src/HOL/Number_Theory/Euclidean_Algorithm.thy
 author haftmann Wed Jan 04 21:28:29 2017 +0100 (2017-01-04) changeset 64785 ae0bbc8e45ad parent 64784 5cb5e7ecb284 child 64786 340db65fd2c1 permissions -rw-r--r--
moved euclidean ring to HOL
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 context euclidean_semiring
13 begin
15 function gcd_eucl :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
16 where
17   "gcd_eucl a b = (if b = 0 then normalize a else gcd_eucl b (a mod b))"
18   by pat_completeness simp
19 termination
20   by (relation "measure (euclidean_size \<circ> snd)") (simp_all add: mod_size_less)
22 declare gcd_eucl.simps [simp del]
24 lemma gcd_eucl_induct [case_names zero mod]:
25   assumes H1: "\<And>b. P b 0"
26   and H2: "\<And>a b. b \<noteq> 0 \<Longrightarrow> P b (a mod b) \<Longrightarrow> P a b"
27   shows "P a b"
28 proof (induct a b rule: gcd_eucl.induct)
29   case ("1" a b)
30   show ?case
31   proof (cases "b = 0")
32     case True then show "P a b" by simp (rule H1)
33   next
34     case False
35     then have "P b (a mod b)"
36       by (rule "1.hyps")
37     with \<open>b \<noteq> 0\<close> show "P a b"
38       by (blast intro: H2)
39   qed
40 qed
42 definition lcm_eucl :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
43 where
44   "lcm_eucl a b = normalize (a * b) div gcd_eucl a b"
46 definition Lcm_eucl :: "'a set \<Rightarrow> 'a" \<comment> \<open>
47   Somewhat complicated definition of Lcm that has the advantage of working
48   for infinite sets as well\<close>
49 where
50   "Lcm_eucl A = (if \<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) then
51      let l = SOME l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l =
52        (LEAST n. \<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n)
53        in normalize l
54       else 0)"
56 definition Gcd_eucl :: "'a set \<Rightarrow> 'a"
57 where
58   "Gcd_eucl A = Lcm_eucl {d. \<forall>a\<in>A. d dvd a}"
60 declare Lcm_eucl_def Gcd_eucl_def [code del]
62 lemma gcd_eucl_0:
63   "gcd_eucl a 0 = normalize a"
64   by (simp add: gcd_eucl.simps [of a 0])
66 lemma gcd_eucl_0_left:
67   "gcd_eucl 0 a = normalize a"
68   by (simp_all add: gcd_eucl_0 gcd_eucl.simps [of 0 a])
70 lemma gcd_eucl_non_0:
71   "b \<noteq> 0 \<Longrightarrow> gcd_eucl a b = gcd_eucl b (a mod b)"
72   by (simp add: gcd_eucl.simps [of a b] gcd_eucl.simps [of b 0])
74 lemma gcd_eucl_dvd1 [iff]: "gcd_eucl a b dvd a"
75   and gcd_eucl_dvd2 [iff]: "gcd_eucl a b dvd b"
76   by (induct a b rule: gcd_eucl_induct)
77      (simp_all add: gcd_eucl_0 gcd_eucl_non_0 dvd_mod_iff)
79 lemma normalize_gcd_eucl [simp]:
80   "normalize (gcd_eucl a b) = gcd_eucl a b"
81   by (induct a b rule: gcd_eucl_induct) (simp_all add: gcd_eucl_0 gcd_eucl_non_0)
83 lemma gcd_eucl_greatest:
84   fixes k a b :: 'a
85   shows "k dvd a \<Longrightarrow> k dvd b \<Longrightarrow> k dvd gcd_eucl a b"
86 proof (induct a b rule: gcd_eucl_induct)
87   case (zero a) from zero(1) show ?case by (rule dvd_trans) (simp add: gcd_eucl_0)
88 next
89   case (mod a b)
90   then show ?case
91     by (simp add: gcd_eucl_non_0 dvd_mod_iff)
92 qed
94 lemma gcd_euclI:
95   fixes gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
96   assumes "d dvd a" "d dvd b" "normalize d = d"
