src/HOL/Computational_Algebra/Euclidean_Algorithm.thy
 author haftmann Thu Apr 06 21:37:13 2017 +0200 (2017-04-06) changeset 65417 fc41a5650fb1 parent 65398 src/HOL/Number_Theory/Euclidean_Algorithm.thy@a14fa655b48c child 65435 378175f44328 permissions -rw-r--r--
session containing computational algebra
```     1 (*  Title:      HOL/Number_Theory/Euclidean_Algorithm.thy
```
```     2     Author:     Manuel Eberl, TU Muenchen
```
```     3 *)
```
```     4
```
```     5 section \<open>Abstract euclidean algorithm in euclidean (semi)rings\<close>
```
```     6
```
```     7 theory Euclidean_Algorithm
```
```     8   imports Factorial_Ring
```
```     9 begin
```
```    10
```
```    11 subsection \<open>Generic construction of the (simple) euclidean algorithm\<close>
```
```    12
```
```    13 context euclidean_semiring
```
```    14 begin
```
```    15
```
```    16 context
```
```    17 begin
```
```    18
```
```    19 qualified function gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
```
```    20   where "gcd a b = (if b = 0 then normalize a else gcd b (a mod b))"
```
```    21   by pat_completeness simp
```
```    22 termination
```
```    23   by (relation "measure (euclidean_size \<circ> snd)") (simp_all add: mod_size_less)
```
```    24
```
```    25 declare gcd.simps [simp del]
```
```    26
```
```    27 lemma eucl_induct [case_names zero mod]:
```
```    28   assumes H1: "\<And>b. P b 0"
```
```    29   and H2: "\<And>a b. b \<noteq> 0 \<Longrightarrow> P b (a mod b) \<Longrightarrow> P a b"
```
```    30   shows "P a b"
```
```    31 proof (induct a b rule: gcd.induct)
```
```    32   case (1 a b)
```
```    33   show ?case
```
```    34   proof (cases "b = 0")
```
```    35     case True then show "P a b" by simp (rule H1)
```
```    36   next
```
```    37     case False
```
```    38     then have "P b (a mod b)"
```
```    39       by (rule "1.hyps")
```
```    40     with \<open>b \<noteq> 0\<close> show "P a b"
```
```    41       by (blast intro: H2)
```
```    42   qed
```
```    43 qed
```
```    44
```
```    45 qualified lemma gcd_0:
```
```    46   "gcd a 0 = normalize a"
```
```    47   by (simp add: gcd.simps [of a 0])
```
```    48
```
```    49 qualified lemma gcd_mod:
```
```    50   "a \<noteq> 0 \<Longrightarrow> gcd a (b mod a) = gcd b a"
```
```    51   by (simp add: gcd.simps [of b 0] gcd.simps [of b a])
```
```    52
```
```    53 qualified definition lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
```
```    54   where "lcm a b = normalize (a * b) div gcd a b"
```
```    55
```
```    56 qualified definition Lcm :: "'a set \<Rightarrow> 'a" \<comment>
```
```    57     \<open>Somewhat complicated definition of Lcm that has the advantage of working
```
```    58     for infinite sets as well\<close>
```
```    59   where
```
```    60   [code del]: "Lcm A = (if \<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) then
```
```    61      let l = SOME l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l =
```
```    62        (LEAST n. \<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n)
```
```    63        in normalize l
```
```    64       else 0)"
```
```    65
```
```    66 qualified definition Gcd :: "'a set \<Rightarrow> 'a"
```
```    67   where [code del]: "Gcd A = Lcm {d. \<forall>a\<in>A. d dvd a}"
```
```    68
```
```    69 end
```
```    70
```
```    71 lemma semiring_gcd:
```
```    72   "class.semiring_gcd one zero times gcd lcm
```
```    73     divide plus minus unit_factor normalize"
```
```    74 proof
```
```    75   show "gcd a b dvd a"
```
```    76     and "gcd a b dvd b" for a b
```
```    77     by (induct a b rule: eucl_induct)
```
```    78       (simp_all add: local.gcd_0 local.gcd_mod dvd_mod_iff)
```
```    79 next
```
```    80   show "c dvd a \<Longrightarrow> c dvd b \<Longrightarrow> c dvd gcd a b" for a b c
```
```    81   proof (induct a b rule: eucl_induct)
```
```    82     case (zero a) from \<open>c dvd a\<close> show ?case
```
```    83       by (rule dvd_trans) (simp add: local.gcd_0)
```
```    84   next
```
```    85     case (mod a b)
```
```    86     then show ?case
```
```    87       by (simp add: local.gcd_mod dvd_mod_iff)
```
```    88   qed
```
```    89 next
```
```    90   show "normalize (gcd a b) = gcd a b" for a b
```
```    91     by (induct a b rule: eucl_induct)
