src/HOL/Number_Theory/Euclidean_Algorithm.thy
 author haftmann Fri Jun 12 08:53:23 2015 +0200 (2015-06-12) changeset 60432 68d75cff8809 parent 60431 db9c67b760f1 child 60433 720f210c5b1d permissions -rw-r--r--
given up trivial definition
 haftmann@58023  1 (* Author: Manuel Eberl *)  haftmann@58023  2 wenzelm@58889  3 section {* Abstract euclidean algorithm *}  haftmann@58023  4 haftmann@58023  5 theory Euclidean_Algorithm  haftmann@58023  6 imports Complex_Main  haftmann@58023  7 begin  haftmann@58023  8 haftmann@58023  9 context semiring_div  haftmann@58023  10 begin  haftmann@58023  11 haftmann@59061  12 abbreviation is_unit :: "'a \ bool"  haftmann@58023  13 where  haftmann@60430  14  "is_unit a \ a dvd 1"  haftmann@58023  15 haftmann@58023  16 definition associated :: "'a \ 'a \ bool"  haftmann@58023  17 where  haftmann@60430  18  "associated a b \ a dvd b \ b dvd a"  haftmann@58023  19 haftmann@58023  20 lemma unit_prod [intro]:  haftmann@60430  21  "is_unit a \ is_unit b \ is_unit (a * b)"  haftmann@59061  22  by (subst mult_1_left [of 1, symmetric], rule mult_dvd_mono)  haftmann@58023  23 haftmann@60432  24 lemma unit_divide_1:  haftmann@60432  25  "is_unit b \ a div b = a * divide 1 b"  haftmann@60432  26  by (simp add: div_mult_swap)  haftmann@58023  27 haftmann@60432  28 lemma unit_divide_1_divide_1 [simp]:  haftmann@60432  29  "is_unit a \ divide 1 (divide 1 a) = a"  haftmann@58023  30  by (metis div_mult_mult1_if div_mult_self1_is_id dvd_mult_div_cancel mult_1_right)  haftmann@58023  31 haftmann@60432  32 lemma inv_imp_eq_divide_1:  haftmann@60432  33  "a * b = 1 \ divide 1 a = b"  haftmann@60432  34  by (metis dvd_mult_div_cancel dvd_mult_right mult_1_right mult.left_commute one_dvd)  haftmann@58023  35 haftmann@60432  36 lemma unit_divide_1_unit [simp, intro]:  haftmann@60430  37  assumes "is_unit a"  haftmann@60432  38  shows "is_unit (divide 1 a)"  haftmann@58023  39 proof -  haftmann@60432  40  from assms have "1 = divide 1 a * a" by simp  haftmann@60432  41  then show "is_unit (divide 1 a)" by (rule dvdI)  haftmann@58023  42 qed  haftmann@58023  43 haftmann@58023  44 lemma mult_unit_dvd_iff:  haftmann@60430  45  "is_unit b \ a * b dvd c \ a dvd c"  haftmann@58023  46 proof  haftmann@60430  47  assume "is_unit b" "a * b dvd c"  haftmann@60430  48  then show "a dvd c" by (simp add: dvd_mult_left)  haftmann@58023  49 next  haftmann@60430  50  assume "is_unit b" "a dvd c"  haftmann@60430  51  then obtain k where "c = a * k" unfolding dvd_def by blast  haftmann@60432  52  with is_unit b have "c = (a * b) * (divide 1 b * k)"  haftmann@58023  53  by (simp add: mult_ac)  haftmann@60430  54  then show "a * b dvd c" by (rule dvdI)  haftmann@58023  55 qed  haftmann@58023  56 haftmann@58023  57 lemma div_unit_dvd_iff:  haftmann@60430  58  "is_unit b \ a div b dvd c \ a dvd c"  haftmann@60432  59  by (subst unit_divide_1) (assumption, simp add: mult_unit_dvd_iff)  haftmann@58023  60 haftmann@58023  61 lemma dvd_mult_unit_iff:  haftmann@60430  62  "is_unit b \ a dvd c * b \ a dvd c"  haftmann@58023  63 proof  haftmann@60430  64  assume "is_unit b" and "a dvd c * b"  haftmann@60432  65  have "c * b dvd c * (b * divide 1 b)" by (subst mult_assoc [symmetric]) simp  haftmann@60432  66  also from is_unit b have "b * divide 1 b = 1" by simp  haftmann@60430  67  finally have "c * b dvd c" by simp  haftmann@60430  68  with a dvd c * b show "a dvd c" by (rule dvd_trans)  haftmann@58023  69 next  haftmann@60430  70  assume "a dvd c"  haftmann@60430  71  then show "a dvd c * b" by simp  haftmann@58023  72 qed  haftmann@58023  73 haftmann@58023  74 lemma dvd_div_unit_iff:  haftmann@60430  75  "is_unit b \ a dvd c div b \ a dvd c"  haftmann@60432  76  by (subst unit_divide_1) (assumption, simp add: dvd_mult_unit_iff)  haftmann@58023  77 haftmann@58023  78 lemmas unit_dvd_iff = mult_unit_dvd_iff div_unit_dvd_iff dvd_mult_unit_iff dvd_div_unit_iff  haftmann@58023  79 haftmann@58023  80 lemma unit_div [intro]:  haftmann@60430  81  "is_unit a \ is_unit b \ is_unit (a div b)"  haftmann@60432  82  by (subst unit_divide_1) (assumption, rule unit_prod, simp_all)  haftmann@58023  83 haftmann@58023  84 lemma unit_div_mult_swap:  haftmann@60430  85  "is_unit c \ a * (b div c) = a * b div c"  haftmann@60432  86  by (simp only: unit_divide_1 [of _ b] unit_divide_1 [of _ "a*b"] ac_simps)  haftmann@58023  87 haftmann@58023  88 lemma unit_div_commute:  haftmann@60430  89  "is_unit b \ a div b * c = a * c div b"  haftmann@60432  90  by (simp only: unit_divide_1 [of _ a] unit_divide_1 [of _ "a*c"] ac_simps)  haftmann@58023  91 haftmann@58023  92 lemma unit_imp_dvd [dest]:  haftmann@60430  93  "is_unit b \ b dvd a"  haftmann@59061  94  by (rule dvd_trans [of _ 1]) simp_all  haftmann@58023  95 haftmann@58023  96 lemma dvd_unit_imp_unit:  haftmann@60430  97  "is_unit b \ a dvd b \ is_unit a"  haftmann@59061  98  by (rule dvd_trans)  haftmann@58023  99 haftmann@60432  100 lemma unit_divide_1'1:  haftmann@60430  101  assumes "is_unit b"  haftmann@60432  102  shows "a div (b * c) = a * divide 1 b div c"  haftmann@58023  103 proof -  haftmann@60432  104  from assms have "a div (b * c) = a * (divide 1 b * b) div (b * c)"  haftmann@58023  105  by simp  haftmann@60432  106  also have "... = b * (a * divide 1 b) div (b * c)"  haftmann@58023  107  by (simp only: mult_ac)  haftmann@60432  108  also have "... = a * divide 1 b div c"  haftmann@60430  109  by (cases "b = 0", simp, rule div_mult_mult1)  haftmann@58023  110  finally show ?thesis .  haftmann@58023  111 qed  haftmann@58023  112 haftmann@58023  113 lemma associated_comm:  haftmann@60430  114  "associated a b \ associated b a"  haftmann@58023  115  by (simp add: associated_def)  haftmann@58023  116 haftmann@58023  117 lemma associated_0 [simp]:  haftmann@58023  118  "associated 0 b \ b = 0"  haftmann@58023  119  "associated a 0 \ a = 0"  haftmann@58023  120  unfolding associated_def by simp_all  haftmann@58023  121 haftmann@58023  122 lemma associated_unit:  haftmann@60430  123  "is_unit a \ associated a b \ is_unit b"  haftmann@59061  124  unfolding associated_def using dvd_unit_imp_unit by auto  haftmann@58023  125 haftmann@58023  126 lemma is_unit_1 [simp]:  haftmann@58023  127  "is_unit 1"  haftmann@59061  128  by simp  haftmann@58023  129 haftmann@58023  130 lemma not_is_unit_0 [simp]:  haftmann@58023  131  "\ is_unit 0"  haftmann@59061  132  by auto  haftmann@58023  133 haftmann@58023  134 lemma unit_mult_left_cancel:  haftmann@60430  135  assumes "is_unit a"  haftmann@60430  136  shows "(a * b) = (a * c) \ b = c"  haftmann@58023  137 proof -  haftmann@60430  138  from assms have "a \ 0" by auto  haftmann@58023  139  then show ?thesis by (metis div_mult_self1_is_id)  haftmann@58023  140 qed  haftmann@58023  141 haftmann@58023  142 lemma unit_mult_right_cancel:  haftmann@60430  143  "is_unit a \ (b * a) = (c * a) \ b = c"  haftmann@58023  144  by (simp add: ac_simps unit_mult_left_cancel)  haftmann@58023  145 haftmann@58023  146 lemma unit_div_cancel:  haftmann@60430  147  "is_unit a \ (b div a) = (c div a) \ b = c"  haftmann@60432  148  apply (subst unit_divide_1[of _ b], assumption)  haftmann@60432  149  apply (subst unit_divide_1[of _ c], assumption)  haftmann@60432  150  apply (rule unit_mult_right_cancel, erule unit_divide_1_unit)  haftmann@58023  151  done  haftmann@58023  152 haftmann@58023  153 lemma unit_eq_div1:  haftmann@60430  154  "is_unit b \ a div b = c \ a = c * b"  haftmann@60432  155  apply (subst unit_divide_1, assumption)  haftmann@58023  156  apply (subst unit_mult_right_cancel[symmetric], assumption)  haftmann@60432  157  apply (subst mult_assoc, subst dvd_div_mult_self, assumption, simp)  haftmann@58023  158  done  haftmann@58023  159 haftmann@58023  160 lemma unit_eq_div2:  haftmann@60430  161  "is_unit b \ a = c div b \ a * b = c"  haftmann@58023  162  by (subst (1 2) eq_commute, simp add: unit_eq_div1, subst eq_commute, rule refl)  haftmann@58023  163 haftmann@58023  164 lemma associated_iff_div_unit:  haftmann@60430  165  "associated a b \ (\c. is_unit c \ a = c * b)"  haftmann@58023  166 proof  haftmann@60430  167  assume "associated a b"  haftmann@60430  168  show "\c. is_unit c \ a = c * b"  haftmann@60430  169  proof (cases "a = 0")  haftmann@60430  170  assume "a = 0"  haftmann@60430  171  then show "\c. is_unit c \ a = c * b" using associated a b  haftmann@58023  172  by (intro exI[of _ 1], simp add: associated_def)  haftmann@58023  173  next  haftmann@60430  174  assume [simp]: "a \ 0"  haftmann@60430  175  hence [simp]: "a dvd b" "b dvd a" using associated a b  haftmann@58023  176  unfolding associated_def by simp_all  haftmann@60430  177  hence "1 = a div b * (b div a)"  haftmann@59009  178  by (simp add: div_mult_swap)  haftmann@60430  179  hence "is_unit (a div b)" ..  haftmann@60430  180  moreover have "a = (a div b) * b" by simp  haftmann@58023  181  ultimately show ?thesis by blast  haftmann@58023  182  qed  haftmann@58023  183 next  haftmann@60430  184  assume "\c. is_unit c \ a = c * b"  haftmann@60430  185  then obtain c where "is_unit c" and "a = c * b" by blast  haftmann@60432  186  hence "b = a * divide 1 c" by (simp add: algebra_simps)  haftmann@60430  187  hence "a dvd b" by simp  haftmann@60430  188  moreover from a = c * b have "b dvd a" by simp  haftmann@60430  189  ultimately show "associated a b" unfolding associated_def by simp  haftmann@58023  190 qed  haftmann@58023  191 haftmann@58023  192 lemmas unit_simps = mult_unit_dvd_iff div_unit_dvd_iff dvd_mult_unit_iff  haftmann@58023  193  dvd_div_unit_iff unit_div_mult_swap unit_div_commute  haftmann@58023  194  unit_mult_left_cancel unit_mult_right_cancel unit_div_cancel  haftmann@58023  195  unit_eq_div1 unit_eq_div2  haftmann@58023  196 haftmann@58023  197 end  haftmann@58023  198 haftmann@58023  199 context ring_div  haftmann@58023  200 begin  haftmann@58023  201 haftmann@58023  202 lemma is_unit_neg [simp]:  haftmann@60430  203  "is_unit (- a) \ is_unit a"  haftmann@59061  204  by simp  haftmann@58023  205 haftmann@58023  206 lemma is_unit_neg_1 [simp]:  haftmann@58023  207  "is_unit (-1)"  haftmann@59061  208  by simp  haftmann@58023  209 haftmann@58023  210 end  haftmann@58023  211 haftmann@58023  212 lemma is_unit_nat [simp]:  haftmann@60430  213  "is_unit (a::nat) \ a = 1"  haftmann@59061  214  by simp  haftmann@58023  215 haftmann@58023  216 lemma is_unit_int:  haftmann@60430  217  "is_unit (a::int) \ a = 1 \ a = -1"  haftmann@59061  218  by auto  haftmann@58023  219 haftmann@58023  220 text {*  haftmann@58023  221  A Euclidean semiring is a semiring upon which the Euclidean algorithm can be  haftmann@58023  222  implemented. It must provide:  haftmann@58023  223  \begin{itemize}  haftmann@58023  224  \item division with remainder  haftmann@58023  225  \item a size function such that @{term "size (a mod b) < size b"}  haftmann@58023  226  for any @{term "b \ 0"}  haftmann@58023  227  \item a normalisation factor such that two associated numbers are equal iff  haftmann@58023  228  they are the same when divided by their normalisation factors.  haftmann@58023  229  \end{itemize}  haftmann@58023  230  The existence of these functions makes it possible to derive gcd and lcm functions  haftmann@58023  231  for any Euclidean semiring.  