src/HOL/Number_Theory/Euclidean_Algorithm.thy
 author haftmann Sat Jun 27 20:26:33 2015 +0200 (2015-06-27) changeset 60600 87fbfea0bd0a parent 60599 f8bb070dc98b child 60634 e3b6e516608b permissions -rw-r--r--
simplified termination criterion for euclidean algorithm (again)
     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> 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"

   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 gcd_eucl b (a mod b))"

   178   by pat_completeness simp

   179 termination

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

   181

   182 declare gcd_eucl.simps [simp del]

   183

   184 lemma gcd_eucl_induct [case_names zero mod]:

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

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

   187   shows "P a b"

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

   189   case ("1" a b)

   190   show ?case

   191   proof (cases "b = 0")

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

   193   next

   194     case False

   195     then have "P b (a mod b)"

   196       by (rule "1.hyps")

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

   198       by (blast intro: H2)

   199   qed

   200 qed

   201

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

   203 where

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

   205

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

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

   208   for infinite sets as well\<close>

   209 where

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

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

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

   213        in l div normalization_factor l

   214       else 0)"

   215

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

   217 where

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

   219

   220 lemma gcd_eucl_0:

   221   "gcd_eucl a 0 = a div normalization_factor a"

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

   223

   224 lemma gcd_eucl_0_left:

   225   "gcd_eucl 0 a = a div normalization_factor a"

   226   by (simp_all add: gcd_eucl_0 gcd_eucl.simps [of 0 a])

   227

   228 lemma gcd_eucl_non_0:

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

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

   231

   232 end

   233

   234 class euclidean_ring = euclidean_semiring + idom

   235 begin

   236

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

   238   "euclid_ext a b =

   239      (if b = 0 then

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

   241       else

   242         case euclid_ext b (a mod b) of

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

   244   by pat_completeness simp

   245 termination

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

   247

   248 declare euclid_ext.simps [simp del]

   249

   250 lemma euclid_ext_0:

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

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

   253

   254 lemma euclid_ext_left_0:

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

   256   by (simp add: euclid_ext_0 euclid_ext.simps [of 0 a])

   257

   258 lemma euclid_ext_non_0:

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

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

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

   262

   263 lemma euclid_ext_code [code]:

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

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

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

   267

   268 lemma euclid_ext_correct:

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

   270 proof (induct a b rule: gcd_eucl_induct)

   271   case (zero a) then show ?case

   272     by (simp add: euclid_ext_0 ac_simps)

   273 next

   274   case (mod a b)

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

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

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

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

   279     by (simp add: algebra_simps)

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

   281   finally show ?case

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

   283 qed

   284

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

   286 where

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

   288

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

   290   by (simp add: euclid_ext'_def euclid_ext_0)

   291

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

   293   by (simp add: euclid_ext'_def euclid_ext_left_0)

   294

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

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

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

   298

   299 end

   300

   301 class euclidean_semiring_gcd = euclidean_semiring + gcd + Gcd +

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

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

   304 begin

   305

   306 lemma gcd_0_left:

   307   "gcd 0 a = a div normalization_factor a"

   308   unfolding gcd_gcd_eucl by (fact gcd_eucl_0_left)

   309

   310 lemma gcd_0:

   311   "gcd a 0 = a div normalization_factor a"

   312   unfolding gcd_gcd_eucl by (fact gcd_eucl_0)

   313

   314 lemma gcd_non_0:

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

   316   unfolding gcd_gcd_eucl by (fact gcd_eucl_non_0)

   317

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

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

   320   by (induct a b rule: gcd_eucl_induct)

   321     (simp_all add: gcd_0 gcd_non_0 dvd_mod_iff)

   322

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

   324   by (rule dvd_trans, assumption, rule gcd_dvd1)

   325

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

   327   by (rule dvd_trans, assumption, rule gcd_dvd2)

   328

   329 lemma gcd_greatest:

   330   fixes k a b :: 'a

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

   332 proof (induct a b rule: gcd_eucl_induct)

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

   334 next

   335   case (mod a b)

   336   then show ?case

   337     by (simp add: gcd_non_0 dvd_mod_iff)

   338 qed

   339

   340 lemma dvd_gcd_iff:

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

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

   343

   344 lemmas gcd_greatest_iff = dvd_gcd_iff

   345

   346 lemma gcd_zero [simp]:

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

   348   by (metis dvd_0_left dvd_refl gcd_dvd1 gcd_dvd2 gcd_greatest)+

   349

   350 lemma normalization_factor_gcd [simp]:

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

   352   by (induct a b rule: gcd_eucl_induct)

   353     (auto simp add: gcd_0 gcd_non_0)

   354

   355 lemma gcdI:

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

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

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

   359

   360 sublocale gcd!: abel_semigroup gcd

   361 proof

   362   fix a b c

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

   364   proof (rule gcdI)

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

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

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

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

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

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

   371       by (rule gcd_greatest)

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

   373       by auto

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

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

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

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

   378       by (intro gcd_greatest)

   379   qed

   380 next

   381   fix a b

   382   show "gcd a b = gcd b a"

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

   384 qed

   385

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

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

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

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

   390

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

   392   using mult_dvd_mono [of 1] by auto

   393

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

   395   by (rule sym, rule gcdI, simp_all)

   396

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

   398   by (rule sym, rule gcdI, simp_all)

   399

   400 lemma gcd_proj2_if_dvd:

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

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

   403

   404 lemma gcd_proj1_if_dvd:

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

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

   407

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

   409 proof

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

   411   show "m dvd n"

   412   proof (cases "m = 0")

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

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

   415       by (simp add: unit_eq_div2)

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

   417   qed (insert A, simp)

   418 next

   419   assume "m dvd n"

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

   421 qed

   422

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

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

   425

   426 lemma gcd_mod1 [simp]:

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

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

   429

   430 lemma gcd_mod2 [simp]:

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

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

   433

   434 lemma gcd_mult_distrib':

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

   436 proof (cases "c = 0")

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

   438 next

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

   440   show ?thesis

   441   proof (induct a b rule: gcd_eucl_induct)

   442     case (zero a) show ?case

   443     proof (cases "a = 0")

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

   445     next

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

   447       then show ?thesis

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

   449     qed

   450     case (mod a b)

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

   452   qed

   453 qed

   454

   455 lemma gcd_mult_distrib:

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

   457 proof-

   458   let ?nf = "normalization_factor"

   459   from gcd_mult_distrib'

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

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

   462     by (metis dvd_div_mult dvd_eq_mod_eq_0 mod_0 normalization_factor_dvd)

   463   finally show ?thesis

   464     by simp

   465 qed

   466

   467 lemma euclidean_size_gcd_le1 [simp]:

