src/HOL/Number_Theory/Euclidean_Algorithm.thy
author haftmann
Sat Dec 17 15:22:14 2016 +0100 (2016-12-17)
changeset 64592 7759f1766189
parent 64243 aee949f6642d
child 64784 5cb5e7ecb284
permissions -rw-r--r--
more fine-grained type class hierarchy for div and mod
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(* Author: Manuel Eberl *)
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section \<open>Abstract euclidean algorithm\<close>
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theory Euclidean_Algorithm
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imports "~~/src/HOL/GCD" Factorial_Ring
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begin
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text \<open>
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  A Euclidean semiring is a semiring upon which the Euclidean algorithm can be
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  implemented. It must provide:
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  \begin{itemize}
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  \item division with remainder
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  \item a size function such that @{term "size (a mod b) < size b"} 
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        for any @{term "b \<noteq> 0"}
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  \end{itemize}
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  The existence of these functions makes it possible to derive gcd and lcm functions 
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  for any Euclidean semiring.
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\<close> 
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class euclidean_semiring = semidom_modulo + normalization_semidom + 
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  fixes euclidean_size :: "'a \<Rightarrow> nat"
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  assumes size_0 [simp]: "euclidean_size 0 = 0"
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  assumes mod_size_less: 
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    "b \<noteq> 0 \<Longrightarrow> euclidean_size (a mod b) < euclidean_size b"
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  assumes size_mult_mono:
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    "b \<noteq> 0 \<Longrightarrow> euclidean_size a \<le> euclidean_size (a * b)"
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begin
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lemma euclidean_size_normalize [simp]:
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  "euclidean_size (normalize a) = euclidean_size a"
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proof (cases "a = 0")
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  case True
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  then show ?thesis
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    by simp
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next
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  case [simp]: False
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  have "euclidean_size (normalize a) \<le> euclidean_size (normalize a * unit_factor a)"
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    by (rule size_mult_mono) simp
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  moreover have "euclidean_size a \<le> euclidean_size (a * (1 div unit_factor a))"
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    by (rule size_mult_mono) simp
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  ultimately show ?thesis
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    by simp
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qed
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lemma euclidean_division:
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  fixes a :: 'a and b :: 'a
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  assumes "b \<noteq> 0"
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  obtains s and t where "a = s * b + t" 
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    and "euclidean_size t < euclidean_size b"
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proof -
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  from div_mult_mod_eq [of a b] 
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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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  hence "b mod a \<noteq> 0" using mod_0_imp_dvd[of b a] by blast
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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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lemma size_mult_mono': "b \<noteq> 0 \<Longrightarrow> euclidean_size a \<le> euclidean_size (b * a)"
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  by (subst mult.commute) (rule size_mult_mono)
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lemma euclidean_size_times_unit:
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  assumes "is_unit a"
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  shows   "euclidean_size (a * b) = euclidean_size b"
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proof (rule antisym)
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  from assms have [simp]: "a \<noteq> 0" by auto
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  thus "euclidean_size (a * b) \<ge> euclidean_size b" by (rule size_mult_mono')
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  from assms have "is_unit (1 div a)" by simp
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  hence "1 div a \<noteq> 0" by (intro notI) simp_all
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  hence "euclidean_size (a * b) \<le> euclidean_size ((1 div a) * (a * b))"
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    by (rule size_mult_mono')
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  also from assms have "(1 div a) * (a * b) = b"
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    by (simp add: algebra_simps unit_div_mult_swap)
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  finally show "euclidean_size (a * b) \<le> euclidean_size b" .
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qed
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lemma euclidean_size_unit: "is_unit a \<Longrightarrow> euclidean_size a = euclidean_size 1"
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  using euclidean_size_times_unit[of a 1] by simp
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lemma unit_iff_euclidean_size: 
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  "is_unit a \<longleftrightarrow> euclidean_size a = euclidean_size 1 \<and> a \<noteq> 0"
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proof safe
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  assume A: "a \<noteq> 0" and B: "euclidean_size a = euclidean_size 1"
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  show "is_unit a" by (rule dvd_euclidean_size_eq_imp_dvd[OF A _ B]) simp_all
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qed (auto intro: euclidean_size_unit)
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lemma euclidean_size_times_nonunit:
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  assumes "a \<noteq> 0" "b \<noteq> 0" "\<not>is_unit a"
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  shows   "euclidean_size b < euclidean_size (a * b)"
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proof (rule ccontr)
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  assume "\<not>euclidean_size b < euclidean_size (a * b)"
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  with size_mult_mono'[OF assms(1), of b] 
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    have eq: "euclidean_size (a * b) = euclidean_size b" by simp
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  have "a * b dvd b"
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    by (rule dvd_euclidean_size_eq_imp_dvd[OF _ _ eq]) (insert assms, simp_all)
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  hence "a * b dvd 1 * b" by simp
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  with \<open>b \<noteq> 0\<close> have "is_unit a" by (subst (asm) dvd_times_right_cancel_iff)
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  with assms(3) show False by contradiction
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qed
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lemma dvd_imp_size_le:
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  assumes "a dvd b" "b \<noteq> 0" 
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  shows   "euclidean_size a \<le> euclidean_size b"
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  using assms by (auto elim!: dvdE simp: size_mult_mono)
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lemma dvd_proper_imp_size_less:
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  assumes "a dvd b" "\<not>b dvd a" "b \<noteq> 0" 
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  shows   "euclidean_size a < euclidean_size b"
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proof -
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  from assms(1) obtain c where "b = a * c" by (erule dvdE)
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  hence z: "b = c * a" by (simp add: mult.commute)
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  from z assms have "\<not>is_unit c" by (auto simp: mult.commute mult_unit_dvd_iff)
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  with z assms show ?thesis
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    by (auto intro!: euclidean_size_times_nonunit simp: )
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qed
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function gcd_eucl :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
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where
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  "gcd_eucl a b = (if b = 0 then normalize a else gcd_eucl b (a mod b))"
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  by pat_completeness simp
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termination
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  by (relation "measure (euclidean_size \<circ> snd)") (simp_all add: mod_size_less)
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declare gcd_eucl.simps [simp del]
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lemma gcd_eucl_induct [case_names zero mod]:
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  assumes H1: "\<And>b. P b 0"
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  and H2: "\<And>a b. b \<noteq> 0 \<Longrightarrow> P b (a mod b) \<Longrightarrow> P a b"
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  shows "P a b"
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proof (induct a b rule: gcd_eucl.induct)
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  case ("1" a b)
