src/HOL/Number_Theory/Euclidean_Algorithm.thy
 author wenzelm Sun Nov 02 18:21:45 2014 +0100 (2014-11-02) changeset 58889 5b7a9633cfa8 parent 58023 62826b36ac5e child 58953 2e19b392d9e3 permissions -rw-r--r--
modernized header uniformly as section;
1 (* Author: Manuel Eberl *)
3 section {* Abstract euclidean algorithm *}
5 theory Euclidean_Algorithm
6 imports Complex_Main
7 begin
9 lemma finite_int_set_iff_bounded_le:
10   "finite (N::int set) = (\<exists>m\<ge>0. \<forall>n\<in>N. abs n \<le> m)"
11 proof
12   assume "finite (N::int set)"
13   hence "finite (nat  abs  N)" by (intro finite_imageI)
14   hence "\<exists>m. \<forall>n\<in>natabsN. n \<le> m" by (simp add: finite_nat_set_iff_bounded_le)
15   then obtain m :: nat where "\<forall>n\<in>N. nat (abs n) \<le> nat (int m)" by auto
16   then show "\<exists>m\<ge>0. \<forall>n\<in>N. abs n \<le> m" by (intro exI[of _ "int m"]) (auto simp: nat_le_eq_zle)
17 next
18   assume "\<exists>m\<ge>0. \<forall>n\<in>N. abs n \<le> m"
19   then obtain m where "m \<ge> 0" and "\<forall>n\<in>N. abs n \<le> m" by blast
20   hence "\<forall>n\<in>N. nat (abs n) \<le> nat m" by (auto simp: nat_le_eq_zle)
21   hence "\<forall>n\<in>natabsN. n \<le> nat m" by (auto simp: nat_le_eq_zle)
22   hence A: "finite ((nat \<circ> abs)N)" unfolding o_def
23       by (subst finite_nat_set_iff_bounded_le) blast
24   {
25     assume "\<not>finite N"
26     from pigeonhole_infinite[OF this A] obtain x
27        where "x \<in> N" and B: "~finite {a\<in>N. nat (abs a) = nat (abs x)}"
28        unfolding o_def by blast
29     have "{a\<in>N. nat (abs a) = nat (abs x)} \<subseteq> {x, -x}" by auto
30     hence "finite {a\<in>N. nat (abs a) = nat (abs x)}" by (rule finite_subset) simp
31     with B have False by contradiction
32   }
33   then show "finite N" by blast
34 qed
36 context semiring_div
37 begin
39 lemma dvd_setprod [intro]:
40   assumes "finite A" and "x \<in> A"
41   shows "f x dvd setprod f A"
42 proof
43   from finite A have "setprod f (insert x (A - {x})) = f x * setprod f (A - {x})"
44     by (intro setprod.insert) auto
45   also from x \<in> A have "insert x (A - {x}) = A" by blast
46   finally show "setprod f A = f x * setprod f (A - {x})" .
47 qed
49 lemma dvd_mult_cancel_left:
50   assumes "a \<noteq> 0" and "a * b dvd a * c"
51   shows "b dvd c"
52 proof-
53   from assms(2) obtain k where "a * c = a * b * k" unfolding dvd_def by blast
54   hence "c * a = b * k * a" by (simp add: ac_simps)
55   hence "c * (a div a) = b * k * (a div a)" by (simp add: div_mult_swap)
56   also from a \<noteq> 0 have "a div a = 1" by simp
57   finally show ?thesis by simp
58 qed
60 lemma dvd_mult_cancel_right:
61   "a \<noteq> 0 \<Longrightarrow> b * a dvd c * a \<Longrightarrow> b dvd c"
62   by (subst (asm) (1 2) ac_simps, rule dvd_mult_cancel_left)
64 lemma nonzero_pow_nonzero:
65   "a \<noteq> 0 \<Longrightarrow> a ^ n \<noteq> 0"
66   by (induct n) (simp_all add: no_zero_divisors)
68 lemma zero_pow_zero: "n \<noteq> 0 \<Longrightarrow> 0 ^ n = 0"
69   by (cases n, simp_all)
71 lemma pow_zero_iff:
72   "n \<noteq> 0 \<Longrightarrow> a^n = 0 \<longleftrightarrow> a = 0"
73   using nonzero_pow_nonzero zero_pow_zero by auto
75 end
77 context semiring_div
78 begin
80 definition ring_inv :: "'a \<Rightarrow> 'a"
81 where
82   "ring_inv x = 1 div x"
84 definition is_unit :: "'a \<Rightarrow> bool"
85 where
86   "is_unit x \<longleftrightarrow> x dvd 1"
88 definition associated :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
89 where
90   "associated x y \<longleftrightarrow> x dvd y \<and> y dvd x"
92 lemma unit_prod [intro]:
93   "is_unit x \<Longrightarrow> is_unit y \<Longrightarrow> is_unit (x * y)"
94   unfolding is_unit_def by (subst mult_1_left [of 1, symmetric], rule mult_dvd_mono)
96 lemma unit_ring_inv:
97   "is_unit y \<Longrightarrow> x div y = x * ring_inv y"
98   by (simp add: div_mult_swap ring_inv_def is_unit_def)
100 lemma unit_ring_inv_ring_inv [simp]:
101   "is_unit x \<Longrightarrow> ring_inv (ring_inv x) = x"
102   unfolding is_unit_def ring_inv_def
103   by (metis div_mult_mult1_if div_mult_self1_is_id dvd_mult_div_cancel mult_1_right)
105 lemma inv_imp_eq_ring_inv:
106   "a * b = 1 \<Longrightarrow> ring_inv a = b"
107   by (metis dvd_mult_div_cancel dvd_mult_right mult_1_right mult.left_commute one_dvd ring_inv_def)
109 lemma ring_inv_is_inv1 [simp]:
110   "is_unit a \<Longrightarrow> a * ring_inv a = 1"
111   unfolding is_unit_def ring_inv_def by (simp add: dvd_mult_div_cancel)
113 lemma ring_inv_is_inv2 [simp]:
114   "is_unit a \<Longrightarrow> ring_inv a * a = 1"
115   by (simp add: ac_simps)
117 lemma unit_ring_inv_unit [simp, intro]:
118   assumes "is_unit x"
119   shows "is_unit (ring_inv x)"
120 proof -
121   from assms have "1 = ring_inv x * x" by simp
122   then show "is_unit (ring_inv x)" unfolding is_unit_def by (rule dvdI)
123 qed
125 lemma mult_unit_dvd_iff:
126   "is_unit y \<Longrightarrow> x * y dvd z \<longleftrightarrow> x dvd z"
127 proof
128   assume "is_unit y" "x * y dvd z"
129   then show "x dvd z" by (simp add: dvd_mult_left)
130 next
131   assume "is_unit y" "x dvd z"
132   then obtain k where "z = x * k" unfolding dvd_def by blast
133   with is_unit y have "z = (x * y) * (ring_inv y * k)"
134       by (simp add: mult_ac)
135   then show "x * y dvd z" by (rule dvdI)
136 qed
138 lemma div_unit_dvd_iff:
139   "is_unit y \<Longrightarrow> x div y dvd z \<longleftrightarrow> x dvd z"
140   by (subst unit_ring_inv) (assumption, simp add: mult_unit_dvd_iff)
142 lemma dvd_mult_unit_iff:
143   "is_unit y \<Longrightarrow> x dvd z * y \<longleftrightarrow> x dvd z"
144 proof
145   assume "is_unit y" and "x dvd z * y"
146   have "z * y dvd z * (y * ring_inv y)" by (subst mult_assoc [symmetric]) simp
147   also from is_unit y have "y * ring_inv y = 1" by simp
148   finally have "z * y dvd z" by simp
149   with x dvd z * y show "x dvd z" by (rule dvd_trans)
150 next
151   assume "x dvd z"
152   then show "x dvd z * y" by simp
153 qed
155 lemma dvd_div_unit_iff:
156   "is_unit y \<Longrightarrow> x dvd z div y \<longleftrightarrow> x dvd z"
157   by (subst unit_ring_inv) (assumption, simp add: dvd_mult_unit_iff)
159 lemmas unit_dvd_iff = mult_unit_dvd_iff div_unit_dvd_iff dvd_mult_unit_iff dvd_div_unit_iff
161 lemma unit_div [intro]:
162   "is_unit x \<Longrightarrow> is_unit y \<Longrightarrow> is_unit (x div y)"
163   by (subst unit_ring_inv) (assumption, rule unit_prod, simp_all)
165 lemma unit_div_mult_swap:
166   "is_unit z \<Longrightarrow> x * (y div z) = x * y div z"
167   by (simp only: unit_ring_inv [of _ y] unit_ring_inv [of _ "x*y"] ac_simps)
169 lemma unit_div_commute:
170   "is_unit y \<Longrightarrow> x div y * z = x * z div y"
171   by (simp only: unit_ring_inv [of _ x] unit_ring_inv [of _ "x*z"] ac_simps)
173 lemma unit_imp_dvd [dest]:
174   "is_unit y \<Longrightarrow> y dvd x"
175   by (rule dvd_trans [of _ 1]) (simp_all add: is_unit_def)
177 lemma dvd_unit_imp_unit:
178   "is_unit y \<Longrightarrow> x dvd y \<Longrightarrow> is_unit x"
179   by (unfold is_unit_def) (rule dvd_trans)
181 lemma ring_inv_0 [simp]:
182   "ring_inv 0 = 0"
183   unfolding ring_inv_def by simp
185 lemma unit_ring_inv'1:
186   assumes "is_unit y"
187   shows "x div (y * z) = x * ring_inv y div z"
188 proof -
189   from assms have "x div (y * z) = x * (ring_inv y * y) div (y * z)"
190     by simp
191   also have "... = y * (x * ring_inv y) div (y * z)"
192     by (simp only: mult_ac)
193   also have "... = x * ring_inv y div z"
194     by (cases "y = 0", simp, rule div_mult_mult1)
195   finally show ?thesis .
