src/HOL/Number_Theory/Euclidean_Algorithm.thy
 author haftmann Tue Oct 11 16:44:13 2016 +0200 (2016-10-11) changeset 64163 62c9e5c05928 parent 63947 559f0882d6a6 child 64164 38c407446400 permissions -rw-r--r--
stripped dependency on pragmatic type class semiring_div
1 (* Author: Manuel Eberl *)
3 section \<open>Abstract euclidean algorithm\<close>
5 theory Euclidean_Algorithm
6 imports "~~/src/HOL/GCD" Factorial_Ring
7 begin
9 class divide_modulo = semidom_divide + modulo +
10   assumes div_mod_equality: "((a div b) * b + a mod b) + c = a + c"
11 begin
13 lemma zero_mod_left [simp]: "0 mod a = 0"
14   using div_mod_equality[of 0 a 0] by simp
16 lemma dvd_mod_iff [simp]:
17   assumes "k dvd n"
18   shows   "(k dvd m mod n) = (k dvd m)"
19 proof -
20   thm div_mod_equality
21   from assms have "(k dvd m mod n) \<longleftrightarrow> (k dvd ((m div n) * n + m mod n))"
23   also have "(m div n) * n + m mod n = m"
24     using div_mod_equality[of m n 0] by simp
25   finally show ?thesis .
26 qed
28 lemma mod_0_imp_dvd:
29   assumes "a mod b = 0"
30   shows   "b dvd a"
31 proof -
32   have "b dvd ((a div b) * b)" by simp
33   also have "(a div b) * b = a"
34     using div_mod_equality[of a b 0] by (simp add: assms)
35   finally show ?thesis .
36 qed
38 end
42 text \<open>
43   A Euclidean semiring is a semiring upon which the Euclidean algorithm can be
44   implemented. It must provide:
45   \begin{itemize}
46   \item division with remainder
47   \item a size function such that @{term "size (a mod b) < size b"}
48         for any @{term "b \<noteq> 0"}
49   \end{itemize}
50   The existence of these functions makes it possible to derive gcd and lcm functions
51   for any Euclidean semiring.
52 \<close>
53 class euclidean_semiring = divide_modulo + normalization_semidom +
54   fixes euclidean_size :: "'a \<Rightarrow> nat"
55   assumes size_0 [simp]: "euclidean_size 0 = 0"
56   assumes mod_size_less:
57     "b \<noteq> 0 \<Longrightarrow> euclidean_size (a mod b) < euclidean_size b"
58   assumes size_mult_mono:
59     "b \<noteq> 0 \<Longrightarrow> euclidean_size a \<le> euclidean_size (a * b)"
60 begin
62 lemma euclidean_size_normalize [simp]:
63   "euclidean_size (normalize a) = euclidean_size a"
64 proof (cases "a = 0")
65   case True
66   then show ?thesis
67     by simp
68 next
69   case [simp]: False
70   have "euclidean_size (normalize a) \<le> euclidean_size (normalize a * unit_factor a)"
71     by (rule size_mult_mono) simp
72   moreover have "euclidean_size a \<le> euclidean_size (a * (1 div unit_factor a))"
73     by (rule size_mult_mono) simp
74   ultimately show ?thesis
75     by simp
76 qed
78 lemma euclidean_division:
79   fixes a :: 'a and b :: 'a
80   assumes "b \<noteq> 0"
81   obtains s and t where "a = s * b + t"
82     and "euclidean_size t < euclidean_size b"
83 proof -
84   from div_mod_equality [of a b 0]
85      have "a = a div b * b + a mod b" by simp
86   with that and assms show ?thesis by (auto simp add: mod_size_less)
87 qed
89 lemma zero_mod_left [simp]: "0 mod a = 0"
90   using div_mod_equality[of 0 a 0] by simp
92 lemma dvd_mod_iff [simp]:
93   assumes "k dvd n"
94   shows   "(k dvd m mod n) = (k dvd m)"
95 proof -
96   thm div_mod_equality
97   from assms have "(k dvd m mod n) \<longleftrightarrow> (k dvd ((m div n) * n + m mod n))"
99   also have "(m div n) * n + m mod n = m"
100     using div_mod_equality[of m n 0] by simp
101   finally show ?thesis .
102 qed
104 lemma mod_0_imp_dvd:
105   assumes "a mod b = 0"
106   shows   "b dvd a"
107 proof -
108   have "b dvd ((a div b) * b)" by simp
109   also have "(a div b) * b = a"
110     using div_mod_equality[of a b 0] by (simp add: assms)
111   finally show ?thesis .
