src/HOL/GCD.thy
author haftmann
Sun Oct 08 22:28:19 2017 +0200 (22 months ago)
changeset 66796 ea9b2e5ca9fc
parent 65811 2653f1cd8775
child 66803 dd8922885a68
permissions -rw-r--r--
tuned proofs
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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 add: 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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begin
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abbreviation coprime :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
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  where "coprime x y \<equiv> gcd x y = 1"
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end
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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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begin
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abbreviation GREATEST_COMMON_DIVISOR :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a"
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  where "GREATEST_COMMON_DIVISOR A f \<equiv> Gcd (f ` A)"
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abbreviation LEAST_COMMON_MULTIPLE :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a"
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  where "LEAST_COMMON_MULTIPLE A f \<equiv> Lcm (f ` A)"
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end
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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. B"   \<rightleftharpoons> "GCD x. GCD y. B"
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  "GCD x. B"     \<rightleftharpoons> "CONST GREATEST_COMMON_DIVISOR CONST UNIV (\<lambda>x. B)"
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  "GCD x. B"     \<rightleftharpoons> "GCD x \<in> CONST UNIV. B"
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  "GCD x\<in>A. B"   \<rightleftharpoons> "CONST GREATEST_COMMON_DIVISOR A (\<lambda>x. B)"
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  "LCM x y. B"   \<rightleftharpoons> "LCM x. LCM y. B"
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  "LCM x. B"     \<rightleftharpoons> "CONST LEAST_COMMON_MULTIPLE CONST UNIV (\<lambda>x. B)"
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  "LCM x. B"     \<rightleftharpoons> "LCM x \<in> CONST UNIV. B"
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  "LCM x\<in>A. B"   \<rightleftharpoons> "CONST LEAST_COMMON_MULTIPLE A (\<lambda>x. B)"
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print_translation \<open>
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  [Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax GREATEST_COMMON_DIVISOR} @{syntax_const "_GCD"},
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    Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax LEAST_COMMON_MULTIPLE} @{syntax_const "_LCM"}]
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\<close> \<comment> \<open>to avoid eta-contraction of body\<close>
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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 [simp]: "is_unit (gcd a b) \<longleftrightarrow> coprime a b"
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  by (cases "a = 0 \<and> b = 0") (auto simp add: 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 "coprime 1 a" 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 coprime_1_left: "coprime 1 a"
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  by (fact gcd.bottom_left_bottom)
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lemma coprime_1_right: "coprime a 1"
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  by (fact gcd.bottom_right_bottom)
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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)
wenzelm@63489
   309
  moreover from False * have "gcd (c * a) (c * b) dvd c * gcd a b"
haftmann@60686
   310
    by (metis div_dvd_iff_mult dvd_mult_left gcd_dvd1 gcd_dvd2 gcd_greatest mult_commute)
haftmann@60686
   311
  ultimately have "normalize (gcd (c * a) (c * b)) = normalize (c * gcd a b)"
haftmann@60688
   312
    by (auto intro: associated_eqI)
wenzelm@63489
   313
  then show ?thesis
wenzelm@63489
   314
    by (simp add: normalize_mult)
haftmann@60686
   315
qed
haftmann@60686
   316
wenzelm@63489
   317
lemma gcd_mult_right: "gcd (a * c) (b * c) = gcd b a * normalize c"
haftmann@60686
   318
  using gcd_mult_left [of c a b] by (simp add: ac_simps)
haftmann@60686
   319
wenzelm@63489
   320
lemma mult_gcd_left: "c * gcd a b = unit_factor c * gcd (c * a) (c * b)"
haftmann@60686
   321
  by (simp add: gcd_mult_left mult.assoc [symmetric])
haftmann@60686
   322
wenzelm@63489
   323
lemma mult_gcd_right: "gcd a b * c = gcd (a * c) (b * c) * unit_factor c"
haftmann@60686
   324
  using mult_gcd_left [of c a b] by (simp add: ac_simps)
haftmann@60686
   325
wenzelm@63489
   326
lemma dvd_lcm1 [iff]: "a dvd lcm a b"
haftmann@60686
   327
proof -
haftmann@60686
   328
  have "normalize (a * b) div gcd a b = normalize a * (normalize b div gcd a b)"
haftmann@60686
   329
    by (simp add: lcm_gcd normalize_mult div_mult_swap)
haftmann@60686
   330
  then show ?thesis
haftmann@60686
   331
    by (simp add: lcm_gcd)
haftmann@60686
   332
qed
wenzelm@63489
   333
wenzelm@63489
   334
lemma dvd_lcm2 [iff]: "b dvd lcm a b"
haftmann@60686
   335
proof -
haftmann@60686
   336
  have "normalize (a * b) div gcd a b = normalize b * (normalize a div gcd a b)"
haftmann@60686
   337
    by (simp add: lcm_gcd normalize_mult div_mult_swap ac_simps)
haftmann@60686
   338
  then show ?thesis
haftmann@60686
   339
    by (simp add: lcm_gcd)
haftmann@60686
   340
qed
haftmann@60686
   341
wenzelm@63489
   342
lemma dvd_lcmI1: "a dvd b \<Longrightarrow> a dvd lcm b c"
wenzelm@63489
   343
  by (rule dvd_trans) (assumption, blast)
wenzelm@63489
   344
wenzelm@63489
   345
lemma dvd_lcmI2: "a dvd c \<Longrightarrow> a dvd lcm b c"
haftmann@60689
   346
  by (rule dvd_trans) (assumption, blast)
haftmann@60689
   347
wenzelm@63489
   348
lemma lcm_dvdD1: "lcm a b dvd c \<Longrightarrow> a dvd c"
haftmann@62345
   349
  using dvd_lcm1 [of a b] by (blast intro: dvd_trans)
haftmann@62345
   350
wenzelm@63489
   351
lemma lcm_dvdD2: "lcm a b dvd c \<Longrightarrow> b dvd c"
haftmann@62345
   352
  using dvd_lcm2 [of a b] by (blast intro: dvd_trans)
haftmann@62345
   353
haftmann@60686
   354
lemma lcm_least:
haftmann@60686
   355
  assumes "a dvd c" and "b dvd c"
haftmann@60686
   356
  shows "lcm a b dvd c"
haftmann@60686
   357
proof (cases "c = 0")
wenzelm@63489
   358
  case True
wenzelm@63489
   359
  then show ?thesis by simp
haftmann@60686
   360
next
wenzelm@63489
   361
  case False
wenzelm@63489
   362
  then have *: "is_unit (unit_factor c)"
wenzelm@63489
   363
    by simp
haftmann@60686
   364
  show ?thesis
haftmann@60686
   365
  proof (cases "gcd a b = 0")
wenzelm@63489
   366
    case True
wenzelm@63489
   367
    with assms show ?thesis by simp
haftmann@60686
   368
  next
wenzelm@63489
   369
    case False
wenzelm@63489
   370
    then have "a \<noteq> 0 \<or> b \<noteq> 0"
wenzelm@63489
   371
      by simp
haftmann@60686
   372
    with \<open>c \<noteq> 0\<close> assms have "a * b dvd a * c" "a * b dvd c * b"
haftmann@60686
   373
      by (simp_all add: mult_dvd_mono)
haftmann@60686
   374
    then have "normalize (a * b) dvd gcd (a * c) (b * c)"
haftmann@60686
   375
      by (auto intro: gcd_greatest simp add: ac_simps)
haftmann@60686
   376
    then have "normalize (a * b) dvd gcd (a * c) (b * c) * unit_factor c"
wenzelm@63489
   377
      using * by (simp add: dvd_mult_unit_iff)
haftmann@60686
   378
    then have "normalize (a * b) dvd gcd a b * c"
haftmann@60686
   379
      by (simp add: mult_gcd_right [of a b c])
haftmann@60686
   380
    then have "normalize (a * b) div gcd a b dvd c"
haftmann@60686
   381
      using False by (simp add: div_dvd_iff_mult ac_simps)
wenzelm@63489
   382
    then show ?thesis
wenzelm@63489
   383
      by (simp add: lcm_gcd)
haftmann@60686
   384
  qed
haftmann@60686
   385
qed
haftmann@60686
   386
wenzelm@63489
   387
lemma lcm_least_iff [simp]: "lcm a b dvd c \<longleftrightarrow> a dvd c \<and> b dvd c"
haftmann@60686
   388
  by (blast intro!: lcm_least intro: dvd_trans)
haftmann@60686
   389
wenzelm@63489
   390
lemma normalize_lcm [simp]: "normalize (lcm a b) = lcm a b"
haftmann@60686
   391
  by (simp add: lcm_gcd dvd_normalize_div)
haftmann@60686
   392
wenzelm@63489
   393
lemma lcm_0_left [simp]: "lcm 0 a = 0"
wenzelm@63489
   394
  by (simp add: lcm_gcd)
wenzelm@63489
   395
wenzelm@63489
   396
lemma lcm_0_right [simp]: "lcm a 0 = 0"
haftmann@60686
   397
  by (simp add: lcm_gcd)
wenzelm@63489
   398
wenzelm@63489
   399
lemma lcm_eq_0_iff: "lcm a b = 0 \<longleftrightarrow> a = 0 \<or> b = 0"
wenzelm@63489
   400
  (is "?P \<longleftrightarrow> ?Q")
haftmann@60686
   401
proof
wenzelm@63489
   402
  assume ?P
wenzelm@63489
   403
  then have "0 dvd lcm a b"
wenzelm@63489
   404
    by simp
haftmann@60686
   405
  then have "0 dvd normalize (a * b) div gcd a b"
haftmann@60686
   406
    by (simp add: lcm_gcd)
haftmann@60686
   407
  then have "0 * gcd a b dvd normalize (a * b)"
haftmann@60686
   408
    using dvd_div_iff_mult [of "gcd a b" _ 0] by (cases "gcd a b = 0") simp_all
haftmann@60686
   409
  then have "normalize (a * b) = 0"
haftmann@60686
   410
    by simp
wenzelm@63489
   411
  then show ?Q
wenzelm@63489
   412
    by simp
haftmann@60686
   413
next
wenzelm@63489
   414
  assume ?Q
wenzelm@63489
   415
  then show ?P
wenzelm@63489
   416
    by auto
haftmann@60686
   417
qed
haftmann@60686
   418
wenzelm@63489
   419
lemma lcm_eq_1_iff [simp]: "lcm a b = 1 \<longleftrightarrow> is_unit a \<and> is_unit b"
haftmann@61913
   420
  by (auto intro: associated_eqI)
haftmann@61913
   421
wenzelm@63489
   422
lemma unit_factor_lcm: "unit_factor (lcm a b) = (if a = 0 \<or> b = 0 then 0 else 1)"
haftmann@60686
   423
  by (simp add: unit_factor_gcd dvd_unit_factor_div lcm_gcd)
haftmann@60686
   424
wenzelm@61605
   425
sublocale lcm: abel_semigroup lcm
haftmann@60686
   426
proof
haftmann@60686
   427
  fix a b c
haftmann@60686
   428
  show "lcm a b = lcm b a"
haftmann@60686
   429
    by (simp add: lcm_gcd ac_simps normalize_mult dvd_normalize_div)
haftmann@60688
   430
  have "lcm (lcm a b) c dvd lcm a (lcm b c)"
haftmann@60688
   431
    and "lcm a (lcm b c) dvd lcm (lcm a b) c"
haftmann@60688
   432
    by (auto intro: lcm_least
haftmann@60686
   433
      dvd_trans [of b "lcm b c" "lcm a (lcm b c)"]
haftmann@60686
   434
      dvd_trans [of c "lcm b c" "lcm a (lcm b c)"]
haftmann@60686
   435
      dvd_trans [of a "lcm a b" "lcm (lcm a b) c"]
haftmann@60686
   436
      dvd_trans [of b "lcm a b" "lcm (lcm a b) c"])
haftmann@60686
   437
  then show "lcm (lcm a b) c = lcm a (lcm b c)"
haftmann@60688
   438
    by (rule associated_eqI) simp_all
haftmann@60686
   439
qed
haftmann@60686
   440
haftmann@64850
   441
sublocale lcm: bounded_quasi_semilattice lcm 1 0 normalize
haftmann@64850
   442
proof
haftmann@64850
   443
  show "lcm a a = normalize a" for a
haftmann@64850
   444
  proof -
haftmann@64850
   445
    have "lcm a a dvd a"
haftmann@64850
   446
      by (rule lcm_least) simp_all
haftmann@64850
   447
    then show ?thesis
haftmann@64850
   448
      by (auto intro: associated_eqI)
haftmann@64850
   449
  qed
haftmann@64850
   450
  show "lcm (normalize a) b = lcm a b" for a b
haftmann@64850
   451
    using dvd_lcm1 [of "normalize a" b] unfolding normalize_dvd_iff
haftmann@60688
   452
    by (auto intro: associated_eqI)
haftmann@64850
   453
  show "lcm 1 a = normalize a" for a
haftmann@64850
   454
    by (rule associated_eqI) simp_all
haftmann@64850
   455
qed simp_all
haftmann@64850
   456
haftmann@64850
   457
lemma lcm_self: "lcm a a = normalize a"
haftmann@64850
   458
  by (fact lcm.idem_normalize)
haftmann@64850
   459
haftmann@64850
   460
lemma lcm_left_idem: "lcm a (lcm a b) = lcm a b"
haftmann@64850
   461
  by (fact lcm.left_idem)
haftmann@64850
   462
haftmann@64850
   463
lemma lcm_right_idem: "lcm (lcm a b) b = lcm a b"
haftmann@64850
   464
  by (fact lcm.right_idem)
haftmann@61913
   465
wenzelm@63489
   466
lemma gcd_mult_lcm [simp]: "gcd a b * lcm a b = normalize a * normalize b"
haftmann@60686
   467
  by (simp add: lcm_gcd normalize_mult)
haftmann@60686
   468
wenzelm@63489
   469
lemma lcm_mult_gcd [simp]: "lcm a b * gcd a b = normalize a * normalize b"
wenzelm@63489
   470
  using gcd_mult_lcm [of a b] by (simp add: ac_simps)
haftmann@60686
   471
haftmann@60686
   472
lemma gcd_lcm:
haftmann@60686
   473
  assumes "a \<noteq> 0" and "b \<noteq> 0"
haftmann@60686
   474
  shows "gcd a b = normalize (a * b) div lcm a b"
haftmann@60686
   475
proof -
haftmann@60686
   476
  from assms have "lcm a b \<noteq> 0"
haftmann@60686
   477
    by (simp add: lcm_eq_0_iff)
wenzelm@63489
   478
  have "gcd a b * lcm a b = normalize a * normalize b"
wenzelm@63489
   479
    by simp
haftmann@60686
   480
  then have "gcd a b * lcm a b div lcm a b = normalize (a * b) div lcm a b"
haftmann@60686
   481
    by (simp_all add: normalize_mult)
haftmann@60686
   482
  with \<open>lcm a b \<noteq> 0\<close> show ?thesis
haftmann@64240
   483
    using nonzero_mult_div_cancel_right [of "lcm a b" "gcd a b"] by simp
haftmann@60686
   484
qed
haftmann@60686
   485
haftmann@64850
   486
lemma lcm_1_left: "lcm 1 a = normalize a"
haftmann@64850
   487
  by (fact lcm.top_left_normalize)
haftmann@64850
   488
haftmann@64850
   489
lemma lcm_1_right: "lcm a 1 = normalize a"
haftmann@64850
   490
  by (fact lcm.top_right_normalize)
wenzelm@63489
   491
wenzelm@63489
   492
lemma lcm_mult_left: "lcm (c * a) (c * b) = normalize c * lcm a b"
haftmann@60686
   493
  by (cases "c = 0")
haftmann@60686
   494
    (simp_all add: gcd_mult_right lcm_gcd div_mult_swap normalize_mult ac_simps,
haftmann@60686
   495
      simp add: dvd_div_mult2_eq mult.left_commute [of "normalize c", symmetric])
haftmann@60686
   496
wenzelm@63489
   497
lemma lcm_mult_right: "lcm (a * c) (b * c) = lcm b a * normalize c"
haftmann@60686
   498
  using lcm_mult_left [of c a b] by (simp add: ac_simps)
haftmann@60686
   499
wenzelm@63489
   500
lemma mult_lcm_left: "c * lcm a b = unit_factor c * lcm (c * a) (c * b)"
haftmann@60686
   501
  by (simp add: lcm_mult_left mult.assoc [symmetric])
haftmann@60686
   502
wenzelm@63489
   503
lemma mult_lcm_right: "lcm a b * c = lcm (a * c) (b * c) * unit_factor c"
haftmann@60686
   504
  using mult_lcm_left [of c a b] by (simp add: ac_simps)
eberlm@62429
   505
eberlm@62429
   506
lemma gcdI:
wenzelm@63489
   507
  assumes "c dvd a" and "c dvd b"
wenzelm@63489
   508
    and greatest: "\<And>d. d dvd a \<Longrightarrow> d dvd b \<Longrightarrow> d dvd c"
eberlm@62429
   509
    and "normalize c = c"
eberlm@62429
   510
  shows "c = gcd a b"
eberlm@62429
   511
  by (rule associated_eqI) (auto simp: assms intro: gcd_greatest)
eberlm@62429
   512
wenzelm@63489
   513
lemma gcd_unique:
wenzelm@63489
   514
  "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
   515
  by rule (auto intro: gcdI simp: gcd_greatest)
eberlm@62429
   516
eberlm@62429
   517
lemma gcd_dvd_prod: "gcd a b dvd k * b"
eberlm@62429
   518
  using mult_dvd_mono [of 1] by auto
eberlm@62429
   519
eberlm@62429
   520
lemma gcd_proj2_if_dvd: "b dvd a \<Longrightarrow> gcd a b = normalize b"
eberlm@62429
   521
  by (rule gcdI [symmetric]) simp_all
eberlm@62429
   522
eberlm@62429
   523
lemma gcd_proj1_if_dvd: "a dvd b \<Longrightarrow> gcd a b = normalize a"
eberlm@62429
   524
  by (rule gcdI [symmetric]) simp_all
eberlm@62429
   525
