src/HOL/Number_Theory/Euclidean_Algorithm.thy
 author haftmann Fri Jun 19 07:53:35 2015 +0200 (2015-06-19) changeset 60517 f16e4fb20652 parent 60516 0826b7025d07 child 60526 fad653acf58f permissions -rw-r--r--
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
     1 (* Author: Manuel Eberl *)

     2

     3 section {* Abstract euclidean algorithm *}

     4

     5 theory Euclidean_Algorithm

     6 imports Complex_Main

     7 begin

     8

     9 text {*

    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   \item a normalization factor such that two associated numbers are equal iff

    17         they are the same when divd by their normalization factors.

    18   \end{itemize}

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

    20   for any Euclidean semiring.

    21 *}

    22 class euclidean_semiring = semiring_div +

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

    24   fixes normalization_factor :: "'a \<Rightarrow> 'a"

    25   assumes mod_size_less [simp]:

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

    27   assumes size_mult_mono:

    28     "b \<noteq> 0 \<Longrightarrow> euclidean_size (a * b) \<ge> euclidean_size a"

    29   assumes normalization_factor_is_unit [intro,simp]:

    30     "a \<noteq> 0 \<Longrightarrow> is_unit (normalization_factor a)"

    31   assumes normalization_factor_mult: "normalization_factor (a * b) =

    32     normalization_factor a * normalization_factor b"

    33   assumes normalization_factor_unit: "is_unit a \<Longrightarrow> normalization_factor a = a"

    34   assumes normalization_factor_0 [simp]: "normalization_factor 0 = 0"

    35 begin

    36

    37 lemma normalization_factor_dvd [simp]:

    38   "a \<noteq> 0 \<Longrightarrow> normalization_factor a dvd b"

    39   by (rule unit_imp_dvd, simp)

    40

    41 lemma normalization_factor_1 [simp]:

    42   "normalization_factor 1 = 1"

    43   by (simp add: normalization_factor_unit)

    44

    45 lemma normalization_factor_0_iff [simp]:

    46   "normalization_factor a = 0 \<longleftrightarrow> a = 0"

    47 proof

    48   assume "normalization_factor a = 0"

    49   hence "\<not> is_unit (normalization_factor a)"

    50     by simp

    51   then show "a = 0" by auto

    52 qed simp

    53

    54 lemma normalization_factor_pow:

    55   "normalization_factor (a ^ n) = normalization_factor a ^ n"

    56   by (induct n) (simp_all add: normalization_factor_mult power_Suc2)

    57

    58 lemma normalization_correct [simp]:

    59   "normalization_factor (a div normalization_factor a) = (if a = 0 then 0 else 1)"

    60 proof (cases "a = 0", simp)

    61   assume "a \<noteq> 0"

    62   let ?nf = "normalization_factor"

    63   from normalization_factor_is_unit[OF a \<noteq> 0] have "?nf a \<noteq> 0"

    64     by auto

    65   have "?nf (a div ?nf a) * ?nf (?nf a) = ?nf (a div ?nf a * ?nf a)"

    66     by (simp add: normalization_factor_mult)

    67   also have "a div ?nf a * ?nf a = a" using a \<noteq> 0

    68     by simp

    69   also have "?nf (?nf a) = ?nf a" using a \<noteq> 0

    70     normalization_factor_is_unit normalization_factor_unit by simp

    71   finally have "normalization_factor (a div normalization_factor a) = 1"

    72     using ?nf a \<noteq> 0 by (metis div_mult_self2_is_id div_self)

    73   with a \<noteq> 0 show ?thesis by simp

    74 qed

    75

    76 lemma normalization_0_iff [simp]:

    77   "a div normalization_factor a = 0 \<longleftrightarrow> a = 0"

    78   by (cases "a = 0", simp, subst unit_eq_div1, blast, simp)

    79

    80 lemma mult_div_normalization [simp]:

    81   "b * (1 div normalization_factor a) = b div normalization_factor a"

    82   by (cases "a = 0") simp_all

    83

    84 lemma associated_iff_normed_eq:

    85   "associated a b \<longleftrightarrow> a div normalization_factor a = b div normalization_factor b"

    86 proof (cases "b = 0", simp, cases "a = 0", metis associated_0(1) normalization_0_iff, rule iffI)

    87   let ?nf = normalization_factor

    88   assume "a \<noteq> 0" "b \<noteq> 0" "a div ?nf a = b div ?nf b"

    89   hence "a = b * (?nf a div ?nf b)"

    90     apply (subst (asm) unit_eq_div1, blast, subst (asm) unit_div_commute, blast)

    91     apply (subst div_mult_swap, simp, simp)

    92     done

    93   with a \<noteq> 0 b \<noteq> 0 have "\<exists>c. is_unit c \<and> a = c * b"

    94     by (intro exI[of _ "?nf a div ?nf b"], force simp: mult_ac)

    95   then obtain c where "is_unit c" and "a = c * b" by blast

    96   then show "associated a b" by (rule is_unit_associatedI)

    97 next

    98   let ?nf = normalization_factor

    99   assume "a \<noteq> 0" "b \<noteq> 0" "associated a b"

   100   then obtain c where "is_unit c" and "a = c * b" by (blast elim: associated_is_unitE)

   101   then show "a div ?nf a = b div ?nf b"

   102     apply (simp only: a = c * b normalization_factor_mult normalization_factor_unit)

   103     apply (rule div_mult_mult1, force)

   104     done

   105   qed

   106

   107 lemma normed_associated_imp_eq:

   108   "associated a b \<Longrightarrow> normalization_factor a \<in> {0, 1} \<Longrightarrow> normalization_factor b \<in> {0, 1} \<Longrightarrow> a = b"

   109   by (simp add: associated_iff_normed_eq, elim disjE, simp_all)

   110

   111 lemmas normalization_factor_dvd_iff [simp] =

   112   unit_dvd_iff [OF normalization_factor_is_unit]

   113

   114 lemma euclidean_division:

   115   fixes a :: 'a and b :: 'a

   116   assumes "b \<noteq> 0"

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

   118     and "euclidean_size t < euclidean_size b"

   119 proof -

   120   from div_mod_equality[of a b 0]

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

   122   with that and assms show ?thesis by force

   123 qed

   124

   125 lemma dvd_euclidean_size_eq_imp_dvd:

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

   127   shows "a dvd b"

   128 proof (subst dvd_eq_mod_eq_0, rule ccontr)

   129   assume "b mod a \<noteq> 0"

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

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

   132     with b mod a \<noteq> 0 have "c \<noteq> 0" by auto

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

   134       using size_mult_mono by force

   135   moreover from a \<noteq> 0 have "euclidean_size (b mod a) < euclidean_size a"

   136       using mod_size_less by blast

   137   ultimately show False using size_eq by simp

   138 qed

   139

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

   141 where

   142   "gcd_eucl a b = (if b = 0 then a div normalization_factor a else gcd_eucl b (a mod b))"

   143   by (pat_completeness, simp)

   144 termination by (relation "measure (euclidean_size \<circ> snd)", simp_all)

   145

   146 declare gcd_eucl.simps [simp del]

   147

   148 lemma gcd_induct: "\<lbrakk>\<And>b. P b 0; \<And>a b. 0 \<noteq> b \<Longrightarrow> P b (a mod b) \<Longrightarrow> P a b\<rbrakk> \<Longrightarrow> P a b"