97           "\<And>k. k dvd a \<Longrightarrow> k dvd b \<Longrightarrow> k dvd d"
98   shows   "gcd_eucl a b = d"
99   by (rule associated_eqI) (simp_all add: gcd_eucl_greatest assms)
101 lemma eq_gcd_euclI:
102   fixes gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
103   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"
104           "\<And>a b k. k dvd a \<Longrightarrow> k dvd b \<Longrightarrow> k dvd gcd a b"
105   shows   "gcd = gcd_eucl"
106   by (intro ext, rule associated_eqI) (simp_all add: gcd_eucl_greatest assms)
108 lemma gcd_eucl_zero [simp]:
109   "gcd_eucl a b = 0 \<longleftrightarrow> a = 0 \<and> b = 0"
110   by (metis dvd_0_left dvd_refl gcd_eucl_dvd1 gcd_eucl_dvd2 gcd_eucl_greatest)+
113 lemma dvd_Lcm_eucl [simp]: "a \<in> A \<Longrightarrow> a dvd Lcm_eucl A"
114   and Lcm_eucl_least: "(\<And>a. a \<in> A \<Longrightarrow> a dvd b) \<Longrightarrow> Lcm_eucl A dvd b"
115   and unit_factor_Lcm_eucl [simp]:
116           "unit_factor (Lcm_eucl A) = (if Lcm_eucl A = 0 then 0 else 1)"
117 proof -
118   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>
119     unit_factor (Lcm_eucl A) = (if Lcm_eucl A = 0 then 0 else 1)" (is ?thesis)
120   proof (cases "\<exists>l. l \<noteq>  0 \<and> (\<forall>a\<in>A. a dvd l)")
121     case False
122     hence "Lcm_eucl A = 0" by (auto simp: Lcm_eucl_def)
123     with False show ?thesis by auto
124   next
125     case True
126     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
127     define n where "n = (LEAST n. \<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n)"
128     define l where "l = (SOME l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n)"
129     have "\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"
130       apply (subst n_def)
131       apply (rule LeastI[of _ "euclidean_size l\<^sub>0"])
132       apply (rule exI[of _ l\<^sub>0])
133       apply (simp add: l\<^sub>0_props)
134       done
135     from someI_ex[OF this] have "l \<noteq> 0" and "\<forall>a\<in>A. a dvd l" and "euclidean_size l = n"
136       unfolding l_def by simp_all
137     {
138       fix l' assume "\<forall>a\<in>A. a dvd l'"
139       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)
140       moreover from \<open>l \<noteq> 0\<close> have "gcd_eucl l l' \<noteq> 0" by simp
141       ultimately have "\<exists>b. b \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd b) \<and>
142                           euclidean_size b = euclidean_size (gcd_eucl l l')"
143         by (intro exI[of _ "gcd_eucl l l'"], auto)
144       hence "euclidean_size (gcd_eucl l l') \<ge> n" by (subst n_def) (rule Least_le)
145       moreover have "euclidean_size (gcd_eucl l l') \<le> n"
146       proof -
147         have "gcd_eucl l l' dvd l" by simp
148         then obtain a where "l = gcd_eucl l l' * a" unfolding dvd_def by blast
149         with \<open>l \<noteq> 0\<close> have "a \<noteq> 0" by auto
150         hence "euclidean_size (gcd_eucl l l') \<le> euclidean_size (gcd_eucl l l' * a)"
151           by (rule size_mult_mono)
152         also have "gcd_eucl l l' * a = l" using \<open>l = gcd_eucl l l' * a\<close> ..
153         also note \<open>euclidean_size l = n\<close>
154         finally show "euclidean_size (gcd_eucl l l') \<le> n" .
