src/HOL/Number_Theory/Euclidean_Algorithm.thy
 author haftmann Mon Jan 09 19:13:49 2017 +0100 (2017-01-09) changeset 64850 fc9265882329 parent 64848 c50db2128048 child 64911 f0e07600de47 permissions -rw-r--r--
gcd/lcm on finite sets
1 (*  Title:      HOL/Number_Theory/Euclidean_Algorithm.thy
2     Author:     Manuel Eberl, TU Muenchen
3 *)
5 section \<open>Abstract euclidean algorithm in euclidean (semi)rings\<close>
7 theory Euclidean_Algorithm
8   imports "~~/src/HOL/GCD"
9     "~~/src/HOL/Number_Theory/Factorial_Ring"
10 begin
12 subsection \<open>Generic construction of the (simple) euclidean algorithm\<close>
14 context euclidean_semiring
15 begin
17 context
18 begin
20 qualified function gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
21   where "gcd a b = (if b = 0 then normalize a else gcd b (a mod b))"
22   by pat_completeness simp
23 termination
24   by (relation "measure (euclidean_size \<circ> snd)") (simp_all add: mod_size_less)
26 declare gcd.simps [simp del]
28 lemma eucl_induct [case_names zero mod]:
29   assumes H1: "\<And>b. P b 0"
30   and H2: "\<And>a b. b \<noteq> 0 \<Longrightarrow> P b (a mod b) \<Longrightarrow> P a b"
31   shows "P a b"
32 proof (induct a b rule: gcd.induct)
33   case (1 a b)
34   show ?case
35   proof (cases "b = 0")
36     case True then show "P a b" by simp (rule H1)
37   next
38     case False
39     then have "P b (a mod b)"
40       by (rule "1.hyps")
41     with \<open>b \<noteq> 0\<close> show "P a b"
42       by (blast intro: H2)
43   qed
44 qed
46 qualified lemma gcd_0:
47   "gcd a 0 = normalize a"
48   by (simp add: gcd.simps [of a 0])
50 qualified lemma gcd_mod:
51   "a \<noteq> 0 \<Longrightarrow> gcd a (b mod a) = gcd b a"
52   by (simp add: gcd.simps [of b 0] gcd.simps [of b a])
54 qualified definition lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
55   where "lcm a b = normalize (a * b) div gcd a b"
57 qualified definition Lcm :: "'a set \<Rightarrow> 'a" \<comment>
58     \<open>Somewhat complicated definition of Lcm that has the advantage of working
59     for infinite sets as well\<close>
60   where
61   [code del]: "Lcm A = (if \<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) then
62      let l = SOME l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l =
63        (LEAST n. \<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n)
64        in normalize l
65       else 0)"
67 qualified definition Gcd :: "'a set \<Rightarrow> 'a"
68   where [code del]: "Gcd A = Lcm {d. \<forall>a\<in>A. d dvd a}"
70 end
72 lemma semiring_gcd:
73   "class.semiring_gcd one zero times gcd lcm
74     divide plus minus unit_factor normalize"
75 proof
76   show "gcd a b dvd a"
77     and "gcd a b dvd b" for a b
78     by (induct a b rule: eucl_induct)
79       (simp_all add: local.gcd_0 local.gcd_mod dvd_mod_iff)
80 next
81   show "c dvd a \<Longrightarrow> c dvd b \<Longrightarrow> c dvd gcd a b" for a b c
82   proof (induct a b rule: eucl_induct)
83     case (zero a) from \<open>c dvd a\<close> show ?case
84       by (rule dvd_trans) (simp add: local.gcd_0)
85   next
86     case (mod a b)
87     then show ?case
88       by (simp add: local.gcd_mod dvd_mod_iff)
89   qed
90 next
91   show "normalize (gcd a b) = gcd a b" for a b
92     by (induct a b rule: eucl_induct)
93       (simp_all add: local.gcd_0 local.gcd_mod)
94 next
95   show "lcm a b = normalize (a * b) div gcd a b" for a b
96     by (fact local.lcm_def)
97 qed
99 interpretation semiring_gcd one zero times gcd lcm
