src/HOL/Number_Theory/Euclidean_Algorithm.thy
 author haftmann Sat Jun 27 20:20:36 2015 +0200 (2015-06-27) changeset 60599 f8bb070dc98b parent 60598 78ca5674c66a child 60600 87fbfea0bd0a permissions -rw-r--r--
tuned proof
     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" (is "?P \<longleftrightarrow> ?Q")

    86 proof (cases "a = 0 \<or> b = 0")

    87   case True then show ?thesis by (auto dest: sym)

    88 next

    89   case False then have "a \<noteq> 0" and "b \<noteq> 0" by auto

    90   show ?thesis

    91   proof

    92     assume ?Q

    93     from \<open>a \<noteq> 0\<close> \<open>b \<noteq> 0\<close>

    94     have "is_unit (normalization_factor a div normalization_factor b)"

    95       by auto

    96     moreover from \<open>?Q\<close> \<open>a \<noteq> 0\<close> \<open>b \<noteq> 0\<close>

    97     have "a = (normalization_factor a div normalization_factor b) * b"

    98       by (simp add: ac_simps div_mult_swap unit_eq_div1)

    99     ultimately show "associated a b" by (rule is_unit_associatedI)

   100   next

   101     assume ?P

   102     then obtain c where "is_unit c" and "a = c * b"

   103       by (blast elim: associated_is_unitE)

   104     then show ?Q

   105       by (auto simp add: normalization_factor_mult normalization_factor_unit)

   106   qed

   107 qed

   108

   109 lemma normed_associated_imp_eq:

   110   "associated a b \<Longrightarrow> normalization_factor a \<in> {0, 1} \<Longrightarrow> normalization_factor b \<in> {0, 1} \<Longrightarrow> a = b"

   111   by (simp add: associated_iff_normed_eq, elim disjE, simp_all)

   112

   113 lemma normed_dvd [iff]:

   114   "a div normalization_factor a dvd a"

   115 proof (cases "a = 0")

   116   case True then show ?thesis by simp

   117 next

   118   case False

   119   then have "a = a div normalization_factor a * normalization_factor a"

   120     by (auto intro: unit_div_mult_self)

   121   then show ?thesis ..

   122 qed

   123

   124 lemma dvd_normed [iff]:

   125   "a dvd a div normalization_factor a"

   126 proof (cases "a = 0")

   127   case True then show ?thesis by simp

   128 next

   129   case False

   130   then have "a div normalization_factor a = a * (1 div normalization_factor a)"

   131     by (auto intro: unit_mult_div_div)

   132   then show ?thesis ..

   133 qed

   134

   135 lemma associated_normed:

   136   "associated (a div normalization_factor a) a"

   137   by (rule associatedI) simp_all

   138

   139 lemma normalization_factor_dvd' [simp]:

   140   "normalization_factor a dvd a"

   141   by (cases "a = 0", simp_all)

   142

   143 lemmas normalization_factor_dvd_iff [simp] =

   144   unit_dvd_iff [OF normalization_factor_is_unit]

   145

   146 lemma euclidean_division:

   147   fixes a :: 'a and b :: 'a

   148   assumes "b \<noteq> 0" and "\<not> b dvd a"

   149   obtains s and t where "a = s * b + t"

   150     and "euclidean_size t < euclidean_size b"

   151 proof -

   152   from div_mod_equality [of a b 0]

   153      have "a = a div b * b + a mod b" by simp

   154   with that and assms show ?thesis by (auto simp add: mod_size_less)

   155 qed

   156

   157 lemma dvd_euclidean_size_eq_imp_dvd:

   158   assumes "a \<noteq> 0" and b_dvd_a: "b dvd a" and size_eq: "euclidean_size a = euclidean_size b"

   159   shows "a dvd b"

   160 proof (rule ccontr)

   161   assume "\<not> a dvd b"

   162   then have "b mod a \<noteq> 0" by (simp add: mod_eq_0_iff_dvd)

   163   from b_dvd_a have b_dvd_mod: "b dvd b mod a" by (simp add: dvd_mod_iff)

   164   from b_dvd_mod obtain c where "b mod a = b * c" unfolding dvd_def by blast

   165     with \<open>b mod a \<noteq> 0\<close> have "c \<noteq> 0" by auto

   166   with \<open>b mod a = b * c\<close> have "euclidean_size (b mod a) \<ge> euclidean_size b"

   167       using size_mult_mono by force

   168   moreover from \<open>\<not> a dvd b\<close> and \<open>a \<noteq> 0\<close>

   169   have "euclidean_size (b mod a) < euclidean_size a"

   170       using mod_size_less by blast

   171   ultimately show False using size_eq by simp

   172 qed

   173

   174 function gcd_eucl :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"

   175 where

   176   "gcd_eucl a b = (if b = 0 then a div normalization_factor a

   177     else if b dvd a then b div normalization_factor b

   178     else gcd_eucl b (a mod b))"

   179   by pat_completeness simp

   180 termination

   181   by (relation "measure (euclidean_size \<circ> snd)") (simp_all add: mod_size_less)

   182

   183 declare gcd_eucl.simps [simp del]

   184

   185 lemma gcd_eucl_induct [case_names zero mod]:

   186   assumes H1: "\<And>b. P b 0"

   187   and H2: "\<And>a b. b \<noteq> 0 \<Longrightarrow> P b (a mod b) \<Longrightarrow> P a b"

   188   shows "P a b"

   189 proof (induct a b rule: gcd_eucl.induct)

   190   case ("1" a b)

   191   show ?case

   192   proof (cases "b = 0")

   193     case True then show "P a b" by simp (rule H1)

   194   next

   195     case False

   196     have "P b (a mod b)"

   197     proof (cases "b dvd a")

   198       case False with \<open>b \<noteq> 0\<close> show "P b (a mod b)"

   199         by (rule "1.hyps")

   200     next

   201       case True then have "a mod b = 0"

   202         by (simp add: mod_eq_0_iff_dvd)

   203       then show "P b (a mod b)" by simp (rule H1)

   204     qed

   205     with \<open>b \<noteq> 0\<close> show "P a b"

   206       by (blast intro: H2)

   207   qed

   208 qed

   209

   210 definition lcm_eucl :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"

   211 where

   212   "lcm_eucl a b = a * b div (gcd_eucl a b * normalization_factor (a * b))"

   213

   214 definition Lcm_eucl :: "'a set \<Rightarrow> 'a" -- \<open>

   215   Somewhat complicated definition of Lcm that has the advantage of working

   216   for infinite sets as well\<close>

   217 where

   218   "Lcm_eucl A = (if \<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) then

   219      let l = SOME l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l =

   220        (LEAST n. \<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n)

   221        in l div normalization_factor l

   222       else 0)"

   223

   224 definition Gcd_eucl :: "'a set \<Rightarrow> 'a"

   225 where

   226   "Gcd_eucl A = Lcm_eucl {d. \<forall>a\<in>A. d dvd a}"

   227

   228 lemma gcd_eucl_0:

   229   "gcd_eucl a 0 = a div normalization_factor a"

   230   by (simp add: gcd_eucl.simps [of a 0])

   231

   232 lemma gcd_eucl_0_left:

   233   "gcd_eucl 0 a = a div normalization_factor a"

   234   by (simp add: gcd_eucl.simps [of 0 a])

   235

   236 lemma gcd_eucl_non_0:

   237   "b \<noteq> 0 \<Longrightarrow> gcd_eucl a b = gcd_eucl b (a mod b)"

   238   by (cases "b dvd a")

   239     (simp_all add: gcd_eucl.simps [of a b] gcd_eucl.simps [of b 0])

   240

   241 lemma gcd_eucl_code [code]:

   242   "gcd_eucl a b = (if b = 0 then a div normalization_factor a else gcd_eucl b (a mod b))"

   243   by (auto simp add: gcd_eucl_non_0 gcd_eucl_0 gcd_eucl_0_left)

   244

   245 end

   246

   247 class euclidean_ring = euclidean_semiring + idom

   248 begin

   249

   250 function euclid_ext :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a" where

   251   "euclid_ext a b =

   252      (if b = 0 then

   253         let c = 1 div normalization_factor a in (c, 0, a * c)

   254       else if b dvd a then

   255         let c = 1 div normalization_factor b in (0, c, b * c)

   256       else

   257         case euclid_ext b (a mod b) of

   258             (s, t, c) \<Rightarrow> (t, s - t * (a div b), c))"

   259   by pat_completeness simp

   260 termination

   261   by (relation "measure (euclidean_size \<circ> snd)") (simp_all add: mod_size_less)

   262

   263 declare euclid_ext.simps [simp del]

   264

   265 lemma euclid_ext_0:

   266   "euclid_ext a 0 = (1 div normalization_factor a, 0, a div normalization_factor a)"

   267   by (simp add: euclid_ext.simps [of a 0])

   268

   269 lemma euclid_ext_left_0:

   270   "euclid_ext 0 a = (0, 1 div normalization_factor a, a div normalization_factor a)"

   271   by (simp add: euclid_ext.simps [of 0 a])

   272

   273 lemma euclid_ext_non_0:

   274   "b \<noteq> 0 \<Longrightarrow> euclid_ext a b = (case euclid_ext b (a mod b) of

   275     (s, t, c) \<Rightarrow> (t, s - t * (a div b), c))"

   276   by (cases "b dvd a")

   277     (simp_all add: euclid_ext.simps [of a b] euclid_ext.simps [of b 0])

   278

   279 lemma euclid_ext_code [code]:

   280   "euclid_ext a b = (if b = 0 then (1 div normalization_factor a, 0, a div normalization_factor a)

   281     else let (s, t, c) = euclid_ext b (a mod b) in  (t, s - t * (a div b), c))"

   282   by (simp add: euclid_ext.simps [of a b] euclid_ext.simps [of b 0])

   283

   284 lemma euclid_ext_correct:

   285   "case euclid_ext a b of (s, t, c) \<Rightarrow> s * a + t * b = c"

   286 proof (induct a b rule: gcd_eucl_induct)

   287   case (zero a) then show ?case

   288     by (simp add: euclid_ext_0 ac_simps)

   289 next

   290   case (mod a b)

   291   obtain s t c where stc: "euclid_ext b (a mod b) = (s,t,c)"

   292     by (cases "euclid_ext b (a mod b)") blast

   293   with mod have "c = s * b + t * (a mod b)" by simp

   294   also have "... = t * ((a div b) * b + a mod b) + (s - t * (a div b)) * b"

   295     by (simp add: algebra_simps)

   296   also have "(a div b) * b + a mod b = a" using mod_div_equality .

