src/HOL/Number_Theory/Euclidean_Algorithm.thy
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 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@60685  6 imports Main "~~/src/HOL/GCD" "~~/src/HOL/Library/Polynomial"  haftmann@58023  7 begin  haftmann@60634  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@58023  16  \end{itemize}  haftmann@58023  17  The existence of these functions makes it possible to derive gcd and lcm functions  haftmann@58023  18  for any Euclidean semiring.  wenzelm@60526  19 \  haftmann@60634  20 class euclidean_semiring = semiring_div + normalization_semidom +  haftmann@58023  21  fixes euclidean_size :: "'a \ nat"  haftmann@60569  22  assumes mod_size_less:  haftmann@60600  23  "b \ 0 \ euclidean_size (a mod b) < euclidean_size b"  haftmann@58023  24  assumes size_mult_mono:  haftmann@60634  25  "b \ 0 \ euclidean_size a \ euclidean_size (a * b)"  haftmann@58023  26 begin  haftmann@58023  27 haftmann@58023  28 lemma euclidean_division:  haftmann@58023  29  fixes a :: 'a and b :: 'a  haftmann@60600  30  assumes "b \ 0"  haftmann@58023  31  obtains s and t where "a = s * b + t"  haftmann@58023  32  and "euclidean_size t < euclidean_size b"  haftmann@58023  33 proof -  haftmann@60569  34  from div_mod_equality [of a b 0]  haftmann@58023  35  have "a = a div b * b + a mod b" by simp  haftmann@60569  36  with that and assms show ?thesis by (auto simp add: mod_size_less)  haftmann@58023  37 qed  haftmann@58023  38 haftmann@58023  39 lemma dvd_euclidean_size_eq_imp_dvd:  haftmann@58023  40  assumes "a \ 0" and b_dvd_a: "b dvd a" and size_eq: "euclidean_size a = euclidean_size b"  haftmann@58023  41  shows "a dvd b"  haftmann@60569  42 proof (rule ccontr)  haftmann@60569  43  assume "\ a dvd b"  haftmann@60569  44  then have "b mod a \ 0" by (simp add: mod_eq_0_iff_dvd)  haftmann@58023  45  from b_dvd_a have b_dvd_mod: "b dvd b mod a" by (simp add: dvd_mod_iff)  haftmann@58023  46  from b_dvd_mod obtain c where "b mod a = b * c" unfolding dvd_def by blast  wenzelm@60526  47  with \b mod a \ 0\ have "c \ 0" by auto  wenzelm@60526  48  with \b mod a = b * c\ have "euclidean_size (b mod a) \ euclidean_size b"  haftmann@58023  49  using size_mult_mono by force  haftmann@60569  50  moreover from \\ a dvd b\ and \a \ 0\  haftmann@60569  51  have "euclidean_size (b mod a) < euclidean_size a"  haftmann@58023  52  using mod_size_less by blast  haftmann@58023  53  ultimately show False using size_eq by simp  haftmann@58023  54 qed  haftmann@58023  55 haftmann@58023  56 function gcd_eucl :: "'a \ 'a \ 'a"  haftmann@58023  57 where  haftmann@60634  58  "gcd_eucl a b = (if b = 0 then normalize a else gcd_eucl b (a mod b))"  haftmann@60572  59  by pat_completeness simp  haftmann@60569  60 termination  haftmann@60569  61  by (relation "measure (euclidean_size \ snd)") (simp_all add: mod_size_less)  haftmann@58023  62 haftmann@58023  63 declare gcd_eucl.simps [simp del]  haftmann@58023  64 haftmann@60569  65 lemma gcd_eucl_induct [case_names zero mod]:  haftmann@60569  66  assumes H1: "\b. P b 0"  haftmann@60569  67  and H2: "\a b. b \ 0 \ P b (a mod b) \ P a b"  haftmann@60569  68  shows "P a b"  haftmann@58023  69 proof (induct a b rule: gcd_eucl.induct)  haftmann@60569  70  case ("1" a b)  haftmann@60569  71  show ?case  haftmann@60569  72  proof (cases "b = 0")  haftmann@60569  73  case True then show "P a b" by simp (rule H1)  haftmann@60569  74  next  haftmann@60569  75  case False  haftmann@60600  76  then have "P b (a mod b)"  haftmann@60600  77  by (rule "1.hyps")  haftmann@60569  78  with \b \ 0\ show "P a b"  haftmann@60569  79  by (blast intro: H2)  haftmann@60569  80  qed  haftmann@58023  81 qed  haftmann@58023  82 haftmann@58023  83 definition lcm_eucl :: "'a \ 'a \ 'a"  haftmann@58023  84 where  haftmann@60634  85  "lcm_eucl a b = normalize (a * b) div gcd_eucl a b"  haftmann@58023  86 haftmann@60572  87 definition Lcm_eucl :: "'a set \ 'a" -- \  haftmann@60572  88  Somewhat complicated definition of Lcm that has the advantage of working  haftmann@60572  89  for infinite sets as well\  haftmann@58023  90 where  haftmann@60430  91  "Lcm_eucl A = (if \l. l \ 0 \ (\a\A. a dvd l) then  haftmann@60430  92  let l = SOME l. l \ 0 \ (\a\A. a dvd l) \ euclidean_size l =  haftmann@60430  93  (LEAST n. \l. l \ 0 \ (\a\A. a dvd l) \ euclidean_size l = n)  haftmann@60634  94  in normalize l  haftmann@58023  95  else 0)"  haftmann@58023  96 haftmann@58023  97 definition Gcd_eucl :: "'a set \ 'a"  haftmann@58023  98 where  haftmann@58023  99  "Gcd_eucl A = Lcm_eucl {d. \a\A. d dvd a}"  haftmann@58023  100 haftmann@60572  101 lemma gcd_eucl_0:  haftmann@60634  102  "gcd_eucl a 0 = normalize a"  haftmann@60572  103  by (simp add: gcd_eucl.simps [of a 0])  haftmann@60572  104 haftmann@60572  105 lemma gcd_eucl_0_left:  haftmann@60634  106  "gcd_eucl 0 a = normalize a"  haftmann@60600  107  by (simp_all add: gcd_eucl_0 gcd_eucl.simps [of 0 a])  haftmann@60572  108 haftmann@60572  109 lemma gcd_eucl_non_0:  haftmann@60572  110  "b \ 0 \ gcd_eucl a b = gcd_eucl b (a mod b)"  haftmann@60600  111  by (simp add: gcd_eucl.simps [of a b] gcd_eucl.simps [of b 0])  haftmann@60572  112 haftmann@58023  113 end  haftmann@58023  114 haftmann@60598  115 class euclidean_ring = euclidean_semiring + idom  haftmann@60598  116 begin  haftmann@60598  117 haftmann@60598  118 function euclid_ext :: "'a \ 'a \ 'a \ 'a \ 'a" where  haftmann@60598  119  "euclid_ext a b =  haftmann@60598  120  (if b = 0 then  haftmann@60634  121  (1 div unit_factor a, 0, normalize a)  haftmann@60598  122  else  haftmann@60598  123  case euclid_ext b (a mod b) of  haftmann@60598  124  (s, t, c) \ (t, s - t * (a div b), c))"  haftmann@60598  125  by pat_completeness simp  haftmann@60598  126 termination  haftmann@60598  127  by (relation "measure (euclidean_size \ snd)") (simp_all add: mod_size_less)  haftmann@60598  128 haftmann@60598  129 declare euclid_ext.simps [simp del]  haftmann@60598  130 haftmann@60598  131 lemma euclid_ext_0:  haftmann@60634  132  "euclid_ext a 0 = (1 div unit_factor a, 0, normalize a)"  haftmann@60598  133  by (simp add: euclid_ext.simps [of a 0])  haftmann@60598  134 haftmann@60598  135 lemma euclid_ext_left_0:  haftmann@60634  136  "euclid_ext 0 a = (0, 1 div unit_factor a, normalize a)"  haftmann@60600  137  by (simp add: euclid_ext_0 euclid_ext.simps [of 0 a])  haftmann@60598  138 haftmann@60598  139 lemma euclid_ext_non_0:  haftmann@60598  140  "b \ 0 \ euclid_ext a b = (case euclid_ext b (a mod b) of  haftmann@60598  141  (s, t, c) \ (t, s - t * (a div b), c))"  haftmann@60600  142  by (simp add: euclid_ext.simps [of a b] euclid_ext.simps [of b 0])  haftmann@60598  143 haftmann@60598  144 lemma euclid_ext_code [code]:  haftmann@60634  145  "euclid_ext a b = (if b = 0 then (1 div unit_factor a, 0, normalize a)  haftmann@60598  146  else let (s, t, c) = euclid_ext b (a mod b) in (t, s - t * (a div b), c))"  haftmann@60598  147  by (simp add: euclid_ext.simps [of a b] euclid_ext.simps [of b 0])  haftmann@60598  148 haftmann@60598  149 lemma euclid_ext_correct:  haftmann@60598  150  "case euclid_ext a b of (s, t, c) \ s * a + t * b = c"  haftmann@60598  151 proof (induct a b rule: gcd_eucl_induct)  haftmann@60598  152  case (zero a) then show ?case  haftmann@60598  153  by (simp add: euclid_ext_0 ac_simps)  haftmann@60598  154 next  haftmann@60598  155  case (mod a b)  haftmann@60598  156  obtain s t c where stc: "euclid_ext b (a mod b) = (s,t,c)"  haftmann@60598  157  by (cases "euclid_ext b (a mod b)") blast  haftmann@60598  158  with mod have "c = s * b + t * (a mod b)" by simp  haftmann@60598  159  also have "... = t * ((a div b) * b + a mod b) + (s - t * (a div b)) * b"  haftmann@60598  160  by (simp add: algebra_simps)  haftmann@60598  161  also have "(a div b) * b + a mod b = a" using mod_div_equality .  