155       qed
156       ultimately have *: "euclidean_size l = euclidean_size (gcd_eucl l l')"
157         by (intro le_antisym, simp_all add: \<open>euclidean_size l = n\<close>)
158       from \<open>l \<noteq> 0\<close> have "l dvd gcd_eucl l l'"
159         by (rule dvd_euclidean_size_eq_imp_dvd) (auto simp add: *)
160       hence "l dvd l'" by (rule dvd_trans[OF _ gcd_eucl_dvd2])
161     }
163     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>
164       have "(\<forall>a\<in>A. a dvd normalize l) \<and>
165         (\<forall>l'. (\<forall>a\<in>A. a dvd l') \<longrightarrow> normalize l dvd l') \<and>
166         unit_factor (normalize l) =
167         (if normalize l = 0 then 0 else 1)"
168       by (auto simp: unit_simps)
169     also from True have "normalize l = Lcm_eucl A"
170       by (simp add: Lcm_eucl_def Let_def n_def l_def)
171     finally show ?thesis .
172   qed
173   note A = this
175   {fix a assume "a \<in> A" then show "a dvd Lcm_eucl A" using A by blast}
176   {fix b assume "\<And>a. a \<in> A \<Longrightarrow> a dvd b" then show "Lcm_eucl A dvd b" using A by blast}
177   from A show "unit_factor (Lcm_eucl A) = (if Lcm_eucl A = 0 then 0 else 1)" by blast
178 qed
180 lemma normalize_Lcm_eucl [simp]:
181   "normalize (Lcm_eucl A) = Lcm_eucl A"
182 proof (cases "Lcm_eucl A = 0")
183   case True then show ?thesis by simp
184 next
185   case False
186   have "unit_factor (Lcm_eucl A) * normalize (Lcm_eucl A) = Lcm_eucl A"
187     by (fact unit_factor_mult_normalize)
188   with False show ?thesis by simp
189 qed
191 lemma eq_Lcm_euclI:
192   fixes lcm :: "'a set \<Rightarrow> 'a"
193   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"
194           "\<And>A. normalize (lcm A) = lcm A" shows "lcm = Lcm_eucl"
195   by (intro ext, rule associated_eqI) (auto simp: assms intro: Lcm_eucl_least)
197 lemma Gcd_eucl_dvd: "a \<in> A \<Longrightarrow> Gcd_eucl A dvd a"
198   unfolding Gcd_eucl_def by (auto intro: Lcm_eucl_least)
200 lemma Gcd_eucl_greatest: "(\<And>x. x \<in> A \<Longrightarrow> d dvd x) \<Longrightarrow> d dvd Gcd_eucl A"
201   unfolding Gcd_eucl_def by auto
203 lemma normalize_Gcd_eucl [simp]: "normalize (Gcd_eucl A) = Gcd_eucl A"
204   by (simp add: Gcd_eucl_def)
206 lemma Lcm_euclI:
207   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"
208   shows   "Lcm_eucl A = d"
209 proof -
210   have "normalize (Lcm_eucl A) = normalize d"
211     by (intro associatedI) (auto intro: dvd_Lcm_eucl Lcm_eucl_least assms)
212   thus ?thesis by (simp add: assms)
213 qed
215 lemma Gcd_euclI:
216   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"
217   shows   "Gcd_eucl A = d"
218 proof -
219   have "normalize (Gcd_eucl A) = normalize d"
220     by (intro associatedI) (auto intro: Gcd_eucl_dvd Gcd_eucl_greatest assms)
221   thus ?thesis by (simp add: assms)
222 qed
224 lemmas lcm_gcd_eucl_facts =
225   gcd_eucl_dvd1 gcd_eucl_dvd2 gcd_eucl_greatest normalize_gcd_eucl lcm_eucl_def
226   Gcd_eucl_def Gcd_eucl_dvd Gcd_eucl_greatest normalize_Gcd_eucl
227   dvd_Lcm_eucl Lcm_eucl_least normalize_Lcm_eucl
229 lemma normalized_factors_product:
