src/HOL/GCD.thy
author Andreas Lochbihler
Fri Dec 18 11:14:28 2015 +0100 (2015-12-18)
changeset 61856 4b1b85f38944
parent 61799 4cf66f21b764
child 61913 58b153bfa737
permissions -rw-r--r--
add gcd instance for integer and serialisation to target language operations
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(*  Authors:    Christophe Tabacznyj, Lawrence C. Paulson, Amine Chaieb,
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                Thomas M. Rasmussen, Jeremy Avigad, Tobias Nipkow
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This file deals with the functions gcd and lcm.  Definitions and
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lemmas are proved uniformly for the natural numbers and integers.
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This file combines and revises a number of prior developments.
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The original theories "GCD" and "Primes" were by Christophe Tabacznyj
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and Lawrence C. Paulson, based on @{cite davenport92}. They introduced
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gcd, lcm, and prime for the natural numbers.
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The original theory "IntPrimes" was by Thomas M. Rasmussen, and
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extended gcd, lcm, primes to the integers. Amine Chaieb provided
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another extension of the notions to the integers, and added a number
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of results to "Primes" and "GCD". IntPrimes also defined and developed
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the congruence relations on the integers. The notion was extended to
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the natural numbers by Chaieb.
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Jeremy Avigad combined all of these, made everything uniform for the
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natural numbers and the integers, and added a number of new theorems.
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Tobias Nipkow cleaned up a lot.
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*)
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section \<open>Greatest common divisor and least common multiple\<close>
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theory GCD
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imports Main
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begin
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subsection \<open>GCD and LCM definitions\<close>
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class gcd = zero + one + dvd +
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  fixes gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
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    and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
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begin
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abbreviation coprime :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
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  where "coprime x y \<equiv> gcd x y = 1"
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end
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class Gcd = gcd +
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  fixes Gcd :: "'a set \<Rightarrow> 'a"
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    and Lcm :: "'a set \<Rightarrow> 'a"
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class semiring_gcd = normalization_semidom + gcd +
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  assumes gcd_dvd1 [iff]: "gcd a b dvd a"
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    and gcd_dvd2 [iff]: "gcd a b dvd b"
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    and gcd_greatest: "c dvd a \<Longrightarrow> c dvd b \<Longrightarrow> c dvd gcd a b"
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    and normalize_gcd [simp]: "normalize (gcd a b) = gcd a b"
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    and lcm_gcd: "lcm a b = normalize (a * b) div gcd a b"
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begin    
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lemma gcd_greatest_iff [simp]:
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  "a dvd gcd b c \<longleftrightarrow> a dvd b \<and> a dvd c"
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  by (blast intro!: gcd_greatest intro: dvd_trans)
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lemma gcd_dvdI1:
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  "a dvd c \<Longrightarrow> gcd a b dvd c"
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  by (rule dvd_trans) (rule gcd_dvd1)
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lemma gcd_dvdI2:
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  "b dvd c \<Longrightarrow> gcd a b dvd c"
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  by (rule dvd_trans) (rule gcd_dvd2)
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lemma gcd_0_left [simp]:
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  "gcd 0 a = normalize a"
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  by (rule associated_eqI) simp_all
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lemma gcd_0_right [simp]:
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  "gcd a 0 = normalize a"
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  by (rule associated_eqI) simp_all
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lemma gcd_eq_0_iff [simp]:
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  "gcd a b = 0 \<longleftrightarrow> a = 0 \<and> b = 0" (is "?P \<longleftrightarrow> ?Q")
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proof
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  assume ?P then have "0 dvd gcd a b" by simp
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  then have "0 dvd a" and "0 dvd b" by (blast intro: dvd_trans)+
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  then show ?Q by simp
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next
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  assume ?Q then show ?P by simp
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qed
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lemma unit_factor_gcd:
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  "unit_factor (gcd a b) = (if a = 0 \<and> b = 0 then 0 else 1)"
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proof (cases "gcd a b = 0")
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  case True then show ?thesis by simp
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next
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  case False
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  have "unit_factor (gcd a b) * normalize (gcd a b) = gcd a b"
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    by (rule unit_factor_mult_normalize)
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  then have "unit_factor (gcd a b) * gcd a b = gcd a b"
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    by simp
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  then have "unit_factor (gcd a b) * gcd a b div gcd a b = gcd a b div gcd a b"
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    by simp
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  with False show ?thesis by simp
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qed
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lemma is_unit_gcd [simp]:
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  "is_unit (gcd a b) \<longleftrightarrow> coprime a b"
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  by (cases "a = 0 \<and> b = 0") (auto simp add: unit_factor_gcd dest: is_unit_unit_factor)
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sublocale gcd: abel_semigroup gcd
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proof
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  fix a b c
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  show "gcd a b = gcd b a"
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    by (rule associated_eqI) simp_all
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  from gcd_dvd1 have "gcd (gcd a b) c dvd a"
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    by (rule dvd_trans) simp
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  moreover from gcd_dvd1 have "gcd (gcd a b) c dvd b"
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    by (rule dvd_trans) simp
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  ultimately have P1: "gcd (gcd a b) c dvd gcd a (gcd b c)"
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    by (auto intro!: gcd_greatest)
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  from gcd_dvd2 have "gcd a (gcd b c) dvd b"
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    by (rule dvd_trans) simp
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  moreover from gcd_dvd2 have "gcd a (gcd b c) dvd c"
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    by (rule dvd_trans) simp
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  ultimately have P2: "gcd a (gcd b c) dvd gcd (gcd a b) c"
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    by (auto intro!: gcd_greatest)
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  from P1 P2 show "gcd (gcd a b) c = gcd a (gcd b c)"
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    by (rule associated_eqI) simp_all
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qed
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lemma gcd_self [simp]:
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  "gcd a a = normalize a"
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proof -
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  have "a dvd gcd a a"
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    by (rule gcd_greatest) simp_all
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  then show ?thesis
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    by (auto intro: associated_eqI)
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qed
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lemma coprime_1_left [simp]:
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  "coprime 1 a"
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  by (rule associated_eqI) simp_all
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lemma coprime_1_right [simp]:
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  "coprime a 1"
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  using coprime_1_left [of a] by (simp add: ac_simps)
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lemma gcd_mult_left:
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  "gcd (c * a) (c * b) = normalize c * gcd a b"
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proof (cases "c = 0")
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  case True then show ?thesis by simp
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next
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  case False
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  then have "c * gcd a b dvd gcd (c * a) (c * b)"
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    by (auto intro: gcd_greatest)
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  moreover from calculation False have "gcd (c * a) (c * b) dvd c * gcd a b"
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    by (metis div_dvd_iff_mult dvd_mult_left gcd_dvd1 gcd_dvd2 gcd_greatest mult_commute)
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  ultimately have "normalize (gcd (c * a) (c * b)) = normalize (c * gcd a b)"
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    by (auto intro: associated_eqI)
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  then show ?thesis by (simp add: normalize_mult)
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qed
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lemma gcd_mult_right:
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  "gcd (a * c) (b * c) = gcd b a * normalize c"
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  using gcd_mult_left [of c a b] by (simp add: ac_simps)
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lemma mult_gcd_left:
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  "c * gcd a b = unit_factor c * gcd (c * a) (c * b)"
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  by (simp add: gcd_mult_left mult.assoc [symmetric])
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lemma mult_gcd_right:
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  "gcd a b * c = gcd (a * c) (b * c) * unit_factor c"
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  using mult_gcd_left [of c a b] by (simp add: ac_simps)
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lemma dvd_lcm1 [iff]:
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  "a dvd lcm a b"
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proof -
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  have "normalize (a * b) div gcd a b = normalize a * (normalize b div gcd a b)"
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    by (simp add: lcm_gcd normalize_mult div_mult_swap)
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  then show ?thesis
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    by (simp add: lcm_gcd)
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qed
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lemma dvd_lcm2 [iff]:
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  "b dvd lcm a b"
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proof -
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  have "normalize (a * b) div gcd a b = normalize b * (normalize a div gcd a b)"
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    by (simp add: lcm_gcd normalize_mult div_mult_swap ac_simps)
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  then show ?thesis
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    by (simp add: lcm_gcd)
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qed
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lemma dvd_lcmI1:
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  "a dvd b \<Longrightarrow> a dvd lcm b c"
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  by (rule dvd_trans) (assumption, blast) 
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lemma dvd_lcmI2:
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  "a dvd c \<Longrightarrow> a dvd lcm b c"
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  by (rule dvd_trans) (assumption, blast)
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lemma lcm_least:
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  assumes "a dvd c" and "b dvd c"
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  shows "lcm a b dvd c"
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proof (cases "c = 0")
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  case True then show ?thesis by simp
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next
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  case False then have U: "is_unit (unit_factor c)" by simp
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  show ?thesis
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  proof (cases "gcd a b = 0")
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    case True with assms show ?thesis by simp
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  next
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    case False then have "a \<noteq> 0 \<or> b \<noteq> 0" by simp
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    with \<open>c \<noteq> 0\<close> assms have "a * b dvd a * c" "a * b dvd c * b"
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      by (simp_all add: mult_dvd_mono)
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    then have "normalize (a * b) dvd gcd (a * c) (b * c)"
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      by (auto intro: gcd_greatest simp add: ac_simps)
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    then have "normalize (a * b) dvd gcd (a * c) (b * c) * unit_factor c"
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      using U by (simp add: dvd_mult_unit_iff)
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    then have "normalize (a * b) dvd gcd a b * c"
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      by (simp add: mult_gcd_right [of a b c])
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    then have "normalize (a * b) div gcd a b dvd c"
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      using False by (simp add: div_dvd_iff_mult ac_simps)
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    then show ?thesis by (simp add: lcm_gcd)
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  qed
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qed
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lemma lcm_least_iff [simp]:
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  "lcm a b dvd c \<longleftrightarrow> a dvd c \<and> b dvd c"
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  by (blast intro!: lcm_least intro: dvd_trans)
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lemma normalize_lcm [simp]:
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  "normalize (lcm a b) = lcm a b"
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  by (simp add: lcm_gcd dvd_normalize_div)
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lemma lcm_0_left [simp]:
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  "lcm 0 a = 0"
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  by (simp add: lcm_gcd)
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lemma lcm_0_right [simp]:
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  "lcm a 0 = 0"
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  by (simp add: lcm_gcd)
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lemma lcm_eq_0_iff:
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  "lcm a b = 0 \<longleftrightarrow> a = 0 \<or> b = 0" (is "?P \<longleftrightarrow> ?Q")
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proof
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  assume ?P then have "0 dvd lcm a b" by simp
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  then have "0 dvd normalize (a * b) div gcd a b"
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    by (simp add: lcm_gcd)
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  then have "0 * gcd a b dvd normalize (a * b)"
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    using dvd_div_iff_mult [of "gcd a b" _ 0] by (cases "gcd a b = 0") simp_all
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  then have "normalize (a * b) = 0"
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    by simp
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  then show ?Q by simp
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next
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  assume ?Q then show ?P by auto
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qed
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lemma unit_factor_lcm :
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  "unit_factor (lcm a b) = (if a = 0 \<or> b = 0 then 0 else 1)"
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  by (simp add: unit_factor_gcd dvd_unit_factor_div lcm_gcd)
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sublocale lcm: abel_semigroup lcm
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proof
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  fix a b c
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  show "lcm a b = lcm b a"
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    by (simp add: lcm_gcd ac_simps normalize_mult dvd_normalize_div)
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  have "lcm (lcm a b) c dvd lcm a (lcm b c)"
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    and "lcm a (lcm b c) dvd lcm (lcm a b) c"
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    by (auto intro: lcm_least
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      dvd_trans [of b "lcm b c" "lcm a (lcm b c)"]
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      dvd_trans [of c "lcm b c" "lcm a (lcm b c)"]
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      dvd_trans [of a "lcm a b" "lcm (lcm a b) c"]
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      dvd_trans [of b "lcm a b" "lcm (lcm a b) c"])
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  then show "lcm (lcm a b) c = lcm a (lcm b c)"
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    by (rule associated_eqI) simp_all
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qed
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lemma lcm_self [simp]:
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  "lcm a a = normalize a"
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proof -
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  have "lcm a a dvd a"
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    by (rule lcm_least) simp_all
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  then show ?thesis
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    by (auto intro: associated_eqI)
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qed
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lemma gcd_mult_lcm [simp]:
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  "gcd a b * lcm a b = normalize a * normalize b"
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  by (simp add: lcm_gcd normalize_mult)
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lemma lcm_mult_gcd [simp]:
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  "lcm a b * gcd a b = normalize a * normalize b"
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  using gcd_mult_lcm [of a b] by (simp add: ac_simps) 
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lemma gcd_lcm:
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  assumes "a \<noteq> 0" and "b \<noteq> 0"
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  shows "gcd a b = normalize (a * b) div lcm a b"
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proof -
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  from assms have "lcm a b \<noteq> 0"
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    by (simp add: lcm_eq_0_iff)
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  have "gcd a b * lcm a b = normalize a * normalize b" by simp
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  then have "gcd a b * lcm a b div lcm a b = normalize (a * b) div lcm a b"
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    by (simp_all add: normalize_mult)
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  with \<open>lcm a b \<noteq> 0\<close> show ?thesis
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    using nonzero_mult_divide_cancel_right [of "lcm a b" "gcd a b"] by simp
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qed
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lemma lcm_1_left [simp]:
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  "lcm 1 a = normalize a"
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  by (simp add: lcm_gcd)
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lemma lcm_1_right [simp]:
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  "lcm a 1 = normalize a"
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  by (simp add: lcm_gcd)
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lemma lcm_mult_left:
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   314
  "lcm (c * a) (c * b) = normalize c * lcm a b"
haftmann@60686
   315
  by (cases "c = 0")
haftmann@60686
   316
    (simp_all add: gcd_mult_right lcm_gcd div_mult_swap normalize_mult ac_simps,
haftmann@60686
   317
      simp add: dvd_div_mult2_eq mult.left_commute [of "normalize c", symmetric])
haftmann@60686
   318
haftmann@60686
   319
lemma lcm_mult_right:
haftmann@60686
   320
  "lcm (a * c) (b * c) = lcm b a * normalize c"
haftmann@60686
   321
  using lcm_mult_left [of c a b] by (simp add: ac_simps)
haftmann@60686
   322
haftmann@60686
   323
lemma mult_lcm_left:
haftmann@60686
   324
  "c * lcm a b = unit_factor c * lcm (c * a) (c * b)"
haftmann@60686
   325
  by (simp add: lcm_mult_left mult.assoc [symmetric])
haftmann@60686
   326
haftmann@60686
   327
lemma mult_lcm_right:
haftmann@60686
   328
  "lcm a b * c = lcm (a * c) (b * c) * unit_factor c"
haftmann@60686
   329
  using mult_lcm_left [of c a b] by (simp add: ac_simps)
haftmann@60686
   330
  
haftmann@60686
   331
end
haftmann@60686
   332
haftmann@60686
   333
class semiring_Gcd = semiring_gcd + Gcd +
haftmann@60686
   334
  assumes Gcd_dvd: "a \<in> A \<Longrightarrow> Gcd A dvd a"
haftmann@60686
   335
    and Gcd_greatest: "(\<And>b. b \<in> A \<Longrightarrow> a dvd b) \<Longrightarrow> a dvd Gcd A"
haftmann@60686
   336
    and normalize_Gcd [simp]: "normalize (Gcd A) = Gcd A"
haftmann@60686
   337
begin
haftmann@60686
   338
haftmann@60686
   339
lemma Gcd_empty [simp]:
haftmann@60686
   340
  "Gcd {} = 0"
haftmann@60686
   341
  by (rule dvd_0_left, rule Gcd_greatest) simp
haftmann@60686
   342
haftmann@60686
   343
lemma Gcd_0_iff [simp]:
haftmann@60687
   344
  "Gcd A = 0 \<longleftrightarrow> A \<subseteq> {0}" (is "?P \<longleftrightarrow> ?Q")
haftmann@60686
   345
proof
haftmann@60686
   346
  assume ?P
haftmann@60686
   347
  show ?Q
haftmann@60686
   348
  proof
haftmann@60686
   349
    fix a
haftmann@60686
   350
    assume "a \<in> A"
haftmann@60686
   351
    then have "Gcd A dvd a" by (rule Gcd_dvd)
haftmann@60687
   352
    with \<open>?P\<close> have "a = 0" by simp
haftmann@60687
   353
    then show "a \<in> {0}" by simp
haftmann@60686
   354
  qed
haftmann@60686
   355
next
haftmann@60686
   356
  assume ?Q
haftmann@60686
   357
  have "0 dvd Gcd A"
haftmann@60686
   358
  proof (rule Gcd_greatest)
haftmann@60686
   359
    fix a
haftmann@60686
   360
    assume "a \<in> A"
haftmann@60687
   361
    with \<open>?Q\<close> have "a = 0" by auto
haftmann@60686
   362
    then show "0 dvd a" by simp
haftmann@60686
   363
  qed
haftmann@60686
   364
  then show ?P by simp
haftmann@60686
   365
qed
haftmann@60686
   366
haftmann@60686
   367
lemma unit_factor_Gcd:
haftmann@60686
   368
  "unit_factor (Gcd A) = (if \<forall>a\<in>A. a = 0 then 0 else 1)"
haftmann@60686
   369
proof (cases "Gcd A = 0")
haftmann@60687
   370
  case True then show ?thesis by auto
haftmann@60686
   371
next
haftmann@60686
   372
  case False
haftmann@60686
   373
  from unit_factor_mult_normalize
haftmann@60686
   374
  have "unit_factor (Gcd A) * normalize (Gcd A) = Gcd A" .
