src/HOL/GCD.thy
author haftmann
Sun Mar 10 15:16:45 2019 +0000 (6 weeks ago)
changeset 69906 55534affe445
parent 69785 9e326f6f8a24
permissions -rw-r--r--
migrated from Nums to Zarith as library for OCaml integer arithmetic
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(*  Title:      HOL/GCD.thy
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    Author:     Christophe Tabacznyj
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    Author:     Lawrence C. Paulson
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    Author:     Amine Chaieb
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    Author:     Thomas M. Rasmussen
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    Author:     Jeremy Avigad
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    Author:     Tobias Nipkow
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This file deals with the functions gcd and lcm.  Definitions and
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lemmas are proved uniformly for the natural numbers and integers.
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This file combines and revises a number of prior developments.
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The original theories "GCD" and "Primes" were by Christophe Tabacznyj
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and Lawrence C. Paulson, based on @{cite davenport92}. They introduced
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gcd, lcm, and prime for the natural numbers.
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The original theory "IntPrimes" was by Thomas M. Rasmussen, and
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extended gcd, lcm, primes to the integers. Amine Chaieb provided
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another extension of the notions to the integers, and added a number
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of results to "Primes" and "GCD". IntPrimes also defined and developed
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the congruence relations on the integers. The notion was extended to
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the natural numbers by Chaieb.
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Jeremy Avigad combined all of these, made everything uniform for the
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natural numbers and the integers, and added a number of new theorems.
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Tobias Nipkow cleaned up a lot.
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*)
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section \<open>Greatest common divisor and least common multiple\<close>
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theory GCD
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  imports Groups_List 
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begin
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subsection \<open>Abstract bounded quasi semilattices as common foundation\<close>
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locale bounded_quasi_semilattice = abel_semigroup +
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  fixes top :: 'a  ("\<^bold>\<top>") and bot :: 'a  ("\<^bold>\<bottom>")
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    and normalize :: "'a \<Rightarrow> 'a"
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  assumes idem_normalize [simp]: "a \<^bold>* a = normalize a"
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    and normalize_left_idem [simp]: "normalize a \<^bold>* b = a \<^bold>* b"
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    and normalize_idem [simp]: "normalize (a \<^bold>* b) = a \<^bold>* b"
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    and normalize_top [simp]: "normalize \<^bold>\<top> = \<^bold>\<top>"
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    and normalize_bottom [simp]: "normalize \<^bold>\<bottom> = \<^bold>\<bottom>"
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    and top_left_normalize [simp]: "\<^bold>\<top> \<^bold>* a = normalize a"
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    and bottom_left_bottom [simp]: "\<^bold>\<bottom> \<^bold>* a = \<^bold>\<bottom>"
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begin
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lemma left_idem [simp]:
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  "a \<^bold>* (a \<^bold>* b) = a \<^bold>* b"
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  using assoc [of a a b, symmetric] by simp
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lemma right_idem [simp]:
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  "(a \<^bold>* b) \<^bold>* b = a \<^bold>* b"
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  using left_idem [of b a] by (simp add: ac_simps)
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lemma comp_fun_idem: "comp_fun_idem f"
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  by standard (simp_all add: fun_eq_iff ac_simps)
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interpretation comp_fun_idem f
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  by (fact comp_fun_idem)
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lemma top_right_normalize [simp]:
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  "a \<^bold>* \<^bold>\<top> = normalize a"
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  using top_left_normalize [of a] by (simp add: ac_simps)
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lemma bottom_right_bottom [simp]:
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  "a \<^bold>* \<^bold>\<bottom> = \<^bold>\<bottom>"
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  using bottom_left_bottom [of a] by (simp add: ac_simps)
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lemma normalize_right_idem [simp]:
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  "a \<^bold>* normalize b = a \<^bold>* b"
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  using normalize_left_idem [of b a] by (simp add: ac_simps)
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end
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locale bounded_quasi_semilattice_set = bounded_quasi_semilattice
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begin
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interpretation comp_fun_idem f
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  by (fact comp_fun_idem)
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definition F :: "'a set \<Rightarrow> 'a"
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where
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  eq_fold: "F A = (if finite A then Finite_Set.fold f \<^bold>\<top> A else \<^bold>\<bottom>)"
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lemma infinite [simp]:
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  "infinite A \<Longrightarrow> F A = \<^bold>\<bottom>"
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  by (simp add: eq_fold)
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lemma set_eq_fold [code]:
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  "F (set xs) = fold f xs \<^bold>\<top>"
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  by (simp add: eq_fold fold_set_fold)
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lemma empty [simp]:
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  "F {} = \<^bold>\<top>"
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  by (simp add: eq_fold)
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lemma insert [simp]:
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  "F (insert a A) = a \<^bold>* F A"
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  by (cases "finite A") (simp_all add: eq_fold)
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lemma normalize [simp]:
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  "normalize (F A) = F A"
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  by (induct A rule: infinite_finite_induct) simp_all
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lemma in_idem:
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  assumes "a \<in> A"
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  shows "a \<^bold>* F A = F A"
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  using assms by (induct A rule: infinite_finite_induct)
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    (auto simp: left_commute [of a])
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lemma union:
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  "F (A \<union> B) = F A \<^bold>* F B"
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  by (induct A rule: infinite_finite_induct)
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    (simp_all add: ac_simps)
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lemma remove:
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  assumes "a \<in> A"
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  shows "F A = a \<^bold>* F (A - {a})"
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proof -
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  from assms obtain B where "A = insert a B" and "a \<notin> B"
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    by (blast dest: mk_disjoint_insert)
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  with assms show ?thesis by simp
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qed
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lemma insert_remove:
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  "F (insert a A) = a \<^bold>* F (A - {a})"
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  by (cases "a \<in> A") (simp_all add: insert_absorb remove)
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lemma subset:
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  assumes "B \<subseteq> A"
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  shows "F B \<^bold>* F A = F A"
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  using assms by (simp add: union [symmetric] Un_absorb1)
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end
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subsection \<open>Abstract GCD and LCM\<close>
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class gcd = zero + one + dvd +
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  fixes gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
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    and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
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class Gcd = gcd +
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  fixes Gcd :: "'a set \<Rightarrow> 'a"
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    and Lcm :: "'a set \<Rightarrow> 'a"
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syntax
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  "_GCD1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3GCD _./ _)" [0, 10] 10)
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  "_GCD"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3GCD _\<in>_./ _)" [0, 0, 10] 10)
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  "_LCM1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3LCM _./ _)" [0, 10] 10)
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  "_LCM"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3LCM _\<in>_./ _)" [0, 0, 10] 10)
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translations
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  "GCD x y. f"   \<rightleftharpoons> "GCD x. GCD y. f"
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  "GCD x. f"     \<rightleftharpoons> "CONST Gcd (CONST range (\<lambda>x. f))"
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  "GCD x\<in>A. f"   \<rightleftharpoons> "CONST Gcd ((\<lambda>x. f) ` A)"
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  "LCM x y. f"   \<rightleftharpoons> "LCM x. LCM y. f"
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  "LCM x. f"     \<rightleftharpoons> "CONST Lcm (CONST range (\<lambda>x. f))"
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  "LCM x\<in>A. f"   \<rightleftharpoons> "CONST Lcm ((\<lambda>x. f) ` A)"
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class semiring_gcd = normalization_semidom + gcd +
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  assumes 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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    and gcd_greatest: "c dvd a \<Longrightarrow> c dvd b \<Longrightarrow> c dvd gcd a b"
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    and normalize_gcd [simp]: "normalize (gcd a b) = gcd a b"
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    and lcm_gcd: "lcm a b = normalize (a * b) div gcd a b"
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begin
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lemma gcd_greatest_iff [simp]: "a dvd gcd b c \<longleftrightarrow> a dvd b \<and> a dvd c"
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  by (blast intro!: gcd_greatest intro: dvd_trans)
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lemma gcd_dvdI1: "a dvd c \<Longrightarrow> gcd a b dvd c"
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  by (rule dvd_trans) (rule gcd_dvd1)
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lemma gcd_dvdI2: "b dvd c \<Longrightarrow> gcd a b dvd c"
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  by (rule dvd_trans) (rule gcd_dvd2)
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lemma dvd_gcdD1: "a dvd gcd b c \<Longrightarrow> a dvd b"
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  using gcd_dvd1 [of b c] by (blast intro: dvd_trans)
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lemma dvd_gcdD2: "a dvd gcd b c \<Longrightarrow> a dvd c"
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  using gcd_dvd2 [of b c] by (blast intro: dvd_trans)
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lemma gcd_0_left [simp]: "gcd 0 a = normalize a"
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  by (rule associated_eqI) simp_all
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lemma gcd_0_right [simp]: "gcd a 0 = normalize a"
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  by (rule associated_eqI) simp_all
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lemma gcd_eq_0_iff [simp]: "gcd a b = 0 \<longleftrightarrow> a = 0 \<and> b = 0"
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  (is "?P \<longleftrightarrow> ?Q")
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proof
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  assume ?P
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  then have "0 dvd gcd a b"
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    by simp
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  then have "0 dvd a" and "0 dvd b"
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    by (blast intro: dvd_trans)+
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  then show ?Q
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    by simp
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next
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  assume ?Q
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  then show ?P
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    by simp
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qed
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lemma unit_factor_gcd: "unit_factor (gcd a b) = (if a = 0 \<and> b = 0 then 0 else 1)"
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proof (cases "gcd a b = 0")
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  case True
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  then show ?thesis by simp
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next
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  case False
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  have "unit_factor (gcd a b) * normalize (gcd a b) = gcd a b"
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    by (rule unit_factor_mult_normalize)
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  then have "unit_factor (gcd a b) * gcd a b = gcd a b"
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    by simp
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  then have "unit_factor (gcd a b) * gcd a b div gcd a b = gcd a b div gcd a b"
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    by simp
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  with False show ?thesis
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    by simp
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qed
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lemma is_unit_gcd_iff [simp]:
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  "is_unit (gcd a b) \<longleftrightarrow> gcd a b = 1"
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  by (cases "a = 0 \<and> b = 0") (auto simp: unit_factor_gcd dest: is_unit_unit_factor)
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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 a b = gcd b a"
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    by (rule associated_eqI) simp_all
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  from gcd_dvd1 have "gcd (gcd a b) c dvd a"
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    by (rule dvd_trans) simp
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  moreover from gcd_dvd1 have "gcd (gcd a b) c dvd b"
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    by (rule dvd_trans) simp
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  ultimately have P1: "gcd (gcd a b) c dvd gcd a (gcd b c)"
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    by (auto intro!: gcd_greatest)
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  from gcd_dvd2 have "gcd a (gcd b c) dvd b"
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    by (rule dvd_trans) simp
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  moreover from gcd_dvd2 have "gcd a (gcd b c) dvd c"
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    by (rule dvd_trans) simp
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  ultimately have P2: "gcd a (gcd b c) dvd gcd (gcd a b) c"
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    by (auto intro!: gcd_greatest)
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  from P1 P2 show "gcd (gcd a b) c = gcd a (gcd b c)"
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    by (rule associated_eqI) simp_all
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qed
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sublocale gcd: bounded_quasi_semilattice gcd 0 1 normalize
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proof
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  show "gcd a a = normalize a" for a
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  proof -
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    have "a dvd gcd a a"
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      by (rule gcd_greatest) simp_all
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    then show ?thesis
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      by (auto intro: associated_eqI)
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  qed
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  show "gcd (normalize a) b = gcd a b" for a b
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    using gcd_dvd1 [of "normalize a" b]
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    by (auto intro: associated_eqI)
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  show "gcd 1 a = 1" for a
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    by (rule associated_eqI) simp_all
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qed simp_all
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lemma gcd_self: "gcd a a = normalize a"
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  by (fact gcd.idem_normalize)
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lemma gcd_left_idem: "gcd a (gcd a b) = gcd a b"
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  by (fact gcd.left_idem)
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lemma gcd_right_idem: "gcd (gcd a b) b = gcd a b"
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  by (fact gcd.right_idem)
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lemma gcd_mult_left: "gcd (c * a) (c * b) = normalize c * gcd a b"
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proof (cases "c = 0")
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  case True
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  then show ?thesis by simp
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next
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  case False
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  then have *: "c * gcd a b dvd gcd (c * a) (c * b)"
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    by (auto intro: gcd_greatest)
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  moreover from False * have "gcd (c * a) (c * b) dvd c * gcd a b"
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    by (metis div_dvd_iff_mult dvd_mult_left gcd_dvd1 gcd_dvd2 gcd_greatest mult_commute)
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  ultimately have "normalize (gcd (c * a) (c * b)) = normalize (c * gcd a b)"
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    by (auto intro: associated_eqI)
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  then show ?thesis
