src/HOL/Number_Theory/Euclidean_Algorithm.thy
 author wenzelm Thu Jun 25 23:33:47 2015 +0200 (2015-06-25) changeset 60580 7e741e22d7fc parent 60526 fad653acf58f child 60582 d694f217ee41 permissions -rw-r--r--
tuned proofs;
 haftmann@58023  1 (* Author: Manuel Eberl *)  haftmann@58023  2 wenzelm@60526  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@60433  8   wenzelm@60526  9 text \  haftmann@58023  10  A Euclidean semiring is a semiring upon which the Euclidean algorithm can be  haftmann@58023  11  implemented. It must provide:  haftmann@58023  12  \begin{itemize}  haftmann@58023  13  \item division with remainder  haftmann@58023  14  \item a size function such that @{term "size (a mod b) < size b"}  haftmann@58023  15  for any @{term "b \ 0"}  haftmann@60438  16  \item a normalization factor such that two associated numbers are equal iff  haftmann@60438  17  they are the same when divd by their normalization factors.  haftmann@58023  18  \end{itemize}  haftmann@58023  19  The existence of these functions makes it possible to derive gcd and lcm functions  haftmann@58023  20  for any Euclidean semiring.  wenzelm@60526  21 \  haftmann@58023  22 class euclidean_semiring = semiring_div +  haftmann@58023  23  fixes euclidean_size :: "'a \ nat"  haftmann@60438  24  fixes normalization_factor :: "'a \ 'a"  haftmann@58023  25  assumes mod_size_less [simp]:  haftmann@58023  26  "b \ 0 \ euclidean_size (a mod b) < euclidean_size b"  haftmann@58023  27  assumes size_mult_mono:  haftmann@58023  28  "b \ 0 \ euclidean_size (a * b) \ euclidean_size a"  haftmann@60438  29  assumes normalization_factor_is_unit [intro,simp]:  haftmann@60438  30  "a \ 0 \ is_unit (normalization_factor a)"  haftmann@60438  31  assumes normalization_factor_mult: "normalization_factor (a * b) =  haftmann@60438  32  normalization_factor a * normalization_factor b"  haftmann@60438  33  assumes normalization_factor_unit: "is_unit a \ normalization_factor a = a"  haftmann@60438  34  assumes normalization_factor_0 [simp]: "normalization_factor 0 = 0"  haftmann@58023  35 begin  haftmann@58023  36 haftmann@60438  37 lemma normalization_factor_dvd [simp]:  haftmann@60438  38  "a \ 0 \ normalization_factor a dvd b"  haftmann@58023  39  by (rule unit_imp_dvd, simp)  haftmann@58023  40   haftmann@60438  41 lemma normalization_factor_1 [simp]:  haftmann@60438  42  "normalization_factor 1 = 1"  haftmann@60438  43  by (simp add: normalization_factor_unit)  haftmann@58023  44 haftmann@60438  45 lemma normalization_factor_0_iff [simp]:  haftmann@60438  46  "normalization_factor a = 0 \ a = 0"  haftmann@58023  47 proof  haftmann@60438  48  assume "normalization_factor a = 0"  haftmann@60438  49  hence "\ is_unit (normalization_factor a)"  haftmann@60433  50  by simp  haftmann@60433  51  then show "a = 0" by auto  haftmann@60433  52 qed simp  haftmann@58023  53 haftmann@60438  54 lemma normalization_factor_pow:  haftmann@60438  55  "normalization_factor (a ^ n) = normalization_factor a ^ n"  haftmann@60438  56  by (induct n) (simp_all add: normalization_factor_mult power_Suc2)  haftmann@58023  57 haftmann@60438  58 lemma normalization_correct [simp]:  haftmann@60438  59  "normalization_factor (a div normalization_factor a) = (if a = 0 then 0 else 1)"  haftmann@60430  60 proof (cases "a = 0", simp)  haftmann@60430  61  assume "a \ 0"  haftmann@60438  62  let ?nf = "normalization_factor"  wenzelm@60526  63  from normalization_factor_is_unit[OF \a \ 0\] have "?nf a \ 0"  haftmann@60433  64  by auto  haftmann@60430  65  have "?nf (a div ?nf a) * ?nf (?nf a) = ?nf (a div ?nf a * ?nf a)"  haftmann@60438  66  by (simp add: normalization_factor_mult)  wenzelm@60526  67  also have "a div ?nf a * ?nf a = a" using \a \ 0\  haftmann@59009  68  by simp  wenzelm@60526  69  also have "?nf (?nf a) = ?nf a" using \a \ 0\  haftmann@60438  70  normalization_factor_is_unit normalization_factor_unit by simp  haftmann@60438  71  finally have "normalization_factor (a div normalization_factor a) = 1"  wenzelm@60526  72  using \?nf a \ 0\ by (metis div_mult_self2_is_id div_self)  wenzelm@60526  73  with \a \ 0\ show ?thesis by simp  haftmann@58023  74 qed  haftmann@58023  75 haftmann@60438  76 lemma normalization_0_iff [simp]:  haftmann@60438  77  "a div normalization_factor a = 0 \ a = 0"  haftmann@60430  78  by (cases "a = 0", simp, subst unit_eq_div1, blast, simp)  haftmann@58023  79 haftmann@60438  80 lemma mult_div_normalization [simp]:  haftmann@60438  81  "b * (1 div normalization_factor a) = b div normalization_factor a"  haftmann@60433  82  by (cases "a = 0") simp_all  haftmann@60433  83 haftmann@58023  84 lemma associated_iff_normed_eq:  haftmann@60438  85  "associated a b \ a div normalization_factor a = b div normalization_factor b"  haftmann@60438  86 proof (cases "b = 0", simp, cases "a = 0", metis associated_0(1) normalization_0_iff, rule iffI)  haftmann@60438  87  let ?nf = normalization_factor  haftmann@58023  88  assume "a \ 0" "b \ 0" "a div ?nf a = b div ?nf b"  haftmann@58023  89  hence "a = b * (?nf a div ?nf b)"  haftmann@58023  90  apply (subst (asm) unit_eq_div1, blast, subst (asm) unit_div_commute, blast)  haftmann@58023  91  apply (subst div_mult_swap, simp, simp)  haftmann@58023  92  done  wenzelm@60526  93  with \a \ 0\ \b \ 0\ have "\c. is_unit c \ a = c * b"  haftmann@58023  94  by (intro exI[of _ "?nf a div ?nf b"], force simp: mult_ac)  haftmann@60436  95  then obtain c where "is_unit c" and "a = c * b" by blast  haftmann@60436  96  then show "associated a b" by (rule is_unit_associatedI)  haftmann@58023  97 next  haftmann@60438  98  let ?nf = normalization_factor  haftmann@58023  99  assume "a \ 0" "b \ 0" "associated a b"  haftmann@60436  100  then obtain c where "is_unit c" and "a = c * b" by (blast elim: associated_is_unitE)  haftmann@58023  101  then show "a div ?nf a = b div ?nf b"  wenzelm@60526  102  apply (simp only: \a = c * b\ normalization_factor_mult normalization_factor_unit)  haftmann@58023  103  apply (rule div_mult_mult1, force)  haftmann@58023  104  done  haftmann@58023  105  qed  haftmann@58023  106 haftmann@58023  107 lemma normed_associated_imp_eq:  haftmann@60438  108  "associated a b \ normalization_factor a \ {0, 1} \ normalization_factor b \ {0, 1} \ a = b"  haftmann@58023  109  by (simp add: associated_iff_normed_eq, elim disjE, simp_all)  haftmann@58023  110   haftmann@60438  111 lemmas normalization_factor_dvd_iff [simp] =  haftmann@60438  112  unit_dvd_iff [OF normalization_factor_is_unit]  haftmann@58023  113 haftmann@58023  114 lemma euclidean_division:  haftmann@58023  115  fixes a :: 'a and b :: 'a  haftmann@58023  116  assumes "b \ 0"  haftmann@58023  117  obtains s and t where "a = s * b + t"  haftmann@58023  118  and "euclidean_size t < euclidean_size b"  haftmann@58023  119 proof -  haftmann@58023  120  from div_mod_equality[of a b 0]  haftmann@58023  121  have "a = a div b * b + a mod b" by simp  haftmann@58023  122  with that and assms show ?thesis by force  haftmann@58023  123 qed  haftmann@58023  124 haftmann@58023  125 lemma dvd_euclidean_size_eq_imp_dvd:  haftmann@58023  126  assumes "a \ 0" and b_dvd_a: "b dvd a" and size_eq: "euclidean_size a = euclidean_size b"  haftmann@58023  127  shows "a dvd b"  haftmann@58023  128 proof (subst dvd_eq_mod_eq_0, rule ccontr)  haftmann@58023  129  assume "b mod a \ 0"  haftmann@58023  130  from b_dvd_a have b_dvd_mod: "b dvd b mod a" by (simp add: dvd_mod_iff)  haftmann@58023  131  from b_dvd_mod obtain c where "b mod a = b * c" unfolding dvd_def by blast  wenzelm@60526  132  with \b mod a \ 0\ have "c \ 0" by auto  wenzelm@60526  133  with \b mod a = b * c\ have "euclidean_size (b mod a) \ euclidean_size b"  haftmann@58023  134  using size_mult_mono by force  wenzelm@60526  135  moreover from \a \ 0\ have "euclidean_size (b mod a) < euclidean_size a"  haftmann@58023  136  using mod_size_less by blast  haftmann@58023  137  ultimately show False using size_eq by simp  haftmann@58023  138 qed  haftmann@58023  139 haftmann@58023  140 function gcd_eucl :: "'a \ 'a \ 'a"  haftmann@58023  141 where  haftmann@60438  142  "gcd_eucl a b = (if b = 0 then a div normalization_factor a else gcd_eucl b (a mod b))"  haftmann@58023  143  by (pat_completeness, simp)  haftmann@58023  144 termination by (relation "measure (euclidean_size \ snd)", simp_all)  haftmann@58023  145 haftmann@58023  146 declare gcd_eucl.simps [simp del]  haftmann@58023  147 haftmann@58023  148 lemma gcd_induct: "\\b. P b 0; \a b. 0 \ b \ P b (a mod b) \ P a b\ \ P a b"  haftmann@58023  149 proof (induct a b rule: gcd_eucl.induct)  haftmann@58023  150  case ("1" m n)  haftmann@58023  151  then show ?case by (cases "n = 0") auto  haftmann@58023  152 qed  haftmann@58023  153 haftmann@58023  154 definition lcm_eucl :: "'a \ 'a \ 'a"  haftmann@58023  155 where  haftmann@60438  156  "lcm_eucl a b = a * b div (gcd_eucl a b * normalization_factor (a * b))"  haftmann@58023  157 haftmann@58023  158  (* Somewhat complicated definition of Lcm that has the advantage of working  haftmann@58023  159  for infinite sets as well *)  haftmann@58023  160 haftmann@58023  161 definition Lcm_eucl :: "'a set \ 'a"  haftmann@58023  162 where  haftmann@60430  163  "Lcm_eucl A = (if \l. l \ 0 \ (\a\A. a dvd l) then  haftmann@60430  164  let l = SOME l. l \ 0 \ (\a\A. a dvd l) \ euclidean_size l =  haftmann@60430  165  (LEAST n. \l. l \ 0 \ (\a\A. a dvd l) \ euclidean_size l = n)  haftmann@60438  166  in l div normalization_factor l  haftmann@58023  167  else 0)"  haftmann@58023  168 haftmann@58023  169 definition Gcd_eucl :: "'a set \ 'a"  haftmann@58023  170 where  haftmann@58023  171  "Gcd_eucl A = Lcm_eucl {d. \a\A. d dvd a}"  haftmann@58023  172 haftmann@58023  173 end  haftmann@58023  174 haftmann@58023  175 class euclidean_semiring_gcd = euclidean_semiring + gcd + Gcd +  haftmann@58023  176  assumes gcd_gcd_eucl: "gcd = gcd_eucl" and lcm_lcm_eucl: "lcm = lcm_eucl"  haftmann@58023  177  assumes Gcd_Gcd_eucl: "Gcd = Gcd_eucl" and Lcm_Lcm_eucl: "Lcm = Lcm_eucl"  haftmann@58023  178 begin  haftmann@58023  179 haftmann@58023  180 lemma gcd_red:  haftmann@60430  181  "gcd a b = gcd b (a mod b)"  haftmann@58023  182  by (metis gcd_eucl.simps mod_0 mod_by_0 gcd_gcd_eucl)  haftmann@58023  183 haftmann@58023  184 lemma gcd_non_0:  haftmann@60430  185  "b \ 0 \ gcd a b = gcd b (a mod b)"  haftmann@58023  186  by (rule gcd_red)  haftmann@58023  187 haftmann@58023  188 lemma gcd_0_left:  