   297   finally show ?case

   298     by (subst euclid_ext.simps) (simp add: stc mod ac_simps)

   299 qed

   300

   301 definition euclid_ext' :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a"

   302 where

   303   "euclid_ext' a b = (case euclid_ext a b of (s, t, _) \<Rightarrow> (s, t))"

   304

   305 lemma euclid_ext'_0: "euclid_ext' a 0 = (1 div normalization_factor a, 0)"

   306   by (simp add: euclid_ext'_def euclid_ext_0)

   307

   308 lemma euclid_ext'_left_0: "euclid_ext' 0 a = (0, 1 div normalization_factor a)"

   309   by (simp add: euclid_ext'_def euclid_ext_left_0)

   310

   311 lemma euclid_ext'_non_0: "b \<noteq> 0 \<Longrightarrow> euclid_ext' a b = (snd (euclid_ext' b (a mod b)),

   312   fst (euclid_ext' b (a mod b)) - snd (euclid_ext' b (a mod b)) * (a div b))"

   313   by (simp add: euclid_ext'_def euclid_ext_non_0 split_def)

   314

   315 end

   316

   317 class euclidean_semiring_gcd = euclidean_semiring + gcd + Gcd +

   318   assumes gcd_gcd_eucl: "gcd = gcd_eucl" and lcm_lcm_eucl: "lcm = lcm_eucl"

   319   assumes Gcd_Gcd_eucl: "Gcd = Gcd_eucl" and Lcm_Lcm_eucl: "Lcm = Lcm_eucl"

   320 begin

   321

   322 lemma gcd_0_left:

   323   "gcd 0 a = a div normalization_factor a"

   324   unfolding gcd_gcd_eucl by (fact gcd_eucl_0_left)

   325

   326 lemma gcd_0:

   327   "gcd a 0 = a div normalization_factor a"

   328   unfolding gcd_gcd_eucl by (fact gcd_eucl_0)

   329

   330 lemma gcd_non_0:

   331   "b \<noteq> 0 \<Longrightarrow> gcd a b = gcd b (a mod b)"

   332   unfolding gcd_gcd_eucl by (fact gcd_eucl_non_0)

   333

   334 lemma gcd_dvd1 [iff]: "gcd a b dvd a"

   335   and gcd_dvd2 [iff]: "gcd a b dvd b"

   336   by (induct a b rule: gcd_eucl_induct)

   337     (simp_all add: gcd_0 gcd_non_0 dvd_mod_iff)

   338

   339 lemma dvd_gcd_D1: "k dvd gcd m n \<Longrightarrow> k dvd m"

   340   by (rule dvd_trans, assumption, rule gcd_dvd1)

   341

   342 lemma dvd_gcd_D2: "k dvd gcd m n \<Longrightarrow> k dvd n"

   343   by (rule dvd_trans, assumption, rule gcd_dvd2)

   344

   345 lemma gcd_greatest:

   346   fixes k a b :: 'a

   347   shows "k dvd a \<Longrightarrow> k dvd b \<Longrightarrow> k dvd gcd a b"

   348 proof (induct a b rule: gcd_eucl_induct)

   349   case (zero a) from zero(1) show ?case by (rule dvd_trans) (simp add: gcd_0)

   350 next

   351   case (mod a b)

   352   then show ?case

   353     by (simp add: gcd_non_0 dvd_mod_iff)

   354 qed

   355

   356 lemma dvd_gcd_iff:

   357   "k dvd gcd a b \<longleftrightarrow> k dvd a \<and> k dvd b"

   358   by (blast intro!: gcd_greatest intro: dvd_trans)

   359

   360 lemmas gcd_greatest_iff = dvd_gcd_iff

   361

   362 lemma gcd_zero [simp]:

   363   "gcd a b = 0 \<longleftrightarrow> a = 0 \<and> b = 0"

   364   by (metis dvd_0_left dvd_refl gcd_dvd1 gcd_dvd2 gcd_greatest)+

   365

   366 lemma normalization_factor_gcd [simp]:

   367   "normalization_factor (gcd a b) = (if a = 0 \<and> b = 0 then 0 else 1)" (is "?f a b = ?g a b")

   368   by (induct a b rule: gcd_eucl_induct)

   369     (auto simp add: gcd_0 gcd_non_0)

   370

   371 lemma gcdI:

   372   "k dvd a \<Longrightarrow> k dvd b \<Longrightarrow> (\<And>l. l dvd a \<Longrightarrow> l dvd b \<Longrightarrow> l dvd k)

   373     \<Longrightarrow> normalization_factor k = (if k = 0 then 0 else 1) \<Longrightarrow> k = gcd a b"

   374   by (intro normed_associated_imp_eq) (auto simp: associated_def intro: gcd_greatest)

   375

   376 sublocale gcd!: abel_semigroup gcd

   377 proof

   378   fix a b c

   379   show "gcd (gcd a b) c = gcd a (gcd b c)"

   380   proof (rule gcdI)

   381     have "gcd (gcd a b) c dvd gcd a b" "gcd a b dvd a" by simp_all

   382     then show "gcd (gcd a b) c dvd a" by (rule dvd_trans)

   383     have "gcd (gcd a b) c dvd gcd a b" "gcd a b dvd b" by simp_all

   384     hence "gcd (gcd a b) c dvd b" by (rule dvd_trans)

   385     moreover have "gcd (gcd a b) c dvd c" by simp

   386     ultimately show "gcd (gcd a b) c dvd gcd b c"

   387       by (rule gcd_greatest)

   388     show "normalization_factor (gcd (gcd a b) c) =  (if gcd (gcd a b) c = 0 then 0 else 1)"

   389       by auto

   390     fix l assume "l dvd a" and "l dvd gcd b c"

   391     with dvd_trans[OF _ gcd_dvd1] and dvd_trans[OF _ gcd_dvd2]

   392       have "l dvd b" and "l dvd c" by blast+

   393     with \<open>l dvd a\<close> show "l dvd gcd (gcd a b) c"

   394       by (intro gcd_greatest)

   395   qed

   396 next

   397   fix a b

   398   show "gcd a b = gcd b a"

   399     by (rule gcdI) (simp_all add: gcd_greatest)

   400 qed

   401

   402 lemma gcd_unique: "d dvd a \<and> d dvd b \<and>

   403     normalization_factor d = (if d = 0 then 0 else 1) \<and>

   404     (\<forall>e. e dvd a \<and> e dvd b \<longrightarrow> e dvd d) \<longleftrightarrow> d = gcd a b"

   405   by (rule, auto intro: gcdI simp: gcd_greatest)

   406

   407 lemma gcd_dvd_prod: "gcd a b dvd k * b"

   408   using mult_dvd_mono [of 1] by auto

   409

   410 lemma gcd_1_left [simp]: "gcd 1 a = 1"

   411   by (rule sym, rule gcdI, simp_all)

   412

   413 lemma gcd_1 [simp]: "gcd a 1 = 1"

   414   by (rule sym, rule gcdI, simp_all)

   415

   416 lemma gcd_proj2_if_dvd:

   417   "b dvd a \<Longrightarrow> gcd a b = b div normalization_factor b"

   418   by (cases "b = 0", simp_all add: dvd_eq_mod_eq_0 gcd_non_0 gcd_0)

   419

   420 lemma gcd_proj1_if_dvd:

   421   "a dvd b \<Longrightarrow> gcd a b = a div normalization_factor a"

   422   by (subst gcd.commute, simp add: gcd_proj2_if_dvd)

   423

   424 lemma gcd_proj1_iff: "gcd m n = m div normalization_factor m \<longleftrightarrow> m dvd n"

   425 proof

   426   assume A: "gcd m n = m div normalization_factor m"

   427   show "m dvd n"

   428   proof (cases "m = 0")

   429     assume [simp]: "m \<noteq> 0"

   430     from A have B: "m = gcd m n * normalization_factor m"

   431       by (simp add: unit_eq_div2)

   432     show ?thesis by (subst B, simp add: mult_unit_dvd_iff)

   433   qed (insert A, simp)

   434 next

   435   assume "m dvd n"

   436   then show "gcd m n = m div normalization_factor m" by (rule gcd_proj1_if_dvd)

   437 qed

   438

   439 lemma gcd_proj2_iff: "gcd m n = n div normalization_factor n \<longleftrightarrow> n dvd m"

   440   by (subst gcd.commute, simp add: gcd_proj1_iff)

   441

   442 lemma gcd_mod1 [simp]:

   443   "gcd (a mod b) b = gcd a b"

   444   by (rule gcdI, metis dvd_mod_iff gcd_dvd1 gcd_dvd2, simp_all add: gcd_greatest dvd_mod_iff)

   445

   446 lemma gcd_mod2 [simp]:

   447   "gcd a (b mod a) = gcd a b"

   448   by (rule gcdI, simp, metis dvd_mod_iff gcd_dvd1 gcd_dvd2, simp_all add: gcd_greatest dvd_mod_iff)

   449

   450 lemma gcd_mult_distrib':

   451   "c div normalization_factor c * gcd a b = gcd (c * a) (c * b)"

   452 proof (cases "c = 0")

   453   case True then show ?thesis by (simp_all add: gcd_0)

   454 next

   455   case False then have [simp]: "is_unit (normalization_factor c)" by simp

   456   show ?thesis

   457   proof (induct a b rule: gcd_eucl_induct)

   458     case (zero a) show ?case

   459     proof (cases "a = 0")

   460       case True then show ?thesis by (simp add: gcd_0)

   461     next

   462       case False then have "is_unit (normalization_factor a)" by simp

   463       then show ?thesis

   464         by (simp add: gcd_0 unit_div_commute unit_div_mult_swap normalization_factor_mult is_unit_div_mult2_eq)

   465     qed

   466     case (mod a b)

   467     then show ?case by (simp add: mult_mod_right gcd.commute)

   468   qed

   469 qed

   470

   471 lemma gcd_mult_distrib:

   472   "k * gcd a b = gcd (k*a) (k*b) * normalization_factor k"

   473 proof-

   474   let ?nf = "normalization_factor"

   475   from gcd_mult_distrib'

   476     have "gcd (k*a) (k*b) = k div ?nf k * gcd a b" ..

