src/HOL/GCD.thy
author haftmann
Wed Feb 17 21:51:57 2016 +0100 (2016-02-17)
changeset 62348 9a5f43dac883
parent 62347 2230b7047376
child 62349 7c23469b5118
permissions -rw-r--r--
dropped various legacy fact bindings
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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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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)"
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    using dvd_div_iff_mult [of "gcd a b" _ 0] by (cases "gcd a b = 0") simp_all
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  then have "normalize (a * b) = 0"
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    by simp
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  then show ?Q by simp
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next
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  assume ?Q then show ?P by auto
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qed
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lemma lcm_eq_1_iff [simp]:
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  "lcm a b = 1 \<longleftrightarrow> is_unit a \<and> is_unit b"
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  by (auto intro: associated_eqI)
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lemma unit_factor_lcm :
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  "unit_factor (lcm a b) = (if a = 0 \<or> b = 0 then 0 else 1)"
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  by (simp add: unit_factor_gcd dvd_unit_factor_div lcm_gcd)
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sublocale lcm: abel_semigroup lcm
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proof
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  fix a b c
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  show "lcm a b = lcm b a"
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    by (simp add: lcm_gcd ac_simps normalize_mult dvd_normalize_div)
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  have "lcm (lcm a b) c dvd lcm a (lcm b c)"
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    and "lcm a (lcm b c) dvd lcm (lcm a b) c"
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    by (auto intro: lcm_least
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      dvd_trans [of b "lcm b c" "lcm a (lcm b c)"]
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      dvd_trans [of c "lcm b c" "lcm a (lcm b c)"]
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      dvd_trans [of a "lcm a b" "lcm (lcm a b) c"]
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      dvd_trans [of b "lcm a b" "lcm (lcm a b) c"])
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  then show "lcm (lcm a b) c = lcm a (lcm b c)"
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    by (rule associated_eqI) simp_all
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qed
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lemma lcm_self [simp]:
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  "lcm a a = normalize a"
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proof -
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  have "lcm a a dvd a"
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    by (rule lcm_least) 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 lcm_left_idem [simp]:
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  "lcm a (lcm a b) = lcm a b"
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   314
  by (auto intro: associated_eqI)
haftmann@61913
   315
haftmann@61913
   316
lemma lcm_right_idem [simp]:
haftmann@61913
   317
  "lcm (lcm a b) b = lcm a b"
haftmann@61913
   318
  unfolding lcm.commute [of a] lcm.commute [of "lcm b a"] by simp
haftmann@61913
   319
haftmann@60686
   320
lemma gcd_mult_lcm [simp]:
haftmann@60686
   321
  "gcd a b * lcm a b = normalize a * normalize b"
haftmann@60686
   322
  by (simp add: lcm_gcd normalize_mult)
haftmann@60686
   323
haftmann@60686
   324
lemma lcm_mult_gcd [simp]:
haftmann@60686
   325
  "lcm a b * gcd a b = normalize a * normalize b"
haftmann@60686
   326
  using gcd_mult_lcm [of a b] by (simp add: ac_simps) 
haftmann@60686
   327
haftmann@60686
   328
lemma gcd_lcm:
haftmann@60686
   329
  assumes "a \<noteq> 0" and "b \<noteq> 0"
haftmann@60686
   330
  shows "gcd a b = normalize (a * b) div lcm a b"
haftmann@60686
   331
proof -
haftmann@60686
   332
  from assms have "lcm a b \<noteq> 0"
haftmann@60686
   333
    by (simp add: lcm_eq_0_iff)
haftmann@60686
   334
  have "gcd a b * lcm a b = normalize a * normalize b" by simp
haftmann@60686
   335
  then have "gcd a b * lcm a b div lcm a b = normalize (a * b) div lcm a b"
haftmann@60686
   336
    by (simp_all add: normalize_mult)
haftmann@60686
   337
  with \<open>lcm a b \<noteq> 0\<close> show ?thesis
haftmann@60686
   338
    using nonzero_mult_divide_cancel_right [of "lcm a b" "gcd a b"] by simp
haftmann@60686
   339
qed
haftmann@60686
   340
haftmann@60686
   341
lemma lcm_1_left [simp]:
haftmann@60686
   342
  "lcm 1 a = normalize a"
haftmann@60686
   343
  by (simp add: lcm_gcd)
haftmann@60686
   344
haftmann@60686
   345
lemma lcm_1_right [simp]:
haftmann@60686
   346
  "lcm a 1 = normalize a"
haftmann@60686
   347
  by (simp add: lcm_gcd)
haftmann@60686
   348
  
haftmann@60686
   349
lemma lcm_mult_left:
haftmann@60686
   350
  "lcm (c * a) (c * b) = normalize c * lcm a b"
haftmann@60686
   351
  by (cases "c = 0")
haftmann@60686
   352
    (simp_all add: gcd_mult_right lcm_gcd div_mult_swap normalize_mult ac_simps,
haftmann@60686
   353
      simp add: dvd_div_mult2_eq mult.left_commute [of "normalize c", symmetric])
haftmann@60686
   354
haftmann@60686
   355
lemma lcm_mult_right:
haftmann@60686
   356
  "lcm (a * c) (b * c) = lcm b a * normalize c"
haftmann@60686
   357
  using lcm_mult_left [of c a b] by (simp add: ac_simps)
haftmann@60686
   358
haftmann@60686
   359
lemma mult_lcm_left:
haftmann@60686
   360
  "c * lcm a b = unit_factor c * lcm (c * a) (c * b)"
haftmann@60686
   361
  by (simp add: lcm_mult_left mult.assoc [symmetric])
haftmann@60686
   362
haftmann@60686
   363
lemma mult_lcm_right:
haftmann@60686
   364
  "lcm a b * c = lcm (a * c) (b * c) * unit_factor c"
haftmann@60686
   365
  using mult_lcm_left [of c a b] by (simp add: ac_simps)
haftmann@60686
   366
  
haftmann@60686
   367
end
haftmann@60686
   368
haftmann@62345
   369
class ring_gcd = comm_ring_1 + semiring_gcd
haftmann@62345
   370
haftmann@60686
   371
class semiring_Gcd = semiring_gcd + Gcd +
haftmann@60686
   372
  assumes Gcd_dvd: "a \<in> A \<Longrightarrow> Gcd A dvd a"
haftmann@60686
   373
    and Gcd_greatest: "(\<And>b. b \<in> A \<Longrightarrow> a dvd b) \<Longrightarrow> a dvd Gcd A"
haftmann@60686
   374
    and normalize_Gcd [simp]: "normalize (Gcd A) = Gcd A"
haftmann@62345
   375
  assumes dvd_Lcm: "a \<in> A \<Longrightarrow> a dvd Lcm A"
haftmann@62345
   376
    and Lcm_least: "(\<And>b. b \<in> A \<Longrightarrow> b dvd a) \<Longrightarrow> Lcm A dvd a"
haftmann@62345
   377
    and normalize_Lcm [simp]: "normalize (Lcm A) = Lcm A"
haftmann@60686
   378
begin
haftmann@60686
   379
haftmann@62345
   380
lemma Lcm_Gcd:
haftmann@62345
   381
  "Lcm A = Gcd {b. \<forall>a\<in>A. a dvd b}"
haftmann@62345
   382
  by (rule associated_eqI) (auto intro: Gcd_dvd dvd_Lcm Gcd_greatest Lcm_least)
haftmann@62345
   383
haftmann@62345
   384
lemma Gcd_Lcm:
haftmann@62345
   385
  "Gcd A = Lcm {b. \<forall>a\<in>A. b dvd a}"
haftmann@62345
   386
  by (rule associated_eqI) (auto intro: Gcd_dvd dvd_Lcm Gcd_greatest Lcm_least)
haftmann@62345
   387
haftmann@60686
   388
lemma Gcd_empty [simp]:
haftmann@60686
   389
  "Gcd {} = 0"
haftmann@60686
   390
  by (rule dvd_0_left, rule Gcd_greatest) simp
haftmann@60686
   391
haftmann@62345
   392
lemma Lcm_empty [simp]:
haftmann@62345
   393
  "Lcm {} = 1"
haftmann@62345
   394
  by (auto intro: associated_eqI Lcm_least)
haftmann@62345
   395
haftmann@62345
   396
lemma Gcd_insert [simp]:
haftmann@62345
   397
  "Gcd (insert a A) = gcd a (Gcd A)"
haftmann@62345
   398
proof -
haftmann@62345
   399
  have "Gcd (insert a A) dvd gcd a (Gcd A)"
haftmann@62345
   400
    by (auto intro: Gcd_dvd Gcd_greatest)
haftmann@62345
   401
  moreover have "gcd a (Gcd A) dvd Gcd (insert a A)"
haftmann@62345
   402
  proof (rule Gcd_greatest)
haftmann@62345
   403
    fix b
haftmann@62345
   404
    assume "b \<in> insert a A"
haftmann@62345
   405
    then show "gcd a (Gcd A) dvd b"
haftmann@62345
   406
    proof
haftmann@62345
   407
      assume "b = a" then show ?thesis by simp
haftmann@62345
   408
    next
haftmann@62345
   409
      assume "b \<in> A"
haftmann@62345
   410
      then have "Gcd A dvd b" by (rule Gcd_dvd)
haftmann@62345
   411
      moreover have "gcd a (Gcd A) dvd Gcd A" by simp
haftmann@62345
   412
      ultimately show ?thesis by (blast intro: dvd_trans)
haftmann@62345
   413
    qed
haftmann@62345
   414
  qed
haftmann@62345
   415
  ultimately show ?thesis
haftmann@62345
   416
    by (auto intro: associated_eqI)
haftmann@62345
   417
qed
haftmann@62345
   418
haftmann@62345
   419
lemma Lcm_insert [simp]:
haftmann@62345
   420
  "Lcm (insert a A) = lcm a (Lcm A)"
haftmann@62345
   421
proof (rule sym)
haftmann@62345
   422
  have "lcm a (Lcm A) dvd Lcm (insert a A)"
haftmann@62345
   423
    by (auto intro: dvd_Lcm Lcm_least)
haftmann@62345
   424
  moreover have "Lcm (insert a A) dvd lcm a (Lcm A)"
haftmann@62345
   425
  proof (rule Lcm_least)
haftmann@62345
   426
    fix b
haftmann@62345
   427
    assume "b \<in> insert a A"
haftmann@62345
   428
    then show "b dvd lcm a (Lcm A)"
haftmann@62345
   429
    proof
haftmann@62345
   430
      assume "b = a" then show ?thesis by simp
haftmann@62345
   431
    next
haftmann@62345
   432
      assume "b \<in> A"
haftmann@62345
   433
      then have "b dvd Lcm A" by (rule dvd_Lcm)
haftmann@62345
   434
      moreover have "Lcm A dvd lcm a (Lcm A)" by simp
haftmann@62345
   435
      ultimately show ?thesis by (blast intro: dvd_trans)
haftmann@62345
   436
    qed
haftmann@62345
   437
  qed
haftmann@62345
   438
  ultimately show "lcm a (Lcm A) = Lcm (insert a A)"
haftmann@62345
   439
    by (rule associated_eqI) (simp_all add: lcm_eq_0_iff)
haftmann@62345
   440
qed
haftmann@62345
   441
haftmann@60686
   442
lemma Gcd_0_iff [simp]:
haftmann@60687
   443
  "Gcd A = 0 \<longleftrightarrow> A \<subseteq> {0}" (is "?P \<longleftrightarrow> ?Q")
haftmann@60686
   444
proof
haftmann@60686
   445
  assume ?P
haftmann@60686
   446
  show ?Q
haftmann@60686
   447
  proof
haftmann@60686
   448
    fix a
haftmann@60686
   449
    assume "a \<in> A"
haftmann@60686
   450
    then have "Gcd A dvd a" by (rule Gcd_dvd)
haftmann@60687
   451
    with \<open>?P\<close> have "a = 0" by simp
haftmann@60687
   452
    then show "a \<in> {0}" by simp
haftmann@60686
   453
  qed
haftmann@60686
   454
next
haftmann@60686
   455
  assume ?Q
haftmann@60686
   456
  have "0 dvd Gcd A"
haftmann@60686
   457
  proof (rule Gcd_greatest)
haftmann@60686
   458
    fix a
haftmann@60686
   459
    assume "a \<in> A"
haftmann@60687
   460
    with \<open>?Q\<close> have "a = 0" by auto
haftmann@60686
   461
    then show "0 dvd a" by simp
haftmann@60686
   462
  qed
haftmann@60686
   463
  then show ?P by simp
haftmann@60686
   464
qed
haftmann@60686
   465
haftmann@60686
   466
lemma Lcm_1_iff [simp]:
haftmann@60686
   467
  "Lcm A = 1 \<longleftrightarrow> (\<forall>a\<in>A. is_unit a)" (is "?P \<longleftrightarrow> ?Q")
haftmann@60686
   468
proof
haftmann@60686
   469
  assume ?P
haftmann@60686
   470
  show ?Q
haftmann@60686
   471
  proof
haftmann@60686
   472
    fix a
haftmann@60686
   473
    assume "a \<in> A"
haftmann@60686
   474
    then have "a dvd Lcm A"
haftmann@60686
   475
      by (rule dvd_Lcm)
haftmann@60686
   476
    with \<open>?P\<close> show "is_unit a"
haftmann@60686
   477
      by simp
haftmann@60686
   478
  qed
haftmann@60686
   479
next
haftmann@60686
   480
  assume ?Q
haftmann@60686
   481
  then have "is_unit (Lcm A)"
haftmann@60686
   482
    by (blast intro: Lcm_least)
haftmann@60686
   483
  then have "normalize (Lcm A) = 1"
haftmann@60686
   484
    by (rule is_unit_normalize)
haftmann@60686
   485
  then show ?P
haftmann@60686
   486
    by simp
haftmann@60686
   487
qed
haftmann@60686
   488
haftmann@62345
   489
lemma unit_factor_Gcd:
haftmann@62345
   490
  "unit_factor (Gcd A) = (if \<forall>a\<in>A. a = 0 then 0 else 1)"
haftmann@62345
   491
proof (cases "Gcd A = 0")
haftmann@62345
   492
  case True then show ?thesis by auto
haftmann@62345
   493
next
haftmann@62345
   494
  case False
haftmann@62345
   495
  from unit_factor_mult_normalize
haftmann@62345
   496
  have "unit_factor (Gcd A) * normalize (Gcd A) = Gcd A" .