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  show ?case
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  proof (cases "b = 0")
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    case True then show "P a b" by simp (rule H1)
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  next
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    case False
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    then have "P b (a mod b)"
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      by (rule "1.hyps")
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    with \<open>b \<noteq> 0\<close> show "P a b"
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      by (blast intro: H2)
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  qed
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qed
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definition lcm_eucl :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
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where
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  "lcm_eucl a b = normalize (a * b) div gcd_eucl a b"
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definition Lcm_eucl :: "'a set \<Rightarrow> 'a" \<comment> \<open>
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  Somewhat complicated definition of Lcm that has the advantage of working
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  for infinite sets as well\<close>
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where
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  "Lcm_eucl A = (if \<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) then
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     let l = SOME l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l =
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       (LEAST n. \<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n)
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       in normalize l 
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      else 0)"
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definition Gcd_eucl :: "'a set \<Rightarrow> 'a"
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where
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  "Gcd_eucl A = Lcm_eucl {d. \<forall>a\<in>A. d dvd a}"
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declare Lcm_eucl_def Gcd_eucl_def [code del]
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lemma gcd_eucl_0:
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  "gcd_eucl a 0 = normalize a"
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  by (simp add: gcd_eucl.simps [of a 0])
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lemma gcd_eucl_0_left:
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  "gcd_eucl 0 a = normalize a"
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  by (simp_all add: gcd_eucl_0 gcd_eucl.simps [of 0 a])
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lemma gcd_eucl_non_0:
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  "b \<noteq> 0 \<Longrightarrow> gcd_eucl a b = gcd_eucl b (a mod b)"
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  by (simp add: gcd_eucl.simps [of a b] gcd_eucl.simps [of b 0])
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lemma gcd_eucl_dvd1 [iff]: "gcd_eucl a b dvd a"
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  and gcd_eucl_dvd2 [iff]: "gcd_eucl a b dvd b"
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  by (induct a b rule: gcd_eucl_induct)
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     (simp_all add: gcd_eucl_0 gcd_eucl_non_0 dvd_mod_iff)
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lemma normalize_gcd_eucl [simp]:
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  "normalize (gcd_eucl a b) = gcd_eucl a b"
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  by (induct a b rule: gcd_eucl_induct) (simp_all add: gcd_eucl_0 gcd_eucl_non_0)
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lemma gcd_eucl_greatest:
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  fixes k a b :: 'a
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  shows "k dvd a \<Longrightarrow> k dvd b \<Longrightarrow> k dvd gcd_eucl a b"
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proof (induct a b rule: gcd_eucl_induct)
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  case (zero a) from zero(1) show ?case by (rule dvd_trans) (simp add: gcd_eucl_0)
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next
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  case (mod a b)
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  then show ?case
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    by (simp add: gcd_eucl_non_0 dvd_mod_iff)
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qed
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lemma gcd_euclI:
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  fixes gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
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  assumes "d dvd a" "d dvd b" "normalize d = d"
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          "\<And>k. k dvd a \<Longrightarrow> k dvd b \<Longrightarrow> k dvd d"
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  shows   "gcd_eucl a b = d"
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  by (rule associated_eqI) (simp_all add: gcd_eucl_greatest assms)
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lemma eq_gcd_euclI:
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  fixes gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
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  assumes "\<And>a b. gcd a b dvd a" "\<And>a b. gcd a b dvd b" "\<And>a b. normalize (gcd a b) = gcd a b"
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          "\<And>a b k. k dvd a \<Longrightarrow> k dvd b \<Longrightarrow> k dvd gcd a b"
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  shows   "gcd = gcd_eucl"
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  by (intro ext, rule associated_eqI) (simp_all add: gcd_eucl_greatest assms)
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lemma gcd_eucl_zero [simp]:
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  "gcd_eucl a b = 0 \<longleftrightarrow> a = 0 \<and> b = 0"
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  by (metis dvd_0_left dvd_refl gcd_eucl_dvd1 gcd_eucl_dvd2 gcd_eucl_greatest)+
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lemma dvd_Lcm_eucl [simp]: "a \<in> A \<Longrightarrow> a dvd Lcm_eucl A"
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  and Lcm_eucl_least: "(\<And>a. a \<in> A \<Longrightarrow> a dvd b) \<Longrightarrow> Lcm_eucl A dvd b"
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  and unit_factor_Lcm_eucl [simp]: 
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          "unit_factor (Lcm_eucl A) = (if Lcm_eucl A = 0 then 0 else 1)"
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proof -
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  have "(\<forall>a\<in>A. a dvd Lcm_eucl A) \<and> (\<forall>l'. (\<forall>a\<in>A. a dvd l') \<longrightarrow> Lcm_eucl A dvd l') \<and>
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    unit_factor (Lcm_eucl A) = (if Lcm_eucl A = 0 then 0 else 1)" (is ?thesis)
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  proof (cases "\<exists>l. l \<noteq>  0 \<and> (\<forall>a\<in>A. a dvd l)")
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    case False
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    hence "Lcm_eucl A = 0" by (auto simp: Lcm_eucl_def)
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    with False show ?thesis by auto
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  next
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    case True
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    then obtain l\<^sub>0 where l\<^sub>0_props: "l\<^sub>0 \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l\<^sub>0)" by blast
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    define n where "n = (LEAST n. \<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n)"
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    define l where "l = (SOME l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n)"
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    have "\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"
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      apply (subst n_def)
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      apply (rule LeastI[of _ "euclidean_size l\<^sub>0"])
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      apply (rule exI[of _ l\<^sub>0])
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      apply (simp add: l\<^sub>0_props)
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      done
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    from someI_ex[OF this] have "l \<noteq> 0" and "\<forall>a\<in>A. a dvd l" and "euclidean_size l = n" 
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      unfolding l_def by simp_all
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    {
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      fix l' assume "\<forall>a\<in>A. a dvd l'"
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      with \<open>\<forall>a\<in>A. a dvd l\<close> have "\<forall>a\<in>A. a dvd gcd_eucl l l'" by (auto intro: gcd_eucl_greatest)
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      moreover from \<open>l \<noteq> 0\<close> have "gcd_eucl l l' \<noteq> 0" by simp
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      ultimately have "\<exists>b. b \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd b) \<and> 
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                          euclidean_size b = euclidean_size (gcd_eucl l l')"
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        by (intro exI[of _ "gcd_eucl l l'"], auto)
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      hence "euclidean_size (gcd_eucl l l') \<ge> n" by (subst n_def) (rule Least_le)
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      moreover have "euclidean_size (gcd_eucl l l') \<le> n"
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      proof -
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        have "gcd_eucl l l' dvd l" by simp
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        then obtain a where "l = gcd_eucl l l' * a" unfolding dvd_def by blast
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        with \<open>l \<noteq> 0\<close> have "a \<noteq> 0" by auto
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        hence "euclidean_size (gcd_eucl l l') \<le> euclidean_size (gcd_eucl l l' * a)"
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          by (rule size_mult_mono)
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        also have "gcd_eucl l l' * a = l" using \<open>l = gcd_eucl l l' * a\<close> ..
