src/HOL/Number_Theory/Euclidean_Algorithm.thy
author haftmann
Sat Jun 27 20:20:34 2015 +0200 (2015-06-27)
changeset 60598 78ca5674c66a
parent 60582 d694f217ee41
child 60599 f8bb070dc98b
permissions -rw-r--r--
rings follow immediately their corresponding semirings
     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"
     7 begin
     8   
     9 text \<open>
    10   A Euclidean semiring is a semiring upon which the Euclidean algorithm can be
    11   implemented. It must provide:
    12   \begin{itemize}
    13   \item division with remainder
    14   \item a size function such that @{term "size (a mod b) < size b"} 
    15         for any @{term "b \<noteq> 0"}
    16   \item a normalization factor such that two associated numbers are equal iff 
    17         they are the same when divd by their normalization factors.
    18   \end{itemize}
    19   The existence of these functions makes it possible to derive gcd and lcm functions 
    20   for any Euclidean semiring.
    21 \<close> 
    22 class euclidean_semiring = semiring_div + 
    23   fixes euclidean_size :: "'a \<Rightarrow> nat"
    24   fixes normalization_factor :: "'a \<Rightarrow> 'a"
    25   assumes mod_size_less: 
    26     "b \<noteq> 0 \<Longrightarrow> \<not> b dvd a \<Longrightarrow> euclidean_size (a mod b) < euclidean_size b"
    27   assumes size_mult_mono:
    28     "b \<noteq> 0 \<Longrightarrow> euclidean_size (a * b) \<ge> euclidean_size a"
    29   assumes normalization_factor_is_unit [intro,simp]: 
    30     "a \<noteq> 0 \<Longrightarrow> is_unit (normalization_factor a)"
    31   assumes normalization_factor_mult: "normalization_factor (a * b) = 
    32     normalization_factor a * normalization_factor b"
    33   assumes normalization_factor_unit: "is_unit a \<Longrightarrow> normalization_factor a = a"
    34   assumes normalization_factor_0 [simp]: "normalization_factor 0 = 0"
    35 begin
    36 
    37 lemma normalization_factor_dvd [simp]:
    38   "a \<noteq> 0 \<Longrightarrow> normalization_factor a dvd b"
    39   by (rule unit_imp_dvd, simp)
    40     
    41 lemma normalization_factor_1 [simp]:
    42   "normalization_factor 1 = 1"
    43   by (simp add: normalization_factor_unit)
    44 
    45 lemma normalization_factor_0_iff [simp]:
    46   "normalization_factor a = 0 \<longleftrightarrow> a = 0"
    47 proof
    48   assume "normalization_factor a = 0"
    49   hence "\<not> is_unit (normalization_factor a)"
    50     by simp
    51   then show "a = 0" by auto
    52 qed simp
    53 
    54 lemma normalization_factor_pow:
    55   "normalization_factor (a ^ n) = normalization_factor a ^ n"
    56   by (induct n) (simp_all add: normalization_factor_mult power_Suc2)
    57 
    58 lemma normalization_correct [simp]:
    59   "normalization_factor (a div normalization_factor a) = (if a = 0 then 0 else 1)"
    60 proof (cases "a = 0", simp)
    61   assume "a \<noteq> 0"
    62   let ?nf = "normalization_factor"
    63   from normalization_factor_is_unit[OF \<open>a \<noteq> 0\<close>] have "?nf a \<noteq> 0"
    64     by auto
    65   have "?nf (a div ?nf a) * ?nf (?nf a) = ?nf (a div ?nf a * ?nf a)" 
    66     by (simp add: normalization_factor_mult)
    67   also have "a div ?nf a * ?nf a = a" using \<open>a \<noteq> 0\<close>
    68     by simp
    69   also have "?nf (?nf a) = ?nf a" using \<open>a \<noteq> 0\<close> 
    70     normalization_factor_is_unit normalization_factor_unit by simp
    71   finally have "normalization_factor (a div normalization_factor a) = 1"  
    72     using \<open>?nf a \<noteq> 0\<close> by (metis div_mult_self2_is_id div_self)
    73   with \<open>a \<noteq> 0\<close> show ?thesis by simp
    74 qed
    75 
    76 lemma normalization_0_iff [simp]:
    77   "a div normalization_factor a = 0 \<longleftrightarrow> a = 0"
    78   by (cases "a = 0", simp, subst unit_eq_div1, blast, simp)
    79 
    80 lemma mult_div_normalization [simp]:
    81   "b * (1 div normalization_factor a) = b div normalization_factor a"
    82   by (cases "a = 0") simp_all
    83 
    84 lemma associated_iff_normed_eq:
    85   "associated a b \<longleftrightarrow> a div normalization_factor a = b div normalization_factor b"
    86 proof (cases "b = 0", simp, cases "a = 0", metis associated_0(1) normalization_0_iff, rule iffI)
    87   let ?nf = normalization_factor
    88   assume "a \<noteq> 0" "b \<noteq> 0" "a div ?nf a = b div ?nf b"
    89   hence "a = b * (?nf a div ?nf b)"
    90     apply (subst (asm) unit_eq_div1, blast, subst (asm) unit_div_commute, blast)
    91     apply (subst div_mult_swap, simp, simp)
    92     done
    93   with \<open>a \<noteq> 0\<close> \<open>b \<noteq> 0\<close> have "\<exists>c. is_unit c \<and> a = c * b"
    94     by (intro exI[of _ "?nf a div ?nf b"], force simp: mult_ac)
    95   then obtain c where "is_unit c" and "a = c * b" by blast
    96   then show "associated a b" by (rule is_unit_associatedI) 
    97 next
    98   let ?nf = normalization_factor
    99   assume "a \<noteq> 0" "b \<noteq> 0" "associated a b"
   100   then obtain c where "is_unit c" and "a = c * b" by (blast elim: associated_is_unitE)
   101   then show "a div ?nf a = b div ?nf b"
   102     apply (simp only: \<open>a = c * b\<close> normalization_factor_mult normalization_factor_unit)
   103     apply (rule div_mult_mult1, force)
   104     done
   105   qed
   106 
   107 lemma normed_associated_imp_eq:
   108   "associated a b \<Longrightarrow> normalization_factor a \<in> {0, 1} \<Longrightarrow> normalization_factor b \<in> {0, 1} \<Longrightarrow> a = b"
   109   by (simp add: associated_iff_normed_eq, elim disjE, simp_all)
   110 
   111 lemma normed_dvd [iff]:
   112   "a div normalization_factor a dvd a"
   113 proof (cases "a = 0")
   114   case True then show ?thesis by simp
   115 next
   116   case False
   117   then have "a = a div normalization_factor a * normalization_factor a"
   118     by (auto intro: unit_div_mult_self)
   119   then show ?thesis ..
   120 qed
   121 
   122 lemma dvd_normed [iff]:
   123   "a dvd a div normalization_factor a"
   124 proof (cases "a = 0")
   125   case True then show ?thesis by simp
   126 next
   127   case False
   128   then have "a div normalization_factor a = a * (1 div normalization_factor a)"
   129     by (auto intro: unit_mult_div_div)
   130   then show ?thesis ..
   131 qed
   132 
   133 lemma associated_normed:
   134   "associated (a div normalization_factor a) a"
   135   by (rule associatedI) simp_all
   136 
   137 lemma normalization_factor_dvd' [simp]:
   138   "normalization_factor a dvd a"
   139   by (cases "a = 0", simp_all)
   140 
   141 lemmas normalization_factor_dvd_iff [simp] =
   142   unit_dvd_iff [OF normalization_factor_is_unit]
   143 
   144 lemma euclidean_division:
   145   fixes a :: 'a and b :: 'a
   146   assumes "b \<noteq> 0" and "\<not> b dvd a"
   147   obtains s and t where "a = s * b + t" 
   148     and "euclidean_size t < euclidean_size b"
   149 proof -
   150   from div_mod_equality [of a b 0] 
   151      have "a = a div b * b + a mod b" by simp
   152   with that and assms show ?thesis by (auto simp add: mod_size_less)
   153 qed
   154 
   155 lemma dvd_euclidean_size_eq_imp_dvd:
   156   assumes "a \<noteq> 0" and b_dvd_a: "b dvd a" and size_eq: "euclidean_size a = euclidean_size b"
   157   shows "a dvd b"
   158 proof (rule ccontr)
   159   assume "\<not> a dvd b"
   160   then have "b mod a \<noteq> 0" by (simp add: mod_eq_0_iff_dvd)
   161   from b_dvd_a have b_dvd_mod: "b dvd b mod a" by (simp add: dvd_mod_iff)
   162   from b_dvd_mod obtain c where "b mod a = b * c" unfolding dvd_def by blast
   163     with \<open>b mod a \<noteq> 0\<close> have "c \<noteq> 0" by auto
   164   with \<open>b mod a = b * c\<close> have "euclidean_size (b mod a) \<ge> euclidean_size b"
   165       using size_mult_mono by force
   166   moreover from \<open>\<not> a dvd b\<close> and \<open>a \<noteq> 0\<close>
   167   have "euclidean_size (b mod a) < euclidean_size a"
   168       using mod_size_less by blast
   169   ultimately show False using size_eq by simp
   170 qed
   171 
   172 function gcd_eucl :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
   173 where
   174   "gcd_eucl a b = (if b = 0 then a div normalization_factor a
   175     else if b dvd a then b div normalization_factor b
   176     else gcd_eucl b (a mod b))"
   177   by pat_completeness simp
   178 termination
   179   by (relation "measure (euclidean_size \<circ> snd)") (simp_all add: mod_size_less)
   180 
   181 declare gcd_eucl.simps [simp del]
   182 
   183 lemma gcd_eucl_induct [case_names zero mod]:
   184   assumes H1: "\<And>b. P b 0"
   185   and H2: "\<And>a b. b \<noteq> 0 \<Longrightarrow> P b (a mod b) \<Longrightarrow> P a b"
   186   shows "P a b"
   187 proof (induct a b rule: gcd_eucl.induct)
   188   case ("1" a b)
   189   show ?case
   190   proof (cases "b = 0")
   191     case True then show "P a b" by simp (rule H1)
   192   next
   193     case False
   194     have "P b (a mod b)"
   195     proof (cases "b dvd a")
   196       case False with \<open>b \<noteq> 0\<close> show "P b (a mod b)"
   197         by (rule "1.hyps")
   198     next
   199       case True then have "a mod b = 0"
   200         by (simp add: mod_eq_0_iff_dvd)
   201       then show "P b (a mod b)" by simp (rule H1)
   202     qed
   203     with \<open>b \<noteq> 0\<close> show "P a b"
   204       by (blast intro: H2)
   205   qed
   206 qed
   207 
   208 definition lcm_eucl :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
   209 where
   210   "lcm_eucl a b = a * b div (gcd_eucl a b * normalization_factor (a * b))"
   211 
   212 definition Lcm_eucl :: "'a set \<Rightarrow> 'a" -- \<open>
   213   Somewhat complicated definition of Lcm that has the advantage of working
   214   for infinite sets as well\<close>
   215 where
   216   "Lcm_eucl A = (if \<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) then
   217      let l = SOME l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l =
   218        (LEAST n. \<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n)
   219        in l div normalization_factor l
   220       else 0)"
   221 
   222 definition Gcd_eucl :: "'a set \<Rightarrow> 'a"
   223 where
   224   "Gcd_eucl A = Lcm_eucl {d. \<forall>a\<in>A. d dvd a}"
   225 
   226 lemma gcd_eucl_0:
   227   "gcd_eucl a 0 = a div normalization_factor a"
   228   by (simp add: gcd_eucl.simps [of a 0])
   229 
   230 lemma gcd_eucl_0_left:
   231   "gcd_eucl 0 a = a div normalization_factor a"
   232   by (simp add: gcd_eucl.simps [of 0 a])