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    by (simp add: normalize_mult)
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qed
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lemma gcd_mult_right: "gcd (a * c) (b * c) = gcd b a * normalize c"
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  using gcd_mult_left [of c a b] by (simp add: ac_simps)
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lemma mult_gcd_left: "c * gcd a b = unit_factor c * gcd (c * a) (c * b)"
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  by (simp add: gcd_mult_left mult.assoc [symmetric])
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lemma mult_gcd_right: "gcd a b * c = gcd (a * c) (b * c) * unit_factor c"
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  using mult_gcd_left [of c a b] by (simp add: ac_simps)
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lemma dvd_lcm1 [iff]: "a dvd lcm a b"
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proof -
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  have "normalize (a * b) div gcd a b = normalize a * (normalize b div gcd a b)"
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    by (simp add: lcm_gcd normalize_mult div_mult_swap)
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  then show ?thesis
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    by (simp add: lcm_gcd)
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qed
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lemma dvd_lcm2 [iff]: "b dvd lcm a b"
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proof -
haftmann@60686
   310
  have "normalize (a * b) div gcd a b = normalize b * (normalize a div gcd a b)"
haftmann@60686
   311
    by (simp add: lcm_gcd normalize_mult div_mult_swap ac_simps)
haftmann@60686
   312
  then show ?thesis
haftmann@60686
   313
    by (simp add: lcm_gcd)
haftmann@60686
   314
qed
haftmann@60686
   315
wenzelm@63489
   316
lemma dvd_lcmI1: "a dvd b \<Longrightarrow> a dvd lcm b c"
wenzelm@63489
   317
  by (rule dvd_trans) (assumption, blast)
wenzelm@63489
   318
wenzelm@63489
   319
lemma dvd_lcmI2: "a dvd c \<Longrightarrow> a dvd lcm b c"
haftmann@60689
   320
  by (rule dvd_trans) (assumption, blast)
haftmann@60689
   321
wenzelm@63489
   322
lemma lcm_dvdD1: "lcm a b dvd c \<Longrightarrow> a dvd c"
haftmann@62345
   323
  using dvd_lcm1 [of a b] by (blast intro: dvd_trans)
haftmann@62345
   324
wenzelm@63489
   325
lemma lcm_dvdD2: "lcm a b dvd c \<Longrightarrow> b dvd c"
haftmann@62345
   326
  using dvd_lcm2 [of a b] by (blast intro: dvd_trans)
haftmann@62345
   327
haftmann@60686
   328
lemma lcm_least:
haftmann@60686
   329
  assumes "a dvd c" and "b dvd c"
haftmann@60686
   330
  shows "lcm a b dvd c"
haftmann@60686
   331
proof (cases "c = 0")
wenzelm@63489
   332
  case True
wenzelm@63489
   333
  then show ?thesis by simp
haftmann@60686
   334
next
wenzelm@63489
   335
  case False
wenzelm@63489
   336
  then have *: "is_unit (unit_factor c)"
wenzelm@63489
   337
    by simp
haftmann@60686
   338
  show ?thesis
haftmann@60686
   339
  proof (cases "gcd a b = 0")
wenzelm@63489
   340
    case True
wenzelm@63489
   341
    with assms show ?thesis by simp
haftmann@60686
   342
  next
wenzelm@63489
   343
    case False
wenzelm@63489
   344
    then have "a \<noteq> 0 \<or> b \<noteq> 0"
wenzelm@63489
   345
      by simp
haftmann@60686
   346
    with \<open>c \<noteq> 0\<close> assms have "a * b dvd a * c" "a * b dvd c * b"
haftmann@60686
   347
      by (simp_all add: mult_dvd_mono)
haftmann@60686
   348
    then have "normalize (a * b) dvd gcd (a * c) (b * c)"
haftmann@60686
   349
      by (auto intro: gcd_greatest simp add: ac_simps)
haftmann@60686
   350
    then have "normalize (a * b) dvd gcd (a * c) (b * c) * unit_factor c"
wenzelm@63489
   351
      using * by (simp add: dvd_mult_unit_iff)
haftmann@60686
   352
    then have "normalize (a * b) dvd gcd a b * c"
haftmann@60686
   353
      by (simp add: mult_gcd_right [of a b c])
haftmann@60686
   354
    then have "normalize (a * b) div gcd a b dvd c"
haftmann@60686
   355
      using False by (simp add: div_dvd_iff_mult ac_simps)
wenzelm@63489
   356
    then show ?thesis
wenzelm@63489
   357
      by (simp add: lcm_gcd)
haftmann@60686
   358
  qed
haftmann@60686
   359
qed
haftmann@60686
   360
wenzelm@63489
   361
lemma lcm_least_iff [simp]: "lcm a b dvd c \<longleftrightarrow> a dvd c \<and> b dvd c"
haftmann@60686
   362
  by (blast intro!: lcm_least intro: dvd_trans)
haftmann@60686
   363
wenzelm@63489
   364
lemma normalize_lcm [simp]: "normalize (lcm a b) = lcm a b"
haftmann@60686
   365
  by (simp add: lcm_gcd dvd_normalize_div)
haftmann@60686
   366
wenzelm@63489
   367
lemma lcm_0_left [simp]: "lcm 0 a = 0"
wenzelm@63489
   368
  by (simp add: lcm_gcd)
wenzelm@63489
   369
wenzelm@63489
   370
lemma lcm_0_right [simp]: "lcm a 0 = 0"
haftmann@60686
   371
  by (simp add: lcm_gcd)
wenzelm@63489
   372
wenzelm@63489
   373
lemma lcm_eq_0_iff: "lcm a b = 0 \<longleftrightarrow> a = 0 \<or> b = 0"
wenzelm@63489
   374
  (is "?P \<longleftrightarrow> ?Q")
haftmann@60686
   375
proof
wenzelm@63489
   376
  assume ?P
wenzelm@63489
   377
  then have "0 dvd lcm a b"
wenzelm@63489
   378
    by simp
haftmann@60686
   379
  then have "0 dvd normalize (a * b) div gcd a b"
haftmann@60686
   380
    by (simp add: lcm_gcd)
haftmann@60686
   381
  then have "0 * gcd a b dvd normalize (a * b)"
haftmann@60686
   382
    using dvd_div_iff_mult [of "gcd a b" _ 0] by (cases "gcd a b = 0") simp_all
haftmann@60686
   383
  then have "normalize (a * b) = 0"
haftmann@60686
   384
    by simp
wenzelm@63489
   385
  then show ?Q
wenzelm@63489
   386
    by simp
haftmann@60686
   387
next
wenzelm@63489
   388
  assume ?Q
wenzelm@63489
   389
  then show ?P
wenzelm@63489
   390
    by auto
haftmann@60686
   391
qed
haftmann@60686
   392
wenzelm@63489
   393
lemma lcm_eq_1_iff [simp]: "lcm a b = 1 \<longleftrightarrow> is_unit a \<and> is_unit b"
haftmann@61913
   394
  by (auto intro: associated_eqI)
haftmann@61913
   395
wenzelm@63489
   396
lemma unit_factor_lcm: "unit_factor (lcm a b) = (if a = 0 \<or> b = 0 then 0 else 1)"
haftmann@60686
   397
  by (simp add: unit_factor_gcd dvd_unit_factor_div lcm_gcd)
haftmann@60686
   398
wenzelm@61605
   399
sublocale lcm: abel_semigroup lcm
haftmann@60686
   400
proof
haftmann@60686
   401
  fix a b c
haftmann@60686
   402
  show "lcm a b = lcm b a"
haftmann@60686
   403
    by (simp add: lcm_gcd ac_simps normalize_mult dvd_normalize_div)
haftmann@60688
   404
  have "lcm (lcm a b) c dvd lcm a (lcm b c)"
haftmann@60688
   405
    and "lcm a (lcm b c) dvd lcm (lcm a b) c"
haftmann@60688
   406
    by (auto intro: lcm_least
haftmann@60686
   407
      dvd_trans [of b "lcm b c" "lcm a (lcm b c)"]
haftmann@60686
   408
      dvd_trans [of c "lcm b c" "lcm a (lcm b c)"]
haftmann@60686
   409
      dvd_trans [of a "lcm a b" "lcm (lcm a b) c"]
haftmann@60686
   410
      dvd_trans [of b "lcm a b" "lcm (lcm a b) c"])
haftmann@60686
   411
  then show "lcm (lcm a b) c = lcm a (lcm b c)"
haftmann@60688
   412
    by (rule associated_eqI) simp_all
haftmann@60686
   413
qed
haftmann@60686
   414
haftmann@64850
   415
sublocale lcm: bounded_quasi_semilattice lcm 1 0 normalize
haftmann@64850
   416
proof
haftmann@64850
   417
  show "lcm a a = normalize a" for a
haftmann@64850
   418
  proof -
haftmann@64850
   419
    have "lcm a a dvd a"
haftmann@64850
   420
      by (rule lcm_least) simp_all
haftmann@64850
   421
    then show ?thesis
haftmann@64850
   422
      by (auto intro: associated_eqI)
haftmann@64850
   423
  qed
haftmann@64850
   424
  show "lcm (normalize a) b = lcm a b" for a b
haftmann@64850
   425
    using dvd_lcm1 [of "normalize a" b] unfolding normalize_dvd_iff
haftmann@60688
   426
    by (auto intro: associated_eqI)
haftmann@64850
   427
  show "lcm 1 a = normalize a" for a
haftmann@64850
   428
    by (rule associated_eqI) simp_all
haftmann@64850
   429
qed simp_all
haftmann@64850
   430
haftmann@64850
   431
lemma lcm_self: "lcm a a = normalize a"
haftmann@64850
   432
  by (fact lcm.idem_normalize)
haftmann@64850
   433
haftmann@64850
   434
lemma lcm_left_idem: "lcm a (lcm a b) = lcm a b"
haftmann@64850
   435
  by (fact lcm.left_idem)
haftmann@64850
   436
haftmann@64850
   437
lemma lcm_right_idem: "lcm (lcm a b) b = lcm a b"
haftmann@64850
   438
  by (fact lcm.right_idem)
haftmann@61913
   439
wenzelm@63489
   440
lemma gcd_mult_lcm [simp]: "gcd a b * lcm a b = normalize a * normalize b"
haftmann@60686
   441
  by (simp add: lcm_gcd normalize_mult)
haftmann@60686
   442
wenzelm@63489
   443
lemma lcm_mult_gcd [simp]: "lcm a b * gcd a b = normalize a * normalize b"
wenzelm@63489
   444
  using gcd_mult_lcm [of a b] by (simp add: ac_simps)
haftmann@60686
   445
haftmann@60686
   446
lemma gcd_lcm:
haftmann@60686
   447
  assumes "a \<noteq> 0" and "b \<noteq> 0"
haftmann@60686
   448
  shows "gcd a b = normalize (a * b) div lcm a b"
haftmann@60686
   449
proof -
haftmann@60686
   450
  from assms have "lcm a b \<noteq> 0"
haftmann@60686
   451
    by (simp add: lcm_eq_0_iff)
wenzelm@63489
   452
  have "gcd a b * lcm a b = normalize a * normalize b"
wenzelm@63489
   453
    by simp
haftmann@60686
   454
  then have "gcd a b * lcm a b div lcm a b = normalize (a * b) div lcm a b"
haftmann@60686
   455
    by (simp_all add: normalize_mult)
haftmann@60686
   456
  with \<open>lcm a b \<noteq> 0\<close> show ?thesis
haftmann@64240
   457
    using nonzero_mult_div_cancel_right [of "lcm a b" "gcd a b"] by simp
haftmann@60686
   458
qed
haftmann@60686
   459
haftmann@64850
   460
lemma lcm_1_left: "lcm 1 a = normalize a"
haftmann@64850
   461
  by (fact lcm.top_left_normalize)
haftmann@64850
   462
haftmann@64850
   463
lemma lcm_1_right: "lcm a 1 = normalize a"
haftmann@64850
   464
  by (fact lcm.top_right_normalize)
wenzelm@63489
   465
wenzelm@63489
   466
lemma lcm_mult_left: "lcm (c * a) (c * b) = normalize c * lcm a b"
haftmann@60686
   467
  by (cases "c = 0")
haftmann@60686
   468
    (simp_all add: gcd_mult_right lcm_gcd div_mult_swap normalize_mult ac_simps,
haftmann@60686
   469
      simp add: dvd_div_mult2_eq mult.left_commute [of "normalize c", symmetric])
haftmann@60686
   470
wenzelm@63489
   471
lemma lcm_mult_right: "lcm (a * c) (b * c) = lcm b a * normalize c"
haftmann@60686
   472
  using lcm_mult_left [of c a b] by (simp add: ac_simps)
haftmann@60686
   473
wenzelm@63489
   474
lemma mult_lcm_left: "c * lcm a b = unit_factor c * lcm (c * a) (c * b)"
haftmann@60686
   475
  by (simp add: lcm_mult_left mult.assoc [symmetric])
haftmann@60686
   476
wenzelm@63489
   477
lemma mult_lcm_right: "lcm a b * c = lcm (a * c) (b * c) * unit_factor c"
haftmann@60686
   478
  using mult_lcm_left [of c a b] by (simp add: ac_simps)
eberlm@62429
   479
eberlm@62429
   480
lemma gcdI:
wenzelm@63489
   481
  assumes "c dvd a" and "c dvd b"
wenzelm@63489
   482
    and greatest: "\<And>d. d dvd a \<Longrightarrow> d dvd b \<Longrightarrow> d dvd c"
eberlm@62429
   483
    and "normalize c = c"
eberlm@62429
   484
  shows "c = gcd a b"
eberlm@62429
   485
  by (rule associated_eqI) (auto simp: assms intro: gcd_greatest)
eberlm@62429
   486
wenzelm@63489
   487
lemma gcd_unique:
wenzelm@63489
   488
  "d dvd a \<and> d dvd b \<and> normalize d = d \<and> (\<forall>e. e dvd a \<and> e dvd b \<longrightarrow> e dvd d) \<longleftrightarrow> d = gcd a b"
eberlm@62429
   489
  by rule (auto intro: gcdI simp: gcd_greatest)
eberlm@62429
   490
eberlm@62429
   491
lemma gcd_dvd_prod: "gcd a b dvd k * b"
eberlm@62429
   492
  using mult_dvd_mono [of 1] by auto
eberlm@62429
   493
eberlm@62429
   494
lemma gcd_proj2_if_dvd: "b dvd a \<Longrightarrow> gcd a b = normalize b"
eberlm@62429
   495
  by (rule gcdI [symmetric]) simp_all
eberlm@62429
   496
eberlm@62429
   497
lemma gcd_proj1_if_dvd: "a dvd b \<Longrightarrow> gcd a b = normalize a"
eberlm@62429
   498
  by (rule gcdI [symmetric]) simp_all
eberlm@62429
   499
eberlm@62429
   500
lemma gcd_proj1_iff: "gcd m n = normalize m \<longleftrightarrow> m dvd n"
eberlm@62429
   501
proof
wenzelm@63489
   502
  assume *: "gcd m n = normalize m"
eberlm@62429
   503
  show "m dvd n"
eberlm@62429
   504
  proof (cases "m = 0")
wenzelm@63489
   505
    case True
wenzelm@63489
   506
    with * show ?thesis by simp
wenzelm@63489
   507
  next
wenzelm@63489
   508
    case [simp]: False
wenzelm@63489
   509
    from * have **: "m = gcd m n * unit_factor m"
eberlm@62429
   510
      by (simp add: unit_eq_div2)
wenzelm@63489
   511
    show ?thesis
wenzelm@63489
   512
      by (subst **) (simp add: mult_unit_dvd_iff)
wenzelm@63489
   513
  qed
eberlm@62429
   514
next
eberlm@62429
   515
  assume "m dvd n"
wenzelm@63489
   516
  then show "gcd m n = normalize m"
wenzelm@63489
   517
    by (rule gcd_proj1_if_dvd)
eberlm@62429
   518
qed
wenzelm@63489
   519
eberlm@62429
   520
lemma gcd_proj2_iff: "gcd m n = normalize n \<longleftrightarrow> n dvd m"
eberlm@62429
   521
  using gcd_proj1_iff [of n m] by (simp add: ac_simps)
eberlm@62429
   522
eberlm@62429
   523
lemma gcd_mult_distrib': "normalize c * gcd a b = gcd (c * a) (c * b)"
eberlm@62429
   524
  by (rule gcdI) (auto simp: normalize_mult gcd_greatest mult_dvd_mono gcd_mult_left[symmetric])
eberlm@62429
   525
wenzelm@63489
   526
lemma gcd_mult_distrib: "k * gcd a b = gcd (k * a) (k * b) * unit_factor k"
eberlm@62429
   527
proof-
eberlm@62429
   528
  have "normalize k * gcd a b = gcd (k * a) (k * b)"
eberlm@62429
   529
    by (simp add: gcd_mult_distrib')
eberlm@62429
   530
  then have "normalize k * gcd a b * unit_factor k = gcd (k * a) (k * b) * unit_factor k"
eberlm@62429
   531
    by simp
eberlm@62429
   532
  then have "normalize k * unit_factor k * gcd a b  = gcd (k * a) (k * b) * unit_factor k"
eberlm@62429
   533
    by (simp only: ac_simps)
eberlm@62429
   534
  then show ?thesis
eberlm@62429
   535
    by simp
eberlm@62429
   536
qed
eberlm@62429
   537
wenzelm@63489
   538
lemma lcm_mult_unit1: "is_unit a \<Longrightarrow> lcm (b * a) c = lcm b c"
eberlm@62429
   539
  by (rule associated_eqI) (simp_all add: mult_unit_dvd_iff dvd_lcmI1)
eberlm@62429
   540
wenzelm@63489
   541
lemma lcm_mult_unit2: "is_unit a \<Longrightarrow> lcm b (c * a) = lcm b c"
eberlm@62429
   542
  using lcm_mult_unit1 [of a c b] by (simp add: ac_simps)
eberlm@62429
   543
eberlm@62429
   544
lemma lcm_div_unit1:
eberlm@62429
   545
  "is_unit a \<Longrightarrow> lcm (b div a) c = lcm b c"
wenzelm@63489
   546
  by (erule is_unitE [of _ b]) (simp add: lcm_mult_unit1)
wenzelm@63489
   547
wenzelm@63489
   548
lemma lcm_div_unit2: "is_unit a \<Longrightarrow> lcm b (c div a) = lcm b c"
eberlm@62429
   549
  by (erule is_unitE [of _ c]) (simp add: lcm_mult_unit2)
eberlm@62429
   550
haftmann@64850
   551
lemma normalize_lcm_left: "lcm (normalize a) b = lcm a b"
haftmann@64850
   552
  by (fact lcm.normalize_left_idem)
haftmann@64850
   553
haftmann@64850
   554
lemma normalize_lcm_right: "lcm a (normalize b) = lcm a b"
haftmann@64850
   555
  by (fact lcm.normalize_right_idem)
eberlm@62429
   556
lp15@68708
   557
lemma gcd_mult_unit1: 
lp15@68708
   558
  assumes "is_unit a" shows "gcd (b * a) c = gcd b c"
lp15@68708
   559
proof -
lp15@68708
   560
  have "gcd (b * a) c dvd b"
lp15@68708
   561
    using assms local.dvd_mult_unit_iff by blast
lp15@68708
   562
  then show ?thesis
lp15@68708
   563
    by (rule gcdI) simp_all
lp15@68708
   564
qed
eberlm@62429
   565
eberlm@62429
   566
lemma gcd_mult_unit2: "is_unit a \<Longrightarrow> gcd b (c * a) = gcd b c"
lp15@68708
   567
  using gcd.commute gcd_mult_unit1 by auto
eberlm@62429
   568
eberlm@62429
   569
lemma gcd_div_unit1: "is_unit a \<Longrightarrow> gcd (b div a) c = gcd b c"
eberlm@62429
   570
  by (erule is_unitE [of _ b]) (simp add: gcd_mult_unit1)
eberlm@62429
   571
eberlm@62429
   572
lemma gcd_div_unit2: "is_unit a \<Longrightarrow> gcd b (c div a) = gcd b c"
eberlm@62429
   573
  by (erule is_unitE [of _ c]) (simp add: gcd_mult_unit2)
eberlm@62429
   574
haftmann@64850
   575
lemma normalize_gcd_left: "gcd (normalize a) b = gcd a b"
haftmann@64850
   576
  by (fact gcd.normalize_left_idem)
haftmann@64850
   577
haftmann@64850
   578
lemma normalize_gcd_right: "gcd a (normalize b) = gcd a b"
haftmann@64850
   579
  by (fact gcd.normalize_right_idem)
eberlm@62429
   580
eberlm@62429
   581
lemma comp_fun_idem_gcd: "comp_fun_idem gcd"
eberlm@62429
   582
  by standard (simp_all add: fun_eq_iff ac_simps)
eberlm@62429
   583
eberlm@62429
   584
lemma comp_fun_idem_lcm: "comp_fun_idem lcm"
eberlm@62429
   585
  by standard (simp_all add: fun_eq_iff ac_simps)
eberlm@62429
   586
wenzelm@63489
   587
lemma gcd_dvd_antisym: "gcd a b dvd gcd c d \<Longrightarrow> gcd c d dvd gcd a b \<Longrightarrow> gcd a b = gcd c d"
eberlm@62429
   588
proof (rule gcdI)
wenzelm@63489
   589
  assume *: "gcd a b dvd gcd c d"
wenzelm@63489
   590
    and **: "gcd c d dvd gcd a b"
wenzelm@63489
   591
  have "gcd c d dvd c"
wenzelm@63489
   592
    by simp
wenzelm@63489
   593
  with * show "gcd a b dvd c"
wenzelm@63489
   594
    by (rule dvd_trans)
wenzelm@63489
   595
  have "gcd c d dvd d"
wenzelm@63489
   596
    by simp
wenzelm@63489
   597
  with * show "gcd a b dvd d"
wenzelm@63489
   598
    by (rule dvd_trans)
eberlm@62429
   599
  show "normalize (gcd a b) = gcd a b"
eberlm@62429
   600
    by simp
eberlm@62429
   601
  fix l assume "l dvd c" and "l dvd d"
wenzelm@63489
   602
  then have "l dvd gcd c d"
wenzelm@63489
   603
    by (rule gcd_greatest)
wenzelm@63489
   604
  from this and ** show "l dvd gcd a b"
wenzelm@63489
   605
    by (rule dvd_trans)
eberlm@62429
   606
qed
eberlm@62429
   607
eberlm@62429
   608
lemma gcd_add1 [simp]: "gcd (m + n) n = gcd m n"
eberlm@62429
   609
  by (rule gcdI [symmetric]) (simp_all add: dvd_add_left_iff)
eberlm@62429
   610
eberlm@62429
   611
lemma gcd_add2 [simp]: "gcd m (m + n) = gcd m n"
eberlm@62429
   612
  using gcd_add1 [of n m] by (simp add: ac_simps)
eberlm@62429
   613
eberlm@62429
   614
lemma gcd_add_mult: "gcd m (k * m + n) = gcd m n"
eberlm@62429
   615
  by (rule gcdI [symmetric]) (simp_all add: dvd_add_right_iff)
eberlm@62429
   616
wenzelm@63489
   617
lemma lcm_gcd_prod: "lcm a b * gcd a b = normalize (a * b)"
eberlm@62429
   618
  by (simp add: lcm_gcd)
eberlm@62429
   619
eberlm@62429
   620
declare unit_factor_lcm [simp]
eberlm@62429
   621
eberlm@62429
   622
lemma lcmI:
eberlm@62429
   623
  assumes "a dvd c" and "b dvd c" and "\<And>d. a dvd d \<Longrightarrow> b dvd d \<Longrightarrow> c dvd d"
eberlm@62429
   624
    and "normalize c = c"
eberlm@62429
   625
  shows "c = lcm a b"
eberlm@62429
   626
  by (rule associated_eqI) (auto simp: assms intro: lcm_least)
eberlm@62429
   627
wenzelm@63489
   628
lemma gcd_dvd_lcm [simp]: "gcd a b dvd lcm a b"
eberlm@62429
   629
  using gcd_dvd2 by (rule dvd_lcmI2)
eberlm@62429
   630
eberlm@62429
   631
lemmas lcm_0 = lcm_0_right
eberlm@62429
   632
eberlm@62429
   633
lemma lcm_unique:
wenzelm@63489
   634
  "a dvd d \<and> b dvd d \<and> normalize d = d \<and> (\<forall>e. a dvd e \<and> b dvd e \<longrightarrow> d dvd e) \<longleftrightarrow> d = lcm a b"
eberlm@62429
   635
  by rule (auto intro: lcmI simp: lcm_least lcm_eq_0_iff)
eberlm@62429
   636
lp15@68708
   637
lemma lcm_proj1_if_dvd:
lp15@68708
   638
  assumes "b dvd a" shows "lcm a b = normalize a"
lp15@68708
   639
proof (cases "a = 0")
lp15@68708
   640
  case False
lp15@68708
   641
  then show ?thesis
lp15@68708
   642
    using assms gcd_proj2_if_dvd lcm_mult_gcd local.lcm_gcd by auto
lp15@68708
   643
qed auto
wenzelm@63489
   644
wenzelm@63489
   645
lemma lcm_proj2_if_dvd: "a dvd b \<Longrightarrow> lcm a b = normalize b"
eberlm@62429
   646
  using lcm_proj1_if_dvd [of a b] by (simp add: ac_simps)
eberlm@62429
   647
wenzelm@63489
   648
lemma lcm_proj1_iff: "lcm m n = normalize m \<longleftrightarrow> n dvd m"
eberlm@62429
   649
proof
wenzelm@63489
   650
  assume *: "lcm m n = normalize m"
eberlm@62429
   651
  show "n dvd m"
eberlm@62429
   652
  proof (cases "m = 0")
wenzelm@63489
   653
    case True
wenzelm@63489
   654
    then show ?thesis by simp
wenzelm@63489
   655
  next
wenzelm@63489
   656
    case [simp]: False
wenzelm@63489
   657
    from * have **: "m = lcm m n * unit_factor m"
eberlm@62429
   658
      by (simp add: unit_eq_div2)
wenzelm@63489
   659
    show ?thesis by (subst **) simp
wenzelm@63489
   660
  qed
eberlm@62429
   661
next
eberlm@62429
   662
  assume "n dvd m"
wenzelm@63489
   663
  then show "lcm m n = normalize m"
wenzelm@63489
   664
    by (rule lcm_proj1_if_dvd)
eberlm@62429
   665
qed
eberlm@62429
   666
wenzelm@63489
   667
lemma lcm_proj2_iff: "lcm m n = normalize n \<longleftrightarrow> m dvd n"
eberlm@62429
   668
  using lcm_proj1_iff [of n m] by (simp add: ac_simps)
eberlm@62429
   669
haftmann@64850
   670
lemma lcm_mult_distrib': "normalize c * lcm a b = lcm (c * a) (c * b)"
haftmann@64850
   671
  by (rule lcmI) (auto simp: normalize_mult lcm_least mult_dvd_mono lcm_mult_left [symmetric])
haftmann@64850
   672
haftmann@64850
   673
lemma lcm_mult_distrib: "k * lcm a b = lcm (k * a) (k * b) * unit_factor k"
haftmann@64850
   674
proof-
haftmann@64850
   675
  have "normalize k * lcm a b = lcm (k * a) (k * b)"
haftmann@64850
   676
    by (simp add: lcm_mult_distrib')
haftmann@64850
   677
  then have "normalize k * lcm a b * unit_factor k = lcm (k * a) (k * b) * unit_factor k"
haftmann@64850
   678
    by simp
haftmann@64850
   679
  then have "normalize k * unit_factor k * lcm a b  = lcm (k * a) (k * b) * unit_factor k"
haftmann@64850
   680
    by (simp only: ac_simps)
haftmann@64850
   681
  then show ?thesis
haftmann@64850
   682
    by simp