230   "{p. p dvd a * b \<and> normalize p = p} =
231      (\<lambda>(x,y). x * y) ` ({p. p dvd a \<and> normalize p = p} \<times> {p. p dvd b \<and> normalize p = p})"
232 proof safe
233   fix p assume p: "p dvd a * b" "normalize p = p"
234   interpret semiring_gcd 1 0 "op *" gcd_eucl lcm_eucl "op div" "op +" "op -" normalize unit_factor
235     by standard (rule lcm_gcd_eucl_facts; assumption)+
236   from dvd_productE[OF p(1)] guess x y . note xy = this
237   define x' y' where "x' = normalize x" and "y' = normalize y"
238   have "p = x' * y'"
239     by (subst p(2) [symmetric]) (simp add: xy x'_def y'_def normalize_mult)
240   moreover from xy have "normalize x' = x'" "normalize y' = y'" "x' dvd a" "y' dvd b"
241     by (simp_all add: x'_def y'_def)
242   ultimately show "p \<in> (\<lambda>(x, y). x * y) `
243                      ({p. p dvd a \<and> normalize p = p} \<times> {p. p dvd b \<and> normalize p = p})"
244     by blast
245 qed (auto simp: normalize_mult mult_dvd_mono)
248 subclass factorial_semiring
249 proof (standard, rule factorial_semiring_altI_aux)
250   fix x assume "x \<noteq> 0"
251   thus "finite {p. p dvd x \<and> normalize p = p}"
252   proof (induction "euclidean_size x" arbitrary: x rule: less_induct)
253     case (less x)
254     show ?case
255     proof (cases "\<exists>y. y dvd x \<and> \<not>x dvd y \<and> \<not>is_unit y")
256       case False
257       have "{p. p dvd x \<and> normalize p = p} \<subseteq> {1, normalize x}"
258       proof
259         fix p assume p: "p \<in> {p. p dvd x \<and> normalize p = p}"
260         with False have "is_unit p \<or> x dvd p" by blast
261         thus "p \<in> {1, normalize x}"
262         proof (elim disjE)
263           assume "is_unit p"
264           hence "normalize p = 1" by (simp add: is_unit_normalize)
265           with p show ?thesis by simp
266         next
267           assume "x dvd p"
268           with p have "normalize p = normalize x" by (intro associatedI) simp_all
269           with p show ?thesis by simp
270         qed
271       qed
272       moreover have "finite \<dots>" by simp
273       ultimately show ?thesis by (rule finite_subset)
275     next
276       case True
277       then obtain y where y: "y dvd x" "\<not>x dvd y" "\<not>is_unit y" by blast
278       define z where "z = x div y"
279       let ?fctrs = "\<lambda>x. {p. p dvd x \<and> normalize p = p}"
280       from y have x: "x = y * z" by (simp add: z_def)
281       with less.prems have "y \<noteq> 0" "z \<noteq> 0" by auto
282       from x y have "\<not>is_unit z" by (auto simp: mult_unit_dvd_iff)
283       have "?fctrs x = (\<lambda>(p,p'). p * p') ` (?fctrs y \<times> ?fctrs z)"
284         by (subst x) (rule normalized_factors_product)
285       also have "\<not>y * z dvd y * 1" "\<not>y * z dvd 1 * z"
286         by (subst dvd_times_left_cancel_iff dvd_times_right_cancel_iff; fact)+
287       hence "finite ((\<lambda>(p,p'). p * p') ` (?fctrs y \<times> ?fctrs z))"
288         by (intro finite_imageI finite_cartesian_product less dvd_proper_imp_size_less)
289            (auto simp: x)
290       finally show ?thesis .
