src/HOL/Number_Theory/Euclidean_Algorithm.thy
 author haftmann Thu Jun 25 15:01:43 2015 +0200 (2015-06-25) changeset 60572 718b1ba06429 parent 60571 c9fdf2080447 child 60582 d694f217ee41 permissions -rw-r--r--
streamlined definitions and primitive lemma of euclidean algorithm, including code generation
     1 (* Author: Manuel Eberl *)

     2

     3 section \<open>Abstract euclidean algorithm\<close>

     4

     5 theory Euclidean_Algorithm

     6 imports Complex_Main "~~/src/HOL/Library/Polynomial"

     7 begin

     8

     9 text \<open>

    10   A Euclidean semiring is a semiring upon which the Euclidean algorithm can be

    11   implemented. It must provide:

    12   \begin{itemize}

    13   \item division with remainder

    14   \item a size function such that @{term "size (a mod b) < size b"}

    15         for any @{term "b \<noteq> 0"}

    16   \item a normalization factor such that two associated numbers are equal iff

    17         they are the same when divd by their normalization factors.

    18   \end{itemize}

    19   The existence of these functions makes it possible to derive gcd and lcm functions

    20   for any Euclidean semiring.

    21 \<close>

    22 class euclidean_semiring = semiring_div +

    23   fixes euclidean_size :: "'a \<Rightarrow> nat"

    24   fixes normalization_factor :: "'a \<Rightarrow> 'a"

    25   assumes mod_size_less:

    26     "b \<noteq> 0 \<Longrightarrow> \<not> b dvd a \<Longrightarrow> euclidean_size (a mod b) < euclidean_size b"

    27   assumes size_mult_mono:

    28     "b \<noteq> 0 \<Longrightarrow> euclidean_size (a * b) \<ge> euclidean_size a"

    29   assumes normalization_factor_is_unit [intro,simp]:

    30     "a \<noteq> 0 \<Longrightarrow> is_unit (normalization_factor a)"

    31   assumes normalization_factor_mult: "normalization_factor (a * b) =

    32     normalization_factor a * normalization_factor b"

    33   assumes normalization_factor_unit: "is_unit a \<Longrightarrow> normalization_factor a = a"

    34   assumes normalization_factor_0 [simp]: "normalization_factor 0 = 0"

    35 begin

    36

    37 lemma normalization_factor_dvd [simp]:

    38   "a \<noteq> 0 \<Longrightarrow> normalization_factor a dvd b"

    39   by (rule unit_imp_dvd, simp)

    40

    41 lemma normalization_factor_1 [simp]:

    42   "normalization_factor 1 = 1"

    43   by (simp add: normalization_factor_unit)

    44

    45 lemma normalization_factor_0_iff [simp]:

    46   "normalization_factor a = 0 \<longleftrightarrow> a = 0"

    47 proof

    48   assume "normalization_factor a = 0"

    49   hence "\<not> is_unit (normalization_factor a)"

    50     by simp

    51   then show "a = 0" by auto

    52 qed simp

    53

    54 lemma normalization_factor_pow:

    55   "normalization_factor (a ^ n) = normalization_factor a ^ n"

    56   by (induct n) (simp_all add: normalization_factor_mult power_Suc2)

    57

    58 lemma normalization_correct [simp]:

    59   "normalization_factor (a div normalization_factor a) = (if a = 0 then 0 else 1)"

    60 proof (cases "a = 0", simp)

    61   assume "a \<noteq> 0"

    62   let ?nf = "normalization_factor"

    63   from normalization_factor_is_unit[OF \<open>a \<noteq> 0\<close>] have "?nf a \<noteq> 0"

    64     by auto

    65   have "?nf (a div ?nf a) * ?nf (?nf a) = ?nf (a div ?nf a * ?nf a)"

    66     by (simp add: normalization_factor_mult)

    67   also have "a div ?nf a * ?nf a = a" using \<open>a \<noteq> 0\<close>

    68     by simp

    69   also have "?nf (?nf a) = ?nf a" using \<open>a \<noteq> 0\<close>

    70     normalization_factor_is_unit normalization_factor_unit by simp

    71   finally have "normalization_factor (a div normalization_factor a) = 1"

    72     using \<open>?nf a \<noteq> 0\<close> by (metis div_mult_self2_is_id div_self)

    73   with \<open>a \<noteq> 0\<close> show ?thesis by simp

    74 qed

    75

    76 lemma normalization_0_iff [simp]:

    77   "a div normalization_factor a = 0 \<longleftrightarrow> a = 0"

    78   by (cases "a = 0", simp, subst unit_eq_div1, blast, simp)

    79

    80 lemma mult_div_normalization [simp]:

    81   "b * (1 div normalization_factor a) = b div normalization_factor a"

    82   by (cases "a = 0") simp_all

    83

    84 lemma associated_iff_normed_eq:

    85   "associated a b \<longleftrightarrow> a div normalization_factor a = b div normalization_factor b"

    86 proof (cases "b = 0", simp, cases "a = 0", metis associated_0(1) normalization_0_iff, rule iffI)

    87   let ?nf = normalization_factor

    88   assume "a \<noteq> 0" "b \<noteq> 0" "a div ?nf a = b div ?nf b"

    89   hence "a = b * (?nf a div ?nf b)"

    90     apply (subst (asm) unit_eq_div1, blast, subst (asm) unit_div_commute, blast)

    91     apply (subst div_mult_swap, simp, simp)

    92     done

    93   with \<open>a \<noteq> 0\<close> \<open>b \<noteq> 0\<close> have "\<exists>c. is_unit c \<and> a = c * b"

    94     by (intro exI[of _ "?nf a div ?nf b"], force simp: mult_ac)

    95   then obtain c where "is_unit c" and "a = c * b" by blast

    96   then show "associated a b" by (rule is_unit_associatedI)

    97 next

    98   let ?nf = normalization_factor

    99   assume "a \<noteq> 0" "b \<noteq> 0" "associated a b"

   100   then obtain c where "is_unit c" and "a = c * b" by (blast elim: associated_is_unitE)

   101   then show "a div ?nf a = b div ?nf b"

   102     apply (simp only: \<open>a = c * b\<close> normalization_factor_mult normalization_factor_unit)

   103     apply (rule div_mult_mult1, force)

   104     done

   105   qed

   106

   107 lemma normed_associated_imp_eq:

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

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

   110

   111 lemma normed_dvd [iff]:

   112   "a div normalization_factor a dvd a"

   113 proof (cases "a = 0")

   114   case True then show ?thesis by simp

   115 next

   116   case False

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

   118     by (auto intro: unit_div_mult_self)

   119   then show ?thesis ..

   120 qed

   121

   122 lemma dvd_normed [iff]:

   123   "a dvd a div normalization_factor a"

   124 proof (cases "a = 0")

   125   case True then show ?thesis by simp

   126 next

   127   case False

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

   129     by (auto intro: unit_mult_div_div)

   130   then show ?thesis ..

