src/HOL/Number_Theory/Euclidean_Algorithm.thy
author wenzelm
Thu May 26 17:51:22 2016 +0200 (2016-05-26)
changeset 63167 0909deb8059b
parent 63040 eb4ddd18d635
child 63498 a3fe3250d05d
permissions -rw-r--r--
isabelle update_cartouches -c -t;
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(* Author: Manuel Eberl *)
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section \<open>Abstract euclidean algorithm\<close>
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theory Euclidean_Algorithm
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imports "~~/src/HOL/GCD" "~~/src/HOL/Library/Polynomial"
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begin
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text \<open>
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  A Euclidean semiring is a semiring upon which the Euclidean algorithm can be
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  implemented. It must provide:
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  \begin{itemize}
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  \item division with remainder
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  \item a size function such that @{term "size (a mod b) < size b"} 
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        for any @{term "b \<noteq> 0"}
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  \end{itemize}
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  The existence of these functions makes it possible to derive gcd and lcm functions 
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  for any Euclidean semiring.
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\<close> 
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class euclidean_semiring = semiring_div + normalization_semidom + 
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  fixes euclidean_size :: "'a \<Rightarrow> nat"
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  assumes size_0 [simp]: "euclidean_size 0 = 0"
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  assumes mod_size_less: 
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    "b \<noteq> 0 \<Longrightarrow> euclidean_size (a mod b) < euclidean_size b"
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  assumes size_mult_mono:
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    "b \<noteq> 0 \<Longrightarrow> euclidean_size a \<le> euclidean_size (a * b)"
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begin
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lemma euclidean_division:
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  fixes a :: 'a and b :: 'a
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  assumes "b \<noteq> 0"
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  obtains s and t where "a = s * b + t" 
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    and "euclidean_size t < euclidean_size b"
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proof -
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  from div_mod_equality [of a b 0] 
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     have "a = a div b * b + a mod b" by simp
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  with that and assms show ?thesis by (auto simp add: mod_size_less)
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qed
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lemma dvd_euclidean_size_eq_imp_dvd:
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  assumes "a \<noteq> 0" and b_dvd_a: "b dvd a" and size_eq: "euclidean_size a = euclidean_size b"
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  shows "a dvd b"
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proof (rule ccontr)
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  assume "\<not> a dvd b"
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  then have "b mod a \<noteq> 0" by (simp add: mod_eq_0_iff_dvd)
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  from b_dvd_a have b_dvd_mod: "b dvd b mod a" by (simp add: dvd_mod_iff)
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  from b_dvd_mod obtain c where "b mod a = b * c" unfolding dvd_def by blast
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    with \<open>b mod a \<noteq> 0\<close> have "c \<noteq> 0" by auto
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  with \<open>b mod a = b * c\<close> have "euclidean_size (b mod a) \<ge> euclidean_size b"
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      using size_mult_mono by force
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  moreover from \<open>\<not> a dvd b\<close> and \<open>a \<noteq> 0\<close>
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  have "euclidean_size (b mod a) < euclidean_size a"
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      using mod_size_less by blast
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  ultimately show False using size_eq by simp
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qed
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function gcd_eucl :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
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where
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  "gcd_eucl a b = (if b = 0 then normalize a else gcd_eucl b (a mod b))"
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  by pat_completeness simp
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termination
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  by (relation "measure (euclidean_size \<circ> snd)") (simp_all add: mod_size_less)
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declare gcd_eucl.simps [simp del]
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lemma gcd_eucl_induct [case_names zero mod]:
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  assumes H1: "\<And>b. P b 0"
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  and H2: "\<And>a b. b \<noteq> 0 \<Longrightarrow> P b (a mod b) \<Longrightarrow> P a b"
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  shows "P a b"
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proof (induct a b rule: gcd_eucl.induct)
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  case ("1" a b)
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  show ?case
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  proof (cases "b = 0")
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    case True then show "P a b" by simp (rule H1)
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  next
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    case False
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    then have "P b (a mod b)"
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      by (rule "1.hyps")
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    with \<open>b \<noteq> 0\<close> show "P a b"
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      by (blast intro: H2)
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  qed
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qed
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definition lcm_eucl :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
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where
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  "lcm_eucl a b = normalize (a * b) div gcd_eucl a b"
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definition Lcm_eucl :: "'a set \<Rightarrow> 'a" \<comment> \<open>
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  Somewhat complicated definition of Lcm that has the advantage of working
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  for infinite sets as well\<close>
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where
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  "Lcm_eucl A = (if \<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) then
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     let l = SOME l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l =
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       (LEAST n. \<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n)
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       in normalize l 
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      else 0)"