291     qed
292   qed
293 next
294   interpret semiring_gcd 1 0 "op *" gcd_eucl lcm_eucl "op div" "op +" "op -" normalize unit_factor
295     by standard (rule lcm_gcd_eucl_facts; assumption)+
296   fix p assume p: "irreducible p"
297   thus "prime_elem p" by (rule irreducible_imp_prime_elem_gcd)
298 qed
300 lemma gcd_eucl_eq_gcd_factorial: "gcd_eucl = gcd_factorial"
301   by (intro ext gcd_euclI gcd_lcm_factorial)
303 lemma lcm_eucl_eq_lcm_factorial: "lcm_eucl = lcm_factorial"
304   by (intro ext) (simp add: lcm_eucl_def lcm_factorial_gcd_factorial gcd_eucl_eq_gcd_factorial)
306 lemma Gcd_eucl_eq_Gcd_factorial: "Gcd_eucl = Gcd_factorial"
307   by (intro ext Gcd_euclI gcd_lcm_factorial)
309 lemma Lcm_eucl_eq_Lcm_factorial: "Lcm_eucl = Lcm_factorial"
310   by (intro ext Lcm_euclI gcd_lcm_factorial)
312 lemmas eucl_eq_factorial =
313   gcd_eucl_eq_gcd_factorial lcm_eucl_eq_lcm_factorial
314   Gcd_eucl_eq_Gcd_factorial Lcm_eucl_eq_Lcm_factorial
316 end
318 context euclidean_ring
319 begin
321 function euclid_ext_aux :: "'a \<Rightarrow> _" where
322   "euclid_ext_aux r' r s' s t' t = (
323      if r = 0 then let c = 1 div unit_factor r' in (s' * c, t' * c, normalize r')
324      else let q = r' div r
325           in  euclid_ext_aux r (r' mod r) s (s' - q * s) t (t' - q * t))"
326 by auto
327 termination by (relation "measure (\<lambda>(_,b,_,_,_,_). euclidean_size b)") (simp_all add: mod_size_less)
329 declare euclid_ext_aux.simps [simp del]
331 lemma euclid_ext_aux_correct:
332   assumes "gcd_eucl r' r = gcd_eucl a b"
333   assumes "s' * a + t' * b = r'"
334   assumes "s * a + t * b = r"
335   shows   "case euclid_ext_aux r' r s' s t' t of (x,y,c) \<Rightarrow>
336              x * a + y * b = c \<and> c = gcd_eucl a b" (is "?P (euclid_ext_aux r' r s' s t' t)")
337 using assms
338 proof (induction r' r s' s t' t rule: euclid_ext_aux.induct)
339   case (1 r' r s' s t' t)
340   show ?case
341   proof (cases "r = 0")
342     case True
343     hence "euclid_ext_aux r' r s' s t' t =
344              (s' div unit_factor r', t' div unit_factor r', normalize r')"
345       by (subst euclid_ext_aux.simps) (simp add: Let_def)
346     also have "?P \<dots>"
347     proof safe
348       have "s' div unit_factor r' * a + t' div unit_factor r' * b =
349                 (s' * a + t' * b) div unit_factor r'"
350         by (cases "r' = 0") (simp_all add: unit_div_commute)
351       also have "s' * a + t' * b = r'" by fact
352       also have "\<dots> div unit_factor r' = normalize r'" by simp
353       finally show "s' div unit_factor r' * a + t' div unit_factor r' * b = normalize r'" .
354     next
355       from "1.prems" True show "normalize r' = gcd_eucl a b" by (simp add: gcd_eucl_0)
356     qed
357     finally show ?thesis .
358   next
359     case False
360     hence "euclid_ext_aux r' r s' s t' t =
361              euclid_ext_aux r (r' mod r) s (s' - r' div r * s) t (t' - r' div r * t)"
362       by (subst euclid_ext_aux.simps) (simp add: Let_def)
363     also from "1.prems" False have "?P \<dots>"
364     proof (intro "1.IH")
365       have "(s' - r' div r * s) * a + (t' - r' div r * t) * b =
366               (s' * a + t' * b) - r' div r * (s * a + t * b)" by (simp add: algebra_simps)
367       also have "s' * a + t' * b = r'" by fact
368       also have "s * a + t * b = r" by fact
369       also have "r' - r' div r * r = r' mod r" using div_mult_mod_eq [of r' r]
370         by (simp add: algebra_simps)
371       finally show "(s' - r' div r * s) * a + (t' - r' div r * t) * b = r' mod r" .
