src/HOL/Number_Theory/Euclidean_Algorithm.thy
 author eberlm Mon Aug 08 17:47:51 2016 +0200 (2016-08-08) changeset 63633 2accfb71e33b parent 63498 a3fe3250d05d child 63924 f91766530e13 permissions -rw-r--r--
is_prime -> prime
     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_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 lemma size_mult_mono': "b \<noteq> 0 \<Longrightarrow> euclidean_size a \<le> euclidean_size (b * a)"

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

    59

    60 lemma euclidean_size_times_unit:

    61   assumes "is_unit a"

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

    63 proof (rule antisym)

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

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

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

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

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

    69     by (rule size_mult_mono')

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

    71     by (simp add: algebra_simps unit_div_mult_swap)

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

    73 qed

    74

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

    76   using euclidean_size_times_unit[of x 1] by simp

    77

    78 lemma unit_iff_euclidean_size:

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

    80 proof safe

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

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

    83 qed (auto intro: euclidean_size_unit)

    84

    85 lemma euclidean_size_times_nonunit:

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

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

    88 proof (rule ccontr)

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

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

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

    92   have "a * b dvd b"

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

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

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

    96   with assms(3) show False by contradiction

    97 qed

    98

    99 lemma dvd_imp_size_le:

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

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

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

   103

   104 lemma dvd_proper_imp_size_less:

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

   106   shows   "euclidean_size x < euclidean_size y"

   107 proof -

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

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

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

   111   with z assms show ?thesis

   112     by (auto intro!: euclidean_size_times_nonunit simp: )

   113 qed

   114

   115 lemma irreducible_normalized_divisors:

   116   assumes "irreducible x" "y dvd x" "normalize y = y"

   117   shows   "y = 1 \<or> y = normalize x"

   118 proof -

   119   from assms have "is_unit y \<or> x dvd y" by (auto simp: irreducible_altdef)

   120   thus ?thesis

   121   proof (elim disjE)

   122     assume "is_unit y"

   123     hence "normalize y = 1" by (simp add: is_unit_normalize)

   124     with assms show ?thesis by simp

   125   next

   126     assume "x dvd y"

   127     with \<open>y dvd x\<close> have "normalize y = normalize x" by (rule associatedI)

   128     with assms show ?thesis by simp

   129   qed

   130 qed

   131

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

   133 where

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

   135   by pat_completeness simp

   136 termination

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

   138

   139 declare gcd_eucl.simps [simp del]

   140

   141 lemma gcd_eucl_induct [case_names zero mod]:

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

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

   144   shows "P a b"

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

   146   case ("1" a b)

   147   show ?case

   148   proof (cases "b = 0")

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

   150   next

   151     case False

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

   153       by (rule "1.hyps")

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

   155       by (blast intro: H2)

   156   qed

   157 qed

   158

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

   160 where

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

   162

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

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

   165   for infinite sets as well\<close>

   166 where

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

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

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

   170        in normalize l

   171       else 0)"

   172

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

   174 where

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

   176

   177 declare Lcm_eucl_def Gcd_eucl_def [code del]

   178

   179 lemma gcd_eucl_0:

   180   "gcd_eucl a 0 = normalize a"

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

   182

   183 lemma gcd_eucl_0_left:

   184   "gcd_eucl 0 a = normalize a"

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

   186

   187 lemma gcd_eucl_non_0:

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

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

   190

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

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

   193   by (induct a b rule: gcd_eucl_induct)

   194      (simp_all add: gcd_eucl_0 gcd_eucl_non_0 dvd_mod_iff)

   195

   196 lemma normalize_gcd_eucl [simp]:

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

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

   199

   200 lemma gcd_eucl_greatest:

   201   fixes k a b :: 'a

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

   203 proof (induct a b rule: gcd_eucl_induct)

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

   205 next

   206   case (mod a b)

   207   then show ?case

   208     by (simp add: gcd_eucl_non_0 dvd_mod_iff)

