated_iff_dvd)
490   then have "a * c dvd b * d" "b * d dvd a * c" by (fast intro: mult_dvd_mono)+
491   then show ?lhs by (simp add: associated_iff_dvd)
492 next
493   assume ?lhs
494   then have dvd: "a * c dvd b * d" "b * d dvd a * c" by (simp_all add: associated_iff_dvd)
495   then have "a dvd b * d" by (metis dvd_mult_left)
496   hence "a dvd b" by (simp add: coprime_dvd_mult_iff)
497   moreover from dvd have "b dvd a * c" by (metis dvd_mult_left)
498   hence "b dvd a" by (simp add: coprime_dvd_mult_iff)
499   moreover from dvd have "c dvd d * b"
500     by (auto dest: dvd_mult_right simp add: ac_simps)
501   hence "c dvd d" by (simp add: coprime_dvd_mult_iff gcd.commute)
502   moreover from dvd have "d dvd c * a"
503     by (auto dest: dvd_mult_right simp add: ac_simps)
504   hence "d dvd c" by (simp add: coprime_dvd_mult_iff gcd.commute)
505   ultimately show ?rhs by (simp add: associated_iff_dvd)
506 qed
508 lemma gcd_add1 [simp]:
509   "gcd (m + n) n = gcd m n"
510   by (cases "n = 0", simp_all add: gcd_non_0)
512 lemma gcd_add2 [simp]:
513   "gcd m (m + n) = gcd m n"
514   using gcd_add1 [of n m] by (simp add: ac_simps)
517   "gcd m (k * m + n) = gcd m n"
518 proof -
519   have "gcd m ((k * m + n) mod m) = gcd m (k * m + n)"
520     by (fact gcd_mod2)
521   then show ?thesis by simp
522 qed
524 lemma coprimeI: "(\<And>l. \<lbrakk>l dvd a; l dvd b\<rbrakk> \<Longrightarrow> l dvd 1) \<Longrightarrow> gcd a b = 1"
525   by (rule sym, rule gcdI, simp_all)
527 lemma coprime: "gcd a b = 1 \<longleftrightarrow> (\<forall>d. d dvd a \<and> d dvd b \<longleftrightarrow> is_unit d)"
528   by (auto intro: coprimeI gcd_greatest dvd_gcd_D1 dvd_gcd_D2)
530 lemma div_gcd_coprime:
531   assumes nz: "a \<noteq> 0 \<or> b \<noteq> 0"
532   defines [simp]: "d \<equiv> gcd a b"
533   defines [simp]: "a' \<equiv> a div d" and [simp]: "b' \<equiv> b div d"
534   shows "gcd a' b' = 1"
535 proof (rule coprimeI)
536   fix l assume "l dvd a'" "l dvd b'"
537   then obtain s t where "a' = l * s" "b' = l * t" unfolding dvd_def by blast
538   moreover have "a = a' * d" "b = b' * d" by simp_all
539   ultimately have "a = (l * d) * s" "b = (l * d) * t"
540     by (simp_all only: ac_simps)
541   hence "l*d dvd a" and "l*d dvd b" by (simp_all only: dvd_triv_left)
542   hence "l*d dvd d" by (simp add: gcd_greatest)
543   then obtain u where "d = l * d * u" ..
544   then have "d * (l * u) = d" by (simp add: ac_simps)
545   moreover from nz have "d \<noteq> 0" by simp
546   with div_mult_self1_is_id have "d * (l * u) div d = l * u" .
547   ultimately have "1 = l * u"
548     using \<open>d \<noteq> 0\<close> by simp
549   then show "l dvd 1" ..
550 qed
552 lemma coprime_mult:
553   assumes da: "gcd d a = 1" and db: "gcd d b = 1"
554   shows "gcd d (a * b) = 1"
555   apply (subst gcd.commute)
556   using da apply (subst gcd_mult_cancel)
557   apply (subst gcd.commute, assumption)
558   apply (subst gcd.commute, rule db)
559   done
561 lemma coprime_lmult:
562   assumes dab: "gcd d (a * b) = 1"
563   shows "gcd d a = 1"
564 proof (rule coprimeI)
565   fix l assume "l dvd d" and "l dvd a"
566   hence "l dvd a * b" by simp
567   with \<open>l dvd d\<close> and dab show "l dvd 1" by (auto intro: gcd_greatest)
568 qed
570 lemma coprime_rmult:
571   assumes dab: "gcd d (a * b) = 1"
572   shows "gcd d b = 1"
573 proof (rule coprimeI)
574   fix l assume "l dvd d" and "l dvd b"
575   hence "l dvd a * b" by simp
576   with \<open>l dvd d\<close> and dab show "l dvd 1" by (auto intro: gcd_greatest)
577 qed
579 lemma coprime_mul_eq: "gcd d (a * b) = 1 \<longleftrightarrow> gcd d a = 1 \<and> gcd d b = 1"
580   using coprime_rmult[of d a b] coprime_lmult[of d a b] coprime_mult[of d a b] by blast
582 lemma gcd_coprime:
583   assumes c: "gcd a b \<noteq> 0" and a: "a = a' * gcd a b" and b: "b = b' * gcd a b"
584   shows "gcd a' b' = 1"
585 proof -
586   from c have "a \<noteq> 0 \<or> b \<noteq> 0" by simp
587   with div_gcd_coprime have "gcd (a div gcd a b) (b div gcd a b) = 1" .
588   also from assms have "a div gcd a b = a'" by (metis div_mult_self2_is_id)+
589   also from assms have "b div gcd a b = b'" by (metis div_mult_self2_is_id)+
590   finally show ?thesis .
