src/HOL/Number_Theory/Euclidean_Algorithm.thy
author haftmann
Thu Jun 25 15:01:42 2015 +0200 (2015-06-25)
changeset 60571 c9fdf2080447
parent 60569 f2f1f6860959
child 60572 718b1ba06429
permissions -rw-r--r--
euclidean algorithm on polynomials
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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 Complex_Main "~~/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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  \item a normalization factor such that two associated numbers are equal iff 
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        they are the same when divd by their normalization factors.
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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 + 
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  fixes euclidean_size :: "'a \<Rightarrow> nat"
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  fixes normalization_factor :: "'a \<Rightarrow> 'a"
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  assumes mod_size_less: 
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    "b \<noteq> 0 \<Longrightarrow> \<not> b dvd a \<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 * b) \<ge> euclidean_size a"
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  assumes normalization_factor_is_unit [intro,simp]: 
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    "a \<noteq> 0 \<Longrightarrow> is_unit (normalization_factor a)"
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  assumes normalization_factor_mult: "normalization_factor (a * b) = 
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    normalization_factor a * normalization_factor b"
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  assumes normalization_factor_unit: "is_unit a \<Longrightarrow> normalization_factor a = a"
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  assumes normalization_factor_0 [simp]: "normalization_factor 0 = 0"
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begin
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lemma normalization_factor_dvd [simp]:
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  "a \<noteq> 0 \<Longrightarrow> normalization_factor a dvd b"
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  by (rule unit_imp_dvd, simp)
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lemma normalization_factor_1 [simp]:
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  "normalization_factor 1 = 1"
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  by (simp add: normalization_factor_unit)
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lemma normalization_factor_0_iff [simp]:
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  "normalization_factor a = 0 \<longleftrightarrow> a = 0"
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proof
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  assume "normalization_factor a = 0"
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  hence "\<not> is_unit (normalization_factor a)"
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    by simp
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  then show "a = 0" by auto
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qed simp
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lemma normalization_factor_pow:
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  "normalization_factor (a ^ n) = normalization_factor a ^ n"
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  by (induct n) (simp_all add: normalization_factor_mult power_Suc2)
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lemma normalization_correct [simp]:
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  "normalization_factor (a div normalization_factor a) = (if a = 0 then 0 else 1)"
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proof (cases "a = 0", simp)
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  assume "a \<noteq> 0"
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  let ?nf = "normalization_factor"
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  from normalization_factor_is_unit[OF \<open>a \<noteq> 0\<close>] have "?nf a \<noteq> 0"
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    by auto
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  have "?nf (a div ?nf a) * ?nf (?nf a) = ?nf (a div ?nf a * ?nf a)" 
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    by (simp add: normalization_factor_mult)
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  also have "a div ?nf a * ?nf a = a" using \<open>a \<noteq> 0\<close>
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    by simp
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  also have "?nf (?nf a) = ?nf a" using \<open>a \<noteq> 0\<close> 
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    normalization_factor_is_unit normalization_factor_unit by simp
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  finally have "normalization_factor (a div normalization_factor a) = 1"  
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    using \<open>?nf a \<noteq> 0\<close> by (metis div_mult_self2_is_id div_self)
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  with \<open>a \<noteq> 0\<close> show ?thesis by simp
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qed
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lemma normalization_0_iff [simp]:
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  "a div normalization_factor a = 0 \<longleftrightarrow> a = 0"
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  by (cases "a = 0", simp, subst unit_eq_div1, blast, simp)
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lemma mult_div_normalization [simp]:
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  "b * (1 div normalization_factor a) = b div normalization_factor a"
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  by (cases "a = 0") simp_all
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lemma associated_iff_normed_eq:
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  "associated a b \<longleftrightarrow> a div normalization_factor a = b div normalization_factor b"
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proof (cases "b = 0", simp, cases "a = 0", metis associated_0(1) normalization_0_iff, rule iffI)
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  let ?nf = normalization_factor
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  assume "a \<noteq> 0" "b \<noteq> 0" "a div ?nf a = b div ?nf b"
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  hence "a = b * (?nf a div ?nf b)"
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    apply (subst (asm) unit_eq_div1, blast, subst (asm) unit_div_commute, blast)
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    apply (subst div_mult_swap, simp, simp)
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    done
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  with \<open>a \<noteq> 0\<close> \<open>b \<noteq> 0\<close> have "\<exists>c. is_unit c \<and> a = c * b"
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    by (intro exI[of _ "?nf a div ?nf b"], force simp: mult_ac)
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  then obtain c where "is_unit c" and "a = c * b" by blast
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  then show "associated a b" by (rule is_unit_associatedI) 
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next
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  let ?nf = normalization_factor
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  assume "a \<noteq> 0" "b \<noteq> 0" "associated a b"
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  then obtain c where "is_unit c" and "a = c * b" by (blast elim: associated_is_unitE)
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  then show "a div ?nf a = b div ?nf b"
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    apply (simp only: \<open>a = c * b\<close> normalization_factor_mult normalization_factor_unit)
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    apply (rule div_mult_mult1, force)
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    done
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  qed
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lemma normed_associated_imp_eq:
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  "associated a b \<Longrightarrow> normalization_factor a \<in> {0, 1} \<Longrightarrow> normalization_factor b \<in> {0, 1} \<Longrightarrow> a = b"
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  by (simp add: associated_iff_normed_eq, elim disjE, simp_all)
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lemma normed_dvd [iff]:
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  "a div normalization_factor a dvd a"
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proof (cases "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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  then have "a = a div normalization_factor a * normalization_factor a"
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    by (auto intro: unit_div_mult_self)
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  then show ?thesis ..
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qed
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lemma dvd_normed [iff]:
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  "a dvd a div normalization_factor a"
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proof (cases "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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  then have "a div normalization_factor a = a * (1 div normalization_factor a)"
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    by (auto intro: unit_mult_div_div)
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  then show ?thesis ..
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qed
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lemma associated_normed:
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  "associated (a div normalization_factor a) a"
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  by (rule associatedI) simp_all
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lemma normalization_factor_dvd' [simp]:
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  "normalization_factor a dvd a"
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  by (cases "a = 0", simp_all)
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lemmas normalization_factor_dvd_iff [simp] =
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  unit_dvd_iff [OF normalization_factor_is_unit]
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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" and "\<not> b dvd a"
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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 a div normalization_factor a
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    else if b dvd a then b div normalization_factor b
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    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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    have "P b (a mod b)"
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    proof (cases "b dvd a")
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      case False with \<open>b \<noteq> 0\<close> show "P b (a mod b)"
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        by (rule "1.hyps")
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    next
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      case True then have "a mod b = 0"
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        by (simp add: mod_eq_0_iff_dvd)
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      then show "P b (a mod b)" by simp (rule H1)
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    qed
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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 = a * b div (gcd_eucl a b * normalization_factor (a * b))"
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  (* Somewhat complicated definition of Lcm that has the advantage of working
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     for infinite sets as well *)
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definition Lcm_eucl :: "'a set \<Rightarrow> 'a"
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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 l div normalization_factor 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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end
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class euclidean_semiring_gcd = euclidean_semiring + gcd + Gcd +
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  assumes gcd_gcd_eucl: "gcd = gcd_eucl" and lcm_lcm_eucl: "lcm = lcm_eucl"
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  assumes Gcd_Gcd_eucl: "Gcd = Gcd_eucl" and Lcm_Lcm_eucl: "Lcm = Lcm_eucl"
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begin
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lemma gcd_red:
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  "gcd a b = gcd b (a mod b)"
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  by (cases "b dvd a")
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    (auto simp add: gcd_gcd_eucl gcd_eucl.simps [of a b] gcd_eucl.simps [of 0 a] gcd_eucl.simps [of b 0])
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lemma gcd_non_0:
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  "b \<noteq> 0 \<Longrightarrow> gcd a b = gcd b (a mod b)"
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  by (rule gcd_red)
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lemma gcd_0_left:
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  "gcd 0 a = a div normalization_factor a"
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   by (simp only: gcd_gcd_eucl, subst gcd_eucl.simps, subst gcd_eucl.simps, simp add: Let_def)
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lemma gcd_0:
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  "gcd a 0 = a div normalization_factor a"
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  by (simp only: gcd_gcd_eucl, subst gcd_eucl.simps, simp add: Let_def)
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lemma gcd_dvd1 [iff]: "gcd a b dvd a"
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  and gcd_dvd2 [iff]: "gcd a b dvd b"
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  by (induct a b rule: gcd_eucl_induct)
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    (simp_all add: gcd_0 gcd_non_0 dvd_mod_iff)
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lemma dvd_gcd_D1: "k dvd gcd m n \<Longrightarrow> k dvd m"
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  by (rule dvd_trans, assumption, rule gcd_dvd1)
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lemma dvd_gcd_D2: "k dvd gcd m n \<Longrightarrow> k dvd n"
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  by (rule dvd_trans, assumption, rule gcd_dvd2)
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lemma gcd_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 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_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_non_0 dvd_mod_iff)
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qed
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lemma dvd_gcd_iff:
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  "k dvd gcd a b \<longleftrightarrow> k dvd a \<and> k dvd b"
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  by (blast intro!: gcd_greatest intro: dvd_trans)
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lemmas gcd_greatest_iff = dvd_gcd_iff
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lemma gcd_zero [simp]:
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  "gcd a b = 0 \<longleftrightarrow> a = 0 \<and> b = 0"
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  by (metis dvd_0_left dvd_refl gcd_dvd1 gcd_dvd2 gcd_greatest)+
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lemma normalization_factor_gcd [simp]:
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  "normalization_factor (gcd a b) = (if a = 0 \<and> b = 0 then 0 else 1)" (is "?f a b = ?g a b")
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  by (induct a b rule: gcd_eucl_induct)
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    (auto simp add: gcd_0 gcd_non_0)
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lemma gcdI:
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  "k dvd a \<Longrightarrow> k dvd b \<Longrightarrow> (\<And>l. l dvd a \<Longrightarrow> l dvd b \<Longrightarrow> l dvd k)
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    \<Longrightarrow> normalization_factor k = (if k = 0 then 0 else 1) \<Longrightarrow> k = gcd a b"
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  by (intro normed_associated_imp_eq) (auto simp: associated_def intro: gcd_greatest)
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sublocale gcd!: abel_semigroup gcd
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proof
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  fix a b c 
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  show "gcd (gcd a b) c = gcd a (gcd b c)"
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  proof (rule gcdI)
haftmann@60430
   298
    have "gcd (gcd a b) c dvd gcd a b" "gcd a b dvd a" by simp_all
haftmann@60430
   299
    then show "gcd (gcd a b) c dvd a" by (rule dvd_trans)
haftmann@60430
   300
    have "gcd (gcd a b) c dvd gcd a b" "gcd a b dvd b" by simp_all
haftmann@60430
   301
    hence "gcd (gcd a b) c dvd b" by (rule dvd_trans)
haftmann@60430
   302
    moreover have "gcd (gcd a b) c dvd c" by simp
haftmann@60430
   303
    ultimately show "gcd (gcd a b) c dvd gcd b c"
haftmann@58023
   304
      by (rule gcd_greatest)
haftmann@60438
   305
    show "normalization_factor (gcd (gcd a b) c) =  (if gcd (gcd a b) c = 0 then 0 else 1)"
haftmann@58023
   306
      by auto
haftmann@60430
   307
    fix l assume "l dvd a" and "l dvd gcd b c"
haftmann@58023
   308
    with dvd_trans[OF _ gcd_dvd1] and dvd_trans[OF _ gcd_dvd2]
haftmann@60430
   309
      have "l dvd b" and "l dvd c" by blast+
wenzelm@60526
   310
    with \<open>l dvd a\<close> show "l dvd gcd (gcd a b) c"
haftmann@58023
   311
      by (intro gcd_greatest)
haftmann@58023
   312
  qed
haftmann@58023
   313
next
haftmann@60430
   314
  fix a b
haftmann@60430
   315
  show "gcd a b = gcd b a"
haftmann@58023
   316
    by (rule gcdI) (simp_all add: gcd_greatest)
haftmann@58023
   317
qed
haftmann@58023
   318
haftmann@58023
   319
lemma gcd_unique: "d dvd a \<and> d dvd b \<and> 
haftmann@60438
   320
    normalization_factor d = (if d = 0 then 0 else 1) \<and>
haftmann@58023
   321
    (\<forall>e. e dvd a \<and> e dvd b \<longrightarrow> e dvd d) \<longleftrightarrow> d = gcd a b"
haftmann@58023
   322
  by (rule, auto intro: gcdI simp: gcd_greatest)
haftmann@58023
   323
haftmann@58023
   324
lemma gcd_dvd_prod: "gcd a b dvd k * b"
haftmann@58023
   325
  using mult_dvd_mono [of 1] by auto
haftmann@58023
   326
haftmann@60430
   327
lemma gcd_1_left [simp]: "gcd 1 a = 1"
haftmann@58023
   328
  by (rule sym, rule gcdI, simp_all)
haftmann@58023
   329
haftmann@60430
   330
lemma gcd_1 [simp]: "gcd a 1 = 1"
haftmann@58023
   331
  by (rule sym, rule gcdI, simp_all)
haftmann@58023
   332
haftmann@58023
   333
lemma gcd_proj2_if_dvd: 
haftmann@60438
   334
  "b dvd a \<Longrightarrow> gcd a b = b div normalization_factor b"
haftmann@60430
   335
  by (cases "b = 0", simp_all add: dvd_eq_mod_eq_0 gcd_non_0 gcd_0)
haftmann@58023
   336
haftmann@58023
   337
lemma gcd_proj1_if_dvd: 
haftmann@60438
   338
  "a dvd b \<Longrightarrow> gcd a b = a div normalization_factor a"
haftmann@58023
   339
  by (subst gcd.commute, simp add: gcd_proj2_if_dvd)
haftmann@58023
   340
haftmann@60438
   341
lemma gcd_proj1_iff: "gcd m n = m div normalization_factor m \<longleftrightarrow> m dvd n"
haftmann@58023
   342
proof
haftmann@60438
   343
  assume A: "gcd m n = m div normalization_factor m"
haftmann@58023
   344
  show "m dvd n"
haftmann@58023
   345
  proof (cases "m = 0")
haftmann@58023
   346
    assume [simp]: "m \<noteq> 0"
haftmann@60438
   347
    from A have B: "m = gcd m n * normalization_factor m"
haftmann@58023
   348
      by (simp add: unit_eq_div2)
haftmann@58023
   349
    show ?thesis by (subst B, simp add: mult_unit_dvd_iff)
haftmann@58023
   350
  qed (insert A, simp)
haftmann@58023
   351
next
haftmann@58023
   352
  assume "m dvd n"
haftmann@60438
   353
  then show "gcd m n = m div normalization_factor m" by (rule gcd_proj1_if_dvd)
haftmann@58023
   354
qed
haftmann@58023
   355
  
haftmann@60438
   356
lemma gcd_proj2_iff: "gcd m n = n div normalization_factor n \<longleftrightarrow> n dvd m"
haftmann@58023
   357
  by (subst gcd.commute, simp add: gcd_proj1_iff)
haftmann@58023
   358
haftmann@58023
   359
lemma gcd_mod1 [simp]:
haftmann@60430
   360
  "gcd (a mod b) b = gcd a b"
haftmann@58023
   361
  by (rule gcdI, metis dvd_mod_iff gcd_dvd1 gcd_dvd2, simp_all add: gcd_greatest dvd_mod_iff)
haftmann@58023
   362
haftmann@58023
   363
lemma gcd_mod2 [simp]:
haftmann@60430
   364
  "gcd a (b mod a) = gcd a b"
haftmann@58023
   365
  by (rule gcdI, simp, metis dvd_mod_iff gcd_dvd1 gcd_dvd2, simp_all add: gcd_greatest dvd_mod_iff)
haftmann@58023
   366
         
haftmann@58023
   367
lemma gcd_mult_distrib': 
haftmann@60569
   368
  "c div normalization_factor c * gcd a b = gcd (c * a) (c * b)"
haftmann@60569
   369
proof (cases "c = 0")
haftmann@60569
   370
  case True then show ?thesis by (simp_all add: gcd_0)
haftmann@60569
   371
next
haftmann@60569
   372
  case False then have [simp]: "is_unit (normalization_factor c)" by simp
haftmann@60569
   373
  show ?thesis
haftmann@60569
   374
  proof (induct a b rule: gcd_eucl_induct)
haftmann@60569
   375
    case (zero a) show ?case
haftmann@60569
   376
    proof (cases "a = 0")
haftmann@60569
   377
      case True then show ?thesis by (simp add: gcd_0)
haftmann@60569
   378
    next
haftmann@60569
   379
      case False then have "is_unit (normalization_factor a)" by simp
haftmann@60569
   380
      then show ?thesis
haftmann@60569
   381
        by (simp add: gcd_0 unit_div_commute unit_div_mult_swap normalization_factor_mult is_unit_div_mult2_eq)
haftmann@60569
   382
    qed
haftmann@60569
   383
    case (mod a b)
haftmann@60569
   384
    then show ?case by (simp add: mult_mod_right gcd.commute)
haftmann@58023
   385
  qed
haftmann@58023
   386
qed
haftmann@58023
   387
haftmann@58023
   388
lemma gcd_mult_distrib:
haftmann@60438
   389
  "k * gcd a b = gcd (k*a) (k*b) * normalization_factor k"
haftmann@58023
   390
proof-
haftmann@60438
   391
  let ?nf = "normalization_factor"
haftmann@58023
   392
  from gcd_mult_distrib' 
haftmann@60430
   393
    have "gcd (k*a) (k*b) = k div ?nf k * gcd a b" ..
