src/HOL/Number_Theory/Euclidean_Algorithm.thy
author haftmann
Fri Jun 19 07:53:33 2015 +0200 (2015-06-19)
changeset 60516 0826b7025d07
parent 60439 b765e08f8bc0
child 60517 f16e4fb20652
permissions -rw-r--r--
generalized some theorems about integral domains and moved to HOL theories
     1 (* Author: Manuel Eberl *)
     2 
     3 section {* Abstract euclidean algorithm *}
     4 
     5 theory Euclidean_Algorithm
     6 imports Complex_Main
     7 begin
     8 
     9 context semidom_divide
    10 begin
    11 
    12 lemma dvd_div_mult_self [simp]:
    13   "a dvd b \<Longrightarrow> b div a * a = b"
    14   by (cases "a = 0") (auto elim: dvdE simp add: ac_simps)
    15 
    16 lemma dvd_mult_div_cancel [simp]:
    17   "a dvd b \<Longrightarrow> a * (b div a) = b"
    18   using dvd_div_mult_self [of a b] by (simp add: ac_simps)
    19   
    20 lemma div_mult_swap:
    21   assumes "c dvd b"
    22   shows "a * (b div c) = (a * b) div c"
    23 proof (cases "c = 0")
    24   case True then show ?thesis by simp
    25 next
    26   case False from assms obtain d where "b = c * d" ..
    27   moreover from False have "a * divide (d * c) c = ((a * d) * c) div c"
    28     by simp
    29   ultimately show ?thesis by (simp add: ac_simps)
    30 qed
    31 
    32 lemma dvd_div_mult:
    33   assumes "c dvd b"
    34   shows "b div c * a = (b * a) div c"
    35   using assms div_mult_swap [of c b a] by (simp add: ac_simps)
    36 
    37   
    38 text \<open>Units: invertible elements in a ring\<close>
    39 
    40 abbreviation is_unit :: "'a \<Rightarrow> bool"
    41 where
    42   "is_unit a \<equiv> a dvd 1"
    43 
    44 lemma not_is_unit_0 [simp]:
    45   "\<not> is_unit 0"
    46   by simp
    47 
    48 lemma unit_imp_dvd [dest]: 
    49   "is_unit b \<Longrightarrow> b dvd a"
    50   by (rule dvd_trans [of _ 1]) simp_all
    51 
    52 lemma unit_dvdE:
    53   assumes "is_unit a"
    54   obtains c where "a \<noteq> 0" and "b = a * c"
    55 proof -
    56   from assms have "a dvd b" by auto
    57   then obtain c where "b = a * c" ..
    58   moreover from assms have "a \<noteq> 0" by auto
    59   ultimately show thesis using that by blast
    60 qed
    61 
    62 lemma dvd_unit_imp_unit:
    63   "a dvd b \<Longrightarrow> is_unit b \<Longrightarrow> is_unit a"
    64   by (rule dvd_trans)
    65 
    66 lemma unit_div_1_unit [simp, intro]:
    67   assumes "is_unit a"
    68   shows "is_unit (1 div a)"
    69 proof -
    70   from assms have "1 = 1 div a * a" by simp
    71   then show "is_unit (1 div a)" by (rule dvdI)
    72 qed
    73 
    74 lemma is_unitE [elim?]:
    75   assumes "is_unit a"
    76   obtains b where "a \<noteq> 0" and "b \<noteq> 0"
    77     and "is_unit b" and "1 div a = b" and "1 div b = a"
    78     and "a * b = 1" and "c div a = c * b"
    79 proof (rule that)
    80   def b \<equiv> "1 div a"
    81   then show "1 div a = b" by simp
    82   from b_def `is_unit a` show "is_unit b" by simp
    83   from `is_unit a` and `is_unit b` show "a \<noteq> 0" and "b \<noteq> 0" by auto
    84   from b_def `is_unit a` show "a * b = 1" by simp
    85   then have "1 = a * b" ..
    86   with b_def `b \<noteq> 0` show "1 div b = a" by simp
    87   from `is_unit a` have "a dvd c" ..
    88   then obtain d where "c = a * d" ..
    89   with `a \<noteq> 0` `a * b = 1` show "c div a = c * b"
    90     by (simp add: mult.assoc mult.left_commute [of a])
    91 qed
    92 
    93 lemma unit_prod [intro]:
    94   "is_unit a \<Longrightarrow> is_unit b \<Longrightarrow> is_unit (a * b)"
    95   by (subst mult_1_left [of 1, symmetric]) (rule mult_dvd_mono) 
    96   
    97 lemma unit_div [intro]:
    98   "is_unit a \<Longrightarrow> is_unit b \<Longrightarrow> is_unit (a div b)"
    99   by (erule is_unitE [of b a]) (simp add: ac_simps unit_prod)
   100 
   101 lemma mult_unit_dvd_iff:
   102   assumes "is_unit b"
   103   shows "a * b dvd c \<longleftrightarrow> a dvd c"
   104 proof
   105   assume "a * b dvd c"
   106   with assms show "a dvd c"
   107     by (simp add: dvd_mult_left)
   108 next
   109   assume "a dvd c"
   110   then obtain k where "c = a * k" ..
   111   with assms have "c = (a * b) * (1 div b * k)"
   112     by (simp add: mult_ac)
   113   then show "a * b dvd c" by (rule dvdI)
   114 qed
   115 
   116 lemma dvd_mult_unit_iff:
   117   assumes "is_unit b"
   118   shows "a dvd c * b \<longleftrightarrow> a dvd c"
   119 proof
   120   assume "a dvd c * b"
   121   with assms have "c * b dvd c * (b * (1 div b))"
   122     by (subst mult_assoc [symmetric]) simp
   123   also from `is_unit b` have "b * (1 div b) = 1" by (rule is_unitE) simp
   124   finally have "c * b dvd c" by simp
   125   with `a dvd c * b` show "a dvd c" by (rule dvd_trans)
   126 next
   127   assume "a dvd c"
   128   then show "a dvd c * b" by simp
   129 qed
   130 
   131 lemma div_unit_dvd_iff:
   132   "is_unit b \<Longrightarrow> a div b dvd c \<longleftrightarrow> a dvd c"
   133   by (erule is_unitE [of _ a]) (auto simp add: mult_unit_dvd_iff)
   134 
   135 lemma dvd_div_unit_iff:
   136   "is_unit b \<Longrightarrow> a dvd c div b \<longleftrightarrow> a dvd c"
   137   by (erule is_unitE [of _ c]) (simp add: dvd_mult_unit_iff)
   138 
   139 lemmas unit_dvd_iff = mult_unit_dvd_iff div_unit_dvd_iff
   140   dvd_mult_unit_iff dvd_div_unit_iff -- \<open>FIXME consider fact collection\<close>
   141 
   142 lemma unit_mult_div_div [simp]:
   143   "is_unit a \<Longrightarrow> b * (1 div a) = b div a"
   144   by (erule is_unitE [of _ b]) simp
   145 
   146 lemma unit_div_mult_self [simp]:
   147   "is_unit a \<Longrightarrow> b div a * a = b"
   148   by (rule dvd_div_mult_self) auto
   149 
   150 lemma unit_div_1_div_1 [simp]:
   151   "is_unit a \<Longrightarrow> 1 div (1 div a) = a"
   152   by (erule is_unitE) simp
   153 
   154 lemma unit_div_mult_swap:
   155   "is_unit c \<Longrightarrow> a * (b div c) = (a * b) div c"
   156   by (erule unit_dvdE [of _ b]) (simp add: mult.left_commute [of _ c])
   157 
   158 lemma unit_div_commute:
   159   "is_unit b \<Longrightarrow> (a div b) * c = (a * c) div b"
   160   using unit_div_mult_swap [of b c a] by (simp add: ac_simps)
   161 
   162 lemma unit_eq_div1:
   163   "is_unit b \<Longrightarrow> a div b = c \<longleftrightarrow> a = c * b"
   164   by (auto elim: is_unitE)
   165 
   166 lemma unit_eq_div2:
   167   "is_unit b \<Longrightarrow> a = c div b \<longleftrightarrow> a * b = c"
   168   using unit_eq_div1 [of b c a] by auto
   169 
   170 lemma unit_mult_left_cancel:
   171   assumes "is_unit a"
   172   shows "a * b = a * c \<longleftrightarrow> b = c" (is "?P \<longleftrightarrow> ?Q")
   173   using assms mult_cancel_left [of a b c] by auto 
   174 
   175 lemma unit_mult_right_cancel:
   176   "is_unit a \<Longrightarrow> b * a = c * a \<longleftrightarrow> b = c"
   177   using unit_mult_left_cancel [of a b c] by (auto simp add: ac_simps)
   178 
   179 lemma unit_div_cancel:
   180   assumes "is_unit a"
   181   shows "b div a = c div a \<longleftrightarrow> b = c"
   182 proof -
   183   from assms have "is_unit (1 div a)" by simp
   184   then have "b * (1 div a) = c * (1 div a) \<longleftrightarrow> b = c"
   185     by (rule unit_mult_right_cancel)
   186   with assms show ?thesis by simp
   187 qed
   188   
   189 
   190 text \<open>Associated elements in a ring – an equivalence relation induced by the quasi-order divisibility \<close>
   191 
   192 definition associated :: "'a \<Rightarrow> 'a \<Rightarrow> bool" 
   193 where
   194   "associated a b \<longleftrightarrow> a dvd b \<and> b dvd a"
   195 
   196 lemma associatedI:
   197   "a dvd b \<Longrightarrow> b dvd a \<Longrightarrow> associated a b"
   198   by (simp add: associated_def)
   199 
   200 lemma associatedD1:
   201   "associated a b \<Longrightarrow> a dvd b"
   202   by (simp add: associated_def)
   203 
   204 lemma associatedD2:
   205   "associated a b \<Longrightarrow> b dvd a"
   206   by (simp add: associated_def)
   207 
   208 lemma associated_refl [simp]:
   209   "associated a a"
   210   by (auto intro: associatedI)
   211 
   212 lemma associated_sym:
   213   "associated b a \<longleftrightarrow> associated a b"
   214   by (auto intro: associatedI dest: associatedD1 associatedD2)
   215 
   216 lemma associated_trans:
   217   "associated a b \<Longrightarrow> associated b c \<Longrightarrow> associated a c"
   218   by (auto intro: associatedI dvd_trans dest: associatedD1 associatedD2)
   219 
   220 lemma equivp_associated:
   221   "equivp associated"
   222 proof (rule equivpI)
   223   show "reflp associated"
   224     by (rule reflpI) simp
   225   show "symp associated"
   226     by (rule sympI) (simp add: associated_sym)
   227   show "transp associated"
   228     by (rule transpI) (fact associated_trans)
   229 qed
   230 
   231 lemma associated_0 [simp]:
   232   "associated 0 b \<longleftrightarrow> b = 0"
   233   "associated a 0 \<longleftrightarrow> a = 0"
   234   by (auto dest: associatedD1 associatedD2)
   235 
   236 lemma associated_unit:
   237   "associated a b \<Longrightarrow> is_unit a \<Longrightarrow> is_unit b"
   238   using dvd_unit_imp_unit by (auto dest!: associatedD1 associatedD2)
   239 
   240 lemma is_unit_associatedI:
   241   assumes "is_unit c" and "a = c * b"
   242   shows "associated a b"
   243 proof (rule associatedI)
   244   from `a = c * b` show "b dvd a" by auto
   245   from `is_unit c` obtain d where "c * d = 1" by (rule is_unitE)
   246   moreover from `a = c * b` have "d * a = d * (c * b)" by simp
   247   ultimately have "b = a * d" by (simp add: ac_simps)
   248   then show "a dvd b" ..
   249 qed
   250 
   251 lemma associated_is_unitE:
   252   assumes "associated a b"
   253   obtains c where "is_unit c" and "a = c * b"
   254 proof (cases "b = 0")
   255   case True with assms have "is_unit 1" and "a = 1 * b" by simp_all
   256   with that show thesis .
