src/HOL/GCD.thy
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1 (*  Title:      HOL/GCD.thy
2     Author:     Christophe Tabacznyj
3     Author:     Lawrence C. Paulson
4     Author:     Amine Chaieb
5     Author:     Thomas M. Rasmussen
7     Author:     Tobias Nipkow
9 This file deals with the functions gcd and lcm.  Definitions and
10 lemmas are proved uniformly for the natural numbers and integers.
12 This file combines and revises a number of prior developments.
14 The original theories "GCD" and "Primes" were by Christophe Tabacznyj
15 and Lawrence C. Paulson, based on @{cite davenport92}. They introduced
16 gcd, lcm, and prime for the natural numbers.
18 The original theory "IntPrimes" was by Thomas M. Rasmussen, and
19 extended gcd, lcm, primes to the integers. Amine Chaieb provided
20 another extension of the notions to the integers, and added a number
21 of results to "Primes" and "GCD". IntPrimes also defined and developed
22 the congruence relations on the integers. The notion was extended to
23 the natural numbers by Chaieb.
25 Jeremy Avigad combined all of these, made everything uniform for the
26 natural numbers and the integers, and added a number of new theorems.
28 Tobias Nipkow cleaned up a lot.
29 *)
31 section \<open>Greatest common divisor and least common multiple\<close>
33 theory GCD
34   imports Groups_List
35 begin
37 subsection \<open>Abstract bounded quasi semilattices as common foundation\<close>
39 locale bounded_quasi_semilattice = abel_semigroup +
40   fixes top :: 'a  ("\<^bold>\<top>") and bot :: 'a  ("\<^bold>\<bottom>")
41     and normalize :: "'a \<Rightarrow> 'a"
42   assumes idem_normalize [simp]: "a \<^bold>* a = normalize a"
43     and normalize_left_idem [simp]: "normalize a \<^bold>* b = a \<^bold>* b"
44     and normalize_idem [simp]: "normalize (a \<^bold>* b) = a \<^bold>* b"
45     and normalize_top [simp]: "normalize \<^bold>\<top> = \<^bold>\<top>"
46     and normalize_bottom [simp]: "normalize \<^bold>\<bottom> = \<^bold>\<bottom>"
47     and top_left_normalize [simp]: "\<^bold>\<top> \<^bold>* a = normalize a"
48     and bottom_left_bottom [simp]: "\<^bold>\<bottom> \<^bold>* a = \<^bold>\<bottom>"
49 begin
51 lemma left_idem [simp]:
52   "a \<^bold>* (a \<^bold>* b) = a \<^bold>* b"
53   using assoc [of a a b, symmetric] by simp
55 lemma right_idem [simp]:
56   "(a \<^bold>* b) \<^bold>* b = a \<^bold>* b"
57   using left_idem [of b a] by (simp add: ac_simps)
59 lemma comp_fun_idem: "comp_fun_idem f"
60   by standard (simp_all add: fun_eq_iff ac_simps)
62 interpretation comp_fun_idem f
63   by (fact comp_fun_idem)
65 lemma top_right_normalize [simp]:
66   "a \<^bold>* \<^bold>\<top> = normalize a"
67   using top_left_normalize [of a] by (simp add: ac_simps)
69 lemma bottom_right_bottom [simp]:
70   "a \<^bold>* \<^bold>\<bottom> = \<^bold>\<bottom>"
71   using bottom_left_bottom [of a] by (simp add: ac_simps)
73 lemma normalize_right_idem [simp]:
74   "a \<^bold>* normalize b = a \<^bold>* b"
75   using normalize_left_idem [of b a] by (simp add: ac_simps)
77 end
79 locale bounded_quasi_semilattice_set = bounded_quasi_semilattice
80 begin
82 interpretation comp_fun_idem f
83   by (fact comp_fun_idem)
85 definition F :: "'a set \<Rightarrow> 'a"
86 where
87   eq_fold: "F A = (if finite A then Finite_Set.fold f \<^bold>\<top> A else \<^bold>\<bottom>)"
89 lemma infinite [simp]:
90   "infinite A \<Longrightarrow> F A = \<^bold>\<bottom>"
93 lemma set_eq_fold [code]:
94   "F (set xs) = fold f xs \<^bold>\<top>"
95   by (simp add: eq_fold fold_set_fold)
97 lemma empty [simp]:
98   "F {} = \<^bold>\<top>"
101 lemma insert [simp]:
102   "F (insert a A) = a \<^bold>* F A"
103   by (cases "finite A") (simp_all add: eq_fold)
105 lemma normalize [simp]:
106   "normalize (F A) = F A"
107   by (induct A rule: infinite_finite_induct) simp_all
109 lemma in_idem:
110   assumes "a \<in> A"
111   shows "a \<^bold>* F A = F A"
112   using assms by (induct A rule: infinite_finite_induct)
113     (auto simp add: left_commute [of a])
115 lemma union:
116   "F (A \<union> B) = F A \<^bold>* F B"
117   by (induct A rule: infinite_finite_induct)
120 lemma remove:
121   assumes "a \<in> A"
122   shows "F A = a \<^bold>* F (A - {a})"
123 proof -
124   from assms obtain B where "A = insert a B" and "a \<notin> B"
125     by (blast dest: mk_disjoint_insert)
126   with assms show ?thesis by simp
127 qed
129 lemma insert_remove:
130   "F (insert a A) = a \<^bold>* F (A - {a})"
131   by (cases "a \<in> A") (simp_all add: insert_absorb remove)
133 lemma subset:
134   assumes "B \<subseteq> A"
135   shows "F B \<^bold>* F A = F A"
136   using assms by (simp add: union [symmetric] Un_absorb1)
138 end
140 subsection \<open>Abstract GCD and LCM\<close>
142 class gcd = zero + one + dvd +
143   fixes gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
144     and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
145 begin
147 abbreviation coprime :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
148   where "coprime x y \<equiv> gcd x y = 1"
150 end
152 class Gcd = gcd +
153   fixes Gcd :: "'a set \<Rightarrow> 'a"
154     and Lcm :: "'a set \<Rightarrow> 'a"
155 begin
157 abbreviation GREATEST_COMMON_DIVISOR :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a"
158   where "GREATEST_COMMON_DIVISOR A f \<equiv> Gcd (f ` A)"
160 abbreviation LEAST_COMMON_MULTIPLE :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a"
161   where "LEAST_COMMON_MULTIPLE A f \<equiv> Lcm (f ` A)"
163 end
165 syntax
166   "_GCD1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3GCD _./ _)" [0, 10] 10)
167   "_GCD"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3GCD _\<in>_./ _)" [0, 0, 10] 10)
168   "_LCM1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3LCM _./ _)" [0, 10] 10)
169   "_LCM"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3LCM _\<in>_./ _)" [0, 0, 10] 10)
170 translations
171   "GCD x y. B"   \<rightleftharpoons> "GCD x. GCD y. B"
172   "GCD x. B"     \<rightleftharpoons> "CONST GREATEST_COMMON_DIVISOR CONST UNIV (\<lambda>x. B)"
173   "GCD x. B"     \<rightleftharpoons> "GCD x \<in> CONST UNIV. B"
174   "GCD x\<in>A. B"   \<rightleftharpoons> "CONST GREATEST_COMMON_DIVISOR A (\<lambda>x. B)"
175   "LCM x y. B"   \<rightleftharpoons> "LCM x. LCM y. B"
176   "LCM x. B"     \<rightleftharpoons> "CONST LEAST_COMMON_MULTIPLE CONST UNIV (\<lambda>x. B)"
177   "LCM x. B"     \<rightleftharpoons> "LCM x \<in> CONST UNIV. B"
178   "LCM x\<in>A. B"   \<rightleftharpoons> "CONST LEAST_COMMON_MULTIPLE A (\<lambda>x. B)"
180 print_translation \<open>
181   [Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax GREATEST_COMMON_DIVISOR} @{syntax_const "_GCD"},
182     Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax LEAST_COMMON_MULTIPLE} @{syntax_const "_LCM"}]
183 \<close> \<comment> \<open>to avoid eta-contraction of body\<close>
185 class semiring_gcd = normalization_semidom + gcd +
186   assumes gcd_dvd1 [iff]: "gcd a b dvd a"
187     and gcd_dvd2 [iff]: "gcd a b dvd b"
188     and gcd_greatest: "c dvd a \<Longrightarrow> c dvd b \<Longrightarrow> c dvd gcd a b"
189     and normalize_gcd [simp]: "normalize (gcd a b) = gcd a b"
190     and lcm_gcd: "lcm a b = normalize (a * b) div gcd a b"
191 begin
193 lemma gcd_greatest_iff [simp]: "a dvd gcd b c \<longleftrightarrow> a dvd b \<and> a dvd c"
194   by (blast intro!: gcd_greatest intro: dvd_trans)
196 lemma gcd_dvdI1: "a dvd c \<Longrightarrow> gcd a b dvd c"
197   by (rule dvd_trans) (rule gcd_dvd1)
199 lemma gcd_dvdI2: "b dvd c \<Longrightarrow> gcd a b dvd c"
200   by (rule dvd_trans) (rule gcd_dvd2)
202 lemma dvd_gcdD1: "a dvd gcd b c \<Longrightarrow> a dvd b"
203   using gcd_dvd1 [of b c] by (blast intro: dvd_trans)
205 lemma dvd_gcdD2: "a dvd gcd b c \<Longrightarrow> a dvd c"
206   using gcd_dvd2 [of b c] by (blast intro: dvd_trans)
208 lemma gcd_0_left [simp]: "gcd 0 a = normalize a"
209   by (rule associated_eqI) simp_all
211 lemma gcd_0_right [simp]: "gcd a 0 = normalize a"
212   by (rule associated_eqI) simp_all
214 lemma gcd_eq_0_iff [simp]: "gcd a b = 0 \<longleftrightarrow> a = 0 \<and> b = 0"
215   (is "?P \<longleftrightarrow> ?Q")
216 proof
217   assume ?P
218   then have "0 dvd gcd a b"
219     by simp
220   then have "0 dvd a" and "0 dvd b"
221     by (blast intro: dvd_trans)+
222   then show ?Q
223     by simp
224 next
225   assume ?Q
226   then show ?P
227     by simp
228 qed
230 lemma unit_factor_gcd: "unit_factor (gcd a b) = (if a = 0 \<and> b = 0 then 0 else 1)"
231 proof (cases "gcd a b = 0")
232   case True
233   then show ?thesis by simp
234 next
235   case False
236   have "unit_factor (gcd a b) * normalize (gcd a b) = gcd a b"
237     by (rule unit_factor_mult_normalize)
238   then have "unit_factor (gcd a b) * gcd a b = gcd a b"
239     by simp
240   then have "unit_factor (gcd a b) * gcd a b div gcd a b = gcd a b div gcd a b"
241     by simp
242   with False show ?thesis
243     by simp
244 qed
246 lemma is_unit_gcd [simp]: "is_unit (gcd a b) \<longleftrightarrow> coprime a b"
247   by (cases "a = 0 \<and> b = 0") (auto simp add: unit_factor_gcd dest: is_unit_unit_factor)
249 sublocale gcd: abel_semigroup gcd
250 proof
251   fix a b c
252   show "gcd a b = gcd b a"
253     by (rule associated_eqI) simp_all
254   from gcd_dvd1 have "gcd (gcd a b) c dvd a"
255     by (rule dvd_trans) simp
256   moreover from gcd_dvd1 have "gcd (gcd a b) c dvd b"
257     by (rule dvd_trans) simp
258   ultimately have P1: "gcd (gcd a b) c dvd gcd a (gcd b c)"
259     by (auto intro!: gcd_greatest)
260   from gcd_dvd2 have "gcd a (gcd b c) dvd b"
261     by (rule dvd_trans) simp
262   moreover from gcd_dvd2 have "gcd a (gcd b c) dvd c"
263     by (rule dvd_trans) simp
264   ultimately have P2: "gcd a (gcd b c) dvd gcd (gcd a b) c"
265     by (auto intro!: gcd_greatest)
266   from P1 P2 show "gcd (gcd a b) c = gcd a (gcd b c)"
267     by (rule associated_eqI) simp_all
268 qed
270 sublocale gcd: bounded_quasi_semilattice gcd 0 1 normalize
271 proof
272   show "gcd a a = normalize a" for a
273   proof -
274     have "a dvd gcd a a"
275       by (rule gcd_greatest) simp_all
276     then show ?thesis
277       by (auto intro: associated_eqI)
278   qed
279   show "gcd (normalize a) b = gcd a b" for a b
280     using gcd_dvd1 [of "normalize a" b]
281     by (auto intro: associated_eqI)
282   show "coprime 1 a" for a
283     by (rule associated_eqI) simp_all
284 qed simp_all
286 lemma gcd_self: "gcd a a = normalize a"
287   by (fact gcd.idem_normalize)
289 lemma gcd_left_idem: "gcd a (gcd a b) = gcd a b"
290   by (fact gcd.left_idem)
292 lemma gcd_right_idem: "gcd (gcd a b) b = gcd a b"
293   by (fact gcd.right_idem)
295 lemma coprime_1_left: "coprime 1 a"
296   by (fact gcd.bottom_left_bottom)
298 lemma coprime_1_right: "coprime a 1"
299   by (fact gcd.bottom_right_bottom)
301 lemma gcd_mult_left: "gcd (c * a) (c * b) = normalize c * gcd a b"
302 proof (cases "c = 0")
303   case True
304   then show ?thesis by simp
305 next
306   case False
307   then have *: "c * gcd a b dvd gcd (c * a) (c * b)"
308     by (auto intro: gcd_greatest)
309   moreover from False * have "gcd (c * a) (c * b) dvd c * gcd a b"
310     by (metis div_dvd_iff_mult dvd_mult_left gcd_dvd1 gcd_dvd2 gcd_greatest mult_commute)
311   ultimately have "normalize (gcd (c * a) (c * b)) = normalize (c * gcd a b)"
312     by (auto intro: associated_eqI)
313   then show ?thesis
315 qed
317 lemma gcd_mult_right: "gcd (a * c) (b * c) = gcd b a * normalize c"
318   using gcd_mult_left [of c a b] by (simp add: ac_simps)
320 lemma mult_gcd_left: "c * gcd a b = unit_factor c * gcd (c * a) (c * b)"
321   by (simp add: gcd_mult_left mult.assoc [symmetric])
323 lemma mult_gcd_right: "gcd a b * c = gcd (a * c) (b * c) * unit_factor c"
324   using mult_gcd_left [of c a b] by (simp add: ac_simps)
326 lemma dvd_lcm1 [iff]: "a dvd lcm a b"
327 proof -
328   have "normalize (a * b) div gcd a b = normalize a * (normalize b div gcd a b)"
329     by (simp add: lcm_gcd normalize_mult div_mult_swap)
330   then show ?thesis
332 qed
334 lemma dvd_lcm2 [iff]: "b dvd lcm a b"
335 proof -
336   have "normalize (a * b) div gcd a b = normalize b * (normalize a div gcd a b)"
337     by (simp add: lcm_gcd normalize_mult div_mult_swap ac_simps)
338   then show ?thesis
340 qed
342 lemma dvd_lcmI1: "a dvd b \<Longrightarrow> a dvd lcm b c"
343   by (rule dvd_trans) (assumption, blast)
345 lemma dvd_lcmI2: "a dvd c \<Longrightarrow> a dvd lcm b c"
346   by (rule dvd_trans) (assumption, blast)
348 lemma lcm_dvdD1: "lcm a b dvd c \<Longrightarrow> a dvd c"
349   using dvd_lcm1 [of a b] by (blast intro: dvd_trans)
351 lemma lcm_dvdD2: "lcm a b dvd c \<Longrightarrow> b dvd c"
352   using dvd_lcm2 [of a b] by (blast intro: dvd_trans)
354 lemma lcm_least:
355   assumes "a dvd c" and "b dvd c"
356   shows "lcm a b dvd c"
357 proof (cases "c = 0")
358   case True
359   then show ?thesis by simp
360 next
361   case False
362   then have *: "is_unit (unit_factor c)"
363     by simp
364   show ?thesis
365   proof (cases "gcd a b = 0")
366     case True
367     with assms show ?thesis by simp
368   next
369     case False
370     then have "a \<noteq> 0 \<or> b \<noteq> 0"
371       by simp
372     with \<open>c \<noteq> 0\<close> assms have "a * b dvd a * c" "a * b dvd c * b"
374     then have "normalize (a * b) dvd gcd (a * c) (b * c)"