eberlm@62429
   526
lemma gcd_proj1_iff: "gcd m n = normalize m \<longleftrightarrow> m dvd n"
eberlm@62429
   527
proof
wenzelm@63489
   528
  assume *: "gcd m n = normalize m"
eberlm@62429
   529
  show "m dvd n"
eberlm@62429
   530
  proof (cases "m = 0")
wenzelm@63489
   531
    case True
wenzelm@63489
   532
    with * show ?thesis by simp
wenzelm@63489
   533
  next
wenzelm@63489
   534
    case [simp]: False
wenzelm@63489
   535
    from * have **: "m = gcd m n * unit_factor m"
eberlm@62429
   536
      by (simp add: unit_eq_div2)
wenzelm@63489
   537
    show ?thesis
wenzelm@63489
   538
      by (subst **) (simp add: mult_unit_dvd_iff)
wenzelm@63489
   539
  qed
eberlm@62429
   540
next
eberlm@62429
   541
  assume "m dvd n"
wenzelm@63489
   542
  then show "gcd m n = normalize m"
wenzelm@63489
   543
    by (rule gcd_proj1_if_dvd)
eberlm@62429
   544
qed
wenzelm@63489
   545
eberlm@62429
   546
lemma gcd_proj2_iff: "gcd m n = normalize n \<longleftrightarrow> n dvd m"
eberlm@62429
   547
  using gcd_proj1_iff [of n m] by (simp add: ac_simps)
eberlm@62429
   548
eberlm@62429
   549
lemma gcd_mult_distrib': "normalize c * gcd a b = gcd (c * a) (c * b)"
eberlm@62429
   550
  by (rule gcdI) (auto simp: normalize_mult gcd_greatest mult_dvd_mono gcd_mult_left[symmetric])
eberlm@62429
   551
wenzelm@63489
   552
lemma gcd_mult_distrib: "k * gcd a b = gcd (k * a) (k * b) * unit_factor k"
eberlm@62429
   553
proof-
eberlm@62429
   554
  have "normalize k * gcd a b = gcd (k * a) (k * b)"
eberlm@62429
   555
    by (simp add: gcd_mult_distrib')
eberlm@62429
   556
  then have "normalize k * gcd a b * unit_factor k = gcd (k * a) (k * b) * unit_factor k"
eberlm@62429
   557
    by simp
eberlm@62429
   558
  then have "normalize k * unit_factor k * gcd a b  = gcd (k * a) (k * b) * unit_factor k"
eberlm@62429
   559
    by (simp only: ac_simps)
eberlm@62429
   560
  then show ?thesis
eberlm@62429
   561
    by simp
eberlm@62429
   562
qed
eberlm@62429
   563
wenzelm@63489
   564
lemma lcm_mult_unit1: "is_unit a \<Longrightarrow> lcm (b * a) c = lcm b c"
eberlm@62429
   565
  by (rule associated_eqI) (simp_all add: mult_unit_dvd_iff dvd_lcmI1)
eberlm@62429
   566
wenzelm@63489
   567
lemma lcm_mult_unit2: "is_unit a \<Longrightarrow> lcm b (c * a) = lcm b c"
eberlm@62429
   568
  using lcm_mult_unit1 [of a c b] by (simp add: ac_simps)
eberlm@62429
   569
eberlm@62429
   570
lemma lcm_div_unit1:
eberlm@62429
   571
  "is_unit a \<Longrightarrow> lcm (b div a) c = lcm b c"
wenzelm@63489
   572
  by (erule is_unitE [of _ b]) (simp add: lcm_mult_unit1)
wenzelm@63489
   573
wenzelm@63489
   574
lemma lcm_div_unit2: "is_unit a \<Longrightarrow> lcm b (c div a) = lcm b c"
eberlm@62429
   575
  by (erule is_unitE [of _ c]) (simp add: lcm_mult_unit2)
eberlm@62429
   576
haftmann@64850
   577
lemma normalize_lcm_left: "lcm (normalize a) b = lcm a b"
haftmann@64850
   578
  by (fact lcm.normalize_left_idem)
haftmann@64850
   579
haftmann@64850
   580
lemma normalize_lcm_right: "lcm a (normalize b) = lcm a b"
haftmann@64850
   581
  by (fact lcm.normalize_right_idem)
eberlm@62429
   582
eberlm@62429
   583
lemma gcd_mult_unit1: "is_unit a \<Longrightarrow> gcd (b * a) c = gcd b c"
eberlm@62429
   584
  apply (rule gcdI)
wenzelm@63489
   585
     apply simp_all
wenzelm@63489
   586
  apply (rule dvd_trans)
wenzelm@63489
   587
   apply (rule gcd_dvd1)
wenzelm@63489
   588
  apply (simp add: unit_simps)
eberlm@62429
   589
  done
eberlm@62429
   590
eberlm@62429
   591
lemma gcd_mult_unit2: "is_unit a \<Longrightarrow> gcd b (c * a) = gcd b c"
wenzelm@63489
   592
  apply (subst gcd.commute)
wenzelm@63489
   593
  apply (subst gcd_mult_unit1)
wenzelm@63489
   594
   apply assumption
wenzelm@63489
   595
  apply (rule gcd.commute)
wenzelm@63489
   596
  done
eberlm@62429
   597
eberlm@62429
   598
lemma gcd_div_unit1: "is_unit a \<Longrightarrow> gcd (b div a) c = gcd b c"
eberlm@62429
   599
  by (erule is_unitE [of _ b]) (simp add: gcd_mult_unit1)
eberlm@62429
   600
eberlm@62429
   601
lemma gcd_div_unit2: "is_unit a \<Longrightarrow> gcd b (c div a) = gcd b c"
eberlm@62429
   602
  by (erule is_unitE [of _ c]) (simp add: gcd_mult_unit2)
eberlm@62429
   603
haftmann@64850
   604
lemma normalize_gcd_left: "gcd (normalize a) b = gcd a b"
haftmann@64850
   605
  by (fact gcd.normalize_left_idem)
haftmann@64850
   606
haftmann@64850
   607
lemma normalize_gcd_right: "gcd a (normalize b) = gcd a b"
haftmann@64850
   608
  by (fact gcd.normalize_right_idem)
eberlm@62429
   609
eberlm@62429
   610
lemma comp_fun_idem_gcd: "comp_fun_idem gcd"
eberlm@62429
   611
  by standard (simp_all add: fun_eq_iff ac_simps)
eberlm@62429
   612
eberlm@62429
   613
lemma comp_fun_idem_lcm: "comp_fun_idem lcm"
eberlm@62429
   614
  by standard (simp_all add: fun_eq_iff ac_simps)
eberlm@62429
   615
wenzelm@63489
   616
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
   617
proof (rule gcdI)
wenzelm@63489
   618
  assume *: "gcd a b dvd gcd c d"
wenzelm@63489
   619
    and **: "gcd c d dvd gcd a b"
wenzelm@63489
   620
  have "gcd c d dvd c"
wenzelm@63489
   621
    by simp
wenzelm@63489
   622
  with * show "gcd a b dvd c"
wenzelm@63489
   623
    by (rule dvd_trans)
wenzelm@63489
   624
  have "gcd c d dvd d"
wenzelm@63489
   625
    by simp
wenzelm@63489
   626
  with * show "gcd a b dvd d"
wenzelm@63489
   627
    by (rule dvd_trans)
eberlm@62429
   628
  show "normalize (gcd a b) = gcd a b"
eberlm@62429
   629
    by simp
eberlm@62429
   630
  fix l assume "l dvd c" and "l dvd d"
wenzelm@63489
   631
  then have "l dvd gcd c d"
wenzelm@63489
   632
    by (rule gcd_greatest)
wenzelm@63489
   633
  from this and ** show "l dvd gcd a b"
wenzelm@63489
   634
    by (rule dvd_trans)
eberlm@62429
   635
qed
eberlm@62429
   636
eberlm@62429
   637
lemma coprime_dvd_mult:
eberlm@62429
   638
  assumes "coprime a b" and "a dvd c * b"
eberlm@62429
   639
  shows "a dvd c"
eberlm@62429
   640
proof (cases "c = 0")
wenzelm@63489
   641
  case True
wenzelm@63489
   642
  then show ?thesis by simp
eberlm@62429
   643
next
eberlm@62429
   644
  case False
wenzelm@63489
   645
  then have unit: "is_unit (unit_factor c)"
wenzelm@63489
   646
    by simp
eberlm@62429
   647
  from \<open>coprime a b\<close> mult_gcd_left [of c a b]
eberlm@62429
   648
  have "gcd (c * a) (c * b) * unit_factor c = c"
eberlm@62429
   649
    by (simp add: ac_simps)
eberlm@62429
   650
  moreover from \<open>a dvd c * b\<close> have "a dvd gcd (c * a) (c * b) * unit_factor c"
eberlm@62429
   651
    by (simp add: dvd_mult_unit_iff unit)
wenzelm@63489
   652
  ultimately show ?thesis
wenzelm@63489
   653
    by simp
eberlm@62429
   654
qed
eberlm@62429
   655
wenzelm@63489
   656
lemma coprime_dvd_mult_iff: "coprime a c \<Longrightarrow> a dvd b * c \<longleftrightarrow> a dvd b"
wenzelm@63489
   657
  by (auto intro: coprime_dvd_mult)
wenzelm@63489
   658
wenzelm@63489
   659
lemma gcd_mult_cancel: "coprime c b \<Longrightarrow> gcd (c * a) b = gcd a b"
eberlm@62429
   660
  apply (rule associated_eqI)
wenzelm@63489
   661
     apply (rule gcd_greatest)
wenzelm@63489
   662
      apply (rule_tac b = c in coprime_dvd_mult)
wenzelm@63489
   663
       apply (simp add: gcd.assoc)
wenzelm@63489
   664
       apply (simp_all add: ac_simps)
eberlm@62429
   665
  done
eberlm@62429
   666
eberlm@62429
   667
lemma coprime_crossproduct:
wenzelm@63489
   668
  fixes a b c d :: 'a
eberlm@62429
   669
  assumes "coprime a d" and "coprime b c"
wenzelm@63489
   670
  shows "normalize a * normalize c = normalize b * normalize d \<longleftrightarrow>
wenzelm@63489
   671
    normalize a = normalize b \<and> normalize c = normalize d"
wenzelm@63489
   672
    (is "?lhs \<longleftrightarrow> ?rhs")
eberlm@62429
   673
proof
wenzelm@63489
   674
  assume ?rhs
wenzelm@63489
   675
  then show ?lhs by simp
eberlm@62429
   676
next
eberlm@62429
   677
  assume ?lhs
eberlm@62429
   678
  from \<open>?lhs\<close> have "normalize a dvd normalize b * normalize d"
eberlm@62429
   679
    by (auto intro: dvdI dest: sym)
eberlm@62429
   680
  with \<open>coprime a d\<close> have "a dvd b"
eberlm@62429
   681
    by (simp add: coprime_dvd_mult_iff normalize_mult [symmetric])
eberlm@62429
   682
  from \<open>?lhs\<close> have "normalize b dvd normalize a * normalize c"
eberlm@62429
   683
    by (auto intro: dvdI dest: sym)
eberlm@62429
   684
  with \<open>coprime b c\<close> have "b dvd a"
eberlm@62429
   685
    by (simp add: coprime_dvd_mult_iff normalize_mult [symmetric])
eberlm@62429
   686
  from \<open>?lhs\<close> have "normalize c dvd normalize d * normalize b"
eberlm@62429
   687
    by (auto intro: dvdI dest: sym simp add: mult.commute)
eberlm@62429
   688
  with \<open>coprime b c\<close> have "c dvd d"
eberlm@62429
   689
    by (simp add: coprime_dvd_mult_iff gcd.commute normalize_mult [symmetric])
eberlm@62429
   690
  from \<open>?lhs\<close> have "normalize d dvd normalize c * normalize a"
eberlm@62429
   691
    by (auto intro: dvdI dest: sym simp add: mult.commute)
eberlm@62429
   692
  with \<open>coprime a d\<close> have "d dvd c"
eberlm@62429
   693
    by (simp add: coprime_dvd_mult_iff gcd.commute normalize_mult [symmetric])
eberlm@62429
   694
  from \<open>a dvd b\<close> \<open>b dvd a\<close> have "normalize a = normalize b"
eberlm@62429
   695
    by (rule associatedI)
eberlm@62429
   696
  moreover from \<open>c dvd d\<close> \<open>d dvd c\<close> have "normalize c = normalize d"
eberlm@62429
   697
    by (rule associatedI)
eberlm@62429
   698
  ultimately show ?rhs ..
eberlm@62429
   699
qed
eberlm@62429
   700
eberlm@62429
   701
lemma gcd_add1 [simp]: "gcd (m + n) n = gcd m n"
eberlm@62429
   702
  by (rule gcdI [symmetric]) (simp_all add: dvd_add_left_iff)
eberlm@62429
   703
eberlm@62429
   704
lemma gcd_add2 [simp]: "gcd m (m + n) = gcd m n"
eberlm@62429
   705
  using gcd_add1 [of n m] by (simp add: ac_simps)
eberlm@62429
   706
eberlm@62429
   707
lemma gcd_add_mult: "gcd m (k * m + n) = gcd m n"
eberlm@62429
   708
  by (rule gcdI [symmetric]) (simp_all add: dvd_add_right_iff)
eberlm@62429
   709
wenzelm@63489
   710
lemma coprimeI: "(\<And>l. l dvd a \<Longrightarrow> l dvd b \<Longrightarrow> l dvd 1) \<Longrightarrow> gcd a b = 1"
wenzelm@63489
   711
  by (rule sym, rule gcdI) simp_all
eberlm@62429
   712
eberlm@62429
   713
lemma coprime: "gcd a b = 1 \<longleftrightarrow> (\<forall>d. d dvd a \<and> d dvd b \<longleftrightarrow> is_unit d)"
eberlm@62429
   714
  by (auto intro: coprimeI gcd_greatest dvd_gcdD1 dvd_gcdD2)
eberlm@62429
   715
eberlm@62429
   716
lemma div_gcd_coprime:
eberlm@62429
   717
  assumes nz: "a \<noteq> 0 \<or> b \<noteq> 0"
eberlm@62429
   718
  shows "coprime (a div gcd a b) (b div gcd a b)"
eberlm@62429
   719
proof -
eberlm@62429
   720
  let ?g = "gcd a b"
eberlm@62429
   721
  let ?a' = "a div ?g"
eberlm@62429
   722
  let ?b' = "b div ?g"
eberlm@62429
   723
  let ?g' = "gcd ?a' ?b'"
wenzelm@63489
   724
  have dvdg: "?g dvd a" "?g dvd b"
wenzelm@63489
   725
    by simp_all
wenzelm@63489
   726
  have dvdg': "?g' dvd ?a'" "?g' dvd ?b'"
wenzelm@63489
   727
    by simp_all
eberlm@62429
   728
  from dvdg dvdg' obtain ka kb ka' kb' where
wenzelm@63489
   729
    kab: "a = ?g * ka" "b = ?g * kb" "?a' = ?g' * ka'" "?b' = ?g' * kb'"
eberlm@62429
   730
    unfolding dvd_def by blast
eberlm@62429
   731
  from this [symmetric] have "?g * ?a' = (?g * ?g') * ka'" "?g * ?b' = (?g * ?g') * kb'"
eberlm@62429
   732
    by (simp_all add: mult.assoc mult.left_commute [of "gcd a b"])
eberlm@62429
   733
  then have dvdgg':"?g * ?g' dvd a" "?g* ?g' dvd b"
wenzelm@63489
   734
    by (auto simp add: dvd_mult_div_cancel [OF dvdg(1)] dvd_mult_div_cancel [OF dvdg(2)] dvd_def)
wenzelm@63489
   735
  have "?g \<noteq> 0"
wenzelm@63489
   736
    using nz by simp
eberlm@62429
   737
  moreover from gcd_greatest [OF dvdgg'] have "?g * ?g' dvd ?g" .
wenzelm@63489
   738
  ultimately show ?thesis
wenzelm@63489
   739
    using dvd_times_left_cancel_iff [of "gcd a b" _ 1] by simp
eberlm@62429
   740
qed
eberlm@62429
   741
eberlm@62429
   742
lemma divides_mult:
eberlm@62429
   743
  assumes "a dvd c" and nr: "b dvd c" and "coprime a b"
eberlm@62429
   744
  shows "a * b dvd c"
wenzelm@63489
   745
proof -
eberlm@62429
   746
  from \<open>b dvd c\<close> obtain b' where"c = b * b'" ..
eberlm@62429
   747
  with \<open>a dvd c\<close> have "a dvd b' * b"
eberlm@62429
   748
    by (simp add: ac_simps)
eberlm@62429
   749
  with \<open>coprime a b\<close> have "a dvd b'"
eberlm@62429
   750
    by (simp add: coprime_dvd_mult_iff)
eberlm@62429
   751
  then obtain a' where "b' = a * a'" ..
eberlm@62429
   752
  with \<open>c = b * b'\<close> have "c = (a * b) * a'"
eberlm@62429
   753
    by (simp add: ac_simps)
eberlm@62429
   754
  then show ?thesis ..
eberlm@62429
   755
qed
eberlm@62429
   756
eberlm@62429
   757
lemma coprime_lmult:
wenzelm@63489
   758
  assumes dab: "gcd d (a * b) = 1"
eberlm@62429
   759
  shows "gcd d a = 1"
eberlm@62429
   760
proof (rule coprimeI)
wenzelm@63489
   761
  fix l
wenzelm@63489
   762
  assume "l dvd d" and "l dvd a"
wenzelm@63489
   763
  then have "l dvd a * b"
wenzelm@63489
   764
    by simp
wenzelm@63489
   765
  with \<open>l dvd d\<close> and dab show "l dvd 1"
wenzelm@63489
   766
    by (auto intro: gcd_greatest)
eberlm@62429
   767
qed
eberlm@62429
   768
eberlm@62429
   769
lemma coprime_rmult:
eberlm@62429
   770
  assumes dab: "gcd d (a * b) = 1"
eberlm@62429
   771
  shows "gcd d b = 1"
eberlm@62429
   772
proof (rule coprimeI)
wenzelm@63489
   773
  fix l
wenzelm@63489
   774
  assume "l dvd d" and "l dvd b"
wenzelm@63489
   775
  then have "l dvd a * b"
wenzelm@63489
   776
    by simp
wenzelm@63489
   777
  with \<open>l dvd d\<close> and dab show "l dvd 1"
wenzelm@63489
   778
    by (auto intro: gcd_greatest)
eberlm@62429
   779
qed
eberlm@62429
   780
eberlm@62429
   781
lemma coprime_mult:
wenzelm@63489
   782
  assumes "coprime d a"
wenzelm@63489
   783
    and "coprime d b"
eberlm@62429
   784
  shows "coprime d (a * b)"
eberlm@62429
   785
  apply (subst gcd.commute)
wenzelm@63489
   786
  using assms(1) apply (subst gcd_mult_cancel)
wenzelm@63489
   787
   apply (subst gcd.commute)
wenzelm@63489
   788
   apply assumption
wenzelm@63489
   789
  apply (subst gcd.commute)
wenzelm@63489
   790
  apply (rule assms(2))
eberlm@62429
   791
  done
eberlm@62429
   792
eberlm@62429
   793
lemma coprime_mul_eq: "gcd d (a * b) = 1 \<longleftrightarrow> gcd d a = 1 \<and> gcd d b = 1"
wenzelm@63489
   794
  using coprime_rmult[of d a b] coprime_lmult[of d a b] coprime_mult[of d a b]
wenzelm@63489
   795
  by blast
eberlm@62429
   796
haftmann@64591
   797
lemma coprime_mul_eq':
haftmann@64591
   798
  "coprime (a * b) d \<longleftrightarrow> coprime a d \<and> coprime b d"
haftmann@64591
   799
  using coprime_mul_eq [of d a b] by (simp add: gcd.commute)
haftmann@64591
   800
eberlm@62429
   801
lemma gcd_coprime:
wenzelm@63489
   802
  assumes c: "gcd a b \<noteq> 0"
wenzelm@63489
   803
    and a: "a = a' * gcd a b"
wenzelm@63489
   804
    and b: "b = b' * gcd a b"
eberlm@62429
   805
  shows "gcd a' b' = 1"
eberlm@62429
   806
proof -
wenzelm@63489
   807
  from c have "a \<noteq> 0 \<or> b \<noteq> 0"
wenzelm@63489
   808
    by simp
eberlm@62429
   809
  with div_gcd_coprime have "gcd (a div gcd a b) (b div gcd a b) = 1" .
wenzelm@63489
   810
  also from assms have "a div gcd a b = a'"
wenzelm@63489
   811
    using dvd_div_eq_mult local.gcd_dvd1 by blast
wenzelm@63489
   812
  also from assms have "b div gcd a b = b'"
wenzelm@63489
   813
    using dvd_div_eq_mult local.gcd_dvd1 by blast
eberlm@62429
   814
  finally show ?thesis .