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

   150   case ("1" m n)

   151     then show ?case by (cases "n = 0") auto

   152 qed

   153

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

   155 where

   156   "lcm_eucl a b = a * b div (gcd_eucl a b * normalization_factor (a * b))"

   157

   158   (* Somewhat complicated definition of Lcm that has the advantage of working

   159      for infinite sets as well *)

   160

   161 definition Lcm_eucl :: "'a set \<Rightarrow> 'a"

   162 where

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

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

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

   166        in l div normalization_factor l

   167       else 0)"

   168

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

   170 where

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

   172

   173 end

   174

   175 class euclidean_semiring_gcd = euclidean_semiring + gcd + Gcd +

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

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

   178 begin

   179

   180 lemma gcd_red:

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

   182   by (metis gcd_eucl.simps mod_0 mod_by_0 gcd_gcd_eucl)

   183

   184 lemma gcd_non_0:

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

   186   by (rule gcd_red)

   187

   188 lemma gcd_0_left:

   189   "gcd 0 a = a div normalization_factor a"

   190    by (simp only: gcd_gcd_eucl, subst gcd_eucl.simps, subst gcd_eucl.simps, simp add: Let_def)

   191

   192 lemma gcd_0:

   193   "gcd a 0 = a div normalization_factor a"

   194   by (simp only: gcd_gcd_eucl, subst gcd_eucl.simps, simp add: Let_def)

   195

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

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

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

   199   fix a b :: 'a

   200   assume IH1: "b \<noteq> 0 \<Longrightarrow> gcd b (a mod b) dvd b"

   201   assume IH2: "b \<noteq> 0 \<Longrightarrow> gcd b (a mod b) dvd (a mod b)"

   202

   203   have "gcd a b dvd a \<and> gcd a b dvd b"

   204   proof (cases "b = 0")

   205     case True

   206       then show ?thesis by (cases "a = 0", simp_all add: gcd_0)

   207   next

   208     case False

   209       with IH1 and IH2 show ?thesis by (simp add: gcd_non_0 dvd_mod_iff)

   210   qed

   211   then show "gcd a b dvd a" "gcd a b dvd b" by simp_all

   212 qed

   213

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

   215   by (rule dvd_trans, assumption, rule gcd_dvd1)

   216

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

   218   by (rule dvd_trans, assumption, rule gcd_dvd2)

   219

   220 lemma gcd_greatest:

   221   fixes k a b :: 'a

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

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

   224   case (1 a b)

   225   show ?case

   226     proof (cases "b = 0")

   227       assume "b = 0"

   228       with 1 show ?thesis by (cases "a = 0", simp_all add: gcd_0)

   229     next

   230       assume "b \<noteq> 0"

   231       with 1 show ?thesis by (simp add: gcd_non_0 dvd_mod_iff)

   232     qed

   233 qed

   234

   235 lemma dvd_gcd_iff:

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

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

   238

   239 lemmas gcd_greatest_iff = dvd_gcd_iff

   240

   241 lemma gcd_zero [simp]:

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

   243   by (metis dvd_0_left dvd_refl gcd_dvd1 gcd_dvd2 gcd_greatest)+

   244

   245 lemma normalization_factor_gcd [simp]:

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

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

   248   fix a b :: 'a

   249   assume IH: "b \<noteq> 0 \<Longrightarrow> ?f b (a mod b) = ?g b (a mod b)"

   250   then show "?f a b = ?g a b" by (cases "b = 0", auto simp: gcd_non_0 gcd_0)

   251 qed

   252

   253 lemma gcdI:

   254   "k dvd a \<Longrightarrow> k dvd b \<Longrightarrow> (\<And>l. l dvd a \<Longrightarrow> l dvd b \<Longrightarrow> l dvd k)

   255     \<Longrightarrow> normalization_factor k = (if k = 0 then 0 else 1) \<Longrightarrow> k = gcd a b"

   256   by (intro normed_associated_imp_eq) (auto simp: associated_def intro: gcd_greatest)

   257

   258 sublocale gcd!: abel_semigroup gcd

   259 proof

   260   fix a b c

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

   262   proof (rule gcdI)

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

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

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

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

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

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

   269       by (rule gcd_greatest)

   270     show "normalization_factor (gcd (gcd a b) c) =  (if gcd (gcd a b) c = 0 then 0 else 1)"

   271       by auto

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

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

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

   275     with l dvd a show "l dvd gcd (gcd a b) c"

   276       by (intro gcd_greatest)

   277   qed

   278 next

   279   fix a b

   280   show "gcd a b = gcd b a"

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

   282 qed

   283

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

   285     normalization_factor d = (if d = 0 then 0 else 1) \<and>

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

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

   288

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

   290   using mult_dvd_mono [of 1] by auto

   291

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

   293   by (rule sym, rule gcdI, simp_all)

   294

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

   296   by (rule sym, rule gcdI, simp_all)

   297

   298 lemma gcd_proj2_if_dvd:

   299   "b dvd a \<Longrightarrow> gcd a b = b div normalization_factor b"

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

   301

   302 lemma gcd_proj1_if_dvd:

   303   "a dvd b \<Longrightarrow> gcd a b = a div normalization_factor a"

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

   305

   306 lemma gcd_proj1_iff: "gcd m n = m div normalization_factor m \<longleftrightarrow> m dvd n"

   307 proof

   308   assume A: "gcd m n = m div normalization_factor m"

   309   show "m dvd n"

   310   proof (cases "m = 0")

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

   312     from A have B: "m = gcd m n * normalization_factor m"

   313       by (simp add: unit_eq_div2)

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

   315   qed (insert A, simp)

   316 next

   317   assume "m dvd n"

   318   then show "gcd m n = m div normalization_factor m" by (rule gcd_proj1_if_dvd)

   319 qed

   320

   321 lemma gcd_proj2_iff: "gcd m n = n div normalization_factor n \<longleftrightarrow> n dvd m"

   322   by (subst gcd.commute, simp add: gcd_proj1_iff)

   323

   324 lemma gcd_mod1 [simp]:

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

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

   327

   328 lemma gcd_mod2 [simp]:

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

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

   331

   332 lemma normalization_factor_dvd' [simp]:

   333   "normalization_factor a dvd a"

   334   by (cases "a = 0", simp_all)

   335

   336 lemma gcd_mult_distrib':

   337   "k div normalization_factor k * gcd a b = gcd (k*a) (k*b)"

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

   339   case (1 a b)

   340   show ?case

   341   proof (cases "b = 0")

   342     case True

   343     then show ?thesis by (simp add: normalization_factor_mult gcd_0 algebra_simps div_mult_div_if_dvd)

   344   next

   345     case False

   346     hence "k div normalization_factor k * gcd a b =  gcd (k * b) (k * (a mod b))"

   347       using 1 by (subst gcd_red, simp)

   348     also have "... = gcd (k * a) (k * b)"

   349       by (simp add: mult_mod_right gcd.commute)

   350     finally show ?thesis .

   351   qed

   352 qed

   353

   354 lemma gcd_mult_distrib:

   355   "k * gcd a b = gcd (k*a) (k*b) * normalization_factor k"

   356 proof-

   357   let ?nf = "normalization_factor"

   358   from gcd_mult_distrib'

   359     have "gcd (k*a) (k*b) = k div ?nf k * gcd a b" ..