196 qed
198 lemma associated_comm:
199   "associated x y \<Longrightarrow> associated y x"
200   by (simp add: associated_def)
202 lemma associated_0 [simp]:
203   "associated 0 b \<longleftrightarrow> b = 0"
204   "associated a 0 \<longleftrightarrow> a = 0"
205   unfolding associated_def by simp_all
207 lemma associated_unit:
208   "is_unit x \<Longrightarrow> associated x y \<Longrightarrow> is_unit y"
209   unfolding associated_def by (fast dest: dvd_unit_imp_unit)
211 lemma is_unit_1 [simp]:
212   "is_unit 1"
213   unfolding is_unit_def by simp
215 lemma not_is_unit_0 [simp]:
216   "\<not> is_unit 0"
217   unfolding is_unit_def by auto
219 lemma unit_mult_left_cancel:
220   assumes "is_unit x"
221   shows "(x * y) = (x * z) \<longleftrightarrow> y = z"
222 proof -
223   from assms have "x \<noteq> 0" by auto
224   then show ?thesis by (metis div_mult_self1_is_id)
225 qed
227 lemma unit_mult_right_cancel:
228   "is_unit x \<Longrightarrow> (y * x) = (z * x) \<longleftrightarrow> y = z"
229   by (simp add: ac_simps unit_mult_left_cancel)
231 lemma unit_div_cancel:
232   "is_unit x \<Longrightarrow> (y div x) = (z div x) \<longleftrightarrow> y = z"
233   apply (subst unit_ring_inv[of _ y], assumption)
234   apply (subst unit_ring_inv[of _ z], assumption)
235   apply (rule unit_mult_right_cancel, erule unit_ring_inv_unit)
236   done
238 lemma unit_eq_div1:
239   "is_unit y \<Longrightarrow> x div y = z \<longleftrightarrow> x = z * y"
240   apply (subst unit_ring_inv, assumption)
241   apply (subst unit_mult_right_cancel[symmetric], assumption)
242   apply (subst mult_assoc, subst ring_inv_is_inv2, assumption, simp)
243   done
245 lemma unit_eq_div2:
246   "is_unit y \<Longrightarrow> x = z div y \<longleftrightarrow> x * y = z"
247   by (subst (1 2) eq_commute, simp add: unit_eq_div1, subst eq_commute, rule refl)
249 lemma associated_iff_div_unit:
250   "associated x y \<longleftrightarrow> (\<exists>z. is_unit z \<and> x = z * y)"
251 proof
252   assume "associated x y"
253   show "\<exists>z. is_unit z \<and> x = z * y"
254   proof (cases "x = 0")
255     assume "x = 0"
256     then show "\<exists>z. is_unit z \<and> x = z * y" using associated x y
257         by (intro exI[of _ 1], simp add: associated_def)
258   next
259     assume [simp]: "x \<noteq> 0"
260     hence [simp]: "x dvd y" "y dvd x" using associated x y
261         unfolding associated_def by simp_all
262     hence "1 = x div y * (y div x)"
263       by (simp add: div_mult_swap dvd_div_mult_self)
264     hence "is_unit (x div y)" unfolding is_unit_def by (rule dvdI)
265     moreover have "x = (x div y) * y" by (simp add: dvd_div_mult_self)
266     ultimately show ?thesis by blast
267   qed
268 next
269   assume "\<exists>z. is_unit z \<and> x = z * y"
270   then obtain z where "is_unit z" and "x = z * y" by blast
271   hence "y = x * ring_inv z" by (simp add: algebra_simps)
272   hence "x dvd y" by simp
273   moreover from x = z * y have "y dvd x" by simp
274   ultimately show "associated x y" unfolding associated_def by simp
275 qed
277 lemmas unit_simps = mult_unit_dvd_iff div_unit_dvd_iff dvd_mult_unit_iff
278   dvd_div_unit_iff unit_div_mult_swap unit_div_commute
279   unit_mult_left_cancel unit_mult_right_cancel unit_div_cancel
280   unit_eq_div1 unit_eq_div2
282 end
284 context ring_div
285 begin
287 lemma is_unit_neg [simp]:
288   "is_unit (- x) \<Longrightarrow> is_unit x"
289   unfolding is_unit_def by simp
291 lemma is_unit_neg_1 [simp]:
292   "is_unit (-1)"
293   unfolding is_unit_def by simp
295 end
297 lemma is_unit_nat [simp]:
298   "is_unit (x::nat) \<longleftrightarrow> x = 1"
299   unfolding is_unit_def by simp
301 lemma is_unit_int:
302   "is_unit (x::int) \<longleftrightarrow> x = 1 \<or> x = -1"
303   unfolding is_unit_def by auto
305 text {*
306   A Euclidean semiring is a semiring upon which the Euclidean algorithm can be
307   implemented. It must provide:
308   \begin{itemize}
309   \item division with remainder
310   \item a size function such that @{term "size (a mod b) < size b"}
311         for any @{term "b \<noteq> 0"}
312   \item a normalisation factor such that two associated numbers are equal iff
313         they are the same when divided by their normalisation factors.
314   \end{itemize}
315   The existence of these functions makes it possible to derive gcd and lcm functions
316   for any Euclidean semiring.
317 *}
318 class euclidean_semiring = semiring_div +
319   fixes euclidean_size :: "'a \<Rightarrow> nat"
320   fixes normalisation_factor :: "'a \<Rightarrow> 'a"
321   assumes mod_size_less [simp]:
322     "b \<noteq> 0 \<Longrightarrow> euclidean_size (a mod b) < euclidean_size b"
323   assumes size_mult_mono:
324     "b \<noteq> 0 \<Longrightarrow> euclidean_size (a * b) \<ge> euclidean_size a"
325   assumes normalisation_factor_is_unit [intro,simp]:
326     "a \<noteq> 0 \<Longrightarrow> is_unit (normalisation_factor a)"
327   assumes normalisation_factor_mult: "normalisation_factor (a * b) =
328     normalisation_factor a * normalisation_factor b"
329   assumes normalisation_factor_unit: "is_unit x \<Longrightarrow> normalisation_factor x = x"
330   assumes normalisation_factor_0 [simp]: "normalisation_factor 0 = 0"
331 begin
333 lemma normalisation_factor_dvd [simp]:
334   "a \<noteq> 0 \<Longrightarrow> normalisation_factor a dvd b"
335   by (rule unit_imp_dvd, simp)
337 lemma normalisation_factor_1 [simp]:
338   "normalisation_factor 1 = 1"
339   by (simp add: normalisation_factor_unit)
341 lemma normalisation_factor_0_iff [simp]:
342   "normalisation_factor x = 0 \<longleftrightarrow> x = 0"
343 proof
344   assume "normalisation_factor x = 0"
345   hence "\<not> is_unit (normalisation_factor x)"
346     by (metis not_is_unit_0)
347   then show "x = 0" by force
348 next
349   assume "x = 0"
350   then show "normalisation_factor x = 0" by simp
351 qed
353 lemma normalisation_factor_pow:
354   "normalisation_factor (x ^ n) = normalisation_factor x ^ n"
355   by (induct n) (simp_all add: normalisation_factor_mult power_Suc2)
357 lemma normalisation_correct [simp]:
358   "normalisation_factor (x div normalisation_factor x) = (if x = 0 then 0 else 1)"
359 proof (cases "x = 0", simp)
360   assume "x \<noteq> 0"
361   let ?nf = "normalisation_factor"
362   from normalisation_factor_is_unit[OF x \<noteq> 0] have "?nf x \<noteq> 0"
363     by (metis not_is_unit_0)
364   have "?nf (x div ?nf x) * ?nf (?nf x) = ?nf (x div ?nf x * ?nf x)"
365     by (simp add: normalisation_factor_mult)
366   also have "x div ?nf x * ?nf x = x" using x \<noteq> 0
367     by (simp add: dvd_div_mult_self)
368   also have "?nf (?nf x) = ?nf x" using x \<noteq> 0
369     normalisation_factor_is_unit normalisation_factor_unit by simp
370   finally show ?thesis using x \<noteq> 0 and ?nf x \<noteq> 0
371     by (metis div_mult_self2_is_id div_self)
372 qed
374 lemma normalisation_0_iff [simp]:
375   "x div normalisation_factor x = 0 \<longleftrightarrow> x = 0"
376   by (cases "x = 0", simp, subst unit_eq_div1, blast, simp)
378 lemma associated_iff_normed_eq:
379   "associated a b \<longleftrightarrow> a div normalisation_factor a = b div normalisation_factor b"
380 proof (cases "b = 0", simp, cases "a = 0", metis associated_0(1) normalisation_0_iff, rule iffI)
381   let ?nf = normalisation_factor
382   assume "a \<noteq> 0" "b \<noteq> 0" "a div ?nf a = b div ?nf b"
383   hence "a = b * (?nf a div ?nf b)"
384     apply (subst (asm) unit_eq_div1, blast, subst (asm) unit_div_commute, blast)
385     apply (subst div_mult_swap, simp, simp)
386     done
387   with a \<noteq> 0 b \<noteq> 0 have "\<exists>z. is_unit z \<and> a = z * b"
388     by (intro exI[of _ "?nf a div ?nf b"], force simp: mult_ac)
389   with associated_iff_div_unit show "associated a b" by simp
390 next
391   let ?nf = normalisation_factor
392   assume "a \<noteq> 0" "b \<noteq> 0" "associated a b"
393   with associated_iff_div_unit obtain z where "is_unit z" and "a = z * b" by blast
394   then show "a div ?nf a = b div ?nf b"
395     apply (simp only: a = z * b normalisation_factor_mult normalisation_factor_unit)
396     apply (rule div_mult_mult1, force)
397     done
398   qed
400 lemma normed_associated_imp_eq:
401   "associated a b \<Longrightarrow> normalisation_factor a \<in> {0, 1} \<Longrightarrow> normalisation_factor b \<in> {0, 1} \<Longrightarrow> a = b"
402   by (simp add: associated_iff_normed_eq, elim disjE, simp_all)
404 lemmas normalisation_factor_dvd_iff [simp] =
405   unit_dvd_iff [OF normalisation_factor_is_unit]
407 lemma euclidean_division:
408   fixes a :: 'a and b :: 'a
409   assumes "b \<noteq> 0"
410   obtains s and t where "a = s * b + t"
411     and "euclidean_size t < euclidean_size b"
412 proof -
413   from div_mod_equality[of a b 0]
414      have "a = a div b * b + a mod b" by simp
415   with that and assms show ?thesis by force
416 qed
418 lemma dvd_euclidean_size_eq_imp_dvd:
419   assumes "a \<noteq> 0" and b_dvd_a: "b dvd a" and size_eq: "euclidean_size a = euclidean_size b"
420   shows "a dvd b"
421 proof (subst dvd_eq_mod_eq_0, rule ccontr)
422   assume "b mod a \<noteq> 0"
423   from b_dvd_a have b_dvd_mod: "b dvd b mod a" by (simp add: dvd_mod_iff)
424   from b_dvd_mod obtain c where "b mod a = b * c" unfolding dvd_def by blast
425     with b mod a \<noteq> 0 have "c \<noteq> 0" by auto
426   with b mod a = b * c have "euclidean_size (b mod a) \<ge> euclidean_size b"
427       using size_mult_mono by force
428   moreover from a \<noteq> 0 have "euclidean_size (b mod a) < euclidean_size a"
429       using mod_size_less by blast
430   ultimately show False using size_eq by simp
431 qed
433 function gcd_eucl :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
434 where
435   "gcd_eucl a b = (if b = 0 then a div normalisation_factor a else gcd_eucl b (a mod b))"
436   by (pat_completeness, simp)
437 termination by (relation "measure (euclidean_size \<circ> snd)", simp_all)
439 declare gcd_eucl.simps [simp del]
441 lemma gcd_induct: "\<lbrakk>\<And>b. P b 0; \<And>a b. 0 \<noteq> b \<Longrightarrow> P b (a mod b) \<Longrightarrow> P a b\<rbrakk> \<Longrightarrow> P a b"
442 proof (induct a b rule: gcd_eucl.induct)
443   case ("1" m n)
444     then show ?case by (cases "n = 0") auto
445 qed
447 definition lcm_eucl :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
448 where
449   "lcm_eucl a b = a * b div (gcd_eucl a b * normalisation_factor (a * b))"
451   (* Somewhat complicated definition of Lcm that has the advantage of working
452      for infinite sets as well *)
454 definition Lcm_eucl :: "'a set \<Rightarrow> 'a"