haftmann@58023  232 *}  haftmann@58023  233 class euclidean_semiring = semiring_div +  haftmann@58023  234  fixes euclidean_size :: "'a \ nat"  haftmann@58023  235  fixes normalisation_factor :: "'a \ 'a"  haftmann@58023  236  assumes mod_size_less [simp]:  haftmann@58023  237  "b \ 0 \ euclidean_size (a mod b) < euclidean_size b"  haftmann@58023  238  assumes size_mult_mono:  haftmann@58023  239  "b \ 0 \ euclidean_size (a * b) \ euclidean_size a"  haftmann@58023  240  assumes normalisation_factor_is_unit [intro,simp]:  haftmann@58023  241  "a \ 0 \ is_unit (normalisation_factor a)"  haftmann@58023  242  assumes normalisation_factor_mult: "normalisation_factor (a * b) =  haftmann@58023  243  normalisation_factor a * normalisation_factor b"  haftmann@60430  244  assumes normalisation_factor_unit: "is_unit a \ normalisation_factor a = a"  haftmann@58023  245  assumes normalisation_factor_0 [simp]: "normalisation_factor 0 = 0"  haftmann@58023  246 begin  haftmann@58023  247 haftmann@58023  248 lemma normalisation_factor_dvd [simp]:  haftmann@58023  249  "a \ 0 \ normalisation_factor a dvd b"  haftmann@58023  250  by (rule unit_imp_dvd, simp)  haftmann@58023  251   haftmann@58023  252 lemma normalisation_factor_1 [simp]:  haftmann@58023  253  "normalisation_factor 1 = 1"  haftmann@58023  254  by (simp add: normalisation_factor_unit)  haftmann@58023  255 haftmann@58023  256 lemma normalisation_factor_0_iff [simp]:  haftmann@60430  257  "normalisation_factor a = 0 \ a = 0"  haftmann@58023  258 proof  haftmann@60430  259  assume "normalisation_factor a = 0"  haftmann@60430  260  hence "\ is_unit (normalisation_factor a)"  haftmann@58023  261  by (metis not_is_unit_0)  haftmann@60430  262  then show "a = 0" by force  haftmann@58023  263 next  haftmann@60430  264  assume "a = 0"  haftmann@60430  265  then show "normalisation_factor a = 0" by simp  haftmann@58023  266 qed  haftmann@58023  267 haftmann@58023  268 lemma normalisation_factor_pow:  haftmann@60430  269  "normalisation_factor (a ^ n) = normalisation_factor a ^ n"  haftmann@58023  270  by (induct n) (simp_all add: normalisation_factor_mult power_Suc2)  haftmann@58023  271 haftmann@58023  272 lemma normalisation_correct [simp]:  haftmann@60430  273  "normalisation_factor (a div normalisation_factor a) = (if a = 0 then 0 else 1)"  haftmann@60430  274 proof (cases "a = 0", simp)  haftmann@60430  275  assume "a \ 0"  haftmann@58023  276  let ?nf = "normalisation_factor"  haftmann@60430  277  from normalisation_factor_is_unit[OF a \ 0] have "?nf a \ 0"  haftmann@58023  278  by (metis not_is_unit_0)  haftmann@60430  279  have "?nf (a div ?nf a) * ?nf (?nf a) = ?nf (a div ?nf a * ?nf a)"  haftmann@58023  280  by (simp add: normalisation_factor_mult)  haftmann@60430  281  also have "a div ?nf a * ?nf a = a" using a \ 0  haftmann@59009  282  by simp  haftmann@60430  283  also have "?nf (?nf a) = ?nf a" using a \ 0  haftmann@58023  284  normalisation_factor_is_unit normalisation_factor_unit by simp  haftmann@60430  285  finally show ?thesis using a \ 0 and ?nf a \ 0  haftmann@58023  286  by (metis div_mult_self2_is_id div_self)  haftmann@58023  287 qed  haftmann@58023  288 haftmann@58023  289 lemma normalisation_0_iff [simp]:  haftmann@60430  290  "a div normalisation_factor a = 0 \ a = 0"  haftmann@60430  291  by (cases "a = 0", simp, subst unit_eq_div1, blast, simp)  haftmann@58023  292 haftmann@58023  293 lemma associated_iff_normed_eq:  haftmann@58023  294  "associated a b \ a div normalisation_factor a = b div normalisation_factor b"  haftmann@58023  295 proof (cases "b = 0", simp, cases "a = 0", metis associated_0(1) normalisation_0_iff, rule iffI)  haftmann@58023  296  let ?nf = normalisation_factor  haftmann@58023  297  assume "a \ 0" "b \ 0" "a div ?nf a = b div ?nf b"  haftmann@58023  298  hence "a = b * (?nf a div ?nf b)"  haftmann@58023  299  apply (subst (asm) unit_eq_div1, blast, subst (asm) unit_div_commute, blast)  haftmann@58023  300  apply (subst div_mult_swap, simp, simp)  haftmann@58023  301  done  haftmann@60430  302  with a \ 0 b \ 0 have "\c. is_unit c \ a = c * b"  haftmann@58023  303  by (intro exI[of _ "?nf a div ?nf b"], force simp: mult_ac)  haftmann@58023  304  with associated_iff_div_unit show "associated a b" by simp  haftmann@58023  305 next  haftmann@58023  306  let ?nf = normalisation_factor  haftmann@58023  307  assume "a \ 0" "b \ 0" "associated a b"  haftmann@60430  308  with associated_iff_div_unit obtain c where "is_unit c" and "a = c * b" by blast  haftmann@58023  309  then show "a div ?nf a = b div ?nf b"  haftmann@60430  310  apply (simp only: a = c * b normalisation_factor_mult normalisation_factor_unit)  haftmann@58023  311  apply (rule div_mult_mult1, force)  haftmann@58023  312  done  haftmann@58023  313  qed  haftmann@58023  314 haftmann@58023  315 lemma normed_associated_imp_eq:  haftmann@58023  316  "associated a b \ normalisation_factor a \ {0, 1} \ normalisation_factor b \ {0, 1} \ a = b"  haftmann@58023  317  by (simp add: associated_iff_normed_eq, elim disjE, simp_all)  haftmann@58023  318   haftmann@58023  319 lemmas normalisation_factor_dvd_iff [simp] =  haftmann@58023  320  unit_dvd_iff [OF normalisation_factor_is_unit]  haftmann@58023  321 haftmann@58023  322 lemma euclidean_division:  haftmann@58023  323  fixes a :: 'a and b :: 'a  haftmann@58023  324  assumes "b \ 0"  haftmann@58023  325  obtains s and t where "a = s * b + t"  haftmann@58023  326  and "euclidean_size t < euclidean_size b"  haftmann@58023  327 proof -  haftmann@58023  328  from div_mod_equality[of a b 0]  haftmann@58023  329  have "a = a div b * b + a mod b" by simp  haftmann@58023  330  with that and assms show ?thesis by force  haftmann@58023  331 qed  haftmann@58023  332 haftmann@58023  333 lemma dvd_euclidean_size_eq_imp_dvd:  haftmann@58023  334  assumes "a \ 0" and b_dvd_a: "b dvd a" and size_eq: "euclidean_size a = euclidean_size b"  haftmann@58023  335  shows "a dvd b"  haftmann@58023  336 proof (subst dvd_eq_mod_eq_0, rule ccontr)  haftmann@58023  337  assume "b mod a \ 0"  haftmann@58023  338  from b_dvd_a have b_dvd_mod: "b dvd b mod a" by (simp add: dvd_mod_iff)  haftmann@58023  339  from b_dvd_mod obtain c where "b mod a = b * c" unfolding dvd_def by blast  haftmann@58023  340  with b mod a \ 0 have "c \ 0" by auto  haftmann@58023  341  with b mod a = b * c have "euclidean_size (b mod a) \ euclidean_size b"  haftmann@58023  342  using size_mult_mono by force  haftmann@58023  343  moreover from a \ 0 have "euclidean_size (b mod a) < euclidean_size a"  haftmann@58023  344  using mod_size_less by blast  haftmann@58023  345  ultimately show False using size_eq by simp  haftmann@58023  346 qed  haftmann@58023  347 haftmann@58023  348 function gcd_eucl :: "'a \ 'a \ 'a"  haftmann@58023  349 where  haftmann@58023  350  "gcd_eucl a b = (if b = 0 then a div normalisation_factor a else gcd_eucl b (a mod b))"  haftmann@58023  351  by (pat_completeness, simp)  haftmann@58023  352 termination by (relation "measure (euclidean_size \ snd)", simp_all)  haftmann@58023  353 haftmann@58023  354 declare gcd_eucl.simps [simp del]  haftmann@58023  355 haftmann@58023  356 lemma gcd_induct: "\\b. P b 0; \a b. 0 \ b \ P b (a mod b) \ P a b\ \ P a b"  haftmann@58023  357 proof (induct a b rule: gcd_eucl.induct)  haftmann@58023  358  case ("1" m n)  haftmann@58023  359  then show ?case by (cases "n = 0") auto  haftmann@58023  360 qed  haftmann@58023  361 haftmann@58023  362 definition lcm_eucl :: "'a \ 'a \ 'a"  haftmann@58023  363 where  haftmann@58023  364  "lcm_eucl a b = a * b div (gcd_eucl a b * normalisation_factor (a * b))"  haftmann@58023  365 haftmann@58023  366  (* Somewhat complicated definition of Lcm that has the advantage of working  haftmann@58023  367  for infinite sets as well *)  haftmann@58023  368 haftmann@58023  369 definition Lcm_eucl :: "'a set \ 'a"  haftmann@58023  370 where  haftmann@60430  371  "Lcm_eucl A = (if \l. l \ 0 \ (\a\A. a dvd l) then  haftmann@60430  372  let l = SOME l. l \ 0 \ (\a\A. a dvd l) \ euclidean_size l =  haftmann@60430  373  (LEAST n. \l. l \ 0 \ (\a\A. a dvd l) \ euclidean_size l = n)  haftmann@58023  374  in l div normalisation_factor l  haftmann@58023  375  else 0)"  haftmann@58023  376 haftmann@58023  377 definition Gcd_eucl :: "'a set \ 'a"  haftmann@58023  378 where  haftmann@58023  379  "Gcd_eucl A = Lcm_eucl {d. \a\A. d dvd a}"  haftmann@58023  380 haftmann@58023  381 end  haftmann@58023  382 haftmann@58023  383 class euclidean_semiring_gcd = euclidean_semiring + gcd + Gcd +  haftmann@58023  384  assumes gcd_gcd_eucl: "gcd = gcd_eucl" and lcm_lcm_eucl: "lcm = lcm_eucl"  haftmann@58023  385  assumes Gcd_Gcd_eucl: "Gcd = Gcd_eucl" and Lcm_Lcm_eucl: "Lcm = Lcm_eucl"  haftmann@58023  386 begin  haftmann@58023  387 haftmann@58023  388 lemma gcd_red:  haftmann@60430  389  "gcd a b = gcd b (a mod b)"  haftmann@58023  390  by (metis gcd_eucl.simps mod_0 mod_by_0 gcd_gcd_eucl)  haftmann@58023  391 haftmann@58023  392 lemma gcd_non_0:  haftmann@60430  393  "b \ 0 \ gcd a b = gcd b (a mod b)"  haftmann@58023  394  by (rule gcd_red)  haftmann@58023  395 haftmann@58023  396 lemma gcd_0_left:  haftmann@60430  397  "gcd 0 a = a div normalisation_factor a"  haftmann@58023  398  by (simp only: gcd_gcd_eucl, subst gcd_eucl.simps, subst gcd_eucl.simps, simp add: Let_def)  haftmann@58023  399 haftmann@58023  400 lemma gcd_0:  haftmann@60430  401  "gcd a 0 = a div normalisation_factor a"  haftmann@58023  402  by (simp only: gcd_gcd_eucl, subst gcd_eucl.simps, simp add: Let_def)  haftmann@58023  403 haftmann@60430  404 lemma gcd_dvd1 [iff]: "gcd a b dvd a"  haftmann@60430  405  and gcd_dvd2 [iff]: "gcd a b dvd b"  haftmann@60430  406 proof (induct a b rule: gcd_eucl.induct)  haftmann@60430  407  fix a b :: 'a  haftmann@60430  408  assume IH1: "b \ 0 \ gcd b (a mod b) dvd b"  haftmann@60430  409  assume IH2: "b \ 0 \ gcd b (a mod b) dvd (a mod b)"  haftmann@58023  410   haftmann@60430  411  have "gcd a b dvd a \ gcd a b dvd b"  haftmann@60430  412  proof (cases "b = 0")  haftmann@58023  413  case True  haftmann@60430  414  then show ?thesis by (cases "a = 0", simp_all add: gcd_0)  haftmann@58023  415  next  haftmann@58023  416  case False  haftmann@58023  417  with IH1 and IH2 show ?thesis by (simp add: gcd_non_0 dvd_mod_iff)  haftmann@58023  418  qed  haftmann@60430  419  then show "gcd a b dvd a" "gcd a b dvd b" by simp_all  haftmann@58023  420 qed  haftmann@58023  421 haftmann@58023  422 lemma dvd_gcd_D1: "k dvd gcd m n \ k dvd m"  haftmann@58023  423  by (rule dvd_trans, assumption, rule gcd_dvd1)  haftmann@58023  424 haftmann@58023  425 lemma dvd_gcd_D2: "k dvd gcd m n \ k dvd n"  haftmann@58023  426  by (rule dvd_trans, assumption, rule gcd_dvd2)  haftmann@58023  427 haftmann@58023  428 lemma gcd_greatest:  haftmann@60430  429  fixes k a b :: 'a  haftmann@60430  430  shows "k dvd a \ k dvd b \ k dvd gcd a b"  haftmann@60430  431 proof (induct a b rule: gcd_eucl.induct)  haftmann@60430  432  case (1 a b)  haftmann@58023  433  show ?case  haftmann@60430  434  proof (cases "b = 0")  haftmann@60430  435  assume "b = 0"  haftmann@60430  436  with 1 show ?thesis by (cases "a = 0", simp_all add: gcd_0)  haftmann@58023  437  next  haftmann@60430  438  assume "b \ 0"  haftmann@58023  439  with 1 show ?thesis by (simp add: gcd_non_0 dvd_mod_iff)  haftmann@58023  440  qed  haftmann@58023  441 qed  haftmann@58023  442 haftmann@58023  443 lemma dvd_gcd_iff:  haftmann@60430  444  "k dvd gcd a b \ k dvd a \ k dvd b"  haftmann@58023  445  by (blast intro!: gcd_greatest intro: dvd_trans)  haftmann@58023  446 haftmann@58023  447 lemmas gcd_greatest_iff = dvd_gcd_iff  haftmann@58023  448 haftmann@58023  449 lemma gcd_zero [simp]:  haftmann@60430  450  "gcd a b = 0 \ a = 0 \ b = 0"  haftmann@58023  451  by (metis dvd_0_left dvd_refl gcd_dvd1 gcd_dvd2 gcd_greatest)+  haftmann@58023  452 haftmann@58023  453 lemma normalisation_factor_gcd [simp]:  haftmann@60430  454  "normalisation_factor (gcd a b) = (if a = 0 \ b = 0 then 0 else 1)" (is "?f a b = ?g a b")  haftmann@60430  455 proof (induct a b rule: gcd_eucl.induct)  haftmann@60430  456  fix a b :: 'a  haftmann@60430  457  assume IH: "b \ 0 \ ?f b (a mod b) = ?g b (a mod b)"  haftmann@60430  458  then show "?f a b = ?g a b" by (cases "b = 0", auto simp: gcd_non_0 gcd_0)  haftmann@58023  459 qed  haftmann@58023  460 haftmann@58023  461 lemma gcdI:  haftmann@60430  462  "k dvd a \ k dvd b \ (\l. l dvd a \ l dvd b \ l dvd k)  haftmann@60430  463  \ normalisation_factor k = (if k = 0 then 0 else 1) \ k = gcd a b"  haftmann@58023  464  by (intro normed_associated_imp_eq) (auto simp: associated_def intro: gcd_greatest)  haftmann@58023  465 haftmann@58023  466 sublocale gcd!: abel_semigroup gcd  haftmann@58023  467 proof  haftmann@60430  468  fix a b c  haftmann@60430  469  show "gcd (gcd a b) c = gcd a (gcd b c)"  haftmann@58023  470  proof (rule gcdI)  haftmann@60430  471  have "gcd (gcd a b) c dvd gcd a b" "gcd a b dvd a" by simp_all  haftmann@60430  472  then show "gcd (gcd a b) c dvd a" by (rule dvd_trans)  haftmann@60430  473  have "gcd (gcd a b) c dvd gcd a b" "gcd a b dvd b" by simp_all  haftmann@60430  474  hence "gcd (gcd a b) c dvd b" by (rule dvd_trans)  haftmann@60430  475  moreover have "gcd (gcd a b) c dvd c" by simp  haftmann@60430  476  ultimately show "gcd (gcd a b) c dvd gcd b c"  haftmann@58023  477  by (rule gcd_greatest)  haftmann@60430  478  show "normalisation_factor (gcd (gcd a b) c) = (if gcd (gcd a b) c = 0 then 0 else 1)"  haftmann@58023  479  by auto  haftmann@60430  480  fix l assume "l dvd a" and "l dvd gcd b c"  haftmann@58023  481  with dvd_trans[OF _ gcd_dvd1] and dvd_trans[OF _ gcd_dvd2]  haftmann@60430  482  have "l dvd b" and "l dvd c" by blast+  haftmann@60430  483  with l dvd a show "l dvd gcd (gcd a b) c"  haftmann@58023  484  by (intro gcd_greatest)  haftmann@58023  485  qed  haftmann@58023  486 next  haftmann@60430  487  fix a b  haftmann@60430  488  show "gcd a b = gcd b a"  haftmann@58023  489  by (rule gcdI) (simp_all add: gcd_greatest)  haftmann@58023  490 qed  haftmann@58023  491 haftmann@58023  492 lemma gcd_unique: "d dvd a \ d dvd b \  haftmann@58023  493  normalisation_factor d = (if d = 0 then 0 else 1) \  haftmann@58023  494  (\e. e dvd a \ e dvd b \ e dvd d) \ d = gcd a b"  haftmann@58023  495  by (rule, auto intro: gcdI simp: gcd_greatest)  haftmann@58023  496 haftmann@58023  497 lemma gcd_dvd_prod: "gcd a b dvd k * b"  haftmann@58023  498  using mult_dvd_mono [of 1] by auto  haftmann@58023  499 haftmann@60430  500 lemma gcd_1_left [simp]: "gcd 1 a = 1"  haftmann@58023  501  by (rule sym, rule gcdI, simp_all)  haftmann@58023  502 haftmann@60430  503 lemma gcd_1 [simp]: "gcd a 1 = 1"  haftmann@58023  504  by (rule sym, rule gcdI, simp_all)  haftmann@58023  505 haftmann@58023  506 lemma gcd_proj2_if_dvd:  haftmann@60430  507  "b dvd a \ gcd a b = b div normalisation_factor b"  haftmann@60430  508  by (cases "b = 0", simp_all add: dvd_eq_mod_eq_0 gcd_non_0 gcd_0)  haftmann@58023  509 haftmann@58023  510 lemma gcd_proj1_if_dvd:  haftmann@60430  511  "a dvd b \ gcd a b = a div normalisation_factor a"  haftmann@58023  512  by (subst gcd.commute, simp add: gcd_proj2_if_dvd)  haftmann@58023  513 haftmann@58023  514 lemma gcd_proj1_iff: "gcd m n = m div normalisation_factor m \ m dvd n"  haftmann@58023  515 proof  haftmann@58023  516  assume A: "gcd m n = m div normalisation_factor m"  haftmann@58023  517  show "m dvd n"  haftmann@58023  518  proof (cases "m = 0")  haftmann@58023  519  assume [simp]: "m \ 0"  haftmann@58023  520  from A have B: "m = gcd m n * normalisation_factor m"  haftmann@58023  521  by (simp add: unit_eq_div2)  haftmann@58023  522  show ?thesis by (subst B, simp add: mult_unit_dvd_iff)  haftmann@58023  523  qed (insert A, simp)  haftmann@58023  524 next  haftmann@58023  525  assume "m dvd n"  haftmann@58023  526  then show "gcd m n = m div normalisation_factor m" by (rule gcd_proj1_if_dvd)  haftmann@58023  527 qed  haftmann@58023  528   haftmann@58023  529 lemma gcd_proj2_iff: "gcd m n = n div normalisation_factor n \ n dvd m"  haftmann@58023  530  by (subst gcd.commute, simp add: gcd_proj1_iff)  haftmann@58023  531 haftmann@58023  532 lemma gcd_mod1 [simp]:  haftmann@60430  533  "gcd (a mod b) b = gcd a b"  haftmann@58023  534  by (rule gcdI, metis dvd_mod_iff gcd_dvd1 gcd_dvd2, simp_all add: gcd_greatest dvd_mod_iff)  haftmann@58023  535 haftmann@58023  536 lemma gcd_mod2 [simp]:  haftmann@60430  537  "gcd a (b mod a) = gcd a b"  haftmann@58023  538  by (rule gcdI, simp, metis dvd_mod_iff gcd_dvd1 gcd_dvd2, simp_all add: gcd_greatest dvd_mod_iff)  haftmann@58023  539   haftmann@58023  540 lemma normalisation_factor_dvd' [simp]:  haftmann@60430  541  "normalisation_factor a dvd a"  haftmann@60430  542  by (cases "a = 0", simp_all)  haftmann@58023  543 haftmann@58023  544 lemma gcd_mult_distrib':  haftmann@60430  545  "k div normalisation_factor k * gcd a b = gcd (k*a) (k*b)"  haftmann@60430  546 proof (induct a b rule: gcd_eucl.induct)  haftmann@60430  547  case (1 a b)  haftmann@58023  548  show ?case  haftmann@60430  549  proof (cases "b = 0")  haftmann@58023  550  case True  haftmann@58023  551  then show ?thesis by (simp add: normalisation_factor_mult gcd_0 algebra_simps div_mult_div_if_dvd)  haftmann@58023  552  next  haftmann@58023  553  case False  haftmann@60430  554  hence "k div normalisation_factor k * gcd a b = gcd (k * b) (k * (a mod b))"  haftmann@58023  555  using 1 by (subst gcd_red, simp)  haftmann@60430  556  also have "... = gcd (k * a) (k * b)"  haftmann@58023  557  by (simp add: mult_mod_right gcd.commute)  haftmann@58023  558  finally show ?thesis .  haftmann@58023  559  qed  haftmann@58023  560 qed  haftmann@58023  561 haftmann@58023  562 lemma gcd_mult_distrib:  haftmann@60430  563  "k * gcd a b = gcd (k*a) (k*b) * normalisation_factor k"  haftmann@58023  564 proof-  haftmann@58023  565  let ?nf = "normalisation_factor"  haftmann@58023  566  from gcd_mult_distrib'  haftmann@60430  567  have "gcd (k*a) (k*b) = k div ?nf k * gcd a b" ..  haftmann@60430  568  also have "... = k * gcd a b div ?nf k"  haftmann@58023  569  by (metis dvd_div_mult dvd_eq_mod_eq_0 mod_0 normalisation_factor_dvd)  haftmann@58023  570  finally show ?thesis  haftmann@59009  571  by simp  haftmann@58023  572 qed  haftmann@58023  573 haftmann@58023  574 lemma euclidean_size_gcd_le1 [simp]:  haftmann@58023  575  assumes "a \ 0"  haftmann@58023  576  shows "euclidean_size (gcd a b) \ euclidean_size a"  haftmann@58023  577 proof -  haftmann@58023  578  have "gcd a b dvd a" by (rule gcd_dvd1)  haftmann@58023  579  then obtain c where A: "a = gcd a b * c" unfolding dvd_def by blast  haftmann@58023  580  with a \ 0 show ?thesis by (subst (2) A, intro size_mult_mono) auto  haftmann@58023  581 qed  haftmann@58023  582 haftmann@58023  583 lemma euclidean_size_gcd_le2 [simp]:  haftmann@58023  584  "b \ 0 \ euclidean_size (gcd a b) \ euclidean_size b"  haftmann@58023  585  by (subst gcd.commute, rule euclidean_size_gcd_le1)  haftmann@58023  586 haftmann@58023  587 lemma euclidean_size_gcd_less1:  haftmann@58023  588  assumes "a \ 0" and "\a dvd b"  haftmann@58023  589  shows "euclidean_size (gcd a b) < euclidean_size a"  haftmann@58023  590 proof (rule ccontr)  haftmann@58023  591  assume "\euclidean_size (gcd a b) < euclidean_size a"  haftmann@58023  592  with a \ 0 have "euclidean_size (gcd a b) = euclidean_size a"  haftmann@58023  593  by (intro le_antisym, simp_all)  haftmann@58023  594  with assms have "a dvd gcd a b" by (auto intro: dvd_euclidean_size_eq_imp_dvd)  haftmann@58023  595  hence "a dvd b" using dvd_gcd_D2 by blast  haftmann@58023  596  with \a dvd b show False by contradiction  haftmann@58023  597 qed  haftmann@58023  598 haftmann@58023  599 lemma euclidean_size_gcd_less2:  haftmann@58023  600  assumes "b \ 0" and "\b dvd a"  haftmann@58023  601  shows "euclidean_size (gcd a b) < euclidean_size b"  haftmann@58023  602  using assms by (subst gcd.commute, rule euclidean_size_gcd_less1)  haftmann@58023  603 haftmann@60430  604 lemma gcd_mult_unit1: "is_unit a \ gcd (b * a) c = gcd b c"  haftmann@58023  605  apply (rule gcdI)  haftmann@58023  606  apply (rule dvd_trans, rule gcd_dvd1, simp add: unit_simps)  haftmann@58023  607  apply (rule gcd_dvd2)  haftmann@58023  608  apply (rule gcd_greatest, simp add: unit_simps, assumption)  haftmann@58023  609  apply (subst normalisation_factor_gcd, simp add: gcd_0)  haftmann@58023  610  done  haftmann@58023  611 haftmann@60430  612 lemma gcd_mult_unit2: "is_unit a \ gcd b (c * a) = gcd b c"  haftmann@58023  613  by (subst gcd.commute, subst gcd_mult_unit1, assumption, rule gcd.commute)  haftmann@58023  614 haftmann@60430  615 lemma gcd_div_unit1: "is_unit a \ gcd (b div a) c = gcd b c"  haftmann@60432  616  by (subst unit_divide_1) (simp_all add: gcd_mult_unit1)  haftmann@58023  617 haftmann@60430  618 lemma gcd_div_unit2: "is_unit a \ gcd b (c div a) = gcd b c"  haftmann@60432  619  by (subst unit_divide_1) (simp_all add: gcd_mult_unit2)  haftmann@58023  620 haftmann@60430  621 lemma gcd_idem: "gcd a a = a div normalisation_factor a"  haftmann@60430  622  by (cases "a = 0") (simp add: gcd_0_left, rule sym, rule gcdI, simp_all)  haftmann@58023  623 haftmann@60430  624 lemma gcd_right_idem: "gcd (gcd a b) b = gcd a b"  haftmann@58023  625  apply (rule gcdI)  haftmann@58023  626  apply (simp add: ac_simps)  haftmann@58023  627  apply (rule gcd_dvd2)  haftmann@58023  628  apply (rule gcd_greatest, erule (1) gcd_greatest, assumption)  haftmann@59009  629  apply simp  haftmann@58023  630  done  haftmann@58023  631 haftmann@60430  632 lemma gcd_left_idem: "gcd a (gcd a b) = gcd a b"  haftmann@58023  633  apply (rule gcdI)  haftmann@58023  634  apply simp  haftmann@58023  635  apply (rule dvd_trans, rule gcd_dvd2, rule gcd_dvd2)  haftmann@58023  636  apply (rule gcd_greatest, assumption, erule gcd_greatest, assumption)  haftmann@59009  637  apply simp  haftmann@58023  638  done  haftmann@58023  639 haftmann@58023  640 lemma comp_fun_idem_gcd: "comp_fun_idem gcd"  haftmann@58023  641 proof  haftmann@58023  642  fix a b show "gcd a \ gcd b = gcd b \ gcd a"  haftmann@58023  643  by (simp add: fun_eq_iff ac_simps)  haftmann@58023  644 next  haftmann@58023  645  fix a show "gcd a \ gcd a = gcd a"  haftmann@58023  646  by (simp add: fun_eq_iff gcd_left_idem)  haftmann@58023  647 qed  haftmann@58023  648 haftmann@58023  649 lemma coprime_dvd_mult:  haftmann@60430  650  assumes "gcd c b = 1" and "c dvd a * b"  haftmann@60430  651  shows "c dvd a"  haftmann@58023  652 proof -  haftmann@58023  653  let ?nf = "normalisation_factor"  haftmann@60430  654  from assms gcd_mult_distrib [of a c b]  haftmann@60430  655  have A: "a = gcd (a * c) (a * b) * ?nf a" by simp  haftmann@60430  656  from c dvd a * b show ?thesis by (subst A, simp_all add: gcd_greatest)  haftmann@58023  657 qed  haftmann@58023  658 haftmann@58023  659 lemma coprime_dvd_mult_iff:  haftmann@60430  660  "gcd c b = 1 \ (c dvd a * b) = (c dvd a)"  haftmann@58023  661  by (rule, rule coprime_dvd_mult, simp_all)  haftmann@58023  662 haftmann@58023  663 lemma gcd_dvd_antisym:  haftmann@58023  664  "gcd a b dvd gcd c d \ gcd c d dvd gcd a b \ gcd a b = gcd c d"  haftmann@58023  665 proof (rule gcdI)  haftmann@58023  666  assume A: "gcd a b dvd gcd c d" and B: "gcd c d dvd gcd a b"  haftmann@58023  667  have "gcd c d dvd c" by simp  haftmann@58023  668  with A show "gcd a b dvd c" by (rule dvd_trans)  haftmann@58023  669  have "gcd c d dvd d" by simp  haftmann@58023  670  with A show "gcd a b dvd d" by (rule dvd_trans)  haftmann@58023  671  show "normalisation_factor (gcd a b) = (if gcd a b = 0 then 0 else 1)"  haftmann@59009  672  by simp  haftmann@58023  