src/HOL/Number_Theory/Euclidean_Algorithm.thy
author haftmann
Thu Jul 02 10:06:47 2015 +0200 (2015-07-02)
changeset 60634 e3b6e516608b
parent 60600 87fbfea0bd0a
child 60685 cb21b7022b00
permissions -rw-r--r--
separate (semi)ring with normalization
     1 (* Author: Manuel Eberl *)
     2 
     3 section \<open>Abstract euclidean algorithm\<close>
     4 
     5 theory Euclidean_Algorithm
     6 imports Complex_Main "~~/src/HOL/Library/Polynomial" "~~/src/HOL/Number_Theory/Normalization_Semidom"
     7 begin
     8 
     9 lemma is_unit_polyE:
    10   assumes "is_unit p"
    11   obtains a where "p = monom a 0" and "a \<noteq> 0"
    12 proof -
    13   obtain a q where "p = pCons a q" by (cases p)
    14   with assms have "p = [:a:]" and "a \<noteq> 0"
    15     by (simp_all add: is_unit_pCons_iff)
    16   with that show thesis by (simp add: monom_0)
    17 qed
    18 
    19 instantiation poly :: (field) normalization_semidom
    20 begin
    21 
    22 definition normalize_poly :: "'a poly \<Rightarrow> 'a poly"
    23   where "normalize_poly p = smult (1 / coeff p (degree p)) p"
    24 
    25 definition unit_factor_poly :: "'a poly \<Rightarrow> 'a poly"
    26   where "unit_factor_poly p = monom (coeff p (degree p)) 0"
    27 
    28 instance
    29 proof
    30   fix p :: "'a poly"
    31   show "unit_factor p * normalize p = p"
    32     by (simp add: normalize_poly_def unit_factor_poly_def)
    33       (simp only: mult_smult_left [symmetric] smult_monom, simp)
    34 next
    35   show "normalize 0 = (0::'a poly)"
    36     by (simp add: normalize_poly_def)
    37 next
    38   show "unit_factor 0 = (0::'a poly)"
    39     by (simp add: unit_factor_poly_def)
    40 next
    41   fix p :: "'a poly"
    42   assume "is_unit p"
    43   then obtain a where "p = monom a 0" and "a \<noteq> 0"
    44     by (rule is_unit_polyE)
    45   then show "normalize p = 1"
    46     by (auto simp add: normalize_poly_def smult_monom degree_monom_eq)
    47 next
    48   fix p q :: "'a poly"
    49   assume "q \<noteq> 0"
    50   from \<open>q \<noteq> 0\<close> have "is_unit (monom (coeff q (degree q)) 0)"
    51     by (auto intro: is_unit_monom_0)
    52   then show "is_unit (unit_factor q)"
    53     by (simp add: unit_factor_poly_def)
    54 next
    55   fix p q :: "'a poly"
    56   have "monom (coeff (p * q) (degree (p * q))) 0 =
    57     monom (coeff p (degree p)) 0 * monom (coeff q (degree q)) 0"
    58     by (simp add: monom_0 coeff_degree_mult)
    59   then show "unit_factor (p * q) =
    60     unit_factor p * unit_factor q"
    61     by (simp add: unit_factor_poly_def)
    62 qed
    63 
    64 end
    65 
    66 text \<open>
    67   A Euclidean semiring is a semiring upon which the Euclidean algorithm can be
    68   implemented. It must provide:
    69   \begin{itemize}
    70   \item division with remainder
    71   \item a size function such that @{term "size (a mod b) < size b"} 
    72         for any @{term "b \<noteq> 0"}
    73   \end{itemize}
    74   The existence of these functions makes it possible to derive gcd and lcm functions 
    75   for any Euclidean semiring.
    76 \<close> 
    77 class euclidean_semiring = semiring_div + normalization_semidom + 
    78   fixes euclidean_size :: "'a \<Rightarrow> nat"
    79   assumes mod_size_less: 
    80     "b \<noteq> 0 \<Longrightarrow> euclidean_size (a mod b) < euclidean_size b"
    81   assumes size_mult_mono:
    82     "b \<noteq> 0 \<Longrightarrow> euclidean_size a \<le> euclidean_size (a * b)"
    83 begin
    84 
    85 lemma euclidean_division:
    86   fixes a :: 'a and b :: 'a
    87   assumes "b \<noteq> 0"
    88   obtains s and t where "a = s * b + t" 
    89     and "euclidean_size t < euclidean_size b"
    90 proof -
    91   from div_mod_equality [of a b 0] 
    92      have "a = a div b * b + a mod b" by simp
    93   with that and assms show ?thesis by (auto simp add: mod_size_less)
    94 qed
    95 
    96 lemma dvd_euclidean_size_eq_imp_dvd:
    97   assumes "a \<noteq> 0" and b_dvd_a: "b dvd a" and size_eq: "euclidean_size a = euclidean_size b"
    98   shows "a dvd b"
    99 proof (rule ccontr)
   100   assume "\<not> a dvd b"
   101   then have "b mod a \<noteq> 0" by (simp add: mod_eq_0_iff_dvd)
   102   from b_dvd_a have b_dvd_mod: "b dvd b mod a" by (simp add: dvd_mod_iff)
   103   from b_dvd_mod obtain c where "b mod a = b * c" unfolding dvd_def by blast
   104     with \<open>b mod a \<noteq> 0\<close> have "c \<noteq> 0" by auto
   105   with \<open>b mod a = b * c\<close> have "euclidean_size (b mod a) \<ge> euclidean_size b"
   106       using size_mult_mono by force
   107   moreover from \<open>\<not> a dvd b\<close> and \<open>a \<noteq> 0\<close>
   108   have "euclidean_size (b mod a) < euclidean_size a"
   109       using mod_size_less by blast
   110   ultimately show False using size_eq by simp
   111 qed
   112 
   113 function gcd_eucl :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
   114 where
   115   "gcd_eucl a b = (if b = 0 then normalize a else gcd_eucl b (a mod b))"
   116   by pat_completeness simp
   117 termination
   118   by (relation "measure (euclidean_size \<circ> snd)") (simp_all add: mod_size_less)
   119 
   120 declare gcd_eucl.simps [simp del]
   121 
   122 lemma gcd_eucl_induct [case_names zero mod]:
   123   assumes H1: "\<And>b. P b 0"
   124   and H2: "\<And>a b. b \<noteq> 0 \<Longrightarrow> P b (a mod b) \<Longrightarrow> P a b"
   125   shows "P a b"
   126 proof (induct a b rule: gcd_eucl.induct)
   127   case ("1" a b)
   128   show ?case
   129   proof (cases "b = 0")
   130     case True then show "P a b" by simp (rule H1)
   131   next
   132     case False
   133     then have "P b (a mod b)"
   134       by (rule "1.hyps")
   135     with \<open>b \<noteq> 0\<close> show "P a b"
   136       by (blast intro: H2)
   137   qed
   138 qed
   139 
   140 definition lcm_eucl :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
   141 where
   142   "lcm_eucl a b = normalize (a * b) div gcd_eucl a b"
   143 
   144 definition Lcm_eucl :: "'a set \<Rightarrow> 'a" -- \<open>
   145   Somewhat complicated definition of Lcm that has the advantage of working
   146   for infinite sets as well\<close>
   147 where
   148   "Lcm_eucl A = (if \<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) then
   149      let l = SOME l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l =
   150        (LEAST n. \<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n)
   151        in normalize l 
   152       else 0)"
   153 
   154 definition Gcd_eucl :: "'a set \<Rightarrow> 'a"
   155 where
   156   "Gcd_eucl A = Lcm_eucl {d. \<forall>a\<in>A. d dvd a}"
   157 
   158 lemma gcd_eucl_0:
   159   "gcd_eucl a 0 = normalize a"
   160   by (simp add: gcd_eucl.simps [of a 0])
   161 
   162 lemma gcd_eucl_0_left:
   163   "gcd_eucl 0 a = normalize a"
   164   by (simp_all add: gcd_eucl_0 gcd_eucl.simps [of 0 a])
   165 
   166 lemma gcd_eucl_non_0:
   167   "b \<noteq> 0 \<Longrightarrow> gcd_eucl a b = gcd_eucl b (a mod b)"
   168   by (simp add: gcd_eucl.simps [of a b] gcd_eucl.simps [of b 0])
   169 
   170 end
   171 
   172 class euclidean_ring = euclidean_semiring + idom
   173 begin
   174 
   175 function euclid_ext :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a" where
   176   "euclid_ext a b = 
   177      (if b = 0 then 
   178         (1 div unit_factor a, 0, normalize a)
   179       else
   180         case euclid_ext b (a mod b) of
   181             (s, t, c) \<Rightarrow> (t, s - t * (a div b), c))"
   182   by pat_completeness simp
   183 termination
   184   by (relation "measure (euclidean_size \<circ> snd)") (simp_all add: mod_size_less)
   185 
   186 declare euclid_ext.simps [simp del]
   187 
   188 lemma euclid_ext_0: 
   189   "euclid_ext a 0 = (1 div unit_factor a, 0, normalize a)"
   190   by (simp add: euclid_ext.simps [of a 0])
   191 
   192 lemma euclid_ext_left_0: 
   193   "euclid_ext 0 a = (0, 1 div unit_factor a, normalize a)"
   194   by (simp add: euclid_ext_0 euclid_ext.simps [of 0 a])
   195 
   196 lemma euclid_ext_non_0: 
   197   "b \<noteq> 0 \<Longrightarrow> euclid_ext a b = (case euclid_ext b (a mod b) of
   198     (s, t, c) \<Rightarrow> (t, s - t * (a div b), c))"
   199   by (simp add: euclid_ext.simps [of a b] euclid_ext.simps [of b 0])
   200 
   201 lemma euclid_ext_code [code]:
   202   "euclid_ext a b = (if b = 0 then (1 div unit_factor a, 0, normalize a)
   203     else let (s, t, c) = euclid_ext b (a mod b) in  (t, s - t * (a div b), c))"
   204   by (simp add: euclid_ext.simps [of a b] euclid_ext.simps [of b 0])
   205 
   206 lemma euclid_ext_correct:
   207   "case euclid_ext a b of (s, t, c) \<Rightarrow> s * a + t * b = c"
   208 proof (induct a b rule: gcd_eucl_induct)
   209   case (zero a) then show ?case
   210     by (simp add: euclid_ext_0 ac_simps)
   211 next
   212   case (mod a b)
   213   obtain s t c where stc: "euclid_ext b (a mod b) = (s,t,c)"
   214     by (cases "euclid_ext b (a mod b)") blast
   215   with mod have "c = s * b + t * (a mod b)" by simp
   216   also have "... = t * ((a div b) * b + a mod b) + (s - t * (a div b)) * b"
   217     by (simp add: algebra_simps) 
   218   also have "(a div b) * b + a mod b = a" using mod_div_equality .
