src/HOL/Number_Theory/Euclidean_Algorithm.thy
 author haftmann Wed Jul 08 14:01:34 2015 +0200 (2015-07-08) changeset 60685 cb21b7022b00 parent 60634 e3b6e516608b child 60686 ea5bc46c11e6 permissions -rw-r--r--
moved normalization and unit_factor into Main HOL corpus
     1 (* Author: Manuel Eberl *)

     2

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

     4

     5 theory Euclidean_Algorithm

     6 imports Main "~~/src/HOL/GCD" "~~/src/HOL/Library/Polynomial"

     7 begin

     8

     9 text \<open>

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

    11   implemented. It must provide:

    12   \begin{itemize}

    13   \item division with remainder

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

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

    16   \end{itemize}

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

    18   for any Euclidean semiring.

    19 \<close>

    20 class euclidean_semiring = semiring_div + normalization_semidom +

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

    22   assumes mod_size_less:

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

    24   assumes size_mult_mono:

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

    26 begin

    27

    28 lemma euclidean_division:

    29   fixes a :: 'a and b :: 'a

    30   assumes "b \<noteq> 0"

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

    32     and "euclidean_size t < euclidean_size b"

    33 proof -

    34   from div_mod_equality [of a b 0]

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

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

    37 qed

    38

    39 lemma dvd_euclidean_size_eq_imp_dvd:

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

    41   shows "a dvd b"

    42 proof (rule ccontr)

    43   assume "\<not> a dvd b"

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

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

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

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

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

    49       using size_mult_mono by force

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

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

    52       using mod_size_less by blast

    53   ultimately show False using size_eq by simp

    54 qed

    55

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

    57 where

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

    59   by pat_completeness simp

    60 termination

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

    62

    63 declare gcd_eucl.simps [simp del]

    64

    65 lemma gcd_eucl_induct [case_names zero mod]:

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

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

    68   shows "P a b"

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

    70   case ("1" a b)

    71   show ?case

    72   proof (cases "b = 0")

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

    74   next

    75     case False

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

    77       by (rule "1.hyps")

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

    79       by (blast intro: H2)

    80   qed

    81 qed

    82

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

    84 where

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

    86

    87 definition Lcm_eucl :: "'a set \<Rightarrow> 'a" -- \<open>

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

    89   for infinite sets as well\<close>

    90 where

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

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

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

    94        in normalize l

    95       else 0)"

    96

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

    98 where

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

   100

   101 lemma gcd_eucl_0:

   102   "gcd_eucl a 0 = normalize a"

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

   104

   105 lemma gcd_eucl_0_left:

   106   "gcd_eucl 0 a = normalize a"

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

   108

   109 lemma gcd_eucl_non_0:

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

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

   112

   113 end

   114

   115 class euclidean_ring = euclidean_semiring + idom

   116 begin

   117

   118 function euclid_ext :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a" where

   119   "euclid_ext a b =

   120      (if b = 0 then

   121         (1 div unit_factor a, 0, normalize a)

   122       else

   123         case euclid_ext b (a mod b) of

   124             (s, t, c) \<Rightarrow> (t, s - t * (a div b), c))"

   125   by pat_completeness simp

   126 termination

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

   128

   129 declare euclid_ext.simps [simp del]

   130

   131 lemma euclid_ext_0:

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

   133   by (simp add: euclid_ext.simps [of a 0])

   134

   135 lemma euclid_ext_left_0:

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

   137   by (simp add: euclid_ext_0 euclid_ext.simps [of 0 a])

   138

   139 lemma euclid_ext_non_0:

   140   "b \<noteq> 0 \<Longrightarrow> euclid_ext a b = (case euclid_ext b (a mod b) of

   141     (s, t, c) \<Rightarrow> (t, s - t * (a div b), c))"

   142   by (simp add: euclid_ext.simps [of a b] euclid_ext.simps [of b 0])

   143

   144 lemma euclid_ext_code [code]:

   145   "euclid_ext a b = (if b = 0 then (1 div unit_factor a, 0, normalize a)

   146     else let (s, t, c) = euclid_ext b (a mod b) in  (t, s - t * (a div b), c))"

   147   by (simp add: euclid_ext.simps [of a b] euclid_ext.simps [of b 0])

   148

   149 lemma euclid_ext_correct:

   150   "case euclid_ext a b of (s, t, c) \<Rightarrow> s * a + t * b = c"

   151 proof (induct a b rule: gcd_eucl_induct)

   152   case (zero a) then show ?case

   153     by (simp add: euclid_ext_0 ac_simps)

   154 next

   155   case (mod a b)

   156   obtain s t c where stc: "euclid_ext b (a mod b) = (s,t,c)"

   157     by (cases "euclid_ext b (a mod b)") blast

   158   with mod have "c = s * b + t * (a mod b)" by simp

   159   also have "... = t * ((a div b) * b + a mod b) + (s - t * (a div b)) * b"

   160     by (simp add: algebra_simps)

   161   also have "(a div b) * b + a mod b = a" using mod_div_equality .

   162   finally show ?case

   163     by (subst euclid_ext.simps) (simp add: stc mod ac_simps)

   164 qed

   165

   166 definition euclid_ext' :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a"

   167 where

   168   "euclid_ext' a b = (case euclid_ext a b of (s, t, _) \<Rightarrow> (s, t))"

   169

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

   171   by (simp add: euclid_ext'_def euclid_ext_0)

   172

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

   174   by (simp add: euclid_ext'_def euclid_ext_left_0)

   175

   176 lemma euclid_ext'_non_0: "b \<noteq> 0 \<Longrightarrow> euclid_ext' a b = (snd (euclid_ext' b (a mod b)),

   177   fst (euclid_ext' b (a mod b)) - snd (euclid_ext' b (a mod b)) * (a div b))"

   178   by (simp add: euclid_ext'_def euclid_ext_non_0 split_def)

   179

   180 end

   181

   182 class euclidean_semiring_gcd = euclidean_semiring + gcd + Gcd +

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

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

   185 begin

   186

   187 lemma gcd_0_left:

   188   "gcd 0 a = normalize a"

   189   unfolding gcd_gcd_eucl by (fact gcd_eucl_0_left)

   190

   191 lemma gcd_0:

   192   "gcd a 0 = normalize a"

   193   unfolding gcd_gcd_eucl by (fact gcd_eucl_0)

   194

   195 lemma gcd_non_0:

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

   197   unfolding gcd_gcd_eucl by (fact gcd_eucl_non_0)

   198

   199 lemma gcd_dvd1 [iff]: "gcd a b dvd a"

   200   and gcd_dvd2 [iff]: "gcd a b dvd b"

   201   by (induct a b rule: gcd_eucl_induct)

   202     (simp_all add: gcd_0 gcd_non_0 dvd_mod_iff)

   203

   204 lemma dvd_gcd_D1: "k dvd gcd m n \<Longrightarrow> k dvd m"

   205   by (rule dvd_trans, assumption, rule gcd_dvd1)

   206

   207 lemma dvd_gcd_D2: "k dvd gcd m n \<Longrightarrow> k dvd n"

   208   by (rule dvd_trans, assumption, rule gcd_dvd2)

   209

   210 lemma gcd_greatest:

   211   fixes k a b :: 'a

   212   shows "k dvd a \<Longrightarrow> k dvd b \<Longrightarrow> k dvd gcd a b"

   213 proof (induct a b rule: gcd_eucl_induct)

   214   case (zero a) from zero(1) show ?case by (rule dvd_trans) (simp add: gcd_0)

   215 next

   216   case (mod a b)

   217   then show ?case

   218     by (simp add: gcd_non_0 dvd_mod_iff)

   219 qed

   220

   221 lemma dvd_gcd_iff:

   222   "k dvd gcd a b \<longleftrightarrow> k dvd a \<and> k dvd b"

   223   by (blast intro!: gcd_greatest intro: dvd_trans)

   224

   225 lemmas gcd_greatest_iff = dvd_gcd_iff

   226

   227 lemma gcd_zero [simp]:

   228   "gcd a b = 0 \<longleftrightarrow> a = 0 \<and> b = 0"

   229   by (metis dvd_0_left dvd_refl gcd_dvd1 gcd_dvd2 gcd_greatest)+

   230

   231 lemma unit_factor_gcd [simp]:

   232   "unit_factor (gcd a b) = (if a = 0 \<and> b = 0 then 0 else 1)" (is "?f a b = ?g a b")

   233   by (induct a b rule: gcd_eucl_induct)

   234     (auto simp add: gcd_0 gcd_non_0)

   235

   236 lemma gcdI:

   237   assumes "c dvd a" and "c dvd b" and greatest: "\<And>d. d dvd a \<Longrightarrow> d dvd b \<Longrightarrow> d dvd c"

   238     and "unit_factor c = (if c = 0 then 0 else 1)"

   239   shows "c = gcd a b"

   240   by (rule associated_eqI) (auto simp: assms associated_def intro: gcd_greatest)

   241

   242 sublocale gcd!: abel_semigroup gcd

   243 proof

   244   fix a b c

   245   show "gcd (gcd a b) c = gcd a (gcd b c)"

   246   proof (rule gcdI)

   247     have "gcd (gcd a b) c dvd gcd a b" "gcd a b dvd a" by simp_all

   248     then show "gcd (gcd a b) c dvd a" by (rule dvd_trans)

   249     have "gcd (gcd a b) c dvd gcd a b" "gcd a b dvd b" by simp_all

   250     hence "gcd (gcd a b) c dvd b" by (rule dvd_trans)

   251     moreover have "gcd (gcd a b) c dvd c" by simp

   252     ultimately show "gcd (gcd a b) c dvd gcd b c"

   253       by (rule gcd_greatest)