   131 qed

   132

   133 lemma associated_normed:

   134   "associated (a div normalization_factor a) a"

   135   by (rule associatedI) simp_all

   136

   137 lemma normalization_factor_dvd' [simp]:

   138   "normalization_factor a dvd a"

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

   140

   141 lemmas normalization_factor_dvd_iff [simp] =

   142   unit_dvd_iff [OF normalization_factor_is_unit]

   143

   144 lemma euclidean_division:

   145   fixes a :: 'a and b :: 'a

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

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

   148     and "euclidean_size t < euclidean_size b"

   149 proof -

   150   from div_mod_equality [of a b 0]

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

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

   153 qed

   154

   155 lemma dvd_euclidean_size_eq_imp_dvd:

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

   157   shows "a dvd b"

   158 proof (rule ccontr)

   159   assume "\<not> a dvd b"

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

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

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

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

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

   165       using size_mult_mono by force

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

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

   168       using mod_size_less by blast

   169   ultimately show False using size_eq by simp

   170 qed

   171

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

   173 where

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

   175     else if b dvd a then b div normalization_factor b

   176     else gcd_eucl b (a mod b))"

   177   by pat_completeness simp

   178 termination

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

   180

   181 declare gcd_eucl.simps [simp del]

   182

   183 lemma gcd_eucl_induct [case_names zero mod]:

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

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

   186   shows "P a b"

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

   188   case ("1" a b)

   189   show ?case

   190   proof (cases "b = 0")

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

   192   next

   193     case False

   194     have "P b (a mod b)"

   195     proof (cases "b dvd a")

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

   197         by (rule "1.hyps")

   198     next

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

   200         by (simp add: mod_eq_0_iff_dvd)

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

   202     qed

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

   204       by (blast intro: H2)

   205   qed

   206 qed

   207

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

   209 where

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

   211

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

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

   214   for infinite sets as well\<close>

   215 where

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

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

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

   219        in l div normalization_factor l

   220       else 0)"

   221

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

   223 where

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

   225

   226 lemma gcd_eucl_0:

   227   "gcd_eucl a 0 = a div normalization_factor a"

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

   229

   230 lemma gcd_eucl_0_left:

   231   "gcd_eucl 0 a = a div normalization_factor a"

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

   233

   234 lemma gcd_eucl_non_0:

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

   236   by (cases "b dvd a")

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

   238

   239 lemma gcd_eucl_code [code]:

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

   241   by (auto simp add: gcd_eucl_non_0 gcd_eucl_0 gcd_eucl_0_left)

   242

   243 end

   244

   245 class euclidean_semiring_gcd = euclidean_semiring + gcd + Gcd +

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

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

   248 begin

   249

   250 lemma gcd_0_left:

   251   "gcd 0 a = a div normalization_factor a"

   252   unfolding gcd_gcd_eucl by (fact gcd_eucl_0_left)

   253

   254 lemma gcd_0:

   255   "gcd a 0 = a div normalization_factor a"

   256   unfolding gcd_gcd_eucl by (fact gcd_eucl_0)

   257

   258 lemma gcd_non_0:

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

   260   unfolding gcd_gcd_eucl by (fact gcd_eucl_non_0)

   261

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

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

   264   by (induct a b rule: gcd_eucl_induct)

   265     (simp_all add: gcd_0 gcd_non_0 dvd_mod_iff)

   266

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

   268   by (rule dvd_trans, assumption, rule gcd_dvd1)

   269

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

   271   by (rule dvd_trans, assumption, rule gcd_dvd2)

   272

   273 lemma gcd_greatest:

   274   fixes k a b :: 'a

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

   276 proof (induct a b rule: gcd_eucl_induct)

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

   278 next

   279   case (mod a b)

   280   then show ?case

   281     by (simp add: gcd_non_0 dvd_mod_iff)

   282 qed

   283

   284 lemma dvd_gcd_iff:

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

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

   287

   288 lemmas gcd_greatest_iff = dvd_gcd_iff

   289

   290 lemma gcd_zero [simp]:

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

   292   by (metis dvd_0_left dvd_refl gcd_dvd1 gcd_dvd2 gcd_greatest)+

   293

   294 lemma normalization_factor_gcd [simp]:

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

   296   by (induct a b rule: gcd_eucl_induct)

   297     (auto simp add: gcd_0 gcd_non_0)

   298

   299 lemma gcdI:

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

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

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

   303

   304 sublocale gcd!: abel_semigroup gcd

   305 proof

   306   fix a b c

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

   308   proof (rule gcdI)

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

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

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

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

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

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

   315       by (rule gcd_greatest)

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

   317       by auto

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

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

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

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

   322       by (intro gcd_greatest)

   323   qed

   324 next

   325   fix a b

   326   show "gcd a b = gcd b a"

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

   328 qed

   329

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

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

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

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

   334

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

   336   using mult_dvd_mono [of 1] by auto

   337

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

   339   by (rule sym, rule gcdI, simp_all)

   340

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

   342   by (rule sym, rule gcdI, simp_all)

   343

   344 lemma gcd_proj2_if_dvd:

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

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

   347

   348 lemma gcd_proj1_if_dvd:

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

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

   351

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

   353 proof

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

   355   show "m dvd n"

   356   proof (cases "m = 0")

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

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

   359       by (simp add: unit_eq_div2)

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

   361   qed (insert A, simp)

   362 next

   363   assume "m dvd n"

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

   365 qed

   366

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

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

   369

   370 lemma gcd_mod1 [simp]:

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

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

   373

   374 lemma gcd_mod2 [simp]:

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

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

   377

   378 lemma gcd_mult_distrib':

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

   380 proof (cases "c = 0")

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

   382 next

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

   384   show ?thesis

   385   proof (induct a b rule: gcd_eucl_induct)

   386     case (zero a) show ?case

   387     proof (cases "a = 0")

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

   389     next

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

   391       then show ?thesis

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

   393     qed

   394     case (mod a b)

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

   396   qed

   397 qed

   398

   399 lemma gcd_mult_distrib:

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

   401 proof-

   402   let ?nf = "normalization_factor"

   403   from gcd_mult_distrib'

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

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

   406     by (metis dvd_div_mult dvd_eq_mod_eq_0 mod_0 normalization_factor_dvd)

   407   finally show ?thesis

   408     by simp

   409 qed

   410

   411 lemma euclidean_size_gcd_le1 [simp]:

   412   assumes "a \<noteq> 0"

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

   414 proof -

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

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

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

   418 qed

   419

   420 lemma euclidean_size_gcd_le2 [simp]:

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

   422   by (subst gcd.commute, rule euclidean_size_gcd_le1)

   423

   424 lemma euclidean_size_gcd_less1:

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

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

   427 proof (rule ccontr)

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

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

   430     by (intro le_antisym, simp_all)

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

   432   hence "a dvd b" using dvd_gcd_D2 by blast

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

   434 qed

   435

   436 lemma euclidean_size_gcd_less2:

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

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

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

   440

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

   442   apply (rule gcdI)

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

   444   apply (rule gcd_dvd2)

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

   446   apply (subst normalization_factor_gcd, simp add: gcd_0)

   447   done

   448

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

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

   451

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

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

   454

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

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

   457

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

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

   460

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

   462   apply (rule gcdI)

   463   apply (simp add: ac_simps)

   464   apply (rule gcd_dvd2)

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

   466   apply simp

   467   done

   468

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

   470   apply (rule gcdI)

   471   apply simp

   472   apply (rule dvd_trans, rule gcd_dvd2, rule gcd_dvd2)

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

   474   apply simp

   475   done

   476

   477 lemma comp_fun_idem_gcd: "comp_fun_idem gcd"

   478 proof

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

   480     by (simp add: fun_eq_iff ac_simps)

   481 next

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

   483     by (simp add: fun_eq_iff gcd_left_idem)

   484 qed

   485

   486 lemma coprime_dvd_mult:

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

   488   shows "c dvd a"

   489 proof -

   490   let ?nf = "normalization_factor"

   491   from assms gcd_mult_distrib [of a c b]

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

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

   494 qed

   495

   496 lemma coprime_dvd_mult_iff:

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

   498   by (rule, rule coprime_dvd_mult, simp_all)

   499

   500 lemma gcd_dvd_antisym:

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

   502 proof (rule gcdI)