haftmann@60438  189  "gcd 0 a = a div normalization_factor a"  haftmann@58023  190  by (simp only: gcd_gcd_eucl, subst gcd_eucl.simps, subst gcd_eucl.simps, simp add: Let_def)  haftmann@58023  191 haftmann@58023  192 lemma gcd_0:  haftmann@60438  193  "gcd a 0 = a div normalization_factor a"  haftmann@58023  194  by (simp only: gcd_gcd_eucl, subst gcd_eucl.simps, simp add: Let_def)  haftmann@58023  195 haftmann@60430  196 lemma gcd_dvd1 [iff]: "gcd a b dvd a"  haftmann@60430  197  and gcd_dvd2 [iff]: "gcd a b dvd b"  haftmann@60430  198 proof (induct a b rule: gcd_eucl.induct)  haftmann@60430  199  fix a b :: 'a  haftmann@60430  200  assume IH1: "b \ 0 \ gcd b (a mod b) dvd b"  haftmann@60430  201  assume IH2: "b \ 0 \ gcd b (a mod b) dvd (a mod b)"  haftmann@58023  202   haftmann@60430  203  have "gcd a b dvd a \ gcd a b dvd b"  haftmann@60430  204  proof (cases "b = 0")  haftmann@58023  205  case True  haftmann@60430  206  then show ?thesis by (cases "a = 0", simp_all add: gcd_0)  haftmann@58023  207  next  haftmann@58023  208  case False  haftmann@58023  209  with IH1 and IH2 show ?thesis by (simp add: gcd_non_0 dvd_mod_iff)  haftmann@58023  210  qed  haftmann@60430  211  then show "gcd a b dvd a" "gcd a b dvd b" by simp_all  haftmann@58023  212 qed  haftmann@58023  213 haftmann@58023  214 lemma dvd_gcd_D1: "k dvd gcd m n \ k dvd m"  haftmann@58023  215  by (rule dvd_trans, assumption, rule gcd_dvd1)  haftmann@58023  216 haftmann@58023  217 lemma dvd_gcd_D2: "k dvd gcd m n \ k dvd n"  haftmann@58023  218  by (rule dvd_trans, assumption, rule gcd_dvd2)  haftmann@58023  219 haftmann@58023  220 lemma gcd_greatest:  haftmann@60430  221  fixes k a b :: 'a  haftmann@60430  222  shows "k dvd a \ k dvd b \ k dvd gcd a b"  haftmann@60430  223 proof (induct a b rule: gcd_eucl.induct)  haftmann@60430  224  case (1 a b)  haftmann@58023  225  show ?case  haftmann@60430  226  proof (cases "b = 0")  haftmann@60430  227  assume "b = 0"  haftmann@60430  228  with 1 show ?thesis by (cases "a = 0", simp_all add: gcd_0)  haftmann@58023  229  next  haftmann@60430  230  assume "b \ 0"  haftmann@58023  231  with 1 show ?thesis by (simp add: gcd_non_0 dvd_mod_iff)  haftmann@58023  232  qed  haftmann@58023  233 qed  haftmann@58023  234 haftmann@58023  235 lemma dvd_gcd_iff:  haftmann@60430  236  "k dvd gcd a b \ k dvd a \ k dvd b"  haftmann@58023  237  by (blast intro!: gcd_greatest intro: dvd_trans)  haftmann@58023  238 haftmann@58023  239 lemmas gcd_greatest_iff = dvd_gcd_iff  haftmann@58023  240 haftmann@58023  241 lemma gcd_zero [simp]:  haftmann@60430  242  "gcd a b = 0 \ a = 0 \ b = 0"  haftmann@58023  243  by (metis dvd_0_left dvd_refl gcd_dvd1 gcd_dvd2 gcd_greatest)+  haftmann@58023  244 haftmann@60438  245 lemma normalization_factor_gcd [simp]:  haftmann@60438  246  "normalization_factor (gcd a b) = (if a = 0 \ b = 0 then 0 else 1)" (is "?f a b = ?g a b")  haftmann@60430  247 proof (induct a b rule: gcd_eucl.induct)  haftmann@60430  248  fix a b :: 'a  haftmann@60430  249  assume IH: "b \ 0 \ ?f b (a mod b) = ?g b (a mod b)"  haftmann@60430  250  then show "?f a b = ?g a b" by (cases "b = 0", auto simp: gcd_non_0 gcd_0)  haftmann@58023  251 qed  haftmann@58023  252 haftmann@58023  253 lemma gcdI:  haftmann@60430  254  "k dvd a \ k dvd b \ (\l. l dvd a \ l dvd b \ l dvd k)  haftmann@60438  255  \ normalization_factor k = (if k = 0 then 0 else 1) \ k = gcd a b"  haftmann@58023  256  by (intro normed_associated_imp_eq) (auto simp: associated_def intro: gcd_greatest)  haftmann@58023  257 haftmann@58023  258 sublocale gcd!: abel_semigroup gcd  haftmann@58023  259 proof  haftmann@60430  260  fix a b c  haftmann@60430  261  show "gcd (gcd a b) c = gcd a (gcd b c)"  haftmann@58023  262  proof (rule gcdI)  haftmann@60430  263  have "gcd (gcd a b) c dvd gcd a b" "gcd a b dvd a" by simp_all  haftmann@60430  264  then show "gcd (gcd a b) c dvd a" by (rule dvd_trans)  haftmann@60430  265  have "gcd (gcd a b) c dvd gcd a b" "gcd a b dvd b" by simp_all  haftmann@60430  266  hence "gcd (gcd a b) c dvd b" by (rule dvd_trans)  haftmann@60430  267  moreover have "gcd (gcd a b) c dvd c" by simp  haftmann@60430  268  ultimately show "gcd (gcd a b) c dvd gcd b c"  haftmann@58023  269  by (rule gcd_greatest)  haftmann@60438  270  show "normalization_factor (gcd (gcd a b) c) = (if gcd (gcd a b) c = 0 then 0 else 1)"  haftmann@58023  271  by auto  haftmann@60430  272  fix l assume "l dvd a" and "l dvd gcd b c"  haftmann@58023  273  with dvd_trans[OF _ gcd_dvd1] and dvd_trans[OF _ gcd_dvd2]  haftmann@60430  274  have "l dvd b" and "l dvd c" by blast+  wenzelm@60526  275  with \l dvd a\ show "l dvd gcd (gcd a b) c"  haftmann@58023  276  by (intro gcd_greatest)  haftmann@58023  277  qed  haftmann@58023  278 next  haftmann@60430  279  fix a b  haftmann@60430  280  show "gcd a b = gcd b a"  haftmann@58023  281  by (rule gcdI) (simp_all add: gcd_greatest)  haftmann@58023  282 qed  haftmann@58023  283 haftmann@58023  284 lemma gcd_unique: "d dvd a \ d dvd b \  haftmann@60438  285  normalization_factor d = (if d = 0 then 0 else 1) \  haftmann@58023  286  (\e. e dvd a \ e dvd b \ e dvd d) \ d = gcd a b"  haftmann@58023  287  by (rule, auto intro: gcdI simp: gcd_greatest)  haftmann@58023  288 haftmann@58023  289 lemma gcd_dvd_prod: "gcd a b dvd k * b"  haftmann@58023  290  using mult_dvd_mono [of 1] by auto  haftmann@58023  291 haftmann@60430  292 lemma gcd_1_left [simp]: "gcd 1 a = 1"  haftmann@58023  293  by (rule sym, rule gcdI, simp_all)  haftmann@58023  294 haftmann@60430  295 lemma gcd_1 [simp]: "gcd a 1 = 1"  haftmann@58023  296  by (rule sym, rule gcdI, simp_all)  haftmann@58023  297 haftmann@58023  298 lemma gcd_proj2_if_dvd:  haftmann@60438  299  "b dvd a \ gcd a b = b div normalization_factor b"  haftmann@60430  300  by (cases "b = 0", simp_all add: dvd_eq_mod_eq_0 gcd_non_0 gcd_0)  haftmann@58023  301 haftmann@58023  302 lemma gcd_proj1_if_dvd:  haftmann@60438  303  "a dvd b \ gcd a b = a div normalization_factor a"  haftmann@58023  304  by (subst gcd.commute, simp add: gcd_proj2_if_dvd)  haftmann@58023  305 haftmann@60438  306 lemma gcd_proj1_iff: "gcd m n = m div normalization_factor m \ m dvd n"  haftmann@58023  307 proof  haftmann@60438  308  assume A: "gcd m n = m div normalization_factor m"  haftmann@58023  309  show "m dvd n"  haftmann@58023  310  proof (cases "m = 0")  haftmann@58023  311  assume [simp]: "m \ 0"  haftmann@60438  312  from A have B: "m = gcd m n * normalization_factor m"  haftmann@58023  313  by (simp add: unit_eq_div2)  haftmann@58023  314  show ?thesis by (subst B, simp add: mult_unit_dvd_iff)  haftmann@58023  315  qed (insert A, simp)  haftmann@58023  316 next  haftmann@58023  317  assume "m dvd n"  haftmann@60438  318  then show "gcd m n = m div normalization_factor m" by (rule gcd_proj1_if_dvd)  haftmann@58023  319 qed  haftmann@58023  320   haftmann@60438  321 lemma gcd_proj2_iff: "gcd m n = n div normalization_factor n \ n dvd m"  haftmann@58023  322  by (subst gcd.commute, simp add: gcd_proj1_iff)  haftmann@58023  323 haftmann@58023  324 lemma gcd_mod1 [simp]:  haftmann@60430  325  "gcd (a mod b) b = gcd a b"  haftmann@58023  326  by (rule gcdI, metis dvd_mod_iff gcd_dvd1 gcd_dvd2, simp_all add: gcd_greatest dvd_mod_iff)  haftmann@58023  327 haftmann@58023  328 lemma gcd_mod2 [simp]:  haftmann@60430  329  "gcd a (b mod a) = gcd a b"  haftmann@58023  330  by (rule gcdI, simp, metis dvd_mod_iff gcd_dvd1 gcd_dvd2, simp_all add: gcd_greatest dvd_mod_iff)  haftmann@58023  331   haftmann@60438  332 lemma normalization_factor_dvd' [simp]:  haftmann@60438  333  "normalization_factor a dvd a"  haftmann@60430  334  by (cases "a = 0", simp_all)  haftmann@58023  335 haftmann@58023  336 lemma gcd_mult_distrib':  haftmann@60438  337  "k div normalization_factor k * gcd a b = gcd (k*a) (k*b)"  haftmann@60430  338 proof (induct a b rule: gcd_eucl.induct)  haftmann@60430  339  case (1 a b)  haftmann@58023  340  show ?case  haftmann@60430  341  proof (cases "b = 0")  haftmann@58023  342  case True  haftmann@60438  343  then show ?thesis by (simp add: normalization_factor_mult gcd_0 algebra_simps div_mult_div_if_dvd)  haftmann@58023  344  next  haftmann@58023  345  case False  haftmann@60438  346  hence "k div normalization_factor k * gcd a b = gcd (k * b) (k * (a mod b))"  haftmann@58023  347  using 1 by (subst gcd_red, simp)  haftmann@60430  348  also have "... = gcd (k * a) (k * b)"  haftmann@58023  349  by (simp add: mult_mod_right gcd.commute)  haftmann@58023  350  finally show ?thesis .  haftmann@58023  351  qed  haftmann@58023  352 qed  haftmann@58023  353 haftmann@58023  354 lemma gcd_mult_distrib:  haftmann@60438  355  "k * gcd a b = gcd (k*a) (k*b) * normalization_factor k"  haftmann@58023  356 proof-  haftmann@60438  357  let ?nf = "normalization_factor"  haftmann@58023  358  from gcd_mult_distrib'  haftmann@60430  359  have "gcd (k*a) (k*b) = k div ?nf k * gcd a b" ..  