src/HOL/GCD.thy
author haftmann
Wed Jul 08 14:01:39 2015 +0200 (2015-07-08)
changeset 60686 ea5bc46c11e6
parent 60597 2da9b632069b
child 60687 33dbbcb6a8a3
permissions -rw-r--r--
more algebraic properties for gcd/lcm
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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 {* Greatest common divisor and least common multiple *}
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theory GCD
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imports Main
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begin
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context semidom_divide
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begin
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lemma divide_1 [simp]:
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  "a div 1 = a"
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  using nonzero_mult_divide_cancel_left [of 1 a] by simp
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lemma dvd_mult_cancel_left [simp]:
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  assumes "a \<noteq> 0"
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  shows "a * b dvd a * c \<longleftrightarrow> b dvd c" (is "?P \<longleftrightarrow> ?Q")
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proof
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  assume ?P then obtain d where "a * c = a * b * d" ..
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  with assms have "c = b * d" by (simp add: ac_simps)
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  then show ?Q ..
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next
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  assume ?Q then obtain d where "c = b * d" .. 
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  then have "a * c = a * b * d" by (simp add: ac_simps)
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  then show ?P ..
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qed
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lemma dvd_mult_cancel_right [simp]:
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  assumes "a \<noteq> 0"
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  shows "b * a dvd c * a \<longleftrightarrow> b dvd c" (is "?P \<longleftrightarrow> ?Q")
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using dvd_mult_cancel_left [of a b c] assms by (simp add: ac_simps)
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lemma div_dvd_iff_mult:
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  assumes "b \<noteq> 0" and "b dvd a"
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  shows "a div b dvd c \<longleftrightarrow> a dvd c * b"
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proof -
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  from \<open>b dvd a\<close> obtain d where "a = b * d" ..
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  with \<open>b \<noteq> 0\<close> show ?thesis by (simp add: ac_simps)
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qed
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lemma dvd_div_iff_mult:
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  assumes "c \<noteq> 0" and "c dvd b"
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  shows "a dvd b div c \<longleftrightarrow> a * c dvd b"
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proof -
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  from \<open>c dvd b\<close> obtain d where "b = c * d" ..
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  with \<open>c \<noteq> 0\<close> show ?thesis by (simp add: mult.commute [of a])
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qed
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end
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context algebraic_semidom
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begin
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lemma associated_1 [simp]:
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  "associated 1 a \<longleftrightarrow> is_unit a"
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  "associated a 1 \<longleftrightarrow> is_unit a"
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  by (auto simp add: associated_def)
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end
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declare One_nat_def [simp del]
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subsection {* GCD and LCM definitions *}
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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 [iff]:
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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_0_left [simp]:
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  "gcd 0 a = normalize a"
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proof (rule associated_eqI)
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  show "associated (gcd 0 a) (normalize a)"
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    by (auto intro!: associatedI gcd_greatest)
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  show "unit_factor (gcd 0 a) = 1" if "gcd 0 a \<noteq> 0"
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  proof -
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    from that have "unit_factor (normalize (gcd 0 a)) = 1"
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      by (rule unit_factor_normalize)
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    then show ?thesis by simp
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  qed
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qed simp
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lemma gcd_0_right [simp]:
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  "gcd a 0 = normalize a"
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proof (rule associated_eqI)
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  show "associated (gcd a 0) (normalize a)"
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    by (auto intro!: associatedI gcd_greatest)
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  show "unit_factor (gcd a 0) = 1" if "gcd a 0 \<noteq> 0"
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  proof -
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    from that have "unit_factor (normalize (gcd a 0)) = 1"
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      by (rule unit_factor_normalize)
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    then show ?thesis by simp
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  qed
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qed simp
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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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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) (auto intro: associatedI gcd_greatest simp add: unit_factor_gcd)
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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 have "associated (gcd (gcd a b) c) (gcd a (gcd b c))"
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    by (rule associatedI)
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  then show "gcd (gcd a b) c = gcd a (gcd b c)"
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    by (rule associated_eqI) (simp_all add: unit_factor_gcd)
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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 have "associated (gcd a a) (normalize a)"
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    by (auto intro: associatedI)
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  then show ?thesis
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    by (auto intro: associated_eqI simp add: unit_factor_gcd)
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qed
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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 add: unit_factor_gcd)
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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 - (rule associated_eqI, auto intro: associated_eqI associatedI simp add: unit_factor_gcd)
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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 lcm_dvd1 [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 lcm_dvd2 [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 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 [iff]:
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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 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
haftmann@60686
   318
  fix a b c
haftmann@60686
   319
  show "lcm a b = lcm b a"
haftmann@60686
   320
    by (simp add: lcm_gcd ac_simps normalize_mult dvd_normalize_div)
haftmann@60686
   321
  have "associated (lcm (lcm a b) c) (lcm a (lcm b c))"
haftmann@60686
   322
    by (auto intro!: associatedI lcm_least
haftmann@60686
   323
      dvd_trans [of b "lcm b c" "lcm a (lcm b c)"]
haftmann@60686
   324
      dvd_trans [of c "lcm b c" "lcm a (lcm b c)"]
haftmann@60686
   325
      dvd_trans [of a "lcm a b" "lcm (lcm a b) c"]
haftmann@60686
   326
      dvd_trans [of b "lcm a b" "lcm (lcm a b) c"])
haftmann@60686
   327
  then show "lcm (lcm a b) c = lcm a (lcm b c)"
haftmann@60686
   328
    by (rule associated_eqI) (simp_all add: unit_factor_lcm lcm_eq_0_iff)
haftmann@60686
   329
qed
haftmann@60686
   330
haftmann@60686
   331
lemma lcm_self [simp]:
haftmann@60686
   332
  "lcm a a = normalize a"
haftmann@60686
   333
proof -
haftmann@60686
   334
  have "lcm a a dvd a"
haftmann@60686
   335
    by (rule lcm_least) simp_all
haftmann@60686
   336
  then have "associated (lcm a a) (normalize a)"
haftmann@60686
   337
    by (auto intro: associatedI)
haftmann@60686
   338
  then show ?thesis
haftmann@60686
   339
    by (rule associated_eqI) (auto simp add: unit_factor_lcm)
haftmann@60686
   340
qed
haftmann@60686
   341
haftmann@60686
   342
lemma gcd_mult_lcm [simp]:
haftmann@60686
   343
  "gcd a b * lcm a b = normalize a * normalize b"
haftmann@60686
   344
  by (simp add: lcm_gcd normalize_mult)
haftmann@60686
   345
haftmann@60686
   346
lemma lcm_mult_gcd [simp]:
haftmann@60686
   347
  "lcm a b * gcd a b = normalize a * normalize b"
haftmann@60686
   348
  using gcd_mult_lcm [of a b] by (simp add: ac_simps) 
haftmann@60686
   349
haftmann@60686
   350
lemma gcd_lcm:
haftmann@60686
   351
  assumes "a \<noteq> 0" and "b \<noteq> 0"
haftmann@60686
   352
  shows "gcd a b = normalize (a * b) div lcm a b"
haftmann@60686
   353
proof -
haftmann@60686
   354
  from assms have "lcm a b \<noteq> 0"
haftmann@60686
   355
    by (simp add: lcm_eq_0_iff)
haftmann@60686
   356
  have "gcd a b * lcm a b = normalize a * normalize b" by simp
haftmann@60686
   357
  then have "gcd a b * lcm a b div lcm a b = normalize (a * b) div lcm a b"
haftmann@60686
   358
    by (simp_all add: normalize_mult)
haftmann@60686
   359
  with \<open>lcm a b \<noteq> 0\<close> show ?thesis
haftmann@60686
   360
    using nonzero_mult_divide_cancel_right [of "lcm a b" "gcd a b"] by simp
haftmann@60686
   361
qed
haftmann@60686
   362
haftmann@60686
   363
lemma lcm_1_left [simp]:
haftmann@60686
   364
  "lcm 1 a = normalize a"
haftmann@60686
   365
  by (simp add: lcm_gcd)
haftmann@60686
   366
haftmann@60686
   367
lemma lcm_1_right [simp]:
haftmann@60686
   368
  "lcm a 1 = normalize a"
haftmann@60686
   369
  by (simp add: lcm_gcd)
haftmann@60686
   370
  
haftmann@60686
   371
lemma lcm_mult_left:
haftmann@60686
   372
  "lcm (c * a) (c * b) = normalize c * lcm a b"
haftmann@60686
   373
  by (cases "c = 0")
haftmann@60686
   374
    (simp_all add: gcd_mult_right lcm_gcd div_mult_swap normalize_mult ac_simps,
haftmann@60686
   375
      simp add: dvd_div_mult2_eq mult.left_commute [of "normalize c", symmetric])
haftmann@60686
   376
haftmann@60686
   377
lemma lcm_mult_right:
haftmann@60686
   378
  "lcm (a * c) (b * c) = lcm b a * normalize c"
haftmann@60686
   379
  using lcm_mult_left [of c a b] by (simp add: ac_simps)
haftmann@60686
   380
haftmann@60686
   381
lemma mult_lcm_left:
haftmann@60686
   382
  "c * lcm a b = unit_factor c * lcm (c * a) (c * b)"
haftmann@60686
   383
  by (simp add: lcm_mult_left mult.assoc [symmetric])
haftmann@60686
   384
haftmann@60686
   385
lemma mult_lcm_right:
haftmann@60686
   386
  "lcm a b * c = lcm (a * c) (b * c) * unit_factor c"
haftmann@60686
   387
  using mult_lcm_left [of c a b] by (simp add: ac_simps)
haftmann@60686
   388
  
haftmann@60686
   389
end
haftmann@60686
   390
haftmann@60686
   391
class semiring_Gcd = semiring_gcd + Gcd +
haftmann@60686
   392
  assumes Gcd_dvd: "a \<in> A \<Longrightarrow> Gcd A dvd a"
haftmann@60686
   393
    and Gcd_greatest: "(\<And>b. b \<in> A \<Longrightarrow> a dvd b) \<Longrightarrow> a dvd Gcd A"
haftmann@60686
   394
    and normalize_Gcd [simp]: "normalize (Gcd A) = Gcd A"
haftmann@60686
   395
begin
haftmann@60686
   396
haftmann@60686
   397
lemma Gcd_empty [simp]:
haftmann@60686
   398
  "Gcd {} = 0"
haftmann@60686
   399
  by (rule dvd_0_left, rule Gcd_greatest) simp
haftmann@60686
   400
haftmann@60686
   401
lemma Gcd_0_iff [simp]:
haftmann@60686
   402
  "Gcd A = 0 \<longleftrightarrow> (\<forall>a\<in>A. a = 0)" (is "?P \<longleftrightarrow> ?Q")
haftmann@60686
   403
proof
haftmann@60686
   404
  assume ?P
haftmann@60686
   405
  show ?Q
haftmann@60686
   406
  proof
haftmann@60686
   407
    fix a
haftmann@60686
   408
    assume "a \<in> A"
haftmann@60686
   409
    then have "Gcd A dvd a" by (rule Gcd_dvd)
haftmann@60686
   410
    with \<open>?P\<close> show "a = 0" by simp
haftmann@60686
   411
  qed
haftmann@60686
   412
next
haftmann@60686
   413
  assume ?Q
haftmann@60686
   414
  have "0 dvd Gcd A"
haftmann@60686
   415
  proof (rule Gcd_greatest)
haftmann@60686
   416
    fix a
haftmann@60686
   417
    assume "a \<in> A"
haftmann@60686
   418
    with \<open>?Q\<close> have "a = 0" by simp
haftmann@60686
   419
    then show "0 dvd a" by simp
haftmann@60686
   420
  qed
haftmann@60686
   421
  then show ?P by simp
haftmann@60686
   422
qed
haftmann@60686
   423
haftmann@60686
   424
lemma unit_factor_Gcd:
haftmann@60686
   425
  "unit_factor (Gcd A) = (if \<forall>a\<in>A. a = 0 then 0 else 1)"
haftmann@60686
   426
proof (cases "Gcd A = 0")
haftmann@60686
   427
  case True then show ?thesis by simp
haftmann@60686
   428
next
haftmann@60686
   429
  case False
haftmann@60686
   430
  from unit_factor_mult_normalize
haftmann@60686
   431
  have "unit_factor (Gcd A) * normalize (Gcd A) = Gcd A" .
haftmann@60686
   432
  then have "unit_factor (Gcd A) * Gcd A = Gcd A" by simp
haftmann@60686
   433
  then have "unit_factor (Gcd A) * Gcd A div Gcd A = Gcd A div Gcd A" by simp
haftmann@60686
   434
  with False have "unit_factor (Gcd A) = 1" by simp
haftmann@60686
   435
  with False show ?thesis by simp
haftmann@60686
   436
qed
haftmann@60686
   437
haftmann@60686
   438
lemma Gcd_UNIV [simp]:
haftmann@60686
   439
  "Gcd UNIV = 1"
haftmann@60686
   440
  by (rule associated_eqI) (auto intro: Gcd_dvd simp add: unit_factor_Gcd)
haftmann@60686
   441
haftmann@60686
   442
lemma Gcd_eq_1_I:
haftmann@60686
   443
  assumes "is_unit a" and "a \<in> A"
haftmann@60686
   444
  shows "Gcd A = 1"
haftmann@60686
   445
proof -
haftmann@60686
   446
  from assms have "is_unit (Gcd A)"
haftmann@60686
   447
    by (blast intro: Gcd_dvd dvd_unit_imp_unit)
haftmann@60686
   448
  then have "normalize (Gcd A) = 1"
haftmann@60686
   449
    by (rule is_unit_normalize)
haftmann@60686
   450
  then show ?thesis
haftmann@60686
   451
    by simp
haftmann@60686
   452
qed
haftmann@60686
   453
haftmann@60686
   454
lemma Gcd_insert [simp]:
haftmann@60686
   455
  "Gcd (insert a A) = gcd a (Gcd A)"
haftmann@60686
   456
proof -
haftmann@60686
   457
  have "Gcd (insert a A) dvd gcd a (Gcd A)"
haftmann@60686
   458
    by (auto intro: Gcd_dvd Gcd_greatest)
haftmann@60686
   459
  moreover have "gcd a (Gcd A) dvd Gcd (insert a A)"
haftmann@60686
   460
  proof (rule Gcd_greatest)
haftmann@60686
   461
    fix b
haftmann@60686
   462
    assume "b \<in> insert a A"
haftmann@60686
   463
    then show "gcd a (Gcd A) dvd b"
haftmann@60686
   464
    proof
haftmann@60686
   465
      assume "b = a" then show ?thesis by simp
haftmann@60686
   466
    next
haftmann@60686
   467
      assume "b \<in> A"
haftmann@60686
   468
      then have "Gcd A dvd b" by (rule Gcd_dvd)
