src/HOL/Number_Theory/Euclidean_Algorithm.thy
 author eberlm Thu Feb 25 16:44:53 2016 +0100 (2016-02-25) changeset 62422 4aa35fd6c152 parent 62353 7f927120b5a2 child 62425 d0936b500bf5 permissions -rw-r--r--
Tuned Euclidean rings
 haftmann@58023  1 (* Author: Manuel Eberl *)  haftmann@58023  2 wenzelm@60526  3 section \Abstract euclidean algorithm\  haftmann@58023  4 haftmann@58023  5 theory Euclidean_Algorithm  haftmann@60685  6 imports Main "~~/src/HOL/GCD" "~~/src/HOL/Library/Polynomial"  haftmann@58023  7 begin  haftmann@60634  8 wenzelm@60526  9 text \  haftmann@58023  10  A Euclidean semiring is a semiring upon which the Euclidean algorithm can be  haftmann@58023  11  implemented. It must provide:  haftmann@58023  12  \begin{itemize}  haftmann@58023  13  \item division with remainder  haftmann@58023  14  \item a size function such that @{term "size (a mod b) < size b"}  haftmann@58023  15  for any @{term "b \ 0"}  haftmann@58023  16  \end{itemize}  haftmann@58023  17  The existence of these functions makes it possible to derive gcd and lcm functions  haftmann@58023  18  for any Euclidean semiring.  wenzelm@60526  19 \  haftmann@60634  20 class euclidean_semiring = semiring_div + normalization_semidom +  haftmann@58023  21  fixes euclidean_size :: "'a \ nat"  eberlm@62422  22  assumes size_0 [simp]: "euclidean_size 0 = 0"  haftmann@60569  23  assumes mod_size_less:  haftmann@60600  24  "b \ 0 \ euclidean_size (a mod b) < euclidean_size b"  haftmann@58023  25  assumes size_mult_mono:  haftmann@60634  26  "b \ 0 \ euclidean_size a \ euclidean_size (a * b)"  haftmann@58023  27 begin  haftmann@58023  28 haftmann@58023  29 lemma euclidean_division:  haftmann@58023  30  fixes a :: 'a and b :: 'a  haftmann@60600  31  assumes "b \ 0"  haftmann@58023  32  obtains s and t where "a = s * b + t"  haftmann@58023  33  and "euclidean_size t < euclidean_size b"  haftmann@58023  34 proof -  haftmann@60569  35  from div_mod_equality [of a b 0]  haftmann@58023  36  have "a = a div b * b + a mod b" by simp  haftmann@60569  37  with that and assms show ?thesis by (auto simp add: mod_size_less)  haftmann@58023  38 qed  haftmann@58023  39 haftmann@58023  40 lemma dvd_euclidean_size_eq_imp_dvd:  haftmann@58023  41  assumes "a \ 0" and b_dvd_a: "b dvd a" and size_eq: "euclidean_size a = euclidean_size b"  haftmann@58023  42  shows "a dvd b"  haftmann@60569  43 proof (rule ccontr)  haftmann@60569  44  assume "\ a dvd b"  haftmann@60569  45  then have "b mod a \ 0" by (simp add: mod_eq_0_iff_dvd)  haftmann@58023  46  from b_dvd_a have b_dvd_mod: "b dvd b mod a" by (simp add: dvd_mod_iff)  haftmann@58023  47  from b_dvd_mod obtain c where "b mod a = b * c" unfolding dvd_def by blast  wenzelm@60526  48  with \b mod a \ 0\ have "c \ 0" by auto  wenzelm@60526  49  with \b mod a = b * c\ have "euclidean_size (b mod a) \ euclidean_size b"  haftmann@58023  50  using size_mult_mono by force  haftmann@60569  51  moreover from \\ a dvd b\ and \a \ 0\  haftmann@60569  52  have "euclidean_size (b mod a) < euclidean_size a"  haftmann@58023  53  using mod_size_less by blast  haftmann@58023  54  ultimately show False using size_eq by simp  haftmann@58023  55 qed  haftmann@58023  56 haftmann@58023  57 function gcd_eucl :: "'a \ 'a \ 'a"  haftmann@58023  58 where  haftmann@60634  59  "gcd_eucl a b = (if b = 0 then normalize a else gcd_eucl b (a mod b))"  haftmann@60572  60  by pat_completeness simp  haftmann@60569  61 termination  haftmann@60569  62  by (relation "measure (euclidean_size \ snd)") (simp_all add: mod_size_less)  haftmann@58023  63 haftmann@58023  64 declare gcd_eucl.simps [simp del]  haftmann@58023  65 haftmann@60569  66 lemma gcd_eucl_induct [case_names zero mod]:  haftmann@60569  67  assumes H1: "\b. P b 0"  haftmann@60569  68  and H2: "\a b. b \ 0 \ P b (a mod b) \ P a b"  haftmann@60569  69  shows "P a b"  haftmann@58023  70 proof (induct a b rule: gcd_eucl.induct)  haftmann@60569  71  case ("1" a b)  haftmann@60569  72  show ?case  haftmann@60569  73  proof (cases "b = 0")  haftmann@60569  74  case True then show "P a b" by simp (rule H1)  haftmann@60569  75  next  haftmann@60569  76  case False  haftmann@60600  77  then have "P b (a mod b)"  haftmann@60600  78  by (rule "1.hyps")  haftmann@60569  79  with \b \ 0\ show "P a b"  haftmann@60569  80  by (blast intro: H2)  haftmann@60569  81  qed  haftmann@58023  82 qed  haftmann@58023  83 haftmann@58023  84 definition lcm_eucl :: "'a \ 'a \ 'a"  haftmann@58023  85 where  haftmann@60634  86  "lcm_eucl a b = normalize (a * b) div gcd_eucl a b"  haftmann@58023  87 haftmann@60572  88 definition Lcm_eucl :: "'a set \ 'a" -- \  haftmann@60572  89  Somewhat complicated definition of Lcm that has the advantage of working  haftmann@60572  90  for infinite sets as well\  haftmann@58023  91 where  haftmann@60430  92  "Lcm_eucl A = (if \l. l \ 0 \ (\a\A. a dvd l) then  haftmann@60430  93  let l = SOME l. l \ 0 \ (\a\A. a dvd l) \ euclidean_size l =  haftmann@60430  94  (LEAST n. \l. l \ 0 \ (\a\A. a dvd l) \ euclidean_size l = n)  haftmann@60634  95  in normalize l  haftmann@58023  96  else 0)"  haftmann@58023  97 haftmann@58023  98 definition Gcd_eucl :: "'a set \ 'a"  haftmann@58023  99 where  haftmann@58023  100  "Gcd_eucl A = Lcm_eucl {d. \a\A. d dvd a}"  haftmann@58023  101 haftmann@60572  102 lemma gcd_eucl_0:  haftmann@60634  103  "gcd_eucl a 0 = normalize a"  haftmann@60572  104  by (simp add: gcd_eucl.simps [of a 0])  haftmann@60572  105 haftmann@60572  106 lemma gcd_eucl_0_left:  haftmann@60634  107  "gcd_eucl 0 a = normalize a"  haftmann@60600  108  by (simp_all add: gcd_eucl_0 gcd_eucl.simps [of 0 a])  haftmann@60572  109 haftmann@60572  110 lemma gcd_eucl_non_0:  haftmann@60572  111  "b \ 0 \ gcd_eucl a b = gcd_eucl b (a mod b)"  haftmann@60600  112  by (simp add: gcd_eucl.simps [of a b] gcd_eucl.simps [of b 0])  haftmann@60572  113 eberlm@62422  114 lemma gcd_eucl_dvd1 [iff]: "gcd_eucl a b dvd a"  eberlm@62422  115  and gcd_eucl_dvd2 [iff]: "gcd_eucl a b dvd b"  eberlm@62422  116  by (induct a b rule: gcd_eucl_induct)  eberlm@62422  117  (simp_all add: gcd_eucl_0 gcd_eucl_non_0 dvd_mod_iff)  eberlm@62422  118 eberlm@62422  119 lemma normalize_gcd_eucl [simp]:  eberlm@62422  120  "normalize (gcd_eucl a b) = gcd_eucl a b"  eberlm@62422  121  by (induct a b rule: gcd_eucl_induct) (simp_all add: gcd_eucl_0 gcd_eucl_non_0)  eberlm@62422  122   eberlm@62422  123 lemma gcd_eucl_greatest:  eberlm@62422  124  fixes k a b :: 'a  eberlm@62422  125  shows "k dvd a \ k dvd b \ k dvd gcd_eucl a b"  eberlm@62422  126 proof (induct a b rule: gcd_eucl_induct)  eberlm@62422  127  case (zero a) from zero(1) show ?case by (rule dvd_trans) (simp add: gcd_eucl_0)  eberlm@62422  128 next  eberlm@62422  129  case (mod a b)  eberlm@62422  130  then show ?case  eberlm@62422  131  by (simp add: gcd_eucl_non_0 dvd_mod_iff)  eberlm@62422  132 qed  eberlm@62422  133 eberlm@62422  134 lemma eq_gcd_euclI:  eberlm@62422  135  fixes gcd :: "'a \ 'a \ 'a"  eberlm@62422  136  assumes "\a b. gcd a b dvd a" "\a b. gcd a b dvd b" "\a b. normalize (gcd a b) = gcd a b"  eberlm@62422  137  "\a b k. k dvd a \ k dvd b \ k dvd gcd a b"  eberlm@62422  138  shows "gcd = gcd_eucl"  eberlm@62422  139  by (intro ext, rule associated_eqI) (simp_all add: gcd_eucl_greatest assms)  eberlm@62422  140 eberlm@62422  141 lemma gcd_eucl_zero [simp]:  eberlm@62422  142  "gcd_eucl a b = 0 \ a = 0 \ b = 0"  eberlm@62422  143  by (metis dvd_0_left dvd_refl gcd_eucl_dvd1 gcd_eucl_dvd2 gcd_eucl_greatest)+  eberlm@62422  144 eberlm@62422  145   eberlm@62422  146 lemma dvd_Lcm_eucl [simp]: "a \ A \ a dvd Lcm_eucl A"  eberlm@62422  147  and Lcm_eucl_least: "(\a. a \ A \ a dvd b) \ Lcm_eucl A dvd b"  eberlm@62422  148  and unit_factor_Lcm_eucl [simp]:  eberlm@62422  149  "unit_factor (Lcm_eucl A) = (if Lcm_eucl A = 0 then 0 else 1)"  eberlm@62422  150 proof -  eberlm@62422  151  have "(\a\A. a dvd