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

   504   have "gcd c d dvd c" by simp

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

   506   have "gcd c d dvd d" by simp

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

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

   509     by simp

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

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

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

   513 qed

   514

   515 lemma gcd_mult_cancel:

   516   assumes "gcd k n = 1"

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

   518 proof (rule gcd_dvd_antisym)

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

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

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

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

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

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

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

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

   527 qed

   528

   529 lemma coprime_crossproduct:

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

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

   532 proof

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

   534 next

   535   assume ?lhs

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

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

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

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

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

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

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

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

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

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

   546   ultimately show ?rhs unfolding associated_def by simp

   547 qed

   548

   549 lemma gcd_add1 [simp]:

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

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

   552

   553 lemma gcd_add2 [simp]:

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

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

   556

   557 lemma gcd_add_mult:

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

   559 proof -

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

   561     by (fact gcd_mod2)

   562   then show ?thesis by simp

   563 qed

   564

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

   566   by (rule sym, rule gcdI, simp_all)

   567

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

   569   by (auto intro: coprimeI gcd_greatest dvd_gcd_D1 dvd_gcd_D2)

   570

   571 lemma div_gcd_coprime:

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

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

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

   575   shows "gcd a' b' = 1"

   576 proof (rule coprimeI)

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

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

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

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

   581     by (simp_all only: ac_simps)

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

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

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

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

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

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

   588   ultimately have "1 = l * u"

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

   590   then show "l dvd 1" ..

   591 qed

   592

   593 lemma coprime_mult:

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

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

   596   apply (subst gcd.commute)

   597   using da apply (subst gcd_mult_cancel)

   598   apply (subst gcd.commute, assumption)

   599   apply (subst gcd.commute, rule db)

   600   done

   601

   602 lemma coprime_lmult:

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

   604   shows "gcd d a = 1"

   605 proof (rule coprimeI)

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

   607   hence "l dvd a * b" by simp

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

   609 qed

   610

   611 lemma coprime_rmult:

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

   613   shows "gcd d b = 1"

   614 proof (rule coprimeI)

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

   616   hence "l dvd a * b" by simp

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

   618 qed

   619

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

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

   622

   623 lemma gcd_coprime:

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

   625   shows "gcd a' b' = 1"

   626 proof -

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

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

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

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

   631   finally show ?thesis .

   632 qed

   633

   634 lemma coprime_power:

   635   assumes "0 < n"

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

   637 using assms proof (induct n)

   638   case (Suc n) then show ?case

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

   640 qed simp

   641

   642 lemma gcd_coprime_exists:

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

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

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

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

   647   apply (insert nz, auto intro: div_gcd_coprime)

   648   done

   649

   650 lemma coprime_exp:

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

   652   by (induct n, simp_all add: coprime_mult)

   653

   654 lemma coprime_exp2 [intro]:

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

   656   apply (rule coprime_exp)

   657   apply (subst gcd.commute)

   658   apply (rule coprime_exp)

   659   apply (subst gcd.commute)

   660   apply assumption

   661   done

   662

   663 lemma gcd_exp:

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

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

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

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

   668 next

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

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

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

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

   673   also note gcd_mult_distrib

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

   675     by (simp add: normalization_factor_pow A)

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

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

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

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

   680   finally show ?thesis by simp

   681 qed

   682

   683 lemma coprime_common_divisor:

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

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

   686   apply simp

   687   apply (erule (1) gcd_greatest)

   688   done

   689

   690 lemma division_decomp:

   691   assumes dc: "a dvd b * c"

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

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

   694   assume "gcd a b = 0"

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

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

   697   then show ?thesis by blast

   698 next

   699   let ?d = "gcd a b"

   700   assume "?d \<noteq> 0"

   701   from gcd_coprime_exists[OF this]

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

   703     by blast

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

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

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

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

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

   709   with coprime_dvd_mult[OF ab'(3)]

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

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

   712   then show ?thesis by blast

   713 qed

   714

   715 lemma pow_divs_pow:

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

   717   shows "a dvd b"

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

   719   assume "gcd a b = 0"

   720   then show ?thesis by simp

   721 next

   722   let ?d = "gcd a b"

   723   assume "?d \<noteq> 0"

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

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

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

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

   728     by blast

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

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

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

   732     by (simp only: power_mult_distrib ac_simps)

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

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

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

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

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

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

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

   740 qed

   741

   742 lemma pow_divs_eq [simp]:

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

   744   by (auto intro: pow_divs_pow dvd_power_same)

   745

   746 lemma divs_mult:

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

   748   shows "m * n dvd r"

   749 proof -

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

   751     unfolding dvd_def by blast

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

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

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

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

   756   then show ?thesis unfolding dvd_def by blast

   757 qed

   758

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

   760   by (subst add_commute, simp)

   761

   762 lemma setprod_coprime [rule_format]:

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

   764   apply (cases "finite A")

   765   apply (induct set: finite)

   766   apply (auto simp add: gcd_mult_cancel)

   767   done

   768

   769 lemma coprime_divisors:

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

   771   shows "gcd d e = 1"

   772 proof -

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

   774     unfolding dvd_def by blast

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

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

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

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

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

   780 qed

   781

   782 lemma invertible_coprime:

   783   assumes "a * b mod m = 1"

   784   shows "coprime a m"

   785 proof -

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

   787     by simp

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

   789     by simp

   790   then have "coprime m a"

   791     by (rule coprime_lmult)

   792   then show ?thesis

   793     by (simp add: ac_simps)

   794 qed

   795

   796 lemma lcm_gcd:

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

   798   by (simp only: lcm_lcm_eucl gcd_gcd_eucl lcm_eucl_def)

   799

   800 lemma lcm_gcd_prod:

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

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

   803   let ?nf = normalization_factor

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

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

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

   807     by (simp add: mult_ac)

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

   809     by (simp add: div_mult_swap mult.commute)

   810   finally show ?thesis .

   811 qed (auto simp add: lcm_gcd)

   812

   813 lemma lcm_dvd1 [iff]:

   814   "a dvd lcm a b"

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

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

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

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

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

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

   821     by (simp add: div_mult_swap unit_div_commute)

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

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

   824     by (subst (asm) div_mult_self2_is_id, simp_all)

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

   826     by (metis div_mult_swap gcd_dvd2 mult_assoc)

   827   finally show ?thesis by (rule dvdI)

   828 qed (auto simp add: lcm_gcd)

   829

   830 lemma lcm_least:

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

   832 proof (cases "k = 0")

   833   let ?nf = normalization_factor

   834   assume "k \<noteq> 0"

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

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

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

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

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

   840     unfolding dvd_def by blast

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   856     by (simp add: algebra_simps)

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

   858     by (metis div_mult_self2_is_id)

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

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

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

   862     by (simp add: algebra_simps)

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

   864     by (metis mult.commute div_mult_self2_is_id)

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

   866     by (metis div_mult_self2_is_id mult_assoc)

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

   868     by (simp add: unit_simps)

   869   finally show ?thesis by (rule dvdI)

   870 qed simp

   871

   872 lemma lcm_zero:

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

   874 proof -

   875   let ?nf = normalization_factor

   876   {

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

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

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

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

   881   } moreover {

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

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

   884   }

   885   ultimately show ?thesis by blast

   886 qed

   887

   888 lemmas lcm_0_iff = lcm_zero

   889

   890 lemma gcd_lcm:

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

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

   893 proof-

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

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

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

   897   hence "is_unit ?c" by simp

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

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

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

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

   902   finally show ?thesis .