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        also note \<open>euclidean_size l = n\<close>
eberlm@62422
   271
        finally show "euclidean_size (gcd_eucl l l') \<le> n" .
eberlm@62422
   272
      qed
eberlm@62422
   273
      ultimately have *: "euclidean_size l = euclidean_size (gcd_eucl l l')" 
eberlm@62422
   274
        by (intro le_antisym, simp_all add: \<open>euclidean_size l = n\<close>)
eberlm@62422
   275
      from \<open>l \<noteq> 0\<close> have "l dvd gcd_eucl l l'"
eberlm@62422
   276
        by (rule dvd_euclidean_size_eq_imp_dvd) (auto simp add: *)
eberlm@62422
   277
      hence "l dvd l'" by (rule dvd_trans[OF _ gcd_eucl_dvd2])
eberlm@62422
   278
    }
eberlm@62422
   279
eberlm@62422
   280
    with \<open>(\<forall>a\<in>A. a dvd l)\<close> and unit_factor_is_unit[OF \<open>l \<noteq> 0\<close>] and \<open>l \<noteq> 0\<close>
eberlm@62422
   281
      have "(\<forall>a\<in>A. a dvd normalize l) \<and> 
eberlm@62422
   282
        (\<forall>l'. (\<forall>a\<in>A. a dvd l') \<longrightarrow> normalize l dvd l') \<and>
eberlm@62422
   283
        unit_factor (normalize l) = 
eberlm@62422
   284
        (if normalize l = 0 then 0 else 1)"
eberlm@62422
   285
      by (auto simp: unit_simps)
eberlm@62422
   286
    also from True have "normalize l = Lcm_eucl A"
eberlm@62422
   287
      by (simp add: Lcm_eucl_def Let_def n_def l_def)
eberlm@62422
   288
    finally show ?thesis .
eberlm@62422
   289
  qed
eberlm@62422
   290
  note A = this
eberlm@62422
   291
eberlm@62422
   292
  {fix a assume "a \<in> A" then show "a dvd Lcm_eucl A" using A by blast}
eberlm@62422
   293
  {fix b assume "\<And>a. a \<in> A \<Longrightarrow> a dvd b" then show "Lcm_eucl A dvd b" using A by blast}
eberlm@62422
   294
  from A show "unit_factor (Lcm_eucl A) = (if Lcm_eucl A = 0 then 0 else 1)" by blast
eberlm@62422
   295
qed
eberlm@63498
   296
eberlm@62422
   297
lemma normalize_Lcm_eucl [simp]:
eberlm@62422
   298
  "normalize (Lcm_eucl A) = Lcm_eucl A"
eberlm@62422
   299
proof (cases "Lcm_eucl A = 0")
eberlm@62422
   300
  case True then show ?thesis by simp
eberlm@62422
   301
next
eberlm@62422
   302
  case False
eberlm@62422
   303
  have "unit_factor (Lcm_eucl A) * normalize (Lcm_eucl A) = Lcm_eucl A"
eberlm@62422
   304
    by (fact unit_factor_mult_normalize)
eberlm@62422
   305
  with False show ?thesis by simp
eberlm@62422
   306
qed
eberlm@62422
   307
eberlm@62422
   308
lemma eq_Lcm_euclI:
eberlm@62422
   309
  fixes lcm :: "'a set \<Rightarrow> 'a"
eberlm@62422
   310
  assumes "\<And>A a. a \<in> A \<Longrightarrow> a dvd lcm A" and "\<And>A c. (\<And>a. a \<in> A \<Longrightarrow> a dvd c) \<Longrightarrow> lcm A dvd c"
eberlm@62422
   311
          "\<And>A. normalize (lcm A) = lcm A" shows "lcm = Lcm_eucl"
eberlm@62422
   312
  by (intro ext, rule associated_eqI) (auto simp: assms intro: Lcm_eucl_least)  
eberlm@62422
   313
haftmann@64177
   314
lemma Gcd_eucl_dvd: "a \<in> A \<Longrightarrow> Gcd_eucl A dvd a"
eberlm@63498
   315
  unfolding Gcd_eucl_def by (auto intro: Lcm_eucl_least)
eberlm@63498
   316
eberlm@63498
   317
lemma Gcd_eucl_greatest: "(\<And>x. x \<in> A \<Longrightarrow> d dvd x) \<Longrightarrow> d dvd Gcd_eucl A"
eberlm@63498
   318
  unfolding Gcd_eucl_def by auto
eberlm@63498
   319
eberlm@63498
   320
lemma normalize_Gcd_eucl [simp]: "normalize (Gcd_eucl A) = Gcd_eucl A"
eberlm@63498
   321
  by (simp add: Gcd_eucl_def)
eberlm@63498
   322
eberlm@63498
   323
lemma Lcm_euclI:
eberlm@63498
   324
  assumes "\<And>x. x \<in> A \<Longrightarrow> x dvd d" "\<And>d'. (\<And>x. x \<in> A \<Longrightarrow> x dvd d') \<Longrightarrow> d dvd d'" "normalize d = d"
eberlm@63498
   325
  shows   "Lcm_eucl A = d"
eberlm@63498
   326
proof -
eberlm@63498
   327
  have "normalize (Lcm_eucl A) = normalize d"
eberlm@63498
   328
    by (intro associatedI) (auto intro: dvd_Lcm_eucl Lcm_eucl_least assms)
eberlm@63498
   329
  thus ?thesis by (simp add: assms)
eberlm@63498
   330
qed
eberlm@63498
   331
eberlm@63498
   332
lemma Gcd_euclI:
eberlm@63498
   333
  assumes "\<And>x. x \<in> A \<Longrightarrow> d dvd x" "\<And>d'. (\<And>x. x \<in> A \<Longrightarrow> d' dvd x) \<Longrightarrow> d' dvd d" "normalize d = d"
eberlm@63498
   334
  shows   "Gcd_eucl A = d"
eberlm@63498
   335
proof -
eberlm@63498
   336
  have "normalize (Gcd_eucl A) = normalize d"
eberlm@63498
   337
    by (intro associatedI) (auto intro: Gcd_eucl_dvd Gcd_eucl_greatest assms)
eberlm@63498
   338
  thus ?thesis by (simp add: assms)
eberlm@63498