Lcm_eucl A) \ (\l'. (\a\A. a dvd l') \ Lcm_eucl A dvd l') \  eberlm@62422  152  unit_factor (Lcm_eucl A) = (if Lcm_eucl A = 0 then 0 else 1)" (is ?thesis)  eberlm@62422  153  proof (cases "\l. l \ 0 \ (\a\A. a dvd l)")  eberlm@62422  154  case False  eberlm@62422  155  hence "Lcm_eucl A = 0" by (auto simp: Lcm_eucl_def)  eberlm@62422  156  with False show ?thesis by auto  eberlm@62422  157  next  eberlm@62422  158  case True  eberlm@62422  159  then obtain l\<^sub>0 where l\<^sub>0_props: "l\<^sub>0 \ 0 \ (\a\A. a dvd l\<^sub>0)" by blast  eberlm@62422  160  def n \ "LEAST n. \l. l \ 0 \ (\a\A. a dvd l) \ euclidean_size l = n"  eberlm@62422  161  def l \ "SOME l. l \ 0 \ (\a\A. a dvd l) \ euclidean_size l = n"  eberlm@62422  162  have "\l. l \ 0 \ (\a\A. a dvd l) \ euclidean_size l = n"  eberlm@62422  163  apply (subst n_def)  eberlm@62422  164  apply (rule LeastI[of _ "euclidean_size l\<^sub>0"])  eberlm@62422  165  apply (rule exI[of _ l\<^sub>0])  eberlm@62422  166  apply (simp add: l\<^sub>0_props)  eberlm@62422  167  done  eberlm@62422  168  from someI_ex[OF this] have "l \ 0" and "\a\A. a dvd l" and "euclidean_size l = n"  eberlm@62422  169  unfolding l_def by simp_all  eberlm@62422  170  {  eberlm@62422  171  fix l' assume "\a\A. a dvd l'"  eberlm@62422  172  with \\a\A. a dvd l\ have "\a\A. a dvd gcd_eucl l l'" by (auto intro: gcd_eucl_greatest)  eberlm@62422  173  moreover from \l \ 0\ have "gcd_eucl l l' \ 0" by simp  eberlm@62422  174  ultimately have "\b. b \ 0 \ (\a\A. a dvd b) \  eberlm@62422  175  euclidean_size b = euclidean_size (gcd_eucl l l')"  eberlm@62422  176  by (intro exI[of _ "gcd_eucl l l'"], auto)  eberlm@62422  177  hence "euclidean_size (gcd_eucl l l') \ n" by (subst n_def) (rule Least_le)  eberlm@62422  178  moreover have "euclidean_size (gcd_eucl l l') \ n"  eberlm@62422  179  proof -  eberlm@62422  180  have "gcd_eucl l l' dvd l" by simp  eberlm@62422  181  then obtain a where "l = gcd_eucl l l' * a" unfolding dvd_def by blast  eberlm@62422  182  with \l \ 0\ have "a \ 0" by auto  eberlm@62422  183  hence "euclidean_size (gcd_eucl l l') \ euclidean_size (gcd_eucl l l' * a)"  eberlm@62422  184  by (rule size_mult_mono)  eberlm@62422  185  also have "gcd_eucl l l' * a = l" using \l = gcd_eucl l l' * a\ ..  eberlm@62422  186  also note \euclidean_size l = n\  eberlm@62422  187  finally show "euclidean_size (gcd_eucl l l') \ n" .  eberlm@62422  188  qed  eberlm@62422  189  ultimately have *: "euclidean_size l = euclidean_size (gcd_eucl l l')"  eberlm@62422  190  by (intro le_antisym, simp_all add: \euclidean_size l = n\)  eberlm@62422  191  from \l \ 0\ have "l dvd gcd_eucl l l'"  eberlm@62422  192  by (rule dvd_euclidean_size_eq_imp_dvd) (auto simp add: *)  eberlm@62422  193  hence "l dvd l'" by (rule dvd_trans[OF _ gcd_eucl_dvd2])  eberlm@62422  194  }  eberlm@62422  195 eberlm@62422  196  with \(\a\A. a dvd l)\ and unit_factor_is_unit[OF \l \ 0\] and \l \ 0\  eberlm@62422  197  have "(\a\A. a dvd normalize l) \  eberlm@62422  198  (\l'. (\a\A. a dvd l') \ normalize l dvd l') \  eberlm@62422  199  unit_factor (normalize l) =  eberlm@62422  200  (if normalize l = 0 then 0 else 1)"  eberlm@62422  201  by (auto simp: unit_simps)  eberlm@62422  202  also from True have "normalize l = Lcm_eucl A"  eberlm@62422  203  by (simp add: Lcm_eucl_def Let_def n_def l_def)  eberlm@62422  204  finally show ?thesis .  eberlm@62422  205  qed  eberlm@62422  206  note A = this  eberlm@62422  207 eberlm@62422  208  {fix a assume "a \ A" then show "a dvd Lcm_eucl A" using A by blast}  eberlm@62422  209  {fix b assume "\a. a \ A \ a dvd b" then show "Lcm_eucl A dvd b" using A by blast}  eberlm@62422  210  from A show "unit_factor (Lcm_eucl A) = (if Lcm_eucl A = 0 then 0 else 1)" by blast  eberlm@62422  211 qed  eberlm@62422  212   eberlm@62422  213 lemma normalize_Lcm_eucl [simp]:  eberlm@62422  214  "normalize (Lcm_eucl A) = Lcm_eucl A"  eberlm@62422  215 proof (cases "Lcm_eucl A = 0")  eberlm@62422  216  case True then show ?thesis by simp  eberlm@62422  217 next  eberlm@62422  218  case False  eberlm@62422  219  have "unit_factor (Lcm_eucl A) * normalize (Lcm_eucl A) = Lcm_eucl A"  eberlm@62422  220  by (fact unit_factor_mult_normalize)  eberlm@62422  221  with False show ?thesis by simp  eberlm@62422  222 qed  eberlm@62422  223 eberlm@62422  224 lemma eq_Lcm_euclI:  eberlm@62422  225  fixes lcm :: "'a set \ 'a"  eberlm@62422  226  assumes "\A a. a \ A \ a dvd lcm A" and "\A c. (\a. a \ A \ a dvd c) \ lcm A dvd c"  eberlm@62422  227  "\A. normalize (lcm A) = lcm A" shows "lcm = Lcm_eucl"  eberlm@62422  228  by (intro ext, rule associated_eqI) (auto simp: assms intro: Lcm_eucl_least)  eberlm@62422  229 haftmann@58023  230 end  haftmann@58023  231 haftmann@60598  232 class euclidean_ring = euclidean_semiring + idom  haftmann@60598  233 begin  haftmann@60598  234 haftmann@60598  235 function euclid_ext :: "'a \ 'a \ 'a \ 'a \ 'a" where  haftmann@60598  236  "euclid_ext a b =  haftmann@60598  237  (if b = 0 then  haftmann@60634  238  (1 div unit_factor a, 0, normalize a)  haftmann@60598  239  else  haftmann@60598  240  case euclid_ext b (a mod b) of  haftmann@60598  241  (s, t, c) \ (t, s - t * (a div b), c))"  haftmann@60598  242  by pat_completeness simp  haftmann@60598  243 termination  haftmann@60598  244  by (relation "measure (euclidean_size \ snd)") (simp_all add: mod_size_less)  haftmann@60598  245 haftmann@60598  246 declare euclid_ext.simps [simp del]  haftmann@60598  247 haftmann@60598  248 lemma euclid_ext_0:  haftmann@60634  249  "euclid_ext a 0 = (1 div unit_factor a, 0, normalize a)"  haftmann@60598  250  by (simp add: euclid_ext.simps [of a 0])  haftmann@60598  251 haftmann@60598  252 lemma euclid_ext_left_0:  haftmann@60634  253  "euclid_ext 0 a = (0, 1 div unit_factor a, normalize a)"  haftmann@60600  254  by (simp add: euclid_ext_0 euclid_ext.simps [of 0 a])  haftmann@60598  255 haftmann@60598  256 lemma euclid_ext_non_0:  haftmann@60598  257  "b \ 0 \ euclid_ext a b = (case euclid_ext b (a mod b) of  haftmann@60598  258  (s, t, c) \ (t, s - t * (a div b), c))"  haftmann@60600  259  by (simp add: euclid_ext.simps [of a b] euclid_ext.simps [of b 0])  haftmann@60598  260 haftmann@60598  261 lemma euclid_ext_code [code]:  haftmann@60634  262  "euclid_ext a b = (if b = 0 then (1 div unit_factor a, 0, normalize a)  haftmann@60598  263  else let (s, t, c) = euclid_ext b (a mod b) in (t, s - t * (a div b), c))"  haftmann@60598  264  by (simp add: euclid_ext.simps [of a b] euclid_ext.simps [of b 0])  haftmann@60598  265 haftmann@60598  266 lemma euclid_ext_correct:  haftmann@60598  267  "case euclid_ext a b of (s, t, c) \ s * a + t * b = c"  haftmann@60598  268 proof (induct a b rule: gcd_eucl_induct)  haftmann@60598  269  case (zero a) then show ?case  haftmann@60598  270  by (simp add: euclid_ext_0 ac_simps)  haftmann@60598  271 next  haftmann@60598  272  case (mod a b)  haftmann@60598  273  obtain s t c where stc: "euclid_ext b (a mod b) = (s,t,c)"  haftmann@60598  274  by (cases "euclid_ext b (a mod b)") blast  haftmann@60598  275  with mod have "c = s * b + t * (a mod b)" by simp  haftmann@60598  276  also have "... = t * ((a div b) * b + a mod b) + (s - t * (a div b)) * b"  haftmann@60598  277  by (simp add: algebra_simps)  haftmann@60598  278  also have "(a div b) * b + a mod b = a" using mod_div_equality .  