src/HOL/GCD.thy
author wenzelm
Wed May 25 11:49:40 2016 +0200 (2016-05-25)
changeset 63145 703edebd1d92
parent 63025 92680537201f
child 63359 99b51ba8da1c
permissions -rw-r--r--
isabelle update_cartouches -c -t;
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(*  Authors:    Christophe Tabacznyj, Lawrence C. Paulson, Amine Chaieb,
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                Thomas M. Rasmussen, Jeremy Avigad, 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 Main
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begin
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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
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  "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
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  "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]:
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  "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:
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  "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:
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  "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:
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  "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:
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  "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]:
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  "gcd 0 a = normalize a"
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  by (rule associated_eqI) simp_all
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lemma gcd_0_right [simp]:
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  "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]:
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  "gcd a b = 0 \<longleftrightarrow> a = 0 \<and> b = 0" (is "?P \<longleftrightarrow> ?Q")
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proof
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  assume ?P then have "0 dvd gcd a b" by simp
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  then have "0 dvd a" and "0 dvd b" by (blast intro: dvd_trans)+
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  then show ?Q by simp
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next
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  assume ?Q then show ?P by simp
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qed
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lemma unit_factor_gcd:
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  "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 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 by simp
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qed
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lemma is_unit_gcd [simp]:
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  "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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lemma gcd_self [simp]:
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  "gcd a a = normalize 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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lemma gcd_left_idem [simp]:
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  "gcd a (gcd a b) = gcd a b"
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  by (auto intro: associated_eqI)
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lemma gcd_right_idem [simp]:
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  "gcd (gcd a b) b = gcd a b"
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  unfolding gcd.commute [of a] gcd.commute [of "gcd b a"] by simp
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lemma coprime_1_left [simp]:
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  "coprime 1 a"
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  by (rule associated_eqI) simp_all
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lemma coprime_1_right [simp]:
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  "coprime a 1"
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  using coprime_1_left [of a] by (simp add: ac_simps)
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lemma gcd_mult_left:
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  "gcd (c * a) (c * b) = normalize c * gcd a b"
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proof (cases "c = 0")
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  case True then show ?thesis by simp
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next
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  case False
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  then have "c * gcd a b dvd gcd (c * a) (c * b)"
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    by (auto intro: gcd_greatest)
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  moreover from calculation False have "gcd (c * a) (c * b) dvd c * gcd a b"
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    by (metis div_dvd_iff_mult dvd_mult_left gcd_dvd1 gcd_dvd2 gcd_greatest mult_commute)
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  ultimately have "normalize (gcd (c * a) (c * b)) = normalize (c * gcd a b)"
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    by (auto intro: associated_eqI)
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  then show ?thesis by (simp add: normalize_mult)
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qed
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lemma gcd_mult_right:
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  "gcd (a * c) (b * c) = gcd b a * normalize c"
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  using gcd_mult_left [of c a b] by (simp add: ac_simps)
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lemma mult_gcd_left:
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  "c * gcd a b = unit_factor c * gcd (c * a) (c * b)"
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  by (simp add: gcd_mult_left mult.assoc [symmetric])
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lemma mult_gcd_right:
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  "gcd a b * c = gcd (a * c) (b * c) * unit_factor c"
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  using mult_gcd_left [of c a b] by (simp add: ac_simps)
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lemma dvd_lcm1 [iff]:
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  "a dvd lcm a b"
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proof -
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  have "normalize (a * b) div gcd a b = normalize a * (normalize b div gcd a b)"
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    by (simp add: lcm_gcd normalize_mult div_mult_swap)
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  then show ?thesis
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    by (simp add: lcm_gcd)
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qed
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lemma dvd_lcm2 [iff]:
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  "b dvd lcm a b"
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proof -
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  have "normalize (a * b) div gcd a b = normalize b * (normalize a div gcd a b)"
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    by (simp add: lcm_gcd normalize_mult div_mult_swap ac_simps)
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  then show ?thesis
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    by (simp add: lcm_gcd)
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qed
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lemma dvd_lcmI1:
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  "a dvd b \<Longrightarrow> a dvd lcm b c"
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  by (rule dvd_trans) (assumption, blast) 
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lemma dvd_lcmI2:
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  "a dvd c \<Longrightarrow> a dvd lcm b c"
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  by (rule dvd_trans) (assumption, blast)
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lemma lcm_dvdD1:
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  "lcm a b dvd c \<Longrightarrow> a dvd c"
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  using dvd_lcm1 [of a b] by (blast intro: dvd_trans)
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lemma lcm_dvdD2:
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  "lcm a b dvd c \<Longrightarrow> b dvd c"
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  using dvd_lcm2 [of a b] by (blast intro: dvd_trans)
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lemma lcm_least:
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  assumes "a dvd c" and "b dvd c"
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  shows "lcm a b dvd c"
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proof (cases "c = 0")
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  case True then show ?thesis by simp
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next
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  case False then have U: "is_unit (unit_factor c)" by simp
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  show ?thesis
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  proof (cases "gcd a b = 0")
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    case True with assms show ?thesis by simp
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  next
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    case False then have "a \<noteq> 0 \<or> b \<noteq> 0" by simp
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    with \<open>c \<noteq> 0\<close> assms have "a * b dvd a * c" "a * b dvd c * b"
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      by (simp_all add: mult_dvd_mono)
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    then have "normalize (a * b) dvd gcd (a * c) (b * c)"
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      by (auto intro: gcd_greatest simp add: ac_simps)
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    then have "normalize (a * b) dvd gcd (a * c) (b * c) * unit_factor c"
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      using U by (simp add: dvd_mult_unit_iff)
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    then have "normalize (a * b) dvd gcd a b * c"
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      by (simp add: mult_gcd_right [of a b c])
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    then have "normalize (a * b) div gcd a b dvd c"
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      using False by (simp add: div_dvd_iff_mult ac_simps)
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    then show ?thesis by (simp add: lcm_gcd)
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  qed
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qed
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lemma lcm_least_iff [simp]:
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  "lcm a b dvd c \<longleftrightarrow> a dvd c \<and> b dvd c"
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  by (blast intro!: lcm_least intro: dvd_trans)
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lemma normalize_lcm [simp]:
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  "normalize (lcm a b) = lcm a b"
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  by (simp add: lcm_gcd dvd_normalize_div)
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lemma lcm_0_left [simp]:
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  "lcm 0 a = 0"
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  by (simp add: lcm_gcd)
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lemma lcm_0_right [simp]:
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  "lcm a 0 = 0"
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  by (simp add: lcm_gcd)
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lemma lcm_eq_0_iff:
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  "lcm a b = 0 \<longleftrightarrow> a = 0 \<or> b = 0" (is "?P \<longleftrightarrow> ?Q")
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proof
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  assume ?P then have "0 dvd lcm a b" by simp
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  then have "0 dvd normalize (a * b) div gcd a b"
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    by (simp add: lcm_gcd)
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  then have "0 * gcd a b dvd normalize (a * b)"
haftmann@60686
   303
    using dvd_div_iff_mult [of "gcd a b" _ 0] by (cases "gcd a b = 0") simp_all
haftmann@60686
   304
  then have "normalize (a * b) = 0"
haftmann@60686
   305
    by simp
haftmann@60686
   306
  then show ?Q by simp
haftmann@60686
   307
next
haftmann@60686
   308
  assume ?Q then show ?P by auto
haftmann@60686
   309
qed
haftmann@60686
   310
haftmann@61913
   311
lemma lcm_eq_1_iff [simp]:
haftmann@61913
   312
  "lcm a b = 1 \<longleftrightarrow> is_unit a \<and> is_unit b"
haftmann@61913
   313
  by (auto intro: associated_eqI)
haftmann@61913
   314
haftmann@60686
   315
lemma unit_factor_lcm :
haftmann@60686
   316
  "unit_factor (lcm a b) = (if a = 0 \<or> b = 0 then 0 else 1)"
haftmann@60686
   317
  by (simp add: unit_factor_gcd dvd_unit_factor_div lcm_gcd)
haftmann@60686
   318
wenzelm@61605
   319
sublocale lcm: abel_semigroup lcm
haftmann@60686
   320
proof
haftmann@60686
   321
  fix a b c
haftmann@60686
   322
  show "lcm a b = lcm b a"
haftmann@60686
   323
    by (simp add: lcm_gcd ac_simps normalize_mult dvd_normalize_div)
haftmann@60688
   324
  have "lcm (lcm a b) c dvd lcm a (lcm b c)"
haftmann@60688
   325
    and "lcm a (lcm b c) dvd lcm (lcm a b) c"
haftmann@60688
   326
    by (auto intro: lcm_least
haftmann@60686
   327
      dvd_trans [of b "lcm b c" "lcm a (lcm b c)"]
haftmann@60686
   328
      dvd_trans [of c "lcm b c" "lcm a (lcm b c)"]
haftmann@60686
   329
      dvd_trans [of a "lcm a b" "lcm (lcm a b) c"]
haftmann@60686
   330
      dvd_trans [of b "lcm a b" "lcm (lcm a b) c"])
haftmann@60686
   331
  then show "lcm (lcm a b) c = lcm a (lcm b c)"
haftmann@60688
   332
    by (rule associated_eqI) simp_all
haftmann@60686
   333
qed
haftmann@60686
   334
haftmann@60686
   335
lemma lcm_self [simp]:
haftmann@60686
   336
  "lcm a a = normalize a"
haftmann@60686
   337
proof -
haftmann@60686
   338
  have "lcm a a dvd a"
haftmann@60686
   339
    by (rule lcm_least) simp_all
haftmann@60686
   340
  then show ?thesis
haftmann@60688
   341
    by (auto intro: associated_eqI)
haftmann@60686
   342
qed
haftmann@60686
   343
haftmann@61913
   344
lemma lcm_left_idem [simp]:
haftmann@61913
   345
  "lcm a (lcm a b) = lcm a b"
haftmann@61913
   346
  by (auto intro: associated_eqI)
haftmann@61913
   347
haftmann@61913
   348
lemma lcm_right_idem [simp]:
haftmann@61913
   349
  "lcm (lcm a b) b = lcm a b"
haftmann@61913
   350
  unfolding lcm.commute [of a] lcm.commute [of "lcm b a"] by simp
haftmann@61913
   351
haftmann@60686
   352
lemma gcd_mult_lcm [simp]:
haftmann@60686
   353
  "gcd a b * lcm a b = normalize a * normalize b"
haftmann@60686
   354
  by (simp add: lcm_gcd normalize_mult)
haftmann@60686
   355
haftmann@60686
   356
lemma lcm_mult_gcd [simp]:
haftmann@60686
   357
  "lcm a b * gcd a b = normalize a * normalize b"
haftmann@60686
   358
  using gcd_mult_lcm [of a b] by (simp add: ac_simps) 
haftmann@60686
   359
haftmann@60686
   360
lemma gcd_lcm:
haftmann@60686
   361
  assumes "a \<noteq> 0" and "b \<noteq> 0"
haftmann@60686
   362
  shows "gcd a b = normalize (a * b) div lcm a b"
haftmann@60686
   363
proof -
haftmann@60686
   364
  from assms have "lcm a b \<noteq> 0"
haftmann@60686
   365
    by (simp add: lcm_eq_0_iff)
haftmann@60686
   366
  have "gcd a b * lcm a b = normalize a * normalize b" by simp
haftmann@60686
   367
  then have "gcd a b * lcm a b div lcm a b = normalize (a * b) div lcm a b"
haftmann@60686
   368
    by (simp_all add: normalize_mult)
haftmann@60686
   369
  with \<open>lcm a b \<noteq> 0\<close> show ?thesis
haftmann@60686
   370
    using nonzero_mult_divide_cancel_right [of "lcm a b" "gcd a b"] by simp
haftmann@60686
   371
qed
haftmann@60686
   372
haftmann@60686
   373
lemma lcm_1_left [simp]:
haftmann@60686
   374
  "lcm 1 a = normalize a"
haftmann@60686
   375
  by (simp add: lcm_gcd)
haftmann@60686
   376
haftmann@60686
   377
lemma lcm_1_right [simp]:
haftmann@60686
   378
  "lcm a 1 = normalize a"
haftmann@60686
   379
  by (simp add: lcm_gcd)
haftmann@60686
   380
  
haftmann@60686
   381
lemma lcm_mult_left:
haftmann@60686
   382
  "lcm (c * a) (c * b) = normalize c * lcm a b"
haftmann@60686
   383
  by (cases "c = 0")
haftmann@60686
   384
    (simp_all add: gcd_mult_right lcm_gcd div_mult_swap normalize_mult ac_simps,
haftmann@60686
   385
      simp add: dvd_div_mult2_eq mult.left_commute [of "normalize c", symmetric])
haftmann@60686
   386
haftmann@60686
   387
lemma lcm_mult_right:
haftmann@60686
   388
  "lcm (a * c) (b * c) = lcm b a * normalize c"
haftmann@60686