haftmann@60430  360  also have "... = k * gcd a b div ?nf k"  haftmann@60438  361  by (metis dvd_div_mult dvd_eq_mod_eq_0 mod_0 normalization_factor_dvd)  haftmann@58023  362  finally show ?thesis  haftmann@59009  363  by simp  haftmann@58023  364 qed  haftmann@58023  365 haftmann@58023  366 lemma euclidean_size_gcd_le1 [simp]:  haftmann@58023  367  assumes "a \ 0"  haftmann@58023  368  shows "euclidean_size (gcd a b) \ euclidean_size a"  haftmann@58023  369 proof -  haftmann@58023  370  have "gcd a b dvd a" by (rule gcd_dvd1)  haftmann@58023  371  then obtain c where A: "a = gcd a b * c" unfolding dvd_def by blast  wenzelm@60526  372  with \a \ 0\ show ?thesis by (subst (2) A, intro size_mult_mono) auto  haftmann@58023  373 qed  haftmann@58023  374 haftmann@58023  375 lemma euclidean_size_gcd_le2 [simp]:  haftmann@58023  376  "b \ 0 \ euclidean_size (gcd a b) \ euclidean_size b"  haftmann@58023  377  by (subst gcd.commute, rule euclidean_size_gcd_le1)  haftmann@58023  378 haftmann@58023  379 lemma euclidean_size_gcd_less1:  haftmann@58023  380  assumes "a \ 0" and "\a dvd b"  haftmann@58023  381  shows "euclidean_size (gcd a b) < euclidean_size a"  haftmann@58023  382 proof (rule ccontr)  haftmann@58023  383  assume "\euclidean_size (gcd a b) < euclidean_size a"  wenzelm@60526  384  with \a \ 0\ have "euclidean_size (gcd a b) = euclidean_size a"  haftmann@58023  385  by (intro le_antisym, simp_all)  haftmann@58023  386  with assms have "a dvd gcd a b" by (auto intro: dvd_euclidean_size_eq_imp_dvd)  haftmann@58023  387  hence "a dvd b" using dvd_gcd_D2 by blast  wenzelm@60526  388  with \\a dvd b\ show False by contradiction  haftmann@58023  389 qed  haftmann@58023  390 haftmann@58023  391 lemma euclidean_size_gcd_less2:  haftmann@58023  392  assumes "b \ 0" and "\b dvd a"  haftmann@58023  393  shows "euclidean_size (gcd a b) < euclidean_size b"  haftmann@58023  394  using assms by (subst gcd.commute, rule euclidean_size_gcd_less1)  haftmann@58023  395 haftmann@60430  396 lemma gcd_mult_unit1: "is_unit a \ gcd (b * a) c = gcd b c"  haftmann@58023  397  apply (rule gcdI)  haftmann@58023  398  apply (rule dvd_trans, rule gcd_dvd1, simp add: unit_simps)  haftmann@58023  399  apply (rule gcd_dvd2)  haftmann@58023  400  apply (rule gcd_greatest, simp add: unit_simps, assumption)  haftmann@60438  401  apply (subst normalization_factor_gcd, simp add: gcd_0)  haftmann@58023  402  done  haftmann@58023  403 haftmann@60430  404 lemma gcd_mult_unit2: "is_unit a \ gcd b (c * a) = gcd b c"  haftmann@58023  405  by (subst gcd.commute, subst gcd_mult_unit1, assumption, rule gcd.commute)  haftmann@58023  406 haftmann@60430  407 lemma gcd_div_unit1: "is_unit a \ gcd (b div a) c = gcd b c"  haftmann@60433  408  by (erule is_unitE [of _ b]) (simp add: gcd_mult_unit1)  haftmann@58023  409 haftmann@60430  410 lemma gcd_div_unit2: "is_unit a \ gcd b (c div a) = gcd b c"  haftmann@60433  411  by (erule is_unitE [of _ c]) (simp add: gcd_mult_unit2)  haftmann@58023  412 haftmann@60438  413 lemma gcd_idem: "gcd a a = a div normalization_factor a"  haftmann@60430  414  by (cases "a = 0") (simp add: gcd_0_left, rule sym, rule gcdI, simp_all)  haftmann@58023  415 haftmann@60430  416 lemma gcd_right_idem: "gcd (gcd a b) b = gcd a b"  haftmann@58023  417  apply (rule gcdI)  haftmann@58023  418  apply (simp add: ac_simps)  haftmann@58023  419  apply (rule gcd_dvd2)  haftmann@58023  420  apply (rule gcd_greatest, erule (1) gcd_greatest, assumption)  haftmann@59009  421  apply simp  haftmann@58023  422  done  haftmann@58023  423 haftmann@60430  424 lemma gcd_left_idem: "gcd a (gcd a b) = gcd a b"  haftmann@58023  425  apply (rule gcdI)  haftmann@58023  426  apply simp  haftmann@58023  427  apply (rule dvd_trans, rule gcd_dvd2, rule gcd_dvd2)  haftmann@58023  428  apply (rule gcd_greatest, assumption, erule gcd_greatest, assumption)  haftmann@59009  429  apply simp  haftmann@58023  430  done  haftmann@58023  431 haftmann@58023  432 lemma comp_fun_idem_gcd: "comp_fun_idem gcd"  haftmann@58023  433 proof  haftmann@58023  434  fix a b show "gcd a \ gcd b = gcd b \ gcd a"  haftmann@58023  435  by (simp add: fun_eq_iff ac_simps)  haftmann@58023  436 next  haftmann@58023  437  fix a show "gcd a \ gcd a = gcd a"  haftmann@58023  438  by (simp add: fun_eq_iff gcd_left_idem)  haftmann@58023  439 qed  haftmann@58023  440 haftmann@58023  441 lemma coprime_dvd_mult:  haftmann@60430  442  assumes "gcd c b = 1" and "c dvd a * b"  haftmann@60430  443  shows "c dvd a"  haftmann@58023  444 proof -  haftmann@60438  445  let ?nf = "normalization_factor"  haftmann@60430  446  from assms gcd_mult_distrib [of a c b]  haftmann@60430  447  have A: "a = gcd (a * c) (a * b) * ?nf a" by simp  wenzelm@60526  448  from \c dvd a * b\ show ?thesis by (subst A, simp_all add: gcd_greatest)  haftmann@58023  449 qed  haftmann@58023  450 haftmann@58023  451 lemma coprime_dvd_mult_iff:  haftmann@60430  452  "gcd c b = 1 \ (c dvd a * b) = (c dvd a)"  haftmann@58023  453  by (rule, rule coprime_dvd_mult, simp_all)  haftmann@58023  454 haftmann@58023  455 lemma gcd_dvd_antisym:  haftmann@58023  456  "gcd a b dvd gcd c d \ gcd c d dvd gcd a b \ gcd a b = gcd c d"  haftmann@58023  457 proof (rule gcdI)  haftmann@58023  458  assume A: "gcd a b dvd gcd c d" and B: "gcd c d dvd gcd a b"  haftmann@58023  459  have "gcd c d dvd c" by simp  haftmann@58023  460  with A show "gcd a b dvd c" by (rule dvd_trans)  haftmann@58023  461  have "gcd c d dvd d" by simp  haftmann@58023  462  with A show "gcd a b dvd d" by (rule dvd_trans)  haftmann@60438  463  show "normalization_factor (gcd a b) = (if gcd a b = 0 then 0 else 1)"  haftmann@59009  464  by simp  haftmann@58023  465  fix l assume "l dvd c" and "l dvd d"  haftmann@58023  466  hence "l dvd gcd c d" by (rule gcd_greatest)  haftmann@58023  467  from this and B show "l dvd gcd a b" by (rule dvd_trans)  haftmann@58023  468 qed  haftmann@58023  469 haftmann@58023  470 lemma gcd_mult_cancel:  haftmann@58023  471  assumes "gcd k n = 1"  haftmann@58023  472  shows "gcd (k * m) n = gcd m n"  haftmann@58023  473 proof (rule gcd_dvd_antisym)  haftmann@58023  474  have "gcd (gcd (k * m) n) k = gcd (gcd k n) (k * m)" by (simp add: ac_simps)  wenzelm@60526  475  also note \gcd k n = 1\  haftmann@58023  476  finally have "gcd (gcd (k * m) n) k = 1" by simp  haftmann@58023  477  hence "gcd (k * m) n dvd m" by (rule coprime_dvd_mult, simp add: ac_simps)  haftmann@58023  478  moreover have "gcd (k * m) n dvd n" by simp  haftmann@58023  479  ultimately show "gcd (k * m) n dvd gcd m n" by (rule gcd_greatest)  haftmann@58023  480  have "gcd m n dvd (k * m)" and "gcd m n dvd n" by simp_all  haftmann@58023  481  then show "gcd m n dvd gcd (k * m) n" by (rule gcd_greatest)  haftmann@58023  482 qed  haftmann@58023  483 haftmann@58023  484 lemma coprime_crossproduct:  haftmann@58023  485  assumes [simp]: "gcd a d = 1" "gcd b c = 1"  haftmann@58023  486  shows "associated (a * c) (b * d) \ associated a b \ associated c d" (is "?lhs \ ?rhs")  haftmann@58023  487 proof  haftmann@58023  488  assume ?rhs then show ?lhs unfolding associated_def by (fast intro: mult_dvd_mono)  haftmann@58023  489 next  haftmann@58023  490  assume ?lhs  wenzelm@60526  491  from \?lhs\ have "a dvd b * d" unfolding associated_def by (metis dvd_mult_left)  haftmann@58023  492  hence "a dvd b" by (simp add: coprime_dvd_mult_iff)  wenzelm@60526  493  moreover from \?lhs\ have "b dvd a * c" unfolding associated_def by (metis dvd_mult_left)  haftmann@58023  494  hence "b dvd a" by (simp add: coprime_dvd_mult_iff)  wenzelm@60526  495  moreover from \?lhs\ have "c dvd d * b"  haftmann@59009  496  unfolding associated_def by (auto dest: dvd_mult_right simp add: ac_simps)  haftmann@58023  497  hence "c dvd d" by (simp add: coprime_dvd_mult_iff gcd.commute)  wenzelm@60526  498  moreover from \?lhs\ have "d dvd c * a"  haftmann@59009  499  unfolding associated_def by (auto dest: dvd_mult_right simp add: ac_simps)  haftmann@58023  500  hence "d dvd c" by (simp add: coprime_dvd_mult_iff gcd.commute)  haftmann@58023  501  ultimately show ?rhs unfolding associated_def by simp  haftmann@58023  502 qed  haftmann@58023  503 haftmann@58023  504 lemma gcd_add1 [simp]:  haftmann@58023  505  "gcd (m + n) n = gcd m n"  haftmann@58023  506  by (cases "n = 0", simp_all add: gcd_non_0)  haftmann@58023  507 haftmann@58023  508 lemma gcd_add2 [simp]:  haftmann@58023  509  "gcd m (m + n) = gcd m n"  haftmann@58023  510  using gcd_add1 [of n m] by (simp add: ac_simps)  haftmann@58023  511 haftmann@58023  512 lemma gcd_add_mult: "gcd m (k * m + n) = gcd m n"  haftmann@58023  513  by (subst gcd.commute, subst gcd_red, simp)  haftmann@58023  514 haftmann@60430  515 lemma coprimeI: "(\l. \l dvd a; l dvd b\ \ l dvd 1) \ gcd a b = 1"  haftmann@58023  516  by (rule sym, rule gcdI, simp_all)  haftmann@58023  517 haftmann@58023  518 lemma coprime: "gcd a b = 1 \ (\d. d dvd a \ d dvd b \ is_unit d)"  haftmann@59061  519  by (auto intro: coprimeI gcd_greatest dvd_gcd_D1 dvd_gcd_D2)  haftmann@58023  520 haftmann@58023  521 lemma div_gcd_coprime:  haftmann@58023  522  assumes nz: "a \ 0 \ b \ 0"  haftmann@58023  523  defines [simp]: "d \ gcd a b"  haftmann@58023  524  defines [simp]: "a' \ a div d" and [simp]: "b' \ b div d"  haftmann@58023  525  shows "gcd a' b' = 1"  haftmann@58023  526 proof (rule coprimeI)  haftmann@58023  527  fix l assume "l dvd a'" "l dvd b'"  haftmann@58023  528  then obtain s t where "a' = l * s" "b' = l * t" unfolding dvd_def by blast  haftmann@59009  529  moreover have "a = a' * d" "b = b' * d" by simp_all  haftmann@58023  530  ultimately have "a = (l * d) * s" "b = (l * d) * t"  haftmann@59009  531  by (simp_all only: ac_simps)  haftmann@58023  532  hence "l*d dvd a" and "l*d dvd b" by (simp_all only: dvd_triv_left)  haftmann@58023  533  hence "l*d dvd d" by (simp add: gcd_greatest)  haftmann@59009  534  then obtain u where "d = l * d * u" ..  haftmann@59009  535  then have "d * (l * u) = d" by (simp add: ac_simps)  haftmann@59009  536  moreover from nz have "d \ 0" by simp  haftmann@59009  537  with div_mult_self1_is_id have "d * (l * u) div d = l * u" .  haftmann@59009  538  ultimately have "1 = l * u"  wenzelm@60526  539  using \d \ 0\ by simp  haftmann@59009  540  then show "l dvd 1" ..  haftmann@58023  541 qed  haftmann@58023  542 haftmann@58023  543 lemma coprime_mult:  haftmann@58023  544  assumes da: "gcd d a = 1" and db: "gcd d b = 1"  haftmann@58023  545  shows "gcd d (a * b) = 1"  haftmann@58023  546  apply (subst gcd.commute)  haftmann@58023  547  using da apply (subst gcd_mult_cancel)  haftmann@58023  548  apply (subst gcd.commute, assumption)  haftmann@58023  549  apply (subst gcd.commute, rule db)  haftmann@58023  550  done  haftmann@58023  551 haftmann@58023  552 lemma coprime_lmult:  haftmann@58023  553  assumes dab: "gcd d (a * b) = 1"  haftmann@58023  554  shows "gcd d a = 1"  haftmann@58023  555 proof (rule coprimeI)  haftmann@58023  556  fix l assume "l dvd d" and "l dvd a"  haftmann@58023  557  hence "l dvd a * b" by simp  wenzelm@60526  558  with \l dvd d\ and dab show "l dvd 1" by (auto intro: gcd_greatest)  haftmann@58023  559 qed  haftmann@58023  560 haftmann@58023  561 lemma coprime_rmult:  haftmann@58023  562  assumes dab: "gcd d (a * b) = 1"  haftmann@58023  563  shows "gcd d b = 1"  haftmann@58023  564 proof (rule coprimeI)  haftmann@58023  565  fix l assume "l dvd d" and "l dvd b"  haftmann@58023  566  hence "l dvd a * b" by simp  wenzelm@60526  567  with \l dvd d\ and dab show "l dvd 1" by (auto intro: gcd_greatest)  haftmann@58023  568 qed  haftmann@58023  569 haftmann@58023  570 lemma coprime_mul_eq: "gcd d (a * b) = 1 \ gcd d a = 1 \ gcd d b = 1"  haftmann@58023  571  using coprime_rmult[of d a b] coprime_lmult[of d a b] coprime_mult[of d a b] by blast  haftmann@58023  572 haftmann@58023  573 lemma gcd_coprime:  haftmann@60430  574  assumes c: "gcd a b \ 0" and a: "a = a' * gcd a b" and b: "b = b' * gcd a b"  haftmann@58023  575  shows "gcd a' b' = 1"  haftmann@58023  576 proof -  haftmann@60430  577  from c have "a \ 0 \ b \ 0" by simp  haftmann@58023  578  with div_gcd_coprime have "gcd (a div gcd a b) (b div gcd a b) = 1" .  