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definition Gcd_eucl :: "'a set \<Rightarrow> 'a"
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where
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  "Gcd_eucl A = Lcm_eucl {d. \<forall>a\<in>A. d dvd a}"
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declare Lcm_eucl_def Gcd_eucl_def [code del]
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lemma gcd_eucl_0:
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  "gcd_eucl a 0 = normalize a"
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  by (simp add: gcd_eucl.simps [of a 0])
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lemma gcd_eucl_0_left:
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  "gcd_eucl 0 a = normalize a"
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  by (simp_all add: gcd_eucl_0 gcd_eucl.simps [of 0 a])
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lemma gcd_eucl_non_0:
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  "b \<noteq> 0 \<Longrightarrow> gcd_eucl a b = gcd_eucl b (a mod b)"
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  by (simp add: gcd_eucl.simps [of a b] gcd_eucl.simps [of b 0])
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lemma gcd_eucl_dvd1 [iff]: "gcd_eucl a b dvd a"
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  and gcd_eucl_dvd2 [iff]: "gcd_eucl a b dvd b"
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  by (induct a b rule: gcd_eucl_induct)
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     (simp_all add: gcd_eucl_0 gcd_eucl_non_0 dvd_mod_iff)
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lemma normalize_gcd_eucl [simp]:
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  "normalize (gcd_eucl a b) = gcd_eucl a b"
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  by (induct a b rule: gcd_eucl_induct) (simp_all add: gcd_eucl_0 gcd_eucl_non_0)
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lemma gcd_eucl_greatest:
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  fixes k a b :: 'a
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  shows "k dvd a \<Longrightarrow> k dvd b \<Longrightarrow> k dvd gcd_eucl a b"
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proof (induct a b rule: gcd_eucl_induct)
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  case (zero a) from zero(1) show ?case by (rule dvd_trans) (simp add: gcd_eucl_0)
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next
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  case (mod a b)
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  then show ?case
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    by (simp add: gcd_eucl_non_0 dvd_mod_iff)
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qed
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lemma eq_gcd_euclI:
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  fixes gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
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  assumes "\<And>a b. gcd a b dvd a" "\<And>a b. gcd a b dvd b" "\<And>a b. normalize (gcd a b) = gcd a b"
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          "\<And>a b k. k dvd a \<Longrightarrow> k dvd b \<Longrightarrow> k dvd gcd a b"
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  shows   "gcd = gcd_eucl"
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  by (intro ext, rule associated_eqI) (simp_all add: gcd_eucl_greatest assms)
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lemma gcd_eucl_zero [simp]:
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  "gcd_eucl a b = 0 \<longleftrightarrow> a = 0 \<and> b = 0"
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  by (metis dvd_0_left dvd_refl gcd_eucl_dvd1 gcd_eucl_dvd2 gcd_eucl_greatest)+
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lemma dvd_Lcm_eucl [simp]: "a \<in> A \<Longrightarrow> a dvd Lcm_eucl A"
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  and Lcm_eucl_least: "(\<And>a. a \<in> A \<Longrightarrow> a dvd b) \<Longrightarrow> Lcm_eucl A dvd b"
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  and unit_factor_Lcm_eucl [simp]: 
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          "unit_factor (Lcm_eucl A) = (if Lcm_eucl A = 0 then 0 else 1)"
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proof -
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  have "(\<forall>a\<in>A. a dvd Lcm_eucl A) \<and> (\<forall>l'. (\<forall>a\<in>A. a dvd l') \<longrightarrow> Lcm_eucl A dvd l') \<and>
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    unit_factor (Lcm_eucl A) = (if Lcm_eucl A = 0 then 0 else 1)" (is ?thesis)
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  proof (cases "\<exists>l. l \<noteq>  0 \<and> (\<forall>a\<in>A. a dvd l)")
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    case False
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    hence "Lcm_eucl A = 0" by (auto simp: Lcm_eucl_def)
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    with False show ?thesis by auto
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  next
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    case True
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    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
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    define n where "n = (LEAST n. \<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n)"
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    define l where "l = (SOME l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n)"
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    have "\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"
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      apply (subst n_def)
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      apply (rule LeastI[of _ "euclidean_size l\<^sub>0"])
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      apply (rule exI[of _ l\<^sub>0])
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      apply (simp add: l\<^sub>0_props)
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      done
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    from someI_ex[OF this] have "l \<noteq> 0" and "\<forall>a\<in>A. a dvd l" and "euclidean_size l = n" 
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      unfolding l_def by simp_all
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    {
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      fix l' assume "\<forall>a\<in>A. a dvd l'"
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      with \<open>\<forall>a\<in>A. a dvd l\<close> have "\<forall>a\<in>A. a dvd gcd_eucl l l'" by (auto intro: gcd_eucl_greatest)
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      moreover from \<open>l \<noteq> 0\<close> have "gcd_eucl l l' \<noteq> 0" by simp
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      ultimately have "\<exists>b. b \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd b) \<and> 
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                          euclidean_size b = euclidean_size (gcd_eucl l l')"
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        by (intro exI[of _ "gcd_eucl l l'"], auto)
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      hence "euclidean_size (gcd_eucl l l') \<ge> n" by (subst n_def) (rule Least_le)
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      moreover have "euclidean_size (gcd_eucl l l') \<le> n"
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      proof -
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        have "gcd_eucl l l' dvd l" by simp
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        then obtain a where "l = gcd_eucl l l' * a" unfolding dvd_def by blast
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        with \<open>l \<noteq> 0\<close> have "a \<noteq> 0" by auto
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        hence "euclidean_size (gcd_eucl l l') \<le> euclidean_size (gcd_eucl l l' * a)"
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          by (rule size_mult_mono)
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        also have "gcd_eucl l l' * a = l" using \<open>l = gcd_eucl l l' * a\<close> ..