haftmann@62345
   497
  then have "unit_factor (Gcd A) * Gcd A = Gcd A" by simp
haftmann@62345
   498
  then have "unit_factor (Gcd A) * Gcd A div Gcd A = Gcd A div Gcd A" by simp
haftmann@62345
   499
  with False have "unit_factor (Gcd A) = 1" by simp
haftmann@62345
   500
  with False show ?thesis by auto
haftmann@62345
   501
qed
haftmann@62345
   502
haftmann@62345
   503
lemma unit_factor_Lcm:
haftmann@62345
   504
  "unit_factor (Lcm A) = (if Lcm A = 0 then 0 else 1)"
haftmann@62345
   505
proof (cases "Lcm A = 0")
haftmann@62345
   506
  case True then show ?thesis by simp
haftmann@62345
   507
next
haftmann@62345
   508
  case False
haftmann@62345
   509
  with unit_factor_normalize have "unit_factor (normalize (Lcm A)) = 1"
haftmann@62345
   510
    by blast
haftmann@62345
   511
  with False show ?thesis
haftmann@62345
   512
    by simp
haftmann@62345
   513
qed
haftmann@62345
   514
haftmann@62345
   515
lemma Gcd_eq_1_I:
haftmann@62345
   516
  assumes "is_unit a" and "a \<in> A"
haftmann@62345
   517
  shows "Gcd A = 1"
haftmann@62345
   518
proof -
haftmann@62345
   519
  from assms have "is_unit (Gcd A)"
haftmann@62345
   520
    by (blast intro: Gcd_dvd dvd_unit_imp_unit)
haftmann@62345
   521
  then have "normalize (Gcd A) = 1"
haftmann@62345
   522
    by (rule is_unit_normalize)
haftmann@62345
   523
  then show ?thesis
haftmann@62345
   524
    by simp
haftmann@62345
   525
qed
haftmann@62345
   526
haftmann@60686
   527
lemma Lcm_eq_0_I:
haftmann@60686
   528
  assumes "0 \<in> A"
haftmann@60686
   529
  shows "Lcm A = 0"
haftmann@60686
   530
proof -
haftmann@60686
   531
  from assms have "0 dvd Lcm A"
haftmann@60686
   532
    by (rule dvd_Lcm)
haftmann@60686
   533
  then show ?thesis
haftmann@60686
   534
    by simp
haftmann@60686
   535
qed
haftmann@60686
   536
haftmann@62345
   537
lemma Gcd_UNIV [simp]:
haftmann@62345
   538
  "Gcd UNIV = 1"
haftmann@62345
   539
  using dvd_refl by (rule Gcd_eq_1_I) simp
haftmann@62345
   540
haftmann@61929
   541
lemma Lcm_UNIV [simp]:
haftmann@61929
   542
  "Lcm UNIV = 0"
haftmann@61929
   543
  by (rule Lcm_eq_0_I) simp
haftmann@60686
   544
haftmann@61929
   545
lemma Lcm_0_iff:
haftmann@61929
   546
  assumes "finite A"
haftmann@61929
   547
  shows "Lcm A = 0 \<longleftrightarrow> 0 \<in> A"
haftmann@61929
   548
proof (cases "A = {}")
haftmann@61929
   549
  case True then show ?thesis by simp
haftmann@61929
   550
next
haftmann@61929
   551
  case False with assms show ?thesis
haftmann@61929
   552
    by (induct A rule: finite_ne_induct)
haftmann@61929
   553
      (auto simp add: lcm_eq_0_iff)
haftmann@60686
   554
qed
haftmann@61929
   555
haftmann@62345
   556
lemma dvd_Gcd: \<comment> \<open>FIXME remove\<close>
haftmann@62345
   557
  "\<forall>b\<in>A. a dvd b \<Longrightarrow> a dvd Gcd A"
haftmann@62345
   558
  by (blast intro: Gcd_greatest)
haftmann@62345
   559
haftmann@62345
   560
lemma Gcd_set [code_unfold]:
haftmann@62345
   561
  "Gcd (set as) = foldr gcd as 0"
haftmann@62345
   562
  by (induct as) simp_all
haftmann@62345
   563
haftmann@60686
   564
lemma Lcm_set [code_unfold]:
haftmann@60686
   565
  "Lcm (set as) = foldr lcm as 1"
haftmann@60686
   566
  by (induct as) simp_all
haftmann@59008
   567
haftmann@62345
   568
lemma Gcd_image_normalize [simp]:
haftmann@62345
   569
  "Gcd (normalize ` A) = Gcd A"
haftmann@62345
   570
proof -
haftmann@62345
   571
  have "Gcd (normalize ` A) dvd a" if "a \<in> A" for a
haftmann@62345
   572
  proof -
haftmann@62345
   573
    from that obtain B where "A = insert a B" by blast
haftmann@62345
   574
    moreover have " gcd (normalize a) (Gcd (normalize ` B)) dvd normalize a"
haftmann@62345
   575
      by (rule gcd_dvd1)
haftmann@62345
   576
    ultimately show "Gcd (normalize ` A) dvd a"
haftmann@62345
   577
      by simp
haftmann@62345
   578
  qed
haftmann@62345
   579
  then have "Gcd (normalize ` A) dvd Gcd A" and "Gcd A dvd Gcd (normalize ` A)"
haftmann@62345
   580
    by (auto intro!: Gcd_greatest intro: Gcd_dvd)
haftmann@62345
   581
  then show ?thesis
haftmann@62345
   582
    by (auto intro: associated_eqI)
haftmann@62345
   583
qed
haftmann@62345
   584
haftmann@62346
   585
lemma Gcd_eqI:
haftmann@62346
   586
  assumes "normalize a = a"
haftmann@62346
   587
  assumes "\<And>b. b \<in> A \<Longrightarrow> a dvd b"
haftmann@62346
   588
    and "\<And>c. (\<And>b. b \<in> A \<Longrightarrow> c dvd b) \<Longrightarrow> c dvd a"
haftmann@62346
   589
  shows "Gcd A = a"
haftmann@62346
   590
  using assms by (blast intro: associated_eqI Gcd_greatest Gcd_dvd normalize_Gcd)
haftmann@62346
   591
haftmann@62346
   592
lemma Lcm_eqI:
haftmann@62346
   593
  assumes "normalize a = a"
haftmann@62346
   594
  assumes "\<And>b. b \<in> A \<Longrightarrow> b dvd a"
haftmann@62346
   595
    and "\<And>c. (\<And>b. b \<in> A \<Longrightarrow> b dvd c) \<Longrightarrow> a dvd c"
haftmann@62346
   596
  shows "Lcm A = a"
haftmann@62346
   597
  using assms by (blast intro: associated_eqI Lcm_least dvd_Lcm normalize_Lcm)
haftmann@62346
   598
haftmann@62345
   599
end  
haftmann@62345
   600
haftmann@62345
   601
haftmann@62345
   602
subsection \<open>GCD and LCM on @{typ nat} and @{typ int}\<close>
haftmann@59008
   603
huffman@31706
   604
instantiation nat :: gcd
huffman@31706
   605
begin
wenzelm@21256
   606
haftmann@62345
   607
fun gcd_nat  :: "nat \<Rightarrow> nat \<Rightarrow> nat"
haftmann@62345
   608
where "gcd_nat x y =
haftmann@62345
   609
  (if y = 0 then x else gcd y (x mod y))"
huffman@31706
   610
haftmann@62345
   611
definition lcm_nat :: "nat \<Rightarrow> nat \<Rightarrow> nat"
huffman@31706
   612
where
huffman@31706
   613
  "lcm_nat x y = x * y div (gcd x y)"
huffman@31706
   614
huffman@31706
   615
instance proof qed
huffman@31706
   616
huffman@31706
   617
end
huffman@31706
   618
huffman@31706
   619
instantiation int :: gcd
huffman@31706
   620
begin
wenzelm@21256
   621
haftmann@62345
   622
definition gcd_int  :: "int \<Rightarrow> int \<Rightarrow> int"
haftmann@62345
   623
  where "gcd_int x y = int (gcd (nat \<bar>x\<bar>) (nat \<bar>y\<bar>))"
haftmann@23687
   624
haftmann@62345
   625
definition lcm_int :: "int \<Rightarrow> int \<Rightarrow> int"
haftmann@62345
   626
  where "lcm_int x y = int (lcm (nat \<bar>x\<bar>) (nat \<bar>y\<bar>))"
haftmann@23687
   627
wenzelm@61944
   628
instance ..
huffman@31706
   629
huffman@31706
   630
end
haftmann@23687
   631
haftmann@62345
   632
text \<open>Transfer setup\<close>
huffman@31706
   633
huffman@31706
   634
lemma transfer_nat_int_gcd:
huffman@31706
   635
  "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> gcd (nat x) (nat y) = nat (gcd x y)"
huffman@31706
   636
  "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> lcm (nat x) (nat y) = nat (lcm x y)"
haftmann@32479
   637
  unfolding gcd_int_def lcm_int_def
huffman@31706
   638
  by auto
haftmann@23687
   639
huffman@31706
   640
lemma transfer_nat_int_gcd_closures:
huffman@31706
   641
  "x >= (0::int) \<Longrightarrow> y >= 0 \<Longrightarrow> gcd x y >= 0"
huffman@31706
   642
  "x >= (0::int) \<Longrightarrow> y >= 0 \<Longrightarrow> lcm x y >= 0"
huffman@31706
   643
  by (auto simp add: gcd_int_def lcm_int_def)
huffman@31706
   644
haftmann@35644
   645
declare transfer_morphism_nat_int[transfer add return:
huffman@31706
   646
    transfer_nat_int_gcd transfer_nat_int_gcd_closures]
huffman@31706
   647
huffman@31706
   648
lemma transfer_int_nat_gcd:
huffman@31706
   649
  "gcd (int x) (int y) = int (gcd x y)"
huffman@31706
   650
  "lcm (int x) (int y) = int (lcm x y)"
haftmann@32479
   651
  by (unfold gcd_int_def lcm_int_def, auto)
huffman@31706
   652
huffman@31706
   653
lemma transfer_int_nat_gcd_closures:
huffman@31706
   654
  "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> gcd x y >= 0"
huffman@31706
   655
  "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> lcm x y >= 0"
huffman@31706
   656
  by (auto simp add: gcd_int_def lcm_int_def)
huffman@31706
   657
haftmann@35644
   658
declare transfer_morphism_int_nat[transfer add return:
huffman@31706
   659
    transfer_int_nat_gcd transfer_int_nat_gcd_closures]
huffman@31706
   660
nipkow@31952
   661
lemma gcd_nat_induct:
haftmann@23687
   662
  fixes m n :: nat
haftmann@23687
   663
  assumes "\<And>m. P m 0"
haftmann@23687
   664
    and "\<And>m n. 0 < n \<Longrightarrow> P n (m mod n) \<Longrightarrow> P m n"
haftmann@23687
   665
  shows "P m n"
huffman@31706
   666
  apply (rule gcd_nat.induct)
huffman@31706
   667
  apply (case_tac "y = 0")
huffman@31706
   668
  using assms apply simp_all
huffman@31706
   669
done
huffman@31706
   670
huffman@31706
   671
(* specific to int *)
huffman@31706
   672
haftmann@62346
   673
lemma gcd_eq_int_iff:
haftmann@62346
   674
  "gcd k l = int n \<longleftrightarrow> gcd (nat \<bar>k\<bar>) (nat \<bar>l\<bar>) = n"
haftmann@62346
   675
  by (simp add: gcd_int_def)
haftmann@62346
   676
haftmann@62346
   677
lemma lcm_eq_int_iff:
haftmann@62346
   678
  "lcm k l = int n \<longleftrightarrow> lcm (nat \<bar>k\<bar>) (nat \<bar>l\<bar>) = n"
haftmann@62346
   679
  by (simp add: lcm_int_def)
haftmann@62346
   680
nipkow@31952
   681
lemma gcd_neg1_int [simp]: "gcd (-x::int) y = gcd x y"
huffman@31706
   682
  by (simp add: gcd_int_def)
huffman@31706
   683
nipkow@31952
   684
lemma gcd_neg2_int [simp]: "gcd (x::int) (-y) = gcd x y"
huffman@31706
   685
  by (simp add: gcd_int_def)
huffman@31706
   686
wenzelm@61944
   687
lemma abs_gcd_int[simp]: "\<bar>gcd (x::int) y\<bar> = gcd x y"
nipkow@31813
   688
by(simp add: gcd_int_def)
nipkow@31813
   689
wenzelm@61944
   690
lemma gcd_abs_int: "gcd (x::int) y = gcd \<bar>x\<bar> \<bar>y\<bar>"
nipkow@31813
   691
by (simp add: gcd_int_def)
nipkow@31813
   692
wenzelm@61944
   693
lemma gcd_abs1_int[simp]: "gcd \<bar>x\<bar> (y::int) = gcd x y"
nipkow@31952
   694
by (metis abs_idempotent gcd_abs_int)
nipkow@31813
   695
wenzelm@61944
   696
lemma gcd_abs2_int[simp]: "gcd x \<bar>y::int\<bar> = gcd x y"
nipkow@31952
   697
by (metis abs_idempotent gcd_abs_int)
huffman@31706
   698
nipkow@31952
   699
lemma gcd_cases_int:
huffman@31706
   700
  fixes x :: int and y
huffman@31706
   701
  assumes "x >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> P (gcd x y)"
huffman@31706
   702
      and "x >= 0 \<Longrightarrow> y <= 0 \<Longrightarrow> P (gcd x (-y))"
huffman@31706
   703
      and "x <= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> P (gcd (-x) y)"
huffman@31706
   704
      and "x <= 0 \<Longrightarrow> y <= 0 \<Longrightarrow> P (gcd (-x) (-y))"
huffman@31706
   705
  shows "P (gcd x y)"
wenzelm@61944
   706
  by (insert assms, auto, arith)
wenzelm@21256
   707
nipkow@31952
   708
lemma gcd_ge_0_int [simp]: "gcd (x::int) y >= 0"
huffman@31706
   709
  by (simp add: gcd_int_def)
huffman@31706
   710
nipkow@31952
   711
lemma lcm_neg1_int: "lcm (-x::int) y = lcm x y"
huffman@31706
   712
  by (simp add: lcm_int_def)
huffman@31706
   713
nipkow@31952
   714
lemma lcm_neg2_int: "lcm (x::int) (-y) = lcm x y"
huffman@31706
   715
  by (simp add: lcm_int_def)
huffman@31706
   716
wenzelm@61944
   717
lemma lcm_abs_int: "lcm (x::int) y = lcm \<bar>x\<bar> \<bar>y\<bar>"
huffman@31706
   718
  by (simp add: lcm_int_def)
wenzelm@21256
   719
wenzelm@61944
   720
lemma abs_lcm_int [simp]: "\<bar>lcm i j::int\<bar> = lcm i j"
wenzelm@61944
   721
  by (simp add:lcm_int_def)
nipkow@31814
   722
wenzelm@61944
   723
lemma lcm_abs1_int[simp]: "lcm \<bar>x\<bar> (y::int) = lcm x y"
wenzelm@61944
   724
  by (metis abs_idempotent lcm_int_def)
nipkow@31814
   725
wenzelm@61944
   726
lemma lcm_abs2_int[simp]: "lcm x \<bar>y::int\<bar> = lcm x y"
wenzelm@61944
   727
  by (metis abs_idempotent lcm_int_def)
nipkow@31814
   728
nipkow@31952
   729
lemma lcm_cases_int:
huffman@31706
   730
  fixes x :: int and y
huffman@31706
   731
  assumes "x >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> P (lcm x y)"
huffman@31706
   732
      and "x >= 0 \<Longrightarrow> y <= 0 \<Longrightarrow> P (lcm x (-y))"
huffman@31706
   733
      and "x <= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> P (lcm (-x) y)"
huffman@31706
   734
      and "x <= 0 \<Longrightarrow> y <= 0 \<Longrightarrow> P (lcm (-x) (-y))"
huffman@31706
   735
  shows "P (lcm x y)"
wenzelm@41550
   736
  using assms by (auto simp add: lcm_neg1_int lcm_neg2_int) arith
huffman@31706
   737
nipkow@31952
   738
lemma lcm_ge_0_int [simp]: "lcm (x::int) y >= 0"
huffman@31706
   739
  by (simp add: lcm_int_def)
huffman@31706
   740
haftmann@54867
   741
lemma gcd_0_nat: "gcd (x::nat) 0 = x"
haftmann@23687
   742
  by simp
haftmann@23687
   743
wenzelm@61944
   744
lemma gcd_0_int [simp]: "gcd (x::int) 0 = \<bar>x\<bar>"
huffman@31706
   745
  by (unfold gcd_int_def, auto)
huffman@31706
   746
haftmann@54867
   747
lemma gcd_0_left_nat: "gcd 0 (x::nat) = x"
haftmann@23687
   748
  by simp
haftmann@23687
   749
wenzelm@61944
   750
lemma gcd_0_left_int [simp]: "gcd 0 (x::int) = \<bar>x\<bar>"
huffman@31706
   751
  by (unfold gcd_int_def, auto)
huffman@31706
   752
nipkow@31952
   753
lemma gcd_red_nat: "gcd (x::nat) y = gcd y (x mod y)"
huffman@31706
   754
  by (case_tac "y = 0", auto)
huffman@31706
   755
huffman@31706
   756
(* weaker, but useful for the simplifier *)
huffman@31706
   757
haftmann@62345
   758
lemma gcd_non_0_nat: "y \<noteq> (0::nat) \<Longrightarrow> gcd (x::nat) y = gcd y (x mod y)"
huffman@31706
   759
  by simp
huffman@31706
   760
nipkow@31952
   761
lemma gcd_1_nat [simp]: "gcd (m::nat) 1 = 1"
wenzelm@21263
   762
  by simp
wenzelm@21256
   763
nipkow@31952
   764
lemma gcd_Suc_0 [simp]: "gcd (m::nat) (Suc 0) = Suc 0"
haftmann@60690
   765
  by simp
huffman@31706
   766
nipkow@31952
   767
lemma gcd_1_int [simp]: "gcd (m::int) 1 = 1"
huffman@31706
   768
  by (simp add: gcd_int_def)
huffman@30082
   769
nipkow@31952
   770
lemma gcd_idem_nat: "gcd (x::nat) x = x"
nipkow@31798
   771
by simp
huffman@31706
   772
wenzelm@61944
   773
lemma gcd_idem_int: "gcd (x::int) x = \<bar>x\<bar>"
nipkow@31813
   774
by (auto simp add: gcd_int_def)
huffman@31706
   775
huffman@31706
   776
declare gcd_nat.simps [simp del]
wenzelm@21256
   777
wenzelm@60758
   778
text \<open>
wenzelm@61799
   779
  \medskip @{term "gcd m n"} divides \<open>m\<close> and \<open>n\<close>.  The
wenzelm@21256
   780
  conjunctions don't seem provable separately.