   254     show "unit_factor (gcd (gcd a b) c) =  (if gcd (gcd a b) c = 0 then 0 else 1)"

   255       by auto

   256     fix l assume "l dvd a" and "l dvd gcd b c"

   257     with dvd_trans[OF _ gcd_dvd1] and dvd_trans[OF _ gcd_dvd2]

   258       have "l dvd b" and "l dvd c" by blast+

   259     with \<open>l dvd a\<close> show "l dvd gcd (gcd a b) c"

   260       by (intro gcd_greatest)

   261   qed

   262 next

   263   fix a b

   264   show "gcd a b = gcd b a"

   265     by (rule gcdI) (simp_all add: gcd_greatest)

   266 qed

   267

   268 lemma gcd_unique: "d dvd a \<and> d dvd b \<and>

   269     unit_factor d = (if d = 0 then 0 else 1) \<and>

   270     (\<forall>e. e dvd a \<and> e dvd b \<longrightarrow> e dvd d) \<longleftrightarrow> d = gcd a b"

   271   by (rule, auto intro: gcdI simp: gcd_greatest)

   272

   273 lemma gcd_dvd_prod: "gcd a b dvd k * b"

   274   using mult_dvd_mono [of 1] by auto

   275

   276 lemma gcd_1_left [simp]: "gcd 1 a = 1"

   277   by (rule sym, rule gcdI, simp_all)

   278

   279 lemma gcd_1 [simp]: "gcd a 1 = 1"

   280   by (rule sym, rule gcdI, simp_all)

   281

   282 lemma gcd_proj2_if_dvd:

   283   "b dvd a \<Longrightarrow> gcd a b = normalize b"

   284   by (cases "b = 0", simp_all add: dvd_eq_mod_eq_0 gcd_non_0 gcd_0)

   285

   286 lemma gcd_proj1_if_dvd:

   287   "a dvd b \<Longrightarrow> gcd a b = normalize a"

   288   by (subst gcd.commute, simp add: gcd_proj2_if_dvd)

   289

   290 lemma gcd_proj1_iff: "gcd m n = normalize m \<longleftrightarrow> m dvd n"

   291 proof

   292   assume A: "gcd m n = normalize m"

   293   show "m dvd n"

   294   proof (cases "m = 0")

   295     assume [simp]: "m \<noteq> 0"

   296     from A have B: "m = gcd m n * unit_factor m"

   297       by (simp add: unit_eq_div2)

   298     show ?thesis by (subst B, simp add: mult_unit_dvd_iff)

   299   qed (insert A, simp)

   300 next

   301   assume "m dvd n"

   302   then show "gcd m n = normalize m" by (rule gcd_proj1_if_dvd)

   303 qed

   304

   305 lemma gcd_proj2_iff: "gcd m n = normalize n \<longleftrightarrow> n dvd m"

   306   using gcd_proj1_iff [of n m] by (simp add: ac_simps)

   307

   308 lemma gcd_mod1 [simp]:

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

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

   311

   312 lemma gcd_mod2 [simp]:

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

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

   315

   316 lemma gcd_mult_distrib':

   317   "normalize c * gcd a b = gcd (c * a) (c * b)"

   318 proof (cases "c = 0")

   319   case True then show ?thesis by (simp_all add: gcd_0)

   320 next

   321   case False then have [simp]: "is_unit (unit_factor c)" by simp

   322   show ?thesis

   323   proof (induct a b rule: gcd_eucl_induct)

   324     case (zero a) show ?case

   325     proof (cases "a = 0")

   326       case True then show ?thesis by (simp add: gcd_0)

   327     next

   328       case False

   329       then show ?thesis by (simp add: gcd_0 normalize_mult)

   330     qed

   331     case (mod a b)

   332     then show ?case by (simp add: mult_mod_right gcd.commute)

   333   qed

   334 qed

   335

   336 lemma gcd_mult_distrib:

   337   "k * gcd a b = gcd (k * a) (k * b) * unit_factor k"

   338 proof-

   339   have "normalize k * gcd a b = gcd (k * a) (k * b)"

   340     by (simp add: gcd_mult_distrib')

   341   then have "normalize k * gcd a b * unit_factor k = gcd (k * a) (k * b) * unit_factor k"

   342     by simp

   343   then have "normalize k * unit_factor k * gcd a b  = gcd (k * a) (k * b) * unit_factor k"

   344     by (simp only: ac_simps)

   345   then show ?thesis

   346     by simp

   347 qed

   348

   349 lemma euclidean_size_gcd_le1 [simp]:

   350   assumes "a \<noteq> 0"

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

   352 proof -

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

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

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

   356 qed

   357

   358 lemma euclidean_size_gcd_le2 [simp]:

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

   360   by (subst gcd.commute, rule euclidean_size_gcd_le1)

   361

   362 lemma euclidean_size_gcd_less1:

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

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

   365 proof (rule ccontr)

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

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

   368     by (intro le_antisym, simp_all)

   369   with assms have "a dvd gcd a b" by (auto intro: dvd_euclidean_size_eq_imp_dvd)

   370   hence "a dvd b" using dvd_gcd_D2 by blast

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

   372 qed

   373

   374 lemma euclidean_size_gcd_less2:

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

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

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

   378

   379 lemma gcd_mult_unit1: "is_unit a \<Longrightarrow> gcd (b * a) c = gcd b c"

   380   apply (rule gcdI)

   381   apply (rule dvd_trans, rule gcd_dvd1, simp add: unit_simps)

   382   apply (rule gcd_dvd2)

   383   apply (rule gcd_greatest, simp add: unit_simps, assumption)

   384   apply (subst unit_factor_gcd, simp add: gcd_0)

   385   done

   386

   387 lemma gcd_mult_unit2: "is_unit a \<Longrightarrow> gcd b (c * a) = gcd b c"

   388   by (subst gcd.commute, subst gcd_mult_unit1, assumption, rule gcd.commute)

   389

   390 lemma gcd_div_unit1: "is_unit a \<Longrightarrow> gcd (b div a) c = gcd b c"

   391   by (erule is_unitE [of _ b]) (simp add: gcd_mult_unit1)

   392

   393 lemma gcd_div_unit2: "is_unit a \<Longrightarrow> gcd b (c div a) = gcd b c"

   394   by (erule is_unitE [of _ c]) (simp add: gcd_mult_unit2)

   395

   396 lemma normalize_gcd_left [simp]:

   397   "gcd (normalize a) b = gcd a b"

   398 proof (cases "a = 0")

   399   case True then show ?thesis

   400     by simp

   401 next

   402   case False then have "is_unit (unit_factor a)"

   403     by simp

   404   moreover have "normalize a = a div unit_factor a"

   405     by simp

   406   ultimately show ?thesis

   407     by (simp only: gcd_div_unit1)

   408 qed

   409

   410 lemma normalize_gcd_right [simp]:

   411   "gcd a (normalize b) = gcd a b"

   412   using normalize_gcd_left [of b a] by (simp add: ac_simps)

   413

   414 lemma gcd_idem: "gcd a a = normalize a"

   415   by (cases "a = 0") (simp add: gcd_0_left, rule sym, rule gcdI, simp_all)

   416

   417 lemma gcd_right_idem: "gcd (gcd a b) b = gcd a b"

   418   apply (rule gcdI)

   419   apply (simp add: ac_simps)

   420   apply (rule gcd_dvd2)

   421   apply (rule gcd_greatest, erule (1) gcd_greatest, assumption)

   422   apply simp

   423   done

   424

   425 lemma gcd_left_idem: "gcd a (gcd a b) = gcd a b"

   426   apply (rule gcdI)

   427   apply simp

   428   apply (rule dvd_trans, rule gcd_dvd2, rule gcd_dvd2)

   429   apply (rule gcd_greatest, assumption, erule gcd_greatest, assumption)

   430   apply simp

   431   done

   432

   433 lemma comp_fun_idem_gcd: "comp_fun_idem gcd"

   434 proof

   435   fix a b show "gcd a \<circ> gcd b = gcd b \<circ> gcd a"

   436     by (simp add: fun_eq_iff ac_simps)

   437 next

   438   fix a show "gcd a \<circ> gcd a = gcd a"

   439     by (simp add: fun_eq_iff gcd_left_idem)

   440 qed

   441

   442 lemma coprime_dvd_mult:

   443   assumes "gcd c b = 1" and "c dvd a * b"

   444   shows "c dvd a"

   445 proof -

   446   let ?nf = "unit_factor"

   447   from assms gcd_mult_distrib [of a c b]

   448     have A: "a = gcd (a * c) (a * b) * ?nf a" by simp

   449   from \<open>c dvd a * b\<close> show ?thesis by (subst A, simp_all add: gcd_greatest)

   450 qed

   451

   452 lemma coprime_dvd_mult_iff:

   453   "gcd c b = 1 \<Longrightarrow> (c dvd a * b) = (c dvd a)"

   454   by (rule, rule coprime_dvd_mult, simp_all)

   455

   456 lemma gcd_dvd_antisym:

   457   "gcd a b dvd gcd c d \<Longrightarrow> gcd c d dvd gcd a b \<Longrightarrow> gcd a b = gcd c d"

   458 proof (rule gcdI)

   459   assume A: "gcd a b dvd gcd c d" and B: "gcd c d dvd gcd a b"

   460   have "gcd c d dvd c" by simp

   461   with A show "gcd a b dvd c" by (rule dvd_trans)

   462   have "gcd c d dvd d" by simp

   463   with A show "gcd a b dvd d" by (rule dvd_trans)

   464   show "unit_factor (gcd a b) = (if gcd a b = 0 then 0 else 1)"

   465     by simp

   466   fix l assume "l dvd c" and "l dvd d"