haftmann@60686
   375
  then have "unit_factor (Gcd A) * Gcd A = Gcd A" by simp
haftmann@60686
   376
  then have "unit_factor (Gcd A) * Gcd A div Gcd A = Gcd A div Gcd A" by simp
haftmann@60686
   377
  with False have "unit_factor (Gcd A) = 1" by simp
haftmann@60687
   378
  with False show ?thesis by auto
haftmann@60686
   379
qed
haftmann@60686
   380
haftmann@60686
   381
lemma Gcd_UNIV [simp]:
haftmann@60686
   382
  "Gcd UNIV = 1"
haftmann@60686
   383
  by (rule associated_eqI) (auto intro: Gcd_dvd simp add: unit_factor_Gcd)
haftmann@60686
   384
haftmann@60686
   385
lemma Gcd_eq_1_I:
haftmann@60686
   386
  assumes "is_unit a" and "a \<in> A"
haftmann@60686
   387
  shows "Gcd A = 1"
haftmann@60686
   388
proof -
haftmann@60686
   389
  from assms have "is_unit (Gcd A)"
haftmann@60686
   390
    by (blast intro: Gcd_dvd dvd_unit_imp_unit)
haftmann@60686
   391
  then have "normalize (Gcd A) = 1"
haftmann@60686
   392
    by (rule is_unit_normalize)
haftmann@60686
   393
  then show ?thesis
haftmann@60686
   394
    by simp
haftmann@60686
   395
qed
haftmann@60686
   396
haftmann@60686
   397
lemma Gcd_insert [simp]:
haftmann@60686
   398
  "Gcd (insert a A) = gcd a (Gcd A)"
haftmann@60686
   399
proof -
haftmann@60686
   400
  have "Gcd (insert a A) dvd gcd a (Gcd A)"
haftmann@60686
   401
    by (auto intro: Gcd_dvd Gcd_greatest)
haftmann@60686
   402
  moreover have "gcd a (Gcd A) dvd Gcd (insert a A)"
haftmann@60686
   403
  proof (rule Gcd_greatest)
haftmann@60686
   404
    fix b
haftmann@60686
   405
    assume "b \<in> insert a A"
haftmann@60686
   406
    then show "gcd a (Gcd A) dvd b"
haftmann@60686
   407
    proof
haftmann@60686
   408
      assume "b = a" then show ?thesis by simp
haftmann@60686
   409
    next
haftmann@60686
   410
      assume "b \<in> A"
haftmann@60686
   411
      then have "Gcd A dvd b" by (rule Gcd_dvd)
haftmann@60686
   412
      moreover have "gcd a (Gcd A) dvd Gcd A" by simp
haftmann@60686
   413
      ultimately show ?thesis by (blast intro: dvd_trans)
haftmann@60686
   414
    qed
haftmann@60686
   415
  qed
haftmann@60688
   416
  ultimately show ?thesis
haftmann@60688
   417
    by (auto intro: associated_eqI)
haftmann@60686
   418
qed
haftmann@60686
   419
wenzelm@61799
   420
lemma dvd_Gcd: \<comment> \<open>FIXME remove\<close>
haftmann@60686
   421
  "\<forall>b\<in>A. a dvd b \<Longrightarrow> a dvd Gcd A"
haftmann@60686
   422
  by (blast intro: Gcd_greatest)
haftmann@60686
   423
haftmann@60686
   424
lemma Gcd_set [code_unfold]:
haftmann@60686
   425
  "Gcd (set as) = foldr gcd as 0"
haftmann@60686
   426
  by (induct as) simp_all
haftmann@60686
   427
haftmann@60686
   428
end  
haftmann@60686
   429
haftmann@60686
   430
class semiring_Lcm = semiring_Gcd +
haftmann@60686
   431
  assumes Lcm_Gcd: "Lcm A = Gcd {b. \<forall>a\<in>A. a dvd b}"
haftmann@60686
   432
begin
haftmann@60686
   433
haftmann@60686
   434
lemma dvd_Lcm:
haftmann@60686
   435
  assumes "a \<in> A"
haftmann@60686
   436
  shows "a dvd Lcm A"
haftmann@60686
   437
  using assms by (auto intro: Gcd_greatest simp add: Lcm_Gcd)
haftmann@60686
   438
haftmann@60686
   439
lemma Gcd_image_normalize [simp]:
haftmann@60686
   440
  "Gcd (normalize ` A) = Gcd A"
haftmann@60686
   441
proof -
haftmann@60686
   442
  have "Gcd (normalize ` A) dvd a" if "a \<in> A" for a
haftmann@60686
   443
  proof -
haftmann@60686
   444
    from that obtain B where "A = insert a B" by blast
haftmann@60686
   445
    moreover have " gcd (normalize a) (Gcd (normalize ` B)) dvd normalize a"
haftmann@60686
   446
      by (rule gcd_dvd1)
haftmann@60686
   447
    ultimately show "Gcd (normalize ` A) dvd a"
haftmann@60686
   448
      by simp
haftmann@60686
   449
  qed
haftmann@60688
   450
  then have "Gcd (normalize ` A) dvd Gcd A" and "Gcd A dvd Gcd (normalize ` A)"
haftmann@60688
   451
    by (auto intro!: Gcd_greatest intro: Gcd_dvd)
haftmann@60686
   452
  then show ?thesis
haftmann@60688
   453
    by (auto intro: associated_eqI)
haftmann@60686
   454
qed
haftmann@60686
   455
haftmann@60686
   456
lemma Lcm_least:
haftmann@60686
   457
  assumes "\<And>b. b \<in> A \<Longrightarrow> b dvd a"
haftmann@60686
   458
  shows "Lcm A dvd a"
haftmann@60686
   459
  using assms by (auto intro: Gcd_dvd simp add: Lcm_Gcd)
haftmann@60686
   460
haftmann@60686
   461
lemma normalize_Lcm [simp]:
haftmann@60686
   462
  "normalize (Lcm A) = Lcm A"
haftmann@60686
   463
  by (simp add: Lcm_Gcd)
haftmann@60686
   464
haftmann@60686
   465
lemma unit_factor_Lcm:
haftmann@60686
   466
  "unit_factor (Lcm A) = (if Lcm A = 0 then 0 else 1)"
haftmann@60686
   467
proof (cases "Lcm A = 0")
haftmann@60686
   468
  case True then show ?thesis by simp
haftmann@60686
   469
next
haftmann@60686
   470
  case False
haftmann@60686
   471
  with unit_factor_normalize have "unit_factor (normalize (Lcm A)) = 1"
haftmann@60686
   472
    by blast
haftmann@60686
   473
  with False show ?thesis
haftmann@60686
   474
    by simp
haftmann@60686
   475
qed
haftmann@60686
   476
  
haftmann@60686
   477
lemma Lcm_empty [simp]:
haftmann@60686
   478
  "Lcm {} = 1"
haftmann@60686
   479
  by (simp add: Lcm_Gcd)
haftmann@60686
   480
haftmann@60686
   481
lemma Lcm_1_iff [simp]:
haftmann@60686
   482
  "Lcm A = 1 \<longleftrightarrow> (\<forall>a\<in>A. is_unit a)" (is "?P \<longleftrightarrow> ?Q")
haftmann@60686
   483
proof
haftmann@60686
   484
  assume ?P
haftmann@60686
   485
  show ?Q
haftmann@60686
   486
  proof
haftmann@60686
   487
    fix a
haftmann@60686
   488
    assume "a \<in> A"
haftmann@60686
   489
    then have "a dvd Lcm A"
haftmann@60686
   490
      by (rule dvd_Lcm)
haftmann@60686
   491
    with \<open>?P\<close> show "is_unit a"
haftmann@60686
   492
      by simp
haftmann@60686
   493
  qed
haftmann@60686
   494
next
haftmann@60686
   495
  assume ?Q
haftmann@60686
   496
  then have "is_unit (Lcm A)"
haftmann@60686
   497
    by (blast intro: Lcm_least)
haftmann@60686
   498
  then have "normalize (Lcm A) = 1"
haftmann@60686
   499
    by (rule is_unit_normalize)
haftmann@60686
   500
  then show ?P
haftmann@60686
   501
    by simp
haftmann@60686
   502
qed
haftmann@60686
   503
haftmann@60686
   504
lemma Lcm_UNIV [simp]:
haftmann@60686
   505
  "Lcm UNIV = 0"
haftmann@60686
   506
proof -
haftmann@60686
   507
  have "0 dvd Lcm UNIV"
haftmann@60686
   508
    by (rule dvd_Lcm) simp
haftmann@60686
   509
  then show ?thesis
haftmann@60686
   510
    by simp
haftmann@60686
   511
qed
haftmann@60686
   512
haftmann@60686
   513
lemma Lcm_eq_0_I:
haftmann@60686
   514
  assumes "0 \<in> A"
haftmann@60686
   515
  shows "Lcm A = 0"
haftmann@60686
   516
proof -
haftmann@60686
   517
  from assms have "0 dvd Lcm A"
haftmann@60686
   518
    by (rule dvd_Lcm)
haftmann@60686
   519
  then show ?thesis
haftmann@60686
   520
    by simp
haftmann@60686
   521
qed
haftmann@60686
   522
haftmann@60686
   523
lemma Gcd_Lcm:
haftmann@60686
   524
  "Gcd A = Lcm {b. \<forall>a\<in>A. b dvd a}"
haftmann@60686
   525
  by (rule associated_eqI) (auto intro: associatedI Gcd_dvd dvd_Lcm Gcd_greatest Lcm_least
haftmann@60686
   526
    simp add: unit_factor_Gcd unit_factor_Lcm)
haftmann@60686
   527
haftmann@60686
   528
lemma Lcm_insert [simp]:
haftmann@60686
   529
  "Lcm (insert a A) = lcm a (Lcm A)"
haftmann@60686
   530
proof (rule sym)
haftmann@60686
   531
  have "lcm a (Lcm A) dvd Lcm (insert a A)"
haftmann@60686
   532
    by (auto intro: dvd_Lcm Lcm_least)
haftmann@60686
   533
  moreover have "Lcm (insert a A) dvd lcm a (Lcm A)"
haftmann@60686
   534
  proof (rule Lcm_least)
haftmann@60686
   535
    fix b
haftmann@60686
   536
    assume "b \<in> insert a A"
haftmann@60686
   537
    then show "b dvd lcm a (Lcm A)"
haftmann@60686
   538
    proof
haftmann@60686
   539
      assume "b = a" then show ?thesis by simp
haftmann@60686
   540
    next
haftmann@60686
   541
      assume "b \<in> A"
haftmann@60686
   542
      then have "b dvd Lcm A" by (rule dvd_Lcm)
haftmann@60686
   543
      moreover have "Lcm A dvd lcm a (Lcm A)" by simp
haftmann@60686
   544
      ultimately show ?thesis by (blast intro: dvd_trans)
haftmann@60686
   545
    qed
haftmann@60686
   546
  qed
haftmann@60688
   547
  ultimately show "lcm a (Lcm A) = Lcm (insert a A)"
haftmann@60688
   548
    by (rule associated_eqI) (simp_all add: lcm_eq_0_iff)
haftmann@60686
   549
qed
haftmann@60686
   550
  
haftmann@60686
   551
lemma Lcm_set [code_unfold]:
haftmann@60686
   552
  "Lcm (set as) = foldr lcm as 1"
haftmann@60686
   553
  by (induct as) simp_all
haftmann@60686
   554
  
haftmann@60686
   555
end
haftmann@59008
   556
haftmann@59008
   557
class ring_gcd = comm_ring_1 + semiring_gcd
haftmann@59008
   558
huffman@31706
   559
instantiation nat :: gcd
huffman@31706
   560
begin
wenzelm@21256
   561
huffman@31706
   562
fun
huffman@31706
   563
  gcd_nat  :: "nat \<Rightarrow> nat \<Rightarrow> nat"
huffman@31706
   564
where
huffman@31706
   565
  "gcd_nat x y =
huffman@31706
   566
   (if y = 0 then x else gcd y (x mod y))"
huffman@31706
   567
huffman@31706
   568
definition
huffman@31706
   569
  lcm_nat :: "nat \<Rightarrow> nat \<Rightarrow> nat"
huffman@31706
   570
where
huffman@31706
   571
  "lcm_nat x y = x * y div (gcd x y)"
huffman@31706
   572
huffman@31706
   573
instance proof qed
huffman@31706
   574
huffman@31706
   575
end
huffman@31706
   576
huffman@31706
   577
instantiation int :: gcd
huffman@31706
   578
begin
wenzelm@21256
   579
huffman@31706
   580
definition
huffman@31706
   581
  gcd_int  :: "int \<Rightarrow> int \<Rightarrow> int"
huffman@31706
   582
where
huffman@31706
   583
  "gcd_int x y = int (gcd (nat (abs x)) (nat (abs y)))"
haftmann@23687
   584
huffman@31706
   585
definition
huffman@31706
   586
  lcm_int :: "int \<Rightarrow> int \<Rightarrow> int"
huffman@31706
   587
where
huffman@31706
   588
  "lcm_int x y = int (lcm (nat (abs x)) (nat (abs y)))"
haftmann@23687
   589
huffman@31706
   590
instance proof qed
huffman@31706
   591
huffman@31706
   592
end
haftmann@23687
   593
haftmann@23687
   594
wenzelm@60758
   595
subsection \<open>Transfer setup\<close>
huffman@31706
   596
huffman@31706
   597
lemma transfer_nat_int_gcd:
huffman@31706
   598
  "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> gcd (nat x) (nat y) = nat (gcd x y)"
huffman@31706
   599
  "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> lcm (nat x) (nat y) = nat (lcm x y)"
haftmann@32479
   600
  unfolding gcd_int_def lcm_int_def
huffman@31706
   601
  by auto
haftmann@23687
   602
huffman@31706
   603
lemma transfer_nat_int_gcd_closures:
huffman@31706
   604
  "x >= (0::int) \<Longrightarrow> y >= 0 \<Longrightarrow> gcd x y >= 0"
huffman@31706
   605
  "x >= (0::int) \<Longrightarrow> y >= 0 \<Longrightarrow> lcm x y >= 0"
huffman@31706
   606
  by (auto simp add: gcd_int_def lcm_int_def)
huffman@31706
   607
haftmann@35644
   608
declare transfer_morphism_nat_int[transfer add return:
huffman@31706
   609
    transfer_nat_int_gcd transfer_nat_int_gcd_closures]
huffman@31706
   610
huffman@31706
   611
lemma transfer_int_nat_gcd:
huffman@31706
   612
  "gcd (int x) (int y) = int (gcd x y)"
huffman@31706
   613
  "lcm (int x) (int y) = int (lcm x y)"
haftmann@32479
   614
  by (unfold gcd_int_def lcm_int_def, auto)
huffman@31706
   615
huffman@31706
   616
lemma transfer_int_nat_gcd_closures:
huffman@31706
   617
  "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> gcd x y >= 0"
huffman@31706
   618
  "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> lcm x y >= 0"
huffman@31706
   619
  by (auto simp add: gcd_int_def lcm_int_def)
huffman@31706
