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
 haftmann@58023  1 (* Author: Manuel Eberl *)  haftmann@58023  2 wenzelm@58889  3 section {* Abstract euclidean algorithm *}  haftmann@58023  4 haftmann@58023  5 theory Euclidean_Algorithm  haftmann@58023  6 imports Complex_Main  haftmann@58023  7 begin  haftmann@58023  8 haftmann@60436  9 context semidom_divide  haftmann@60436  10 begin  haftmann@60436  11 haftmann@60437  12 lemma dvd_div_mult_self [simp]:  haftmann@60437  13  "a dvd b \ b div a * a = b"  haftmann@60437  14  by (cases "a = 0") (auto elim: dvdE simp add: ac_simps)  haftmann@60437  15 haftmann@60437  16 lemma dvd_mult_div_cancel [simp]:  haftmann@60437  17  "a dvd b \ a * (b div a) = b"  haftmann@60437  18  using dvd_div_mult_self [of a b] by (simp add: ac_simps)  haftmann@60437  19   haftmann@60437  20 lemma div_mult_swap:  haftmann@60437  21  assumes "c dvd b"  haftmann@60437  22  shows "a * (b div c) = (a * b) div c"  haftmann@60437  23 proof (cases "c = 0")  haftmann@60437  24  case True then show ?thesis by simp  haftmann@60437  25 next  haftmann@60437  26  case False from assms obtain d where "b = c * d" ..  haftmann@60437  27  moreover from False have "a * divide (d * c) c = ((a * d) * c) div c"  haftmann@60437  28  by simp  haftmann@60437  29  ultimately show ?thesis by (simp add: ac_simps)  haftmann@60437  30 qed  haftmann@60437  31 haftmann@60437  32 lemma dvd_div_mult:  haftmann@60437  33  assumes "c dvd b"  haftmann@60437  34  shows "b div c * a = (b * a) div c"  haftmann@60437  35  using assms div_mult_swap [of c b a] by (simp add: ac_simps)  haftmann@60437  36 haftmann@60437  37   haftmann@60433  38 text \Units: invertible elements in a ring\  haftmann@60433  39 haftmann@59061  40 abbreviation is_unit :: "'a \ bool"  haftmann@58023  41 where  haftmann@60430  42  "is_unit a \ a dvd 1"  haftmann@58023  43 haftmann@60433  44 lemma not_is_unit_0 [simp]:  haftmann@60433  45  "\ is_unit 0"  haftmann@60433  46  by simp  haftmann@60433  47 haftmann@60433  48 lemma unit_imp_dvd [dest]:  haftmann@60433  49  "is_unit b \ b dvd a"  haftmann@60433  50  by (rule dvd_trans [of _ 1]) simp_all  haftmann@60433  51 haftmann@60433  52 lemma unit_dvdE:  haftmann@60433  53  assumes "is_unit a"  haftmann@60433  54  obtains c where "a \ 0" and "b = a * c"  haftmann@60433  55 proof -  haftmann@60433  56  from assms have "a dvd b" by auto  haftmann@60433  57  then obtain c where "b = a * c" ..  haftmann@60433  58  moreover from assms have "a \ 0" by auto  haftmann@60433  59  ultimately show thesis using that by blast  haftmann@60433  60 qed  haftmann@60433  61 haftmann@60433  62 lemma dvd_unit_imp_unit:  haftmann@60433  63  "a dvd b \ is_unit b \ is_unit a"  haftmann@60433  64  by (rule dvd_trans)  haftmann@60433  65 haftmann@60433  66 lemma unit_div_1_unit [simp, intro]:  haftmann@60433  67  assumes "is_unit a"  haftmann@60433  68  shows "is_unit (1 div a)"  haftmann@60433  69 proof -  haftmann@60433  70  from assms have "1 = 1 div a * a" by simp  haftmann@60433  71  then show "is_unit (1 div a)" by (rule dvdI)  haftmann@60433  72 qed  haftmann@60433  73 haftmann@60433  74 lemma is_unitE [elim?]:  haftmann@60433  75  assumes "is_unit a"  haftmann@60433  76  obtains b where "a \ 0" and "b \ 0"  haftmann@60433  77  and "is_unit b" and "1 div a = b" and "1 div b = a"  haftmann@60433  78  and "a * b = 1" and "c div a = c * b"  haftmann@60433  79 proof (rule that)  haftmann@60433  80  def b \ "1 div a"  haftmann@60433  81  then show "1 div a = b" by simp  haftmann@60433  82  from b_def is_unit a show "is_unit b" by simp  haftmann@60433  83  from is_unit a and is_unit b show "a \ 0" and "b \ 0" by auto  haftmann@60433  84  from b_def is_unit a show "a * b = 1" by simp  haftmann@60433  85  then have "1 = a * b" ..  haftmann@60433  86  with b_def b \ 0 show "1 div b = a" by simp  haftmann@60433  87  from is_unit a have "a dvd c" ..  haftmann@60433  88  then obtain d where "c = a * d" ..  haftmann@60433  89  with a \ 0 a * b = 1 show "c div a = c * b"  haftmann@60433  90  by (simp add: mult.assoc mult.left_commute [of a])  haftmann@60433  91 qed  haftmann@58023  92 haftmann@58023  93 lemma unit_prod [intro]:  haftmann@60430  94  "is_unit a \ is_unit b \ is_unit (a * b)"  haftmann@60433  95  by (subst mult_1_left [of 1, symmetric]) (rule mult_dvd_mono)  haftmann@60433  96   haftmann@60433  97 lemma unit_div [intro]:  haftmann@60433  98  "is_unit a \ is_unit b \ is_unit (a div b)"  haftmann@60433  99  by (erule is_unitE [of b a]) (simp add: ac_simps unit_prod)  haftmann@58023  100 haftmann@58023  101 lemma mult_unit_dvd_iff:  haftmann@60433  102  assumes "is_unit b"  haftmann@60433  103  shows "a * b dvd c \ a dvd c"  haftmann@58023  104 proof  haftmann@60433  105  assume "a * b dvd c"  haftmann@60433  106  with assms show "a dvd c"  haftmann@60433  107  by (simp add: dvd_mult_left)  haftmann@58023  108 next  haftmann@60433  109  assume "a dvd c"  haftmann@60433  110  then obtain k where "c = a * k" ..  haftmann@60433  111  with assms have "c = (a * b) * (1 div b * k)"  haftmann@60433  112  by (simp add: mult_ac)  haftmann@60430  113  then show "a * b dvd c" by (rule dvdI)  haftmann@58023  114 qed  haftmann@58023  115 haftmann@58023  116 lemma dvd_mult_unit_iff:  haftmann@60433  117  assumes "is_unit b"  haftmann@60433  118  shows "a dvd c * b \ a dvd c"  haftmann@58023  119 proof  haftmann@60433  120  assume "a dvd c * b"  haftmann@60433  121  with assms have "c * b dvd c * (b * (1 div b))"  haftmann@60433  122  by (subst mult_assoc [symmetric]) simp  haftmann@60433  123  also from is_unit b have "b * (1 div b) = 1" by (rule is_unitE) simp  haftmann@60430  124  finally have "c * b dvd c" by simp  haftmann@60430  125  with a dvd c * b show "a dvd c" by (rule dvd_trans)  haftmann@58023  126 next  haftmann@60430  127  assume "a dvd c"  haftmann@60430  128  then show "a dvd c * b" by simp  haftmann@58023  129 qed  haftmann@58023  130 haftmann@60433  131 lemma div_unit_dvd_iff:  haftmann@60433  132  "is_unit b \ a div b dvd c \ a dvd c"  haftmann@60433  133  by (erule is_unitE [of _ a]) (auto simp add: mult_unit_dvd_iff)  haftmann@60433  134 haftmann@58023  135 lemma dvd_div_unit_iff:  haftmann@60430  136  "is_unit b \ a dvd c div b \ a dvd c"  haftmann@60433  137  by (erule is_unitE [of _ c]) (simp add: dvd_mult_unit_iff)  haftmann@58023  138 haftmann@60433  139 lemmas unit_dvd_iff = mult_unit_dvd_iff div_unit_dvd_iff  haftmann@60433  140  dvd_mult_unit_iff dvd_div_unit_iff -- \FIXME consider fact collection\  haftmann@58023  141 haftmann@60433  142 lemma unit_mult_div_div [simp]:  haftmann@60433  143  "is_unit a \ b * (1 div a) = b div a"  haftmann@60433  144  by (erule is_unitE [of _ b]) simp  haftmann@60433  145 haftmann@60433  146 lemma unit_div_mult_self [simp]:  haftmann@60433  147  "is_unit a \ b div a * a = b"  haftmann@60433  148  by (rule dvd_div_mult_self) auto  haftmann@60433  149 haftmann@60433  150 lemma unit_div_1_div_1 [simp]:  haftmann@60433  151  "is_unit a \ 1 div (1 div a) = a"  haftmann@60433  152  by (erule is_unitE) simp  haftmann@58023  153 haftmann@58023  154 lemma unit_div_mult_swap:  haftmann@60433  155  "is_unit c \ a * (b div c) = (a * b) div c"  haftmann@60433  156  by (erule unit_dvdE [of _ b]) (simp add: mult.left_commute [of _ c])  haftmann@58023  157 haftmann@58023  158 lemma unit_div_commute:  haftmann@60433  159  "is_unit b \ (a div b) * c = (a * c) div b"  haftmann@60433  160  using unit_div_mult_swap [of b c a] by (simp add: ac_simps)  haftmann@58023  161 haftmann@60433  162 lemma unit_eq_div1:  haftmann@60433  163  "is_unit b \ a div b = c \ a = c * b"  haftmann@60433  164  by (auto elim: is_unitE)  haftmann@58023  165 haftmann@60433  166 lemma unit_eq_div2:  haftmann@60433  167  "is_unit b \ a = c div b \ a * b = c"  haftmann@60433  168  using unit_eq_div1 [of b c a] by auto  haftmann@60433  169 haftmann@60433  170 lemma unit_mult_left_cancel:  haftmann@60433  171  assumes "is_unit a"  haftmann@60433  172  shows "a * b = a * c \ b = c" (is "?P \ ?Q")  haftmann@60436  173  using assms mult_cancel_left [of a b c] by auto  haftmann@58023  174 haftmann@60433  175 lemma unit_mult_right_cancel:  haftmann@60433  176  "is_unit a \ b * a = c * a \ b = c"  haftmann@60433  177  using unit_mult_left_cancel [of a b c] by (auto simp add: ac_simps)  haftmann@60433  178 haftmann@60433  179 lemma unit_div_cancel:  haftmann@60433  180  assumes "is_unit a"  haftmann@60433  181  shows "b div a = c div a \ b = c"  haftmann@60433  182 proof -  haftmann@60433  183  from assms have "is_unit (1 div a)" by simp  haftmann@60433  184  then have "b * (1 div a) = c * (1 div a) \ b = c"  haftmann@60433  185  by (rule unit_mult_right_cancel)  haftmann@60433  186  with assms show ?thesis by simp  haftmann@60433  187 qed  haftmann@60433  188   haftmann@60433  189 haftmann@60433  190 text \Associated elements in a ring â€“ an equivalence relation induced by the quasi-order divisibility \  haftmann@60433  191 haftmann@60433  192 definition associated :: "'a \ 'a \ bool"  haftmann@60433  193 where  haftmann@60433  194  "associated a b \ a dvd b \ b dvd a"  haftmann@60433  195 haftmann@60433  196 lemma associatedI:  haftmann@60433  197  "a dvd b \ b dvd a \ associated a b"  haftmann@60433  198  by (simp add: associated_def)  haftmann@60433  199 haftmann@60433  200 lemma associatedD1:  haftmann@60433  201  "associated a b \ a dvd b"  haftmann@58023  202  by (simp add: associated_def)  haftmann@58023  203 haftmann@60433  204 lemma associatedD2:  haftmann@60433  205  "associated a b \ b dvd a"  haftmann@60433  206  by (simp add: associated_def)  haftmann@60433  207 haftmann@60433  208 lemma associated_refl [simp]:  haftmann@60433  209  "associated a a"  haftmann@60433  210  by (auto intro: associatedI)  haftmann@60433  211 haftmann@60433  212 lemma associated_sym:  haftmann@60433  213  "associated b a \ associated a b"  haftmann@60433  214  by (auto intro: associatedI dest: associatedD1 associatedD2)  haftmann@60433  215 haftmann@60433  216 lemma associated_trans:  haftmann@60433  217  "associated a b \ associated b c \ associated a c"  haftmann@60433  218  by (auto intro: associatedI dvd_trans dest: associatedD1 associatedD2)  haftmann@60433  219 haftmann@60433  220 lemma equivp_associated:  haftmann@60433  221  "equivp associated"  haftmann@60433  222 proof (rule equivpI)  haftmann@60433  223  show "reflp associated"  haftmann@60433  224  by (rule reflpI) simp  haftmann@60433  225  show "symp associated"  haftmann@60433  226  by (rule sympI) (simp add: associated_sym)  haftmann@60433  227  show "transp associated"  haftmann@60433  228  by (rule transpI) (fact associated_trans)  haftmann@60433  229 qed  haftmann@60433  230 haftmann@58023  231 lemma associated_0 [simp]:  haftmann@58023  232  "associated 0 b \ b = 0"  haftmann@58023  233  "associated a 0 \ a = 0"  haftmann@60433  234  by (auto dest: associatedD1 associatedD2)  haftmann@58023  235 haftmann@58023  236 lemma associated_unit:  haftmann@60433  237  "associated a b \ is_unit a \ is_unit b"  haftmann@60433  238  using dvd_unit_imp_unit by (auto dest!: associatedD1 associatedD2)  haftmann@58023  239 haftmann@60436  240 lemma is_unit_associatedI:  haftmann@60436  241  assumes "is_unit c" and "a = c * b"  haftmann@60436  242  shows "associated a b"  haftmann@60436  243 proof (rule associatedI)  haftmann@60436  244  from a = c * b show "b dvd a" by auto  haftmann@60436  245  from is_unit c obtain d where "c * d = 1" by (rule is_unitE)  haftmann@60436  246  moreover from a = c * b have "d * a = d * (c * b)" by simp  haftmann@60436  247  ultimately have "b = a * d" by (simp add: ac_simps)  haftmann@60436  248  then show "a dvd b" ..  haftmann@58023  249 qed  haftmann@58023  250 haftmann@60436  251 lemma associated_is_unitE:  haftmann@60436  252  assumes "associated a b"  haftmann@60436  253  obtains c where "is_unit c" and "a = c * b"  haftmann@60436  254 proof (cases "b = 0")  haftmann@60436  255  case True with assms have "is_unit 1" and "a = 1 * b" by simp_all  haftmann@60436  256  with that show thesis .  