src/HOL/Number_Theory/Euclidean_Algorithm.thy
author Manuel Eberl <eberlm@in.tum.de>
Sun Feb 28 12:05:52 2016 +0100 (2016-02-28)
changeset 62442 26e4be6a680f
parent 62429 25271ff79171
child 62457 a3c7bd201da7
permissions -rw-r--r--
More efficient Extended Euclidean Algorithm
     1 (* Author: Manuel Eberl *)
     2 
     3 section \<open>Abstract euclidean algorithm\<close>
     4 
     5 theory Euclidean_Algorithm
     6 imports "~~/src/HOL/GCD" "~~/src/HOL/Library/Polynomial"
     7 begin
     8 
     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
    28 
    29 lemma euclidean_division:
    30   fixes a :: 'a and b :: 'a
    31   assumes "b \<noteq> 0"
    32   obtains s and t where "a = s * b + t" 
    33     and "euclidean_size t < euclidean_size b"
    34 proof -
    35   from div_mod_equality [of a b 0] 
    36      have "a = a div b * b + a mod b" by simp
    37   with that and assms show ?thesis by (auto simp add: mod_size_less)
    38 qed
    39 
    40 lemma dvd_euclidean_size_eq_imp_dvd:
    41   assumes "a \<noteq> 0" and b_dvd_a: "b dvd a" and size_eq: "euclidean_size a = euclidean_size b"
    42   shows "a dvd b"
    43 proof (rule ccontr)
    44   assume "\<not> a dvd b"
    45   then have "b mod a \<noteq> 0" by (simp add: mod_eq_0_iff_dvd)
    46   from b_dvd_a have b_dvd_mod: "b dvd b mod a" by (simp add: dvd_mod_iff)
    47   from b_dvd_mod obtain c where "b mod a = b * c" unfolding dvd_def by blast
    48     with \<open>b mod a \<noteq> 0\<close> have "c \<noteq> 0" by auto
    49   with \<open>b mod a = b * c\<close> have "euclidean_size (b mod a) \<ge> euclidean_size b"
    50       using size_mult_mono by force
    51   moreover from \<open>\<not> a dvd b\<close> and \<open>a \<noteq> 0\<close>
    52   have "euclidean_size (b mod a) < euclidean_size a"
    53       using mod_size_less by blast
    54   ultimately show False using size_eq by simp
    55 qed
    56 
    57 function gcd_eucl :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
    58 where
    59   "gcd_eucl a b = (if b = 0 then normalize a else gcd_eucl b (a mod b))"
    60   by pat_completeness simp
    61 termination
    62   by (relation "measure (euclidean_size \<circ> snd)") (simp_all add: mod_size_less)
    63 
    64 declare gcd_eucl.simps [simp del]
    65 
    66 lemma gcd_eucl_induct [case_names zero mod]:
    67   assumes H1: "\<And>b. P b 0"
    68   and H2: "\<And>a b. b \<noteq> 0 \<Longrightarrow> P b (a mod b) \<Longrightarrow> P a b"
    69   shows "P a b"
    70 proof (induct a b rule: gcd_eucl.induct)
    71   case ("1" a b)
    72   show ?case
    73   proof (cases "b = 0")
    74     case True then show "P a b" by simp (rule H1)
    75   next
    76     case False
    77     then have "P b (a mod b)"
    78       by (rule "1.hyps")
    79     with \<open>b \<noteq> 0\<close> show "P a b"
    80       by (blast intro: H2)
    81   qed
    82 qed
    83 
    84 definition lcm_eucl :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
    85 where
    86   "lcm_eucl a b = normalize (a * b) div gcd_eucl a b"
    87 