455 where
456   "Lcm_eucl A = (if \<exists>l. l \<noteq> 0 \<and> (\<forall>x\<in>A. x dvd l) then
457      let l = SOME l. l \<noteq> 0 \<and> (\<forall>x\<in>A. x dvd l) \<and> euclidean_size l =
458        (LEAST n. \<exists>l. l \<noteq> 0 \<and> (\<forall>x\<in>A. x dvd l) \<and> euclidean_size l = n)
459        in l div normalisation_factor l
460       else 0)"
462 definition Gcd_eucl :: "'a set \<Rightarrow> 'a"
463 where
464   "Gcd_eucl A = Lcm_eucl {d. \<forall>a\<in>A. d dvd a}"
466 end
468 class euclidean_semiring_gcd = euclidean_semiring + gcd + Gcd +
469   assumes gcd_gcd_eucl: "gcd = gcd_eucl" and lcm_lcm_eucl: "lcm = lcm_eucl"
470   assumes Gcd_Gcd_eucl: "Gcd = Gcd_eucl" and Lcm_Lcm_eucl: "Lcm = Lcm_eucl"
471 begin
473 lemma gcd_red:
474   "gcd x y = gcd y (x mod y)"
475   by (metis gcd_eucl.simps mod_0 mod_by_0 gcd_gcd_eucl)
477 lemma gcd_non_0:
478   "y \<noteq> 0 \<Longrightarrow> gcd x y = gcd y (x mod y)"
479   by (rule gcd_red)
481 lemma gcd_0_left:
482   "gcd 0 x = x div normalisation_factor x"
483    by (simp only: gcd_gcd_eucl, subst gcd_eucl.simps, subst gcd_eucl.simps, simp add: Let_def)
485 lemma gcd_0:
486   "gcd x 0 = x div normalisation_factor x"
487   by (simp only: gcd_gcd_eucl, subst gcd_eucl.simps, simp add: Let_def)
489 lemma gcd_dvd1 [iff]: "gcd x y dvd x"
490   and gcd_dvd2 [iff]: "gcd x y dvd y"
491 proof (induct x y rule: gcd_eucl.induct)
492   fix x y :: 'a
493   assume IH1: "y \<noteq> 0 \<Longrightarrow> gcd y (x mod y) dvd y"
494   assume IH2: "y \<noteq> 0 \<Longrightarrow> gcd y (x mod y) dvd (x mod y)"
496   have "gcd x y dvd x \<and> gcd x y dvd y"
497   proof (cases "y = 0")
498     case True
499       then show ?thesis by (cases "x = 0", simp_all add: gcd_0)
500   next
501     case False
502       with IH1 and IH2 show ?thesis by (simp add: gcd_non_0 dvd_mod_iff)
503   qed
504   then show "gcd x y dvd x" "gcd x y dvd y" by simp_all
505 qed
507 lemma dvd_gcd_D1: "k dvd gcd m n \<Longrightarrow> k dvd m"
508   by (rule dvd_trans, assumption, rule gcd_dvd1)
510 lemma dvd_gcd_D2: "k dvd gcd m n \<Longrightarrow> k dvd n"
511   by (rule dvd_trans, assumption, rule gcd_dvd2)
513 lemma gcd_greatest:
514   fixes k x y :: 'a
515   shows "k dvd x \<Longrightarrow> k dvd y \<Longrightarrow> k dvd gcd x y"
516 proof (induct x y rule: gcd_eucl.induct)
517   case (1 x y)
518   show ?case
519     proof (cases "y = 0")
520       assume "y = 0"
521       with 1 show ?thesis by (cases "x = 0", simp_all add: gcd_0)
522     next
523       assume "y \<noteq> 0"
524       with 1 show ?thesis by (simp add: gcd_non_0 dvd_mod_iff)
525     qed
526 qed
528 lemma dvd_gcd_iff:
529   "k dvd gcd x y \<longleftrightarrow> k dvd x \<and> k dvd y"
530   by (blast intro!: gcd_greatest intro: dvd_trans)
532 lemmas gcd_greatest_iff = dvd_gcd_iff
534 lemma gcd_zero [simp]:
535   "gcd x y = 0 \<longleftrightarrow> x = 0 \<and> y = 0"
536   by (metis dvd_0_left dvd_refl gcd_dvd1 gcd_dvd2 gcd_greatest)+
538 lemma normalisation_factor_gcd [simp]:
539   "normalisation_factor (gcd x y) = (if x = 0 \<and> y = 0 then 0 else 1)" (is "?f x y = ?g x y")
540 proof (induct x y rule: gcd_eucl.induct)
541   fix x y :: 'a
542   assume IH: "y \<noteq> 0 \<Longrightarrow> ?f y (x mod y) = ?g y (x mod y)"
543   then show "?f x y = ?g x y" by (cases "y = 0", auto simp: gcd_non_0 gcd_0)
544 qed
546 lemma gcdI:
547   "k dvd x \<Longrightarrow> k dvd y \<Longrightarrow> (\<And>l. l dvd x \<Longrightarrow> l dvd y \<Longrightarrow> l dvd k)
548     \<Longrightarrow> normalisation_factor k = (if k = 0 then 0 else 1) \<Longrightarrow> k = gcd x y"
549   by (intro normed_associated_imp_eq) (auto simp: associated_def intro: gcd_greatest)
551 sublocale gcd!: abel_semigroup gcd
552 proof
553   fix x y z
554   show "gcd (gcd x y) z = gcd x (gcd y z)"
555   proof (rule gcdI)
556     have "gcd (gcd x y) z dvd gcd x y" "gcd x y dvd x" by simp_all
557     then show "gcd (gcd x y) z dvd x" by (rule dvd_trans)
558     have "gcd (gcd x y) z dvd gcd x y" "gcd x y dvd y" by simp_all
559     hence "gcd (gcd x y) z dvd y" by (rule dvd_trans)
560     moreover have "gcd (gcd x y) z dvd z" by simp
561     ultimately show "gcd (gcd x y) z dvd gcd y z"
562       by (rule gcd_greatest)
563     show "normalisation_factor (gcd (gcd x y) z) =  (if gcd (gcd x y) z = 0 then 0 else 1)"
564       by auto
565     fix l assume "l dvd x" and "l dvd gcd y z"
566     with dvd_trans[OF _ gcd_dvd1] and dvd_trans[OF _ gcd_dvd2]
567       have "l dvd y" and "l dvd z" by blast+
568     with l dvd x show "l dvd gcd (gcd x y) z"
569       by (intro gcd_greatest)
570   qed
571 next
572   fix x y
573   show "gcd x y = gcd y x"
574     by (rule gcdI) (simp_all add: gcd_greatest)
575 qed
577 lemma gcd_unique: "d dvd a \<and> d dvd b \<and>
578     normalisation_factor d = (if d = 0 then 0 else 1) \<and>
579     (\<forall>e. e dvd a \<and> e dvd b \<longrightarrow> e dvd d) \<longleftrightarrow> d = gcd a b"
580   by (rule, auto intro: gcdI simp: gcd_greatest)
582 lemma gcd_dvd_prod: "gcd a b dvd k * b"
583   using mult_dvd_mono [of 1] by auto
585 lemma gcd_1_left [simp]: "gcd 1 x = 1"
586   by (rule sym, rule gcdI, simp_all)
588 lemma gcd_1 [simp]: "gcd x 1 = 1"
589   by (rule sym, rule gcdI, simp_all)
591 lemma gcd_proj2_if_dvd:
592   "y dvd x \<Longrightarrow> gcd x y = y div normalisation_factor y"
593   by (cases "y = 0", simp_all add: dvd_eq_mod_eq_0 gcd_non_0 gcd_0)
595 lemma gcd_proj1_if_dvd:
596   "x dvd y \<Longrightarrow> gcd x y = x div normalisation_factor x"
597   by (subst gcd.commute, simp add: gcd_proj2_if_dvd)
599 lemma gcd_proj1_iff: "gcd m n = m div normalisation_factor m \<longleftrightarrow> m dvd n"
600 proof
601   assume A: "gcd m n = m div normalisation_factor m"
602   show "m dvd n"
603   proof (cases "m = 0")
604     assume [simp]: "m \<noteq> 0"
605     from A have B: "m = gcd m n * normalisation_factor m"
606       by (simp add: unit_eq_div2)
607     show ?thesis by (subst B, simp add: mult_unit_dvd_iff)
608   qed (insert A, simp)
609 next
610   assume "m dvd n"
611   then show "gcd m n = m div normalisation_factor m" by (rule gcd_proj1_if_dvd)
612 qed
614 lemma gcd_proj2_iff: "gcd m n = n div normalisation_factor n \<longleftrightarrow> n dvd m"
615   by (subst gcd.commute, simp add: gcd_proj1_iff)
617 lemma gcd_mod1 [simp]:
618   "gcd (x mod y) y = gcd x y"
619   by (rule gcdI, metis dvd_mod_iff gcd_dvd1 gcd_dvd2, simp_all add: gcd_greatest dvd_mod_iff)
621 lemma gcd_mod2 [simp]:
622   "gcd x (y mod x) = gcd x y"
623   by (rule gcdI, simp, metis dvd_mod_iff gcd_dvd1 gcd_dvd2, simp_all add: gcd_greatest dvd_mod_iff)
625 lemma normalisation_factor_dvd' [simp]:
626   "normalisation_factor x dvd x"
627   by (cases "x = 0", simp_all)
629 lemma gcd_mult_distrib':
630   "k div normalisation_factor k * gcd x y = gcd (k*x) (k*y)"
631 proof (induct x y rule: gcd_eucl.induct)
632   case (1 x y)
633   show ?case
634   proof (cases "y = 0")
635     case True
636     then show ?thesis by (simp add: normalisation_factor_mult gcd_0 algebra_simps div_mult_div_if_dvd)
637   next
638     case False
639     hence "k div normalisation_factor k * gcd x y =  gcd (k * y) (k * (x mod y))"
640       using 1 by (subst gcd_red, simp)
641     also have "... = gcd (k * x) (k * y)"
642       by (simp add: mult_mod_right gcd.commute)
643     finally show ?thesis .
644   qed
645 qed
647 lemma gcd_mult_distrib:
648   "k * gcd x y = gcd (k*x) (k*y) * normalisation_factor k"
649 proof-
650   let ?nf = "normalisation_factor"
651   from gcd_mult_distrib'
652     have "gcd (k*x) (k*y) = k div ?nf k * gcd x y" ..
653   also have "... = k * gcd x y div ?nf k"
654     by (metis dvd_div_mult dvd_eq_mod_eq_0 mod_0 normalisation_factor_dvd)
655   finally show ?thesis
656     by (simp add: ac_simps dvd_mult_div_cancel)
657 qed
659 lemma euclidean_size_gcd_le1 [simp]:
660   assumes "a \<noteq> 0"
661   shows "euclidean_size (gcd a b) \<le> euclidean_size a"
662 proof -
663    have "gcd a b dvd a" by (rule gcd_dvd1)
664    then obtain c where A: "a = gcd a b * c" unfolding dvd_def by blast
665    with a \<noteq> 0 show ?thesis by (subst (2) A, intro size_mult_mono) auto
666 qed
668 lemma euclidean_size_gcd_le2 [simp]:
669   "b \<noteq> 0 \<Longrightarrow> euclidean_size (gcd a b) \<le> euclidean_size b"
670   by (subst gcd.commute, rule euclidean_size_gcd_le1)
672 lemma euclidean_size_gcd_less1:
673   assumes "a \<noteq> 0" and "\<not>a dvd b"
674   shows "euclidean_size (gcd a b) < euclidean_size a"
675 proof (rule ccontr)
676   assume "\<not>euclidean_size (gcd a b) < euclidean_size a"
677   with a \<noteq> 0 have "euclidean_size (gcd a b) = euclidean_size a"
678     by (intro le_antisym, simp_all)
679   with assms have "a dvd gcd a b" by (auto intro: dvd_euclidean_size_eq_imp_dvd)
680   hence "a dvd b" using dvd_gcd_D2 by blast
681   with \<not>a dvd b show False by contradiction
682 qed
684 lemma euclidean_size_gcd_less2:
685   assumes "b \<noteq> 0" and "\<not>b dvd a"
686   shows "euclidean_size (gcd a b) < euclidean_size b"
687   using assms by (subst gcd.commute, rule euclidean_size_gcd_less1)
689 lemma gcd_mult_unit1: "is_unit a \<Longrightarrow> gcd (x*a) y = gcd x y"
690   apply (rule gcdI)
691   apply (rule dvd_trans, rule gcd_dvd1, simp add: unit_simps)
692   apply (rule gcd_dvd2)
693   apply (rule gcd_greatest, simp add: unit_simps, assumption)
694   apply (subst normalisation_factor_gcd, simp add: gcd_0)
695   done
697 lemma gcd_mult_unit2: "is_unit a \<Longrightarrow> gcd x (y*a) = gcd x y"
698   by (subst gcd.commute, subst gcd_mult_unit1, assumption, rule gcd.commute)
700 lemma gcd_div_unit1: "is_unit a \<Longrightarrow> gcd (x div a) y = gcd x y"
701   by (simp add: unit_ring_inv gcd_mult_unit1)
703 lemma gcd_div_unit2: "is_unit a \<Longrightarrow> gcd x (y div a) = gcd x y"
704   by (simp add: unit_ring_inv gcd_mult_unit2)
706 lemma gcd_idem: "gcd x x = x div normalisation_factor x"
707   by (cases "x = 0") (simp add: gcd_0_left, rule sym, rule gcdI, simp_all)
709 lemma gcd_right_idem: "gcd (gcd p q) q = gcd p q"
710   apply (rule gcdI)
711   apply (simp add: ac_simps)
712   apply (rule gcd_dvd2)
713   apply (rule gcd_greatest, erule (1) gcd_greatest, assumption)
714   apply (simp add: gcd_zero)
715   done
717 lemma gcd_left_idem: "gcd p (gcd p q) = gcd p q"
718   apply (rule gcdI)
719   apply simp
720   apply (rule dvd_trans, rule gcd_dvd2, rule gcd_dvd2)
721   apply (rule gcd_greatest, assumption, erule gcd_greatest, assumption)
722   apply (simp add: gcd_zero)
723   done
725 lemma comp_fun_idem_gcd: "comp_fun_idem gcd"
726 proof