haftmann@60436  257 next  haftmann@60436  258  case False  haftmann@60436  259  from assms have "a dvd b" and "b dvd a" by (auto dest: associatedD1 associatedD2)  haftmann@60436  260  then obtain c d where "b = a * d" and "a = b * c" by (blast elim: dvdE)  haftmann@60436  261  then have "a = c * b" and "(c * d) * b = 1 * b" by (simp_all add: ac_simps)  haftmann@60436  262  with False have "c * d = 1" using mult_cancel_right [of "c * d" b 1] by simp  haftmann@60436  263  then have "is_unit c" by auto  haftmann@60436  264  with a = c * b that show thesis by blast  haftmann@60436  265 qed  haftmann@60436  266   haftmann@58023  267 lemmas unit_simps = mult_unit_dvd_iff div_unit_dvd_iff dvd_mult_unit_iff  haftmann@58023  268  dvd_div_unit_iff unit_div_mult_swap unit_div_commute  haftmann@58023  269  unit_mult_left_cancel unit_mult_right_cancel unit_div_cancel  haftmann@58023  270  unit_eq_div1 unit_eq_div2  haftmann@58023  271 haftmann@58023  272 end  haftmann@58023  273 haftmann@58023  274 lemma is_unit_int:  haftmann@60433  275  "is_unit (k::int) \ k = 1 \ k = - 1"  haftmann@59061  276  by auto  haftmann@58023  277 haftmann@60433  278   haftmann@58023  279 text {*  haftmann@58023  280  A Euclidean semiring is a semiring upon which the Euclidean algorithm can be  haftmann@58023  281  implemented. It must provide:  haftmann@58023  282  \begin{itemize}  haftmann@58023  283  \item division with remainder  haftmann@58023  284  \item a size function such that @{term "size (a mod b) < size b"}  haftmann@58023  285  for any @{term "b \ 0"}  haftmann@60438  286  \item a normalization factor such that two associated numbers are equal iff  haftmann@60438  287  they are the same when divd by their normalization factors.  haftmann@58023  288  \end{itemize}  haftmann@58023  289  The existence of these functions makes it possible to derive gcd and lcm functions  haftmann@58023  290  for any Euclidean semiring.  haftmann@58023  291 *}  haftmann@58023  292 class euclidean_semiring = semiring_div +  haftmann@58023  293  fixes euclidean_size :: "'a \ nat"  haftmann@60438  294  fixes normalization_factor :: "'a \ 'a"  haftmann@58023  295  assumes mod_size_less [simp]:  haftmann@58023  296  "b \ 0 \ euclidean_size (a mod b) < euclidean_size b"  haftmann@58023  297  assumes size_mult_mono:  haftmann@58023  298  "b \ 0 \ euclidean_size (a * b) \ euclidean_size a"  haftmann@60438  299  assumes normalization_factor_is_unit [intro,simp]:  haftmann@60438  300  "a \ 0 \ is_unit (normalization_factor a)"  haftmann@60438  301  assumes normalization_factor_mult: "normalization_factor (a * b) =  haftmann@60438  302  normalization_factor a * normalization_factor b"  haftmann@60438  303  assumes normalization_factor_unit: "is_unit a \ normalization_factor a = a"  haftmann@60438  304  assumes normalization_factor_0 [simp]: "normalization_factor 0 = 0"  haftmann@58023  305 begin  haftmann@58023  306 haftmann@60438  307 lemma normalization_factor_dvd [simp]:  haftmann@60438  308  "a \ 0 \ normalization_factor a dvd b"  haftmann@58023  309  by (rule unit_imp_dvd, simp)  haftmann@58023  310   haftmann@60438  311 lemma normalization_factor_1 [simp]:  haftmann@60438  312  "normalization_factor 1 = 1"  haftmann@60438  313  by (simp add: normalization_factor_unit)  haftmann@58023  314 haftmann@60438  315 lemma normalization_factor_0_iff [simp]:  haftmann@60438  316  "normalization_factor a = 0 \ a = 0"  haftmann@58023  317 proof  haftmann@60438  318  assume "normalization_factor a = 0"  haftmann@60438  319  hence "\ is_unit (normalization_factor a)"  haftmann@60433  320  by simp  haftmann@60433  321  then show "a = 0" by auto  haftmann@60433  322 qed simp  haftmann@58023  323 haftmann@60438  324 lemma normalization_factor_pow:  haftmann@60438  325  "normalization_factor (a ^ n) = normalization_factor a ^ n"  haftmann@60438  326  by (induct n) (simp_all add: normalization_factor_mult power_Suc2)  haftmann@58023  327 haftmann@60438  328 lemma normalization_correct [simp]:  haftmann@60438  329  "normalization_factor (a div normalization_factor a) = (if a = 0 then 0 else 1)"  haftmann@60430  330 proof (cases "a = 0", simp)  haftmann@60430  331  assume "a \ 0"  haftmann@60438  332  let ?nf = "normalization_factor"  haftmann@60438  333  from normalization_factor_is_unit[OF a \ 0] have "?nf a \ 0"  haftmann@60433  334  by auto  haftmann@60430  335  have "?nf (a div ?nf a) * ?nf (?nf a) = ?nf (a div ?nf a * ?nf a)"  haftmann@60438  336  by (simp add: normalization_factor_mult)  haftmann@60430  337  also have "a div ?nf a * ?nf a = a" using a \ 0  haftmann@59009  338  by simp  haftmann@60430  339  also have "?nf (?nf a) = ?nf a" using a \ 0  haftmann@60438  340  normalization_factor_is_unit normalization_factor_unit by simp  haftmann@60438  341  finally have "normalization_factor (a div normalization_factor a) = 1"  haftmann@60433  342  using ?nf a \ 0 by (metis div_mult_self2_is_id div_self)  haftmann@60433  343  with a \ 0 show ?thesis by simp  haftmann@58023  344 qed  haftmann@58023  345 haftmann@60438  346 lemma normalization_0_iff [simp]:  haftmann@60438  347  "a div normalization_factor a = 0 \ a = 0"  haftmann@60430  348  by (cases "a = 0", simp, subst unit_eq_div1, blast, simp)  haftmann@58023  349 haftmann@60438  350 lemma mult_div_normalization [simp]:  haftmann@60438  351  "b * (1 div normalization_factor a) = b div normalization_factor a"  haftmann@60433  352  by (cases "a = 0") simp_all  haftmann@60433  353 haftmann@58023  354 lemma associated_iff_normed_eq:  haftmann@60438  355  "associated a b \ a div normalization_factor a = b div normalization_factor b"  haftmann@60438  356 proof (cases "b = 0", simp, cases "a = 0", metis associated_0(1) normalization_0_iff, rule iffI)  haftmann@60438  357  let ?nf = normalization_factor  haftmann@58023  358  assume "a \ 0" "b \ 0" "a div ?nf a = b div ?nf b"  haftmann@58023  359  hence "a = b * (?nf a div ?nf b)"  haftmann@58023  360  apply (subst (asm) unit_eq_div1, blast, subst (asm) unit_div_commute, blast)  haftmann@58023  361  apply (subst div_mult_swap, simp, simp)  haftmann@58023  362  done  haftmann@60430  363  with a \ 0 b \ 0 have "\c. is_unit c \ a = c * b"  haftmann@58023  364  by (intro exI[of _ "?nf a div ?nf b"], force simp: mult_ac)  haftmann@60436  365  then obtain c where "is_unit c" and "a = c * b" by blast  haftmann@60436  366  then show "associated a b" by (rule is_unit_associatedI)  haftmann@58023  367 next  haftmann@60438  368  let ?nf = normalization_factor  haftmann@58023  369  assume "a \ 0" "b \ 0" "associated a b"  haftmann@60436  370  then obtain c where "is_unit c" and "a = c * b" by (blast elim: associated_is_unitE)  haftmann@58023  371  then show "a div ?nf a = b div ?nf b"  haftmann@60438  372  apply (simp only: a = c * b normalization_factor_mult normalization_factor_unit)  haftmann@58023  373  apply (rule div_mult_mult1, force)  haftmann@58023  374  done  haftmann@58023  375  qed  haftmann@58023  376 haftmann@58023  377 lemma normed_associated_imp_eq:  haftmann@60438  378  "associated a b \ normalization_factor a \ {0, 1} \ normalization_factor b \ {0, 1} \ a = b"  haftmann@58023  379  by (simp add: associated_iff_normed_eq, elim disjE, simp_all)  haftmann@58023  380   haftmann@60438  381 lemmas normalization_factor_dvd_iff [simp] =  haftmann@60438  382  unit_dvd_iff [OF normalization_factor_is_unit]  haftmann@58023  383 haftmann@58023  384 lemma euclidean_division:  haftmann@58023  385  fixes a :: 'a and b :: 'a  haftmann@58023  386  assumes "b \ 0"  haftmann@58023  387  obtains s and t where "a = s * b + t"  haftmann@58023  388  and "euclidean_size t < euclidean_size b"  haftmann@58023  389 proof -  haftmann@58023  390  from div_mod_equality[of a b 0]  haftmann@58023  391  have "a = a div b * b + a mod b" by simp  haftmann@58023  392  with that and assms show ?thesis by force  haftmann@58023  393 qed  haftmann@58023  394 haftmann@58023  395 lemma dvd_euclidean_size_eq_imp_dvd:  haftmann@58023  396  assumes "a \ 0" and b_dvd_a: "b dvd a" and size_eq: "euclidean_size a = euclidean_size b"  haftmann@58023  397  shows "a dvd b"  haftmann@58023  398 proof (subst dvd_eq_mod_eq_0, rule ccontr)  haftmann@58023  399  assume "b mod a \ 0"  haftmann@58023  400  from b_dvd_a have b_dvd_mod: "b dvd b mod a" by (simp add: dvd_mod_iff)  haftmann@58023  401  from b_dvd_mod obtain c where "b mod a = b * c" unfolding dvd_def by blast  haftmann@58023  402  with b mod a \ 0 have "c \ 0" by auto  haftmann@58023  403  with b mod a = b * c have "euclidean_size (b mod a) \ euclidean_size b"  haftmann@58023  404  using size_mult_mono by force  haftmann@58023  405  moreover from a \ 0 have "euclidean_size (b mod a) < euclidean_size a"  haftmann@58023  406  using mod_size_less by blast  haftmann@58023  407  ultimately show False using size_eq by simp  haftmann@58023  408 qed  haftmann@58023  409 haftmann@58023  410 function gcd_eucl :: "'a \ 'a \ 'a"  haftmann@58023  411 where  haftmann@60438  412  "gcd_eucl a b = (if b = 0 then a div normalization_factor a else gcd_eucl b (a mod b))"  haftmann@58023  413  by (pat_completeness, simp)  haftmann@58023  414 termination by (relation "measure (euclidean_size \ snd)", simp_all)  haftmann@58023  415 haftmann@58023  416 declare gcd_eucl.simps [simp del]  haftmann@58023  417 haftmann@58023  418 lemma gcd_induct: "\\b. P b 0; \a b. 0 \ b \ P b (a mod b) \ P a b\ \ P a b"  haftmann@58023  419 proof (induct a b rule: gcd_eucl.induct)  haftmann@58023  420  case ("1" m n)  haftmann@58023  421  then show ?case by (cases "n = 0") auto  haftmann@58023  422 qed  haftmann@58023  423 haftmann@58023  424 definition lcm_eucl :: "'a \ 'a \ 'a"  haftmann@58023  425 where  haftmann@60438  426  "lcm_eucl a b = a * b div (gcd_eucl a b * normalization_factor (a * b))"  haftmann@58023  427 haftmann@58023  428  (* Somewhat complicated definition of Lcm that has the advantage of working  haftmann@58023  429  for infinite sets as well *)  haftmann@58023  430 haftmann@58023  431 definition Lcm_eucl :: "'a set \ 'a"  haftmann@58023  432 where  haftmann@60430  433  "Lcm_eucl A = (if \l. l \ 0 \ (\a\A. a dvd l) then  haftmann@60430  434  let l = SOME l. l \ 0 \ (\a\A. a dvd l) \ euclidean_size l =  haftmann@60430  435  (LEAST n. \l. l \ 0 \ (\a\A. a dvd l) \ euclidean_size l = n)  haftmann@60438  436  in l div normalization_factor l  haftmann@58023  437  else 0)"  haftmann@58023  438 haftmann@58023  439 definition Gcd_eucl :: "'a set \ 'a"  haftmann@58023  440 where  haftmann@58023  441  "Gcd_eucl A = Lcm_eucl {d. \a\A. d dvd a}"  haftmann@58023  442 haftmann@58023  443 end  haftmann@58023  444 haftmann@58023  445 class euclidean_semiring_gcd = euclidean_semiring + gcd + Gcd +  haftmann@58023  446  assumes gcd_gcd_eucl: "gcd = gcd_eucl" and lcm_lcm_eucl: "lcm = lcm_eucl"  haftmann@58023  447  assumes Gcd_Gcd_eucl: "Gcd = Gcd_eucl" and Lcm_Lcm_eucl: "Lcm = Lcm_eucl"  haftmann@58023  448 begin  haftmann@58023  449 haftmann@58023  450 lemma gcd_red:  haftmann@60430  451  "gcd a b = gcd b (a mod b)"  haftmann@58023  452  by (metis gcd_eucl.simps mod_0 mod_by_0 gcd_gcd_eucl)  haftmann@58023  453 haftmann@58023  454 lemma gcd_non_0:  haftmann@60430  455  "b \ 0 \ gcd a b = gcd b (a mod b)"  haftmann@58023  456  by (rule gcd_red)  haftmann@58023  457 haftmann@58023  458 lemma gcd_0_left:  haftmann@60438  459  "gcd 0 a = a div normalization_factor a"  haftmann@58023  460  by (simp only: gcd_gcd_eucl, subst gcd_eucl.simps, subst gcd_eucl.simps, simp add: Let_def)  haftmann@58023  461 haftmann@58023  462 lemma gcd_0:  haftmann@60438  463  "gcd a 0 = a div normalization_factor a"  haftmann@58023  464  by (simp only: gcd_gcd_eucl, subst gcd_eucl.simps, simp add: Let_def)  haftmann@58023  465 haftmann@60430  466 lemma gcd_dvd1 [iff]: "gcd a b dvd a"  haftmann@60430  467  and gcd_dvd2 [iff]: "gcd a b dvd b"  haftmann@60430  468 proof (induct a b rule: gcd_eucl.induct)  haftmann@60430  469  fix a b :: 'a  haftmann@60430  470  assume IH1: "b \ 0 \ gcd b (a mod b) dvd b"  haftmann@60430  471  assume IH2: "b \ 0 \ gcd b (a mod b) dvd (a mod b)"  haftmann@58023  472   