    88 definition Lcm_eucl :: "'a set \<Rightarrow> 'a" -- \<open>
    89   Somewhat complicated definition of Lcm that has the advantage of working
    90   for infinite sets as well\<close>
    91 where
    92   "Lcm_eucl A = (if \<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) then
    93      let l = SOME l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l =
    94        (LEAST n. \<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n)
    95        in normalize l 
    96       else 0)"
    97 
    98 definition Gcd_eucl :: "'a set \<Rightarrow> 'a"
    99 where
   100   "Gcd_eucl A = Lcm_eucl {d. \<forall>a\<in>A. d dvd a}"
   101 
   102 declare Lcm_eucl_def Gcd_eucl_def [code del]
   103 
   104 lemma gcd_eucl_0:
   105   "gcd_eucl a 0 = normalize a"
   106   by (simp add: gcd_eucl.simps [of a 0])
   107 
   108 lemma gcd_eucl_0_left:
   109   "gcd_eucl 0 a = normalize a"
   110   by (simp_all add: gcd_eucl_0 gcd_eucl.simps [of 0 a])
   111 
   112 lemma gcd_eucl_non_0:
   113   "b \<noteq> 0 \<Longrightarrow> gcd_eucl a b = gcd_eucl b (a mod b)"
   114   by (simp add: gcd_eucl.simps [of a b] gcd_eucl.simps [of b 0])
   115 
   116 lemma gcd_eucl_dvd1 [iff]: "gcd_eucl a b dvd a"
   117   and gcd_eucl_dvd2 [iff]: "gcd_eucl a b dvd b"
   118   by (induct a b rule: gcd_eucl_induct)
   119      (simp_all add: gcd_eucl_0 gcd_eucl_non_0 dvd_mod_iff)
   120 
   121 lemma normalize_gcd_eucl [simp]:
   122   "normalize (gcd_eucl a b) = gcd_eucl a b"
   123   by (induct a b rule: gcd_eucl_induct) (simp_all add: gcd_eucl_0 gcd_eucl_non_0)
   124      
   125 lemma gcd_eucl_greatest:
   126   fixes k a b :: 'a
   127   shows "k dvd a \<Longrightarrow> k dvd b \<Longrightarrow> k dvd gcd_eucl a b"
   128 proof (induct a b rule: gcd_eucl_induct)
   129   case (zero a) from zero(1) show ?case by (rule dvd_trans) (simp add: gcd_eucl_0)
   130 next
   131   case (mod a b)
   132   then show ?case
   133     by (simp add: gcd_eucl_non_0 dvd_mod_iff)
   134 qed
   135 
   136 lemma eq_gcd_euclI:
   137   fixes gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
   138   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"
   139           "\<And>a b k. k dvd a \<Longrightarrow> k dvd b \<Longrightarrow> k dvd gcd a b"
   140   shows   "gcd = gcd_eucl"
   141   by (intro ext, rule associated_eqI) (simp_all add: gcd_eucl_greatest assms)
   142 
   143 lemma gcd_eucl_zero [simp]:
   144   "gcd_eucl a b = 0 \<longleftrightarrow> a = 0 \<and> b = 0"
   145   by (metis dvd_0_left dvd_refl gcd_eucl_dvd1 gcd_eucl_dvd2 gcd_eucl_greatest)+
   146 
   147   
   148 lemma dvd_Lcm_eucl [simp]: "a \<in> A \<Longrightarrow> a dvd Lcm_eucl A"
   149   and Lcm_eucl_least: "(\<And>a. a \<in> A \<Longrightarrow> a dvd b) \<Longrightarrow> Lcm_eucl A dvd b"
   150   and unit_factor_Lcm_eucl [simp]: 
   151           "unit_factor (Lcm_eucl A) = (if Lcm_eucl A = 0 then 0 else 1)"
   152 proof -
   153   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>
   154     unit_factor (Lcm_eucl A) = (if Lcm_eucl A = 0 then 0 else 1)" (is ?thesis)