haftmann@60598  162  finally show ?case  haftmann@60598  163  by (subst euclid_ext.simps) (simp add: stc mod ac_simps)  haftmann@60598  164 qed  haftmann@60598  165 haftmann@60598  166 definition euclid_ext' :: "'a \ 'a \ 'a \ 'a"  haftmann@60598  167 where  haftmann@60598  168  "euclid_ext' a b = (case euclid_ext a b of (s, t, _) \ (s, t))"  haftmann@60598  169 haftmann@60634  170 lemma euclid_ext'_0: "euclid_ext' a 0 = (1 div unit_factor a, 0)"  haftmann@60598  171  by (simp add: euclid_ext'_def euclid_ext_0)  haftmann@60598  172 haftmann@60634  173 lemma euclid_ext'_left_0: "euclid_ext' 0 a = (0, 1 div unit_factor a)"  haftmann@60598  174  by (simp add: euclid_ext'_def euclid_ext_left_0)  haftmann@60598  175   haftmann@60598  176 lemma euclid_ext'_non_0: "b \ 0 \ euclid_ext' a b = (snd (euclid_ext' b (a mod b)),  haftmann@60598  177  fst (euclid_ext' b (a mod b)) - snd (euclid_ext' b (a mod b)) * (a div b))"  haftmann@60598  178  by (simp add: euclid_ext'_def euclid_ext_non_0 split_def)  haftmann@60598  179 haftmann@60598  180 end  haftmann@60598  181 haftmann@58023  182 class euclidean_semiring_gcd = euclidean_semiring + gcd + Gcd +  haftmann@58023  183  assumes gcd_gcd_eucl: "gcd = gcd_eucl" and lcm_lcm_eucl: "lcm = lcm_eucl"  haftmann@58023  184  assumes Gcd_Gcd_eucl: "Gcd = Gcd_eucl" and Lcm_Lcm_eucl: "Lcm = Lcm_eucl"  haftmann@58023  185 begin  haftmann@58023  186 haftmann@58023  187 lemma gcd_0_left:  haftmann@60634  188  "gcd 0 a = normalize a"  haftmann@60572  189  unfolding gcd_gcd_eucl by (fact gcd_eucl_0_left)  haftmann@58023  190 haftmann@58023  191 lemma gcd_0:  haftmann@60634  192  "gcd a 0 = normalize a"  haftmann@60572  193  unfolding gcd_gcd_eucl by (fact gcd_eucl_0)  haftmann@58023  194 haftmann@58023  195 lemma gcd_non_0:  haftmann@60430  196  "b \ 0 \ gcd a b = gcd b (a mod b)"  haftmann@60572  197  unfolding gcd_gcd_eucl by (fact gcd_eucl_non_0)  haftmann@58023  198 haftmann@60430  199 lemma gcd_dvd1 [iff]: "gcd a b dvd a"  haftmann@60430  200  and gcd_dvd2 [iff]: "gcd a b dvd b"  haftmann@60569  201  by (induct a b rule: gcd_eucl_induct)  haftmann@60569  202  (simp_all add: gcd_0 gcd_non_0 dvd_mod_iff)  haftmann@60569  203   haftmann@58023  204 lemma dvd_gcd_D1: "k dvd gcd m n \ k dvd m"  haftmann@58023  205  by (rule dvd_trans, assumption, rule gcd_dvd1)  haftmann@58023  206 haftmann@58023  207 lemma dvd_gcd_D2: "k dvd gcd m n \ k dvd n"  haftmann@58023  208  by (rule dvd_trans, assumption, rule gcd_dvd2)  haftmann@58023  209 haftmann@58023  210 lemma gcd_greatest:  haftmann@60430  211  fixes k a b :: 'a  haftmann@60430  212  shows "k dvd a \ k dvd b \ k dvd gcd a b"  haftmann@60569  213 proof (induct a b rule: gcd_eucl_induct)  haftmann@60569  214  case (zero a) from zero(1) show ?case by (rule dvd_trans) (simp add: gcd_0)  haftmann@60569  215 next  haftmann@60569  216  case (mod a b)  haftmann@60569  217  then show ?case  haftmann@60569  218  by (simp add: gcd_non_0 dvd_mod_iff)  haftmann@58023  219 qed  haftmann@58023  220 haftmann@58023  221 lemma dvd_gcd_iff:  haftmann@60430  222  "k dvd gcd a b \ k dvd a \ k dvd b"  haftmann@58023  223  by (blast intro!: gcd_greatest intro: dvd_trans)  haftmann@58023  224 haftmann@58023  225 lemmas gcd_greatest_iff = dvd_gcd_iff  haftmann@58023  226 haftmann@58023  227 lemma gcd_zero [simp]:  haftmann@60430  228  "gcd a b = 0 \ a = 0 \ b = 0"  haftmann@58023  229  by (metis dvd_0_left dvd_refl gcd_dvd1 gcd_dvd2 gcd_greatest)+  haftmann@58023  230 haftmann@60688  231 lemma normalize_gcd [simp]:  haftmann@60688  232  "normalize (gcd a b) = gcd a b"  haftmann@60688  233  by (induct a b rule: gcd_eucl_induct) (simp_all add: gcd_0 gcd_non_0)  haftmann@58023  234 haftmann@58023  235 lemma gcdI:  haftmann@60634  236  assumes "c dvd a" and "c dvd b" and greatest: "\d. d dvd a \ d dvd b \ d dvd c"  haftmann@60688  237  and "normalize c = c"  haftmann@60634  238  shows "c = gcd a b"  haftmann@60688  239  by (rule associated_eqI) (auto simp: assms intro: gcd_greatest)  haftmann@58023  240 wenzelm@61605  241 sublocale gcd: abel_semigroup gcd  haftmann@58023  242 proof  haftmann@60430  243  fix a b c  haftmann@60430  244  show "gcd (gcd a b) c = gcd a (gcd b c)"  haftmann@58023  245  proof (rule gcdI)  haftmann@60430  246  have "gcd (gcd a b) c dvd gcd a b" "gcd a b dvd a" by simp_all  haftmann@60430  247  then show "gcd (gcd a b) c dvd a" by (rule dvd_trans)  haftmann@60430  248  have "gcd (gcd a b) c dvd gcd a b" "gcd a b dvd b" by simp_all  haftmann@60430  249  hence "gcd (gcd a b) c dvd b" by (rule dvd_trans)  haftmann@60430  250  moreover have "gcd (gcd a b) c dvd c" by simp  haftmann@60430  251  ultimately show "gcd (gcd a b) c dvd gcd b c"  haftmann@58023  252  by (rule gcd_greatest)  haftmann@60688  253  show "normalize (gcd (gcd a b) c) = gcd (gcd a b) c"  haftmann@58023  254  by auto  haftmann@60430  255  fix l assume "l dvd a" and "l dvd gcd b c"  haftmann@60688  256  with dvd_trans [OF _ gcd_dvd1] and dvd_trans [OF _ gcd_dvd2]  haftmann@60430  257  have "l dvd b" and "l dvd c" by blast+  wenzelm@60526  258  with \l dvd a\ show "l dvd gcd (gcd a b) c"  haftmann@58023  259  by (intro gcd_greatest)  haftmann@58023  260  qed  haftmann@58023  261 next  haftmann@60430  262  fix a b  haftmann@60430  263  show "gcd a b = gcd b a"  haftmann@58023  264  by (rule gcdI) (simp_all add: gcd_greatest)  haftmann@58023  265 qed  haftmann@58023  266 haftmann@58023  267 lemma gcd_unique: "d dvd a \ d dvd b \  haftmann@60688  268  normalize d = d \  haftmann@58023  269  (\e. e dvd a \ e dvd b \ e dvd d) \ d = gcd a b"  haftmann@60688  270  by rule (auto intro: gcdI simp: gcd_greatest)  haftmann@58023  271 haftmann@58023  272 lemma gcd_dvd_prod: "gcd a b dvd k * b"  haftmann@58023  273  using mult_dvd_mono [of 1] by auto  haftmann@58023  274 haftmann@60430  275 lemma gcd_1_left [simp]: "gcd 1 a = 1"  haftmann@58023  276  by (rule sym, rule gcdI, simp_all)  haftmann@58023  277 haftmann@60430  278 lemma gcd_1 [simp]: "gcd a 1 = 1"  haftmann@58023  279  by (rule sym, rule gcdI, simp_all)  haftmann@58023  280 haftmann@58023  281 lemma gcd_proj2_if_dvd:  haftmann@60634  282  "b dvd a \ gcd a b = normalize b"  haftmann@60430  283  by (cases "b = 0", simp_all add: dvd_eq_mod_eq_0 gcd_non_0 gcd_0)  haftmann@58023  284 haftmann@58023  285 lemma gcd_proj1_if_dvd:  haftmann@60634  286  "a dvd b \ gcd a b = normalize a"  haftmann@58023  287  by (subst gcd.commute, simp add: gcd_proj2_if_dvd)  haftmann@58023  288 haftmann@60634  289 lemma gcd_proj1_iff: "gcd m n = normalize m \ m dvd n"  haftmann@58023  290 proof  haftmann@60634  291  assume A: "gcd m n = normalize m"  haftmann@58023  292  show "m dvd n"  haftmann@58023  293  proof (cases "m = 0")  haftmann@58023  294  assume [simp]: "m \ 0"  haftmann@60634  295  from A have B: "m = gcd m n * unit_factor m"  haftmann@58023  296  by (simp add: unit_eq_div2)  haftmann@58023  297  show ?thesis by (subst B, simp add: mult_unit_dvd_iff)  haftmann@58023  298  qed (insert A, simp)  haftmann@58023  299 next  haftmann@58023  300  assume "m dvd n"  haftmann@60634  301  then show "gcd m n = normalize m" by (rule gcd_proj1_if_dvd)  haftmann@58023  302 qed  haftmann@58023  303   haftmann@60634  304 lemma gcd_proj2_iff: "gcd m n = normalize n \ n dvd m"  haftmann@60634  305  using gcd_proj1_iff [of n m] by (simp add: ac_simps)  haftmann@58023  306 haftmann@58023  307 lemma gcd_mod1 [simp]:  haftmann@60430  308  "gcd (a mod b) b = gcd a b"  haftmann@58023  309  by (rule gcdI, metis dvd_mod_iff gcd_dvd1 gcd_dvd2, simp_all add: gcd_greatest dvd_mod_iff)  haftmann@58023  310 haftmann@58023  311 lemma gcd_mod2 [simp]:  haftmann@60430  312  "gcd a (b mod a) = gcd a b"  haftmann@58023  313  by (rule gcdI, simp, metis dvd_mod_iff gcd_dvd1 gcd_dvd2, simp_all add: gcd_greatest dvd_mod_iff)  haftmann@58023  314   haftmann@58023  315 lemma gcd_mult_distrib':  haftmann@60634  316  "normalize c * gcd a b = gcd (c * a) (c * b)"  haftmann@60569  317 proof (cases "c = 0")  haftmann@60569  318  case True then show ?thesis by (simp_all add: gcd_0)  haftmann@60569  319 next  haftmann@60634  320  case False then have [simp]: "is_unit (unit_factor c)" by simp  haftmann@60569  321  show ?thesis  haftmann@60569  322  proof (induct a b rule: gcd_eucl_induct)  haftmann@60569  323  case (zero a) show ?case  haftmann@60569  324  proof (cases "a = 0")  haftmann@60569  325  case True then show ?thesis by (simp add: gcd_0)  haftmann@60569  326  next  haftmann@60634  327  case False  haftmann@60634  328  then show ?thesis by (simp add: gcd_0 normalize_mult)  haftmann@60569  329  qed  haftmann@60569  330  case (mod a b)  haftmann@60569  331  then show ?case by (simp add: mult_mod_right gcd.commute)  haftmann@58023  332  qed  haftmann@58023  333 qed  haftmann@58023  334 haftmann@58023  335 lemma gcd_mult_distrib:  haftmann@60634  336  "k * gcd a b = gcd (k * a) (k * b) * unit_factor k"  haftmann@58023  337 proof-  haftmann@60634  338  have "normalize k * gcd a b = gcd (k * a) (k * b)"  haftmann@60634  339  by (simp add: gcd_mult_distrib')  haftmann@60634  340  then have "normalize k * gcd a b * unit_factor k = gcd (k * a) (k * b) * unit_factor k"  haftmann@60634  341  by simp  haftmann@60634  342  then have "normalize k * unit_factor k * gcd a b = gcd (k * a) (k * b) * unit_factor k"  haftmann@60634  343  by (simp only: ac_simps)  haftmann@60634  344  then show ?thesis  haftmann@59009  345  by simp  haftmann@58023  346 qed  haftmann@58023  347 haftmann@58023  348 lemma euclidean_size_gcd_le1 [simp]:  haftmann@58023  349  assumes "a \ 0"  haftmann@58023  350  shows "euclidean_size (gcd a b) \ euclidean_size a"  haftmann@58023  351 proof -  haftmann@58023  352  have "gcd a b dvd a" by (rule gcd_dvd1)  haftmann@58023  353  then obtain c where A: "a = gcd a b * c" unfolding dvd_def by blast  wenzelm@60526  354  with \a \ 0\ show ?thesis by (subst (2) A, intro size_mult_mono) auto  haftmann@58023  355 qed  haftmann@58023  