src/HOL/Number_Theory/Euclidean_Algorithm.thy
 author haftmann Mon Sep 26 07:56:54 2016 +0200 (2016-09-26) changeset 63947 559f0882d6a6 parent 63924 f91766530e13 child 64163 62c9e5c05928 permissions -rw-r--r--
more lemmas
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_div + 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 euclidean_size_normalize [simp]:
30   "euclidean_size (normalize a) = euclidean_size a"
31 proof (cases "a = 0")
32   case True
33   then show ?thesis
34     by simp
35 next
36   case [simp]: False
37   have "euclidean_size (normalize a) \<le> euclidean_size (normalize a * unit_factor a)"
38     by (rule size_mult_mono) simp
39   moreover have "euclidean_size a \<le> euclidean_size (a * (1 div unit_factor a))"
40     by (rule size_mult_mono) simp
41   ultimately show ?thesis
42     by simp
43 qed
45 lemma euclidean_division:
46   fixes a :: 'a and b :: 'a
47   assumes "b \<noteq> 0"
48   obtains s and t where "a = s * b + t"
49     and "euclidean_size t < euclidean_size b"
50 proof -
51   from div_mod_equality [of a b 0]
52      have "a = a div b * b + a mod b" by simp
53   with that and assms show ?thesis by (auto simp add: mod_size_less)
54 qed
56 lemma dvd_euclidean_size_eq_imp_dvd:
57   assumes "a \<noteq> 0" and b_dvd_a: "b dvd a" and size_eq: "euclidean_size a = euclidean_size b"
58   shows "a dvd b"
59 proof (rule ccontr)
60   assume "\<not> a dvd b"
61   then have "b mod a \<noteq> 0" by (simp add: mod_eq_0_iff_dvd)
62   from b_dvd_a have b_dvd_mod: "b dvd b mod a" by (simp add: dvd_mod_iff)
63   from b_dvd_mod obtain c where "b mod a = b * c" unfolding dvd_def by blast
64     with \<open>b mod a \<noteq> 0\<close> have "c \<noteq> 0" by auto
65   with \<open>b mod a = b * c\<close> have "euclidean_size (b mod a) \<ge> euclidean_size b"
66       using size_mult_mono by force
67   moreover from \<open>\<not> a dvd b\<close> and \<open>a \<noteq> 0\<close>
68   have "euclidean_size (b mod a) < euclidean_size a"
69       using mod_size_less by blast
70   ultimately show False using size_eq by simp
71 qed
73 lemma size_mult_mono': "b \<noteq> 0 \<Longrightarrow> euclidean_size a \<le> euclidean_size (b * a)"
74   by (subst mult.commute) (rule size_mult_mono)
76 lemma euclidean_size_times_unit:
77   assumes "is_unit a"
78   shows   "euclidean_size (a * b) = euclidean_size b"
79 proof (rule antisym)
80   from assms have [simp]: "a \<noteq> 0" by auto
81   thus "euclidean_size (a * b) \<ge> euclidean_size b" by (rule size_mult_mono')
82   from assms have "is_unit (1 div a)" by simp
83   hence "1 div a \<noteq> 0" by (intro notI) simp_all
84   hence "euclidean_size (a * b) \<le> euclidean_size ((1 div a) * (a * b))"
85     by (rule size_mult_mono')
86   also from assms have "(1 div a) * (a * b) = b"
87     by (simp add: algebra_simps unit_div_mult_swap)
88   finally show "euclidean_size (a * b) \<le> euclidean_size b" .