   389
  using lcm_mult_left [of c a b] by (simp add: ac_simps)
haftmann@60686
   390
haftmann@60686
   391
lemma mult_lcm_left:
haftmann@60686
   392
  "c * lcm a b = unit_factor c * lcm (c * a) (c * b)"
haftmann@60686
   393
  by (simp add: lcm_mult_left mult.assoc [symmetric])
haftmann@60686
   394
haftmann@60686
   395
lemma mult_lcm_right:
haftmann@60686
   396
  "lcm a b * c = lcm (a * c) (b * c) * unit_factor c"
haftmann@60686
   397
  using mult_lcm_left [of c a b] by (simp add: ac_simps)
eberlm@62429
   398
eberlm@62429
   399
lemma gcdI:
eberlm@62429
   400
  assumes "c dvd a" and "c dvd b" and greatest: "\<And>d. d dvd a \<Longrightarrow> d dvd b \<Longrightarrow> d dvd c"
eberlm@62429
   401
    and "normalize c = c"
eberlm@62429
   402
  shows "c = gcd a b"
eberlm@62429
   403
  by (rule associated_eqI) (auto simp: assms intro: gcd_greatest)
eberlm@62429
   404
eberlm@62429
   405
lemma gcd_unique: "d dvd a \<and> d dvd b \<and> 
eberlm@62429
   406
    normalize d = d \<and>
eberlm@62429
   407
    (\<forall>e. e dvd a \<and> e dvd b \<longrightarrow> e dvd d) \<longleftrightarrow> d = gcd a b"
eberlm@62429
   408
  by rule (auto intro: gcdI simp: gcd_greatest)
eberlm@62429
   409
eberlm@62429
   410
lemma gcd_dvd_prod: "gcd a b dvd k * b"
eberlm@62429
   411
  using mult_dvd_mono [of 1] by auto
eberlm@62429
   412
eberlm@62429
   413
lemma gcd_proj2_if_dvd: "b dvd a \<Longrightarrow> gcd a b = normalize b"
eberlm@62429
   414
  by (rule gcdI [symmetric]) simp_all
eberlm@62429
   415
eberlm@62429
   416
lemma gcd_proj1_if_dvd: "a dvd b \<Longrightarrow> gcd a b = normalize a"
eberlm@62429
   417
  by (rule gcdI [symmetric]) simp_all
eberlm@62429
   418
eberlm@62429
   419
lemma gcd_proj1_iff: "gcd m n = normalize m \<longleftrightarrow> m dvd n"
eberlm@62429
   420
proof
eberlm@62429
   421
  assume A: "gcd m n = normalize m"
eberlm@62429
   422
  show "m dvd n"
eberlm@62429
   423
  proof (cases "m = 0")
eberlm@62429
   424
    assume [simp]: "m \<noteq> 0"
eberlm@62429
   425
    from A have B: "m = gcd m n * unit_factor m"
eberlm@62429
   426
      by (simp add: unit_eq_div2)
eberlm@62429
   427
    show ?thesis by (subst B, simp add: mult_unit_dvd_iff)
eberlm@62429
   428
  qed (insert A, simp)
eberlm@62429
   429
next
eberlm@62429
   430
  assume "m dvd n"
eberlm@62429
   431
  then show "gcd m n = normalize m" by (rule gcd_proj1_if_dvd)
eberlm@62429
   432
qed
haftmann@60686
   433
  
eberlm@62429
   434
lemma gcd_proj2_iff: "gcd m n = normalize n \<longleftrightarrow> n dvd m"
eberlm@62429
   435
  using gcd_proj1_iff [of n m] by (simp add: ac_simps)
eberlm@62429
   436
eberlm@62429
   437
lemma gcd_mult_distrib': "normalize c * gcd a b = gcd (c * a) (c * b)"
eberlm@62429
   438
  by (rule gcdI) (auto simp: normalize_mult gcd_greatest mult_dvd_mono gcd_mult_left[symmetric])
eberlm@62429
   439
eberlm@62429
   440
lemma gcd_mult_distrib:
eberlm@62429
   441
  "k * gcd a b = gcd (k * a) (k * b) * unit_factor k"
eberlm@62429
   442
proof-
eberlm@62429
   443
  have "normalize k * gcd a b = gcd (k * a) (k * b)"
eberlm@62429
   444
    by (simp add: gcd_mult_distrib')
eberlm@62429
   445
  then have "normalize k * gcd a b * unit_factor k = gcd (k * a) (k * b) * unit_factor k"
eberlm@62429
   446
    by simp
eberlm@62429
   447
  then have "normalize k * unit_factor k * gcd a b  = gcd (k * a) (k * b) * unit_factor k"
eberlm@62429
   448
    by (simp only: ac_simps)
eberlm@62429
   449
  then show ?thesis
eberlm@62429
   450
    by simp
eberlm@62429
   451
qed
eberlm@62429
   452
eberlm@62429
   453
lemma lcm_mult_unit1:
eberlm@62429
   454
  "is_unit a \<Longrightarrow> lcm (b * a) c = lcm b c"
eberlm@62429
   455
  by (rule associated_eqI) (simp_all add: mult_unit_dvd_iff dvd_lcmI1)
eberlm@62429
   456
eberlm@62429
   457
lemma lcm_mult_unit2:
eberlm@62429
   458
  "is_unit a \<Longrightarrow> lcm b (c * a) = lcm b c"
eberlm@62429
   459
  using lcm_mult_unit1 [of a c b] by (simp add: ac_simps)
eberlm@62429
   460
eberlm@62429
   461
lemma lcm_div_unit1:
eberlm@62429
   462
  "is_unit a \<Longrightarrow> lcm (b div a) c = lcm b c"
eberlm@62429
   463
  by (erule is_unitE [of _ b]) (simp add: lcm_mult_unit1) 
eberlm@62429
   464
eberlm@62429
   465
lemma lcm_div_unit2:
eberlm@62429
   466
  "is_unit a \<Longrightarrow> lcm b (c div a) = lcm b c"
eberlm@62429
   467
  by (erule is_unitE [of _ c]) (simp add: lcm_mult_unit2)
eberlm@62429
   468
eberlm@62429
   469
lemma normalize_lcm_left [simp]:
eberlm@62429
   470
  "lcm (normalize a) b = lcm a b"
eberlm@62429
   471
proof (cases "a = 0")
eberlm@62429
   472
  case True then show ?thesis
eberlm@62429
   473
    by simp
eberlm@62429
   474
next
eberlm@62429
   475
  case False then have "is_unit (unit_factor a)"
eberlm@62429
   476
    by simp
eberlm@62429
   477
  moreover have "normalize a = a div unit_factor a"
eberlm@62429
   478
    by simp
eberlm@62429
   479
  ultimately show ?thesis
eberlm@62429
   480
    by (simp only: lcm_div_unit1)
eberlm@62429
   481
qed
eberlm@62429
   482
eberlm@62429
   483
lemma normalize_lcm_right [simp]:
eberlm@62429
   484
  "lcm a (normalize b) = lcm a b"
eberlm@62429
   485
  using normalize_lcm_left [of b a] by (simp add: ac_simps)
eberlm@62429
   486
eberlm@62429
   487
lemma gcd_mult_unit1: "is_unit a \<Longrightarrow> gcd (b * a) c = gcd b c"
eberlm@62429
   488
  apply (rule gcdI)
eberlm@62429
   489
  apply simp_all
eberlm@62429
   490
  apply (rule dvd_trans, rule gcd_dvd1, simp add: unit_simps)
eberlm@62429
   491
  done
eberlm@62429
   492
eberlm@62429
   493
lemma gcd_mult_unit2: "is_unit a \<Longrightarrow> gcd b (c * a) = gcd b c"
eberlm@62429
   494
  by (subst gcd.commute, subst gcd_mult_unit1, assumption, rule gcd.commute)
eberlm@62429
   495
eberlm@62429
   496
lemma gcd_div_unit1: "is_unit a \<Longrightarrow> gcd (b div a) c = gcd b c"
eberlm@62429
   497
  by (erule is_unitE [of _ b]) (simp add: gcd_mult_unit1)
eberlm@62429
   498
eberlm@62429
   499
lemma gcd_div_unit2: "is_unit a \<Longrightarrow> gcd b (c div a) = gcd b c"
eberlm@62429
   500
  by (erule is_unitE [of _ c]) (simp add: gcd_mult_unit2)
eberlm@62429
   501
eberlm@62429
   502
lemma normalize_gcd_left [simp]:
eberlm@62429
   503
  "gcd (normalize a) b = gcd a b"
eberlm@62429
   504
proof (cases "a = 0")
eberlm@62429
   505
  case True then show ?thesis
eberlm@62429
   506
    by simp
eberlm@62429
   507
next
eberlm@62429
   508
  case False then have "is_unit (unit_factor a)"
eberlm@62429
   509
    by simp
eberlm@62429
   510
  moreover have "normalize a = a div unit_factor a"
eberlm@62429
   511
    by simp
eberlm@62429
   512
  ultimately show ?thesis
eberlm@62429
   513
    by (simp only: gcd_div_unit1)
eberlm@62429
   514
qed
eberlm@62429
   515
eberlm@62429
   516
lemma normalize_gcd_right [simp]:
eberlm@62429
   517
  "gcd a (normalize b) = gcd a b"
eberlm@62429
   518
  using normalize_gcd_left [of b a] by (simp add: ac_simps)
eberlm@62429
   519
eberlm@62429
   520
lemma comp_fun_idem_gcd: "comp_fun_idem gcd"
eberlm@62429
   521
  by standard (simp_all add: fun_eq_iff ac_simps)
eberlm@62429
   522
eberlm@62429
   523
lemma comp_fun_idem_lcm: "comp_fun_idem lcm"
eberlm@62429
   524
  by standard (simp_all add: fun_eq_iff ac_simps)
eberlm@62429
   525
eberlm@62429
   526
lemma gcd_dvd_antisym:
eberlm@62429
   527
  "gcd a b dvd gcd c d \<Longrightarrow> gcd c d dvd gcd a b \<Longrightarrow> gcd a b = gcd c d"
eberlm@62429
   528
proof (rule gcdI)
eberlm@62429
   529
  assume A: "gcd a b dvd gcd c d" and B: "gcd c d dvd gcd a b"
eberlm@62429
   530
  have "gcd c d dvd c" by simp
eberlm@62429
   531
  with A show "gcd a b dvd c" by (rule dvd_trans)
eberlm@62429
   532
  have "gcd c d dvd d" by simp
eberlm@62429
   533
  with A show "gcd a b dvd d" by (rule dvd_trans)
eberlm@62429
   534
  show "normalize (gcd a b) = gcd a b"
eberlm@62429
   535
    by simp
eberlm@62429
   536
  fix l assume "l dvd c" and "l dvd d"
eberlm@62429
   537
  hence "l dvd gcd c d" by (rule gcd_greatest)
eberlm@62429
   538
  from this and B show "l dvd gcd a b" by (rule dvd_trans)
eberlm@62429
   539
qed
eberlm@62429
   540
eberlm@62429
   541
lemma coprime_dvd_mult:
eberlm@62429
   542
  assumes "coprime a b" and "a dvd c * b"
eberlm@62429
   543
  shows "a dvd c"
eberlm@62429
   544
proof (cases "c = 0")
eberlm@62429
   545
  case True then show ?thesis by simp
eberlm@62429
   546
next
eberlm@62429
   547
  case False
eberlm@62429
   548
  then have unit: "is_unit (unit_factor c)" by simp
eberlm@62429
   549
  from \<open>coprime a b\<close> mult_gcd_left [of c a b]
eberlm@62429
   550
  have "gcd (c * a) (c * b) * unit_factor c = c"
eberlm@62429
   551
    by (simp add: ac_simps)
eberlm@62429
   552
  moreover from \<open>a dvd c * b\<close> have "a dvd gcd (c * a) (c * b) * unit_factor c"
eberlm@62429
   553
    by (simp add: dvd_mult_unit_iff unit)
eberlm@62429
   554
  ultimately show ?thesis by simp
eberlm@62429
   555
qed
eberlm@62429
   556
eberlm@62429
   557
lemma coprime_dvd_mult_iff:
eberlm@62429
   558
  assumes "coprime a c"
eberlm@62429
   559
  shows "a dvd b * c \<longleftrightarrow> a dvd b"
eberlm@62429
   560
  using assms by (auto intro: coprime_dvd_mult)
eberlm@62429
   561
eberlm@62429
   562
lemma gcd_mult_cancel:
eberlm@62429
   563
  "coprime c b \<Longrightarrow> gcd (c * a) b = gcd a b"
eberlm@62429
   564
  apply (rule associated_eqI)
eberlm@62429
   565
  apply (rule gcd_greatest)
eberlm@62429
   566
  apply (rule_tac b = c in coprime_dvd_mult)
eberlm@62429
   567
  apply (simp add: gcd.assoc)
eberlm@62429
   568
  apply (simp_all add: ac_simps)
eberlm@62429
   569
  done
eberlm@62429
   570
eberlm@62429
   571
lemma coprime_crossproduct:
eberlm@62429
   572
  fixes a b c d
eberlm@62429
   573
  assumes "coprime a d" and "coprime b c"
eberlm@62429
   574
  shows "normalize a * normalize c = normalize b * normalize d
eberlm@62429
   575
    \<longleftrightarrow> normalize a = normalize b \<and> normalize c = normalize d" (is "?lhs \<longleftrightarrow> ?rhs")
eberlm@62429
   576
proof
eberlm@62429
   577
  assume ?rhs then show ?lhs by simp
eberlm@62429
   578
next
eberlm@62429
   579
  assume ?lhs
eberlm@62429
   580
  from \<open>?lhs\<close> have "normalize a dvd normalize b * normalize d"
eberlm@62429
   581
    by (auto intro: dvdI dest: sym)
eberlm@62429
   582
  with \<open>coprime a d\<close> have "a dvd b"
eberlm@62429
   583
    by (simp add: coprime_dvd_mult_iff normalize_mult [symmetric])
eberlm@62429
   584
  from \<open>?lhs\<close> have "normalize b dvd normalize a * normalize c"
eberlm@62429
   585
    by (auto intro: dvdI dest: sym)
eberlm@62429
   586
  with \<open>coprime b c\<close> have "b dvd a"
eberlm@62429
   587
    by (simp add: coprime_dvd_mult_iff normalize_mult [symmetric])
eberlm@62429
   588
  from \<open>?lhs\<close> have "normalize c dvd normalize d * normalize b"
eberlm@62429
   589
    by (auto intro: dvdI dest: sym simp add: mult.commute)
eberlm@62429
   590
  with \<open>coprime b c\<close> have "c dvd d"
eberlm@62429
   591
    by (simp add: coprime_dvd_mult_iff gcd.commute normalize_mult [symmetric])
eberlm@62429
   592
  from \<open>?lhs\<close> have "normalize d dvd normalize c * normalize a"
eberlm@62429
   593
    by (auto intro: dvdI dest: sym simp add: mult.commute)
eberlm@62429
   594
  with \<open>coprime a d\<close> have "d dvd c"
eberlm@62429
   595
    by (simp add: coprime_dvd_mult_iff gcd.commute normalize_mult [symmetric])
eberlm@62429
   596
  from \<open>a dvd b\<close> \<open>b dvd a\<close> have "normalize a = normalize b"
eberlm@62429
   597
    by (rule associatedI)
eberlm@62429
   598
  moreover from \<open>c dvd d\<close> \<open>d dvd c\<close> have "normalize c = normalize d"
eberlm@62429
   599
    by (rule associatedI)
eberlm@62429
   600
  ultimately show ?rhs ..
eberlm@62429
   601
qed
eberlm@62429
   602
eberlm@62429
   603
lemma gcd_add1 [simp]: "gcd (m + n) n = gcd m n"
eberlm@62429
   604
  by (rule gcdI [symmetric]) (simp_all add: dvd_add_left_iff)
eberlm@62429
   605
eberlm@62429
   606
lemma gcd_add2 [simp]: "gcd m (m + n) = gcd m n"
eberlm@62429
   607
  using gcd_add1 [of n m] by (simp add: ac_simps)
eberlm@62429
   608
eberlm@62429
   609
lemma gcd_add_mult: "gcd m (k * m + n) = gcd m n"
eberlm@62429
   610
  by (rule gcdI [symmetric]) (simp_all add: dvd_add_right_iff)
eberlm@62429
   611
eberlm@62429
   612
lemma coprimeI: "(\<And>l. \<lbrakk>l dvd a; l dvd b\<rbrakk> \<Longrightarrow> l dvd 1) \<Longrightarrow> gcd a b = 1"
eberlm@62429
   613
  by (rule sym, rule gcdI, simp_all)
eberlm@62429
   614
eberlm@62429
   615
lemma coprime: "gcd a b = 1 \<longleftrightarrow> (\<forall>d. d dvd a \<and> d dvd b \<longleftrightarrow> is_unit d)"
eberlm@62429
   616
  by (auto intro: coprimeI gcd_greatest dvd_gcdD1 dvd_gcdD2)
eberlm@62429
   617
eberlm@62429
   618
lemma div_gcd_coprime:
eberlm@62429
   619
  assumes nz: "a \<noteq> 0 \<or> b \<noteq> 0"
eberlm@62429
   620
  shows "coprime (a div gcd a b) (b div gcd a b)"
eberlm@62429
   621
proof -
eberlm@62429
   622
  let ?g = "gcd a b"
eberlm@62429
   623
  let ?a' = "a div ?g"
eberlm@62429
   624
  let ?b' = "b div ?g"
eberlm@62429
   625
  let ?g' = "gcd ?a' ?b'"
eberlm@62429
   626
  have dvdg: "?g dvd a" "?g dvd b" by simp_all
eberlm@62429
   627
  have dvdg': "?g' dvd ?a'" "?g' dvd ?b'" by simp_all
eberlm@62429
   628
  from dvdg dvdg' obtain ka kb ka' kb' where
eberlm@62429
   629
      kab: "a = ?g * ka" "b = ?g * kb" "?a' = ?g' * ka'" "?b' = ?g' * kb'"
eberlm@62429
   630
    unfolding dvd_def by blast
eberlm@62429
   631
  from this [symmetric] have "?g * ?a' = (?g * ?g') * ka'" "?g * ?b' = (?g * ?g') * kb'"
eberlm@62429
   632
    by (simp_all add: mult.assoc mult.left_commute [of "gcd a b"])
eberlm@62429
   633
  then have dvdgg':"?g * ?g' dvd a" "?g* ?g' dvd b"
eberlm@62429
   634
    by (auto simp add: dvd_mult_div_cancel [OF dvdg(1)]
eberlm@62429
   635
      dvd_mult_div_cancel [OF dvdg(2)] dvd_def)
eberlm@62429
   636
  have "?g \<noteq> 0" using nz by simp
eberlm@62429
   637
  moreover from gcd_greatest [OF dvdgg'] have "?g * ?g' dvd ?g" .
eberlm@62429
   638
  thm dvd_mult_cancel_left
eberlm@62429
   639
  ultimately show ?thesis using dvd_times_left_cancel_iff [of "gcd a b" _ 1] by simp
eberlm@62429
   640
qed
eberlm@62429
   641
eberlm@62429
   642
eberlm@62429
   643
lemma divides_mult:
eberlm@62429
   644
  assumes "a dvd c" and nr: "b dvd c" and "coprime a b"
eberlm@62429
   645
  shows "a * b dvd c"
eberlm@62429
   646
proof-
eberlm@62429
   647
  from \<open>b dvd c\<close> obtain b' where"c = b * b'" ..
eberlm@62429
   648
  with \<open>a dvd c\<close> have "a dvd b' * b"
eberlm@62429
   649
    by (simp add: ac_simps)
eberlm@62429
   650
  with \<open>coprime a b\<close> have "a dvd b'"
eberlm@62429
   651
    by (simp add: coprime_dvd_mult_iff)
eberlm@62429
   652
  then obtain a' where "b' = a * a'" ..
eberlm@62429
   653
  with \<open>c = b * b'\<close> have "c = (a * b) * a'"
eberlm@62429
   654
    by (simp add: ac_simps)
eberlm@62429
   655
  then show ?thesis ..
eberlm@62429
   656
qed
eberlm@62429
   657
eberlm@62429
   658
lemma coprime_lmult:
eberlm@62429
   659
  assumes dab: "gcd d (a * b) = 1" 
eberlm@62429
   660
  shows "gcd d a = 1"
eberlm@62429
   661
proof (rule coprimeI)
eberlm@62429
   662
  fix l assume "l dvd d" and "l dvd a"
eberlm@62429
   663
  hence "l dvd a * b" by simp
eberlm@62429
   664
  with \<open>l dvd d\<close> and dab show "l dvd 1" by (auto intro: gcd_greatest)
eberlm@62429
   665
qed
eberlm@62429
   666
eberlm@62429
   667
lemma coprime_rmult:
eberlm@62429
   668
  assumes dab: "gcd d (a * b) = 1"
eberlm@62429
   669
  shows "gcd d b = 1"
eberlm@62429
   670
proof (rule coprimeI)
eberlm@62429
   671
  fix l assume "l dvd d" and "l dvd b"
eberlm@62429
   672
  hence "l dvd a * b" by simp
eberlm@62429
   673
  with \<open>l dvd d\<close> and dab show "l dvd 1" by (auto intro: gcd_greatest)
eberlm@62429
   674
qed
eberlm@62429
   675
eberlm@62429
   676
lemma coprime_mult:
eberlm@62429
   677
  assumes da: "coprime d a" and db: "coprime d b"
eberlm@62429
   678
  shows "coprime d (a * b)"
eberlm@62429
   679
  apply (subst gcd.commute)
eberlm@62429
   680
  using da apply (subst gcd_mult_cancel)
eberlm@62429
   681
  apply (subst gcd.commute, assumption)
eberlm@62429
   682
  apply (subst gcd.commute, rule db)
eberlm@62429
   683
  done
eberlm@62429
   684
eberlm@62429
   685
lemma coprime_mul_eq: "gcd d (a * b) = 1 \<longleftrightarrow> gcd d a = 1 \<and> gcd d b = 1"
eberlm@62429
   686
  using coprime_rmult[of d a b] coprime_lmult[of d a b] coprime_mult[of d a b] by blast
eberlm@62429
   687
eberlm@62429
   688
lemma gcd_coprime:
eberlm@62429
   689
  assumes c: "gcd a b \<noteq> 0" and a: "a = a' * gcd a b" and b: "b = b' * gcd a b"
eberlm@62429
   690
  shows "gcd a' b' = 1"
eberlm@62429
   691
proof -
eberlm@62429
   692
  from c have "a \<noteq> 0 \<or> b \<noteq> 0" by simp
eberlm@62429
   693
  with div_gcd_coprime have "gcd (a div gcd a b) (b div gcd a b) = 1" .
eberlm@62429
   694
  also from assms have "a div gcd a b = a'" using dvd_div_eq_mult local.gcd_dvd1 by blast
eberlm@62429
   695
  also from assms have "b div gcd a b = b'" using dvd_div_eq_mult local.gcd_dvd1 by blast
eberlm@62429
   696
  finally show ?thesis .