haftmann@60686
   469
      moreover have "gcd a (Gcd A) dvd Gcd A" by simp
haftmann@60686
   470
      ultimately show ?thesis by (blast intro: dvd_trans)
haftmann@60686
   471
    qed
haftmann@60686
   472
  qed
haftmann@60686
   473
  ultimately have "associated (Gcd (insert a A)) (gcd a (Gcd A))"
haftmann@60686
   474
    by (rule associatedI)
haftmann@60686
   475
  then show ?thesis
haftmann@60686
   476
    by (rule associated_eqI) (simp_all add: unit_factor_gcd unit_factor_Gcd)
haftmann@60686
   477
qed
haftmann@60686
   478
haftmann@60686
   479
lemma dvd_Gcd: -- \<open>FIXME remove\<close>
haftmann@60686
   480
  "\<forall>b\<in>A. a dvd b \<Longrightarrow> a dvd Gcd A"
haftmann@60686
   481
  by (blast intro: Gcd_greatest)
haftmann@60686
   482
haftmann@60686
   483
lemma Gcd_set [code_unfold]:
haftmann@60686
   484
  "Gcd (set as) = foldr gcd as 0"
haftmann@60686
   485
  by (induct as) simp_all
haftmann@60686
   486
haftmann@60686
   487
end  
haftmann@60686
   488
haftmann@60686
   489
class semiring_Lcm = semiring_Gcd +
haftmann@60686
   490
  assumes Lcm_Gcd: "Lcm A = Gcd {b. \<forall>a\<in>A. a dvd b}"
haftmann@60686
   491
begin
haftmann@60686
   492
haftmann@60686
   493
lemma dvd_Lcm:
haftmann@60686
   494
  assumes "a \<in> A"
haftmann@60686
   495
  shows "a dvd Lcm A"
haftmann@60686
   496
  using assms by (auto intro: Gcd_greatest simp add: Lcm_Gcd)
haftmann@60686
   497
haftmann@60686
   498
lemma Gcd_image_normalize [simp]:
haftmann@60686
   499
  "Gcd (normalize ` A) = Gcd A"
haftmann@60686
   500
proof -
haftmann@60686
   501
  have "Gcd (normalize ` A) dvd a" if "a \<in> A" for a
haftmann@60686
   502
  proof -
haftmann@60686
   503
    from that obtain B where "A = insert a B" by blast
haftmann@60686
   504
    moreover have " gcd (normalize a) (Gcd (normalize ` B)) dvd normalize a"
haftmann@60686
   505
      by (rule gcd_dvd1)
haftmann@60686
   506
    ultimately show "Gcd (normalize ` A) dvd a"
haftmann@60686
   507
      by simp
haftmann@60686
   508
  qed
haftmann@60686
   509
  then have "associated (Gcd (normalize ` A)) (Gcd A)"
haftmann@60686
   510
    by (auto intro!: associatedI Gcd_greatest intro: Gcd_dvd)
haftmann@60686
   511
  then show ?thesis
haftmann@60686
   512
    by (rule associated_eqI) (simp_all add: unit_factor_Gcd)
haftmann@60686
   513
qed
haftmann@60686
   514
haftmann@60686
   515
lemma Lcm_least:
haftmann@60686
   516
  assumes "\<And>b. b \<in> A \<Longrightarrow> b dvd a"
haftmann@60686
   517
  shows "Lcm A dvd a"
haftmann@60686
   518
  using assms by (auto intro: Gcd_dvd simp add: Lcm_Gcd)
haftmann@60686
   519
haftmann@60686
   520
lemma normalize_Lcm [simp]:
haftmann@60686
   521
  "normalize (Lcm A) = Lcm A"
haftmann@60686
   522
  by (simp add: Lcm_Gcd)
haftmann@60686
   523
haftmann@60686
   524
lemma unit_factor_Lcm:
haftmann@60686
   525
  "unit_factor (Lcm A) = (if Lcm A = 0 then 0 else 1)"
haftmann@60686
   526
proof (cases "Lcm A = 0")
haftmann@60686
   527
  case True then show ?thesis by simp
haftmann@60686
   528
next
haftmann@60686
   529
  case False
haftmann@60686
   530
  with unit_factor_normalize have "unit_factor (normalize (Lcm A)) = 1"
haftmann@60686
   531
    by blast
haftmann@60686
   532
  with False show ?thesis
haftmann@60686
   533
    by simp
haftmann@60686
   534
qed
haftmann@60686
   535
  
haftmann@60686
   536
lemma Lcm_empty [simp]:
haftmann@60686
   537
  "Lcm {} = 1"
haftmann@60686
   538
  by (simp add: Lcm_Gcd)
haftmann@60686
   539
haftmann@60686
   540
lemma Lcm_1_iff [simp]:
haftmann@60686
   541
  "Lcm A = 1 \<longleftrightarrow> (\<forall>a\<in>A. is_unit a)" (is "?P \<longleftrightarrow> ?Q")
haftmann@60686
   542
proof
haftmann@60686
   543
  assume ?P
haftmann@60686
   544
  show ?Q
haftmann@60686
   545
  proof
haftmann@60686
   546
    fix a
haftmann@60686
   547
    assume "a \<in> A"
haftmann@60686
   548
    then have "a dvd Lcm A"
haftmann@60686
   549
      by (rule dvd_Lcm)
haftmann@60686
   550
    with \<open>?P\<close> show "is_unit a"
haftmann@60686
   551
      by simp
haftmann@60686
   552
  qed
haftmann@60686
   553
next
haftmann@60686
   554
  assume ?Q
haftmann@60686
   555
  then have "is_unit (Lcm A)"
haftmann@60686
   556
    by (blast intro: Lcm_least)
haftmann@60686
   557
  then have "normalize (Lcm A) = 1"
haftmann@60686
   558
    by (rule is_unit_normalize)
haftmann@60686
   559
  then show ?P
haftmann@60686
   560
    by simp
haftmann@60686
   561
qed
haftmann@60686
   562
haftmann@60686
   563
lemma Lcm_UNIV [simp]:
haftmann@60686
   564
  "Lcm UNIV = 0"
haftmann@60686
   565
proof -
haftmann@60686
   566
  have "0 dvd Lcm UNIV"
haftmann@60686
   567
    by (rule dvd_Lcm) simp
haftmann@60686
   568
  then show ?thesis
haftmann@60686
   569
    by simp
haftmann@60686
   570
qed
haftmann@60686
   571
haftmann@60686
   572
lemma Lcm_eq_0_I:
haftmann@60686
   573
  assumes "0 \<in> A"
haftmann@60686
   574
  shows "Lcm A = 0"
haftmann@60686
   575
proof -
haftmann@60686
   576
  from assms have "0 dvd Lcm A"
haftmann@60686
   577
    by (rule dvd_Lcm)
haftmann@60686
   578
  then show ?thesis
haftmann@60686
   579
    by simp
haftmann@60686
   580
qed
haftmann@60686
   581
haftmann@60686
   582
lemma Gcd_Lcm:
haftmann@60686
   583
  "Gcd A = Lcm {b. \<forall>a\<in>A. b dvd a}"
haftmann@60686
   584
  by (rule associated_eqI) (auto intro: associatedI Gcd_dvd dvd_Lcm Gcd_greatest Lcm_least
haftmann@60686
   585
    simp add: unit_factor_Gcd unit_factor_Lcm)
haftmann@60686
   586
haftmann@60686
   587
lemma Lcm_insert [simp]:
haftmann@60686
   588
  "Lcm (insert a A) = lcm a (Lcm A)"
haftmann@60686
   589
proof (rule sym)
haftmann@60686
   590
  have "lcm a (Lcm A) dvd Lcm (insert a A)"
haftmann@60686
   591
    by (auto intro: dvd_Lcm Lcm_least)
haftmann@60686
   592
  moreover have "Lcm (insert a A) dvd lcm a (Lcm A)"
haftmann@60686
   593
  proof (rule Lcm_least)
haftmann@60686
   594
    fix b
haftmann@60686
   595
    assume "b \<in> insert a A"
haftmann@60686
   596
    then show "b dvd lcm a (Lcm A)"
haftmann@60686
   597
    proof
haftmann@60686
   598
      assume "b = a" then show ?thesis by simp
haftmann@60686
   599
    next
haftmann@60686
   600
      assume "b \<in> A"
haftmann@60686
   601
      then have "b dvd Lcm A" by (rule dvd_Lcm)
haftmann@60686
   602
      moreover have "Lcm A dvd lcm a (Lcm A)" by simp
haftmann@60686
   603
      ultimately show ?thesis by (blast intro: dvd_trans)
haftmann@60686
   604
    qed
haftmann@60686
   605
  qed
haftmann@60686
   606
  ultimately have "associated (lcm a (Lcm A)) (Lcm (insert a A))"
haftmann@60686
   607
    by (rule associatedI)
haftmann@60686
   608
  then show "lcm a (Lcm A) = Lcm (insert a A)"
haftmann@60686
   609
    by (rule associated_eqI) (simp_all add: unit_factor_lcm unit_factor_Lcm lcm_eq_0_iff)
haftmann@60686
   610
qed
haftmann@60686
   611
  
haftmann@60686
   612
lemma Lcm_set [code_unfold]:
haftmann@60686
   613
  "Lcm (set as) = foldr lcm as 1"
haftmann@60686
   614
  by (induct as) simp_all
haftmann@60686
   615
  
haftmann@60686
   616
end
haftmann@59008
   617
haftmann@59008
   618
class ring_gcd = comm_ring_1 + semiring_gcd
haftmann@59008
   619
huffman@31706
   620
instantiation nat :: gcd
huffman@31706
   621
begin
wenzelm@21256
   622
huffman@31706
   623
fun
huffman@31706
   624
  gcd_nat  :: "nat \<Rightarrow> nat \<Rightarrow> nat"
huffman@31706
   625
where
huffman@31706
   626
  "gcd_nat x y =
huffman@31706
   627
   (if y = 0 then x else gcd y (x mod y))"
huffman@31706
   628
huffman@31706
   629
definition
huffman@31706
   630
  lcm_nat :: "nat \<Rightarrow> nat \<Rightarrow> nat"
huffman@31706
   631
where
huffman@31706
   632
  "lcm_nat x y = x * y div (gcd x y)"
huffman@31706
   633
huffman@31706
   634
instance proof qed
huffman@31706
   635
huffman@31706
   636
end
huffman@31706
   637
huffman@31706
   638
instantiation int :: gcd
huffman@31706
   639
begin
wenzelm@21256
   640
huffman@31706
   641
definition
huffman@31706
   642
  gcd_int  :: "int \<Rightarrow> int \<Rightarrow> int"
huffman@31706
   643
where
huffman@31706
   644
  "gcd_int x y = int (gcd (nat (abs x)) (nat (abs y)))"
haftmann@23687
   645
huffman@31706
   646
definition
huffman@31706
   647
  lcm_int :: "int \<Rightarrow> int \<Rightarrow> int"
huffman@31706
   648
where
huffman@31706
   649
  "lcm_int x y = int (lcm (nat (abs x)) (nat (abs y)))"
haftmann@23687
   650
huffman@31706
   651
instance proof qed
huffman@31706
   652
huffman@31706
   653
end
haftmann@23687
   654
haftmann@23687
   655
haftmann@34030
   656
subsection {* Transfer setup *}
huffman@31706
   657
huffman@31706
   658
lemma transfer_nat_int_gcd:
huffman@31706
   659
  "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> gcd (nat x) (nat y) = nat (gcd x y)"
huffman@31706
   660
  "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> lcm (nat x) (nat y) = nat (lcm x y)"
haftmann@32479
   661
  unfolding gcd_int_def lcm_int_def
huffman@31706
   662
  by auto
haftmann@23687
   663
huffman@31706
   664
lemma transfer_nat_int_gcd_closures:
huffman@31706
   665
  "x >= (0::int) \<Longrightarrow> y >= 0 \<Longrightarrow> gcd x y >= 0"
huffman@31706
   666
  "x >= (0::int) \<Longrightarrow> y >= 0 \<Longrightarrow> lcm x y >= 0"
huffman@31706
   667
  by (auto simp add: gcd_int_def lcm_int_def)
huffman@31706
   668
haftmann@35644
   669
declare transfer_morphism_nat_int[transfer add return:
huffman@31706
   670
    transfer_nat_int_gcd transfer_nat_int_gcd_closures]
huffman@31706
   671
huffman@31706
   672
lemma transfer_int_nat_gcd:
huffman@31706
   673
  "gcd (int x) (int y) = int (gcd x y)"
huffman@31706
   674
  "lcm (int x) (int y) = int (lcm x y)"
haftmann@32479
   675
  by (unfold gcd_int_def lcm_int_def, auto)
huffman@31706
   676
huffman@31706
   677
lemma transfer_int_nat_gcd_closures:
huffman@31706
   678
  "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> gcd x y >= 0"
huffman@31706
   679
  "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> lcm x y >= 0"
huffman@31706
   680
  by (auto simp add: gcd_int_def lcm_int_def)
huffman@31706
   681
haftmann@35644
   682
declare transfer_morphism_int_nat[transfer add return:
huffman@31706
   683
    transfer_int_nat_gcd transfer_int_nat_gcd_closures]
huffman@31706
   684
huffman@31706
   685
haftmann@34030
   686
subsection {* GCD properties *}
huffman@31706
   687
huffman@31706
   688
(* was gcd_induct *)
nipkow@31952
   689
lemma gcd_nat_induct:
haftmann@23687
   690
  fixes m n :: nat
haftmann@23687
   691
  assumes "\<And>m. P m 0"
haftmann@23687
   692
    and "\<And>m n. 0 < n \<Longrightarrow> P n (m mod n) \<Longrightarrow> P m n"
haftmann@23687
   693
  shows "P m n"
huffman@31706
   694
  apply (rule gcd_nat.induct)
huffman@31706
   695
  apply (case_tac "y = 0")
huffman@31706
   696
  using assms apply simp_all
huffman@31706
   697
done
huffman@31706
   698
huffman@31706
   699
(* specific to int *)
huffman@31706
   700
nipkow@31952
   701
lemma gcd_neg1_int [simp]: "gcd (-x::int) y = gcd x y"
huffman@31706
   702
  by (simp add: gcd_int_def)
huffman@31706
   703
nipkow@31952
   704
lemma gcd_neg2_int [simp]: "gcd (x::int) (-y) = gcd x y"
huffman@31706
   705
  by (simp add: gcd_int_def)
huffman@31706
   706
haftmann@54489
   707
lemma gcd_neg_numeral_1_int [simp]:
haftmann@54489
   708
  "gcd (- numeral n :: int) x = gcd (numeral n) x"
haftmann@54489
   709
  by (fact gcd_neg1_int)
haftmann@54489
   710
haftmann@54489
   711
lemma gcd_neg_numeral_2_int [simp]:
haftmann@54489
   712
  "gcd x (- numeral n :: int) = gcd x (numeral n)"
haftmann@54489
   713
  by (fact gcd_neg2_int)
haftmann@54489
   714
nipkow@31813
   715
lemma abs_gcd_int[simp]: "abs(gcd (x::int) y) = gcd x y"
nipkow@31813
   716
by(simp add: gcd_int_def)
nipkow@31813
   717
nipkow@31952
   718
lemma gcd_abs_int: "gcd (x::int) y = gcd (abs x) (abs y)"
nipkow@31813
   719
by (simp add: gcd_int_def)
nipkow@31813
   720
nipkow@31813
   721
lemma gcd_abs1_int[simp]: "gcd (abs x) (y::int) = gcd x y"
nipkow@31952
   722
by (metis abs_idempotent gcd_abs_int)
nipkow@31813
   723
nipkow@31813
   724
lemma gcd_abs2_int[simp]: "gcd x (abs y::int) = gcd x y"
nipkow@31952
   725
by (metis abs_idempotent gcd_abs_int)
huffman@31706
   726
nipkow@31952
   727
lemma gcd_cases_int:
huffman@31706
   728
  fixes x :: int and y
huffman@31706
   729
  assumes "x >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> P (gcd x y)"
huffman@31706
   730
      and "x >= 0 \<Longrightarrow> y <= 0 \<Longrightarrow> P (gcd x (-y))"
huffman@31706
   731
      and "x <= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> P (gcd (-x) y)"
huffman@31706
   732
      and "x <= 0 \<Longrightarrow> y <= 0 \<Longrightarrow> P (gcd (-x) (-y))"
huffman@31706
   733
  shows "P (gcd x y)"
huffman@35216
   734
by (insert assms, auto, arith)
wenzelm@21256
   735
nipkow@31952
   736
lemma gcd_ge_0_int [simp]: "gcd (x::int) y >= 0"
huffman@31706
   737
  by (simp add: gcd_int_def)
huffman@31706
   738
nipkow@31952
   739
lemma lcm_neg1_int: "lcm (-x::int) y = lcm x y"
huffman@31706
   740
  by (simp add: lcm_int_def)
huffman@31706
   741
nipkow@31952
   742
lemma lcm_neg2_int: "lcm (x::int) (-y) = lcm x y"
huffman@31706
   743
  by (simp add: lcm_int_def)
huffman@31706
   744
nipkow@31952
   745
lemma lcm_abs_int: "lcm (x::int) y = lcm (abs x) (abs y)"
huffman@31706
   746
  by (simp add: lcm_int_def)
wenzelm@21256
   747
nipkow@31814
   748
lemma abs_lcm_int [simp]: "abs (lcm i j::int) = lcm i j"
nipkow@31814
   749
by(simp add:lcm_int_def)
nipkow@31814
   750
nipkow@31814
   751
lemma lcm_abs1_int[simp]: "lcm (abs x) (y::int) = lcm x y"
nipkow@31814
   752
by (metis abs_idempotent lcm_int_def)
nipkow@31814
   753
nipkow@31814
   754
lemma lcm_abs2_int[simp]: "lcm x (abs y::int) = lcm x y"
nipkow@31814
   755
by (metis abs_idempotent lcm_int_def)
nipkow@31814
   756
nipkow@31952
   757
lemma lcm_cases_int:
huffman@31706
   758
  fixes x :: int and y
huffman@31706
   759
  assumes "x >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> P (lcm x y)"
huffman@31706
   760
      and "x >= 0 \<Longrightarrow> y <= 0 \<Longrightarrow> P (lcm x (-y))"
huffman@31706
   761
      and "x <= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> P (lcm (-x) y)"
huffman@31706
   762
      and "x <= 0 \<Longrightarrow> y <= 0 \<Longrightarrow> P (lcm (-x) (-y))"
huffman@31706
   763
  shows "P (lcm x y)"
wenzelm@41550
   764
  using assms by (auto simp add: lcm_neg1_int lcm_neg2_int) arith
huffman@31706
   765
nipkow@31952
   766
lemma lcm_ge_0_int [simp]: "lcm (x::int) y >= 0"
huffman@31706
   767
  by (simp add: lcm_int_def)
huffman@31706
   768
huffman@31706
   769
(* was gcd_0, etc. *)
haftmann@54867
   770
lemma gcd_0_nat: "gcd (x::nat) 0 = x"
haftmann@23687
   771
  by simp
haftmann@23687
   772
huffman@31706
   773
(* was igcd_0, etc. *)
nipkow@31952
   774
lemma gcd_0_int [simp]: "gcd (x::int) 0 = abs x"
huffman@31706
   775
  by (unfold gcd_int_def, auto)
huffman@31706
   776
haftmann@54867
   777
lemma gcd_0_left_nat: "gcd 0 (x::nat) = x"
haftmann@23687
   778
  by simp
haftmann@23687
   779
nipkow@31952
   780
lemma gcd_0_left_int [simp]: "gcd 0 (x::int) = abs x"
huffman@31706
   781
  by (unfold gcd_int_def, auto)
huffman@31706
   782
nipkow@31952
   783
lemma gcd_red_nat: "gcd (x::nat) y = gcd y (x mod y)"
huffman@31706
   784
  by (case_tac "y = 0", auto)
huffman@31706
   785
huffman@31706
   786
(* weaker, but useful for the simplifier *)
huffman@31706
   787
nipkow@31952
   788
lemma gcd_non_0_nat: "y ~= (0::nat) \<Longrightarrow> gcd (x::nat) y = gcd y (x mod y)"
huffman@31706
   789
  by simp
huffman@31706
   790
nipkow@31952
   791
lemma gcd_1_nat [simp]: "gcd (m::nat) 1 = 1"
wenzelm@21263
   792
  by simp
wenzelm@21256
   793
nipkow@31952
   794
lemma gcd_Suc_0 [simp]: "gcd (m::nat) (Suc 0) = Suc 0"
huffman@31706
   795
  by (simp add: One_nat_def)
huffman@31706
   796
nipkow@31952
   797
lemma gcd_1_int [simp]: "gcd (m::int) 1 = 1"
huffman@31706
   798
  by (simp add: gcd_int_def)
huffman@30082
   799
nipkow@31952
   800
lemma gcd_idem_nat: "gcd (x::nat) x = x"
nipkow@31798
   801
by simp
huffman@31706
   802
nipkow@31952
   803
lemma gcd_idem_int: "gcd (x::int) x = abs x"
nipkow@31813
   804
by (auto simp add: gcd_int_def)
huffman@31706
   805
huffman@31706
   806
declare gcd_nat.simps [simp del]
wenzelm@21256
   807
wenzelm@21256
   808
text {*
haftmann@27556
   809
  \medskip @{term "gcd m n"} divides @{text m} and @{text n}.  The
wenzelm@21256
   810
  conjunctions don't seem provable separately.