   233 
   234 lemma gcd_eucl_non_0:
   235   "b \<noteq> 0 \<Longrightarrow> gcd_eucl a b = gcd_eucl b (a mod b)"
   236   by (cases "b dvd a")
   237     (simp_all add: gcd_eucl.simps [of a b] gcd_eucl.simps [of b 0])
   238 
   239 lemma gcd_eucl_code [code]:
   240   "gcd_eucl a b = (if b = 0 then a div normalization_factor a else gcd_eucl b (a mod b))"
   241   by (auto simp add: gcd_eucl_non_0 gcd_eucl_0 gcd_eucl_0_left) 
   242 
   243 end
   244 
   245 class euclidean_ring = euclidean_semiring + idom
   246 begin
   247 
   248 function euclid_ext :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a" where
   249   "euclid_ext a b = 
   250      (if b = 0 then 
   251         let c = 1 div normalization_factor a in (c, 0, a * c)
   252       else if b dvd a then
   253         let c = 1 div normalization_factor b in (0, c, b * c)
   254       else
   255         case euclid_ext b (a mod b) of
   256             (s, t, c) \<Rightarrow> (t, s - t * (a div b), c))"
   257   by pat_completeness simp
   258 termination
   259   by (relation "measure (euclidean_size \<circ> snd)") (simp_all add: mod_size_less)
   260 
   261 declare euclid_ext.simps [simp del]
   262 
   263 lemma euclid_ext_0: 
   264   "euclid_ext a 0 = (1 div normalization_factor a, 0, a div normalization_factor a)"
   265   by (simp add: euclid_ext.simps [of a 0])
   266 
   267 lemma euclid_ext_left_0: 
   268   "euclid_ext 0 a = (0, 1 div normalization_factor a, a div normalization_factor a)"
   269   by (simp add: euclid_ext.simps [of 0 a])
   270 
   271 lemma euclid_ext_non_0: 
   272   "b \<noteq> 0 \<Longrightarrow> euclid_ext a b = (case euclid_ext b (a mod b) of
   273     (s, t, c) \<Rightarrow> (t, s - t * (a div b), c))"
   274   by (cases "b dvd a")
   275     (simp_all add: euclid_ext.simps [of a b] euclid_ext.simps [of b 0])
   276 
   277 lemma euclid_ext_code [code]:
   278   "euclid_ext a b = (if b = 0 then (1 div normalization_factor a, 0, a div normalization_factor a)
   279     else let (s, t, c) = euclid_ext b (a mod b) in  (t, s - t * (a div b), c))"
   280   by (simp add: euclid_ext.simps [of a b] euclid_ext.simps [of b 0])
   281 
   282 lemma euclid_ext_correct:
   283   "case euclid_ext a b of (s, t, c) \<Rightarrow> s * a + t * b = c"
   284 proof (induct a b rule: gcd_eucl_induct)
   285   case (zero a) then show ?case
   286     by (simp add: euclid_ext_0 ac_simps)
   287 next
   288   case (mod a b)
   289   obtain s t c where stc: "euclid_ext b (a mod b) = (s,t,c)"
   290     by (cases "euclid_ext b (a mod b)") blast
   291   with mod have "c = s * b + t * (a mod b)" by simp
   292   also have "... = t * ((a div b) * b + a mod b) + (s - t * (a div b)) * b"
   293     by (simp add: algebra_simps) 
   294   also have "(a div b) * b + a mod b = a" using mod_div_equality .
   295   finally show ?case
   296     by (subst euclid_ext.simps) (simp add: stc mod ac_simps)
   297 qed
   298 
   299 definition euclid_ext' :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a"
   300 where
   301   "euclid_ext' a b = (case euclid_ext a b of (s, t, _) \<Rightarrow> (s, t))"
   302 
   303 lemma euclid_ext'_0: "euclid_ext' a 0 = (1 div normalization_factor a, 0)" 
   304   by (simp add: euclid_ext'_def euclid_ext_0)
   305 
   306 lemma euclid_ext'_left_0: "euclid_ext' 0 a = (0, 1 div normalization_factor a)" 
   307   by (simp add: euclid_ext'_def euclid_ext_left_0)
   308   
   309 lemma euclid_ext'_non_0: "b \<noteq> 0 \<Longrightarrow> euclid_ext' a b = (snd (euclid_ext' b (a mod b)),
   310   fst (euclid_ext' b (a mod b)) - snd (euclid_ext' b (a mod b)) * (a div b))"
   311   by (simp add: euclid_ext'_def euclid_ext_non_0 split_def)
   312 
   313 end
   314 
   315 class euclidean_semiring_gcd = euclidean_semiring + gcd + Gcd +
   316   assumes gcd_gcd_eucl: "gcd = gcd_eucl" and lcm_lcm_eucl: "lcm = lcm_eucl"
   317   assumes Gcd_Gcd_eucl: "Gcd = Gcd_eucl" and Lcm_Lcm_eucl: "Lcm = Lcm_eucl"
   318 begin
   319 
   320 lemma gcd_0_left:
   321   "gcd 0 a = a div normalization_factor a"
   322   unfolding gcd_gcd_eucl by (fact gcd_eucl_0_left)
   323 
   324 lemma gcd_0:
   325   "gcd a 0 = a div normalization_factor a"
   326   unfolding gcd_gcd_eucl by (fact gcd_eucl_0)
   327 
   328 lemma gcd_non_0:
   329   "b \<noteq> 0 \<Longrightarrow> gcd a b = gcd b (a mod b)"
   330   unfolding gcd_gcd_eucl by (fact gcd_eucl_non_0)
   331 
   332 lemma gcd_dvd1 [iff]: "gcd a b dvd a"
   333   and gcd_dvd2 [iff]: "gcd a b dvd b"
   334   by (induct a b rule: gcd_eucl_induct)
   335     (simp_all add: gcd_0 gcd_non_0 dvd_mod_iff)
   336     
   337 lemma dvd_gcd_D1: "k dvd gcd m n \<Longrightarrow> k dvd m"
   338   by (rule dvd_trans, assumption, rule gcd_dvd1)
   339 
   340 lemma dvd_gcd_D2: "k dvd gcd m n \<Longrightarrow> k dvd n"
   341   by (rule dvd_trans, assumption, rule gcd_dvd2)
   342 
   343 lemma gcd_greatest:
   344   fixes k a b :: 'a
   345   shows "k dvd a \<Longrightarrow> k dvd b \<Longrightarrow> k dvd gcd a b"
   346 proof (induct a b rule: gcd_eucl_induct)
   347   case (zero a) from zero(1) show ?case by (rule dvd_trans) (simp add: gcd_0)
   348 next
   349   case (mod a b)
   350   then show ?case
   351     by (simp add: gcd_non_0 dvd_mod_iff)
   352 qed
   353 
   354 lemma dvd_gcd_iff:
   355   "k dvd gcd a b \<longleftrightarrow> k dvd a \<and> k dvd b"
   356   by (blast intro!: gcd_greatest intro: dvd_trans)
   357 
   358 lemmas gcd_greatest_iff = dvd_gcd_iff
   359 
   360 lemma gcd_zero [simp]:
   361   "gcd a b = 0 \<longleftrightarrow> a = 0 \<and> b = 0"
   362   by (metis dvd_0_left dvd_refl gcd_dvd1 gcd_dvd2 gcd_greatest)+
   363 
   364 lemma normalization_factor_gcd [simp]:
   365   "normalization_factor (gcd a b) = (if a = 0 \<and> b = 0 then 0 else 1)" (is "?f a b = ?g a b")
   366   by (induct a b rule: gcd_eucl_induct)
   367     (auto simp add: gcd_0 gcd_non_0)
   368 
   369 lemma gcdI:
   370   "k dvd a \<Longrightarrow> k dvd b \<Longrightarrow> (\<And>l. l dvd a \<Longrightarrow> l dvd b \<Longrightarrow> l dvd k)
   371     \<Longrightarrow> normalization_factor k = (if k = 0 then 0 else 1) \<Longrightarrow> k = gcd a b"
   372   by (intro normed_associated_imp_eq) (auto simp: associated_def intro: gcd_greatest)
   373 
   374 sublocale gcd!: abel_semigroup gcd
   375 proof
   376   fix a b c 
   377   show "gcd (gcd a b) c = gcd a (gcd b c)"
   378   proof (rule gcdI)
   379     have "gcd (gcd a b) c dvd gcd a b" "gcd a b dvd a" by simp_all
   380     then show "gcd (gcd a b) c dvd a" by (rule dvd_trans)
   381     have "gcd (gcd a b) c dvd gcd a b" "gcd a b dvd b" by simp_all
   382     hence "gcd (gcd a b) c dvd b" by (rule dvd_trans)
   383     moreover have "gcd (gcd a b) c dvd c" by simp
   384     ultimately show "gcd (gcd a b) c dvd gcd b c"
   385       by (rule gcd_greatest)
   386     show "normalization_factor (gcd (gcd a b) c) =  (if gcd (gcd a b) c = 0 then 0 else 1)"
   387       by auto
   388     fix l assume "l dvd a" and "l dvd gcd b c"
   389     with dvd_trans[OF _ gcd_dvd1] and dvd_trans[OF _ gcd_dvd2]
   390       have "l dvd b" and "l dvd c" by blast+
   391     with \<open>l dvd a\<close> show "l dvd gcd (gcd a b) c"
   392       by (intro gcd_greatest)
   393   qed
   394 next
   395   fix a b
   396   show "gcd a b = gcd b a"
   397     by (rule gcdI) (simp_all add: gcd_greatest)
   398 qed
   399 
   400 lemma gcd_unique: "d dvd a \<and> d dvd b \<and> 
   401     normalization_factor d = (if d = 0 then 0 else 1) \<and>
   402     (\<forall>e. e dvd a \<and> e dvd b \<longrightarrow> e dvd d) \<longleftrightarrow> d = gcd a b"
   403   by (rule, auto intro: gcdI simp: gcd_greatest)
   404 
   405 lemma gcd_dvd_prod: "gcd a b dvd k * b"
   406   using mult_dvd_mono [of 1] by auto
   407 
   408 lemma gcd_1_left [simp]: "gcd 1 a = 1"
   409   by (rule sym, rule gcdI, simp_all)
   410 
   411 lemma gcd_1 [simp]: "gcd a 1 = 1"
   412   by (rule sym, rule gcdI, simp_all)
   413 
   414 lemma gcd_proj2_if_dvd: 
   415   "b dvd a \<Longrightarrow> gcd a b = b div normalization_factor b"
   416   by (cases "b = 0", simp_all add: dvd_eq_mod_eq_0 gcd_non_0 gcd_0)
   417 
   418 lemma gcd_proj1_if_dvd: 
   419   "a dvd b \<Longrightarrow> gcd a b = a div normalization_factor a"
   420   by (subst gcd.commute, simp add: gcd_proj2_if_dvd)
   421 
   422 lemma gcd_proj1_iff: "gcd m n = m div normalization_factor m \<longleftrightarrow> m dvd n"
   423 proof
   424   assume A: "gcd m n = m div normalization_factor m"
   425   show "m dvd n"
   426   proof (cases "m = 0")
   427     assume [simp]: "m \<noteq> 0"
   428     from A have B: "m = gcd m n * normalization_factor m"
   429       by (simp add: unit_eq_div2)
   430     show ?thesis by (subst B, simp add: mult_unit_dvd_iff)
   431   qed (insert A, simp)
   432 next
   433   assume "m dvd n"
   434   then show "gcd m n = m div normalization_factor m" by (rule gcd_proj1_if_dvd)
   435 qed
   436   
   437 lemma gcd_proj2_iff: "gcd m n = n div normalization_factor n \<longleftrightarrow> n dvd m"
   438   by (subst gcd.commute, simp add: gcd_proj1_iff)
   439 
   440 lemma gcd_mod1 [simp]:
   441   "gcd (a mod b) b = gcd a b"
   442   by (rule gcdI, metis dvd_mod_iff gcd_dvd1 gcd_dvd2, simp_all add: gcd_greatest dvd_mod_iff)
   443 
   444 lemma gcd_mod2 [simp]:
   445   "gcd a (b mod a) = gcd a b"
   446   by (rule gcdI, simp, metis dvd_mod_iff gcd_dvd1 gcd_dvd2, simp_all add: gcd_greatest dvd_mod_iff)
   447          
   448 lemma gcd_mult_distrib': 
   449   "c div normalization_factor c * gcd a b = gcd (c * a) (c * b)"
   450 proof (cases "c = 0")
   451   case True then show ?thesis by (simp_all add: gcd_0)
   452 next
   453   case False then have [simp]: "is_unit (normalization_factor c)" by simp
   454   show ?thesis
   455   proof (induct a b rule: gcd_eucl_induct)
   456     case (zero a) show ?case
   457     proof (cases "a = 0")
   458       case True then show ?thesis by (simp add: gcd_0)
   459     next
   460       case False then have "is_unit (normalization_factor a)" by simp
   461       then show ?thesis
   462         by (simp add: gcd_0 unit_div_commute unit_div_mult_swap normalization_factor_mult is_unit_div_mult2_eq)
   463     qed
   464     case (mod a b)
   465     then show ?case by (simp add: mult_mod_right gcd.commute)
   466   qed
   467 qed
   468 
   469 lemma gcd_mult_distrib:
   470   "k * gcd a b = gcd (k*a) (k*b) * normalization_factor k"
   471 proof-
   472   let ?nf = "normalization_factor"
   473   from gcd_mult_distrib' 
   474     have "gcd (k*a) (k*b) = k div ?nf k * gcd a b" ..