eberlm@62429
   697
qed
eberlm@62429
   698
eberlm@62429
   699
lemma coprime_power:
eberlm@62429
   700
  assumes "0 < n"
eberlm@62429
   701
  shows "gcd a (b ^ n) = 1 \<longleftrightarrow> gcd a b = 1"
eberlm@62429
   702
using assms proof (induct n)
eberlm@62429
   703
  case (Suc n) then show ?case
eberlm@62429
   704
    by (cases n) (simp_all add: coprime_mul_eq)
eberlm@62429
   705
qed simp
eberlm@62429
   706
eberlm@62429
   707
lemma gcd_coprime_exists:
eberlm@62429
   708
  assumes nz: "gcd a b \<noteq> 0"
eberlm@62429
   709
  shows "\<exists>a' b'. a = a' * gcd a b \<and> b = b' * gcd a b \<and> gcd a' b' = 1"
eberlm@62429
   710
  apply (rule_tac x = "a div gcd a b" in exI)
eberlm@62429
   711
  apply (rule_tac x = "b div gcd a b" in exI)
eberlm@62429
   712
  apply (insert nz, auto intro: div_gcd_coprime)
eberlm@62429
   713
  done
eberlm@62429
   714
eberlm@62429
   715
lemma coprime_exp:
eberlm@62429
   716
  "gcd d a = 1 \<Longrightarrow> gcd d (a^n) = 1"
eberlm@62429
   717
  by (induct n, simp_all add: coprime_mult)
eberlm@62429
   718
eberlm@62429
   719
lemma coprime_exp_left:
eberlm@62429
   720
  assumes "coprime a b"
eberlm@62429
   721
  shows "coprime (a ^ n) b"
eberlm@62429
   722
  using assms by (induct n) (simp_all add: gcd_mult_cancel)
eberlm@62429
   723
eberlm@62429
   724
lemma coprime_exp2:
eberlm@62429
   725
  assumes "coprime a b"
eberlm@62429
   726
  shows "coprime (a ^ n) (b ^ m)"
eberlm@62429
   727
proof (rule coprime_exp_left)
eberlm@62429
   728
  from assms show "coprime a (b ^ m)"
eberlm@62429
   729
    by (induct m) (simp_all add: gcd_mult_cancel gcd.commute [of a])
eberlm@62429
   730
qed
eberlm@62429
   731
eberlm@62429
   732
lemma gcd_exp:
eberlm@62429
   733
  "gcd (a ^ n) (b ^ n) = gcd a b ^ n"
eberlm@62429
   734
proof (cases "a = 0 \<and> b = 0")
eberlm@62429
   735
  case True
eberlm@62429
   736
  then show ?thesis by (cases n) simp_all
eberlm@62429
   737
next
eberlm@62429
   738
  case False
eberlm@62429
   739
  then have "1 = gcd ((a div gcd a b) ^ n) ((b div gcd a b) ^ n)"
eberlm@62429
   740
    using coprime_exp2[OF div_gcd_coprime[of a b], of n n, symmetric] by simp
eberlm@62429
   741
  then have "gcd a b ^ n = gcd a b ^ n * ..." by simp
eberlm@62429
   742
  also note gcd_mult_distrib
eberlm@62429
   743
  also have "unit_factor (gcd a b ^ n) = 1"
eberlm@62429
   744
    using False by (auto simp add: unit_factor_power unit_factor_gcd)
eberlm@62429
   745
  also have "(gcd a b)^n * (a div gcd a b)^n = a^n"
eberlm@62429
   746
    by (subst ac_simps, subst div_power, simp, rule dvd_div_mult_self, rule dvd_power_same, simp)
eberlm@62429
   747
  also have "(gcd a b)^n * (b div gcd a b)^n = b^n"
eberlm@62429
   748
    by (subst ac_simps, subst div_power, simp, rule dvd_div_mult_self, rule dvd_power_same, simp)
eberlm@62429
   749
  finally show ?thesis by simp
eberlm@62429
   750
qed
eberlm@62429
   751
eberlm@62429
   752
lemma coprime_common_divisor: 
eberlm@62429
   753
  "gcd a b = 1 \<Longrightarrow> a dvd a \<Longrightarrow> a dvd b \<Longrightarrow> is_unit a"
eberlm@62429
   754
  apply (subgoal_tac "a dvd gcd a b")
eberlm@62429
   755
  apply simp
eberlm@62429
   756
  apply (erule (1) gcd_greatest)
eberlm@62429
   757
  done
eberlm@62429
   758
eberlm@62429
   759
lemma division_decomp: 
eberlm@62429
   760
  assumes dc: "a dvd b * c"
eberlm@62429
   761
  shows "\<exists>b' c'. a = b' * c' \<and> b' dvd b \<and> c' dvd c"
eberlm@62429
   762
proof (cases "gcd a b = 0")
eberlm@62429
   763
  assume "gcd a b = 0"
eberlm@62429
   764
  hence "a = 0 \<and> b = 0" by simp
eberlm@62429
   765
  hence "a = 0 * c \<and> 0 dvd b \<and> c dvd c" by simp
eberlm@62429
   766
  then show ?thesis by blast
eberlm@62429
   767
next
eberlm@62429
   768
  let ?d = "gcd a b"
eberlm@62429
   769
  assume "?d \<noteq> 0"
eberlm@62429
   770
  from gcd_coprime_exists[OF this]
eberlm@62429
   771
    obtain a' b' where ab': "a = a' * ?d" "b = b' * ?d" "gcd a' b' = 1"
eberlm@62429
   772
    by blast
eberlm@62429
   773
  from ab'(1) have "a' dvd a" unfolding dvd_def by blast
eberlm@62429
   774
  with dc have "a' dvd b*c" using dvd_trans[of a' a "b*c"] by simp
eberlm@62429
   775
  from dc ab'(1,2) have "a'*?d dvd (b'*?d) * c" by simp
eberlm@62429
   776
  hence "?d * a' dvd ?d * (b' * c)" by (simp add: mult_ac)
eberlm@62429
   777
  with \<open>?d \<noteq> 0\<close> have "a' dvd b' * c" by simp
eberlm@62429
   778
  with coprime_dvd_mult[OF ab'(3)] 
eberlm@62429
   779
    have "a' dvd c" by (subst (asm) ac_simps, blast)
eberlm@62429
   780
  with ab'(1) have "a = ?d * a' \<and> ?d dvd b \<and> a' dvd c" by (simp add: mult_ac)
eberlm@62429
   781
  then show ?thesis by blast
eberlm@62429
   782
qed
eberlm@62429
   783
eberlm@62429
   784
lemma pow_divs_pow:
eberlm@62429
   785
  assumes ab: "a ^ n dvd b ^ n" and n: "n \<noteq> 0"
eberlm@62429
   786
  shows "a dvd b"
eberlm@62429
   787
proof (cases "gcd a b = 0")
eberlm@62429
   788
  assume "gcd a b = 0"
eberlm@62429
   789
  then show ?thesis by simp
eberlm@62429
   790
next
eberlm@62429
   791
  let ?d = "gcd a b"
eberlm@62429
   792
  assume "?d \<noteq> 0"
eberlm@62429
   793
  from n obtain m where m: "n = Suc m" by (cases n, simp_all)
eberlm@62429
   794
  from \<open>?d \<noteq> 0\<close> have zn: "?d ^ n \<noteq> 0" by (rule power_not_zero)
eberlm@62429
   795
  from gcd_coprime_exists[OF \<open>?d \<noteq> 0\<close>]
eberlm@62429
   796
    obtain a' b' where ab': "a = a' * ?d" "b = b' * ?d" "gcd a' b' = 1"
eberlm@62429
   797
    by blast
eberlm@62429
   798
  from ab have "(a' * ?d) ^ n dvd (b' * ?d) ^ n"
eberlm@62429
   799
    by (simp add: ab'(1,2)[symmetric])
eberlm@62429
   800
  hence "?d^n * a'^n dvd ?d^n * b'^n"
eberlm@62429
   801
    by (simp only: power_mult_distrib ac_simps)
eberlm@62429
   802
  with zn have "a'^n dvd b'^n" by simp
eberlm@62429
   803
  hence "a' dvd b'^n" using dvd_trans[of a' "a'^n" "b'^n"] by (simp add: m)
eberlm@62429
   804
  hence "a' dvd b'^m * b'" by (simp add: m ac_simps)
eberlm@62429
   805
  with coprime_dvd_mult[OF coprime_exp[OF ab'(3), of m]]
eberlm@62429
   806
    have "a' dvd b'" by (subst (asm) ac_simps, blast)
eberlm@62429
   807
  hence "a'*?d dvd b'*?d" by (rule mult_dvd_mono, simp)
eberlm@62429
   808
  with ab'(1,2) show ?thesis by simp
eberlm@62429
   809
qed
eberlm@62429
   810
eberlm@62429
   811
lemma pow_divs_eq [simp]:
eberlm@62429
   812
  "n \<noteq> 0 \<Longrightarrow> a ^ n dvd b ^ n \<longleftrightarrow> a dvd b"
eberlm@62429
   813
  by (auto intro: pow_divs_pow dvd_power_same)
eberlm@62429
   814
eberlm@62429
   815
lemma coprime_plus_one [simp]: "gcd (n + 1) n = 1"
eberlm@62429
   816
  by (subst add_commute, simp)
eberlm@62429
   817
eberlm@62429
   818
lemma setprod_coprime [rule_format]:
eberlm@62429
   819
  "(\<forall>i\<in>A. gcd (f i) a = 1) \<longrightarrow> gcd (\<Prod>i\<in>A. f i) a = 1"
eberlm@62429
   820
  apply (cases "finite A")
eberlm@62429
   821
  apply (induct set: finite)
eberlm@62429
   822
  apply (auto simp add: gcd_mult_cancel)
eberlm@62429
   823
  done
eberlm@62429
   824
  
eberlm@62429
   825
lemma listprod_coprime:
eberlm@62429
   826
  "(\<And>x. x \<in> set xs \<Longrightarrow> coprime x y) \<Longrightarrow> coprime (listprod xs) y" 
eberlm@62429
   827
  by (induction xs) (simp_all add: gcd_mult_cancel)
eberlm@62429
   828
eberlm@62429
   829
lemma coprime_divisors: 
eberlm@62429
   830
  assumes "d dvd a" "e dvd b" "gcd a b = 1"
eberlm@62429
   831
  shows "gcd d e = 1" 
eberlm@62429
   832
proof -
eberlm@62429
   833
  from assms obtain k l where "a = d * k" "b = e * l"
eberlm@62429
   834
    unfolding dvd_def by blast
eberlm@62429
   835
  with assms have "gcd (d * k) (e * l) = 1" by simp
eberlm@62429
   836
  hence "gcd (d * k) e = 1" by (rule coprime_lmult)
eberlm@62429
   837
  also have "gcd (d * k) e = gcd e (d * k)" by (simp add: ac_simps)
eberlm@62429
   838
  finally have "gcd e d = 1" by (rule coprime_lmult)
eberlm@62429
   839
  then show ?thesis by (simp add: ac_simps)
eberlm@62429
   840
qed
eberlm@62429
   841
eberlm@62429
   842
lemma lcm_gcd_prod:
eberlm@62429
   843
  "lcm a b * gcd a b = normalize (a * b)"
eberlm@62429
   844
  by (simp add: lcm_gcd)
eberlm@62429
   845
eberlm@62429
   846
declare unit_factor_lcm [simp]
eberlm@62429
   847
eberlm@62429
   848
lemma lcmI:
eberlm@62429
   849
  assumes "a dvd c" and "b dvd c" and "\<And>d. a dvd d \<Longrightarrow> b dvd d \<Longrightarrow> c dvd d"
eberlm@62429
   850
    and "normalize c = c"
eberlm@62429
   851
  shows "c = lcm a b"
eberlm@62429
   852
  by (rule associated_eqI) (auto simp: assms intro: lcm_least)
eberlm@62429
   853
eberlm@62429
   854
lemma gcd_dvd_lcm [simp]:
eberlm@62429
   855
  "gcd a b dvd lcm a b"
eberlm@62429
   856
  using gcd_dvd2 by (rule dvd_lcmI2)
eberlm@62429
   857
eberlm@62429
   858
lemmas lcm_0 = lcm_0_right
eberlm@62429
   859
eberlm@62429
   860
lemma lcm_unique:
eberlm@62429
   861
  "a dvd d \<and> b dvd d \<and> 
eberlm@62429
   862
  normalize d = d \<and>
eberlm@62429
   863
  (\<forall>e. a dvd e \<and> b dvd e \<longrightarrow> d dvd e) \<longleftrightarrow> d = lcm a b"
eberlm@62429
   864
  by rule (auto intro: lcmI simp: lcm_least lcm_eq_0_iff)
eberlm@62429
   865
eberlm@62429
   866
lemma lcm_coprime:
eberlm@62429
   867
  "gcd a b = 1 \<Longrightarrow> lcm a b = normalize (a * b)"
eberlm@62429
   868
  by (subst lcm_gcd) simp
eberlm@62429
   869
eberlm@62429
   870
lemma lcm_proj1_if_dvd: 
eberlm@62429
   871
  "b dvd a \<Longrightarrow> lcm a b = normalize a"
eberlm@62429
   872
  by (cases "a = 0") (simp, rule sym, rule lcmI, simp_all)
eberlm@62429
   873
eberlm@62429
   874
lemma lcm_proj2_if_dvd: 
eberlm@62429
   875
  "a dvd b \<Longrightarrow> lcm a b = normalize b"
eberlm@62429
   876
  using lcm_proj1_if_dvd [of a b] by (simp add: ac_simps)
eberlm@62429
   877
eberlm@62429
   878
lemma lcm_proj1_iff:
eberlm@62429
   879
  "lcm m n = normalize m \<longleftrightarrow> n dvd m"
eberlm@62429
   880
proof
eberlm@62429
   881
  assume A: "lcm m n = normalize m"
eberlm@62429
   882
  show "n dvd m"
eberlm@62429
   883
  proof (cases "m = 0")
eberlm@62429
   884
    assume [simp]: "m \<noteq> 0"
eberlm@62429
   885
    from A have B: "m = lcm m n * unit_factor m"
eberlm@62429
   886
      by (simp add: unit_eq_div2)
eberlm@62429
   887
    show ?thesis by (subst B, simp)
eberlm@62429
   888
  qed simp
eberlm@62429
   889
next
eberlm@62429
   890
  assume "n dvd m"
eberlm@62429
   891
  then show "lcm m n = normalize m" by (rule lcm_proj1_if_dvd)
eberlm@62429
   892
qed
eberlm@62429
   893
eberlm@62429
   894
lemma lcm_proj2_iff:
eberlm@62429
   895
  "lcm m n = normalize n \<longleftrightarrow> m dvd n"
eberlm@62429
   896
  using lcm_proj1_iff [of n m] by (simp add: ac_simps)
eberlm@62429
   897
haftmann@60686
   898
end
haftmann@60686
   899
haftmann@62345
   900
class ring_gcd = comm_ring_1 + semiring_gcd
eberlm@62429
   901
begin
eberlm@62429
   902
eberlm@62429
   903
lemma coprime_minus_one: "coprime (n - 1) n"
eberlm@62429
   904
  using coprime_plus_one[of "n - 1"] by (simp add: gcd.commute)
eberlm@62429
   905
eberlm@62429
   906
lemma gcd_neg1 [simp]:
eberlm@62429
   907
  "gcd (-a) b = gcd a b"
eberlm@62429
   908
  by (rule sym, rule gcdI, simp_all add: gcd_greatest)
eberlm@62429
   909
eberlm@62429
   910
lemma gcd_neg2 [simp]:
eberlm@62429
   911
  "gcd a (-b) = gcd a b"
eberlm@62429
   912
  by (rule sym, rule gcdI, simp_all add: gcd_greatest)
eberlm@62429
   913
eberlm@62429
   914
lemma gcd_neg_numeral_1 [simp]:
eberlm@62429
   915
  "gcd (- numeral n) a = gcd (numeral n) a"
eberlm@62429
   916
  by (fact gcd_neg1)
eberlm@62429
   917
eberlm@62429
   918
lemma gcd_neg_numeral_2 [simp]:
eberlm@62429
   919
  "gcd a (- numeral n) = gcd a (numeral n)"
eberlm@62429
   920
  by (fact gcd_neg2)
eberlm@62429
   921
eberlm@62429
   922
lemma gcd_diff1: "gcd (m - n) n = gcd m n"
eberlm@62429
   923
  by (subst diff_conv_add_uminus, subst gcd_neg2[symmetric],  subst gcd_add1, simp)
eberlm@62429
   924
eberlm@62429
   925
lemma gcd_diff2: "gcd (n - m) n = gcd m n"
eberlm@62429
   926
  by (subst gcd_neg1[symmetric], simp only: minus_diff_eq gcd_diff1)
eberlm@62429
   927
eberlm@62429
   928
lemma lcm_neg1 [simp]: "lcm (-a) b = lcm a b"
eberlm@62429
   929
  by (rule sym, rule lcmI, simp_all add: lcm_least lcm_eq_0_iff)
eberlm@62429
   930
eberlm@62429
   931
lemma lcm_neg2 [simp]: "lcm a (-b) = lcm a b"
eberlm@62429
   932
  by (rule sym, rule lcmI, simp_all add: lcm_least lcm_eq_0_iff)
eberlm@62429
   933
eberlm@62429
   934
lemma lcm_neg_numeral_1 [simp]: "lcm (- numeral n) a = lcm (numeral n) a"
eberlm@62429
   935
  by (fact lcm_neg1)
eberlm@62429
   936
eberlm@62429
   937
lemma lcm_neg_numeral_2 [simp]: "lcm a (- numeral n) = lcm a (numeral n)"
eberlm@62429
   938
  by (fact lcm_neg2)
eberlm@62429
   939
eberlm@62429
   940
end
haftmann@62345
   941
haftmann@60686
   942
class semiring_Gcd = semiring_gcd + Gcd +
haftmann@60686
   943
  assumes Gcd_dvd: "a \<in> A \<Longrightarrow> Gcd A dvd a"
haftmann@60686
   944
    and Gcd_greatest: "(\<And>b. b \<in> A \<Longrightarrow> a dvd b) \<Longrightarrow> a dvd Gcd A"
haftmann@60686
   945
    and normalize_Gcd [simp]: "normalize (Gcd A) = Gcd A"
haftmann@62345
   946
  assumes dvd_Lcm: "a \<in> A \<Longrightarrow> a dvd Lcm A"
haftmann@62345
   947
    and Lcm_least: "(\<And>b. b \<in> A \<Longrightarrow> b dvd a) \<Longrightarrow> Lcm A dvd a"
haftmann@62345
   948
    and normalize_Lcm [simp]: "normalize (Lcm A) = Lcm A"
haftmann@60686
   949
begin
haftmann@60686
   950
haftmann@62345
   951
lemma Lcm_Gcd:
haftmann@62345
   952
  "Lcm A = Gcd {b. \<forall>a\<in>A. a dvd b}"
haftmann@62345
   953
  by (rule associated_eqI) (auto intro: Gcd_dvd dvd_Lcm Gcd_greatest Lcm_least)
haftmann@62345
   954
haftmann@62345
   955
lemma Gcd_Lcm:
haftmann@62345
   956
  "Gcd A = Lcm {b. \<forall>a\<in>A. b dvd a}"
haftmann@62345
   957
  by (rule associated_eqI) (auto intro: Gcd_dvd dvd_Lcm Gcd_greatest Lcm_least)
haftmann@62345
   958
haftmann@60686
   959
lemma Gcd_empty [simp]:
haftmann@60686
   960
  "Gcd {} = 0"
haftmann@60686
   961
  by (rule dvd_0_left, rule Gcd_greatest) simp
haftmann@60686
   962
haftmann@62345
   963
lemma Lcm_empty [simp]:
haftmann@62345
   964
  "Lcm {} = 1"
haftmann@62345
   965
  by (auto intro: associated_eqI Lcm_least)
haftmann@62345
   966
haftmann@62345
   967
lemma Gcd_insert [simp]:
haftmann@62345
   968
  "Gcd (insert a A) = gcd a (Gcd A)"
haftmann@62345
   969
proof -
haftmann@62345
   970
  have "Gcd (insert a A) dvd gcd a (Gcd A)"
haftmann@62345
   971
    by (auto intro: Gcd_dvd Gcd_greatest)
haftmann@62345
   972
  moreover have "gcd a (Gcd A) dvd Gcd (insert a A)"
haftmann@62345
   973
  proof (rule Gcd_greatest)
haftmann@62345
   974
    fix b
haftmann@62345
   975
    assume "b \<in> insert a A"
haftmann@62345
   976
    then show "gcd a (Gcd A) dvd b"
haftmann@62345
   977
    proof
haftmann@62345
   978
      assume "b = a" then show ?thesis by simp
haftmann@62345
   979
    next
haftmann@62345
   980
      assume "b \<in> A"
haftmann@62345
   981
      then have "Gcd A dvd b" by (rule Gcd_dvd)
haftmann@62345
   982
      moreover have "gcd a (Gcd A) dvd Gcd A" by simp
haftmann@62345
   983
      ultimately show ?thesis by (blast intro: dvd_trans)
haftmann@62345
   984
    qed
haftmann@62345
   985
  qed
haftmann@62345
   986
  ultimately show ?thesis
haftmann@62345
   987
    by (auto intro: associated_eqI)
haftmann@62345
   988
qed
haftmann@62345
   989
haftmann@62345
   990
lemma Lcm_insert [simp]:
haftmann@62345
   991
  "Lcm (insert a A) = lcm a (Lcm A)"
haftmann@62345
   992
proof (rule sym)