   257 next
   258   case False
   259   from assms have "a dvd b" and "b dvd a" by (auto dest: associatedD1 associatedD2)
   260   then obtain c d where "b = a * d" and "a = b * c" by (blast elim: dvdE)
   261   then have "a = c * b" and "(c * d) * b = 1 * b" by (simp_all add: ac_simps)
   262   with False have "c * d = 1" using mult_cancel_right [of "c * d" b 1] by simp
   263   then have "is_unit c" by auto
   264   with `a = c * b` that show thesis by blast
   265 qed
   266   
   267 lemmas unit_simps = mult_unit_dvd_iff div_unit_dvd_iff dvd_mult_unit_iff 
   268   dvd_div_unit_iff unit_div_mult_swap unit_div_commute
   269   unit_mult_left_cancel unit_mult_right_cancel unit_div_cancel 
   270   unit_eq_div1 unit_eq_div2
   271 
   272 end
   273 
   274 lemma is_unit_int:
   275   "is_unit (k::int) \<longleftrightarrow> k = 1 \<or> k = - 1"
   276   by auto
   277 
   278   
   279 text {*
   280   A Euclidean semiring is a semiring upon which the Euclidean algorithm can be
   281   implemented. It must provide:
   282   \begin{itemize}
   283   \item division with remainder
   284   \item a size function such that @{term "size (a mod b) < size b"} 
   285         for any @{term "b \<noteq> 0"}
   286   \item a normalization factor such that two associated numbers are equal iff 
   287         they are the same when divd by their normalization factors.
   288   \end{itemize}
   289   The existence of these functions makes it possible to derive gcd and lcm functions 
   290   for any Euclidean semiring.
   291 *} 
   292 class euclidean_semiring = semiring_div + 
   293   fixes euclidean_size :: "'a \<Rightarrow> nat"
   294   fixes normalization_factor :: "'a \<Rightarrow> 'a"
   295   assumes mod_size_less [simp]: 
   296     "b \<noteq> 0 \<Longrightarrow> euclidean_size (a mod b) < euclidean_size b"
   297   assumes size_mult_mono:
   298     "b \<noteq> 0 \<Longrightarrow> euclidean_size (a * b) \<ge> euclidean_size a"
   299   assumes normalization_factor_is_unit [intro,simp]: 
   300     "a \<noteq> 0 \<Longrightarrow> is_unit (normalization_factor a)"
   301   assumes normalization_factor_mult: "normalization_factor (a * b) = 
   302     normalization_factor a * normalization_factor b"
   303   assumes normalization_factor_unit: "is_unit a \<Longrightarrow> normalization_factor a = a"
   304   assumes normalization_factor_0 [simp]: "normalization_factor 0 = 0"
   305 begin
   306 
   307 lemma normalization_factor_dvd [simp]:
   308   "a \<noteq> 0 \<Longrightarrow> normalization_factor a dvd b"
   309   by (rule unit_imp_dvd, simp)
   310     
   311 lemma normalization_factor_1 [simp]:
   312   "normalization_factor 1 = 1"
   313   by (simp add: normalization_factor_unit)
   314 
   315 lemma normalization_factor_0_iff [simp]:
   316   "normalization_factor a = 0 \<longleftrightarrow> a = 0"
   317 proof
   318   assume "normalization_factor a = 0"
   319   hence "\<not> is_unit (normalization_factor a)"
   320     by simp
   321   then show "a = 0" by auto
   322 qed simp
   323 
   324 lemma normalization_factor_pow:
   325   "normalization_factor (a ^ n) = normalization_factor a ^ n"
   326   by (induct n) (simp_all add: normalization_factor_mult power_Suc2)
   327 
   328 lemma normalization_correct [simp]:
   329   "normalization_factor (a div normalization_factor a) = (if a = 0 then 0 else 1)"
   330 proof (cases "a = 0", simp)
   331   assume "a \<noteq> 0"
   332   let ?nf = "normalization_factor"
   333   from normalization_factor_is_unit[OF `a \<noteq> 0`] have "?nf a \<noteq> 0"
   334     by auto
   335   have "?nf (a div ?nf a) * ?nf (?nf a) = ?nf (a div ?nf a * ?nf a)" 
   336     by (simp add: normalization_factor_mult)
   337   also have "a div ?nf a * ?nf a = a" using `a \<noteq> 0`
   338     by simp
   339   also have "?nf (?nf a) = ?nf a" using `a \<noteq> 0` 
   340     normalization_factor_is_unit normalization_factor_unit by simp
   341   finally have "normalization_factor (a div normalization_factor a) = 1"  
   342     using `?nf a \<noteq> 0` by (metis div_mult_self2_is_id div_self)
   343   with `a \<noteq> 0` show ?thesis by simp
   344 qed
   345 
   346 lemma normalization_0_iff [simp]:
   347   "a div normalization_factor a = 0 \<longleftrightarrow> a = 0"
   348   by (cases "a = 0", simp, subst unit_eq_div1, blast, simp)
   349 
   350 lemma mult_div_normalization [simp]:
   351   "b * (1 div normalization_factor a) = b div normalization_factor a"
   352   by (cases "a = 0") simp_all
   353 
   354 lemma associated_iff_normed_eq:
   355   "associated a b \<longleftrightarrow> a div normalization_factor a = b div normalization_factor b"
   356 proof (cases "b = 0", simp, cases "a = 0", metis associated_0(1) normalization_0_iff, rule iffI)
   357   let ?nf = normalization_factor
   358   assume "a \<noteq> 0" "b \<noteq> 0" "a div ?nf a = b div ?nf b"
   359   hence "a = b * (?nf a div ?nf b)"
   360     apply (subst (asm) unit_eq_div1, blast, subst (asm) unit_div_commute, blast)
   361     apply (subst div_mult_swap, simp, simp)
   362     done
   363   with `a \<noteq> 0` `b \<noteq> 0` have "\<exists>c. is_unit c \<and> a = c * b"
   364     by (intro exI[of _ "?nf a div ?nf b"], force simp: mult_ac)
   365   then obtain c where "is_unit c" and "a = c * b" by blast
   366   then show "associated a b" by (rule is_unit_associatedI) 
   367 next
   368   let ?nf = normalization_factor
   369   assume "a \<noteq> 0" "b \<noteq> 0" "associated a b"
   370   then obtain c where "is_unit c" and "a = c * b" by (blast elim: associated_is_unitE)
   371   then show "a div ?nf a = b div ?nf b"
   372     apply (simp only: `a = c * b` normalization_factor_mult normalization_factor_unit)
   373     apply (rule div_mult_mult1, force)
   374     done
   375   qed
   376 
   377 lemma normed_associated_imp_eq:
   378   "associated a b \<Longrightarrow> normalization_factor a \<in> {0, 1} \<Longrightarrow> normalization_factor b \<in> {0, 1} \<Longrightarrow> a = b"
   379   by (simp add: associated_iff_normed_eq, elim disjE, simp_all)
   380     
   381 lemmas normalization_factor_dvd_iff [simp] =
   382   unit_dvd_iff [OF normalization_factor_is_unit]
   383 
   384 lemma euclidean_division:
   385   fixes a :: 'a and b :: 'a
   386   assumes "b \<noteq> 0"
   387   obtains s and t where "a = s * b + t" 
   388     and "euclidean_size t < euclidean_size b"
   389 proof -
   390   from div_mod_equality[of a b 0] 
   391      have "a = a div b * b + a mod b" by simp
   392   with that and assms show ?thesis by force
   393 qed
   394 
   395 lemma dvd_euclidean_size_eq_imp_dvd:
   396   assumes "a \<noteq> 0" and b_dvd_a: "b dvd a" and size_eq: "euclidean_size a = euclidean_size b"
   397   shows "a dvd b"
   398 proof (subst dvd_eq_mod_eq_0, rule ccontr)
   399   assume "b mod a \<noteq> 0"
   400   from b_dvd_a have b_dvd_mod: "b dvd b mod a" by (simp add: dvd_mod_iff)
   401   from b_dvd_mod obtain c where "b mod a = b * c" unfolding dvd_def by blast
   402     with `b mod a \<noteq> 0` have "c \<noteq> 0" by auto
   403   with `b mod a = b * c` have "euclidean_size (b mod a) \<ge> euclidean_size b"
   404       using size_mult_mono by force
   405   moreover from `a \<noteq> 0` have "euclidean_size (b mod a) < euclidean_size a"
   406       using mod_size_less by blast
   407   ultimately show False using size_eq by simp
   408 qed
   409 
   410 function gcd_eucl :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
   411 where
   412   "gcd_eucl a b = (if b = 0 then a div normalization_factor a else gcd_eucl b (a mod b))"
   413   by (pat_completeness, simp)
   414 termination by (relation "measure (euclidean_size \<circ> snd)", simp_all)
   415 
   416 declare gcd_eucl.simps [simp del]
   417 
   418 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"
   419 proof (induct a b rule: gcd_eucl.induct)
   420   case ("1" m n)
   421     then show ?case by (cases "n = 0") auto
   422 qed
   423 
   424 definition lcm_eucl :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
   425 where
   426   "lcm_eucl a b = a * b div (gcd_eucl a b * normalization_factor (a * b))"
   427 
   428   (* Somewhat complicated definition of Lcm that has the advantage of working
   429      for infinite sets as well *)
   430 
   431 definition Lcm_eucl :: "'a set \<Rightarrow> 'a"
   432 where
   433   "Lcm_eucl A = (if \<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) then
   434      let l = SOME l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l =
   435        (LEAST n. \<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n)
   436        in l div normalization_factor l
   437       else 0)"
   438 
   439 definition Gcd_eucl :: "'a set \<Rightarrow> 'a"
   440 where
   441   "Gcd_eucl A = Lcm_eucl {d. \<forall>a\<in>A. d dvd a}"
   442 
   443 end
   444 
   445 class euclidean_semiring_gcd = euclidean_semiring + gcd + Gcd +
   446   assumes gcd_gcd_eucl: "gcd = gcd_eucl" and lcm_lcm_eucl: "lcm = lcm_eucl"
   447   assumes Gcd_Gcd_eucl: "Gcd = Gcd_eucl" and Lcm_Lcm_eucl: "Lcm = Lcm_eucl"
   448 begin
   449 
   450 lemma gcd_red:
   451   "gcd a b = gcd b (a mod b)"
   452   by (metis gcd_eucl.simps mod_0 mod_by_0 gcd_gcd_eucl)
   453 
   454 lemma gcd_non_0:
   455   "b \<noteq> 0 \<Longrightarrow> gcd a b = gcd b (a mod b)"
   456   by (rule gcd_red)
   457 
   458 lemma gcd_0_left:
   459   "gcd 0 a = a div normalization_factor a"
   460    by (simp only: gcd_gcd_eucl, subst gcd_eucl.simps, subst gcd_eucl.simps, simp add: Let_def)
   461 
   462 lemma gcd_0:
   463   "gcd a 0 = a div normalization_factor a"
   464   by (simp only: gcd_gcd_eucl, subst gcd_eucl.simps, simp add: Let_def)
   465 
   466 lemma gcd_dvd1 [iff]: "gcd a b dvd a"