haftmann@64850
   683
qed
haftmann@64850
   684
eberlm@69785
   685
lemma gcd_mono: "a dvd c \<Longrightarrow> b dvd d \<Longrightarrow> gcd a b dvd gcd c d"
eberlm@69785
   686
  by (simp add: gcd_dvdI1 gcd_dvdI2)
eberlm@69785
   687
eberlm@69785
   688
lemma lcm_mono: "a dvd c \<Longrightarrow> b dvd d \<Longrightarrow> lcm a b dvd lcm c d"
eberlm@69785
   689
  by (simp add: dvd_lcmI1 dvd_lcmI2)
eberlm@69785
   690
haftmann@63924
   691
lemma dvd_productE:
haftmann@67051
   692
  assumes "p dvd a * b"
haftmann@63924
   693
  obtains x y where "p = x * y" "x dvd a" "y dvd b"
haftmann@63924
   694
proof (cases "a = 0")
haftmann@63924
   695
  case True
haftmann@63924
   696
  thus ?thesis by (intro that[of p 1]) simp_all
haftmann@63924
   697
next
haftmann@63924
   698
  case False
haftmann@63924
   699
  define x y where "x = gcd a p" and "y = p div x"
haftmann@63924
   700
  have "p = x * y" by (simp add: x_def y_def)
haftmann@63924
   701
  moreover have "x dvd a" by (simp add: x_def)
haftmann@63924
   702
  moreover from assms have "p dvd gcd (b * a) (b * p)"
haftmann@63924
   703
    by (intro gcd_greatest) (simp_all add: mult.commute)
haftmann@63924
   704
  hence "p dvd b * gcd a p" by (simp add: gcd_mult_distrib)
wenzelm@65552
   705
  with False have "y dvd b"
haftmann@63924
   706
    by (simp add: x_def y_def div_dvd_iff_mult assms)
haftmann@63924
   707
  ultimately show ?thesis by (rule that)
haftmann@63924
   708
qed
haftmann@63924
   709
haftmann@60686
   710
end
haftmann@60686
   711
haftmann@62345
   712
class ring_gcd = comm_ring_1 + semiring_gcd
eberlm@62429
   713
begin
eberlm@62429
   714
wenzelm@63489
   715
lemma gcd_neg1 [simp]: "gcd (-a) b = gcd a b"
wenzelm@63489
   716
  by (rule sym, rule gcdI) (simp_all add: gcd_greatest)
wenzelm@63489
   717
wenzelm@63489
   718
lemma gcd_neg2 [simp]: "gcd a (-b) = gcd a b"
wenzelm@63489
   719
  by (rule sym, rule gcdI) (simp_all add: gcd_greatest)
wenzelm@63489
   720
wenzelm@63489
   721
lemma gcd_neg_numeral_1 [simp]: "gcd (- numeral n) a = gcd (numeral n) a"
eberlm@62429
   722
  by (fact gcd_neg1)
eberlm@62429
   723
wenzelm@63489
   724
lemma gcd_neg_numeral_2 [simp]: "gcd a (- numeral n) = gcd a (numeral n)"
eberlm@62429
   725
  by (fact gcd_neg2)
eberlm@62429
   726
eberlm@62429
   727
lemma gcd_diff1: "gcd (m - n) n = gcd m n"
wenzelm@63489
   728
  by (subst diff_conv_add_uminus, subst gcd_neg2[symmetric], subst gcd_add1, simp)
eberlm@62429
   729
eberlm@62429
   730
lemma gcd_diff2: "gcd (n - m) n = gcd m n"
wenzelm@63489
   731
  by (subst gcd_neg1[symmetric]) (simp only: minus_diff_eq gcd_diff1)
eberlm@62429
   732
eberlm@62429
   733
lemma lcm_neg1 [simp]: "lcm (-a) b = lcm a b"
wenzelm@63489
   734
  by (rule sym, rule lcmI) (simp_all add: lcm_least lcm_eq_0_iff)
eberlm@62429
   735
eberlm@62429
   736
lemma lcm_neg2 [simp]: "lcm a (-b) = lcm a b"
wenzelm@63489
   737
  by (rule sym, rule lcmI) (simp_all add: lcm_least lcm_eq_0_iff)
eberlm@62429
   738
eberlm@62429
   739
lemma lcm_neg_numeral_1 [simp]: "lcm (- numeral n) a = lcm (numeral n) a"
eberlm@62429
   740
  by (fact lcm_neg1)
eberlm@62429
   741
eberlm@62429
   742
lemma lcm_neg_numeral_2 [simp]: "lcm a (- numeral n) = lcm a (numeral n)"
eberlm@62429
   743
  by (fact lcm_neg2)
eberlm@62429
   744
eberlm@62429
   745
end
haftmann@62345
   746
haftmann@60686
   747
class semiring_Gcd = semiring_gcd + Gcd +
haftmann@60686
   748
  assumes Gcd_dvd: "a \<in> A \<Longrightarrow> Gcd A dvd a"
haftmann@60686
   749
    and Gcd_greatest: "(\<And>b. b \<in> A \<Longrightarrow> a dvd b) \<Longrightarrow> a dvd Gcd A"
haftmann@60686
   750
    and normalize_Gcd [simp]: "normalize (Gcd A) = Gcd A"
haftmann@62345
   751
  assumes dvd_Lcm: "a \<in> A \<Longrightarrow> a dvd Lcm A"
haftmann@62345
   752
    and Lcm_least: "(\<And>b. b \<in> A \<Longrightarrow> b dvd a) \<Longrightarrow> Lcm A dvd a"
haftmann@62345
   753
    and normalize_Lcm [simp]: "normalize (Lcm A) = Lcm A"
haftmann@60686
   754
begin
haftmann@60686
   755
wenzelm@63489
   756
lemma Lcm_Gcd: "Lcm A = Gcd {b. \<forall>a\<in>A. a dvd b}"
haftmann@62345
   757
  by (rule associated_eqI) (auto intro: Gcd_dvd dvd_Lcm Gcd_greatest Lcm_least)
haftmann@62345
   758
wenzelm@63489
   759
lemma Gcd_Lcm: "Gcd A = Lcm {b. \<forall>a\<in>A. b dvd a}"
haftmann@62345
   760
  by (rule associated_eqI) (auto intro: Gcd_dvd dvd_Lcm Gcd_greatest Lcm_least)
haftmann@62345
   761
wenzelm@63489
   762
lemma Gcd_empty [simp]: "Gcd {} = 0"
haftmann@60686
   763
  by (rule dvd_0_left, rule Gcd_greatest) simp
haftmann@60686
   764
wenzelm@63489
   765
lemma Lcm_empty [simp]: "Lcm {} = 1"
haftmann@62345
   766
  by (auto intro: associated_eqI Lcm_least)
haftmann@62345
   767
wenzelm@63489
   768
lemma Gcd_insert [simp]: "Gcd (insert a A) = gcd a (Gcd A)"
haftmann@62345
   769
proof -
haftmann@62345
   770
  have "Gcd (insert a A) dvd gcd a (Gcd A)"
haftmann@62345
   771
    by (auto intro: Gcd_dvd Gcd_greatest)
haftmann@62345
   772
  moreover have "gcd a (Gcd A) dvd Gcd (insert a A)"
haftmann@62345
   773
  proof (rule Gcd_greatest)
haftmann@62345
   774
    fix b
haftmann@62345
   775
    assume "b \<in> insert a A"
haftmann@62345
   776
    then show "gcd a (Gcd A) dvd b"
haftmann@62345
   777
    proof
wenzelm@63489
   778
      assume "b = a"
wenzelm@63489
   779
      then show ?thesis
wenzelm@63489
   780
        by simp
haftmann@62345
   781
    next
haftmann@62345
   782
      assume "b \<in> A"
wenzelm@63489
   783
      then have "Gcd A dvd b"
wenzelm@63489
   784
        by (rule Gcd_dvd)
wenzelm@63489
   785
      moreover have "gcd a (Gcd A) dvd Gcd A"
wenzelm@63489
   786
        by simp
wenzelm@63489
   787
      ultimately show ?thesis
wenzelm@63489
   788
        by (blast intro: dvd_trans)
haftmann@62345
   789
    qed
haftmann@62345
   790
  qed
haftmann@62345
   791
  ultimately show ?thesis
haftmann@62345
   792
    by (auto intro: associated_eqI)
haftmann@62345
   793
qed
haftmann@62345
   794
wenzelm@63489
   795
lemma Lcm_insert [simp]: "Lcm (insert a A) = lcm a (Lcm A)"
haftmann@62345
   796
proof (rule sym)
haftmann@62345
   797
  have "lcm a (Lcm A) dvd Lcm (insert a A)"
haftmann@62345
   798
    by (auto intro: dvd_Lcm Lcm_least)
haftmann@62345
   799
  moreover have "Lcm (insert a A) dvd lcm a (Lcm A)"
haftmann@62345
   800
  proof (rule Lcm_least)
haftmann@62345
   801
    fix b
haftmann@62345
   802
    assume "b \<in> insert a A"
haftmann@62345
   803
    then show "b dvd lcm a (Lcm A)"
haftmann@62345
   804
    proof
wenzelm@63489
   805
      assume "b = a"
wenzelm@63489
   806
      then show ?thesis by simp
haftmann@62345
   807
    next
haftmann@62345
   808
      assume "b \<in> A"
wenzelm@63489
   809
      then have "b dvd Lcm A"
wenzelm@63489
   810
        by (rule dvd_Lcm)
wenzelm@63489
   811
      moreover have "Lcm A dvd lcm a (Lcm A)"
wenzelm@63489
   812
        by simp
wenzelm@63489
   813
      ultimately show ?thesis
wenzelm@63489
   814
        by (blast intro: dvd_trans)
haftmann@62345
   815
    qed
haftmann@62345
   816
  qed
haftmann@62345
   817
  ultimately show "lcm a (Lcm A) = Lcm (insert a A)"
haftmann@62345
   818
    by (rule associated_eqI) (simp_all add: lcm_eq_0_iff)
haftmann@62345
   819
qed
haftmann@62345
   820
eberlm@62429
   821
lemma LcmI:
wenzelm@63489
   822
  assumes "\<And>a. a \<in> A \<Longrightarrow> a dvd b"
wenzelm@63489
   823
    and "\<And>c. (\<And>a. a \<in> A \<Longrightarrow> a dvd c) \<Longrightarrow> b dvd c"
wenzelm@63489
   824
    and "normalize b = b"
wenzelm@63489
   825
  shows "b = Lcm A"
eberlm@62429
   826
  by (rule associated_eqI) (auto simp: assms dvd_Lcm intro: Lcm_least)
eberlm@62429
   827
wenzelm@63489
   828
lemma Lcm_subset: "A \<subseteq> B \<Longrightarrow> Lcm A dvd Lcm B"
eberlm@62429
   829
  by (blast intro: Lcm_least dvd_Lcm)
eberlm@62429
   830
wenzelm@63489
   831
lemma Lcm_Un: "Lcm (A \<union> B) = lcm (Lcm A) (Lcm B)"
lp15@68708
   832
proof -
lp15@68708
   833
  have "\<And>d. \<lbrakk>Lcm A dvd d; Lcm B dvd d\<rbrakk> \<Longrightarrow> Lcm (A \<union> B) dvd d"
lp15@68708
   834
    by (meson UnE local.Lcm_least local.dvd_Lcm local.dvd_trans)
lp15@68708
   835
  then show ?thesis
lp15@68708
   836
    by (meson Lcm_subset local.lcm_unique local.normalize_Lcm sup.cobounded1 sup.cobounded2)
lp15@68708
   837
qed
wenzelm@63489
   838
wenzelm@63489
   839
lemma Gcd_0_iff [simp]: "Gcd A = 0 \<longleftrightarrow> A \<subseteq> {0}"
wenzelm@63489
   840
  (is "?P \<longleftrightarrow> ?Q")
haftmann@60686
   841
proof
haftmann@60686
   842
  assume ?P
haftmann@60686
   843
  show ?Q
haftmann@60686
   844
  proof
haftmann@60686
   845
    fix a
haftmann@60686
   846
    assume "a \<in> A"
wenzelm@63489
   847
    then have "Gcd A dvd a"
wenzelm@63489
   848
      by (rule Gcd_dvd)
wenzelm@63489
   849
    with \<open>?P\<close> have "a = 0"
wenzelm@63489
   850
      by simp
wenzelm@63489
   851
    then show "a \<in> {0}"
wenzelm@63489
   852
      by simp
haftmann@60686
   853
  qed
haftmann@60686
   854
next
haftmann@60686
   855
  assume ?Q
haftmann@60686
   856
  have "0 dvd Gcd A"
haftmann@60686
   857
  proof (rule Gcd_greatest)
haftmann@60686
   858
    fix a
haftmann@60686
   859
    assume "a \<in> A"
wenzelm@63489
   860
    with \<open>?Q\<close> have "a = 0"
wenzelm@63489
   861
      by auto
wenzelm@63489
   862
    then show "0 dvd a"
wenzelm@63489
   863
      by simp
haftmann@60686
   864
  qed
wenzelm@63489
   865
  then show ?P
wenzelm@63489
   866
    by simp
haftmann@60686
   867
qed
haftmann@60686
   868
wenzelm@63489
   869
lemma Lcm_1_iff [simp]: "Lcm A = 1 \<longleftrightarrow> (\<forall>a\<in>A. is_unit a)"
wenzelm@63489
   870
  (is "?P \<longleftrightarrow> ?Q")
haftmann@60686
   871
proof
haftmann@60686
   872
  assume ?P
haftmann@60686
   873
  show ?Q
haftmann@60686
   874
  proof
haftmann@60686
   875
    fix a
haftmann@60686
   876
    assume "a \<in> A"
haftmann@60686
   877
    then have "a dvd Lcm A"
haftmann@60686
   878
      by (rule dvd_Lcm)
haftmann@60686
   879
    with \<open>?P\<close> show "is_unit a"
haftmann@60686
   880
      by simp
haftmann@60686
   881
  qed
haftmann@60686
   882
next
haftmann@60686
   883
  assume ?Q
haftmann@60686
   884
  then have "is_unit (Lcm A)"
haftmann@60686
   885
    by (blast intro: Lcm_least)
haftmann@60686
   886
  then have "normalize (Lcm A) = 1"
haftmann@60686
   887
    by (rule is_unit_normalize)
haftmann@60686
   888
  then show ?P
haftmann@60686
   889
    by simp
haftmann@60686
   890
qed
haftmann@60686
   891
wenzelm@63489
   892
lemma unit_factor_Lcm: "unit_factor (Lcm A) = (if Lcm A = 0 then 0 else 1)"
haftmann@62345
   893
proof (cases "Lcm A = 0")
wenzelm@63489
   894
  case True
wenzelm@63489
   895
  then show ?thesis
wenzelm@63489
   896
    by simp
haftmann@62345
   897
next
haftmann@62345
   898
  case False
haftmann@62345
   899
  with unit_factor_normalize have "unit_factor (normalize (Lcm A)) = 1"
haftmann@62345
   900
    by blast
haftmann@62345
   901
  with False show ?thesis
haftmann@62345
   902
    by simp
haftmann@62345
   903
qed
haftmann@62345
   904
eberlm@62429
   905
lemma unit_factor_Gcd: "unit_factor (Gcd A) = (if Gcd A = 0 then 0 else 1)"
wenzelm@63489
   906
  by (simp add: Gcd_Lcm unit_factor_Lcm)
eberlm@62429
   907
eberlm@62429
   908
lemma GcdI:
wenzelm@63489
   909
  assumes "\<And>a. a \<in> A \<Longrightarrow> b dvd a"
wenzelm@63489
   910
    and "\<And>c. (\<And>a. a \<in> A \<Longrightarrow> c dvd a) \<Longrightarrow> c dvd b"
eberlm@62429
   911
    and "normalize b = b"
eberlm@62429
   912
  shows "b = Gcd A"
eberlm@62429
   913
  by (rule associated_eqI) (auto simp: assms Gcd_dvd intro: Gcd_greatest)
eberlm@62429
   914
haftmann@62345
   915
lemma Gcd_eq_1_I:
haftmann@62345
   916
  assumes "is_unit a" and "a \<in> A"
haftmann@62345
   917
  shows "Gcd A = 1"
haftmann@62345
   918
proof -
haftmann@62345
   919
  from assms have "is_unit (Gcd A)"
haftmann@62345
   920
    by (blast intro: Gcd_dvd dvd_unit_imp_unit)
haftmann@62345
   921
  then have "normalize (Gcd A) = 1"
haftmann@62345
   922
    by (rule is_unit_normalize)
haftmann@62345
   923
  then show ?thesis
haftmann@62345
   924
    by simp
haftmann@62345
   925
qed
haftmann@62345
   926
haftmann@60686
   927
lemma Lcm_eq_0_I:
haftmann@60686
   928
  assumes "0 \<in> A"
haftmann@60686
   929
  shows "Lcm A = 0"
haftmann@60686
   930
proof -
haftmann@60686
   931
  from assms have "0 dvd Lcm A"
haftmann@60686
   932
    by (rule dvd_Lcm)
haftmann@60686
   933
  then show ?thesis
haftmann@60686
   934
    by simp
haftmann@60686
   935
qed
haftmann@60686
   936
wenzelm@63489
   937
lemma Gcd_UNIV [simp]: "Gcd UNIV = 1"
haftmann@62345
   938
  using dvd_refl by (rule Gcd_eq_1_I) simp
haftmann@62345
   939
wenzelm@63489
   940
lemma Lcm_UNIV [simp]: "Lcm UNIV = 0"
haftmann@61929
   941
  by (rule Lcm_eq_0_I) simp
haftmann@60686
   942
haftmann@61929
   943
lemma Lcm_0_iff:
haftmann@61929
   944
  assumes "finite A"
haftmann@61929
   945
  shows "Lcm A = 0 \<longleftrightarrow> 0 \<in> A"
haftmann@61929
   946
proof (cases "A = {}")
wenzelm@63489
   947
  case True
wenzelm@63489
   948
  then show ?thesis by simp
haftmann@61929
   949
next
wenzelm@63489
   950
  case False
wenzelm@63489
   951
  with assms show ?thesis
lp15@68708
   952
    by (induct A rule: finite_ne_induct) (auto simp: lcm_eq_0_iff)
haftmann@60686
   953
qed
haftmann@61929
   954
wenzelm@63489
   955
lemma Gcd_image_normalize [simp]: "Gcd (normalize ` A) = Gcd A"
haftmann@62345
   956
proof -
haftmann@62345
   957
  have "Gcd (normalize ` A) dvd a" if "a \<in> A" for a
haftmann@62345
   958
  proof -
wenzelm@63489
   959
    from that obtain B where "A = insert a B"
wenzelm@63489
   960
      by blast
haftmann@62350
   961
    moreover have "gcd (normalize a) (Gcd (normalize ` B)) dvd normalize a"
haftmann@62345
   962
      by (rule gcd_dvd1)
haftmann@62345
   963
    ultimately show "Gcd (normalize ` A) dvd a"
haftmann@62345
   964
      by simp
haftmann@62345
   965
  qed
haftmann@62345
   966
  then have "Gcd (normalize ` A) dvd Gcd A" and "Gcd A dvd Gcd (normalize ` A)"
haftmann@62345
   967
    by (auto intro!: Gcd_greatest intro: Gcd_dvd)
haftmann@62345
   968
  then show ?thesis
haftmann@62345
   969
    by (auto intro: associated_eqI)
haftmann@62345
   970
qed
haftmann@62345
   971
haftmann@62346
   972
lemma Gcd_eqI:
haftmann@62346
   973
  assumes "normalize a = a"
haftmann@62346
   974
  assumes "\<And>b. b \<in> A \<Longrightarrow> a dvd b"
haftmann@62346
   975
    and "\<And>c. (\<And>b. b \<in> A \<Longrightarrow> c dvd b) \<Longrightarrow> c dvd a"
haftmann@62346
   976
  shows "Gcd A = a"
haftmann@62346
   977
  using assms by (blast intro: associated_eqI Gcd_greatest Gcd_dvd normalize_Gcd)
haftmann@62346
   978
wenzelm@63489
   979
lemma dvd_GcdD: "x dvd Gcd A \<Longrightarrow> y \<in> A \<Longrightarrow> x dvd y"
wenzelm@63489
   980
  using Gcd_dvd dvd_trans by blast
wenzelm@63489
   981
wenzelm@63489
   982
lemma dvd_Gcd_iff: "x dvd Gcd A \<longleftrightarrow> (\<forall>y\<in>A. x dvd y)"
eberlm@63359
   983
  by (blast dest: dvd_GcdD intro: Gcd_greatest)
eberlm@63359
   984
nipkow@69064
   985
lemma Gcd_mult: "Gcd ((*) c ` A) = normalize c * Gcd A"
eberlm@63359
   986
proof (cases "c = 0")
wenzelm@63489
   987
  case True
wenzelm@63489
   988
  then show ?thesis by auto
wenzelm@63489
   989
next
eberlm@63359
   990
  case [simp]: False
nipkow@69064
   991
  have "Gcd ((*) c ` A) div c dvd Gcd A"
eberlm@63359
   992
    by (intro Gcd_greatest, subst div_dvd_iff_mult)
eberlm@63359
   993
       (auto intro!: Gcd_greatest Gcd_dvd simp: mult.commute[of _ c])
nipkow@69064
   994
  then have "Gcd ((*) c ` A) dvd c * Gcd A"
eberlm@63359
   995
    by (subst (asm) div_dvd_iff_mult) (auto intro: Gcd_greatest simp: mult_ac)
eberlm@63359
   996
  also have "c * Gcd A = (normalize c * Gcd A) * unit_factor c"
eberlm@63359
   997
    by (subst unit_factor_mult_normalize [symmetric]) (simp only: mult_ac)
nipkow@69064
   998
  also have "Gcd ((*) c ` A) dvd \<dots> \<longleftrightarrow> Gcd ((*) c ` A) dvd normalize c * Gcd A"
eberlm@63359
   999
    by (simp add: dvd_mult_unit_iff)
nipkow@69064
  1000
  finally have "Gcd ((*) c ` A) dvd normalize c * Gcd A" .
nipkow@69064
  1001
  moreover have "normalize c * Gcd A dvd Gcd ((*) c ` A)"
eberlm@63359
  1002
    by (intro Gcd_greatest) (auto intro: mult_dvd_mono Gcd_dvd)
nipkow@69064
  1003
  ultimately have "normalize (Gcd ((*) c ` A)) = normalize (normalize c * Gcd A)"
eberlm@63359
  1004
    by (rule associatedI)
wenzelm@63489
  1005
  then show ?thesis
wenzelm@63489
  1006
    by (simp add: normalize_mult)
wenzelm@63489
  1007
qed
eberlm@63359
  1008
haftmann@62346
  1009
lemma Lcm_eqI:
haftmann@62346
  1010
  assumes "normalize a = a"
wenzelm@63489
  1011
    and "\<And>b. b \<in> A \<Longrightarrow> b dvd a"
haftmann@62346
  1012
    and "\<And>c. (\<And>b. b \<in> A \<Longrightarrow> b dvd c) \<Longrightarrow> a dvd c"
haftmann@62346
  1013
  shows "Lcm A = a"
haftmann@62346
  1014
  using assms by (blast intro: associated_eqI Lcm_least dvd_Lcm normalize_Lcm)
haftmann@62346
  1015
wenzelm@63489
  1016
lemma Lcm_dvdD: "Lcm A dvd x \<Longrightarrow> y \<in> A \<Longrightarrow> y dvd x"
wenzelm@63489
  1017
  using dvd_Lcm dvd_trans by blast
wenzelm@63489
  1018
wenzelm@63489
  1019
lemma Lcm_dvd_iff: "Lcm A dvd x \<longleftrightarrow> (\<forall>y\<in>A. y dvd x)"
eberlm@63359
  1020
  by (blast dest: Lcm_dvdD intro: Lcm_least)
eberlm@63359
  1021
wenzelm@63489
  1022
lemma Lcm_mult:
eberlm@63359
  1023
  assumes "A \<noteq> {}"
nipkow@69064
  1024
  shows "Lcm ((*) c ` A) = normalize c * Lcm A"
eberlm@63359
  1025
proof (cases "c = 0")
eberlm@63359
  1026
  case True
nipkow@69064
  1027
  with assms have "(*) c ` A = {0}"
wenzelm@63489
  1028
    by auto
wenzelm@63489
  1029
  with True show ?thesis by auto
eberlm@63359
  1030
next
eberlm@63359
  1031
  case [simp]: False
wenzelm@63489
  1032
  from assms obtain x where x: "x \<in> A"
wenzelm@63489
  1033
    by blast
wenzelm@63489
  1034
  have "c dvd c * x"
wenzelm@63489
  1035
    by simp
nipkow@69064
  1036
  also from x have "c * x dvd Lcm ((*) c ` A)"
wenzelm@63489
  1037
    by (intro dvd_Lcm) auto
nipkow@69064
  1038
  finally have dvd: "c dvd Lcm ((*) c ` A)" .
haftmann@69768
  1039
  moreover have "Lcm A dvd Lcm ((*) c ` A) div c"
eberlm@63359
  1040
    by (intro Lcm_least dvd_mult_imp_div)
wenzelm@63489
  1041
      (auto intro!: Lcm_least dvd_Lcm simp: mult.commute[of _ c])
haftmann@69768
  1042
  ultimately have "c * Lcm A dvd Lcm ((*) c ` A)"
haftmann@69768
  1043
    by auto
eberlm@63359
  1044
  also have "c * Lcm A = (normalize c * Lcm A) * unit_factor c"
eberlm@63359
  1045
    by (subst unit_factor_mult_normalize [symmetric]) (simp only: mult_ac)
nipkow@69064
  1046
  also have "\<dots> dvd Lcm ((*) c ` A) \<longleftrightarrow> normalize c * Lcm A dvd Lcm ((*) c ` A)"
eberlm@63359
  1047
    by (simp add: mult_unit_dvd_iff)
nipkow@69064
  1048
  finally have "normalize c * Lcm A dvd Lcm ((*) c ` A)" .
nipkow@69064
  1049
  moreover have "Lcm ((*) c ` A) dvd normalize c * Lcm A"
eberlm@63359
  1050
    by (intro Lcm_least) (auto intro: mult_dvd_mono dvd_Lcm)
nipkow@69064
  1051
  ultimately have "normalize (normalize c * Lcm A) = normalize (Lcm ((*) c ` A))"
eberlm@63359
  1052
    by (rule associatedI)
wenzelm@63489
  1053
  then show ?thesis
wenzelm@63489
  1054
    by (simp add: normalize_mult)
eberlm@62429
  1055
qed
eberlm@62429
  1056
wenzelm@63489
  1057
lemma Lcm_no_units: "Lcm A = Lcm (A - {a. is_unit a})"
wenzelm@63489
  1058
proof -
wenzelm@63489
  1059
  have "(A - {a. is_unit a}) \<union> {a\<in>A. is_unit a} = A"
wenzelm@63489
  1060
    by blast
wenzelm@63489
  1061
  then have "Lcm A = lcm (Lcm (A - {a. is_unit a})) (Lcm {a\<in>A. is_unit a})"
wenzelm@63489
  1062
    by (simp add: Lcm_Un [symmetric])
wenzelm@63489
  1063
  also have "Lcm {a\<in>A. is_unit a} = 1"
wenzelm@63489
  1064
    by simp
wenzelm@63489
  1065
  finally show ?thesis
wenzelm@63489
  1066
    by simp
wenzelm@63489
  1067
qed
wenzelm@63489
  1068
wenzelm@63489
  1069
lemma Lcm_0_iff': "Lcm A = 0 \<longleftrightarrow> (\<nexists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l))"
eberlm@62429
  1070
  by (metis Lcm_least dvd_0_left dvd_Lcm)
eberlm@62429
  1071
wenzelm@63489
  1072
lemma Lcm_no_multiple: "(\<forall>m. m \<noteq> 0 \<longrightarrow> (\<exists>a\<in>A. \<not> a dvd m)) \<Longrightarrow> Lcm A = 0"
eberlm@62429
  1073
  by (auto simp: Lcm_0_iff')
eberlm@62429
  1074
wenzelm@63489
  1075
lemma Lcm_singleton [simp]: "Lcm {a} = normalize a"
eberlm@62429
  1076
  by simp
eberlm@62429
  1077
wenzelm@63489
  1078
lemma Lcm_2 [simp]: "Lcm {a, b} = lcm a b"
eberlm@62429
  1079
  by simp
eberlm@62429
  1080
wenzelm@63489
  1081
lemma Gcd_1: "1 \<in> A \<Longrightarrow> Gcd A = 1"
eberlm@62429
  1082
  by (auto intro!: Gcd_eq_1_I)
eberlm@62429
  1083
eberlm@62429
  1084
lemma Gcd_singleton [simp]: "Gcd {a} = normalize a"
eberlm@62429
  1085
  by simp
eberlm@62429
  1086
wenzelm@63489
  1087
lemma Gcd_2 [simp]: "Gcd {a, b} = gcd a b"
eberlm@62429
  1088
  by simp
eberlm@62429
  1089
eberlm@69785
  1090
lemma Gcd_mono:
eberlm@69785
  1091
  assumes "\<And>x. x \<in> A \<Longrightarrow> f x dvd g x"
eberlm@69785
  1092
  shows   "(GCD x\<in>A. f x) dvd (GCD x\<in>A. g x)"
eberlm@69785
  1093
proof (intro Gcd_greatest, safe)
eberlm@69785
  1094
  fix x assume "x \<in> A"
eberlm@69785
  1095
  hence "(GCD x\<in>A. f x) dvd f x"
eberlm@69785
  1096
    by (intro Gcd_dvd) auto
eberlm@69785
  1097
  also have "f x dvd g x"
eberlm@69785
  1098
    using \<open>x \<in> A\<close> assms by blast
eberlm@69785
  1099
  finally show "(GCD x\<in>A. f x) dvd \<dots>" .