   219   finally show ?case
   220     by (subst euclid_ext.simps) (simp add: stc mod ac_simps)
   221 qed
   222 
   223 definition euclid_ext' :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a"
   224 where
   225   "euclid_ext' a b = (case euclid_ext a b of (s, t, _) \<Rightarrow> (s, t))"
   226 
   227 lemma euclid_ext'_0: "euclid_ext' a 0 = (1 div unit_factor a, 0)" 
   228   by (simp add: euclid_ext'_def euclid_ext_0)
   229 
   230 lemma euclid_ext'_left_0: "euclid_ext' 0 a = (0, 1 div unit_factor a)" 
   231   by (simp add: euclid_ext'_def euclid_ext_left_0)
   232   
   233 lemma euclid_ext'_non_0: "b \<noteq> 0 \<Longrightarrow> euclid_ext' a b = (snd (euclid_ext' b (a mod b)),
   234   fst (euclid_ext' b (a mod b)) - snd (euclid_ext' b (a mod b)) * (a div b))"
   235   by (simp add: euclid_ext'_def euclid_ext_non_0 split_def)
   236 
   237 end
   238 
   239 class euclidean_semiring_gcd = euclidean_semiring + gcd + Gcd +
   240   assumes gcd_gcd_eucl: "gcd = gcd_eucl" and lcm_lcm_eucl: "lcm = lcm_eucl"
   241   assumes Gcd_Gcd_eucl: "Gcd = Gcd_eucl" and Lcm_Lcm_eucl: "Lcm = Lcm_eucl"
   242 begin
   243 
   244 lemma gcd_0_left:
   245   "gcd 0 a = normalize a"
   246   unfolding gcd_gcd_eucl by (fact gcd_eucl_0_left)
   247 
   248 lemma gcd_0:
   249   "gcd a 0 = normalize a"
   250   unfolding gcd_gcd_eucl by (fact gcd_eucl_0)
   251 
   252 lemma gcd_non_0:
   253   "b \<noteq> 0 \<Longrightarrow> gcd a b = gcd b (a mod b)"
   254   unfolding gcd_gcd_eucl by (fact gcd_eucl_non_0)
   255 
   256 lemma gcd_dvd1 [iff]: "gcd a b dvd a"
   257   and gcd_dvd2 [iff]: "gcd a b dvd b"
   258   by (induct a b rule: gcd_eucl_induct)
   259     (simp_all add: gcd_0 gcd_non_0 dvd_mod_iff)
   260     
   261 lemma dvd_gcd_D1: "k dvd gcd m n \<Longrightarrow> k dvd m"
   262   by (rule dvd_trans, assumption, rule gcd_dvd1)
   263 
   264 lemma dvd_gcd_D2: "k dvd gcd m n \<Longrightarrow> k dvd n"
   265   by (rule dvd_trans, assumption, rule gcd_dvd2)
   266 
   267 lemma gcd_greatest:
   268   fixes k a b :: 'a
   269   shows "k dvd a \<Longrightarrow> k dvd b \<Longrightarrow> k dvd gcd a b"
   270 proof (induct a b rule: gcd_eucl_induct)
   271   case (zero a) from zero(1) show ?case by (rule dvd_trans) (simp add: gcd_0)
   272 next
   273   case (mod a b)
   274   then show ?case
   275     by (simp add: gcd_non_0 dvd_mod_iff)
   276 qed
   277 
   278 lemma dvd_gcd_iff:
   279   "k dvd gcd a b \<longleftrightarrow> k dvd a \<and> k dvd b"
   280   by (blast intro!: gcd_greatest intro: dvd_trans)
   281 
   282 lemmas gcd_greatest_iff = dvd_gcd_iff
   283 
   284 lemma gcd_zero [simp]:
   285   "gcd a b = 0 \<longleftrightarrow> a = 0 \<and> b = 0"
   286   by (metis dvd_0_left dvd_refl gcd_dvd1 gcd_dvd2 gcd_greatest)+
   287 
   288 lemma unit_factor_gcd [simp]:
   289   "unit_factor (gcd a b) = (if a = 0 \<and> b = 0 then 0 else 1)" (is "?f a b = ?g a b")
   290   by (induct a b rule: gcd_eucl_induct)
   291     (auto simp add: gcd_0 gcd_non_0)
   292 
   293 lemma gcdI:
   294   assumes "c dvd a" and "c dvd b" and greatest: "\<And>d. d dvd a \<Longrightarrow> d dvd b \<Longrightarrow> d dvd c"
   295     and "unit_factor c = (if c = 0 then 0 else 1)"
   296   shows "c = gcd a b"
   297   by (rule associated_eqI) (auto simp: assms associated_def intro: gcd_greatest)
   298 
   299 sublocale gcd!: abel_semigroup gcd
   300 proof
   301   fix a b c 
   302   show "gcd (gcd a b) c = gcd a (gcd b c)"
   303   proof (rule gcdI)
   304     have "gcd (gcd a b) c dvd gcd a b" "gcd a b dvd a" by simp_all
   305     then show "gcd (gcd a b) c dvd a" by (rule dvd_trans)
   306     have "gcd (gcd a b) c dvd gcd a b" "gcd a b dvd b" by simp_all
   307     hence "gcd (gcd a b) c dvd b" by (rule dvd_trans)
   308     moreover have "gcd (gcd a b) c dvd c" by simp
   309     ultimately show "gcd (gcd a b) c dvd gcd b c"
   310       by (rule gcd_greatest)
   311     show "unit_factor (gcd (gcd a b) c) =  (if gcd (gcd a b) c = 0 then 0 else 1)"
   312       by auto
   313     fix l assume "l dvd a" and "l dvd gcd b c"
   314     with dvd_trans[OF _ gcd_dvd1] and dvd_trans[OF _ gcd_dvd2]
   315       have "l dvd b" and "l dvd c" by blast+
   316     with \<open>l dvd a\<close> show "l dvd gcd (gcd a b) c"
   317       by (intro gcd_greatest)
   318   qed
   319 next
   320   fix a b
   321   show "gcd a b = gcd b a"
   322     by (rule gcdI) (simp_all add: gcd_greatest)
   323 qed
   324 
   325 lemma gcd_unique: "d dvd a \<and> d dvd b \<and> 
   326     unit_factor d = (if d = 0 then 0 else 1) \<and>
   327     (\<forall>e. e dvd a \<and> e dvd b \<longrightarrow> e dvd d) \<longleftrightarrow> d = gcd a b"
   328   by (rule, auto intro: gcdI simp: gcd_greatest)
   329 
   330 lemma gcd_dvd_prod: "gcd a b dvd k * b"
   331   using mult_dvd_mono [of 1] by auto
   332 
   333 lemma gcd_1_left [simp]: "gcd 1 a = 1"
   334   by (rule sym, rule gcdI, simp_all)
   335 
   336 lemma gcd_1 [simp]: "gcd a 1 = 1"
   337   by (rule sym, rule gcdI, simp_all)
   338 
   339 lemma gcd_proj2_if_dvd: 
   340   "b dvd a \<Longrightarrow> gcd a b = normalize b"
   341   by (cases "b = 0", simp_all add: dvd_eq_mod_eq_0 gcd_non_0 gcd_0)
   342 
   343 lemma gcd_proj1_if_dvd: 
   344   "a dvd b \<Longrightarrow> gcd a b = normalize a"
   345   by (subst gcd.commute, simp add: gcd_proj2_if_dvd)
   346 
   347 lemma gcd_proj1_iff: "gcd m n = normalize m \<longleftrightarrow> m dvd n"
   348 proof
   349   assume A: "gcd m n = normalize m"
   350   show "m dvd n"
   351   proof (cases "m = 0")
   352     assume [simp]: "m \<noteq> 0"
   353     from A have B: "m = gcd m n * unit_factor m"
   354       by (simp add: unit_eq_div2)
   355     show ?thesis by (subst B, simp add: mult_unit_dvd_iff)
   356   qed (insert A, simp)
   357 next
   358   assume "m dvd n"
   359   then show "gcd m n = normalize m" by (rule gcd_proj1_if_dvd)
   360 qed
   361   
   362 lemma gcd_proj2_iff: "gcd m n = normalize n \<longleftrightarrow> n dvd m"
   363   using gcd_proj1_iff [of n m] by (simp add: ac_simps)
   364 
   365 lemma gcd_mod1 [simp]:
   366   "gcd (a mod b) b = gcd a b"
   367   by (rule gcdI, metis dvd_mod_iff gcd_dvd1 gcd_dvd2, simp_all add: gcd_greatest dvd_mod_iff)
   368 
   369 lemma gcd_mod2 [simp]:
   370   "gcd a (b mod a) = gcd a b"
   371   by (rule gcdI, simp, metis dvd_mod_iff gcd_dvd1 gcd_dvd2, simp_all add: gcd_greatest dvd_mod_iff)
   372          
   373 lemma gcd_mult_distrib': 
   374   "normalize c * gcd a b = gcd (c * a) (c * b)"
   375 proof (cases "c = 0")
   376   case True then show ?thesis by (simp_all add: gcd_0)
   377 next
   378   case False then have [simp]: "is_unit (unit_factor c)" by simp
   379   show ?thesis
   380   proof (induct a b rule: gcd_eucl_induct)
   381     case (zero a) show ?case
   382     proof (cases "a = 0")
   383       case True then show ?thesis by (simp add: gcd_0)
   384     next
   385       case False
   386       then show ?thesis by (simp add: gcd_0 normalize_mult)
   387     qed
   388     case (mod a b)
   389     then show ?case by (simp add: mult_mod_right gcd.commute)
   390   qed
   391 qed
   392 
   393 lemma gcd_mult_distrib:
   394   "k * gcd a b = gcd (k * a) (k * b) * unit_factor k"
   395 proof-
   396   have "normalize k * gcd a b = gcd (k * a) (k * b)"
   397     by (simp add: gcd_mult_distrib')
   398   then have "normalize k * gcd a b * unit_factor k = gcd (k * a) (k * b) * unit_factor k"
   399     by simp
   400   then have "normalize k * unit_factor k * gcd a b  = gcd (k * a) (k * b) * unit_factor k"
   401     by (simp only: ac_simps)
   402   then show ?thesis
   403     by simp
   404 qed
   405 
   406 lemma euclidean_size_gcd_le1 [simp]:
   407   assumes "a \<noteq> 0"
   408   shows "euclidean_size (gcd a b) \<le> euclidean_size a"
   409 proof -
   410    have "gcd a b dvd a" by (rule gcd_dvd1)
   411    then obtain c where A: "a = gcd a b * c" unfolding dvd_def by blast
   412    with \<open>a \<noteq> 0\<close> show ?thesis by (subst (2) A, intro size_mult_mono) auto
   413 qed
   414 
   415 lemma euclidean_size_gcd_le2 [simp]:
   416   "b \<noteq> 0 \<Longrightarrow> euclidean_size (gcd a b) \<le> euclidean_size b"
   417   by (subst gcd.commute, rule euclidean_size_gcd_le1)
   418 
   419 lemma euclidean_size_gcd_less1:
   420   assumes "a \<noteq> 0" and "\<not>a dvd b"
   421   shows "euclidean_size (gcd a b) < euclidean_size a"
   422 proof (rule ccontr)
   423   assume "\<not>euclidean_size (gcd a b) < euclidean_size a"
   424   with \<open>a \<noteq> 0\<close> have "euclidean_size (gcd a b) = euclidean_size a"
   425     by (intro le_antisym, simp_all)
   426   with assms have "a dvd gcd a b" by (auto intro: dvd_euclidean_size_eq_imp_dvd)
   427   hence "a dvd b" using dvd_gcd_D2 by blast
   428   with \<open>\<not>a dvd b\<close> show False by contradiction
   429 qed
   430 
   431 lemma euclidean_size_gcd_less2:
   432   assumes "b \<noteq> 0" and "\<not>b dvd a"
   433   shows "euclidean_size (gcd a b) < euclidean_size b"
   434   using assms by (subst gcd.commute, rule euclidean_size_gcd_less1)
   435 
   436 lemma gcd_mult_unit1: "is_unit a \<Longrightarrow> gcd (b * a) c = gcd b c"
   437   apply (rule gcdI)
   438   apply (rule dvd_trans, rule gcd_dvd1, simp add: unit_simps)
   439   apply (rule gcd_dvd2)
   440   apply (rule gcd_greatest, simp add: unit_simps, assumption)
   441   apply (subst unit_factor_gcd, simp add: gcd_0)
   442   done
   443 
   444 lemma gcd_mult_unit2: "is_unit a \<Longrightarrow> gcd b (c * a) = gcd b c"
   445   by (subst gcd.commute, subst gcd_mult_unit1, assumption, rule gcd.commute)
   446 
   447 lemma gcd_div_unit1: "is_unit a \<Longrightarrow> gcd (b div a) c = gcd b c"
   448   by (erule is_unitE [of _ b]) (simp add: gcd_mult_unit1)
   449 
   450 lemma gcd_div_unit2: "is_unit a \<Longrightarrow> gcd b (c div a) = gcd b c"
   451   by (erule is_unitE [of _ c]) (simp add: gcd_mult_unit2)
   452 
   453 lemma normalize_gcd_left [simp]:
   454   "gcd (normalize a) b = gcd a b"
   455 proof (cases "a = 0")
   456   case True then show ?thesis
   457     by simp
   458 next
   459   case False then have "is_unit (unit_factor a)"
   460     by simp
   461   moreover have "normalize a = a div unit_factor a"
   462     by simp
   463   ultimately show ?thesis
   464     by (simp only: gcd_div_unit1)
   465 qed
   466 
   467 lemma normalize_gcd_right [simp]:
   468   "gcd a (normalize b) = gcd a b"
   469   using normalize_gcd_left [of b a] by (simp add: ac_simps)
   470 
   471 lemma gcd_idem: "gcd a a = normalize a"
   472   by (cases "a = 0") (simp add: gcd_0_left, rule sym, rule gcdI, simp_all)
   473 
   474 lemma gcd_right_idem: "gcd (gcd a b) b = gcd a b"
   475   apply (rule gcdI)
   476   apply (simp add: ac_simps)
   477   apply (rule gcd_dvd2)
   478   apply (rule gcd_greatest, erule (1) gcd_greatest, assumption)
   479   apply simp
   480   done
   481 
   482 lemma gcd_left_idem: "gcd a (gcd a b) = gcd a b"
   483   apply (rule gcdI)
   484   apply simp
   485   apply (rule dvd_trans, rule gcd_dvd2, rule gcd_dvd2)
   486   apply (rule gcd_greatest, assumption, erule gcd_greatest, assumption)
   487   apply simp
   488   done
   489 
   490 lemma comp_fun_idem_gcd: "comp_fun_idem gcd"
   491 proof
   492   fix a b show "gcd a \<circ> gcd b = gcd b \<circ> gcd a"
   493     by (simp add: fun_eq_iff ac_simps)
   494 next
   495   fix a show "gcd a \<circ> gcd a = gcd a"
   496     by (simp add: fun_eq_iff gcd_left_idem)
   497 qed
   498 
   499 lemma coprime_dvd_mult:
   500   assumes "gcd c b = 1" and "c dvd a * b"
   501   shows "c dvd a"
   502 proof -
   503   let ?nf = "unit_factor"
   504   from assms gcd_mult_distrib [of a c b] 
   505     have A: "a = gcd (a * c) (a * b) * ?nf a" by simp
   506   from \<open>c dvd a * b\<close> show ?thesis by (subst A, simp_all add: gcd_greatest)
   507 qed
   508 
   509 lemma coprime_dvd_mult_iff:
   510   "gcd c b = 1 \<Longrightarrow> (c dvd a * b) = (c dvd a)"
   511   by (rule, rule coprime_dvd_mult, simp_all)
   512 
   513 lemma gcd_dvd_antisym:
   514   "gcd a b dvd gcd c d \<Longrightarrow> gcd c d dvd gcd a b \<Longrightarrow> gcd a b = gcd c d"
   515 proof (rule gcdI)
   516   assume A: "gcd a b dvd gcd c d" and B: "gcd c d dvd gcd a b"
   517   have "gcd c d dvd c" by simp
   518   with A show "gcd a b dvd c" by (rule dvd_trans)
   519   have "gcd c d dvd d" by simp
   520   with A show "gcd a b dvd d" by (rule dvd_trans)
   521   show "unit_factor (gcd a b) = (if gcd a b = 0 then 0 else 1)"
   522     by simp
   523   fix l assume "l dvd c" and "l dvd d"
   524   hence "l dvd gcd c d" by (rule gcd_greatest)
   525   from this and B show "l dvd gcd a b" by (rule dvd_trans)
   526 qed
   527 
   528 lemma gcd_mult_cancel:
   529   assumes "gcd k n = 1"
   530   shows "gcd (k * m) n = gcd m n"
   531 proof (rule gcd_dvd_antisym)
   532   have "gcd (gcd (k * m) n) k = gcd (gcd k n) (k * m)" by (simp add: ac_simps)
   533   also note \<open>gcd k n = 1\<close>
   534   finally have "gcd (gcd (k * m) n) k = 1" by simp
   535   hence "gcd (k * m) n dvd m" by (rule coprime_dvd_mult, simp add: ac_simps)
   536   moreover have "gcd (k * m) n dvd n" by simp
   537   ultimately show "gcd (k * m) n dvd gcd m n" by (rule gcd_greatest)
   538   have "gcd m n dvd (k * m)" and "gcd m n dvd n" by simp_all
   539   then show "gcd m n dvd gcd (k * m) n" by (rule gcd_greatest)
   540 qed
   541 
   542 lemma coprime_crossproduct:
   543   assumes [simp]: "gcd a d = 1" "gcd b c = 1"
   544   shows "associated (a * c) (b * d) \<longleftrightarrow> associated a b \<and> associated c d" (is "?lhs \<longleftrightarrow> ?rhs")
   545 proof
   546   assume ?rhs then show ?lhs unfolding associated_def by (fast intro: mult_dvd_mono)
   547 next
   548   assume ?lhs
   549   from \<open>?lhs\<close> have "a dvd b * d" unfolding associated_def by (metis dvd_mult_left) 
   550   hence "a dvd b" by (simp add: coprime_dvd_mult_iff)
   551   moreover from \<open>?lhs\<close> have "b dvd a * c" unfolding associated_def by (metis dvd_mult_left) 
   552   hence "b dvd a" by (simp add: coprime_dvd_mult_iff)
   553   moreover from \<open>?lhs\<close> have "c dvd d * b" 
   554     unfolding associated_def by (auto dest: dvd_mult_right simp add: ac_simps)
   555   hence "c dvd d" by (simp add: coprime_dvd_mult_iff gcd.commute)
   556   moreover from \<open>?lhs\<close> have "d dvd c * a"
   557     unfolding associated_def by (auto dest: dvd_mult_right simp add: ac_simps)
   558   hence "d dvd c" by (simp add: coprime_dvd_mult_iff gcd.commute)
   559   ultimately show ?rhs unfolding associated_def by simp
   560 qed
   561 
   562 lemma gcd_add1 [simp]:
   563   "gcd (m + n) n = gcd m n"
   564   by (cases "n = 0", simp_all add: gcd_non_0)
   565 
   566 lemma gcd_add2 [simp]:
   567   "gcd m (m + n) = gcd m n"
   568   using gcd_add1 [of n m] by (simp add: ac_simps)
   569 
   570 lemma gcd_add_mult:
   571   "gcd m (k * m + n) = gcd m n"
   572 proof -
   573   have "gcd m ((k * m + n) mod m) = gcd m (k * m + n)"
   574     by (fact gcd_mod2)
   575   then show ?thesis by simp 
   576 qed
   577 
   578 lemma coprimeI: "(\<And>l. \<lbrakk>l dvd a; l dvd b\<rbrakk> \<Longrightarrow> l dvd 1) \<Longrightarrow> gcd a b = 1"
   579   by (rule sym, rule gcdI, simp_all)
   580 
   581 lemma coprime: "gcd a b = 1 \<longleftrightarrow> (\<forall>d. d dvd a \<and> d dvd b \<longleftrightarrow> is_unit d)"
   582   by (auto intro: coprimeI gcd_greatest dvd_gcd_D1 dvd_gcd_D2)
   583 
   584 lemma div_gcd_coprime:
   585   assumes nz: "a \<noteq> 0 \<or> b \<noteq> 0"
   586   defines [simp]: "d \<equiv> gcd a b"
   587   defines [simp]: "a' \<equiv> a div d" and [simp]: "b' \<equiv> b div d"
   588   shows "gcd a' b' = 1"
   589 proof (rule coprimeI)
   590   fix l assume "l dvd a'" "l dvd b'"
   591   then obtain s t where "a' = l * s" "b' = l * t" unfolding dvd_def by blast
   592   moreover have "a = a' * d" "b = b' * d" by simp_all
   593   ultimately have "a = (l * d) * s" "b = (l * d) * t"
   594     by (simp_all only: ac_simps)
   595   hence "l*d dvd a" and "l*d dvd b" by (simp_all only: dvd_triv_left)
   596   hence "l*d dvd d" by (simp add: gcd_greatest)
   597   then obtain u where "d = l * d * u" ..
   598   then have "d * (l * u) = d" by (simp add: ac_simps)
   599   moreover from nz have "d \<noteq> 0" by simp
   600   with div_mult_self1_is_id have "d * (l * u) div d = l * u" . 
   601   ultimately have "1 = l * u"
   602     using \<open>d \<noteq> 0\<close> by simp
   603   then show "l dvd 1" ..