haftmann@60430
   394
  also have "... = k * gcd a b div ?nf k"
haftmann@60438
   395
    by (metis dvd_div_mult dvd_eq_mod_eq_0 mod_0 normalization_factor_dvd)
haftmann@58023
   396
  finally show ?thesis
haftmann@59009
   397
    by simp
haftmann@58023
   398
qed
haftmann@58023
   399
haftmann@58023
   400
lemma euclidean_size_gcd_le1 [simp]:
haftmann@58023
   401
  assumes "a \<noteq> 0"
haftmann@58023
   402
  shows "euclidean_size (gcd a b) \<le> euclidean_size a"
haftmann@58023
   403
proof -
haftmann@58023
   404
   have "gcd a b dvd a" by (rule gcd_dvd1)
haftmann@58023
   405
   then obtain c where A: "a = gcd a b * c" unfolding dvd_def by blast
wenzelm@60526
   406
   with \<open>a \<noteq> 0\<close> show ?thesis by (subst (2) A, intro size_mult_mono) auto
haftmann@58023
   407
qed
haftmann@58023
   408
haftmann@58023
   409
lemma euclidean_size_gcd_le2 [simp]:
haftmann@58023
   410
  "b \<noteq> 0 \<Longrightarrow> euclidean_size (gcd a b) \<le> euclidean_size b"
haftmann@58023
   411
  by (subst gcd.commute, rule euclidean_size_gcd_le1)
haftmann@58023
   412
haftmann@58023
   413
lemma euclidean_size_gcd_less1:
haftmann@58023
   414
  assumes "a \<noteq> 0" and "\<not>a dvd b"
haftmann@58023
   415
  shows "euclidean_size (gcd a b) < euclidean_size a"
haftmann@58023
   416
proof (rule ccontr)
haftmann@58023
   417
  assume "\<not>euclidean_size (gcd a b) < euclidean_size a"
wenzelm@60526
   418
  with \<open>a \<noteq> 0\<close> have "euclidean_size (gcd a b) = euclidean_size a"
haftmann@58023
   419
    by (intro le_antisym, simp_all)
haftmann@58023
   420
  with assms have "a dvd gcd a b" by (auto intro: dvd_euclidean_size_eq_imp_dvd)
haftmann@58023
   421
  hence "a dvd b" using dvd_gcd_D2 by blast
wenzelm@60526
   422
  with \<open>\<not>a dvd b\<close> show False by contradiction
haftmann@58023
   423
qed
haftmann@58023
   424
haftmann@58023
   425
lemma euclidean_size_gcd_less2:
haftmann@58023
   426
  assumes "b \<noteq> 0" and "\<not>b dvd a"
haftmann@58023
   427
  shows "euclidean_size (gcd a b) < euclidean_size b"
haftmann@58023
   428
  using assms by (subst gcd.commute, rule euclidean_size_gcd_less1)
haftmann@58023
   429
haftmann@60430
   430
lemma gcd_mult_unit1: "is_unit a \<Longrightarrow> gcd (b * a) c = gcd b c"
haftmann@58023
   431
  apply (rule gcdI)
haftmann@58023
   432
  apply (rule dvd_trans, rule gcd_dvd1, simp add: unit_simps)
haftmann@58023
   433
  apply (rule gcd_dvd2)
haftmann@58023
   434
  apply (rule gcd_greatest, simp add: unit_simps, assumption)
haftmann@60438
   435
  apply (subst normalization_factor_gcd, simp add: gcd_0)
haftmann@58023
   436
  done
haftmann@58023
   437
haftmann@60430
   438
lemma gcd_mult_unit2: "is_unit a \<Longrightarrow> gcd b (c * a) = gcd b c"
haftmann@58023
   439
  by (subst gcd.commute, subst gcd_mult_unit1, assumption, rule gcd.commute)
haftmann@58023
   440
haftmann@60430
   441
lemma gcd_div_unit1: "is_unit a \<Longrightarrow> gcd (b div a) c = gcd b c"
haftmann@60433
   442
  by (erule is_unitE [of _ b]) (simp add: gcd_mult_unit1)
haftmann@58023
   443
haftmann@60430
   444
lemma gcd_div_unit2: "is_unit a \<Longrightarrow> gcd b (c div a) = gcd b c"
haftmann@60433
   445
  by (erule is_unitE [of _ c]) (simp add: gcd_mult_unit2)
haftmann@58023
   446
haftmann@60438
   447
lemma gcd_idem: "gcd a a = a div normalization_factor a"
haftmann@60430
   448
  by (cases "a = 0") (simp add: gcd_0_left, rule sym, rule gcdI, simp_all)
haftmann@58023
   449
haftmann@60430
   450
lemma gcd_right_idem: "gcd (gcd a b) b = gcd a b"
haftmann@58023
   451
  apply (rule gcdI)
haftmann@58023
   452
  apply (simp add: ac_simps)
haftmann@58023
   453
  apply (rule gcd_dvd2)
haftmann@58023
   454
  apply (rule gcd_greatest, erule (1) gcd_greatest, assumption)
haftmann@59009
   455
  apply simp
haftmann@58023
   456
  done
haftmann@58023
   457
haftmann@60430
   458
lemma gcd_left_idem: "gcd a (gcd a b) = gcd a b"
haftmann@58023
   459
  apply (rule gcdI)
haftmann@58023
   460
  apply simp
haftmann@58023
   461
  apply (rule dvd_trans, rule gcd_dvd2, rule gcd_dvd2)
haftmann@58023
   462
  apply (rule gcd_greatest, assumption, erule gcd_greatest, assumption)
haftmann@59009
   463
  apply simp
haftmann@58023
   464
  done
haftmann@58023
   465
haftmann@58023
   466
lemma comp_fun_idem_gcd: "comp_fun_idem gcd"
haftmann@58023
   467
proof
haftmann@58023
   468
  fix a b show "gcd a \<circ> gcd b = gcd b \<circ> gcd a"
haftmann@58023
   469
    by (simp add: fun_eq_iff ac_simps)
haftmann@58023
   470
next
haftmann@58023
   471
  fix a show "gcd a \<circ> gcd a = gcd a"
haftmann@58023
   472
    by (simp add: fun_eq_iff gcd_left_idem)
haftmann@58023
   473
qed
haftmann@58023
   474
haftmann@58023
   475
lemma coprime_dvd_mult:
haftmann@60430
   476
  assumes "gcd c b = 1" and "c dvd a * b"
haftmann@60430
   477
  shows "c dvd a"
haftmann@58023
   478
proof -
haftmann@60438
   479
  let ?nf = "normalization_factor"
haftmann@60430
   480
  from assms gcd_mult_distrib [of a c b] 
haftmann@60430
   481
    have A: "a = gcd (a * c) (a * b) * ?nf a" by simp
wenzelm@60526
   482
  from \<open>c dvd a * b\<close> show ?thesis by (subst A, simp_all add: gcd_greatest)
haftmann@58023
   483
qed
haftmann@58023
   484
haftmann@58023
   485
lemma coprime_dvd_mult_iff:
haftmann@60430
   486
  "gcd c b = 1 \<Longrightarrow> (c dvd a * b) = (c dvd a)"
haftmann@58023
   487
  by (rule, rule coprime_dvd_mult, simp_all)
haftmann@58023
   488
haftmann@58023
   489
lemma gcd_dvd_antisym:
haftmann@58023
   490
  "gcd a b dvd gcd c d \<Longrightarrow> gcd c d dvd gcd a b \<Longrightarrow> gcd a b = gcd c d"
haftmann@58023
   491
proof (rule gcdI)
haftmann@58023
   492
  assume A: "gcd a b dvd gcd c d" and B: "gcd c d dvd gcd a b"
haftmann@58023
   493
  have "gcd c d dvd c" by simp
haftmann@58023
   494
  with A show "gcd a b dvd c" by (rule dvd_trans)
haftmann@58023
   495
  have "gcd c d dvd d" by simp
haftmann@58023
   496
  with A show "gcd a b dvd d" by (rule dvd_trans)
haftmann@60438
   497
  show "normalization_factor (gcd a b) = (if gcd a b = 0 then 0 else 1)"
haftmann@59009
   498
    by simp
haftmann@58023
   499
  fix l assume "l dvd c" and "l dvd d"
haftmann@58023
   500
  hence "l dvd gcd c d" by (rule gcd_greatest)
haftmann@58023
   501
  from this and B show "l dvd gcd a b" by (rule dvd_trans)
haftmann@58023
   502
qed
haftmann@58023
   503
haftmann@58023
   504
lemma gcd_mult_cancel:
haftmann@58023
   505
  assumes "gcd k n = 1"
haftmann@58023
   506
  shows "gcd (k * m) n = gcd m n"
haftmann@58023
   507
proof (rule gcd_dvd_antisym)
haftmann@58023
   508
  have "gcd (gcd (k * m) n) k = gcd (gcd k n) (k * m)" by (simp add: ac_simps)
wenzelm@60526
   509
  also note \<open>gcd k n = 1\<close>
haftmann@58023
   510
  finally have "gcd (gcd (k * m) n) k = 1" by simp
haftmann@58023
   511
  hence "gcd (k * m) n dvd m" by (rule coprime_dvd_mult, simp add: ac_simps)
haftmann@58023
   512
  moreover have "gcd (k * m) n dvd n" by simp
haftmann@58023
   513
  ultimately show "gcd (k * m) n dvd gcd m n" by (rule gcd_greatest)
haftmann@58023
   514
  have "gcd m n dvd (k * m)" and "gcd m n dvd n" by simp_all
haftmann@58023
   515
  then show "gcd m n dvd gcd (k * m) n" by (rule gcd_greatest)
haftmann@58023
   516
qed
haftmann@58023
   517
haftmann@58023
   518
lemma coprime_crossproduct:
haftmann@58023
   519
  assumes [simp]: "gcd a d = 1" "gcd b c = 1"
haftmann@58023
   520
  shows "associated (a * c) (b * d) \<longleftrightarrow> associated a b \<and> associated c d" (is "?lhs \<longleftrightarrow> ?rhs")
haftmann@58023
   521
proof
haftmann@58023
   522
  assume ?rhs then show ?lhs unfolding associated_def by (fast intro: mult_dvd_mono)
haftmann@58023
   523
next
haftmann@58023
   524
  assume ?lhs
wenzelm@60526
   525
  from \<open>?lhs\<close> have "a dvd b * d" unfolding associated_def by (metis dvd_mult_left) 
haftmann@58023
   526
  hence "a dvd b" by (simp add: coprime_dvd_mult_iff)
wenzelm@60526
   527
  moreover from \<open>?lhs\<close> have "b dvd a * c" unfolding associated_def by (metis dvd_mult_left) 
haftmann@58023
   528
  hence "b dvd a" by (simp add: coprime_dvd_mult_iff)
wenzelm@60526
   529
  moreover from \<open>?lhs\<close> have "c dvd d * b" 
haftmann@59009
   530
    unfolding associated_def by (auto dest: dvd_mult_right simp add: ac_simps)
haftmann@58023
   531
  hence "c dvd d" by (simp add: coprime_dvd_mult_iff gcd.commute)
wenzelm@60526
   532
  moreover from \<open>?lhs\<close> have "d dvd c * a"
haftmann@59009
   533
    unfolding associated_def by (auto dest: dvd_mult_right simp add: ac_simps)
haftmann@58023
   534
  hence "d dvd c" by (simp add: coprime_dvd_mult_iff gcd.commute)
haftmann@58023
   535
  ultimately show ?rhs unfolding associated_def by simp
haftmann@58023
   536
qed
haftmann@58023
   537
haftmann@58023
   538
lemma gcd_add1 [simp]:
haftmann@58023
   539
  "gcd (m + n) n = gcd m n"
haftmann@58023
   540
  by (cases "n = 0", simp_all add: gcd_non_0)
haftmann@58023
   541
haftmann@58023
   542
lemma gcd_add2 [simp]:
haftmann@58023
   543
  "gcd m (m + n) = gcd m n"
haftmann@58023
   544
  using gcd_add1 [of n m] by (simp add: ac_simps)
haftmann@58023
   545
haftmann@58023
   546
lemma gcd_add_mult: "gcd m (k * m + n) = gcd m n"
haftmann@58023
   547
  by (subst gcd.commute, subst gcd_red, simp)
haftmann@58023
   548
haftmann@60430
   549
lemma coprimeI: "(\<And>l. \<lbrakk>l dvd a; l dvd b\<rbrakk> \<Longrightarrow> l dvd 1) \<Longrightarrow> gcd a b = 1"
haftmann@58023
   550
  by (rule sym, rule gcdI, simp_all)
haftmann@58023
   551
haftmann@58023
   552
lemma coprime: "gcd a b = 1 \<longleftrightarrow> (\<forall>d. d dvd a \<and> d dvd b \<longleftrightarrow> is_unit d)"
haftmann@59061
   553
  by (auto intro: coprimeI gcd_greatest dvd_gcd_D1 dvd_gcd_D2)
haftmann@58023
   554
haftmann@58023
   555
lemma div_gcd_coprime:
haftmann@58023
   556
  assumes nz: "a \<noteq> 0 \<or> b \<noteq> 0"
haftmann@58023
   557
  defines [simp]: "d \<equiv> gcd a b"
haftmann@58023
   558
  defines [simp]: "a' \<equiv> a div d" and [simp]: "b' \<equiv> b div d"
haftmann@58023
   559
  shows "gcd a' b' = 1"
haftmann@58023
   560
proof (rule coprimeI)
haftmann@58023
   561
  fix l assume "l dvd a'" "l dvd b'"
haftmann@58023
   562
  then obtain s t where "a' = l * s" "b' = l * t" unfolding dvd_def by blast
haftmann@59009
   563
  moreover have "a = a' * d" "b = b' * d" by simp_all
haftmann@58023
   564
  ultimately have "a = (l * d) * s" "b = (l * d) * t"
haftmann@59009
   565
    by (simp_all only: ac_simps)
haftmann@58023
   566
  hence "l*d dvd a" and "l*d dvd b" by (simp_all only: dvd_triv_left)
haftmann@58023
   567
  hence "l*d dvd d" by (simp add: gcd_greatest)
haftmann@59009
   568
  then obtain u where "d = l * d * u" ..