   468   assumes "a \<noteq> 0"

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

   470 proof -

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

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

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

   474 qed

   475

   476 lemma euclidean_size_gcd_le2 [simp]:

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

   478   by (subst gcd.commute, rule euclidean_size_gcd_le1)

   479

   480 lemma euclidean_size_gcd_less1:

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

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

   483 proof (rule ccontr)

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

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

   486     by (intro le_antisym, simp_all)

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

   488   hence "a dvd b" using dvd_gcd_D2 by blast

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

   490 qed

   491

   492 lemma euclidean_size_gcd_less2:

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

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

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

   496

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

   498   apply (rule gcdI)

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

   500   apply (rule gcd_dvd2)

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

   502   apply (subst normalization_factor_gcd, simp add: gcd_0)

   503   done

   504

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

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

   507

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

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

   510

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

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

   513

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

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

   516

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

   518   apply (rule gcdI)

   519   apply (simp add: ac_simps)

   520   apply (rule gcd_dvd2)

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

   522   apply simp

   523   done

   524

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

   526   apply (rule gcdI)

   527   apply simp

   528   apply (rule dvd_trans, rule gcd_dvd2, rule gcd_dvd2)

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

   530   apply simp

   531   done

   532

   533 lemma comp_fun_idem_gcd: "comp_fun_idem gcd"

   534 proof

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

   536     by (simp add: fun_eq_iff ac_simps)

   537 next

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

   539     by (simp add: fun_eq_iff gcd_left_idem)

   540 qed

   541

   542 lemma coprime_dvd_mult:

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

   544   shows "c dvd a"

   545 proof -

   546   let ?nf = "normalization_factor"

   547   from assms gcd_mult_distrib [of a c b]

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

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

   550 qed

   551

   552 lemma coprime_dvd_mult_iff:

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

   554   by (rule, rule coprime_dvd_mult, simp_all)

   555

   556 lemma gcd_dvd_antisym:

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

   558 proof (rule gcdI)

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

   560   have "gcd c d dvd c" by simp

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

   562   have "gcd c d dvd d" by simp

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

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

   565     by simp

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

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

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

   569 qed

   570

   571 lemma gcd_mult_cancel:

   572   assumes "gcd k n = 1"

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

   574 proof (rule gcd_dvd_antisym)

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

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

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

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

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

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

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

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

   583 qed

   584

   585 lemma coprime_crossproduct:

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

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

   588 proof

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

   590 next

   591   assume ?lhs

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

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

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

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

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

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

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

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

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

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

   602   ultimately show ?rhs unfolding associated_def by simp

   603 qed

   604

   605 lemma gcd_add1 [simp]:

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

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

   608

   609 lemma gcd_add2 [simp]:

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

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

   612

   613 lemma gcd_add_mult:

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

   615 proof -

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

   617     by (fact gcd_mod2)

   618   then show ?thesis by simp

   619 qed

   620

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

   622   by (rule sym, rule gcdI, simp_all)

   623

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

   625   by (auto intro: coprimeI gcd_greatest dvd_gcd_D1 dvd_gcd_D2)

   626

   627 lemma div_gcd_coprime:

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

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

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

   631   shows "gcd a' b' = 1"

   632 proof (rule coprimeI)

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

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

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

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

   637     by (simp_all only: ac_simps)

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

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

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

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

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

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

   644   ultimately have "1 = l * u"

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

   646   then show "l dvd 1" ..

   647 qed

   648

   649 lemma coprime_mult:

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

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

   652   apply (subst gcd.commute)

   653   using da apply (subst gcd_mult_cancel)

   654   apply (subst gcd.commute, assumption)

   655   apply (subst gcd.commute, rule db)

   656   done

   657

   658 lemma coprime_lmult:

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

   660   shows "gcd d a = 1"

   661 proof (rule coprimeI)

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

   663   hence "l dvd a * b" by simp

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

   665 qed

   666

   667 lemma coprime_rmult:

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

   669   shows "gcd d b = 1"

   670 proof (rule coprimeI)

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

   672   hence "l dvd a * b" by simp

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

   674 qed

   675

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

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

   678

   679 lemma gcd_coprime:

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

   681   shows "gcd a' b' = 1"

   682 proof -

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

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

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

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

   687   finally show ?thesis .

   688 qed

   689

   690 lemma coprime_power:

   691   assumes "0 < n"

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

   693 using assms proof (induct n)

   694   case (Suc n) then show ?case

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

   696 qed simp

   697

   698 lemma gcd_coprime_exists:

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

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

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

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

   703   apply (insert nz, auto intro: div_gcd_coprime)

   704   done

   705

   706 lemma coprime_exp:

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

   708   by (induct n, simp_all add: coprime_mult)

   709

   710 lemma coprime_exp2 [intro]:

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

   712   apply (rule coprime_exp)

   713   apply (subst gcd.commute)

   714   apply (rule coprime_exp)

   715   apply (subst gcd.commute)

   716   apply assumption

   717   done

   718

   719 lemma gcd_exp:

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

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

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

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

   724 next

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

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

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

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

   729   also note gcd_mult_distrib

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

   731     by (simp add: normalization_factor_pow A)

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

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

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

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

   736   finally show ?thesis by simp

   737 qed

   738

   739 lemma coprime_common_divisor:

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

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

   742   apply simp

   743   apply (erule (1) gcd_greatest)

   744   done

   745

   746 lemma division_decomp:

   747   assumes dc: "a dvd b * c"

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

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

   750   assume "gcd a b = 0"

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

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

   753   then show ?thesis by blast

   754 next

   755   let ?d = "gcd a b"

   756   assume "?d \<noteq> 0"

   757   from gcd_coprime_exists[OF this]

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

   759     by blast

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

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

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

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

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

   765   with coprime_dvd_mult[OF ab'(3)]

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

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

   768   then show ?thesis by blast

   769 qed

   770

   771 lemma pow_divs_pow:

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

   773   shows "a dvd b"

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

   775   assume "gcd a b = 0"

   776   then show ?thesis by simp

   777 next

   778   let ?d = "gcd a b"

   779   assume "?d \<noteq> 0"

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

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

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

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

   784     by blast

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

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

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

   788     by (simp only: power_mult_distrib ac_simps)

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

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

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

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

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

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

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

   796 qed

   797

   798 lemma pow_divs_eq [simp]:

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

   800   by (auto intro: pow_divs_pow dvd_power_same)

   801

   802 lemma divs_mult:

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

   804   shows "m * n dvd r"

   805 proof -

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

   807     unfolding dvd_def by blast

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

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

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

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

   812   then show ?thesis unfolding dvd_def by blast

   813 qed

   814

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

   816   by (subst add_commute, simp)