727   fix a b show "gcd a \<circ> gcd b = gcd b \<circ> gcd a"
728     by (simp add: fun_eq_iff ac_simps)
729 next
730   fix a show "gcd a \<circ> gcd a = gcd a"
731     by (simp add: fun_eq_iff gcd_left_idem)
732 qed
734 lemma coprime_dvd_mult:
735   assumes "gcd k n = 1" and "k dvd m * n"
736   shows "k dvd m"
737 proof -
738   let ?nf = "normalisation_factor"
739   from assms gcd_mult_distrib [of m k n]
740     have A: "m = gcd (m * k) (m * n) * ?nf m" by simp
741   from k dvd m * n show ?thesis by (subst A, simp_all add: gcd_greatest)
742 qed
744 lemma coprime_dvd_mult_iff:
745   "gcd k n = 1 \<Longrightarrow> (k dvd m * n) = (k dvd m)"
746   by (rule, rule coprime_dvd_mult, simp_all)
748 lemma gcd_dvd_antisym:
749   "gcd a b dvd gcd c d \<Longrightarrow> gcd c d dvd gcd a b \<Longrightarrow> gcd a b = gcd c d"
750 proof (rule gcdI)
751   assume A: "gcd a b dvd gcd c d" and B: "gcd c d dvd gcd a b"
752   have "gcd c d dvd c" by simp
753   with A show "gcd a b dvd c" by (rule dvd_trans)
754   have "gcd c d dvd d" by simp
755   with A show "gcd a b dvd d" by (rule dvd_trans)
756   show "normalisation_factor (gcd a b) = (if gcd a b = 0 then 0 else 1)"
757     by (simp add: gcd_zero)
758   fix l assume "l dvd c" and "l dvd d"
759   hence "l dvd gcd c d" by (rule gcd_greatest)
760   from this and B show "l dvd gcd a b" by (rule dvd_trans)
761 qed
763 lemma gcd_mult_cancel:
764   assumes "gcd k n = 1"
765   shows "gcd (k * m) n = gcd m n"
766 proof (rule gcd_dvd_antisym)
767   have "gcd (gcd (k * m) n) k = gcd (gcd k n) (k * m)" by (simp add: ac_simps)
768   also note gcd k n = 1
769   finally have "gcd (gcd (k * m) n) k = 1" by simp
770   hence "gcd (k * m) n dvd m" by (rule coprime_dvd_mult, simp add: ac_simps)
771   moreover have "gcd (k * m) n dvd n" by simp
772   ultimately show "gcd (k * m) n dvd gcd m n" by (rule gcd_greatest)
773   have "gcd m n dvd (k * m)" and "gcd m n dvd n" by simp_all
774   then show "gcd m n dvd gcd (k * m) n" by (rule gcd_greatest)
775 qed
777 lemma coprime_crossproduct:
778   assumes [simp]: "gcd a d = 1" "gcd b c = 1"
779   shows "associated (a * c) (b * d) \<longleftrightarrow> associated a b \<and> associated c d" (is "?lhs \<longleftrightarrow> ?rhs")
780 proof
781   assume ?rhs then show ?lhs unfolding associated_def by (fast intro: mult_dvd_mono)
782 next
783   assume ?lhs
784   from ?lhs have "a dvd b * d" unfolding associated_def by (metis dvd_mult_left)
785   hence "a dvd b" by (simp add: coprime_dvd_mult_iff)
786   moreover from ?lhs have "b dvd a * c" unfolding associated_def by (metis dvd_mult_left)
787   hence "b dvd a" by (simp add: coprime_dvd_mult_iff)
788   moreover from ?lhs have "c dvd d * b"
789     unfolding associated_def by (metis dvd_mult_right ac_simps)
790   hence "c dvd d" by (simp add: coprime_dvd_mult_iff gcd.commute)
791   moreover from ?lhs have "d dvd c * a"
792     unfolding associated_def by (metis dvd_mult_right ac_simps)
793   hence "d dvd c" by (simp add: coprime_dvd_mult_iff gcd.commute)
794   ultimately show ?rhs unfolding associated_def by simp
795 qed
797 lemma gcd_add1 [simp]:
798   "gcd (m + n) n = gcd m n"
799   by (cases "n = 0", simp_all add: gcd_non_0)
801 lemma gcd_add2 [simp]:
802   "gcd m (m + n) = gcd m n"
803   using gcd_add1 [of n m] by (simp add: ac_simps)
805 lemma gcd_add_mult: "gcd m (k * m + n) = gcd m n"
806   by (subst gcd.commute, subst gcd_red, simp)
808 lemma coprimeI: "(\<And>l. \<lbrakk>l dvd x; l dvd y\<rbrakk> \<Longrightarrow> l dvd 1) \<Longrightarrow> gcd x y = 1"
809   by (rule sym, rule gcdI, simp_all)
811 lemma coprime: "gcd a b = 1 \<longleftrightarrow> (\<forall>d. d dvd a \<and> d dvd b \<longleftrightarrow> is_unit d)"
812   by (auto simp: is_unit_def intro: coprimeI gcd_greatest dvd_gcd_D1 dvd_gcd_D2)
814 lemma div_gcd_coprime:
815   assumes nz: "a \<noteq> 0 \<or> b \<noteq> 0"
816   defines [simp]: "d \<equiv> gcd a b"
817   defines [simp]: "a' \<equiv> a div d" and [simp]: "b' \<equiv> b div d"
818   shows "gcd a' b' = 1"
819 proof (rule coprimeI)
820   fix l assume "l dvd a'" "l dvd b'"
821   then obtain s t where "a' = l * s" "b' = l * t" unfolding dvd_def by blast
822   moreover have "a = a' * d" "b = b' * d" by (simp_all add: dvd_div_mult_self)
823   ultimately have "a = (l * d) * s" "b = (l * d) * t"
824     by (metis ac_simps)+
825   hence "l*d dvd a" and "l*d dvd b" by (simp_all only: dvd_triv_left)
826   hence "l*d dvd d" by (simp add: gcd_greatest)
827   then obtain u where "u * l * d = d" unfolding dvd_def
828     by (metis ac_simps mult_assoc)
829   moreover from nz have "d \<noteq> 0" by (simp add: gcd_zero)
830   ultimately have "u * l = 1"
831     by (metis div_mult_self1_is_id div_self ac_simps)
832   then show "l dvd 1" by force
833 qed
835 lemma coprime_mult:
836   assumes da: "gcd d a = 1" and db: "gcd d b = 1"
837   shows "gcd d (a * b) = 1"
838   apply (subst gcd.commute)
839   using da apply (subst gcd_mult_cancel)
840   apply (subst gcd.commute, assumption)
841   apply (subst gcd.commute, rule db)
842   done
844 lemma coprime_lmult:
845   assumes dab: "gcd d (a * b) = 1"
846   shows "gcd d a = 1"
847 proof (rule coprimeI)
848   fix l assume "l dvd d" and "l dvd a"
849   hence "l dvd a * b" by simp
850   with l dvd d and dab show "l dvd 1" by (auto intro: gcd_greatest)
851 qed
853 lemma coprime_rmult:
854   assumes dab: "gcd d (a * b) = 1"
855   shows "gcd d b = 1"
856 proof (rule coprimeI)
857   fix l assume "l dvd d" and "l dvd b"
858   hence "l dvd a * b" by simp
859   with l dvd d and dab show "l dvd 1" by (auto intro: gcd_greatest)
860 qed
862 lemma coprime_mul_eq: "gcd d (a * b) = 1 \<longleftrightarrow> gcd d a = 1 \<and> gcd d b = 1"
863   using coprime_rmult[of d a b] coprime_lmult[of d a b] coprime_mult[of d a b] by blast
865 lemma gcd_coprime:
866   assumes z: "gcd a b \<noteq> 0" and a: "a = a' * gcd a b" and b: "b = b' * gcd a b"
867   shows "gcd a' b' = 1"
868 proof -
869   from z have "a \<noteq> 0 \<or> b \<noteq> 0" by (simp add: gcd_zero)
870   with div_gcd_coprime have "gcd (a div gcd a b) (b div gcd a b) = 1" .
871   also from assms have "a div gcd a b = a'" by (metis div_mult_self2_is_id)+
872   also from assms have "b div gcd a b = b'" by (metis div_mult_self2_is_id)+
873   finally show ?thesis .
874 qed
876 lemma coprime_power:
877   assumes "0 < n"
878   shows "gcd a (b ^ n) = 1 \<longleftrightarrow> gcd a b = 1"
879 using assms proof (induct n)
880   case (Suc n) then show ?case
881     by (cases n) (simp_all add: coprime_mul_eq)
882 qed simp
884 lemma gcd_coprime_exists:
885   assumes nz: "gcd a b \<noteq> 0"
886   shows "\<exists>a' b'. a = a' * gcd a b \<and> b = b' * gcd a b \<and> gcd a' b' = 1"
887   apply (rule_tac x = "a div gcd a b" in exI)
888   apply (rule_tac x = "b div gcd a b" in exI)
889   apply (insert nz, auto simp add: dvd_div_mult gcd_0_left  gcd_zero intro: div_gcd_coprime)
890   done
892 lemma coprime_exp:
893   "gcd d a = 1 \<Longrightarrow> gcd d (a^n) = 1"
894   by (induct n, simp_all add: coprime_mult)
896 lemma coprime_exp2 [intro]:
897   "gcd a b = 1 \<Longrightarrow> gcd (a^n) (b^m) = 1"
898   apply (rule coprime_exp)
899   apply (subst gcd.commute)
900   apply (rule coprime_exp)
901   apply (subst gcd.commute)
902   apply assumption
903   done
905 lemma gcd_exp:
906   "gcd (a^n) (b^n) = (gcd a b) ^ n"
907 proof (cases "a = 0 \<and> b = 0")
908   assume "a = 0 \<and> b = 0"
909   then show ?thesis by (cases n, simp_all add: gcd_0_left)
910 next
911   assume A: "\<not>(a = 0 \<and> b = 0)"
912   hence "1 = gcd ((a div gcd a b)^n) ((b div gcd a b)^n)"
913     using div_gcd_coprime by (subst sym, auto simp: div_gcd_coprime)
914   hence "(gcd a b) ^ n = (gcd a b) ^ n * ..." by simp
915   also note gcd_mult_distrib
916   also have "normalisation_factor ((gcd a b)^n) = 1"
917     by (simp add: normalisation_factor_pow A)
918   also have "(gcd a b)^n * (a div gcd a b)^n = a^n"
919     by (subst ac_simps, subst div_power, simp, rule dvd_div_mult_self, rule dvd_power_same, simp)
920   also have "(gcd a b)^n * (b div gcd a b)^n = b^n"
921     by (subst ac_simps, subst div_power, simp, rule dvd_div_mult_self, rule dvd_power_same, simp)
922   finally show ?thesis by simp
923 qed
925 lemma coprime_common_divisor:
926   "gcd a b = 1 \<Longrightarrow> x dvd a \<Longrightarrow> x dvd b \<Longrightarrow> is_unit x"
927   apply (subgoal_tac "x dvd gcd a b")
928   apply (simp add: is_unit_def)
929   apply (erule (1) gcd_greatest)
930   done
932 lemma division_decomp:
933   assumes dc: "a dvd b * c"
934   shows "\<exists>b' c'. a = b' * c' \<and> b' dvd b \<and> c' dvd c"
935 proof (cases "gcd a b = 0")
936   assume "gcd a b = 0"
937   hence "a = 0 \<and> b = 0" by (simp add: gcd_zero)
938   hence "a = 0 * c \<and> 0 dvd b \<and> c dvd c" by simp
939   then show ?thesis by blast
940 next
941   let ?d = "gcd a b"
942   assume "?d \<noteq> 0"
943   from gcd_coprime_exists[OF this]
944     obtain a' b' where ab': "a = a' * ?d" "b = b' * ?d" "gcd a' b' = 1"
945     by blast
946   from ab'(1) have "a' dvd a" unfolding dvd_def by blast
947   with dc have "a' dvd b*c" using dvd_trans[of a' a "b*c"] by simp
948   from dc ab'(1,2) have "a'*?d dvd (b'*?d) * c" by simp
949   hence "?d * a' dvd ?d * (b' * c)" by (simp add: mult_ac)
950   with ?d \<noteq> 0 have "a' dvd b' * c" by (rule dvd_mult_cancel_left)
951   with coprime_dvd_mult[OF ab'(3)]
952     have "a' dvd c" by (subst (asm) ac_simps, blast)
953   with ab'(1) have "a = ?d * a' \<and> ?d dvd b \<and> a' dvd c" by (simp add: mult_ac)
954   then show ?thesis by blast
955 qed
957 lemma pow_divides_pow:
958   assumes ab: "a ^ n dvd b ^ n" and n: "n \<noteq> 0"
959   shows "a dvd b"
960 proof (cases "gcd a b = 0")
961   assume "gcd a b = 0"
962   then show ?thesis by (simp add: gcd_zero)
963 next
964   let ?d = "gcd a b"
965   assume "?d \<noteq> 0"
966   from n obtain m where m: "n = Suc m" by (cases n, simp_all)
967   from ?d \<noteq> 0 have zn: "?d ^ n \<noteq> 0" by (rule nonzero_pow_nonzero)
968   from gcd_coprime_exists[OF ?d \<noteq> 0]
969     obtain a' b' where ab': "a = a' * ?d" "b = b' * ?d" "gcd a' b' = 1"
970     by blast
971   from ab have "(a' * ?d) ^ n dvd (b' * ?d) ^ n"
972     by (simp add: ab'(1,2)[symmetric])
973   hence "?d^n * a'^n dvd ?d^n * b'^n"
974     by (simp only: power_mult_distrib ac_simps)
975   with zn have "a'^n dvd b'^n" by (rule dvd_mult_cancel_left)
976   hence "a' dvd b'^n" using dvd_trans[of a' "a'^n" "b'^n"] by (simp add: m)
977   hence "a' dvd b'^m * b'" by (simp add: m ac_simps)
978   with coprime_dvd_mult[OF coprime_exp[OF ab'(3), of m]]