haftmann@59009
   569
  then have "d * (l * u) = d" by (simp add: ac_simps)
haftmann@59009
   570
  moreover from nz have "d \<noteq> 0" by simp
haftmann@59009
   571
  with div_mult_self1_is_id have "d * (l * u) div d = l * u" . 
haftmann@59009
   572
  ultimately have "1 = l * u"
wenzelm@60526
   573
    using \<open>d \<noteq> 0\<close> by simp
haftmann@59009
   574
  then show "l dvd 1" ..
haftmann@58023
   575
qed
haftmann@58023
   576
haftmann@58023
   577
lemma coprime_mult: 
haftmann@58023
   578
  assumes da: "gcd d a = 1" and db: "gcd d b = 1"
haftmann@58023
   579
  shows "gcd d (a * b) = 1"
haftmann@58023
   580
  apply (subst gcd.commute)
haftmann@58023
   581
  using da apply (subst gcd_mult_cancel)
haftmann@58023
   582
  apply (subst gcd.commute, assumption)
haftmann@58023
   583
  apply (subst gcd.commute, rule db)
haftmann@58023
   584
  done
haftmann@58023
   585
haftmann@58023
   586
lemma coprime_lmult:
haftmann@58023
   587
  assumes dab: "gcd d (a * b) = 1" 
haftmann@58023
   588
  shows "gcd d a = 1"
haftmann@58023
   589
proof (rule coprimeI)
haftmann@58023
   590
  fix l assume "l dvd d" and "l dvd a"
haftmann@58023
   591
  hence "l dvd a * b" by simp
wenzelm@60526
   592
  with \<open>l dvd d\<close> and dab show "l dvd 1" by (auto intro: gcd_greatest)
haftmann@58023
   593
qed
haftmann@58023
   594
haftmann@58023
   595
lemma coprime_rmult:
haftmann@58023
   596
  assumes dab: "gcd d (a * b) = 1"
haftmann@58023
   597
  shows "gcd d b = 1"
haftmann@58023
   598
proof (rule coprimeI)
haftmann@58023
   599
  fix l assume "l dvd d" and "l dvd b"
haftmann@58023
   600
  hence "l dvd a * b" by simp
wenzelm@60526
   601
  with \<open>l dvd d\<close> and dab show "l dvd 1" by (auto intro: gcd_greatest)
haftmann@58023
   602
qed
haftmann@58023
   603
haftmann@58023
   604
lemma coprime_mul_eq: "gcd d (a * b) = 1 \<longleftrightarrow> gcd d a = 1 \<and> gcd d b = 1"
haftmann@58023
   605
  using coprime_rmult[of d a b] coprime_lmult[of d a b] coprime_mult[of d a b] by blast
haftmann@58023
   606
haftmann@58023
   607
lemma gcd_coprime:
haftmann@60430
   608
  assumes c: "gcd a b \<noteq> 0" and a: "a = a' * gcd a b" and b: "b = b' * gcd a b"
haftmann@58023
   609
  shows "gcd a' b' = 1"
haftmann@58023
   610
proof -
haftmann@60430
   611
  from c have "a \<noteq> 0 \<or> b \<noteq> 0" by simp
haftmann@58023
   612
  with div_gcd_coprime have "gcd (a div gcd a b) (b div gcd a b) = 1" .
haftmann@58023
   613
  also from assms have "a div gcd a b = a'" by (metis div_mult_self2_is_id)+
haftmann@58023
   614
  also from assms have "b div gcd a b = b'" by (metis div_mult_self2_is_id)+
haftmann@58023
   615
  finally show ?thesis .
haftmann@58023
   616
qed
haftmann@58023
   617
haftmann@58023
   618
lemma coprime_power:
haftmann@58023
   619
  assumes "0 < n"
haftmann@58023
   620
  shows "gcd a (b ^ n) = 1 \<longleftrightarrow> gcd a b = 1"
haftmann@58023
   621
using assms proof (induct n)
haftmann@58023
   622
  case (Suc n) then show ?case
haftmann@58023
   623
    by (cases n) (simp_all add: coprime_mul_eq)
haftmann@58023
   624
qed simp
haftmann@58023
   625
haftmann@58023
   626
lemma gcd_coprime_exists:
haftmann@58023
   627
  assumes nz: "gcd a b \<noteq> 0"
haftmann@58023
   628
  shows "\<exists>a' b'. a = a' * gcd a b \<and> b = b' * gcd a b \<and> gcd a' b' = 1"
haftmann@58023
   629
  apply (rule_tac x = "a div gcd a b" in exI)
haftmann@58023
   630
  apply (rule_tac x = "b div gcd a b" in exI)
haftmann@59009
   631
  apply (insert nz, auto intro: div_gcd_coprime)
haftmann@58023
   632
  done
haftmann@58023
   633
haftmann@58023
   634
lemma coprime_exp:
haftmann@58023
   635
  "gcd d a = 1 \<Longrightarrow> gcd d (a^n) = 1"
haftmann@58023
   636
  by (induct n, simp_all add: coprime_mult)
haftmann@58023
   637
haftmann@58023
   638
lemma coprime_exp2 [intro]:
haftmann@58023
   639
  "gcd a b = 1 \<Longrightarrow> gcd (a^n) (b^m) = 1"
haftmann@58023
   640
  apply (rule coprime_exp)
haftmann@58023
   641
  apply (subst gcd.commute)
haftmann@58023
   642
  apply (rule coprime_exp)
haftmann@58023
   643
  apply (subst gcd.commute)
haftmann@58023
   644
  apply assumption
haftmann@58023
   645
  done
haftmann@58023
   646
haftmann@58023
   647
lemma gcd_exp:
haftmann@58023
   648
  "gcd (a^n) (b^n) = (gcd a b) ^ n"
haftmann@58023
   649
proof (cases "a = 0 \<and> b = 0")
haftmann@58023
   650
  assume "a = 0 \<and> b = 0"
haftmann@58023
   651
  then show ?thesis by (cases n, simp_all add: gcd_0_left)
haftmann@58023
   652
next
haftmann@58023
   653
  assume A: "\<not>(a = 0 \<and> b = 0)"
haftmann@58023
   654
  hence "1 = gcd ((a div gcd a b)^n) ((b div gcd a b)^n)"
haftmann@58023
   655
    using div_gcd_coprime by (subst sym, auto simp: div_gcd_coprime)
haftmann@58023
   656
  hence "(gcd a b) ^ n = (gcd a b) ^ n * ..." by simp
haftmann@58023
   657
  also note gcd_mult_distrib
haftmann@60438
   658
  also have "normalization_factor ((gcd a b)^n) = 1"
haftmann@60438
   659
    by (simp add: normalization_factor_pow A)
haftmann@58023
   660
  also have "(gcd a b)^n * (a div gcd a b)^n = a^n"
haftmann@58023
   661
    by (subst ac_simps, subst div_power, simp, rule dvd_div_mult_self, rule dvd_power_same, simp)
haftmann@58023
   662
  also have "(gcd a b)^n * (b div gcd a b)^n = b^n"
haftmann@58023
   663
    by (subst ac_simps, subst div_power, simp, rule dvd_div_mult_self, rule dvd_power_same, simp)
haftmann@58023
   664
  finally show ?thesis by simp
haftmann@58023
   665
qed
haftmann@58023
   666
haftmann@58023
   667
lemma coprime_common_divisor: 
haftmann@60430
   668
  "gcd a b = 1 \<Longrightarrow> a dvd a \<Longrightarrow> a dvd b \<Longrightarrow> is_unit a"
haftmann@60430
   669
  apply (subgoal_tac "a dvd gcd a b")
haftmann@59061
   670
  apply simp
haftmann@58023
   671
  apply (erule (1) gcd_greatest)
haftmann@58023
   672
  done
haftmann@58023
   673
haftmann@58023
   674
lemma division_decomp: 
haftmann@58023
   675
  assumes dc: "a dvd b * c"
haftmann@58023
   676
  shows "\<exists>b' c'. a = b' * c' \<and> b' dvd b \<and> c' dvd c"
haftmann@58023
   677
proof (cases "gcd a b = 0")
haftmann@58023
   678
  assume "gcd a b = 0"
haftmann@59009
   679
  hence "a = 0 \<and> b = 0" by simp
haftmann@58023
   680
  hence "a = 0 * c \<and> 0 dvd b \<and> c dvd c" by simp
haftmann@58023
   681
  then show ?thesis by blast
haftmann@58023
   682
next
haftmann@58023
   683
  let ?d = "gcd a b"
haftmann@58023
   684
  assume "?d \<noteq> 0"
haftmann@58023
   685
  from gcd_coprime_exists[OF this]
haftmann@58023
   686
    obtain a' b' where ab': "a = a' * ?d" "b = b' * ?d" "gcd a' b' = 1"
haftmann@58023
   687
    by blast
haftmann@58023
   688
  from ab'(1) have "a' dvd a" unfolding dvd_def by blast
haftmann@58023
   689
  with dc have "a' dvd b*c" using dvd_trans[of a' a "b*c"] by simp
haftmann@58023
   690
  from dc ab'(1,2) have "a'*?d dvd (b'*?d) * c" by simp
haftmann@58023
   691
  hence "?d * a' dvd ?d * (b' * c)" by (simp add: mult_ac)
wenzelm@60526
   692
  with \<open>?d \<noteq> 0\<close> have "a' dvd b' * c" by simp
haftmann@58023
   693
  with coprime_dvd_mult[OF ab'(3)] 
haftmann@58023
   694
    have "a' dvd c" by (subst (asm) ac_simps, blast)
haftmann@58023
   695
  with ab'(1) have "a = ?d * a' \<and> ?d dvd b \<and> a' dvd c" by (simp add: mult_ac)
haftmann@58023
   696
  then show ?thesis by blast
haftmann@58023
   697
qed
haftmann@58023
   698
haftmann@60433
   699
lemma pow_divs_pow:
haftmann@58023
   700
  assumes ab: "a ^ n dvd b ^ n" and n: "n \<noteq> 0"
haftmann@58023
   701
  shows "a dvd b"
haftmann@58023
   702
proof (cases "gcd a b = 0")
haftmann@58023
   703
  assume "gcd a b = 0"
haftmann@59009
   704
  then show ?thesis by simp
haftmann@58023
   705
next
haftmann@58023
   706
  let ?d = "gcd a b"
haftmann@58023
   707
  assume "?d \<noteq> 0"
haftmann@58023
   708
  from n obtain m where m: "n = Suc m" by (cases n, simp_all)
wenzelm@60526
   709
  from \<open>?d \<noteq> 0\<close> have zn: "?d ^ n \<noteq> 0" by (rule power_not_zero)
wenzelm@60526
   710
  from gcd_coprime_exists[OF \<open>?d \<noteq> 0\<close>]
haftmann@58023
   711
    obtain a' b' where ab': "a = a' * ?d" "b = b' * ?d" "gcd a' b' = 1"
haftmann@58023
   712
    by blast
haftmann@58023
   713
  from ab have "(a' * ?d) ^ n dvd (b' * ?d) ^ n"
haftmann@58023
   714
    by (simp add: ab'(1,2)[symmetric])
haftmann@58023
   715
  hence "?d^n * a'^n dvd ?d^n * b'^n"
haftmann@58023
   716
    by (simp only: power_mult_distrib ac_simps)
haftmann@59009
   717
  with zn have "a'^n dvd b'^n" by simp
haftmann@58023
   718
  hence "a' dvd b'^n" using dvd_trans[of a' "a'^n" "b'^n"] by (simp add: m)
haftmann@58023
   719
  hence "a' dvd b'^m * b'" by (simp add: m ac_simps)
haftmann@58023
   720
  with coprime_dvd_mult[OF coprime_exp[OF ab'(3), of m]]
haftmann@58023
   721
    have "a' dvd b'" by (subst (asm) ac_simps, blast)
haftmann@58023
   722
  hence "a'*?d dvd b'*?d" by (rule mult_dvd_mono, simp)
haftmann@58023
   723
  with ab'(1,2) show ?thesis by simp
haftmann@58023
   724
qed
haftmann@58023
   725
haftmann@60433
   726
lemma pow_divs_eq [simp]:
haftmann@58023
   727
  "n \<noteq> 0 \<Longrightarrow> a ^ n dvd b ^ n \<longleftrightarrow> a dvd b"
haftmann@60433
   728
  by (auto intro: pow_divs_pow dvd_power_same)
haftmann@58023
   729
haftmann@60433
   730
lemma divs_mult:
haftmann@58023
   731
  assumes mr: "m dvd r" and nr: "n dvd r" and mn: "gcd m n = 1"
haftmann@58023
   732
  shows "m * n dvd r"
haftmann@58023
   733
proof -
haftmann@58023
   734
  from mr nr obtain m' n' where m': "r = m*m'" and n': "r = n*n'"
haftmann@58023
   735
    unfolding dvd_def by blast
haftmann@58023
   736