haftmann@62345
   993
  have "lcm a (Lcm A) dvd Lcm (insert a A)"
haftmann@62345
   994
    by (auto intro: dvd_Lcm Lcm_least)
haftmann@62345
   995
  moreover have "Lcm (insert a A) dvd lcm a (Lcm A)"
haftmann@62345
   996
  proof (rule Lcm_least)
haftmann@62345
   997
    fix b
haftmann@62345
   998
    assume "b \<in> insert a A"
haftmann@62345
   999
    then show "b dvd lcm a (Lcm A)"
haftmann@62345
  1000
    proof
haftmann@62345
  1001
      assume "b = a" then show ?thesis by simp
haftmann@62345
  1002
    next
haftmann@62345
  1003
      assume "b \<in> A"
haftmann@62345
  1004
      then have "b dvd Lcm A" by (rule dvd_Lcm)
haftmann@62345
  1005
      moreover have "Lcm A dvd lcm a (Lcm A)" by simp
haftmann@62345
  1006
      ultimately show ?thesis by (blast intro: dvd_trans)
haftmann@62345
  1007
    qed
haftmann@62345
  1008
  qed
haftmann@62345
  1009
  ultimately show "lcm a (Lcm A) = Lcm (insert a A)"
haftmann@62345
  1010
    by (rule associated_eqI) (simp_all add: lcm_eq_0_iff)
haftmann@62345
  1011
qed
haftmann@62345
  1012
eberlm@62429
  1013
lemma LcmI:
eberlm@62429
  1014
  assumes "\<And>a. a \<in> A \<Longrightarrow> a dvd b" and "\<And>c. (\<And>a. a \<in> A \<Longrightarrow> a dvd c) \<Longrightarrow> b dvd c"
eberlm@62429
  1015
    and "normalize b = b" shows "b = Lcm A"
eberlm@62429
  1016
  by (rule associated_eqI) (auto simp: assms dvd_Lcm intro: Lcm_least)
eberlm@62429
  1017
eberlm@62429
  1018
lemma Lcm_subset:
eberlm@62429
  1019
  "A \<subseteq> B \<Longrightarrow> Lcm A dvd Lcm B"
eberlm@62429
  1020
  by (blast intro: Lcm_least dvd_Lcm)
eberlm@62429
  1021
eberlm@62429
  1022
lemma Lcm_Un:
eberlm@62429
  1023
  "Lcm (A \<union> B) = lcm (Lcm A) (Lcm B)"
eberlm@62429
  1024
  apply (rule lcmI)
eberlm@62429
  1025
  apply (blast intro: Lcm_subset)
eberlm@62429
  1026
  apply (blast intro: Lcm_subset)
eberlm@62429
  1027
  apply (intro Lcm_least ballI, elim UnE)
eberlm@62429
  1028
  apply (rule dvd_trans, erule dvd_Lcm, assumption)
eberlm@62429
  1029
  apply (rule dvd_trans, erule dvd_Lcm, assumption)
eberlm@62429
  1030
  apply simp
eberlm@62429
  1031
  done
eberlm@62429
  1032
  
eberlm@62429
  1033
haftmann@60686
  1034
lemma Gcd_0_iff [simp]:
haftmann@60687
  1035
  "Gcd A = 0 \<longleftrightarrow> A \<subseteq> {0}" (is "?P \<longleftrightarrow> ?Q")
haftmann@60686
  1036
proof
haftmann@60686
  1037
  assume ?P
haftmann@60686
  1038
  show ?Q
haftmann@60686
  1039
  proof
haftmann@60686
  1040
    fix a
haftmann@60686
  1041
    assume "a \<in> A"
haftmann@60686
  1042
    then have "Gcd A dvd a" by (rule Gcd_dvd)
haftmann@60687
  1043
    with \<open>?P\<close> have "a = 0" by simp
haftmann@60687
  1044
    then show "a \<in> {0}" by simp
haftmann@60686
  1045
  qed
haftmann@60686
  1046
next
haftmann@60686
  1047
  assume ?Q
haftmann@60686
  1048
  have "0 dvd Gcd A"
haftmann@60686
  1049
  proof (rule Gcd_greatest)
haftmann@60686
  1050
    fix a
haftmann@60686
  1051
    assume "a \<in> A"
haftmann@60687
  1052
    with \<open>?Q\<close> have "a = 0" by auto
haftmann@60686
  1053
    then show "0 dvd a" by simp
haftmann@60686
  1054
  qed
haftmann@60686
  1055
  then show ?P by simp
haftmann@60686
  1056
qed
haftmann@60686
  1057
haftmann@60686
  1058
lemma Lcm_1_iff [simp]:
haftmann@60686
  1059
  "Lcm A = 1 \<longleftrightarrow> (\<forall>a\<in>A. is_unit a)" (is "?P \<longleftrightarrow> ?Q")
haftmann@60686
  1060
proof
haftmann@60686
  1061
  assume ?P
haftmann@60686
  1062
  show ?Q
haftmann@60686
  1063
  proof
haftmann@60686
  1064
    fix a
haftmann@60686
  1065
    assume "a \<in> A"
haftmann@60686
  1066
    then have "a dvd Lcm A"
haftmann@60686
  1067
      by (rule dvd_Lcm)
haftmann@60686
  1068
    with \<open>?P\<close> show "is_unit a"
haftmann@60686
  1069
      by simp
haftmann@60686
  1070
  qed
haftmann@60686
  1071
next
haftmann@60686
  1072
  assume ?Q
haftmann@60686
  1073
  then have "is_unit (Lcm A)"
haftmann@60686
  1074
    by (blast intro: Lcm_least)
haftmann@60686
  1075
  then have "normalize (Lcm A) = 1"
haftmann@60686
  1076
    by (rule is_unit_normalize)
haftmann@60686
  1077
  then show ?P
haftmann@60686
  1078
    by simp
haftmann@60686
  1079
qed
haftmann@60686
  1080
haftmann@62345
  1081
lemma unit_factor_Lcm:
haftmann@62345
  1082
  "unit_factor (Lcm A) = (if Lcm A = 0 then 0 else 1)"
haftmann@62345
  1083
proof (cases "Lcm A = 0")
haftmann@62345
  1084
  case True then show ?thesis by simp
haftmann@62345
  1085
next
haftmann@62345
  1086
  case False
haftmann@62345
  1087
  with unit_factor_normalize have "unit_factor (normalize (Lcm A)) = 1"
haftmann@62345
  1088
    by blast
haftmann@62345
  1089
  with False show ?thesis
haftmann@62345
  1090
    by simp
haftmann@62345
  1091
qed
haftmann@62345
  1092
eberlm@62429
  1093
lemma unit_factor_Gcd: "unit_factor (Gcd A) = (if Gcd A = 0 then 0 else 1)"
eberlm@62429
  1094
proof -
eberlm@62429
  1095
  show "unit_factor (Gcd A) = (if Gcd A = 0 then 0 else 1)"
eberlm@62429
  1096
    by (simp add: Gcd_Lcm unit_factor_Lcm)
eberlm@62429
  1097
qed
eberlm@62429
  1098
eberlm@62429
  1099
lemma GcdI:
eberlm@62429
  1100
  assumes "\<And>a. a \<in> A \<Longrightarrow> b dvd a" and "\<And>c. (\<And>a. a \<in> A \<Longrightarrow> c dvd a) \<Longrightarrow> c dvd b"
eberlm@62429
  1101
    and "normalize b = b"
eberlm@62429
  1102
  shows "b = Gcd A"
eberlm@62429
  1103
  by (rule associated_eqI) (auto simp: assms Gcd_dvd intro: Gcd_greatest)
eberlm@62429
  1104
haftmann@62345
  1105
lemma Gcd_eq_1_I:
haftmann@62345
  1106
  assumes "is_unit a" and "a \<in> A"
haftmann@62345
  1107
  shows "Gcd A = 1"
haftmann@62345
  1108
proof -
haftmann@62345
  1109
  from assms have "is_unit (Gcd A)"
haftmann@62345
  1110
    by (blast intro: Gcd_dvd dvd_unit_imp_unit)
haftmann@62345
  1111
  then have "normalize (Gcd A) = 1"
haftmann@62345
  1112
    by (rule is_unit_normalize)
haftmann@62345
  1113
  then show ?thesis
haftmann@62345
  1114
    by simp
haftmann@62345
  1115
qed
haftmann@62345
  1116
haftmann@60686
  1117
lemma Lcm_eq_0_I:
haftmann@60686
  1118
  assumes "0 \<in> A"
haftmann@60686
  1119
  shows "Lcm A = 0"
haftmann@60686
  1120
proof -
haftmann@60686
  1121
  from assms have "0 dvd Lcm A"
haftmann@60686
  1122
    by (rule dvd_Lcm)
haftmann@60686
  1123
  then show ?thesis
haftmann@60686
  1124
    by simp
haftmann@60686
  1125
qed
haftmann@60686
  1126
haftmann@62345
  1127
lemma Gcd_UNIV [simp]:
haftmann@62345
  1128
  "Gcd UNIV = 1"
haftmann@62345
  1129
  using dvd_refl by (rule Gcd_eq_1_I) simp
haftmann@62345
  1130
haftmann@61929
  1131
lemma Lcm_UNIV [simp]:
haftmann@61929
  1132
  "Lcm UNIV = 0"
haftmann@61929
  1133
  by (rule Lcm_eq_0_I) simp
haftmann@60686
  1134
haftmann@61929
  1135
lemma Lcm_0_iff:
haftmann@61929
  1136
  assumes "finite A"
haftmann@61929
  1137
  shows "Lcm A = 0 \<longleftrightarrow> 0 \<in> A"
haftmann@61929
  1138
proof (cases "A = {}")
haftmann@61929
  1139
  case True then show ?thesis by simp
haftmann@61929
  1140
next
haftmann@61929
  1141
  case False with assms show ?thesis
haftmann@61929
  1142
    by (induct A rule: finite_ne_induct)
haftmann@61929
  1143
      (auto simp add: lcm_eq_0_iff)
haftmann@60686
  1144
qed
haftmann@61929
  1145
eberlm@62429
  1146
lemma Gcd_finite:
eberlm@62429
  1147
  assumes "finite A"
eberlm@62429
  1148
  shows "Gcd A = Finite_Set.fold gcd 0 A"
eberlm@62429
  1149
  by (induct rule: finite.induct[OF \<open>finite A\<close>])
eberlm@62429
  1150
     (simp_all add: comp_fun_idem.fold_insert_idem[OF comp_fun_idem_gcd])
eberlm@62429
  1151
eberlm@62429
  1152
lemma Gcd_set [code_unfold]: "Gcd (set as) = foldl gcd 0 as"
eberlm@62429
  1153
  by (simp add: Gcd_finite comp_fun_idem.fold_set_fold[OF comp_fun_idem_gcd] 
eberlm@62429
  1154
                foldl_conv_fold gcd.commute)
eberlm@62429
  1155
eberlm@62429
  1156
lemma Lcm_finite:
eberlm@62429
  1157
  assumes "finite A"
eberlm@62429
  1158
  shows "Lcm A = Finite_Set.fold lcm 1 A"
eberlm@62429
  1159
  by (induct rule: finite.induct[OF \<open>finite A\<close>])
eberlm@62429
  1160
     (simp_all add: comp_fun_idem.fold_insert_idem[OF comp_fun_idem_lcm])
haftmann@62345
  1161
haftmann@60686
  1162
lemma Lcm_set [code_unfold]:
eberlm@62429
  1163
  "Lcm (set as) = foldl lcm 1 as"
eberlm@62429
  1164
  by (simp add: Lcm_finite comp_fun_idem.fold_set_fold[OF comp_fun_idem_lcm] 
eberlm@62429
  1165
                foldl_conv_fold lcm.commute)
haftmann@59008
  1166
haftmann@62345
  1167
lemma Gcd_image_normalize [simp]:
haftmann@62345
  1168
  "Gcd (normalize ` A) = Gcd A"
haftmann@62345
  1169
proof -
haftmann@62345
  1170
  have "Gcd (normalize ` A) dvd a" if "a \<in> A" for a
haftmann@62345
  1171
  proof -
haftmann@62345
  1172
    from that obtain B where "A = insert a B" by blast
haftmann@62350
  1173
    moreover have "gcd (normalize a) (Gcd (normalize ` B)) dvd normalize a"
haftmann@62345
  1174
      by (rule gcd_dvd1)
haftmann@62345
  1175
    ultimately show "Gcd (normalize ` A) dvd a"
haftmann@62345
  1176
      by simp
haftmann@62345
  1177
  qed
haftmann@62345
  1178
  then have "Gcd (normalize ` A) dvd Gcd A" and "Gcd A dvd Gcd (normalize ` A)"
haftmann@62345
  1179
    by (auto intro!: Gcd_greatest intro: Gcd_dvd)
haftmann@62345
  1180
  then show ?thesis
haftmann@62345
  1181
    by (auto intro: associated_eqI)
haftmann@62345
  1182
qed
haftmann@62345
  1183
haftmann@62346
  1184
lemma Gcd_eqI:
haftmann@62346
  1185
  assumes "normalize a = a"
haftmann@62346
  1186
  assumes "\<And>b. b \<in> A \<Longrightarrow> a dvd b"
haftmann@62346
  1187
    and "\<And>c. (\<And>b. b \<in> A \<Longrightarrow> c dvd b) \<Longrightarrow> c dvd a"
haftmann@62346
  1188
  shows "Gcd A = a"
haftmann@62346
  1189
  using assms by (blast intro: associated_eqI Gcd_greatest Gcd_dvd normalize_Gcd)
haftmann@62346
  1190
haftmann@62346
  1191
lemma Lcm_eqI:
haftmann@62346
  1192
  assumes "normalize a = a"
haftmann@62346
  1193
  assumes "\<And>b. b \<in> A \<Longrightarrow> b dvd a"
haftmann@62346
  1194
    and "\<And>c. (\<And>b. b \<in> A \<Longrightarrow> b dvd c) \<Longrightarrow> a dvd c"
haftmann@62346
  1195
  shows "Lcm A = a"
haftmann@62346
  1196
  using assms by (blast intro: associated_eqI Lcm_least dvd_Lcm normalize_Lcm)
haftmann@62346
  1197
eberlm@62429
  1198
eberlm@62429
  1199
lemma Lcm_no_units:
eberlm@62429
  1200
  "Lcm A = Lcm (A - {a. is_unit a})"
eberlm@62429
  1201
proof -
eberlm@62429
  1202
  have "(A - {a. is_unit a}) \<union> {a\<in>A. is_unit a} = A" by blast
eberlm@62429
  1203
  hence "Lcm A = lcm (Lcm (A - {a. is_unit a})) (Lcm {a\<in>A. is_unit a})"
eberlm@62429
  1204
    by (simp add: Lcm_Un [symmetric])
eberlm@62429
  1205
  also have "Lcm {a\<in>A. is_unit a} = 1" by (simp add: Lcm_1_iff)
eberlm@62429
  1206
  finally show ?thesis by simp
eberlm@62429
  1207
qed
eberlm@62429
  1208
eberlm@62429
  1209
lemma Lcm_0_iff': "Lcm A = 0 \<longleftrightarrow> \<not>(\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l))"
eberlm@62429
  1210
  by (metis Lcm_least dvd_0_left dvd_Lcm)
eberlm@62429
  1211
eberlm@62429
  1212
lemma Lcm_no_multiple: "(\<forall>m. m \<noteq> 0 \<longrightarrow> (\<exists>a\<in>A. \<not>a dvd m)) \<Longrightarrow> Lcm A = 0"
eberlm@62429
  1213
  by (auto simp: Lcm_0_iff')
eberlm@62429
  1214
eberlm@62429
  1215
lemma Lcm_singleton [simp]:
eberlm@62429
  1216
  "Lcm {a} = normalize a"
eberlm@62429
  1217
  by simp
eberlm@62429
  1218
eberlm@62429
  1219
lemma Lcm_2 [simp]:
eberlm@62429
  1220
  "Lcm {a,b} = lcm a b"
eberlm@62429
  1221
  by simp
eberlm@62429
  1222
eberlm@62429
  1223
lemma Lcm_coprime:
eberlm@62429
  1224
  assumes "finite A" and "A \<noteq> {}" 
eberlm@62429
  1225
  assumes "\<And>a b. a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<noteq> b \<Longrightarrow> gcd a b = 1"
eberlm@62429
  1226
  shows "Lcm A = normalize (\<Prod>A)"
eberlm@62429
  1227
using assms proof (induct rule: finite_ne_induct)
eberlm@62429
  1228
  case (insert a A)
eberlm@62429
  1229
  have "Lcm (insert a A) = lcm a (Lcm A)" by simp
eberlm@62429
  1230
  also from insert have "Lcm A = normalize (\<Prod>A)" by blast
eberlm@62429
  1231
  also have "lcm a \<dots> = lcm a (\<Prod>A)" by (cases "\<Prod>A = 0") (simp_all add: lcm_div_unit2)
eberlm@62429
  1232
  also from insert have "gcd a (\<Prod>A) = 1" by (subst gcd.commute, intro setprod_coprime) auto
eberlm@62429
  1233
  with insert have "lcm a (\<Prod>A) = normalize (\<Prod>(insert a A))"
eberlm@62429
  1234
    by (simp add: lcm_coprime)
eberlm@62429
  1235
  finally show ?case .
eberlm@62429
  1236
qed simp
eberlm@62429
  1237
      
eberlm@62429
  1238
lemma Lcm_coprime':
eberlm@62429
  1239
  "card A \<noteq> 0 \<Longrightarrow> (\<And>a b. a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<noteq> b \<Longrightarrow> gcd a b = 1)
eberlm@62429
  1240
    \<Longrightarrow> Lcm A = normalize (\<Prod>A)"
eberlm@62429
  1241
  by (rule Lcm_coprime) (simp_all add: card_eq_0_iff)
eberlm@62429
  1242
eberlm@62429
  1243
lemma Gcd_1:
eberlm@62429
  1244
  "1 \<in> A \<Longrightarrow> Gcd A = 1"
eberlm@62429
  1245
  by (auto intro!: Gcd_eq_1_I)
eberlm@62429
  1246
eberlm@62429
  1247
lemma Gcd_singleton [simp]: "Gcd {a} = normalize a"
eberlm@62429
  1248
  by simp
eberlm@62429
  1249
eberlm@62429
  1250
lemma Gcd_2 [simp]: "Gcd {a,b} = gcd a b"
eberlm@62429
  1251
  by simp
eberlm@62429
  1252
eberlm@62429
  1253
eberlm@62429
  1254
definition pairwise_coprime where
eberlm@62429
  1255
  "pairwise_coprime A = (\<forall>x y. x \<in> A \<and> y \<in> A \<and> x \<noteq> y \<longrightarrow> coprime x y)"
eberlm@62429
  1256
eberlm@62429
  1257
lemma pairwise_coprimeI [intro?]:
eberlm@62429
  1258
  "(\<And>x y. x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> x \<noteq> y \<Longrightarrow> coprime x y) \<Longrightarrow> pairwise_coprime A"
eberlm@62429
  1259
  by (simp add: pairwise_coprime_def)
eberlm@62429
  1260
eberlm@62429
  1261
lemma pairwise_coprimeD:
eberlm@62429
  1262
  "pairwise_coprime A \<Longrightarrow> x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> x \<noteq> y \<Longrightarrow> coprime x y"
eberlm@62429
  1263
  by (simp add: pairwise_coprime_def)
eberlm@62429
  1264
eberlm@62429
  1265
lemma pairwise_coprime_subset: "pairwise_coprime A \<Longrightarrow> B \<subseteq> A \<Longrightarrow> pairwise_coprime B"
eberlm@62429
  1266
  by (force simp: pairwise_coprime_def)
eberlm@62429
  1267
haftmann@62350
  1268
end
haftmann@62345
  1269
haftmann@62345
  1270
subsection \<open>GCD and LCM on @{typ nat} and @{typ int}\<close>
haftmann@59008
  1271
huffman@31706
  1272
instantiation nat :: gcd
huffman@31706
  1273
begin
wenzelm@21256
  1274
haftmann@62345
  1275
fun gcd_nat  :: "nat \<Rightarrow> nat \<Rightarrow> nat"
haftmann@62345
  1276
where "gcd_nat x y =
haftmann@62345
  1277
  (if y = 0 then x else gcd y (x mod y))"
huffman@31706
  1278
haftmann@62345
  1279
definition lcm_nat :: "nat \<Rightarrow> nat \<Rightarrow> nat"
huffman@31706
  1280
where
huffman@31706
  1281
  "lcm_nat x y = x * y div (gcd x y)"
huffman@31706
  1282
huffman@31706
  1283
instance proof qed
huffman@31706
  1284
huffman@31706
  1285
end
huffman@31706
  1286
huffman@31706
  1287
instantiation int :: gcd
huffman@31706
  1288
begin
wenzelm@21256
  1289
haftmann@62345
  1290
definition gcd_int  :: "int \<Rightarrow> int \<Rightarrow> int"
haftmann@62345
  1291
  where "gcd_int x y = int (gcd (nat \<bar>x\<bar>) (nat \<bar>y\<bar>))"
haftmann@23687
  1292
haftmann@62345
  1293
definition lcm_int :: "int \<Rightarrow> int \<Rightarrow> int"
haftmann@62345
  1294
  where "lcm_int x y = int (lcm (nat \<bar>x\<bar>) (nat \<bar>y\<bar>))"
haftmann@23687
  1295
wenzelm@61944
  1296
instance ..