   817

   818 lemma setprod_coprime [rule_format]:

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

   820   apply (cases "finite A")

   821   apply (induct set: finite)

   822   apply (auto simp add: gcd_mult_cancel)

   823   done

   824

   825 lemma coprime_divisors:

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

   827   shows "gcd d e = 1"

   828 proof -

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

   830     unfolding dvd_def by blast

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

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

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

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

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

   836 qed

   837

   838 lemma invertible_coprime:

   839   assumes "a * b mod m = 1"

   840   shows "coprime a m"

   841 proof -

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

   843     by simp

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

   845     by simp

   846   then have "coprime m a"

   847     by (rule coprime_lmult)

   848   then show ?thesis

   849     by (simp add: ac_simps)

   850 qed

   851

   852 lemma lcm_gcd:

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

   854   by (simp only: lcm_lcm_eucl gcd_gcd_eucl lcm_eucl_def)

   855

   856 lemma lcm_gcd_prod:

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

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

   859   let ?nf = normalization_factor

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

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

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

   863     by (simp add: mult_ac)

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

   865     by (simp add: div_mult_swap mult.commute)

   866   finally show ?thesis .

   867 qed (auto simp add: lcm_gcd)

   868

   869 lemma lcm_dvd1 [iff]:

   870   "a dvd lcm a b"

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

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

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

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

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

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

   877     by (simp add: div_mult_swap unit_div_commute)

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

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

   880     by (subst (asm) div_mult_self2_is_id, simp_all)

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

   882     by (metis div_mult_swap gcd_dvd2 mult_assoc)

   883   finally show ?thesis by (rule dvdI)

   884 qed (auto simp add: lcm_gcd)

   885

   886 lemma lcm_least:

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

   888 proof (cases "k = 0")

   889   let ?nf = normalization_factor

   890   assume "k \<noteq> 0"

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

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

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

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

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

   896     unfolding dvd_def by blast

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   912     by (simp add: algebra_simps)

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

   914     by (metis div_mult_self2_is_id)

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

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

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

   918     by (simp add: algebra_simps)

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

   920     by (metis mult.commute div_mult_self2_is_id)

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

   922     by (metis div_mult_self2_is_id mult_assoc)

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

   924     by (simp add: unit_simps)

   925   finally show ?thesis by (rule dvdI)

   926 qed simp

   927

   928 lemma lcm_zero:

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

   930 proof -

   931   let ?nf = normalization_factor

   932   {

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

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

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

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

   937   } moreover {

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

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

   940   }

   941   ultimately show ?thesis by blast

   942 qed

   943

   944 lemmas lcm_0_iff = lcm_zero

   945

   946 lemma gcd_lcm:

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

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

   949 proof-

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

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

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

   953   hence "is_unit ?c" by simp

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

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

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

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

   958   finally show ?thesis .

   959 qed

   960

   961 lemma normalization_factor_lcm [simp]:

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

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

   964   case True then show ?thesis

   965     by (auto simp add: lcm_gcd)

   966 next

   967   case False

   968   let ?nf = normalization_factor

   969   from lcm_gcd_prod[of a b]

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

   971     by (metis div_by_0 div_self normalization_correct normalization_factor_0 normalization_factor_mult)

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

   973     by simp

   974   finally show ?thesis using False by simp

   975 qed

   976

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

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

   979

   980 lemma lcmI:

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

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

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

   984

   985 sublocale lcm!: abel_semigroup lcm

   986 proof

   987   fix a b c

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

   989   proof (rule lcmI)

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

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

   992

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

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

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

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

   997

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

   999     have "b dvd lcm b c" by simp

  1000     from this and \<open>lcm b c dvd l\<close> have "b dvd l" by (rule dvd_trans)

  1001     have "c dvd lcm b c" by simp

  1002     from this and \<open>lcm b c dvd l\<close> have "c dvd l" by (rule dvd_trans)

  1003     from \<open>a dvd l\<close> and \<open>b dvd l\<close> have "lcm a b dvd l" by (rule lcm_least)

  1004     from this and \<open>c dvd l\<close> show "lcm (lcm a b) c dvd l" by (rule lcm_least)

  1005   qed (simp add: lcm_zero)

  1006 next

  1007   fix a b

  1008   show "lcm a b = lcm b a"

  1009     by (simp add: lcm_gcd ac_simps)

  1010 qed

  1011

  1012 lemma dvd_lcm_D1:

  1013   "lcm m n dvd k \<Longrightarrow> m dvd k"

  1014   by (rule dvd_trans, rule lcm_dvd1, assumption)

  1015

  1016 lemma dvd_lcm_D2:

  1017   "lcm m n dvd k \<Longrightarrow> n dvd k"

  1018   by (rule dvd_trans, rule lcm_dvd2, assumption)

  1019

  1020 lemma gcd_dvd_lcm [simp]:

  1021   "gcd a b dvd lcm a b"

  1022   by (metis dvd_trans gcd_dvd2 lcm_dvd2)

  1023

  1024 lemma lcm_1_iff:

  1025   "lcm a b = 1 \<longleftrightarrow> is_unit a \<and> is_unit b"

  1026 proof

  1027   assume "lcm a b = 1"

  1028   then show "is_unit a \<and> is_unit b" by auto

  1029 next

  1030   assume "is_unit a \<and> is_unit b"

  1031   hence "a dvd 1" and "b dvd 1" by simp_all

  1032   hence "is_unit (lcm a b)" by (rule lcm_least)

  1033   hence "lcm a b = normalization_factor (lcm a b)"

  1034     by (subst normalization_factor_unit, simp_all)

  1035   also have "\<dots> = 1" using \<open>is_unit a \<and> is_unit b\<close>

  1036     by auto

  1037   finally show "lcm a b = 1" .

