src/HOL/Number_Theory/Euclidean_Algorithm.thy
 author haftmann Sun Oct 16 09:31:05 2016 +0200 (2016-10-16) changeset 64242 93c6f0da5c70 parent 64240 eabf80376aab child 64243 aee949f6642d permissions -rw-r--r--
more standardized theorem names for facts involving the div and mod identity
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 text \<open>
10   A Euclidean semiring is a semiring upon which the Euclidean algorithm can be
11   implemented. It must provide:
12   \begin{itemize}
13   \item division with remainder
14   \item a size function such that @{term "size (a mod b) < size b"}
15         for any @{term "b \<noteq> 0"}
16   \end{itemize}
17   The existence of these functions makes it possible to derive gcd and lcm functions
18   for any Euclidean semiring.
19 \<close>
20 class euclidean_semiring = semiring_modulo + normalization_semidom +
21   fixes euclidean_size :: "'a \<Rightarrow> nat"
22   assumes size_0 [simp]: "euclidean_size 0 = 0"
23   assumes mod_size_less:
24     "b \<noteq> 0 \<Longrightarrow> euclidean_size (a mod b) < euclidean_size b"
25   assumes size_mult_mono:
26     "b \<noteq> 0 \<Longrightarrow> euclidean_size a \<le> euclidean_size (a * b)"
27 begin
29 lemma mod_0 [simp]: "0 mod a = 0"
30   using div_mult_mod_eq [of 0 a] by simp
32 lemma dvd_mod_iff:
33   assumes "k dvd n"
34   shows   "(k dvd m mod n) = (k dvd m)"
35 proof -
36   from assms have "(k dvd m mod n) \<longleftrightarrow> (k dvd ((m div n) * n + m mod n))"
38   also have "(m div n) * n + m mod n = m"
39     using div_mult_mod_eq [of m n] by simp
40   finally show ?thesis .
41 qed
43 lemma mod_0_imp_dvd:
44   assumes "a mod b = 0"
45   shows   "b dvd a"
46 proof -
47   have "b dvd ((a div b) * b)" by simp
48   also have "(a div b) * b = a"
49     using div_mult_mod_eq [of a b] by (simp add: assms)
50   finally show ?thesis .
51 qed
53 lemma euclidean_size_normalize [simp]:
54   "euclidean_size (normalize a) = euclidean_size a"
55 proof (cases "a = 0")
56   case True
57   then show ?thesis
58     by simp
59 next
60   case [simp]: False
61   have "euclidean_size (normalize a) \<le> euclidean_size (normalize a * unit_factor a)"
62     by (rule size_mult_mono) simp
63   moreover have "euclidean_size a \<le> euclidean_size (a * (1 div unit_factor a))"
64     by (rule size_mult_mono) simp
65   ultimately show ?thesis
66     by simp
67 qed
69 lemma euclidean_division:
70   fixes a :: 'a and b :: 'a
71   assumes "b \<noteq> 0"
72   obtains s and t where "a = s * b + t"
73     and "euclidean_size t < euclidean_size b"
74 proof -
75   from div_mult_mod_eq [of a b]
76      have "a = a div b * b + a mod b" by simp
77   with that and assms show ?thesis by (auto simp add: mod_size_less)
78 qed
80 lemma dvd_euclidean_size_eq_imp_dvd:
81   assumes "a \<noteq> 0" and b_dvd_a: "b dvd a" and size_eq: "euclidean_size a = euclidean_size b"
82   shows "a dvd b"
83 proof (rule ccontr)
84   assume "\<not> a dvd b"
85   hence "b mod a \<noteq> 0" using mod_0_imp_dvd[of b a] by blast
86   then have "b mod a \<noteq> 0" by (simp add: mod_eq_0_iff_dvd)
87   from b_dvd_a have b_dvd_mod: "b dvd b mod a" by (simp add: dvd_mod_iff)
88   from b_dvd_mod obtain c where "b mod a = b * c" unfolding dvd_def by blast
89     with \<open>b mod a \<noteq> 0\<close> have "c \<noteq> 0" by auto
90   with \<open>b mod a = b * c\<close> have "euclidean_size (b mod a) \<ge> euclidean_size b"
91       using size_mult_mono by force
92   moreover from \<open>\<not> a dvd b\<close> and \<open>a \<noteq> 0\<close>
93   have "euclidean_size (b mod a) < euclidean_size a"
94       using mod_size_less by blast
95   ultimately show False using size_eq by simp
96 qed
98 lemma size_mult_mono': "b \<noteq> 0 \<Longrightarrow> euclidean_size a \<le> euclidean_size (b * a)"
99   by (subst mult.commute) (rule size_mult_mono)
101 lemma euclidean_size_times_unit:
102   assumes "is_unit a"
103   shows   "euclidean_size (a * b) = euclidean_size b"
104 proof (rule antisym)
105   from assms have [simp]: "a \<noteq> 0" by auto
106   thus "euclidean_size (a * b) \<ge> euclidean_size b" by (rule size_mult_mono')
107   from assms have "is_unit (1 div a)" by simp
108   hence "1 div a \<noteq> 0" by (intro notI) simp_all
109   hence "euclidean_size (a * b) \<le> euclidean_size ((1 div a) * (a * b))"
110     by (rule size_mult_mono')
111   also from assms have "(1 div a) * (a * b) = b"
112     by (simp add: algebra_simps unit_div_mult_swap)
113   finally show "euclidean_size (a * b) \<le> euclidean_size b" .