356 haftmann@58023  357 lemma euclidean_size_gcd_le2 [simp]:  haftmann@58023  358  "b \ 0 \ euclidean_size (gcd a b) \ euclidean_size b"  haftmann@58023  359  by (subst gcd.commute, rule euclidean_size_gcd_le1)  haftmann@58023  360 haftmann@58023  361 lemma euclidean_size_gcd_less1:  haftmann@58023  362  assumes "a \ 0" and "\a dvd b"  haftmann@58023  363  shows "euclidean_size (gcd a b) < euclidean_size a"  haftmann@58023  364 proof (rule ccontr)  haftmann@58023  365  assume "\euclidean_size (gcd a b) < euclidean_size a"  wenzelm@60526  366  with \a \ 0\ have "euclidean_size (gcd a b) = euclidean_size a"  haftmann@58023  367  by (intro le_antisym, simp_all)  haftmann@58023  368  with assms have "a dvd gcd a b" by (auto intro: dvd_euclidean_size_eq_imp_dvd)  haftmann@58023  369  hence "a dvd b" using dvd_gcd_D2 by blast  wenzelm@60526  370  with \\a dvd b\ show False by contradiction  haftmann@58023  371 qed  haftmann@58023  372 haftmann@58023  373 lemma euclidean_size_gcd_less2:  haftmann@58023  374  assumes "b \ 0" and "\b dvd a"  haftmann@58023  375  shows "euclidean_size (gcd a b) < euclidean_size b"  haftmann@58023  376  using assms by (subst gcd.commute, rule euclidean_size_gcd_less1)  haftmann@58023  377 haftmann@60430  378 lemma gcd_mult_unit1: "is_unit a \ gcd (b * a) c = gcd b c"  haftmann@58023  379  apply (rule gcdI)  haftmann@60688  380  apply simp_all  haftmann@58023  381  apply (rule dvd_trans, rule gcd_dvd1, simp add: unit_simps)  haftmann@58023  382  apply (rule gcd_greatest, simp add: unit_simps, assumption)  haftmann@58023  383  done  haftmann@58023  384 haftmann@60430  385 lemma gcd_mult_unit2: "is_unit a \ gcd b (c * a) = gcd b c"  haftmann@58023  386  by (subst gcd.commute, subst gcd_mult_unit1, assumption, rule gcd.commute)  haftmann@58023  387 haftmann@60430  388 lemma gcd_div_unit1: "is_unit a \ gcd (b div a) c = gcd b c"  haftmann@60433  389  by (erule is_unitE [of _ b]) (simp add: gcd_mult_unit1)  haftmann@58023  390 haftmann@60430  391 lemma gcd_div_unit2: "is_unit a \ gcd b (c div a) = gcd b c"  haftmann@60433  392  by (erule is_unitE [of _ c]) (simp add: gcd_mult_unit2)  haftmann@58023  393 haftmann@60634  394 lemma normalize_gcd_left [simp]:  haftmann@60634  395  "gcd (normalize a) b = gcd a b"  haftmann@60634  396 proof (cases "a = 0")  haftmann@60634  397  case True then show ?thesis  haftmann@60634  398  by simp  haftmann@60634  399 next  haftmann@60634  400  case False then have "is_unit (unit_factor a)"  haftmann@60634  401  by simp  haftmann@60634  402  moreover have "normalize a = a div unit_factor a"  haftmann@60634  403  by simp  haftmann@60634  404  ultimately show ?thesis  haftmann@60634  405  by (simp only: gcd_div_unit1)  haftmann@60634  406 qed  haftmann@60634  407 haftmann@60634  408 lemma normalize_gcd_right [simp]:  haftmann@60634  409  "gcd a (normalize b) = gcd a b"  haftmann@60634  410  using normalize_gcd_left [of b a] by (simp add: ac_simps)  haftmann@60634  411 haftmann@60634  412 lemma gcd_idem: "gcd a a = normalize a"  haftmann@60430  413  by (cases "a = 0") (simp add: gcd_0_left, rule sym, rule gcdI, simp_all)  haftmann@58023  414 haftmann@60430  415 lemma gcd_right_idem: "gcd (gcd a b) b = gcd a b"  haftmann@58023  416  apply (rule gcdI)  haftmann@58023  417  apply (simp add: ac_simps)  haftmann@58023  418  apply (rule gcd_dvd2)  haftmann@58023  419  apply (rule gcd_greatest, erule (1) gcd_greatest, assumption)  haftmann@59009  420  apply simp  haftmann@58023  421  done  haftmann@58023  422 haftmann@60430  423 lemma gcd_left_idem: "gcd a (gcd a b) = gcd a b"  haftmann@58023  424  apply (rule gcdI)  haftmann@58023  425  apply simp  haftmann@58023  426  apply (rule dvd_trans, rule gcd_dvd2, rule gcd_dvd2)  haftmann@58023  427  apply (rule gcd_greatest, assumption, erule gcd_greatest, assumption)  haftmann@59009  428  apply simp  haftmann@58023  429  done  haftmann@58023  430 haftmann@58023  431 lemma comp_fun_idem_gcd: "comp_fun_idem gcd"  haftmann@58023  432 proof  haftmann@58023  433  fix a b show "gcd a \ gcd b = gcd b \ gcd a"  haftmann@58023  434  by (simp add: fun_eq_iff ac_simps)  haftmann@58023  435 next  haftmann@58023  436  fix a show "gcd a \ gcd a = gcd a"  haftmann@58023  437  by (simp add: fun_eq_iff gcd_left_idem)  haftmann@58023  438 qed  haftmann@58023  439 haftmann@58023  440 lemma coprime_dvd_mult:  haftmann@60430  441  assumes "gcd c b = 1" and "c dvd a * b"  haftmann@60430  442  shows "c dvd a"  haftmann@58023  443 proof -  haftmann@60634  444  let ?nf = "unit_factor"  haftmann@60430  445  from assms gcd_mult_distrib [of a c b]  haftmann@60430  446  have A: "a = gcd (a * c) (a * b) * ?nf a" by simp  wenzelm@60526  447  from \c dvd a * b\ show ?thesis by (subst A, simp_all add: gcd_greatest)  haftmann@58023  448 qed  haftmann@58023  449 haftmann@58023  450 lemma coprime_dvd_mult_iff:  haftmann@60430  451  "gcd c b = 1 \ (c dvd a * b) = (c dvd a)"  haftmann@58023  452  by (rule, rule coprime_dvd_mult, simp_all)  haftmann@58023  453 haftmann@58023  454 lemma gcd_dvd_antisym:  haftmann@58023  455  "gcd a b dvd gcd c d \ gcd c d dvd gcd a b \ gcd a b = gcd c d"  haftmann@58023  456 proof (rule gcdI)  haftmann@58023  457  assume A: "gcd a b dvd gcd c d" and B: "gcd c d dvd gcd a b"  haftmann@58023  458  have "gcd c d dvd c" by simp  haftmann@58023  459  with A show "gcd a b dvd c" by (rule dvd_trans)  haftmann@58023  460  have "gcd c d dvd d" by simp  haftmann@58023  461  with A show "gcd a b dvd d" by (rule dvd_trans)  haftmann@60688  462  show "normalize (gcd a b) = gcd a b"  haftmann@59009  463  by simp  haftmann@58023  464  fix l assume "l dvd c" and "l dvd d"  haftmann@58023  465  hence "l dvd gcd c d" by (rule gcd_greatest)  haftmann@58023  466  from this and B show "l dvd gcd a b" by (rule dvd_trans)  haftmann@58023  467 qed  haftmann@58023  468 haftmann@58023  469 lemma gcd_mult_cancel:  haftmann@58023  470  assumes "gcd k n = 1"  haftmann@58023  471  shows "gcd (k * m) n = gcd m n"  haftmann@58023  472 proof (rule gcd_dvd_antisym)  haftmann@58023  473  have "gcd (gcd (k * m) n) k = gcd (gcd k n) (k * m)" by (simp add: ac_simps)  wenzelm@60526  474  also note \gcd k n = 1\  haftmann@58023  475  finally have "gcd (gcd (k * m) n) k = 1" by simp  haftmann@58023  476  hence "gcd (k * m) n dvd m" by (rule coprime_dvd_mult, simp add: ac_simps)  haftmann@58023  477  moreover have "gcd (k * m) n dvd n" by simp  haftmann@58023  478  ultimately show "gcd (k * m) n dvd gcd m n" by (rule gcd_greatest)  haftmann@58023  479  have "gcd m n dvd (k * m)" and "gcd m n dvd n" by simp_all  haftmann@58023  480  then show "gcd m n dvd gcd (k * m) n" by (rule gcd_greatest)  haftmann@58023  481 qed  haftmann@58023  482 haftmann@58023  483 lemma coprime_crossproduct:  haftmann@58023  484  assumes [simp]: "gcd a d = 1" "gcd b c = 1"  haftmann@60688  485  shows "normalize (a * c) = normalize (b * d) \ normalize a = normalize b \ normalize c = normalize d"  haftmann@60688  486  (is "?lhs \ ?rhs")  haftmann@58023  487 proof  haftmann@60688  488  assume ?rhs  haftmann@60688  489  then have "a dvd b" "b dvd a" "c dvd d" "d dvd c" by (simp_all add: associated_iff_dvd)  haftmann@60688  490  then have "a * c dvd b * d" "b * d dvd a * c" by (fast intro: mult_dvd_mono)+  haftmann@60688  491  then show ?lhs by (simp add: associated_iff_dvd)  haftmann@58023  492 next  haftmann@58023  493  assume ?lhs  haftmann@60688  494  then have dvd: "a * c dvd b * d" "b * d dvd a * c" by (simp_all add: associated_iff_dvd)  haftmann@60688  495  then have "a dvd b * d" by (metis dvd_mult_left)  haftmann@58023  496  hence "a dvd b" by (simp add: coprime_dvd_mult_iff)  haftmann@60688  497  moreover from dvd have "b dvd a * c" by (metis dvd_mult_left)  haftmann@58023  498  hence "b dvd a" by (simp add: coprime_dvd_mult_iff)  haftmann@60688  499  moreover from dvd have "c dvd d * b"  haftmann@60688  500  by (auto dest: dvd_mult_right simp add: ac_simps)  haftmann@58023  501  hence "c dvd d" by (simp add: coprime_dvd_mult_iff gcd.commute)  haftmann@60688  502  moreover from dvd have "d dvd c * a"  haftmann@60688  503  by (auto dest: dvd_mult_right simp add: ac_simps)  haftmann@58023  504  hence "d dvd c" by (simp add: coprime_dvd_mult_iff gcd.commute)  haftmann@60688  505  ultimately show ?rhs by (simp add: associated_iff_dvd)  haftmann@58023  506 qed  haftmann@58023  507 haftmann@58023  508 lemma gcd_add1 [simp]:  haftmann@58023  509  "gcd (m + n) n = gcd m n"  haftmann@58023  510  by (cases "n = 0", simp_all add: gcd_non_0)  haftmann@58023  511 haftmann@58023  512 lemma gcd_add2 [simp]:  haftmann@58023  513  "gcd m (m + n) = gcd m n"  haftmann@58023  514  using gcd_add1 [of n m] by (simp add: ac_simps)  haftmann@58023  515 haftmann@60572  516 lemma gcd_add_mult:  haftmann@60572  517  "gcd m (k * m + n) = gcd m n"  haftmann@60572  518 proof -  haftmann@60572  519  have "gcd m ((k * m + n) mod m) = gcd m (k * m + n)"  haftmann@60572  520  by (fact gcd_mod2)  haftmann@60572  521  then show ?thesis by simp  haftmann@60572  522 qed  haftmann@58023  523 haftmann@60430  524 lemma coprimeI: "(\l. \l dvd a; l dvd b\ \ l dvd 1) \ gcd a b = 1"  haftmann@58023  525  by (rule sym, rule gcdI, simp_all)  haftmann@58023  526 haftmann@58023  527 lemma coprime: "gcd a b = 1 \ (\d. d dvd a \ d dvd b \ is_unit d)"  haftmann@59061  528  by (auto intro: coprimeI gcd_greatest dvd_gcd_D1 dvd_gcd_D2)  haftmann@58023  529 haftmann@58023  530 lemma div_gcd_coprime:  haftmann@58023  531  assumes nz: "a \ 0 \ b \ 0"  haftmann@58023  532  defines [simp]: "d \ gcd a b"  haftmann@58023  533  defines [simp]: "a' \ a div d" and [simp]: "b' \ b div d"  haftmann@58023  534  shows "gcd a' b' = 1"  haftmann@58023  535 proof (rule coprimeI)  haftmann@58023  536  fix l assume "l dvd a'" "l dvd b'"  haftmann@58023  537  then obtain s t where "a' = l * s" "b' = l * t" unfolding dvd_def by blast  haftmann@59009  538  moreover have "a = a' * d" "b = b' * d" by simp_all  haftmann@58023  539  ultimately have "a = (l * d) * s" "b = (l * d) * t"  haftmann@59009  540  by (simp_all only: ac_simps)  haftmann@58023  541  hence "l*d dvd a" and "l*d dvd b" by (simp_all only: dvd_triv_left)  haftmann@58023  542  hence "l*d dvd d" by (simp add: gcd_greatest)  haftmann@59009  543  then obtain u where "d = l * d * u" ..  haftmann@59009  544  then have "d * (l * u) = d" by (simp add: ac_simps)  haftmann@59009  545  moreover from nz have "d \ 0" by simp  haftmann@59009  546  with div_mult_self1_is_id have "d * (l * u) div d = l * u" .  haftmann@59009  547  ultimately have "1 = l * u"  wenzelm@60526  548  using \d \ 0\ by simp  haftmann@59009  549  then show "l dvd 1" ..  haftmann@58023  550 qed  haftmann@58023  551 haftmann@58023  552 lemma coprime_mult:  haftmann@58023  553  assumes da: "gcd d a = 1" and db: "gcd d b = 1"  haftmann@58023  554  shows "gcd d (a * b) = 1"  haftmann@58023  555  apply (subst gcd.commute)  haftmann@58023  556  using da apply (subst gcd_mult_cancel)  haftmann@58023  557  apply (subst gcd.commute, assumption)  haftmann@58023  558  apply (subst gcd.commute, rule db)  haftmann@58023  559  done  haftmann@58023  560 haftmann@58023  561 lemma coprime_lmult:  haftmann@58023  562  assumes dab: "gcd d (a * b) = 1"  haftmann@58023  563  shows "gcd d a = 1"  haftmann@58023  564 proof (rule coprimeI)  haftmann@58023  565  fix l assume "l dvd d" and "l dvd a"  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_rmult:  haftmann@58023  571  assumes dab: "gcd d (a * b) = 1"  haftmann@58023  572  shows "gcd d b = 1"  haftmann@58023  573 proof (rule coprimeI)  haftmann@58023  574  fix l assume "l dvd d" and "l dvd b"  haftmann@58023  575  hence "l dvd a * b" by simp  wenzelm@60526  576  with \l dvd d\ and dab show "l dvd 1" by (auto intro: gcd_greatest)  haftmann@58023  577 qed  haftmann@58023  578 haftmann@58023  579 lemma coprime_mul_eq: "gcd d (a * b) = 1 \ gcd d a = 1 \ gcd d b = 1"  haftmann@58023  580  using coprime_rmult[of d a b] coprime_lmult[of d a b] coprime_mult[of d a b] by blast  haftmann@58023  581 haftmann@58023  582 lemma gcd_coprime:  haftmann@60430  583  assumes c: "gcd a b \ 0" and a: "a = a' * gcd a b" and b: "b = b' * gcd a b"  haftmann@58023  584  shows "gcd a' b' = 1"  haftmann@58023  585 proof -  haftmann@60430  586  from c have "a \ 0 \ b \ 0" by simp  haftmann@58023  587  with div_gcd_coprime have "gcd (a div gcd a b) (b div gcd a b) = 1" .  haftmann@58023  588  also from assms have "a div gcd a b = a'" by (metis div_mult_self2_is_id)+  haftmann@58023  589  also from assms have "b div gcd a b = b'" by (metis div_mult_self2_is_id)+  haftmann@58023  590  finally show ?thesis .  haftmann@58023  591 qed  haftmann@58023  592 haftmann@58023  593 lemma coprime_power:  haftmann@58023  594  assumes "0 < n"  haftmann@58023  595  shows "gcd a (b ^ n) = 1 \ gcd a b = 1"  haftmann@58023  596 using assms proof (induct n)  haftmann@58023  597  case (Suc n) then show ?case  haftmann@58023  598  by (cases n) (simp_all add: coprime_mul_eq)  haftmann@58023  599 qed simp  haftmann@58023  600 haftmann@58023  601 lemma gcd_coprime_exists:  haftmann@58023  602  assumes nz: "gcd a b \ 0"  haftmann@58023  603  shows "\a' b'. a = a' * gcd a b \ b = b' * gcd a b \ gcd a' b' = 1"  haftmann@58023  604  apply (rule_tac x = "a div gcd a b" in exI)  haftmann@58023  605  apply (rule_tac x = "b div gcd a b" in exI)  haftmann@59009  606  apply (insert nz, auto intro: div_gcd_coprime)  haftmann@58023  607  done  haftmann@58023  608 haftmann@58023  609 lemma coprime_exp:  haftmann@58023  610  "gcd d a = 1 \ gcd d (a^n) = 1"  haftmann@58023  611  by (induct n, simp_all add: coprime_mult)  haftmann@58023  612 haftmann@58023  613 lemma coprime_exp2 [intro]:  haftmann@58023  614  "gcd a b = 1 \ gcd (a^n) (b^m) = 1"  haftmann@58023  615  apply (rule coprime_exp)  haftmann@58023  616  apply (subst gcd.commute)  haftmann@58023  617  apply (rule coprime_exp)  haftmann@58023  618  apply (subst gcd.commute)  haftmann@58023  619  apply assumption  haftmann@58023  620  done  haftmann@58023  621 haftmann@60688  622 lemma lcm_gcd:  haftmann@60688  623  "lcm a b = normalize (a * b) div gcd a b"  haftmann@60688  624  by (simp add: lcm_lcm_eucl gcd_gcd_eucl lcm_eucl_def)  haftmann@60688  625 haftmann@60688  626 subclass semiring_gcd  haftmann@60688  627  apply standard  haftmann@60688  628  using gcd_right_idem  haftmann@60688  629  apply (simp_all add: lcm_gcd gcd_greatest_iff gcd_proj1_if_dvd)  haftmann@60688  630  done  haftmann@60688  631 haftmann@58023  632 lemma gcd_exp:  haftmann@60688  633  "gcd (a ^ n) (b ^ n) = gcd a b ^ n"  haftmann@58023  634 proof (cases "a = 0 \ b = 0")  haftmann@60688  635  case True  haftmann@60688  636  then show ?thesis by (cases n) simp_all  haftmann@58023  637 next  haftmann@60688  638  case False  haftmann@60688  639  then have "1 = gcd ((a div gcd a b) ^ n) ((b div gcd a b) ^ n)"  haftmann@60688  640  using div_gcd_coprime by (subst sym) (auto simp: div_gcd_coprime)  haftmann@60688  641  then have "gcd a b ^ n = gcd a b ^ n * ..." by simp  haftmann@58023  642  also note gcd_mult_distrib  haftmann@60688  643  also have "unit_factor (gcd a b ^ n) = 1"  haftmann@60688  644  using False by (auto simp add: unit_factor_power unit_factor_gcd)  haftmann@58023  645  also have "(gcd a b)^n * (a div gcd a b)^n = a^n"  haftmann@58023  646  by (subst ac_simps, subst div_power, simp, rule dvd_div_mult_self, rule dvd_power_same, simp)  haftmann@58023  647  also have "(gcd a b)^n * (b div gcd a b)^n = b^n"  haftmann@58023  648  by (subst ac_simps, subst div_power, simp, rule dvd_div_mult_self, rule dvd_power_same, simp)  haftmann@58023  649  finally show ?thesis by simp  haftmann@58023  650 qed  haftmann@58023  651 haftmann@58023  652 lemma coprime_common_divisor:  haftmann@60430  653  "gcd a b = 1 \ a dvd a \ a dvd b \ is_unit a"  haftmann@60430  654  apply (subgoal_tac "a dvd gcd a b")  haftmann@59061  655  apply simp  haftmann@58023  656  apply (erule (1) gcd_greatest)  haftmann@58023  657  done  haftmann@58023  658 haftmann@58023  659 lemma division_decomp:  haftmann@58023  660  assumes dc: "a dvd b * c"  haftmann@58023  661  shows "\b' c'. a = b' * c' \ b' dvd b \ c' dvd c"  haftmann@58023  662 proof (cases "gcd a b = 0")  haftmann@58023  663  assume "gcd a b = 0"  haftmann@59009  664  hence "a = 0 \ b = 0" by simp  haftmann@58023  665  hence "a = 0 * c \ 0 dvd b \ c dvd c" by simp  haftmann@58023  666  then show ?thesis by blast  haftmann@58023  667 next  haftmann@58023  668  let ?d = "gcd a b"  haftmann@58023  669  assume "?d \ 0"  haftmann@58023  670  from gcd_coprime_exists[OF this]  haftmann@58023  671  obtain a' b' where ab': "a = a' * ?d" "b = b' * ?d" "gcd a' b' = 1"  haftmann@58023  672  by blast  haftmann@58023  673  from ab'(1) have "a' dvd a" unfolding dvd_def by blast  haftmann@58023  674  with dc have "a' dvd b*c" using dvd_trans[of a' a "b*c"] by simp  haftmann@58023  675  from dc ab'(1,2) have "a'*?d dvd (b'*?d) * c" by simp  haftmann@58023  676  hence "?d * a' dvd ?d * (b' * c)" by (simp add: mult_ac)  wenzelm@60526  677  with \?d \ 0\ have "a' dvd b' * c" by simp  haftmann@58023  678  with coprime_dvd_mult[OF ab'(3)]  haftmann@58023  679  have "a' dvd c" by (subst (asm) ac_simps, blast)  haftmann@58023  680  with ab'(1) have "a = ?d * a' \ ?d dvd b \ a' dvd c" by (simp add: mult_ac)  haftmann@58023  681  then show ?thesis by blast  haftmann@58023  682 qed  haftmann@58023  683 haftmann@60433  684 lemma pow_divs_pow:  haftmann@58023  685  assumes ab: "a ^ n dvd b ^ n" and n: "n \ 0"  haftmann@58023  686  shows "a dvd b"  haftmann@58023  687 proof (cases "gcd a b = 0")  haftmann@58023  688  assume "gcd a b = 0"  haftmann@59009  689  then show ?thesis by simp  haftmann@58023  690 next  haftmann@58023  691  let ?d = "gcd a b"  haftmann@58023  692  assume "?d \ 0"  