375       by (auto intro: gcd_greatest simp add: ac_simps)
376     then have "normalize (a * b) dvd gcd (a * c) (b * c) * unit_factor c"
377       using * by (simp add: dvd_mult_unit_iff)
378     then have "normalize (a * b) dvd gcd a b * c"
379       by (simp add: mult_gcd_right [of a b c])
380     then have "normalize (a * b) div gcd a b dvd c"
381       using False by (simp add: div_dvd_iff_mult ac_simps)
382     then show ?thesis
384   qed
385 qed
387 lemma lcm_least_iff [simp]: "lcm a b dvd c \<longleftrightarrow> a dvd c \<and> b dvd c"
388   by (blast intro!: lcm_least intro: dvd_trans)
390 lemma normalize_lcm [simp]: "normalize (lcm a b) = lcm a b"
391   by (simp add: lcm_gcd dvd_normalize_div)
393 lemma lcm_0_left [simp]: "lcm 0 a = 0"
396 lemma lcm_0_right [simp]: "lcm a 0 = 0"
399 lemma lcm_eq_0_iff: "lcm a b = 0 \<longleftrightarrow> a = 0 \<or> b = 0"
400   (is "?P \<longleftrightarrow> ?Q")
401 proof
402   assume ?P
403   then have "0 dvd lcm a b"
404     by simp
405   then have "0 dvd normalize (a * b) div gcd a b"
407   then have "0 * gcd a b dvd normalize (a * b)"
408     using dvd_div_iff_mult [of "gcd a b" _ 0] by (cases "gcd a b = 0") simp_all
409   then have "normalize (a * b) = 0"
410     by simp
411   then show ?Q
412     by simp
413 next
414   assume ?Q
415   then show ?P
416     by auto
417 qed
419 lemma lcm_eq_1_iff [simp]: "lcm a b = 1 \<longleftrightarrow> is_unit a \<and> is_unit b"
420   by (auto intro: associated_eqI)
422 lemma unit_factor_lcm: "unit_factor (lcm a b) = (if a = 0 \<or> b = 0 then 0 else 1)"
423   by (simp add: unit_factor_gcd dvd_unit_factor_div lcm_gcd)
425 sublocale lcm: abel_semigroup lcm
426 proof
427   fix a b c
428   show "lcm a b = lcm b a"
429     by (simp add: lcm_gcd ac_simps normalize_mult dvd_normalize_div)
430   have "lcm (lcm a b) c dvd lcm a (lcm b c)"
431     and "lcm a (lcm b c) dvd lcm (lcm a b) c"
432     by (auto intro: lcm_least
433       dvd_trans [of b "lcm b c" "lcm a (lcm b c)"]
434       dvd_trans [of c "lcm b c" "lcm a (lcm b c)"]
435       dvd_trans [of a "lcm a b" "lcm (lcm a b) c"]
436       dvd_trans [of b "lcm a b" "lcm (lcm a b) c"])
437   then show "lcm (lcm a b) c = lcm a (lcm b c)"
438     by (rule associated_eqI) simp_all
439 qed
441 sublocale lcm: bounded_quasi_semilattice lcm 1 0 normalize
442 proof
443   show "lcm a a = normalize a" for a
444   proof -
445     have "lcm a a dvd a"
446       by (rule lcm_least) simp_all
447     then show ?thesis
448       by (auto intro: associated_eqI)
449   qed
450   show "lcm (normalize a) b = lcm a b" for a b
451     using dvd_lcm1 [of "normalize a" b] unfolding normalize_dvd_iff
452     by (auto intro: associated_eqI)
453   show "lcm 1 a = normalize a" for a
454     by (rule associated_eqI) simp_all
455 qed simp_all
457 lemma lcm_self: "lcm a a = normalize a"
458   by (fact lcm.idem_normalize)
460 lemma lcm_left_idem: "lcm a (lcm a b) = lcm a b"
461   by (fact lcm.left_idem)
463 lemma lcm_right_idem: "lcm (lcm a b) b = lcm a b"
464   by (fact lcm.right_idem)
466 lemma gcd_mult_lcm [simp]: "gcd a b * lcm a b = normalize a * normalize b"
467   by (simp add: lcm_gcd normalize_mult)
469 lemma lcm_mult_gcd [simp]: "lcm a b * gcd a b = normalize a * normalize b"
470   using gcd_mult_lcm [of a b] by (simp add: ac_simps)
472 lemma gcd_lcm:
473   assumes "a \<noteq> 0" and "b \<noteq> 0"
474   shows "gcd a b = normalize (a * b) div lcm a b"
475 proof -
476   from assms have "lcm a b \<noteq> 0"
478   have "gcd a b * lcm a b = normalize a * normalize b"
479     by simp
480   then have "gcd a b * lcm a b div lcm a b = normalize (a * b) div lcm a b"
482   with \<open>lcm a b \<noteq> 0\<close> show ?thesis
483     using nonzero_mult_div_cancel_right [of "lcm a b" "gcd a b"] by simp
484 qed
486 lemma lcm_1_left: "lcm 1 a = normalize a"
487   by (fact lcm.top_left_normalize)
489 lemma lcm_1_right: "lcm a 1 = normalize a"
490   by (fact lcm.top_right_normalize)
492 lemma lcm_mult_left: "lcm (c * a) (c * b) = normalize c * lcm a b"
493   by (cases "c = 0")
494     (simp_all add: gcd_mult_right lcm_gcd div_mult_swap normalize_mult ac_simps,
495       simp add: dvd_div_mult2_eq mult.left_commute [of "normalize c", symmetric])
497 lemma lcm_mult_right: "lcm (a * c) (b * c) = lcm b a * normalize c"
498   using lcm_mult_left [of c a b] by (simp add: ac_simps)
500 lemma mult_lcm_left: "c * lcm a b = unit_factor c * lcm (c * a) (c * b)"
501   by (simp add: lcm_mult_left mult.assoc [symmetric])
503 lemma mult_lcm_right: "lcm a b * c = lcm (a * c) (b * c) * unit_factor c"
504   using mult_lcm_left [of c a b] by (simp add: ac_simps)
506 lemma gcdI:
507   assumes "c dvd a" and "c dvd b"
508     and greatest: "\<And>d. d dvd a \<Longrightarrow> d dvd b \<Longrightarrow> d dvd c"
509     and "normalize c = c"
510   shows "c = gcd a b"
511   by (rule associated_eqI) (auto simp: assms intro: gcd_greatest)
513 lemma gcd_unique:
514   "d dvd a \<and> d dvd b \<and> normalize d = d \<and> (\<forall>e. e dvd a \<and> e dvd b \<longrightarrow> e dvd d) \<longleftrightarrow> d = gcd a b"
515   by rule (auto intro: gcdI simp: gcd_greatest)
517 lemma gcd_dvd_prod: "gcd a b dvd k * b"
518   using mult_dvd_mono [of 1] by auto
520 lemma gcd_proj2_if_dvd: "b dvd a \<Longrightarrow> gcd a b = normalize b"
521   by (rule gcdI [symmetric]) simp_all
523 lemma gcd_proj1_if_dvd: "a dvd b \<Longrightarrow> gcd a b = normalize a"
524   by (rule gcdI [symmetric]) simp_all
526 lemma gcd_proj1_iff: "gcd m n = normalize m \<longleftrightarrow> m dvd n"
527 proof
528   assume *: "gcd m n = normalize m"
529   show "m dvd n"
530   proof (cases "m = 0")
531     case True
532     with * show ?thesis by simp
533   next
534     case [simp]: False
535     from * have **: "m = gcd m n * unit_factor m"
537     show ?thesis
538       by (subst **) (simp add: mult_unit_dvd_iff)
539   qed
540 next
541   assume "m dvd n"
542   then show "gcd m n = normalize m"
543     by (rule gcd_proj1_if_dvd)
544 qed
546 lemma gcd_proj2_iff: "gcd m n = normalize n \<longleftrightarrow> n dvd m"
547   using gcd_proj1_iff [of n m] by (simp add: ac_simps)
549 lemma gcd_mult_distrib': "normalize c * gcd a b = gcd (c * a) (c * b)"
550   by (rule gcdI) (auto simp: normalize_mult gcd_greatest mult_dvd_mono gcd_mult_left[symmetric])
552 lemma gcd_mult_distrib: "k * gcd a b = gcd (k * a) (k * b) * unit_factor k"
553 proof-
554   have "normalize k * gcd a b = gcd (k * a) (k * b)"
556   then have "normalize k * gcd a b * unit_factor k = gcd (k * a) (k * b) * unit_factor k"
557     by simp
558   then have "normalize k * unit_factor k * gcd a b  = gcd (k * a) (k * b) * unit_factor k"
559     by (simp only: ac_simps)
560   then show ?thesis
561     by simp
562 qed
564 lemma lcm_mult_unit1: "is_unit a \<Longrightarrow> lcm (b * a) c = lcm b c"
565   by (rule associated_eqI) (simp_all add: mult_unit_dvd_iff dvd_lcmI1)
567 lemma lcm_mult_unit2: "is_unit a \<Longrightarrow> lcm b (c * a) = lcm b c"
568   using lcm_mult_unit1 [of a c b] by (simp add: ac_simps)
570 lemma lcm_div_unit1:
571   "is_unit a \<Longrightarrow> lcm (b div a) c = lcm b c"
572   by (erule is_unitE [of _ b]) (simp add: lcm_mult_unit1)
574 lemma lcm_div_unit2: "is_unit a \<Longrightarrow> lcm b (c div a) = lcm b c"
575   by (erule is_unitE [of _ c]) (simp add: lcm_mult_unit2)
577 lemma normalize_lcm_left: "lcm (normalize a) b = lcm a b"
578   by (fact lcm.normalize_left_idem)
580 lemma normalize_lcm_right: "lcm a (normalize b) = lcm a b"
581   by (fact lcm.normalize_right_idem)
583 lemma gcd_mult_unit1: "is_unit a \<Longrightarrow> gcd (b * a) c = gcd b c"
584   apply (rule gcdI)
585      apply simp_all
586   apply (rule dvd_trans)
587    apply (rule gcd_dvd1)
589   done
591 lemma gcd_mult_unit2: "is_unit a \<Longrightarrow> gcd b (c * a) = gcd b c"
592   apply (subst gcd.commute)
593   apply (subst gcd_mult_unit1)
594    apply assumption
595   apply (rule gcd.commute)
596   done
598 lemma gcd_div_unit1: "is_unit a \<Longrightarrow> gcd (b div a) c = gcd b c"
599   by (erule is_unitE [of _ b]) (simp add: gcd_mult_unit1)
601 lemma gcd_div_unit2: "is_unit a \<Longrightarrow> gcd b (c div a) = gcd b c"
602   by (erule is_unitE [of _ c]) (simp add: gcd_mult_unit2)
604 lemma normalize_gcd_left: "gcd (normalize a) b = gcd a b"
605   by (fact gcd.normalize_left_idem)
607 lemma normalize_gcd_right: "gcd a (normalize b) = gcd a b"
608   by (fact gcd.normalize_right_idem)
610 lemma comp_fun_idem_gcd: "comp_fun_idem gcd"
611   by standard (simp_all add: fun_eq_iff ac_simps)
613 lemma comp_fun_idem_lcm: "comp_fun_idem lcm"
614   by standard (simp_all add: fun_eq_iff ac_simps)
616 lemma gcd_dvd_antisym: "gcd a b dvd gcd c d \<Longrightarrow> gcd c d dvd gcd a b \<Longrightarrow> gcd a b = gcd c d"
617 proof (rule gcdI)
618   assume *: "gcd a b dvd gcd c d"
619     and **: "gcd c d dvd gcd a b"
620   have "gcd c d dvd c"
621     by simp
622   with * show "gcd a b dvd c"
623     by (rule dvd_trans)
624   have "gcd c d dvd d"
625     by simp
626   with * show "gcd a b dvd d"
627     by (rule dvd_trans)
628   show "normalize (gcd a b) = gcd a b"
629     by simp
630   fix l assume "l dvd c" and "l dvd d"
631   then have "l dvd gcd c d"
632     by (rule gcd_greatest)
633   from this and ** show "l dvd gcd a b"
634     by (rule dvd_trans)
635 qed
637 lemma coprime_dvd_mult:
638   assumes "coprime a b" and "a dvd c * b"
639   shows "a dvd c"
640 proof (cases "c = 0")
641   case True
642   then show ?thesis by simp
643 next
644   case False
645   then have unit: "is_unit (unit_factor c)"
646     by simp
647   from \<open>coprime a b\<close> mult_gcd_left [of c a b]
648   have "gcd (c * a) (c * b) * unit_factor c = c"
650   moreover from \<open>a dvd c * b\<close> have "a dvd gcd (c * a) (c * b) * unit_factor c"
651     by (simp add: dvd_mult_unit_iff unit)
652   ultimately show ?thesis
653     by simp
654 qed
656 lemma coprime_dvd_mult_iff: "coprime a c \<Longrightarrow> a dvd b * c \<longleftrightarrow> a dvd b"
657   by (auto intro: coprime_dvd_mult)
659 lemma gcd_mult_cancel: "coprime c b \<Longrightarrow> gcd (c * a) b = gcd a b"
660   apply (rule associated_eqI)
661      apply (rule gcd_greatest)
662       apply (rule_tac b = c in coprime_dvd_mult)
665   done
667 lemma coprime_crossproduct:
668   fixes a b c d :: 'a
669   assumes "coprime a d" and "coprime b c"
670   shows "normalize a * normalize c = normalize b * normalize d \<longleftrightarrow>
671     normalize a = normalize b \<and> normalize c = normalize d"
672     (is "?lhs \<longleftrightarrow> ?rhs")
673 proof
674   assume ?rhs
675   then show ?lhs by simp
676 next
677   assume ?lhs
678   from \<open>?lhs\<close> have "normalize a dvd normalize b * normalize d"
679     by (auto intro: dvdI dest: sym)
680   with \<open>coprime a d\<close> have "a dvd b"
681     by (simp add: coprime_dvd_mult_iff normalize_mult [symmetric])
682   from \<open>?lhs\<close> have "normalize b dvd normalize a * normalize c"
683     by (auto intro: dvdI dest: sym)
684   with \<open>coprime b c\<close> have "b dvd a"
685     by (simp add: coprime_dvd_mult_iff normalize_mult [symmetric])
686   from \<open>?lhs\<close> have "normalize c dvd normalize d * normalize b"
687     by (auto intro: dvdI dest: sym simp add: mult.commute)
688   with \<open>coprime b c\<close> have "c dvd d"
689     by (simp add: coprime_dvd_mult_iff gcd.commute normalize_mult [symmetric])
690   from \<open>?lhs\<close> have "normalize d dvd normalize c * normalize a"
691     by (auto intro: dvdI dest: sym simp add: mult.commute)
692   with \<open>coprime a d\<close> have "d dvd c"
693     by (simp add: coprime_dvd_mult_iff gcd.commute normalize_mult [symmetric])
694   from \<open>a dvd b\<close> \<open>b dvd a\<close> have "normalize a = normalize b"
695     by (rule associatedI)
696   moreover from \<open>c dvd d\<close> \<open>d dvd c\<close> have "normalize c = normalize d"
697     by (rule associatedI)
698   ultimately show ?rhs ..
699 qed
701 lemma gcd_add1 [simp]: "gcd (m + n) n = gcd m n"
704 lemma gcd_add2 [simp]: "gcd m (m + n) = gcd m n"
707 lemma gcd_add_mult: "gcd m (k * m + n) = gcd m n"
710 lemma coprimeI: "(\<And>l. l dvd a \<Longrightarrow> l dvd b \<Longrightarrow> l dvd 1) \<Longrightarrow> gcd a b = 1"
711   by (rule sym, rule gcdI) simp_all
713 lemma coprime: "gcd a b = 1 \<longleftrightarrow> (\<forall>d. d dvd a \<and> d dvd b \<longleftrightarrow> is_unit d)"
714   by (auto intro: coprimeI gcd_greatest dvd_gcdD1 dvd_gcdD2)
716 lemma div_gcd_coprime:
717   assumes nz: "a \<noteq> 0 \<or> b \<noteq> 0"
718   shows "coprime (a div gcd a b) (b div gcd a b)"
719 proof -
720   let ?g = "gcd a b"
721   let ?a' = "a div ?g"
722   let ?b' = "b div ?g"
723   let ?g' = "gcd ?a' ?b'"
724   have dvdg: "?g dvd a" "?g dvd b"
725     by simp_all
726   have dvdg': "?g' dvd ?a'" "?g' dvd ?b'"
727     by simp_all
728   from dvdg dvdg' obtain ka kb ka' kb' where
729     kab: "a = ?g * ka" "b = ?g * kb" "?a' = ?g' * ka'" "?b' = ?g' * kb'"
730     unfolding dvd_def by blast
731   from this [symmetric] have "?g * ?a' = (?g * ?g') * ka'" "?g * ?b' = (?g * ?g') * kb'"
732     by (simp_all add: mult.assoc mult.left_commute [of "gcd a b"])
733   then have dvdgg':"?g * ?g' dvd a" "?g* ?g' dvd b"
734     by (auto simp add: dvd_mult_div_cancel [OF dvdg(1)] dvd_mult_div_cancel [OF dvdg(2)] dvd_def)
735   have "?g \<noteq> 0"
736     using nz by simp
737   moreover from gcd_greatest [OF dvdgg'] have "?g * ?g' dvd ?g" .