114 qed
116 lemma euclidean_size_unit: "is_unit a \<Longrightarrow> euclidean_size a = euclidean_size 1"
117   using euclidean_size_times_unit[of a 1] by simp
119 lemma unit_iff_euclidean_size:
120   "is_unit a \<longleftrightarrow> euclidean_size a = euclidean_size 1 \<and> a \<noteq> 0"
121 proof safe
122   assume A: "a \<noteq> 0" and B: "euclidean_size a = euclidean_size 1"
123   show "is_unit a" by (rule dvd_euclidean_size_eq_imp_dvd[OF A _ B]) simp_all
124 qed (auto intro: euclidean_size_unit)
126 lemma euclidean_size_times_nonunit:
127   assumes "a \<noteq> 0" "b \<noteq> 0" "\<not>is_unit a"
128   shows   "euclidean_size b < euclidean_size (a * b)"
129 proof (rule ccontr)
130   assume "\<not>euclidean_size b < euclidean_size (a * b)"
131   with size_mult_mono'[OF assms(1), of b]
132     have eq: "euclidean_size (a * b) = euclidean_size b" by simp
133   have "a * b dvd b"
134     by (rule dvd_euclidean_size_eq_imp_dvd[OF _ _ eq]) (insert assms, simp_all)
135   hence "a * b dvd 1 * b" by simp
136   with \<open>b \<noteq> 0\<close> have "is_unit a" by (subst (asm) dvd_times_right_cancel_iff)
137   with assms(3) show False by contradiction
138 qed
140 lemma dvd_imp_size_le:
141   assumes "a dvd b" "b \<noteq> 0"
142   shows   "euclidean_size a \<le> euclidean_size b"
143   using assms by (auto elim!: dvdE simp: size_mult_mono)
145 lemma dvd_proper_imp_size_less:
146   assumes "a dvd b" "\<not>b dvd a" "b \<noteq> 0"
147   shows   "euclidean_size a < euclidean_size b"
148 proof -
149   from assms(1) obtain c where "b = a * c" by (erule dvdE)
150   hence z: "b = c * a" by (simp add: mult.commute)
151   from z assms have "\<not>is_unit c" by (auto simp: mult.commute mult_unit_dvd_iff)
152   with z assms show ?thesis
153     by (auto intro!: euclidean_size_times_nonunit simp: )
154 qed
156 function gcd_eucl :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
157 where
158   "gcd_eucl a b = (if b = 0 then normalize a else gcd_eucl b (a mod b))"
159   by pat_completeness simp
160 termination
161   by (relation "measure (euclidean_size \<circ> snd)") (simp_all add: mod_size_less)
163 declare gcd_eucl.simps [simp del]
165 lemma gcd_eucl_induct [case_names zero mod]:
166   assumes H1: "\<And>b. P b 0"
167   and H2: "\<And>a b. b \<noteq> 0 \<Longrightarrow> P b (a mod b) \<Longrightarrow> P a b"
168   shows "P a b"
169 proof (induct a b rule: gcd_eucl.induct)
170   case ("1" a b)
171   show ?case
172   proof (cases "b = 0")
173     case True then show "P a b" by simp (rule H1)
174   next
175     case False
176     then have "P b (a mod b)"
177       by (rule "1.hyps")
178     with \<open>b \<noteq> 0\<close> show "P a b"
179       by (blast intro: H2)
180   qed
181 qed
183 definition lcm_eucl :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
184 where
185   "lcm_eucl a b = normalize (a * b) div gcd_eucl a b"
187 definition Lcm_eucl :: "'a set \<Rightarrow> 'a" \<comment> \<open>
188   Somewhat complicated definition of Lcm that has the advantage of working
189   for infinite sets as well\<close>
190 where
191   "Lcm_eucl A = (if \<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) then
192      let l = SOME l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l =
193        (LEAST n. \<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n)
194        in normalize l
195       else 0)"
197 definition Gcd_eucl :: "'a set \<Rightarrow> 'a"
198 where
199   "Gcd_eucl A = Lcm_eucl {d. \<forall>a\<in>A. d dvd a}"
201 declare Lcm_eucl_def Gcd_eucl_def [code del]
203 lemma gcd_eucl_0:
204   "gcd_eucl a 0 = normalize a"
205   by (simp add: gcd_eucl.simps [of a 0])
207 lemma gcd_eucl_0_left:
208   "gcd_eucl 0 a = normalize a"
209   by (simp_all add: gcd_eucl_0 gcd_eucl.simps [of 0 a])
211 lemma gcd_eucl_non_0:
212   "b \<noteq> 0 \<Longrightarrow> gcd_eucl a b = gcd_eucl b (a mod b)"
213   by (simp add: gcd_eucl.simps [of a b] gcd_eucl.simps [of b 0])
215 lemma gcd_eucl_dvd1 [iff]: "gcd_eucl a b dvd a"