   155   proof (cases "\<exists>l. l \<noteq>  0 \<and> (\<forall>a\<in>A. a dvd l)")
   156     case False
   157     hence "Lcm_eucl A = 0" by (auto simp: Lcm_eucl_def)
   158     with False show ?thesis by auto
   159   next
   160     case True
   161     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
   162     def n \<equiv> "LEAST n. \<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"
   163     def l \<equiv> "SOME l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"
   164     have "\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"
   165       apply (subst n_def)
   166       apply (rule LeastI[of _ "euclidean_size l\<^sub>0"])
   167       apply (rule exI[of _ l\<^sub>0])
   168       apply (simp add: l\<^sub>0_props)
   169       done
   170     from someI_ex[OF this] have "l \<noteq> 0" and "\<forall>a\<in>A. a dvd l" and "euclidean_size l = n" 
   171       unfolding l_def by simp_all
   172     {
   173       fix l' assume "\<forall>a\<in>A. a dvd l'"
   174       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)
   175       moreover from \<open>l \<noteq> 0\<close> have "gcd_eucl l l' \<noteq> 0" by simp
   176       ultimately have "\<exists>b. b \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd b) \<and> 
   177                           euclidean_size b = euclidean_size (gcd_eucl l l')"
   178         by (intro exI[of _ "gcd_eucl l l'"], auto)
   179       hence "euclidean_size (gcd_eucl l l') \<ge> n" by (subst n_def) (rule Least_le)
   180       moreover have "euclidean_size (gcd_eucl l l') \<le> n"
   181       proof -
   182         have "gcd_eucl l l' dvd l" by simp
   183         then obtain a where "l = gcd_eucl l l' * a" unfolding dvd_def by blast
   184         with \<open>l \<noteq> 0\<close> have "a \<noteq> 0" by auto
   185         hence "euclidean_size (gcd_eucl l l') \<le> euclidean_size (gcd_eucl l l' * a)"
   186           by (rule size_mult_mono)
   187         also have "gcd_eucl l l' * a = l" using \<open>l = gcd_eucl l l' * a\<close> ..
   188         also note \<open>euclidean_size l = n\<close>
   189         finally show "euclidean_size (gcd_eucl l l') \<le> n" .
   190       qed
   191       ultimately have *: "euclidean_size l = euclidean_size (gcd_eucl l l')" 
   192         by (intro le_antisym, simp_all add: \<open>euclidean_size l = n\<close>)
   193       from \<open>l \<noteq> 0\<close> have "l dvd gcd_eucl l l'"
   194         by (rule dvd_euclidean_size_eq_imp_dvd) (auto simp add: *)
   195       hence "l dvd l'" by (rule dvd_trans[OF _ gcd_eucl_dvd2])
   196     }
   197 
   198     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>
   199       have "(\<forall>a\<in>A. a dvd normalize l) \<and> 
   200         (\<forall>l'. (\<forall>a\<in>A. a dvd l') \<longrightarrow> normalize l dvd l') \<and>
   201         unit_factor (normalize l) = 
   202         (if normalize l = 0 then 0 else 1)"
   203       by (auto simp: unit_simps)
   204     also from True have "normalize l = Lcm_eucl A"
   205       by (simp add: Lcm_eucl_def Let_def n_def l_def)
   206     finally show ?thesis .