haftmann@58023  579  also from assms have "a div gcd a b = a'" by (metis div_mult_self2_is_id)+  haftmann@58023  580  also from assms have "b div gcd a b = b'" by (metis div_mult_self2_is_id)+  haftmann@58023  581  finally show ?thesis .  haftmann@58023  582 qed  haftmann@58023  583 haftmann@58023  584 lemma coprime_power:  haftmann@58023  585  assumes "0 < n"  haftmann@58023  586  shows "gcd a (b ^ n) = 1 \ gcd a b = 1"  haftmann@58023  587 using assms proof (induct n)  haftmann@58023  588  case (Suc n) then show ?case  haftmann@58023  589  by (cases n) (simp_all add: coprime_mul_eq)  haftmann@58023  590 qed simp  haftmann@58023  591 haftmann@58023  592 lemma gcd_coprime_exists:  haftmann@58023  593  assumes nz: "gcd a b \ 0"  haftmann@58023  594  shows "\a' b'. a = a' * gcd a b \ b = b' * gcd a b \ gcd a' b' = 1"  haftmann@58023  595  apply (rule_tac x = "a div gcd a b" in exI)  haftmann@58023  596  apply (rule_tac x = "b div gcd a b" in exI)  haftmann@59009  597  apply (insert nz, auto intro: div_gcd_coprime)  haftmann@58023  598  done  haftmann@58023  599 haftmann@58023  600 lemma coprime_exp:  haftmann@58023  601  "gcd d a = 1 \ gcd d (a^n) = 1"  haftmann@58023  602  by (induct n, simp_all add: coprime_mult)  haftmann@58023  603 haftmann@58023  604 lemma coprime_exp2 [intro]:  haftmann@58023  605  "gcd a b = 1 \ gcd (a^n) (b^m) = 1"  haftmann@58023  606  apply (rule coprime_exp)  haftmann@58023  607  apply (subst gcd.commute)  haftmann@58023  608  apply (rule coprime_exp)  haftmann@58023  609  apply (subst gcd.commute)  haftmann@58023  610  apply assumption  haftmann@58023  611  done  haftmann@58023  612 haftmann@58023  613 lemma gcd_exp:  haftmann@58023  614  "gcd (a^n) (b^n) = (gcd a b) ^ n"  haftmann@58023  615 proof (cases "a = 0 \ b = 0")  haftmann@58023  616  assume "a = 0 \ b = 0"  haftmann@58023  617  then show ?thesis by (cases n, simp_all add: gcd_0_left)  haftmann@58023  618 next  haftmann@58023  619  assume A: "$$a = 0 \ b = 0)"  haftmann@58023  620  hence "1 = gcd ((a div gcd a b)^n) ((b div gcd a b)^n)"  haftmann@58023  621  using div_gcd_coprime by (subst sym, auto simp: div_gcd_coprime)  haftmann@58023  622  hence "(gcd a b) ^ n = (gcd a b) ^ n * ..." by simp  haftmann@58023  623  also note gcd_mult_distrib  haftmann@60438  624  also have "normalization_factor ((gcd a b)^n) = 1"  haftmann@60438  625  by (simp add: normalization_factor_pow A)  haftmann@58023  626  also have "(gcd a b)^n * (a div gcd a b)^n = a^n"  haftmann@58023  627  by (subst ac_simps, subst div_power, simp, rule dvd_div_mult_self, rule dvd_power_same, simp)  haftmann@58023  628  also have "(gcd a b)^n * (b div gcd a b)^n = b^n"  haftmann@58023  629  by (subst ac_simps, subst div_power, simp, rule dvd_div_mult_self, rule dvd_power_same, simp)  haftmann@58023  630  finally show ?thesis by simp  haftmann@58023  631 qed  haftmann@58023  632 haftmann@58023  633 lemma coprime_common_divisor:  haftmann@60430  634  "gcd a b = 1 \ a dvd a \ a dvd b \ is_unit a"  haftmann@60430  635  apply (subgoal_tac "a dvd gcd a b")  haftmann@59061  636  apply simp  haftmann@58023  637  apply (erule (1) gcd_greatest)  haftmann@58023  638  done  haftmann@58023  639 haftmann@58023  640 lemma division_decomp:  haftmann@58023  641  assumes dc: "a dvd b * c"  haftmann@58023  642  shows "\b' c'. a = b' * c' \ b' dvd b \ c' dvd c"  haftmann@58023  643 proof (cases "gcd a b = 0")  haftmann@58023  644  assume "gcd a b = 0"  haftmann@59009  645  hence "a = 0 \ b = 0" by simp  haftmann@58023  646  hence "a = 0 * c \ 0 dvd b \ c dvd c" by simp  haftmann@58023  647  then show ?thesis by blast  haftmann@58023  648 next  haftmann@58023  649  let ?d = "gcd a b"  haftmann@58023  650  assume "?d \ 0"  haftmann@58023  651  from gcd_coprime_exists[OF this]  haftmann@58023  652  obtain a' b' where ab': "a = a' * ?d" "b = b' * ?d" "gcd a' b' = 1"  haftmann@58023  653  by blast  haftmann@58023  654  from ab'(1) have "a' dvd a" unfolding dvd_def by blast  haftmann@58023  655  with dc have "a' dvd b*c" using dvd_trans[of a' a "b*c"] by simp  haftmann@58023  656  from dc ab'(1,2) have "a'*?d dvd (b'*?d) * c" by simp  haftmann@58023  657  hence "?d * a' dvd ?d * (b' * c)" by (simp add: mult_ac)  wenzelm@60526  658  with \?d \ 0\ have "a' dvd b' * c" by simp  haftmann@58023  659  with coprime_dvd_mult[OF ab'(3)]  haftmann@58023  660  have "a' dvd c" by (subst (asm) ac_simps, blast)  haftmann@58023  661  with ab'(1) have "a = ?d * a' \ ?d dvd b \ a' dvd c" by (simp add: mult_ac)  haftmann@58023  662  then show ?thesis by blast  haftmann@58023  663 qed  haftmann@58023  664 haftmann@60433  665 lemma pow_divs_pow:  haftmann@58023  666  assumes ab: "a ^ n dvd b ^ n" and n: "n \ 0"  haftmann@58023  667  shows "a dvd b"  haftmann@58023  668 proof (cases "gcd a b = 0")  haftmann@58023  669  assume "gcd a b = 0"  haftmann@59009  670  then show ?thesis by simp  haftmann@58023  671 next  haftmann@58023  672  let ?d = "gcd a b"  haftmann@58023  673  assume "?d \ 0"  haftmann@58023  674  from n obtain m where m: "n = Suc m" by (cases n, simp_all)  wenzelm@60526  675  from \?d \ 0\ have zn: "?d ^ n \ 0" by (rule power_not_zero)  wenzelm@60526  676  from gcd_coprime_exists[OF \?d \ 0\]  haftmann@58023  677  obtain a' b' where ab': "a = a' * ?d" "b = b' * ?d" "gcd a' b' = 1"  haftmann@58023  678  by blast  haftmann@58023  679  from ab have "(a' * ?d) ^ n dvd (b' * ?d) ^ n"  haftmann@58023  680  by (simp add: ab'(1,2)[symmetric])  haftmann@58023  681  hence "?d^n * a'^n dvd ?d^n * b'^n"  haftmann@58023  682  by (simp only: power_mult_distrib ac_simps)  haftmann@59009  683  with zn have "a'^n dvd b'^n" by simp  haftmann@58023  684  hence "a' dvd b'^n" using dvd_trans[of a' "a'^n" "b'^n"] by (simp add: m)  haftmann@58023  685  hence "a' dvd b'^m * b'" by (simp add: m ac_simps)  haftmann@58023  686  with coprime_dvd_mult[OF coprime_exp[OF ab'(3), of m]]  haftmann@58023  687  have "a' dvd b'" by (subst (asm) ac_simps, blast)  haftmann@58023  688  hence "a'*?d dvd b'*?d" by (rule mult_dvd_mono, simp)  haftmann@58023  689  with ab'(1,2) show ?thesis by simp  haftmann@58023  690 qed  haftmann@58023  691 haftmann@60433  692 lemma pow_divs_eq [simp]:  haftmann@58023  693  "n \ 0 \ a ^ n dvd b ^ n \ a dvd b"  haftmann@60433  694  by (auto intro: pow_divs_pow dvd_power_same)  haftmann@58023  695 haftmann@60433  696 lemma divs_mult:  haftmann@58023  697  assumes mr: "m dvd r" and nr: "n dvd r" and mn: "gcd m n = 1"  haftmann@58023  698  shows "m * n dvd r"  haftmann@58023  699 proof -  haftmann@58023  700  from mr nr obtain m' n' where m': "r = m*m'" and n': "r = n*n'"  haftmann@58023  701  unfolding dvd_def by blast  haftmann@58023  702  from mr n' have "m dvd n'*n" by (simp add: ac_simps)  haftmann@58023  703  hence "m dvd n'" using coprime_dvd_mult_iff[OF mn] by simp  haftmann@58023  704  then obtain k where k: "n' = m*k" unfolding dvd_def by blast  haftmann@58023  705  with n' have "r = m * n * k" by (simp add: mult_ac)  haftmann@58023  706  then show ?thesis unfolding dvd_def by blast  haftmann@58023  707 qed  haftmann@58023  708 haftmann@58023  709 lemma coprime_plus_one [simp]: "gcd (n + 1) n = 1"  haftmann@58023  710  by (subst add_commute, simp)  haftmann@58023  711 haftmann@58023  712 lemma setprod_coprime [rule_format]:  haftmann@60430  713  "(\i\A. gcd (f i) a = 1) \ gcd (\i\A. f i) a = 1"  haftmann@58023  714  apply (cases "finite A")  haftmann@58023  715  apply (induct set: finite)  haftmann@58023  716  apply (auto simp add: gcd_mult_cancel)  haftmann@58023  717  done  haftmann@58023  718 haftmann@58023  719 lemma coprime_divisors:  haftmann@58023  720  assumes "d dvd a" "e dvd b" "gcd a b = 1"  haftmann@58023  721  shows "gcd d e = 1"  haftmann@58023  722 proof -  haftmann@58023  723  from assms obtain k l where "a = d * k" "b = e * l"  haftmann@58023  724  unfolding dvd_def by blast  haftmann@58023  725  with assms have "gcd (d * k) (e * l) = 1" by simp  haftmann@58023  726  hence "gcd (d * k) e = 1" by (rule coprime_lmult)  haftmann@58023  727  also have "gcd (d * k) e = gcd e (d * k)" by (simp add: ac_simps)  haftmann@58023  728  finally have "gcd e d = 1" by (rule coprime_lmult)  haftmann@58023  729  then show ?thesis by (simp add: ac_simps)  haftmann@58023  730 qed  haftmann@58023  731 haftmann@58023  732 lemma invertible_coprime:  haftmann@60430  733  assumes "a * b mod m = 1"  haftmann@60430  734  shows "coprime a m"  haftmann@59009  735 proof -  haftmann@60430  736  from assms have "coprime m (a * b mod m)"  haftmann@59009  737  by simp  haftmann@60430  738  then have "coprime m (a * b)"  haftmann@59009  739  by simp  haftmann@60430  740  then have "coprime m a"  haftmann@59009  741  by (rule coprime_lmult)  haftmann@59009  742  then show ?thesis  haftmann@59009  743  by (simp add: ac_simps)  haftmann@59009  744 qed  haftmann@58023  745 haftmann@58023  746 lemma lcm_gcd:  haftmann@60438  747  "lcm a b = a * b div (gcd a b * normalization_factor (a*b))"  haftmann@58023  748  by (simp only: lcm_lcm_eucl gcd_gcd_eucl lcm_eucl_def)  haftmann@58023  749 haftmann@58023  750 lemma lcm_gcd_prod:  haftmann@60438  751  "lcm a b * gcd a b = a * b div normalization_factor (a*b)"  haftmann@58023  752 proof (cases "a * b = 0")  haftmann@60438  753  let ?nf = normalization_factor  haftmann@58023  754  assume "a * b \ 0"  haftmann@58953  755  hence "gcd a b \ 0" by simp  haftmann@58023  756  from lcm_gcd have "lcm a b * gcd a b = gcd a b * (a * b div (?nf (a*b) * gcd a b))"  haftmann@58023  757  by (simp add: mult_ac)  wenzelm@60526  758  also from \a * b \ 0\ have "... = a * b div ?nf (a*b)"  haftmann@60432  759  by (simp add: div_mult_swap mult.commute)  haftmann@58023  760  finally show ?thesis .  