haftmann@60430  473  have "gcd a b dvd a \ gcd a b dvd b"  haftmann@60430  474  proof (cases "b = 0")  haftmann@58023  475  case True  haftmann@60430  476  then show ?thesis by (cases "a = 0", simp_all add: gcd_0)  haftmann@58023  477  next  haftmann@58023  478  case False  haftmann@58023  479  with IH1 and IH2 show ?thesis by (simp add: gcd_non_0 dvd_mod_iff)  haftmann@58023  480  qed  haftmann@60430  481  then show "gcd a b dvd a" "gcd a b dvd b" by simp_all  haftmann@58023  482 qed  haftmann@58023  483 haftmann@58023  484 lemma dvd_gcd_D1: "k dvd gcd m n \ k dvd m"  haftmann@58023  485  by (rule dvd_trans, assumption, rule gcd_dvd1)  haftmann@58023  486 haftmann@58023  487 lemma dvd_gcd_D2: "k dvd gcd m n \ k dvd n"  haftmann@58023  488  by (rule dvd_trans, assumption, rule gcd_dvd2)  haftmann@58023  489 haftmann@58023  490 lemma gcd_greatest:  haftmann@60430  491  fixes k a b :: 'a  haftmann@60430  492  shows "k dvd a \ k dvd b \ k dvd gcd a b"  haftmann@60430  493 proof (induct a b rule: gcd_eucl.induct)  haftmann@60430  494  case (1 a b)  haftmann@58023  495  show ?case  haftmann@60430  496  proof (cases "b = 0")  haftmann@60430  497  assume "b = 0"  haftmann@60430  498  with 1 show ?thesis by (cases "a = 0", simp_all add: gcd_0)  haftmann@58023  499  next  haftmann@60430  500  assume "b \ 0"  haftmann@58023  501  with 1 show ?thesis by (simp add: gcd_non_0 dvd_mod_iff)  haftmann@58023  502  qed  haftmann@58023  503 qed  haftmann@58023  504 haftmann@58023  505 lemma dvd_gcd_iff:  haftmann@60430  506  "k dvd gcd a b \ k dvd a \ k dvd b"  haftmann@58023  507  by (blast intro!: gcd_greatest intro: dvd_trans)  haftmann@58023  508 haftmann@58023  509 lemmas gcd_greatest_iff = dvd_gcd_iff  haftmann@58023  510 haftmann@58023  511 lemma gcd_zero [simp]:  haftmann@60430  512  "gcd a b = 0 \ a = 0 \ b = 0"  haftmann@58023  513  by (metis dvd_0_left dvd_refl gcd_dvd1 gcd_dvd2 gcd_greatest)+  haftmann@58023  514 haftmann@60438  515 lemma normalization_factor_gcd [simp]:  haftmann@60438  516  "normalization_factor (gcd a b) = (if a = 0 \ b = 0 then 0 else 1)" (is "?f a b = ?g a b")  haftmann@60430  517 proof (induct a b rule: gcd_eucl.induct)  haftmann@60430  518  fix a b :: 'a  haftmann@60430  519  assume IH: "b \ 0 \ ?f b (a mod b) = ?g b (a mod b)"  haftmann@60430  520  then show "?f a b = ?g a b" by (cases "b = 0", auto simp: gcd_non_0 gcd_0)  haftmann@58023  521 qed  haftmann@58023  522 haftmann@58023  523 lemma gcdI:  haftmann@60430  524  "k dvd a \ k dvd b \ (\l. l dvd a \ l dvd b \ l dvd k)  haftmann@60438  525  \ normalization_factor k = (if k = 0 then 0 else 1) \ k = gcd a b"  haftmann@58023  526  by (intro normed_associated_imp_eq) (auto simp: associated_def intro: gcd_greatest)  haftmann@58023  527 haftmann@58023  528 sublocale gcd!: abel_semigroup gcd  haftmann@58023  529 proof  haftmann@60430  530  fix a b c  haftmann@60430  531  show "gcd (gcd a b) c = gcd a (gcd b c)"  haftmann@58023  532  proof (rule gcdI)  haftmann@60430  533  have "gcd (gcd a b) c dvd gcd a b" "gcd a b dvd a" by simp_all  haftmann@60430  534  then show "gcd (gcd a b) c dvd a" by (rule dvd_trans)  haftmann@60430  535  have "gcd (gcd a b) c dvd gcd a b" "gcd a b dvd b" by simp_all  haftmann@60430  536  hence "gcd (gcd a b) c dvd b" by (rule dvd_trans)  haftmann@60430  537  moreover have "gcd (gcd a b) c dvd c" by simp  haftmann@60430  538  ultimately show "gcd (gcd a b) c dvd gcd b c"  haftmann@58023  539  by (rule gcd_greatest)  haftmann@60438  540  show "normalization_factor (gcd (gcd a b) c) = (if gcd (gcd a b) c = 0 then 0 else 1)"  haftmann@58023  541  by auto  haftmann@60430  542  fix l assume "l dvd a" and "l dvd gcd b c"  haftmann@58023  543  with dvd_trans[OF _ gcd_dvd1] and dvd_trans[OF _ gcd_dvd2]  haftmann@60430  544  have "l dvd b" and "l dvd c" by blast+  haftmann@60430  545  with l dvd a show "l dvd gcd (gcd a b) c"  haftmann@58023  546  by (intro gcd_greatest)  haftmann@58023  547  qed  haftmann@58023  548 next  haftmann@60430  549  fix a b  haftmann@60430  550  show "gcd a b = gcd b a"  haftmann@58023  551  by (rule gcdI) (simp_all add: gcd_greatest)  haftmann@58023  552 qed  haftmann@58023  553 haftmann@58023  554 lemma gcd_unique: "d dvd a \ d dvd b \  haftmann@60438  555  normalization_factor d = (if d = 0 then 0 else 1) \  haftmann@58023  556  (\e. e dvd a \ e dvd b \ e dvd d) \ d = gcd a b"  haftmann@58023  557  by (rule, auto intro: gcdI simp: gcd_greatest)  haftmann@58023  558 haftmann@58023  559 lemma gcd_dvd_prod: "gcd a b dvd k * b"  haftmann@58023  560  using mult_dvd_mono [of 1] by auto  haftmann@58023  561 haftmann@60430  562 lemma gcd_1_left [simp]: "gcd 1 a = 1"  haftmann@58023  563  by (rule sym, rule gcdI, simp_all)  haftmann@58023  564 haftmann@60430  565 lemma gcd_1 [simp]: "gcd a 1 = 1"  haftmann@58023  566  by (rule sym, rule gcdI, simp_all)  haftmann@58023  567 haftmann@58023  568 lemma gcd_proj2_if_dvd:  haftmann@60438  569  "b dvd a \ gcd a b = b div normalization_factor b"  haftmann@60430  570  by (cases "b = 0", simp_all add: dvd_eq_mod_eq_0 gcd_non_0 gcd_0)  haftmann@58023  571 haftmann@58023  572 lemma gcd_proj1_if_dvd:  haftmann@60438  573  "a dvd b \ gcd a b = a div normalization_factor a"  haftmann@58023  574  by (subst gcd.commute, simp add: gcd_proj2_if_dvd)  haftmann@58023  575 haftmann@60438  576 lemma gcd_proj1_iff: "gcd m n = m div normalization_factor m \ m dvd n"  haftmann@58023  577 proof  haftmann@60438  578  assume A: "gcd m n = m div normalization_factor m"  haftmann@58023  579  show "m dvd n"  haftmann@58023  580  proof (cases "m = 0")  haftmann@58023  581  assume [simp]: "m \ 0"  haftmann@60438  582  from A have B: "m = gcd m n * normalization_factor m"  haftmann@58023  583  by (simp add: unit_eq_div2)  haftmann@58023  584  show ?thesis by (subst B, simp add: mult_unit_dvd_iff)  haftmann@58023  585  qed (insert A, simp)  haftmann@58023  586 next  haftmann@58023  587  assume "m dvd n"  haftmann@60438  588  then show "gcd m n = m div normalization_factor m" by (rule gcd_proj1_if_dvd)  haftmann@58023  589 qed  haftmann@58023  590   haftmann@60438  591 lemma gcd_proj2_iff: "gcd m n = n div normalization_factor n \ n dvd m"  haftmann@58023  592  by (subst gcd.commute, simp add: gcd_proj1_iff)  haftmann@58023  593 haftmann@58023  594 lemma gcd_mod1 [simp]:  haftmann@60430  595  "gcd (a mod b) b = gcd a b"  haftmann@58023  596  by (rule gcdI, metis dvd_mod_iff gcd_dvd1 gcd_dvd2, simp_all add: gcd_greatest dvd_mod_iff)  haftmann@58023  597 haftmann@58023  598 lemma gcd_mod2 [simp]:  haftmann@60430  599  "gcd a (b mod a) = gcd a b"  haftmann@58023  600  by (rule gcdI, simp, metis dvd_mod_iff gcd_dvd1 gcd_dvd2, simp_all add: gcd_greatest dvd_mod_iff)  haftmann@58023  601   haftmann@60438  602 lemma normalization_factor_dvd' [simp]:  haftmann@60438  603  "normalization_factor a dvd a"  haftmann@60430  604  by (cases "a = 0", simp_all)  haftmann@58023  605 haftmann@58023  606 lemma gcd_mult_distrib':  haftmann@60438  607  "k div normalization_factor k * gcd a b = gcd (k*a) (k*b)"  haftmann@60430  608 proof (induct a b rule: gcd_eucl.induct)  haftmann@60430  609  case (1 a b)  haftmann@58023  610  show ?case  haftmann@60430  611  proof (cases "b = 0")  haftmann@58023  612  case True  haftmann@60438  613  then show ?thesis by (simp add: normalization_factor_mult gcd_0 algebra_simps div_mult_div_if_dvd)  haftmann@58023  614  next  haftmann@58023  615  case False  haftmann@60438  616  hence "k div normalization_factor k * gcd a b = gcd (k * b) (k * (a mod b))"  haftmann@58023  617  using 1 by (subst gcd_red, simp)  haftmann@60430  618  also have "... = gcd (k * a) (k * b)"  haftmann@58023  619  by (simp add: mult_mod_right gcd.commute)  haftmann@58023  620  finally show ?thesis .  haftmann@58023  621  qed  haftmann@58023  622 qed  haftmann@58023  623 haftmann@58023  624 lemma gcd_mult_distrib:  haftmann@60438  625  "k * gcd a b = gcd (k*a) (k*b) * normalization_factor k"  haftmann@58023  626 proof-  haftmann@60438  627  let ?nf = "normalization_factor"  haftmann@58023  628  from gcd_mult_distrib'  haftmann@60430  629  have "gcd (k*a) (k*b) = k div ?nf k * gcd a b" ..  haftmann@60430  630  also have "... = k * gcd a b div ?nf k"  haftmann@60438  631  by (metis dvd_div_mult dvd_eq_mod_eq_0 mod_0 normalization_factor_dvd)  haftmann@58023  632  finally show ?thesis  haftmann@59009  633  by simp  haftmann@58023  634 qed  haftmann@58023  635 haftmann@58023  636 lemma euclidean_size_gcd_le1 [simp]:  haftmann@58023  637  assumes "a \ 0"  haftmann@58023  638  shows "euclidean_size (gcd a b) \ euclidean_size a"  haftmann@58023  639 proof -  haftmann@58023  640  have "gcd a b dvd a" by (rule gcd_dvd1)  haftmann@58023  641  then obtain c where A: "a = gcd a b * c" unfolding dvd_def by blast  haftmann@58023  642  with a \ 0 show ?thesis by (subst (2) A, intro size_mult_mono) auto  haftmann@58023  643 qed  haftmann@58023  644 haftmann@58023  645 lemma euclidean_size_gcd_le2 [simp]:  haftmann@58023  646  "b \ 0 \ euclidean_size (gcd a b) \ euclidean_size b"  haftmann@58023  647  by (subst gcd.commute, rule euclidean_size_gcd_le1)  haftmann@58023  648 haftmann@58023  649 lemma euclidean_size_gcd_less1:  haftmann@58023  650  assumes "a \ 0" and "\a dvd b"  haftmann@58023  651  shows "euclidean_size (gcd a b) < euclidean_size a"  haftmann@58023  652 proof (rule ccontr)  haftmann@58023  653  assume "\euclidean_size (gcd a b) < euclidean_size a"  haftmann@58023  654  with a \ 0 have "euclidean_size (gcd a b) = euclidean_size a"  haftmann@58023  655  by (intro le_antisym, simp_all)  haftmann@58023  656  with assms have "a dvd gcd a b" by (auto intro: dvd_euclidean_size_eq_imp_dvd)  haftmann@58023  657  hence "a dvd b" using dvd_gcd_D2 by blast  haftmann@58023  658  with \a dvd b show False by contradiction  haftmann@58023  659 qed  haftmann@58023  660 haftmann@58023  661 lemma euclidean_size_gcd_less2:  haftmann@58023  662  assumes "b \ 0" and "\b dvd a"  haftmann@58023  663  shows "euclidean_size (gcd a b) < euclidean_size b"  haftmann@58023  664  using assms by (subst gcd.commute, rule euclidean_size_gcd_less1)  haftmann@58023  665 haftmann@60430  666 lemma gcd_mult_unit1: "is_unit a \ gcd (b * a) c = gcd b c"  haftmann@58023  667  apply (rule gcdI)  haftmann@58023  668  apply (rule dvd_trans, rule gcd_dvd1, simp add: unit_simps)  haftmann@58023  669  apply (rule gcd_dvd2)  haftmann@58023  670  apply (rule gcd_greatest, simp add: unit_simps, assumption)  haftmann@60438  671  apply (subst normalization_factor_gcd, simp add: gcd_0)  haftmann@58023  672  done  haftmann@58023  673 haftmann@60430  674 lemma gcd_mult_unit2: "is_unit a \ gcd b (c * a) = gcd b c"  haftmann@58023  675  by (subst gcd.commute, subst gcd_mult_unit1, assumption, rule gcd.commute)  haftmann@58023  676 haftmann@60430  677 lemma gcd_div_unit1: "is_unit a \ gcd (b div a) c = gcd b c"  haftmann@60433  678  by (erule is_unitE [of _ b]) (simp add: gcd_mult_unit1)  haftmann@58023  679 haftmann@60430  680 lemma gcd_div_unit2: "is_unit a \ gcd b (c div a) = gcd b c"  haftmann@60433  681  by (erule is_unitE [of _ c]) (simp add: gcd_mult_unit2)  haftmann@58023  682 haftmann@60438  683 lemma gcd_idem: "gcd a a = a div normalization_factor a"  haftmann@60430  684  by (cases "a = 0") (simp add: gcd_0_left, rule sym, rule gcdI, simp_all)  haftmann@58023  685 haftmann@60430  686 lemma gcd_right_idem: "gcd (gcd a b) b = gcd a b"  haftmann@58023  687  apply (rule gcdI)  haftmann@58023  688  apply (simp add: ac_simps)  haftmann@58023  689  apply (rule gcd_dvd2)  haftmann@58023  690  apply (rule gcd_greatest, erule (1) gcd_greatest, assumption)  haftmann@59009  691  apply simp  haftmann@58023  692  done  haftmann@58023  693 haftmann@60430  694 lemma gcd_left_idem: "gcd a (gcd a b) = gcd a b"  haftmann@58023  695  apply (rule gcdI)  haftmann@58023  696  apply simp  haftmann@58023  697  apply (rule dvd_trans, rule gcd_dvd2, rule gcd_dvd2)  haftmann@58023  698  apply (rule gcd_greatest, assumption, erule gcd_greatest, assumption)  haftmann@59009  699  apply simp  haftmann@58023  700  done  haftmann@58023  701 