src/HOL/Number_Theory/Euclidean_Algorithm.thy
 author haftmann Tue Oct 11 16:44:13 2016 +0200 (2016-10-11) changeset 64163 62c9e5c05928 parent 63947 559f0882d6a6 child 64164 38c407446400 permissions -rw-r--r--
stripped dependency on pragmatic type class semiring_div
     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 class divide_modulo = semidom_divide + modulo +

    10   assumes div_mod_equality: "((a div b) * b + a mod b) + c = a + c"

    11 begin

    12

    13 lemma zero_mod_left [simp]: "0 mod a = 0"

    14   using div_mod_equality[of 0 a 0] by simp

    15

    16 lemma dvd_mod_iff [simp]:

    17   assumes "k dvd n"

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

    19 proof -

    20   thm div_mod_equality

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

    22     by (simp add: dvd_add_right_iff)

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

    24     using div_mod_equality[of m n 0] by simp

    25   finally show ?thesis .

    26 qed

    27

    28 lemma mod_0_imp_dvd:

    29   assumes "a mod b = 0"

    30   shows   "b dvd a"

    31 proof -

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

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

    34     using div_mod_equality[of a b 0] by (simp add: assms)

    35   finally show ?thesis .

    36 qed

    37

    38 end

    39

    40

    41

    42 text \<open>

    43   A Euclidean semiring is a semiring upon which the Euclidean algorithm can be

    44   implemented. It must provide:

    45   \begin{itemize}

    46   \item division with remainder

    47   \item a size function such that @{term "size (a mod b) < size b"}

    48         for any @{term "b \<noteq> 0"}

    49   \end{itemize}

    50   The existence of these functions makes it possible to derive gcd and lcm functions

    51   for any Euclidean semiring.

    52 \<close>

    53 class euclidean_semiring = divide_modulo + normalization_semidom +

    54   fixes euclidean_size :: "'a \<Rightarrow> nat"

    55   assumes size_0 [simp]: "euclidean_size 0 = 0"

    56   assumes mod_size_less:

    57     "b \<noteq> 0 \<Longrightarrow> euclidean_size (a mod b) < euclidean_size b"

    58   assumes size_mult_mono:

    59     "b \<noteq> 0 \<Longrightarrow> euclidean_size a \<le> euclidean_size (a * b)"

    60 begin

    61

    62 lemma euclidean_size_normalize [simp]:

    63   "euclidean_size (normalize a) = euclidean_size a"

    64 proof (cases "a = 0")

    65   case True

    66   then show ?thesis

    67     by simp

    68 next

    69   case [simp]: False

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

    71     by (rule size_mult_mono) simp

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

    73     by (rule size_mult_mono) simp

    74   ultimately show ?thesis

    75     by simp

    76 qed

    77

    78 lemma euclidean_division:

    79   fixes a :: 'a and b :: 'a

    80   assumes "b \<noteq> 0"

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

    82     and "euclidean_size t < euclidean_size b"

    83 proof -

    84   from div_mod_equality [of a b 0]

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

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

    87 qed

    88

    89 lemma zero_mod_left [simp]: "0 mod a = 0"

    90   using div_mod_equality[of 0 a 0] by simp

    91

    92 lemma dvd_mod_iff [simp]:

    93   assumes "k dvd n"

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

    95 proof -

    96   thm div_mod_equality

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

    98     by (simp add: dvd_add_right_iff)

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

   100     using div_mod_equality[of m n 0] by simp

   101   finally show ?thesis .

   102 qed

   103

   104 lemma mod_0_imp_dvd:

   105   assumes "a mod b = 0"

   106   shows   "b dvd a"

   107 proof -

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

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

   110     using div_mod_equality[of a b 0] by (simp add: assms)

   111   finally show ?thesis .