   339
qed
eberlm@63498
   340
  
eberlm@63498
   341
lemmas lcm_gcd_eucl_facts = 
eberlm@63498
   342
  gcd_eucl_dvd1 gcd_eucl_dvd2 gcd_eucl_greatest normalize_gcd_eucl lcm_eucl_def
eberlm@63498
   343
  Gcd_eucl_def Gcd_eucl_dvd Gcd_eucl_greatest normalize_Gcd_eucl
eberlm@63498
   344
  dvd_Lcm_eucl Lcm_eucl_least normalize_Lcm_eucl
eberlm@63498
   345
eberlm@63498
   346
lemma normalized_factors_product:
eberlm@63498
   347
  "{p. p dvd a * b \<and> normalize p = p} = 
eberlm@63498
   348
     (\<lambda>(x,y). x * y) ` ({p. p dvd a \<and> normalize p = p} \<times> {p. p dvd b \<and> normalize p = p})"
eberlm@63498
   349
proof safe
eberlm@63498
   350
  fix p assume p: "p dvd a * b" "normalize p = p"
eberlm@63498
   351
  interpret semiring_gcd 1 0 "op *" gcd_eucl lcm_eucl "op div" "op +" "op -" normalize unit_factor
eberlm@63498
   352
    by standard (rule lcm_gcd_eucl_facts; assumption)+
eberlm@63498
   353
  from dvd_productE[OF p(1)] guess x y . note xy = this
eberlm@63498
   354
  define x' y' where "x' = normalize x" and "y' = normalize y"
eberlm@63498
   355
  have "p = x' * y'"
eberlm@63498
   356
    by (subst p(2) [symmetric]) (simp add: xy x'_def y'_def normalize_mult)
eberlm@63498
   357
  moreover from xy have "normalize x' = x'" "normalize y' = y'" "x' dvd a" "y' dvd b" 
eberlm@63498
   358
    by (simp_all add: x'_def y'_def)
eberlm@63498
   359
  ultimately show "p \<in> (\<lambda>(x, y). x * y) ` 
eberlm@63498
   360
                     ({p. p dvd a \<and> normalize p = p} \<times> {p. p dvd b \<and> normalize p = p})"
eberlm@63498
   361
    by blast
eberlm@63498
   362
qed (auto simp: normalize_mult mult_dvd_mono)
eberlm@63498
   363
eberlm@63498
   364
eberlm@63498
   365
subclass factorial_semiring
eberlm@63498
   366
proof (standard, rule factorial_semiring_altI_aux)
eberlm@63498
   367
  fix x assume "x \<noteq> 0"
eberlm@63498
   368
  thus "finite {p. p dvd x \<and> normalize p = p}"
eberlm@63498
   369
  proof (induction "euclidean_size x" arbitrary: x rule: less_induct)
eberlm@63498
   370
    case (less x)
eberlm@63498
   371
    show ?case
eberlm@63498
   372
    proof (cases "\<exists>y. y dvd x \<and> \<not>x dvd y \<and> \<not>is_unit y")
eberlm@63498
   373
      case False
eberlm@63498
   374
      have "{p. p dvd x \<and> normalize p = p} \<subseteq> {1, normalize x}"
eberlm@63498
   375
      proof
eberlm@63498
   376
        fix p assume p: "p \<in> {p. p dvd x \<and> normalize p = p}"
eberlm@63498
   377
        with False have "is_unit p \<or> x dvd p" by blast
eberlm@63498
   378
        thus "p \<in> {1, normalize x}"
eberlm@63498
   379
        proof (elim disjE)
eberlm@63498
   380
          assume "is_unit p"
eberlm@63498
   381
          hence "normalize p = 1" by (simp add: is_unit_normalize)
eberlm@63498
   382
          with p show ?thesis by simp
eberlm@63498
   383
        next
eberlm@63498
   384
          assume "x dvd p"
eberlm@63498
   385
          with p have "normalize p = normalize x" by (intro associatedI) simp_all
eberlm@63498
   386
          with p show ?thesis by simp
eberlm@63498
   387
        qed
eberlm@63498
   388
      qed
eberlm@63498
   389
      moreover have "finite \<dots>" by simp
eberlm@63498
   390
      ultimately show ?thesis by (rule finite_subset)
eberlm@63498
   391
      
eberlm@63498
   392
    next
eberlm@63498
   393
      case True
eberlm@63498
   394
      then obtain y where y: "y dvd x" "\<not>x dvd y" "\<not>is_unit y" by blast
eberlm@63498
   395
      define z where "z = x div y"
eberlm@63498
   396
      let ?fctrs = "\<lambda>x. {p. p dvd x \<and> normalize p = p}"
eberlm@63498
   397
      from y have x: "x = y * z" by (simp add: z_def)
eberlm@63498
   398
      with less.prems have "y \<noteq> 0" "z \<noteq> 0" by auto
eberlm@63498
   399
      from x y have "\<not>is_unit z" by (auto simp: mult_unit_dvd_iff)
eberlm@63498
   400
      have "?fctrs x = (\<lambda>(p,p'). p * p') ` (?fctrs y \<times> ?fctrs z)"
eberlm@63498
   401
        by (subst x) (rule normalized_factors_product)
eberlm@63498
   402
      also have "\<not>y * z dvd y * 1" "\<not>y * z dvd 1 * z"
eberlm@63498
   403
        by (subst dvd_times_left_cancel_iff dvd_times_right_cancel_iff; fact)+
eberlm@63498
   404
      hence "finite ((\<lambda>(p,p'). p * p') ` (?fctrs y \<times> ?fctrs z))"
eberlm@63498
   405
        by (intro finite_imageI finite_cartesian_product less dvd_proper_imp_size_less)
eberlm@63498
   406
           (auto simp: x)
eberlm@63498
   407
      finally show ?thesis .