haftmann@58023  693  from n obtain m where m: "n = Suc m" by (cases n, simp_all)  wenzelm@60526  694  from \?d \ 0\ have zn: "?d ^ n \ 0" by (rule power_not_zero)  wenzelm@60526  695  from gcd_coprime_exists[OF \?d \ 0\]  haftmann@58023  696  obtain a' b' where ab': "a = a' * ?d" "b = b' * ?d" "gcd a' b' = 1"  haftmann@58023  697  by blast  haftmann@58023  698  from ab have "(a' * ?d) ^ n dvd (b' * ?d) ^ n"  haftmann@58023  699  by (simp add: ab'(1,2)[symmetric])  haftmann@58023  700  hence "?d^n * a'^n dvd ?d^n * b'^n"  haftmann@58023  701  by (simp only: power_mult_distrib ac_simps)  haftmann@59009  702  with zn have "a'^n dvd b'^n" by simp  haftmann@58023  703  hence "a' dvd b'^n" using dvd_trans[of a' "a'^n" "b'^n"] by (simp add: m)  haftmann@58023  704  hence "a' dvd b'^m * b'" by (simp add: m ac_simps)  haftmann@58023  705  with coprime_dvd_mult[OF coprime_exp[OF ab'(3), of m]]  haftmann@58023  706  have "a' dvd b'" by (subst (asm) ac_simps, blast)  haftmann@58023  707  hence "a'*?d dvd b'*?d" by (rule mult_dvd_mono, simp)  haftmann@58023  708  with ab'(1,2) show ?thesis by simp  haftmann@58023  709 qed  haftmann@58023  710 haftmann@60433  711 lemma pow_divs_eq [simp]:  haftmann@58023  712  "n \ 0 \ a ^ n dvd b ^ n \ a dvd b"  haftmann@60433  713  by (auto intro: pow_divs_pow dvd_power_same)  haftmann@58023  714 haftmann@60433  715 lemma divs_mult:  haftmann@58023  716  assumes mr: "m dvd r" and nr: "n dvd r" and mn: "gcd m n = 1"  haftmann@58023  717  shows "m * n dvd r"  haftmann@58023  718 proof -  haftmann@58023  719  from mr nr obtain m' n' where m': "r = m*m'" and n': "r = n*n'"  haftmann@58023  720  unfolding dvd_def by blast  haftmann@58023  721  from mr n' have "m dvd n'*n" by (simp add: ac_simps)  haftmann@58023  722  hence "m dvd n'" using coprime_dvd_mult_iff[OF mn] by simp  haftmann@58023  723  then obtain k where k: "n' = m*k" unfolding dvd_def by blast  haftmann@58023  724  with n' have "r = m * n * k" by (simp add: mult_ac)  haftmann@58023  725  then show ?thesis unfolding dvd_def by blast  haftmann@58023  726 qed  haftmann@58023  727 haftmann@58023  728 lemma coprime_plus_one [simp]: "gcd (n + 1) n = 1"  haftmann@58023  729  by (subst add_commute, simp)  haftmann@58023  730 haftmann@58023  731 lemma setprod_coprime [rule_format]:  haftmann@60430  732  "(\i\A. gcd (f i) a = 1) \ gcd (\i\A. f i) a = 1"  haftmann@58023  733  apply (cases "finite A")  haftmann@58023  734  apply (induct set: finite)  haftmann@58023  735  apply (auto simp add: gcd_mult_cancel)  haftmann@58023  736  done  haftmann@58023  737 haftmann@58023  738 lemma coprime_divisors:  haftmann@58023  739  assumes "d dvd a" "e dvd b" "gcd a b = 1"  haftmann@58023  740  shows "gcd d e = 1"  haftmann@58023  741 proof -  haftmann@58023  742  from assms obtain k l where "a = d * k" "b = e * l"  haftmann@58023  743  unfolding dvd_def by blast  haftmann@58023  744  with assms have "gcd (d * k) (e * l) = 1" by simp  haftmann@58023  745  hence "gcd (d * k) e = 1" by (rule coprime_lmult)  haftmann@58023  746  also have "gcd (d * k) e = gcd e (d * k)" by (simp add: ac_simps)  haftmann@58023  747  finally have "gcd e d = 1" by (rule coprime_lmult)  haftmann@58023  748  then show ?thesis by (simp add: ac_simps)  haftmann@58023  749 qed  haftmann@58023  750 haftmann@58023  751 lemma invertible_coprime:  haftmann@60430  752  assumes "a * b mod m = 1"  haftmann@60430  753  shows "coprime a m"  haftmann@59009  754 proof -  haftmann@60430  755  from assms have "coprime m (a * b mod m)"  haftmann@59009  756  by simp  haftmann@60430  757  then have "coprime m (a * b)"  haftmann@59009  758  by simp  haftmann@60430  759  then have "coprime m a"  haftmann@59009  760  by (rule coprime_lmult)  haftmann@59009  761  then show ?thesis  haftmann@59009  762  by (simp add: ac_simps)  haftmann@59009  763 qed  haftmann@58023  764 haftmann@58023  765 lemma lcm_gcd_prod:  haftmann@60634  766  "lcm a b * gcd a b = normalize (a * b)"  haftmann@60634  767  by (simp add: lcm_gcd)  haftmann@58023  768 haftmann@58023  769 lemma lcm_zero:  haftmann@58023  770  "lcm a b = 0 \ a = 0 \ b = 0"  haftmann@60687  771  by (fact lcm_eq_0_iff)  haftmann@58023  772 haftmann@58023  773 lemmas lcm_0_iff = lcm_zero  haftmann@58023  774 haftmann@58023  775 lemma gcd_lcm:  haftmann@58023  776  assumes "lcm a b \ 0"  haftmann@60634  777  shows "gcd a b = normalize (a * b) div lcm a b"  haftmann@60634  778 proof -  haftmann@60634  779  have "lcm a b * gcd a b = normalize (a * b)"  haftmann@60634  780  by (fact lcm_gcd_prod)  haftmann@60634  781  with assms show ?thesis  haftmann@60634  782  by (metis nonzero_mult_divide_cancel_left)  haftmann@58023  783 qed  haftmann@58023  784 haftmann@60687  785 declare unit_factor_lcm [simp]  haftmann@58023  786 haftmann@58023  787 lemma lcmI:  haftmann@60634  788  assumes "a dvd c" and "b dvd c" and "\d. a dvd d \ b dvd d \ c dvd d"  haftmann@60688  789  and "normalize c = c"  haftmann@60634  790  shows "c = lcm a b"  haftmann@60688  791  by (rule associated_eqI) (auto simp: assms intro: lcm_least)  haftmann@58023  792 wenzelm@61605  793 sublocale lcm: abel_semigroup lcm ..  haftmann@58023  794 haftmann@58023  795 lemma dvd_lcm_D1:  haftmann@58023  796  "lcm m n dvd k \ m dvd k"  haftmann@60690  797  by (rule dvd_trans, rule dvd_lcm1, assumption)  haftmann@58023  798 haftmann@58023  799 lemma dvd_lcm_D2:  haftmann@58023  800  "lcm m n dvd k \ n dvd k"  haftmann@60690  801  by (rule dvd_trans, rule dvd_lcm2, assumption)  haftmann@58023  802 haftmann@58023  803 lemma gcd_dvd_lcm [simp]:  haftmann@58023  804  "gcd a b dvd lcm a b"  haftmann@60690  805  using gcd_dvd2 by (rule dvd_lcmI2)  haftmann@58023  806 haftmann@58023  807 lemma lcm_1_iff:  haftmann@58023  808  "lcm a b = 1 \ is_unit a \ is_unit b"  haftmann@58023  809 proof  haftmann@58023  810  assume "lcm a b = 1"  haftmann@59061  811  then show "is_unit a \ is_unit b" by auto  haftmann@58023  812 next  haftmann@58023  813  assume "is_unit a \ is_unit b"  haftmann@59061  814  hence "a dvd 1" and "b dvd 1" by simp_all  haftmann@59061  815  hence "is_unit (lcm a b)" by (rule lcm_least)  haftmann@60634  816  hence "lcm a b = unit_factor (lcm a b)"  haftmann@60634  817  by (blast intro: sym is_unit_unit_factor)  wenzelm@60526  818  also have "\ = 1" using \is_unit a \ is_unit b\  haftmann@59061  819  by auto  haftmann@58023  820  finally show "lcm a b = 1" .  haftmann@58023  821 qed  haftmann@58023  822 haftmann@60687  823 lemma lcm_0:  haftmann@60430  824  "lcm a 0 = 0"  haftmann@60687  825  by (fact lcm_0_right)  haftmann@58023  826 haftmann@58023  827 lemma lcm_unique:  haftmann@58023  828  "a dvd d \ b dvd d \  haftmann@60688  829  normalize d = d \  haftmann@58023  830  (\e. a dvd e \ b dvd e \ d dvd e) \ d = lcm a b"  haftmann@60688  831  by rule (auto intro: lcmI simp: lcm_least lcm_zero)  haftmann@58023  832 haftmann@58023  833 lemma lcm_coprime:  haftmann@60634  834  "gcd a b = 1 \ lcm a b = normalize (a * b)"  haftmann@58023  835  by (subst lcm_gcd) simp  haftmann@58023  836 haftmann@58023  837 lemma lcm_proj1_if_dvd:  haftmann@60634  838  "b dvd a \ lcm a b = normalize a"  haftmann@60430  839  by (cases "a = 0") (simp, rule sym, rule lcmI, simp_all)  haftmann@58023  840 haftmann@58023  841 lemma lcm_proj2_if_dvd:  haftmann@60634  842  "a dvd b \ lcm a b = normalize b"  haftmann@60430  843  using lcm_proj1_if_dvd [of a b] by (simp add: ac_simps)  haftmann@58023  844 haftmann@58023  845 lemma lcm_proj1_iff:  haftmann@60634  846  "lcm m n = normalize m \ n dvd m"  haftmann@58023  847 proof  haftmann@60634  848  assume A: "lcm m n = normalize m"  haftmann@58023  849  show "n dvd m"  haftmann@58023  850  proof (cases "m = 0")  haftmann@58023  851  assume [simp]: "m \ 0"  haftmann@60634  852  from A have B: "m = lcm m n * unit_factor m"  haftmann@58023  853  by (simp add: unit_eq_div2)  haftmann@58023  854  show ?thesis by (subst B, simp)  haftmann@58023  855  qed simp  haftmann@58023  856 next  haftmann@58023  857  assume "n dvd m"  haftmann@60634  858  then show "lcm m n = normalize m" by (rule lcm_proj1_if_dvd)  haftmann@58023  859 qed  haftmann@58023  860 haftmann@58023  861 lemma lcm_proj2_iff:  haftmann@60634  862  "lcm m n = normalize n \ m dvd n"  haftmann@58023  863  using lcm_proj1_iff [of n m] by (simp add: ac_simps)  haftmann@58023  864 haftmann@58023  865 lemma euclidean_size_lcm_le1:  haftmann@58023  866  assumes "a \ 0" and "b \ 0"  haftmann@58023  867  shows "euclidean_size a \ euclidean_size (lcm a b)"  haftmann@58023  868 proof -  haftmann@60690  869  have "a dvd lcm a b" by (rule dvd_lcm1)  haftmann@60690  870  then obtain c where A: "lcm a b = a * c" ..  