216   and gcd_eucl_dvd2 [iff]: "gcd_eucl a b dvd b"
217   by (induct a b rule: gcd_eucl_induct)
218      (simp_all add: gcd_eucl_0 gcd_eucl_non_0 dvd_mod_iff)
220 lemma normalize_gcd_eucl [simp]:
221   "normalize (gcd_eucl a b) = gcd_eucl a b"
222   by (induct a b rule: gcd_eucl_induct) (simp_all add: gcd_eucl_0 gcd_eucl_non_0)
224 lemma gcd_eucl_greatest:
225   fixes k a b :: 'a
226   shows "k dvd a \<Longrightarrow> k dvd b \<Longrightarrow> k dvd gcd_eucl a b"
227 proof (induct a b rule: gcd_eucl_induct)
228   case (zero a) from zero(1) show ?case by (rule dvd_trans) (simp add: gcd_eucl_0)
229 next
230   case (mod a b)
231   then show ?case
232     by (simp add: gcd_eucl_non_0 dvd_mod_iff)
233 qed
235 lemma gcd_euclI:
236   fixes gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
237   assumes "d dvd a" "d dvd b" "normalize d = d"
238           "\<And>k. k dvd a \<Longrightarrow> k dvd b \<Longrightarrow> k dvd d"
239   shows   "gcd_eucl a b = d"
240   by (rule associated_eqI) (simp_all add: gcd_eucl_greatest assms)
242 lemma eq_gcd_euclI:
243   fixes gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
244   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"
245           "\<And>a b k. k dvd a \<Longrightarrow> k dvd b \<Longrightarrow> k dvd gcd a b"
246   shows   "gcd = gcd_eucl"
247   by (intro ext, rule associated_eqI) (simp_all add: gcd_eucl_greatest assms)
249 lemma gcd_eucl_zero [simp]:
250   "gcd_eucl a b = 0 \<longleftrightarrow> a = 0 \<and> b = 0"
251   by (metis dvd_0_left dvd_refl gcd_eucl_dvd1 gcd_eucl_dvd2 gcd_eucl_greatest)+
254 lemma dvd_Lcm_eucl [simp]: "a \<in> A \<Longrightarrow> a dvd Lcm_eucl A"
255   and Lcm_eucl_least: "(\<And>a. a \<in> A \<Longrightarrow> a dvd b) \<Longrightarrow> Lcm_eucl A dvd b"
256   and unit_factor_Lcm_eucl [simp]:
257           "unit_factor (Lcm_eucl A) = (if Lcm_eucl A = 0 then 0 else 1)"
258 proof -
259   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>
260     unit_factor (Lcm_eucl A) = (if Lcm_eucl A = 0 then 0 else 1)" (is ?thesis)
261   proof (cases "\<exists>l. l \<noteq>  0 \<and> (\<forall>a\<in>A. a dvd l)")
262     case False
263     hence "Lcm_eucl A = 0" by (auto simp: Lcm_eucl_def)
264     with False show ?thesis by auto
265   next
266     case True
267     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
268     define n where "n = (LEAST n. \<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n)"
269     define l where "l = (SOME l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n)"
270     have "\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"
271       apply (subst n_def)
272       apply (rule LeastI[of _ "euclidean_size l\<^sub>0"])
273       apply (rule exI[of _ l\<^sub>0])
274       apply (simp add: l\<^sub>0_props)
275       done
276     from someI_ex[OF this] have "l \<noteq> 0" and "\<forall>a\<in>A. a dvd l" and "euclidean_size l = n"
277       unfolding l_def by simp_all
278     {
279       fix l' assume "\<forall>a\<in>A. a dvd l'"
280       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)
281       moreover from \<open>l \<noteq> 0\<close> have "gcd_eucl l l' \<noteq> 0" by simp
282       ultimately have "\<exists>b. b \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd b) \<and>
283                           euclidean_size b = euclidean_size (gcd_eucl l l')"
284         by (intro exI[of _ "gcd_eucl l l'"], auto)
285       hence "euclidean_size (gcd_eucl l l') \<ge> n" by (subst n_def) (rule Least_le)
286       moreover have "euclidean_size (gcd_eucl l l') \<le> n"
287       proof -
288         have "gcd_eucl l l' dvd l" by simp
289         then obtain a where "l = gcd_eucl l l' * a" unfolding dvd_def by blast
290         with \<open>l \<noteq> 0\<close> have "a \<noteq> 0" by auto
291         hence "euclidean_size (gcd_eucl l l') \<le> euclidean_size (gcd_eucl l l' * a)"
292           by (rule size_mult_mono)
293         also have "gcd_eucl l l' * a = l" using \<open>l = gcd_eucl l l' * a\<close> ..