wenzelm@60758
   781
\<close>
wenzelm@21256
   782
haftmann@59008
   783
instance nat :: semiring_gcd
haftmann@59008
   784
proof
haftmann@59008
   785
  fix m n :: nat
haftmann@59008
   786
  show "gcd m n dvd m" and "gcd m n dvd n"
haftmann@59008
   787
  proof (induct m n rule: gcd_nat_induct)
haftmann@59008
   788
    fix m n :: nat
haftmann@59008
   789
    assume "gcd n (m mod n) dvd m mod n" and "gcd n (m mod n) dvd n"
haftmann@59008
   790
    then have "gcd n (m mod n) dvd m"
haftmann@59008
   791
      by (rule dvd_mod_imp_dvd)
haftmann@59008
   792
    moreover assume "0 < n"
haftmann@59008
   793
    ultimately show "gcd m n dvd m"
haftmann@59008
   794
      by (simp add: gcd_non_0_nat)
haftmann@59008
   795
  qed (simp_all add: gcd_0_nat gcd_non_0_nat)
haftmann@59008
   796
next
haftmann@59008
   797
  fix m n k :: nat
haftmann@59008
   798
  assume "k dvd m" and "k dvd n"
haftmann@59008
   799
  then show "k dvd gcd m n"
haftmann@59008
   800
    by (induct m n rule: gcd_nat_induct) (simp_all add: gcd_non_0_nat dvd_mod gcd_0_nat)
haftmann@60686
   801
qed (simp_all add: lcm_nat_def)
lp15@59667
   802
haftmann@59008
   803
instance int :: ring_gcd
haftmann@60686
   804
  by standard
haftmann@60686
   805
    (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
   806
nipkow@31952
   807
lemma gcd_le1_nat [simp]: "a \<noteq> 0 \<Longrightarrow> gcd (a::nat) b \<le> a"
huffman@31706
   808
  by (rule dvd_imp_le, auto)
huffman@31706
   809
nipkow@31952
   810
lemma gcd_le2_nat [simp]: "b \<noteq> 0 \<Longrightarrow> gcd (a::nat) b \<le> b"
huffman@31706
   811
  by (rule dvd_imp_le, auto)
huffman@31706
   812
nipkow@31952
   813
lemma gcd_le1_int [simp]: "a > 0 \<Longrightarrow> gcd (a::int) b \<le> a"
huffman@31706
   814
  by (rule zdvd_imp_le, auto)
wenzelm@21256
   815
nipkow@31952
   816
lemma gcd_le2_int [simp]: "b > 0 \<Longrightarrow> gcd (a::int) b \<le> b"
huffman@31706
   817
  by (rule zdvd_imp_le, auto)
huffman@31706
   818
nipkow@31952
   819
lemma gcd_pos_nat [simp]: "(gcd (m::nat) n > 0) = (m ~= 0 | n ~= 0)"
haftmann@62344
   820
  by (insert gcd_eq_0_iff [of m n], arith)
wenzelm@21256
   821
nipkow@31952
   822
lemma gcd_pos_int [simp]: "(gcd (m::int) n > 0) = (m ~= 0 | n ~= 0)"
haftmann@62344
   823
  by (insert gcd_eq_0_iff [of m n], insert gcd_ge_0_int [of m n], arith)
huffman@31706
   824
nipkow@31952
   825
lemma gcd_unique_nat: "(d::nat) dvd a \<and> d dvd b \<and>
huffman@31706
   826
    (\<forall>e. e dvd a \<and> e dvd b \<longrightarrow> e dvd d) \<longleftrightarrow> d = gcd a b"
huffman@31706
   827
  apply auto
nipkow@33657
   828
  apply (rule dvd_antisym)
haftmann@59008
   829
  apply (erule (1) gcd_greatest)
huffman@31706
   830
  apply auto
huffman@31706
   831
done
wenzelm@21256
   832
nipkow@31952
   833
lemma gcd_unique_int: "d >= 0 & (d::int) dvd a \<and> d dvd b \<and>
huffman@31706
   834
    (\<forall>e. e dvd a \<and> e dvd b \<longrightarrow> e dvd d) \<longleftrightarrow> d = gcd a b"
nipkow@33657
   835
apply (case_tac "d = 0")
nipkow@33657
   836
 apply simp
nipkow@33657
   837
apply (rule iffI)
nipkow@33657
   838
 apply (rule zdvd_antisym_nonneg)
haftmann@59008
   839
 apply (auto intro: gcd_greatest)
huffman@31706
   840
done
huffman@30082
   841
haftmann@61913
   842
interpretation gcd_nat:
haftmann@62344
   843
  semilattice_neutr_order gcd "0::nat" Rings.dvd "\<lambda>m n. m dvd n \<and> m \<noteq> n"
haftmann@62344
   844
  by standard (auto simp add: gcd_unique_nat [symmetric] intro: dvd_antisym dvd_trans)
nipkow@31798
   845
wenzelm@61944
   846
lemma gcd_proj1_if_dvd_int [simp]: "x dvd y \<Longrightarrow> gcd (x::int) y = \<bar>x\<bar>"
haftmann@54867
   847
  by (metis abs_dvd_iff gcd_0_left_int gcd_abs_int gcd_unique_int)
nipkow@31798
   848
wenzelm@61944
   849
lemma gcd_proj2_if_dvd_int [simp]: "y dvd x \<Longrightarrow> gcd (x::int) y = \<bar>y\<bar>"
haftmann@62344
   850
  by (metis gcd_proj1_if_dvd_int gcd.commute)
nipkow@31798
   851
wenzelm@60758
   852
text \<open>
wenzelm@21256
   853
  \medskip Multiplication laws
wenzelm@60758
   854
\<close>
wenzelm@21256
   855
nipkow@31952
   856
lemma gcd_mult_distrib_nat: "(k::nat) * gcd m n = gcd (k * m) (k * n)"
wenzelm@61799
   857
    \<comment> \<open>@{cite \<open>page 27\<close> davenport92}\<close>
nipkow@31952
   858
  apply (induct m n rule: gcd_nat_induct)
huffman@31706
   859
  apply simp
wenzelm@21256
   860
  apply (case_tac "k = 0")
huffman@45270
   861
  apply (simp_all add: gcd_non_0_nat)
huffman@31706
   862
done
wenzelm@21256
   863
wenzelm@61944
   864
lemma gcd_mult_distrib_int: "\<bar>k::int\<bar> * gcd m n = gcd (k * m) (k * n)"
nipkow@31952
   865
  apply (subst (1 2) gcd_abs_int)
nipkow@31813
   866
  apply (subst (1 2) abs_mult)
nipkow@31952
   867
  apply (rule gcd_mult_distrib_nat [transferred])
huffman@31706
   868
  apply auto
huffman@31706
   869
done
wenzelm@21256
   870
haftmann@60690
   871
context semiring_gcd
haftmann@60690
   872
begin
wenzelm@21256
   873
haftmann@60690
   874
lemma coprime_dvd_mult:
haftmann@60690
   875
  assumes "coprime a b" and "a dvd c * b"
haftmann@60690
   876
  shows "a dvd c"
haftmann@60690
   877
proof (cases "c = 0")
haftmann@60690
   878
  case True then show ?thesis by simp
haftmann@60690
   879
next
haftmann@60690
   880
  case False
haftmann@60690
   881
  then have unit: "is_unit (unit_factor c)" by simp
haftmann@60690
   882
  from \<open>coprime a b\<close> mult_gcd_left [of c a b]
haftmann@60690
   883
  have "gcd (c * a) (c * b) * unit_factor c = c"
haftmann@60690
   884
    by (simp add: ac_simps)
haftmann@60690
   885
  moreover from \<open>a dvd c * b\<close> have "a dvd gcd (c * a) (c * b) * unit_factor c"
haftmann@60690
   886
    by (simp add: dvd_mult_unit_iff unit)
haftmann@60690
   887
  ultimately show ?thesis by simp
haftmann@60690
   888
qed
haftmann@60690
   889
haftmann@62344
   890
lemma coprime_dvd_mult_iff:
haftmann@62344
   891
  assumes "coprime a c"
haftmann@62344
   892
  shows "a dvd b * c \<longleftrightarrow> a dvd b"
haftmann@62344
   893
  using assms by (auto intro: coprime_dvd_mult)
haftmann@60690
   894
haftmann@60690
   895
lemma gcd_mult_cancel:
haftmann@60690
   896
  "coprime c b \<Longrightarrow> gcd (c * a) b = gcd a b"
haftmann@60690
   897
  apply (rule associated_eqI)
haftmann@59008
   898
  apply (rule gcd_greatest)
haftmann@60690
   899
  apply (rule_tac b = c in coprime_dvd_mult)
haftmann@60690
   900
  apply (simp add: gcd.assoc)
haftmann@60690
   901
  apply (simp_all add: ac_simps)
haftmann@60690
   902
  done
wenzelm@21256
   903
haftmann@62344
   904
lemma coprime_crossproduct:
haftmann@62344
   905
  fixes a b c d
haftmann@35368
   906
  assumes "coprime a d" and "coprime b c"
haftmann@62344
   907
  shows "normalize a * normalize c = normalize b * normalize d
haftmann@62344
   908
    \<longleftrightarrow> normalize a = normalize b \<and> normalize c = normalize d" (is "?lhs \<longleftrightarrow> ?rhs")
haftmann@35368
   909
proof
haftmann@35368
   910
  assume ?rhs then show ?lhs by simp
haftmann@35368
   911
next
haftmann@35368
   912
  assume ?lhs
haftmann@62344
   913
  from \<open>?lhs\<close> have "normalize a dvd normalize b * normalize d"
haftmann@62344
   914
    by (auto intro: dvdI dest: sym)
haftmann@62344
   915
  with \<open>coprime a d\<close> have "a dvd b"
haftmann@62344
   916
    by (simp add: coprime_dvd_mult_iff normalize_mult [symmetric])
haftmann@62344
   917
  from \<open>?lhs\<close> have "normalize b dvd normalize a * normalize c"
haftmann@62344
   918
    by (auto intro: dvdI dest: sym)
haftmann@62344
   919
  with \<open>coprime b c\<close> have "b dvd a"
haftmann@62344
   920
    by (simp add: coprime_dvd_mult_iff normalize_mult [symmetric])
haftmann@62344
   921
  from \<open>?lhs\<close> have "normalize c dvd normalize d * normalize b"
haftmann@62344
   922
    by (auto intro: dvdI dest: sym simp add: mult.commute)
haftmann@62344
   923
  with \<open>coprime b c\<close> have "c dvd d"
haftmann@62344
   924
    by (simp add: coprime_dvd_mult_iff gcd.commute normalize_mult [symmetric])
haftmann@62344
   925
  from \<open>?lhs\<close> have "normalize d dvd normalize c * normalize a"
haftmann@62344
   926
    by (auto intro: dvdI dest: sym simp add: mult.commute)
haftmann@62344
   927
  with \<open>coprime a d\<close> have "d dvd c"
haftmann@62344
   928
    by (simp add: coprime_dvd_mult_iff gcd.commute normalize_mult [symmetric])
haftmann@62344
   929
  from \<open>a dvd b\<close> \<open>b dvd a\<close> have "normalize a = normalize b"
haftmann@62344
   930
    by (rule associatedI)
haftmann@62344
   931
  moreover from \<open>c dvd d\<close> \<open>d dvd c\<close> have "normalize c = normalize d"
haftmann@62344
   932
    by (rule associatedI)
haftmann@35368
   933
  ultimately show ?rhs ..
haftmann@35368
   934
qed
haftmann@35368
   935
haftmann@62344
   936
end
haftmann@62344
   937
haftmann@62344
   938
lemma coprime_crossproduct_nat:
haftmann@62344
   939
  fixes a b c d :: nat
haftmann@62344
   940
  assumes "coprime a d" and "coprime b c"
haftmann@62344
   941
  shows "a * c = b * d \<longleftrightarrow> a = b \<and> c = d"
haftmann@62344
   942
  using assms coprime_crossproduct [of a d b c] by simp
haftmann@62344
   943
haftmann@35368
   944
lemma coprime_crossproduct_int:
haftmann@35368
   945
  fixes a b c d :: int
haftmann@35368
   946
  assumes "coprime a d" and "coprime b c"
haftmann@35368
   947
  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
   948
  using assms coprime_crossproduct [of a d b c] by simp
haftmann@35368
   949
wenzelm@60758
   950
text \<open>\medskip Addition laws\<close>
wenzelm@21256
   951
nipkow@31952
   952
lemma gcd_add1_nat [simp]: "gcd ((m::nat) + n) n = gcd m n"
huffman@31706
   953
  apply (case_tac "n = 0")
nipkow@31952
   954
  apply (simp_all add: gcd_non_0_nat)
haftmann@62344
   955
  done
huffman@31706
   956
nipkow@31952
   957
lemma gcd_add2_nat [simp]: "gcd (m::nat) (m + n) = gcd m n"
haftmann@62344
   958
  apply (subst (1 2) gcd.commute)
haftmann@57512
   959
  apply (subst add.commute)
huffman@31706
   960
  apply simp
haftmann@62344
   961
  done
huffman@31706
   962
huffman@31706
   963
(* to do: add the other variations? *)
huffman@31706
   964
nipkow@31952
   965
lemma gcd_diff1_nat: "(m::nat) >= n \<Longrightarrow> gcd (m - n) n = gcd m n"
haftmann@62344
   966
  by (subst gcd_add1_nat [symmetric]) auto
huffman@31706
   967
nipkow@31952
   968
lemma gcd_diff2_nat: "(n::nat) >= m \<Longrightarrow> gcd (n - m) n = gcd m n"
haftmann@62344
   969
  apply (subst gcd.commute)
nipkow@31952
   970
  apply (subst gcd_diff1_nat [symmetric])
huffman@31706
   971
  apply auto
haftmann@62344
   972
  apply (subst gcd.commute)
nipkow@31952
   973
  apply (subst gcd_diff1_nat)
huffman@31706
   974
  apply assumption
haftmann@62344
   975
  apply (rule gcd.commute)
haftmann@62344
   976
  done
huffman@31706
   977
nipkow@31952
   978
lemma gcd_non_0_int: "(y::int) > 0 \<Longrightarrow> gcd x y = gcd y (x mod y)"
huffman@31706
   979
  apply (frule_tac b = y and a = x in pos_mod_sign)
huffman@31706
   980
  apply (simp del: pos_mod_sign add: gcd_int_def abs_if nat_mod_distrib)
nipkow@31952
   981
  apply (auto simp add: gcd_non_0_nat nat_mod_distrib [symmetric]
huffman@31706
   982
    zmod_zminus1_eq_if)
huffman@31706
   983
  apply (frule_tac a = x in pos_mod_bound)
haftmann@62344
   984
  apply (subst (1 2) gcd.commute)
nipkow@31952
   985
  apply (simp del: pos_mod_bound add: nat_diff_distrib gcd_diff2_nat
huffman@31706
   986
    nat_le_eq_zle)
haftmann@62344
   987
  done
wenzelm@21256
   988
nipkow@31952
   989
lemma gcd_red_int: "gcd (x::int) y = gcd y (x mod y)"
huffman@31706
   990
  apply (case_tac "y = 0")
huffman@31706
   991
  apply force
huffman@31706
   992
  apply (case_tac "y > 0")
nipkow@31952
   993
  apply (subst gcd_non_0_int, auto)
nipkow@31952
   994
  apply (insert gcd_non_0_int [of "-y" "-x"])
huffman@35216
   995
  apply auto
huffman@31706
   996
done
huffman@31706
   997
nipkow@31952
   998
lemma gcd_add1_int [simp]: "gcd ((m::int) + n) n = gcd m n"
haftmann@57512
   999
by (metis gcd_red_int mod_add_self1 add.commute)
huffman@31706
  1000
nipkow@31952
  1001
lemma gcd_add2_int [simp]: "gcd m ((m::int) + n) = gcd m n"
haftmann@62344
  1002
by (metis gcd_add1_int gcd.commute add.commute)
wenzelm@21256
  1003
nipkow@31952
  1004
lemma gcd_add_mult_nat: "gcd (m::nat) (k * m + n) = gcd m n"
haftmann@62344
  1005
by (metis mod_mult_self3 gcd.commute gcd_red_nat)
wenzelm@21256
  1006
nipkow@31952
  1007
lemma gcd_add_mult_int: "gcd (m::int) (k * m + n) = gcd m n"
haftmann@62344
  1008
by (metis gcd.commute gcd_red_int mod_mult_self1 add.commute)
nipkow@31798
  1009
wenzelm@21256
  1010
huffman@31706
  1011
(* to do: differences, and all variations of addition rules
huffman@31706
  1012
    as simplification rules for nat and int *)
huffman@31706
  1013
haftmann@60689
  1014
lemma gcd_dvd_prod_nat: "gcd (m::nat) n dvd k * n"
haftmann@23687
  1015
  using mult_dvd_mono [of 1] by auto
chaieb@22027
  1016
huffman@31706
  1017
(* to do: add the three variations of these, and for ints? *)
huffman@31706
  1018
nipkow@31992
  1019
lemma finite_divisors_nat[simp]:
nipkow@31992
  1020
  assumes "(m::nat) ~= 0" shows "finite{d. d dvd m}"
nipkow@31734
  1021
proof-
wenzelm@60512
  1022
  have "finite{d. d <= m}"
wenzelm@60512
  1023
    by (blast intro: bounded_nat_set_is_finite)
nipkow@31734
  1024
  from finite_subset[OF _ this] show ?thesis using assms
wenzelm@60512
  1025
    by (metis Collect_mono dvd_imp_le neq0_conv)
nipkow@31734
  1026
qed
nipkow@31734
  1027
nipkow@31995
  1028
lemma finite_divisors_int[simp]:
nipkow@31734
  1029
  assumes "(i::int) ~= 0" shows "finite{d. d dvd i}"
nipkow@31734
  1030
proof-
wenzelm@61944
  1031
  have "{d. \<bar>d\<bar> <= \<bar>i\<bar>} = {- \<bar>i\<bar> .. \<bar>i\<bar>}" by(auto simp:abs_if)
wenzelm@61944
  1032
  hence "finite {d. \<bar>d\<bar> <= \<bar>i\<bar>}" by simp
nipkow@31734
  1033
  from finite_subset[OF _ this] show ?thesis using assms
wenzelm@60512
  1034
    by (simp add: dvd_imp_le_int subset_iff)
nipkow@31734
  1035
qed
nipkow@31734
  1036
nipkow@31995
  1037
lemma Max_divisors_self_nat[simp]: "n\<noteq>0 \<Longrightarrow> Max{d::nat. d dvd n} = n"
nipkow@31995
  1038
apply(rule antisym)
nipkow@44890
  1039
 apply (fastforce intro: Max_le_iff[THEN iffD2] simp: dvd_imp_le)
nipkow@31995
  1040
apply simp
nipkow@31995
  1041
done
nipkow@31995
  1042
wenzelm@61944
  1043
lemma Max_divisors_self_int[simp]: "n\<noteq>0 \<Longrightarrow> Max{d::int. d dvd n} = \<bar>n\<bar>"
nipkow@31995
  1044
apply(rule antisym)
haftmann@44278
  1045
 apply(rule Max_le_iff [THEN iffD2])
haftmann@44278
  1046
  apply (auto intro: abs_le_D1 dvd_imp_le_int)
nipkow@31995
  1047
done
nipkow@31995
  1048
nipkow@31734
  1049
lemma gcd_is_Max_divisors_nat:
nipkow@31734
  1050
  "m ~= 0 \<Longrightarrow> n ~= 0 \<Longrightarrow> gcd (m::nat) n = (Max {d. d dvd m & d dvd n})"
nipkow@31734
  1051
apply(rule Max_eqI[THEN sym])
nipkow@31995
  1052
  apply (metis finite_Collect_conjI finite_divisors_nat)
nipkow@31734
  1053
 apply simp
haftmann@62344
  1054
 apply(metis Suc_diff_1 Suc_neq_Zero dvd_imp_le gcd_greatest_iff gcd_pos_nat)
nipkow@31734
  1055
apply simp
nipkow@31734
  1056
done
nipkow@31734
  1057
nipkow@31734
  1058
lemma gcd_is_Max_divisors_int:
nipkow@31734
  1059
  "m ~= 0 ==> n ~= 0 ==> gcd (m::int) n = (Max {d. d dvd m & d dvd n})"
nipkow@31734
  1060
apply(rule Max_eqI[THEN sym])
nipkow@31995
  1061
  apply (metis finite_Collect_conjI finite_divisors_int)
nipkow@31734
  1062
 apply simp
haftmann@62344
  1063
 apply (metis gcd_greatest_iff gcd_pos_int zdvd_imp_le)
nipkow@31734
  1064
apply simp
nipkow@31734
  1065
done
nipkow@31734
  1066
haftmann@34030
  1067
lemma gcd_code_int [code]:
haftmann@34030
  1068
  "gcd k l = \<bar>if l = (0::int) then k else gcd l (\<bar>k\<bar> mod \<bar>l\<bar>)\<bar>"
haftmann@34030
  1069
  by (simp add: gcd_int_def nat_mod_distrib gcd_non_0_nat)
haftmann@34030
  1070
chaieb@22027
  1071
wenzelm@60758
  1072
subsection \<open>Coprimality\<close>
huffman@31706
  1073
haftmann@60690
  1074
context semiring_gcd
haftmann@60690
  1075
begin
haftmann@60690
  1076
haftmann@60690
  1077
lemma div_gcd_coprime:
haftmann@60690
  1078
  assumes nz: "a \<noteq> 0 \<or> b \<noteq> 0"
huffman@31706
  1079
  shows "coprime (a div gcd a b) (b div gcd a b)"
wenzelm@22367
  1080
proof -
haftmann@27556
  1081
  let ?g = "gcd a b"
chaieb@22027
  1082
  let ?a' = "a div ?g"
chaieb@22027
  1083
  let ?b' = "b div ?g"
haftmann@27556
  1084
  let ?g' = "gcd ?a' ?b'"
chaieb@22027
  1085
  have dvdg: "?g dvd a" "?g dvd b" by simp_all
chaieb@22027
  1086
  have dvdg': "?g' dvd ?a'" "?g' dvd ?b'" by simp_all
wenzelm@22367
  1087
  from dvdg dvdg' obtain ka kb ka' kb' where
wenzelm@22367
  1088
      kab: "a = ?g * ka" "b = ?g * kb" "?a' = ?g' * ka'" "?b' = ?g' * kb'"
chaieb@22027
  1089
    unfolding dvd_def by blast
haftmann@58834
  1090
  from this [symmetric] have "?g * ?a' = (?g * ?g') * ka'" "?g * ?b' = (?g * ?g') * kb'"
haftmann@58834
  1091
    by (simp_all add: mult.assoc mult.left_commute [of "gcd a b"])
wenzelm@22367
  1092
  then have dvdgg':"?g * ?g' dvd a" "?g* ?g' dvd b"
wenzelm@22367
  1093
    by (auto simp add: dvd_mult_div_cancel [OF dvdg(1)]
wenzelm@22367
  1094
      dvd_mult_div_cancel [OF dvdg(2)] dvd_def)
huffman@35216
  1095
  have "?g \<noteq> 0" using nz by simp
haftmann@60690
  1096
  moreover from gcd_greatest [OF dvdgg'] have "?g * ?g' dvd ?g" .