   620
haftmann@35644
   621
declare transfer_morphism_int_nat[transfer add return:
huffman@31706
   622
    transfer_int_nat_gcd transfer_int_nat_gcd_closures]
huffman@31706
   623
huffman@31706
   624
wenzelm@60758
   625
subsection \<open>GCD properties\<close>
huffman@31706
   626
huffman@31706
   627
(* was gcd_induct *)
nipkow@31952
   628
lemma gcd_nat_induct:
haftmann@23687
   629
  fixes m n :: nat
haftmann@23687
   630
  assumes "\<And>m. P m 0"
haftmann@23687
   631
    and "\<And>m n. 0 < n \<Longrightarrow> P n (m mod n) \<Longrightarrow> P m n"
haftmann@23687
   632
  shows "P m n"
huffman@31706
   633
  apply (rule gcd_nat.induct)
huffman@31706
   634
  apply (case_tac "y = 0")
huffman@31706
   635
  using assms apply simp_all
huffman@31706
   636
done
huffman@31706
   637
huffman@31706
   638
(* specific to int *)
huffman@31706
   639
nipkow@31952
   640
lemma gcd_neg1_int [simp]: "gcd (-x::int) y = gcd x y"
huffman@31706
   641
  by (simp add: gcd_int_def)
huffman@31706
   642
nipkow@31952
   643
lemma gcd_neg2_int [simp]: "gcd (x::int) (-y) = gcd x y"
huffman@31706
   644
  by (simp add: gcd_int_def)
huffman@31706
   645
haftmann@54489
   646
lemma gcd_neg_numeral_1_int [simp]:
haftmann@54489
   647
  "gcd (- numeral n :: int) x = gcd (numeral n) x"
haftmann@54489
   648
  by (fact gcd_neg1_int)
haftmann@54489
   649
haftmann@54489
   650
lemma gcd_neg_numeral_2_int [simp]:
haftmann@54489
   651
  "gcd x (- numeral n :: int) = gcd x (numeral n)"
haftmann@54489
   652
  by (fact gcd_neg2_int)
haftmann@54489
   653
nipkow@31813
   654
lemma abs_gcd_int[simp]: "abs(gcd (x::int) y) = gcd x y"
nipkow@31813
   655
by(simp add: gcd_int_def)
nipkow@31813
   656
nipkow@31952
   657
lemma gcd_abs_int: "gcd (x::int) y = gcd (abs x) (abs y)"
nipkow@31813
   658
by (simp add: gcd_int_def)
nipkow@31813
   659
nipkow@31813
   660
lemma gcd_abs1_int[simp]: "gcd (abs x) (y::int) = gcd x y"
nipkow@31952
   661
by (metis abs_idempotent gcd_abs_int)
nipkow@31813
   662
nipkow@31813
   663
lemma gcd_abs2_int[simp]: "gcd x (abs y::int) = gcd x y"
nipkow@31952
   664
by (metis abs_idempotent gcd_abs_int)
huffman@31706
   665
nipkow@31952
   666
lemma gcd_cases_int:
huffman@31706
   667
  fixes x :: int and y
huffman@31706
   668
  assumes "x >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> P (gcd x y)"
huffman@31706
   669
      and "x >= 0 \<Longrightarrow> y <= 0 \<Longrightarrow> P (gcd x (-y))"
huffman@31706
   670
      and "x <= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> P (gcd (-x) y)"
huffman@31706
   671
      and "x <= 0 \<Longrightarrow> y <= 0 \<Longrightarrow> P (gcd (-x) (-y))"
huffman@31706
   672
  shows "P (gcd x y)"
huffman@35216
   673
by (insert assms, auto, arith)
wenzelm@21256
   674
nipkow@31952
   675
lemma gcd_ge_0_int [simp]: "gcd (x::int) y >= 0"
huffman@31706
   676
  by (simp add: gcd_int_def)
huffman@31706
   677
nipkow@31952
   678
lemma lcm_neg1_int: "lcm (-x::int) y = lcm x y"
huffman@31706
   679
  by (simp add: lcm_int_def)
huffman@31706
   680
nipkow@31952
   681
lemma lcm_neg2_int: "lcm (x::int) (-y) = lcm x y"
huffman@31706
   682
  by (simp add: lcm_int_def)
huffman@31706
   683
nipkow@31952
   684
lemma lcm_abs_int: "lcm (x::int) y = lcm (abs x) (abs y)"
huffman@31706
   685
  by (simp add: lcm_int_def)
wenzelm@21256
   686
nipkow@31814
   687
lemma abs_lcm_int [simp]: "abs (lcm i j::int) = lcm i j"
nipkow@31814
   688
by(simp add:lcm_int_def)
nipkow@31814
   689
nipkow@31814
   690
lemma lcm_abs1_int[simp]: "lcm (abs x) (y::int) = lcm x y"
nipkow@31814
   691
by (metis abs_idempotent lcm_int_def)
nipkow@31814
   692
nipkow@31814
   693
lemma lcm_abs2_int[simp]: "lcm x (abs y::int) = lcm x y"
nipkow@31814
   694
by (metis abs_idempotent lcm_int_def)
nipkow@31814
   695
nipkow@31952
   696
lemma lcm_cases_int:
huffman@31706
   697
  fixes x :: int and y
huffman@31706
   698
  assumes "x >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> P (lcm x y)"
huffman@31706
   699
      and "x >= 0 \<Longrightarrow> y <= 0 \<Longrightarrow> P (lcm x (-y))"
huffman@31706
   700
      and "x <= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> P (lcm (-x) y)"
huffman@31706
   701
      and "x <= 0 \<Longrightarrow> y <= 0 \<Longrightarrow> P (lcm (-x) (-y))"
huffman@31706
   702
  shows "P (lcm x y)"
wenzelm@41550
   703
  using assms by (auto simp add: lcm_neg1_int lcm_neg2_int) arith
huffman@31706
   704
nipkow@31952
   705
lemma lcm_ge_0_int [simp]: "lcm (x::int) y >= 0"
huffman@31706
   706
  by (simp add: lcm_int_def)
huffman@31706
   707
huffman@31706
   708
(* was gcd_0, etc. *)
haftmann@54867
   709
lemma gcd_0_nat: "gcd (x::nat) 0 = x"
haftmann@23687
   710
  by simp
haftmann@23687
   711
huffman@31706
   712
(* was igcd_0, etc. *)
nipkow@31952
   713
lemma gcd_0_int [simp]: "gcd (x::int) 0 = abs x"
huffman@31706
   714
  by (unfold gcd_int_def, auto)
huffman@31706
   715
haftmann@54867
   716
lemma gcd_0_left_nat: "gcd 0 (x::nat) = x"
haftmann@23687
   717
  by simp
haftmann@23687
   718
nipkow@31952
   719
lemma gcd_0_left_int [simp]: "gcd 0 (x::int) = abs x"
huffman@31706
   720
  by (unfold gcd_int_def, auto)
huffman@31706
   721
nipkow@31952
   722
lemma gcd_red_nat: "gcd (x::nat) y = gcd y (x mod y)"
huffman@31706
   723
  by (case_tac "y = 0", auto)
huffman@31706
   724
huffman@31706
   725
(* weaker, but useful for the simplifier *)
huffman@31706
   726
nipkow@31952
   727
lemma gcd_non_0_nat: "y ~= (0::nat) \<Longrightarrow> gcd (x::nat) y = gcd y (x mod y)"
huffman@31706
   728
  by simp
huffman@31706
   729
nipkow@31952
   730
lemma gcd_1_nat [simp]: "gcd (m::nat) 1 = 1"
wenzelm@21263
   731
  by simp
wenzelm@21256
   732
nipkow@31952
   733
lemma gcd_Suc_0 [simp]: "gcd (m::nat) (Suc 0) = Suc 0"
haftmann@60690
   734
  by simp
huffman@31706
   735
nipkow@31952
   736
lemma gcd_1_int [simp]: "gcd (m::int) 1 = 1"
huffman@31706
   737
  by (simp add: gcd_int_def)
huffman@30082
   738
nipkow@31952
   739
lemma gcd_idem_nat: "gcd (x::nat) x = x"
nipkow@31798
   740
by simp
huffman@31706
   741
nipkow@31952
   742
lemma gcd_idem_int: "gcd (x::int) x = abs x"
nipkow@31813
   743
by (auto simp add: gcd_int_def)
huffman@31706
   744
huffman@31706
   745
declare gcd_nat.simps [simp del]
wenzelm@21256
   746
wenzelm@60758
   747
text \<open>
wenzelm@61799
   748
  \medskip @{term "gcd m n"} divides \<open>m\<close> and \<open>n\<close>.  The
wenzelm@21256
   749
  conjunctions don't seem provable separately.
wenzelm@60758
   750
\<close>
wenzelm@21256
   751
haftmann@59008
   752
instance nat :: semiring_gcd
haftmann@59008
   753
proof
haftmann@59008
   754
  fix m n :: nat
haftmann@59008
   755
  show "gcd m n dvd m" and "gcd m n dvd n"
haftmann@59008
   756
  proof (induct m n rule: gcd_nat_induct)
haftmann@59008
   757
    fix m n :: nat
haftmann@59008
   758
    assume "gcd n (m mod n) dvd m mod n" and "gcd n (m mod n) dvd n"
haftmann@59008
   759
    then have "gcd n (m mod n) dvd m"
haftmann@59008
   760
      by (rule dvd_mod_imp_dvd)
haftmann@59008
   761
    moreover assume "0 < n"
haftmann@59008
   762
    ultimately show "gcd m n dvd m"
haftmann@59008
   763
      by (simp add: gcd_non_0_nat)
haftmann@59008
   764
  qed (simp_all add: gcd_0_nat gcd_non_0_nat)
haftmann@59008
   765
next
haftmann@59008
   766
  fix m n k :: nat
haftmann@59008
   767
  assume "k dvd m" and "k dvd n"
haftmann@59008
   768
  then show "k dvd gcd m n"
haftmann@59008
   769
    by (induct m n rule: gcd_nat_induct) (simp_all add: gcd_non_0_nat dvd_mod gcd_0_nat)
haftmann@60686
   770
qed (simp_all add: lcm_nat_def)
lp15@59667
   771
haftmann@59008
   772
instance int :: ring_gcd
haftmann@60686
   773
  by standard
haftmann@60686
   774
    (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
   775
nipkow@31730
   776
lemma dvd_gcd_D1_nat: "k dvd gcd m n \<Longrightarrow> (k::nat) dvd m"
haftmann@59008
   777
  by (metis gcd_dvd1 dvd_trans)
nipkow@31730
   778
nipkow@31730
   779
lemma dvd_gcd_D2_nat: "k dvd gcd m n \<Longrightarrow> (k::nat) dvd n"
haftmann@59008
   780
  by (metis gcd_dvd2 dvd_trans)
nipkow@31730
   781
nipkow@31730
   782
lemma dvd_gcd_D1_int: "i dvd gcd m n \<Longrightarrow> (i::int) dvd m"
haftmann@59008
   783
  by (metis gcd_dvd1 dvd_trans)
nipkow@31730
   784
nipkow@31730
   785
lemma dvd_gcd_D2_int: "i dvd gcd m n \<Longrightarrow> (i::int) dvd n"
haftmann@59008
   786
  by (metis gcd_dvd2 dvd_trans)
nipkow@31730
   787
nipkow@31952
   788
lemma gcd_le1_nat [simp]: "a \<noteq> 0 \<Longrightarrow> gcd (a::nat) b \<le> a"
huffman@31706
   789
  by (rule dvd_imp_le, auto)
huffman@31706
   790
nipkow@31952
   791
lemma gcd_le2_nat [simp]: "b \<noteq> 0 \<Longrightarrow> gcd (a::nat) b \<le> b"
huffman@31706
   792
  by (rule dvd_imp_le, auto)
huffman@31706
   793
nipkow@31952
   794
lemma gcd_le1_int [simp]: "a > 0 \<Longrightarrow> gcd (a::int) b \<le> a"
huffman@31706
   795
  by (rule zdvd_imp_le, auto)
wenzelm@21256
   796
nipkow@31952
   797
lemma gcd_le2_int [simp]: "b > 0 \<Longrightarrow> gcd (a::int) b \<le> b"
huffman@31706
   798
  by (rule zdvd_imp_le, auto)
huffman@31706
   799
haftmann@60689
   800
lemma gcd_greatest_iff_nat:
haftmann@60689
   801
  "(k dvd gcd (m::nat) n) = (k dvd m & k dvd n)"
haftmann@60689
   802
  by (fact gcd_greatest_iff)
haftmann@60689
   803
haftmann@60689
   804
lemma gcd_greatest_iff_int:
haftmann@60689
   805
  "((k::int) dvd gcd m n) = (k dvd m & k dvd n)"
haftmann@60689
   806
  by (fact gcd_greatest_iff)
huffman@31706
   807
haftmann@60689
   808
lemma gcd_zero_nat: 
haftmann@60689
   809
  "(gcd (m::nat) n = 0) = (m = 0 & n = 0)"
haftmann@60689
   810
  by (fact gcd_eq_0_iff)
wenzelm@21256
   811
haftmann@60689
   812
lemma gcd_zero_int [simp]:
haftmann@60689
   813
  "(gcd (m::int) n = 0) = (m = 0 & n = 0)"
haftmann@60689
   814
  by (fact gcd_eq_0_iff)
wenzelm@21256
   815
nipkow@31952
   816
lemma gcd_pos_nat [simp]: "(gcd (m::nat) n > 0) = (m ~= 0 | n ~= 0)"
nipkow@31952
   817
  by (insert gcd_zero_nat [of m n], arith)
wenzelm@21256
   818
nipkow@31952
   819
lemma gcd_pos_int [simp]: "(gcd (m::int) n > 0) = (m ~= 0 | n ~= 0)"
nipkow@31952
   820
  by (insert gcd_zero_int [of m n], insert gcd_ge_0_int [of m n], arith)
huffman@31706
   821
nipkow@31952
   822
lemma gcd_unique_nat: "(d::nat) dvd a \<and> d dvd b \<and>
huffman@31706
   823
    (\<forall>e. e dvd a \<and> e dvd b \<longrightarrow> e dvd d) \<longleftrightarrow> d = gcd a b"
huffman@31706
   824
  apply auto
nipkow@33657
   825
  apply (rule dvd_antisym)
haftmann@59008
   826
  apply (erule (1) gcd_greatest)
huffman@31706
   827
  apply auto
huffman@31706
   828
done
wenzelm@21256
   829
nipkow@31952
   830
lemma gcd_unique_int: "d >= 0 & (d::int) dvd a \<and> d dvd b \<and>
huffman@31706
   831
    (\<forall>e. e dvd a \<and> e dvd b \<longrightarrow> e dvd d) \<longleftrightarrow> d = gcd a b"
nipkow@33657
   832
apply (case_tac "d = 0")
nipkow@33657
   833
 apply simp
nipkow@33657
   834
apply (rule iffI)
nipkow@33657
   835
 apply (rule zdvd_antisym_nonneg)
haftmann@59008
   836
 apply (auto intro: gcd_greatest)
huffman@31706
   837
done
huffman@30082
   838
haftmann@54867
   839
interpretation gcd_nat: abel_semigroup "gcd :: nat \<Rightarrow> nat \<Rightarrow> nat"
haftmann@54867
   840
  + 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
   841
apply standard
haftmann@60686
   842
apply (auto intro: dvd_antisym dvd_trans)[2]
haftmann@59545
   843
apply (metis dvd.dual_order.refl gcd_unique_nat)+
haftmann@54867
   844
done
haftmann@54867
   845
haftmann@60686
   846
interpretation gcd_int: abel_semigroup "gcd :: int \<Rightarrow> int \<Rightarrow> int" ..
