src/HOL/Number_Theory/Euclidean_Algorithm.thy
 author nipkow Mon Oct 17 11:46:22 2016 +0200 (2016-10-17) changeset 64267 b9a1486e79be parent 64243 aee949f6642d child 64592 7759f1766189 permissions -rw-r--r--
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     1 (* Author: Manuel Eberl *)

     2

     3 section \<open>Abstract euclidean algorithm\<close>

     4

     5 theory Euclidean_Algorithm

     6 imports "~~/src/HOL/GCD" Factorial_Ring

     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_modulo + 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 mod_0 [simp]: "0 mod a = 0"

    30   using div_mult_mod_eq [of 0 a] by simp

    31

    32 lemma dvd_mod_iff:

    33   assumes "k dvd n"

    34   shows   "(k dvd m mod n) = (k dvd m)"

    35 proof -

    36   from assms have "(k dvd m mod n) \<longleftrightarrow> (k dvd ((m div n) * n + m mod n))"

    37     by (simp add: dvd_add_right_iff)

    38   also have "(m div n) * n + m mod n = m"

    39     using div_mult_mod_eq [of m n] by simp

    40   finally show ?thesis .

    41 qed

    42

    43 lemma mod_0_imp_dvd:

    44   assumes "a mod b = 0"

    45   shows   "b dvd a"

    46 proof -

    47   have "b dvd ((a div b) * b)" by simp

    48   also have "(a div b) * b = a"

    49     using div_mult_mod_eq [of a b] by (simp add: assms)

    50   finally show ?thesis .

    51 qed

    52

    53 lemma euclidean_size_normalize [simp]:

    54   "euclidean_size (normalize a) = euclidean_size a"

    55 proof (cases "a = 0")

    56   case True

    57   then show ?thesis

    58     by simp

    59 next

    60   case [simp]: False

    61   have "euclidean_size (normalize a) \<le> euclidean_size (normalize a * unit_factor a)"

    62     by (rule size_mult_mono) simp

    63   moreover have "euclidean_size a \<le> euclidean_size (a * (1 div unit_factor a))"

    64     by (rule size_mult_mono) simp

    65   ultimately show ?thesis

    66     by simp

    67 qed

    68

    69 lemma euclidean_division:

    70   fixes a :: 'a and b :: 'a

    71   assumes "b \<noteq> 0"

    72   obtains s and t where "a = s * b + t"

    73     and "euclidean_size t < euclidean_size b"

    74 proof -

    75   from div_mult_mod_eq [of a b]

    76      have "a = a div b * b + a mod b" by simp

    77   with that and assms show ?thesis by (auto simp add: mod_size_less)

    78 qed

    79

    80 lemma dvd_euclidean_size_eq_imp_dvd:

    81   assumes "a \<noteq> 0" and b_dvd_a: "b dvd a" and size_eq: "euclidean_size a = euclidean_size b"

    82   shows "a dvd b"

    83 proof (rule ccontr)

    84   assume "\<not> a dvd b"

    85   hence "b mod a \<noteq> 0" using mod_0_imp_dvd[of b a] by blast

    86   then have "b mod a \<noteq> 0" by (simp add: mod_eq_0_iff_dvd)

    87   from b_dvd_a have b_dvd_mod: "b dvd b mod a" by (simp add: dvd_mod_iff)

    88   from b_dvd_mod obtain c where "b mod a = b * c" unfolding dvd_def by blast

    89     with \<open>b mod a \<noteq> 0\<close> have "c \<noteq> 0" by auto

    90   with \<open>b mod a = b * c\<close> have "euclidean_size (b mod a) \<ge> euclidean_size b"

    91       using size_mult_mono by force

    92   moreover from \<open>\<not> a dvd b\<close> and \<open>a \<noteq> 0\<close>

    93   have "euclidean_size (b mod a) < euclidean_size a"

    94       using mod_size_less by blast

    95   ultimately show False using size_eq by simp

    96 qed

    97

    98 lemma size_mult_mono': "b \<noteq> 0 \<Longrightarrow> euclidean_size a \<le> euclidean_size (b * a)"

    99   by (subst mult.commute) (rule size_mult_mono)

   100

   101 lemma euclidean_size_times_unit:

   102   assumes "is_unit a"

   103   shows   "euclidean_size (a * b) = euclidean_size b"

   104 proof (rule antisym)

   105   from assms have [simp]: "a \<noteq> 0" by auto

   106   thus "euclidean_size (a * b) \<ge> euclidean_size b" by (rule size_mult_mono')

   107   from assms have "is_unit (1 div a)" by simp

   108   hence "1 div a \<noteq> 0" by (intro notI) simp_all

   109   hence "euclidean_size (a * b) \<le> euclidean_size ((1 div a) * (a * b))"

   110     by (rule size_mult_mono')

   111   also from assms have "(1 div a) * (a * b) = b"

   112     by (simp add: algebra_simps unit_div_mult_swap)

   113   finally show "euclidean_size (a * b) \<le> euclidean_size b" .