   112 qed

   113

   114 lemma dvd_euclidean_size_eq_imp_dvd:

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

   116   shows "a dvd b"

   117 proof (rule ccontr)

   118   assume "\<not> a dvd b"

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

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

   121   from b_dvd_a have b_dvd_mod: "b dvd b mod a" by simp

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

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

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

   125       using size_mult_mono by force

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

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

   128       using mod_size_less by blast

   129   ultimately show False using size_eq by simp

   130 qed

   131

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

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

   134

   135 lemma euclidean_size_times_unit:

   136   assumes "is_unit a"

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

   138 proof (rule antisym)

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

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

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

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

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

   144     by (rule size_mult_mono')

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

   146     by (simp add: algebra_simps unit_div_mult_swap)

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

   148 qed

   149

   150 lemma euclidean_size_unit: "is_unit x \<Longrightarrow> euclidean_size x = euclidean_size 1"

   151   using euclidean_size_times_unit[of x 1] by simp

   152

   153 lemma unit_iff_euclidean_size:

   154   "is_unit x \<longleftrightarrow> euclidean_size x = euclidean_size 1 \<and> x \<noteq> 0"

   155 proof safe

   156   assume A: "x \<noteq> 0" and B: "euclidean_size x = euclidean_size 1"

   157   show "is_unit x" by (rule dvd_euclidean_size_eq_imp_dvd[OF A _ B]) simp_all

   158 qed (auto intro: euclidean_size_unit)

   159

   160 lemma euclidean_size_times_nonunit:

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

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

   163 proof (rule ccontr)

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

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

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

   167   have "a * b dvd b"

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

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

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

   171   with assms(3) show False by contradiction

   172 qed

   173

   174 lemma dvd_imp_size_le:

   175   assumes "x dvd y" "y \<noteq> 0"

   176   shows   "euclidean_size x \<le> euclidean_size y"

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

   178

   179 lemma dvd_proper_imp_size_less:

   180   assumes "x dvd y" "\<not>y dvd x" "y \<noteq> 0"

   181   shows   "euclidean_size x < euclidean_size y"

   182 proof -

   183   from assms(1) obtain z where "y = x * z" by (erule dvdE)

   184   hence z: "y = z * x" by (simp add: mult.commute)

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

   186   with z assms show ?thesis

   187     by (auto intro!: euclidean_size_times_nonunit simp: )

   188 qed

   189

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

   191 where

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

   193   by pat_completeness simp

   194 termination

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

   196

   197 declare gcd_eucl.simps [simp del]

   198

   199 lemma gcd_eucl_induct [case_names zero mod]:

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

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

   202   shows "P a b"

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

   204   case ("1" a b)

   205   show ?case

   206   proof (cases "b = 0")

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

   208   next

   209     case False

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

   211       by (rule "1.hyps")

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

   213       by (blast intro: H2)

   214   qed

   215 qed

   216

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

   218 where

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

   220

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

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

   223   for infinite sets as well\<close>

   224 where

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

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

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

   228        in normalize l

   229       else 0)"

   230

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

   232 where

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

   234

   235 declare Lcm_eucl_def Gcd_eucl_def [code del]

   236

   237 lemma gcd_eucl_0:

   238   "gcd_eucl a 0 = normalize a"

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

   240

   241 lemma gcd_eucl_0_left:

   242   "gcd_eucl 0 a = normalize a"

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

   244

   245 lemma gcd_eucl_non_0:

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

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

   248

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

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

   251   by (induct a b rule: gcd_eucl_induct)

   252      (simp_all add: gcd_eucl_0 gcd_eucl_non_0 dvd_mod_iff)

   253

   254 lemma normalize_gcd_eucl [simp]:

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

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

   257

   258 lemma gcd_eucl_greatest:

   259   fixes k a b :: 'a

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

   261 proof (induct a b rule: gcd_eucl_induct)

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

   263 next

   264   case (mod a b)

   265   then show ?case

   266     by (simp add: gcd_eucl_non_0 dvd_mod_iff)

   267 qed

   268

   269 lemma gcd_euclI:

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

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

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

   273   shows   "gcd_eucl a b = d"

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

   275

   276 lemma eq_gcd_euclI:

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

   278   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"

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

   280   shows   "gcd = gcd_eucl"

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

   282

   283 lemma gcd_eucl_zero [simp]:

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

   285   by (metis dvd_0_left dvd_refl gcd_eucl_dvd1 gcd_eucl_dvd2 gcd_eucl_greatest)+