wenzelm@60526  871  with \a \ 0\ and \b \ 0\ have "c \ 0" by (auto simp: lcm_zero)  haftmann@58023  872  then show ?thesis by (subst A, intro size_mult_mono)  haftmann@58023  873 qed  haftmann@58023  874 haftmann@58023  875 lemma euclidean_size_lcm_le2:  haftmann@58023  876  "a \ 0 \ b \ 0 \ euclidean_size b \ euclidean_size (lcm a b)"  haftmann@58023  877  using euclidean_size_lcm_le1 [of b a] by (simp add: ac_simps)  haftmann@58023  878 haftmann@58023  879 lemma euclidean_size_lcm_less1:  haftmann@58023  880  assumes "b \ 0" and "\b dvd a"  haftmann@58023  881  shows "euclidean_size a < euclidean_size (lcm a b)"  haftmann@58023  882 proof (rule ccontr)  haftmann@58023  883  from assms have "a \ 0" by auto  haftmann@58023  884  assume "\euclidean_size a < euclidean_size (lcm a b)"  wenzelm@60526  885  with \a \ 0\ and \b \ 0\ have "euclidean_size (lcm a b) = euclidean_size a"  haftmann@58023  886  by (intro le_antisym, simp, intro euclidean_size_lcm_le1)  haftmann@58023  887  with assms have "lcm a b dvd a"  haftmann@58023  888  by (rule_tac dvd_euclidean_size_eq_imp_dvd) (auto simp: lcm_zero)  haftmann@58023  889  hence "b dvd a" by (rule dvd_lcm_D2)  wenzelm@60526  890  with \\b dvd a\ show False by contradiction  haftmann@58023  891 qed  haftmann@58023  892 haftmann@58023  893 lemma euclidean_size_lcm_less2:  haftmann@58023  894  assumes "a \ 0" and "\a dvd b"  haftmann@58023  895  shows "euclidean_size b < euclidean_size (lcm a b)"  haftmann@58023  896  using assms euclidean_size_lcm_less1 [of a b] by (simp add: ac_simps)  haftmann@58023  897 haftmann@58023  898 lemma lcm_mult_unit1:  haftmann@60430  899  "is_unit a \ lcm (b * a) c = lcm b c"  haftmann@60690  900  by (rule associated_eqI) (simp_all add: mult_unit_dvd_iff dvd_lcmI1)  haftmann@58023  901 haftmann@58023  902 lemma lcm_mult_unit2:  haftmann@60430  903  "is_unit a \ lcm b (c * a) = lcm b c"  haftmann@60430  904  using lcm_mult_unit1 [of a c b] by (simp add: ac_simps)  haftmann@58023  905 haftmann@58023  906 lemma lcm_div_unit1:  haftmann@60430  907  "is_unit a \ lcm (b div a) c = lcm b c"  haftmann@60433  908  by (erule is_unitE [of _ b]) (simp add: lcm_mult_unit1)  haftmann@58023  909 haftmann@58023  910 lemma lcm_div_unit2:  haftmann@60430  911  "is_unit a \ lcm b (c div a) = lcm b c"  haftmann@60433  912  by (erule is_unitE [of _ c]) (simp add: lcm_mult_unit2)  haftmann@58023  913 haftmann@60634  914 lemma normalize_lcm_left [simp]:  haftmann@60634  915  "lcm (normalize a) b = lcm a b"  haftmann@60634  916 proof (cases "a = 0")  haftmann@60634  917  case True then show ?thesis  haftmann@60634  918  by simp  haftmann@60634  919 next  haftmann@60634  920  case False then have "is_unit (unit_factor a)"  haftmann@60634  921  by simp  haftmann@60634  922  moreover have "normalize a = a div unit_factor a"  haftmann@60634  923  by simp  haftmann@60634  924  ultimately show ?thesis  haftmann@60634  925  by (simp only: lcm_div_unit1)  haftmann@60634  926 qed  haftmann@60634  927 haftmann@60634  928 lemma normalize_lcm_right [simp]:  haftmann@60634  929  "lcm a (normalize b) = lcm a b"  haftmann@60634  930  using normalize_lcm_left [of b a] by (simp add: ac_simps)  haftmann@60634  931 haftmann@58023  932 lemma lcm_left_idem:  haftmann@60430  933  "lcm a (lcm a b) = lcm a b"  haftmann@60690  934  by (rule associated_eqI) simp_all  haftmann@58023  935 haftmann@58023  936 lemma lcm_right_idem:  haftmann@60430  937  "lcm (lcm a b) b = lcm a b"  haftmann@60690  938  by (rule associated_eqI) simp_all  haftmann@58023  939 haftmann@58023  940 lemma comp_fun_idem_lcm: "comp_fun_idem lcm"  haftmann@58023  941 proof  haftmann@58023  942  fix a b show "lcm a \ lcm b = lcm b \ lcm a"  haftmann@58023  943  by (simp add: fun_eq_iff ac_simps)  haftmann@58023  944 next  haftmann@58023  945  fix a show "lcm a \ lcm a = lcm a" unfolding o_def  haftmann@58023  946  by (intro ext, simp add: lcm_left_idem)  haftmann@58023  947 qed  haftmann@58023  948 haftmann@60430  949 lemma dvd_Lcm [simp]: "a \ A \ a dvd Lcm A"  haftmann@60634  950  and Lcm_least: "(\a. a \ A \ a dvd b) \ Lcm A dvd b"  haftmann@60634  951  and unit_factor_Lcm [simp]:  haftmann@60634  952  "unit_factor (Lcm A) = (if Lcm A = 0 then 0 else 1)"  haftmann@58023  953 proof -  haftmann@60430  954  have "(\a\A. a dvd Lcm A) \ (\l'. (\a\A. a dvd l') \ Lcm A dvd l') \  haftmann@60634  955  unit_factor (Lcm A) = (if Lcm A = 0 then 0 else 1)" (is ?thesis)  haftmann@60430  956  proof (cases "\l. l \ 0 \ (\a\A. a dvd l)")  haftmann@58023  957  case False  haftmann@58023  958  hence "Lcm A = 0" by (auto simp: Lcm_Lcm_eucl Lcm_eucl_def)  haftmann@58023  959  with False show ?thesis by auto  haftmann@58023  960  next  haftmann@58023  961  case True  haftmann@60430  962  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  963  def n \ "LEAST n. \l. l \ 0 \ (\a\A. a dvd l) \ euclidean_size l = n"  haftmann@60430  964  def l \ "SOME l. l \ 0 \ (\a\A. a dvd l) \ euclidean_size l = n"  haftmann@60430  965  have "\l. l \ 0 \ (\a\A. a dvd l) \ euclidean_size l = n"  haftmann@58023  966  apply (subst n_def)  haftmann@58023  967  apply (rule LeastI[of _ "euclidean_size l\<^sub>0"])  haftmann@58023  968  apply (rule exI[of _ l\<^sub>0])  haftmann@58023  969  apply (simp add: l\<^sub>0_props)  haftmann@58023  970  done  haftmann@60430  971  from someI_ex[OF this] have "l \ 0" and "\a\A. a dvd l" and "euclidean_size l = n"  haftmann@58023  972  unfolding l_def by simp_all  haftmann@58023  973  {  haftmann@60430  974  fix l' assume "\a\A. a dvd l'"  wenzelm@60526  975  with \\a\A. a dvd l\ have "\a\A. a dvd gcd l l'" by (auto intro: gcd_greatest)  wenzelm@60526  976  moreover from \l \ 0\ have "gcd l l' \ 0" by simp  haftmann@60430  977  ultimately have "\b. b \ 0 \ (\a\A. a dvd b) \ euclidean_size b = euclidean_size (gcd l l')"  haftmann@58023  978  by (intro exI[of _ "gcd l l'"], auto)  haftmann@58023  979  hence "euclidean_size (gcd l l') \ n" by (subst n_def) (rule Least_le)  haftmann@58023  980  moreover have "euclidean_size (gcd l l') \ n"  haftmann@58023  981  proof -  haftmann@58023  982  have "gcd l l' dvd l" by simp  haftmann@58023  983  then obtain a where "l = gcd l l' * a" unfolding dvd_def by blast  wenzelm@60526  984  with \l \ 0\ have "a \ 0" by auto  haftmann@58023  985  hence "euclidean_size (gcd l l') \ euclidean_size (gcd l l' * a)"  haftmann@58023  986  by (rule size_mult_mono)  wenzelm@60526  987  also have "gcd l l' * a = l" using \l = gcd l l' * a\ ..  wenzelm@60526  988  also note \euclidean_size l = n\  haftmann@58023  989  finally show "euclidean_size (gcd l l') \ n" .  haftmann@58023  990  qed  haftmann@60690  991  ultimately have *: "euclidean_size l = euclidean_size (gcd l l')"  wenzelm@60526  992  by (intro le_antisym, simp_all add: \euclidean_size l = n\)  haftmann@60690  993  from \l \ 0\ have "l dvd gcd l l'"  haftmann@60690  994  by (rule dvd_euclidean_size_eq_imp_dvd) (auto simp add: *)  haftmann@58023  995  hence "l dvd l'" by (blast dest: dvd_gcd_D2)  haftmann@58023  996  }  haftmann@58023  997 haftmann@60634  998  with \(\a\A. a dvd l)\ and unit_factor_is_unit[OF \l \ 0\] and \l \ 0\  haftmann@60634  999  have "(\a\A. a dvd normalize l) \  haftmann@60634  1000  (\l'. (\a\A. a dvd l') \ normalize l dvd l') \  haftmann@60634  1001  unit_factor (normalize l) =  haftmann@60634  1002  (if normalize l = 0 then 0 else 1)"  haftmann@58023  1003  by (auto simp: unit_simps)  haftmann@60634  1004  also from True have "normalize l = Lcm A"  haftmann@58023  1005  by (simp add: Lcm_Lcm_eucl Lcm_eucl_def Let_def n_def l_def)  haftmann@58023  1006  finally show ?thesis .  haftmann@58023  1007  qed  haftmann@58023  1008  note A = this  haftmann@58023  1009 haftmann@60430  1010  {fix a assume "a \ A" then show "a dvd Lcm A" using A by blast}  haftmann@60634  1011  {fix b assume "\a. a \ A \ a dvd b" then show "Lcm A dvd b" using A by blast}  haftmann@60634  1012  from A show "unit_factor (Lcm A) = (if Lcm A = 0 then 0 else 1)" by blast  haftmann@58023  1013 qed  haftmann@60634  1014 haftmann@60634  1015 lemma normalize_Lcm [simp]:  haftmann@60634  1016  "normalize (Lcm A) = Lcm A"  haftmann@60688  1017 proof (cases "Lcm A = 0")  haftmann@60688  1018  case True then show ?thesis by simp  haftmann@60688  1019 next  haftmann@60688  1020  case False  haftmann@60688  1021  have "unit_factor (Lcm A) * normalize (Lcm A) = Lcm A"  haftmann@60688  1022  by (fact unit_factor_mult_normalize)  haftmann@60688  1023  with False show ?thesis by simp  haftmann@60688  1024 qed  haftmann@60634  1025 haftmann@58023  1026 lemma LcmI:  haftmann@60634  1027  assumes "\a. a \ A \ a dvd b" and "\c. (\a. a \ A \ a dvd c) \ b dvd c"  haftmann@60688  1028  and "normalize b = b" shows "b = Lcm A"  haftmann@60688  1029  by (rule associated_eqI) (auto simp: assms intro: Lcm_least)  haftmann@58023  1030 haftmann@58023  1031 lemma Lcm_subset:  haftmann@58023  1032  "A \ B \ Lcm A dvd Lcm B"  haftmann@60634  1033  by (blast intro: Lcm_least dvd_Lcm)  haftmann@58023  1034 haftmann@58023  1035 lemma Lcm_Un:  haftmann@58023  1036  "Lcm (A \ B) = lcm (Lcm A) (Lcm B)"  haftmann@58023  1037  apply (rule lcmI)  