wenzelm@21256
   811
*}
wenzelm@21256
   812
haftmann@59008
   813
instance nat :: semiring_gcd
haftmann@59008
   814
proof
haftmann@59008
   815
  fix m n :: nat
haftmann@59008
   816
  show "gcd m n dvd m" and "gcd m n dvd n"
haftmann@59008
   817
  proof (induct m n rule: gcd_nat_induct)
haftmann@59008
   818
    fix m n :: nat
haftmann@59008
   819
    assume "gcd n (m mod n) dvd m mod n" and "gcd n (m mod n) dvd n"
haftmann@59008
   820
    then have "gcd n (m mod n) dvd m"
haftmann@59008
   821
      by (rule dvd_mod_imp_dvd)
haftmann@59008
   822
    moreover assume "0 < n"
haftmann@59008
   823
    ultimately show "gcd m n dvd m"
haftmann@59008
   824
      by (simp add: gcd_non_0_nat)
haftmann@59008
   825
  qed (simp_all add: gcd_0_nat gcd_non_0_nat)
haftmann@59008
   826
next
haftmann@59008
   827
  fix m n k :: nat
haftmann@59008
   828
  assume "k dvd m" and "k dvd n"
haftmann@59008
   829
  then show "k dvd gcd m n"
haftmann@59008
   830
    by (induct m n rule: gcd_nat_induct) (simp_all add: gcd_non_0_nat dvd_mod gcd_0_nat)
haftmann@60686
   831
qed (simp_all add: lcm_nat_def)
lp15@59667
   832
haftmann@59008
   833
instance int :: ring_gcd
haftmann@60686
   834
  by standard
haftmann@60686
   835
    (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
   836
nipkow@31730
   837
lemma dvd_gcd_D1_nat: "k dvd gcd m n \<Longrightarrow> (k::nat) dvd m"
haftmann@59008
   838
  by (metis gcd_dvd1 dvd_trans)
nipkow@31730
   839
nipkow@31730
   840
lemma dvd_gcd_D2_nat: "k dvd gcd m n \<Longrightarrow> (k::nat) dvd n"
haftmann@59008
   841
  by (metis gcd_dvd2 dvd_trans)
nipkow@31730
   842
nipkow@31730
   843
lemma dvd_gcd_D1_int: "i dvd gcd m n \<Longrightarrow> (i::int) dvd m"
haftmann@59008
   844
  by (metis gcd_dvd1 dvd_trans)
nipkow@31730
   845
nipkow@31730
   846
lemma dvd_gcd_D2_int: "i dvd gcd m n \<Longrightarrow> (i::int) dvd n"
haftmann@59008
   847
  by (metis gcd_dvd2 dvd_trans)
nipkow@31730
   848
nipkow@31952
   849
lemma gcd_le1_nat [simp]: "a \<noteq> 0 \<Longrightarrow> gcd (a::nat) b \<le> a"
huffman@31706
   850
  by (rule dvd_imp_le, auto)
huffman@31706
   851
nipkow@31952
   852
lemma gcd_le2_nat [simp]: "b \<noteq> 0 \<Longrightarrow> gcd (a::nat) b \<le> b"
huffman@31706
   853
  by (rule dvd_imp_le, auto)
huffman@31706
   854
nipkow@31952
   855
lemma gcd_le1_int [simp]: "a > 0 \<Longrightarrow> gcd (a::int) b \<le> a"
huffman@31706
   856
  by (rule zdvd_imp_le, auto)
wenzelm@21256
   857
nipkow@31952
   858
lemma gcd_le2_int [simp]: "b > 0 \<Longrightarrow> gcd (a::int) b \<le> b"
huffman@31706
   859
  by (rule zdvd_imp_le, auto)
huffman@31706
   860
nipkow@31952
   861
lemma gcd_greatest_iff_nat [iff]: "(k dvd gcd (m::nat) n) =
huffman@31706
   862
    (k dvd m & k dvd n)"
haftmann@59008
   863
  by (blast intro!: gcd_greatest intro: dvd_trans)
huffman@31706
   864
nipkow@31952
   865
lemma gcd_greatest_iff_int: "((k::int) dvd gcd m n) = (k dvd m & k dvd n)"
haftmann@59008
   866
  by (blast intro!: gcd_greatest intro: dvd_trans)
wenzelm@21256
   867
nipkow@31952
   868
lemma gcd_zero_nat [simp]: "(gcd (m::nat) n = 0) = (m = 0 & n = 0)"
nipkow@31952
   869
  by (simp only: dvd_0_left_iff [symmetric] gcd_greatest_iff_nat)
wenzelm@21256
   870
nipkow@31952
   871
lemma gcd_zero_int [simp]: "(gcd (m::int) n = 0) = (m = 0 & n = 0)"
huffman@31706
   872
  by (auto simp add: gcd_int_def)
wenzelm@21256
   873
nipkow@31952
   874
lemma gcd_pos_nat [simp]: "(gcd (m::nat) n > 0) = (m ~= 0 | n ~= 0)"
nipkow@31952
   875
  by (insert gcd_zero_nat [of m n], arith)
wenzelm@21256
   876
nipkow@31952
   877
lemma gcd_pos_int [simp]: "(gcd (m::int) n > 0) = (m ~= 0 | n ~= 0)"
nipkow@31952
   878
  by (insert gcd_zero_int [of m n], insert gcd_ge_0_int [of m n], arith)
huffman@31706
   879
nipkow@31952
   880
lemma gcd_unique_nat: "(d::nat) dvd a \<and> d dvd b \<and>
huffman@31706
   881
    (\<forall>e. e dvd a \<and> e dvd b \<longrightarrow> e dvd d) \<longleftrightarrow> d = gcd a b"
huffman@31706
   882
  apply auto
nipkow@33657
   883
  apply (rule dvd_antisym)
haftmann@59008
   884
  apply (erule (1) gcd_greatest)
huffman@31706
   885
  apply auto
huffman@31706
   886
done
wenzelm@21256
   887
nipkow@31952
   888
lemma gcd_unique_int: "d >= 0 & (d::int) dvd a \<and> d dvd b \<and>
huffman@31706
   889
    (\<forall>e. e dvd a \<and> e dvd b \<longrightarrow> e dvd d) \<longleftrightarrow> d = gcd a b"
nipkow@33657
   890
apply (case_tac "d = 0")
nipkow@33657
   891
 apply simp
nipkow@33657
   892
apply (rule iffI)
nipkow@33657
   893
 apply (rule zdvd_antisym_nonneg)
haftmann@59008
   894
 apply (auto intro: gcd_greatest)
huffman@31706
   895
done
huffman@30082
   896
haftmann@54867
   897
interpretation gcd_nat: abel_semigroup "gcd :: nat \<Rightarrow> nat \<Rightarrow> nat"
haftmann@54867
   898
  + gcd_nat: semilattice_neutr_order "gcd :: nat \<Rightarrow> nat \<Rightarrow> nat" 0 "op dvd" "(\<lambda>m n. m dvd n \<and> \<not> n dvd m)"
haftmann@60686
   899
apply standard
haftmann@60686
   900
apply (auto intro: dvd_antisym dvd_trans)[2]
haftmann@59545
   901
apply (metis dvd.dual_order.refl gcd_unique_nat)+
haftmann@54867
   902
done
haftmann@54867
   903
haftmann@60686
   904
interpretation gcd_int: abel_semigroup "gcd :: int \<Rightarrow> int \<Rightarrow> int" ..
haftmann@54867
   905
haftmann@60686
   906
lemmas gcd_assoc_nat = gcd.assoc [where ?'a = nat]
haftmann@60686
   907
lemmas gcd_commute_nat = gcd.commute [where ?'a = nat]
haftmann@60686
   908
lemmas gcd_left_commute_nat = gcd.left_commute [where ?'a = nat]
haftmann@60686
   909
lemmas gcd_assoc_int = gcd.assoc [where ?'a = int]
haftmann@60686
   910
lemmas gcd_commute_int = gcd.commute [where ?'a = int]
haftmann@60686
   911
lemmas gcd_left_commute_int = gcd.left_commute [where ?'a = int]
haftmann@54867
   912
haftmann@54867
   913
lemmas gcd_ac_nat = gcd_assoc_nat gcd_commute_nat gcd_left_commute_nat
haftmann@54867
   914
haftmann@54867
   915
lemmas gcd_ac_int = gcd_assoc_int gcd_commute_int gcd_left_commute_int
haftmann@54867
   916
nipkow@31798
   917
lemma gcd_proj1_if_dvd_nat [simp]: "(x::nat) dvd y \<Longrightarrow> gcd x y = x"
haftmann@54867
   918
  by (fact gcd_nat.absorb1)
nipkow@31798
   919
nipkow@31798
   920
lemma gcd_proj2_if_dvd_nat [simp]: "(y::nat) dvd x \<Longrightarrow> gcd x y = y"
haftmann@54867
   921
  by (fact gcd_nat.absorb2)
nipkow@31798
   922
haftmann@54867
   923
lemma gcd_proj1_if_dvd_int [simp]: "x dvd y \<Longrightarrow> gcd (x::int) y = abs x"
haftmann@54867
   924
  by (metis abs_dvd_iff gcd_0_left_int gcd_abs_int gcd_unique_int)
nipkow@31798
   925
haftmann@54867
   926
lemma gcd_proj2_if_dvd_int [simp]: "y dvd x \<Longrightarrow> gcd (x::int) y = abs y"
haftmann@54867
   927
  by (metis gcd_proj1_if_dvd_int gcd_commute_int)
nipkow@31798
   928
wenzelm@21256
   929
text {*
wenzelm@21256
   930
  \medskip Multiplication laws
wenzelm@21256
   931
*}
wenzelm@21256
   932
nipkow@31952
   933
lemma gcd_mult_distrib_nat: "(k::nat) * gcd m n = gcd (k * m) (k * n)"
wenzelm@58623
   934
    -- {* @{cite \<open>page 27\<close> davenport92} *}
nipkow@31952
   935
  apply (induct m n rule: gcd_nat_induct)
huffman@31706
   936
  apply simp
wenzelm@21256
   937
  apply (case_tac "k = 0")
huffman@45270
   938
  apply (simp_all add: gcd_non_0_nat)
huffman@31706
   939
done
wenzelm@21256
   940
nipkow@31952
   941
lemma gcd_mult_distrib_int: "abs (k::int) * gcd m n = gcd (k * m) (k * n)"
nipkow@31952
   942
  apply (subst (1 2) gcd_abs_int)
nipkow@31813
   943
  apply (subst (1 2) abs_mult)
nipkow@31952
   944
  apply (rule gcd_mult_distrib_nat [transferred])
huffman@31706
   945
  apply auto
huffman@31706
   946
done
wenzelm@21256
   947
nipkow@31952
   948
lemma coprime_dvd_mult_nat: "coprime (k::nat) n \<Longrightarrow> k dvd m * n \<Longrightarrow> k dvd m"
nipkow@31952
   949
  apply (insert gcd_mult_distrib_nat [of m k n])
wenzelm@21256
   950
  apply simp
wenzelm@21256
   951
  apply (erule_tac t = m in ssubst)
wenzelm@21256
   952
  apply simp
wenzelm@21256
   953
  done
wenzelm@21256
   954
nipkow@31952
   955
lemma coprime_dvd_mult_int:
nipkow@31813
   956
  "coprime (k::int) n \<Longrightarrow> k dvd m * n \<Longrightarrow> k dvd m"
nipkow@31813
   957
apply (subst abs_dvd_iff [symmetric])
nipkow@31813
   958
apply (subst dvd_abs_iff [symmetric])
nipkow@31952
   959
apply (subst (asm) gcd_abs_int)
nipkow@31952
   960
apply (rule coprime_dvd_mult_nat [transferred])
nipkow@31813
   961
    prefer 4 apply assumption
nipkow@31813
   962
   apply auto
nipkow@31813
   963
apply (subst abs_mult [symmetric], auto)
huffman@31706
   964
done
huffman@31706
   965
nipkow@31952
   966
lemma coprime_dvd_mult_iff_nat: "coprime (k::nat) n \<Longrightarrow>
huffman@31706
   967
    (k dvd m * n) = (k dvd m)"
nipkow@31952
   968
  by (auto intro: coprime_dvd_mult_nat)
huffman@31706
   969
nipkow@31952
   970
lemma coprime_dvd_mult_iff_int: "coprime (k::int) n \<Longrightarrow>
huffman@31706
   971
    (k dvd m * n) = (k dvd m)"
nipkow@31952
   972
  by (auto intro: coprime_dvd_mult_int)
huffman@31706
   973
nipkow@31952
   974
lemma gcd_mult_cancel_nat: "coprime k n \<Longrightarrow> gcd ((k::nat) * m) n = gcd m n"
nipkow@33657
   975
  apply (rule dvd_antisym)
haftmann@59008
   976
  apply (rule gcd_greatest)
nipkow@31952
   977
  apply (rule_tac n = k in coprime_dvd_mult_nat)
nipkow@31952
   978
  apply (simp add: gcd_assoc_nat)
nipkow@31952
   979
  apply (simp add: gcd_commute_nat)
haftmann@57512
   980
  apply (simp_all add: mult.commute)
huffman@31706
   981
done
wenzelm@21256
   982
nipkow@31952
   983
lemma gcd_mult_cancel_int:
nipkow@31813
   984
  "coprime (k::int) n \<Longrightarrow> gcd (k * m) n = gcd m n"
nipkow@31952
   985
apply (subst (1 2) gcd_abs_int)
nipkow@31813
   986
apply (subst abs_mult)
nipkow@31952
   987
apply (rule gcd_mult_cancel_nat [transferred], auto)
huffman@31706
   988
done
wenzelm@21256
   989
haftmann@35368
   990
lemma coprime_crossproduct_nat:
haftmann@35368
   991
  fixes a b c d :: nat
haftmann@35368
   992
  assumes "coprime a d" and "coprime b c"
haftmann@35368
   993
  shows "a * c = b * d \<longleftrightarrow> a = b \<and> c = d" (is "?lhs \<longleftrightarrow> ?rhs")
haftmann@35368
   994
proof
haftmann@35368
   995
  assume ?rhs then show ?lhs by simp
haftmann@35368
   996
next
haftmann@35368
   997
  assume ?lhs
haftmann@35368
   998
  from `?lhs` have "a dvd b * d" by (auto intro: dvdI dest: sym)
haftmann@35368
   999
  with `coprime a d` have "a dvd b" by (simp add: coprime_dvd_mult_iff_nat)
haftmann@35368
  1000
  from `?lhs` have "b dvd a * c" by (auto intro: dvdI dest: sym)
haftmann@35368
  1001
  with `coprime b c` have "b dvd a" by (simp add: coprime_dvd_mult_iff_nat)
haftmann@57512
  1002
  from `?lhs` have "c dvd d * b" by (auto intro: dvdI dest: sym simp add: mult.commute)
haftmann@35368
  1003
  with `coprime b c` have "c dvd d" by (simp add: coprime_dvd_mult_iff_nat gcd_commute_nat)
haftmann@57512
  1004
  from `?lhs` have "d dvd c * a" by (auto intro: dvdI dest: sym simp add: mult.commute)
haftmann@35368
  1005
  with `coprime a d` have "d dvd c" by (simp add: coprime_dvd_mult_iff_nat gcd_commute_nat)
haftmann@35368
  1006
  from `a dvd b` `b dvd a` have "a = b" by (rule Nat.dvd.antisym)
haftmann@35368
  1007
  moreover from `c dvd d` `d dvd c` have "c = d" by (rule Nat.dvd.antisym)
haftmann@35368
  1008
  ultimately show ?rhs ..