89 qed
91 lemma euclidean_size_unit: "is_unit x \<Longrightarrow> euclidean_size x = euclidean_size 1"
92   using euclidean_size_times_unit[of x 1] by simp
94 lemma unit_iff_euclidean_size:
95   "is_unit x \<longleftrightarrow> euclidean_size x = euclidean_size 1 \<and> x \<noteq> 0"
96 proof safe
97   assume A: "x \<noteq> 0" and B: "euclidean_size x = euclidean_size 1"
98   show "is_unit x" by (rule dvd_euclidean_size_eq_imp_dvd[OF A _ B]) simp_all
99 qed (auto intro: euclidean_size_unit)
101 lemma euclidean_size_times_nonunit:
102   assumes "a \<noteq> 0" "b \<noteq> 0" "\<not>is_unit a"
103   shows   "euclidean_size b < euclidean_size (a * b)"
104 proof (rule ccontr)
105   assume "\<not>euclidean_size b < euclidean_size (a * b)"
106   with size_mult_mono'[OF assms(1), of b]
107     have eq: "euclidean_size (a * b) = euclidean_size b" by simp
108   have "a * b dvd b"
109     by (rule dvd_euclidean_size_eq_imp_dvd[OF _ _ eq]) (insert assms, simp_all)
110   hence "a * b dvd 1 * b" by simp
111   with \<open>b \<noteq> 0\<close> have "is_unit a" by (subst (asm) dvd_times_right_cancel_iff)
112   with assms(3) show False by contradiction
113 qed
115 lemma dvd_imp_size_le:
116   assumes "x dvd y" "y \<noteq> 0"
117   shows   "euclidean_size x \<le> euclidean_size y"
118   using assms by (auto elim!: dvdE simp: size_mult_mono)
120 lemma dvd_proper_imp_size_less:
121   assumes "x dvd y" "\<not>y dvd x" "y \<noteq> 0"
122   shows   "euclidean_size x < euclidean_size y"
123 proof -
124   from assms(1) obtain z where "y = x * z" by (erule dvdE)
125   hence z: "y = z * x" by (simp add: mult.commute)
126   from z assms have "\<not>is_unit z" by (auto simp: mult.commute mult_unit_dvd_iff)
127   with z assms show ?thesis
128     by (auto intro!: euclidean_size_times_nonunit simp: )
129 qed
131 function gcd_eucl :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
132 where
133   "gcd_eucl a b = (if b = 0 then normalize a else gcd_eucl b (a mod b))"
134   by pat_completeness simp
135 termination
136   by (relation "measure (euclidean_size \<circ> snd)") (simp_all add: mod_size_less)
138 declare gcd_eucl.simps [simp del]
140 lemma gcd_eucl_induct [case_names zero mod]:
141   assumes H1: "\<And>b. P b 0"
142   and H2: "\<And>a b. b \<noteq> 0 \<Longrightarrow> P b (a mod b) \<Longrightarrow> P a b"
143   shows "P a b"
144 proof (induct a b rule: gcd_eucl.induct)
145   case ("1" a b)
146   show ?case
147   proof (cases "b = 0")
148     case True then show "P a b" by simp (rule H1)
149   next
150     case False
151     then have "P b (a mod b)"
152       by (rule "1.hyps")
153     with \<open>b \<noteq> 0\<close> show "P a b"
154       by (blast intro: H2)
155   qed
156 qed
158 definition lcm_eucl :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
159 where
160   "lcm_eucl a b = normalize (a * b) div gcd_eucl a b"
162 definition Lcm_eucl :: "'a set \<Rightarrow> 'a" \<comment> \<open>
163   Somewhat complicated definition of Lcm that has the advantage of working
164   for infinite sets as well\<close>
165 where
166   "Lcm_eucl A = (if \<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) then
167      let l = SOME l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l =
168        (LEAST n. \<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n)
169        in normalize l
170       else 0)"
172 definition Gcd_eucl :: "'a set \<Rightarrow> 'a"
173 where
174   "Gcd_eucl A = Lcm_eucl {d. \<forall>a\<in>A. d dvd a}"
176 declare Lcm_eucl_def Gcd_eucl_def [code del]
178 lemma gcd_eucl_0:
179   "gcd_eucl a 0 = normalize a"
180   by (simp add: gcd_eucl.simps [of a 0])
182 lemma gcd_eucl_0_left:
183   "gcd_eucl 0 a = normalize a"
184   by (simp_all add: gcd_eucl_0 gcd_eucl.simps [of 0 a])
186 lemma gcd_eucl_non_0:
187   "b \<noteq> 0 \<Longrightarrow> gcd_eucl a b = gcd_eucl b (a mod b)"
188   by (simp add: gcd_eucl.simps [of a b] gcd_eucl.simps [of b 0])
190 lemma gcd_eucl_dvd1 [iff]: "gcd_eucl a b dvd a"
191   and gcd_eucl_dvd2 [iff]: "gcd_eucl a b dvd b"
192   by (induct a b rule: gcd_eucl_induct)
193      (simp_all add: gcd_eucl_0 gcd_eucl_non_0 dvd_mod_iff)
195 lemma normalize_gcd_eucl [simp]:
196   "normalize (gcd_eucl a b) = gcd_eucl a b"
197   by (induct a b rule: gcd_eucl_induct) (simp_all add: gcd_eucl_0 gcd_eucl_non_0)
199 lemma gcd_eucl_greatest:
200   fixes k a b :: 'a
201   shows "k dvd a \<Longrightarrow> k dvd b \<Longrightarrow> k dvd gcd_eucl a b"
202 proof (induct a b rule: gcd_eucl_induct)