   903 qed

   904

   905 lemma normalization_factor_lcm [simp]:

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

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

   908   case True then show ?thesis

   909     by (auto simp add: lcm_gcd)

   910 next

   911   case False

   912   let ?nf = normalization_factor

   913   from lcm_gcd_prod[of a b]

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

   915     by (metis div_by_0 div_self normalization_correct normalization_factor_0 normalization_factor_mult)

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

   917     by simp

   918   finally show ?thesis using False by simp

   919 qed

   920

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

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

   923

   924 lemma lcmI:

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

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

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

   928

   929 sublocale lcm!: abel_semigroup lcm

   930 proof

   931   fix a b c

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

   933   proof (rule lcmI)

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

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

   936

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

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

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

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

   941

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

   943     have "b dvd lcm b c" by simp

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

   945     have "c dvd lcm b c" by simp

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

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

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

   949   qed (simp add: lcm_zero)

   950 next

   951   fix a b

   952   show "lcm a b = lcm b a"

   953     by (simp add: lcm_gcd ac_simps)

   954 qed

   955

   956 lemma dvd_lcm_D1:

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

   958   by (rule dvd_trans, rule lcm_dvd1, assumption)

   959

   960 lemma dvd_lcm_D2:

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

   962   by (rule dvd_trans, rule lcm_dvd2, assumption)

   963

   964 lemma gcd_dvd_lcm [simp]:

   965   "gcd a b dvd lcm a b"

   966   by (metis dvd_trans gcd_dvd2 lcm_dvd2)

   967

   968 lemma lcm_1_iff:

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

   970 proof

   971   assume "lcm a b = 1"

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

   973 next

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

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

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

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

   978     by (subst normalization_factor_unit, simp_all)

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

   980     by auto

   981   finally show "lcm a b = 1" .

   982 qed

   983

   984 lemma lcm_0_left [simp]:

   985   "lcm 0 a = 0"

   986   by (rule sym, rule lcmI, simp_all)

   987

   988 lemma lcm_0 [simp]:

   989   "lcm a 0 = 0"

   990   by (rule sym, rule lcmI, simp_all)

   991

   992 lemma lcm_unique:

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

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

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

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

   997

   998 lemma dvd_lcm_I1 [simp]:

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

  1000   by (metis lcm_dvd1 dvd_trans)

  1001

  1002 lemma dvd_lcm_I2 [simp]:

  1003   "k dvd n \<Longrightarrow> k dvd lcm m n"

  1004   by (metis lcm_dvd2 dvd_trans)

  1005

  1006 lemma lcm_1_left [simp]:

  1007   "lcm 1 a = a div normalization_factor a"

  1008   by (cases "a = 0") (simp, rule sym, rule lcmI, simp_all)

  1009

  1010 lemma lcm_1_right [simp]:

  1011   "lcm a 1 = a div normalization_factor a"

  1012   using lcm_1_left [of a] by (simp add: ac_simps)

  1013

  1014 lemma lcm_coprime:

  1015   "gcd a b = 1 \<Longrightarrow> lcm a b = a * b div normalization_factor (a*b)"

  1016   by (subst lcm_gcd) simp

  1017

  1018 lemma lcm_proj1_if_dvd:

  1019   "b dvd a \<Longrightarrow> lcm a b = a div normalization_factor a"

  1020   by (cases "a = 0") (simp, rule sym, rule lcmI, simp_all)

  1021

  1022 lemma lcm_proj2_if_dvd:

  1023   "a dvd b \<Longrightarrow> lcm a b = b div normalization_factor b"

  1024   using lcm_proj1_if_dvd [of a b] by (simp add: ac_simps)

  1025

  1026 lemma lcm_proj1_iff:

  1027   "lcm m n = m div normalization_factor m \<longleftrightarrow> n dvd m"

  1028 proof

  1029   assume A: "lcm m n = m div normalization_factor m"

  1030   show "n dvd m"

  1031   proof (cases "m = 0")

  1032     assume [simp]: "m \<noteq> 0"

  1033     from A have B: "m = lcm m n * normalization_factor m"

  1034       by (simp add: unit_eq_div2)

  1035     show ?thesis by (subst B, simp)

  1036   qed simp

  1037 next

  1038   assume "n dvd m"

  1039   then show "lcm m n = m div normalization_factor m" by (rule lcm_proj1_if_dvd)

  1040 qed

  1041

  1042 lemma lcm_proj2_iff:

  1043   "lcm m n = n div normalization_factor n \<longleftrightarrow> m dvd n"

  1044   using lcm_proj1_iff [of n m] by (simp add: ac_simps)

  1045

  1046 lemma euclidean_size_lcm_le1:

  1047   assumes "a \<noteq> 0" and "b \<noteq> 0"

  1048   shows "euclidean_size a \<le> euclidean_size (lcm a b)"

  1049 proof -

  1050   have "a dvd lcm a b" by (rule lcm_dvd1)

  1051   then obtain c where A: "lcm a b = a * c" unfolding dvd_def by blast

  1052   with \<open>a \<noteq> 0\<close> and \<open>b \<noteq> 0\<close> have "c \<noteq> 0" by (auto simp: lcm_zero)

  1053   then show ?thesis by (subst A, intro size_mult_mono)

  1054 qed

  1055

  1056 lemma euclidean_size_lcm_le2:

  1057   "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> euclidean_size b \<le> euclidean_size (lcm a b)"

  1058   using euclidean_size_lcm_le1 [of b a] by (simp add: ac_simps)

  1059

  1060 lemma euclidean_size_lcm_less1:

  1061   assumes "b \<noteq> 0" and "\<not>b dvd a"

  1062   shows "euclidean_size a < euclidean_size (lcm a b)"

  1063 proof (rule ccontr)

  1064   from assms have "a \<noteq> 0" by auto

  1065   assume "\<not>euclidean_size a < euclidean_size (lcm a b)"

  1066   with \<open>a \<noteq> 0\<close> and \<open>b \<noteq> 0\<close> have "euclidean_size (lcm a b) = euclidean_size a"

  1067     by (intro le_antisym, simp, intro euclidean_size_lcm_le1)

  1068   with assms have "lcm a b dvd a"

  1069     by (rule_tac dvd_euclidean_size_eq_imp_dvd) (auto simp: lcm_zero)

  1070   hence "b dvd a" by (rule dvd_lcm_D2)

  1071   with \<open>\<not>b dvd a\<close> show False by contradiction

  1072 qed

  1073

  1074 lemma euclidean_size_lcm_less2:

  1075   assumes "a \<noteq> 0" and "\<not>a dvd b"

  1076   shows "euclidean_size b < euclidean_size (lcm a b)"

  1077   using assms euclidean_size_lcm_less1 [of a b] by (simp add: ac_simps)

  1078

  1079 lemma lcm_mult_unit1:

  1080   "is_unit a \<Longrightarrow> lcm (b * a) c = lcm b c"

  1081   apply (rule lcmI)

  1082   apply (rule dvd_trans[of _ "b * a"], simp, rule lcm_dvd1)

  1083   apply (rule lcm_dvd2)

  1084   apply (rule lcm_least, simp add: unit_simps, assumption)

  1085   apply (subst normalization_factor_lcm, simp add: lcm_zero)

  1086   done

  1087

  1088 lemma lcm_mult_unit2:

  1089   "is_unit a \<Longrightarrow> lcm b (c * a) = lcm b c"

  1090   using lcm_mult_unit1 [of a c b] by (simp add: ac_simps)

  1091

  1092 lemma lcm_div_unit1:

  1093   "is_unit a \<Longrightarrow> lcm (b div a) c = lcm b c"

  1094   by (erule is_unitE [of _ b]) (simp add: lcm_mult_unit1)

  1095

  1096 lemma lcm_div_unit2:

  1097   "is_unit a \<Longrightarrow> lcm b (c div a) = lcm b c"

  1098   by (erule is_unitE [of _ c]) (simp add: lcm_mult_unit2)

  1099

  1100 lemma lcm_left_idem:

  1101   "lcm a (lcm a b) = lcm a b"

  1102   apply (rule lcmI)

  1103   apply simp

  1104   apply (subst lcm.assoc [symmetric], rule lcm_dvd2)

  1105   apply (rule lcm_least, assumption)

  1106   apply (erule (1) lcm_least)

  1107   apply (auto simp: lcm_zero)

  1108   done

  1109

  1110 lemma lcm_right_idem:

  1111   "lcm (lcm a b) b = lcm a b"

  1112   apply (rule lcmI)

  1113   apply (subst lcm.assoc, rule lcm_dvd1)

  1114   apply (rule lcm_dvd2)

  1115   apply (rule lcm_least, erule (1) lcm_least, assumption)

  1116   apply (auto simp: lcm_zero)

  1117   done

  1118

  1119 lemma comp_fun_idem_lcm: "comp_fun_idem lcm"

  1120 proof

  1121   fix a b show "lcm a \<circ> lcm b = lcm b \<circ> lcm a"

  1122     by (simp add: fun_eq_iff ac_simps)

  1123 next

  1124   fix a show "lcm a \<circ> lcm a = lcm a" unfolding o_def

  1125     by (intro ext, simp add: lcm_left_idem)

  1126 qed

  1127

  1128 lemma dvd_Lcm [simp]: "a \<in> A \<Longrightarrow> a dvd Lcm A"

  1129   and Lcm_dvd [simp]: "(\<forall>a\<in>A. a dvd l') \<Longrightarrow> Lcm A dvd l'"

  1130   and normalization_factor_Lcm [simp]:

  1131           "normalization_factor (Lcm A) = (if Lcm A = 0 then 0 else 1)"

  1132 proof -

  1133   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>

  1134     normalization_factor (Lcm A) = (if Lcm A = 0 then 0 else 1)" (is ?thesis)

  1135   proof (cases "\<exists>l. l \<noteq>  0 \<and> (\<forall>a\<in>A. a dvd l)")

  1136     case False

  1137     hence "Lcm A = 0" by (auto simp: Lcm_Lcm_eucl Lcm_eucl_def)

  1138     with False show ?thesis by auto

  1139   next

  1140     case True

  1141     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

  1142     def n \<equiv> "LEAST n. \<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"

  1143     def l \<equiv> "SOME l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"

  1144     have "\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"

  1145       apply (subst n_def)

  1146       apply (rule LeastI[of _ "euclidean_size l\<^sub>0"])

  1147       apply (rule exI[of _ l\<^sub>0])

  1148       apply (simp add: l\<^sub>0_props)

  1149       done

  1150     from someI_ex[OF this] have "l \<noteq> 0" and "\<forall>a\<in>A. a dvd l" and "euclidean_size l = n"

  1151       unfolding l_def by simp_all

  1152     {

  1153       fix l' assume "\<forall>a\<in>A. a dvd l'"

  1154       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)

  1155       moreover from \<open>l \<noteq> 0\<close> have "gcd l l' \<noteq> 0" by simp

  1156       ultimately have "\<exists>b. b \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd b) \<and> euclidean_size b = euclidean_size (gcd l l')"

  1157         by (intro exI[of _ "gcd l l'"], auto)

  1158       hence "euclidean_size (gcd l l') \<ge> n" by (subst n_def) (rule Least_le)

  1159       moreover have "euclidean_size (gcd l l') \<le> n"

  1160       proof -

  1161         have "gcd l l' dvd l" by simp

  1162         then obtain a where "l = gcd l l' * a" unfolding dvd_def by blast

  1163         with \<open>l \<noteq> 0\<close> have "a \<noteq> 0" by auto

  1164         hence "euclidean_size (gcd l l') \<le> euclidean_size (gcd l l' * a)"

  1165           by (rule size_mult_mono)

  1166         also have "gcd l l' * a = l" using \<open>l = gcd l l' * a\<close> ..

  1167         also note \<open>euclidean_size l = n\<close>

  1168         finally show "euclidean_size (gcd l l') \<le> n" .

  1169       qed

  1170       ultimately have "euclidean_size l = euclidean_size (gcd l l')"

  1171         by (intro le_antisym, simp_all add: \<open>euclidean_size l = n\<close>)

  1172       with \<open>l \<noteq> 0\<close> have "l dvd gcd l l'" by (blast intro: dvd_euclidean_size_eq_imp_dvd)

  1173       hence "l dvd l'" by (blast dest: dvd_gcd_D2)

  1174     }

  1175

  1176     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>

  1177       have "(\<forall>a\<in>A. a dvd l div normalization_factor l) \<and>

  1178         (\<forall>l'. (\<forall>a\<in>A. a dvd l') \<longrightarrow> l div normalization_factor l dvd l') \<and>

  1179         normalization_factor (l div normalization_factor l) =

  1180         (if l div normalization_factor l = 0 then 0 else 1)"

  1181       by (auto simp: unit_simps)

  1182     also from True have "l div normalization_factor l = Lcm A"

  1183       by (simp add: Lcm_Lcm_eucl Lcm_eucl_def Let_def n_def l_def)

  1184     finally show ?thesis .

  1185   qed

  1186   note A = this

  1187

  1188   {fix a assume "a \<in> A" then show "a dvd Lcm A" using A by blast}

  1189   {fix l' assume "\<forall>a\<in>A. a dvd l'" then show "Lcm A dvd l'" using A by blast}

  1190   from A show "normalization_factor (Lcm A) = (if Lcm A = 0 then 0 else 1)" by blast

  1191 qed

  1192

  1193 lemma LcmI:

  1194   "(\<And>a. a\<in>A \<Longrightarrow> a dvd l) \<Longrightarrow> (\<And>l'. (\<forall>a\<in>A. a dvd l') \<Longrightarrow> l dvd l') \<Longrightarrow>

  1195       normalization_factor l = (if l = 0 then 0 else 1) \<Longrightarrow> l = Lcm A"

  1196   by (intro normed_associated_imp_eq)

  1197     (auto intro: Lcm_dvd dvd_Lcm simp: associated_def)

  1198

  1199 lemma Lcm_subset:

  1200   "A \<subseteq> B \<Longrightarrow> Lcm A dvd Lcm B"

  1201   by (blast intro: Lcm_dvd dvd_Lcm)

  1202

  1203 lemma Lcm_Un:

  1204   "Lcm (A \<union> B) = lcm (Lcm A) (Lcm B)"

  1205   apply (rule lcmI)

  1206   apply (blast intro: Lcm_subset)

  1207   apply (blast intro: Lcm_subset)

  1208   apply (intro Lcm_dvd ballI, elim UnE)

  1209   apply (rule dvd_trans, erule dvd_Lcm, assumption)

  1210   apply (rule dvd_trans, erule dvd_Lcm, assumption)

  1211   apply simp

  1212   done

  1213

  1214 lemma Lcm_1_iff:

  1215   "Lcm A = 1 \<longleftrightarrow> (\<forall>a\<in>A. is_unit a)"

  1216 proof

  1217   assume "Lcm A = 1"

  1218   then show "\<forall>a\<in>A. is_unit a" by auto

  1219 qed (rule LcmI [symmetric], auto)

  1220

  1221 lemma Lcm_no_units:

  1222   "Lcm A = Lcm (A - {a. is_unit a})"

