src/HOL/Number_Theory/Euclidean_Algorithm.thy
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 haftmann@58023  1 (* Author: Manuel Eberl *)  haftmann@58023  2 wenzelm@60526  3 section \Abstract euclidean algorithm\  haftmann@58023  4 haftmann@58023  5 theory Euclidean_Algorithm  eberlm@62429  6 imports "~~/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 wenzelm@63167  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 eberlm@62428  102 declare Lcm_eucl_def Gcd_eucl_def [code del]  eberlm@62428  103 haftmann@60572  104 lemma gcd_eucl_0:  haftmann@60634  105  "gcd_eucl a 0 = normalize a"  haftmann@60572  106  by (simp add: gcd_eucl.simps [of a 0])  haftmann@60572  107 haftmann@60572  108 lemma gcd_eucl_0_left:  haftmann@60634  109  "gcd_eucl 0 a = normalize a"  haftmann@60600  110  by (simp_all add: gcd_eucl_0 gcd_eucl.simps [of 0 a])  haftmann@60572  111 haftmann@60572  112 lemma gcd_eucl_non_0:  haftmann@60572  113  "b \ 0 \ gcd_eucl a b = gcd_eucl b (a mod b)"  haftmann@60600  114  by (simp add: gcd_eucl.simps [of a b] gcd_eucl.simps [of b 0])  haftmann@60572  115 eberlm@62422  116 lemma gcd_eucl_dvd1 [iff]: "gcd_eucl a b dvd a"  eberlm@62422  117  and gcd_eucl_dvd2 [iff]: "gcd_eucl a b dvd b"  eberlm@62422  118  by (induct a b rule: gcd_eucl_induct)  eberlm@62422  119  (simp_all add: gcd_eucl_0 gcd_eucl_non_0 dvd_mod_iff)  eberlm@62422  120 eberlm@62422  121 lemma normalize_gcd_eucl [simp]:  eberlm@62422  122  "normalize (gcd_eucl a b) = gcd_eucl a b"  eberlm@62422  123  by (induct a b rule: gcd_eucl_induct) (simp_all add: gcd_eucl_0 gcd_eucl_non_0)  eberlm@62422  124   eberlm@62422  125 lemma gcd_eucl_greatest:  eberlm@62422  126  fixes k a b :: 'a  eberlm@62422  127  shows "k dvd a \ k dvd b \ k dvd gcd_eucl a b"  eberlm@62422  128 proof (induct a b rule: gcd_eucl_induct)  eberlm@62422  129  case (zero a) from zero(1) show ?case by (rule dvd_trans) (simp add: gcd_eucl_0)  eberlm@62422  130 next  eberlm@62422  131  case (mod a b)  eberlm@62422  132  then show ?case  eberlm@62422  133  by (simp add: gcd_eucl_non_0 dvd_mod_iff)  eberlm@62422  134 qed  eberlm@62422  135 eberlm@62422  136 lemma eq_gcd_euclI:  eberlm@62422  137  fixes gcd :: "'a \ 'a \ 'a"  eberlm@62422  138  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  139  "\a b k. k dvd a \ k dvd b \ k dvd gcd a b"  eberlm@62422  140  shows "gcd = gcd_eucl"  eberlm@62422  141  by (intro ext, rule associated_eqI) (simp_all add: gcd_eucl_greatest assms)  eberlm@62422  142 eberlm@62422  143 lemma gcd_eucl_zero [simp]:  eberlm@62422  144  "gcd_eucl a b = 0 \ a = 0 \ b = 0"  eberlm@62422  145  by (metis dvd_0_left dvd_refl gcd_eucl_dvd1 gcd_eucl_dvd2 gcd_eucl_greatest)+  eberlm@62422  146 eberlm@62422  147   eberlm@62422  148 lemma dvd_Lcm_eucl [simp]: "a \ A \ a dvd Lcm_eucl A"  eberlm@62422  149  and Lcm_eucl_least: "(\a. a \ A \ a dvd b) \ Lcm_eucl A dvd b"  eberlm@62422  150  and unit_factor_Lcm_eucl [simp]:  eberlm@62422  151  "unit_factor (Lcm_eucl A) = (if Lcm_eucl A = 0 then 0 else 1)"  eberlm@62422  152 proof -  eberlm@62422  153  have "(\a\A. a dvd Lcm_eucl A) \ (\l'. (\a\A. a dvd l') \ Lcm_eucl A dvd l') \  eberlm@62422  154  unit_factor (Lcm_eucl A) = (if Lcm_eucl A = 0 then 0 else 1)" (is ?thesis)  eberlm@62422  155  proof (cases "\l. l \ 0 \ (\a\A. a dvd l)")  eberlm@62422  156  case False  eberlm@62422  