haftmann@60690
  1097
  thm dvd_mult_cancel_left
haftmann@60690
  1098
  ultimately show ?thesis using dvd_times_left_cancel_iff [of "gcd a b" _ 1] by simp
chaieb@22027
  1099
qed
chaieb@22027
  1100
haftmann@62345
  1101
lemma coprime:
haftmann@62345
  1102
  "coprime a b \<longleftrightarrow> (\<forall>d. d dvd a \<and> d dvd b \<longleftrightarrow> is_unit d)" (is "?P \<longleftrightarrow> ?Q")
haftmann@62345
  1103
proof
haftmann@62345
  1104
  assume ?P then show ?Q by auto
haftmann@62345
  1105
next
haftmann@62345
  1106
  assume ?Q
haftmann@62345
  1107
  then have "is_unit (gcd a b) \<longleftrightarrow> gcd a b dvd a \<and> gcd a b dvd b"
haftmann@62345
  1108
    by blast
haftmann@62345
  1109
  then have "is_unit (gcd a b)"
haftmann@62345
  1110
    by simp
haftmann@62345
  1111
  then show ?P
haftmann@62345
  1112
    by simp
haftmann@62345
  1113
qed
haftmann@62345
  1114
haftmann@60690
  1115
end
haftmann@60690
  1116
haftmann@62345
  1117
lemma coprime_nat:
haftmann@62345
  1118
  "coprime (a::nat) b \<longleftrightarrow> (\<forall>d. d dvd a \<and> d dvd b \<longleftrightarrow> d = 1)"
haftmann@62345
  1119
  using coprime [of a b] by simp
huffman@31706
  1120
nipkow@31952
  1121
lemma coprime_Suc_0_nat:
haftmann@62345
  1122
  "coprime (a::nat) b \<longleftrightarrow> (\<forall>d. d dvd a \<and> d dvd b \<longleftrightarrow> d = Suc 0)"
haftmann@60690
  1123
  using coprime_nat by simp
huffman@31706
  1124
haftmann@62345
  1125
lemma coprime_int:
haftmann@62345
  1126
  "coprime (a::int) b \<longleftrightarrow> (\<forall>d. d \<ge> 0 \<and> d dvd a \<and> d dvd b \<longleftrightarrow> d = 1)"
nipkow@31952
  1127
  using gcd_unique_int [of 1 a b]
huffman@31706
  1128
  apply clarsimp
huffman@31706
  1129
  apply (erule subst)
huffman@31706
  1130
  apply (rule iffI)
huffman@31706
  1131
  apply force
lp15@61649
  1132
  using abs_dvd_iff abs_ge_zero apply blast
wenzelm@59807
  1133
  done
huffman@31706
  1134
nipkow@31952
  1135
lemma gcd_coprime_nat:
huffman@31706
  1136
  assumes z: "gcd (a::nat) b \<noteq> 0" and a: "a = a' * gcd a b" and
huffman@31706
  1137
    b: "b = b' * gcd a b"
huffman@31706
  1138
  shows    "coprime a' b'"
huffman@31706
  1139
huffman@31706
  1140
  apply (subgoal_tac "a' = a div gcd a b")
huffman@31706
  1141
  apply (erule ssubst)
huffman@31706
  1142
  apply (subgoal_tac "b' = b div gcd a b")
huffman@31706
  1143
  apply (erule ssubst)
haftmann@62344
  1144
  apply (rule div_gcd_coprime)
wenzelm@41550
  1145
  using z apply force
huffman@31706
  1146
  apply (subst (1) b)
huffman@31706
  1147
  using z apply force
huffman@31706
  1148
  apply (subst (1) a)
huffman@31706
  1149
  using z apply force
wenzelm@41550
  1150
  done
huffman@31706
  1151
nipkow@31952
  1152
lemma gcd_coprime_int:
huffman@31706
  1153
  assumes z: "gcd (a::int) b \<noteq> 0" and a: "a = a' * gcd a b" and
huffman@31706
  1154
    b: "b = b' * gcd a b"
huffman@31706
  1155
  shows    "coprime a' b'"
huffman@31706
  1156
  apply (subgoal_tac "a' = a div gcd a b")
huffman@31706
  1157
  apply (erule ssubst)
huffman@31706
  1158
  apply (subgoal_tac "b' = b div gcd a b")
huffman@31706
  1159
  apply (erule ssubst)
haftmann@62344
  1160
  apply (rule div_gcd_coprime)
wenzelm@41550
  1161
  using z apply force
huffman@31706
  1162
  apply (subst (1) b)
huffman@31706
  1163
  using z apply force
huffman@31706
  1164
  apply (subst (1) a)
huffman@31706
  1165
  using z apply force
wenzelm@41550
  1166
  done
huffman@31706
  1167
haftmann@60690
  1168
context semiring_gcd
haftmann@60690
  1169
begin
huffman@31706
  1170
haftmann@60690
  1171
lemma coprime_mult:
haftmann@60690
  1172
  assumes da: "coprime d a" and db: "coprime d b"
haftmann@60690
  1173
  shows "coprime d (a * b)"
haftmann@60690
  1174
  apply (subst gcd.commute)
haftmann@60690
  1175
  using da apply (subst gcd_mult_cancel)
haftmann@60690
  1176
  apply (subst gcd.commute, assumption)
haftmann@60690
  1177
  apply (subst gcd.commute, rule db)
haftmann@60690
  1178
  done
huffman@31706
  1179
haftmann@60690
  1180
end
haftmann@60690
  1181
nipkow@31952
  1182
lemma coprime_lmult_nat:
huffman@31706
  1183
  assumes dab: "coprime (d::nat) (a * b)" shows "coprime d a"
huffman@31706
  1184
proof -
huffman@31706
  1185
  have "gcd d a dvd gcd d (a * b)"
haftmann@59008
  1186
    by (rule gcd_greatest, auto)
huffman@31706
  1187
  with dab show ?thesis
huffman@31706
  1188
    by auto
huffman@31706
  1189
qed
huffman@31706
  1190
nipkow@31952
  1191
lemma coprime_lmult_int:
nipkow@31798
  1192
  assumes "coprime (d::int) (a * b)" shows "coprime d a"
huffman@31706
  1193
proof -
huffman@31706
  1194
  have "gcd d a dvd gcd d (a * b)"
haftmann@59008
  1195
    by (rule gcd_greatest, auto)
nipkow@31798
  1196
  with assms show ?thesis
huffman@31706
  1197
    by auto
huffman@31706
  1198
qed
huffman@31706
  1199
nipkow@31952
  1200
lemma coprime_rmult_nat:
nipkow@31798
  1201
  assumes "coprime (d::nat) (a * b)" shows "coprime d b"
huffman@31706
  1202
proof -
huffman@31706
  1203
  have "gcd d b dvd gcd d (a * b)"
haftmann@59008
  1204
    by (rule gcd_greatest, auto intro: dvd_mult)
nipkow@31798
  1205
  with assms show ?thesis
huffman@31706
  1206
    by auto
huffman@31706
  1207
qed
huffman@31706
  1208
nipkow@31952
  1209
lemma coprime_rmult_int:
huffman@31706
  1210
  assumes dab: "coprime (d::int) (a * b)" shows "coprime d b"
huffman@31706
  1211
proof -
huffman@31706
  1212
  have "gcd d b dvd gcd d (a * b)"
haftmann@59008
  1213
    by (rule gcd_greatest, auto intro: dvd_mult)
huffman@31706
  1214
  with dab show ?thesis
huffman@31706
  1215
    by auto
huffman@31706
  1216
qed
huffman@31706
  1217
nipkow@31952
  1218
lemma coprime_mul_eq_nat: "coprime (d::nat) (a * b) \<longleftrightarrow>
huffman@31706
  1219
    coprime d a \<and>  coprime d b"
nipkow@31952
  1220
  using coprime_rmult_nat[of d a b] coprime_lmult_nat[of d a b]
haftmann@62344
  1221
    coprime_mult [of d a b]
huffman@31706
  1222
  by blast
huffman@31706
  1223
nipkow@31952
  1224
lemma coprime_mul_eq_int: "coprime (d::int) (a * b) \<longleftrightarrow>
huffman@31706
  1225
    coprime d a \<and>  coprime d b"
nipkow@31952
  1226
  using coprime_rmult_int[of d a b] coprime_lmult_int[of d a b]
haftmann@62344
  1227
    coprime_mult [of d a b]
huffman@31706
  1228
  by blast
huffman@31706
  1229
noschinl@52397
  1230
lemma coprime_power_int:
noschinl@52397
  1231
  assumes "0 < n" shows "coprime (a :: int) (b ^ n) \<longleftrightarrow> coprime a b"
noschinl@52397
  1232
  using assms
noschinl@52397
  1233
proof (induct n)
noschinl@52397
  1234
  case (Suc n) then show ?case
noschinl@52397
  1235
    by (cases n) (simp_all add: coprime_mul_eq_int)
noschinl@52397
  1236
qed simp
noschinl@52397
  1237
nipkow@31952
  1238
lemma gcd_coprime_exists_nat:
huffman@31706
  1239
    assumes nz: "gcd (a::nat) b \<noteq> 0"
huffman@31706
  1240
    shows "\<exists>a' b'. a = a' * gcd a b \<and> b = b' * gcd a b \<and> coprime a' b'"
huffman@31706
  1241
  apply (rule_tac x = "a div gcd a b" in exI)
huffman@31706
  1242
  apply (rule_tac x = "b div gcd a b" in exI)
haftmann@62344
  1243
  using nz apply (auto simp add: div_gcd_coprime dvd_div_mult)
huffman@31706
  1244
done
huffman@31706
  1245
nipkow@31952
  1246
lemma gcd_coprime_exists_int:
huffman@31706
  1247
    assumes nz: "gcd (a::int) b \<noteq> 0"
huffman@31706
  1248
    shows "\<exists>a' b'. a = a' * gcd a b \<and> b = b' * gcd a b \<and> coprime a' b'"
huffman@31706
  1249
  apply (rule_tac x = "a div gcd a b" in exI)
huffman@31706
  1250
  apply (rule_tac x = "b div gcd a b" in exI)
haftmann@62344
  1251
  using nz apply (auto simp add: div_gcd_coprime)
huffman@31706
  1252
done
huffman@31706
  1253
nipkow@31952
  1254
lemma coprime_exp_nat: "coprime (d::nat) a \<Longrightarrow> coprime d (a^n)"
haftmann@62344
  1255
  by (induct n) (simp_all add: coprime_mult)
huffman@31706
  1256
nipkow@31952
  1257
lemma coprime_exp_int: "coprime (d::int) a \<Longrightarrow> coprime d (a^n)"
haftmann@62344
  1258
  by (induct n) (simp_all add: coprime_mult)
huffman@31706
  1259
haftmann@60690
  1260
context semiring_gcd
haftmann@60690
  1261
begin
haftmann@60690
  1262
haftmann@60690
  1263
lemma coprime_exp_left:
haftmann@60690
  1264
  assumes "coprime a b"
haftmann@60690
  1265
  shows "coprime (a ^ n) b"
haftmann@60690
  1266
  using assms by (induct n) (simp_all add: gcd_mult_cancel)
haftmann@60690
  1267
haftmann@60690
  1268
lemma coprime_exp2:
haftmann@60690
  1269
  assumes "coprime a b"
haftmann@60690
  1270
  shows "coprime (a ^ n) (b ^ m)"
haftmann@60690
  1271
proof (rule coprime_exp_left)
haftmann@60690
  1272
  from assms show "coprime a (b ^ m)"
haftmann@60690
  1273
    by (induct m) (simp_all add: gcd_mult_cancel gcd.commute [of a])
haftmann@60690
  1274
qed
haftmann@60690
  1275
haftmann@60690
  1276
end
haftmann@60690
  1277
haftmann@60690
  1278
lemma gcd_exp_nat:
haftmann@60690
  1279
  "gcd ((a :: nat) ^ n) (b ^ n) = gcd a b ^ n"
haftmann@60690
  1280
proof (cases "a = 0 \<and> b = 0")
haftmann@60690
  1281
  case True then show ?thesis by (cases "n > 0") (simp_all add: zero_power)
haftmann@60690
  1282
next
haftmann@60690
  1283
  case False
haftmann@60690
  1284
  then have "coprime (a div gcd a b) (b div gcd a b)"
haftmann@60690
  1285
    by (auto simp: div_gcd_coprime)
haftmann@60690
  1286
  then have "coprime ((a div gcd a b) ^ n) ((b div gcd a b) ^ n)"
haftmann@60690
  1287
    by (simp add: coprime_exp2)
haftmann@60690
  1288
  then have "gcd ((a div gcd a b)^n * (gcd a b)^n)
huffman@31706
  1289
      ((b div gcd a b)^n * (gcd a b)^n) = (gcd a b)^n"
lp15@60162
  1290
    by (metis gcd_mult_distrib_nat mult.commute mult.right_neutral)
huffman@31706
  1291
  also have "(a div gcd a b)^n * (gcd a b)^n = a^n"
lp15@60162
  1292
    by (metis dvd_div_mult_self gcd_unique_nat power_mult_distrib)
huffman@31706
  1293
  also have "(b div gcd a b)^n * (gcd a b)^n = b^n"
lp15@60162
  1294
    by (metis dvd_div_mult_self gcd_unique_nat power_mult_distrib)
huffman@31706
  1295
  finally show ?thesis .