  from mr n' have "m dvd n'*n" by (simp add: ac_simps)
haftmann@58023
   737
  hence "m dvd n'" using coprime_dvd_mult_iff[OF mn] by simp
haftmann@58023
   738
  then obtain k where k: "n' = m*k" unfolding dvd_def by blast
haftmann@58023
   739
  with n' have "r = m * n * k" by (simp add: mult_ac)
haftmann@58023
   740
  then show ?thesis unfolding dvd_def by blast
haftmann@58023
   741
qed
haftmann@58023
   742
haftmann@58023
   743
lemma coprime_plus_one [simp]: "gcd (n + 1) n = 1"
haftmann@58023
   744
  by (subst add_commute, simp)
haftmann@58023
   745
haftmann@58023
   746
lemma setprod_coprime [rule_format]:
haftmann@60430
   747
  "(\<forall>i\<in>A. gcd (f i) a = 1) \<longrightarrow> gcd (\<Prod>i\<in>A. f i) a = 1"
haftmann@58023
   748
  apply (cases "finite A")
haftmann@58023
   749
  apply (induct set: finite)
haftmann@58023
   750
  apply (auto simp add: gcd_mult_cancel)
haftmann@58023
   751
  done
haftmann@58023
   752
haftmann@58023
   753
lemma coprime_divisors: 
haftmann@58023
   754
  assumes "d dvd a" "e dvd b" "gcd a b = 1"
haftmann@58023
   755
  shows "gcd d e = 1" 
haftmann@58023
   756
proof -
haftmann@58023
   757
  from assms obtain k l where "a = d * k" "b = e * l"
haftmann@58023
   758
    unfolding dvd_def by blast
haftmann@58023
   759
  with assms have "gcd (d * k) (e * l) = 1" by simp
haftmann@58023
   760
  hence "gcd (d * k) e = 1" by (rule coprime_lmult)
haftmann@58023
   761
  also have "gcd (d * k) e = gcd e (d * k)" by (simp add: ac_simps)
haftmann@58023
   762
  finally have "gcd e d = 1" by (rule coprime_lmult)
haftmann@58023
   763
  then show ?thesis by (simp add: ac_simps)
haftmann@58023
   764
qed
haftmann@58023
   765
haftmann@58023
   766
lemma invertible_coprime:
haftmann@60430
   767
  assumes "a * b mod m = 1"
haftmann@60430
   768
  shows "coprime a m"
haftmann@59009
   769
proof -
haftmann@60430
   770
  from assms have "coprime m (a * b mod m)"
haftmann@59009
   771
    by simp
haftmann@60430
   772
  then have "coprime m (a * b)"
haftmann@59009
   773
    by simp
haftmann@60430
   774
  then have "coprime m a"
haftmann@59009
   775
    by (rule coprime_lmult)
haftmann@59009
   776
  then show ?thesis
haftmann@59009
   777
    by (simp add: ac_simps)
haftmann@59009
   778
qed
haftmann@58023
   779
haftmann@58023
   780
lemma lcm_gcd:
haftmann@60438
   781
  "lcm a b = a * b div (gcd a b * normalization_factor (a*b))"
haftmann@58023
   782
  by (simp only: lcm_lcm_eucl gcd_gcd_eucl lcm_eucl_def)
haftmann@58023
   783
haftmann@58023
   784
lemma lcm_gcd_prod:
haftmann@60438
   785
  "lcm a b * gcd a b = a * b div normalization_factor (a*b)"
haftmann@58023
   786
proof (cases "a * b = 0")
haftmann@60438
   787
  let ?nf = normalization_factor
haftmann@58023
   788
  assume "a * b \<noteq> 0"
haftmann@58953
   789
  hence "gcd a b \<noteq> 0" by simp
haftmann@58023
   790
  from lcm_gcd have "lcm a b * gcd a b = gcd a b * (a * b div (?nf (a*b) * gcd a b))" 
haftmann@58023
   791
    by (simp add: mult_ac)
wenzelm@60526
   792
  also from \<open>a * b \<noteq> 0\<close> have "... = a * b div ?nf (a*b)"
haftmann@60432
   793
    by (simp add: div_mult_swap mult.commute)
haftmann@58023
   794
  finally show ?thesis .
haftmann@58953
   795
qed (auto simp add: lcm_gcd)
haftmann@58023
   796
haftmann@58023
   797
lemma lcm_dvd1 [iff]:
haftmann@60430
   798
  "a dvd lcm a b"
haftmann@60430
   799
proof (cases "a*b = 0")
haftmann@60430
   800
  assume "a * b \<noteq> 0"
haftmann@60430
   801
  hence "gcd a b \<noteq> 0" by simp
haftmann@60438
   802
  let ?c = "1 div normalization_factor (a * b)"
wenzelm@60526
   803
  from \<open>a * b \<noteq> 0\<close> have [simp]: "is_unit (normalization_factor (a * b))" by simp
haftmann@60430
   804
  from lcm_gcd_prod[of a b] have "lcm a b * gcd a b = a * ?c * b"
haftmann@60432
   805
    by (simp add: div_mult_swap unit_div_commute)
haftmann@60430
   806
  hence "lcm a b * gcd a b div gcd a b = a * ?c * b div gcd a b" by simp
wenzelm@60526
   807
  with \<open>gcd a b \<noteq> 0\<close> have "lcm a b = a * ?c * b div gcd a b"
haftmann@58023
   808
    by (subst (asm) div_mult_self2_is_id, simp_all)
haftmann@60430
   809
  also have "... = a * (?c * b div gcd a b)"
haftmann@58023
   810
    by (metis div_mult_swap gcd_dvd2 mult_assoc)
haftmann@58023
   811
  finally show ?thesis by (rule dvdI)
haftmann@58953
   812
qed (auto simp add: lcm_gcd)
haftmann@58023
   813
haftmann@58023
   814
lemma lcm_least:
haftmann@58023
   815
  "\<lbrakk>a dvd k; b dvd k\<rbrakk> \<Longrightarrow> lcm a b dvd k"
haftmann@58023
   816
proof (cases "k = 0")
haftmann@60438
   817
  let ?nf = normalization_factor
haftmann@58023
   818
  assume "k \<noteq> 0"
haftmann@58023
   819
  hence "is_unit (?nf k)" by simp
haftmann@58023
   820
  hence "?nf k \<noteq> 0" by (metis not_is_unit_0)
haftmann@58023
   821
  assume A: "a dvd k" "b dvd k"
wenzelm@60526
   822
  hence "gcd a b \<noteq> 0" using \<open>k \<noteq> 0\<close> by auto
haftmann@58023
   823
  from A obtain r s where ar: "k = a * r" and bs: "k = b * s" 
haftmann@58023
   824
    unfolding dvd_def by blast
wenzelm@60526
   825
  with \<open>k \<noteq> 0\<close> have "r * s \<noteq> 0"
haftmann@58953
   826
    by auto (drule sym [of 0], simp)
haftmann@58023
   827
  hence "is_unit (?nf (r * s))" by simp
haftmann@58023
   828
  let ?c = "?nf k div ?nf (r*s)"
wenzelm@60526
   829
  from \<open>is_unit (?nf k)\<close> and \<open>is_unit (?nf (r * s))\<close> have "is_unit ?c" by (rule unit_div)
haftmann@58023
   830
  hence "?c \<noteq> 0" using not_is_unit_0 by fast 
haftmann@58023
   831
  from ar bs have "k * k * gcd s r = ?nf k * k * gcd (k * s) (k * r)"
haftmann@58953
   832
    by (subst mult_assoc, subst gcd_mult_distrib[of k s r], simp only: ac_simps)
haftmann@58023
   833
  also have "... = ?nf k * k * gcd ((r*s) * a) ((r*s) * b)"
wenzelm@60526
   834
    by (subst (3) \<open>k = a * r\<close>, subst (3) \<open>k = b * s\<close>, simp add: algebra_simps)
wenzelm@60526
   835
  also have "... = ?c * r*s * k * gcd a b" using \<open>r * s \<noteq> 0\<close>
haftmann@58023
   836
    by (subst gcd_mult_distrib'[symmetric], simp add: algebra_simps unit_simps)
haftmann@58023
   837
  finally have "(a*r) * (b*s) * gcd s r = ?c * k * r * s * gcd a b"
haftmann@58023
   838
    by (subst ar[symmetric], subst bs[symmetric], simp add: mult_ac)
haftmann@58023
   839
  hence "a * b * gcd s r * (r * s) = ?c * k * gcd a b * (r * s)"
haftmann@58023
   840
    by (simp add: algebra_simps)
wenzelm@60526
   841
  hence "?c * k * gcd a b = a * b * gcd s r" using \<open>r * s \<noteq> 0\<close>
haftmann@58023
   842
    by (metis div_mult_self2_is_id)
haftmann@58023
   843
  also have "... = lcm a b * gcd a b * gcd s r * ?nf (a*b)"
haftmann@58023
   844
    by (subst lcm_gcd_prod[of a b], metis gcd_mult_distrib gcd_mult_distrib') 
haftmann@58023
   845
  also have "... = lcm a b * gcd s r * ?nf (a*b) * gcd a b"
haftmann@58023
   846
    by (simp add: algebra_simps)
wenzelm@60526
   847
  finally have "k * ?c = lcm a b * gcd s r * ?nf (a*b)" using \<open>gcd a b \<noteq> 0\<close>
haftmann@58023
   848
    by (metis mult.commute div_mult_self2_is_id)
wenzelm@60526
   849
  hence "k = lcm a b * (gcd s r * ?nf (a*b)) div ?c" using \<open>?c \<noteq> 0\<close>
haftmann@58023
   850
    by (metis div_mult_self2_is_id mult_assoc) 
wenzelm@60526
   851
  also have "... = lcm a b * (gcd s r * ?nf (a*b) div ?c)" using \<open>is_unit ?c\<close>
haftmann@58023
   852
    by (simp add: unit_simps)
haftmann@58023
   853
  finally show ?thesis by (rule dvdI)
haftmann@58023
   854
qed simp
haftmann@58023
   855
haftmann@58023
   856
lemma lcm_zero:
haftmann@58023
   857
  "lcm a b = 0 \<longleftrightarrow> a = 0 \<or> b = 0"
haftmann@58023
   858
proof -
haftmann@60438
   859
  let ?nf = normalization_factor
haftmann@58023
   860
  {
haftmann@58023
   861
    assume "a \<noteq> 0" "b \<noteq> 0"
haftmann@58023
   862
    hence "a * b div ?nf (a * b) \<noteq> 0" by (simp add: no_zero_divisors)
wenzelm@60526
   863
    moreover from \<open>a \<noteq> 0\<close> and \<open>b \<noteq> 0\<close> have "gcd a b \<noteq> 0" by simp
haftmann@58023
   864
    ultimately have "lcm a b \<noteq> 0" using lcm_gcd_prod[of a b] by (intro notI, simp)
haftmann@58023
   865
  } moreover {
haftmann@58023
   866
    assume "a = 0 \<or> b = 0"
haftmann@58023
   867
    hence "lcm a b = 0" by (elim disjE, simp_all add: lcm_gcd)
haftmann@58023
   868
  }
haftmann@58023
   869
  ultimately show ?thesis by blast
haftmann@58023
   870
qed
haftmann@58023
   871
haftmann@58023
   872
lemmas lcm_0_iff = lcm_zero
haftmann@58023
   873
haftmann@58023
   874
lemma gcd_lcm: 
haftmann@58023
   875
  assumes "lcm a b \<noteq> 0"
haftmann@60438
   876
  shows "gcd a b = a * b div (lcm a b * normalization_factor (a * b))"
haftmann@58023
   877
proof-
haftmann@59009
   878
  from assms have "gcd a b \<noteq> 0" by (simp add: lcm_zero)
haftmann@60438
   879
  let ?c = "normalization_factor (a * b)"
wenzelm@60526
   880
  from \<open>lcm a b \<noteq> 0\<close> have "?c \<noteq> 0" by (intro notI, simp add: lcm_zero no_zero_divisors)
haftmann@58023
   881
  hence "is_unit ?c" by simp
haftmann@58023
   882
  from lcm_gcd_prod [of a b] have "gcd a b = a * b div ?c div lcm a b"
wenzelm@60526
   883
    by (subst (2) div_mult_self2_is_id[OF \<open>lcm a b \<noteq> 0\<close>, symmetric], simp add: mult_ac)
wenzelm@60526
   884
  also from \<open>is_unit ?c\<close> have "... = a * b div (lcm a b * ?c)"
wenzelm@60526
   885
    by (metis \<open>?c \<noteq> 0\<close> div_mult_mult1 dvd_mult_div_cancel mult_commute normalization_factor_dvd')
haftmann@60433
   886
  finally show ?thesis .