   477   also have "... = k * gcd a b div ?nf k"

   478     by (metis dvd_div_mult dvd_eq_mod_eq_0 mod_0 normalization_factor_dvd)

   479   finally show ?thesis

   480     by simp

   481 qed

   482

   483 lemma euclidean_size_gcd_le1 [simp]:

   484   assumes "a \<noteq> 0"

   485   shows "euclidean_size (gcd a b) \<le> euclidean_size a"

   486 proof -

   487    have "gcd a b dvd a" by (rule gcd_dvd1)

   488    then obtain c where A: "a = gcd a b * c" unfolding dvd_def by blast

   489    with \<open>a \<noteq> 0\<close> show ?thesis by (subst (2) A, intro size_mult_mono) auto

   490 qed

   491

   492 lemma euclidean_size_gcd_le2 [simp]:

   493   "b \<noteq> 0 \<Longrightarrow> euclidean_size (gcd a b) \<le> euclidean_size b"

   494   by (subst gcd.commute, rule euclidean_size_gcd_le1)

   495

   496 lemma euclidean_size_gcd_less1:

   497   assumes "a \<noteq> 0" and "\<not>a dvd b"

   498   shows "euclidean_size (gcd a b) < euclidean_size a"

   499 proof (rule ccontr)

   500   assume "\<not>euclidean_size (gcd a b) < euclidean_size a"

   501   with \<open>a \<noteq> 0\<close> have "euclidean_size (gcd a b) = euclidean_size a"

   502     by (intro le_antisym, simp_all)

   503   with assms have "a dvd gcd a b" by (auto intro: dvd_euclidean_size_eq_imp_dvd)

   504   hence "a dvd b" using dvd_gcd_D2 by blast

   505   with \<open>\<not>a dvd b\<close> show False by contradiction

   506 qed

   507

   508 lemma euclidean_size_gcd_less2:

   509   assumes "b \<noteq> 0" and "\<not>b dvd a"

   510   shows "euclidean_size (gcd a b) < euclidean_size b"

   511   using assms by (subst gcd.commute, rule euclidean_size_gcd_less1)

   512

   513 lemma gcd_mult_unit1: "is_unit a \<Longrightarrow> gcd (b * a) c = gcd b c"

   514   apply (rule gcdI)

   515   apply (rule dvd_trans, rule gcd_dvd1, simp add: unit_simps)

   516   apply (rule gcd_dvd2)

   517   apply (rule gcd_greatest, simp add: unit_simps, assumption)

   518   apply (subst normalization_factor_gcd, simp add: gcd_0)

   519   done

   520

   521 lemma gcd_mult_unit2: "is_unit a \<Longrightarrow> gcd b (c * a) = gcd b c"

   522   by (subst gcd.commute, subst gcd_mult_unit1, assumption, rule gcd.commute)

   523

   524 lemma gcd_div_unit1: "is_unit a \<Longrightarrow> gcd (b div a) c = gcd b c"

   525   by (erule is_unitE [of _ b]) (simp add: gcd_mult_unit1)

   526

   527 lemma gcd_div_unit2: "is_unit a \<Longrightarrow> gcd b (c div a) = gcd b c"

   528   by (erule is_unitE [of _ c]) (simp add: gcd_mult_unit2)

   529

   530 lemma gcd_idem: "gcd a a = a div normalization_factor a"

   531   by (cases "a = 0") (simp add: gcd_0_left, rule sym, rule gcdI, simp_all)

   532

   533 lemma gcd_right_idem: "gcd (gcd a b) b = gcd a b"

   534   apply (rule gcdI)

   535   apply (simp add: ac_simps)

   536   apply (rule gcd_dvd2)

   537   apply (rule gcd_greatest, erule (1) gcd_greatest, assumption)

   538   apply simp

   539   done

   540

   541 lemma gcd_left_idem: "gcd a (gcd a b) = gcd a b"

   542   apply (rule gcdI)

   543   apply simp

   544   apply (rule dvd_trans, rule gcd_dvd2, rule gcd_dvd2)

   545   apply (rule gcd_greatest, assumption, erule gcd_greatest, assumption)

   546   apply simp

   547   done

   548

   549 lemma comp_fun_idem_gcd: "comp_fun_idem gcd"

   550 proof

   551   fix a b show "gcd a \<circ> gcd b = gcd b \<circ> gcd a"

   552     by (simp add: fun_eq_iff ac_simps)

   553 next

   554   fix a show "gcd a \<circ> gcd a = gcd a"

   555     by (simp add: fun_eq_iff gcd_left_idem)

   556 qed

   557

   558 lemma coprime_dvd_mult:

   559   assumes "gcd c b = 1" and "c dvd a * b"

   560   shows "c dvd a"

   561 proof -

   562   let ?nf = "normalization_factor"

   563   from assms gcd_mult_distrib [of a c b]

   564     have A: "a = gcd (a * c) (a * b) * ?nf a" by simp

   565   from \<open>c dvd a * b\<close> show ?thesis by (subst A, simp_all add: gcd_greatest)

   566 qed

   567

   568 lemma coprime_dvd_mult_iff:

   569   "gcd c b = 1 \<Longrightarrow> (c dvd a * b) = (c dvd a)"

   570   by (rule, rule coprime_dvd_mult, simp_all)

   571

   572 lemma gcd_dvd_antisym:

   573   "gcd a b dvd gcd c d \<Longrightarrow> gcd c d dvd gcd a b \<Longrightarrow> gcd a b = gcd c d"

   574 proof (rule gcdI)

   575   assume A: "gcd a b dvd gcd c d" and B: "gcd c d dvd gcd a b"

   576   have "gcd c d dvd c" by simp

   577   with A show "gcd a b dvd c" by (rule dvd_trans)

   578   have "gcd c d dvd d" by simp

   579   with A show "gcd a b dvd d" by (rule dvd_trans)

   580   show "normalization_factor (gcd a b) = (if gcd a b = 0 then 0 else 1)"

   581     by simp

   582   fix l assume "l dvd c" and "l dvd d"

   583   hence "l dvd gcd c d" by (rule gcd_greatest)

   584   from this and B show "l dvd gcd a b" by (rule dvd_trans)

   585 qed

   586

   587 lemma gcd_mult_cancel:

   588   assumes "gcd k n = 1"

   589   shows "gcd (k * m) n = gcd m n"

   590 proof (rule gcd_dvd_antisym)

   591   have "gcd (gcd (k * m) n) k = gcd (gcd k n) (k * m)" by (simp add: ac_simps)

   592   also note \<open>gcd k n = 1\<close>

   593   finally have "gcd (gcd (k * m) n) k = 1" by simp

   594   hence "gcd (k * m) n dvd m" by (rule coprime_dvd_mult, simp add: ac_simps)

   595   moreover have "gcd (k * m) n dvd n" by simp

   596   ultimately show "gcd (k * m) n dvd gcd m n" by (rule gcd_greatest)

   597   have "gcd m n dvd (k * m)" and "gcd m n dvd n" by simp_all

   598   then show "gcd m n dvd gcd (k * m) n" by (rule gcd_greatest)

   599 qed

   600

   601 lemma coprime_crossproduct:

   602   assumes [simp]: "gcd a d = 1" "gcd b c = 1"

   603   shows "associated (a * c) (b * d) \<longleftrightarrow> associated a b \<and> associated c d" (is "?lhs \<longleftrightarrow> ?rhs")

   604 proof

   605   assume ?rhs then show ?lhs unfolding associated_def by (fast intro: mult_dvd_mono)

   606 next

   607   assume ?lhs

   608   from \<open>?lhs\<close> have "a dvd b * d" unfolding associated_def by (metis dvd_mult_left)

   609   hence "a dvd b" by (simp add: coprime_dvd_mult_iff)

   610   moreover from \<open>?lhs\<close> have "b dvd a * c" unfolding associated_def by (metis dvd_mult_left)

   611   hence "b dvd a" by (simp add: coprime_dvd_mult_iff)

   612   moreover from \<open>?lhs\<close> have "c dvd d * b"

   613     unfolding associated_def by (auto dest: dvd_mult_right simp add: ac_simps)

   614   hence "c dvd d" by (simp add: coprime_dvd_mult_iff gcd.commute)

   615   moreover from \<open>?lhs\<close> have "d dvd c * a"

   616     unfolding associated_def by (auto dest: dvd_mult_right simp add: ac_simps)

   617   hence "d dvd c" by (simp add: coprime_dvd_mult_iff gcd.commute)

   618   ultimately show ?rhs unfolding associated_def by simp

   619 qed

   620

   621 lemma gcd_add1 [simp]:

   622   "gcd (m + n) n = gcd m n"

   623   by (cases "n = 0", simp_all add: gcd_non_0)

   624

   625 lemma gcd_add2 [simp]:

   626   "gcd m (m + n) = gcd m n"

   627   using gcd_add1 [of n m] by (simp add: ac_simps)

   628

   629 lemma gcd_add_mult:

   630   "gcd m (k * m + n) = gcd m n"

   631 proof -

   632   have "gcd m ((k * m + n) mod m) = gcd m (k * m + n)"

   633     by (fact gcd_mod2)

   634   then show ?thesis by simp

   635 qed

   636

   637 lemma coprimeI: "(\<And>l. \<lbrakk>l dvd a; l dvd b\<rbrakk> \<Longrightarrow> l dvd 1) \<Longrightarrow> gcd a b = 1"

   638   by (rule sym, rule gcdI, simp_all)

   639

   640 lemma coprime: "gcd a b = 1 \<longleftrightarrow> (\<forall>d. d dvd a \<and> d dvd b \<longleftrightarrow> is_unit d)"

   641   by (auto intro: coprimeI gcd_greatest dvd_gcd_D1 dvd_gcd_D2)

   642

   643 lemma div_gcd_coprime:

   644   assumes nz: "a \<noteq> 0 \<or> b \<noteq> 0"

   645   defines [simp]: "d \<equiv> gcd a b"

   646   defines [simp]: "a' \<equiv> a div d" and [simp]: "b' \<equiv> b div d"

   647   shows "gcd a' b' = 1"

   648 proof (rule coprimeI)

   649   fix l assume "l dvd a'" "l dvd b'"

   650   then obtain s t where "a' = l * s" "b' = l * t" unfolding dvd_def by blast

   651   moreover have "a = a' * d" "b = b' * d" by simp_all

   652   ultimately have "a = (l * d) * s" "b = (l * d) * t"

   653     by (simp_all only: ac_simps)

   654   hence "l*d dvd a" and "l*d dvd b" by (simp_all only: dvd_triv_left)

   655   hence "l*d dvd d" by (simp add: gcd_greatest)

   656   then obtain u where "d = l * d * u" ..

   657   then have "d * (l * u) = d" by (simp add: ac_simps)

   658   moreover from nz have "d \<noteq> 0" by simp

   659   with div_mult_self1_is_id have "d * (l * u) div d = l * u" .

   660   ultimately have "1 = l * u"

   661     using \<open>d \<noteq> 0\<close> by simp

   662   then show "l dvd 1" ..

   663 qed

   664

   665 lemma coprime_mult:

   666   assumes da: "gcd d a = 1" and db: "gcd d b = 1"

   667   shows "gcd d (a * b) = 1"

   668   apply (subst gcd.commute)

   669   using da apply (subst gcd_mult_cancel)

   670   apply (subst gcd.commute, assumption)

   671   apply (subst gcd.commute, rule db)

   672   done

   673

   674 lemma coprime_lmult:

   675   assumes dab: "gcd d (a * b) = 1"

   676   shows "gcd d a = 1"

   677 proof (rule coprimeI)

   678   fix l assume "l dvd d" and "l dvd a"

   679   hence "l dvd a * b" by simp

   680   with \<open>l dvd d\<close> and dab show "l dvd 1" by (auto intro: gcd_greatest)

   681 qed

   682

   683 lemma coprime_rmult:

   684   assumes dab: "gcd d (a * b) = 1"

   685   shows "gcd d b = 1"

   686 proof (rule coprimeI)

   687   fix l assume "l dvd d" and "l dvd b"

   688   hence "l dvd a * b" by simp

   689   with \<open>l dvd d\<close> and dab show "l dvd 1" by (auto intro: gcd_greatest)

   690 qed

   691

   692 lemma coprime_mul_eq: "gcd d (a * b) = 1 \<longleftrightarrow> gcd d a = 1 \<and> gcd d b = 1"

   693   using coprime_rmult[of d a b] coprime_lmult[of d a b] coprime_mult[of d a b] by blast

   694

   695 lemma gcd_coprime:

   696   assumes c: "gcd a b \<noteq> 0" and a: "a = a' * gcd a b" and b: "b = b' * gcd a b"

   697   shows "gcd a' b' = 1"

   698 proof -

   699   from c have "a \<noteq> 0 \<or> b \<noteq> 0" by simp

   700   with div_gcd_coprime have "gcd (a div gcd a b) (b div gcd a b) = 1" .