979     have "a' dvd b'" by (subst (asm) ac_simps, blast)
980   hence "a'*?d dvd b'*?d" by (rule mult_dvd_mono, simp)
981   with ab'(1,2) show ?thesis by simp
982 qed
984 lemma pow_divides_eq [simp]:
985   "n \<noteq> 0 \<Longrightarrow> a ^ n dvd b ^ n \<longleftrightarrow> a dvd b"
986   by (auto intro: pow_divides_pow dvd_power_same)
988 lemma divides_mult:
989   assumes mr: "m dvd r" and nr: "n dvd r" and mn: "gcd m n = 1"
990   shows "m * n dvd r"
991 proof -
992   from mr nr obtain m' n' where m': "r = m*m'" and n': "r = n*n'"
993     unfolding dvd_def by blast
994   from mr n' have "m dvd n'*n" by (simp add: ac_simps)
995   hence "m dvd n'" using coprime_dvd_mult_iff[OF mn] by simp
996   then obtain k where k: "n' = m*k" unfolding dvd_def by blast
997   with n' have "r = m * n * k" by (simp add: mult_ac)
998   then show ?thesis unfolding dvd_def by blast
999 qed
1001 lemma coprime_plus_one [simp]: "gcd (n + 1) n = 1"
1002   by (subst add_commute, simp)
1004 lemma setprod_coprime [rule_format]:
1005   "(\<forall>i\<in>A. gcd (f i) x = 1) \<longrightarrow> gcd (\<Prod>i\<in>A. f i) x = 1"
1006   apply (cases "finite A")
1007   apply (induct set: finite)
1008   apply (auto simp add: gcd_mult_cancel)
1009   done
1011 lemma coprime_divisors:
1012   assumes "d dvd a" "e dvd b" "gcd a b = 1"
1013   shows "gcd d e = 1"
1014 proof -
1015   from assms obtain k l where "a = d * k" "b = e * l"
1016     unfolding dvd_def by blast
1017   with assms have "gcd (d * k) (e * l) = 1" by simp
1018   hence "gcd (d * k) e = 1" by (rule coprime_lmult)
1019   also have "gcd (d * k) e = gcd e (d * k)" by (simp add: ac_simps)
1020   finally have "gcd e d = 1" by (rule coprime_lmult)
1021   then show ?thesis by (simp add: ac_simps)
1022 qed
1024 lemma invertible_coprime:
1025   "x * y mod m = 1 \<Longrightarrow> gcd x m = 1"
1026   by (metis coprime_lmult gcd_1 ac_simps gcd_red)
1028 lemma lcm_gcd:
1029   "lcm a b = a * b div (gcd a b * normalisation_factor (a*b))"
1030   by (simp only: lcm_lcm_eucl gcd_gcd_eucl lcm_eucl_def)
1032 lemma lcm_gcd_prod:
1033   "lcm a b * gcd a b = a * b div normalisation_factor (a*b)"
1034 proof (cases "a * b = 0")
1035   let ?nf = normalisation_factor
1036   assume "a * b \<noteq> 0"
1037   hence "gcd a b \<noteq> 0" by (auto simp add: gcd_zero)
1038   from lcm_gcd have "lcm a b * gcd a b = gcd a b * (a * b div (?nf (a*b) * gcd a b))"
1039     by (simp add: mult_ac)
1040   also from a * b \<noteq> 0 have "... = a * b div ?nf (a*b)"
1041     by (simp_all add: unit_ring_inv'1 dvd_mult_div_cancel unit_ring_inv)
1042   finally show ?thesis .
1043 qed (simp add: lcm_gcd)
1045 lemma lcm_dvd1 [iff]:
1046   "x dvd lcm x y"
1047 proof (cases "x*y = 0")
1048   assume "x * y \<noteq> 0"
1049   hence "gcd x y \<noteq> 0" by (auto simp: gcd_zero)
1050   let ?c = "ring_inv (normalisation_factor (x*y))"
1051   from x * y \<noteq> 0 have [simp]: "is_unit (normalisation_factor (x*y))" by simp
1052   from lcm_gcd_prod[of x y] have "lcm x y * gcd x y = x * ?c * y"
1053     by (simp add: mult_ac unit_ring_inv)
1054   hence "lcm x y * gcd x y div gcd x y = x * ?c * y div gcd x y" by simp
1055   with gcd x y \<noteq> 0 have "lcm x y = x * ?c * y div gcd x y"
1056     by (subst (asm) div_mult_self2_is_id, simp_all)
1057   also have "... = x * (?c * y div gcd x y)"
1058     by (metis div_mult_swap gcd_dvd2 mult_assoc)
1059   finally show ?thesis by (rule dvdI)
1060 qed (simp add: lcm_gcd)
1062 lemma lcm_least:
1063   "\<lbrakk>a dvd k; b dvd k\<rbrakk> \<Longrightarrow> lcm a b dvd k"
1064 proof (cases "k = 0")
1065   let ?nf = normalisation_factor
1066   assume "k \<noteq> 0"
1067   hence "is_unit (?nf k)" by simp
1068   hence "?nf k \<noteq> 0" by (metis not_is_unit_0)
1069   assume A: "a dvd k" "b dvd k"
1070   hence "gcd a b \<noteq> 0" using k \<noteq> 0 by (auto simp add: gcd_zero)
1071   from A obtain r s where ar: "k = a * r" and bs: "k = b * s"
1072     unfolding dvd_def by blast
1073   with k \<noteq> 0 have "r * s \<noteq> 0"
1074     by (intro notI) (drule divisors_zero, elim disjE, simp_all)
1075   hence "is_unit (?nf (r * s))" by simp
1076   let ?c = "?nf k div ?nf (r*s)"
1077   from is_unit (?nf k) and is_unit (?nf (r * s)) have "is_unit ?c" by (rule unit_div)
1078   hence "?c \<noteq> 0" using not_is_unit_0 by fast
1079   from ar bs have "k * k * gcd s r = ?nf k * k * gcd (k * s) (k * r)"
1080     by (subst mult_assoc, subst gcd_mult_distrib[of k s r], simp only: ac_simps mult_assoc)
1081   also have "... = ?nf k * k * gcd ((r*s) * a) ((r*s) * b)"
1082     by (subst (3) k = a * r, subst (3) k = b * s, simp add: algebra_simps)
1083   also have "... = ?c * r*s * k * gcd a b" using r * s \<noteq> 0
1084     by (subst gcd_mult_distrib'[symmetric], simp add: algebra_simps unit_simps)
1085   finally have "(a*r) * (b*s) * gcd s r = ?c * k * r * s * gcd a b"
1086     by (subst ar[symmetric], subst bs[symmetric], simp add: mult_ac)
1087   hence "a * b * gcd s r * (r * s) = ?c * k * gcd a b * (r * s)"
1088     by (simp add: algebra_simps)
1089   hence "?c * k * gcd a b = a * b * gcd s r" using r * s \<noteq> 0
1090     by (metis div_mult_self2_is_id)
1091   also have "... = lcm a b * gcd a b * gcd s r * ?nf (a*b)"
1092     by (subst lcm_gcd_prod[of a b], metis gcd_mult_distrib gcd_mult_distrib')
1093   also have "... = lcm a b * gcd s r * ?nf (a*b) * gcd a b"
1094     by (simp add: algebra_simps)
1095   finally have "k * ?c = lcm a b * gcd s r * ?nf (a*b)" using gcd a b \<noteq> 0
1096     by (metis mult.commute div_mult_self2_is_id)
1097   hence "k = lcm a b * (gcd s r * ?nf (a*b)) div ?c" using ?c \<noteq> 0
1098     by (metis div_mult_self2_is_id mult_assoc)
1099   also have "... = lcm a b * (gcd s r * ?nf (a*b) div ?c)" using is_unit ?c
1100     by (simp add: unit_simps)
1101   finally show ?thesis by (rule dvdI)
1102 qed simp
1104 lemma lcm_zero:
1105   "lcm a b = 0 \<longleftrightarrow> a = 0 \<or> b = 0"
1106 proof -
1107   let ?nf = normalisation_factor
1108   {
1109     assume "a \<noteq> 0" "b \<noteq> 0"
1110     hence "a * b div ?nf (a * b) \<noteq> 0" by (simp add: no_zero_divisors)
1111     moreover from a \<noteq> 0 and b \<noteq> 0 have "gcd a b \<noteq> 0" by (simp add: gcd_zero)
1112     ultimately have "lcm a b \<noteq> 0" using lcm_gcd_prod[of a b] by (intro notI, simp)
1113   } moreover {
1114     assume "a = 0 \<or> b = 0"
1115     hence "lcm a b = 0" by (elim disjE, simp_all add: lcm_gcd)
1116   }
1117   ultimately show ?thesis by blast
1118 qed
1120 lemmas lcm_0_iff = lcm_zero
1122 lemma gcd_lcm:
1123   assumes "lcm a b \<noteq> 0"
1124   shows "gcd a b = a * b div (lcm a b * normalisation_factor (a * b))"
1125 proof-
1126   from assms have "gcd a b \<noteq> 0" by (simp add: gcd_zero lcm_zero)
1127   let ?c = "normalisation_factor (a*b)"
1128   from lcm a b \<noteq> 0 have "?c \<noteq> 0" by (intro notI, simp add: lcm_zero no_zero_divisors)
1129   hence "is_unit ?c" by simp
1130   from lcm_gcd_prod [of a b] have "gcd a b = a * b div ?c div lcm a b"
1131     by (subst (2) div_mult_self2_is_id[OF lcm a b \<noteq> 0, symmetric], simp add: mult_ac)
1132   also from is_unit ?c have "... = a * b div (?c * lcm a b)"
1133     by (simp only: unit_ring_inv'1 unit_ring_inv)
1134   finally show ?thesis by (simp only: ac_simps)
1135 qed
1137 lemma normalisation_factor_lcm [simp]:
1138   "normalisation_factor (lcm a b) = (if a = 0 \<or> b = 0 then 0 else 1)"
1139 proof (cases "a = 0 \<or> b = 0")
1140   case True then show ?thesis
1141     by (simp add: lcm_gcd) (metis div_0 ac_simps mult_zero_left normalisation_factor_0)
1142 next
1143   case False
1144   let ?nf = normalisation_factor
1145   from lcm_gcd_prod[of a b]
1146     have "?nf (lcm a b) * ?nf (gcd a b) = ?nf (a*b) div ?nf (a*b)"
1147     by (metis div_by_0 div_self normalisation_correct normalisation_factor_0 normalisation_factor_mult)
1148   also have "... = (if a*b = 0 then 0 else 1)"
1149     by (cases "a*b = 0", simp, subst div_self, metis dvd_0_left normalisation_factor_dvd, simp)
1150   finally show ?thesis using False by (simp add: no_zero_divisors)
1151 qed
1153 lemma lcm_dvd2 [iff]: "y dvd lcm x y"
1154   using lcm_dvd1 [of y x] by (simp add: lcm_gcd ac_simps)
1156 lemma lcmI:
1157   "\<lbrakk>x dvd k; y dvd k; \<And>l. x dvd l \<Longrightarrow> y dvd l \<Longrightarrow> k dvd l;
1158     normalisation_factor k = (if k = 0 then 0 else 1)\<rbrakk> \<Longrightarrow> k = lcm x y"
1159   by (intro normed_associated_imp_eq) (auto simp: associated_def intro: lcm_least)
1161 sublocale lcm!: abel_semigroup lcm
1162 proof
1163   fix x y z
1164   show "lcm (lcm x y) z = lcm x (lcm y z)"
1165   proof (rule lcmI)
1166     have "x dvd lcm x y" and "lcm x y dvd lcm (lcm x y) z" by simp_all
1167     then show "x dvd lcm (lcm x y) z" by (rule dvd_trans)
1169     have "y dvd lcm x y" and "lcm x y dvd lcm (lcm x y) z" by simp_all
1170     hence "y dvd lcm (lcm x y) z" by (rule dvd_trans)
1171     moreover have "z dvd lcm (lcm x y) z" by simp
1172     ultimately show "lcm y z dvd lcm (lcm x y) z" by (rule lcm_least)
1174     fix l assume "x dvd l" and "lcm y z dvd l"
1175     have "y dvd lcm y z" by simp
1176     from this and lcm y z dvd l have "y dvd l" by (rule dvd_trans)
1177     have "z dvd lcm y z" by simp
1178     from this and lcm y z dvd l have "z dvd l" by (rule dvd_trans)
1179     from x dvd l and y dvd l have "lcm x y dvd l" by (rule lcm_least)
1180     from this and z dvd l show "lcm (lcm x y) z dvd l" by (rule lcm_least)
1181   qed (simp add: lcm_zero)
1182 next
1183   fix x y
1184   show "lcm x y = lcm y x"
1185     by (simp add: lcm_gcd ac_simps)
1186 qed
1188 lemma dvd_lcm_D1:
1189   "lcm m n dvd k \<Longrightarrow> m dvd k"
1190   by (rule dvd_trans, rule lcm_dvd1, assumption)
1192 lemma dvd_lcm_D2:
1193   "lcm m n dvd k \<Longrightarrow> n dvd k"
1194   by (rule dvd_trans, rule lcm_dvd2, assumption)
1196 lemma gcd_dvd_lcm [simp]:
1197   "gcd a b dvd lcm a b"
1198   by (metis dvd_trans gcd_dvd2 lcm_dvd2)
1200 lemma lcm_1_iff:
1201   "lcm a b = 1 \<longleftrightarrow> is_unit a \<and> is_unit b"
1202 proof
1203   assume "lcm a b = 1"
1204   then show "is_unit a \<and> is_unit b" unfolding is_unit_def by auto
1205 next
1206   assume "is_unit a \<and> is_unit b"
1207   hence "a dvd 1" and "b dvd 1" unfolding is_unit_def by simp_all
1208   hence "is_unit (lcm a b)" unfolding is_unit_def by (rule lcm_least)
1209   hence "lcm a b = normalisation_factor (lcm a b)"
1210     by (subst normalisation_factor_unit, simp_all)
1211   also have "\<dots> = 1" using is_unit a \<and> is_unit b by (auto simp add: is_unit_def)
1212   finally show "lcm a b = 1" .