   475   also have "... = k * gcd a b div ?nf k"
   476     by (metis dvd_div_mult dvd_eq_mod_eq_0 mod_0 normalization_factor_dvd)
   477   finally show ?thesis
   478     by simp
   479 qed
   480 
   481 lemma euclidean_size_gcd_le1 [simp]:
   482   assumes "a \<noteq> 0"
   483   shows "euclidean_size (gcd a b) \<le> euclidean_size a"
   484 proof -
   485    have "gcd a b dvd a" by (rule gcd_dvd1)
   486    then obtain c where A: "a = gcd a b * c" unfolding dvd_def by blast
   487    with \<open>a \<noteq> 0\<close> show ?thesis by (subst (2) A, intro size_mult_mono) auto
   488 qed
   489 
   490 lemma euclidean_size_gcd_le2 [simp]:
   491   "b \<noteq> 0 \<Longrightarrow> euclidean_size (gcd a b) \<le> euclidean_size b"
   492   by (subst gcd.commute, rule euclidean_size_gcd_le1)
   493 
   494 lemma euclidean_size_gcd_less1:
   495   assumes "a \<noteq> 0" and "\<not>a dvd b"
   496   shows "euclidean_size (gcd a b) < euclidean_size a"
   497 proof (rule ccontr)
   498   assume "\<not>euclidean_size (gcd a b) < euclidean_size a"
   499   with \<open>a \<noteq> 0\<close> have "euclidean_size (gcd a b) = euclidean_size a"
   500     by (intro le_antisym, simp_all)
   501   with assms have "a dvd gcd a b" by (auto intro: dvd_euclidean_size_eq_imp_dvd)
   502   hence "a dvd b" using dvd_gcd_D2 by blast
   503   with \<open>\<not>a dvd b\<close> show False by contradiction
   504 qed
   505 
   506 lemma euclidean_size_gcd_less2:
   507   assumes "b \<noteq> 0" and "\<not>b dvd a"
   508   shows "euclidean_size (gcd a b) < euclidean_size b"
   509   using assms by (subst gcd.commute, rule euclidean_size_gcd_less1)
   510 
   511 lemma gcd_mult_unit1: "is_unit a \<Longrightarrow> gcd (b * a) c = gcd b c"
   512   apply (rule gcdI)
   513   apply (rule dvd_trans, rule gcd_dvd1, simp add: unit_simps)
   514   apply (rule gcd_dvd2)
   515   apply (rule gcd_greatest, simp add: unit_simps, assumption)
   516   apply (subst normalization_factor_gcd, simp add: gcd_0)
   517   done
   518 
   519 lemma gcd_mult_unit2: "is_unit a \<Longrightarrow> gcd b (c * a) = gcd b c"
   520   by (subst gcd.commute, subst gcd_mult_unit1, assumption, rule gcd.commute)
   521 
   522 lemma gcd_div_unit1: "is_unit a \<Longrightarrow> gcd (b div a) c = gcd b c"
   523   by (erule is_unitE [of _ b]) (simp add: gcd_mult_unit1)
   524 
   525 lemma gcd_div_unit2: "is_unit a \<Longrightarrow> gcd b (c div a) = gcd b c"
   526   by (erule is_unitE [of _ c]) (simp add: gcd_mult_unit2)
   527 
   528 lemma gcd_idem: "gcd a a = a div normalization_factor a"
   529   by (cases "a = 0") (simp add: gcd_0_left, rule sym, rule gcdI, simp_all)
   530 
   531 lemma gcd_right_idem: "gcd (gcd a b) b = gcd a b"
   532   apply (rule gcdI)
   533   apply (simp add: ac_simps)
   534   apply (rule gcd_dvd2)
   535   apply (rule gcd_greatest, erule (1) gcd_greatest, assumption)
   536   apply simp
   537   done
   538 
   539 lemma gcd_left_idem: "gcd a (gcd a b) = gcd a b"
   540   apply (rule gcdI)
   541   apply simp
   542   apply (rule dvd_trans, rule gcd_dvd2, rule gcd_dvd2)
   543   apply (rule gcd_greatest, assumption, erule gcd_greatest, assumption)
   544   apply simp
   545   done
   546 
   547 lemma comp_fun_idem_gcd: "comp_fun_idem gcd"
   548 proof
   549   fix a b show "gcd a \<circ> gcd b = gcd b \<circ> gcd a"
   550     by (simp add: fun_eq_iff ac_simps)
   551 next
   552   fix a show "gcd a \<circ> gcd a = gcd a"
   553     by (simp add: fun_eq_iff gcd_left_idem)
   554 qed
   555 
   556 lemma coprime_dvd_mult:
   557   assumes "gcd c b = 1" and "c dvd a * b"
   558   shows "c dvd a"
   559 proof -
   560   let ?nf = "normalization_factor"
   561   from assms gcd_mult_distrib [of a c b] 
   562     have A: "a = gcd (a * c) (a * b) * ?nf a" by simp
   563   from \<open>c dvd a * b\<close> show ?thesis by (subst A, simp_all add: gcd_greatest)
   564 qed
   565 
   566 lemma coprime_dvd_mult_iff:
   567   "gcd c b = 1 \<Longrightarrow> (c dvd a * b) = (c dvd a)"
   568   by (rule, rule coprime_dvd_mult, simp_all)
   569 
   570 lemma gcd_dvd_antisym:
   571   "gcd a b dvd gcd c d \<Longrightarrow> gcd c d dvd gcd a b \<Longrightarrow> gcd a b = gcd c d"
   572 proof (rule gcdI)
   573   assume A: "gcd a b dvd gcd c d" and B: "gcd c d dvd gcd a b"
   574   have "gcd c d dvd c" by simp
   575   with A show "gcd a b dvd c" by (rule dvd_trans)
   576   have "gcd c d dvd d" by simp
   577   with A show "gcd a b dvd d" by (rule dvd_trans)
   578   show "normalization_factor (gcd a b) = (if gcd a b = 0 then 0 else 1)"
   579     by simp
   580   fix l assume "l dvd c" and "l dvd d"
   581   hence "l dvd gcd c d" by (rule gcd_greatest)
   582   from this and B show "l dvd gcd a b" by (rule dvd_trans)
   583 qed
   584 
   585 lemma gcd_mult_cancel:
   586   assumes "gcd k n = 1"
   587   shows "gcd (k * m) n = gcd m n"
   588 proof (rule gcd_dvd_antisym)
   589   have "gcd (gcd (k * m) n) k = gcd (gcd k n) (k * m)" by (simp add: ac_simps)
   590   also note \<open>gcd k n = 1\<close>
   591   finally have "gcd (gcd (k * m) n) k = 1" by simp
   592   hence "gcd (k * m) n dvd m" by (rule coprime_dvd_mult, simp add: ac_simps)
   593   moreover have "gcd (k * m) n dvd n" by simp
   594   ultimately show "gcd (k * m) n dvd gcd m n" by (rule gcd_greatest)
   595   have "gcd m n dvd (k * m)" and "gcd m n dvd n" by simp_all
   596   then show "gcd m n dvd gcd (k * m) n" by (rule gcd_greatest)
   597 qed
   598 
   599 lemma coprime_crossproduct:
   600   assumes [simp]: "gcd a d = 1" "gcd b c = 1"
   601   shows "associated (a * c) (b * d) \<longleftrightarrow> associated a b \<and> associated c d" (is "?lhs \<longleftrightarrow> ?rhs")
   602 proof
   603   assume ?rhs then show ?lhs unfolding associated_def by (fast intro: mult_dvd_mono)
   604 next
   605   assume ?lhs
   606   from \<open>?lhs\<close> have "a dvd b * d" unfolding associated_def by (metis dvd_mult_left) 
   607   hence "a dvd b" by (simp add: coprime_dvd_mult_iff)
   608   moreover from \<open>?lhs\<close> have "b dvd a * c" unfolding associated_def by (metis dvd_mult_left) 
   609   hence "b dvd a" by (simp add: coprime_dvd_mult_iff)
   610   moreover from \<open>?lhs\<close> have "c dvd d * b" 
   611     unfolding associated_def by (auto dest: dvd_mult_right simp add: ac_simps)
   612   hence "c dvd d" by (simp add: coprime_dvd_mult_iff gcd.commute)
   613   moreover from \<open>?lhs\<close> have "d dvd c * a"
   614     unfolding associated_def by (auto dest: dvd_mult_right simp add: ac_simps)
   615   hence "d dvd c" by (simp add: coprime_dvd_mult_iff gcd.commute)
   616   ultimately show ?rhs unfolding associated_def by simp
   617 qed
   618 
   619 lemma gcd_add1 [simp]:
   620   "gcd (m + n) n = gcd m n"
   621   by (cases "n = 0", simp_all add: gcd_non_0)
   622 
   623 lemma gcd_add2 [simp]:
   624   "gcd m (m + n) = gcd m n"
   625   using gcd_add1 [of n m] by (simp add: ac_simps)
   626 
   627 lemma gcd_add_mult:
   628   "gcd m (k * m + n) = gcd m n"
   629 proof -
   630   have "gcd m ((k * m + n) mod m) = gcd m (k * m + n)"
   631     by (fact gcd_mod2)
   632   then show ?thesis by simp 
   633 qed
   634 
   635 lemma coprimeI: "(\<And>l. \<lbrakk>l dvd a; l dvd b\<rbrakk> \<Longrightarrow> l dvd 1) \<Longrightarrow> gcd a b = 1"
   636   by (rule sym, rule gcdI, simp_all)
   637 
   638 lemma coprime: "gcd a b = 1 \<longleftrightarrow> (\<forall>d. d dvd a \<and> d dvd b \<longleftrightarrow> is_unit d)"
   639   by (auto intro: coprimeI gcd_greatest dvd_gcd_D1 dvd_gcd_D2)
   640 
   641 lemma div_gcd_coprime:
   642   assumes nz: "a \<noteq> 0 \<or> b \<noteq> 0"
   643   defines [simp]: "d \<equiv> gcd a b"
   644   defines [simp]: "a' \<equiv> a div d" and [simp]: "b' \<equiv> b div d"
   645   shows "gcd a' b' = 1"
   646 proof (rule coprimeI)
   647   fix l assume "l dvd a'" "l dvd b'"
   648   then obtain s t where "a' = l * s" "b' = l * t" unfolding dvd_def by blast
   649   moreover have "a = a' * d" "b = b' * d" by simp_all
   650   ultimately have "a = (l * d) * s" "b = (l * d) * t"
   651     by (simp_all only: ac_simps)
   652   hence "l*d dvd a" and "l*d dvd b" by (simp_all only: dvd_triv_left)
   653   hence "l*d dvd d" by (simp add: gcd_greatest)
   654   then obtain u where "d = l * d * u" ..