huffman@31706
  1296
qed
huffman@31706
  1297
nipkow@31952
  1298
lemma gcd_exp_int: "gcd ((a::int)^n) (b^n) = (gcd a b)^n"
nipkow@31952
  1299
  apply (subst (1 2) gcd_abs_int)
huffman@31706
  1300
  apply (subst (1 2) power_abs)
nipkow@31952
  1301
  apply (rule gcd_exp_nat [where n = n, transferred])
huffman@31706
  1302
  apply auto
huffman@31706
  1303
done
huffman@31706
  1304
nipkow@31952
  1305
lemma division_decomp_nat: assumes dc: "(a::nat) dvd b * c"
huffman@31706
  1306
  shows "\<exists>b' c'. a = b' * c' \<and> b' dvd b \<and> c' dvd c"
huffman@31706
  1307
proof-
huffman@31706
  1308
  let ?g = "gcd a b"
huffman@31706
  1309
  {assume "?g = 0" with dc have ?thesis by auto}
huffman@31706
  1310
  moreover
huffman@31706
  1311
  {assume z: "?g \<noteq> 0"
nipkow@31952
  1312
    from gcd_coprime_exists_nat[OF z]
huffman@31706
  1313
    obtain a' b' where ab': "a = a' * ?g" "b = b' * ?g" "coprime a' b'"
huffman@31706
  1314
      by blast
huffman@31706
  1315
    have thb: "?g dvd b" by auto
huffman@31706
  1316
    from ab'(1) have "a' dvd a"  unfolding dvd_def by blast
huffman@31706
  1317
    with dc have th0: "a' dvd b*c" using dvd_trans[of a' a "b*c"] by simp
huffman@31706
  1318
    from dc ab'(1,2) have "a'*?g dvd (b'*?g) *c" by auto
haftmann@57512
  1319
    hence "?g*a' dvd ?g * (b' * c)" by (simp add: mult.assoc)
huffman@31706
  1320
    with z have th_1: "a' dvd b' * c" by auto
haftmann@62344
  1321
    from coprime_dvd_mult [OF ab'(3)] th_1
haftmann@57512
  1322
    have thc: "a' dvd c" by (subst (asm) mult.commute, blast)
huffman@31706
  1323
    from ab' have "a = ?g*a'" by algebra
huffman@31706
  1324
    with thb thc have ?thesis by blast }
huffman@31706
  1325
  ultimately show ?thesis by blast
huffman@31706
  1326
qed
huffman@31706
  1327
nipkow@31952
  1328
lemma division_decomp_int: assumes dc: "(a::int) dvd b * c"
huffman@31706
  1329
  shows "\<exists>b' c'. a = b' * c' \<and> b' dvd b \<and> c' dvd c"
huffman@31706
  1330
proof-
huffman@31706
  1331
  let ?g = "gcd a b"
huffman@31706
  1332
  {assume "?g = 0" with dc have ?thesis by auto}
huffman@31706
  1333
  moreover
huffman@31706
  1334
  {assume z: "?g \<noteq> 0"
nipkow@31952
  1335
    from gcd_coprime_exists_int[OF z]
huffman@31706
  1336
    obtain a' b' where ab': "a = a' * ?g" "b = b' * ?g" "coprime a' b'"
huffman@31706
  1337
      by blast
huffman@31706
  1338
    have thb: "?g dvd b" by auto
huffman@31706
  1339
    from ab'(1) have "a' dvd a"  unfolding dvd_def by blast
huffman@31706
  1340
    with dc have th0: "a' dvd b*c"
huffman@31706
  1341
      using dvd_trans[of a' a "b*c"] by simp
huffman@31706
  1342
    from dc ab'(1,2) have "a'*?g dvd (b'*?g) *c" by auto
haftmann@60690
  1343
    hence "?g*a' dvd ?g * (b' * c)" by (simp add: ac_simps)
huffman@31706
  1344
    with z have th_1: "a' dvd b' * c" by auto
haftmann@62344
  1345
    from coprime_dvd_mult [OF ab'(3)] th_1
haftmann@57512
  1346
    have thc: "a' dvd c" by (subst (asm) mult.commute, blast)
huffman@31706
  1347
    from ab' have "a = ?g*a'" by algebra
huffman@31706
  1348
    with thb thc have ?thesis by blast }
huffman@31706
  1349
  ultimately show ?thesis by blast
chaieb@27669
  1350
qed
chaieb@27669
  1351
nipkow@31952
  1352
lemma pow_divides_pow_nat:
huffman@31706
  1353
  assumes ab: "(a::nat) ^ n dvd b ^n" and n:"n \<noteq> 0"
huffman@31706
  1354
  shows "a dvd b"
huffman@31706
  1355
proof-
huffman@31706
  1356
  let ?g = "gcd a b"
huffman@31706
  1357
  from n obtain m where m: "n = Suc m" by (cases n, simp_all)
huffman@31706
  1358
  {assume "?g = 0" with ab n have ?thesis by auto }
huffman@31706
  1359
  moreover
huffman@31706
  1360
  {assume z: "?g \<noteq> 0"
huffman@35216
  1361
    hence zn: "?g ^ n \<noteq> 0" using n by simp
nipkow@31952
  1362
    from gcd_coprime_exists_nat[OF z]
huffman@31706
  1363
    obtain a' b' where ab': "a = a' * ?g" "b = b' * ?g" "coprime a' b'"
huffman@31706
  1364
      by blast
huffman@31706
  1365
    from ab have "(a' * ?g) ^ n dvd (b' * ?g)^n"
huffman@31706
  1366
      by (simp add: ab'(1,2)[symmetric])
huffman@31706
  1367
    hence "?g^n*a'^n dvd ?g^n *b'^n"
haftmann@57512
  1368
      by (simp only: power_mult_distrib mult.commute)
haftmann@58787
  1369
    then have th0: "a'^n dvd b'^n"
haftmann@58787
  1370
      using zn by auto
huffman@31706
  1371
    have "a' dvd a'^n" by (simp add: m)
huffman@31706
  1372
    with th0 have "a' dvd b'^n" using dvd_trans[of a' "a'^n" "b'^n"] by simp
haftmann@57512
  1373
    hence th1: "a' dvd b'^m * b'" by (simp add: m mult.commute)
haftmann@62344
  1374
    from coprime_dvd_mult [OF coprime_exp_nat [OF ab'(3), of m]] th1
haftmann@57512
  1375
    have "a' dvd b'" by (subst (asm) mult.commute, blast)
huffman@31706
  1376
    hence "a'*?g dvd b'*?g" by simp
huffman@31706
  1377
    with ab'(1,2)  have ?thesis by simp }
huffman@31706
  1378
  ultimately show ?thesis by blast
huffman@31706
  1379
qed
huffman@31706
  1380
nipkow@31952
  1381
lemma pow_divides_pow_int:
huffman@31706
  1382
  assumes ab: "(a::int) ^ n dvd b ^n" and n:"n \<noteq> 0"
huffman@31706
  1383
  shows "a dvd b"
chaieb@27669
  1384
proof-
huffman@31706
  1385
  let ?g = "gcd a b"
huffman@31706
  1386
  from n obtain m where m: "n = Suc m" by (cases n, simp_all)
huffman@31706
  1387
  {assume "?g = 0" with ab n have ?thesis by auto }
huffman@31706
  1388
  moreover
huffman@31706
  1389
  {assume z: "?g \<noteq> 0"
huffman@35216
  1390
    hence zn: "?g ^ n \<noteq> 0" using n by simp
nipkow@31952
  1391
    from gcd_coprime_exists_int[OF z]
huffman@31706
  1392
    obtain a' b' where ab': "a = a' * ?g" "b = b' * ?g" "coprime a' b'"
huffman@31706
  1393
      by blast
huffman@31706
  1394
    from ab have "(a' * ?g) ^ n dvd (b' * ?g)^n"
huffman@31706
  1395
      by (simp add: ab'(1,2)[symmetric])
huffman@31706
  1396
    hence "?g^n*a'^n dvd ?g^n *b'^n"
haftmann@57512
  1397
      by (simp only: power_mult_distrib mult.commute)
huffman@31706
  1398
    with zn z n have th0:"a'^n dvd b'^n" by auto
huffman@31706
  1399
    have "a' dvd a'^n" by (simp add: m)
huffman@31706
  1400
    with th0 have "a' dvd b'^n"
huffman@31706
  1401
      using dvd_trans[of a' "a'^n" "b'^n"] by simp
haftmann@60596
  1402
    hence th1: "a' dvd b'^m * b'" by (simp add: m mult.commute)
haftmann@62344
  1403
    from coprime_dvd_mult [OF coprime_exp_int [OF ab'(3), of m]] th1
haftmann@57512
  1404
    have "a' dvd b'" by (subst (asm) mult.commute, blast)
huffman@31706
  1405
    hence "a'*?g dvd b'*?g" by simp
huffman@31706
  1406
    with ab'(1,2)  have ?thesis by simp }
huffman@31706
  1407
  ultimately show ?thesis by blast
huffman@31706
  1408
qed
huffman@31706
  1409
nipkow@31952
  1410
lemma pow_divides_eq_nat [simp]: "n ~= 0 \<Longrightarrow> ((a::nat)^n dvd b^n) = (a dvd b)"
nipkow@31952
  1411
  by (auto intro: pow_divides_pow_nat dvd_power_same)
huffman@31706
  1412
nipkow@31952
  1413
lemma pow_divides_eq_int [simp]: "n ~= 0 \<Longrightarrow> ((a::int)^n dvd b^n) = (a dvd b)"
nipkow@31952
  1414
  by (auto intro: pow_divides_pow_int dvd_power_same)
huffman@31706
  1415
nipkow@31952
  1416
lemma divides_mult_nat:
huffman@31706
  1417
  assumes mr: "(m::nat) dvd r" and nr: "n dvd r" and mn:"coprime m n"
huffman@31706
  1418
  shows "m * n dvd r"
huffman@31706
  1419
proof-
huffman@31706
  1420
  from mr nr obtain m' n' where m': "r = m*m'" and n': "r = n*n'"
huffman@31706
  1421
    unfolding dvd_def by blast
haftmann@57512
  1422
  from mr n' have "m dvd n'*n" by (simp add: mult.commute)
haftmann@62344
  1423
  hence "m dvd n'" using coprime_dvd_mult_iff [OF mn] by simp
huffman@31706
  1424
  then obtain k where k: "n' = m*k" unfolding dvd_def by blast
huffman@31706
  1425
  from n' k show ?thesis unfolding dvd_def by auto
huffman@31706
  1426
qed
huffman@31706
  1427
nipkow@31952
  1428
lemma divides_mult_int:
huffman@31706
  1429
  assumes mr: "(m::int) dvd r" and nr: "n dvd r" and mn:"coprime m n"
huffman@31706
  1430
  shows "m * n dvd r"
huffman@31706
  1431
proof-
huffman@31706
  1432
  from mr nr obtain m' n' where m': "r = m*m'" and n': "r = n*n'"
huffman@31706
  1433
    unfolding dvd_def by blast
haftmann@57512
  1434
  from mr n' have "m dvd n'*n" by (simp add: mult.commute)
haftmann@62344
  1435
  hence "m dvd n'" using coprime_dvd_mult_iff [OF mn] by simp
huffman@31706
  1436
  then obtain k where k: "n' = m*k" unfolding dvd_def by blast
huffman@31706
  1437
  from n' k show ?thesis unfolding dvd_def by auto
chaieb@27669
  1438
qed
chaieb@27669
  1439
nipkow@31952
  1440
lemma coprime_plus_one_nat [simp]: "coprime ((n::nat) + 1) n"
haftmann@60690
  1441
  by (simp add: gcd.commute del: One_nat_def)
huffman@31706
  1442
nipkow@31952
  1443
lemma coprime_Suc_nat [simp]: "coprime (Suc n) n"
haftmann@60690
  1444
  using coprime_plus_one_nat by simp
huffman@31706
  1445
nipkow@31952
  1446
lemma coprime_plus_one_int [simp]: "coprime ((n::int) + 1) n"
haftmann@60686
  1447
  by (simp add: gcd.commute)
huffman@31706
  1448
nipkow@31952
  1449
lemma coprime_minus_one_nat: "(n::nat) \<noteq> 0 \<Longrightarrow> coprime (n - 1) n"
nipkow@31952
  1450
  using coprime_plus_one_nat [of "n - 1"]
haftmann@62344
  1451
    gcd.commute [of "n - 1" n] by auto
huffman@31706
  1452
nipkow@31952
  1453
lemma coprime_minus_one_int: "coprime ((n::int) - 1) n"
nipkow@31952
  1454
  using coprime_plus_one_int [of "n - 1"]
haftmann@62344
  1455
    gcd.commute [of "n - 1" n] by auto
huffman@31706
  1456
haftmann@62344
  1457
lemma setprod_coprime_nat:
haftmann@62344
  1458
  fixes x :: nat
haftmann@62344
  1459
  shows "(\<And>i. i \<in> A \<Longrightarrow> coprime (f i) x) \<Longrightarrow> coprime (\<Prod>i\<in>A. f i) x"
haftmann@62344
  1460
  by (induct A rule: infinite_finite_induct)
haftmann@62344
  1461
    (auto simp add: gcd_mult_cancel One_nat_def [symmetric] simp del: One_nat_def)
huffman@31706
  1462
haftmann@62344
  1463
lemma setprod_coprime_int:
haftmann@62344
  1464
  fixes x :: int
haftmann@62344
  1465
  shows "(\<And>i. i \<in> A \<Longrightarrow> coprime (f i) x) \<Longrightarrow> coprime (\<Prod>i\<in>A. f i) x"
haftmann@62344
  1466
  by (induct A rule: infinite_finite_induct)
haftmann@62344
  1467
    (auto simp add: gcd_mult_cancel)
huffman@31706
  1468
lp15@60162
  1469
lemma coprime_common_divisor_nat: 
haftmann@60686
  1470
  "coprime (a::nat) b \<Longrightarrow> x dvd a \<Longrightarrow> x dvd b \<Longrightarrow> x = 1"
haftmann@62344
  1471
  by (metis gcd_greatest_iff nat_dvd_1_iff_1)
huffman@31706
  1472
lp15@60162
  1473
lemma coprime_common_divisor_int:
wenzelm@61944
  1474
  "coprime (a::int) b \<Longrightarrow> x dvd a \<Longrightarrow> x dvd b \<Longrightarrow> \<bar>x\<bar> = 1"
haftmann@60686
  1475
  using gcd_greatest_iff [of x a b] by auto
huffman@31706
  1476
lp15@60162
  1477
lemma coprime_divisors_nat:
lp15@60162
  1478
    "(d::int) dvd a \<Longrightarrow> e dvd b \<Longrightarrow> coprime a b \<Longrightarrow> coprime d e"
lp15@60162
  1479
  by (meson coprime_int dvd_trans gcd_dvd1 gcd_dvd2 gcd_ge_0_int)
huffman@31706
  1480
nipkow@31952
  1481
lemma invertible_coprime_nat: "(x::nat) * y mod m = 1 \<Longrightarrow> coprime x m"
haftmann@62344
  1482
by (metis coprime_lmult_nat gcd_1_nat gcd.commute gcd_red_nat)
huffman@31706
  1483
nipkow@31952
  1484
lemma invertible_coprime_int: "(x::int) * y mod m = 1 \<Longrightarrow> coprime x m"
haftmann@62344
  1485
by (metis coprime_lmult_int gcd_1_int gcd.commute gcd_red_int)
huffman@31706
  1486
huffman@31706
  1487
wenzelm@60758
  1488
subsection \<open>Bezout's theorem\<close>
huffman@31706
  1489
huffman@31706
  1490
(* Function bezw returns a pair of witnesses to Bezout's theorem --
huffman@31706
  1491
   see the theorems that follow the definition. *)
huffman@31706
  1492
fun
huffman@31706
  1493
  bezw  :: "nat \<Rightarrow> nat \<Rightarrow> int * int"
huffman@31706
  1494
where
huffman@31706
  1495
  "bezw x y =
huffman@31706
  1496
  (if y = 0 then (1, 0) else
huffman@31706
  1497
      (snd (bezw y (x mod y)),
huffman@31706
  1498
       fst (bezw y (x mod y)) - snd (bezw y (x mod y)) * int(x div y)))"
huffman@31706
  1499
huffman@31706
  1500
lemma bezw_0 [simp]: "bezw x 0 = (1, 0)" by simp
huffman@31706
  1501
huffman@31706
  1502
lemma bezw_non_0: "y > 0 \<Longrightarrow> bezw x y = (snd (bezw y (x mod y)),
huffman@31706
  1503
       fst (bezw y (x mod y)) - snd (bezw y (x mod y)) * int(x div y))"
huffman@31706
  1504
  by simp
huffman@31706
  1505
huffman@31706
  1506
declare bezw.simps [simp del]
huffman@31706
  1507
huffman@31706
  1508
lemma bezw_aux [rule_format]:
huffman@31706
  1509
    "fst (bezw x y) * int x + snd (bezw x y) * int y = int (gcd x y)"
nipkow@31952
  1510
proof (induct x y rule: gcd_nat_induct)
huffman@31706
  1511
  fix m :: nat
huffman@31706
  1512
  show "fst (bezw m 0) * int m + snd (bezw m 0) * int 0 = int (gcd m 0)"
huffman@31706
  1513
    by auto
huffman@31706
  1514
  next fix m :: nat and n
huffman@31706
  1515
    assume ngt0: "n > 0" and
huffman@31706
  1516
      ih: "fst (bezw n (m mod n)) * int n +
huffman@31706
  1517
        snd (bezw n (m mod n)) * int (m mod n) =
huffman@31706
  1518
        int (gcd n (m mod n))"
huffman@31706
  1519
    thus "fst (bezw m n) * int m + snd (bezw m n) * int n = int (gcd m n)"
nipkow@31952
  1520
      apply (simp add: bezw_non_0 gcd_non_0_nat)
huffman@31706
  1521
      apply (erule subst)
haftmann@36350
  1522
      apply (simp add: field_simps)
huffman@31706
  1523
      apply (subst mod_div_equality [of m n, symmetric])
huffman@31706
  1524
      (* applying simp here undoes the last substitution!