372     qed (auto simp: gcd_eucl_non_0 algebra_simps minus_mod_eq_div_mult [symmetric])
373     finally show ?thesis .
374   qed
375 qed
377 definition euclid_ext where
378   "euclid_ext a b = euclid_ext_aux a b 1 0 0 1"
380 lemma euclid_ext_0:
381   "euclid_ext a 0 = (1 div unit_factor a, 0, normalize a)"
382   by (simp add: euclid_ext_def euclid_ext_aux.simps)
384 lemma euclid_ext_left_0:
385   "euclid_ext 0 a = (0, 1 div unit_factor a, normalize a)"
386   by (simp add: euclid_ext_def euclid_ext_aux.simps)
388 lemma euclid_ext_correct':
389   "case euclid_ext a b of (x,y,c) \<Rightarrow> x * a + y * b = c \<and> c = gcd_eucl a b"
390   unfolding euclid_ext_def by (rule euclid_ext_aux_correct) simp_all
392 lemma euclid_ext_gcd_eucl:
393   "(case euclid_ext a b of (x,y,c) \<Rightarrow> c) = gcd_eucl a b"
394   using euclid_ext_correct'[of a b] by (simp add: case_prod_unfold)
396 definition euclid_ext' where
397   "euclid_ext' a b = (case euclid_ext a b of (x, y, _) \<Rightarrow> (x, y))"
399 lemma euclid_ext'_correct':
400   "case euclid_ext' a b of (x,y) \<Rightarrow> x * a + y * b = gcd_eucl a b"
401   using euclid_ext_correct'[of a b] by (simp add: case_prod_unfold euclid_ext'_def)
403 lemma euclid_ext'_0: "euclid_ext' a 0 = (1 div unit_factor a, 0)"
404   by (simp add: euclid_ext'_def euclid_ext_0)
406 lemma euclid_ext'_left_0: "euclid_ext' 0 a = (0, 1 div unit_factor a)"
407   by (simp add: euclid_ext'_def euclid_ext_left_0)
409 end
411 class euclidean_semiring_gcd = euclidean_semiring + gcd + Gcd +
412   assumes gcd_gcd_eucl: "gcd = gcd_eucl" and lcm_lcm_eucl: "lcm = lcm_eucl"
413   assumes Gcd_Gcd_eucl: "Gcd = Gcd_eucl" and Lcm_Lcm_eucl: "Lcm = Lcm_eucl"
414 begin
416 subclass semiring_gcd
417   by standard (simp_all add: gcd_gcd_eucl gcd_eucl_greatest lcm_lcm_eucl lcm_eucl_def)
419 subclass semiring_Gcd
420   by standard (auto simp: Gcd_Gcd_eucl Lcm_Lcm_eucl Gcd_eucl_def intro: Lcm_eucl_least)
422 subclass factorial_semiring_gcd
423 proof
424   fix a b
425   show "gcd a b = gcd_factorial a b"
426     by (rule sym, rule gcdI) (rule gcd_lcm_factorial; assumption)+
427   thus "lcm a b = lcm_factorial a b"
428     by (simp add: lcm_factorial_gcd_factorial lcm_gcd)
429 next
430   fix A
431   show "Gcd A = Gcd_factorial A"
432     by (rule sym, rule GcdI) (rule gcd_lcm_factorial; assumption)+
433   show "Lcm A = Lcm_factorial A"
434     by (rule sym, rule LcmI) (rule gcd_lcm_factorial; assumption)+
435 qed
437 lemma gcd_non_0:
438   "b \<noteq> 0 \<Longrightarrow> gcd a b = gcd b (a mod b)"
439   unfolding gcd_gcd_eucl by (fact gcd_eucl_non_0)
441 lemmas gcd_0 = gcd_0_right
442 lemmas dvd_gcd_iff = gcd_greatest_iff
443 lemmas gcd_greatest_iff = dvd_gcd_iff
445 lemma gcd_mod1 [simp]:
446   "gcd (a mod b) b = gcd a b"
447   by (rule gcdI, metis dvd_mod_iff gcd_dvd1 gcd_dvd2, simp_all add: gcd_greatest dvd_mod_iff)
449 lemma gcd_mod2 [simp]:
450   "gcd a (b mod a) = gcd a b"
451   by (rule gcdI, simp, metis dvd_mod_iff gcd_dvd1 gcd_dvd2, simp_all add: gcd_greatest dvd_mod_iff)
453 lemma euclidean_size_gcd_le1 [simp]:
454   assumes "a \<noteq> 0"
455   shows "euclidean_size (gcd a b) \<le> euclidean_size a"
456 proof -
457    have "gcd a b dvd a" by (rule gcd_dvd1)