haftmann@58023  702 lemma comp_fun_idem_gcd: "comp_fun_idem gcd"  haftmann@58023  703 proof  haftmann@58023  704  fix a b show "gcd a \ gcd b = gcd b \ gcd a"  haftmann@58023  705  by (simp add: fun_eq_iff ac_simps)  haftmann@58023  706 next  haftmann@58023  707  fix a show "gcd a \ gcd a = gcd a"  haftmann@58023  708  by (simp add: fun_eq_iff gcd_left_idem)  haftmann@58023  709 qed  haftmann@58023  710 haftmann@58023  711 lemma coprime_dvd_mult:  haftmann@60430  712  assumes "gcd c b = 1" and "c dvd a * b"  haftmann@60430  713  shows "c dvd a"  haftmann@58023  714 proof -  haftmann@60438  715  let ?nf = "normalization_factor"  haftmann@60430  716  from assms gcd_mult_distrib [of a c b]  haftmann@60430  717  have A: "a = gcd (a * c) (a * b) * ?nf a" by simp  haftmann@60430  718  from c dvd a * b show ?thesis by (subst A, simp_all add: gcd_greatest)  haftmann@58023  719 qed  haftmann@58023  720 haftmann@58023  721 lemma coprime_dvd_mult_iff:  haftmann@60430  722  "gcd c b = 1 \ (c dvd a * b) = (c dvd a)"  haftmann@58023  723  by (rule, rule coprime_dvd_mult, simp_all)  haftmann@58023  724 haftmann@58023  725 lemma gcd_dvd_antisym:  haftmann@58023  726  "gcd a b dvd gcd c d \ gcd c d dvd gcd a b \ gcd a b = gcd c d"  haftmann@58023  727 proof (rule gcdI)  haftmann@58023  728  assume A: "gcd a b dvd gcd c d" and B: "gcd c d dvd gcd a b"  haftmann@58023  729  have "gcd c d dvd c" by simp  haftmann@58023  730  with A show "gcd a b dvd c" by (rule dvd_trans)  haftmann@58023  731  have "gcd c d dvd d" by simp  haftmann@58023  732  with A show "gcd a b dvd d" by (rule dvd_trans)  haftmann@60438  733  show "normalization_factor (gcd a b) = (if gcd a b = 0 then 0 else 1)"  haftmann@59009  734  by simp  haftmann@58023  735  fix l assume "l dvd c" and "l dvd d"  haftmann@58023  736  hence "l dvd gcd c d" by (rule gcd_greatest)  haftmann@58023  737  from this and B show "l dvd gcd a b" by (rule dvd_trans)  haftmann@58023  738 qed  haftmann@58023  739 haftmann@58023  740 lemma gcd_mult_cancel:  haftmann@58023  741  assumes "gcd k n = 1"  haftmann@58023  742  shows "gcd (k * m) n = gcd m n"  haftmann@58023  743 proof (rule gcd_dvd_antisym)  haftmann@58023  744  have "gcd (gcd (k * m) n) k = gcd (gcd k n) (k * m)" by (simp add: ac_simps)  haftmann@58023  745  also note gcd k n = 1  haftmann@58023  746  finally have "gcd (gcd (k * m) n) k = 1" by simp  haftmann@58023  747  hence "gcd (k * m) n dvd m" by (rule coprime_dvd_mult, simp add: ac_simps)  haftmann@58023  748  moreover have "gcd (k * m) n dvd n" by simp  haftmann@58023  749  ultimately show "gcd (k * m) n dvd gcd m n" by (rule gcd_greatest)  haftmann@58023  750  have "gcd m n dvd (k * m)" and "gcd m n dvd n" by simp_all  haftmann@58023  751  then show "gcd m n dvd gcd (k * m) n" by (rule gcd_greatest)  haftmann@58023  752 qed  haftmann@58023  753 haftmann@58023  754 lemma coprime_crossproduct:  haftmann@58023  755  assumes [simp]: "gcd a d = 1" "gcd b c = 1"  haftmann@58023  756  shows "associated (a * c) (b * d) \ associated a b \ associated c d" (is "?lhs \ ?rhs")  haftmann@58023  757 proof  haftmann@58023  758  assume ?rhs then show ?lhs unfolding associated_def by (fast intro: mult_dvd_mono)  haftmann@58023  759 next  haftmann@58023  760  assume ?lhs  haftmann@58023  761  from ?lhs have "a dvd b * d" unfolding associated_def by (metis dvd_mult_left)  haftmann@58023  762  hence "a dvd b" by (simp add: coprime_dvd_mult_iff)  haftmann@58023  763  moreover from ?lhs have "b dvd a * c" unfolding associated_def by (metis dvd_mult_left)  haftmann@58023  764  hence "b dvd a" by (simp add: coprime_dvd_mult_iff)  haftmann@58023  765  moreover from ?lhs have "c dvd d * b"  haftmann@59009  766  unfolding associated_def by (auto dest: dvd_mult_right simp add: ac_simps)  haftmann@58023  767  hence "c dvd d" by (simp add: coprime_dvd_mult_iff gcd.commute)  haftmann@58023  768  moreover from ?lhs have "d dvd c * a"  haftmann@59009  769  unfolding associated_def by (auto dest: dvd_mult_right simp add: ac_simps)  haftmann@58023  770  hence "d dvd c" by (simp add: coprime_dvd_mult_iff gcd.commute)  haftmann@58023  771  ultimately show ?rhs unfolding associated_def by simp  haftmann@58023  772 qed  haftmann@58023  773 haftmann@58023  774 lemma gcd_add1 [simp]:  haftmann@58023  775  "gcd (m + n) n = gcd m n"  haftmann@58023  776  by (cases "n = 0", simp_all add: gcd_non_0)  haftmann@58023  777 haftmann@58023  778 lemma gcd_add2 [simp]:  haftmann@58023  779  "gcd m (m + n) = gcd m n"  haftmann@58023  780  using gcd_add1 [of n m] by (simp add: ac_simps)  haftmann@58023  781 haftmann@58023  782 lemma gcd_add_mult: "gcd m (k * m + n) = gcd m n"  haftmann@58023  783  by (subst gcd.commute, subst gcd_red, simp)  haftmann@58023  784 haftmann@60430  785 lemma coprimeI: "(\l. \l dvd a; l dvd b\ \ l dvd 1) \ gcd a b = 1"  haftmann@58023  786  by (rule sym, rule gcdI, simp_all)  haftmann@58023  787 haftmann@58023  788 lemma coprime: "gcd a b = 1 \ (\d. d dvd a \ d dvd b \ is_unit d)"  haftmann@59061  789  by (auto intro: coprimeI gcd_greatest dvd_gcd_D1 dvd_gcd_D2)  haftmann@58023  790 haftmann@58023  791 lemma div_gcd_coprime:  haftmann@58023  792  assumes nz: "a \ 0 \ b \ 0"  haftmann@58023  793  defines [simp]: "d \ gcd a b"  haftmann@58023  794  defines [simp]: "a' \ a div d" and [simp]: "b' \ b div d"  haftmann@58023  795  shows "gcd a' b' = 1"  haftmann@58023  796 proof (rule coprimeI)  haftmann@58023  797  fix l assume "l dvd a'" "l dvd b'"  haftmann@58023  798  then obtain s t where "a' = l * s" "b' = l * t" unfolding dvd_def by blast  haftmann@59009  799  moreover have "a = a' * d" "b = b' * d" by simp_all  haftmann@58023  800  ultimately have "a = (l * d) * s" "b = (l * d) * t"  haftmann@59009  801  by (simp_all only: ac_simps)  haftmann@58023  802  hence "l*d dvd a" and "l*d dvd b" by (simp_all only: dvd_triv_left)  haftmann@58023  803  hence "l*d dvd d" by (simp add: gcd_greatest)  haftmann@59009  804  then obtain u where "d = l * d * u" ..  haftmann@59009  805  then have "d * (l * u) = d" by (simp add: ac_simps)  haftmann@59009  806  moreover from nz have "d \ 0" by simp  haftmann@59009  807  with div_mult_self1_is_id have "d * (l * u) div d = l * u" .  haftmann@59009  808  ultimately have "1 = l * u"  haftmann@59009  809  using d \ 0 by simp  haftmann@59009  810  then show "l dvd 1" ..  haftmann@58023  811 qed  haftmann@58023  812 haftmann@58023  813 lemma coprime_mult:  haftmann@58023  814  assumes da: "gcd d a = 1" and db: "gcd d b = 1"  haftmann@58023  815  shows "gcd d (a * b) = 1"  haftmann@58023  816  apply (subst gcd.commute)  haftmann@58023  817  using da apply (subst gcd_mult_cancel)  haftmann@58023  818  apply (subst gcd.commute, assumption)  haftmann@58023  819  apply (subst gcd.commute, rule db)  haftmann@58023  820  done  haftmann@58023  821 haftmann@58023  822 lemma coprime_lmult:  haftmann@58023  823  assumes dab: "gcd d (a * b) = 1"  haftmann@58023  824  shows "gcd d a = 1"  haftmann@58023  825 proof (rule coprimeI)  haftmann@58023  826  fix l assume "l dvd d" and "l dvd a"  haftmann@58023  827  hence "l dvd a * b" by simp  haftmann@58023  828  with l dvd d and dab show "l dvd 1" by (auto intro: gcd_greatest)  haftmann@58023  829 qed  haftmann@58023  830 haftmann@58023  831 lemma coprime_rmult:  haftmann@58023  832  assumes dab: "gcd d (a * b) = 1"  haftmann@58023  833  shows "gcd d b = 1"  haftmann@58023  834 proof (rule coprimeI)  haftmann@58023  835  fix l assume "l dvd d" and "l dvd b"  haftmann@58023  836  hence "l dvd a * b" by simp  haftmann@58023  837  with l dvd d and dab show "l dvd 1" by (auto intro: gcd_greatest)  haftmann@58023  838 qed  haftmann@58023  839 haftmann@58023  840 lemma coprime_mul_eq: "gcd d (a * b) = 1 \ gcd d a = 1 \ gcd d b = 1"  haftmann@58023  841  using coprime_rmult[of d a b] coprime_lmult[of d a b] coprime_mult[of d a b] by blast  haftmann@58023  842 haftmann@58023  843 lemma gcd_coprime:  haftmann@60430  844  assumes c: "gcd a b \ 0" and a: "a = a' * gcd a b" and b: "b = b' * gcd a b"  haftmann@58023  845  shows "gcd a' b' = 1"  haftmann@58023  846 proof -  haftmann@60430  847  from c have "a \ 0 \ b \ 0" by simp  haftmann@58023  848  with div_gcd_coprime have "gcd (a div gcd a b) (b div gcd a b) = 1" .  haftmann@58023  849  also from assms have "a div gcd a b = a'" by (metis div_mult_self2_is_id)+  haftmann@58023  850  also from assms have "b div gcd a b = b'" by (metis div_mult_self2_is_id)+  haftmann@58023  851  finally show ?thesis .  haftmann@58023  852 qed  haftmann@58023  853 haftmann@58023  854 lemma coprime_power:  haftmann@58023  855  assumes "0 < n"  haftmann@58023  856  shows "gcd a (b ^ n) = 1 \ gcd a b = 1"  haftmann@58023  857 using assms proof (induct n)  haftmann@58023  858  case (Suc n) then show ?case  haftmann@58023  859  by (cases n) (simp_all add: coprime_mul_eq)  haftmann@58023  860 qed simp  haftmann@58023  861 haftmann@58023  862 lemma gcd_coprime_exists:  haftmann@58023  863  assumes nz: "gcd a b \ 0"  haftmann@58023  864  shows "\a' b'. a = a' * gcd a b \ b = b' * gcd a b \ gcd a' b' = 1"  haftmann@58023  865  apply (rule_tac x = "a div gcd a b" in exI)  haftmann@58023  866  apply (rule_tac x = "b div gcd a b" in exI)  haftmann@59009  867  apply (insert nz, auto intro: div_gcd_coprime)  haftmann@58023  868  done  haftmann@58023  869 haftmann@58023  870 lemma coprime_exp:  haftmann@58023  871  "gcd d a = 1 \ gcd d (a^n) = 1"  haftmann@58023  872  by (induct n, simp_all add: coprime_mult)  haftmann@58023  873 haftmann@58023  874 lemma coprime_exp2 [intro]:  haftmann@58023  875  "gcd a b = 1 \ gcd (a^n) (b^m) = 1"  haftmann@58023  876  apply (rule coprime_exp)  haftmann@58023  877  apply (subst gcd.commute)  haftmann@58023  878  apply (rule coprime_exp)  haftmann@58023  879  apply (subst gcd.commute)  haftmann@58023  880  apply assumption  haftmann@58023  881  done  haftmann@58023  882 haftmann@58023  883 lemma gcd_exp:  haftmann@58023  884  "gcd (a^n) (b^n) = (gcd a b) ^ n"  haftmann@58023  885 proof (cases "a = 0 \ b = 0")  haftmann@58023  886  assume "a = 0 \ b = 0"  haftmann@58023  887  then show ?thesis by (cases n, simp_all add: gcd_0_left)  haftmann@58023  888 next  haftmann@58023  889  assume A: "\(a = 0 \ b = 0)"  haftmann@58023  890  hence "1 = gcd ((a div gcd a b)^n) ((b div gcd a b)^n)"  haftmann@58023  891  using div_gcd_coprime by (subst sym, auto simp: div_gcd_coprime)  haftmann@58023  892  hence "(gcd a b) ^ n = (gcd a b) ^ n * ..." by simp  haftmann@58023  893  also note gcd_mult_distrib  haftmann@60438  894  also have "normalization_factor ((gcd a b)^n) = 1"  haftmann@60438  895  by (simp add: normalization_factor_pow A)  haftmann@58023  896  also have "(gcd a b)^n * (a div gcd a b)^n = a^n"  haftmann@58023  897  by (subst ac_simps, subst div_power, simp, rule dvd_div_mult_self, rule dvd_power_same, simp)  haftmann@58023  898  also have "(gcd a b)^n * (b div gcd a b)^n = b^n"  haftmann@58023  899  by (subst ac_simps, subst div_power, simp, rule dvd_div_mult_self, rule dvd_power_same, simp)  haftmann@58023  900  finally show ?thesis by simp  haftmann@58023  901 qed  haftmann@58023  902 haftmann@58023  903 lemma coprime_common_divisor:  haftmann@60430  904  "gcd a b = 1 \ a dvd a \ a dvd b \ is_unit a"  haftmann@60430  905  apply (subgoal_tac "a dvd gcd a b")  haftmann@59061  906  apply simp  haftmann@58023  907  apply (erule (1) gcd_greatest)  haftmann@58023  908  done  haftmann@58023  909 haftmann@58023  910 lemma division_decomp:  haftmann@58023  911  assumes dc: "a dvd b * c"  haftmann@58023  912  shows "\b' c'. a = b' * c' \ b' dvd b \ c' dvd c"  haftmann@58023  913 proof (cases "gcd a b = 0")  haftmann@58023  914  assume "gcd a b = 0"  haftmann@59009  915  hence "a = 0 \ b = 0" by simp  haftmann@58023  916  hence "a = 0 * c \ 0 dvd b \ c dvd c" by simp  haftmann@58023  917  then show ?thesis by blast  haftmann@58023  918 next  haftmann@58023  919  let ?d = "gcd a b"  haftmann@58023  920  assume "?d \ 0"  haftmann@58023  921  from gcd_coprime_exists[OF this]  haftmann@58023  922  obtain a' b' where ab': "a = a' * ?d" "b = b' * ?d" "gcd a' b' = 