294         also note \<open>euclidean_size l = n\<close>
295         finally show "euclidean_size (gcd_eucl l l') \<le> n" .
296       qed
297       ultimately have *: "euclidean_size l = euclidean_size (gcd_eucl l l')"
298         by (intro le_antisym, simp_all add: \<open>euclidean_size l = n\<close>)
299       from \<open>l \<noteq> 0\<close> have "l dvd gcd_eucl l l'"
300         by (rule dvd_euclidean_size_eq_imp_dvd) (auto simp add: *)
301       hence "l dvd l'" by (rule dvd_trans[OF _ gcd_eucl_dvd2])
302     }
304     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>
305       have "(\<forall>a\<in>A. a dvd normalize l) \<and>
306         (\<forall>l'. (\<forall>a\<in>A. a dvd l') \<longrightarrow> normalize l dvd l') \<and>
307         unit_factor (normalize l) =
308         (if normalize l = 0 then 0 else 1)"
309       by (auto simp: unit_simps)
310     also from True have "normalize l = Lcm_eucl A"
311       by (simp add: Lcm_eucl_def Let_def n_def l_def)
312     finally show ?thesis .
313   qed
314   note A = this
316   {fix a assume "a \<in> A" then show "a dvd Lcm_eucl A" using A by blast}
317   {fix b assume "\<And>a. a \<in> A \<Longrightarrow> a dvd b" then show "Lcm_eucl A dvd b" using A by blast}
318   from A show "unit_factor (Lcm_eucl A) = (if Lcm_eucl A = 0 then 0 else 1)" by blast
319 qed
321 lemma normalize_Lcm_eucl [simp]:
322   "normalize (Lcm_eucl A) = Lcm_eucl A"
323 proof (cases "Lcm_eucl A = 0")
324   case True then show ?thesis by simp
325 next
326   case False
327   have "unit_factor (Lcm_eucl A) * normalize (Lcm_eucl A) = Lcm_eucl A"
328     by (fact unit_factor_mult_normalize)
329   with False show ?thesis by simp
330 qed
332 lemma eq_Lcm_euclI:
333   fixes lcm :: "'a set \<Rightarrow> 'a"
334   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"
335           "\<And>A. normalize (lcm A) = lcm A" shows "lcm = Lcm_eucl"
336   by (intro ext, rule associated_eqI) (auto simp: assms intro: Lcm_eucl_least)
338 lemma Gcd_eucl_dvd: "a \<in> A \<Longrightarrow> Gcd_eucl A dvd a"
339   unfolding Gcd_eucl_def by (auto intro: Lcm_eucl_least)
341 lemma Gcd_eucl_greatest: "(\<And>x. x \<in> A \<Longrightarrow> d dvd x) \<Longrightarrow> d dvd Gcd_eucl A"
342   unfolding Gcd_eucl_def by auto
344 lemma normalize_Gcd_eucl [simp]: "normalize (Gcd_eucl A) = Gcd_eucl A"
345   by (simp add: Gcd_eucl_def)
347 lemma Lcm_euclI:
348   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"
349   shows   "Lcm_eucl A = d"
350 proof -
351   have "normalize (Lcm_eucl A) = normalize d"
352     by (intro associatedI) (auto intro: dvd_Lcm_eucl Lcm_eucl_least assms)
353   thus ?thesis by (simp add: assms)
354 qed
356 lemma Gcd_euclI:
357   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"
358   shows   "Gcd_eucl A = d"
359 proof -
360   have "normalize (Gcd_eucl A) = normalize d"
361     by (intro associatedI) (auto intro: Gcd_eucl_dvd Gcd_eucl_greatest assms)
362   thus ?thesis by (simp add: assms)
363 qed
365 lemmas lcm_gcd_eucl_facts =
366   gcd_eucl_dvd1 gcd_eucl_dvd2 gcd_eucl_greatest normalize_gcd_eucl lcm_eucl_def
367   Gcd_eucl_def Gcd_eucl_dvd Gcd_eucl_greatest normalize_Gcd_eucl
368   dvd_Lcm_eucl Lcm_eucl_least normalize_Lcm_eucl
370 lemma normalized_factors_product:
371   "{p. p dvd a * b \<and> normalize p = p} =
372      (\<lambda>(x,y). x * y)  ({p. p dvd a \<and> normalize p = p} \<times> {p. p dvd b \<and> normalize p = p})"
373 proof safe
374   fix p assume p: "p dvd a * b" "normalize p = p"
375   interpret semiring_gcd 1 0 "op *" gcd_eucl lcm_eucl "op div" "op +" "op -" normalize unit_factor
376     by standard (rule lcm_gcd_eucl_facts; assumption)+