```
```    92       (simp_all add: local.gcd_0 local.gcd_mod)
```
```    93 next
```
```    94   show "lcm a b = normalize (a * b) div gcd a b" for a b
```
```    95     by (fact local.lcm_def)
```
```    96 qed
```
```    97
```
```    98 interpretation semiring_gcd one zero times gcd lcm
```
```    99   divide plus minus unit_factor normalize
```
```   100   by (fact semiring_gcd)
```
```   101
```
```   102 lemma semiring_Gcd:
```
```   103   "class.semiring_Gcd one zero times gcd lcm Gcd Lcm
```
```   104     divide plus minus unit_factor normalize"
```
```   105 proof -
```
```   106   show ?thesis
```
```   107   proof
```
```   108     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
```
```   109     proof (cases "\<exists>l. l \<noteq>  0 \<and> (\<forall>a\<in>A. a dvd l)")
```
```   110       case False
```
```   111       then have "Lcm A = 0"
```
```   112         by (auto simp add: local.Lcm_def)
```
```   113       with False show ?thesis
```
```   114         by auto
```
```   115     next
```
```   116       case True
```
```   117       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
```
```   118       define n where "n = (LEAST n. \<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n)"
```
```   119       define l where "l = (SOME l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n)"
```
```   120       have "\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"
```
```   121         apply (subst n_def)
```
```   122         apply (rule LeastI [of _ "euclidean_size l\<^sub>0"])
```
```   123         apply (rule exI [of _ l\<^sub>0])
```
```   124         apply (simp add: l\<^sub>0_props)
```
```   125         done
```
```   126       from someI_ex [OF this] have "l \<noteq> 0" and "\<forall>a\<in>A. a dvd l"
```
```   127         and "euclidean_size l = n"
```
```   128         unfolding l_def by simp_all
```
```   129       {
```
```   130         fix l' assume "\<forall>a\<in>A. a dvd l'"
```
```   131         with \<open>\<forall>a\<in>A. a dvd l\<close> have "\<forall>a\<in>A. a dvd gcd l l'"
```
```   132           by (auto intro: gcd_greatest)
```
```   133         moreover from \<open>l \<noteq> 0\<close> have "gcd l l' \<noteq> 0"
```
```   134           by simp
```
```   135         ultimately have "\<exists>b. b \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd b) \<and>
```
```   136           euclidean_size b = euclidean_size (gcd l l')"
```
```   137           by (intro exI [of _ "gcd l l'"], auto)
```
```   138         then have "euclidean_size (gcd l l') \<ge> n"
```
```   139           by (subst n_def) (rule Least_le)
```
```   140         moreover have "euclidean_size (gcd l l') \<le> n"
```
```   141         proof -
```
```   142           have "gcd l l' dvd l"
```
```   143             by simp
```
```   144           then obtain a where "l = gcd l l' * a" ..
```
```   145           with \<open>l \<noteq> 0\<close> have "a \<noteq> 0"
```
```   146             by auto
```
```   147           hence "euclidean_size (gcd l l') \<le> euclidean_size (gcd l l' * a)"
```
```   148             by (rule size_mult_mono)
```
```   149           also have "gcd l l' * a = l" using \<open>l = gcd l l' * a\<close> ..
```
```   150           also note \<open>euclidean_size l = n\<close>
```
```   151           finally show "euclidean_size (gcd l l') \<le> n" .
```
```   152         qed
```
```   153         ultimately have *: "euclidean_size l = euclidean_size (gcd l l')"
```
```   154           by (intro le_antisym, simp_all add: \<open>euclidean_size l = n\<close>)
```
```   155         from \<open>l \<noteq> 0\<close> have "l dvd gcd l l'"
```
```   156           by (rule dvd_euclidean_size_eq_imp_dvd) (auto simp add: *)
```
```   157         hence "l dvd l'" by (rule dvd_trans [OF _ gcd_dvd2])
```
```   158       }
```
```   159       with \<open>\<forall>a\<in>A. a dvd l\<close> and \<open>l \<noteq> 0\<close>
```
```   160         have "(\<forall>a\<in>A. a dvd normalize l) \<and>
```
```   161           (\<forall>l'. (\<forall>a\<in>A. a dvd l') \<longrightarrow> normalize l dvd l')"
```
```   162         by auto
```
```   163       also from True have "normalize l = Lcm A"
```
```   164         by (simp add: local.Lcm_def Let_def n_def l_def)
```
```   165       finally show ?thesis .