  1223 proof -

  1224   have "(A - {a. is_unit a}) \<union> {a\<in>A. is_unit a} = A" by blast

  1225   hence "Lcm A = lcm (Lcm (A - {a. is_unit a})) (Lcm {a\<in>A. is_unit a})"

  1226     by (simp add: Lcm_Un[symmetric])

  1227   also have "Lcm {a\<in>A. is_unit a} = 1" by (simp add: Lcm_1_iff)

  1228   finally show ?thesis by simp

  1229 qed

  1230

  1231 lemma Lcm_empty [simp]:

  1232   "Lcm {} = 1"

  1233   by (simp add: Lcm_1_iff)

  1234

  1235 lemma Lcm_eq_0 [simp]:

  1236   "0 \<in> A \<Longrightarrow> Lcm A = 0"

  1237   by (drule dvd_Lcm) simp

  1238

  1239 lemma Lcm0_iff':

  1240   "Lcm A = 0 \<longleftrightarrow> \<not>(\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l))"

  1241 proof

  1242   assume "Lcm A = 0"

  1243   show "\<not>(\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l))"

  1244   proof

  1245     assume ex: "\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l)"

  1246     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

  1247     def n \<equiv> "LEAST n. \<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"

  1248     def l \<equiv> "SOME l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"

  1249     have "\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"

  1250       apply (subst n_def)

  1251       apply (rule LeastI[of _ "euclidean_size l\<^sub>0"])

  1252       apply (rule exI[of _ l\<^sub>0])

  1253       apply (simp add: l\<^sub>0_props)

  1254       done

  1255     from someI_ex[OF this] have "l \<noteq> 0" unfolding l_def by simp_all

  1256     hence "l div normalization_factor l \<noteq> 0" by simp

  1257     also from ex have "l div normalization_factor l = Lcm A"

  1258        by (simp only: Lcm_Lcm_eucl Lcm_eucl_def n_def l_def if_True Let_def)

  1259     finally show False using \<open>Lcm A = 0\<close> by contradiction

  1260   qed

  1261 qed (simp only: Lcm_Lcm_eucl Lcm_eucl_def if_False)

  1262

  1263 lemma Lcm0_iff [simp]:

  1264   "finite A \<Longrightarrow> Lcm A = 0 \<longleftrightarrow> 0 \<in> A"

  1265 proof -

  1266   assume "finite A"

  1267   have "0 \<in> A \<Longrightarrow> Lcm A = 0"  by (intro dvd_0_left dvd_Lcm)

  1268   moreover {

  1269     assume "0 \<notin> A"

  1270     hence "\<Prod>A \<noteq> 0"

  1271       apply (induct rule: finite_induct[OF \<open>finite A\<close>])

  1272       apply simp

  1273       apply (subst setprod.insert, assumption, assumption)

  1274       apply (rule no_zero_divisors)

  1275       apply blast+

  1276       done

  1277     moreover from \<open>finite A\<close> have "\<forall>a\<in>A. a dvd \<Prod>A" by blast

  1278     ultimately have "\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l)" by blast

  1279     with Lcm0_iff' have "Lcm A \<noteq> 0" by simp

  1280   }

  1281   ultimately show "Lcm A = 0 \<longleftrightarrow> 0 \<in> A" by blast

  1282 qed

  1283

  1284 lemma Lcm_no_multiple:

  1285   "(\<forall>m. m \<noteq> 0 \<longrightarrow> (\<exists>a\<in>A. \<not>a dvd m)) \<Longrightarrow> Lcm A = 0"

  1286 proof -

  1287   assume "\<forall>m. m \<noteq> 0 \<longrightarrow> (\<exists>a\<in>A. \<not>a dvd m)"

  1288   hence "\<not>(\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l))" by blast

  1289   then show "Lcm A = 0" by (simp only: Lcm_Lcm_eucl Lcm_eucl_def if_False)

  1290 qed

  1291

  1292 lemma Lcm_insert [simp]:

  1293   "Lcm (insert a A) = lcm a (Lcm A)"

  1294 proof (rule lcmI)

  1295   fix l assume "a dvd l" and "Lcm A dvd l"

  1296   hence "\<forall>a\<in>A. a dvd l" by (blast intro: dvd_trans dvd_Lcm)

  1297   with \<open>a dvd l\<close> show "Lcm (insert a A) dvd l" by (force intro: Lcm_dvd)

  1298 qed (auto intro: Lcm_dvd dvd_Lcm)

  1299

  1300 lemma Lcm_finite:

  1301   assumes "finite A"

  1302   shows "Lcm A = Finite_Set.fold lcm 1 A"

  1303   by (induct rule: finite.induct[OF \<open>finite A\<close>])

  1304     (simp_all add: comp_fun_idem.fold_insert_idem[OF comp_fun_idem_lcm])

  1305

  1306 lemma Lcm_set [code_unfold]:

  1307   "Lcm (set xs) = fold lcm xs 1"

  1308   using comp_fun_idem.fold_set_fold[OF comp_fun_idem_lcm] Lcm_finite by (simp add: ac_simps)

  1309

  1310 lemma Lcm_singleton [simp]:

  1311   "Lcm {a} = a div normalization_factor a"

  1312   by simp

  1313

  1314 lemma Lcm_2 [simp]:

  1315   "Lcm {a,b} = lcm a b"

  1316   by (simp only: Lcm_insert Lcm_empty lcm_1_right)

  1317     (cases "b = 0", simp, rule lcm_div_unit2, simp)

  1318

  1319 lemma Lcm_coprime:

  1320   assumes "finite A" and "A \<noteq> {}"

  1321   assumes "\<And>a b. a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<noteq> b \<Longrightarrow> gcd a b = 1"

  1322   shows "Lcm A = \<Prod>A div normalization_factor (\<Prod>A)"

  1323 using assms proof (induct rule: finite_ne_induct)

  1324   case (insert a A)

  1325   have "Lcm (insert a A) = lcm a (Lcm A)" by simp

  1326   also from insert have "Lcm A = \<Prod>A div normalization_factor (\<Prod>A)" by blast

  1327   also have "lcm a \<dots> = lcm a (\<Prod>A)" by (cases "\<Prod>A = 0") (simp_all add: lcm_div_unit2)

  1328   also from insert have "gcd a (\<Prod>A) = 1" by (subst gcd.commute, intro setprod_coprime) auto

  1329   with insert have "lcm a (\<Prod>A) = \<Prod>(insert a A) div normalization_factor (\<Prod>(insert a A))"

  1330     by (simp add: lcm_coprime)

  1331   finally show ?case .

  1332 qed simp

  1333

  1334 lemma Lcm_coprime':

  1335   "card A \<noteq> 0 \<Longrightarrow> (\<And>a b. a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<noteq> b \<Longrightarrow> gcd a b = 1)

  1336     \<Longrightarrow> Lcm A = \<Prod>A div normalization_factor (\<Prod>A)"

  1337   by (rule Lcm_coprime) (simp_all add: card_eq_0_iff)

  1338

  1339 lemma Gcd_Lcm:

  1340   "Gcd A = Lcm {d. \<forall>a\<in>A. d dvd a}"

  1341   by (simp add: Gcd_Gcd_eucl Lcm_Lcm_eucl Gcd_eucl_def)

  1342

  1343 lemma Gcd_dvd [simp]: "a \<in> A \<Longrightarrow> Gcd A dvd a"

  1344   and dvd_Gcd [simp]: "(\<forall>a\<in>A. g' dvd a) \<Longrightarrow> g' dvd Gcd A"

  1345   and normalization_factor_Gcd [simp]:

  1346     "normalization_factor (Gcd A) = (if Gcd A = 0 then 0 else 1)"

  1347 proof -

  1348   fix a assume "a \<in> A"