591 qed
593 lemma coprime_power:
594   assumes "0 < n"
595   shows "gcd a (b ^ n) = 1 \<longleftrightarrow> gcd a b = 1"
596 using assms proof (induct n)
597   case (Suc n) then show ?case
598     by (cases n) (simp_all add: coprime_mul_eq)
599 qed simp
601 lemma gcd_coprime_exists:
602   assumes nz: "gcd a b \<noteq> 0"
603   shows "\<exists>a' b'. a = a' * gcd a b \<and> b = b' * gcd a b \<and> gcd a' b' = 1"
604   apply (rule_tac x = "a div gcd a b" in exI)
605   apply (rule_tac x = "b div gcd a b" in exI)
606   apply (insert nz, auto intro: div_gcd_coprime)
607   done
609 lemma coprime_exp:
610   "gcd d a = 1 \<Longrightarrow> gcd d (a^n) = 1"
611   by (induct n, simp_all add: coprime_mult)
613 lemma coprime_exp2 [intro]:
614   "gcd a b = 1 \<Longrightarrow> gcd (a^n) (b^m) = 1"
615   apply (rule coprime_exp)
616   apply (subst gcd.commute)
617   apply (rule coprime_exp)
618   apply (subst gcd.commute)
619   apply assumption
620   done
622 lemma lcm_gcd:
623   "lcm a b = normalize (a * b) div gcd a b"
624   by (simp add: lcm_lcm_eucl gcd_gcd_eucl lcm_eucl_def)
626 subclass semiring_gcd
627   apply standard
628   using gcd_right_idem
629   apply (simp_all add: lcm_gcd gcd_greatest_iff gcd_proj1_if_dvd)
630   done
632 lemma gcd_exp:
633   "gcd (a ^ n) (b ^ n) = gcd a b ^ n"
634 proof (cases "a = 0 \<and> b = 0")
635   case True
636   then show ?thesis by (cases n) simp_all
637 next
638   case False
639   then have "1 = gcd ((a div gcd a b) ^ n) ((b div gcd a b) ^ n)"
640     using div_gcd_coprime by (subst sym) (auto simp: div_gcd_coprime)
641   then have "gcd a b ^ n = gcd a b ^ n * ..." by simp
642   also note gcd_mult_distrib
643   also have "unit_factor (gcd a b ^ n) = 1"
644     using False by (auto simp add: unit_factor_power unit_factor_gcd)
645   also have "(gcd a b)^n * (a div gcd a b)^n = a^n"
646     by (subst ac_simps, subst div_power, simp, rule dvd_div_mult_self, rule dvd_power_same, simp)
647   also have "(gcd a b)^n * (b div gcd a b)^n = b^n"
648     by (subst ac_simps, subst div_power, simp, rule dvd_div_mult_self, rule dvd_power_same, simp)
649   finally show ?thesis by simp
650 qed
652 lemma coprime_common_divisor:
653   "gcd a b = 1 \<Longrightarrow> a dvd a \<Longrightarrow> a dvd b \<Longrightarrow> is_unit a"
654   apply (subgoal_tac "a dvd gcd a b")
655   apply simp
656   apply (erule (1) gcd_greatest)
657   done
659 lemma division_decomp:
660   assumes dc: "a dvd b * c"
661   shows "\<exists>b' c'. a = b' * c' \<and> b' dvd b \<and> c' dvd c"
662 proof (cases "gcd a b = 0")
663   assume "gcd a b = 0"
664   hence "a = 0 \<and> b = 0" by simp
665   hence "a = 0 * c \<and> 0 dvd b \<and> c dvd c" by simp
666   then show ?thesis by blast
667 next
668   let ?d = "gcd a b"
669   assume "?d \<noteq> 0"
670   from gcd_coprime_exists[OF this]
671     obtain a' b' where ab': "a = a' * ?d" "b = b' * ?d" "gcd a' b' = 1"
672     by blast
673   from ab'(1) have "a' dvd a" unfolding dvd_def by blast
674   with dc have "a' dvd b*c" using dvd_trans[of a' a "b*c"] by simp
675   from dc ab'(1,2) have "a'*?d dvd (b'*?d) * c" by simp
676   hence "?d * a' dvd ?d * (b' * c)" by (simp add: mult_ac)
677   with \<open>?d \<noteq> 0\<close> have "a' dvd b' * c" by simp
678   with coprime_dvd_mult[OF ab'(3)]
679     have "a' dvd c" by (subst (asm) ac_simps, blast)
680   with ab'(1) have "a = ?d * a' \<and> ?d dvd b \<and> a' dvd c" by (simp add: mult_ac)
681   then show ?thesis by blast
682 qed
684 lemma pow_divs_pow:
685   assumes ab: "a ^ n dvd b ^ n" and n: "n \<noteq> 0"
686   shows "a dvd b"
687 proof (cases "gcd a b = 0")
688   assume "gcd a b = 0"
689   then show ?thesis by simp
690 next
691   let ?d = "gcd a b"
692   assume "?d \<noteq> 0"
693   from n obtain m where m: "n = Suc m" by (cases n, simp_all)
694   from \<open>?d \<noteq> 0\<close> have zn: "?d ^ n \<noteq> 0" by (rule power_not_zero)
695   from gcd_coprime_exists[OF \<open>?d \<noteq> 0\<close>]
696     obtain a' b' where ab': "a = a' * ?d" "b = b' * ?d" "gcd a' b' = 1"
697     by blast
698   from ab have "(a' * ?d) ^ n dvd (b' * ?d) ^ n"
699     by (simp add: ab'(1,2)[symmetric])