haftmann@60598  279  finally show ?case  haftmann@60598  280  by (subst euclid_ext.simps) (simp add: stc mod ac_simps)  haftmann@60598  281 qed  haftmann@60598  282 haftmann@60598  283 definition euclid_ext' :: "'a \ 'a \ 'a \ 'a"  haftmann@60598  284 where  haftmann@60598  285  "euclid_ext' a b = (case euclid_ext a b of (s, t, _) \ (s, t))"  haftmann@60598  286 haftmann@60634  287 lemma euclid_ext'_0: "euclid_ext' a 0 = (1 div unit_factor a, 0)"  haftmann@60598  288  by (simp add: euclid_ext'_def euclid_ext_0)  haftmann@60598  289 haftmann@60634  290 lemma euclid_ext'_left_0: "euclid_ext' 0 a = (0, 1 div unit_factor a)"  haftmann@60598  291  by (simp add: euclid_ext'_def euclid_ext_left_0)  haftmann@60598  292   haftmann@60598  293 lemma euclid_ext'_non_0: "b \ 0 \ euclid_ext' a b = (snd (euclid_ext' b (a mod b)),  haftmann@60598  294  fst (euclid_ext' b (a mod b)) - snd (euclid_ext' b (a mod b)) * (a div b))"  haftmann@60598  295  by (simp add: euclid_ext'_def euclid_ext_non_0 split_def)  haftmann@60598  296 haftmann@60598  297 end  haftmann@60598  298 haftmann@58023  299 class euclidean_semiring_gcd = euclidean_semiring + gcd + Gcd +  haftmann@58023  300  assumes gcd_gcd_eucl: "gcd = gcd_eucl" and lcm_lcm_eucl: "lcm = lcm_eucl"  haftmann@58023  301  assumes Gcd_Gcd_eucl: "Gcd = Gcd_eucl" and Lcm_Lcm_eucl: "Lcm = Lcm_eucl"  haftmann@58023  302 begin  haftmann@58023  303 eberlm@62422  304 subclass semiring_gcd  eberlm@62422  305  by standard (simp_all add: gcd_gcd_eucl gcd_eucl_greatest lcm_lcm_eucl lcm_eucl_def)  haftmann@58023  306 eberlm@62422  307 subclass semiring_Gcd  eberlm@62422  308  by standard (auto simp: Gcd_Gcd_eucl Lcm_Lcm_eucl Gcd_eucl_def intro: Lcm_eucl_least)  eberlm@62422  309   haftmann@58023  310 haftmann@58023  311 lemma gcd_non_0:  haftmann@60430  312  "b \ 0 \ gcd a b = gcd b (a mod b)"  haftmann@60572  313  unfolding gcd_gcd_eucl by (fact gcd_eucl_non_0)  haftmann@58023  314 eberlm@62422  315 lemmas gcd_0 = gcd_0_right  eberlm@62422  316 lemmas dvd_gcd_iff = gcd_greatest_iff  haftmann@58023  317 haftmann@58023  318 lemmas gcd_greatest_iff = dvd_gcd_iff  haftmann@58023  319 haftmann@58023  320 lemma gcdI:  haftmann@60634  321  assumes "c dvd a" and "c dvd b" and greatest: "\d. d dvd a \ d dvd b \ d dvd c"  haftmann@60688  322  and "normalize c = c"  haftmann@60634  323  shows "c = gcd a b"  haftmann@60688  324  by (rule associated_eqI) (auto simp: assms intro: gcd_greatest)  haftmann@58023  325 haftmann@58023  326 lemma gcd_unique: "d dvd a \ d dvd b \  haftmann@60688  327  normalize d = d \  haftmann@58023  328  (\e. e dvd a \ e dvd b \ e dvd d) \ d = gcd a b"  haftmann@60688  329  by rule (auto intro: gcdI simp: gcd_greatest)  haftmann@58023  330 haftmann@58023  331 lemma gcd_dvd_prod: "gcd a b dvd k * b"  haftmann@58023  332  using mult_dvd_mono [of 1] by auto  haftmann@58023  333 haftmann@58023  334 lemma gcd_proj2_if_dvd:  haftmann@60634  335  "b dvd a \ gcd a b = normalize b"  eberlm@62422  336  by (cases "b = 0", simp_all add: dvd_eq_mod_eq_0 gcd_non_0)  haftmann@58023  337 haftmann@58023  338 lemma gcd_proj1_if_dvd:  haftmann@60634  339  "a dvd b \ gcd a b = normalize a"  haftmann@58023  340  by (subst gcd.commute, simp add: gcd_proj2_if_dvd)  haftmann@58023  341 haftmann@60634  342 lemma gcd_proj1_iff: "gcd m n = normalize m \ m dvd n"  haftmann@58023  343 proof  haftmann@60634  344  assume A: "gcd m n = normalize m"  haftmann@58023  345  show "m dvd n"  haftmann@58023  346  proof (cases "m = 0")  haftmann@58023  347  assume [simp]: "m \ 0"  haftmann@60634  348  from A have B: "m = gcd m n * unit_factor m"  haftmann@58023  349  by (simp add: unit_eq_div2)  haftmann@58023  350  show ?thesis by (subst B, simp add: mult_unit_dvd_iff)  haftmann@58023  351  qed (insert A, simp)  haftmann@58023  352 next  haftmann@58023  353  assume "m dvd n"  haftmann@60634  354  then show "gcd m n = normalize m" by (rule gcd_proj1_if_dvd)  haftmann@58023  355 qed  haftmann@58023  356   haftmann@60634  357 lemma gcd_proj2_iff: "gcd m n = normalize n \ n dvd m"  haftmann@60634  358  using gcd_proj1_iff [of n m] by (simp add: ac_simps)  haftmann@58023  359 haftmann@58023  360 lemma gcd_mod1 [simp]:  haftmann@60430  361  "gcd (a mod b) b = gcd a b"  haftmann@58023  362  by (rule gcdI, metis dvd_mod_iff gcd_dvd1 gcd_dvd2, simp_all add: gcd_greatest dvd_mod_iff)  haftmann@58023  363 haftmann@58023  364 lemma gcd_mod2 [simp]:  haftmann@60430  365  "gcd a (b mod a) = gcd a b"  haftmann@58023  366  by (rule gcdI, simp, metis dvd_mod_iff gcd_dvd1 gcd_dvd2, simp_all add: gcd_greatest dvd_mod_iff)  haftmann@58023  367   haftmann@58023  368 lemma gcd_mult_distrib':  haftmann@60634  369  "normalize c * gcd a b = gcd (c * a) (c * b)"  haftmann@60569  370 proof (cases "c = 0")  eberlm@62422  371  case True then show ?thesis by simp_all  haftmann@60569  372 next  haftmann@60634  373  case False then have [simp]: "is_unit (unit_factor c)" by simp  haftmann@60569  374  show ?thesis  haftmann@60569  375  proof (induct a b rule: gcd_eucl_induct)  haftmann@60569  376  case (zero a) show ?case  haftmann@60569  377  proof (cases "a = 0")  eberlm@62422  378  case True then show ?thesis by simp  haftmann@60569  379  next  haftmann@60634  380  case False  eberlm@62422  381  then show ?thesis by (simp add: normalize_mult)  haftmann@60569  382  qed  haftmann@60569  383  case (mod a b)  haftmann@60569  384  then show ?case by (simp add: mult_mod_right gcd.commute)  haftmann@58023  385  qed  haftmann@58023  386 qed  haftmann@58023  387 haftmann@58023  388 lemma gcd_mult_distrib:  haftmann@60634  389  "k * gcd a b = gcd (k * a) (k * b) * unit_factor k"  haftmann@58023  390 proof-  haftmann@60634  391  have "normalize k * gcd a b = gcd (k * a) (k * b)"  haftmann@60634  392  by (simp add: gcd_mult_distrib')  haftmann@60634  393  then have "normalize k * gcd a b * unit_factor k = gcd (k * a) (k * b) * unit_factor k"  haftmann@60634  394  by simp  haftmann@60634  395  then have "normalize k * unit_factor k * gcd a b = gcd (k * a) (k * b) * unit_factor k"  haftmann@60634  396  by (simp only: ac_simps)  haftmann@60634  397  then show ?thesis  haftmann@59009  398  by simp  haftmann@58023  399 qed  haftmann@58023  400 haftmann@58023  401 lemma euclidean_size_gcd_le1 [simp]:  haftmann@58023  402  assumes "a \ 0"  haftmann@58023  403  shows "euclidean_size (gcd a b) \ euclidean_size a"  haftmann@58023  404 proof -  haftmann@58023  405  have "gcd a b dvd a" by (rule gcd_dvd1)  haftmann@58023  406  then obtain c where A: "a = gcd a b * c" unfolding dvd_def by blast  wenzelm@60526  407  with \a \ 0\ show ?thesis by (subst (2) A, intro size_mult_mono) auto  haftmann@58023  408 qed  haftmann@58023  409 haftmann@58023  410 lemma euclidean_size_gcd_le2 [simp]:  haftmann@58023  411  "b \ 0 \ euclidean_size (gcd a b) \ euclidean_size b"  haftmann@58023  412  by (subst gcd.commute, rule euclidean_size_gcd_le1)  haftmann@58023  413 haftmann@58023  414 lemma euclidean_size_gcd_less1:  haftmann@58023  415  assumes "a \ 0" and "\a dvd b"  haftmann@58023  416  shows "euclidean_size (gcd a b) < euclidean_size a"  haftmann@58023  417 proof (rule ccontr)  haftmann@58023  418  assume "\euclidean_size (gcd a b) < euclidean_size a"  eberlm@62422  419  with \a \ 0\ have A: "euclidean_size (gcd a b) = euclidean_size a"  haftmann@58023  420  by (intro le_antisym, simp_all)  eberlm@62422  421  have "a dvd gcd a b"  eberlm@62422  422  by (rule dvd_euclidean_size_eq_imp_dvd) (simp_all add: assms A)  eberlm@62422  423  hence "a dvd b" using dvd_gcdD2 by blast  wenzelm@60526  424  with \\a dvd b\ show False by contradiction  haftmann@58023  425 qed  haftmann@58023  426 haftmann@58023  427 lemma euclidean_size_gcd_less2:  haftmann@58023  428  assumes "b \ 0" and "\b dvd a"  haftmann@58023  429  shows "euclidean_size (gcd a b) < euclidean_size b"  haftmann@58023  430  using assms by (subst gcd.commute, rule euclidean_size_gcd_less1)  haftmann@58023  431 haftmann@60430  432 lemma gcd_mult_unit1: "is_unit a \ gcd (b * a) c = gcd b c"  haftmann@58023  433  apply (rule gcdI)  haftmann@60688  434  apply simp_all  haftmann@58023  435  apply (rule dvd_trans, rule gcd_dvd1, simp add: unit_simps)  haftmann@58023  436  done  haftmann@58023  437 haftmann@60430  438 lemma gcd_mult_unit2: "is_unit a \ gcd b (c * a) = gcd b c"  haftmann@58023  439  by (subst gcd.commute, subst gcd_mult_unit1, assumption, rule gcd.commute)  haftmann@58023  440 haftmann@60430  