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        also note \<open>euclidean_size l = n\<close>
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        finally show "euclidean_size (gcd_eucl l l') \<le> n" .
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      qed
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      ultimately have *: "euclidean_size l = euclidean_size (gcd_eucl l l')" 
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        by (intro le_antisym, simp_all add: \<open>euclidean_size l = n\<close>)
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      from \<open>l \<noteq> 0\<close> have "l dvd gcd_eucl l l'"
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        by (rule dvd_euclidean_size_eq_imp_dvd) (auto simp add: *)
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      hence "l dvd l'" by (rule dvd_trans[OF _ gcd_eucl_dvd2])
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    }
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    with \<open>(\<forall>a\<in>A. a dvd l)\<close> and unit_factor_is_unit[OF \<open>l \<noteq> 0\<close>] and \<open>l \<noteq> 0\<close>
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      have "(\<forall>a\<in>A. a dvd normalize l) \<and> 
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        (\<forall>l'. (\<forall>a\<in>A. a dvd l') \<longrightarrow> normalize l dvd l') \<and>
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        unit_factor (normalize l) = 
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        (if normalize l = 0 then 0 else 1)"
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      by (auto simp: unit_simps)
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    also from True have "normalize l = Lcm_eucl A"
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      by (simp add: Lcm_eucl_def Let_def n_def l_def)
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    finally show ?thesis .
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  qed
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  note A = this
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  {fix a assume "a \<in> A" then show "a dvd Lcm_eucl A" using A by blast}
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  {fix b assume "\<And>a. a \<in> A \<Longrightarrow> a dvd b" then show "Lcm_eucl A dvd b" using A by blast}
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  from A show "unit_factor (Lcm_eucl A) = (if Lcm_eucl A = 0 then 0 else 1)" by blast
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qed
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lemma normalize_Lcm_eucl [simp]:
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  "normalize (Lcm_eucl A) = Lcm_eucl A"
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proof (cases "Lcm_eucl A = 0")
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  case True then show ?thesis by simp
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next
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  case False
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  have "unit_factor (Lcm_eucl A) * normalize (Lcm_eucl A) = Lcm_eucl A"
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    by (fact unit_factor_mult_normalize)
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  with False show ?thesis by simp
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qed
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lemma eq_Lcm_euclI:
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  fixes lcm :: "'a set \<Rightarrow> 'a"
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  assumes "\<And>A a. a \<in> A \<Longrightarrow> a dvd lcm A" and "\<And>A c. (\<And>a. a \<in> A \<Longrightarrow> a dvd c) \<Longrightarrow> lcm A dvd c"
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          "\<And>A. normalize (lcm A) = lcm A" shows "lcm = Lcm_eucl"
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  by (intro ext, rule associated_eqI) (auto simp: assms intro: Lcm_eucl_least)  
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end
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class euclidean_ring = euclidean_semiring + idom
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begin
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subclass ring_div ..
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function euclid_ext_aux :: "'a \<Rightarrow> _" where
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  "euclid_ext_aux r' r s' s t' t = (
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     if r = 0 then let c = 1 div unit_factor r' in (s' * c, t' * c, normalize r')
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     else let q = r' div r
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          in  euclid_ext_aux r (r' mod r) s (s' - q * s) t (t' - q * t))"
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by auto
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termination by (relation "measure (\<lambda>(_,b,_,_,_,_). euclidean_size b)") (simp_all add: mod_size_less)
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declare euclid_ext_aux.simps [simp del]
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lemma euclid_ext_aux_correct:
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  assumes "gcd_eucl r' r = gcd_eucl x y"
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  assumes "s' * x + t' * y = r'"
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  assumes "s * x + t * y = r"
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  shows   "case euclid_ext_aux r' r s' s t' t of (a,b,c) \<Rightarrow>
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             a * x + b * y = c \<and> c = gcd_eucl x y" (is "?P (euclid_ext_aux r' r s' s t' t)")
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using assms
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proof (induction r' r s' s t' t rule: euclid_ext_aux.induct)
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  case (1 r' r s' s t' t)
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  show ?case
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  proof (cases "r = 0")
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    case True
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    hence "euclid_ext_aux r' r s' s t' t = 
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             (s' div unit_factor r', t' div unit_factor r', normalize r')"
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      by (subst euclid_ext_aux.simps) (simp add: Let_def)
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    also have "?P \<dots>"
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    proof safe
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      have "s' div unit_factor r' * x + t' div unit_factor r' * y = 
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                (s' * x + t' * y) div unit_factor r'"
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        by (cases "r' = 0") (simp_all add: unit_div_commute)
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      also have "s' * x + t' * y = r'" by fact
eberlm@62442
   270
      also have "\<dots> div unit_factor r' = normalize r'" by simp
eberlm@62442
   271
      finally show "s' div unit_factor r' * x + t' div unit_factor r' * y = normalize r'" .