eberlm@69785
  1100
qed
eberlm@69785
  1101
eberlm@69785
  1102
lemma Lcm_mono:
eberlm@69785
  1103
  assumes "\<And>x. x \<in> A \<Longrightarrow> f x dvd g x"
eberlm@69785
  1104
  shows   "(LCM x\<in>A. f x) dvd (LCM x\<in>A. g x)"
eberlm@69785
  1105
proof (intro Lcm_least, safe)
eberlm@69785
  1106
  fix x assume "x \<in> A"
eberlm@69785
  1107
  hence "f x dvd g x" by (rule assms)
eberlm@69785
  1108
  also have "g x dvd (LCM x\<in>A. g x)"
eberlm@69785
  1109
    using \<open>x \<in> A\<close> by (intro dvd_Lcm) auto
eberlm@69785
  1110
  finally show "f x dvd \<dots>" .
eberlm@69785
  1111
qed
eberlm@69785
  1112
haftmann@62350
  1113
end
haftmann@62345
  1114
wenzelm@65552
  1115
haftmann@64850
  1116
subsection \<open>An aside: GCD and LCM on finite sets for incomplete gcd rings\<close>
haftmann@64850
  1117
haftmann@64850
  1118
context semiring_gcd
haftmann@64850
  1119
begin
haftmann@64850
  1120
haftmann@64850
  1121
sublocale Gcd_fin: bounded_quasi_semilattice_set gcd 0 1 normalize
haftmann@64850
  1122
defines
nipkow@69038
  1123
  Gcd_fin ("Gcd\<^sub>f\<^sub>i\<^sub>n") = "Gcd_fin.F :: 'a set \<Rightarrow> 'a" ..
haftmann@64850
  1124
haftmann@64850
  1125
abbreviation gcd_list :: "'a list \<Rightarrow> 'a"
haftmann@64850
  1126
  where "gcd_list xs \<equiv> Gcd\<^sub>f\<^sub>i\<^sub>n (set xs)"
haftmann@64850
  1127
haftmann@64850
  1128
sublocale Lcm_fin: bounded_quasi_semilattice_set lcm 1 0 normalize
haftmann@64850
  1129
defines
nipkow@69038
  1130
  Lcm_fin ("Lcm\<^sub>f\<^sub>i\<^sub>n") = Lcm_fin.F ..
haftmann@64850
  1131
haftmann@64850
  1132
abbreviation lcm_list :: "'a list \<Rightarrow> 'a"
haftmann@64850
  1133
  where "lcm_list xs \<equiv> Lcm\<^sub>f\<^sub>i\<^sub>n (set xs)"
wenzelm@65552
  1134
haftmann@64850
  1135
lemma Gcd_fin_dvd:
haftmann@64850
  1136
  "a \<in> A \<Longrightarrow> Gcd\<^sub>f\<^sub>i\<^sub>n A dvd a"
wenzelm@65552
  1137
  by (induct A rule: infinite_finite_induct)
haftmann@64850
  1138
    (auto intro: dvd_trans)
haftmann@64850
  1139
haftmann@64850
  1140
lemma dvd_Lcm_fin:
haftmann@64850
  1141
  "a \<in> A \<Longrightarrow> a dvd Lcm\<^sub>f\<^sub>i\<^sub>n A"
wenzelm@65552
  1142
  by (induct A rule: infinite_finite_induct)
haftmann@64850
  1143
    (auto intro: dvd_trans)
haftmann@64850
  1144
haftmann@64850
  1145
lemma Gcd_fin_greatest:
haftmann@64850
  1146
  "a dvd Gcd\<^sub>f\<^sub>i\<^sub>n A" if "finite A" and "\<And>b. b \<in> A \<Longrightarrow> a dvd b"
haftmann@64850
  1147
  using that by (induct A) simp_all
haftmann@64850
  1148
haftmann@64850
  1149
lemma Lcm_fin_least:
haftmann@64850
  1150
  "Lcm\<^sub>f\<^sub>i\<^sub>n A dvd a" if "finite A" and "\<And>b. b \<in> A \<Longrightarrow> b dvd a"
haftmann@64850
  1151
  using that by (induct A) simp_all
haftmann@64850
  1152
haftmann@64850
  1153
lemma gcd_list_greatest:
haftmann@64850
  1154
  "a dvd gcd_list bs" if "\<And>b. b \<in> set bs \<Longrightarrow> a dvd b"
haftmann@64850
  1155
  by (rule Gcd_fin_greatest) (simp_all add: that)
haftmann@64850
  1156
haftmann@64850
  1157
lemma lcm_list_least:
haftmann@64850
  1158
  "lcm_list bs dvd a" if "\<And>b. b \<in> set bs \<Longrightarrow> b dvd a"
haftmann@64850
  1159
  by (rule Lcm_fin_least) (simp_all add: that)
haftmann@64850
  1160
haftmann@64850
  1161
lemma dvd_Gcd_fin_iff:
haftmann@64850
  1162
  "b dvd Gcd\<^sub>f\<^sub>i\<^sub>n A \<longleftrightarrow> (\<forall>a\<in>A. b dvd a)" if "finite A"
haftmann@64850
  1163
  using that by (auto intro: Gcd_fin_greatest Gcd_fin_dvd dvd_trans [of b "Gcd\<^sub>f\<^sub>i\<^sub>n A"])
haftmann@64850
  1164
haftmann@64850
  1165
lemma dvd_gcd_list_iff:
haftmann@64850
  1166
  "b dvd gcd_list xs \<longleftrightarrow> (\<forall>a\<in>set xs. b dvd a)"
haftmann@64850
  1167
  by (simp add: dvd_Gcd_fin_iff)
wenzelm@65552
  1168
haftmann@64850
  1169
lemma Lcm_fin_dvd_iff:
haftmann@64850
  1170
  "Lcm\<^sub>f\<^sub>i\<^sub>n A dvd b  \<longleftrightarrow> (\<forall>a\<in>A. a dvd b)" if "finite A"
haftmann@64850
  1171
  using that by (auto intro: Lcm_fin_least dvd_Lcm_fin dvd_trans [of _ "Lcm\<^sub>f\<^sub>i\<^sub>n A" b])
haftmann@64850
  1172
haftmann@64850
  1173
lemma lcm_list_dvd_iff:
haftmann@64850
  1174
  "lcm_list xs dvd b  \<longleftrightarrow> (\<forall>a\<in>set xs. a dvd b)"
haftmann@64850
  1175
  by (simp add: Lcm_fin_dvd_iff)
haftmann@64850
  1176
haftmann@64850
  1177
lemma Gcd_fin_mult:
haftmann@64850
  1178
  "Gcd\<^sub>f\<^sub>i\<^sub>n (image (times b) A) = normalize b * Gcd\<^sub>f\<^sub>i\<^sub>n A" if "finite A"
haftmann@64850
  1179
using that proof induct
haftmann@64850
  1180
  case empty
haftmann@64850
  1181
  then show ?case
haftmann@64850
  1182
    by simp
haftmann@64850
  1183
next
haftmann@64850
  1184
  case (insert a A)
haftmann@64850
  1185
  have "gcd (b * a) (b * Gcd\<^sub>f\<^sub>i\<^sub>n A) = gcd (b * a) (normalize (b * Gcd\<^sub>f\<^sub>i\<^sub>n A))"
haftmann@64850
  1186
    by simp
haftmann@64850
  1187
  also have "\<dots> = gcd (b * a) (normalize b * Gcd\<^sub>f\<^sub>i\<^sub>n A)"
haftmann@64850
  1188
    by (simp add: normalize_mult)
haftmann@64850
  1189
  finally show ?case
haftmann@64850
  1190
    using insert by (simp add: gcd_mult_distrib')
haftmann@64850
  1191
qed
haftmann@64850
  1192
haftmann@64850
  1193
lemma Lcm_fin_mult:
haftmann@64850
  1194
  "Lcm\<^sub>f\<^sub>i\<^sub>n (image (times b) A) = normalize b * Lcm\<^sub>f\<^sub>i\<^sub>n A" if "A \<noteq> {}"
haftmann@64850
  1195
proof (cases "b = 0")
haftmann@64850
  1196
  case True
haftmann@64850
  1197
  moreover from that have "times 0 ` A = {0}"
haftmann@64850
  1198
    by auto
haftmann@64850
  1199
  ultimately show ?thesis
haftmann@64850
  1200
    by simp
haftmann@64850
  1201
next
haftmann@64850
  1202
  case False
haftmann@64850
  1203
  show ?thesis proof (cases "finite A")
haftmann@64850
  1204
    case False
haftmann@66936
  1205
    moreover have "inj_on (times b) A"
haftmann@66936
  1206
      using \<open>b \<noteq> 0\<close> by (rule inj_on_mult)
haftmann@64850
  1207
    ultimately have "infinite (times b ` A)"
haftmann@64850
  1208
      by (simp add: finite_image_iff)
haftmann@64850
  1209
    with False show ?thesis
haftmann@64850
  1210
      by simp
haftmann@64850
  1211
  next
haftmann@64850
  1212
    case True
haftmann@64850
  1213
    then show ?thesis using that proof (induct A rule: finite_ne_induct)
haftmann@64850
  1214
      case (singleton a)
haftmann@64850
  1215
      then show ?case
haftmann@64850
  1216
        by (simp add: normalize_mult)
haftmann@64850
  1217
    next
haftmann@64850
  1218
      case (insert a A)
haftmann@64850
  1219
      have "lcm (b * a) (b * Lcm\<^sub>f\<^sub>i\<^sub>n A) = lcm (b * a) (normalize (b * Lcm\<^sub>f\<^sub>i\<^sub>n A))"
haftmann@64850
  1220
        by simp
haftmann@64850
  1221
      also have "\<dots> = lcm (b * a) (normalize b * Lcm\<^sub>f\<^sub>i\<^sub>n A)"
haftmann@64850
  1222
        by (simp add: normalize_mult)
haftmann@64850
  1223
      finally show ?case
haftmann@64850
  1224
        using insert by (simp add: lcm_mult_distrib')
haftmann@64850
  1225
    qed
haftmann@64850
  1226
  qed
haftmann@64850
  1227
qed
haftmann@64850
  1228
haftmann@65811
  1229
lemma unit_factor_Gcd_fin:
haftmann@65811
  1230
  "unit_factor (Gcd\<^sub>f\<^sub>i\<^sub>n A) = of_bool (Gcd\<^sub>f\<^sub>i\<^sub>n A \<noteq> 0)"
haftmann@65811
  1231
  by (rule normalize_idem_imp_unit_factor_eq) simp
haftmann@65811
  1232
haftmann@65811
  1233
lemma unit_factor_Lcm_fin:
haftmann@65811
  1234
  "unit_factor (Lcm\<^sub>f\<^sub>i\<^sub>n A) = of_bool (Lcm\<^sub>f\<^sub>i\<^sub>n A \<noteq> 0)"
haftmann@65811
  1235
  by (rule normalize_idem_imp_unit_factor_eq) simp
haftmann@65811
  1236
haftmann@65811
  1237
lemma is_unit_Gcd_fin_iff [simp]:
haftmann@65811
  1238
  "is_unit (Gcd\<^sub>f\<^sub>i\<^sub>n A) \<longleftrightarrow> Gcd\<^sub>f\<^sub>i\<^sub>n A = 1"
haftmann@65811
  1239
  by (rule normalize_idem_imp_is_unit_iff) simp
haftmann@65811
  1240
haftmann@65811
  1241
lemma is_unit_Lcm_fin_iff [simp]:
haftmann@65811
  1242
  "is_unit (Lcm\<^sub>f\<^sub>i\<^sub>n A) \<longleftrightarrow> Lcm\<^sub>f\<^sub>i\<^sub>n A = 1"
haftmann@65811
  1243
  by (rule normalize_idem_imp_is_unit_iff) simp
haftmann@65811
  1244
 
haftmann@65811
  1245
lemma Gcd_fin_0_iff:
haftmann@65811
  1246
  "Gcd\<^sub>f\<^sub>i\<^sub>n A = 0 \<longleftrightarrow> A \<subseteq> {0} \<and> finite A"
haftmann@65811
  1247
  by (induct A rule: infinite_finite_induct) simp_all
haftmann@65811
  1248
haftmann@65811
  1249
lemma Lcm_fin_0_iff:
haftmann@65811
  1250
  "Lcm\<^sub>f\<^sub>i\<^sub>n A = 0 \<longleftrightarrow> 0 \<in> A" if "finite A"
lp15@68708
  1251
  using that by (induct A) (auto simp: lcm_eq_0_iff)
haftmann@65811
  1252
haftmann@65811
  1253
lemma Lcm_fin_1_iff:
haftmann@65811
  1254
  "Lcm\<^sub>f\<^sub>i\<^sub>n A = 1 \<longleftrightarrow> (\<forall>a\<in>A. is_unit a) \<and> finite A"
haftmann@65811
  1255
  by (induct A rule: infinite_finite_induct) simp_all
haftmann@65811
  1256
haftmann@64850
  1257
end
haftmann@64850
  1258
haftmann@64850
  1259
context semiring_Gcd
haftmann@64850
  1260
begin
haftmann@64850
  1261
haftmann@64850
  1262
lemma Gcd_fin_eq_Gcd [simp]:
haftmann@64850
  1263
  "Gcd\<^sub>f\<^sub>i\<^sub>n A = Gcd A" if "finite A" for A :: "'a set"
haftmann@64850
  1264
  using that by induct simp_all
haftmann@64850
  1265
haftmann@64850
  1266
lemma Gcd_set_eq_fold [code_unfold]:
haftmann@64850
  1267
  "Gcd (set xs) = fold gcd xs 0"
haftmann@64850
  1268
  by (simp add: Gcd_fin.set_eq_fold [symmetric])
haftmann@64850
  1269
haftmann@64850
  1270
lemma Lcm_fin_eq_Lcm [simp]:
haftmann@64850
  1271
  "Lcm\<^sub>f\<^sub>i\<^sub>n A = Lcm A" if "finite A" for A :: "'a set"
haftmann@64850
  1272
  using that by induct simp_all
haftmann@64850
  1273
haftmann@64850
  1274
lemma Lcm_set_eq_fold [code_unfold]:
haftmann@64850
  1275
  "Lcm (set xs) = fold lcm xs 1"
haftmann@64850
  1276
  by (simp add: Lcm_fin.set_eq_fold [symmetric])
haftmann@64850
  1277
haftmann@64850
  1278
end
wenzelm@63489
  1279
haftmann@67051
  1280
haftmann@67051
  1281
subsection \<open>Coprimality\<close>
haftmann@67051
  1282
haftmann@67051
  1283
context semiring_gcd
haftmann@67051
  1284
begin
haftmann@67051
  1285
haftmann@67051
  1286
lemma coprime_imp_gcd_eq_1 [simp]:
haftmann@67051
  1287
  "gcd a b = 1" if "coprime a b"
haftmann@67051
  1288
proof -
haftmann@67051
  1289
  define t r s where "t = gcd a b" and "r = a div t" and "s = b div t"
haftmann@67051
  1290
  then have "a = t * r" and "b = t * s"
haftmann@67051
  1291
    by simp_all
haftmann@67051
  1292
  with that have "coprime (t * r) (t * s)"
haftmann@67051
  1293
    by simp
haftmann@67051
  1294
  then show ?thesis
haftmann@67051
  1295
    by (simp add: t_def)
haftmann@67051
  1296
qed
haftmann@67051
  1297
haftmann@68270
  1298
lemma gcd_eq_1_imp_coprime [dest!]:
haftmann@67051
  1299
  "coprime a b" if "gcd a b = 1"
haftmann@67051
  1300
proof (rule coprimeI)
haftmann@67051
  1301
  fix c
haftmann@67051
  1302
  assume "c dvd a" and "c dvd b"
haftmann@67051
  1303
  then have "c dvd gcd a b"
haftmann@67051
  1304
    by (rule gcd_greatest)
haftmann@67051
  1305
  with that show "is_unit c"
haftmann@67051
  1306
    by simp
haftmann@67051
  1307
qed
haftmann@67051
  1308
haftmann@67051
  1309
lemma coprime_iff_gcd_eq_1 [presburger, code]:
haftmann@67051
  1310
  "coprime a b \<longleftrightarrow> gcd a b = 1"
haftmann@67051
  1311
  by rule (simp_all add: gcd_eq_1_imp_coprime)
haftmann@67051
  1312
haftmann@67051
  1313
lemma is_unit_gcd [simp]:
haftmann@67051
  1314
  "is_unit (gcd a b) \<longleftrightarrow> coprime a b"
haftmann@67051
  1315
  by (simp add: coprime_iff_gcd_eq_1)
haftmann@67051
  1316
haftmann@67051
  1317
lemma coprime_add_one_left [simp]: "coprime (a + 1) a"
haftmann@67051
  1318
  by (simp add: gcd_eq_1_imp_coprime ac_simps)
haftmann@67051
  1319
haftmann@67051
  1320
lemma coprime_add_one_right [simp]: "coprime a (a + 1)"
haftmann@67051
  1321
  using coprime_add_one_left [of a] by (simp add: ac_simps)
haftmann@67051
  1322
haftmann@67051
  1323
lemma coprime_mult_left_iff [simp]:
haftmann@67051
  1324
  "coprime (a * b) c \<longleftrightarrow> coprime a c \<and> coprime b c"
haftmann@67051
  1325
proof
haftmann@67051
  1326
  assume "coprime (a * b) c"
haftmann@67051
  1327
  with coprime_common_divisor [of "a * b" c]
haftmann@67051
  1328
  have *: "is_unit d" if "d dvd a * b" and "d dvd c" for d
haftmann@67051
  1329
    using that by blast
haftmann@67051
  1330
  have "coprime a c"
haftmann@67051
  1331
    by (rule coprimeI, rule *) simp_all
haftmann@67051
  1332
  moreover have "coprime b c"
haftmann@67051
  1333
    by (rule coprimeI, rule *) simp_all
haftmann@67051
  1334
  ultimately show "coprime a c \<and> coprime b c" ..
haftmann@67051
  1335
next
haftmann@67051
  1336
  assume "coprime a c \<and> coprime b c"
haftmann@67051
  1337
  then have "coprime a c" "coprime b c"
haftmann@67051
  1338
    by simp_all
haftmann@67051
  1339
  show "coprime (a * b) c"
haftmann@67051
  1340
  proof (rule coprimeI)
haftmann@67051
  1341
    fix d
haftmann@67051
  1342
    assume "d dvd a * b"
haftmann@67051
  1343
    then obtain r s where d: "d = r * s" "r dvd a" "s dvd b"
haftmann@67051
  1344
      by (rule dvd_productE)
haftmann@67051
  1345
    assume "d dvd c"
haftmann@67051
  1346
    with d have "r * s dvd c"
haftmann@67051
  1347
      by simp
haftmann@67051
  1348
    then have "r dvd c" "s dvd c"
haftmann@67051
  1349
      by (auto intro: dvd_mult_left dvd_mult_right)
haftmann@67051
  1350
    from \<open>coprime a c\<close> \<open>r dvd a\<close> \<open>r dvd c\<close>
haftmann@67051
  1351
    have "is_unit r"
haftmann@67051
  1352
      by (rule coprime_common_divisor)
haftmann@67051
  1353
    moreover from \<open>coprime b c\<close> \<open>s dvd b\<close> \<open>s dvd c\<close>
haftmann@67051
  1354
    have "is_unit s"
haftmann@67051
  1355
      by (rule coprime_common_divisor)
haftmann@67051
  1356
    ultimately show "is_unit d"
haftmann@67051
  1357
      by (simp add: d is_unit_mult_iff)
haftmann@67051
  1358
  qed
haftmann@67051
  1359
qed
haftmann@67051
  1360
haftmann@67051
  1361
lemma coprime_mult_right_iff [simp]:
haftmann@67051
  1362
  "coprime c (a * b) \<longleftrightarrow> coprime c a \<and> coprime c b"
haftmann@67051
  1363
  using coprime_mult_left_iff [of a b c] by (simp add: ac_simps)
haftmann@67051
  1364
haftmann@67051
  1365
lemma coprime_power_left_iff [simp]:
haftmann@67051
  1366
  "coprime (a ^ n) b \<longleftrightarrow> coprime a b \<or> n = 0"
haftmann@67051
  1367
proof (cases "n = 0")
haftmann@67051
  1368
  case True
haftmann@67051
  1369
  then show ?thesis
haftmann@67051
  1370
    by simp
haftmann@67051
  1371
next
haftmann@67051
  1372
  case False
haftmann@67051
  1373
  then have "n > 0"
haftmann@67051
  1374
    by simp
haftmann@67051
  1375
  then show ?thesis
haftmann@67051
  1376
    by (induction n rule: nat_induct_non_zero) simp_all
haftmann@67051
  1377
qed
haftmann@67051
  1378
haftmann@67051
  1379
lemma coprime_power_right_iff [simp]:
haftmann@67051
  1380
  "coprime a (b ^ n) \<longleftrightarrow> coprime a b \<or> n = 0"
haftmann@67051
  1381
  using coprime_power_left_iff [of b n a] by (simp add: ac_simps)
haftmann@67051
  1382
haftmann@67051
  1383
lemma prod_coprime_left:
haftmann@67051
  1384
  "coprime (\<Prod>i\<in>A. f i) a" if "\<And>i. i \<in> A \<Longrightarrow> coprime (f i) a"
haftmann@67051
  1385
  using that by (induct A rule: infinite_finite_induct) simp_all
haftmann@67051
  1386
haftmann@67051
  1387
lemma prod_coprime_right:
haftmann@67051
  1388
  "coprime a (\<Prod>i\<in>A. f i)" if "\<And>i. i \<in> A \<Longrightarrow> coprime a (f i)"
haftmann@67051
  1389
  using that prod_coprime_left [of A f a] by (simp add: ac_simps)
haftmann@67051
  1390
haftmann@67051
  1391
lemma prod_list_coprime_left:
haftmann@67051
  1392
  "coprime (prod_list xs) a" if "\<And>x. x \<in> set xs \<Longrightarrow> coprime x a"
haftmann@67051
  1393
  using that by (induct xs) simp_all
haftmann@67051
  1394
haftmann@67051
  1395
lemma prod_list_coprime_right:
haftmann@67051
  1396
  "coprime a (prod_list xs)" if "\<And>x. x \<in> set xs \<Longrightarrow> coprime a x"
haftmann@67051
  1397
  using that prod_list_coprime_left [of xs a] by (simp add: ac_simps)
haftmann@67051
  1398
haftmann@67051
  1399
lemma coprime_dvd_mult_left_iff:
haftmann@67051
  1400
  "a dvd b * c \<longleftrightarrow> a dvd b" if "coprime a c"
haftmann@67051
  1401
proof
haftmann@67051
  1402
  assume "a dvd b"
haftmann@67051
  1403
  then show "a dvd b * c"
haftmann@67051
  1404
    by simp
haftmann@67051
  1405
next
haftmann@67051
  1406
  assume "a dvd b * c"
haftmann@67051
  1407
  show "a dvd b"
haftmann@67051
  1408
  proof (cases "b = 0")
haftmann@67051
  1409
    case True
haftmann@67051
  1410
    then show ?thesis
haftmann@67051
  1411
      by simp
haftmann@67051
  1412
  next
haftmann@67051
  1413
    case False
haftmann@67051
  1414
    then have unit: "is_unit (unit_factor b)"
haftmann@67051
  1415
      by simp
haftmann@67051
  1416
    from \<open>coprime a c\<close> mult_gcd_left [of b a c]
haftmann@67051
  1417
    have "gcd (b * a) (b * c) * unit_factor b = b"
haftmann@67051
  1418
      by (simp add: ac_simps)
haftmann@67051
  1419
    moreover from \<open>a dvd b * c\<close>
haftmann@67051
  1420
    have "a dvd gcd (b * a) (b * c) * unit_factor b"
haftmann@67051
  1421
      by (simp add: dvd_mult_unit_iff unit)
haftmann@67051
  1422
    ultimately show ?thesis
haftmann@67051
  1423
      by simp
haftmann@67051
  1424
  qed
haftmann@67051
  1425
qed
haftmann@67051
  1426
haftmann@67051
  1427
lemma coprime_dvd_mult_right_iff:
haftmann@67051
  1428
  "a dvd c * b \<longleftrightarrow> a dvd b" if "coprime a c"
haftmann@67051
  1429
  using that coprime_dvd_mult_left_iff [of a c b] by (simp add: ac_simps)
haftmann@67051
  1430
haftmann@67051
  1431
lemma divides_mult:
haftmann@67051
  1432
  "a * b dvd c" if "a dvd c" and "b dvd c" and "coprime a b"
haftmann@67051
  1433
proof -
haftmann@67051
  1434
  from \<open>b dvd c\<close> obtain b' where "c = b * b'" ..