eberlm@63498
   408
    qed
eberlm@63498
   409
  qed
eberlm@63498
   410
next
eberlm@63498
   411
  interpret semiring_gcd 1 0 "op *" gcd_eucl lcm_eucl "op div" "op +" "op -" normalize unit_factor
eberlm@63498
   412
    by standard (rule lcm_gcd_eucl_facts; assumption)+
eberlm@63498
   413
  fix p assume p: "irreducible p"
eberlm@63633
   414
  thus "prime_elem p" by (rule irreducible_imp_prime_elem_gcd)
eberlm@63498
   415
qed
eberlm@63498
   416
eberlm@63498
   417
lemma gcd_eucl_eq_gcd_factorial: "gcd_eucl = gcd_factorial"
eberlm@63498
   418
  by (intro ext gcd_euclI gcd_lcm_factorial)
eberlm@63498
   419
eberlm@63498
   420
lemma lcm_eucl_eq_lcm_factorial: "lcm_eucl = lcm_factorial"
eberlm@63498
   421
  by (intro ext) (simp add: lcm_eucl_def lcm_factorial_gcd_factorial gcd_eucl_eq_gcd_factorial)
eberlm@63498
   422
eberlm@63498
   423
lemma Gcd_eucl_eq_Gcd_factorial: "Gcd_eucl = Gcd_factorial"
eberlm@63498
   424
  by (intro ext Gcd_euclI gcd_lcm_factorial)
eberlm@63498
   425
eberlm@63498
   426
lemma Lcm_eucl_eq_Lcm_factorial: "Lcm_eucl = Lcm_factorial"
eberlm@63498
   427
  by (intro ext Lcm_euclI gcd_lcm_factorial)
eberlm@63498
   428
eberlm@63498
   429
lemmas eucl_eq_factorial = 
eberlm@63498
   430
  gcd_eucl_eq_gcd_factorial lcm_eucl_eq_lcm_factorial 
eberlm@63498
   431
  Gcd_eucl_eq_Gcd_factorial Lcm_eucl_eq_Lcm_factorial
eberlm@63498
   432
  
haftmann@58023
   433
end
haftmann@58023
   434
haftmann@60598
   435
class euclidean_ring = euclidean_semiring + idom
haftmann@60598
   436
begin
haftmann@60598
   437
eberlm@62442
   438
function euclid_ext_aux :: "'a \<Rightarrow> _" where
eberlm@62442
   439
  "euclid_ext_aux r' r s' s t' t = (
eberlm@62442
   440
     if r = 0 then let c = 1 div unit_factor r' in (s' * c, t' * c, normalize r')
eberlm@62442
   441
     else let q = r' div r
eberlm@62442
   442
          in  euclid_ext_aux r (r' mod r) s (s' - q * s) t (t' - q * t))"
eberlm@62442
   443
by auto
eberlm@62442
   444
termination by (relation "measure (\<lambda>(_,b,_,_,_,_). euclidean_size b)") (simp_all add: mod_size_less)
eberlm@62442
   445
eberlm@62442
   446
declare euclid_ext_aux.simps [simp del]
haftmann@60598
   447
eberlm@62442
   448
lemma euclid_ext_aux_correct:
haftmann@64177
   449
  assumes "gcd_eucl r' r = gcd_eucl a b"
haftmann@64177
   450
  assumes "s' * a + t' * b = r'"
haftmann@64177
   451
  assumes "s * a + t * b = r"
haftmann@64177
   452
  shows   "case euclid_ext_aux r' r s' s t' t of (x,y,c) \<Rightarrow>
haftmann@64177
   453
             x * a + y * b = c \<and> c = gcd_eucl a b" (is "?P (euclid_ext_aux r' r s' s t' t)")
eberlm@62442
   454
using assms
eberlm@62442
   455
proof (induction r' r s' s t' t rule: euclid_ext_aux.induct)
eberlm@62442
   456
  case (1 r' r s' s t' t)
eberlm@62442
   457
  show ?case
eberlm@62442
   458
  proof (cases "r = 0")
eberlm@62442
   459
    case True
eberlm@62442
   460
    hence "euclid_ext_aux r' r s' s t' t = 
eberlm@62442
   461
             (s' div unit_factor r', t' div unit_factor r', normalize r')"
eberlm@62442
   462
      by (subst euclid_ext_aux.simps) (simp add: Let_def)
eberlm@62442
   463
    also have "?P \<dots>"
eberlm@62442
   464
    proof safe
haftmann@64177
   465
      have "s' div unit_factor r' * a + t' div unit_factor r' * b = 
haftmann@64177
   466
                (s' * a + t' * b) div unit_factor r'"
eberlm@62442
   467
        by (cases "r' = 0") (simp_all add: unit_div_commute)
haftmann@64177
   468
      also have "s' * a + t' * b = r'" by fact
eberlm@62442
   469
      also have "\<dots> div unit_factor r' = normalize r'" by simp
haftmann@64177
   470
      finally show "s' div unit_factor r' * a + t' div unit_factor r' * b = normalize r'" .
eberlm@62442
   471
    next
haftmann@64177
   472
      from "1.prems" True show "normalize r' = gcd_eucl a b" by (simp add: gcd_eucl_0)
eberlm@62442
   473
    qed
eberlm@62442
   474
    finally show ?thesis .