haftmann@54867
   847
haftmann@60686
   848
lemmas gcd_assoc_nat = gcd.assoc [where ?'a = nat]
haftmann@60686
   849
lemmas gcd_commute_nat = gcd.commute [where ?'a = nat]
haftmann@60686
   850
lemmas gcd_left_commute_nat = gcd.left_commute [where ?'a = nat]
haftmann@60686
   851
lemmas gcd_assoc_int = gcd.assoc [where ?'a = int]
haftmann@60686
   852
lemmas gcd_commute_int = gcd.commute [where ?'a = int]
haftmann@60686
   853
lemmas gcd_left_commute_int = gcd.left_commute [where ?'a = int]
haftmann@54867
   854
haftmann@54867
   855
lemmas gcd_ac_nat = gcd_assoc_nat gcd_commute_nat gcd_left_commute_nat
haftmann@54867
   856
haftmann@54867
   857
lemmas gcd_ac_int = gcd_assoc_int gcd_commute_int gcd_left_commute_int
haftmann@54867
   858
nipkow@31798
   859
lemma gcd_proj1_if_dvd_nat [simp]: "(x::nat) dvd y \<Longrightarrow> gcd x y = x"
haftmann@54867
   860
  by (fact gcd_nat.absorb1)
nipkow@31798
   861
nipkow@31798
   862
lemma gcd_proj2_if_dvd_nat [simp]: "(y::nat) dvd x \<Longrightarrow> gcd x y = y"
haftmann@54867
   863
  by (fact gcd_nat.absorb2)
nipkow@31798
   864
haftmann@54867
   865
lemma gcd_proj1_if_dvd_int [simp]: "x dvd y \<Longrightarrow> gcd (x::int) y = abs x"
haftmann@54867
   866
  by (metis abs_dvd_iff gcd_0_left_int gcd_abs_int gcd_unique_int)
nipkow@31798
   867
haftmann@54867
   868
lemma gcd_proj2_if_dvd_int [simp]: "y dvd x \<Longrightarrow> gcd (x::int) y = abs y"
haftmann@54867
   869
  by (metis gcd_proj1_if_dvd_int gcd_commute_int)
nipkow@31798
   870
wenzelm@60758
   871
text \<open>
wenzelm@21256
   872
  \medskip Multiplication laws
wenzelm@60758
   873
\<close>
wenzelm@21256
   874
nipkow@31952
   875
lemma gcd_mult_distrib_nat: "(k::nat) * gcd m n = gcd (k * m) (k * n)"
wenzelm@61799
   876
    \<comment> \<open>@{cite \<open>page 27\<close> davenport92}\<close>
nipkow@31952
   877
  apply (induct m n rule: gcd_nat_induct)
huffman@31706
   878
  apply simp
wenzelm@21256
   879
  apply (case_tac "k = 0")
huffman@45270
   880
  apply (simp_all add: gcd_non_0_nat)
huffman@31706
   881
done
wenzelm@21256
   882
nipkow@31952
   883
lemma gcd_mult_distrib_int: "abs (k::int) * gcd m n = gcd (k * m) (k * n)"
nipkow@31952
   884
  apply (subst (1 2) gcd_abs_int)
nipkow@31813
   885
  apply (subst (1 2) abs_mult)
nipkow@31952
   886
  apply (rule gcd_mult_distrib_nat [transferred])
huffman@31706
   887
  apply auto
huffman@31706
   888
done
wenzelm@21256
   889
haftmann@60690
   890
context semiring_gcd
haftmann@60690
   891
begin
wenzelm@21256
   892
haftmann@60690
   893
lemma coprime_dvd_mult:
haftmann@60690
   894
  assumes "coprime a b" and "a dvd c * b"
haftmann@60690
   895
  shows "a dvd c"
haftmann@60690
   896
proof (cases "c = 0")
haftmann@60690
   897
  case True then show ?thesis by simp
haftmann@60690
   898
next
haftmann@60690
   899
  case False
haftmann@60690
   900
  then have unit: "is_unit (unit_factor c)" by simp
haftmann@60690
   901
  from \<open>coprime a b\<close> mult_gcd_left [of c a b]
haftmann@60690
   902
  have "gcd (c * a) (c * b) * unit_factor c = c"
haftmann@60690
   903
    by (simp add: ac_simps)
haftmann@60690
   904
  moreover from \<open>a dvd c * b\<close> have "a dvd gcd (c * a) (c * b) * unit_factor c"
haftmann@60690
   905
    by (simp add: dvd_mult_unit_iff unit)
haftmann@60690
   906
  ultimately show ?thesis by simp
haftmann@60690
   907
qed
haftmann@60690
   908
haftmann@60690
   909
end
haftmann@60690
   910
haftmann@60690
   911
lemmas coprime_dvd_mult_nat = coprime_dvd_mult [where ?'a = nat]
haftmann@60690
   912
lemmas coprime_dvd_mult_int = coprime_dvd_mult [where ?'a = int]
huffman@31706
   913
nipkow@31952
   914
lemma coprime_dvd_mult_iff_nat: "coprime (k::nat) n \<Longrightarrow>
huffman@31706
   915
    (k dvd m * n) = (k dvd m)"
nipkow@31952
   916
  by (auto intro: coprime_dvd_mult_nat)
huffman@31706
   917
nipkow@31952
   918
lemma coprime_dvd_mult_iff_int: "coprime (k::int) n \<Longrightarrow>
huffman@31706
   919
    (k dvd m * n) = (k dvd m)"
nipkow@31952
   920
  by (auto intro: coprime_dvd_mult_int)
huffman@31706
   921
haftmann@60690
   922
context semiring_gcd
haftmann@60690
   923
begin
haftmann@60690
   924
haftmann@60690
   925
lemma gcd_mult_cancel:
haftmann@60690
   926
  "coprime c b \<Longrightarrow> gcd (c * a) b = gcd a b"
haftmann@60690
   927
  apply (rule associated_eqI)
haftmann@59008
   928
  apply (rule gcd_greatest)
haftmann@60690
   929
  apply (rule_tac b = c in coprime_dvd_mult)
haftmann@60690
   930
  apply (simp add: gcd.assoc)
haftmann@60690
   931
  apply (simp_all add: ac_simps)
haftmann@60690
   932
  done
wenzelm@21256
   933
haftmann@60690
   934
end  
haftmann@60690
   935
haftmann@60690
   936
lemmas gcd_mult_cancel_nat = gcd_mult_cancel [where ?'a = nat] 
haftmann@60690
   937
lemmas gcd_mult_cancel_int = gcd_mult_cancel [where ?'a = int] 
wenzelm@21256
   938
haftmann@35368
   939
lemma coprime_crossproduct_nat:
haftmann@35368
   940
  fixes a b c d :: nat
haftmann@35368
   941
  assumes "coprime a d" and "coprime b c"
haftmann@35368
   942
  shows "a * c = b * d \<longleftrightarrow> a = b \<and> c = d" (is "?lhs \<longleftrightarrow> ?rhs")
haftmann@35368
   943
proof
haftmann@35368
   944
  assume ?rhs then show ?lhs by simp
haftmann@35368
   945
next
haftmann@35368
   946
  assume ?lhs
wenzelm@60758
   947
  from \<open>?lhs\<close> have "a dvd b * d" by (auto intro: dvdI dest: sym)
wenzelm@60758
   948
  with \<open>coprime a d\<close> have "a dvd b" by (simp add: coprime_dvd_mult_iff_nat)
wenzelm@60758
   949
  from \<open>?lhs\<close> have "b dvd a * c" by (auto intro: dvdI dest: sym)
wenzelm@60758
   950
  with \<open>coprime b c\<close> have "b dvd a" by (simp add: coprime_dvd_mult_iff_nat)
wenzelm@60758
   951
  from \<open>?lhs\<close> have "c dvd d * b" by (auto intro: dvdI dest: sym simp add: mult.commute)
wenzelm@60758
   952
  with \<open>coprime b c\<close> have "c dvd d" by (simp add: coprime_dvd_mult_iff_nat gcd_commute_nat)
wenzelm@60758
   953
  from \<open>?lhs\<close> have "d dvd c * a" by (auto intro: dvdI dest: sym simp add: mult.commute)
wenzelm@60758
   954
  with \<open>coprime a d\<close> have "d dvd c" by (simp add: coprime_dvd_mult_iff_nat gcd_commute_nat)
wenzelm@60758
   955
  from \<open>a dvd b\<close> \<open>b dvd a\<close> have "a = b" by (rule Nat.dvd.antisym)
wenzelm@60758
   956
  moreover from \<open>c dvd d\<close> \<open>d dvd c\<close> have "c = d" by (rule Nat.dvd.antisym)
haftmann@35368
   957
  ultimately show ?rhs ..
haftmann@35368
   958
qed
haftmann@35368
   959
haftmann@35368
   960
lemma coprime_crossproduct_int:
haftmann@35368
   961
  fixes a b c d :: int
haftmann@35368
   962
  assumes "coprime a d" and "coprime b c"
haftmann@35368
   963
  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
   964
  using assms by (intro coprime_crossproduct_nat [transferred]) auto
haftmann@35368
   965
wenzelm@60758
   966
text \<open>\medskip Addition laws\<close>
wenzelm@21256
   967
nipkow@31952
   968
lemma gcd_add1_nat [simp]: "gcd ((m::nat) + n) n = gcd m n"
huffman@31706
   969
  apply (case_tac "n = 0")
nipkow@31952
   970
  apply (simp_all add: gcd_non_0_nat)
huffman@31706
   971
done
huffman@31706
   972
nipkow@31952
   973
lemma gcd_add2_nat [simp]: "gcd (m::nat) (m + n) = gcd m n"
nipkow@31952
   974
  apply (subst (1 2) gcd_commute_nat)
haftmann@57512
   975
  apply (subst add.commute)
huffman@31706
   976
  apply simp
huffman@31706
   977
done
huffman@31706
   978
huffman@31706
   979
(* to do: add the other variations? *)
huffman@31706
   980
nipkow@31952
   981
lemma gcd_diff1_nat: "(m::nat) >= n \<Longrightarrow> gcd (m - n) n = gcd m n"
nipkow@31952
   982
  by (subst gcd_add1_nat [symmetric], auto)
huffman@31706
   983
nipkow@31952
   984
lemma gcd_diff2_nat: "(n::nat) >= m \<Longrightarrow> gcd (n - m) n = gcd m n"
nipkow@31952
   985
  apply (subst gcd_commute_nat)
nipkow@31952
   986
  apply (subst gcd_diff1_nat [symmetric])
huffman@31706
   987
  apply auto
nipkow@31952
   988
  apply (subst gcd_commute_nat)
nipkow@31952
   989
  apply (subst gcd_diff1_nat)
huffman@31706
   990
  apply assumption
nipkow@31952
   991
  apply (rule gcd_commute_nat)
huffman@31706
   992
done
huffman@31706
   993
nipkow@31952
   994
lemma gcd_non_0_int: "(y::int) > 0 \<Longrightarrow> gcd x y = gcd y (x mod y)"
huffman@31706
   995
  apply (frule_tac b = y and a = x in pos_mod_sign)
huffman@31706
   996
  apply (simp del: pos_mod_sign add: gcd_int_def abs_if nat_mod_distrib)
nipkow@31952
   997
  apply (auto simp add: gcd_non_0_nat nat_mod_distrib [symmetric]
huffman@31706
   998
    zmod_zminus1_eq_if)
huffman@31706
   999
  apply (frule_tac a = x in pos_mod_bound)
nipkow@31952
  1000
  apply (subst (1 2) gcd_commute_nat)
nipkow@31952
  1001
  apply (simp del: pos_mod_bound add: nat_diff_distrib gcd_diff2_nat
huffman@31706
  1002
    nat_le_eq_zle)
huffman@31706
  1003
done
wenzelm@21256
  1004
nipkow@31952
  1005
lemma gcd_red_int: "gcd (x::int) y = gcd y (x mod y)"
huffman@31706
  1006
  apply (case_tac "y = 0")
huffman@31706
  1007
  apply force
huffman@31706
  1008
  apply (case_tac "y > 0")
nipkow@31952
  1009
  apply (subst gcd_non_0_int, auto)
nipkow@31952
  1010
  apply (insert gcd_non_0_int [of "-y" "-x"])
huffman@35216
  1011
  apply auto
huffman@31706
  1012
done
huffman@31706
  1013
nipkow@31952
  1014
lemma gcd_add1_int [simp]: "gcd ((m::int) + n) n = gcd m n"
haftmann@57512
  1015
by (metis gcd_red_int mod_add_self1 add.commute)
huffman@31706
  1016
nipkow@31952
  1017
lemma gcd_add2_int [simp]: "gcd m ((m::int) + n) = gcd m n"
haftmann@57512
  1018
by (metis gcd_add1_int gcd_commute_int add.commute)
wenzelm@21256
  1019
nipkow@31952
  1020
lemma gcd_add_mult_nat: "gcd (m::nat) (k * m + n) = gcd m n"
nipkow@31952
  1021
by (metis mod_mult_self3 gcd_commute_nat gcd_red_nat)
wenzelm@21256
  1022
nipkow@31952
  1023
lemma gcd_add_mult_int: "gcd (m::int) (k * m + n) = gcd m n"
haftmann@57512
  1024
by (metis gcd_commute_int gcd_red_int mod_mult_self1 add.commute)
nipkow@31798
  1025
wenzelm@21256
  1026
huffman@31706
  1027
(* to do: differences, and all variations of addition rules
huffman@31706
  1028
    as simplification rules for nat and int *)
huffman@31706
  1029
haftmann@60689
  1030
lemma gcd_dvd_prod_nat: "gcd (m::nat) n dvd k * n"
haftmann@23687
  1031
  using mult_dvd_mono [of 1] by auto
chaieb@22027
  1032
huffman@31706
  1033
(* to do: add the three variations of these, and for ints? *)
huffman@31706
  1034
nipkow@31992
  1035
lemma finite_divisors_nat[simp]:
nipkow@31992
  1036
  assumes "(m::nat) ~= 0" shows "finite{d. d dvd m}"
nipkow@31734
  1037
proof-
wenzelm@60512
  1038
  have "finite{d. d <= m}"
wenzelm@60512
  1039
    by (blast intro: bounded_nat_set_is_finite)
nipkow@31734
  1040
  from finite_subset[OF _ this] show ?thesis using assms
wenzelm@60512
  1041
    by (metis Collect_mono dvd_imp_le neq0_conv)
nipkow@31734
  1042
qed
nipkow@31734
  1043
nipkow@31995
  1044
lemma finite_divisors_int[simp]:
nipkow@31734
  1045
  assumes "(i::int) ~= 0" shows "finite{d. d dvd i}"
nipkow@31734
  1046
proof-
nipkow@31734
  1047
  have "{d. abs d <= abs i} = {- abs i .. abs i}" by(auto simp:abs_if)
nipkow@31734
  1048
  hence "finite{d. abs d <= abs i}" by simp
nipkow@31734
  1049
  from finite_subset[OF _ this] show ?thesis using assms
wenzelm@60512
  1050
    by (simp add: dvd_imp_le_int subset_iff)
nipkow@31734
  1051
qed
nipkow@31734
  1052
nipkow@31995
  1053
lemma Max_divisors_self_nat[simp]: "n\<noteq>0 \<Longrightarrow> Max{d::nat. d dvd n} = n"
nipkow@31995
  1054
apply(rule antisym)
nipkow@44890
  1055
 apply (fastforce intro: Max_le_iff[THEN iffD2] simp: dvd_imp_le)
nipkow@31995
  1056
apply simp
nipkow@31995
  1057
done
nipkow@31995
  1058
nipkow@31995
  1059
lemma Max_divisors_self_int[simp]: "n\<noteq>0 \<Longrightarrow> Max{d::int. d dvd n} = abs n"
nipkow@31995
  1060
apply(rule antisym)
haftmann@44278
  1061
 apply(rule Max_le_iff [THEN iffD2])
haftmann@44278
  1062
  apply (auto intro: abs_le_D1 dvd_imp_le_int)
nipkow@31995
  1063
done
nipkow@31995
  1064
nipkow@31734
  1065
lemma gcd_is_Max_divisors_nat:
nipkow@31734
  1066
  "m ~= 0 \<Longrightarrow> n ~= 0 \<Longrightarrow> gcd (m::nat) n = (Max {d. d dvd m & d dvd n})"
nipkow@31734
  1067
apply(rule Max_eqI[THEN sym])
nipkow@31995
  1068
  apply (metis finite_Collect_conjI finite_divisors_nat)
nipkow@31734
  1069
 apply simp
nipkow@31952
  1070
 apply(metis Suc_diff_1 Suc_neq_Zero dvd_imp_le gcd_greatest_iff_nat gcd_pos_nat)
nipkow@31734
  1071
apply simp
nipkow@31734
  1072
done
nipkow@31734
  1073
nipkow@31734
  1074
lemma gcd_is_Max_divisors_int:
nipkow@31734
  1075
  "m ~= 0 ==> n ~= 0 ==> gcd (m::int) n = (Max {d. d dvd m & d dvd n})"
nipkow@31734
  1076
apply(rule Max_eqI[THEN sym])
nipkow@31995
  1077
  apply (metis finite_Collect_conjI finite_divisors_int)
nipkow@31734
  1078
 apply simp
nipkow@31952
  1079
 apply (metis gcd_greatest_iff_int gcd_pos_int zdvd_imp_le)
nipkow@31734
  1080
apply simp
nipkow@31734
  1081
done
nipkow@31734
  1082
haftmann@34030
  1083
lemma gcd_code_int [code]:
haftmann@34030
  1084
  "gcd k l = \<bar>if l = (0::int) then k else gcd l (\<bar>k\<bar> mod \<bar>l\<bar>)\<bar>"
haftmann@34030
  1085
  by (simp add: gcd_int_def nat_mod_distrib gcd_non_0_nat)
haftmann@34030
  1086
chaieb@22027
  1087
wenzelm@60758
  1088
subsection \<open>Coprimality\<close>
huffman@31706
  1089
haftmann@60690
  1090
context semiring_gcd
haftmann@60690
  1091
begin
haftmann@60690
  1092
haftmann@60690
  1093
lemma div_gcd_coprime:
haftmann@60690
  1094
  assumes nz: "a \<noteq> 0 \<or> b \<noteq> 0"
huffman@31706
  1095
  shows "coprime (a div gcd a b) (b div gcd a b)"
wenzelm@22367
  1096
proof -
haftmann@27556
  1097
  let ?g = "gcd a b"
chaieb@22027
  1098
  let ?a' = "a div ?g"
chaieb@22027
  1099
  let ?b' = "b div ?g"
haftmann@27556
  1100
  let ?g' = "gcd ?a' ?b'"
chaieb@22027
  1101
  have dvdg: "?g dvd a" "?g dvd b" by simp_all
chaieb@22027
  1102
  have dvdg': "?g' dvd ?a'" "?g' dvd ?b'" by simp_all
wenzelm@22367
  1103
  from dvdg dvdg' obtain ka kb ka' kb' where
wenzelm@22367
  1104
      kab: "a = ?g * ka" "b = ?g * kb" "?a' = ?g' * ka'" "?b' = ?g' * kb'"
chaieb@22027
  1105
    unfolding dvd_def by blast
haftmann@58834
  1106
  from this [symmetric] have "?g * ?a' = (?g * ?g') * ka'" "?g * ?b' = (?g * ?g') * kb'"
haftmann@58834
  1107
    by (simp_all add: mult.assoc mult.left_commute [of "gcd a b"])
wenzelm@22367
  1108
  then have dvdgg':"?g * ?g' dvd a" "?g* ?g' dvd b"
wenzelm@22367
  1109
    by (auto simp add: dvd_mult_div_cancel [OF dvdg(1)]
wenzelm@22367
  1110
      dvd_mult_div_cancel [OF dvdg(2)] dvd_def)
huffman@35216
  1111
  have "?g \<noteq> 0" using nz by simp
haftmann@60690
  1112
  moreover from gcd_greatest [OF dvdgg'] have "?g * ?g' dvd ?g" .