haftmann@58023
   887
qed
haftmann@58023
   888
haftmann@60438
   889
lemma normalization_factor_lcm [simp]:
haftmann@60438
   890
  "normalization_factor (lcm a b) = (if a = 0 \<or> b = 0 then 0 else 1)"
haftmann@58023
   891
proof (cases "a = 0 \<or> b = 0")
haftmann@58023
   892
  case True then show ?thesis
haftmann@58953
   893
    by (auto simp add: lcm_gcd) 
haftmann@58023
   894
next
haftmann@58023
   895
  case False
haftmann@60438
   896
  let ?nf = normalization_factor
haftmann@58023
   897
  from lcm_gcd_prod[of a b] 
haftmann@58023
   898
    have "?nf (lcm a b) * ?nf (gcd a b) = ?nf (a*b) div ?nf (a*b)"
haftmann@60438
   899
    by (metis div_by_0 div_self normalization_correct normalization_factor_0 normalization_factor_mult)
haftmann@58023
   900
  also have "... = (if a*b = 0 then 0 else 1)"
haftmann@58953
   901
    by simp
haftmann@58953
   902
  finally show ?thesis using False by simp
haftmann@58023
   903
qed
haftmann@58023
   904
haftmann@60430
   905
lemma lcm_dvd2 [iff]: "b dvd lcm a b"
haftmann@60430
   906
  using lcm_dvd1 [of b a] by (simp add: lcm_gcd ac_simps)
haftmann@58023
   907
haftmann@58023
   908
lemma lcmI:
haftmann@60430
   909
  "\<lbrakk>a dvd k; b dvd k; \<And>l. a dvd l \<Longrightarrow> b dvd l \<Longrightarrow> k dvd l;
haftmann@60438
   910
    normalization_factor k = (if k = 0 then 0 else 1)\<rbrakk> \<Longrightarrow> k = lcm a b"
haftmann@58023
   911
  by (intro normed_associated_imp_eq) (auto simp: associated_def intro: lcm_least)
haftmann@58023
   912
haftmann@58023
   913
sublocale lcm!: abel_semigroup lcm
haftmann@58023
   914
proof
haftmann@60430
   915
  fix a b c
haftmann@60430
   916
  show "lcm (lcm a b) c = lcm a (lcm b c)"
haftmann@58023
   917
  proof (rule lcmI)
haftmann@60430
   918
    have "a dvd lcm a b" and "lcm a b dvd lcm (lcm a b) c" by simp_all
haftmann@60430
   919
    then show "a dvd lcm (lcm a b) c" by (rule dvd_trans)
haftmann@58023
   920
    
haftmann@60430
   921
    have "b dvd lcm a b" and "lcm a b dvd lcm (lcm a b) c" by simp_all
haftmann@60430
   922
    hence "b dvd lcm (lcm a b) c" by (rule dvd_trans)
haftmann@60430
   923
    moreover have "c dvd lcm (lcm a b) c" by simp
haftmann@60430
   924
    ultimately show "lcm b c dvd lcm (lcm a b) c" by (rule lcm_least)
haftmann@58023
   925
haftmann@60430
   926
    fix l assume "a dvd l" and "lcm b c dvd l"
haftmann@60430
   927
    have "b dvd lcm b c" by simp
wenzelm@60526
   928
    from this and \<open>lcm b c dvd l\<close> have "b dvd l" by (rule dvd_trans)
haftmann@60430
   929
    have "c dvd lcm b c" by simp
wenzelm@60526
   930
    from this and \<open>lcm b c dvd l\<close> have "c dvd l" by (rule dvd_trans)
wenzelm@60526
   931
    from \<open>a dvd l\<close> and \<open>b dvd l\<close> have "lcm a b dvd l" by (rule lcm_least)
wenzelm@60526
   932
    from this and \<open>c dvd l\<close> show "lcm (lcm a b) c dvd l" by (rule lcm_least)
haftmann@58023
   933
  qed (simp add: lcm_zero)
haftmann@58023
   934
next
haftmann@60430
   935
  fix a b
haftmann@60430
   936
  show "lcm a b = lcm b a"
haftmann@58023
   937
    by (simp add: lcm_gcd ac_simps)
haftmann@58023
   938
qed
haftmann@58023
   939
haftmann@58023
   940
lemma dvd_lcm_D1:
haftmann@58023
   941
  "lcm m n dvd k \<Longrightarrow> m dvd k"
haftmann@58023
   942
  by (rule dvd_trans, rule lcm_dvd1, assumption)
haftmann@58023
   943
haftmann@58023
   944
lemma dvd_lcm_D2:
haftmann@58023
   945
  "lcm m n dvd k \<Longrightarrow> n dvd k"
haftmann@58023
   946
  by (rule dvd_trans, rule lcm_dvd2, assumption)
haftmann@58023
   947
haftmann@58023
   948
lemma gcd_dvd_lcm [simp]:
haftmann@58023
   949
  "gcd a b dvd lcm a b"
haftmann@58023
   950
  by (metis dvd_trans gcd_dvd2 lcm_dvd2)
haftmann@58023
   951
haftmann@58023
   952
lemma lcm_1_iff:
haftmann@58023
   953
  "lcm a b = 1 \<longleftrightarrow> is_unit a \<and> is_unit b"
haftmann@58023
   954
proof
haftmann@58023
   955
  assume "lcm a b = 1"
haftmann@59061
   956
  then show "is_unit a \<and> is_unit b" by auto
haftmann@58023
   957
next
haftmann@58023
   958
  assume "is_unit a \<and> is_unit b"
haftmann@59061
   959
  hence "a dvd 1" and "b dvd 1" by simp_all
haftmann@59061
   960
  hence "is_unit (lcm a b)" by (rule lcm_least)
haftmann@60438
   961
  hence "lcm a b = normalization_factor (lcm a b)"
haftmann@60438
   962
    by (subst normalization_factor_unit, simp_all)
wenzelm@60526
   963
  also have "\<dots> = 1" using \<open>is_unit a \<and> is_unit b\<close>
haftmann@59061
   964
    by auto
haftmann@58023
   965
  finally show "lcm a b = 1" .
haftmann@58023
   966
qed
haftmann@58023
   967
haftmann@58023
   968
lemma lcm_0_left [simp]:
haftmann@60430
   969
  "lcm 0 a = 0"
haftmann@58023
   970
  by (rule sym, rule lcmI, simp_all)
haftmann@58023
   971
haftmann@58023
   972
lemma lcm_0 [simp]:
haftmann@60430
   973
  "lcm a 0 = 0"
haftmann@58023
   974
  by (rule sym, rule lcmI, simp_all)
haftmann@58023
   975
haftmann@58023
   976
lemma lcm_unique:
haftmann@58023
   977
  "a dvd d \<and> b dvd d \<and> 
haftmann@60438
   978
  normalization_factor d = (if d = 0 then 0 else 1) \<and>
haftmann@58023
   979
  (\<forall>e. a dvd e \<and> b dvd e \<longrightarrow> d dvd e) \<longleftrightarrow> d = lcm a b"
haftmann@58023
   980
  by (rule, auto intro: lcmI simp: lcm_least lcm_zero)
haftmann@58023
   981
haftmann@58023
   982
lemma dvd_lcm_I1 [simp]:
haftmann@58023
   983
  "k dvd m \<Longrightarrow> k dvd lcm m n"
haftmann@58023
   984
  by (metis lcm_dvd1 dvd_trans)
haftmann@58023
   985
haftmann@58023
   986
lemma dvd_lcm_I2 [simp]:
haftmann@58023
   987
  "k dvd n \<Longrightarrow> k dvd lcm m n"
haftmann@58023
   988
  by (metis lcm_dvd2 dvd_trans)
haftmann@58023
   989
haftmann@58023
   990
lemma lcm_1_left [simp]:
haftmann@60438
   991
  "lcm 1 a = a div normalization_factor a"
haftmann@60430
   992
  by (cases "a = 0") (simp, rule sym, rule lcmI, simp_all)
haftmann@58023
   993
haftmann@58023
   994
lemma lcm_1_right [simp]:
haftmann@60438
   995
  "lcm a 1 = a div normalization_factor a"
haftmann@60430
   996
  using lcm_1_left [of a] by (simp add: ac_simps)
haftmann@58023
   997
haftmann@58023
   998
lemma lcm_coprime:
haftmann@60438
   999
  "gcd a b = 1 \<Longrightarrow> lcm a b = a * b div normalization_factor (a*b)"
haftmann@58023
  1000
  by (subst lcm_gcd) simp
haftmann@58023
  1001
haftmann@58023
  1002
lemma lcm_proj1_if_dvd: 
haftmann@60438
  1003
  "b dvd a \<Longrightarrow> lcm a b = a div normalization_factor a"
haftmann@60430
  1004
  by (cases "a = 0") (simp, rule sym, rule lcmI, simp_all)
haftmann@58023
  1005
haftmann@58023
  1006
lemma lcm_proj2_if_dvd: 
haftmann@60438
  1007
  "a dvd b \<Longrightarrow> lcm a b = b div normalization_factor b"
haftmann@60430
  1008
  using lcm_proj1_if_dvd [of a b] by (simp add: ac_simps)
haftmann@58023
  1009
haftmann@58023
  1010
lemma lcm_proj1_iff:
haftmann@60438
  1011
  "lcm m n = m div normalization_factor m \<longleftrightarrow> n dvd m"
haftmann@58023
  1012
proof
haftmann@60438
  1013
  assume A: "lcm m n = m div normalization_factor m"
haftmann@58023
  1014
  show "n dvd m"
haftmann@58023
  1015
  proof (cases "m = 0")
haftmann@58023
  1016
    assume [simp]: "m \<noteq> 0"
haftmann@60438
  1017
    from A have B: "m = lcm m n * normalization_factor m"
haftmann@58023
  1018
      by (simp add: unit_eq_div2)
haftmann@58023
  1019
    show ?thesis by (subst B, simp)
haftmann@58023
  1020
  qed simp
haftmann@58023
  1021
next
haftmann@58023
  1022
  assume "n dvd m"
haftmann@60438
  1023
  then show "lcm m n = m div normalization_factor m" by (rule lcm_proj1_if_dvd)
haftmann@58023
  1024
qed
haftmann@58023
  1025
haftmann@58023
  1026
lemma lcm_proj2_iff:
haftmann@60438
  1027
  "lcm m n = n div normalization_factor n \<longleftrightarrow> m dvd n"
haftmann@58023
  1028
  using lcm_proj1_iff [of n m] by (simp add: ac_simps)
haftmann@58023
  1029
haftmann@58023
  1030
lemma euclidean_size_lcm_le1: 
haftmann@58023
  1031
  assumes "a \<noteq> 0" and "b \<noteq> 0"
haftmann@58023
  1032
  shows "euclidean_size a \<le> euclidean_size (lcm a b)"
haftmann@58023
  1033
proof -
haftmann@58023
  1034
  have "a dvd lcm a b" by (rule lcm_dvd1)
haftmann@58023
  1035
  then obtain c where A: "lcm a b = a * c" unfolding dvd_def by blast
wenzelm@60526
  1036
  with \<open>a \<noteq> 0\<close> and \<open>b \<noteq> 0\<close> have "c \<noteq> 0" by (auto simp: lcm_zero)
haftmann@58023
  1037
  then show ?thesis by (subst A, intro size_mult_mono)
haftmann@58023
  1038
qed
haftmann@58023
  1039
haftmann@58023
  1040
lemma euclidean_size_lcm_le2:
haftmann@58023
  1041
  "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> euclidean_size b \<le> euclidean_size (lcm a b)"
haftmann@58023
  1042
  using euclidean_size_lcm_le1 [of b a] by (simp add: ac_simps)
haftmann@58023
  1043
haftmann@58023
  1044
lemma euclidean_size_lcm_less1:
haftmann@58023
  1045
  assumes "b \<noteq> 0" and "\<not>b dvd a"
haftmann@58023
  1046
  shows "euclidean_size a < euclidean_size (lcm a b)"
haftmann@58023
  1047
proof (rule ccontr)
haftmann@58023
  1048
  from assms have "a \<noteq> 0" by auto
haftmann@58023
  1049
  assume "\<not>euclidean_size a < euclidean_size (lcm a b)"
wenzelm@60526
  1050
  with \<open>a \<noteq> 0\<close> and \<open>b \<noteq> 0\<close> have "euclidean_size (lcm a b) = euclidean_size a"
haftmann@58023
  1051
    by (intro le_antisym, simp, intro euclidean_size_lcm_le1)
haftmann@58023
  1052
  with assms have "lcm a b dvd a" 
haftmann@58023
  1053
    by (rule_tac dvd_euclidean_size_eq_imp_dvd) (auto simp: lcm_zero)
haftmann@58023
  1054
  hence "b dvd a" by (rule dvd_lcm_D2)
wenzelm@60526
  1055
  with \<open>\<not>b dvd a\<close> show False by contradiction
haftmann@58023
  1056
qed
haftmann@58023
  1057
haftmann@58023
  1058
lemma euclidean_size_lcm_less2:
haftmann@58023
  1059
  assumes "a \<noteq> 0" and "\<not>a dvd b"
haftmann@58023
  1060
  shows "euclidean_size b < euclidean_size (lcm a b)"
haftmann@58023
  1061
  using assms euclidean_size_lcm_less1 [of a b] by (simp add: ac_simps)
haftmann@58023
  1062
haftmann@58023
  1063
lemma lcm_mult_unit1:
haftmann@60430
  1064
  "is_unit a \<Longrightarrow> lcm (b * a) c = lcm b c"
haftmann@58023
  1065
  apply (rule lcmI)
haftmann@60430
  1066
  apply (rule dvd_trans[of _ "b * a"], simp, rule lcm_dvd1)
haftmann@58023
  1067
  apply (rule lcm_dvd2)
haftmann@58023
  1068
