src/HOL/Number_Theory/Euclidean_Algorithm.thy
 author haftmann Sat Dec 17 15:22:14 2016 +0100 (2016-12-17) changeset 64592 7759f1766189 parent 64243 aee949f6642d child 64784 5cb5e7ecb284 permissions -rw-r--r--
more fine-grained type class hierarchy for div and mod
     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 = semidom_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 euclidean_size_normalize [simp]:

    30   "euclidean_size (normalize a) = euclidean_size a"

    31 proof (cases "a = 0")

    32   case True

    33   then show ?thesis

    34     by simp

    35 next

    36   case [simp]: False

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

    38     by (rule size_mult_mono) simp

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

    40     by (rule size_mult_mono) simp

    41   ultimately show ?thesis

    42     by simp

    43 qed

    44

    45 lemma euclidean_division:

    46   fixes a :: 'a and b :: 'a

    47   assumes "b \<noteq> 0"

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

    49     and "euclidean_size t < euclidean_size b"

    50 proof -

    51   from div_mult_mod_eq [of a b]

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

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

    54 qed

    55

    56 lemma dvd_euclidean_size_eq_imp_dvd:

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

    58   shows "a dvd b"

    59 proof (rule ccontr)

    60   assume "\<not> a dvd b"

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

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

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

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

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

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

    67       using size_mult_mono by force

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

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

    70       using mod_size_less by blast

    71   ultimately show False using size_eq by simp

    72 qed

    73

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

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

    76

    77 lemma euclidean_size_times_unit:

    78   assumes "is_unit a"

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

    80 proof (rule antisym)

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

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

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

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

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

    86     by (rule size_mult_mono')

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

    88     by (simp add: algebra_simps unit_div_mult_swap)

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

    90 qed

    91

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

    93   using euclidean_size_times_unit[of a 1] by simp

    94

    95 lemma unit_iff_euclidean_size:

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

    97 proof safe

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

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

   100 qed (auto intro: euclidean_size_unit)

   101

   102 lemma euclidean_size_times_nonunit:

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

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

   105 proof (rule ccontr)

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

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

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

   109   have "a * b dvd b"

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

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

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

   113   with assms(3) show False by contradiction

   114 qed

   115

   116 lemma dvd_imp_size_le:

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

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

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

   120

   121 lemma dvd_proper_imp_size_less:

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

   123   shows   "euclidean_size a < euclidean_size b"

   124 proof -

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

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

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

   128   with z assms show ?thesis

   129     by (auto intro!: euclidean_size_times_nonunit simp: )

   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: "a \<in> A \<Longrightarrow> Gcd_eucl A dvd a"

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

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

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

   441      else let q = r' div r

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

   443 by auto

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

   445

   446 declare euclid_ext_aux.simps [simp del]

   447

   448 lemma euclid_ext_aux_correct:

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

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

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

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

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

   454 using assms

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

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

   457   show ?case

   458   proof (cases "r = 0")

   459     case True

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

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

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

   463     also have "?P \<dots>"

   464     proof safe

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

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

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

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

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

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

   471     next

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

   473     qed

   474     finally show ?thesis .

   475   next

   476     case False

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

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

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

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

   481     proof (intro "1.IH")

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

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

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

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

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

   487         by (simp add: algebra_simps)

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

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

   490     finally show ?thesis .

   491   qed

   492 qed

   493

   494 definition euclid_ext where

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

   496

   497 lemma euclid_ext_0:

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

   499   by (simp add: euclid_ext_def euclid_ext_aux.simps)

   500

   501 lemma euclid_ext_left_0:

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

   503   by (simp add: euclid_ext_def euclid_ext_aux.simps)

   504

   505 lemma euclid_ext_correct':

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

   507   unfolding euclid_ext_def by (rule euclid_ext_aux_correct) simp_all

   508

   509 lemma euclid_ext_gcd_eucl:

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

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

   512

   513 definition euclid_ext' where

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

   515

   516 lemma euclid_ext'_correct':

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

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

   519

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

   521   by (simp add: euclid_ext'_def euclid_ext_0)

   522

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

   524   by (simp add: euclid_ext'_def euclid_ext_left_0)