   467   and gcd_dvd2 [iff]: "gcd a b dvd b"
   468 proof (induct a b rule: gcd_eucl.induct)
   469   fix a b :: 'a
   470   assume IH1: "b \<noteq> 0 \<Longrightarrow> gcd b (a mod b) dvd b"
   471   assume IH2: "b \<noteq> 0 \<Longrightarrow> gcd b (a mod b) dvd (a mod b)"
   472   
   473   have "gcd a b dvd a \<and> gcd a b dvd b"
   474   proof (cases "b = 0")
   475     case True
   476       then show ?thesis by (cases "a = 0", simp_all add: gcd_0)
   477   next
   478     case False
   479       with IH1 and IH2 show ?thesis by (simp add: gcd_non_0 dvd_mod_iff)
   480   qed
   481   then show "gcd a b dvd a" "gcd a b dvd b" by simp_all
   482 qed
   483 
   484 lemma dvd_gcd_D1: "k dvd gcd m n \<Longrightarrow> k dvd m"
   485   by (rule dvd_trans, assumption, rule gcd_dvd1)
   486 
   487 lemma dvd_gcd_D2: "k dvd gcd m n \<Longrightarrow> k dvd n"
   488   by (rule dvd_trans, assumption, rule gcd_dvd2)
   489 
   490 lemma gcd_greatest:
   491   fixes k a b :: 'a
   492   shows "k dvd a \<Longrightarrow> k dvd b \<Longrightarrow> k dvd gcd a b"
   493 proof (induct a b rule: gcd_eucl.induct)
   494   case (1 a b)
   495   show ?case
   496     proof (cases "b = 0")
   497       assume "b = 0"
   498       with 1 show ?thesis by (cases "a = 0", simp_all add: gcd_0)
   499     next
   500       assume "b \<noteq> 0"
   501       with 1 show ?thesis by (simp add: gcd_non_0 dvd_mod_iff) 
   502     qed
   503 qed
   504 
   505 lemma dvd_gcd_iff:
   506   "k dvd gcd a b \<longleftrightarrow> k dvd a \<and> k dvd b"
   507   by (blast intro!: gcd_greatest intro: dvd_trans)
   508 
   509 lemmas gcd_greatest_iff = dvd_gcd_iff
   510 
   511 lemma gcd_zero [simp]:
   512   "gcd a b = 0 \<longleftrightarrow> a = 0 \<and> b = 0"
   513   by (metis dvd_0_left dvd_refl gcd_dvd1 gcd_dvd2 gcd_greatest)+
   514 
   515 lemma normalization_factor_gcd [simp]:
   516   "normalization_factor (gcd a b) = (if a = 0 \<and> b = 0 then 0 else 1)" (is "?f a b = ?g a b")
   517 proof (induct a b rule: gcd_eucl.induct)
   518   fix a b :: 'a
   519   assume IH: "b \<noteq> 0 \<Longrightarrow> ?f b (a mod b) = ?g b (a mod b)"
   520   then show "?f a b = ?g a b" by (cases "b = 0", auto simp: gcd_non_0 gcd_0)
   521 qed
   522 
   523 lemma gcdI:
   524   "k dvd a \<Longrightarrow> k dvd b \<Longrightarrow> (\<And>l. l dvd a \<Longrightarrow> l dvd b \<Longrightarrow> l dvd k)
   525     \<Longrightarrow> normalization_factor k = (if k = 0 then 0 else 1) \<Longrightarrow> k = gcd a b"
   526   by (intro normed_associated_imp_eq) (auto simp: associated_def intro: gcd_greatest)
   527 
   528 sublocale gcd!: abel_semigroup gcd
   529 proof
   530   fix a b c 
   531   show "gcd (gcd a b) c = gcd a (gcd b c)"
   532   proof (rule gcdI)
   533     have "gcd (gcd a b) c dvd gcd a b" "gcd a b dvd a" by simp_all
   534     then show "gcd (gcd a b) c dvd a" by (rule dvd_trans)
   535     have "gcd (gcd a b) c dvd gcd a b" "gcd a b dvd b" by simp_all
   536     hence "gcd (gcd a b) c dvd b" by (rule dvd_trans)
   537     moreover have "gcd (gcd a b) c dvd c" by simp
   538     ultimately show "gcd (gcd a b) c dvd gcd b c"
   539       by (rule gcd_greatest)
   540     show "normalization_factor (gcd (gcd a b) c) =  (if gcd (gcd a b) c = 0 then 0 else 1)"
   541       by auto
   542     fix l assume "l dvd a" and "l dvd gcd b c"
   543     with dvd_trans[OF _ gcd_dvd1] and dvd_trans[OF _ gcd_dvd2]
   544       have "l dvd b" and "l dvd c" by blast+
   545     with `l dvd a` show "l dvd gcd (gcd a b) c"
   546       by (intro gcd_greatest)
   547   qed
   548 next
   549   fix a b
   550   show "gcd a b = gcd b a"
   551     by (rule gcdI) (simp_all add: gcd_greatest)
   552 qed
   553 
   554 lemma gcd_unique: "d dvd a \<and> d dvd b \<and> 
   555     normalization_factor d = (if d = 0 then 0 else 1) \<and>
   556     (\<forall>e. e dvd a \<and> e dvd b \<longrightarrow> e dvd d) \<longleftrightarrow> d = gcd a b"
   557   by (rule, auto intro: gcdI simp: gcd_greatest)
   558 
   559 lemma gcd_dvd_prod: "gcd a b dvd k * b"
   560   using mult_dvd_mono [of 1] by auto
   561 
   562 lemma gcd_1_left [simp]: "gcd 1 a = 1"
   563   by (rule sym, rule gcdI, simp_all)
   564 
   565 lemma gcd_1 [simp]: "gcd a 1 = 1"
   566   by (rule sym, rule gcdI, simp_all)
   567 
   568 lemma gcd_proj2_if_dvd: 
   569   "b dvd a \<Longrightarrow> gcd a b = b div normalization_factor b"
   570   by (cases "b = 0", simp_all add: dvd_eq_mod_eq_0 gcd_non_0 gcd_0)
   571 
   572 lemma gcd_proj1_if_dvd: 
   573   "a dvd b \<Longrightarrow> gcd a b = a div normalization_factor a"
   574   by (subst gcd.commute, simp add: gcd_proj2_if_dvd)
   575 
   576 lemma gcd_proj1_iff: "gcd m n = m div normalization_factor m \<longleftrightarrow> m dvd n"
   577 proof
   578   assume A: "gcd m n = m div normalization_factor m"
   579   show "m dvd n"
   580   proof (cases "m = 0")
   581     assume [simp]: "m \<noteq> 0"
   582     from A have B: "m = gcd m n * normalization_factor m"
   583       by (simp add: unit_eq_div2)
   584     show ?thesis by (subst B, simp add: mult_unit_dvd_iff)
   585   qed (insert A, simp)
   586 next
   587   assume "m dvd n"
   588   then show "gcd m n = m div normalization_factor m" by (rule gcd_proj1_if_dvd)
   589 qed
   590   
   591 lemma gcd_proj2_iff: "gcd m n = n div normalization_factor n \<longleftrightarrow> n dvd m"
   592   by (subst gcd.commute, simp add: gcd_proj1_iff)
   593 
   594 lemma gcd_mod1 [simp]:
   595   "gcd (a mod b) b = gcd a b"
   596   by (rule gcdI, metis dvd_mod_iff gcd_dvd1 gcd_dvd2, simp_all add: gcd_greatest dvd_mod_iff)
   597 
   598 lemma gcd_mod2 [simp]:
   599   "gcd a (b mod a) = gcd a b"
   600   by (rule gcdI, simp, metis dvd_mod_iff gcd_dvd1 gcd_dvd2, simp_all add: gcd_greatest dvd_mod_iff)
   601          
   602 lemma normalization_factor_dvd' [simp]:
   603   "normalization_factor a dvd a"
   604   by (cases "a = 0", simp_all)
   605 
   606 lemma gcd_mult_distrib': 
   607   "k div normalization_factor k * gcd a b = gcd (k*a) (k*b)"
   608 proof (induct a b rule: gcd_eucl.induct)
   609   case (1 a b)
   610   show ?case
   611   proof (cases "b = 0")
   612     case True
   613     then show ?thesis by (simp add: normalization_factor_mult gcd_0 algebra_simps div_mult_div_if_dvd)
   614   next
   615     case False
   616     hence "k div normalization_factor k * gcd a b =  gcd (k * b) (k * (a mod b))" 
   617       using 1 by (subst gcd_red, simp)
   618     also have "... = gcd (k * a) (k * b)"
   619       by (simp add: mult_mod_right gcd.commute)
   620     finally show ?thesis .
   621   qed
   622 qed
   623 
   624 lemma gcd_mult_distrib:
   625   "k * gcd a b = gcd (k*a) (k*b) * normalization_factor k"
   626 proof-
   627   let ?nf = "normalization_factor"
   628   from gcd_mult_distrib' 
   629     have "gcd (k*a) (k*b) = k div ?nf k * gcd a b" ..
   630   also have "... = k * gcd a b div ?nf k"
   631     by (metis dvd_div_mult dvd_eq_mod_eq_0 mod_0 normalization_factor_dvd)
   632   finally show ?thesis
   633     by simp
   634 qed
   635 
   636 lemma euclidean_size_gcd_le1 [simp]:
   637   assumes "a \<noteq> 0"
   638   shows "euclidean_size (gcd a b) \<le> euclidean_size a"
   639 proof -
   640    have "gcd a b dvd a" by (rule gcd_dvd1)
   641    then obtain c where A: "a = gcd a b * c" unfolding dvd_def by blast
   642    with `a \<noteq> 0` show ?thesis by (subst (2) A, intro size_mult_mono) auto
   643 qed
   644 
   645 lemma euclidean_size_gcd_le2 [simp]:
   646   "b \<noteq> 0 \<Longrightarrow> euclidean_size (gcd a b) \<le> euclidean_size b"
   647   by (subst gcd.commute, rule euclidean_size_gcd_le1)
   648 
   649 lemma euclidean_size_gcd_less1:
   650   assumes "a \<noteq> 0" and "\<not>a dvd b"
   651   shows "euclidean_size (gcd a b) < euclidean_size a"
   652 proof (rule ccontr)
   653   assume "\<not>euclidean_size (gcd a b) < euclidean_size a"
   654   with `a \<noteq> 0` have "euclidean_size (gcd a b) = euclidean_size a"
   655     by (intro le_antisym, simp_all)
   656   with assms have "a dvd gcd a b" by (auto intro: dvd_euclidean_size_eq_imp_dvd)
   657   hence "a dvd b" using dvd_gcd_D2 by blast
   658   with `\<not>a dvd b` show False by contradiction
   659 qed
   660 
   661 lemma euclidean_size_gcd_less2:
   662   assumes "b \<noteq> 0" and "\<not>b dvd a"
   663   shows "euclidean_size (gcd a b) < euclidean_size b"
   664   using assms by (subst gcd.commute, rule euclidean_size_gcd_less1)
   665 
   666 lemma gcd_mult_unit1: "is_unit a \<Longrightarrow> gcd (b * a) c = gcd b c"
   667   apply (rule gcdI)
   668   apply (rule dvd_trans, rule gcd_dvd1, simp add: unit_simps)
   669   apply (rule gcd_dvd2)
   670   apply (rule gcd_greatest, simp add: unit_simps, assumption)
   671   apply (subst normalization_factor_gcd, simp add: gcd_0)
   672   done
   673 
   674 lemma gcd_mult_unit2: "is_unit a \<Longrightarrow> gcd b (c * a) = gcd b c"
   675   by (subst gcd.commute, subst gcd_mult_unit1, assumption, rule gcd.commute)
   676 
   677 lemma gcd_div_unit1: "is_unit a \<Longrightarrow> gcd (b div a) c = gcd b c"
   678   by (erule is_unitE [of _ b]) (simp add: gcd_mult_unit1)
   679 
   680 lemma gcd_div_unit2: "is_unit a \<Longrightarrow> gcd b (c div a) = gcd b c"
   681   by (erule is_unitE [of _ c]) (simp add: gcd_mult_unit2)
   682 
   683 lemma gcd_idem: "gcd a a = a div normalization_factor a"
   684   by (cases "a = 0") (simp add: gcd_0_left, rule sym, rule gcdI, simp_all)
   685 
   686 lemma gcd_right_idem: "gcd (gcd a b) b = gcd a b"
   687   apply (rule gcdI)
   688   apply (simp add: ac_simps)
   689   apply (rule gcd_dvd2)
   690   apply (rule gcd_greatest, erule (1) gcd_greatest, assumption)
   691   apply simp
   692   done
   693 