haftmann@35368
  1009
qed
haftmann@35368
  1010
haftmann@35368
  1011
lemma coprime_crossproduct_int:
haftmann@35368
  1012
  fixes a b c d :: int
haftmann@35368
  1013
  assumes "coprime a d" and "coprime b c"
haftmann@35368
  1014
  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@35368
  1015
  using assms by (intro coprime_crossproduct_nat [transferred]) auto
haftmann@35368
  1016
wenzelm@21256
  1017
text {* \medskip Addition laws *}
wenzelm@21256
  1018
nipkow@31952
  1019
lemma gcd_add1_nat [simp]: "gcd ((m::nat) + n) n = gcd m n"
huffman@31706
  1020
  apply (case_tac "n = 0")
nipkow@31952
  1021
  apply (simp_all add: gcd_non_0_nat)
huffman@31706
  1022
done
huffman@31706
  1023
nipkow@31952
  1024
lemma gcd_add2_nat [simp]: "gcd (m::nat) (m + n) = gcd m n"
nipkow@31952
  1025
  apply (subst (1 2) gcd_commute_nat)
haftmann@57512
  1026
  apply (subst add.commute)
huffman@31706
  1027
  apply simp
huffman@31706
  1028
done
huffman@31706
  1029
huffman@31706
  1030
(* to do: add the other variations? *)
huffman@31706
  1031
nipkow@31952
  1032
lemma gcd_diff1_nat: "(m::nat) >= n \<Longrightarrow> gcd (m - n) n = gcd m n"
nipkow@31952
  1033
  by (subst gcd_add1_nat [symmetric], auto)
huffman@31706
  1034
nipkow@31952
  1035
lemma gcd_diff2_nat: "(n::nat) >= m \<Longrightarrow> gcd (n - m) n = gcd m n"
nipkow@31952
  1036
  apply (subst gcd_commute_nat)
nipkow@31952
  1037
  apply (subst gcd_diff1_nat [symmetric])
huffman@31706
  1038
  apply auto
nipkow@31952
  1039
  apply (subst gcd_commute_nat)
nipkow@31952
  1040
  apply (subst gcd_diff1_nat)
huffman@31706
  1041
  apply assumption
nipkow@31952
  1042
  apply (rule gcd_commute_nat)
huffman@31706
  1043
done
huffman@31706
  1044
nipkow@31952
  1045
lemma gcd_non_0_int: "(y::int) > 0 \<Longrightarrow> gcd x y = gcd y (x mod y)"
huffman@31706
  1046
  apply (frule_tac b = y and a = x in pos_mod_sign)
huffman@31706
  1047
  apply (simp del: pos_mod_sign add: gcd_int_def abs_if nat_mod_distrib)
nipkow@31952
  1048
  apply (auto simp add: gcd_non_0_nat nat_mod_distrib [symmetric]
huffman@31706
  1049
    zmod_zminus1_eq_if)
huffman@31706
  1050
  apply (frule_tac a = x in pos_mod_bound)
nipkow@31952
  1051
  apply (subst (1 2) gcd_commute_nat)
nipkow@31952
  1052
  apply (simp del: pos_mod_bound add: nat_diff_distrib gcd_diff2_nat
huffman@31706
  1053
    nat_le_eq_zle)
huffman@31706
  1054
done
wenzelm@21256
  1055
nipkow@31952
  1056
lemma gcd_red_int: "gcd (x::int) y = gcd y (x mod y)"
huffman@31706
  1057
  apply (case_tac "y = 0")
huffman@31706
  1058
  apply force
huffman@31706
  1059
  apply (case_tac "y > 0")
nipkow@31952
  1060
  apply (subst gcd_non_0_int, auto)
nipkow@31952
  1061
  apply (insert gcd_non_0_int [of "-y" "-x"])
huffman@35216
  1062
  apply auto
huffman@31706
  1063
done
huffman@31706
  1064
nipkow@31952
  1065
lemma gcd_add1_int [simp]: "gcd ((m::int) + n) n = gcd m n"
haftmann@57512
  1066
by (metis gcd_red_int mod_add_self1 add.commute)
huffman@31706
  1067
nipkow@31952
  1068
lemma gcd_add2_int [simp]: "gcd m ((m::int) + n) = gcd m n"
haftmann@57512
  1069
by (metis gcd_add1_int gcd_commute_int add.commute)
wenzelm@21256
  1070
nipkow@31952
  1071
lemma gcd_add_mult_nat: "gcd (m::nat) (k * m + n) = gcd m n"
nipkow@31952
  1072
by (metis mod_mult_self3 gcd_commute_nat gcd_red_nat)
wenzelm@21256
  1073
nipkow@31952
  1074
lemma gcd_add_mult_int: "gcd (m::int) (k * m + n) = gcd m n"
haftmann@57512
  1075
by (metis gcd_commute_int gcd_red_int mod_mult_self1 add.commute)
nipkow@31798
  1076
wenzelm@21256
  1077
huffman@31706
  1078
(* to do: differences, and all variations of addition rules
huffman@31706
  1079
    as simplification rules for nat and int *)
huffman@31706
  1080
nipkow@31798
  1081
(* FIXME remove iff *)
nipkow@31952
  1082
lemma gcd_dvd_prod_nat [iff]: "gcd (m::nat) n dvd k * n"
haftmann@23687
  1083
  using mult_dvd_mono [of 1] by auto
chaieb@22027
  1084
huffman@31706
  1085
(* to do: add the three variations of these, and for ints? *)
huffman@31706
  1086
nipkow@31992
  1087
lemma finite_divisors_nat[simp]:
nipkow@31992
  1088
  assumes "(m::nat) ~= 0" shows "finite{d. d dvd m}"
nipkow@31734
  1089
proof-
wenzelm@60512
  1090
  have "finite{d. d <= m}"
wenzelm@60512
  1091
    by (blast intro: bounded_nat_set_is_finite)
nipkow@31734
  1092
  from finite_subset[OF _ this] show ?thesis using assms
wenzelm@60512
  1093
    by (metis Collect_mono dvd_imp_le neq0_conv)
nipkow@31734
  1094
qed
nipkow@31734
  1095
nipkow@31995
  1096
lemma finite_divisors_int[simp]:
nipkow@31734
  1097
  assumes "(i::int) ~= 0" shows "finite{d. d dvd i}"
nipkow@31734
  1098
proof-
nipkow@31734
  1099
  have "{d. abs d <= abs i} = {- abs i .. abs i}" by(auto simp:abs_if)
nipkow@31734
  1100
  hence "finite{d. abs d <= abs i}" by simp
nipkow@31734
  1101
  from finite_subset[OF _ this] show ?thesis using assms
wenzelm@60512
  1102
    by (simp add: dvd_imp_le_int subset_iff)
nipkow@31734
  1103
qed
nipkow@31734
  1104
nipkow@31995
  1105
lemma Max_divisors_self_nat[simp]: "n\<noteq>0 \<Longrightarrow> Max{d::nat. d dvd n} = n"
nipkow@31995
  1106
apply(rule antisym)
nipkow@44890
  1107
 apply (fastforce intro: Max_le_iff[THEN iffD2] simp: dvd_imp_le)
nipkow@31995
  1108
apply simp
nipkow@31995
  1109
done
nipkow@31995
  1110
nipkow@31995
  1111
lemma Max_divisors_self_int[simp]: "n\<noteq>0 \<Longrightarrow> Max{d::int. d dvd n} = abs n"
nipkow@31995
  1112
apply(rule antisym)
haftmann@44278
  1113
 apply(rule Max_le_iff [THEN iffD2])
haftmann@44278
  1114
  apply (auto intro: abs_le_D1 dvd_imp_le_int)
nipkow@31995
  1115
done
nipkow@31995
  1116
nipkow@31734
  1117
lemma gcd_is_Max_divisors_nat:
nipkow@31734
  1118
  "m ~= 0 \<Longrightarrow> n ~= 0 \<Longrightarrow> gcd (m::nat) n = (Max {d. d dvd m & d dvd n})"
nipkow@31734
  1119
apply(rule Max_eqI[THEN sym])
nipkow@31995
  1120
  apply (metis finite_Collect_conjI finite_divisors_nat)
nipkow@31734
  1121
 apply simp
nipkow@31952
  1122
 apply(metis Suc_diff_1 Suc_neq_Zero dvd_imp_le gcd_greatest_iff_nat gcd_pos_nat)
nipkow@31734
  1123
apply simp
nipkow@31734
  1124
done
nipkow@31734
  1125
nipkow@31734
  1126
lemma gcd_is_Max_divisors_int:
nipkow@31734
  1127
  "m ~= 0 ==> n ~= 0 ==> gcd (m::int) n = (Max {d. d dvd m & d dvd n})"
nipkow@31734
  1128
apply(rule Max_eqI[THEN sym])
nipkow@31995
  1129
  apply (metis finite_Collect_conjI finite_divisors_int)
nipkow@31734
  1130
 apply simp
nipkow@31952
  1131
 apply (metis gcd_greatest_iff_int gcd_pos_int zdvd_imp_le)
nipkow@31734
  1132
apply simp
nipkow@31734
  1133
done
nipkow@31734
  1134
haftmann@34030
  1135
lemma gcd_code_int [code]:
haftmann@34030
  1136
  "gcd k l = \<bar>if l = (0::int) then k else gcd l (\<bar>k\<bar> mod \<bar>l\<bar>)\<bar>"
haftmann@34030
  1137
  by (simp add: gcd_int_def nat_mod_distrib gcd_non_0_nat)
haftmann@34030
  1138
chaieb@22027
  1139
huffman@31706
  1140
subsection {* Coprimality *}
huffman@31706
  1141
nipkow@31952
  1142
lemma div_gcd_coprime_nat:
huffman@31706
  1143
  assumes nz: "(a::nat) \<noteq> 0 \<or> b \<noteq> 0"
huffman@31706
  1144
  shows "coprime (a div gcd a b) (b div gcd a b)"
wenzelm@22367
  1145
proof -
haftmann@27556
  1146
  let ?g = "gcd a b"
chaieb@22027
  1147
  let ?a' = "a div ?g"
chaieb@22027
  1148
  let ?b' = "b div ?g"
haftmann@27556
  1149
  let ?g' = "gcd ?a' ?b'"
chaieb@22027
  1150
  have dvdg: "?g dvd a" "?g dvd b" by simp_all
chaieb@22027
  1151
  have dvdg': "?g' dvd ?a'" "?g' dvd ?b'" by simp_all
wenzelm@22367
  1152
  from dvdg dvdg' obtain ka kb ka' kb' where
wenzelm@22367
  1153
      kab: "a = ?g * ka" "b = ?g * kb" "?a' = ?g' * ka'" "?b' = ?g' * kb'"
chaieb@22027
  1154
    unfolding dvd_def by blast
haftmann@58834
  1155
  from this [symmetric] have "?g * ?a' = (?g * ?g') * ka'" "?g * ?b' = (?g * ?g') * kb'"
haftmann@58834
  1156
    by (simp_all add: mult.assoc mult.left_commute [of "gcd a b"])
wenzelm@22367
  1157
  then have dvdgg':"?g * ?g' dvd a" "?g* ?g' dvd b"
wenzelm@22367
  1158
    by (auto simp add: dvd_mult_div_cancel [OF dvdg(1)]
wenzelm@22367
  1159
      dvd_mult_div_cancel [OF dvdg(2)] dvd_def)
huffman@35216
  1160
  have "?g \<noteq> 0" using nz by simp
huffman@31706
  1161
  then have gp: "?g > 0" by arith
haftmann@59008
  1162
  from gcd_greatest [OF dvdgg'] have "?g * ?g' dvd ?g" .
wenzelm@22367
  1163
  with dvd_mult_cancel1 [OF gp] show "?g' = 1" by simp
chaieb@22027
  1164
qed
chaieb@22027
  1165
nipkow@31952
  1166
lemma div_gcd_coprime_int:
huffman@31706
  1167
  assumes nz: "(a::int) \<noteq> 0 \<or> b \<noteq> 0"
huffman@31706
  1168
  shows "coprime (a div gcd a b) (b div gcd a b)"
nipkow@31952
  1169
apply (subst (1 2 3) gcd_abs_int)
nipkow@31813
  1170
apply (subst (1 2) abs_div)
nipkow@31813
  1171
  apply simp
nipkow@31813
  1172
 apply simp
nipkow@31813
  1173
apply(subst (1 2) abs_gcd_int)
nipkow@31952
  1174
apply (rule div_gcd_coprime_nat [transferred])
nipkow@31952
  1175
using nz apply (auto simp add: gcd_abs_int [symmetric])
huffman@31706
  1176
done
huffman@31706
  1177
nipkow@31952
  1178
lemma coprime_nat: "coprime (a::nat) b \<longleftrightarrow> (\<forall>d. d dvd a \<and> d dvd b \<longleftrightarrow> d = 1)"
nipkow@31952
  1179
  using gcd_unique_nat[of 1 a b, simplified] by auto
huffman@31706
  1180
nipkow@31952
  1181
lemma coprime_Suc_0_nat:
huffman@31706
  1182
    "coprime (a::nat) b \<longleftrightarrow> (\<forall>d. d dvd a \<and> d dvd b \<longleftrightarrow> d = Suc 0)"
nipkow@31952
  1183
  using coprime_nat by (simp add: One_nat_def)
huffman@31706
  1184
nipkow@31952
  1185
lemma coprime_int: "coprime (a::int) b \<longleftrightarrow>
huffman@31706
  1186
    (\<forall>d. d >= 0 \<and> d dvd a \<and> d dvd b \<longleftrightarrow> d = 1)"
nipkow@31952
  1187
  using gcd_unique_int [of 1 a b]
huffman@31706
  1188
  apply clarsimp
huffman@31706
  1189
  apply (erule subst)
huffman@31706
  1190
  apply (rule iffI)
huffman@31706
  1191
  apply force
wenzelm@59807
  1192
  apply (drule_tac x = "abs e" for e in exI)
wenzelm@59807
  1193
  apply (case_tac "e >= 0" for e :: int)
huffman@31706
  1194
  apply force
huffman@31706
  1195
  apply force
wenzelm@59807
  1196
  done
huffman@31706
  1197
nipkow@31952
  1198
lemma gcd_coprime_nat:
huffman@31706
  1199
  assumes z: "gcd (a::nat) b \<noteq> 0" and a: "a = a' * gcd a b" and
huffman@31706
  1200
    b: "b = b' * gcd a b"
huffman@31706
  1201
  shows    "coprime a' b'"
huffman@31706
  1202
huffman@31706
  1203
  apply (subgoal_tac "a' = a div gcd a b")
huffman@31706
  1204
  apply (erule ssubst)
huffman@31706
  1205
  apply (subgoal_tac "b' = b div gcd a b")
huffman@31706
  1206
  apply (erule ssubst)
nipkow@31952
  1207
  apply (rule div_gcd_coprime_nat)
wenzelm@41550
  1208
  using z apply force
huffman@31706
  1209
  apply (subst (1) b)
huffman@31706
  1210
  using z apply force
huffman@31706
  1211
  apply (subst (1) a)
huffman@31706
  1212
  using z apply force
wenzelm@41550
  1213
  done
huffman@31706
  1214
nipkow@31952
  1215
lemma gcd_coprime_int:
huffman@31706
  1216
  assumes z: "gcd (a::int) b \<noteq> 0" and a: "a = a' * gcd a b" and
huffman@31706
  1217
    b: "b = b' * gcd a b"
huffman@31706
  1218
  shows    "coprime a' b'"
huffman@31706
  1219
huffman@31706
  1220
  apply (subgoal_tac "a' = a div gcd a b")
huffman@31706
  1221
  apply (erule ssubst)
huffman@31706
  1222
  apply (subgoal_tac "b' = b div gcd a b")
huffman@31706
  1223
  apply (erule ssubst)
nipkow@31952
  1224
  apply (rule div_gcd_coprime_int)
wenzelm@41550
  1225
  using z apply force
huffman@31706
  1226
  apply (subst (1) b)
huffman@31706
  1227
  using z apply force
huffman@31706
  1228
  apply (subst (1) a)
huffman@31706
  1229
  using z apply force
wenzelm@41550
  1230
  done
huffman@31706
  1231
nipkow@31952
  1232
lemma coprime_mult_nat: assumes da: "coprime (d::nat) a" and db: "coprime d b"
huffman@31706
  1233
    shows "coprime d (a * b)"
nipkow@31952
  1234
  apply (subst gcd_commute_nat)
nipkow@31952
  1235
  using da apply (subst gcd_mult_cancel_nat)
nipkow@31952
  1236
  apply (subst gcd_commute_nat, assumption)
nipkow@31952
  1237
  apply (subst gcd_commute_nat, rule db)
huffman@31706
  1238
done
huffman@31706
  1239
nipkow@31952
  1240
lemma coprime_mult_int: assumes da: "coprime (d::int) a" and db: "coprime d b"
huffman@31706
  1241
    shows "coprime d (a * b)"
nipkow@31952
  1242
  apply (subst gcd_commute_int)
nipkow@31952
  1243
  using da apply (subst gcd_mult_cancel_int)
nipkow@31952
  1244
  apply (subst gcd_commute_int, assumption)
nipkow@31952
  1245
  apply (subst gcd_commute_int, rule db)
huffman@31706
  1246
done
huffman@31706
  1247
nipkow@31952
  1248
lemma coprime_lmult_nat:
huffman@31706
  1249
  assumes dab: "coprime (d::nat) (a * b)" shows "coprime d a"
huffman@31706
  1250
proof -
huffman@31706
  1251
  have "gcd d a dvd gcd d (a * b)"
haftmann@59008
  1252
    by (rule gcd_greatest, auto)
huffman@31706
  1253
  with dab show ?thesis
huffman@31706
  1254
    by auto
huffman@31706
  1255
qed
huffman@31706
  1256
nipkow@31952
  1257
lemma coprime_lmult_int:
nipkow@31798
  1258
  assumes "coprime (d::int) (a * b)" shows "coprime d a"
huffman@31706
  1259
proof -
huffman@31706
  1260
  have "gcd d a dvd gcd d (a * b)"
haftmann@59008
  1261
    by (rule gcd_greatest, auto)
nipkow@31798
  1262
  with assms show ?thesis
huffman@31706
  1263
    by auto
huffman@31706
  1264
qed
huffman@31706
  1265
nipkow@31952
  1266
lemma coprime_rmult_nat:
nipkow@31798
  1267
  assumes "coprime (d::nat) (a * b)" shows "coprime d b"
huffman@31706
  1268
proof -
huffman@31706
  1269
  have "gcd d b dvd gcd d (a * b)"
haftmann@59008
  1270
    by (rule gcd_greatest, auto intro: dvd_mult)
nipkow@31798
  1271
  with assms show ?thesis
huffman@31706
  1272
    by auto
huffman@31706
  1273
qed
huffman@31706
  1274
nipkow@31952
  1275
lemma coprime_rmult_int:
huffman@31706
  1276
  assumes dab: "coprime (d::int) (a * b)" shows "coprime d b"
huffman@31706
  1277
proof -
huffman@31706
  1278
  have "gcd d b dvd gcd d (a * b)"
haftmann@59008
  1279
    by (rule gcd_greatest, auto intro: dvd_mult)
huffman@31706
  1280
  with dab show ?thesis
huffman@31706
  1281
    by auto
huffman@31706
  1282
qed
huffman@31706
  1283
nipkow@31952
  1284
lemma coprime_mul_eq_nat: "coprime (d::nat) (a * b) \<longleftrightarrow>
huffman@31706
  1285
    coprime d a \<and>  coprime d b"
nipkow@31952
  1286
  using coprime_rmult_nat[of d a b] coprime_lmult_nat[of d a b]
nipkow@31952
  1287
    coprime_mult_nat[of d a b]
huffman@31706
  1288
  by blast
huffman@31706
  1289
nipkow@31952
  1290
lemma coprime_mul_eq_int: "coprime (d::int) (a * b) \<longleftrightarrow>
huffman@31706
  1291
    coprime d a \<and>  coprime d b"
nipkow@31952
  1292
  using coprime_rmult_int[of d a b] coprime_lmult_int[of d a b]
nipkow@31952
  1293
    coprime_mult_int[of d a b]
huffman@31706
  1294
  by blast
huffman@31706
  1295
noschinl@52397
  1296
lemma coprime_power_int:
noschinl@52397
  1297
  assumes "0 < n" shows "coprime (a :: int) (b ^ n) \<longleftrightarrow> coprime a b"
noschinl@52397
  1298
  using assms
noschinl@52397
  1299
proof (induct n)
noschinl@52397
  1300
  case (Suc n) then show ?case
noschinl@52397
  1301
    by (cases n) (simp_all add: coprime_mul_eq_int)
noschinl@52397
  1302
qed simp
noschinl@52397
  1303
nipkow@31952
  1304
lemma gcd_coprime_exists_nat:
huffman@31706
  1305
    assumes nz: "gcd (a::nat) b \<noteq> 0"
huffman@31706
  1306
    shows "\<exists>a' b'. a = a' * gcd a b \<and> b = b' * gcd a b \<and> coprime a' b'"
huffman@31706
  1307
  apply (rule_tac x = "a div gcd a b" in exI)
huffman@31706
  1308
  apply (rule_tac x = "b div gcd a b" in exI)
nipkow@31952
  1309
  using nz apply (auto simp add: div_gcd_coprime_nat dvd_div_mult)
huffman@31706
  1310
done
huffman@31706
  1311
nipkow@31952
  1312
lemma gcd_coprime_exists_int:
huffman@31706
  1313
    assumes nz: "gcd (a::int) b \<noteq> 0"
huffman@31706
  1314
    shows "\<exists>a' b'. a = a' * gcd a b \<and> b = b' * gcd a b \<and> coprime a' b'"
huffman@31706
  1315
  apply (rule_tac x = "a div gcd a b" in exI)
huffman@31706
  1316
  apply (rule_tac x = "b div gcd a b" in exI)
haftmann@59008
  1317
  using nz apply (auto simp add: div_gcd_coprime_int)
huffman@31706
  1318
done
huffman@31706
  1319
nipkow@31952
  1320
lemma coprime_exp_nat: "coprime (d::nat) a \<Longrightarrow> coprime d (a^n)"
haftmann@60596
  1321
  by (induct n) (simp_all add: coprime_mult_nat)
huffman@31706
  1322
nipkow@31952
  1323
lemma coprime_exp_int: "coprime (d::int) a \<Longrightarrow> coprime d (a^n)"
haftmann@60596
  1324
  by (induct n) (simp_all add: coprime_mult_int)
huffman@31706
  1325
nipkow@31952
  1326
lemma coprime_exp2_nat [intro]: "coprime (a::nat) b \<Longrightarrow> coprime (a^n) (b^m)"
haftmann@60686
  1327
  by (simp add: coprime_exp_nat ac_simps)
huffman@31706
  1328
nipkow@31952
  1329
lemma coprime_exp2_int [intro]: "coprime (a::int) b \<Longrightarrow> coprime (a^n) (b^m)"
haftmann@60686
  1330
  by (simp add: coprime_exp_int ac_simps)
huffman@31706
  1331
nipkow@31952
  1332
lemma gcd_exp_nat: "gcd ((a::nat)^n) (b^n) = (gcd a b)^n"
huffman@31706
  1333
proof (cases)
huffman@31706
  1334
  assume "a = 0 & b = 0"
huffman@31706
  1335
  thus ?thesis by simp
huffman@31706
  1336
  next assume "~(a = 0 & b = 0)"
huffman@31706
  1337
  hence "coprime ((a div gcd a b)^n) ((b div gcd a b)^n)"
nipkow@31952
  1338
    by (auto simp:div_gcd_coprime_nat)
huffman@31706
  1339
  hence "gcd ((a div gcd a b)^n * (gcd a b)^n)
huffman@31706
  1340
      ((b div gcd a b)^n * (gcd a b)^n) = (gcd a b)^n"
lp15@60162
  1341
    by (metis gcd_mult_distrib_nat mult.commute mult.right_neutral)
huffman@31706
  1342
  also have "(a div gcd a b)^n * (gcd a b)^n = a^n"
lp15@60162
  1343
    by (metis dvd_div_mult_self gcd_unique_nat power_mult_distrib)
huffman@31706
  1344
  also have "(b div gcd a b)^n * (gcd a b)^n = b^n"
lp15@60162
  1345
    by (metis dvd_div_mult_self gcd_unique_nat power_mult_distrib)
huffman@31706
  1346
  finally show ?thesis .