   209 qed

   210

   211 lemma gcd_euclI:

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

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

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

   215   shows   "gcd_eucl a b = d"

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

   217

   218 lemma eq_gcd_euclI:

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

   220   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"

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

   222   shows   "gcd = gcd_eucl"

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

   224

   225 lemma gcd_eucl_zero [simp]:

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

   227   by (metis dvd_0_left dvd_refl gcd_eucl_dvd1 gcd_eucl_dvd2 gcd_eucl_greatest)+

   228

   229

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

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

   232   and unit_factor_Lcm_eucl [simp]:

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

   234 proof -

   235   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>

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

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

   238     case False

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

   240     with False show ?thesis by auto

   241   next

   242     case True

   243     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

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

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

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

   247       apply (subst n_def)

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

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

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

   251       done

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

   253       unfolding l_def by simp_all

   254     {

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

   256       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)

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

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

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

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

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

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

   263       proof -

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

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

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

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

   268           by (rule size_mult_mono)

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

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

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

   272       qed

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

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

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

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

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

   278     }

   279

   280     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>

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

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

   283         unit_factor (normalize l) =

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

   285       by (auto simp: unit_simps)

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

   287       by (simp add: Lcm_eucl_def Let_def n_def l_def)

   288     finally show ?thesis .

   289   qed

   290   note A = this

   291

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

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

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

   295 qed

   296

   297 lemma normalize_Lcm_eucl [simp]:

   298   "normalize (Lcm_eucl A) = Lcm_eucl A"

   299 proof (cases "Lcm_eucl A = 0")

   300   case True then show ?thesis by simp

   301 next

   302   case False

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

   304     by (fact unit_factor_mult_normalize)

   305   with False show ?thesis by simp

   306 qed

   307

   308 lemma eq_Lcm_euclI:

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

   310   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"

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

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

   313

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

   315   unfolding Gcd_eucl_def by (auto intro: Lcm_eucl_least)

   316

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

   318   unfolding Gcd_eucl_def by auto

   319

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

   321   by (simp add: Gcd_eucl_def)

   322

   323 lemma Lcm_euclI:

   324   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"

   325   shows   "Lcm_eucl A = d"

   326 proof -

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

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

   329   thus ?thesis by (simp add: assms)

   330 qed

   331

   332 lemma Gcd_euclI:

   333   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"

   334   shows   "Gcd_eucl A = d"

   335 proof -

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

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

   338   thus ?thesis by (simp add: assms)

   339 qed

   340

   341 lemmas lcm_gcd_eucl_facts =

   342   gcd_eucl_dvd1 gcd_eucl_dvd2 gcd_eucl_greatest normalize_gcd_eucl lcm_eucl_def

   343   Gcd_eucl_def Gcd_eucl_dvd Gcd_eucl_greatest normalize_Gcd_eucl

   344   dvd_Lcm_eucl Lcm_eucl_least normalize_Lcm_eucl

   345

   346 lemma normalized_factors_product:

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

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

   349 proof safe

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

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

   352     by standard (rule lcm_gcd_eucl_facts; assumption)+

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

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

   355   have "p = x' * y'"

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

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

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

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

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

   361     by blast

   362 qed (auto simp: normalize_mult mult_dvd_mono)

   363

   364

   365 subclass factorial_semiring

   366 proof (standard, rule factorial_semiring_altI_aux)

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

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

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

   370     case (less x)

   371     show ?case

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

   373       case False

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

   375       proof

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

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

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

   379         proof (elim disjE)

   380           assume "is_unit p"

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

   382           with p show ?thesis by simp

   383         next

   384           assume "x dvd p"

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

   386           with p show ?thesis by simp

   387         qed

   388       qed

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

   390       ultimately show ?thesis by (rule finite_subset)

   391

   392     next

   393       case True

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

   395       define z where "z = x div y"

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

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

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

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

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

   401         by (subst x) (rule normalized_factors_product)

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

   403         by (subst dvd_times_left_cancel_iff dvd_times_right_cancel_iff; fact)+

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

   405         by (intro finite_imageI finite_cartesian_product less dvd_proper_imp_size_less)

   406            (auto simp: x)

   407       finally show ?thesis .