   207   qed
   208   note A = this
   209 
   210   {fix a assume "a \<in> A" then show "a dvd Lcm_eucl A" using A by blast}
   211   {fix b assume "\<And>a. a \<in> A \<Longrightarrow> a dvd b" then show "Lcm_eucl A dvd b" using A by blast}
   212   from A show "unit_factor (Lcm_eucl A) = (if Lcm_eucl A = 0 then 0 else 1)" by blast
   213 qed
   214   
   215 lemma normalize_Lcm_eucl [simp]:
   216   "normalize (Lcm_eucl A) = Lcm_eucl A"
   217 proof (cases "Lcm_eucl A = 0")
   218   case True then show ?thesis by simp
   219 next
   220   case False
   221   have "unit_factor (Lcm_eucl A) * normalize (Lcm_eucl A) = Lcm_eucl A"
   222     by (fact unit_factor_mult_normalize)
   223   with False show ?thesis by simp
   224 qed
   225 
   226 lemma eq_Lcm_euclI:
   227   fixes lcm :: "'a set \<Rightarrow> 'a"
   228   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"
   229           "\<And>A. normalize (lcm A) = lcm A" shows "lcm = Lcm_eucl"
   230   by (intro ext, rule associated_eqI) (auto simp: assms intro: Lcm_eucl_least)  
   231 
   232 end
   233 
   234 class euclidean_ring = euclidean_semiring + idom
   235 begin
   236 
   237 function euclid_ext_aux :: "'a \<Rightarrow> _" where
   238   "euclid_ext_aux r' r s' s t' t = (
   239      if r = 0 then let c = 1 div unit_factor r' in (s' * c, t' * c, normalize r')
   240      else let q = r' div r
   241           in  euclid_ext_aux r (r' mod r) s (s' - q * s) t (t' - q * t))"
   242 by auto
   243 termination by (relation "measure (\<lambda>(_,b,_,_,_,_). euclidean_size b)") (simp_all add: mod_size_less)
   244 
   245 declare euclid_ext_aux.simps [simp del]
   246 
   247 lemma euclid_ext_aux_correct:
   248   assumes "gcd_eucl r' r = gcd_eucl x y"
   249   assumes "s' * x + t' * y = r'"
   250   assumes "s * x + t * y = r"
   251   shows   "case euclid_ext_aux r' r s' s t' t of (a,b,c) \<Rightarrow>
   252              a * x + b * y = c \<and> c = gcd_eucl x y" (is "?P (euclid_ext_aux r' r s' s t' t)")
   253 using assms
   254 proof (induction r' r s' s t' t rule: euclid_ext_aux.induct)
   255   case (1 r' r s' s t' t)
   256   show ?case
   257   proof (cases "r = 0")
   258     case True
   259     hence "euclid_ext_aux r' r s' s t' t = 
   260              (s' div unit_factor r', t' div unit_factor r', normalize r')"
   261       by (subst euclid_ext_aux.simps) (simp add: Let_def)
   262     also have "?P \<dots>"
   263     proof safe
   264       have "s' div unit_factor r' * x + t' div unit_factor r' * y = 
   265                 (s' * x + t' * y) div unit_factor r'"
   266         by (cases "r' = 0") (simp_all add: unit_div_commute)
   267       also have "s' * x + t' * y = r'" by fact
   268       also have "\<dots> div unit_factor r' = normalize r'" by simp
   269       finally show "s' div unit_factor r' * x + t' div unit_factor r' * y = normalize r'" .
   270     next
   271       from "1.prems" True show "normalize r' = gcd_eucl x y" by (simp add: gcd_eucl_0)
   272     qed
   273     finally show ?thesis .
   274   next
   275     case False
   276     hence "euclid_ext_aux r' r s' s t' t = 
   277              euclid_ext_aux r (r' mod r) s (s' - r' div r * s) t (t' - r' div r * t)"
   278       by (subst euclid_ext_aux.simps) (simp add: Let_def)
   279     also from "1.prems" False have "?P \<dots>"
   280     proof (intro "1.IH")
   281       have "(s' - r' div r * s) * x + (t' - r' div r * t) * y =
   282               (s' * x + t' * y) - r' div r * (s * x + t * y)" by (simp add: algebra_simps)
   283       also have "s' * x + t' * y = r'" by fact
   284       also have "s * x + t * y = r" by fact
   285       also have "r' - r' div r * r = r' mod r" using mod_div_equality[of r' r]
   286         by (simp add: algebra_simps)
   287       finally show "(s' - r' div r * s) * x + (t' - r' div r * t) * y = r' mod r" .
   288     qed (auto simp: gcd_eucl_non_0 algebra_simps div_mod_equality')
   289     finally show ?thesis .