157  hence "Lcm_eucl A = 0" by (auto simp: Lcm_eucl_def)  eberlm@62422  158  with False show ?thesis by auto  eberlm@62422  159  next  eberlm@62422  160  case True  eberlm@62422  161  then obtain l\<^sub>0 where l\<^sub>0_props: "l\<^sub>0 \ 0 \ (\a\A. a dvd l\<^sub>0)" by blast  wenzelm@63040  162  define n where "n = (LEAST n. \l. l \ 0 \ (\a\A. a dvd l) \ euclidean_size l = n)"  wenzelm@63040  163  define l where "l = (SOME l. l \ 0 \ (\a\A. a dvd l) \ euclidean_size l = n)"  eberlm@62422  164  have "\l. l \ 0 \ (\a\A. a dvd l) \ euclidean_size l = n"  eberlm@62422  165  apply (subst n_def)  eberlm@62422  166  apply (rule LeastI[of _ "euclidean_size l\<^sub>0"])  eberlm@62422  167  apply (rule exI[of _ l\<^sub>0])  eberlm@62422  168  apply (simp add: l\<^sub>0_props)  eberlm@62422  169  done  eberlm@62422  170  from someI_ex[OF this] have "l \ 0" and "\a\A. a dvd l" and "euclidean_size l = n"  eberlm@62422  171  unfolding l_def by simp_all  eberlm@62422  172  {  eberlm@62422  173  fix l' assume "\a\A. a dvd l'"  eberlm@62422  174  with \\a\A. a dvd l\ have "\a\A. a dvd gcd_eucl l l'" by (auto intro: gcd_eucl_greatest)  eberlm@62422  175  moreover from \l \ 0\ have "gcd_eucl l l' \ 0" by simp  eberlm@62422  176  ultimately have "\b. b \ 0 \ (\a\A. a dvd b) \  eberlm@62422  177  euclidean_size b = euclidean_size (gcd_eucl l l')"  eberlm@62422  178  by (intro exI[of _ "gcd_eucl l l'"], auto)  eberlm@62422  179  hence "euclidean_size (gcd_eucl l l') \ n" by (subst n_def) (rule Least_le)  eberlm@62422  180  moreover have "euclidean_size (gcd_eucl l l') \ n"  eberlm@62422  181  proof -  eberlm@62422  182  have "gcd_eucl l l' dvd l" by simp  eberlm@62422  183  then obtain a where "l = gcd_eucl l l' * a" unfolding dvd_def by blast  eberlm@62422  184  with \l \ 0\ have "a \ 0" by auto  eberlm@62422  185  hence "euclidean_size (gcd_eucl l l') \ euclidean_size (gcd_eucl l l' * a)"  eberlm@62422  186  by (rule size_mult_mono)  eberlm@62422  187  also have "gcd_eucl l l' * a = l" using \l = gcd_eucl l l' * a\ ..  eberlm@62422  188  also note \euclidean_size l = n\  eberlm@62422  189  finally show "euclidean_size (gcd_eucl l l') \ n" .  eberlm@62422  190  qed  eberlm@62422  191  ultimately have *: "euclidean_size l = euclidean_size (gcd_eucl l l')"  eberlm@62422  192  by (intro le_antisym, simp_all add: \euclidean_size l = n\)  eberlm@62422  193  from \l \ 0\ have "l dvd gcd_eucl l l'"  eberlm@62422  194  by (rule dvd_euclidean_size_eq_imp_dvd) (auto simp add: *)  eberlm@62422  195  hence "l dvd l'" by (rule dvd_trans[OF _ gcd_eucl_dvd2])  eberlm@62422  196  }  eberlm@62422  197 eberlm@62422  198  with \(\a\A. a dvd l)\ and unit_factor_is_unit[OF \l \ 0\] and \l \ 0\  eberlm@62422  199  have "(\a\A. a dvd normalize l) \  eberlm@62422  200  (\l'. (\a\A. a dvd l') \ normalize l dvd l') \  eberlm@62422  201  unit_factor (normalize l) =  eberlm@62422  202  (if normalize l = 0 then 0 else 1)"  eberlm@62422  203  by (auto simp: unit_simps)  eberlm@62422  204  also from True have "normalize l = Lcm_eucl A"  eberlm@62422  205  by (simp add: Lcm_eucl_def Let_def n_def l_def)  eberlm@62422  206  finally show ?thesis .  