   604 qed
   605 
   606 lemma coprime_mult: 
   607   assumes da: "gcd d a = 1" and db: "gcd d b = 1"
   608   shows "gcd d (a * b) = 1"
   609   apply (subst gcd.commute)
   610   using da apply (subst gcd_mult_cancel)
   611   apply (subst gcd.commute, assumption)
   612   apply (subst gcd.commute, rule db)
   613   done
   614 
   615 lemma coprime_lmult:
   616   assumes dab: "gcd d (a * b) = 1" 
   617   shows "gcd d a = 1"
   618 proof (rule coprimeI)
   619   fix l assume "l dvd d" and "l dvd a"
   620   hence "l dvd a * b" by simp
   621   with \<open>l dvd d\<close> and dab show "l dvd 1" by (auto intro: gcd_greatest)
   622 qed
   623 
   624 lemma coprime_rmult:
   625   assumes dab: "gcd d (a * b) = 1"
   626   shows "gcd d b = 1"
   627 proof (rule coprimeI)
   628   fix l assume "l dvd d" and "l dvd b"
   629   hence "l dvd a * b" by simp
   630   with \<open>l dvd d\<close> and dab show "l dvd 1" by (auto intro: gcd_greatest)
   631 qed
   632 
   633 lemma coprime_mul_eq: "gcd d (a * b) = 1 \<longleftrightarrow> gcd d a = 1 \<and> gcd d b = 1"
   634   using coprime_rmult[of d a b] coprime_lmult[of d a b] coprime_mult[of d a b] by blast
   635 
   636 lemma gcd_coprime:
   637   assumes c: "gcd a b \<noteq> 0" and a: "a = a' * gcd a b" and b: "b = b' * gcd a b"
   638   shows "gcd a' b' = 1"
   639 proof -
   640   from c have "a \<noteq> 0 \<or> b \<noteq> 0" by simp
   641   with div_gcd_coprime have "gcd (a div gcd a b) (b div gcd a b) = 1" .
   642   also from assms have "a div gcd a b = a'" by (metis div_mult_self2_is_id)+
   643   also from assms have "b div gcd a b = b'" by (metis div_mult_self2_is_id)+
   644   finally show ?thesis .
   645 qed
   646 
   647 lemma coprime_power:
   648   assumes "0 < n"
   649   shows "gcd a (b ^ n) = 1 \<longleftrightarrow> gcd a b = 1"
   650 using assms proof (induct n)
   651   case (Suc n) then show ?case
   652     by (cases n) (simp_all add: coprime_mul_eq)
   653 qed simp
   654 
   655 lemma gcd_coprime_exists:
   656   assumes nz: "gcd a b \<noteq> 0"
   657   shows "\<exists>a' b'. a = a' * gcd a b \<and> b = b' * gcd a b \<and> gcd a' b' = 1"
   658   apply (rule_tac x = "a div gcd a b" in exI)
   659   apply (rule_tac x = "b div gcd a b" in exI)
   660   apply (insert nz, auto intro: div_gcd_coprime)
   661   done
   662 
   663 lemma coprime_exp:
   664   "gcd d a = 1 \<Longrightarrow> gcd d (a^n) = 1"
   665   by (induct n, simp_all add: coprime_mult)
   666 
   667 lemma coprime_exp2 [intro]:
   668   "gcd a b = 1 \<Longrightarrow> gcd (a^n) (b^m) = 1"
   669   apply (rule coprime_exp)
   670   apply (subst gcd.commute)
   671   apply (rule coprime_exp)
   672   apply (subst gcd.commute)
   673   apply assumption
   674   done
   675 
   676 lemma gcd_exp:
   677   "gcd (a^n) (b^n) = (gcd a b) ^ n"
   678 proof (cases "a = 0 \<and> b = 0")
   679   assume "a = 0 \<and> b = 0"
   680   then show ?thesis by (cases n, simp_all add: gcd_0_left)
   681 next
   682   assume A: "\<not>(a = 0 \<and> b = 0)"
   683   hence "1 = gcd ((a div gcd a b)^n) ((b div gcd a b)^n)"
   684     using div_gcd_coprime by (subst sym, auto simp: div_gcd_coprime)
   685   hence "(gcd a b) ^ n = (gcd a b) ^ n * ..." by simp
   686   also note gcd_mult_distrib
   687   also have "unit_factor ((gcd a b)^n) = 1"
   688     by (simp add: unit_factor_power A)
   689   also have "(gcd a b)^n * (a div gcd a b)^n = a^n"
   690     by (subst ac_simps, subst div_power, simp, rule dvd_div_mult_self, rule dvd_power_same, simp)
   691   also have "(gcd a b)^n * (b div gcd a b)^n = b^n"
   692     by (subst ac_simps, subst div_power, simp, rule dvd_div_mult_self, rule dvd_power_same, simp)
   693   finally show ?thesis by simp
   694 qed
   695 
   696 lemma coprime_common_divisor: 
   697   "gcd a b = 1 \<Longrightarrow> a dvd a \<Longrightarrow> a dvd b \<Longrightarrow> is_unit a"
   698   apply (subgoal_tac "a dvd gcd a b")
   699   apply simp
   700   apply (erule (1) gcd_greatest)
   701   done
   702 
   703 lemma division_decomp: 
   704   assumes dc: "a dvd b * c"
   705   shows "\<exists>b' c'. a = b' * c' \<and> b' dvd b \<and> c' dvd c"
   706 proof (cases "gcd a b = 0")
   707   assume "gcd a b = 0"
   708   hence "a = 0 \<and> b = 0" by simp
   709   hence "a = 0 * c \<and> 0 dvd b \<and> c dvd c" by simp
   710   then show ?thesis by blast
   711 next
   712   let ?d = "gcd a b"
   713   assume "?d \<noteq> 0"
   714   from gcd_coprime_exists[OF this]
   715     obtain a' b' where ab': "a = a' * ?d" "b = b' * ?d" "gcd a' b' = 1"
   716     by blast
   717   from ab'(1) have "a' dvd a" unfolding dvd_def by blast
   718   with dc have "a' dvd b*c" using dvd_trans[of a' a "b*c"] by simp
   719   from dc ab'(1,2) have "a'*?d dvd (b'*?d) * c" by simp
   720   hence "?d * a' dvd ?d * (b' * c)" by (simp add: mult_ac)
   721   with \<open>?d \<noteq> 0\<close> have "a' dvd b' * c" by simp
   722   with coprime_dvd_mult[OF ab'(3)] 
   723     have "a' dvd c" by (subst (asm) ac_simps, blast)
   724   with ab'(1) have "a = ?d * a' \<and> ?d dvd b \<and> a' dvd c" by (simp add: mult_ac)
   725   then show ?thesis by blast
   726 qed
   727 
   728 lemma pow_divs_pow:
   729   assumes ab: "a ^ n dvd b ^ n" and n: "n \<noteq> 0"
   730   shows "a dvd b"
   731 proof (cases "gcd a b = 0")
   732   assume "gcd a b = 0"
   733   then show ?thesis by simp
   734 next
   735   let ?d = "gcd a b"
   736   assume "?d \<noteq> 0"
   737   from n obtain m where m: "n = Suc m" by (cases n, simp_all)
   738   from \<open>?d \<noteq> 0\<close> have zn: "?d ^ n \<noteq> 0" by (rule power_not_zero)
   739   from gcd_coprime_exists[OF \<open>?d \<noteq> 0\<close>]
   740     obtain a' b' where ab': "a = a' * ?d" "b = b' * ?d" "gcd a' b' = 1"
   741     by blast
   742   from ab have "(a' * ?d) ^ n dvd (b' * ?d) ^ n"
   743     by (simp add: ab'(1,2)[symmetric])
   744   hence "?d^n * a'^n dvd ?d^n * b'^n"
   745     by (simp only: power_mult_distrib ac_simps)
   746   with zn have "a'^n dvd b'^n" by simp
   747   hence "a' dvd b'^n" using dvd_trans[of a' "a'^n" "b'^n"] by (simp add: m)
   748   hence "a' dvd b'^m * b'" by (simp add: m ac_simps)
   749   with coprime_dvd_mult[OF coprime_exp[OF ab'(3), of m]]
   750     have "a' dvd b'" by (subst (asm) ac_simps, blast)
   751   hence "a'*?d dvd b'*?d" by (rule mult_dvd_mono, simp)
   752   with ab'(1,2) show ?thesis by simp
   753 qed
   754 
   755 lemma pow_divs_eq [simp]:
   756   "n \<noteq> 0 \<Longrightarrow> a ^ n dvd b ^ n \<longleftrightarrow> a dvd b"
   757   by (auto intro: pow_divs_pow dvd_power_same)
   758 
   759 lemma divs_mult:
   760   assumes mr: "m dvd r" and nr: "n dvd r" and mn: "gcd m n = 1"
   761   shows "m * n dvd r"
   762 proof -
   763   from mr nr obtain m' n' where m': "r = m*m'" and n': "r = n*n'"
   764     unfolding dvd_def by blast
   765   from mr n' have "m dvd n'*n" by (simp add: ac_simps)
   766   hence "m dvd n'" using coprime_dvd_mult_iff[OF mn] by simp
   767   then obtain k where k: "n' = m*k" unfolding dvd_def by blast
   768   with n' have "r = m * n * k" by (simp add: mult_ac)
   769   then show ?thesis unfolding dvd_def by blast
   770 qed
   771 
   772 lemma coprime_plus_one [simp]: "gcd (n + 1) n = 1"
   773   by (subst add_commute, simp)
   774 
   775 lemma setprod_coprime [rule_format]:
   776   "(\<forall>i\<in>A. gcd (f i) a = 1) \<longrightarrow> gcd (\<Prod>i\<in>A. f i) a = 1"
   777   apply (cases "finite A")
   778   apply (induct set: finite)
   779   apply (auto simp add: gcd_mult_cancel)
   780   done
   781 
   782 lemma coprime_divisors: 
   783   assumes "d dvd a" "e dvd b" "gcd a b = 1"
   784   shows "gcd d e = 1" 
   785 proof -
   786   from assms obtain k l where "a = d * k" "b = e * l"
   787     unfolding dvd_def by blast
   788   with assms have "gcd (d * k) (e * l) = 1" by simp
   789   hence "gcd (d * k) e = 1" by (rule coprime_lmult)
   790   also have "gcd (d * k) e = gcd e (d * k)" by (simp add: ac_simps)
   791   finally have "gcd e d = 1" by (rule coprime_lmult)