   290   qed
   291 qed
   292 
   293 definition euclid_ext where
   294   "euclid_ext a b = euclid_ext_aux a b 1 0 0 1"
   295 
   296 lemma euclid_ext_0: 
   297   "euclid_ext a 0 = (1 div unit_factor a, 0, normalize a)"
   298   by (simp add: euclid_ext_def euclid_ext_aux.simps)
   299 
   300 lemma euclid_ext_left_0: 
   301   "euclid_ext 0 a = (0, 1 div unit_factor a, normalize a)"
   302   by (simp add: euclid_ext_def euclid_ext_aux.simps)
   303 
   304 lemma euclid_ext_correct':
   305   "case euclid_ext x y of (a,b,c) \<Rightarrow> a * x + b * y = c \<and> c = gcd_eucl x y"
   306   unfolding euclid_ext_def by (rule euclid_ext_aux_correct) simp_all
   307 
   308 definition euclid_ext' where
   309   "euclid_ext' x y = (case euclid_ext x y of (a, b, _) \<Rightarrow> (a, b))"
   310 
   311 lemma euclid_ext'_correct':
   312   "case euclid_ext' x y of (a,b) \<Rightarrow> a * x + b * y = gcd_eucl x y"
   313   using euclid_ext_correct'[of x y] by (simp add: case_prod_unfold euclid_ext'_def)
   314 
   315 lemma euclid_ext'_0: "euclid_ext' a 0 = (1 div unit_factor a, 0)" 
   316   by (simp add: euclid_ext'_def euclid_ext_0)
   317 
   318 lemma euclid_ext'_left_0: "euclid_ext' 0 a = (0, 1 div unit_factor a)" 
   319   by (simp add: euclid_ext'_def euclid_ext_left_0)
   320 
   321 end
   322 
   323 class euclidean_semiring_gcd = euclidean_semiring + gcd + Gcd +
   324   assumes gcd_gcd_eucl: "gcd = gcd_eucl" and lcm_lcm_eucl: "lcm = lcm_eucl"
   325   assumes Gcd_Gcd_eucl: "Gcd = Gcd_eucl" and Lcm_Lcm_eucl: "Lcm = Lcm_eucl"
   326 begin
   327 
   328 subclass semiring_gcd
   329   by standard (simp_all add: gcd_gcd_eucl gcd_eucl_greatest lcm_lcm_eucl lcm_eucl_def)
   330 
   331 subclass semiring_Gcd
   332   by standard (auto simp: Gcd_Gcd_eucl Lcm_Lcm_eucl Gcd_eucl_def intro: Lcm_eucl_least)
   333   
   334 lemma gcd_non_0:
   335   "b \<noteq> 0 \<Longrightarrow> gcd a b = gcd b (a mod b)"
   336   unfolding gcd_gcd_eucl by (fact gcd_eucl_non_0)
   337 
   338 lemmas gcd_0 = gcd_0_right
   339 lemmas dvd_gcd_iff = gcd_greatest_iff
   340 lemmas gcd_greatest_iff = dvd_gcd_iff
   341 
   342 lemma gcd_mod1 [simp]:
   343   "gcd (a mod b) b = gcd a b"
   344   by (rule gcdI, metis dvd_mod_iff gcd_dvd1 gcd_dvd2, simp_all add: gcd_greatest dvd_mod_iff)
   345 
   346 lemma gcd_mod2 [simp]:
   347   "gcd a (b mod a) = gcd a b"
   348   by (rule gcdI, simp, metis dvd_mod_iff gcd_dvd1 gcd_dvd2, simp_all add: gcd_greatest dvd_mod_iff)
   349          
   350 lemma euclidean_size_gcd_le1 [simp]:
   351   assumes "a \<noteq> 0"
   352   shows "euclidean_size (gcd a b) \<le> euclidean_size a"
   353 proof -
   354    have "gcd a b dvd a" by (rule gcd_dvd1)
   355    then obtain c where A: "a = gcd a b * c" unfolding dvd_def by blast
   356    with \<open>a \<noteq> 0\<close> show ?thesis by (subst (2) A, intro size_mult_mono) auto
   357 qed
   358 
   359 lemma euclidean_size_gcd_le2 [simp]:
   360   "b \<noteq> 0 \<Longrightarrow> euclidean_size (gcd a b) \<le> euclidean_size b"
   361   by (subst gcd.commute, rule euclidean_size_gcd_le1)
   362 
   363 lemma euclidean_size_gcd_less1:
   364   assumes "a \<noteq> 0" and "\<not>a dvd b"