   525

   526 end

   527

   528 class euclidean_semiring_gcd = euclidean_semiring + gcd + Gcd +

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

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

   531 begin

   532

   533 subclass semiring_gcd

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

   535

   536 subclass semiring_Gcd

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

   538

   539 subclass factorial_semiring_gcd

   540 proof

   541   fix a b

   542   show "gcd a b = gcd_factorial a b"

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

   544   thus "lcm a b = lcm_factorial a b"

   545     by (simp add: lcm_factorial_gcd_factorial lcm_gcd)

   546 next

   547   fix A

   548   show "Gcd A = Gcd_factorial A"

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

   550   show "Lcm A = Lcm_factorial A"

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

   552 qed

   553

   554 lemma gcd_non_0:

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

   556   unfolding gcd_gcd_eucl by (fact gcd_eucl_non_0)

   557

   558 lemmas gcd_0 = gcd_0_right

   559 lemmas dvd_gcd_iff = gcd_greatest_iff

   560 lemmas gcd_greatest_iff = dvd_gcd_iff

   561

   562 lemma gcd_mod1 [simp]:

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

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

   565

   566 lemma gcd_mod2 [simp]:

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

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

   569

   570 lemma euclidean_size_gcd_le1 [simp]:

   571   assumes "a \<noteq> 0"

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

   573 proof -

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

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

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

   577 qed

   578

   579 lemma euclidean_size_gcd_le2 [simp]:

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

   581   by (subst gcd.commute, rule euclidean_size_gcd_le1)

   582

   583 lemma euclidean_size_gcd_less1:

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

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

   586 proof (rule ccontr)

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

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

   589     by (intro le_antisym, simp_all)

   590   have "a dvd gcd a b"

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

   592   hence "a dvd b" using dvd_gcdD2 by blast

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

   594 qed

   595

   596 lemma euclidean_size_gcd_less2:

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

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

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

   600

   601 lemma euclidean_size_lcm_le1:

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

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

   604 proof -

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

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

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

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

   609 qed

   610

   611 lemma euclidean_size_lcm_le2:

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

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

   614

   615 lemma euclidean_size_lcm_less1:

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

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

   618 proof (rule ccontr)

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

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

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

   622     by (intro le_antisym, simp, intro euclidean_size_lcm_le1)

   623   with assms have "lcm a b dvd a"

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

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

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

   627 qed

   628

   629 lemma euclidean_size_lcm_less2:

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

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

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

   633

   634 lemma Lcm_eucl_set [code]:

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

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

   637

   638 lemma Gcd_eucl_set [code]:

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

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

   641

   642 end

   643

   644

   645 text \<open>

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

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

   648 \<close>

   649

   650 class euclidean_ring_gcd = euclidean_semiring_gcd + idom

   651 begin

   652

   653 subclass euclidean_ring ..

   654 subclass ring_gcd ..

   655 subclass factorial_ring_gcd ..

   656

   657 lemma euclid_ext_gcd [simp]:

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

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

   660

   661 lemma euclid_ext_gcd' [simp]:

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

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

   664

   665 lemma euclid_ext_correct:

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

   667   using euclid_ext_correct'[of a b]

   668   by (simp add: gcd_gcd_eucl case_prod_unfold)

   669

   670 lemma euclid_ext'_correct:

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

   672   using euclid_ext_correct'[of a b]

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

   674

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

   676   using euclid_ext'_correct by blast

   677

   678 end

   679

   680

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

   682

   683 instantiation nat :: euclidean_semiring

   684 begin

   685

   686 definition [simp]:

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

   688

   689 instance by standard simp_all

   690

   691 end

   692

   693

   694 instantiation int :: euclidean_ring

   695 begin

   696

   697 definition [simp]:

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

   699

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

   701

   702 end

   703

   704 instance nat :: euclidean_semiring_gcd

   705 proof

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

   707     by (simp_all add: eq_gcd_euclI eq_Lcm_euclI)

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

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

   710 qed

   711

   712 instance int :: euclidean_ring_gcd

   713 proof

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

   715     by (simp_all add: eq_gcd_euclI eq_Lcm_euclI)

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

   717     by (intro ext, simp add: lcm_eucl_def lcm_altdef_int

   718           semiring_Gcd_class.Gcd_Lcm Gcd_eucl_def abs_mult)+

   719 qed

   720

   721 end