haftmann@67051
  1435
  with \<open>a dvd c\<close> have "a dvd b' * b"
haftmann@67051
  1436
    by (simp add: ac_simps)
haftmann@67051
  1437
  with \<open>coprime a b\<close> have "a dvd b'"
haftmann@67051
  1438
    by (simp add: coprime_dvd_mult_left_iff)
haftmann@67051
  1439
  then obtain a' where "b' = a * a'" ..
haftmann@67051
  1440
  with \<open>c = b * b'\<close> have "c = (a * b) * a'"
haftmann@67051
  1441
    by (simp add: ac_simps)
haftmann@67051
  1442
  then show ?thesis ..
haftmann@67051
  1443
qed
haftmann@67051
  1444
haftmann@67051
  1445
lemma div_gcd_coprime:
haftmann@67051
  1446
  assumes "a \<noteq> 0 \<or> b \<noteq> 0"
haftmann@67051
  1447
  shows "coprime (a div gcd a b) (b div gcd a b)"
haftmann@67051
  1448
proof -
haftmann@67051
  1449
  let ?g = "gcd a b"
haftmann@67051
  1450
  let ?a' = "a div ?g"
haftmann@67051
  1451
  let ?b' = "b div ?g"
haftmann@67051
  1452
  let ?g' = "gcd ?a' ?b'"
haftmann@67051
  1453
  have dvdg: "?g dvd a" "?g dvd b"
haftmann@67051
  1454
    by simp_all
haftmann@67051
  1455
  have dvdg': "?g' dvd ?a'" "?g' dvd ?b'"
haftmann@67051
  1456
    by simp_all
haftmann@67051
  1457
  from dvdg dvdg' obtain ka kb ka' kb' where
haftmann@67051
  1458
    kab: "a = ?g * ka" "b = ?g * kb" "?a' = ?g' * ka'" "?b' = ?g' * kb'"
haftmann@67051
  1459
    unfolding dvd_def by blast
haftmann@67051
  1460
  from this [symmetric] have "?g * ?a' = (?g * ?g') * ka'" "?g * ?b' = (?g * ?g') * kb'"
haftmann@67051
  1461
    by (simp_all add: mult.assoc mult.left_commute [of "gcd a b"])
haftmann@67051
  1462
  then have dvdgg':"?g * ?g' dvd a" "?g* ?g' dvd b"
lp15@68708
  1463
    by (auto simp: dvd_mult_div_cancel [OF dvdg(1)] dvd_mult_div_cancel [OF dvdg(2)] dvd_def)
haftmann@67051
  1464
  have "?g \<noteq> 0"
haftmann@67051
  1465
    using assms by simp
haftmann@67051
  1466
  moreover from gcd_greatest [OF dvdgg'] have "?g * ?g' dvd ?g" .
haftmann@67051
  1467
  ultimately show ?thesis
haftmann@67051
  1468
    using dvd_times_left_cancel_iff [of "gcd a b" _ 1]
haftmann@67051
  1469
    by simp (simp only: coprime_iff_gcd_eq_1)
haftmann@67051
  1470
qed
haftmann@67051
  1471
haftmann@67051
  1472
lemma gcd_coprime:
haftmann@67051
  1473
  assumes c: "gcd a b \<noteq> 0"
haftmann@67051
  1474
    and a: "a = a' * gcd a b"
haftmann@67051
  1475
    and b: "b = b' * gcd a b"
haftmann@67051
  1476
  shows "coprime a' b'"
haftmann@67051
  1477
proof -
haftmann@67051
  1478
  from c have "a \<noteq> 0 \<or> b \<noteq> 0"
haftmann@67051
  1479
    by simp
haftmann@67051
  1480
  with div_gcd_coprime have "coprime (a div gcd a b) (b div gcd a b)" .
haftmann@67051
  1481
  also from assms have "a div gcd a b = a'"
haftmann@67051
  1482
    using dvd_div_eq_mult local.gcd_dvd1 by blast
haftmann@67051
  1483
  also from assms have "b div gcd a b = b'"
haftmann@67051
  1484
    using dvd_div_eq_mult local.gcd_dvd1 by blast
haftmann@67051
  1485
  finally show ?thesis .
haftmann@67051
  1486
qed
haftmann@67051
  1487
haftmann@67051
  1488
lemma gcd_coprime_exists:
haftmann@67051
  1489
  assumes "gcd a b \<noteq> 0"
haftmann@67051
  1490
  shows "\<exists>a' b'. a = a' * gcd a b \<and> b = b' * gcd a b \<and> coprime a' b'"
lp15@68708
  1491
proof -
lp15@68708
  1492
  have "coprime (a div gcd a b) (b div gcd a b)"
lp15@68708
  1493
    using assms div_gcd_coprime by auto
lp15@68708
  1494
  then show ?thesis
lp15@68708
  1495
    by force
lp15@68708
  1496
qed
haftmann@67051
  1497
haftmann@67051
  1498
lemma pow_divides_pow_iff [simp]:
haftmann@67051
  1499
  "a ^ n dvd b ^ n \<longleftrightarrow> a dvd b" if "n > 0"
haftmann@67051
  1500
proof (cases "gcd a b = 0")
haftmann@67051
  1501
  case True
haftmann@67051
  1502
  then show ?thesis
haftmann@67051
  1503
    by simp
haftmann@67051
  1504
next
haftmann@67051
  1505
  case False
haftmann@67051
  1506
  show ?thesis
haftmann@67051
  1507
  proof
haftmann@67051
  1508
    let ?d = "gcd a b"
haftmann@67051
  1509
    from \<open>n > 0\<close> obtain m where m: "n = Suc m"
haftmann@67051
  1510
      by (cases n) simp_all
haftmann@67051
  1511
    from False have zn: "?d ^ n \<noteq> 0"
haftmann@67051
  1512
      by (rule power_not_zero)
haftmann@67051
  1513
    from gcd_coprime_exists [OF False]
haftmann@67051
  1514
    obtain a' b' where ab': "a = a' * ?d" "b = b' * ?d" "coprime a' b'"
haftmann@67051
  1515
      by blast
haftmann@67051
  1516
    assume "a ^ n dvd b ^ n"
haftmann@67051
  1517
    then have "(a' * ?d) ^ n dvd (b' * ?d) ^ n"
haftmann@67051
  1518
      by (simp add: ab'(1,2)[symmetric])
haftmann@67051
  1519
    then have "?d^n * a'^n dvd ?d^n * b'^n"
haftmann@67051
  1520
      by (simp only: power_mult_distrib ac_simps)
haftmann@67051
  1521
    with zn have "a' ^ n dvd b' ^ n"
haftmann@67051
  1522
      by simp
haftmann@67051
  1523
    then have "a' dvd b' ^ n"
haftmann@67051
  1524
      using dvd_trans[of a' "a'^n" "b'^n"] by (simp add: m)
haftmann@67051
  1525
    then have "a' dvd b' ^ m * b'"
haftmann@67051
  1526
      by (simp add: m ac_simps)
haftmann@67051
  1527
    moreover have "coprime a' (b' ^ n)"
haftmann@67051
  1528
      using \<open>coprime a' b'\<close> by simp
haftmann@67051
  1529
    then have "a' dvd b'"
haftmann@67051
  1530
      using \<open>a' dvd b' ^ n\<close> coprime_dvd_mult_left_iff dvd_mult by blast
haftmann@67051
  1531
    then have "a' * ?d dvd b' * ?d"
haftmann@67051
  1532
      by (rule mult_dvd_mono) simp
haftmann@67051
  1533
    with ab'(1,2) show "a dvd b"
haftmann@67051
  1534
      by simp
haftmann@67051
  1535
  next
haftmann@67051
  1536
    assume "a dvd b"
haftmann@67051
  1537
    with \<open>n > 0\<close> show "a ^ n dvd b ^ n"
haftmann@67051
  1538
      by (induction rule: nat_induct_non_zero)
haftmann@67051
  1539
        (simp_all add: mult_dvd_mono)
haftmann@67051
  1540
  qed
haftmann@67051
  1541
qed
haftmann@67051
  1542
haftmann@67051
  1543
lemma coprime_crossproduct:
haftmann@67051
  1544
  fixes a b c d :: 'a
haftmann@67051
  1545
  assumes "coprime a d" and "coprime b c"
haftmann@67051
  1546
  shows "normalize a * normalize c = normalize b * normalize d \<longleftrightarrow>
haftmann@67051
  1547
    normalize a = normalize b \<and> normalize c = normalize d"
haftmann@67051
  1548
    (is "?lhs \<longleftrightarrow> ?rhs")
haftmann@67051
  1549
proof
haftmann@67051
  1550
  assume ?rhs
haftmann@67051
  1551
  then show ?lhs by simp
haftmann@67051
  1552
next
haftmann@67051
  1553
  assume ?lhs
haftmann@67051
  1554
  from \<open>?lhs\<close> have "normalize a dvd normalize b * normalize d"
haftmann@67051
  1555
    by (auto intro: dvdI dest: sym)
haftmann@67051
  1556
  with \<open>coprime a d\<close> have "a dvd b"
haftmann@67051
  1557
    by (simp add: coprime_dvd_mult_left_iff normalize_mult [symmetric])
haftmann@67051
  1558
  from \<open>?lhs\<close> have "normalize b dvd normalize a * normalize c"
haftmann@67051
  1559
    by (auto intro: dvdI dest: sym)
haftmann@67051
  1560
  with \<open>coprime b c\<close> have "b dvd a"
haftmann@67051
  1561
    by (simp add: coprime_dvd_mult_left_iff normalize_mult [symmetric])
haftmann@67051
  1562
  from \<open>?lhs\<close> have "normalize c dvd normalize d * normalize b"
haftmann@67051
  1563
    by (auto intro: dvdI dest: sym simp add: mult.commute)
haftmann@67051
  1564
  with \<open>coprime b c\<close> have "c dvd d"
haftmann@67051
  1565
    by (simp add: coprime_dvd_mult_left_iff coprime_commute normalize_mult [symmetric])
haftmann@67051
  1566
  from \<open>?lhs\<close> have "normalize d dvd normalize c * normalize a"
haftmann@67051
  1567
    by (auto intro: dvdI dest: sym simp add: mult.commute)
haftmann@67051
  1568
  with \<open>coprime a d\<close> have "d dvd c"
haftmann@67051
  1569
    by (simp add: coprime_dvd_mult_left_iff coprime_commute normalize_mult [symmetric])
haftmann@67051
  1570
  from \<open>a dvd b\<close> \<open>b dvd a\<close> have "normalize a = normalize b"
haftmann@67051
  1571
    by (rule associatedI)
haftmann@67051
  1572
  moreover from \<open>c dvd d\<close> \<open>d dvd c\<close> have "normalize c = normalize d"
haftmann@67051
  1573
    by (rule associatedI)
haftmann@67051
  1574
  ultimately show ?rhs ..
haftmann@67051
  1575
qed
haftmann@67051
  1576
haftmann@67051
  1577
lemma coprime_crossproduct':
haftmann@67051
  1578
  fixes a b c d
haftmann@67051
  1579
  assumes "b \<noteq> 0"
haftmann@67051
  1580
  assumes unit_factors: "unit_factor b = unit_factor d"
haftmann@67051
  1581
  assumes coprime: "coprime a b" "coprime c d"
haftmann@67051
  1582
  shows "a * d = b * c \<longleftrightarrow> a = c \<and> b = d"
haftmann@67051
  1583
proof safe
haftmann@67051
  1584
  assume eq: "a * d = b * c"
haftmann@67051
  1585
  hence "normalize a * normalize d = normalize c * normalize b"
haftmann@67051
  1586
    by (simp only: normalize_mult [symmetric] mult_ac)
haftmann@67051
  1587
  with coprime have "normalize b = normalize d"
haftmann@67051
  1588
    by (subst (asm) coprime_crossproduct) simp_all
haftmann@67051
  1589
  from this and unit_factors show "b = d"
haftmann@67051
  1590
    by (rule normalize_unit_factor_eqI)
haftmann@67051
  1591
  from eq have "a * d = c * d" by (simp only: \<open>b = d\<close> mult_ac)
haftmann@67051
  1592
  with \<open>b \<noteq> 0\<close> \<open>b = d\<close> show "a = c" by simp
haftmann@67051
  1593
qed (simp_all add: mult_ac)
haftmann@67051
  1594
haftmann@67051
  1595
lemma gcd_mult_left_left_cancel:
haftmann@67051
  1596
  "gcd (c * a) b = gcd a b" if "coprime b c"
haftmann@67051
  1597
proof -
haftmann@67051
  1598
  have "coprime (gcd b (a * c)) c"
haftmann@67051
  1599
    by (rule coprimeI) (auto intro: that coprime_common_divisor)
haftmann@67051
  1600
  then have "gcd b (a * c) dvd a"
haftmann@67051
  1601
    using coprime_dvd_mult_left_iff [of "gcd b (a * c)" c a]
haftmann@67051
  1602
    by simp
haftmann@67051
  1603
  then show ?thesis
haftmann@67051
  1604
    by (auto intro: associated_eqI simp add: ac_simps)
haftmann@67051
  1605
qed
haftmann@67051
  1606
haftmann@67051
  1607
lemma gcd_mult_left_right_cancel:
haftmann@67051
  1608
  "gcd (a * c) b = gcd a b" if "coprime b c"
haftmann@67051
  1609
  using that gcd_mult_left_left_cancel [of b c a]
haftmann@67051
  1610
  by (simp add: ac_simps)
haftmann@67051
  1611
haftmann@67051
  1612
lemma gcd_mult_right_left_cancel:
haftmann@67051
  1613
  "gcd a (c * b) = gcd a b" if "coprime a c"
haftmann@67051
  1614
  using that gcd_mult_left_left_cancel [of a c b]
haftmann@67051
  1615
  by (simp add: ac_simps)
haftmann@67051
  1616
haftmann@67051
  1617
lemma gcd_mult_right_right_cancel:
haftmann@67051
  1618
  "gcd a (b * c) = gcd a b" if "coprime a c"
haftmann@67051
  1619
  using that gcd_mult_right_left_cancel [of a c b]
haftmann@67051
  1620
  by (simp add: ac_simps)
haftmann@67051
  1621
haftmann@67051
  1622
lemma gcd_exp [simp]:
haftmann@67051
  1623
  "gcd (a ^ n) (b ^ n) = gcd a b ^ n"
haftmann@67051
  1624
proof (cases "a = 0 \<and> b = 0 \<or> n = 0")
haftmann@67051
  1625
  case True
haftmann@67051
  1626
  then show ?thesis
haftmann@67051
  1627
    by (cases n) simp_all
haftmann@67051
  1628
next
haftmann@67051
  1629
  case False
haftmann@67051
  1630
  then have "coprime (a div gcd a b) (b div gcd a b)" and "n > 0"
haftmann@67051
  1631
    by (auto intro: div_gcd_coprime)
haftmann@67051
  1632
  then have "coprime ((a div gcd a b) ^ n) ((b div gcd a b) ^ n)"
haftmann@67051
  1633
    by simp
haftmann@67051
  1634
  then have "1 = gcd ((a div gcd a b) ^ n) ((b div gcd a b) ^ n)"
haftmann@67051
  1635
    by simp
haftmann@67051
  1636
  then have "gcd a b ^ n = gcd a b ^ n * \<dots>"
haftmann@67051
  1637
    by simp
haftmann@67051
  1638
  also note gcd_mult_distrib
haftmann@67051
  1639
  also have "unit_factor (gcd a b ^ n) = 1"
lp15@68708
  1640
    using False by (auto simp: unit_factor_power unit_factor_gcd)
haftmann@67051
  1641
  also have "(gcd a b) ^ n * (a div gcd a b) ^ n = a ^ n"
haftmann@67051
  1642
    by (simp add: ac_simps div_power dvd_power_same)
haftmann@67051
  1643
  also have "(gcd a b) ^ n * (b div gcd a b) ^ n = b ^ n"
haftmann@67051
  1644
    by (simp add: ac_simps div_power dvd_power_same)
haftmann@67051
  1645
  finally show ?thesis by simp
haftmann@67051
  1646
qed
haftmann@67051
  1647
haftmann@67051
  1648
lemma division_decomp:
haftmann@67051
  1649
  assumes "a dvd b * c"
haftmann@67051
  1650
  shows "\<exists>b' c'. a = b' * c' \<and> b' dvd b \<and> c' dvd c"
haftmann@67051
  1651
proof (cases "gcd a b = 0")
haftmann@67051
  1652
  case True
haftmann@67051
  1653
  then have "a = 0 \<and> b = 0"
haftmann@67051
  1654
    by simp
haftmann@67051
  1655
  then have "a = 0 * c \<and> 0 dvd b \<and> c dvd c"
haftmann@67051
  1656
    by simp
haftmann@67051
  1657
  then show ?thesis by blast
haftmann@67051
  1658
next
haftmann@67051
  1659
  case False
haftmann@67051
  1660
  let ?d = "gcd a b"
haftmann@67051
  1661
  from gcd_coprime_exists [OF False]
haftmann@67051
  1662
    obtain a' b' where ab': "a = a' * ?d" "b = b' * ?d" "coprime a' b'"
haftmann@67051
  1663
    by blast
haftmann@67051
  1664
  from ab'(1) have "a' dvd a" ..
haftmann@67051
  1665
  with assms have "a' dvd b * c"
haftmann@67051
  1666
    using dvd_trans [of a' a "b * c"] by simp
haftmann@67051
  1667
  from assms ab'(1,2) have "a' * ?d dvd (b' * ?d) * c"
haftmann@67051
  1668
    by simp
haftmann@67051
  1669
  then have "?d * a' dvd ?d * (b' * c)"
haftmann@67051
  1670
    by (simp add: mult_ac)
haftmann@67051
  1671
  with \<open>?d \<noteq> 0\<close> have "a' dvd b' * c"
haftmann@67051
  1672
    by simp
haftmann@67051
  1673
  then have "a' dvd c"
haftmann@67051
  1674
    using \<open>coprime a' b'\<close> by (simp add: coprime_dvd_mult_right_iff)
haftmann@67051
  1675
  with ab'(1) have "a = ?d * a' \<and> ?d dvd b \<and> a' dvd c"
haftmann@67051
  1676
    by (simp add: ac_simps)
haftmann@67051
  1677
  then show ?thesis by blast
haftmann@67051
  1678
qed
haftmann@67051
  1679
haftmann@67051
  1680
lemma lcm_coprime: "coprime a b \<Longrightarrow> lcm a b = normalize (a * b)"
haftmann@67051
  1681
  by (subst lcm_gcd) simp
haftmann@67051
  1682
haftmann@67051
  1683
end
haftmann@67051
  1684
haftmann@67051
  1685
context ring_gcd
haftmann@67051
  1686
begin
haftmann@67051
  1687
haftmann@67051
  1688
lemma coprime_minus_left_iff [simp]:
haftmann@67051
  1689
  "coprime (- a) b \<longleftrightarrow> coprime a b"
haftmann@67051
  1690
  by (rule; rule coprimeI) (auto intro: coprime_common_divisor)
haftmann@67051
  1691
haftmann@67051
  1692
lemma coprime_minus_right_iff [simp]:
haftmann@67051
  1693
  "coprime a (- b) \<longleftrightarrow> coprime a b"
haftmann@67051
  1694
  using coprime_minus_left_iff [of b a] by (simp add: ac_simps)
haftmann@67051
  1695
haftmann@67051
  1696
lemma coprime_diff_one_left [simp]: "coprime (a - 1) a"
haftmann@67051
  1697
  using coprime_add_one_right [of "a - 1"] by simp
haftmann@67051
  1698
haftmann@67051
  1699
lemma coprime_doff_one_right [simp]: "coprime a (a - 1)"
haftmann@67051
  1700
  using coprime_diff_one_left [of a] by (simp add: ac_simps)
haftmann@67051
  1701
haftmann@67051
  1702
end
haftmann@67051
  1703
haftmann@67051
  1704
context semiring_Gcd
haftmann@67051
  1705
begin
haftmann@67051
  1706
haftmann@67051
  1707
lemma Lcm_coprime:
haftmann@67051
  1708
  assumes "finite A"
haftmann@67051
  1709
    and "A \<noteq> {}"
haftmann@67051
  1710
    and "\<And>a b. a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<noteq> b \<Longrightarrow> coprime a b"
haftmann@67051
  1711
  shows "Lcm A = normalize (\<Prod>A)"
haftmann@67051
  1712
  using assms
haftmann@67051
  1713
proof (induct rule: finite_ne_induct)
haftmann@67051
  1714
  case singleton
haftmann@67051
  1715
  then show ?case by simp
haftmann@67051
  1716
next
haftmann@67051
  1717
  case (insert a A)
haftmann@67051
  1718
  have "Lcm (insert a A) = lcm a (Lcm A)"
haftmann@67051
  1719
    by simp
haftmann@67051
  1720
  also from insert have "Lcm A = normalize (\<Prod>A)"
haftmann@67051
  1721
    by blast
haftmann@67051
  1722
  also have "lcm a \<dots> = lcm a (\<Prod>A)"
haftmann@67051
  1723
    by (cases "\<Prod>A = 0") (simp_all add: lcm_div_unit2)
haftmann@67051
  1724
  also from insert have "coprime a (\<Prod>A)"
haftmann@67051
  1725
    by (subst coprime_commute, intro prod_coprime_left) auto
haftmann@67051
  1726
  with insert have "lcm a (\<Prod>A) = normalize (\<Prod>(insert a A))"
haftmann@67051
  1727
    by (simp add: lcm_coprime)
haftmann@67051
  1728
  finally show ?case .