   114 qed

   115

   116 lemma euclidean_size_unit: "is_unit a \<Longrightarrow> euclidean_size a = euclidean_size 1"

   117   using euclidean_size_times_unit[of a 1] by simp

   118

   119 lemma unit_iff_euclidean_size:

   120   "is_unit a \<longleftrightarrow> euclidean_size a = euclidean_size 1 \<and> a \<noteq> 0"

   121 proof safe

   122   assume A: "a \<noteq> 0" and B: "euclidean_size a = euclidean_size 1"

   123   show "is_unit a" by (rule dvd_euclidean_size_eq_imp_dvd[OF A _ B]) simp_all

   124 qed (auto intro: euclidean_size_unit)

   125

   126 lemma euclidean_size_times_nonunit:

   127   assumes "a \<noteq> 0" "b \<noteq> 0" "\<not>is_unit a"

   128   shows   "euclidean_size b < euclidean_size (a * b)"

   129 proof (rule ccontr)

   130   assume "\<not>euclidean_size b < euclidean_size (a * b)"

   131   with size_mult_mono'[OF assms(1), of b]

   132     have eq: "euclidean_size (a * b) = euclidean_size b" by simp

   133   have "a * b dvd b"

   134     by (rule dvd_euclidean_size_eq_imp_dvd[OF _ _ eq]) (insert assms, simp_all)

   135   hence "a * b dvd 1 * b" by simp

   136   with \<open>b \<noteq> 0\<close> have "is_unit a" by (subst (asm) dvd_times_right_cancel_iff)

   137   with assms(3) show False by contradiction

   138 qed

   139

   140 lemma dvd_imp_size_le:

   141   assumes "a dvd b" "b \<noteq> 0"

   142   shows   "euclidean_size a \<le> euclidean_size b"

   143   using assms by (auto elim!: dvdE simp: size_mult_mono)

   144

   145 lemma dvd_proper_imp_size_less:

   146   assumes "a dvd b" "\<not>b dvd a" "b \<noteq> 0"

   147   shows   "euclidean_size a < euclidean_size b"

   148 proof -

   149   from assms(1) obtain c where "b = a * c" by (erule dvdE)

   150   hence z: "b = c * a" by (simp add: mult.commute)

   151   from z assms have "\<not>is_unit c" by (auto simp: mult.commute mult_unit_dvd_iff)

   152   with z assms show ?thesis

   153     by (auto intro!: euclidean_size_times_nonunit simp: )

   154 qed

   155

   156 function gcd_eucl :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"

   157 where

   158   "gcd_eucl a b = (if b = 0 then normalize a else gcd_eucl b (a mod b))"

   159   by pat_completeness simp

   160 termination

   161   by (relation "measure (euclidean_size \<circ> snd)") (simp_all add: mod_size_less)

   162

   163 declare gcd_eucl.simps [simp del]

   164

   165 lemma gcd_eucl_induct [case_names zero mod]:

   166   assumes H1: "\<And>b. P b 0"

   167   and H2: "\<And>a b. b \<noteq> 0 \<Longrightarrow> P b (a mod b) \<Longrightarrow> P a b"

   168   shows "P a b"

   169 proof (induct a b rule: gcd_eucl.induct)

   170   case ("1" a b)

   171   show ?case

   172   proof (cases "b = 0")

   173     case True then show "P a b" by simp (rule H1)

   174   next

   175     case False

   176     then have "P b (a mod b)"

   177       by (rule "1.hyps")

   178     with \<open>b \<noteq> 0\<close> show "P a b"

   179       by (blast intro: H2)

   180   qed

   181 qed

   182

   183 definition lcm_eucl :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"

   184 where

   185   "lcm_eucl a b = normalize (a * b) div gcd_eucl a b"

   186

   187 definition Lcm_eucl :: "'a set \<Rightarrow> 'a" \<comment> \<open>

   188   Somewhat complicated definition of Lcm that has the advantage of working

   189   for infinite sets as well\<close>

   190 where

   191   "Lcm_eucl A = (if \<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) then

   192      let l = SOME l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l =

   193        (LEAST n. \<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n)

   194        in normalize l

   195       else 0)"

   196

   197 definition Gcd_eucl :: "'a set \<Rightarrow> 'a"

   198 where

   199   "Gcd_eucl A = Lcm_eucl {d. \<forall>a\<in>A. d dvd a}"

   200

   201 declare Lcm_eucl_def Gcd_eucl_def [code del]

   202

   203 lemma gcd_eucl_0:

   204   "gcd_eucl a 0 = normalize a"

   205   by (simp add: gcd_eucl.simps [of a 0])

   206

   207 lemma gcd_eucl_0_left:

   208   "gcd_eucl 0 a = normalize a"

   209   by (simp_all add: gcd_eucl_0 gcd_eucl.simps [of 0 a])

   210

   211 lemma gcd_eucl_non_0:

   212   "b \<noteq> 0 \<Longrightarrow> gcd_eucl a b = gcd_eucl b (a mod b)"

   213   by (simp add: gcd_eucl.simps [of a b] gcd_eucl.simps [of b 0])

   214

   215 lemma gcd_eucl_dvd1 [iff]: "gcd_eucl a b dvd a"

   216   and gcd_eucl_dvd2 [iff]: "gcd_eucl a b dvd b"