eberlm@62442
   272
    next
eberlm@62442
   273
      from "1.prems" True show "normalize r' = gcd_eucl x y" by (simp add: gcd_eucl_0)
eberlm@62442
   274
    qed
eberlm@62442
   275
    finally show ?thesis .
eberlm@62442
   276
  next
eberlm@62442
   277
    case False
eberlm@62442
   278
    hence "euclid_ext_aux r' r s' s t' t = 
eberlm@62442
   279
             euclid_ext_aux r (r' mod r) s (s' - r' div r * s) t (t' - r' div r * t)"
eberlm@62442
   280
      by (subst euclid_ext_aux.simps) (simp add: Let_def)
eberlm@62442
   281
    also from "1.prems" False have "?P \<dots>"
eberlm@62442
   282
    proof (intro "1.IH")
eberlm@62442
   283
      have "(s' - r' div r * s) * x + (t' - r' div r * t) * y =
eberlm@62442
   284
              (s' * x + t' * y) - r' div r * (s * x + t * y)" by (simp add: algebra_simps)
eberlm@62442
   285
      also have "s' * x + t' * y = r'" by fact
eberlm@62442
   286
      also have "s * x + t * y = r" by fact
eberlm@62442
   287
      also have "r' - r' div r * r = r' mod r" using mod_div_equality[of r' r]
eberlm@62442
   288
        by (simp add: algebra_simps)
eberlm@62442
   289
      finally show "(s' - r' div r * s) * x + (t' - r' div r * t) * y = r' mod r" .
eberlm@62442
   290
    qed (auto simp: gcd_eucl_non_0 algebra_simps div_mod_equality')
eberlm@62442
   291
    finally show ?thesis .
eberlm@62442
   292
  qed
eberlm@62442
   293
qed
eberlm@62442
   294
eberlm@62442
   295
definition euclid_ext where
eberlm@62442
   296
  "euclid_ext a b = euclid_ext_aux a b 1 0 0 1"
haftmann@60598
   297
haftmann@60598
   298
lemma euclid_ext_0: 
haftmann@60634
   299
  "euclid_ext a 0 = (1 div unit_factor a, 0, normalize a)"
eberlm@62442
   300
  by (simp add: euclid_ext_def euclid_ext_aux.simps)
haftmann@60598
   301
haftmann@60598
   302
lemma euclid_ext_left_0: 
haftmann@60634
   303
  "euclid_ext 0 a = (0, 1 div unit_factor a, normalize a)"
eberlm@62442
   304
  by (simp add: euclid_ext_def euclid_ext_aux.simps)
haftmann@60598
   305
eberlm@62442
   306
lemma euclid_ext_correct':
eberlm@62442
   307
  "case euclid_ext x y of (a,b,c) \<Rightarrow> a * x + b * y = c \<and> c = gcd_eucl x y"
eberlm@62442
   308
  unfolding euclid_ext_def by (rule euclid_ext_aux_correct) simp_all
haftmann@60598
   309
eberlm@62457
   310
lemma euclid_ext_gcd_eucl:
eberlm@62457
   311
  "(case euclid_ext x y of (a,b,c) \<Rightarrow> c) = gcd_eucl x y"
eberlm@62457
   312
  using euclid_ext_correct'[of x y] by (simp add: case_prod_unfold)
eberlm@62457
   313
eberlm@62442
   314
definition euclid_ext' where
eberlm@62442
   315
  "euclid_ext' x y = (case euclid_ext x y of (a, b, _) \<Rightarrow> (a, b))"
haftmann@60598
   316
eberlm@62442
   317
lemma euclid_ext'_correct':
eberlm@62442
   318
  "case euclid_ext' x y of (a,b) \<Rightarrow> a * x + b * y = gcd_eucl x y"
eberlm@62442
   319
  using euclid_ext_correct'[of x y] by (simp add: case_prod_unfold euclid_ext'_def)
haftmann@60598
   320
haftmann@60634
   321
lemma euclid_ext'_0: "euclid_ext' a 0 = (1 div unit_factor a, 0)" 
haftmann@60598
   322
  by (simp add: euclid_ext'_def euclid_ext_0)
haftmann@60598
   323
haftmann@60634
   324
lemma euclid_ext'_left_0: "euclid_ext' 0 a = (0, 1 div unit_factor a)" 
haftmann@60598
   325
  by (simp add: euclid_ext'_def euclid_ext_left_0)
haftmann@60598
   326
haftmann@60598
   327
end
haftmann@60598
   328
haftmann@58023
   329
class euclidean_semiring_gcd = euclidean_semiring + gcd + Gcd +
haftmann@58023
   330
  assumes gcd_gcd_eucl: "gcd = gcd_eucl" and lcm_lcm_eucl: "lcm = lcm_eucl"
haftmann@58023
   331