458    then obtain c where A: "a = gcd a b * c" unfolding dvd_def by blast
459    with \<open>a \<noteq> 0\<close> show ?thesis by (subst (2) A, intro size_mult_mono) auto
460 qed
462 lemma euclidean_size_gcd_le2 [simp]:
463   "b \<noteq> 0 \<Longrightarrow> euclidean_size (gcd a b) \<le> euclidean_size b"
464   by (subst gcd.commute, rule euclidean_size_gcd_le1)
466 lemma euclidean_size_gcd_less1:
467   assumes "a \<noteq> 0" and "\<not>a dvd b"
468   shows "euclidean_size (gcd a b) < euclidean_size a"
469 proof (rule ccontr)
470   assume "\<not>euclidean_size (gcd a b) < euclidean_size a"
471   with \<open>a \<noteq> 0\<close> have A: "euclidean_size (gcd a b) = euclidean_size a"
472     by (intro le_antisym, simp_all)
473   have "a dvd gcd a b"
474     by (rule dvd_euclidean_size_eq_imp_dvd) (simp_all add: assms A)
475   hence "a dvd b" using dvd_gcdD2 by blast
476   with \<open>\<not>a dvd b\<close> show False by contradiction
477 qed
479 lemma euclidean_size_gcd_less2:
480   assumes "b \<noteq> 0" and "\<not>b dvd a"
481   shows "euclidean_size (gcd a b) < euclidean_size b"
482   using assms by (subst gcd.commute, rule euclidean_size_gcd_less1)
484 lemma euclidean_size_lcm_le1:
485   assumes "a \<noteq> 0" and "b \<noteq> 0"
486   shows "euclidean_size a \<le> euclidean_size (lcm a b)"
487 proof -
488   have "a dvd lcm a b" by (rule dvd_lcm1)
489   then obtain c where A: "lcm a b = a * c" ..
490   with \<open>a \<noteq> 0\<close> and \<open>b \<noteq> 0\<close> have "c \<noteq> 0" by (auto simp: lcm_eq_0_iff)
491   then show ?thesis by (subst A, intro size_mult_mono)
492 qed
494 lemma euclidean_size_lcm_le2:
495   "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> euclidean_size b \<le> euclidean_size (lcm a b)"
496   using euclidean_size_lcm_le1 [of b a] by (simp add: ac_simps)
498 lemma euclidean_size_lcm_less1:
499   assumes "b \<noteq> 0" and "\<not>b dvd a"
500   shows "euclidean_size a < euclidean_size (lcm a b)"
501 proof (rule ccontr)
502   from assms have "a \<noteq> 0" by auto
503   assume "\<not>euclidean_size a < euclidean_size (lcm a b)"
504   with \<open>a \<noteq> 0\<close> and \<open>b \<noteq> 0\<close> have "euclidean_size (lcm a b) = euclidean_size a"
505     by (intro le_antisym, simp, intro euclidean_size_lcm_le1)
506   with assms have "lcm a b dvd a"
507     by (rule_tac dvd_euclidean_size_eq_imp_dvd) (auto simp: lcm_eq_0_iff)
508   hence "b dvd a" by (rule lcm_dvdD2)
509   with \<open>\<not>b dvd a\<close> show False by contradiction
510 qed
512 lemma euclidean_size_lcm_less2:
513   assumes "a \<noteq> 0" and "\<not>a dvd b"
514   shows "euclidean_size b < euclidean_size (lcm a b)"
515   using assms euclidean_size_lcm_less1 [of a b] by (simp add: ac_simps)
517 lemma Lcm_eucl_set [code]:
518   "Lcm_eucl (set xs) = foldl lcm_eucl 1 xs"
519   by (simp add: Lcm_Lcm_eucl [symmetric] lcm_lcm_eucl Lcm_set)
521 lemma Gcd_eucl_set [code]:
522   "Gcd_eucl (set xs) = foldl gcd_eucl 0 xs"
523   by (simp add: Gcd_Gcd_eucl [symmetric] gcd_gcd_eucl Gcd_set)
525 end
528 text \<open>
529   A Euclidean ring is a Euclidean semiring with additive inverses. It provides a
530   few more lemmas; in particular, Bezout's lemma holds for any Euclidean ring.