eberlm@62429
   815
qed
eberlm@62429
   816
eberlm@62429
   817
lemma coprime_power:
eberlm@62429
   818
  assumes "0 < n"
eberlm@62429
   819
  shows "gcd a (b ^ n) = 1 \<longleftrightarrow> gcd a b = 1"
wenzelm@63489
   820
  using assms
wenzelm@63489
   821
proof (induct n)
wenzelm@63489
   822
  case 0
wenzelm@63489
   823
  then show ?case by simp
wenzelm@63489
   824
next
wenzelm@63489
   825
  case (Suc n)
wenzelm@63489
   826
  then show ?case
eberlm@62429
   827
    by (cases n) (simp_all add: coprime_mul_eq)
wenzelm@63489
   828
qed
eberlm@62429
   829
eberlm@62429
   830
lemma gcd_coprime_exists:
wenzelm@63489
   831
  assumes "gcd a b \<noteq> 0"
eberlm@62429
   832
  shows "\<exists>a' b'. a = a' * gcd a b \<and> b = b' * gcd a b \<and> gcd a' b' = 1"
eberlm@62429
   833
  apply (rule_tac x = "a div gcd a b" in exI)
eberlm@62429
   834
  apply (rule_tac x = "b div gcd a b" in exI)
wenzelm@63489
   835
  using assms
wenzelm@63489
   836
  apply (auto intro: div_gcd_coprime)
eberlm@62429
   837
  done
eberlm@62429
   838
wenzelm@63489
   839
lemma coprime_exp: "gcd d a = 1 \<Longrightarrow> gcd d (a^n) = 1"
wenzelm@63489
   840
  by (induct n) (simp_all add: coprime_mult)
wenzelm@63489
   841
wenzelm@63489
   842
lemma coprime_exp_left: "coprime a b \<Longrightarrow> coprime (a ^ n) b"
wenzelm@63489
   843
  by (induct n) (simp_all add: gcd_mult_cancel)
eberlm@62429
   844
eberlm@62429
   845
lemma coprime_exp2:
eberlm@62429
   846
  assumes "coprime a b"
eberlm@62429
   847
  shows "coprime (a ^ n) (b ^ m)"
eberlm@62429
   848
proof (rule coprime_exp_left)
eberlm@62429
   849
  from assms show "coprime a (b ^ m)"
eberlm@62429
   850
    by (induct m) (simp_all add: gcd_mult_cancel gcd.commute [of a])
eberlm@62429
   851
qed
eberlm@62429
   852
wenzelm@63489
   853
lemma gcd_exp: "gcd (a ^ n) (b ^ n) = gcd a b ^ n"
eberlm@62429
   854
proof (cases "a = 0 \<and> b = 0")
eberlm@62429
   855
  case True
wenzelm@63489
   856
  then show ?thesis
wenzelm@63489
   857
    by (cases n) simp_all
eberlm@62429
   858
next
eberlm@62429
   859
  case False
eberlm@62429
   860
  then have "1 = gcd ((a div gcd a b) ^ n) ((b div gcd a b) ^ n)"
eberlm@62429
   861
    using coprime_exp2[OF div_gcd_coprime[of a b], of n n, symmetric] by simp
wenzelm@63489
   862
  then have "gcd a b ^ n = gcd a b ^ n * \<dots>"
wenzelm@63489
   863
    by simp
eberlm@62429
   864
  also note gcd_mult_distrib
eberlm@62429
   865
  also have "unit_factor (gcd a b ^ n) = 1"
eberlm@62429
   866
    using False by (auto simp add: unit_factor_power unit_factor_gcd)
eberlm@62429
   867
  also have "(gcd a b)^n * (a div gcd a b)^n = a^n"
wenzelm@63489
   868
    apply (subst ac_simps)
wenzelm@63489
   869
    apply (subst div_power)
wenzelm@63489
   870
     apply simp
wenzelm@63489
   871
    apply (rule dvd_div_mult_self)
wenzelm@63489
   872
    apply (rule dvd_power_same)
wenzelm@63489
   873
    apply simp
wenzelm@63489
   874
    done
eberlm@62429
   875
  also have "(gcd a b)^n * (b div gcd a b)^n = b^n"
wenzelm@63489
   876
    apply (subst ac_simps)
wenzelm@63489
   877
    apply (subst div_power)
wenzelm@63489
   878
     apply simp
wenzelm@63489
   879
    apply (rule dvd_div_mult_self)
wenzelm@63489
   880
    apply (rule dvd_power_same)
wenzelm@63489
   881
    apply simp
wenzelm@63489
   882
    done
eberlm@62429
   883
  finally show ?thesis by simp
eberlm@62429
   884
qed
eberlm@62429
   885
wenzelm@63489
   886
lemma coprime_common_divisor: "gcd a b = 1 \<Longrightarrow> a dvd a \<Longrightarrow> a dvd b \<Longrightarrow> is_unit a"
eberlm@62429
   887
  apply (subgoal_tac "a dvd gcd a b")
wenzelm@63489
   888
   apply simp
eberlm@62429
   889
  apply (erule (1) gcd_greatest)
eberlm@62429
   890
  done
eberlm@62429
   891
wenzelm@63489
   892
lemma division_decomp:
wenzelm@63489
   893
  assumes "a dvd b * c"
eberlm@62429
   894
  shows "\<exists>b' c'. a = b' * c' \<and> b' dvd b \<and> c' dvd c"
eberlm@62429
   895
proof (cases "gcd a b = 0")
wenzelm@63489
   896
  case True
wenzelm@63489
   897
  then have "a = 0 \<and> b = 0"
wenzelm@63489
   898
    by simp
wenzelm@63489
   899
  then have "a = 0 * c \<and> 0 dvd b \<and> c dvd c"
wenzelm@63489
   900
    by simp
eberlm@62429
   901
  then show ?thesis by blast
eberlm@62429
   902
next
wenzelm@63489
   903
  case False
eberlm@62429
   904
  let ?d = "gcd a b"
wenzelm@63489
   905
  from gcd_coprime_exists [OF False]
eberlm@62429
   906
    obtain a' b' where ab': "a = a' * ?d" "b = b' * ?d" "gcd a' b' = 1"
eberlm@62429
   907
    by blast
wenzelm@63489
   908
  from ab'(1) have "a' dvd a"
wenzelm@63489
   909
    unfolding dvd_def by blast
wenzelm@63489
   910
  with assms have "a' dvd b * c"
haftmann@65555
   911
    using dvd_trans [of a' a "b * c"] by simp
wenzelm@63489
   912
  from assms ab'(1,2) have "a' * ?d dvd (b' * ?d) * c"
wenzelm@63489
   913
    by simp
wenzelm@63489
   914
  then have "?d * a' dvd ?d * (b' * c)"
wenzelm@63489
   915
    by (simp add: mult_ac)
wenzelm@63489
   916
  with \<open>?d \<noteq> 0\<close> have "a' dvd b' * c"
wenzelm@63489
   917
    by simp
wenzelm@63489
   918
  with coprime_dvd_mult[OF ab'(3)] have "a' dvd c"
wenzelm@63489
   919
    by (subst (asm) ac_simps) blast
wenzelm@63489
   920
  with ab'(1) have "a = ?d * a' \<and> ?d dvd b \<and> a' dvd c"
wenzelm@63489
   921
    by (simp add: mult_ac)
eberlm@62429
   922
  then show ?thesis by blast
eberlm@62429
   923
qed
eberlm@62429
   924
eberlm@62429
   925
lemma pow_divs_pow:
eberlm@62429
   926
  assumes ab: "a ^ n dvd b ^ n" and n: "n \<noteq> 0"
eberlm@62429
   927
  shows "a dvd b"
eberlm@62429
   928
proof (cases "gcd a b = 0")
wenzelm@63489
   929
  case True
eberlm@62429
   930
  then show ?thesis by simp
eberlm@62429
   931
next
wenzelm@63489
   932
  case False
eberlm@62429
   933
  let ?d = "gcd a b"
wenzelm@63489
   934
  from n obtain m where m: "n = Suc m"
wenzelm@63489
   935
    by (cases n) simp_all
wenzelm@63489
   936
  from False have zn: "?d ^ n \<noteq> 0"
wenzelm@63489
   937
    by (rule power_not_zero)
wenzelm@63489
   938
  from gcd_coprime_exists [OF False]
wenzelm@63489
   939
  obtain a' b' where ab': "a = a' * ?d" "b = b' * ?d" "gcd a' b' = 1"
eberlm@62429
   940
    by blast
eberlm@62429
   941
  from ab have "(a' * ?d) ^ n dvd (b' * ?d) ^ n"
eberlm@62429
   942
    by (simp add: ab'(1,2)[symmetric])
wenzelm@63489
   943
  then have "?d^n * a'^n dvd ?d^n * b'^n"
eberlm@62429
   944
    by (simp only: power_mult_distrib ac_simps)
wenzelm@63489
   945
  with zn have "a'^n dvd b'^n"
wenzelm@63489
   946
    by simp
wenzelm@63489
   947
  then have "a' dvd b'^n"
wenzelm@63489
   948
    using dvd_trans[of a' "a'^n" "b'^n"] by (simp add: m)
wenzelm@63489
   949
  then have "a' dvd b'^m * b'"
wenzelm@63489
   950
    by (simp add: m ac_simps)
eberlm@62429
   951
  with coprime_dvd_mult[OF coprime_exp[OF ab'(3), of m]]
wenzelm@63489
   952
  have "a' dvd b'" by (subst (asm) ac_simps) blast
wenzelm@63489
   953
  then have "a' * ?d dvd b' * ?d"
wenzelm@63489
   954
    by (rule mult_dvd_mono) simp
wenzelm@63489
   955
  with ab'(1,2) show ?thesis
wenzelm@63489
   956
    by simp
eberlm@62429
   957
qed
eberlm@62429
   958
wenzelm@63489
   959
lemma pow_divs_eq [simp]: "n \<noteq> 0 \<Longrightarrow> a ^ n dvd b ^ n \<longleftrightarrow> a dvd b"
eberlm@62429
   960
  by (auto intro: pow_divs_pow dvd_power_same)
eberlm@62429
   961
eberlm@62429
   962
lemma coprime_plus_one [simp]: "gcd (n + 1) n = 1"
wenzelm@63489
   963
  by (subst add_commute) simp
wenzelm@63489
   964
nipkow@64272
   965
lemma prod_coprime [rule_format]: "(\<forall>i\<in>A. gcd (f i) a = 1) \<longrightarrow> gcd (\<Prod>i\<in>A. f i) a = 1"
wenzelm@63915
   966
  by (induct A rule: infinite_finite_induct) (auto simp add: gcd_mult_cancel)
wenzelm@63489
   967
nipkow@63882
   968
lemma prod_list_coprime: "(\<And>x. x \<in> set xs \<Longrightarrow> coprime x y) \<Longrightarrow> coprime (prod_list xs) y"
wenzelm@63489
   969
  by (induct xs) (simp_all add: gcd_mult_cancel)
wenzelm@63489
   970
wenzelm@63489
   971
lemma coprime_divisors:
eberlm@62429
   972
  assumes "d dvd a" "e dvd b" "gcd a b = 1"
wenzelm@63489
   973
  shows "gcd d e = 1"
eberlm@62429
   974
proof -
eberlm@62429
   975
  from assms obtain k l where "a = d * k" "b = e * l"
eberlm@62429
   976
    unfolding dvd_def by blast
wenzelm@63489
   977
  with assms have "gcd (d * k) (e * l) = 1"
wenzelm@63489
   978
    by simp
wenzelm@63489
   979
  then have "gcd (d * k) e = 1"
wenzelm@63489
   980
    by (rule coprime_lmult)
wenzelm@63489
   981
  also have "gcd (d * k) e = gcd e (d * k)"
wenzelm@63489
   982
    by (simp add: ac_simps)
wenzelm@63489
   983
  finally have "gcd e d = 1"
wenzelm@63489
   984
    by (rule coprime_lmult)
wenzelm@63489
   985
  then show ?thesis
wenzelm@63489
   986
    by (simp add: ac_simps)
eberlm@62429
   987
qed
eberlm@62429
   988
wenzelm@63489
   989
lemma lcm_gcd_prod: "lcm a b * gcd a b = normalize (a * b)"
eberlm@62429
   990
  by (simp add: lcm_gcd)
eberlm@62429
   991
eberlm@62429
   992
declare unit_factor_lcm [simp]
eberlm@62429
   993
eberlm@62429
   994
lemma lcmI:
eberlm@62429
   995
  assumes "a dvd c" and "b dvd c" and "\<And>d. a dvd d \<Longrightarrow> b dvd d \<Longrightarrow> c dvd d"
eberlm@62429
   996
    and "normalize c = c"
eberlm@62429
   997
  shows "c = lcm a b"
eberlm@62429
   998
  by (rule associated_eqI) (auto simp: assms intro: lcm_least)
eberlm@62429
   999
wenzelm@63489
  1000
lemma gcd_dvd_lcm [simp]: "gcd a b dvd lcm a b"
eberlm@62429
  1001
  using gcd_dvd2 by (rule dvd_lcmI2)
eberlm@62429
  1002
eberlm@62429
  1003
lemmas lcm_0 = lcm_0_right
eberlm@62429
  1004
eberlm@62429
  1005
lemma lcm_unique:
wenzelm@63489
  1006
  "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
  1007
  by rule (auto intro: lcmI simp: lcm_least lcm_eq_0_iff)
eberlm@62429
  1008
wenzelm@63489
  1009
lemma lcm_coprime: "gcd a b = 1 \<Longrightarrow> lcm a b = normalize (a * b)"
eberlm@62429
  1010
  by (subst lcm_gcd) simp
eberlm@62429
  1011
wenzelm@63489
  1012
lemma lcm_proj1_if_dvd: "b dvd a \<Longrightarrow> lcm a b = normalize a"
wenzelm@63489
  1013
  apply (cases "a = 0")
wenzelm@63489
  1014
   apply simp
wenzelm@63489
  1015
  apply (rule sym)
wenzelm@63489
  1016
  apply (rule lcmI)
wenzelm@63489
  1017
     apply simp_all
wenzelm@63489
  1018
  done
wenzelm@63489
  1019
wenzelm@63489
  1020
lemma lcm_proj2_if_dvd: "a dvd b \<Longrightarrow> lcm a b = normalize b"
eberlm@62429
  1021
  using lcm_proj1_if_dvd [of a b] by (simp add: ac_simps)
eberlm@62429
  1022
wenzelm@63489
  1023
lemma lcm_proj1_iff: "lcm m n = normalize m \<longleftrightarrow> n dvd m"
eberlm@62429
  1024
proof
wenzelm@63489
  1025
  assume *: "lcm m n = normalize m"
eberlm@62429
  1026
  show "n dvd m"
eberlm@62429
  1027
  proof (cases "m = 0")
wenzelm@63489
  1028
    case True
wenzelm@63489
  1029
    then show ?thesis by simp
wenzelm@63489
  1030
  next
wenzelm@63489
  1031
    case [simp]: False
wenzelm@63489
  1032
    from * have **: "m = lcm m n * unit_factor m"
eberlm@62429
  1033
      by (simp add: unit_eq_div2)
wenzelm@63489
  1034
    show ?thesis by (subst **) simp
wenzelm@63489
  1035
  qed
eberlm@62429
  1036
next
eberlm@62429
  1037
  assume "n dvd m"
wenzelm@63489
  1038
  then show "lcm m n = normalize m"
wenzelm@63489
  1039
    by (rule lcm_proj1_if_dvd)
eberlm@62429
  1040
qed
eberlm@62429
  1041
wenzelm@63489
  1042
lemma lcm_proj2_iff: "lcm m n = normalize n \<longleftrightarrow> m dvd n"
eberlm@62429
  1043
  using lcm_proj1_iff [of n m] by (simp add: ac_simps)
eberlm@62429
  1044
haftmann@64850
  1045
lemma lcm_mult_distrib': "normalize c * lcm a b = lcm (c * a) (c * b)"
haftmann@64850
  1046
  by (rule lcmI) (auto simp: normalize_mult lcm_least mult_dvd_mono lcm_mult_left [symmetric])
haftmann@64850
  1047
haftmann@64850
  1048
lemma lcm_mult_distrib: "k * lcm a b = lcm (k * a) (k * b) * unit_factor k"
haftmann@64850
  1049
proof-
haftmann@64850
  1050
  have "normalize k * lcm a b = lcm (k * a) (k * b)"
haftmann@64850
  1051
    by (simp add: lcm_mult_distrib')
haftmann@64850
  1052
  then have "normalize k * lcm a b * unit_factor k = lcm (k * a) (k * b) * unit_factor k"
haftmann@64850
  1053
    by simp
haftmann@64850
  1054
  then have "normalize k * unit_factor k * lcm a b  = lcm (k * a) (k * b) * unit_factor k"
haftmann@64850
  1055
    by (simp only: ac_simps)
haftmann@64850
  1056
  then show ?thesis
haftmann@64850
  1057
    by simp
haftmann@64850
  1058
qed
haftmann@64850
  1059
haftmann@63924
  1060
lemma dvd_productE:
haftmann@63924
  1061
  assumes "p dvd (a * b)"
haftmann@63924
  1062
  obtains x y where "p = x * y" "x dvd a" "y dvd b"
haftmann@63924
  1063
proof (cases "a = 0")
haftmann@63924
  1064
  case True
haftmann@63924
  1065
  thus ?thesis by (intro that[of p 1]) simp_all
haftmann@63924
  1066
next
haftmann@63924
  1067
  case False
haftmann@63924
  1068
  define x y where "x = gcd a p" and "y = p div x"
haftmann@63924
  1069
  have "p = x * y" by (simp add: x_def y_def)
haftmann@63924
  1070
  moreover have "x dvd a" by (simp add: x_def)
haftmann@63924
  1071
  moreover from assms have "p dvd gcd (b * a) (b * p)"
haftmann@63924
  1072
    by (intro gcd_greatest) (simp_all add: mult.commute)
haftmann@63924
  1073
  hence "p dvd b * gcd a p" by (simp add: gcd_mult_distrib)
wenzelm@65552
  1074
  with False have "y dvd b"
haftmann@63924
  1075
    by (simp add: x_def y_def div_dvd_iff_mult assms)
haftmann@63924
  1076
  ultimately show ?thesis by (rule that)
haftmann@63924
  1077
qed
haftmann@63924
  1078
haftmann@64591
  1079
lemma coprime_crossproduct':
haftmann@64591
  1080
  fixes a b c d
haftmann@64591
  1081
  assumes "b \<noteq> 0"
haftmann@64591
  1082
  assumes unit_factors: "unit_factor b = unit_factor d"
haftmann@64591
  1083
  assumes coprime: "coprime a b" "coprime c d"
haftmann@64591
  1084
  shows "a * d = b * c \<longleftrightarrow> a = c \<and> b = d"
haftmann@64591
  1085
proof safe
haftmann@64591
  1086
  assume eq: "a * d = b * c"
haftmann@64591
  1087
  hence "normalize a * normalize d = normalize c * normalize b"
haftmann@64591
  1088
    by (simp only: normalize_mult [symmetric] mult_ac)
haftmann@64591
  1089
  with coprime have "normalize b = normalize d"
haftmann@64591
  1090
    by (subst (asm) coprime_crossproduct) simp_all
haftmann@64591
  1091
  from this and unit_factors show "b = d"
haftmann@64591
  1092
    by (rule normalize_unit_factor_eqI)
haftmann@64591
  1093
  from eq have "a * d = c * d" by (simp only: \<open>b = d\<close> mult_ac)
haftmann@64591
  1094
  with \<open>b \<noteq> 0\<close> \<open>b = d\<close> show "a = c" by simp
haftmann@64591
  1095
qed (simp_all add: mult_ac)
haftmann@64591
  1096
haftmann@60686
  1097
end
haftmann@60686
  1098
haftmann@62345
  1099
class ring_gcd = comm_ring_1 + semiring_gcd
eberlm@62429
  1100
begin
eberlm@62429
  1101
eberlm@62429
  1102
lemma coprime_minus_one: "coprime (n - 1) n"
eberlm@62429
  1103
  using coprime_plus_one[of "n - 1"] by (simp add: gcd.commute)
eberlm@62429
  1104
wenzelm@63489
  1105
lemma gcd_neg1 [simp]: "gcd (-a) b = gcd a b"
wenzelm@63489
  1106
  by (rule sym, rule gcdI) (simp_all add: gcd_greatest)
wenzelm@63489
  1107
wenzelm@63489
  1108
lemma gcd_neg2 [simp]: "gcd a (-b) = gcd a b"
wenzelm@63489
  1109
  by (rule sym, rule gcdI) (simp_all add: gcd_greatest)
wenzelm@63489
  1110
wenzelm@63489
  1111
lemma gcd_neg_numeral_1 [simp]: "gcd (- numeral n) a = gcd (numeral n) a"
eberlm@62429
  1112
  by (fact gcd_neg1)
eberlm@62429
  1113
wenzelm@63489
  1114
lemma gcd_neg_numeral_2 [simp]: "gcd a (- numeral n) = gcd a (numeral n)"
eberlm@62429
  1115
  by (fact gcd_neg2)
eberlm@62429
  1116
eberlm@62429
  1117
lemma gcd_diff1: "gcd (m - n) n = gcd m n"
wenzelm@63489
  1118
  by (subst diff_conv_add_uminus, subst gcd_neg2[symmetric], subst gcd_add1, simp)
eberlm@62429
  1119
eberlm@62429
  1120
lemma gcd_diff2: "gcd (n - m) n = gcd m n"
wenzelm@63489
  1121
  by (subst gcd_neg1[symmetric]) (simp only: minus_diff_eq gcd_diff1)
eberlm@62429
  1122
eberlm@62429
  1123
lemma lcm_neg1 [simp]: "lcm (-a) b = lcm a b"
wenzelm@63489
  1124
  by (rule sym, rule lcmI) (simp_all add: lcm_least lcm_eq_0_iff)
eberlm@62429
  1125
eberlm@62429
  1126
lemma lcm_neg2 [simp]: "lcm a (-b) = lcm a b"
wenzelm@63489
  1127
  by (rule sym, rule lcmI) (simp_all add: lcm_least lcm_eq_0_iff)
eberlm@62429
  1128
eberlm@62429
  1129
lemma lcm_neg_numeral_1 [simp]: "lcm (- numeral n) a = lcm (numeral n) a"
eberlm@62429
  1130
  by (fact lcm_neg1)
eberlm@62429
  1131
eberlm@62429
  1132
lemma lcm_neg_numeral_2 [simp]: "lcm a (- numeral n) = lcm a (numeral n)"
eberlm@62429
  1133
  by (fact lcm_neg2)
eberlm@62429
  1134
eberlm@62429
  1135
end
haftmann@62345
  1136
haftmann@60686
  1137
class semiring_Gcd = semiring_gcd + Gcd +
haftmann@60686
  1138
  assumes Gcd_dvd: "a \<in> A \<Longrightarrow> Gcd A dvd a"
haftmann@60686
  1139
    and Gcd_greatest: "(\<And>b. b \<in> A \<Longrightarrow> a dvd b) \<Longrightarrow> a dvd Gcd A"
haftmann@60686
  1140
    and normalize_Gcd [simp]: "normalize (Gcd A) = Gcd A"
haftmann@62345
  1141
  assumes dvd_Lcm: "a \<in> A \<Longrightarrow> a dvd Lcm A"
haftmann@62345
  1142
    and Lcm_least: "(\<And>b. b \<in> A \<Longrightarrow> b dvd a) \<Longrightarrow> Lcm A dvd a"
haftmann@62345
  1143
    and normalize_Lcm [simp]: "normalize (Lcm A) = Lcm A"
haftmann@60686
  1144
begin
haftmann@60686
  1145
wenzelm@63489
  1146
lemma Lcm_Gcd: "Lcm A = Gcd {b. \<forall>a\<in>A. a dvd b}"
haftmann@62345
  1147
  by (rule associated_eqI) (auto intro: Gcd_dvd dvd_Lcm Gcd_greatest Lcm_least)
haftmann@62345
  1148
wenzelm@63489
  1149
lemma Gcd_Lcm: "Gcd A = Lcm {b. \<forall>a\<in>A. b dvd a}"
haftmann@62345
  1150
  by (rule associated_eqI) (auto intro: Gcd_dvd dvd_Lcm Gcd_greatest Lcm_least)