   655   then have "d * (l * u) = d" by (simp add: ac_simps)
   656   moreover from nz have "d \<noteq> 0" by simp
   657   with div_mult_self1_is_id have "d * (l * u) div d = l * u" . 
   658   ultimately have "1 = l * u"
   659     using \<open>d \<noteq> 0\<close> by simp
   660   then show "l dvd 1" ..
   661 qed
   662 
   663 lemma coprime_mult: 
   664   assumes da: "gcd d a = 1" and db: "gcd d b = 1"
   665   shows "gcd d (a * b) = 1"
   666   apply (subst gcd.commute)
   667   using da apply (subst gcd_mult_cancel)
   668   apply (subst gcd.commute, assumption)
   669   apply (subst gcd.commute, rule db)
   670   done
   671 
   672 lemma coprime_lmult:
   673   assumes dab: "gcd d (a * b) = 1" 
   674   shows "gcd d a = 1"
   675 proof (rule coprimeI)
   676   fix l assume "l dvd d" and "l dvd a"
   677   hence "l dvd a * b" by simp
   678   with \<open>l dvd d\<close> and dab show "l dvd 1" by (auto intro: gcd_greatest)
   679 qed
   680 
   681 lemma coprime_rmult:
   682   assumes dab: "gcd d (a * b) = 1"
   683   shows "gcd d b = 1"
   684 proof (rule coprimeI)
   685   fix l assume "l dvd d" and "l dvd b"
   686   hence "l dvd a * b" by simp
   687   with \<open>l dvd d\<close> and dab show "l dvd 1" by (auto intro: gcd_greatest)
   688 qed
   689 
   690 lemma coprime_mul_eq: "gcd d (a * b) = 1 \<longleftrightarrow> gcd d a = 1 \<and> gcd d b = 1"
   691   using coprime_rmult[of d a b] coprime_lmult[of d a b] coprime_mult[of d a b] by blast
   692 
   693 lemma gcd_coprime:
   694   assumes c: "gcd a b \<noteq> 0" and a: "a = a' * gcd a b" and b: "b = b' * gcd a b"
   695   shows "gcd a' b' = 1"
   696 proof -
   697   from c have "a \<noteq> 0 \<or> b \<noteq> 0" by simp
   698   with div_gcd_coprime have "gcd (a div gcd a b) (b div gcd a b) = 1" .
   699   also from assms have "a div gcd a b = a'" by (metis div_mult_self2_is_id)+
   700   also from assms have "b div gcd a b = b'" by (metis div_mult_self2_is_id)+
   701   finally show ?thesis .
   702 qed
   703 
   704 lemma coprime_power:
   705   assumes "0 < n"
   706   shows "gcd a (b ^ n) = 1 \<longleftrightarrow> gcd a b = 1"
   707 using assms proof (induct n)
   708   case (Suc n) then show ?case
   709     by (cases n) (simp_all add: coprime_mul_eq)
   710 qed simp
   711 
   712 lemma gcd_coprime_exists:
   713   assumes nz: "gcd a b \<noteq> 0"
   714   shows "\<exists>a' b'. a = a' * gcd a b \<and> b = b' * gcd a b \<and> gcd a' b' = 1"
   715   apply (rule_tac x = "a div gcd a b" in exI)
   716   apply (rule_tac x = "b div gcd a b" in exI)
   717   apply (insert nz, auto intro: div_gcd_coprime)
   718   done
   719 
   720 lemma coprime_exp:
   721   "gcd d a = 1 \<Longrightarrow> gcd d (a^n) = 1"
   722   by (induct n, simp_all add: coprime_mult)
   723 
   724 lemma coprime_exp2 [intro]:
   725   "gcd a b = 1 \<Longrightarrow> gcd (a^n) (b^m) = 1"
   726   apply (rule coprime_exp)
   727   apply (subst gcd.commute)
   728   apply (rule coprime_exp)
   729   apply (subst gcd.commute)
   730   apply assumption
   731   done
   732 
   733 lemma gcd_exp:
   734   "gcd (a^n) (b^n) = (gcd a b) ^ n"
   735 proof (cases "a = 0 \<and> b = 0")
   736   assume "a = 0 \<and> b = 0"
   737   then show ?thesis by (cases n, simp_all add: gcd_0_left)
   738 next
   739   assume A: "\<not>(a = 0 \<and> b = 0)"
   740   hence "1 = gcd ((a div gcd a b)^n) ((b div gcd a b)^n)"
   741     using div_gcd_coprime by (subst sym, auto simp: div_gcd_coprime)
   742   hence "(gcd a b) ^ n = (gcd a b) ^ n * ..." by simp
   743   also note gcd_mult_distrib
   744   also have "normalization_factor ((gcd a b)^n) = 1"
   745     by (simp add: normalization_factor_pow A)
   746   also have "(gcd a b)^n * (a div gcd a b)^n = a^n"
   747     by (subst ac_simps, subst div_power, simp, rule dvd_div_mult_self, rule dvd_power_same, simp)
   748   also have "(gcd a b)^n * (b div gcd a b)^n = b^n"
   749     by (subst ac_simps, subst div_power, simp, rule dvd_div_mult_self, rule dvd_power_same, simp)
   750   finally show ?thesis by simp
   751 qed
   752 
   753 lemma coprime_common_divisor: 
   754   "gcd a b = 1 \<Longrightarrow> a dvd a \<Longrightarrow> a dvd b \<Longrightarrow> is_unit a"
   755   apply (subgoal_tac "a dvd gcd a b")
   756   apply simp
   757   apply (erule (1) gcd_greatest)
   758   done
   759 
   760 lemma division_decomp: 
   761   assumes dc: "a dvd b * c"
   762   shows "\<exists>b' c'. a = b' * c' \<and> b' dvd b \<and> c' dvd c"
   763 proof (cases "gcd a b = 0")
   764   assume "gcd a b = 0"
   765   hence "a = 0 \<and> b = 0" by simp
   766   hence "a = 0 * c \<and> 0 dvd b \<and> c dvd c" by simp
   767   then show ?thesis by blast
   768 next
   769   let ?d = "gcd a b"
   770   assume "?d \<noteq> 0"
   771   from gcd_coprime_exists[OF this]
   772     obtain a' b' where ab': "a = a' * ?d" "b = b' * ?d" "gcd a' b' = 1"
   773     by blast
   774   from ab'(1) have "a' dvd a" unfolding dvd_def by blast
   775   with dc have "a' dvd b*c" using dvd_trans[of a' a "b*c"] by simp
   776   from dc ab'(1,2) have "a'*?d dvd (b'*?d) * c" by simp
   777   hence "?d * a' dvd ?d * (b' * c)" by (simp add: mult_ac)
   778   with \<open>?d \<noteq> 0\<close> have "a' dvd b' * c" by simp
   779   with coprime_dvd_mult[OF ab'(3)] 
   780     have "a' dvd c" by (subst (asm) ac_simps, blast)
   781   with ab'(1) have "a = ?d * a' \<and> ?d dvd b \<and> a' dvd c" by (simp add: mult_ac)
   782   then show ?thesis by blast
   783 qed
   784 
   785 lemma pow_divs_pow:
   786   assumes ab: "a ^ n dvd b ^ n" and n: "n \<noteq> 0"
   787   shows "a dvd b"
   788 proof (cases "gcd a b = 0")
   789   assume "gcd a b = 0"
   790   then show ?thesis by simp
   791 next
   792   let ?d = "gcd a b"
   793   assume "?d \<noteq> 0"
   794   from n obtain m where m: "n = Suc m" by (cases n, simp_all)
   795   from \<open>?d \<noteq> 0\<close> have zn: "?d ^ n \<noteq> 0" by (rule power_not_zero)
   796   from gcd_coprime_exists[OF \<open>?d \<noteq> 0\<close>]
   797     obtain a' b' where ab': "a = a' * ?d" "b = b' * ?d" "gcd a' b' = 1"
   798     by blast
   799   from ab have "(a' * ?d) ^ n dvd (b' * ?d) ^ n"
   800     by (simp add: ab'(1,2)[symmetric])
   801   hence "?d^n * a'^n dvd ?d^n * b'^n"
   802     by (simp only: power_mult_distrib ac_simps)
   803   with zn have "a'^n dvd b'^n" by simp
   804   hence "a' dvd b'^n" using dvd_trans[of a' "a'^n" "b'^n"] by (simp add: m)
   805   hence "a' dvd b'^m * b'" by (simp add: m ac_simps)
   806   with coprime_dvd_mult[OF coprime_exp[OF ab'(3), of m]]
   807     have "a' dvd b'" by (subst (asm) ac_simps, blast)
   808   hence "a'*?d dvd b'*?d" by (rule mult_dvd_mono, simp)
   809   with ab'(1,2) show ?thesis by simp
   810 qed
   811 
   812 lemma pow_divs_eq [simp]:
   813   "n \<noteq> 0 \<Longrightarrow> a ^ n dvd b ^ n \<longleftrightarrow> a dvd b"
   814   by (auto intro: pow_divs_pow dvd_power_same)
   815 
   816 lemma divs_mult:
   817   assumes mr: "m dvd r" and nr: "n dvd r" and mn: "gcd m n = 1"
   818   shows "m * n dvd r"
   819 proof -
   820   from mr nr obtain m' n' where m': "r = m*m'" and n': "r = n*n'"
   821     unfolding dvd_def by blast
   822   from mr n' have "m dvd n'*n" by (simp add: ac_simps)
   823   hence "m dvd n'" using coprime_dvd_mult_iff[OF mn] by simp
   824   then obtain k where k: "n' = m*k" unfolding dvd_def by blast
   825   with n' have "r = m * n * k" by (simp add: mult_ac)
   826   then show ?thesis unfolding dvd_def by blast
   827 qed
   828 
   829 lemma coprime_plus_one [simp]: "gcd (n + 1) n = 1"
   830   by (subst add_commute, simp)
   831 
   832 lemma setprod_coprime [rule_format]:
   833   "(\<forall>i\<in>A. gcd (f i) a = 1) \<longrightarrow> gcd (\<Prod>i\<in>A. f i) a = 1"
   834   apply (cases "finite A")
   835   apply (induct set: finite)
   836   apply (auto simp add: gcd_mult_cancel)
   837   done
   838 
   839 lemma coprime_divisors: 
   840   assumes "d dvd a" "e dvd b" "gcd a b = 1"
   841   shows "gcd d e = 1" 
   842 proof -
   843   from assms obtain k l where "a = d * k" "b = e * l"
   844     unfolding dvd_def by blast
   845   with assms have "gcd (d * k) (e * l) = 1" by simp
   846   hence "gcd (d * k) e = 1" by (rule coprime_lmult)
   847   also have "gcd (d * k) e = gcd e (d * k)" by (simp add: ac_simps)
   848   finally have "gcd e d = 1" by (rule coprime_lmult)
   849   then show ?thesis by (simp add: ac_simps)
   850 qed
   851 
   852 lemma invertible_coprime:
   853   assumes "a * b mod m = 1"
   854   shows "coprime a m"
   855 proof -
   856   from assms have "coprime m (a * b mod m)"
   857     by simp
   858   then have "coprime m (a * b)"
   859     by simp
   860   then have "coprime m a"
   861     by (rule coprime_lmult)
   862   then show ?thesis
   863     by (simp add: ac_simps)
   864 qed
   865 
   866 lemma lcm_gcd:
   867   "lcm a b = a * b div (gcd a b * normalization_factor (a*b))"
   868   by (simp only: lcm_lcm_eucl gcd_gcd_eucl lcm_eucl_def)
   869 
   870 lemma lcm_gcd_prod:
   871   "lcm a b * gcd a b = a * b div normalization_factor (a*b)"
   872 proof (cases "a * b = 0")
   873   let ?nf = normalization_factor
   874   assume "a * b \<noteq> 0"
   875   hence "gcd a b \<noteq> 0" by simp
   876   from lcm_gcd have "lcm a b * gcd a b = gcd a b * (a * b div (?nf (a*b) * gcd a b))" 
   877     by (simp add: mult_ac)
   878   also from \<open>a * b \<noteq> 0\<close> have "... = a * b div ?nf (a*b)"
   879     by (simp add: div_mult_swap mult.commute)
   880   finally show ?thesis .