haftmann@60690
  1113
  thm dvd_mult_cancel_left
haftmann@60690
  1114
  ultimately show ?thesis using dvd_times_left_cancel_iff [of "gcd a b" _ 1] by simp
chaieb@22027
  1115
qed
chaieb@22027
  1116
haftmann@60690
  1117
end
haftmann@60690
  1118
haftmann@60690
  1119
lemmas div_gcd_coprime_nat = div_gcd_coprime [where ?'a = nat]
haftmann@60690
  1120
lemmas div_gcd_coprime_int = div_gcd_coprime [where ?'a = int]
huffman@31706
  1121
nipkow@31952
  1122
lemma coprime_nat: "coprime (a::nat) b \<longleftrightarrow> (\<forall>d. d dvd a \<and> d dvd b \<longleftrightarrow> d = 1)"
nipkow@31952
  1123
  using gcd_unique_nat[of 1 a b, simplified] by auto
huffman@31706
  1124
nipkow@31952
  1125
lemma coprime_Suc_0_nat:
huffman@31706
  1126
    "coprime (a::nat) b \<longleftrightarrow> (\<forall>d. d dvd a \<and> d dvd b \<longleftrightarrow> d = Suc 0)"
haftmann@60690
  1127
  using coprime_nat by simp
huffman@31706
  1128
nipkow@31952
  1129
lemma coprime_int: "coprime (a::int) b \<longleftrightarrow>
huffman@31706
  1130
    (\<forall>d. d >= 0 \<and> d dvd a \<and> d dvd b \<longleftrightarrow> d = 1)"
nipkow@31952
  1131
  using gcd_unique_int [of 1 a b]
huffman@31706
  1132
  apply clarsimp
huffman@31706
  1133
  apply (erule subst)
huffman@31706
  1134
  apply (rule iffI)
huffman@31706
  1135
  apply force
lp15@61649
  1136
  using abs_dvd_iff abs_ge_zero apply blast
wenzelm@59807
  1137
  done
huffman@31706
  1138
nipkow@31952
  1139
lemma gcd_coprime_nat:
huffman@31706
  1140
  assumes z: "gcd (a::nat) b \<noteq> 0" and a: "a = a' * gcd a b" and
huffman@31706
  1141
    b: "b = b' * gcd a b"
huffman@31706
  1142
  shows    "coprime a' b'"
huffman@31706
  1143
huffman@31706
  1144
  apply (subgoal_tac "a' = a div gcd a b")
huffman@31706
  1145
  apply (erule ssubst)
huffman@31706
  1146
  apply (subgoal_tac "b' = b div gcd a b")
huffman@31706
  1147
  apply (erule ssubst)
nipkow@31952
  1148
  apply (rule div_gcd_coprime_nat)
wenzelm@41550
  1149
  using z apply force
huffman@31706
  1150
  apply (subst (1) b)
huffman@31706
  1151
  using z apply force
huffman@31706
  1152
  apply (subst (1) a)
huffman@31706
  1153
  using z apply force
wenzelm@41550
  1154
  done
huffman@31706
  1155
nipkow@31952
  1156
lemma gcd_coprime_int:
huffman@31706
  1157
  assumes z: "gcd (a::int) b \<noteq> 0" and a: "a = a' * gcd a b" and
huffman@31706
  1158
    b: "b = b' * gcd a b"
huffman@31706
  1159
  shows    "coprime a' b'"
huffman@31706
  1160
huffman@31706
  1161
  apply (subgoal_tac "a' = a div gcd a b")
huffman@31706
  1162
  apply (erule ssubst)
huffman@31706
  1163
  apply (subgoal_tac "b' = b div gcd a b")
huffman@31706
  1164
  apply (erule ssubst)
nipkow@31952
  1165
  apply (rule div_gcd_coprime_int)
wenzelm@41550
  1166
  using z apply force
huffman@31706
  1167
  apply (subst (1) b)
huffman@31706
  1168
  using z apply force
huffman@31706
  1169
  apply (subst (1) a)
huffman@31706
  1170
  using z apply force
wenzelm@41550
  1171
  done
huffman@31706
  1172
haftmann@60690
  1173
context semiring_gcd
haftmann@60690
  1174
begin
huffman@31706
  1175
haftmann@60690
  1176
lemma coprime_mult:
haftmann@60690
  1177
  assumes da: "coprime d a" and db: "coprime d b"
haftmann@60690
  1178
  shows "coprime d (a * b)"
haftmann@60690
  1179
  apply (subst gcd.commute)
haftmann@60690
  1180
  using da apply (subst gcd_mult_cancel)
haftmann@60690
  1181
  apply (subst gcd.commute, assumption)
haftmann@60690
  1182
  apply (subst gcd.commute, rule db)
haftmann@60690
  1183
  done
huffman@31706
  1184
haftmann@60690
  1185
end
haftmann@60690
  1186
haftmann@60690
  1187
lemmas coprime_mult_nat = coprime_mult [where ?'a = nat]
haftmann@60690
  1188
lemmas coprime_mult_int = coprime_mult [where ?'a = int]
haftmann@60690
  1189
  
nipkow@31952
  1190
lemma coprime_lmult_nat:
huffman@31706
  1191
  assumes dab: "coprime (d::nat) (a * b)" shows "coprime d a"
huffman@31706
  1192
proof -
huffman@31706
  1193
  have "gcd d a dvd gcd d (a * b)"
haftmann@59008
  1194
    by (rule gcd_greatest, auto)
huffman@31706
  1195
  with dab show ?thesis
huffman@31706
  1196
    by auto
huffman@31706
  1197
qed
huffman@31706
  1198
nipkow@31952
  1199
lemma coprime_lmult_int:
nipkow@31798
  1200
  assumes "coprime (d::int) (a * b)" shows "coprime d a"
huffman@31706
  1201
proof -
huffman@31706
  1202
  have "gcd d a dvd gcd d (a * b)"
haftmann@59008
  1203
    by (rule gcd_greatest, auto)
nipkow@31798
  1204
  with assms show ?thesis
huffman@31706
  1205
    by auto
huffman@31706
  1206
qed
huffman@31706
  1207
nipkow@31952
  1208
lemma coprime_rmult_nat:
nipkow@31798
  1209
  assumes "coprime (d::nat) (a * b)" shows "coprime d b"
huffman@31706
  1210
proof -
huffman@31706
  1211
  have "gcd d b dvd gcd d (a * b)"
haftmann@59008
  1212
    by (rule gcd_greatest, auto intro: dvd_mult)
nipkow@31798
  1213
  with assms show ?thesis
huffman@31706
  1214
    by auto
huffman@31706
  1215
qed
huffman@31706
  1216
nipkow@31952
  1217
lemma coprime_rmult_int:
huffman@31706
  1218
  assumes dab: "coprime (d::int) (a * b)" shows "coprime d b"
huffman@31706
  1219
proof -
huffman@31706
  1220
  have "gcd d b dvd gcd d (a * b)"
haftmann@59008
  1221
    by (rule gcd_greatest, auto intro: dvd_mult)
huffman@31706
  1222
  with dab show ?thesis
huffman@31706
  1223
    by auto
huffman@31706
  1224
qed
huffman@31706
  1225
nipkow@31952
  1226
lemma coprime_mul_eq_nat: "coprime (d::nat) (a * b) \<longleftrightarrow>
huffman@31706
  1227
    coprime d a \<and>  coprime d b"
nipkow@31952
  1228
  using coprime_rmult_nat[of d a b] coprime_lmult_nat[of d a b]
nipkow@31952
  1229
    coprime_mult_nat[of d a b]
huffman@31706
  1230
  by blast
huffman@31706
  1231
nipkow@31952
  1232
lemma coprime_mul_eq_int: "coprime (d::int) (a * b) \<longleftrightarrow>
huffman@31706
  1233
    coprime d a \<and>  coprime d b"
nipkow@31952
  1234
  using coprime_rmult_int[of d a b] coprime_lmult_int[of d a b]
nipkow@31952
  1235
    coprime_mult_int[of d a b]
huffman@31706
  1236
  by blast
huffman@31706
  1237
noschinl@52397
  1238
lemma coprime_power_int:
noschinl@52397
  1239
  assumes "0 < n" shows "coprime (a :: int) (b ^ n) \<longleftrightarrow> coprime a b"
noschinl@52397
  1240
  using assms
noschinl@52397
  1241
proof (induct n)
noschinl@52397
  1242
  case (Suc n) then show ?case
noschinl@52397
  1243
    by (cases n) (simp_all add: coprime_mul_eq_int)
noschinl@52397
  1244
qed simp
noschinl@52397
  1245
nipkow@31952
  1246
lemma gcd_coprime_exists_nat:
huffman@31706
  1247
    assumes nz: "gcd (a::nat) b \<noteq> 0"
huffman@31706
  1248
    shows "\<exists>a' b'. a = a' * gcd a b \<and> b = b' * gcd a b \<and> coprime a' b'"
huffman@31706
  1249
  apply (rule_tac x = "a div gcd a b" in exI)
huffman@31706
  1250
  apply (rule_tac x = "b div gcd a b" in exI)
nipkow@31952
  1251
  using nz apply (auto simp add: div_gcd_coprime_nat dvd_div_mult)
huffman@31706
  1252
done
huffman@31706
  1253
nipkow@31952
  1254
lemma gcd_coprime_exists_int:
huffman@31706
  1255
    assumes nz: "gcd (a::int) b \<noteq> 0"
huffman@31706
  1256
    shows "\<exists>a' b'. a = a' * gcd a b \<and> b = b' * gcd a b \<and> coprime a' b'"
huffman@31706
  1257
  apply (rule_tac x = "a div gcd a b" in exI)
huffman@31706
  1258
  apply (rule_tac x = "b div gcd a b" in exI)
haftmann@59008
  1259
  using nz apply (auto simp add: div_gcd_coprime_int)
huffman@31706
  1260
done
huffman@31706
  1261
nipkow@31952
  1262
lemma coprime_exp_nat: "coprime (d::nat) a \<Longrightarrow> coprime d (a^n)"
haftmann@60596
  1263
  by (induct n) (simp_all add: coprime_mult_nat)
huffman@31706
  1264
nipkow@31952
  1265
lemma coprime_exp_int: "coprime (d::int) a \<Longrightarrow> coprime d (a^n)"
haftmann@60596
  1266
  by (induct n) (simp_all add: coprime_mult_int)
huffman@31706
  1267
haftmann@60690
  1268
context semiring_gcd
haftmann@60690
  1269
begin
haftmann@60690
  1270
haftmann@60690
  1271
lemma coprime_exp_left:
haftmann@60690
  1272
  assumes "coprime a b"
haftmann@60690
  1273
  shows "coprime (a ^ n) b"
haftmann@60690
  1274
  using assms by (induct n) (simp_all add: gcd_mult_cancel)
haftmann@60690
  1275
haftmann@60690
  1276
lemma coprime_exp2:
haftmann@60690
  1277
  assumes "coprime a b"
haftmann@60690
  1278
  shows "coprime (a ^ n) (b ^ m)"
haftmann@60690
  1279
proof (rule coprime_exp_left)
haftmann@60690
  1280
  from assms show "coprime a (b ^ m)"
haftmann@60690
  1281
    by (induct m) (simp_all add: gcd_mult_cancel gcd.commute [of a])
haftmann@60690
  1282
qed
haftmann@60690
  1283
haftmann@60690
  1284
end
haftmann@60690
  1285
nipkow@31952
  1286
lemma coprime_exp2_nat [intro]: "coprime (a::nat) b \<Longrightarrow> coprime (a^n) (b^m)"
haftmann@60690
  1287
  by (fact coprime_exp2)
huffman@31706
  1288
nipkow@31952
  1289
lemma coprime_exp2_int [intro]: "coprime (a::int) b \<Longrightarrow> coprime (a^n) (b^m)"
haftmann@60690
  1290
  by (fact coprime_exp2)
huffman@31706
  1291
haftmann@60690
  1292
lemma gcd_exp_nat:
haftmann@60690
  1293
  "gcd ((a :: nat) ^ n) (b ^ n) = gcd a b ^ n"
haftmann@60690
  1294
proof (cases "a = 0 \<and> b = 0")
haftmann@60690
  1295
  case True then show ?thesis by (cases "n > 0") (simp_all add: zero_power)
haftmann@60690
  1296
next
haftmann@60690
  1297
  case False
haftmann@60690
  1298
  then have "coprime (a div gcd a b) (b div gcd a b)"
haftmann@60690
  1299
    by (auto simp: div_gcd_coprime)
haftmann@60690
  1300
  then have "coprime ((a div gcd a b) ^ n) ((b div gcd a b) ^ n)"
haftmann@60690
  1301
    by (simp add: coprime_exp2)
haftmann@60690
  1302
  then have "gcd ((a div gcd a b)^n * (gcd a b)^n)
huffman@31706
  1303
      ((b div gcd a b)^n * (gcd a b)^n) = (gcd a b)^n"
lp15@60162
  1304
    by (metis gcd_mult_distrib_nat mult.commute mult.right_neutral)
huffman@31706
  1305
  also have "(a div gcd a b)^n * (gcd a b)^n = a^n"
lp15@60162
  1306
    by (metis dvd_div_mult_self gcd_unique_nat power_mult_distrib)
huffman@31706
  1307
  also have "(b div gcd a b)^n * (gcd a b)^n = b^n"
lp15@60162
  1308
    by (metis dvd_div_mult_self gcd_unique_nat power_mult_distrib)
huffman@31706
  1309
  finally show ?thesis .