1"  haftmann@58023  923  by blast  haftmann@58023  924  from ab'(1) have "a' dvd a" unfolding dvd_def by blast  haftmann@58023  925  with dc have "a' dvd b*c" using dvd_trans[of a' a "b*c"] by simp  haftmann@58023  926  from dc ab'(1,2) have "a'*?d dvd (b'*?d) * c" by simp  haftmann@58023  927  hence "?d * a' dvd ?d * (b' * c)" by (simp add: mult_ac)  haftmann@59009  928  with ?d \ 0 have "a' dvd b' * c" by simp  haftmann@58023  929  with coprime_dvd_mult[OF ab'(3)]  haftmann@58023  930  have "a' dvd c" by (subst (asm) ac_simps, blast)  haftmann@58023  931  with ab'(1) have "a = ?d * a' \ ?d dvd b \ a' dvd c" by (simp add: mult_ac)  haftmann@58023  932  then show ?thesis by blast  haftmann@58023  933 qed  haftmann@58023  934 haftmann@60433  935 lemma pow_divs_pow:  haftmann@58023  936  assumes ab: "a ^ n dvd b ^ n" and n: "n \ 0"  haftmann@58023  937  shows "a dvd b"  haftmann@58023  938 proof (cases "gcd a b = 0")  haftmann@58023  939  assume "gcd a b = 0"  haftmann@59009  940  then show ?thesis by simp  haftmann@58023  941 next  haftmann@58023  942  let ?d = "gcd a b"  haftmann@58023  943  assume "?d \ 0"  haftmann@58023  944  from n obtain m where m: "n = Suc m" by (cases n, simp_all)  haftmann@59009  945  from ?d \ 0 have zn: "?d ^ n \ 0" by (rule power_not_zero)  haftmann@58023  946  from gcd_coprime_exists[OF ?d \ 0]  haftmann@58023  947  obtain a' b' where ab': "a = a' * ?d" "b = b' * ?d" "gcd a' b' = 1"  haftmann@58023  948  by blast  haftmann@58023  949  from ab have "(a' * ?d) ^ n dvd (b' * ?d) ^ n"  haftmann@58023  950  by (simp add: ab'(1,2)[symmetric])  haftmann@58023  951  hence "?d^n * a'^n dvd ?d^n * b'^n"  haftmann@58023  952  by (simp only: power_mult_distrib ac_simps)  haftmann@59009  953  with zn have "a'^n dvd b'^n" by simp  haftmann@58023  954  hence "a' dvd b'^n" using dvd_trans[of a' "a'^n" "b'^n"] by (simp add: m)  haftmann@58023  955  hence "a' dvd b'^m * b'" by (simp add: m ac_simps)  haftmann@58023  956  with coprime_dvd_mult[OF coprime_exp[OF ab'(3), of m]]  haftmann@58023  957  have "a' dvd b'" by (subst (asm) ac_simps, blast)  haftmann@58023  958  hence "a'*?d dvd b'*?d" by (rule mult_dvd_mono, simp)  haftmann@58023  959  with ab'(1,2) show ?thesis by simp  haftmann@58023  960 qed  haftmann@58023  961 haftmann@60433  962 lemma pow_divs_eq [simp]:  haftmann@58023  963  "n \ 0 \ a ^ n dvd b ^ n \ a dvd b"  haftmann@60433  964  by (auto intro: pow_divs_pow dvd_power_same)  haftmann@58023  965 haftmann@60433  966 lemma divs_mult:  haftmann@58023  967  assumes mr: "m dvd r" and nr: "n dvd r" and mn: "gcd m n = 1"  haftmann@58023  968  shows "m * n dvd r"  haftmann@58023  969 proof -  haftmann@58023  970  from mr nr obtain m' n' where m': "r = m*m'" and n': "r = n*n'"  haftmann@58023  971  unfolding dvd_def by blast  haftmann@58023  972  from mr n' have "m dvd n'*n" by (simp add: ac_simps)  haftmann@58023  973  hence "m dvd n'" using coprime_dvd_mult_iff[OF mn] by simp  haftmann@58023  974  then obtain k where k: "n' = m*k" unfolding dvd_def by blast  haftmann@58023  975  with n' have "r = m * n * k" by (simp add: mult_ac)  haftmann@58023  976  then show ?thesis unfolding dvd_def by blast  haftmann@58023  977 qed  haftmann@58023  978 haftmann@58023  979 lemma coprime_plus_one [simp]: "gcd (n + 1) n = 1"  haftmann@58023  980  by (subst add_commute, simp)  haftmann@58023  981 haftmann@58023  982 lemma setprod_coprime [rule_format]:  haftmann@60430  983  "(\i\A. gcd (f i) a = 1) \ gcd (\i\A. f i) a = 1"  haftmann@58023  984  apply (cases "finite A")  haftmann@58023  985  apply (induct set: finite)  haftmann@58023  986  apply (auto simp add: gcd_mult_cancel)  haftmann@58023  987  done  haftmann@58023  988 haftmann@58023  989 lemma coprime_divisors:  haftmann@58023  990  assumes "d dvd a" "e dvd b" "gcd a b = 1"  haftmann@58023  991  shows "gcd d e = 1"  haftmann@58023  992 proof -  haftmann@58023  993  from assms obtain k l where "a = d * k" "b = e * l"  haftmann@58023  994  unfolding dvd_def by blast  haftmann@58023  995  with assms have "gcd (d * k) (e * l) = 1" by simp  haftmann@58023  996  hence "gcd (d * k) e = 1" by (rule coprime_lmult)  haftmann@58023  997  also have "gcd (d * k) e = gcd e (d * k)" by (simp add: ac_simps)  haftmann@58023  998  finally have "gcd e d = 1" by (rule coprime_lmult)  haftmann@58023  999  then show ?thesis by (simp add: ac_simps)  haftmann@58023  1000 qed  haftmann@58023  1001 haftmann@58023  1002 lemma invertible_coprime:  haftmann@60430  1003  assumes "a * b mod m = 1"  haftmann@60430  1004  shows "coprime a m"  haftmann@59009  1005 proof -  haftmann@60430  1006  from assms have "coprime m (a * b mod m)"  haftmann@59009  1007  by simp  haftmann@60430  1008  then have "coprime m (a * b)"  haftmann@59009  1009  by simp  haftmann@60430  1010  then have "coprime m a"  haftmann@59009  1011  by (rule coprime_lmult)  haftmann@59009  1012  then show ?thesis  haftmann@59009  1013  by (simp add: ac_simps)  haftmann@59009  1014 qed  haftmann@58023  1015 haftmann@58023  1016 lemma lcm_gcd:  haftmann@60438  1017  "lcm a b = a * b div (gcd a b * normalization_factor (a*b))"  haftmann@58023  1018  by (simp only: lcm_lcm_eucl gcd_gcd_eucl lcm_eucl_def)  haftmann@58023  1019 haftmann@58023  1020 lemma lcm_gcd_prod:  haftmann@60438  1021  "lcm a b * gcd a b = a * b div normalization_factor (a*b)"  haftmann@58023  1022 proof (cases "a * b = 0")  haftmann@60438  1023  let ?nf = normalization_factor  haftmann@58023  1024  assume "a * b \ 0"  haftmann@58953  1025  hence "gcd a b \ 0" by simp  haftmann@58023  1026  from lcm_gcd have "lcm a b * gcd a b = gcd a b * (a * b div (?nf (a*b) * gcd a b))"  haftmann@58023  1027  by (simp add: mult_ac)  haftmann@60432  1028  also from a * b \ 0 have "... = a * b div ?nf (a*b)"  haftmann@60432  1029  by (simp add: div_mult_swap mult.commute)  haftmann@58023  1030  finally show ?thesis .  haftmann@58953  1031 qed (auto simp add: lcm_gcd)  haftmann@58023  1032 haftmann@58023  1033 lemma lcm_dvd1 [iff]:  haftmann@60430  1034  "a dvd lcm a b"  haftmann@60430  1035 proof (cases "a*b = 0")  haftmann@60430  1036  assume "a * b \ 0"  haftmann@60430  1037  hence "gcd a b \ 0" by simp  haftmann@60438  1038  let ?c = "1 div normalization_factor (a * b)"  haftmann@60438  1039  from a * b \ 0 have [simp]: "is_unit (normalization_factor (a * b))" by simp  haftmann@60430  1040  from lcm_gcd_prod[of a b] have "lcm a b * gcd a b = a * ?c * b"  haftmann@60432  1041  by (simp add: div_mult_swap unit_div_commute)  haftmann@60430  1042  hence "lcm a b * gcd a b div gcd a b = a * ?c * b div gcd a b" by simp  haftmann@60430  1043  with gcd a b \ 0 have "lcm a b = a * ?c * b div gcd a b"  haftmann@58023  1044  by (subst (asm) div_mult_self2_is_id, simp_all)  haftmann@60430  1045  also have "... = a * (?c * b div gcd a b)"  haftmann@58023  1046  by (metis div_mult_swap gcd_dvd2 mult_assoc)  haftmann@58023  1047  finally show ?thesis by (rule dvdI)  haftmann@58953  1048 qed (auto simp add: lcm_gcd)  haftmann@58023  1049 haftmann@58023  1050 lemma lcm_least:  haftmann@58023  1051  "\a dvd k; b dvd k\ \ lcm a b dvd k"  haftmann@58023  1052 proof (cases "k = 0")  haftmann@60438  1053  let ?nf = normalization_factor  haftmann@58023  1054  assume "k \ 0"  haftmann@58023  1055  hence "is_unit (?nf k)" by simp  haftmann@58023  1056  hence "?nf k \ 0" by (metis not_is_unit_0)  haftmann@58023  1057  assume A: "a dvd k" "b dvd k"  haftmann@58953  1058  hence "gcd a b \ 0" using k \ 0 by auto  haftmann@58023  1059  from A obtain r s where ar: "k = a * r" and bs: "k = b * s"  haftmann@58023  1060  unfolding dvd_def by blast  haftmann@58953  1061  with k \ 0 have "r * s \ 0"  haftmann@58953  1062  by auto (drule sym [of 0], simp)  haftmann@58023  1063  hence "is_unit (?nf (r * s))" by simp  haftmann@58023  1064  let ?c = "?nf k div ?nf (r*s)"  haftmann@58023  1065  from is_unit (?nf k) and is_unit (?nf (r * s)) have "is_unit ?c" by (rule unit_div)  haftmann@58023  1066  hence "?c \ 0" using not_is_unit_0 by fast  haftmann@58023  1067  from ar bs have "k * k * gcd s r = ?nf k * k * gcd (k * s) (k * r)"  haftmann@58953  1068  by (subst mult_assoc, subst gcd_mult_distrib[of k s r], simp only: ac_simps)  haftmann@58023  1069  also have "... = ?nf k * k * gcd ((r*s) * a) ((r*s) * b)"  haftmann@58023  1070  by (subst (3) k = a * r, subst (3) k = b * s, simp add: algebra_simps)  haftmann@58023  1071  also have "... = ?c * r*s * k * gcd a b" using r * s \ 0  haftmann@58023  1072  by (subst gcd_mult_distrib'[symmetric], simp add: algebra_simps unit_simps)  haftmann@58023  1073  finally have "(a*r) * (b*s) * gcd s r = ?c * k * r * s * gcd a b"  haftmann@58023  1074  by (subst ar[symmetric], subst bs[symmetric], simp add: mult_ac)  haftmann@58023  1075  hence "a * b * gcd s r * (r * s) = ?c * k * gcd a b * (r * s)"  haftmann@58023  1076  by (simp add: algebra_simps)  haftmann@58023  1077  hence "?c * k * gcd a b = a * b * gcd s r" using r * s \ 0  haftmann@58023  1078  by (metis div_mult_self2_is_id)  haftmann@58023  1079  also have "... = lcm a b * gcd a b * gcd s r * ?nf (a*b)"  haftmann@58023  1080  by (subst lcm_gcd_prod[of a b], metis gcd_mult_distrib gcd_mult_distrib')  haftmann@58023  1081  also have "... = lcm a b * gcd s r * ?nf (a*b) * gcd a b"  haftmann@58023  1082  by (simp add: algebra_simps)  haftmann@58023  1083  finally have "k * ?c = lcm a b * gcd s r * ?nf (a*b)" using gcd a b \ 0  haftmann@58023  1084  by (metis mult.commute div_mult_self2_is_id)  haftmann@58023  1085  hence "k = lcm a b * (gcd s r * ?nf (a*b)) div ?c" using ?c \ 0  haftmann@58023  1086  by (metis div_mult_self2_is_id mult_assoc)  haftmann@58023  1087  also have "... = lcm a b * (gcd s r * ?nf (a*b) div ?c)" using is_unit ?c  haftmann@58023  1088  by (simp add: unit_simps)  haftmann@58023  1089  finally show ?thesis by (rule dvdI)  haftmann@58023  1090 qed simp  haftmann@58023  1091 haftmann@58023  1092 lemma lcm_zero:  haftmann@58023  1093  "lcm a b = 0 \ a = 0 \ b = 0"  haftmann@58023  1094 proof -  haftmann@60438  1095  let ?nf = normalization_factor  haftmann@58023  1096  {  haftmann@58023  1097  assume "a \ 0" "b \ 0"  haftmann@58023  1098  hence "a * b div ?nf (a * b) \ 0" by (simp add: no_zero_divisors)  haftmann@59009  1099  moreover from a \ 0 and b \ 0 have "gcd a b \ 0" by simp  haftmann@58023  1100  ultimately have "lcm a b \ 0" using lcm_gcd_prod[of a b] by (intro notI, simp)  haftmann@58023  1101  } moreover {  haftmann@58023  1102  assume "a = 0 \ b = 0"  haftmann@58023  1103  hence "lcm a b = 0" by (elim disjE, simp_all add: lcm_gcd)  haftmann@58023  1104  }  haftmann@58023  1105  ultimately show ?thesis by blast  haftmann@58023  1106 qed  haftmann@58023  1107 haftmann@58023  1108 lemmas lcm_0_iff = lcm_zero  haftmann@58023  1109 haftmann@58023  1110 lemma gcd_lcm:  haftmann@58023  1111  assumes "lcm a b \ 0"  haftmann@60438  1112  shows "gcd a b = a * b div (lcm a b * normalization_factor (a * b))"  haftmann@58023  1113 proof-  haftmann@59009  1114  from assms have "gcd a b \ 0" by (simp add: lcm_zero)  haftmann@60438  1115  let ?c = "normalization_factor (a * b)"  haftmann@58023  1116  from lcm a b \ 0 have "?c \ 0" by (intro notI, simp add: lcm_zero no_zero_divisors)  haftmann@58023  1117  hence "is_unit ?c" by simp  haftmann@58023  1118  from lcm_gcd_prod [of a b] have "gcd a b = a * b div ?c div lcm a b"  haftmann@58023  1119  by (subst (2) div_mult_self2_is_id[OF lcm a b \ 0, symmetric], simp add: mult_ac)  haftmann@60433  1120  also from is_unit ?c have "... = a * b div (lcm a b * ?c)"  haftmann@60438  1121  by (metis ?c \ 0 div_mult_mult1 dvd_mult_div_cancel mult_commute normalization_factor_dvd')  haftmann@60433  1122  finally show ?thesis .  