huffman@31706
  1297
huffman@31706
  1298
end
haftmann@23687
  1299
haftmann@62345
  1300
text \<open>Transfer setup\<close>
huffman@31706
  1301
huffman@31706
  1302
lemma transfer_nat_int_gcd:
huffman@31706
  1303
  "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> gcd (nat x) (nat y) = nat (gcd x y)"
huffman@31706
  1304
  "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> lcm (nat x) (nat y) = nat (lcm x y)"
haftmann@32479
  1305
  unfolding gcd_int_def lcm_int_def
huffman@31706
  1306
  by auto
haftmann@23687
  1307
huffman@31706
  1308
lemma transfer_nat_int_gcd_closures:
huffman@31706
  1309
  "x >= (0::int) \<Longrightarrow> y >= 0 \<Longrightarrow> gcd x y >= 0"
huffman@31706
  1310
  "x >= (0::int) \<Longrightarrow> y >= 0 \<Longrightarrow> lcm x y >= 0"
huffman@31706
  1311
  by (auto simp add: gcd_int_def lcm_int_def)
huffman@31706
  1312
haftmann@35644
  1313
declare transfer_morphism_nat_int[transfer add return:
huffman@31706
  1314
    transfer_nat_int_gcd transfer_nat_int_gcd_closures]
huffman@31706
  1315
huffman@31706
  1316
lemma transfer_int_nat_gcd:
huffman@31706
  1317
  "gcd (int x) (int y) = int (gcd x y)"
huffman@31706
  1318
  "lcm (int x) (int y) = int (lcm x y)"
haftmann@32479
  1319
  by (unfold gcd_int_def lcm_int_def, auto)
huffman@31706
  1320
huffman@31706
  1321
lemma transfer_int_nat_gcd_closures:
huffman@31706
  1322
  "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> gcd x y >= 0"
huffman@31706
  1323
  "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> lcm x y >= 0"
huffman@31706
  1324
  by (auto simp add: gcd_int_def lcm_int_def)
huffman@31706
  1325
haftmann@35644
  1326
declare transfer_morphism_int_nat[transfer add return:
huffman@31706
  1327
    transfer_int_nat_gcd transfer_int_nat_gcd_closures]
huffman@31706
  1328
nipkow@31952
  1329
lemma gcd_nat_induct:
haftmann@23687
  1330
  fixes m n :: nat
haftmann@23687
  1331
  assumes "\<And>m. P m 0"
haftmann@23687
  1332
    and "\<And>m n. 0 < n \<Longrightarrow> P n (m mod n) \<Longrightarrow> P m n"
haftmann@23687
  1333
  shows "P m n"
huffman@31706
  1334
  apply (rule gcd_nat.induct)
huffman@31706
  1335
  apply (case_tac "y = 0")
huffman@31706
  1336
  using assms apply simp_all
huffman@31706
  1337
done
huffman@31706
  1338
huffman@31706
  1339
(* specific to int *)
huffman@31706
  1340
haftmann@62346
  1341
lemma gcd_eq_int_iff:
haftmann@62346
  1342
  "gcd k l = int n \<longleftrightarrow> gcd (nat \<bar>k\<bar>) (nat \<bar>l\<bar>) = n"
haftmann@62346
  1343
  by (simp add: gcd_int_def)
haftmann@62346
  1344
haftmann@62346
  1345
lemma lcm_eq_int_iff:
haftmann@62346
  1346
  "lcm k l = int n \<longleftrightarrow> lcm (nat \<bar>k\<bar>) (nat \<bar>l\<bar>) = n"
haftmann@62346
  1347
  by (simp add: lcm_int_def)
haftmann@62346
  1348
nipkow@31952
  1349
lemma gcd_neg1_int [simp]: "gcd (-x::int) y = gcd x y"
huffman@31706
  1350
  by (simp add: gcd_int_def)
huffman@31706
  1351
nipkow@31952
  1352
lemma gcd_neg2_int [simp]: "gcd (x::int) (-y) = gcd x y"
huffman@31706
  1353
  by (simp add: gcd_int_def)
huffman@31706
  1354
haftmann@62353
  1355
lemma abs_gcd_int [simp]: "\<bar>gcd (x::int) y\<bar> = gcd x y"
nipkow@31813
  1356
by(simp add: gcd_int_def)
nipkow@31813
  1357
wenzelm@61944
  1358
lemma gcd_abs_int: "gcd (x::int) y = gcd \<bar>x\<bar> \<bar>y\<bar>"
nipkow@31813
  1359
by (simp add: gcd_int_def)
nipkow@31813
  1360
haftmann@62353
  1361
lemma gcd_abs1_int [simp]: "gcd \<bar>x\<bar> (y::int) = gcd x y"
nipkow@31952
  1362
by (metis abs_idempotent gcd_abs_int)
nipkow@31813
  1363
haftmann@62353
  1364
lemma gcd_abs2_int [simp]: "gcd x \<bar>y::int\<bar> = gcd x y"
nipkow@31952
  1365
by (metis abs_idempotent gcd_abs_int)
huffman@31706
  1366
nipkow@31952
  1367
lemma gcd_cases_int:
huffman@31706
  1368
  fixes x :: int and y
huffman@31706
  1369
  assumes "x >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> P (gcd x y)"
huffman@31706
  1370
      and "x >= 0 \<Longrightarrow> y <= 0 \<Longrightarrow> P (gcd x (-y))"
huffman@31706
  1371
      and "x <= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> P (gcd (-x) y)"
huffman@31706
  1372
      and "x <= 0 \<Longrightarrow> y <= 0 \<Longrightarrow> P (gcd (-x) (-y))"
huffman@31706
  1373
  shows "P (gcd x y)"
wenzelm@61944
  1374
  by (insert assms, auto, arith)
wenzelm@21256
  1375
nipkow@31952
  1376
lemma gcd_ge_0_int [simp]: "gcd (x::int) y >= 0"
huffman@31706
  1377
  by (simp add: gcd_int_def)
huffman@31706
  1378
nipkow@31952
  1379
lemma lcm_neg1_int: "lcm (-x::int) y = lcm x y"
huffman@31706
  1380
  by (simp add: lcm_int_def)
huffman@31706
  1381
nipkow@31952
  1382
lemma lcm_neg2_int: "lcm (x::int) (-y) = lcm x y"
huffman@31706
  1383
  by (simp add: lcm_int_def)
huffman@31706
  1384
wenzelm@61944
  1385
lemma lcm_abs_int: "lcm (x::int) y = lcm \<bar>x\<bar> \<bar>y\<bar>"
huffman@31706
  1386
  by (simp add: lcm_int_def)
wenzelm@21256
  1387
wenzelm@61944
  1388
lemma abs_lcm_int [simp]: "\<bar>lcm i j::int\<bar> = lcm i j"
wenzelm@61944
  1389
  by (simp add:lcm_int_def)
nipkow@31814
  1390
haftmann@62353
  1391
lemma lcm_abs1_int [simp]: "lcm \<bar>x\<bar> (y::int) = lcm x y"
wenzelm@61944
  1392
  by (metis abs_idempotent lcm_int_def)
nipkow@31814
  1393
haftmann@62353
  1394
lemma lcm_abs2_int [simp]: "lcm x \<bar>y::int\<bar> = lcm x y"
wenzelm@61944
  1395
  by (metis abs_idempotent lcm_int_def)
nipkow@31814
  1396
nipkow@31952
  1397
lemma lcm_cases_int:
huffman@31706
  1398
  fixes x :: int and y
huffman@31706
  1399
  assumes "x >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> P (lcm x y)"
huffman@31706
  1400
      and "x >= 0 \<Longrightarrow> y <= 0 \<Longrightarrow> P (lcm x (-y))"
huffman@31706
  1401
      and "x <= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> P (lcm (-x) y)"
huffman@31706
  1402
      and "x <= 0 \<Longrightarrow> y <= 0 \<Longrightarrow> P (lcm (-x) (-y))"
huffman@31706
  1403
  shows "P (lcm x y)"
wenzelm@41550
  1404
  using assms by (auto simp add: lcm_neg1_int lcm_neg2_int) arith
huffman@31706
  1405
nipkow@31952
  1406
lemma lcm_ge_0_int [simp]: "lcm (x::int) y >= 0"
huffman@31706
  1407
  by (simp add: lcm_int_def)
huffman@31706
  1408
haftmann@54867
  1409
lemma gcd_0_nat: "gcd (x::nat) 0 = x"
haftmann@23687
  1410
  by simp
haftmann@23687
  1411
wenzelm@61944
  1412
lemma gcd_0_int [simp]: "gcd (x::int) 0 = \<bar>x\<bar>"
huffman@31706
  1413
  by (unfold gcd_int_def, auto)
huffman@31706
  1414
haftmann@54867
  1415
lemma gcd_0_left_nat: "gcd 0 (x::nat) = x"
haftmann@23687
  1416
  by simp
haftmann@23687
  1417
wenzelm@61944
  1418
lemma gcd_0_left_int [simp]: "gcd 0 (x::int) = \<bar>x\<bar>"
huffman@31706
  1419
  by (unfold gcd_int_def, auto)
huffman@31706
  1420
nipkow@31952
  1421
lemma gcd_red_nat: "gcd (x::nat) y = gcd y (x mod y)"
huffman@31706
  1422
  by (case_tac "y = 0", auto)
huffman@31706
  1423
huffman@31706
  1424
(* weaker, but useful for the simplifier *)
huffman@31706
  1425
haftmann@62345
  1426
lemma gcd_non_0_nat: "y \<noteq> (0::nat) \<Longrightarrow> gcd (x::nat) y = gcd y (x mod y)"
huffman@31706
  1427
  by simp
huffman@31706
  1428
nipkow@31952
  1429
lemma gcd_1_nat [simp]: "gcd (m::nat) 1 = 1"
wenzelm@21263
  1430
  by simp
wenzelm@21256
  1431
nipkow@31952
  1432
lemma gcd_Suc_0 [simp]: "gcd (m::nat) (Suc 0) = Suc 0"
haftmann@60690
  1433
  by simp
huffman@31706
  1434
nipkow@31952
  1435
lemma gcd_1_int [simp]: "gcd (m::int) 1 = 1"
huffman@31706
  1436
  by (simp add: gcd_int_def)
huffman@30082
  1437
nipkow@31952
  1438
lemma gcd_idem_nat: "gcd (x::nat) x = x"
nipkow@31798
  1439
by simp
huffman@31706
  1440
wenzelm@61944
  1441
lemma gcd_idem_int: "gcd (x::int) x = \<bar>x\<bar>"
nipkow@31813
  1442
by (auto simp add: gcd_int_def)
huffman@31706
  1443
huffman@31706
  1444
declare gcd_nat.simps [simp del]
wenzelm@21256
  1445
wenzelm@60758
  1446
text \<open>
wenzelm@61799
  1447
  \medskip @{term "gcd m n"} divides \<open>m\<close> and \<open>n\<close>.  The
wenzelm@21256
  1448
  conjunctions don't seem provable separately.
wenzelm@60758
  1449
\<close>
wenzelm@21256
  1450
haftmann@59008
  1451
instance nat :: semiring_gcd
haftmann@59008
  1452
proof
haftmann@59008
  1453
  fix m n :: nat
haftmann@59008
  1454
  show "gcd m n dvd m" and "gcd m n dvd n"
haftmann@59008
  1455
  proof (induct m n rule: gcd_nat_induct)
haftmann@59008
  1456
    fix m n :: nat
haftmann@59008
  1457
    assume "gcd n (m mod n) dvd m mod n" and "gcd n (m mod n) dvd n"
haftmann@59008
  1458
    then have "gcd n (m mod n) dvd m"
haftmann@59008
  1459
      by (rule dvd_mod_imp_dvd)
haftmann@59008
  1460
    moreover assume "0 < n"
haftmann@59008
  1461
    ultimately show "gcd m n dvd m"
haftmann@59008
  1462
      by (simp add: gcd_non_0_nat)
haftmann@59008
  1463
  qed (simp_all add: gcd_0_nat gcd_non_0_nat)
haftmann@59008
  1464
next
haftmann@59008
  1465
  fix m n k :: nat
haftmann@59008
  1466
  assume "k dvd m" and "k dvd n"
haftmann@59008
  1467
  then show "k dvd gcd m n"
haftmann@59008
  1468
    by (induct m n rule: gcd_nat_induct) (simp_all add: gcd_non_0_nat dvd_mod gcd_0_nat)
haftmann@60686
  1469
qed (simp_all add: lcm_nat_def)
lp15@59667
  1470
haftmann@59008
  1471
instance int :: ring_gcd
haftmann@60686
  1472
  by standard
haftmann@60686
  1473
    (simp_all add: dvd_int_unfold_dvd_nat gcd_int_def lcm_int_def zdiv_int nat_abs_mult_distrib [symmetric] lcm_gcd gcd_greatest)
lp15@59667
  1474
nipkow@31952
  1475
lemma gcd_le1_nat [simp]: "a \<noteq> 0 \<Longrightarrow> gcd (a::nat) b \<le> a"
huffman@31706
  1476
  by (rule dvd_imp_le, auto)
huffman@31706
  1477
nipkow@31952
  1478
lemma gcd_le2_nat [simp]: "b \<noteq> 0 \<Longrightarrow> gcd (a::nat) b \<le> b"
huffman@31706
  1479
  by (rule dvd_imp_le, auto)
huffman@31706
  1480
nipkow@31952
  1481
lemma gcd_le1_int [simp]: "a > 0 \<Longrightarrow> gcd (a::int) b \<le> a"
huffman@31706
  1482
  by (rule zdvd_imp_le, auto)
wenzelm@21256
  1483
nipkow@31952
  1484
lemma gcd_le2_int [simp]: "b > 0 \<Longrightarrow> gcd (a::int) b \<le> b"
huffman@31706
  1485
  by (rule zdvd_imp_le, auto)
huffman@31706
  1486
nipkow@31952
  1487
lemma gcd_pos_nat [simp]: "(gcd (m::nat) n > 0) = (m ~= 0 | n ~= 0)"
haftmann@62344
  1488
  by (insert gcd_eq_0_iff [of m n], arith)
wenzelm@21256
  1489
nipkow@31952
  1490
lemma gcd_pos_int [simp]: "(gcd (m::int) n > 0) = (m ~= 0 | n ~= 0)"
haftmann@62344
  1491
  by (insert gcd_eq_0_iff [of m n], insert gcd_ge_0_int [of m n], arith)
huffman@31706
  1492
nipkow@31952
  1493
lemma gcd_unique_nat: "(d::nat) dvd a \<and> d dvd b \<and>
huffman@31706
  1494
    (\<forall>e. e dvd a \<and> e dvd b \<longrightarrow> e dvd d) \<longleftrightarrow> d = gcd a b"
huffman@31706
  1495
  apply auto
nipkow@33657
  1496
  apply (rule dvd_antisym)
haftmann@59008
  1497
  apply (erule (1) gcd_greatest)
huffman@31706
  1498
  apply auto
huffman@31706
  1499
done
wenzelm@21256
  1500
nipkow@31952
  1501
lemma gcd_unique_int: "d >= 0 & (d::int) dvd a \<and> d dvd b \<and>
huffman@31706
  1502
    (\<forall>e. e dvd a \<and> e dvd b \<longrightarrow> e dvd d) \<longleftrightarrow> d = gcd a b"
nipkow@33657
  1503
apply (case_tac "d = 0")
nipkow@33657
  1504
 apply simp
nipkow@33657
  1505
apply (rule iffI)
nipkow@33657
  1506
 apply (rule zdvd_antisym_nonneg)
haftmann@59008
  1507
 apply (auto intro: gcd_greatest)
huffman@31706
  1508
done
huffman@30082
  1509
haftmann@61913
  1510
interpretation gcd_nat:
haftmann@62344
  1511
  semilattice_neutr_order gcd "0::nat" Rings.dvd "\<lambda>m n. m dvd n \<and> m \<noteq> n"
haftmann@62344
  1512
  by standard (auto simp add: gcd_unique_nat [symmetric] intro: dvd_antisym dvd_trans)
nipkow@31798
  1513
wenzelm@61944
  1514
lemma gcd_proj1_if_dvd_int [simp]: "x dvd y \<Longrightarrow> gcd (x::int) y = \<bar>x\<bar>"
haftmann@54867
  1515
  by (metis abs_dvd_iff gcd_0_left_int gcd_abs_int gcd_unique_int)
nipkow@31798
  1516
wenzelm@61944
  1517
lemma gcd_proj2_if_dvd_int [simp]: "y dvd x \<Longrightarrow> gcd (x::int) y = \<bar>y\<bar>"
haftmann@62344
  1518
  by (metis gcd_proj1_if_dvd_int gcd.commute)
nipkow@31798
  1519
wenzelm@60758
  1520
text \<open>
wenzelm@21256
  1521
  \medskip Multiplication laws
wenzelm@60758
  1522
\<close>
wenzelm@21256
  1523
nipkow@31952
  1524
lemma gcd_mult_distrib_nat: "(k::nat) * gcd m n = gcd (k * m) (k * n)"
wenzelm@61799
  1525
    \<comment> \<open>@{cite \<open>page 27\<close> davenport92}\<close>
nipkow@31952
  1526
  apply (induct m n rule: gcd_nat_induct)
huffman@31706
  1527
  apply simp
wenzelm@21256
  1528
  apply (case_tac "k = 0")
huffman@45270
  1529
  apply (simp_all add: gcd_non_0_nat)
huffman@31706
  1530
done
wenzelm@21256
  1531
wenzelm@61944
  1532
lemma gcd_mult_distrib_int: "\<bar>k::int\<bar> * gcd m n = gcd (k * m) (k * n)"
nipkow@31952
  1533
  apply (subst (1 2) gcd_abs_int)
nipkow@31813
  1534
  apply (subst (1 2) abs_mult)
nipkow@31952
  1535
  apply (rule gcd_mult_distrib_nat [transferred])
huffman@31706
  1536
  apply auto
huffman@31706
  1537
done
wenzelm@21256
  1538
haftmann@62344
  1539
lemma coprime_crossproduct_nat:
haftmann@62344
  1540
  fixes a b c d :: nat
haftmann@62344
  1541
  assumes "coprime a d" and "coprime b c"
haftmann@62344
  1542
  shows "a * c = b * d \<longleftrightarrow> a = b \<and> c = d"
haftmann@62344
  1543
  using assms coprime_crossproduct [of a d b c] by simp
haftmann@62344
  1544
haftmann@35368
  1545
lemma coprime_crossproduct_int:
haftmann@35368
  1546
  fixes a b c d :: int
haftmann@35368
  1547
  assumes "coprime a d" and "coprime b c"
haftmann@35368
  1548
  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
  1549
  using assms coprime_crossproduct [of a d b c] by simp
haftmann@35368
  1550
wenzelm@60758
  1551
text \<open>\medskip Addition laws\<close>
wenzelm@21256
  1552
huffman@31706
  1553
(* to do: add the other variations? *)
huffman@31706
  1554
nipkow@31952
  1555