  1038 qed

  1039

  1040 lemma lcm_0_left [simp]:

  1041   "lcm 0 a = 0"

  1042   by (rule sym, rule lcmI, simp_all)

  1043

  1044 lemma lcm_0 [simp]:

  1045   "lcm a 0 = 0"

  1046   by (rule sym, rule lcmI, simp_all)

  1047

  1048 lemma lcm_unique:

  1049   "a dvd d \<and> b dvd d \<and>

  1050   normalization_factor d = (if d = 0 then 0 else 1) \<and>

  1051   (\<forall>e. a dvd e \<and> b dvd e \<longrightarrow> d dvd e) \<longleftrightarrow> d = lcm a b"

  1052   by (rule, auto intro: lcmI simp: lcm_least lcm_zero)

  1053

  1054 lemma dvd_lcm_I1 [simp]:

  1055   "k dvd m \<Longrightarrow> k dvd lcm m n"

  1056   by (metis lcm_dvd1 dvd_trans)

  1057

  1058 lemma dvd_lcm_I2 [simp]:

  1059   "k dvd n \<Longrightarrow> k dvd lcm m n"

  1060   by (metis lcm_dvd2 dvd_trans)

  1061

  1062 lemma lcm_1_left [simp]:

  1063   "lcm 1 a = a div normalization_factor a"

  1064   by (cases "a = 0") (simp, rule sym, rule lcmI, simp_all)

  1065

  1066 lemma lcm_1_right [simp]:

  1067   "lcm a 1 = a div normalization_factor a"

  1068   using lcm_1_left [of a] by (simp add: ac_simps)

  1069

  1070 lemma lcm_coprime:

  1071   "gcd a b = 1 \<Longrightarrow> lcm a b = a * b div normalization_factor (a*b)"

  1072   by (subst lcm_gcd) simp

  1073

  1074 lemma lcm_proj1_if_dvd:

  1075   "b dvd a \<Longrightarrow> lcm a b = a div normalization_factor a"

  1076   by (cases "a = 0") (simp, rule sym, rule lcmI, simp_all)

  1077

  1078 lemma lcm_proj2_if_dvd:

  1079   "a dvd b \<Longrightarrow> lcm a b = b div normalization_factor b"

  1080   using lcm_proj1_if_dvd [of a b] by (simp add: ac_simps)

  1081

  1082 lemma lcm_proj1_iff:

  1083   "lcm m n = m div normalization_factor m \<longleftrightarrow> n dvd m"

  1084 proof

  1085   assume A: "lcm m n = m div normalization_factor m"

  1086   show "n dvd m"

  1087   proof (cases "m = 0")

  1088     assume [simp]: "m \<noteq> 0"

  1089     from A have B: "m = lcm m n * normalization_factor m"

  1090       by (simp add: unit_eq_div2)

  1091     show ?thesis by (subst B, simp)

  1092   qed simp

  1093 next

  1094   assume "n dvd m"

  1095   then show "lcm m n = m div normalization_factor m" by (rule lcm_proj1_if_dvd)

  1096 qed

  1097

  1098 lemma lcm_proj2_iff:

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

  1100   using lcm_proj1_iff [of n m] by (simp add: ac_simps)

  1101

  1102 lemma euclidean_size_lcm_le1:

  1103   assumes "a \<noteq> 0" and "b \<noteq> 0"

  1104   shows "euclidean_size a \<le> euclidean_size (lcm a b)"

  1105 proof -

  1106   have "a dvd lcm a b" by (rule lcm_dvd1)

  1107   then obtain c where A: "lcm a b = a * c" unfolding dvd_def by blast

  1108   with \<open>a \<noteq> 0\<close> and \<open>b \<noteq> 0\<close> have "c \<noteq> 0" by (auto simp: lcm_zero)

  1109   then show ?thesis by (subst A, intro size_mult_mono)

  1110 qed

  1111

  1112 lemma euclidean_size_lcm_le2:

  1113   "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> euclidean_size b \<le> euclidean_size (lcm a b)"

  1114   using euclidean_size_lcm_le1 [of b a] by (simp add: ac_simps)

  1115

  1116 lemma euclidean_size_lcm_less1:

  1117   assumes "b \<noteq> 0" and "\<not>b dvd a"

  1118   shows "euclidean_size a < euclidean_size (lcm a b)"

  1119 proof (rule ccontr)

  1120   from assms have "a \<noteq> 0" by auto

  1121   assume "\<not>euclidean_size a < euclidean_size (lcm a b)"

  1122   with \<open>a \<noteq> 0\<close> and \<open>b \<noteq> 0\<close> have "euclidean_size (lcm a b) = euclidean_size a"

  1123     by (intro le_antisym, simp, intro euclidean_size_lcm_le1)

  1124   with assms have "lcm a b dvd a"

  1125     by (rule_tac dvd_euclidean_size_eq_imp_dvd) (auto simp: lcm_zero)

  1126   hence "b dvd a" by (rule dvd_lcm_D2)

  1127   with \<open>\<not>b dvd a\<close> show False by contradiction

  1128 qed

  1129

  1130 lemma euclidean_size_lcm_less2:

  1131   assumes "a \<noteq> 0" and "\<not>a dvd b"

  1132   shows "euclidean_size b < euclidean_size (lcm a b)"

  1133   using assms euclidean_size_lcm_less1 [of a b] by (simp add: ac_simps)

  1134

  1135 lemma lcm_mult_unit1:

  1136   "is_unit a \<Longrightarrow> lcm (b * a) c = lcm b c"

  1137   apply (rule lcmI)

  1138   apply (rule dvd_trans[of _ "b * a"], simp, rule lcm_dvd1)

  1139   apply (rule lcm_dvd2)

  1140   apply (rule lcm_least, simp add: unit_simps, assumption)

  1141   apply (subst normalization_factor_lcm, simp add: lcm_zero)

  1142   done

  1143

  1144 lemma lcm_mult_unit2:

  1145   "is_unit a \<Longrightarrow> lcm b (c * a) = lcm b c"

  1146   using lcm_mult_unit1 [of a c b] by (simp add: ac_simps)

  1147

  1148 lemma lcm_div_unit1:

  1149   "is_unit a \<Longrightarrow> lcm (b div a) c = lcm b c"

  1150   by (erule is_unitE [of _ b]) (simp add: lcm_mult_unit1)

  1151

  1152 lemma lcm_div_unit2:

  1153   "is_unit a \<Longrightarrow> lcm b (c div a) = lcm b c"

  1154   by (erule is_unitE [of _ c]) (simp add: lcm_mult_unit2)

  1155

  1156 lemma lcm_left_idem:

  1157   "lcm a (lcm a b) = lcm a b"

  1158   apply (rule lcmI)

  1159   apply simp

  1160   apply (subst lcm.assoc [symmetric], rule lcm_dvd2)

  1161   apply (rule lcm_least, assumption)

  1162   apply (erule (1) lcm_least)

  1163   apply (auto simp: lcm_zero)

  1164   done

  1165

  1166 lemma lcm_right_idem:

  1167   "lcm (lcm a b) b = lcm a b"

  1168   apply (rule lcmI)

  1169   apply (subst lcm.assoc, rule lcm_dvd1)

  1170   apply (rule lcm_dvd2)

  1171   apply (rule lcm_least, erule (1) lcm_least, assumption)

  1172   apply (auto simp: lcm_zero)

  1173   done

  1174

  1175 lemma comp_fun_idem_lcm: "comp_fun_idem lcm"