haftmann@62345
  1151
wenzelm@63489
  1152
lemma Gcd_empty [simp]: "Gcd {} = 0"
haftmann@60686
  1153
  by (rule dvd_0_left, rule Gcd_greatest) simp
haftmann@60686
  1154
wenzelm@63489
  1155
lemma Lcm_empty [simp]: "Lcm {} = 1"
haftmann@62345
  1156
  by (auto intro: associated_eqI Lcm_least)
haftmann@62345
  1157
wenzelm@63489
  1158
lemma Gcd_insert [simp]: "Gcd (insert a A) = gcd a (Gcd A)"
haftmann@62345
  1159
proof -
haftmann@62345
  1160
  have "Gcd (insert a A) dvd gcd a (Gcd A)"
haftmann@62345
  1161
    by (auto intro: Gcd_dvd Gcd_greatest)
haftmann@62345
  1162
  moreover have "gcd a (Gcd A) dvd Gcd (insert a A)"
haftmann@62345
  1163
  proof (rule Gcd_greatest)
haftmann@62345
  1164
    fix b
haftmann@62345
  1165
    assume "b \<in> insert a A"
haftmann@62345
  1166
    then show "gcd a (Gcd A) dvd b"
haftmann@62345
  1167
    proof
wenzelm@63489
  1168
      assume "b = a"
wenzelm@63489
  1169
      then show ?thesis
wenzelm@63489
  1170
        by simp
haftmann@62345
  1171
    next
haftmann@62345
  1172
      assume "b \<in> A"
wenzelm@63489
  1173
      then have "Gcd A dvd b"
wenzelm@63489
  1174
        by (rule Gcd_dvd)
wenzelm@63489
  1175
      moreover have "gcd a (Gcd A) dvd Gcd A"
wenzelm@63489
  1176
        by simp
wenzelm@63489
  1177
      ultimately show ?thesis
wenzelm@63489
  1178
        by (blast intro: dvd_trans)
haftmann@62345
  1179
    qed
haftmann@62345
  1180
  qed
haftmann@62345
  1181
  ultimately show ?thesis
haftmann@62345
  1182
    by (auto intro: associated_eqI)
haftmann@62345
  1183
qed
haftmann@62345
  1184
wenzelm@63489
  1185
lemma Lcm_insert [simp]: "Lcm (insert a A) = lcm a (Lcm A)"
haftmann@62345
  1186
proof (rule sym)
haftmann@62345
  1187
  have "lcm a (Lcm A) dvd Lcm (insert a A)"
haftmann@62345
  1188
    by (auto intro: dvd_Lcm Lcm_least)
haftmann@62345
  1189
  moreover have "Lcm (insert a A) dvd lcm a (Lcm A)"
haftmann@62345
  1190
  proof (rule Lcm_least)
haftmann@62345
  1191
    fix b
haftmann@62345
  1192
    assume "b \<in> insert a A"
haftmann@62345
  1193
    then show "b dvd lcm a (Lcm A)"
haftmann@62345
  1194
    proof
wenzelm@63489
  1195
      assume "b = a"
wenzelm@63489
  1196
      then show ?thesis by simp
haftmann@62345
  1197
    next
haftmann@62345
  1198
      assume "b \<in> A"
wenzelm@63489
  1199
      then have "b dvd Lcm A"
wenzelm@63489
  1200
        by (rule dvd_Lcm)
wenzelm@63489
  1201
      moreover have "Lcm A dvd lcm a (Lcm A)"
wenzelm@63489
  1202
        by simp
wenzelm@63489
  1203
      ultimately show ?thesis
wenzelm@63489
  1204
        by (blast intro: dvd_trans)
haftmann@62345
  1205
    qed
haftmann@62345
  1206
  qed
haftmann@62345
  1207
  ultimately show "lcm a (Lcm A) = Lcm (insert a A)"
haftmann@62345
  1208
    by (rule associated_eqI) (simp_all add: lcm_eq_0_iff)
haftmann@62345
  1209
qed
haftmann@62345
  1210
eberlm@62429
  1211
lemma LcmI:
wenzelm@63489
  1212
  assumes "\<And>a. a \<in> A \<Longrightarrow> a dvd b"
wenzelm@63489
  1213
    and "\<And>c. (\<And>a. a \<in> A \<Longrightarrow> a dvd c) \<Longrightarrow> b dvd c"
wenzelm@63489
  1214
    and "normalize b = b"
wenzelm@63489
  1215
  shows "b = Lcm A"
eberlm@62429
  1216
  by (rule associated_eqI) (auto simp: assms dvd_Lcm intro: Lcm_least)
eberlm@62429
  1217
wenzelm@63489
  1218
lemma Lcm_subset: "A \<subseteq> B \<Longrightarrow> Lcm A dvd Lcm B"
eberlm@62429
  1219
  by (blast intro: Lcm_least dvd_Lcm)
eberlm@62429
  1220
wenzelm@63489
  1221
lemma Lcm_Un: "Lcm (A \<union> B) = lcm (Lcm A) (Lcm B)"
eberlm@62429
  1222
  apply (rule lcmI)
wenzelm@63489
  1223
     apply (blast intro: Lcm_subset)
wenzelm@63489
  1224
    apply (blast intro: Lcm_subset)
wenzelm@63489
  1225
   apply (intro Lcm_least ballI, elim UnE)
wenzelm@63489
  1226
    apply (rule dvd_trans, erule dvd_Lcm, assumption)
wenzelm@63489
  1227
   apply (rule dvd_trans, erule dvd_Lcm, assumption)
eberlm@62429
  1228
  apply simp
eberlm@62429
  1229
  done
wenzelm@63489
  1230
wenzelm@63489
  1231
lemma Gcd_0_iff [simp]: "Gcd A = 0 \<longleftrightarrow> A \<subseteq> {0}"
wenzelm@63489
  1232
  (is "?P \<longleftrightarrow> ?Q")
haftmann@60686
  1233
proof
haftmann@60686
  1234
  assume ?P
haftmann@60686
  1235
  show ?Q
haftmann@60686
  1236
  proof
haftmann@60686
  1237
    fix a
haftmann@60686
  1238
    assume "a \<in> A"
wenzelm@63489
  1239
    then have "Gcd A dvd a"
wenzelm@63489
  1240
      by (rule Gcd_dvd)
wenzelm@63489
  1241
    with \<open>?P\<close> have "a = 0"
wenzelm@63489
  1242
      by simp
wenzelm@63489
  1243
    then show "a \<in> {0}"
wenzelm@63489
  1244
      by simp
haftmann@60686
  1245
  qed
haftmann@60686
  1246
next
haftmann@60686
  1247
  assume ?Q
haftmann@60686
  1248
  have "0 dvd Gcd A"
haftmann@60686
  1249
  proof (rule Gcd_greatest)
haftmann@60686
  1250
    fix a
haftmann@60686
  1251
    assume "a \<in> A"
wenzelm@63489
  1252
    with \<open>?Q\<close> have "a = 0"
wenzelm@63489
  1253
      by auto
wenzelm@63489
  1254
    then show "0 dvd a"
wenzelm@63489
  1255
      by simp
haftmann@60686
  1256
  qed
wenzelm@63489
  1257
  then show ?P
wenzelm@63489
  1258
    by simp
haftmann@60686
  1259
qed
haftmann@60686
  1260
wenzelm@63489
  1261
lemma Lcm_1_iff [simp]: "Lcm A = 1 \<longleftrightarrow> (\<forall>a\<in>A. is_unit a)"
wenzelm@63489
  1262
  (is "?P \<longleftrightarrow> ?Q")
haftmann@60686
  1263
proof
haftmann@60686
  1264
  assume ?P
haftmann@60686
  1265
  show ?Q
haftmann@60686
  1266
  proof
haftmann@60686
  1267
    fix a
haftmann@60686
  1268
    assume "a \<in> A"
haftmann@60686
  1269
    then have "a dvd Lcm A"
haftmann@60686
  1270
      by (rule dvd_Lcm)
haftmann@60686
  1271
    with \<open>?P\<close> show "is_unit a"
haftmann@60686
  1272
      by simp
haftmann@60686
  1273
  qed
haftmann@60686
  1274
next
haftmann@60686
  1275
  assume ?Q
haftmann@60686
  1276
  then have "is_unit (Lcm A)"
haftmann@60686
  1277
    by (blast intro: Lcm_least)
haftmann@60686
  1278
  then have "normalize (Lcm A) = 1"
haftmann@60686
  1279
    by (rule is_unit_normalize)
haftmann@60686
  1280
  then show ?P
haftmann@60686
  1281
    by simp
haftmann@60686
  1282
qed
haftmann@60686
  1283
wenzelm@63489
  1284
lemma unit_factor_Lcm: "unit_factor (Lcm A) = (if Lcm A = 0 then 0 else 1)"
haftmann@62345
  1285
proof (cases "Lcm A = 0")
wenzelm@63489
  1286
  case True
wenzelm@63489
  1287
  then show ?thesis
wenzelm@63489
  1288
    by simp
haftmann@62345
  1289
next
haftmann@62345
  1290
  case False
haftmann@62345
  1291
  with unit_factor_normalize have "unit_factor (normalize (Lcm A)) = 1"
haftmann@62345
  1292
    by blast
haftmann@62345
  1293
  with False show ?thesis
haftmann@62345
  1294
    by simp
haftmann@62345
  1295
qed
haftmann@62345
  1296
eberlm@62429
  1297
lemma unit_factor_Gcd: "unit_factor (Gcd A) = (if Gcd A = 0 then 0 else 1)"
wenzelm@63489
  1298
  by (simp add: Gcd_Lcm unit_factor_Lcm)
eberlm@62429
  1299
eberlm@62429
  1300
lemma GcdI:
wenzelm@63489
  1301
  assumes "\<And>a. a \<in> A \<Longrightarrow> b dvd a"
wenzelm@63489
  1302
    and "\<And>c. (\<And>a. a \<in> A \<Longrightarrow> c dvd a) \<Longrightarrow> c dvd b"
eberlm@62429
  1303
    and "normalize b = b"
eberlm@62429
  1304
  shows "b = Gcd A"
eberlm@62429
  1305
  by (rule associated_eqI) (auto simp: assms Gcd_dvd intro: Gcd_greatest)
eberlm@62429
  1306
haftmann@62345
  1307
lemma Gcd_eq_1_I:
haftmann@62345
  1308
  assumes "is_unit a" and "a \<in> A"
haftmann@62345
  1309
  shows "Gcd A = 1"
haftmann@62345
  1310
proof -
haftmann@62345
  1311
  from assms have "is_unit (Gcd A)"
haftmann@62345
  1312
    by (blast intro: Gcd_dvd dvd_unit_imp_unit)
haftmann@62345
  1313
  then have "normalize (Gcd A) = 1"
haftmann@62345
  1314
    by (rule is_unit_normalize)
haftmann@62345
  1315
  then show ?thesis
haftmann@62345
  1316
    by simp
haftmann@62345
  1317
qed
haftmann@62345
  1318
haftmann@60686
  1319
lemma Lcm_eq_0_I:
haftmann@60686
  1320
  assumes "0 \<in> A"
haftmann@60686
  1321
  shows "Lcm A = 0"
haftmann@60686
  1322
proof -
haftmann@60686
  1323
  from assms have "0 dvd Lcm A"
haftmann@60686
  1324
    by (rule dvd_Lcm)
haftmann@60686
  1325
  then show ?thesis
haftmann@60686
  1326
    by simp
haftmann@60686
  1327
qed
haftmann@60686
  1328
wenzelm@63489
  1329
lemma Gcd_UNIV [simp]: "Gcd UNIV = 1"
haftmann@62345
  1330
  using dvd_refl by (rule Gcd_eq_1_I) simp
haftmann@62345
  1331
wenzelm@63489
  1332
lemma Lcm_UNIV [simp]: "Lcm UNIV = 0"
haftmann@61929
  1333
  by (rule Lcm_eq_0_I) simp
haftmann@60686
  1334
haftmann@61929
  1335
lemma Lcm_0_iff:
haftmann@61929
  1336
  assumes "finite A"
haftmann@61929
  1337
  shows "Lcm A = 0 \<longleftrightarrow> 0 \<in> A"
haftmann@61929
  1338
proof (cases "A = {}")
wenzelm@63489
  1339
  case True
wenzelm@63489
  1340
  then show ?thesis by simp
haftmann@61929
  1341
next
wenzelm@63489
  1342
  case False
wenzelm@63489
  1343
  with assms show ?thesis
wenzelm@63489
  1344
    by (induct A rule: finite_ne_induct) (auto simp add: lcm_eq_0_iff)
haftmann@60686
  1345
qed
haftmann@61929
  1346
wenzelm@63489
  1347
lemma Gcd_image_normalize [simp]: "Gcd (normalize ` A) = Gcd A"
haftmann@62345
  1348
proof -
haftmann@62345
  1349
  have "Gcd (normalize ` A) dvd a" if "a \<in> A" for a
haftmann@62345
  1350
  proof -
wenzelm@63489
  1351
    from that obtain B where "A = insert a B"
wenzelm@63489
  1352
      by blast
haftmann@62350
  1353
    moreover have "gcd (normalize a) (Gcd (normalize ` B)) dvd normalize a"
haftmann@62345
  1354
      by (rule gcd_dvd1)
haftmann@62345
  1355
    ultimately show "Gcd (normalize ` A) dvd a"
haftmann@62345
  1356
      by simp
haftmann@62345
  1357
  qed
haftmann@62345
  1358
  then have "Gcd (normalize ` A) dvd Gcd A" and "Gcd A dvd Gcd (normalize ` A)"
haftmann@62345
  1359
    by (auto intro!: Gcd_greatest intro: Gcd_dvd)
haftmann@62345
  1360
  then show ?thesis
haftmann@62345
  1361
    by (auto intro: associated_eqI)
haftmann@62345
  1362
qed
haftmann@62345
  1363
haftmann@62346
  1364
lemma Gcd_eqI:
haftmann@62346
  1365
  assumes "normalize a = a"
haftmann@62346
  1366
  assumes "\<And>b. b \<in> A \<Longrightarrow> a dvd b"
haftmann@62346
  1367
    and "\<And>c. (\<And>b. b \<in> A \<Longrightarrow> c dvd b) \<Longrightarrow> c dvd a"
haftmann@62346
  1368
  shows "Gcd A = a"
haftmann@62346
  1369
  using assms by (blast intro: associated_eqI Gcd_greatest Gcd_dvd normalize_Gcd)
haftmann@62346
  1370
wenzelm@63489
  1371
lemma dvd_GcdD: "x dvd Gcd A \<Longrightarrow> y \<in> A \<Longrightarrow> x dvd y"
wenzelm@63489
  1372
  using Gcd_dvd dvd_trans by blast
wenzelm@63489
  1373
wenzelm@63489
  1374
lemma dvd_Gcd_iff: "x dvd Gcd A \<longleftrightarrow> (\<forall>y\<in>A. x dvd y)"
eberlm@63359
  1375
  by (blast dest: dvd_GcdD intro: Gcd_greatest)
eberlm@63359
  1376
eberlm@63359
  1377
lemma Gcd_mult: "Gcd (op * c ` A) = normalize c * Gcd A"
eberlm@63359
  1378
proof (cases "c = 0")
wenzelm@63489
  1379
  case True
wenzelm@63489
  1380
  then show ?thesis by auto
wenzelm@63489
  1381
next
eberlm@63359
  1382
  case [simp]: False
eberlm@63359
  1383
  have "Gcd (op * c ` A) div c dvd Gcd A"
eberlm@63359
  1384
    by (intro Gcd_greatest, subst div_dvd_iff_mult)
eberlm@63359
  1385
       (auto intro!: Gcd_greatest Gcd_dvd simp: mult.commute[of _ c])
wenzelm@63489
  1386
  then have "Gcd (op * c ` A) dvd c * Gcd A"
eberlm@63359
  1387
    by (subst (asm) div_dvd_iff_mult) (auto intro: Gcd_greatest simp: mult_ac)
eberlm@63359
  1388
  also have "c * Gcd A = (normalize c * Gcd A) * unit_factor c"
eberlm@63359
  1389
    by (subst unit_factor_mult_normalize [symmetric]) (simp only: mult_ac)
eberlm@63359
  1390
  also have "Gcd (op * c ` A) dvd \<dots> \<longleftrightarrow> Gcd (op * c ` A) dvd normalize c * Gcd A"
eberlm@63359
  1391
    by (simp add: dvd_mult_unit_iff)
eberlm@63359
  1392
  finally have "Gcd (op * c ` A) dvd normalize c * Gcd A" .
eberlm@63359
  1393
  moreover have "normalize c * Gcd A dvd Gcd (op * c ` A)"
eberlm@63359
  1394
    by (intro Gcd_greatest) (auto intro: mult_dvd_mono Gcd_dvd)
eberlm@63359
  1395
  ultimately have "normalize (Gcd (op * c ` A)) = normalize (normalize c * Gcd A)"
eberlm@63359
  1396
    by (rule associatedI)
wenzelm@63489
  1397
  then show ?thesis
wenzelm@63489
  1398
    by (simp add: normalize_mult)
wenzelm@63489
  1399
qed
eberlm@63359
  1400
haftmann@62346
  1401
lemma Lcm_eqI:
haftmann@62346
  1402
  assumes "normalize a = a"
wenzelm@63489
  1403
    and "\<And>b. b \<in> A \<Longrightarrow> b dvd a"
haftmann@62346
  1404
    and "\<And>c. (\<And>b. b \<in> A \<Longrightarrow> b dvd c) \<Longrightarrow> a dvd c"
haftmann@62346
  1405
  shows "Lcm A = a"
haftmann@62346
  1406
  using assms by (blast intro: associated_eqI Lcm_least dvd_Lcm normalize_Lcm)
haftmann@62346
  1407
wenzelm@63489
  1408
lemma Lcm_dvdD: "Lcm A dvd x \<Longrightarrow> y \<in> A \<Longrightarrow> y dvd x"
wenzelm@63489
  1409
  using dvd_Lcm dvd_trans by blast
wenzelm@63489
  1410
wenzelm@63489
  1411
lemma Lcm_dvd_iff: "Lcm A dvd x \<longleftrightarrow> (\<forall>y\<in>A. y dvd x)"
eberlm@63359
  1412
  by (blast dest: Lcm_dvdD intro: Lcm_least)
eberlm@63359
  1413
wenzelm@63489
  1414
lemma Lcm_mult:
eberlm@63359
  1415
  assumes "A \<noteq> {}"
wenzelm@63489
  1416
  shows "Lcm (op * c ` A) = normalize c * Lcm A"
eberlm@63359
  1417
proof (cases "c = 0")
eberlm@63359
  1418
  case True
wenzelm@63489
  1419
  with assms have "op * c ` A = {0}"
wenzelm@63489
  1420
    by auto
wenzelm@63489
  1421
  with True show ?thesis by auto
eberlm@63359
  1422
next
eberlm@63359
  1423
  case [simp]: False
wenzelm@63489
  1424
  from assms obtain x where x: "x \<in> A"
wenzelm@63489
  1425
    by blast
wenzelm@63489
  1426
  have "c dvd c * x"
wenzelm@63489
  1427
    by simp
wenzelm@63489
  1428
  also from x have "c * x dvd Lcm (op * c ` A)"
wenzelm@63489
  1429
    by (intro dvd_Lcm) auto
eberlm@63359
  1430
  finally have dvd: "c dvd Lcm (op * c ` A)" .
eberlm@63359
  1431
eberlm@63359
  1432
  have "Lcm A dvd Lcm (op * c ` A) div c"
eberlm@63359
  1433
    by (intro Lcm_least dvd_mult_imp_div)
wenzelm@63489
  1434
      (auto intro!: Lcm_least dvd_Lcm simp: mult.commute[of _ c])
wenzelm@63489
  1435
  then have "c * Lcm A dvd Lcm (op * c ` A)"
eberlm@63359
  1436
    by (subst (asm) dvd_div_iff_mult) (auto intro!: Lcm_least simp: mult_ac dvd)
eberlm@63359
  1437
  also have "c * Lcm A = (normalize c * Lcm A) * unit_factor c"
eberlm@63359
  1438
    by (subst unit_factor_mult_normalize [symmetric]) (simp only: mult_ac)
eberlm@63359
  1439
  also have "\<dots> dvd Lcm (op * c ` A) \<longleftrightarrow> normalize c * Lcm A dvd Lcm (op * c ` A)"
eberlm@63359
  1440
    by (simp add: mult_unit_dvd_iff)
eberlm@63359
  1441
  finally have "normalize c * Lcm A dvd Lcm (op * c ` A)" .
eberlm@63359
  1442
  moreover have "Lcm (op * c ` A) dvd normalize c * Lcm A"
eberlm@63359
  1443
    by (intro Lcm_least) (auto intro: mult_dvd_mono dvd_Lcm)
eberlm@63359
  1444
  ultimately have "normalize (normalize c * Lcm A) = normalize (Lcm (op * c ` A))"
eberlm@63359
  1445
    by (rule associatedI)
wenzelm@63489
  1446
  then show ?thesis
wenzelm@63489
  1447
    by (simp add: normalize_mult)
eberlm@62429
  1448
qed
eberlm@62429
  1449
wenzelm@63489
  1450
lemma Lcm_no_units: "Lcm A = Lcm (A - {a. is_unit a})"
wenzelm@63489
  1451
proof -
wenzelm@63489
  1452
  have "(A - {a. is_unit a}) \<union> {a\<in>A. is_unit a} = A"
wenzelm@63489
  1453
    by blast
wenzelm@63489
  1454
  then have "Lcm A = lcm (Lcm (A - {a. is_unit a})) (Lcm {a\<in>A. is_unit a})"
wenzelm@63489
  1455
    by (simp add: Lcm_Un [symmetric])
wenzelm@63489
  1456
  also have "Lcm {a\<in>A. is_unit a} = 1"
wenzelm@63489
  1457
    by simp
wenzelm@63489
  1458
  finally show ?thesis
wenzelm@63489
  1459
    by simp
wenzelm@63489
  1460
qed
wenzelm@63489
  1461
wenzelm@63489
  1462
lemma Lcm_0_iff': "Lcm A = 0 \<longleftrightarrow> (\<nexists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l))"
eberlm@62429
  1463
  by (metis Lcm_least dvd_0_left dvd_Lcm)
eberlm@62429
  1464
wenzelm@63489
  1465
lemma Lcm_no_multiple: "(\<forall>m. m \<noteq> 0 \<longrightarrow> (\<exists>a\<in>A. \<not> a dvd m)) \<Longrightarrow> Lcm A = 0"
eberlm@62429
  1466
  by (auto simp: Lcm_0_iff')
eberlm@62429
  1467
wenzelm@63489
  1468
lemma Lcm_singleton [simp]: "Lcm {a} = normalize a"
eberlm@62429
  1469
  by simp
eberlm@62429
  1470
wenzelm@63489
  1471
lemma Lcm_2 [simp]: "Lcm {a, b} = lcm a b"
eberlm@62429
  1472
  by simp
eberlm@62429
  1473
eberlm@62429
  1474
lemma Lcm_coprime:
wenzelm@63489
  1475
  assumes "finite A"
wenzelm@63489
  1476
    and "A \<noteq> {}"
wenzelm@63489
  1477
    and "\<And>a b. a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<noteq> b \<Longrightarrow> gcd a b = 1"
eberlm@62429
  1478
  shows "Lcm A = normalize (\<Prod>A)"
wenzelm@63489
  1479
  using assms
wenzelm@63489
  1480
proof (induct rule: finite_ne_induct)
wenzelm@63489
  1481
  case singleton
wenzelm@63489
  1482
  then show ?case by simp
wenzelm@63489
  1483
next
eberlm@62429
  1484
  case (insert a A)
wenzelm@63489
  1485
  have "Lcm (insert a A) = lcm a (Lcm A)"
wenzelm@63489
  1486
    by simp
wenzelm@63489
  1487
  also from insert have "Lcm A = normalize (\<Prod>A)"
wenzelm@63489
  1488
    by blast
wenzelm@63489
  1489
  also have "lcm a \<dots> = lcm a (\<Prod>A)"
wenzelm@63489
  1490
    by (cases "\<Prod>A = 0") (simp_all add: lcm_div_unit2)
wenzelm@63489
  1491
  also from insert have "gcd a (\<Prod>A) = 1"
nipkow@64272
  1492
    by (subst gcd.commute, intro prod_coprime) auto
eberlm@62429
  1493
  with insert have "lcm a (\<Prod>A) = normalize (\<Prod>(insert a A))"
eberlm@62429
  1494
    by (simp add: lcm_coprime)
eberlm@62429
  1495
  finally show ?case .