203   case (zero a) from zero(1) show ?case by (rule dvd_trans) (simp add: gcd_eucl_0)
204 next
205   case (mod a b)
206   then show ?case
207     by (simp add: gcd_eucl_non_0 dvd_mod_iff)
208 qed
210 lemma gcd_euclI:
211   fixes gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
212   assumes "d dvd a" "d dvd b" "normalize d = d"
213           "\<And>k. k dvd a \<Longrightarrow> k dvd b \<Longrightarrow> k dvd d"
214   shows   "gcd_eucl a b = d"
215   by (rule associated_eqI) (simp_all add: gcd_eucl_greatest assms)
217 lemma eq_gcd_euclI:
218   fixes gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
219   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"
220           "\<And>a b k. k dvd a \<Longrightarrow> k dvd b \<Longrightarrow> k dvd gcd a b"
221   shows   "gcd = gcd_eucl"
222   by (intro ext, rule associated_eqI) (simp_all add: gcd_eucl_greatest assms)
224 lemma gcd_eucl_zero [simp]:
225   "gcd_eucl a b = 0 \<longleftrightarrow> a = 0 \<and> b = 0"
226   by (metis dvd_0_left dvd_refl gcd_eucl_dvd1 gcd_eucl_dvd2 gcd_eucl_greatest)+
229 lemma dvd_Lcm_eucl [simp]: "a \<in> A \<Longrightarrow> a dvd Lcm_eucl A"
230   and Lcm_eucl_least: "(\<And>a. a \<in> A \<Longrightarrow> a dvd b) \<Longrightarrow> Lcm_eucl A dvd b"
231   and unit_factor_Lcm_eucl [simp]:
232           "unit_factor (Lcm_eucl A) = (if Lcm_eucl A = 0 then 0 else 1)"
233 proof -
234   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>
235     unit_factor (Lcm_eucl A) = (if Lcm_eucl A = 0 then 0 else 1)" (is ?thesis)
236   proof (cases "\<exists>l. l \<noteq>  0 \<and> (\<forall>a\<in>A. a dvd l)")
237     case False
238     hence "Lcm_eucl A = 0" by (auto simp: Lcm_eucl_def)
239     with False show ?thesis by auto
240   next
241     case True
242     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
243     define n where "n = (LEAST n. \<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n)"
244     define l where "l = (SOME l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n)"
245     have "\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"
246       apply (subst n_def)
247       apply (rule LeastI[of _ "euclidean_size l\<^sub>0"])
248       apply (rule exI[of _ l\<^sub>0])
249       apply (simp add: l\<^sub>0_props)
250       done
251     from someI_ex[OF this] have "l \<noteq> 0" and "\<forall>a\<in>A. a dvd l" and "euclidean_size l = n"
252       unfolding l_def by simp_all
253     {
254       fix l' assume "\<forall>a\<in>A. a dvd l'"
255       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)
256       moreover from \<open>l \<noteq> 0\<close> have "gcd_eucl l l' \<noteq> 0" by simp
257       ultimately have "\<exists>b. b \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd b) \<and>
258                           euclidean_size b = euclidean_size (gcd_eucl l l')"
259         by (intro exI[of _ "gcd_eucl l l'"], auto)
260       hence "euclidean_size (gcd_eucl l l') \<ge> n" by (subst n_def) (rule Least_le)
261       moreover have "euclidean_size (gcd_eucl l l') \<le> n"
262       proof -
263         have "gcd_eucl l l' dvd l" by simp
264         then obtain a where "l = gcd_eucl l l' * a" unfolding dvd_def by blast
265         with \<open>l \<noteq> 0\<close> have "a \<noteq> 0" by auto
266         hence "euclidean_size (gcd_eucl l l') \<le> euclidean_size (gcd_eucl l l' * a)"
267           by (rule size_mult_mono)
268         also have "gcd_eucl l l' * a = l" using \<open>l = gcd_eucl l l' * a\<close> ..
269         also note \<open>euclidean_size l = n\<close>
270         finally show "euclidean_size (gcd_eucl l l') \<le> n" .
271       qed
272       ultimately have *: "euclidean_size l = euclidean_size (gcd_eucl l l')"
273         by (intro le_antisym, simp_all add: \<open>euclidean_size l = n\<close>)
274       from \<open>l \<noteq> 0\<close> have "l dvd gcd_eucl l l'"
275         by (rule dvd_euclidean_size_eq_imp_dvd) (auto simp add: *)
276       hence "l dvd l'" by (rule dvd_trans[OF _ gcd_eucl_dvd2])
277     }
279     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>
280       have "(\<forall>a\<in>A. a dvd normalize l) \<and>
281         (\<forall>l'. (\<forall>a\<in>A. a dvd l') \<longrightarrow> normalize l dvd l') \<and>
282         unit_factor (normalize l) =
283         (if normalize l = 0 then 0 else 1)"
284       by (auto simp: unit_simps)
285     also from True have "normalize l = Lcm_eucl A"
286       by (simp add: Lcm_eucl_def Let_def n_def l_def)
287     finally show ?thesis .