100   divide plus minus unit_factor normalize
101   by (fact semiring_gcd)
103 lemma semiring_Gcd:
104   "class.semiring_Gcd one zero times gcd lcm Gcd Lcm
105     divide plus minus unit_factor normalize"
106 proof -
107   show ?thesis
108   proof
109     have "(\<forall>a\<in>A. a dvd Lcm A) \<and> (\<forall>b. (\<forall>a\<in>A. a dvd b) \<longrightarrow> Lcm A dvd b)" for A
110     proof (cases "\<exists>l. l \<noteq>  0 \<and> (\<forall>a\<in>A. a dvd l)")
111       case False
112       then have "Lcm A = 0"
113         by (auto simp add: local.Lcm_def)
114       with False show ?thesis
115         by auto
116     next
117       case True
118       then obtain l\<^sub>0 where l\<^sub>0_props: "l\<^sub>0 \<noteq> 0" "\<forall>a\<in>A. a dvd l\<^sub>0" by blast
119       define n where "n = (LEAST n. \<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n)"
120       define l where "l = (SOME l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n)"
121       have "\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"
122         apply (subst n_def)
123         apply (rule LeastI [of _ "euclidean_size l\<^sub>0"])
124         apply (rule exI [of _ l\<^sub>0])
125         apply (simp add: l\<^sub>0_props)
126         done
127       from someI_ex [OF this] have "l \<noteq> 0" and "\<forall>a\<in>A. a dvd l"
128         and "euclidean_size l = n"
129         unfolding l_def by simp_all
130       {
131         fix l' assume "\<forall>a\<in>A. a dvd l'"
132         with \<open>\<forall>a\<in>A. a dvd l\<close> have "\<forall>a\<in>A. a dvd gcd l l'"
133           by (auto intro: gcd_greatest)
134         moreover from \<open>l \<noteq> 0\<close> have "gcd l l' \<noteq> 0"
135           by simp
136         ultimately have "\<exists>b. b \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd b) \<and>
137           euclidean_size b = euclidean_size (gcd l l')"
138           by (intro exI [of _ "gcd l l'"], auto)
139         then have "euclidean_size (gcd l l') \<ge> n"
140           by (subst n_def) (rule Least_le)
141         moreover have "euclidean_size (gcd l l') \<le> n"
142         proof -
143           have "gcd l l' dvd l"
144             by simp
145           then obtain a where "l = gcd l l' * a" ..
146           with \<open>l \<noteq> 0\<close> have "a \<noteq> 0"
147             by auto
148           hence "euclidean_size (gcd l l') \<le> euclidean_size (gcd l l' * a)"
149             by (rule size_mult_mono)
150           also have "gcd l l' * a = l" using \<open>l = gcd l l' * a\<close> ..
151           also note \<open>euclidean_size l = n\<close>
152           finally show "euclidean_size (gcd l l') \<le> n" .
153         qed
154         ultimately have *: "euclidean_size l = euclidean_size (gcd l l')"
155           by (intro le_antisym, simp_all add: \<open>euclidean_size l = n\<close>)
156         from \<open>l \<noteq> 0\<close> have "l dvd gcd l l'"
157           by (rule dvd_euclidean_size_eq_imp_dvd) (auto simp add: *)
158         hence "l dvd l'" by (rule dvd_trans [OF _ gcd_dvd2])
159       }
160       with \<open>\<forall>a\<in>A. a dvd l\<close> and \<open>l \<noteq> 0\<close>
161         have "(\<forall>a\<in>A. a dvd normalize l) \<and>
162           (\<forall>l'. (\<forall>a\<in>A. a dvd l') \<longrightarrow> normalize l dvd l')"
163         by auto
164       also from True have "normalize l = Lcm A"
165         by (simp add: local.Lcm_def Let_def n_def l_def)
166       finally show ?thesis .