   701   also from assms have "a div gcd a b = a'" by (metis div_mult_self2_is_id)+

   702   also from assms have "b div gcd a b = b'" by (metis div_mult_self2_is_id)+

   703   finally show ?thesis .

   704 qed

   705

   706 lemma coprime_power:

   707   assumes "0 < n"

   708   shows "gcd a (b ^ n) = 1 \<longleftrightarrow> gcd a b = 1"

   709 using assms proof (induct n)

   710   case (Suc n) then show ?case

   711     by (cases n) (simp_all add: coprime_mul_eq)

   712 qed simp

   713

   714 lemma gcd_coprime_exists:

   715   assumes nz: "gcd a b \<noteq> 0"

   716   shows "\<exists>a' b'. a = a' * gcd a b \<and> b = b' * gcd a b \<and> gcd a' b' = 1"

   717   apply (rule_tac x = "a div gcd a b" in exI)

   718   apply (rule_tac x = "b div gcd a b" in exI)

   719   apply (insert nz, auto intro: div_gcd_coprime)

   720   done

   721

   722 lemma coprime_exp:

   723   "gcd d a = 1 \<Longrightarrow> gcd d (a^n) = 1"

   724   by (induct n, simp_all add: coprime_mult)

   725

   726 lemma coprime_exp2 [intro]:

   727   "gcd a b = 1 \<Longrightarrow> gcd (a^n) (b^m) = 1"

   728   apply (rule coprime_exp)

   729   apply (subst gcd.commute)

   730   apply (rule coprime_exp)

   731   apply (subst gcd.commute)

   732   apply assumption

   733   done

   734

   735 lemma gcd_exp:

   736   "gcd (a^n) (b^n) = (gcd a b) ^ n"

   737 proof (cases "a = 0 \<and> b = 0")

   738   assume "a = 0 \<and> b = 0"

   739   then show ?thesis by (cases n, simp_all add: gcd_0_left)

   740 next

   741   assume A: "\<not>(a = 0 \<and> b = 0)"

   742   hence "1 = gcd ((a div gcd a b)^n) ((b div gcd a b)^n)"

   743     using div_gcd_coprime by (subst sym, auto simp: div_gcd_coprime)

   744   hence "(gcd a b) ^ n = (gcd a b) ^ n * ..." by simp

   745   also note gcd_mult_distrib

   746   also have "normalization_factor ((gcd a b)^n) = 1"

   747     by (simp add: normalization_factor_pow A)

   748   also have "(gcd a b)^n * (a div gcd a b)^n = a^n"

   749     by (subst ac_simps, subst div_power, simp, rule dvd_div_mult_self, rule dvd_power_same, simp)

   750   also have "(gcd a b)^n * (b div gcd a b)^n = b^n"

   751     by (subst ac_simps, subst div_power, simp, rule dvd_div_mult_self, rule dvd_power_same, simp)

   752   finally show ?thesis by simp

   753 qed

   754

   755 lemma coprime_common_divisor:

   756   "gcd a b = 1 \<Longrightarrow> a dvd a \<Longrightarrow> a dvd b \<Longrightarrow> is_unit a"

   757   apply (subgoal_tac "a dvd gcd a b")

   758   apply simp

   759   apply (erule (1) gcd_greatest)

   760   done

   761

   762 lemma division_decomp:

   763   assumes dc: "a dvd b * c"

   764   shows "\<exists>b' c'. a = b' * c' \<and> b' dvd b \<and> c' dvd c"

   765 proof (cases "gcd a b = 0")

   766   assume "gcd a b = 0"

   767   hence "a = 0 \<and> b = 0" by simp

   768   hence "a = 0 * c \<and> 0 dvd b \<and> c dvd c" by simp

   769   then show ?thesis by blast

   770 next

   771   let ?d = "gcd a b"

   772   assume "?d \<noteq> 0"

   773   from gcd_coprime_exists[OF this]

   774     obtain a' b' where ab': "a = a' * ?d" "b = b' * ?d" "gcd a' b' = 1"

   775     by blast

   776   from ab'(1) have "a' dvd a" unfolding dvd_def by blast

   777   with dc have "a' dvd b*c" using dvd_trans[of a' a "b*c"] by simp

   778   from dc ab'(1,2) have "a'*?d dvd (b'*?d) * c" by simp

   779   hence "?d * a' dvd ?d * (b' * c)" by (simp add: mult_ac)

   780   with \<open>?d \<noteq> 0\<close> have "a' dvd b' * c" by simp

   781   with coprime_dvd_mult[OF ab'(3)]

   782     have "a' dvd c" by (subst (asm) ac_simps, blast)

   783   with ab'(1) have "a = ?d * a' \<and> ?d dvd b \<and> a' dvd c" by (simp add: mult_ac)

   784   then show ?thesis by blast

   785 qed

   786

   787 lemma pow_divs_pow:

   788   assumes ab: "a ^ n dvd b ^ n" and n: "n \<noteq> 0"

   789   shows "a dvd b"

   790 proof (cases "gcd a b = 0")

   791   assume "gcd a b = 0"

   792   then show ?thesis by simp

   793 next

   794   let ?d = "gcd a b"

   795   assume "?d \<noteq> 0"

   796   from n obtain m where m: "n = Suc m" by (cases n, simp_all)

   797   from \<open>?d \<noteq> 0\<close> have zn: "?d ^ n \<noteq> 0" by (rule power_not_zero)

   798   from gcd_coprime_exists[OF \<open>?d \<noteq> 0\<close>]

   799     obtain a' b' where ab': "a = a' * ?d" "b = b' * ?d" "gcd a' b' = 1"

   800     by blast

   801   from ab have "(a' * ?d) ^ n dvd (b' * ?d) ^ n"

   802     by (simp add: ab'(1,2)[symmetric])

   803   hence "?d^n * a'^n dvd ?d^n * b'^n"

   804     by (simp only: power_mult_distrib ac_simps)

   805   with zn have "a'^n dvd b'^n" by simp

   806   hence "a' dvd b'^n" using dvd_trans[of a' "a'^n" "b'^n"] by (simp add: m)

   807   hence "a' dvd b'^m * b'" by (simp add: m ac_simps)

   808   with coprime_dvd_mult[OF coprime_exp[OF ab'(3), of m]]

   809     have "a' dvd b'" by (subst (asm) ac_simps, blast)

   810   hence "a'*?d dvd b'*?d" by (rule mult_dvd_mono, simp)

   811   with ab'(1,2) show ?thesis by simp

   812 qed

   813

   814 lemma pow_divs_eq [simp]:

   815   "n \<noteq> 0 \<Longrightarrow> a ^ n dvd b ^ n \<longleftrightarrow> a dvd b"

   816   by (auto intro: pow_divs_pow dvd_power_same)

   817

   818 lemma divs_mult:

   819   assumes mr: "m dvd r" and nr: "n dvd r" and mn: "gcd m n = 1"

   820   shows "m * n dvd r"

   821 proof -

   822   from mr nr obtain m' n' where m': "r = m*m'" and n': "r = n*n'"

   823     unfolding dvd_def by blast

   824   from mr n' have "m dvd n'*n" by (simp add: ac_simps)

   825   hence "m dvd n'" using coprime_dvd_mult_iff[OF mn] by simp

   826   then obtain k where k: "n' = m*k" unfolding dvd_def by blast

   827   with n' have "r = m * n * k" by (simp add: mult_ac)

   828   then show ?thesis unfolding dvd_def by blast

   829 qed

   830

   831 lemma coprime_plus_one [simp]: "gcd (n + 1) n = 1"

   832   by (subst add_commute, simp)

   833

   834 lemma setprod_coprime [rule_format]:

   835   "(\<forall>i\<in>A. gcd (f i) a = 1) \<longrightarrow> gcd (\<Prod>i\<in>A. f i) a = 1"

   836   apply (cases "finite A")

   837   apply (induct set: finite)

   838   apply (auto simp add: gcd_mult_cancel)

   839   done

   840

   841 lemma coprime_divisors:

   842   assumes "d dvd a" "e dvd b" "gcd a b = 1"

   843   shows "gcd d e = 1"

   844 proof -

   845   from assms obtain k l where "a = d * k" "b = e * l"

   846     unfolding dvd_def by blast

   847   with assms have "gcd (d * k) (e * l) = 1" by simp

   848   hence "gcd (d * k) e = 1" by (rule coprime_lmult)

   849   also have "gcd (d * k) e = gcd e (d * k)" by (simp add: ac_simps)

   850   finally have "gcd e d = 1" by (rule coprime_lmult)

   851   then show ?thesis by (simp add: ac_simps)

   852 qed

   853

   854 lemma invertible_coprime:

   855   assumes "a * b mod m = 1"

   856   shows "coprime a m"

   857 proof -

   858   from assms have "coprime m (a * b mod m)"

   859     by simp

   860   then have "coprime m (a * b)"

   861     by simp

   862   then have "coprime m a"

   863     by (rule coprime_lmult)

   864   then show ?thesis

   865     by (simp add: ac_simps)

   866 qed

   867

   868 lemma lcm_gcd:

   869   "lcm a b = a * b div (gcd a b * normalization_factor (a*b))"

   870   by (simp only: lcm_lcm_eucl gcd_gcd_eucl lcm_eucl_def)

   871

   872 lemma lcm_gcd_prod:

   873   "lcm a b * gcd a b = a * b div normalization_factor (a*b)"

   874 proof (cases "a * b = 0")

   875   let ?nf = normalization_factor

   876   assume "a * b \<noteq> 0"

   877   hence "gcd a b \<noteq> 0" by simp

   878   from lcm_gcd have "lcm a b * gcd a b = gcd a b * (a * b div (?nf (a*b) * gcd a b))"

   879     by (simp add: mult_ac)

   880   also from \<open>a * b \<noteq> 0\<close> have "... = a * b div ?nf (a*b)"

   881     by (simp add: div_mult_swap mult.commute)

   882   finally show ?thesis .