377   from dvd_productE[OF p(1)] guess x y . note xy = this
378   define x' y' where "x' = normalize x" and "y' = normalize y"
379   have "p = x' * y'"
380     by (subst p(2) [symmetric]) (simp add: xy x'_def y'_def normalize_mult)
381   moreover from xy have "normalize x' = x'" "normalize y' = y'" "x' dvd a" "y' dvd b"
382     by (simp_all add: x'_def y'_def)
383   ultimately show "p \<in> (\<lambda>(x, y). x * y)
384                      ({p. p dvd a \<and> normalize p = p} \<times> {p. p dvd b \<and> normalize p = p})"
385     by blast
386 qed (auto simp: normalize_mult mult_dvd_mono)
389 subclass factorial_semiring
390 proof (standard, rule factorial_semiring_altI_aux)
391   fix x assume "x \<noteq> 0"
392   thus "finite {p. p dvd x \<and> normalize p = p}"
393   proof (induction "euclidean_size x" arbitrary: x rule: less_induct)
394     case (less x)
395     show ?case
396     proof (cases "\<exists>y. y dvd x \<and> \<not>x dvd y \<and> \<not>is_unit y")
397       case False
398       have "{p. p dvd x \<and> normalize p = p} \<subseteq> {1, normalize x}"
399       proof
400         fix p assume p: "p \<in> {p. p dvd x \<and> normalize p = p}"
401         with False have "is_unit p \<or> x dvd p" by blast
402         thus "p \<in> {1, normalize x}"
403         proof (elim disjE)
404           assume "is_unit p"
405           hence "normalize p = 1" by (simp add: is_unit_normalize)
406           with p show ?thesis by simp
407         next
408           assume "x dvd p"
409           with p have "normalize p = normalize x" by (intro associatedI) simp_all
410           with p show ?thesis by simp
411         qed
412       qed
413       moreover have "finite \<dots>" by simp
414       ultimately show ?thesis by (rule finite_subset)
416     next
417       case True
418       then obtain y where y: "y dvd x" "\<not>x dvd y" "\<not>is_unit y" by blast
419       define z where "z = x div y"
420       let ?fctrs = "\<lambda>x. {p. p dvd x \<and> normalize p = p}"
421       from y have x: "x = y * z" by (simp add: z_def)
422       with less.prems have "y \<noteq> 0" "z \<noteq> 0" by auto
423       from x y have "\<not>is_unit z" by (auto simp: mult_unit_dvd_iff)
424       have "?fctrs x = (\<lambda>(p,p'). p * p')  (?fctrs y \<times> ?fctrs z)"
425         by (subst x) (rule normalized_factors_product)
426       also have "\<not>y * z dvd y * 1" "\<not>y * z dvd 1 * z"
427         by (subst dvd_times_left_cancel_iff dvd_times_right_cancel_iff; fact)+
428       hence "finite ((\<lambda>(p,p'). p * p')  (?fctrs y \<times> ?fctrs z))"
429         by (intro finite_imageI finite_cartesian_product less dvd_proper_imp_size_less)
430            (auto simp: x)
431       finally show ?thesis .
432     qed
433   qed
434 next
435   interpret semiring_gcd 1 0 "op *" gcd_eucl lcm_eucl "op div" "op +" "op -" normalize unit_factor
436     by standard (rule lcm_gcd_eucl_facts; assumption)+
437   fix p assume p: "irreducible p"
438   thus "prime_elem p" by (rule irreducible_imp_prime_elem_gcd)
439 qed
441 lemma gcd_eucl_eq_gcd_factorial: "gcd_eucl = gcd_factorial"
442   by (intro ext gcd_euclI gcd_lcm_factorial)
444 lemma lcm_eucl_eq_lcm_factorial: "lcm_eucl = lcm_factorial"
445   by (intro ext) (simp add: lcm_eucl_def lcm_factorial_gcd_factorial gcd_eucl_eq_gcd_factorial)
447 lemma Gcd_eucl_eq_Gcd_factorial: "Gcd_eucl = Gcd_factorial"
448   by (intro ext Gcd_euclI gcd_lcm_factorial)
450 lemma Lcm_eucl_eq_Lcm_factorial: "Lcm_eucl = Lcm_factorial"
451   by (intro ext Lcm_euclI gcd_lcm_factorial)
453 lemmas eucl_eq_factorial =
454   gcd_eucl_eq_gcd_factorial lcm_eucl_eq_lcm_factorial