112 qed
114 lemma dvd_euclidean_size_eq_imp_dvd:
115   assumes "a \<noteq> 0" and b_dvd_a: "b dvd a" and size_eq: "euclidean_size a = euclidean_size b"
116   shows "a dvd b"
117 proof (rule ccontr)
118   assume "\<not> a dvd b"
119   hence "b mod a \<noteq> 0" using mod_0_imp_dvd[of b a] by blast
120   then have "b mod a \<noteq> 0" by (simp add: mod_eq_0_iff_dvd)
121   from b_dvd_a have b_dvd_mod: "b dvd b mod a" by simp
122   from b_dvd_mod obtain c where "b mod a = b * c" unfolding dvd_def by blast
123     with \<open>b mod a \<noteq> 0\<close> have "c \<noteq> 0" by auto
124   with \<open>b mod a = b * c\<close> have "euclidean_size (b mod a) \<ge> euclidean_size b"
125       using size_mult_mono by force
126   moreover from \<open>\<not> a dvd b\<close> and \<open>a \<noteq> 0\<close>
127   have "euclidean_size (b mod a) < euclidean_size a"
128       using mod_size_less by blast
129   ultimately show False using size_eq by simp
130 qed
132 lemma size_mult_mono': "b \<noteq> 0 \<Longrightarrow> euclidean_size a \<le> euclidean_size (b * a)"
133   by (subst mult.commute) (rule size_mult_mono)
135 lemma euclidean_size_times_unit:
136   assumes "is_unit a"
137   shows   "euclidean_size (a * b) = euclidean_size b"
138 proof (rule antisym)
139   from assms have [simp]: "a \<noteq> 0" by auto
140   thus "euclidean_size (a * b) \<ge> euclidean_size b" by (rule size_mult_mono')
141   from assms have "is_unit (1 div a)" by simp
142   hence "1 div a \<noteq> 0" by (intro notI) simp_all
143   hence "euclidean_size (a * b) \<le> euclidean_size ((1 div a) * (a * b))"
144     by (rule size_mult_mono')
145   also from assms have "(1 div a) * (a * b) = b"
146     by (simp add: algebra_simps unit_div_mult_swap)
147   finally show "euclidean_size (a * b) \<le> euclidean_size b" .
148 qed
150 lemma euclidean_size_unit: "is_unit x \<Longrightarrow> euclidean_size x = euclidean_size 1"
151   using euclidean_size_times_unit[of x 1] by simp
153 lemma unit_iff_euclidean_size:
154   "is_unit x \<longleftrightarrow> euclidean_size x = euclidean_size 1 \<and> x \<noteq> 0"
155 proof safe
156   assume A: "x \<noteq> 0" and B: "euclidean_size x = euclidean_size 1"
157   show "is_unit x" by (rule dvd_euclidean_size_eq_imp_dvd[OF A _ B]) simp_all
158 qed (auto intro: euclidean_size_unit)
160 lemma euclidean_size_times_nonunit:
161   assumes "a \<noteq> 0" "b \<noteq> 0" "\<not>is_unit a"
162   shows   "euclidean_size b < euclidean_size (a * b)"
163 proof (rule ccontr)
164   assume "\<not>euclidean_size b < euclidean_size (a * b)"
165   with size_mult_mono'[OF assms(1), of b]
166     have eq: "euclidean_size (a * b) = euclidean_size b" by simp
167   have "a * b dvd b"
168     by (rule dvd_euclidean_size_eq_imp_dvd[OF _ _ eq]) (insert assms, simp_all)
169   hence "a * b dvd 1 * b" by simp
170   with \<open>b \<noteq> 0\<close> have "is_unit a" by (subst (asm) dvd_times_right_cancel_iff)
171   with assms(3) show False by contradiction
172 qed
174 lemma dvd_imp_size_le:
175   assumes "x dvd y" "y \<noteq> 0"
176   shows   "euclidean_size x \<le> euclidean_size y"
177   using assms by (auto elim!: dvdE simp: size_mult_mono)
179 lemma dvd_proper_imp_size_less:
180   assumes "x dvd y" "\<not>y dvd x" "y \<noteq> 0"
181   shows   "euclidean_size x < euclidean_size y"
182 proof -
183   from assms(1) obtain z where "y = x * z" by (erule dvdE)
184   hence z: "y = z * x" by (simp add: mult.commute)
185   from z assms have "\<not>is_unit z" by (auto simp: mult.commute mult_unit_dvd_iff)
186   with z assms show ?thesis
187     by (auto intro!: euclidean_size_times_nonunit simp: )
188 qed
190 function gcd_eucl :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
191 where
192   "gcd_eucl a b = (if b = 0 then normalize a else gcd_eucl b (a mod b))"
193   by pat_completeness simp
194 termination
195   by (relation "measure (euclidean_size \<circ> snd)") (simp_all add: mod_size_less)
197 declare gcd_eucl.simps [simp del]
199 lemma gcd_eucl_induct [case_names zero mod]:
200   assumes H1: "\<And>b. P b 0"
201   and H2: "\<And>a b. b \<noteq> 0 \<Longrightarrow> P b (a mod b) \<Longrightarrow> P a b"
202   shows "P a b"
203 proof (induct a b rule: gcd_eucl.induct)
204   case ("1" a b)
205   show ?case
206   proof (cases "b = 0")
207     case True then show "P a b" by simp (rule H1)
208   next
209     case False
210     then have "P b (a mod b)"
211       by (rule "1.hyps")
212     with \<open>b \<noteq> 0\<close> show "P a b"
213       by (blast intro: H2)
214   qed
215 qed
217 definition lcm_eucl :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
218 where
219   "lcm_eucl a b = normalize (a * b) div gcd_eucl a b"
221 definition Lcm_eucl :: "'a set \<Rightarrow> 'a" \<comment> \<open>
222   Somewhat complicated definition of Lcm that has the advantage of working
223   for infinite sets as well\<close>
224 where
225   "Lcm_eucl A = (if \<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) then
226      let l = SOME l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l =