haftmann@58023  1038  apply (blast intro: Lcm_subset)  haftmann@58023  1039  apply (blast intro: Lcm_subset)  haftmann@60634  1040  apply (intro Lcm_least ballI, elim UnE)  haftmann@58023  1041  apply (rule dvd_trans, erule dvd_Lcm, assumption)  haftmann@58023  1042  apply (rule dvd_trans, erule dvd_Lcm, assumption)  haftmann@58023  1043  apply simp  haftmann@58023  1044  done  haftmann@58023  1045 haftmann@58023  1046 lemma Lcm_1_iff:  haftmann@60430  1047  "Lcm A = 1 \ (\a\A. is_unit a)"  haftmann@58023  1048 proof  haftmann@58023  1049  assume "Lcm A = 1"  haftmann@60430  1050  then show "\a\A. is_unit a" by auto  haftmann@58023  1051 qed (rule LcmI [symmetric], auto)  haftmann@58023  1052 haftmann@58023  1053 lemma Lcm_no_units:  haftmann@60430  1054  "Lcm A = Lcm (A - {a. is_unit a})"  haftmann@58023  1055 proof -  haftmann@60430  1056  have "(A - {a. is_unit a}) \ {a\A. is_unit a} = A" by blast  haftmann@60430  1057  hence "Lcm A = lcm (Lcm (A - {a. is_unit a})) (Lcm {a\A. is_unit a})"  haftmann@60634  1058  by (simp add: Lcm_Un [symmetric])  haftmann@60430  1059  also have "Lcm {a\A. is_unit a} = 1" by (simp add: Lcm_1_iff)  haftmann@58023  1060  finally show ?thesis by simp  haftmann@58023  1061 qed  haftmann@58023  1062 haftmann@58023  1063 lemma Lcm_empty [simp]:  haftmann@58023  1064  "Lcm {} = 1"  haftmann@58023  1065  by (simp add: Lcm_1_iff)  haftmann@58023  1066 haftmann@58023  1067 lemma Lcm_eq_0 [simp]:  haftmann@58023  1068  "0 \ A \ Lcm A = 0"  haftmann@58023  1069  by (drule dvd_Lcm) simp  haftmann@58023  1070 haftmann@58023  1071 lemma Lcm0_iff':  haftmann@60430  1072  "Lcm A = 0 \ \(\l. l \ 0 \ (\a\A. a dvd l))"  haftmann@58023  1073 proof  haftmann@58023  1074  assume "Lcm A = 0"  haftmann@60430  1075  show "\(\l. l \ 0 \ (\a\A. a dvd l))"  haftmann@58023  1076  proof  haftmann@60430  1077  assume ex: "\l. l \ 0 \ (\a\A. a dvd l)"  haftmann@60430  1078  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  1079  def n \ "LEAST n. \l. l \ 0 \ (\a\A. a dvd l) \ euclidean_size l = n"  haftmann@60430  1080  def l \ "SOME l. l \ 0 \ (\a\A. a dvd l) \ euclidean_size l = n"  haftmann@60430  1081  have "\l. l \ 0 \ (\a\A. a dvd l) \ euclidean_size l = n"  haftmann@58023  1082  apply (subst n_def)  haftmann@58023  1083  apply (rule LeastI[of _ "euclidean_size l\<^sub>0"])  haftmann@58023  1084  apply (rule exI[of _ l\<^sub>0])  haftmann@58023  1085  apply (simp add: l\<^sub>0_props)  haftmann@58023  1086  done  haftmann@58023  1087  from someI_ex[OF this] have "l \ 0" unfolding l_def by simp_all  haftmann@60634  1088  hence "normalize l \ 0" by simp  haftmann@60634  1089  also from ex have "normalize l = Lcm A"  haftmann@58023  1090  by (simp only: Lcm_Lcm_eucl Lcm_eucl_def n_def l_def if_True Let_def)  wenzelm@60526  1091  finally show False using \Lcm A = 0\ by contradiction  haftmann@58023  1092  qed  haftmann@58023  1093 qed (simp only: Lcm_Lcm_eucl Lcm_eucl_def if_False)  haftmann@58023  1094 haftmann@58023  1095 lemma Lcm0_iff [simp]:  haftmann@58023  1096  "finite A \ Lcm A = 0 \ 0 \ A"  haftmann@58023  1097 proof -  haftmann@58023  1098  assume "finite A"  haftmann@58023  1099  have "0 \ A \ Lcm A = 0" by (intro dvd_0_left dvd_Lcm)  haftmann@58023  1100  moreover {  haftmann@58023  1101  assume "0 \ A"  haftmann@58023  1102  hence "\A \ 0"  wenzelm@60526  1103  apply (induct rule: finite_induct[OF \finite A\])  haftmann@58023  1104  apply simp  haftmann@58023  1105  apply (subst setprod.insert, assumption, assumption)  haftmann@58023  1106  apply (rule no_zero_divisors)  haftmann@58023  1107  apply blast+  haftmann@58023  1108  done  wenzelm@60526  1109  moreover from \finite A\ have "\a\A. a dvd \A" by blast  haftmann@60430  1110  ultimately have "\l. l \ 0 \ (\a\A. a dvd l)" by blast  haftmann@58023  1111  with Lcm0_iff' have "Lcm A \ 0" by simp  haftmann@58023  1112  }  haftmann@58023  1113  ultimately show "Lcm A = 0 \ 0 \ A" by blast  haftmann@58023  1114 qed  haftmann@58023  1115 haftmann@58023  1116 lemma Lcm_no_multiple:  haftmann@60430  1117  "(\m. m \ 0 \ (\a\A. \a dvd m)) \ Lcm A = 0"  haftmann@58023  1118 proof -  haftmann@60430  1119  assume "\m. m \ 0 \ (\a\A. \a dvd m)"  haftmann@60430  1120  hence "\(\l. l \ 0 \ (\a\A. a dvd l))" by blast  haftmann@58023  1121  then show "Lcm A = 0" by (simp only: Lcm_Lcm_eucl Lcm_eucl_def if_False)  haftmann@58023  1122 qed  haftmann@58023  1123 haftmann@58023  1124 lemma Lcm_insert [simp]:  haftmann@58023  1125  "Lcm (insert a A) = lcm a (Lcm A)"  haftmann@58023  1126 proof (rule lcmI)  haftmann@58023  1127  fix l assume "a dvd l" and "Lcm A dvd l"  haftmann@60687  1128  then have "\a\A. a dvd l" by (auto intro: dvd_trans [of _ "Lcm A" l])  haftmann@60634  1129  with \a dvd l\ show "Lcm (insert a A) dvd l" by (force intro: Lcm_least)  haftmann@60634  1130 qed (auto intro: Lcm_least dvd_Lcm)  haftmann@58023  1131   haftmann@58023  1132 lemma Lcm_finite:  haftmann@58023  1133  assumes "finite A"  haftmann@58023  1134  shows "Lcm A = Finite_Set.fold lcm 1 A"  wenzelm@60526  1135  by (induct rule: finite.induct[OF \finite A\])  haftmann@58023  1136  (simp_all add: comp_fun_idem.fold_insert_idem[OF comp_fun_idem_lcm])  haftmann@58023  1137 haftmann@60431  1138 lemma Lcm_set [code_unfold]:  haftmann@58023  1139  "Lcm (set xs) = fold lcm xs 1"  haftmann@58023  1140  using comp_fun_idem.fold_set_fold[OF comp_fun_idem_lcm] Lcm_finite by (simp add: ac_simps)  haftmann@58023  1141 haftmann@58023  1142 lemma Lcm_singleton [simp]:  haftmann@60634  1143  "Lcm {a} = normalize a"  haftmann@58023  1144  by simp  haftmann@58023  1145 haftmann@58023  1146 lemma Lcm_2 [simp]:  haftmann@58023  1147  "Lcm {a,b} = lcm a b"  haftmann@60634  1148  by simp  haftmann@58023  1149 haftmann@58023  1150 lemma Lcm_coprime:  haftmann@58023  1151  assumes "finite A" and "A \ {}"  haftmann@58023  1152  assumes "\a b. a \ A \ b \ A \ a \ b \ gcd a b = 1"  haftmann@60634  1153  shows "Lcm A = normalize (\A)"  haftmann@58023  1154 using assms proof (induct rule: finite_ne_induct)  haftmann@58023  1155  case (insert a A)  haftmann@58023  1156  have "Lcm (insert a A) = lcm a (Lcm A)" by simp  haftmann@60634  1157  also from insert have "Lcm A = normalize (\A)" by blast  haftmann@58023  1158  also have "lcm a \ = lcm a (\A)" by (cases "\A = 0") (simp_all add: lcm_div_unit2)  haftmann@58023  1159  also from insert have "gcd a (\A) = 1" by (subst gcd.commute, intro setprod_coprime) auto  haftmann@60634  1160  with insert have "lcm a (\A) = normalize (\(insert a A))"  haftmann@58023  1161  by (simp add: lcm_coprime)  haftmann@58023  1162  finally show ?case .  haftmann@58023  1163 qed simp  haftmann@58023  1164   haftmann@58023  1165 lemma Lcm_coprime':  haftmann@58023  1166  "card A \ 0 \ (\a b. a \ A \ b \ A \ a \ b \ gcd a b = 1)  haftmann@60634  1167  \ Lcm A = normalize (\A)"  haftmann@58023  1168  by (rule Lcm_coprime) (simp_all add: card_eq_0_iff)  haftmann@58023  1169 haftmann@58023  1170 lemma Gcd_Lcm:  haftmann@60430  1171  "Gcd A = Lcm {d. \a\A. d dvd a}"  haftmann@58023  1172  by (simp add: Gcd_Gcd_eucl Lcm_Lcm_eucl Gcd_eucl_def)  haftmann@58023  1173 haftmann@60430  1174 lemma Gcd_dvd [simp]: "a \ A \ Gcd A dvd a"  haftmann@60634  1175  and Gcd_greatest: "(\a. a \ A \ b dvd a) \ b dvd Gcd A"  haftmann@60634  1176  and unit_factor_Gcd [simp]:  haftmann@60634  1177  "unit_factor (Gcd A) = (if Gcd A = 0 then 0 else 1)"  haftmann@58023  1178 proof -  haftmann@60430  1179  fix a assume "a \ A"  haftmann@60634  1180  hence "Lcm {d. \a\A. d dvd a} dvd a" by (intro Lcm_least) blast  haftmann@60430  1181  then show "Gcd A dvd a" by (simp add: Gcd_Lcm)  haftmann@58023  1182 next  haftmann@60634  1183  fix g' assume "\a. a \ A \ g' dvd a"  haftmann@60430  1184  hence "g' dvd Lcm {d. \a\A. d dvd a}" by (intro dvd_Lcm) blast  haftmann@58023  1185  then show "g' dvd Gcd A" by (simp add: Gcd_Lcm)  haftmann@58023  1186 next  haftmann@60634  1187  show "unit_factor (Gcd A) = (if Gcd A = 0 then 0 else 1)"  haftmann@59009  1188  by (simp add: Gcd_Lcm)  haftmann@58023  1189 qed  haftmann@58023  1190 haftmann@60634  1191 lemma normalize_Gcd [simp]:  haftmann@60634  1192  "normalize (Gcd A) = Gcd A"  haftmann@60688  1193 proof (cases "Gcd A = 0")  haftmann@60688  1194  case True then show ?thesis by simp  haftmann@60688  1195 next  haftmann@60688  1196  case False  haftmann@60688  1197  have "unit_factor (Gcd A) * normalize (Gcd A) = Gcd A"  haftmann@60688  1198  by (fact unit_factor_mult_normalize)  haftmann@60688  1199  with False show ?thesis by simp  haftmann@60688  1200 qed  haftmann@60634  1201 haftmann@60687  1202 subclass semiring_Gcd  haftmann@60687  1203  by standard (simp_all add: Gcd_greatest)  haftmann@60687  1204 haftmann@58023  1205 lemma GcdI:  haftmann@60634  1206  assumes "\a. a \ A \ b dvd a" and "\c. (\a. a \ A \ c dvd a) \ c dvd b"  haftmann@60688  1207  and "normalize b = b"  haftmann@60634  