   408     qed

   409   qed

   410 next

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

   412     by standard (rule lcm_gcd_eucl_facts; assumption)+

   413   fix p assume p: "irreducible p"

   414   thus "prime_elem p" by (rule irreducible_imp_prime_elem_gcd)

   415 qed

   416

   417 lemma gcd_eucl_eq_gcd_factorial: "gcd_eucl = gcd_factorial"

   418   by (intro ext gcd_euclI gcd_lcm_factorial)

   419

   420 lemma lcm_eucl_eq_lcm_factorial: "lcm_eucl = lcm_factorial"

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

   422

   423 lemma Gcd_eucl_eq_Gcd_factorial: "Gcd_eucl = Gcd_factorial"

   424   by (intro ext Gcd_euclI gcd_lcm_factorial)

   425

   426 lemma Lcm_eucl_eq_Lcm_factorial: "Lcm_eucl = Lcm_factorial"

   427   by (intro ext Lcm_euclI gcd_lcm_factorial)

   428

   429 lemmas eucl_eq_factorial =

   430   gcd_eucl_eq_gcd_factorial lcm_eucl_eq_lcm_factorial

   431   Gcd_eucl_eq_Gcd_factorial Lcm_eucl_eq_Lcm_factorial

   432

   433 end

   434

   435 class euclidean_ring = euclidean_semiring + idom

   436 begin

   437

   438 subclass ring_div ..

   439

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

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

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

   443      else let q = r' div r

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

   445 by auto

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

   447

   448 declare euclid_ext_aux.simps [simp del]

   449

   450 lemma euclid_ext_aux_correct:

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

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

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

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

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

   456 using assms

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

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

   459   show ?case

   460   proof (cases "r = 0")

   461     case True

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

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

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

   465     also have "?P \<dots>"

   466     proof safe

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

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

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

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

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

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

   473     next

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

   475     qed

   476     finally show ?thesis .

   477   next

   478     case False

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

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

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

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

   483     proof (intro "1.IH")

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

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

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

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

   488       also have "r' - r' div r * r = r' mod r" using mod_div_equality[of r' r]

   489         by (simp add: algebra_simps)

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

   491     qed (auto simp: gcd_eucl_non_0 algebra_simps div_mod_equality')

   492     finally show ?thesis .

   493   qed

   494 qed

   495

   496 definition euclid_ext where

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

   498

   499 lemma euclid_ext_0:

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

   501   by (simp add: euclid_ext_def euclid_ext_aux.simps)

   502

   503 lemma euclid_ext_left_0:

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

   505   by (simp add: euclid_ext_def euclid_ext_aux.simps)

   506

   507 lemma euclid_ext_correct':

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

   509   unfolding euclid_ext_def by (rule euclid_ext_aux_correct) simp_all

   510

   511 lemma euclid_ext_gcd_eucl:

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

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

   514

   515 definition euclid_ext' where

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

   517

   518 lemma euclid_ext'_correct':

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

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

   521

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

   523   by (simp add: euclid_ext'_def euclid_ext_0)

   524

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

   526   by (simp add: euclid_ext'_def euclid_ext_left_0)

   527

   528 end

   529

   530 class euclidean_semiring_gcd = euclidean_semiring + gcd + Gcd +

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

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

   533 begin

   534

   535 subclass semiring_gcd

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

   537

   538 subclass semiring_Gcd

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

   540

   541 subclass factorial_semiring_gcd

   542 proof

   543   fix a b

   544   show "gcd a b = gcd_factorial a b"

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

   546   thus "lcm a b = lcm_factorial a b"