   467   hence "l dvd gcd c d" by (rule gcd_greatest)

   468   from this and B show "l dvd gcd a b" by (rule dvd_trans)

   469 qed

   470

   471 lemma gcd_mult_cancel:

   472   assumes "gcd k n = 1"

   473   shows "gcd (k * m) n = gcd m n"

   474 proof (rule gcd_dvd_antisym)

   475   have "gcd (gcd (k * m) n) k = gcd (gcd k n) (k * m)" by (simp add: ac_simps)

   476   also note \<open>gcd k n = 1\<close>

   477   finally have "gcd (gcd (k * m) n) k = 1" by simp

   478   hence "gcd (k * m) n dvd m" by (rule coprime_dvd_mult, simp add: ac_simps)

   479   moreover have "gcd (k * m) n dvd n" by simp

   480   ultimately show "gcd (k * m) n dvd gcd m n" by (rule gcd_greatest)

   481   have "gcd m n dvd (k * m)" and "gcd m n dvd n" by simp_all

   482   then show "gcd m n dvd gcd (k * m) n" by (rule gcd_greatest)

   483 qed

   484

   485 lemma coprime_crossproduct:

   486   assumes [simp]: "gcd a d = 1" "gcd b c = 1"

   487   shows "associated (a * c) (b * d) \<longleftrightarrow> associated a b \<and> associated c d" (is "?lhs \<longleftrightarrow> ?rhs")

   488 proof

   489   assume ?rhs then show ?lhs unfolding associated_def by (fast intro: mult_dvd_mono)

   490 next

   491   assume ?lhs

   492   from \<open>?lhs\<close> have "a dvd b * d" unfolding associated_def by (metis dvd_mult_left)

   493   hence "a dvd b" by (simp add: coprime_dvd_mult_iff)

   494   moreover from \<open>?lhs\<close> have "b dvd a * c" unfolding associated_def by (metis dvd_mult_left)

   495   hence "b dvd a" by (simp add: coprime_dvd_mult_iff)

   496   moreover from \<open>?lhs\<close> have "c dvd d * b"

   497     unfolding associated_def by (auto dest: dvd_mult_right simp add: ac_simps)

   498   hence "c dvd d" by (simp add: coprime_dvd_mult_iff gcd.commute)

   499   moreover from \<open>?lhs\<close> have "d dvd c * a"

   500     unfolding associated_def by (auto dest: dvd_mult_right simp add: ac_simps)

   501   hence "d dvd c" by (simp add: coprime_dvd_mult_iff gcd.commute)

   502   ultimately show ?rhs unfolding associated_def by simp

   503 qed

   504

   505 lemma gcd_add1 [simp]:

   506   "gcd (m + n) n = gcd m n"

   507   by (cases "n = 0", simp_all add: gcd_non_0)

   508

   509 lemma gcd_add2 [simp]:

   510   "gcd m (m + n) = gcd m n"

   511   using gcd_add1 [of n m] by (simp add: ac_simps)

   512

   513 lemma gcd_add_mult:

   514   "gcd m (k * m + n) = gcd m n"

   515 proof -

   516   have "gcd m ((k * m + n) mod m) = gcd m (k * m + n)"

   517     by (fact gcd_mod2)

   518   then show ?thesis by simp

   519 qed

   520

   521 lemma coprimeI: "(\<And>l. \<lbrakk>l dvd a; l dvd b\<rbrakk> \<Longrightarrow> l dvd 1) \<Longrightarrow> gcd a b = 1"

   522   by (rule sym, rule gcdI, simp_all)

   523

   524 lemma coprime: "gcd a b = 1 \<longleftrightarrow> (\<forall>d. d dvd a \<and> d dvd b \<longleftrightarrow> is_unit d)"

   525   by (auto intro: coprimeI gcd_greatest dvd_gcd_D1 dvd_gcd_D2)

   526

   527 lemma div_gcd_coprime:

   528   assumes nz: "a \<noteq> 0 \<or> b \<noteq> 0"

   529   defines [simp]: "d \<equiv> gcd a b"

   530   defines [simp]: "a' \<equiv> a div d" and [simp]: "b' \<equiv> b div d"

   531   shows "gcd a' b' = 1"

   532 proof (rule coprimeI)

   533   fix l assume "l dvd a'" "l dvd b'"

   534   then obtain s t where "a' = l * s" "b' = l * t" unfolding dvd_def by blast

   535   moreover have "a = a' * d" "b = b' * d" by simp_all

   536   ultimately have "a = (l * d) * s" "b = (l * d) * t"

   537     by (simp_all only: ac_simps)

   538   hence "l*d dvd a" and "l*d dvd b" by (simp_all only: dvd_triv_left)

   539   hence "l*d dvd d" by (simp add: gcd_greatest)

   540   then obtain u where "d = l * d * u" ..

   541   then have "d * (l * u) = d" by (simp add: ac_simps)

   542   moreover from nz have "d \<noteq> 0" by simp

   543   with div_mult_self1_is_id have "d * (l * u) div d = l * u" .

   544   ultimately have "1 = l * u"

   545     using \<open>d \<noteq> 0\<close> by simp

   546   then show "l dvd 1" ..

   547 qed

   548

   549 lemma coprime_mult:

   550   assumes da: "gcd d a = 1" and db: "gcd d b = 1"

   551   shows "gcd d (a * b) = 1"

   552   apply (subst gcd.commute)

   553   using da apply (subst gcd_mult_cancel)

   554   apply (subst gcd.commute, assumption)

   555   apply (subst gcd.commute, rule db)

   556   done

   557

   558 lemma coprime_lmult:

   559   assumes dab: "gcd d (a * b) = 1"

   560   shows "gcd d a = 1"

   561 proof (rule coprimeI)

   562   fix l assume "l dvd d" and "l dvd a"

   563   hence "l dvd a * b" by simp

   564   with \<open>l dvd d\<close> and dab show "l dvd 1" by (auto intro: gcd_greatest)

   565 qed

   566

   567 lemma coprime_rmult:

   568   assumes dab: "gcd d (a * b) = 1"

   569   shows "gcd d b = 1"

   570 proof (rule coprimeI)

   571   fix l assume "l dvd d" and "l dvd b"

   572   hence "l dvd a * b" by simp

   573   with \<open>l dvd d\<close> and dab show "l dvd 1" by (auto intro: gcd_greatest)

   574 qed

   575

   576 lemma coprime_mul_eq: "gcd d (a * b) = 1 \<longleftrightarrow> gcd d a = 1 \<and> gcd d b = 1"

   577   using coprime_rmult[of d a b] coprime_lmult[of d a b] coprime_mult[of d a b] by blast

   578

   579 lemma gcd_coprime:

   580   assumes c: "gcd a b \<noteq> 0" and a: "a = a' * gcd a b" and b: "b = b' * gcd a b"

   581   shows "gcd a' b' = 1"

   582 proof -

   583   from c have "a \<noteq> 0 \<or> b \<noteq> 0" by simp

   584   with div_gcd_coprime have "gcd (a div gcd a b) (b div gcd a b) = 1" .

   585   also from assms have "a div gcd a b = a'" by (metis div_mult_self2_is_id)+

   586   also from assms have "b div gcd a b = b'" by (metis div_mult_self2_is_id)+

   587   finally show ?thesis .

   588 qed

   589

   590 lemma coprime_power:

   591   assumes "0 < n"

   592   shows "gcd a (b ^ n) = 1 \<longleftrightarrow> gcd a b = 1"

   593 using assms proof (induct n)

   594   case (Suc n) then show ?case

   595     by (cases n) (simp_all add: coprime_mul_eq)

   596 qed simp

   597

   598 lemma gcd_coprime_exists:

   599   assumes nz: "gcd a b \<noteq> 0"

   600   shows "\<exists>a' b'. a = a' * gcd a b \<and> b = b' * gcd a b \<and> gcd a' b' = 1"

   601   apply (rule_tac x = "a div gcd a b" in exI)

   602   apply (rule_tac x = "b div gcd a b" in exI)

   603   apply (insert nz, auto intro: div_gcd_coprime)

   604   done

   605

   606 lemma coprime_exp:

   607   "gcd d a = 1 \<Longrightarrow> gcd d (a^n) = 1"

   608   by (induct n, simp_all add: coprime_mult)

   609

   610 lemma coprime_exp2 [intro]:

   611   "gcd a b = 1 \<Longrightarrow> gcd (a^n) (b^m) = 1"

   612   apply (rule coprime_exp)

   613   apply (subst gcd.commute)

   614   apply (rule coprime_exp)

   615   apply (subst gcd.commute)

   616   apply assumption

   617   done

   618

   619 lemma gcd_exp:

   620   "gcd (a^n) (b^n) = (gcd a b) ^ n"

   621 proof (cases "a = 0 \<and> b = 0")

   622   assume "a = 0 \<and> b = 0"

   623   then show ?thesis by (cases n, simp_all add: gcd_0_left)

   624 next

   625   assume A: "\<not>(a = 0 \<and> b = 0)"

   626   hence "1 = gcd ((a div gcd a b)^n) ((b div gcd a b)^n)"

   627     using div_gcd_coprime by (subst sym, auto simp: div_gcd_coprime)

   628   hence "(gcd a b) ^ n = (gcd a b) ^ n * ..." by simp

   629   also note gcd_mult_distrib

   630   also have "unit_factor ((gcd a b)^n) = 1"

   631     by (simp add: unit_factor_power A)

   632   also have "(gcd a b)^n * (a div gcd a b)^n = a^n"

   633     by (subst ac_simps, subst div_power, simp, rule dvd_div_mult_self, rule dvd_power_same, simp)