288   qed
289   note A = this
291   {fix a assume "a \<in> A" then show "a dvd Lcm_eucl A" using A by blast}
292   {fix b assume "\<And>a. a \<in> A \<Longrightarrow> a dvd b" then show "Lcm_eucl A dvd b" using A by blast}
293   from A show "unit_factor (Lcm_eucl A) = (if Lcm_eucl A = 0 then 0 else 1)" by blast
294 qed
296 lemma normalize_Lcm_eucl [simp]:
297   "normalize (Lcm_eucl A) = Lcm_eucl A"
298 proof (cases "Lcm_eucl A = 0")
299   case True then show ?thesis by simp
300 next
301   case False
302   have "unit_factor (Lcm_eucl A) * normalize (Lcm_eucl A) = Lcm_eucl A"
303     by (fact unit_factor_mult_normalize)
304   with False show ?thesis by simp
305 qed
307 lemma eq_Lcm_euclI:
308   fixes lcm :: "'a set \<Rightarrow> 'a"
309   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"
310           "\<And>A. normalize (lcm A) = lcm A" shows "lcm = Lcm_eucl"
311   by (intro ext, rule associated_eqI) (auto simp: assms intro: Lcm_eucl_least)
313 lemma Gcd_eucl_dvd: "x \<in> A \<Longrightarrow> Gcd_eucl A dvd x"
314   unfolding Gcd_eucl_def by (auto intro: Lcm_eucl_least)
316 lemma Gcd_eucl_greatest: "(\<And>x. x \<in> A \<Longrightarrow> d dvd x) \<Longrightarrow> d dvd Gcd_eucl A"
317   unfolding Gcd_eucl_def by auto
319 lemma normalize_Gcd_eucl [simp]: "normalize (Gcd_eucl A) = Gcd_eucl A"
320   by (simp add: Gcd_eucl_def)
322 lemma Lcm_euclI:
323   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"
324   shows   "Lcm_eucl A = d"
325 proof -
326   have "normalize (Lcm_eucl A) = normalize d"
327     by (intro associatedI) (auto intro: dvd_Lcm_eucl Lcm_eucl_least assms)
328   thus ?thesis by (simp add: assms)
329 qed
331 lemma Gcd_euclI:
332   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"
333   shows   "Gcd_eucl A = d"
334 proof -
335   have "normalize (Gcd_eucl A) = normalize d"
336     by (intro associatedI) (auto intro: Gcd_eucl_dvd Gcd_eucl_greatest assms)
337   thus ?thesis by (simp add: assms)
338 qed
340 lemmas lcm_gcd_eucl_facts =
341   gcd_eucl_dvd1 gcd_eucl_dvd2 gcd_eucl_greatest normalize_gcd_eucl lcm_eucl_def
342   Gcd_eucl_def Gcd_eucl_dvd Gcd_eucl_greatest normalize_Gcd_eucl
343   dvd_Lcm_eucl Lcm_eucl_least normalize_Lcm_eucl
345 lemma normalized_factors_product:
346   "{p. p dvd a * b \<and> normalize p = p} =
347      (\<lambda>(x,y). x * y)  ({p. p dvd a \<and> normalize p = p} \<times> {p. p dvd b \<and> normalize p = p})"
348 proof safe
349   fix p assume p: "p dvd a * b" "normalize p = p"
350   interpret semiring_gcd 1 0 "op *" gcd_eucl lcm_eucl "op div" "op +" "op -" normalize unit_factor
351     by standard (rule lcm_gcd_eucl_facts; assumption)+
352   from dvd_productE[OF p(1)] guess x y . note xy = this
353   define x' y' where "x' = normalize x" and "y' = normalize y"
354   have "p = x' * y'"
355     by (subst p(2) [symmetric]) (simp add: xy x'_def y'_def normalize_mult)
356   moreover from xy have "normalize x' = x'" "normalize y' = y'" "x' dvd a" "y' dvd b"
357     by (simp_all add: x'_def y'_def)
358   ultimately show "p \<in> (\<lambda>(x, y). x * y)
359                      ({p. p dvd a \<and> normalize p = p} \<times> {p. p dvd b \<and> normalize p = p})"
360     by blast
361 qed (auto simp: normalize_mult mult_dvd_mono)
364 subclass factorial_semiring
365 proof (standard, rule factorial_semiring_altI_aux)
366   fix x assume "x \<noteq> 0"
367   thus "finite {p. p dvd x \<and> normalize p = p}"
368   proof (induction "euclidean_size x" arbitrary: x rule: less_induct)
369     case (less x)
370     show ?case
371     proof (cases "\<exists>y. y dvd x \<and> \<not>x dvd y \<and> \<not>is_unit y")
372       case False
373       have "{p. p dvd x \<and> normalize p = p} \<subseteq> {1, normalize x}"
374       proof
375         fix p assume p: "p \<in> {p. p dvd x \<and> normalize p = p}"
376         with False have "is_unit p \<or> x dvd p" by blast
377         thus "p \<in> {1, normalize x}"
378         proof (elim disjE)
379           assume "is_unit p"
380           hence "normalize p = 1" by (simp add: is_unit_normalize)