eberlm@62442
   475
  next
eberlm@62442
   476
    case False
eberlm@62442
   477
    hence "euclid_ext_aux r' r s' s t' t = 
eberlm@62442
   478
             euclid_ext_aux r (r' mod r) s (s' - r' div r * s) t (t' - r' div r * t)"
eberlm@62442
   479
      by (subst euclid_ext_aux.simps) (simp add: Let_def)
eberlm@62442
   480
    also from "1.prems" False have "?P \<dots>"
eberlm@62442
   481
    proof (intro "1.IH")
haftmann@64177
   482
      have "(s' - r' div r * s) * a + (t' - r' div r * t) * b =
haftmann@64177
   483
              (s' * a + t' * b) - r' div r * (s * a + t * b)" by (simp add: algebra_simps)
haftmann@64177
   484
      also have "s' * a + t' * b = r'" by fact
haftmann@64177
   485
      also have "s * a + t * b = r" by fact
haftmann@64242
   486
      also have "r' - r' div r * r = r' mod r" using div_mult_mod_eq [of r' r]
eberlm@62442
   487
        by (simp add: algebra_simps)
haftmann@64177
   488
      finally show "(s' - r' div r * s) * a + (t' - r' div r * t) * b = r' mod r" .
haftmann@64243
   489
    qed (auto simp: gcd_eucl_non_0 algebra_simps minus_mod_eq_div_mult [symmetric])
eberlm@62442
   490
    finally show ?thesis .
eberlm@62442
   491
  qed
eberlm@62442
   492
qed
eberlm@62442
   493
eberlm@62442
   494
definition euclid_ext where
eberlm@62442
   495
  "euclid_ext a b = euclid_ext_aux a b 1 0 0 1"
haftmann@60598
   496
haftmann@60598
   497
lemma euclid_ext_0: 
haftmann@60634
   498
  "euclid_ext a 0 = (1 div unit_factor a, 0, normalize a)"
eberlm@62442
   499
  by (simp add: euclid_ext_def euclid_ext_aux.simps)
haftmann@60598
   500
haftmann@60598
   501
lemma euclid_ext_left_0: 
haftmann@60634
   502
  "euclid_ext 0 a = (0, 1 div unit_factor a, normalize a)"
eberlm@62442
   503
  by (simp add: euclid_ext_def euclid_ext_aux.simps)
haftmann@60598
   504
eberlm@62442
   505
lemma euclid_ext_correct':
haftmann@64177
   506
  "case euclid_ext a b of (x,y,c) \<Rightarrow> x * a + y * b = c \<and> c = gcd_eucl a b"
eberlm@62442
   507
  unfolding euclid_ext_def by (rule euclid_ext_aux_correct) simp_all
haftmann@60598
   508
eberlm@62457
   509
lemma euclid_ext_gcd_eucl:
haftmann@64177
   510
  "(case euclid_ext a b of (x,y,c) \<Rightarrow> c) = gcd_eucl a b"
haftmann@64177
   511
  using euclid_ext_correct'[of a b] by (simp add: case_prod_unfold)
eberlm@62457
   512
eberlm@62442
   513
definition euclid_ext' where
haftmann@64177
   514
  "euclid_ext' a b = (case euclid_ext a b of (x, y, _) \<Rightarrow> (x, y))"
haftmann@60598
   515
eberlm@62442
   516
lemma euclid_ext'_correct':
haftmann@64177
   517
  "case euclid_ext' a b of (x,y) \<Rightarrow> x * a + y * b = gcd_eucl a b"
haftmann@64177
   518
  using euclid_ext_correct'[of a b] by (simp add: case_prod_unfold euclid_ext'_def)
haftmann@60598
   519
haftmann@60634
   520
lemma euclid_ext'_0: "euclid_ext' a 0 = (1 div unit_factor a, 0)" 
haftmann@60598
   521
  by (simp add: euclid_ext'_def euclid_ext_0)
haftmann@60598
   522
haftmann@60634
   523
lemma euclid_ext'_left_0: "euclid_ext' 0 a = (0, 1 div unit_factor a)" 
haftmann@60598
   524
  by (simp add: euclid_ext'_def euclid_ext_left_0)
haftmann@60598
   525
haftmann@60598
   526
end
haftmann@60598
   527
haftmann@58023
   528
class euclidean_semiring_gcd = euclidean_semiring + gcd + Gcd +
haftmann@58023
   529
  assumes gcd_gcd_eucl: "gcd = gcd_eucl" and lcm_lcm_eucl: "lcm = lcm_eucl"
haftmann@58023
   530
  assumes Gcd_Gcd_eucl: "Gcd = Gcd_eucl" and Lcm_Lcm_eucl: "Lcm = Lcm_eucl"
haftmann@58023
   531
begin
haftmann@58023
   532
eberlm@62422
   533
subclass semiring_gcd
eberlm@62422
   534
  by standard (simp_all add: gcd_gcd_eucl gcd_eucl_greatest lcm_lcm_eucl lcm_eucl_def)
haftmann@58023
   535
eberlm@62422
   536
subclass semiring_Gcd
eberlm@62422
   537
  by standard (auto simp: Gcd_Gcd_eucl Lcm_Lcm_eucl Gcd_eucl_def intro: Lcm_eucl_least)
eberlm@63498
   538
eberlm@63498
   539
subclass factorial_semiring_gcd
eberlm@63498
   540
proof
eberlm@63498
   541
  fix a b
eberlm@63498
   542
  show "gcd a b = gcd_factorial a b"
eberlm@63498
   543
    by (rule sym, rule gcdI) (rule gcd_lcm_factorial; assumption)+
eberlm@63498
   544
  thus "lcm a b = lcm_factorial a b"
eberlm@63498
   545
    by (simp add: lcm_factorial_gcd_factorial lcm_gcd)
eberlm@63498
   546
next
eberlm@63498
   547
  fix A 
eberlm@63498
   548
  show "Gcd A = Gcd_factorial A"
eberlm@63498
   549
    by (rule sym, rule GcdI) (rule gcd_lcm_factorial; assumption)+
eberlm@63498
   550
  show "Lcm A = Lcm_factorial A"
eberlm@63498
   551
    by (rule sym, rule LcmI) (rule gcd_lcm_factorial; assumption)+
eberlm@63498
   552
qed
eberlm@63498
   553
haftmann@58023
   554
lemma gcd_non_0:
haftmann@60430
   555
  "b \<noteq> 0 \<Longrightarrow> gcd a b = gcd b (a mod b)"
haftmann@60572
   556
  unfolding gcd_gcd_eucl by (fact gcd_eucl_non_0)
haftmann@58023
   557
eberlm@62422
   558
lemmas gcd_0 = gcd_0_right
eberlm@62422
   559
lemmas dvd_gcd_iff = gcd_greatest_iff
haftmann@58023
   560
lemmas gcd_greatest_iff = dvd_gcd_iff
haftmann@58023
   561
haftmann@58023
   562
lemma gcd_mod1 [simp]:
haftmann@60430
   563
  "gcd (a mod b) b = gcd a b"
haftmann@58023
   564