   694 lemma gcd_left_idem: "gcd a (gcd a b) = gcd a b"
   695   apply (rule gcdI)
   696   apply simp
   697   apply (rule dvd_trans, rule gcd_dvd2, rule gcd_dvd2)
   698   apply (rule gcd_greatest, assumption, erule gcd_greatest, assumption)
   699   apply simp
   700   done
   701 
   702 lemma comp_fun_idem_gcd: "comp_fun_idem gcd"
   703 proof
   704   fix a b show "gcd a \<circ> gcd b = gcd b \<circ> gcd a"
   705     by (simp add: fun_eq_iff ac_simps)
   706 next
   707   fix a show "gcd a \<circ> gcd a = gcd a"
   708     by (simp add: fun_eq_iff gcd_left_idem)
   709 qed
   710 
   711 lemma coprime_dvd_mult:
   712   assumes "gcd c b = 1" and "c dvd a * b"
   713   shows "c dvd a"
   714 proof -
   715   let ?nf = "normalization_factor"
   716   from assms gcd_mult_distrib [of a c b] 
   717     have A: "a = gcd (a * c) (a * b) * ?nf a" by simp
   718   from `c dvd a * b` show ?thesis by (subst A, simp_all add: gcd_greatest)
   719 qed
   720 
   721 lemma coprime_dvd_mult_iff:
   722   "gcd c b = 1 \<Longrightarrow> (c dvd a * b) = (c dvd a)"
   723   by (rule, rule coprime_dvd_mult, simp_all)
   724 
   725 lemma gcd_dvd_antisym:
   726   "gcd a b dvd gcd c d \<Longrightarrow> gcd c d dvd gcd a b \<Longrightarrow> gcd a b = gcd c d"
   727 proof (rule gcdI)
   728   assume A: "gcd a b dvd gcd c d" and B: "gcd c d dvd gcd a b"
   729   have "gcd c d dvd c" by simp
   730   with A show "gcd a b dvd c" by (rule dvd_trans)
   731   have "gcd c d dvd d" by simp
   732   with A show "gcd a b dvd d" by (rule dvd_trans)
   733   show "normalization_factor (gcd a b) = (if gcd a b = 0 then 0 else 1)"
   734     by simp
   735   fix l assume "l dvd c" and "l dvd d"
   736   hence "l dvd gcd c d" by (rule gcd_greatest)
   737   from this and B show "l dvd gcd a b" by (rule dvd_trans)
   738 qed
   739 
   740 lemma gcd_mult_cancel:
   741   assumes "gcd k n = 1"
   742   shows "gcd (k * m) n = gcd m n"
   743 proof (rule gcd_dvd_antisym)
   744   have "gcd (gcd (k * m) n) k = gcd (gcd k n) (k * m)" by (simp add: ac_simps)
   745   also note `gcd k n = 1`
   746   finally have "gcd (gcd (k * m) n) k = 1" by simp
   747   hence "gcd (k * m) n dvd m" by (rule coprime_dvd_mult, simp add: ac_simps)
   748   moreover have "gcd (k * m) n dvd n" by simp
   749   ultimately show "gcd (k * m) n dvd gcd m n" by (rule gcd_greatest)
   750   have "gcd m n dvd (k * m)" and "gcd m n dvd n" by simp_all
   751   then show "gcd m n dvd gcd (k * m) n" by (rule gcd_greatest)
   752 qed
   753 
   754 lemma coprime_crossproduct:
   755   assumes [simp]: "gcd a d = 1" "gcd b c = 1"
   756   shows "associated (a * c) (b * d) \<longleftrightarrow> associated a b \<and> associated c d" (is "?lhs \<longleftrightarrow> ?rhs")
   757 proof
   758   assume ?rhs then show ?lhs unfolding associated_def by (fast intro: mult_dvd_mono)
   759 next
   760   assume ?lhs
   761   from `?lhs` have "a dvd b * d" unfolding associated_def by (metis dvd_mult_left) 
   762   hence "a dvd b" by (simp add: coprime_dvd_mult_iff)
   763   moreover from `?lhs` have "b dvd a * c" unfolding associated_def by (metis dvd_mult_left) 
   764   hence "b dvd a" by (simp add: coprime_dvd_mult_iff)
   765   moreover from `?lhs` have "c dvd d * b" 
   766     unfolding associated_def by (auto dest: dvd_mult_right simp add: ac_simps)
   767   hence "c dvd d" by (simp add: coprime_dvd_mult_iff gcd.commute)
   768   moreover from `?lhs` have "d dvd c * a"
   769     unfolding associated_def by (auto dest: dvd_mult_right simp add: ac_simps)
   770   hence "d dvd c" by (simp add: coprime_dvd_mult_iff gcd.commute)
   771   ultimately show ?rhs unfolding associated_def by simp
   772 qed
   773 
   774 lemma gcd_add1 [simp]:
   775   "gcd (m + n) n = gcd m n"
   776   by (cases "n = 0", simp_all add: gcd_non_0)
   777 
   778 lemma gcd_add2 [simp]:
   779   "gcd m (m + n) = gcd m n"
   780   using gcd_add1 [of n m] by (simp add: ac_simps)
   781 
   782 lemma gcd_add_mult: "gcd m (k * m + n) = gcd m n"
   783   by (subst gcd.commute, subst gcd_red, simp)
   784 
   785 lemma coprimeI: "(\<And>l. \<lbrakk>l dvd a; l dvd b\<rbrakk> \<Longrightarrow> l dvd 1) \<Longrightarrow> gcd a b = 1"
   786   by (rule sym, rule gcdI, simp_all)
   787 
   788 lemma coprime: "gcd a b = 1 \<longleftrightarrow> (\<forall>d. d dvd a \<and> d dvd b \<longleftrightarrow> is_unit d)"
   789   by (auto intro: coprimeI gcd_greatest dvd_gcd_D1 dvd_gcd_D2)
   790 
   791 lemma div_gcd_coprime:
   792   assumes nz: "a \<noteq> 0 \<or> b \<noteq> 0"
   793   defines [simp]: "d \<equiv> gcd a b"
   794   defines [simp]: "a' \<equiv> a div d" and [simp]: "b' \<equiv> b div d"
   795   shows "gcd a' b' = 1"
   796 proof (rule coprimeI)
   797   fix l assume "l dvd a'" "l dvd b'"
   798   then obtain s t where "a' = l * s" "b' = l * t" unfolding dvd_def by blast
   799   moreover have "a = a' * d" "b = b' * d" by simp_all
   800   ultimately have "a = (l * d) * s" "b = (l * d) * t"
   801     by (simp_all only: ac_simps)
   802   hence "l*d dvd a" and "l*d dvd b" by (simp_all only: dvd_triv_left)
   803   hence "l*d dvd d" by (simp add: gcd_greatest)
   804   then obtain u where "d = l * d * u" ..
   805   then have "d * (l * u) = d" by (simp add: ac_simps)
   806   moreover from nz have "d \<noteq> 0" by simp
   807   with div_mult_self1_is_id have "d * (l * u) div d = l * u" . 
   808   ultimately have "1 = l * u"
   809     using `d \<noteq> 0` by simp
   810   then show "l dvd 1" ..
   811 qed
   812 
   813 lemma coprime_mult: 
   814   assumes da: "gcd d a = 1" and db: "gcd d b = 1"
   815   shows "gcd d (a * b) = 1"
   816   apply (subst gcd.commute)
   817   using da apply (subst gcd_mult_cancel)
   818   apply (subst gcd.commute, assumption)
   819   apply (subst gcd.commute, rule db)
   820   done
   821 
   822 lemma coprime_lmult:
   823   assumes dab: "gcd d (a * b) = 1" 
   824   shows "gcd d a = 1"
   825 proof (rule coprimeI)
   826   fix l assume "l dvd d" and "l dvd a"
   827   hence "l dvd a * b" by simp
   828   with `l dvd d` and dab show "l dvd 1" by (auto intro: gcd_greatest)
   829 qed
   830 
   831 lemma coprime_rmult:
   832   assumes dab: "gcd d (a * b) = 1"
   833   shows "gcd d b = 1"
   834 proof (rule coprimeI)
   835   fix l assume "l dvd d" and "l dvd b"
   836   hence "l dvd a * b" by simp
   837   with `l dvd d` and dab show "l dvd 1" by (auto intro: gcd_greatest)
   838 qed
   839 
   840 lemma coprime_mul_eq: "gcd d (a * b) = 1 \<longleftrightarrow> gcd d a = 1 \<and> gcd d b = 1"
   841   using coprime_rmult[of d a b] coprime_lmult[of d a b] coprime_mult[of d a b] by blast
   842 
   843 lemma gcd_coprime:
   844   assumes c: "gcd a b \<noteq> 0" and a: "a = a' * gcd a b" and b: "b = b' * gcd a b"
   845   shows "gcd a' b' = 1"
   846 proof -
   847   from c have "a \<noteq> 0 \<or> b \<noteq> 0" by simp
   848   with div_gcd_coprime have "gcd (a div gcd a b) (b div gcd a b) = 1" .
   849   also from assms have "a div gcd a b = a'" by (metis div_mult_self2_is_id)+
   850   also from assms have "b div gcd a b = b'" by (metis div_mult_self2_is_id)+
   851   finally show ?thesis .
   852 qed
   853 
   854 lemma coprime_power:
   855   assumes "0 < n"
   856   shows "gcd a (b ^ n) = 1 \<longleftrightarrow> gcd a b = 1"
   857 using assms proof (induct n)
   858   case (Suc n) then show ?case
   859     by (cases n) (simp_all add: coprime_mul_eq)
   860 qed simp
   861 
   862 lemma gcd_coprime_exists:
   863   assumes nz: "gcd a b \<noteq> 0"
   864   shows "\<exists>a' b'. a = a' * gcd a b \<and> b = b' * gcd a b \<and> gcd a' b' = 1"
   865   apply (rule_tac x = "a div gcd a b" in exI)
   866   apply (rule_tac x = "b div gcd a b" in exI)
   867   apply (insert nz, auto intro: div_gcd_coprime)
   868   done
   869 
   870 lemma coprime_exp:
   871   "gcd d a = 1 \<Longrightarrow> gcd d (a^n) = 1"
   872   by (induct n, simp_all add: coprime_mult)
   873 
   874 lemma coprime_exp2 [intro]:
   875   "gcd a b = 1 \<Longrightarrow> gcd (a^n) (b^m) = 1"
   876   apply (rule coprime_exp)
   877   apply (subst gcd.commute)
   878   apply (rule coprime_exp)
   879   apply (subst gcd.commute)
   880   apply assumption
   881   done
   882 
   883 lemma gcd_exp:
   884   "gcd (a^n) (b^n) = (gcd a b) ^ n"
   885 proof (cases "a = 0 \<and> b = 0")
   886   assume "a = 0 \<and> b = 0"
   887   then show ?thesis by (cases n, simp_all add: gcd_0_left)
   888 next
   889   assume A: "\<not>(a = 0 \<and> b = 0)"
   890   hence "1 = gcd ((a div gcd a b)^n) ((b div gcd a b)^n)"
   891     using div_gcd_coprime by (subst sym, auto simp: div_gcd_coprime)
   892   hence "(gcd a b) ^ n = (gcd a b) ^ n * ..." by simp
   893   also note gcd_mult_distrib
   894   also have "normalization_factor ((gcd a b)^n) = 1"
   895     by (simp add: normalization_factor_pow A)
   896   also have "(gcd a b)^n * (a div gcd a b)^n = a^n"
   897     by (subst ac_simps, subst div_power, simp, rule dvd_div_mult_self, rule dvd_power_same, simp)
   898   also have "(gcd a b)^n * (b div gcd a b)^n = b^n"
   899     by (subst ac_simps, subst div_power, simp, rule dvd_div_mult_self, rule dvd_power_same, simp)
   900   finally show ?thesis by simp
   901 qed
   902 
   903 lemma coprime_common_divisor: 
   904   "gcd a b = 1 \<Longrightarrow> a dvd a \<Longrightarrow> a dvd b \<Longrightarrow> is_unit a"
   905   apply (subgoal_tac "a dvd gcd a b")
   906   apply simp
   907   apply (erule (1) gcd_greatest)
   908   done
   909 
   910 lemma division_decomp: 
   911   assumes dc: "a dvd b * c"
   912   shows "\<exists>b' c'. a = b' * c' \<and> b' dvd b \<and> c' dvd c"
   913 proof (cases "gcd a b = 0")
   914   assume "gcd a b = 0"
   915   hence "a = 0 \<and> b = 0" by simp