673  fix l assume "l dvd c" and "l dvd d"  haftmann@58023  674  hence "l dvd gcd c d" by (rule gcd_greatest)  haftmann@58023  675  from this and B show "l dvd gcd a b" by (rule dvd_trans)  haftmann@58023  676 qed  haftmann@58023  677 haftmann@58023  678 lemma gcd_mult_cancel:  haftmann@58023  679  assumes "gcd k n = 1"  haftmann@58023  680  shows "gcd (k * m) n = gcd m n"  haftmann@58023  681 proof (rule gcd_dvd_antisym)  haftmann@58023  682  have "gcd (gcd (k * m) n) k = gcd (gcd k n) (k * m)" by (simp add: ac_simps)  haftmann@58023  683  also note gcd k n = 1  haftmann@58023  684  finally have "gcd (gcd (k * m) n) k = 1" by simp  haftmann@58023  685  hence "gcd (k * m) n dvd m" by (rule coprime_dvd_mult, simp add: ac_simps)  haftmann@58023  686  moreover have "gcd (k * m) n dvd n" by simp  haftmann@58023  687  ultimately show "gcd (k * m) n dvd gcd m n" by (rule gcd_greatest)  haftmann@58023  688  have "gcd m n dvd (k * m)" and "gcd m n dvd n" by simp_all  haftmann@58023  689  then show "gcd m n dvd gcd (k * m) n" by (rule gcd_greatest)  haftmann@58023  690 qed  haftmann@58023  691 haftmann@58023  692 lemma coprime_crossproduct:  haftmann@58023  693  assumes [simp]: "gcd a d = 1" "gcd b c = 1"  haftmann@58023  694  shows "associated (a * c) (b * d) \ associated a b \ associated c d" (is "?lhs \ ?rhs")  haftmann@58023  695 proof  haftmann@58023  696  assume ?rhs then show ?lhs unfolding associated_def by (fast intro: mult_dvd_mono)  haftmann@58023  697 next  haftmann@58023  698  assume ?lhs  haftmann@58023  699  from ?lhs have "a dvd b * d" unfolding associated_def by (metis dvd_mult_left)  haftmann@58023  700  hence "a dvd b" by (simp add: coprime_dvd_mult_iff)  haftmann@58023  701  moreover from ?lhs have "b dvd a * c" unfolding associated_def by (metis dvd_mult_left)  haftmann@58023  702  hence "b dvd a" by (simp add: coprime_dvd_mult_iff)  haftmann@58023  703  moreover from ?lhs have "c dvd d * b"  haftmann@59009  704  unfolding associated_def by (auto dest: dvd_mult_right simp add: ac_simps)  haftmann@58023  705  hence "c dvd d" by (simp add: coprime_dvd_mult_iff gcd.commute)  haftmann@58023  706  moreover from ?lhs have "d dvd c * a"  haftmann@59009  707  unfolding associated_def by (auto dest: dvd_mult_right simp add: ac_simps)  haftmann@58023  708  hence "d dvd c" by (simp add: coprime_dvd_mult_iff gcd.commute)  haftmann@58023  709  ultimately show ?rhs unfolding associated_def by simp  haftmann@58023  710 qed  haftmann@58023  711 haftmann@58023  712 lemma gcd_add1 [simp]:  haftmann@58023  713  "gcd (m + n) n = gcd m n"  haftmann@58023  714  by (cases "n = 0", simp_all add: gcd_non_0)  haftmann@58023  715 haftmann@58023  716 lemma gcd_add2 [simp]:  haftmann@58023  717  "gcd m (m + n) = gcd m n"  haftmann@58023  718  using gcd_add1 [of n m] by (simp add: ac_simps)  haftmann@58023  719 haftmann@58023  720 lemma gcd_add_mult: "gcd m (k * m + n) = gcd m n"  haftmann@58023  721  by (subst gcd.commute, subst gcd_red, simp)  haftmann@58023  722 haftmann@60430  723 lemma coprimeI: "(\l. \l dvd a; l dvd b\ \ l dvd 1) \ gcd a b = 1"  haftmann@58023  724  by (rule sym, rule gcdI, simp_all)  haftmann@58023  725 haftmann@58023  726 lemma coprime: "gcd a b = 1 \ (\d. d dvd a \ d dvd b \ is_unit d)"  haftmann@59061  727  by (auto intro: coprimeI gcd_greatest dvd_gcd_D1 dvd_gcd_D2)  haftmann@58023  728 haftmann@58023  729 lemma div_gcd_coprime:  haftmann@58023  730  assumes nz: "a \ 0 \ b \ 0"  haftmann@58023  731  defines [simp]: "d \ gcd a b"  haftmann@58023  732  defines [simp]: "a' \ a div d" and [simp]: "b' \ b div d"  haftmann@58023  733  shows "gcd a' b' = 1"  haftmann@58023  734 proof (rule coprimeI)  haftmann@58023  735  fix l assume "l dvd a'" "l dvd b'"  haftmann@58023  736  then obtain s t where "a' = l * s" "b' = l * t" unfolding dvd_def by blast  haftmann@59009  737  moreover have "a = a' * d" "b = b' * d" by simp_all  haftmann@58023  738  ultimately have "a = (l * d) * s" "b = (l * d) * t"  haftmann@59009  739  by (simp_all only: ac_simps)  haftmann@58023  740  hence "l*d dvd a" and "l*d dvd b" by (simp_all only: dvd_triv_left)  haftmann@58023  741  hence "l*d dvd d" by (simp add: gcd_greatest)  haftmann@59009  742  then obtain u where "d = l * d * u" ..  haftmann@59009  743  then have "d * (l * u) = d" by (simp add: ac_simps)  haftmann@59009  744  moreover from nz have "d \ 0" by simp  haftmann@59009  745  with div_mult_self1_is_id have "d * (l * u) div d = l * u" .  haftmann@59009  746  ultimately have "1 = l * u"  haftmann@59009  747  using d \ 0 by simp  haftmann@59009  748  then show "l dvd 1" ..  haftmann@58023  749 qed  haftmann@58023  750 haftmann@58023  751 lemma coprime_mult:  haftmann@58023  752  assumes da: "gcd d a = 1" and db: "gcd d b = 1"  haftmann@58023  753  shows "gcd d (a * b) = 1"  haftmann@58023  754  apply (subst gcd.commute)  haftmann@58023  755  using da apply (subst gcd_mult_cancel)  haftmann@58023  756  apply (subst gcd.commute, assumption)  haftmann@58023  757  apply (subst gcd.commute, rule db)  haftmann@58023  758  done  haftmann@58023  759 haftmann@58023  760 lemma coprime_lmult:  haftmann@58023  761  assumes dab: "gcd d (a * b) = 1"  haftmann@58023  762  shows "gcd d a = 1"  haftmann@58023  763 proof (rule coprimeI)  haftmann@58023  764  fix l assume "l dvd d" and "l dvd a"  haftmann@58023  765  hence "l dvd a * b" by simp  haftmann@58023  766  with l dvd d and dab show "l dvd 1" by (auto intro: gcd_greatest)  haftmann@58023  767 qed  haftmann@58023  768 haftmann@58023  769 lemma coprime_rmult:  haftmann@58023  770  assumes dab: "gcd d (a * b) = 1"  haftmann@58023  771  shows "gcd d b = 1"  haftmann@58023  772 proof (rule coprimeI)  haftmann@58023  773  fix l assume "l dvd d" and "l dvd b"  haftmann@58023  774  hence "l dvd a * b" by simp  haftmann@58023  775  with l dvd d and dab show "l dvd 1" by (auto intro: gcd_greatest)  haftmann@58023  776 qed  haftmann@58023  777 haftmann@58023  778 lemma coprime_mul_eq: "gcd d (a * b) = 1 \ gcd d a = 1 \ gcd d b = 1"  haftmann@58023  779  using coprime_rmult[of d a b] coprime_lmult[of d a b] coprime_mult[of d a b] by blast  haftmann@58023  780 haftmann@58023  781 lemma gcd_coprime:  haftmann@60430  782  assumes c: "gcd a b \ 0" and a: "a = a' * gcd a b" and b: "b = b' * gcd a b"  haftmann@58023  783  shows "gcd a' b' = 1"  haftmann@58023  784 proof -  haftmann@60430  785  from c have "a \ 0 \ b \ 0" by simp  haftmann@58023  786  with div_gcd_coprime have "gcd (a div gcd a b) (b div gcd a b) = 1" .  haftmann@58023  787  also from assms have "a div gcd a b = a'" by (metis div_mult_self2_is_id)+  haftmann@58023  788  also from assms have "b div gcd a b = b'" by (metis div_mult_self2_is_id)+  haftmann@58023  789  finally show ?thesis .  haftmann@58023  790 qed  haftmann@58023  791 haftmann@58023  792 lemma coprime_power:  haftmann@58023  793  assumes "0 < n"  haftmann@58023  794  shows "gcd a (b ^ n) = 1 \ gcd a b = 1"  haftmann@58023  795 using assms proof (induct n)  haftmann@58023  796  case (Suc n) then show ?case  haftmann@58023  797  by (cases n) (simp_all add: coprime_mul_eq)  haftmann@58023  798 qed simp  haftmann@58023  799 haftmann@58023  800 lemma gcd_coprime_exists:  haftmann@58023  801  assumes nz: "gcd a b \ 0"  haftmann@58023  802  shows "\a' b'. a = a' * gcd a b \ b = b' * gcd a b \ gcd a' b' = 1"  haftmann@58023  803  apply (rule_tac x = "a div gcd a b" in exI)  haftmann@58023  804  apply (rule_tac x = "b div gcd a b" in exI)  haftmann@59009  805  apply (insert nz, auto intro: div_gcd_coprime)  haftmann@58023  806  done  haftmann@58023  807 haftmann@58023  808 lemma coprime_exp:  haftmann@58023  809  "gcd d a = 1 \ gcd d (a^n) = 1"  haftmann@58023  810  by (induct n, simp_all add: coprime_mult)  haftmann@58023  811 haftmann@58023  812 lemma coprime_exp2 [intro]:  haftmann@58023  813  "gcd a b = 1 \ gcd (a^n) (b^m) = 1"  haftmann@58023  814  apply (rule coprime_exp)  haftmann@58023  815  apply (subst gcd.commute)  haftmann@58023  816  apply (rule coprime_exp)  haftmann@58023  817  apply (subst gcd.commute)  haftmann@58023  818  apply assumption  haftmann@58023  819  done  haftmann@58023  820 haftmann@58023  821 lemma gcd_exp:  haftmann@58023  822  "gcd (a^n) (b^n) = (gcd a b) ^ n"  haftmann@58023  823 proof (cases "a = 0 \ b = 0")  haftmann@58023  824  assume "a = 0 \ b = 0"  haftmann@58023  825  then show ?thesis by (cases n, simp_all add: gcd_0_left)  haftmann@58023  826 next  haftmann@58023  827  assume A: "\(a = 0 \ b = 0)"  haftmann@58023  828  hence "1 = gcd ((a div gcd a b)^n) ((b div gcd a b)^n)"  haftmann@58023  829  using div_gcd_coprime by (subst sym, auto simp: div_gcd_coprime)  haftmann@58023  830  hence "(gcd a b) ^ n = (gcd a b) ^ n * ..." by simp  haftmann@58023  831  also note gcd_mult_distrib  haftmann@58023  832  also have "normalisation_factor ((gcd a b)^n) = 1"  haftmann@58023  833  by (simp add: normalisation_factor_pow A)  haftmann@58023  834  also have "(gcd a b)^n * (a div gcd a b)^n = a^n"  haftmann@58023  835  by (subst ac_simps, subst div_power, simp, rule dvd_div_mult_self, rule dvd_power_same, simp)  haftmann@58023  836  also have "(gcd a b)^n * (b div gcd a b)^n = b^n"  haftmann@58023  837  by (subst ac_simps, subst div_power, simp, rule dvd_div_mult_self, rule dvd_power_same, simp)  haftmann@58023  838  finally show ?thesis by simp  haftmann@58023  839 qed  haftmann@58023  840 haftmann@58023  841 lemma coprime_common_divisor:  haftmann@60430  842  "gcd a b = 1 \ a dvd a \ a dvd b \ is_unit a"  haftmann@60430  843  apply (subgoal_tac "a dvd gcd a b")  haftmann@59061  844  apply simp  haftmann@58023  845  apply (erule (1) gcd_greatest)  haftmann@58023  846  done  haftmann@58023  847 haftmann@58023  848 lemma division_decomp:  haftmann@58023  849  assumes dc: "a dvd b * c"  haftmann@58023  850  shows "\b' c'. a = b' * c' \ b' dvd b \ c' dvd c"  haftmann@58023  851 proof (cases "gcd a b = 0")  haftmann@58023  852  assume "gcd a b = 0"  haftmann@59009  853  hence "a = 0 \ b = 0" by simp  haftmann@58023  854  hence "a = 0 * c \ 0 dvd b \ c dvd c" by simp  haftmann@58023  855  then show ?thesis by blast  haftmann@58023  856 next  haftmann@58023  857  let ?d = "gcd a b"  haftmann@58023  858  assume "?d \ 0"  haftmann@58023  859  from gcd_coprime_exists[OF this]  haftmann@58023  860  obtain a' b' where ab': "a = a' * ?d" "b = b' * ?d" "gcd a' b' = 1"  haftmann@58023  861  by blast  haftmann@58023  862  from ab'(1) have "a' dvd a" unfolding dvd_def by blast  haftmann@58023  863  with dc have "a' dvd b*c" using dvd_trans[of a' a "b*c"] by simp  haftmann@58023  864  from dc ab'(1,2) have "a'*?d dvd (b'*?d) * c" by simp  haftmann@58023  865  hence "?d * a' dvd ?d * (b' * c)" by (simp add: mult_ac)  haftmann@59009  866  with ?d \ 0 have "a' dvd b' * c" by simp  haftmann@58023  867  with coprime_dvd_mult[OF ab'(3)]  haftmann@58023  868  have "a' dvd c" by (subst (asm) ac_simps, blast)  haftmann@58023  869  with ab'(1) have "a = ?d * a' \ ?d dvd b \ a' dvd c" by (simp add: mult_ac)  haftmann@58023  870  then show ?thesis by blast  haftmann@58023  871 qed  haftmann@58023  872 haftmann@58023  873 lemma pow_divides_pow:  haftmann@58023  874  assumes ab: "a ^ n dvd b ^ n" and n: "n \ 0"  haftmann@58023  875  shows "a dvd b"  haftmann@58023  876 proof (cases "gcd a b = 0")  haftmann@58023  877  assume "gcd a b = 0"  haftmann@59009  878  then show ?thesis by simp  haftmann@58023  879 next  haftmann@58023  880  let ?d = "gcd a b"  haftmann@58023  881  assume "?d \ 0"  haftmann@58023  882  from n obtain m where m: "n = Suc m" by (cases n, simp_all)  haftmann@59009  883  from ?d \ 0 have zn: "?d ^ n \ 0" by (rule power_not_zero)  haftmann@58023  884  from