haftmann@67051
  1729
qed
haftmann@67051
  1730
haftmann@67051
  1731
lemma Lcm_coprime':
haftmann@67051
  1732
  "card A \<noteq> 0 \<Longrightarrow>
haftmann@67051
  1733
    (\<And>a b. a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<noteq> b \<Longrightarrow> coprime a b) \<Longrightarrow>
haftmann@67051
  1734
    Lcm A = normalize (\<Prod>A)"
haftmann@67051
  1735
  by (rule Lcm_coprime) (simp_all add: card_eq_0_iff)
haftmann@67051
  1736
haftmann@67051
  1737
end
haftmann@67051
  1738
haftmann@67051
  1739
wenzelm@69593
  1740
subsection \<open>GCD and LCM on \<^typ>\<open>nat\<close> and \<^typ>\<open>int\<close>\<close>
haftmann@59008
  1741
huffman@31706
  1742
instantiation nat :: gcd
huffman@31706
  1743
begin
wenzelm@21256
  1744
haftmann@62345
  1745
fun gcd_nat  :: "nat \<Rightarrow> nat \<Rightarrow> nat"
wenzelm@63489
  1746
  where "gcd_nat x y = (if y = 0 then x else gcd y (x mod y))"
huffman@31706
  1747
haftmann@62345
  1748
definition lcm_nat :: "nat \<Rightarrow> nat \<Rightarrow> nat"
wenzelm@63489
  1749
  where "lcm_nat x y = x * y div (gcd x y)"
wenzelm@63489
  1750
wenzelm@63489
  1751
instance ..
huffman@31706
  1752
huffman@31706
  1753
end
huffman@31706
  1754
huffman@31706
  1755
instantiation int :: gcd
huffman@31706
  1756
begin
wenzelm@21256
  1757
haftmann@62345
  1758
definition gcd_int  :: "int \<Rightarrow> int \<Rightarrow> int"
haftmann@62345
  1759
  where "gcd_int x y = int (gcd (nat \<bar>x\<bar>) (nat \<bar>y\<bar>))"
haftmann@23687
  1760
haftmann@62345
  1761
definition lcm_int :: "int \<Rightarrow> int \<Rightarrow> int"
haftmann@62345
  1762
  where "lcm_int x y = int (lcm (nat \<bar>x\<bar>) (nat \<bar>y\<bar>))"
haftmann@23687
  1763
wenzelm@61944
  1764
instance ..
huffman@31706
  1765
huffman@31706
  1766
end
haftmann@23687
  1767
haftmann@67118
  1768
lemma gcd_int_int_eq [simp]:
haftmann@67118
  1769
  "gcd (int m) (int n) = int (gcd m n)"
haftmann@67118
  1770
  by (simp add: gcd_int_def)
haftmann@67118
  1771
haftmann@67118
  1772
lemma gcd_nat_abs_left_eq [simp]:
haftmann@67118
  1773
  "gcd (nat \<bar>k\<bar>) n = nat (gcd k (int n))"
haftmann@67118
  1774
  by (simp add: gcd_int_def)
haftmann@67118
  1775
haftmann@67118
  1776
lemma gcd_nat_abs_right_eq [simp]:
haftmann@67118
  1777
  "gcd n (nat \<bar>k\<bar>) = nat (gcd (int n) k)"
haftmann@67118
  1778
  by (simp add: gcd_int_def)
haftmann@67118
  1779
haftmann@67118
  1780
lemma abs_gcd_int [simp]:
haftmann@67118
  1781
  "\<bar>gcd x y\<bar> = gcd x y"
haftmann@67118
  1782
  for x y :: int
haftmann@67118
  1783
  by (simp only: gcd_int_def)
haftmann@67118
  1784
haftmann@67118
  1785
lemma gcd_abs1_int [simp]:
haftmann@67118
  1786
  "gcd \<bar>x\<bar> y = gcd x y"
haftmann@67118
  1787
  for x y :: int
haftmann@67118
  1788
  by (simp only: gcd_int_def) simp
haftmann@67118
  1789
haftmann@67118
  1790
lemma gcd_abs2_int [simp]:
haftmann@67118
  1791
  "gcd x \<bar>y\<bar> = gcd x y"
haftmann@67118
  1792
  for x y :: int
haftmann@67118
  1793
  by (simp only: gcd_int_def) simp
haftmann@67118
  1794
haftmann@67118
  1795
lemma lcm_int_int_eq [simp]:
haftmann@67118
  1796
  "lcm (int m) (int n) = int (lcm m n)"
haftmann@67118
  1797
  by (simp add: lcm_int_def)
haftmann@67118
  1798
haftmann@67118
  1799
lemma lcm_nat_abs_left_eq [simp]:
haftmann@67118
  1800
  "lcm (nat \<bar>k\<bar>) n = nat (lcm k (int n))"
haftmann@67118
  1801
  by (simp add: lcm_int_def)
haftmann@67118
  1802
haftmann@67118
  1803
lemma lcm_nat_abs_right_eq [simp]:
haftmann@67118
  1804
  "lcm n (nat \<bar>k\<bar>) = nat (lcm (int n) k)"
haftmann@67118
  1805
  by (simp add: lcm_int_def)
haftmann@67118
  1806
haftmann@67118
  1807
lemma lcm_abs1_int [simp]:
haftmann@67118
  1808
  "lcm \<bar>x\<bar> y = lcm x y"
haftmann@67118
  1809
  for x y :: int
haftmann@67118
  1810
  by (simp only: lcm_int_def) simp
haftmann@67118
  1811
haftmann@67118
  1812
lemma lcm_abs2_int [simp]:
haftmann@67118
  1813
  "lcm x \<bar>y\<bar> = lcm x y"
haftmann@67118
  1814
  for x y :: int
haftmann@67118
  1815
  by (simp only: lcm_int_def) simp
haftmann@67118
  1816
haftmann@67118
  1817
lemma abs_lcm_int [simp]: "\<bar>lcm i j\<bar> = lcm i j"
haftmann@67118
  1818
  for i j :: int
haftmann@67118
  1819
  by (simp only: lcm_int_def)
haftmann@67118
  1820
lp15@68708
  1821
lemma gcd_nat_induct [case_names base step]:
haftmann@23687
  1822
  fixes m n :: nat
haftmann@23687
  1823
  assumes "\<And>m. P m 0"
haftmann@23687
  1824
    and "\<And>m n. 0 < n \<Longrightarrow> P n (m mod n) \<Longrightarrow> P m n"
haftmann@23687
  1825
  shows "P m n"
lp15@68708
  1826
proof (induction m n rule: gcd_nat.induct)
lp15@68708
  1827
  case (1 x y)
lp15@68708
  1828
  then show ?case
lp15@68708
  1829
    using assms neq0_conv by blast
lp15@68708
  1830
qed
wenzelm@63489
  1831
wenzelm@63489
  1832
lemma gcd_neg1_int [simp]: "gcd (- x) y = gcd x y"
wenzelm@63489
  1833
  for x y :: int
haftmann@67118
  1834
  by (simp only: gcd_int_def) simp
huffman@31706
  1835
wenzelm@63489
  1836
lemma gcd_neg2_int [simp]: "gcd x (- y) = gcd x y"
wenzelm@63489
  1837
  for x y :: int
haftmann@67118
  1838
  by (simp only: gcd_int_def) simp
huffman@31706
  1839
nipkow@31952
  1840
lemma gcd_cases_int:
wenzelm@63489
  1841
  fixes x y :: int
wenzelm@63489
  1842
  assumes "x \<ge> 0 \<Longrightarrow> y \<ge> 0 \<Longrightarrow> P (gcd x y)"
wenzelm@63489
  1843
    and "x \<ge> 0 \<Longrightarrow> y \<le> 0 \<Longrightarrow> P (gcd x (- y))"
wenzelm@63489
  1844
    and "x \<le> 0 \<Longrightarrow> y \<ge> 0 \<Longrightarrow> P (gcd (- x) y)"
wenzelm@63489
  1845
    and "x \<le> 0 \<Longrightarrow> y \<le> 0 \<Longrightarrow> P (gcd (- x) (- y))"
huffman@31706
  1846
  shows "P (gcd x y)"
wenzelm@63489
  1847
  using assms by auto arith
wenzelm@21256
  1848
nipkow@31952
  1849
lemma gcd_ge_0_int [simp]: "gcd (x::int) y >= 0"
wenzelm@63489
  1850
  for x y :: int
huffman@31706
  1851
  by (simp add: gcd_int_def)
huffman@31706
  1852
wenzelm@63489
  1853
lemma lcm_neg1_int: "lcm (- x) y = lcm x y"
wenzelm@63489
  1854
  for x y :: int
haftmann@67118
  1855
  by (simp only: lcm_int_def) simp
huffman@31706
  1856
wenzelm@63489
  1857
lemma lcm_neg2_int: "lcm x (- y) = lcm x y"
wenzelm@63489
  1858
  for x y :: int
haftmann@67118
  1859
  by (simp only: lcm_int_def) simp
nipkow@31814
  1860
nipkow@31952
  1861
lemma lcm_cases_int:
wenzelm@63489
  1862
  fixes x y :: int
wenzelm@63489
  1863
  assumes "x \<ge> 0 \<Longrightarrow> y \<ge> 0 \<Longrightarrow> P (lcm x y)"
wenzelm@63489
  1864
    and "x \<ge> 0 \<Longrightarrow> y \<le> 0 \<Longrightarrow> P (lcm x (- y))"
wenzelm@63489
  1865
    and "x \<le> 0 \<Longrightarrow> y \<ge> 0 \<Longrightarrow> P (lcm (- x) y)"
wenzelm@63489
  1866
    and "x \<le> 0 \<Longrightarrow> y \<le> 0 \<Longrightarrow> P (lcm (- x) (- y))"
huffman@31706
  1867
  shows "P (lcm x y)"
lp15@68708
  1868
  using assms by (auto simp: lcm_neg1_int lcm_neg2_int) arith
huffman@31706
  1869
wenzelm@63489
  1870
lemma lcm_ge_0_int [simp]: "lcm x y \<ge> 0"
wenzelm@63489
  1871
  for x y :: int
haftmann@67118
  1872
  by (simp only: lcm_int_def)
huffman@31706
  1873
wenzelm@63489
  1874
lemma gcd_0_nat: "gcd x 0 = x"
wenzelm@63489
  1875
  for x :: nat
haftmann@23687
  1876
  by simp
haftmann@23687
  1877
wenzelm@63489
  1878
lemma gcd_0_int [simp]: "gcd x 0 = \<bar>x\<bar>"
wenzelm@63489
  1879
  for x :: int
wenzelm@63489
  1880
  by (auto simp: gcd_int_def)
wenzelm@63489
  1881
wenzelm@63489
  1882
lemma gcd_0_left_nat: "gcd 0 x = x"
wenzelm@63489
  1883
  for x :: nat
haftmann@23687
  1884
  by simp
haftmann@23687
  1885
wenzelm@63489
  1886
lemma gcd_0_left_int [simp]: "gcd 0 x = \<bar>x\<bar>"
wenzelm@63489
  1887
  for x :: int
haftmann@67118
  1888
  by (auto simp: gcd_int_def)
wenzelm@63489
  1889
wenzelm@63489
  1890
lemma gcd_red_nat: "gcd x y = gcd y (x mod y)"
wenzelm@63489
  1891
  for x y :: nat
wenzelm@63489
  1892
  by (cases "y = 0") auto
wenzelm@63489
  1893
wenzelm@63489
  1894
wenzelm@63489
  1895
text \<open>Weaker, but useful for the simplifier.\<close>
wenzelm@63489
  1896
wenzelm@63489
  1897
lemma gcd_non_0_nat: "y \<noteq> 0 \<Longrightarrow> gcd x y = gcd y (x mod y)"
wenzelm@63489
  1898
  for x y :: nat
wenzelm@21263
  1899
  by simp
wenzelm@21256
  1900
wenzelm@63489
  1901
lemma gcd_1_nat [simp]: "gcd m 1 = 1"
wenzelm@63489
  1902
  for m :: nat
haftmann@60690
  1903
  by simp
huffman@31706
  1904
wenzelm@63489
  1905
lemma gcd_Suc_0 [simp]: "gcd m (Suc 0) = Suc 0"
wenzelm@63489
  1906
  for m :: nat
wenzelm@63489
  1907
  by simp
wenzelm@63489
  1908
wenzelm@63489
  1909
lemma gcd_1_int [simp]: "gcd m 1 = 1"
wenzelm@63489
  1910
  for m :: int
huffman@31706
  1911
  by (simp add: gcd_int_def)
huffman@30082
  1912
wenzelm@63489
  1913
lemma gcd_idem_nat: "gcd x x = x"
wenzelm@63489
  1914
  for x :: nat
wenzelm@63489
  1915
  by simp
wenzelm@63489
  1916
wenzelm@63489
  1917
lemma gcd_idem_int: "gcd x x = \<bar>x\<bar>"
wenzelm@63489
  1918
  for x :: int
lp15@68708
  1919
  by (auto simp: gcd_int_def)
huffman@31706
  1920
huffman@31706
  1921
declare gcd_nat.simps [simp del]
wenzelm@21256
  1922
wenzelm@60758
  1923
text \<open>
wenzelm@69593
  1924
  \<^medskip> \<^term>\<open>gcd m n\<close> divides \<open>m\<close> and \<open>n\<close>.
wenzelm@63489
  1925
  The conjunctions don't seem provable separately.
wenzelm@60758
  1926
\<close>
wenzelm@21256
  1927
haftmann@59008
  1928
instance nat :: semiring_gcd
haftmann@59008
  1929
proof
haftmann@59008
  1930
  fix m n :: nat
haftmann@59008
  1931
  show "gcd m n dvd m" and "gcd m n dvd n"
haftmann@59008
  1932
  proof (induct m n rule: gcd_nat_induct)
lp15@68708
  1933
    case (step m n)
haftmann@59008
  1934
    then have "gcd n (m mod n) dvd m"
lp15@68708
  1935
      by (metis dvd_mod_imp_dvd)
lp15@68708
  1936
    with step show "gcd m n dvd m"
haftmann@59008
  1937
      by (simp add: gcd_non_0_nat)
haftmann@59008
  1938
  qed (simp_all add: gcd_0_nat gcd_non_0_nat)
haftmann@59008
  1939
next
haftmann@59008
  1940
  fix m n k :: nat
haftmann@59008
  1941
  assume "k dvd m" and "k dvd n"
haftmann@59008
  1942
  then show "k dvd gcd m n"
haftmann@59008
  1943
    by (induct m n rule: gcd_nat_induct) (simp_all add: gcd_non_0_nat dvd_mod gcd_0_nat)
haftmann@60686
  1944
qed (simp_all add: lcm_nat_def)
lp15@59667
  1945
haftmann@59008
  1946
instance int :: ring_gcd
haftmann@67118
  1947
proof
haftmann@67118
  1948
  fix k l r :: int
haftmann@67118
  1949
  show "gcd k l dvd k" "gcd k l dvd l"
haftmann@67118
  1950
    using gcd_dvd1 [of "nat \<bar>k\<bar>" "nat \<bar>l\<bar>"]
haftmann@67118
  1951
      gcd_dvd2 [of "nat \<bar>k\<bar>" "nat \<bar>l\<bar>"]
haftmann@67118
  1952
    by simp_all
haftmann@67118
  1953
  show "lcm k l = normalize (k * l) div gcd k l"
haftmann@67118
  1954
    using lcm_gcd [of "nat \<bar>k\<bar>" "nat \<bar>l\<bar>"]
haftmann@67118
  1955
    by (simp add: nat_eq_iff of_nat_div abs_mult)
haftmann@67118
  1956
  assume "r dvd k" "r dvd l"
haftmann@67118
  1957
  then show "r dvd gcd k l"
haftmann@67118
  1958
    using gcd_greatest [of "nat \<bar>r\<bar>" "nat \<bar>k\<bar>" "nat \<bar>l\<bar>"]
haftmann@67118
  1959
    by simp
haftmann@67118
  1960
qed simp
wenzelm@63489
  1961
wenzelm@63489
  1962
lemma gcd_le1_nat [simp]: "a \<noteq> 0 \<Longrightarrow> gcd a b \<le> a"
wenzelm@63489
  1963
  for a b :: nat
wenzelm@63489
  1964
  by (rule dvd_imp_le) auto
wenzelm@63489
  1965
wenzelm@63489
  1966
lemma gcd_le2_nat [simp]: "b \<noteq> 0 \<Longrightarrow> gcd a b \<le> b"
wenzelm@63489
  1967
  for a b :: nat
wenzelm@63489
  1968
  by (rule dvd_imp_le) auto
wenzelm@63489
  1969
wenzelm@63489
  1970
lemma gcd_le1_int [simp]: "a > 0 \<Longrightarrow> gcd a b \<le> a"
wenzelm@63489
  1971
  for a b :: int
wenzelm@63489
  1972
  by (rule zdvd_imp_le) auto
wenzelm@63489
  1973
wenzelm@63489
  1974
lemma gcd_le2_int [simp]: "b > 0 \<Longrightarrow> gcd a b \<le> b"
wenzelm@63489
  1975
  for a b :: int
wenzelm@63489
  1976
  by (rule zdvd_imp_le) auto
wenzelm@63489
  1977
wenzelm@63489
  1978
lemma gcd_pos_nat [simp]: "gcd m n > 0 \<longleftrightarrow> m \<noteq> 0 \<or> n \<noteq> 0"
wenzelm@63489
  1979
  for m n :: nat
wenzelm@63489
  1980
  using gcd_eq_0_iff [of m n] by arith
wenzelm@63489
  1981
wenzelm@63489
  1982
lemma gcd_pos_int [simp]: "gcd m n > 0 \<longleftrightarrow> m \<noteq> 0 \<or> n \<noteq> 0"
wenzelm@63489
  1983
  for m n :: int
wenzelm@63489
  1984
  using gcd_eq_0_iff [of m n] gcd_ge_0_int [of m n] by arith
wenzelm@63489
  1985
wenzelm@63489
  1986
lemma gcd_unique_nat: "d dvd a \<and> d dvd b \<and> (\<forall>e. e dvd a \<and> e dvd b \<longrightarrow> e dvd d) \<longleftrightarrow> d = gcd a b"
wenzelm@63489
  1987
  for d a :: nat
lp15@68708
  1988
  using gcd_unique by fastforce
wenzelm@63489
  1989
wenzelm@63489
  1990
lemma gcd_unique_int:
wenzelm@63489
  1991
  "d \<ge> 0 \<and> d dvd a \<and> d dvd b \<and> (\<forall>e. e dvd a \<and> e dvd b \<longrightarrow> e dvd d) \<longleftrightarrow> d = gcd a b"
wenzelm@63489
  1992
  for d a :: int
lp15@68708
  1993
  using zdvd_antisym_nonneg by auto
huffman@30082
  1994
haftmann@61913
  1995
interpretation gcd_nat:
haftmann@62344
  1996
  semilattice_neutr_order gcd "0::nat" Rings.dvd "\<lambda>m n. m dvd n \<and> m \<noteq> n"
lp15@68708
  1997
  by standard (auto simp: gcd_unique_nat [symmetric] intro: dvd_antisym dvd_trans)
nipkow@31798
  1998
wenzelm@63489
  1999
lemma gcd_proj1_if_dvd_int [simp]: "x dvd y \<Longrightarrow> gcd x y = \<bar>x\<bar>"
wenzelm@63489
  2000
  for x y :: int
haftmann@67118
  2001
  by (metis abs_dvd_iff gcd_0_left_int gcd_unique_int)
nipkow@31798
  2002
wenzelm@63489
  2003
lemma gcd_proj2_if_dvd_int [simp]: "y dvd x \<Longrightarrow> gcd x y = \<bar>y\<bar>"
wenzelm@63489
  2004
  for x y :: int
haftmann@62344
  2005
  by (metis gcd_proj1_if_dvd_int gcd.commute)
nipkow@31798
  2006
wenzelm@63489
  2007
wenzelm@63489
  2008
text \<open>\<^medskip> Multiplication laws.\<close>
wenzelm@63489
  2009
wenzelm@63489
  2010
lemma gcd_mult_distrib_nat: "k * gcd m n = gcd (k * m) (k * n)"
wenzelm@63489
  2011
  for k m n :: nat
wenzelm@63489
  2012
  \<comment> \<open>@{cite \<open>page 27\<close> davenport92}\<close>
lp15@68708
  2013
proof (induct m n rule: gcd_nat_induct)
lp15@68708
  2014
  case (step m n)
lp15@68708
  2015
  then show ?case
lp15@68708
  2016
    by (metis gcd_mult_distrib' gcd_red_nat)
lp15@68708
  2017
qed auto
wenzelm@63489
  2018
wenzelm@63489
  2019
lemma gcd_mult_distrib_int: "\<bar>k\<bar> * gcd m n = gcd (k * m) (k * n)"
wenzelm@63489
  2020
  for k m n :: int
haftmann@67118
  2021
  using gcd_mult_distrib' [of k m n] by simp
wenzelm@21256
  2022
wenzelm@63489
  2023
text \<open>\medskip Addition laws.\<close>
wenzelm@63489
  2024
wenzelm@63489
  2025
(* TODO: add the other variations? *)
wenzelm@63489
  2026
wenzelm@63489
  2027
lemma gcd_diff1_nat: "m \<ge> n \<Longrightarrow> gcd (m - n) n = gcd m n"
wenzelm@63489
  2028
  for m n :: nat
eberlm@62429
  2029
  by (subst gcd_add1 [symmetric]) auto
huffman@31706
  2030
wenzelm@63489
  2031
lemma gcd_diff2_nat: "n \<ge> m \<Longrightarrow> gcd (n - m) n = gcd m n"
wenzelm@63489
  2032
  for m n :: nat
lp15@68708
  2033
  by (metis gcd.commute gcd_add2 gcd_diff1_nat le_add_diff_inverse2)
lp15@68708
  2034
lp15@68708
  2035
lemma gcd_non_0_int: 
lp15@68708
  2036
  fixes x y :: int
lp15@68708
  2037
  assumes "y > 0" shows "gcd x y = gcd y (x mod y)"
lp15@68708
  2038
proof (cases "x mod y = 0")
lp15@68708
  2039
  case False
lp15@68708
  2040
  then have neg: "x mod y = y - (- x) mod y"
lp15@68708
  2041
    by (simp add: zmod_zminus1_eq_if)
lp15@68708
  2042
  have xy: "0 \<le> x mod y" 
lp15@68708
  2043
    by (simp add: assms)
lp15@68708
  2044
  show ?thesis
lp15@68708
  2045
  proof (cases "x < 0")
lp15@68708
  2046
    case True
lp15@68708
  2047
    have "nat (- x mod y) \<le> nat y"
lp15@68708
  2048
      by (simp add: assms dual_order.order_iff_strict)
lp15@68708
  2049
    moreover have "gcd (nat (- x)) (nat y) = gcd (nat (- x mod y)) (nat y)"
lp15@68708
  2050
      using True assms gcd_non_0_nat nat_mod_distrib by auto
lp15@68708
  2051
    ultimately have "gcd (nat (- x)) (nat y) = gcd (nat y) (nat (x mod y))"
lp15@68708
  2052
      using assms 
lp15@68708
  2053
      by (simp add: neg nat_diff_distrib') (metis gcd.commute gcd_diff2_nat)
lp15@68708
  2054
    with assms \<open>0 \<le> x mod y\<close> show ?thesis
lp15@68708
  2055
      by (simp add: True dual_order.order_iff_strict gcd_int_def)
lp15@68708
  2056
  next
lp15@68708
  2057
    case False
lp15@68708
  2058
    with assms xy have "gcd (nat x) (nat y) = gcd (nat y) (nat x mod nat y)"
lp15@68708
  2059
      using gcd_red_nat by blast
lp15@68708
  2060
    with False assms show ?thesis
lp15@68708
  2061
      by (simp add: gcd_int_def nat_mod_distrib)
lp15@68708
  2062
  qed
lp15@68708
  2063
qed (use assms in auto)
wenzelm@21256
  2064
wenzelm@63489
  2065
lemma gcd_red_int: "gcd x y = gcd y (x mod y)"
wenzelm@63489
  2066
  for x y :: int