   881 qed (auto simp add: lcm_gcd)
   882 
   883 lemma lcm_dvd1 [iff]:
   884   "a dvd lcm a b"
   885 proof (cases "a*b = 0")
   886   assume "a * b \<noteq> 0"
   887   hence "gcd a b \<noteq> 0" by simp
   888   let ?c = "1 div normalization_factor (a * b)"
   889   from \<open>a * b \<noteq> 0\<close> have [simp]: "is_unit (normalization_factor (a * b))" by simp
   890   from lcm_gcd_prod[of a b] have "lcm a b * gcd a b = a * ?c * b"
   891     by (simp add: div_mult_swap unit_div_commute)
   892   hence "lcm a b * gcd a b div gcd a b = a * ?c * b div gcd a b" by simp
   893   with \<open>gcd a b \<noteq> 0\<close> have "lcm a b = a * ?c * b div gcd a b"
   894     by (subst (asm) div_mult_self2_is_id, simp_all)
   895   also have "... = a * (?c * b div gcd a b)"
   896     by (metis div_mult_swap gcd_dvd2 mult_assoc)
   897   finally show ?thesis by (rule dvdI)
   898 qed (auto simp add: lcm_gcd)
   899 
   900 lemma lcm_least:
   901   "\<lbrakk>a dvd k; b dvd k\<rbrakk> \<Longrightarrow> lcm a b dvd k"
   902 proof (cases "k = 0")
   903   let ?nf = normalization_factor
   904   assume "k \<noteq> 0"
   905   hence "is_unit (?nf k)" by simp
   906   hence "?nf k \<noteq> 0" by (metis not_is_unit_0)
   907   assume A: "a dvd k" "b dvd k"
   908   hence "gcd a b \<noteq> 0" using \<open>k \<noteq> 0\<close> by auto
   909   from A obtain r s where ar: "k = a * r" and bs: "k = b * s" 
   910     unfolding dvd_def by blast
   911   with \<open>k \<noteq> 0\<close> have "r * s \<noteq> 0"
   912     by auto (drule sym [of 0], simp)
   913   hence "is_unit (?nf (r * s))" by simp
   914   let ?c = "?nf k div ?nf (r*s)"
   915   from \<open>is_unit (?nf k)\<close> and \<open>is_unit (?nf (r * s))\<close> have "is_unit ?c" by (rule unit_div)
   916   hence "?c \<noteq> 0" using not_is_unit_0 by fast 
   917   from ar bs have "k * k * gcd s r = ?nf k * k * gcd (k * s) (k * r)"
   918     by (subst mult_assoc, subst gcd_mult_distrib[of k s r], simp only: ac_simps)
   919   also have "... = ?nf k * k * gcd ((r*s) * a) ((r*s) * b)"
   920     by (subst (3) \<open>k = a * r\<close>, subst (3) \<open>k = b * s\<close>, simp add: algebra_simps)
   921   also have "... = ?c * r*s * k * gcd a b" using \<open>r * s \<noteq> 0\<close>
   922     by (subst gcd_mult_distrib'[symmetric], simp add: algebra_simps unit_simps)
   923   finally have "(a*r) * (b*s) * gcd s r = ?c * k * r * s * gcd a b"
   924     by (subst ar[symmetric], subst bs[symmetric], simp add: mult_ac)
   925   hence "a * b * gcd s r * (r * s) = ?c * k * gcd a b * (r * s)"
   926     by (simp add: algebra_simps)
   927   hence "?c * k * gcd a b = a * b * gcd s r" using \<open>r * s \<noteq> 0\<close>
   928     by (metis div_mult_self2_is_id)
   929   also have "... = lcm a b * gcd a b * gcd s r * ?nf (a*b)"
   930     by (subst lcm_gcd_prod[of a b], metis gcd_mult_distrib gcd_mult_distrib') 
   931   also have "... = lcm a b * gcd s r * ?nf (a*b) * gcd a b"
   932     by (simp add: algebra_simps)
   933   finally have "k * ?c = lcm a b * gcd s r * ?nf (a*b)" using \<open>gcd a b \<noteq> 0\<close>
   934     by (metis mult.commute div_mult_self2_is_id)
   935   hence "k = lcm a b * (gcd s r * ?nf (a*b)) div ?c" using \<open>?c \<noteq> 0\<close>
   936     by (metis div_mult_self2_is_id mult_assoc) 
   937   also have "... = lcm a b * (gcd s r * ?nf (a*b) div ?c)" using \<open>is_unit ?c\<close>
   938     by (simp add: unit_simps)
   939   finally show ?thesis by (rule dvdI)
   940 qed simp
   941 
   942 lemma lcm_zero:
   943   "lcm a b = 0 \<longleftrightarrow> a = 0 \<or> b = 0"
   944 proof -
   945   let ?nf = normalization_factor
   946   {
   947     assume "a \<noteq> 0" "b \<noteq> 0"
   948     hence "a * b div ?nf (a * b) \<noteq> 0" by (simp add: no_zero_divisors)
   949     moreover from \<open>a \<noteq> 0\<close> and \<open>b \<noteq> 0\<close> have "gcd a b \<noteq> 0" by simp
   950     ultimately have "lcm a b \<noteq> 0" using lcm_gcd_prod[of a b] by (intro notI, simp)
   951   } moreover {
   952     assume "a = 0 \<or> b = 0"
   953     hence "lcm a b = 0" by (elim disjE, simp_all add: lcm_gcd)
   954   }
   955   ultimately show ?thesis by blast
   956 qed
   957 
   958 lemmas lcm_0_iff = lcm_zero
   959 
   960 lemma gcd_lcm: 
   961   assumes "lcm a b \<noteq> 0"
   962   shows "gcd a b = a * b div (lcm a b * normalization_factor (a * b))"
   963 proof-
   964   from assms have "gcd a b \<noteq> 0" by (simp add: lcm_zero)
   965   let ?c = "normalization_factor (a * b)"
   966   from \<open>lcm a b \<noteq> 0\<close> have "?c \<noteq> 0" by (intro notI, simp add: lcm_zero no_zero_divisors)
   967   hence "is_unit ?c" by simp
   968   from lcm_gcd_prod [of a b] have "gcd a b = a * b div ?c div lcm a b"
   969     by (subst (2) div_mult_self2_is_id[OF \<open>lcm a b \<noteq> 0\<close>, symmetric], simp add: mult_ac)
   970   also from \<open>is_unit ?c\<close> have "... = a * b div (lcm a b * ?c)"
   971     by (metis \<open>?c \<noteq> 0\<close> div_mult_mult1 dvd_mult_div_cancel mult_commute normalization_factor_dvd')
   972   finally show ?thesis .
   973 qed
   974 
   975 lemma normalization_factor_lcm [simp]:
   976   "normalization_factor (lcm a b) = (if a = 0 \<or> b = 0 then 0 else 1)"
   977 proof (cases "a = 0 \<or> b = 0")
   978   case True then show ?thesis
   979     by (auto simp add: lcm_gcd) 
   980 next
   981   case False
   982   let ?nf = normalization_factor
   983   from lcm_gcd_prod[of a b] 
   984     have "?nf (lcm a b) * ?nf (gcd a b) = ?nf (a*b) div ?nf (a*b)"
   985     by (metis div_by_0 div_self normalization_correct normalization_factor_0 normalization_factor_mult)
   986   also have "... = (if a*b = 0 then 0 else 1)"
   987     by simp
   988   finally show ?thesis using False by simp
   989 qed
   990 
   991 lemma lcm_dvd2 [iff]: "b dvd lcm a b"
   992   using lcm_dvd1 [of b a] by (simp add: lcm_gcd ac_simps)
   993 
   994 lemma lcmI:
   995   "\<lbrakk>a dvd k; b dvd k; \<And>l. a dvd l \<Longrightarrow> b dvd l \<Longrightarrow> k dvd l;
   996     normalization_factor k = (if k = 0 then 0 else 1)\<rbrakk> \<Longrightarrow> k = lcm a b"
   997   by (intro normed_associated_imp_eq) (auto simp: associated_def intro: lcm_least)
   998 
   999 sublocale lcm!: abel_semigroup lcm
  1000 proof
  1001   fix a b c
  1002   show "lcm (lcm a b) c = lcm a (lcm b c)"
  1003   proof (rule lcmI)
  1004     have "a dvd lcm a b" and "lcm a b dvd lcm (lcm a b) c" by simp_all
  1005     then show "a dvd lcm (lcm a b) c" by (rule dvd_trans)
  1006     
  1007     have "b dvd lcm a b" and "lcm a b dvd lcm (lcm a b) c" by simp_all
  1008     hence "b dvd lcm (lcm a b) c" by (rule dvd_trans)
  1009     moreover have "c dvd lcm (lcm a b) c" by simp
  1010     ultimately show "lcm b c dvd lcm (lcm a b) c" by (rule lcm_least)
  1011 
  1012     fix l assume "a dvd l" and "lcm b c dvd l"
  1013     have "b dvd lcm b c" by simp
  1014     from this and \<open>lcm b c dvd l\<close> have "b dvd l" by (rule dvd_trans)
  1015     have "c dvd lcm b c" by simp
  1016     from this and \<open>lcm b c dvd l\<close> have "c dvd l" by (rule dvd_trans)
  1017     from \<open>a dvd l\<close> and \<open>b dvd l\<close> have "lcm a b dvd l" by (rule lcm_least)
  1018     from this and \<open>c dvd l\<close> show "lcm (lcm a b) c dvd l" by (rule lcm_least)
  1019   qed (simp add: lcm_zero)
  1020 next
  1021   fix a b
  1022   show "lcm a b = lcm b a"
  1023     by (simp add: lcm_gcd ac_simps)
  1024 qed
  1025 
  1026 lemma dvd_lcm_D1:
  1027   "lcm m n dvd k \<Longrightarrow> m dvd k"
  1028   by (rule dvd_trans, rule lcm_dvd1, assumption)
  1029 
  1030 lemma dvd_lcm_D2:
  1031   "lcm m n dvd k \<Longrightarrow> n dvd k"
  1032   by (rule dvd_trans, rule lcm_dvd2, assumption)
  1033 
  1034 lemma gcd_dvd_lcm [simp]:
  1035   "gcd a b dvd lcm a b"
  1036   by (metis dvd_trans gcd_dvd2 lcm_dvd2)
  1037 
  1038 lemma lcm_1_iff:
  1039   "lcm a b = 1 \<longleftrightarrow> is_unit a \<and> is_unit b"
  1040 proof
  1041   assume "lcm a b = 1"
  1042   then show "is_unit a \<and> is_unit b" by auto
  1043 next
  1044   assume "is_unit a \<and> is_unit b"
  1045   hence "a dvd 1" and "b dvd 1" by simp_all
  1046   hence "is_unit (lcm a b)" by (rule lcm_least)
  1047   hence "lcm a b = normalization_factor (lcm a b)"
  1048     by (subst normalization_factor_unit, simp_all)
  1049   also have "\<dots> = 1" using \<open>is_unit a \<and> is_unit b\<close>
  1050     by auto
  1051   finally show "lcm a b = 1" .