   634   also have "(gcd a b)^n * (b div gcd a b)^n = b^n"

   635     by (subst ac_simps, subst div_power, simp, rule dvd_div_mult_self, rule dvd_power_same, simp)

   636   finally show ?thesis by simp

   637 qed

   638

   639 lemma coprime_common_divisor:

   640   "gcd a b = 1 \<Longrightarrow> a dvd a \<Longrightarrow> a dvd b \<Longrightarrow> is_unit a"

   641   apply (subgoal_tac "a dvd gcd a b")

   642   apply simp

   643   apply (erule (1) gcd_greatest)

   644   done

   645

   646 lemma division_decomp:

   647   assumes dc: "a dvd b * c"

   648   shows "\<exists>b' c'. a = b' * c' \<and> b' dvd b \<and> c' dvd c"

   649 proof (cases "gcd a b = 0")

   650   assume "gcd a b = 0"

   651   hence "a = 0 \<and> b = 0" by simp

   652   hence "a = 0 * c \<and> 0 dvd b \<and> c dvd c" by simp

   653   then show ?thesis by blast

   654 next

   655   let ?d = "gcd a b"

   656   assume "?d \<noteq> 0"

   657   from gcd_coprime_exists[OF this]

   658     obtain a' b' where ab': "a = a' * ?d" "b = b' * ?d" "gcd a' b' = 1"

   659     by blast

   660   from ab'(1) have "a' dvd a" unfolding dvd_def by blast

   661   with dc have "a' dvd b*c" using dvd_trans[of a' a "b*c"] by simp

   662   from dc ab'(1,2) have "a'*?d dvd (b'*?d) * c" by simp

   663   hence "?d * a' dvd ?d * (b' * c)" by (simp add: mult_ac)

   664   with \<open>?d \<noteq> 0\<close> have "a' dvd b' * c" by simp

   665   with coprime_dvd_mult[OF ab'(3)]

   666     have "a' dvd c" by (subst (asm) ac_simps, blast)

   667   with ab'(1) have "a = ?d * a' \<and> ?d dvd b \<and> a' dvd c" by (simp add: mult_ac)

   668   then show ?thesis by blast

   669 qed

   670

   671 lemma pow_divs_pow:

   672   assumes ab: "a ^ n dvd b ^ n" and n: "n \<noteq> 0"

   673   shows "a dvd b"

   674 proof (cases "gcd a b = 0")

   675   assume "gcd a b = 0"

   676   then show ?thesis by simp

   677 next

   678   let ?d = "gcd a b"

   679   assume "?d \<noteq> 0"

   680   from n obtain m where m: "n = Suc m" by (cases n, simp_all)

   681   from \<open>?d \<noteq> 0\<close> have zn: "?d ^ n \<noteq> 0" by (rule power_not_zero)

   682   from gcd_coprime_exists[OF \<open>?d \<noteq> 0\<close>]

   683     obtain a' b' where ab': "a = a' * ?d" "b = b' * ?d" "gcd a' b' = 1"

   684     by blast

   685   from ab have "(a' * ?d) ^ n dvd (b' * ?d) ^ n"

   686     by (simp add: ab'(1,2)[symmetric])

   687   hence "?d^n * a'^n dvd ?d^n * b'^n"

   688     by (simp only: power_mult_distrib ac_simps)

   689   with zn have "a'^n dvd b'^n" by simp

   690   hence "a' dvd b'^n" using dvd_trans[of a' "a'^n" "b'^n"] by (simp add: m)

   691   hence "a' dvd b'^m * b'" by (simp add: m ac_simps)

   692   with coprime_dvd_mult[OF coprime_exp[OF ab'(3), of m]]

   693     have "a' dvd b'" by (subst (asm) ac_simps, blast)

   694   hence "a'*?d dvd b'*?d" by (rule mult_dvd_mono, simp)

   695   with ab'(1,2) show ?thesis by simp

   696 qed

   697

   698 lemma pow_divs_eq [simp]:

   699   "n \<noteq> 0 \<Longrightarrow> a ^ n dvd b ^ n \<longleftrightarrow> a dvd b"

   700   by (auto intro: pow_divs_pow dvd_power_same)

   701

   702 lemma divs_mult:

   703   assumes mr: "m dvd r" and nr: "n dvd r" and mn: "gcd m n = 1"

   704   shows "m * n dvd r"

   705 proof -

   706   from mr nr obtain m' n' where m': "r = m*m'" and n': "r = n*n'"

   707     unfolding dvd_def by blast

   708   from mr n' have "m dvd n'*n" by (simp add: ac_simps)

   709   hence "m dvd n'" using coprime_dvd_mult_iff[OF mn] by simp

   710   then obtain k where k: "n' = m*k" unfolding dvd_def by blast

   711   with n' have "r = m * n * k" by (simp add: mult_ac)

   712   then show ?thesis unfolding dvd_def by blast

   713 qed

   714

   715 lemma coprime_plus_one [simp]: "gcd (n + 1) n = 1"

   716   by (subst add_commute, simp)

   717

   718 lemma setprod_coprime [rule_format]:

   719   "(\<forall>i\<in>A. gcd (f i) a = 1) \<longrightarrow> gcd (\<Prod>i\<in>A. f i) a = 1"

   720   apply (cases "finite A")

   721   apply (induct set: finite)

   722   apply (auto simp add: gcd_mult_cancel)

   723   done

   724

   725 lemma coprime_divisors:

   726   assumes "d dvd a" "e dvd b" "gcd a b = 1"

   727   shows "gcd d e = 1"

   728 proof -

   729   from assms obtain k l where "a = d * k" "b = e * l"

   730     unfolding dvd_def by blast

   731   with assms have "gcd (d * k) (e * l) = 1" by simp

   732   hence "gcd (d * k) e = 1" by (rule coprime_lmult)

   733   also have "gcd (d * k) e = gcd e (d * k)" by (simp add: ac_simps)

   734   finally have "gcd e d = 1" by (rule coprime_lmult)

   735   then show ?thesis by (simp add: ac_simps)

   736 qed

   737

   738 lemma invertible_coprime:

   739   assumes "a * b mod m = 1"

   740   shows "coprime a m"

   741 proof -

   742   from assms have "coprime m (a * b mod m)"

   743     by simp

   744   then have "coprime m (a * b)"

   745     by simp

   746   then have "coprime m a"

   747     by (rule coprime_lmult)

   748   then show ?thesis

   749     by (simp add: ac_simps)

   750 qed

   751

   752 lemma lcm_gcd:

   753   "lcm a b = normalize (a * b) div gcd a b"

   754   by (simp add: lcm_lcm_eucl gcd_gcd_eucl lcm_eucl_def)

   755

   756 lemma lcm_gcd_prod:

   757   "lcm a b * gcd a b = normalize (a * b)"

   758   by (simp add: lcm_gcd)

   759

   760 lemma lcm_dvd1 [iff]:

   761   "a dvd lcm a b"

   762 proof (cases "a*b = 0")

   763   assume "a * b \<noteq> 0"

   764   hence "gcd a b \<noteq> 0" by simp

   765   let ?c = "1 div unit_factor (a * b)"

   766   from \<open>a * b \<noteq> 0\<close> have [simp]: "is_unit (unit_factor (a * b))" by simp

   767   from lcm_gcd_prod[of a b] have "lcm a b * gcd a b = a * ?c * b"

   768     by (simp add: div_mult_swap unit_div_commute)

   769   hence "lcm a b * gcd a b div gcd a b = a * ?c * b div gcd a b" by simp

   770   with \<open>gcd a b \<noteq> 0\<close> have "lcm a b = a * ?c * b div gcd a b"

   771     by (subst (asm) div_mult_self2_is_id, simp_all)

   772   also have "... = a * (?c * b div gcd a b)"

   773     by (metis div_mult_swap gcd_dvd2 mult_assoc)

   774   finally show ?thesis by (rule dvdI)

   775 qed (auto simp add: lcm_gcd)

   776

   777 lemma lcm_least:

   778   "\<lbrakk>a dvd k; b dvd k\<rbrakk> \<Longrightarrow> lcm a b dvd k"

   779 proof (cases "k = 0")

   780   let ?nf = unit_factor

   781   assume "k \<noteq> 0"

   782   hence "is_unit (?nf k)" by simp

   783   hence "?nf k \<noteq> 0" by (metis not_is_unit_0)

   784   assume A: "a dvd k" "b dvd k"

   785   hence "gcd a b \<noteq> 0" using \<open>k \<noteq> 0\<close> by auto

   786   from A obtain r s where ar: "k = a * r" and bs: "k = b * s"

   787     unfolding dvd_def by blast

   788   with \<open>k \<noteq> 0\<close> have "r * s \<noteq> 0"

   789     by auto (drule sym [of 0], simp)

   790   hence "is_unit (?nf (r * s))" by simp

   791   let ?c = "?nf k div ?nf (r*s)"

   792   from \<open>is_unit (?nf k)\<close> and \<open>is_unit (?nf (r * s))\<close> have "is_unit ?c" by (rule unit_div)

   793   hence "?c \<noteq> 0" using not_is_unit_0 by fast

   794   from ar bs have "k * k * gcd s r = ?nf k * k * gcd (k * s) (k * r)"

   795     by (subst mult_assoc, subst gcd_mult_distrib[of k s r], simp only: ac_simps)

   796   also have "... = ?nf k * k * gcd ((r*s) * a) ((r*s) * b)"

   797     by (subst (3) \<open>k = a * r\<close>, subst (3) \<open>k = b * s\<close>, simp add: algebra_simps)

   798   also have "... = ?c * r*s * k * gcd a b" using \<open>r * s \<noteq> 0\<close>