441 lemma gcd_div_unit1: "is_unit a \ gcd (b div a) c = gcd b c"  haftmann@60433  442  by (erule is_unitE [of _ b]) (simp add: gcd_mult_unit1)  haftmann@58023  443 haftmann@60430  444 lemma gcd_div_unit2: "is_unit a \ gcd b (c div a) = gcd b c"  haftmann@60433  445  by (erule is_unitE [of _ c]) (simp add: gcd_mult_unit2)  haftmann@58023  446 haftmann@60634  447 lemma normalize_gcd_left [simp]:  haftmann@60634  448  "gcd (normalize a) b = gcd a b"  haftmann@60634  449 proof (cases "a = 0")  haftmann@60634  450  case True then show ?thesis  haftmann@60634  451  by simp  haftmann@60634  452 next  haftmann@60634  453  case False then have "is_unit (unit_factor a)"  haftmann@60634  454  by simp  haftmann@60634  455  moreover have "normalize a = a div unit_factor a"  haftmann@60634  456  by simp  haftmann@60634  457  ultimately show ?thesis  haftmann@60634  458  by (simp only: gcd_div_unit1)  haftmann@60634  459 qed  haftmann@60634  460 haftmann@60634  461 lemma normalize_gcd_right [simp]:  haftmann@60634  462  "gcd a (normalize b) = gcd a b"  haftmann@60634  463  using normalize_gcd_left [of b a] by (simp add: ac_simps)  haftmann@60634  464 haftmann@60634  465 lemma gcd_idem: "gcd a a = normalize a"  eberlm@62422  466  by (cases "a = 0") (simp, rule sym, rule gcdI, simp_all)  haftmann@58023  467 haftmann@60430  468 lemma gcd_right_idem: "gcd (gcd a b) b = gcd a b"  haftmann@58023  469  apply (rule gcdI)  haftmann@58023  470  apply (simp add: ac_simps)  haftmann@58023  471  apply (rule gcd_dvd2)  haftmann@58023  472  apply (rule gcd_greatest, erule (1) gcd_greatest, assumption)  haftmann@59009  473  apply simp  haftmann@58023  474  done  haftmann@58023  475 haftmann@60430  476 lemma gcd_left_idem: "gcd a (gcd a b) = gcd a b"  haftmann@58023  477  apply (rule gcdI)  haftmann@58023  478  apply simp  haftmann@58023  479  apply (rule dvd_trans, rule gcd_dvd2, rule gcd_dvd2)  haftmann@58023  480  apply (rule gcd_greatest, assumption, erule gcd_greatest, assumption)  haftmann@59009  481  apply simp  haftmann@58023  482  done  haftmann@58023  483 haftmann@58023  484 lemma comp_fun_idem_gcd: "comp_fun_idem gcd"  haftmann@58023  485 proof  haftmann@58023  486  fix a b show "gcd a \ gcd b = gcd b \ gcd a"  haftmann@58023  487  by (simp add: fun_eq_iff ac_simps)  haftmann@58023  488 next  haftmann@58023  489  fix a show "gcd a \ gcd a = gcd a"  haftmann@58023  490  by (simp add: fun_eq_iff gcd_left_idem)  haftmann@58023  491 qed  haftmann@58023  492 haftmann@58023  493 lemma gcd_dvd_antisym:  haftmann@58023  494  "gcd a b dvd gcd c d \ gcd c d dvd gcd a b \ gcd a b = gcd c d"  haftmann@58023  495 proof (rule gcdI)  haftmann@58023  496  assume A: "gcd a b dvd gcd c d" and B: "gcd c d dvd gcd a b"  haftmann@58023  497  have "gcd c d dvd c" by simp  haftmann@58023  498  with A show "gcd a b dvd c" by (rule dvd_trans)  haftmann@58023  499  have "gcd c d dvd d" by simp  haftmann@58023  500  with A show "gcd a b dvd d" by (rule dvd_trans)  haftmann@60688  501  show "normalize (gcd a b) = gcd a b"  haftmann@59009  502  by simp  haftmann@58023  503  fix l assume "l dvd c" and "l dvd d"  haftmann@58023  504  hence "l dvd gcd c d" by (rule gcd_greatest)  haftmann@58023  505  from this and B show "l dvd gcd a b" by (rule dvd_trans)  haftmann@58023  506 qed  haftmann@58023  507 haftmann@58023  508 lemma coprime_crossproduct:  haftmann@58023  509  assumes [simp]: "gcd a d = 1" "gcd b c = 1"  haftmann@60688  510  shows "normalize (a * c) = normalize (b * d) \ normalize a = normalize b \ normalize c = normalize d"  haftmann@60688  511  (is "?lhs \ ?rhs")  haftmann@58023  512 proof  haftmann@60688  513  assume ?rhs  haftmann@60688  514  then have "a dvd b" "b dvd a" "c dvd d" "d dvd c" by (simp_all add: associated_iff_dvd)  haftmann@60688  515  then have "a * c dvd b * d" "b * d dvd a * c" by (fast intro: mult_dvd_mono)+  haftmann@60688  516  then show ?lhs by (simp add: associated_iff_dvd)  haftmann@58023  517 next  haftmann@58023  518  assume ?lhs  haftmann@60688  519  then have dvd: "a * c dvd b * d" "b * d dvd a * c" by (simp_all add: associated_iff_dvd)  haftmann@60688  520  then have "a dvd b * d" by (metis dvd_mult_left)  haftmann@58023  521  hence "a dvd b" by (simp add: coprime_dvd_mult_iff)  haftmann@60688  522  moreover from dvd have "b dvd a * c" by (metis dvd_mult_left)  haftmann@58023  523  hence "b dvd a" by (simp add: coprime_dvd_mult_iff)  haftmann@60688  524  moreover from dvd have "c dvd d * b"  haftmann@60688  525  by (auto dest: dvd_mult_right simp add: ac_simps)  haftmann@58023  526  hence "c dvd d" by (simp add: coprime_dvd_mult_iff gcd.commute)  haftmann@60688  527  moreover from dvd have "d dvd c * a"  haftmann@60688  528  by (auto dest: dvd_mult_right simp add: ac_simps)  haftmann@58023  529  hence "d dvd c" by (simp add: coprime_dvd_mult_iff gcd.commute)  haftmann@60688  530  ultimately show ?rhs by (simp add: associated_iff_dvd)  haftmann@58023  531 qed  haftmann@58023  532 haftmann@58023  533 lemma gcd_add1 [simp]:  haftmann@58023  534  "gcd (m + n) n = gcd m n"  haftmann@58023  535  by (cases "n = 0", simp_all add: gcd_non_0)  haftmann@58023  536 haftmann@58023  537 lemma gcd_add2 [simp]:  haftmann@58023  538  "gcd m (m + n) = gcd m n"  haftmann@58023  539  using gcd_add1 [of n m] by (simp add: ac_simps)  haftmann@58023  540 haftmann@60572  541 lemma gcd_add_mult:  haftmann@60572  542  "gcd m (k * m + n) = gcd m n"  haftmann@60572  543 proof -  haftmann@60572  544  have "gcd m ((k * m + n) mod m) = gcd m (k * m + n)"  haftmann@60572  545  by (fact gcd_mod2)  haftmann@60572  546  then show ?thesis by simp  haftmann@60572  547 qed  haftmann@58023  548 haftmann@60430  549 lemma coprimeI: "(\l. \l dvd a; l dvd b\ \ l dvd 1) \ gcd a b = 1"  haftmann@58023  550  by (rule sym, rule gcdI, simp_all)  haftmann@58023  551 haftmann@58023  552 lemma coprime: "gcd a b = 1 \ (\d. d dvd a \ d dvd b \ is_unit d)"  eberlm@62422  553  by (auto intro: coprimeI gcd_greatest dvd_gcdD1 dvd_gcdD2)  haftmann@58023  554 haftmann@58023  555 lemma div_gcd_coprime:  haftmann@58023  556  assumes nz: "a \ 0 \ b \ 0"  haftmann@58023  557  defines [simp]: "d \ gcd a b"  haftmann@58023  558  defines [simp]: "a' \ a div d" and [simp]: "b' \ b div d"  haftmann@58023  559  shows "gcd a' b' = 1"  haftmann@58023  560 proof (rule coprimeI)  haftmann@58023  561  fix l assume "l dvd a'" "l dvd b'"  haftmann@58023  562  then obtain s t where "a' = l * s" "b' = l * t" unfolding dvd_def by blast  haftmann@59009  563  moreover have "a = a' * d" "b = b' * d" by simp_all  haftmann@58023  564  ultimately have "a = (l * d) * s" "b = (l * d) * t"  haftmann@59009  565  by (simp_all only: ac_simps)  haftmann@58023  566  hence "l*d dvd a" and "l*d dvd b" by (simp_all only: dvd_triv_left)  haftmann@58023  567  hence "l*d dvd d" by (simp add: gcd_greatest)  haftmann@59009  568  then obtain u where "d = l * d * u" ..  haftmann@59009  569  then have "d * (l * u) = d" by (simp add: ac_simps)  haftmann@59009  570  moreover from nz have "d \ 0" by simp  haftmann@59009  571  with div_mult_self1_is_id have "d * (l * u) div d = l * u" .  haftmann@59009  572  ultimately have "1 = l * u"  wenzelm@60526  573  using \d \ 0\ by simp  haftmann@59009  574  then show "l dvd 1" ..  