  1052 qed
  1053 
  1054 lemma lcm_0_left [simp]:
  1055   "lcm 0 a = 0"
  1056   by (rule sym, rule lcmI, simp_all)
  1057 
  1058 lemma lcm_0 [simp]:
  1059   "lcm a 0 = 0"
  1060   by (rule sym, rule lcmI, simp_all)
  1061 
  1062 lemma lcm_unique:
  1063   "a dvd d \<and> b dvd d \<and> 
  1064   normalization_factor d = (if d = 0 then 0 else 1) \<and>
  1065   (\<forall>e. a dvd e \<and> b dvd e \<longrightarrow> d dvd e) \<longleftrightarrow> d = lcm a b"
  1066   by (rule, auto intro: lcmI simp: lcm_least lcm_zero)
  1067 
  1068 lemma dvd_lcm_I1 [simp]:
  1069   "k dvd m \<Longrightarrow> k dvd lcm m n"
  1070   by (metis lcm_dvd1 dvd_trans)
  1071 
  1072 lemma dvd_lcm_I2 [simp]:
  1073   "k dvd n \<Longrightarrow> k dvd lcm m n"
  1074   by (metis lcm_dvd2 dvd_trans)
  1075 
  1076 lemma lcm_1_left [simp]:
  1077   "lcm 1 a = a div normalization_factor a"
  1078   by (cases "a = 0") (simp, rule sym, rule lcmI, simp_all)
  1079 
  1080 lemma lcm_1_right [simp]:
  1081   "lcm a 1 = a div normalization_factor a"
  1082   using lcm_1_left [of a] by (simp add: ac_simps)
  1083 
  1084 lemma lcm_coprime:
  1085   "gcd a b = 1 \<Longrightarrow> lcm a b = a * b div normalization_factor (a*b)"
  1086   by (subst lcm_gcd) simp
  1087 
  1088 lemma lcm_proj1_if_dvd: 
  1089   "b dvd a \<Longrightarrow> lcm a b = a div normalization_factor a"
  1090   by (cases "a = 0") (simp, rule sym, rule lcmI, simp_all)
  1091 
  1092 lemma lcm_proj2_if_dvd: 
  1093   "a dvd b \<Longrightarrow> lcm a b = b div normalization_factor b"
  1094   using lcm_proj1_if_dvd [of a b] by (simp add: ac_simps)
  1095 
  1096 lemma lcm_proj1_iff:
  1097   "lcm m n = m div normalization_factor m \<longleftrightarrow> n dvd m"
  1098 proof
  1099   assume A: "lcm m n = m div normalization_factor m"
  1100   show "n dvd m"
  1101   proof (cases "m = 0")
  1102     assume [simp]: "m \<noteq> 0"
  1103     from A have B: "m = lcm m n * normalization_factor m"
  1104       by (simp add: unit_eq_div2)
  1105     show ?thesis by (subst B, simp)
  1106   qed simp
  1107 next
  1108   assume "n dvd m"
  1109   then show "lcm m n = m div normalization_factor m" by (rule lcm_proj1_if_dvd)
  1110 qed
  1111 
  1112 lemma lcm_proj2_iff:
  1113   "lcm m n = n div normalization_factor n \<longleftrightarrow> m dvd n"
  1114   using lcm_proj1_iff [of n m] by (simp add: ac_simps)
  1115 
  1116 lemma euclidean_size_lcm_le1: 
  1117   assumes "a \<noteq> 0" and "b \<noteq> 0"
  1118   shows "euclidean_size a \<le> euclidean_size (lcm a b)"
  1119 proof -
  1120   have "a dvd lcm a b" by (rule lcm_dvd1)
  1121   then obtain c where A: "lcm a b = a * c" unfolding dvd_def by blast
  1122   with \<open>a \<noteq> 0\<close> and \<open>b \<noteq> 0\<close> have "c \<noteq> 0" by (auto simp: lcm_zero)
  1123   then show ?thesis by (subst A, intro size_mult_mono)
  1124 qed
  1125 
  1126 lemma euclidean_size_lcm_le2:
  1127   "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> euclidean_size b \<le> euclidean_size (lcm a b)"
  1128   using euclidean_size_lcm_le1 [of b a] by (simp add: ac_simps)
  1129 
  1130 lemma euclidean_size_lcm_less1:
  1131   assumes "b \<noteq> 0" and "\<not>b dvd a"
  1132   shows "euclidean_size a < euclidean_size (lcm a b)"
  1133 proof (rule ccontr)
  1134   from assms have "a \<noteq> 0" by auto
  1135   assume "\<not>euclidean_size a < euclidean_size (lcm a b)"
  1136   with \<open>a \<noteq> 0\<close> and \<open>b \<noteq> 0\<close> have "euclidean_size (lcm a b) = euclidean_size a"
  1137     by (intro le_antisym, simp, intro euclidean_size_lcm_le1)
  1138   with assms have "lcm a b dvd a" 
  1139     by (rule_tac dvd_euclidean_size_eq_imp_dvd) (auto simp: lcm_zero)
  1140   hence "b dvd a" by (rule dvd_lcm_D2)
  1141   with \<open>\<not>b dvd a\<close> show False by contradiction
  1142 qed
  1143 
  1144 lemma euclidean_size_lcm_less2:
  1145   assumes "a \<noteq> 0" and "\<not>a dvd b"
  1146   shows "euclidean_size b < euclidean_size (lcm a b)"
  1147   using assms euclidean_size_lcm_less1 [of a b] by (simp add: ac_simps)
  1148 
  1149 lemma lcm_mult_unit1:
  1150   "is_unit a \<Longrightarrow> lcm (b * a) c = lcm b c"
  1151   apply (rule lcmI)
  1152   apply (rule dvd_trans[of _ "b * a"], simp, rule lcm_dvd1)
  1153   apply (rule lcm_dvd2)
  1154   apply (rule lcm_least, simp add: unit_simps, assumption)
  1155   apply (subst normalization_factor_lcm, simp add: lcm_zero)
  1156   done
  1157 
  1158 lemma lcm_mult_unit2:
  1159   "is_unit a \<Longrightarrow> lcm b (c * a) = lcm b c"
  1160   using lcm_mult_unit1 [of a c b] by (simp add: ac_simps)
  1161 
  1162 lemma lcm_div_unit1:
  1163   "is_unit a \<Longrightarrow> lcm (b div a) c = lcm b c"
  1164   by (erule is_unitE [of _ b]) (simp add: lcm_mult_unit1) 
  1165 
  1166 lemma lcm_div_unit2:
  1167   "is_unit a \<Longrightarrow> lcm b (c div a) = lcm b c"
  1168   by (erule is_unitE [of _ c]) (simp add: lcm_mult_unit2)
  1169 
  1170 lemma lcm_left_idem:
  1171   "lcm a (lcm a b) = lcm a b"
  1172   apply (rule lcmI)
  1173   apply simp
  1174   apply (subst lcm.assoc [symmetric], rule lcm_dvd2)
  1175   apply (rule lcm_least, assumption)
  1176   apply (erule (1) lcm_least)
  1177   apply (auto simp: lcm_zero)
  1178   done
  1179 
  1180 lemma lcm_right_idem:
  1181   "lcm (lcm a b) b = lcm a b"
  1182   apply (rule lcmI)
  1183   apply (subst lcm.assoc, rule lcm_dvd1)
  1184   apply (rule lcm_dvd2)
  1185   apply (rule lcm_least, erule (1) lcm_least, assumption)
  1186   apply (auto simp: lcm_zero)
  1187   done
  1188 
  1189 lemma comp_fun_idem_lcm: "comp_fun_idem lcm"
  1190 proof
  1191   fix a b show "lcm a \<circ> lcm b = lcm b \<circ> lcm a"
  1192     by (simp add: fun_eq_iff ac_simps)
  1193 next
  1194   fix a show "lcm a \<circ> lcm a = lcm a" unfolding o_def
  1195     by (intro ext, simp add: lcm_left_idem)
  1196 qed
  1197 
  1198 lemma dvd_Lcm [simp]: "a \<in> A \<Longrightarrow> a dvd Lcm A"
  1199   and Lcm_dvd [simp]: "(\<forall>a\<in>A. a dvd l') \<Longrightarrow> Lcm A dvd l'"
  1200   and normalization_factor_Lcm [simp]: 
  1201           "normalization_factor (Lcm A) = (if Lcm A = 0 then 0 else 1)"
  1202 proof -
  1203   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>
  1204     normalization_factor (Lcm A) = (if Lcm A = 0 then 0 else 1)" (is ?thesis)
  1205   proof (cases "\<exists>l. l \<noteq>  0 \<and> (\<forall>a\<in>A. a dvd l)")
  1206     case False
  1207     hence "Lcm A = 0" by (auto simp: Lcm_Lcm_eucl Lcm_eucl_def)
  1208     with False show ?thesis by auto
  1209   next
  1210     case True
  1211     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
  1212     def n \<equiv> "LEAST n. \<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"
  1213     def l \<equiv> "SOME l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"
  1214     have "\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"
  1215       apply (subst n_def)
  1216       apply (rule LeastI[of _ "euclidean_size l\<^sub>0"])
  1217       apply (rule exI[of _ l\<^sub>0])
  1218       apply (simp add: l\<^sub>0_props)
  1219       done
  1220     from someI_ex[OF this] have "l \<noteq> 0" and "\<forall>a\<in>A. a dvd l" and "euclidean_size l = n" 
  1221       unfolding l_def by simp_all
  1222     {
  1223       fix l' assume "\<forall>a\<in>A. a dvd l'"
  1224       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)
  1225       moreover from \<open>l \<noteq> 0\<close> have "gcd l l' \<noteq> 0" by simp
  1226       ultimately have "\<exists>b. b \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd b) \<and> euclidean_size b = euclidean_size (gcd l l')"
  1227         by (intro exI[of _ "gcd l l'"], auto)
  1228       hence "euclidean_size (gcd l l') \<ge> n" by (subst n_def) (rule Least_le)
  1229       moreover have "euclidean_size (gcd l l') \<le> n"
  1230       proof -
  1231         have "gcd l l' dvd l" by simp
  1232         then obtain a where "l = gcd l l' * a" unfolding dvd_def by blast
  1233         with \<open>l \<noteq> 0\<close> have "a \<noteq> 0" by auto
  1234         hence "euclidean_size (gcd l l') \<le> euclidean_size (gcd l l' * a)"
  1235           by (rule size_mult_mono)
  1236         also have "gcd l l' * a = l" using \<open>l = gcd l l' * a\<close> ..