738   ultimately show ?thesis
739     using dvd_times_left_cancel_iff [of "gcd a b" _ 1] by simp
740 qed
742 lemma divides_mult:
743   assumes "a dvd c" and nr: "b dvd c" and "coprime a b"
744   shows "a * b dvd c"
745 proof -
746   from \<open>b dvd c\<close> obtain b' where"c = b * b'" ..
747   with \<open>a dvd c\<close> have "a dvd b' * b"
749   with \<open>coprime a b\<close> have "a dvd b'"
751   then obtain a' where "b' = a * a'" ..
752   with \<open>c = b * b'\<close> have "c = (a * b) * a'"
754   then show ?thesis ..
755 qed
757 lemma coprime_lmult:
758   assumes dab: "gcd d (a * b) = 1"
759   shows "gcd d a = 1"
760 proof (rule coprimeI)
761   fix l
762   assume "l dvd d" and "l dvd a"
763   then have "l dvd a * b"
764     by simp
765   with \<open>l dvd d\<close> and dab show "l dvd 1"
766     by (auto intro: gcd_greatest)
767 qed
769 lemma coprime_rmult:
770   assumes dab: "gcd d (a * b) = 1"
771   shows "gcd d b = 1"
772 proof (rule coprimeI)
773   fix l
774   assume "l dvd d" and "l dvd b"
775   then have "l dvd a * b"
776     by simp
777   with \<open>l dvd d\<close> and dab show "l dvd 1"
778     by (auto intro: gcd_greatest)
779 qed
781 lemma coprime_mult:
782   assumes "coprime d a"
783     and "coprime d b"
784   shows "coprime d (a * b)"
785   apply (subst gcd.commute)
786   using assms(1) apply (subst gcd_mult_cancel)
787    apply (subst gcd.commute)
788    apply assumption
789   apply (subst gcd.commute)
790   apply (rule assms(2))
791   done
793 lemma coprime_mul_eq: "gcd d (a * b) = 1 \<longleftrightarrow> gcd d a = 1 \<and> gcd d b = 1"
794   using coprime_rmult[of d a b] coprime_lmult[of d a b] coprime_mult[of d a b]
795   by blast
797 lemma coprime_mul_eq':
798   "coprime (a * b) d \<longleftrightarrow> coprime a d \<and> coprime b d"
799   using coprime_mul_eq [of d a b] by (simp add: gcd.commute)
801 lemma gcd_coprime:
802   assumes c: "gcd a b \<noteq> 0"
803     and a: "a = a' * gcd a b"
804     and b: "b = b' * gcd a b"
805   shows "gcd a' b' = 1"
806 proof -
807   from c have "a \<noteq> 0 \<or> b \<noteq> 0"
808     by simp
809   with div_gcd_coprime have "gcd (a div gcd a b) (b div gcd a b) = 1" .
810   also from assms have "a div gcd a b = a'"
811     using dvd_div_eq_mult local.gcd_dvd1 by blast
812   also from assms have "b div gcd a b = b'"
813     using dvd_div_eq_mult local.gcd_dvd1 by blast
814   finally show ?thesis .
815 qed
817 lemma coprime_power:
818   assumes "0 < n"
819   shows "gcd a (b ^ n) = 1 \<longleftrightarrow> gcd a b = 1"
820   using assms
821 proof (induct n)
822   case 0
823   then show ?case by simp
824 next
825   case (Suc n)
826   then show ?case
827     by (cases n) (simp_all add: coprime_mul_eq)
828 qed
830 lemma gcd_coprime_exists:
831   assumes "gcd a b \<noteq> 0"
832   shows "\<exists>a' b'. a = a' * gcd a b \<and> b = b' * gcd a b \<and> gcd a' b' = 1"
833   apply (rule_tac x = "a div gcd a b" in exI)
834   apply (rule_tac x = "b div gcd a b" in exI)
835   using assms
836   apply (auto intro: div_gcd_coprime)
837   done
839 lemma coprime_exp: "gcd d a = 1 \<Longrightarrow> gcd d (a^n) = 1"
840   by (induct n) (simp_all add: coprime_mult)
842 lemma coprime_exp_left: "coprime a b \<Longrightarrow> coprime (a ^ n) b"
843   by (induct n) (simp_all add: gcd_mult_cancel)
845 lemma coprime_exp2:
846   assumes "coprime a b"
847   shows "coprime (a ^ n) (b ^ m)"
848 proof (rule coprime_exp_left)
849   from assms show "coprime a (b ^ m)"
850     by (induct m) (simp_all add: gcd_mult_cancel gcd.commute [of a])
851 qed
853 lemma gcd_exp: "gcd (a ^ n) (b ^ n) = gcd a b ^ n"
854 proof (cases "a = 0 \<and> b = 0")
855   case True
856   then show ?thesis
857     by (cases n) simp_all
858 next
859   case False
860   then have "1 = gcd ((a div gcd a b) ^ n) ((b div gcd a b) ^ n)"
861     using coprime_exp2[OF div_gcd_coprime[of a b], of n n, symmetric] by simp
862   then have "gcd a b ^ n = gcd a b ^ n * \<dots>"
863     by simp
864   also note gcd_mult_distrib
865   also have "unit_factor (gcd a b ^ n) = 1"
866     using False by (auto simp add: unit_factor_power unit_factor_gcd)
867   also have "(gcd a b)^n * (a div gcd a b)^n = a^n"
868     apply (subst ac_simps)
869     apply (subst div_power)
870      apply simp
871     apply (rule dvd_div_mult_self)
872     apply (rule dvd_power_same)
873     apply simp
874     done
875   also have "(gcd a b)^n * (b div gcd a b)^n = b^n"
876     apply (subst ac_simps)
877     apply (subst div_power)
878      apply simp
879     apply (rule dvd_div_mult_self)
880     apply (rule dvd_power_same)
881     apply simp
882     done
883   finally show ?thesis by simp
884 qed
886 lemma coprime_common_divisor: "gcd a b = 1 \<Longrightarrow> a dvd a \<Longrightarrow> a dvd b \<Longrightarrow> is_unit a"
887   apply (subgoal_tac "a dvd gcd a b")
888    apply simp
889   apply (erule (1) gcd_greatest)
890   done
892 lemma division_decomp:
893   assumes "a dvd b * c"
894   shows "\<exists>b' c'. a = b' * c' \<and> b' dvd b \<and> c' dvd c"
895 proof (cases "gcd a b = 0")
896   case True
897   then have "a = 0 \<and> b = 0"
898     by simp
899   then have "a = 0 * c \<and> 0 dvd b \<and> c dvd c"
900     by simp
901   then show ?thesis by blast
902 next
903   case False
904   let ?d = "gcd a b"
905   from gcd_coprime_exists [OF False]
906     obtain a' b' where ab': "a = a' * ?d" "b = b' * ?d" "gcd a' b' = 1"
907     by blast
908   from ab'(1) have "a' dvd a"
909     unfolding dvd_def by blast
910   with assms have "a' dvd b * c"
911     using dvd_trans [of a' a "b * c"] by simp
912   from assms ab'(1,2) have "a' * ?d dvd (b' * ?d) * c"
913     by simp
914   then have "?d * a' dvd ?d * (b' * c)"
916   with \<open>?d \<noteq> 0\<close> have "a' dvd b' * c"
917     by simp
918   with coprime_dvd_mult[OF ab'(3)] have "a' dvd c"
919     by (subst (asm) ac_simps) blast
920   with ab'(1) have "a = ?d * a' \<and> ?d dvd b \<and> a' dvd c"
922   then show ?thesis by blast
923 qed
925 lemma pow_divs_pow:
926   assumes ab: "a ^ n dvd b ^ n" and n: "n \<noteq> 0"
927   shows "a dvd b"
928 proof (cases "gcd a b = 0")
929   case True
930   then show ?thesis by simp
931 next
932   case False
933   let ?d = "gcd a b"
934   from n obtain m where m: "n = Suc m"
935     by (cases n) simp_all
936   from False have zn: "?d ^ n \<noteq> 0"
937     by (rule power_not_zero)
938   from gcd_coprime_exists [OF False]
939   obtain a' b' where ab': "a = a' * ?d" "b = b' * ?d" "gcd a' b' = 1"
940     by blast
941   from ab have "(a' * ?d) ^ n dvd (b' * ?d) ^ n"
943   then have "?d^n * a'^n dvd ?d^n * b'^n"
944     by (simp only: power_mult_distrib ac_simps)
945   with zn have "a'^n dvd b'^n"
946     by simp
947   then have "a' dvd b'^n"
948     using dvd_trans[of a' "a'^n" "b'^n"] by (simp add: m)
949   then have "a' dvd b'^m * b'"
950     by (simp add: m ac_simps)
951   with coprime_dvd_mult[OF coprime_exp[OF ab'(3), of m]]
952   have "a' dvd b'" by (subst (asm) ac_simps) blast
953   then have "a' * ?d dvd b' * ?d"
954     by (rule mult_dvd_mono) simp
955   with ab'(1,2) show ?thesis
956     by simp
957 qed
959 lemma pow_divs_eq [simp]: "n \<noteq> 0 \<Longrightarrow> a ^ n dvd b ^ n \<longleftrightarrow> a dvd b"
960   by (auto intro: pow_divs_pow dvd_power_same)
962 lemma coprime_plus_one [simp]: "gcd (n + 1) n = 1"
965 lemma prod_coprime [rule_format]: "(\<forall>i\<in>A. gcd (f i) a = 1) \<longrightarrow> gcd (\<Prod>i\<in>A. f i) a = 1"
966   by (induct A rule: infinite_finite_induct) (auto simp add: gcd_mult_cancel)
968 lemma prod_list_coprime: "(\<And>x. x \<in> set xs \<Longrightarrow> coprime x y) \<Longrightarrow> coprime (prod_list xs) y"
969   by (induct xs) (simp_all add: gcd_mult_cancel)
971 lemma coprime_divisors:
972   assumes "d dvd a" "e dvd b" "gcd a b = 1"
973   shows "gcd d e = 1"
974 proof -
975   from assms obtain k l where "a = d * k" "b = e * l"
976     unfolding dvd_def by blast
977   with assms have "gcd (d * k) (e * l) = 1"
978     by simp
979   then have "gcd (d * k) e = 1"
980     by (rule coprime_lmult)
981   also have "gcd (d * k) e = gcd e (d * k)"
983   finally have "gcd e d = 1"
984     by (rule coprime_lmult)
985   then show ?thesis
987 qed
989 lemma lcm_gcd_prod: "lcm a b * gcd a b = normalize (a * b)"
992 declare unit_factor_lcm [simp]
994 lemma lcmI:
995   assumes "a dvd c" and "b dvd c" and "\<And>d. a dvd d \<Longrightarrow> b dvd d \<Longrightarrow> c dvd d"
996     and "normalize c = c"
997   shows "c = lcm a b"
998   by (rule associated_eqI) (auto simp: assms intro: lcm_least)
1000 lemma gcd_dvd_lcm [simp]: "gcd a b dvd lcm a b"
1001   using gcd_dvd2 by (rule dvd_lcmI2)
1003 lemmas lcm_0 = lcm_0_right
1005 lemma lcm_unique:
1006   "a dvd d \<and> b dvd d \<and> normalize d = d \<and> (\<forall>e. a dvd e \<and> b dvd e \<longrightarrow> d dvd e) \<longleftrightarrow> d = lcm a b"
1007   by rule (auto intro: lcmI simp: lcm_least lcm_eq_0_iff)
1009 lemma lcm_coprime: "gcd a b = 1 \<Longrightarrow> lcm a b = normalize (a * b)"
1010   by (subst lcm_gcd) simp
1012 lemma lcm_proj1_if_dvd: "b dvd a \<Longrightarrow> lcm a b = normalize a"
1013   apply (cases "a = 0")
1014    apply simp
1015   apply (rule sym)
1016   apply (rule lcmI)
1017      apply simp_all
1018   done
1020 lemma lcm_proj2_if_dvd: "a dvd b \<Longrightarrow> lcm a b = normalize b"
1021   using lcm_proj1_if_dvd [of a b] by (simp add: ac_simps)
1023 lemma lcm_proj1_iff: "lcm m n = normalize m \<longleftrightarrow> n dvd m"
1024 proof
1025   assume *: "lcm m n = normalize m"
1026   show "n dvd m"
1027   proof (cases "m = 0")
1028     case True
1029     then show ?thesis by simp
1030   next
1031     case [simp]: False
1032     from * have **: "m = lcm m n * unit_factor m"
1034     show ?thesis by (subst **) simp
1035   qed
1036 next
1037   assume "n dvd m"
1038   then show "lcm m n = normalize m"
1039     by (rule lcm_proj1_if_dvd)
1040 qed
1042 lemma lcm_proj2_iff: "lcm m n = normalize n \<longleftrightarrow> m dvd n"
1043   using lcm_proj1_iff [of n m] by (simp add: ac_simps)
1045 lemma lcm_mult_distrib': "normalize c * lcm a b = lcm (c * a) (c * b)"
1046   by (rule lcmI) (auto simp: normalize_mult lcm_least mult_dvd_mono lcm_mult_left [symmetric])
1048 lemma lcm_mult_distrib: "k * lcm a b = lcm (k * a) (k * b) * unit_factor k"
1049 proof-
1050   have "normalize k * lcm a b = lcm (k * a) (k * b)"
1052   then have "normalize k * lcm a b * unit_factor k = lcm (k * a) (k * b) * unit_factor k"
1053     by simp
1054   then have "normalize k * unit_factor k * lcm a b  = lcm (k * a) (k * b) * unit_factor k"
1055     by (simp only: ac_simps)
1056   then show ?thesis
1057     by simp
1058 qed
1060 lemma dvd_productE:
1061   assumes "p dvd (a * b)"
1062   obtains x y where "p = x * y" "x dvd a" "y dvd b"
1063 proof (cases "a = 0")
1064   case True
1065   thus ?thesis by (intro that[of p 1]) simp_all
1066 next
1067   case False
1068   define x y where "x = gcd a p" and "y = p div x"
1069   have "p = x * y" by (simp add: x_def y_def)
1070   moreover have "x dvd a" by (simp add: x_def)
1071   moreover from assms have "p dvd gcd (b * a) (b * p)"
1072     by (intro gcd_greatest) (simp_all add: mult.commute)
1073   hence "p dvd b * gcd a p" by (simp add: gcd_mult_distrib)
1074   with False have "y dvd b"
1075     by (simp add: x_def y_def div_dvd_iff_mult assms)
1076   ultimately show ?thesis by (rule that)
1077 qed
1079 lemma coprime_crossproduct':
1080   fixes a b c d
1081   assumes "b \<noteq> 0"
1082   assumes unit_factors: "unit_factor b = unit_factor d"
1083   assumes coprime: "coprime a b" "coprime c d"
1084   shows "a * d = b * c \<longleftrightarrow> a = c \<and> b = d"
1085 proof safe
1086   assume eq: "a * d = b * c"
1087   hence "normalize a * normalize d = normalize c * normalize b"
1088     by (simp only: normalize_mult [symmetric] mult_ac)
1089   with coprime have "normalize b = normalize d"
1090     by (subst (asm) coprime_crossproduct) simp_all
1091   from this and unit_factors show "b = d"
1092     by (rule normalize_unit_factor_eqI)