haftmann@58023  575 qed  haftmann@58023  576 haftmann@58023  577 lemma coprime_lmult:  haftmann@58023  578  assumes dab: "gcd d (a * b) = 1"  haftmann@58023  579  shows "gcd d a = 1"  haftmann@58023  580 proof (rule coprimeI)  haftmann@58023  581  fix l assume "l dvd d" and "l dvd a"  haftmann@58023  582  hence "l dvd a * b" by simp  wenzelm@60526  583  with \l dvd d\ and dab show "l dvd 1" by (auto intro: gcd_greatest)  haftmann@58023  584 qed  haftmann@58023  585 haftmann@58023  586 lemma coprime_rmult:  haftmann@58023  587  assumes dab: "gcd d (a * b) = 1"  haftmann@58023  588  shows "gcd d b = 1"  haftmann@58023  589 proof (rule coprimeI)  haftmann@58023  590  fix l assume "l dvd d" and "l dvd b"  haftmann@58023  591  hence "l dvd a * b" by simp  wenzelm@60526  592  with \l dvd d\ and dab show "l dvd 1" by (auto intro: gcd_greatest)  haftmann@58023  593 qed  haftmann@58023  594 haftmann@58023  595 lemma coprime_mul_eq: "gcd d (a * b) = 1 \ gcd d a = 1 \ gcd d b = 1"  haftmann@58023  596  using coprime_rmult[of d a b] coprime_lmult[of d a b] coprime_mult[of d a b] by blast  haftmann@58023  597 haftmann@58023  598 lemma gcd_coprime:  haftmann@60430  599  assumes c: "gcd a b \ 0" and a: "a = a' * gcd a b" and b: "b = b' * gcd a b"  haftmann@58023  600  shows "gcd a' b' = 1"  haftmann@58023  601 proof -  haftmann@60430  602  from c have "a \ 0 \ b \ 0" by simp  haftmann@58023  603  with div_gcd_coprime have "gcd (a div gcd a b) (b div gcd a b) = 1" .  haftmann@58023  604  also from assms have "a div gcd a b = a'" by (metis div_mult_self2_is_id)+  haftmann@58023  605  also from assms have "b div gcd a b = b'" by (metis div_mult_self2_is_id)+  haftmann@58023  606  finally show ?thesis .  haftmann@58023  607 qed  haftmann@58023  608 haftmann@58023  609 lemma coprime_power:  haftmann@58023  610  assumes "0 < n"  haftmann@58023  611  shows "gcd a (b ^ n) = 1 \ gcd a b = 1"  haftmann@58023  612 using assms proof (induct n)  haftmann@58023  613  case (Suc n) then show ?case  haftmann@58023  614  by (cases n) (simp_all add: coprime_mul_eq)  haftmann@58023  615 qed simp  haftmann@58023  616 haftmann@58023  617 lemma gcd_coprime_exists:  haftmann@58023  618  assumes nz: "gcd a b \ 0"  haftmann@58023  619  shows "\a' b'. a = a' * gcd a b \ b = b' * gcd a b \ gcd a' b' = 1"  haftmann@58023  620  apply (rule_tac x = "a div gcd a b" in exI)  haftmann@58023  621  apply (rule_tac x = "b div gcd a b" in exI)  haftmann@59009  622  apply (insert nz, auto intro: div_gcd_coprime)  haftmann@58023  623  done  haftmann@58023  624 haftmann@58023  625 lemma coprime_exp:  haftmann@58023  626  "gcd d a = 1 \ gcd d (a^n) = 1"  haftmann@58023  627  by (induct n, simp_all add: coprime_mult)  haftmann@58023  628 haftmann@58023  629 lemma gcd_exp:  haftmann@60688  630  "gcd (a ^ n) (b ^ n) = gcd a b ^ n"  haftmann@58023  631 proof (cases "a = 0 \ b = 0")  haftmann@60688  632  case True  haftmann@60688  633  then show ?thesis by (cases n) simp_all  haftmann@58023  634 next  haftmann@60688  635  case False  haftmann@60688  636  then have "1 = gcd ((a div gcd a b) ^ n) ((b div gcd a b) ^ n)"  eberlm@62422  637  using coprime_exp2[OF div_gcd_coprime[of a b], of n n, symmetric] by simp  haftmann@60688  638  then have "gcd a b ^ n = gcd a b ^ n * ..." by simp  haftmann@58023  639  also note gcd_mult_distrib  haftmann@60688  640  also have "unit_factor (gcd a b ^ n) = 1"  haftmann@60688  641  using False by (auto simp add: unit_factor_power unit_factor_gcd)  haftmann@58023  642  also have "(gcd a b)^n * (a div gcd a b)^n = a^n"  haftmann@58023  643  by (subst ac_simps, subst div_power, simp, rule dvd_div_mult_self, rule dvd_power_same, simp)  haftmann@58023  644  also have "(gcd a b)^n * (b div gcd a b)^n = b^n"  haftmann@58023  645  by (subst ac_simps, subst div_power, simp, rule dvd_div_mult_self, rule dvd_power_same, simp)  haftmann@58023  646  finally show ?thesis by simp  haftmann@58023  647 qed  haftmann@58023  648 haftmann@58023  649 lemma coprime_common_divisor:  haftmann@60430  650  "gcd a b = 1 \ a dvd a \ a dvd b \ is_unit a"  haftmann@60430  651  apply (subgoal_tac "a dvd gcd a b")  haftmann@59061  652  apply simp  haftmann@58023  653  apply (erule (1) gcd_greatest)  haftmann@58023  654  done  haftmann@58023  655 haftmann@58023  656 lemma division_decomp:  haftmann@58023  657  assumes dc: "a dvd b * c"  haftmann@58023  658  shows "\b' c'. a = b' * c' \ b' dvd b \ c' dvd c"  haftmann@58023  659 proof (cases "gcd a b = 0")  haftmann@58023  660  assume "gcd a b = 0"  haftmann@59009  661  hence "a = 0 \ b = 0" by simp  haftmann@58023  662  hence "a = 0 * c \ 0 dvd b \ c dvd c" by simp  haftmann@58023  663  then show ?thesis by blast  haftmann@58023  664 next  haftmann@58023  665  let ?d = "gcd a b"  haftmann@58023  666  assume "?d \ 0"  haftmann@58023  667  from gcd_coprime_exists[OF this]  haftmann@58023  668  obtain a' b' where ab': "a = a' * ?d" "b = b' * ?d" "gcd a' b' = 1"  haftmann@58023  669  by blast  haftmann@58023  670  from ab'(1) have "a' dvd a" unfolding dvd_def by blast  haftmann@58023  671  with dc have "a' dvd b*c" using dvd_trans[of a' a "b*c"] by simp  haftmann@58023  672  from dc ab'(1,2) have "a'*?d dvd (b'*?d) * c" by simp  haftmann@58023  673  hence "?d * a' dvd ?d * (b' * c)" by (simp add: mult_ac)  wenzelm@60526  674  with \?d \ 0\ have "a' dvd b' * c" by simp  haftmann@58023  675  with coprime_dvd_mult[OF ab'(3)]  haftmann@58023  676  have "a' dvd c" by (subst (asm) ac_simps, blast)  haftmann@58023  677  with ab'(1) have "a = ?d * a' \ ?d dvd b \ a' dvd c" by (simp add: mult_ac)  haftmann@58023  678  then show ?thesis by blast  haftmann@58023  679 qed  haftmann@58023  680 haftmann@60433  681 lemma pow_divs_pow:  haftmann@58023  682  assumes ab: "a ^ n dvd b ^ n" and n: "n \ 0"  haftmann@58023  683  shows "a dvd b"  haftmann@58023  684 proof (cases "gcd a b = 0")  haftmann@58023  685  assume "gcd a b = 0"  haftmann@59009  686  then show ?thesis by simp  haftmann@58023  687 next  haftmann@58023  688  let ?d = "gcd a b"  haftmann@58023  689  assume "?d \ 0"  haftmann@58023  690  from n obtain m where m: "n = Suc m" by (cases n, simp_all)  wenzelm@60526  691  from \?d \ 0\ have zn: "?d ^ n \ 0" by (rule power_not_zero)  wenzelm@60526  692  from gcd_coprime_exists[OF \?d \ 0\]  haftmann@58023  693  obtain a' b' where ab': "a = a' * ?d" "b = b' * ?d" "gcd a' b' = 1"  haftmann@58023  694  by blast  haftmann@58023  695  from ab have "(a' * ?d) ^ n dvd (b' * ?d) ^ n"  haftmann@58023  696  by (simp add: ab'(1,2)[symmetric])  haftmann@58023  697  hence "?d^n * a'^n dvd ?d^n * b'^n"  haftmann@58023  698  by (simp only: power_mult_distrib ac_simps)  haftmann@59009  699  with zn have "a'^n dvd b'^n" by simp  haftmann@58023  700  hence "a' dvd b'^n" using dvd_trans[of a' "a'^n" "b'^n"] by (simp add: m)  haftmann@58023  701  hence "a' dvd b'^m * b'" by (simp add: m ac_simps)  haftmann@58023  702  with coprime_dvd_mult[OF coprime_exp[OF ab'(3), of m]]  haftmann@58023  703  have "a' dvd b'" by (subst (asm) ac_simps, blast)  haftmann@58023  704  hence "a'*?d dvd b'*?d" by (rule mult_dvd_mono, simp)  haftmann@58023  705  with ab'(1,2) show ?thesis by simp  haftmann@58023  706 qed  haftmann@58023  707 haftmann@60433  708 lemma pow_divs_eq [simp]:  haftmann@58023  709  "n \ 0 \ a ^ n dvd b ^ n \ a dvd b"  haftmann@60433  710  by (auto intro: pow_divs_pow dvd_power_same)  haftmann@58023  711 eberlm@62422  712 lemmas divs_mult = divides_mult  haftmann@58023  713 haftmann@58023  714 lemma coprime_plus_one [simp]: "gcd (n + 1) n = 1"  haftmann@58023  715  by (subst add_commute, simp)  haftmann@58023  716 haftmann@58023  717 lemma setprod_coprime [rule_format]:  haftmann@60430  718  "(\i\A. gcd (f i) a = 1) \ gcd (\i\A. f i) a = 1"  haftmann@58023  719  apply (cases "finite A")  haftmann@58023  720  apply (induct set: finite)  haftmann@58023  721  apply (auto simp add: gcd_mult_cancel)  haftmann@58023  722  done  eberlm@62422  723   eberlm@62422  724 lemma listprod_coprime:  eberlm@62422  725  "(\x. x \ set xs \ coprime x y) \ coprime (listprod xs) y"  eberlm@62422  726  by (induction xs) (simp_all add: gcd_mult_cancel)  haftmann@58023  727 haftmann@58023  728 lemma coprime_divisors:  haftmann@58023  729  assumes "d dvd a" "e dvd b" "gcd a b = 1"  haftmann@58023  730  shows "gcd d e = 1"  haftmann@58023  731 proof -  haftmann@58023  732  from assms obtain k l where "a = d * k" "b = e * l"  haftmann@58023  733  unfolding dvd_def by blast  haftmann@58023  734  with assms have "gcd (d * k) (e * l) = 1" by simp  haftmann@58023  735  hence "gcd (d * k) e = 1" by (rule coprime_lmult)  haftmann@58023  736  also have "gcd (d * k) e = gcd e (d * k)" by (simp add: ac_simps)  haftmann@58023  737  finally have "gcd e d = 1" by (rule coprime_lmult)  haftmann@58023  738  then show ?thesis by (simp add: ac_simps)  haftmann@58023  739 qed  haftmann@58023  740 haftmann@58023  741 lemma invertible_coprime:  haftmann@60430  742  assumes "a * b mod m = 1"  haftmann@60430  743  shows "coprime a m"  haftmann@59009  744 proof -  haftmann@60430  745  from assms have "coprime m (a * b mod m)"  haftmann@59009  746  by simp  haftmann@60430  747  then have "coprime m (a * b)"  haftmann@59009  748  by simp  haftmann@60430  749  then have "coprime m a"  haftmann@59009  750  by (rule coprime_lmult)  haftmann@59009  751  then show ?thesis  haftmann@59009  752  by (simp add: ac_simps)  haftmann@59009  753 qed  haftmann@58023  754 haftmann@58023  755 lemma lcm_gcd_prod:  haftmann@60634  756  "lcm a b * gcd a b = normalize (a * b)"  haftmann@60634  757  by (simp add: lcm_gcd)  haftmann@58023  758 haftmann@58023  759 lemma lcm_zero:  haftmann@58023  760  "lcm a b = 0 \ a = 0 \ b = 0"  haftmann@60687  761  by (fact lcm_eq_0_iff)  haftmann@58023  762 haftmann@58023  763 lemmas lcm_0_iff = lcm_zero  haftmann@58023  764 haftmann@58023  765 lemma gcd_lcm:  haftmann@58023  766  assumes "lcm a b \ 0"  haftmann@60634  767  shows "gcd a b = normalize (a * b) div lcm a b"  haftmann@60634  768 proof -  haftmann@60634  769  have "lcm a b * gcd a b = normalize (a * b)"  haftmann@60634  770  by (fact lcm_gcd_prod)  haftmann@60634  771  with assms show ?thesis  haftmann@60634  772  by (metis nonzero_mult_divide_cancel_left)  haftmann@58023  773 qed  haftmann@58023  774 haftmann@60687  775 declare unit_factor_lcm [simp]  haftmann@58023  776 haftmann@58023  777 lemma lcmI:  haftmann@60634  778  assumes "a dvd c" and "b dvd c" and "\d. a dvd d \ b dvd d \ c dvd d"  haftmann@60688  779  and "normalize c = c"  haftmann@60634  780  shows "c = lcm a b"  haftmann@60688  781  by (rule associated_eqI) (auto simp: assms intro: lcm_least)  haftmann@58023  782 haftmann@58023  783 lemma gcd_dvd_lcm [simp]:  haftmann@58023  784  "gcd a b dvd lcm a b"  haftmann@60690  785  using gcd_dvd2 by (rule dvd_lcmI2)  haftmann@58023  786 eberlm@62422  787 lemmas lcm_0 = lcm_0_right  haftmann@58023  788 haftmann@58023  789 lemma lcm_unique:  haftmann@58023  790  "a dvd d \ b dvd d \  haftmann@60688  791  normalize d = d \  haftmann@58023  792  (\e. a dvd e \ b dvd e \ d dvd e) \ d = lcm a b"  haftmann@60688  793  by rule (auto intro: lcmI simp: lcm_least lcm_zero)  haftmann@58023  794 haftmann@58023  795 lemma lcm_coprime:  haftmann@60634  796  "gcd a b = 1 \ lcm a b = normalize (a * b)"  haftmann@58023  797  by (subst lcm_gcd) simp  haftmann@58023  798 haftmann@58023  799 lemma lcm_proj1_if_dvd:  haftmann@60634  800  "b dvd a \ lcm a b = normalize a"  haftmann@60430  801  by (cases "a = 0") (simp, rule sym, rule lcmI, simp_all)  haftmann@58023  802 haftmann@58023  803 lemma lcm_proj2_if_dvd:  haftmann@60634  804  "a dvd b \ lcm a b = normalize b"  haftmann@60430  805  using lcm_proj1_if_dvd [of a b] by (simp add: ac_simps)  haftmann@58023  806 haftmann@58023  807 lemma lcm_proj1_iff:  haftmann@60634  808  "lcm m n = normalize m \ n dvd m"  haftmann@58023  809 proof  haftmann@60634  810  assume A: "lcm m n = normalize m"  haftmann@58023  811  show "n dvd m"  haftmann@58023  812  proof (cases "m = 0")  haftmann@58023  813  assume [simp]: "m \ 0"  haftmann@60634  814  from A have B: "m = lcm m n * unit_factor m"  haftmann@58023  815  by (simp add: unit_eq_div2)  haftmann@58023  816  show ?thesis by (subst B, simp)  haftmann@58023  817  qed simp  haftmann@58023  818 next  haftmann@58023  819  assume "n dvd m"  haftmann@60634  820  then show "lcm m n = normalize m" by (rule lcm_proj1_if_dvd)  haftmann@58023  821 qed  haftmann@58023  822 haftmann@58023  823 lemma lcm_proj2_iff:  haftmann@60634  824  "lcm m n = normalize n \ m dvd n"  haftmann@58023  825  using lcm_proj1_iff [of n m] by (simp add: ac_simps)  haftmann@58023  826 haftmann@58023  827 lemma euclidean_size_lcm_le1:  haftmann@58023  828  assumes "a \ 0" and "b \ 0"  haftmann@58023  829  shows "euclidean_size a \ euclidean_size (lcm a b)"  haftmann@58023  830 proof -  haftmann@60690  831  have "a dvd lcm a b" by (rule dvd_lcm1)  haftmann@60690  832  then obtain c where A: "lcm a b = a * c" ..  wenzelm@60526  833  with \a \ 0\ and \b \ 0\ have "c \ 0" by (auto simp: lcm_zero)  haftmann@58023  834  then show ?thesis by (subst A, intro size_mult_mono)  haftmann@58023  835 qed  haftmann@58023  836 haftmann@58023  837 lemma euclidean_size_lcm_le2:  haftmann@58023  838  "a \ 0 \ b \ 0 \ euclidean_size b \ euclidean_size (lcm a b)"  haftmann@58023  839  using euclidean_size_lcm_le1 [of b a] by (simp add: ac_simps)  haftmann@58023  840 haftmann@58023  841 lemma euclidean_size_lcm_less1:  haftmann@58023  842  assumes "b \ 0" and "\b dvd a"  haftmann@58023  843  shows "euclidean_size a < euclidean_size (lcm a b)"  haftmann@58023  844 proof (rule ccontr)  haftmann@58023  845  from assms have "a \ 0" by auto  haftmann@58023  846  assume "\euclidean_size a < euclidean_size (lcm a b)"  wenzelm@60526  847  with \a \ 0\ and \b \ 0\ have "euclidean_size (lcm a b) = euclidean_size a"  haftmann@58023  848  by (intro le_antisym, simp, intro euclidean_size_lcm_le1)  haftmann@58023  849  with assms have "lcm a b dvd a"  haftmann@58023  850  by (rule_tac dvd_euclidean_size_eq_imp_dvd) (auto simp: lcm_zero)  eberlm@62422  851  hence "b dvd a" by (rule lcm_dvdD2)  wenzelm@60526  852  with \\b dvd a\ show False by contradiction  haftmann@58023  853 qed  haftmann@58023  854 haftmann@58023  855 lemma euclidean_size_lcm_less2:  haftmann@58023  856  assumes "a \ 0" and "\a dvd b"  haftmann@58023  857  shows "euclidean_size b < euclidean_size (lcm a b)"  haftmann@58023  858  using assms euclidean_size_lcm_less1 [of a b] by (simp add: ac_simps)  haftmann@58023  859 haftmann@58023  860 lemma lcm_mult_unit1:  haftmann@60430  861  "is_unit a \ lcm (b * a) c = lcm b c"  haftmann@60690  862  by (rule associated_eqI) (simp_all add: mult_unit_dvd_iff dvd_lcmI1)  haftmann@58023  863 haftmann@58023  864 lemma lcm_mult_unit2:  haftmann@60430  865  "is_unit a \ lcm b (c * a) = lcm b c"  haftmann@60430  866  using lcm_mult_unit1 [of a c b] by (simp add: ac_simps)  haftmann@58023  867 haftmann@58023  868 lemma lcm_div_unit1:  haftmann@60430  869  "is_unit a \ lcm (b div a) c = lcm b c"  haftmann@60433  870  by (erule is_unitE [of _ b]) (simp add: lcm_mult_unit1)  haftmann@58023  871 haftmann@58023  872 lemma lcm_div_unit2:  haftmann@60430  873  "is_unit a \ lcm b (c div a) = lcm b c"  haftmann@60433  874  by (erule is_unitE [of _ c]) (simp add: lcm_mult_unit2)  haftmann@58023  875 haftmann@60634  876 lemma normalize_lcm_left [simp]:  haftmann@60634  877  "lcm (normalize a) b = lcm a b"  haftmann@60634  878 proof (cases "a = 0")  haftmann@60634  879  case True then show ?thesis  haftmann@60634  880  by simp  haftmann@60634  881 next  haftmann@60634  882  case False then have "is_unit (unit_factor a)"  haftmann@60634  883  by simp  haftmann@60634  884  moreover have "normalize a = a div unit_factor a"  haftmann@60634  885  by simp  haftmann@60634  886  ultimately show ?thesis  haftmann@60634  887  by (simp only: lcm_div_unit1)  haftmann@60634  888 qed  haftmann@60634  889 haftmann@60634  890 lemma normalize_lcm_right [simp]:  haftmann@60634  891  "lcm a (normalize b) = lcm a b"  haftmann@60634  892  using normalize_lcm_left [of b a] by (simp add: ac_simps)  haftmann@60634  893 haftmann@58023  894 lemma LcmI:  haftmann@60634  895  assumes "\a. a \ A \ a dvd b" and "\c. (\a. a \ A \ a dvd c) \ b dvd c"  haftmann@60688  896  and "normalize b = b" shows "b = Lcm A"  eberlm@62422  897  by (rule associated_eqI) (auto simp: assms dvd_Lcm intro: Lcm_least)  haftmann@58023  898 