381           with p show ?thesis by simp
382         next
383           assume "x dvd p"
384           with p have "normalize p = normalize x" by (intro associatedI) simp_all
385           with p show ?thesis by simp
386         qed
387       qed
388       moreover have "finite \<dots>" by simp
389       ultimately show ?thesis by (rule finite_subset)
391     next
392       case True
393       then obtain y where y: "y dvd x" "\<not>x dvd y" "\<not>is_unit y" by blast
394       define z where "z = x div y"
395       let ?fctrs = "\<lambda>x. {p. p dvd x \<and> normalize p = p}"
396       from y have x: "x = y * z" by (simp add: z_def)
397       with less.prems have "y \<noteq> 0" "z \<noteq> 0" by auto
398       from x y have "\<not>is_unit z" by (auto simp: mult_unit_dvd_iff)
399       have "?fctrs x = (\<lambda>(p,p'). p * p')  (?fctrs y \<times> ?fctrs z)"
400         by (subst x) (rule normalized_factors_product)
401       also have "\<not>y * z dvd y * 1" "\<not>y * z dvd 1 * z"
402         by (subst dvd_times_left_cancel_iff dvd_times_right_cancel_iff; fact)+
403       hence "finite ((\<lambda>(p,p'). p * p')  (?fctrs y \<times> ?fctrs z))"
404         by (intro finite_imageI finite_cartesian_product less dvd_proper_imp_size_less)
405            (auto simp: x)
406       finally show ?thesis .
407     qed
408   qed
409 next
410   interpret semiring_gcd 1 0 "op *" gcd_eucl lcm_eucl "op div" "op +" "op -" normalize unit_factor
411     by standard (rule lcm_gcd_eucl_facts; assumption)+
412   fix p assume p: "irreducible p"
413   thus "prime_elem p" by (rule irreducible_imp_prime_elem_gcd)
414 qed
416 lemma gcd_eucl_eq_gcd_factorial: "gcd_eucl = gcd_factorial"
417   by (intro ext gcd_euclI gcd_lcm_factorial)
419 lemma lcm_eucl_eq_lcm_factorial: "lcm_eucl = lcm_factorial"
420   by (intro ext) (simp add: lcm_eucl_def lcm_factorial_gcd_factorial gcd_eucl_eq_gcd_factorial)
422 lemma Gcd_eucl_eq_Gcd_factorial: "Gcd_eucl = Gcd_factorial"
423   by (intro ext Gcd_euclI gcd_lcm_factorial)
425 lemma Lcm_eucl_eq_Lcm_factorial: "Lcm_eucl = Lcm_factorial"
426   by (intro ext Lcm_euclI gcd_lcm_factorial)
428 lemmas eucl_eq_factorial =
429   gcd_eucl_eq_gcd_factorial lcm_eucl_eq_lcm_factorial
430   Gcd_eucl_eq_Gcd_factorial Lcm_eucl_eq_Lcm_factorial
432 end
434 class euclidean_ring = euclidean_semiring + idom
435 begin
437 subclass ring_div ..
439 function euclid_ext_aux :: "'a \<Rightarrow> _" where
440   "euclid_ext_aux r' r s' s t' t = (
441      if r = 0 then let c = 1 div unit_factor r' in (s' * c, t' * c, normalize r')
442      else let q = r' div r
443           in  euclid_ext_aux r (r' mod r) s (s' - q * s) t (t' - q * t))"
444 by auto
445 termination by (relation "measure (\<lambda>(_,b,_,_,_,_). euclidean_size b)") (simp_all add: mod_size_less)
447 declare euclid_ext_aux.simps [simp del]
449 lemma euclid_ext_aux_correct:
450   assumes "gcd_eucl r' r = gcd_eucl x y"
451   assumes "s' * x + t' * y = r'"
452   assumes "s * x + t * y = r"
453   shows   "case euclid_ext_aux r' r s' s t' t of (a,b,c) \<Rightarrow>
454              a * x + b * y = c \<and> c = gcd_eucl x y" (is "?P (euclid_ext_aux r' r s' s t' t)")
455 using assms
456 proof (induction r' r s' s t' t rule: euclid_ext_aux.induct)
457   case (1 r' r s' s t' t)
458   show ?case
459   proof (cases "r = 0")
460     case True
461     hence "euclid_ext_aux r' r s' s t' t =
462              (s' div unit_factor r', t' div unit_factor r', normalize r')"
463       by (subst euclid_ext_aux.simps) (simp add: Let_def)
464     also have "?P \<dots>"
465     proof safe
466       have "s' div unit_factor r' * x + t' div unit_factor r' * y =
467                 (s' * x + t' * y) div unit_factor r'"
468         by (cases "r' = 0") (simp_all add: unit_div_commute)
469       also have "s' * x + t' * y = r'" by fact
470       also have "\<dots> div unit_factor r' = normalize r'" by simp
471       finally show "s' div unit_factor r' * x + t' div unit_factor r' * y = normalize r'" .