   799     by (subst gcd_mult_distrib'[symmetric], simp add: algebra_simps unit_simps)

   800   finally have "(a*r) * (b*s) * gcd s r = ?c * k * r * s * gcd a b"

   801     by (subst ar[symmetric], subst bs[symmetric], simp add: mult_ac)

   802   hence "a * b * gcd s r * (r * s) = ?c * k * gcd a b * (r * s)"

   803     by (simp add: algebra_simps)

   804   hence "?c * k * gcd a b = a * b * gcd s r" using \<open>r * s \<noteq> 0\<close>

   805     by (metis div_mult_self2_is_id)

   806   also have "... = lcm a b * gcd a b * gcd s r * ?nf (a*b)"

   807     by (subst lcm_gcd_prod[of a b], metis gcd_mult_distrib gcd_mult_distrib')

   808   also have "... = lcm a b * gcd s r * ?nf (a*b) * gcd a b"

   809     by (simp add: algebra_simps)

   810   finally have "k * ?c = lcm a b * gcd s r * ?nf (a*b)" using \<open>gcd a b \<noteq> 0\<close>

   811     by (metis mult.commute div_mult_self2_is_id)

   812   hence "k = lcm a b * (gcd s r * ?nf (a*b)) div ?c" using \<open>?c \<noteq> 0\<close>

   813     by (metis div_mult_self2_is_id mult_assoc)

   814   also have "... = lcm a b * (gcd s r * ?nf (a*b) div ?c)" using \<open>is_unit ?c\<close>

   815     by (simp add: unit_simps)

   816   finally show ?thesis by (rule dvdI)

   817 qed simp

   818

   819 lemma lcm_zero:

   820   "lcm a b = 0 \<longleftrightarrow> a = 0 \<or> b = 0"

   821 proof -

   822   let ?nf = unit_factor

   823   {

   824     assume "a \<noteq> 0" "b \<noteq> 0"

   825     hence "a * b div ?nf (a * b) \<noteq> 0" by (simp add: no_zero_divisors)

   826     moreover from \<open>a \<noteq> 0\<close> and \<open>b \<noteq> 0\<close> have "gcd a b \<noteq> 0" by simp

   827     ultimately have "lcm a b \<noteq> 0" using lcm_gcd_prod[of a b] by (intro notI, simp)

   828   } moreover {

   829     assume "a = 0 \<or> b = 0"

   830     hence "lcm a b = 0" by (elim disjE, simp_all add: lcm_gcd)

   831   }

   832   ultimately show ?thesis by blast

   833 qed

   834

   835 lemmas lcm_0_iff = lcm_zero

   836

   837 lemma gcd_lcm:

   838   assumes "lcm a b \<noteq> 0"

   839   shows "gcd a b = normalize (a * b) div lcm a b"

   840 proof -

   841   have "lcm a b * gcd a b = normalize (a * b)"

   842     by (fact lcm_gcd_prod)

   843   with assms show ?thesis

   844     by (metis nonzero_mult_divide_cancel_left)

   845 qed

   846

   847 lemma unit_factor_lcm [simp]:

   848   "unit_factor (lcm a b) = (if a = 0 \<or> b = 0 then 0 else 1)"

   849   by (simp add: dvd_unit_factor_div lcm_gcd)

   850

   851 lemma lcm_dvd2 [iff]: "b dvd lcm a b"

   852   using lcm_dvd1 [of b a] by (simp add: lcm_gcd ac_simps)

   853

   854 lemma lcmI:

   855   assumes "a dvd c" and "b dvd c" and "\<And>d. a dvd d \<Longrightarrow> b dvd d \<Longrightarrow> c dvd d"

   856     and "unit_factor c = (if c = 0 then 0 else 1)"

   857   shows "c = lcm a b"

   858   by (rule associated_eqI)

   859     (auto simp: assms associated_def intro: lcm_least, simp_all add: lcm_gcd)

   860

   861 sublocale lcm!: abel_semigroup lcm

   862 proof

   863   fix a b c

   864   show "lcm (lcm a b) c = lcm a (lcm b c)"

   865   proof (rule lcmI)

   866     have "a dvd lcm a b" and "lcm a b dvd lcm (lcm a b) c" by simp_all

   867     then show "a dvd lcm (lcm a b) c" by (rule dvd_trans)

   868

   869     have "b dvd lcm a b" and "lcm a b dvd lcm (lcm a b) c" by simp_all

   870     hence "b dvd lcm (lcm a b) c" by (rule dvd_trans)

   871     moreover have "c dvd lcm (lcm a b) c" by simp

   872     ultimately show "lcm b c dvd lcm (lcm a b) c" by (rule lcm_least)

   873

   874     fix l assume "a dvd l" and "lcm b c dvd l"

   875     have "b dvd lcm b c" by simp

   876     from this and \<open>lcm b c dvd l\<close> have "b dvd l" by (rule dvd_trans)

   877     have "c dvd lcm b c" by simp

   878     from this and \<open>lcm b c dvd l\<close> have "c dvd l" by (rule dvd_trans)

   879     from \<open>a dvd l\<close> and \<open>b dvd l\<close> have "lcm a b dvd l" by (rule lcm_least)

   880     from this and \<open>c dvd l\<close> show "lcm (lcm a b) c dvd l" by (rule lcm_least)

   881   qed (simp add: lcm_zero)

   882 next

   883   fix a b

   884   show "lcm a b = lcm b a"

   885     by (simp add: lcm_gcd ac_simps)

   886 qed

   887

   888 lemma dvd_lcm_D1:

   889   "lcm m n dvd k \<Longrightarrow> m dvd k"

   890   by (rule dvd_trans, rule lcm_dvd1, assumption)

   891

   892 lemma dvd_lcm_D2:

   893   "lcm m n dvd k \<Longrightarrow> n dvd k"

   894   by (rule dvd_trans, rule lcm_dvd2, assumption)

   895

   896 lemma gcd_dvd_lcm [simp]:

   897   "gcd a b dvd lcm a b"

   898   by (metis dvd_trans gcd_dvd2 lcm_dvd2)

   899

   900 lemma lcm_1_iff:

   901   "lcm a b = 1 \<longleftrightarrow> is_unit a \<and> is_unit b"

   902 proof

   903   assume "lcm a b = 1"

   904   then show "is_unit a \<and> is_unit b" by auto

   905 next

   906   assume "is_unit a \<and> is_unit b"

   907   hence "a dvd 1" and "b dvd 1" by simp_all

   908   hence "is_unit (lcm a b)" by (rule lcm_least)

   909   hence "lcm a b = unit_factor (lcm a b)"

   910     by (blast intro: sym is_unit_unit_factor)

   911   also have "\<dots> = 1" using \<open>is_unit a \<and> is_unit b\<close>

   912     by auto

   913   finally show "lcm a b = 1" .

   914 qed

   915

   916 lemma lcm_0_left [simp]:

   917   "lcm 0 a = 0"

   918   by (rule sym, rule lcmI, simp_all)

   919

   920 lemma lcm_0 [simp]:

   921   "lcm a 0 = 0"

   922   by (rule sym, rule lcmI, simp_all)

   923

   924 lemma lcm_unique:

   925   "a dvd d \<and> b dvd d \<and>

   926   unit_factor d = (if d = 0 then 0 else 1) \<and>

   927   (\<forall>e. a dvd e \<and> b dvd e \<longrightarrow> d dvd e) \<longleftrightarrow> d = lcm a b"

   928   by (rule, auto intro: lcmI simp: lcm_least lcm_zero)

   929

   930 lemma dvd_lcm_I1 [simp]:

   931   "k dvd m \<Longrightarrow> k dvd lcm m n"

   932   by (metis lcm_dvd1 dvd_trans)

   933

   934 lemma dvd_lcm_I2 [simp]:

   935   "k dvd n \<Longrightarrow> k dvd lcm m n"

   936   by (metis lcm_dvd2 dvd_trans)

   937

   938 lemma lcm_1_left [simp]:

   939   "lcm 1 a = normalize a"

   940   by (cases "a = 0") (simp, rule sym, rule lcmI, simp_all)

   941

   942 lemma lcm_1_right [simp]:

   943   "lcm a 1 = normalize a"

   944   using lcm_1_left [of a] by (simp add: ac_simps)

   945

   946 lemma lcm_coprime:

   947   "gcd a b = 1 \<Longrightarrow> lcm a b = normalize (a * b)"

   948   by (subst lcm_gcd) simp

   949

   950 lemma lcm_proj1_if_dvd:

   951   "b dvd a \<Longrightarrow> lcm a b = normalize a"

   952   by (cases "a = 0") (simp, rule sym, rule lcmI, simp_all)

   953

   954 lemma lcm_proj2_if_dvd:

   955   "a dvd b \<Longrightarrow> lcm a b = normalize b"

   956   using lcm_proj1_if_dvd [of a b] by (simp add: ac_simps)

   957

   958 lemma lcm_proj1_iff:

   959   "lcm m n = normalize m \<longleftrightarrow> n dvd m"

   960 proof

   961   assume A: "lcm m n = normalize m"

   962   show "n dvd m"

   963   proof (cases "m = 0")

   964     assume [simp]: "m \<noteq> 0"

   965     from A have B: "m = lcm m n * unit_factor m"

   966       by (simp add: unit_eq_div2)

   967     show ?thesis by (subst B, simp)

   968   qed simp

   969 next

   970   assume "n dvd m"

   971   then show "lcm m n = normalize m" by (rule lcm_proj1_if_dvd)

   972 qed

   973

   974 lemma lcm_proj2_iff:

   975   "lcm m n = normalize n \<longleftrightarrow> m dvd n"

   976   using lcm_proj1_iff [of n m] by (simp add: ac_simps)