wenzelm@63489
  1496
qed
wenzelm@63489
  1497
eberlm@62429
  1498
lemma Lcm_coprime':
wenzelm@63489
  1499
  "card A \<noteq> 0 \<Longrightarrow>
wenzelm@63489
  1500
    (\<And>a b. a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<noteq> b \<Longrightarrow> gcd a b = 1) \<Longrightarrow>
wenzelm@63489
  1501
    Lcm A = normalize (\<Prod>A)"
eberlm@62429
  1502
  by (rule Lcm_coprime) (simp_all add: card_eq_0_iff)
eberlm@62429
  1503
wenzelm@63489
  1504
lemma Gcd_1: "1 \<in> A \<Longrightarrow> Gcd A = 1"
eberlm@62429
  1505
  by (auto intro!: Gcd_eq_1_I)
eberlm@62429
  1506
eberlm@62429
  1507
lemma Gcd_singleton [simp]: "Gcd {a} = normalize a"
eberlm@62429
  1508
  by simp
eberlm@62429
  1509
wenzelm@63489
  1510
lemma Gcd_2 [simp]: "Gcd {a, b} = gcd a b"
eberlm@62429
  1511
  by simp
eberlm@62429
  1512
eberlm@62429
  1513
wenzelm@63489
  1514
definition pairwise_coprime
wenzelm@63489
  1515
  where "pairwise_coprime A = (\<forall>x y. x \<in> A \<and> y \<in> A \<and> x \<noteq> y \<longrightarrow> coprime x y)"
eberlm@62429
  1516
eberlm@62429
  1517
lemma pairwise_coprimeI [intro?]:
eberlm@62429
  1518
  "(\<And>x y. x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> x \<noteq> y \<Longrightarrow> coprime x y) \<Longrightarrow> pairwise_coprime A"
eberlm@62429
  1519
  by (simp add: pairwise_coprime_def)
eberlm@62429
  1520
eberlm@62429
  1521
lemma pairwise_coprimeD:
eberlm@62429
  1522
  "pairwise_coprime A \<Longrightarrow> x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> x \<noteq> y \<Longrightarrow> coprime x y"
eberlm@62429
  1523
  by (simp add: pairwise_coprime_def)
eberlm@62429
  1524
eberlm@62429
  1525
lemma pairwise_coprime_subset: "pairwise_coprime A \<Longrightarrow> B \<subseteq> A \<Longrightarrow> pairwise_coprime B"
eberlm@62429
  1526
  by (force simp: pairwise_coprime_def)
eberlm@62429
  1527
haftmann@62350
  1528
end
haftmann@62345
  1529
wenzelm@65552
  1530
haftmann@64850
  1531
subsection \<open>An aside: GCD and LCM on finite sets for incomplete gcd rings\<close>
haftmann@64850
  1532
haftmann@64850
  1533
context semiring_gcd
haftmann@64850
  1534
begin
haftmann@64850
  1535
haftmann@64850
  1536
sublocale Gcd_fin: bounded_quasi_semilattice_set gcd 0 1 normalize
haftmann@64850
  1537
defines
haftmann@64850
  1538
  Gcd_fin ("Gcd\<^sub>f\<^sub>i\<^sub>n _" [900] 900) = "Gcd_fin.F :: 'a set \<Rightarrow> 'a" ..
haftmann@64850
  1539
haftmann@64850
  1540
abbreviation gcd_list :: "'a list \<Rightarrow> 'a"
haftmann@64850
  1541
  where "gcd_list xs \<equiv> Gcd\<^sub>f\<^sub>i\<^sub>n (set xs)"
haftmann@64850
  1542
haftmann@64850
  1543
sublocale Lcm_fin: bounded_quasi_semilattice_set lcm 1 0 normalize
haftmann@64850
  1544
defines
haftmann@64850
  1545
  Lcm_fin ("Lcm\<^sub>f\<^sub>i\<^sub>n _" [900] 900) = Lcm_fin.F ..
haftmann@64850
  1546
haftmann@64850
  1547
abbreviation lcm_list :: "'a list \<Rightarrow> 'a"
haftmann@64850
  1548
  where "lcm_list xs \<equiv> Lcm\<^sub>f\<^sub>i\<^sub>n (set xs)"
wenzelm@65552
  1549
haftmann@64850
  1550
lemma Gcd_fin_dvd:
haftmann@64850
  1551
  "a \<in> A \<Longrightarrow> Gcd\<^sub>f\<^sub>i\<^sub>n A dvd a"
wenzelm@65552
  1552
  by (induct A rule: infinite_finite_induct)
haftmann@64850
  1553
    (auto intro: dvd_trans)
haftmann@64850
  1554
haftmann@64850
  1555
lemma dvd_Lcm_fin:
haftmann@64850
  1556
  "a \<in> A \<Longrightarrow> a dvd Lcm\<^sub>f\<^sub>i\<^sub>n A"
wenzelm@65552
  1557
  by (induct A rule: infinite_finite_induct)
haftmann@64850
  1558
    (auto intro: dvd_trans)
haftmann@64850
  1559
haftmann@64850
  1560
lemma Gcd_fin_greatest:
haftmann@64850
  1561
  "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
  1562
  using that by (induct A) simp_all
haftmann@64850
  1563
haftmann@64850
  1564
lemma Lcm_fin_least:
haftmann@64850
  1565
  "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
  1566
  using that by (induct A) simp_all
haftmann@64850
  1567
haftmann@64850
  1568
lemma gcd_list_greatest:
haftmann@64850
  1569
  "a dvd gcd_list bs" if "\<And>b. b \<in> set bs \<Longrightarrow> a dvd b"
haftmann@64850
  1570
  by (rule Gcd_fin_greatest) (simp_all add: that)
haftmann@64850
  1571
haftmann@64850
  1572
lemma lcm_list_least:
haftmann@64850
  1573
  "lcm_list bs dvd a" if "\<And>b. b \<in> set bs \<Longrightarrow> b dvd a"
haftmann@64850
  1574
  by (rule Lcm_fin_least) (simp_all add: that)
haftmann@64850
  1575
haftmann@64850
  1576
lemma dvd_Gcd_fin_iff:
haftmann@64850
  1577
  "b dvd Gcd\<^sub>f\<^sub>i\<^sub>n A \<longleftrightarrow> (\<forall>a\<in>A. b dvd a)" if "finite A"
haftmann@64850
  1578
  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
  1579
haftmann@64850
  1580
lemma dvd_gcd_list_iff:
haftmann@64850
  1581
  "b dvd gcd_list xs \<longleftrightarrow> (\<forall>a\<in>set xs. b dvd a)"
haftmann@64850
  1582
  by (simp add: dvd_Gcd_fin_iff)
wenzelm@65552
  1583
haftmann@64850
  1584
lemma Lcm_fin_dvd_iff:
haftmann@64850
  1585
  "Lcm\<^sub>f\<^sub>i\<^sub>n A dvd b  \<longleftrightarrow> (\<forall>a\<in>A. a dvd b)" if "finite A"
haftmann@64850
  1586
  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
  1587
haftmann@64850
  1588
lemma lcm_list_dvd_iff:
haftmann@64850
  1589
  "lcm_list xs dvd b  \<longleftrightarrow> (\<forall>a\<in>set xs. a dvd b)"
haftmann@64850
  1590
  by (simp add: Lcm_fin_dvd_iff)
haftmann@64850
  1591
haftmann@64850
  1592
lemma Gcd_fin_mult:
haftmann@64850
  1593
  "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
  1594
using that proof induct
haftmann@64850
  1595
  case empty
haftmann@64850
  1596
  then show ?case
haftmann@64850
  1597
    by simp
haftmann@64850
  1598
next
haftmann@64850
  1599
  case (insert a A)
haftmann@64850
  1600
  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
  1601
    by simp
haftmann@64850
  1602
  also have "\<dots> = gcd (b * a) (normalize b * Gcd\<^sub>f\<^sub>i\<^sub>n A)"
haftmann@64850
  1603
    by (simp add: normalize_mult)
haftmann@64850
  1604
  finally show ?case
haftmann@64850
  1605
    using insert by (simp add: gcd_mult_distrib')
haftmann@64850
  1606
qed
haftmann@64850
  1607
haftmann@64850
  1608
lemma Lcm_fin_mult:
haftmann@64850
  1609
  "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
  1610
proof (cases "b = 0")
haftmann@64850
  1611
  case True
haftmann@64850
  1612
  moreover from that have "times 0 ` A = {0}"
haftmann@64850
  1613
    by auto
haftmann@64850
  1614
  ultimately show ?thesis
haftmann@64850
  1615
    by simp
haftmann@64850
  1616
next
haftmann@64850
  1617
  case False
haftmann@64850
  1618
  then have "inj (times b)"
haftmann@64850
  1619
    by (rule inj_times)
haftmann@64850
  1620
  show ?thesis proof (cases "finite A")
haftmann@64850
  1621
    case False
haftmann@64850
  1622
    moreover from \<open>inj (times b)\<close>
haftmann@64850
  1623
    have "inj_on (times b) A"
haftmann@64850
  1624
      by (rule inj_on_subset) simp
haftmann@64850
  1625
    ultimately have "infinite (times b ` A)"
haftmann@64850
  1626
      by (simp add: finite_image_iff)
haftmann@64850
  1627
    with False show ?thesis
haftmann@64850
  1628
      by simp
haftmann@64850
  1629
  next
haftmann@64850
  1630
    case True
haftmann@64850
  1631
    then show ?thesis using that proof (induct A rule: finite_ne_induct)
haftmann@64850
  1632
      case (singleton a)
haftmann@64850
  1633
      then show ?case
haftmann@64850
  1634
        by (simp add: normalize_mult)
haftmann@64850
  1635
    next
haftmann@64850
  1636
      case (insert a A)
haftmann@64850
  1637
      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
  1638
        by simp
haftmann@64850
  1639
      also have "\<dots> = lcm (b * a) (normalize b * Lcm\<^sub>f\<^sub>i\<^sub>n A)"
haftmann@64850
  1640
        by (simp add: normalize_mult)
haftmann@64850
  1641
      finally show ?case
haftmann@64850
  1642
        using insert by (simp add: lcm_mult_distrib')
haftmann@64850
  1643
    qed
haftmann@64850
  1644
  qed
haftmann@64850
  1645
qed
haftmann@64850
  1646
haftmann@65811
  1647
lemma unit_factor_Gcd_fin:
haftmann@65811
  1648
  "unit_factor (Gcd\<^sub>f\<^sub>i\<^sub>n A) = of_bool (Gcd\<^sub>f\<^sub>i\<^sub>n A \<noteq> 0)"
haftmann@65811
  1649
  by (rule normalize_idem_imp_unit_factor_eq) simp
haftmann@65811
  1650
haftmann@65811
  1651
lemma unit_factor_Lcm_fin:
haftmann@65811
  1652
  "unit_factor (Lcm\<^sub>f\<^sub>i\<^sub>n A) = of_bool (Lcm\<^sub>f\<^sub>i\<^sub>n A \<noteq> 0)"
haftmann@65811
  1653
  by (rule normalize_idem_imp_unit_factor_eq) simp
haftmann@65811
  1654
haftmann@65811
  1655
lemma is_unit_Gcd_fin_iff [simp]:
haftmann@65811
  1656
  "is_unit (Gcd\<^sub>f\<^sub>i\<^sub>n A) \<longleftrightarrow> Gcd\<^sub>f\<^sub>i\<^sub>n A = 1"
haftmann@65811
  1657
  by (rule normalize_idem_imp_is_unit_iff) simp
haftmann@65811
  1658
haftmann@65811
  1659
lemma is_unit_Lcm_fin_iff [simp]:
haftmann@65811
  1660
  "is_unit (Lcm\<^sub>f\<^sub>i\<^sub>n A) \<longleftrightarrow> Lcm\<^sub>f\<^sub>i\<^sub>n A = 1"
haftmann@65811
  1661
  by (rule normalize_idem_imp_is_unit_iff) simp
haftmann@65811
  1662
 
haftmann@65811
  1663
lemma Gcd_fin_0_iff:
haftmann@65811
  1664
  "Gcd\<^sub>f\<^sub>i\<^sub>n A = 0 \<longleftrightarrow> A \<subseteq> {0} \<and> finite A"
haftmann@65811
  1665
  by (induct A rule: infinite_finite_induct) simp_all
haftmann@65811
  1666
haftmann@65811
  1667
lemma Lcm_fin_0_iff:
haftmann@65811
  1668
  "Lcm\<^sub>f\<^sub>i\<^sub>n A = 0 \<longleftrightarrow> 0 \<in> A" if "finite A"
haftmann@65811
  1669
  using that by (induct A) (auto simp add: lcm_eq_0_iff)
haftmann@65811
  1670
haftmann@65811
  1671
lemma Lcm_fin_1_iff:
haftmann@65811
  1672
  "Lcm\<^sub>f\<^sub>i\<^sub>n A = 1 \<longleftrightarrow> (\<forall>a\<in>A. is_unit a) \<and> finite A"
haftmann@65811
  1673
  by (induct A rule: infinite_finite_induct) simp_all
haftmann@65811
  1674
haftmann@64850
  1675
end
haftmann@64850
  1676
haftmann@64850
  1677
context semiring_Gcd
haftmann@64850
  1678
begin
haftmann@64850
  1679
haftmann@64850
  1680
lemma Gcd_fin_eq_Gcd [simp]:
haftmann@64850
  1681
  "Gcd\<^sub>f\<^sub>i\<^sub>n A = Gcd A" if "finite A" for A :: "'a set"
haftmann@64850
  1682
  using that by induct simp_all
haftmann@64850
  1683
haftmann@64850
  1684
lemma Gcd_set_eq_fold [code_unfold]:
haftmann@64850
  1685
  "Gcd (set xs) = fold gcd xs 0"
haftmann@64850
  1686
  by (simp add: Gcd_fin.set_eq_fold [symmetric])
haftmann@64850
  1687
haftmann@64850
  1688
lemma Lcm_fin_eq_Lcm [simp]:
haftmann@64850
  1689
  "Lcm\<^sub>f\<^sub>i\<^sub>n A = Lcm A" if "finite A" for A :: "'a set"
haftmann@64850
  1690
  using that by induct simp_all
haftmann@64850
  1691
haftmann@64850
  1692
lemma Lcm_set_eq_fold [code_unfold]:
haftmann@64850
  1693
  "Lcm (set xs) = fold lcm xs 1"
haftmann@64850
  1694
  by (simp add: Lcm_fin.set_eq_fold [symmetric])
haftmann@64850
  1695
haftmann@64850
  1696
end
wenzelm@63489
  1697
haftmann@62345
  1698
subsection \<open>GCD and LCM on @{typ nat} and @{typ int}\<close>
haftmann@59008
  1699
huffman@31706
  1700
instantiation nat :: gcd
huffman@31706
  1701
begin
wenzelm@21256
  1702
haftmann@62345
  1703
fun gcd_nat  :: "nat \<Rightarrow> nat \<Rightarrow> nat"
wenzelm@63489
  1704
  where "gcd_nat x y = (if y = 0 then x else gcd y (x mod y))"
huffman@31706
  1705
haftmann@62345
  1706
definition lcm_nat :: "nat \<Rightarrow> nat \<Rightarrow> nat"
wenzelm@63489
  1707
  where "lcm_nat x y = x * y div (gcd x y)"
wenzelm@63489
  1708
wenzelm@63489
  1709
instance ..
huffman@31706
  1710
huffman@31706
  1711
end
huffman@31706
  1712
huffman@31706
  1713
instantiation int :: gcd
huffman@31706
  1714
begin
wenzelm@21256
  1715
haftmann@62345
  1716
definition gcd_int  :: "int \<Rightarrow> int \<Rightarrow> int"
haftmann@62345
  1717
  where "gcd_int x y = int (gcd (nat \<bar>x\<bar>) (nat \<bar>y\<bar>))"
haftmann@23687
  1718
haftmann@62345
  1719
definition lcm_int :: "int \<Rightarrow> int \<Rightarrow> int"
haftmann@62345
  1720
  where "lcm_int x y = int (lcm (nat \<bar>x\<bar>) (nat \<bar>y\<bar>))"
haftmann@23687
  1721
wenzelm@61944
  1722
instance ..