  1176 proof

  1177   fix a b show "lcm a \<circ> lcm b = lcm b \<circ> lcm a"

  1178     by (simp add: fun_eq_iff ac_simps)

  1179 next

  1180   fix a show "lcm a \<circ> lcm a = lcm a" unfolding o_def

  1181     by (intro ext, simp add: lcm_left_idem)

  1182 qed

  1183

  1184 lemma dvd_Lcm [simp]: "a \<in> A \<Longrightarrow> a dvd Lcm A"

  1185   and Lcm_dvd [simp]: "(\<forall>a\<in>A. a dvd l') \<Longrightarrow> Lcm A dvd l'"

  1186   and normalization_factor_Lcm [simp]:

  1187           "normalization_factor (Lcm A) = (if Lcm A = 0 then 0 else 1)"

  1188 proof -

  1189   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>

  1190     normalization_factor (Lcm A) = (if Lcm A = 0 then 0 else 1)" (is ?thesis)

  1191   proof (cases "\<exists>l. l \<noteq>  0 \<and> (\<forall>a\<in>A. a dvd l)")

  1192     case False

  1193     hence "Lcm A = 0" by (auto simp: Lcm_Lcm_eucl Lcm_eucl_def)

  1194     with False show ?thesis by auto

  1195   next

  1196     case True

  1197     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

  1198     def n \<equiv> "LEAST n. \<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"

  1199     def l \<equiv> "SOME l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"

  1200     have "\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"

  1201       apply (subst n_def)

  1202       apply (rule LeastI[of _ "euclidean_size l\<^sub>0"])

  1203       apply (rule exI[of _ l\<^sub>0])

  1204       apply (simp add: l\<^sub>0_props)

  1205       done

  1206     from someI_ex[OF this] have "l \<noteq> 0" and "\<forall>a\<in>A. a dvd l" and "euclidean_size l = n"

  1207       unfolding l_def by simp_all

  1208     {

  1209       fix l' assume "\<forall>a\<in>A. a dvd l'"

  1210       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)

  1211       moreover from \<open>l \<noteq> 0\<close> have "gcd l l' \<noteq> 0" by simp

  1212       ultimately have "\<exists>b. b \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd b) \<and> euclidean_size b = euclidean_size (gcd l l')"

  1213         by (intro exI[of _ "gcd l l'"], auto)

  1214       hence "euclidean_size (gcd l l') \<ge> n" by (subst n_def) (rule Least_le)

  1215       moreover have "euclidean_size (gcd l l') \<le> n"

  1216       proof -

  1217         have "gcd l l' dvd l" by simp

  1218         then obtain a where "l = gcd l l' * a" unfolding dvd_def by blast

  1219         with \<open>l \<noteq> 0\<close> have "a \<noteq> 0" by auto

  1220         hence "euclidean_size (gcd l l') \<le> euclidean_size (gcd l l' * a)"

  1221           by (rule size_mult_mono)

  1222         also have "gcd l l' * a = l" using \<open>l = gcd l l' * a\<close> ..

  1223         also note \<open>euclidean_size l = n\<close>

  1224         finally show "euclidean_size (gcd l l') \<le> n" .

  1225       qed

  1226       ultimately have "euclidean_size l = euclidean_size (gcd l l')"

  1227         by (intro le_antisym, simp_all add: \<open>euclidean_size l = n\<close>)

  1228       with \<open>l \<noteq> 0\<close> have "l dvd gcd l l'" by (blast intro: dvd_euclidean_size_eq_imp_dvd)

  1229       hence "l dvd l'" by (blast dest: dvd_gcd_D2)

  1230     }

  1231

  1232     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>

  1233       have "(\<forall>a\<in>A. a dvd l div normalization_factor l) \<and>

  1234         (\<forall>l'. (\<forall>a\<in>A. a dvd l') \<longrightarrow> l div normalization_factor l dvd l') \<and>

  1235         normalization_factor (l div normalization_factor l) =

  1236         (if l div normalization_factor l = 0 then 0 else 1)"

  1237       by (auto simp: unit_simps)

  1238     also from True have "l div normalization_factor l = Lcm A"

  1239       by (simp add: Lcm_Lcm_eucl Lcm_eucl_def Let_def n_def l_def)

  1240     finally show ?thesis .

  1241   qed

  1242   note A = this

  1243

  1244   {fix a assume "a \<in> A" then show "a dvd Lcm A" using A by blast}

  1245   {fix l' assume "\<forall>a\<in>A. a dvd l'" then show "Lcm A dvd l'" using A by blast}

  1246   from A show "normalization_factor (Lcm A) = (if Lcm A = 0 then 0 else 1)" by blast

  1247 qed

  1248

  1249 lemma LcmI:

  1250   "(\<And>a. a\<in>A \<Longrightarrow> a dvd l) \<Longrightarrow> (\<And>l'. (\<forall>a\<in>A. a dvd l') \<Longrightarrow> l dvd l') \<Longrightarrow>

  1251       normalization_factor l = (if l = 0 then 0 else 1) \<Longrightarrow> l = Lcm A"

  1252   by (intro normed_associated_imp_eq)

  1253     (auto intro: Lcm_dvd dvd_Lcm simp: associated_def)

  1254

  1255 lemma Lcm_subset:

  1256   "A \<subseteq> B \<Longrightarrow> Lcm A dvd Lcm B"

  1257   by (blast intro: Lcm_dvd dvd_Lcm)

  1258

  1259 lemma Lcm_Un:

  1260   "Lcm (A \<union> B) = lcm (Lcm A) (Lcm B)"

  1261   apply (rule lcmI)

  1262   apply (blast intro: Lcm_subset)

  1263   apply (blast intro: Lcm_subset)

  1264   apply (intro Lcm_dvd ballI, elim UnE)

  1265   apply (rule dvd_trans, erule dvd_Lcm, assumption)

  1266   apply (rule dvd_trans, erule dvd_Lcm, assumption)

  1267   apply simp

  1268   done

  1269

  1270 lemma Lcm_1_iff:

  1271   "Lcm A = 1 \<longleftrightarrow> (\<forall>a\<in>A. is_unit a)"

  1272 proof

  1273   assume "Lcm A = 1"

  1274   then show "\<forall>a\<in>A. is_unit a" by auto

  1275 qed (rule LcmI [symmetric], auto)

  1276

  1277 lemma Lcm_no_units:

  1278   "Lcm A = Lcm (A - {a. is_unit a})"

  1279 proof -

  1280   have "(A - {a. is_unit a}) \<union> {a\<in>A. is_unit a} = A" by blast

  1281   hence "Lcm A = lcm (Lcm (A - {a. is_unit a})) (Lcm {a\<in>A. is_unit a})"