haftmann@58953  761 qed (auto simp add: lcm_gcd)  haftmann@58023  762 haftmann@58023  763 lemma lcm_dvd1 [iff]:  haftmann@60430  764  "a dvd lcm a b"  haftmann@60430  765 proof (cases "a*b = 0")  haftmann@60430  766  assume "a * b \ 0"  haftmann@60430  767  hence "gcd a b \ 0" by simp  haftmann@60438  768  let ?c = "1 div normalization_factor (a * b)"  wenzelm@60526  769  from \a * b \ 0\ have [simp]: "is_unit (normalization_factor (a * b))" by simp  haftmann@60430  770  from lcm_gcd_prod[of a b] have "lcm a b * gcd a b = a * ?c * b"  haftmann@60432  771  by (simp add: div_mult_swap unit_div_commute)  haftmann@60430  772  hence "lcm a b * gcd a b div gcd a b = a * ?c * b div gcd a b" by simp  wenzelm@60526  773  with \gcd a b \ 0\ have "lcm a b = a * ?c * b div gcd a b"  haftmann@58023  774  by (subst (asm) div_mult_self2_is_id, simp_all)  haftmann@60430  775  also have "... = a * (?c * b div gcd a b)"  haftmann@58023  776  by (metis div_mult_swap gcd_dvd2 mult_assoc)  haftmann@58023  777  finally show ?thesis by (rule dvdI)  haftmann@58953  778 qed (auto simp add: lcm_gcd)  haftmann@58023  779 haftmann@58023  780 lemma lcm_least:  haftmann@58023  781  "\a dvd k; b dvd k\ \ lcm a b dvd k"  haftmann@58023  782 proof (cases "k = 0")  haftmann@60438  783  let ?nf = normalization_factor  haftmann@58023  784  assume "k \ 0"  haftmann@58023  785  hence "is_unit (?nf k)" by simp  haftmann@58023  786  hence "?nf k \ 0" by (metis not_is_unit_0)  haftmann@58023  787  assume A: "a dvd k" "b dvd k"  wenzelm@60526  788  hence "gcd a b \ 0" using \k \ 0\ by auto  haftmann@58023  789  from A obtain r s where ar: "k = a * r" and bs: "k = b * s"  haftmann@58023  790  unfolding dvd_def by blast  wenzelm@60526  791  with \k \ 0\ have "r * s \ 0"  haftmann@58953  792  by auto (drule sym [of 0], simp)  haftmann@58023  793  hence "is_unit (?nf (r * s))" by simp  haftmann@58023  794  let ?c = "?nf k div ?nf (r*s)"  wenzelm@60526  795  from \is_unit (?nf k)\ and \is_unit (?nf (r * s))\ have "is_unit ?c" by (rule unit_div)  haftmann@58023  796  hence "?c \ 0" using not_is_unit_0 by fast  haftmann@58023  797  from ar bs have "k * k * gcd s r = ?nf k * k * gcd (k * s) (k * r)"  haftmann@58953  798  by (subst mult_assoc, subst gcd_mult_distrib[of k s r], simp only: ac_simps)  haftmann@58023  799  also have "... = ?nf k * k * gcd ((r*s) * a) ((r*s) * b)"  wenzelm@60526  800  by (subst (3) \k = a * r\, subst (3) \k = b * s\, simp add: algebra_simps)  wenzelm@60526  801  also have "... = ?c * r*s * k * gcd a b" using \r * s \ 0\  haftmann@58023  802  by (subst gcd_mult_distrib'[symmetric], simp add: algebra_simps unit_simps)  haftmann@58023  803  finally have "(a*r) * (b*s) * gcd s r = ?c * k * r * s * gcd a b"  haftmann@58023  804  by (subst ar[symmetric], subst bs[symmetric], simp add: mult_ac)  haftmann@58023  805  hence "a * b * gcd s r * (r * s) = ?c * k * gcd a b * (r * s)"  haftmann@58023  806  by (simp add: algebra_simps)  wenzelm@60526  807  hence "?c * k * gcd a b = a * b * gcd s r" using \r * s \ 0\  haftmann@58023  808  by (metis div_mult_self2_is_id)  haftmann@58023  809  also have "... = lcm a b * gcd a b * gcd s r * ?nf (a*b)"  haftmann@58023  810  by (subst lcm_gcd_prod[of a b], metis gcd_mult_distrib gcd_mult_distrib')  haftmann@58023  811  also have "... = lcm a b * gcd s r * ?nf (a*b) * gcd a b"  haftmann@58023  812  by (simp add: algebra_simps)  wenzelm@60526  813  finally have "k * ?c = lcm a b * gcd s r * ?nf (a*b)" using \gcd a b \ 0\  haftmann@58023  814  by (metis mult.commute div_mult_self2_is_id)  wenzelm@60526  815  hence "k = lcm a b * (gcd s r * ?nf (a*b)) div ?c" using \?c \ 0\  haftmann@58023  816  by (metis div_mult_self2_is_id mult_assoc)  wenzelm@60526  817  also have "... = lcm a b * (gcd s r * ?nf (a*b) div ?c)" using \is_unit ?c\  haftmann@58023  818  by (simp add: unit_simps)  haftmann@58023  819  finally show ?thesis by (rule dvdI)  haftmann@58023  820 qed simp  haftmann@58023  821 haftmann@58023  822 lemma lcm_zero:  haftmann@58023  823  "lcm a b = 0 \ a = 0 \ b = 0"  haftmann@58023  824 proof -  haftmann@60438  825  let ?nf = normalization_factor  haftmann@58023  826  {  haftmann@58023  827  assume "a \ 0" "b \ 0"  haftmann@58023  828  hence "a * b div ?nf (a * b) \ 0" by (simp add: no_zero_divisors)  wenzelm@60526  829  moreover from \a \ 0\ and \b \ 0\ have "gcd a b \ 0" by simp  haftmann@58023  830  ultimately have "lcm a b \ 0" using lcm_gcd_prod[of a b] by (intro notI, simp)  haftmann@58023  831  } moreover {  haftmann@58023  832  assume "a = 0 \ b = 0"  haftmann@58023  833  hence "lcm a b = 0" by (elim disjE, simp_all add: lcm_gcd)  haftmann@58023  834  }  haftmann@58023  835  ultimately show ?thesis by blast  haftmann@58023  836 qed  haftmann@58023  837 haftmann@58023  838 lemmas lcm_0_iff = lcm_zero  haftmann@58023  839 haftmann@58023  840 lemma gcd_lcm:  haftmann@58023  841  assumes "lcm a b \ 0"  haftmann@60438  842  shows "gcd a b = a * b div (lcm a b * normalization_factor (a * b))"  haftmann@58023  843 proof-  haftmann@59009  844  from assms have "gcd a b \ 0" by (simp add: lcm_zero)  haftmann@60438  845  let ?c = "normalization_factor (a * b)"  wenzelm@60526  846  from \lcm a b \ 0\ have "?c \ 0" by (intro notI, simp add: lcm_zero no_zero_divisors)  haftmann@58023  847  hence "is_unit ?c" by simp  haftmann@58023  848  from lcm_gcd_prod [of a b] have "gcd a b = a * b div ?c div lcm a b"  wenzelm@60526  849  by (subst (2) div_mult_self2_is_id[OF \lcm a b \ 0\, symmetric], simp add: mult_ac)  wenzelm@60526  850  also from \is_unit ?c\ have "... = a * b div (lcm a b * ?c)"  wenzelm@60526  851  by (metis \?c \ 0\ div_mult_mult1 dvd_mult_div_cancel mult_commute normalization_factor_dvd')  haftmann@60433  852  finally show ?thesis .  haftmann@58023  853 qed  haftmann@58023  854 haftmann@60438  855 lemma normalization_factor_lcm [simp]:  haftmann@60438  856  "normalization_factor (lcm a b) = (if a = 0 \ b = 0 then 0 else 1)"  haftmann@58023  857 proof (cases "a = 0 \ b = 0")  haftmann@58023  858  case True then show ?thesis  haftmann@58953  859  by (auto simp add: lcm_gcd)  haftmann@58023  860 next  haftmann@58023  861  case False  haftmann@60438  862  let ?nf = normalization_factor  haftmann@58023  863  from lcm_gcd_prod[of a b]  haftmann@58023  864  have "?nf (lcm a b) * ?nf (gcd a b) = ?nf (a*b) div ?nf (a*b)"  haftmann@60438  865  by (metis div_by_0 div_self normalization_correct normalization_factor_0 normalization_factor_mult)  haftmann@58023  866  also have "... = (if a*b = 0 then 0 else 1)"  haftmann@58953  867  by simp  haftmann@58953  868  finally show ?thesis using False by simp  haftmann@58023  869 qed  haftmann@58023  870 haftmann@60430  871 lemma lcm_dvd2 [iff]: "b dvd lcm a b"  haftmann@60430  872  using lcm_dvd1 [of b a] by (simp add: lcm_gcd ac_simps)  haftmann@58023  873 haftmann@58023  874 lemma lcmI:  haftmann@60430  875  "\a dvd k; b dvd k; \l. a dvd l \ b dvd l \ k dvd l;  haftmann@60438  876  normalization_factor k = (if k = 0 then 0 else 1)\ \ k = lcm a b"  haftmann@58023  877  by (intro normed_associated_imp_eq) (auto simp: associated_def intro: lcm_least)  haftmann@58023  878 haftmann@58023  879 sublocale lcm!: abel_semigroup lcm  haftmann@58023  880 proof  haftmann@60430  881  fix a b c  haftmann@60430  882  show "lcm (lcm a b) c = lcm a (lcm b c)"  haftmann@58023  883  proof (rule lcmI)  haftmann@60430  884  have "a dvd lcm a b" and "lcm a b dvd lcm (lcm a b) c" by simp_all  haftmann@60430  885  then show "a dvd lcm (lcm a b) c" by (rule dvd_trans)  haftmann@58023  886   haftmann@60430  887  have "b dvd lcm a b" and "lcm a b dvd lcm (lcm a b) c" by simp_all  haftmann@60430  888  hence "b dvd lcm (lcm a b) c" by (rule dvd_trans)  haftmann@60430  889  moreover have "c dvd lcm (lcm a b) c" by simp  haftmann@60430  890  ultimately show "lcm b c dvd lcm (lcm a b) c" by (rule lcm_least)  haftmann@58023  891 haftmann@60430  892  fix l assume "a dvd l" and "lcm b c dvd l"  haftmann@60430  893  have "b dvd lcm b c" by simp  wenzelm@60526  894  from this and \lcm b c dvd l\ have "b dvd l" by (rule dvd_trans)  haftmann@60430  895  have "c dvd lcm b c" by simp  wenzelm@60526  896  from this and \lcm b c dvd l\ have "c dvd l" by (rule dvd_trans)  wenzelm@60526  897  from \a dvd l\ and \b dvd l\ have "lcm a b dvd l" by (rule lcm_least)  wenzelm@60526  898  from this and \c dvd l\ show "lcm (lcm a b) c dvd l" by (rule lcm_least)  haftmann@58023  899  qed (simp add: lcm_zero)  haftmann@58023  900 next  haftmann@60430  901  fix a b  haftmann@60430  902  show "lcm a b = lcm b a"  haftmann@58023  903  by (simp add: lcm_gcd ac_simps)  haftmann@58023  904 qed  haftmann@58023  905 haftmann@58023  906 lemma dvd_lcm_D1:  haftmann@58023  907  "lcm m n dvd k \ m dvd k"  haftmann@58023  908  by (rule dvd_trans, rule lcm_dvd1, assumption)  haftmann@58023  909 haftmann@58023  910 lemma dvd_lcm_D2:  haftmann@58023  911  "lcm m n dvd k \ n dvd k"  haftmann@58023  912  by (rule dvd_trans, rule lcm_dvd2, assumption)  haftmann@58023  913 haftmann@58023  914 lemma gcd_dvd_lcm [simp]:  haftmann@58023  915  "gcd a b dvd lcm a b"  haftmann@58023  916  by (metis dvd_trans gcd_dvd2 lcm_dvd2)  haftmann@58023  917 haftmann@58023  918 lemma lcm_1_iff:  haftmann@58023  919  "lcm a b = 1 \ is_unit a \ is_unit b"  haftmann@58023  920 proof  haftmann@58023  921  assume "lcm a b = 1"  haftmann@59061  922  then show "is_unit a \ is_unit b" by auto  haftmann@58023  923 next  haftmann@58023  924  assume "is_unit a \ is_unit b"  haftmann@59061  925  hence "a dvd 1" and "b dvd 1" by simp_all  haftmann@59061  926  hence "is_unit (lcm a b)" by (rule lcm_least)  haftmann@60438  927  hence "lcm a b = normalization_factor (lcm a b)"  haftmann@60438  928  by (subst normalization_factor_unit, simp_all)  wenzelm@60526  929  also have "\ = 1" using \is_unit a \ is_unit b\  haftmann@59061  930  by auto  haftmann@58023  931  finally show "lcm a b = 1" .  