```
```   166     qed
```
```   167     then show dvd_Lcm: "a \<in> A \<Longrightarrow> a dvd Lcm A"
```
```   168       and Lcm_least: "(\<And>a. a \<in> A \<Longrightarrow> a dvd b) \<Longrightarrow> Lcm A dvd b" for A and a b
```
```   169       by auto
```
```   170     show "a \<in> A \<Longrightarrow> Gcd A dvd a" for A and a
```
```   171       by (auto simp add: local.Gcd_def intro: Lcm_least)
```
```   172     show "(\<And>a. a \<in> A \<Longrightarrow> b dvd a) \<Longrightarrow> b dvd Gcd A" for A and b
```
```   173       by (auto simp add: local.Gcd_def intro: dvd_Lcm)
```
```   174     show [simp]: "normalize (Lcm A) = Lcm A" for A
```
```   175       by (simp add: local.Lcm_def)
```
```   176     show "normalize (Gcd A) = Gcd A" for A
```
```   177       by (simp add: local.Gcd_def)
```
```   178   qed
```
```   179 qed
```
```   180
```
```   181 interpretation semiring_Gcd one zero times gcd lcm Gcd Lcm
```
```   182     divide plus minus unit_factor normalize
```
```   183   by (fact semiring_Gcd)
```
```   184
```
```   185 subclass factorial_semiring
```
```   186 proof -
```
```   187   show "class.factorial_semiring divide plus minus zero times one
```
```   188      unit_factor normalize"
```
```   189   proof (standard, rule factorial_semiring_altI_aux) \<comment> \<open>FIXME rule\<close>
```
```   190     fix x assume "x \<noteq> 0"
```
```   191     thus "finite {p. p dvd x \<and> normalize p = p}"
```
```   192     proof (induction "euclidean_size x" arbitrary: x rule: less_induct)
```
```   193       case (less x)
```
```   194       show ?case
```
```   195       proof (cases "\<exists>y. y dvd x \<and> \<not>x dvd y \<and> \<not>is_unit y")
```
```   196         case False
```
```   197         have "{p. p dvd x \<and> normalize p = p} \<subseteq> {1, normalize x}"
```
```   198         proof
```
```   199           fix p assume p: "p \<in> {p. p dvd x \<and> normalize p = p}"
```
```   200           with False have "is_unit p \<or> x dvd p" by blast
```
```   201           thus "p \<in> {1, normalize x}"
```
```   202           proof (elim disjE)
```
```   203             assume "is_unit p"
```
```   204             hence "normalize p = 1" by (simp add: is_unit_normalize)
```
```   205             with p show ?thesis by simp
```
```   206           next
```
```   207             assume "x dvd p"
```
```   208             with p have "normalize p = normalize x" by (intro associatedI) simp_all
```
```   209             with p show ?thesis by simp
```
```   210           qed
```
```   211         qed
```
```   212         moreover have "finite \<dots>" by simp
```
```   213         ultimately show ?thesis by (rule finite_subset)
```
```   214       next
```
```   215         case True
```
```   216         then obtain y where y: "y dvd x" "\<not>x dvd y" "\<not>is_unit y" by blast
```
```   217         define z where "z = x div y"
```
```   218         let ?fctrs = "\<lambda>x. {p. p dvd x \<and> normalize p = p}"
```
```   219         from y have x: "x = y * z" by (simp add: z_def)
```
```   220         with less.prems have "y \<noteq> 0" "z \<noteq> 0" by auto
```
```   221         have normalized_factors_product:
```
```   222           "{p. p dvd a * b \<and> normalize p = p} =
```
```   223              (\<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
```
```   224         proof safe
```
```   225           fix p assume p: "p dvd a * b" "normalize p = p"
```
```   226           from dvd_productE[OF p(1)] guess x y . note xy = this
```
```   227           define x' y' where "x' = normalize x" and "y' = normalize y"
```
```   228           have "p = x' * y'"
```
```   229             by (subst p(2) [symmetric]) (simp add: xy x'_def y'_def normalize_mult)
```
```   230           moreover from xy have "normalize x' = x'" "normalize y' = y'" "x' dvd a" "y' dvd b"
```
```   231             by (simp_all add: x'_def y'_def)
```
```   232           ultimately show "p \<in> (\<lambda>(x, y). x * y) `
```
```   233             ({p. p dvd a \<and> normalize p = p} \<times> {p. p dvd b \<and> normalize p = p})"
```
```   234             by blast
```
```   235         qed (auto simp: normalize_mult mult_dvd_mono)
```
```   236         from x y have "\<not>is_unit z" by (auto simp: mult_unit_dvd_iff)
```
```   237         have "?fctrs x = (\<lambda>(p,p'). p * p') ` (?fctrs y \<times> ?fctrs z)"
```
```   238           by (subst x) (rule normalized_factors_product)
```
```   239         also have "\<not>y * z dvd y * 1" "\<not>y * z dvd 1 * z"
```
```   240           by (subst dvd_times_left_cancel_iff dvd_times_right_cancel_iff; fact)+
```
```   241         hence "finite ((\<lambda>(p,p'). p * p') ` (?fctrs y \<times> ?fctrs z))"
```
```   242           by (intro finite_imageI finite_cartesian_product less dvd_proper_imp_size_less)
```
```   243              (auto simp: x)
```
```   244         finally show ?thesis .