   286

   287

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

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

   290   and unit_factor_Lcm_eucl [simp]:

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

   292 proof -

   293   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>

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

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

   296     case False

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

   298     with False show ?thesis by auto

   299   next

   300     case True

   301     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

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

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

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

   305       apply (subst n_def)

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

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

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

   309       done

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

   311       unfolding l_def by simp_all

   312     {

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

   314       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)

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

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

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

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

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

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

   321       proof -

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

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

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

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

   326           by (rule size_mult_mono)

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

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

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

   330       qed

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

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

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

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

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

   336     }

   337

   338     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>

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

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

   341         unit_factor (normalize l) =

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

   343       by (auto simp: unit_simps)

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

   345       by (simp add: Lcm_eucl_def Let_def n_def l_def)

   346     finally show ?thesis .

   347   qed

   348   note A = this

   349

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

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

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

   353 qed

   354

   355 lemma normalize_Lcm_eucl [simp]:

   356   "normalize (Lcm_eucl A) = Lcm_eucl A"

   357 proof (cases "Lcm_eucl A = 0")

   358   case True then show ?thesis by simp

   359 next

   360   case False

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

   362     by (fact unit_factor_mult_normalize)

   363   with False show ?thesis by simp

   364 qed

   365

   366 lemma eq_Lcm_euclI:

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

   368   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"

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

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

   371

   372 lemma Gcd_eucl_dvd: "x \<in> A \<Longrightarrow> Gcd_eucl A dvd x"

   373   unfolding Gcd_eucl_def by (auto intro: Lcm_eucl_least)

   374

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

   376   unfolding Gcd_eucl_def by auto

   377

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

   379   by (simp add: Gcd_eucl_def)

   380

   381 lemma Lcm_euclI:

   382   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"

   383   shows   "Lcm_eucl A = d"

   384 proof -

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

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

   387   thus ?thesis by (simp add: assms)

   388 qed

   389

   390 lemma Gcd_euclI:

   391   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"

   392   shows   "Gcd_eucl A = d"

   393 proof -

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

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

   396   thus ?thesis by (simp add: assms)

   397 qed

   398

   399 lemmas lcm_gcd_eucl_facts =

   400   gcd_eucl_dvd1 gcd_eucl_dvd2 gcd_eucl_greatest normalize_gcd_eucl lcm_eucl_def

   401   Gcd_eucl_def Gcd_eucl_dvd Gcd_eucl_greatest normalize_Gcd_eucl

   402   dvd_Lcm_eucl Lcm_eucl_least normalize_Lcm_eucl

   403

   404 lemma normalized_factors_product:

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

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

   407 proof safe

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

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

   410     by standard (rule lcm_gcd_eucl_facts; assumption)+

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

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

   413   have "p = x' * y'"

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

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

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

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

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

   419     by blast

   420 qed (auto simp: normalize_mult mult_dvd_mono)

   421

   422

   423 subclass factorial_semiring

   424 proof (standard, rule factorial_semiring_altI_aux)

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

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

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

   428     case (less x)

   429     show ?case

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

   431       case False

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

   433       proof

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

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

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

   437         proof (elim disjE)

   438           assume "is_unit p"

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

   440           with p show ?thesis by simp

   441         next

   442           assume "x dvd p"

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

   444           with p show ?thesis by simp

   445         qed

   446       qed

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

   448       ultimately show ?thesis by (rule finite_subset)

   449

   450     next

   451       case True

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

   453       define z where "z = x div y"

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

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

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

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

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

   459         by (subst x) (rule normalized_factors_product)

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

   461         by (subst dvd_times_left_cancel_iff dvd_times_right_cancel_iff; fact)+

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

   463         by (intro finite_imageI finite_cartesian_product less dvd_proper_imp_size_less)

   464            (auto simp: x)

   465       finally show ?thesis .