   977

   978 lemma euclidean_size_lcm_le1:

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

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

   981 proof -

   982   have "a dvd lcm a b" by (rule lcm_dvd1)

   983   then obtain c where A: "lcm a b = a * c" unfolding dvd_def by blast

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

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

   986 qed

   987

   988 lemma euclidean_size_lcm_le2:

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

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

   991

   992 lemma euclidean_size_lcm_less1:

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

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

   995 proof (rule ccontr)

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

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

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

   999     by (intro le_antisym, simp, intro euclidean_size_lcm_le1)

  1000   with assms have "lcm a b dvd a"

  1001     by (rule_tac dvd_euclidean_size_eq_imp_dvd) (auto simp: lcm_zero)

  1002   hence "b dvd a" by (rule dvd_lcm_D2)

  1003   with \<open>\<not>b dvd a\<close> show False by contradiction

  1004 qed

  1005

  1006 lemma euclidean_size_lcm_less2:

  1007   assumes "a \<noteq> 0" and "\<not>a dvd b"

  1008   shows "euclidean_size b < euclidean_size (lcm a b)"

  1009   using assms euclidean_size_lcm_less1 [of a b] by (simp add: ac_simps)

  1010

  1011 lemma lcm_mult_unit1:

  1012   "is_unit a \<Longrightarrow> lcm (b * a) c = lcm b c"

  1013   apply (rule lcmI)

  1014   apply (rule dvd_trans[of _ "b * a"], simp, rule lcm_dvd1)

  1015   apply (rule lcm_dvd2)

  1016   apply (rule lcm_least, simp add: unit_simps, assumption)

  1017   apply (subst unit_factor_lcm, simp add: lcm_zero)

  1018   done

  1019

  1020 lemma lcm_mult_unit2:

  1021   "is_unit a \<Longrightarrow> lcm b (c * a) = lcm b c"

  1022   using lcm_mult_unit1 [of a c b] by (simp add: ac_simps)

  1023

  1024 lemma lcm_div_unit1:

  1025   "is_unit a \<Longrightarrow> lcm (b div a) c = lcm b c"

  1026   by (erule is_unitE [of _ b]) (simp add: lcm_mult_unit1)

  1027

  1028 lemma lcm_div_unit2:

  1029   "is_unit a \<Longrightarrow> lcm b (c div a) = lcm b c"

  1030   by (erule is_unitE [of _ c]) (simp add: lcm_mult_unit2)

  1031

  1032 lemma normalize_lcm_left [simp]:

  1033   "lcm (normalize a) b = lcm a b"

  1034 proof (cases "a = 0")

  1035   case True then show ?thesis

  1036     by simp

  1037 next

  1038   case False then have "is_unit (unit_factor a)"

  1039     by simp

  1040   moreover have "normalize a = a div unit_factor a"

  1041     by simp

  1042   ultimately show ?thesis

  1043     by (simp only: lcm_div_unit1)

  1044 qed

  1045

  1046 lemma normalize_lcm_right [simp]:

  1047   "lcm a (normalize b) = lcm a b"

  1048   using normalize_lcm_left [of b a] by (simp add: ac_simps)

  1049

  1050 lemma lcm_left_idem:

  1051   "lcm a (lcm a b) = lcm a b"

  1052   apply (rule lcmI)

  1053   apply simp

  1054   apply (subst lcm.assoc [symmetric], rule lcm_dvd2)

  1055   apply (rule lcm_least, assumption)

  1056   apply (erule (1) lcm_least)

  1057   apply (auto simp: lcm_zero)

  1058   done

  1059

  1060 lemma lcm_right_idem:

  1061   "lcm (lcm a b) b = lcm a b"

  1062   apply (rule lcmI)

  1063   apply (subst lcm.assoc, rule lcm_dvd1)

  1064   apply (rule lcm_dvd2)

  1065   apply (rule lcm_least, erule (1) lcm_least, assumption)

  1066   apply (auto simp: lcm_zero)

  1067   done

  1068

  1069 lemma comp_fun_idem_lcm: "comp_fun_idem lcm"

  1070 proof

  1071   fix a b show "lcm a \<circ> lcm b = lcm b \<circ> lcm a"

  1072     by (simp add: fun_eq_iff ac_simps)

  1073 next

  1074   fix a show "lcm a \<circ> lcm a = lcm a" unfolding o_def

  1075     by (intro ext, simp add: lcm_left_idem)

  1076 qed

  1077

  1078 lemma dvd_Lcm [simp]: "a \<in> A \<Longrightarrow> a dvd Lcm A"

  1079   and Lcm_least: "(\<And>a. a \<in> A \<Longrightarrow> a dvd b) \<Longrightarrow> Lcm A dvd b"

  1080   and unit_factor_Lcm [simp]:

  1081           "unit_factor (Lcm A) = (if Lcm A = 0 then 0 else 1)"

  1082 proof -

  1083   have "(\<forall>a\<in>A. a dvd Lcm A) \<and> (\<forall>l'. (\<forall>a\<in>A. a dvd l') \<longrightarrow> Lcm A dvd l') \<and>

  1084     unit_factor (Lcm A) = (if Lcm A = 0 then 0 else 1)" (is ?thesis)

  1085   proof (cases "\<exists>l. l \<noteq>  0 \<and> (\<forall>a\<in>A. a dvd l)")

  1086     case False

  1087     hence "Lcm A = 0" by (auto simp: Lcm_Lcm_eucl Lcm_eucl_def)

  1088     with False show ?thesis by auto

  1089   next

  1090     case True

  1091     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

  1092     def n \<equiv> "LEAST n. \<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"

  1093     def l \<equiv> "SOME l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"

  1094     have "\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"

  1095       apply (subst n_def)

  1096       apply (rule LeastI[of _ "euclidean_size l\<^sub>0"])

  1097       apply (rule exI[of _ l\<^sub>0])

  1098       apply (simp add: l\<^sub>0_props)

  1099       done

  1100     from someI_ex[OF this] have "l \<noteq> 0" and "\<forall>a\<in>A. a dvd l" and "euclidean_size l = n"

  1101       unfolding l_def by simp_all

  1102     {

  1103       fix l' assume "\<forall>a\<in>A. a dvd l'"

  1104       with \<open>\<forall>a\<in>A. a dvd l\<close> have "\<forall>a\<in>A. a dvd gcd l l'" by (auto intro: gcd_greatest)

  1105       moreover from \<open>l \<noteq> 0\<close> have "gcd l l' \<noteq> 0" by simp

  1106       ultimately have "\<exists>b. b \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd b) \<and> euclidean_size b = euclidean_size (gcd l l')"

  1107         by (intro exI[of _ "gcd l l'"], auto)

  1108       hence "euclidean_size (gcd l l') \<ge> n" by (subst n_def) (rule Least_le)

  1109       moreover have "euclidean_size (gcd l l') \<le> n"

  1110       proof -

  1111         have "gcd l l' dvd l" by simp

  1112         then obtain a where "l = gcd l l' * a" unfolding dvd_def by blast

  1113         with \<open>l \<noteq> 0\<close> have "a \<noteq> 0" by auto

  1114         hence "euclidean_size (gcd l l') \<le> euclidean_size (gcd l l' * a)"

  1115           by (rule size_mult_mono)

  1116         also have "gcd l l' * a = l" using \<open>l = gcd l l' * a\<close> ..

  1117         also note \<open>euclidean_size l = n\<close>

  1118         finally show "euclidean_size (gcd l l') \<le> n" .

  1119       qed

  1120       ultimately have "euclidean_size l = euclidean_size (gcd l l')"

  1121         by (intro le_antisym, simp_all add: \<open>euclidean_size l = n\<close>)

  1122       with \<open>l \<noteq> 0\<close> have "l dvd gcd l l'" by (blast intro: dvd_euclidean_size_eq_imp_dvd)

  1123       hence "l dvd l'" by (blast dest: dvd_gcd_D2)

  1124     }

  1125

  1126     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>

  1127       have "(\<forall>a\<in>A. a dvd normalize l) \<and>

  1128         (\<forall>l'. (\<forall>a\<in>A. a dvd l') \<longrightarrow> normalize l dvd l') \<and>

  1129         unit_factor (normalize l) =

  1130         (if normalize l = 0 then 0 else 1)"

  1131       by (auto simp: unit_simps)

  1132     also from True have "normalize l = Lcm A"

  1133       by (simp add: Lcm_Lcm_eucl Lcm_eucl_def Let_def n_def l_def)

  1134     finally show ?thesis .