```
```   245       qed
```
```   246     qed
```
```   247   next
```
```   248     fix p
```
```   249     assume "irreducible p"
```
```   250     then show "prime_elem p"
```
```   251       by (rule irreducible_imp_prime_elem_gcd)
```
```   252   qed
```
```   253 qed
```
```   254
```
```   255 lemma Gcd_eucl_set [code]:
```
```   256   "Gcd (set xs) = fold gcd xs 0"
```
```   257   by (fact Gcd_set_eq_fold)
```
```   258
```
```   259 lemma Lcm_eucl_set [code]:
```
```   260   "Lcm (set xs) = fold lcm xs 1"
```
```   261   by (fact Lcm_set_eq_fold)
```
```   262
```
```   263 end
```
```   264
```
```   265 hide_const (open) gcd lcm Gcd Lcm
```
```   266
```
```   267 lemma prime_elem_int_abs_iff [simp]:
```
```   268   fixes p :: int
```
```   269   shows "prime_elem \<bar>p\<bar> \<longleftrightarrow> prime_elem p"
```
```   270   using prime_elem_normalize_iff [of p] by simp
```
```   271
```
```   272 lemma prime_elem_int_minus_iff [simp]:
```
```   273   fixes p :: int
```
```   274   shows "prime_elem (- p) \<longleftrightarrow> prime_elem p"
```
```   275   using prime_elem_normalize_iff [of "- p"] by simp
```
```   276
```
```   277 lemma prime_int_iff:
```
```   278   fixes p :: int
```
```   279   shows "prime p \<longleftrightarrow> p > 0 \<and> prime_elem p"
```
```   280   by (auto simp add: prime_def dest: prime_elem_not_zeroI)
```
```   281
```
```   282
```
```   283 subsection \<open>The (simple) euclidean algorithm as gcd computation\<close>
```
```   284
```
```   285 class euclidean_semiring_gcd = euclidean_semiring + gcd + Gcd +
```
```   286   assumes gcd_eucl: "Euclidean_Algorithm.gcd = GCD.gcd"
```
```   287     and lcm_eucl: "Euclidean_Algorithm.lcm = GCD.lcm"
```
```   288   assumes Gcd_eucl: "Euclidean_Algorithm.Gcd = GCD.Gcd"
```
```   289     and Lcm_eucl: "Euclidean_Algorithm.Lcm = GCD.Lcm"
```
```   290 begin
```
```   291
```
```   292 subclass semiring_gcd
```
```   293   unfolding gcd_eucl [symmetric] lcm_eucl [symmetric]
```
```   294   by (fact semiring_gcd)
```
```   295
```
```   296 subclass semiring_Gcd
```
```   297   unfolding  gcd_eucl [symmetric] lcm_eucl [symmetric]
```
```   298     Gcd_eucl [symmetric] Lcm_eucl [symmetric]
```
```   299   by (fact semiring_Gcd)
```
```   300
```
```   301 subclass factorial_semiring_gcd
```
```   302 proof
```
```   303   show "gcd a b = gcd_factorial a b" for a b
```
```   304     apply (rule sym)
```
```   305     apply (rule gcdI)
```
```   306        apply (fact gcd_lcm_factorial)+
```
```   307     done
```
```   308   then show "lcm a b = lcm_factorial a b" for a b
```
```   309     by (simp add: lcm_factorial_gcd_factorial lcm_gcd)
```
```   310   show "Gcd A = Gcd_factorial A" for A
```
```   311     apply (rule sym)
```
```   312     apply (rule GcdI)
```
```   313        apply (fact gcd_lcm_factorial)+
```
```   314     done
```
```   315   show "Lcm A = Lcm_factorial A" for A
```
```   316     apply (rule sym)
```
```   317     apply (rule LcmI)
```
```   318        apply (fact gcd_lcm_factorial)+
```
```   319     done
```
```   320 qed
```
```   321
```
```   322 lemma gcd_mod_right [simp]:
```
```   323   "a \<noteq> 0 \<Longrightarrow> gcd a (b mod a) = gcd a b"
```
```   324   unfolding gcd.commute [of a b]
```
```   325   by (simp add: gcd_eucl [symmetric] local.gcd_mod)
```
```   326
```
```   327 lemma gcd_mod_left [simp]:
```
```   328   "b \<noteq> 0 \<Longrightarrow> gcd (a mod b) b = gcd a b"
```
```   329   by (drule gcd_mod_right [of _ a]) (simp add: gcd.commute)
```
```   330
```
```   331 lemma euclidean_size_gcd_le1 [simp]:
```
```   332   assumes "a \<noteq> 0"
```
```   333   shows "euclidean_size (gcd a b) \<le> euclidean_size a"
```
```   334 proof -
```
```   335   from gcd_dvd1 obtain c where A: "a = gcd a b * c" ..