huffman@31706
  1525
         what is procedure cancel_div_mod? *)
hoelzl@58776
  1526
      apply (simp only: NO_MATCH_def field_simps of_nat_add of_nat_mult)
huffman@31706
  1527
      done
huffman@31706
  1528
qed
huffman@31706
  1529
nipkow@31952
  1530
lemma bezout_int:
huffman@31706
  1531
  fixes x y
huffman@31706
  1532
  shows "EX u v. u * (x::int) + v * y = gcd x y"
huffman@31706
  1533
proof -
huffman@31706
  1534
  have bezout_aux: "!!x y. x \<ge> (0::int) \<Longrightarrow> y \<ge> 0 \<Longrightarrow>
huffman@31706
  1535
      EX u v. u * x + v * y = gcd x y"
huffman@31706
  1536
    apply (rule_tac x = "fst (bezw (nat x) (nat y))" in exI)
huffman@31706
  1537
    apply (rule_tac x = "snd (bezw (nat x) (nat y))" in exI)
huffman@31706
  1538
    apply (unfold gcd_int_def)
huffman@31706
  1539
    apply simp
huffman@31706
  1540
    apply (subst bezw_aux [symmetric])
huffman@31706
  1541
    apply auto
huffman@31706
  1542
    done
huffman@31706
  1543
  have "(x \<ge> 0 \<and> y \<ge> 0) | (x \<ge> 0 \<and> y \<le> 0) | (x \<le> 0 \<and> y \<ge> 0) |
huffman@31706
  1544
      (x \<le> 0 \<and> y \<le> 0)"
huffman@31706
  1545
    by auto
huffman@31706
  1546
  moreover have "x \<ge> 0 \<Longrightarrow> y \<ge> 0 \<Longrightarrow> ?thesis"
huffman@31706
  1547
    by (erule (1) bezout_aux)
huffman@31706
  1548
  moreover have "x >= 0 \<Longrightarrow> y <= 0 \<Longrightarrow> ?thesis"
huffman@31706
  1549
    apply (insert bezout_aux [of x "-y"])
huffman@31706
  1550
    apply auto
huffman@31706
  1551
    apply (rule_tac x = u in exI)
huffman@31706
  1552
    apply (rule_tac x = "-v" in exI)
nipkow@31952
  1553
    apply (subst gcd_neg2_int [symmetric])
huffman@31706
  1554
    apply auto
huffman@31706
  1555
    done
huffman@31706
  1556
  moreover have "x <= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> ?thesis"
huffman@31706
  1557
    apply (insert bezout_aux [of "-x" y])
huffman@31706
  1558
    apply auto
huffman@31706
  1559
    apply (rule_tac x = "-u" in exI)
huffman@31706
  1560
    apply (rule_tac x = v in exI)
nipkow@31952
  1561
    apply (subst gcd_neg1_int [symmetric])
huffman@31706
  1562
    apply auto
huffman@31706
  1563
    done
huffman@31706
  1564
  moreover have "x <= 0 \<Longrightarrow> y <= 0 \<Longrightarrow> ?thesis"
huffman@31706
  1565
    apply (insert bezout_aux [of "-x" "-y"])
huffman@31706
  1566
    apply auto
huffman@31706
  1567
    apply (rule_tac x = "-u" in exI)
huffman@31706
  1568
    apply (rule_tac x = "-v" in exI)
nipkow@31952
  1569
    apply (subst gcd_neg1_int [symmetric])
nipkow@31952
  1570
    apply (subst gcd_neg2_int [symmetric])
huffman@31706
  1571
    apply auto
huffman@31706
  1572
    done
huffman@31706
  1573
  ultimately show ?thesis by blast
huffman@31706
  1574
qed
huffman@31706
  1575
wenzelm@60758
  1576
text \<open>versions of Bezout for nat, by Amine Chaieb\<close>
huffman@31706
  1577
huffman@31706
  1578
lemma ind_euclid:
huffman@31706
  1579
  assumes c: " \<forall>a b. P (a::nat) b \<longleftrightarrow> P b a" and z: "\<forall>a. P a 0"
huffman@31706
  1580
  and add: "\<forall>a b. P a b \<longrightarrow> P a (a + b)"
chaieb@27669
  1581
  shows "P a b"
berghofe@34915
  1582
proof(induct "a + b" arbitrary: a b rule: less_induct)
berghofe@34915
  1583
  case less
chaieb@27669
  1584
  have "a = b \<or> a < b \<or> b < a" by arith
chaieb@27669
  1585
  moreover {assume eq: "a= b"
huffman@31706
  1586
    from add[rule_format, OF z[rule_format, of a]] have "P a b" using eq
huffman@31706
  1587
    by simp}
chaieb@27669
  1588
  moreover
chaieb@27669
  1589
  {assume lt: "a < b"
berghofe@34915
  1590
    hence "a + b - a < a + b \<or> a = 0" by arith
chaieb@27669
  1591
    moreover
chaieb@27669
  1592
    {assume "a =0" with z c have "P a b" by blast }
chaieb@27669
  1593
    moreover
berghofe@34915
  1594
    {assume "a + b - a < a + b"
berghofe@34915
  1595
      also have th0: "a + b - a = a + (b - a)" using lt by arith
berghofe@34915
  1596
      finally have "a + (b - a) < a + b" .
berghofe@34915
  1597
      then have "P a (a + (b - a))" by (rule add[rule_format, OF less])
berghofe@34915
  1598
      then have "P a b" by (simp add: th0[symmetric])}
chaieb@27669
  1599
    ultimately have "P a b" by blast}
chaieb@27669
  1600
  moreover
chaieb@27669
  1601
  {assume lt: "a > b"
berghofe@34915
  1602
    hence "b + a - b < a + b \<or> b = 0" by arith
chaieb@27669
  1603
    moreover
chaieb@27669
  1604
    {assume "b =0" with z c have "P a b" by blast }
chaieb@27669
  1605
    moreover
berghofe@34915
  1606
    {assume "b + a - b < a + b"
berghofe@34915
  1607
      also have th0: "b + a - b = b + (a - b)" using lt by arith
berghofe@34915
  1608
      finally have "b + (a - b) < a + b" .
berghofe@34915
  1609
      then have "P b (b + (a - b))" by (rule add[rule_format, OF less])
berghofe@34915
  1610
      then have "P b a" by (simp add: th0[symmetric])
chaieb@27669
  1611
      hence "P a b" using c by blast }
chaieb@27669
  1612
    ultimately have "P a b" by blast}
chaieb@27669
  1613
ultimately  show "P a b" by blast
chaieb@27669
  1614
qed
chaieb@27669
  1615
nipkow@31952
  1616
lemma bezout_lemma_nat:
huffman@31706
  1617
  assumes ex: "\<exists>(d::nat) x y. d dvd a \<and> d dvd b \<and>
huffman@31706
  1618
    (a * x = b * y + d \<or> b * x = a * y + d)"
huffman@31706
  1619
  shows "\<exists>d x y. d dvd a \<and> d dvd a + b \<and>
huffman@31706
  1620
    (a * x = (a + b) * y + d \<or> (a + b) * x = a * y + d)"
huffman@31706
  1621
  using ex
huffman@31706
  1622
  apply clarsimp
huffman@35216
  1623
  apply (rule_tac x="d" in exI, simp)
huffman@31706
  1624
  apply (case_tac "a * x = b * y + d" , simp_all)
huffman@31706
  1625
  apply (rule_tac x="x + y" in exI)
huffman@31706
  1626
  apply (rule_tac x="y" in exI)
huffman@31706
  1627
  apply algebra
huffman@31706
  1628
  apply (rule_tac x="x" in exI)
huffman@31706
  1629
  apply (rule_tac x="x + y" in exI)
huffman@31706
  1630
  apply algebra
chaieb@27669
  1631
done
chaieb@27669
  1632
nipkow@31952
  1633
lemma bezout_add_nat: "\<exists>(d::nat) x y. d dvd a \<and> d dvd b \<and>
huffman@31706
  1634
    (a * x = b * y + d \<or> b * x = a * y + d)"
huffman@31706
  1635
  apply(induct a b rule: ind_euclid)
huffman@31706
  1636
  apply blast
huffman@31706
  1637
  apply clarify
huffman@35216
  1638
  apply (rule_tac x="a" in exI, simp)
huffman@31706
  1639
  apply clarsimp
huffman@31706
  1640
  apply (rule_tac x="d" in exI)
huffman@35216
  1641
  apply (case_tac "a * x = b * y + d", simp_all)
huffman@31706
  1642
  apply (rule_tac x="x+y" in exI)
huffman@31706
  1643
  apply (rule_tac x="y" in exI)
huffman@31706
  1644
  apply algebra
huffman@31706
  1645
  apply (rule_tac x="x" in exI)
huffman@31706
  1646
  apply (rule_tac x="x+y" in exI)
huffman@31706
  1647
  apply algebra
chaieb@27669
  1648
done
chaieb@27669
  1649
nipkow@31952
  1650
lemma bezout1_nat: "\<exists>(d::nat) x y. d dvd a \<and> d dvd b \<and>
huffman@31706
  1651
    (a * x - b * y = d \<or> b * x - a * y = d)"
nipkow@31952
  1652
  using bezout_add_nat[of a b]
huffman@31706
  1653
  apply clarsimp
huffman@31706
  1654
  apply (rule_tac x="d" in exI, simp)
huffman@31706
  1655
  apply (rule_tac x="x" in exI)
huffman@31706
  1656
  apply (rule_tac x="y" in exI)
huffman@31706
  1657
  apply auto
chaieb@27669
  1658
done
chaieb@27669
  1659
nipkow@31952
  1660
lemma bezout_add_strong_nat: assumes nz: "a \<noteq> (0::nat)"
chaieb@27669
  1661
  shows "\<exists>d x y. d dvd a \<and> d dvd b \<and> a * x = b * y + d"
chaieb@27669
  1662
proof-
huffman@31706
  1663
 from nz have ap: "a > 0" by simp
nipkow@31952
  1664
 from bezout_add_nat[of a b]
huffman@31706
  1665
 have "(\<exists>d x y. d dvd a \<and> d dvd b \<and> a * x = b * y + d) \<or>
huffman@31706
  1666
   (\<exists>d x y. d dvd a \<and> d dvd b \<and> b * x = a * y + d)" by blast
chaieb@27669
  1667
 moreover
huffman@31706
  1668
    {fix d x y assume H: "d dvd a" "d dvd b" "a * x = b * y + d"
huffman@31706
  1669
     from H have ?thesis by blast }
chaieb@27669
  1670
 moreover
chaieb@27669
  1671
 {fix d x y assume H: "d dvd a" "d dvd b" "b * x = a * y + d"
chaieb@27669
  1672
   {assume b0: "b = 0" with H  have ?thesis by simp}
huffman@31706
  1673
   moreover
chaieb@27669
  1674
   {assume b: "b \<noteq> 0" hence bp: "b > 0" by simp
huffman@31706
  1675
     from b dvd_imp_le [OF H(2)] have "d < b \<or> d = b"
huffman@31706
  1676
       by auto
chaieb@27669
  1677
     moreover
chaieb@27669
  1678
     {assume db: "d=b"
wenzelm@41550
  1679
       with nz H have ?thesis apply simp
wenzelm@32960
  1680
         apply (rule exI[where x = b], simp)
wenzelm@32960
  1681
         apply (rule exI[where x = b])
wenzelm@32960
  1682
        by (rule exI[where x = "a - 1"], simp add: diff_mult_distrib2)}
chaieb@27669
  1683
    moreover
huffman@31706
  1684
    {assume db: "d < b"
wenzelm@41550
  1685
        {assume "x=0" hence ?thesis using nz H by simp }
wenzelm@32960
  1686
        moreover
wenzelm@32960
  1687
        {assume x0: "x \<noteq> 0" hence xp: "x > 0" by simp
wenzelm@32960
  1688
          from db have "d \<le> b - 1" by simp
wenzelm@32960
  1689
          hence "d*b \<le> b*(b - 1)" by simp
wenzelm@32960
  1690
          with xp mult_mono[of "1" "x" "d*b" "b*(b - 1)"]
wenzelm@32960
  1691
          have dble: "d*b \<le> x*b*(b - 1)" using bp by simp
wenzelm@32960
  1692
          from H (3) have "d + (b - 1) * (b*x) = d + (b - 1) * (a*y + d)"
huffman@31706
  1693
            by simp
wenzelm@32960
  1694
          hence "d + (b - 1) * a * y + (b - 1) * d = d + (b - 1) * b * x"
haftmann@57512
  1695
            by (simp only: mult.assoc distrib_left)
wenzelm@32960
  1696
          hence "a * ((b - 1) * y) + d * (b - 1 + 1) = d + x*b*(b - 1)"
huffman@31706
  1697
            by algebra
wenzelm@32960
  1698
          hence "a * ((b - 1) * y) = d + x*b*(b - 1) - d*b" using bp by simp
wenzelm@32960
  1699
          hence "a * ((b - 1) * y) = d + (x*b*(b - 1) - d*b)"
wenzelm@32960
  1700
            by (simp only: diff_add_assoc[OF dble, of d, symmetric])
wenzelm@32960
  1701
          hence "a * ((b - 1) * y) = b*(x*(b - 1) - d) + d"
haftmann@59008
  1702
            by (simp only: diff_mult_distrib2 ac_simps)
wenzelm@32960
  1703
          hence ?thesis using H(1,2)
wenzelm@32960
  1704
            apply -
wenzelm@32960
  1705
            apply (rule exI[where x=d], simp)
wenzelm@32960
  1706
            apply (rule exI[where x="(b - 1) * y"])
wenzelm@32960
  1707
            by (rule exI[where x="x*(b - 1) - d"], simp)}
wenzelm@32960
  1708
        ultimately have ?thesis by blast}
chaieb@27669
  1709
    ultimately have ?thesis by blast}
chaieb@27669
  1710
  ultimately have ?thesis by blast}
chaieb@27669
  1711
 ultimately show ?thesis by blast
chaieb@27669
  1712
qed
chaieb@27669
  1713
nipkow@31952
  1714
lemma bezout_nat: assumes a: "(a::nat) \<noteq> 0"
chaieb@27669
  1715
  shows "\<exists>x y. a * x = b * y + gcd a b"
chaieb@27669
  1716
proof-
chaieb@27669
  1717
  let ?g = "gcd a b"
nipkow@31952
  1718
  from bezout_add_strong_nat[OF a, of b]
chaieb@27669
  1719
  obtain d x y where d: "d dvd a" "d dvd b" "a * x = b * y + d" by blast
chaieb@27669
  1720
  from d(1,2) have "d dvd ?g" by simp
chaieb@27669
  1721
  then obtain k where k: "?g = d*k" unfolding dvd_def by blast
huffman@31706
  1722
  from d(3) have "a * x * k = (b * y + d) *k " by auto
chaieb@27669
  1723
  hence "a * (x * k) = b * (y*k) + ?g" by (algebra add: k)
chaieb@27669
  1724
  thus ?thesis by blast
chaieb@27669
  1725
qed
chaieb@27669
  1726
huffman@31706
  1727
haftmann@62345
  1728
subsection \<open>LCM properties  on @{typ nat} and @{typ int}\<close>
huffman@31706
  1729
wenzelm@61944
  1730
lemma lcm_altdef_int [code]: "lcm (a::int) b = \<bar>a\<bar> * \<bar>b\<bar> div gcd a b"
haftmann@62344
  1731
  by (simp add: lcm_int_def lcm_nat_def zdiv_int gcd_int_def)
huffman@31706
  1732
nipkow@31952
  1733
lemma prod_gcd_lcm_nat: "(m::nat) * n = gcd m n * lcm m n"
huffman@31706
  1734
  unfolding lcm_nat_def
nipkow@31952
  1735
  by (simp add: dvd_mult_div_cancel [OF gcd_dvd_prod_nat])
huffman@31706
  1736
wenzelm@61944
  1737
lemma prod_gcd_lcm_int: "\<bar>m::int\<bar> * \<bar>n\<bar> = gcd m n * lcm m n"
huffman@31706
  1738
  unfolding lcm_int_def gcd_int_def
haftmann@62348
  1739
  apply (subst of_nat_mult [symmetric])
nipkow@31952
  1740
  apply (subst prod_gcd_lcm_nat [symmetric])
huffman@31706
  1741
  apply (subst nat_abs_mult_distrib [symmetric])
huffman@31706
  1742
  apply (simp, simp add: abs_mult)
huffman@31706
  1743
done
huffman@31706
  1744
nipkow@31952
  1745
lemma lcm_0_nat [simp]: "lcm (m::nat) 0 = 0"
huffman@31706
  1746
  unfolding lcm_nat_def by simp
huffman@31706
  1747
nipkow@31952
  1748
lemma lcm_0_int [simp]: "lcm (m::int) 0 = 0"
huffman@31706
  1749
  unfolding lcm_int_def by simp
huffman@31706
  1750
nipkow@31952
  1751
lemma lcm_0_left_nat [simp]: "lcm (0::nat) n = 0"
huffman@31706
  1752
  unfolding lcm_nat_def by simp
chaieb@27669
  1753
nipkow@31952
  1754
lemma lcm_0_left_int [simp]: "lcm (0::int) n = 0"
huffman@31706
  1755
  unfolding lcm_int_def by simp
huffman@31706
  1756
nipkow@31952
  1757
lemma lcm_pos_nat:
nipkow@31798