   792   then show ?thesis by (simp add: ac_simps)
   793 qed
   794 
   795 lemma invertible_coprime:
   796   assumes "a * b mod m = 1"
   797   shows "coprime a m"
   798 proof -
   799   from assms have "coprime m (a * b mod m)"
   800     by simp
   801   then have "coprime m (a * b)"
   802     by simp
   803   then have "coprime m a"
   804     by (rule coprime_lmult)
   805   then show ?thesis
   806     by (simp add: ac_simps)
   807 qed
   808 
   809 lemma lcm_gcd:
   810   "lcm a b = normalize (a * b) div gcd a b"
   811   by (simp add: lcm_lcm_eucl gcd_gcd_eucl lcm_eucl_def)
   812 
   813 lemma lcm_gcd_prod:
   814   "lcm a b * gcd a b = normalize (a * b)"
   815   by (simp add: lcm_gcd)
   816 
   817 lemma lcm_dvd1 [iff]:
   818   "a dvd lcm a b"
   819 proof (cases "a*b = 0")
   820   assume "a * b \<noteq> 0"
   821   hence "gcd a b \<noteq> 0" by simp
   822   let ?c = "1 div unit_factor (a * b)"
   823   from \<open>a * b \<noteq> 0\<close> have [simp]: "is_unit (unit_factor (a * b))" by simp
   824   from lcm_gcd_prod[of a b] have "lcm a b * gcd a b = a * ?c * b"
   825     by (simp add: div_mult_swap unit_div_commute)
   826   hence "lcm a b * gcd a b div gcd a b = a * ?c * b div gcd a b" by simp
   827   with \<open>gcd a b \<noteq> 0\<close> have "lcm a b = a * ?c * b div gcd a b"
   828     by (subst (asm) div_mult_self2_is_id, simp_all)
   829   also have "... = a * (?c * b div gcd a b)"
   830     by (metis div_mult_swap gcd_dvd2 mult_assoc)
   831   finally show ?thesis by (rule dvdI)
   832 qed (auto simp add: lcm_gcd)
   833 
   834 lemma lcm_least:
   835   "\<lbrakk>a dvd k; b dvd k\<rbrakk> \<Longrightarrow> lcm a b dvd k"
   836 proof (cases "k = 0")
   837   let ?nf = unit_factor
   838   assume "k \<noteq> 0"
   839   hence "is_unit (?nf k)" by simp
   840   hence "?nf k \<noteq> 0" by (metis not_is_unit_0)
   841   assume A: "a dvd k" "b dvd k"
   842   hence "gcd a b \<noteq> 0" using \<open>k \<noteq> 0\<close> by auto
   843   from A obtain r s where ar: "k = a * r" and bs: "k = b * s" 
   844     unfolding dvd_def by blast
   845   with \<open>k \<noteq> 0\<close> have "r * s \<noteq> 0"
   846     by auto (drule sym [of 0], simp)
   847   hence "is_unit (?nf (r * s))" by simp
   848   let ?c = "?nf k div ?nf (r*s)"
   849   from \<open>is_unit (?nf k)\<close> and \<open>is_unit (?nf (r * s))\<close> have "is_unit ?c" by (rule unit_div)
   850   hence "?c \<noteq> 0" using not_is_unit_0 by fast 
   851   from ar bs have "k * k * gcd s r = ?nf k * k * gcd (k * s) (k * r)"
   852     by (subst mult_assoc, subst gcd_mult_distrib[of k s r], simp only: ac_simps)
   853   also have "... = ?nf k * k * gcd ((r*s) * a) ((r*s) * b)"
   854     by (subst (3) \<open>k = a * r\<close>, subst (3) \<open>k = b * s\<close>, simp add: algebra_simps)
   855   also have "... = ?c * r*s * k * gcd a b" using \<open>r * s \<noteq> 0\<close>
   856     by (subst gcd_mult_distrib'[symmetric], simp add: algebra_simps unit_simps)
   857   finally have "(a*r) * (b*s) * gcd s r = ?c * k * r * s * gcd a b"
   858     by (subst ar[symmetric], subst bs[symmetric], simp add: mult_ac)
   859   hence "a * b * gcd s r * (r * s) = ?c * k * gcd a b * (r * s)"
   860     by (simp add: algebra_simps)
   861   hence "?c * k * gcd a b = a * b * gcd s r" using \<open>r * s \<noteq> 0\<close>
   862     by (metis div_mult_self2_is_id)
   863   also have "... = lcm a b * gcd a b * gcd s r * ?nf (a*b)"
   864     by (subst lcm_gcd_prod[of a b], metis gcd_mult_distrib gcd_mult_distrib') 
   865   also have "... = lcm a b * gcd s r * ?nf (a*b) * gcd a b"
   866     by (simp add: algebra_simps)
   867   finally have "k * ?c = lcm a b * gcd s r * ?nf (a*b)" using \<open>gcd a b \<noteq> 0\<close>
   868     by (metis mult.commute div_mult_self2_is_id)
   869   hence "k = lcm a b * (gcd s r * ?nf (a*b)) div ?c" using \<open>?c \<noteq> 0\<close>
   870     by (metis div_mult_self2_is_id mult_assoc) 
   871   also have "... = lcm a b * (gcd s r * ?nf (a*b) div ?c)" using \<open>is_unit ?c\<close>
   872     by (simp add: unit_simps)
   873   finally show ?thesis by (rule dvdI)
   874 qed simp
   875 
   876 lemma lcm_zero:
   877   "lcm a b = 0 \<longleftrightarrow> a = 0 \<or> b = 0"
   878 proof -
   879   let ?nf = unit_factor
   880   {
   881     assume "a \<noteq> 0" "b \<noteq> 0"
   882     hence "a * b div ?nf (a * b) \<noteq> 0" by (simp add: no_zero_divisors)
   883     moreover from \<open>a \<noteq> 0\<close> and \<open>b \<noteq> 0\<close> have "gcd a b \<noteq> 0" by simp
   884     ultimately have "lcm a b \<noteq> 0" using lcm_gcd_prod[of a b] by (intro notI, simp)
   885   } moreover {
   886     assume "a = 0 \<or> b = 0"
   887     hence "lcm a b = 0" by (elim disjE, simp_all add: lcm_gcd)
   888   }
   889   ultimately show ?thesis by blast
   890 qed
   891 
   892 lemmas lcm_0_iff = lcm_zero
   893 
   894 lemma gcd_lcm: 
   895   assumes "lcm a b \<noteq> 0"
   896   shows "gcd a b = normalize (a * b) div lcm a b"
   897 proof -
   898   have "lcm a b * gcd a b = normalize (a * b)"
   899     by (fact lcm_gcd_prod)
   900   with assms show ?thesis
   901     by (metis nonzero_mult_divide_cancel_left)
   902 qed
   903 
   904 lemma unit_factor_lcm [simp]:
   905   "unit_factor (lcm a b) = (if a = 0 \<or> b = 0 then 0 else 1)"
   906   by (simp add: dvd_unit_factor_div lcm_gcd)
   907 
   908 lemma lcm_dvd2 [iff]: "b dvd lcm a b"
   909   using lcm_dvd1 [of b a] by (simp add: lcm_gcd ac_simps)
   910 
   911 lemma lcmI:
   912   assumes "a dvd c" and "b dvd c" and "\<And>d. a dvd d \<Longrightarrow> b dvd d \<Longrightarrow> c dvd d"
   913     and "unit_factor c = (if c = 0 then 0 else 1)"
   914   shows "c = lcm a b"
   915   by (rule associated_eqI) (auto simp: assms associated_def intro: lcm_least)
   916 
   917 sublocale lcm!: abel_semigroup lcm
   918 proof
   919   fix a b c
   920   show "lcm (lcm a b) c = lcm a (lcm b c)"
   921   proof (rule lcmI)
   922     have "a dvd lcm a b" and "lcm a b dvd lcm (lcm a b) c" by simp_all
   923     then show "a dvd lcm (lcm a b) c" by (rule dvd_trans)
   924     
   925     have "b dvd lcm a b" and "lcm a b dvd lcm (lcm a b) c" by simp_all
   926     hence "b dvd lcm (lcm a b) c" by (rule dvd_trans)
   927     moreover have "c dvd lcm (lcm a b) c" by simp
   928     ultimately show "lcm b c dvd lcm (lcm a b) c" by (rule lcm_least)
   929 
   930     fix l assume "a dvd l" and "lcm b c dvd l"
   931     have "b dvd lcm b c" by simp
   932     from this and \<open>lcm b c dvd l\<close> have "b dvd l" by (rule dvd_trans)
   933     have "c dvd lcm b c" by simp
   934     from this and \<open>lcm b c dvd l\<close> have "c dvd l" by (rule dvd_trans)
   935     from \<open>a dvd l\<close> and \<open>b dvd l\<close> have "lcm a b dvd l" by (rule lcm_least)
   936     from this and \<open>c dvd l\<close> show "lcm (lcm a b) c dvd l" by (rule lcm_least)
   937   qed (simp add: lcm_zero)
   938 next
   939   fix a b
   940   show "lcm a b = lcm b a"
   941     by (simp add: lcm_gcd ac_simps)
   942 qed
   943 
   944 lemma dvd_lcm_D1:
   945   "lcm m n dvd k \<Longrightarrow> m dvd k"
   946   by (rule dvd_trans, rule lcm_dvd1, assumption)
   947 
   948 lemma dvd_lcm_D2:
   949   "lcm m n dvd k \<Longrightarrow> n dvd k"
   950   by (rule dvd_trans, rule lcm_dvd2, assumption)
   951 
   952 lemma gcd_dvd_lcm [simp]:
   953   "gcd a b dvd lcm a b"
   954   by (metis dvd_trans gcd_dvd2 lcm_dvd2)
   955 
   956 lemma lcm_1_iff:
   957   "lcm a b = 1 \<longleftrightarrow> is_unit a \<and> is_unit b"
   958 proof
   959   assume "lcm a b = 1"
   960   then show "is_unit a \<and> is_unit b" by auto
   961 next
   962   assume "is_unit a \<and> is_unit b"
   963   hence "a dvd 1" and "b dvd 1" by simp_all
   964   hence "is_unit (lcm a b)" by (rule lcm_least)
   965   hence "lcm a b = unit_factor (lcm a b)"
   966     by (blast intro: sym is_unit_unit_factor)
   967   also have "\<dots> = 1" using \<open>is_unit a \<and> is_unit b\<close>
   968     by auto
   969   finally show "lcm a b = 1" .