   360   also have "... = k * gcd a b div ?nf k"

   361     by (metis dvd_div_mult dvd_eq_mod_eq_0 mod_0 normalization_factor_dvd)

   362   finally show ?thesis

   363     by simp

   364 qed

   365

   366 lemma euclidean_size_gcd_le1 [simp]:

   367   assumes "a \<noteq> 0"

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

   369 proof -

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

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

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

   373 qed

   374

   375 lemma euclidean_size_gcd_le2 [simp]:

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

   377   by (subst gcd.commute, rule euclidean_size_gcd_le1)

   378

   379 lemma euclidean_size_gcd_less1:

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

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

   382 proof (rule ccontr)

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

   384   with a \<noteq> 0 have "euclidean_size (gcd a b) = euclidean_size a"

   385     by (intro le_antisym, simp_all)

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

   387   hence "a dvd b" using dvd_gcd_D2 by blast

   388   with \<not>a dvd b show False by contradiction

   389 qed

   390

   391 lemma euclidean_size_gcd_less2:

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

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

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

   395

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

   397   apply (rule gcdI)

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

   399   apply (rule gcd_dvd2)

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

   401   apply (subst normalization_factor_gcd, simp add: gcd_0)

   402   done

   403

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

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

   406

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

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

   409

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

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

   412

   413 lemma gcd_idem: "gcd a a = a div normalization_factor a"

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

   415

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

   417   apply (rule gcdI)

   418   apply (simp add: ac_simps)

   419   apply (rule gcd_dvd2)

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

   421   apply simp

   422   done

   423

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

   425   apply (rule gcdI)

   426   apply simp

   427   apply (rule dvd_trans, rule gcd_dvd2, rule gcd_dvd2)

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

   429   apply simp

   430   done

   431

   432 lemma comp_fun_idem_gcd: "comp_fun_idem gcd"

   433 proof

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

   435     by (simp add: fun_eq_iff ac_simps)

   436 next

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

   438     by (simp add: fun_eq_iff gcd_left_idem)

   439 qed

   440

   441 lemma coprime_dvd_mult:

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

   443   shows "c dvd a"

   444 proof -

   445   let ?nf = "normalization_factor"

   446   from assms gcd_mult_distrib [of a c b]

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

   448   from c dvd a * b show ?thesis by (subst A, simp_all add: gcd_greatest)

   449 qed

   450

   451 lemma coprime_dvd_mult_iff:

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

   453   by (rule, rule coprime_dvd_mult, simp_all)

   454

   455 lemma gcd_dvd_antisym:

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

   457 proof (rule gcdI)

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

   459   have "gcd c d dvd c" by simp

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

   461   have "gcd c d dvd d" by simp

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

   463   show "normalization_factor (gcd a b) = (if gcd a b = 0 then 0 else 1)"

   464     by simp

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

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

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

   468 qed

   469

   470 lemma gcd_mult_cancel:

   471   assumes "gcd k n = 1"

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

   473 proof (rule gcd_dvd_antisym)

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

   475   also note gcd k n = 1

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

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

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

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

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

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

   482 qed

   483

   484 lemma coprime_crossproduct:

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

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

   487 proof

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

   489 next

   490   assume ?lhs

   491   from ?lhs have "a dvd b * d" unfolding associated_def by (metis dvd_mult_left)

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

   493   moreover from ?lhs have "b dvd a * c" unfolding associated_def by (metis dvd_mult_left)

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

   495   moreover from ?lhs have "c dvd d * b"

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

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

   498   moreover from ?lhs have "d dvd c * a"

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

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

   501   ultimately show ?rhs unfolding associated_def by simp

   502 qed

   503

   504 lemma gcd_add1 [simp]:

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

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

   507

   508 lemma gcd_add2 [simp]:

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

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

   511

   512 lemma gcd_add_mult: "gcd m (k * m + n) = gcd m n"

   513   by (subst gcd.commute, subst gcd_red, simp)

   514

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

   516   by (rule sym, rule gcdI, simp_all)

   517

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

   519   by (auto intro: coprimeI gcd_greatest dvd_gcd_D1 dvd_gcd_D2)

   520

   521 lemma div_gcd_coprime:

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

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

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

   525   shows "gcd a' b' = 1"

   526 proof (rule coprimeI)

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

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

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

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

   531     by (simp_all only: ac_simps)

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

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

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

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

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

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

   538   ultimately have "1 = l * u"

   539     using d \<noteq> 0 by simp

   540   then show "l dvd 1" ..

   541 qed

   542

   543 lemma coprime_mult:

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

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

   546   apply (subst gcd.commute)

   547   using da apply (subst gcd_mult_cancel)

   548   apply (subst gcd.commute, assumption)

   549   apply (subst gcd.commute, rule db)

   550   done

   551

   552 lemma coprime_lmult:

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

   554   shows "gcd d a = 1"

   555 proof (rule coprimeI)

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

   557   hence "l dvd a * b" by simp

   558   with l dvd d and dab show "l dvd 1" by (auto intro: gcd_greatest)

   559 qed

   560

   561 lemma coprime_rmult:

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

   563   shows "gcd d b = 1"

   564 proof (rule coprimeI)

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

   566   hence "l dvd a * b" by simp

   567   with l dvd d and dab show "l dvd 1" by (auto intro: gcd_greatest)

   568 qed

   569

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

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

   572

   573 lemma gcd_coprime:

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

   575   shows "gcd a' b' = 1"

   576 proof -

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

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

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

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

   581   finally show ?thesis .

   582 qed

   583

   584 lemma coprime_power:

   585   assumes "0 < n"

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

   587 using assms proof (induct n)

   588   case (Suc n) then show ?case

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

   590 qed simp

   591

   592 lemma gcd_coprime_exists:

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

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

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

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

   597   apply (insert nz, auto intro: div_gcd_coprime)

   598   done

   599

   600 lemma coprime_exp:

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

   602   by (induct n, simp_all add: coprime_mult)

   603

   604 lemma coprime_exp2 [intro]:

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

   606   apply (rule coprime_exp)

   607   apply (subst gcd.commute)

   608   apply (rule coprime_exp)

   609   apply (subst gcd.commute)

   610   apply assumption

   611   done

   612

   613 lemma gcd_exp:

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

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

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

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

   618 next

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

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

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

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

   623   also note gcd_mult_distrib

   624   also have "normalization_factor ((gcd a b)^n) = 1"

   625     by (simp add: normalization_factor_pow A)

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

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

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

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

   630   finally show ?thesis by simp

   631 qed

   632

   633 lemma coprime_common_divisor:

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

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

   636   apply simp

   637   apply (erule (1) gcd_greatest)

   638   done

   639

   640 lemma division_decomp:

   641   assumes dc: "a dvd b * c"

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

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

   644   assume "gcd a b = 0"

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

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

   647   then show ?thesis by blast

   648 next

   649   let ?d = "gcd a b"

   650   assume "?d \<noteq> 0"

   651   from gcd_coprime_exists[OF this]

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

   653     by blast

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

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

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

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

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

   659   with coprime_dvd_mult[OF ab'(3)]

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

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

   662   then show ?thesis by blast

   663 qed

   664

   665 lemma pow_divs_pow:

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

   667   shows "a dvd b"

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

   669   assume "gcd a b = 0"

   670   then show ?thesis by simp

   671 next

   672   let ?d = "gcd a b"