eberlm@62422  207  qed  eberlm@62422  208  note A = this  eberlm@62422  209 eberlm@62422  210  {fix a assume "a \ A" then show "a dvd Lcm_eucl A" using A by blast}  eberlm@62422  211  {fix b assume "\a. a \ A \ a dvd b" then show "Lcm_eucl A dvd b" using A by blast}  eberlm@62422  212  from A show "unit_factor (Lcm_eucl A) = (if Lcm_eucl A = 0 then 0 else 1)" by blast  eberlm@62422  213 qed  eberlm@62422  214   eberlm@62422  215 lemma normalize_Lcm_eucl [simp]:  eberlm@62422  216  "normalize (Lcm_eucl A) = Lcm_eucl A"  eberlm@62422  217 proof (cases "Lcm_eucl A = 0")  eberlm@62422  218  case True then show ?thesis by simp  eberlm@62422  219 next  eberlm@62422  220  case False  eberlm@62422  221  have "unit_factor (Lcm_eucl A) * normalize (Lcm_eucl A) = Lcm_eucl A"  eberlm@62422  222  by (fact unit_factor_mult_normalize)  eberlm@62422  223  with False show ?thesis by simp  eberlm@62422  224 qed  eberlm@62422  225 eberlm@62422  226 lemma eq_Lcm_euclI:  eberlm@62422  227  fixes lcm :: "'a set \ 'a"  eberlm@62422  228  assumes "\A a. a \ A \ a dvd lcm A" and "\A c. (\a. a \ A \ a dvd c) \ lcm A dvd c"  eberlm@62422  229  "\A. normalize (lcm A) = lcm A" shows "lcm = Lcm_eucl"  eberlm@62422  230  by (intro ext, rule associated_eqI) (auto simp: assms intro: Lcm_eucl_least)  eberlm@62422  231 haftmann@58023  232 end  haftmann@58023  233 haftmann@60598  234 class euclidean_ring = euclidean_semiring + idom  haftmann@60598  235 begin  haftmann@60598  236 eberlm@62457  237 subclass ring_div ..  eberlm@62457  238 eberlm@62442  239 function euclid_ext_aux :: "'a \ _" where  eberlm@62442  240  "euclid_ext_aux r' r s' s t' t = (  eberlm@62442  241  if r = 0 then let c = 1 div unit_factor r' in (s' * c, t' * c, normalize r')  eberlm@62442  242  else let q = r' div r  eberlm@62442  243  in euclid_ext_aux r (r' mod r) s (s' - q * s) t (t' - q * t))"  eberlm@62442  244 by auto  eberlm@62442  245 termination by (relation "measure (\(_,b,_,_,_,_). euclidean_size b)") (simp_all add: mod_size_less)  eberlm@62442  246 eberlm@62442  247 declare euclid_ext_aux.simps [simp del]  haftmann@60598  248 eberlm@62442  249 lemma euclid_ext_aux_correct:  eberlm@62442  250  assumes "gcd_eucl r' r = gcd_eucl x y"  eberlm@62442  251  assumes "s' * x + t' * y = r'"  eberlm@62442  252  assumes "s * x + t * y = r"  eberlm@62442  253  shows "case euclid_ext_aux r' r s' s t' t of (a,b,c) \  eberlm@62442  254  a * x + b * y = c \ c = gcd_eucl x y" (is "?P (euclid_ext_aux r' r s' s t' t)")  eberlm@62442  255 using assms  eberlm@62442  256 proof (induction r' r s' s t' t rule: euclid_ext_aux.induct)  eberlm@62442  257  case (1 r' r s' s t' t)  eberlm@62442  258  show ?case  eberlm@62442  259  proof (cases "r = 0")  eberlm@62442  260  case True  eberlm@62442  261  hence "euclid_ext_aux r' r s' s t' t =  eberlm@62442  262  (s' div unit_factor r', t' div unit_factor r', normalize r')"  eberlm@62442  263  by (subst euclid_ext_aux.simps) (simp add: Let_def)  eberlm@62442  264  also have "?P \"  eberlm@62442  265  proof safe  eberlm@62442  266  have "s' div unit_factor r' * x + t' div unit_factor r' * y =  eberlm@62442  267  (s' * x + t' * y) div unit_factor r'"  eberlm@62442  268  by (cases "r' = 0") (simp_all add: unit_div_commute)  eberlm@62442  269  also have "s' * x + t' * y = r'" by fact  eberlm@62442  270  also have "\ div unit_factor r' = normalize r'" by simp  eberlm@62442  271  finally show "s' div unit_factor r' * x + t' div unit_factor r' * y = normalize r'" .  