  assumes Gcd_Gcd_eucl: "Gcd = Gcd_eucl" and Lcm_Lcm_eucl: "Lcm = Lcm_eucl"
haftmann@58023
   332
begin
haftmann@58023
   333
eberlm@62422
   334
subclass semiring_gcd
eberlm@62422
   335
  by standard (simp_all add: gcd_gcd_eucl gcd_eucl_greatest lcm_lcm_eucl lcm_eucl_def)
haftmann@58023
   336
eberlm@62422
   337
subclass semiring_Gcd
eberlm@62422
   338
  by standard (auto simp: Gcd_Gcd_eucl Lcm_Lcm_eucl Gcd_eucl_def intro: Lcm_eucl_least)
eberlm@62422
   339
  
haftmann@58023
   340
lemma gcd_non_0:
haftmann@60430
   341
  "b \<noteq> 0 \<Longrightarrow> gcd a b = gcd b (a mod b)"
haftmann@60572
   342
  unfolding gcd_gcd_eucl by (fact gcd_eucl_non_0)
haftmann@58023
   343
eberlm@62422
   344
lemmas gcd_0 = gcd_0_right
eberlm@62422
   345
lemmas dvd_gcd_iff = gcd_greatest_iff
haftmann@58023
   346
lemmas gcd_greatest_iff = dvd_gcd_iff
haftmann@58023
   347
haftmann@58023
   348
lemma gcd_mod1 [simp]:
haftmann@60430
   349
  "gcd (a mod b) b = gcd a b"
haftmann@58023
   350
  by (rule gcdI, metis dvd_mod_iff gcd_dvd1 gcd_dvd2, simp_all add: gcd_greatest dvd_mod_iff)
haftmann@58023
   351
haftmann@58023
   352
lemma gcd_mod2 [simp]:
haftmann@60430
   353
  "gcd a (b mod a) = gcd a b"
haftmann@58023
   354
  by (rule gcdI, simp, metis dvd_mod_iff gcd_dvd1 gcd_dvd2, simp_all add: gcd_greatest dvd_mod_iff)
haftmann@58023
   355
         
haftmann@58023
   356
lemma euclidean_size_gcd_le1 [simp]:
haftmann@58023
   357
  assumes "a \<noteq> 0"
haftmann@58023
   358
  shows "euclidean_size (gcd a b) \<le> euclidean_size a"
haftmann@58023
   359
proof -
haftmann@58023
   360
   have "gcd a b dvd a" by (rule gcd_dvd1)
haftmann@58023
   361
   then obtain c where A: "a = gcd a b * c" unfolding dvd_def by blast
wenzelm@60526
   362
   with \<open>a \<noteq> 0\<close> show ?thesis by (subst (2) A, intro size_mult_mono) auto
haftmann@58023
   363
qed
haftmann@58023
   364
haftmann@58023
   365
lemma euclidean_size_gcd_le2 [simp]:
haftmann@58023
   366
  "b \<noteq> 0 \<Longrightarrow> euclidean_size (gcd a b) \<le> euclidean_size b"
haftmann@58023
   367
  by (subst gcd.commute, rule euclidean_size_gcd_le1)
haftmann@58023
   368
haftmann@58023
   369
lemma euclidean_size_gcd_less1:
haftmann@58023
   370
  assumes "a \<noteq> 0" and "\<not>a dvd b"
haftmann@58023
   371
  shows "euclidean_size (gcd a b) < euclidean_size a"
haftmann@58023
   372
proof (rule ccontr)
haftmann@58023
   373
  assume "\<not>euclidean_size (gcd a b) < euclidean_size a"
eberlm@62422
   374
  with \<open>a \<noteq> 0\<close> have A: "euclidean_size (gcd a b) = euclidean_size a"
haftmann@58023
   375
    by (intro le_antisym, simp_all)
eberlm@62422
   376
  have "a dvd gcd a b"
eberlm@62422
   377
    by (rule dvd_euclidean_size_eq_imp_dvd) (simp_all add: assms A)
eberlm@62422
   378
  hence "a dvd b" using dvd_gcdD2 by blast
wenzelm@60526
   379
  with \<open>\<not>a dvd b\<close> show False by contradiction
haftmann@58023
   380
qed
haftmann@58023
   381
haftmann@58023
   382
lemma euclidean_size_gcd_less2:
haftmann@58023
   383
  assumes "b \<noteq> 0" and "\<not>b dvd a"
haftmann@58023
   384
  shows "euclidean_size (gcd a b) < euclidean_size b"
haftmann@58023
   385
  using assms by (subst gcd.commute, rule euclidean_size_gcd_less1)
haftmann@58023
   386
haftmann@58023
   387
lemma euclidean_size_lcm_le1: 
haftmann@58023
   388
  assumes "a \<noteq> 0" and "b \<noteq> 0"
haftmann@58023
   389
  shows "euclidean_size a \<le> euclidean_size (lcm a b)"
haftmann@58023
   390
proof -
haftmann@60690
   391
  have "a dvd lcm a b" by (rule dvd_lcm1)
haftmann@60690
   392
  then obtain c where A: "lcm a b = a * c" ..