huffman@31706
  1723
huffman@31706
  1724
end
haftmann@23687
  1725
haftmann@62345
  1726
text \<open>Transfer setup\<close>
huffman@31706
  1727
huffman@31706
  1728
lemma transfer_nat_int_gcd:
wenzelm@63489
  1729
  "x \<ge> 0 \<Longrightarrow> y \<ge> 0 \<Longrightarrow> gcd (nat x) (nat y) = nat (gcd x y)"
wenzelm@63489
  1730
  "x \<ge> 0 \<Longrightarrow> y \<ge> 0 \<Longrightarrow> lcm (nat x) (nat y) = nat (lcm x y)"
wenzelm@63489
  1731
  for x y :: int
wenzelm@63489
  1732
  unfolding gcd_int_def lcm_int_def by auto
haftmann@23687
  1733
huffman@31706
  1734
lemma transfer_nat_int_gcd_closures:
wenzelm@63489
  1735
  "x \<ge> 0 \<Longrightarrow> y \<ge> 0 \<Longrightarrow> gcd x y \<ge> 0"
wenzelm@63489
  1736
  "x \<ge> 0 \<Longrightarrow> y \<ge> 0 \<Longrightarrow> lcm x y \<ge> 0"
wenzelm@63489
  1737
  for x y :: int
huffman@31706
  1738
  by (auto simp add: gcd_int_def lcm_int_def)
huffman@31706
  1739
wenzelm@63489
  1740
declare transfer_morphism_nat_int
wenzelm@63489
  1741
  [transfer add return: transfer_nat_int_gcd transfer_nat_int_gcd_closures]
huffman@31706
  1742
huffman@31706
  1743
lemma transfer_int_nat_gcd:
huffman@31706
  1744
  "gcd (int x) (int y) = int (gcd x y)"
huffman@31706
  1745
  "lcm (int x) (int y) = int (lcm x y)"
wenzelm@63489
  1746
  by (auto simp: gcd_int_def lcm_int_def)
huffman@31706
  1747
huffman@31706
  1748
lemma transfer_int_nat_gcd_closures:
huffman@31706
  1749
  "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> gcd x y >= 0"
huffman@31706
  1750
  "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> lcm x y >= 0"
wenzelm@63489
  1751
  by (auto simp: gcd_int_def lcm_int_def)
wenzelm@63489
  1752
wenzelm@63489
  1753
declare transfer_morphism_int_nat
wenzelm@63489
  1754
  [transfer add return: transfer_int_nat_gcd transfer_int_nat_gcd_closures]
huffman@31706
  1755
nipkow@31952
  1756
lemma gcd_nat_induct:
haftmann@23687
  1757
  fixes m n :: nat
haftmann@23687
  1758
  assumes "\<And>m. P m 0"
haftmann@23687
  1759
    and "\<And>m n. 0 < n \<Longrightarrow> P n (m mod n) \<Longrightarrow> P m n"
haftmann@23687
  1760
  shows "P m n"
huffman@31706
  1761
  apply (rule gcd_nat.induct)
huffman@31706
  1762
  apply (case_tac "y = 0")
wenzelm@63489
  1763
  using assms
wenzelm@63489
  1764
   apply simp_all
wenzelm@63489
  1765
  done
wenzelm@63489
  1766
wenzelm@63489
  1767
wenzelm@63489
  1768
text \<open>Specific to \<open>int\<close>.\<close>
wenzelm@63489
  1769
wenzelm@63489
  1770
lemma gcd_eq_int_iff: "gcd k l = int n \<longleftrightarrow> gcd (nat \<bar>k\<bar>) (nat \<bar>l\<bar>) = n"
haftmann@62346
  1771
  by (simp add: gcd_int_def)
haftmann@62346
  1772
wenzelm@63489
  1773
lemma lcm_eq_int_iff: "lcm k l = int n \<longleftrightarrow> lcm (nat \<bar>k\<bar>) (nat \<bar>l\<bar>) = n"
haftmann@62346
  1774
  by (simp add: lcm_int_def)
haftmann@62346
  1775
wenzelm@63489
  1776
lemma gcd_neg1_int [simp]: "gcd (- x) y = gcd x y"
wenzelm@63489
  1777
  for x y :: int
huffman@31706
  1778
  by (simp add: gcd_int_def)
huffman@31706
  1779
wenzelm@63489
  1780
lemma gcd_neg2_int [simp]: "gcd x (- y) = gcd x y"
wenzelm@63489
  1781
  for x y :: int
wenzelm@63489
  1782
  by (simp add: gcd_int_def)
wenzelm@63489
  1783
wenzelm@63489
  1784
lemma abs_gcd_int [simp]: "\<bar>gcd x y\<bar> = gcd x y"
wenzelm@63489
  1785
  for x y :: int
huffman@31706
  1786
  by (simp add: gcd_int_def)
huffman@31706
  1787
wenzelm@63489
  1788
lemma gcd_abs_int: "gcd x y = gcd \<bar>x\<bar> \<bar>y\<bar>"
wenzelm@63489
  1789
  for x y :: int
wenzelm@63489
  1790
  by (simp add: gcd_int_def)
wenzelm@63489
  1791
wenzelm@63489
  1792
lemma gcd_abs1_int [simp]: "gcd \<bar>x\<bar> y = gcd x y"
wenzelm@63489
  1793
  for x y :: int
wenzelm@63489
  1794
  by (metis abs_idempotent gcd_abs_int)
wenzelm@63489
  1795
wenzelm@63489
  1796
lemma gcd_abs2_int [simp]: "gcd x \<bar>y\<bar> = gcd x y"
wenzelm@63489
  1797
  for x y :: int
wenzelm@63489
  1798
  by (metis abs_idempotent gcd_abs_int)
huffman@31706
  1799
nipkow@31952
  1800
lemma gcd_cases_int:
wenzelm@63489
  1801
  fixes x y :: int
wenzelm@63489
  1802
  assumes "x \<ge> 0 \<Longrightarrow> y \<ge> 0 \<Longrightarrow> P (gcd x y)"
wenzelm@63489
  1803
    and "x \<ge> 0 \<Longrightarrow> y \<le> 0 \<Longrightarrow> P (gcd x (- y))"
wenzelm@63489
  1804
    and "x \<le> 0 \<Longrightarrow> y \<ge> 0 \<Longrightarrow> P (gcd (- x) y)"
wenzelm@63489
  1805
    and "x \<le> 0 \<Longrightarrow> y \<le> 0 \<Longrightarrow> P (gcd (- x) (- y))"
huffman@31706
  1806
  shows "P (gcd x y)"
wenzelm@63489
  1807
  using assms by auto arith
wenzelm@21256
  1808
nipkow@31952
  1809
lemma gcd_ge_0_int [simp]: "gcd (x::int) y >= 0"
wenzelm@63489
  1810
  for x y :: int
huffman@31706
  1811
  by (simp add: gcd_int_def)
huffman@31706
  1812
wenzelm@63489
  1813
lemma lcm_neg1_int: "lcm (- x) y = lcm x y"
wenzelm@63489
  1814
  for x y :: int
huffman@31706
  1815
  by (simp add: lcm_int_def)
huffman@31706
  1816
wenzelm@63489
  1817
lemma lcm_neg2_int: "lcm x (- y) = lcm x y"
wenzelm@63489
  1818
  for x y :: int
huffman@31706
  1819
  by (simp add: lcm_int_def)
huffman@31706
  1820
wenzelm@63489
  1821
lemma lcm_abs_int: "lcm x y = lcm \<bar>x\<bar> \<bar>y\<bar>"
wenzelm@63489
  1822
  for x y :: int
huffman@31706
  1823
  by (simp add: lcm_int_def)
wenzelm@21256
  1824
wenzelm@63489
  1825
lemma abs_lcm_int [simp]: "\<bar>lcm i j\<bar> = lcm i j"
wenzelm@63489
  1826
  for i j :: int
wenzelm@61944
  1827
  by (simp add:lcm_int_def)
nipkow@31814
  1828
wenzelm@63489
  1829
lemma lcm_abs1_int [simp]: "lcm \<bar>x\<bar> y = lcm x y"
wenzelm@63489
  1830
  for x y :: int
wenzelm@61944
  1831
  by (metis abs_idempotent lcm_int_def)
nipkow@31814
  1832
wenzelm@63489
  1833
lemma lcm_abs2_int [simp]: "lcm x \<bar>y\<bar> = lcm x y"
wenzelm@63489
  1834
  for x y :: int
wenzelm@61944
  1835
  by (metis abs_idempotent lcm_int_def)
nipkow@31814
  1836
nipkow@31952
  1837
lemma lcm_cases_int:
wenzelm@63489
  1838
  fixes x y :: int
wenzelm@63489
  1839
  assumes "x \<ge> 0 \<Longrightarrow> y \<ge> 0 \<Longrightarrow> P (lcm x y)"
wenzelm@63489
  1840
    and "x \<ge> 0 \<Longrightarrow> y \<le> 0 \<Longrightarrow> P (lcm x (- y))"
wenzelm@63489
  1841
    and "x \<le> 0 \<Longrightarrow> y \<ge> 0 \<Longrightarrow> P (lcm (- x) y)"
wenzelm@63489
  1842
    and "x \<le> 0 \<Longrightarrow> y \<le> 0 \<Longrightarrow> P (lcm (- x) (- y))"
huffman@31706
  1843
  shows "P (lcm x y)"
wenzelm@41550
  1844
  using assms by (auto simp add: lcm_neg1_int lcm_neg2_int) arith
huffman@31706
  1845
wenzelm@63489
  1846
lemma lcm_ge_0_int [simp]: "lcm x y \<ge> 0"
wenzelm@63489
  1847
  for x y :: int
huffman@31706
  1848
  by (simp add: lcm_int_def)
huffman@31706
  1849
wenzelm@63489
  1850
lemma gcd_0_nat: "gcd x 0 = x"
wenzelm@63489
  1851
  for x :: nat
haftmann@23687
  1852
  by simp
haftmann@23687
  1853
wenzelm@63489
  1854
lemma gcd_0_int [simp]: "gcd x 0 = \<bar>x\<bar>"
wenzelm@63489
  1855
  for x :: int
wenzelm@63489
  1856
  by (auto simp: gcd_int_def)
wenzelm@63489
  1857
wenzelm@63489
  1858
lemma gcd_0_left_nat: "gcd 0 x = x"
wenzelm@63489
  1859
  for x :: nat
haftmann@23687
  1860
  by simp
haftmann@23687
  1861
wenzelm@63489
  1862
lemma gcd_0_left_int [simp]: "gcd 0 x = \<bar>x\<bar>"
wenzelm@63489
  1863
  for x :: int
wenzelm@63489
  1864
  by (auto simp:gcd_int_def)
wenzelm@63489
  1865
wenzelm@63489
  1866
lemma gcd_red_nat: "gcd x y = gcd y (x mod y)"
wenzelm@63489
  1867
  for x y :: nat
wenzelm@63489
  1868
  by (cases "y = 0") auto
wenzelm@63489
  1869
wenzelm@63489
  1870
wenzelm@63489
  1871
text \<open>Weaker, but useful for the simplifier.\<close>
wenzelm@63489
  1872
wenzelm@63489
  1873
lemma gcd_non_0_nat: "y \<noteq> 0 \<Longrightarrow> gcd x y = gcd y (x mod y)"
wenzelm@63489
  1874
  for x y :: nat
wenzelm@21263
  1875
  by simp
wenzelm@21256
  1876
wenzelm@63489
  1877
lemma gcd_1_nat [simp]: "gcd m 1 = 1"
wenzelm@63489
  1878
  for m :: nat
haftmann@60690
  1879
  by simp
huffman@31706
  1880
wenzelm@63489
  1881
lemma gcd_Suc_0 [simp]: "gcd m (Suc 0) = Suc 0"
wenzelm@63489
  1882
  for m :: nat
wenzelm@63489
  1883
  by simp
wenzelm@63489
  1884
wenzelm@63489
  1885
lemma gcd_1_int [simp]: "gcd m 1 = 1"
wenzelm@63489
  1886
  for m :: int
huffman@31706
  1887
  by (simp add: gcd_int_def)
huffman@30082
  1888
wenzelm@63489
  1889
lemma gcd_idem_nat: "gcd x x = x"
wenzelm@63489
  1890
  for x :: nat
wenzelm@63489
  1891
  by simp
wenzelm@63489
  1892
wenzelm@63489
  1893
lemma gcd_idem_int: "gcd x x = \<bar>x\<bar>"
wenzelm@63489
  1894
  for x :: int
wenzelm@63489
  1895
  by (auto simp add: gcd_int_def)
huffman@31706
  1896
huffman@31706
  1897
declare gcd_nat.simps [simp del]
wenzelm@21256
  1898
wenzelm@60758
  1899
text \<open>
wenzelm@63489
  1900
  \<^medskip> @{term "gcd m n"} divides \<open>m\<close> and \<open>n\<close>.
wenzelm@63489
  1901
  The conjunctions don't seem provable separately.
wenzelm@60758
  1902
\<close>
wenzelm@21256
  1903
haftmann@59008
  1904
instance nat :: semiring_gcd
haftmann@59008
  1905
proof
haftmann@59008
  1906
  fix m n :: nat
haftmann@59008
  1907
  show "gcd m n dvd m" and "gcd m n dvd n"
haftmann@59008
  1908
  proof (induct m n rule: gcd_nat_induct)
haftmann@59008
  1909
    fix m n :: nat
wenzelm@63489
  1910
    assume "gcd n (m mod n) dvd m mod n"
wenzelm@63489
  1911
      and "gcd n (m mod n) dvd n"
haftmann@59008
  1912
    then have "gcd n (m mod n) dvd m"
haftmann@59008
  1913
      by (rule dvd_mod_imp_dvd)
haftmann@59008
  1914
    moreover assume "0 < n"
haftmann@59008
  1915
    ultimately show "gcd m n dvd m"
haftmann@59008
  1916
      by (simp add: gcd_non_0_nat)
haftmann@59008
  1917
  qed (simp_all add: gcd_0_nat gcd_non_0_nat)
haftmann@59008
  1918
next
haftmann@59008
  1919
  fix m n k :: nat
haftmann@59008
  1920
  assume "k dvd m" and "k dvd n"
haftmann@59008
  1921
  then show "k dvd gcd m n"
haftmann@59008
  1922
    by (induct m n rule: gcd_nat_induct) (simp_all add: gcd_non_0_nat dvd_mod gcd_0_nat)
haftmann@60686
  1923
qed (simp_all add: lcm_nat_def)
lp15@59667
  1924
haftmann@59008
  1925
instance int :: ring_gcd
haftmann@60686
  1926
  by standard
wenzelm@63489
  1927
    (simp_all add: dvd_int_unfold_dvd_nat gcd_int_def lcm_int_def
wenzelm@63489
  1928
      zdiv_int nat_abs_mult_distrib [symmetric] lcm_gcd gcd_greatest)
wenzelm@63489
  1929
wenzelm@63489
  1930
lemma gcd_le1_nat [simp]: "a \<noteq> 0 \<Longrightarrow> gcd a b \<le> a"
wenzelm@63489
  1931
  for a b :: nat
wenzelm@63489
  1932
  by (rule dvd_imp_le) auto
wenzelm@63489
  1933
wenzelm@63489
  1934
lemma gcd_le2_nat [simp]: "b \<noteq> 0 \<Longrightarrow> gcd a b \<le> b"
wenzelm@63489
  1935
  for a b :: nat
wenzelm@63489
  1936
  by (rule dvd_imp_le) auto
wenzelm@63489
  1937
wenzelm@63489
  1938
lemma gcd_le1_int [simp]: "a > 0 \<Longrightarrow> gcd a b \<le> a"
wenzelm@63489
  1939
  for a b :: int
wenzelm@63489
  1940
  by (rule zdvd_imp_le) auto
wenzelm@63489
  1941
wenzelm@63489
  1942
lemma gcd_le2_int [simp]: "b > 0 \<Longrightarrow> gcd a b \<le> b"
wenzelm@63489
  1943
  for a b :: int
wenzelm@63489
  1944
  by (rule zdvd_imp_le) auto
wenzelm@63489
  1945
wenzelm@63489
  1946
lemma gcd_pos_nat [simp]: "gcd m n > 0 \<longleftrightarrow> m \<noteq> 0 \<or> n \<noteq> 0"
wenzelm@63489
  1947
  for m n :: nat
wenzelm@63489
  1948
  using gcd_eq_0_iff [of m n] by arith
wenzelm@63489
  1949
wenzelm@63489
  1950
lemma gcd_pos_int [simp]: "gcd m n > 0 \<longleftrightarrow> m \<noteq> 0 \<or> n \<noteq> 0"
wenzelm@63489
  1951
  for m n :: int
wenzelm@63489
  1952
  using gcd_eq_0_iff [of m n] gcd_ge_0_int [of m n] by arith
wenzelm@63489
  1953
wenzelm@63489
  1954
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
  1955
  for d a :: nat
huffman@31706
  1956
  apply auto
nipkow@33657
  1957
  apply (rule dvd_antisym)
wenzelm@63489
  1958
   apply (erule (1) gcd_greatest)
huffman@31706
  1959
  apply auto
wenzelm@63489
  1960
  done
wenzelm@63489
  1961
wenzelm@63489
  1962
lemma gcd_unique_int:
wenzelm@63489
  1963
  "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
  1964
  for d a :: int
wenzelm@63489
  1965
  apply (cases "d = 0")
wenzelm@63489
  1966
   apply simp
wenzelm@63489
  1967
  apply (rule iffI)
wenzelm@63489
  1968
   apply (rule zdvd_antisym_nonneg)
wenzelm@63489
  1969
      apply (auto intro: gcd_greatest)
wenzelm@63489
  1970
  done
huffman@30082
  1971
haftmann@61913
  1972
interpretation gcd_nat:
haftmann@62344
  1973
  semilattice_neutr_order gcd "0::nat" Rings.dvd "\<lambda>m n. m dvd n \<and> m \<noteq> n"
haftmann@62344
  1974
  by standard (auto simp add: gcd_unique_nat [symmetric] intro: dvd_antisym dvd_trans)
nipkow@31798
  1975
wenzelm@63489
  1976
lemma gcd_proj1_if_dvd_int [simp]: "x dvd y \<Longrightarrow> gcd x y = \<bar>x\<bar>"
wenzelm@63489
  1977
  for x y :: int
haftmann@54867
  1978
  by (metis abs_dvd_iff gcd_0_left_int gcd_abs_int gcd_unique_int)
nipkow@31798
  1979
wenzelm@63489
  1980
lemma gcd_proj2_if_dvd_int [simp]: "y dvd x \<Longrightarrow> gcd x y = \<bar>y\<bar>"
wenzelm@63489
  1981
  for x y :: int
haftmann@62344
  1982
  by (metis gcd_proj1_if_dvd_int gcd.commute)
nipkow@31798
  1983
wenzelm@63489
  1984
wenzelm@63489
  1985
text \<open>\<^medskip> Multiplication laws.\<close>
wenzelm@63489
  1986
wenzelm@63489
  1987
lemma gcd_mult_distrib_nat: "k * gcd m n = gcd (k * m) (k * n)"
wenzelm@63489
  1988
  for k m n :: nat
wenzelm@63489
  1989
  \<comment> \<open>@{cite \<open>page 27\<close> davenport92}\<close>
nipkow@31952
  1990
  apply (induct m n rule: gcd_nat_induct)
wenzelm@63489
  1991
   apply simp
wenzelm@63489
  1992
  apply (cases "k = 0")
wenzelm@63489
  1993
   apply (simp_all add: gcd_non_0_nat)
wenzelm@63489
  1994
  done
wenzelm@63489
  1995
wenzelm@63489
  1996
lemma gcd_mult_distrib_int: "\<bar>k\<bar> * gcd m n = gcd (k * m) (k * n)"
wenzelm@63489
  1997
  for k m n :: int
haftmann@66796
  1998
  by (simp add: gcd_int_def abs_mult nat_mult_distrib gcd_mult_distrib_nat [symmetric])
wenzelm@21256
  1999
haftmann@62344
  2000
lemma coprime_crossproduct_nat:
haftmann@62344
  2001
  fixes a b c d :: nat
haftmann@62344
  2002
  assumes "coprime a d" and "coprime b c"
haftmann@62344
  2003
  shows "a * c = b * d \<longleftrightarrow> a = b \<and> c = d"
haftmann@62344
  2004
  using assms coprime_crossproduct [of a d b c] by simp
haftmann@62344
  2005
haftmann@35368
  2006
lemma coprime_crossproduct_int:
haftmann@35368
  2007
  fixes a b c d :: int
haftmann@35368
  2008
  assumes "coprime a d" and "coprime b c"
haftmann@35368
  2009
  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@62344
  2010
  using assms coprime_crossproduct [of a d b c] by simp
haftmann@35368
  2011
wenzelm@63489
  2012
wenzelm@63489
  2013
text \<open>\medskip Addition laws.\<close>
wenzelm@63489
  2014
wenzelm@63489
  2015
(* TODO: add the other variations? *)
wenzelm@63489
  2016
wenzelm@63489
  2017
lemma gcd_diff1_nat: "m \<ge> n \<Longrightarrow> gcd (m - n) n = gcd m n"
wenzelm@63489
  2018
  for m n :: nat
eberlm@62429
  2019
  by (subst gcd_add1 [symmetric]) auto
huffman@31706
  2020
wenzelm@63489
  2021
lemma gcd_diff2_nat: "n \<ge> m \<Longrightarrow> gcd (n - m) n = gcd m n"
wenzelm@63489
  2022
  for m n :: nat
haftmann@62344
  2023
  apply (subst gcd.commute)
nipkow@31952
  2024
  apply (subst gcd_diff1_nat [symmetric])
wenzelm@63489
  2025
   apply auto
haftmann@62344
  2026
  apply (subst gcd.commute)
nipkow@31952
  2027
  apply (subst gcd_diff1_nat)
wenzelm@63489
  2028
   apply assumption
haftmann@62344
  2029
  apply (rule gcd.commute)
haftmann@62344
  2030
  done
huffman@31706
  2031
wenzelm@63489
  2032
lemma gcd_non_0_int: "y > 0 \<Longrightarrow> gcd x y = gcd y (x mod y)"
wenzelm@63489
  2033
  for x y :: int
huffman@31706
  2034
  apply (frule_tac b = y and a = x in pos_mod_sign)
huffman@31706
  2035
  apply (simp del: pos_mod_sign add: gcd_int_def abs_if nat_mod_distrib)
wenzelm@63489
  2036
  apply (auto simp add: gcd_non_0_nat nat_mod_distrib [symmetric] zmod_zminus1_eq_if)
huffman@31706
  2037
  apply (frule_tac a = x in pos_mod_bound)
haftmann@62344
  2038
  apply (subst (1 2) gcd.commute)
wenzelm@63489
  2039
  apply (simp del: pos_mod_bound add: nat_diff_distrib gcd_diff2_nat nat_le_eq_zle)
haftmann@62344
  2040
  done
wenzelm@21256
  2041
wenzelm@63489
  2042
lemma gcd_red_int: "gcd x y = gcd y (x mod y)"
wenzelm@63489
  2043
  for x y :: int
wenzelm@63489
  2044
  apply (cases "y = 0")
wenzelm@63489
  2045
   apply force
wenzelm@63489
  2046
  apply (cases "y > 0")
wenzelm@63489
  2047
   apply (subst gcd_non_0_int, auto)
wenzelm@63489
  2048
  apply (insert gcd_non_0_int [of "- y" "- x"])
huffman@35216
  2049
  apply auto
wenzelm@63489
  2050
  done
wenzelm@63489
  2051
wenzelm@63489
  2052
(* TODO: differences, and all variations of addition rules
huffman@31706
  2053
    as simplification rules for nat and int *)
huffman@31706
  2054
wenzelm@63489
  2055
(* TODO: add the three variations of these, and for ints? *)
wenzelm@63489
  2056
wenzelm@63489
  2057
lemma finite_divisors_nat [simp]: (* FIXME move *)
haftmann@62353
  2058
  fixes m :: nat
wenzelm@63489
  2059
  assumes "m > 0"
haftmann@62353
  2060
  shows "finite {d. d dvd m}"
nipkow@31734
  2061
proof-
haftmann@62353
  2062
  from assms have "{d. d dvd m} \<subseteq> {d. d \<le> m}"
haftmann@62353
  2063
    by (auto dest: dvd_imp_le)
haftmann@62353
  2064
  then show ?thesis
haftmann@62353
  2065
    using finite_Collect_le_nat by (rule finite_subset)
nipkow@31734
  2066
qed
nipkow@31734
  2067
haftmann@62353
  2068
lemma finite_divisors_int [simp]:
haftmann@62353
  2069
  fixes i :: int
haftmann@62353
  2070
  assumes "i \<noteq> 0"
haftmann@62353
  2071
  shows "finite {d. d dvd i}"
haftmann@62353
  2072
proof -
haftmann@62353
  2073
  have "{d. \<bar>d\<bar> \<le> \<bar>i\<bar>} = {- \<bar>i\<bar>..\<bar>i\<bar>}"
haftmann@62353
  2074
    by (auto simp: abs_if)
wenzelm@63489
  2075
  then have "finite {d. \<bar>d\<bar> \<le> \<bar>i\<bar>}"
haftmann@62353
  2076
    by simp
wenzelm@63489
  2077
  from finite_subset [OF _ this] show ?thesis
wenzelm@63489
  2078
    using assms by (simp add: dvd_imp_le_int subset_iff)
nipkow@31734
  2079
qed
nipkow@31734
  2080
wenzelm@63489
  2081
lemma Max_divisors_self_nat [simp]: "n \<noteq> 0 \<Longrightarrow> Max {d::nat. d dvd n} = n"
wenzelm@63489
  2082
  apply (rule antisym)
wenzelm@63489
  2083
   apply (fastforce intro: Max_le_iff[THEN iffD2] simp: dvd_imp_le)
wenzelm@63489
  2084
  apply simp
wenzelm@63489
  2085
  done
wenzelm@63489
  2086
wenzelm@63489
  2087
lemma Max_divisors_self_int [simp]: "n \<noteq> 0 \<Longrightarrow> Max {d::int. d dvd n} = \<bar>n\<bar>"
wenzelm@63489
  2088
  apply (rule antisym)
wenzelm@63489
  2089
   apply (rule Max_le_iff [THEN iffD2])
wenzelm@63489
  2090
     apply (auto intro: abs_le_D1 dvd_imp_le_int)
wenzelm@63489
  2091
  done
wenzelm@63489
  2092
wenzelm@63489
  2093
lemma gcd_is_Max_divisors_nat: "m > 0 \<Longrightarrow> n > 0 \<Longrightarrow> gcd m n = Max {d. d dvd m \<and> d dvd n}"
wenzelm@63489
  2094
  for m n :: nat
wenzelm@63489
  2095
  apply (rule Max_eqI[THEN sym])
wenzelm@63489
  2096
    apply (metis finite_Collect_conjI finite_divisors_nat)
wenzelm@63489
  2097
   apply simp
wenzelm@63489
  2098
   apply (metis Suc_diff_1 Suc_neq_Zero dvd_imp_le gcd_greatest_iff gcd_pos_nat)
wenzelm@63489
  2099
  apply simp
wenzelm@63489
  2100
  done
wenzelm@63489
  2101
wenzelm@63489
  2102
lemma gcd_is_Max_divisors_int: "m \<noteq> 0 \<Longrightarrow> n \<noteq> 0 \<Longrightarrow> gcd m n = Max {d. d dvd m \<and> d dvd n}"
wenzelm@63489
  2103
  for m n :: int
wenzelm@63489
  2104
  apply (rule Max_eqI[THEN sym])
wenzelm@63489
  2105
    apply (metis finite_Collect_conjI finite_divisors_int)
wenzelm@63489
  2106
   apply simp
wenzelm@63489
  2107
   apply (metis gcd_greatest_iff gcd_pos_int zdvd_imp_le)
wenzelm@63489
  2108
  apply simp
wenzelm@63489
  2109
  done
wenzelm@63489
  2110
wenzelm@63489
  2111
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
  2112
  for k l :: int
haftmann@34030
  2113
  by (simp add: gcd_int_def nat_mod_distrib gcd_non_0_nat)
haftmann@34030
  2114
chaieb@22027
  2115
wenzelm@60758
  2116
subsection \<open>Coprimality\<close>
huffman@31706
  2117
wenzelm@63489
  2118
lemma coprime_nat: "coprime a b \<longleftrightarrow> (\<forall>d. d dvd a \<and> d dvd b \<longleftrightarrow> d = 1)"
wenzelm@63489
  2119
  for a b :: nat
haftmann@62345
  2120
  using coprime [of a b] by simp
huffman@31706
  2121
wenzelm@63489
  2122
lemma coprime_Suc_0_nat: "coprime a b \<longleftrightarrow> (\<forall>d. d dvd a \<and> d dvd b \<longleftrightarrow> d = Suc 0)"
wenzelm@63489
  2123
  for a b :: nat
haftmann@60690
  2124
  using coprime_nat by simp
huffman@31706
  2125
wenzelm@63489
  2126
lemma coprime_int: "coprime a b \<longleftrightarrow> (\<forall>d. d \<ge> 0 \<and> d dvd a \<and> d dvd b \<longleftrightarrow> d = 1)"
wenzelm@63489
  2127
  for a b :: int
nipkow@31952
  2128
  using gcd_unique_int [of 1 a b]
huffman@31706
  2129
  apply clarsimp
huffman@31706
  2130
  apply (erule subst)
huffman@31706
  2131
  apply (rule iffI)
wenzelm@63489
  2132
   apply force
lp15@61649
  2133
  using abs_dvd_iff abs_ge_zero apply blast
wenzelm@59807
  2134
  done
huffman@31706
  2135
wenzelm@63489
  2136
lemma pow_divides_eq_nat [simp]: "n > 0 \<Longrightarrow> a^n dvd b^n \<longleftrightarrow> a dvd b"
wenzelm@63489
  2137
  for a b n :: nat
eberlm@62429
  2138
  using pow_divs_eq[of n] by simp
huffman@31706
  2139
nipkow@31952
  2140
lemma coprime_Suc_nat [simp]: "coprime (Suc n) n"
eberlm@62429
  2141
  using coprime_plus_one[of n] by simp
huffman@31706
  2142
wenzelm@63489
  2143
lemma coprime_minus_one_nat: "n \<noteq> 0 \<Longrightarrow> coprime (n - 1) n"
wenzelm@63489
  2144
  for n :: nat
eberlm@62429
  2145
  using coprime_Suc_nat [of "n - 1"] gcd.commute [of "n - 1" n] by auto
huffman@31706
  2146
wenzelm@63489
  2147
lemma coprime_common_divisor_nat: "coprime a b \<Longrightarrow> x dvd a \<Longrightarrow> x dvd b \<Longrightarrow> x = 1"
wenzelm@63489
  2148
  for a b :: nat
haftmann@62344
  2149
  by (metis gcd_greatest_iff nat_dvd_1_iff_1)
huffman@31706
  2150
wenzelm@63489
  2151
lemma coprime_common_divisor_int: "coprime a b \<Longrightarrow> x dvd a \<Longrightarrow> x dvd b \<Longrightarrow> \<bar>x\<bar> = 1"
wenzelm@63489
  2152
  for a b :: int
haftmann@60686
  2153
  using gcd_greatest_iff [of x a b] by auto
huffman@31706
  2154
wenzelm@63489
  2155
lemma invertible_coprime_nat: "x * y mod m = 1 \<Longrightarrow> coprime x m"
wenzelm@63489
  2156
  for m x y :: nat
wenzelm@63489
  2157
  by (metis coprime_lmult gcd_1_nat gcd.commute gcd_red_nat)
wenzelm@63489
  2158
wenzelm@63489
  2159
lemma invertible_coprime_int: "x * y mod m = 1 \<Longrightarrow> coprime x m"
wenzelm@63489
  2160
  for m x y :: int
wenzelm@63489
  2161
  by (metis coprime_lmult gcd_1_int gcd.commute gcd_red_int)
huffman@31706
  2162
huffman@31706
  2163
wenzelm@60758
  2164
subsection \<open>Bezout's theorem\<close>
huffman@31706
  2165
wenzelm@63489
  2166
text \<open>
wenzelm@63489
  2167
  Function \<open>bezw\<close> returns a pair of witnesses to Bezout's theorem --
wenzelm@63489
  2168
  see the theorems that follow the definition.