   365   shows "euclidean_size (gcd a b) < euclidean_size a"
   366 proof (rule ccontr)
   367   assume "\<not>euclidean_size (gcd a b) < euclidean_size a"
   368   with \<open>a \<noteq> 0\<close> have A: "euclidean_size (gcd a b) = euclidean_size a"
   369     by (intro le_antisym, simp_all)
   370   have "a dvd gcd a b"
   371     by (rule dvd_euclidean_size_eq_imp_dvd) (simp_all add: assms A)
   372   hence "a dvd b" using dvd_gcdD2 by blast
   373   with \<open>\<not>a dvd b\<close> show False by contradiction
   374 qed
   375 
   376 lemma euclidean_size_gcd_less2:
   377   assumes "b \<noteq> 0" and "\<not>b dvd a"
   378   shows "euclidean_size (gcd a b) < euclidean_size b"
   379   using assms by (subst gcd.commute, rule euclidean_size_gcd_less1)
   380 
   381 lemma euclidean_size_lcm_le1: 
   382   assumes "a \<noteq> 0" and "b \<noteq> 0"
   383   shows "euclidean_size a \<le> euclidean_size (lcm a b)"
   384 proof -
   385   have "a dvd lcm a b" by (rule dvd_lcm1)
   386   then obtain c where A: "lcm a b = a * c" ..
   387   with \<open>a \<noteq> 0\<close> and \<open>b \<noteq> 0\<close> have "c \<noteq> 0" by (auto simp: lcm_eq_0_iff)
   388   then show ?thesis by (subst A, intro size_mult_mono)
   389 qed
   390 
   391 lemma euclidean_size_lcm_le2:
   392   "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> euclidean_size b \<le> euclidean_size (lcm a b)"
   393   using euclidean_size_lcm_le1 [of b a] by (simp add: ac_simps)
   394 
   395 lemma euclidean_size_lcm_less1:
   396   assumes "b \<noteq> 0" and "\<not>b dvd a"
   397   shows "euclidean_size a < euclidean_size (lcm a b)"
   398 proof (rule ccontr)
   399   from assms have "a \<noteq> 0" by auto
   400   assume "\<not>euclidean_size a < euclidean_size (lcm a b)"
   401   with \<open>a \<noteq> 0\<close> and \<open>b \<noteq> 0\<close> have "euclidean_size (lcm a b) = euclidean_size a"
   402     by (intro le_antisym, simp, intro euclidean_size_lcm_le1)
   403   with assms have "lcm a b dvd a" 
   404     by (rule_tac dvd_euclidean_size_eq_imp_dvd) (auto simp: lcm_eq_0_iff)
   405   hence "b dvd a" by (rule lcm_dvdD2)
   406   with \<open>\<not>b dvd a\<close> show False by contradiction
   407 qed
   408 
   409 lemma euclidean_size_lcm_less2:
   410   assumes "a \<noteq> 0" and "\<not>a dvd b"
   411   shows "euclidean_size b < euclidean_size (lcm a b)"
   412   using assms euclidean_size_lcm_less1 [of a b] by (simp add: ac_simps)
   413 
   414 lemma Lcm_eucl_set [code]:
   415   "Lcm_eucl (set xs) = foldl lcm_eucl 1 xs"
   416   by (simp add: Lcm_Lcm_eucl [symmetric] lcm_lcm_eucl Lcm_set)
   417 
   418 lemma Gcd_eucl_set [code]:
   419   "Gcd_eucl (set xs) = foldl gcd_eucl 0 xs"
   420   by (simp add: Gcd_Gcd_eucl [symmetric] gcd_gcd_eucl Gcd_set)
   421 
   422 end
   423 
   424 text \<open>
   425   A Euclidean ring is a Euclidean semiring with additive inverses. It provides a 
   426   few more lemmas; in particular, Bezout's lemma holds for any Euclidean ring.
   427 \<close>
   428 
   429 class euclidean_ring_gcd = euclidean_semiring_gcd + idom
   430 begin
   431 
   432 subclass euclidean_ring ..