eberlm@62429
   393
  with \<open>a \<noteq> 0\<close> and \<open>b \<noteq> 0\<close> have "c \<noteq> 0" by (auto simp: lcm_eq_0_iff)
haftmann@58023
   394
  then show ?thesis by (subst A, intro size_mult_mono)
haftmann@58023
   395
qed
haftmann@58023
   396
haftmann@58023
   397
lemma euclidean_size_lcm_le2:
haftmann@58023
   398
  "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> euclidean_size b \<le> euclidean_size (lcm a b)"
haftmann@58023
   399
  using euclidean_size_lcm_le1 [of b a] by (simp add: ac_simps)
haftmann@58023
   400
haftmann@58023
   401
lemma euclidean_size_lcm_less1:
haftmann@58023
   402
  assumes "b \<noteq> 0" and "\<not>b dvd a"
haftmann@58023
   403
  shows "euclidean_size a < euclidean_size (lcm a b)"
haftmann@58023
   404
proof (rule ccontr)
haftmann@58023
   405
  from assms have "a \<noteq> 0" by auto
haftmann@58023
   406
  assume "\<not>euclidean_size a < euclidean_size (lcm a b)"
wenzelm@60526
   407
  with \<open>a \<noteq> 0\<close> and \<open>b \<noteq> 0\<close> have "euclidean_size (lcm a b) = euclidean_size a"
haftmann@58023
   408
    by (intro le_antisym, simp, intro euclidean_size_lcm_le1)
haftmann@58023
   409
  with assms have "lcm a b dvd a" 
eberlm@62429
   410
    by (rule_tac dvd_euclidean_size_eq_imp_dvd) (auto simp: lcm_eq_0_iff)
eberlm@62422
   411
  hence "b dvd a" by (rule lcm_dvdD2)
wenzelm@60526
   412
  with \<open>\<not>b dvd a\<close> show False by contradiction
haftmann@58023
   413
qed
haftmann@58023
   414
haftmann@58023
   415
lemma euclidean_size_lcm_less2:
haftmann@58023
   416
  assumes "a \<noteq> 0" and "\<not>a dvd b"
haftmann@58023
   417
  shows "euclidean_size b < euclidean_size (lcm a b)"
haftmann@58023
   418
  using assms euclidean_size_lcm_less1 [of a b] by (simp add: ac_simps)
haftmann@58023
   419
eberlm@62428
   420
lemma Lcm_eucl_set [code]:
eberlm@62428
   421
  "Lcm_eucl (set xs) = foldl lcm_eucl 1 xs"
eberlm@62428
   422
  by (simp add: Lcm_Lcm_eucl [symmetric] lcm_lcm_eucl Lcm_set)
haftmann@58023
   423
eberlm@62428
   424
lemma Gcd_eucl_set [code]:
eberlm@62428
   425
  "Gcd_eucl (set xs) = foldl gcd_eucl 0 xs"
eberlm@62428
   426
  by (simp add: Gcd_Gcd_eucl [symmetric] gcd_gcd_eucl Gcd_set)
haftmann@58023
   427
haftmann@58023
   428
end
haftmann@58023
   429
wenzelm@60526
   430
text \<open>
haftmann@58023
   431
  A Euclidean ring is a Euclidean semiring with additive inverses. It provides a 
haftmann@58023
   432
  few more lemmas; in particular, Bezout's lemma holds for any Euclidean ring.
wenzelm@60526
   433
\<close>
haftmann@58023
   434
haftmann@58023
   435
class euclidean_ring_gcd = euclidean_semiring_gcd + idom
haftmann@58023
   436
begin
haftmann@58023
   437
haftmann@58023
   438
subclass euclidean_ring ..
haftmann@60439
   439
subclass ring_gcd ..