  1758
  "(m::nat) > 0 \<Longrightarrow> n>0 \<Longrightarrow> lcm m n > 0"
nipkow@31952
  1759
by (metis gr0I mult_is_0 prod_gcd_lcm_nat)
chaieb@27669
  1760
nipkow@31952
  1761
lemma lcm_pos_int:
nipkow@31798
  1762
  "(m::int) ~= 0 \<Longrightarrow> n ~= 0 \<Longrightarrow> lcm m n > 0"
nipkow@31952
  1763
  apply (subst lcm_abs_int)
nipkow@31952
  1764
  apply (rule lcm_pos_nat [transferred])
nipkow@31798
  1765
  apply auto
haftmann@62344
  1766
  done
haftmann@23687
  1767
nipkow@31952
  1768
lemma dvd_pos_nat:
haftmann@23687
  1769
  fixes n m :: nat
haftmann@23687
  1770
  assumes "n > 0" and "m dvd n"
haftmann@23687
  1771
  shows "m > 0"
haftmann@62344
  1772
  using assms by (cases m) auto
nipkow@31729
  1773
nipkow@31952
  1774
lemma lcm_unique_nat: "(a::nat) dvd d \<and> b dvd d \<and>
huffman@31706
  1775
    (\<forall>e. a dvd e \<and> b dvd e \<longrightarrow> d dvd e) \<longleftrightarrow> d = lcm a b"
haftmann@62344
  1776
  by (auto intro: dvd_antisym lcm_least)
chaieb@27568
  1777
nipkow@31952
  1778
lemma lcm_unique_int: "d >= 0 \<and> (a::int) dvd d \<and> b dvd d \<and>
huffman@31706
  1779
    (\<forall>e. a dvd e \<and> b dvd e \<longrightarrow> d dvd e) \<longleftrightarrow> d = lcm a b"
haftmann@62344
  1780
  using lcm_least zdvd_antisym_nonneg by auto
haftmann@34973
  1781
nipkow@31798
  1782
lemma lcm_proj2_if_dvd_nat [simp]: "(x::nat) dvd y \<Longrightarrow> lcm x y = y"
huffman@31706
  1783
  apply (rule sym)
nipkow@31952
  1784
  apply (subst lcm_unique_nat [symmetric])
huffman@31706
  1785
  apply auto
huffman@31706
  1786
done
huffman@31706
  1787
wenzelm@61944
  1788
lemma lcm_proj2_if_dvd_int [simp]: "(x::int) dvd y \<Longrightarrow> lcm x y = \<bar>y\<bar>"
huffman@31706
  1789
  apply (rule sym)
nipkow@31952
  1790
  apply (subst lcm_unique_int [symmetric])
huffman@31706
  1791
  apply auto
huffman@31706
  1792
done
huffman@31706
  1793
nipkow@31798
  1794
lemma lcm_proj1_if_dvd_nat [simp]: "(x::nat) dvd y \<Longrightarrow> lcm y x = y"
haftmann@62344
  1795
by (subst lcm.commute, erule lcm_proj2_if_dvd_nat)
huffman@31706
  1796
wenzelm@61944
  1797
lemma lcm_proj1_if_dvd_int [simp]: "(x::int) dvd y \<Longrightarrow> lcm y x = \<bar>y\<bar>"
haftmann@62344
  1798
by (subst lcm.commute, erule lcm_proj2_if_dvd_int)
huffman@31706
  1799
nipkow@31992
  1800
lemma lcm_proj1_iff_nat[simp]: "lcm m n = (m::nat) \<longleftrightarrow> n dvd m"
nipkow@31992
  1801
by (metis lcm_proj1_if_dvd_nat lcm_unique_nat)
nipkow@31992
  1802
nipkow@31992
  1803
lemma lcm_proj2_iff_nat[simp]: "lcm m n = (n::nat) \<longleftrightarrow> m dvd n"
nipkow@31992
  1804
by (metis lcm_proj2_if_dvd_nat lcm_unique_nat)
nipkow@31992
  1805
wenzelm@61944
  1806
lemma lcm_proj1_iff_int[simp]: "lcm m n = \<bar>m::int\<bar> \<longleftrightarrow> n dvd m"
nipkow@31992
  1807
by (metis dvd_abs_iff lcm_proj1_if_dvd_int lcm_unique_int)
nipkow@31992
  1808
wenzelm@61944
  1809
lemma lcm_proj2_iff_int[simp]: "lcm m n = \<bar>n::int\<bar> \<longleftrightarrow> m dvd n"
nipkow@31992
  1810
by (metis dvd_abs_iff lcm_proj2_if_dvd_int lcm_unique_int)
chaieb@27568
  1811
haftmann@61913
  1812
lemma (in semiring_gcd) comp_fun_idem_gcd:
haftmann@61913
  1813
  "comp_fun_idem gcd"
haftmann@61913
  1814
  by standard (simp_all add: fun_eq_iff ac_simps)
haftmann@61913
  1815
haftmann@61913
  1816
lemma (in semiring_gcd) comp_fun_idem_lcm:
haftmann@61913
  1817
  "comp_fun_idem lcm"
haftmann@61913
  1818
  by standard (simp_all add: fun_eq_iff ac_simps)
haftmann@61913
  1819
haftmann@61913
  1820
lemma lcm_1_iff_nat [simp]: "lcm (m::nat) n = 1 \<longleftrightarrow> m=1 \<and> n=1"
haftmann@61913
  1821
  by (simp only: lcm_eq_1_iff) simp
haftmann@61913
  1822
  
haftmann@61913
  1823
lemma lcm_1_iff_int [simp]: "lcm (m::int) n = 1 \<longleftrightarrow> (m=1 \<or> m = -1) \<and> (n=1 \<or> n = -1)"
haftmann@61913
  1824
  by auto
nipkow@31995
  1825
haftmann@34030
  1826
haftmann@62345
  1827
subsection \<open>The complete divisibility lattice on @{typ nat} and @{typ int}\<close>
nipkow@32112
  1828
wenzelm@60758
  1829
text\<open>Lifting gcd and lcm to sets (Gcd/Lcm).
huffman@45264
  1830
Gcd is defined via Lcm to facilitate the proof that we have a complete lattice.
wenzelm@60758
  1831
\<close>
huffman@45264
  1832
haftmann@62345
  1833
instantiation nat :: semiring_Gcd
nipkow@32112
  1834
begin
nipkow@32112
  1835
haftmann@62344
  1836
interpretation semilattice_neutr_set lcm "1::nat"
haftmann@62344
  1837
  by standard simp_all
haftmann@54867
  1838
haftmann@61929
  1839
definition
haftmann@61929
  1840
  "Lcm (M::nat set) = (if finite M then F M else 0)"
haftmann@51489
  1841
haftmann@51489
  1842
lemma Lcm_nat_empty:
haftmann@51489
  1843
  "Lcm {} = (1::nat)"
haftmann@60690
  1844
  by (simp add: Lcm_nat_def del: One_nat_def)
haftmann@51489
  1845
haftmann@51489
  1846
lemma Lcm_nat_insert:
haftmann@51489
  1847
  "Lcm (insert n M) = lcm (n::nat) (Lcm M)"
haftmann@61929
  1848
  by (cases "finite M") (auto simp add: Lcm_nat_def simp del: One_nat_def)
haftmann@61929
  1849
haftmann@61929
  1850
lemma Lcm_nat_infinite:
haftmann@61929
  1851
  "infinite M \<Longrightarrow> Lcm M = (0::nat)"
haftmann@61929
  1852
  by (simp add: Lcm_nat_def)
haftmann@61929
  1853
haftmann@61929
  1854
lemma dvd_Lcm_nat [simp]:
haftmann@61929
  1855
  fixes M :: "nat set"
haftmann@61929
  1856
  assumes "m \<in> M"
haftmann@61929
  1857
  shows "m dvd Lcm M"
haftmann@61929
  1858
proof -
haftmann@61929
  1859
  from assms have "insert m M = M" by auto
haftmann@61929
  1860
  moreover have "m dvd Lcm (insert m M)"
haftmann@61929
  1861
    by (simp add: Lcm_nat_insert)
haftmann@61929
  1862
  ultimately show ?thesis by simp
haftmann@61929
  1863
qed
haftmann@61929
  1864
haftmann@61929
  1865
lemma Lcm_dvd_nat [simp]:
haftmann@61929
  1866
  fixes M :: "nat set"
haftmann@61929
  1867
  assumes "\<forall>m\<in>M. m dvd n"
haftmann@61929
  1868
  shows "Lcm M dvd n"
haftmann@61929
  1869
proof (cases "n = 0")
haftmann@61929
  1870
  case True then show ?thesis by simp
haftmann@61929
  1871
next
haftmann@61929
  1872
  case False
haftmann@61929
  1873
  then have "finite {d. d dvd n}" by (rule finite_divisors_nat)
haftmann@61929
  1874
  moreover have "M \<subseteq> {d. d dvd n}" using assms by fast
haftmann@61929
  1875
  ultimately have "finite M" by (rule rev_finite_subset)
haftmann@61929
  1876
  then show ?thesis using assms by (induct M) (simp_all add: Lcm_nat_empty Lcm_nat_insert)
haftmann@61929
  1877
qed
nipkow@32112
  1878
huffman@45264
  1879
definition
huffman@45264
  1880
  "Gcd (M::nat set) = Lcm {d. \<forall>m\<in>M. d dvd m}"
nipkow@32112
  1881
haftmann@62345
  1882
instance proof
haftmann@61929
  1883
  show "Gcd N dvd n" if "n \<in> N" for N and n :: nat
haftmann@61929
  1884
  using that by (induct N rule: infinite_finite_induct)
haftmann@61929
  1885
    (auto simp add: Gcd_nat_def)
haftmann@61929
  1886
  show "n dvd Gcd N" if "\<And>m. m \<in> N \<Longrightarrow> n dvd m" for N and n :: nat
haftmann@61929
  1887
  using that by (induct N rule: infinite_finite_induct)
haftmann@61929
  1888
    (auto simp add: Gcd_nat_def)
haftmann@62345
  1889
  show "n dvd Lcm N" if "n \<in> N" for N and n ::nat
haftmann@62345
  1890
  using that by (induct N rule: infinite_finite_induct)
haftmann@62345
  1891
    auto
haftmann@62345
  1892
  show "Lcm N dvd n" if "\<And>m. m \<in> N \<Longrightarrow> m dvd n" for N and n ::nat
haftmann@62345
  1893
  using that by (induct N rule: infinite_finite_induct)
haftmann@62345
  1894
    auto
haftmann@62345
  1895
qed simp_all
nipkow@32112
  1896
haftmann@62345
  1897
end
haftmann@61913
  1898
haftmann@62346
  1899
lemma Gcd_nat_eq_one:
haftmann@62346
  1900
  "1 \<in> N \<Longrightarrow> Gcd N = (1::nat)"
haftmann@62346
  1901
  by (rule Gcd_eq_1_I) auto
haftmann@62346
  1902
wenzelm@60758
  1903
text\<open>Alternative characterizations of Gcd:\<close>
nipkow@32112
  1904
nipkow@32112
  1905
lemma Gcd_eq_Max: "finite(M::nat set) \<Longrightarrow> M \<noteq> {} \<Longrightarrow> 0 \<notin> M \<Longrightarrow> Gcd M = Max(\<Inter>m\<in>M. {d. d dvd m})"
nipkow@32112
  1906
apply(rule antisym)
nipkow@32112
  1907
 apply(rule Max_ge)
nipkow@32112
  1908
  apply (metis all_not_in_conv finite_divisors_nat finite_INT)
haftmann@60596
  1909
 apply (simp add: Gcd_dvd)
nipkow@32112
  1910
apply (rule Max_le_iff[THEN iffD2])
nipkow@32112
  1911
  apply (metis all_not_in_conv finite_divisors_nat finite_INT)
nipkow@44890
  1912
 apply fastforce
nipkow@32112
  1913
apply clarsimp
haftmann@60596
  1914
apply (metis Gcd_dvd Max_in dvd_0_left dvd_Gcd dvd_imp_le linorder_antisym_conv3 not_less0)
nipkow@32112
  1915
done
nipkow@32112
  1916
nipkow@32112
  1917
lemma Gcd_remove0_nat: "finite M \<Longrightarrow> Gcd M = Gcd (M - {0::nat})"
nipkow@32112
  1918
apply(induct pred:finite)
nipkow@32112
  1919
 apply simp
nipkow@32112
  1920
apply(case_tac "x=0")
nipkow@32112
  1921
 apply simp
nipkow@32112
  1922
apply(subgoal_tac "insert x F - {0} = insert x (F - {0})")
nipkow@32112
  1923
 apply simp
nipkow@32112
  1924
apply blast
nipkow@32112
  1925
done
nipkow@32112
  1926
nipkow@32112
  1927
lemma Lcm_in_lcm_closed_set_nat:
nipkow@32112
  1928
  "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
  1929
apply(induct rule:finite_linorder_min_induct)
nipkow@32112
  1930
 apply simp
nipkow@32112
  1931
apply simp
nipkow@32112
  1932
apply(subgoal_tac "ALL m n :: nat. m:A \<longrightarrow> n:A \<longrightarrow> lcm m n : A")
nipkow@32112
  1933
 apply simp
nipkow@32112
  1934
 apply(case_tac "A={}")
nipkow@32112
  1935
  apply simp
nipkow@32112
  1936
 apply simp
nipkow@32112
  1937
apply (metis lcm_pos_nat lcm_unique_nat linorder_neq_iff nat_dvd_not_less not_less0)
nipkow@32112
  1938
done
nipkow@32112
  1939
nipkow@32112
  1940
lemma Lcm_eq_Max_nat:
nipkow@32112
  1941
  "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
  1942
apply(rule antisym)
nipkow@32112
  1943
 apply(rule Max_ge, assumption)
nipkow@32112
  1944
 apply(erule (2) Lcm_in_lcm_closed_set_nat)
haftmann@62345
  1945
apply (auto simp add: not_le Lcm_0_iff dvd_imp_le leD le_neq_trans)
nipkow@32112
  1946
done
nipkow@32112
  1947
nipkow@34222
  1948
lemma mult_inj_if_coprime_nat:
nipkow@34222
  1949
  "inj_on f A \<Longrightarrow> inj_on g B \<Longrightarrow> ALL a:A. ALL b:B. coprime (f a) (g b)
nipkow@34222
  1950
   \<Longrightarrow> inj_on (%(a,b). f a * g b::nat) (A \<times> B)"
haftmann@61913
  1951
  by (auto simp add: inj_on_def coprime_crossproduct_nat simp del: One_nat_def)
nipkow@34222
  1952
wenzelm@60758
  1953
text\<open>Nitpick:\<close>
nipkow@34222
  1954
blanchet@41792
  1955
lemma gcd_eq_nitpick_gcd [nitpick_unfold]: "gcd x y = Nitpick.nat_gcd x y"
blanchet@41792
  1956
by (induct x y rule: nat_gcd.induct)
blanchet@41792
  1957
   (simp add: gcd_nat.simps Nitpick.nat_gcd.simps)
blanchet@33197
  1958
blanchet@41792
  1959
lemma lcm_eq_nitpick_lcm [nitpick_unfold]: "lcm x y = Nitpick.nat_lcm x y"
blanchet@33197
  1960
by (simp only: lcm_nat_def Nitpick.nat_lcm_def gcd_eq_nitpick_gcd)
blanchet@33197
  1961
haftmann@54867
  1962
wenzelm@60758
  1963
subsubsection \<open>Setwise gcd and lcm for integers\<close>
huffman@45264
  1964
haftmann@62345
  1965
instantiation int :: semiring_Gcd
huffman@45264
  1966
begin
huffman@45264
  1967
huffman@45264
  1968
definition
haftmann@62346
  1969
  "Lcm M = int (Lcm m\<in>M. (nat \<circ> abs) m)"
huffman@45264
  1970
huffman@45264
  1971
definition
haftmann@62346
  1972
  "Gcd M = int (Gcd m\<in>M. (nat \<circ> abs) m)"
huffman@45264
  1973
haftmann@62345
  1974
instance by standard
haftmann@62345
  1975
  (auto intro!: Gcd_dvd Gcd_greatest simp add: Gcd_int_def
haftmann@62345
  1976
    Lcm_int_def int_dvd_iff dvd_int_iff dvd_int_unfold_dvd_nat [symmetric])
haftmann@62345
  1977
haftmann@62345
  1978
end
haftmann@62345
  1979
haftmann@62346
  1980
lemma abs_Gcd [simp]:
haftmann@62346
  1981
  fixes K :: "int set"
haftmann@62346
  1982
  shows "\<bar>Gcd K\<bar> = Gcd K"
haftmann@62346
  1983
  using normalize_Gcd [of K] by simp
haftmann@62346
  1984
haftmann@62346
  1985
lemma abs_Lcm [simp]:
haftmann@62346
  1986
  fixes K :: "int set"
haftmann@62346
  1987
  shows "\<bar>Lcm K\<bar> = Lcm K"
haftmann@62346
  1988
  using normalize_Lcm [of K] by simp
haftmann@62346
  1989
haftmann@62346
  1990
lemma Gcm_eq_int_iff:
haftmann@62346
  1991
  "Gcd K = int n \<longleftrightarrow> Gcd ((nat \<circ> abs) ` K) = n"
haftmann@62346
  1992
  by (simp add: Gcd_int_def comp_def image_image)
haftmann@62346
  1993
haftmann@62346
  1994
lemma Lcm_eq_int_iff:
haftmann@62346
  1995
  "Lcm K = int n \<longleftrightarrow> Lcm ((nat \<circ> abs) ` K) = n"
haftmann@62346
  1996
  by (simp add: Lcm_int_def comp_def image_image)
haftmann@62346
  1997
haftmann@62345
  1998
haftmann@62345
  1999
subsection \<open>GCD and LCM on @{typ integer}\<close>
haftmann@62345
  2000
haftmann@62345
  2001
instantiation integer :: gcd
haftmann@62345
  2002
begin
haftmann@62345
  2003
haftmann@62345
  2004
context
haftmann@62345
  2005
  includes integer.lifting
haftmann@62345
  2006
begin
haftmann@62345
  2007
haftmann@62345
  2008
lift_definition gcd_integer :: "integer \<Rightarrow> integer \<Rightarrow> integer"
haftmann@62345
  2009
  is gcd .