167     qed
168     then show dvd_Lcm: "a \<in> A \<Longrightarrow> a dvd Lcm A"
169       and Lcm_least: "(\<And>a. a \<in> A \<Longrightarrow> a dvd b) \<Longrightarrow> Lcm A dvd b" for A and a b
170       by auto
171     show "a \<in> A \<Longrightarrow> Gcd A dvd a" for A and a
172       by (auto simp add: local.Gcd_def intro: Lcm_least)
173     show "(\<And>a. a \<in> A \<Longrightarrow> b dvd a) \<Longrightarrow> b dvd Gcd A" for A and b
174       by (auto simp add: local.Gcd_def intro: dvd_Lcm)
175     show [simp]: "normalize (Lcm A) = Lcm A" for A
176       by (simp add: local.Lcm_def)
177     show "normalize (Gcd A) = Gcd A" for A
178       by (simp add: local.Gcd_def)
179   qed
180 qed
182 interpretation semiring_Gcd one zero times gcd lcm Gcd Lcm
183     divide plus minus unit_factor normalize
184   by (fact semiring_Gcd)
186 subclass factorial_semiring
187 proof -
188   show "class.factorial_semiring divide plus minus zero times one
189      unit_factor normalize"
190   proof (standard, rule factorial_semiring_altI_aux) -- \<open>FIXME rule\<close>
191     fix x assume "x \<noteq> 0"
192     thus "finite {p. p dvd x \<and> normalize p = p}"
193     proof (induction "euclidean_size x" arbitrary: x rule: less_induct)
194       case (less x)
195       show ?case
196       proof (cases "\<exists>y. y dvd x \<and> \<not>x dvd y \<and> \<not>is_unit y")
197         case False
198         have "{p. p dvd x \<and> normalize p = p} \<subseteq> {1, normalize x}"
199         proof
200           fix p assume p: "p \<in> {p. p dvd x \<and> normalize p = p}"
201           with False have "is_unit p \<or> x dvd p" by blast
202           thus "p \<in> {1, normalize x}"
203           proof (elim disjE)
204             assume "is_unit p"
205             hence "normalize p = 1" by (simp add: is_unit_normalize)
206             with p show ?thesis by simp
207           next
208             assume "x dvd p"
209             with p have "normalize p = normalize x" by (intro associatedI) simp_all
210             with p show ?thesis by simp
211           qed
212         qed
213         moreover have "finite \<dots>" by simp
214         ultimately show ?thesis by (rule finite_subset)
215       next
216         case True
217         then obtain y where y: "y dvd x" "\<not>x dvd y" "\<not>is_unit y" by blast
218         define z where "z = x div y"
219         let ?fctrs = "\<lambda>x. {p. p dvd x \<and> normalize p = p}"
220         from y have x: "x = y * z" by (simp add: z_def)
221         with less.prems have "y \<noteq> 0" "z \<noteq> 0" by auto
222         have normalized_factors_product:
223           "{p. p dvd a * b \<and> normalize p = p} =
224              (\<lambda>(x,y). x * y) ` ({p. p dvd a \<and> normalize p = p} \<times> {p. p dvd b \<and> normalize p = p})" for a b
225         proof safe
226           fix p assume p: "p dvd a * b" "normalize p = p"
227           from dvd_productE[OF p(1)] guess x y . note xy = this
228           define x' y' where "x' = normalize x" and "y' = normalize y"
229           have "p = x' * y'"
230             by (subst p(2) [symmetric]) (simp add: xy x'_def y'_def normalize_mult)
231           moreover from xy have "normalize x' = x'" "normalize y' = y'" "x' dvd a" "y' dvd b"
232             by (simp_all add: x'_def y'_def)
233           ultimately show "p \<in> (\<lambda>(x, y). x * y) `
234             ({p. p dvd a \<and> normalize p = p} \<times> {p. p dvd b \<and> normalize p = p})"
235             by blast
236         qed (auto simp: normalize_mult mult_dvd_mono)
237         from x y have "\<not>is_unit z" by (auto simp: mult_unit_dvd_iff)
238         have "?fctrs x = (\<lambda>(p,p'). p * p') ` (?fctrs y \<times> ?fctrs z)"
239           by (subst x) (rule normalized_factors_product)
240         also have "\<not>y * z dvd y * 1" "\<not>y * z dvd 1 * z"
241           by (subst dvd_times_left_cancel_iff dvd_times_right_cancel_iff; fact)+
242         hence "finite ((\<lambda>(p,p'). p * p') ` (?fctrs y \<times> ?fctrs z))"
243           by (intro finite_imageI finite_cartesian_product less dvd_proper_imp_size_less)
244              (auto simp: x)
245         finally show ?thesis .