   916   hence "a = 0 * c \<and> 0 dvd b \<and> c dvd c" by simp
   917   then show ?thesis by blast
   918 next
   919   let ?d = "gcd a b"
   920   assume "?d \<noteq> 0"
   921   from gcd_coprime_exists[OF this]
   922     obtain a' b' where ab': "a = a' * ?d" "b = b' * ?d" "gcd a' b' = 1"
   923     by blast
   924   from ab'(1) have "a' dvd a" unfolding dvd_def by blast
   925   with dc have "a' dvd b*c" using dvd_trans[of a' a "b*c"] by simp
   926   from dc ab'(1,2) have "a'*?d dvd (b'*?d) * c" by simp
   927   hence "?d * a' dvd ?d * (b' * c)" by (simp add: mult_ac)
   928   with `?d \<noteq> 0` have "a' dvd b' * c" by simp
   929   with coprime_dvd_mult[OF ab'(3)] 
   930     have "a' dvd c" by (subst (asm) ac_simps, blast)
   931   with ab'(1) have "a = ?d * a' \<and> ?d dvd b \<and> a' dvd c" by (simp add: mult_ac)
   932   then show ?thesis by blast
   933 qed
   934 
   935 lemma pow_divs_pow:
   936   assumes ab: "a ^ n dvd b ^ n" and n: "n \<noteq> 0"
   937   shows "a dvd b"
   938 proof (cases "gcd a b = 0")
   939   assume "gcd a b = 0"
   940   then show ?thesis by simp
   941 next
   942   let ?d = "gcd a b"
   943   assume "?d \<noteq> 0"
   944   from n obtain m where m: "n = Suc m" by (cases n, simp_all)
   945   from `?d \<noteq> 0` have zn: "?d ^ n \<noteq> 0" by (rule power_not_zero)
   946   from gcd_coprime_exists[OF `?d \<noteq> 0`]
   947     obtain a' b' where ab': "a = a' * ?d" "b = b' * ?d" "gcd a' b' = 1"
   948     by blast
   949   from ab have "(a' * ?d) ^ n dvd (b' * ?d) ^ n"
   950     by (simp add: ab'(1,2)[symmetric])
   951   hence "?d^n * a'^n dvd ?d^n * b'^n"
   952     by (simp only: power_mult_distrib ac_simps)
   953   with zn have "a'^n dvd b'^n" by simp
   954   hence "a' dvd b'^n" using dvd_trans[of a' "a'^n" "b'^n"] by (simp add: m)
   955   hence "a' dvd b'^m * b'" by (simp add: m ac_simps)
   956   with coprime_dvd_mult[OF coprime_exp[OF ab'(3), of m]]
   957     have "a' dvd b'" by (subst (asm) ac_simps, blast)
   958   hence "a'*?d dvd b'*?d" by (rule mult_dvd_mono, simp)
   959   with ab'(1,2) show ?thesis by simp
   960 qed
   961 
   962 lemma pow_divs_eq [simp]:
   963   "n \<noteq> 0 \<Longrightarrow> a ^ n dvd b ^ n \<longleftrightarrow> a dvd b"
   964   by (auto intro: pow_divs_pow dvd_power_same)
   965 
   966 lemma divs_mult:
   967   assumes mr: "m dvd r" and nr: "n dvd r" and mn: "gcd m n = 1"
   968   shows "m * n dvd r"
   969 proof -
   970   from mr nr obtain m' n' where m': "r = m*m'" and n': "r = n*n'"
   971     unfolding dvd_def by blast
   972   from mr n' have "m dvd n'*n" by (simp add: ac_simps)
   973   hence "m dvd n'" using coprime_dvd_mult_iff[OF mn] by simp
   974   then obtain k where k: "n' = m*k" unfolding dvd_def by blast
   975   with n' have "r = m * n * k" by (simp add: mult_ac)
   976   then show ?thesis unfolding dvd_def by blast
   977 qed
   978 
   979 lemma coprime_plus_one [simp]: "gcd (n + 1) n = 1"
   980   by (subst add_commute, simp)
   981 
   982 lemma setprod_coprime [rule_format]:
   983   "(\<forall>i\<in>A. gcd (f i) a = 1) \<longrightarrow> gcd (\<Prod>i\<in>A. f i) a = 1"
   984   apply (cases "finite A")
   985   apply (induct set: finite)
   986   apply (auto simp add: gcd_mult_cancel)
   987   done
   988 
   989 lemma coprime_divisors: 
   990   assumes "d dvd a" "e dvd b" "gcd a b = 1"
   991   shows "gcd d e = 1" 
   992 proof -
   993   from assms obtain k l where "a = d * k" "b = e * l"
   994     unfolding dvd_def by blast
   995   with assms have "gcd (d * k) (e * l) = 1" by simp
   996   hence "gcd (d * k) e = 1" by (rule coprime_lmult)
   997   also have "gcd (d * k) e = gcd e (d * k)" by (simp add: ac_simps)
   998   finally have "gcd e d = 1" by (rule coprime_lmult)
   999   then show ?thesis by (simp add: ac_simps)
  1000 qed
  1001 
  1002 lemma invertible_coprime:
  1003   assumes "a * b mod m = 1"
  1004   shows "coprime a m"
  1005 proof -
  1006   from assms have "coprime m (a * b mod m)"
  1007     by simp
  1008   then have "coprime m (a * b)"
  1009     by simp
  1010   then have "coprime m a"
  1011     by (rule coprime_lmult)
  1012   then show ?thesis
  1013     by (simp add: ac_simps)
  1014 qed
  1015 
  1016 lemma lcm_gcd:
  1017   "lcm a b = a * b div (gcd a b * normalization_factor (a*b))"
  1018   by (simp only: lcm_lcm_eucl gcd_gcd_eucl lcm_eucl_def)
  1019 
  1020 lemma lcm_gcd_prod:
  1021   "lcm a b * gcd a b = a * b div normalization_factor (a*b)"
  1022 proof (cases "a * b = 0")
  1023   let ?nf = normalization_factor
  1024   assume "a * b \<noteq> 0"
  1025   hence "gcd a b \<noteq> 0" by simp
  1026   from lcm_gcd have "lcm a b * gcd a b = gcd a b * (a * b div (?nf (a*b) * gcd a b))" 
  1027     by (simp add: mult_ac)
  1028   also from `a * b \<noteq> 0` have "... = a * b div ?nf (a*b)"
  1029     by (simp add: div_mult_swap mult.commute)
  1030   finally show ?thesis .
  1031 qed (auto simp add: lcm_gcd)
  1032 
  1033 lemma lcm_dvd1 [iff]:
  1034   "a dvd lcm a b"
  1035 proof (cases "a*b = 0")
  1036   assume "a * b \<noteq> 0"
  1037   hence "gcd a b \<noteq> 0" by simp
  1038   let ?c = "1 div normalization_factor (a * b)"
  1039   from `a * b \<noteq> 0` have [simp]: "is_unit (normalization_factor (a * b))" by simp
  1040   from lcm_gcd_prod[of a b] have "lcm a b * gcd a b = a * ?c * b"
  1041     by (simp add: div_mult_swap unit_div_commute)
  1042   hence "lcm a b * gcd a b div gcd a b = a * ?c * b div gcd a b" by simp
  1043   with `gcd a b \<noteq> 0` have "lcm a b = a * ?c * b div gcd a b"
  1044     by (subst (asm) div_mult_self2_is_id, simp_all)
  1045   also have "... = a * (?c * b div gcd a b)"
  1046     by (metis div_mult_swap gcd_dvd2 mult_assoc)
  1047   finally show ?thesis by (rule dvdI)
  1048 qed (auto simp add: lcm_gcd)
  1049 
  1050 lemma lcm_least:
  1051   "\<lbrakk>a dvd k; b dvd k\<rbrakk> \<Longrightarrow> lcm a b dvd k"
  1052 proof (cases "k = 0")
  1053   let ?nf = normalization_factor
  1054   assume "k \<noteq> 0"
  1055   hence "is_unit (?nf k)" by simp
  1056   hence "?nf k \<noteq> 0" by (metis not_is_unit_0)
  1057   assume A: "a dvd k" "b dvd k"
  1058   hence "gcd a b \<noteq> 0" using `k \<noteq> 0` by auto
  1059   from A obtain r s where ar: "k = a * r" and bs: "k = b * s" 
  1060     unfolding dvd_def by blast
  1061   with `k \<noteq> 0` have "r * s \<noteq> 0"
  1062     by auto (drule sym [of 0], simp)
  1063   hence "is_unit (?nf (r * s))" by simp
  1064   let ?c = "?nf k div ?nf (r*s)"
  1065   from `is_unit (?nf k)` and `is_unit (?nf (r * s))` have "is_unit ?c" by (rule unit_div)
  1066   hence "?c \<noteq> 0" using not_is_unit_0 by fast 
  1067   from ar bs have "k * k * gcd s r = ?nf k * k * gcd (k * s) (k * r)"
  1068     by (subst mult_assoc, subst gcd_mult_distrib[of k s r], simp only: ac_simps)
  1069   also have "... = ?nf k * k * gcd ((r*s) * a) ((r*s) * b)"
  1070     by (subst (3) `k = a * r`, subst (3) `k = b * s`, simp add: algebra_simps)
  1071   also have "... = ?c * r*s * k * gcd a b" using `r * s \<noteq> 0`
  1072     by (subst gcd_mult_distrib'[symmetric], simp add: algebra_simps unit_simps)
  1073   finally have "(a*r) * (b*s) * gcd s r = ?c * k * r * s * gcd a b"
  1074     by (subst ar[symmetric], subst bs[symmetric], simp add: mult_ac)
  1075   hence "a * b * gcd s r * (r * s) = ?c * k * gcd a b * (r * s)"
  1076     by (simp add: algebra_simps)
  1077   hence "?c * k * gcd a b = a * b * gcd s r" using `r * s \<noteq> 0`
  1078     by (metis div_mult_self2_is_id)
  1079   also have "... = lcm a b * gcd a b * gcd s r * ?nf (a*b)"
  1080     by (subst lcm_gcd_prod[of a b], metis gcd_mult_distrib gcd_mult_distrib') 
  1081   also have "... = lcm a b * gcd s r * ?nf (a*b) * gcd a b"
  1082     by (simp add: algebra_simps)
  1083   finally have "k * ?c = lcm a b * gcd s r * ?nf (a*b)" using `gcd a b \<noteq> 0`
  1084     by (metis mult.commute div_mult_self2_is_id)
  1085   hence "k = lcm a b * (gcd s r * ?nf (a*b)) div ?c" using `?c \<noteq> 0`
  1086     by (metis div_mult_self2_is_id mult_assoc) 
  1087   also have "... = lcm a b * (gcd s r * ?nf (a*b) div ?c)" using `is_unit ?c`
  1088     by (simp add: unit_simps)
  1089   finally show ?thesis by (rule dvdI)
  1090 qed simp
  1091 
  1092 lemma lcm_zero:
  1093   "lcm a b = 0 \<longleftrightarrow> a = 0 \<or> b = 0"
  1094 proof -
  1095   let ?nf = normalization_factor
  1096   {
  1097     assume "a \<noteq> 0" "b \<noteq> 0"
  1098     hence "a * b div ?nf (a * b) \<noteq> 0" by (simp add: no_zero_divisors)
  1099     moreover from `a \<noteq> 0` and `b \<noteq> 0` have "gcd a b \<noteq> 0" by simp
  1100     ultimately have "lcm a b \<noteq> 0" using lcm_gcd_prod[of a b] by (intro notI, simp)
  1101   } moreover {
  1102     assume "a = 0 \<or> b = 0"
  1103     hence "lcm a b = 0" by (elim disjE, simp_all add: lcm_gcd)
  1104   }
  1105   ultimately show ?thesis by blast
  1106 qed
  1107 
  1108 lemmas lcm_0_iff = lcm_zero
  1109 
  1110 lemma gcd_lcm: 
  1111   assumes "lcm a b \<noteq> 0"
  1112   shows "gcd a b = a * b div (lcm a b * normalization_factor (a * b))"
  1113 proof-
  1114   from assms have "gcd a b \<noteq> 0" by (simp add: lcm_zero)
  1115   let ?c = "normalization_factor (a * b)"
  1116   from `lcm a b \<noteq> 0` have "?c \<noteq> 0" by (intro notI, simp add: lcm_zero no_zero_divisors)
  1117   hence "is_unit ?c" by simp
  1118   from lcm_gcd_prod [of a b] have "gcd a b = a * b div ?c div lcm a b"
  1119     by (subst (2) div_mult_self2_is_id[OF `lcm a b \<noteq> 0`, symmetric], simp add: mult_ac)
  1120   also from `is_unit ?c` have "... = a * b div (lcm a b * ?c)"
  1121     by (metis `?c \<noteq> 0` div_mult_mult1 dvd_mult_div_cancel mult_commute normalization_factor_dvd')
  1122   finally show ?thesis .