   217   by (induct a b rule: gcd_eucl_induct)

   218      (simp_all add: gcd_eucl_0 gcd_eucl_non_0 dvd_mod_iff)

   219

   220 lemma normalize_gcd_eucl [simp]:

   221   "normalize (gcd_eucl a b) = gcd_eucl a b"

   222   by (induct a b rule: gcd_eucl_induct) (simp_all add: gcd_eucl_0 gcd_eucl_non_0)

   223

   224 lemma gcd_eucl_greatest:

   225   fixes k a b :: 'a

   226   shows "k dvd a \<Longrightarrow> k dvd b \<Longrightarrow> k dvd gcd_eucl a b"

   227 proof (induct a b rule: gcd_eucl_induct)

   228   case (zero a) from zero(1) show ?case by (rule dvd_trans) (simp add: gcd_eucl_0)

   229 next

   230   case (mod a b)

   231   then show ?case

   232     by (simp add: gcd_eucl_non_0 dvd_mod_iff)

   233 qed

   234

   235 lemma gcd_euclI:

   236   fixes gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"

   237   assumes "d dvd a" "d dvd b" "normalize d = d"

   238           "\<And>k. k dvd a \<Longrightarrow> k dvd b \<Longrightarrow> k dvd d"

   239   shows   "gcd_eucl a b = d"

   240   by (rule associated_eqI) (simp_all add: gcd_eucl_greatest assms)

   241

   242 lemma eq_gcd_euclI:

   243   fixes gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"

   244   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"

   245           "\<And>a b k. k dvd a \<Longrightarrow> k dvd b \<Longrightarrow> k dvd gcd a b"

   246   shows   "gcd = gcd_eucl"

   247   by (intro ext, rule associated_eqI) (simp_all add: gcd_eucl_greatest assms)

   248

   249 lemma gcd_eucl_zero [simp]:

   250   "gcd_eucl a b = 0 \<longleftrightarrow> a = 0 \<and> b = 0"

   251   by (metis dvd_0_left dvd_refl gcd_eucl_dvd1 gcd_eucl_dvd2 gcd_eucl_greatest)+

   252

   253

   254 lemma dvd_Lcm_eucl [simp]: "a \<in> A \<Longrightarrow> a dvd Lcm_eucl A"

   255   and Lcm_eucl_least: "(\<And>a. a \<in> A \<Longrightarrow> a dvd b) \<Longrightarrow> Lcm_eucl A dvd b"

   256   and unit_factor_Lcm_eucl [simp]:

   257           "unit_factor (Lcm_eucl A) = (if Lcm_eucl A = 0 then 0 else 1)"

   258 proof -

   259   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>

   260     unit_factor (Lcm_eucl A) = (if Lcm_eucl A = 0 then 0 else 1)" (is ?thesis)

   261   proof (cases "\<exists>l. l \<noteq>  0 \<and> (\<forall>a\<in>A. a dvd l)")

   262     case False

   263     hence "Lcm_eucl A = 0" by (auto simp: Lcm_eucl_def)

   264     with False show ?thesis by auto

   265   next

   266     case True

   267     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

   268     define n where "n = (LEAST n. \<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n)"

   269     define l where "l = (SOME l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n)"

   270     have "\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"

   271       apply (subst n_def)

   272       apply (rule LeastI[of _ "euclidean_size l\<^sub>0"])

   273       apply (rule exI[of _ l\<^sub>0])

   274       apply (simp add: l\<^sub>0_props)

   275       done

   276     from someI_ex[OF this] have "l \<noteq> 0" and "\<forall>a\<in>A. a dvd l" and "euclidean_size l = n"

   277       unfolding l_def by simp_all

   278     {

   279       fix l' assume "\<forall>a\<in>A. a dvd l'"

   280       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)

   281       moreover from \<open>l \<noteq> 0\<close> have "gcd_eucl l l' \<noteq> 0" by simp

   282       ultimately have "\<exists>b. b \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd b) \<and>

   283                           euclidean_size b = euclidean_size (gcd_eucl l l')"

   284         by (intro exI[of _ "gcd_eucl l l'"], auto)

   285       hence "euclidean_size (gcd_eucl l l') \<ge> n" by (subst n_def) (rule Least_le)

   286       moreover have "euclidean_size (gcd_eucl l l') \<le> n"

   287       proof -

   288         have "gcd_eucl l l' dvd l" by simp

   289         then obtain a where "l = gcd_eucl l l' * a" unfolding dvd_def by blast

   290         with \<open>l \<noteq> 0\<close> have "a \<noteq> 0" by auto

   291         hence "euclidean_size (gcd_eucl l l') \<le> euclidean_size (gcd_eucl l l' * a)"

   292           by (rule size_mult_mono)

   293         also have "gcd_eucl l l' * a = l" using \<open>l = gcd_eucl l l' * a\<close> ..

   294         also note \<open>euclidean_size l = n\<close>

   295         finally show "euclidean_size (gcd_eucl l l') \<le> n" .