455   Gcd_eucl_eq_Gcd_factorial Lcm_eucl_eq_Lcm_factorial
457 end
459 class euclidean_ring = euclidean_semiring + idom
460 begin
462 function euclid_ext_aux :: "'a \<Rightarrow> _" where
463   "euclid_ext_aux r' r s' s t' t = (
464      if r = 0 then let c = 1 div unit_factor r' in (s' * c, t' * c, normalize r')
465      else let q = r' div r
466           in  euclid_ext_aux r (r' mod r) s (s' - q * s) t (t' - q * t))"
467 by auto
468 termination by (relation "measure (\<lambda>(_,b,_,_,_,_). euclidean_size b)") (simp_all add: mod_size_less)
470 declare euclid_ext_aux.simps [simp del]
472 lemma euclid_ext_aux_correct:
473   assumes "gcd_eucl r' r = gcd_eucl a b"
474   assumes "s' * a + t' * b = r'"
475   assumes "s * a + t * b = r"
476   shows   "case euclid_ext_aux r' r s' s t' t of (x,y,c) \<Rightarrow>
477              x * a + y * b = c \<and> c = gcd_eucl a b" (is "?P (euclid_ext_aux r' r s' s t' t)")
478 using assms
479 proof (induction r' r s' s t' t rule: euclid_ext_aux.induct)
480   case (1 r' r s' s t' t)
481   show ?case
482   proof (cases "r = 0")
483     case True
484     hence "euclid_ext_aux r' r s' s t' t =
485              (s' div unit_factor r', t' div unit_factor r', normalize r')"
486       by (subst euclid_ext_aux.simps) (simp add: Let_def)
487     also have "?P \<dots>"
488     proof safe
489       have "s' div unit_factor r' * a + t' div unit_factor r' * b =
490                 (s' * a + t' * b) div unit_factor r'"
491         by (cases "r' = 0") (simp_all add: unit_div_commute)
492       also have "s' * a + t' * b = r'" by fact
493       also have "\<dots> div unit_factor r' = normalize r'" by simp
494       finally show "s' div unit_factor r' * a + t' div unit_factor r' * b = normalize r'" .
495     next
496       from "1.prems" True show "normalize r' = gcd_eucl a b" by (simp add: gcd_eucl_0)
497     qed
498     finally show ?thesis .
499   next
500     case False
501     hence "euclid_ext_aux r' r s' s t' t =
502              euclid_ext_aux r (r' mod r) s (s' - r' div r * s) t (t' - r' div r * t)"
503       by (subst euclid_ext_aux.simps) (simp add: Let_def)
504     also from "1.prems" False have "?P \<dots>"
505     proof (intro "1.IH")
506       have "(s' - r' div r * s) * a + (t' - r' div r * t) * b =
507               (s' * a + t' * b) - r' div r * (s * a + t * b)" by (simp add: algebra_simps)
508       also have "s' * a + t' * b = r'" by fact
509       also have "s * a + t * b = r" by fact
510       also have "r' - r' div r * r = r' mod r" using div_mult_mod_eq [of r' r]
511         by (simp add: algebra_simps)
512       finally show "(s' - r' div r * s) * a + (t' - r' div r * t) * b = r' mod r" .
513     qed (auto simp: gcd_eucl_non_0 algebra_simps div_mod_equality')
514     finally show ?thesis .
515   qed
516 qed
518 definition euclid_ext where
519   "euclid_ext a b = euclid_ext_aux a b 1 0 0 1"
521 lemma euclid_ext_0:
522   "euclid_ext a 0 = (1 div unit_factor a, 0, normalize a)"
523   by (simp add: euclid_ext_def euclid_ext_aux.simps)
525 lemma euclid_ext_left_0:
526   "euclid_ext 0 a = (0, 1 div unit_factor a, normalize a)"
527   by (simp add: euclid_ext_def euclid_ext_aux.simps)
529 lemma euclid_ext_correct':
530   "case euclid_ext a b of (x,y,c) \<Rightarrow> x * a + y * b = c \<and> c = gcd_eucl a b"
531   unfolding euclid_ext_def by (rule euclid_ext_aux_correct) simp_all
533 lemma euclid_ext_gcd_eucl:
534   "(case euclid_ext a b of (x,y,c) \<Rightarrow> c) = gcd_eucl a b"
535   using euclid_ext_correct'[of a b] by (simp add: case_prod_unfold)
537 definition euclid_ext' where
538   "euclid_ext' a b = (case euclid_ext a b of (x, y, _) \<Rightarrow> (x, y))"