   547     by (simp add: lcm_factorial_gcd_factorial lcm_gcd)

   548 next

   549   fix A

   550   show "Gcd A = Gcd_factorial A"

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

   552   show "Lcm A = Lcm_factorial A"

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

   554 qed

   555

   556 lemma gcd_non_0:

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

   558   unfolding gcd_gcd_eucl by (fact gcd_eucl_non_0)

   559

   560 lemmas gcd_0 = gcd_0_right

   561 lemmas dvd_gcd_iff = gcd_greatest_iff

   562 lemmas gcd_greatest_iff = dvd_gcd_iff

   563

   564 lemma gcd_mod1 [simp]:

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

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

   567

   568 lemma gcd_mod2 [simp]:

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

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

   571

   572 lemma euclidean_size_gcd_le1 [simp]:

   573   assumes "a \<noteq> 0"

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

   575 proof -

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

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

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

   579 qed

   580

   581 lemma euclidean_size_gcd_le2 [simp]:

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

   583   by (subst gcd.commute, rule euclidean_size_gcd_le1)

   584

   585 lemma euclidean_size_gcd_less1:

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

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

   588 proof (rule ccontr)

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

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

   591     by (intro le_antisym, simp_all)

   592   have "a dvd gcd a b"

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

   594   hence "a dvd b" using dvd_gcdD2 by blast

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

   596 qed

   597

   598 lemma euclidean_size_gcd_less2:

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

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

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

   602

   603 lemma euclidean_size_lcm_le1:

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

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

   606 proof -

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

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

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

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

   611 qed

   612

   613 lemma euclidean_size_lcm_le2:

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

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

   616

   617 lemma euclidean_size_lcm_less1:

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

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

   620 proof (rule ccontr)

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

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

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

   624     by (intro le_antisym, simp, intro euclidean_size_lcm_le1)

   625   with assms have "lcm a b dvd a"

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

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

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

   629 qed

   630

   631 lemma euclidean_size_lcm_less2:

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

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

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

   635

   636 lemma Lcm_eucl_set [code]:

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

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

   639

   640 lemma Gcd_eucl_set [code]:

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

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

   643

   644 end

   645

   646

   647 text \<open>

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

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

   650 \<close>

   651

   652 class euclidean_ring_gcd = euclidean_semiring_gcd + idom

   653 begin

   654

   655 subclass euclidean_ring ..

   656 subclass ring_gcd ..

   657 subclass factorial_ring_gcd ..

   658

   659 lemma euclid_ext_gcd [simp]:

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

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

   662

   663 lemma euclid_ext_gcd' [simp]:

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

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

   666

   667 lemma euclid_ext_correct:

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

   669   using euclid_ext_correct'[of x y]

   670   by (simp add: gcd_gcd_eucl case_prod_unfold)

   671

   672 lemma euclid_ext'_correct:

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

   674   using euclid_ext_correct'[of a b]

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

   676

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

   678   using euclid_ext'_correct by blast

   679

   680 end

   681

   682

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

   684

   685 instantiation nat :: euclidean_semiring

   686 begin

   687

   688 definition [simp]:

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

   690

   691 instance by standard simp_all

   692

   693 end

   694

   695

   696 instantiation int :: euclidean_ring

   697 begin

   698

   699 definition [simp]:

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

   701

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

   703

   704 end

   705

   706 instance nat :: euclidean_semiring_gcd

   707 proof

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

   709     by (simp_all add: eq_gcd_euclI eq_Lcm_euclI)

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

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

   712 qed

   713

   714 instance int :: euclidean_ring_gcd

   715 proof

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

   717     by (simp_all add: eq_gcd_euclI eq_Lcm_euclI)

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

   719     by (intro ext, simp add: lcm_eucl_def lcm_altdef_int

   720           semiring_Gcd_class.Gcd_Lcm Gcd_eucl_def abs_mult)+

   721 qed

   722

   723 end