  by (rule gcdI, metis dvd_mod_iff gcd_dvd1 gcd_dvd2, simp_all add: gcd_greatest dvd_mod_iff)
haftmann@58023
   565
haftmann@58023
   566
lemma gcd_mod2 [simp]:
haftmann@60430
   567
  "gcd a (b mod a) = gcd a b"
haftmann@58023
   568
  by (rule gcdI, simp, metis dvd_mod_iff gcd_dvd1 gcd_dvd2, simp_all add: gcd_greatest dvd_mod_iff)
haftmann@58023
   569
         
haftmann@58023
   570
lemma euclidean_size_gcd_le1 [simp]:
haftmann@58023
   571
  assumes "a \<noteq> 0"
haftmann@58023
   572
  shows "euclidean_size (gcd a b) \<le> euclidean_size a"
haftmann@58023
   573
proof -
haftmann@58023
   574
   have "gcd a b dvd a" by (rule gcd_dvd1)
haftmann@58023
   575
   then obtain c where A: "a = gcd a b * c" unfolding dvd_def by blast
wenzelm@60526
   576
   with \<open>a \<noteq> 0\<close> show ?thesis by (subst (2) A, intro size_mult_mono) auto
haftmann@58023
   577
qed
haftmann@58023
   578
haftmann@58023
   579
lemma euclidean_size_gcd_le2 [simp]:
haftmann@58023
   580
  "b \<noteq> 0 \<Longrightarrow> euclidean_size (gcd a b) \<le> euclidean_size b"
haftmann@58023
   581
  by (subst gcd.commute, rule euclidean_size_gcd_le1)
haftmann@58023
   582
haftmann@58023
   583
lemma euclidean_size_gcd_less1:
haftmann@58023
   584
  assumes "a \<noteq> 0" and "\<not>a dvd b"
haftmann@58023
   585
  shows "euclidean_size (gcd a b) < euclidean_size a"
haftmann@58023
   586
proof (rule ccontr)
haftmann@58023
   587
  assume "\<not>euclidean_size (gcd a b) < euclidean_size a"
eberlm@62422
   588
  with \<open>a \<noteq> 0\<close> have A: "euclidean_size (gcd a b) = euclidean_size a"
haftmann@58023
   589
    by (intro le_antisym, simp_all)
eberlm@62422
   590
  have "a dvd gcd a b"
eberlm@62422
   591
    by (rule dvd_euclidean_size_eq_imp_dvd) (simp_all add: assms A)
eberlm@62422
   592
  hence "a dvd b" using dvd_gcdD2 by blast
wenzelm@60526
   593
  with \<open>\<not>a dvd b\<close> show False by contradiction
haftmann@58023
   594
qed
haftmann@58023
   595
haftmann@58023
   596
lemma euclidean_size_gcd_less2:
haftmann@58023
   597
  assumes "b \<noteq> 0" and "\<not>b dvd a"
haftmann@58023
   598
  shows "euclidean_size (gcd a b) < euclidean_size b"
haftmann@58023
   599
  using assms by (subst gcd.commute, rule euclidean_size_gcd_less1)
haftmann@58023
   600
haftmann@58023
   601
lemma euclidean_size_lcm_le1: 
haftmann@58023
   602
  assumes "a \<noteq> 0" and "b \<noteq> 0"
haftmann@58023
   603
  shows "euclidean_size a \<le> euclidean_size (lcm a b)"
haftmann@58023
   604
proof -
haftmann@60690
   605
  have "a dvd lcm a b" by (rule dvd_lcm1)
haftmann@60690
   606
  then obtain c where A: "lcm a b = a * c" ..
eberlm@62429
   607
  with \<open>a \<noteq> 0\<close> and \<open>b \<noteq> 0\<close> have "c \<noteq> 0" by (auto simp: lcm_eq_0_iff)
haftmann@58023
   608
  then show ?thesis by (subst A, intro size_mult_mono)
haftmann@58023
   609
qed
haftmann@58023
   610
haftmann@58023
   611
lemma euclidean_size_lcm_le2:
haftmann@58023
   612
  "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> euclidean_size b \<le> euclidean_size (lcm a b)"
haftmann@58023
   613
  using euclidean_size_lcm_le1 [of b a] by (simp add: ac_simps)
haftmann@58023
   614
haftmann@58023
   615
lemma euclidean_size_lcm_less1:
haftmann@58023
   616
  assumes "b \<noteq> 0" and "\<not>b dvd a"
haftmann@58023
   617
  shows "euclidean_size a < euclidean_size (lcm a b)"
haftmann@58023
   618
proof (rule ccontr)
haftmann@58023
   619
  from assms have "a \<noteq> 0" by auto
haftmann@58023
   620
  assume "\<not>euclidean_size a < euclidean_size (lcm a b)"
wenzelm@60526
   621
  with \<open>a \<noteq> 0\<close> and \<open>b \<noteq> 0\<close> have "euclidean_size (lcm a b) = euclidean_size a"
haftmann@58023
   622
    by (intro le_antisym, simp, intro euclidean_size_lcm_le1)
haftmann@58023
   623
  with assms have "lcm a b dvd a" 
eberlm@62429
   624
    by (rule_tac dvd_euclidean_size_eq_imp_dvd) (auto simp: lcm_eq_0_iff)
eberlm@62422
   625
  hence "b dvd a" by (rule lcm_dvdD2)
wenzelm@60526
   626
  with \<open>\<not>b dvd a\<close> show False by contradiction
haftmann@58023
   627
qed
haftmann@58023
   628
haftmann@58023
   629
lemma euclidean_size_lcm_less2:
haftmann@58023
   630
  assumes "a \<noteq> 0" and "\<not>a dvd b"
haftmann@58023
   631
  shows "euclidean_size b < euclidean_size (lcm a b)"
haftmann@58023
   632
  using assms euclidean_size_lcm_less1 [of a b] by (simp add: ac_simps)
haftmann@58023
   633
eberlm@62428
   634
lemma Lcm_eucl_set [code]:
eberlm@62428
   635
  "Lcm_eucl (set xs) = foldl lcm_eucl 1 xs"
eberlm@62428
   636
  by (simp add: Lcm_Lcm_eucl [symmetric] lcm_lcm_eucl Lcm_set)
haftmann@58023
   637
eberlm@62428
   638
lemma Gcd_eucl_set [code]:
eberlm@62428
   639
  "Gcd_eucl (set xs) = foldl gcd_eucl 0 xs"
eberlm@62428
   640
  by (simp add: Gcd_Gcd_eucl [symmetric] gcd_gcd_eucl Gcd_set)
haftmann@58023
   641
haftmann@58023
   642
end
haftmann@58023
   643
eberlm@63498
   644
wenzelm@60526
   645
text \<open>
haftmann@58023
   646
  A Euclidean ring is a Euclidean semiring with additive inverses. It provides a 
haftmann@58023
   647
  few more lemmas; in particular, Bezout's lemma holds for any Euclidean ring.