540 lemma euclid_ext'_correct':
541   "case euclid_ext' a b of (x,y) \<Rightarrow> x * a + y * b = gcd_eucl a b"
542   using euclid_ext_correct'[of a b] by (simp add: case_prod_unfold euclid_ext'_def)
544 lemma euclid_ext'_0: "euclid_ext' a 0 = (1 div unit_factor a, 0)"
545   by (simp add: euclid_ext'_def euclid_ext_0)
547 lemma euclid_ext'_left_0: "euclid_ext' 0 a = (0, 1 div unit_factor a)"
548   by (simp add: euclid_ext'_def euclid_ext_left_0)
550 end
552 class euclidean_semiring_gcd = euclidean_semiring + gcd + Gcd +
553   assumes gcd_gcd_eucl: "gcd = gcd_eucl" and lcm_lcm_eucl: "lcm = lcm_eucl"
554   assumes Gcd_Gcd_eucl: "Gcd = Gcd_eucl" and Lcm_Lcm_eucl: "Lcm = Lcm_eucl"
555 begin
557 subclass semiring_gcd
558   by standard (simp_all add: gcd_gcd_eucl gcd_eucl_greatest lcm_lcm_eucl lcm_eucl_def)
560 subclass semiring_Gcd
561   by standard (auto simp: Gcd_Gcd_eucl Lcm_Lcm_eucl Gcd_eucl_def intro: Lcm_eucl_least)
563 subclass factorial_semiring_gcd
564 proof
565   fix a b
566   show "gcd a b = gcd_factorial a b"
567     by (rule sym, rule gcdI) (rule gcd_lcm_factorial; assumption)+
568   thus "lcm a b = lcm_factorial a b"
569     by (simp add: lcm_factorial_gcd_factorial lcm_gcd)
570 next
571   fix A
572   show "Gcd A = Gcd_factorial A"
573     by (rule sym, rule GcdI) (rule gcd_lcm_factorial; assumption)+
574   show "Lcm A = Lcm_factorial A"
575     by (rule sym, rule LcmI) (rule gcd_lcm_factorial; assumption)+
576 qed
578 lemma gcd_non_0:
579   "b \<noteq> 0 \<Longrightarrow> gcd a b = gcd b (a mod b)"
580   unfolding gcd_gcd_eucl by (fact gcd_eucl_non_0)
582 lemmas gcd_0 = gcd_0_right
583 lemmas dvd_gcd_iff = gcd_greatest_iff
584 lemmas gcd_greatest_iff = dvd_gcd_iff
586 lemma gcd_mod1 [simp]:
587   "gcd (a mod b) b = gcd a b"
588   by (rule gcdI, metis dvd_mod_iff gcd_dvd1 gcd_dvd2, simp_all add: gcd_greatest dvd_mod_iff)
590 lemma gcd_mod2 [simp]:
591   "gcd a (b mod a) = gcd a b"
592   by (rule gcdI, simp, metis dvd_mod_iff gcd_dvd1 gcd_dvd2, simp_all add: gcd_greatest dvd_mod_iff)
594 lemma euclidean_size_gcd_le1 [simp]:
595   assumes "a \<noteq> 0"
596   shows "euclidean_size (gcd a b) \<le> euclidean_size a"
597 proof -
598    have "gcd a b dvd a" by (rule gcd_dvd1)
599    then obtain c where A: "a = gcd a b * c" unfolding dvd_def by blast
600    with \<open>a \<noteq> 0\<close> show ?thesis by (subst (2) A, intro size_mult_mono) auto
601 qed
603 lemma euclidean_size_gcd_le2 [simp]:
604   "b \<noteq> 0 \<Longrightarrow> euclidean_size (gcd a b) \<le> euclidean_size b"
605   by (subst gcd.commute, rule euclidean_size_gcd_le1)
607 lemma euclidean_size_gcd_less1:
608   assumes "a \<noteq> 0" and "\<not>a dvd b"
609   shows "euclidean_size (gcd a b) < euclidean_size a"
610 proof (rule ccontr)
611   assume "\<not>euclidean_size (gcd a b) < euclidean_size a"
612   with \<open>a \<noteq> 0\<close> have A: "euclidean_size (gcd a b) = euclidean_size a"
613     by (intro le_antisym, simp_all)
614   have "a dvd gcd a b"
615     by (rule dvd_euclidean_size_eq_imp_dvd) (simp_all add: assms A)
616   hence "a dvd b" using dvd_gcdD2 by blast
617   with \<open>\<not>a dvd b\<close> show False by contradiction
618 qed
620 lemma euclidean_size_gcd_less2:
621   assumes "b \<noteq> 0" and "\<not>b dvd a"
622   shows "euclidean_size (gcd a b) < euclidean_size b"
623   using assms by (subst gcd.commute, rule euclidean_size_gcd_less1)
625 lemma euclidean_size_lcm_le1:
626   assumes "a \<noteq> 0" and "b \<noteq> 0"
627   shows "euclidean_size a \<le> euclidean_size (lcm a b)"
628 proof -
629   have "a dvd lcm a b" by (rule dvd_lcm1)
630   then obtain c where A: "lcm a b = a * c" ..