  1349   hence "Lcm {d. \<forall>a\<in>A. d dvd a} dvd a" by (intro Lcm_dvd) blast

  1350   then show "Gcd A dvd a" by (simp add: Gcd_Lcm)

  1351 next

  1352   fix g' assume "\<forall>a\<in>A. g' dvd a"

  1353   hence "g' dvd Lcm {d. \<forall>a\<in>A. d dvd a}" by (intro dvd_Lcm) blast

  1354   then show "g' dvd Gcd A" by (simp add: Gcd_Lcm)

  1355 next

  1356   show "normalization_factor (Gcd A) = (if Gcd A = 0 then 0 else 1)"

  1357     by (simp add: Gcd_Lcm)

  1358 qed

  1359

  1360 lemma GcdI:

  1361   "(\<And>a. a\<in>A \<Longrightarrow> l dvd a) \<Longrightarrow> (\<And>l'. (\<forall>a\<in>A. l' dvd a) \<Longrightarrow> l' dvd l) \<Longrightarrow>

  1362     normalization_factor l = (if l = 0 then 0 else 1) \<Longrightarrow> l = Gcd A"

  1363   by (intro normed_associated_imp_eq)

  1364     (auto intro: Gcd_dvd dvd_Gcd simp: associated_def)

  1365

  1366 lemma Lcm_Gcd:

  1367   "Lcm A = Gcd {m. \<forall>a\<in>A. a dvd m}"

  1368   by (rule LcmI[symmetric]) (auto intro: dvd_Gcd Gcd_dvd)

  1369

  1370 lemma Gcd_0_iff:

  1371   "Gcd A = 0 \<longleftrightarrow> A \<subseteq> {0}"

  1372   apply (rule iffI)

  1373   apply (rule subsetI, drule Gcd_dvd, simp)

  1374   apply (auto intro: GcdI[symmetric])

  1375   done

  1376

  1377 lemma Gcd_empty [simp]:

  1378   "Gcd {} = 0"

  1379   by (simp add: Gcd_0_iff)

  1380

  1381 lemma Gcd_1:

  1382   "1 \<in> A \<Longrightarrow> Gcd A = 1"

  1383   by (intro GcdI[symmetric]) (auto intro: Gcd_dvd dvd_Gcd)

  1384

  1385 lemma Gcd_insert [simp]:

  1386   "Gcd (insert a A) = gcd a (Gcd A)"

  1387 proof (rule gcdI)

  1388   fix l assume "l dvd a" and "l dvd Gcd A"

  1389   hence "\<forall>a\<in>A. l dvd a" by (blast intro: dvd_trans Gcd_dvd)

  1390   with \<open>l dvd a\<close> show "l dvd Gcd (insert a A)" by (force intro: Gcd_dvd)

  1391 qed auto

  1392

  1393 lemma Gcd_finite:

  1394   assumes "finite A"

  1395   shows "Gcd A = Finite_Set.fold gcd 0 A"

  1396   by (induct rule: finite.induct[OF \<open>finite A\<close>])

  1397     (simp_all add: comp_fun_idem.fold_insert_idem[OF comp_fun_idem_gcd])

  1398

  1399 lemma Gcd_set [code_unfold]:

  1400   "Gcd (set xs) = fold gcd xs 0"

  1401   using comp_fun_idem.fold_set_fold[OF comp_fun_idem_gcd] Gcd_finite by (simp add: ac_simps)

  1402

  1403 lemma Gcd_singleton [simp]: "Gcd {a} = a div normalization_factor a"

  1404   by (simp add: gcd_0)

  1405

  1406 lemma Gcd_2 [simp]: "Gcd {a,b} = gcd a b"

  1407   by (simp only: Gcd_insert Gcd_empty gcd_0) (cases "b = 0", simp, rule gcd_div_unit2, simp)

  1408

  1409 subclass semiring_gcd

  1410   by unfold_locales (simp_all add: gcd_greatest_iff)

  1411

  1412 end

  1413

  1414 text \<open>

  1415   A Euclidean ring is a Euclidean semiring with additive inverses. It provides a

  1416   few more lemmas; in particular, Bezout's lemma holds for any Euclidean ring.

  1417 \<close>

  1418

  1419 class euclidean_ring = euclidean_semiring + idom

  1420 begin

  1421

  1422 function euclid_ext :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a" where

  1423   "euclid_ext a b =

  1424      (if b = 0 then

  1425         let c = 1 div normalization_factor a in (c, 0, a * c)

  1426       else if b dvd a then

  1427         let c = 1 div normalization_factor b in (0, c, b * c)

  1428       else

  1429         case euclid_ext b (a mod b) of

  1430             (s, t, c) \<Rightarrow> (t, s - t * (a div b), c))"

  1431   by pat_completeness simp

  1432 termination

  1433   by (relation "measure (euclidean_size \<circ> snd)") (simp_all add: mod_size_less)

  1434

  1435 declare euclid_ext.simps [simp del]

  1436

  1437 lemma euclid_ext_0:

  1438   "euclid_ext a 0 = (1 div normalization_factor a, 0, a div normalization_factor a)"

  1439   by (simp add: euclid_ext.simps [of a 0])

  1440

  1441 lemma euclid_ext_left_0:

  1442   "euclid_ext 0 a = (0, 1 div normalization_factor a, a div normalization_factor a)"

  1443   by (simp add: euclid_ext.simps [of 0 a])

  1444

  1445 lemma euclid_ext_non_0:

  1446   "b \<noteq> 0 \<Longrightarrow> euclid_ext a b = (case euclid_ext b (a mod b) of

  1447     (s, t, c) \<Rightarrow> (t, s - t * (a div b), c))"

  1448   by (cases "b dvd a")

  1449     (simp_all add: euclid_ext.simps [of a b] euclid_ext.simps [of b 0])

  1450

  1451 lemma euclid_ext_code [code]:

  1452   "euclid_ext a b = (if b = 0 then (1 div normalization_factor a, 0, a div normalization_factor a)

  1453     else let (s, t, c) = euclid_ext b (a mod b) in  (t, s - t * (a div b), c))"

  1454   by (simp add: euclid_ext.simps [of a b] euclid_ext.simps [of b 0])

  1455

  1456 lemma euclid_ext_correct:

  1457   "case euclid_ext a b of (s, t, c) \<Rightarrow> s * a + t * b = c"

  1458 proof (induct a b rule: gcd_eucl_induct)

  1459   case (zero a) then show ?case

  1460     by (simp add: euclid_ext_0 ac_simps)

  1461 next

  1462   case (mod a b)

  1463   obtain s t c where stc: "euclid_ext b (a mod b) = (s,t,c)"

  1464     by (cases "euclid_ext b (a mod b)") blast

  1465   with mod have "c = s * b + t * (a mod b)" by simp

  1466   also have "... = t * ((a div b) * b + a mod b) + (s - t * (a div b)) * b"

  1467     by (simp add: algebra_simps)

  1468   also have "(a div b) * b + a mod b = a" using mod_div_equality .