700   hence "?d^n * a'^n dvd ?d^n * b'^n"
701     by (simp only: power_mult_distrib ac_simps)
702   with zn have "a'^n dvd b'^n" by simp
703   hence "a' dvd b'^n" using dvd_trans[of a' "a'^n" "b'^n"] by (simp add: m)
704   hence "a' dvd b'^m * b'" by (simp add: m ac_simps)
705   with coprime_dvd_mult[OF coprime_exp[OF ab'(3), of m]]
706     have "a' dvd b'" by (subst (asm) ac_simps, blast)
707   hence "a'*?d dvd b'*?d" by (rule mult_dvd_mono, simp)
708   with ab'(1,2) show ?thesis by simp
709 qed
711 lemma pow_divs_eq [simp]:
712   "n \<noteq> 0 \<Longrightarrow> a ^ n dvd b ^ n \<longleftrightarrow> a dvd b"
713   by (auto intro: pow_divs_pow dvd_power_same)
715 lemma divs_mult:
716   assumes mr: "m dvd r" and nr: "n dvd r" and mn: "gcd m n = 1"
717   shows "m * n dvd r"
718 proof -
719   from mr nr obtain m' n' where m': "r = m*m'" and n': "r = n*n'"
720     unfolding dvd_def by blast
721   from mr n' have "m dvd n'*n" by (simp add: ac_simps)
722   hence "m dvd n'" using coprime_dvd_mult_iff[OF mn] by simp
723   then obtain k where k: "n' = m*k" unfolding dvd_def by blast
724   with n' have "r = m * n * k" by (simp add: mult_ac)
725   then show ?thesis unfolding dvd_def by blast
726 qed
728 lemma coprime_plus_one [simp]: "gcd (n + 1) n = 1"
729   by (subst add_commute, simp)
731 lemma setprod_coprime [rule_format]:
732   "(\<forall>i\<in>A. gcd (f i) a = 1) \<longrightarrow> gcd (\<Prod>i\<in>A. f i) a = 1"
733   apply (cases "finite A")
734   apply (induct set: finite)
735   apply (auto simp add: gcd_mult_cancel)
736   done
738 lemma coprime_divisors:
739   assumes "d dvd a" "e dvd b" "gcd a b = 1"
740   shows "gcd d e = 1"
741 proof -
742   from assms obtain k l where "a = d * k" "b = e * l"
743     unfolding dvd_def by blast
744   with assms have "gcd (d * k) (e * l) = 1" by simp
745   hence "gcd (d * k) e = 1" by (rule coprime_lmult)
746   also have "gcd (d * k) e = gcd e (d * k)" by (simp add: ac_simps)
747   finally have "gcd e d = 1" by (rule coprime_lmult)
748   then show ?thesis by (simp add: ac_simps)
749 qed
751 lemma invertible_coprime:
752   assumes "a * b mod m = 1"
753   shows "coprime a m"
754 proof -
755   from assms have "coprime m (a * b mod m)"
756     by simp
757   then have "coprime m (a * b)"
758     by simp
759   then have "coprime m a"
760     by (rule coprime_lmult)
761   then show ?thesis
762     by (simp add: ac_simps)
763 qed
765 lemma lcm_gcd_prod:
766   "lcm a b * gcd a b = normalize (a * b)"
767   by (simp add: lcm_gcd)
769 lemma lcm_zero:
770   "lcm a b = 0 \<longleftrightarrow> a = 0 \<or> b = 0"
771   by (fact lcm_eq_0_iff)
773 lemmas lcm_0_iff = lcm_zero
775 lemma gcd_lcm:
776   assumes "lcm a b \<noteq> 0"
777   shows "gcd a b = normalize (a * b) div lcm a b"
778 proof -
779   have "lcm a b * gcd a b = normalize (a * b)"
780     by (fact lcm_gcd_prod)
781   with assms show ?thesis
782     by (metis nonzero_mult_divide_cancel_left)
783 qed
785 declare unit_factor_lcm [simp]
787 lemma lcmI:
788   assumes "a dvd c" and "b dvd c" and "\<And>d. a dvd d \<Longrightarrow> b dvd d \<Longrightarrow> c dvd d"
789     and "normalize c = c"
790   shows "c = lcm a b"
791   by (rule associated_eqI) (auto simp: assms intro: lcm_least)
793 sublocale lcm: abel_semigroup lcm ..
795 lemma dvd_lcm_D1:
796   "lcm m n dvd k \<Longrightarrow> m dvd k"
797   by (rule dvd_trans, rule dvd_lcm1, assumption)
799 lemma dvd_lcm_D2:
800   "lcm m n dvd k \<Longrightarrow> n dvd k"
801   by (rule dvd_trans, rule dvd_lcm2, assumption)
803 lemma gcd_dvd_lcm [simp]:
804   "gcd a b dvd lcm a b"
805   using gcd_dvd2 by (rule dvd_lcmI2)
807 lemma lcm_1_iff:
808   "lcm a b = 1 \<longleftrightarrow> is_unit a \<and> is_unit b"
809 proof
810   assume "lcm a b = 1"
811   then show "is_unit a \<and> is_unit b" by auto
812 next
813   assume "is_unit a \<and> is_unit b"
814   hence "a dvd 1" and "b dvd 1" by simp_all
815   hence "is_unit (lcm a b)" by (rule lcm_least)
816   hence "lcm a b = unit_factor (lcm a b)"
817     by (blast intro: sym is_unit_unit_factor)
818   also have "\<dots> = 1" using \<open>is_unit a \<and> is_unit b\<close>
819     by auto
820   finally show "lcm a b = 1" .