haftmann@58023  932 qed  haftmann@58023  933 haftmann@58023  934 lemma lcm_0_left [simp]:  haftmann@60430  935  "lcm 0 a = 0"  haftmann@58023  936  by (rule sym, rule lcmI, simp_all)  haftmann@58023  937 haftmann@58023  938 lemma lcm_0 [simp]:  haftmann@60430  939  "lcm a 0 = 0"  haftmann@58023  940  by (rule sym, rule lcmI, simp_all)  haftmann@58023  941 haftmann@58023  942 lemma lcm_unique:  haftmann@58023  943  "a dvd d \ b dvd d \  haftmann@60438  944  normalization_factor d = (if d = 0 then 0 else 1) \  haftmann@58023  945  (\e. a dvd e \ b dvd e \ d dvd e) \ d = lcm a b"  haftmann@58023  946  by (rule, auto intro: lcmI simp: lcm_least lcm_zero)  haftmann@58023  947 haftmann@58023  948 lemma dvd_lcm_I1 [simp]:  haftmann@58023  949  "k dvd m \ k dvd lcm m n"  haftmann@58023  950  by (metis lcm_dvd1 dvd_trans)  haftmann@58023  951 haftmann@58023  952 lemma dvd_lcm_I2 [simp]:  haftmann@58023  953  "k dvd n \ k dvd lcm m n"  haftmann@58023  954  by (metis lcm_dvd2 dvd_trans)  haftmann@58023  955 haftmann@58023  956 lemma lcm_1_left [simp]:  haftmann@60438  957  "lcm 1 a = a div normalization_factor a"  haftmann@60430  958  by (cases "a = 0") (simp, rule sym, rule lcmI, simp_all)  haftmann@58023  959 haftmann@58023  960 lemma lcm_1_right [simp]:  haftmann@60438  961  "lcm a 1 = a div normalization_factor a"  haftmann@60430  962  using lcm_1_left [of a] by (simp add: ac_simps)  haftmann@58023  963 haftmann@58023  964 lemma lcm_coprime:  haftmann@60438  965  "gcd a b = 1 \ lcm a b = a * b div normalization_factor (a*b)"  haftmann@58023  966  by (subst lcm_gcd) simp  haftmann@58023  967 haftmann@58023  968 lemma lcm_proj1_if_dvd:  haftmann@60438  969  "b dvd a \ lcm a b = a div normalization_factor a"  haftmann@60430  970  by (cases "a = 0") (simp, rule sym, rule lcmI, simp_all)  haftmann@58023  971 haftmann@58023  972 lemma lcm_proj2_if_dvd:  haftmann@60438  973  "a dvd b \ lcm a b = b div normalization_factor b"  haftmann@60430  974  using lcm_proj1_if_dvd [of a b] by (simp add: ac_simps)  haftmann@58023  975 haftmann@58023  976 lemma lcm_proj1_iff:  haftmann@60438  977  "lcm m n = m div normalization_factor m \ n dvd m"  haftmann@58023  978 proof  haftmann@60438  979  assume A: "lcm m n = m div normalization_factor m"  haftmann@58023  980  show "n dvd m"  haftmann@58023  981  proof (cases "m = 0")  haftmann@58023  982  assume [simp]: "m \ 0"  haftmann@60438  983  from A have B: "m = lcm m n * normalization_factor m"  haftmann@58023  984  by (simp add: unit_eq_div2)  haftmann@58023  985  show ?thesis by (subst B, simp)  haftmann@58023  986  qed simp  haftmann@58023  987 next  haftmann@58023  988  assume "n dvd m"  haftmann@60438  989  then show "lcm m n = m div normalization_factor m" by (rule lcm_proj1_if_dvd)  haftmann@58023  990 qed  haftmann@58023  991 haftmann@58023  992 lemma lcm_proj2_iff:  haftmann@60438  993  "lcm m n = n div normalization_factor n \ m dvd n"  haftmann@58023  994  using lcm_proj1_iff [of n m] by (simp add: ac_simps)  haftmann@58023  995 haftmann@58023  996 lemma euclidean_size_lcm_le1:  haftmann@58023  997  assumes "a \ 0" and "b \ 0"  haftmann@58023  998  shows "euclidean_size a \ euclidean_size (lcm a b)"  haftmann@58023  999 proof -  haftmann@58023  1000  have "a dvd lcm a b" by (rule lcm_dvd1)  haftmann@58023  1001  then obtain c where A: "lcm a b = a * c" unfolding dvd_def by blast  wenzelm@60526  1002  with \a \ 0\ and \b \ 0\ have "c \ 0" by (auto simp: lcm_zero)  haftmann@58023  1003  then show ?thesis by (subst A, intro size_mult_mono)  haftmann@58023  1004 qed  haftmann@58023  1005 haftmann@58023  1006 lemma euclidean_size_lcm_le2:  haftmann@58023  1007  "a \ 0 \ b \ 0 \ euclidean_size b \ euclidean_size (lcm a b)"  haftmann@58023  1008  using euclidean_size_lcm_le1 [of b a] by (simp add: ac_simps)  haftmann@58023  1009 haftmann@58023  1010 lemma euclidean_size_lcm_less1:  haftmann@58023  1011  assumes "b \ 0" and "\b dvd a"  haftmann@58023  1012  shows "euclidean_size a < euclidean_size (lcm a b)"  haftmann@58023  1013 proof (rule ccontr)  haftmann@58023  1014  from assms have "a \ 0" by auto  haftmann@58023  1015  assume "\euclidean_size a < euclidean_size (lcm a b)"  wenzelm@60526  1016  with \a \ 0\ and \b \ 0\ have "euclidean_size (lcm a b) = euclidean_size a"  haftmann@58023  1017  by (intro le_antisym, simp, intro euclidean_size_lcm_le1)  haftmann@58023  1018  with assms have "lcm a b dvd a"  haftmann@58023  1019  by (rule_tac dvd_euclidean_size_eq_imp_dvd) (auto simp: lcm_zero)  haftmann@58023  1020  hence "b dvd a" by (rule dvd_lcm_D2)  wenzelm@60526  1021  with \\b dvd a\ show False by contradiction  haftmann@58023  1022 qed  haftmann@58023  1023 haftmann@58023  1024 lemma euclidean_size_lcm_less2:  haftmann@58023  1025  assumes "a \ 0" and "\a dvd b"  haftmann@58023  1026  shows "euclidean_size b < euclidean_size (lcm a b)"  haftmann@58023  1027  using assms euclidean_size_lcm_less1 [of a b] by (simp add: ac_simps)  haftmann@58023  1028 haftmann@58023  1029 lemma lcm_mult_unit1:  haftmann@60430  1030  "is_unit a \ lcm (b * a) c = lcm b c"  haftmann@58023  1031  apply (rule lcmI)  haftmann@60430  1032  apply (rule dvd_trans[of _ "b * a"], simp, rule lcm_dvd1)  haftmann@58023  1033  apply (rule lcm_dvd2)  haftmann@58023  1034  apply (rule lcm_least, simp add: unit_simps, assumption)  haftmann@60438  1035  apply (subst normalization_factor_lcm, simp add: lcm_zero)  haftmann@58023  1036  done  haftmann@58023  1037 haftmann@58023  1038 lemma lcm_mult_unit2:  haftmann@60430  1039  "is_unit a \ lcm b (c * a) = lcm b c"  haftmann@60430  1040  using lcm_mult_unit1 [of a c b] by (simp add: ac_simps)  haftmann@58023  1041 haftmann@58023  1042 lemma lcm_div_unit1:  haftmann@60430  1043  "is_unit a \ lcm (b div a) c = lcm b c"  haftmann@60433  1044  by (erule is_unitE [of _ b]) (simp add: lcm_mult_unit1)  haftmann@58023  1045 haftmann@58023  1046 lemma lcm_div_unit2:  haftmann@60430  1047  "is_unit a \ lcm b (c div a) = lcm b c"  haftmann@60433  1048  by (erule is_unitE [of _ c]) (simp add: lcm_mult_unit2)  haftmann@58023  1049 haftmann@58023  1050 lemma lcm_left_idem:  haftmann@60430  1051  "lcm a (lcm a b) = lcm a b"  haftmann@58023  1052  apply (rule lcmI)  haftmann@58023  1053  apply simp  haftmann@58023  1054  apply (subst lcm.assoc [symmetric], rule lcm_dvd2)  haftmann@58023  1055  apply (rule lcm_least, assumption)  haftmann@58023  1056  apply (erule (1) lcm_least)  haftmann@58023  1057  apply (auto simp: lcm_zero)  haftmann@58023  1058  done  haftmann@58023  1059 haftmann@58023  1060 lemma lcm_right_idem:  haftmann@60430  1061  "lcm (lcm a b) b = lcm a b"  haftmann@58023  1062  apply (rule lcmI)  haftmann@58023  1063  apply (subst lcm.assoc, rule lcm_dvd1)  haftmann@58023  1064  apply (rule lcm_dvd2)  haftmann@58023  1065  apply (rule lcm_least, erule (1) lcm_least, assumption)  haftmann@58023  1066  apply (auto simp: lcm_zero)  haftmann@58023  1067  done  haftmann@58023  1068 haftmann@58023  1069 lemma comp_fun_idem_lcm: "comp_fun_idem lcm"  haftmann@58023  1070 proof  haftmann@58023  1071  fix a b show "lcm a \ lcm b = lcm b \ lcm a"  haftmann@58023  1072  by (simp add: fun_eq_iff ac_simps)  haftmann@58023  1073 next  haftmann@58023  1074  fix a show "lcm a \ lcm a = lcm a" unfolding o_def  haftmann@58023  1075  by (intro ext, simp add: lcm_left_idem)  haftmann@58023  1076 qed  haftmann@58023  1077 haftmann@60430  1078 lemma dvd_Lcm [simp]: "a \ A \ a dvd Lcm A"  haftmann@60430  1079  and Lcm_dvd [simp]: "(\a\A. a dvd l') \ Lcm A dvd l'"  haftmann@60438  1080  and normalization_factor_Lcm [simp]:  haftmann@60438  1081  "normalization_factor (Lcm A) = (if Lcm A = 0 then 0 else 1)"  haftmann@58023  1082 proof -  haftmann@60430  1083  have "(\a\A. a dvd Lcm A) \ (\l'. (\a\A. a dvd l') \ Lcm A dvd l') \  haftmann@60438  1084  normalization_factor (Lcm A) = (if Lcm A = 0 then 0 else 1)" (is ?thesis)  haftmann@60430  1085  proof (cases "\l. l \ 0 \ (\a\A. a dvd l)")  haftmann@58023  1086  case False  haftmann@58023  1087  hence "Lcm A = 0" by (auto simp: Lcm_Lcm_eucl Lcm_eucl_def)  haftmann@58023  1088  with False show ?thesis by auto  haftmann@58023  1089  next  haftmann@58023  1090  case True  haftmann@60430  1091  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  1092  def n \ "LEAST n. \l. l \ 0 \ (\a\A. a dvd l) \ euclidean_size l = n"  haftmann@60430  1093  def l \ "SOME l. l \ 0 \ (\a\A. a dvd l) \ euclidean_size l = n"  haftmann@60430  1094  have "\l. l \ 0 \ (\a\A. a dvd l) \ euclidean_size l = n"  haftmann@58023  1095  apply (subst n_def)  haftmann@58023  1096  apply (rule LeastI[of _ "euclidean_size l\<^sub>0"])  haftmann@58023  1097  apply (rule exI[of _ l\<^sub>0])  haftmann@58023  1098  apply (simp add: l\<^sub>0_props)  haftmann@58023  1099  done  haftmann@60430  1100  from someI_ex[OF this] have "l \ 0" and "\a\A. a dvd l" and "euclidean_size l = n"  haftmann@58023  1101  unfolding l_def by simp_all  haftmann@58023  1102  {  haftmann@60430  1103  fix l' assume "\a\A. a dvd l'"  wenzelm@60526  1104  with \\a\A. a dvd l\ have "\a\A. a dvd gcd l l'" by (auto intro: gcd_greatest)  wenzelm@60526  1105  moreover from \l \ 0\ have "gcd l l' \ 0" by simp  haftmann@60430  1106  ultimately have "\b. b \ 0 \ (\a\A. a dvd b) \ euclidean_size b = euclidean_size (gcd l l')"  haftmann@58023  1107  by (intro exI[of _ "gcd l l'"], auto)  haftmann@58023  1108  hence "euclidean_size (gcd l l') \ n" by (subst n_def) (rule Least_le)  haftmann@58023  1109  moreover have "euclidean_size (gcd l l') \ n"  haftmann@58023  1110  proof -  haftmann@58023  1111  have "gcd l l' dvd l" by simp  haftmann@58023  1112  then obtain a where "l = gcd l l' * a" unfolding dvd_def by blast  wenzelm@60526  1113  with \l \ 0\ have "a \ 0" by auto  haftmann@58023  1114  hence "euclidean_size (gcd l l') \ euclidean_size (gcd l l' * a)"  haftmann@58023  1115  by (rule size_mult_mono)  wenzelm@60526  1116  also have "gcd l l' * a = l" using \l = gcd l l' * a\ ..  wenzelm@60526  1117  also note \euclidean_size l = n\  haftmann@58023  1118  finally show "euclidean_size (gcd l l') \ n" .  haftmann@58023  1119  qed  haftmann@58023  1120  ultimately have "euclidean_size l = euclidean_size (gcd l l')"  wenzelm@60526  1121  by (intro le_antisym, simp_all add: \euclidean_size l = n$$  wenzelm@60526  1122  with \l \ 0\ have "l dvd gcd l l'" by (blast intro: dvd_euclidean_size_eq_imp_dvd)  haftmann@58023  1123  hence "l dvd l'" by (blast dest: dvd_gcd_D2)  haftmann@58023  1124  }  haftmann@58023  1125 wenzelm@60526  1126  with \(\a\A. a dvd l)\ and normalization_factor_is_unit[OF \l \ 0\] and \l \ 0\  haftmann@60438  1127  have "(\a\A. a dvd l div normalization_factor l) \  haftmann@60438  1128  (\l'. (\a\A. a dvd l') \ l div normalization_factor l dvd l') \  haftmann@60438  1129  normalization_factor (l div normalization_factor l) =  haftmann@60438  1130  (if l div normalization_factor l = 0 then 0 else 1)"  haftmann@58023  1131  by (auto simp: unit_simps)  haftmann@60438  1132  also from True have "l div normalization_factor l = Lcm A"  haftmann@58023  1133  by (simp add: Lcm_Lcm_eucl Lcm_eucl_def Let_def n_def l_def)  haftmann@58023  1134  finally show ?thesis .  