lemma gcd_diff1_nat: "(m::nat) >= n \<Longrightarrow> gcd (m - n) n = gcd m n"
eberlm@62429
  1556
  by (subst gcd_add1 [symmetric]) auto
huffman@31706
  1557
nipkow@31952
  1558
lemma gcd_diff2_nat: "(n::nat) >= m \<Longrightarrow> gcd (n - m) n = gcd m n"
haftmann@62344
  1559
  apply (subst gcd.commute)
nipkow@31952
  1560
  apply (subst gcd_diff1_nat [symmetric])
huffman@31706
  1561
  apply auto
haftmann@62344
  1562
  apply (subst gcd.commute)
nipkow@31952
  1563
  apply (subst gcd_diff1_nat)
huffman@31706
  1564
  apply assumption
haftmann@62344
  1565
  apply (rule gcd.commute)
haftmann@62344
  1566
  done
huffman@31706
  1567
nipkow@31952
  1568
lemma gcd_non_0_int: "(y::int) > 0 \<Longrightarrow> gcd x y = gcd y (x mod y)"
huffman@31706
  1569
  apply (frule_tac b = y and a = x in pos_mod_sign)
huffman@31706
  1570
  apply (simp del: pos_mod_sign add: gcd_int_def abs_if nat_mod_distrib)
nipkow@31952
  1571
  apply (auto simp add: gcd_non_0_nat nat_mod_distrib [symmetric]
huffman@31706
  1572
    zmod_zminus1_eq_if)
huffman@31706
  1573
  apply (frule_tac a = x in pos_mod_bound)
haftmann@62344
  1574
  apply (subst (1 2) gcd.commute)
nipkow@31952
  1575
  apply (simp del: pos_mod_bound add: nat_diff_distrib gcd_diff2_nat
huffman@31706
  1576
    nat_le_eq_zle)
haftmann@62344
  1577
  done
wenzelm@21256
  1578
nipkow@31952
  1579
lemma gcd_red_int: "gcd (x::int) y = gcd y (x mod y)"
huffman@31706
  1580
  apply (case_tac "y = 0")
huffman@31706
  1581
  apply force
huffman@31706
  1582
  apply (case_tac "y > 0")
nipkow@31952
  1583
  apply (subst gcd_non_0_int, auto)
nipkow@31952
  1584
  apply (insert gcd_non_0_int [of "-y" "-x"])
huffman@35216
  1585
  apply auto
huffman@31706
  1586
done
huffman@31706
  1587
huffman@31706
  1588
(* to do: differences, and all variations of addition rules
huffman@31706
  1589
    as simplification rules for nat and int *)
huffman@31706
  1590
huffman@31706
  1591
(* to do: add the three variations of these, and for ints? *)
huffman@31706
  1592
wenzelm@63145
  1593
lemma finite_divisors_nat [simp]: \<comment> \<open>FIXME move\<close>
haftmann@62353
  1594
  fixes m :: nat
haftmann@62353
  1595
  assumes "m > 0" 
haftmann@62353
  1596
  shows "finite {d. d dvd m}"
nipkow@31734
  1597
proof-
haftmann@62353
  1598
  from assms have "{d. d dvd m} \<subseteq> {d. d \<le> m}"
haftmann@62353
  1599
    by (auto dest: dvd_imp_le)
haftmann@62353
  1600
  then show ?thesis
haftmann@62353
  1601
    using finite_Collect_le_nat by (rule finite_subset)
nipkow@31734
  1602
qed
nipkow@31734
  1603
haftmann@62353
  1604
lemma finite_divisors_int [simp]:
haftmann@62353
  1605
  fixes i :: int
haftmann@62353
  1606
  assumes "i \<noteq> 0"
haftmann@62353
  1607
  shows "finite {d. d dvd i}"
haftmann@62353
  1608
proof -
haftmann@62353
  1609
  have "{d. \<bar>d\<bar> \<le> \<bar>i\<bar>} = {- \<bar>i\<bar>..\<bar>i\<bar>}"
haftmann@62353
  1610
    by (auto simp: abs_if)
haftmann@62353
  1611
  then have "finite {d. \<bar>d\<bar> <= \<bar>i\<bar>}"
haftmann@62353
  1612
    by simp
haftmann@62353
  1613
  from finite_subset [OF _ this] show ?thesis using assms
wenzelm@60512
  1614
    by (simp add: dvd_imp_le_int subset_iff)
nipkow@31734
  1615
qed
nipkow@31734
  1616
haftmann@62353
  1617
lemma Max_divisors_self_nat [simp]: "n\<noteq>0 \<Longrightarrow> Max{d::nat. d dvd n} = n"
nipkow@31995
  1618
apply(rule antisym)
nipkow@44890
  1619
 apply (fastforce intro: Max_le_iff[THEN iffD2] simp: dvd_imp_le)
nipkow@31995
  1620
apply simp
nipkow@31995
  1621
done
nipkow@31995
  1622
haftmann@62353
  1623
lemma Max_divisors_self_int [simp]: "n\<noteq>0 \<Longrightarrow> Max{d::int. d dvd n} = \<bar>n\<bar>"
nipkow@31995
  1624
apply(rule antisym)
haftmann@44278
  1625
 apply(rule Max_le_iff [THEN iffD2])
haftmann@44278
  1626
  apply (auto intro: abs_le_D1 dvd_imp_le_int)
nipkow@31995
  1627
done
nipkow@31995
  1628
nipkow@31734
  1629
lemma gcd_is_Max_divisors_nat:
haftmann@62353
  1630
  "m > 0 \<Longrightarrow> n > 0 \<Longrightarrow> gcd (m::nat) n = Max {d. d dvd m \<and> d dvd n}"
nipkow@31734
  1631
apply(rule Max_eqI[THEN sym])
nipkow@31995
  1632
  apply (metis finite_Collect_conjI finite_divisors_nat)
nipkow@31734
  1633
 apply simp
haftmann@62344
  1634
 apply(metis Suc_diff_1 Suc_neq_Zero dvd_imp_le gcd_greatest_iff gcd_pos_nat)
nipkow@31734
  1635
apply simp
nipkow@31734
  1636
done
nipkow@31734
  1637
nipkow@31734
  1638
lemma gcd_is_Max_divisors_int:
nipkow@31734
  1639
  "m ~= 0 ==> n ~= 0 ==> gcd (m::int) n = (Max {d. d dvd m & d dvd n})"
nipkow@31734
  1640
apply(rule Max_eqI[THEN sym])
nipkow@31995
  1641
  apply (metis finite_Collect_conjI finite_divisors_int)
nipkow@31734
  1642
 apply simp
haftmann@62344
  1643
 apply (metis gcd_greatest_iff gcd_pos_int zdvd_imp_le)
nipkow@31734
  1644
apply simp
nipkow@31734
  1645
done
nipkow@31734
  1646
haftmann@34030
  1647
lemma gcd_code_int [code]:
haftmann@34030
  1648
  "gcd k l = \<bar>if l = (0::int) then k else gcd l (\<bar>k\<bar> mod \<bar>l\<bar>)\<bar>"
haftmann@34030
  1649
  by (simp add: gcd_int_def nat_mod_distrib gcd_non_0_nat)
haftmann@34030
  1650
chaieb@22027
  1651
wenzelm@60758
  1652
subsection \<open>Coprimality\<close>
huffman@31706
  1653
haftmann@62345
  1654
lemma coprime_nat:
haftmann@62345
  1655
  "coprime (a::nat) b \<longleftrightarrow> (\<forall>d. d dvd a \<and> d dvd b \<longleftrightarrow> d = 1)"
haftmann@62345
  1656
  using coprime [of a b] by simp
huffman@31706
  1657
nipkow@31952
  1658
lemma coprime_Suc_0_nat:
haftmann@62345
  1659
  "coprime (a::nat) b \<longleftrightarrow> (\<forall>d. d dvd a \<and> d dvd b \<longleftrightarrow> d = Suc 0)"
haftmann@60690
  1660
  using coprime_nat by simp
huffman@31706
  1661
haftmann@62345
  1662
lemma coprime_int:
haftmann@62345
  1663
  "coprime (a::int) b \<longleftrightarrow> (\<forall>d. d \<ge> 0 \<and> d dvd a \<and> d dvd b \<longleftrightarrow> d = 1)"
nipkow@31952
  1664
  using gcd_unique_int [of 1 a b]
huffman@31706
  1665
  apply clarsimp
huffman@31706
  1666
  apply (erule subst)
huffman@31706
  1667
  apply (rule iffI)
huffman@31706
  1668
  apply force
lp15@61649
  1669
  using abs_dvd_iff abs_ge_zero apply blast
wenzelm@59807
  1670
  done
huffman@31706
  1671
haftmann@62353
  1672
lemma pow_divides_eq_nat [simp]:
haftmann@62353
  1673
  "n > 0 \<Longrightarrow> (a::nat) ^ n dvd b ^ n \<longleftrightarrow> a dvd b"
eberlm@62429
  1674
  using pow_divs_eq[of n] by simp
huffman@31706
  1675
nipkow@31952
  1676
lemma coprime_Suc_nat [simp]: "coprime (Suc n) n"
eberlm@62429
  1677
  using coprime_plus_one[of n] by simp
huffman@31706
  1678
nipkow@31952
  1679
lemma coprime_minus_one_nat: "(n::nat) \<noteq> 0 \<Longrightarrow> coprime (n - 1) n"
eberlm@62429
  1680
  using coprime_Suc_nat [of "n - 1"] gcd.commute [of "n - 1" n] by auto
huffman@31706
  1681
lp15@60162
  1682
lemma coprime_common_divisor_nat: 
haftmann@60686
  1683
  "coprime (a::nat) b \<Longrightarrow> x dvd a \<Longrightarrow> x dvd b \<Longrightarrow> x = 1"
haftmann@62344
  1684
  by (metis gcd_greatest_iff nat_dvd_1_iff_1)
huffman@31706
  1685
lp15@60162
  1686
lemma coprime_common_divisor_int:
wenzelm@61944
  1687
  "coprime (a::int) b \<Longrightarrow> x dvd a \<Longrightarrow> x dvd b \<Longrightarrow> \<bar>x\<bar> = 1"
haftmann@60686
  1688
  using gcd_greatest_iff [of x a b] by auto
huffman@31706
  1689
nipkow@31952
  1690
lemma invertible_coprime_nat: "(x::nat) * y mod m = 1 \<Longrightarrow> coprime x m"
eberlm@62429
  1691
by (metis coprime_lmult gcd_1_nat gcd.commute gcd_red_nat)
huffman@31706
  1692
nipkow@31952
  1693
lemma invertible_coprime_int: "(x::int) * y mod m = 1 \<Longrightarrow> coprime x m"
eberlm@62429
  1694
by (metis coprime_lmult gcd_1_int gcd.commute gcd_red_int)
huffman@31706
  1695
huffman@31706
  1696
wenzelm@60758
  1697
subsection \<open>Bezout's theorem\<close>
huffman@31706
  1698
huffman@31706
  1699
(* Function bezw returns a pair of witnesses to Bezout's theorem --
huffman@31706
  1700
   see the theorems that follow the definition. *)
huffman@31706
  1701
fun
huffman@31706
  1702
  bezw  :: "nat \<Rightarrow> nat \<Rightarrow> int * int"
huffman@31706
  1703
where
huffman@31706
  1704
  "bezw x y =
huffman@31706
  1705
  (if y = 0 then (1, 0) else
huffman@31706
  1706
      (snd (bezw y (x mod y)),
huffman@31706
  1707
       fst (bezw y (x mod y)) - snd (bezw y (x mod y)) * int(x div y)))"
huffman@31706
  1708
huffman@31706
  1709
lemma bezw_0 [simp]: "bezw x 0 = (1, 0)" by simp
huffman@31706
  1710
huffman@31706
  1711
lemma bezw_non_0: "y > 0 \<Longrightarrow> bezw x y = (snd (bezw y (x mod y)),
huffman@31706
  1712
       fst (bezw y (x mod y)) - snd (bezw y (x mod y)) * int(x div y))"
huffman@31706
  1713
  by simp
huffman@31706
  1714
huffman@31706
  1715
declare bezw.simps [simp del]
huffman@31706
  1716
huffman@31706
  1717
lemma bezw_aux [rule_format]:
huffman@31706
  1718
    "fst (bezw x y) * int x + snd (bezw x y) * int y = int (gcd x y)"
nipkow@31952
  1719
proof (induct x y rule: gcd_nat_induct)
huffman@31706
  1720
  fix m :: nat
huffman@31706
  1721
  show "fst (bezw m 0) * int m + snd (bezw m 0) * int 0 = int (gcd m 0)"
huffman@31706
  1722
    by auto
huffman@31706
  1723
  next fix m :: nat and n
huffman@31706
  1724
    assume ngt0: "n > 0" and
huffman@31706
  1725
      ih: "fst (bezw n (m mod n)) * int n +
huffman@31706
  1726
        snd (bezw n (m mod n)) * int (m mod n) =
huffman@31706
  1727
        int (gcd n (m mod n))"
huffman@31706
  1728
    thus "fst (bezw m n) * int m + snd (bezw m n) * int n = int (gcd m n)"
nipkow@31952
  1729
      apply (simp add: bezw_non_0 gcd_non_0_nat)
huffman@31706
  1730
      apply (erule subst)
haftmann@36350
  1731
      apply (simp add: field_simps)
huffman@31706
  1732
      apply (subst mod_div_equality [of m n, symmetric])
huffman@31706
  1733
      (* applying simp here undoes the last substitution!
huffman@31706
  1734
         what is procedure cancel_div_mod? *)
hoelzl@58776
  1735
      apply (simp only: NO_MATCH_def field_simps of_nat_add of_nat_mult)
huffman@31706
  1736
      done
huffman@31706
  1737
qed
huffman@31706
  1738
nipkow@31952
  1739
lemma bezout_int:
huffman@31706
  1740
  fixes x y
huffman@31706
  1741
  shows "EX u v. u * (x::int) + v * y = gcd x y"
huffman@31706
  1742
proof -
huffman@31706
  1743
  have bezout_aux: "!!x y. x \<ge> (0::int) \<Longrightarrow> y \<ge> 0 \<Longrightarrow>
huffman@31706
  1744
      EX u v. u * x + v * y = gcd x y"
huffman@31706
  1745
    apply (rule_tac x = "fst (bezw (nat x) (nat y))" in exI)
huffman@31706
  1746
    apply (rule_tac x = "snd (bezw (nat x) (nat y))" in exI)
huffman@31706
  1747
    apply (unfold gcd_int_def)
huffman@31706
  1748
    apply simp
huffman@31706
  1749
    apply (subst bezw_aux [symmetric])
huffman@31706
  1750
    apply auto
huffman@31706
  1751
    done
huffman@31706
  1752
  have "(x \<ge> 0 \<and> y \<ge> 0) | (x \<ge> 0 \<and> y \<le> 0) | (x \<le> 0 \<and> y \<ge> 0) |
huffman@31706
  1753
      (x \<le> 0 \<and> y \<le> 0)"
huffman@31706
  1754
    by auto
huffman@31706
  1755
  moreover have "x \<ge> 0 \<Longrightarrow> y \<ge> 0 \<Longrightarrow> ?thesis"
huffman@31706
  1756
    by (erule (1) bezout_aux)
huffman@31706
  1757
  moreover have "x >= 0 \<Longrightarrow> y <= 0 \<Longrightarrow> ?thesis"
huffman@31706
  1758
    apply (insert bezout_aux [of x "-y"])
huffman@31706
  1759
    apply auto
huffman@31706
  1760
    apply (rule_tac x = u in exI)
huffman@31706
  1761
    apply (rule_tac x = "-v" in exI)
nipkow@31952
  1762
    apply (subst gcd_neg2_int [symmetric])
huffman@31706
  1763
    apply auto
huffman@31706
  1764
    done
huffman@31706
  1765
  moreover have "x <= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> ?thesis"
huffman@31706
  1766
    apply (insert bezout_aux [of "-x" y])
huffman@31706
  1767
    apply auto
huffman@31706
  1768
    apply (rule_tac x = "-u" in exI)
huffman@31706
  1769
    apply (rule_tac x = v in exI)
nipkow@31952
  1770
    apply (subst gcd_neg1_int [symmetric])
huffman@31706
  1771
    apply auto
huffman@31706
  1772
    done
huffman@31706
  1773
  moreover have "x <= 0 \<Longrightarrow> y <= 0 \<Longrightarrow> ?thesis"
huffman@31706
  1774
    apply (insert bezout_aux [of "-x" "-y"])
huffman@31706
  1775
    apply auto
huffman@31706
  1776
    apply (rule_tac x = "-u" in exI)
huffman@31706
  1777
    apply (rule_tac x = "-v" in exI)
nipkow@31952
  1778
    apply (subst gcd_neg1_int [symmetric])
nipkow@31952
  1779
    apply (subst gcd_neg2_int [symmetric])
huffman@31706
  1780
    apply auto
huffman@31706
  1781
    done
huffman@31706
  1782
  ultimately show ?thesis by blast
huffman@31706
  1783
qed
huffman@31706
  1784
wenzelm@60758
  1785
text \<open>versions of Bezout for nat, by Amine Chaieb\<close>
huffman@31706
  1786
huffman@31706
  1787
lemma ind_euclid:
huffman@31706
  1788
  assumes c: " \<forall>a b. P (a::nat) b \<longleftrightarrow> P b a" and z: "\<forall>a. P a 0"
huffman@31706
  1789
  and add: "\<forall>a b. P a b \<longrightarrow> P a (a + b)"
chaieb@27669
  1790
  shows "P a b"
berghofe@34915
  1791
proof(induct "a + b" arbitrary: a b rule: less_induct)
berghofe@34915
  1792
  case less
chaieb@27669
  1793
  have "a = b \<or> a < b \<or> b < a" by arith
chaieb@27669
  1794
  moreover {assume eq: "a= b"
huffman@31706
  1795
    from add[rule_format, OF z[rule_format, of a]] have "P a b" using eq
huffman@31706
  1796
    by simp}
chaieb@27669
  1797
  moreover
chaieb@27669
  1798
  {assume lt: "a < b"
berghofe@34915
  1799
    hence "a + b - a < a + b \<or> a = 0" by arith
chaieb@27669
  1800
    moreover
chaieb@27669
  1801
    {assume "a =0" with z c have "P a b" by blast }
chaieb@27669
  1802
    moreover
berghofe@34915
  1803
    {assume "a + b - a < a + b"
berghofe@34915
  1804
      also have th0: "a + b - a = a + (b - a)" using lt by arith
berghofe@34915
  1805
      finally have "a + (b - a) < a + b" .