  1237         also note \<open>euclidean_size l = n\<close>
  1238         finally show "euclidean_size (gcd l l') \<le> n" .
  1239       qed
  1240       ultimately have "euclidean_size l = euclidean_size (gcd l l')" 
  1241         by (intro le_antisym, simp_all add: \<open>euclidean_size l = n\<close>)
  1242       with \<open>l \<noteq> 0\<close> have "l dvd gcd l l'" by (blast intro: dvd_euclidean_size_eq_imp_dvd)
  1243       hence "l dvd l'" by (blast dest: dvd_gcd_D2)
  1244     }
  1245 
  1246     with \<open>(\<forall>a\<in>A. a dvd l)\<close> and normalization_factor_is_unit[OF \<open>l \<noteq> 0\<close>] and \<open>l \<noteq> 0\<close>
  1247       have "(\<forall>a\<in>A. a dvd l div normalization_factor l) \<and> 
  1248         (\<forall>l'. (\<forall>a\<in>A. a dvd l') \<longrightarrow> l div normalization_factor l dvd l') \<and>
  1249         normalization_factor (l div normalization_factor l) = 
  1250         (if l div normalization_factor l = 0 then 0 else 1)"
  1251       by (auto simp: unit_simps)
  1252     also from True have "l div normalization_factor l = Lcm A"
  1253       by (simp add: Lcm_Lcm_eucl Lcm_eucl_def Let_def n_def l_def)
  1254     finally show ?thesis .
  1255   qed
  1256   note A = this
  1257 
  1258   {fix a assume "a \<in> A" then show "a dvd Lcm A" using A by blast}
  1259   {fix l' assume "\<forall>a\<in>A. a dvd l'" then show "Lcm A dvd l'" using A by blast}
  1260   from A show "normalization_factor (Lcm A) = (if Lcm A = 0 then 0 else 1)" by blast
  1261 qed
  1262     
  1263 lemma LcmI:
  1264   "(\<And>a. a\<in>A \<Longrightarrow> a dvd l) \<Longrightarrow> (\<And>l'. (\<forall>a\<in>A. a dvd l') \<Longrightarrow> l dvd l') \<Longrightarrow>
  1265       normalization_factor l = (if l = 0 then 0 else 1) \<Longrightarrow> l = Lcm A"
  1266   by (intro normed_associated_imp_eq)
  1267     (auto intro: Lcm_dvd dvd_Lcm simp: associated_def)
  1268 
  1269 lemma Lcm_subset:
  1270   "A \<subseteq> B \<Longrightarrow> Lcm A dvd Lcm B"
  1271   by (blast intro: Lcm_dvd dvd_Lcm)
  1272 
  1273 lemma Lcm_Un:
  1274   "Lcm (A \<union> B) = lcm (Lcm A) (Lcm B)"
  1275   apply (rule lcmI)
  1276   apply (blast intro: Lcm_subset)
  1277   apply (blast intro: Lcm_subset)
  1278   apply (intro Lcm_dvd ballI, elim UnE)
  1279   apply (rule dvd_trans, erule dvd_Lcm, assumption)
  1280   apply (rule dvd_trans, erule dvd_Lcm, assumption)
  1281   apply simp
  1282   done
  1283 
  1284 lemma Lcm_1_iff:
  1285   "Lcm A = 1 \<longleftrightarrow> (\<forall>a\<in>A. is_unit a)"
  1286 proof
  1287   assume "Lcm A = 1"
  1288   then show "\<forall>a\<in>A. is_unit a" by auto
  1289 qed (rule LcmI [symmetric], auto)
  1290 
  1291 lemma Lcm_no_units:
  1292   "Lcm A = Lcm (A - {a. is_unit a})"
  1293 proof -
  1294   have "(A - {a. is_unit a}) \<union> {a\<in>A. is_unit a} = A" by blast
  1295   hence "Lcm A = lcm (Lcm (A - {a. is_unit a})) (Lcm {a\<in>A. is_unit a})"
  1296     by (simp add: Lcm_Un[symmetric])
  1297   also have "Lcm {a\<in>A. is_unit a} = 1" by (simp add: Lcm_1_iff)
  1298   finally show ?thesis by simp
  1299 qed
  1300 
  1301 lemma Lcm_empty [simp]:
  1302   "Lcm {} = 1"
  1303   by (simp add: Lcm_1_iff)
  1304 
  1305 lemma Lcm_eq_0 [simp]:
  1306   "0 \<in> A \<Longrightarrow> Lcm A = 0"
  1307   by (drule dvd_Lcm) simp
  1308 
  1309 lemma Lcm0_iff':
  1310   "Lcm A = 0 \<longleftrightarrow> \<not>(\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l))"
  1311 proof
  1312   assume "Lcm A = 0"
  1313   show "\<not>(\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l))"
  1314   proof
  1315     assume ex: "\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l)"
  1316     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
  1317     def n \<equiv> "LEAST n. \<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"
  1318     def l \<equiv> "SOME l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"
  1319     have "\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"
  1320       apply (subst n_def)
  1321       apply (rule LeastI[of _ "euclidean_size l\<^sub>0"])
  1322       apply (rule exI[of _ l\<^sub>0])
  1323       apply (simp add: l\<^sub>0_props)
  1324       done
  1325     from someI_ex[OF this] have "l \<noteq> 0" unfolding l_def by simp_all
  1326     hence "l div normalization_factor l \<noteq> 0" by simp
  1327     also from ex have "l div normalization_factor l = Lcm A"
  1328        by (simp only: Lcm_Lcm_eucl Lcm_eucl_def n_def l_def if_True Let_def)
  1329     finally show False using \<open>Lcm A = 0\<close> by contradiction
  1330   qed
  1331 qed (simp only: Lcm_Lcm_eucl Lcm_eucl_def if_False)
  1332 
  1333 lemma Lcm0_iff [simp]:
  1334   "finite A \<Longrightarrow> Lcm A = 0 \<longleftrightarrow> 0 \<in> A"
  1335 proof -
  1336   assume "finite A"
  1337   have "0 \<in> A \<Longrightarrow> Lcm A = 0"  by (intro dvd_0_left dvd_Lcm)
  1338   moreover {
  1339     assume "0 \<notin> A"
  1340     hence "\<Prod>A \<noteq> 0" 
  1341       apply (induct rule: finite_induct[OF \<open>finite A\<close>]) 
  1342       apply simp
  1343       apply (subst setprod.insert, assumption, assumption)
  1344       apply (rule no_zero_divisors)
  1345       apply blast+
  1346       done
  1347     moreover from \<open>finite A\<close> have "\<forall>a\<in>A. a dvd \<Prod>A" by blast
  1348     ultimately have "\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l)" by blast
  1349     with Lcm0_iff' have "Lcm A \<noteq> 0" by simp
  1350   }
  1351   ultimately show "Lcm A = 0 \<longleftrightarrow> 0 \<in> A" by blast
  1352 qed
  1353 
  1354 lemma Lcm_no_multiple:
  1355   "(\<forall>m. m \<noteq> 0 \<longrightarrow> (\<exists>a\<in>A. \<not>a dvd m)) \<Longrightarrow> Lcm A = 0"
  1356 proof -
  1357   assume "\<forall>m. m \<noteq> 0 \<longrightarrow> (\<exists>a\<in>A. \<not>a dvd m)"
  1358   hence "\<not>(\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l))" by blast
  1359   then show "Lcm A = 0" by (simp only: Lcm_Lcm_eucl Lcm_eucl_def if_False)
  1360 qed
  1361 
  1362 lemma Lcm_insert [simp]:
  1363   "Lcm (insert a A) = lcm a (Lcm A)"
  1364 proof (rule lcmI)
  1365   fix l assume "a dvd l" and "Lcm A dvd l"
  1366   hence "\<forall>a\<in>A. a dvd l" by (blast intro: dvd_trans dvd_Lcm)
  1367   with \<open>a dvd l\<close> show "Lcm (insert a A) dvd l" by (force intro: Lcm_dvd)
  1368 qed (auto intro: Lcm_dvd dvd_Lcm)
  1369  
  1370 lemma Lcm_finite:
  1371   assumes "finite A"
  1372   shows "Lcm A = Finite_Set.fold lcm 1 A"
  1373   by (induct rule: finite.induct[OF \<open>finite A\<close>])
  1374     (simp_all add: comp_fun_idem.fold_insert_idem[OF comp_fun_idem_lcm])
  1375 
  1376 lemma Lcm_set [code_unfold]:
  1377   "Lcm (set xs) = fold lcm xs 1"
  1378   using comp_fun_idem.fold_set_fold[OF comp_fun_idem_lcm] Lcm_finite by (simp add: ac_simps)
  1379 
  1380 lemma Lcm_singleton [simp]:
  1381   "Lcm {a} = a div normalization_factor a"
  1382   by simp
  1383 
  1384 lemma Lcm_2 [simp]:
  1385   "Lcm {a,b} = lcm a b"
  1386   by (simp only: Lcm_insert Lcm_empty lcm_1_right)
  1387     (cases "b = 0", simp, rule lcm_div_unit2, simp)
  1388 
  1389 lemma Lcm_coprime:
  1390   assumes "finite A" and "A \<noteq> {}" 
  1391   assumes "\<And>a b. a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<noteq> b \<Longrightarrow> gcd a b = 1"
  1392   shows "Lcm A = \<Prod>A div normalization_factor (\<Prod>A)"
  1393 using assms proof (induct rule: finite_ne_induct)
  1394   case (insert a A)
  1395   have "Lcm (insert a A) = lcm a (Lcm A)" by simp
  1396   also from insert have "Lcm A = \<Prod>A div normalization_factor (\<Prod>A)" by blast
  1397   also have "lcm a \<dots> = lcm a (\<Prod>A)" by (cases "\<Prod>A = 0") (simp_all add: lcm_div_unit2)
  1398   also from insert have "gcd a (\<Prod>A) = 1" by (subst gcd.commute, intro setprod_coprime) auto
  1399   with insert have "lcm a (\<Prod>A) = \<Prod>(insert a A) div normalization_factor (\<Prod>(insert a A))"
  1400     by (simp add: lcm_coprime)
  1401   finally show ?case .