haftmann@58023  1135  qed  haftmann@58023  1136  note A = this  haftmann@58023  1137 haftmann@60430  1138  {fix a assume "a \ A" then show "a dvd Lcm A" using A by blast}  haftmann@60430  1139  {fix l' assume "\a\A. a dvd l'" then show "Lcm A dvd l'" using A by blast}  haftmann@60438  1140  from A show "normalization_factor (Lcm A) = (if Lcm A = 0 then 0 else 1)" by blast  haftmann@58023  1141 qed  haftmann@58023  1142   haftmann@58023  1143 lemma LcmI:  haftmann@60430  1144  "(\a. a\A \ a dvd l) \ (\l'. (\a\A. a dvd l') \ l dvd l') \  haftmann@60438  1145  normalization_factor l = (if l = 0 then 0 else 1) \ l = Lcm A"  haftmann@58023  1146  by (intro normed_associated_imp_eq)  haftmann@58023  1147  (auto intro: Lcm_dvd dvd_Lcm simp: associated_def)  haftmann@58023  1148 haftmann@58023  1149 lemma Lcm_subset:  haftmann@58023  1150  "A \ B \ Lcm A dvd Lcm B"  haftmann@58023  1151  by (blast intro: Lcm_dvd dvd_Lcm)  haftmann@58023  1152 haftmann@58023  1153 lemma Lcm_Un:  haftmann@58023  1154  "Lcm (A \ B) = lcm (Lcm A) (Lcm B)"  haftmann@58023  1155  apply (rule lcmI)  haftmann@58023  1156  apply (blast intro: Lcm_subset)  haftmann@58023  1157  apply (blast intro: Lcm_subset)  haftmann@58023  1158  apply (intro Lcm_dvd ballI, elim UnE)  haftmann@58023  1159  apply (rule dvd_trans, erule dvd_Lcm, assumption)  haftmann@58023  1160  apply (rule dvd_trans, erule dvd_Lcm, assumption)  haftmann@58023  1161  apply simp  haftmann@58023  1162  done  haftmann@58023  1163 haftmann@58023  1164 lemma Lcm_1_iff:  haftmann@60430  1165  "Lcm A = 1 \ (\a\A. is_unit a)"  haftmann@58023  1166 proof  haftmann@58023  1167  assume "Lcm A = 1"  haftmann@60430  1168  then show "\a\A. is_unit a" by auto  haftmann@58023  1169 qed (rule LcmI [symmetric], auto)  haftmann@58023  1170 haftmann@58023  1171 lemma Lcm_no_units:  haftmann@60430  1172  "Lcm A = Lcm (A - {a. is_unit a})"  haftmann@58023  1173 proof -  haftmann@60430  1174  have "(A - {a. is_unit a}) \ {a\A. is_unit a} = A" by blast  haftmann@60430  1175  hence "Lcm A = lcm (Lcm (A - {a. is_unit a})) (Lcm {a\A. is_unit a})"  haftmann@58023  1176  by (simp add: Lcm_Un[symmetric])  haftmann@60430  1177  also have "Lcm {a\A. is_unit a} = 1" by (simp add: Lcm_1_iff)  haftmann@58023  1178  finally show ?thesis by simp  haftmann@58023  1179 qed  haftmann@58023  1180 haftmann@58023  1181 lemma Lcm_empty [simp]:  haftmann@58023  1182  "Lcm {} = 1"  haftmann@58023  1183  by (simp add: Lcm_1_iff)  haftmann@58023  1184 haftmann@58023  1185 lemma Lcm_eq_0 [simp]:  haftmann@58023  1186  "0 \ A \ Lcm A = 0"  haftmann@58023  1187  by (drule dvd_Lcm) simp  haftmann@58023  1188 haftmann@58023  1189 lemma Lcm0_iff':  haftmann@60430  1190  "Lcm A = 0 \ \(\l. l \ 0 \ (\a\A. a dvd l))"  haftmann@58023  1191 proof  haftmann@58023  1192  assume "Lcm A = 0"  haftmann@60430  1193  show "\(\l. l \ 0 \ (\a\A. a dvd l))"  haftmann@58023  1194  proof  haftmann@60430  1195  assume ex: "\l. l \ 0 \ (\a\A. a dvd l)"  haftmann@60430  1196  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  1197  def n \ "LEAST n. \l. l \ 0 \ (\a\A. a dvd l) \ euclidean_size l = n"  haftmann@60430  1198  def l \ "SOME l. l \ 0 \ (\a\A. a dvd l) \ euclidean_size l = n"  haftmann@60430  1199  have "\l. l \ 0 \ (\a\A. a dvd l) \ euclidean_size l = n"  haftmann@58023  1200  apply (subst n_def)  haftmann@58023  1201  apply (rule LeastI[of _ "euclidean_size l\<^sub>0"])  haftmann@58023  1202  apply (rule exI[of _ l\<^sub>0])  haftmann@58023  1203  apply (simp add: l\<^sub>0_props)  haftmann@58023  1204  done  haftmann@58023  1205  from someI_ex[OF this] have "l \ 0" unfolding l_def by simp_all  haftmann@60438  1206  hence "l div normalization_factor l \ 0" by simp  haftmann@60438  1207  also from ex have "l div normalization_factor l = Lcm A"  haftmann@58023  1208  by (simp only: Lcm_Lcm_eucl Lcm_eucl_def n_def l_def if_True Let_def)  wenzelm@60526  1209  finally show False using \Lcm A = 0\ by contradiction  haftmann@58023  1210  qed  haftmann@58023  1211 qed (simp only: Lcm_Lcm_eucl Lcm_eucl_def if_False)  haftmann@58023  1212 haftmann@58023  1213 lemma Lcm0_iff [simp]:  haftmann@58023  1214  "finite A \ Lcm A = 0 \ 0 \ A"  haftmann@58023  1215 proof -  haftmann@58023  1216  assume "finite A"  haftmann@58023  1217  have "0 \ A \ Lcm A = 0" by (intro dvd_0_left dvd_Lcm)  haftmann@58023  1218  moreover {  haftmann@58023  1219  assume "0 \ A"  haftmann@58023  1220  hence "\A \ 0"  wenzelm@60526  1221  apply (induct rule: finite_induct[OF \finite A\])  haftmann@58023  1222  apply simp  haftmann@58023  1223  apply (subst setprod.insert, assumption, assumption)  haftmann@58023  1224  apply (rule no_zero_divisors)  haftmann@58023  1225  apply blast+  haftmann@58023  1226  done  wenzelm@60526  1227  moreover from \finite A\ have "\a\A. a dvd \A" by blast  haftmann@60430  1228  ultimately have "\l. l \ 0 \ (\a\A. a dvd l)" by blast  haftmann@58023  1229  with Lcm0_iff' have "Lcm A \ 0" by simp  haftmann@58023  1230  }  haftmann@58023  1231  ultimately show "Lcm A = 0 \ 0 \ A" by blast  haftmann@58023  1232 qed  haftmann@58023  1233 haftmann@58023  1234 lemma Lcm_no_multiple:  haftmann@60430  1235  "(\m. m \ 0 \ (\a\A. \a dvd m)) \ Lcm A = 0"  haftmann@58023  1236 proof -  haftmann@60430  1237  assume "\m. m \ 0 \ (\a\A. \a dvd m)"  haftmann@60430  1238  hence "\(\l. l \ 0 \ (\a\A. a dvd l))" by blast  haftmann@58023  1239  then show "Lcm A = 0" by (simp only: Lcm_Lcm_eucl Lcm_eucl_def if_False)  haftmann@58023  1240 qed  haftmann@58023  1241 haftmann@58023  1242 lemma Lcm_insert [simp]:  haftmann@58023  1243  "Lcm (insert a A) = lcm a (Lcm A)"  haftmann@58023  1244 proof (rule lcmI)  haftmann@58023  1245  fix l assume "a dvd l" and "Lcm A dvd l"  haftmann@60430  1246  hence "\a\A. a dvd l" by (blast intro: dvd_trans dvd_Lcm)  wenzelm@60526  1247  with \a dvd l\ show "Lcm (insert a A) dvd l" by (force intro: Lcm_dvd)  haftmann@58023  1248 qed (auto intro: Lcm_dvd dvd_Lcm)  haftmann@58023  1249   haftmann@58023  1250 lemma Lcm_finite:  haftmann@58023  1251  assumes "finite A"  haftmann@58023  1252  shows "Lcm A = Finite_Set.fold lcm 1 A"  wenzelm@60526  1253  by (induct rule: finite.induct[OF \finite A\])  haftmann@58023  1254  (simp_all add: comp_fun_idem.fold_insert_idem[OF comp_fun_idem_lcm])  haftmann@58023  1255 haftmann@60431  1256 lemma Lcm_set [code_unfold]:  haftmann@58023  1257  "Lcm (set xs) = fold lcm xs 1"  haftmann@58023  1258  using comp_fun_idem.fold_set_fold[OF comp_fun_idem_lcm] Lcm_finite by (simp add: ac_simps)  haftmann@58023  1259 haftmann@58023  1260 lemma Lcm_singleton [simp]:  haftmann@60438  1261  "Lcm {a} = a div normalization_factor a"  haftmann@58023  1262  by simp  haftmann@58023  1263 haftmann@58023  1264 lemma Lcm_2 [simp]:  haftmann@58023  1265  "Lcm {a,b} = lcm a b"  haftmann@58023  1266  by (simp only: Lcm_insert Lcm_empty lcm_1_right)  haftmann@58023  1267  (cases "b = 0", simp, rule lcm_div_unit2, simp)  haftmann@58023  1268 haftmann@58023  1269 lemma Lcm_coprime:  haftmann@58023  1270  assumes "finite A" and "A \ {}"  haftmann@58023  1271  assumes "\a b. a \ A \ b \ A \ a \ b \ gcd a b = 1"  haftmann@60438  1272  shows "Lcm A = \A div normalization_factor (\A)"  haftmann@58023  1273 using assms proof (induct rule: finite_ne_induct)  haftmann@58023  1274  case (insert a A)  haftmann@58023  1275  have "Lcm (insert a A) = lcm a (Lcm A)" by simp  haftmann@60438  1276  also from insert have "Lcm A = \A div normalization_factor (\A)" by blast  haftmann@58023  1277  also have "lcm a \ = lcm a (\A)" by (cases "\A = 0") (simp_all add: lcm_div_unit2)  haftmann@58023  1278  also from insert have "gcd a (\A) = 1" by (subst gcd.commute, intro setprod_coprime) auto  haftmann@60438  1279  with insert have "lcm a (\A) = \(insert a A) div normalization_factor (\(insert a A))"  haftmann@58023  1280  by (simp add: lcm_coprime)  haftmann@58023  1281  finally show ?case .  haftmann@58023  1282 qed simp  haftmann@58023  1283   haftmann@58023  1284 lemma Lcm_coprime':  haftmann@58023  1285  "card A \ 0 \ (\a b. a \ A \ b \ A \ a \ b \ gcd a b = 1)  haftmann@60438  1286  \ Lcm A = \A div normalization_factor (\A)"  haftmann@58023  1287  by (rule Lcm_coprime) (simp_all add: card_eq_0_iff)  haftmann@58023  1288 haftmann@58023  1289 lemma Gcd_Lcm:  haftmann@60430  1290  "Gcd A = Lcm {d. \a\A. d dvd a}"  haftmann@58023  1291  by (simp add: Gcd_Gcd_eucl Lcm_Lcm_eucl Gcd_eucl_def)  haftmann@58023  1292 haftmann@60430  1293 lemma Gcd_dvd [simp]: "a \ A \ Gcd A dvd a"  haftmann@60430  1294  and dvd_Gcd [simp]: "(\a\A. g' dvd a) \ g' dvd Gcd A"  haftmann@60438  1295  and normalization_factor_Gcd [simp]:  haftmann@60438  1296  "normalization_factor (Gcd A) = (if Gcd A = 0 then 0 else 1)"  haftmann@58023  1297 proof -  haftmann@60430  1298  fix a assume "a \ A"  haftmann@60430  1299  hence "Lcm {d. \a\A. d dvd a} dvd a" by (intro Lcm_dvd) blast  haftmann@60430  1300  then show "Gcd A dvd a" by (simp add: Gcd_Lcm)  haftmann@58023  1301 next  haftmann@60430  1302  fix g' assume "\a\A. g' dvd a"  haftmann@60430  1303  hence "g' dvd Lcm {d. \a\A. d dvd a}" by (intro dvd_Lcm) blast  haftmann@58023  1304  then show "g' dvd Gcd A" by (simp add: Gcd_Lcm)  haftmann@58023  1305 next  haftmann@60438  1306  show "normalization_factor (Gcd A) = (if Gcd A = 0 then 0 else 1)"  haftmann@59009  1307  by (simp add: Gcd_Lcm)  haftmann@58023  1308 qed  haftmann@58023  1309 haftmann@58023  1310 lemma GcdI:  haftmann@60430  1311  "(\a. a\A \ l dvd a) \ (\l'. (\a\A. l' dvd a) \ l' dvd l) \  haftmann@60438  1312  normalization_factor l = (if l = 0 then 0 else 1) \ l = Gcd A"  haftmann@58023  1313  by (intro normed_associated_imp_eq)  haftmann@58023  1314  (auto intro: Gcd_dvd dvd_Gcd simp: associated_def)  haftmann@58023  1315 haftmann@58023  1316 lemma Lcm_Gcd:  haftmann@60430  1317  "Lcm A = Gcd {m. \a\A. a dvd m}"  haftmann@58023  1318  by (rule LcmI[symmetric]) (auto intro: dvd_Gcd Gcd_dvd)  haftmann@58023  1319 haftmann@58023  1320 lemma Gcd_0_iff:  haftmann@58023  1321  "Gcd A = 0 \ A \ {0}"  haftmann@58023  1322  apply (rule iffI)  haftmann@58023  1323  apply (rule subsetI, drule Gcd_dvd, simp)  haftmann@58023  1324  apply (auto intro: GcdI[symmetric])  haftmann@58023  1325  done  haftmann@58023  1326 haftmann@58023  1327 lemma Gcd_empty [simp]:  haftmann@58023  1328  "Gcd {} = 0"  haftmann@58023  1329  by (simp add: Gcd_0_iff)  haftmann@58023  1330 haftmann@58023  1331 lemma Gcd_1:  haftmann@58023  1332  "1 \ A \ Gcd A = 1"  haftmann@58023  1333  by (intro GcdI[symmetric]) (auto intro: Gcd_dvd dvd_Gcd)  haftmann@58023  1334 haftmann@58023  1335 lemma Gcd_insert [simp]:  haftmann@58023  1336  "Gcd (insert a A) = gcd a (Gcd A)"  haftmann@58023  1337 proof (rule gcdI)  haftmann@58023  1338  