```
```   336   with assms have "c \<noteq> 0"
```
```   337     by auto
```
```   338   moreover from this
```
```   339   have "euclidean_size (gcd a b) \<le> euclidean_size (gcd a b * c)"
```
```   340     by (rule size_mult_mono)
```
```   341   with A show ?thesis
```
```   342     by simp
```
```   343 qed
```
```   344
```
```   345 lemma euclidean_size_gcd_le2 [simp]:
```
```   346   "b \<noteq> 0 \<Longrightarrow> euclidean_size (gcd a b) \<le> euclidean_size b"
```
```   347   by (subst gcd.commute, rule euclidean_size_gcd_le1)
```
```   348
```
```   349 lemma euclidean_size_gcd_less1:
```
```   350   assumes "a \<noteq> 0" and "\<not> a dvd b"
```
```   351   shows "euclidean_size (gcd a b) < euclidean_size a"
```
```   352 proof (rule ccontr)
```
```   353   assume "\<not>euclidean_size (gcd a b) < euclidean_size a"
```
```   354   with \<open>a \<noteq> 0\<close> have A: "euclidean_size (gcd a b) = euclidean_size a"
```
```   355     by (intro le_antisym, simp_all)
```
```   356   have "a dvd gcd a b"
```
```   357     by (rule dvd_euclidean_size_eq_imp_dvd) (simp_all add: assms A)
```
```   358   hence "a dvd b" using dvd_gcdD2 by blast
```
```   359   with \<open>\<not> a dvd b\<close> show False by contradiction
```
```   360 qed
```
```   361
```
```   362 lemma euclidean_size_gcd_less2:
```
```   363   assumes "b \<noteq> 0" and "\<not> b dvd a"
```
```   364   shows "euclidean_size (gcd a b) < euclidean_size b"
```
```   365   using assms by (subst gcd.commute, rule euclidean_size_gcd_less1)
```
```   366
```
```   367 lemma euclidean_size_lcm_le1:
```
```   368   assumes "a \<noteq> 0" and "b \<noteq> 0"
```
```   369   shows "euclidean_size a \<le> euclidean_size (lcm a b)"
```
```   370 proof -
```
```   371   have "a dvd lcm a b" by (rule dvd_lcm1)
```
```   372   then obtain c where A: "lcm a b = a * c" ..
```
```   373   with \<open>a \<noteq> 0\<close> and \<open>b \<noteq> 0\<close> have "c \<noteq> 0" by (auto simp: lcm_eq_0_iff)
```
```   374   then show ?thesis by (subst A, intro size_mult_mono)
```
```   375 qed
```
```   376
```
```   377 lemma euclidean_size_lcm_le2:
```
```   378   "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> euclidean_size b \<le> euclidean_size (lcm a b)"
```
```   379   using euclidean_size_lcm_le1 [of b a] by (simp add: ac_simps)
```
```   380
```
```   381 lemma euclidean_size_lcm_less1:
```
```   382   assumes "b \<noteq> 0" and "\<not> b dvd a"
```
```   383   shows "euclidean_size a < euclidean_size (lcm a b)"
```
```   384 proof (rule ccontr)
```
```   385   from assms have "a \<noteq> 0" by auto
```
```   386   assume "\<not>euclidean_size a < euclidean_size (lcm a b)"
```
```   387   with \<open>a \<noteq> 0\<close> and \<open>b \<noteq> 0\<close> have "euclidean_size (lcm a b) = euclidean_size a"
```
```   388     by (intro le_antisym, simp, intro euclidean_size_lcm_le1)
```
```   389   with assms have "lcm a b dvd a"
```
```   390     by (rule_tac dvd_euclidean_size_eq_imp_dvd) (auto simp: lcm_eq_0_iff)
```
```   391   hence "b dvd a" by (rule lcm_dvdD2)
```
```   392   with \<open>\<not>b dvd a\<close> show False by contradiction
```
```   393 qed
```
```   394
```
```   395 lemma euclidean_size_lcm_less2:
```
```   396   assumes "a \<noteq> 0" and "\<not> a dvd b"
```
```   397   shows "euclidean_size b < euclidean_size (lcm a b)"
```
```   398   using assms euclidean_size_lcm_less1 [of a b] by (simp add: ac_simps)
```
```   399
```
```   400 end
```
```   401
```
```   402 lemma factorial_euclidean_semiring_gcdI:
```
```   403   "OFCLASS('a::{factorial_semiring_gcd, euclidean_semiring}, euclidean_semiring_gcd_class)"
```
```   404 proof
```
```   405   interpret semiring_Gcd 1 0 times
```
```   406     Euclidean_Algorithm.gcd Euclidean_Algorithm.lcm
```
```   407     Euclidean_Algorithm.Gcd Euclidean_Algorithm.Lcm
```
```   408     divide plus minus unit_factor normalize
```
```   409     rewrites "dvd.dvd op * = Rings.dvd"
```
```   410     by (fact semiring_Gcd) (simp add: dvd.dvd_def dvd_def fun_eq_iff)
```