  apply (rule lcm_least, simp add: unit_simps, assumption)
haftmann@60438
  1069
  apply (subst normalization_factor_lcm, simp add: lcm_zero)
haftmann@58023
  1070
  done
haftmann@58023
  1071
haftmann@58023
  1072
lemma lcm_mult_unit2:
haftmann@60430
  1073
  "is_unit a \<Longrightarrow> lcm b (c * a) = lcm b c"
haftmann@60430
  1074
  using lcm_mult_unit1 [of a c b] by (simp add: ac_simps)
haftmann@58023
  1075
haftmann@58023
  1076
lemma lcm_div_unit1:
haftmann@60430
  1077
  "is_unit a \<Longrightarrow> lcm (b div a) c = lcm b c"
haftmann@60433
  1078
  by (erule is_unitE [of _ b]) (simp add: lcm_mult_unit1) 
haftmann@58023
  1079
haftmann@58023
  1080
lemma lcm_div_unit2:
haftmann@60430
  1081
  "is_unit a \<Longrightarrow> lcm b (c div a) = lcm b c"
haftmann@60433
  1082
  by (erule is_unitE [of _ c]) (simp add: lcm_mult_unit2)
haftmann@58023
  1083
haftmann@58023
  1084
lemma lcm_left_idem:
haftmann@60430
  1085
  "lcm a (lcm a b) = lcm a b"
haftmann@58023
  1086
  apply (rule lcmI)
haftmann@58023
  1087
  apply simp
haftmann@58023
  1088
  apply (subst lcm.assoc [symmetric], rule lcm_dvd2)
haftmann@58023
  1089
  apply (rule lcm_least, assumption)
haftmann@58023
  1090
  apply (erule (1) lcm_least)
haftmann@58023
  1091
  apply (auto simp: lcm_zero)
haftmann@58023
  1092
  done
haftmann@58023
  1093
haftmann@58023
  1094
lemma lcm_right_idem:
haftmann@60430
  1095
  "lcm (lcm a b) b = lcm a b"
haftmann@58023
  1096
  apply (rule lcmI)
haftmann@58023
  1097
  apply (subst lcm.assoc, rule lcm_dvd1)
haftmann@58023
  1098
  apply (rule lcm_dvd2)
haftmann@58023
  1099
  apply (rule lcm_least, erule (1) lcm_least, assumption)
haftmann@58023
  1100
  apply (auto simp: lcm_zero)
haftmann@58023
  1101
  done
haftmann@58023
  1102
haftmann@58023
  1103
lemma comp_fun_idem_lcm: "comp_fun_idem lcm"
haftmann@58023
  1104
proof
haftmann@58023
  1105
  fix a b show "lcm a \<circ> lcm b = lcm b \<circ> lcm a"
haftmann@58023
  1106
    by (simp add: fun_eq_iff ac_simps)
haftmann@58023
  1107
next
haftmann@58023
  1108
  fix a show "lcm a \<circ> lcm a = lcm a" unfolding o_def
haftmann@58023
  1109
    by (intro ext, simp add: lcm_left_idem)
haftmann@58023
  1110
qed
haftmann@58023
  1111
haftmann@60430
  1112
lemma dvd_Lcm [simp]: "a \<in> A \<Longrightarrow> a dvd Lcm A"
haftmann@60430
  1113
  and Lcm_dvd [simp]: "(\<forall>a\<in>A. a dvd l') \<Longrightarrow> Lcm A dvd l'"
haftmann@60438
  1114
  and normalization_factor_Lcm [simp]: 
haftmann@60438
  1115
          "normalization_factor (Lcm A) = (if Lcm A = 0 then 0 else 1)"
haftmann@58023
  1116
proof -
haftmann@60430
  1117
  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>
haftmann@60438
  1118
    normalization_factor (Lcm A) = (if Lcm A = 0 then 0 else 1)" (is ?thesis)
haftmann@60430
  1119
  proof (cases "\<exists>l. l \<noteq>  0 \<and> (\<forall>a\<in>A. a dvd l)")
haftmann@58023
  1120
    case False
haftmann@58023
  1121
    hence "Lcm A = 0" by (auto simp: Lcm_Lcm_eucl Lcm_eucl_def)
haftmann@58023
  1122
    with False show ?thesis by auto
haftmann@58023
  1123
  next
haftmann@58023
  1124
    case True
haftmann@60430
  1125
    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
haftmann@60430
  1126
    def n \<equiv> "LEAST n. \<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"
haftmann@60430
  1127
    def l \<equiv> "SOME l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"
haftmann@60430
  1128
    have "\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"
haftmann@58023
  1129
      apply (subst n_def)
haftmann@58023
  1130
      apply (rule LeastI[of _ "euclidean_size l\<^sub>0"])
haftmann@58023
  1131
      apply (rule exI[of _ l\<^sub>0])
haftmann@58023
  1132
      apply (simp add: l\<^sub>0_props)
haftmann@58023
  1133
      done
haftmann@60430
  1134
    from someI_ex[OF this] have "l \<noteq> 0" and "\<forall>a\<in>A. a dvd l" and "euclidean_size l = n" 
haftmann@58023
  1135
      unfolding l_def by simp_all
haftmann@58023
  1136
    {
haftmann@60430
  1137
      fix l' assume "\<forall>a\<in>A. a dvd l'"
wenzelm@60526
  1138
      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)
wenzelm@60526
  1139
      moreover from \<open>l \<noteq> 0\<close> have "gcd l l' \<noteq> 0" by simp
haftmann@60430
  1140
      ultimately have "\<exists>b. b \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd b) \<and> euclidean_size b = euclidean_size (gcd l l')"
haftmann@58023
  1141
        by (intro exI[of _ "gcd l l'"], auto)
haftmann@58023
  1142
      hence "euclidean_size (gcd l l') \<ge> n" by (subst n_def) (rule Least_le)
haftmann@58023
  1143
      moreover have "euclidean_size (gcd l l') \<le> n"
haftmann@58023
  1144
      proof -
haftmann@58023
  1145
        have "gcd l l' dvd l" by simp
haftmann@58023
  1146
        then obtain a where "l = gcd l l' * a" unfolding dvd_def by blast
wenzelm@60526
  1147
        with \<open>l \<noteq> 0\<close> have "a \<noteq> 0" by auto
haftmann@58023
  1148
        hence "euclidean_size (gcd l l') \<le> euclidean_size (gcd l l' * a)"
haftmann@58023
  1149
          by (rule size_mult_mono)
wenzelm@60526
  1150
        also have "gcd l l' * a = l" using \<open>l = gcd l l' * a\<close> ..
wenzelm@60526
  1151
        also note \<open>euclidean_size l = n\<close>
haftmann@58023
  1152
        finally show "euclidean_size (gcd l l') \<le> n" .
haftmann@58023
  1153
      qed
haftmann@58023
  1154
      ultimately have "euclidean_size l = euclidean_size (gcd l l')" 
wenzelm@60526
  1155
        by (intro le_antisym, simp_all add: \<open>euclidean_size l = n\<close>)
wenzelm@60526
  1156
      with \<open>l \<noteq> 0\<close> have "l dvd gcd l l'" by (blast intro: dvd_euclidean_size_eq_imp_dvd)
haftmann@58023
  1157
      hence "l dvd l'" by (blast dest: dvd_gcd_D2)
haftmann@58023
  1158
    }
haftmann@58023
  1159
wenzelm@60526
  1160
    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>
haftmann@60438
  1161
      have "(\<forall>a\<in>A. a dvd l div normalization_factor l) \<and> 
haftmann@60438
  1162
        (\<forall>l'. (\<forall>a\<in>A. a dvd l') \<longrightarrow> l div normalization_factor l dvd l') \<and>
haftmann@60438
  1163
        normalization_factor (l div normalization_factor l) = 
haftmann@60438
  1164
        (if l div normalization_factor l = 0 then 0 else 1)"
haftmann@58023
  1165
      by (auto simp: unit_simps)
haftmann@60438
  1166
    also from True have "l div normalization_factor l = Lcm A"
haftmann@58023
  1167
      by (simp add: Lcm_Lcm_eucl Lcm_eucl_def Let_def n_def l_def)
haftmann@58023
  1168
    finally show ?thesis .
haftmann@58023
  1169
  qed
haftmann@58023
  1170
  note A = this
haftmann@58023
  1171
haftmann@60430
  1172
  {fix a assume "a \<in> A" then show "a dvd Lcm A" using A by blast}
haftmann@60430
  1173
  {fix l' assume "\<forall>a\<in>A. a dvd l'" then show "Lcm A dvd l'" using A by blast}
haftmann@60438
  1174
  from A show "normalization_factor (Lcm A) = (if Lcm A = 0 then 0 else 1)" by blast
haftmann@58023
  1175
qed
haftmann@58023
  1176
    
haftmann@58023
  1177
lemma LcmI:
haftmann@60430
  1178
  "(\<And>a. a\<in>A \<Longrightarrow> a dvd l) \<Longrightarrow> (\<And>l'. (\<forall>a\<in>A. a dvd l') \<Longrightarrow> l dvd l') \<Longrightarrow>
haftmann@60438
  1179
      normalization_factor l = (if l = 0 then 0 else 1) \<Longrightarrow> l = Lcm A"
haftmann@58023
  1180
  by (intro normed_associated_imp_eq)
haftmann@58023
  1181
    (auto intro: Lcm_dvd dvd_Lcm simp: associated_def)
haftmann@58023
  1182
haftmann@58023
  1183
lemma Lcm_subset:
haftmann@58023
  1184
  "A \<subseteq> B \<Longrightarrow> Lcm A dvd Lcm B"
haftmann@58023
  1185
  by (blast intro: Lcm_dvd dvd_Lcm)
haftmann@58023
  1186
haftmann@58023
  1187
lemma Lcm_Un:
haftmann@58023
  1188
  "Lcm (A \<union> B) = lcm (Lcm A) (Lcm B)"
haftmann@58023
  1189
  apply (rule lcmI)
haftmann@58023
  1190
  apply (blast intro: Lcm_subset)
haftmann@58023
  1191
  apply (blast intro: Lcm_subset)
haftmann@58023
  1192
  apply (intro Lcm_dvd ballI, elim UnE)
haftmann@58023
  1193
  apply (rule dvd_trans, erule dvd_Lcm, assumption)
haftmann@58023
  1194
  apply (rule dvd_trans, erule dvd_Lcm, assumption)
haftmann@58023
  1195
  apply simp
haftmann@58023
  1196
  done
haftmann@58023
  1197
haftmann@58023
  1198
lemma Lcm_1_iff:
haftmann@60430
  1199
  "Lcm A = 1 \<longleftrightarrow> (\<forall>a\<in>A. is_unit a)"
haftmann@58023
  1200
proof
haftmann@58023
  1201
  assume "Lcm A = 1"
haftmann@60430
  1202
  then show "\<forall>a\<in>A. is_unit a" by auto
haftmann@58023
  1203
qed (rule LcmI [symmetric], auto)
haftmann@58023
  1204
haftmann@58023
  1205
lemma Lcm_no_units:
haftmann@60430
  1206
  "Lcm A = Lcm (A - {a. is_unit a})"
haftmann@58023
  1207
proof -
haftmann@60430
  1208
  have "(A - {a. is_unit a}) \<union> {a\<in>A. is_unit a} = A" by blast
haftmann@60430
  1209
  hence "Lcm A = lcm (Lcm (A - {a. is_unit a})) (Lcm {a\<in>A. is_unit a})"
haftmann@58023
  1210
    by (simp add: Lcm_Un[symmetric])
haftmann@60430
  1211
  also have "Lcm {a\<in>A. is_unit a} = 1" by (simp add: Lcm_1_iff)
haftmann@58023
  1212
  finally show ?thesis by simp
haftmann@58023
  1213
qed
haftmann@58023
  1214
haftmann@58023
  1215
lemma Lcm_empty [simp]:
haftmann@58023
  1216
  "Lcm {} = 1"
haftmann@58023
  1217
  by (simp add: Lcm_1_iff)
haftmann@58023
  1218
haftmann@58023
  1219
lemma Lcm_eq_0 [simp]:
haftmann@58023
  1220
  "0 \<in> A \<Longrightarrow> Lcm A = 0"
haftmann@58023
  1221
  by (drule dvd_Lcm) simp
haftmann@58023
  1222
haftmann@58023
  1223
lemma Lcm0_iff':
haftmann@60430
  1224
  "Lcm A = 0 \<longleftrightarrow> \<not>(\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l))"
haftmann@58023
  1225
proof
haftmann@58023
  1226
  assume "Lcm A = 0"
haftmann@60430
  1227
  show "\<not>(\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l))"
haftmann@58023
  1228
  proof
haftmann@60430
  1229
    assume ex: "\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l)"
haftmann@60430
  1230
    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
haftmann@60430
  1231
    def n \<equiv> "LEAST n. \<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"
haftmann@60430
  1232
    def l \<equiv> "SOME l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"
haftmann@60430
  1233
    have "\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"
haftmann@58023
  1234
      apply (subst n_def)
haftmann@58023
  1235
      apply (rule LeastI[of _ "euclidean_size l\<^sub>0"])