huffman@31706
  1310
qed
huffman@31706
  1311
nipkow@31952
  1312
lemma gcd_exp_int: "gcd ((a::int)^n) (b^n) = (gcd a b)^n"
nipkow@31952
  1313
  apply (subst (1 2) gcd_abs_int)
huffman@31706
  1314
  apply (subst (1 2) power_abs)
nipkow@31952
  1315
  apply (rule gcd_exp_nat [where n = n, transferred])
huffman@31706
  1316
  apply auto
huffman@31706
  1317
done
huffman@31706
  1318
nipkow@31952
  1319
lemma division_decomp_nat: assumes dc: "(a::nat) dvd b * c"
huffman@31706
  1320
  shows "\<exists>b' c'. a = b' * c' \<and> b' dvd b \<and> c' dvd c"
huffman@31706
  1321
proof-
huffman@31706
  1322
  let ?g = "gcd a b"
huffman@31706
  1323
  {assume "?g = 0" with dc have ?thesis by auto}
huffman@31706
  1324
  moreover
huffman@31706
  1325
  {assume z: "?g \<noteq> 0"
nipkow@31952
  1326
    from gcd_coprime_exists_nat[OF z]
huffman@31706
  1327
    obtain a' b' where ab': "a = a' * ?g" "b = b' * ?g" "coprime a' b'"
huffman@31706
  1328
      by blast
huffman@31706
  1329
    have thb: "?g dvd b" by auto
huffman@31706
  1330
    from ab'(1) have "a' dvd a"  unfolding dvd_def by blast
huffman@31706
  1331
    with dc have th0: "a' dvd b*c" using dvd_trans[of a' a "b*c"] by simp
huffman@31706
  1332
    from dc ab'(1,2) have "a'*?g dvd (b'*?g) *c" by auto
haftmann@57512
  1333
    hence "?g*a' dvd ?g * (b' * c)" by (simp add: mult.assoc)
huffman@31706
  1334
    with z have th_1: "a' dvd b' * c" by auto
nipkow@31952
  1335
    from coprime_dvd_mult_nat[OF ab'(3)] th_1
haftmann@57512
  1336
    have thc: "a' dvd c" by (subst (asm) mult.commute, blast)
huffman@31706
  1337
    from ab' have "a = ?g*a'" by algebra
huffman@31706
  1338
    with thb thc have ?thesis by blast }
huffman@31706
  1339
  ultimately show ?thesis by blast
huffman@31706
  1340
qed
huffman@31706
  1341
nipkow@31952
  1342
lemma division_decomp_int: assumes dc: "(a::int) dvd b * c"
huffman@31706
  1343
  shows "\<exists>b' c'. a = b' * c' \<and> b' dvd b \<and> c' dvd c"
huffman@31706
  1344
proof-
huffman@31706
  1345
  let ?g = "gcd a b"
huffman@31706
  1346
  {assume "?g = 0" with dc have ?thesis by auto}
huffman@31706
  1347
  moreover
huffman@31706
  1348
  {assume z: "?g \<noteq> 0"
nipkow@31952
  1349
    from gcd_coprime_exists_int[OF z]
huffman@31706
  1350
    obtain a' b' where ab': "a = a' * ?g" "b = b' * ?g" "coprime a' b'"
huffman@31706
  1351
      by blast
huffman@31706
  1352
    have thb: "?g dvd b" by auto
huffman@31706
  1353
    from ab'(1) have "a' dvd a"  unfolding dvd_def by blast
huffman@31706
  1354
    with dc have th0: "a' dvd b*c"
huffman@31706
  1355
      using dvd_trans[of a' a "b*c"] by simp
huffman@31706
  1356
    from dc ab'(1,2) have "a'*?g dvd (b'*?g) *c" by auto
haftmann@60690
  1357
    hence "?g*a' dvd ?g * (b' * c)" by (simp add: ac_simps)
huffman@31706
  1358
    with z have th_1: "a' dvd b' * c" by auto
nipkow@31952
  1359
    from coprime_dvd_mult_int[OF ab'(3)] th_1
haftmann@57512
  1360
    have thc: "a' dvd c" by (subst (asm) mult.commute, blast)
huffman@31706
  1361
    from ab' have "a = ?g*a'" by algebra
huffman@31706
  1362
    with thb thc have ?thesis by blast }
huffman@31706
  1363
  ultimately show ?thesis by blast
chaieb@27669
  1364
qed
chaieb@27669
  1365
nipkow@31952
  1366
lemma pow_divides_pow_nat:
huffman@31706
  1367
  assumes ab: "(a::nat) ^ n dvd b ^n" and n:"n \<noteq> 0"
huffman@31706
  1368
  shows "a dvd b"
huffman@31706
  1369
proof-
huffman@31706
  1370
  let ?g = "gcd a b"
huffman@31706
  1371
  from n obtain m where m: "n = Suc m" by (cases n, simp_all)
huffman@31706
  1372
  {assume "?g = 0" with ab n have ?thesis by auto }
huffman@31706
  1373
  moreover
huffman@31706
  1374
  {assume z: "?g \<noteq> 0"
huffman@35216
  1375
    hence zn: "?g ^ n \<noteq> 0" using n by simp
nipkow@31952
  1376
    from gcd_coprime_exists_nat[OF z]
huffman@31706
  1377
    obtain a' b' where ab': "a = a' * ?g" "b = b' * ?g" "coprime a' b'"
huffman@31706
  1378
      by blast
huffman@31706
  1379
    from ab have "(a' * ?g) ^ n dvd (b' * ?g)^n"
huffman@31706
  1380
      by (simp add: ab'(1,2)[symmetric])
huffman@31706
  1381
    hence "?g^n*a'^n dvd ?g^n *b'^n"
haftmann@57512
  1382
      by (simp only: power_mult_distrib mult.commute)
haftmann@58787
  1383
    then have th0: "a'^n dvd b'^n"
haftmann@58787
  1384
      using zn by auto
huffman@31706
  1385
    have "a' dvd a'^n" by (simp add: m)
huffman@31706
  1386
    with th0 have "a' dvd b'^n" using dvd_trans[of a' "a'^n" "b'^n"] by simp
haftmann@57512
  1387
    hence th1: "a' dvd b'^m * b'" by (simp add: m mult.commute)
nipkow@31952
  1388
    from coprime_dvd_mult_nat[OF coprime_exp_nat [OF ab'(3), of m]] th1
haftmann@57512
  1389
    have "a' dvd b'" by (subst (asm) mult.commute, blast)
huffman@31706
  1390
    hence "a'*?g dvd b'*?g" by simp
huffman@31706
  1391
    with ab'(1,2)  have ?thesis by simp }
huffman@31706
  1392
  ultimately show ?thesis by blast
huffman@31706
  1393
qed
huffman@31706
  1394
nipkow@31952
  1395
lemma pow_divides_pow_int:
huffman@31706
  1396
  assumes ab: "(a::int) ^ n dvd b ^n" and n:"n \<noteq> 0"
huffman@31706
  1397
  shows "a dvd b"
chaieb@27669
  1398
proof-
huffman@31706
  1399
  let ?g = "gcd a b"
huffman@31706
  1400
  from n obtain m where m: "n = Suc m" by (cases n, simp_all)
huffman@31706
  1401
  {assume "?g = 0" with ab n have ?thesis by auto }
huffman@31706
  1402
  moreover
huffman@31706
  1403
  {assume z: "?g \<noteq> 0"
huffman@35216
  1404
    hence zn: "?g ^ n \<noteq> 0" using n by simp
nipkow@31952
  1405
    from gcd_coprime_exists_int[OF z]
huffman@31706
  1406
    obtain a' b' where ab': "a = a' * ?g" "b = b' * ?g" "coprime a' b'"
huffman@31706
  1407
      by blast
huffman@31706
  1408
    from ab have "(a' * ?g) ^ n dvd (b' * ?g)^n"
huffman@31706
  1409
      by (simp add: ab'(1,2)[symmetric])
huffman@31706
  1410
    hence "?g^n*a'^n dvd ?g^n *b'^n"
haftmann@57512
  1411
      by (simp only: power_mult_distrib mult.commute)
huffman@31706
  1412
    with zn z n have th0:"a'^n dvd b'^n" by auto
huffman@31706
  1413
    have "a' dvd a'^n" by (simp add: m)
huffman@31706
  1414
    with th0 have "a' dvd b'^n"
huffman@31706
  1415
      using dvd_trans[of a' "a'^n" "b'^n"] by simp
haftmann@60596
  1416
    hence th1: "a' dvd b'^m * b'" by (simp add: m mult.commute)
nipkow@31952
  1417
    from coprime_dvd_mult_int[OF coprime_exp_int [OF ab'(3), of m]] th1
haftmann@57512
  1418
    have "a' dvd b'" by (subst (asm) mult.commute, blast)
huffman@31706
  1419
    hence "a'*?g dvd b'*?g" by simp
huffman@31706
  1420
    with ab'(1,2)  have ?thesis by simp }
huffman@31706
  1421
  ultimately show ?thesis by blast
huffman@31706
  1422
qed
huffman@31706
  1423
nipkow@31952
  1424
lemma pow_divides_eq_nat [simp]: "n ~= 0 \<Longrightarrow> ((a::nat)^n dvd b^n) = (a dvd b)"
nipkow@31952
  1425
  by (auto intro: pow_divides_pow_nat dvd_power_same)
huffman@31706
  1426
nipkow@31952
  1427
lemma pow_divides_eq_int [simp]: "n ~= 0 \<Longrightarrow> ((a::int)^n dvd b^n) = (a dvd b)"
nipkow@31952
  1428
  by (auto intro: pow_divides_pow_int dvd_power_same)
huffman@31706
  1429
nipkow@31952
  1430
lemma divides_mult_nat:
huffman@31706
  1431
  assumes mr: "(m::nat) dvd r" and nr: "n dvd r" and mn:"coprime m n"
huffman@31706
  1432
  shows "m * n dvd r"
huffman@31706
  1433
proof-
huffman@31706
  1434
  from mr nr obtain m' n' where m': "r = m*m'" and n': "r = n*n'"
huffman@31706
  1435
    unfolding dvd_def by blast
haftmann@57512
  1436
  from mr n' have "m dvd n'*n" by (simp add: mult.commute)
nipkow@31952
  1437
  hence "m dvd n'" using coprime_dvd_mult_iff_nat[OF mn] by simp
huffman@31706
  1438
  then obtain k where k: "n' = m*k" unfolding dvd_def by blast
huffman@31706
  1439
  from n' k show ?thesis unfolding dvd_def by auto
huffman@31706
  1440
qed
huffman@31706
  1441
nipkow@31952
  1442
lemma divides_mult_int:
huffman@31706
  1443
  assumes mr: "(m::int) dvd r" and nr: "n dvd r" and mn:"coprime m n"
huffman@31706
  1444
  shows "m * n dvd r"
huffman@31706
  1445
proof-
huffman@31706
  1446
  from mr nr obtain m' n' where m': "r = m*m'" and n': "r = n*n'"
huffman@31706
  1447
    unfolding dvd_def by blast
haftmann@57512
  1448
  from mr n' have "m dvd n'*n" by (simp add: mult.commute)
nipkow@31952
  1449
  hence "m dvd n'" using coprime_dvd_mult_iff_int[OF mn] by simp
huffman@31706
  1450
  then obtain k where k: "n' = m*k" unfolding dvd_def by blast
huffman@31706
  1451
  from n' k show ?thesis unfolding dvd_def by auto
chaieb@27669
  1452
qed
chaieb@27669
  1453
nipkow@31952
  1454
lemma coprime_plus_one_nat [simp]: "coprime ((n::nat) + 1) n"
haftmann@60690
  1455
  by (simp add: gcd.commute del: One_nat_def)
huffman@31706
  1456
nipkow@31952
  1457
lemma coprime_Suc_nat [simp]: "coprime (Suc n) n"
haftmann@60690
  1458
  using coprime_plus_one_nat by simp
huffman@31706
  1459
nipkow@31952
  1460
lemma coprime_plus_one_int [simp]: "coprime ((n::int) + 1) n"
haftmann@60686
  1461
  by (simp add: gcd.commute)
huffman@31706
  1462
nipkow@31952
  1463
lemma coprime_minus_one_nat: "(n::nat) \<noteq> 0 \<Longrightarrow> coprime (n - 1) n"
nipkow@31952
  1464
  using coprime_plus_one_nat [of "n - 1"]
nipkow@31952
  1465
    gcd_commute_nat [of "n - 1" n] by auto
huffman@31706
  1466
nipkow@31952
  1467
lemma coprime_minus_one_int: "coprime ((n::int) - 1) n"
nipkow@31952
  1468
  using coprime_plus_one_int [of "n - 1"]
nipkow@31952
  1469
    gcd_commute_int [of "n - 1" n] by auto
huffman@31706
  1470
nipkow@31952
  1471
lemma setprod_coprime_nat [rule_format]:
huffman@31706
  1472
    "(ALL i: A. coprime (f i) (x::nat)) --> coprime (PROD i:A. f i) x"
huffman@31706
  1473
  apply (case_tac "finite A")
huffman@31706
  1474
  apply (induct set: finite)
nipkow@31952
  1475
  apply (auto simp add: gcd_mult_cancel_nat)
huffman@31706
  1476
done
huffman@31706
  1477
nipkow@31952
  1478
lemma setprod_coprime_int [rule_format]:
huffman@31706
  1479
    "(ALL i: A. coprime (f i) (x::int)) --> coprime (PROD i:A. f i) x"
huffman@31706
  1480
  apply (case_tac "finite A")
huffman@31706
  1481
  apply (induct set: finite)
nipkow@31952
  1482
  apply (auto simp add: gcd_mult_cancel_int)
huffman@31706
  1483
done
huffman@31706
  1484
lp15@60162
  1485
lemma coprime_common_divisor_nat: 
haftmann@60686
  1486
  "coprime (a::nat) b \<Longrightarrow> x dvd a \<Longrightarrow> x dvd b \<Longrightarrow> x = 1"
lp15@60162
  1487
  by (metis gcd_greatest_iff_nat nat_dvd_1_iff_1)
huffman@31706
  1488
lp15@60162
  1489
lemma coprime_common_divisor_int:
haftmann@60686
  1490
  "coprime (a::int) b \<Longrightarrow> x dvd a \<Longrightarrow> x dvd b \<Longrightarrow> abs x = 1"
haftmann@60686
  1491
  using gcd_greatest_iff [of x a b] by auto
huffman@31706
  1492
lp15@60162
  1493
lemma coprime_divisors_nat:
lp15@60162
  1494
    "(d::int) dvd a \<Longrightarrow> e dvd b \<Longrightarrow> coprime a b \<Longrightarrow> coprime d e"
lp15@60162
  1495
  by (meson coprime_int dvd_trans gcd_dvd1 gcd_dvd2 gcd_ge_0_int)
huffman@31706
  1496
nipkow@31952
  1497
lemma invertible_coprime_nat: "(x::nat) * y mod m = 1 \<Longrightarrow> coprime x m"
lp15@60162
  1498
by (metis coprime_lmult_nat gcd_1_nat gcd_commute_nat gcd_red_nat)
huffman@31706
  1499
nipkow@31952
  1500
lemma invertible_coprime_int: "(x::int) * y mod m = 1 \<Longrightarrow> coprime x m"
lp15@60162
  1501
by (metis coprime_lmult_int gcd_1_int gcd_commute_int gcd_red_int)
huffman@31706
  1502
huffman@31706
  1503
wenzelm@60758
  1504
subsection \<open>Bezout's theorem\<close>
huffman@31706
  1505
huffman@31706
  1506
(* Function bezw returns a pair of witnesses to Bezout's theorem --
huffman@31706
  1507
   see the theorems that follow the definition. *)
huffman@31706
  1508
fun
huffman@31706
  1509
  bezw  :: "nat \<Rightarrow> nat \<Rightarrow> int * int"
huffman@31706
  1510
where
huffman@31706
  1511
  "bezw x y =
huffman@31706
  1512
  (if y = 0 then (1, 0) else
huffman@31706
  1513
      (snd (bezw y (x mod y)),
huffman@31706
  1514
       fst (bezw y (x mod y)) - snd (bezw y (x mod y)) * int(x div y)))"
huffman@31706
  1515
huffman@31706
  1516
lemma bezw_0 [simp]: "bezw x 0 = (1, 0)" by simp
huffman@31706
  1517
huffman@31706
  1518
lemma bezw_non_0: "y > 0 \<Longrightarrow> bezw x y = (snd (bezw y (x mod y)),
huffman@31706
  1519
       fst (bezw y (x mod y)) - snd (bezw y (x mod y)) * int(x div y))"
huffman@31706
  1520
  by simp
huffman@31706
  1521
huffman@31706
  1522
declare bezw.simps [simp del]
huffman@31706
  1523
huffman@31706
  1524
lemma bezw_aux [rule_format]:
huffman@31706
  1525
    "fst (bezw x y) * int x + snd (bezw x y) * int y = int (gcd x y)"
nipkow@31952
  1526
proof (induct x y rule: gcd_nat_induct)
huffman@31706
  1527
  fix m :: nat
huffman@31706
  1528
  show "fst (bezw m 0) * int m + snd (bezw m 0) * int 0 = int (gcd m 0)"
huffman@31706
  1529
    by auto
huffman@31706
  1530
  next fix m :: nat and n
huffman@31706
  1531
    assume ngt0: "n > 0" and
huffman@31706
  1532
      ih: "fst (bezw n (m mod n)) * int n +
huffman@31706
  1533
        snd (bezw n (m mod n)) * int (m mod n) =
huffman@31706
  1534
        int (gcd n (m mod n))"
huffman@31706
  1535
    thus "fst (bezw m n) * int m + snd (bezw m n) * int n = int (gcd m n)"
nipkow@31952
  1536
      apply (simp add: bezw_non_0 gcd_non_0_nat)
huffman@31706
  1537
      apply (erule subst)
haftmann@36350
  1538
      apply (simp add: field_simps)
huffman@31706
  1539
      apply (subst mod_div_equality [of m n, symmetric])
huffman@31706
  1540
      (* applying simp here undoes the last substitution!