472     next
473       from "1.prems" True show "normalize r' = gcd_eucl x y" by (simp add: gcd_eucl_0)
474     qed
475     finally show ?thesis .
476   next
477     case False
478     hence "euclid_ext_aux r' r s' s t' t =
479              euclid_ext_aux r (r' mod r) s (s' - r' div r * s) t (t' - r' div r * t)"
480       by (subst euclid_ext_aux.simps) (simp add: Let_def)
481     also from "1.prems" False have "?P \<dots>"
482     proof (intro "1.IH")
483       have "(s' - r' div r * s) * x + (t' - r' div r * t) * y =
484               (s' * x + t' * y) - r' div r * (s * x + t * y)" by (simp add: algebra_simps)
485       also have "s' * x + t' * y = r'" by fact
486       also have "s * x + t * y = r" by fact
487       also have "r' - r' div r * r = r' mod r" using mod_div_equality[of r' r]
488         by (simp add: algebra_simps)
489       finally show "(s' - r' div r * s) * x + (t' - r' div r * t) * y = r' mod r" .
490     qed (auto simp: gcd_eucl_non_0 algebra_simps div_mod_equality')
491     finally show ?thesis .
492   qed
493 qed
495 definition euclid_ext where
496   "euclid_ext a b = euclid_ext_aux a b 1 0 0 1"
498 lemma euclid_ext_0:
499   "euclid_ext a 0 = (1 div unit_factor a, 0, normalize a)"
500   by (simp add: euclid_ext_def euclid_ext_aux.simps)
502 lemma euclid_ext_left_0:
503   "euclid_ext 0 a = (0, 1 div unit_factor a, normalize a)"
504   by (simp add: euclid_ext_def euclid_ext_aux.simps)
506 lemma euclid_ext_correct':
507   "case euclid_ext x y of (a,b,c) \<Rightarrow> a * x + b * y = c \<and> c = gcd_eucl x y"
508   unfolding euclid_ext_def by (rule euclid_ext_aux_correct) simp_all
510 lemma euclid_ext_gcd_eucl:
511   "(case euclid_ext x y of (a,b,c) \<Rightarrow> c) = gcd_eucl x y"
512   using euclid_ext_correct'[of x y] by (simp add: case_prod_unfold)
514 definition euclid_ext' where
515   "euclid_ext' x y = (case euclid_ext x y of (a, b, _) \<Rightarrow> (a, b))"
517 lemma euclid_ext'_correct':
518   "case euclid_ext' x y of (a,b) \<Rightarrow> a * x + b * y = gcd_eucl x y"
519   using euclid_ext_correct'[of x y] by (simp add: case_prod_unfold euclid_ext'_def)
521 lemma euclid_ext'_0: "euclid_ext' a 0 = (1 div unit_factor a, 0)"
522   by (simp add: euclid_ext'_def euclid_ext_0)
524 lemma euclid_ext'_left_0: "euclid_ext' 0 a = (0, 1 div unit_factor a)"
525   by (simp add: euclid_ext'_def euclid_ext_left_0)
527 end
529 class euclidean_semiring_gcd = euclidean_semiring + gcd + Gcd +
530   assumes gcd_gcd_eucl: "gcd = gcd_eucl" and lcm_lcm_eucl: "lcm = lcm_eucl"
531   assumes Gcd_Gcd_eucl: "Gcd = Gcd_eucl" and Lcm_Lcm_eucl: "Lcm = Lcm_eucl"
532 begin
534 subclass semiring_gcd
535   by standard (simp_all add: gcd_gcd_eucl gcd_eucl_greatest lcm_lcm_eucl lcm_eucl_def)
537 subclass semiring_Gcd