   433 subclass ring_gcd ..
   434 
   435 lemma euclid_ext_gcd [simp]:
   436   "(case euclid_ext a b of (_, _ , t) \<Rightarrow> t) = gcd a b"
   437   using euclid_ext_correct'[of a b] by (simp add: case_prod_unfold Let_def gcd_gcd_eucl)
   438 
   439 lemma euclid_ext_gcd' [simp]:
   440   "euclid_ext a b = (r, s, t) \<Longrightarrow> t = gcd a b"
   441   by (insert euclid_ext_gcd[of a b], drule (1) subst, simp)
   442 
   443 lemma euclid_ext_correct:
   444   "case euclid_ext x y of (a,b,c) \<Rightarrow> a * x + b * y = c \<and> c = gcd x y"
   445   using euclid_ext_correct'[of x y]
   446   by (simp add: gcd_gcd_eucl case_prod_unfold)
   447   
   448 lemma euclid_ext'_correct:
   449   "fst (euclid_ext' a b) * a + snd (euclid_ext' a b) * b = gcd a b"
   450   using euclid_ext_correct'[of a b]
   451   by (simp add: gcd_gcd_eucl case_prod_unfold euclid_ext'_def)
   452 
   453 lemma bezout: "\<exists>s t. s * a + t * b = gcd a b"
   454   using euclid_ext'_correct by blast
   455 
   456 end
   457 
   458 
   459 subsection \<open>Typical instances\<close>
   460 
   461 instantiation nat :: euclidean_semiring
   462 begin
   463 
   464 definition [simp]:
   465   "euclidean_size_nat = (id :: nat \<Rightarrow> nat)"
   466 
   467 instance proof
   468 qed simp_all
   469 
   470 end
   471 
   472 
   473 instantiation int :: euclidean_ring
   474 begin
   475 
   476 definition [simp]:
   477   "euclidean_size_int = (nat \<circ> abs :: int \<Rightarrow> nat)"
   478 
   479 instance
   480 by standard (auto simp add: abs_mult nat_mult_distrib split: abs_split)
   481 
   482 end
   483 
   484 
   485 instantiation poly :: (field) euclidean_ring
   486 begin
   487 
   488 definition euclidean_size_poly :: "'a poly \<Rightarrow> nat"
   489   where "euclidean_size p = (if p = 0 then 0 else 2 ^ degree p)"
   490 
   491 lemma euclidean_size_poly_0 [simp]:
   492   "euclidean_size (0::'a poly) = 0"
   493   by (simp add: euclidean_size_poly_def)
   494 
   495 lemma euclidean_size_poly_not_0 [simp]:
   496   "p \<noteq> 0 \<Longrightarrow> euclidean_size p = 2 ^ degree p"
   497   by (simp add: euclidean_size_poly_def)
   498 
   499 instance
   500 proof
   501   fix p q :: "'a poly"
   502   assume "q \<noteq> 0"
   503   then have "p mod q = 0 \<or> degree (p mod q) < degree q"
   504     by (rule degree_mod_less [of q p])  
   505   with \<open>q \<noteq> 0\<close> show "euclidean_size (p mod q) < euclidean_size q"
   506     by (cases "p mod q = 0") simp_all
   507 next
   508   fix p q :: "'a poly"
   509   assume "q \<noteq> 0"
   510   from \<open>q \<noteq> 0\<close> have "degree p \<le> degree (p * q)"
   511     by (rule degree_mult_right_le)
   512   with \<open>q \<noteq> 0\<close> show "euclidean_size p \<le> euclidean_size (p * q)"
   513     by (cases "p = 0") simp_all
   514 qed simp
   515 
   516 end
   517 
   518 
   519 instance nat :: euclidean_semiring_gcd
   520 proof
   521   show [simp]: "gcd = (gcd_eucl :: nat \<Rightarrow> _)" "Lcm = (Lcm_eucl :: nat set \<Rightarrow> _)"
   522     by (simp_all add: eq_gcd_euclI eq_Lcm_euclI)