631   with \<open>a \<noteq> 0\<close> and \<open>b \<noteq> 0\<close> have "c \<noteq> 0" by (auto simp: lcm_eq_0_iff)
632   then show ?thesis by (subst A, intro size_mult_mono)
633 qed
635 lemma euclidean_size_lcm_le2:
636   "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> euclidean_size b \<le> euclidean_size (lcm a b)"
637   using euclidean_size_lcm_le1 [of b a] by (simp add: ac_simps)
639 lemma euclidean_size_lcm_less1:
640   assumes "b \<noteq> 0" and "\<not>b dvd a"
641   shows "euclidean_size a < euclidean_size (lcm a b)"
642 proof (rule ccontr)
643   from assms have "a \<noteq> 0" by auto
644   assume "\<not>euclidean_size a < euclidean_size (lcm a b)"
645   with \<open>a \<noteq> 0\<close> and \<open>b \<noteq> 0\<close> have "euclidean_size (lcm a b) = euclidean_size a"
646     by (intro le_antisym, simp, intro euclidean_size_lcm_le1)
647   with assms have "lcm a b dvd a"
648     by (rule_tac dvd_euclidean_size_eq_imp_dvd) (auto simp: lcm_eq_0_iff)
649   hence "b dvd a" by (rule lcm_dvdD2)
650   with \<open>\<not>b dvd a\<close> show False by contradiction
651 qed
653 lemma euclidean_size_lcm_less2:
654   assumes "a \<noteq> 0" and "\<not>a dvd b"
655   shows "euclidean_size b < euclidean_size (lcm a b)"
656   using assms euclidean_size_lcm_less1 [of a b] by (simp add: ac_simps)
658 lemma Lcm_eucl_set [code]:
659   "Lcm_eucl (set xs) = foldl lcm_eucl 1 xs"
660   by (simp add: Lcm_Lcm_eucl [symmetric] lcm_lcm_eucl Lcm_set)
662 lemma Gcd_eucl_set [code]:
663   "Gcd_eucl (set xs) = foldl gcd_eucl 0 xs"
664   by (simp add: Gcd_Gcd_eucl [symmetric] gcd_gcd_eucl Gcd_set)
666 end
669 text \<open>
670   A Euclidean ring is a Euclidean semiring with additive inverses. It provides a
671   few more lemmas; in particular, Bezout's lemma holds for any Euclidean ring.
672 \<close>
674 class euclidean_ring_gcd = euclidean_semiring_gcd + idom
675 begin
677 subclass euclidean_ring ..
678 subclass ring_gcd ..
679 subclass factorial_ring_gcd ..
681 lemma euclid_ext_gcd [simp]:
682   "(case euclid_ext a b of (_, _ , t) \<Rightarrow> t) = gcd a b"
683   using euclid_ext_correct'[of a b] by (simp add: case_prod_unfold Let_def gcd_gcd_eucl)
685 lemma euclid_ext_gcd' [simp]:
686   "euclid_ext a b = (r, s, t) \<Longrightarrow> t = gcd a b"
687   by (insert euclid_ext_gcd[of a b], drule (1) subst, simp)
689 lemma euclid_ext_correct:
690   "case euclid_ext a b of (x,y,c) \<Rightarrow> x * a + y * b = c \<and> c = gcd a b"
691   using euclid_ext_correct'[of a b]
692   by (simp add: gcd_gcd_eucl case_prod_unfold)
694 lemma euclid_ext'_correct:
695   "fst (euclid_ext' a b) * a + snd (euclid_ext' a b) * b = gcd a b"
696   using euclid_ext_correct'[of a b]
697   by (simp add: gcd_gcd_eucl case_prod_unfold euclid_ext'_def)
699 lemma bezout: "\<exists>s t. s * a + t * b = gcd a b"
700   using euclid_ext'_correct by blast
702 end
705 subsection \<open>Typical instances\<close>
707 instantiation nat :: euclidean_semiring
708 begin
710 definition [simp]:
711   "euclidean_size_nat = (id :: nat \<Rightarrow> nat)"
713 instance by standard simp_all
715 end
718 instantiation int :: euclidean_ring
719 begin
721 definition [simp]:
722   "euclidean_size_int = (nat \<circ> abs :: int \<Rightarrow> nat)"
724 instance by standard (auto simp add: abs_mult nat_mult_distrib split: abs_split)
726 end
728 instance nat :: euclidean_semiring_gcd
729 proof
730   show [simp]: "gcd = (gcd_eucl :: nat \<Rightarrow> _)" "Lcm = (Lcm_eucl :: nat set \<Rightarrow> _)"
731     by (simp_all add: eq_gcd_euclI eq_Lcm_euclI)
732   show "lcm = (lcm_eucl :: nat \<Rightarrow> _)" "Gcd = (Gcd_eucl :: nat set \<Rightarrow> _)"
733     by (intro ext, simp add: lcm_eucl_def lcm_nat_def Gcd_nat_def Gcd_eucl_def)+
734 qed
736 instance int :: euclidean_ring_gcd
737 proof
738   show [simp]: "gcd = (gcd_eucl :: int \<Rightarrow> _)" "Lcm = (Lcm_eucl :: int set \<Rightarrow> _)"
739     by (simp_all add: eq_gcd_euclI eq_Lcm_euclI)
740   show "lcm = (lcm_eucl :: int \<Rightarrow> _)" "Gcd = (Gcd_eucl :: int set \<Rightarrow> _)"
741     by (intro ext, simp add: lcm_eucl_def lcm_altdef_int
742           semiring_Gcd_class.Gcd_Lcm Gcd_eucl_def abs_mult)+
743 qed
745 end