1093   from eq have "a * d = c * d" by (simp only: \<open>b = d\<close> mult_ac)
1094   with \<open>b \<noteq> 0\<close> \<open>b = d\<close> show "a = c" by simp
1097 end
1099 class ring_gcd = comm_ring_1 + semiring_gcd
1100 begin
1102 lemma coprime_minus_one: "coprime (n - 1) n"
1103   using coprime_plus_one[of "n - 1"] by (simp add: gcd.commute)
1105 lemma gcd_neg1 [simp]: "gcd (-a) b = gcd a b"
1106   by (rule sym, rule gcdI) (simp_all add: gcd_greatest)
1108 lemma gcd_neg2 [simp]: "gcd a (-b) = gcd a b"
1109   by (rule sym, rule gcdI) (simp_all add: gcd_greatest)
1111 lemma gcd_neg_numeral_1 [simp]: "gcd (- numeral n) a = gcd (numeral n) a"
1112   by (fact gcd_neg1)
1114 lemma gcd_neg_numeral_2 [simp]: "gcd a (- numeral n) = gcd a (numeral n)"
1115   by (fact gcd_neg2)
1117 lemma gcd_diff1: "gcd (m - n) n = gcd m n"
1120 lemma gcd_diff2: "gcd (n - m) n = gcd m n"
1121   by (subst gcd_neg1[symmetric]) (simp only: minus_diff_eq gcd_diff1)
1123 lemma lcm_neg1 [simp]: "lcm (-a) b = lcm a b"
1124   by (rule sym, rule lcmI) (simp_all add: lcm_least lcm_eq_0_iff)
1126 lemma lcm_neg2 [simp]: "lcm a (-b) = lcm a b"
1127   by (rule sym, rule lcmI) (simp_all add: lcm_least lcm_eq_0_iff)
1129 lemma lcm_neg_numeral_1 [simp]: "lcm (- numeral n) a = lcm (numeral n) a"
1130   by (fact lcm_neg1)
1132 lemma lcm_neg_numeral_2 [simp]: "lcm a (- numeral n) = lcm a (numeral n)"
1133   by (fact lcm_neg2)
1135 end
1137 class semiring_Gcd = semiring_gcd + Gcd +
1138   assumes Gcd_dvd: "a \<in> A \<Longrightarrow> Gcd A dvd a"
1139     and Gcd_greatest: "(\<And>b. b \<in> A \<Longrightarrow> a dvd b) \<Longrightarrow> a dvd Gcd A"
1140     and normalize_Gcd [simp]: "normalize (Gcd A) = Gcd A"
1141   assumes dvd_Lcm: "a \<in> A \<Longrightarrow> a dvd Lcm A"
1142     and Lcm_least: "(\<And>b. b \<in> A \<Longrightarrow> b dvd a) \<Longrightarrow> Lcm A dvd a"
1143     and normalize_Lcm [simp]: "normalize (Lcm A) = Lcm A"
1144 begin
1146 lemma Lcm_Gcd: "Lcm A = Gcd {b. \<forall>a\<in>A. a dvd b}"
1147   by (rule associated_eqI) (auto intro: Gcd_dvd dvd_Lcm Gcd_greatest Lcm_least)
1149 lemma Gcd_Lcm: "Gcd A = Lcm {b. \<forall>a\<in>A. b dvd a}"
1150   by (rule associated_eqI) (auto intro: Gcd_dvd dvd_Lcm Gcd_greatest Lcm_least)
1152 lemma Gcd_empty [simp]: "Gcd {} = 0"
1153   by (rule dvd_0_left, rule Gcd_greatest) simp
1155 lemma Lcm_empty [simp]: "Lcm {} = 1"
1156   by (auto intro: associated_eqI Lcm_least)
1158 lemma Gcd_insert [simp]: "Gcd (insert a A) = gcd a (Gcd A)"
1159 proof -
1160   have "Gcd (insert a A) dvd gcd a (Gcd A)"
1161     by (auto intro: Gcd_dvd Gcd_greatest)
1162   moreover have "gcd a (Gcd A) dvd Gcd (insert a A)"
1163   proof (rule Gcd_greatest)
1164     fix b
1165     assume "b \<in> insert a A"
1166     then show "gcd a (Gcd A) dvd b"
1167     proof
1168       assume "b = a"
1169       then show ?thesis
1170         by simp
1171     next
1172       assume "b \<in> A"
1173       then have "Gcd A dvd b"
1174         by (rule Gcd_dvd)
1175       moreover have "gcd a (Gcd A) dvd Gcd A"
1176         by simp
1177       ultimately show ?thesis
1178         by (blast intro: dvd_trans)
1179     qed
1180   qed
1181   ultimately show ?thesis
1182     by (auto intro: associated_eqI)
1183 qed
1185 lemma Lcm_insert [simp]: "Lcm (insert a A) = lcm a (Lcm A)"
1186 proof (rule sym)
1187   have "lcm a (Lcm A) dvd Lcm (insert a A)"
1188     by (auto intro: dvd_Lcm Lcm_least)
1189   moreover have "Lcm (insert a A) dvd lcm a (Lcm A)"
1190   proof (rule Lcm_least)
1191     fix b
1192     assume "b \<in> insert a A"
1193     then show "b dvd lcm a (Lcm A)"
1194     proof
1195       assume "b = a"
1196       then show ?thesis by simp
1197     next
1198       assume "b \<in> A"
1199       then have "b dvd Lcm A"
1200         by (rule dvd_Lcm)
1201       moreover have "Lcm A dvd lcm a (Lcm A)"
1202         by simp
1203       ultimately show ?thesis
1204         by (blast intro: dvd_trans)
1205     qed
1206   qed
1207   ultimately show "lcm a (Lcm A) = Lcm (insert a A)"
1208     by (rule associated_eqI) (simp_all add: lcm_eq_0_iff)
1209 qed
1211 lemma LcmI:
1212   assumes "\<And>a. a \<in> A \<Longrightarrow> a dvd b"
1213     and "\<And>c. (\<And>a. a \<in> A \<Longrightarrow> a dvd c) \<Longrightarrow> b dvd c"
1214     and "normalize b = b"
1215   shows "b = Lcm A"
1216   by (rule associated_eqI) (auto simp: assms dvd_Lcm intro: Lcm_least)
1218 lemma Lcm_subset: "A \<subseteq> B \<Longrightarrow> Lcm A dvd Lcm B"
1219   by (blast intro: Lcm_least dvd_Lcm)
1221 lemma Lcm_Un: "Lcm (A \<union> B) = lcm (Lcm A) (Lcm B)"
1222   apply (rule lcmI)
1223      apply (blast intro: Lcm_subset)
1224     apply (blast intro: Lcm_subset)
1225    apply (intro Lcm_least ballI, elim UnE)
1226     apply (rule dvd_trans, erule dvd_Lcm, assumption)
1227    apply (rule dvd_trans, erule dvd_Lcm, assumption)
1228   apply simp
1229   done
1231 lemma Gcd_0_iff [simp]: "Gcd A = 0 \<longleftrightarrow> A \<subseteq> {0}"
1232   (is "?P \<longleftrightarrow> ?Q")
1233 proof
1234   assume ?P
1235   show ?Q
1236   proof
1237     fix a
1238     assume "a \<in> A"
1239     then have "Gcd A dvd a"
1240       by (rule Gcd_dvd)
1241     with \<open>?P\<close> have "a = 0"
1242       by simp
1243     then show "a \<in> {0}"
1244       by simp
1245   qed
1246 next
1247   assume ?Q
1248   have "0 dvd Gcd A"
1249   proof (rule Gcd_greatest)
1250     fix a
1251     assume "a \<in> A"
1252     with \<open>?Q\<close> have "a = 0"
1253       by auto
1254     then show "0 dvd a"
1255       by simp
1256   qed
1257   then show ?P
1258     by simp
1259 qed
1261 lemma Lcm_1_iff [simp]: "Lcm A = 1 \<longleftrightarrow> (\<forall>a\<in>A. is_unit a)"
1262   (is "?P \<longleftrightarrow> ?Q")
1263 proof
1264   assume ?P
1265   show ?Q
1266   proof
1267     fix a
1268     assume "a \<in> A"
1269     then have "a dvd Lcm A"
1270       by (rule dvd_Lcm)
1271     with \<open>?P\<close> show "is_unit a"
1272       by simp
1273   qed
1274 next
1275   assume ?Q
1276   then have "is_unit (Lcm A)"
1277     by (blast intro: Lcm_least)
1278   then have "normalize (Lcm A) = 1"
1279     by (rule is_unit_normalize)
1280   then show ?P
1281     by simp
1282 qed
1284 lemma unit_factor_Lcm: "unit_factor (Lcm A) = (if Lcm A = 0 then 0 else 1)"
1285 proof (cases "Lcm A = 0")
1286   case True
1287   then show ?thesis
1288     by simp
1289 next
1290   case False
1291   with unit_factor_normalize have "unit_factor (normalize (Lcm A)) = 1"
1292     by blast
1293   with False show ?thesis
1294     by simp
1295 qed
1297 lemma unit_factor_Gcd: "unit_factor (Gcd A) = (if Gcd A = 0 then 0 else 1)"
1298   by (simp add: Gcd_Lcm unit_factor_Lcm)
1300 lemma GcdI:
1301   assumes "\<And>a. a \<in> A \<Longrightarrow> b dvd a"
1302     and "\<And>c. (\<And>a. a \<in> A \<Longrightarrow> c dvd a) \<Longrightarrow> c dvd b"
1303     and "normalize b = b"
1304   shows "b = Gcd A"
1305   by (rule associated_eqI) (auto simp: assms Gcd_dvd intro: Gcd_greatest)
1307 lemma Gcd_eq_1_I:
1308   assumes "is_unit a" and "a \<in> A"
1309   shows "Gcd A = 1"
1310 proof -
1311   from assms have "is_unit (Gcd A)"
1312     by (blast intro: Gcd_dvd dvd_unit_imp_unit)
1313   then have "normalize (Gcd A) = 1"
1314     by (rule is_unit_normalize)
1315   then show ?thesis
1316     by simp
1317 qed
1319 lemma Lcm_eq_0_I:
1320   assumes "0 \<in> A"
1321   shows "Lcm A = 0"
1322 proof -
1323   from assms have "0 dvd Lcm A"
1324     by (rule dvd_Lcm)
1325   then show ?thesis
1326     by simp
1327 qed
1329 lemma Gcd_UNIV [simp]: "Gcd UNIV = 1"
1330   using dvd_refl by (rule Gcd_eq_1_I) simp
1332 lemma Lcm_UNIV [simp]: "Lcm UNIV = 0"
1333   by (rule Lcm_eq_0_I) simp
1335 lemma Lcm_0_iff:
1336   assumes "finite A"
1337   shows "Lcm A = 0 \<longleftrightarrow> 0 \<in> A"
1338 proof (cases "A = {}")
1339   case True
1340   then show ?thesis by simp
1341 next
1342   case False
1343   with assms show ?thesis
1344     by (induct A rule: finite_ne_induct) (auto simp add: lcm_eq_0_iff)
1345 qed
1347 lemma Gcd_image_normalize [simp]: "Gcd (normalize ` A) = Gcd A"
1348 proof -
1349   have "Gcd (normalize ` A) dvd a" if "a \<in> A" for a
1350   proof -
1351     from that obtain B where "A = insert a B"
1352       by blast
1353     moreover have "gcd (normalize a) (Gcd (normalize ` B)) dvd normalize a"
1354       by (rule gcd_dvd1)
1355     ultimately show "Gcd (normalize ` A) dvd a"
1356       by simp
1357   qed
1358   then have "Gcd (normalize ` A) dvd Gcd A" and "Gcd A dvd Gcd (normalize ` A)"
1359     by (auto intro!: Gcd_greatest intro: Gcd_dvd)
1360   then show ?thesis
1361     by (auto intro: associated_eqI)
1362 qed
1364 lemma Gcd_eqI:
1365   assumes "normalize a = a"
1366   assumes "\<And>b. b \<in> A \<Longrightarrow> a dvd b"
1367     and "\<And>c. (\<And>b. b \<in> A \<Longrightarrow> c dvd b) \<Longrightarrow> c dvd a"
1368   shows "Gcd A = a"
1369   using assms by (blast intro: associated_eqI Gcd_greatest Gcd_dvd normalize_Gcd)
1371 lemma dvd_GcdD: "x dvd Gcd A \<Longrightarrow> y \<in> A \<Longrightarrow> x dvd y"
1372   using Gcd_dvd dvd_trans by blast
1374 lemma dvd_Gcd_iff: "x dvd Gcd A \<longleftrightarrow> (\<forall>y\<in>A. x dvd y)"
1375   by (blast dest: dvd_GcdD intro: Gcd_greatest)
1377 lemma Gcd_mult: "Gcd (op * c ` A) = normalize c * Gcd A"
1378 proof (cases "c = 0")
1379   case True
1380   then show ?thesis by auto
1381 next
1382   case [simp]: False
1383   have "Gcd (op * c ` A) div c dvd Gcd A"
1384     by (intro Gcd_greatest, subst div_dvd_iff_mult)
1385        (auto intro!: Gcd_greatest Gcd_dvd simp: mult.commute[of _ c])
1386   then have "Gcd (op * c ` A) dvd c * Gcd A"
1387     by (subst (asm) div_dvd_iff_mult) (auto intro: Gcd_greatest simp: mult_ac)
1388   also have "c * Gcd A = (normalize c * Gcd A) * unit_factor c"
1389     by (subst unit_factor_mult_normalize [symmetric]) (simp only: mult_ac)
1390   also have "Gcd (op * c ` A) dvd \<dots> \<longleftrightarrow> Gcd (op * c ` A) dvd normalize c * Gcd A"
1392   finally have "Gcd (op * c ` A) dvd normalize c * Gcd A" .
1393   moreover have "normalize c * Gcd A dvd Gcd (op * c ` A)"
1394     by (intro Gcd_greatest) (auto intro: mult_dvd_mono Gcd_dvd)
1395   ultimately have "normalize (Gcd (op * c ` A)) = normalize (normalize c * Gcd A)"
1396     by (rule associatedI)
1397   then show ?thesis
1399 qed
1401 lemma Lcm_eqI:
1402   assumes "normalize a = a"
1403     and "\<And>b. b \<in> A \<Longrightarrow> b dvd a"
1404     and "\<And>c. (\<And>b. b \<in> A \<Longrightarrow> b dvd c) \<Longrightarrow> a dvd c"
1405   shows "Lcm A = a"
1406   using assms by (blast intro: associated_eqI Lcm_least dvd_Lcm normalize_Lcm)
1408 lemma Lcm_dvdD: "Lcm A dvd x \<Longrightarrow> y \<in> A \<Longrightarrow> y dvd x"
1409   using dvd_Lcm dvd_trans by blast
1411 lemma Lcm_dvd_iff: "Lcm A dvd x \<longleftrightarrow> (\<forall>y\<in>A. y dvd x)"
1412   by (blast dest: Lcm_dvdD intro: Lcm_least)
1414 lemma Lcm_mult:
1415   assumes "A \<noteq> {}"
1416   shows "Lcm (op * c ` A) = normalize c * Lcm A"
1417 proof (cases "c = 0")
1418   case True
1419   with assms have "op * c ` A = {0}"
1420     by auto
1421   with True show ?thesis by auto
1422 next
1423   case [simp]: False
1424   from assms obtain x where x: "x \<in> A"
1425     by blast
1426   have "c dvd c * x"
1427     by simp
1428   also from x have "c * x dvd Lcm (op * c ` A)"
1429     by (intro dvd_Lcm) auto
1430   finally have dvd: "c dvd Lcm (op * c ` A)" .
1432   have "Lcm A dvd Lcm (op * c ` A) div c"
1433     by (intro Lcm_least dvd_mult_imp_div)
1434       (auto intro!: Lcm_least dvd_Lcm simp: mult.commute[of _ c])
1435   then have "c * Lcm A dvd Lcm (op * c ` A)"
1436     by (subst (asm) dvd_div_iff_mult) (auto intro!: Lcm_least simp: mult_ac dvd)
1437   also have "c * Lcm A = (normalize c * Lcm A) * unit_factor c"
1438     by (subst unit_factor_mult_normalize [symmetric]) (simp only: mult_ac)
1439   also have "\<dots> dvd Lcm (op * c ` A) \<longleftrightarrow> normalize c * Lcm A dvd Lcm (op * c ` A)"
1441   finally have "normalize c * Lcm A dvd Lcm (op * c ` A)" .