haftmann@58023  1123 qed  haftmann@58023  1124 haftmann@60438  1125 lemma normalization_factor_lcm [simp]:  haftmann@60438  1126  "normalization_factor (lcm a b) = (if a = 0 \ b = 0 then 0 else 1)"  haftmann@58023  1127 proof (cases "a = 0 \ b = 0")  haftmann@58023  1128  case True then show ?thesis  haftmann@58953  1129  by (auto simp add: lcm_gcd)  haftmann@58023  1130 next  haftmann@58023  1131  case False  haftmann@60438  1132  let ?nf = normalization_factor  haftmann@58023  1133  from lcm_gcd_prod[of a b]  haftmann@58023  1134  have "?nf (lcm a b) * ?nf (gcd a b) = ?nf (a*b) div ?nf (a*b)"  haftmann@60438  1135  by (metis div_by_0 div_self normalization_correct normalization_factor_0 normalization_factor_mult)  haftmann@58023  1136  also have "... = (if a*b = 0 then 0 else 1)"  haftmann@58953  1137  by simp  haftmann@58953  1138  finally show ?thesis using False by simp  haftmann@58023  1139 qed  haftmann@58023  1140 haftmann@60430  1141 lemma lcm_dvd2 [iff]: "b dvd lcm a b"  haftmann@60430  1142  using lcm_dvd1 [of b a] by (simp add: lcm_gcd ac_simps)  haftmann@58023  1143 haftmann@58023  1144 lemma lcmI:  haftmann@60430  1145  "\a dvd k; b dvd k; \l. a dvd l \ b dvd l \ k dvd l;  haftmann@60438  1146  normalization_factor k = (if k = 0 then 0 else 1)\ \ k = lcm a b"  haftmann@58023  1147  by (intro normed_associated_imp_eq) (auto simp: associated_def intro: lcm_least)  haftmann@58023  1148 haftmann@58023  1149 sublocale lcm!: abel_semigroup lcm  haftmann@58023  1150 proof  haftmann@60430  1151  fix a b c  haftmann@60430  1152  show "lcm (lcm a b) c = lcm a (lcm b c)"  haftmann@58023  1153  proof (rule lcmI)  haftmann@60430  1154  have "a dvd lcm a b" and "lcm a b dvd lcm (lcm a b) c" by simp_all  haftmann@60430  1155  then show "a dvd lcm (lcm a b) c" by (rule dvd_trans)  haftmann@58023  1156   haftmann@60430  1157  have "b dvd lcm a b" and "lcm a b dvd lcm (lcm a b) c" by simp_all  haftmann@60430  1158  hence "b dvd lcm (lcm a b) c" by (rule dvd_trans)  haftmann@60430  1159  moreover have "c dvd lcm (lcm a b) c" by simp  haftmann@60430  1160  ultimately show "lcm b c dvd lcm (lcm a b) c" by (rule lcm_least)  haftmann@58023  1161 haftmann@60430  1162  fix l assume "a dvd l" and "lcm b c dvd l"  haftmann@60430  1163  have "b dvd lcm b c" by simp  haftmann@60430  1164  from this and lcm b c dvd l have "b dvd l" by (rule dvd_trans)  haftmann@60430  1165  have "c dvd lcm b c" by simp  haftmann@60430  1166  from this and lcm b c dvd l have "c dvd l" by (rule dvd_trans)  haftmann@60430  1167  from a dvd l and b dvd l have "lcm a b dvd l" by (rule lcm_least)  haftmann@60430  1168  from this and c dvd l show "lcm (lcm a b) c dvd l" by (rule lcm_least)  haftmann@58023  1169  qed (simp add: lcm_zero)  haftmann@58023  1170 next  haftmann@60430  1171  fix a b  haftmann@60430  1172  show "lcm a b = lcm b a"  haftmann@58023  1173  by (simp add: lcm_gcd ac_simps)  haftmann@58023  1174 qed  haftmann@58023  1175 haftmann@58023  1176 lemma dvd_lcm_D1:  haftmann@58023  1177  "lcm m n dvd k \ m dvd k"  haftmann@58023  1178  by (rule dvd_trans, rule lcm_dvd1, assumption)  haftmann@58023  1179 haftmann@58023  1180 lemma dvd_lcm_D2:  haftmann@58023  1181  "lcm m n dvd k \ n dvd k"  haftmann@58023  1182  by (rule dvd_trans, rule lcm_dvd2, assumption)  haftmann@58023  1183 haftmann@58023  1184 lemma gcd_dvd_lcm [simp]:  haftmann@58023  1185  "gcd a b dvd lcm a b"  haftmann@58023  1186  by (metis dvd_trans gcd_dvd2 lcm_dvd2)  haftmann@58023  1187 haftmann@58023  1188 lemma lcm_1_iff:  haftmann@58023  1189  "lcm a b = 1 \ is_unit a \ is_unit b"  haftmann@58023  1190 proof  haftmann@58023  1191  assume "lcm a b = 1"  haftmann@59061  1192  then show "is_unit a \ is_unit b" by auto  haftmann@58023  1193 next  haftmann@58023  1194  assume "is_unit a \ is_unit b"  haftmann@59061  1195  hence "a dvd 1" and "b dvd 1" by simp_all  haftmann@59061  1196  hence "is_unit (lcm a b)" by (rule lcm_least)  haftmann@60438  1197  hence "lcm a b = normalization_factor (lcm a b)"  haftmann@60438  1198  by (subst normalization_factor_unit, simp_all)  haftmann@59061  1199  also have "\ = 1" using is_unit a \ is_unit b  haftmann@59061  1200  by auto  haftmann@58023  1201  finally show "lcm a b = 1" .  haftmann@58023  1202 qed  haftmann@58023  1203 haftmann@58023  1204 lemma lcm_0_left [simp]:  haftmann@60430  1205  "lcm 0 a = 0"  haftmann@58023  1206  by (rule sym, rule lcmI, simp_all)  haftmann@58023  1207 haftmann@58023  1208 lemma lcm_0 [simp]:  haftmann@60430  1209  "lcm a 0 = 0"  haftmann@58023  1210  by (rule sym, rule lcmI, simp_all)  haftmann@58023  1211 haftmann@58023  1212 lemma lcm_unique:  haftmann@58023  1213  "a dvd d \ b dvd d \  haftmann@60438  1214  normalization_factor d = (if d = 0 then 0 else 1) \  haftmann@58023  1215  (\e. a dvd e \ b dvd e \ d dvd e) \ d = lcm a b"  haftmann@58023  1216  by (rule, auto intro: lcmI simp: lcm_least lcm_zero)  haftmann@58023  1217 haftmann@58023  1218 lemma dvd_lcm_I1 [simp]:  haftmann@58023  1219  "k dvd m \ k dvd lcm m n"  haftmann@58023  1220  by (metis lcm_dvd1 dvd_trans)  haftmann@58023  1221 haftmann@58023  1222 lemma dvd_lcm_I2 [simp]:  haftmann@58023  1223  "k dvd n \ k dvd lcm m n"  haftmann@58023  1224  by (metis lcm_dvd2 dvd_trans)  haftmann@58023  1225 haftmann@58023  1226 lemma lcm_1_left [simp]:  haftmann@60438  1227  "lcm 1 a = a div normalization_factor a"  haftmann@60430  1228  by (cases "a = 0") (simp, rule sym, rule lcmI, simp_all)  haftmann@58023  1229 haftmann@58023  1230 lemma lcm_1_right [simp]:  haftmann@60438  1231  "lcm a 1 = a div normalization_factor a"  haftmann@60430  1232  using lcm_1_left [of a] by (simp add: ac_simps)  haftmann@58023  1233 haftmann@58023  1234 lemma lcm_coprime:  haftmann@60438  1235  "gcd a b = 1 \ lcm a b = a * b div normalization_factor (a*b)"  haftmann@58023  1236  by (subst lcm_gcd) simp  haftmann@58023  1237 haftmann@58023  1238 lemma lcm_proj1_if_dvd:  haftmann@60438  1239  "b dvd a \ lcm a b = a div normalization_factor a"  haftmann@60430  1240  by (cases "a = 0") (simp, rule sym, rule lcmI, simp_all)  haftmann@58023  1241 haftmann@58023  1242 lemma lcm_proj2_if_dvd:  haftmann@60438  1243  "a dvd b \ lcm a b = b div normalization_factor b"  haftmann@60430  1244  using lcm_proj1_if_dvd [of a b] by (simp add: ac_simps)  haftmann@58023  1245 haftmann@58023  1246 lemma lcm_proj1_iff:  haftmann@60438  1247  "lcm m n = m div normalization_factor m \ n dvd m"  haftmann@58023  1248 proof  haftmann@60438  1249  assume A: "lcm m n = m div normalization_factor m"  haftmann@58023  1250  show "n dvd m"  haftmann@58023  1251  proof (cases "m = 0")  haftmann@58023  1252  assume [simp]: "m \ 0"  haftmann@60438  1253  from A have B: "m = lcm m n * normalization_factor m"  haftmann@58023  1254  by (simp add: unit_eq_div2)  haftmann@58023  1255  show ?thesis by (subst B, simp)  haftmann@58023  1256  qed simp  haftmann@58023  1257 next  haftmann@58023  1258  assume "n dvd m"  haftmann@60438  1259  then show "lcm m n = m div normalization_factor m" by (rule lcm_proj1_if_dvd)  haftmann@58023  1260 qed  haftmann@58023  1261 haftmann@58023  1262 lemma lcm_proj2_iff:  haftmann@60438  1263  "lcm m n = n div normalization_factor n \ m dvd n"  haftmann@58023  1264  using lcm_proj1_iff [of n m] by (simp add: ac_simps)  haftmann@58023  1265 haftmann@58023  1266 lemma euclidean_size_lcm_le1:  haftmann@58023  1267  assumes "a \ 0" and "b \ 0"  haftmann@58023  1268  shows "euclidean_size a \ euclidean_size (lcm a b)"  haftmann@58023  1269 proof -  haftmann@58023  1270  have "a dvd lcm a b" by (rule lcm_dvd1)  haftmann@58023  1271  then obtain c where A: "lcm a b = a * c" unfolding dvd_def by blast  haftmann@58023  1272  with a \ 0 and b \ 0 have "c \ 0" by (auto simp: lcm_zero)  haftmann@58023  1273  then show ?thesis by (subst A, intro size_mult_mono)  haftmann@58023  1274 qed  haftmann@58023  1275 haftmann@58023  1276 lemma euclidean_size_lcm_le2:  haftmann@58023  1277  "a \ 0 \ b \ 0 \ euclidean_size b \ euclidean_size (lcm a b)"  haftmann@58023  1278  using euclidean_size_lcm_le1 [of b a] by (simp add: ac_simps)  haftmann@58023  1279 haftmann@58023  1280 lemma euclidean_size_lcm_less1:  haftmann@58023  1281  assumes "b \ 0" and "\b dvd a"  haftmann@58023  1282  shows "euclidean_size a < euclidean_size (lcm a b)"  haftmann@58023  1283 proof (rule ccontr)  haftmann@58023  1284  from assms have "a \ 0" by auto  haftmann@58023  1285  assume "\euclidean_size a < euclidean_size (lcm a b)"  haftmann@58023  1286  with a \ 0 and b \ 0 have "euclidean_size (lcm a b) = euclidean_size a"  haftmann@58023  1287  by (intro le_antisym, simp, intro euclidean_size_lcm_le1)  haftmann@58023  1288  with assms have "lcm a b dvd a"  haftmann@58023  1289  by (rule_tac dvd_euclidean_size_eq_imp_dvd) (auto simp: lcm_zero)  haftmann@58023  1290  hence "b dvd a" by (rule dvd_lcm_D2)  haftmann@58023  1291  with \b dvd a show False by contradiction  haftmann@58023  1292 qed  haftmann@58023  1293 haftmann@58023  1294 lemma euclidean_size_lcm_less2:  haftmann@58023  1295  assumes "a \ 0" and "\a dvd b"  haftmann@58023  1296  shows "euclidean_size b < euclidean_size (lcm a b)"  haftmann@58023  1297  using assms euclidean_size_lcm_less1 [of a b] by (simp add: ac_simps)  haftmann@58023  1298 haftmann@58023  1299 lemma lcm_mult_unit1:  haftmann@60430  1300  "is_unit a \ lcm (b * a) c = lcm b c"  haftmann@58023  1301  apply (rule lcmI)  haftmann@60430  1302  apply (rule dvd_trans[of _ "b * a"], simp, rule lcm_dvd1)  haftmann@58023  1303  apply (rule lcm_dvd2)  haftmann@58023  1304  apply (rule lcm_least, simp add: unit_simps, assumption)  haftmann@60438  1305  apply (subst normalization_factor_lcm, simp add: lcm_zero)  haftmann@58023  1306  done  haftmann@58023  1307 haftmann@58023  1308 lemma lcm_mult_unit2:  haftmann@60430  1309  "is_unit a \ lcm b (c * a) = lcm b c"  haftmann@60430  1310  using lcm_mult_unit1 [of a c b] by (simp add: ac_simps)  haftmann@58023  1311 haftmann@58023  1312 lemma lcm_div_unit1:  haftmann@60430  1313  "is_unit a \ lcm (b div a) c = lcm b c"  haftmann@60433  1314  by (erule is_unitE [of _ b]) (simp add: lcm_mult_unit1)  haftmann@58023  1315 haftmann@58023  1316 lemma lcm_div_unit2:  haftmann@60430  1317  "is_unit a \ lcm b (c div a) = lcm b c"  haftmann@60433  1318  by (erule is_unitE [of _ c]) (simp add: lcm_mult_unit2)  haftmann@58023  1319 haftmann@58023  1320 lemma lcm_left_idem:  haftmann@60430  1321  "lcm a (lcm a b) = lcm a b"  haftmann@58023  1322  apply (rule lcmI)  haftmann@58023  1323  apply simp  haftmann@58023  1324  apply (subst lcm.assoc [symmetric], rule lcm_dvd2)  haftmann@58023  1325  apply (rule lcm_least, assumption)  haftmann@58023  1326  apply (erule (1) lcm_least)  haftmann@58023  1327  apply (auto simp: lcm_zero)  haftmann@58023  1328  done  haftmann@58023  1329 haftmann@58023  1330 lemma lcm_right_idem:  haftmann@60430  1331  "lcm (lcm a b) b = lcm a b"  haftmann@58023  1332  apply (rule lcmI)  haftmann@58023  1333  apply (subst lcm.assoc, rule lcm_dvd1)  haftmann@58023  1334  apply (rule lcm_dvd2)  haftmann@58023  1335  apply (rule lcm_least, erule (1) lcm_least, assumption)  haftmann@58023  1336  apply (auto simp: lcm_zero)  haftmann@58023  1337  done  haftmann@58023  1338 haftmann@58023  1339 lemma comp_fun_idem_lcm: "comp_fun_idem lcm"  haftmann@58023  1340 proof  haftmann@58023  1341  fix a b show "lcm a \ lcm b = lcm b \ lcm a"  haftmann@58023  1342  by (simp add: fun_eq_iff ac_simps)  haftmann@58023  1343 next  haftmann@58023  1344  fix a show "lcm a \ lcm a = lcm a" unfolding o_def  haftmann@58023  1345  by (intro ext, simp add: lcm_left_idem)  haftmann@58023  1346 qed  haftmann@58023  1347 haftmann@60430  1348 lemma dvd_Lcm [simp]: "a \ A \ a dvd Lcm A"  haftmann@60430  1349  and Lcm_dvd [simp]: "(\a\A. a dvd l') \ Lcm A dvd l'"  haftmann@60438  1350  and normalization_factor_Lcm [simp]:  haftmann@60438  1351  "normalization_factor (Lcm A) = (if Lcm A = 0 then 0 else 1)"  haftmann@58023  1352 proof -  haftmann@60430  1353  have "(\a\A. a dvd Lcm A) \ (\l'. (\a\A. a dvd l') \ Lcm A dvd l') \  haftmann@60438  1354  normalization_factor (Lcm A) = (if Lcm A = 0 then 0 else 1)" (is ?thesis)  haftmann@60430  1355  proof (cases "\l. l \ 0 \ (\a\A. a dvd l)")  haftmann@58023  1356  case False  haftmann@58023  1357  hence "Lcm A = 0" by (auto simp: Lcm_Lcm_eucl Lcm_eucl_def)  haftmann@58023  1358  with False show ?thesis by auto  haftmann@58023  1359  next  haftmann@58023  1360  case True  