haftmann@58023
  1236
      apply (rule exI[of _ l\<^sub>0])
haftmann@58023
  1237
      apply (simp add: l\<^sub>0_props)
haftmann@58023
  1238
      done
haftmann@58023
  1239
    from someI_ex[OF this] have "l \<noteq> 0" unfolding l_def by simp_all
haftmann@60438
  1240
    hence "l div normalization_factor l \<noteq> 0" by simp
haftmann@60438
  1241
    also from ex have "l div normalization_factor l = Lcm A"
haftmann@58023
  1242
       by (simp only: Lcm_Lcm_eucl Lcm_eucl_def n_def l_def if_True Let_def)
wenzelm@60526
  1243
    finally show False using \<open>Lcm A = 0\<close> by contradiction
haftmann@58023
  1244
  qed
haftmann@58023
  1245
qed (simp only: Lcm_Lcm_eucl Lcm_eucl_def if_False)
haftmann@58023
  1246
haftmann@58023
  1247
lemma Lcm0_iff [simp]:
haftmann@58023
  1248
  "finite A \<Longrightarrow> Lcm A = 0 \<longleftrightarrow> 0 \<in> A"
haftmann@58023
  1249
proof -
haftmann@58023
  1250
  assume "finite A"
haftmann@58023
  1251
  have "0 \<in> A \<Longrightarrow> Lcm A = 0"  by (intro dvd_0_left dvd_Lcm)
haftmann@58023
  1252
  moreover {
haftmann@58023
  1253
    assume "0 \<notin> A"
haftmann@58023
  1254
    hence "\<Prod>A \<noteq> 0" 
wenzelm@60526
  1255
      apply (induct rule: finite_induct[OF \<open>finite A\<close>]) 
haftmann@58023
  1256
      apply simp
haftmann@58023
  1257
      apply (subst setprod.insert, assumption, assumption)
haftmann@58023
  1258
      apply (rule no_zero_divisors)
haftmann@58023
  1259
      apply blast+
haftmann@58023
  1260
      done
wenzelm@60526
  1261
    moreover from \<open>finite A\<close> have "\<forall>a\<in>A. a dvd \<Prod>A" by blast
haftmann@60430
  1262
    ultimately have "\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l)" by blast
haftmann@58023
  1263
    with Lcm0_iff' have "Lcm A \<noteq> 0" by simp
haftmann@58023
  1264
  }
haftmann@58023
  1265
  ultimately show "Lcm A = 0 \<longleftrightarrow> 0 \<in> A" by blast
haftmann@58023
  1266
qed
haftmann@58023
  1267
haftmann@58023
  1268
lemma Lcm_no_multiple:
haftmann@60430
  1269
  "(\<forall>m. m \<noteq> 0 \<longrightarrow> (\<exists>a\<in>A. \<not>a dvd m)) \<Longrightarrow> Lcm A = 0"
haftmann@58023
  1270
proof -
haftmann@60430
  1271
  assume "\<forall>m. m \<noteq> 0 \<longrightarrow> (\<exists>a\<in>A. \<not>a dvd m)"
haftmann@60430
  1272
  hence "\<not>(\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l))" by blast
haftmann@58023
  1273
  then show "Lcm A = 0" by (simp only: Lcm_Lcm_eucl Lcm_eucl_def if_False)
haftmann@58023
  1274
qed
haftmann@58023
  1275
haftmann@58023
  1276
lemma Lcm_insert [simp]:
haftmann@58023
  1277
  "Lcm (insert a A) = lcm a (Lcm A)"
haftmann@58023
  1278
proof (rule lcmI)
haftmann@58023
  1279
  fix l assume "a dvd l" and "Lcm A dvd l"
haftmann@60430
  1280
  hence "\<forall>a\<in>A. a dvd l" by (blast intro: dvd_trans dvd_Lcm)
wenzelm@60526
  1281
  with \<open>a dvd l\<close> show "Lcm (insert a A) dvd l" by (force intro: Lcm_dvd)
haftmann@58023
  1282
qed (auto intro: Lcm_dvd dvd_Lcm)
haftmann@58023
  1283
 
haftmann@58023
  1284
lemma Lcm_finite:
haftmann@58023
  1285
  assumes "finite A"
haftmann@58023
  1286
  shows "Lcm A = Finite_Set.fold lcm 1 A"
wenzelm@60526
  1287
  by (induct rule: finite.induct[OF \<open>finite A\<close>])
haftmann@58023
  1288
    (simp_all add: comp_fun_idem.fold_insert_idem[OF comp_fun_idem_lcm])
haftmann@58023
  1289
haftmann@60431
  1290
lemma Lcm_set [code_unfold]:
haftmann@58023
  1291
  "Lcm (set xs) = fold lcm xs 1"
haftmann@58023
  1292
  using comp_fun_idem.fold_set_fold[OF comp_fun_idem_lcm] Lcm_finite by (simp add: ac_simps)
haftmann@58023
  1293
haftmann@58023
  1294
lemma Lcm_singleton [simp]:
haftmann@60438
  1295
  "Lcm {a} = a div normalization_factor a"
haftmann@58023
  1296
  by simp
haftmann@58023
  1297
haftmann@58023
  1298
lemma Lcm_2 [simp]:
haftmann@58023
  1299
  "Lcm {a,b} = lcm a b"
haftmann@58023
  1300
  by (simp only: Lcm_insert Lcm_empty lcm_1_right)
haftmann@58023
  1301
    (cases "b = 0", simp, rule lcm_div_unit2, simp)
haftmann@58023
  1302
haftmann@58023
  1303
lemma Lcm_coprime:
haftmann@58023
  1304
  assumes "finite A" and "A \<noteq> {}" 
haftmann@58023
  1305
  assumes "\<And>a b. a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<noteq> b \<Longrightarrow> gcd a b = 1"
haftmann@60438
  1306
  shows "Lcm A = \<Prod>A div normalization_factor (\<Prod>A)"
haftmann@58023
  1307
using assms proof (induct rule: finite_ne_induct)
haftmann@58023
  1308
  case (insert a A)
haftmann@58023
  1309
  have "Lcm (insert a A) = lcm a (Lcm A)" by simp
haftmann@60438
  1310
  also from insert have "Lcm A = \<Prod>A div normalization_factor (\<Prod>A)" by blast
haftmann@58023
  1311
  also have "lcm a \<dots> = lcm a (\<Prod>A)" by (cases "\<Prod>A = 0") (simp_all add: lcm_div_unit2)
haftmann@58023
  1312
  also from insert have "gcd a (\<Prod>A) = 1" by (subst gcd.commute, intro setprod_coprime) auto
haftmann@60438
  1313
  with insert have "lcm a (\<Prod>A) = \<Prod>(insert a A) div normalization_factor (\<Prod>(insert a A))"
haftmann@58023
  1314
    by (simp add: lcm_coprime)
haftmann@58023
  1315
  finally show ?case .
haftmann@58023
  1316
qed simp
haftmann@58023
  1317
      
haftmann@58023
  1318
lemma Lcm_coprime':
haftmann@58023
  1319
  "card A \<noteq> 0 \<Longrightarrow> (\<And>a b. a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<noteq> b \<Longrightarrow> gcd a b = 1)
haftmann@60438
  1320
    \<Longrightarrow> Lcm A = \<Prod>A div normalization_factor (\<Prod>A)"
haftmann@58023
  1321
  by (rule Lcm_coprime) (simp_all add: card_eq_0_iff)
haftmann@58023
  1322
haftmann@58023
  1323
lemma Gcd_Lcm:
haftmann@60430
  1324
  "Gcd A = Lcm {d. \<forall>a\<in>A. d dvd a}"
haftmann@58023
  1325
  by (simp add: Gcd_Gcd_eucl Lcm_Lcm_eucl Gcd_eucl_def)
haftmann@58023
  1326
haftmann@60430
  1327
lemma Gcd_dvd [simp]: "a \<in> A \<Longrightarrow> Gcd A dvd a"
haftmann@60430
  1328
  and dvd_Gcd [simp]: "(\<forall>a\<in>A. g' dvd a) \<Longrightarrow> g' dvd Gcd A"
haftmann@60438
  1329
  and normalization_factor_Gcd [simp]: 
haftmann@60438
  1330
    "normalization_factor (Gcd A) = (if Gcd A = 0 then 0 else 1)"
haftmann@58023
  1331
proof -
haftmann@60430
  1332
  fix a assume "a \<in> A"
haftmann@60430
  1333
  hence "Lcm {d. \<forall>a\<in>A. d dvd a} dvd a" by (intro Lcm_dvd) blast
haftmann@60430
  1334
  then show "Gcd A dvd a" by (simp add: Gcd_Lcm)
haftmann@58023
  1335
next
haftmann@60430
  1336
  fix g' assume "\<forall>a\<in>A. g' dvd a"
haftmann@60430
  1337
  hence "g' dvd Lcm {d. \<forall>a\<in>A. d dvd a}" by (intro dvd_Lcm) blast
haftmann@58023
  1338
  then show "g' dvd Gcd A" by (simp add: Gcd_Lcm)
haftmann@58023
  1339
next
haftmann@60438
  1340
  show "normalization_factor (Gcd A) = (if Gcd A = 0 then 0 else 1)"
haftmann@59009
  1341
    by (simp add: Gcd_Lcm)
haftmann@58023
  1342
qed
haftmann@58023
  1343
haftmann@58023
  1344
lemma GcdI:
haftmann@60430
  1345
  "(\<And>a. a\<in>A \<Longrightarrow> l dvd a) \<Longrightarrow> (\<And>l'. (\<forall>a\<in>A. l' dvd a) \<Longrightarrow> l' dvd l) \<Longrightarrow>
haftmann@60438
  1346
    normalization_factor l = (if l = 0 then 0 else 1) \<Longrightarrow> l = Gcd A"
haftmann@58023
  1347
  by (intro normed_associated_imp_eq)
haftmann@58023
  1348
    (auto intro: Gcd_dvd dvd_Gcd simp: associated_def)
haftmann@58023
  1349
haftmann@58023
  1350
lemma Lcm_Gcd:
haftmann@60430
  1351
  "Lcm A = Gcd {m. \<forall>a\<in>A. a dvd m}"
haftmann@58023
  1352
  by (rule LcmI[symmetric]) (auto intro: dvd_Gcd Gcd_dvd)
haftmann@58023
  1353
haftmann@58023
  1354
lemma Gcd_0_iff:
haftmann@58023
  1355
  "Gcd A = 0 \<longleftrightarrow> A \<subseteq> {0}"
haftmann@58023
  1356
  apply (rule iffI)
haftmann@58023
  1357
  apply (rule subsetI, drule Gcd_dvd, simp)
haftmann@58023
  1358
  apply (auto intro: GcdI[symmetric])
haftmann@58023
  1359
  done
haftmann@58023
  1360
haftmann@58023
  1361
lemma Gcd_empty [simp]:
haftmann@58023
  1362
  "Gcd {} = 0"
haftmann@58023
  1363
  by (simp add: Gcd_0_iff)
haftmann@58023
  1364
haftmann@58023
  1365
lemma Gcd_1:
haftmann@58023
  1366
  "1 \<in> A \<Longrightarrow> Gcd A = 1"
haftmann@58023
  1367
  by (intro GcdI[symmetric]) (auto intro: Gcd_dvd dvd_Gcd)
haftmann@58023
  1368
haftmann@58023
  1369
lemma Gcd_insert [simp]:
haftmann@58023
  1370
  "Gcd (insert a A) = gcd a (Gcd A)"
haftmann@58023
  1371
proof (rule gcdI)
haftmann@58023
  1372
  fix l assume "l dvd a" and "l dvd Gcd A"
haftmann@60430
  1373
  hence "\<forall>a\<in>A. l dvd a" by (blast intro: dvd_trans Gcd_dvd)
wenzelm@60526
  1374
  with \<open>l dvd a\<close> show "l dvd Gcd (insert a A)" by (force intro: Gcd_dvd)
haftmann@59009
  1375
qed auto
haftmann@58023
  1376
haftmann@58023
  1377
lemma Gcd_finite:
haftmann@58023
  1378
  assumes "finite A"
haftmann@58023
  1379
  shows "Gcd A = Finite_Set.fold gcd 0 A"
wenzelm@60526
  1380
  by (induct rule: finite.induct[OF \<open>finite A\<close>])
haftmann@58023
  1381
    (simp_all add: comp_fun_idem.fold_insert_idem[OF comp_fun_idem_gcd])
haftmann@58023
  1382
haftmann@60431
  1383
lemma Gcd_set [code_unfold]:
haftmann@58023
  1384
  "Gcd (set xs) = fold gcd xs 0"
haftmann@58023
  1385
  using comp_fun_idem.fold_set_fold[OF comp_fun_idem_gcd] Gcd_finite by (simp add: ac_simps)
haftmann@58023
  1386
haftmann@60438
  1387
lemma Gcd_singleton [simp]: "Gcd {a} = a div normalization_factor a"
haftmann@58023
  1388
  by (simp add: gcd_0)
haftmann@58023
  1389
haftmann@58023
  1390
lemma Gcd_2 [simp]: "Gcd {a,b} = gcd a b"
haftmann@58023
  1391
  by (simp only: Gcd_insert Gcd_empty gcd_0) (cases "b = 0", simp, rule gcd_div_unit2, simp)
haftmann@58023
  1392
haftmann@60439
  1393
subclass semiring_gcd
haftmann@60439
  1394
  by unfold_locales (simp_all add: gcd_greatest_iff)
haftmann@60439
  1395
  
haftmann@58023
  1396
end
haftmann@58023
  1397
wenzelm@60526
  1398
text \<open>
haftmann@58023
  1399
  A Euclidean ring is a Euclidean semiring with additive inverses. It provides a 
haftmann@58023
  1400
  few more lemmas; in particular, Bezout's lemma holds for any Euclidean ring.
wenzelm@60526
  1401
\<close>
haftmann@58023
  1402
haftmann@58023
  1403
class euclidean_ring = euclidean_semiring + idom
haftmann@58023
  1404
haftmann@58023
  1405
class euclidean_ring_gcd = euclidean_semiring_gcd + idom
haftmann@58023
  1406
begin
haftmann@58023
  1407
haftmann@58023
  1408
subclass euclidean_ring ..