821 qed
823 lemma lcm_0:
824   "lcm a 0 = 0"
825   by (fact lcm_0_right)
827 lemma lcm_unique:
828   "a dvd d \<and> b dvd d \<and>
829   normalize d = d \<and>
830   (\<forall>e. a dvd e \<and> b dvd e \<longrightarrow> d dvd e) \<longleftrightarrow> d = lcm a b"
831   by rule (auto intro: lcmI simp: lcm_least lcm_zero)
833 lemma lcm_coprime:
834   "gcd a b = 1 \<Longrightarrow> lcm a b = normalize (a * b)"
835   by (subst lcm_gcd) simp
837 lemma lcm_proj1_if_dvd:
838   "b dvd a \<Longrightarrow> lcm a b = normalize a"
839   by (cases "a = 0") (simp, rule sym, rule lcmI, simp_all)
841 lemma lcm_proj2_if_dvd:
842   "a dvd b \<Longrightarrow> lcm a b = normalize b"
843   using lcm_proj1_if_dvd [of a b] by (simp add: ac_simps)
845 lemma lcm_proj1_iff:
846   "lcm m n = normalize m \<longleftrightarrow> n dvd m"
847 proof
848   assume A: "lcm m n = normalize m"
849   show "n dvd m"
850   proof (cases "m = 0")
851     assume [simp]: "m \<noteq> 0"
852     from A have B: "m = lcm m n * unit_factor m"
853       by (simp add: unit_eq_div2)
854     show ?thesis by (subst B, simp)
855   qed simp
856 next
857   assume "n dvd m"
858   then show "lcm m n = normalize m" by (rule lcm_proj1_if_dvd)
859 qed
861 lemma lcm_proj2_iff:
862   "lcm m n = normalize n \<longleftrightarrow> m dvd n"
863   using lcm_proj1_iff [of n m] by (simp add: ac_simps)
865 lemma euclidean_size_lcm_le1:
866   assumes "a \<noteq> 0" and "b \<noteq> 0"
867   shows "euclidean_size a \<le> euclidean_size (lcm a b)"
868 proof -
869   have "a dvd lcm a b" by (rule dvd_lcm1)
870   then obtain c where A: "lcm a b = a * c" ..
871   with \<open>a \<noteq> 0\<close> and \<open>b \<noteq> 0\<close> have "c \<noteq> 0" by (auto simp: lcm_zero)
872   then show ?thesis by (subst A, intro size_mult_mono)
873 qed
875 lemma euclidean_size_lcm_le2:
876   "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> euclidean_size b \<le> euclidean_size (lcm a b)"
877   using euclidean_size_lcm_le1 [of b a] by (simp add: ac_simps)
879 lemma euclidean_size_lcm_less1:
880   assumes "b \<noteq> 0" and "\<not>b dvd a"
881   shows "euclidean_size a < euclidean_size (lcm a b)"
882 proof (rule ccontr)
883   from assms have "a \<noteq> 0" by auto
884   assume "\<not>euclidean_size a < euclidean_size (lcm a b)"
885   with \<open>a \<noteq> 0\<close> and \<open>b \<noteq> 0\<close> have "euclidean_size (lcm a b) = euclidean_size a"
886     by (intro le_antisym, simp, intro euclidean_size_lcm_le1)
887   with assms have "lcm a b dvd a"
888     by (rule_tac dvd_euclidean_size_eq_imp_dvd) (auto simp: lcm_zero)
889   hence "b dvd a" by (rule dvd_lcm_D2)
890   with \<open>\<not>b dvd a\<close> show False by contradiction
891 qed
893 lemma euclidean_size_lcm_less2:
894   assumes "a \<noteq> 0" and "\<not>a dvd b"
895   shows "euclidean_size b < euclidean_size (lcm a b)"
896   using assms euclidean_size_lcm_less1 [of a b] by (simp add: ac_simps)
898 lemma lcm_mult_unit1:
899   "is_unit a \<Longrightarrow> lcm (b * a) c = lcm b c"
900   by (rule associated_eqI) (simp_all add: mult_unit_dvd_iff dvd_lcmI1)
902 lemma lcm_mult_unit2:
903   "is_unit a \<Longrightarrow> lcm b (c * a) = lcm b c"
904   using lcm_mult_unit1 [of a c b] by (simp add: ac_simps)
906 lemma lcm_div_unit1:
907   "is_unit a \<Longrightarrow> lcm (b div a) c = lcm b c"
908   by (erule is_unitE [of _ b]) (simp add: lcm_mult_unit1)
910 lemma lcm_div_unit2:
911   "is_unit a \<Longrightarrow> lcm b (c div a) = lcm b c"
912   by (erule is_unitE [of _ c]) (simp add: lcm_mult_unit2)
914 lemma normalize_lcm_left [simp]:
915   "lcm (normalize a) b = lcm a b"
916 proof (cases "a = 0")
917   case True then show ?thesis
918     by simp
919 next
920   case False then have "is_unit (unit_factor a)"
921     by simp
922   moreover have "normalize a = a div unit_factor a"
923     by simp
924   ultimately show ?thesis
925     by (simp only: lcm_div_unit1)
926 qed
928 lemma normalize_lcm_right [simp]:
929   "lcm a (normalize b) = lcm a b"
930   using normalize_lcm_left [of b a] by (simp add: ac_simps)
932 lemma lcm_left_idem:
933   "lcm a (lcm a b) = lcm a b"
934   by (rule associated_eqI) simp_all
936 lemma lcm_right_idem:
937   "lcm (lcm a b) b = lcm a b"
938   by (rule associated_eqI) simp_all
940 lemma comp_fun_idem_lcm: "comp_fun_idem lcm"
941 proof
942   fix a b show "lcm a \<circ> lcm b = lcm b \<circ> lcm a"
943     by (simp add: fun_eq_iff ac_simps)
944 next
945   fix a show "lcm a \<circ> lcm a = lcm a" unfolding o_def
946     by (intro ext, simp add: lcm_left_idem)
947 qed
949 lemma dvd_Lcm [simp]: "a \<in> A \<Longrightarrow> a dvd Lcm A"
950   and Lcm_least: "(\<And>a. a \<in> A \<Longrightarrow> a dvd b) \<Longrightarrow> Lcm A dvd b"
951   and unit_factor_Lcm [simp]:
952           "unit_factor (Lcm A) = (if Lcm A = 0 then 0 else 1)"
953 proof -