   673   assume "?d \<noteq> 0"

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

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

   676   from gcd_coprime_exists[OF ?d \<noteq> 0]

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

   678     by blast

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

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

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

   682     by (simp only: power_mult_distrib ac_simps)

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

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

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

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

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

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

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

   690 qed

   691

   692 lemma pow_divs_eq [simp]:

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

   694   by (auto intro: pow_divs_pow dvd_power_same)

   695

   696 lemma divs_mult:

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

   698   shows "m * n dvd r"

   699 proof -

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

   701     unfolding dvd_def by blast

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

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

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

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

   706   then show ?thesis unfolding dvd_def by blast

   707 qed

   708

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

   710   by (subst add_commute, simp)

   711

   712 lemma setprod_coprime [rule_format]:

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

   714   apply (cases "finite A")

   715   apply (induct set: finite)

   716   apply (auto simp add: gcd_mult_cancel)

   717   done

   718

   719 lemma coprime_divisors:

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

   721   shows "gcd d e = 1"

   722 proof -

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

   724     unfolding dvd_def by blast

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

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

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

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

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

   730 qed

   731

   732 lemma invertible_coprime:

   733   assumes "a * b mod m = 1"

   734   shows "coprime a m"

   735 proof -

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

   737     by simp

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

   739     by simp

   740   then have "coprime m a"

   741     by (rule coprime_lmult)

   742   then show ?thesis

   743     by (simp add: ac_simps)

   744 qed

   745

   746 lemma lcm_gcd:

   747   "lcm a b = a * b div (gcd a b * normalization_factor (a*b))"

   748   by (simp only: lcm_lcm_eucl gcd_gcd_eucl lcm_eucl_def)

   749

   750 lemma lcm_gcd_prod:

   751   "lcm a b * gcd a b = a * b div normalization_factor (a*b)"

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

   753   let ?nf = normalization_factor

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

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

   756   from lcm_gcd have "lcm a b * gcd a b = gcd a b * (a * b div (?nf (a*b) * gcd a b))"

   757     by (simp add: mult_ac)

   758   also from a * b \<noteq> 0 have "... = a * b div ?nf (a*b)"

   759     by (simp add: div_mult_swap mult.commute)

   760   finally show ?thesis .

   761 qed (auto simp add: lcm_gcd)

   762

   763 lemma lcm_dvd1 [iff]:

   764   "a dvd lcm a b"

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

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

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

   768   let ?c = "1 div normalization_factor (a * b)"

   769   from a * b \<noteq> 0 have [simp]: "is_unit (normalization_factor (a * b))" by simp

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

   771     by (simp add: div_mult_swap unit_div_commute)

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

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

   774     by (subst (asm) div_mult_self2_is_id, simp_all)

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

   776     by (metis div_mult_swap gcd_dvd2 mult_assoc)

   777   finally show ?thesis by (rule dvdI)

   778 qed (auto simp add: lcm_gcd)

   779

   780 lemma lcm_least:

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

   782 proof (cases "k = 0")

   783   let ?nf = normalization_factor

   784   assume "k \<noteq> 0"

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

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

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

   788   hence "gcd a b \<noteq> 0" using k \<noteq> 0 by auto

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

   790     unfolding dvd_def by blast

   791   with k \<noteq> 0 have "r * s \<noteq> 0"

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

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

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

   795   from is_unit (?nf k) and is_unit (?nf (r * s)) have "is_unit ?c" by (rule unit_div)

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

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

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

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

   800     by (subst (3) k = a * r, subst (3) k = b * s, simp add: algebra_simps)

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

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

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

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

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

   806     by (simp add: algebra_simps)

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

   808     by (metis div_mult_self2_is_id)

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

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

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

   812     by (simp add: algebra_simps)

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

   814     by (metis mult.commute div_mult_self2_is_id)

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

   816     by (metis div_mult_self2_is_id mult_assoc)

   817   also have "... = lcm a b * (gcd s r * ?nf (a*b) div ?c)" using is_unit ?c

   818     by (simp add: unit_simps)

   819   finally show ?thesis by (rule dvdI)

   820 qed simp

   821

   822 lemma lcm_zero:

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

   824 proof -

   825   let ?nf = normalization_factor

   826   {

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

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

   829     moreover from a \<noteq> 0 and b \<noteq> 0 have "gcd a b \<noteq> 0" by simp

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

   831   } moreover {

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

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

   834   }

   835   ultimately show ?thesis by blast

   836 qed

   837

   838 lemmas lcm_0_iff = lcm_zero

   839

   840 lemma gcd_lcm:

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

   842   shows "gcd a b = a * b div (lcm a b * normalization_factor (a * b))"

   843 proof-

   844   from assms have "gcd a b \<noteq> 0" by (simp add: lcm_zero)

   845   let ?c = "normalization_factor (a * b)"

   846   from lcm a b \<noteq> 0 have "?c \<noteq> 0" by (intro notI, simp add: lcm_zero no_zero_divisors)

   847   hence "is_unit ?c" by simp

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

   849     by (subst (2) div_mult_self2_is_id[OF lcm a b \<noteq> 0, symmetric], simp add: mult_ac)

   850   also from is_unit ?c have "... = a * b div (lcm a b * ?c)"

   851     by (metis ?c \<noteq> 0 div_mult_mult1 dvd_mult_div_cancel mult_commute normalization_factor_dvd')

   852   finally show ?thesis .

   853 qed

   854

   855 lemma normalization_factor_lcm [simp]:

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

   857 proof (cases "a = 0 \<or> b = 0")

   858   case True then show ?thesis

   859     by (auto simp add: lcm_gcd)

   860 next

   861   case False

   862   let ?nf = normalization_factor

   863   from lcm_gcd_prod[of a b]

   864     have "?nf (lcm a b) * ?nf (gcd a b) = ?nf (a*b) div ?nf (a*b)"

   865     by (metis div_by_0 div_self normalization_correct normalization_factor_0 normalization_factor_mult)

   866   also have "... = (if a*b = 0 then 0 else 1)"

   867     by simp

   868   finally show ?thesis using False by simp

   869 qed

   870

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

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

   873

   874 lemma lcmI:

   875   "\<lbrakk>a dvd k; b dvd k; \<And>l. a dvd l \<Longrightarrow> b dvd l \<Longrightarrow> k dvd l;

   876     normalization_factor k = (if k = 0 then 0 else 1)\<rbrakk> \<Longrightarrow> k = lcm a b"

   877   by (intro normed_associated_imp_eq) (auto simp: associated_def intro: lcm_least)

   878

   879 sublocale lcm!: abel_semigroup lcm

   880 proof

   881   fix a b c

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

   883   proof (rule lcmI)

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

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

   886

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

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

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

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

   891

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

   893     have "b dvd lcm b c" by simp

   894     from this and lcm b c dvd l have "b dvd l" by (rule dvd_trans)

   895     have "c dvd lcm b c" by simp

   896     from this and lcm b c dvd l have "c dvd l" by (rule dvd_trans)

   897     from a dvd l and b dvd l have "lcm a b dvd l" by (rule lcm_least)

   898     from this and c dvd l show "lcm (lcm a b) c dvd l" by (rule lcm_least)

   899   qed (simp add: lcm_zero)

   900 next

   901   fix a b

   902   show "lcm a b = lcm b a"

   903     by (simp add: lcm_gcd ac_simps)

   904 qed

   905

   906 lemma dvd_lcm_D1:

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

   908   by (rule dvd_trans, rule lcm_dvd1, assumption)

   909

   910 lemma dvd_lcm_D2:

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

   912   by (rule dvd_trans, rule lcm_dvd2, assumption)

   913

   914 lemma gcd_dvd_lcm [simp]:

   915   "gcd a b dvd lcm a b"

   916   by (metis dvd_trans gcd_dvd2 lcm_dvd2)

   917

   918 lemma lcm_1_iff:

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

   920 proof

   921   assume "lcm a b = 1"

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

   923 next

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

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

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

   927   hence "lcm a b = normalization_factor (lcm a b)"

   928     by (subst normalization_factor_unit, simp_all)

   929   also have "\<dots> = 1" using is_unit a \<and> is_unit b

   930     by auto

   931   finally show "lcm a b = 1" .