  1282     by (simp add: Lcm_Un[symmetric])

  1283   also have "Lcm {a\<in>A. is_unit a} = 1" by (simp add: Lcm_1_iff)

  1284   finally show ?thesis by simp

  1285 qed

  1286

  1287 lemma Lcm_empty [simp]:

  1288   "Lcm {} = 1"

  1289   by (simp add: Lcm_1_iff)

  1290

  1291 lemma Lcm_eq_0 [simp]:

  1292   "0 \<in> A \<Longrightarrow> Lcm A = 0"

  1293   by (drule dvd_Lcm) simp

  1294

  1295 lemma Lcm0_iff':

  1296   "Lcm A = 0 \<longleftrightarrow> \<not>(\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l))"

  1297 proof

  1298   assume "Lcm A = 0"

  1299   show "\<not>(\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l))"

  1300   proof

  1301     assume ex: "\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l)"

  1302     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

  1303     def n \<equiv> "LEAST n. \<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"

  1304     def l \<equiv> "SOME l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"

  1305     have "\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"

  1306       apply (subst n_def)

  1307       apply (rule LeastI[of _ "euclidean_size l\<^sub>0"])

  1308       apply (rule exI[of _ l\<^sub>0])

  1309       apply (simp add: l\<^sub>0_props)

  1310       done

  1311     from someI_ex[OF this] have "l \<noteq> 0" unfolding l_def by simp_all

  1312     hence "l div normalization_factor l \<noteq> 0" by simp

  1313     also from ex have "l div normalization_factor l = Lcm A"

  1314        by (simp only: Lcm_Lcm_eucl Lcm_eucl_def n_def l_def if_True Let_def)

  1315     finally show False using \<open>Lcm A = 0\<close> by contradiction

  1316   qed

  1317 qed (simp only: Lcm_Lcm_eucl Lcm_eucl_def if_False)

  1318

  1319 lemma Lcm0_iff [simp]:

  1320   "finite A \<Longrightarrow> Lcm A = 0 \<longleftrightarrow> 0 \<in> A"

  1321 proof -

  1322   assume "finite A"

  1323   have "0 \<in> A \<Longrightarrow> Lcm A = 0"  by (intro dvd_0_left dvd_Lcm)

  1324   moreover {

  1325     assume "0 \<notin> A"

  1326     hence "\<Prod>A \<noteq> 0"

  1327       apply (induct rule: finite_induct[OF \<open>finite A\<close>])

  1328       apply simp

  1329       apply (subst setprod.insert, assumption, assumption)

  1330       apply (rule no_zero_divisors)

  1331       apply blast+

  1332       done

  1333     moreover from \<open>finite A\<close> have "\<forall>a\<in>A. a dvd \<Prod>A" by blast

  1334     ultimately have "\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l)" by blast

  1335     with Lcm0_iff' have "Lcm A \<noteq> 0" by simp

  1336   }

  1337   ultimately show "Lcm A = 0 \<longleftrightarrow> 0 \<in> A" by blast

  1338 qed

  1339

  1340 lemma Lcm_no_multiple:

  1341   "(\<forall>m. m \<noteq> 0 \<longrightarrow> (\<exists>a\<in>A. \<not>a dvd m)) \<Longrightarrow> Lcm A = 0"

  1342 proof -

  1343   assume "\<forall>m. m \<noteq> 0 \<longrightarrow> (\<exists>a\<in>A. \<not>a dvd m)"

  1344   hence "\<not>(\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l))" by blast

  1345   then show "Lcm A = 0" by (simp only: Lcm_Lcm_eucl Lcm_eucl_def if_False)

  1346 qed

  1347

  1348 lemma Lcm_insert [simp]:

  1349   "Lcm (insert a A) = lcm a (Lcm A)"

  1350 proof (rule lcmI)

  1351   fix l assume "a dvd l" and "Lcm A dvd l"

  1352   hence "\<forall>a\<in>A. a dvd l" by (blast intro: dvd_trans dvd_Lcm)

  1353   with \<open>a dvd l\<close> show "Lcm (insert a A) dvd l" by (force intro: Lcm_dvd)

  1354 qed (auto intro: Lcm_dvd dvd_Lcm)

  1355

  1356 lemma Lcm_finite:

  1357   assumes "finite A"

  1358   shows "Lcm A = Finite_Set.fold lcm 1 A"

  1359   by (induct rule: finite.induct[OF \<open>finite A\<close>])

  1360     (simp_all add: comp_fun_idem.fold_insert_idem[OF comp_fun_idem_lcm])

  1361

  1362 lemma Lcm_set [code_unfold]:

  1363   "Lcm (set xs) = fold lcm xs 1"

  1364   using comp_fun_idem.fold_set_fold[OF comp_fun_idem_lcm] Lcm_finite by (simp add: ac_simps)

  1365

  1366 lemma Lcm_singleton [simp]:

  1367   "Lcm {a} = a div normalization_factor a"

  1368   by simp

  1369

  1370 lemma Lcm_2 [simp]:

  1371   "Lcm {a,b} = lcm a b"

  1372   by (simp only: Lcm_insert Lcm_empty lcm_1_right)

  1373     (cases "b = 0", simp, rule lcm_div_unit2, simp)

  1374

  1375 lemma Lcm_coprime:

  1376   assumes "finite A" and "A \<noteq> {}"

  1377   assumes "\<And>a b. a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<noteq> b \<Longrightarrow> gcd a b = 1"

  1378   shows "Lcm A = \<Prod>A div normalization_factor (\<Prod>A)"

  1379 using assms proof (induct rule: finite_ne_induct)

  1380   case (insert a A)

  1381   have "Lcm (insert a A) = lcm a (Lcm A)" by simp

  1382   also from insert have "Lcm A = \<Prod>A div normalization_factor (\<Prod>A)" by blast

  1383   also have "lcm a \<dots> = lcm a (\<Prod>A)" by (cases "\<Prod>A = 0") (simp_all add: lcm_div_unit2)

  1384   also from insert have "gcd a (\<Prod>A) = 1" by (subst gcd.commute, intro setprod_coprime) auto

  1385   with insert have "lcm a (\<Prod>A) = \<Prod>(insert a A) div normalization_factor (\<Prod>(insert a A))"

  1386     by (simp add: lcm_coprime)

  1387   finally show ?case .