   883 qed (auto simp add: lcm_gcd)

   884

   885 lemma lcm_dvd1 [iff]:

   886   "a dvd lcm a b"

   887 proof (cases "a*b = 0")

   888   assume "a * b \<noteq> 0"

   889   hence "gcd a b \<noteq> 0" by simp

   890   let ?c = "1 div normalization_factor (a * b)"

   891   from \<open>a * b \<noteq> 0\<close> have [simp]: "is_unit (normalization_factor (a * b))" by simp

   892   from lcm_gcd_prod[of a b] have "lcm a b * gcd a b = a * ?c * b"

   893     by (simp add: div_mult_swap unit_div_commute)

   894   hence "lcm a b * gcd a b div gcd a b = a * ?c * b div gcd a b" by simp

   895   with \<open>gcd a b \<noteq> 0\<close> have "lcm a b = a * ?c * b div gcd a b"

   896     by (subst (asm) div_mult_self2_is_id, simp_all)

   897   also have "... = a * (?c * b div gcd a b)"

   898     by (metis div_mult_swap gcd_dvd2 mult_assoc)

   899   finally show ?thesis by (rule dvdI)

   900 qed (auto simp add: lcm_gcd)

   901

   902 lemma lcm_least:

   903   "\<lbrakk>a dvd k; b dvd k\<rbrakk> \<Longrightarrow> lcm a b dvd k"

   904 proof (cases "k = 0")

   905   let ?nf = normalization_factor

   906   assume "k \<noteq> 0"

   907   hence "is_unit (?nf k)" by simp

   908   hence "?nf k \<noteq> 0" by (metis not_is_unit_0)

   909   assume A: "a dvd k" "b dvd k"

   910   hence "gcd a b \<noteq> 0" using \<open>k \<noteq> 0\<close> by auto

   911   from A obtain r s where ar: "k = a * r" and bs: "k = b * s"

   912     unfolding dvd_def by blast

   913   with \<open>k \<noteq> 0\<close> have "r * s \<noteq> 0"

   914     by auto (drule sym [of 0], simp)

   915   hence "is_unit (?nf (r * s))" by simp

   916   let ?c = "?nf k div ?nf (r*s)"

   917   from \<open>is_unit (?nf k)\<close> and \<open>is_unit (?nf (r * s))\<close> have "is_unit ?c" by (rule unit_div)

   918   hence "?c \<noteq> 0" using not_is_unit_0 by fast

   919   from ar bs have "k * k * gcd s r = ?nf k * k * gcd (k * s) (k * r)"

   920     by (subst mult_assoc, subst gcd_mult_distrib[of k s r], simp only: ac_simps)

   921   also have "... = ?nf k * k * gcd ((r*s) * a) ((r*s) * b)"

   922     by (subst (3) \<open>k = a * r\<close>, subst (3) \<open>k = b * s\<close>, simp add: algebra_simps)

   923   also have "... = ?c * r*s * k * gcd a b" using \<open>r * s \<noteq> 0\<close>

   924     by (subst gcd_mult_distrib'[symmetric], simp add: algebra_simps unit_simps)

   925   finally have "(a*r) * (b*s) * gcd s r = ?c * k * r * s * gcd a b"

   926     by (subst ar[symmetric], subst bs[symmetric], simp add: mult_ac)

   927   hence "a * b * gcd s r * (r * s) = ?c * k * gcd a b * (r * s)"

   928     by (simp add: algebra_simps)

   929   hence "?c * k * gcd a b = a * b * gcd s r" using \<open>r * s \<noteq> 0\<close>

   930     by (metis div_mult_self2_is_id)

   931   also have "... = lcm a b * gcd a b * gcd s r * ?nf (a*b)"

   932     by (subst lcm_gcd_prod[of a b], metis gcd_mult_distrib gcd_mult_distrib')

   933   also have "... = lcm a b * gcd s r * ?nf (a*b) * gcd a b"

   934     by (simp add: algebra_simps)

   935   finally have "k * ?c = lcm a b * gcd s r * ?nf (a*b)" using \<open>gcd a b \<noteq> 0\<close>

   936     by (metis mult.commute div_mult_self2_is_id)

   937   hence "k = lcm a b * (gcd s r * ?nf (a*b)) div ?c" using \<open>?c \<noteq> 0\<close>

   938     by (metis div_mult_self2_is_id mult_assoc)

   939   also have "... = lcm a b * (gcd s r * ?nf (a*b) div ?c)" using \<open>is_unit ?c\<close>

   940     by (simp add: unit_simps)

   941   finally show ?thesis by (rule dvdI)

   942 qed simp

   943

   944 lemma lcm_zero:

   945   "lcm a b = 0 \<longleftrightarrow> a = 0 \<or> b = 0"

   946 proof -

   947   let ?nf = normalization_factor

   948   {

   949     assume "a \<noteq> 0" "b \<noteq> 0"

   950     hence "a * b div ?nf (a * b) \<noteq> 0" by (simp add: no_zero_divisors)

   951     moreover from \<open>a \<noteq> 0\<close> and \<open>b \<noteq> 0\<close> have "gcd a b \<noteq> 0" by simp

   952     ultimately have "lcm a b \<noteq> 0" using lcm_gcd_prod[of a b] by (intro notI, simp)

   953   } moreover {

   954     assume "a = 0 \<or> b = 0"

   955     hence "lcm a b = 0" by (elim disjE, simp_all add: lcm_gcd)

   956   }

   957   ultimately show ?thesis by blast

   958 qed

   959

   960 lemmas lcm_0_iff = lcm_zero

   961

   962 lemma gcd_lcm:

   963   assumes "lcm a b \<noteq> 0"

   964   shows "gcd a b = a * b div (lcm a b * normalization_factor (a * b))"

   965 proof-

   966   from assms have "gcd a b \<noteq> 0" by (simp add: lcm_zero)

   967   let ?c = "normalization_factor (a * b)"

   968   from \<open>lcm a b \<noteq> 0\<close> have "?c \<noteq> 0" by (intro notI, simp add: lcm_zero no_zero_divisors)

   969   hence "is_unit ?c" by simp

   970   from lcm_gcd_prod [of a b] have "gcd a b = a * b div ?c div lcm a b"

   971     by (subst (2) div_mult_self2_is_id[OF \<open>lcm a b \<noteq> 0\<close>, symmetric], simp add: mult_ac)

   972   also from \<open>is_unit ?c\<close> have "... = a * b div (lcm a b * ?c)"

   973     by (metis \<open>?c \<noteq> 0\<close> div_mult_mult1 dvd_mult_div_cancel mult_commute normalization_factor_dvd')

   974   finally show ?thesis .

   975 qed

   976

   977 lemma normalization_factor_lcm [simp]:

   978   "normalization_factor (lcm a b) = (if a = 0 \<or> b = 0 then 0 else 1)"

   979 proof (cases "a = 0 \<or> b = 0")

   980   case True then show ?thesis

   981     by (auto simp add: lcm_gcd)

   982 next

   983   case False

   984   let ?nf = normalization_factor

   985   from lcm_gcd_prod[of a b]

   986     have "?nf (lcm a b) * ?nf (gcd a b) = ?nf (a*b) div ?nf (a*b)"

   987     by (metis div_by_0 div_self normalization_correct normalization_factor_0 normalization_factor_mult)

   988   also have "... = (if a*b = 0 then 0 else 1)"

   989     by simp

   990   finally show ?thesis using False by simp

   991 qed

   992

   993 lemma lcm_dvd2 [iff]: "b dvd lcm a b"

   994   using lcm_dvd1 [of b a] by (simp add: lcm_gcd ac_simps)

   995

   996 lemma lcmI:

   997   "\<lbrakk>a dvd k; b dvd k; \<And>l. a dvd l \<Longrightarrow> b dvd l \<Longrightarrow> k dvd l;

   998     normalization_factor k = (if k = 0 then 0 else 1)\<rbrakk> \<Longrightarrow> k = lcm a b"

   999   by (intro normed_associated_imp_eq) (auto simp: associated_def intro: lcm_least)

  1000

  1001 sublocale lcm!: abel_semigroup lcm

  1002 proof

  1003   fix a b c

  1004   show "lcm (lcm a b) c = lcm a (lcm b c)"

  1005   proof (rule lcmI)

  1006     have "a dvd lcm a b" and "lcm a b dvd lcm (lcm a b) c" by simp_all

  1007     then show "a dvd lcm (lcm a b) c" by (rule dvd_trans)

  1008

  1009     have "b dvd lcm a b" and "lcm a b dvd lcm (lcm a b) c" by simp_all

  1010     hence "b dvd lcm (lcm a b) c" by (rule dvd_trans)

  1011     moreover have "c dvd lcm (lcm a b) c" by simp

  1012     ultimately show "lcm b c dvd lcm (lcm a b) c" by (rule lcm_least)

  1013

  1014     fix l assume "a dvd l" and "lcm b c dvd l"

  1015     have "b dvd lcm b c" by simp

  1016     from this and \<open>lcm b c dvd l\<close> have "b dvd l" by (rule dvd_trans)

  1017     have "c dvd lcm b c" by simp

  1018     from this and \<open>lcm b c dvd l\<close> have "c dvd l" by (rule dvd_trans)

  1019     from \<open>a dvd l\<close> and \<open>b dvd l\<close> have "lcm a b dvd l" by (rule lcm_least)

  1020     from this and \<open>c dvd l\<close> show "lcm (lcm a b) c dvd l" by (rule lcm_least)

  1021   qed (simp add: lcm_zero)

  1022 next

  1023   fix a b

  1024   show "lcm a b = lcm b a"

  1025     by (simp add: lcm_gcd ac_simps)

  1026 qed

  1027

  1028 lemma dvd_lcm_D1:

  1029   "lcm m n dvd k \<Longrightarrow> m dvd k"

  1030   by (rule dvd_trans, rule lcm_dvd1, assumption)

  1031

  1032 lemma dvd_lcm_D2:

  1033   "lcm m n dvd k \<Longrightarrow> n dvd k"

  1034   by (rule dvd_trans, rule lcm_dvd2, assumption)

  1035

  1036 lemma gcd_dvd_lcm [simp]:

  1037   "gcd a b dvd lcm a b"

  1038   by (metis dvd_trans gcd_dvd2 lcm_dvd2)

  1039

  1040 lemma lcm_1_iff:

  1041   "lcm a b = 1 \<longleftrightarrow> is_unit a \<and> is_unit b"

  1042 proof

  1043   assume "lcm a b = 1"

  1044   then show "is_unit a \<and> is_unit b" by auto

  1045 next

  1046   assume "is_unit a \<and> is_unit b"

  1047   hence "a dvd 1" and "b dvd 1" by simp_all

  1048   hence "is_unit (lcm a b)" by (rule lcm_least)

  1049   hence "lcm a b = normalization_factor (lcm a b)"

  1050     by (subst normalization_factor_unit, simp_all)

  1051   also have "\<dots> = 1" using \<open>is_unit a \<and> is_unit b\<close>

  1052     by auto

  1053   finally show "lcm a b = 1" .