227        (LEAST n. \<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n)
228        in normalize l
229       else 0)"
231 definition Gcd_eucl :: "'a set \<Rightarrow> 'a"
232 where
233   "Gcd_eucl A = Lcm_eucl {d. \<forall>a\<in>A. d dvd a}"
235 declare Lcm_eucl_def Gcd_eucl_def [code del]
237 lemma gcd_eucl_0:
238   "gcd_eucl a 0 = normalize a"
239   by (simp add: gcd_eucl.simps [of a 0])
241 lemma gcd_eucl_0_left:
242   "gcd_eucl 0 a = normalize a"
243   by (simp_all add: gcd_eucl_0 gcd_eucl.simps [of 0 a])
245 lemma gcd_eucl_non_0:
246   "b \<noteq> 0 \<Longrightarrow> gcd_eucl a b = gcd_eucl b (a mod b)"
247   by (simp add: gcd_eucl.simps [of a b] gcd_eucl.simps [of b 0])
249 lemma gcd_eucl_dvd1 [iff]: "gcd_eucl a b dvd a"
250   and gcd_eucl_dvd2 [iff]: "gcd_eucl a b dvd b"
251   by (induct a b rule: gcd_eucl_induct)
252      (simp_all add: gcd_eucl_0 gcd_eucl_non_0 dvd_mod_iff)
254 lemma normalize_gcd_eucl [simp]:
255   "normalize (gcd_eucl a b) = gcd_eucl a b"
256   by (induct a b rule: gcd_eucl_induct) (simp_all add: gcd_eucl_0 gcd_eucl_non_0)
258 lemma gcd_eucl_greatest:
259   fixes k a b :: 'a
260   shows "k dvd a \<Longrightarrow> k dvd b \<Longrightarrow> k dvd gcd_eucl a b"
261 proof (induct a b rule: gcd_eucl_induct)
262   case (zero a) from zero(1) show ?case by (rule dvd_trans) (simp add: gcd_eucl_0)
263 next
264   case (mod a b)
265   then show ?case
266     by (simp add: gcd_eucl_non_0 dvd_mod_iff)
267 qed
269 lemma gcd_euclI:
270   fixes gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
271   assumes "d dvd a" "d dvd b" "normalize d = d"
272           "\<And>k. k dvd a \<Longrightarrow> k dvd b \<Longrightarrow> k dvd d"
273   shows   "gcd_eucl a b = d"
274   by (rule associated_eqI) (simp_all add: gcd_eucl_greatest assms)
276 lemma eq_gcd_euclI:
277   fixes gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
278   assumes "\<And>a b. gcd a b dvd a" "\<And>a b. gcd a b dvd b" "\<And>a b. normalize (gcd a b) = gcd a b"
279           "\<And>a b k. k dvd a \<Longrightarrow> k dvd b \<Longrightarrow> k dvd gcd a b"
280   shows   "gcd = gcd_eucl"
281   by (intro ext, rule associated_eqI) (simp_all add: gcd_eucl_greatest assms)
283 lemma gcd_eucl_zero [simp]:
284   "gcd_eucl a b = 0 \<longleftrightarrow> a = 0 \<and> b = 0"
285   by (metis dvd_0_left dvd_refl gcd_eucl_dvd1 gcd_eucl_dvd2 gcd_eucl_greatest)+
288 lemma dvd_Lcm_eucl [simp]: "a \<in> A \<Longrightarrow> a dvd Lcm_eucl A"
289   and Lcm_eucl_least: "(\<And>a. a \<in> A \<Longrightarrow> a dvd b) \<Longrightarrow> Lcm_eucl A dvd b"
290   and unit_factor_Lcm_eucl [simp]:
291           "unit_factor (Lcm_eucl A) = (if Lcm_eucl A = 0 then 0 else 1)"
292 proof -
293   have "(\<forall>a\<in>A. a dvd Lcm_eucl A) \<and> (\<forall>l'. (\<forall>a\<in>A. a dvd l') \<longrightarrow> Lcm_eucl A dvd l') \<and>
294     unit_factor (Lcm_eucl A) = (if Lcm_eucl A = 0 then 0 else 1)" (is ?thesis)
295   proof (cases "\<exists>l. l \<noteq>  0 \<and> (\<forall>a\<in>A. a dvd l)")
296     case False
297     hence "Lcm_eucl A = 0" by (auto simp: Lcm_eucl_def)
298     with False show ?thesis by auto
299   next
300     case True
301     then obtain l\<^sub>0 where l\<^sub>0_props: "l\<^sub>0 \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l\<^sub>0)" by blast
302     define n where "n = (LEAST n. \<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n)"
303     define l where "l = (SOME l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n)"
304     have "\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"
305       apply (subst n_def)
306       apply (rule LeastI[of _ "euclidean_size l\<^sub>0"])
307       apply (rule exI[of _ l\<^sub>0])
308       apply (simp add: l\<^sub>0_props)
309       done
310     from someI_ex[OF this] have "l \<noteq> 0" and "\<forall>a\<in>A. a dvd l" and "euclidean_size l = n"
311       unfolding l_def by simp_all
312     {
313       fix l' assume "\<forall>a\<in>A. a dvd l'"
314       with \<open>\<forall>a\<in>A. a dvd l\<close> have "\<forall>a\<in>A. a dvd gcd_eucl l l'" by (auto intro: gcd_eucl_greatest)
315       moreover from \<open>l \<noteq> 0\<close> have "gcd_eucl l l' \<noteq> 0" by simp
316       ultimately have "\<exists>b. b \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd b) \<and>
317                           euclidean_size b = euclidean_size (gcd_eucl l l')"
318         by (intro exI[of _ "gcd_eucl l l'"], auto)
319       hence "euclidean_size (gcd_eucl l l') \<ge> n" by (subst n_def) (rule Least_le)
320       moreover have "euclidean_size (gcd_eucl l l') \<le> n"
321       proof -
322         have "gcd_eucl l l' dvd l" by simp
323         then obtain a where "l = gcd_eucl l l' * a" unfolding dvd_def by blast
324         with \<open>l \<noteq> 0\<close> have "a \<noteq> 0" by auto
325         hence "euclidean_size (gcd_eucl l l') \<le> euclidean_size (gcd_eucl l l' * a)"
326           by (rule size_mult_mono)
327         also have "gcd_eucl l l' * a = l" using \<open>l = gcd_eucl l l' * a\<close> ..