1442   moreover have "Lcm (op * c ` A) dvd normalize c * Lcm A"
1443     by (intro Lcm_least) (auto intro: mult_dvd_mono dvd_Lcm)
1444   ultimately have "normalize (normalize c * Lcm A) = normalize (Lcm (op * c ` A))"
1445     by (rule associatedI)
1446   then show ?thesis
1448 qed
1450 lemma Lcm_no_units: "Lcm A = Lcm (A - {a. is_unit a})"
1451 proof -
1452   have "(A - {a. is_unit a}) \<union> {a\<in>A. is_unit a} = A"
1453     by blast
1454   then have "Lcm A = lcm (Lcm (A - {a. is_unit a})) (Lcm {a\<in>A. is_unit a})"
1455     by (simp add: Lcm_Un [symmetric])
1456   also have "Lcm {a\<in>A. is_unit a} = 1"
1457     by simp
1458   finally show ?thesis
1459     by simp
1460 qed
1462 lemma Lcm_0_iff': "Lcm A = 0 \<longleftrightarrow> (\<nexists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l))"
1463   by (metis Lcm_least dvd_0_left dvd_Lcm)
1465 lemma Lcm_no_multiple: "(\<forall>m. m \<noteq> 0 \<longrightarrow> (\<exists>a\<in>A. \<not> a dvd m)) \<Longrightarrow> Lcm A = 0"
1466   by (auto simp: Lcm_0_iff')
1468 lemma Lcm_singleton [simp]: "Lcm {a} = normalize a"
1469   by simp
1471 lemma Lcm_2 [simp]: "Lcm {a, b} = lcm a b"
1472   by simp
1474 lemma Lcm_coprime:
1475   assumes "finite A"
1476     and "A \<noteq> {}"
1477     and "\<And>a b. a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<noteq> b \<Longrightarrow> gcd a b = 1"
1478   shows "Lcm A = normalize (\<Prod>A)"
1479   using assms
1480 proof (induct rule: finite_ne_induct)
1481   case singleton
1482   then show ?case by simp
1483 next
1484   case (insert a A)
1485   have "Lcm (insert a A) = lcm a (Lcm A)"
1486     by simp
1487   also from insert have "Lcm A = normalize (\<Prod>A)"
1488     by blast
1489   also have "lcm a \<dots> = lcm a (\<Prod>A)"
1490     by (cases "\<Prod>A = 0") (simp_all add: lcm_div_unit2)
1491   also from insert have "gcd a (\<Prod>A) = 1"
1492     by (subst gcd.commute, intro prod_coprime) auto
1493   with insert have "lcm a (\<Prod>A) = normalize (\<Prod>(insert a A))"
1495   finally show ?case .
1496 qed
1498 lemma Lcm_coprime':
1499   "card A \<noteq> 0 \<Longrightarrow>
1500     (\<And>a b. a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<noteq> b \<Longrightarrow> gcd a b = 1) \<Longrightarrow>
1501     Lcm A = normalize (\<Prod>A)"
1502   by (rule Lcm_coprime) (simp_all add: card_eq_0_iff)
1504 lemma Gcd_1: "1 \<in> A \<Longrightarrow> Gcd A = 1"
1505   by (auto intro!: Gcd_eq_1_I)
1507 lemma Gcd_singleton [simp]: "Gcd {a} = normalize a"
1508   by simp
1510 lemma Gcd_2 [simp]: "Gcd {a, b} = gcd a b"
1511   by simp
1514 definition pairwise_coprime
1515   where "pairwise_coprime A = (\<forall>x y. x \<in> A \<and> y \<in> A \<and> x \<noteq> y \<longrightarrow> coprime x y)"
1517 lemma pairwise_coprimeI [intro?]:
1518   "(\<And>x y. x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> x \<noteq> y \<Longrightarrow> coprime x y) \<Longrightarrow> pairwise_coprime A"
1521 lemma pairwise_coprimeD:
1522   "pairwise_coprime A \<Longrightarrow> x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> x \<noteq> y \<Longrightarrow> coprime x y"
1525 lemma pairwise_coprime_subset: "pairwise_coprime A \<Longrightarrow> B \<subseteq> A \<Longrightarrow> pairwise_coprime B"
1526   by (force simp: pairwise_coprime_def)
1528 end
1531 subsection \<open>An aside: GCD and LCM on finite sets for incomplete gcd rings\<close>
1533 context semiring_gcd
1534 begin
1536 sublocale Gcd_fin: bounded_quasi_semilattice_set gcd 0 1 normalize
1537 defines
1538   Gcd_fin ("Gcd\<^sub>f\<^sub>i\<^sub>n _" [900] 900) = "Gcd_fin.F :: 'a set \<Rightarrow> 'a" ..
1540 abbreviation gcd_list :: "'a list \<Rightarrow> 'a"
1541   where "gcd_list xs \<equiv> Gcd\<^sub>f\<^sub>i\<^sub>n (set xs)"
1543 sublocale Lcm_fin: bounded_quasi_semilattice_set lcm 1 0 normalize
1544 defines
1545   Lcm_fin ("Lcm\<^sub>f\<^sub>i\<^sub>n _" [900] 900) = Lcm_fin.F ..
1547 abbreviation lcm_list :: "'a list \<Rightarrow> 'a"
1548   where "lcm_list xs \<equiv> Lcm\<^sub>f\<^sub>i\<^sub>n (set xs)"
1550 lemma Gcd_fin_dvd:
1551   "a \<in> A \<Longrightarrow> Gcd\<^sub>f\<^sub>i\<^sub>n A dvd a"
1552   by (induct A rule: infinite_finite_induct)
1553     (auto intro: dvd_trans)
1555 lemma dvd_Lcm_fin:
1556   "a \<in> A \<Longrightarrow> a dvd Lcm\<^sub>f\<^sub>i\<^sub>n A"
1557   by (induct A rule: infinite_finite_induct)
1558     (auto intro: dvd_trans)
1560 lemma Gcd_fin_greatest:
1561   "a dvd Gcd\<^sub>f\<^sub>i\<^sub>n A" if "finite A" and "\<And>b. b \<in> A \<Longrightarrow> a dvd b"
1562   using that by (induct A) simp_all
1564 lemma Lcm_fin_least:
1565   "Lcm\<^sub>f\<^sub>i\<^sub>n A dvd a" if "finite A" and "\<And>b. b \<in> A \<Longrightarrow> b dvd a"
1566   using that by (induct A) simp_all
1568 lemma gcd_list_greatest:
1569   "a dvd gcd_list bs" if "\<And>b. b \<in> set bs \<Longrightarrow> a dvd b"
1570   by (rule Gcd_fin_greatest) (simp_all add: that)
1572 lemma lcm_list_least:
1573   "lcm_list bs dvd a" if "\<And>b. b \<in> set bs \<Longrightarrow> b dvd a"
1574   by (rule Lcm_fin_least) (simp_all add: that)
1576 lemma dvd_Gcd_fin_iff:
1577   "b dvd Gcd\<^sub>f\<^sub>i\<^sub>n A \<longleftrightarrow> (\<forall>a\<in>A. b dvd a)" if "finite A"
1578   using that by (auto intro: Gcd_fin_greatest Gcd_fin_dvd dvd_trans [of b "Gcd\<^sub>f\<^sub>i\<^sub>n A"])
1580 lemma dvd_gcd_list_iff:
1581   "b dvd gcd_list xs \<longleftrightarrow> (\<forall>a\<in>set xs. b dvd a)"
1584 lemma Lcm_fin_dvd_iff:
1585   "Lcm\<^sub>f\<^sub>i\<^sub>n A dvd b  \<longleftrightarrow> (\<forall>a\<in>A. a dvd b)" if "finite A"
1586   using that by (auto intro: Lcm_fin_least dvd_Lcm_fin dvd_trans [of _ "Lcm\<^sub>f\<^sub>i\<^sub>n A" b])
1588 lemma lcm_list_dvd_iff:
1589   "lcm_list xs dvd b  \<longleftrightarrow> (\<forall>a\<in>set xs. a dvd b)"
1592 lemma Gcd_fin_mult:
1593   "Gcd\<^sub>f\<^sub>i\<^sub>n (image (times b) A) = normalize b * Gcd\<^sub>f\<^sub>i\<^sub>n A" if "finite A"
1594 using that proof induct
1595   case empty
1596   then show ?case
1597     by simp
1598 next
1599   case (insert a A)
1600   have "gcd (b * a) (b * Gcd\<^sub>f\<^sub>i\<^sub>n A) = gcd (b * a) (normalize (b * Gcd\<^sub>f\<^sub>i\<^sub>n A))"
1601     by simp
1602   also have "\<dots> = gcd (b * a) (normalize b * Gcd\<^sub>f\<^sub>i\<^sub>n A)"
1604   finally show ?case
1605     using insert by (simp add: gcd_mult_distrib')
1606 qed
1608 lemma Lcm_fin_mult:
1609   "Lcm\<^sub>f\<^sub>i\<^sub>n (image (times b) A) = normalize b * Lcm\<^sub>f\<^sub>i\<^sub>n A" if "A \<noteq> {}"
1610 proof (cases "b = 0")
1611   case True
1612   moreover from that have "times 0 ` A = {0}"
1613     by auto
1614   ultimately show ?thesis
1615     by simp
1616 next
1617   case False
1618   then have "inj (times b)"
1619     by (rule inj_times)
1620   show ?thesis proof (cases "finite A")
1621     case False
1622     moreover from \<open>inj (times b)\<close>
1623     have "inj_on (times b) A"
1624       by (rule inj_on_subset) simp
1625     ultimately have "infinite (times b ` A)"
1627     with False show ?thesis
1628       by simp
1629   next
1630     case True
1631     then show ?thesis using that proof (induct A rule: finite_ne_induct)
1632       case (singleton a)
1633       then show ?case
1635     next
1636       case (insert a A)
1637       have "lcm (b * a) (b * Lcm\<^sub>f\<^sub>i\<^sub>n A) = lcm (b * a) (normalize (b * Lcm\<^sub>f\<^sub>i\<^sub>n A))"
1638         by simp
1639       also have "\<dots> = lcm (b * a) (normalize b * Lcm\<^sub>f\<^sub>i\<^sub>n A)"
1641       finally show ?case
1642         using insert by (simp add: lcm_mult_distrib')
1643     qed
1644   qed
1645 qed
1647 lemma unit_factor_Gcd_fin:
1648   "unit_factor (Gcd\<^sub>f\<^sub>i\<^sub>n A) = of_bool (Gcd\<^sub>f\<^sub>i\<^sub>n A \<noteq> 0)"
1649   by (rule normalize_idem_imp_unit_factor_eq) simp
1651 lemma unit_factor_Lcm_fin:
1652   "unit_factor (Lcm\<^sub>f\<^sub>i\<^sub>n A) = of_bool (Lcm\<^sub>f\<^sub>i\<^sub>n A \<noteq> 0)"
1653   by (rule normalize_idem_imp_unit_factor_eq) simp
1655 lemma is_unit_Gcd_fin_iff [simp]:
1656   "is_unit (Gcd\<^sub>f\<^sub>i\<^sub>n A) \<longleftrightarrow> Gcd\<^sub>f\<^sub>i\<^sub>n A = 1"
1657   by (rule normalize_idem_imp_is_unit_iff) simp
1659 lemma is_unit_Lcm_fin_iff [simp]:
1660   "is_unit (Lcm\<^sub>f\<^sub>i\<^sub>n A) \<longleftrightarrow> Lcm\<^sub>f\<^sub>i\<^sub>n A = 1"
1661   by (rule normalize_idem_imp_is_unit_iff) simp
1663 lemma Gcd_fin_0_iff:
1664   "Gcd\<^sub>f\<^sub>i\<^sub>n A = 0 \<longleftrightarrow> A \<subseteq> {0} \<and> finite A"
1665   by (induct A rule: infinite_finite_induct) simp_all
1667 lemma Lcm_fin_0_iff:
1668   "Lcm\<^sub>f\<^sub>i\<^sub>n A = 0 \<longleftrightarrow> 0 \<in> A" if "finite A"
1669   using that by (induct A) (auto simp add: lcm_eq_0_iff)
1671 lemma Lcm_fin_1_iff:
1672   "Lcm\<^sub>f\<^sub>i\<^sub>n A = 1 \<longleftrightarrow> (\<forall>a\<in>A. is_unit a) \<and> finite A"
1673   by (induct A rule: infinite_finite_induct) simp_all
1675 end
1677 context semiring_Gcd
1678 begin
1680 lemma Gcd_fin_eq_Gcd [simp]:
1681   "Gcd\<^sub>f\<^sub>i\<^sub>n A = Gcd A" if "finite A" for A :: "'a set"
1682   using that by induct simp_all
1684 lemma Gcd_set_eq_fold [code_unfold]:
1685   "Gcd (set xs) = fold gcd xs 0"
1686   by (simp add: Gcd_fin.set_eq_fold [symmetric])
1688 lemma Lcm_fin_eq_Lcm [simp]:
1689   "Lcm\<^sub>f\<^sub>i\<^sub>n A = Lcm A" if "finite A" for A :: "'a set"
1690   using that by induct simp_all
1692 lemma Lcm_set_eq_fold [code_unfold]:
1693   "Lcm (set xs) = fold lcm xs 1"
1694   by (simp add: Lcm_fin.set_eq_fold [symmetric])
1696 end
1698 subsection \<open>GCD and LCM on @{typ nat} and @{typ int}\<close>
1700 instantiation nat :: gcd
1701 begin
1703 fun gcd_nat  :: "nat \<Rightarrow> nat \<Rightarrow> nat"
1704   where "gcd_nat x y = (if y = 0 then x else gcd y (x mod y))"
1706 definition lcm_nat :: "nat \<Rightarrow> nat \<Rightarrow> nat"
1707   where "lcm_nat x y = x * y div (gcd x y)"
1709 instance ..
1711 end
1713 instantiation int :: gcd
1714 begin
1716 definition gcd_int  :: "int \<Rightarrow> int \<Rightarrow> int"
1717   where "gcd_int x y = int (gcd (nat \<bar>x\<bar>) (nat \<bar>y\<bar>))"
1719 definition lcm_int :: "int \<Rightarrow> int \<Rightarrow> int"
1720   where "lcm_int x y = int (lcm (nat \<bar>x\<bar>) (nat \<bar>y\<bar>))"
1722 instance ..