   466     qed

   467   qed

   468 next

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

   470     by standard (rule lcm_gcd_eucl_facts; assumption)+

   471   fix p assume p: "irreducible p"

   472   thus "prime_elem p" by (rule irreducible_imp_prime_elem_gcd)

   473 qed

   474

   475 lemma gcd_eucl_eq_gcd_factorial: "gcd_eucl = gcd_factorial"

   476   by (intro ext gcd_euclI gcd_lcm_factorial)

   477

   478 lemma lcm_eucl_eq_lcm_factorial: "lcm_eucl = lcm_factorial"

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

   480

   481 lemma Gcd_eucl_eq_Gcd_factorial: "Gcd_eucl = Gcd_factorial"

   482   by (intro ext Gcd_euclI gcd_lcm_factorial)

   483

   484 lemma Lcm_eucl_eq_Lcm_factorial: "Lcm_eucl = Lcm_factorial"

   485   by (intro ext Lcm_euclI gcd_lcm_factorial)

   486

   487 lemmas eucl_eq_factorial =

   488   gcd_eucl_eq_gcd_factorial lcm_eucl_eq_lcm_factorial

   489   Gcd_eucl_eq_Gcd_factorial Lcm_eucl_eq_Lcm_factorial

   490

   491 end

   492

   493 class euclidean_ring = euclidean_semiring + idom

   494 begin

   495

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

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

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

   499      else let q = r' div r

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

   501 by auto

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

   503

   504 declare euclid_ext_aux.simps [simp del]

   505

   506 lemma euclid_ext_aux_correct:

   507   assumes "gcd_eucl r' r = gcd_eucl x y"

   508   assumes "s' * x + t' * y = r'"

   509   assumes "s * x + t * y = r"

   510   shows   "case euclid_ext_aux r' r s' s t' t of (a,b,c) \<Rightarrow>

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

   512 using assms

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

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

   515   show ?case

   516   proof (cases "r = 0")

   517     case True

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

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

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

   521     also have "?P \<dots>"

   522     proof safe

   523       have "s' div unit_factor r' * x + t' div unit_factor r' * y =

   524                 (s' * x + t' * y) div unit_factor r'"

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

   526       also have "s' * x + t' * y = r'" by fact

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

   528       finally show "s' div unit_factor r' * x + t' div unit_factor r' * y = normalize r'" .

   529     next

   530       from "1.prems" True show "normalize r' = gcd_eucl x y" by (simp add: gcd_eucl_0)

   531     qed

   532     finally show ?thesis .

   533   next

   534     case False

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

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

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

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

   539     proof (intro "1.IH")

   540       have "(s' - r' div r * s) * x + (t' - r' div r * t) * y =

   541               (s' * x + t' * y) - r' div r * (s * x + t * y)" by (simp add: algebra_simps)

   542       also have "s' * x + t' * y = r'" by fact

   543       also have "s * x + t * y = r" by fact

   544       also have "r' - r' div r * r = r' mod r" using div_mod_equality[of r' r]

   545         by (simp add: algebra_simps)

   546       finally show "(s' - r' div r * s) * x + (t' - r' div r * t) * y = r' mod r" .

   547     qed (auto simp: gcd_eucl_non_0 algebra_simps div_mod_equality')

   548     finally show ?thesis .

   549   qed

   550 qed

   551

   552 definition euclid_ext where

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

   554

   555 lemma euclid_ext_0:

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

   557   by (simp add: euclid_ext_def euclid_ext_aux.simps)

   558

   559 lemma euclid_ext_left_0:

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

   561   by (simp add: euclid_ext_def euclid_ext_aux.simps)

   562

   563 lemma euclid_ext_correct':

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

   565   unfolding euclid_ext_def by (rule euclid_ext_aux_correct) simp_all

   566

   567 lemma euclid_ext_gcd_eucl:

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

   569   using euclid_ext_correct'[of x y] by (simp add: case_prod_unfold)

   570

   571 definition euclid_ext' where

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

   573

   574 lemma euclid_ext'_correct':

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

   576   using euclid_ext_correct'[of x y] by (simp add: case_prod_unfold euclid_ext'_def)

   577

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

   579   by (simp add: euclid_ext'_def euclid_ext_0)

   580

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

   582   by (simp add: euclid_ext'_def euclid_ext_left_0)

   583

   584 end

   585

   586 class euclidean_semiring_gcd = euclidean_semiring + gcd + Gcd +

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

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

   589 begin

   590

   591 subclass semiring_gcd

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

   593

   594 subclass semiring_Gcd

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

   596

   597 subclass factorial_semiring_gcd

   598 proof

   599   fix a b

   600   show "gcd a b = gcd_factorial a b"