246       qed
247     qed
248   next
249     fix p
250     assume "irreducible p"
251     then show "prime_elem p"
252       by (rule irreducible_imp_prime_elem_gcd)
253   qed
254 qed
256 lemma Gcd_eucl_set [code]:
257   "Gcd (set xs) = fold gcd xs 0"
258   by (fact Gcd_set_eq_fold)
260 lemma Lcm_eucl_set [code]:
261   "Lcm (set xs) = fold lcm xs 1"
262   by (fact Lcm_set_eq_fold)
264 end
266 hide_const (open) gcd lcm Gcd Lcm
268 lemma prime_elem_int_abs_iff [simp]:
269   fixes p :: int
270   shows "prime_elem \<bar>p\<bar> \<longleftrightarrow> prime_elem p"
271   using prime_elem_normalize_iff [of p] by simp
273 lemma prime_elem_int_minus_iff [simp]:
274   fixes p :: int
275   shows "prime_elem (- p) \<longleftrightarrow> prime_elem p"
276   using prime_elem_normalize_iff [of "- p"] by simp
278 lemma prime_int_iff:
279   fixes p :: int
280   shows "prime p \<longleftrightarrow> p > 0 \<and> prime_elem p"
281   by (auto simp add: prime_def dest: prime_elem_not_zeroI)
284 subsection \<open>The (simple) euclidean algorithm as gcd computation\<close>
286 class euclidean_semiring_gcd = euclidean_semiring + gcd + Gcd +
287   assumes gcd_eucl: "Euclidean_Algorithm.gcd = GCD.gcd"
288     and lcm_eucl: "Euclidean_Algorithm.lcm = GCD.lcm"
289   assumes Gcd_eucl: "Euclidean_Algorithm.Gcd = GCD.Gcd"
290     and Lcm_eucl: "Euclidean_Algorithm.Lcm = GCD.Lcm"
291 begin
293 subclass semiring_gcd
294   unfolding gcd_eucl [symmetric] lcm_eucl [symmetric]
295   by (fact semiring_gcd)
297 subclass semiring_Gcd
298   unfolding  gcd_eucl [symmetric] lcm_eucl [symmetric]
299     Gcd_eucl [symmetric] Lcm_eucl [symmetric]
300   by (fact semiring_Gcd)
302 subclass factorial_semiring_gcd
303 proof
304   show "gcd a b = gcd_factorial a b" for a b
305     apply (rule sym)
306     apply (rule gcdI)
307        apply (fact gcd_lcm_factorial)+
308     done
309   then show "lcm a b = lcm_factorial a b" for a b
310     by (simp add: lcm_factorial_gcd_factorial lcm_gcd)
311   show "Gcd A = Gcd_factorial A" for A
312     apply (rule sym)
313     apply (rule GcdI)
314        apply (fact gcd_lcm_factorial)+
315     done
316   show "Lcm A = Lcm_factorial A" for A
317     apply (rule sym)
318     apply (rule LcmI)
319        apply (fact gcd_lcm_factorial)+
320     done
321 qed
323 lemma gcd_mod_right [simp]:
324   "a \<noteq> 0 \<Longrightarrow> gcd a (b mod a) = gcd a b"
325   unfolding gcd.commute [of a b]
326   by (simp add: gcd_eucl [symmetric] local.gcd_mod)
328 lemma gcd_mod_left [simp]:
329   "b \<noteq> 0 \<Longrightarrow> gcd (a mod b) b = gcd a b"
330   by (drule gcd_mod_right [of _ a]) (simp add: gcd.commute)
332 lemma euclidean_size_gcd_le1 [simp]:
333   assumes "a \<noteq> 0"
334   shows "euclidean_size (gcd a b) \<le> euclidean_size a"
335 proof -
336   from gcd_dvd1 obtain c where A: "a = gcd a b * c" ..