huffman@31706
  1541
         what is procedure cancel_div_mod? *)
hoelzl@58776
  1542
      apply (simp only: NO_MATCH_def field_simps of_nat_add of_nat_mult)
huffman@31706
  1543
      done
huffman@31706
  1544
qed
huffman@31706
  1545
nipkow@31952
  1546
lemma bezout_int:
huffman@31706
  1547
  fixes x y
huffman@31706
  1548
  shows "EX u v. u * (x::int) + v * y = gcd x y"
huffman@31706
  1549
proof -
huffman@31706
  1550
  have bezout_aux: "!!x y. x \<ge> (0::int) \<Longrightarrow> y \<ge> 0 \<Longrightarrow>
huffman@31706
  1551
      EX u v. u * x + v * y = gcd x y"
huffman@31706
  1552
    apply (rule_tac x = "fst (bezw (nat x) (nat y))" in exI)
huffman@31706
  1553
    apply (rule_tac x = "snd (bezw (nat x) (nat y))" in exI)
huffman@31706
  1554
    apply (unfold gcd_int_def)
huffman@31706
  1555
    apply simp
huffman@31706
  1556
    apply (subst bezw_aux [symmetric])
huffman@31706
  1557
    apply auto
huffman@31706
  1558
    done
huffman@31706
  1559
  have "(x \<ge> 0 \<and> y \<ge> 0) | (x \<ge> 0 \<and> y \<le> 0) | (x \<le> 0 \<and> y \<ge> 0) |
huffman@31706
  1560
      (x \<le> 0 \<and> y \<le> 0)"
huffman@31706
  1561
    by auto
huffman@31706
  1562
  moreover have "x \<ge> 0 \<Longrightarrow> y \<ge> 0 \<Longrightarrow> ?thesis"
huffman@31706
  1563
    by (erule (1) bezout_aux)
huffman@31706
  1564
  moreover have "x >= 0 \<Longrightarrow> y <= 0 \<Longrightarrow> ?thesis"
huffman@31706
  1565
    apply (insert bezout_aux [of x "-y"])
huffman@31706
  1566
    apply auto
huffman@31706
  1567
    apply (rule_tac x = u in exI)
huffman@31706
  1568
    apply (rule_tac x = "-v" in exI)
nipkow@31952
  1569
    apply (subst gcd_neg2_int [symmetric])
huffman@31706
  1570
    apply auto
huffman@31706
  1571
    done
huffman@31706
  1572
  moreover have "x <= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> ?thesis"
huffman@31706
  1573
    apply (insert bezout_aux [of "-x" y])
huffman@31706
  1574
    apply auto
huffman@31706
  1575
    apply (rule_tac x = "-u" in exI)
huffman@31706
  1576
    apply (rule_tac x = v in exI)
nipkow@31952
  1577
    apply (subst gcd_neg1_int [symmetric])
huffman@31706
  1578
    apply auto
huffman@31706
  1579
    done
huffman@31706
  1580
  moreover have "x <= 0 \<Longrightarrow> y <= 0 \<Longrightarrow> ?thesis"
huffman@31706
  1581
    apply (insert bezout_aux [of "-x" "-y"])
huffman@31706
  1582
    apply auto
huffman@31706
  1583
    apply (rule_tac x = "-u" in exI)
huffman@31706
  1584
    apply (rule_tac x = "-v" in exI)
nipkow@31952
  1585
    apply (subst gcd_neg1_int [symmetric])
nipkow@31952
  1586
    apply (subst gcd_neg2_int [symmetric])
huffman@31706
  1587
    apply auto
huffman@31706
  1588
    done
huffman@31706
  1589
  ultimately show ?thesis by blast
huffman@31706
  1590
qed
huffman@31706
  1591
wenzelm@60758
  1592
text \<open>versions of Bezout for nat, by Amine Chaieb\<close>
huffman@31706
  1593
huffman@31706
  1594
lemma ind_euclid:
huffman@31706
  1595
  assumes c: " \<forall>a b. P (a::nat) b \<longleftrightarrow> P b a" and z: "\<forall>a. P a 0"
huffman@31706
  1596
  and add: "\<forall>a b. P a b \<longrightarrow> P a (a + b)"
chaieb@27669
  1597
  shows "P a b"
berghofe@34915
  1598
proof(induct "a + b" arbitrary: a b rule: less_induct)
berghofe@34915
  1599
  case less
chaieb@27669
  1600
  have "a = b \<or> a < b \<or> b < a" by arith
chaieb@27669
  1601
  moreover {assume eq: "a= b"
huffman@31706
  1602
    from add[rule_format, OF z[rule_format, of a]] have "P a b" using eq
huffman@31706
  1603
    by simp}
chaieb@27669
  1604
  moreover
chaieb@27669
  1605
  {assume lt: "a < b"
berghofe@34915
  1606
    hence "a + b - a < a + b \<or> a = 0" by arith
chaieb@27669
  1607
    moreover
chaieb@27669
  1608
    {assume "a =0" with z c have "P a b" by blast }
chaieb@27669
  1609
    moreover
berghofe@34915
  1610
    {assume "a + b - a < a + b"
berghofe@34915
  1611
      also have th0: "a + b - a = a + (b - a)" using lt by arith
berghofe@34915
  1612
      finally have "a + (b - a) < a + b" .
berghofe@34915
  1613
      then have "P a (a + (b - a))" by (rule add[rule_format, OF less])
berghofe@34915
  1614
      then have "P a b" by (simp add: th0[symmetric])}
chaieb@27669
  1615
    ultimately have "P a b" by blast}
chaieb@27669
  1616
  moreover
chaieb@27669
  1617
  {assume lt: "a > b"
berghofe@34915
  1618
    hence "b + a - b < a + b \<or> b = 0" by arith
chaieb@27669
  1619
    moreover
chaieb@27669
  1620
    {assume "b =0" with z c have "P a b" by blast }
chaieb@27669
  1621
    moreover
berghofe@34915
  1622
    {assume "b + a - b < a + b"
berghofe@34915
  1623
      also have th0: "b + a - b = b + (a - b)" using lt by arith
berghofe@34915
  1624
      finally have "b + (a - b) < a + b" .
berghofe@34915
  1625
      then have "P b (b + (a - b))" by (rule add[rule_format, OF less])
berghofe@34915
  1626
      then have "P b a" by (simp add: th0[symmetric])
chaieb@27669
  1627
      hence "P a b" using c by blast }
chaieb@27669
  1628
    ultimately have "P a b" by blast}
chaieb@27669
  1629
ultimately  show "P a b" by blast
chaieb@27669
  1630
qed
chaieb@27669
  1631
nipkow@31952
  1632
lemma bezout_lemma_nat:
huffman@31706
  1633
  assumes ex: "\<exists>(d::nat) x y. d dvd a \<and> d dvd b \<and>
huffman@31706
  1634
    (a * x = b * y + d \<or> b * x = a * y + d)"
huffman@31706
  1635
  shows "\<exists>d x y. d dvd a \<and> d dvd a + b \<and>
huffman@31706
  1636
    (a * x = (a + b) * y + d \<or> (a + b) * x = a * y + d)"
huffman@31706
  1637
  using ex
huffman@31706
  1638
  apply clarsimp
huffman@35216
  1639
  apply (rule_tac x="d" in exI, simp)
huffman@31706
  1640
  apply (case_tac "a * x = b * y + d" , simp_all)
huffman@31706
  1641
  apply (rule_tac x="x + y" in exI)
huffman@31706
  1642
  apply (rule_tac x="y" in exI)
huffman@31706
  1643
  apply algebra
huffman@31706
  1644
  apply (rule_tac x="x" in exI)
huffman@31706
  1645
  apply (rule_tac x="x + y" in exI)
huffman@31706
  1646
  apply algebra
chaieb@27669
  1647
done
chaieb@27669
  1648
nipkow@31952
  1649
lemma bezout_add_nat: "\<exists>(d::nat) x y. d dvd a \<and> d dvd b \<and>
huffman@31706
  1650
    (a * x = b * y + d \<or> b * x = a * y + d)"
huffman@31706
  1651
  apply(induct a b rule: ind_euclid)
huffman@31706
  1652
  apply blast
huffman@31706
  1653
  apply clarify
huffman@35216
  1654
  apply (rule_tac x="a" in exI, simp)
huffman@31706
  1655
  apply clarsimp
huffman@31706
  1656
  apply (rule_tac x="d" in exI)
huffman@35216
  1657
  apply (case_tac "a * x = b * y + d", simp_all)
huffman@31706
  1658
  apply (rule_tac x="x+y" in exI)
huffman@31706
  1659
  apply (rule_tac x="y" in exI)
huffman@31706
  1660
  apply algebra
huffman@31706
  1661
  apply (rule_tac x="x" in exI)
huffman@31706
  1662
  apply (rule_tac x="x+y" in exI)
huffman@31706
  1663
  apply algebra
chaieb@27669
  1664
done
chaieb@27669
  1665
nipkow@31952
  1666
lemma bezout1_nat: "\<exists>(d::nat) x y. d dvd a \<and> d dvd b \<and>
huffman@31706
  1667
    (a * x - b * y = d \<or> b * x - a * y = d)"
nipkow@31952
  1668
  using bezout_add_nat[of a b]
huffman@31706
  1669
  apply clarsimp
huffman@31706
  1670
  apply (rule_tac x="d" in exI, simp)
huffman@31706
  1671
  apply (rule_tac x="x" in exI)
huffman@31706
  1672
  apply (rule_tac x="y" in exI)
huffman@31706
  1673
  apply auto
chaieb@27669
  1674
done
chaieb@27669
  1675
nipkow@31952
  1676
lemma bezout_add_strong_nat: assumes nz: "a \<noteq> (0::nat)"
chaieb@27669
  1677
  shows "\<exists>d x y. d dvd a \<and> d dvd b \<and> a * x = b * y + d"
chaieb@27669
  1678
proof-
huffman@31706
  1679
 from nz have ap: "a > 0" by simp
nipkow@31952
  1680
 from bezout_add_nat[of a b]
huffman@31706
  1681
 have "(\<exists>d x y. d dvd a \<and> d dvd b \<and> a * x = b * y + d) \<or>
huffman@31706
  1682
   (\<exists>d x y. d dvd a \<and> d dvd b \<and> b * x = a * y + d)" by blast
chaieb@27669
  1683
 moreover
huffman@31706
  1684
    {fix d x y assume H: "d dvd a" "d dvd b" "a * x = b * y + d"
huffman@31706
  1685
     from H have ?thesis by blast }
chaieb@27669
  1686
 moreover
chaieb@27669
  1687
 {fix d x y assume H: "d dvd a" "d dvd b" "b * x = a * y + d"
chaieb@27669
  1688
   {assume b0: "b = 0" with H  have ?thesis by simp}
huffman@31706
  1689
   moreover
chaieb@27669
  1690
   {assume b: "b \<noteq> 0" hence bp: "b > 0" by simp
huffman@31706
  1691
     from b dvd_imp_le [OF H(2)] have "d < b \<or> d = b"
huffman@31706
  1692
       by auto
chaieb@27669
  1693
     moreover
chaieb@27669
  1694
     {assume db: "d=b"
wenzelm@41550
  1695
       with nz H have ?thesis apply simp
wenzelm@32960
  1696
         apply (rule exI[where x = b], simp)
wenzelm@32960
  1697
         apply (rule exI[where x = b])
wenzelm@32960
  1698
        by (rule exI[where x = "a - 1"], simp add: diff_mult_distrib2)}
chaieb@27669
  1699
    moreover
huffman@31706
  1700
    {assume db: "d < b"
wenzelm@41550
  1701
        {assume "x=0" hence ?thesis using nz H by simp }
wenzelm@32960
  1702
        moreover
wenzelm@32960
  1703
        {assume x0: "x \<noteq> 0" hence xp: "x > 0" by simp
wenzelm@32960
  1704
          from db have "d \<le> b - 1" by simp
wenzelm@32960
  1705
          hence "d*b \<le> b*(b - 1)" by simp
wenzelm@32960
  1706
          with xp mult_mono[of "1" "x" "d*b" "b*(b - 1)"]
wenzelm@32960
  1707
          have dble: "d*b \<le> x*b*(b - 1)" using bp by simp
wenzelm@32960
  1708
          from H (3) have "d + (b - 1) * (b*x) = d + (b - 1) * (a*y + d)"
huffman@31706
  1709
            by simp
wenzelm@32960
  1710
          hence "d + (b - 1) * a * y + (b - 1) * d = d + (b - 1) * b * x"
haftmann@57512
  1711
            by (simp only: mult.assoc distrib_left)
wenzelm@32960
  1712
          hence "a * ((b - 1) * y) + d * (b - 1 + 1) = d + x*b*(b - 1)"
huffman@31706
  1713
            by algebra
wenzelm@32960
  1714
          hence "a * ((b - 1) * y) = d + x*b*(b - 1) - d*b" using bp by simp
wenzelm@32960
  1715
          hence "a * ((b - 1) * y) = d + (x*b*(b - 1) - d*b)"
wenzelm@32960
  1716
            by (simp only: diff_add_assoc[OF dble, of d, symmetric])
wenzelm@32960
  1717
          hence "a * ((b - 1) * y) = b*(x*(b - 1) - d) + d"
haftmann@59008
  1718
            by (simp only: diff_mult_distrib2 ac_simps)
wenzelm@32960
  1719
          hence ?thesis using H(1,2)
wenzelm@32960
  1720
            apply -
wenzelm@32960
  1721
            apply (rule exI[where x=d], simp)
wenzelm@32960
  1722
            apply (rule exI[where x="(b - 1) * y"])
wenzelm@32960
  1723
            by (rule exI[where x="x*(b - 1) - d"], simp)}
wenzelm@32960
  1724
        ultimately have ?thesis by blast}
chaieb@27669
  1725
    ultimately have ?thesis by blast}
chaieb@27669
  1726
  ultimately have ?thesis by blast}
chaieb@27669
  1727
 ultimately show ?thesis by blast
chaieb@27669
  1728
qed
chaieb@27669
  1729
nipkow@31952
  1730
lemma bezout_nat: assumes a: "(a::nat) \<noteq> 0"
chaieb@27669
  1731
  shows "\<exists>x y. a * x = b * y + gcd a b"
chaieb@27669
  1732
proof-
chaieb@27669
  1733
  let ?g = "gcd a b"
nipkow@31952
  1734
  from bezout_add_strong_nat[OF a, of b]
chaieb@27669
  1735
  obtain d x y where d: "d dvd a" "d dvd b" "a * x = b * y + d" by blast
chaieb@27669
  1736
  from d(1,2) have "d dvd ?g" by simp
chaieb@27669
  1737
  then obtain k where k: "?g = d*k" unfolding dvd_def by blast
huffman@31706
  1738
  from d(3) have "a * x * k = (b * y + d) *k " by auto
chaieb@27669
  1739
  hence "a * (x * k) = b * (y*k) + ?g" by (algebra add: k)
chaieb@27669
  1740
  thus ?thesis by blast
chaieb@27669
  1741
qed
chaieb@27669
  1742
huffman@31706
  1743
wenzelm@60758
  1744
subsection \<open>LCM properties\<close>
huffman@31706
  1745
haftmann@34030
  1746
lemma lcm_altdef_int [code]: "lcm (a::int) b = (abs a) * (abs b) div gcd a b"
huffman@31706
  1747
  by (simp add: lcm_int_def lcm_nat_def zdiv_int
huffman@44821
  1748
    of_nat_mult gcd_int_def)
huffman@31706
  1749
nipkow@31952
  1750
lemma prod_gcd_lcm_nat: "(m::nat) * n = gcd m n * lcm m n"
huffman@31706
  1751
  unfolding lcm_nat_def
nipkow@31952
  1752
  by (simp add: dvd_mult_div_cancel [OF gcd_dvd_prod_nat])
huffman@31706
  1753
nipkow@31952
  1754
lemma prod_gcd_lcm_int: "abs(m::int) * abs n = gcd m n * lcm m n"
huffman@31706
  1755
  unfolding lcm_int_def gcd_int_def
huffman@31706
  1756
  apply (subst int_mult [symmetric])
nipkow@31952
  1757
  apply (subst prod_gcd_lcm_nat [symmetric])
huffman@31706
  1758
  apply (subst nat_abs_mult_distrib [symmetric])
huffman@31706
  1759
  apply (simp, simp add: abs_mult)
huffman@31706
  1760
done
huffman@31706
  1761
nipkow@31952
  1762
lemma lcm_0_nat [simp]: "lcm (m::nat) 0 = 0"
huffman@31706
  1763
  unfolding lcm_nat_def by simp
huffman@31706
  1764
nipkow@31952
  1765
lemma lcm_0_int [simp]: "lcm (m::int) 0 = 0"
huffman@31706
  1766
  unfolding lcm_int_def by simp
huffman@31706
  1767
nipkow@31952
  1768
lemma lcm_0_left_nat [simp]: "lcm (0::nat) n = 0"
huffman@31706
  1769
  unfolding lcm_nat_def by simp
chaieb@27669
  1770
nipkow@31952
  1771
lemma lcm_0_left_int [simp]: "lcm (0::int) n = 0"
huffman@31706
  1772
  unfolding lcm_int_def by simp
huffman@31706
  1773
nipkow@31952
  1774
lemma lcm_pos_nat:
nipkow@31798
  1775
  "(m::nat) > 0 \<Longrightarrow> n>0 \<Longrightarrow> lcm m n > 0"
nipkow@31952
  1776
by (metis gr0I mult_is_0 prod_gcd_lcm_nat)
chaieb@27669
  1777
nipkow@31952
  1778
lemma lcm_pos_int:
nipkow@31798
  1779
  "(m::int) ~= 0 \<Longrightarrow> n ~= 0 \<Longrightarrow> lcm m n > 0"
nipkow@31952
  1780
  apply (subst lcm_abs_int)
nipkow@31952
  1781
  apply (rule lcm_pos_nat [transferred])
nipkow@31798
  1782
  apply auto
huffman@31706
  1783
done
haftmann@23687
  1784
nipkow@31952
  1785
lemma dvd_pos_nat:
haftmann@23687
  1786
  fixes n m :: nat
haftmann@23687
  1787
  assumes "n > 0" and "m dvd n"
haftmann@23687
  1788
  shows "m > 0"
haftmann@23687
  1789
using assms by (cases m) auto
haftmann@23687
  1790
nipkow@31952
  1791
lemma lcm_least_nat:
huffman@31706
  1792
  assumes "(m::nat) dvd k" and "n dvd k"
haftmann@27556
  1793
  shows "lcm m n dvd k"
haftmann@60686
  1794
  using assms by (rule lcm_least)
haftmann@23687
  1795
nipkow@31952
  1796
lemma lcm_least_int:
nipkow@31798
  1797
  "(m::int) dvd k \<Longrightarrow> n dvd k \<Longrightarrow> lcm m n dvd k"
haftmann@60686
  1798
  by (rule lcm_least)
huffman@31706
  1799
nipkow@31952
  1800
lemma lcm_dvd1_nat: "(m::nat) dvd lcm m n"
haftmann@60689
  1801
  by (fact dvd_lcm1)
haftmann@23687
  1802
nipkow@31952
  1803
lemma lcm_dvd1_int: "(m::int) dvd lcm m n"
haftmann@60689
  1804
  by (fact dvd_lcm1)
huffman@31706
  1805
nipkow@31952
  1806
lemma lcm_dvd2_nat: "(n::nat) dvd lcm m n"
haftmann@60689
  1807
  by (fact dvd_lcm2)
huffman@31706
  1808
nipkow@31952
  1809
lemma lcm_dvd2_int: "(n::int) dvd lcm m n"
haftmann@60689
  1810
  by (fact dvd_lcm2)
huffman@31706
  1811
nipkow@31730
  1812
lemma dvd_lcm_I1_nat[simp]: "(k::nat) dvd m \<Longrightarrow> k dvd lcm m n"
nipkow@31952
  1813
by(metis lcm_dvd1_nat dvd_trans)
nipkow@31729
  1814
nipkow@31730
  1815
lemma dvd_lcm_I2_nat[simp]: "(k::nat) dvd n \<Longrightarrow> k dvd lcm m n"
nipkow@31952
  1816
by(metis lcm_dvd2_nat dvd_trans)
nipkow@31729
  1817
nipkow@31730
  1818
lemma dvd_lcm_I1_int[simp]: "(i::int) dvd m \<Longrightarrow> i dvd lcm m n"
nipkow@31952
  1819
by(metis lcm_dvd1_int dvd_trans)
nipkow@31729
  1820
nipkow@31730
  1821
lemma dvd_lcm_I2_int[simp]: "(i::int) dvd n \<Longrightarrow> i dvd lcm m n"
nipkow@31952
  1822
by(metis lcm_dvd2_int dvd_trans)
nipkow@31729
  1823
nipkow@31952
  1824
lemma lcm_unique_nat: "(a::nat) dvd d \<and> b dvd d \<and>
huffman@31706
  1825
    (\<forall>e. a dvd e \<and> b dvd e \<longrightarrow> d dvd e) \<longleftrightarrow> d = lcm a b"
nipkow@33657
  1826
  by (auto intro: dvd_antisym lcm_least_nat lcm_dvd1_nat lcm_dvd2_nat)
chaieb@27568
  1827
nipkow@31952
  1828
lemma lcm_unique_int: "d >= 0 \<and> (a::int) dvd d \<and> b dvd d \<and>
huffman@31706
  1829
    (\<forall>e. a dvd e \<and> b dvd e \<longrightarrow> d dvd e) \<longleftrightarrow> d = lcm a b"
wenzelm@60357
  1830
  using lcm_least_int zdvd_antisym_nonneg by auto
huffman@31706
  1831
haftmann@37770
  1832
interpretation lcm_nat: abel_semigroup "lcm :: nat \<Rightarrow> nat \<Rightarrow> nat"
haftmann@54867
  1833
  + lcm_nat: semilattice_neutr "lcm :: nat \<Rightarrow> nat \<Rightarrow> nat" 1
haftmann@60690
  1834
  by standard (simp_all del: One_nat_def)
haftmann@60686
  1835
haftmann@60686
  1836
interpretation lcm_int: abel_semigroup "lcm :: int \<Rightarrow> int \<Rightarrow> int" ..