1724 end
1726 text \<open>Transfer setup\<close>
1728 lemma transfer_nat_int_gcd:
1729   "x \<ge> 0 \<Longrightarrow> y \<ge> 0 \<Longrightarrow> gcd (nat x) (nat y) = nat (gcd x y)"
1730   "x \<ge> 0 \<Longrightarrow> y \<ge> 0 \<Longrightarrow> lcm (nat x) (nat y) = nat (lcm x y)"
1731   for x y :: int
1732   unfolding gcd_int_def lcm_int_def by auto
1734 lemma transfer_nat_int_gcd_closures:
1735   "x \<ge> 0 \<Longrightarrow> y \<ge> 0 \<Longrightarrow> gcd x y \<ge> 0"
1736   "x \<ge> 0 \<Longrightarrow> y \<ge> 0 \<Longrightarrow> lcm x y \<ge> 0"
1737   for x y :: int
1738   by (auto simp add: gcd_int_def lcm_int_def)
1740 declare transfer_morphism_nat_int
1741   [transfer add return: transfer_nat_int_gcd transfer_nat_int_gcd_closures]
1743 lemma transfer_int_nat_gcd:
1744   "gcd (int x) (int y) = int (gcd x y)"
1745   "lcm (int x) (int y) = int (lcm x y)"
1746   by (auto simp: gcd_int_def lcm_int_def)
1748 lemma transfer_int_nat_gcd_closures:
1749   "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> gcd x y >= 0"
1750   "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> lcm x y >= 0"
1751   by (auto simp: gcd_int_def lcm_int_def)
1753 declare transfer_morphism_int_nat
1754   [transfer add return: transfer_int_nat_gcd transfer_int_nat_gcd_closures]
1756 lemma gcd_nat_induct:
1757   fixes m n :: nat
1758   assumes "\<And>m. P m 0"
1759     and "\<And>m n. 0 < n \<Longrightarrow> P n (m mod n) \<Longrightarrow> P m n"
1760   shows "P m n"
1761   apply (rule gcd_nat.induct)
1762   apply (case_tac "y = 0")
1763   using assms
1764    apply simp_all
1765   done
1768 text \<open>Specific to \<open>int\<close>.\<close>
1770 lemma gcd_eq_int_iff: "gcd k l = int n \<longleftrightarrow> gcd (nat \<bar>k\<bar>) (nat \<bar>l\<bar>) = n"
1773 lemma lcm_eq_int_iff: "lcm k l = int n \<longleftrightarrow> lcm (nat \<bar>k\<bar>) (nat \<bar>l\<bar>) = n"
1776 lemma gcd_neg1_int [simp]: "gcd (- x) y = gcd x y"
1777   for x y :: int
1780 lemma gcd_neg2_int [simp]: "gcd x (- y) = gcd x y"
1781   for x y :: int
1784 lemma abs_gcd_int [simp]: "\<bar>gcd x y\<bar> = gcd x y"
1785   for x y :: int
1788 lemma gcd_abs_int: "gcd x y = gcd \<bar>x\<bar> \<bar>y\<bar>"
1789   for x y :: int
1792 lemma gcd_abs1_int [simp]: "gcd \<bar>x\<bar> y = gcd x y"
1793   for x y :: int
1794   by (metis abs_idempotent gcd_abs_int)
1796 lemma gcd_abs2_int [simp]: "gcd x \<bar>y\<bar> = gcd x y"
1797   for x y :: int
1798   by (metis abs_idempotent gcd_abs_int)
1800 lemma gcd_cases_int:
1801   fixes x y :: int
1802   assumes "x \<ge> 0 \<Longrightarrow> y \<ge> 0 \<Longrightarrow> P (gcd x y)"
1803     and "x \<ge> 0 \<Longrightarrow> y \<le> 0 \<Longrightarrow> P (gcd x (- y))"
1804     and "x \<le> 0 \<Longrightarrow> y \<ge> 0 \<Longrightarrow> P (gcd (- x) y)"
1805     and "x \<le> 0 \<Longrightarrow> y \<le> 0 \<Longrightarrow> P (gcd (- x) (- y))"
1806   shows "P (gcd x y)"
1807   using assms by auto arith
1809 lemma gcd_ge_0_int [simp]: "gcd (x::int) y >= 0"
1810   for x y :: int
1813 lemma lcm_neg1_int: "lcm (- x) y = lcm x y"
1814   for x y :: int
1817 lemma lcm_neg2_int: "lcm x (- y) = lcm x y"
1818   for x y :: int
1821 lemma lcm_abs_int: "lcm x y = lcm \<bar>x\<bar> \<bar>y\<bar>"
1822   for x y :: int
1825 lemma abs_lcm_int [simp]: "\<bar>lcm i j\<bar> = lcm i j"
1826   for i j :: int
1829 lemma lcm_abs1_int [simp]: "lcm \<bar>x\<bar> y = lcm x y"
1830   for x y :: int
1831   by (metis abs_idempotent lcm_int_def)
1833 lemma lcm_abs2_int [simp]: "lcm x \<bar>y\<bar> = lcm x y"
1834   for x y :: int
1835   by (metis abs_idempotent lcm_int_def)
1837 lemma lcm_cases_int:
1838   fixes x y :: int
1839   assumes "x \<ge> 0 \<Longrightarrow> y \<ge> 0 \<Longrightarrow> P (lcm x y)"
1840     and "x \<ge> 0 \<Longrightarrow> y \<le> 0 \<Longrightarrow> P (lcm x (- y))"
1841     and "x \<le> 0 \<Longrightarrow> y \<ge> 0 \<Longrightarrow> P (lcm (- x) y)"
1842     and "x \<le> 0 \<Longrightarrow> y \<le> 0 \<Longrightarrow> P (lcm (- x) (- y))"
1843   shows "P (lcm x y)"
1844   using assms by (auto simp add: lcm_neg1_int lcm_neg2_int) arith
1846 lemma lcm_ge_0_int [simp]: "lcm x y \<ge> 0"
1847   for x y :: int
1850 lemma gcd_0_nat: "gcd x 0 = x"
1851   for x :: nat
1852   by simp
1854 lemma gcd_0_int [simp]: "gcd x 0 = \<bar>x\<bar>"
1855   for x :: int
1856   by (auto simp: gcd_int_def)
1858 lemma gcd_0_left_nat: "gcd 0 x = x"
1859   for x :: nat
1860   by simp
1862 lemma gcd_0_left_int [simp]: "gcd 0 x = \<bar>x\<bar>"
1863   for x :: int
1864   by (auto simp:gcd_int_def)
1866 lemma gcd_red_nat: "gcd x y = gcd y (x mod y)"
1867   for x y :: nat
1868   by (cases "y = 0") auto
1871 text \<open>Weaker, but useful for the simplifier.\<close>
1873 lemma gcd_non_0_nat: "y \<noteq> 0 \<Longrightarrow> gcd x y = gcd y (x mod y)"
1874   for x y :: nat
1875   by simp
1877 lemma gcd_1_nat [simp]: "gcd m 1 = 1"
1878   for m :: nat
1879   by simp
1881 lemma gcd_Suc_0 [simp]: "gcd m (Suc 0) = Suc 0"
1882   for m :: nat
1883   by simp
1885 lemma gcd_1_int [simp]: "gcd m 1 = 1"
1886   for m :: int
1889 lemma gcd_idem_nat: "gcd x x = x"
1890   for x :: nat
1891   by simp
1893 lemma gcd_idem_int: "gcd x x = \<bar>x\<bar>"
1894   for x :: int
1895   by (auto simp add: gcd_int_def)
1897 declare gcd_nat.simps [simp del]
1899 text \<open>
1900   \<^medskip> @{term "gcd m n"} divides \<open>m\<close> and \<open>n\<close>.
1901   The conjunctions don't seem provable separately.
1902 \<close>
1904 instance nat :: semiring_gcd
1905 proof
1906   fix m n :: nat
1907   show "gcd m n dvd m" and "gcd m n dvd n"
1908   proof (induct m n rule: gcd_nat_induct)
1909     fix m n :: nat
1910     assume "gcd n (m mod n) dvd m mod n"
1911       and "gcd n (m mod n) dvd n"
1912     then have "gcd n (m mod n) dvd m"
1913       by (rule dvd_mod_imp_dvd)
1914     moreover assume "0 < n"
1915     ultimately show "gcd m n dvd m"
1917   qed (simp_all add: gcd_0_nat gcd_non_0_nat)
1918 next
1919   fix m n k :: nat
1920   assume "k dvd m" and "k dvd n"
1921   then show "k dvd gcd m n"
1922     by (induct m n rule: gcd_nat_induct) (simp_all add: gcd_non_0_nat dvd_mod gcd_0_nat)
1925 instance int :: ring_gcd
1926   by standard
1927     (simp_all add: dvd_int_unfold_dvd_nat gcd_int_def lcm_int_def
1928       zdiv_int nat_abs_mult_distrib [symmetric] lcm_gcd gcd_greatest)
1930 lemma gcd_le1_nat [simp]: "a \<noteq> 0 \<Longrightarrow> gcd a b \<le> a"
1931   for a b :: nat
1932   by (rule dvd_imp_le) auto
1934 lemma gcd_le2_nat [simp]: "b \<noteq> 0 \<Longrightarrow> gcd a b \<le> b"
1935   for a b :: nat
1936   by (rule dvd_imp_le) auto
1938 lemma gcd_le1_int [simp]: "a > 0 \<Longrightarrow> gcd a b \<le> a"
1939   for a b :: int
1940   by (rule zdvd_imp_le) auto
1942 lemma gcd_le2_int [simp]: "b > 0 \<Longrightarrow> gcd a b \<le> b"
1943   for a b :: int
1944   by (rule zdvd_imp_le) auto
1946 lemma gcd_pos_nat [simp]: "gcd m n > 0 \<longleftrightarrow> m \<noteq> 0 \<or> n \<noteq> 0"
1947   for m n :: nat
1948   using gcd_eq_0_iff [of m n] by arith
1950 lemma gcd_pos_int [simp]: "gcd m n > 0 \<longleftrightarrow> m \<noteq> 0 \<or> n \<noteq> 0"
1951   for m n :: int
1952   using gcd_eq_0_iff [of m n] gcd_ge_0_int [of m n] by arith
1954 lemma gcd_unique_nat: "d dvd a \<and> d dvd b \<and> (\<forall>e. e dvd a \<and> e dvd b \<longrightarrow> e dvd d) \<longleftrightarrow> d = gcd a b"
1955   for d a :: nat
1956   apply auto
1957   apply (rule dvd_antisym)
1958    apply (erule (1) gcd_greatest)
1959   apply auto
1960   done
1962 lemma gcd_unique_int:
1963   "d \<ge> 0 \<and> d dvd a \<and> d dvd b \<and> (\<forall>e. e dvd a \<and> e dvd b \<longrightarrow> e dvd d) \<longleftrightarrow> d = gcd a b"
1964   for d a :: int
1965   apply (cases "d = 0")
1966    apply simp
1967   apply (rule iffI)
1968    apply (rule zdvd_antisym_nonneg)
1969       apply (auto intro: gcd_greatest)
1970   done
1972 interpretation gcd_nat:
1973   semilattice_neutr_order gcd "0::nat" Rings.dvd "\<lambda>m n. m dvd n \<and> m \<noteq> n"
1974   by standard (auto simp add: gcd_unique_nat [symmetric] intro: dvd_antisym dvd_trans)
1976 lemma gcd_proj1_if_dvd_int [simp]: "x dvd y \<Longrightarrow> gcd x y = \<bar>x\<bar>"
1977   for x y :: int
1978   by (metis abs_dvd_iff gcd_0_left_int gcd_abs_int gcd_unique_int)
1980 lemma gcd_proj2_if_dvd_int [simp]: "y dvd x \<Longrightarrow> gcd x y = \<bar>y\<bar>"
1981   for x y :: int
1982   by (metis gcd_proj1_if_dvd_int gcd.commute)
1985 text \<open>\<^medskip> Multiplication laws.\<close>
1987 lemma gcd_mult_distrib_nat: "k * gcd m n = gcd (k * m) (k * n)"
1988   for k m n :: nat
1989   \<comment> \<open>@{cite \<open>page 27\<close> davenport92}\<close>
1990   apply (induct m n rule: gcd_nat_induct)
1991    apply simp
1992   apply (cases "k = 0")
1994   done
1996 lemma gcd_mult_distrib_int: "\<bar>k\<bar> * gcd m n = gcd (k * m) (k * n)"
1997   for k m n :: int
1998   apply (subst (1 2) gcd_abs_int)
1999   apply (subst (1 2) abs_mult)
2000   apply (rule gcd_mult_distrib_nat [transferred])
2001     apply auto
2002   done
2004 lemma coprime_crossproduct_nat:
2005   fixes a b c d :: nat
2006   assumes "coprime a d" and "coprime b c"
2007   shows "a * c = b * d \<longleftrightarrow> a = b \<and> c = d"
2008   using assms coprime_crossproduct [of a d b c] by simp
2010 lemma coprime_crossproduct_int:
2011   fixes a b c d :: int
2012   assumes "coprime a d" and "coprime b c"
2013   shows "\<bar>a\<bar> * \<bar>c\<bar> = \<bar>b\<bar> * \<bar>d\<bar> \<longleftrightarrow> \<bar>a\<bar> = \<bar>b\<bar> \<and> \<bar>c\<bar> = \<bar>d\<bar>"
2014   using assms coprime_crossproduct [of a d b c] by simp
2019 (* TODO: add the other variations? *)
2021 lemma gcd_diff1_nat: "m \<ge> n \<Longrightarrow> gcd (m - n) n = gcd m n"
2022   for m n :: nat
2023   by (subst gcd_add1 [symmetric]) auto
2025 lemma gcd_diff2_nat: "n \<ge> m \<Longrightarrow> gcd (n - m) n = gcd m n"
2026   for m n :: nat
2027   apply (subst gcd.commute)
2028   apply (subst gcd_diff1_nat [symmetric])
2029    apply auto
2030   apply (subst gcd.commute)
2031   apply (subst gcd_diff1_nat)
2032    apply assumption
2033   apply (rule gcd.commute)
2034   done
2036 lemma gcd_non_0_int: "y > 0 \<Longrightarrow> gcd x y = gcd y (x mod y)"
2037   for x y :: int
2038   apply (frule_tac b = y and a = x in pos_mod_sign)
2039   apply (simp del: pos_mod_sign add: gcd_int_def abs_if nat_mod_distrib)
2040   apply (auto simp add: gcd_non_0_nat nat_mod_distrib [symmetric] zmod_zminus1_eq_if)
2041   apply (frule_tac a = x in pos_mod_bound)
2042   apply (subst (1 2) gcd.commute)
2043   apply (simp del: pos_mod_bound add: nat_diff_distrib gcd_diff2_nat nat_le_eq_zle)
2044   done
2046 lemma gcd_red_int: "gcd x y = gcd y (x mod y)"
2047   for x y :: int
2048   apply (cases "y = 0")
2049    apply force
2050   apply (cases "y > 0")
2051    apply (subst gcd_non_0_int, auto)
2052   apply (insert gcd_non_0_int [of "- y" "- x"])
2053   apply auto
2054   done
2056 (* TODO: differences, and all variations of addition rules
2057     as simplification rules for nat and int *)
2059 (* TODO: add the three variations of these, and for ints? *)
2061 lemma finite_divisors_nat [simp]: (* FIXME move *)
2062   fixes m :: nat
2063   assumes "m > 0"
2064   shows "finite {d. d dvd m}"
2065 proof-
2066   from assms have "{d. d dvd m} \<subseteq> {d. d \<le> m}"
2067     by (auto dest: dvd_imp_le)
2068   then show ?thesis
2069     using finite_Collect_le_nat by (rule finite_subset)
2070 qed
2072 lemma finite_divisors_int [simp]:
2073   fixes i :: int
2074   assumes "i \<noteq> 0"
2075   shows "finite {d. d dvd i}"
2076 proof -
2077   have "{d. \<bar>d\<bar> \<le> \<bar>i\<bar>} = {- \<bar>i\<bar>..\<bar>i\<bar>}"
2078     by (auto simp: abs_if)
2079   then have "finite {d. \<bar>d\<bar> \<le> \<bar>i\<bar>}"
2080     by simp
2081   from finite_subset [OF _ this] show ?thesis
2082     using assms by (simp add: dvd_imp_le_int subset_iff)
2083 qed
2085 lemma Max_divisors_self_nat [simp]: "n \<noteq> 0 \<Longrightarrow> Max {d::nat. d dvd n} = n"
2086   apply (rule antisym)
2087    apply (fastforce intro: Max_le_iff[THEN iffD2] simp: dvd_imp_le)
2088   apply simp
2089   done
2091 lemma Max_divisors_self_int [simp]: "n \<noteq> 0 \<Longrightarrow> Max {d::int. d dvd n} = \<bar>n\<bar>"
2092   apply (rule antisym)
2093    apply (rule Max_le_iff [THEN iffD2])
2094      apply (auto intro: abs_le_D1 dvd_imp_le_int)
2095   done
2097 lemma gcd_is_Max_divisors_nat: "m > 0 \<Longrightarrow> n > 0 \<Longrightarrow> gcd m n = Max {d. d dvd m \<and> d dvd n}"
2098   for m n :: nat
2099   apply (rule Max_eqI[THEN sym])
2100     apply (metis finite_Collect_conjI finite_divisors_nat)
2101    apply simp
2102    apply (metis Suc_diff_1 Suc_neq_Zero dvd_imp_le gcd_greatest_iff gcd_pos_nat)
2103   apply simp
2104   done
2106 lemma gcd_is_Max_divisors_int: "m \<noteq> 0 \<Longrightarrow> n \<noteq> 0 \<Longrightarrow> gcd m n = Max {d. d dvd m \<and> d dvd n}"
2107   for m n :: int
2108   apply (rule Max_eqI[THEN sym])
2109     apply (metis finite_Collect_conjI finite_divisors_int)
2110    apply simp
2111    apply (metis gcd_greatest_iff gcd_pos_int zdvd_imp_le)
2112   apply simp
2113   done
2115 lemma gcd_code_int [code]: "gcd k l = \<bar>if l = 0 then k else gcd l (\<bar>k\<bar> mod \<bar>l\<bar>)\<bar>"
2116   for k l :: int
2117   by (simp add: gcd_int_def nat_mod_distrib gcd_non_0_nat)
2120 subsection \<open>Coprimality\<close>
2122 lemma coprime_nat: "coprime a b \<longleftrightarrow> (\<forall>d. d dvd a \<and> d dvd b \<longleftrightarrow> d = 1)"
2123   for a b :: nat
2124   using coprime [of a b] by simp
2126 lemma coprime_Suc_0_nat: "coprime a b \<longleftrightarrow> (\<forall>d. d dvd a \<and> d dvd b \<longleftrightarrow> d = Suc 0)"
2127   for a b :: nat
2128   using coprime_nat by simp
2130 lemma coprime_int: "coprime a b \<longleftrightarrow> (\<forall>d. d \<ge> 0 \<and> d dvd a \<and> d dvd b \<longleftrightarrow> d = 1)"
2131   for a b :: int
2132   using gcd_unique_int [of 1 a b]
2133   apply clarsimp
2134   apply (erule subst)
2135   apply (rule iffI)
2136    apply force
2137   using abs_dvd_iff abs_ge_zero apply blast
2138   done
2140 lemma pow_divides_eq_nat [simp]: "n > 0 \<Longrightarrow> a^n dvd b^n \<longleftrightarrow> a dvd b"
2141   for a b n :: nat
2142   using pow_divs_eq[of n] by simp
2144 lemma coprime_Suc_nat [simp]: "coprime (Suc n) n"
2145   using coprime_plus_one[of n] by simp
2147 lemma coprime_minus_one_nat: "n \<noteq> 0 \<Longrightarrow> coprime (n - 1) n"
2148   for n :: nat
2149   using coprime_Suc_nat [of "n - 1"] gcd.commute [of "n - 1" n] by auto
2151 lemma coprime_common_divisor_nat: "coprime a b \<Longrightarrow> x dvd a \<Longrightarrow> x dvd b \<Longrightarrow> x = 1"
2152   for a b :: nat
2153   by (metis gcd_greatest_iff nat_dvd_1_iff_1)
2155 lemma coprime_common_divisor_int: "coprime a b \<Longrightarrow> x dvd a \<Longrightarrow> x dvd b \<Longrightarrow> \<bar>x\<bar> = 1"
2156   for a b :: int
2157   using gcd_greatest_iff [of x a b] by auto
2159 lemma invertible_coprime_nat: "x * y mod m = 1 \<Longrightarrow> coprime x m"
2160   for m x y :: nat
2161   by (metis coprime_lmult gcd_1_nat gcd.commute gcd_red_nat)
2163 lemma invertible_coprime_int: "x * y mod m = 1 \<Longrightarrow> coprime x m"
2164   for m x y :: int
2165   by (metis coprime_lmult gcd_1_int gcd.commute gcd_red_int)
2168 subsection \<open>Bezout's theorem\<close>
2170 text \<open>
2171   Function \<open>bezw\<close> returns a pair of witnesses to Bezout's theorem --
2172   see the theorems that follow the definition.