  1054 qed

  1055

  1056 lemma lcm_0_left [simp]:

  1057   "lcm 0 a = 0"

  1058   by (rule sym, rule lcmI, simp_all)

  1059

  1060 lemma lcm_0 [simp]:

  1061   "lcm a 0 = 0"

  1062   by (rule sym, rule lcmI, simp_all)

  1063

  1064 lemma lcm_unique:

  1065   "a dvd d \<and> b dvd d \<and>

  1066   normalization_factor d = (if d = 0 then 0 else 1) \<and>

  1067   (\<forall>e. a dvd e \<and> b dvd e \<longrightarrow> d dvd e) \<longleftrightarrow> d = lcm a b"

  1068   by (rule, auto intro: lcmI simp: lcm_least lcm_zero)

  1069

  1070 lemma dvd_lcm_I1 [simp]:

  1071   "k dvd m \<Longrightarrow> k dvd lcm m n"

  1072   by (metis lcm_dvd1 dvd_trans)

  1073

  1074 lemma dvd_lcm_I2 [simp]:

  1075   "k dvd n \<Longrightarrow> k dvd lcm m n"

  1076   by (metis lcm_dvd2 dvd_trans)

  1077

  1078 lemma lcm_1_left [simp]:

  1079   "lcm 1 a = a div normalization_factor a"

  1080   by (cases "a = 0") (simp, rule sym, rule lcmI, simp_all)

  1081

  1082 lemma lcm_1_right [simp]:

  1083   "lcm a 1 = a div normalization_factor a"

  1084   using lcm_1_left [of a] by (simp add: ac_simps)

  1085

  1086 lemma lcm_coprime:

  1087   "gcd a b = 1 \<Longrightarrow> lcm a b = a * b div normalization_factor (a*b)"

  1088   by (subst lcm_gcd) simp

  1089

  1090 lemma lcm_proj1_if_dvd:

  1091   "b dvd a \<Longrightarrow> lcm a b = a div normalization_factor a"

  1092   by (cases "a = 0") (simp, rule sym, rule lcmI, simp_all)

  1093

  1094 lemma lcm_proj2_if_dvd:

  1095   "a dvd b \<Longrightarrow> lcm a b = b div normalization_factor b"

  1096   using lcm_proj1_if_dvd [of a b] by (simp add: ac_simps)

  1097

  1098 lemma lcm_proj1_iff:

  1099   "lcm m n = m div normalization_factor m \<longleftrightarrow> n dvd m"

  1100 proof

  1101   assume A: "lcm m n = m div normalization_factor m"

  1102   show "n dvd m"

  1103   proof (cases "m = 0")

  1104     assume [simp]: "m \<noteq> 0"

  1105     from A have B: "m = lcm m n * normalization_factor m"

  1106       by (simp add: unit_eq_div2)

  1107     show ?thesis by (subst B, simp)

  1108   qed simp

  1109 next

  1110   assume "n dvd m"

  1111   then show "lcm m n = m div normalization_factor m" by (rule lcm_proj1_if_dvd)

  1112 qed

  1113

  1114 lemma lcm_proj2_iff:

  1115   "lcm m n = n div normalization_factor n \<longleftrightarrow> m dvd n"

  1116   using lcm_proj1_iff [of n m] by (simp add: ac_simps)

  1117

  1118 lemma euclidean_size_lcm_le1:

  1119   assumes "a \<noteq> 0" and "b \<noteq> 0"

  1120   shows "euclidean_size a \<le> euclidean_size (lcm a b)"

  1121 proof -

  1122   have "a dvd lcm a b" by (rule lcm_dvd1)

  1123   then obtain c where A: "lcm a b = a * c" unfolding dvd_def by blast

  1124   with \<open>a \<noteq> 0\<close> and \<open>b \<noteq> 0\<close> have "c \<noteq> 0" by (auto simp: lcm_zero)

  1125   then show ?thesis by (subst A, intro size_mult_mono)

  1126 qed

  1127

  1128 lemma euclidean_size_lcm_le2:

  1129   "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> euclidean_size b \<le> euclidean_size (lcm a b)"

  1130   using euclidean_size_lcm_le1 [of b a] by (simp add: ac_simps)

  1131

  1132 lemma euclidean_size_lcm_less1:

  1133   assumes "b \<noteq> 0" and "\<not>b dvd a"

  1134   shows "euclidean_size a < euclidean_size (lcm a b)"

  1135 proof (rule ccontr)

  1136   from assms have "a \<noteq> 0" by auto

  1137   assume "\<not>euclidean_size a < euclidean_size (lcm a b)"

  1138   with \<open>a \<noteq> 0\<close> and \<open>b \<noteq> 0\<close> have "euclidean_size (lcm a b) = euclidean_size a"

  1139     by (intro le_antisym, simp, intro euclidean_size_lcm_le1)

  1140   with assms have "lcm a b dvd a"

  1141     by (rule_tac dvd_euclidean_size_eq_imp_dvd) (auto simp: lcm_zero)

  1142   hence "b dvd a" by (rule dvd_lcm_D2)

  1143   with \<open>\<not>b dvd a\<close> show False by contradiction

  1144 qed

  1145

  1146 lemma euclidean_size_lcm_less2:

  1147   assumes "a \<noteq> 0" and "\<not>a dvd b"

  1148   shows "euclidean_size b < euclidean_size (lcm a b)"

  1149   using assms euclidean_size_lcm_less1 [of a b] by (simp add: ac_simps)

  1150

  1151 lemma lcm_mult_unit1:

  1152   "is_unit a \<Longrightarrow> lcm (b * a) c = lcm b c"

  1153   apply (rule lcmI)

  1154   apply (rule dvd_trans[of _ "b * a"], simp, rule lcm_dvd1)

  1155   apply (rule lcm_dvd2)

  1156   apply (rule lcm_least, simp add: unit_simps, assumption)

  1157   apply (subst normalization_factor_lcm, simp add: lcm_zero)

  1158   done

  1159

  1160 lemma lcm_mult_unit2:

  1161   "is_unit a \<Longrightarrow> lcm b (c * a) = lcm b c"

  1162   using lcm_mult_unit1 [of a c b] by (simp add: ac_simps)

  1163

  1164 lemma lcm_div_unit1:

  1165   "is_unit a \<Longrightarrow> lcm (b div a) c = lcm b c"

  1166   by (erule is_unitE [of _ b]) (simp add: lcm_mult_unit1)

  1167

  1168 lemma lcm_div_unit2:

  1169   "is_unit a \<Longrightarrow> lcm b (c div a) = lcm b c"

  1170   by (erule is_unitE [of _ c]) (simp add: lcm_mult_unit2)

  1171

  1172 lemma lcm_left_idem:

  1173   "lcm a (lcm a b) = lcm a b"

  1174   apply (rule lcmI)

  1175   apply simp

  1176   apply (subst lcm.assoc [symmetric], rule lcm_dvd2)

  1177   apply (rule lcm_least, assumption)

  1178   apply (erule (1) lcm_least)

  1179   apply (auto simp: lcm_zero)

  1180   done

  1181

  1182 lemma lcm_right_idem:

  1183   "lcm (lcm a b) b = lcm a b"

  1184   apply (rule lcmI)

  1185   apply (subst lcm.assoc, rule lcm_dvd1)

  1186   apply (rule lcm_dvd2)

  1187   apply (rule lcm_least, erule (1) lcm_least, assumption)

  1188   apply (auto simp: lcm_zero)

  1189   done

  1190

  1191 lemma comp_fun_idem_lcm: "comp_fun_idem lcm"

  1192 proof

  1193   fix a b show "lcm a \<circ> lcm b = lcm b \<circ> lcm a"

  1194     by (simp add: fun_eq_iff ac_simps)

  1195 next

  1196   fix a show "lcm a \<circ> lcm a = lcm a" unfolding o_def

  1197     by (intro ext, simp add: lcm_left_idem)

  1198 qed

  1199

  1200 lemma dvd_Lcm [simp]: "a \<in> A \<Longrightarrow> a dvd Lcm A"

  1201   and Lcm_dvd [simp]: "(\<forall>a\<in>A. a dvd l') \<Longrightarrow> Lcm A dvd l'"

  1202   and normalization_factor_Lcm [simp]:

  1203           "normalization_factor (Lcm A) = (if Lcm A = 0 then 0 else 1)"

  1204 proof -

  1205   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>

  1206     normalization_factor (Lcm A) = (if Lcm A = 0 then 0 else 1)" (is ?thesis)

  1207   proof (cases "\<exists>l. l \<noteq>  0 \<and> (\<forall>a\<in>A. a dvd l)")

  1208     case False

  1209     hence "Lcm A = 0" by (auto simp: Lcm_Lcm_eucl Lcm_eucl_def)

  1210     with False show ?thesis by auto

  1211   next

  1212     case True

  1213     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

  1214     def n \<equiv> "LEAST n. \<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"

  1215     def l \<equiv> "SOME l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"

  1216     have "\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"

  1217       apply (subst n_def)

  1218       apply (rule LeastI[of _ "euclidean_size l\<^sub>0"])

  1219       apply (rule exI[of _ l\<^sub>0])

  1220       apply (simp add: l\<^sub>0_props)

  1221       done

  1222     from someI_ex[OF this] have "l \<noteq> 0" and "\<forall>a\<in>A. a dvd l" and "euclidean_size l = n"

  1223       unfolding l_def by simp_all

  1224     {

  1225       fix l' assume "\<forall>a\<in>A. a dvd l'"

  1226       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)

  1227       moreover from \<open>l \<noteq> 0\<close> have "gcd l l' \<noteq> 0" by simp

  1228       ultimately have "\<exists>b. b \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd b) \<and> euclidean_size b = euclidean_size (gcd l l')"

  1229         by (intro exI[of _ "gcd l l'"], auto)

  1230       hence "euclidean_size (gcd l l') \<ge> n" by (subst n_def) (rule Least_le)

  1231       moreover have "euclidean_size (gcd l l') \<le> n"

  1232       proof -

  1233         have "gcd l l' dvd l" by simp

  1234         then obtain a where "l = gcd l l' * a" unfolding dvd_def by blast

  1235         with \<open>l \<noteq> 0\<close> have "a \<noteq> 0" by auto

  1236         hence "euclidean_size (gcd l l') \<le> euclidean_size (gcd l l' * a)"

  1237           by (rule size_mult_mono)

  1238         also have "gcd l l' * a = l" using \<open>l = gcd l l' * a\<close> ..

  1239         also note \<open>euclidean_size l = n\<close>

  1240         finally show "euclidean_size (gcd l l') \<le> n" .