328         also note \<open>euclidean_size l = n\<close>
329         finally show "euclidean_size (gcd_eucl l l') \<le> n" .
330       qed
331       ultimately have *: "euclidean_size l = euclidean_size (gcd_eucl l l')"
332         by (intro le_antisym, simp_all add: \<open>euclidean_size l = n\<close>)
333       from \<open>l \<noteq> 0\<close> have "l dvd gcd_eucl l l'"
334         by (rule dvd_euclidean_size_eq_imp_dvd) (auto simp add: *)
335       hence "l dvd l'" by (rule dvd_trans[OF _ gcd_eucl_dvd2])
336     }
338     with \<open>(\<forall>a\<in>A. a dvd l)\<close> and unit_factor_is_unit[OF \<open>l \<noteq> 0\<close>] and \<open>l \<noteq> 0\<close>
339       have "(\<forall>a\<in>A. a dvd normalize l) \<and>
340         (\<forall>l'. (\<forall>a\<in>A. a dvd l') \<longrightarrow> normalize l dvd l') \<and>
341         unit_factor (normalize l) =
342         (if normalize l = 0 then 0 else 1)"
343       by (auto simp: unit_simps)
344     also from True have "normalize l = Lcm_eucl A"
345       by (simp add: Lcm_eucl_def Let_def n_def l_def)
346     finally show ?thesis .
347   qed
348   note A = this
350   {fix a assume "a \<in> A" then show "a dvd Lcm_eucl A" using A by blast}
351   {fix b assume "\<And>a. a \<in> A \<Longrightarrow> a dvd b" then show "Lcm_eucl A dvd b" using A by blast}
352   from A show "unit_factor (Lcm_eucl A) = (if Lcm_eucl A = 0 then 0 else 1)" by blast
353 qed
355 lemma normalize_Lcm_eucl [simp]:
356   "normalize (Lcm_eucl A) = Lcm_eucl A"
357 proof (cases "Lcm_eucl A = 0")
358   case True then show ?thesis by simp
359 next
360   case False
361   have "unit_factor (Lcm_eucl A) * normalize (Lcm_eucl A) = Lcm_eucl A"
362     by (fact unit_factor_mult_normalize)
363   with False show ?thesis by simp
364 qed
366 lemma eq_Lcm_euclI:
367   fixes lcm :: "'a set \<Rightarrow> 'a"
368   assumes "\<And>A a. a \<in> A \<Longrightarrow> a dvd lcm A" and "\<And>A c. (\<And>a. a \<in> A \<Longrightarrow> a dvd c) \<Longrightarrow> lcm A dvd c"
369           "\<And>A. normalize (lcm A) = lcm A" shows "lcm = Lcm_eucl"
370   by (intro ext, rule associated_eqI) (auto simp: assms intro: Lcm_eucl_least)
372 lemma Gcd_eucl_dvd: "x \<in> A \<Longrightarrow> Gcd_eucl A dvd x"
373   unfolding Gcd_eucl_def by (auto intro: Lcm_eucl_least)
375 lemma Gcd_eucl_greatest: "(\<And>x. x \<in> A \<Longrightarrow> d dvd x) \<Longrightarrow> d dvd Gcd_eucl A"
376   unfolding Gcd_eucl_def by auto
378 lemma normalize_Gcd_eucl [simp]: "normalize (Gcd_eucl A) = Gcd_eucl A"
379   by (simp add: Gcd_eucl_def)
381 lemma Lcm_euclI:
382   assumes "\<And>x. x \<in> A \<Longrightarrow> x dvd d" "\<And>d'. (\<And>x. x \<in> A \<Longrightarrow> x dvd d') \<Longrightarrow> d dvd d'" "normalize d = d"
383   shows   "Lcm_eucl A = d"
384 proof -
385   have "normalize (Lcm_eucl A) = normalize d"
386     by (intro associatedI) (auto intro: dvd_Lcm_eucl Lcm_eucl_least assms)
387   thus ?thesis by (simp add: assms)
388 qed
390 lemma Gcd_euclI:
391   assumes "\<And>x. x \<in> A \<Longrightarrow> d dvd x" "\<And>d'. (\<And>x. x \<in> A \<Longrightarrow> d' dvd x) \<Longrightarrow> d' dvd d" "normalize d = d"
392   shows   "Gcd_eucl A = d"
393 proof -
394   have "normalize (Gcd_eucl A) = normalize d"
395     by (intro associatedI) (auto intro: Gcd_eucl_dvd Gcd_eucl_greatest assms)
396   thus ?thesis by (simp add: assms)
397 qed
399 lemmas lcm_gcd_eucl_facts =