wenzelm@60526
   648
\<close>
haftmann@58023
   649
haftmann@58023
   650
class euclidean_ring_gcd = euclidean_semiring_gcd + idom
haftmann@58023
   651
begin
haftmann@58023
   652
haftmann@58023
   653
subclass euclidean_ring ..
haftmann@60439
   654
subclass ring_gcd ..
eberlm@63498
   655
subclass factorial_ring_gcd ..
haftmann@60439
   656
haftmann@60572
   657
lemma euclid_ext_gcd [simp]:
haftmann@60572
   658
  "(case euclid_ext a b of (_, _ , t) \<Rightarrow> t) = gcd a b"
eberlm@62442
   659
  using euclid_ext_correct'[of a b] by (simp add: case_prod_unfold Let_def gcd_gcd_eucl)
haftmann@60572
   660
haftmann@60572
   661
lemma euclid_ext_gcd' [simp]:
haftmann@60572
   662
  "euclid_ext a b = (r, s, t) \<Longrightarrow> t = gcd a b"
haftmann@60572
   663
  by (insert euclid_ext_gcd[of a b], drule (1) subst, simp)
eberlm@62442
   664
eberlm@62442
   665
lemma euclid_ext_correct:
haftmann@64177
   666
  "case euclid_ext a b of (x,y,c) \<Rightarrow> x * a + y * b = c \<and> c = gcd a b"
haftmann@64177
   667
  using euclid_ext_correct'[of a b]
eberlm@62442
   668
  by (simp add: gcd_gcd_eucl case_prod_unfold)
haftmann@60572
   669
  
haftmann@60572
   670
lemma euclid_ext'_correct:
haftmann@60572
   671
  "fst (euclid_ext' a b) * a + snd (euclid_ext' a b) * b = gcd a b"
eberlm@62442
   672
  using euclid_ext_correct'[of a b]
eberlm@62442
   673
  by (simp add: gcd_gcd_eucl case_prod_unfold euclid_ext'_def)
haftmann@60572
   674
haftmann@60572
   675
lemma bezout: "\<exists>s t. s * a + t * b = gcd a b"
haftmann@60572
   676
  using euclid_ext'_correct by blast
haftmann@60572
   677
haftmann@60572
   678
end
haftmann@58023
   679
haftmann@58023
   680
haftmann@60572
   681
subsection \<open>Typical instances\<close>
haftmann@58023
   682
haftmann@58023
   683
instantiation nat :: euclidean_semiring
haftmann@58023
   684
begin
haftmann@58023
   685
haftmann@58023
   686
definition [simp]:
haftmann@58023
   687
  "euclidean_size_nat = (id :: nat \<Rightarrow> nat)"
haftmann@58023
   688
eberlm@63498
   689
instance by standard simp_all
haftmann@58023
   690
haftmann@58023
   691
end
haftmann@58023
   692
eberlm@62422
   693
haftmann@58023
   694
instantiation int :: euclidean_ring
haftmann@58023
   695
begin
haftmann@58023
   696
haftmann@58023
   697
definition [simp]:
haftmann@58023
   698
  "euclidean_size_int = (nat \<circ> abs :: int \<Rightarrow> nat)"
haftmann@58023
   699
eberlm@63498
   700
instance by standard (auto simp add: abs_mult nat_mult_distrib split: abs_split)
haftmann@58023
   701
haftmann@58023
   702
end
haftmann@58023
   703
eberlm@62422
   704
instance nat :: euclidean_semiring_gcd
eberlm@62422
   705
proof
eberlm@62422
   706
  show [simp]: "gcd = (gcd_eucl :: nat \<Rightarrow> _)" "Lcm = (Lcm_eucl :: nat set \<Rightarrow> _)"
eberlm@62422
   707
    by (simp_all add: eq_gcd_euclI eq_Lcm_euclI)
eberlm@62422
   708
  show "lcm = (lcm_eucl :: nat \<Rightarrow> _)" "Gcd = (Gcd_eucl :: nat set \<Rightarrow> _)"
eberlm@62422
   709
    by (intro ext, simp add: lcm_eucl_def lcm_nat_def Gcd_nat_def Gcd_eucl_def)+
eberlm@62422
   710
qed
eberlm@62422
   711
eberlm@62422
   712
instance int :: euclidean_ring_gcd
eberlm@62422
   713
proof
eberlm@62422
   714
  show [simp]: "gcd = (gcd_eucl :: int \<Rightarrow> _)" "Lcm = (Lcm_eucl :: int set \<Rightarrow> _)"
eberlm@62422
   715
    by (simp_all add: eq_gcd_euclI eq_Lcm_euclI)
eberlm@62422
   716
  show "lcm = (lcm_eucl :: int \<Rightarrow> _)" "Gcd = (Gcd_eucl :: int set \<Rightarrow> _)"
eberlm@62422
   717
    by (intro ext, simp add: lcm_eucl_def lcm_altdef_int 
eberlm@62422
   718
          semiring_Gcd_class.Gcd_Lcm Gcd_eucl_def abs_mult)+
eberlm@62422
   719
qed
eberlm@62422
   720
haftmann@63924
   721
end