haftmann@60430  1361  then obtain l\<^sub>0 where l\<^sub>0_props: "l\<^sub>0 \ 0 \ (\a\A. a dvd l\<^sub>0)" by blast  haftmann@60430  1362  def n \ "LEAST n. \l. l \ 0 \ (\a\A. a dvd l) \ euclidean_size l = n"  haftmann@60430  1363  def l \ "SOME l. l \ 0 \ (\a\A. a dvd l) \ euclidean_size l = n"  haftmann@60430  1364  have "\l. l \ 0 \ (\a\A. a dvd l) \ euclidean_size l = n"  haftmann@58023  1365  apply (subst n_def)  haftmann@58023  1366  apply (rule LeastI[of _ "euclidean_size l\<^sub>0"])  haftmann@58023  1367  apply (rule exI[of _ l\<^sub>0])  haftmann@58023  1368  apply (simp add: l\<^sub>0_props)  haftmann@58023  1369  done  haftmann@60430  1370  from someI_ex[OF this] have "l \ 0" and "\a\A. a dvd l" and "euclidean_size l = n"  haftmann@58023  1371  unfolding l_def by simp_all  haftmann@58023  1372  {  haftmann@60430  1373  fix l' assume "\a\A. a dvd l'"  haftmann@60430  1374  with \a\A. a dvd l have "\a\A. a dvd gcd l l'" by (auto intro: gcd_greatest)  haftmann@59009  1375  moreover from l \ 0 have "gcd l l' \ 0" by simp  haftmann@60430  1376  ultimately have "\b. b \ 0 \ (\a\A. a dvd b) \ euclidean_size b = euclidean_size (gcd l l')"  haftmann@58023  1377  by (intro exI[of _ "gcd l l'"], auto)  haftmann@58023  1378  hence "euclidean_size (gcd l l') \ n" by (subst n_def) (rule Least_le)  haftmann@58023  1379  moreover have "euclidean_size (gcd l l') \ n"  haftmann@58023  1380  proof -  haftmann@58023  1381  have "gcd l l' dvd l" by simp  haftmann@58023  1382  then obtain a where "l = gcd l l' * a" unfolding dvd_def by blast  haftmann@58023  1383  with l \ 0 have "a \ 0" by auto  haftmann@58023  1384  hence "euclidean_size (gcd l l') \ euclidean_size (gcd l l' * a)"  haftmann@58023  1385  by (rule size_mult_mono)  haftmann@58023  1386  also have "gcd l l' * a = l" using l = gcd l l' * a ..  haftmann@58023  1387  also note euclidean_size l = n  haftmann@58023  1388  finally show "euclidean_size (gcd l l') \ n" .  haftmann@58023  1389  qed  haftmann@58023  1390  ultimately have "euclidean_size l = euclidean_size (gcd l l')"  haftmann@58023  1391  by (intro le_antisym, simp_all add: euclidean_size l = n)  haftmann@58023  1392  with l \ 0 have "l dvd gcd l l'" by (blast intro: dvd_euclidean_size_eq_imp_dvd)  haftmann@58023  1393  hence "l dvd l'" by (blast dest: dvd_gcd_D2)  haftmann@58023  1394  }  haftmann@58023  1395 haftmann@60438  1396  with (\a\A. a dvd l) and normalization_factor_is_unit[OF l \ 0] and l \ 0  haftmann@60438  1397  have "(\a\A. a dvd l div normalization_factor l) \  haftmann@60438  1398  (\l'. (\a\A. a dvd l') \ l div normalization_factor l dvd l') \  haftmann@60438  1399  normalization_factor (l div normalization_factor l) =  haftmann@60438  1400  (if l div normalization_factor l = 0 then 0 else 1)"  haftmann@58023  1401  by (auto simp: unit_simps)  haftmann@60438  1402  also from True have "l div normalization_factor l = Lcm A"  haftmann@58023  1403  by (simp add: Lcm_Lcm_eucl Lcm_eucl_def Let_def n_def l_def)  haftmann@58023  1404  finally show ?thesis .  haftmann@58023  1405  qed  haftmann@58023  1406  note A = this  haftmann@58023  1407 haftmann@60430  1408  {fix a assume "a \ A" then show "a dvd Lcm A" using A by blast}  haftmann@60430  1409  {fix l' assume "\a\A. a dvd l'" then show "Lcm A dvd l'" using A by blast}  haftmann@60438  1410  from A show "normalization_factor (Lcm A) = (if Lcm A = 0 then 0 else 1)" by blast  haftmann@58023  1411 qed  haftmann@58023  1412   haftmann@58023  1413 lemma LcmI:  haftmann@60430  1414  "(\a. a\A \ a dvd l) \ (\l'. (\a\A. a dvd l') \ l dvd l') \  haftmann@60438  1415  normalization_factor l = (if l = 0 then 0 else 1) \ l = Lcm A"  haftmann@58023  1416  by (intro normed_associated_imp_eq)  haftmann@58023  1417  (auto intro: Lcm_dvd dvd_Lcm simp: associated_def)  haftmann@58023  1418 haftmann@58023  1419 lemma Lcm_subset:  haftmann@58023  1420  "A \ B \ Lcm A dvd Lcm B"  haftmann@58023  1421  by (blast intro: Lcm_dvd dvd_Lcm)  haftmann@58023  1422 haftmann@58023  1423 lemma Lcm_Un:  haftmann@58023  1424  "Lcm (A \ B) = lcm (Lcm A) (Lcm B)"  haftmann@58023  1425  apply (rule lcmI)  haftmann@58023  1426  apply (blast intro: Lcm_subset)  haftmann@58023  1427  apply (blast intro: Lcm_subset)  haftmann@58023  1428  apply (intro Lcm_dvd ballI, elim UnE)  haftmann@58023  1429  apply (rule dvd_trans, erule dvd_Lcm, assumption)  haftmann@58023  1430  apply (rule dvd_trans, erule dvd_Lcm, assumption)  haftmann@58023  1431  apply simp  haftmann@58023  1432  done  haftmann@58023  1433 haftmann@58023  1434 lemma Lcm_1_iff:  haftmann@60430  1435  "Lcm A = 1 \ (\a\A. is_unit a)"  haftmann@58023  1436 proof  haftmann@58023  1437  assume "Lcm A = 1"  haftmann@60430  1438  then show "\a\A. is_unit a" by auto  haftmann@58023  1439 qed (rule LcmI [symmetric], auto)  haftmann@58023  1440 haftmann@58023  1441 lemma Lcm_no_units:  haftmann@60430  1442  "Lcm A = Lcm (A - {a. is_unit a})"  haftmann@58023  1443 proof -  haftmann@60430  1444  have "(A - {a. is_unit a}) \ {a\A. is_unit a} = A" by blast  haftmann@60430  1445  hence "Lcm A = lcm (Lcm (A - {a. is_unit a})) (Lcm {a\A. is_unit a})"  haftmann@58023  1446  by (simp add: Lcm_Un[symmetric])  haftmann@60430  1447  also have "Lcm {a\A. is_unit a} = 1" by (simp add: Lcm_1_iff)  haftmann@58023  1448  finally show ?thesis by simp  haftmann@58023  1449 qed  haftmann@58023  1450 haftmann@58023  1451 lemma Lcm_empty [simp]:  haftmann@58023  1452  "Lcm {} = 1"  haftmann@58023  1453  by (simp add: Lcm_1_iff)  haftmann@58023  1454 haftmann@58023  1455 lemma Lcm_eq_0 [simp]:  haftmann@58023  1456  "0 \ A \ Lcm A = 0"  haftmann@58023  1457  by (drule dvd_Lcm) simp  haftmann@58023  1458 haftmann@58023  1459 lemma Lcm0_iff':  haftmann@60430  1460  "Lcm A = 0 \ \(\l. l \ 0 \ (\a\A. a dvd l))"  haftmann@58023  1461 proof  haftmann@58023  1462  assume "Lcm A = 0"  haftmann@60430  1463  show "\(\l. l \ 0 \ (\a\A. a dvd l))"  haftmann@58023  1464  proof  haftmann@60430  1465  assume ex: "\l. l \ 0 \ (\a\A. a dvd l)"  haftmann@60430  1466  then obtain l\<^sub>0 where l\<^sub>0_props: "l\<^sub>0 \ 0 \ (\a\A. a dvd l\<^sub>0)" by blast  haftmann@60430  1467  def n \ "LEAST n. \l. l \ 0 \ (\a\A. a dvd l) \ euclidean_size l = n"  haftmann@60430  1468  def l \ "SOME l. l \ 0 \ (\a\A. a dvd l) \ euclidean_size l = n"  haftmann@60430  1469  have "\l. l \ 0 \ (\a\A. a dvd l) \ euclidean_size l = n"  haftmann@58023  1470  apply (subst n_def)  haftmann@58023  1471  apply (rule LeastI[of _ "euclidean_size l\<^sub>0"])  haftmann@58023  1472  apply (rule exI[of _ l\<^sub>0])  haftmann@58023  1473  apply (simp add: l\<^sub>0_props)  haftmann@58023  1474  done  haftmann@58023  1475  from someI_ex[OF this] have "l \ 0" unfolding l_def by simp_all  haftmann@60438  1476  hence "l div normalization_factor l \ 0" by simp  haftmann@60438  1477  also from ex have "l div normalization_factor l = Lcm A"  haftmann@58023  1478  by (simp only: Lcm_Lcm_eucl Lcm_eucl_def n_def l_def if_True Let_def)  haftmann@58023  1479  finally show False using Lcm A = 0 by contradiction  haftmann@58023  1480  qed  haftmann@58023  1481 qed (simp only: Lcm_Lcm_eucl Lcm_eucl_def if_False)  haftmann@58023  1482 haftmann@58023  1483 lemma Lcm0_iff [simp]:  haftmann@58023  1484  "finite A \ Lcm A = 0 \ 0 \ A"  haftmann@58023  1485 proof -  haftmann@58023  1486  assume "finite A"  haftmann@58023  1487  have "0 \ A \ Lcm A = 0" by (intro dvd_0_left dvd_Lcm)  haftmann@58023  1488  moreover {  haftmann@58023  1489  assume "0 \ A"  haftmann@58023  1490  hence "\A \ 0"  haftmann@58023  1491  apply (induct rule: finite_induct[OF finite A])  haftmann@58023  1492  apply simp  haftmann@58023  1493  apply (subst setprod.insert, assumption, assumption)  haftmann@58023  1494  apply (rule no_zero_divisors)  haftmann@58023  1495  apply blast+  haftmann@58023  1496  done  haftmann@60430  1497  moreover from finite A have "\a\A. a dvd \A" by blast  haftmann@60430  1498  ultimately have "\l. l \ 0 \ (\a\A. a dvd l)" by blast  haftmann@58023  1499  with Lcm0_iff' have "Lcm A \ 0" by simp  haftmann@58023  1500  }  haftmann@58023  1501  ultimately show "Lcm A = 0 \ 0 \ A" by blast  haftmann@58023  1502 qed  haftmann@58023  1503 haftmann@58023  1504 lemma Lcm_no_multiple:  haftmann@60430  1505  "(\m. m \ 0 \ (\a\A. \a dvd m)) \ Lcm A = 0"  haftmann@58023  1506 proof -  haftmann@60430  1507  assume "\m. m \ 0 \ (\a\A. \a dvd m)"  haftmann@60430  1508  hence "\(\l. l \ 0 \ (\a\A. a dvd l))" by blast  haftmann@58023  1509  then show "Lcm A = 0" by (simp only: Lcm_Lcm_eucl Lcm_eucl_def if_False)  haftmann@58023  1510 qed  haftmann@58023  1511 haftmann@58023  1512 lemma Lcm_insert [simp]:  haftmann@58023  1513  "Lcm (insert a A) = lcm a (Lcm A)"  haftmann@58023  1514 proof (rule lcmI)  haftmann@58023  1515  fix l assume "a dvd l" and "Lcm A dvd l"  haftmann@60430  1516  hence "\a\A. a dvd l" by (blast intro: dvd_trans dvd_Lcm)  haftmann@58023  1517  with a dvd l show "Lcm (insert a A) dvd l" by (force intro: Lcm_dvd)  haftmann@58023  1518 qed (auto intro: Lcm_dvd dvd_Lcm)  haftmann@58023  1519   haftmann@58023  1520 lemma Lcm_finite:  haftmann@58023  1521  assumes "finite A"  haftmann@58023  1522  shows "Lcm A = Finite_Set.fold lcm 1 A"  haftmann@58023  1523  by (induct rule: finite.induct[OF finite A])  haftmann@58023  1524  (simp_all add: comp_fun_idem.fold_insert_idem[OF comp_fun_idem_lcm])  haftmann@58023  1525 haftmann@60431  1526 lemma Lcm_set [code_unfold]:  haftmann@58023  1527  "Lcm (set xs) = fold lcm xs 1"  haftmann@58023  1528  using comp_fun_idem.fold_set_fold[OF comp_fun_idem_lcm] Lcm_finite by (simp add: ac_simps)  haftmann@58023  1529 haftmann@58023  1530 lemma Lcm_singleton [simp]:  haftmann@60438  1531  "Lcm {a} = a div normalization_factor a"  haftmann@58023  1532  by simp  haftmann@58023  1533 haftmann@58023  1534 lemma Lcm_2 [simp]:  haftmann@58023  1535  "Lcm {a,b} = lcm a b"  haftmann@58023  1536  by (simp only: Lcm_insert Lcm_empty lcm_1_right)  haftmann@58023  1537  (cases "b = 0", simp, rule lcm_div_unit2, simp)  haftmann@58023  1538 haftmann@58023  1539 lemma Lcm_coprime:  haftmann@58023  1540  assumes "finite A" and "A \ {}"  haftmann@58023  1541  assumes "\a b. a \ A \ b \ A \ a \ b \ gcd a b = 1"  haftmann@60438  1542  shows "Lcm A = \A div normalization_factor (\A)"  haftmann@58023  1543 using assms proof (induct rule: finite_ne_induct)  haftmann@58023  1544  case (insert a A)  haftmann@58023  1545  have "Lcm (insert a A) = lcm a (Lcm A)" by simp  haftmann@60438  1546  also from insert have "Lcm A = \A div normalization_factor (\A)" by blast  haftmann@58023  1547  also have "lcm a \ = lcm a (\A)" by (cases "\A = 0") (simp_all add: lcm_div_unit2)  haftmann@58023  1548  also from insert have "gcd a (\A) = 1" by (subst gcd.commute, intro setprod_coprime) auto  haftmann@60438  1549  with insert have "lcm a (\A) = \(insert a A) div normalization_factor (\(insert a A))"  haftmann@58023  1550  by (simp add: lcm_coprime)  haftmann@58023  1551  finally show ?case .  