berghofe@34915
  1806
      then have "P a (a + (b - a))" by (rule add[rule_format, OF less])
berghofe@34915
  1807
      then have "P a b" by (simp add: th0[symmetric])}
chaieb@27669
  1808
    ultimately have "P a b" by blast}
chaieb@27669
  1809
  moreover
chaieb@27669
  1810
  {assume lt: "a > b"
berghofe@34915
  1811
    hence "b + a - b < a + b \<or> b = 0" by arith
chaieb@27669
  1812
    moreover
chaieb@27669
  1813
    {assume "b =0" with z c have "P a b" by blast }
chaieb@27669
  1814
    moreover
berghofe@34915
  1815
    {assume "b + a - b < a + b"
berghofe@34915
  1816
      also have th0: "b + a - b = b + (a - b)" using lt by arith
berghofe@34915
  1817
      finally have "b + (a - b) < a + b" .
berghofe@34915
  1818
      then have "P b (b + (a - b))" by (rule add[rule_format, OF less])
berghofe@34915
  1819
      then have "P b a" by (simp add: th0[symmetric])
chaieb@27669
  1820
      hence "P a b" using c by blast }
chaieb@27669
  1821
    ultimately have "P a b" by blast}
chaieb@27669
  1822
ultimately  show "P a b" by blast
chaieb@27669
  1823
qed
chaieb@27669
  1824
nipkow@31952
  1825
lemma bezout_lemma_nat:
huffman@31706
  1826
  assumes ex: "\<exists>(d::nat) x y. d dvd a \<and> d dvd b \<and>
huffman@31706
  1827
    (a * x = b * y + d \<or> b * x = a * y + d)"
huffman@31706
  1828
  shows "\<exists>d x y. d dvd a \<and> d dvd a + b \<and>
huffman@31706
  1829
    (a * x = (a + b) * y + d \<or> (a + b) * x = a * y + d)"
huffman@31706
  1830
  using ex
huffman@31706
  1831
  apply clarsimp
huffman@35216
  1832
  apply (rule_tac x="d" in exI, simp)
huffman@31706
  1833
  apply (case_tac "a * x = b * y + d" , simp_all)
huffman@31706
  1834
  apply (rule_tac x="x + y" in exI)
huffman@31706
  1835
  apply (rule_tac x="y" in exI)
huffman@31706
  1836
  apply algebra
huffman@31706
  1837
  apply (rule_tac x="x" in exI)
huffman@31706
  1838
  apply (rule_tac x="x + y" in exI)
huffman@31706
  1839
  apply algebra
chaieb@27669
  1840
done
chaieb@27669
  1841
nipkow@31952
  1842
lemma bezout_add_nat: "\<exists>(d::nat) x y. d dvd a \<and> d dvd b \<and>
huffman@31706
  1843
    (a * x = b * y + d \<or> b * x = a * y + d)"
huffman@31706
  1844
  apply(induct a b rule: ind_euclid)
huffman@31706
  1845
  apply blast
huffman@31706
  1846
  apply clarify
huffman@35216
  1847
  apply (rule_tac x="a" in exI, simp)
huffman@31706
  1848
  apply clarsimp
huffman@31706
  1849
  apply (rule_tac x="d" in exI)
huffman@35216
  1850
  apply (case_tac "a * x = b * y + d", simp_all)
huffman@31706
  1851
  apply (rule_tac x="x+y" in exI)
huffman@31706
  1852
  apply (rule_tac x="y" in exI)
huffman@31706
  1853
  apply algebra
huffman@31706
  1854
  apply (rule_tac x="x" in exI)
huffman@31706
  1855
  apply (rule_tac x="x+y" in exI)
huffman@31706
  1856
  apply algebra
chaieb@27669
  1857
done
chaieb@27669
  1858
nipkow@31952
  1859
lemma bezout1_nat: "\<exists>(d::nat) x y. d dvd a \<and> d dvd b \<and>
huffman@31706
  1860
    (a * x - b * y = d \<or> b * x - a * y = d)"
nipkow@31952
  1861
  using bezout_add_nat[of a b]
huffman@31706
  1862
  apply clarsimp
huffman@31706
  1863
  apply (rule_tac x="d" in exI, simp)
huffman@31706
  1864
  apply (rule_tac x="x" in exI)
huffman@31706
  1865
  apply (rule_tac x="y" in exI)
huffman@31706
  1866
  apply auto
chaieb@27669
  1867
done
chaieb@27669
  1868
nipkow@31952
  1869
lemma bezout_add_strong_nat: assumes nz: "a \<noteq> (0::nat)"
chaieb@27669
  1870
  shows "\<exists>d x y. d dvd a \<and> d dvd b \<and> a * x = b * y + d"
chaieb@27669
  1871
proof-
huffman@31706
  1872
 from nz have ap: "a > 0" by simp
nipkow@31952
  1873
 from bezout_add_nat[of a b]
huffman@31706
  1874
 have "(\<exists>d x y. d dvd a \<and> d dvd b \<and> a * x = b * y + d) \<or>
huffman@31706
  1875
   (\<exists>d x y. d dvd a \<and> d dvd b \<and> b * x = a * y + d)" by blast
chaieb@27669
  1876
 moreover
huffman@31706
  1877
    {fix d x y assume H: "d dvd a" "d dvd b" "a * x = b * y + d"
huffman@31706
  1878
     from H have ?thesis by blast }
chaieb@27669
  1879
 moreover
chaieb@27669
  1880
 {fix d x y assume H: "d dvd a" "d dvd b" "b * x = a * y + d"
chaieb@27669
  1881
   {assume b0: "b = 0" with H  have ?thesis by simp}
huffman@31706
  1882
   moreover
chaieb@27669
  1883
   {assume b: "b \<noteq> 0" hence bp: "b > 0" by simp
huffman@31706
  1884
     from b dvd_imp_le [OF H(2)] have "d < b \<or> d = b"
huffman@31706
  1885
       by auto
chaieb@27669
  1886
     moreover
chaieb@27669
  1887
     {assume db: "d=b"
wenzelm@41550
  1888
       with nz H have ?thesis apply simp
wenzelm@32960
  1889
         apply (rule exI[where x = b], simp)
wenzelm@32960
  1890
         apply (rule exI[where x = b])
wenzelm@32960
  1891
        by (rule exI[where x = "a - 1"], simp add: diff_mult_distrib2)}
chaieb@27669
  1892
    moreover
huffman@31706
  1893
    {assume db: "d < b"
wenzelm@41550
  1894
        {assume "x=0" hence ?thesis using nz H by simp }
wenzelm@32960
  1895
        moreover
wenzelm@32960
  1896
        {assume x0: "x \<noteq> 0" hence xp: "x > 0" by simp
wenzelm@32960
  1897
          from db have "d \<le> b - 1" by simp
wenzelm@32960
  1898
          hence "d*b \<le> b*(b - 1)" by simp
wenzelm@32960
  1899
          with xp mult_mono[of "1" "x" "d*b" "b*(b - 1)"]
wenzelm@32960
  1900
          have dble: "d*b \<le> x*b*(b - 1)" using bp by simp
wenzelm@32960
  1901
          from H (3) have "d + (b - 1) * (b*x) = d + (b - 1) * (a*y + d)"
huffman@31706
  1902
            by simp
wenzelm@32960
  1903
          hence "d + (b - 1) * a * y + (b - 1) * d = d + (b - 1) * b * x"
haftmann@57512
  1904
            by (simp only: mult.assoc distrib_left)
wenzelm@32960
  1905
          hence "a * ((b - 1) * y) + d * (b - 1 + 1) = d + x*b*(b - 1)"
huffman@31706
  1906
            by algebra
wenzelm@32960
  1907
          hence "a * ((b - 1) * y) = d + x*b*(b - 1) - d*b" using bp by simp
wenzelm@32960
  1908
          hence "a * ((b - 1) * y) = d + (x*b*(b - 1) - d*b)"
wenzelm@32960
  1909
            by (simp only: diff_add_assoc[OF dble, of d, symmetric])
wenzelm@32960
  1910
          hence "a * ((b - 1) * y) = b*(x*(b - 1) - d) + d"
haftmann@59008
  1911
            by (simp only: diff_mult_distrib2 ac_simps)
wenzelm@32960
  1912
          hence ?thesis using H(1,2)
wenzelm@32960
  1913
            apply -
wenzelm@32960
  1914
            apply (rule exI[where x=d], simp)
wenzelm@32960
  1915
            apply (rule exI[where x="(b - 1) * y"])
wenzelm@32960
  1916
            by (rule exI[where x="x*(b - 1) - d"], simp)}
wenzelm@32960
  1917
        ultimately have ?thesis by blast}
chaieb@27669
  1918
    ultimately have ?thesis by blast}
chaieb@27669
  1919
  ultimately have ?thesis by blast}
chaieb@27669
  1920
 ultimately show ?thesis by blast
chaieb@27669
  1921
qed
chaieb@27669
  1922
nipkow@31952
  1923
lemma bezout_nat: assumes a: "(a::nat) \<noteq> 0"
chaieb@27669
  1924
  shows "\<exists>x y. a * x = b * y + gcd a b"
chaieb@27669
  1925
proof-
chaieb@27669
  1926
  let ?g = "gcd a b"
nipkow@31952
  1927
  from bezout_add_strong_nat[OF a, of b]
chaieb@27669
  1928
  obtain d x y where d: "d dvd a" "d dvd b" "a * x = b * y + d" by blast
chaieb@27669
  1929
  from d(1,2) have "d dvd ?g" by simp
chaieb@27669
  1930
  then obtain k where k: "?g = d*k" unfolding dvd_def by blast
huffman@31706
  1931
  from d(3) have "a * x * k = (b * y + d) *k " by auto
chaieb@27669
  1932
  hence "a * (x * k) = b * (y*k) + ?g" by (algebra add: k)
chaieb@27669
  1933
  thus ?thesis by blast
chaieb@27669
  1934
qed
chaieb@27669
  1935
huffman@31706
  1936
haftmann@62345
  1937
subsection \<open>LCM properties  on @{typ nat} and @{typ int}\<close>
huffman@31706
  1938
wenzelm@61944
  1939
lemma lcm_altdef_int [code]: "lcm (a::int) b = \<bar>a\<bar> * \<bar>b\<bar> div gcd a b"
haftmann@62344
  1940
  by (simp add: lcm_int_def lcm_nat_def zdiv_int gcd_int_def)
huffman@31706
  1941
nipkow@31952
  1942
lemma prod_gcd_lcm_nat: "(m::nat) * n = gcd m n * lcm m n"
huffman@31706
  1943
  unfolding lcm_nat_def
eberlm@62429
  1944
  by (simp add: dvd_mult_div_cancel [OF gcd_dvd_prod])
huffman@31706
  1945
wenzelm@61944
  1946
lemma prod_gcd_lcm_int: "\<bar>m::int\<bar> * \<bar>n\<bar> = gcd m n * lcm m n"
huffman@31706
  1947
  unfolding lcm_int_def gcd_int_def
haftmann@62348
  1948
  apply (subst of_nat_mult [symmetric])
nipkow@31952
  1949
  apply (subst prod_gcd_lcm_nat [symmetric])
huffman@31706
  1950
  apply (subst nat_abs_mult_distrib [symmetric])
huffman@31706
  1951
  apply (simp, simp add: abs_mult)
huffman@31706
  1952
done
huffman@31706
  1953
nipkow@31952
  1954
lemma lcm_pos_nat:
nipkow@31798
  1955
  "(m::nat) > 0 \<Longrightarrow> n>0 \<Longrightarrow> lcm m n > 0"
nipkow@31952
  1956
by (metis gr0I mult_is_0 prod_gcd_lcm_nat)
chaieb@27669
  1957
nipkow@31952
  1958
lemma lcm_pos_int:
nipkow@31798
  1959
  "(m::int) ~= 0 \<Longrightarrow> n ~= 0 \<Longrightarrow> lcm m n > 0"
nipkow@31952
  1960
  apply (subst lcm_abs_int)
nipkow@31952
  1961
  apply (rule lcm_pos_nat [transferred])
nipkow@31798
  1962
  apply auto
haftmann@62344
  1963
  done
haftmann@23687
  1964
wenzelm@63145
  1965
lemma dvd_pos_nat: \<comment> \<open>FIXME move\<close>
haftmann@23687
  1966
  fixes n m :: nat
haftmann@23687
  1967
  assumes "n > 0" and "m dvd n"
haftmann@23687
  1968
  shows "m > 0"
haftmann@62344
  1969
  using assms by (cases m) auto
nipkow@31729
  1970
nipkow@31952
  1971
lemma lcm_unique_nat: "(a::nat) dvd d \<and> b dvd d \<and>
huffman@31706
  1972
    (\<forall>e. a dvd e \<and> b dvd e \<longrightarrow> d dvd e) \<longleftrightarrow> d = lcm a b"
haftmann@62344
  1973
  by (auto intro: dvd_antisym lcm_least)
chaieb@27568
  1974
nipkow@31952
  1975
lemma lcm_unique_int: "d >= 0 \<and> (a::int) dvd d \<and> b dvd d \<and>
huffman@31706
  1976
    (\<forall>e. a dvd e \<and> b dvd e \<longrightarrow> d dvd e) \<longleftrightarrow> d = lcm a b"
haftmann@62344
  1977
  using lcm_least zdvd_antisym_nonneg by auto
haftmann@34973
  1978
nipkow@31798
  1979
lemma lcm_proj2_if_dvd_nat [simp]: "(x::nat) dvd y \<Longrightarrow> lcm x y = y"
huffman@31706
  1980
  apply (rule sym)
nipkow@31952
  1981
  apply (subst lcm_unique_nat [symmetric])
huffman@31706
  1982
  apply auto
huffman@31706
  1983
done
huffman@31706
  1984
wenzelm@61944
  1985
lemma lcm_proj2_if_dvd_int [simp]: "(x::int) dvd y \<Longrightarrow> lcm x y = \<bar>y\<bar>"
huffman@31706
  1986
  apply (rule sym)
nipkow@31952
  1987
  apply (subst lcm_unique_int [symmetric])
huffman@31706
  1988
  apply auto
huffman@31706
  1989
done
huffman@31706
  1990
nipkow@31798
  1991
lemma lcm_proj1_if_dvd_nat [simp]: "(x::nat) dvd y \<Longrightarrow> lcm y x = y"
haftmann@62344
  1992
by (subst lcm.commute, erule lcm_proj2_if_dvd_nat)
huffman@31706
  1993
wenzelm@61944
  1994
lemma lcm_proj1_if_dvd_int [simp]: "(x::int) dvd y \<Longrightarrow> lcm y x = \<bar>y\<bar>"
haftmann@62344
  1995
by (subst lcm.commute, erule lcm_proj2_if_dvd_int)
huffman@31706
  1996
haftmann@62353
  1997
lemma lcm_proj1_iff_nat [simp]: "lcm m n = (m::nat) \<longleftrightarrow> n dvd m"
nipkow@31992
  1998
by (metis lcm_proj1_if_dvd_nat lcm_unique_nat)
nipkow@31992
  1999
haftmann@62353
  2000
lemma lcm_proj2_iff_nat [simp]: "lcm m n = (n::nat) \<longleftrightarrow> m dvd n"
nipkow@31992
  2001
by (metis lcm_proj2_if_dvd_nat lcm_unique_nat)
nipkow@31992
  2002
haftmann@62353
  2003
lemma lcm_proj1_iff_int [simp]: "lcm m n = \<bar>m::int\<bar> \<longleftrightarrow> n dvd m"
nipkow@31992
  2004
by (metis dvd_abs_iff lcm_proj1_if_dvd_int lcm_unique_int)
nipkow@31992
  2005
haftmann@62353
  2006
lemma lcm_proj2_iff_int [simp]: "lcm m n = \<bar>n::int\<bar> \<longleftrightarrow> m dvd n"
nipkow@31992
  2007
by (metis dvd_abs_iff lcm_proj2_if_dvd_int lcm_unique_int)
chaieb@27568
  2008
haftmann@62353
  2009
lemma lcm_1_iff_nat [simp]:
haftmann@62353
  2010
  "lcm (m::nat) n = Suc 0 \<longleftrightarrow> m = Suc 0 \<and> n = Suc 0"
haftmann@62353
  2011
  using lcm_eq_1_iff [of m n] by simp
haftmann@61913
  2012
  
haftmann@62353
  2013
lemma lcm_1_iff_int [simp]:
haftmann@62353
  2014
  "lcm (m::int) n = 1 \<longleftrightarrow> (m=1 \<or> m = -1) \<and> (n=1 \<or> n = -1)"
haftmann@61913
  2015
  by auto
nipkow@31995
  2016
haftmann@34030
  2017
haftmann@62345
  2018
subsection \<open>The complete divisibility lattice on @{typ nat} and @{typ int}\<close>
nipkow@32112
  2019
wenzelm@60758
  2020
text\<open>Lifting gcd and lcm to sets (Gcd/Lcm).
huffman@45264
  2021
Gcd is defined via Lcm to facilitate the proof that we have a complete lattice.