337   with assms have "c \<noteq> 0"
338     by auto
339   moreover from this
340   have "euclidean_size (gcd a b) \<le> euclidean_size (gcd a b * c)"
341     by (rule size_mult_mono)
342   with A show ?thesis
343     by simp
344 qed
346 lemma euclidean_size_gcd_le2 [simp]:
347   "b \<noteq> 0 \<Longrightarrow> euclidean_size (gcd a b) \<le> euclidean_size b"
348   by (subst gcd.commute, rule euclidean_size_gcd_le1)
350 lemma euclidean_size_gcd_less1:
351   assumes "a \<noteq> 0" and "\<not> a dvd b"
352   shows "euclidean_size (gcd a b) < euclidean_size a"
353 proof (rule ccontr)
354   assume "\<not>euclidean_size (gcd a b) < euclidean_size a"
355   with \<open>a \<noteq> 0\<close> have A: "euclidean_size (gcd a b) = euclidean_size a"
356     by (intro le_antisym, simp_all)
357   have "a dvd gcd a b"
358     by (rule dvd_euclidean_size_eq_imp_dvd) (simp_all add: assms A)
359   hence "a dvd b" using dvd_gcdD2 by blast
360   with \<open>\<not> a dvd b\<close> show False by contradiction
361 qed
363 lemma euclidean_size_gcd_less2:
364   assumes "b \<noteq> 0" and "\<not> b dvd a"
365   shows "euclidean_size (gcd a b) < euclidean_size b"
366   using assms by (subst gcd.commute, rule euclidean_size_gcd_less1)
368 lemma euclidean_size_lcm_le1:
369   assumes "a \<noteq> 0" and "b \<noteq> 0"
370   shows "euclidean_size a \<le> euclidean_size (lcm a b)"
371 proof -
372   have "a dvd lcm a b" by (rule dvd_lcm1)
373   then obtain c where A: "lcm a b = a * c" ..
374   with \<open>a \<noteq> 0\<close> and \<open>b \<noteq> 0\<close> have "c \<noteq> 0" by (auto simp: lcm_eq_0_iff)
375   then show ?thesis by (subst A, intro size_mult_mono)
376 qed
378 lemma euclidean_size_lcm_le2:
379   "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> euclidean_size b \<le> euclidean_size (lcm a b)"
380   using euclidean_size_lcm_le1 [of b a] by (simp add: ac_simps)
382 lemma euclidean_size_lcm_less1:
383   assumes "b \<noteq> 0" and "\<not> b dvd a"
384   shows "euclidean_size a < euclidean_size (lcm a b)"
385 proof (rule ccontr)
386   from assms have "a \<noteq> 0" by auto
387   assume "\<not>euclidean_size a < euclidean_size (lcm a b)"
388   with \<open>a \<noteq> 0\<close> and \<open>b \<noteq> 0\<close> have "euclidean_size (lcm a b) = euclidean_size a"
389     by (intro le_antisym, simp, intro euclidean_size_lcm_le1)
390   with assms have "lcm a b dvd a"
391     by (rule_tac dvd_euclidean_size_eq_imp_dvd) (auto simp: lcm_eq_0_iff)
392   hence "b dvd a" by (rule lcm_dvdD2)
393   with \<open>\<not>b dvd a\<close> show False by contradiction
394 qed
396 lemma euclidean_size_lcm_less2:
397   assumes "a \<noteq> 0" and "\<not> a dvd b"
398   shows "euclidean_size b < euclidean_size (lcm a b)"
399   using assms euclidean_size_lcm_less1 [of a b] by (simp add: ac_simps)
401 end
403 lemma factorial_euclidean_semiring_gcdI:
404   "OFCLASS('a::{factorial_semiring_gcd, euclidean_semiring}, euclidean_semiring_gcd_class)"
405 proof
406   interpret semiring_Gcd 1 0 times
407     Euclidean_Algorithm.gcd Euclidean_Algorithm.lcm
408     Euclidean_Algorithm.Gcd Euclidean_Algorithm.Lcm
409     divide plus minus unit_factor normalize
410     rewrites "dvd.dvd op * = Rings.dvd"
411     by (fact semiring_Gcd) (simp add: dvd.dvd_def dvd_def fun_eq_iff)