eberlm@62442  272  next  eberlm@62442  273  from "1.prems" True show "normalize r' = gcd_eucl x y" by (simp add: gcd_eucl_0)  eberlm@62442  274  qed  eberlm@62442  275  finally show ?thesis .  eberlm@62442  276  next  eberlm@62442  277  case False  eberlm@62442  278  hence "euclid_ext_aux r' r s' s t' t =  eberlm@62442  279  euclid_ext_aux r (r' mod r) s (s' - r' div r * s) t (t' - r' div r * t)"  eberlm@62442  280  by (subst euclid_ext_aux.simps) (simp add: Let_def)  eberlm@62442  281  also from "1.prems" False have "?P \"  eberlm@62442  282  proof (intro "1.IH")  eberlm@62442  283  have "(s' - r' div r * s) * x + (t' - r' div r * t) * y =  eberlm@62442  284  (s' * x + t' * y) - r' div r * (s * x + t * y)" by (simp add: algebra_simps)  eberlm@62442  285  also have "s' * x + t' * y = r'" by fact  eberlm@62442  286  also have "s * x + t * y = r" by fact  eberlm@62442  287  also have "r' - r' div r * r = r' mod r" using mod_div_equality[of r' r]  eberlm@62442  288  by (simp add: algebra_simps)  eberlm@62442  289  finally show "(s' - r' div r * s) * x + (t' - r' div r * t) * y = r' mod r" .  eberlm@62442  290  qed (auto simp: gcd_eucl_non_0 algebra_simps div_mod_equality')  eberlm@62442  291  finally show ?thesis .  eberlm@62442  292  qed  eberlm@62442  293 qed  eberlm@62442  294 eberlm@62442  295 definition euclid_ext where  eberlm@62442  296  "euclid_ext a b = euclid_ext_aux a b 1 0 0 1"  haftmann@60598  297 haftmann@60598  298 lemma euclid_ext_0:  haftmann@60634  299  "euclid_ext a 0 = (1 div unit_factor a, 0, normalize a)"  eberlm@62442  300  by (simp add: euclid_ext_def euclid_ext_aux.simps)  haftmann@60598  301 haftmann@60598  302 lemma euclid_ext_left_0:  haftmann@60634  303  "euclid_ext 0 a = (0, 1 div unit_factor a, normalize a)"  eberlm@62442  304  by (simp add: euclid_ext_def euclid_ext_aux.simps)  haftmann@60598  305 eberlm@62442  306 lemma euclid_ext_correct':  eberlm@62442  307  "case euclid_ext x y of (a,b,c) \ a * x + b * y = c \ c = gcd_eucl x y"  eberlm@62442  308  unfolding euclid_ext_def by (rule euclid_ext_aux_correct) simp_all  haftmann@60598  309 eberlm@62457  310 lemma euclid_ext_gcd_eucl:  eberlm@62457  311  "(case euclid_ext x y of (a,b,c) \ c) = gcd_eucl x y"  eberlm@62457  312  using euclid_ext_correct'[of x y] by (simp add: case_prod_unfold)  eberlm@62457  313 eberlm@62442  314 definition euclid_ext' where  eberlm@62442  315  "euclid_ext' x y = (case euclid_ext x y of (a, b, _) \ (a, b))"  haftmann@60598  316 eberlm@62442  317 lemma euclid_ext'_correct':  eberlm@62442  318  "case euclid_ext' x y of (a,b) \ a * x + b * y = gcd_eucl x y"  eberlm@62442  319  using euclid_ext_correct'[of x y] by (simp add: case_prod_unfold euclid_ext'_def)  haftmann@60598  320 haftmann@60634  321 lemma euclid_ext'_0: "euclid_ext' a 0 = (1 div unit_factor a, 0)"  haftmann@60598  322  by (simp add: euclid_ext'_def euclid_ext_0)  haftmann@60598  323 haftmann@60634  324 lemma euclid_ext'_left_0: "euclid_ext' 0 a = (0, 1 div unit_factor a)"  haftmann@60598  325  by (simp add: euclid_ext'_def euclid_ext_left_0)  haftmann@60598  326 haftmann@60598  327 end  haftmann@60598  328 haftmann@58023  329 class euclidean_semiring_gcd = euclidean_semiring + gcd + Gcd +  haftmann@58023  330  assumes gcd_gcd_eucl: "gcd = gcd_eucl" and lcm_lcm_eucl: "lcm = lcm_eucl"  haftmann@58023  331  assumes Gcd_Gcd_eucl: "Gcd = Gcd_eucl" and