fix l assume "l dvd a" and "l dvd Gcd A"  haftmann@60430  1339  hence "\a\A. l dvd a" by (blast intro: dvd_trans Gcd_dvd)  wenzelm@60526  1340  with \l dvd a\ show "l dvd Gcd (insert a A)" by (force intro: Gcd_dvd)  haftmann@59009  1341 qed auto  haftmann@58023  1342 haftmann@58023  1343 lemma Gcd_finite:  haftmann@58023  1344  assumes "finite A"  haftmann@58023  1345  shows "Gcd A = Finite_Set.fold gcd 0 A"  wenzelm@60526  1346  by (induct rule: finite.induct[OF \finite A\])  haftmann@58023  1347  (simp_all add: comp_fun_idem.fold_insert_idem[OF comp_fun_idem_gcd])  haftmann@58023  1348 haftmann@60431  1349 lemma Gcd_set [code_unfold]:  haftmann@58023  1350  "Gcd (set xs) = fold gcd xs 0"  haftmann@58023  1351  using comp_fun_idem.fold_set_fold[OF comp_fun_idem_gcd] Gcd_finite by (simp add: ac_simps)  haftmann@58023  1352 haftmann@60438  1353 lemma Gcd_singleton [simp]: "Gcd {a} = a div normalization_factor a"  haftmann@58023  1354  by (simp add: gcd_0)  haftmann@58023  1355 haftmann@58023  1356 lemma Gcd_2 [simp]: "Gcd {a,b} = gcd a b"  haftmann@58023  1357  by (simp only: Gcd_insert Gcd_empty gcd_0) (cases "b = 0", simp, rule gcd_div_unit2, simp)  haftmann@58023  1358 haftmann@60439  1359 subclass semiring_gcd  haftmann@60439  1360  by unfold_locales (simp_all add: gcd_greatest_iff)  haftmann@60439  1361   haftmann@58023  1362 end  haftmann@58023  1363 wenzelm@60526  1364 text \  haftmann@58023  1365  A Euclidean ring is a Euclidean semiring with additive inverses. It provides a  haftmann@58023  1366  few more lemmas; in particular, Bezout's lemma holds for any Euclidean ring.  wenzelm@60526  1367 \  haftmann@58023  1368 haftmann@58023  1369 class euclidean_ring = euclidean_semiring + idom  haftmann@58023  1370 haftmann@58023  1371 class euclidean_ring_gcd = euclidean_semiring_gcd + idom  haftmann@58023  1372 begin  haftmann@58023  1373 haftmann@58023  1374 subclass euclidean_ring ..  haftmann@58023  1375 haftmann@60439  1376 subclass ring_gcd ..  haftmann@60439  1377 haftmann@58023  1378 lemma gcd_neg1 [simp]:  haftmann@60430  1379  "gcd (-a) b = gcd a b"  haftmann@59009  1380  by (rule sym, rule gcdI, simp_all add: gcd_greatest)  haftmann@58023  1381 haftmann@58023  1382 lemma gcd_neg2 [simp]:  haftmann@60430  1383  "gcd a (-b) = gcd a b"  haftmann@59009  1384  by (rule sym, rule gcdI, simp_all add: gcd_greatest)  haftmann@58023  1385 haftmann@58023  1386 lemma gcd_neg_numeral_1 [simp]:  haftmann@60430  1387  "gcd (- numeral n) a = gcd (numeral n) a"  haftmann@58023  1388  by (fact gcd_neg1)  haftmann@58023  1389 haftmann@58023  1390 lemma gcd_neg_numeral_2 [simp]:  haftmann@60430  1391  "gcd a (- numeral n) = gcd a (numeral n)"  haftmann@58023  1392  by (fact gcd_neg2)  haftmann@58023  1393 haftmann@58023  1394 lemma gcd_diff1: "gcd (m - n) n = gcd m n"  haftmann@58023  1395  by (subst diff_conv_add_uminus, subst gcd_neg2[symmetric], subst gcd_add1, simp)  haftmann@58023  1396 haftmann@58023  1397 lemma gcd_diff2: "gcd (n - m) n = gcd m n"  haftmann@58023  1398  by (subst gcd_neg1[symmetric], simp only: minus_diff_eq gcd_diff1)  haftmann@58023  1399 haftmann@58023  1400 lemma coprime_minus_one [simp]: "gcd (n - 1) n = 1"  haftmann@58023  1401 proof -  haftmann@58023  1402  have "gcd (n - 1) n = gcd n (n - 1)" by (fact gcd.commute)  haftmann@58023  1403  also have "\ = gcd ((n - 1) + 1) (n - 1)" by simp  haftmann@58023  1404  also have "\ = 1" by (rule coprime_plus_one)  haftmann@58023  1405  finally show ?thesis .  haftmann@58023  1406 qed  haftmann@58023  1407 haftmann@60430  1408 lemma lcm_neg1 [simp]: "lcm (-a) b = lcm a b"  haftmann@58023  1409  by (rule sym, rule lcmI, simp_all add: lcm_least lcm_zero)  haftmann@58023  1410 haftmann@60430  1411 lemma lcm_neg2 [simp]: "lcm a (-b) = lcm a b"  haftmann@58023  1412  by (rule sym, rule lcmI, simp_all add: lcm_least lcm_zero)  haftmann@58023  1413 haftmann@60430  1414 lemma lcm_neg_numeral_1 [simp]: "lcm (- numeral n) a = lcm (numeral n) a"  haftmann@58023  1415  by (fact lcm_neg1)  haftmann@58023  1416 haftmann@60430  1417 lemma lcm_neg_numeral_2 [simp]: "lcm a (- numeral n) = lcm a (numeral n)"  haftmann@58023  1418  by (fact lcm_neg2)  haftmann@58023  1419 haftmann@58023  1420 function euclid_ext :: "'a \ 'a \ 'a \ 'a \ 'a" where  haftmann@58023  1421  "euclid_ext a b =  haftmann@58023  1422  (if b = 0 then  haftmann@60438  1423  let c = 1 div normalization_factor a in (c, 0, a * c)  haftmann@58023  1424  else  haftmann@58023  1425  case euclid_ext b (a mod b) of  haftmann@58023  1426  (s,t,c) \ (t, s - t * (a div b), c))"  haftmann@58023  1427  by (pat_completeness, simp)  haftmann@58023  1428  termination by (relation "measure (euclidean_size \ snd)", simp_all)  haftmann@58023  1429 haftmann@58023  1430 declare euclid_ext.simps [simp del]  haftmann@58023  1431 haftmann@58023  1432 lemma euclid_ext_0:  haftmann@60438  1433  "euclid_ext a 0 = (1 div normalization_factor a, 0, a div normalization_factor a)"  haftmann@60433  1434  by (subst euclid_ext.simps) (simp add: Let_def)  haftmann@58023  1435 haftmann@58023  1436 lemma euclid_ext_non_0:  haftmann@58023  1437  "b \ 0 \ euclid_ext a b = (case euclid_ext b (a mod b) of  haftmann@58023  1438  (s,t,c) \ (t, s - t * (a div b), c))"  haftmann@60433  1439  by (subst euclid_ext.simps) simp  haftmann@58023  1440 haftmann@58023  1441 definition euclid_ext' :: "'a \ 'a \ 'a \ 'a"  haftmann@58023  1442 where  haftmann@58023  1443  "euclid_ext' a b = (case euclid_ext a b of (s, t, _) \ (s, t))"  haftmann@58023  1444 haftmann@58023  1445 lemma euclid_ext_gcd [simp]:  haftmann@58023  1446  "(case euclid_ext a b of (_,_,t) \ t) = gcd a b"  haftmann@58023  1447 proof (induct a b rule: euclid_ext.induct)  haftmann@58023  1448  case (1 a b)  haftmann@58023  1449  then show ?case  haftmann@58023  1450  proof (cases "b = 0")  haftmann@58023  1451  case True  haftmann@60433  1452  then show ?thesis by  haftmann@60433  1453  (simp add: euclid_ext_0 unit_div mult_ac unit_simps gcd_0)  haftmann@58023  1454  next  haftmann@58023  1455  case False with 1 show ?thesis  haftmann@58023  1456  by (simp add: euclid_ext_non_0 ac_simps split: prod.split prod.split_asm)  haftmann@58023  1457  qed  haftmann@58023  1458 qed  haftmann@58023  1459 haftmann@58023  1460 lemma euclid_ext_gcd' [simp]:  haftmann@58023  1461  "euclid_ext a b = (r, s, t) \ t = gcd a b"  haftmann@58023  1462  by (insert euclid_ext_gcd[of a b], drule (1) subst, simp)  haftmann@58023  1463 haftmann@58023  1464 lemma euclid_ext_correct:  haftmann@60430  1465  "case euclid_ext a b of (s,t,c) \ s*a + t*b = c"  haftmann@60430  1466 proof (induct a b rule: euclid_ext.induct)  haftmann@60430  1467  case (1 a b)  haftmann@58023  1468  show ?case  haftmann@60430  1469  proof (cases "b = 0")  haftmann@58023  1470  case True  haftmann@58023  1471  then show ?thesis by (simp add: euclid_ext_0 mult_ac)  haftmann@58023  1472  next  haftmann@58023  1473  case False  haftmann@60430  1474  obtain s t c where stc: "euclid_ext b (a mod b) = (s,t,c)"  haftmann@60430  1475  by (cases "euclid_ext b (a mod b)", blast)  haftmann@60430  1476  from 1 have "c = s * b + t * (a mod b)" by (simp add: stc False)  haftmann@60430  1477  also have "... = t*((a div b)*b + a mod b) + (s - t * (a div b))*b"  haftmann@58023  1478  by (simp add: algebra_simps)  haftmann@60430  1479  also have "(a div b)*b + a mod b = a" using mod_div_equality .  haftmann@58023  1480  finally show ?thesis  haftmann@58023  1481  by (subst euclid_ext.simps, simp add: False stc)  haftmann@58023  1482  qed  haftmann@58023  1483 qed  haftmann@58023  1484 haftmann@58023  1485 lemma euclid_ext'_correct:  haftmann@58023  1486  "fst (euclid_ext' a b) * a + snd (euclid_ext' a b) * b = gcd a b"  haftmann@58023  1487 proof-  haftmann@58023  1488  obtain s t c where "euclid_ext a b = (s,t,c)"  haftmann@58023  1489  by (cases "euclid_ext a b", blast)  haftmann@58023  1490  with euclid_ext_correct[of a b] euclid_ext_gcd[of a b]  haftmann@58023  1491  show ?thesis unfolding euclid_ext'_def by simp  haftmann@58023  1492 qed  haftmann@58023  1493 haftmann@60430  1494 lemma bezout: "\s t. s * a + t * b = gcd a b"  haftmann@58023  1495  using euclid_ext'_correct by blast  haftmann@58023  1496 haftmann@60438  1497 lemma euclid_ext'_0 [simp]: "euclid_ext' a 0 = (1 div normalization_factor a, 0)"  haftmann@58023  1498  by (simp add: bezw_def euclid_ext'_def euclid_ext_0)  haftmann@58023  1499 haftmann@60430  1500 lemma euclid_ext'_non_0: "b \ 0 \ euclid_ext' a b = (snd (euclid_ext' b (a mod b)),  haftmann@60430  1501  fst (euclid_ext' b (a mod b)) - snd (euclid_ext' b (a mod b)) * (a div b))"  haftmann@60430  1502  by (cases "euclid_ext b (a mod b)")  haftmann@58023  1503  (simp add: euclid_ext'_def euclid_ext_non_0)  haftmann@58023  1504   haftmann@58023  1505 end  haftmann@58023  1506 haftmann@58023  1507 instantiation nat :: euclidean_semiring  haftmann@58023  1508 begin  haftmann@58023  1509 haftmann@58023  1510 definition [simp]:  haftmann@58023  1511  "euclidean_size_nat = (id :: nat \ nat)"  haftmann@58023  1512 haftmann@58023  1513 definition [simp]:  haftmann@60438  1514  "normalization_factor_nat (n::nat) = (if n = 0 then 0 else 1 :: nat)"  haftmann@58023  1515 haftmann@58023  1516 instance proof  haftmann@59061  1517 qed simp_all  haftmann@58023  1518 haftmann@58023  1519 end  haftmann@58023  1520 haftmann@58023  1521 instantiation int :: euclidean_ring  haftmann@58023  1522 begin  haftmann@58023  1523 haftmann@58023  1524 definition [simp]:  haftmann@58023  1525  "euclidean_size_int = (nat \ abs :: int \ nat)"  haftmann@58023  1526 haftmann@58023  1527 definition [simp]:  haftmann@60438  1528  "normalization_factor_int = (sgn :: int \ int)"  haftmann@58023  1529 wenzelm@60580  1530 instance  wenzelm@60580  1531 proof (default, goals)  wenzelm@60580  1532  case 2  wenzelm@60580  1533  then show ?case by (auto simp add: abs_mult nat_mult_distrib)  haftmann@58023  1534 next  wenzelm@60580  1535  case 3  wenzelm@60580  1536  then show ?case by (simp add: zsgn_def)  haftmann@58023  1537 next  wenzelm@60580  1538  case 5  wenzelm@60580  1539  then show ?case by (auto simp: zsgn_def)  haftmann@58023  1540 next  wenzelm@60580  1541  case 6  wenzelm@60580  1542  then show ?case by (auto split: abs_split simp: zsgn_def)  haftmann@58023  1543 qed (auto simp: sgn_times split: abs_split)  haftmann@58023  1544 haftmann@58023  1545 end  haftmann@58023  1546 haftmann@58023  1547 end