1213 qed
1215 lemma lcm_0_left [simp]:
1216   "lcm 0 x = 0"
1217   by (rule sym, rule lcmI, simp_all)
1219 lemma lcm_0 [simp]:
1220   "lcm x 0 = 0"
1221   by (rule sym, rule lcmI, simp_all)
1223 lemma lcm_unique:
1224   "a dvd d \<and> b dvd d \<and>
1225   normalisation_factor d = (if d = 0 then 0 else 1) \<and>
1226   (\<forall>e. a dvd e \<and> b dvd e \<longrightarrow> d dvd e) \<longleftrightarrow> d = lcm a b"
1227   by (rule, auto intro: lcmI simp: lcm_least lcm_zero)
1229 lemma dvd_lcm_I1 [simp]:
1230   "k dvd m \<Longrightarrow> k dvd lcm m n"
1231   by (metis lcm_dvd1 dvd_trans)
1233 lemma dvd_lcm_I2 [simp]:
1234   "k dvd n \<Longrightarrow> k dvd lcm m n"
1235   by (metis lcm_dvd2 dvd_trans)
1237 lemma lcm_1_left [simp]:
1238   "lcm 1 x = x div normalisation_factor x"
1239   by (cases "x = 0") (simp, rule sym, rule lcmI, simp_all)
1241 lemma lcm_1_right [simp]:
1242   "lcm x 1 = x div normalisation_factor x"
1243   by (simp add: ac_simps)
1245 lemma lcm_coprime:
1246   "gcd a b = 1 \<Longrightarrow> lcm a b = a * b div normalisation_factor (a*b)"
1247   by (subst lcm_gcd) simp
1249 lemma lcm_proj1_if_dvd:
1250   "y dvd x \<Longrightarrow> lcm x y = x div normalisation_factor x"
1251   by (cases "x = 0") (simp, rule sym, rule lcmI, simp_all)
1253 lemma lcm_proj2_if_dvd:
1254   "x dvd y \<Longrightarrow> lcm x y = y div normalisation_factor y"
1255   using lcm_proj1_if_dvd [of x y] by (simp add: ac_simps)
1257 lemma lcm_proj1_iff:
1258   "lcm m n = m div normalisation_factor m \<longleftrightarrow> n dvd m"
1259 proof
1260   assume A: "lcm m n = m div normalisation_factor m"
1261   show "n dvd m"
1262   proof (cases "m = 0")
1263     assume [simp]: "m \<noteq> 0"
1264     from A have B: "m = lcm m n * normalisation_factor m"
1265       by (simp add: unit_eq_div2)
1266     show ?thesis by (subst B, simp)
1267   qed simp
1268 next
1269   assume "n dvd m"
1270   then show "lcm m n = m div normalisation_factor m" by (rule lcm_proj1_if_dvd)
1271 qed
1273 lemma lcm_proj2_iff:
1274   "lcm m n = n div normalisation_factor n \<longleftrightarrow> m dvd n"
1275   using lcm_proj1_iff [of n m] by (simp add: ac_simps)
1277 lemma euclidean_size_lcm_le1:
1278   assumes "a \<noteq> 0" and "b \<noteq> 0"
1279   shows "euclidean_size a \<le> euclidean_size (lcm a b)"
1280 proof -
1281   have "a dvd lcm a b" by (rule lcm_dvd1)
1282   then obtain c where A: "lcm a b = a * c" unfolding dvd_def by blast
1283   with a \<noteq> 0 and b \<noteq> 0 have "c \<noteq> 0" by (auto simp: lcm_zero)
1284   then show ?thesis by (subst A, intro size_mult_mono)
1285 qed
1287 lemma euclidean_size_lcm_le2:
1288   "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> euclidean_size b \<le> euclidean_size (lcm a b)"
1289   using euclidean_size_lcm_le1 [of b a] by (simp add: ac_simps)
1291 lemma euclidean_size_lcm_less1:
1292   assumes "b \<noteq> 0" and "\<not>b dvd a"
1293   shows "euclidean_size a < euclidean_size (lcm a b)"
1294 proof (rule ccontr)
1295   from assms have "a \<noteq> 0" by auto
1296   assume "\<not>euclidean_size a < euclidean_size (lcm a b)"
1297   with a \<noteq> 0 and b \<noteq> 0 have "euclidean_size (lcm a b) = euclidean_size a"
1298     by (intro le_antisym, simp, intro euclidean_size_lcm_le1)
1299   with assms have "lcm a b dvd a"
1300     by (rule_tac dvd_euclidean_size_eq_imp_dvd) (auto simp: lcm_zero)
1301   hence "b dvd a" by (rule dvd_lcm_D2)
1302   with \<not>b dvd a show False by contradiction
1303 qed
1305 lemma euclidean_size_lcm_less2:
1306   assumes "a \<noteq> 0" and "\<not>a dvd b"
1307   shows "euclidean_size b < euclidean_size (lcm a b)"
1308   using assms euclidean_size_lcm_less1 [of a b] by (simp add: ac_simps)
1310 lemma lcm_mult_unit1:
1311   "is_unit a \<Longrightarrow> lcm (x*a) y = lcm x y"
1312   apply (rule lcmI)
1313   apply (rule dvd_trans[of _ "x*a"], simp, rule lcm_dvd1)
1314   apply (rule lcm_dvd2)
1315   apply (rule lcm_least, simp add: unit_simps, assumption)
1316   apply (subst normalisation_factor_lcm, simp add: lcm_zero)
1317   done
1319 lemma lcm_mult_unit2:
1320   "is_unit a \<Longrightarrow> lcm x (y*a) = lcm x y"
1321   using lcm_mult_unit1 [of a y x] by (simp add: ac_simps)
1323 lemma lcm_div_unit1:
1324   "is_unit a \<Longrightarrow> lcm (x div a) y = lcm x y"
1325   by (simp add: unit_ring_inv lcm_mult_unit1)
1327 lemma lcm_div_unit2:
1328   "is_unit a \<Longrightarrow> lcm x (y div a) = lcm x y"
1329   by (simp add: unit_ring_inv lcm_mult_unit2)
1331 lemma lcm_left_idem:
1332   "lcm p (lcm p q) = lcm p q"
1333   apply (rule lcmI)
1334   apply simp
1335   apply (subst lcm.assoc [symmetric], rule lcm_dvd2)
1336   apply (rule lcm_least, assumption)
1337   apply (erule (1) lcm_least)
1338   apply (auto simp: lcm_zero)
1339   done
1341 lemma lcm_right_idem:
1342   "lcm (lcm p q) q = lcm p q"
1343   apply (rule lcmI)
1344   apply (subst lcm.assoc, rule lcm_dvd1)
1345   apply (rule lcm_dvd2)
1346   apply (rule lcm_least, erule (1) lcm_least, assumption)
1347   apply (auto simp: lcm_zero)
1348   done
1350 lemma comp_fun_idem_lcm: "comp_fun_idem lcm"
1351 proof
1352   fix a b show "lcm a \<circ> lcm b = lcm b \<circ> lcm a"
1353     by (simp add: fun_eq_iff ac_simps)
1354 next
1355   fix a show "lcm a \<circ> lcm a = lcm a" unfolding o_def
1356     by (intro ext, simp add: lcm_left_idem)
1357 qed
1359 lemma dvd_Lcm [simp]: "x \<in> A \<Longrightarrow> x dvd Lcm A"
1360   and Lcm_dvd [simp]: "(\<forall>x\<in>A. x dvd l') \<Longrightarrow> Lcm A dvd l'"
1361   and normalisation_factor_Lcm [simp]:
1362           "normalisation_factor (Lcm A) = (if Lcm A = 0 then 0 else 1)"
1363 proof -
1364   have "(\<forall>x\<in>A. x dvd Lcm A) \<and> (\<forall>l'. (\<forall>x\<in>A. x dvd l') \<longrightarrow> Lcm A dvd l') \<and>
1365     normalisation_factor (Lcm A) = (if Lcm A = 0 then 0 else 1)" (is ?thesis)
1366   proof (cases "\<exists>l. l \<noteq>  0 \<and> (\<forall>x\<in>A. x dvd l)")
1367     case False
1368     hence "Lcm A = 0" by (auto simp: Lcm_Lcm_eucl Lcm_eucl_def)
1369     with False show ?thesis by auto
1370   next
1371     case True
1372     then obtain l\<^sub>0 where l\<^sub>0_props: "l\<^sub>0 \<noteq> 0 \<and> (\<forall>x\<in>A. x dvd l\<^sub>0)" by blast
1373     def n \<equiv> "LEAST n. \<exists>l. l \<noteq> 0 \<and> (\<forall>x\<in>A. x dvd l) \<and> euclidean_size l = n"
1374     def l \<equiv> "SOME l. l \<noteq> 0 \<and> (\<forall>x\<in>A. x dvd l) \<and> euclidean_size l = n"
1375     have "\<exists>l. l \<noteq> 0 \<and> (\<forall>x\<in>A. x dvd l) \<and> euclidean_size l = n"
1376       apply (subst n_def)
1377       apply (rule LeastI[of _ "euclidean_size l\<^sub>0"])
1378       apply (rule exI[of _ l\<^sub>0])
1379       apply (simp add: l\<^sub>0_props)
1380       done
1381     from someI_ex[OF this] have "l \<noteq> 0" and "\<forall>x\<in>A. x dvd l" and "euclidean_size l = n"
1382       unfolding l_def by simp_all
1383     {
1384       fix l' assume "\<forall>x\<in>A. x dvd l'"
1385       with \<forall>x\<in>A. x dvd l have "\<forall>x\<in>A. x dvd gcd l l'" by (auto intro: gcd_greatest)
1386       moreover from l \<noteq> 0 have "gcd l l' \<noteq> 0" by (simp add: gcd_zero)
1387       ultimately have "\<exists>b. b \<noteq> 0 \<and> (\<forall>x\<in>A. x dvd b) \<and> euclidean_size b = euclidean_size (gcd l l')"
1388         by (intro exI[of _ "gcd l l'"], auto)
1389       hence "euclidean_size (gcd l l') \<ge> n" by (subst n_def) (rule Least_le)
1390       moreover have "euclidean_size (gcd l l') \<le> n"
1391       proof -
1392         have "gcd l l' dvd l" by simp
1393         then obtain a where "l = gcd l l' * a" unfolding dvd_def by blast
1394         with l \<noteq> 0 have "a \<noteq> 0" by auto
1395         hence "euclidean_size (gcd l l') \<le> euclidean_size (gcd l l' * a)"
1396           by (rule size_mult_mono)
1397         also have "gcd l l' * a = l" using l = gcd l l' * a ..