Lcm_Lcm_eucl: "Lcm = Lcm_eucl"  haftmann@58023  332 begin  haftmann@58023  333 eberlm@62422  334 subclass semiring_gcd  eberlm@62422  335  by standard (simp_all add: gcd_gcd_eucl gcd_eucl_greatest lcm_lcm_eucl lcm_eucl_def)  haftmann@58023  336 eberlm@62422  337 subclass semiring_Gcd  eberlm@62422  338  by standard (auto simp: Gcd_Gcd_eucl Lcm_Lcm_eucl Gcd_eucl_def intro: Lcm_eucl_least)  eberlm@62422  339   haftmann@58023  340 lemma gcd_non_0:  haftmann@60430  341  "b \ 0 \ gcd a b = gcd b (a mod b)"  haftmann@60572  342  unfolding gcd_gcd_eucl by (fact gcd_eucl_non_0)  haftmann@58023  343 eberlm@62422  344 lemmas gcd_0 = gcd_0_right  eberlm@62422  345 lemmas dvd_gcd_iff = gcd_greatest_iff  haftmann@58023  346 lemmas gcd_greatest_iff = dvd_gcd_iff  haftmann@58023  347 haftmann@58023  348 lemma gcd_mod1 [simp]:  haftmann@60430  349  "gcd (a mod b) b = gcd a b"  haftmann@58023  350  by (rule gcdI, metis dvd_mod_iff gcd_dvd1 gcd_dvd2, simp_all add: gcd_greatest dvd_mod_iff)  haftmann@58023  351 haftmann@58023  352 lemma gcd_mod2 [simp]:  haftmann@60430  353  "gcd a (b mod a) = gcd a b"  haftmann@58023  354  by (rule gcdI, simp, metis dvd_mod_iff gcd_dvd1 gcd_dvd2, simp_all add: gcd_greatest dvd_mod_iff)  haftmann@58023  355   haftmann@58023  356 lemma euclidean_size_gcd_le1 [simp]:  haftmann@58023  357  assumes "a \ 0"  haftmann@58023  358  shows "euclidean_size (gcd a b) \ euclidean_size a"  haftmann@58023  359 proof -  haftmann@58023  360  have "gcd a b dvd a" by (rule gcd_dvd1)  haftmann@58023  361  then obtain c where A: "a = gcd a b * c" unfolding dvd_def by blast  wenzelm@60526  362  with \a \ 0\ show ?thesis by (subst (2) A, intro size_mult_mono) auto  haftmann@58023  363 qed  haftmann@58023  364 haftmann@58023  365 lemma euclidean_size_gcd_le2 [simp]:  haftmann@58023  366  "b \ 0 \ euclidean_size (gcd a b) \ euclidean_size b"  haftmann@58023  367  by (subst gcd.commute, rule euclidean_size_gcd_le1)  haftmann@58023  368 haftmann@58023  369 lemma euclidean_size_gcd_less1:  haftmann@58023  370  assumes "a \ 0" and "\a dvd b"  haftmann@58023  371  shows "euclidean_size (gcd a b) < euclidean_size a"  haftmann@58023  372 proof (rule ccontr)  haftmann@58023  373  assume "\euclidean_size (gcd a b) < euclidean_size a"  eberlm@62422  374  with \a \ 0\ have A: "euclidean_size (gcd a b) = euclidean_size a"  haftmann@58023  375  by (intro le_antisym, simp_all)  eberlm@62422  376  have "a dvd gcd a b"  eberlm@62422  377  by (rule dvd_euclidean_size_eq_imp_dvd) (simp_all add: assms A)  eberlm@62422  378  hence "a dvd b" using dvd_gcdD2 by blast  wenzelm@60526  379  with \\a dvd b\ show False by contradiction  haftmann@58023  380 qed  haftmann@58023  381 haftmann@58023  382 lemma euclidean_size_gcd_less2:  haftmann@58023  383  assumes "b \ 0" and "\b dvd a"  haftmann@58023  384  shows "euclidean_size (gcd a b) < euclidean_size b"  haftmann@58023  385  using assms by (subst gcd.commute, rule euclidean_size_gcd_less1)  haftmann@58023  386 haftmann@58023  387 lemma euclidean_size_lcm_le1:  haftmann@58023  388  assumes "a \ 0" and "b \ 0"  haftmann@58023  389  shows "euclidean_size a \ euclidean_size (lcm a b)"  haftmann@58023  390 proof -  haftmann@60690  391  have "a dvd lcm a b" by (rule dvd_lcm1)  haftmann@60690  392  then obtain c where A: "lcm a b = a * c" ..  