lp15@68708
  2067
proof (cases y "0::int" rule: linorder_cases)
lp15@68708
  2068
  case less
lp15@68708
  2069
  with gcd_non_0_int [of "- y" "- x"] show ?thesis
lp15@68708
  2070
    by auto
lp15@68708
  2071
next
lp15@68708
  2072
  case greater
lp15@68708
  2073
  with gcd_non_0_int [of y x] show ?thesis
lp15@68708
  2074
    by auto
lp15@68708
  2075
qed auto
wenzelm@63489
  2076
wenzelm@63489
  2077
(* TODO: differences, and all variations of addition rules
huffman@31706
  2078
    as simplification rules for nat and int *)
huffman@31706
  2079
wenzelm@63489
  2080
(* TODO: add the three variations of these, and for ints? *)
wenzelm@63489
  2081
wenzelm@63489
  2082
lemma finite_divisors_nat [simp]: (* FIXME move *)
haftmann@62353
  2083
  fixes m :: nat
wenzelm@63489
  2084
  assumes "m > 0"
haftmann@62353
  2085
  shows "finite {d. d dvd m}"
nipkow@31734
  2086
proof-
haftmann@62353
  2087
  from assms have "{d. d dvd m} \<subseteq> {d. d \<le> m}"
haftmann@62353
  2088
    by (auto dest: dvd_imp_le)
haftmann@62353
  2089
  then show ?thesis
haftmann@62353
  2090
    using finite_Collect_le_nat by (rule finite_subset)
nipkow@31734
  2091
qed
nipkow@31734
  2092
haftmann@62353
  2093
lemma finite_divisors_int [simp]:
haftmann@62353
  2094
  fixes i :: int
haftmann@62353
  2095
  assumes "i \<noteq> 0"
haftmann@62353
  2096
  shows "finite {d. d dvd i}"
haftmann@62353
  2097
proof -
haftmann@62353
  2098
  have "{d. \<bar>d\<bar> \<le> \<bar>i\<bar>} = {- \<bar>i\<bar>..\<bar>i\<bar>}"
haftmann@62353
  2099
    by (auto simp: abs_if)
wenzelm@63489
  2100
  then have "finite {d. \<bar>d\<bar> \<le> \<bar>i\<bar>}"
haftmann@62353
  2101
    by simp
wenzelm@63489
  2102
  from finite_subset [OF _ this] show ?thesis
wenzelm@63489
  2103
    using assms by (simp add: dvd_imp_le_int subset_iff)
nipkow@31734
  2104
qed
nipkow@31734
  2105
wenzelm@63489
  2106
lemma Max_divisors_self_nat [simp]: "n \<noteq> 0 \<Longrightarrow> Max {d::nat. d dvd n} = n"
lp15@68708
  2107
  by (fastforce intro: antisym Max_le_iff[THEN iffD2] simp: dvd_imp_le)
lp15@68708
  2108
lp15@68708
  2109
lemma Max_divisors_self_int [simp]: 
lp15@68708
  2110
  assumes "n \<noteq> 0" shows "Max {d::int. d dvd n} = \<bar>n\<bar>"
lp15@68708
  2111
proof (rule antisym)
lp15@68708
  2112
  show "Max {d. d dvd n} \<le> \<bar>n\<bar>"
lp15@68708
  2113
    using assms by (auto intro: abs_le_D1 dvd_imp_le_int intro!: Max_le_iff [THEN iffD2])
lp15@68708
  2114
qed (simp add: assms)
lp15@68708
  2115
lp15@68708
  2116
lemma gcd_is_Max_divisors_nat:
lp15@68708
  2117
  fixes m n :: nat
lp15@68708
  2118
  assumes "n > 0" shows "gcd m n = Max {d. d dvd m \<and> d dvd n}"
lp15@68708
  2119
proof (rule Max_eqI[THEN sym], simp_all)
lp15@68708
  2120
  show "finite {d. d dvd m \<and> d dvd n}"
lp15@68708
  2121
    by (simp add: \<open>n > 0\<close>)
lp15@68708
  2122
  show "\<And>y. y dvd m \<and> y dvd n \<Longrightarrow> y \<le> gcd m n"
lp15@68708
  2123
    by (simp add: \<open>n > 0\<close> dvd_imp_le)
lp15@68708
  2124
qed
lp15@68708
  2125
lp15@68708
  2126
lemma gcd_is_Max_divisors_int:
lp15@68708
  2127
  fixes m n :: int
lp15@68708
  2128
  assumes "n \<noteq> 0" shows "gcd m n = Max {d. d dvd m \<and> d dvd n}"
lp15@68708
  2129
proof (rule Max_eqI[THEN sym], simp_all)
lp15@68708
  2130
  show "finite {d. d dvd m \<and> d dvd n}"
lp15@68708
  2131
    by (simp add: \<open>n \<noteq> 0\<close>)
lp15@68708
  2132
  show "\<And>y. y dvd m \<and> y dvd n \<Longrightarrow> y \<le> gcd m n"
lp15@68708
  2133
    by (simp add: \<open>n \<noteq> 0\<close> zdvd_imp_le)
lp15@68708
  2134
qed
wenzelm@63489
  2135
wenzelm@63489
  2136
lemma gcd_code_int [code]: "gcd k l = \<bar>if l = 0 then k else gcd l (\<bar>k\<bar> mod \<bar>l\<bar>)\<bar>"
wenzelm@63489
  2137
  for k l :: int
haftmann@67118
  2138
  using gcd_red_int [of "\<bar>k\<bar>" "\<bar>l\<bar>"] by simp
haftmann@34030
  2139
haftmann@67051
  2140
lemma coprime_Suc_left_nat [simp]:
haftmann@67051
  2141
  "coprime (Suc n) n"
haftmann@67051
  2142
  using coprime_add_one_left [of n] by simp
haftmann@67051
  2143
haftmann@67051
  2144
lemma coprime_Suc_right_nat [simp]:
haftmann@67051
  2145
  "coprime n (Suc n)"
haftmann@67051
  2146
  using coprime_Suc_left_nat [of n] by (simp add: ac_simps)
haftmann@67051
  2147
haftmann@67051
  2148
lemma coprime_diff_one_left_nat [simp]:
haftmann@67051
  2149
  "coprime (n - 1) n" if "n > 0" for n :: nat
haftmann@67051
  2150
  using that coprime_Suc_right_nat [of "n - 1"] by simp
haftmann@67051
  2151
haftmann@67051
  2152
lemma coprime_diff_one_right_nat [simp]:
haftmann@67051
  2153
  "coprime n (n - 1)" if "n > 0" for n :: nat
haftmann@67051
  2154
  using that coprime_diff_one_left_nat [of n] by (simp add: ac_simps)
haftmann@67051
  2155
haftmann@67051
  2156
lemma coprime_crossproduct_nat:
haftmann@67051
  2157
  fixes a b c d :: nat
haftmann@67051
  2158
  assumes "coprime a d" and "coprime b c"
haftmann@67051
  2159
  shows "a * c = b * d \<longleftrightarrow> a = b \<and> c = d"
haftmann@67051
  2160
  using assms coprime_crossproduct [of a d b c] by simp
haftmann@67051
  2161
haftmann@67051
  2162
lemma coprime_crossproduct_int:
haftmann@67051
  2163
  fixes a b c d :: int
haftmann@67051
  2164
  assumes "coprime a d" and "coprime b c"
haftmann@67051
  2165
  shows "\<bar>a\<bar> * \<bar>c\<bar> = \<bar>b\<bar> * \<bar>d\<bar> \<longleftrightarrow> \<bar>a\<bar> = \<bar>b\<bar> \<and> \<bar>c\<bar> = \<bar>d\<bar>"
haftmann@67051
  2166
  using assms coprime_crossproduct [of a d b c] by simp
huffman@31706
  2167
huffman@31706
  2168
wenzelm@60758
  2169
subsection \<open>Bezout's theorem\<close>
huffman@31706
  2170
wenzelm@63489
  2171
text \<open>
wenzelm@63489
  2172
  Function \<open>bezw\<close> returns a pair of witnesses to Bezout's theorem --
wenzelm@63489
  2173
  see the theorems that follow the definition.
wenzelm@63489
  2174
\<close>
wenzelm@63489
  2175
wenzelm@63489
  2176
fun bezw :: "nat \<Rightarrow> nat \<Rightarrow> int * int"
wenzelm@63489
  2177
  where "bezw x y =
wenzelm@63489
  2178
    (if y = 0 then (1, 0)
wenzelm@63489
  2179
     else
huffman@31706
  2180
      (snd (bezw y (x mod y)),
huffman@31706
  2181
       fst (bezw y (x mod y)) - snd (bezw y (x mod y)) * int(x div y)))"
huffman@31706
  2182
wenzelm@63489
  2183
lemma bezw_0 [simp]: "bezw x 0 = (1, 0)"
wenzelm@63489
  2184
  by simp
wenzelm@63489
  2185
wenzelm@63489
  2186
lemma bezw_non_0:
wenzelm@63489
  2187
  "y > 0 \<Longrightarrow> bezw x y =
wenzelm@63489
  2188
    (snd (bezw y (x mod y)), fst (bezw y (x mod y)) - snd (bezw y (x mod y)) * int(x div y))"
huffman@31706
  2189
  by simp
huffman@31706
  2190
huffman@31706
  2191
declare bezw.simps [simp del]
huffman@31706
  2192
lp15@68708
  2193
lp15@68708
  2194
lemma bezw_aux: "int (gcd x y) = fst (bezw x y) * int x + snd (bezw x y) * int y"
nipkow@31952
  2195
proof (induct x y rule: gcd_nat_induct)
lp15@68708
  2196
  case (step m n)
lp15@68708
  2197
  then have "fst (bezw m n) * int m + snd (bezw m n) * int n - int (gcd m n) 
lp15@68708
  2198
             = int m * snd (bezw n (m mod n)) -
lp15@68708
  2199
               (int (m mod n) * snd (bezw n (m mod n)) + int n * (int (m div n) * snd (bezw n (m mod n))))"
lp15@68708
  2200
    by (simp add: bezw_non_0 gcd_non_0_nat field_simps)
lp15@68708
  2201
  also have "\<dots> = int m * snd (bezw n (m mod n)) - (int (m mod n) + int (n * (m div n))) * snd (bezw n (m mod n))"
lp15@68708
  2202
    by (simp add: distrib_right)
lp15@68708
  2203
  also have "\<dots> = 0"
lp15@68708
  2204
    by (metis cancel_comm_monoid_add_class.diff_cancel mod_mult_div_eq of_nat_add)
lp15@68708
  2205
  finally show ?case
lp15@68708
  2206
    by simp
lp15@68708
  2207
qed auto
lp15@68708
  2208
huffman@31706
  2209
wenzelm@63489
  2210
lemma bezout_int: "\<exists>u v. u * x + v * y = gcd x y"
wenzelm@63489
  2211
  for x y :: int
huffman@31706
  2212
proof -
wenzelm@63489
  2213
  have aux: "x \<ge> 0 \<Longrightarrow> y \<ge> 0 \<Longrightarrow> \<exists>u v. u * x + v * y = gcd x y" for x y :: int
huffman@31706
  2214
    apply (rule_tac x = "fst (bezw (nat x) (nat y))" in exI)
huffman@31706
  2215
    apply (rule_tac x = "snd (bezw (nat x) (nat y))" in exI)
lp15@68708
  2216
    by (simp add: bezw_aux gcd_int_def)
wenzelm@63489
  2217
  consider "x \<ge> 0" "y \<ge> 0" | "x \<ge> 0" "y \<le> 0" | "x \<le> 0" "y \<ge> 0" | "x \<le> 0" "y \<le> 0"
lp15@68708
  2218
    using linear by blast
wenzelm@63489
  2219
  then show ?thesis
wenzelm@63489
  2220
  proof cases
wenzelm@63489
  2221
    case 1
wenzelm@63489
  2222
    then show ?thesis by (rule aux)
wenzelm@63489
  2223
  next
wenzelm@63489
  2224
    case 2
wenzelm@63489
  2225
    then show ?thesis
lp15@68708
  2226
      using aux [of x "-y"]
lp15@68708
  2227
      by (metis gcd_neg2_int mult.commute mult_minus_right neg_0_le_iff_le)
wenzelm@63489
  2228
  next
wenzelm@63489
  2229
    case 3
wenzelm@63489
  2230
    then show ?thesis
lp15@68708
  2231
      using aux [of "-x" y]
lp15@68708
  2232
      by (metis gcd.commute gcd_neg2_int mult.commute mult_minus_right neg_0_le_iff_le)
wenzelm@63489
  2233
  next
wenzelm@63489
  2234
    case 4
wenzelm@63489
  2235
    then show ?thesis
lp15@68708
  2236
      using aux [of "-x" "-y"]
lp15@68708
  2237
      by (metis diff_0 diff_ge_0_iff_ge gcd_neg1_int gcd_neg2_int mult.commute mult_minus_right)
wenzelm@63489
  2238
  qed
huffman@31706
  2239
qed
huffman@31706
  2240
wenzelm@63489
  2241
wenzelm@63489
  2242
text \<open>Versions of Bezout for \<open>nat\<close>, by Amine Chaieb.\<close>
huffman@31706
  2243
lp15@68708
  2244
lemma Euclid_induct [case_names swap zero add]:
wenzelm@63489
  2245
  fixes P :: "nat \<Rightarrow> nat \<Rightarrow> bool"
lp15@68708
  2246
  assumes c: "\<And>a b. P a b \<longleftrightarrow> P b a"
lp15@68708
  2247
    and z: "\<And>a. P a 0"
lp15@68708
  2248
    and add: "\<And>a b. P a b \<longrightarrow> P a (a + b)"
chaieb@27669
  2249
  shows "P a b"
wenzelm@63489
  2250
proof (induct "a + b" arbitrary: a b rule: less_induct)
berghofe@34915
  2251
  case less
wenzelm@63489
  2252
  consider (eq) "a = b" | (lt) "a < b" "a + b - a < a + b" | "b = 0" | "b + a - b < a + b"
wenzelm@63489
  2253
    by arith
wenzelm@63489
  2254
  show ?case
wenzelm@63489
  2255
  proof (cases a b rule: linorder_cases)
wenzelm@63489
  2256
    case equal
wenzelm@63489
  2257
    with add [rule_format, OF z [rule_format, of a]] show ?thesis by simp
wenzelm@63489
  2258
  next
wenzelm@63489
  2259
    case lt: less
wenzelm@63489
  2260
    then consider "a = 0" | "a + b - a < a + b" by arith
wenzelm@63489
  2261
    then show ?thesis
wenzelm@63489
  2262
    proof cases
wenzelm@63489
  2263
      case 1
wenzelm@63489
  2264
      with z c show ?thesis by blast
wenzelm@63489
  2265
    next
wenzelm@63489
  2266
      case 2
wenzelm@63489
  2267
      also have *: "a + b - a = a + (b - a)" using lt by arith
berghofe@34915
  2268
      finally have "a + (b - a) < a + b" .
wenzelm@63489
  2269
      then have "P a (a + (b - a))" by (rule add [rule_format, OF less])
wenzelm@63489
  2270
      then show ?thesis by (simp add: *[symmetric])
wenzelm@63489
  2271
    qed
wenzelm@63489
  2272
  next
wenzelm@63489
  2273
    case gt: greater
wenzelm@63489
  2274
    then consider "b = 0" | "b + a - b < a + b" by arith
wenzelm@63489
  2275
    then show ?thesis
wenzelm@63489
  2276
    proof cases
wenzelm@63489
  2277
      case 1
wenzelm@63489
  2278
      with z c show ?thesis by blast
wenzelm@63489
  2279
    next
wenzelm@63489
  2280
      case 2
wenzelm@63489
  2281
      also have *: "b + a - b = b + (a - b)" using gt by arith
berghofe@34915
  2282
      finally have "b + (a - b) < a + b" .
wenzelm@63489
  2283
      then have "P b (b + (a - b))" by (rule add [rule_format, OF less])
wenzelm@63489
  2284
      then have "P b a" by (simp add: *[symmetric])
wenzelm@63489
  2285
      with c show ?thesis by blast
wenzelm@63489
  2286
    qed
wenzelm@63489
  2287
  qed
chaieb@27669
  2288
qed
chaieb@27669
  2289
nipkow@31952
  2290
lemma bezout_lemma_nat:
lp15@68708
  2291
  fixes d::nat
lp15@68708
  2292
  shows "\<lbrakk>d dvd a; d dvd b; a * x = b * y + d \<or> b * x = a * y + d\<rbrakk>
lp15@68708
  2293
    \<Longrightarrow> \<exists>x y. d dvd a \<and> d dvd a + b \<and> (a * x = (a + b) * y + d \<or> (a + b) * x = a * y + d)"
huffman@31706
  2294
  apply auto
lp15@68708
  2295
  apply (metis add_mult_distrib2 left_add_mult_distrib)
lp15@68708
  2296
  apply (rule_tac x=x in exI)
lp15@68708
  2297
  by (metis add_mult_distrib2 mult.commute add.assoc)
lp15@68708
  2298
lp15@68708
  2299
lemma bezout_add_nat: 
lp15@68708
  2300
  "\<exists>(d::nat) x y. d dvd a \<and> d dvd b \<and> (a * x = b * y + d \<or> b * x = a * y + d)"
lp15@68708
  2301
proof (induct a b rule: Euclid_induct)
lp15@68708
  2302
  case (swap a b)
lp15@68708
  2303
  then show ?case
lp15@68708
  2304
    by blast
lp15@68708
  2305
next
lp15@68708
  2306
  case (zero a)
lp15@68708
  2307
  then show ?case
lp15@68708
  2308
    by fastforce    
lp15@68708
  2309
next
lp15@68708
  2310
  case (add a b)
lp15@68708
  2311
  then show ?case
lp15@68708
  2312
    by (meson bezout_lemma_nat)
lp15@68708
  2313
qed
lp15@68708
  2314
lp15@68708
  2315
lemma bezout1_nat: "\<exists>(d::nat) x y. d dvd a \<and> d dvd b \<and> (a * x - b * y = d \<or> b * x - a * y = d)"
lp15@68708
  2316
  using bezout_add_nat[of a b]  by (metis add_diff_cancel_left')
wenzelm@63489
  2317
wenzelm@63489
  2318
lemma bezout_add_strong_nat:
wenzelm@63489
  2319
  fixes a b :: nat
wenzelm@63489
  2320
  assumes a: "a \<noteq> 0"
chaieb@27669
  2321
  shows "\<exists>d x y. d dvd a \<and> d dvd b \<and> a * x = b * y + d"
wenzelm@63489
  2322
proof -
wenzelm@63489
  2323
  consider d x y where "d dvd a" "d dvd b" "a * x = b * y + d"
lp15@68708
  2324
         | d x y where "d dvd a" "d dvd b" "b * x = a * y + d"
wenzelm@63489
  2325
    using bezout_add_nat [of a b] by blast
wenzelm@63489
  2326
  then show ?thesis
wenzelm@63489
  2327
  proof cases
wenzelm@63489
  2328
    case 1
wenzelm@63489
  2329
    then show ?thesis by blast
wenzelm@63489
  2330
  next
wenzelm@63489
  2331
    case H: 2
wenzelm@63489
  2332
    show ?thesis
wenzelm@63489
  2333
    proof (cases "b = 0")
wenzelm@63489
  2334
      case True
wenzelm@63489
  2335
      with H show ?thesis by simp
wenzelm@63489
  2336
    next
wenzelm@63489
  2337
      case False
wenzelm@63489
  2338
      then have bp: "b > 0" by simp
wenzelm@63489
  2339
      with dvd_imp_le [OF H(2)] consider "d = b" | "d < b"
wenzelm@63489
  2340
        by atomize_elim auto
wenzelm@63489
  2341
      then show ?thesis
wenzelm@63489
  2342
      proof cases
wenzelm@63489
  2343
        case 1
wenzelm@63489
  2344
        with a H show ?thesis
lp15@68708
  2345
          by (metis Suc_pred add.commute mult.commute mult_Suc_right neq0_conv)
wenzelm@63489
  2346
      next
wenzelm@63489
  2347
        case 2
wenzelm@63489
  2348
        show ?thesis
wenzelm@63489
  2349
        proof (cases "x = 0")
wenzelm@63489
  2350
          case True
wenzelm@63489
  2351
          with a H show ?thesis by simp
wenzelm@63489
  2352
        next
wenzelm@63489
  2353
          case x0: False
wenzelm@63489
  2354
          then have xp: "x > 0" by simp
wenzelm@63489
  2355
          from \<open>d < b\<close> have "d \<le> b - 1" by simp
wenzelm@63489
  2356
          then have "d * b \<le> b * (b - 1)" by simp
wenzelm@63489
  2357
          with xp mult_mono[of "1" "x" "d * b" "b * (b - 1)"]
wenzelm@63489
  2358
          have dble: "d * b \<le> x * b * (b - 1)" using bp by simp
wenzelm@63489
  2359
          from H(3) have "d + (b - 1) * (b * x) = d + (b - 1) * (a * y + d)"
huffman@31706
  2360
            by simp
wenzelm@63489
  2361
          then have "d + (b - 1) * a * y + (b - 1) * d = d + (b - 1) * b * x"
haftmann@57512
  2362
            by (simp only: mult.assoc distrib_left)
wenzelm@63489
  2363
          then have "a * ((b - 1) * y) + d * (b - 1 + 1) = d + x * b * (b - 1)"
huffman@31706
  2364
            by algebra
wenzelm@63489
  2365
          then have "a * ((b - 1) * y) = d + x * b * (b - 1) - d * b"
wenzelm@63489
  2366
            using bp by simp
wenzelm@63489
  2367
          then have "a * ((b - 1) * y) = d + (x * b * (b - 1) - d * b)"
wenzelm@32960
  2368
            by (simp only: diff_add_assoc[OF dble, of d, symmetric])
wenzelm@63489
  2369
          then have "a * ((b - 1) * y) = b * (x * (b - 1) - d) + d"
haftmann@59008
  2370
            by (simp only: diff_mult_distrib2 ac_simps)
wenzelm@63489
  2371
          with H(1,2) show ?thesis
lp15@68708
  2372
            by blast
wenzelm@63489
  2373
        qed
wenzelm@63489
  2374
      qed
wenzelm@63489
  2375
    qed
wenzelm@63489
  2376
  qed
chaieb@27669
  2377
qed
chaieb@27669
  2378
wenzelm@63489
  2379
lemma bezout_nat:
wenzelm@63489
  2380
  fixes a :: nat
wenzelm@63489
  2381
  assumes a: "a \<noteq> 0"
chaieb@27669
  2382
  shows "\<exists>x y. a * x = b * y + gcd a b"
wenzelm@63489