wenzelm@63489
  2169
\<close>
wenzelm@63489
  2170
wenzelm@63489
  2171
fun bezw :: "nat \<Rightarrow> nat \<Rightarrow> int * int"
wenzelm@63489
  2172
  where "bezw x y =
wenzelm@63489
  2173
    (if y = 0 then (1, 0)
wenzelm@63489
  2174
     else
huffman@31706
  2175
      (snd (bezw y (x mod y)),
huffman@31706
  2176
       fst (bezw y (x mod y)) - snd (bezw y (x mod y)) * int(x div y)))"
huffman@31706
  2177
wenzelm@63489
  2178
lemma bezw_0 [simp]: "bezw x 0 = (1, 0)"
wenzelm@63489
  2179
  by simp
wenzelm@63489
  2180
wenzelm@63489
  2181
lemma bezw_non_0:
wenzelm@63489
  2182
  "y > 0 \<Longrightarrow> bezw x y =
wenzelm@63489
  2183
    (snd (bezw y (x mod y)), fst (bezw y (x mod y)) - snd (bezw y (x mod y)) * int(x div y))"
huffman@31706
  2184
  by simp
huffman@31706
  2185
huffman@31706
  2186
declare bezw.simps [simp del]
huffman@31706
  2187
wenzelm@63489
  2188
lemma bezw_aux: "fst (bezw x y) * int x + snd (bezw x y) * int y = int (gcd x y)"
nipkow@31952
  2189
proof (induct x y rule: gcd_nat_induct)
huffman@31706
  2190
  fix m :: nat
huffman@31706
  2191
  show "fst (bezw m 0) * int m + snd (bezw m 0) * int 0 = int (gcd m 0)"
huffman@31706
  2192
    by auto
wenzelm@63489
  2193
next
wenzelm@63489
  2194
  fix m n :: nat
wenzelm@63489
  2195
  assume ngt0: "n > 0"
wenzelm@63489
  2196
    and ih: "fst (bezw n (m mod n)) * int n + snd (bezw n (m mod n)) * int (m mod n) =
wenzelm@63489
  2197
      int (gcd n (m mod n))"
wenzelm@63489
  2198
  then show "fst (bezw m n) * int m + snd (bezw m n) * int n = int (gcd m n)"
wenzelm@63489
  2199
    apply (simp add: bezw_non_0 gcd_non_0_nat)
wenzelm@63489
  2200
    apply (erule subst)
wenzelm@63489
  2201
    apply (simp add: field_simps)
haftmann@64242
  2202
    apply (subst div_mult_mod_eq [of m n, symmetric])
wenzelm@63489
  2203
      (* applying simp here undoes the last substitution! what is procedure cancel_div_mod? *)
wenzelm@63489
  2204
    apply (simp only: NO_MATCH_def field_simps of_nat_add of_nat_mult)
wenzelm@63489
  2205
    done
huffman@31706
  2206
qed
huffman@31706
  2207
wenzelm@63489
  2208
lemma bezout_int: "\<exists>u v. u * x + v * y = gcd x y"
wenzelm@63489
  2209
  for x y :: int
huffman@31706
  2210
proof -
wenzelm@63489
  2211
  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
  2212
    apply (rule_tac x = "fst (bezw (nat x) (nat y))" in exI)
huffman@31706
  2213
    apply (rule_tac x = "snd (bezw (nat x) (nat y))" in exI)
huffman@31706
  2214
    apply (unfold gcd_int_def)
huffman@31706
  2215
    apply simp
huffman@31706
  2216
    apply (subst bezw_aux [symmetric])
huffman@31706
  2217
    apply auto
huffman@31706
  2218
    done
wenzelm@63489
  2219
  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"
wenzelm@63489
  2220
    by atomize_elim auto
wenzelm@63489
  2221
  then show ?thesis
wenzelm@63489
  2222
  proof cases
wenzelm@63489
  2223
    case 1
wenzelm@63489
  2224
    then show ?thesis by (rule aux)
wenzelm@63489
  2225
  next
wenzelm@63489
  2226
    case 2
wenzelm@63489
  2227
    then show ?thesis
wenzelm@63489
  2228
      apply -
wenzelm@63489
  2229
      apply (insert aux [of x "-y"])
wenzelm@63489
  2230
      apply auto
wenzelm@63489
  2231
      apply (rule_tac x = u in exI)
wenzelm@63489
  2232
      apply (rule_tac x = "-v" in exI)
wenzelm@63489
  2233
      apply (subst gcd_neg2_int [symmetric])
wenzelm@63489
  2234
      apply auto
wenzelm@63489
  2235
      done
wenzelm@63489
  2236
  next
wenzelm@63489
  2237
    case 3
wenzelm@63489
  2238
    then show ?thesis
wenzelm@63489
  2239
      apply -
wenzelm@63489
  2240
      apply (insert aux [of "-x" y])
wenzelm@63489
  2241
      apply auto
wenzelm@63489
  2242
      apply (rule_tac x = "-u" in exI)
wenzelm@63489
  2243
      apply (rule_tac x = v in exI)
wenzelm@63489
  2244
      apply (subst gcd_neg1_int [symmetric])
wenzelm@63489
  2245
      apply auto
wenzelm@63489
  2246
      done
wenzelm@63489
  2247
  next
wenzelm@63489
  2248
    case 4
wenzelm@63489
  2249
    then show ?thesis
wenzelm@63489
  2250
      apply -
wenzelm@63489
  2251
      apply (insert aux [of "-x" "-y"])
wenzelm@63489
  2252
      apply auto
wenzelm@63489
  2253
      apply (rule_tac x = "-u" in exI)
wenzelm@63489
  2254
      apply (rule_tac x = "-v" in exI)
wenzelm@63489
  2255
      apply (subst gcd_neg1_int [symmetric])
wenzelm@63489
  2256
      apply (subst gcd_neg2_int [symmetric])
wenzelm@63489
  2257
      apply auto
wenzelm@63489
  2258
      done
wenzelm@63489
  2259
  qed
huffman@31706
  2260
qed
huffman@31706
  2261
wenzelm@63489
  2262
wenzelm@63489
  2263
text \<open>Versions of Bezout for \<open>nat\<close>, by Amine Chaieb.\<close>
huffman@31706
  2264
huffman@31706
  2265
lemma ind_euclid:
wenzelm@63489
  2266
  fixes P :: "nat \<Rightarrow> nat \<Rightarrow> bool"
wenzelm@63489
  2267
  assumes c: " \<forall>a b. P a b \<longleftrightarrow> P b a"
wenzelm@63489
  2268
    and z: "\<forall>a. P a 0"
wenzelm@63489
  2269
    and add: "\<forall>a b. P a b \<longrightarrow> P a (a + b)"
chaieb@27669
  2270
  shows "P a b"
wenzelm@63489
  2271
proof (induct "a + b" arbitrary: a b rule: less_induct)
berghofe@34915
  2272
  case less
wenzelm@63489
  2273
  consider (eq) "a = b" | (lt) "a < b" "a + b - a < a + b" | "b = 0" | "b + a - b < a + b"
wenzelm@63489
  2274
    by arith
wenzelm@63489
  2275
  show ?case
wenzelm@63489
  2276
  proof (cases a b rule: linorder_cases)
wenzelm@63489
  2277
    case equal
wenzelm@63489
  2278
    with add [rule_format, OF z [rule_format, of a]] show ?thesis by simp
wenzelm@63489
  2279
  next
wenzelm@63489
  2280
    case lt: less
wenzelm@63489
  2281
    then consider "a = 0" | "a + b - a < a + b" by arith
wenzelm@63489
  2282
    then show ?thesis
wenzelm@63489
  2283
    proof cases
wenzelm@63489
  2284
      case 1
wenzelm@63489
  2285
      with z c show ?thesis by blast
wenzelm@63489
  2286
    next
wenzelm@63489
  2287
      case 2
wenzelm@63489
  2288
      also have *: "a + b - a = a + (b - a)" using lt by arith
berghofe@34915
  2289
      finally have "a + (b - a) < a + b" .
wenzelm@63489
  2290
      then have "P a (a + (b - a))" by (rule add [rule_format, OF less])
wenzelm@63489
  2291
      then show ?thesis by (simp add: *[symmetric])
wenzelm@63489
  2292
    qed
wenzelm@63489
  2293
  next
wenzelm@63489
  2294
    case gt: greater
wenzelm@63489
  2295
    then consider "b = 0" | "b + a - b < a + b" by arith
wenzelm@63489
  2296
    then show ?thesis
wenzelm@63489
  2297
    proof cases
wenzelm@63489
  2298
      case 1
wenzelm@63489
  2299
      with z c show ?thesis by blast
wenzelm@63489
  2300
    next
wenzelm@63489
  2301
      case 2
wenzelm@63489
  2302
      also have *: "b + a - b = b + (a - b)" using gt by arith
berghofe@34915
  2303
      finally have "b + (a - b) < a + b" .
wenzelm@63489
  2304
      then have "P b (b + (a - b))" by (rule add [rule_format, OF less])
wenzelm@63489
  2305
      then have "P b a" by (simp add: *[symmetric])
wenzelm@63489
  2306
      with c show ?thesis by blast
wenzelm@63489
  2307
    qed
wenzelm@63489
  2308
  qed
chaieb@27669
  2309
qed
chaieb@27669
  2310
nipkow@31952
  2311
lemma bezout_lemma_nat:
huffman@31706
  2312
  assumes ex: "\<exists>(d::nat) x y. d dvd a \<and> d dvd b \<and>
huffman@31706
  2313
    (a * x = b * y + d \<or> b * x = a * y + d)"
huffman@31706
  2314
  shows "\<exists>d x y. d dvd a \<and> d dvd a + b \<and>
huffman@31706
  2315
    (a * x = (a + b) * y + d \<or> (a + b) * x = a * y + d)"
huffman@31706
  2316
  using ex
huffman@31706
  2317
  apply clarsimp
wenzelm@63489
  2318
  apply (rule_tac x="d" in exI)
wenzelm@63489
  2319
  apply simp
wenzelm@63489
  2320
  apply (case_tac "a * x = b * y + d")
wenzelm@63489
  2321
   apply simp_all
wenzelm@63489
  2322
   apply (rule_tac x="x + y" in exI)
wenzelm@63489
  2323
   apply (rule_tac x="y" in exI)
wenzelm@63489
  2324
   apply algebra
huffman@31706
  2325
  apply (rule_tac x="x" in exI)
huffman@31706
  2326
  apply (rule_tac x="x + y" in exI)
huffman@31706
  2327
  apply algebra
wenzelm@63489
  2328
  done
chaieb@27669
  2329
nipkow@31952
  2330
lemma bezout_add_nat: "\<exists>(d::nat) x y. d dvd a \<and> d dvd b \<and>
huffman@31706
  2331
    (a * x = b * y + d \<or> b * x = a * y + d)"
wenzelm@63489
  2332
  apply (induct a b rule: ind_euclid)
wenzelm@63489
  2333
    apply blast
wenzelm@63489
  2334
   apply clarify
wenzelm@63489
  2335
   apply (rule_tac x="a" in exI)
wenzelm@63489
  2336
   apply simp
huffman@31706
  2337
  apply clarsimp
huffman@31706
  2338
  apply (rule_tac x="d" in exI)
wenzelm@63489
  2339
  apply (case_tac "a * x = b * y + d")
wenzelm@63489
  2340
   apply simp_all
wenzelm@63489
  2341
   apply (rule_tac x="x+y" in exI)
wenzelm@63489
  2342
   apply (rule_tac x="y" in exI)
wenzelm@63489
  2343
   apply algebra
huffman@31706
  2344
  apply (rule_tac x="x" in exI)
huffman@31706
  2345
  apply (rule_tac x="x+y" in exI)
huffman@31706
  2346
  apply algebra
wenzelm@63489
  2347
  done
chaieb@27669
  2348
nipkow@31952
  2349
lemma bezout1_nat: "\<exists>(d::nat) x y. d dvd a \<and> d dvd b \<and>
huffman@31706
  2350
    (a * x - b * y = d \<or> b * x - a * y = d)"
nipkow@31952
  2351
  using bezout_add_nat[of a b]
huffman@31706
  2352
  apply clarsimp
wenzelm@63489
  2353
  apply (rule_tac x="d" in exI)
wenzelm@63489
  2354
  apply simp
huffman@31706
  2355
  apply (rule_tac x="x" in exI)
huffman@31706
  2356
  apply (rule_tac x="y" in exI)
huffman@31706
  2357
  apply auto
wenzelm@63489
  2358
  done
wenzelm@63489
  2359
wenzelm@63489
  2360
lemma bezout_add_strong_nat:
wenzelm@63489
  2361
  fixes a b :: nat
wenzelm@63489
  2362
  assumes a: "a \<noteq> 0"
chaieb@27669
  2363
  shows "\<exists>d x y. d dvd a \<and> d dvd b \<and> a * x = b * y + d"
wenzelm@63489
  2364
proof -
wenzelm@63489
  2365
  consider d x y where "d dvd a" "d dvd b" "a * x = b * y + d"
wenzelm@63489
  2366
    | d x y where "d dvd a" "d dvd b" "b * x = a * y + d"
wenzelm@63489
  2367
    using bezout_add_nat [of a b] by blast
wenzelm@63489
  2368
  then show ?thesis
wenzelm@63489
  2369
  proof cases
wenzelm@63489
  2370
    case 1
wenzelm@63489
  2371
    then show ?thesis by blast
wenzelm@63489
  2372
  next
wenzelm@63489
  2373
    case H: 2
wenzelm@63489
  2374
    show ?thesis
wenzelm@63489
  2375
    proof (cases "b = 0")
wenzelm@63489
  2376
      case True
wenzelm@63489
  2377
      with H show ?thesis by simp
wenzelm@63489
  2378
    next
wenzelm@63489
  2379
      case False
wenzelm@63489
  2380
      then have bp: "b > 0" by simp
wenzelm@63489
  2381
      with dvd_imp_le [OF H(2)] consider "d = b" | "d < b"
wenzelm@63489
  2382
        by atomize_elim auto
wenzelm@63489
  2383
      then show ?thesis
wenzelm@63489
  2384
      proof cases
wenzelm@63489
  2385
        case 1
wenzelm@63489
  2386
        with a H show ?thesis
wenzelm@63489
  2387
          apply simp
wenzelm@63489
  2388
          apply (rule exI[where x = b])
wenzelm@63489
  2389
          apply simp
wenzelm@63489
  2390
          apply (rule exI[where x = b])
wenzelm@63489
  2391
          apply (rule exI[where x = "a - 1"])
wenzelm@63489
  2392
          apply (simp add: diff_mult_distrib2)
wenzelm@63489
  2393
          done
wenzelm@63489
  2394
      next
wenzelm@63489
  2395
        case 2
wenzelm@63489
  2396
        show ?thesis
wenzelm@63489
  2397
        proof (cases "x = 0")
wenzelm@63489
  2398
          case True
wenzelm@63489
  2399
          with a H show ?thesis by simp
wenzelm@63489
  2400
        next
wenzelm@63489
  2401
          case x0: False
wenzelm@63489
  2402
          then have xp: "x > 0" by simp
wenzelm@63489
  2403
          from \<open>d < b\<close> have "d \<le> b - 1" by simp
wenzelm@63489
  2404
          then have "d * b \<le> b * (b - 1)" by simp
wenzelm@63489
  2405
          with xp mult_mono[of "1" "x" "d * b" "b * (b - 1)"]
wenzelm@63489
  2406
          have dble: "d * b \<le> x * b * (b - 1)" using bp by simp
wenzelm@63489
  2407
          from H(3) have "d + (b - 1) * (b * x) = d + (b - 1) * (a * y + d)"
huffman@31706
  2408
            by simp
wenzelm@63489
  2409
          then have "d + (b - 1) * a * y + (b - 1) * d = d + (b - 1) * b * x"
haftmann@57512
  2410
            by (simp only: mult.assoc distrib_left)
wenzelm@63489
  2411
          then have "a * ((b - 1) * y) + d * (b - 1 + 1) = d + x * b * (b - 1)"
huffman@31706
  2412
            by algebra
wenzelm@63489
  2413
          then have "a * ((b - 1) * y) = d + x * b * (b - 1) - d * b"
wenzelm@63489
  2414
            using bp by simp
wenzelm@63489
  2415
          then have "a * ((b - 1) * y) = d + (x * b * (b - 1) - d * b)"
wenzelm@32960
  2416
            by (simp only: diff_add_assoc[OF dble, of d, symmetric])
wenzelm@63489
  2417
          then have "a * ((b - 1) * y) = b * (x * (b - 1) - d) + d"
haftmann@59008
  2418
            by (simp only: diff_mult_distrib2 ac_simps)
wenzelm@63489
  2419
          with H(1,2) show ?thesis
wenzelm@32960
  2420
            apply -
wenzelm@63489
  2421
            apply (rule exI [where x = d])
wenzelm@63489
  2422
            apply simp
wenzelm@63489
  2423
            apply (rule exI [where x = "(b - 1) * y"])
wenzelm@63489
  2424
            apply (rule exI [where x = "x * (b - 1) - d"])
wenzelm@63489
  2425
            apply simp
wenzelm@63489
  2426
            done
wenzelm@63489
  2427
        qed
wenzelm@63489
  2428
      qed
wenzelm@63489
  2429
    qed
wenzelm@63489
  2430
  qed
chaieb@27669
  2431
qed
chaieb@27669
  2432
wenzelm@63489
  2433
lemma bezout_nat:
wenzelm@63489
  2434
  fixes a :: nat
wenzelm@63489
  2435
  assumes a: "a \<noteq> 0"
chaieb@27669
  2436
  shows "\<exists>x y. a * x = b * y + gcd a b"
wenzelm@63489
  2437
proof -
wenzelm@63489
  2438
  obtain d x y where d: "d dvd a" "d dvd b" and eq: "a * x = b * y + d"
wenzelm@63489
  2439
    using bezout_add_strong_nat [OF a, of b] by blast
wenzelm@63489