   296       qed

   297       ultimately have *: "euclidean_size l = euclidean_size (gcd_eucl l l')"

   298         by (intro le_antisym, simp_all add: \<open>euclidean_size l = n\<close>)

   299       from \<open>l \<noteq> 0\<close> have "l dvd gcd_eucl l l'"

   300         by (rule dvd_euclidean_size_eq_imp_dvd) (auto simp add: *)

   301       hence "l dvd l'" by (rule dvd_trans[OF _ gcd_eucl_dvd2])

   302     }

   303

   304     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>

   305       have "(\<forall>a\<in>A. a dvd normalize l) \<and>

   306         (\<forall>l'. (\<forall>a\<in>A. a dvd l') \<longrightarrow> normalize l dvd l') \<and>

   307         unit_factor (normalize l) =

   308         (if normalize l = 0 then 0 else 1)"

   309       by (auto simp: unit_simps)

   310     also from True have "normalize l = Lcm_eucl A"

   311       by (simp add: Lcm_eucl_def Let_def n_def l_def)

   312     finally show ?thesis .

   313   qed

   314   note A = this

   315

   316   {fix a assume "a \<in> A" then show "a dvd Lcm_eucl A" using A by blast}

   317   {fix b assume "\<And>a. a \<in> A \<Longrightarrow> a dvd b" then show "Lcm_eucl A dvd b" using A by blast}

   318   from A show "unit_factor (Lcm_eucl A) = (if Lcm_eucl A = 0 then 0 else 1)" by blast

   319 qed

   320

   321 lemma normalize_Lcm_eucl [simp]:

   322   "normalize (Lcm_eucl A) = Lcm_eucl A"

   323 proof (cases "Lcm_eucl A = 0")

   324   case True then show ?thesis by simp

   325 next

   326   case False

   327   have "unit_factor (Lcm_eucl A) * normalize (Lcm_eucl A) = Lcm_eucl A"

   328     by (fact unit_factor_mult_normalize)

   329   with False show ?thesis by simp

   330 qed

   331

   332 lemma eq_Lcm_euclI:

   333   fixes lcm :: "'a set \<Rightarrow> 'a"

   334   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"

   335           "\<And>A. normalize (lcm A) = lcm A" shows "lcm = Lcm_eucl"

   336   by (intro ext, rule associated_eqI) (auto simp: assms intro: Lcm_eucl_least)

   337

   338 lemma Gcd_eucl_dvd: "a \<in> A \<Longrightarrow> Gcd_eucl A dvd a"

   339   unfolding Gcd_eucl_def by (auto intro: Lcm_eucl_least)

   340

   341 lemma Gcd_eucl_greatest: "(\<And>x. x \<in> A \<Longrightarrow> d dvd x) \<Longrightarrow> d dvd Gcd_eucl A"

   342   unfolding Gcd_eucl_def by auto

   343

   344 lemma normalize_Gcd_eucl [simp]: "normalize (Gcd_eucl A) = Gcd_eucl A"

   345   by (simp add: Gcd_eucl_def)

   346

   347 lemma Lcm_euclI:

   348   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"

   349   shows   "Lcm_eucl A = d"

   350 proof -

   351   have "normalize (Lcm_eucl A) = normalize d"

   352     by (intro associatedI) (auto intro: dvd_Lcm_eucl Lcm_eucl_least assms)

   353   thus ?thesis by (simp add: assms)

   354 qed

   355

   356 lemma Gcd_euclI:

   357   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"

   358   shows   "Gcd_eucl A = d"

   359 proof -

   360   have "normalize (Gcd_eucl A) = normalize d"

   361     by (intro associatedI) (auto intro: Gcd_eucl_dvd Gcd_eucl_greatest assms)

   362   thus ?thesis by (simp add: assms)

   363 qed

   364

   365 lemmas lcm_gcd_eucl_facts =

   366   gcd_eucl_dvd1 gcd_eucl_dvd2 gcd_eucl_greatest normalize_gcd_eucl lcm_eucl_def

   367   Gcd_eucl_def Gcd_eucl_dvd Gcd_eucl_greatest normalize_Gcd_eucl

   368   dvd_Lcm_eucl Lcm_eucl_least normalize_Lcm_eucl

   369

   370 lemma normalized_factors_product:

   371   "{p. p dvd a * b \<and> normalize p = p} =

   372      (\<lambda>(x,y). x * y)  ({p. p dvd a \<and> normalize p = p} \<times> {p. p dvd b \<and> normalize p = p})"

   373 proof safe

   374   fix p assume p: "p dvd a * b" "normalize p = p"

   375   interpret semiring_gcd 1 0 "op *" gcd_eucl lcm_eucl "op div" "op +" "op -" normalize unit_factor

   376     by standard (rule lcm_gcd_eucl_facts; assumption)+

   377   from dvd_productE[OF p(1)] guess x y . note xy = this

   378   define x' y' where "x' = normalize x" and "y' = normalize y"

   379   have "p = x' * y'"

   380     by (subst p(2) [symmetric]) (simp add: xy x'_def y'_def normalize_mult)

   381   moreover from xy have "normalize x' = x'" "normalize y' = y'" "x' dvd a" "y' dvd b"

   382     by (simp_all add: x'_def y'_def)

   383   ultimately show "p \<in> (\<lambda>(x, y). x * y) 