   932 qed

   933

   934 lemma lcm_0_left [simp]:

   935   "lcm 0 a = 0"

   936   by (rule sym, rule lcmI, simp_all)

   937

   938 lemma lcm_0 [simp]:

   939   "lcm a 0 = 0"

   940   by (rule sym, rule lcmI, simp_all)

   941

   942 lemma lcm_unique:

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

   944   normalization_factor d = (if d = 0 then 0 else 1) \<and>

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

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

   947

   948 lemma dvd_lcm_I1 [simp]:

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

   950   by (metis lcm_dvd1 dvd_trans)

   951

   952 lemma dvd_lcm_I2 [simp]:

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

   954   by (metis lcm_dvd2 dvd_trans)

   955

   956 lemma lcm_1_left [simp]:

   957   "lcm 1 a = a div normalization_factor a"

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

   959

   960 lemma lcm_1_right [simp]:

   961   "lcm a 1 = a div normalization_factor a"

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

   963

   964 lemma lcm_coprime:

   965   "gcd a b = 1 \<Longrightarrow> lcm a b = a * b div normalization_factor (a*b)"

   966   by (subst lcm_gcd) simp

   967

   968 lemma lcm_proj1_if_dvd:

   969   "b dvd a \<Longrightarrow> lcm a b = a div normalization_factor a"

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

   971

   972 lemma lcm_proj2_if_dvd:

   973   "a dvd b \<Longrightarrow> lcm a b = b div normalization_factor b"

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

   975

   976 lemma lcm_proj1_iff:

   977   "lcm m n = m div normalization_factor m \<longleftrightarrow> n dvd m"

   978 proof

   979   assume A: "lcm m n = m div normalization_factor m"

   980   show "n dvd m"

   981   proof (cases "m = 0")

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

   983     from A have B: "m = lcm m n * normalization_factor m"

   984       by (simp add: unit_eq_div2)

   985     show ?thesis by (subst B, simp)

   986   qed simp

   987 next

   988   assume "n dvd m"

   989   then show "lcm m n = m div normalization_factor m" by (rule lcm_proj1_if_dvd)

   990 qed

   991

   992 lemma lcm_proj2_iff:

   993   "lcm m n = n div normalization_factor n \<longleftrightarrow> m dvd n"

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

   995

   996 lemma euclidean_size_lcm_le1:

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

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

   999 proof -

  1000   have "a dvd lcm a b" by (rule lcm_dvd1)

  1001   then obtain c where A: "lcm a b = a * c" unfolding dvd_def by blast

  1002   with a \<noteq> 0 and b \<noteq> 0 have "c \<noteq> 0" by (auto simp: lcm_zero)

  1003   then show ?thesis by (subst A, intro size_mult_mono)

  1004 qed

  1005

  1006 lemma euclidean_size_lcm_le2:

  1007   "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> euclidean_size b \<le> euclidean_size (lcm a b)"

  1008   using euclidean_size_lcm_le1 [of b a] by (simp add: ac_simps)

  1009

  1010 lemma euclidean_size_lcm_less1:

  1011   assumes "b \<noteq> 0" and "\<not>b dvd a"

  1012   shows "euclidean_size a < euclidean_size (lcm a b)"

  1013 proof (rule ccontr)

  1014   from assms have "a \<noteq> 0" by auto

  1015   assume "\<not>euclidean_size a < euclidean_size (lcm a b)"

  1016   with a \<noteq> 0 and b \<noteq> 0 have "euclidean_size (lcm a b) = euclidean_size a"

  1017     by (intro le_antisym, simp, intro euclidean_size_lcm_le1)

  1018   with assms have "lcm a b dvd a"

  1019     by (rule_tac dvd_euclidean_size_eq_imp_dvd) (auto simp: lcm_zero)

  1020   hence "b dvd a" by (rule dvd_lcm_D2)

  1021   with \<not>b dvd a show False by contradiction

  1022 qed

  1023

  1024 lemma euclidean_size_lcm_less2:

  1025   assumes "a \<noteq> 0" and "\<not>a dvd b"

  1026   shows "euclidean_size b < euclidean_size (lcm a b)"

  1027   using assms euclidean_size_lcm_less1 [of a b] by (simp add: ac_simps)

  1028

  1029 lemma lcm_mult_unit1:

  1030   "is_unit a \<Longrightarrow> lcm (b * a) c = lcm b c"

  1031   apply (rule lcmI)

  1032   apply (rule dvd_trans[of _ "b * a"], simp, rule lcm_dvd1)

  1033   apply (rule lcm_dvd2)

  1034   apply (rule lcm_least, simp add: unit_simps, assumption)

  1035   apply (subst normalization_factor_lcm, simp add: lcm_zero)

  1036   done

  1037

  1038 lemma lcm_mult_unit2:

  1039   "is_unit a \<Longrightarrow> lcm b (c * a) = lcm b c"

  1040   using lcm_mult_unit1 [of a c b] by (simp add: ac_simps)

  1041

  1042 lemma lcm_div_unit1:

  1043   "is_unit a \<Longrightarrow> lcm (b div a) c = lcm b c"

  1044   by (erule is_unitE [of _ b]) (simp add: lcm_mult_unit1)

  1045

  1046 lemma lcm_div_unit2:

  1047   "is_unit a \<Longrightarrow> lcm b (c div a) = lcm b c"

  1048   by (erule is_unitE [of _ c]) (simp add: lcm_mult_unit2)

  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_dvd [simp]: "(\<forall>a\<in>A. a dvd l') \<Longrightarrow> Lcm A dvd l'"

  1080   and normalization_factor_Lcm [simp]:

  1081           "normalization_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     normalization_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 \<forall>a\<in>A. a dvd l have "\<forall>a\<in>A. a dvd gcd l l'" by (auto intro: gcd_greatest)

  1105       moreover from l \<noteq> 0 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 l \<noteq> 0 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 l = gcd l l' * a ..