   970 qed
   971 
   972 lemma lcm_0_left [simp]:
   973   "lcm 0 a = 0"
   974   by (rule sym, rule lcmI, simp_all)
   975 
   976 lemma lcm_0 [simp]:
   977   "lcm a 0 = 0"
   978   by (rule sym, rule lcmI, simp_all)
   979 
   980 lemma lcm_unique:
   981   "a dvd d \<and> b dvd d \<and> 
   982   unit_factor d = (if d = 0 then 0 else 1) \<and>
   983   (\<forall>e. a dvd e \<and> b dvd e \<longrightarrow> d dvd e) \<longleftrightarrow> d = lcm a b"
   984   by (rule, auto intro: lcmI simp: lcm_least lcm_zero)
   985 
   986 lemma dvd_lcm_I1 [simp]:
   987   "k dvd m \<Longrightarrow> k dvd lcm m n"
   988   by (metis lcm_dvd1 dvd_trans)
   989 
   990 lemma dvd_lcm_I2 [simp]:
   991   "k dvd n \<Longrightarrow> k dvd lcm m n"
   992   by (metis lcm_dvd2 dvd_trans)
   993 
   994 lemma lcm_1_left [simp]:
   995   "lcm 1 a = normalize a"
   996   by (cases "a = 0") (simp, rule sym, rule lcmI, simp_all)
   997 
   998 lemma lcm_1_right [simp]:
   999   "lcm a 1 = normalize a"
  1000   using lcm_1_left [of a] by (simp add: ac_simps)
  1001 
  1002 lemma lcm_coprime:
  1003   "gcd a b = 1 \<Longrightarrow> lcm a b = normalize (a * b)"
  1004   by (subst lcm_gcd) simp
  1005 
  1006 lemma lcm_proj1_if_dvd: 
  1007   "b dvd a \<Longrightarrow> lcm a b = normalize a"
  1008   by (cases "a = 0") (simp, rule sym, rule lcmI, simp_all)
  1009 
  1010 lemma lcm_proj2_if_dvd: 
  1011   "a dvd b \<Longrightarrow> lcm a b = normalize b"
  1012   using lcm_proj1_if_dvd [of a b] by (simp add: ac_simps)
  1013 
  1014 lemma lcm_proj1_iff:
  1015   "lcm m n = normalize m \<longleftrightarrow> n dvd m"
  1016 proof
  1017   assume A: "lcm m n = normalize m"
  1018   show "n dvd m"
  1019   proof (cases "m = 0")
  1020     assume [simp]: "m \<noteq> 0"
  1021     from A have B: "m = lcm m n * unit_factor m"
  1022       by (simp add: unit_eq_div2)
  1023     show ?thesis by (subst B, simp)
  1024   qed simp
  1025 next
  1026   assume "n dvd m"
  1027   then show "lcm m n = normalize m" by (rule lcm_proj1_if_dvd)
  1028 qed
  1029 
  1030 lemma lcm_proj2_iff:
  1031   "lcm m n = normalize n \<longleftrightarrow> m dvd n"
  1032   using lcm_proj1_iff [of n m] by (simp add: ac_simps)
  1033 
  1034 lemma euclidean_size_lcm_le1: 
  1035   assumes "a \<noteq> 0" and "b \<noteq> 0"
  1036   shows "euclidean_size a \<le> euclidean_size (lcm a b)"
  1037 proof -
  1038   have "a dvd lcm a b" by (rule lcm_dvd1)
  1039   then obtain c where A: "lcm a b = a * c" unfolding dvd_def by blast
  1040   with \<open>a \<noteq> 0\<close> and \<open>b \<noteq> 0\<close> have "c \<noteq> 0" by (auto simp: lcm_zero)
  1041   then show ?thesis by (subst A, intro size_mult_mono)
  1042 qed
  1043 
  1044 lemma euclidean_size_lcm_le2:
  1045   "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> euclidean_size b \<le> euclidean_size (lcm a b)"
  1046   using euclidean_size_lcm_le1 [of b a] by (simp add: ac_simps)
  1047 
  1048 lemma euclidean_size_lcm_less1:
  1049   assumes "b \<noteq> 0" and "\<not>b dvd a"
  1050   shows "euclidean_size a < euclidean_size (lcm a b)"
  1051 proof (rule ccontr)
  1052   from assms have "a \<noteq> 0" by auto
  1053   assume "\<not>euclidean_size a < euclidean_size (lcm a b)"
  1054   with \<open>a \<noteq> 0\<close> and \<open>b \<noteq> 0\<close> have "euclidean_size (lcm a b) = euclidean_size a"
  1055     by (intro le_antisym, simp, intro euclidean_size_lcm_le1)
  1056   with assms have "lcm a b dvd a" 
  1057     by (rule_tac dvd_euclidean_size_eq_imp_dvd) (auto simp: lcm_zero)
  1058   hence "b dvd a" by (rule dvd_lcm_D2)
  1059   with \<open>\<not>b dvd a\<close> show False by contradiction
  1060 qed
  1061 
  1062 lemma euclidean_size_lcm_less2:
  1063   assumes "a \<noteq> 0" and "\<not>a dvd b"
  1064   shows "euclidean_size b < euclidean_size (lcm a b)"
  1065   using assms euclidean_size_lcm_less1 [of a b] by (simp add: ac_simps)
  1066 
  1067 lemma lcm_mult_unit1:
  1068   "is_unit a \<Longrightarrow> lcm (b * a) c = lcm b c"
  1069   apply (rule lcmI)
  1070   apply (rule dvd_trans[of _ "b * a"], simp, rule lcm_dvd1)
  1071   apply (rule lcm_dvd2)
  1072   apply (rule lcm_least, simp add: unit_simps, assumption)
  1073   apply (subst unit_factor_lcm, simp add: lcm_zero)
  1074   done
  1075 
  1076 lemma lcm_mult_unit2:
  1077   "is_unit a \<Longrightarrow> lcm b (c * a) = lcm b c"
  1078   using lcm_mult_unit1 [of a c b] by (simp add: ac_simps)
  1079 
  1080 lemma lcm_div_unit1:
  1081   "is_unit a \<Longrightarrow> lcm (b div a) c = lcm b c"
  1082   by (erule is_unitE [of _ b]) (simp add: lcm_mult_unit1) 
  1083 
  1084 lemma lcm_div_unit2:
  1085   "is_unit a \<Longrightarrow> lcm b (c div a) = lcm b c"
  1086   by (erule is_unitE [of _ c]) (simp add: lcm_mult_unit2)
  1087 
  1088 lemma normalize_lcm_left [simp]:
  1089   "lcm (normalize a) b = lcm a b"
  1090 proof (cases "a = 0")
  1091   case True then show ?thesis
  1092     by simp
  1093 next
  1094   case False then have "is_unit (unit_factor a)"
  1095     by simp
  1096   moreover have "normalize a = a div unit_factor a"
  1097     by simp
  1098   ultimately show ?thesis
  1099     by (simp only: lcm_div_unit1)
  1100 qed
  1101 
  1102 lemma normalize_lcm_right [simp]:
  1103   "lcm a (normalize b) = lcm a b"
  1104   using normalize_lcm_left [of b a] by (simp add: ac_simps)
  1105 
  1106 lemma lcm_left_idem:
  1107   "lcm a (lcm a b) = lcm a b"
  1108   apply (rule lcmI)
  1109   apply simp
  1110   apply (subst lcm.assoc [symmetric], rule lcm_dvd2)
  1111   apply (rule lcm_least, assumption)
  1112   apply (erule (1) lcm_least)
  1113   apply (auto simp: lcm_zero)
  1114   done
  1115 
  1116 lemma lcm_right_idem:
  1117   "lcm (lcm a b) b = lcm a b"
  1118   apply (rule lcmI)
  1119   apply (subst lcm.assoc, rule lcm_dvd1)
  1120   apply (rule lcm_dvd2)
  1121   apply (rule lcm_least, erule (1) lcm_least, assumption)
  1122   apply (auto simp: lcm_zero)
  1123   done
  1124 
  1125 lemma comp_fun_idem_lcm: "comp_fun_idem lcm"
  1126 proof
  1127   fix a b show "lcm a \<circ> lcm b = lcm b \<circ> lcm a"
  1128     by (simp add: fun_eq_iff ac_simps)
  1129 next
  1130   fix a show "lcm a \<circ> lcm a = lcm a" unfolding o_def
  1131     by (intro ext, simp add: lcm_left_idem)
  1132 qed
  1133 
  1134 lemma dvd_Lcm [simp]: "a \<in> A \<Longrightarrow> a dvd Lcm A"
  1135   and Lcm_least: "(\<And>a. a \<in> A \<Longrightarrow> a dvd b) \<Longrightarrow> Lcm A dvd b"
  1136   and unit_factor_Lcm [simp]: 
  1137           "unit_factor (Lcm A) = (if Lcm A = 0 then 0 else 1)"
  1138 proof -
  1139   have "(\<forall>a\<in>A. a dvd Lcm A) \<and> (\<forall>l'. (\<forall>a\<in>A. a dvd l') \<longrightarrow> Lcm A dvd l') \<and>
  1140     unit_factor (Lcm A) = (if Lcm A = 0 then 0 else 1)" (is ?thesis)
  1141   proof (cases "\<exists>l. l \<noteq>  0 \<and> (\<forall>a\<in>A. a dvd l)")
  1142     case False
  1143     hence "Lcm A = 0" by (auto simp: Lcm_Lcm_eucl Lcm_eucl_def)
  1144     with False show ?thesis by auto
  1145   next
  1146     case True
  1147     then obtain l\<^sub>0 where l\<^sub>0_props: "l\<^sub>0 \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l\<^sub>0)" by blast
  1148     def n \<equiv> "LEAST n. \<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"
  1149     def l \<equiv> "SOME l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"
  1150     have "\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"
  1151       apply (subst n_def)
  1152       apply (rule LeastI[of _ "euclidean_size l\<^sub>0"])
  1153       apply (rule exI[of _ l\<^sub>0])
  1154       apply (simp add: l\<^sub>0_props)
  1155       done
  1156     from someI_ex[OF this] have "l \<noteq> 0" and "\<forall>a\<in>A. a dvd l" and "euclidean_size l = n" 
  1157       unfolding l_def by simp_all
  1158     {
  1159       fix l' assume "\<forall>a\<in>A. a dvd l'"
  1160       with \<open>\<forall>a\<in>A. a dvd l\<close> have "\<forall>a\<in>A. a dvd gcd l l'" by (auto intro: gcd_greatest)
  1161       moreover from \<open>l \<noteq> 0\<close> have "gcd l l' \<noteq> 0" by simp
  1162       ultimately have "\<exists>b. b \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd b) \<and> euclidean_size b = euclidean_size (gcd l l')"
  1163         by (intro exI[of _ "gcd l l'"], auto)
  1164       hence "euclidean_size (gcd l l') \<ge> n" by (subst n_def) (rule Least_le)
  1165       moreover have "euclidean_size (gcd l l') \<le> n"
  1166       proof -
  1167         have "gcd l l' dvd l" by simp
  1168         then obtain a where "l = gcd l l' * a" unfolding dvd_def by blast
  1169         with \<open>l \<noteq> 0\<close> have "a \<noteq> 0" by auto
  1170         hence "euclidean_size (gcd l l') \<le> euclidean_size (gcd l l' * a)"
  1171           by (rule size_mult_mono)
  1172         also have "gcd l l' * a = l" using \<open>l = gcd l l' * a\<close> ..
  1173         also note \<open>euclidean_size l = n\<close>
  1174         finally show "euclidean_size (gcd l l') \<le> n" .