huffman@31706
  1347
qed
huffman@31706
  1348
nipkow@31952
  1349
lemma gcd_exp_int: "gcd ((a::int)^n) (b^n) = (gcd a b)^n"
nipkow@31952
  1350
  apply (subst (1 2) gcd_abs_int)
huffman@31706
  1351
  apply (subst (1 2) power_abs)
nipkow@31952
  1352
  apply (rule gcd_exp_nat [where n = n, transferred])
huffman@31706
  1353
  apply auto
huffman@31706
  1354
done
huffman@31706
  1355
nipkow@31952
  1356
lemma division_decomp_nat: assumes dc: "(a::nat) dvd b * c"
huffman@31706
  1357
  shows "\<exists>b' c'. a = b' * c' \<and> b' dvd b \<and> c' dvd c"
huffman@31706
  1358
proof-
huffman@31706
  1359
  let ?g = "gcd a b"
huffman@31706
  1360
  {assume "?g = 0" with dc have ?thesis by auto}
huffman@31706
  1361
  moreover
huffman@31706
  1362
  {assume z: "?g \<noteq> 0"
nipkow@31952
  1363
    from gcd_coprime_exists_nat[OF z]
huffman@31706
  1364
    obtain a' b' where ab': "a = a' * ?g" "b = b' * ?g" "coprime a' b'"
huffman@31706
  1365
      by blast
huffman@31706
  1366
    have thb: "?g dvd b" by auto
huffman@31706
  1367
    from ab'(1) have "a' dvd a"  unfolding dvd_def by blast
huffman@31706
  1368
    with dc have th0: "a' dvd b*c" using dvd_trans[of a' a "b*c"] by simp
huffman@31706
  1369
    from dc ab'(1,2) have "a'*?g dvd (b'*?g) *c" by auto
haftmann@57512
  1370
    hence "?g*a' dvd ?g * (b' * c)" by (simp add: mult.assoc)
huffman@31706
  1371
    with z have th_1: "a' dvd b' * c" by auto
nipkow@31952
  1372
    from coprime_dvd_mult_nat[OF ab'(3)] th_1
haftmann@57512
  1373
    have thc: "a' dvd c" by (subst (asm) mult.commute, blast)
huffman@31706
  1374
    from ab' have "a = ?g*a'" by algebra
huffman@31706
  1375
    with thb thc have ?thesis by blast }
huffman@31706
  1376
  ultimately show ?thesis by blast
huffman@31706
  1377
qed
huffman@31706
  1378
nipkow@31952
  1379
lemma division_decomp_int: assumes dc: "(a::int) dvd b * c"
huffman@31706
  1380
  shows "\<exists>b' c'. a = b' * c' \<and> b' dvd b \<and> c' dvd c"
huffman@31706
  1381
proof-
huffman@31706
  1382
  let ?g = "gcd a b"
huffman@31706
  1383
  {assume "?g = 0" with dc have ?thesis by auto}
huffman@31706
  1384
  moreover
huffman@31706
  1385
  {assume z: "?g \<noteq> 0"
nipkow@31952
  1386
    from gcd_coprime_exists_int[OF z]
huffman@31706
  1387
    obtain a' b' where ab': "a = a' * ?g" "b = b' * ?g" "coprime a' b'"
huffman@31706
  1388
      by blast
huffman@31706
  1389
    have thb: "?g dvd b" by auto
huffman@31706
  1390
    from ab'(1) have "a' dvd a"  unfolding dvd_def by blast
huffman@31706
  1391
    with dc have th0: "a' dvd b*c"
huffman@31706
  1392
      using dvd_trans[of a' a "b*c"] by simp
huffman@31706
  1393
    from dc ab'(1,2) have "a'*?g dvd (b'*?g) *c" by auto
haftmann@57512
  1394
    hence "?g*a' dvd ?g * (b' * c)" by (simp add: mult.assoc)
huffman@31706
  1395
    with z have th_1: "a' dvd b' * c" by auto
nipkow@31952
  1396
    from coprime_dvd_mult_int[OF ab'(3)] th_1
haftmann@57512
  1397
    have thc: "a' dvd c" by (subst (asm) mult.commute, blast)
huffman@31706
  1398
    from ab' have "a = ?g*a'" by algebra
huffman@31706
  1399
    with thb thc have ?thesis by blast }
huffman@31706
  1400
  ultimately show ?thesis by blast
chaieb@27669
  1401
qed
chaieb@27669
  1402
nipkow@31952
  1403
lemma pow_divides_pow_nat:
huffman@31706
  1404
  assumes ab: "(a::nat) ^ n dvd b ^n" and n:"n \<noteq> 0"
huffman@31706
  1405
  shows "a dvd b"
huffman@31706
  1406
proof-
huffman@31706
  1407
  let ?g = "gcd a b"
huffman@31706
  1408
  from n obtain m where m: "n = Suc m" by (cases n, simp_all)
huffman@31706
  1409
  {assume "?g = 0" with ab n have ?thesis by auto }
huffman@31706
  1410
  moreover
huffman@31706
  1411
  {assume z: "?g \<noteq> 0"
huffman@35216
  1412
    hence zn: "?g ^ n \<noteq> 0" using n by simp
nipkow@31952
  1413
    from gcd_coprime_exists_nat[OF z]
huffman@31706
  1414
    obtain a' b' where ab': "a = a' * ?g" "b = b' * ?g" "coprime a' b'"
huffman@31706
  1415
      by blast
huffman@31706
  1416
    from ab have "(a' * ?g) ^ n dvd (b' * ?g)^n"
huffman@31706
  1417
      by (simp add: ab'(1,2)[symmetric])
huffman@31706
  1418
    hence "?g^n*a'^n dvd ?g^n *b'^n"
haftmann@57512
  1419
      by (simp only: power_mult_distrib mult.commute)
haftmann@58787
  1420
    then have th0: "a'^n dvd b'^n"
haftmann@58787
  1421
      using zn by auto
huffman@31706
  1422
    have "a' dvd a'^n" by (simp add: m)
huffman@31706
  1423
    with th0 have "a' dvd b'^n" using dvd_trans[of a' "a'^n" "b'^n"] by simp
haftmann@57512
  1424
    hence th1: "a' dvd b'^m * b'" by (simp add: m mult.commute)
nipkow@31952
  1425
    from coprime_dvd_mult_nat[OF coprime_exp_nat [OF ab'(3), of m]] th1
haftmann@57512
  1426
    have "a' dvd b'" by (subst (asm) mult.commute, blast)
huffman@31706
  1427
    hence "a'*?g dvd b'*?g" by simp
huffman@31706
  1428
    with ab'(1,2)  have ?thesis by simp }
huffman@31706
  1429
  ultimately show ?thesis by blast
huffman@31706
  1430
qed
huffman@31706
  1431
nipkow@31952
  1432
lemma pow_divides_pow_int:
huffman@31706
  1433
  assumes ab: "(a::int) ^ n dvd b ^n" and n:"n \<noteq> 0"
huffman@31706
  1434
  shows "a dvd b"
chaieb@27669
  1435
proof-
huffman@31706
  1436
  let ?g = "gcd a b"
huffman@31706
  1437
  from n obtain m where m: "n = Suc m" by (cases n, simp_all)
huffman@31706
  1438
  {assume "?g = 0" with ab n have ?thesis by auto }
huffman@31706
  1439
  moreover
huffman@31706
  1440
  {assume z: "?g \<noteq> 0"
huffman@35216
  1441
    hence zn: "?g ^ n \<noteq> 0" using n by simp
nipkow@31952
  1442
    from gcd_coprime_exists_int[OF z]
huffman@31706
  1443
    obtain a' b' where ab': "a = a' * ?g" "b = b' * ?g" "coprime a' b'"
huffman@31706
  1444
      by blast
huffman@31706
  1445
    from ab have "(a' * ?g) ^ n dvd (b' * ?g)^n"
huffman@31706
  1446
      by (simp add: ab'(1,2)[symmetric])
huffman@31706
  1447
    hence "?g^n*a'^n dvd ?g^n *b'^n"
haftmann@57512
  1448
      by (simp only: power_mult_distrib mult.commute)
huffman@31706
  1449
    with zn z n have th0:"a'^n dvd b'^n" by auto
huffman@31706
  1450
    have "a' dvd a'^n" by (simp add: m)
huffman@31706
  1451
    with th0 have "a' dvd b'^n"
huffman@31706
  1452
      using dvd_trans[of a' "a'^n" "b'^n"] by simp
haftmann@60596
  1453
    hence th1: "a' dvd b'^m * b'" by (simp add: m mult.commute)
nipkow@31952
  1454
    from coprime_dvd_mult_int[OF coprime_exp_int [OF ab'(3), of m]] th1
haftmann@57512
  1455
    have "a' dvd b'" by (subst (asm) mult.commute, blast)
huffman@31706
  1456
    hence "a'*?g dvd b'*?g" by simp
huffman@31706
  1457
    with ab'(1,2)  have ?thesis by simp }
huffman@31706
  1458
  ultimately show ?thesis by blast
huffman@31706
  1459
qed
huffman@31706
  1460
nipkow@31952
  1461
lemma pow_divides_eq_nat [simp]: "n ~= 0 \<Longrightarrow> ((a::nat)^n dvd b^n) = (a dvd b)"
nipkow@31952
  1462
  by (auto intro: pow_divides_pow_nat dvd_power_same)
huffman@31706
  1463
nipkow@31952
  1464
lemma pow_divides_eq_int [simp]: "n ~= 0 \<Longrightarrow> ((a::int)^n dvd b^n) = (a dvd b)"
nipkow@31952
  1465
  by (auto intro: pow_divides_pow_int dvd_power_same)
huffman@31706
  1466
nipkow@31952
  1467
lemma divides_mult_nat:
huffman@31706
  1468
  assumes mr: "(m::nat) dvd r" and nr: "n dvd r" and mn:"coprime m n"
huffman@31706
  1469
  shows "m * n dvd r"
huffman@31706
  1470
proof-
huffman@31706
  1471
  from mr nr obtain m' n' where m': "r = m*m'" and n': "r = n*n'"
huffman@31706
  1472
    unfolding dvd_def by blast
haftmann@57512
  1473
  from mr n' have "m dvd n'*n" by (simp add: mult.commute)
nipkow@31952
  1474
  hence "m dvd n'" using coprime_dvd_mult_iff_nat[OF mn] by simp
huffman@31706
  1475
  then obtain k where k: "n' = m*k" unfolding dvd_def by blast
huffman@31706
  1476
  from n' k show ?thesis unfolding dvd_def by auto
huffman@31706
  1477
qed
huffman@31706
  1478
nipkow@31952
  1479
lemma divides_mult_int:
huffman@31706
  1480
  assumes mr: "(m::int) dvd r" and nr: "n dvd r" and mn:"coprime m n"
huffman@31706
  1481
  shows "m * n dvd r"
huffman@31706
  1482
proof-
huffman@31706
  1483
  from mr nr obtain m' n' where m': "r = m*m'" and n': "r = n*n'"
huffman@31706
  1484
    unfolding dvd_def by blast
haftmann@57512
  1485
  from mr n' have "m dvd n'*n" by (simp add: mult.commute)
nipkow@31952
  1486
  hence "m dvd n'" using coprime_dvd_mult_iff_int[OF mn] by simp
huffman@31706
  1487
  then obtain k where k: "n' = m*k" unfolding dvd_def by blast
huffman@31706
  1488
  from n' k show ?thesis unfolding dvd_def by auto
chaieb@27669
  1489
qed
chaieb@27669
  1490
nipkow@31952
  1491
lemma coprime_plus_one_nat [simp]: "coprime ((n::nat) + 1) n"
haftmann@60686
  1492
  by (simp add: gcd.commute)
huffman@31706
  1493
nipkow@31952
  1494
lemma coprime_Suc_nat [simp]: "coprime (Suc n) n"
nipkow@31952
  1495
  using coprime_plus_one_nat by (simp add: One_nat_def)
huffman@31706
  1496
nipkow@31952
  1497
lemma coprime_plus_one_int [simp]: "coprime ((n::int) + 1) n"
haftmann@60686
  1498
  by (simp add: gcd.commute)
huffman@31706
  1499
nipkow@31952
  1500
lemma coprime_minus_one_nat: "(n::nat) \<noteq> 0 \<Longrightarrow> coprime (n - 1) n"
nipkow@31952
  1501
  using coprime_plus_one_nat [of "n - 1"]
nipkow@31952
  1502
    gcd_commute_nat [of "n - 1" n] by auto
huffman@31706
  1503
nipkow@31952
  1504
lemma coprime_minus_one_int: "coprime ((n::int) - 1) n"
nipkow@31952
  1505
  using coprime_plus_one_int [of "n - 1"]
nipkow@31952
  1506
    gcd_commute_int [of "n - 1" n] by auto
huffman@31706
  1507
nipkow@31952
  1508
lemma setprod_coprime_nat [rule_format]:
huffman@31706
  1509
    "(ALL i: A. coprime (f i) (x::nat)) --> coprime (PROD i:A. f i) x"
huffman@31706
  1510
  apply (case_tac "finite A")
huffman@31706
  1511
  apply (induct set: finite)
nipkow@31952
  1512
  apply (auto simp add: gcd_mult_cancel_nat)
huffman@31706
  1513
done
huffman@31706
  1514
nipkow@31952
  1515
lemma setprod_coprime_int [rule_format]:
huffman@31706
  1516
    "(ALL i: A. coprime (f i) (x::int)) --> coprime (PROD i:A. f i) x"
huffman@31706
  1517
  apply (case_tac "finite A")
huffman@31706
  1518
  apply (induct set: finite)
nipkow@31952
  1519
  apply (auto simp add: gcd_mult_cancel_int)
huffman@31706
  1520
done
huffman@31706
  1521
lp15@60162
  1522
lemma coprime_common_divisor_nat: 
haftmann@60686
  1523
  "coprime (a::nat) b \<Longrightarrow> x dvd a \<Longrightarrow> x dvd b \<Longrightarrow> x = 1"
lp15@60162
  1524
  by (metis gcd_greatest_iff_nat nat_dvd_1_iff_1)
huffman@31706
  1525
lp15@60162
  1526
lemma coprime_common_divisor_int:
haftmann@60686
  1527
  "coprime (a::int) b \<Longrightarrow> x dvd a \<Longrightarrow> x dvd b \<Longrightarrow> abs x = 1"
haftmann@60686
  1528
  using gcd_greatest_iff [of x a b] by auto
huffman@31706
  1529
lp15@60162
  1530
lemma coprime_divisors_nat:
lp15@60162
  1531
    "(d::int) dvd a \<Longrightarrow> e dvd b \<Longrightarrow> coprime a b \<Longrightarrow> coprime d e"
lp15@60162
  1532
  by (meson coprime_int dvd_trans gcd_dvd1 gcd_dvd2 gcd_ge_0_int)
huffman@31706
  1533
nipkow@31952
  1534
lemma invertible_coprime_nat: "(x::nat) * y mod m = 1 \<Longrightarrow> coprime x m"
lp15@60162
  1535
by (metis coprime_lmult_nat gcd_1_nat gcd_commute_nat gcd_red_nat)
huffman@31706
  1536
nipkow@31952
  1537
lemma invertible_coprime_int: "(x::int) * y mod m = 1 \<Longrightarrow> coprime x m"
lp15@60162
  1538
by (metis coprime_lmult_int gcd_1_int gcd_commute_int gcd_red_int)
huffman@31706
  1539
huffman@31706
  1540
huffman@31706
  1541
subsection {* Bezout's theorem *}
huffman@31706
  1542
huffman@31706
  1543
(* Function bezw returns a pair of witnesses to Bezout's theorem --
huffman@31706
  1544
   see the theorems that follow the definition. *)
huffman@31706
  1545
fun
huffman@31706
  1546
  bezw  :: "nat \<Rightarrow> nat \<Rightarrow> int * int"
huffman@31706
  1547
where
huffman@31706
  1548
  "bezw x y =
huffman@31706
  1549
  (if y = 0 then (1, 0) else
huffman@31706
  1550
      (snd (bezw y (x mod y)),
huffman@31706
  1551
       fst (bezw y (x mod y)) - snd (bezw y (x mod y)) * int(x div y)))"
huffman@31706
  1552
huffman@31706
  1553
lemma bezw_0 [simp]: "bezw x 0 = (1, 0)" by simp
huffman@31706
  1554
huffman@31706
  1555
lemma bezw_non_0: "y > 0 \<Longrightarrow> bezw x y = (snd (bezw y (x mod y)),
huffman@31706
  1556
       fst (bezw y (x mod y)) - snd (bezw y (x mod y)) * int(x div y))"
huffman@31706
  1557
  by simp
huffman@31706
  1558
huffman@31706
  1559
declare bezw.simps [simp del]
huffman@31706
  1560
huffman@31706
  1561
lemma bezw_aux [rule_format]:
huffman@31706
  1562
    "fst (bezw x y) * int x + snd (bezw x y) * int y = int (gcd x y)"
nipkow@31952
  1563
proof (induct x y rule: gcd_nat_induct)
huffman@31706
  1564
  fix m :: nat
huffman@31706
  1565
  show "fst (bezw m 0) * int m + snd (bezw m 0) * int 0 = int (gcd m 0)"
huffman@31706
  1566
    by auto
huffman@31706
  1567
  next fix m :: nat and n
huffman@31706
  1568
    assume ngt0: "n > 0" and
huffman@31706
  1569
      ih: "fst (bezw n (m mod n)) * int n +
huffman@31706
  1570
        snd (bezw n (m mod n)) * int (m mod n) =
huffman@31706
  1571
        int (gcd n (m mod n))"
huffman@31706
  1572
    thus "fst (bezw m n) * int m + snd (bezw m n) * int n = int (gcd m n)"
nipkow@31952
  1573
      apply (simp add: bezw_non_0 gcd_non_0_nat)
huffman@31706
  1574
      apply (erule subst)
haftmann@36350
  1575
      apply (simp add: field_simps)
huffman@31706
  1576
      apply (subst mod_div_equality [of m n, symmetric])
huffman@31706
  1577
      (* applying simp here undoes the last substitution!