  1135   qed

  1136   note A = this

  1137

  1138   {fix a assume "a \<in> A" then show "a dvd Lcm A" using A by blast}

  1139   {fix b assume "\<And>a. a \<in> A \<Longrightarrow> a dvd b" then show "Lcm A dvd b" using A by blast}

  1140   from A show "unit_factor (Lcm A) = (if Lcm A = 0 then 0 else 1)" by blast

  1141 qed

  1142

  1143 lemma normalize_Lcm [simp]:

  1144   "normalize (Lcm A) = Lcm A"

  1145   by (cases "Lcm A = 0") (auto intro: associated_eqI)

  1146

  1147 lemma LcmI:

  1148   assumes "\<And>a. a \<in> A \<Longrightarrow> a dvd b" and "\<And>c. (\<And>a. a \<in> A \<Longrightarrow> a dvd c) \<Longrightarrow> b dvd c"

  1149     and "unit_factor b = (if b = 0 then 0 else 1)" shows "b = Lcm A"

  1150   by (rule associated_eqI) (auto simp: assms associated_def intro: Lcm_least)

  1151

  1152 lemma Lcm_subset:

  1153   "A \<subseteq> B \<Longrightarrow> Lcm A dvd Lcm B"

  1154   by (blast intro: Lcm_least dvd_Lcm)

  1155

  1156 lemma Lcm_Un:

  1157   "Lcm (A \<union> B) = lcm (Lcm A) (Lcm B)"

  1158   apply (rule lcmI)

  1159   apply (blast intro: Lcm_subset)

  1160   apply (blast intro: Lcm_subset)

  1161   apply (intro Lcm_least ballI, elim UnE)

  1162   apply (rule dvd_trans, erule dvd_Lcm, assumption)

  1163   apply (rule dvd_trans, erule dvd_Lcm, assumption)

  1164   apply simp

  1165   done

  1166

  1167 lemma Lcm_1_iff:

  1168   "Lcm A = 1 \<longleftrightarrow> (\<forall>a\<in>A. is_unit a)"

  1169 proof

  1170   assume "Lcm A = 1"

  1171   then show "\<forall>a\<in>A. is_unit a" by auto

  1172 qed (rule LcmI [symmetric], auto)

  1173

  1174 lemma Lcm_no_units:

  1175   "Lcm A = Lcm (A - {a. is_unit a})"

  1176 proof -

  1177   have "(A - {a. is_unit a}) \<union> {a\<in>A. is_unit a} = A" by blast

  1178   hence "Lcm A = lcm (Lcm (A - {a. is_unit a})) (Lcm {a\<in>A. is_unit a})"

  1179     by (simp add: Lcm_Un [symmetric])

  1180   also have "Lcm {a\<in>A. is_unit a} = 1" by (simp add: Lcm_1_iff)

  1181   finally show ?thesis by simp

  1182 qed

  1183

  1184 lemma Lcm_empty [simp]:

  1185   "Lcm {} = 1"

  1186   by (simp add: Lcm_1_iff)

  1187

  1188 lemma Lcm_eq_0 [simp]:

  1189   "0 \<in> A \<Longrightarrow> Lcm A = 0"

  1190   by (drule dvd_Lcm) simp

  1191

  1192 lemma Lcm0_iff':

  1193   "Lcm A = 0 \<longleftrightarrow> \<not>(\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l))"

  1194 proof

  1195   assume "Lcm A = 0"

  1196   show "\<not>(\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l))"

  1197   proof

  1198     assume ex: "\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l)"

  1199     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

  1200     def n \<equiv> "LEAST n. \<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"

  1201     def l \<equiv> "SOME l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"

  1202     have "\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"

  1203       apply (subst n_def)

  1204       apply (rule LeastI[of _ "euclidean_size l\<^sub>0"])

  1205       apply (rule exI[of _ l\<^sub>0])

  1206       apply (simp add: l\<^sub>0_props)

  1207       done

  1208     from someI_ex[OF this] have "l \<noteq> 0" unfolding l_def by simp_all

  1209     hence "normalize l \<noteq> 0" by simp

  1210     also from ex have "normalize l = Lcm A"

  1211        by (simp only: Lcm_Lcm_eucl Lcm_eucl_def n_def l_def if_True Let_def)

  1212     finally show False using \<open>Lcm A = 0\<close> by contradiction

  1213   qed

  1214 qed (simp only: Lcm_Lcm_eucl Lcm_eucl_def if_False)

  1215

  1216 lemma Lcm0_iff [simp]:

  1217   "finite A \<Longrightarrow> Lcm A = 0 \<longleftrightarrow> 0 \<in> A"

  1218 proof -

  1219   assume "finite A"

  1220   have "0 \<in> A \<Longrightarrow> Lcm A = 0"  by (intro dvd_0_left dvd_Lcm)

  1221   moreover {

  1222     assume "0 \<notin> A"

  1223     hence "\<Prod>A \<noteq> 0"

  1224       apply (induct rule: finite_induct[OF \<open>finite A\<close>])

  1225       apply simp

  1226       apply (subst setprod.insert, assumption, assumption)

  1227       apply (rule no_zero_divisors)

  1228       apply blast+

  1229       done

  1230     moreover from \<open>finite A\<close> have "\<forall>a\<in>A. a dvd \<Prod>A" by blast

  1231     ultimately have "\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l)" by blast

  1232     with Lcm0_iff' have "Lcm A \<noteq> 0" by simp

  1233   }

  1234   ultimately show "Lcm A = 0 \<longleftrightarrow> 0 \<in> A" by blast

  1235 qed

  1236

  1237 lemma Lcm_no_multiple:

  1238   "(\<forall>m. m \<noteq> 0 \<longrightarrow> (\<exists>a\<in>A. \<not>a dvd m)) \<Longrightarrow> Lcm A = 0"

  1239 proof -

  1240   assume "\<forall>m. m \<noteq> 0 \<longrightarrow> (\<exists>a\<in>A. \<not>a dvd m)"

  1241   hence "\<not>(\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l))" by blast

  1242   then show "Lcm A = 0" by (simp only: Lcm_Lcm_eucl Lcm_eucl_def if_False)

  1243 qed

  1244

  1245 lemma Lcm_insert [simp]:

  1246   "Lcm (insert a A) = lcm a (Lcm A)"

  1247 proof (rule lcmI)

  1248   fix l assume "a dvd l" and "Lcm A dvd l"

  1249   hence "\<forall>a\<in>A. a dvd l" by (blast intro: dvd_trans dvd_Lcm)

  1250   with \<open>a dvd l\<close> show "Lcm (insert a A) dvd l" by (force intro: Lcm_least)

  1251 qed (auto intro: Lcm_least dvd_Lcm)

  1252

  1253 lemma Lcm_finite:

  1254   assumes "finite A"

  1255   shows "Lcm A = Finite_Set.fold lcm 1 A"

  1256   by (induct rule: finite.induct[OF \<open>finite A\<close>])

  1257     (simp_all add: comp_fun_idem.fold_insert_idem[OF comp_fun_idem_lcm])

  1258

  1259 lemma Lcm_set [code_unfold]:

  1260   "Lcm (set xs) = fold lcm xs 1"

  1261   using comp_fun_idem.fold_set_fold[OF comp_fun_idem_lcm] Lcm_finite by (simp add: ac_simps)

  1262

  1263 lemma Lcm_singleton [simp]:

  1264   "Lcm {a} = normalize a"

  1265   by simp

  1266

  1267 lemma Lcm_2 [simp]:

  1268   "Lcm {a,b} = lcm a b"

  1269   by simp

  1270

  1271 lemma Lcm_coprime:

  1272   assumes "finite A" and "A \<noteq> {}"

  1273   assumes "\<And>a b. a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<noteq> b \<Longrightarrow> gcd a b = 1"

  1274   shows "Lcm A = normalize (\<Prod>A)"

  1275 using assms proof (induct rule: finite_ne_induct)

  1276   case (insert a A)

  1277   have "Lcm (insert a A) = lcm a (Lcm A)" by simp

  1278   also from insert have "Lcm A = normalize (\<Prod>A)" by blast

  1279   also have "lcm a \<dots> = lcm a (\<Prod>A)" by (cases "\<Prod>A = 0") (simp_all add: lcm_div_unit2)

  1280   also from insert have "gcd a (\<Prod>A) = 1" by (subst gcd.commute, intro setprod_coprime) auto

  1281   with insert have "lcm a (\<Prod>A) = normalize (\<Prod>(insert a A))"

  1282     by (simp add: lcm_coprime)

  1283   finally show ?case .

  1284 qed simp

  1285

  1286 lemma Lcm_coprime':

  1287   "card A \<noteq> 0 \<Longrightarrow> (\<And>a b. a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<noteq> b \<Longrightarrow> gcd a b = 1)

  1288     \<Longrightarrow> Lcm A = normalize (\<Prod>A)"

  1289   by (rule Lcm_coprime) (simp_all add: card_eq_0_iff)

  1290

  1291 lemma Gcd_Lcm:

  1292   "Gcd A = Lcm {d. \<forall>a\<in>A. d dvd a}"

  1293   by (simp add: Gcd_Gcd_eucl Lcm_Lcm_eucl Gcd_eucl_def)

  1294

  1295 lemma Gcd_dvd [simp]: "a \<in> A \<Longrightarrow> Gcd A dvd a"

  1296   and Gcd_greatest: "(\<And>a. a \<in> A \<Longrightarrow> b dvd a) \<Longrightarrow> b dvd Gcd A"

  1297   and unit_factor_Gcd [simp]:

  1298     "unit_factor (Gcd A) = (if Gcd A = 0 then 0 else 1)"

  1299 proof -

  1300   fix a assume "a \<in> A"

  1301   hence "Lcm {d. \<forall>a\<in>A. d dvd a} dvd a" by (intro Lcm_least) blast

  1302   then show "Gcd A dvd a" by (simp add: Gcd_Lcm)

  1303 next

  1304   fix g' assume "\<And>a. a \<in> A \<Longrightarrow> g' dvd a"

  1305   hence "g' dvd Lcm {d. \<forall>a\<in>A. d dvd a}" by (intro dvd_Lcm) blast

  1306   then show "g' dvd Gcd A" by (simp add: Gcd_Lcm)

  1307 next

  1308   show "unit_factor (Gcd A) = (if Gcd A = 0 then 0 else 1)"

  1309     by (simp add: Gcd_Lcm)

  1310 qed

  1311

  1312 lemma normalize_Gcd [simp]:

  1313   "normalize (Gcd A) = Gcd A"

  1314   by (cases "Gcd A = 0") (auto intro: associated_eqI)

  1315

  1316 lemma GcdI:

  1317   assumes "\<And>a. a \<in> A \<Longrightarrow> b dvd a" and "\<And>c. (\<And>a. a \<in> A \<Longrightarrow> c dvd a) \<Longrightarrow> c dvd b"