  1175       qed
  1176       ultimately have "euclidean_size l = euclidean_size (gcd l l')" 
  1177         by (intro le_antisym, simp_all add: \<open>euclidean_size l = n\<close>)
  1178       with \<open>l \<noteq> 0\<close> have "l dvd gcd l l'" by (blast intro: dvd_euclidean_size_eq_imp_dvd)
  1179       hence "l dvd l'" by (blast dest: dvd_gcd_D2)
  1180     }
  1181 
  1182     with \<open>(\<forall>a\<in>A. a dvd l)\<close> and unit_factor_is_unit[OF \<open>l \<noteq> 0\<close>] and \<open>l \<noteq> 0\<close>
  1183       have "(\<forall>a\<in>A. a dvd normalize l) \<and> 
  1184         (\<forall>l'. (\<forall>a\<in>A. a dvd l') \<longrightarrow> normalize l dvd l') \<and>
  1185         unit_factor (normalize l) = 
  1186         (if normalize l = 0 then 0 else 1)"
  1187       by (auto simp: unit_simps)
  1188     also from True have "normalize l = Lcm A"
  1189       by (simp add: Lcm_Lcm_eucl Lcm_eucl_def Let_def n_def l_def)
  1190     finally show ?thesis .
  1191   qed
  1192   note A = this
  1193 
  1194   {fix a assume "a \<in> A" then show "a dvd Lcm A" using A by blast}
  1195   {fix b assume "\<And>a. a \<in> A \<Longrightarrow> a dvd b" then show "Lcm A dvd b" using A by blast}
  1196   from A show "unit_factor (Lcm A) = (if Lcm A = 0 then 0 else 1)" by blast
  1197 qed
  1198 
  1199 lemma normalize_Lcm [simp]:
  1200   "normalize (Lcm A) = Lcm A"
  1201   by (cases "Lcm A = 0") (auto intro: associated_eqI)
  1202 
  1203 lemma LcmI:
  1204   assumes "\<And>a. a \<in> A \<Longrightarrow> a dvd b" and "\<And>c. (\<And>a. a \<in> A \<Longrightarrow> a dvd c) \<Longrightarrow> b dvd c"
  1205     and "unit_factor b = (if b = 0 then 0 else 1)" shows "b = Lcm A"
  1206   by (rule associated_eqI) (auto simp: assms associated_def intro: Lcm_least)
  1207 
  1208 lemma Lcm_subset:
  1209   "A \<subseteq> B \<Longrightarrow> Lcm A dvd Lcm B"
  1210   by (blast intro: Lcm_least dvd_Lcm)
  1211 
  1212 lemma Lcm_Un:
  1213   "Lcm (A \<union> B) = lcm (Lcm A) (Lcm B)"
  1214   apply (rule lcmI)
  1215   apply (blast intro: Lcm_subset)
  1216   apply (blast intro: Lcm_subset)
  1217   apply (intro Lcm_least ballI, elim UnE)
  1218   apply (rule dvd_trans, erule dvd_Lcm, assumption)
  1219   apply (rule dvd_trans, erule dvd_Lcm, assumption)
  1220   apply simp
  1221   done
  1222 
  1223 lemma Lcm_1_iff:
  1224   "Lcm A = 1 \<longleftrightarrow> (\<forall>a\<in>A. is_unit a)"
  1225 proof
  1226   assume "Lcm A = 1"
  1227   then show "\<forall>a\<in>A. is_unit a" by auto
  1228 qed (rule LcmI [symmetric], auto)
  1229 
  1230 lemma Lcm_no_units:
  1231   "Lcm A = Lcm (A - {a. is_unit a})"
  1232 proof -
  1233   have "(A - {a. is_unit a}) \<union> {a\<in>A. is_unit a} = A" by blast
  1234   hence "Lcm A = lcm (Lcm (A - {a. is_unit a})) (Lcm {a\<in>A. is_unit a})"
  1235     by (simp add: Lcm_Un [symmetric])
  1236   also have "Lcm {a\<in>A. is_unit a} = 1" by (simp add: Lcm_1_iff)
  1237   finally show ?thesis by simp
  1238 qed
  1239 
  1240 lemma Lcm_empty [simp]:
  1241   "Lcm {} = 1"
  1242   by (simp add: Lcm_1_iff)
  1243 
  1244 lemma Lcm_eq_0 [simp]:
  1245   "0 \<in> A \<Longrightarrow> Lcm A = 0"
  1246   by (drule dvd_Lcm) simp
  1247 
  1248 lemma Lcm0_iff':
  1249   "Lcm A = 0 \<longleftrightarrow> \<not>(\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l))"
  1250 proof
  1251   assume "Lcm A = 0"
  1252   show "\<not>(\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l))"
  1253   proof
  1254     assume ex: "\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l)"
  1255     then obtain l\<^sub>0 where l\<^sub>0_props: "l\<^sub>0 \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l\<^sub>0)" by blast
  1256     def n \<equiv> "LEAST n. \<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"
  1257     def l \<equiv> "SOME l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"
  1258     have "\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"
  1259       apply (subst n_def)
  1260       apply (rule LeastI[of _ "euclidean_size l\<^sub>0"])
  1261       apply (rule exI[of _ l\<^sub>0])
  1262       apply (simp add: l\<^sub>0_props)
  1263       done
  1264     from someI_ex[OF this] have "l \<noteq> 0" unfolding l_def by simp_all
  1265     hence "normalize l \<noteq> 0" by simp
  1266     also from ex have "normalize l = Lcm A"
  1267        by (simp only: Lcm_Lcm_eucl Lcm_eucl_def n_def l_def if_True Let_def)
  1268     finally show False using \<open>Lcm A = 0\<close> by contradiction
  1269   qed
  1270 qed (simp only: Lcm_Lcm_eucl Lcm_eucl_def if_False)
  1271 
  1272 lemma Lcm0_iff [simp]:
  1273   "finite A \<Longrightarrow> Lcm A = 0 \<longleftrightarrow> 0 \<in> A"
  1274 proof -
  1275   assume "finite A"
  1276   have "0 \<in> A \<Longrightarrow> Lcm A = 0"  by (intro dvd_0_left dvd_Lcm)
  1277   moreover {
  1278     assume "0 \<notin> A"
  1279     hence "\<Prod>A \<noteq> 0" 
  1280       apply (induct rule: finite_induct[OF \<open>finite A\<close>]) 
  1281       apply simp
  1282       apply (subst setprod.insert, assumption, assumption)
  1283       apply (rule no_zero_divisors)
  1284       apply blast+
  1285       done
  1286     moreover from \<open>finite A\<close> have "\<forall>a\<in>A. a dvd \<Prod>A" by blast
  1287     ultimately have "\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l)" by blast
  1288     with Lcm0_iff' have "Lcm A \<noteq> 0" by simp
  1289   }
  1290   ultimately show "Lcm A = 0 \<longleftrightarrow> 0 \<in> A" by blast
  1291 qed
  1292 
  1293 lemma Lcm_no_multiple:
  1294   "(\<forall>m. m \<noteq> 0 \<longrightarrow> (\<exists>a\<in>A. \<not>a dvd m)) \<Longrightarrow> Lcm A = 0"
  1295 proof -
  1296   assume "\<forall>m. m \<noteq> 0 \<longrightarrow> (\<exists>a\<in>A. \<not>a dvd m)"
  1297   hence "\<not>(\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l))" by blast
  1298   then show "Lcm A = 0" by (simp only: Lcm_Lcm_eucl Lcm_eucl_def if_False)
  1299 qed
  1300 
  1301 lemma Lcm_insert [simp]:
  1302   "Lcm (insert a A) = lcm a (Lcm A)"
  1303 proof (rule lcmI)
  1304   fix l assume "a dvd l" and "Lcm A dvd l"
  1305   hence "\<forall>a\<in>A. a dvd l" by (blast intro: dvd_trans dvd_Lcm)
  1306   with \<open>a dvd l\<close> show "Lcm (insert a A) dvd l" by (force intro: Lcm_least)
  1307 qed (auto intro: Lcm_least dvd_Lcm)
  1308  
  1309 lemma Lcm_finite:
  1310   assumes "finite A"
  1311   shows "Lcm A = Finite_Set.fold lcm 1 A"
  1312   by (induct rule: finite.induct[OF \<open>finite A\<close>])
  1313     (simp_all add: comp_fun_idem.fold_insert_idem[OF comp_fun_idem_lcm])
  1314 
  1315 lemma Lcm_set [code_unfold]:
  1316   "Lcm (set xs) = fold lcm xs 1"
  1317   using comp_fun_idem.fold_set_fold[OF comp_fun_idem_lcm] Lcm_finite by (simp add: ac_simps)
  1318 
  1319 lemma Lcm_singleton [simp]:
  1320   "Lcm {a} = normalize a"
  1321   by simp
  1322 
  1323 lemma Lcm_2 [simp]:
  1324   "Lcm {a,b} = lcm a b"
  1325   by simp
  1326 
  1327 lemma Lcm_coprime:
  1328   assumes "finite A" and "A \<noteq> {}" 
  1329   assumes "\<And>a b. a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<noteq> b \<Longrightarrow> gcd a b = 1"
  1330   shows "Lcm A = normalize (\<Prod>A)"
  1331 using assms proof (induct rule: finite_ne_induct)
  1332   case (insert a A)
  1333   have "Lcm (insert a A) = lcm a (Lcm A)" by simp
  1334   also from insert have "Lcm A = normalize (\<Prod>A)" by blast
  1335   also have "lcm a \<dots> = lcm a (\<Prod>A)" by (cases "\<Prod>A = 0") (simp_all add: lcm_div_unit2)
  1336   also from insert have "gcd a (\<Prod>A) = 1" by (subst gcd.commute, intro setprod_coprime) auto
  1337   with insert have "lcm a (\<Prod>A) = normalize (\<Prod>(insert a A))"
  1338     by (simp add: lcm_coprime)
  1339   finally show ?case .