gcd_coprime_exists[OF ?d \ 0]  haftmann@58023  885  obtain a' b' where ab': "a = a' * ?d" "b = b' * ?d" "gcd a' b' = 1"  haftmann@58023  886  by blast  haftmann@58023  887  from ab have "(a' * ?d) ^ n dvd (b' * ?d) ^ n"  haftmann@58023  888  by (simp add: ab'(1,2)[symmetric])  haftmann@58023  889  hence "?d^n * a'^n dvd ?d^n * b'^n"  haftmann@58023  890  by (simp only: power_mult_distrib ac_simps)  haftmann@59009  891  with zn have "a'^n dvd b'^n" by simp  haftmann@58023  892  hence "a' dvd b'^n" using dvd_trans[of a' "a'^n" "b'^n"] by (simp add: m)  haftmann@58023  893  hence "a' dvd b'^m * b'" by (simp add: m ac_simps)  haftmann@58023  894  with coprime_dvd_mult[OF coprime_exp[OF ab'(3), of m]]  haftmann@58023  895  have "a' dvd b'" by (subst (asm) ac_simps, blast)  haftmann@58023  896  hence "a'*?d dvd b'*?d" by (rule mult_dvd_mono, simp)  haftmann@58023  897  with ab'(1,2) show ?thesis by simp  haftmann@58023  898 qed  haftmann@58023  899 haftmann@58023  900 lemma pow_divides_eq [simp]:  haftmann@58023  901  "n \ 0 \ a ^ n dvd b ^ n \ a dvd b"  haftmann@58023  902  by (auto intro: pow_divides_pow dvd_power_same)  haftmann@58023  903 haftmann@58023  904 lemma divides_mult:  haftmann@58023  905  assumes mr: "m dvd r" and nr: "n dvd r" and mn: "gcd m n = 1"  haftmann@58023  906  shows "m * n dvd r"  haftmann@58023  907 proof -  haftmann@58023  908  from mr nr obtain m' n' where m': "r = m*m'" and n': "r = n*n'"  haftmann@58023  909  unfolding dvd_def by blast  haftmann@58023  910  from mr n' have "m dvd n'*n" by (simp add: ac_simps)  haftmann@58023  911  hence "m dvd n'" using coprime_dvd_mult_iff[OF mn] by simp  haftmann@58023  912  then obtain k where k: "n' = m*k" unfolding dvd_def by blast  haftmann@58023  913  with n' have "r = m * n * k" by (simp add: mult_ac)  haftmann@58023  914  then show ?thesis unfolding dvd_def by blast  haftmann@58023  915 qed  haftmann@58023  916 haftmann@58023  917 lemma coprime_plus_one [simp]: "gcd (n + 1) n = 1"  haftmann@58023  918  by (subst add_commute, simp)  haftmann@58023  919 haftmann@58023  920 lemma setprod_coprime [rule_format]:  haftmann@60430  921  "(\i\A. gcd (f i) a = 1) \ gcd (\i\A. f i) a = 1"  haftmann@58023  922  apply (cases "finite A")  haftmann@58023  923  apply (induct set: finite)  haftmann@58023  924  apply (auto simp add: gcd_mult_cancel)  haftmann@58023  925  done  haftmann@58023  926 haftmann@58023  927 lemma coprime_divisors:  haftmann@58023  928  assumes "d dvd a" "e dvd b" "gcd a b = 1"  haftmann@58023  929  shows "gcd d e = 1"  haftmann@58023  930 proof -  haftmann@58023  931  from assms obtain k l where "a = d * k" "b = e * l"  haftmann@58023  932  unfolding dvd_def by blast  haftmann@58023  933  with assms have "gcd (d * k) (e * l) = 1" by simp  haftmann@58023  934  hence "gcd (d * k) e = 1" by (rule coprime_lmult)  haftmann@58023  935  also have "gcd (d * k) e = gcd e (d * k)" by (simp add: ac_simps)  haftmann@58023  936  finally have "gcd e d = 1" by (rule coprime_lmult)  haftmann@58023  937  then show ?thesis by (simp add: ac_simps)  haftmann@58023  938 qed  haftmann@58023  939 haftmann@58023  940 lemma invertible_coprime:  haftmann@60430  941  assumes "a * b mod m = 1"  haftmann@60430  942  shows "coprime a m"  haftmann@59009  943 proof -  haftmann@60430  944  from assms have "coprime m (a * b mod m)"  haftmann@59009  945  by simp  haftmann@60430  946  then have "coprime m (a * b)"  haftmann@59009  947  by simp  haftmann@60430  948  then have "coprime m a"  haftmann@59009  949  by (rule coprime_lmult)  haftmann@59009  950  then show ?thesis  haftmann@59009  951  by (simp add: ac_simps)  haftmann@59009  952 qed  haftmann@58023  953 haftmann@58023  954 lemma lcm_gcd:  haftmann@58023  955  "lcm a b = a * b div (gcd a b * normalisation_factor (a*b))"  haftmann@58023  956  by (simp only: lcm_lcm_eucl gcd_gcd_eucl lcm_eucl_def)  haftmann@58023  957 haftmann@58023  958 lemma lcm_gcd_prod:  haftmann@58023  959  "lcm a b * gcd a b = a * b div normalisation_factor (a*b)"  haftmann@58023  960 proof (cases "a * b = 0")  haftmann@58023  961  let ?nf = normalisation_factor  haftmann@58023  962  assume "a * b \ 0"  haftmann@58953  963  hence "gcd a b \ 0" by simp  haftmann@58023  964  from lcm_gcd have "lcm a b * gcd a b = gcd a b * (a * b div (?nf (a*b) * gcd a b))"  haftmann@58023  965  by (simp add: mult_ac)  haftmann@60432  966  also from a * b \ 0 have "... = a * b div ?nf (a*b)"  haftmann@60432  967  by (simp add: div_mult_swap mult.commute)  haftmann@58023  968  finally show ?thesis .  haftmann@58953  969 qed (auto simp add: lcm_gcd)  haftmann@58023  970 haftmann@58023  971 lemma lcm_dvd1 [iff]:  haftmann@60430  972  "a dvd lcm a b"  haftmann@60430  973 proof (cases "a*b = 0")  haftmann@60430  974  assume "a * b \ 0"  haftmann@60430  975  hence "gcd a b \ 0" by simp  haftmann@60432  976  let ?c = "divide 1 (normalisation_factor (a*b))"  haftmann@60430  977  from a * b \ 0 have [simp]: "is_unit (normalisation_factor (a*b))" by simp  haftmann@60430  978  from lcm_gcd_prod[of a b] have "lcm a b * gcd a b = a * ?c * b"  haftmann@60432  979  by (simp add: div_mult_swap unit_div_commute)  haftmann@60430  980  hence "lcm a b * gcd a b div gcd a b = a * ?c * b div gcd a b" by simp  haftmann@60430  981  with gcd a b \ 0 have "lcm a b = a * ?c * b div gcd a b"  haftmann@58023  982  by (subst (asm) div_mult_self2_is_id, simp_all)  haftmann@60430  983  also have "... = a * (?c * b div gcd a b)"  haftmann@58023  984  by (metis div_mult_swap gcd_dvd2 mult_assoc)  haftmann@58023  985  finally show ?thesis by (rule dvdI)  haftmann@58953  986 qed (auto simp add: lcm_gcd)  haftmann@58023  987 haftmann@58023  988 lemma lcm_least:  haftmann@58023  989  "\a dvd k; b dvd k\ \ lcm a b dvd k"  haftmann@58023  990 proof (cases "k = 0")  haftmann@58023  991  let ?nf = normalisation_factor  haftmann@58023  992  assume "k \ 0"  haftmann@58023  993  hence "is_unit (?nf k)" by simp  haftmann@58023  994  hence "?nf k \ 0" by (metis not_is_unit_0)  haftmann@58023  995  assume A: "a dvd k" "b dvd k"  haftmann@58953  996  hence "gcd a b \ 0" using k \ 0 by auto  haftmann@58023  997  from A obtain r s where ar: "k = a * r" and bs: "k = b * s"  haftmann@58023  998  unfolding dvd_def by blast  haftmann@58953  999  with k \ 0 have "r * s \ 0"  haftmann@58953  1000  by auto (drule sym [of 0], simp)  haftmann@58023  1001  hence "is_unit (?nf (r * s))" by simp  haftmann@58023  1002  let ?c = "?nf k div ?nf (r*s)"  haftmann@58023  1003  from is_unit (?nf k) and is_unit (?nf (r * s)) have "is_unit ?c" by (rule unit_div)  haftmann@58023  1004  hence "?c \ 0" using not_is_unit_0 by fast  haftmann@58023  1005  from ar bs have "k * k * gcd s r = ?nf k * k * gcd (k * s) (k * r)"  haftmann@58953  1006  by (subst mult_assoc, subst gcd_mult_distrib[of k s r], simp only: ac_simps)  haftmann@58023  1007  also have "... = ?nf k * k * gcd ((r*s) * a) ((r*s) * b)"  haftmann@58023  1008  by (subst (3) k = a * r, subst (3) k = b * s, simp add: algebra_simps)  haftmann@58023  1009  also have "... = ?c * r*s * k * gcd a b" using r * s \ 0  haftmann@58023  1010  by (subst gcd_mult_distrib'[symmetric], simp add: algebra_simps unit_simps)  haftmann@58023  1011  finally have "(a*r) * (b*s) * gcd s r = ?c * k * r * s * gcd a b"  haftmann@58023  1012  by (subst ar[symmetric], subst bs[symmetric], simp add: mult_ac)  haftmann@58023  1013  hence "a * b * gcd s r * (r * s) = ?c * k * gcd a b * (r * s)"  haftmann@58023  1014  by (simp add: algebra_simps)  haftmann@58023  1015  hence "?c * k * gcd a b = a * b * gcd s r" using r * s \ 0  haftmann@58023  1016  by (metis div_mult_self2_is_id)  haftmann@58023  1017  also have "... = lcm a b * gcd a b * gcd s r * ?nf (a*b)"  haftmann@58023  1018  by (subst lcm_gcd_prod[of a b], metis gcd_mult_distrib gcd_mult_distrib')  haftmann@58023  1019  also have "... = lcm a b * gcd s r * ?nf (a*b) * gcd a b"  haftmann@58023  1020  by (simp add: algebra_simps)  haftmann@58023  1021  finally have "k * ?c = lcm a b * gcd s r * ?nf (a*b)" using gcd a b \ 0  haftmann@58023  1022  by (metis mult.commute div_mult_self2_is_id)  haftmann@58023  1023  hence "k = lcm a b * (gcd s r * ?nf (a*b)) div ?c" using ?c \ 0  haftmann@58023  1024  by (metis div_mult_self2_is_id mult_assoc)  haftmann@58023  1025  also have "... = lcm a b * (gcd s r * ?nf (a*b) div ?c)" using is_unit ?c  haftmann@58023  1026  by (simp add: unit_simps)  haftmann@58023  1027  finally show ?thesis by (rule dvdI)  haftmann@58023  1028 qed simp  haftmann@58023  1029 haftmann@58023  1030 lemma lcm_zero:  haftmann@58023  1031  "lcm a b = 0 \ a = 0 \ b = 0"  haftmann@58023  1032 proof -  haftmann@58023  1033  let ?nf = normalisation_factor  haftmann@58023  1034  {  haftmann@58023  1035  assume "a \ 0" "b \ 0"  haftmann@58023  1036  hence "a * b div ?nf (a * b) \ 0" by (simp add: no_zero_divisors)  haftmann@59009  1037  moreover from a \ 0 and b \ 0 have "gcd a b \ 0" by simp  haftmann@58023  1038  ultimately have "lcm a b \ 0" using lcm_gcd_prod[of a b] by (intro notI, simp)  haftmann@58023  1039  } moreover {  haftmann@58023  1040  assume "a = 0 \ b = 0"  haftmann@58023  1041  hence "lcm a b = 0" by (elim disjE, simp_all add: lcm_gcd)  haftmann@58023  1042  }  haftmann@58023  1043  ultimately show ?thesis by blast  haftmann@58023  1044 qed  haftmann@58023  1045 haftmann@58023  1046 lemmas lcm_0_iff = lcm_zero  haftmann@58023  1047 haftmann@58023  1048 lemma gcd_lcm:  haftmann@58023  1049  assumes "lcm a b \ 0"  haftmann@58023  1050  shows "gcd a b = a * b div (lcm a b * normalisation_factor (a * b))"  haftmann@58023  1051 proof-  haftmann@59009  1052  from assms have "gcd a b \ 0" by (simp add: lcm_zero)  haftmann@58023  1053  let ?c = "normalisation_factor (a*b)"  haftmann@58023  1054  from lcm a b \ 0 have "?c \ 0" by (intro notI, simp add: lcm_zero no_zero_divisors)  haftmann@58023  1055  hence "is_unit ?c" by simp  haftmann@58023  1056  from lcm_gcd_prod [of a b] have "gcd a b = a * b div ?c div lcm a b"  haftmann@58023  1057  by (subst (2) div_mult_self2_is_id[OF lcm a b \ 0, symmetric], simp add: mult_ac)  haftmann@58023  1058  also from is_unit ?c have "... = a * b div (?c * lcm a b)"  haftmann@60432  1059  by (metis local.unit_divide_1 local.unit_divide_1'1)  haftmann@58023  1060  finally show ?thesis by (simp only: ac_simps)  haftmann@58023  1061 qed  haftmann@58023  1062 haftmann@58023  1063 lemma normalisation_factor_lcm [simp]:  haftmann@58023  1064  "normalisation_factor (lcm a b) = (if a = 0 \ b = 0 then 0 else 1)"  haftmann@58023  1065 proof (cases "a = 0 \ b = 0")  haftmann@58023  1066  case True then show ?thesis  haftmann@58953  1067  by (auto simp add: lcm_gcd)  haftmann@58023  1068 next  haftmann@58023  1069  case False  haftmann@58023  1070  let ?nf = normalisation_factor  haftmann@58023  1071  from lcm_gcd_prod[of a b]  haftmann@58023  1072  have "?nf (lcm a b) * ?nf (gcd a b) = ?nf (a*b) div ?nf (a*b)"  haftmann@58023  1073  by (metis div_by_0 div_self normalisation_correct normalisation_factor_0 normalisation_factor_mult)  haftmann@58023  1074  also have "... = (if a*b = 0 then 0 else 1)"  haftmann@58953  1075  by simp  haftmann@58953  1076  finally show ?thesis using False by simp  haftmann@58023  1077 qed  haftmann@58023  1078 haftmann@60430  1079 lemma lcm_dvd2 [iff]: "b dvd lcm a b"  haftmann@60430  1080  using lcm_dvd1 [of b a] by (simp add: lcm_gcd ac_simps)  haftmann@58023  1081 haftmann@58023  1082 lemma lcmI:  haftmann@60430  1083  "\a dvd k; b dvd k; \l. a dvd l \ b dvd l \ k dvd l;  haftmann@60430  1084  normalisation_factor k = (if k = 0 then 0 else 1)\ \ k = lcm a b"  haftmann@58023  1085  by (intro normed_associated_imp_eq) (auto simp: associated_def