  1123 qed
  1124 
  1125 lemma normalization_factor_lcm [simp]:
  1126   "normalization_factor (lcm a b) = (if a = 0 \<or> b = 0 then 0 else 1)"
  1127 proof (cases "a = 0 \<or> b = 0")
  1128   case True then show ?thesis
  1129     by (auto simp add: lcm_gcd) 
  1130 next
  1131   case False
  1132   let ?nf = normalization_factor
  1133   from lcm_gcd_prod[of a b] 
  1134     have "?nf (lcm a b) * ?nf (gcd a b) = ?nf (a*b) div ?nf (a*b)"
  1135     by (metis div_by_0 div_self normalization_correct normalization_factor_0 normalization_factor_mult)
  1136   also have "... = (if a*b = 0 then 0 else 1)"
  1137     by simp
  1138   finally show ?thesis using False by simp
  1139 qed
  1140 
  1141 lemma lcm_dvd2 [iff]: "b dvd lcm a b"
  1142   using lcm_dvd1 [of b a] by (simp add: lcm_gcd ac_simps)
  1143 
  1144 lemma lcmI:
  1145   "\<lbrakk>a dvd k; b dvd k; \<And>l. a dvd l \<Longrightarrow> b dvd l \<Longrightarrow> k dvd l;
  1146     normalization_factor k = (if k = 0 then 0 else 1)\<rbrakk> \<Longrightarrow> k = lcm a b"
  1147   by (intro normed_associated_imp_eq) (auto simp: associated_def intro: lcm_least)
  1148 
  1149 sublocale lcm!: abel_semigroup lcm
  1150 proof
  1151   fix a b c
  1152   show "lcm (lcm a b) c = lcm a (lcm b c)"
  1153   proof (rule lcmI)
  1154     have "a dvd lcm a b" and "lcm a b dvd lcm (lcm a b) c" by simp_all
  1155     then show "a dvd lcm (lcm a b) c" by (rule dvd_trans)
  1156     
  1157     have "b dvd lcm a b" and "lcm a b dvd lcm (lcm a b) c" by simp_all
  1158     hence "b dvd lcm (lcm a b) c" by (rule dvd_trans)
  1159     moreover have "c dvd lcm (lcm a b) c" by simp
  1160     ultimately show "lcm b c dvd lcm (lcm a b) c" by (rule lcm_least)
  1161 
  1162     fix l assume "a dvd l" and "lcm b c dvd l"
  1163     have "b dvd lcm b c" by simp
  1164     from this and `lcm b c dvd l` have "b dvd l" by (rule dvd_trans)
  1165     have "c dvd lcm b c" by simp
  1166     from this and `lcm b c dvd l` have "c dvd l" by (rule dvd_trans)
  1167     from `a dvd l` and `b dvd l` have "lcm a b dvd l" by (rule lcm_least)
  1168     from this and `c dvd l` show "lcm (lcm a b) c dvd l" by (rule lcm_least)
  1169   qed (simp add: lcm_zero)
  1170 next
  1171   fix a b
  1172   show "lcm a b = lcm b a"
  1173     by (simp add: lcm_gcd ac_simps)
  1174 qed
  1175 
  1176 lemma dvd_lcm_D1:
  1177   "lcm m n dvd k \<Longrightarrow> m dvd k"
  1178   by (rule dvd_trans, rule lcm_dvd1, assumption)
  1179 
  1180 lemma dvd_lcm_D2:
  1181   "lcm m n dvd k \<Longrightarrow> n dvd k"
  1182   by (rule dvd_trans, rule lcm_dvd2, assumption)
  1183 
  1184 lemma gcd_dvd_lcm [simp]:
  1185   "gcd a b dvd lcm a b"
  1186   by (metis dvd_trans gcd_dvd2 lcm_dvd2)
  1187 
  1188 lemma lcm_1_iff:
  1189   "lcm a b = 1 \<longleftrightarrow> is_unit a \<and> is_unit b"
  1190 proof
  1191   assume "lcm a b = 1"
  1192   then show "is_unit a \<and> is_unit b" by auto
  1193 next
  1194   assume "is_unit a \<and> is_unit b"
  1195   hence "a dvd 1" and "b dvd 1" by simp_all
  1196   hence "is_unit (lcm a b)" by (rule lcm_least)
  1197   hence "lcm a b = normalization_factor (lcm a b)"
  1198     by (subst normalization_factor_unit, simp_all)
  1199   also have "\<dots> = 1" using `is_unit a \<and> is_unit b`
  1200     by auto
  1201   finally show "lcm a b = 1" .
  1202 qed
  1203 
  1204 lemma lcm_0_left [simp]:
  1205   "lcm 0 a = 0"
  1206   by (rule sym, rule lcmI, simp_all)
  1207 
  1208 lemma lcm_0 [simp]:
  1209   "lcm a 0 = 0"
  1210   by (rule sym, rule lcmI, simp_all)
  1211 
  1212 lemma lcm_unique:
  1213   "a dvd d \<and> b dvd d \<and> 
  1214   normalization_factor d = (if d = 0 then 0 else 1) \<and>
  1215   (\<forall>e. a dvd e \<and> b dvd e \<longrightarrow> d dvd e) \<longleftrightarrow> d = lcm a b"
  1216   by (rule, auto intro: lcmI simp: lcm_least lcm_zero)
  1217 
  1218 lemma dvd_lcm_I1 [simp]:
  1219   "k dvd m \<Longrightarrow> k dvd lcm m n"
  1220   by (metis lcm_dvd1 dvd_trans)
  1221 
  1222 lemma dvd_lcm_I2 [simp]:
  1223   "k dvd n \<Longrightarrow> k dvd lcm m n"
  1224   by (metis lcm_dvd2 dvd_trans)
  1225 
  1226 lemma lcm_1_left [simp]:
  1227   "lcm 1 a = a div normalization_factor a"
  1228   by (cases "a = 0") (simp, rule sym, rule lcmI, simp_all)
  1229 
  1230 lemma lcm_1_right [simp]:
  1231   "lcm a 1 = a div normalization_factor a"
  1232   using lcm_1_left [of a] by (simp add: ac_simps)
  1233 
  1234 lemma lcm_coprime:
  1235   "gcd a b = 1 \<Longrightarrow> lcm a b = a * b div normalization_factor (a*b)"
  1236   by (subst lcm_gcd) simp
  1237 
  1238 lemma lcm_proj1_if_dvd: 
  1239   "b dvd a \<Longrightarrow> lcm a b = a div normalization_factor a"
  1240   by (cases "a = 0") (simp, rule sym, rule lcmI, simp_all)
  1241 
  1242 lemma lcm_proj2_if_dvd: 
  1243   "a dvd b \<Longrightarrow> lcm a b = b div normalization_factor b"
  1244   using lcm_proj1_if_dvd [of a b] by (simp add: ac_simps)
  1245 
  1246 lemma lcm_proj1_iff:
  1247   "lcm m n = m div normalization_factor m \<longleftrightarrow> n dvd m"
  1248 proof
  1249   assume A: "lcm m n = m div normalization_factor m"
  1250   show "n dvd m"
  1251   proof (cases "m = 0")
  1252     assume [simp]: "m \<noteq> 0"
  1253     from A have B: "m = lcm m n * normalization_factor m"
  1254       by (simp add: unit_eq_div2)
  1255     show ?thesis by (subst B, simp)
  1256   qed simp
  1257 next
  1258   assume "n dvd m"
  1259   then show "lcm m n = m div normalization_factor m" by (rule lcm_proj1_if_dvd)
  1260 qed
  1261 
  1262 lemma lcm_proj2_iff:
  1263   "lcm m n = n div normalization_factor n \<longleftrightarrow> m dvd n"
  1264   using lcm_proj1_iff [of n m] by (simp add: ac_simps)
  1265 
  1266 lemma euclidean_size_lcm_le1: 
  1267   assumes "a \<noteq> 0" and "b \<noteq> 0"
  1268   shows "euclidean_size a \<le> euclidean_size (lcm a b)"
  1269 proof -
  1270   have "a dvd lcm a b" by (rule lcm_dvd1)
  1271   then obtain c where A: "lcm a b = a * c" unfolding dvd_def by blast
  1272   with `a \<noteq> 0` and `b \<noteq> 0` have "c \<noteq> 0" by (auto simp: lcm_zero)
  1273   then show ?thesis by (subst A, intro size_mult_mono)
  1274 qed
  1275 
  1276 lemma euclidean_size_lcm_le2:
  1277   "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> euclidean_size b \<le> euclidean_size (lcm a b)"
  1278   using euclidean_size_lcm_le1 [of b a] by (simp add: ac_simps)
  1279 
  1280 lemma euclidean_size_lcm_less1:
  1281   assumes "b \<noteq> 0" and "\<not>b dvd a"
  1282   shows "euclidean_size a < euclidean_size (lcm a b)"
  1283 proof (rule ccontr)
  1284   from assms have "a \<noteq> 0" by auto
  1285   assume "\<not>euclidean_size a < euclidean_size (lcm a b)"
  1286   with `a \<noteq> 0` and `b \<noteq> 0` have "euclidean_size (lcm a b) = euclidean_size a"
  1287     by (intro le_antisym, simp, intro euclidean_size_lcm_le1)
  1288   with assms have "lcm a b dvd a" 
  1289     by (rule_tac dvd_euclidean_size_eq_imp_dvd) (auto simp: lcm_zero)
  1290   hence "b dvd a" by (rule dvd_lcm_D2)
  1291   with `\<not>b dvd a` show False by contradiction
  1292 qed
  1293 
  1294 lemma euclidean_size_lcm_less2:
  1295   assumes "a \<noteq> 0" and "\<not>a dvd b"
  1296   shows "euclidean_size b < euclidean_size (lcm a b)"
  1297   using assms euclidean_size_lcm_less1 [of a b] by (simp add: ac_simps)
  1298 
  1299 lemma lcm_mult_unit1:
  1300   "is_unit a \<Longrightarrow> lcm (b * a) c = lcm b c"
  1301   apply (rule lcmI)
  1302   apply (rule dvd_trans[of _ "b * a"], simp, rule lcm_dvd1)
  1303   apply (rule lcm_dvd2)
  1304   apply (rule lcm_least, simp add: unit_simps, assumption)
  1305   apply (subst normalization_factor_lcm, simp add: lcm_zero)
  1306   done
  1307 
  1308 lemma lcm_mult_unit2:
  1309   "is_unit a \<Longrightarrow> lcm b (c * a) = lcm b c"
  1310   using lcm_mult_unit1 [of a c b] by (simp add: ac_simps)
  1311 
  1312 lemma lcm_div_unit1:
  1313   "is_unit a \<Longrightarrow> lcm (b div a) c = lcm b c"
  1314   by (erule is_unitE [of _ b]) (simp add: lcm_mult_unit1) 
  1315 
  1316 lemma lcm_div_unit2:
  1317   "is_unit a \<Longrightarrow> lcm b (c div a) = lcm b c"
  1318   by (erule is_unitE [of _ c]) (simp add: lcm_mult_unit2)
  1319 
  1320 lemma lcm_left_idem:
  1321   "lcm a (lcm a b) = lcm a b"
  1322   apply (rule lcmI)
  1323   apply simp
  1324   apply (subst lcm.assoc [symmetric], rule lcm_dvd2)
  1325   apply (rule lcm_least, assumption)
  1326   apply (erule (1) lcm_least)
  1327   apply (auto simp: lcm_zero)
  1328   done
  1329 
  1330 lemma lcm_right_idem:
  1331   "lcm (lcm a b) b = lcm a b"
  1332   apply (rule lcmI)
  1333   apply (subst lcm.assoc, rule lcm_dvd1)
  1334   apply (rule lcm_dvd2)
  1335   apply (rule lcm_least, erule (1) lcm_least, assumption)
  1336   apply (auto simp: lcm_zero)
  1337   done
  1338 
  1339 lemma comp_fun_idem_lcm: "comp_fun_idem lcm"
  1340 proof
  1341   fix a b show "lcm a \<circ> lcm b = lcm b \<circ> lcm a"
  1342     by (simp add: fun_eq_iff ac_simps)
  1343 next
  1344   fix a show "lcm a \<circ> lcm a = lcm a" unfolding o_def
  1345     by (intro ext, simp add: lcm_left_idem)
  1346 qed
  1347 
  1348 lemma dvd_Lcm [simp]: "a \<in> A \<Longrightarrow> a dvd Lcm A"
  1349   and Lcm_dvd [simp]: "(\<forall>a\<in>A. a dvd l') \<Longrightarrow> Lcm A dvd l'"
  1350   and normalization_factor_Lcm [simp]: 
  1351           "normalization_factor (Lcm A) = (if Lcm A = 0 then 0 else 1)"
  1352 proof -
  1353   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>
  1354     normalization_factor (Lcm A) = (if Lcm A = 0 then 0 else 1)" (is ?thesis)
  1355   proof (cases "\<exists>l. l \<noteq>  0 \<and> (\<forall>a\<in>A. a dvd l)")
  1356     case False
  1357     hence "Lcm A = 0" by (auto simp: Lcm_Lcm_eucl Lcm_eucl_def)
  1358     with False show ?thesis by auto
  1359   next
  1360     case True
  1361     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
  1362     def n \<equiv> "LEAST n. \<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"
  1363     def l \<equiv> "SOME l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"
  1364     have "\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"
  1365       apply (subst n_def)
  1366       apply (rule LeastI[of _ "euclidean_size l\<^sub>0"])
  1367       apply (rule exI[of _ l\<^sub>0])
  1368       apply (simp add: l\<^sub>0_props)
  1369       done
  1370     from someI_ex[OF this] have "l \<noteq> 0" and "\<forall>a\<in>A. a dvd l" and "euclidean_size l = n" 
  1371       unfolding l_def by simp_all
  1372     {
  1373       fix l' assume "\<forall>a\<in>A. a dvd l'"
  1374       with `\<forall>a\<in>A. a dvd l` have "\<forall>a\<in>A. a dvd gcd l l'" by (auto intro: gcd_greatest)
  1375       moreover from `l \<noteq> 0` have "gcd l l' \<noteq> 0" by simp
  1376       ultimately have "\<exists>b. b \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd b) \<and> euclidean_size b = euclidean_size (gcd l l')"
  1377         by (intro exI[of _ "gcd l l'"], auto)
  1378       hence "euclidean_size (gcd l l') \<ge> n" by (subst n_def) (rule Least_le)
  1379       moreover have "euclidean_size (gcd l l') \<le> n"
  1380       proof -
  1381         have "gcd l l' dvd l" by simp
  1382         then obtain a where "l = gcd l l' * a" unfolding dvd_def by blast
  1383         with `l \<noteq> 0` have "a \<noteq> 0" by auto
  1384         hence "euclidean_size (gcd l l') \<le> euclidean_size (gcd l l' * a)"
  1385           by (rule size_mult_mono)
  1386         also have "gcd l l' * a = l" using `l = gcd l l' * a` ..