  1241       qed

  1242       ultimately have "euclidean_size l = euclidean_size (gcd l l')"

  1243         by (intro le_antisym, simp_all add: \<open>euclidean_size l = n\<close>)

  1244       with \<open>l \<noteq> 0\<close> have "l dvd gcd l l'" by (blast intro: dvd_euclidean_size_eq_imp_dvd)

  1245       hence "l dvd l'" by (blast dest: dvd_gcd_D2)

  1246     }

  1247

  1248     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>

  1249       have "(\<forall>a\<in>A. a dvd l div normalization_factor l) \<and>

  1250         (\<forall>l'. (\<forall>a\<in>A. a dvd l') \<longrightarrow> l div normalization_factor l dvd l') \<and>

  1251         normalization_factor (l div normalization_factor l) =

  1252         (if l div normalization_factor l = 0 then 0 else 1)"

  1253       by (auto simp: unit_simps)

  1254     also from True have "l div normalization_factor l = Lcm A"

  1255       by (simp add: Lcm_Lcm_eucl Lcm_eucl_def Let_def n_def l_def)

  1256     finally show ?thesis .

  1257   qed

  1258   note A = this

  1259

  1260   {fix a assume "a \<in> A" then show "a dvd Lcm A" using A by blast}

  1261   {fix l' assume "\<forall>a\<in>A. a dvd l'" then show "Lcm A dvd l'" using A by blast}

  1262   from A show "normalization_factor (Lcm A) = (if Lcm A = 0 then 0 else 1)" by blast

  1263 qed

  1264

  1265 lemma LcmI:

  1266   "(\<And>a. a\<in>A \<Longrightarrow> a dvd l) \<Longrightarrow> (\<And>l'. (\<forall>a\<in>A. a dvd l') \<Longrightarrow> l dvd l') \<Longrightarrow>

  1267       normalization_factor l = (if l = 0 then 0 else 1) \<Longrightarrow> l = Lcm A"

  1268   by (intro normed_associated_imp_eq)

  1269     (auto intro: Lcm_dvd dvd_Lcm simp: associated_def)

  1270

  1271 lemma Lcm_subset:

  1272   "A \<subseteq> B \<Longrightarrow> Lcm A dvd Lcm B"

  1273   by (blast intro: Lcm_dvd dvd_Lcm)

  1274

  1275 lemma Lcm_Un:

  1276   "Lcm (A \<union> B) = lcm (Lcm A) (Lcm B)"

  1277   apply (rule lcmI)

  1278   apply (blast intro: Lcm_subset)

  1279   apply (blast intro: Lcm_subset)

  1280   apply (intro Lcm_dvd ballI, elim UnE)

  1281   apply (rule dvd_trans, erule dvd_Lcm, assumption)

  1282   apply (rule dvd_trans, erule dvd_Lcm, assumption)

  1283   apply simp

  1284   done

  1285

  1286 lemma Lcm_1_iff:

  1287   "Lcm A = 1 \<longleftrightarrow> (\<forall>a\<in>A. is_unit a)"

  1288 proof

  1289   assume "Lcm A = 1"

  1290   then show "\<forall>a\<in>A. is_unit a" by auto

  1291 qed (rule LcmI [symmetric], auto)

  1292

  1293 lemma Lcm_no_units:

  1294   "Lcm A = Lcm (A - {a. is_unit a})"

  1295 proof -

  1296   have "(A - {a. is_unit a}) \<union> {a\<in>A. is_unit a} = A" by blast

  1297   hence "Lcm A = lcm (Lcm (A - {a. is_unit a})) (Lcm {a\<in>A. is_unit a})"

  1298     by (simp add: Lcm_Un[symmetric])

  1299   also have "Lcm {a\<in>A. is_unit a} = 1" by (simp add: Lcm_1_iff)

  1300   finally show ?thesis by simp

  1301 qed

  1302

  1303 lemma Lcm_empty [simp]:

  1304   "Lcm {} = 1"

  1305   by (simp add: Lcm_1_iff)

  1306

  1307 lemma Lcm_eq_0 [simp]:

  1308   "0 \<in> A \<Longrightarrow> Lcm A = 0"

  1309   by (drule dvd_Lcm) simp

  1310

  1311 lemma Lcm0_iff':

  1312   "Lcm A = 0 \<longleftrightarrow> \<not>(\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l))"

  1313 proof

  1314   assume "Lcm A = 0"

  1315   show "\<not>(\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l))"

  1316   proof

  1317     assume ex: "\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l)"

  1318     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

  1319     def n \<equiv> "LEAST n. \<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"

  1320     def l \<equiv> "SOME l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"

  1321     have "\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"

  1322       apply (subst n_def)

  1323       apply (rule LeastI[of _ "euclidean_size l\<^sub>0"])

  1324       apply (rule exI[of _ l\<^sub>0])

  1325       apply (simp add: l\<^sub>0_props)

  1326       done

  1327     from someI_ex[OF this] have "l \<noteq> 0" unfolding l_def by simp_all

  1328     hence "l div normalization_factor l \<noteq> 0" by simp

  1329     also from ex have "l div normalization_factor l = Lcm A"

  1330        by (simp only: Lcm_Lcm_eucl Lcm_eucl_def n_def l_def if_True Let_def)

  1331     finally show False using \<open>Lcm A = 0\<close> by contradiction

  1332   qed

  1333 qed (simp only: Lcm_Lcm_eucl Lcm_eucl_def if_False)

  1334

  1335 lemma Lcm0_iff [simp]:

  1336   "finite A \<Longrightarrow> Lcm A = 0 \<longleftrightarrow> 0 \<in> A"

  1337 proof -

  1338   assume "finite A"

  1339   have "0 \<in> A \<Longrightarrow> Lcm A = 0"  by (intro dvd_0_left dvd_Lcm)

  1340   moreover {

  1341     assume "0 \<notin> A"

  1342     hence "\<Prod>A \<noteq> 0"

  1343       apply (induct rule: finite_induct[OF \<open>finite A\<close>])

  1344       apply simp

  1345       apply (subst setprod.insert, assumption, assumption)

  1346       apply (rule no_zero_divisors)

  1347       apply blast+

  1348       done

  1349     moreover from \<open>finite A\<close> have "\<forall>a\<in>A. a dvd \<Prod>A" by blast

  1350     ultimately have "\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l)" by blast

  1351     with Lcm0_iff' have "Lcm A \<noteq> 0" by simp

  1352   }

  1353   ultimately show "Lcm A = 0 \<longleftrightarrow> 0 \<in> A" by blast

  1354 qed

  1355

  1356 lemma Lcm_no_multiple:

  1357   "(\<forall>m. m \<noteq> 0 \<longrightarrow> (\<exists>a\<in>A. \<not>a dvd m)) \<Longrightarrow> Lcm A = 0"

  1358 proof -

  1359   assume "\<forall>m. m \<noteq> 0 \<longrightarrow> (\<exists>a\<in>A. \<not>a dvd m)"

  1360   hence "\<not>(\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l))" by blast

  1361   then show "Lcm A = 0" by (simp only: Lcm_Lcm_eucl Lcm_eucl_def if_False)

  1362 qed

  1363

  1364 lemma Lcm_insert [simp]:

  1365   "Lcm (insert a A) = lcm a (Lcm A)"

  1366 proof (rule lcmI)

  1367   fix l assume "a dvd l" and "Lcm A dvd l"

  1368   hence "\<forall>a\<in>A. a dvd l" by (blast intro: dvd_trans dvd_Lcm)

  1369   with \<open>a dvd l\<close> show "Lcm (insert a A) dvd l" by (force intro: Lcm_dvd)

  1370 qed (auto intro: Lcm_dvd dvd_Lcm)

  1371

  1372 lemma Lcm_finite:

  1373   assumes "finite A"

  1374   shows "Lcm A = Finite_Set.fold lcm 1 A"

  1375   by (induct rule: finite.induct[OF \<open>finite A\<close>])

  1376     (simp_all add: comp_fun_idem.fold_insert_idem[OF comp_fun_idem_lcm])

  1377

  1378 lemma Lcm_set [code_unfold]:

  1379   "Lcm (set xs) = fold lcm xs 1"

  1380   using comp_fun_idem.fold_set_fold[OF comp_fun_idem_lcm] Lcm_finite by (simp add: ac_simps)

  1381

  1382 lemma Lcm_singleton [simp]:

  1383   "Lcm {a} = a div normalization_factor a"

  1384   by simp

  1385

  1386 lemma Lcm_2 [simp]:

  1387   "Lcm {a,b} = lcm a b"

  1388   by (simp only: Lcm_insert Lcm_empty lcm_1_right)

  1389     (cases "b = 0", simp, rule lcm_div_unit2, simp)

  1390

  1391 lemma Lcm_coprime:

  1392   assumes "finite A" and "A \<noteq> {}"

  1393   assumes "\<And>a b. a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<noteq> b \<Longrightarrow> gcd a b = 1"

  1394   shows "Lcm A = \<Prod>A div normalization_factor (\<Prod>A)"

  1395 using assms proof (induct rule: finite_ne_induct)

  1396   case (insert a A)

  1397   have "Lcm (insert a A) = lcm a (Lcm A)" by simp

  1398   also from insert have "Lcm A = \<Prod>A div normalization_factor (\<Prod>A)" by blast

  1399   also have "lcm a \<dots> = lcm a (\<Prod>A)" by (cases "\<Prod>A = 0") (simp_all add: lcm_div_unit2)

  1400   also from insert have "gcd a (\<Prod>A) = 1" by (subst gcd.commute, intro setprod_coprime) auto

  1401   with insert have "lcm a (\<Prod>A) = \<Prod>(insert a A) div normalization_factor (\<Prod>(insert a A))"

  1402     by (simp add: lcm_coprime)

  1403   finally show ?case .