  1318     and "unit_factor b = (if b = 0 then 0 else 1)"

  1319   shows "b = Gcd A"

  1320   by (rule associated_eqI) (auto simp: assms associated_def intro: Gcd_greatest)

  1321

  1322 lemma Lcm_Gcd:

  1323   "Lcm A = Gcd {m. \<forall>a\<in>A. a dvd m}"

  1324   by (rule LcmI[symmetric]) (auto intro: dvd_Gcd Gcd_greatest)

  1325

  1326 lemma Gcd_0_iff:

  1327   "Gcd A = 0 \<longleftrightarrow> A \<subseteq> {0}"

  1328   apply (rule iffI)

  1329   apply (rule subsetI, drule Gcd_dvd, simp)

  1330   apply (auto intro: GcdI[symmetric])

  1331   done

  1332

  1333 lemma Gcd_empty [simp]:

  1334   "Gcd {} = 0"

  1335   by (simp add: Gcd_0_iff)

  1336

  1337 lemma Gcd_1:

  1338   "1 \<in> A \<Longrightarrow> Gcd A = 1"

  1339   by (intro GcdI[symmetric]) (auto intro: Gcd_dvd dvd_Gcd)

  1340

  1341 lemma Gcd_insert [simp]:

  1342   "Gcd (insert a A) = gcd a (Gcd A)"

  1343 proof (rule gcdI)

  1344   fix l assume "l dvd a" and "l dvd Gcd A"

  1345   hence "\<forall>a\<in>A. l dvd a" by (blast intro: dvd_trans Gcd_dvd)

  1346   with \<open>l dvd a\<close> show "l dvd Gcd (insert a A)" by (force intro: Gcd_dvd Gcd_greatest)

  1347 qed (auto intro: Gcd_greatest)

  1348

  1349 lemma Gcd_finite:

  1350   assumes "finite A"

  1351   shows "Gcd A = Finite_Set.fold gcd 0 A"

  1352   by (induct rule: finite.induct[OF \<open>finite A\<close>])

  1353     (simp_all add: comp_fun_idem.fold_insert_idem[OF comp_fun_idem_gcd])

  1354

  1355 lemma Gcd_set [code_unfold]:

  1356   "Gcd (set xs) = fold gcd xs 0"

  1357   using comp_fun_idem.fold_set_fold[OF comp_fun_idem_gcd] Gcd_finite by (simp add: ac_simps)

  1358

  1359 lemma Gcd_singleton [simp]: "Gcd {a} = normalize a"

  1360   by (simp add: gcd_0)

  1361

  1362 lemma Gcd_2 [simp]: "Gcd {a,b} = gcd a b"

  1363   by (simp add: gcd_0)

  1364

  1365 subclass semiring_gcd

  1366   by unfold_locales (simp_all add: gcd_greatest_iff)

  1367

  1368 end

  1369

  1370 text \<open>

  1371   A Euclidean ring is a Euclidean semiring with additive inverses. It provides a

  1372   few more lemmas; in particular, Bezout's lemma holds for any Euclidean ring.

  1373 \<close>

  1374

  1375 class euclidean_ring_gcd = euclidean_semiring_gcd + idom

  1376 begin

  1377

  1378 subclass euclidean_ring ..

  1379

  1380 subclass ring_gcd ..

  1381

  1382 lemma euclid_ext_gcd [simp]:

  1383   "(case euclid_ext a b of (_, _ , t) \<Rightarrow> t) = gcd a b"

  1384   by (induct a b rule: gcd_eucl_induct)

  1385     (simp_all add: euclid_ext_0 gcd_0 euclid_ext_non_0 ac_simps split: prod.split prod.split_asm)

  1386

  1387 lemma euclid_ext_gcd' [simp]:

  1388   "euclid_ext a b = (r, s, t) \<Longrightarrow> t = gcd a b"

  1389   by (insert euclid_ext_gcd[of a b], drule (1) subst, simp)

  1390

  1391 lemma euclid_ext'_correct:

  1392   "fst (euclid_ext' a b) * a + snd (euclid_ext' a b) * b = gcd a b"

  1393 proof-

  1394   obtain s t c where "euclid_ext a b = (s,t,c)"

  1395     by (cases "euclid_ext a b", blast)

  1396   with euclid_ext_correct[of a b] euclid_ext_gcd[of a b]

  1397     show ?thesis unfolding euclid_ext'_def by simp

  1398 qed

  1399

  1400 lemma bezout: "\<exists>s t. s * a + t * b = gcd a b"

  1401   using euclid_ext'_correct by blast

  1402

  1403 lemma gcd_neg1 [simp]:

  1404   "gcd (-a) b = gcd a b"

  1405   by (rule sym, rule gcdI, simp_all add: gcd_greatest)

  1406

  1407 lemma gcd_neg2 [simp]:

  1408   "gcd a (-b) = gcd a b"

  1409   by (rule sym, rule gcdI, simp_all add: gcd_greatest)

  1410

  1411 lemma gcd_neg_numeral_1 [simp]:

  1412   "gcd (- numeral n) a = gcd (numeral n) a"

  1413   by (fact gcd_neg1)

  1414

  1415 lemma gcd_neg_numeral_2 [simp]:

  1416   "gcd a (- numeral n) = gcd a (numeral n)"

  1417   by (fact gcd_neg2)

  1418

  1419 lemma gcd_diff1: "gcd (m - n) n = gcd m n"

  1420   by (subst diff_conv_add_uminus, subst gcd_neg2[symmetric],  subst gcd_add1, simp)

  1421

  1422 lemma gcd_diff2: "gcd (n - m) n = gcd m n"

  1423   by (subst gcd_neg1[symmetric], simp only: minus_diff_eq gcd_diff1)

  1424

  1425 lemma coprime_minus_one [simp]: "gcd (n - 1) n = 1"

  1426 proof -

  1427   have "gcd (n - 1) n = gcd n (n - 1)" by (fact gcd.commute)

  1428   also have "\<dots> = gcd ((n - 1) + 1) (n - 1)" by simp

  1429   also have "\<dots> = 1" by (rule coprime_plus_one)

  1430   finally show ?thesis .

  1431 qed

  1432

  1433 lemma lcm_neg1 [simp]: "lcm (-a) b = lcm a b"

  1434   by (rule sym, rule lcmI, simp_all add: lcm_least lcm_zero)

  1435

  1436 lemma lcm_neg2 [simp]: "lcm a (-b) = lcm a b"

  1437   by (rule sym, rule lcmI, simp_all add: lcm_least lcm_zero)

  1438

  1439 lemma lcm_neg_numeral_1 [simp]: "lcm (- numeral n) a = lcm (numeral n) a"

  1440   by (fact lcm_neg1)

  1441

  1442 lemma lcm_neg_numeral_2 [simp]: "lcm a (- numeral n) = lcm a (numeral n)"

  1443   by (fact lcm_neg2)

  1444

  1445 end

  1446

  1447

  1448 subsection \<open>Typical instances\<close>

  1449

  1450 instantiation nat :: euclidean_semiring

  1451 begin

  1452

  1453 definition [simp]:

  1454   "euclidean_size_nat = (id :: nat \<Rightarrow> nat)"

  1455

  1456 definition [simp]:

  1457   "unit_factor_nat (n::nat) = (if n = 0 then 0 else 1 :: nat)"

  1458

  1459 instance proof

  1460 qed simp_all

  1461

  1462 end

  1463

  1464 instantiation int :: euclidean_ring

  1465 begin

  1466

  1467 definition [simp]:

  1468   "euclidean_size_int = (nat \<circ> abs :: int \<Rightarrow> nat)"

  1469

  1470 definition [simp]:

  1471   "unit_factor_int = (sgn :: int \<Rightarrow> int)"

  1472

  1473 instance

  1474 by standard (auto simp add: abs_mult nat_mult_distrib sgn_times split: abs_split)

  1475

  1476 end

  1477

  1478 instantiation poly :: (field) euclidean_ring

  1479 begin

  1480

  1481 definition euclidean_size_poly :: "'a poly \<Rightarrow> nat"

  1482   where "euclidean_size p = (if p = 0 then 0 else Suc (degree p))"

  1483

  1484 lemma euclidenan_size_poly_minus_one_degree [simp]:

  1485   "euclidean_size p - 1 = degree p"

  1486   by (simp add: euclidean_size_poly_def)

  1487

  1488 lemma euclidean_size_poly_0 [simp]:

  1489   "euclidean_size (0::'a poly) = 0"

  1490   by (simp add: euclidean_size_poly_def)

  1491

  1492 lemma euclidean_size_poly_not_0 [simp]:

  1493   "p \<noteq> 0 \<Longrightarrow> euclidean_size p = Suc (degree p)"

  1494   by (simp add: euclidean_size_poly_def)

  1495

  1496 instance

  1497 proof

  1498   fix p q :: "'a poly"

  1499   assume "q \<noteq> 0"

  1500   then have "p mod q = 0 \<or> degree (p mod q) < degree q"

  1501     by (rule degree_mod_less [of q p])

  1502   with \<open>q \<noteq> 0\<close> show "euclidean_size (p mod q) < euclidean_size q"

  1503     by (cases "p mod q = 0") simp_all

  1504 next

  1505   fix p q :: "'a poly"

  1506   assume "q \<noteq> 0"

  1507   from \<open>q \<noteq> 0\<close> have "degree p \<le> degree (p * q)"

  1508     by (rule degree_mult_right_le)

  1509   with \<open>q \<noteq> 0\<close> show "euclidean_size p \<le> euclidean_size (p * q)"

  1510     by (cases "p = 0") simp_all

  1511 qed

  1512

  1513 end

  1514

  1515 end