  1387         also note `euclidean_size l = n`
  1388         finally show "euclidean_size (gcd l l') \<le> n" .
  1389       qed
  1390       ultimately have "euclidean_size l = euclidean_size (gcd l l')" 
  1391         by (intro le_antisym, simp_all add: `euclidean_size l = n`)
  1392       with `l \<noteq> 0` have "l dvd gcd l l'" by (blast intro: dvd_euclidean_size_eq_imp_dvd)
  1393       hence "l dvd l'" by (blast dest: dvd_gcd_D2)
  1394     }
  1395 
  1396     with `(\<forall>a\<in>A. a dvd l)` and normalization_factor_is_unit[OF `l \<noteq> 0`] and `l \<noteq> 0`
  1397       have "(\<forall>a\<in>A. a dvd l div normalization_factor l) \<and> 
  1398         (\<forall>l'. (\<forall>a\<in>A. a dvd l') \<longrightarrow> l div normalization_factor l dvd l') \<and>
  1399         normalization_factor (l div normalization_factor l) = 
  1400         (if l div normalization_factor l = 0 then 0 else 1)"
  1401       by (auto simp: unit_simps)
  1402     also from True have "l div normalization_factor l = Lcm A"
  1403       by (simp add: Lcm_Lcm_eucl Lcm_eucl_def Let_def n_def l_def)
  1404     finally show ?thesis .
  1405   qed
  1406   note A = this
  1407 
  1408   {fix a assume "a \<in> A" then show "a dvd Lcm A" using A by blast}
  1409   {fix l' assume "\<forall>a\<in>A. a dvd l'" then show "Lcm A dvd l'" using A by blast}
  1410   from A show "normalization_factor (Lcm A) = (if Lcm A = 0 then 0 else 1)" by blast
  1411 qed
  1412     
  1413 lemma LcmI:
  1414   "(\<And>a. a\<in>A \<Longrightarrow> a dvd l) \<Longrightarrow> (\<And>l'. (\<forall>a\<in>A. a dvd l') \<Longrightarrow> l dvd l') \<Longrightarrow>
  1415       normalization_factor l = (if l = 0 then 0 else 1) \<Longrightarrow> l = Lcm A"
  1416   by (intro normed_associated_imp_eq)
  1417     (auto intro: Lcm_dvd dvd_Lcm simp: associated_def)
  1418 
  1419 lemma Lcm_subset:
  1420   "A \<subseteq> B \<Longrightarrow> Lcm A dvd Lcm B"
  1421   by (blast intro: Lcm_dvd dvd_Lcm)
  1422 
  1423 lemma Lcm_Un:
  1424   "Lcm (A \<union> B) = lcm (Lcm A) (Lcm B)"
  1425   apply (rule lcmI)
  1426   apply (blast intro: Lcm_subset)
  1427   apply (blast intro: Lcm_subset)
  1428   apply (intro Lcm_dvd ballI, elim UnE)
  1429   apply (rule dvd_trans, erule dvd_Lcm, assumption)
  1430   apply (rule dvd_trans, erule dvd_Lcm, assumption)
  1431   apply simp
  1432   done
  1433 
  1434 lemma Lcm_1_iff:
  1435   "Lcm A = 1 \<longleftrightarrow> (\<forall>a\<in>A. is_unit a)"
  1436 proof
  1437   assume "Lcm A = 1"
  1438   then show "\<forall>a\<in>A. is_unit a" by auto
  1439 qed (rule LcmI [symmetric], auto)
  1440 
  1441 lemma Lcm_no_units:
  1442   "Lcm A = Lcm (A - {a. is_unit a})"
  1443 proof -
  1444   have "(A - {a. is_unit a}) \<union> {a\<in>A. is_unit a} = A" by blast
  1445   hence "Lcm A = lcm (Lcm (A - {a. is_unit a})) (Lcm {a\<in>A. is_unit a})"
  1446     by (simp add: Lcm_Un[symmetric])
  1447   also have "Lcm {a\<in>A. is_unit a} = 1" by (simp add: Lcm_1_iff)
  1448   finally show ?thesis by simp
  1449 qed
  1450 
  1451 lemma Lcm_empty [simp]:
  1452   "Lcm {} = 1"
  1453   by (simp add: Lcm_1_iff)
  1454 
  1455 lemma Lcm_eq_0 [simp]:
  1456   "0 \<in> A \<Longrightarrow> Lcm A = 0"
  1457   by (drule dvd_Lcm) simp
  1458 
  1459 lemma Lcm0_iff':
  1460   "Lcm A = 0 \<longleftrightarrow> \<not>(\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l))"
  1461 proof
  1462   assume "Lcm A = 0"
  1463   show "\<not>(\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l))"
  1464   proof
  1465     assume ex: "\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l)"
  1466     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
  1467     def n \<equiv> "LEAST n. \<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"
  1468     def l \<equiv> "SOME l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"
  1469     have "\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"
  1470       apply (subst n_def)
  1471       apply (rule LeastI[of _ "euclidean_size l\<^sub>0"])
  1472       apply (rule exI[of _ l\<^sub>0])
  1473       apply (simp add: l\<^sub>0_props)
  1474       done
  1475     from someI_ex[OF this] have "l \<noteq> 0" unfolding l_def by simp_all
  1476     hence "l div normalization_factor l \<noteq> 0" by simp
  1477     also from ex have "l div normalization_factor l = Lcm A"
  1478        by (simp only: Lcm_Lcm_eucl Lcm_eucl_def n_def l_def if_True Let_def)
  1479     finally show False using `Lcm A = 0` by contradiction
  1480   qed
  1481 qed (simp only: Lcm_Lcm_eucl Lcm_eucl_def if_False)
  1482 
  1483 lemma Lcm0_iff [simp]:
  1484   "finite A \<Longrightarrow> Lcm A = 0 \<longleftrightarrow> 0 \<in> A"
  1485 proof -
  1486   assume "finite A"
  1487   have "0 \<in> A \<Longrightarrow> Lcm A = 0"  by (intro dvd_0_left dvd_Lcm)
  1488   moreover {
  1489     assume "0 \<notin> A"
  1490     hence "\<Prod>A \<noteq> 0" 
  1491       apply (induct rule: finite_induct[OF `finite A`]) 
  1492       apply simp
  1493       apply (subst setprod.insert, assumption, assumption)
  1494       apply (rule no_zero_divisors)
  1495       apply blast+
  1496       done
  1497     moreover from `finite A` have "\<forall>a\<in>A. a dvd \<Prod>A" by blast
  1498     ultimately have "\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l)" by blast
  1499     with Lcm0_iff' have "Lcm A \<noteq> 0" by simp
  1500   }
  1501   ultimately show "Lcm A = 0 \<longleftrightarrow> 0 \<in> A" by blast
  1502 qed
  1503 
  1504 lemma Lcm_no_multiple:
  1505   "(\<forall>m. m \<noteq> 0 \<longrightarrow> (\<exists>a\<in>A. \<not>a dvd m)) \<Longrightarrow> Lcm A = 0"
  1506 proof -
  1507   assume "\<forall>m. m \<noteq> 0 \<longrightarrow> (\<exists>a\<in>A. \<not>a dvd m)"
  1508   hence "\<not>(\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l))" by blast
  1509   then show "Lcm A = 0" by (simp only: Lcm_Lcm_eucl Lcm_eucl_def if_False)
  1510 qed
  1511 
  1512 lemma Lcm_insert [simp]:
  1513   "Lcm (insert a A) = lcm a (Lcm A)"
  1514 proof (rule lcmI)
  1515   fix l assume "a dvd l" and "Lcm A dvd l"
  1516   hence "\<forall>a\<in>A. a dvd l" by (blast intro: dvd_trans dvd_Lcm)
  1517   with `a dvd l` show "Lcm (insert a A) dvd l" by (force intro: Lcm_dvd)
  1518 qed (auto intro: Lcm_dvd dvd_Lcm)
  1519  
  1520 lemma Lcm_finite:
  1521   assumes "finite A"
  1522   shows "Lcm A = Finite_Set.fold lcm 1 A"
  1523   by (induct rule: finite.induct[OF `finite A`])
  1524     (simp_all add: comp_fun_idem.fold_insert_idem[OF comp_fun_idem_lcm])
  1525 
  1526 lemma Lcm_set [code_unfold]:
  1527   "Lcm (set xs) = fold lcm xs 1"
  1528   using comp_fun_idem.fold_set_fold[OF comp_fun_idem_lcm] Lcm_finite by (simp add: ac_simps)
  1529 
  1530 lemma Lcm_singleton [simp]:
  1531   "Lcm {a} = a div normalization_factor a"
  1532   by simp
  1533 
  1534 lemma Lcm_2 [simp]:
  1535   "Lcm {a,b} = lcm a b"
  1536   by (simp only: Lcm_insert Lcm_empty lcm_1_right)
  1537     (cases "b = 0", simp, rule lcm_div_unit2, simp)
  1538 
  1539 lemma Lcm_coprime:
  1540   assumes "finite A" and "A \<noteq> {}" 
  1541   assumes "\<And>a b. a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<noteq> b \<Longrightarrow> gcd a b = 1"
  1542   shows "Lcm A = \<Prod>A div normalization_factor (\<Prod>A)"
  1543 using assms proof (induct rule: finite_ne_induct)
  1544   case (insert a A)
  1545   have "Lcm (insert a A) = lcm a (Lcm A)" by simp
  1546   also from insert have "Lcm A = \<Prod>A div normalization_factor (\<Prod>A)" by blast
  1547   also have "lcm a \<dots> = lcm a (\<Prod>A)" by (cases "\<Prod>A = 0") (simp_all add: lcm_div_unit2)
  1548   also from insert have "gcd a (\<Prod>A) = 1" by (subst gcd.commute, intro setprod_coprime) auto
  1549   with insert have "lcm a (\<Prod>A) = \<Prod>(insert a A) div normalization_factor (\<Prod>(insert a A))"
  1550     by (simp add: lcm_coprime)
  1551   finally show ?case .