538   by standard (auto simp: Gcd_Gcd_eucl Lcm_Lcm_eucl Gcd_eucl_def intro: Lcm_eucl_least)
540 subclass factorial_semiring_gcd
541 proof
542   fix a b
543   show "gcd a b = gcd_factorial a b"
544     by (rule sym, rule gcdI) (rule gcd_lcm_factorial; assumption)+
545   thus "lcm a b = lcm_factorial a b"
546     by (simp add: lcm_factorial_gcd_factorial lcm_gcd)
547 next
548   fix A
549   show "Gcd A = Gcd_factorial A"
550     by (rule sym, rule GcdI) (rule gcd_lcm_factorial; assumption)+
551   show "Lcm A = Lcm_factorial A"
552     by (rule sym, rule LcmI) (rule gcd_lcm_factorial; assumption)+
553 qed
555 lemma gcd_non_0:
556   "b \<noteq> 0 \<Longrightarrow> gcd a b = gcd b (a mod b)"
557   unfolding gcd_gcd_eucl by (fact gcd_eucl_non_0)
559 lemmas gcd_0 = gcd_0_right
560 lemmas dvd_gcd_iff = gcd_greatest_iff
561 lemmas gcd_greatest_iff = dvd_gcd_iff
563 lemma gcd_mod1 [simp]:
564   "gcd (a mod b) b = gcd a b"
565   by (rule gcdI, metis dvd_mod_iff gcd_dvd1 gcd_dvd2, simp_all add: gcd_greatest dvd_mod_iff)
567 lemma gcd_mod2 [simp]:
568   "gcd a (b mod a) = gcd a b"
569   by (rule gcdI, simp, metis dvd_mod_iff gcd_dvd1 gcd_dvd2, simp_all add: gcd_greatest dvd_mod_iff)
571 lemma euclidean_size_gcd_le1 [simp]:
572   assumes "a \<noteq> 0"
573   shows "euclidean_size (gcd a b) \<le> euclidean_size a"
574 proof -
575    have "gcd a b dvd a" by (rule gcd_dvd1)
576    then obtain c where A: "a = gcd a b * c" unfolding dvd_def by blast
577    with \<open>a \<noteq> 0\<close> show ?thesis by (subst (2) A, intro size_mult_mono) auto
578 qed
580 lemma euclidean_size_gcd_le2 [simp]:
581   "b \<noteq> 0 \<Longrightarrow> euclidean_size (gcd a b) \<le> euclidean_size b"
582   by (subst gcd.commute, rule euclidean_size_gcd_le1)
584 lemma euclidean_size_gcd_less1:
585   assumes "a \<noteq> 0" and "\<not>a dvd b"
586   shows "euclidean_size (gcd a b) < euclidean_size a"
587 proof (rule ccontr)
588   assume "\<not>euclidean_size (gcd a b) < euclidean_size a"
589   with \<open>a \<noteq> 0\<close> have A: "euclidean_size (gcd a b) = euclidean_size a"
590     by (intro le_antisym, simp_all)
591   have "a dvd gcd a b"
592     by (rule dvd_euclidean_size_eq_imp_dvd) (simp_all add: assms A)
593   hence "a dvd b" using dvd_gcdD2 by blast
594   with \<open>\<not>a dvd b\<close> show False by contradiction
595 qed
597 lemma euclidean_size_gcd_less2:
598   assumes "b \<noteq> 0" and "\<not>b dvd a"
599   shows "euclidean_size (gcd a b) < euclidean_size b"
600   using assms by (subst gcd.commute, rule euclidean_size_gcd_less1)
602 lemma euclidean_size_lcm_le1:
603   assumes "a \<noteq> 0" and "b \<noteq> 0"
604   shows "euclidean_size a \<le> euclidean_size (lcm a b)"
605 proof -
606   have "a dvd lcm a b" by (rule dvd_lcm1)
607   then obtain c where A: "lcm a b = a * c" ..