haftmann@58023  899 lemma Lcm_subset:  haftmann@58023  900  "A \ B \ Lcm A dvd Lcm B"  haftmann@60634  901  by (blast intro: Lcm_least dvd_Lcm)  haftmann@58023  902 haftmann@58023  903 lemma Lcm_Un:  haftmann@58023  904  "Lcm (A \ B) = lcm (Lcm A) (Lcm B)"  haftmann@58023  905  apply (rule lcmI)  haftmann@58023  906  apply (blast intro: Lcm_subset)  haftmann@58023  907  apply (blast intro: Lcm_subset)  haftmann@60634  908  apply (intro Lcm_least ballI, elim UnE)  haftmann@58023  909  apply (rule dvd_trans, erule dvd_Lcm, assumption)  haftmann@58023  910  apply (rule dvd_trans, erule dvd_Lcm, assumption)  haftmann@58023  911  apply simp  haftmann@58023  912  done  haftmann@58023  913 haftmann@58023  914 lemma Lcm_no_units:  haftmann@60430  915  "Lcm A = Lcm (A - {a. is_unit a})"  haftmann@58023  916 proof -  haftmann@60430  917  have "(A - {a. is_unit a}) \ {a\A. is_unit a} = A" by blast  haftmann@60430  918  hence "Lcm A = lcm (Lcm (A - {a. is_unit a})) (Lcm {a\A. is_unit a})"  haftmann@60634  919  by (simp add: Lcm_Un [symmetric])  haftmann@60430  920  also have "Lcm {a\A. is_unit a} = 1" by (simp add: Lcm_1_iff)  haftmann@58023  921  finally show ?thesis by simp  haftmann@58023  922 qed  haftmann@58023  923 haftmann@62353  924 lemma Lcm_0_iff':  haftmann@60430  925  "Lcm A = 0 \ \(\l. l \ 0 \ (\a\A. a dvd l))"  haftmann@58023  926 proof  haftmann@58023  927  assume "Lcm A = 0"  haftmann@60430  928  show "\(\l. l \ 0 \ (\a\A. a dvd l))"  haftmann@58023  929  proof  haftmann@60430  930  assume ex: "\l. l \ 0 \ (\a\A. a dvd l)"  haftmann@60430  931  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  932  def n \ "LEAST n. \l. l \ 0 \ (\a\A. a dvd l) \ euclidean_size l = n"  haftmann@60430  933  def l \ "SOME l. l \ 0 \ (\a\A. a dvd l) \ euclidean_size l = n"  haftmann@60430  934  have "\l. l \ 0 \ (\a\A. a dvd l) \ euclidean_size l = n"  haftmann@58023  935  apply (subst n_def)  haftmann@58023  936  apply (rule LeastI[of _ "euclidean_size l\<^sub>0"])  haftmann@58023  937  apply (rule exI[of _ l\<^sub>0])  haftmann@58023  938  apply (simp add: l\<^sub>0_props)  haftmann@58023  939  done  haftmann@58023  940  from someI_ex[OF this] have "l \ 0" unfolding l_def by simp_all  haftmann@60634  941  hence "normalize l \ 0" by simp  haftmann@60634  942  also from ex have "normalize l = Lcm A"  haftmann@58023  943  by (simp only: Lcm_Lcm_eucl Lcm_eucl_def n_def l_def if_True Let_def)  wenzelm@60526  944  finally show False using \Lcm A = 0\ by contradiction  haftmann@58023  945  qed  haftmann@58023  946 qed (simp only: Lcm_Lcm_eucl Lcm_eucl_def if_False)  haftmann@58023  947 haftmann@58023  948 lemma Lcm_no_multiple:  haftmann@60430  949  "(\m. m \ 0 \ (\a\A. \a dvd m)) \ Lcm A = 0"  haftmann@58023  950 proof -  haftmann@60430  951  assume "\m. m \ 0 \ (\a\A. \a dvd m)"  haftmann@60430  952  hence "\(\l. l \ 0 \ (\a\A. a dvd l))" by blast  haftmann@58023  953  then show "Lcm A = 0" by (simp only: Lcm_Lcm_eucl Lcm_eucl_def if_False)  haftmann@58023  954 qed  haftmann@58023  955 haftmann@58023  956 lemma Lcm_finite:  haftmann@58023  957  assumes "finite A"  haftmann@58023  958  shows "Lcm A = Finite_Set.fold lcm 1 A"  wenzelm@60526  959  by (induct rule: finite.induct[OF \finite A\])  haftmann@58023  960  (simp_all add: comp_fun_idem.fold_insert_idem[OF comp_fun_idem_lcm])  haftmann@58023  961 haftmann@60431  962 lemma Lcm_set [code_unfold]:  haftmann@58023  963  "Lcm (set xs) = fold lcm xs 1"  haftmann@58023  964  using comp_fun_idem.fold_set_fold[OF comp_fun_idem_lcm] Lcm_finite by (simp add: ac_simps)  haftmann@58023  965 haftmann@58023  966 lemma Lcm_singleton [simp]:  haftmann@60634  967  "Lcm {a} = normalize a"  haftmann@58023  968  by simp  haftmann@58023  969 haftmann@58023  970 lemma Lcm_2 [simp]:  haftmann@58023  971  "Lcm {a,b} = lcm a b"  haftmann@60634  972  by simp  haftmann@58023  973 haftmann@58023  974 lemma Lcm_coprime:  haftmann@58023  975  assumes "finite A" and "A \ {}"  haftmann@58023  976  assumes "\a b. a \ A \ b \ A \ a \ b \ gcd a b = 1"  haftmann@60634  977  shows "Lcm A = normalize (\A)"  haftmann@58023  978 using assms proof (induct rule: finite_ne_induct)  haftmann@58023  979  case (insert a A)  haftmann@58023  980  have "Lcm (insert a A) = lcm a (Lcm A)" by simp  haftmann@60634  981  also from insert have "Lcm A = normalize (\A)" by blast  haftmann@58023  982  also have "lcm a \ = lcm a (\A)" by (cases "\A = 0") (simp_all add: lcm_div_unit2)  haftmann@58023  983  also from insert have "gcd a (\A) = 1" by (subst gcd.commute, intro setprod_coprime) auto  haftmann@60634  984  with insert have "lcm a (\A) = normalize (\(insert a A))"  haftmann@58023  985  by (simp add: lcm_coprime)  haftmann@58023  986  finally show ?case .  haftmann@58023  987 qed simp  haftmann@58023  988   haftmann@58023  989 lemma Lcm_coprime':  haftmann@58023  990  "card A \ 0 \ (\a b. a \ A \ b \ A \ a \ b \ gcd a b = 1)  haftmann@60634  991  \ Lcm A = normalize (\A)"  haftmann@58023  992  by (rule Lcm_coprime) (simp_all add: card_eq_0_iff)  haftmann@58023  993 eberlm@62422  994 lemma unit_factor_Gcd [simp]: "unit_factor (Gcd A) = (if Gcd A = 0 then 0 else 1)"  haftmann@58023  995 proof -  haftmann@60634  996  show "unit_factor (Gcd A) = (if Gcd A = 0 then 0 else 1)"  eberlm@62422  997  by (simp add: Gcd_Lcm unit_factor_Lcm)  haftmann@58023  998 qed  haftmann@58023  999 haftmann@58023  1000 lemma GcdI:  haftmann@60634  1001  assumes "\a. a \ A \ b dvd a" and "\c. (\a. a \ A \ c dvd a) \ c dvd b"  haftmann@60688  1002  and "normalize b = b"  haftmann@60634  1003  shows "b = Gcd A"  eberlm@62422  1004  by (rule associated_eqI) (auto simp: assms Gcd_dvd intro: Gcd_greatest)  haftmann@58023  1005 haftmann@58023  1006 lemma Gcd_1:  haftmann@58023  1007  "1 \ A \ Gcd A = 1"  haftmann@60687  1008  by (auto intro!: Gcd_eq_1_I)  haftmann@58023  1009 haftmann@58023  1010 lemma Gcd_finite:  haftmann@58023  1011  assumes "finite A"  haftmann@58023  1012  shows "Gcd A = Finite_Set.fold gcd 0 A"  wenzelm@60526  1013  by (induct rule: finite.induct[OF \finite A\])  haftmann@58023  1014  (simp_all add: comp_fun_idem.fold_insert_idem[OF comp_fun_idem_gcd])  haftmann@58023  1015 haftmann@60431  1016 lemma Gcd_set [code_unfold]:  haftmann@58023  1017  "Gcd (set xs) = fold gcd xs 0"  haftmann@58023  1018  using comp_fun_idem.fold_set_fold[OF comp_fun_idem_gcd] Gcd_finite by (simp add: ac_simps)  haftmann@58023  1019 haftmann@60634  1020 lemma Gcd_singleton [simp]: "Gcd {a} = normalize a"  haftmann@60687  1021  by simp  haftmann@58023  1022 haftmann@58023  1023 lemma Gcd_2 [simp]: "Gcd {a,b} = gcd a b"  haftmann@60687  1024  by simp  haftmann@60686  1025 eberlm@62422  1026 eberlm@62422  1027 definition pairwise_coprime where  eberlm@62422  1028  "pairwise_coprime A = (\x y. x \ A \ y \ A \ x \ y \ coprime x y)"  eberlm@62422  1029 eberlm@62422  1030 lemma pairwise_coprimeI [intro?]:  eberlm@62422  1031  "(\x y. x \ A \ y \ A \ x \ y \ coprime x y) \ pairwise_coprime A"  eberlm@62422  1032  by (simp add: pairwise_coprime_def)  eberlm@62422  1033 eberlm@62422  1034 lemma pairwise_coprimeD:  eberlm@62422  1035  "pairwise_coprime A \ x \ A \ y \ A \ x \ y \ coprime x y"  eberlm@62422  1036  by (simp add: pairwise_coprime_def)  eberlm@62422  1037 eberlm@62422  1038 lemma pairwise_coprime_subset: "pairwise_coprime A \ B \ A \ pairwise_coprime B"  eberlm@62422  1039  by (force simp: pairwise_coprime_def)  eberlm@62422  1040 haftmann@58023  1041 end  haftmann@58023  1042 wenzelm@60526  1043 text \  haftmann@58023  1044  A Euclidean ring is a Euclidean semiring with additive inverses. It provides a  haftmann@58023  1045  few more lemmas; in particular, Bezout's lemma holds for any Euclidean ring.  wenzelm@60526  1046 \  haftmann@58023  1047 haftmann@58023  1048 class euclidean_ring_gcd = euclidean_semiring_gcd + idom  haftmann@58023  1049 begin  haftmann@58023  1050 haftmann@58023  1051 subclass euclidean_ring ..  