2173 \<close>
2175 fun bezw :: "nat \<Rightarrow> nat \<Rightarrow> int * int"
2176   where "bezw x y =
2177     (if y = 0 then (1, 0)
2178      else
2179       (snd (bezw y (x mod y)),
2180        fst (bezw y (x mod y)) - snd (bezw y (x mod y)) * int(x div y)))"
2182 lemma bezw_0 [simp]: "bezw x 0 = (1, 0)"
2183   by simp
2185 lemma bezw_non_0:
2186   "y > 0 \<Longrightarrow> bezw x y =
2187     (snd (bezw y (x mod y)), fst (bezw y (x mod y)) - snd (bezw y (x mod y)) * int(x div y))"
2188   by simp
2190 declare bezw.simps [simp del]
2192 lemma bezw_aux: "fst (bezw x y) * int x + snd (bezw x y) * int y = int (gcd x y)"
2193 proof (induct x y rule: gcd_nat_induct)
2194   fix m :: nat
2195   show "fst (bezw m 0) * int m + snd (bezw m 0) * int 0 = int (gcd m 0)"
2196     by auto
2197 next
2198   fix m n :: nat
2199   assume ngt0: "n > 0"
2200     and ih: "fst (bezw n (m mod n)) * int n + snd (bezw n (m mod n)) * int (m mod n) =
2201       int (gcd n (m mod n))"
2202   then show "fst (bezw m n) * int m + snd (bezw m n) * int n = int (gcd m n)"
2203     apply (simp add: bezw_non_0 gcd_non_0_nat)
2204     apply (erule subst)
2206     apply (subst div_mult_mod_eq [of m n, symmetric])
2207       (* applying simp here undoes the last substitution! what is procedure cancel_div_mod? *)
2208     apply (simp only: NO_MATCH_def field_simps of_nat_add of_nat_mult)
2209     done
2210 qed
2212 lemma bezout_int: "\<exists>u v. u * x + v * y = gcd x y"
2213   for x y :: int
2214 proof -
2215   have aux: "x \<ge> 0 \<Longrightarrow> y \<ge> 0 \<Longrightarrow> \<exists>u v. u * x + v * y = gcd x y" for x y :: int
2216     apply (rule_tac x = "fst (bezw (nat x) (nat y))" in exI)
2217     apply (rule_tac x = "snd (bezw (nat x) (nat y))" in exI)
2218     apply (unfold gcd_int_def)
2219     apply simp
2220     apply (subst bezw_aux [symmetric])
2221     apply auto
2222     done
2223   consider "x \<ge> 0" "y \<ge> 0" | "x \<ge> 0" "y \<le> 0" | "x \<le> 0" "y \<ge> 0" | "x \<le> 0" "y \<le> 0"
2224     by atomize_elim auto
2225   then show ?thesis
2226   proof cases
2227     case 1
2228     then show ?thesis by (rule aux)
2229   next
2230     case 2
2231     then show ?thesis
2232       apply -
2233       apply (insert aux [of x "-y"])
2234       apply auto
2235       apply (rule_tac x = u in exI)
2236       apply (rule_tac x = "-v" in exI)
2237       apply (subst gcd_neg2_int [symmetric])
2238       apply auto
2239       done
2240   next
2241     case 3
2242     then show ?thesis
2243       apply -
2244       apply (insert aux [of "-x" y])
2245       apply auto
2246       apply (rule_tac x = "-u" in exI)
2247       apply (rule_tac x = v in exI)
2248       apply (subst gcd_neg1_int [symmetric])
2249       apply auto
2250       done
2251   next
2252     case 4
2253     then show ?thesis
2254       apply -
2255       apply (insert aux [of "-x" "-y"])
2256       apply auto
2257       apply (rule_tac x = "-u" in exI)
2258       apply (rule_tac x = "-v" in exI)
2259       apply (subst gcd_neg1_int [symmetric])
2260       apply (subst gcd_neg2_int [symmetric])
2261       apply auto
2262       done
2263   qed
2264 qed
2267 text \<open>Versions of Bezout for \<open>nat\<close>, by Amine Chaieb.\<close>
2269 lemma ind_euclid:
2270   fixes P :: "nat \<Rightarrow> nat \<Rightarrow> bool"
2271   assumes c: " \<forall>a b. P a b \<longleftrightarrow> P b a"
2272     and z: "\<forall>a. P a 0"
2273     and add: "\<forall>a b. P a b \<longrightarrow> P a (a + b)"
2274   shows "P a b"
2275 proof (induct "a + b" arbitrary: a b rule: less_induct)
2276   case less
2277   consider (eq) "a = b" | (lt) "a < b" "a + b - a < a + b" | "b = 0" | "b + a - b < a + b"
2278     by arith
2279   show ?case
2280   proof (cases a b rule: linorder_cases)
2281     case equal
2282     with add [rule_format, OF z [rule_format, of a]] show ?thesis by simp
2283   next
2284     case lt: less
2285     then consider "a = 0" | "a + b - a < a + b" by arith
2286     then show ?thesis
2287     proof cases
2288       case 1
2289       with z c show ?thesis by blast
2290     next
2291       case 2
2292       also have *: "a + b - a = a + (b - a)" using lt by arith
2293       finally have "a + (b - a) < a + b" .
2294       then have "P a (a + (b - a))" by (rule add [rule_format, OF less])
2295       then show ?thesis by (simp add: *[symmetric])
2296     qed
2297   next
2298     case gt: greater
2299     then consider "b = 0" | "b + a - b < a + b" by arith
2300     then show ?thesis
2301     proof cases
2302       case 1
2303       with z c show ?thesis by blast
2304     next
2305       case 2
2306       also have *: "b + a - b = b + (a - b)" using gt by arith
2307       finally have "b + (a - b) < a + b" .
2308       then have "P b (b + (a - b))" by (rule add [rule_format, OF less])
2309       then have "P b a" by (simp add: *[symmetric])
2310       with c show ?thesis by blast
2311     qed
2312   qed
2313 qed
2315 lemma bezout_lemma_nat:
2316   assumes ex: "\<exists>(d::nat) x y. d dvd a \<and> d dvd b \<and>
2317     (a * x = b * y + d \<or> b * x = a * y + d)"
2318   shows "\<exists>d x y. d dvd a \<and> d dvd a + b \<and>
2319     (a * x = (a + b) * y + d \<or> (a + b) * x = a * y + d)"
2320   using ex
2321   apply clarsimp
2322   apply (rule_tac x="d" in exI)
2323   apply simp
2324   apply (case_tac "a * x = b * y + d")
2325    apply simp_all
2326    apply (rule_tac x="x + y" in exI)
2327    apply (rule_tac x="y" in exI)
2328    apply algebra
2329   apply (rule_tac x="x" in exI)
2330   apply (rule_tac x="x + y" in exI)
2331   apply algebra
2332   done
2334 lemma bezout_add_nat: "\<exists>(d::nat) x y. d dvd a \<and> d dvd b \<and>
2335     (a * x = b * y + d \<or> b * x = a * y + d)"
2336   apply (induct a b rule: ind_euclid)
2337     apply blast
2338    apply clarify
2339    apply (rule_tac x="a" in exI)
2340    apply simp
2341   apply clarsimp
2342   apply (rule_tac x="d" in exI)
2343   apply (case_tac "a * x = b * y + d")
2344    apply simp_all
2345    apply (rule_tac x="x+y" in exI)
2346    apply (rule_tac x="y" in exI)
2347    apply algebra
2348   apply (rule_tac x="x" in exI)
2349   apply (rule_tac x="x+y" in exI)
2350   apply algebra
2351   done
2353 lemma bezout1_nat: "\<exists>(d::nat) x y. d dvd a \<and> d dvd b \<and>
2354     (a * x - b * y = d \<or> b * x - a * y = d)"
2356   apply clarsimp
2357   apply (rule_tac x="d" in exI)
2358   apply simp
2359   apply (rule_tac x="x" in exI)
2360   apply (rule_tac x="y" in exI)
2361   apply auto
2362   done
2365   fixes a b :: nat
2366   assumes a: "a \<noteq> 0"
2367   shows "\<exists>d x y. d dvd a \<and> d dvd b \<and> a * x = b * y + d"
2368 proof -
2369   consider d x y where "d dvd a" "d dvd b" "a * x = b * y + d"
2370     | d x y where "d dvd a" "d dvd b" "b * x = a * y + d"
2371     using bezout_add_nat [of a b] by blast
2372   then show ?thesis
2373   proof cases
2374     case 1
2375     then show ?thesis by blast
2376   next
2377     case H: 2
2378     show ?thesis
2379     proof (cases "b = 0")
2380       case True
2381       with H show ?thesis by simp
2382     next
2383       case False
2384       then have bp: "b > 0" by simp
2385       with dvd_imp_le [OF H(2)] consider "d = b" | "d < b"
2386         by atomize_elim auto
2387       then show ?thesis
2388       proof cases
2389         case 1
2390         with a H show ?thesis
2391           apply simp
2392           apply (rule exI[where x = b])
2393           apply simp
2394           apply (rule exI[where x = b])
2395           apply (rule exI[where x = "a - 1"])
2397           done
2398       next
2399         case 2
2400         show ?thesis
2401         proof (cases "x = 0")
2402           case True
2403           with a H show ?thesis by simp
2404         next
2405           case x0: False
2406           then have xp: "x > 0" by simp
2407           from \<open>d < b\<close> have "d \<le> b - 1" by simp
2408           then have "d * b \<le> b * (b - 1)" by simp
2409           with xp mult_mono[of "1" "x" "d * b" "b * (b - 1)"]
2410           have dble: "d * b \<le> x * b * (b - 1)" using bp by simp
2411           from H(3) have "d + (b - 1) * (b * x) = d + (b - 1) * (a * y + d)"
2412             by simp
2413           then have "d + (b - 1) * a * y + (b - 1) * d = d + (b - 1) * b * x"
2414             by (simp only: mult.assoc distrib_left)
2415           then have "a * ((b - 1) * y) + d * (b - 1 + 1) = d + x * b * (b - 1)"
2416             by algebra
2417           then have "a * ((b - 1) * y) = d + x * b * (b - 1) - d * b"
2418             using bp by simp
2419           then have "a * ((b - 1) * y) = d + (x * b * (b - 1) - d * b)"
2420             by (simp only: diff_add_assoc[OF dble, of d, symmetric])
2421           then have "a * ((b - 1) * y) = b * (x * (b - 1) - d) + d"
2422             by (simp only: diff_mult_distrib2 ac_simps)
2423           with H(1,2) show ?thesis
2424             apply -
2425             apply (rule exI [where x = d])
2426             apply simp
2427             apply (rule exI [where x = "(b - 1) * y"])
2428             apply (rule exI [where x = "x * (b - 1) - d"])
2429             apply simp
2430             done
2431         qed
2432       qed
2433     qed
2434   qed
2435 qed
2437 lemma bezout_nat:
2438   fixes a :: nat
2439   assumes a: "a \<noteq> 0"
2440   shows "\<exists>x y. a * x = b * y + gcd a b"
2441 proof -
2442   obtain d x y where d: "d dvd a" "d dvd b" and eq: "a * x = b * y + d"
2443     using bezout_add_strong_nat [OF a, of b] by blast
2444   from d have "d dvd gcd a b"
2445     by simp
2446   then obtain k where k: "gcd a b = d * k"
2447     unfolding dvd_def by blast
2448   from eq have "a * x * k = (b * y + d) * k"
2449     by auto
2450   then have "a * (x * k) = b * (y * k) + gcd a b"
2452   then show ?thesis
2453     by blast
2454 qed
2457 subsection \<open>LCM properties on @{typ nat} and @{typ int}\<close>
2459 lemma lcm_altdef_int [code]: "lcm a b = \<bar>a\<bar> * \<bar>b\<bar> div gcd a b"
2460   for a b :: int
2461   by (simp add: lcm_int_def lcm_nat_def zdiv_int gcd_int_def)
2463 lemma prod_gcd_lcm_nat: "m * n = gcd m n * lcm m n"
2464   for m n :: nat
2465   unfolding lcm_nat_def
2466   by (simp add: dvd_mult_div_cancel [OF gcd_dvd_prod])
2468 lemma prod_gcd_lcm_int: "\<bar>m\<bar> * \<bar>n\<bar> = gcd m n * lcm m n"
2469   for m n :: int
2470   unfolding lcm_int_def gcd_int_def
2471   apply (subst of_nat_mult [symmetric])
2472   apply (subst prod_gcd_lcm_nat [symmetric])
2473   apply (subst nat_abs_mult_distrib [symmetric])
2475   done
2477 lemma lcm_pos_nat: "m > 0 \<Longrightarrow> n > 0 \<Longrightarrow> lcm m n > 0"
2478   for m n :: nat
2479   by (metis gr0I mult_is_0 prod_gcd_lcm_nat)
2481 lemma lcm_pos_int: "m \<noteq> 0 \<Longrightarrow> n \<noteq> 0 \<Longrightarrow> lcm m n > 0"
2482   for m n :: int
2483   apply (subst lcm_abs_int)
2484   apply (rule lcm_pos_nat [transferred])
2485      apply auto
2486   done
2488 lemma dvd_pos_nat: "n > 0 \<Longrightarrow> m dvd n \<Longrightarrow> m > 0"  (* FIXME move *)
2489   for m n :: nat
2490   by (cases m) auto
2492 lemma lcm_unique_nat:
2493   "a dvd d \<and> b dvd d \<and> (\<forall>e. a dvd e \<and> b dvd e \<longrightarrow> d dvd e) \<longleftrightarrow> d = lcm a b"
2494   for a b d :: nat
2495   by (auto intro: dvd_antisym lcm_least)
2497 lemma lcm_unique_int:
2498   "d \<ge> 0 \<and> a dvd d \<and> b dvd d \<and> (\<forall>e. a dvd e \<and> b dvd e \<longrightarrow> d dvd e) \<longleftrightarrow> d = lcm a b"
2499   for a b d :: int
2500   using lcm_least zdvd_antisym_nonneg by auto
2502 lemma lcm_proj2_if_dvd_nat [simp]: "x dvd y \<Longrightarrow> lcm x y = y"
2503   for x y :: nat
2504   apply (rule sym)
2505   apply (subst lcm_unique_nat [symmetric])
2506   apply auto
2507   done
2509 lemma lcm_proj2_if_dvd_int [simp]: "x dvd y \<Longrightarrow> lcm x y = \<bar>y\<bar>"
2510   for x y :: int
2511   apply (rule sym)
2512   apply (subst lcm_unique_int [symmetric])
2513   apply auto
2514   done
2516 lemma lcm_proj1_if_dvd_nat [simp]: "x dvd y \<Longrightarrow> lcm y x = y"
2517   for x y :: nat
2518   by (subst lcm.commute) (erule lcm_proj2_if_dvd_nat)
2520 lemma lcm_proj1_if_dvd_int [simp]: "x dvd y \<Longrightarrow> lcm y x = \<bar>y\<bar>"
2521   for x y :: int
2522   by (subst lcm.commute) (erule lcm_proj2_if_dvd_int)
2524 lemma lcm_proj1_iff_nat [simp]: "lcm m n = m \<longleftrightarrow> n dvd m"
2525   for m n :: nat
2526   by (metis lcm_proj1_if_dvd_nat lcm_unique_nat)
2528 lemma lcm_proj2_iff_nat [simp]: "lcm m n = n \<longleftrightarrow> m dvd n"
2529   for m n :: nat
2530   by (metis lcm_proj2_if_dvd_nat lcm_unique_nat)
2532 lemma lcm_proj1_iff_int [simp]: "lcm m n = \<bar>m\<bar> \<longleftrightarrow> n dvd m"
2533   for m n :: int
2534   by (metis dvd_abs_iff lcm_proj1_if_dvd_int lcm_unique_int)
2536 lemma lcm_proj2_iff_int [simp]: "lcm m n = \<bar>n\<bar> \<longleftrightarrow> m dvd n"
2537   for m n :: int
2538   by (metis dvd_abs_iff lcm_proj2_if_dvd_int lcm_unique_int)
2540 lemma lcm_1_iff_nat [simp]: "lcm m n = Suc 0 \<longleftrightarrow> m = Suc 0 \<and> n = Suc 0"
2541   for m n :: nat
2542   using lcm_eq_1_iff [of m n] by simp
2544 lemma lcm_1_iff_int [simp]: "lcm m n = 1 \<longleftrightarrow> (m = 1 \<or> m = -1) \<and> (n = 1 \<or> n = -1)"
2545   for m n :: int
2546   by auto
2549 subsection \<open>The complete divisibility lattice on @{typ nat} and @{typ int}\<close>