haftmann@60439
   440
haftmann@60572
   441
lemma euclid_ext_gcd [simp]:
haftmann@60572
   442
  "(case euclid_ext a b of (_, _ , t) \<Rightarrow> t) = gcd a b"
eberlm@62442
   443
  using euclid_ext_correct'[of a b] by (simp add: case_prod_unfold Let_def gcd_gcd_eucl)
haftmann@60572
   444
haftmann@60572
   445
lemma euclid_ext_gcd' [simp]:
haftmann@60572
   446
  "euclid_ext a b = (r, s, t) \<Longrightarrow> t = gcd a b"
haftmann@60572
   447
  by (insert euclid_ext_gcd[of a b], drule (1) subst, simp)
eberlm@62442
   448
eberlm@62442
   449
lemma euclid_ext_correct:
eberlm@62442
   450
  "case euclid_ext x y of (a,b,c) \<Rightarrow> a * x + b * y = c \<and> c = gcd x y"
eberlm@62442
   451
  using euclid_ext_correct'[of x y]
eberlm@62442
   452
  by (simp add: gcd_gcd_eucl case_prod_unfold)
haftmann@60572
   453
  
haftmann@60572
   454
lemma euclid_ext'_correct:
haftmann@60572
   455
  "fst (euclid_ext' a b) * a + snd (euclid_ext' a b) * b = gcd a b"
eberlm@62442
   456
  using euclid_ext_correct'[of a b]
eberlm@62442
   457
  by (simp add: gcd_gcd_eucl case_prod_unfold euclid_ext'_def)
haftmann@60572
   458
haftmann@60572
   459
lemma bezout: "\<exists>s t. s * a + t * b = gcd a b"
haftmann@60572
   460
  using euclid_ext'_correct by blast
haftmann@60572
   461
haftmann@60572
   462
end
haftmann@58023
   463
haftmann@58023
   464
haftmann@60572
   465
subsection \<open>Typical instances\<close>
haftmann@58023
   466
haftmann@58023
   467
instantiation nat :: euclidean_semiring
haftmann@58023
   468
begin
haftmann@58023
   469
haftmann@58023
   470
definition [simp]:
haftmann@58023
   471
  "euclidean_size_nat = (id :: nat \<Rightarrow> nat)"
haftmann@58023
   472
haftmann@58023
   473
instance proof
haftmann@59061
   474
qed simp_all
haftmann@58023
   475
haftmann@58023
   476
end
haftmann@58023
   477
eberlm@62422
   478
haftmann@58023
   479
instantiation int :: euclidean_ring
haftmann@58023
   480
begin
haftmann@58023
   481
haftmann@58023
   482
definition [simp]:
haftmann@58023
   483
  "euclidean_size_int = (nat \<circ> abs :: int \<Rightarrow> nat)"
haftmann@58023
   484
wenzelm@60580
   485
instance
haftmann@60686
   486
by standard (auto simp add: abs_mult nat_mult_distrib split: abs_split)
haftmann@58023
   487
haftmann@58023
   488
end
haftmann@58023
   489
eberlm@62422
   490
haftmann@60572
   491
instantiation poly :: (field) euclidean_ring
haftmann@60571
   492
begin
haftmann@60571
   493
haftmann@60571
   494
definition euclidean_size_poly :: "'a poly \<Rightarrow> nat"
eberlm@62422
   495
  where "euclidean_size p = (if p = 0 then 0 else 2 ^ degree p)"
haftmann@60571
   496
haftmann@60600
   497
lemma euclidean_size_poly_0 [simp]:
haftmann@60600
   498
  "euclidean_size (0::'a poly) = 0"
haftmann@60600
   499
  by (simp add: euclidean_size_poly_def)
haftmann@60600
   500
haftmann@60600
   501
lemma euclidean_size_poly_not_0 [simp]:
eberlm@62422
   502
  "p \<noteq> 0 \<Longrightarrow> euclidean_size p = 2 ^ degree p"
haftmann@60600
   503
  by (simp add: euclidean_size_poly_def)
haftmann@60600
   504
haftmann@60571
   505
instance
haftmann@60600
   506
proof
haftmann@60571
   507
  fix p q :: "'a poly"
haftmann@60600
   508
  assume "q \<noteq> 0"
haftmann@60600
   509
  then have "p mod q = 0 \<or> degree (p mod q) < degree q"
haftmann@60600
   510
    by (rule degree_mod_less [of q p])  
haftmann@60600
   511
  with \<open>q \<noteq> 0\<close> show "euclidean_size (p mod q) < euclidean_size q"
haftmann@60600
   512
    by (cases "p mod q = 0") simp_all
haftmann@60571
   513
next
haftmann@60571
   514
  fix p q :: "'a poly"
haftmann@60571
   515
  assume "q \<noteq> 0"
haftmann@60600
   516
  from \<open>q \<noteq> 0\<close> have "degree p \<le> degree (p * q)"
haftmann@60571
   517
    by (rule degree_mult_right_le)
haftmann@60600
   518
  with \<open>q \<noteq> 0\<close> show "euclidean_size p \<le> euclidean_size (p * q)"
haftmann@60600
   519
    by (cases "p = 0") simp_all
eberlm@62422
   520
qed simp
haftmann@60571
   521
haftmann@58023
   522
end
haftmann@60571
   523
eberlm@62422
   524
eberlm@62422
   525
instance nat :: euclidean_semiring_gcd
eberlm@62422
   526
proof
eberlm@62422
   527