   384                      ({p. p dvd a \<and> normalize p = p} \<times> {p. p dvd b \<and> normalize p = p})"

   385     by blast

   386 qed (auto simp: normalize_mult mult_dvd_mono)

   387

   388

   389 subclass factorial_semiring

   390 proof (standard, rule factorial_semiring_altI_aux)

   391   fix x assume "x \<noteq> 0"

   392   thus "finite {p. p dvd x \<and> normalize p = p}"

   393   proof (induction "euclidean_size x" arbitrary: x rule: less_induct)

   394     case (less x)

   395     show ?case

   396     proof (cases "\<exists>y. y dvd x \<and> \<not>x dvd y \<and> \<not>is_unit y")

   397       case False

   398       have "{p. p dvd x \<and> normalize p = p} \<subseteq> {1, normalize x}"

   399       proof

   400         fix p assume p: "p \<in> {p. p dvd x \<and> normalize p = p}"

   401         with False have "is_unit p \<or> x dvd p" by blast

   402         thus "p \<in> {1, normalize x}"

   403         proof (elim disjE)

   404           assume "is_unit p"

   405           hence "normalize p = 1" by (simp add: is_unit_normalize)

   406           with p show ?thesis by simp

   407         next

   408           assume "x dvd p"

   409           with p have "normalize p = normalize x" by (intro associatedI) simp_all

   410           with p show ?thesis by simp

   411         qed

   412       qed

   413       moreover have "finite \<dots>" by simp

   414       ultimately show ?thesis by (rule finite_subset)

   415

   416     next

   417       case True

   418       then obtain y where y: "y dvd x" "\<not>x dvd y" "\<not>is_unit y" by blast

   419       define z where "z = x div y"

   420       let ?fctrs = "\<lambda>x. {p. p dvd x \<and> normalize p = p}"

   421       from y have x: "x = y * z" by (simp add: z_def)

   422       with less.prems have "y \<noteq> 0" "z \<noteq> 0" by auto

   423       from x y have "\<not>is_unit z" by (auto simp: mult_unit_dvd_iff)

   424       have "?fctrs x = (\<lambda>(p,p'). p * p')  (?fctrs y \<times> ?fctrs z)"

   425         by (subst x) (rule normalized_factors_product)

   426       also have "\<not>y * z dvd y * 1" "\<not>y * z dvd 1 * z"

   427         by (subst dvd_times_left_cancel_iff dvd_times_right_cancel_iff; fact)+

   428       hence "finite ((\<lambda>(p,p'). p * p')  (?fctrs y \<times> ?fctrs z))"

   429         by (intro finite_imageI finite_cartesian_product less dvd_proper_imp_size_less)

   430            (auto simp: x)

   431       finally show ?thesis .

   432     qed

   433   qed

   434 next

   435   interpret semiring_gcd 1 0 "op *" gcd_eucl lcm_eucl "op div" "op +" "op -" normalize unit_factor

   436     by standard (rule lcm_gcd_eucl_facts; assumption)+

   437   fix p assume p: "irreducible p"

   438   thus "prime_elem p" by (rule irreducible_imp_prime_elem_gcd)

   439 qed

   440

   441 lemma gcd_eucl_eq_gcd_factorial: "gcd_eucl = gcd_factorial"

   442   by (intro ext gcd_euclI gcd_lcm_factorial)

   443

   444 lemma lcm_eucl_eq_lcm_factorial: "lcm_eucl = lcm_factorial"

   445   by (intro ext) (simp add: lcm_eucl_def lcm_factorial_gcd_factorial gcd_eucl_eq_gcd_factorial)

   446

   447 lemma Gcd_eucl_eq_Gcd_factorial: "Gcd_eucl = Gcd_factorial"

   448   by (intro ext Gcd_euclI gcd_lcm_factorial)

   449

   450 lemma Lcm_eucl_eq_Lcm_factorial: "Lcm_eucl = Lcm_factorial"

   451   by (intro ext Lcm_euclI gcd_lcm_factorial)

   452

   453 lemmas eucl_eq_factorial =

   454   gcd_eucl_eq_gcd_factorial lcm_eucl_eq_lcm_factorial

   455   Gcd_eucl_eq_Gcd_factorial Lcm_eucl_eq_Lcm_factorial

   456

   457 end

   458

   459 class euclidean_ring = euclidean_semiring + idom

   460 begin

   461

   462 function euclid_ext_aux :: "'a \<Rightarrow> _" where

   463   "euclid_ext_aux r' r s' s t' t = (

   464      if r = 0 then let c = 1 div unit_factor r' in (s' * c, t' * c, normalize r')

   465      else let q = r' div r

   466           in  euclid_ext_aux r (r' mod r) s (s' - q * s) t (t' - q * t))"

   467 by auto

   468 termination by (relation "measure (\<lambda>(_,b,_,_,_,_). euclidean_size b)") (simp_all add: mod_size_less)

   469

   470 declare euclid_ext_aux.simps [simp del]

   471

   472 lemma euclid_ext_aux_correct:

   473   assumes "gcd_eucl r' r = gcd_eucl a b"