eberlm@62429  393  with \a \ 0\ and \b \ 0\ have "c \ 0" by (auto simp: lcm_eq_0_iff)  haftmann@58023  394  then show ?thesis by (subst A, intro size_mult_mono)  haftmann@58023  395 qed  haftmann@58023  396 haftmann@58023  397 lemma euclidean_size_lcm_le2:  haftmann@58023  398  "a \ 0 \ b \ 0 \ euclidean_size b \ euclidean_size (lcm a b)"  haftmann@58023  399  using euclidean_size_lcm_le1 [of b a] by (simp add: ac_simps)  haftmann@58023  400 haftmann@58023  401 lemma euclidean_size_lcm_less1:  haftmann@58023  402  assumes "b \ 0" and "\b dvd a"  haftmann@58023  403  shows "euclidean_size a < euclidean_size (lcm a b)"  haftmann@58023  404 proof (rule ccontr)  haftmann@58023  405  from assms have "a \ 0" by auto  haftmann@58023  406  assume "\euclidean_size a < euclidean_size (lcm a b)"  wenzelm@60526  407  with \a \ 0\ and \b \ 0\ have "euclidean_size (lcm a b) = euclidean_size a"  haftmann@58023  408  by (intro le_antisym, simp, intro euclidean_size_lcm_le1)  haftmann@58023  409  with assms have "lcm a b dvd a"  eberlm@62429  410  by (rule_tac dvd_euclidean_size_eq_imp_dvd) (auto simp: lcm_eq_0_iff)  eberlm@62422  411  hence "b dvd a" by (rule lcm_dvdD2)  wenzelm@60526  412  with \\b dvd a\ show False by contradiction  haftmann@58023  413 qed  haftmann@58023  414 haftmann@58023  415 lemma euclidean_size_lcm_less2:  haftmann@58023  416  assumes "a \ 0" and "\a dvd b"  haftmann@58023  417  shows "euclidean_size b < euclidean_size (lcm a b)"  haftmann@58023  418  using assms euclidean_size_lcm_less1 [of a b] by (simp add: ac_simps)  haftmann@58023  419 eberlm@62428  420 lemma Lcm_eucl_set [code]:  eberlm@62428  421  "Lcm_eucl (set xs) = foldl lcm_eucl 1 xs"  eberlm@62428  422  by (simp add: Lcm_Lcm_eucl [symmetric] lcm_lcm_eucl Lcm_set)  haftmann@58023  423 eberlm@62428  424 lemma Gcd_eucl_set [code]:  eberlm@62428  425  "Gcd_eucl (set xs) = foldl gcd_eucl 0 xs"  eberlm@62428  426  by (simp add: Gcd_Gcd_eucl [symmetric] gcd_gcd_eucl Gcd_set)  haftmann@58023  427 haftmann@58023  428 end  haftmann@58023  429 wenzelm@60526  430 text \  haftmann@58023  431  A Euclidean ring is a Euclidean semiring with additive inverses. It provides a  haftmann@58023  432  few more lemmas; in particular, Bezout's lemma holds for any Euclidean ring.  wenzelm@60526  433 \  haftmann@58023  434 haftmann@58023  435 class euclidean_ring_gcd = euclidean_semiring_gcd + idom  haftmann@58023  436 begin  haftmann@58023  437 haftmann@58023  438 subclass euclidean_ring ..  haftmann@60439  439 subclass ring_gcd ..  haftmann@60439  440 haftmann@60572  441 lemma euclid_ext_gcd [simp]:  haftmann@60572  442  "(case euclid_ext a b of (_, _ , t) \ t) = gcd a b"  eberlm@62442  443  using euclid_ext_correct'[of a b] by (simp add: case_prod_unfold Let_def gcd_gcd_eucl)  haftmann@60572  444 haftmann@60572  445 lemma euclid_ext_gcd' [simp]:  haftmann@60572  446  "euclid_ext a b = (r, s, t) \ t = gcd a b"  haftmann@60572  447  by (insert euclid_ext_gcd[of a b], drule (1) subst, simp)  eberlm@62442  448 eberlm@62442  449 lemma euclid_ext_correct:  eberlm@62442  450  "case euclid_ext x y of (a,b,c) \ a * x + b * y = c \ c = gcd x y"  eberlm@62442  451  using euclid_ext_correct'[of x y]  eberlm@62442  452  by (simp add: gcd_gcd_eucl case_prod_unfold)  haftmann@60572  453   haftmann@60572  454 lemma euclid_ext'_correct:  haftmann@60572  455  "fst (euclid_ext' a b) * a + snd (euclid_ext' a b) * b = gcd a b"  eberlm@62442  456  using euclid_ext_correct'[of a b]  eberlm@62442  457  by (simp add: gcd_gcd_eucl case_prod_unfold euclid_ext'_def)  haftmann@60572  458 haftmann@60572  459 lemma bezout: "\s t. s * a + t * b = gcd a b"  haftmann@60572  460  using euclid_ext'_correct by blast  haftmann@60572  461 haftmann@60572  462 end  haftmann@58023  463 haftmann@58023  464 haftmann@60572  465 subsection \Typical instances\  haftmann@58023  466 haftmann@58023  467 instantiation nat :: euclidean_semiring  haftmann@58023  468 begin  haftmann@58023  469 haftmann@58023  470 definition [simp]:  haftmann@58023  471  "euclidean_size_nat = (id :: nat \ nat)"  haftmann@58023  472 haftmann@58023  473 instance proof  haftmann@59061  474 qed simp_all  haftmann@58023  475 haftmann@58023  476 end  haftmann@58023  477 eberlm@62422  478 haftmann@58023  479 instantiation int :: euclidean_ring  haftmann@58023  480 begin  haftmann@58023  481 haftmann@58023  482 definition [simp]:  haftmann@58023  483  "euclidean_size_int = (nat \ abs :: int \ nat)"  haftmann@58023  484 wenzelm@60580  485 instance  haftmann@60686  486 by standard (auto simp add: abs_mult nat_mult_distrib split: abs_split)  haftmann@58023  487 haftmann@58023  488 end  haftmann@58023  489 eberlm@62422  490 haftmann@60572  491 instantiation poly :: (field) euclidean_ring  haftmann@60571  492 begin  haftmann@60571  493 haftmann@60571  494 definition euclidean_size_poly :: "'a poly \ nat"  eberlm@62422  495  where "euclidean_size p = (if p = 0 then 0 else 2 ^ degree p)"  haftmann@60571  496 haftmann@60600  497 lemma euclidean_size_poly_0 [simp]:  haftmann@60600  498  "euclidean_size (0::'a poly) = 0"  haftmann@60600  499  by (simp add: euclidean_size_poly_def)  haftmann@60600  500 haftmann@60600  501 lemma euclidean_size_poly_not_0 [simp]:  eberlm@62422  502  "p \ 0 \ euclidean_size p = 2 ^ degree p"  haftmann@60600  503  by (simp add: euclidean_size_poly_def)  haftmann@60600  504 haftmann@60571  505 instance  haftmann@60600  506 proof  haftmann@60571  507  fix p q :: "'a poly"  haftmann@60600  508  assume "q \ 0"  haftmann@60600  509  then have "p mod q = 0 \ degree (p mod q) < degree q"  haftmann@60600  510  by (rule degree_mod_less [of q p])  haftmann@60600  511  with \q \ 0\ show "euclidean_size (p mod q) < euclidean_size q"  haftmann@60600  512  by (cases "p mod q = 0") simp_all  haftmann@60571  513 next  haftmann@60571  514  fix p q :: "'a poly"  haftmann@60571  515  assume "q \ 0"  haftmann@60600  516  from \q \ 0\ have "degree p \ degree (p * q)"  haftmann@60571  517  by (rule degree_mult_right_le)  haftmann@60600  518  with \q \ 0\ show "euclidean_size p \ euclidean_size (p * q)"  haftmann@60600  519  by (cases "p = 0") simp_all  eberlm@62422  520 qed simp  haftmann@60571  521 haftmann@58023  522 end  haftmann@60571  523 eberlm@62422  524 eberlm@62422  525 instance nat :: euclidean_semiring_gcd  eberlm@62422  526 proof  eberlm@62422  527  show [simp]: "gcd = (gcd_eucl :: nat \ _)" "Lcm = (Lcm_eucl :: nat set \ _)"  eberlm@62422  528  by (simp_all add: eq_gcd_euclI eq_Lcm_euclI)  eberlm@62422  529  show "lcm = (lcm_eucl :: nat \ _)" "Gcd = (Gcd_eucl :: nat set \ _)"  eberlm@62422  530  by (intro ext, simp add: lcm_eucl_def lcm_nat_def Gcd_nat_def Gcd_eucl_def)+  eberlm@62422  531 qed  eberlm@62422  532 