haftmann@62345
  2010
lift_definition lcm_integer :: "integer \<Rightarrow> integer \<Rightarrow> integer"
haftmann@62345
  2011
  is lcm .
haftmann@62345
  2012
haftmann@62345
  2013
end
huffman@45264
  2014
instance ..
haftmann@60686
  2015
wenzelm@21256
  2016
end
huffman@45264
  2017
Andreas@61856
  2018
lifting_update integer.lifting
Andreas@61856
  2019
lifting_forget integer.lifting
Andreas@61856
  2020
haftmann@62345
  2021
context
haftmann@62345
  2022
  includes integer.lifting
haftmann@62345
  2023
begin
Andreas@61856
  2024
Andreas@61856
  2025
lemma gcd_code_integer [code]:
Andreas@61856
  2026
  "gcd k l = \<bar>if l = (0::integer) then k else gcd l (\<bar>k\<bar> mod \<bar>l\<bar>)\<bar>"
haftmann@62345
  2027
  by transfer (fact gcd_code_int)
Andreas@61856
  2028
wenzelm@61944
  2029
lemma lcm_code_integer [code]: "lcm (a::integer) b = \<bar>a\<bar> * \<bar>b\<bar> div gcd a b"
haftmann@62345
  2030
  by transfer (fact lcm_altdef_int)
Andreas@61856
  2031
Andreas@61856
  2032
end
Andreas@61856
  2033
Andreas@61856
  2034
code_printing constant "gcd :: integer \<Rightarrow> _"
Andreas@61856
  2035
  \<rightharpoonup> (OCaml) "Big'_int.gcd'_big'_int"
Andreas@61856
  2036
  and (Haskell) "Prelude.gcd"
Andreas@61856
  2037
  and (Scala) "_.gcd'((_)')"
wenzelm@61975
  2038
  \<comment> \<open>There is no gcd operation in the SML standard library, so no code setup for SML\<close>
Andreas@61856
  2039
haftmann@62344
  2040
text \<open>Some code equations\<close>
haftmann@62344
  2041
haftmann@62344
  2042
lemma Lcm_set_nat [code, code_unfold]:
haftmann@62344
  2043
  "Lcm (set ns) = fold lcm ns (1::nat)"
haftmann@62344
  2044
  using Lcm_set [of ns] by (simp_all add: fun_eq_iff ac_simps foldr_fold [symmetric])
haftmann@62344
  2045
haftmann@62344
  2046
lemma Gcd_set_nat [code]:
haftmann@62344
  2047
  "Gcd (set ns) = fold gcd ns (0::nat)"
haftmann@62344
  2048
  using Gcd_set [of ns] by (simp_all add: fun_eq_iff ac_simps foldr_fold [symmetric])
haftmann@62344
  2049
haftmann@62344
  2050
lemma Lcm_set_int [code, code_unfold]:
haftmann@62344
  2051
  "Lcm (set xs) = fold lcm xs (1::int)"
haftmann@62344
  2052
  using Lcm_set [of xs] by (simp_all add: fun_eq_iff ac_simps foldr_fold [symmetric])
haftmann@62344
  2053
haftmann@62344
  2054
lemma Gcd_set_int [code]:
haftmann@62344
  2055
  "Gcd (set xs) = fold gcd xs (0::int)"
haftmann@62344
  2056
  using Gcd_set [of xs] by (simp_all add: fun_eq_iff ac_simps foldr_fold [symmetric])
haftmann@62344
  2057
haftmann@62344
  2058
text \<open>Fact aliasses\<close>
haftmann@62344
  2059
haftmann@62344
  2060
lemma lcm_0_iff_nat [simp]: "lcm (m::nat) n = 0 \<longleftrightarrow> m = 0 \<or> n= 0"
haftmann@62344
  2061
  by (fact lcm_eq_0_iff)
haftmann@62344
  2062
haftmann@62344
  2063
lemma lcm_0_iff_int [simp]: "lcm (m::int) n = 0 \<longleftrightarrow> m = 0 \<or> n = 0"
haftmann@62344
  2064
  by (fact lcm_eq_0_iff)
haftmann@62344
  2065
haftmann@62344
  2066
lemma dvd_lcm_I1_nat [simp]: "(k::nat) dvd m \<Longrightarrow> k dvd lcm m n"
haftmann@62344
  2067
  by (fact dvd_lcmI1)
haftmann@62344
  2068
haftmann@62344
  2069
lemma dvd_lcm_I2_nat [simp]: "(k::nat) dvd n \<Longrightarrow> k dvd lcm m n"
haftmann@62344
  2070
  by (fact dvd_lcmI2)
haftmann@62344
  2071
haftmann@62344
  2072
lemma dvd_lcm_I1_int [simp]: "(i::int) dvd m \<Longrightarrow> i dvd lcm m n"
haftmann@62344
  2073
  by (fact dvd_lcmI1)
haftmann@62344
  2074
haftmann@62344
  2075
lemma dvd_lcm_I2_int[simp]: "(i::int) dvd n \<Longrightarrow> i dvd lcm m n"
haftmann@62344
  2076
  by (fact dvd_lcmI2)
haftmann@62344
  2077
haftmann@62344
  2078
lemma coprime_dvd_mult_iff_nat: "coprime (k::nat) n \<Longrightarrow>
haftmann@62344
  2079
    (k dvd m * n) = (k dvd m)"
haftmann@62344
  2080
  by (fact coprime_dvd_mult_iff)
haftmann@62344
  2081
haftmann@62344
  2082
lemma coprime_dvd_mult_iff_int: "coprime (k::int) n \<Longrightarrow>
haftmann@62344
  2083
    (k dvd m * n) = (k dvd m)"
haftmann@62344
  2084
  by (fact coprime_dvd_mult_iff)
haftmann@62344
  2085
haftmann@62344
  2086
lemma coprime_exp2_nat [intro]: "coprime (a::nat) b \<Longrightarrow> coprime (a^n) (b^m)"
haftmann@62344
  2087
  by (fact coprime_exp2)
haftmann@62344
  2088
haftmann@62344
  2089
lemma coprime_exp2_int [intro]: "coprime (a::int) b \<Longrightarrow> coprime (a^n) (b^m)"
haftmann@62344
  2090
  by (fact coprime_exp2)
haftmann@62344
  2091
haftmann@62344
  2092
lemmas Gcd_dvd_nat [simp] = Gcd_dvd [where ?'a = nat]
haftmann@62344
  2093
lemmas Gcd_dvd_int [simp] = Gcd_dvd [where ?'a = int]
haftmann@62344
  2094
lemmas dvd_Gcd_nat [simp] = dvd_Gcd [where ?'a = nat]
haftmann@62344
  2095
lemmas dvd_Gcd_int [simp] = dvd_Gcd [where ?'a = int]
haftmann@62344
  2096
haftmann@62344
  2097
lemma dvd_Lcm_int [simp]:
haftmann@62344
  2098
  fixes M :: "int set" assumes "m \<in> M" shows "m dvd Lcm M"
haftmann@62344
  2099
  using assms by (fact dvd_Lcm)
haftmann@62344
  2100
haftmann@62344
  2101
lemma Lcm_empty_nat:
haftmann@62344
  2102
  "Lcm {} = (1::nat)"
haftmann@62344
  2103
  by (fact Lcm_empty)
haftmann@62344
  2104
haftmann@62344
  2105
lemma Lcm_empty_int:
haftmann@62344
  2106
  "Lcm {} = (1::int)"
haftmann@62344
  2107
  by (fact Lcm_empty)
haftmann@62344
  2108
haftmann@62344
  2109
lemma Lcm_insert_nat:
haftmann@62344
  2110
  "Lcm (insert (n::nat) N) = lcm n (Lcm N)"
haftmann@62344
  2111
  by (fact Lcm_insert)
haftmann@62344
  2112
haftmann@62344
  2113
lemma Lcm_insert_int:
haftmann@62344
  2114
  "Lcm (insert (n::int) N) = lcm n (Lcm N)"
haftmann@62344
  2115
  by (fact Lcm_insert)
haftmann@62344
  2116
haftmann@62344
  2117
lemma gcd_neg_numeral_1_int [simp]:
haftmann@62344
  2118
  "gcd (- numeral n :: int) x = gcd (numeral n) x"
haftmann@62344
  2119
  by (fact gcd_neg1_int)
haftmann@62344
  2120
haftmann@62344
  2121
lemma gcd_neg_numeral_2_int [simp]:
haftmann@62344
  2122
  "gcd x (- numeral n :: int) = gcd x (numeral n)"
haftmann@62344
  2123
  by (fact gcd_neg2_int)
haftmann@62344
  2124
haftmann@62344
  2125
lemma gcd_proj1_if_dvd_nat [simp]: "(x::nat) dvd y \<Longrightarrow> gcd x y = x"
haftmann@62344
  2126
  by (fact gcd_nat.absorb1)
haftmann@62344
  2127
haftmann@62344
  2128
lemma gcd_proj2_if_dvd_nat [simp]: "(y::nat) dvd x \<Longrightarrow> gcd x y = y"
haftmann@62344
  2129
  by (fact gcd_nat.absorb2)
haftmann@62344
  2130
haftmann@62344
  2131
lemma comp_fun_idem_gcd_nat: "comp_fun_idem (gcd :: nat\<Rightarrow>nat\<Rightarrow>nat)"
haftmann@62344
  2132
  by (fact comp_fun_idem_gcd)
haftmann@62344
  2133
haftmann@62344
  2134
lemma comp_fun_idem_gcd_int: "comp_fun_idem (gcd :: int\<Rightarrow>int\<Rightarrow>int)"
haftmann@62344
  2135
  by (fact comp_fun_idem_gcd)
haftmann@62344
  2136
haftmann@62344
  2137
lemma comp_fun_idem_lcm_nat: "comp_fun_idem (lcm :: nat\<Rightarrow>nat\<Rightarrow>nat)"
haftmann@62344
  2138
  by (fact comp_fun_idem_lcm)
haftmann@62344
  2139
haftmann@62344
  2140
lemma comp_fun_idem_lcm_int: "comp_fun_idem (lcm :: int\<Rightarrow>int\<Rightarrow>int)"
haftmann@62344
  2141
  by (fact comp_fun_idem_lcm)
haftmann@62344
  2142
haftmann@62345
  2143
lemma Lcm_eq_0 [simp]:
haftmann@62345
  2144
  "finite (M::nat set) \<Longrightarrow> 0 \<in> M \<Longrightarrow> Lcm M = 0"
haftmann@62345
  2145
  by (rule Lcm_eq_0_I)
haftmann@62345
  2146
haftmann@62345
  2147
lemma Lcm0_iff [simp]:
haftmann@62345
  2148
  fixes M :: "nat set"
haftmann@62345
  2149
  assumes "finite M" and "M \<noteq> {}"
haftmann@62345
  2150
  shows "Lcm M = 0 \<longleftrightarrow> 0 \<in> M"
haftmann@62345
  2151
  using assms by (simp add: Lcm_0_iff)
haftmann@62345
  2152
haftmann@62345
  2153
lemma Lcm_dvd_int [simp]:
haftmann@62345
  2154
  fixes M :: "int set"
haftmann@62345
  2155
  assumes "\<forall>m\<in>M. m dvd n" shows "Lcm M dvd n"
haftmann@62345
  2156
  using assms by (auto intro: Lcm_least)
haftmann@62345
  2157
haftmann@62345
  2158
lemma dvd_gcd_D1_nat: "k dvd gcd m n \<Longrightarrow> (k::nat) dvd m"
haftmann@62345
  2159
  by (fact dvd_gcdD1)
haftmann@62345
  2160
haftmann@62345
  2161
lemma dvd_gcd_D2_nat: "k dvd gcd m n \<Longrightarrow> (k::nat) dvd n"
haftmann@62345
  2162
  by (fact dvd_gcdD2)
haftmann@62345
  2163
haftmann@62345
  2164
lemma dvd_gcd_D1_int: "i dvd gcd m n \<Longrightarrow> (i::int) dvd m"
haftmann@62345
  2165
  by (fact dvd_gcdD1)
haftmann@62345
  2166
haftmann@62345
  2167
lemma dvd_gcd_D2_int: "i dvd gcd m n \<Longrightarrow> (i::int) dvd n"
haftmann@62345
  2168
  by (fact dvd_gcdD2)
haftmann@62345
  2169
haftmann@62344
  2170
interpretation dvd:
haftmann@62344
  2171
  order "op dvd" "\<lambda>n m :: nat. n dvd m \<and> m \<noteq> n"
haftmann@62344
  2172
  by standard (auto intro: dvd_refl dvd_trans dvd_antisym)
haftmann@62344
  2173
haftmann@62344
  2174
interpretation gcd_semilattice_nat:
haftmann@62344
  2175
  semilattice_inf gcd Rings.dvd "\<lambda>m n::nat. m dvd n \<and> m \<noteq> n"
haftmann@62344
  2176
  by standard (auto dest: dvd_antisym dvd_trans)
haftmann@62344
  2177
haftmann@62344
  2178
interpretation lcm_semilattice_nat:
haftmann@62344
  2179
  semilattice_sup lcm Rings.dvd "\<lambda>m n::nat. m dvd n \<and> m \<noteq> n"
haftmann@62344
  2180
  by standard simp_all
haftmann@62344
  2181
haftmann@62344
  2182
interpretation gcd_lcm_lattice_nat:
haftmann@62344
  2183
  lattice gcd Rings.dvd "\<lambda>m n::nat. m dvd n \<and> m \<noteq> n" lcm
haftmann@62344
  2184
  ..
haftmann@62344
  2185
haftmann@62344
  2186
interpretation gcd_lcm_complete_lattice_nat:
haftmann@62344
  2187
  complete_lattice Gcd Lcm gcd Rings.dvd "\<lambda>m n. m dvd n \<and> m \<noteq> n" lcm 1 "0::nat"
haftmann@62344
  2188
  by standard (auto simp add: Gcd_nat_def Lcm_nat_empty Lcm_nat_infinite)
haftmann@62344
  2189
Andreas@61856
  2190
end