  1388 qed simp

  1389

  1390 lemma Lcm_coprime':

  1391   "card A \<noteq> 0 \<Longrightarrow> (\<And>a b. a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<noteq> b \<Longrightarrow> gcd a b = 1)

  1392     \<Longrightarrow> Lcm A = \<Prod>A div normalization_factor (\<Prod>A)"

  1393   by (rule Lcm_coprime) (simp_all add: card_eq_0_iff)

  1394

  1395 lemma Gcd_Lcm:

  1396   "Gcd A = Lcm {d. \<forall>a\<in>A. d dvd a}"

  1397   by (simp add: Gcd_Gcd_eucl Lcm_Lcm_eucl Gcd_eucl_def)

  1398

  1399 lemma Gcd_dvd [simp]: "a \<in> A \<Longrightarrow> Gcd A dvd a"

  1400   and dvd_Gcd [simp]: "(\<forall>a\<in>A. g' dvd a) \<Longrightarrow> g' dvd Gcd A"

  1401   and normalization_factor_Gcd [simp]:

  1402     "normalization_factor (Gcd A) = (if Gcd A = 0 then 0 else 1)"

  1403 proof -

  1404   fix a assume "a \<in> A"

  1405   hence "Lcm {d. \<forall>a\<in>A. d dvd a} dvd a" by (intro Lcm_dvd) blast

  1406   then show "Gcd A dvd a" by (simp add: Gcd_Lcm)

  1407 next

  1408   fix g' assume "\<forall>a\<in>A. g' dvd a"

  1409   hence "g' dvd Lcm {d. \<forall>a\<in>A. d dvd a}" by (intro dvd_Lcm) blast

  1410   then show "g' dvd Gcd A" by (simp add: Gcd_Lcm)

  1411 next

  1412   show "normalization_factor (Gcd A) = (if Gcd A = 0 then 0 else 1)"

  1413     by (simp add: Gcd_Lcm)

  1414 qed

  1415

  1416 lemma GcdI:

  1417   "(\<And>a. a\<in>A \<Longrightarrow> l dvd a) \<Longrightarrow> (\<And>l'. (\<forall>a\<in>A. l' dvd a) \<Longrightarrow> l' dvd l) \<Longrightarrow>

  1418     normalization_factor l = (if l = 0 then 0 else 1) \<Longrightarrow> l = Gcd A"

  1419   by (intro normed_associated_imp_eq)

  1420     (auto intro: Gcd_dvd dvd_Gcd simp: associated_def)

  1421

  1422 lemma Lcm_Gcd:

  1423   "Lcm A = Gcd {m. \<forall>a\<in>A. a dvd m}"

  1424   by (rule LcmI[symmetric]) (auto intro: dvd_Gcd Gcd_dvd)

  1425

  1426 lemma Gcd_0_iff:

  1427   "Gcd A = 0 \<longleftrightarrow> A \<subseteq> {0}"

  1428   apply (rule iffI)

  1429   apply (rule subsetI, drule Gcd_dvd, simp)

  1430   apply (auto intro: GcdI[symmetric])

  1431   done

  1432

  1433 lemma Gcd_empty [simp]:

  1434   "Gcd {} = 0"

  1435   by (simp add: Gcd_0_iff)

  1436

  1437 lemma Gcd_1:

  1438   "1 \<in> A \<Longrightarrow> Gcd A = 1"

  1439   by (intro GcdI[symmetric]) (auto intro: Gcd_dvd dvd_Gcd)

  1440

  1441 lemma Gcd_insert [simp]:

  1442   "Gcd (insert a A) = gcd a (Gcd A)"

  1443 proof (rule gcdI)

  1444   fix l assume "l dvd a" and "l dvd Gcd A"

  1445   hence "\<forall>a\<in>A. l dvd a" by (blast intro: dvd_trans Gcd_dvd)

  1446   with \<open>l dvd a\<close> show "l dvd Gcd (insert a A)" by (force intro: Gcd_dvd)

  1447 qed auto

  1448

  1449 lemma Gcd_finite:

  1450   assumes "finite A"

  1451   shows "Gcd A = Finite_Set.fold gcd 0 A"

  1452   by (induct rule: finite.induct[OF \<open>finite A\<close>])

  1453     (simp_all add: comp_fun_idem.fold_insert_idem[OF comp_fun_idem_gcd])

  1454

  1455 lemma Gcd_set [code_unfold]:

  1456   "Gcd (set xs) = fold gcd xs 0"

  1457   using comp_fun_idem.fold_set_fold[OF comp_fun_idem_gcd] Gcd_finite by (simp add: ac_simps)

  1458

  1459 lemma Gcd_singleton [simp]: "Gcd {a} = a div normalization_factor a"

  1460   by (simp add: gcd_0)

  1461

  1462 lemma Gcd_2 [simp]: "Gcd {a,b} = gcd a b"

  1463   by (simp only: Gcd_insert Gcd_empty gcd_0) (cases "b = 0", simp, rule gcd_div_unit2, simp)

  1464

  1465 subclass semiring_gcd

  1466   by unfold_locales (simp_all add: gcd_greatest_iff)

  1467

  1468 end

  1469

  1470 text \<open>

  1471   A Euclidean ring is a Euclidean semiring with additive inverses. It provides a

  1472   few more lemmas; in particular, Bezout's lemma holds for any Euclidean ring.

  1473 \<close>

  1474

  1475 class euclidean_ring_gcd = euclidean_semiring_gcd + idom

  1476 begin

  1477

  1478 subclass euclidean_ring ..

  1479

  1480 subclass ring_gcd ..

  1481

  1482 lemma euclid_ext_gcd [simp]:

  1483   "(case euclid_ext a b of (_, _ , t) \<Rightarrow> t) = gcd a b"

  1484   by (induct a b rule: gcd_eucl_induct)

  1485     (simp_all add: euclid_ext_0 gcd_0 euclid_ext_non_0 ac_simps split: prod.split prod.split_asm)

  1486

  1487 lemma euclid_ext_gcd' [simp]:

  1488   "euclid_ext a b = (r, s, t) \<Longrightarrow> t = gcd a b"

  1489   by (insert euclid_ext_gcd[of a b], drule (1) subst, simp)

  1490

  1491 lemma euclid_ext'_correct:

  1492   "fst (euclid_ext' a b) * a + snd (euclid_ext' a b) * b = gcd a b"

  1493 proof-

  1494   obtain s t c where "euclid_ext a b = (s,t,c)"

  1495     by (cases "euclid_ext a b", blast)

  1496   with euclid_ext_correct[of a b] euclid_ext_gcd[of a b]

  1497     show ?thesis unfolding euclid_ext'_def by simp

  1498 qed

  1499

  1500 lemma bezout: "\<exists>s t. s * a + t * b = gcd a b"

  1501   using euclid_ext'_correct by blast

  1502

  1503 lemma gcd_neg1 [simp]:

  1504   "gcd (-a) b = gcd a b"