412   show [simp]: "Euclidean_Algorithm.gcd = (gcd :: 'a \<Rightarrow> _)"
413   proof (rule ext)+
414     fix a b :: 'a
415     show "Euclidean_Algorithm.gcd a b = gcd a b"
416     proof (induct a b rule: eucl_induct)
417       case zero
418       then show ?case
419         by simp
420     next
421       case (mod a b)
422       moreover have "gcd b (a mod b) = gcd b a"
423         using GCD.gcd_add_mult [of b "a div b" "a mod b", symmetric]
424           by (simp add: div_mult_mod_eq)
425       ultimately show ?case
426         by (simp add: Euclidean_Algorithm.gcd_mod ac_simps)
427     qed
428   qed
429   show [simp]: "Euclidean_Algorithm.Lcm = (Lcm :: 'a set \<Rightarrow> _)"
430     by (auto intro!: Lcm_eqI GCD.dvd_Lcm GCD.Lcm_least)
431   show "Euclidean_Algorithm.lcm = (lcm :: 'a \<Rightarrow> _)"
432     by (simp add: fun_eq_iff Euclidean_Algorithm.lcm_def semiring_gcd_class.lcm_gcd)
433   show "Euclidean_Algorithm.Gcd = (Gcd :: 'a set \<Rightarrow> _)"
434     by (simp add: fun_eq_iff Euclidean_Algorithm.Gcd_def semiring_Gcd_class.Gcd_Lcm)
435 qed
438 subsection \<open>The extended euclidean algorithm\<close>
440 class euclidean_ring_gcd = euclidean_semiring_gcd + idom
441 begin
443 subclass euclidean_ring ..
444 subclass ring_gcd ..
445 subclass factorial_ring_gcd ..
447 function euclid_ext_aux :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> ('a \<times> 'a) \<times> 'a"
448   where "euclid_ext_aux s' s t' t r' r = (
449      if r = 0 then let c = 1 div unit_factor r' in ((s' * c, t' * c), normalize r')
450      else let q = r' div r
451           in euclid_ext_aux s (s' - q * s) t (t' - q * t) r (r' mod r))"
452   by auto
453 termination
454   by (relation "measure (\<lambda>(_, _, _, _, _, b). euclidean_size b)")
455     (simp_all add: mod_size_less)
457 abbreviation (input) euclid_ext :: "'a \<Rightarrow> 'a \<Rightarrow> ('a \<times> 'a) \<times> 'a"
458   where "euclid_ext \<equiv> euclid_ext_aux 1 0 0 1"
460 lemma
461   assumes "gcd r' r = gcd a b"
462   assumes "s' * a + t' * b = r'"
463   assumes "s * a + t * b = r"
464   assumes "euclid_ext_aux s' s t' t r' r = ((x, y), c)"
465   shows euclid_ext_aux_eq_gcd: "c = gcd a b"
466     and euclid_ext_aux_bezout: "x * a + y * b = gcd a b"
467 proof -
468   have "case euclid_ext_aux s' s t' t r' r of ((x, y), c) \<Rightarrow>
469     x * a + y * b = c \<and> c = gcd a b" (is "?P (euclid_ext_aux s' s t' t r' r)")
470     using assms(1-3)
471   proof (induction s' s t' t r' r rule: euclid_ext_aux.induct)
472     case (1 s' s t' t r' r)
473     show ?case
474     proof (cases "r = 0")
475       case True
476       hence "euclid_ext_aux s' s t' t r' r =
477                ((s' div unit_factor r', t' div unit_factor r'), normalize r')"
478         by (subst euclid_ext_aux.simps) (simp add: Let_def)
479       also have "?P \<dots>"
480       proof safe
481         have "s' div unit_factor r' * a + t' div unit_factor r' * b =
482                 (s' * a + t' * b) div unit_factor r'"
483           by (cases "r' = 0") (simp_all add: unit_div_commute)
484         also have "s' * a + t' * b = r'" by fact
485         also have "\<dots> div unit_factor r' = normalize r'" by simp
486         finally show "s' div unit_factor r' * a + t' div unit_factor r' * b = normalize r'" .