954   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>
955     unit_factor (Lcm A) = (if Lcm A = 0 then 0 else 1)" (is ?thesis)
956   proof (cases "\<exists>l. l \<noteq>  0 \<and> (\<forall>a\<in>A. a dvd l)")
957     case False
958     hence "Lcm A = 0" by (auto simp: Lcm_Lcm_eucl Lcm_eucl_def)
959     with False show ?thesis by auto
960   next
961     case True
962     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
963     def n \<equiv> "LEAST n. \<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"
964     def l \<equiv> "SOME l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"
965     have "\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"
966       apply (subst n_def)
967       apply (rule LeastI[of _ "euclidean_size l\<^sub>0"])
968       apply (rule exI[of _ l\<^sub>0])
969       apply (simp add: l\<^sub>0_props)
970       done
971     from someI_ex[OF this] have "l \<noteq> 0" and "\<forall>a\<in>A. a dvd l" and "euclidean_size l = n"
972       unfolding l_def by simp_all
973     {
974       fix l' assume "\<forall>a\<in>A. a dvd l'"
975       with \<open>\<forall>a\<in>A. a dvd l\<close> have "\<forall>a\<in>A. a dvd gcd l l'" by (auto intro: gcd_greatest)
976       moreover from \<open>l \<noteq> 0\<close> have "gcd l l' \<noteq> 0" by simp
977       ultimately have "\<exists>b. b \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd b) \<and> euclidean_size b = euclidean_size (gcd l l')"
978         by (intro exI[of _ "gcd l l'"], auto)
979       hence "euclidean_size (gcd l l') \<ge> n" by (subst n_def) (rule Least_le)
980       moreover have "euclidean_size (gcd l l') \<le> n"
981       proof -
982         have "gcd l l' dvd l" by simp
983         then obtain a where "l = gcd l l' * a" unfolding dvd_def by blast
984         with \<open>l \<noteq> 0\<close> have "a \<noteq> 0" by auto
985         hence "euclidean_size (gcd l l') \<le> euclidean_size (gcd l l' * a)"
986           by (rule size_mult_mono)
987         also have "gcd l l' * a = l" using \<open>l = gcd l l' * a\<close> ..
988         also note \<open>euclidean_size l = n\<close>
989         finally show "euclidean_size (gcd l l') \<le> n" .
990       qed
991       ultimately have *: "euclidean_size l = euclidean_size (gcd l l')"
992         by (intro le_antisym, simp_all add: \<open>euclidean_size l = n\<close>)
993       from \<open>l \<noteq> 0\<close> have "l dvd gcd l l'"
994         by (rule dvd_euclidean_size_eq_imp_dvd) (auto simp add: *)
995       hence "l dvd l'" by (blast dest: dvd_gcd_D2)
996     }
998     with \<open>(\<forall>a\<in>A. a dvd l)\<close> and unit_factor_is_unit[OF \<open>l \<noteq> 0\<close>] and \<open>l \<noteq> 0\<close>
999       have "(\<forall>a\<in>A. a dvd normalize l) \<and>
1000         (\<forall>l'. (\<forall>a\<in>A. a dvd l') \<longrightarrow> normalize l dvd l') \<and>
1001         unit_factor (normalize l) =
1002         (if normalize l = 0 then 0 else 1)"
1003       by (auto simp: unit_simps)
1004     also from True have "normalize l = Lcm A"
1005       by (simp add: Lcm_Lcm_eucl Lcm_eucl_def Let_def n_def l_def)
1006     finally show ?thesis .
1007   qed
1008   note A = this
1010   {fix a assume "a \<in> A" then show "a dvd Lcm A" using A by blast}
1011   {fix b assume "\<And>a. a \<in> A \<Longrightarrow> a dvd b" then show "Lcm A dvd b" using A by blast}
1012   from A show "unit_factor (Lcm A) = (if Lcm A = 0 then 0 else 1)" by blast
1013 qed
1015 lemma normalize_Lcm [simp]:
1016   "normalize (Lcm A) = Lcm A"
1017 proof (cases "Lcm A = 0")
1018   case True then show ?thesis by simp
1019 next
1020   case False
1021   have "unit_factor (Lcm A) * normalize (Lcm A) = Lcm A"
1022     by (fact unit_factor_mult_normalize)
1023   with False show ?thesis by simp
1024 qed
1026 lemma LcmI:
1027   assumes "\<And>a. a \<in> A \<Longrightarrow> a dvd b" and "\<And>c. (\<And>a. a \<in> A \<Longrightarrow> a dvd c) \<Longrightarrow> b dvd c"
1028     and "normalize b = b" shows "b = Lcm A"
1029   by (rule associated_eqI) (auto simp: assms intro: Lcm_least)
1031 lemma Lcm_subset:
1032   "A \<subseteq> B \<Longrightarrow> Lcm A dvd Lcm B"
1033   by (blast intro: Lcm_least dvd_Lcm)
1035 lemma Lcm_Un:
1036   "Lcm (A \<union> B) = lcm (Lcm A) (Lcm B)"
1037   apply (rule lcmI)
1038   apply (blast intro: Lcm_subset)
1039   apply (blast intro: Lcm_subset)
1040   apply (intro Lcm_least ballI, elim UnE)
1041   apply (rule dvd_trans, erule dvd_Lcm, assumption)
1042   apply (rule dvd_trans, erule dvd_Lcm, assumption)
1043   apply simp
1044   done
1046 lemma Lcm_1_iff:
1047   "Lcm A = 1 \<longleftrightarrow> (\<forall>a\<in>A. is_unit a)"
1048 proof
1049   assume "Lcm A = 1"
1050   then show "\<forall>a\<in>A. is_unit a" by auto
1051 qed (rule LcmI [symmetric], auto)
1053 lemma Lcm_no_units:
1054   "Lcm A = Lcm (A - {a. is_unit a})"