  2383
proof -
wenzelm@63489
  2384
  obtain d x y where d: "d dvd a" "d dvd b" and eq: "a * x = b * y + d"
wenzelm@63489
  2385
    using bezout_add_strong_nat [OF a, of b] by blast
wenzelm@63489
  2386
  from d have "d dvd gcd a b"
wenzelm@63489
  2387
    by simp
wenzelm@63489
  2388
  then obtain k where k: "gcd a b = d * k"
wenzelm@63489
  2389
    unfolding dvd_def by blast
wenzelm@63489
  2390
  from eq have "a * x * k = (b * y + d) * k"
wenzelm@63489
  2391
    by auto
wenzelm@63489
  2392
  then have "a * (x * k) = b * (y * k) + gcd a b"
wenzelm@63489
  2393
    by (algebra add: k)
wenzelm@63489
  2394
  then show ?thesis
wenzelm@63489
  2395
    by blast
chaieb@27669
  2396
qed
chaieb@27669
  2397
huffman@31706
  2398
wenzelm@69593
  2399
subsection \<open>LCM properties on \<^typ>\<open>nat\<close> and \<^typ>\<open>int\<close>\<close>
wenzelm@63489
  2400
wenzelm@63489
  2401
lemma lcm_altdef_int [code]: "lcm a b = \<bar>a\<bar> * \<bar>b\<bar> div gcd a b"
wenzelm@63489
  2402
  for a b :: int
haftmann@67118
  2403
  by (simp add: abs_mult lcm_gcd)
haftmann@67118
  2404
  
wenzelm@63489
  2405
lemma prod_gcd_lcm_nat: "m * n = gcd m n * lcm m n"
wenzelm@63489
  2406
  for m n :: nat
lp15@68708
  2407
  by simp
huffman@31706
  2408
wenzelm@63489
  2409
lemma prod_gcd_lcm_int: "\<bar>m\<bar> * \<bar>n\<bar> = gcd m n * lcm m n"
wenzelm@63489
  2410
  for m n :: int
lp15@68708
  2411
  by simp
wenzelm@63489
  2412
wenzelm@63489
  2413
lemma lcm_pos_nat: "m > 0 \<Longrightarrow> n > 0 \<Longrightarrow> lcm m n > 0"
wenzelm@63489
  2414
  for m n :: nat
haftmann@67118
  2415
  using lcm_eq_0_iff [of m n] by auto
wenzelm@63489
  2416
wenzelm@63489
  2417
lemma lcm_pos_int: "m \<noteq> 0 \<Longrightarrow> n \<noteq> 0 \<Longrightarrow> lcm m n > 0"
wenzelm@63489
  2418
  for m n :: int
haftmann@67118
  2419
  by (simp add: less_le lcm_eq_0_iff)
haftmann@23687
  2420
wenzelm@63489
  2421
lemma dvd_pos_nat: "n > 0 \<Longrightarrow> m dvd n \<Longrightarrow> m > 0"  (* FIXME move *)
wenzelm@63489
  2422
  for m n :: nat
lp15@68708
  2423
  by auto
wenzelm@63489
  2424
wenzelm@63489
  2425
lemma lcm_unique_nat:
wenzelm@63489
  2426
  "a dvd d \<and> b dvd d \<and> (\<forall>e. a dvd e \<and> b dvd e \<longrightarrow> d dvd e) \<longleftrightarrow> d = lcm a b"
wenzelm@63489
  2427
  for a b d :: nat
haftmann@62344
  2428
  by (auto intro: dvd_antisym lcm_least)
chaieb@27568
  2429
wenzelm@63489
  2430
lemma lcm_unique_int:
wenzelm@63489
  2431
  "d \<ge> 0 \<and> a dvd d \<and> b dvd d \<and> (\<forall>e. a dvd e \<and> b dvd e \<longrightarrow> d dvd e) \<longleftrightarrow> d = lcm a b"
wenzelm@63489
  2432
  for a b d :: int
haftmann@62344
  2433
  using lcm_least zdvd_antisym_nonneg by auto
haftmann@34973
  2434
wenzelm@63489
  2435
lemma lcm_proj2_if_dvd_nat [simp]: "x dvd y \<Longrightarrow> lcm x y = y"
wenzelm@63489
  2436
  for x y :: nat
lp15@68708
  2437
  by (simp add: lcm_proj2_if_dvd)
wenzelm@63489
  2438
wenzelm@63489
  2439
lemma lcm_proj2_if_dvd_int [simp]: "x dvd y \<Longrightarrow> lcm x y = \<bar>y\<bar>"
wenzelm@63489
  2440
  for x y :: int
lp15@68708
  2441
  by (simp add: lcm_proj2_if_dvd)
wenzelm@63489
  2442
wenzelm@63489
  2443
lemma lcm_proj1_if_dvd_nat [simp]: "x dvd y \<Longrightarrow> lcm y x = y"
wenzelm@63489
  2444
  for x y :: nat
wenzelm@63489
  2445
  by (subst lcm.commute) (erule lcm_proj2_if_dvd_nat)
wenzelm@63489
  2446
wenzelm@63489
  2447
lemma lcm_proj1_if_dvd_int [simp]: "x dvd y \<Longrightarrow> lcm y x = \<bar>y\<bar>"
wenzelm@63489
  2448
  for x y :: int
wenzelm@63489
  2449
  by (subst lcm.commute) (erule lcm_proj2_if_dvd_int)
wenzelm@63489
  2450
wenzelm@63489
  2451
lemma lcm_proj1_iff_nat [simp]: "lcm m n = m \<longleftrightarrow> n dvd m"
wenzelm@63489
  2452
  for m n :: nat
wenzelm@63489
  2453
  by (metis lcm_proj1_if_dvd_nat lcm_unique_nat)
wenzelm@63489
  2454
wenzelm@63489
  2455
lemma lcm_proj2_iff_nat [simp]: "lcm m n = n \<longleftrightarrow> m dvd n"
wenzelm@63489
  2456
  for m n :: nat
wenzelm@63489
  2457
  by (metis lcm_proj2_if_dvd_nat lcm_unique_nat)
wenzelm@63489
  2458
wenzelm@63489
  2459
lemma lcm_proj1_iff_int [simp]: "lcm m n = \<bar>m\<bar> \<longleftrightarrow> n dvd m"
wenzelm@63489
  2460
  for m n :: int
wenzelm@63489
  2461
  by (metis dvd_abs_iff lcm_proj1_if_dvd_int lcm_unique_int)
wenzelm@63489
  2462
wenzelm@63489
  2463
lemma lcm_proj2_iff_int [simp]: "lcm m n = \<bar>n\<bar> \<longleftrightarrow> m dvd n"
wenzelm@63489
  2464
  for m n :: int
wenzelm@63489
  2465
  by (metis dvd_abs_iff lcm_proj2_if_dvd_int lcm_unique_int)
wenzelm@63489
  2466
wenzelm@63489
  2467
lemma lcm_1_iff_nat [simp]: "lcm m n = Suc 0 \<longleftrightarrow> m = Suc 0 \<and> n = Suc 0"
wenzelm@63489
  2468
  for m n :: nat
haftmann@62353
  2469
  using lcm_eq_1_iff [of m n] by simp
wenzelm@63489
  2470
wenzelm@63489
  2471
lemma lcm_1_iff_int [simp]: "lcm m n = 1 \<longleftrightarrow> (m = 1 \<or> m = -1) \<and> (n = 1 \<or> n = -1)"
wenzelm@63489
  2472
  for m n :: int
haftmann@61913
  2473
  by auto
nipkow@31995
  2474
haftmann@34030
  2475
wenzelm@69593
  2476
subsection \<open>The complete divisibility lattice on \<^typ>\<open>nat\<close> and \<^typ>\<open>int\<close>\<close>
nipkow@32112
  2477
wenzelm@63489
  2478
text \<open>
wenzelm@63489
  2479
  Lifting \<open>gcd\<close> and \<open>lcm\<close> to sets (\<open>Gcd\<close> / \<open>Lcm\<close>).
wenzelm@63489
  2480
  \<open>Gcd\<close> is defined via \<open>Lcm\<close> to facilitate the proof that we have a complete lattice.
wenzelm@60758
  2481
\<close>
huffman@45264
  2482
haftmann@62345
  2483
instantiation nat :: semiring_Gcd
nipkow@32112
  2484
begin
nipkow@32112
  2485
haftmann@62344
  2486
interpretation semilattice_neutr_set lcm "1::nat"
haftmann@62344
  2487
  by standard simp_all
haftmann@54867
  2488
wenzelm@63489
  2489
definition "Lcm M = (if finite M then F M else 0)" for M :: "nat set"
wenzelm@63489
  2490
wenzelm@63489
  2491
lemma Lcm_nat_empty: "Lcm {} = (1::nat)"
haftmann@60690
  2492
  by (simp add: Lcm_nat_def del: One_nat_def)
haftmann@51489
  2493
wenzelm@63489
  2494
lemma Lcm_nat_insert: "Lcm (insert n M) = lcm n (Lcm M)" for n :: nat
lp15@68708
  2495
  by (cases "finite M") (auto simp: Lcm_nat_def simp del: One_nat_def)
haftmann@61929
  2496
wenzelm@63489
  2497
lemma Lcm_nat_infinite: "infinite M \<Longrightarrow> Lcm M = 0" for M :: "nat set"
haftmann@61929
  2498
  by (simp add: Lcm_nat_def)
haftmann@61929
  2499
haftmann@61929
  2500
lemma dvd_Lcm_nat [simp]:
haftmann@61929
  2501
  fixes M :: "nat set"
haftmann@61929
  2502
  assumes "m \<in> M"
haftmann@61929
  2503
  shows "m dvd Lcm M"
haftmann@61929
  2504
proof -
wenzelm@63489
  2505
  from assms have "insert m M = M"
wenzelm@63489
  2506
    by auto
haftmann@61929
  2507
  moreover have "m dvd Lcm (insert m M)"
haftmann@61929
  2508
    by (simp add: Lcm_nat_insert)
wenzelm@63489
  2509
  ultimately show ?thesis
wenzelm@63489
  2510
    by simp
haftmann@61929
  2511
qed
haftmann@61929
  2512
haftmann@61929
  2513
lemma Lcm_dvd_nat [simp]:
haftmann@61929
  2514
  fixes M :: "nat set"
haftmann@61929
  2515
  assumes "\<forall>m\<in>M. m dvd n"
haftmann@61929
  2516
  shows "Lcm M dvd n"
haftmann@62353
  2517
proof (cases "n > 0")
wenzelm@63489
  2518
  case False
wenzelm@63489
  2519
  then show ?thesis by simp
haftmann@61929
  2520
next
haftmann@62353
  2521
  case True
wenzelm@63489
  2522
  then have "finite {d. d dvd n}"
wenzelm@63489
  2523
    by (rule finite_divisors_nat)
wenzelm@63489
  2524
  moreover have "M \<subseteq> {d. d dvd n}"
wenzelm@63489
  2525
    using assms by fast
wenzelm@63489
  2526
  ultimately have "finite M"
wenzelm@63489
  2527
    by (rule rev_finite_subset)
wenzelm@63489
  2528
  then show ?thesis
wenzelm@63489
  2529
    using assms by (induct M) (simp_all add: Lcm_nat_empty Lcm_nat_insert)
haftmann@61929
  2530
qed
nipkow@32112
  2531
wenzelm@63489
  2532
definition "Gcd M = Lcm {d. \<forall>m\<in>M. d dvd m}" for M :: "nat set"
wenzelm@63489
  2533
wenzelm@63489
  2534
instance
wenzelm@63489
  2535
proof
wenzelm@63489
  2536
  fix N :: "nat set"
wenzelm@63489
  2537
  fix n :: nat
wenzelm@63489
  2538
  show "Gcd N dvd n" if "n \<in> N"
lp15@68708
  2539
    using that by (induct N rule: infinite_finite_induct) (auto simp: Gcd_nat_def)
wenzelm@63489
  2540
  show "n dvd Gcd N" if "\<And>m. m \<in> N \<Longrightarrow> n dvd m"
lp15@68708
  2541
    using that by (induct N rule: infinite_finite_induct) (auto simp: Gcd_nat_def)
wenzelm@63489
  2542
  show "n dvd Lcm N" if "n \<in> N"
wenzelm@63489
  2543
    using that by (induct N rule: infinite_finite_induct) auto
wenzelm@63489
  2544
  show "Lcm N dvd n" if "\<And>m. m \<in> N \<Longrightarrow> m dvd n"
wenzelm@63489
  2545
    using that by (induct N rule: infinite_finite_induct) auto
wenzelm@63489
  2546
  show "normalize (Gcd N) = Gcd N" and "normalize (Lcm N) = Lcm N"
wenzelm@63489
  2547
    by simp_all
wenzelm@63489
  2548
qed
nipkow@32112
  2549
haftmann@62345
  2550
end
haftmann@61913
  2551
wenzelm@63489
  2552
lemma Gcd_nat_eq_one: "1 \<in> N \<Longrightarrow> Gcd N = 1"
wenzelm@63489
  2553
  for N :: "nat set"
haftmann@62346
  2554
  by (rule Gcd_eq_1_I) auto
haftmann@62346
  2555
wenzelm@63489
  2556
wenzelm@63489
  2557
text \<open>Alternative characterizations of Gcd:\<close>
nipkow@32112
  2558
haftmann@62353
  2559
lemma Gcd_eq_Max:
haftmann@62353
  2560
  fixes M :: "nat set"
haftmann@62353
  2561
  assumes "finite (M::nat set)" and "M \<noteq> {}" and "0 \<notin> M"
haftmann@62353
  2562
  shows "Gcd M = Max (\<Inter>m\<in>M. {d. d dvd m})"
haftmann@62353
  2563
proof (rule antisym)
haftmann@62353
  2564
  from assms obtain m where "m \<in> M" and "m > 0"
haftmann@62353
  2565
    by auto
haftmann@62353
  2566
  from \<open>m > 0\<close> have "finite {d. d dvd m}"
haftmann@62353
  2567
    by (blast intro: finite_divisors_nat)
haftmann@62353
  2568
  with \<open>m \<in> M\<close> have fin: "finite (\<Inter>m\<in>M. {d. d dvd m})"
haftmann@62353
  2569
    by blast
haftmann@62353
  2570
  from fin show "Gcd M \<le> Max (\<Inter>m\<in>M. {d. d dvd m})"
haftmann@62353
  2571
    by (auto intro: Max_ge Gcd_dvd)
haftmann@62353
  2572
  from fin show "Max (\<Inter>m\<in>M. {d. d dvd m}) \<le> Gcd M"
lp15@68708
  2573
  proof (rule Max.boundedI, simp_all)
lp15@68708
  2574
    show "(\<Inter>m\<in>M. {d. d dvd m}) \<noteq> {}"
lp15@68708
  2575
      by auto
lp15@68708
  2576
    show "\<And>a. \<forall>x\<in>M. a dvd x \<Longrightarrow> a \<le> Gcd M"
lp15@68708
  2577
      by (meson Gcd_dvd Gcd_greatest \<open>0 < m\<close> \<open>m \<in> M\<close> dvd_imp_le dvd_pos_nat)
lp15@68708
  2578
  qed
haftmann@62353
  2579
qed
nipkow@32112
  2580
wenzelm@63489
  2581
lemma Gcd_remove0_nat: "finite M \<Longrightarrow> Gcd M = Gcd (M - {0})"
wenzelm@63489
  2582
  for M :: "nat set"
lp15@68708
  2583
proof (induct pred: finite)
lp15@68708
  2584
  case (insert x M)
lp15@68708
  2585
  then show ?case
lp15@68708
  2586
    by (simp add: insert_Diff_if)
lp15@68708
  2587
qed auto
nipkow@32112
  2588
nipkow@32112
  2589
lemma Lcm_in_lcm_closed_set_nat:
lp15@68708
  2590
  fixes M :: "nat set" 
lp15@68708
  2591
  assumes "finite M" "M \<noteq> {}" "\<And>m n. \<lbrakk>m \<in> M; n \<in> M\<rbrakk> \<Longrightarrow> lcm m n \<in> M"
lp15@68708
  2592
  shows "Lcm M \<in> M"
lp15@68708
  2593
  using assms
lp15@68708
  2594
proof (induction M rule: finite_linorder_min_induct)
lp15@68708
  2595
  case (insert x M)
lp15@68708
  2596
  then have "\<And>m n. m \<in> M \<Longrightarrow> n \<in> M \<Longrightarrow> lcm m n \<in> M"
lp15@68708
  2597
    by (metis dvd_lcm1 gr0I insert_iff lcm_pos_nat nat_dvd_not_less)
lp15@68708
  2598
  with insert show ?case
lp15@68708
  2599
    by simp (metis Lcm_nat_empty One_nat_def dvd_1_left dvd_lcm2)
lp15@68708
  2600
qed auto
nipkow@32112
  2601
nipkow@32112
  2602
lemma Lcm_eq_Max_nat:
lp15@68708
  2603
  fixes M :: "nat set" 
lp15@68708
  2604
  assumes M: "finite M" "M \<noteq> {}" "0 \<notin> M" and lcm: "\<And>m n. \<lbrakk>m \<in> M; n \<in> M\<rbrakk> \<Longrightarrow> lcm m n \<in> M"
lp15@68708
  2605
  shows "Lcm M = Max M"
lp15@68708
  2606
proof (rule antisym)
lp15@68708
  2607
  show "Lcm M \<le> Max M"
lp15@68708
  2608
    by (simp add: Lcm_in_lcm_closed_set_nat \<open>finite M\<close> \<open>M \<noteq> {}\<close> lcm)
lp15@68708
  2609
  show "Max M \<le> Lcm M"
lp15@68708
  2610
    by (meson Lcm_0_iff Max_in M dvd_Lcm dvd_imp_le le_0_eq not_le)
lp15@68708
  2611
qed
nipkow@32112
  2612
nipkow@34222
  2613
lemma mult_inj_if_coprime_nat:
lp15@68708
  2614
  "inj_on f A \<Longrightarrow> inj_on g B \<Longrightarrow> (\<And>a b. \<lbrakk>a\<in>A; b\<in>B\<rbrakk> \<Longrightarrow> coprime (f a) (g b)) \<Longrightarrow>
wenzelm@63489
  2615
    inj_on (\<lambda>(a, b). f a * g b) (A \<times> B)"
wenzelm@63489
  2616
  for f :: "'a \<Rightarrow> nat" and g :: "'b \<Rightarrow> nat"
lp15@68708
  2617
  by (auto simp: inj_on_def coprime_crossproduct_nat simp del: One_nat_def)
nipkow@34222
  2618
wenzelm@63489
  2619
wenzelm@63489
  2620
subsubsection \<open>Setwise GCD and LCM for integers\<close>
huffman@45264
  2621
haftmann@67118
  2622
instantiation int :: Gcd
huffman@45264
  2623
begin
huffman@45264
  2624
haftmann@67118
  2625
definition Gcd_int :: "int set \<Rightarrow> int"
haftmann@67118
  2626
  where "Gcd K = int (GCD k\<in>K. (nat \<circ> abs) k)"
haftmann@67118
  2627
haftmann@67118
  2628
definition Lcm_int :: "int set \<Rightarrow> int"
haftmann@67118
  2629
  where "Lcm K = int (LCM k\<in>K. (nat \<circ> abs) k)"
haftmann@67118
  2630
haftmann@67118
  2631
instance ..
haftmann@62345
  2632
haftmann@62345
  2633
end
haftmann@62345
  2634
haftmann@67118
  2635
lemma Gcd_int_eq [simp]:
haftmann@67118
  2636
  "(GCD n\<in>N. int n) = int (Gcd N)"
haftmann@67118
  2637
  by (simp add: Gcd_int_def image_image)
haftmann@67118
  2638
haftmann@67118
  2639
lemma Gcd_nat_abs_eq [simp]:
haftmann@67118
  2640
  "(GCD k\<in>K. nat \<bar>k\<bar>) = nat (Gcd K)"
haftmann@67118
  2641
  by (simp add: Gcd_int_def)
haftmann@67118
  2642
haftmann@67118
  2643
lemma abs_Gcd_eq [simp]:
haftmann@67118
  2644
  "\<bar>Gcd K\<bar> = Gcd K" for K :: "int set"
haftmann@67118
  2645
  by (simp only: Gcd_int_def)
haftmann@67118
  2646
haftmann@67118
  2647
lemma Gcd_int_greater_eq_0 [simp]:
haftmann@67118
  2648
  "Gcd K \<ge> 0"
wenzelm@63489
  2649
  for K :: "int set"
haftmann@67118
  2650
  using abs_ge_zero [of "Gcd K"] by simp
haftmann@67118
  2651
haftmann@67118
  2652
lemma Gcd_abs_eq [simp]:
haftmann@67118
  2653
  "(GCD k\<in>K. \<bar>k\<bar>) = Gcd K"
wenzelm@63489
  2654
  for K :: "int set"
haftmann@67118
  2655
  by (simp only: Gcd_int_def image_image) simp
haftmann@67118
  2656
haftmann@67118
  2657
lemma Lcm_int_eq [simp]:
haftmann@67118
  2658
  "(LCM n\<in>N. int n) = int (Lcm N)"
haftmann@67118
  2659
  by (simp add: Lcm_int_def image_image)
haftmann@67118
  2660
haftmann@67118
  2661
lemma Lcm_nat_abs_eq [simp]:
haftmann@67118
  2662
  "(LCM k\<in>K. nat \<bar>k\<bar>) = nat (Lcm K)"
haftmann@67118
  2663
  by (simp add: Lcm_int_def)
haftmann@67118
  2664
haftmann@67118
  2665
lemma abs_Lcm_eq [simp]:
haftmann@67118
  2666
  "\<bar>Lcm K\<bar> = Lcm K" for K :: "int set"
haftmann@67118
  2667
  by (simp only: Lcm_int_def)
haftmann@67118
  2668
haftmann@67118
  2669
lemma Lcm_int_greater_eq_0 [simp]:
haftmann@67118
  2670
  "Lcm K \<ge> 0"
haftmann@67118
  2671
  for K :: "int set"
haftmann@67118
  2672
  using abs_ge_zero [of "Lcm K"] by simp
haftmann@67118
  2673
haftmann@67118
  2674
lemma Lcm_abs_eq [simp]:
haftmann@67118
  2675
  "(LCM k\<in>K. \<bar>k\<bar>) = Lcm K"
haftmann@67118
  2676
  for K :: "int set"
haftmann@67118
  2677
  by (simp only: Lcm_int_def image_image) simp
haftmann@67118
  2678
haftmann@67118
  2679
instance int :: semiring_Gcd
haftmann@67118
  2680
proof
haftmann@67118
  2681
  fix K :: "int set" and k :: int
haftmann@67118
  2682
  show "Gcd K dvd k" and "k dvd Lcm K" if "k \<in> K"
haftmann@67118
  2683
    using that Gcd_dvd [of "nat \<bar>k\<bar>" "(nat \<circ> abs) ` K"]
haftmann@67118
  2684
      dvd_Lcm [of "nat \<bar>k\<bar>" "(nat \<circ> abs) ` K"]
haftmann@67118
  2685
    by (simp_all add: comp_def)
haftmann@67118
  2686
  show "k dvd Gcd K" if "\<And>l. l \<in> K \<Longrightarrow> k dvd l"
haftmann@67118
  2687
  proof -
haftmann@67118
  2688
    have "nat \<bar>k\<bar> dvd (GCD k\<in>K. nat \<bar>k\<bar>)"
haftmann@67118
  2689
      by (rule Gcd_greatest) (use that in auto)
haftmann@67118
  2690
    then show ?thesis by simp
haftmann@67118
  2691
  qed
haftmann@67118
  2692
  show "Lcm K dvd k" if "\<And>l. l \<in> K \<Longrightarrow> l dvd k"
haftmann@67118
  2693
  proof -
haftmann@67118
  2694
    have "(LCM k\<in>K. nat \<bar>k\<bar>) dvd nat \<bar>k\<bar>"
haftmann@67118
  2695
      by (rule Lcm_least) (use that in auto)
haftmann@67118
  2696
    then show ?thesis by simp
haftmann@67118
  2697
  qed
haftmann@67118
  2698
qed simp_all
haftmann@62346
  2699
haftmann@62345
  2700
w