  1404 qed simp

  1405

  1406 lemma Lcm_coprime':

  1407   "card A \<noteq> 0 \<Longrightarrow> (\<And>a b. a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<noteq> b \<Longrightarrow> gcd a b = 1)

  1408     \<Longrightarrow> Lcm A = \<Prod>A div normalization_factor (\<Prod>A)"

  1409   by (rule Lcm_coprime) (simp_all add: card_eq_0_iff)

  1410

  1411 lemma Gcd_Lcm:

  1412   "Gcd A = Lcm {d. \<forall>a\<in>A. d dvd a}"

  1413   by (simp add: Gcd_Gcd_eucl Lcm_Lcm_eucl Gcd_eucl_def)

  1414

  1415 lemma Gcd_dvd [simp]: "a \<in> A \<Longrightarrow> Gcd A dvd a"

  1416   and dvd_Gcd [simp]: "(\<forall>a\<in>A. g' dvd a) \<Longrightarrow> g' dvd Gcd A"

  1417   and normalization_factor_Gcd [simp]:

  1418     "normalization_factor (Gcd A) = (if Gcd A = 0 then 0 else 1)"

  1419 proof -

  1420   fix a assume "a \<in> A"

  1421   hence "Lcm {d. \<forall>a\<in>A. d dvd a} dvd a" by (intro Lcm_dvd) blast

  1422   then show "Gcd A dvd a" by (simp add: Gcd_Lcm)

  1423 next

  1424   fix g' assume "\<forall>a\<in>A. g' dvd a"

  1425   hence "g' dvd Lcm {d. \<forall>a\<in>A. d dvd a}" by (intro dvd_Lcm) blast

  1426   then show "g' dvd Gcd A" by (simp add: Gcd_Lcm)

  1427 next

  1428   show "normalization_factor (Gcd A) = (if Gcd A = 0 then 0 else 1)"

  1429     by (simp add: Gcd_Lcm)

  1430 qed

  1431

  1432 lemma GcdI:

  1433   "(\<And>a. a\<in>A \<Longrightarrow> l dvd a) \<Longrightarrow> (\<And>l'. (\<forall>a\<in>A. l' dvd a) \<Longrightarrow> l' dvd l) \<Longrightarrow>

  1434     normalization_factor l = (if l = 0 then 0 else 1) \<Longrightarrow> l = Gcd A"

  1435   by (intro normed_associated_imp_eq)

  1436     (auto intro: Gcd_dvd dvd_Gcd simp: associated_def)

  1437

  1438 lemma Lcm_Gcd:

  1439   "Lcm A = Gcd {m. \<forall>a\<in>A. a dvd m}"

  1440   by (rule LcmI[symmetric]) (auto intro: dvd_Gcd Gcd_dvd)

  1441

  1442 lemma Gcd_0_iff:

  1443   "Gcd A = 0 \<longleftrightarrow> A \<subseteq> {0}"

  1444   apply (rule iffI)

  1445   apply (rule subsetI, drule Gcd_dvd, simp)

  1446   apply (auto intro: GcdI[symmetric])

  1447   done

  1448

  1449 lemma Gcd_empty [simp]:

  1450   "Gcd {} = 0"

  1451   by (simp add: Gcd_0_iff)

  1452

  1453 lemma Gcd_1:

  1454   "1 \<in> A \<Longrightarrow> Gcd A = 1"

  1455   by (intro GcdI[symmetric]) (auto intro: Gcd_dvd dvd_Gcd)

  1456

  1457 lemma Gcd_insert [simp]:

  1458   "Gcd (insert a A) = gcd a (Gcd A)"

  1459 proof (rule gcdI)

  1460   fix l assume "l dvd a" and "l dvd Gcd A"

  1461   hence "\<forall>a\<in>A. l dvd a" by (blast intro: dvd_trans Gcd_dvd)

  1462   with \<open>l dvd a\<close> show "l dvd Gcd (insert a A)" by (force intro: Gcd_dvd)

  1463 qed auto

  1464

  1465 lemma Gcd_finite:

  1466   assumes "finite A"

  1467   shows "Gcd A = Finite_Set.fold gcd 0 A"

  1468   by (induct rule: finite.induct[OF \<open>finite A\<close>])

  1469     (simp_all add: comp_fun_idem.fold_insert_idem[OF comp_fun_idem_gcd])

  1470

  1471 lemma Gcd_set [code_unfold]:

  1472   "Gcd (set xs) = fold gcd xs 0"

  1473   using comp_fun_idem.fold_set_fold[OF comp_fun_idem_gcd] Gcd_finite by (simp add: ac_simps)

  1474

  1475 lemma Gcd_singleton [simp]: "Gcd {a} = a div normalization_factor a"

  1476   by (simp add: gcd_0)

  1477

  1478 lemma Gcd_2 [simp]: "Gcd {a,b} = gcd a b"

  1479   by (simp only: Gcd_insert Gcd_empty gcd_0) (cases "b = 0", simp, rule gcd_div_unit2, simp)

  1480

  1481 subclass semiring_gcd

  1482   by unfold_locales (simp_all add: gcd_greatest_iff)

  1483

  1484 end

  1485

  1486 text \<open>

  1487   A Euclidean ring is a Euclidean semiring with additive inverses. It provides a

  1488   few more lemmas; in particular, Bezout's lemma holds for any Euclidean ring.

  1489 \<close>

  1490

  1491 class euclidean_ring_gcd = euclidean_semiring_gcd + idom

  1492 begin

  1493

  1494 subclass euclidean_ring ..

  1495

  1496 subclass ring_gcd ..

  1497

  1498 lemma euclid_ext_gcd [simp]:

  1499   "(case euclid_ext a b of (_, _ , t) \<Rightarrow> t) = gcd a b"

  1500   by (induct a b rule: gcd_eucl_induct)

  1501     (simp_all add: euclid_ext_0 gcd_0 euclid_ext_non_0 ac_simps split: prod.split prod.split_asm)

  1502

  1503 lemma euclid_ext_gcd' [simp]:

  1504   "euclid_ext a b = (r, s, t) \<Longrightarrow> t = gcd a b"

  1505   by (insert euclid_ext_gcd[of a b], drule (1) subst, simp)

  1506

  1507 lemma euclid_ext'_correct:

  1508   "fst (euclid_ext' a b) * a + snd (euclid_ext' a b) * b = gcd a b"

  1509 proof-

  1510   obtain s t c where "euclid_ext a b = (s,t,c)"

  1511     by (cases "euclid_ext a b", blast)

  1512   with euclid_ext_correct[of a b] euclid_ext_gcd[of a b]

  1513     show ?thesis unfolding euclid_ext'_def by simp

  1514 qed

  1515

  1516 lemma bezout: "\<exists>s t. s * a + t * b = gcd a b"

  1517   using euclid_ext'_correct by blast

  1518

  1519 lemma gcd_neg1 [simp]:

  1520   "gcd (-a) b = gcd a b"

  1521   by (rule sym, rule gcdI, simp_all add: gcd_greatest)

  1522

  1523 lemma gcd_neg2 [simp]:

  1524   "gcd a (-b) = gcd a b"

  1525   by (rule sym, rule gcdI, simp_all add: gcd_greatest)

  1526

  1527 lemma gcd_neg_numeral_1 [simp]:

  1528   "gcd (- numeral n) a = gcd (numeral n) a"

  1529   by (fact gcd_neg1)

  1530

  1531 lemma gcd_neg_numeral_2 [simp]:

  1532   "gcd a (- numeral n) = gcd a (numeral n)"

  1533   by (fact gcd_neg2)

  1534

  1535 lemma gcd_diff1: "gcd (m - n) n = gcd m n"

  1536   by (subst diff_conv_add_uminus, subst gcd_neg2[symmetric],  subst gcd_add1, simp)

  1537

  1538 lemma gcd_diff2: "gcd (n - m) n = gcd m n"

  1539   by (subst gcd_neg1[symmetric], simp only: minus_diff_eq gcd_diff1)

  1540

  1541 lemma coprime_minus_one [simp]: "gcd (n - 1) n = 1"

  1542 proof -

  1543   have "gcd (n - 1) n = gcd n (n - 1)" by (fact gcd.commute)

  1544   also have "\<dots> = gcd ((n - 1) + 1) (n - 1)" by simp

  1545   also have "\<dots> = 1" by (rule coprime_plus_one)

  1546   finally show ?thesis .

  1547 qed

  1548

  1549 lemma lcm_neg1 [simp]: "lcm (-a) b = lcm a b"

  1550   by (rule sym, rule lcmI, simp_all add: lcm_least lcm_zero)

  1551

  1552 lemma lcm_neg2 [simp]: "lcm a (-b) = lcm a b"

  1553   by (rule sym, rule lcmI, simp_all add: lcm_least lcm_zero)

  1554

  1555 lemma lcm_neg_numeral_1 [simp]: "lcm (- numeral n) a = lcm (numeral n) a"

  1556   by (fact lcm_neg1)

  1557

  1558 lemma lcm_neg_numeral_2 [simp]: "lcm a (- numeral n) = lcm a (numeral n)"

  1559   by (fact lcm_neg2)

  1560

  1561 end

  1562

  1563

  1564 subsection \<open>Typical instances\<close>

  1565

  1566 instantiation nat :: euclidean_semiring

  1567 begin

  1568

  1569 definition [simp]:

  1570   "euclidean_size_nat = (id :: nat \<Rightarrow> nat)"

  1571

  1572 definition [simp]:

  1573   "normalization_factor_nat (n::nat) = (if n = 0 then 0 else 1 :: nat)"

  1574

  1575 instance proof

  1576 qed simp_all

  1577

  1578 end

  1579

  1580 instantiation int :: euclidean_ring

  1581 begin

  1582

  1583 definition [simp]:

  1584   "euclidean_size_int = (nat \<circ> abs :: int \<Rightarrow> nat)"

  1585

  1586 definition [simp]:

  1587   "normalization_factor_int = (sgn :: int \<Rightarrow> int)"

  1588

  1589 instance

  1590 proof (default, goals)

  1591   case 2

  1592   then show ?case by (auto simp add: abs_mult nat_mult_distrib)

  1593 next

  1594   case 3

  1595   then show ?case by (simp add: zsgn_def)

  1596 next

  1597   case 5

  1598   then show ?case by (auto simp: zsgn_def)

  1599 next

  1600   case 6

  1601   then show ?case by (auto split: abs_split simp: zsgn_def)

  1602 qed (auto simp: sgn_times split: abs_split)

  1603

  1604 end

  1605

  1606 instantiation poly :: (field) euclidean_ring

  1607 begin

  1608

  1609 definition euclidean_size_poly :: "'a poly \<Rightarrow> nat"

  1610   where "euclidean_size = (degree :: 'a poly \<Rightarrow> nat)"

  1611

  1612 definition normalization_factor_poly :: "'a poly \<Rightarrow> 'a poly"

  1613   where "normalization_factor p = monom (coeff p (degree p)) 0"

  1614

  1615 instance

  1616 proof (default, unfold euclidean_size_poly_def normalization_factor_poly_def)

  1617   fix p q :: "'a poly"

  1618   assume "q \<noteq> 0" and "\<not> q dvd p"

  1619   then show "degree (p mod q) < degree q"

  1620     using degree_mod_less [of q p] by (simp add: mod_eq_0_iff_dvd)

  1621 next

  1622   fix p q :: "'a poly"

  1623   assume "q \<noteq> 0"

  1624   from \<open>q \<noteq> 0\<close> show "degree p \<le> degree (p * q)"

  1625     by (rule degree_mult_right_le)

  1626   from \<open>q \<noteq> 0\<close> show "is_unit (monom (coeff q (degree q)) 0)"

  1627     by (auto intro: is_unit_monom_0)

  1628 next

  1629   fix p :: "'a poly"

  1630   show "monom (coeff p (degree p)) 0 = p" if "is_unit p"

  1631     using that by (fact is_unit_monom_trival)

  1632 next

  1633   fix p q :: "'a poly"

  1634   show "monom (coeff (p * q) (degree (p * q))) 0 =

  1635     monom (coeff p (degree p)) 0 * monom (coeff q (degree q)) 0"

  1636     by (simp add: monom_0 coeff_degree_mult)

  1637 next

  1638   show "monom (coeff 0 (degree 0)) 0 = 0"

  1639     by simp

  1640 qed

  1641

  1642 end

  1643

  1644 end