487       next
488         from "1.prems" True show "normalize r' = gcd a b"
489           by simp
490       qed
491       finally show ?thesis .
492     next
493       case False
494       hence "euclid_ext_aux s' s t' t r' r =
495              euclid_ext_aux s (s' - r' div r * s) t (t' - r' div r * t) r (r' mod r)"
496         by (subst euclid_ext_aux.simps) (simp add: Let_def)
497       also from "1.prems" False have "?P \<dots>"
498       proof (intro "1.IH")
499         have "(s' - r' div r * s) * a + (t' - r' div r * t) * b =
500               (s' * a + t' * b) - r' div r * (s * a + t * b)" by (simp add: algebra_simps)
501         also have "s' * a + t' * b = r'" by fact
502         also have "s * a + t * b = r" by fact
503         also have "r' - r' div r * r = r' mod r" using div_mult_mod_eq [of r' r]
504           by (simp add: algebra_simps)
505         finally show "(s' - r' div r * s) * a + (t' - r' div r * t) * b = r' mod r" .
506       qed (auto simp: gcd_mod_right algebra_simps minus_mod_eq_div_mult [symmetric] gcd.commute)
507       finally show ?thesis .
508     qed
509   qed
510   with assms(4) show "c = gcd a b" "x * a + y * b = gcd a b"
511     by simp_all
512 qed
514 declare euclid_ext_aux.simps [simp del]
516 definition bezout_coefficients :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a"
517   where [code]: "bezout_coefficients a b = fst (euclid_ext a b)"
519 lemma bezout_coefficients_0:
520   "bezout_coefficients a 0 = (1 div unit_factor a, 0)"
521   by (simp add: bezout_coefficients_def euclid_ext_aux.simps)
523 lemma bezout_coefficients_left_0:
524   "bezout_coefficients 0 a = (0, 1 div unit_factor a)"
525   by (simp add: bezout_coefficients_def euclid_ext_aux.simps)
527 lemma bezout_coefficients:
528   assumes "bezout_coefficients a b = (x, y)"
529   shows "x * a + y * b = gcd a b"
530   using assms by (simp add: bezout_coefficients_def
531     euclid_ext_aux_bezout [of a b a b 1 0 0 1 x y] prod_eq_iff)
533 lemma bezout_coefficients_fst_snd:
534   "fst (bezout_coefficients a b) * a + snd (bezout_coefficients a b) * b = gcd a b"
535   by (rule bezout_coefficients) simp
537 lemma euclid_ext_eq [simp]:
538   "euclid_ext a b = (bezout_coefficients a b, gcd a b)" (is "?p = ?q")
539 proof
540   show "fst ?p = fst ?q"
541     by (simp add: bezout_coefficients_def)
542   have "snd (euclid_ext_aux 1 0 0 1 a b) = gcd a b"
543     by (rule euclid_ext_aux_eq_gcd [of a b a b 1 0 0 1])
544       (simp_all add: prod_eq_iff)
545   then show "snd ?p = snd ?q"
546     by simp
547 qed
549 declare euclid_ext_eq [symmetric, code_unfold]
551 end
554 subsection \<open>Typical instances\<close>
556 instance nat :: euclidean_semiring_gcd
557 proof
558   interpret semiring_Gcd 1 0 times
559     "Euclidean_Algorithm.gcd" "Euclidean_Algorithm.lcm"
560     "Euclidean_Algorithm.Gcd" "Euclidean_Algorithm.Lcm"