   474   assumes "s' * a + t' * b = r'"

   475   assumes "s * a + t * b = r"

   476   shows   "case euclid_ext_aux r' r s' s t' t of (x,y,c) \<Rightarrow>

   477              x * a + y * b = c \<and> c = gcd_eucl a b" (is "?P (euclid_ext_aux r' r s' s t' t)")

   478 using assms

   479 proof (induction r' r s' s t' t rule: euclid_ext_aux.induct)

   480   case (1 r' r s' s t' t)

   481   show ?case

   482   proof (cases "r = 0")

   483     case True

   484     hence "euclid_ext_aux r' r s' s t' t =

   485              (s' div unit_factor r', t' div unit_factor r', normalize r')"

   486       by (subst euclid_ext_aux.simps) (simp add: Let_def)

   487     also have "?P \<dots>"

   488     proof safe

   489       have "s' div unit_factor r' * a + t' div unit_factor r' * b =

   490                 (s' * a + t' * b) div unit_factor r'"

   491         by (cases "r' = 0") (simp_all add: unit_div_commute)

   492       also have "s' * a + t' * b = r'" by fact

   493       also have "\<dots> div unit_factor r' = normalize r'" by simp

   494       finally show "s' div unit_factor r' * a + t' div unit_factor r' * b = normalize r'" .

   495     next

   496       from "1.prems" True show "normalize r' = gcd_eucl a b" by (simp add: gcd_eucl_0)

   497     qed

   498     finally show ?thesis .

   499   next

   500     case False

   501     hence "euclid_ext_aux r' r s' s t' t =

   502              euclid_ext_aux r (r' mod r) s (s' - r' div r * s) t (t' - r' div r * t)"

   503       by (subst euclid_ext_aux.simps) (simp add: Let_def)

   504     also from "1.prems" False have "?P \<dots>"

   505     proof (intro "1.IH")

   506       have "(s' - r' div r * s) * a + (t' - r' div r * t) * b =

   507               (s' * a + t' * b) - r' div r * (s * a + t * b)" by (simp add: algebra_simps)

   508       also have "s' * a + t' * b = r'" by fact

   509       also have "s * a + t * b = r" by fact

   510       also have "r' - r' div r * r = r' mod r" using div_mult_mod_eq [of r' r]

   511         by (simp add: algebra_simps)

   512       finally show "(s' - r' div r * s) * a + (t' - r' div r * t) * b = r' mod r" .

   513     qed (auto simp: gcd_eucl_non_0 algebra_simps minus_mod_eq_div_mult [symmetric])

   514     finally show ?thesis .

   515   qed

   516 qed

   517

   518 definition euclid_ext where

   519   "euclid_ext a b = euclid_ext_aux a b 1 0 0 1"

   520

   521 lemma euclid_ext_0:

   522   "euclid_ext a 0 = (1 div unit_factor a, 0, normalize a)"

   523   by (simp add: euclid_ext_def euclid_ext_aux.simps)

   524

   525 lemma euclid_ext_left_0:

   526   "euclid_ext 0 a = (0, 1 div unit_factor a, normalize a)"

   527   by (simp add: euclid_ext_def euclid_ext_aux.simps)

   528

   529 lemma euclid_ext_correct':

   530   "case euclid_ext a b of (x,y,c) \<Rightarrow> x * a + y * b = c \<and> c = gcd_eucl a b"

   531   unfolding euclid_ext_def by (rule euclid_ext_aux_correct) simp_all

   532

   533 lemma euclid_ext_gcd_eucl:

   534   "(case euclid_ext a b of (x,y,c) \<Rightarrow> c) = gcd_eucl a b"

   535   using euclid_ext_correct'[of a b] by (simp add: case_prod_unfold)

   536

   537 definition euclid_ext' where

   538   "euclid_ext' a b = (case euclid_ext a b of (x, y, _) \<Rightarrow> (x, y))"

   539

   540 lemma euclid_ext'_correct':

   541   "case euclid_ext' a b of (x,y) \<Rightarrow> x * a + y * b = gcd_eucl a b"

   542   using euclid_ext_correct'[of a b] by (simp add: case_prod_unfold euclid_ext'_def)

   543

   544 lemma euclid_ext'_0: "euclid_ext' a 0 = (1 div unit_factor a, 0)"

   545   by (simp add: euclid_ext'_def euclid_ext_0)

   546

   547 lemma euclid_ext'_left_0: "euclid_ext' 0 a = (0, 1 div unit_factor a)"

   548   by (simp add: euclid_ext'_def euclid_ext_left_0)

   549

   550 end

   551

   552 class euclidean_semiring_gcd = euclidean_semiring + gcd + Gcd +

   553   assumes gcd_gcd_eucl: "gcd = gcd_eucl" and lcm_lcm_eucl: "lcm = lcm_eucl"

   554   assumes Gcd_Gcd_eucl: "Gcd = Gcd_eucl" and Lcm_Lcm_eucl: "Lcm = Lcm_eucl"

   555 begin

   556

   557 subclass semiring_gcd

   558   by standard (simp_all add: gcd_gcd_eucl gcd_eucl_greatest lcm_lcm_eucl lcm_eucl_def)