intro: lcm_least)  haftmann@58023  1086 haftmann@58023  1087 sublocale lcm!: abel_semigroup lcm  haftmann@58023  1088 proof  haftmann@60430  1089  fix a b c  haftmann@60430  1090  show "lcm (lcm a b) c = lcm a (lcm b c)"  haftmann@58023  1091  proof (rule lcmI)  haftmann@60430  1092  have "a dvd lcm a b" and "lcm a b dvd lcm (lcm a b) c" by simp_all  haftmann@60430  1093  then show "a dvd lcm (lcm a b) c" by (rule dvd_trans)  haftmann@58023  1094   haftmann@60430  1095  have "b dvd lcm a b" and "lcm a b dvd lcm (lcm a b) c" by simp_all  haftmann@60430  1096  hence "b dvd lcm (lcm a b) c" by (rule dvd_trans)  haftmann@60430  1097  moreover have "c dvd lcm (lcm a b) c" by simp  haftmann@60430  1098  ultimately show "lcm b c dvd lcm (lcm a b) c" by (rule lcm_least)  haftmann@58023  1099 haftmann@60430  1100  fix l assume "a dvd l" and "lcm b c dvd l"  haftmann@60430  1101  have "b dvd lcm b c" by simp  haftmann@60430  1102  from this and lcm b c dvd l have "b dvd l" by (rule dvd_trans)  haftmann@60430  1103  have "c dvd lcm b c" by simp  haftmann@60430  1104  from this and lcm b c dvd l have "c dvd l" by (rule dvd_trans)  haftmann@60430  1105  from a dvd l and b dvd l have "lcm a b dvd l" by (rule lcm_least)  haftmann@60430  1106  from this and c dvd l show "lcm (lcm a b) c dvd l" by (rule lcm_least)  haftmann@58023  1107  qed (simp add: lcm_zero)  haftmann@58023  1108 next  haftmann@60430  1109  fix a b  haftmann@60430  1110  show "lcm a b = lcm b a"  haftmann@58023  1111  by (simp add: lcm_gcd ac_simps)  haftmann@58023  1112 qed  haftmann@58023  1113 haftmann@58023  1114 lemma dvd_lcm_D1:  haftmann@58023  1115  "lcm m n dvd k \ m dvd k"  haftmann@58023  1116  by (rule dvd_trans, rule lcm_dvd1, assumption)  haftmann@58023  1117 haftmann@58023  1118 lemma dvd_lcm_D2:  haftmann@58023  1119  "lcm m n dvd k \ n dvd k"  haftmann@58023  1120  by (rule dvd_trans, rule lcm_dvd2, assumption)  haftmann@58023  1121 haftmann@58023  1122 lemma gcd_dvd_lcm [simp]:  haftmann@58023  1123  "gcd a b dvd lcm a b"  haftmann@58023  1124  by (metis dvd_trans gcd_dvd2 lcm_dvd2)  haftmann@58023  1125 haftmann@58023  1126 lemma lcm_1_iff:  haftmann@58023  1127  "lcm a b = 1 \ is_unit a \ is_unit b"  haftmann@58023  1128 proof  haftmann@58023  1129  assume "lcm a b = 1"  haftmann@59061  1130  then show "is_unit a \ is_unit b" by auto  haftmann@58023  1131 next  haftmann@58023  1132  assume "is_unit a \ is_unit b"  haftmann@59061  1133  hence "a dvd 1" and "b dvd 1" by simp_all  haftmann@59061  1134  hence "is_unit (lcm a b)" by (rule lcm_least)  haftmann@58023  1135  hence "lcm a b = normalisation_factor (lcm a b)"  haftmann@58023  1136  by (subst normalisation_factor_unit, simp_all)  haftmann@59061  1137  also have "\ = 1" using is_unit a \ is_unit b  haftmann@59061  1138  by auto  haftmann@58023  1139  finally show "lcm a b = 1" .  haftmann@58023  1140 qed  haftmann@58023  1141 haftmann@58023  1142 lemma lcm_0_left [simp]:  haftmann@60430  1143  "lcm 0 a = 0"  haftmann@58023  1144  by (rule sym, rule lcmI, simp_all)  haftmann@58023  1145 haftmann@58023  1146 lemma lcm_0 [simp]:  haftmann@60430  1147  "lcm a 0 = 0"  haftmann@58023  1148  by (rule sym, rule lcmI, simp_all)  haftmann@58023  1149 haftmann@58023  1150 lemma lcm_unique:  haftmann@58023  1151  "a dvd d \ b dvd d \  haftmann@58023  1152  normalisation_factor d = (if d = 0 then 0 else 1) \  haftmann@58023  1153  (\e. a dvd e \ b dvd e \ d dvd e) \ d = lcm a b"  haftmann@58023  1154  by (rule, auto intro: lcmI simp: lcm_least lcm_zero)  haftmann@58023  1155 haftmann@58023  1156 lemma dvd_lcm_I1 [simp]:  haftmann@58023  1157  "k dvd m \ k dvd lcm m n"  haftmann@58023  1158  by (metis lcm_dvd1 dvd_trans)  haftmann@58023  1159 haftmann@58023  1160 lemma dvd_lcm_I2 [simp]:  haftmann@58023  1161  "k dvd n \ k dvd lcm m n"  haftmann@58023  1162  by (metis lcm_dvd2 dvd_trans)  haftmann@58023  1163 haftmann@58023  1164 lemma lcm_1_left [simp]:  haftmann@60430  1165  "lcm 1 a = a div normalisation_factor a"  haftmann@60430  1166  by (cases "a = 0") (simp, rule sym, rule lcmI, simp_all)  haftmann@58023  1167 haftmann@58023  1168 lemma lcm_1_right [simp]:  haftmann@60430  1169  "lcm a 1 = a div normalisation_factor a"  haftmann@60430  1170  using lcm_1_left [of a] by (simp add: ac_simps)  haftmann@58023  1171 haftmann@58023  1172 lemma lcm_coprime:  haftmann@58023  1173  "gcd a b = 1 \ lcm a b = a * b div normalisation_factor (a*b)"  haftmann@58023  1174  by (subst lcm_gcd) simp  haftmann@58023  1175 haftmann@58023  1176 lemma lcm_proj1_if_dvd:  haftmann@60430  1177  "b dvd a \ lcm a b = a div normalisation_factor a"  haftmann@60430  1178  by (cases "a = 0") (simp, rule sym, rule lcmI, simp_all)  haftmann@58023  1179 haftmann@58023  1180 lemma lcm_proj2_if_dvd:  haftmann@60430  1181  "a dvd b \ lcm a b = b div normalisation_factor b"  haftmann@60430  1182  using lcm_proj1_if_dvd [of a b] by (simp add: ac_simps)  haftmann@58023  1183 haftmann@58023  1184 lemma lcm_proj1_iff:  haftmann@58023  1185  "lcm m n = m div normalisation_factor m \ n dvd m"  haftmann@58023  1186 proof  haftmann@58023  1187  assume A: "lcm m n = m div normalisation_factor m"  haftmann@58023  1188  show "n dvd m"  haftmann@58023  1189  proof (cases "m = 0")  haftmann@58023  1190  assume [simp]: "m \ 0"  haftmann@58023  1191  from A have B: "m = lcm m n * normalisation_factor m"  haftmann@58023  1192  by (simp add: unit_eq_div2)  haftmann@58023  1193  show ?thesis by (subst B, simp)  haftmann@58023  1194  qed simp  haftmann@58023  1195 next  haftmann@58023  1196  assume "n dvd m"  haftmann@58023  1197  then show "lcm m n = m div normalisation_factor m" by (rule lcm_proj1_if_dvd)  haftmann@58023  1198 qed  haftmann@58023  1199 haftmann@58023  1200 lemma lcm_proj2_iff:  haftmann@58023  1201  "lcm m n = n div normalisation_factor n \ m dvd n"  haftmann@58023  1202  using lcm_proj1_iff [of n m] by (simp add: ac_simps)  haftmann@58023  1203 haftmann@58023  1204 lemma euclidean_size_lcm_le1:  haftmann@58023  1205  assumes "a \ 0" and "b \ 0"  haftmann@58023  1206  shows "euclidean_size a \ euclidean_size (lcm a b)"  haftmann@58023  1207 proof -  haftmann@58023  1208  have "a dvd lcm a b" by (rule lcm_dvd1)  haftmann@58023  1209  then obtain c where A: "lcm a b = a * c" unfolding dvd_def by blast  haftmann@58023  1210  with a \ 0 and b \ 0 have "c \ 0" by (auto simp: lcm_zero)  haftmann@58023  1211  then show ?thesis by (subst A, intro size_mult_mono)  haftmann@58023  1212 qed  haftmann@58023  1213 haftmann@58023  1214 lemma euclidean_size_lcm_le2:  haftmann@58023  1215  "a \ 0 \ b \ 0 \ euclidean_size b \ euclidean_size (lcm a b)"  haftmann@58023  1216  using euclidean_size_lcm_le1 [of b a] by (simp add: ac_simps)  haftmann@58023  1217 haftmann@58023  1218 lemma euclidean_size_lcm_less1:  haftmann@58023  1219  assumes "b \ 0" and "\b dvd a"  haftmann@58023  1220  shows "euclidean_size a < euclidean_size (lcm a b)"  haftmann@58023  1221 proof (rule ccontr)  haftmann@58023  1222  from assms have "a \ 0" by auto  haftmann@58023  1223  assume "\euclidean_size a < euclidean_size (lcm a b)"  haftmann@58023  1224  with a \ 0 and b \ 0 have "euclidean_size (lcm a b) = euclidean_size a"  haftmann@58023  1225  by (intro le_antisym, simp, intro euclidean_size_lcm_le1)  haftmann@58023  1226  with assms have "lcm a b dvd a"  haftmann@58023  1227  by (rule_tac dvd_euclidean_size_eq_imp_dvd) (auto simp: lcm_zero)  haftmann@58023  1228  hence "b dvd a" by (rule dvd_lcm_D2)  haftmann@58023  1229  with \b dvd a show False by contradiction  haftmann@58023  1230 qed  haftmann@58023  1231 haftmann@58023  1232 lemma euclidean_size_lcm_less2:  haftmann@58023  1233  assumes "a \ 0" and "\a dvd b"  haftmann@58023  1234  shows "euclidean_size b < euclidean_size (lcm a b)"  haftmann@58023  1235  using assms euclidean_size_lcm_less1 [of a b] by (simp add: ac_simps)  haftmann@58023  1236 haftmann@58023  1237 lemma lcm_mult_unit1:  haftmann@60430  1238  "is_unit a \ lcm (b * a) c = lcm b c"  haftmann@58023  1239  apply (rule lcmI)  haftmann@60430  1240  apply (rule dvd_trans[of _ "b * a"], simp, rule lcm_dvd1)  haftmann@58023  1241  apply (rule lcm_dvd2)  haftmann@58023  1242  apply (rule lcm_least, simp add: unit_simps, assumption)  haftmann@58023  1243  apply (subst normalisation_factor_lcm, simp add: lcm_zero)  haftmann@58023  1244  done  haftmann@58023  1245 haftmann@58023  1246 lemma lcm_mult_unit2:  haftmann@60430  1247  "is_unit a \ lcm b (c * a) = lcm b c"  haftmann@60430  1248  using lcm_mult_unit1 [of a c b] by (simp add: ac_simps)  haftmann@58023  1249 haftmann@58023  1250 lemma lcm_div_unit1:  haftmann@60430  1251  "is_unit a \ lcm (b div a) c = lcm b c"  haftmann@60432  1252  by (metis lcm_mult_unit1 local.unit_divide_1 local.unit_divide_1_unit)  haftmann@58023  1253 haftmann@58023  1254 lemma lcm_div_unit2:  haftmann@60430  1255  "is_unit a \ lcm b (c div a) = lcm b c"  haftmann@60432  1256  by (metis lcm_mult_unit2 local.unit_divide_1 local.unit_divide_1_unit)  haftmann@58023  1257 haftmann@58023  1258 lemma lcm_left_idem:  haftmann@60430  1259  "lcm a (lcm a b) = lcm a b"  haftmann@58023  1260  apply (rule lcmI)  haftmann@58023  1261  apply simp  haftmann@58023  1262  apply (subst lcm.assoc [symmetric], rule lcm_dvd2)  haftmann@58023  1263  apply (rule lcm_least, assumption)  haftmann@58023  1264  apply (erule (1) lcm_least)  haftmann@58023  1265  apply (auto simp: lcm_zero)  haftmann@58023  1266  done  haftmann@58023  1267 haftmann@58023  1268 lemma lcm_right_idem:  haftmann@60430  1269  "lcm (lcm a b) b = lcm a b"  haftmann@58023  1270  apply (rule lcmI)  haftmann@58023  1271  apply (subst lcm.assoc, rule lcm_dvd1)  haftmann@58023  1272  apply (rule lcm_dvd2)  haftmann@58023  1273  apply (rule lcm_least, erule (1) lcm_least, assumption)  haftmann@58023  1274  apply (auto simp: lcm_zero)  haftmann@58023  1275  done  haftmann@58023  1276 haftmann@58023  1277 lemma comp_fun_idem_lcm: "comp_fun_idem lcm"  haftmann@58023  1278 proof  haftmann@58023  1279  fix a b show "lcm a \ lcm b = lcm b \ lcm a"  haftmann@58023  1280  by (simp add: fun_eq_iff ac_simps)  haftmann@58023  1281 next  haftmann@58023  1282  fix a show "lcm a \ lcm a = lcm a" unfolding o_def  haftmann@58023  1283  by (intro ext, simp add: lcm_left_idem)  haftmann@58023  1284 qed  haftmann@58023  1285 haftmann@60430  1286 lemma dvd_Lcm [simp]: "a \ A \ a dvd Lcm A"  haftmann@60430  1287  and Lcm_dvd [simp]: "(\a\A. a dvd l') \ Lcm A dvd l'"  haftmann@58023  1288  and normalisation_factor_Lcm [simp]:  haftmann@58023  1289  "normalisation_factor (Lcm A) = (if Lcm A = 0 then 0 else 1)"  haftmann@58023  1290 proof -  haftmann@60430  1291  have "(\a\A. a dvd Lcm A) \ (\l'. (\a\A. a dvd l') \ Lcm A dvd l') \  haftmann@58023  1292  normalisation_factor (Lcm A) = (if Lcm A = 0 then 0 else 1)" (is ?thesis)  haftmann@60430  1293  proof (cases "\l. l \ 0 \ (\a\A. a dvd l)")  haftmann@58023  1294  case False  haftmann@58023  1295  hence "Lcm A = 0" by (auto simp: Lcm_Lcm_eucl Lcm_eucl_def)  haftmann@58023  1296  with False show ?thesis by auto  haftmann@58023  1297  next  haftmann@58023  1298  case True  haftmann@60430  1299  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  1300  def n \ "LEAST n. \l. l \ 0 \ (\a\A. a dvd l) \ euclidean_size l = n"  haftmann@60430  1301  def l \ "SOME l. l \ 0 \ (\a\A. a dvd l) \ euclidean_size l = n"  haftmann@60430  1302  have "\l. l \ 0 \ (\a\A. a dvd l) \ euclidean_size l = n"  haftmann@58023  1303  apply (subst n_def)  haftmann@58023  1304  apply (rule LeastI[of _ "euclidean_size l\<^sub>0"])  haftmann@58023  1305  apply (rule exI[of _ l\<^sub>0])  haftmann@58023  1306  apply (simp add: l\<^sub>0_props)  haftmann@58023  1307  done  haftmann@60430  1308  from someI_ex[OF this] have "l \ 0" and "\a\A. a dvd l" and "euclidean_size l = n"  haftmann@58023  1309  unfolding l_def by simp_all  haftmann@58023  1310  {  haftmann@60430  1311  fix l' assume "\a\A. a dvd l'"  haftmann@60430  1312  with \a\A. a dvd l have "\a\A. a dvd gcd l l'" by (auto intro: gcd_greatest)  haftmann@59009  1313  moreover from l \ 0 have "gcd l l' \ 0" by simp  haftmann@60430  1314  ultimately have "\b. b \ 0 \ (\a\A. a dvd b) \ euclidean_size b = euclidean_size (gcd l l')"  haftmann@58023  1315  by (intro exI[of _ "gcd l l'"], auto)  haftmann@58023  1316  hence "euclidean_size (gcd l l') \ n" by (subst n_def) (rule Least_le)  haftmann@58023  1317  moreover have "euclidean_size (gcd l l') \ n"  haftmann@58023  1318  proof -  haftmann@58023  1319  have "gcd l l' dvd l" by simp  haftmann@58023  1320  then obtain a where "l = gcd l l' * a" unfolding dvd_def by blast  haftmann@58023  1321  with l \ 0 have "a \ 0" by auto  haftmann@58023  1322  hence "euclidean_size (gcd l l') \ euclidean_size (gcd l l' * a)"  haftmann@58023  1323  by (rule size_mult_mono)  haftmann@58023  1324  also have "gcd l l' * a = l" using l = gcd l l' * a ..  