1398         also note euclidean_size l = n
1399         finally show "euclidean_size (gcd l l') \<le> n" .
1400       qed
1401       ultimately have "euclidean_size l = euclidean_size (gcd l l')"
1402         by (intro le_antisym, simp_all add: euclidean_size l = n)
1403       with l \<noteq> 0 have "l dvd gcd l l'" by (blast intro: dvd_euclidean_size_eq_imp_dvd)
1404       hence "l dvd l'" by (blast dest: dvd_gcd_D2)
1405     }
1407     with (\<forall>x\<in>A. x dvd l) and normalisation_factor_is_unit[OF l \<noteq> 0] and l \<noteq> 0
1408       have "(\<forall>x\<in>A. x dvd l div normalisation_factor l) \<and>
1409         (\<forall>l'. (\<forall>x\<in>A. x dvd l') \<longrightarrow> l div normalisation_factor l dvd l') \<and>
1410         normalisation_factor (l div normalisation_factor l) =
1411         (if l div normalisation_factor l = 0 then 0 else 1)"
1412       by (auto simp: unit_simps)
1413     also from True have "l div normalisation_factor l = Lcm A"
1414       by (simp add: Lcm_Lcm_eucl Lcm_eucl_def Let_def n_def l_def)
1415     finally show ?thesis .
1416   qed
1417   note A = this
1419   {fix x assume "x \<in> A" then show "x dvd Lcm A" using A by blast}
1420   {fix l' assume "\<forall>x\<in>A. x dvd l'" then show "Lcm A dvd l'" using A by blast}
1421   from A show "normalisation_factor (Lcm A) = (if Lcm A = 0 then 0 else 1)" by blast
1422 qed
1424 lemma LcmI:
1425   "(\<And>x. x\<in>A \<Longrightarrow> x dvd l) \<Longrightarrow> (\<And>l'. (\<forall>x\<in>A. x dvd l') \<Longrightarrow> l dvd l') \<Longrightarrow>
1426       normalisation_factor l = (if l = 0 then 0 else 1) \<Longrightarrow> l = Lcm A"
1427   by (intro normed_associated_imp_eq)
1428     (auto intro: Lcm_dvd dvd_Lcm simp: associated_def)
1430 lemma Lcm_subset:
1431   "A \<subseteq> B \<Longrightarrow> Lcm A dvd Lcm B"
1432   by (blast intro: Lcm_dvd dvd_Lcm)
1434 lemma Lcm_Un:
1435   "Lcm (A \<union> B) = lcm (Lcm A) (Lcm B)"
1436   apply (rule lcmI)
1437   apply (blast intro: Lcm_subset)
1438   apply (blast intro: Lcm_subset)
1439   apply (intro Lcm_dvd ballI, elim UnE)
1440   apply (rule dvd_trans, erule dvd_Lcm, assumption)
1441   apply (rule dvd_trans, erule dvd_Lcm, assumption)
1442   apply simp
1443   done
1445 lemma Lcm_1_iff:
1446   "Lcm A = 1 \<longleftrightarrow> (\<forall>x\<in>A. is_unit x)"
1447 proof
1448   assume "Lcm A = 1"
1449   then show "\<forall>x\<in>A. is_unit x" unfolding is_unit_def by auto
1450 qed (rule LcmI [symmetric], auto)
1452 lemma Lcm_no_units:
1453   "Lcm A = Lcm (A - {x. is_unit x})"
1454 proof -
1455   have "(A - {x. is_unit x}) \<union> {x\<in>A. is_unit x} = A" by blast
1456   hence "Lcm A = lcm (Lcm (A - {x. is_unit x})) (Lcm {x\<in>A. is_unit x})"
1457     by (simp add: Lcm_Un[symmetric])
1458   also have "Lcm {x\<in>A. is_unit x} = 1" by (simp add: Lcm_1_iff)
1459   finally show ?thesis by simp
1460 qed
1462 lemma Lcm_empty [simp]:
1463   "Lcm {} = 1"
1464   by (simp add: Lcm_1_iff)
1466 lemma Lcm_eq_0 [simp]:
1467   "0 \<in> A \<Longrightarrow> Lcm A = 0"
1468   by (drule dvd_Lcm) simp
1470 lemma Lcm0_iff':
1471   "Lcm A = 0 \<longleftrightarrow> \<not>(\<exists>l. l \<noteq> 0 \<and> (\<forall>x\<in>A. x dvd l))"
1472 proof
1473   assume "Lcm A = 0"
1474   show "\<not>(\<exists>l. l \<noteq> 0 \<and> (\<forall>x\<in>A. x dvd l))"
1475   proof
1476     assume ex: "\<exists>l. l \<noteq> 0 \<and> (\<forall>x\<in>A. x dvd l)"
1477     then obtain l\<^sub>0 where l\<^sub>0_props: "l\<^sub>0 \<noteq> 0 \<and> (\<forall>x\<in>A. x dvd l\<^sub>0)" by blast
1478     def n \<equiv> "LEAST n. \<exists>l. l \<noteq> 0 \<and> (\<forall>x\<in>A. x dvd l) \<and> euclidean_size l = n"
1479     def l \<equiv> "SOME l. l \<noteq> 0 \<and> (\<forall>x\<in>A. x dvd l) \<and> euclidean_size l = n"
1480     have "\<exists>l. l \<noteq> 0 \<and> (\<forall>x\<in>A. x dvd l) \<and> euclidean_size l = n"
1481       apply (subst n_def)
1482       apply (rule LeastI[of _ "euclidean_size l\<^sub>0"])
1483       apply (rule exI[of _ l\<^sub>0])
1484       apply (simp add: l\<^sub>0_props)
1485       done
1486     from someI_ex[OF this] have "l \<noteq> 0" unfolding l_def by simp_all
1487     hence "l div normalisation_factor l \<noteq> 0" by simp
1488     also from ex have "l div normalisation_factor l = Lcm A"
1489        by (simp only: Lcm_Lcm_eucl Lcm_eucl_def n_def l_def if_True Let_def)
1490     finally show False using Lcm A = 0 by contradiction
1491   qed
1492 qed (simp only: Lcm_Lcm_eucl Lcm_eucl_def if_False)
1494 lemma Lcm0_iff [simp]:
1495   "finite A \<Longrightarrow> Lcm A = 0 \<longleftrightarrow> 0 \<in> A"
1496 proof -
1497   assume "finite A"
1498   have "0 \<in> A \<Longrightarrow> Lcm A = 0"  by (intro dvd_0_left dvd_Lcm)
1499   moreover {
1500     assume "0 \<notin> A"
1501     hence "\<Prod>A \<noteq> 0"
1502       apply (induct rule: finite_induct[OF finite A])
1503       apply simp
1504       apply (subst setprod.insert, assumption, assumption)
1505       apply (rule no_zero_divisors)
1506       apply blast+
1507       done
1508     moreover from finite A have "\<forall>x\<in>A. x dvd \<Prod>A" by (intro ballI dvd_setprod)
1509     ultimately have "\<exists>l. l \<noteq> 0 \<and> (\<forall>x\<in>A. x dvd l)" by blast
1510     with Lcm0_iff' have "Lcm A \<noteq> 0" by simp
1511   }
1512   ultimately show "Lcm A = 0 \<longleftrightarrow> 0 \<in> A" by blast
1513 qed
1515 lemma Lcm_no_multiple:
1516   "(\<forall>m. m \<noteq> 0 \<longrightarrow> (\<exists>x\<in>A. \<not>x dvd m)) \<Longrightarrow> Lcm A = 0"
1517 proof -
1518   assume "\<forall>m. m \<noteq> 0 \<longrightarrow> (\<exists>x\<in>A. \<not>x dvd m)"
1519   hence "\<not>(\<exists>l. l \<noteq> 0 \<and> (\<forall>x\<in>A. x dvd l))" by blast
1520   then show "Lcm A = 0" by (simp only: Lcm_Lcm_eucl Lcm_eucl_def if_False)
1521 qed
1523 lemma Lcm_insert [simp]:
1524   "Lcm (insert a A) = lcm a (Lcm A)"
1525 proof (rule lcmI)
1526   fix l assume "a dvd l" and "Lcm A dvd l"
1527   hence "\<forall>x\<in>A. x dvd l" by (blast intro: dvd_trans dvd_Lcm)
1528   with a dvd l show "Lcm (insert a A) dvd l" by (force intro: Lcm_dvd)
1529 qed (auto intro: Lcm_dvd dvd_Lcm)
1531 lemma Lcm_finite:
1532   assumes "finite A"
1533   shows "Lcm A = Finite_Set.fold lcm 1 A"
1534   by (induct rule: finite.induct[OF finite A])
1535     (simp_all add: comp_fun_idem.fold_insert_idem[OF comp_fun_idem_lcm])
1537 lemma Lcm_set [code, code_unfold]:
1538   "Lcm (set xs) = fold lcm xs 1"
1539   using comp_fun_idem.fold_set_fold[OF comp_fun_idem_lcm] Lcm_finite by (simp add: ac_simps)
1541 lemma Lcm_singleton [simp]:
1542   "Lcm {a} = a div normalisation_factor a"
1543   by simp
1545 lemma Lcm_2 [simp]:
1546   "Lcm {a,b} = lcm a b"
1547   by (simp only: Lcm_insert Lcm_empty lcm_1_right)
1548     (cases "b = 0", simp, rule lcm_div_unit2, simp)
1550 lemma Lcm_coprime:
1551   assumes "finite A" and "A \<noteq> {}"
1552   assumes "\<And>a b. a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<noteq> b \<Longrightarrow> gcd a b = 1"
1553   shows "Lcm A = \<Prod>A div normalisation_factor (\<Prod>A)"
1554 using assms proof (induct rule: finite_ne_induct)
1555   case (insert a A)
1556   have "Lcm (insert a A) = lcm a (Lcm A)" by simp
1557   also from insert have "Lcm A = \<Prod>A div normalisation_factor (\<Prod>A)" by blast
1558   also have "lcm a \<dots> = lcm a (\<Prod>A)" by (cases "\<Prod>A = 0") (simp_all add: lcm_div_unit2)
1559   also from insert have "gcd a (\<Prod>A) = 1" by (subst gcd.commute, intro setprod_coprime) auto
1560   with insert have "lcm a (\<Prod>A) = \<Prod>(insert a A) div normalisation_factor (\<Prod>(insert a A))"
1561     by (simp add: lcm_coprime)
1562   finally show ?case .