huffman@31706
  1578
         what is procedure cancel_div_mod? *)
hoelzl@58776
  1579
      apply (simp only: NO_MATCH_def field_simps of_nat_add of_nat_mult)
huffman@31706
  1580
      done
huffman@31706
  1581
qed
huffman@31706
  1582
nipkow@31952
  1583
lemma bezout_int:
huffman@31706
  1584
  fixes x y
huffman@31706
  1585
  shows "EX u v. u * (x::int) + v * y = gcd x y"
huffman@31706
  1586
proof -
huffman@31706
  1587
  have bezout_aux: "!!x y. x \<ge> (0::int) \<Longrightarrow> y \<ge> 0 \<Longrightarrow>
huffman@31706
  1588
      EX u v. u * x + v * y = gcd x y"
huffman@31706
  1589
    apply (rule_tac x = "fst (bezw (nat x) (nat y))" in exI)
huffman@31706
  1590
    apply (rule_tac x = "snd (bezw (nat x) (nat y))" in exI)
huffman@31706
  1591
    apply (unfold gcd_int_def)
huffman@31706
  1592
    apply simp
huffman@31706
  1593
    apply (subst bezw_aux [symmetric])
huffman@31706
  1594
    apply auto
huffman@31706
  1595
    done
huffman@31706
  1596
  have "(x \<ge> 0 \<and> y \<ge> 0) | (x \<ge> 0 \<and> y \<le> 0) | (x \<le> 0 \<and> y \<ge> 0) |
huffman@31706
  1597
      (x \<le> 0 \<and> y \<le> 0)"
huffman@31706
  1598
    by auto
huffman@31706
  1599
  moreover have "x \<ge> 0 \<Longrightarrow> y \<ge> 0 \<Longrightarrow> ?thesis"
huffman@31706
  1600
    by (erule (1) bezout_aux)
huffman@31706
  1601
  moreover have "x >= 0 \<Longrightarrow> y <= 0 \<Longrightarrow> ?thesis"
huffman@31706
  1602
    apply (insert bezout_aux [of x "-y"])
huffman@31706
  1603
    apply auto
huffman@31706
  1604
    apply (rule_tac x = u in exI)
huffman@31706
  1605
    apply (rule_tac x = "-v" in exI)
nipkow@31952
  1606
    apply (subst gcd_neg2_int [symmetric])
huffman@31706
  1607
    apply auto
huffman@31706
  1608
    done
huffman@31706
  1609
  moreover have "x <= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> ?thesis"
huffman@31706
  1610
    apply (insert bezout_aux [of "-x" y])
huffman@31706
  1611
    apply auto
huffman@31706
  1612
    apply (rule_tac x = "-u" in exI)
huffman@31706
  1613
    apply (rule_tac x = v in exI)
nipkow@31952
  1614
    apply (subst gcd_neg1_int [symmetric])
huffman@31706
  1615
    apply auto
huffman@31706
  1616
    done
huffman@31706
  1617
  moreover have "x <= 0 \<Longrightarrow> y <= 0 \<Longrightarrow> ?thesis"
huffman@31706
  1618
    apply (insert bezout_aux [of "-x" "-y"])
huffman@31706
  1619
    apply auto
huffman@31706
  1620
    apply (rule_tac x = "-u" in exI)
huffman@31706
  1621
    apply (rule_tac x = "-v" in exI)
nipkow@31952
  1622
    apply (subst gcd_neg1_int [symmetric])
nipkow@31952
  1623
    apply (subst gcd_neg2_int [symmetric])
huffman@31706
  1624
    apply auto
huffman@31706
  1625
    done
huffman@31706
  1626
  ultimately show ?thesis by blast
huffman@31706
  1627
qed
huffman@31706
  1628
huffman@31706
  1629
text {* versions of Bezout for nat, by Amine Chaieb *}
huffman@31706
  1630
huffman@31706
  1631
lemma ind_euclid:
huffman@31706
  1632
  assumes c: " \<forall>a b. P (a::nat) b \<longleftrightarrow> P b a" and z: "\<forall>a. P a 0"
huffman@31706
  1633
  and add: "\<forall>a b. P a b \<longrightarrow> P a (a + b)"
chaieb@27669
  1634
  shows "P a b"
berghofe@34915
  1635
proof(induct "a + b" arbitrary: a b rule: less_induct)
berghofe@34915
  1636
  case less
chaieb@27669
  1637
  have "a = b \<or> a < b \<or> b < a" by arith
chaieb@27669
  1638
  moreover {assume eq: "a= b"
huffman@31706
  1639
    from add[rule_format, OF z[rule_format, of a]] have "P a b" using eq
huffman@31706
  1640
    by simp}
chaieb@27669
  1641
  moreover
chaieb@27669
  1642
  {assume lt: "a < b"
berghofe@34915
  1643
    hence "a + b - a < a + b \<or> a = 0" by arith
chaieb@27669
  1644
    moreover
chaieb@27669
  1645
    {assume "a =0" with z c have "P a b" by blast }
chaieb@27669
  1646
    moreover
berghofe@34915
  1647
    {assume "a + b - a < a + b"
berghofe@34915
  1648
      also have th0: "a + b - a = a + (b - a)" using lt by arith
berghofe@34915
  1649
      finally have "a + (b - a) < a + b" .
berghofe@34915
  1650
      then have "P a (a + (b - a))" by (rule add[rule_format, OF less])
berghofe@34915
  1651
      then have "P a b" by (simp add: th0[symmetric])}
chaieb@27669
  1652
    ultimately have "P a b" by blast}
chaieb@27669
  1653
  moreover
chaieb@27669
  1654
  {assume lt: "a > b"
berghofe@34915
  1655
    hence "b + a - b < a + b \<or> b = 0" by arith
chaieb@27669
  1656
    moreover
chaieb@27669
  1657
    {assume "b =0" with z c have "P a b" by blast }
chaieb@27669
  1658
    moreover
berghofe@34915
  1659
    {assume "b + a - b < a + b"
berghofe@34915
  1660
      also have th0: "b + a - b = b + (a - b)" using lt by arith
berghofe@34915
  1661
      finally have "b + (a - b) < a + b" .
berghofe@34915
  1662
      then have "P b (b + (a - b))" by (rule add[rule_format, OF less])
berghofe@34915
  1663
      then have "P b a" by (simp add: th0[symmetric])
chaieb@27669
  1664
      hence "P a b" using c by blast }
chaieb@27669
  1665
    ultimately have "P a b" by blast}
chaieb@27669
  1666
ultimately  show "P a b" by blast
chaieb@27669
  1667
qed
chaieb@27669
  1668
nipkow@31952
  1669
lemma bezout_lemma_nat:
huffman@31706
  1670
  assumes ex: "\<exists>(d::nat) x y. d dvd a \<and> d dvd b \<and>
huffman@31706
  1671
    (a * x = b * y + d \<or> b * x = a * y + d)"
huffman@31706
  1672
  shows "\<exists>d x y. d dvd a \<and> d dvd a + b \<and>
huffman@31706
  1673
    (a * x = (a + b) * y + d \<or> (a + b) * x = a * y + d)"
huffman@31706
  1674
  using ex
huffman@31706
  1675
  apply clarsimp
huffman@35216
  1676
  apply (rule_tac x="d" in exI, simp)
huffman@31706
  1677
  apply (case_tac "a * x = b * y + d" , simp_all)
huffman@31706
  1678
  apply (rule_tac x="x + y" in exI)
huffman@31706
  1679
  apply (rule_tac x="y" in exI)
huffman@31706
  1680
  apply algebra
huffman@31706
  1681
  apply (rule_tac x="x" in exI)
huffman@31706
  1682
  apply (rule_tac x="x + y" in exI)
huffman@31706
  1683
  apply algebra
chaieb@27669
  1684
done
chaieb@27669
  1685
nipkow@31952
  1686
lemma bezout_add_nat: "\<exists>(d::nat) x y. d dvd a \<and> d dvd b \<and>
huffman@31706
  1687
    (a * x = b * y + d \<or> b * x = a * y + d)"
huffman@31706
  1688
  apply(induct a b rule: ind_euclid)
huffman@31706
  1689
  apply blast
huffman@31706
  1690
  apply clarify
huffman@35216
  1691
  apply (rule_tac x="a" in exI, simp)
huffman@31706
  1692
  apply clarsimp
huffman@31706
  1693
  apply (rule_tac x="d" in exI)
huffman@35216
  1694
  apply (case_tac "a * x = b * y + d", simp_all)
huffman@31706
  1695
  apply (rule_tac x="x+y" in exI)
huffman@31706
  1696
  apply (rule_tac x="y" in exI)
huffman@31706
  1697
  apply algebra
huffman@31706
  1698
  apply (rule_tac x="x" in exI)
huffman@31706
  1699
  apply (rule_tac x="x+y" in exI)
huffman@31706
  1700
  apply algebra
chaieb@27669
  1701
done
chaieb@27669
  1702
nipkow@31952
  1703
lemma bezout1_nat: "\<exists>(d::nat) x y. d dvd a \<and> d dvd b \<and>
huffman@31706
  1704
    (a * x - b * y = d \<or> b * x - a * y = d)"
nipkow@31952
  1705
  using bezout_add_nat[of a b]
huffman@31706
  1706
  apply clarsimp
huffman@31706
  1707
  apply (rule_tac x="d" in exI, simp)
huffman@31706
  1708
  apply (rule_tac x="x" in exI)
huffman@31706
  1709
  apply (rule_tac x="y" in exI)
huffman@31706
  1710
  apply auto
chaieb@27669
  1711
done
chaieb@27669
  1712
nipkow@31952
  1713
lemma bezout_add_strong_nat: assumes nz: "a \<noteq> (0::nat)"
chaieb@27669
  1714
  shows "\<exists>d x y. d dvd a \<and> d dvd b \<and> a * x = b * y + d"
chaieb@27669
  1715
proof-
huffman@31706
  1716
 from nz have ap: "a > 0" by simp
nipkow@31952
  1717
 from bezout_add_nat[of a b]
huffman@31706
  1718
 have "(\<exists>d x y. d dvd a \<and> d dvd b \<and> a * x = b * y + d) \<or>
huffman@31706
  1719
   (\<exists>d x y. d dvd a \<and> d dvd b \<and> b * x = a * y + d)" by blast
chaieb@27669
  1720
 moreover
huffman@31706
  1721
    {fix d x y assume H: "d dvd a" "d dvd b" "a * x = b * y + d"
huffman@31706
  1722
     from H have ?thesis by blast }
chaieb@27669
  1723
 moreover
chaieb@27669
  1724
 {fix d x y assume H: "d dvd a" "d dvd b" "b * x = a * y + d"
chaieb@27669
  1725
   {assume b0: "b = 0" with H  have ?thesis by simp}
huffman@31706
  1726
   moreover
chaieb@27669
  1727
   {assume b: "b \<noteq> 0" hence bp: "b > 0" by simp
huffman@31706
  1728
     from b dvd_imp_le [OF H(2)] have "d < b \<or> d = b"
huffman@31706
  1729
       by auto
chaieb@27669
  1730
     moreover
chaieb@27669
  1731
     {assume db: "d=b"
wenzelm@41550
  1732
       with nz H have ?thesis apply simp
wenzelm@32960
  1733
         apply (rule exI[where x = b], simp)
wenzelm@32960
  1734
         apply (rule exI[where x = b])
wenzelm@32960
  1735
        by (rule exI[where x = "a - 1"], simp add: diff_mult_distrib2)}
chaieb@27669
  1736
    moreover
huffman@31706
  1737
    {assume db: "d < b"
wenzelm@41550
  1738
        {assume "x=0" hence ?thesis using nz H by simp }
wenzelm@32960
  1739
        moreover
wenzelm@32960
  1740
        {assume x0: "x \<noteq> 0" hence xp: "x > 0" by simp
wenzelm@32960
  1741
          from db have "d \<le> b - 1" by simp
wenzelm@32960
  1742
          hence "d*b \<le> b*(b - 1)" by simp
wenzelm@32960
  1743
          with xp mult_mono[of "1" "x" "d*b" "b*(b - 1)"]
wenzelm@32960
  1744
          have dble: "d*b \<le> x*b*(b - 1)" using bp by simp
wenzelm@32960
  1745
          from H (3) have "d + (b - 1) * (b*x) = d + (b - 1) * (a*y + d)"
huffman@31706
  1746
            by simp
wenzelm@32960
  1747
          hence "d + (b - 1) * a * y + (b - 1) * d = d + (b - 1) * b * x"
haftmann@57512
  1748
            by (simp only: mult.assoc distrib_left)
wenzelm@32960
  1749
          hence "a * ((b - 1) * y) + d * (b - 1 + 1) = d + x*b*(b - 1)"
huffman@31706
  1750
            by algebra
wenzelm@32960
  1751
          hence "a * ((b - 1) * y) = d + x*b*(b - 1) - d*b" using bp by simp
wenzelm@32960
  1752
          hence "a * ((b - 1) * y) = d + (x*b*(b - 1) - d*b)"
wenzelm@32960
  1753
            by (simp only: diff_add_assoc[OF dble, of d, symmetric])
wenzelm@32960
  1754
          hence "a * ((b - 1) * y) = b*(x*(b - 1) - d) + d"
haftmann@59008
  1755
            by (simp only: diff_mult_distrib2 ac_simps)
wenzelm@32960
  1756
          hence ?thesis using H(1,2)
wenzelm@32960
  1757
            apply -
wenzelm@32960
  1758
            apply (rule exI[where x=d], simp)
wenzelm@32960
  1759
            apply (rule exI[where x="(b - 1) * y"])
wenzelm@32960
  1760
            by (rule exI[where x="x*(b - 1) - d"], simp)}
wenzelm@32960
  1761
        ultimately have ?thesis by blast}
chaieb@27669
  1762
    ultimately have ?thesis by blast}
chaieb@27669
  1763
  ultimately have ?thesis by blast}
chaieb@27669
  1764
 ultimately show ?thesis by blast
chaieb@27669
  1765
qed
chaieb@27669
  1766
nipkow@31952
  1767
lemma bezout_nat: assumes a: "(a::nat) \<noteq> 0"
chaieb@27669
  1768
  shows "\<exists>x y. a * x = b * y + gcd a b"
chaieb@27669
  1769
proof-
chaieb@27669
  1770
  let ?g = "gcd a b"
nipkow@31952
  1771
  from bezout_add_strong_nat[OF a, of b]
chaieb@27669
  1772
  obtain d x y where d: "d dvd a" "d dvd b" "a * x = b * y + d" by blast
chaieb@27669
  1773
  from d(1,2) have "d dvd ?g" by simp
chaieb@27669
  1774
  then obtain k where k: "?g = d*k" unfolding dvd_def by blast
huffman@31706
  1775
  from d(3) have "a * x * k = (b * y + d) *k " by auto
chaieb@27669
  1776
  hence "a * (x * k) = b * (y*k) + ?g" by (algebra add: k)
chaieb@27669
  1777
  thus ?thesis by blast
chaieb@27669
  1778
qed
chaieb@27669
  1779
huffman@31706
  1780
haftmann@34030
  1781
subsection {* LCM properties *}
huffman@31706
  1782
haftmann@34030
  1783
lemma lcm_altdef_int [code]: "lcm (a::int) b = (abs a) * (abs b) div gcd a b"
huffman@31706
  1784
  by (simp add: lcm_int_def lcm_nat_def zdiv_int
huffman@44821
  1785
    of_nat_mult gcd_int_def)
huffman@31706
  1786
nipkow@31952
  1787