400   gcd_eucl_dvd1 gcd_eucl_dvd2 gcd_eucl_greatest normalize_gcd_eucl lcm_eucl_def
401   Gcd_eucl_def Gcd_eucl_dvd Gcd_eucl_greatest normalize_Gcd_eucl
402   dvd_Lcm_eucl Lcm_eucl_least normalize_Lcm_eucl
404 lemma normalized_factors_product:
405   "{p. p dvd a * b \<and> normalize p = p} =
406      (\<lambda>(x,y). x * y)  ({p. p dvd a \<and> normalize p = p} \<times> {p. p dvd b \<and> normalize p = p})"
407 proof safe
408   fix p assume p: "p dvd a * b" "normalize p = p"
409   interpret semiring_gcd 1 0 "op *" gcd_eucl lcm_eucl "op div" "op +" "op -" normalize unit_factor
410     by standard (rule lcm_gcd_eucl_facts; assumption)+
411   from dvd_productE[OF p(1)] guess x y . note xy = this
412   define x' y' where "x' = normalize x" and "y' = normalize y"
413   have "p = x' * y'"
414     by (subst p(2) [symmetric]) (simp add: xy x'_def y'_def normalize_mult)
415   moreover from xy have "normalize x' = x'" "normalize y' = y'" "x' dvd a" "y' dvd b"
416     by (simp_all add: x'_def y'_def)
417   ultimately show "p \<in> (\<lambda>(x, y). x * y)
418                      ({p. p dvd a \<and> normalize p = p} \<times> {p. p dvd b \<and> normalize p = p})"
419     by blast
420 qed (auto simp: normalize_mult mult_dvd_mono)
423 subclass factorial_semiring
424 proof (standard, rule factorial_semiring_altI_aux)
425   fix x assume "x \<noteq> 0"
426   thus "finite {p. p dvd x \<and> normalize p = p}"
427   proof (induction "euclidean_size x" arbitrary: x rule: less_induct)
428     case (less x)
429     show ?case
430     proof (cases "\<exists>y. y dvd x \<and> \<not>x dvd y \<and> \<not>is_unit y")
431       case False
432       have "{p. p dvd x \<and> normalize p = p} \<subseteq> {1, normalize x}"
433       proof
434         fix p assume p: "p \<in> {p. p dvd x \<and> normalize p = p}"
435         with False have "is_unit p \<or> x dvd p" by blast
436         thus "p \<in> {1, normalize x}"
437         proof (elim disjE)
438           assume "is_unit p"
439           hence "normalize p = 1" by (simp add: is_unit_normalize)
440           with p show ?thesis by simp
441         next
442           assume "x dvd p"
443           with p have "normalize p = normalize x" by (intro associatedI) simp_all
444           with p show ?thesis by simp
445         qed
446       qed
447       moreover have "finite \<dots>" by simp
448       ultimately show ?thesis by (rule finite_subset)
450     next
451       case True
452       then obtain y where y: "y dvd x" "\<not>x dvd y" "\<not>is_unit y" by blast
453       define z where "z = x div y"
454       let ?fctrs = "\<lambda>x. {p. p dvd x \<and> normalize p = p}"
455       from y have x: "x = y * z" by (simp add: z_def)
456       with less.prems have "y \<noteq> 0" "z \<noteq> 0" by auto
457       from x y have "\<not>is_unit z" by (auto simp: mult_unit_dvd_iff)
458       have "?fctrs x = (\<lambda>(p,p'). p * p')  (?fctrs y \<times> ?fctrs z)"
459         by (subst x) (rule normalized_factors_product)
460       also have "\<not>y * z dvd y * 1" "\<not>y * z dvd 1 * z"
461         by (subst dvd_times_left_cancel_iff dvd_times_right_cancel_iff; fact)+
462       hence "finite ((\<lambda>(p,p'). p * p')  (?fctrs y \<times> ?fctrs z))"
463         by (intro finite_imageI finite_cartesian_product less dvd_proper_imp_size_less)
464            (auto simp: x)
465       finally show ?thesis .