1055 proof -
1056   have "(A - {a. is_unit a}) \<union> {a\<in>A. is_unit a} = A" by blast
1057   hence "Lcm A = lcm (Lcm (A - {a. is_unit a})) (Lcm {a\<in>A. is_unit a})"
1058     by (simp add: Lcm_Un [symmetric])
1059   also have "Lcm {a\<in>A. is_unit a} = 1" by (simp add: Lcm_1_iff)
1060   finally show ?thesis by simp
1061 qed
1063 lemma Lcm_empty [simp]:
1064   "Lcm {} = 1"
1065   by (simp add: Lcm_1_iff)
1067 lemma Lcm_eq_0_I [simp]:
1068   "0 \<in> A \<Longrightarrow> Lcm A = 0"
1069   by (drule dvd_Lcm) simp
1071 lemma Lcm_0_iff':
1072   "Lcm A = 0 \<longleftrightarrow> \<not>(\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l))"
1073 proof
1074   assume "Lcm A = 0"
1075   show "\<not>(\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l))"
1076   proof
1077     assume ex: "\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l)"
1078     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
1079     def n \<equiv> "LEAST n. \<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"
1080     def l \<equiv> "SOME l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"
1081     have "\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"
1082       apply (subst n_def)
1083       apply (rule LeastI[of _ "euclidean_size l\<^sub>0"])
1084       apply (rule exI[of _ l\<^sub>0])
1085       apply (simp add: l\<^sub>0_props)
1086       done
1087     from someI_ex[OF this] have "l \<noteq> 0" unfolding l_def by simp_all
1088     hence "normalize l \<noteq> 0" by simp
1089     also from ex have "normalize l = Lcm A"
1090        by (simp only: Lcm_Lcm_eucl Lcm_eucl_def n_def l_def if_True Let_def)
1091     finally show False using \<open>Lcm A = 0\<close> by contradiction
1092   qed
1093 qed (simp only: Lcm_Lcm_eucl Lcm_eucl_def if_False)
1095 lemma Lcm_0_iff [simp]:
1096   "finite A \<Longrightarrow> Lcm A = 0 \<longleftrightarrow> 0 \<in> A"
1097 proof -
1098   assume "finite A"
1099   have "0 \<in> A \<Longrightarrow> Lcm A = 0"  by (intro dvd_0_left dvd_Lcm)
1100   moreover {
1101     assume "0 \<notin> A"
1102     hence "\<Prod>A \<noteq> 0"
1103       apply (induct rule: finite_induct[OF \<open>finite A\<close>])
1104       apply simp
1105       apply (subst setprod.insert, assumption, assumption)
1106       apply (rule no_zero_divisors)
1107       apply blast+
1108       done
1109     moreover from \<open>finite A\<close> have "\<forall>a\<in>A. a dvd \<Prod>A" by blast
1110     ultimately have "\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l)" by blast
1111     with Lcm_0_iff' have "Lcm A \<noteq> 0" by simp
1112   }
1113   ultimately show "Lcm A = 0 \<longleftrightarrow> 0 \<in> A" by blast
1114 qed
1116 lemma Lcm_no_multiple:
1117   "(\<forall>m. m \<noteq> 0 \<longrightarrow> (\<exists>a\<in>A. \<not>a dvd m)) \<Longrightarrow> Lcm A = 0"
1118 proof -
1119   assume "\<forall>m. m \<noteq> 0 \<longrightarrow> (\<exists>a\<in>A. \<not>a dvd m)"
1120   hence "\<not>(\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l))" by blast
1121   then show "Lcm A = 0" by (simp only: Lcm_Lcm_eucl Lcm_eucl_def if_False)
1122 qed
1124 lemma Lcm_insert [simp]:
1125   "Lcm (insert a A) = lcm a (Lcm A)"
1126 proof (rule lcmI)
1127   fix l assume "a dvd l" and "Lcm A dvd l"
1128   then have "\<forall>a\<in>A. a dvd l" by (auto intro: dvd_trans [of _ "Lcm A" l])
1129   with \<open>a dvd l\<close> show "Lcm (insert a A) dvd l" by (force intro: Lcm_least)
1130 qed (auto intro: Lcm_least dvd_Lcm)
1132 lemma Lcm_finite:
1133   assumes "finite A"
1134   shows "Lcm A = Finite_Set.fold lcm 1 A"
1135   by (induct rule: finite.induct[OF \<open>finite A\<close>])
1136     (simp_all add: comp_fun_idem.fold_insert_idem[OF comp_fun_idem_lcm])
1138 lemma Lcm_set [code_unfold]:
1139   "Lcm (set xs) = fold lcm xs 1"
1140   using comp_fun_idem.fold_set_fold[OF comp_fun_idem_lcm] Lcm_finite by (simp add: ac_simps)
1142 lemma Lcm_singleton [simp]:
1143   "Lcm {a} = normalize a"
1144   by simp
1146 lemma Lcm_2 [simp]:
1147   "Lcm {a,b} = lcm a b"
1148   by simp
1150 lemma Lcm_coprime:
1151   assumes "finite A" and "A \<noteq> {}"
1152   assumes "\<And>a b. a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<noteq> b \<Longrightarrow> gcd a b = 1"
1153   shows "Lcm A = normalize (\<Prod>A)"
1154 using assms proof (induct rule: finite_ne_induct)
1155   case (insert a A)
1156   have "Lcm (insert a A) = lcm a (Lcm A)" by simp
1157   also from insert have "Lcm A = normalize (\<Prod>A)" by blast
1158   also have "lcm a \<dots> = lcm a (\<Prod>A)" by (cases "\<Prod>A = 0") (simp_all add: lcm_div_unit2)
1159   also from insert have "gcd a (\<Prod>A) = 1" by (subst gcd.commute, intro setprod_coprime) auto
1160   with insert have "lcm a (\<Prod>A) = normalize (\<Prod>(insert a A))"
1161     by (simp add: lcm_coprime)
1162   finally show ?case .