   559

   560 subclass semiring_Gcd

   561   by standard (auto simp: Gcd_Gcd_eucl Lcm_Lcm_eucl Gcd_eucl_def intro: Lcm_eucl_least)

   562

   563 subclass factorial_semiring_gcd

   564 proof

   565   fix a b

   566   show "gcd a b = gcd_factorial a b"

   567     by (rule sym, rule gcdI) (rule gcd_lcm_factorial; assumption)+

   568   thus "lcm a b = lcm_factorial a b"

   569     by (simp add: lcm_factorial_gcd_factorial lcm_gcd)

   570 next

   571   fix A

   572   show "Gcd A = Gcd_factorial A"

   573     by (rule sym, rule GcdI) (rule gcd_lcm_factorial; assumption)+

   574   show "Lcm A = Lcm_factorial A"

   575     by (rule sym, rule LcmI) (rule gcd_lcm_factorial; assumption)+

   576 qed

   577

   578 lemma gcd_non_0:

   579   "b \<noteq> 0 \<Longrightarrow> gcd a b = gcd b (a mod b)"

   580   unfolding gcd_gcd_eucl by (fact gcd_eucl_non_0)

   581

   582 lemmas gcd_0 = gcd_0_right

   583 lemmas dvd_gcd_iff = gcd_greatest_iff

   584 lemmas gcd_greatest_iff = dvd_gcd_iff

   585

   586 lemma gcd_mod1 [simp]:

   587   "gcd (a mod b) b = gcd a b"

   588   by (rule gcdI, metis dvd_mod_iff gcd_dvd1 gcd_dvd2, simp_all add: gcd_greatest dvd_mod_iff)

   589

   590 lemma gcd_mod2 [simp]:

   591   "gcd a (b mod a) = gcd a b"

   592   by (rule gcdI, simp, metis dvd_mod_iff gcd_dvd1 gcd_dvd2, simp_all add: gcd_greatest dvd_mod_iff)

   593

   594 lemma euclidean_size_gcd_le1 [simp]:

   595   assumes "a \<noteq> 0"

   596   shows "euclidean_size (gcd a b) \<le> euclidean_size a"

   597 proof -

   598    have "gcd a b dvd a" by (rule gcd_dvd1)

   599    then obtain c where A: "a = gcd a b * c" unfolding dvd_def by blast

   600    with \<open>a \<noteq> 0\<close> show ?thesis by (subst (2) A, intro size_mult_mono) auto

   601 qed

   602

   603 lemma euclidean_size_gcd_le2 [simp]:

   604   "b \<noteq> 0 \<Longrightarrow> euclidean_size (gcd a b) \<le> euclidean_size b"

   605   by (subst gcd.commute, rule euclidean_size_gcd_le1)

   606

   607 lemma euclidean_size_gcd_less1:

   608   assumes "a \<noteq> 0" and "\<not>a dvd b"

   609   shows "euclidean_size (gcd a b) < euclidean_size a"

   610 proof (rule ccontr)

   611   assume "\<not>euclidean_size (gcd a b) < euclidean_size a"

   612   with \<open>a \<noteq> 0\<close> have A: "euclidean_size (gcd a b) = euclidean_size a"

   613     by (intro le_antisym, simp_all)

   614   have "a dvd gcd a b"

   615     by (rule dvd_euclidean_size_eq_imp_dvd) (simp_all add: assms A)

   616   hence "a dvd b" using dvd_gcdD2 by blast

   617   with \<open>\<not>a dvd b\<close> show False by contradiction

   618 qed

   619

   620 lemma euclidean_size_gcd_less2:

   621   assumes "b \<noteq> 0" and "\<not>b dvd a"

   622   shows "euclidean_size (gcd a b) < euclidean_size b"

   623   using assms by (subst gcd.commute, rule euclidean_size_gcd_less1)

   624

   625 lemma euclidean_size_lcm_le1:

   626   assumes "a \<noteq> 0" and "b \<noteq> 0"

   627   shows "euclidean_size a \<le> euclidean_size (lcm a b)"

   628 proof -

   629   have "a dvd lcm a b" by (rule dvd_lcm1)

   630   then obtain c where A: "lcm a b = a * c" ..

   631   with \<open>a \<noteq> 0\<close> and \<open>b \<noteq> 0\<close> have "c \<noteq> 0" by (auto simp: lcm_eq_0_iff)

   632   then show ?thesis by (subst A, intro size_mult_mono)

   633 qed

   634

   635 lemma euclidean_size_lcm_le2:

   636   "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> euclidean_size b \<le> euclidean_size (lcm a b)"

   637   using euclidean_size_lcm_le1 [of b a] by (simp add: ac_simps)

   638

   639 lemma euclidean_size_lcm_less1:

   640   assumes "b \<noteq> 0" and "\<not>b dvd a"

   641   shows "euclidean_size a < euclidean_size (lcm a b)"

   642 proof (rule ccontr)

   643   from assms have "a \<noteq> 0" by auto

   644   assume "\<not>euclidean_size a < euclidean_size (lcm a b)"