2551 text \<open>
2552   Lifting \<open>gcd\<close> and \<open>lcm\<close> to sets (\<open>Gcd\<close> / \<open>Lcm\<close>).
2553   \<open>Gcd\<close> is defined via \<open>Lcm\<close> to facilitate the proof that we have a complete lattice.
2554 \<close>
2556 instantiation nat :: semiring_Gcd
2557 begin
2559 interpretation semilattice_neutr_set lcm "1::nat"
2560   by standard simp_all
2562 definition "Lcm M = (if finite M then F M else 0)" for M :: "nat set"
2564 lemma Lcm_nat_empty: "Lcm {} = (1::nat)"
2565   by (simp add: Lcm_nat_def del: One_nat_def)
2567 lemma Lcm_nat_insert: "Lcm (insert n M) = lcm n (Lcm M)" for n :: nat
2568   by (cases "finite M") (auto simp add: Lcm_nat_def simp del: One_nat_def)
2570 lemma Lcm_nat_infinite: "infinite M \<Longrightarrow> Lcm M = 0" for M :: "nat set"
2573 lemma dvd_Lcm_nat [simp]:
2574   fixes M :: "nat set"
2575   assumes "m \<in> M"
2576   shows "m dvd Lcm M"
2577 proof -
2578   from assms have "insert m M = M"
2579     by auto
2580   moreover have "m dvd Lcm (insert m M)"
2582   ultimately show ?thesis
2583     by simp
2584 qed
2586 lemma Lcm_dvd_nat [simp]:
2587   fixes M :: "nat set"
2588   assumes "\<forall>m\<in>M. m dvd n"
2589   shows "Lcm M dvd n"
2590 proof (cases "n > 0")
2591   case False
2592   then show ?thesis by simp
2593 next
2594   case True
2595   then have "finite {d. d dvd n}"
2596     by (rule finite_divisors_nat)
2597   moreover have "M \<subseteq> {d. d dvd n}"
2598     using assms by fast
2599   ultimately have "finite M"
2600     by (rule rev_finite_subset)
2601   then show ?thesis
2602     using assms by (induct M) (simp_all add: Lcm_nat_empty Lcm_nat_insert)
2603 qed
2605 definition "Gcd M = Lcm {d. \<forall>m\<in>M. d dvd m}" for M :: "nat set"
2607 instance
2608 proof
2609   fix N :: "nat set"
2610   fix n :: nat
2611   show "Gcd N dvd n" if "n \<in> N"
2612     using that by (induct N rule: infinite_finite_induct) (auto simp add: Gcd_nat_def)
2613   show "n dvd Gcd N" if "\<And>m. m \<in> N \<Longrightarrow> n dvd m"
2614     using that by (induct N rule: infinite_finite_induct) (auto simp add: Gcd_nat_def)
2615   show "n dvd Lcm N" if "n \<in> N"
2616     using that by (induct N rule: infinite_finite_induct) auto
2617   show "Lcm N dvd n" if "\<And>m. m \<in> N \<Longrightarrow> m dvd n"
2618     using that by (induct N rule: infinite_finite_induct) auto
2619   show "normalize (Gcd N) = Gcd N" and "normalize (Lcm N) = Lcm N"
2620     by simp_all
2621 qed
2623 end
2625 lemma Gcd_nat_eq_one: "1 \<in> N \<Longrightarrow> Gcd N = 1"
2626   for N :: "nat set"
2627   by (rule Gcd_eq_1_I) auto
2630 text \<open>Alternative characterizations of Gcd:\<close>
2632 lemma Gcd_eq_Max:
2633   fixes M :: "nat set"
2634   assumes "finite (M::nat set)" and "M \<noteq> {}" and "0 \<notin> M"
2635   shows "Gcd M = Max (\<Inter>m\<in>M. {d. d dvd m})"
2636 proof (rule antisym)
2637   from assms obtain m where "m \<in> M" and "m > 0"
2638     by auto
2639   from \<open>m > 0\<close> have "finite {d. d dvd m}"
2640     by (blast intro: finite_divisors_nat)
2641   with \<open>m \<in> M\<close> have fin: "finite (\<Inter>m\<in>M. {d. d dvd m})"
2642     by blast
2643   from fin show "Gcd M \<le> Max (\<Inter>m\<in>M. {d. d dvd m})"
2644     by (auto intro: Max_ge Gcd_dvd)
2645   from fin show "Max (\<Inter>m\<in>M. {d. d dvd m}) \<le> Gcd M"
2646     apply (rule Max.boundedI)
2647      apply auto
2648     apply (meson Gcd_dvd Gcd_greatest \<open>0 < m\<close> \<open>m \<in> M\<close> dvd_imp_le dvd_pos_nat)
2649     done
2650 qed
2652 lemma Gcd_remove0_nat: "finite M \<Longrightarrow> Gcd M = Gcd (M - {0})"
2653   for M :: "nat set"
2654   apply (induct pred: finite)
2655    apply simp
2656   apply (case_tac "x = 0")
2657    apply simp
2658   apply (subgoal_tac "insert x F - {0} = insert x (F - {0})")
2659    apply simp
2660   apply blast
2661   done
2663 lemma Lcm_in_lcm_closed_set_nat:
2664   "finite M \<Longrightarrow> M \<noteq> {} \<Longrightarrow> \<forall>m n. m \<in> M \<longrightarrow> n \<in> M \<longrightarrow> lcm m n \<in> M \<Longrightarrow> Lcm M \<in> M"
2665   for M :: "nat set"
2666   apply (induct rule: finite_linorder_min_induct)
2667    apply simp
2668   apply simp
2669   apply (subgoal_tac "\<forall>m n. m \<in> A \<longrightarrow> n \<in> A \<longrightarrow> lcm m n \<in> A")
2670    apply simp
2671    apply(case_tac "A = {}")
2672     apply simp
2673    apply simp
2674   apply (metis lcm_pos_nat lcm_unique_nat linorder_neq_iff nat_dvd_not_less not_less0)
2675   done
2677 lemma Lcm_eq_Max_nat:
2678   "finite M \<Longrightarrow> M \<noteq> {} \<Longrightarrow> 0 \<notin> M \<Longrightarrow> \<forall>m n. m \<in> M \<longrightarrow> n \<in> M \<longrightarrow> lcm m n \<in> M \<Longrightarrow> Lcm M = Max M"
2679   for M :: "nat set"
2680   apply (rule antisym)
2681    apply (rule Max_ge)
2682     apply assumption
2683    apply (erule (2) Lcm_in_lcm_closed_set_nat)
2684   apply (auto simp add: not_le Lcm_0_iff dvd_imp_le leD le_neq_trans)
2685   done
2687 lemma mult_inj_if_coprime_nat:
2688   "inj_on f A \<Longrightarrow> inj_on g B \<Longrightarrow> \<forall>a\<in>A. \<forall>b\<in>B. coprime (f a) (g b) \<Longrightarrow>
2689     inj_on (\<lambda>(a, b). f a * g b) (A \<times> B)"
2690   for f :: "'a \<Rightarrow> nat" and g :: "'b \<Rightarrow> nat"
2691   by (auto simp add: inj_on_def coprime_crossproduct_nat simp del: One_nat_def)
2694 subsubsection \<open>Setwise GCD and LCM for integers\<close>
2696 instantiation int :: semiring_Gcd
2697 begin
2699 definition "Lcm M = int (LCM m\<in>M. (nat \<circ> abs) m)"
2701 definition "Gcd M = int (GCD m\<in>M. (nat \<circ> abs) m)"
2703 instance
2704   by standard
2705     (auto intro!: Gcd_dvd Gcd_greatest simp add: Gcd_int_def
2706       Lcm_int_def int_dvd_iff dvd_int_iff dvd_int_unfold_dvd_nat [symmetric])
2708 end
2710 lemma abs_Gcd [simp]: "\<bar>Gcd K\<bar> = Gcd K"
2711   for K :: "int set"
2712   using normalize_Gcd [of K] by simp
2714 lemma abs_Lcm [simp]: "\<bar>Lcm K\<bar> = Lcm K"
2715   for K :: "int set"
2716   using normalize_Lcm [of K] by simp
2718 lemma Gcm_eq_int_iff: "Gcd K = int n \<longleftrightarrow> Gcd ((nat \<circ> abs) ` K) = n"
2719   by (simp add: Gcd_int_def comp_def image_image)
2721 lemma Lcm_eq_int_iff: "Lcm K = int n \<longleftrightarrow> Lcm ((nat \<circ> abs) ` K) = n"
2722   by (simp add: Lcm_int_def comp_def image_image)
2725 subsection \<open>GCD and LCM on @{typ integer}\<close>
2727 instantiation integer :: gcd
2728 begin
2730 context
2731   includes integer.lifting
2732 begin
2734 lift_definition gcd_integer :: "integer \<Rightarrow> integer \<Rightarrow> integer" is gcd .
2736 lift_definition lcm_integer :: "integer \<Rightarrow> integer \<Rightarrow> integer" is lcm .
2738 end
2740 instance ..
2742 end
2744 lifting_update integer.lifting
2745 lifting_forget integer.lifting
2747 context
2748   includes integer.lifting
2749 begin
2751 lemma gcd_code_integer [code]: "gcd k l = \<bar>if l = (0::integer) then k else gcd l (\<bar>k\<bar> mod \<bar>l\<bar>)\<bar>"
2752   by transfer (fact gcd_code_int)
2754 lemma lcm_code_integer [code]: "lcm a b = \<bar>a\<bar> * \<bar>b\<bar> div gcd a b"
2755   for a b :: integer
2756   by transfer (fact lcm_altdef_int)
2758 end
2760 code_printing
2761   constant "gcd :: integer \<Rightarrow> _" \<rightharpoonup>
2762     (OCaml) "Big'_int.gcd'_big'_int"
2764   and (Scala) "_.gcd'((_)')"
2765   \<comment> \<open>There is no gcd operation in the SML standard library, so no code setup for SML\<close>
2767 text \<open>Some code equations\<close>
2769 lemmas Gcd_nat_set_eq_fold [code] = Gcd_set_eq_fold [where ?'a = nat]
2770 lemmas Lcm_nat_set_eq_fold [code] = Lcm_set_eq_fold [where ?'a = nat]
2771 lemmas Gcd_int_set_eq_fold [code] = Gcd_set_eq_fold [where ?'a = int]
2772 lemmas Lcm_int_set_eq_fold [code] = Lcm_set_eq_fold [where ?'a = int]
2774 text \<open>Fact aliases.\<close>
2776 lemma lcm_0_iff_nat [simp]: "lcm m n = 0 \<longleftrightarrow> m = 0 \<or> n = 0"
2777   for m n :: nat
2778   by (fact lcm_eq_0_iff)
2780 lemma lcm_0_iff_int [simp]: "lcm m n = 0 \<longleftrightarrow> m = 0 \<or> n = 0"
2781   for m n :: int
2782   by (fact lcm_eq_0_iff)
2784 lemma dvd_lcm_I1_nat [simp]: "k dvd m \<Longrightarrow> k dvd lcm m n"
2785   for k m n :: nat
2786   by (fact dvd_lcmI1)
2788 lemma dvd_lcm_I2_nat [simp]: "k dvd n \<Longrightarrow> k dvd lcm m n"
2789   for k m n :: nat
2790   by (fact dvd_lcmI2)
2792 lemma dvd_lcm_I1_int [simp]: "i dvd m \<Longrightarrow> i dvd lcm m n"
2793   for i m n :: int
2794   by (fact dvd_lcmI1)
2796 lemma dvd_lcm_I2_int [simp]: "i dvd n \<Longrightarrow> i dvd lcm m n"
2797   for i m n :: int
2798   by (fact dvd_lcmI2)
2800 lemma coprime_exp2_nat [intro]: "coprime a b \<Longrightarrow> coprime (a^n) (b^m)"
2801   for a b :: nat
2802   by (fact coprime_exp2)
2804 lemma coprime_exp2_int [intro]: "coprime a b \<Longrightarrow> coprime (a^n) (b^m)"
2805   for a b :: int
2806   by (fact coprime_exp2)
2808 lemmas Gcd_dvd_nat [simp] = Gcd_dvd [where ?'a = nat]
2809 lemmas Gcd_dvd_int [simp] = Gcd_dvd [where ?'a = int]
2810 lemmas Gcd_greatest_nat [simp] = Gcd_greatest [where ?'a = nat]
2811 lemmas Gcd_greatest_int [simp] = Gcd_greatest [where ?'a = int]
2813 lemma dvd_Lcm_int [simp]: "m \<in> M \<Longrightarrow> m dvd Lcm M"
2814   for M :: "int set"
2815   by (fact dvd_Lcm)
2817 lemma gcd_neg_numeral_1_int [simp]: "gcd (- numeral n :: int) x = gcd (numeral n) x"
2818   by (fact gcd_neg1_int)
2820 lemma gcd_neg_numeral_2_int [simp]: "gcd x (- numeral n :: int) = gcd x (numeral n)"
2821   by (fact gcd_neg2_int)
2823 lemma gcd_proj1_if_dvd_nat [simp]: "x dvd y \<Longrightarrow> gcd x y = x"
2824   for x y :: nat
2825   by (fact gcd_nat.absorb1)
2827 lemma gcd_proj2_if_dvd_nat [simp]: "y dvd x \<Longrightarrow> gcd x y = y"
2828   for x y :: nat
2829   by (fact gcd_nat.absorb2)
2831 lemmas Lcm_eq_0_I_nat [simp] = Lcm_eq_0_I [where ?'a = nat]
2832 lemmas Lcm_0_iff_nat [simp] = Lcm_0_iff [where ?'a = nat]
2833 lemmas Lcm_least_int [simp] = Lcm_least [where ?'a = int]
2835 end