  show [simp]: "gcd = (gcd_eucl :: nat \<Rightarrow> _)" "Lcm = (Lcm_eucl :: nat set \<Rightarrow> _)"
eberlm@62422
   528
    by (simp_all add: eq_gcd_euclI eq_Lcm_euclI)
eberlm@62422
   529
  show "lcm = (lcm_eucl :: nat \<Rightarrow> _)" "Gcd = (Gcd_eucl :: nat set \<Rightarrow> _)"
eberlm@62422
   530
    by (intro ext, simp add: lcm_eucl_def lcm_nat_def Gcd_nat_def Gcd_eucl_def)+
eberlm@62422
   531
qed
eberlm@62422
   532
eberlm@62422
   533
instance int :: euclidean_ring_gcd
eberlm@62422
   534
proof
eberlm@62422
   535
  show [simp]: "gcd = (gcd_eucl :: int \<Rightarrow> _)" "Lcm = (Lcm_eucl :: int set \<Rightarrow> _)"
eberlm@62422
   536
    by (simp_all add: eq_gcd_euclI eq_Lcm_euclI)
eberlm@62422
   537
  show "lcm = (lcm_eucl :: int \<Rightarrow> _)" "Gcd = (Gcd_eucl :: int set \<Rightarrow> _)"
eberlm@62422
   538
    by (intro ext, simp add: lcm_eucl_def lcm_altdef_int 
eberlm@62422
   539
          semiring_Gcd_class.Gcd_Lcm Gcd_eucl_def abs_mult)+
eberlm@62422
   540
qed
eberlm@62422
   541
eberlm@62422
   542
eberlm@62422
   543
instantiation poly :: (field) euclidean_ring_gcd
eberlm@62422
   544
begin
eberlm@62422
   545
eberlm@62422
   546
definition gcd_poly :: "'a poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly" where
eberlm@62422
   547
  "gcd_poly = gcd_eucl"
eberlm@62422
   548
  
eberlm@62422
   549
definition lcm_poly :: "'a poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly" where
eberlm@62422
   550
  "lcm_poly = lcm_eucl"
eberlm@62422
   551
  
eberlm@62422
   552
definition Gcd_poly :: "'a poly set \<Rightarrow> 'a poly" where
eberlm@62422
   553
  "Gcd_poly = Gcd_eucl"
eberlm@62422
   554
  
eberlm@62422
   555
definition Lcm_poly :: "'a poly set \<Rightarrow> 'a poly" where
eberlm@62422
   556
  "Lcm_poly = Lcm_eucl"
eberlm@62422
   557
eberlm@62422
   558
instance by standard (simp_all only: gcd_poly_def lcm_poly_def Gcd_poly_def Lcm_poly_def)
eberlm@62422
   559
end
haftmann@60687
   560
eberlm@62425
   561
lemma poly_gcd_monic:
eberlm@62425
   562
  "lead_coeff (gcd x y) = (if x = 0 \<and> y = 0 then 0 else 1)"
eberlm@62425
   563
  using unit_factor_gcd[of x y]
eberlm@62425
   564
  by (simp add: unit_factor_poly_def monom_0 one_poly_def lead_coeff_def split: if_split_asm)
eberlm@62425
   565
eberlm@62425
   566
lemma poly_dvd_antisym:
eberlm@62425
   567
  fixes p q :: "'a::idom poly"
eberlm@62425
   568
  assumes coeff: "coeff p (degree p) = coeff q (degree q)"
eberlm@62425
   569
  assumes dvd1: "p dvd q" and dvd2: "q dvd p" shows "p = q"
eberlm@62425
   570
proof (cases "p = 0")
eberlm@62425
   571
  case True with coeff show "p = q" by simp
eberlm@62425
   572
next
eberlm@62425
   573
  case False with coeff have "q \<noteq> 0" by auto
eberlm@62425
   574
  have degree: "degree p = degree q"
eberlm@62425
   575
    using \<open>p dvd q\<close> \<open>q dvd p\<close> \<open>p \<noteq> 0\<close> \<open>q \<noteq> 0\<close>
eberlm@62425
   576
    by (intro order_antisym dvd_imp_degree_le)
eberlm@62425
   577
eberlm@62425
   578
  from \<open>p dvd q\<close> obtain a where a: "q = p * a" ..
eberlm@62425
   579
  with \<open>q \<noteq> 0\<close> have "a \<noteq> 0" by auto
eberlm@62425
   580
  with degree a \<open>p \<noteq> 0\<close> have "degree a = 0"
eberlm@62425
   581
    by (simp add: degree_mult_eq)
eberlm@62425
   582
  with coeff a show "p = q"
eberlm@62425
   583
    by (cases a, auto split: if_splits)
eberlm@62425
   584
qed
eberlm@62425
   585
eberlm@62425
   586
lemma poly_gcd_unique:
eberlm@62425
   587
  fixes d x y :: "_ poly"
eberlm@62425
   588
  assumes dvd1: "d dvd x" and dvd2: "d dvd y"
eberlm@62425
   589
    and greatest: "\<And>k. k dvd x \<Longrightarrow> k dvd y \<Longrightarrow> k dvd d"
eberlm@62425
   590
    and monic: "coeff d (degree d) = (if x = 0 \<and> y = 0 then 0 else 1)"
eberlm@62425
   591
  shows "d = gcd x y"
eberlm@62425
   592
  using assms by (intro gcdI) (auto simp: normalize_poly_def split: if_split_asm)
eberlm@62425
   593
eberlm@62425
   594
lemma poly_gcd_code [code]:
eberlm@62425
   595
  "gcd x y = (if y = 0 then normalize x else gcd y (x mod (y :: _ poly)))"
eberlm@62425
   596
  by (simp add: gcd_0 gcd_non_0)
eberlm@62425
   597
haftmann@60571
   598
end