haftmann@58023  1325  also note euclidean_size l = n  haftmann@58023  1326  finally show "euclidean_size (gcd l l') \ n" .  haftmann@58023  1327  qed  haftmann@58023  1328  ultimately have "euclidean_size l = euclidean_size (gcd l l')"  haftmann@58023  1329  by (intro le_antisym, simp_all add: euclidean_size l = n)  haftmann@58023  1330  with l \ 0 have "l dvd gcd l l'" by (blast intro: dvd_euclidean_size_eq_imp_dvd)  haftmann@58023  1331  hence "l dvd l'" by (blast dest: dvd_gcd_D2)  haftmann@58023  1332  }  haftmann@58023  1333 haftmann@60430  1334  with (\a\A. a dvd l) and normalisation_factor_is_unit[OF l \ 0] and l \ 0  haftmann@60430  1335  have "(\a\A. a dvd l div normalisation_factor l) \  haftmann@60430  1336  (\l'. (\a\A. a dvd l') \ l div normalisation_factor l dvd l') \  haftmann@58023  1337  normalisation_factor (l div normalisation_factor l) =  haftmann@58023  1338  (if l div normalisation_factor l = 0 then 0 else 1)"  haftmann@58023  1339  by (auto simp: unit_simps)  haftmann@58023  1340  also from True have "l div normalisation_factor l = Lcm A"  haftmann@58023  1341  by (simp add: Lcm_Lcm_eucl Lcm_eucl_def Let_def n_def l_def)  haftmann@58023  1342  finally show ?thesis .  haftmann@58023  1343  qed  haftmann@58023  1344  note A = this  haftmann@58023  1345 haftmann@60430  1346  {fix a assume "a \ A" then show "a dvd Lcm A" using A by blast}  haftmann@60430  1347  {fix l' assume "\a\A. a dvd l'" then show "Lcm A dvd l'" using A by blast}  haftmann@58023  1348  from A show "normalisation_factor (Lcm A) = (if Lcm A = 0 then 0 else 1)" by blast  haftmann@58023  1349 qed  haftmann@58023  1350   haftmann@58023  1351 lemma LcmI:  haftmann@60430  1352  "(\a. a\A \ a dvd l) \ (\l'. (\a\A. a dvd l') \ l dvd l') \  haftmann@58023  1353  normalisation_factor l = (if l = 0 then 0 else 1) \ l = Lcm A"  haftmann@58023  1354  by (intro normed_associated_imp_eq)  haftmann@58023  1355  (auto intro: Lcm_dvd dvd_Lcm simp: associated_def)  haftmann@58023  1356 haftmann@58023  1357 lemma Lcm_subset:  haftmann@58023  1358  "A \ B \ Lcm A dvd Lcm B"  haftmann@58023  1359  by (blast intro: Lcm_dvd dvd_Lcm)  haftmann@58023  1360 haftmann@58023  1361 lemma Lcm_Un:  haftmann@58023  1362  "Lcm (A \ B) = lcm (Lcm A) (Lcm B)"  haftmann@58023  1363  apply (rule lcmI)  haftmann@58023  1364  apply (blast intro: Lcm_subset)  haftmann@58023  1365  apply (blast intro: Lcm_subset)  haftmann@58023  1366  apply (intro Lcm_dvd ballI, elim UnE)  haftmann@58023  1367  apply (rule dvd_trans, erule dvd_Lcm, assumption)  haftmann@58023  1368  apply (rule dvd_trans, erule dvd_Lcm, assumption)  haftmann@58023  1369  apply simp  haftmann@58023  1370  done  haftmann@58023  1371 haftmann@58023  1372 lemma Lcm_1_iff:  haftmann@60430  1373  "Lcm A = 1 \ (\a\A. is_unit a)"  haftmann@58023  1374 proof  haftmann@58023  1375  assume "Lcm A = 1"  haftmann@60430  1376  then show "\a\A. is_unit a" by auto  haftmann@58023  1377 qed (rule LcmI [symmetric], auto)  haftmann@58023  1378 haftmann@58023  1379 lemma Lcm_no_units:  haftmann@60430  1380  "Lcm A = Lcm (A - {a. is_unit a})"  haftmann@58023  1381 proof -  haftmann@60430  1382  have "(A - {a. is_unit a}) \ {a\A. is_unit a} = A" by blast  haftmann@60430  1383  hence "Lcm A = lcm (Lcm (A - {a. is_unit a})) (Lcm {a\A. is_unit a})"  haftmann@58023  1384  by (simp add: Lcm_Un[symmetric])  haftmann@60430  1385  also have "Lcm {a\A. is_unit a} = 1" by (simp add: Lcm_1_iff)  haftmann@58023  1386  finally show ?thesis by simp  haftmann@58023  1387 qed  haftmann@58023  1388 haftmann@58023  1389 lemma Lcm_empty [simp]:  haftmann@58023  1390  "Lcm {} = 1"  haftmann@58023  1391  by (simp add: Lcm_1_iff)  haftmann@58023  1392 haftmann@58023  1393 lemma Lcm_eq_0 [simp]:  haftmann@58023  1394  "0 \ A \ Lcm A = 0"  haftmann@58023  1395  by (drule dvd_Lcm) simp  haftmann@58023  1396 haftmann@58023  1397 lemma Lcm0_iff':  haftmann@60430  1398  "Lcm A = 0 \ \(\l. l \ 0 \ (\a\A. a dvd l))"  haftmann@58023  1399 proof  haftmann@58023  1400  assume "Lcm A = 0"  haftmann@60430  1401  show "\(\l. l \ 0 \ (\a\A. a dvd l))"  haftmann@58023  1402  proof  haftmann@60430  1403  assume ex: "\l. l \ 0 \ (\a\A. a dvd l)"  haftmann@60430  1404  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  1405  def n \ "LEAST n. \l. l \ 0 \ (\a\A. a dvd l) \ euclidean_size l = n"  haftmann@60430  1406  def l \ "SOME l. l \ 0 \ (\a\A. a dvd l) \ euclidean_size l = n"  haftmann@60430  1407  have "\l. l \ 0 \ (\a\A. a dvd l) \ euclidean_size l = n"  haftmann@58023  1408  apply (subst n_def)  haftmann@58023  1409  apply (rule LeastI[of _ "euclidean_size l\<^sub>0"])  haftmann@58023  1410  apply (rule exI[of _ l\<^sub>0])  haftmann@58023  1411  apply (simp add: l\<^sub>0_props)  haftmann@58023  1412  done  haftmann@58023  1413  from someI_ex[OF this] have "l \ 0" unfolding l_def by simp_all  haftmann@58023  1414  hence "l div normalisation_factor l \ 0" by simp  haftmann@58023  1415  also from ex have "l div normalisation_factor l = Lcm A"  haftmann@58023  1416  by (simp only: Lcm_Lcm_eucl Lcm_eucl_def n_def l_def if_True Let_def)  haftmann@58023  1417  finally show False using Lcm A = 0 by contradiction  haftmann@58023  1418  qed  haftmann@58023  1419 qed (simp only: Lcm_Lcm_eucl Lcm_eucl_def if_False)  haftmann@58023  1420 haftmann@58023  1421 lemma Lcm0_iff [simp]:  haftmann@58023  1422  "finite A \ Lcm A = 0 \ 0 \ A"  haftmann@58023  1423 proof -  haftmann@58023  1424  assume "finite A"  haftmann@58023  1425  have "0 \ A \ Lcm A = 0" by (intro dvd_0_left dvd_Lcm)  haftmann@58023  1426  moreover {  haftmann@58023  1427  assume "0 \ A"  haftmann@58023  1428  hence "\A \ 0"  haftmann@58023  1429  apply (induct rule: finite_induct[OF finite A])  haftmann@58023  1430  apply simp  haftmann@58023  1431  apply (subst setprod.insert, assumption, assumption)  haftmann@58023  1432  apply (rule no_zero_divisors)  haftmann@58023  1433  apply blast+  haftmann@58023  1434  done  haftmann@60430  1435  moreover from finite A have "\a\A. a dvd \A" by blast  haftmann@60430  1436  ultimately have "\l. l \ 0 \ (\a\A. a dvd l)" by blast  haftmann@58023  1437  with Lcm0_iff' have "Lcm A \ 0" by simp  haftmann@58023  1438  }  haftmann@58023  1439  ultimately show "Lcm A = 0 \ 0 \ A" by blast  haftmann@58023  1440 qed  haftmann@58023  1441 haftmann@58023  1442 lemma Lcm_no_multiple:  haftmann@60430  1443  "(\m. m \ 0 \ (\a\A. \a dvd m)) \ Lcm A = 0"  haftmann@58023  1444 proof -  haftmann@60430  1445  assume "\m. m \ 0 \ (\a\A. \a dvd m)"  haftmann@60430  1446  hence "\(\l. l \ 0 \ (\a\A. a dvd l))" by blast  haftmann@58023  1447  then show "Lcm A = 0" by (simp only: Lcm_Lcm_eucl Lcm_eucl_def if_False)  haftmann@58023  1448 qed  haftmann@58023  1449 haftmann@58023  1450 lemma Lcm_insert [simp]:  haftmann@58023  1451  "Lcm (insert a A) = lcm a (Lcm A)"  haftmann@58023  1452 proof (rule lcmI)  haftmann@58023  1453  fix l assume "a dvd l" and "Lcm A dvd l"  haftmann@60430  1454  hence "\a\A. a dvd l" by (blast intro: dvd_trans dvd_Lcm)  haftmann@58023  1455  with a dvd l show "Lcm (insert a A) dvd l" by (force intro: Lcm_dvd)  haftmann@58023  1456 qed (auto intro: Lcm_dvd dvd_Lcm)  haftmann@58023  1457   haftmann@58023  1458 lemma Lcm_finite:  haftmann@58023  1459  assumes "finite A"  haftmann@58023  1460  shows "Lcm A = Finite_Set.fold lcm 1 A"  haftmann@58023  1461  by (induct rule: finite.induct[OF finite A])  haftmann@58023  1462  (simp_all add: comp_fun_idem.fold_insert_idem[OF comp_fun_idem_lcm])  haftmann@58023  1463 haftmann@60431  1464 lemma Lcm_set [code_unfold]:  haftmann@58023  1465  "Lcm (set xs) = fold lcm xs 1"  haftmann@58023  1466  using comp_fun_idem.fold_set_fold[OF comp_fun_idem_lcm] Lcm_finite by (simp add: ac_simps)  haftmann@58023  1467 haftmann@58023  1468 lemma Lcm_singleton [simp]:  haftmann@58023  1469  "Lcm {a} = a div normalisation_factor a"  haftmann@58023  1470  by simp  haftmann@58023  1471 haftmann@58023  1472 lemma Lcm_2 [simp]:  haftmann@58023  1473  "Lcm {a,b} = lcm a b"  haftmann@58023  1474  by (simp only: Lcm_insert Lcm_empty lcm_1_right)  haftmann@58023  1475  (cases "b = 0", simp, rule lcm_div_unit2, simp)  haftmann@58023  1476 haftmann@58023  1477 lemma Lcm_coprime:  haftmann@58023  1478  assumes "finite A" and "A \ {}"  haftmann@58023  1479  assumes "\a b. a \ A \ b \ A \ a \ b \ gcd a b = 1"  haftmann@58023  1480  shows "Lcm A = \A div normalisation_factor (\A)"  haftmann@58023  1481 using assms proof (induct rule: finite_ne_induct)  haftmann@58023  1482  case (insert a A)  haftmann@58023  1483  have "Lcm (insert a A) = lcm a (Lcm A)" by simp  haftmann@58023  1484  also from insert have "Lcm A = \A div normalisation_factor (\A)" by blast  haftmann@58023  1485  also have "lcm a \ = lcm a (\A)" by (cases "\A = 0") (simp_all add: lcm_div_unit2)  haftmann@58023  1486  also from insert have "gcd a (\A) = 1" by (subst gcd.commute, intro setprod_coprime) auto  haftmann@58023  1487  with insert have "lcm a (\A) = \(insert a A) div normalisation_factor (\(insert a A))"  haftmann@58023  1488  by (simp add: lcm_coprime)  haftmann@58023  1489  finally show ?case .  