lemma prod_gcd_lcm_nat: "(m::nat) * n = gcd m n * lcm m n"
huffman@31706
  1788
  unfolding lcm_nat_def
nipkow@31952
  1789
  by (simp add: dvd_mult_div_cancel [OF gcd_dvd_prod_nat])
huffman@31706
  1790
nipkow@31952
  1791
lemma prod_gcd_lcm_int: "abs(m::int) * abs n = gcd m n * lcm m n"
huffman@31706
  1792
  unfolding lcm_int_def gcd_int_def
huffman@31706
  1793
  apply (subst int_mult [symmetric])
nipkow@31952
  1794
  apply (subst prod_gcd_lcm_nat [symmetric])
huffman@31706
  1795
  apply (subst nat_abs_mult_distrib [symmetric])
huffman@31706
  1796
  apply (simp, simp add: abs_mult)
huffman@31706
  1797
done
huffman@31706
  1798
nipkow@31952
  1799
lemma lcm_0_nat [simp]: "lcm (m::nat) 0 = 0"
huffman@31706
  1800
  unfolding lcm_nat_def by simp
huffman@31706
  1801
nipkow@31952
  1802
lemma lcm_0_int [simp]: "lcm (m::int) 0 = 0"
huffman@31706
  1803
  unfolding lcm_int_def by simp
huffman@31706
  1804
nipkow@31952
  1805
lemma lcm_0_left_nat [simp]: "lcm (0::nat) n = 0"
huffman@31706
  1806
  unfolding lcm_nat_def by simp
chaieb@27669
  1807
nipkow@31952
  1808
lemma lcm_0_left_int [simp]: "lcm (0::int) n = 0"
huffman@31706
  1809
  unfolding lcm_int_def by simp
huffman@31706
  1810
nipkow@31952
  1811
lemma lcm_pos_nat:
nipkow@31798
  1812
  "(m::nat) > 0 \<Longrightarrow> n>0 \<Longrightarrow> lcm m n > 0"
nipkow@31952
  1813
by (metis gr0I mult_is_0 prod_gcd_lcm_nat)
chaieb@27669
  1814
nipkow@31952
  1815
lemma lcm_pos_int:
nipkow@31798
  1816
  "(m::int) ~= 0 \<Longrightarrow> n ~= 0 \<Longrightarrow> lcm m n > 0"
nipkow@31952
  1817
  apply (subst lcm_abs_int)
nipkow@31952
  1818
  apply (rule lcm_pos_nat [transferred])
nipkow@31798
  1819
  apply auto
huffman@31706
  1820
done
haftmann@23687
  1821
nipkow@31952
  1822
lemma dvd_pos_nat:
haftmann@23687
  1823
  fixes n m :: nat
haftmann@23687
  1824
  assumes "n > 0" and "m dvd n"
haftmann@23687
  1825
  shows "m > 0"
haftmann@23687
  1826
using assms by (cases m) auto
haftmann@23687
  1827
nipkow@31952
  1828
lemma lcm_least_nat:
huffman@31706
  1829
  assumes "(m::nat) dvd k" and "n dvd k"
haftmann@27556
  1830
  shows "lcm m n dvd k"
haftmann@60686
  1831
  using assms by (rule lcm_least)
haftmann@23687
  1832
nipkow@31952
  1833
lemma lcm_least_int:
nipkow@31798
  1834
  "(m::int) dvd k \<Longrightarrow> n dvd k \<Longrightarrow> lcm m n dvd k"
haftmann@60686
  1835
  by (rule lcm_least)
huffman@31706
  1836
nipkow@31952
  1837
lemma lcm_dvd1_nat: "(m::nat) dvd lcm m n"
haftmann@23687
  1838
proof (cases m)
haftmann@23687
  1839
  case 0 then show ?thesis by simp
haftmann@23687
  1840
next
haftmann@23687
  1841
  case (Suc _)
haftmann@23687
  1842
  then have mpos: "m > 0" by simp
haftmann@23687
  1843
  show ?thesis
haftmann@23687
  1844
  proof (cases n)
haftmann@23687
  1845
    case 0 then show ?thesis by simp
haftmann@23687
  1846
  next
haftmann@23687
  1847
    case (Suc _)
haftmann@23687
  1848
    then have npos: "n > 0" by simp
haftmann@27556
  1849
    have "gcd m n dvd n" by simp
haftmann@27556
  1850
    then obtain k where "n = gcd m n * k" using dvd_def by auto
huffman@31706
  1851
    then have "m * n div gcd m n = m * (gcd m n * k) div gcd m n"
haftmann@57514
  1852
      by (simp add: ac_simps)
nipkow@31952
  1853
    also have "\<dots> = m * k" using mpos npos gcd_zero_nat by simp
huffman@31706
  1854
    finally show ?thesis by (simp add: lcm_nat_def)
haftmann@23687
  1855
  qed
haftmann@23687
  1856
qed
haftmann@23687
  1857
nipkow@31952
  1858
lemma lcm_dvd1_int: "(m::int) dvd lcm m n"
nipkow@31952
  1859
  apply (subst lcm_abs_int)
huffman@31706
  1860
  apply (rule dvd_trans)
huffman@31706
  1861
  prefer 2
nipkow@31952
  1862
  apply (rule lcm_dvd1_nat [transferred])
huffman@31706
  1863
  apply auto
huffman@31706
  1864
done
huffman@31706
  1865
nipkow@31952
  1866
lemma lcm_dvd2_nat: "(n::nat) dvd lcm m n"
haftmann@60686
  1867
  by (rule lcm_dvd2)
huffman@31706
  1868
nipkow@31952
  1869
lemma lcm_dvd2_int: "(n::int) dvd lcm m n"
haftmann@60686
  1870
  by (rule lcm_dvd2)
huffman@31706
  1871
nipkow@31730
  1872
lemma dvd_lcm_I1_nat[simp]: "(k::nat) dvd m \<Longrightarrow> k dvd lcm m n"
nipkow@31952
  1873
by(metis lcm_dvd1_nat dvd_trans)
nipkow@31729
  1874
nipkow@31730
  1875
lemma dvd_lcm_I2_nat[simp]: "(k::nat) dvd n \<Longrightarrow> k dvd lcm m n"
nipkow@31952
  1876
by(metis lcm_dvd2_nat dvd_trans)
nipkow@31729
  1877
nipkow@31730
  1878
lemma dvd_lcm_I1_int[simp]: "(i::int) dvd m \<Longrightarrow> i dvd lcm m n"
nipkow@31952
  1879
by(metis lcm_dvd1_int dvd_trans)
nipkow@31729
  1880
nipkow@31730
  1881
lemma dvd_lcm_I2_int[simp]: "(i::int) dvd n \<Longrightarrow> i dvd lcm m n"
nipkow@31952
  1882
by(metis lcm_dvd2_int dvd_trans)
nipkow@31729
  1883
nipkow@31952
  1884
lemma lcm_unique_nat: "(a::nat) dvd d \<and> b dvd d \<and>
huffman@31706
  1885
    (\<forall>e. a dvd e \<and> b dvd e \<longrightarrow> d dvd e) \<longleftrightarrow> d = lcm a b"
nipkow@33657
  1886
  by (auto intro: dvd_antisym lcm_least_nat lcm_dvd1_nat lcm_dvd2_nat)
chaieb@27568
  1887
nipkow@31952
  1888
lemma lcm_unique_int: "d >= 0 \<and> (a::int) dvd d \<and> b dvd d \<and>
huffman@31706
  1889
    (\<forall>e. a dvd e \<and> b dvd e \<longrightarrow> d dvd e) \<longleftrightarrow> d = lcm a b"
wenzelm@60357
  1890
  using lcm_least_int zdvd_antisym_nonneg by auto
huffman@31706
  1891
haftmann@37770
  1892
interpretation lcm_nat: abel_semigroup "lcm :: nat \<Rightarrow> nat \<Rightarrow> nat"
haftmann@54867
  1893
  + lcm_nat: semilattice_neutr "lcm :: nat \<Rightarrow> nat \<Rightarrow> nat" 1
haftmann@60686
  1894
  by standard simp_all
haftmann@60686
  1895
haftmann@60686
  1896
interpretation lcm_int: abel_semigroup "lcm :: int \<Rightarrow> int \<Rightarrow> int" ..
haftmann@34973
  1897
haftmann@60686
  1898
lemmas lcm_assoc_nat = lcm.assoc [where ?'a = nat]
haftmann@60686
  1899
lemmas lcm_commute_nat = lcm.commute [where ?'a = nat]
haftmann@60686
  1900
lemmas lcm_left_commute_nat = lcm.left_commute [where ?'a = nat]
haftmann@60686
  1901
lemmas lcm_assoc_int = lcm.assoc [where ?'a = int]
haftmann@60686
  1902
lemmas lcm_commute_int = lcm.commute [where ?'a = int]
haftmann@60686
  1903
lemmas lcm_left_commute_int = lcm.left_commute [where ?'a = int]
haftmann@34973
  1904
haftmann@34973
  1905
lemmas lcm_ac_nat = lcm_assoc_nat lcm_commute_nat lcm_left_commute_nat
haftmann@34973
  1906
lemmas lcm_ac_int = lcm_assoc_int lcm_commute_int lcm_left_commute_int
haftmann@34973
  1907
nipkow@31798
  1908
lemma lcm_proj2_if_dvd_nat [simp]: "(x::nat) dvd y \<Longrightarrow> lcm x y = y"
huffman@31706
  1909
  apply (rule sym)
nipkow@31952
  1910
  apply (subst lcm_unique_nat [symmetric])
huffman@31706
  1911
  apply auto
huffman@31706
  1912
done
huffman@31706
  1913
nipkow@31798
  1914
lemma lcm_proj2_if_dvd_int [simp]: "(x::int) dvd y \<Longrightarrow> lcm x y = abs y"
huffman@31706
  1915
  apply (rule sym)
nipkow@31952
  1916
  apply (subst lcm_unique_int [symmetric])
huffman@31706
  1917
  apply auto
huffman@31706
  1918
done
huffman@31706
  1919
nipkow@31798
  1920
lemma lcm_proj1_if_dvd_nat [simp]: "(x::nat) dvd y \<Longrightarrow> lcm y x = y"
nipkow@31952
  1921
by (subst lcm_commute_nat, erule lcm_proj2_if_dvd_nat)
huffman@31706
  1922
nipkow@31798
  1923
lemma lcm_proj1_if_dvd_int [simp]: "(x::int) dvd y \<Longrightarrow> lcm y x = abs y"
nipkow@31952
  1924
by (subst lcm_commute_int, erule lcm_proj2_if_dvd_int)
huffman@31706
  1925
nipkow@31992
  1926
lemma lcm_proj1_iff_nat[simp]: "lcm m n = (m::nat) \<longleftrightarrow> n dvd m"
nipkow@31992
  1927
by (metis lcm_proj1_if_dvd_nat lcm_unique_nat)
nipkow@31992
  1928
nipkow@31992
  1929
lemma lcm_proj2_iff_nat[simp]: "lcm m n = (n::nat) \<longleftrightarrow> m dvd n"
nipkow@31992
  1930
by (metis lcm_proj2_if_dvd_nat lcm_unique_nat)
nipkow@31992
  1931
nipkow@31992
  1932
lemma lcm_proj1_iff_int[simp]: "lcm m n = abs(m::int) \<longleftrightarrow> n dvd m"
nipkow@31992
  1933
by (metis dvd_abs_iff lcm_proj1_if_dvd_int lcm_unique_int)
nipkow@31992
  1934
nipkow@31992
  1935
lemma lcm_proj2_iff_int[simp]: "lcm m n = abs(n::int) \<longleftrightarrow> m dvd n"
nipkow@31992
  1936
by (metis dvd_abs_iff lcm_proj2_if_dvd_int lcm_unique_int)
chaieb@27568
  1937
haftmann@42871
  1938
lemma comp_fun_idem_gcd_nat: "comp_fun_idem (gcd :: nat\<Rightarrow>nat\<Rightarrow>nat)"
nipkow@31992
  1939
proof qed (auto simp add: gcd_ac_nat)
nipkow@31992
  1940
haftmann@42871
  1941
lemma comp_fun_idem_gcd_int: "comp_fun_idem (gcd :: int\<Rightarrow>int\<Rightarrow>int)"
nipkow@31992
  1942
proof qed (auto simp add: gcd_ac_int)
nipkow@31992
  1943
haftmann@42871
  1944
lemma comp_fun_idem_lcm_nat: "comp_fun_idem (lcm :: nat\<Rightarrow>nat\<Rightarrow>nat)"
nipkow@31992
  1945
proof qed (auto simp add: lcm_ac_nat)
nipkow@31992
  1946
haftmann@42871
  1947
lemma comp_fun_idem_lcm_int: "comp_fun_idem (lcm :: int\<Rightarrow>int\<Rightarrow>int)"
nipkow@31992
  1948
proof qed (auto simp add: lcm_ac_int)
nipkow@31992
  1949
haftmann@23687
  1950
nipkow@31995
  1951
(* FIXME introduce selimattice_bot/top and derive the following lemmas in there: *)
nipkow@31995
  1952
nipkow@31995
  1953
lemma lcm_0_iff_nat[simp]: "lcm (m::nat) n = 0 \<longleftrightarrow> m=0 \<or> n=0"
nipkow@31995
  1954
by (metis lcm_0_left_nat lcm_0_nat mult_is_0 prod_gcd_lcm_nat)
nipkow@31995
  1955
nipkow@31995
  1956
lemma lcm_0_iff_int[simp]: "lcm (m::int) n = 0 \<longleftrightarrow> m=0 \<or> n=0"
huffman@44766
  1957
by (metis lcm_0_int lcm_0_left_int lcm_pos_int less_le)
nipkow@31995
  1958
nipkow@31995
  1959
lemma lcm_1_iff_nat[simp]: "lcm (m::nat) n = 1 \<longleftrightarrow> m=1 \<and> n=1"
nipkow@31995
  1960
by (metis gcd_1_nat lcm_unique_nat nat_mult_1 prod_gcd_lcm_nat)
nipkow@31995
  1961
nipkow@31995
  1962
lemma lcm_1_iff_int[simp]: "lcm (m::int) n = 1 \<longleftrightarrow> (m=1 \<or> m = -1) \<and> (n=1 \<or> n = -1)"
berghofe@31996
  1963
by (auto simp add: abs_mult_self trans [OF lcm_unique_int eq_commute, symmetric] zmult_eq_1_iff)
nipkow@31995
  1964
haftmann@34030
  1965
huffman@45264
  1966
subsection {* The complete divisibility lattice *}
nipkow@32112
  1967
wenzelm@60580
  1968
interpretation gcd_semilattice_nat: semilattice_inf gcd "op dvd" "(\<lambda>m n::nat. m dvd n \<and> \<not> n dvd m)"
haftmann@60686
  1969
  by standard simp_all
nipkow@32112
  1970
wenzelm@60580
  1971
interpretation lcm_semilattice_nat: semilattice_sup lcm "op dvd" "(\<lambda>m n::nat. m dvd n \<and> \<not> n dvd m)"
haftmann@60686
  1972
  by standard simp_all
nipkow@32112
  1973
wenzelm@60580
  1974
interpretation gcd_lcm_lattice_nat: lattice gcd "op dvd" "(\<lambda>m n::nat. m dvd n & ~ n dvd m)" lcm ..
nipkow@32112
  1975
huffman@45264
  1976
text{* Lifting gcd and lcm to sets (Gcd/Lcm).
huffman@45264
  1977
Gcd is defined via Lcm to facilitate the proof that we have a complete lattice.
nipkow@32112
  1978
*}
huffman@45264
  1979
huffman@45264
  1980
instantiation nat :: Gcd
nipkow@32112
  1981
begin
nipkow@32112
  1982
huffman@45264
  1983
definition
haftmann@51489
  1984
  "Lcm (M::nat set) = (if finite M then semilattice_neutr_set.F lcm 1 M else 0)"
haftmann@51489
  1985
haftmann@54867
  1986
interpretation semilattice_neutr_set lcm "1::nat" ..
haftmann@54867
  1987
haftmann@51489
  1988
lemma Lcm_nat_infinite:
haftmann@51489
  1989
  "\<not> finite M \<Longrightarrow> Lcm M = (0::nat)"
haftmann@51489
  1990
  by (simp add: Lcm_nat_def)
haftmann@51489
  1991
haftmann@51489
  1992
lemma Lcm_nat_empty:
haftmann@51489
  1993
  "Lcm {} = (1::nat)"
haftmann@54867
  1994
  by (simp add: Lcm_nat_def)
haftmann@51489
  1995
haftmann@51489
  1996
lemma Lcm_nat_insert:
haftmann@51489
  1997
  "Lcm (insert n M) = lcm (n::nat) (Lcm M)"
haftmann@54867
  1998
  by (cases "finite M") (simp_all add: Lcm_nat_def Lcm_nat_infinite)
nipkow@32112
  1999
huffman@45264
  2000
definition
huffman@45264
  2001
  "Gcd (M::nat set) = Lcm {d. \<forall>m\<in>M. d dvd m}"
nipkow@32112
  2002
huffman@45264
  2003
instance ..