  1117         also note euclidean_size l = n

  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: euclidean_size l = n)

  1122       with l \<noteq> 0 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 (\<forall>a\<in>A. a dvd l) and normalization_factor_is_unit[OF l \<noteq> 0] and l \<noteq> 0

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

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

  1129         normalization_factor (l div normalization_factor l) =

  1130         (if l div normalization_factor l = 0 then 0 else 1)"

  1131       by (auto simp: unit_simps)

  1132     also from True have "l div normalization_factor 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 l' assume "\<forall>a\<in>A. a dvd l'" then show "Lcm A dvd l'" using A by blast}

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

  1141 qed

  1142

  1143 lemma LcmI:

  1144   "(\<And>a. a\<in>A \<Longrightarrow> a dvd l) \<Longrightarrow> (\<And>l'. (\<forall>a\<in>A. a dvd l') \<Longrightarrow> l dvd l') \<Longrightarrow>

  1145       normalization_factor l = (if l = 0 then 0 else 1) \<Longrightarrow> l = Lcm A"

  1146   by (intro normed_associated_imp_eq)

  1147     (auto intro: Lcm_dvd dvd_Lcm simp: associated_def)

  1148

  1149 lemma Lcm_subset:

  1150   "A \<subseteq> B \<Longrightarrow> Lcm A dvd Lcm B"

  1151   by (blast intro: Lcm_dvd dvd_Lcm)

  1152

  1153 lemma Lcm_Un:

  1154   "Lcm (A \<union> B) = lcm (Lcm A) (Lcm B)"

  1155   apply (rule lcmI)

  1156   apply (blast intro: Lcm_subset)

  1157   apply (blast intro: Lcm_subset)

  1158   apply (intro Lcm_dvd ballI, elim UnE)

  1159   apply (rule dvd_trans, erule dvd_Lcm, assumption)

  1160   apply (rule dvd_trans, erule dvd_Lcm, assumption)

  1161   apply simp

  1162   done

  1163

  1164 lemma Lcm_1_iff:

  1165   "Lcm A = 1 \<longleftrightarrow> (\<forall>a\<in>A. is_unit a)"

  1166 proof

  1167   assume "Lcm A = 1"

  1168   then show "\<forall>a\<in>A. is_unit a" by auto

  1169 qed (rule LcmI [symmetric], auto)

  1170

  1171 lemma Lcm_no_units:

  1172   "Lcm A = Lcm (A - {a. is_unit a})"

  1173 proof -

  1174   have "(A - {a. is_unit a}) \<union> {a\<in>A. is_unit a} = A" by blast

  1175   hence "Lcm A = lcm (Lcm (A - {a. is_unit a})) (Lcm {a\<in>A. is_unit a})"

  1176     by (simp add: Lcm_Un[symmetric])

  1177   also have "Lcm {a\<in>A. is_unit a} = 1" by (simp add: Lcm_1_iff)

  1178   finally show ?thesis by simp

  1179 qed

  1180

  1181 lemma Lcm_empty [simp]:

  1182   "Lcm {} = 1"

  1183   by (simp add: Lcm_1_iff)

  1184

  1185 lemma Lcm_eq_0 [simp]:

  1186   "0 \<in> A \<Longrightarrow> Lcm A = 0"

  1187   by (drule dvd_Lcm) simp

  1188

  1189 lemma Lcm0_iff':

  1190   "Lcm A = 0 \<longleftrightarrow> \<not>(\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l))"

  1191 proof

  1192   assume "Lcm A = 0"

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

  1194   proof

  1195     assume ex: "\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l)"

  1196     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

  1197     def n \<equiv> "LEAST n. \<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"

  1198     def l \<equiv> "SOME l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"

  1199     have "\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"

  1200       apply (subst n_def)

  1201       apply (rule LeastI[of _ "euclidean_size l\<^sub>0"])

  1202       apply (rule exI[of _ l\<^sub>0])

  1203       apply (simp add: l\<^sub>0_props)

  1204       done

  1205     from someI_ex[OF this] have "l \<noteq> 0" unfolding l_def by simp_all

  1206     hence "l div normalization_factor l \<noteq> 0" by simp

  1207     also from ex have "l div normalization_factor l = Lcm A"

  1208        by (simp only: Lcm_Lcm_eucl Lcm_eucl_def n_def l_def if_True Let_def)

  1209     finally show False using Lcm A = 0 by contradiction

  1210   qed

  1211 qed (simp only: Lcm_Lcm_eucl Lcm_eucl_def if_False)

  1212

  1213 lemma Lcm0_iff [simp]:

  1214   "finite A \<Longrightarrow> Lcm A = 0 \<longleftrightarrow> 0 \<in> A"

  1215 proof -

  1216   assume "finite A"

  1217   have "0 \<in> A \<Longrightarrow> Lcm A = 0"  by (intro dvd_0_left dvd_Lcm)

  1218   moreover {

  1219     assume "0 \<notin> A"

  1220     hence "\<Prod>A \<noteq> 0"

  1221       apply (induct rule: finite_induct[OF finite A])

  1222       apply simp

  1223       apply (subst setprod.insert, assumption, assumption)

  1224       apply (rule no_zero_divisors)

  1225       apply blast+

  1226       done

  1227     moreover from finite A have "\<forall>a\<in>A. a dvd \<Prod>A" by blast

  1228     ultimately have "\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l)" by blast

  1229     with Lcm0_iff' have "Lcm A \<noteq> 0" by simp

  1230   }

  1231   ultimately show "Lcm A = 0 \<longleftrightarrow> 0 \<in> A" by blast

  1232 qed

  1233

  1234 lemma Lcm_no_multiple:

  1235   "(\<forall>m. m \<noteq> 0 \<longrightarrow> (\<exists>a\<in>A. \<not>a dvd m)) \<Longrightarrow> Lcm A = 0"

  1236 proof -

  1237   assume "\<forall>m. m \<noteq> 0 \<longrightarrow> (\<exists>a\<in>A. \<not>a dvd m)"

  1238   hence "\<not>(\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l))" by blast

  1239   then show "Lcm A = 0" by (simp only: Lcm_Lcm_eucl Lcm_eucl_def if_False)

  1240 qed

  1241

  1242 lemma Lcm_insert [simp]:

  1243   "Lcm (insert a A) = lcm a (Lcm A)"

  1244 proof (rule lcmI)

  1245   fix l assume "a dvd l" and "Lcm A dvd l"

  1246   hence "\<forall>a\<in>A. a dvd l" by (blast intro: dvd_trans dvd_Lcm)

  1247   with a dvd l show "Lcm (insert a A) dvd l" by (force intro: Lcm_dvd)

  1248 qed (auto intro: Lcm_dvd dvd_Lcm)

  1249

  1250 lemma Lcm_finite:

  1251   assumes "finite A"

  1252   shows "Lcm A = Finite_Set.fold lcm 1 A"

  1253   by (induct rule: finite.induct[OF finite A])

  1254     (simp_all add: comp_fun_idem.fold_insert_idem[OF comp_fun_idem_lcm])

  1255

  1256 lemma Lcm_set [code_unfold]:

  1257   "Lcm (set xs) = fold lcm xs 1"

  1258   using comp_fun_idem.fold_set_fold[OF comp_fun_idem_lcm] Lcm_finite by (simp add: ac_simps)

  1259

  1260 lemma Lcm_singleton [simp]:

  1261   "Lcm {a} = a div normalization_factor a"

  1262   by simp

  1263

  1264 lemma Lcm_2 [simp]:

  1265   "Lcm {a,b} = lcm a b"

  1266   by (simp only: Lcm_insert Lcm_empty lcm_1_right)

  1267     (cases "b = 0", simp, rule lcm_div_unit2, simp)

  1268

  1269 lemma Lcm_coprime:

  1270   assumes "finite A" and "A \<noteq> {}"

  1271   assumes "\<And>a b. a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<noteq> b \<Longrightarrow> gcd a b = 1"

  1272   shows "Lcm A = \<Prod>A div normalization_factor (\<Prod>A)"

  1273 using assms proof (induct rule: finite_ne_induct)

  1274   case (insert a A)

  1275   have "Lcm (insert a A) = lcm a (Lcm A)" by simp

  1276   also from insert have "Lcm A = \<Prod>A div normalization_factor (\<Prod>A)" by blast

  1277   also have "lcm a \<dots> = lcm a (\<Prod>A)" by (cases "\<Prod>A = 0") (simp_all add: lcm_div_unit2)

  1278   also from insert have "gcd a (\<Prod>A) = 1" by (subst gcd.commute, intro setprod_coprime) auto

  1279   with insert have "lcm a (\<Prod>A) = \<Prod>(insert a A) div normalization_factor (\<Prod>(insert a A))"

  1280     by (simp add: lcm_coprime)

  1281   finally show ?case .