1563 qed simp
1565 lemma Lcm_coprime':
1566   "card A \<noteq> 0 \<Longrightarrow> (\<And>a b. a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<noteq> b \<Longrightarrow> gcd a b = 1)
1567     \<Longrightarrow> Lcm A = \<Prod>A div normalisation_factor (\<Prod>A)"
1568   by (rule Lcm_coprime) (simp_all add: card_eq_0_iff)
1570 lemma Gcd_Lcm:
1571   "Gcd A = Lcm {d. \<forall>x\<in>A. d dvd x}"
1572   by (simp add: Gcd_Gcd_eucl Lcm_Lcm_eucl Gcd_eucl_def)
1574 lemma Gcd_dvd [simp]: "x \<in> A \<Longrightarrow> Gcd A dvd x"
1575   and dvd_Gcd [simp]: "(\<forall>x\<in>A. g' dvd x) \<Longrightarrow> g' dvd Gcd A"
1576   and normalisation_factor_Gcd [simp]:
1577     "normalisation_factor (Gcd A) = (if Gcd A = 0 then 0 else 1)"
1578 proof -
1579   fix x assume "x \<in> A"
1580   hence "Lcm {d. \<forall>x\<in>A. d dvd x} dvd x" by (intro Lcm_dvd) blast
1581   then show "Gcd A dvd x" by (simp add: Gcd_Lcm)
1582 next
1583   fix g' assume "\<forall>x\<in>A. g' dvd x"
1584   hence "g' dvd Lcm {d. \<forall>x\<in>A. d dvd x}" by (intro dvd_Lcm) blast
1585   then show "g' dvd Gcd A" by (simp add: Gcd_Lcm)
1586 next
1587   show "normalisation_factor (Gcd A) = (if Gcd A = 0 then 0 else 1)"
1588     by (simp add: Gcd_Lcm normalisation_factor_Lcm)
1589 qed
1591 lemma GcdI:
1592   "(\<And>x. x\<in>A \<Longrightarrow> l dvd x) \<Longrightarrow> (\<And>l'. (\<forall>x\<in>A. l' dvd x) \<Longrightarrow> l' dvd l) \<Longrightarrow>
1593     normalisation_factor l = (if l = 0 then 0 else 1) \<Longrightarrow> l = Gcd A"
1594   by (intro normed_associated_imp_eq)
1595     (auto intro: Gcd_dvd dvd_Gcd simp: associated_def)
1597 lemma Lcm_Gcd:
1598   "Lcm A = Gcd {m. \<forall>x\<in>A. x dvd m}"
1599   by (rule LcmI[symmetric]) (auto intro: dvd_Gcd Gcd_dvd)
1601 lemma Gcd_0_iff:
1602   "Gcd A = 0 \<longleftrightarrow> A \<subseteq> {0}"
1603   apply (rule iffI)
1604   apply (rule subsetI, drule Gcd_dvd, simp)
1605   apply (auto intro: GcdI[symmetric])
1606   done
1608 lemma Gcd_empty [simp]:
1609   "Gcd {} = 0"
1610   by (simp add: Gcd_0_iff)
1612 lemma Gcd_1:
1613   "1 \<in> A \<Longrightarrow> Gcd A = 1"
1614   by (intro GcdI[symmetric]) (auto intro: Gcd_dvd dvd_Gcd)
1616 lemma Gcd_insert [simp]:
1617   "Gcd (insert a A) = gcd a (Gcd A)"
1618 proof (rule gcdI)
1619   fix l assume "l dvd a" and "l dvd Gcd A"
1620   hence "\<forall>x\<in>A. l dvd x" by (blast intro: dvd_trans Gcd_dvd)
1621   with l dvd a show "l dvd Gcd (insert a A)" by (force intro: Gcd_dvd)
1622 qed (auto intro: Gcd_dvd dvd_Gcd simp: normalisation_factor_Gcd)
1624 lemma Gcd_finite:
1625   assumes "finite A"
1626   shows "Gcd A = Finite_Set.fold gcd 0 A"
1627   by (induct rule: finite.induct[OF finite A`])
1628     (simp_all add: comp_fun_idem.fold_insert_idem[OF comp_fun_idem_gcd])
1630 lemma Gcd_set [code, code_unfold]:
1631   "Gcd (set xs) = fold gcd xs 0"
1632   using comp_fun_idem.fold_set_fold[OF comp_fun_idem_gcd] Gcd_finite by (simp add: ac_simps)
1634 lemma Gcd_singleton [simp]: "Gcd {a} = a div normalisation_factor a"
1635   by (simp add: gcd_0)
1637 lemma Gcd_2 [simp]: "Gcd {a,b} = gcd a b"
1638   by (simp only: Gcd_insert Gcd_empty gcd_0) (cases "b = 0", simp, rule gcd_div_unit2, simp)
1640 end
1642 text {*
1643   A Euclidean ring is a Euclidean semiring with additive inverses. It provides a
1644   few more lemmas; in particular, Bezout's lemma holds for any Euclidean ring.
1645 *}
1647 class euclidean_ring = euclidean_semiring + idom
1649 class euclidean_ring_gcd = euclidean_semiring_gcd + idom
1650 begin
1652 subclass euclidean_ring ..
1654 lemma gcd_neg1 [simp]:
1655   "gcd (-x) y = gcd x y"
1656   by (rule sym, rule gcdI, simp_all add: gcd_greatest gcd_zero)
1658 lemma gcd_neg2 [simp]:
1659   "gcd x (-y) = gcd x y"
1660   by (rule sym, rule gcdI, simp_all add: gcd_greatest gcd_zero)
1662 lemma gcd_neg_numeral_1 [simp]:
1663   "gcd (- numeral n) x = gcd (numeral n) x"
1664   by (fact gcd_neg1)
1666 lemma gcd_neg_numeral_2 [simp]:
1667   "gcd x (- numeral n) = gcd x (numeral n)"
1668   by (fact gcd_neg2)
1670 lemma gcd_diff1: "gcd (m - n) n = gcd m n"
1671   by (subst diff_conv_add_uminus, subst gcd_neg2[symmetric],  subst gcd_add1, simp)
1673 lemma gcd_diff2: "gcd (n - m) n = gcd m n"
1674   by (subst gcd_neg1[symmetric], simp only: minus_diff_eq gcd_diff1)
1676 lemma coprime_minus_one [simp]: "gcd (n - 1) n = 1"
1677 proof -
1678   have "gcd (n - 1) n = gcd n (n - 1)" by (fact gcd.commute)
1679   also have "\<dots> = gcd ((n - 1) + 1) (n - 1)" by simp
1680   also have "\<dots> = 1" by (rule coprime_plus_one)
1681   finally show ?thesis .
1682 qed
1684 lemma lcm_neg1 [simp]: "lcm (-x) y = lcm x y"
1685   by (rule sym, rule lcmI, simp_all add: lcm_least lcm_zero)
1687 lemma lcm_neg2 [simp]: "lcm x (-y) = lcm x y"
1688   by (rule sym, rule lcmI, simp_all add: lcm_least lcm_zero)
1690 lemma lcm_neg_numeral_1 [simp]: "lcm (- numeral n) x = lcm (numeral n) x"
1691   by (fact lcm_neg1)
1693 lemma lcm_neg_numeral_2 [simp]: "lcm x (- numeral n) = lcm x (numeral n)"
1694   by (fact lcm_neg2)
1696 function euclid_ext :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a" where
1697   "euclid_ext a b =
1698      (if b = 0 then
1699         let x = ring_inv (normalisation_factor a) in (x, 0, a * x)
1700       else
1701         case euclid_ext b (a mod b) of
1702             (s,t,c) \<Rightarrow> (t, s - t * (a div b), c))"
1703   by (pat_completeness, simp)
1704   termination by (relation "measure (euclidean_size \<circ> snd)", simp_all)
1706 declare euclid_ext.simps [simp del]
1708 lemma euclid_ext_0:
1709   "euclid_ext a 0 = (ring_inv (normalisation_factor a), 0, a * ring_inv (normalisation_factor a))"
1710   by (subst euclid_ext.simps, simp add: Let_def)
1712 lemma euclid_ext_non_0:
1713   "b \<noteq> 0 \<Longrightarrow> euclid_ext a b = (case euclid_ext b (a mod b) of
1714     (s,t,c) \<Rightarrow> (t, s - t * (a div b), c))"
1715   by (subst euclid_ext.simps, simp)
1717 definition euclid_ext' :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a"
1718 where
1719   "euclid_ext' a b = (case euclid_ext a b of (s, t, _) \<Rightarrow> (s, t))"
1721 lemma euclid_ext_gcd [simp]:
1722   "(case euclid_ext a b of (_,_,t) \<Rightarrow> t) = gcd a b"
1723 proof (induct a b rule: euclid_ext.induct)
1724   case (1 a b)
1725   then show ?case
1726   proof (cases "b = 0")
1727     case True
1728       then show ?thesis by (cases "a = 0")
1729         (simp_all add: euclid_ext_0 unit_div mult_ac unit_simps gcd_0)
1730     next
1731     case False with 1 show ?thesis
1732       by (simp add: euclid_ext_non_0 ac_simps split: prod.split prod.split_asm)
1733     qed
1734 qed
1736 lemma euclid_ext_gcd' [simp]:
1737   "euclid_ext a b = (r, s, t) \<Longrightarrow> t = gcd a b"
1738   by (insert euclid_ext_gcd[of a b], drule (1) subst, simp)
1740 lemma euclid_ext_correct:
1741   "case euclid_ext x y of (s,t,c) \<Rightarrow> s*x + t*y = c"
1742 proof (induct x y rule: euclid_ext.induct)
1743   case (1 x y)
1744   show ?case
1745   proof (cases "y = 0")
1746     case True
1747     then show ?thesis by (simp add: euclid_ext_0 mult_ac)
1748   next
1749     case False
1750     obtain s t c where stc: "euclid_ext y (x mod y) = (s,t,c)"
1751       by (cases "euclid_ext y (x mod y)", blast)
1752     from 1 have "c = s * y + t * (x mod y)" by (simp add: stc False)
1753     also have "... = t*((x div y)*y + x mod y) + (s - t * (x div y))*y"
1754       by (simp add: algebra_simps)
1755     also have "(x div y)*y + x mod y = x" using mod_div_equality .
1756     finally show ?thesis
1757       by (subst euclid_ext.simps, simp add: False stc)
1758     qed
1759 qed
1761 lemma euclid_ext'_correct:
1762   "fst (euclid_ext' a b) * a + snd (euclid_ext' a b) * b = gcd a b"
1763 proof-
1764   obtain s t c where "euclid_ext a b = (s,t,c)"
1765     by (cases "euclid_ext a b", blast)
1766   with euclid_ext_correct[of a b] euclid_ext_gcd[of a b]
1767     show ?thesis unfolding euclid_ext'_def by simp
1768 qed
1770 lemma bezout: "\<exists>s t. s * x + t * y = gcd x y"
1771   using euclid_ext'_correct by blast
1773 lemma euclid_ext'_0 [simp]: "euclid_ext' x 0 = (ring_inv (normalisation_factor x), 0)"
1774   by (simp add: bezw_def euclid_ext'_def euclid_ext_0)
1776 lemma euclid_ext'_non_0: "y \<noteq> 0 \<Longrightarrow> euclid_ext' x y = (snd (euclid_ext' y (x mod y)),
1777   fst (euclid_ext' y (x mod y)) - snd (euclid_ext' y (x mod y)) * (x div y))"
1778   by (cases "euclid_ext y (x mod y)")
1779     (simp add: euclid_ext'_def euclid_ext_non_0)
1781 end
1783 instantiation nat :: euclidean_semiring
1784 begin
1786 definition [simp]:
1787   "euclidean_size_nat = (id :: nat \<Rightarrow> nat)"
1789 definition [simp]:
1790   "normalisation_factor_nat (n::nat) = (if n = 0 then 0 else 1 :: nat)"
1792 instance proof
1793 qed (simp_all add: is_unit_def)
1795 end
1797 instantiation int :: euclidean_ring
1798 begin
1800 definition [simp]:
1801   "euclidean_size_int = (nat \<circ> abs :: int \<Rightarrow> nat)"
1803 definition [simp]:
1804   "normalisation_factor_int = (sgn :: int \<Rightarrow> int)"
1806 instance proof
1807   case goal2 then show ?case by (auto simp add: abs_mult nat_mult_distrib)
1808 next
1809   case goal3 then show ?case by (simp add: zsgn_def is_unit_def)
1810 next
1811   case goal5 then show ?case by (auto simp: zsgn_def is_unit_def)
1812 next
1813   case goal6 then show ?case by (auto split: abs_split simp: zsgn_def is_unit_def)
1814 qed (auto simp: sgn_times split: abs_split)
1816 end
1818 end