src/HOL/Number_Theory/Euclidean_Algorithm.thy
 author Manuel Eberl 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