  1282 qed simp

  1283

  1284 lemma Lcm_coprime':

  1285   "card A \<noteq> 0 \<Longrightarrow> (\<And>a b. a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<noteq> b \<Longrightarrow> gcd a b = 1)

  1286     \<Longrightarrow> Lcm A = \<Prod>A div normalization_factor (\<Prod>A)"

  1287   by (rule Lcm_coprime) (simp_all add: card_eq_0_iff)

  1288

  1289 lemma Gcd_Lcm:

  1290   "Gcd A = Lcm {d. \<forall>a\<in>A. d dvd a}"

  1291   by (simp add: Gcd_Gcd_eucl Lcm_Lcm_eucl Gcd_eucl_def)

  1292

  1293 lemma Gcd_dvd [simp]: "a \<in> A \<Longrightarrow> Gcd A dvd a"

  1294   and dvd_Gcd [simp]: "(\<forall>a\<in>A. g' dvd a) \<Longrightarrow> g' dvd Gcd A"

  1295   and normalization_factor_Gcd [simp]:

  1296     "normalization_factor (Gcd A) = (if Gcd A = 0 then 0 else 1)"

  1297 proof -

  1298   fix a assume "a \<in> A"

  1299   hence "Lcm {d. \<forall>a\<in>A. d dvd a} dvd a" by (intro Lcm_dvd) blast

  1300   then show "Gcd A dvd a" by (simp add: Gcd_Lcm)

  1301 next

  1302   fix g' assume "\<forall>a\<in>A. g' dvd a"

  1303   hence "g' dvd Lcm {d. \<forall>a\<in>A. d dvd a}" by (intro dvd_Lcm) blast

  1304   then show "g' dvd Gcd A" by (simp add: Gcd_Lcm)

  1305 next

  1306   show "normalization_factor (Gcd A) = (if Gcd A = 0 then 0 else 1)"

  1307     by (simp add: Gcd_Lcm)

  1308 qed

  1309

  1310 lemma GcdI:

  1311   "(\<And>a. a\<in>A \<Longrightarrow> l dvd a) \<Longrightarrow> (\<And>l'. (\<forall>a\<in>A. l' dvd a) \<Longrightarrow> l' dvd l) \<Longrightarrow>

  1312     normalization_factor l = (if l = 0 then 0 else 1) \<Longrightarrow> l = Gcd A"

  1313   by (intro normed_associated_imp_eq)

  1314     (auto intro: Gcd_dvd dvd_Gcd simp: associated_def)

  1315

  1316 lemma Lcm_Gcd:

  1317   "Lcm A = Gcd {m. \<forall>a\<in>A. a dvd m}"

  1318   by (rule LcmI[symmetric]) (auto intro: dvd_Gcd Gcd_dvd)

  1319

  1320 lemma Gcd_0_iff:

  1321   "Gcd A = 0 \<longleftrightarrow> A \<subseteq> {0}"

  1322   apply (rule iffI)

  1323   apply (rule subsetI, drule Gcd_dvd, simp)

  1324   apply (auto intro: GcdI[symmetric])

  1325   done

  1326

  1327 lemma Gcd_empty [simp]:

  1328   "Gcd {} = 0"

  1329   by (simp add: Gcd_0_iff)

  1330

  1331 lemma Gcd_1:

  1332   "1 \<in> A \<Longrightarrow> Gcd A = 1"

  1333   by (intro GcdI[symmetric]) (auto intro: Gcd_dvd dvd_Gcd)

  1334

  1335 lemma Gcd_insert [simp]:

  1336   "Gcd (insert a A) = gcd a (Gcd A)"

  1337 proof (rule gcdI)

  1338   fix l assume "l dvd a" and "l dvd Gcd A"

  1339   hence "\<forall>a\<in>A. l dvd a" by (blast intro: dvd_trans Gcd_dvd)

  1340   with l dvd a show "l dvd Gcd (insert a A)" by (force intro: Gcd_dvd)

  1341 qed auto

  1342

  1343 lemma Gcd_finite:

  1344   assumes "finite A"

  1345   shows "Gcd A = Finite_Set.fold gcd 0 A"

  1346   by (induct rule: finite.induct[OF finite A])

  1347     (simp_all add: comp_fun_idem.fold_insert_idem[OF comp_fun_idem_gcd])

  1348

  1349 lemma Gcd_set [code_unfold]:

  1350   "Gcd (set xs) = fold gcd xs 0"

  1351   using comp_fun_idem.fold_set_fold[OF comp_fun_idem_gcd] Gcd_finite by (simp add: ac_simps)

  1352

  1353 lemma Gcd_singleton [simp]: "Gcd {a} = a div normalization_factor a"

  1354   by (simp add: gcd_0)

  1355

  1356 lemma Gcd_2 [simp]: "Gcd {a,b} = gcd a b"

  1357   by (simp only: Gcd_insert Gcd_empty gcd_0) (cases "b = 0", simp, rule gcd_div_unit2, simp)

  1358

  1359 subclass semiring_gcd

  1360   by unfold_locales (simp_all add: gcd_greatest_iff)

  1361

  1362 end

  1363

  1364 text {*

  1365   A Euclidean ring is a Euclidean semiring with additive inverses. It provides a

  1366   few more lemmas; in particular, Bezout's lemma holds for any Euclidean ring.

  1367 *}

  1368

  1369 class euclidean_ring = euclidean_semiring + idom

  1370

  1371 class euclidean_ring_gcd = euclidean_semiring_gcd + idom

  1372 begin

  1373

  1374 subclass euclidean_ring ..

  1375

  1376 subclass ring_gcd ..

  1377

  1378 lemma gcd_neg1 [simp]:

  1379   "gcd (-a) b = gcd a b"

  1380   by (rule sym, rule gcdI, simp_all add: gcd_greatest)

  1381

  1382 lemma gcd_neg2 [simp]:

  1383   "gcd a (-b) = gcd a b"

  1384   by (rule sym, rule gcdI, simp_all add: gcd_greatest)

  1385

  1386 lemma gcd_neg_numeral_1 [simp]:

  1387   "gcd (- numeral n) a = gcd (numeral n) a"

  1388   by (fact gcd_neg1)

  1389

  1390 lemma gcd_neg_numeral_2 [simp]:

  1391   "gcd a (- numeral n) = gcd a (numeral n)"

  1392   by (fact gcd_neg2)

  1393

  1394 lemma gcd_diff1: "gcd (m - n) n = gcd m n"

  1395   by (subst diff_conv_add_uminus, subst gcd_neg2[symmetric],  subst gcd_add1, simp)

  1396

  1397 lemma gcd_diff2: "gcd (n - m) n = gcd m n"

  1398   by (subst gcd_neg1[symmetric], simp only: minus_diff_eq gcd_diff1)

  1399

  1400 lemma coprime_minus_one [simp]: "gcd (n - 1) n = 1"

  1401 proof -

  1402   have "gcd (n - 1) n = gcd n (n - 1)" by (fact gcd.commute)

  1403   also have "\<dots> = gcd ((n - 1) + 1) (n - 1)" by simp

  1404   also have "\<dots> = 1" by (rule coprime_plus_one)

  1405   finally show ?thesis .