haftmann@60439  1052 subclass ring_gcd ..  haftmann@60439  1053 haftmann@60572  1054 lemma euclid_ext_gcd [simp]:  haftmann@60572  1055  "(case euclid_ext a b of (_, _ , t) \ t) = gcd a b"  haftmann@60572  1056  by (induct a b rule: gcd_eucl_induct)  haftmann@60686  1057  (simp_all add: euclid_ext_0 euclid_ext_non_0 ac_simps split: prod.split prod.split_asm)  haftmann@60572  1058 haftmann@60572  1059 lemma euclid_ext_gcd' [simp]:  haftmann@60572  1060  "euclid_ext a b = (r, s, t) \ t = gcd a b"  haftmann@60572  1061  by (insert euclid_ext_gcd[of a b], drule (1) subst, simp)  haftmann@60572  1062   haftmann@60572  1063 lemma euclid_ext'_correct:  haftmann@60572  1064  "fst (euclid_ext' a b) * a + snd (euclid_ext' a b) * b = gcd a b"  haftmann@60572  1065 proof-  haftmann@60572  1066  obtain s t c where "euclid_ext a b = (s,t,c)"  haftmann@60572  1067  by (cases "euclid_ext a b", blast)  haftmann@60572  1068  with euclid_ext_correct[of a b] euclid_ext_gcd[of a b]  haftmann@60572  1069  show ?thesis unfolding euclid_ext'_def by simp  haftmann@60572  1070 qed  haftmann@60572  1071 haftmann@60572  1072 lemma bezout: "\s t. s * a + t * b = gcd a b"  haftmann@60572  1073  using euclid_ext'_correct by blast  haftmann@60572  1074 haftmann@58023  1075 lemma gcd_neg1 [simp]:  haftmann@60430  1076  "gcd (-a) b = gcd a b"  haftmann@59009  1077  by (rule sym, rule gcdI, simp_all add: gcd_greatest)  haftmann@58023  1078 haftmann@58023  1079 lemma gcd_neg2 [simp]:  haftmann@60430  1080  "gcd a (-b) = gcd a b"  haftmann@59009  1081  by (rule sym, rule gcdI, simp_all add: gcd_greatest)  haftmann@58023  1082 haftmann@58023  1083 lemma gcd_neg_numeral_1 [simp]:  haftmann@60430  1084  "gcd (- numeral n) a = gcd (numeral n) a"  haftmann@58023  1085  by (fact gcd_neg1)  haftmann@58023  1086 haftmann@58023  1087 lemma gcd_neg_numeral_2 [simp]:  haftmann@60430  1088  "gcd a (- numeral n) = gcd a (numeral n)"  haftmann@58023  1089  by (fact gcd_neg2)  haftmann@58023  1090 haftmann@58023  1091 lemma gcd_diff1: "gcd (m - n) n = gcd m n"  haftmann@58023  1092  by (subst diff_conv_add_uminus, subst gcd_neg2[symmetric], subst gcd_add1, simp)  haftmann@58023  1093 haftmann@58023  1094 lemma gcd_diff2: "gcd (n - m) n = gcd m n"  haftmann@58023  1095  by (subst gcd_neg1[symmetric], simp only: minus_diff_eq gcd_diff1)  haftmann@58023  1096 haftmann@58023  1097 lemma coprime_minus_one [simp]: "gcd (n - 1) n = 1"  haftmann@58023  1098 proof -  haftmann@58023  1099  have "gcd (n - 1) n = gcd n (n - 1)" by (fact gcd.commute)  haftmann@58023  1100  also have "\ = gcd ((n - 1) + 1) (n - 1)" by simp  haftmann@58023  1101  also have "\ = 1" by (rule coprime_plus_one)  haftmann@58023  1102  finally show ?thesis .  haftmann@58023  1103 qed  haftmann@58023  1104 haftmann@60430  1105 lemma lcm_neg1 [simp]: "lcm (-a) b = lcm a b"  haftmann@58023  1106  by (rule sym, rule lcmI, simp_all add: lcm_least lcm_zero)  haftmann@58023  1107 haftmann@60430  1108 lemma lcm_neg2 [simp]: "lcm a (-b) = lcm a b"  haftmann@58023  1109  by (rule sym, rule lcmI, simp_all add: lcm_least lcm_zero)  haftmann@58023  1110 haftmann@60430  1111 lemma lcm_neg_numeral_1 [simp]: "lcm (- numeral n) a = lcm (numeral n) a"  haftmann@58023  1112  by (fact lcm_neg1)  haftmann@58023  1113 haftmann@60430  1114 lemma lcm_neg_numeral_2 [simp]: "lcm a (- numeral n) = lcm a (numeral n)"  haftmann@58023  1115  by (fact lcm_neg2)  haftmann@58023  1116 haftmann@60572  1117 end  haftmann@58023  1118 haftmann@58023  1119 haftmann@60572  1120 subsection \Typical instances\  haftmann@58023  1121 haftmann@58023  1122 instantiation nat :: euclidean_semiring  haftmann@58023  1123 begin  haftmann@58023  1124 haftmann@58023  1125 definition [simp]:  haftmann@58023  1126  "euclidean_size_nat = (id :: nat \ nat)"  haftmann@58023  1127 haftmann@58023  1128 instance proof  haftmann@59061  1129 qed simp_all  haftmann@58023  1130 haftmann@58023  1131 end  haftmann@58023  1132 eberlm@62422  1133 haftmann@58023  1134 instantiation int :: euclidean_ring  haftmann@58023  1135 begin  haftmann@58023  1136 haftmann@58023  1137 definition [simp]:  haftmann@58023  1138  "euclidean_size_int = (nat \ abs :: int \ nat)"  haftmann@58023  1139 wenzelm@60580  1140 instance  haftmann@60686  1141 by standard (auto simp add: abs_mult nat_mult_distrib split: abs_split)  haftmann@58023  1142 haftmann@58023  1143 end  haftmann@58023  1144 eberlm@62422  1145 haftmann@60572  1146 instantiation poly :: (field) euclidean_ring  haftmann@60571  1147 begin  haftmann@60571  1148 haftmann@60571  1149 definition euclidean_size_poly :: "'a poly \ nat"  eberlm@62422  1150  where "euclidean_size p = (if p = 0 then 0 else 2 ^ degree p)"  haftmann@60571  1151 haftmann@60600  1152 lemma euclidean_size_poly_0 [simp]:  haftmann@60600  1153  "euclidean_size (0::'a poly) = 0"  haftmann@60600  1154  by (simp add: euclidean_size_poly_def)  haftmann@60600  1155 haftmann@60600  1156 lemma euclidean_size_poly_not_0 [simp]:  eberlm@62422  1157  "p \ 0 \ euclidean_size p = 2 ^ degree p"  haftmann@60600  1158  by (simp add: euclidean_size_poly_def)  haftmann@60600  1159 haftmann@60571  1160 instance  haftmann@60600  1161 proof  haftmann@60571  1162  fix p q :: "'a poly"  haftmann@60600  1163  assume "q \ 0"  haftmann@60600  1164  then have "p mod q = 0 \ degree (p mod q) < degree q"  haftmann@60600  1165  by (rule degree_mod_less [of q p])  haftmann@60600  1166  with \q \ 0\ show "euclidean_size (p mod q) < euclidean_size q"  haftmann@60600  1167  by (cases "p mod q = 0") simp_all  haftmann@60571  1168 next  haftmann@60571  1169  fix p q :: "'a poly"  haftmann@60571  1170  assume "q \ 0"  haftmann@60600  1171  from \q \ 0\ have "degree p \ degree (p * q)"  haftmann@60571  1172  by (rule degree_mult_right_le)  haftmann@60600  1173  with \q \ 0\ show "euclidean_size p \ euclidean_size (p * q)"  haftmann@60600  1174  by (cases "p = 0") simp_all  eberlm@62422  1175 qed simp  haftmann@60571  1176 haftmann@58023  1177 end  haftmann@60571  1178 eberlm@62422  1179 eberlm@62422  1180 instance nat :: euclidean_semiring_gcd  eberlm@62422  1181 proof  eberlm@62422  1182  show [simp]: "gcd = (gcd_eucl :: nat \ _)" "Lcm = (Lcm_eucl :: nat set \ _)"  eberlm@62422  1183  by (simp_all add: eq_gcd_euclI eq_Lcm_euclI)  eberlm@62422  1184  show "lcm = (lcm_eucl :: nat \ _)" "Gcd = (Gcd_eucl :: nat set \ _)"  eberlm@62422  1185  by (intro ext, simp add: lcm_eucl_def lcm_nat_def Gcd_nat_def Gcd_eucl_def)+  eberlm@62422  1186 qed  eberlm@62422  1187 eberlm@62422  1188 instance int :: euclidean_ring_gcd  eberlm@62422  1189 proof  eberlm@62422  1190  show [simp]: "gcd = (gcd_eucl :: int \ _)" "Lcm = (Lcm_eucl :: int set \ _)"  eberlm@62422  1191  by (simp_all add: eq_gcd_euclI eq_Lcm_euclI)  eberlm@62422  1192  show "lcm = (lcm_eucl :: int \ _)" "Gcd = (Gcd_eucl :: int set \ _)"  eberlm@62422  1193  by (intro ext, simp add: lcm_eucl_def lcm_altdef_int  eberlm@62422  1194  semiring_Gcd_class.Gcd_Lcm Gcd_eucl_def abs_mult)+  eberlm@62422  1195 qed  eberlm@62422  1196 eberlm@62422  1197 eberlm@62422  1198 instantiation poly :: (field) euclidean_ring_gcd  eberlm@62422  1199 begin  eberlm@62422  1200 eberlm@62422  1201 definition gcd_poly :: "'a poly \ 'a poly \ 'a poly" where  eberlm@62422  1202  "gcd_poly = gcd_eucl"  eberlm@62422  1203   eberlm@62422  1204 definition lcm_poly :: "'a poly \ 'a poly \ 'a poly" where  eberlm@62422  1205  "lcm_poly = lcm_eucl"  eberlm@62422  1206   eberlm@62422  1207 definition Gcd_poly :: "'a poly set \ 'a poly" where  eberlm@62422  1208  "Gcd_poly = Gcd_eucl"  eberlm@62422  1209   eberlm@62422  1210 definition Lcm_poly :: "'a poly set \ 'a poly" where  eberlm@62422  1211  "Lcm_poly = Lcm_eucl"  eberlm@62422  1212 eberlm@62422  1213 instance by standard (simp_all only: gcd_poly_def lcm_poly_def Gcd_poly_def Lcm_poly_def)  eberlm@62422  1214 end  haftmann@60687  1215 haftmann@60571  1216 end