  1505   by (rule sym, rule gcdI, simp_all add: gcd_greatest)

  1506

  1507 lemma gcd_neg2 [simp]:

  1508   "gcd a (-b) = gcd a b"

  1509   by (rule sym, rule gcdI, simp_all add: gcd_greatest)

  1510

  1511 lemma gcd_neg_numeral_1 [simp]:

  1512   "gcd (- numeral n) a = gcd (numeral n) a"

  1513   by (fact gcd_neg1)

  1514

  1515 lemma gcd_neg_numeral_2 [simp]:

  1516   "gcd a (- numeral n) = gcd a (numeral n)"

  1517   by (fact gcd_neg2)

  1518

  1519 lemma gcd_diff1: "gcd (m - n) n = gcd m n"

  1520   by (subst diff_conv_add_uminus, subst gcd_neg2[symmetric],  subst gcd_add1, simp)

  1521

  1522 lemma gcd_diff2: "gcd (n - m) n = gcd m n"

  1523   by (subst gcd_neg1[symmetric], simp only: minus_diff_eq gcd_diff1)

  1524

  1525 lemma coprime_minus_one [simp]: "gcd (n - 1) n = 1"

  1526 proof -

  1527   have "gcd (n - 1) n = gcd n (n - 1)" by (fact gcd.commute)

  1528   also have "\<dots> = gcd ((n - 1) + 1) (n - 1)" by simp

  1529   also have "\<dots> = 1" by (rule coprime_plus_one)

  1530   finally show ?thesis .

  1531 qed

  1532

  1533 lemma lcm_neg1 [simp]: "lcm (-a) b = lcm a b"

  1534   by (rule sym, rule lcmI, simp_all add: lcm_least lcm_zero)

  1535

  1536 lemma lcm_neg2 [simp]: "lcm a (-b) = lcm a b"

  1537   by (rule sym, rule lcmI, simp_all add: lcm_least lcm_zero)

  1538

  1539 lemma lcm_neg_numeral_1 [simp]: "lcm (- numeral n) a = lcm (numeral n) a"

  1540   by (fact lcm_neg1)

  1541

  1542 lemma lcm_neg_numeral_2 [simp]: "lcm a (- numeral n) = lcm a (numeral n)"

  1543   by (fact lcm_neg2)

  1544

  1545 end

  1546

  1547

  1548 subsection \<open>Typical instances\<close>

  1549

  1550 instantiation nat :: euclidean_semiring

  1551 begin

  1552

  1553 definition [simp]:

  1554   "euclidean_size_nat = (id :: nat \<Rightarrow> nat)"

  1555

  1556 definition [simp]:

  1557   "normalization_factor_nat (n::nat) = (if n = 0 then 0 else 1 :: nat)"

  1558

  1559 instance proof

  1560 qed simp_all

  1561

  1562 end

  1563

  1564 instantiation int :: euclidean_ring

  1565 begin

  1566

  1567 definition [simp]:

  1568   "euclidean_size_int = (nat \<circ> abs :: int \<Rightarrow> nat)"

  1569

  1570 definition [simp]:

  1571   "normalization_factor_int = (sgn :: int \<Rightarrow> int)"

  1572

  1573 instance

  1574 proof (default, goals)

  1575   case 2

  1576   then show ?case by (auto simp add: abs_mult nat_mult_distrib)

  1577 next

  1578   case 3

  1579   then show ?case by (simp add: zsgn_def)

  1580 next

  1581   case 5

  1582   then show ?case by (auto simp: zsgn_def)

  1583 next

  1584   case 6

  1585   then show ?case by (auto split: abs_split simp: zsgn_def)

  1586 qed (auto simp: sgn_times split: abs_split)

  1587

  1588 end

  1589

  1590 instantiation poly :: (field) euclidean_ring

  1591 begin

  1592

  1593 definition euclidean_size_poly :: "'a poly \<Rightarrow> nat"

  1594   where "euclidean_size p = (if p = 0 then 0 else Suc (degree p))"

  1595

  1596 definition normalization_factor_poly :: "'a poly \<Rightarrow> 'a poly"

  1597   where "normalization_factor p = monom (coeff p (degree p)) 0"

  1598

  1599 lemma euclidean_size_poly_0 [simp]:

  1600   "euclidean_size (0::'a poly) = 0"

  1601   by (simp add: euclidean_size_poly_def)

  1602

  1603 lemma euclidean_size_poly_not_0 [simp]:

  1604   "p \<noteq> 0 \<Longrightarrow> euclidean_size p = Suc (degree p)"

  1605   by (simp add: euclidean_size_poly_def)

  1606

  1607 instance

  1608 proof

  1609   fix p q :: "'a poly"

  1610   assume "q \<noteq> 0"

  1611   then have "p mod q = 0 \<or> degree (p mod q) < degree q"

  1612     by (rule degree_mod_less [of q p])

  1613   with \<open>q \<noteq> 0\<close> show "euclidean_size (p mod q) < euclidean_size q"

  1614     by (cases "p mod q = 0") simp_all

  1615 next

  1616   fix p q :: "'a poly"

  1617   assume "q \<noteq> 0"

  1618   from \<open>q \<noteq> 0\<close> have "degree p \<le> degree (p * q)"

  1619     by (rule degree_mult_right_le)

  1620   with \<open>q \<noteq> 0\<close> show "euclidean_size p \<le> euclidean_size (p * q)"

  1621     by (cases "p = 0") simp_all

  1622   from \<open>q \<noteq> 0\<close> have "is_unit (monom (coeff q (degree q)) 0)"

  1623     by (auto intro: is_unit_monom_0)

  1624   then show "is_unit (normalization_factor q)"

  1625     by (simp add: normalization_factor_poly_def)

  1626 next

  1627   fix p :: "'a poly"

  1628   assume "is_unit p"

  1629   then have "monom (coeff p (degree p)) 0 = p"

  1630     by (fact is_unit_monom_trival)

  1631   then show "normalization_factor p = p"

  1632     by (simp add: normalization_factor_poly_def)

  1633 next

  1634   fix p q :: "'a poly"

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

  1636     monom (coeff p (degree p)) 0 * monom (coeff q (degree q)) 0"

  1637     by (simp add: monom_0 coeff_degree_mult)

  1638   then show "normalization_factor (p * q) =

  1639     normalization_factor p * normalization_factor q"

  1640     by (simp add: normalization_factor_poly_def)

  1641 next

  1642   have "monom (coeff 0 (degree 0)) 0 = 0"

  1643     by simp

  1644   then show "normalization_factor 0 = (0::'a poly)"

  1645     by (simp add: normalization_factor_poly_def)

  1646 qed

  1647

  1648 end

  1649

  1650 end