531 \<close>
533 class euclidean_ring_gcd = euclidean_semiring_gcd + idom
534 begin
536 subclass euclidean_ring ..
537 subclass ring_gcd ..
538 subclass factorial_ring_gcd ..
540 lemma euclid_ext_gcd [simp]:
541   "(case euclid_ext a b of (_, _ , t) \<Rightarrow> t) = gcd a b"
542   using euclid_ext_correct'[of a b] by (simp add: case_prod_unfold Let_def gcd_gcd_eucl)
544 lemma euclid_ext_gcd' [simp]:
545   "euclid_ext a b = (r, s, t) \<Longrightarrow> t = gcd a b"
546   by (insert euclid_ext_gcd[of a b], drule (1) subst, simp)
548 lemma euclid_ext_correct:
549   "case euclid_ext a b of (x,y,c) \<Rightarrow> x * a + y * b = c \<and> c = gcd a b"
550   using euclid_ext_correct'[of a b]
551   by (simp add: gcd_gcd_eucl case_prod_unfold)
553 lemma euclid_ext'_correct:
554   "fst (euclid_ext' a b) * a + snd (euclid_ext' a b) * b = gcd a b"
555   using euclid_ext_correct'[of a b]
556   by (simp add: gcd_gcd_eucl case_prod_unfold euclid_ext'_def)
558 lemma bezout: "\<exists>s t. s * a + t * b = gcd a b"
559   using euclid_ext'_correct by blast
561 end
564 subsection \<open>Typical instances\<close>
566 instance nat :: euclidean_semiring_gcd
567 proof
568   show [simp]: "gcd = (gcd_eucl :: nat \<Rightarrow> _)" "Lcm = (Lcm_eucl :: nat set \<Rightarrow> _)"
569     by (simp_all add: eq_gcd_euclI eq_Lcm_euclI)
570   show "lcm = (lcm_eucl :: nat \<Rightarrow> _)" "Gcd = (Gcd_eucl :: nat set \<Rightarrow> _)"
571     by (intro ext, simp add: lcm_eucl_def lcm_nat_def Gcd_nat_def Gcd_eucl_def)+
572 qed
574 instance int :: euclidean_ring_gcd
575 proof
576   show [simp]: "gcd = (gcd_eucl :: int \<Rightarrow> _)" "Lcm = (Lcm_eucl :: int set \<Rightarrow> _)"
577     by (simp_all add: eq_gcd_euclI eq_Lcm_euclI)
578   show "lcm = (lcm_eucl :: int \<Rightarrow> _)" "Gcd = (Gcd_eucl :: int set \<Rightarrow> _)"
579     by (intro ext, simp add: lcm_eucl_def lcm_altdef_int
580           semiring_Gcd_class.Gcd_Lcm Gcd_eucl_def abs_mult)+
581 qed
583 end