466     qed
467   qed
468 next
469   interpret semiring_gcd 1 0 "op *" gcd_eucl lcm_eucl "op div" "op +" "op -" normalize unit_factor
470     by standard (rule lcm_gcd_eucl_facts; assumption)+
471   fix p assume p: "irreducible p"
472   thus "prime_elem p" by (rule irreducible_imp_prime_elem_gcd)
473 qed
475 lemma gcd_eucl_eq_gcd_factorial: "gcd_eucl = gcd_factorial"
476   by (intro ext gcd_euclI gcd_lcm_factorial)
478 lemma lcm_eucl_eq_lcm_factorial: "lcm_eucl = lcm_factorial"
479   by (intro ext) (simp add: lcm_eucl_def lcm_factorial_gcd_factorial gcd_eucl_eq_gcd_factorial)
481 lemma Gcd_eucl_eq_Gcd_factorial: "Gcd_eucl = Gcd_factorial"
482   by (intro ext Gcd_euclI gcd_lcm_factorial)
484 lemma Lcm_eucl_eq_Lcm_factorial: "Lcm_eucl = Lcm_factorial"
485   by (intro ext Lcm_euclI gcd_lcm_factorial)
487 lemmas eucl_eq_factorial =
488   gcd_eucl_eq_gcd_factorial lcm_eucl_eq_lcm_factorial
489   Gcd_eucl_eq_Gcd_factorial Lcm_eucl_eq_Lcm_factorial
491 end
493 class euclidean_ring = euclidean_semiring + idom
494 begin
496 function euclid_ext_aux :: "'a \<Rightarrow> _" where
497   "euclid_ext_aux r' r s' s t' t = (
498      if r = 0 then let c = 1 div unit_factor r' in (s' * c, t' * c, normalize r')
499      else let q = r' div r
500           in  euclid_ext_aux r (r' mod r) s (s' - q * s) t (t' - q * t))"
501 by auto
502 termination by (relation "measure (\<lambda>(_,b,_,_,_,_). euclidean_size b)") (simp_all add: mod_size_less)
504 declare euclid_ext_aux.simps [simp del]
506 lemma euclid_ext_aux_correct:
507   assumes "gcd_eucl r' r = gcd_eucl x y"
508   assumes "s' * x + t' * y = r'"
509   assumes "s * x + t * y = r"
510   shows   "case euclid_ext_aux r' r s' s t' t of (a,b,c) \<Rightarrow>
511              a * x + b * y = c \<and> c = gcd_eucl x y" (is "?P (euclid_ext_aux r' r s' s t' t)")
512 using assms
513 proof (induction r' r s' s t' t rule: euclid_ext_aux.induct)
514   case (1 r' r s' s t' t)
515   show ?case
516   proof (cases "r = 0")
517     case True
518     hence "euclid_ext_aux r' r s' s t' t =
519              (s' div unit_factor r', t' div unit_factor r', normalize r')"
520       by (subst euclid_ext_aux.simps) (simp add: Let_def)
521     also have "?P \<dots>"
522     proof safe
523       have "s' div unit_factor r' * x + t' div unit_factor r' * y =
524                 (s' * x + t' * y) div unit_factor r'"
525         by (cases "r' = 0") (simp_all add: unit_div_commute)
526       also have "s' * x + t' * y = r'" by fact
527       also have "\<dots> div unit_factor r' = normalize r'" by simp
528       finally show "s' div unit_factor r' * x + t' div unit_factor r' * y = normalize r'" .
529     next
530       from "1.prems" True show "normalize r' = gcd_eucl x y" by (simp add: gcd_eucl_0)
531     qed
532     finally show ?thesis .
533   next
534     case False
535     hence "euclid_ext_aux r' r s' s t' t =
536              euclid_ext_aux r (r' mod r) s (s' - r' div r * s) t (t' - r' div r * t)"
537       by (subst euclid_ext_aux.simps) (simp add: Let_def)
538     also from "1.prems" False have "?P \<dots>"
539     proof (intro "1.IH")
540       have "(s' - r' div r * s) * x + (t' - r' div r * t) * y =
541               (s' * x + t' * y) - r' div r * (s * x + t * y)" by (simp add: algebra_simps)
542       also have "s' * x + t' * y = r'" by fact
543       also have "s * x + t * y = r" by fact
544       also have "r' - r' div r * r = r' mod r" using div_mod_equality[of r' r]
545         by (simp add: algebra_simps)
546       finally show "(s' - r' div r * s) * x + (t' - r' div r * t) * y = r' mod r" .
547     qed (auto simp: gcd_eucl_non_0 algebra_simps div_mod_equality')
548     finally show ?thesis .