  1340 qed simp
  1341       
  1342 lemma Lcm_coprime':
  1343   "card A \<noteq> 0 \<Longrightarrow> (\<And>a b. a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<noteq> b \<Longrightarrow> gcd a b = 1)
  1344     \<Longrightarrow> Lcm A = normalize (\<Prod>A)"
  1345   by (rule Lcm_coprime) (simp_all add: card_eq_0_iff)
  1346 
  1347 lemma Gcd_Lcm:
  1348   "Gcd A = Lcm {d. \<forall>a\<in>A. d dvd a}"
  1349   by (simp add: Gcd_Gcd_eucl Lcm_Lcm_eucl Gcd_eucl_def)
  1350 
  1351 lemma Gcd_dvd [simp]: "a \<in> A \<Longrightarrow> Gcd A dvd a"
  1352   and Gcd_greatest: "(\<And>a. a \<in> A \<Longrightarrow> b dvd a) \<Longrightarrow> b dvd Gcd A"
  1353   and unit_factor_Gcd [simp]: 
  1354     "unit_factor (Gcd A) = (if Gcd A = 0 then 0 else 1)"
  1355 proof -
  1356   fix a assume "a \<in> A"
  1357   hence "Lcm {d. \<forall>a\<in>A. d dvd a} dvd a" by (intro Lcm_least) blast
  1358   then show "Gcd A dvd a" by (simp add: Gcd_Lcm)
  1359 next
  1360   fix g' assume "\<And>a. a \<in> A \<Longrightarrow> g' dvd a"
  1361   hence "g' dvd Lcm {d. \<forall>a\<in>A. d dvd a}" by (intro dvd_Lcm) blast
  1362   then show "g' dvd Gcd A" by (simp add: Gcd_Lcm)
  1363 next
  1364   show "unit_factor (Gcd A) = (if Gcd A = 0 then 0 else 1)"
  1365     by (simp add: Gcd_Lcm)
  1366 qed
  1367 
  1368 lemma normalize_Gcd [simp]:
  1369   "normalize (Gcd A) = Gcd A"
  1370   by (cases "Gcd A = 0") (auto intro: associated_eqI)
  1371 
  1372 lemma GcdI:
  1373   assumes "\<And>a. a \<in> A \<Longrightarrow> b dvd a" and "\<And>c. (\<And>a. a \<in> A \<Longrightarrow> c dvd a) \<Longrightarrow> c dvd b"
  1374     and "unit_factor b = (if b = 0 then 0 else 1)"
  1375   shows "b = Gcd A"
  1376   by (rule associated_eqI) (auto simp: assms associated_def intro: Gcd_greatest)
  1377 
  1378 lemma Lcm_Gcd:
  1379   "Lcm A = Gcd {m. \<forall>a\<in>A. a dvd m}"
  1380   by (rule LcmI[symmetric]) (auto intro: dvd_Gcd Gcd_greatest)
  1381 
  1382 lemma Gcd_0_iff:
  1383   "Gcd A = 0 \<longleftrightarrow> A \<subseteq> {0}"
  1384   apply (rule iffI)
  1385   apply (rule subsetI, drule Gcd_dvd, simp)
  1386   apply (auto intro: GcdI[symmetric])
  1387   done
  1388 
  1389 lemma Gcd_empty [simp]:
  1390   "Gcd {} = 0"
  1391   by (simp add: Gcd_0_iff)
  1392 
  1393 lemma Gcd_1:
  1394   "1 \<in> A \<Longrightarrow> Gcd A = 1"
  1395   by (intro GcdI[symmetric]) (auto intro: Gcd_dvd dvd_Gcd)
  1396 
  1397 lemma Gcd_insert [simp]:
  1398   "Gcd (insert a A) = gcd a (Gcd A)"
  1399 proof (rule gcdI)
  1400   fix l assume "l dvd a" and "l dvd Gcd A"
  1401   hence "\<forall>a\<in>A. l dvd a" by (blast intro: dvd_trans Gcd_dvd)
  1402   with \<open>l dvd a\<close> show "l dvd Gcd (insert a A)" by (force intro: Gcd_dvd Gcd_greatest)
  1403 qed (auto intro: Gcd_greatest)
  1404 
  1405 lemma Gcd_finite:
  1406   assumes "finite A"
  1407   shows "Gcd A = Finite_Set.fold gcd 0 A"
  1408   by (induct rule: finite.induct[OF \<open>finite A\<close>])
  1409     (simp_all add: comp_fun_idem.fold_insert_idem[OF comp_fun_idem_gcd])
  1410 
  1411 lemma Gcd_set [code_unfold]:
  1412   "Gcd (set xs) = fold gcd xs 0"
  1413   using comp_fun_idem.fold_set_fold[OF comp_fun_idem_gcd] Gcd_finite by (simp add: ac_simps)
  1414 
  1415 lemma Gcd_singleton [simp]: "Gcd {a} = normalize a"
  1416   by (simp add: gcd_0)
  1417 
  1418 lemma Gcd_2 [simp]: "Gcd {a,b} = gcd a b"
  1419   by (simp add: gcd_0)
  1420 
  1421 subclass semiring_gcd
  1422   by unfold_locales (simp_all add: gcd_greatest_iff)
  1423   
  1424 end
  1425 
  1426 text \<open>
  1427   A Euclidean ring is a Euclidean semiring with additive inverses. It provides a 
  1428   few more lemmas; in particular, Bezout's lemma holds for any Euclidean ring.
  1429 \<close>
  1430 
  1431 class euclidean_ring_gcd = euclidean_semiring_gcd + idom
  1432 begin
  1433 
  1434 subclass euclidean_ring ..
  1435 
  1436 subclass ring_gcd ..
  1437 
  1438 lemma euclid_ext_gcd [simp]:
  1439   "(case euclid_ext a b of (_, _ , t) \<Rightarrow> t) = gcd a b"
  1440   by (induct a b rule: gcd_eucl_induct)
  1441     (simp_all add: euclid_ext_0 gcd_0 euclid_ext_non_0 ac_simps split: prod.split prod.split_asm)
  1442 
  1443 lemma euclid_ext_gcd' [simp]:
  1444   "euclid_ext a b = (r, s, t) \<Longrightarrow> t = gcd a b"
  1445   by (insert euclid_ext_gcd[of a b], drule (1) subst, simp)
  1446   
  1447 lemma euclid_ext'_correct:
  1448   "fst (euclid_ext' a b) * a + snd (euclid_ext' a b) * b = gcd a b"
  1449 proof-
  1450   obtain s t c where "euclid_ext a b = (s,t,c)"
  1451     by (cases "euclid_ext a b", blast)
  1452   with euclid_ext_correct[of a b] euclid_ext_gcd[of a b]
  1453     show ?thesis unfolding euclid_ext'_def by simp
  1454 qed
  1455 
  1456 lemma bezout: "\<exists>s t. s * a + t * b = gcd a b"
  1457   using euclid_ext'_correct by blast
  1458 
  1459 lemma gcd_neg1 [simp]:
  1460   "gcd (-a) b = gcd a b"
  1461   by (rule sym, rule gcdI, simp_all add: gcd_greatest)
  1462 
  1463 lemma gcd_neg2 [simp]:
  1464   "gcd a (-b) = gcd a b"
  1465   by (rule sym, rule gcdI, simp_all add: gcd_greatest)
  1466 
  1467 lemma gcd_neg_numeral_1 [simp]:
  1468   "gcd (- numeral n) a = gcd (numeral n) a"
  1469   by (fact gcd_neg1)
  1470 
  1471 lemma gcd_neg_numeral_2 [simp]:
  1472   "gcd a (- numeral n) = gcd a (numeral n)"
  1473   by (fact gcd_neg2)
  1474 
  1475 lemma gcd_diff1: "gcd (m - n) n = gcd m n"
  1476   by (subst diff_conv_add_uminus, subst gcd_neg2[symmetric],  subst gcd_add1, simp)
  1477 
  1478 lemma gcd_diff2: "gcd (n - m) n = gcd m n"
  1479   by (subst gcd_neg1[symmetric], simp only: minus_diff_eq gcd_diff1)
  1480 
  1481 lemma coprime_minus_one [simp]: "gcd (n - 1) n = 1"
  1482 proof -
  1483   have "gcd (n - 1) n = gcd n (n - 1)" by (fact gcd.commute)
  1484   also have "\<dots> = gcd ((n - 1) + 1) (n - 1)" by simp
  1485   also have "\<dots> = 1" by (rule coprime_plus_one)
  1486   finally show ?thesis .
  1487 qed
  1488 
  1489 lemma lcm_neg1 [simp]: "lcm (-a) b = lcm a b"
  1490   by (rule sym, rule lcmI, simp_all add: lcm_least lcm_zero)
  1491 
  1492 lemma lcm_neg2 [simp]: "lcm a (-b) = lcm a b"
  1493   by (rule sym, rule lcmI, simp_all add: lcm_least lcm_zero)
  1494 
  1495 lemma lcm_neg_numeral_1 [simp]: "lcm (- numeral n) a = lcm (numeral n) a"
  1496   by (fact lcm_neg1)
  1497 
  1498 lemma lcm_neg_numeral_2 [simp]: "lcm a (- numeral n) = lcm a (numeral n)"
  1499   by (fact lcm_neg2)
  1500 
  1501 end
  1502 
  1503 
  1504 subsection \<open>Typical instances\<close>
  1505 
  1506 instantiation nat :: euclidean_semiring
  1507 begin
  1508 
  1509 definition [simp]:
  1510   "euclidean_size_nat = (id :: nat \<Rightarrow> nat)"
  1511 
  1512 definition [simp]:
  1513   "unit_factor_nat (n::nat) = (if n = 0 then 0 else 1 :: nat)"
  1514 
  1515 instance proof
  1516 qed simp_all
  1517 
  1518 end
  1519 
  1520 instantiation int :: euclidean_ring
  1521 begin
  1522 
  1523 definition [simp]:
  1524   "euclidean_size_int = (nat \<circ> abs :: int \<Rightarrow> nat)"
  1525 
  1526 definition [simp]:
  1527   "unit_factor_int = (sgn :: int \<Rightarrow> int)"
  1528 
  1529 instance
  1530 by standard (auto simp add: abs_mult nat_mult_distrib sgn_times split: abs_split)
  1531 
  1532 end
  1533 
  1534 instantiation poly :: (field) euclidean_ring
  1535 begin
  1536 
  1537 definition euclidean_size_poly :: "'a poly \<Rightarrow> nat"
  1538   where "euclidean_size p = (if p = 0 then 0 else Suc (degree p))"
  1539 
  1540 lemma euclidenan_size_poly_minus_one_degree [simp]:
  1541   "euclidean_size p - 1 = degree p"
  1542   by (simp add: euclidean_size_poly_def)
  1543 
  1544 lemma euclidean_size_poly_0 [simp]:
  1545   "euclidean_size (0::'a poly) = 0"
  1546   by (simp add: euclidean_size_poly_def)
  1547 
  1548 lemma euclidean_size_poly_not_0 [simp]:
  1549   "p \<noteq> 0 \<Longrightarrow> euclidean_size p = Suc (degree p)"
  1550   by (simp add: euclidean_size_poly_def)
  1551 
  1552 instance
  1553 proof
  1554   fix p q :: "'a poly"
  1555   assume "q \<noteq> 0"
  1556   then have "p mod q = 0 \<or> degree (p mod q) < degree q"
  1557     by (rule degree_mod_less [of q p])  
  1558   with \<open>q \<noteq> 0\<close> show "euclidean_size (p mod q) < euclidean_size q"
  1559     by (cases "p mod q = 0") simp_all
  1560 next
  1561   fix p q :: "'a poly"
  1562   assume "q \<noteq> 0"
  1563   from \<open>q \<noteq> 0\<close> have "degree p \<le> degree (p * q)"
  1564     by (rule degree_mult_right_le)
  1565   with \<open>q \<noteq> 0\<close> show "euclidean_size p \<le> euclidean_size (p * q)"
  1566     by (cases "p = 0") simp_all
  1567 qed
  1568 
  1569 end
  1570 
  1571 end