haftmann@51489
  2004
nipkow@32112
  2005
end
nipkow@32112
  2006
huffman@45264
  2007
lemma dvd_Lcm_nat [simp]:
haftmann@51489
  2008
  fixes M :: "nat set"
haftmann@51489
  2009
  assumes "m \<in> M"
haftmann@51489
  2010
  shows "m dvd Lcm M"
haftmann@51489
  2011
proof (cases "finite M")
haftmann@51489
  2012
  case False then show ?thesis by (simp add: Lcm_nat_infinite)
haftmann@51489
  2013
next
haftmann@51489
  2014
  case True then show ?thesis using assms by (induct M) (auto simp add: Lcm_nat_insert)
haftmann@51489
  2015
qed
nipkow@32112
  2016
huffman@45264
  2017
lemma Lcm_dvd_nat [simp]:
haftmann@51489
  2018
  fixes M :: "nat set"
haftmann@51489
  2019
  assumes "\<forall>m\<in>M. m dvd n"
haftmann@51489
  2020
  shows "Lcm M dvd n"
huffman@45264
  2021
proof (cases "n = 0")
huffman@45264
  2022
  assume "n \<noteq> 0"
huffman@45264
  2023
  hence "finite {d. d dvd n}" by (rule finite_divisors_nat)
huffman@45264
  2024
  moreover have "M \<subseteq> {d. d dvd n}" using assms by fast
huffman@45264
  2025
  ultimately have "finite M" by (rule rev_finite_subset)
haftmann@51489
  2026
  then show ?thesis using assms by (induct M) (simp_all add: Lcm_nat_empty Lcm_nat_insert)
huffman@45264
  2027
qed simp
nipkow@32112
  2028
huffman@45264
  2029
interpretation gcd_lcm_complete_lattice_nat:
haftmann@51547
  2030
  complete_lattice Gcd Lcm gcd Rings.dvd "\<lambda>m n. m dvd n \<and> \<not> n dvd m" lcm 1 "0::nat"
wenzelm@60580
  2031
where "Inf.INFIMUM Gcd A f = Gcd (f ` A :: nat set)"
haftmann@56218
  2032
  and "Sup.SUPREMUM Lcm A f = Lcm (f ` A)"
haftmann@51547
  2033
proof -
haftmann@51547
  2034
  show "class.complete_lattice Gcd Lcm gcd Rings.dvd (\<lambda>m n. m dvd n \<and> \<not> n dvd m) lcm 1 (0::nat)"
haftmann@60686
  2035
    by default (auto simp add: Gcd_nat_def Lcm_nat_empty Lcm_nat_infinite)
haftmann@51547
  2036
  then interpret gcd_lcm_complete_lattice_nat:
haftmann@51547
  2037
    complete_lattice Gcd Lcm gcd Rings.dvd "\<lambda>m n. m dvd n \<and> \<not> n dvd m" lcm 1 "0::nat" .
haftmann@56218
  2038
  from gcd_lcm_complete_lattice_nat.INF_def show "Inf.INFIMUM Gcd A f = Gcd (f ` A)" .
haftmann@56218
  2039
  from gcd_lcm_complete_lattice_nat.SUP_def show "Sup.SUPREMUM Lcm A f = Lcm (f ` A)" .
huffman@45264
  2040
qed
nipkow@32112
  2041
haftmann@56166
  2042
declare gcd_lcm_complete_lattice_nat.Inf_image_eq [simp del]
haftmann@56166
  2043
declare gcd_lcm_complete_lattice_nat.Sup_image_eq [simp del]
haftmann@56166
  2044
huffman@45264
  2045
lemma Lcm_empty_nat: "Lcm {} = (1::nat)"
haftmann@54867
  2046
  by (fact Lcm_nat_empty)
huffman@45264
  2047
nipkow@32112
  2048
lemma Lcm_insert_nat [simp]:
nipkow@32112
  2049
  shows "Lcm (insert (n::nat) N) = lcm n (Lcm N)"
huffman@45264
  2050
  by (fact gcd_lcm_complete_lattice_nat.Sup_insert)
nipkow@32112
  2051
nipkow@32112
  2052
lemma Lcm0_iff[simp]: "finite (M::nat set) \<Longrightarrow> M \<noteq> {} \<Longrightarrow> Lcm M = 0 \<longleftrightarrow> 0 : M"
nipkow@32112
  2053
by(induct rule:finite_ne_induct) auto
nipkow@32112
  2054
nipkow@32112
  2055
lemma Lcm_eq_0[simp]: "finite (M::nat set) \<Longrightarrow> 0 : M \<Longrightarrow> Lcm M = 0"
nipkow@32112
  2056
by (metis Lcm0_iff empty_iff)
nipkow@32112
  2057
haftmann@60596
  2058
instance nat :: semiring_Gcd
haftmann@60596
  2059
proof
haftmann@60596
  2060
  show "Gcd N dvd n" if "n \<in> N" for N and n :: nat
haftmann@60596
  2061
    using that by (fact gcd_lcm_complete_lattice_nat.Inf_lower)
haftmann@60596
  2062
next
haftmann@60686
  2063
  show "n dvd Gcd N" if "\<And>m. m \<in> N \<Longrightarrow> n dvd m" for N and n :: nat
haftmann@60596
  2064
    using that by (simp only: gcd_lcm_complete_lattice_nat.Inf_greatest)
haftmann@60596
  2065
next
haftmann@60686
  2066
  show "normalize (Gcd N) = Gcd N" for N :: "nat set"
haftmann@60596
  2067
    by simp
haftmann@60596
  2068
qed
nipkow@32112
  2069
haftmann@60686
  2070
instance nat :: semiring_Lcm
haftmann@60686
  2071
proof
haftmann@60686
  2072
  have uf: "unit_factor (Lcm N) = 1" if "0 < Lcm N" for N :: "nat set"
haftmann@60686
  2073
  proof (cases "finite N")
haftmann@60686
  2074
    case False with that show ?thesis by (simp add: Lcm_nat_infinite)
haftmann@60686
  2075
  next
haftmann@60686
  2076
    case True then show ?thesis
haftmann@60686
  2077
    using that proof (induct N)
haftmann@60686
  2078
      case empty then show ?case by simp
haftmann@60686
  2079
    next
haftmann@60686
  2080
      case (insert n N)
haftmann@60686
  2081
      have "lcm n (Lcm N) \<noteq> 0 \<longleftrightarrow> n \<noteq> 0 \<and> Lcm N \<noteq> 0"
haftmann@60686
  2082
        using lcm_eq_0_iff [of n "Lcm N"] by simp
haftmann@60686
  2083
      then have "lcm n (Lcm N) > 0 \<longleftrightarrow> n > 0 \<and> Lcm N > 0"
haftmann@60686
  2084
        unfolding neq0_conv .
haftmann@60686
  2085
      with insert show ?case
haftmann@60686
  2086
        by (simp add: Lcm_nat_insert unit_factor_lcm)
haftmann@60686
  2087
    qed
haftmann@60686
  2088
  qed
haftmann@60686
  2089
  show "Lcm N = Gcd {m. \<forall>n\<in>N. n dvd m}" for N :: "nat set"
haftmann@60686
  2090
    by (rule associated_eqI) (auto intro!: associatedI Gcd_dvd Gcd_greatest
haftmann@60686
  2091
      simp add: unit_factor_Gcd uf)
haftmann@60686
  2092
qed
haftmann@60686
  2093
huffman@45264
  2094
text{* Alternative characterizations of Gcd: *}
nipkow@32112
  2095
nipkow@32112
  2096
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
  2097
apply(rule antisym)
nipkow@32112
  2098
 apply(rule Max_ge)
nipkow@32112
  2099
  apply (metis all_not_in_conv finite_divisors_nat finite_INT)
haftmann@60596
  2100
 apply (simp add: Gcd_dvd)
nipkow@32112
  2101
apply (rule Max_le_iff[THEN iffD2])
nipkow@32112
  2102
  apply (metis all_not_in_conv finite_divisors_nat finite_INT)
nipkow@44890
  2103
 apply fastforce
nipkow@32112
  2104
apply clarsimp
haftmann@60596
  2105
apply (metis Gcd_dvd Max_in dvd_0_left dvd_Gcd dvd_imp_le linorder_antisym_conv3 not_less0)
nipkow@32112
  2106
done
nipkow@32112
  2107
nipkow@32112
  2108
lemma Gcd_remove0_nat: "finite M \<Longrightarrow> Gcd M = Gcd (M - {0::nat})"
nipkow@32112
  2109
apply(induct pred:finite)
nipkow@32112
  2110
 apply simp
nipkow@32112
  2111
apply(case_tac "x=0")
nipkow@32112
  2112
 apply simp
nipkow@32112
  2113
apply(subgoal_tac "insert x F - {0} = insert x (F - {0})")
nipkow@32112
  2114
 apply simp
nipkow@32112
  2115
apply blast
nipkow@32112
  2116
done
nipkow@32112
  2117
nipkow@32112
  2118
lemma Lcm_in_lcm_closed_set_nat:
nipkow@32112
  2119
  "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
  2120
apply(induct rule:finite_linorder_min_induct)
nipkow@32112
  2121
 apply simp
nipkow@32112
  2122
apply simp
nipkow@32112
  2123
apply(subgoal_tac "ALL m n :: nat. m:A \<longrightarrow> n:A \<longrightarrow> lcm m n : A")
nipkow@32112
  2124
 apply simp
nipkow@32112
  2125
 apply(case_tac "A={}")
nipkow@32112
  2126
  apply simp
nipkow@32112
  2127
 apply simp
nipkow@32112
  2128
apply (metis lcm_pos_nat lcm_unique_nat linorder_neq_iff nat_dvd_not_less not_less0)
nipkow@32112
  2129
done
nipkow@32112
  2130
nipkow@32112
  2131
lemma Lcm_eq_Max_nat:
nipkow@32112
  2132
  "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
  2133
apply(rule antisym)
nipkow@32112
  2134
 apply(rule Max_ge, assumption)
nipkow@32112
  2135
 apply(erule (2) Lcm_in_lcm_closed_set_nat)
nipkow@32112
  2136
apply clarsimp
nipkow@32112
  2137
apply (metis Lcm0_iff dvd_Lcm_nat dvd_imp_le neq0_conv)
nipkow@32112
  2138
done
nipkow@32112
  2139
haftmann@54437
  2140
lemma Lcm_set_nat [code, code_unfold]:
haftmann@45992
  2141
  "Lcm (set ns) = fold lcm ns (1::nat)"
huffman@45264
  2142
  by (fact gcd_lcm_complete_lattice_nat.Sup_set_fold)
nipkow@32112
  2143
haftmann@60597
  2144
lemma Gcd_set_nat [code]:
haftmann@45992
  2145
  "Gcd (set ns) = fold gcd ns (0::nat)"
huffman@45264
  2146
  by (fact gcd_lcm_complete_lattice_nat.Inf_set_fold)
nipkow@34222
  2147
nipkow@34222
  2148
lemma mult_inj_if_coprime_nat:
nipkow@34222
  2149
  "inj_on f A \<Longrightarrow> inj_on g B \<Longrightarrow> ALL a:A. ALL b:B. coprime (f a) (g b)
nipkow@34222
  2150
   \<Longrightarrow> inj_on (%(a,b). f a * g b::nat) (A \<times> B)"
nipkow@34222
  2151
apply(auto simp add:inj_on_def)
huffman@35216
  2152
apply (metis coprime_dvd_mult_iff_nat dvd.neq_le_trans dvd_triv_left)
nipkow@34223
  2153
apply (metis gcd_semilattice_nat.inf_commute coprime_dvd_mult_iff_nat
haftmann@57512
  2154
             dvd.neq_le_trans dvd_triv_right mult.commute)
nipkow@34222
  2155
done
nipkow@34222
  2156
nipkow@34222
  2157
text{* Nitpick: *}
nipkow@34222
  2158
blanchet@41792
  2159
lemma gcd_eq_nitpick_gcd [nitpick_unfold]: "gcd x y = Nitpick.nat_gcd x y"
blanchet@41792
  2160
by (induct x y rule: nat_gcd.induct)
blanchet@41792
  2161
   (simp add: gcd_nat.simps Nitpick.nat_gcd.simps)
blanchet@33197
  2162
blanchet@41792
  2163
lemma lcm_eq_nitpick_lcm [nitpick_unfold]: "lcm x y = Nitpick.nat_lcm x y"
blanchet@33197
  2164
by (simp only: lcm_nat_def Nitpick.nat_lcm_def gcd_eq_nitpick_gcd)
blanchet@33197
  2165
haftmann@54867
  2166
huffman@45264
  2167
subsubsection {* Setwise gcd and lcm for integers *}
huffman@45264
  2168
huffman@45264
  2169
instantiation int :: Gcd
huffman@45264
  2170
begin
huffman@45264
  2171
huffman@45264
  2172
definition
huffman@45264
  2173
  "Lcm M = int (Lcm (nat ` abs ` M))"
huffman@45264
  2174
huffman@45264
  2175
definition
huffman@45264
  2176
  "Gcd M = int (Gcd (nat ` abs ` M))"
huffman@45264
  2177
huffman@45264
  2178
instance ..
haftmann@60686
  2179
wenzelm@21256
  2180
end
huffman@45264
  2181
haftmann@60686
  2182
instance int :: semiring_Gcd
haftmann@60686
  2183
  by standard (auto intro!: Gcd_dvd Gcd_greatest simp add: Gcd_int_def Lcm_int_def int_dvd_iff dvd_int_iff
haftmann@60686
  2184
    dvd_int_unfold_dvd_nat [symmetric])
haftmann@60686
  2185
haftmann@60686
  2186
instance int :: semiring_Lcm
haftmann@60686
  2187
proof
haftmann@60686
  2188
  fix K :: "int set"
haftmann@60686
  2189
  have "{n. \<forall>k\<in>K. nat \<bar>k\<bar> dvd n} = ((\<lambda>k. nat \<bar>k\<bar>) ` {l. \<forall>k\<in>K. k dvd l})"
haftmann@60686
  2190
  proof (rule set_eqI)
haftmann@60686
  2191
    fix n
haftmann@60686
  2192
    have "(\<forall>k\<in>K. nat \<bar>k\<bar> dvd n) \<longleftrightarrow> (\<exists>l. (\<forall>k\<in>K. k dvd l) \<and> n = nat \<bar>l\<bar>)" (is "?P \<longleftrightarrow> ?Q")
haftmann@60686
  2193
    proof
haftmann@60686
  2194
      assume ?P
haftmann@60686
  2195
      then have "(\<forall>k\<in>K. k dvd int n) \<and> n = nat \<bar>int n\<bar>"
haftmann@60686
  2196
        by (auto simp add: dvd_int_unfold_dvd_nat)
haftmann@60686
  2197
      then show ?Q by blast
haftmann@60686
  2198
    next
haftmann@60686
  2199
      assume ?Q then show ?P
haftmann@60686
  2200
        by (auto simp add: dvd_int_unfold_dvd_nat)
haftmann@60686
  2201
    qed
haftmann@60686
  2202
    then show "n \<in> {n. \<forall>k\<in>K. nat \<bar>k\<bar> dvd n} \<longleftrightarrow> n \<in> (\<lambda>k. nat \<bar>k\<bar>) ` {l. \<forall>k\<in>K. k dvd l}"
haftmann@60686
  2203
      by auto
haftmann@60686
  2204
  qed
haftmann@60686
  2205
  then show "Lcm K = Gcd {l. \<forall>k\<in>K. k dvd l}"
haftmann@60686
  2206
    by (simp add: Gcd_int_def Lcm_int_def Lcm_Gcd)
haftmann@60686
  2207
qed
haftmann@60686
  2208
huffman@45264
  2209
lemma Lcm_empty_int [simp]: "Lcm {} = (1::int)"
huffman@45264
  2210
  by (simp add: Lcm_int_def)
huffman@45264
  2211
huffman@45264
  2212
lemma Lcm_insert_int [simp]:
huffman@45264
  2213
  shows "Lcm (insert (n::int) N) = lcm n (Lcm N)"
huffman@45264
  2214
  by (simp add: Lcm_int_def lcm_int_def)
huffman@45264
  2215
huffman@45264
  2216
lemma dvd_int_iff: "x dvd y \<longleftrightarrow> nat (abs x) dvd nat (abs y)"
haftmann@60686
  2217
  by (fact dvd_int_unfold_dvd_nat)
huffman@45264
  2218
huffman@45264
  2219
lemma dvd_Lcm_int [simp]:
huffman@45264
  2220
  fixes M :: "int set" assumes "m \<in> M" shows "m dvd Lcm M"
huffman@45264
  2221
  using assms by (simp add: Lcm_int_def dvd_int_iff)
huffman@45264
  2222
huffman@45264
  2223
lemma Lcm_dvd_int [simp]:
huffman@45264
  2224
  fixes M :: "int set"
huffman@45264
  2225
  assumes "\<forall>m\<in>M. m dvd n" shows "Lcm M dvd n"
huffman@45264
  2226
  using assms by (simp add: Lcm_int_def dvd_int_iff)
huffman@45264
  2227
haftmann@54437
  2228
lemma Lcm_set_int [code, code_unfold]:
haftmann@51547
  2229
  "Lcm (set xs) = fold lcm xs (1::int)"
haftmann@56166
  2230
  by (induct xs rule: rev_induct) (simp_all add: lcm_commute_int)
huffman@45264
  2231
haftmann@60597
  2232
lemma Gcd_set_int [code]:
haftmann@51547
  2233
  "Gcd (set xs) = fold gcd xs (0::int)"
haftmann@56166
  2234
  by (induct xs rule: rev_induct) (simp_all add: gcd_commute_int)
huffman@45264
  2235
haftmann@59008
  2236
haftmann@59008
  2237
text \<open>Fact aliasses\<close>
lp15@59667
  2238
lp15@59667
  2239
lemmas gcd_dvd1_nat = gcd_dvd1 [where ?'a = nat]
haftmann@59008
  2240
  and gcd_dvd2_nat = gcd_dvd2 [where ?'a = nat]
haftmann@59008
  2241
  and gcd_greatest_nat = gcd_greatest [where ?'a = nat]
haftmann@59008
  2242
lp15@59667
  2243
lemmas gcd_dvd1_int = gcd_dvd1 [where ?'a = int]
haftmann@59008
  2244
  and gcd_dvd2_int = gcd_dvd2 [where ?'a = int]
haftmann@59008
  2245
  and gcd_greatest_int = gcd_greatest [where ?'a = int]
haftmann@59008
  2246
haftmann@60596
  2247
lemmas Gcd_dvd_nat [simp] = Gcd_dvd [where ?'a = nat]
haftmann@60596
  2248
  and dvd_Gcd_nat [simp] = dvd_Gcd [where ?'a = nat]
haftmann@60596
  2249
haftmann@60596
  2250
lemmas Gcd_dvd_int [simp] = Gcd_dvd [where ?'a = int]
haftmann@60596
  2251
  and dvd_Gcd_int [simp] = dvd_Gcd [where ?'a = int]
haftmann@60596
  2252
haftmann@60596
  2253
lemmas Gcd_empty_nat = Gcd_empty [where ?'a = nat]
haftmann@60596
  2254
  and Gcd_insert_nat = Gcd_insert [where ?'a = nat]
haftmann@60596
  2255
haftmann@60596
  2256
lemmas Gcd_empty_int = Gcd_empty [where ?'a = int]
haftmann@60596
  2257
  and Gcd_insert_int = Gcd_insert [where ?'a = int]
haftmann@60596
  2258
huffman@45264
  2259
end