549   qed
550 qed
552 definition euclid_ext where
553   "euclid_ext a b = euclid_ext_aux a b 1 0 0 1"
555 lemma euclid_ext_0:
556   "euclid_ext a 0 = (1 div unit_factor a, 0, normalize a)"
557   by (simp add: euclid_ext_def euclid_ext_aux.simps)
559 lemma euclid_ext_left_0:
560   "euclid_ext 0 a = (0, 1 div unit_factor a, normalize a)"
561   by (simp add: euclid_ext_def euclid_ext_aux.simps)
563 lemma euclid_ext_correct':
564   "case euclid_ext x y of (a,b,c) \<Rightarrow> a * x + b * y = c \<and> c = gcd_eucl x y"
565   unfolding euclid_ext_def by (rule euclid_ext_aux_correct) simp_all
567 lemma euclid_ext_gcd_eucl:
568   "(case euclid_ext x y of (a,b,c) \<Rightarrow> c) = gcd_eucl x y"
569   using euclid_ext_correct'[of x y] by (simp add: case_prod_unfold)
571 definition euclid_ext' where
572   "euclid_ext' x y = (case euclid_ext x y of (a, b, _) \<Rightarrow> (a, b))"
574 lemma euclid_ext'_correct':
575   "case euclid_ext' x y of (a,b) \<Rightarrow> a * x + b * y = gcd_eucl x y"
576   using euclid_ext_correct'[of x y] by (simp add: case_prod_unfold euclid_ext'_def)
578 lemma euclid_ext'_0: "euclid_ext' a 0 = (1 div unit_factor a, 0)"
579   by (simp add: euclid_ext'_def euclid_ext_0)
581 lemma euclid_ext'_left_0: "euclid_ext' 0 a = (0, 1 div unit_factor a)"
582   by (simp add: euclid_ext'_def euclid_ext_left_0)
584 end
586 class euclidean_semiring_gcd = euclidean_semiring + gcd + Gcd +
587   assumes gcd_gcd_eucl: "gcd = gcd_eucl" and lcm_lcm_eucl: "lcm = lcm_eucl"
588   assumes Gcd_Gcd_eucl: "Gcd = Gcd_eucl" and Lcm_Lcm_eucl: "Lcm = Lcm_eucl"
589 begin
591 subclass semiring_gcd
592   by standard (simp_all add: gcd_gcd_eucl gcd_eucl_greatest lcm_lcm_eucl lcm_eucl_def)
594 subclass semiring_Gcd
595   by standard (auto simp: Gcd_Gcd_eucl Lcm_Lcm_eucl Gcd_eucl_def intro: Lcm_eucl_least)
597 subclass factorial_semiring_gcd
598 proof
599   fix a b
600   show "gcd a b = gcd_factorial a b"
601     by (rule sym, rule gcdI) (rule gcd_lcm_factorial; assumption)+
602   thus "lcm a b = lcm_factorial a b"
603     by (simp add: lcm_factorial_gcd_factorial lcm_gcd)
604 next
605   fix A
606   show "Gcd A = Gcd_factorial A"
607     by (rule sym, rule GcdI) (rule gcd_lcm_factorial; assumption)+
608   show "Lcm A = Lcm_factorial A"
609     by (rule sym, rule LcmI) (rule gcd_lcm_factorial; assumption)+
610 qed
612 lemma gcd_non_0:
613   "b \<noteq> 0 \<Longrightarrow> gcd a b = gcd b (a mod b)"
614   unfolding gcd_gcd_eucl by (fact gcd_eucl_non_0)
616 lemmas gcd_0 = gcd_0_right
617 lemmas dvd_gcd_iff = gcd_greatest_iff
618 lemmas gcd_greatest_iff = dvd_gcd_iff
620 lemma gcd_mod1 [simp]:
621   "gcd (a mod b) b = gcd a b"
622   by (rule gcdI, metis dvd_mod_iff gcd_dvd1 gcd_dvd2, simp_all add: gcd_greatest dvd_mod_iff)
624 lemma gcd_mod2 [simp]:
625   "gcd a (b mod a) = gcd a b"
626   by (rule gcdI, simp, metis dvd_mod_iff gcd_dvd1 gcd_dvd2, simp_all add: gcd_greatest dvd_mod_iff)
628 lemma euclidean_size_gcd_le1 [simp]:
629   assumes "a \<noteq> 0"
630   shows "euclidean_size (gcd a b) \<le> euclidean_size a"
631 proof -
632    have "gcd a b dvd a" by (rule gcd_dvd1)
633    then obtain c where A: "a = gcd a b * c" unfolding dvd_def by blast
634    with \<open>a \<noteq> 0\<close> show ?thesis by (subst (2) A, intro size_mult_mono) auto
635 qed
637 lemma euclidean_size_gcd_le2 [simp]:
638   "b \<noteq> 0 \<Longrightarrow> euclidean_size (gcd a b) \<le> euclidean_size b"
639   by (subst gcd.commute, rule euclidean_size_gcd_le1)
641 lemma euclidean_size_gcd_less1:
642   assumes "a \<noteq> 0" and "\<not>a dvd b"
643   shows "euclidean_size (gcd a b) < euclidean_size a"
644 proof (rule ccontr)
645   assume "\<not>euclidean_size (gcd a b) < euclidean_size a"
646   with \<open>a \<noteq> 0\<close> have A: "euclidean_size (gcd a b) = euclidean_size a"
647     by (intro le_antisym, simp_all)
648   have "a dvd gcd a b"
649     by (rule dvd_euclidean_size_eq_imp_dvd) (simp_all add: assms A)
650   hence "a dvd b" using dvd_gcdD2 by blast
651   with \<open>\<not>a dvd b\<close> show False by contradiction
652 qed
654 lemma euclidean_size_gcd_less2:
655   assumes "b \<noteq> 0" and "\<not>b dvd a"
656   shows "euclidean_size (gcd a b) < euclidean_size b"
657   using assms by (subst gcd.commute, rule euclidean_size_gcd_less1)
659 lemma euclidean_size_lcm_le1:
660   assumes "a \<noteq> 0" and "b \<noteq> 0"
661   shows "euclidean_size a \<le> euclidean_size (lcm a b)"
662 proof -
663   have "a dvd lcm a b" by (rule dvd_lcm1)
664   then obtain c where A: "lcm a b = a * c" ..