  1552 qed simp
  1553       
  1554 lemma Lcm_coprime':
  1555   "card A \<noteq> 0 \<Longrightarrow> (\<And>a b. a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<noteq> b \<Longrightarrow> gcd a b = 1)
  1556     \<Longrightarrow> Lcm A = \<Prod>A div normalization_factor (\<Prod>A)"
  1557   by (rule Lcm_coprime) (simp_all add: card_eq_0_iff)
  1558 
  1559 lemma Gcd_Lcm:
  1560   "Gcd A = Lcm {d. \<forall>a\<in>A. d dvd a}"
  1561   by (simp add: Gcd_Gcd_eucl Lcm_Lcm_eucl Gcd_eucl_def)
  1562 
  1563 lemma Gcd_dvd [simp]: "a \<in> A \<Longrightarrow> Gcd A dvd a"
  1564   and dvd_Gcd [simp]: "(\<forall>a\<in>A. g' dvd a) \<Longrightarrow> g' dvd Gcd A"
  1565   and normalization_factor_Gcd [simp]: 
  1566     "normalization_factor (Gcd A) = (if Gcd A = 0 then 0 else 1)"
  1567 proof -
  1568   fix a assume "a \<in> A"
  1569   hence "Lcm {d. \<forall>a\<in>A. d dvd a} dvd a" by (intro Lcm_dvd) blast
  1570   then show "Gcd A dvd a" by (simp add: Gcd_Lcm)
  1571 next
  1572   fix g' assume "\<forall>a\<in>A. g' dvd a"
  1573   hence "g' dvd Lcm {d. \<forall>a\<in>A. d dvd a}" by (intro dvd_Lcm) blast
  1574   then show "g' dvd Gcd A" by (simp add: Gcd_Lcm)
  1575 next
  1576   show "normalization_factor (Gcd A) = (if Gcd A = 0 then 0 else 1)"
  1577     by (simp add: Gcd_Lcm)
  1578 qed
  1579 
  1580 lemma GcdI:
  1581   "(\<And>a. a\<in>A \<Longrightarrow> l dvd a) \<Longrightarrow> (\<And>l'. (\<forall>a\<in>A. l' dvd a) \<Longrightarrow> l' dvd l) \<Longrightarrow>
  1582     normalization_factor l = (if l = 0 then 0 else 1) \<Longrightarrow> l = Gcd A"
  1583   by (intro normed_associated_imp_eq)
  1584     (auto intro: Gcd_dvd dvd_Gcd simp: associated_def)
  1585 
  1586 lemma Lcm_Gcd:
  1587   "Lcm A = Gcd {m. \<forall>a\<in>A. a dvd m}"
  1588   by (rule LcmI[symmetric]) (auto intro: dvd_Gcd Gcd_dvd)
  1589 
  1590 lemma Gcd_0_iff:
  1591   "Gcd A = 0 \<longleftrightarrow> A \<subseteq> {0}"
  1592   apply (rule iffI)
  1593   apply (rule subsetI, drule Gcd_dvd, simp)
  1594   apply (auto intro: GcdI[symmetric])
  1595   done
  1596 
  1597 lemma Gcd_empty [simp]:
  1598   "Gcd {} = 0"
  1599   by (simp add: Gcd_0_iff)
  1600 
  1601 lemma Gcd_1:
  1602   "1 \<in> A \<Longrightarrow> Gcd A = 1"
  1603   by (intro GcdI[symmetric]) (auto intro: Gcd_dvd dvd_Gcd)
  1604 
  1605 lemma Gcd_insert [simp]:
  1606   "Gcd (insert a A) = gcd a (Gcd A)"
  1607 proof (rule gcdI)
  1608   fix l assume "l dvd a" and "l dvd Gcd A"
  1609   hence "\<forall>a\<in>A. l dvd a" by (blast intro: dvd_trans Gcd_dvd)
  1610   with `l dvd a` show "l dvd Gcd (insert a A)" by (force intro: Gcd_dvd)
  1611 qed auto
  1612 
  1613 lemma Gcd_finite:
  1614   assumes "finite A"
  1615   shows "Gcd A = Finite_Set.fold gcd 0 A"
  1616   by (induct rule: finite.induct[OF `finite A`])
  1617     (simp_all add: comp_fun_idem.fold_insert_idem[OF comp_fun_idem_gcd])
  1618 
  1619 lemma Gcd_set [code_unfold]:
  1620   "Gcd (set xs) = fold gcd xs 0"
  1621   using comp_fun_idem.fold_set_fold[OF comp_fun_idem_gcd] Gcd_finite by (simp add: ac_simps)
  1622 
  1623 lemma Gcd_singleton [simp]: "Gcd {a} = a div normalization_factor a"
  1624   by (simp add: gcd_0)
  1625 
  1626 lemma Gcd_2 [simp]: "Gcd {a,b} = gcd a b"
  1627   by (simp only: Gcd_insert Gcd_empty gcd_0) (cases "b = 0", simp, rule gcd_div_unit2, simp)
  1628 
  1629 subclass semiring_gcd
  1630   by unfold_locales (simp_all add: gcd_greatest_iff)
  1631   
  1632 end
  1633 
  1634 text {*
  1635   A Euclidean ring is a Euclidean semiring with additive inverses. It provides a 
  1636   few more lemmas; in particular, Bezout's lemma holds for any Euclidean ring.
  1637 *}
  1638 
  1639 class euclidean_ring = euclidean_semiring + idom
  1640 
  1641 class euclidean_ring_gcd = euclidean_semiring_gcd + idom
  1642 begin
  1643 
  1644 subclass euclidean_ring ..
  1645 
  1646 subclass ring_gcd ..
  1647 
  1648 lemma gcd_neg1 [simp]:
  1649   "gcd (-a) b = gcd a b"
  1650   by (rule sym, rule gcdI, simp_all add: gcd_greatest)
  1651 
  1652 lemma gcd_neg2 [simp]:
  1653   "gcd a (-b) = gcd a b"
  1654   by (rule sym, rule gcdI, simp_all add: gcd_greatest)
  1655 
  1656 lemma gcd_neg_numeral_1 [simp]:
  1657   "gcd (- numeral n) a = gcd (numeral n) a"
  1658   by (fact gcd_neg1)
  1659 
  1660 lemma gcd_neg_numeral_2 [simp]:
  1661   "gcd a (- numeral n) = gcd a (numeral n)"
  1662   by (fact gcd_neg2)
  1663 
  1664 lemma gcd_diff1: "gcd (m - n) n = gcd m n"
  1665   by (subst diff_conv_add_uminus, subst gcd_neg2[symmetric],  subst gcd_add1, simp)
  1666 
  1667 lemma gcd_diff2: "gcd (n - m) n = gcd m n"
  1668   by (subst gcd_neg1[symmetric], simp only: minus_diff_eq gcd_diff1)
  1669 
  1670 lemma coprime_minus_one [simp]: "gcd (n - 1) n = 1"
  1671 proof -
  1672   have "gcd (n - 1) n = gcd n (n - 1)" by (fact gcd.commute)
  1673   also have "\<dots> = gcd ((n - 1) + 1) (n - 1)" by simp
  1674   also have "\<dots> = 1" by (rule coprime_plus_one)
  1675   finally show ?thesis .
  1676 qed
  1677 
  1678 lemma lcm_neg1 [simp]: "lcm (-a) b = lcm a b"
  1679   by (rule sym, rule lcmI, simp_all add: lcm_least lcm_zero)
  1680 
  1681 lemma lcm_neg2 [simp]: "lcm a (-b) = lcm a b"
  1682   by (rule sym, rule lcmI, simp_all add: lcm_least lcm_zero)
  1683 
  1684 lemma lcm_neg_numeral_1 [simp]: "lcm (- numeral n) a = lcm (numeral n) a"
  1685   by (fact lcm_neg1)
  1686 
  1687 lemma lcm_neg_numeral_2 [simp]: "lcm a (- numeral n) = lcm a (numeral n)"
  1688   by (fact lcm_neg2)
  1689 
  1690 function euclid_ext :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a" where
  1691   "euclid_ext a b = 
  1692      (if b = 0 then 
  1693         let c = 1 div normalization_factor a in (c, 0, a * c)
  1694       else 
  1695         case euclid_ext b (a mod b) of
  1696             (s,t,c) \<Rightarrow> (t, s - t * (a div b), c))"
  1697   by (pat_completeness, simp)
  1698   termination by (relation "measure (euclidean_size \<circ> snd)", simp_all)
  1699 
  1700 declare euclid_ext.simps [simp del]
  1701 
  1702 lemma euclid_ext_0: 
  1703   "euclid_ext a 0 = (1 div normalization_factor a, 0, a div normalization_factor a)"
  1704   by (subst euclid_ext.simps) (simp add: Let_def)
  1705 
  1706 lemma euclid_ext_non_0:
  1707   "b \<noteq> 0 \<Longrightarrow> euclid_ext a b = (case euclid_ext b (a mod b) of 
  1708     (s,t,c) \<Rightarrow> (t, s - t * (a div b), c))"
  1709   by (subst euclid_ext.simps) simp
  1710 
  1711 definition euclid_ext' :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a"
  1712 where
  1713   "euclid_ext' a b = (case euclid_ext a b of (s, t, _) \<Rightarrow> (s, t))"
  1714 
  1715 lemma euclid_ext_gcd [simp]:
  1716   "(case euclid_ext a b of (_,_,t) \<Rightarrow> t) = gcd a b"
  1717 proof (induct a b rule: euclid_ext.induct)
  1718   case (1 a b)
  1719   then show ?case
  1720   proof (cases "b = 0")
  1721     case True
  1722       then show ?thesis by  
  1723         (simp add: euclid_ext_0 unit_div mult_ac unit_simps gcd_0)
  1724     next
  1725     case False with 1 show ?thesis
  1726       by (simp add: euclid_ext_non_0 ac_simps split: prod.split prod.split_asm)
  1727     qed
  1728 qed
  1729 
  1730 lemma euclid_ext_gcd' [simp]:
  1731   "euclid_ext a b = (r, s, t) \<Longrightarrow> t = gcd a b"
  1732   by (insert euclid_ext_gcd[of a b], drule (1) subst, simp)
  1733 
  1734 lemma euclid_ext_correct:
  1735   "case euclid_ext a b of (s,t,c) \<Rightarrow> s*a + t*b = c"
  1736 proof (induct a b rule: euclid_ext.induct)
  1737   case (1 a b)
  1738   show ?case
  1739   proof (cases "b = 0")
  1740     case True
  1741     then show ?thesis by (simp add: euclid_ext_0 mult_ac)
  1742   next
  1743     case False
  1744     obtain s t c where stc: "euclid_ext b (a mod b) = (s,t,c)"
  1745       by (cases "euclid_ext b (a mod b)", blast)
  1746     from 1 have "c = s * b + t * (a mod b)" by (simp add: stc False)
  1747     also have "... = t*((a div b)*b + a mod b) + (s - t * (a div b))*b"
  1748       by (simp add: algebra_simps) 
  1749     also have "(a div b)*b + a mod b = a" using mod_div_equality .
  1750     finally show ?thesis
  1751       by (subst euclid_ext.simps, simp add: False stc)
  1752     qed
  1753 qed
  1754 
  1755 lemma euclid_ext'_correct:
  1756   "fst (euclid_ext' a b) * a + snd (euclid_ext' a b) * b = gcd a b"
  1757 proof-
  1758   obtain s t c where "euclid_ext a b = (s,t,c)"
  1759     by (cases "euclid_ext a b", blast)
  1760   with euclid_ext_correct[of a b] euclid_ext_gcd[of a b]
  1761     show ?thesis unfolding euclid_ext'_def by simp
  1762 qed
  1763 
  1764 lemma bezout: "\<exists>s t. s * a + t * b = gcd a b"
  1765   using euclid_ext'_correct by blast
  1766 
  1767 lemma euclid_ext'_0 [simp]: "euclid_ext' a 0 = (1 div normalization_factor a, 0)" 
  1768   by (simp add: bezw_def euclid_ext'_def euclid_ext_0)
  1769 
  1770 lemma euclid_ext'_non_0: "b \<noteq> 0 \<Longrightarrow> euclid_ext' a b = (snd (euclid_ext' b (a mod b)),
  1771   fst (euclid_ext' b (a mod b)) - snd (euclid_ext' b (a mod b)) * (a div b))"
  1772   by (cases "euclid_ext b (a mod b)") 
  1773     (simp add: euclid_ext'_def euclid_ext_non_0)
  1774   
  1775 end
  1776 
  1777 instantiation nat :: euclidean_semiring
  1778 begin
  1779 
  1780 definition [simp]:
  1781   "euclidean_size_nat = (id :: nat \<Rightarrow> nat)"
  1782 
  1783 definition [simp]:
  1784   "normalization_factor_nat (n::nat) = (if n = 0 then 0 else 1 :: nat)"
  1785 
  1786 instance proof
  1787 qed simp_all
  1788 
  1789 end
  1790 
  1791 instantiation int :: euclidean_ring
  1792 begin
  1793 
  1794 definition [simp]:
  1795   "euclidean_size_int = (nat \<circ> abs :: int \<Rightarrow> nat)"
  1796 
  1797 definition [simp]:
  1798   "normalization_factor_int = (sgn :: int \<Rightarrow> int)"
  1799 
  1800 instance proof
  1801   case goal2 then show ?case by (auto simp add: abs_mult nat_mult_distrib)
  1802 next
  1803   case goal3 then show ?case by (simp add: zsgn_def)
  1804 next
  1805   case goal5 then show ?case by (auto simp: zsgn_def)
  1806 next
  1807   case goal6 then show ?case by (auto split: abs_split simp: zsgn_def)
  1808 qed (auto simp: sgn_times split: abs_split)
  1809 
  1810 end
  1811 
  1812 end