  1469   finally show ?case

  1470     by (subst euclid_ext.simps) (simp add: stc mod ac_simps)

  1471 qed

  1472

  1473 definition euclid_ext' :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a"

  1474 where

  1475   "euclid_ext' a b = (case euclid_ext a b of (s, t, _) \<Rightarrow> (s, t))"

  1476

  1477 lemma euclid_ext'_0: "euclid_ext' a 0 = (1 div normalization_factor a, 0)"

  1478   by (simp add: euclid_ext'_def euclid_ext_0)

  1479

  1480 lemma euclid_ext'_left_0: "euclid_ext' 0 a = (0, 1 div normalization_factor a)"

  1481   by (simp add: euclid_ext'_def euclid_ext_left_0)

  1482

  1483 lemma euclid_ext'_non_0: "b \<noteq> 0 \<Longrightarrow> euclid_ext' a b = (snd (euclid_ext' b (a mod b)),

  1484   fst (euclid_ext' b (a mod b)) - snd (euclid_ext' b (a mod b)) * (a div b))"

  1485   by (simp add: euclid_ext'_def euclid_ext_non_0 split_def)

  1486

  1487 end

  1488

  1489 class euclidean_ring_gcd = euclidean_semiring_gcd + idom

  1490 begin

  1491

  1492 subclass euclidean_ring ..

  1493

  1494 subclass ring_gcd ..

  1495

  1496 lemma euclid_ext_gcd [simp]:

  1497   "(case euclid_ext a b of (_, _ , t) \<Rightarrow> t) = gcd a b"

  1498   by (induct a b rule: gcd_eucl_induct)

  1499     (simp_all add: euclid_ext_0 gcd_0 euclid_ext_non_0 ac_simps split: prod.split prod.split_asm)

  1500

  1501 lemma euclid_ext_gcd' [simp]:

  1502   "euclid_ext a b = (r, s, t) \<Longrightarrow> t = gcd a b"

  1503   by (insert euclid_ext_gcd[of a b], drule (1) subst, simp)

  1504

  1505 lemma euclid_ext'_correct:

  1506   "fst (euclid_ext' a b) * a + snd (euclid_ext' a b) * b = gcd a b"

  1507 proof-

  1508   obtain s t c where "euclid_ext a b = (s,t,c)"

  1509     by (cases "euclid_ext a b", blast)

  1510   with euclid_ext_correct[of a b] euclid_ext_gcd[of a b]

  1511     show ?thesis unfolding euclid_ext'_def by simp

  1512 qed

  1513

  1514 lemma bezout: "\<exists>s t. s * a + t * b = gcd a b"

  1515   using euclid_ext'_correct by blast

  1516

  1517 lemma gcd_neg1 [simp]:

  1518   "gcd (-a) b = gcd a b"

  1519   by (rule sym, rule gcdI, simp_all add: gcd_greatest)

  1520

  1521 lemma gcd_neg2 [simp]:

  1522   "gcd a (-b) = gcd a b"

  1523   by (rule sym, rule gcdI, simp_all add: gcd_greatest)

  1524

  1525 lemma gcd_neg_numeral_1 [simp]:

  1526   "gcd (- numeral n) a = gcd (numeral n) a"

  1527   by (fact gcd_neg1)

  1528

  1529 lemma gcd_neg_numeral_2 [simp]:

  1530   "gcd a (- numeral n) = gcd a (numeral n)"

  1531   by (fact gcd_neg2)

  1532

  1533 lemma gcd_diff1: "gcd (m - n) n = gcd m n"

  1534   by (subst diff_conv_add_uminus, subst gcd_neg2[symmetric],  subst gcd_add1, simp)

  1535

  1536 lemma gcd_diff2: "gcd (n - m) n = gcd m n"

  1537   by (subst gcd_neg1[symmetric], simp only: minus_diff_eq gcd_diff1)

  1538

  1539 lemma coprime_minus_one [simp]: "gcd (n - 1) n = 1"

  1540 proof -

  1541   have "gcd (n - 1) n = gcd n (n - 1)" by (fact gcd.commute)

  1542   also have "\<dots> = gcd ((n - 1) + 1) (n - 1)" by simp

  1543   also have "\<dots> = 1" by (rule coprime_plus_one)

  1544   finally show ?thesis .

  1545 qed

  1546

  1547 lemma lcm_neg1 [simp]: "lcm (-a) b = lcm a b"

  1548   by (rule sym, rule lcmI, simp_all add: lcm_least lcm_zero)

  1549

  1550 lemma lcm_neg2 [simp]: "lcm a (-b) = lcm a b"

  1551   by (rule sym, rule lcmI, simp_all add: lcm_least lcm_zero)

  1552

  1553 lemma lcm_neg_numeral_1 [simp]: "lcm (- numeral n) a = lcm (numeral n) a"

  1554   by (fact lcm_neg1)

  1555

  1556 lemma lcm_neg_numeral_2 [simp]: "lcm a (- numeral n) = lcm a (numeral n)"

  1557   by (fact lcm_neg2)

  1558

  1559 end

  1560

  1561

  1562 subsection \<open>Typical instances\<close>

  1563

  1564 instantiation nat :: euclidean_semiring

  1565 begin

  1566

  1567 definition [simp]:

  1568   "euclidean_size_nat = (id :: nat \<Rightarrow> nat)"

  1569

  1570 definition [simp]:

  1571   "normalization_factor_nat (n::nat) = (if n = 0 then 0 else 1 :: nat)"

  1572

  1573 instance proof

  1574 qed simp_all

  1575

  1576 end

  1577

  1578 instantiation int :: euclidean_ring

  1579 begin

  1580

  1581 definition [simp]:

  1582   "euclidean_size_int = (nat \<circ> abs :: int \<Rightarrow> nat)"

  1583

  1584 definition [simp]:

  1585   "normalization_factor_int = (sgn :: int \<Rightarrow> int)"

  1586

  1587 instance proof

  1588   case goal2 then show ?case by (auto simp add: abs_mult nat_mult_distrib)

  1589 next

  1590   case goal3 then show ?case by (simp add: zsgn_def)

  1591 next

  1592   case goal5 then show ?case by (auto simp: zsgn_def)

  1593 next

  1594   case goal6 then show ?case by (auto split: abs_split simp: zsgn_def)

  1595 qed (auto simp: sgn_times split: abs_split)

  1596

  1597 end

  1598

  1599 instantiation poly :: (field) euclidean_ring

  1600 begin

  1601

  1602 definition euclidean_size_poly :: "'a poly \<Rightarrow> nat"

  1603   where "euclidean_size = (degree :: 'a poly \<Rightarrow> nat)"

  1604

  1605 definition normalization_factor_poly :: "'a poly \<Rightarrow> 'a poly"

  1606   where "normalization_factor p = monom (coeff p (degree p)) 0"

  1607

  1608 instance

  1609 proof (default, unfold euclidean_size_poly_def normalization_factor_poly_def)

  1610   fix p q :: "'a poly"

  1611   assume "q \<noteq> 0" and "\<not> q dvd p"

  1612   then show "degree (p mod q) < degree q"

  1613     using degree_mod_less [of q p] by (simp add: mod_eq_0_iff_dvd)

  1614 next

  1615   fix p q :: "'a poly"

  1616   assume "q \<noteq> 0"

  1617   from \<open>q \<noteq> 0\<close> show "degree p \<le> degree (p * q)"

  1618     by (rule degree_mult_right_le)

  1619   from \<open>q \<noteq> 0\<close> show "is_unit (monom (coeff q (degree q)) 0)"

  1620     by (auto intro: is_unit_monom_0)

  1621 next

  1622   fix p :: "'a poly"

  1623   show "monom (coeff p (degree p)) 0 = p" if "is_unit p"

  1624     using that by (fact is_unit_monom_trival)

  1625 next

  1626   fix p q :: "'a poly"

  1627   show "monom (coeff (p * q) (degree (p * q))) 0 =

  1628     monom (coeff p (degree p)) 0 * monom (coeff q (degree q)) 0"

  1629     by (simp add: monom_0 coeff_degree_mult)

  1630 next

  1631   show "monom (coeff 0 (degree 0)) 0 = 0"

  1632     by simp

  1633 qed

  1634

  1635 end

  1636

  1637 end