608   with \<open>a \<noteq> 0\<close> and \<open>b \<noteq> 0\<close> have "c \<noteq> 0" by (auto simp: lcm_eq_0_iff)
609   then show ?thesis by (subst A, intro size_mult_mono)
610 qed
612 lemma euclidean_size_lcm_le2:
613   "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> euclidean_size b \<le> euclidean_size (lcm a b)"
614   using euclidean_size_lcm_le1 [of b a] by (simp add: ac_simps)
616 lemma euclidean_size_lcm_less1:
617   assumes "b \<noteq> 0" and "\<not>b dvd a"
618   shows "euclidean_size a < euclidean_size (lcm a b)"
619 proof (rule ccontr)
620   from assms have "a \<noteq> 0" by auto
621   assume "\<not>euclidean_size a < euclidean_size (lcm a b)"
622   with \<open>a \<noteq> 0\<close> and \<open>b \<noteq> 0\<close> have "euclidean_size (lcm a b) = euclidean_size a"
623     by (intro le_antisym, simp, intro euclidean_size_lcm_le1)
624   with assms have "lcm a b dvd a"
625     by (rule_tac dvd_euclidean_size_eq_imp_dvd) (auto simp: lcm_eq_0_iff)
626   hence "b dvd a" by (rule lcm_dvdD2)
627   with \<open>\<not>b dvd a\<close> show False by contradiction
628 qed
630 lemma euclidean_size_lcm_less2:
631   assumes "a \<noteq> 0" and "\<not>a dvd b"
632   shows "euclidean_size b < euclidean_size (lcm a b)"
633   using assms euclidean_size_lcm_less1 [of a b] by (simp add: ac_simps)
635 lemma Lcm_eucl_set [code]:
636   "Lcm_eucl (set xs) = foldl lcm_eucl 1 xs"
637   by (simp add: Lcm_Lcm_eucl [symmetric] lcm_lcm_eucl Lcm_set)
639 lemma Gcd_eucl_set [code]:
640   "Gcd_eucl (set xs) = foldl gcd_eucl 0 xs"
641   by (simp add: Gcd_Gcd_eucl [symmetric] gcd_gcd_eucl Gcd_set)
643 end
646 text \<open>
647   A Euclidean ring is a Euclidean semiring with additive inverses. It provides a
648   few more lemmas; in particular, Bezout's lemma holds for any Euclidean ring.
649 \<close>
651 class euclidean_ring_gcd = euclidean_semiring_gcd + idom
652 begin
654 subclass euclidean_ring ..
655 subclass ring_gcd ..
656 subclass factorial_ring_gcd ..
658 lemma euclid_ext_gcd [simp]:
659   "(case euclid_ext a b of (_, _ , t) \<Rightarrow> t) = gcd a b"
660   using euclid_ext_correct'[of a b] by (simp add: case_prod_unfold Let_def gcd_gcd_eucl)
662 lemma euclid_ext_gcd' [simp]:
663   "euclid_ext a b = (r, s, t) \<Longrightarrow> t = gcd a b"
664   by (insert euclid_ext_gcd[of a b], drule (1) subst, simp)
666 lemma euclid_ext_correct:
667   "case euclid_ext x y of (a,b,c) \<Rightarrow> a * x + b * y = c \<and> c = gcd x y"
668   using euclid_ext_correct'[of x y]
669   by (simp add: gcd_gcd_eucl case_prod_unfold)
671 lemma euclid_ext'_correct:
672   "fst (euclid_ext' a b) * a + snd (euclid_ext' a b) * b = gcd a b"
673   using euclid_ext_correct'[of a b]
674   by (simp add: gcd_gcd_eucl case_prod_unfold euclid_ext'_def)
676 lemma bezout: "\<exists>s t. s * a + t * b = gcd a b"
677   using euclid_ext'_correct by blast
679 end
682 subsection \<open>Typical instances\<close>
684 instantiation nat :: euclidean_semiring
685 begin
687 definition [simp]:
688   "euclidean_size_nat = (id :: nat \<Rightarrow> nat)"
690 instance by standard simp_all
692 end
695 instantiation int :: euclidean_ring
696 begin
698 definition [simp]:
699   "euclidean_size_int = (nat \<circ> abs :: int \<Rightarrow> nat)"
701 instance by standard (auto simp add: abs_mult nat_mult_distrib split: abs_split)
703 end
705 instance nat :: euclidean_semiring_gcd
706 proof
707   show [simp]: "gcd = (gcd_eucl :: nat \<Rightarrow> _)" "Lcm = (Lcm_eucl :: nat set \<Rightarrow> _)"
708     by (simp_all add: eq_gcd_euclI eq_Lcm_euclI)
709   show "lcm = (lcm_eucl :: nat \<Rightarrow> _)" "Gcd = (Gcd_eucl :: nat set \<Rightarrow> _)"
710     by (intro ext, simp add: lcm_eucl_def lcm_nat_def Gcd_nat_def Gcd_eucl_def)+
711 qed
713 instance int :: euclidean_ring_gcd
714 proof
715   show [simp]: "gcd = (gcd_eucl :: int \<Rightarrow> _)" "Lcm = (Lcm_eucl :: int set \<Rightarrow> _)"
716     by (simp_all add: eq_gcd_euclI eq_Lcm_euclI)
717   show "lcm = (lcm_eucl :: int \<Rightarrow> _)" "Gcd = (Gcd_eucl :: int set \<Rightarrow> _)"
718     by (intro ext, simp add: lcm_eucl_def lcm_altdef_int
719           semiring_Gcd_class.Gcd_Lcm Gcd_eucl_def abs_mult)+
720 qed
722 end