  1406 qed

  1407

  1408 lemma lcm_neg1 [simp]: "lcm (-a) b = lcm a b"

  1409   by (rule sym, rule lcmI, simp_all add: lcm_least lcm_zero)

  1410

  1411 lemma lcm_neg2 [simp]: "lcm a (-b) = lcm a b"

  1412   by (rule sym, rule lcmI, simp_all add: lcm_least lcm_zero)

  1413

  1414 lemma lcm_neg_numeral_1 [simp]: "lcm (- numeral n) a = lcm (numeral n) a"

  1415   by (fact lcm_neg1)

  1416

  1417 lemma lcm_neg_numeral_2 [simp]: "lcm a (- numeral n) = lcm a (numeral n)"

  1418   by (fact lcm_neg2)

  1419

  1420 function euclid_ext :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a" where

  1421   "euclid_ext a b =

  1422      (if b = 0 then

  1423         let c = 1 div normalization_factor a in (c, 0, a * c)

  1424       else

  1425         case euclid_ext b (a mod b) of

  1426             (s,t,c) \<Rightarrow> (t, s - t * (a div b), c))"

  1427   by (pat_completeness, simp)

  1428   termination by (relation "measure (euclidean_size \<circ> snd)", simp_all)

  1429

  1430 declare euclid_ext.simps [simp del]

  1431

  1432 lemma euclid_ext_0:

  1433   "euclid_ext a 0 = (1 div normalization_factor a, 0, a div normalization_factor a)"

  1434   by (subst euclid_ext.simps) (simp add: Let_def)

  1435

  1436 lemma euclid_ext_non_0:

  1437   "b \<noteq> 0 \<Longrightarrow> euclid_ext a b = (case euclid_ext b (a mod b) of

  1438     (s,t,c) \<Rightarrow> (t, s - t * (a div b), c))"

  1439   by (subst euclid_ext.simps) simp

  1440

  1441 definition euclid_ext' :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a"

  1442 where

  1443   "euclid_ext' a b = (case euclid_ext a b of (s, t, _) \<Rightarrow> (s, t))"

  1444

  1445 lemma euclid_ext_gcd [simp]:

  1446   "(case euclid_ext a b of (_,_,t) \<Rightarrow> t) = gcd a b"

  1447 proof (induct a b rule: euclid_ext.induct)

  1448   case (1 a b)

  1449   then show ?case

  1450   proof (cases "b = 0")

  1451     case True

  1452       then show ?thesis by

  1453         (simp add: euclid_ext_0 unit_div mult_ac unit_simps gcd_0)

  1454     next

  1455     case False with 1 show ?thesis

  1456       by (simp add: euclid_ext_non_0 ac_simps split: prod.split prod.split_asm)

  1457     qed

  1458 qed

  1459

  1460 lemma euclid_ext_gcd' [simp]:

  1461   "euclid_ext a b = (r, s, t) \<Longrightarrow> t = gcd a b"

  1462   by (insert euclid_ext_gcd[of a b], drule (1) subst, simp)

  1463

  1464 lemma euclid_ext_correct:

  1465   "case euclid_ext a b of (s,t,c) \<Rightarrow> s*a + t*b = c"

  1466 proof (induct a b rule: euclid_ext.induct)

  1467   case (1 a b)

  1468   show ?case

  1469   proof (cases "b = 0")

  1470     case True

  1471     then show ?thesis by (simp add: euclid_ext_0 mult_ac)

  1472   next

  1473     case False

  1474     obtain s t c where stc: "euclid_ext b (a mod b) = (s,t,c)"

  1475       by (cases "euclid_ext b (a mod b)", blast)

  1476     from 1 have "c = s * b + t * (a mod b)" by (simp add: stc False)

  1477     also have "... = t*((a div b)*b + a mod b) + (s - t * (a div b))*b"

  1478       by (simp add: algebra_simps)

  1479     also have "(a div b)*b + a mod b = a" using mod_div_equality .

  1480     finally show ?thesis

  1481       by (subst euclid_ext.simps, simp add: False stc)

  1482     qed

  1483 qed

  1484

  1485 lemma euclid_ext'_correct:

  1486   "fst (euclid_ext' a b) * a + snd (euclid_ext' a b) * b = gcd a b"

  1487 proof-

  1488   obtain s t c where "euclid_ext a b = (s,t,c)"

  1489     by (cases "euclid_ext a b", blast)

  1490   with euclid_ext_correct[of a b] euclid_ext_gcd[of a b]

  1491     show ?thesis unfolding euclid_ext'_def by simp

  1492 qed

  1493

  1494 lemma bezout: "\<exists>s t. s * a + t * b = gcd a b"

  1495   using euclid_ext'_correct by blast

  1496

  1497 lemma euclid_ext'_0 [simp]: "euclid_ext' a 0 = (1 div normalization_factor a, 0)"

  1498   by (simp add: bezw_def euclid_ext'_def euclid_ext_0)

  1499

  1500 lemma euclid_ext'_non_0: "b \<noteq> 0 \<Longrightarrow> euclid_ext' a b = (snd (euclid_ext' b (a mod b)),

  1501   fst (euclid_ext' b (a mod b)) - snd (euclid_ext' b (a mod b)) * (a div b))"

  1502   by (cases "euclid_ext b (a mod b)")

  1503     (simp add: euclid_ext'_def euclid_ext_non_0)

  1504

  1505 end

  1506

  1507 instantiation nat :: euclidean_semiring

  1508 begin

  1509

  1510 definition [simp]:

  1511   "euclidean_size_nat = (id :: nat \<Rightarrow> nat)"

  1512

  1513 definition [simp]:

  1514   "normalization_factor_nat (n::nat) = (if n = 0 then 0 else 1 :: nat)"

  1515

  1516 instance proof

  1517 qed simp_all

  1518

  1519 end

  1520

  1521 instantiation int :: euclidean_ring

  1522 begin

  1523

  1524 definition [simp]:

  1525   "euclidean_size_int = (nat \<circ> abs :: int \<Rightarrow> nat)"

  1526

  1527 definition [simp]:

  1528   "normalization_factor_int = (sgn :: int \<Rightarrow> int)"

  1529

  1530 instance proof

  1531   case goal2 then show ?case by (auto simp add: abs_mult nat_mult_distrib)

  1532 next

  1533   case goal3 then show ?case by (simp add: zsgn_def)

  1534 next

  1535   case goal5 then show ?case by (auto simp: zsgn_def)

  1536 next

  1537   case goal6 then show ?case by (auto split: abs_split simp: zsgn_def)

  1538 qed (auto simp: sgn_times split: abs_split)

  1539

  1540 end

  1541

  1542 end