  2440
  from d have "d dvd gcd a b"
wenzelm@63489
  2441
    by simp
wenzelm@63489
  2442
  then obtain k where k: "gcd a b = d * k"
wenzelm@63489
  2443
    unfolding dvd_def by blast
wenzelm@63489
  2444
  from eq have "a * x * k = (b * y + d) * k"
wenzelm@63489
  2445
    by auto
wenzelm@63489
  2446
  then have "a * (x * k) = b * (y * k) + gcd a b"
wenzelm@63489
  2447
    by (algebra add: k)
wenzelm@63489
  2448
  then show ?thesis
wenzelm@63489
  2449
    by blast
chaieb@27669
  2450
qed
chaieb@27669
  2451
huffman@31706
  2452
wenzelm@63489
  2453
subsection \<open>LCM properties on @{typ nat} and @{typ int}\<close>
wenzelm@63489
  2454
wenzelm@63489
  2455
lemma lcm_altdef_int [code]: "lcm a b = \<bar>a\<bar> * \<bar>b\<bar> div gcd a b"
wenzelm@63489
  2456
  for a b :: int
haftmann@62344
  2457
  by (simp add: lcm_int_def lcm_nat_def zdiv_int gcd_int_def)
huffman@31706
  2458
wenzelm@63489
  2459
lemma prod_gcd_lcm_nat: "m * n = gcd m n * lcm m n"
wenzelm@63489
  2460
  for m n :: nat
huffman@31706
  2461
  unfolding lcm_nat_def
eberlm@62429
  2462
  by (simp add: dvd_mult_div_cancel [OF gcd_dvd_prod])
huffman@31706
  2463
wenzelm@63489
  2464
lemma prod_gcd_lcm_int: "\<bar>m\<bar> * \<bar>n\<bar> = gcd m n * lcm m n"
wenzelm@63489
  2465
  for m n :: int
huffman@31706
  2466
  unfolding lcm_int_def gcd_int_def
haftmann@62348
  2467
  apply (subst of_nat_mult [symmetric])
nipkow@31952
  2468
  apply (subst prod_gcd_lcm_nat [symmetric])
huffman@31706
  2469
  apply (subst nat_abs_mult_distrib [symmetric])
wenzelm@63489
  2470
  apply (simp add: abs_mult)
wenzelm@63489
  2471
  done
wenzelm@63489
  2472
wenzelm@63489
  2473
lemma lcm_pos_nat: "m > 0 \<Longrightarrow> n > 0 \<Longrightarrow> lcm m n > 0"
wenzelm@63489
  2474
  for m n :: nat
wenzelm@63489
  2475
  by (metis gr0I mult_is_0 prod_gcd_lcm_nat)
wenzelm@63489
  2476
wenzelm@63489
  2477
lemma lcm_pos_int: "m \<noteq> 0 \<Longrightarrow> n \<noteq> 0 \<Longrightarrow> lcm m n > 0"
wenzelm@63489
  2478
  for m n :: int
haftmann@66796
  2479
  by (simp add: lcm_int_def lcm_pos_nat)
haftmann@23687
  2480
wenzelm@63489
  2481
lemma dvd_pos_nat: "n > 0 \<Longrightarrow> m dvd n \<Longrightarrow> m > 0"  (* FIXME move *)
wenzelm@63489
  2482
  for m n :: nat
wenzelm@63489
  2483
  by (cases m) auto
wenzelm@63489
  2484
wenzelm@63489
  2485
lemma lcm_unique_nat:
wenzelm@63489
  2486
  "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
  2487
  for a b d :: nat
haftmann@62344
  2488
  by (auto intro: dvd_antisym lcm_least)
chaieb@27568
  2489
wenzelm@63489
  2490
lemma lcm_unique_int:
wenzelm@63489
  2491
  "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
  2492
  for a b d :: int
haftmann@62344
  2493
  using lcm_least zdvd_antisym_nonneg by auto
haftmann@34973
  2494
wenzelm@63489
  2495
lemma lcm_proj2_if_dvd_nat [simp]: "x dvd y \<Longrightarrow> lcm x y = y"
wenzelm@63489
  2496
  for x y :: nat
huffman@31706
  2497
  apply (rule sym)
nipkow@31952
  2498
  apply (subst lcm_unique_nat [symmetric])
huffman@31706
  2499
  apply auto
wenzelm@63489
  2500
  done
wenzelm@63489
  2501
wenzelm@63489
  2502
lemma lcm_proj2_if_dvd_int [simp]: "x dvd y \<Longrightarrow> lcm x y = \<bar>y\<bar>"
wenzelm@63489
  2503
  for x y :: int
huffman@31706
  2504
  apply (rule sym)
nipkow@31952
  2505
  apply (subst lcm_unique_int [symmetric])
huffman@31706
  2506
  apply auto
wenzelm@63489
  2507
  done
wenzelm@63489
  2508
wenzelm@63489
  2509
lemma lcm_proj1_if_dvd_nat [simp]: "x dvd y \<Longrightarrow> lcm y x = y"
wenzelm@63489
  2510
  for x y :: nat
wenzelm@63489
  2511
  by (subst lcm.commute) (erule lcm_proj2_if_dvd_nat)
wenzelm@63489
  2512
wenzelm@63489
  2513
lemma lcm_proj1_if_dvd_int [simp]: "x dvd y \<Longrightarrow> lcm y x = \<bar>y\<bar>"
wenzelm@63489
  2514
  for x y :: int
wenzelm@63489
  2515
  by (subst lcm.commute) (erule lcm_proj2_if_dvd_int)
wenzelm@63489
  2516
wenzelm@63489
  2517
lemma lcm_proj1_iff_nat [simp]: "lcm m n = m \<longleftrightarrow> n dvd m"
wenzelm@63489
  2518
  for m n :: nat
wenzelm@63489
  2519
  by (metis lcm_proj1_if_dvd_nat lcm_unique_nat)
wenzelm@63489
  2520
wenzelm@63489
  2521
lemma lcm_proj2_iff_nat [simp]: "lcm m n = n \<longleftrightarrow> m dvd n"
wenzelm@63489
  2522
  for m n :: nat
wenzelm@63489
  2523
  by (metis lcm_proj2_if_dvd_nat lcm_unique_nat)
wenzelm@63489
  2524
wenzelm@63489
  2525
lemma lcm_proj1_iff_int [simp]: "lcm m n = \<bar>m\<bar> \<longleftrightarrow> n dvd m"
wenzelm@63489
  2526
  for m n :: int
wenzelm@63489
  2527
  by (metis dvd_abs_iff lcm_proj1_if_dvd_int lcm_unique_int)
wenzelm@63489
  2528
wenzelm@63489
  2529
lemma lcm_proj2_iff_int [simp]: "lcm m n = \<bar>n\<bar> \<longleftrightarrow> m dvd n"
wenzelm@63489
  2530
  for m n :: int
wenzelm@63489
  2531
  by (metis dvd_abs_iff lcm_proj2_if_dvd_int lcm_unique_int)
wenzelm@63489
  2532
wenzelm@63489
  2533
lemma lcm_1_iff_nat [simp]: "lcm m n = Suc 0 \<longleftrightarrow> m = Suc 0 \<and> n = Suc 0"
wenzelm@63489
  2534
  for m n :: nat
haftmann@62353
  2535
  using lcm_eq_1_iff [of m n] by simp
wenzelm@63489
  2536
wenzelm@63489
  2537
lemma lcm_1_iff_int [simp]: "lcm m n = 1 \<longleftrightarrow> (m = 1 \<or> m = -1) \<and> (n = 1 \<or> n = -1)"
wenzelm@63489
  2538
  for m n :: int
haftmann@61913
  2539
  by auto
nipkow@31995
  2540
haftmann@34030
  2541
haftmann@62345
  2542
subsection \<open>The complete divisibility lattice on @{typ nat} and @{typ int}\<close>
nipkow@32112
  2543
wenzelm@63489
  2544
text \<open>
wenzelm@63489
  2545
  Lifting \<open>gcd\<close> and \<open>lcm\<close> to sets (\<open>Gcd\<close> / \<open>Lcm\<close>).
wenzelm@63489
  2546
  \<open>Gcd\<close> is defined via \<open>Lcm\<close> to facilitate the proof that we have a complete lattice.
wenzelm@60758
  2547
\<close>
huffman@45264
  2548
haftmann@62345
  2549
instantiation nat :: semiring_Gcd
nipkow@32112
  2550
begin
nipkow@32112
  2551
haftmann@62344
  2552
interpretation semilattice_neutr_set lcm "1::nat"
haftmann@62344
  2553
  by standard simp_all
haftmann@54867
  2554
wenzelm@63489
  2555
definition "Lcm M = (if finite M then F M else 0)" for M :: "nat set"
wenzelm@63489
  2556
wenzelm@63489
  2557
lemma Lcm_nat_empty: "Lcm {} = (1::nat)"
haftmann@60690
  2558
  by (simp add: Lcm_nat_def del: One_nat_def)
haftmann@51489
  2559
wenzelm@63489
  2560
lemma Lcm_nat_insert: "Lcm (insert n M) = lcm n (Lcm M)" for n :: nat
haftmann@61929
  2561
  by (cases "finite M") (auto simp add: Lcm_nat_def simp del: One_nat_def)
haftmann@61929
  2562
wenzelm@63489
  2563
lemma Lcm_nat_infinite: "infinite M \<Longrightarrow> Lcm M = 0" for M :: "nat set"
haftmann@61929
  2564
  by (simp add: Lcm_nat_def)
haftmann@61929
  2565
haftmann@61929
  2566
lemma dvd_Lcm_nat [simp]:
haftmann@61929
  2567
  fixes M :: "nat set"
haftmann@61929
  2568
  assumes "m \<in> M"
haftmann@61929
  2569
  shows "m dvd Lcm M"
haftmann@61929
  2570
proof -
wenzelm@63489
  2571
  from assms have "insert m M = M"
wenzelm@63489
  2572
    by auto
haftmann@61929
  2573
  moreover have "m dvd Lcm (insert m M)"
haftmann@61929
  2574
    by (simp add: Lcm_nat_insert)
wenzelm@63489
  2575
  ultimately show ?thesis
wenzelm@63489
  2576
    by simp
haftmann@61929
  2577
qed
haftmann@61929
  2578
haftmann@61929
  2579
lemma Lcm_dvd_nat [simp]:
haftmann@61929
  2580
  fixes M :: "nat set"
haftmann@61929
  2581
  assumes "\<forall>m\<in>M. m dvd n"
haftmann@61929
  2582
  shows "Lcm M dvd n"
haftmann@62353
  2583
proof (cases "n > 0")
wenzelm@63489
  2584
  case False
wenzelm@63489
  2585
  then show ?thesis by simp
haftmann@61929
  2586
next
haftmann@62353
  2587
  case True
wenzelm@63489
  2588
  then have "finite {d. d dvd n}"
wenzelm@63489
  2589
    by (rule finite_divisors_nat)
wenzelm@63489
  2590
  moreover have "M \<subseteq> {d. d dvd n}"
wenzelm@63489
  2591
    using assms by fast
wenzelm@63489
  2592
  ultimately have "finite M"
wenzelm@63489
  2593
    by (rule rev_finite_subset)
wenzelm@63489
  2594
  then show ?thesis
wenzelm@63489
  2595
    using assms by (induct M) (simp_all add: Lcm_nat_empty Lcm_nat_insert)
haftmann@61929
  2596
qed
nipkow@32112
  2597
wenzelm@63489
  2598
definition "Gcd M = Lcm {d. \<forall>m\<in>M. d dvd m}" for M :: "nat set"
wenzelm@63489
  2599
wenzelm@63489
  2600
instance
wenzelm@63489
  2601
proof
wenzelm@63489
  2602
  fix N :: "nat set"
wenzelm@63489
  2603
  fix n :: nat
wenzelm@63489
  2604
  show "Gcd N dvd n" if "n \<in> N"
wenzelm@63489
  2605
    using that by (induct N rule: infinite_finite_induct) (auto simp add: Gcd_nat_def)
wenzelm@63489
  2606
  show "n dvd Gcd N" if "\<And>m. m \<in> N \<Longrightarrow> n dvd m"
wenzelm@63489
  2607
    using that by (induct N rule: infinite_finite_induct) (auto simp add: Gcd_nat_def)
wenzelm@63489
  2608
  show "n dvd Lcm N" if "n \<in> N"
wenzelm@63489
  2609
    using that by (induct N rule: infinite_finite_induct) auto
wenzelm@63489
  2610
  show "Lcm N dvd n" if "\<And>m. m \<in> N \<Longrightarrow> m dvd n"
wenzelm@63489
  2611
    using that by (induct N rule: infinite_finite_induct) auto
wenzelm@63489
  2612
  show "normalize (Gcd N) = Gcd N" and "normalize (Lcm N) = Lcm N"
wenzelm@63489
  2613
    by simp_all
wenzelm@63489
  2614
qed
nipkow@32112
  2615
haftmann@62345
  2616
end
haftmann@61913
  2617
wenzelm@63489
  2618
lemma Gcd_nat_eq_one: "1 \<in> N \<Longrightarrow> Gcd N = 1"
wenzelm@63489
  2619
  for N :: "nat set"
haftmann@62346
  2620
  by (rule Gcd_eq_1_I) auto
haftmann@62346
  2621
wenzelm@63489
  2622
wenzelm@63489
  2623
text \<open>Alternative characterizations of Gcd:\<close>
nipkow@32112
  2624
haftmann@62353
  2625
lemma Gcd_eq_Max:
haftmann@62353
  2626
  fixes M :: "nat set"
haftmann@62353
  2627
  assumes "finite (M::nat set)" and "M \<noteq> {}" and "0 \<notin> M"
haftmann@62353
  2628
  shows "Gcd M = Max (\<Inter>m\<in>M. {d. d dvd m})"
haftmann@62353
  2629
proof (rule antisym)
haftmann@62353
  2630
  from assms obtain m where "m \<in> M" and "m > 0"
haftmann@62353
  2631
    by auto
haftmann@62353
  2632
  from \<open>m > 0\<close> have "finite {d. d dvd m}"
haftmann@62353
  2633
    by (blast intro: finite_divisors_nat)
haftmann@62353
  2634
  with \<open>m \<in> M\<close> have fin: "finite (\<Inter>m\<in>M. {d. d dvd m})"
haftmann@62353
  2635
    by blast
haftmann@62353
  2636
  from fin show "Gcd M \<le> Max (\<Inter>m\<in>M. {d. d dvd m})"
haftmann@62353
  2637
    by (auto intro: Max_ge Gcd_dvd)
haftmann@62353
  2638
  from fin show "Max (\<Inter>m\<in>M. {d. d dvd m}) \<le> Gcd M"
haftmann@62353
  2639
    apply (rule Max.boundedI)
wenzelm@63489
  2640
     apply auto
haftmann@62353
  2641
    apply (meson Gcd_dvd Gcd_greatest \<open>0 < m\<close> \<open>m \<in> M\<close> dvd_imp_le dvd_pos_nat)
haftmann@62353
  2642
    done
haftmann@62353
  2643
qed
nipkow@32112
  2644
wenzelm@63489
  2645
lemma Gcd_remove0_nat: "finite M \<Longrightarrow> Gcd M = Gcd (M - {0})"
wenzelm@63489
  2646
  for M :: "nat set"
wenzelm@63489
  2647
  apply (induct pred: finite)
wenzelm@63489
  2648
   apply simp
wenzelm@63489
  2649
  apply (case_tac "x = 0")
wenzelm@63489
  2650
   apply simp
wenzelm@63489
  2651
  apply (subgoal_tac "insert x F - {0} = insert x (F - {0})")
wenzelm@63489
  2652
   apply simp
wenzelm@63489
  2653
  apply blast
wenzelm@63489
  2654
  done
nipkow@32112
  2655
nipkow@32112
  2656
lemma Lcm_in_lcm_closed_set_nat:
wenzelm@63489
  2657
  "finite M \<Longrightarrow> M \<noteq> {} \<Longrightarrow> \<forall>m n. m \<in> M \<longrightarrow> n \<in> M \<longrightarrow> lcm m n \<in> M \<Longrightarrow> Lcm M \<in> M"
wenzelm@63489
  2658
  for M :: "nat set"
wenzelm@63489
  2659
  apply (induct rule: finite_linorder_min_induct)
wenzelm@63489
  2660
   apply simp
nipkow@32112
  2661
  apply simp
wenzelm@63489
  2662
  apply (subgoal_tac "\<forall>m n. m \<in> A \<longrightarrow> n \<in> A \<longrightarrow> lcm m n \<in> A")
wenzelm@63489
  2663
   apply simp
wenzelm@63489
  2664
   apply(case_tac "A = {}")
wenzelm@63489
  2665
    apply simp
wenzelm@63489
  2666
   apply simp
wenzelm@63489
  2667
  apply (metis lcm_pos_nat lcm_unique_nat linorder_neq_iff nat_dvd_not_less not_less0)
wenzelm@63489
  2668
  done
nipkow@32112
  2669
nipkow@32112
  2670
lemma Lcm_eq_Max_nat:
wenzelm@63489
  2671
  "finite M \<Longrightarrow> M \<noteq> {} \<Longrightarrow> 0 \<notin> M \<Longrightarrow> \<forall>m n. m \<in> M \<longrightarrow> n \<in> M \<longrightarrow> lcm m n \<in> M \<Longrightarrow> Lcm M = Max M"
wenzelm@63489
  2672
  for M :: "nat set"
wenzelm@63489
  2673
  apply (rule antisym)
wenzelm@63489
  2674
   apply (rule Max_ge)
wenzelm@63489
  2675
    apply assumption
wenzelm@63489
  2676
   apply (erule (2) Lcm_in_lcm_closed_set_nat)
wenzelm@63489
  2677
  apply (auto simp add: not_le Lcm_0_iff dvd_imp_le leD le_neq_trans)
wenzelm@63489
  2678
  done
nipkow@32112
  2679
nipkow@34222
  2680
lemma mult_inj_if_coprime_nat:
wenzelm@63489
  2681
  "inj_on f A \<Longrightarrow> inj_on g B \<Longrightarrow> \<forall>a\<in>A. \<forall>b\<in>B. coprime (f a) (g b) \<Longrightarrow>
wenzelm@63489
  2682
    inj_on (\<lambda>(a, b). f a * g b) (A \<times> B)"
wenzelm@63489
  2683
  for f :: "'a \<Rightarrow> nat" and g :: "'b \<Rightarrow> nat"
haftmann@61913
  2684
  by (auto simp add: inj_on_def coprime_crossproduct_nat simp del: One_nat_def)
nipkow@34222
  2685
wenzelm@63489
  2686
wenzelm@63489
  2687
subsubsection \<open>Setwise GCD and LCM for integers\<close>
huffman@45264
  2688
haftmann@62345
  2689
instantiation int :: semiring_Gcd
huffman@45264
  2690
begin
huffman@45264
  2691
wenzelm@63489
  2692
definition "Lcm M = int (LCM m\<in>M. (nat \<circ> abs) m)"
wenzelm@63489
  2693
wenzelm@63489
  2694
definition "Gcd M = int (GCD m\<in>M. (nat \<circ> abs) m)"
wenzelm@63489
  2695
wenzelm@63489
  2696
instance
wenzelm@63489
  2697
  by standard
wenzelm@63489
  2698
    (auto intro!: Gcd_dvd Gcd_greatest simp add: Gcd_int_def