```   411   show [simp]: "Euclidean_Algorithm.gcd = (gcd :: 'a \<Rightarrow> _)"
```
```   412   proof (rule ext)+
```
```   413     fix a b :: 'a
```
```   414     show "Euclidean_Algorithm.gcd a b = gcd a b"
```
```   415     proof (induct a b rule: eucl_induct)
```
```   416       case zero
```
```   417       then show ?case
```
```   418         by simp
```
```   419     next
```
```   420       case (mod a b)
```
```   421       moreover have "gcd b (a mod b) = gcd b a"
```
```   422         using GCD.gcd_add_mult [of b "a div b" "a mod b", symmetric]
```
```   423           by (simp add: div_mult_mod_eq)
```
```   424       ultimately show ?case
```
```   425         by (simp add: Euclidean_Algorithm.gcd_mod ac_simps)
```
```   426     qed
```
```   427   qed
```
```   428   show [simp]: "Euclidean_Algorithm.Lcm = (Lcm :: 'a set \<Rightarrow> _)"
```
```   429     by (auto intro!: Lcm_eqI GCD.dvd_Lcm GCD.Lcm_least)
```
```   430   show "Euclidean_Algorithm.lcm = (lcm :: 'a \<Rightarrow> _)"
```
```   431     by (simp add: fun_eq_iff Euclidean_Algorithm.lcm_def semiring_gcd_class.lcm_gcd)
```
```   432   show "Euclidean_Algorithm.Gcd = (Gcd :: 'a set \<Rightarrow> _)"
```
```   433     by (simp add: fun_eq_iff Euclidean_Algorithm.Gcd_def semiring_Gcd_class.Gcd_Lcm)
```
```   434 qed
```
```   435
```
```   436
```
```   437 subsection \<open>The extended euclidean algorithm\<close>
```
```   438
```
```   439 class euclidean_ring_gcd = euclidean_semiring_gcd + idom
```
```   440 begin
```
```   441
```
```   442 subclass euclidean_ring ..
```
```   443 subclass ring_gcd ..
```
```   444 subclass factorial_ring_gcd ..
```
```   445
```
```   446 function euclid_ext_aux :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> ('a \<times> 'a) \<times> 'a"
```
```   447   where "euclid_ext_aux s' s t' t r' r = (
```
```   448      if r = 0 then let c = 1 div unit_factor r' in ((s' * c, t' * c), normalize r')
```
```   449      else let q = r' div r
```
```   450           in euclid_ext_aux s (s' - q * s) t (t' - q * t) r (r' mod r))"
```
```   451   by auto
```
```   452 termination
```
```   453   by (relation "measure (\<lambda>(_, _, _, _, _, b). euclidean_size b)")
```
```   454     (simp_all add: mod_size_less)
```
```   455
```
```   456 abbreviation (input) euclid_ext :: "'a \<Rightarrow> 'a \<Rightarrow> ('a \<times> 'a) \<times> 'a"
```
```   457   where "euclid_ext \<equiv> euclid_ext_aux 1 0 0 1"
```
```   458
```
```   459 lemma
```
```   460   assumes "gcd r' r = gcd a b"
```
```   461   assumes "s' * a + t' * b = r'"
```
```   462   assumes "s * a + t * b = r"
```
```   463   assumes "euclid_ext_aux s' s t' t r' r = ((x, y), c)"
```
```   464   shows euclid_ext_aux_eq_gcd: "c = gcd a b"
```
```   465     and euclid_ext_aux_bezout: "x * a + y * b = gcd a b"
```
```   466 proof -
```
```   467   have "case euclid_ext_aux s' s t' t r' r of ((x, y), c) \<Rightarrow>
```
```   468     x * a + y * b = c \<and> c = gcd a b" (is "?P (euclid_ext_aux s' s t' t r' r)")
```
```   469     using assms(1-3)
```
```   470   proof (induction s' s t' t r' r rule: euclid_ext_aux.induct)
```
```   471     case (1 s' s t' t r' r)
```
```   472     show ?case
```
```   473     proof (cases "r = 0")
```
```   474       case True
```
```   475       hence "euclid_ext_aux s' s t' t r' r =
```
```   476                ((s' div unit_factor r', t' div unit_factor r'), normalize r')"
```
```   477         by (subst euclid_ext_aux.simps) (simp add: Let_def)
```
```   478       also have "?P \<dots>"
```
```   479       proof safe
```
```   480         have "s' div unit_factor r' * a + t' div unit_factor r' * b =
```
```   481                 (s' * a + t' * b) div unit_factor r'"
```
```   482           by (cases "r' = 0") (simp_all add: unit_div_commute)
```
```   483         also have "s' * a + t' * b = r'" by fact
```
```   484         also have "\<dots> div unit_factor r' = normalize r'" by simp
```
```   485         finally show "s' div unit_factor r' * a + t' div unit_factor r' * b = normalize r'" .