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

   602   thus "lcm a b = lcm_factorial a b"

   603     by (simp add: lcm_factorial_gcd_factorial lcm_gcd)

   604 next

   605   fix A

   606   show "Gcd A = Gcd_factorial A"

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

   608   show "Lcm A = Lcm_factorial A"

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

   610 qed

   611

   612 lemma gcd_non_0:

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

   614   unfolding gcd_gcd_eucl by (fact gcd_eucl_non_0)

   615

   616 lemmas gcd_0 = gcd_0_right

   617 lemmas dvd_gcd_iff = gcd_greatest_iff

   618 lemmas gcd_greatest_iff = dvd_gcd_iff

   619

   620 lemma gcd_mod1 [simp]:

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

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

   623

   624 lemma gcd_mod2 [simp]:

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

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

   627

   628 lemma euclidean_size_gcd_le1 [simp]:

   629   assumes "a \<noteq> 0"

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

   631 proof -

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

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

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

   635 qed

   636

   637 lemma euclidean_size_gcd_le2 [simp]:

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

   639   by (subst gcd.commute, rule euclidean_size_gcd_le1)

   640

   641 lemma euclidean_size_gcd_less1:

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

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

   644 proof (rule ccontr)

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

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

   647     by (intro le_antisym, simp_all)

   648   have "a dvd gcd a b"

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

   650   hence "a dvd b" using dvd_gcdD2 by blast

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

   652 qed

   653

   654 lemma euclidean_size_gcd_less2:

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

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

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

   658

   659 lemma euclidean_size_lcm_le1:

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

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

   662 proof -

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

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

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

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

   667 qed

   668

   669 lemma euclidean_size_lcm_le2:

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

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

   672

   673 lemma euclidean_size_lcm_less1:

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

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

   676 proof (rule ccontr)

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

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

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

   680     by (intro le_antisym, simp, intro euclidean_size_lcm_le1)

   681   with assms have "lcm a b dvd a"

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

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

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

   685 qed

   686

   687 lemma euclidean_size_lcm_less2:

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

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

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

   691

   692 lemma Lcm_eucl_set [code]:

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

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

   695

   696 lemma Gcd_eucl_set [code]:

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

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

   699

   700 end

   701

   702

   703 text \<open>

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

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

   706 \<close>

   707

   708 class euclidean_ring_gcd = euclidean_semiring_gcd + idom

   709 begin

   710

   711 subclass euclidean_ring ..

   712 subclass ring_gcd ..

   713 subclass factorial_ring_gcd ..

   714

   715 lemma euclid_ext_gcd [simp]:

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

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

   718

   719 lemma euclid_ext_gcd' [simp]:

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

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

   722

   723 lemma euclid_ext_correct:

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

   725   using euclid_ext_correct'[of x y]

   726   by (simp add: gcd_gcd_eucl case_prod_unfold)

   727

   728 lemma euclid_ext'_correct:

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

   730   using euclid_ext_correct'[of a b]

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

   732

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

   734   using euclid_ext'_correct by blast

   735

   736 end

   737

   738

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

   740

   741 instantiation nat :: euclidean_semiring

   742 begin

   743

   744 definition [simp]:

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

   746

   747 instance by standard simp_all

   748

   749 end

   750

   751

   752 instantiation int :: euclidean_ring

   753 begin

   754

   755 definition [simp]:

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

   757

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

   759

   760 end

   761

   762 instance nat :: euclidean_semiring_gcd

   763 proof

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

   765     by (simp_all add: eq_gcd_euclI eq_Lcm_euclI)

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

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

   768 qed

   769

   770 instance int :: euclidean_ring_gcd

   771 proof

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

   773     by (simp_all add: eq_gcd_euclI eq_Lcm_euclI)

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

   775     by (intro ext, simp add: lcm_eucl_def lcm_altdef_int

   776           semiring_Gcd_class.Gcd_Lcm Gcd_eucl_def abs_mult)+

   777 qed

   778

   779 end