wenzelm@60758
  2022
\<close>
huffman@45264
  2023
haftmann@62345
  2024
instantiation nat :: semiring_Gcd
nipkow@32112
  2025
begin
nipkow@32112
  2026
haftmann@62344
  2027
interpretation semilattice_neutr_set lcm "1::nat"
haftmann@62344
  2028
  by standard simp_all
haftmann@54867
  2029
haftmann@61929
  2030
definition
haftmann@61929
  2031
  "Lcm (M::nat set) = (if finite M then F M else 0)"
haftmann@51489
  2032
haftmann@51489
  2033
lemma Lcm_nat_empty:
haftmann@51489
  2034
  "Lcm {} = (1::nat)"
haftmann@60690
  2035
  by (simp add: Lcm_nat_def del: One_nat_def)
haftmann@51489
  2036
haftmann@51489
  2037
lemma Lcm_nat_insert:
haftmann@51489
  2038
  "Lcm (insert n M) = lcm (n::nat) (Lcm M)"
haftmann@61929
  2039
  by (cases "finite M") (auto simp add: Lcm_nat_def simp del: One_nat_def)
haftmann@61929
  2040
haftmann@61929
  2041
lemma Lcm_nat_infinite:
haftmann@61929
  2042
  "infinite M \<Longrightarrow> Lcm M = (0::nat)"
haftmann@61929
  2043
  by (simp add: Lcm_nat_def)
haftmann@61929
  2044
haftmann@61929
  2045
lemma dvd_Lcm_nat [simp]:
haftmann@61929
  2046
  fixes M :: "nat set"
haftmann@61929
  2047
  assumes "m \<in> M"
haftmann@61929
  2048
  shows "m dvd Lcm M"
haftmann@61929
  2049
proof -
haftmann@61929
  2050
  from assms have "insert m M = M" by auto
haftmann@61929
  2051
  moreover have "m dvd Lcm (insert m M)"
haftmann@61929
  2052
    by (simp add: Lcm_nat_insert)
haftmann@61929
  2053
  ultimately show ?thesis by simp
haftmann@61929
  2054
qed
haftmann@61929
  2055
haftmann@61929
  2056
lemma Lcm_dvd_nat [simp]:
haftmann@61929
  2057
  fixes M :: "nat set"
haftmann@61929
  2058
  assumes "\<forall>m\<in>M. m dvd n"
haftmann@61929
  2059
  shows "Lcm M dvd n"
haftmann@62353
  2060
proof (cases "n > 0")
haftmann@62353
  2061
  case False then show ?thesis by simp
haftmann@61929
  2062
next
haftmann@62353
  2063
  case True
haftmann@61929
  2064
  then have "finite {d. d dvd n}" by (rule finite_divisors_nat)
haftmann@61929
  2065
  moreover have "M \<subseteq> {d. d dvd n}" using assms by fast
haftmann@61929
  2066
  ultimately have "finite M" by (rule rev_finite_subset)
haftmann@62353
  2067
  then show ?thesis using assms
haftmann@62353
  2068
    by (induct M) (simp_all add: Lcm_nat_empty Lcm_nat_insert)
haftmann@61929
  2069
qed
nipkow@32112
  2070
huffman@45264
  2071
definition
huffman@45264
  2072
  "Gcd (M::nat set) = Lcm {d. \<forall>m\<in>M. d dvd m}"
nipkow@32112
  2073
haftmann@62345
  2074
instance proof
haftmann@61929
  2075
  show "Gcd N dvd n" if "n \<in> N" for N and n :: nat
haftmann@61929
  2076
  using that by (induct N rule: infinite_finite_induct)
haftmann@61929
  2077
    (auto simp add: Gcd_nat_def)
haftmann@61929
  2078
  show "n dvd Gcd N" if "\<And>m. m \<in> N \<Longrightarrow> n dvd m" for N and n :: nat
haftmann@61929
  2079
  using that by (induct N rule: infinite_finite_induct)
haftmann@61929
  2080
    (auto simp add: Gcd_nat_def)
haftmann@62345
  2081
  show "n dvd Lcm N" if "n \<in> N" for N and n ::nat
haftmann@62345
  2082
  using that by (induct N rule: infinite_finite_induct)
haftmann@62345
  2083
    auto
haftmann@62345
  2084
  show "Lcm N dvd n" if "\<And>m. m \<in> N \<Longrightarrow> m dvd n" for N and n ::nat
haftmann@62345
  2085
  using that by (induct N rule: infinite_finite_induct)
haftmann@62345
  2086
    auto
haftmann@62345
  2087
qed simp_all
nipkow@32112
  2088
haftmann@62345
  2089
end
haftmann@61913
  2090
haftmann@62346
  2091
lemma Gcd_nat_eq_one:
haftmann@62346
  2092
  "1 \<in> N \<Longrightarrow> Gcd N = (1::nat)"
haftmann@62346
  2093
  by (rule Gcd_eq_1_I) auto
haftmann@62346
  2094
wenzelm@60758
  2095
text\<open>Alternative characterizations of Gcd:\<close>
nipkow@32112
  2096
haftmann@62353
  2097
lemma Gcd_eq_Max:
haftmann@62353
  2098
  fixes M :: "nat set"
haftmann@62353
  2099
  assumes "finite (M::nat set)" and "M \<noteq> {}" and "0 \<notin> M"
haftmann@62353
  2100
  shows "Gcd M = Max (\<Inter>m\<in>M. {d. d dvd m})"
haftmann@62353
  2101
proof (rule antisym)
haftmann@62353
  2102
  from assms obtain m where "m \<in> M" and "m > 0"
haftmann@62353
  2103
    by auto
haftmann@62353
  2104
  from \<open>m > 0\<close> have "finite {d. d dvd m}"
haftmann@62353
  2105
    by (blast intro: finite_divisors_nat)
haftmann@62353
  2106
  with \<open>m \<in> M\<close> have fin: "finite (\<Inter>m\<in>M. {d. d dvd m})"
haftmann@62353
  2107
    by blast
haftmann@62353
  2108
  from fin show "Gcd M \<le> Max (\<Inter>m\<in>M. {d. d dvd m})"
haftmann@62353
  2109
    by (auto intro: Max_ge Gcd_dvd)
haftmann@62353
  2110
  from fin show "Max (\<Inter>m\<in>M. {d. d dvd m}) \<le> Gcd M"
haftmann@62353
  2111
    apply (rule Max.boundedI)
haftmann@62353
  2112
    apply auto
haftmann@62353
  2113
    apply (meson Gcd_dvd Gcd_greatest \<open>0 < m\<close> \<open>m \<in> M\<close> dvd_imp_le dvd_pos_nat)
haftmann@62353
  2114
    done
haftmann@62353
  2115
qed
nipkow@32112
  2116
nipkow@32112
  2117
lemma Gcd_remove0_nat: "finite M \<Longrightarrow> Gcd M = Gcd (M - {0::nat})"
nipkow@32112
  2118
apply(induct pred:finite)
nipkow@32112
  2119
 apply simp
nipkow@32112
  2120
apply(case_tac "x=0")
nipkow@32112
  2121
 apply simp
nipkow@32112
  2122
apply(subgoal_tac "insert x F - {0} = insert x (F - {0})")
nipkow@32112
  2123
 apply simp
nipkow@32112
  2124
apply blast
nipkow@32112
  2125
done
nipkow@32112
  2126
nipkow@32112
  2127
lemma Lcm_in_lcm_closed_set_nat:
nipkow@32112
  2128
  "finite M \<Longrightarrow> M \<noteq> {} \<Longrightarrow> ALL m n :: nat. m:M \<longrightarrow> n:M \<longrightarrow> lcm m n : M \<Longrightarrow> Lcm M : M"
nipkow@32112
  2129
apply(induct rule:finite_linorder_min_induct)
nipkow@32112
  2130
 apply simp
nipkow@32112
  2131
apply simp
nipkow@32112
  2132
apply(subgoal_tac "ALL m n :: nat. m:A \<longrightarrow> n:A \<longrightarrow> lcm m n : A")
nipkow@32112
  2133
 apply simp
nipkow@32112
  2134
 apply(case_tac "A={}")
nipkow@32112
  2135
  apply simp
nipkow@32112
  2136
 apply simp
nipkow@32112
  2137
apply (metis lcm_pos_nat lcm_unique_nat linorder_neq_iff nat_dvd_not_less not_less0)
nipkow@32112
  2138
done
nipkow@32112
  2139
nipkow@32112
  2140
lemma Lcm_eq_Max_nat:
nipkow@32112
  2141
  "finite M \<Longrightarrow> M \<noteq> {} \<Longrightarrow> 0 \<notin> M \<Longrightarrow> ALL m n :: nat. m:M \<longrightarrow> n:M \<longrightarrow> lcm m n : M \<Longrightarrow> Lcm M = Max M"
nipkow@32112
  2142
apply(rule antisym)
nipkow@32112
  2143
 apply(rule Max_ge, assumption)
nipkow@32112
  2144
 apply(erule (2) Lcm_in_lcm_closed_set_nat)
haftmann@62345
  2145
apply (auto simp add: not_le Lcm_0_iff dvd_imp_le leD le_neq_trans)
nipkow@32112
  2146
done
nipkow@32112
  2147
nipkow@34222
  2148
lemma mult_inj_if_coprime_nat:
nipkow@34222
  2149
  "inj_on f A \<Longrightarrow> inj_on g B \<Longrightarrow> ALL a:A. ALL b:B. coprime (f a) (g b)
nipkow@34222
  2150
   \<Longrightarrow> inj_on (%(a,b). f a * g b::nat) (A \<times> B)"
haftmann@61913
  2151
  by (auto simp add: inj_on_def coprime_crossproduct_nat simp del: One_nat_def)
nipkow@34222
  2152
wenzelm@60758
  2153
text\<open>Nitpick:\<close>
nipkow@34222
  2154
blanchet@41792
  2155
lemma gcd_eq_nitpick_gcd [nitpick_unfold]: "gcd x y = Nitpick.nat_gcd x y"
blanchet@41792
  2156
by (induct x y rule: nat_gcd.induct)
blanchet@41792
  2157
   (simp add: gcd_nat.simps Nitpick.nat_gcd.simps)
blanchet@33197
  2158
blanchet@41792
  2159
lemma lcm_eq_nitpick_lcm [nitpick_unfold]: "lcm x y = Nitpick.nat_lcm x y"
blanchet@33197
  2160
by (simp only: lcm_nat_def Nitpick.nat_lcm_def gcd_eq_nitpick_gcd)
blanchet@33197
  2161
haftmann@54867
  2162
wenzelm@60758
  2163
subsubsection \<open>Setwise gcd and lcm for integers\<close>
huffman@45264
  2164
haftmann@62345
  2165
instantiation int :: semiring_Gcd
huffman@45264
  2166
begin
huffman@45264
  2167
huffman@45264
  2168
definition
haftmann@63025
  2169
  "Lcm M = int (LCM m\<in>M. (nat \<circ> abs) m)"
huffman@45264
  2170
huffman@45264
  2171
definition
haftmann@63025
  2172
  "Gcd M = int (GCD m\<in>M. (nat \<circ> abs) m)"
huffman@45264
  2173
haftmann@62345
  2174
instance by standard
haftmann@62345
  2175
  (auto intro!: Gcd_dvd Gcd_greatest simp add: Gcd_int_def
haftmann@62345
  2176
    Lcm_int_def int_dvd_iff dvd_int_iff dvd_int_unfold_dvd_nat [symmetric])
haftmann@62345
  2177
haftmann@62345
  2178
end
haftmann@62345
  2179
haftmann@62346
  2180
lemma abs_Gcd [simp]:
haftmann@62346
  2181
  fixes K :: "int set"
haftmann@62346
  2182
  shows "\<bar>Gcd K\<bar> = Gcd K"
haftmann@62346
  2183
  using normalize_Gcd [of K] by simp
haftmann@62346
  2184
haftmann@62346
  2185
lemma abs_Lcm [simp]:
haftmann@62346
  2186
  fixes K :: "int set"
haftmann@62346
  2187
  shows "\<bar>Lcm K\<bar> = Lcm K"
haftmann@62346
  2188
  using normalize_Lcm [of K] by simp
haftmann@62346
  2189
haftmann@62346
  2190
lemma Gcm_eq_int_iff:
haftmann@62346
  2191
  "Gcd K = int n \<longleftrightarrow> Gcd ((nat \<circ> abs) ` K) = n"
haftmann@62346
  2192
  by (simp add: Gcd_int_def comp_def image_image)
haftmann@62346
  2193
haftmann@62346
  2194
lemma Lcm_eq_int_iff:
haftmann@62346
  2195
  "Lcm K = int n \<longleftrightarrow> Lcm ((nat \<circ> abs) ` K) = n"
haftmann@62346
  2196
  by (simp add: Lcm_int_def comp_def image_image)
haftmann@62346
  2197
haftmann@62345
  2198
haftmann@62345
  2199
subsection \<open>GCD and LCM on @{typ integer}\<close>
haftmann@62345
  2200
haftmann@62345
  2201
instantiation integer :: gcd
haftmann@62345
  2202
begin
haftmann@62345
  2203
haftmann@62345
  2204
context
haftmann@62345
  2205
  includes integer.lifting
haftmann@62345
  2206
begin
haftmann@62345
  2207
haftmann@62345
  2208
lift_definition gcd_integer :: "integer \<Rightarrow> integer \<Rightarrow> integer"
haftmann@62345
  2209
  is gcd .
haftmann@62345
  2210
lift_definition lcm_integer :: "integer \<Rightarrow> integer \<Rightarrow> integer"
haftmann@62345
  2211
  is lcm .
haftmann@62345
  2212
haftmann@62345
  2213
end
huffman@45264
  2214
instance ..
haftmann@60686
  2215
wenzelm@21256
  2216
end
huffman@45264
  2217
Andreas@61856
  2218
lifting_update integer.lifting
Andreas@61856
  2219
lifting_forget integer.lifting
Andreas@61856
  2220
haftmann@62345
  2221
context
haftmann@62345
  2222
  includes integer.lifting
haftmann@62345
  2223
begin
Andreas@61856
  2224
Andreas@61856
  2225
lemma gcd_code_integer [code]:
Andreas@61856
  2226
  "gcd k l = \<bar>if l = (0::integer) then k else gcd l (\<bar>k\<bar> mod \<bar>l\<bar>)\<bar>"
haftmann@62345
  2227
  by transfer (fact gcd_code_int)
Andreas@61856
  2228
wenzelm@61944
  2229
lemma lcm_code_integer [code]: "lcm (a::integer) b = \<bar>a\<bar> * \<bar>b\<bar> div gcd a b"
haftmann@62345
  2230
  by transfer (fact lcm_altdef_int)
Andreas@61856
  2231
Andreas@61856
  2232
end
Andreas@61856
  2233
Andreas@61856
  2234
code_printing constant "gcd :: integer \<Rightarrow> _"
Andreas@61856
  2235
  \<rightharpoonup> (OCaml) "Big'_int.gcd'_big'_int"
Andreas@61856
  2236
  and (Haskell) "Prelude.gcd"
Andreas@61856
  2237
  and (Scala) "_.gcd'((_)')"
wenzelm@61975
  2238
  \<comment> \<open>There is no gcd operation in the SML standard library, so no code setup for SML\<close>
Andreas@61856
  2239
haftmann@62344
  2240
text \<open>Some code equations\<close>
haftmann@62344
  2241
eberlm@62429
  2242
lemmas Lcm_set_nat [code, code_unfold] = Lcm_set[where ?'a = nat]
eberlm@62429
  2243
lemmas Gcd_set_nat [code] = Gcd_set[where ?'a = nat]
eberlm@62429
  2244
lemmas Lcm_set_int [code, code_unfold] = Lcm_set[where ?'a = int]
eberlm@62429
  2245
lemmas Gcd_set_int [code] = Gcd_set[where ?'a = int]
haftmann@62344
  2246
haftmann@62344
  2247
haftmann@62344
  2248
text \<open>Fact aliasses\<close>
haftmann@62344
  2249
haftmann@62353
  2250
lemma lcm_0_iff_nat [simp]: "lcm (m::nat) n = 0 \<longleftrightarrow> m = 0 \<or> n = 0"
haftmann@62344
  2251
  by (fact lcm_eq_0_iff)
haftmann@62344
  2252
haftmann@62344
  2253
lemma lcm_0_iff_int [simp]: "lcm (m::int) n = 0 \<longleftrightarrow> m = 0 \<or> n = 0"
haftmann@62344
  2254
  by (fact lcm_eq_0_iff)
haftmann@62344
  2255
haftmann@62344
  2256
lemma dvd_lcm_I1_nat [simp]: "(k::nat) dvd m \<Longrightarrow> k dvd lcm m n"
haftmann@62344
  2257
  by (fact dvd_lcmI1)
haftmann@62344
  2258
haftmann@62344
  2259
lemma dvd_lcm_I2_nat [simp]: "(k::nat) dvd n \<Longrightarrow> k dvd lcm m n"
haftmann@62344
  2260
  by (fact dvd_lcmI2)
haftmann@62344
  2261
haftmann@62344
  2262
lemma dvd_lcm_I1_int [simp]: "(i::int) dvd m \<Longrightarrow> i dvd lcm m n"
haftmann@62344
  2263
  by (fact dvd_lcmI1)
haftmann@62344
  2264
haftmann@62353
  2265
lemma dvd_lcm_I2_int [simp]: "(i::int) dvd n \<Longrightarrow> i dvd lcm m n"
haftmann@62344
  2266
  by (fact dvd_lcmI2)
haftmann@62344
  2267
haftmann@62344
  2268
lemma coprime_exp2_nat [intro]: "coprime (a::nat) b \<Longrightarrow> coprime (a^n) (b^m)"
haftmann@62344
  2269
  by (fact coprime_exp2)
haftmann@62344
  2270
haftmann@62344
  2271
lemma coprime_exp2_int [intro]: "coprime (a::int) b \<Longrightarrow> coprime (a^n) (b^m)"
haftmann@62344
  2272
  by (fact coprime_exp2)
haftmann@62344
  2273
haftmann@62344
  2274
lemmas Gcd_dvd_nat [simp] = Gcd_dvd [where ?'a = nat]
haftmann@62344
  2275
lemmas Gcd_dvd_int [simp] = Gcd_dvd [where ?'a = int]
haftmann@62353
  2276
lemmas Gcd_greatest_nat [simp] = Gcd_greatest [where ?'a = nat]
haftmann@62353
  2277
lemmas Gcd_greatest_int [simp] = Gcd_greatest [where ?'a = int]
haftmann@62344
  2278
haftmann@62344
  2279
lemma dvd_Lcm_int [simp]:
haftmann@62344
  2280
  fixes M :: "int set" assumes "m \<in> M" shows "m dvd Lcm M"
haftmann@62344
  2281
  using assms by (fact dvd_Lcm)
haftmann@62344
  2282
haftmann@62344
  2283
lemma gcd_neg_numeral_1_int [simp]:
haftmann@62344
  2284
  "gcd (- numeral n :: int) x = gcd (numeral n) x"
haftmann@62344
  2285
  by (fact gcd_neg1_int)
haftmann@62344
  2286
haftmann@62344
  2287
lemma gcd_neg_numeral_2_int [simp]:
haftmann@62344
  2288
  "gcd x (- numeral n :: int) = gcd x (numeral n)"
haftmann@62344
  2289
  by (fact gcd_neg2_int)
haftmann@62344
  2290
haftmann@62344
  2291
lemma gcd_proj1_if_dvd_nat [simp]: "(x::nat) dvd y \<Longrightarrow> gcd x y = x"
haftmann@62344
  2292
  by (fact gcd_nat.absorb1)
haftmann@62344
  2293
haftmann@62344
  2294
lemma gcd_proj2_if_dvd_nat [simp]: "(y::nat) dvd x \<Longrightarrow> gcd x y = y"
haftmann@62344
  2295
  by (fact gcd_nat.absorb2)
haftmann@62344
  2296
haftmann@62353
  2297
lemmas Lcm_eq_0_I_nat [simp] = Lcm_eq_0_I [where ?'a = nat]
haftmann@62353
  2298
lemmas Lcm_0_iff_nat [simp] = Lcm_0_iff [where ?'a = nat]
haftmann@62353
  2299
lemmas Lcm_least_int [simp] = Lcm_least [where ?'a = int]
haftmann@62345
  2300
Andreas@61856
  2301
end