561     divide plus minus unit_factor normalize
562     rewrites "dvd.dvd op * = Rings.dvd"
563     by (fact semiring_Gcd) (simp add: dvd.dvd_def dvd_def fun_eq_iff)
564   show [simp]: "(Euclidean_Algorithm.gcd :: nat \<Rightarrow> _) = gcd"
565   proof (rule ext)+
566     fix m n :: nat
567     show "Euclidean_Algorithm.gcd m n = gcd m n"
568     proof (induct m n rule: eucl_induct)
569       case zero
570       then show ?case
571         by simp
572     next
573       case (mod m n)
574       then have "gcd n (m mod n) = gcd n m"
575         using gcd_nat.simps [of m n] by (simp add: ac_simps)
576       with mod show ?case
577         by (simp add: Euclidean_Algorithm.gcd_mod ac_simps)
578     qed
579   qed
580   show [simp]: "(Euclidean_Algorithm.Lcm :: nat set \<Rightarrow> _) = Lcm"
581     by (auto intro!: ext Lcm_eqI)
582   show "(Euclidean_Algorithm.lcm :: nat \<Rightarrow> _) = lcm"
583     by (simp add: fun_eq_iff Euclidean_Algorithm.lcm_def semiring_gcd_class.lcm_gcd)
584   show "(Euclidean_Algorithm.Gcd :: nat set \<Rightarrow> _) = Gcd"
585     by (simp add: fun_eq_iff Euclidean_Algorithm.Gcd_def semiring_Gcd_class.Gcd_Lcm)
586 qed
588 instance int :: euclidean_ring_gcd
589 proof
590   interpret semiring_Gcd 1 0 times
591     "Euclidean_Algorithm.gcd" "Euclidean_Algorithm.lcm"
592     "Euclidean_Algorithm.Gcd" "Euclidean_Algorithm.Lcm"
593     divide plus minus unit_factor normalize
594     rewrites "dvd.dvd op * = Rings.dvd"
595     by (fact semiring_Gcd) (simp add: dvd.dvd_def dvd_def fun_eq_iff)
596   show [simp]: "(Euclidean_Algorithm.gcd :: int \<Rightarrow> _) = gcd"
597   proof (rule ext)+
598     fix k l :: int
599     show "Euclidean_Algorithm.gcd k l = gcd k l"
600     proof (induct k l rule: eucl_induct)
601       case zero
602       then show ?case
603         by simp
604     next
605       case (mod k l)
606       have "gcd l (k mod l) = gcd l k"
607       proof (cases l "0::int" rule: linorder_cases)
608         case less
609         then show ?thesis
610           using gcd_non_0_int [of "- l" "- k"] by (simp add: ac_simps)
611       next
612         case equal
613         with mod show ?thesis
614           by simp
615       next
616         case greater
617         then show ?thesis
618           using gcd_non_0_int [of l k] by (simp add: ac_simps)
619       qed
620       with mod show ?case
621         by (simp add: Euclidean_Algorithm.gcd_mod ac_simps)
622     qed
623   qed
624   show [simp]: "(Euclidean_Algorithm.Lcm :: int set \<Rightarrow> _) = Lcm"
625     by (auto intro!: ext Lcm_eqI)
626   show "(Euclidean_Algorithm.lcm :: int \<Rightarrow> _) = lcm"
627     by (simp add: fun_eq_iff Euclidean_Algorithm.lcm_def semiring_gcd_class.lcm_gcd)
628   show "(Euclidean_Algorithm.Gcd :: int set \<Rightarrow> _) = Gcd"
629     by (simp add: fun_eq_iff Euclidean_Algorithm.Gcd_def semiring_Gcd_class.Gcd_Lcm)
630 qed
632 end