  1402 qed simp
  1403       
  1404 lemma Lcm_coprime':
  1405   "card A \<noteq> 0 \<Longrightarrow> (\<And>a b. a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<noteq> b \<Longrightarrow> gcd a b = 1)
  1406     \<Longrightarrow> Lcm A = \<Prod>A div normalization_factor (\<Prod>A)"
  1407   by (rule Lcm_coprime) (simp_all add: card_eq_0_iff)
  1408 
  1409 lemma Gcd_Lcm:
  1410   "Gcd A = Lcm {d. \<forall>a\<in>A. d dvd a}"
  1411   by (simp add: Gcd_Gcd_eucl Lcm_Lcm_eucl Gcd_eucl_def)
  1412 
  1413 lemma Gcd_dvd [simp]: "a \<in> A \<Longrightarrow> Gcd A dvd a"
  1414   and dvd_Gcd [simp]: "(\<forall>a\<in>A. g' dvd a) \<Longrightarrow> g' dvd Gcd A"
  1415   and normalization_factor_Gcd [simp]: 
  1416     "normalization_factor (Gcd A) = (if Gcd A = 0 then 0 else 1)"
  1417 proof -
  1418   fix a assume "a \<in> A"
  1419   hence "Lcm {d. \<forall>a\<in>A. d dvd a} dvd a" by (intro Lcm_dvd) blast
  1420   then show "Gcd A dvd a" by (simp add: Gcd_Lcm)
  1421 next
  1422   fix g' assume "\<forall>a\<in>A. g' dvd a"
  1423   hence "g' dvd Lcm {d. \<forall>a\<in>A. d dvd a}" by (intro dvd_Lcm) blast
  1424   then show "g' dvd Gcd A" by (simp add: Gcd_Lcm)
  1425 next
  1426   show "normalization_factor (Gcd A) = (if Gcd A = 0 then 0 else 1)"
  1427     by (simp add: Gcd_Lcm)
  1428 qed
  1429 
  1430 lemma GcdI:
  1431   "(\<And>a. a\<in>A \<Longrightarrow> l dvd a) \<Longrightarrow> (\<And>l'. (\<forall>a\<in>A. l' dvd a) \<Longrightarrow> l' dvd l) \<Longrightarrow>
  1432     normalization_factor l = (if l = 0 then 0 else 1) \<Longrightarrow> l = Gcd A"
  1433   by (intro normed_associated_imp_eq)
  1434     (auto intro: Gcd_dvd dvd_Gcd simp: associated_def)
  1435 
  1436 lemma Lcm_Gcd:
  1437   "Lcm A = Gcd {m. \<forall>a\<in>A. a dvd m}"
  1438   by (rule LcmI[symmetric]) (auto intro: dvd_Gcd Gcd_dvd)
  1439 
  1440 lemma Gcd_0_iff:
  1441   "Gcd A = 0 \<longleftrightarrow> A \<subseteq> {0}"
  1442   apply (rule iffI)
  1443   apply (rule subsetI, drule Gcd_dvd, simp)
  1444   apply (auto intro: GcdI[symmetric])
  1445   done
  1446 
  1447 lemma Gcd_empty [simp]:
  1448   "Gcd {} = 0"
  1449   by (simp add: Gcd_0_iff)
  1450 
  1451 lemma Gcd_1:
  1452   "1 \<in> A \<Longrightarrow> Gcd A = 1"
  1453   by (intro GcdI[symmetric]) (auto intro: Gcd_dvd dvd_Gcd)
  1454 
  1455 lemma Gcd_insert [simp]:
  1456   "Gcd (insert a A) = gcd a (Gcd A)"
  1457 proof (rule gcdI)
  1458   fix l assume "l dvd a" and "l dvd Gcd A"
  1459   hence "\<forall>a\<in>A. l dvd a" by (blast intro: dvd_trans Gcd_dvd)
  1460   with \<open>l dvd a\<close> show "l dvd Gcd (insert a A)" by (force intro: Gcd_dvd)
  1461 qed auto
  1462 
  1463 lemma Gcd_finite:
  1464   assumes "finite A"
  1465   shows "Gcd A = Finite_Set.fold gcd 0 A"
  1466   by (induct rule: finite.induct[OF \<open>finite A\<close>])
  1467     (simp_all add: comp_fun_idem.fold_insert_idem[OF comp_fun_idem_gcd])
  1468 
  1469 lemma Gcd_set [code_unfold]:
  1470   "Gcd (set xs) = fold gcd xs 0"
  1471   using comp_fun_idem.fold_set_fold[OF comp_fun_idem_gcd] Gcd_finite by (simp add: ac_simps)
  1472 
  1473 lemma Gcd_singleton [simp]: "Gcd {a} = a div normalization_factor a"
  1474   by (simp add: gcd_0)
  1475 
  1476 lemma Gcd_2 [simp]: "Gcd {a,b} = gcd a b"
  1477   by (simp only: Gcd_insert Gcd_empty gcd_0) (cases "b = 0", simp, rule gcd_div_unit2, simp)
  1478 
  1479 subclass semiring_gcd
  1480   by unfold_locales (simp_all add: gcd_greatest_iff)
  1481   
  1482 end
  1483 
  1484 text \<open>
  1485   A Euclidean ring is a Euclidean semiring with additive inverses. It provides a 
  1486   few more lemmas; in particular, Bezout's lemma holds for any Euclidean ring.
  1487 \<close>
  1488 
  1489 class euclidean_ring_gcd = euclidean_semiring_gcd + idom
  1490 begin
  1491 
  1492 subclass euclidean_ring ..
  1493 
  1494 subclass ring_gcd ..
  1495 
  1496 lemma euclid_ext_gcd [simp]:
  1497   "(case euclid_ext a b of (_, _ , t) \<Rightarrow> t) = gcd a b"
  1498   by (induct a b rule: gcd_eucl_induct)
  1499     (simp_all add: euclid_ext_0 gcd_0 euclid_ext_non_0 ac_simps split: prod.split prod.split_asm)
  1500 
  1501 lemma euclid_ext_gcd' [simp]:
  1502   "euclid_ext a b = (r, s, t) \<Longrightarrow> t = gcd a b"
  1503   by (insert euclid_ext_gcd[of a b], drule (1) subst, simp)
  1504   
  1505 lemma euclid_ext'_correct:
  1506   "fst (euclid_ext' a b) * a + snd (euclid_ext' a b) * b = gcd a b"
  1507 proof-
  1508   obtain s t c where "euclid_ext a b = (s,t,c)"
  1509     by (cases "euclid_ext a b", blast)
  1510   with euclid_ext_correct[of a b] euclid_ext_gcd[of a b]
  1511     show ?thesis unfolding euclid_ext'_def by simp
  1512 qed
  1513 
  1514 lemma bezout: "\<exists>s t. s * a + t * b = gcd a b"
  1515   using euclid_ext'_correct by blast
  1516 
  1517 lemma gcd_neg1 [simp]:
  1518   "gcd (-a) b = gcd a b"
  1519   by (rule sym, rule gcdI, simp_all add: gcd_greatest)
  1520 
  1521 lemma gcd_neg2 [simp]:
  1522   "gcd a (-b) = gcd a b"
  1523   by (rule sym, rule gcdI, simp_all add: gcd_greatest)
  1524 
  1525 lemma gcd_neg_numeral_1 [simp]:
  1526   "gcd (- numeral n) a = gcd (numeral n) a"
  1527   by (fact gcd_neg1)
  1528 
  1529 lemma gcd_neg_numeral_2 [simp]:
  1530   "gcd a (- numeral n) = gcd a (numeral n)"
  1531   by (fact gcd_neg2)
  1532 
  1533 lemma gcd_diff1: "gcd (m - n) n = gcd m n"
  1534   by (subst diff_conv_add_uminus, subst gcd_neg2[symmetric],  subst gcd_add1, simp)
  1535 
  1536 lemma gcd_diff2: "gcd (n - m) n = gcd m n"
  1537   by (subst gcd_neg1[symmetric], simp only: minus_diff_eq gcd_diff1)
  1538 
  1539 lemma coprime_minus_one [simp]: "gcd (n - 1) n = 1"
  1540 proof -
  1541   have "gcd (n - 1) n = gcd n (n - 1)" by (fact gcd.commute)
  1542   also have "\<dots> = gcd ((n - 1) + 1) (n - 1)" by simp
  1543   also have "\<dots> = 1" by (rule coprime_plus_one)
  1544   finally show ?thesis .
  1545 qed
  1546 
  1547 lemma lcm_neg1 [simp]: "lcm (-a) b = lcm a b"
  1548   by (rule sym, rule lcmI, simp_all add: lcm_least lcm_zero)
  1549 
  1550 lemma lcm_neg2 [simp]: "lcm a (-b) = lcm a b"
  1551   by (rule sym, rule lcmI, simp_all add: lcm_least lcm_zero)
  1552 
  1553 lemma lcm_neg_numeral_1 [simp]: "lcm (- numeral n) a = lcm (numeral n) a"
  1554   by (fact lcm_neg1)
  1555 
  1556 lemma lcm_neg_numeral_2 [simp]: "lcm a (- numeral n) = lcm a (numeral n)"
  1557   by (fact lcm_neg2)
  1558 
  1559 end
  1560 
  1561 
  1562 subsection \<open>Typical instances\<close>
  1563 
  1564 instantiation nat :: euclidean_semiring
  1565 begin
  1566 
  1567 definition [simp]:
  1568   "euclidean_size_nat = (id :: nat \<Rightarrow> nat)"
  1569 
  1570 definition [simp]:
  1571   "normalization_factor_nat (n::nat) = (if n = 0 then 0 else 1 :: nat)"
  1572 
  1573 instance proof
  1574 qed simp_all
  1575 
  1576 end
  1577 
  1578 instantiation int :: euclidean_ring
  1579 begin
  1580 
  1581 definition [simp]:
  1582   "euclidean_size_int = (nat \<circ> abs :: int \<Rightarrow> nat)"
  1583 
  1584 definition [simp]:
  1585   "normalization_factor_int = (sgn :: int \<Rightarrow> int)"
  1586 
  1587 instance
  1588 proof (default, goals)
  1589   case 2
  1590   then show ?case by (auto simp add: abs_mult nat_mult_distrib)
  1591 next
  1592   case 3
  1593   then show ?case by (simp add: zsgn_def)
  1594 next
  1595   case 5
  1596   then show ?case by (auto simp: zsgn_def)
  1597 next
  1598   case 6
  1599   then show ?case by (auto split: abs_split simp: zsgn_def)
  1600 qed (auto simp: sgn_times split: abs_split)
  1601 
  1602 end
  1603 
  1604 instantiation poly :: (field) euclidean_ring
  1605 begin
  1606 
  1607 definition euclidean_size_poly :: "'a poly \<Rightarrow> nat"
  1608   where "euclidean_size = (degree :: 'a poly \<Rightarrow> nat)"
  1609 
  1610 definition normalization_factor_poly :: "'a poly \<Rightarrow> 'a poly"
  1611   where "normalization_factor p = monom (coeff p (degree p)) 0"
  1612 
  1613 instance
  1614 proof (default, unfold euclidean_size_poly_def normalization_factor_poly_def)
  1615   fix p q :: "'a poly"
  1616   assume "q \<noteq> 0" and "\<not> q dvd p"
  1617   then show "degree (p mod q) < degree q"
  1618     using degree_mod_less [of q p] by (simp add: mod_eq_0_iff_dvd)
  1619 next
  1620   fix p q :: "'a poly"
  1621   assume "q \<noteq> 0"
  1622   from \<open>q \<noteq> 0\<close> show "degree p \<le> degree (p * q)"
  1623     by (rule degree_mult_right_le)
  1624   from \<open>q \<noteq> 0\<close> show "is_unit (monom (coeff q (degree q)) 0)"
  1625     by (auto intro: is_unit_monom_0)
  1626 next
  1627   fix p :: "'a poly"
  1628   show "monom (coeff p (degree p)) 0 = p" if "is_unit p"
  1629     using that by (fact is_unit_monom_trival)
  1630 next
  1631   fix p q :: "'a poly"
  1632   show "monom (coeff (p * q) (degree (p * q))) 0 =
  1633     monom (coeff p (degree p)) 0 * monom (coeff q (degree q)) 0"
  1634     by (simp add: monom_0 coeff_degree_mult)
  1635 next
  1636   show "monom (coeff 0 (degree 0)) 0 = 0"
  1637     by simp
  1638 qed
  1639 
  1640 end
  1641 
  1642 end