eberlm@62422  533 instance int :: euclidean_ring_gcd  eberlm@62422  534 proof  eberlm@62422  535  show [simp]: "gcd = (gcd_eucl :: int \ _)" "Lcm = (Lcm_eucl :: int set \ _)"  eberlm@62422  536  by (simp_all add: eq_gcd_euclI eq_Lcm_euclI)  eberlm@62422  537  show "lcm = (lcm_eucl :: int \ _)" "Gcd = (Gcd_eucl :: int set \ _)"  eberlm@62422  538  by (intro ext, simp add: lcm_eucl_def lcm_altdef_int  eberlm@62422  539  semiring_Gcd_class.Gcd_Lcm Gcd_eucl_def abs_mult)+  eberlm@62422  540 qed  eberlm@62422  541 eberlm@62422  542 eberlm@62422  543 instantiation poly :: (field) euclidean_ring_gcd  eberlm@62422  544 begin  eberlm@62422  545 eberlm@62422  546 definition gcd_poly :: "'a poly \ 'a poly \ 'a poly" where  eberlm@62422  547  "gcd_poly = gcd_eucl"  eberlm@62422  548   eberlm@62422  549 definition lcm_poly :: "'a poly \ 'a poly \ 'a poly" where  eberlm@62422  550  "lcm_poly = lcm_eucl"  eberlm@62422  551   eberlm@62422  552 definition Gcd_poly :: "'a poly set \ 'a poly" where  eberlm@62422  553  "Gcd_poly = Gcd_eucl"  eberlm@62422  554   eberlm@62422  555 definition Lcm_poly :: "'a poly set \ 'a poly" where  eberlm@62422  556  "Lcm_poly = Lcm_eucl"  eberlm@62422  557 eberlm@62422  558 instance by standard (simp_all only: gcd_poly_def lcm_poly_def Gcd_poly_def Lcm_poly_def)  eberlm@62422  559 end  haftmann@60687  560 eberlm@62425  561 lemma poly_gcd_monic:  eberlm@62425  562  "lead_coeff (gcd x y) = (if x = 0 \ y = 0 then 0 else 1)"  eberlm@62425  563  using unit_factor_gcd[of x y]  eberlm@62425  564  by (simp add: unit_factor_poly_def monom_0 one_poly_def lead_coeff_def split: if_split_asm)  eberlm@62425  565 eberlm@62425  566 lemma poly_dvd_antisym:  eberlm@62425  567  fixes p q :: "'a::idom poly"  eberlm@62425  568  assumes coeff: "coeff p (degree p) = coeff q (degree q)"  eberlm@62425  569  assumes dvd1: "p dvd q" and dvd2: "q dvd p" shows "p = q"  eberlm@62425  570 proof (cases "p = 0")  eberlm@62425  571  case True with coeff show "p = q" by simp  eberlm@62425  572 next  eberlm@62425  573  case False with coeff have "q \ 0" by auto  eberlm@62425  574  have degree: "degree p = degree q"  eberlm@62425  575  using \p dvd q\ \q dvd p\ \p \ 0\ \q \ 0\  eberlm@62425  576  by (intro order_antisym dvd_imp_degree_le)  eberlm@62425  577 eberlm@62425  578  from \p dvd q\ obtain a where a: "q = p * a" ..  eberlm@62425  579  with \q \ 0\ have "a \ 0" by auto  eberlm@62425  580  with degree a \p \ 0\ have "degree a = 0"  eberlm@62425  581  by (simp add: degree_mult_eq)  eberlm@62425  582  with coeff a show "p = q"  eberlm@62425  583  by (cases a, auto split: if_splits)  eberlm@62425  584 qed  eberlm@62425  585 eberlm@62425  586 lemma poly_gcd_unique:  eberlm@62425  587  fixes d x y :: "_ poly"  eberlm@62425  588  assumes dvd1: "d dvd x" and dvd2: "d dvd y"  eberlm@62425  589  and greatest: "\k. k dvd x \ k dvd y \ k dvd d"  eberlm@62425  590  and monic: "coeff d (degree d) = (if x = 0 \ y = 0 then 0 else 1)"  eberlm@62425  591  shows "d = gcd x y"  eberlm@62425  592  using assms by (intro gcdI) (auto simp: normalize_poly_def split: if_split_asm)  eberlm@62425  593 eberlm@62425  594 lemma poly_gcd_code [code]:  eberlm@62425  595  "gcd x y = (if y = 0 then normalize x else gcd y (x mod (y :: _ poly)))"  eberlm@62425  596  by (simp add: gcd_0 gcd_non_0)  eberlm@62425  597 haftmann@60571  598 end