   523   show "lcm = (lcm_eucl :: nat \<Rightarrow> _)" "Gcd = (Gcd_eucl :: nat set \<Rightarrow> _)"
   524     by (intro ext, simp add: lcm_eucl_def lcm_nat_def Gcd_nat_def Gcd_eucl_def)+
   525 qed
   526 
   527 instance int :: euclidean_ring_gcd
   528 proof
   529   show [simp]: "gcd = (gcd_eucl :: int \<Rightarrow> _)" "Lcm = (Lcm_eucl :: int set \<Rightarrow> _)"
   530     by (simp_all add: eq_gcd_euclI eq_Lcm_euclI)
   531   show "lcm = (lcm_eucl :: int \<Rightarrow> _)" "Gcd = (Gcd_eucl :: int set \<Rightarrow> _)"
   532     by (intro ext, simp add: lcm_eucl_def lcm_altdef_int 
   533           semiring_Gcd_class.Gcd_Lcm Gcd_eucl_def abs_mult)+
   534 qed
   535 
   536 
   537 instantiation poly :: (field) euclidean_ring_gcd
   538 begin
   539 
   540 definition gcd_poly :: "'a poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly" where
   541   "gcd_poly = gcd_eucl"
   542   
   543 definition lcm_poly :: "'a poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly" where
   544   "lcm_poly = lcm_eucl"
   545   
   546 definition Gcd_poly :: "'a poly set \<Rightarrow> 'a poly" where
   547   "Gcd_poly = Gcd_eucl"
   548   
   549 definition Lcm_poly :: "'a poly set \<Rightarrow> 'a poly" where
   550   "Lcm_poly = Lcm_eucl"
   551 
   552 instance by standard (simp_all only: gcd_poly_def lcm_poly_def Gcd_poly_def Lcm_poly_def)
   553 end
   554 
   555 lemma poly_gcd_monic:
   556   "lead_coeff (gcd x y) = (if x = 0 \<and> y = 0 then 0 else 1)"
   557   using unit_factor_gcd[of x y]
   558   by (simp add: unit_factor_poly_def monom_0 one_poly_def lead_coeff_def split: if_split_asm)
   559 
   560 lemma poly_dvd_antisym:
   561   fixes p q :: "'a::idom poly"
   562   assumes coeff: "coeff p (degree p) = coeff q (degree q)"
   563   assumes dvd1: "p dvd q" and dvd2: "q dvd p" shows "p = q"
   564 proof (cases "p = 0")
   565   case True with coeff show "p = q" by simp
   566 next
   567   case False with coeff have "q \<noteq> 0" by auto
   568   have degree: "degree p = degree q"
   569     using \<open>p dvd q\<close> \<open>q dvd p\<close> \<open>p \<noteq> 0\<close> \<open>q \<noteq> 0\<close>
   570     by (intro order_antisym dvd_imp_degree_le)
   571 
   572   from \<open>p dvd q\<close> obtain a where a: "q = p * a" ..
   573   with \<open>q \<noteq> 0\<close> have "a \<noteq> 0" by auto
   574   with degree a \<open>p \<noteq> 0\<close> have "degree a = 0"
   575     by (simp add: degree_mult_eq)
   576   with coeff a show "p = q"
   577     by (cases a, auto split: if_splits)
   578 qed
   579 
   580 lemma poly_gcd_unique:
   581   fixes d x y :: "_ poly"
   582   assumes dvd1: "d dvd x" and dvd2: "d dvd y"
   583     and greatest: "\<And>k. k dvd x \<Longrightarrow> k dvd y \<Longrightarrow> k dvd d"
   584     and monic: "coeff d (degree d) = (if x = 0 \<and> y = 0 then 0 else 1)"
   585   shows "d = gcd x y"
   586   using assms by (intro gcdI) (auto simp: normalize_poly_def split: if_split_asm)
   587 
   588 lemma poly_gcd_code [code]:
   589   "gcd x y = (if y = 0 then normalize x else gcd y (x mod (y :: _ poly)))"
   590   by (simp add: gcd_0 gcd_non_0)
   591 
   592 end