haftmann@58023  1552 qed simp  haftmann@58023  1553   haftmann@58023  1554 lemma Lcm_coprime':  haftmann@58023  1555  "card A \ 0 \ (\a b. a \ A \ b \ A \ a \ b \ gcd a b = 1)  haftmann@60438  1556  \ Lcm A = \A div normalization_factor (\A)"  haftmann@58023  1557  by (rule Lcm_coprime) (simp_all add: card_eq_0_iff)  haftmann@58023  1558 haftmann@58023  1559 lemma Gcd_Lcm:  haftmann@60430  1560  "Gcd A = Lcm {d. \a\A. d dvd a}"  haftmann@58023  1561  by (simp add: Gcd_Gcd_eucl Lcm_Lcm_eucl Gcd_eucl_def)  haftmann@58023  1562 haftmann@60430  1563 lemma Gcd_dvd [simp]: "a \ A \ Gcd A dvd a"  haftmann@60430  1564  and dvd_Gcd [simp]: "(\a\A. g' dvd a) \ g' dvd Gcd A"  haftmann@60438  1565  and normalization_factor_Gcd [simp]:  haftmann@60438  1566  "normalization_factor (Gcd A) = (if Gcd A = 0 then 0 else 1)"  haftmann@58023  1567 proof -  haftmann@60430  1568  fix a assume "a \ A"  haftmann@60430  1569  hence "Lcm {d. \a\A. d dvd a} dvd a" by (intro Lcm_dvd) blast  haftmann@60430  1570  then show "Gcd A dvd a" by (simp add: Gcd_Lcm)  haftmann@58023  1571 next  haftmann@60430  1572  fix g' assume "\a\A. g' dvd a"  haftmann@60430  1573  hence "g' dvd Lcm {d. \a\A. d dvd a}" by (intro dvd_Lcm) blast  haftmann@58023  1574  then show "g' dvd Gcd A" by (simp add: Gcd_Lcm)  haftmann@58023  1575 next  haftmann@60438  1576  show "normalization_factor (Gcd A) = (if Gcd A = 0 then 0 else 1)"  haftmann@59009  1577  by (simp add: Gcd_Lcm)  haftmann@58023  1578 qed  haftmann@58023  1579 haftmann@58023  1580 lemma GcdI:  haftmann@60430  1581  "(\a. a\A \ l dvd a) \ (\l'. (\a\A. l' dvd a) \ l' dvd l) \  haftmann@60438  1582  normalization_factor l = (if l = 0 then 0 else 1) \ l = Gcd A"  haftmann@58023  1583  by (intro normed_associated_imp_eq)  haftmann@58023  1584  (auto intro: Gcd_dvd dvd_Gcd simp: associated_def)  haftmann@58023  1585 haftmann@58023  1586 lemma Lcm_Gcd:  haftmann@60430  1587  "Lcm A = Gcd {m. \a\A. a dvd m}"  haftmann@58023  1588  by (rule LcmI[symmetric]) (auto intro: dvd_Gcd Gcd_dvd)  haftmann@58023  1589 haftmann@58023  1590 lemma Gcd_0_iff:  haftmann@58023  1591  "Gcd A = 0 \ A \ {0}"  haftmann@58023  1592  apply (rule iffI)  haftmann@58023  1593  apply (rule subsetI, drule Gcd_dvd, simp)  haftmann@58023  1594  apply (auto intro: GcdI[symmetric])  haftmann@58023  1595  done  haftmann@58023  1596 haftmann@58023  1597 lemma Gcd_empty [simp]:  haftmann@58023  1598  "Gcd {} = 0"  haftmann@58023  1599  by (simp add: Gcd_0_iff)  haftmann@58023  1600 haftmann@58023  1601 lemma Gcd_1:  haftmann@58023  1602  "1 \ A \ Gcd A = 1"  haftmann@58023  1603  by (intro GcdI[symmetric]) (auto intro: Gcd_dvd dvd_Gcd)  haftmann@58023  1604 haftmann@58023  1605 lemma Gcd_insert [simp]:  haftmann@58023  1606  "Gcd (insert a A) = gcd a (Gcd A)"  haftmann@58023  1607 proof (rule gcdI)  haftmann@58023  1608  fix l assume "l dvd a" and "l dvd Gcd A"  haftmann@60430  1609  hence "\a\A. l dvd a" by (blast intro: dvd_trans Gcd_dvd)  haftmann@58023  1610  with l dvd a show "l dvd Gcd (insert a A)" by (force intro: Gcd_dvd)  haftmann@59009  1611 qed auto  haftmann@58023  1612 haftmann@58023  1613 lemma Gcd_finite:  haftmann@58023  1614  assumes "finite A"  haftmann@58023  1615  shows "Gcd A = Finite_Set.fold gcd 0 A"  haftmann@58023  1616  by (induct rule: finite.induct[OF finite A])  haftmann@58023  1617  (simp_all add: comp_fun_idem.fold_insert_idem[OF comp_fun_idem_gcd])  haftmann@58023  1618 haftmann@60431  1619 lemma Gcd_set [code_unfold]:  haftmann@58023  1620  "Gcd (set xs) = fold gcd xs 0"  haftmann@58023  1621  using comp_fun_idem.fold_set_fold[OF comp_fun_idem_gcd] Gcd_finite by (simp add: ac_simps)  haftmann@58023  1622 haftmann@60438  1623 lemma Gcd_singleton [simp]: "Gcd {a} = a div normalization_factor a"  haftmann@58023  1624  by (simp add: gcd_0)  haftmann@58023  1625 haftmann@58023  1626 lemma Gcd_2 [simp]: "Gcd {a,b} = gcd a b"  haftmann@58023  1627  by (simp only: Gcd_insert Gcd_empty gcd_0) (cases "b = 0", simp, rule gcd_div_unit2, simp)  haftmann@58023  1628 haftmann@60439  1629 subclass semiring_gcd  haftmann@60439  1630  by unfold_locales (simp_all add: gcd_greatest_iff)  haftmann@60439  1631   haftmann@58023  1632 end  haftmann@58023  1633 haftmann@58023  1634 text {*  haftmann@58023  1635  A Euclidean ring is a Euclidean semiring with additive inverses. It provides a  haftmann@58023  1636  few more lemmas; in particular, Bezout's lemma holds for any Euclidean ring.  haftmann@58023  1637 *}  haftmann@58023  1638 haftmann@58023  1639 class euclidean_ring = euclidean_semiring + idom  haftmann@58023  1640 haftmann@58023  1641 class euclidean_ring_gcd = euclidean_semiring_gcd + idom  haftmann@58023  1642 begin  haftmann@58023  1643 haftmann@58023  1644 subclass euclidean_ring ..  haftmann@58023  1645 haftmann@60439  1646 subclass ring_gcd ..  haftmann@60439  1647 haftmann@58023  1648 lemma gcd_neg1 [simp]:  haftmann@60430  1649  "gcd (-a) b = gcd a b"  haftmann@59009  1650  by (rule sym, rule gcdI, simp_all add: gcd_greatest)  haftmann@58023  1651 haftmann@58023  1652 lemma gcd_neg2 [simp]:  haftmann@60430  1653  "gcd a (-b) = gcd a b"  haftmann@59009  1654  by (rule sym, rule gcdI, simp_all add: gcd_greatest)  haftmann@58023  1655 haftmann@58023  1656 lemma gcd_neg_numeral_1 [simp]:  haftmann@60430  1657  "gcd (- numeral n) a = gcd (numeral n) a"  haftmann@58023  1658  by (fact gcd_neg1)  haftmann@58023  1659 haftmann@58023  1660 lemma gcd_neg_numeral_2 [simp]:  haftmann@60430  1661  "gcd a (- numeral n) = gcd a (numeral n)"  haftmann@58023  1662  by (fact gcd_neg2)  haftmann@58023  1663 haftmann@58023  1664 lemma gcd_diff1: "gcd (m - n) n = gcd m n"  haftmann@58023  1665  by (subst diff_conv_add_uminus, subst gcd_neg2[symmetric], subst gcd_add1, simp)  haftmann@58023  1666 haftmann@58023  1667 lemma gcd_diff2: "gcd (n - m) n = gcd m n"  haftmann@58023  1668  by (subst gcd_neg1[symmetric], simp only: minus_diff_eq gcd_diff1)  haftmann@58023  1669 haftmann@58023  1670 lemma coprime_minus_one [simp]: "gcd (n - 1) n = 1"  haftmann@58023  1671 proof -  haftmann@58023  1672  have "gcd (n - 1) n = gcd n (n - 1)" by (fact gcd.commute)  haftmann@58023  1673  also have "\ = gcd ((n - 1) + 1) (n - 1)" by simp  haftmann@58023  1674  also have "\ = 1" by (rule coprime_plus_one)  haftmann@58023  1675  finally show ?thesis .  haftmann@58023  1676 qed  haftmann@58023  1677 haftmann@60430  1678 lemma lcm_neg1 [simp]: "lcm (-a) b = lcm a b"  haftmann@58023  1679  by (rule sym, rule lcmI, simp_all add: lcm_least lcm_zero)  haftmann@58023  1680 haftmann@60430  1681 lemma lcm_neg2 [simp]: "lcm a (-b) = lcm a b"  haftmann@58023  1682  by (rule sym, rule lcmI, simp_all add: lcm_least lcm_zero)  haftmann@58023  1683 haftmann@60430  1684 lemma lcm_neg_numeral_1 [simp]: "lcm (- numeral n) a = lcm (numeral n) a"  haftmann@58023  1685  by (fact lcm_neg1)  haftmann@58023  1686 haftmann@60430  1687 lemma lcm_neg_numeral_2 [simp]: "lcm a (- numeral n) = lcm a (numeral n)"  haftmann@58023  1688  by (fact lcm_neg2)  haftmann@58023  1689 haftmann@58023  1690 function euclid_ext :: "'a \ 'a \ 'a \ 'a \ 'a" where  haftmann@58023  1691  "euclid_ext a b =  haftmann@58023  1692  (if b = 0 then  haftmann@60438  1693  let c = 1 div normalization_factor a in (c, 0, a * c)  haftmann@58023  1694  else  haftmann@58023  1695  case euclid_ext b (a mod b) of  haftmann@58023  1696  (s,t,c) \ (t, s - t * (a div b), c))"  haftmann@58023  1697  by (pat_completeness, simp)  haftmann@58023  1698  termination by (relation "measure (euclidean_size \ snd)", simp_all)  haftmann@58023  1699 haftmann@58023  1700 declare euclid_ext.simps [simp del]  haftmann@58023  1701 haftmann@58023  1702 lemma euclid_ext_0:  haftmann@60438  1703  "euclid_ext a 0 = (1 div normalization_factor a, 0, a div normalization_factor a)"  haftmann@60433  1704  by (subst euclid_ext.simps) (simp add: Let_def)  haftmann@58023  1705 haftmann@58023  1706 lemma euclid_ext_non_0:  haftmann@58023  1707  "b \ 0 \ euclid_ext a b = (case euclid_ext b (a mod b) of  haftmann@58023  1708  (s,t,c) \ (t, s - t * (a div b), c))"  haftmann@60433  1709  by (subst euclid_ext.simps) simp  haftmann@58023  1710 haftmann@58023  1711 definition euclid_ext' :: "'a \ 'a \ 'a \ 'a"  haftmann@58023  1712 where  haftmann@58023  1713  "euclid_ext' a b = (case euclid_ext a b of (s, t, _) \ (s, t))"  haftmann@58023  1714 haftmann@58023  1715 lemma euclid_ext_gcd [simp]:  haftmann@58023  1716  "(case euclid_ext a b of (_,_,t) \ t) = gcd a b"  haftmann@58023  1717 proof (induct a b rule: euclid_ext.induct)  haftmann@58023  1718  case (1 a b)  haftmann@58023  1719  then show ?case  haftmann@58023  1720  proof (cases "b = 0")  haftmann@58023  1721  case True  haftmann@60433  1722  then show ?thesis by  haftmann@60433  1723  (simp add: euclid_ext_0 unit_div mult_ac unit_simps gcd_0)  haftmann@58023  1724  next  haftmann@58023  1725  case False with 1 show ?thesis  haftmann@58023  1726  by (simp add: euclid_ext_non_0 ac_simps split: prod.split prod.split_asm)  haftmann@58023  1727  qed  haftmann@58023  1728 qed  haftmann@58023  1729 haftmann@58023  1730 lemma euclid_ext_gcd' [simp]:  haftmann@58023  1731  "euclid_ext a b = (r, s, t) \ t = gcd a b"  haftmann@58023  1732  by (insert euclid_ext_gcd[of a b], drule (1) subst, simp)  haftmann@58023  1733 haftmann@58023  1734 lemma euclid_ext_correct:  haftmann@60430  1735  "case euclid_ext a b of (s,t,c) \ s*a + t*b = c"  haftmann@60430  1736 proof (induct a b rule: euclid_ext.induct)  haftmann@60430  1737  case (1 a b)  haftmann@58023  1738  show ?case  haftmann@60430  1739  proof (cases "b = 0")  haftmann@58023  1740  case True  haftmann@58023  1741  then show ?thesis by (simp add: euclid_ext_0 mult_ac)  haftmann@58023  1742  next  haftmann@58023  1743  case False  haftmann@60430  1744  obtain s t c where stc: "euclid_ext b (a mod b) = (s,t,c)"  haftmann@60430  1745  by (cases "euclid_ext b (a mod b)", blast)  haftmann@60430  1746  from 1 have "c = s * b + t * (a mod b)" by (simp add: stc False)  haftmann@60430  1747  also have "... = t*((a div b)*b + a mod b) + (s - t * (a div b))*b"  haftmann@58023  1748  by (simp add: algebra_simps)  haftmann@60430  1749  also have "(a div b)*b + a mod b = a" using mod_div_equality .  haftmann@58023  1750  finally show ?thesis  haftmann@58023  1751  by (subst euclid_ext.simps, simp add: False stc)  haftmann@58023  1752  qed  haftmann@58023  1753 qed  haftmann@58023  1754 haftmann@58023  1755 lemma euclid_ext'_correct:  haftmann@58023  1756  "fst (euclid_ext' a b) * a + snd (euclid_ext' a b) * b = gcd a b"  haftmann@58023  1757 proof-  haftmann@58023  1758  obtain s t c where "euclid_ext a b = (s,t,c)"  haftmann@58023  1759  by (cases "euclid_ext a b", blast)  haftmann@58023  1760  with euclid_ext_correct[of a b] euclid_ext_gcd[of a b]  haftmann@58023  1761  show ?thesis unfolding euclid_ext'_def by simp  haftmann@58023  1762 qed  haftmann@58023  1763 haftmann@60430  1764 lemma bezout: "\s t. s * a + t * b = gcd a b"  haftmann@58023  1765  using euclid_ext'_correct by blast  haftmann@58023  1766 haftmann@60438  1767 lemma euclid_ext'_0 [simp]: "euclid_ext' a 0 = (1 div normalization_factor a, 0)"  haftmann@58023  1768  by (simp add: bezw_def euclid_ext'_def euclid_ext_0)  haftmann@58023  1769 haftmann@60430  1770 lemma euclid_ext'_non_0: "b \ 0 \ euclid_ext' a b = (snd (euclid_ext' b (a mod b)),  haftmann@60430  1771  fst (euclid_ext' b (a mod b)) - snd (euclid_ext' b (a mod b)) * (a div b))"  haftmann@60430  1772  by (cases "euclid_ext b (a mod b)")  haftmann@58023  1773  (simp add: euclid_ext'_def euclid_ext_non_0)  haftmann@58023  1774   haftmann@58023  1775 end  haftmann@58023  1776 haftmann@58023  1777 instantiation nat :: euclidean_semiring  haftmann@58023  1778 begin  haftmann@58023  1779 haftmann@58023  1780 definition [simp]:  haftmann@58023  1781  "euclidean_size_nat = (id :: nat \ nat)"  haftmann@58023  1782 haftmann@58023  1783 definition [simp]:  haftmann@60438  1784  "normalization_factor_nat (n::nat) = (if n = 0 then 0 else 1 :: nat)"  haftmann@58023  1785 haftmann@58023  1786 instance proof  haftmann@59061  1787 qed simp_all  haftmann@58023  1788 haftmann@58023  1789 end  haftmann@58023  1790 haftmann@58023  1791 instantiation int :: euclidean_ring  haftmann@58023  1792 begin  haftmann@58023  1793 haftmann@58023  1794 definition [simp]:  haftmann@58023  1795  "euclidean_size_int = (nat \ abs :: int \ nat)"  haftmann@58023  1796 haftmann@58023  1797 definition [simp]:  haftmann@60438  1798  "normalization_factor_int = (sgn :: int \ int)"  haftmann@58023  1799 haftmann@58023  1800 instance proof  haftmann@58023  1801  case goal2 then show ?case by (auto simp add: abs_mult nat_mult_distrib)  haftmann@58023  1802 next  haftmann@59061  1803  case goal3 then show ?case by (simp add: zsgn_def)  haftmann@58023  1804 next  haftmann@59061  1805  case goal5 then show ?case by (auto simp: zsgn_def)  haftmann@58023  1806 next  haftmann@59061  1807  case goal6 then show ?case by (auto split: abs_split simp: zsgn_def)  haftmann@58023  1808 qed (auto simp: sgn_times split: abs_split)  haftmann@58023  1809 haftmann@58023  1810 end  haftmann@58023  1811 haftmann@58023  1812 end