haftmann@58023  1490 qed simp  haftmann@58023  1491   haftmann@58023  1492 lemma Lcm_coprime':  haftmann@58023  1493  "card A \ 0 \ (\a b. a \ A \ b \ A \ a \ b \ gcd a b = 1)  haftmann@58023  1494  \ Lcm A = \A div normalisation_factor (\A)"  haftmann@58023  1495  by (rule Lcm_coprime) (simp_all add: card_eq_0_iff)  haftmann@58023  1496 haftmann@58023  1497 lemma Gcd_Lcm:  haftmann@60430  1498  "Gcd A = Lcm {d. \a\A. d dvd a}"  haftmann@58023  1499  by (simp add: Gcd_Gcd_eucl Lcm_Lcm_eucl Gcd_eucl_def)  haftmann@58023  1500 haftmann@60430  1501 lemma Gcd_dvd [simp]: "a \ A \ Gcd A dvd a"  haftmann@60430  1502  and dvd_Gcd [simp]: "(\a\A. g' dvd a) \ g' dvd Gcd A"  haftmann@58023  1503  and normalisation_factor_Gcd [simp]:  haftmann@58023  1504  "normalisation_factor (Gcd A) = (if Gcd A = 0 then 0 else 1)"  haftmann@58023  1505 proof -  haftmann@60430  1506  fix a assume "a \ A"  haftmann@60430  1507  hence "Lcm {d. \a\A. d dvd a} dvd a" by (intro Lcm_dvd) blast  haftmann@60430  1508  then show "Gcd A dvd a" by (simp add: Gcd_Lcm)  haftmann@58023  1509 next  haftmann@60430  1510  fix g' assume "\a\A. g' dvd a"  haftmann@60430  1511  hence "g' dvd Lcm {d. \a\A. d dvd a}" by (intro dvd_Lcm) blast  haftmann@58023  1512  then show "g' dvd Gcd A" by (simp add: Gcd_Lcm)  haftmann@58023  1513 next  haftmann@58023  1514  show "normalisation_factor (Gcd A) = (if Gcd A = 0 then 0 else 1)"  haftmann@59009  1515  by (simp add: Gcd_Lcm)  haftmann@58023  1516 qed  haftmann@58023  1517 haftmann@58023  1518 lemma GcdI:  haftmann@60430  1519  "(\a. a\A \ l dvd a) \ (\l'. (\a\A. l' dvd a) \ l' dvd l) \  haftmann@58023  1520  normalisation_factor l = (if l = 0 then 0 else 1) \ l = Gcd A"  haftmann@58023  1521  by (intro normed_associated_imp_eq)  haftmann@58023  1522  (auto intro: Gcd_dvd dvd_Gcd simp: associated_def)  haftmann@58023  1523 haftmann@58023  1524 lemma Lcm_Gcd:  haftmann@60430  1525  "Lcm A = Gcd {m. \a\A. a dvd m}"  haftmann@58023  1526  by (rule LcmI[symmetric]) (auto intro: dvd_Gcd Gcd_dvd)  haftmann@58023  1527 haftmann@58023  1528 lemma Gcd_0_iff:  haftmann@58023  1529  "Gcd A = 0 \ A \ {0}"  haftmann@58023  1530  apply (rule iffI)  haftmann@58023  1531  apply (rule subsetI, drule Gcd_dvd, simp)  haftmann@58023  1532  apply (auto intro: GcdI[symmetric])  haftmann@58023  1533  done  haftmann@58023  1534 haftmann@58023  1535 lemma Gcd_empty [simp]:  haftmann@58023  1536  "Gcd {} = 0"  haftmann@58023  1537  by (simp add: Gcd_0_iff)  haftmann@58023  1538 haftmann@58023  1539 lemma Gcd_1:  haftmann@58023  1540  "1 \ A \ Gcd A = 1"  haftmann@58023  1541  by (intro GcdI[symmetric]) (auto intro: Gcd_dvd dvd_Gcd)  haftmann@58023  1542 haftmann@58023  1543 lemma Gcd_insert [simp]:  haftmann@58023  1544  "Gcd (insert a A) = gcd a (Gcd A)"  haftmann@58023  1545 proof (rule gcdI)  haftmann@58023  1546  fix l assume "l dvd a" and "l dvd Gcd A"  haftmann@60430  1547  hence "\a\A. l dvd a" by (blast intro: dvd_trans Gcd_dvd)  haftmann@58023  1548  with l dvd a show "l dvd Gcd (insert a A)" by (force intro: Gcd_dvd)  haftmann@59009  1549 qed auto  haftmann@58023  1550 haftmann@58023  1551 lemma Gcd_finite:  haftmann@58023  1552  assumes "finite A"  haftmann@58023  1553  shows "Gcd A = Finite_Set.fold gcd 0 A"  haftmann@58023  1554  by (induct rule: finite.induct[OF finite A])  haftmann@58023  1555  (simp_all add: comp_fun_idem.fold_insert_idem[OF comp_fun_idem_gcd])  haftmann@58023  1556 haftmann@60431  1557 lemma Gcd_set [code_unfold]:  haftmann@58023  1558  "Gcd (set xs) = fold gcd xs 0"  haftmann@58023  1559  using comp_fun_idem.fold_set_fold[OF comp_fun_idem_gcd] Gcd_finite by (simp add: ac_simps)  haftmann@58023  1560 haftmann@58023  1561 lemma Gcd_singleton [simp]: "Gcd {a} = a div normalisation_factor a"  haftmann@58023  1562  by (simp add: gcd_0)  haftmann@58023  1563 haftmann@58023  1564 lemma Gcd_2 [simp]: "Gcd {a,b} = gcd a b"  haftmann@58023  1565  by (simp only: Gcd_insert Gcd_empty gcd_0) (cases "b = 0", simp, rule gcd_div_unit2, simp)  haftmann@58023  1566 haftmann@58023  1567 end  haftmann@58023  1568 haftmann@58023  1569 text {*  haftmann@58023  1570  A Euclidean ring is a Euclidean semiring with additive inverses. It provides a  haftmann@58023  1571  few more lemmas; in particular, Bezout's lemma holds for any Euclidean ring.  haftmann@58023  1572 *}  haftmann@58023  1573 haftmann@58023  1574 class euclidean_ring = euclidean_semiring + idom  haftmann@58023  1575 haftmann@58023  1576 class euclidean_ring_gcd = euclidean_semiring_gcd + idom  haftmann@58023  1577 begin  haftmann@58023  1578 haftmann@58023  1579 subclass euclidean_ring ..  haftmann@58023  1580 haftmann@58023  1581 lemma gcd_neg1 [simp]:  haftmann@60430  1582  "gcd (-a) b = gcd a b"  haftmann@59009  1583  by (rule sym, rule gcdI, simp_all add: gcd_greatest)  haftmann@58023  1584 haftmann@58023  1585 lemma gcd_neg2 [simp]:  haftmann@60430  1586  "gcd a (-b) = gcd a b"  haftmann@59009  1587  by (rule sym, rule gcdI, simp_all add: gcd_greatest)  haftmann@58023  1588 haftmann@58023  1589 lemma gcd_neg_numeral_1 [simp]:  haftmann@60430  1590  "gcd (- numeral n) a = gcd (numeral n) a"  haftmann@58023  1591  by (fact gcd_neg1)  haftmann@58023  1592 haftmann@58023  1593 lemma gcd_neg_numeral_2 [simp]:  haftmann@60430  1594  "gcd a (- numeral n) = gcd a (numeral n)"  haftmann@58023  1595  by (fact gcd_neg2)  haftmann@58023  1596 haftmann@58023  1597 lemma gcd_diff1: "gcd (m - n) n = gcd m n"  haftmann@58023  1598  by (subst diff_conv_add_uminus, subst gcd_neg2[symmetric], subst gcd_add1, simp)  haftmann@58023  1599 haftmann@58023  1600 lemma gcd_diff2: "gcd (n - m) n = gcd m n"  haftmann@58023  1601  by (subst gcd_neg1[symmetric], simp only: minus_diff_eq gcd_diff1)  haftmann@58023  1602 haftmann@58023  1603 lemma coprime_minus_one [simp]: "gcd (n - 1) n = 1"  haftmann@58023  1604 proof -  haftmann@58023  1605  have "gcd (n - 1) n = gcd n (n - 1)" by (fact gcd.commute)  haftmann@58023  1606  also have "\ = gcd ((n - 1) + 1) (n - 1)" by simp  haftmann@58023  1607  also have "\ = 1" by (rule coprime_plus_one)  haftmann@58023  1608  finally show ?thesis .  haftmann@58023  1609 qed  haftmann@58023  1610 haftmann@60430  1611 lemma lcm_neg1 [simp]: "lcm (-a) b = lcm a b"  haftmann@58023  1612  by (rule sym, rule lcmI, simp_all add: lcm_least lcm_zero)  haftmann@58023  1613 haftmann@60430  1614 lemma lcm_neg2 [simp]: "lcm a (-b) = lcm a b"  haftmann@58023  1615  by (rule sym, rule lcmI, simp_all add: lcm_least lcm_zero)  haftmann@58023  1616 haftmann@60430  1617 lemma lcm_neg_numeral_1 [simp]: "lcm (- numeral n) a = lcm (numeral n) a"  haftmann@58023  1618  by (fact lcm_neg1)  haftmann@58023  1619 haftmann@60430  1620 lemma lcm_neg_numeral_2 [simp]: "lcm a (- numeral n) = lcm a (numeral n)"  haftmann@58023  1621  by (fact lcm_neg2)  haftmann@58023  1622 haftmann@58023  1623 function euclid_ext :: "'a \ 'a \ 'a \ 'a \ 'a" where  haftmann@58023  1624  "euclid_ext a b =  haftmann@58023  1625  (if b = 0 then  haftmann@60432  1626  let c = divide 1 (normalisation_factor a) in (c, 0, a * c)  haftmann@58023  1627  else  haftmann@58023  1628  case euclid_ext b (a mod b) of  haftmann@58023  1629  (s,t,c) \ (t, s - t * (a div b), c))"  haftmann@58023  1630  by (pat_completeness, simp)  haftmann@58023  1631  termination by (relation "measure (euclidean_size \ snd)", simp_all)  haftmann@58023  1632 haftmann@58023  1633 declare euclid_ext.simps [simp del]  haftmann@58023  1634 haftmann@58023  1635 lemma euclid_ext_0:  haftmann@60432  1636  "euclid_ext a 0 = (divide 1 (normalisation_factor a), 0, a * divide 1 (normalisation_factor a))"  haftmann@58023  1637  by (subst euclid_ext.simps, simp add: Let_def)  haftmann@58023  1638 haftmann@58023  1639 lemma euclid_ext_non_0:  haftmann@58023  1640  "b \ 0 \ euclid_ext a b = (case euclid_ext b (a mod b) of  haftmann@58023  1641  (s,t,c) \ (t, s - t * (a div b), c))"  haftmann@58023  1642  by (subst euclid_ext.simps, simp)  haftmann@58023  1643 haftmann@58023  1644 definition euclid_ext' :: "'a \ 'a \ 'a \ 'a"  haftmann@58023  1645 where  haftmann@58023  1646  "euclid_ext' a b = (case euclid_ext a b of (s, t, _) \ (s, t))"  haftmann@58023  1647 haftmann@58023  1648 lemma euclid_ext_gcd [simp]:  haftmann@58023  1649  "(case euclid_ext a b of (_,_,t) \ t) = gcd a b"  haftmann@58023  1650 proof (induct a b rule: euclid_ext.induct)  haftmann@58023  1651  case (1 a b)  haftmann@58023  1652  then show ?case  haftmann@58023  1653  proof (cases "b = 0")  haftmann@58023  1654  case True  haftmann@58023  1655  then show ?thesis by (cases "a = 0")  haftmann@58023  1656  (simp_all add: euclid_ext_0 unit_div mult_ac unit_simps gcd_0)  haftmann@58023  1657  next  haftmann@58023  1658  case False with 1 show ?thesis  haftmann@58023  1659  by (simp add: euclid_ext_non_0 ac_simps split: prod.split prod.split_asm)  haftmann@58023  1660  qed  haftmann@58023  1661 qed  haftmann@58023  1662 haftmann@58023  1663 lemma euclid_ext_gcd' [simp]:  haftmann@58023  1664  "euclid_ext a b = (r, s, t) \ t = gcd a b"  haftmann@58023  1665  by (insert euclid_ext_gcd[of a b], drule (1) subst, simp)  haftmann@58023  1666 haftmann@58023  1667 lemma euclid_ext_correct:  haftmann@60430  1668  "case euclid_ext a b of (s,t,c) \ s*a + t*b = c"  haftmann@60430  1669 proof (induct a b rule: euclid_ext.induct)  haftmann@60430  1670  case (1 a b)  haftmann@58023  1671  show ?case  haftmann@60430  1672  proof (cases "b = 0")  haftmann@58023  1673  case True  haftmann@58023  1674  then show ?thesis by (simp add: euclid_ext_0 mult_ac)  haftmann@58023  1675  next  haftmann@58023  1676  case False  haftmann@60430  1677  obtain s t c where stc: "euclid_ext b (a mod b) = (s,t,c)"  haftmann@60430  1678  by (cases "euclid_ext b (a mod b)", blast)  haftmann@60430  1679  from 1 have "c = s * b + t * (a mod b)" by (simp add: stc False)  haftmann@60430  1680  also have "... = t*((a div b)*b + a mod b) + (s - t * (a div b))*b"  haftmann@58023  1681  by (simp add: algebra_simps)  haftmann@60430  1682  also have "(a div b)*b + a mod b = a" using mod_div_equality .  haftmann@58023  1683  finally show ?thesis  haftmann@58023  1684  by (subst euclid_ext.simps, simp add: False stc)  haftmann@58023  1685  qed  haftmann@58023  1686 qed  haftmann@58023  1687 haftmann@58023  1688 lemma euclid_ext'_correct:  haftmann@58023  1689  "fst (euclid_ext' a b) * a + snd (euclid_ext' a b) * b = gcd a b"  haftmann@58023  1690 proof-  haftmann@58023  1691  obtain s t c where "euclid_ext a b = (s,t,c)"  haftmann@58023  1692  by (cases "euclid_ext a b", blast)  haftmann@58023  1693  with euclid_ext_correct[of a b] euclid_ext_gcd[of a b]  haftmann@58023  1694  show ?thesis unfolding euclid_ext'_def by simp  haftmann@58023  1695 qed  haftmann@58023  1696 haftmann@60430  1697 lemma bezout: "\s t. s * a + t * b = gcd a b"  haftmann@58023  1698  using euclid_ext'_correct by blast  haftmann@58023  1699 haftmann@60432  1700 lemma euclid_ext'_0 [simp]: "euclid_ext' a 0 = (divide 1 (normalisation_factor a), 0)"  haftmann@58023  1701  by (simp add: bezw_def euclid_ext'_def euclid_ext_0)  haftmann@58023  1702 haftmann@60430  1703 lemma euclid_ext'_non_0: "b \ 0 \ euclid_ext' a b = (snd (euclid_ext' b (a mod b)),  haftmann@60430  1704  fst (euclid_ext' b (a mod b)) - snd (euclid_ext' b (a mod b)) * (a div b))"  haftmann@60430  1705  by (cases "euclid_ext b (a mod b)")  haftmann@58023  1706  (simp add: euclid_ext'_def euclid_ext_non_0)  haftmann@58023  1707   haftmann@58023  1708 end  haftmann@58023  1709 haftmann@58023  1710 instantiation nat :: euclidean_semiring  haftmann@58023  1711 begin  haftmann@58023  1712 haftmann@58023  1713 definition [simp]:  haftmann@58023  1714  "euclidean_size_nat = (id :: nat \ nat)"  haftmann@58023  1715 haftmann@58023  1716 definition [simp]:  haftmann@58023  1717  "normalisation_factor_nat (n::nat) = (if n = 0 then 0 else 1 :: nat)"  haftmann@58023  1718 haftmann@58023  1719 instance proof  haftmann@59061  1720 qed simp_all  haftmann@58023  1721 haftmann@58023  1722 end  haftmann@58023  1723 haftmann@58023  1724 instantiation int :: euclidean_ring  haftmann@58023  1725 begin  haftmann@58023  1726 haftmann@58023  1727 definition [simp]:  haftmann@58023  1728  "euclidean_size_int = (nat \ abs :: int \ nat)"  haftmann@58023  1729 haftmann@58023  1730 definition [simp]:  haftmann@58023  1731  "normalisation_factor_int = (sgn :: int \ int)"  haftmann@58023  1732 haftmann@58023  1733 instance proof  haftmann@58023  1734  case goal2 then show ?case by (auto simp add: abs_mult nat_mult_distrib)  haftmann@58023  1735 next  haftmann@59061  1736  case goal3 then show ?case by (simp add: zsgn_def)  haftmann@58023  1737 next  haftmann@59061  1738  case goal5 then show ?case by (auto simp: zsgn_def)  haftmann@58023  1739 next  haftmann@59061  1740  case goal6 then show ?case by (auto split: abs_split simp: zsgn_def)  haftmann@58023  1741 qed (auto simp: sgn_times split: abs_split)  haftmann@58023  1742 haftmann@58023  1743 end  haftmann@58023  1744 haftmann@58023  1745 end