665   with \<open>a \<noteq> 0\<close> and \<open>b \<noteq> 0\<close> have "c \<noteq> 0" by (auto simp: lcm_eq_0_iff)
666   then show ?thesis by (subst A, intro size_mult_mono)
667 qed
669 lemma euclidean_size_lcm_le2:
670   "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> euclidean_size b \<le> euclidean_size (lcm a b)"
671   using euclidean_size_lcm_le1 [of b a] by (simp add: ac_simps)
673 lemma euclidean_size_lcm_less1:
674   assumes "b \<noteq> 0" and "\<not>b dvd a"
675   shows "euclidean_size a < euclidean_size (lcm a b)"
676 proof (rule ccontr)
677   from assms have "a \<noteq> 0" by auto
678   assume "\<not>euclidean_size a < euclidean_size (lcm a b)"
679   with \<open>a \<noteq> 0\<close> and \<open>b \<noteq> 0\<close> have "euclidean_size (lcm a b) = euclidean_size a"
680     by (intro le_antisym, simp, intro euclidean_size_lcm_le1)
681   with assms have "lcm a b dvd a"
682     by (rule_tac dvd_euclidean_size_eq_imp_dvd) (auto simp: lcm_eq_0_iff)
683   hence "b dvd a" by (rule lcm_dvdD2)
684   with \<open>\<not>b dvd a\<close> show False by contradiction
685 qed
687 lemma euclidean_size_lcm_less2:
688   assumes "a \<noteq> 0" and "\<not>a dvd b"
689   shows "euclidean_size b < euclidean_size (lcm a b)"
690   using assms euclidean_size_lcm_less1 [of a b] by (simp add: ac_simps)
692 lemma Lcm_eucl_set [code]:
693   "Lcm_eucl (set xs) = foldl lcm_eucl 1 xs"
694   by (simp add: Lcm_Lcm_eucl [symmetric] lcm_lcm_eucl Lcm_set)
696 lemma Gcd_eucl_set [code]:
697   "Gcd_eucl (set xs) = foldl gcd_eucl 0 xs"
698   by (simp add: Gcd_Gcd_eucl [symmetric] gcd_gcd_eucl Gcd_set)
700 end
703 text \<open>
704   A Euclidean ring is a Euclidean semiring with additive inverses. It provides a
705   few more lemmas; in particular, Bezout's lemma holds for any Euclidean ring.
706 \<close>
708 class euclidean_ring_gcd = euclidean_semiring_gcd + idom
709 begin
711 subclass euclidean_ring ..
712 subclass ring_gcd ..
713 subclass factorial_ring_gcd ..
715 lemma euclid_ext_gcd [simp]:
716   "(case euclid_ext a b of (_, _ , t) \<Rightarrow> t) = gcd a b"
717   using euclid_ext_correct'[of a b] by (simp add: case_prod_unfold Let_def gcd_gcd_eucl)
719 lemma euclid_ext_gcd' [simp]:
720   "euclid_ext a b = (r, s, t) \<Longrightarrow> t = gcd a b"
721   by (insert euclid_ext_gcd[of a b], drule (1) subst, simp)
723 lemma euclid_ext_correct:
724   "case euclid_ext x y of (a,b,c) \<Rightarrow> a * x + b * y = c \<and> c = gcd x y"
725   using euclid_ext_correct'[of x y]
726   by (simp add: gcd_gcd_eucl case_prod_unfold)
728 lemma euclid_ext'_correct:
729   "fst (euclid_ext' a b) * a + snd (euclid_ext' a b) * b = gcd a b"
730   using euclid_ext_correct'[of a b]
731   by (simp add: gcd_gcd_eucl case_prod_unfold euclid_ext'_def)
733 lemma bezout: "\<exists>s t. s * a + t * b = gcd a b"
734   using euclid_ext'_correct by blast
736 end
739 subsection \<open>Typical instances\<close>
741 instantiation nat :: euclidean_semiring
742 begin
744 definition [simp]:
745   "euclidean_size_nat = (id :: nat \<Rightarrow> nat)"
747 instance by standard simp_all
749 end
752 instantiation int :: euclidean_ring
753 begin
755 definition [simp]:
756   "euclidean_size_int = (nat \<circ> abs :: int \<Rightarrow> nat)"
758 instance by standard (auto simp add: abs_mult nat_mult_distrib split: abs_split)
760 end
762 instance nat :: euclidean_semiring_gcd
763 proof
764   show [simp]: "gcd = (gcd_eucl :: nat \<Rightarrow> _)" "Lcm = (Lcm_eucl :: nat set \<Rightarrow> _)"
765     by (simp_all add: eq_gcd_euclI eq_Lcm_euclI)
766   show "lcm = (lcm_eucl :: nat \<Rightarrow> _)" "Gcd = (Gcd_eucl :: nat set \<Rightarrow> _)"
767     by (intro ext, simp add: lcm_eucl_def lcm_nat_def Gcd_nat_def Gcd_eucl_def)+
768 qed
770 instance int :: euclidean_ring_gcd
771 proof
772   show [simp]: "gcd = (gcd_eucl :: int \<Rightarrow> _)" "Lcm = (Lcm_eucl :: int set \<Rightarrow> _)"
773     by (simp_all add: eq_gcd_euclI eq_Lcm_euclI)
774   show "lcm = (lcm_eucl :: int \<Rightarrow> _)" "Gcd = (Gcd_eucl :: int set \<Rightarrow> _)"
775     by (intro ext, simp add: lcm_eucl_def lcm_altdef_int
776           semiring_Gcd_class.Gcd_Lcm Gcd_eucl_def abs_mult)+
777 qed
779 end