haftmann@58023
  1409
haftmann@60439
  1410
subclass ring_gcd ..
haftmann@60439
  1411
haftmann@58023
  1412
lemma gcd_neg1 [simp]:
haftmann@60430
  1413
  "gcd (-a) b = gcd a b"
haftmann@59009
  1414
  by (rule sym, rule gcdI, simp_all add: gcd_greatest)
haftmann@58023
  1415
haftmann@58023
  1416
lemma gcd_neg2 [simp]:
haftmann@60430
  1417
  "gcd a (-b) = gcd a b"
haftmann@59009
  1418
  by (rule sym, rule gcdI, simp_all add: gcd_greatest)
haftmann@58023
  1419
haftmann@58023
  1420
lemma gcd_neg_numeral_1 [simp]:
haftmann@60430
  1421
  "gcd (- numeral n) a = gcd (numeral n) a"
haftmann@58023
  1422
  by (fact gcd_neg1)
haftmann@58023
  1423
haftmann@58023
  1424
lemma gcd_neg_numeral_2 [simp]:
haftmann@60430
  1425
  "gcd a (- numeral n) = gcd a (numeral n)"
haftmann@58023
  1426
  by (fact gcd_neg2)
haftmann@58023
  1427
haftmann@58023
  1428
lemma gcd_diff1: "gcd (m - n) n = gcd m n"
haftmann@58023
  1429
  by (subst diff_conv_add_uminus, subst gcd_neg2[symmetric],  subst gcd_add1, simp)
haftmann@58023
  1430
haftmann@58023
  1431
lemma gcd_diff2: "gcd (n - m) n = gcd m n"
haftmann@58023
  1432
  by (subst gcd_neg1[symmetric], simp only: minus_diff_eq gcd_diff1)
haftmann@58023
  1433
haftmann@58023
  1434
lemma coprime_minus_one [simp]: "gcd (n - 1) n = 1"
haftmann@58023
  1435
proof -
haftmann@58023
  1436
  have "gcd (n - 1) n = gcd n (n - 1)" by (fact gcd.commute)
haftmann@58023
  1437
  also have "\<dots> = gcd ((n - 1) + 1) (n - 1)" by simp
haftmann@58023
  1438
  also have "\<dots> = 1" by (rule coprime_plus_one)
haftmann@58023
  1439
  finally show ?thesis .
haftmann@58023
  1440
qed
haftmann@58023
  1441
haftmann@60430
  1442
lemma lcm_neg1 [simp]: "lcm (-a) b = lcm a b"
haftmann@58023
  1443
  by (rule sym, rule lcmI, simp_all add: lcm_least lcm_zero)
haftmann@58023
  1444
haftmann@60430
  1445
lemma lcm_neg2 [simp]: "lcm a (-b) = lcm a b"
haftmann@58023
  1446
  by (rule sym, rule lcmI, simp_all add: lcm_least lcm_zero)
haftmann@58023
  1447
haftmann@60430
  1448
lemma lcm_neg_numeral_1 [simp]: "lcm (- numeral n) a = lcm (numeral n) a"
haftmann@58023
  1449
  by (fact lcm_neg1)
haftmann@58023
  1450
haftmann@60430
  1451
lemma lcm_neg_numeral_2 [simp]: "lcm a (- numeral n) = lcm a (numeral n)"
haftmann@58023
  1452
  by (fact lcm_neg2)
haftmann@58023
  1453
haftmann@58023
  1454
function euclid_ext :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a" where
haftmann@58023
  1455
  "euclid_ext a b = 
haftmann@58023
  1456
     (if b = 0 then 
haftmann@60438
  1457
        let c = 1 div normalization_factor a in (c, 0, a * c)
haftmann@60569
  1458
      else if b dvd a then
haftmann@60569
  1459
        let c = 1 div normalization_factor b in (0, c, b * c)
haftmann@60569
  1460
      else
haftmann@58023
  1461
        case euclid_ext b (a mod b) of
haftmann@60569
  1462
            (s, t, c) \<Rightarrow> (t, s - t * (a div b), c))"
haftmann@58023
  1463
  by (pat_completeness, simp)
haftmann@60569
  1464
  termination by (relation "measure (euclidean_size \<circ> snd)")
haftmann@60569
  1465
    (simp_all add: mod_size_less)
haftmann@58023
  1466
haftmann@58023
  1467
declare euclid_ext.simps [simp del]
haftmann@58023
  1468
haftmann@58023
  1469
lemma euclid_ext_0: 
haftmann@60438
  1470
  "euclid_ext a 0 = (1 div normalization_factor a, 0, a div normalization_factor a)"
haftmann@60433
  1471
  by (subst euclid_ext.simps) (simp add: Let_def)
haftmann@58023
  1472
haftmann@58023
  1473
lemma euclid_ext_non_0:
haftmann@58023
  1474
  "b \<noteq> 0 \<Longrightarrow> euclid_ext a b = (case euclid_ext b (a mod b) of 
haftmann@60569
  1475
    (s, t, c) \<Rightarrow> (t, s - t * (a div b), c))"
haftmann@60569
  1476
  apply (subst euclid_ext.simps)
haftmann@60569
  1477
  apply (auto simp add: split: if_splits)
haftmann@60569
  1478
  apply (subst euclid_ext.simps)
haftmann@60569
  1479
  apply (auto simp add: split: if_splits)
haftmann@60569
  1480
  done
haftmann@58023
  1481
haftmann@58023
  1482
definition euclid_ext' :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a"
haftmann@58023
  1483
where
haftmann@58023
  1484
  "euclid_ext' a b = (case euclid_ext a b of (s, t, _) \<Rightarrow> (s, t))"
haftmann@58023
  1485
haftmann@58023
  1486
lemma euclid_ext_gcd [simp]:
haftmann@60569
  1487
  "(case euclid_ext a b of (_, _ , t) \<Rightarrow> t) = gcd a b"
haftmann@60569
  1488
  by (induct a b rule: gcd_eucl_induct)
haftmann@60569
  1489
    (simp_all add: euclid_ext_0 gcd_0 euclid_ext_non_0 ac_simps split: prod.split prod.split_asm)
haftmann@58023
  1490
haftmann@58023
  1491
lemma euclid_ext_gcd' [simp]:
haftmann@58023
  1492
  "euclid_ext a b = (r, s, t) \<Longrightarrow> t = gcd a b"
haftmann@58023
  1493
  by (insert euclid_ext_gcd[of a b], drule (1) subst, simp)
haftmann@58023
  1494
haftmann@58023
  1495
lemma euclid_ext_correct:
haftmann@60569
  1496
  "case euclid_ext a b of (s, t, c) \<Rightarrow> s * a + t * b = c"
haftmann@60569
  1497
proof (induct a b rule: gcd_eucl_induct)
haftmann@60569
  1498
  case (zero a) then show ?case
haftmann@60569
  1499
    by (simp add: euclid_ext_0 ac_simps)
haftmann@60569
  1500
next
haftmann@60569
  1501
  case (mod a b)
haftmann@60569
  1502
  obtain s t c where stc: "euclid_ext b (a mod b) = (s,t,c)"
haftmann@60569
  1503
    by (cases "euclid_ext b (a mod b)") blast
haftmann@60569
  1504
  with mod have "c = s * b + t * (a mod b)" by simp
haftmann@60569
  1505
  also have "... = t * ((a div b) * b + a mod b) + (s - t * (a div b)) * b"
haftmann@60569
  1506
    by (simp add: algebra_simps) 
haftmann@60569
  1507
  also have "(a div b) * b + a mod b = a" using mod_div_equality .
haftmann@60569
  1508
  finally show ?case
haftmann@60569
  1509
    by (subst euclid_ext.simps) (simp add: stc mod ac_simps)
haftmann@58023
  1510
qed
haftmann@58023
  1511
haftmann@58023
  1512
lemma euclid_ext'_correct:
haftmann@58023
  1513
  "fst (euclid_ext' a b) * a + snd (euclid_ext' a b) * b = gcd a b"
haftmann@58023
  1514
proof-
haftmann@58023
  1515
  obtain s t c where "euclid_ext a b = (s,t,c)"
haftmann@58023
  1516
    by (cases "euclid_ext a b", blast)
haftmann@58023
  1517
  with euclid_ext_correct[of a b] euclid_ext_gcd[of a b]
haftmann@58023
  1518
    show ?thesis unfolding euclid_ext'_def by simp
haftmann@58023
  1519
qed
haftmann@58023
  1520
haftmann@60430
  1521
lemma bezout: "\<exists>s t. s * a + t * b = gcd a b"
haftmann@58023
  1522
  using euclid_ext'_correct by blast
haftmann@58023
  1523
haftmann@60438
  1524
lemma euclid_ext'_0 [simp]: "euclid_ext' a 0 = (1 div normalization_factor a, 0)" 
haftmann@58023
  1525
  by (simp add: bezw_def euclid_ext'_def euclid_ext_0)
haftmann@58023
  1526
haftmann@60430
  1527
lemma euclid_ext'_non_0: "b \<noteq> 0 \<Longrightarrow> euclid_ext' a b = (snd (euclid_ext' b (a mod b)),
haftmann@60430
  1528
  fst (euclid_ext' b (a mod b)) - snd (euclid_ext' b (a mod b)) * (a div b))"
haftmann@60430
  1529
  by (cases "euclid_ext b (a mod b)") 
haftmann@58023
  1530
    (simp add: euclid_ext'_def euclid_ext_non_0)
haftmann@58023
  1531
  
haftmann@58023
  1532
end
haftmann@58023
  1533
haftmann@58023
  1534
instantiation nat :: euclidean_semiring
haftmann@58023
  1535
begin
haftmann@58023
  1536
haftmann@58023
  1537
definition [simp]:
haftmann@58023
  1538
  "euclidean_size_nat = (id :: nat \<Rightarrow> nat)"
haftmann@58023
  1539
haftmann@58023
  1540
definition [simp]:
haftmann@60438
  1541
  "normalization_factor_nat (n::nat) = (if n = 0 then 0 else 1 :: nat)"
haftmann@58023
  1542
haftmann@58023
  1543
instance proof
haftmann@59061
  1544
qed simp_all
haftmann@58023
  1545
haftmann@58023
  1546
end
haftmann@58023
  1547
haftmann@58023
  1548
instantiation int :: euclidean_ring
haftmann@58023
  1549
begin
haftmann@58023
  1550
haftmann@58023
  1551
definition [simp]:
haftmann@58023
  1552
  "euclidean_size_int = (nat \<circ> abs :: int \<Rightarrow> nat)"
haftmann@58023
  1553
haftmann@58023
  1554
definition [simp]:
haftmann@60438
  1555
  "normalization_factor_int = (sgn :: int \<Rightarrow> int)"
haftmann@58023
  1556
haftmann@58023
  1557
instance proof
haftmann@58023
  1558
  case goal2 then show ?case by (auto simp add: abs_mult nat_mult_distrib)
haftmann@58023
  1559
next
haftmann@59061
  1560
  case goal3 then show ?case by (simp add: zsgn_def)
haftmann@58023
  1561
next
haftmann@59061
  1562
  case goal5 then show ?case by (auto simp: zsgn_def)
haftmann@58023
  1563
next
haftmann@59061
  1564
  case goal6 then show ?case by (auto split: abs_split simp: zsgn_def)
haftmann@58023
  1565
qed (auto simp: sgn_times split: abs_split)
haftmann@58023
  1566
haftmann@58023
  1567
end
haftmann@58023
  1568
haftmann@60571
  1569
instantiation poly :: (field) euclidean_semiring
haftmann@60571
  1570
begin
haftmann@60571
  1571
haftmann@60571
  1572
definition euclidean_size_poly :: "'a poly \<Rightarrow> nat"
haftmann@60571
  1573
  where "euclidean_size = (degree :: 'a poly \<Rightarrow> nat)"
haftmann@60571
  1574
haftmann@60571
  1575
definition normalization_factor_poly :: "'a poly \<Rightarrow> 'a poly"
haftmann@60571
  1576
  where "normalization_factor p = monom (coeff p (degree p)) 0"
haftmann@60571
  1577
haftmann@60571
  1578
instance
haftmann@60571
  1579
proof (default, unfold euclidean_size_poly_def normalization_factor_poly_def)
haftmann@60571
  1580
  fix p q :: "'a poly"
haftmann@60571
  1581
  assume "q \<noteq> 0" and "\<not> q dvd p"
haftmann@60571
  1582
  then show "degree (p mod q) < degree q"
haftmann@60571
  1583
    using degree_mod_less [of q p] by (simp add: mod_eq_0_iff_dvd)
haftmann@60571
  1584
next
haftmann@60571
  1585
  fix p q :: "'a poly"
haftmann@60571
  1586
  assume "q \<noteq> 0"
haftmann@60571
  1587
  from \<open>q \<noteq> 0\<close> show "degree p \<le> degree (p * q)"
haftmann@60571
  1588
    by (rule degree_mult_right_le)
haftmann@60571
  1589
  from \<open>q \<noteq> 0\<close> show "is_unit (monom (coeff q (degree q)) 0)"
haftmann@60571
  1590
    by (auto intro: is_unit_monom_0)
haftmann@60571
  1591
next
haftmann@60571
  1592
  fix p :: "'a poly"
haftmann@60571
  1593
  show "monom (coeff p (degree p)) 0 = p" if "is_unit p"
haftmann@60571
  1594
    using that by (fact is_unit_monom_trival)
haftmann@60571
  1595
next
haftmann@60571
  1596
  fix p q :: "'a poly"
haftmann@60571
  1597
  show "monom (coeff (p * q) (degree (p * q))) 0 =
haftmann@60571
  1598
    monom (coeff p (degree p)) 0 * monom (coeff q (degree q)) 0"
haftmann@60571
  1599
    by (simp add: monom_0 coeff_degree_mult)
haftmann@60571
  1600
next
haftmann@60571
  1601
  show "monom (coeff 0 (degree 0)) 0 = 0"
haftmann@60571
  1602
    by simp
haftmann@60571
  1603
qed
haftmann@60571
  1604
haftmann@58023
  1605
end
haftmann@60571
  1606
haftmann@60571
  1607
end