```
```   486       next
```
```   487         from "1.prems" True show "normalize r' = gcd a b"
```
```   488           by simp
```
```   489       qed
```
```   490       finally show ?thesis .
```
```   491     next
```
```   492       case False
```
```   493       hence "euclid_ext_aux s' s t' t r' r =
```
```   494              euclid_ext_aux s (s' - r' div r * s) t (t' - r' div r * t) r (r' mod r)"
```
```   495         by (subst euclid_ext_aux.simps) (simp add: Let_def)
```
```   496       also from "1.prems" False have "?P \<dots>"
```
```   497       proof (intro "1.IH")
```
```   498         have "(s' - r' div r * s) * a + (t' - r' div r * t) * b =
```
```   499               (s' * a + t' * b) - r' div r * (s * a + t * b)" by (simp add: algebra_simps)
```
```   500         also have "s' * a + t' * b = r'" by fact
```
```   501         also have "s * a + t * b = r" by fact
```
```   502         also have "r' - r' div r * r = r' mod r" using div_mult_mod_eq [of r' r]
```
```   503           by (simp add: algebra_simps)
```
```   504         finally show "(s' - r' div r * s) * a + (t' - r' div r * t) * b = r' mod r" .
```
```   505       qed (auto simp: gcd_mod_right algebra_simps minus_mod_eq_div_mult [symmetric] gcd.commute)
```
```   506       finally show ?thesis .
```
```   507     qed
```
```   508   qed
```
```   509   with assms(4) show "c = gcd a b" "x * a + y * b = gcd a b"
```
```   510     by simp_all
```
```   511 qed
```
```   512
```
```   513 declare euclid_ext_aux.simps [simp del]
```
```   514
```
```   515 definition bezout_coefficients :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a"
```
```   516   where [code]: "bezout_coefficients a b = fst (euclid_ext a b)"
```
```   517
```
```   518 lemma bezout_coefficients_0:
```
```   519   "bezout_coefficients a 0 = (1 div unit_factor a, 0)"
```
```   520   by (simp add: bezout_coefficients_def euclid_ext_aux.simps)
```
```   521
```
```   522 lemma bezout_coefficients_left_0:
```
```   523   "bezout_coefficients 0 a = (0, 1 div unit_factor a)"
```
```   524   by (simp add: bezout_coefficients_def euclid_ext_aux.simps)
```
```   525
```
```   526 lemma bezout_coefficients:
```
```   527   assumes "bezout_coefficients a b = (x, y)"
```
```   528   shows "x * a + y * b = gcd a b"
```
```   529   using assms by (simp add: bezout_coefficients_def
```
```   530     euclid_ext_aux_bezout [of a b a b 1 0 0 1 x y] prod_eq_iff)
```
```   531
```
```   532 lemma bezout_coefficients_fst_snd:
```
```   533   "fst (bezout_coefficients a b) * a + snd (bezout_coefficients a b) * b = gcd a b"
```
```   534   by (rule bezout_coefficients) simp
```
```   535
```
```   536 lemma euclid_ext_eq [simp]:
```
```   537   "euclid_ext a b = (bezout_coefficients a b, gcd a b)" (is "?p = ?q")
```
```   538 proof
```
```   539   show "fst ?p = fst ?q"
```
```   540     by (simp add: bezout_coefficients_def)
```
```   541   have "snd (euclid_ext_aux 1 0 0 1 a b) = gcd a b"
```
```   542     by (rule euclid_ext_aux_eq_gcd [of a b a b 1 0 0 1])
```
```   543       (simp_all add: prod_eq_iff)
```
```   544   then show "snd ?p = snd ?q"
```
```   545     by simp
```
```   546 qed
```
```   547
```
```   548 declare euclid_ext_eq [symmetric, code_unfold]
```
```   549
```
```   550 end
```
```   551
```
```   552
```
```   553 subsection \<open>Typical instances\<close>
```
```   554
```
```   555 instance nat :: euclidean_semiring_gcd
```
```   556 proof
```
```   557   interpret semiring_Gcd 1 0 times
```
```   558     "Euclidean_Algorithm.gcd" "Euclidean_Algorithm.lcm"
```
```   559     "Euclidean_Algorithm.Gcd" "Euclidean_Algorithm.Lcm"