   645   with \<open>a \<noteq> 0\<close> and \<open>b \<noteq> 0\<close> have "euclidean_size (lcm a b) = euclidean_size a"

   646     by (intro le_antisym, simp, intro euclidean_size_lcm_le1)

   647   with assms have "lcm a b dvd a"

   648     by (rule_tac dvd_euclidean_size_eq_imp_dvd) (auto simp: lcm_eq_0_iff)

   649   hence "b dvd a" by (rule lcm_dvdD2)

   650   with \<open>\<not>b dvd a\<close> show False by contradiction

   651 qed

   652

   653 lemma euclidean_size_lcm_less2:

   654   assumes "a \<noteq> 0" and "\<not>a dvd b"

   655   shows "euclidean_size b < euclidean_size (lcm a b)"

   656   using assms euclidean_size_lcm_less1 [of a b] by (simp add: ac_simps)

   657

   658 lemma Lcm_eucl_set [code]:

   659   "Lcm_eucl (set xs) = foldl lcm_eucl 1 xs"

   660   by (simp add: Lcm_Lcm_eucl [symmetric] lcm_lcm_eucl Lcm_set)

   661

   662 lemma Gcd_eucl_set [code]:

   663   "Gcd_eucl (set xs) = foldl gcd_eucl 0 xs"

   664   by (simp add: Gcd_Gcd_eucl [symmetric] gcd_gcd_eucl Gcd_set)

   665

   666 end

   667

   668

   669 text \<open>

   670   A Euclidean ring is a Euclidean semiring with additive inverses. It provides a

   671   few more lemmas; in particular, Bezout's lemma holds for any Euclidean ring.

   672 \<close>

   673

   674 class euclidean_ring_gcd = euclidean_semiring_gcd + idom

   675 begin

   676

   677 subclass euclidean_ring ..

   678 subclass ring_gcd ..

   679 subclass factorial_ring_gcd ..

   680

   681 lemma euclid_ext_gcd [simp]:

   682   "(case euclid_ext a b of (_, _ , t) \<Rightarrow> t) = gcd a b"

   683   using euclid_ext_correct'[of a b] by (simp add: case_prod_unfold Let_def gcd_gcd_eucl)

   684

   685 lemma euclid_ext_gcd' [simp]:

   686   "euclid_ext a b = (r, s, t) \<Longrightarrow> t = gcd a b"

   687   by (insert euclid_ext_gcd[of a b], drule (1) subst, simp)

   688

   689 lemma euclid_ext_correct:

   690   "case euclid_ext a b of (x,y,c) \<Rightarrow> x * a + y * b = c \<and> c = gcd a b"

   691   using euclid_ext_correct'[of a b]

   692   by (simp add: gcd_gcd_eucl case_prod_unfold)

   693

   694 lemma euclid_ext'_correct:

   695   "fst (euclid_ext' a b) * a + snd (euclid_ext' a b) * b = gcd a b"

   696   using euclid_ext_correct'[of a b]

   697   by (simp add: gcd_gcd_eucl case_prod_unfold euclid_ext'_def)

   698

   699 lemma bezout: "\<exists>s t. s * a + t * b = gcd a b"

   700   using euclid_ext'_correct by blast

   701

   702 end

   703

   704

   705 subsection \<open>Typical instances\<close>

   706

   707 instantiation nat :: euclidean_semiring

   708 begin

   709

   710 definition [simp]:

   711   "euclidean_size_nat = (id :: nat \<Rightarrow> nat)"

   712

   713 instance by standard simp_all

   714

   715 end

   716

   717

   718 instantiation int :: euclidean_ring

   719 begin

   720

   721 definition [simp]:

   722   "euclidean_size_int = (nat \<circ> abs :: int \<Rightarrow> nat)"

   723

   724 instance by standard (auto simp add: abs_mult nat_mult_distrib split: abs_split)

   725

   726 end

   727

   728 instance nat :: euclidean_semiring_gcd

   729 proof

   730   show [simp]: "gcd = (gcd_eucl :: nat \<Rightarrow> _)" "Lcm = (Lcm_eucl :: nat set \<Rightarrow> _)"

   731     by (simp_all add: eq_gcd_euclI eq_Lcm_euclI)

   732   show "lcm = (lcm_eucl :: nat \<Rightarrow> _)" "Gcd = (Gcd_eucl :: nat set \<Rightarrow> _)"

   733     by (intro ext, simp add: lcm_eucl_def lcm_nat_def Gcd_nat_def Gcd_eucl_def)+

   734 qed

   735

   736 instance int :: euclidean_ring_gcd

   737 proof

   738   show [simp]: "gcd = (gcd_eucl :: int \<Rightarrow> _)" "Lcm = (Lcm_eucl :: int set \<Rightarrow> _)"

   739     by (simp_all add: eq_gcd_euclI eq_Lcm_euclI)

   740   show "lcm = (lcm_eucl :: int \<Rightarrow> _)" "Gcd = (Gcd_eucl :: int set \<Rightarrow> _)"

   741     by (intro ext, simp add: lcm_eucl_def lcm_altdef_int

   742           semiring_Gcd_class.Gcd_Lcm Gcd_eucl_def abs_mult)+

   743 qed

   744

   745 end