1163 qed simp
1165 lemma Lcm_coprime':
1166   "card A \<noteq> 0 \<Longrightarrow> (\<And>a b. a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<noteq> b \<Longrightarrow> gcd a b = 1)
1167     \<Longrightarrow> Lcm A = normalize (\<Prod>A)"
1168   by (rule Lcm_coprime) (simp_all add: card_eq_0_iff)
1170 lemma Gcd_Lcm:
1171   "Gcd A = Lcm {d. \<forall>a\<in>A. d dvd a}"
1172   by (simp add: Gcd_Gcd_eucl Lcm_Lcm_eucl Gcd_eucl_def)
1174 lemma Gcd_dvd [simp]: "a \<in> A \<Longrightarrow> Gcd A dvd a"
1175   and Gcd_greatest: "(\<And>a. a \<in> A \<Longrightarrow> b dvd a) \<Longrightarrow> b dvd Gcd A"
1176   and unit_factor_Gcd [simp]:
1177     "unit_factor (Gcd A) = (if Gcd A = 0 then 0 else 1)"
1178 proof -
1179   fix a assume "a \<in> A"
1180   hence "Lcm {d. \<forall>a\<in>A. d dvd a} dvd a" by (intro Lcm_least) blast
1181   then show "Gcd A dvd a" by (simp add: Gcd_Lcm)
1182 next
1183   fix g' assume "\<And>a. a \<in> A \<Longrightarrow> g' dvd a"
1184   hence "g' dvd Lcm {d. \<forall>a\<in>A. d dvd a}" by (intro dvd_Lcm) blast
1185   then show "g' dvd Gcd A" by (simp add: Gcd_Lcm)
1186 next
1187   show "unit_factor (Gcd A) = (if Gcd A = 0 then 0 else 1)"
1188     by (simp add: Gcd_Lcm)
1189 qed
1191 lemma normalize_Gcd [simp]:
1192   "normalize (Gcd A) = Gcd A"
1193 proof (cases "Gcd A = 0")
1194   case True then show ?thesis by simp
1195 next
1196   case False
1197   have "unit_factor (Gcd A) * normalize (Gcd A) = Gcd A"
1198     by (fact unit_factor_mult_normalize)
1199   with False show ?thesis by simp
1200 qed
1202 subclass semiring_Gcd
1203   by standard (auto intro: Gcd_greatest Lcm_least)
1205 lemma GcdI:
1206   assumes "\<And>a. a \<in> A \<Longrightarrow> b dvd a" and "\<And>c. (\<And>a. a \<in> A \<Longrightarrow> c dvd a) \<Longrightarrow> c dvd b"
1207     and "normalize b = b"
1208   shows "b = Gcd A"
1209   by (rule associated_eqI) (auto simp: assms intro: Gcd_greatest)
1211 lemma Lcm_Gcd:
1212   "Lcm A = Gcd {m. \<forall>a\<in>A. a dvd m}"
1213   by (rule LcmI[symmetric]) (auto intro: Gcd_greatest Gcd_greatest)
1215 lemma Gcd_1:
1216   "1 \<in> A \<Longrightarrow> Gcd A = 1"
1217   by (auto intro!: Gcd_eq_1_I)
1219 lemma Gcd_finite:
1220   assumes "finite A"
1221   shows "Gcd A = Finite_Set.fold gcd 0 A"
1222   by (induct rule: finite.induct[OF \<open>finite A\<close>])
1223     (simp_all add: comp_fun_idem.fold_insert_idem[OF comp_fun_idem_gcd])
1225 lemma Gcd_set [code_unfold]:
1226   "Gcd (set xs) = fold gcd xs 0"
1227   using comp_fun_idem.fold_set_fold[OF comp_fun_idem_gcd] Gcd_finite by (simp add: ac_simps)
1229 lemma Gcd_singleton [simp]: "Gcd {a} = normalize a"
1230   by simp
1232 lemma Gcd_2 [simp]: "Gcd {a,b} = gcd a b"
1233   by simp
1235 end
1237 text \<open>
1238   A Euclidean ring is a Euclidean semiring with additive inverses. It provides a
1239   few more lemmas; in particular, Bezout's lemma holds for any Euclidean ring.
1240 \<close>
1242 class euclidean_ring_gcd = euclidean_semiring_gcd + idom
1243 begin
1245 subclass euclidean_ring ..
1247 subclass ring_gcd ..