```
```   560     divide plus minus unit_factor normalize
```
```   561     rewrites "dvd.dvd op * = Rings.dvd"
```
```   562     by (fact semiring_Gcd) (simp add: dvd.dvd_def dvd_def fun_eq_iff)
```
```   563   show [simp]: "(Euclidean_Algorithm.gcd :: nat \<Rightarrow> _) = gcd"
```
```   564   proof (rule ext)+
```
```   565     fix m n :: nat
```
```   566     show "Euclidean_Algorithm.gcd m n = gcd m n"
```
```   567     proof (induct m n rule: eucl_induct)
```
```   568       case zero
```
```   569       then show ?case
```
```   570         by simp
```
```   571     next
```
```   572       case (mod m n)
```
```   573       then have "gcd n (m mod n) = gcd n m"
```
```   574         using gcd_nat.simps [of m n] by (simp add: ac_simps)
```
```   575       with mod show ?case
```
```   576         by (simp add: Euclidean_Algorithm.gcd_mod ac_simps)
```
```   577     qed
```
```   578   qed
```
```   579   show [simp]: "(Euclidean_Algorithm.Lcm :: nat set \<Rightarrow> _) = Lcm"
```
```   580     by (auto intro!: ext Lcm_eqI)
```
```   581   show "(Euclidean_Algorithm.lcm :: nat \<Rightarrow> _) = lcm"
```
```   582     by (simp add: fun_eq_iff Euclidean_Algorithm.lcm_def semiring_gcd_class.lcm_gcd)
```
```   583   show "(Euclidean_Algorithm.Gcd :: nat set \<Rightarrow> _) = Gcd"
```
```   584     by (simp add: fun_eq_iff Euclidean_Algorithm.Gcd_def semiring_Gcd_class.Gcd_Lcm)
```
```   585 qed
```
```   586
```
```   587 instance int :: euclidean_ring_gcd
```
```   588 proof
```
```   589   interpret semiring_Gcd 1 0 times
```
```   590     "Euclidean_Algorithm.gcd" "Euclidean_Algorithm.lcm"
```
```   591     "Euclidean_Algorithm.Gcd" "Euclidean_Algorithm.Lcm"
```
```   592     divide plus minus unit_factor normalize
```
```   593     rewrites "dvd.dvd op * = Rings.dvd"
```
```   594     by (fact semiring_Gcd) (simp add: dvd.dvd_def dvd_def fun_eq_iff)
```
```   595   show [simp]: "(Euclidean_Algorithm.gcd :: int \<Rightarrow> _) = gcd"
```
```   596   proof (rule ext)+
```
```   597     fix k l :: int
```
```   598     show "Euclidean_Algorithm.gcd k l = gcd k l"
```
```   599     proof (induct k l rule: eucl_induct)
```
```   600       case zero
```
```   601       then show ?case
```
```   602         by simp
```
```   603     next
```
```   604       case (mod k l)
```
```   605       have "gcd l (k mod l) = gcd l k"
```
```   606       proof (cases l "0::int" rule: linorder_cases)
```
```   607         case less
```
```   608         then show ?thesis
```
```   609           using gcd_non_0_int [of "- l" "- k"] by (simp add: ac_simps)
```
```   610       next
```
```   611         case equal
```
```   612         with mod show ?thesis
```
```   613           by simp
```
```   614       next
```
```   615         case greater
```
```   616         then show ?thesis
```
```   617           using gcd_non_0_int [of l k] by (simp add: ac_simps)
```
```   618       qed
```
```   619       with mod show ?case
```
```   620         by (simp add: Euclidean_Algorithm.gcd_mod ac_simps)
```
```   621     qed
```
```   622   qed
```
```   623   show [simp]: "(Euclidean_Algorithm.Lcm :: int set \<Rightarrow> _) = Lcm"
```
```   624     by (auto intro!: ext Lcm_eqI)
```
```   625   show "(Euclidean_Algorithm.lcm :: int \<Rightarrow> _) = lcm"
```
```   626     by (simp add: fun_eq_iff Euclidean_Algorithm.lcm_def semiring_gcd_class.lcm_gcd)
```
```   627   show "(Euclidean_Algorithm.Gcd :: int set \<Rightarrow> _) = Gcd"
```
```   628     by (simp add: fun_eq_iff Euclidean_Algorithm.Gcd_def semiring_Gcd_class.Gcd_Lcm)
```
```   629 qed
```
```   630
```
```   631 end
```