1208  shows "b = Gcd A"  haftmann@60688  1209  by (rule associated_eqI) (auto simp: assms intro: Gcd_greatest)  haftmann@58023  1210 haftmann@58023  1211 lemma Lcm_Gcd:  haftmann@60430  1212  "Lcm A = Gcd {m. \a\A. a dvd m}"  haftmann@60634  1213  by (rule LcmI[symmetric]) (auto intro: dvd_Gcd Gcd_greatest)  haftmann@58023  1214 haftmann@60687  1215 subclass semiring_Lcm  haftmann@60687  1216  by standard (simp add: Lcm_Gcd)  haftmann@58023  1217 haftmann@58023  1218 lemma Gcd_1:  haftmann@58023  1219  "1 \ A \ Gcd A = 1"  haftmann@60687  1220  by (auto intro!: Gcd_eq_1_I)  haftmann@58023  1221 haftmann@58023  1222 lemma Gcd_finite:  haftmann@58023  1223  assumes "finite A"  haftmann@58023  1224  shows "Gcd A = Finite_Set.fold gcd 0 A"  wenzelm@60526  1225  by (induct rule: finite.induct[OF \finite A\])  haftmann@58023  1226  (simp_all add: comp_fun_idem.fold_insert_idem[OF comp_fun_idem_gcd])  haftmann@58023  1227 haftmann@60431  1228 lemma Gcd_set [code_unfold]:  haftmann@58023  1229  "Gcd (set xs) = fold gcd xs 0"  haftmann@58023  1230  using comp_fun_idem.fold_set_fold[OF comp_fun_idem_gcd] Gcd_finite by (simp add: ac_simps)  haftmann@58023  1231 haftmann@60634  1232 lemma Gcd_singleton [simp]: "Gcd {a} = normalize a"  haftmann@60687  1233  by simp  haftmann@58023  1234 haftmann@58023  1235 lemma Gcd_2 [simp]: "Gcd {a,b} = gcd a b"  haftmann@60687  1236  by simp  haftmann@60686  1237 haftmann@58023  1238 end  haftmann@58023  1239 wenzelm@60526  1240 text \  haftmann@58023  1241  A Euclidean ring is a Euclidean semiring with additive inverses. It provides a  haftmann@58023  1242  few more lemmas; in particular, Bezout's lemma holds for any Euclidean ring.  wenzelm@60526  1243 \  haftmann@58023  1244 haftmann@58023  1245 class euclidean_ring_gcd = euclidean_semiring_gcd + idom  haftmann@58023  1246 begin  haftmann@58023  1247 haftmann@58023  1248 subclass euclidean_ring ..  haftmann@58023  1249 haftmann@60439  1250 subclass ring_gcd ..  haftmann@60439  1251 haftmann@60572  1252 lemma euclid_ext_gcd [simp]:  haftmann@60572  1253  "(case euclid_ext a b of (_, _ , t) \ t) = gcd a b"  haftmann@60572  1254  by (induct a b rule: gcd_eucl_induct)  haftmann@60686  1255  (simp_all add: euclid_ext_0 euclid_ext_non_0 ac_simps split: prod.split prod.split_asm)  haftmann@60572  1256 haftmann@60572  1257 lemma euclid_ext_gcd' [simp]:  haftmann@60572  1258  "euclid_ext a b = (r, s, t) \ t = gcd a b"  haftmann@60572  1259  by (insert euclid_ext_gcd[of a b], drule (1) subst, simp)  haftmann@60572  1260   haftmann@60572  1261 lemma euclid_ext'_correct:  haftmann@60572  1262  "fst (euclid_ext' a b) * a + snd (euclid_ext' a b) * b = gcd a b"  haftmann@60572  1263 proof-  haftmann@60572  1264  obtain s t c where "euclid_ext a b = (s,t,c)"  haftmann@60572  1265  by (cases "euclid_ext a b", blast)  haftmann@60572  1266  with euclid_ext_correct[of a b] euclid_ext_gcd[of a b]  haftmann@60572  1267  show ?thesis unfolding euclid_ext'_def by simp  haftmann@60572  1268 qed  haftmann@60572  1269 haftmann@60572  1270 lemma bezout: "\s t. s * a + t * b = gcd a b"  haftmann@60572  1271  using euclid_ext'_correct by blast  haftmann@60572  1272 haftmann@58023  1273 lemma gcd_neg1 [simp]:  haftmann@60430  1274  "gcd (-a) b = gcd a b"  haftmann@59009  1275  by (rule sym, rule gcdI, simp_all add: gcd_greatest)  haftmann@58023  1276 haftmann@58023  1277 lemma gcd_neg2 [simp]:  haftmann@60430  1278  "gcd a (-b) = gcd a b"  haftmann@59009  1279  by (rule sym, rule gcdI, simp_all add: gcd_greatest)  haftmann@58023  1280 haftmann@58023  1281 lemma gcd_neg_numeral_1 [simp]:  haftmann@60430  1282  "gcd (- numeral n) a = gcd (numeral n) a"  haftmann@58023  1283  by (fact gcd_neg1)  haftmann@58023  1284 haftmann@58023  1285 lemma gcd_neg_numeral_2 [simp]:  haftmann@60430  1286  "gcd a (- numeral n) = gcd a (numeral n)"  haftmann@58023  1287  by (fact gcd_neg2)  haftmann@58023  1288 haftmann@58023  1289 lemma gcd_diff1: "gcd (m - n) n = gcd m n"  haftmann@58023  1290  by (subst diff_conv_add_uminus, subst gcd_neg2[symmetric], subst gcd_add1, simp)  haftmann@58023  1291 haftmann@58023  1292 lemma gcd_diff2: "gcd (n - m) n = gcd m n"  haftmann@58023  1293  by (subst gcd_neg1[symmetric], simp only: minus_diff_eq gcd_diff1)  haftmann@58023  1294 haftmann@58023  1295 lemma coprime_minus_one [simp]: "gcd (n - 1) n = 1"  haftmann@58023  1296 proof -  haftmann@58023  1297  have "gcd (n - 1) n = gcd n (n - 1)" by (fact gcd.commute)  haftmann@58023  1298  also have "\ = gcd ((n - 1) + 1) (n - 1)" by simp  haftmann@58023  1299  also have "\ = 1" by (rule coprime_plus_one)  haftmann@58023  1300  finally show ?thesis .  haftmann@58023  1301 qed  haftmann@58023  1302 haftmann@60430  1303 lemma lcm_neg1 [simp]: "lcm (-a) b = lcm a b"  haftmann@58023  1304  by (rule sym, rule lcmI, simp_all add: lcm_least lcm_zero)  haftmann@58023  1305 haftmann@60430  1306 lemma lcm_neg2 [simp]: "lcm a (-b) = lcm a b"  haftmann@58023  1307  by (rule sym, rule lcmI, simp_all add: lcm_least lcm_zero)  haftmann@58023  1308 haftmann@60430  1309 lemma lcm_neg_numeral_1 [simp]: "lcm (- numeral n) a = lcm (numeral n) a"  haftmann@58023  1310  by (fact lcm_neg1)  haftmann@58023  1311 haftmann@60430  1312 lemma lcm_neg_numeral_2 [simp]: "lcm a (- numeral n) = lcm a (numeral n)"  haftmann@58023  1313  by (fact lcm_neg2)  haftmann@58023  1314 haftmann@60572  1315 end  haftmann@58023  1316 haftmann@58023  1317 haftmann@60572  1318 subsection \Typical instances\  haftmann@58023  1319 haftmann@58023  1320 instantiation nat :: euclidean_semiring  haftmann@58023  1321 begin  haftmann@58023  1322 haftmann@58023  1323 definition [simp]:  haftmann@58023  1324  "euclidean_size_nat = (id :: nat \ nat)"  haftmann@58023  1325 haftmann@58023  1326 instance proof  haftmann@59061  1327 qed simp_all  haftmann@58023  1328 haftmann@58023  1329 end  haftmann@58023  1330 haftmann@58023  1331 instantiation int :: euclidean_ring  haftmann@58023  1332 begin  haftmann@58023  1333 haftmann@58023  1334 definition [simp]:  haftmann@58023  1335  "euclidean_size_int = (nat \ abs :: int \ nat)"  haftmann@58023  1336 wenzelm@60580  1337 instance  haftmann@60686  1338 by standard (auto simp add: abs_mult nat_mult_distrib split: abs_split)  haftmann@58023  1339 haftmann@58023  1340 end  haftmann@58023  1341 haftmann@60572  1342 instantiation poly :: (field) euclidean_ring  haftmann@60571  1343 begin  haftmann@60571  1344 haftmann@60571  1345 definition euclidean_size_poly :: "'a poly \ nat"  haftmann@60600  1346  where "euclidean_size p = (if p = 0 then 0 else Suc (degree p))"  haftmann@60571  1347 haftmann@60634  1348 lemma euclidenan_size_poly_minus_one_degree [simp]:  haftmann@60634  1349  "euclidean_size p - 1 = degree p"  haftmann@60634  1350  by (simp add: euclidean_size_poly_def)  haftmann@60571  1351 haftmann@60600  1352 lemma euclidean_size_poly_0 [simp]:  haftmann@60600  1353  "euclidean_size (0::'a poly) = 0"  haftmann@60600  1354  by (simp add: euclidean_size_poly_def)  haftmann@60600  1355 haftmann@60600  1356 lemma euclidean_size_poly_not_0 [simp]:  haftmann@60600  1357  "p \ 0 \ euclidean_size p = Suc (degree p)"  haftmann@60600  1358  by (simp add: euclidean_size_poly_def)  haftmann@60600  1359 haftmann@60571  1360 instance  haftmann@60600  1361 proof  haftmann@60571  1362  fix p q :: "'a poly"  haftmann@60600  1363  assume "q \ 0"  haftmann@60600  1364  then have "p mod q = 0 \ degree (p mod q) < degree q"  haftmann@60600  1365  by (rule degree_mod_less [of q p])  haftmann@60600  1366  with \q \ 0\ show "euclidean_size (p mod q) < euclidean_size q"  haftmann@60600  1367  by (cases "p mod q = 0") simp_all  haftmann@60571  1368 next  haftmann@60571  1369  fix p q :: "'a poly"  haftmann@60571  1370  assume "q \ 0"  haftmann@60600  1371  from \q \ 0\ have "degree p \ degree (p * q)"  haftmann@60571  1372  by (rule degree_mult_right_le)  haftmann@60600  1373  with \q \ 0\ show "euclidean_size p \ euclidean_size (p * q)"  haftmann@60600  1374  by (cases "p = 0") simp_all  haftmann@60571  1375 qed  haftmann@60571  1376 haftmann@58023  1377 end  haftmann@60571  1378 haftmann@60687  1379 (*instance nat :: euclidean_semiring_gcd  haftmann@60687  1380 proof (standard, auto intro!: ext)  haftmann@60687  1381  fix m n :: nat  haftmann@60687  1382  show *: "gcd m n = gcd_eucl m n"  haftmann@60687  1383  proof (induct m n rule: gcd_eucl_induct)  haftmann@60687  1384  case zero then show ?case by (simp add: gcd_eucl_0)  haftmann@60687  1385  next  haftmann@60687  1386  case (mod m n)  haftmann@60687  1387  with gcd_eucl_non_0 [of n m, symmetric]  haftmann@60687  1388  show ?case by (simp add: gcd_non_0_nat)  haftmann@60687  1389  qed  haftmann@60687  1390  show "lcm m n = lcm_eucl m n"  haftmann@60687  1391  by (simp add: lcm_eucl_def lcm_gcd * [symmetric] lcm_nat_def)  haftmann@60687  1392 qed*)  haftmann@60687  1393 haftmann@60571  1394 end