1249 lemma euclid_ext_gcd [simp]:
1250   "(case euclid_ext a b of (_, _ , t) \<Rightarrow> t) = gcd a b"
1251   by (induct a b rule: gcd_eucl_induct)
1252     (simp_all add: euclid_ext_0 euclid_ext_non_0 ac_simps split: prod.split prod.split_asm)
1254 lemma euclid_ext_gcd' [simp]:
1255   "euclid_ext a b = (r, s, t) \<Longrightarrow> t = gcd a b"
1256   by (insert euclid_ext_gcd[of a b], drule (1) subst, simp)
1258 lemma euclid_ext'_correct:
1259   "fst (euclid_ext' a b) * a + snd (euclid_ext' a b) * b = gcd a b"
1260 proof-
1261   obtain s t c where "euclid_ext a b = (s,t,c)"
1262     by (cases "euclid_ext a b", blast)
1263   with euclid_ext_correct[of a b] euclid_ext_gcd[of a b]
1264     show ?thesis unfolding euclid_ext'_def by simp
1265 qed
1267 lemma bezout: "\<exists>s t. s * a + t * b = gcd a b"
1268   using euclid_ext'_correct by blast
1270 lemma gcd_neg1 [simp]:
1271   "gcd (-a) b = gcd a b"
1272   by (rule sym, rule gcdI, simp_all add: gcd_greatest)
1274 lemma gcd_neg2 [simp]:
1275   "gcd a (-b) = gcd a b"
1276   by (rule sym, rule gcdI, simp_all add: gcd_greatest)
1278 lemma gcd_neg_numeral_1 [simp]:
1279   "gcd (- numeral n) a = gcd (numeral n) a"
1280   by (fact gcd_neg1)
1282 lemma gcd_neg_numeral_2 [simp]:
1283   "gcd a (- numeral n) = gcd a (numeral n)"
1284   by (fact gcd_neg2)
1286 lemma gcd_diff1: "gcd (m - n) n = gcd m n"
1287   by (subst diff_conv_add_uminus, subst gcd_neg2[symmetric],  subst gcd_add1, simp)
1289 lemma gcd_diff2: "gcd (n - m) n = gcd m n"
1290   by (subst gcd_neg1[symmetric], simp only: minus_diff_eq gcd_diff1)
1292 lemma coprime_minus_one [simp]: "gcd (n - 1) n = 1"
1293 proof -
1294   have "gcd (n - 1) n = gcd n (n - 1)" by (fact gcd.commute)
1295   also have "\<dots> = gcd ((n - 1) + 1) (n - 1)" by simp
1296   also have "\<dots> = 1" by (rule coprime_plus_one)
1297   finally show ?thesis .
1298 qed
1300 lemma lcm_neg1 [simp]: "lcm (-a) b = lcm a b"
1301   by (rule sym, rule lcmI, simp_all add: lcm_least lcm_zero)
1303 lemma lcm_neg2 [simp]: "lcm a (-b) = lcm a b"
1304   by (rule sym, rule lcmI, simp_all add: lcm_least lcm_zero)
1306 lemma lcm_neg_numeral_1 [simp]: "lcm (- numeral n) a = lcm (numeral n) a"
1307   by (fact lcm_neg1)
1309 lemma lcm_neg_numeral_2 [simp]: "lcm a (- numeral n) = lcm a (numeral n)"
1310   by (fact lcm_neg2)
1312 end
1315 subsection \<open>Typical instances\<close>
1317 instantiation nat :: euclidean_semiring
1318 begin
1320 definition [simp]:
1321   "euclidean_size_nat = (id :: nat \<Rightarrow> nat)"
1323 instance proof
1324 qed simp_all
1326 end
1328 instantiation int :: euclidean_ring
1329 begin
1331 definition [simp]:
1332   "euclidean_size_int = (nat \<circ> abs :: int \<Rightarrow> nat)"
1334 instance
1335 by standard (auto simp add: abs_mult nat_mult_distrib split: abs_split)
1337 end
1339 instantiation poly :: (field) euclidean_ring
1340 begin
1342 definition euclidean_size_poly :: "'a poly \<Rightarrow> nat"
1343   where "euclidean_size p = (if p = 0 then 0 else Suc (degree p))"
1345 lemma euclidenan_size_poly_minus_one_degree [simp]:
1346   "euclidean_size p - 1 = degree p"
1347   by (simp add: euclidean_size_poly_def)
1349 lemma euclidean_size_poly_0 [simp]:
1350   "euclidean_size (0::'a poly) = 0"
1351   by (simp add: euclidean_size_poly_def)
1353 lemma euclidean_size_poly_not_0 [simp]:
1354   "p \<noteq> 0 \<Longrightarrow> euclidean_size p = Suc (degree p)"
1355   by (simp add: euclidean_size_poly_def)
1357 instance
1358 proof
1359   fix p q :: "'a poly"
1360   assume "q \<noteq> 0"
1361   then have "p mod q = 0 \<or> degree (p mod q) < degree q"
1362     by (rule degree_mod_less [of q p])
1363   with \<open>q \<noteq> 0\<close> show "euclidean_size (p mod q) < euclidean_size q"
1364     by (cases "p mod q = 0") simp_all
1365 next
1366   fix p q :: "'a poly"
1367   assume "q \<noteq> 0"
1368   from \<open>q \<noteq> 0\<close> have "degree p \<le> degree (p * q)"
1369     by (rule degree_mult_right_le)
1370   with \<open>q \<noteq> 0\<close> show "euclidean_size p \<le> euclidean_size (p * q)"
1371     by (cases "p = 0") simp_all
1372 qed
1374 end
1376 (*instance nat :: euclidean_semiring_gcd
1377 proof (standard, auto intro!: ext)
1378   fix m n :: nat
1379   show *: "gcd m n = gcd_eucl m n"
1380   proof (induct m n rule: gcd_eucl_induct)
1381     case zero then show ?case by (simp add: gcd_eucl_0)
1382   next
1383     case (mod m n)
1384     with gcd_eucl_non_0 [of n m, symmetric]
1385     show ?case by (simp add: gcd_non_0_nat)
1386   qed
1387   show "lcm m n = lcm_eucl m n"
1388     by (simp add: lcm_eucl_def lcm_gcd * [symmetric] lcm_nat_def)
1389 qed*)
1391 end