haftmann@34973
  1837
haftmann@60686
  1838
lemmas lcm_assoc_nat = lcm.assoc [where ?'a = nat]
haftmann@60686
  1839
lemmas lcm_commute_nat = lcm.commute [where ?'a = nat]
haftmann@60686
  1840
lemmas lcm_left_commute_nat = lcm.left_commute [where ?'a = nat]
haftmann@60686
  1841
lemmas lcm_assoc_int = lcm.assoc [where ?'a = int]
haftmann@60686
  1842
lemmas lcm_commute_int = lcm.commute [where ?'a = int]
haftmann@60686
  1843
lemmas lcm_left_commute_int = lcm.left_commute [where ?'a = int]
haftmann@34973
  1844
haftmann@34973
  1845
lemmas lcm_ac_nat = lcm_assoc_nat lcm_commute_nat lcm_left_commute_nat
haftmann@34973
  1846
lemmas lcm_ac_int = lcm_assoc_int lcm_commute_int lcm_left_commute_int
haftmann@34973
  1847
nipkow@31798
  1848
lemma lcm_proj2_if_dvd_nat [simp]: "(x::nat) dvd y \<Longrightarrow> lcm x y = y"
huffman@31706
  1849
  apply (rule sym)
nipkow@31952
  1850
  apply (subst lcm_unique_nat [symmetric])
huffman@31706
  1851
  apply auto
huffman@31706
  1852
done
huffman@31706
  1853
nipkow@31798
  1854
lemma lcm_proj2_if_dvd_int [simp]: "(x::int) dvd y \<Longrightarrow> lcm x y = abs y"
huffman@31706
  1855
  apply (rule sym)
nipkow@31952
  1856
  apply (subst lcm_unique_int [symmetric])
huffman@31706
  1857
  apply auto
huffman@31706
  1858
done
huffman@31706
  1859
nipkow@31798
  1860
lemma lcm_proj1_if_dvd_nat [simp]: "(x::nat) dvd y \<Longrightarrow> lcm y x = y"
nipkow@31952
  1861
by (subst lcm_commute_nat, erule lcm_proj2_if_dvd_nat)
huffman@31706
  1862
nipkow@31798
  1863
lemma lcm_proj1_if_dvd_int [simp]: "(x::int) dvd y \<Longrightarrow> lcm y x = abs y"
nipkow@31952
  1864
by (subst lcm_commute_int, erule lcm_proj2_if_dvd_int)
huffman@31706
  1865
nipkow@31992
  1866
lemma lcm_proj1_iff_nat[simp]: "lcm m n = (m::nat) \<longleftrightarrow> n dvd m"
nipkow@31992
  1867
by (metis lcm_proj1_if_dvd_nat lcm_unique_nat)
nipkow@31992
  1868
nipkow@31992
  1869
lemma lcm_proj2_iff_nat[simp]: "lcm m n = (n::nat) \<longleftrightarrow> m dvd n"
nipkow@31992
  1870
by (metis lcm_proj2_if_dvd_nat lcm_unique_nat)
nipkow@31992
  1871
nipkow@31992
  1872
lemma lcm_proj1_iff_int[simp]: "lcm m n = abs(m::int) \<longleftrightarrow> n dvd m"
nipkow@31992
  1873
by (metis dvd_abs_iff lcm_proj1_if_dvd_int lcm_unique_int)
nipkow@31992
  1874
nipkow@31992
  1875
lemma lcm_proj2_iff_int[simp]: "lcm m n = abs(n::int) \<longleftrightarrow> m dvd n"
nipkow@31992
  1876
by (metis dvd_abs_iff lcm_proj2_if_dvd_int lcm_unique_int)
chaieb@27568
  1877
haftmann@42871
  1878
lemma comp_fun_idem_gcd_nat: "comp_fun_idem (gcd :: nat\<Rightarrow>nat\<Rightarrow>nat)"
nipkow@31992
  1879
proof qed (auto simp add: gcd_ac_nat)
nipkow@31992
  1880
haftmann@42871
  1881
lemma comp_fun_idem_gcd_int: "comp_fun_idem (gcd :: int\<Rightarrow>int\<Rightarrow>int)"
nipkow@31992
  1882
proof qed (auto simp add: gcd_ac_int)
nipkow@31992
  1883
haftmann@42871
  1884
lemma comp_fun_idem_lcm_nat: "comp_fun_idem (lcm :: nat\<Rightarrow>nat\<Rightarrow>nat)"
nipkow@31992
  1885
proof qed (auto simp add: lcm_ac_nat)
nipkow@31992
  1886
haftmann@42871
  1887
lemma comp_fun_idem_lcm_int: "comp_fun_idem (lcm :: int\<Rightarrow>int\<Rightarrow>int)"
nipkow@31992
  1888
proof qed (auto simp add: lcm_ac_int)
nipkow@31992
  1889
haftmann@23687
  1890
nipkow@31995
  1891
(* FIXME introduce selimattice_bot/top and derive the following lemmas in there: *)
nipkow@31995
  1892
nipkow@31995
  1893
lemma lcm_0_iff_nat[simp]: "lcm (m::nat) n = 0 \<longleftrightarrow> m=0 \<or> n=0"
nipkow@31995
  1894
by (metis lcm_0_left_nat lcm_0_nat mult_is_0 prod_gcd_lcm_nat)
nipkow@31995
  1895
nipkow@31995
  1896
lemma lcm_0_iff_int[simp]: "lcm (m::int) n = 0 \<longleftrightarrow> m=0 \<or> n=0"
huffman@44766
  1897
by (metis lcm_0_int lcm_0_left_int lcm_pos_int less_le)
nipkow@31995
  1898
nipkow@31995
  1899
lemma lcm_1_iff_nat[simp]: "lcm (m::nat) n = 1 \<longleftrightarrow> m=1 \<and> n=1"
nipkow@31995
  1900
by (metis gcd_1_nat lcm_unique_nat nat_mult_1 prod_gcd_lcm_nat)
nipkow@31995
  1901
nipkow@31995
  1902
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
  1903
by (auto simp add: abs_mult_self trans [OF lcm_unique_int eq_commute, symmetric] zmult_eq_1_iff)
nipkow@31995
  1904
haftmann@34030
  1905
wenzelm@60758
  1906
subsection \<open>The complete divisibility lattice\<close>
nipkow@32112
  1907
wenzelm@60580
  1908
interpretation gcd_semilattice_nat: semilattice_inf gcd "op dvd" "(\<lambda>m n::nat. m dvd n \<and> \<not> n dvd m)"
haftmann@60686
  1909
  by standard simp_all
nipkow@32112
  1910
wenzelm@60580
  1911
interpretation lcm_semilattice_nat: semilattice_sup lcm "op dvd" "(\<lambda>m n::nat. m dvd n \<and> \<not> n dvd m)"
haftmann@60686
  1912
  by standard simp_all
nipkow@32112
  1913
wenzelm@60580
  1914
interpretation gcd_lcm_lattice_nat: lattice gcd "op dvd" "(\<lambda>m n::nat. m dvd n & ~ n dvd m)" lcm ..
nipkow@32112
  1915
wenzelm@60758
  1916
text\<open>Lifting gcd and lcm to sets (Gcd/Lcm).
huffman@45264
  1917
Gcd is defined via Lcm to facilitate the proof that we have a complete lattice.
wenzelm@60758
  1918
\<close>
huffman@45264
  1919
huffman@45264
  1920
instantiation nat :: Gcd
nipkow@32112
  1921
begin
nipkow@32112
  1922
huffman@45264
  1923
definition
haftmann@51489
  1924
  "Lcm (M::nat set) = (if finite M then semilattice_neutr_set.F lcm 1 M else 0)"
haftmann@51489
  1925
haftmann@54867
  1926
interpretation semilattice_neutr_set lcm "1::nat" ..
haftmann@54867
  1927
haftmann@51489
  1928
lemma Lcm_nat_infinite:
haftmann@51489
  1929
  "\<not> finite M \<Longrightarrow> Lcm M = (0::nat)"
haftmann@51489
  1930
  by (simp add: Lcm_nat_def)
haftmann@51489
  1931
haftmann@51489
  1932
lemma Lcm_nat_empty:
haftmann@51489
  1933
  "Lcm {} = (1::nat)"
haftmann@60690
  1934
  by (simp add: Lcm_nat_def del: One_nat_def)
haftmann@51489
  1935
haftmann@51489
  1936
lemma Lcm_nat_insert:
haftmann@51489
  1937
  "Lcm (insert n M) = lcm (n::nat) (Lcm M)"
haftmann@60690
  1938
  by (cases "finite M") (simp_all add: Lcm_nat_def Lcm_nat_infinite del: One_nat_def)
nipkow@32112
  1939
huffman@45264
  1940
definition
huffman@45264
  1941
  "Gcd (M::nat set) = Lcm {d. \<forall>m\<in>M. d dvd m}"
nipkow@32112
  1942
huffman@45264
  1943
instance ..
haftmann@51489
  1944
nipkow@32112
  1945
end
nipkow@32112
  1946
huffman@45264
  1947
lemma dvd_Lcm_nat [simp]:
haftmann@51489
  1948
  fixes M :: "nat set"
haftmann@51489
  1949
  assumes "m \<in> M"
haftmann@51489
  1950
  shows "m dvd Lcm M"
haftmann@51489
  1951
proof (cases "finite M")
haftmann@51489
  1952
  case False then show ?thesis by (simp add: Lcm_nat_infinite)
haftmann@51489
  1953
next
haftmann@51489
  1954
  case True then show ?thesis using assms by (induct M) (auto simp add: Lcm_nat_insert)
haftmann@51489
  1955
qed
nipkow@32112
  1956
huffman@45264
  1957
lemma Lcm_dvd_nat [simp]:
haftmann@51489
  1958
  fixes M :: "nat set"
haftmann@51489
  1959
  assumes "\<forall>m\<in>M. m dvd n"
haftmann@51489
  1960
  shows "Lcm M dvd n"
huffman@45264
  1961
proof (cases "n = 0")
huffman@45264
  1962
  assume "n \<noteq> 0"
huffman@45264
  1963
  hence "finite {d. d dvd n}" by (rule finite_divisors_nat)
huffman@45264
  1964
  moreover have "M \<subseteq> {d. d dvd n}" using assms by fast
huffman@45264
  1965
  ultimately have "finite M" by (rule rev_finite_subset)
haftmann@51489
  1966
  then show ?thesis using assms by (induct M) (simp_all add: Lcm_nat_empty Lcm_nat_insert)
huffman@45264
  1967
qed simp
nipkow@32112
  1968
huffman@45264
  1969
interpretation gcd_lcm_complete_lattice_nat:
haftmann@51547
  1970
  complete_lattice Gcd Lcm gcd Rings.dvd "\<lambda>m n. m dvd n \<and> \<not> n dvd m" lcm 1 "0::nat"
ballarin@61566
  1971
rewrites "Inf.INFIMUM Gcd A f = Gcd (f ` A :: nat set)"
haftmann@56218
  1972
  and "Sup.SUPREMUM Lcm A f = Lcm (f ` A)"
haftmann@51547
  1973
proof -
haftmann@51547
  1974
  show "class.complete_lattice Gcd Lcm gcd Rings.dvd (\<lambda>m n. m dvd n \<and> \<not> n dvd m) lcm 1 (0::nat)"
wenzelm@61169
  1975
    by standard (auto simp add: Gcd_nat_def Lcm_nat_empty Lcm_nat_infinite)
haftmann@51547
  1976
  then interpret gcd_lcm_complete_lattice_nat:
haftmann@51547
  1977
    complete_lattice Gcd Lcm gcd Rings.dvd "\<lambda>m n. m dvd n \<and> \<not> n dvd m" lcm 1 "0::nat" .
haftmann@56218
  1978
  from gcd_lcm_complete_lattice_nat.INF_def show "Inf.INFIMUM Gcd A f = Gcd (f ` A)" .
haftmann@56218
  1979
  from gcd_lcm_complete_lattice_nat.SUP_def show "Sup.SUPREMUM Lcm A f = Lcm (f ` A)" .
huffman@45264
  1980
qed
nipkow@32112
  1981
haftmann@56166
  1982
declare gcd_lcm_complete_lattice_nat.Inf_image_eq [simp del]
haftmann@56166
  1983
declare gcd_lcm_complete_lattice_nat.Sup_image_eq [simp del]
haftmann@56166
  1984
huffman@45264
  1985
lemma Lcm_empty_nat: "Lcm {} = (1::nat)"
haftmann@54867
  1986
  by (fact Lcm_nat_empty)
huffman@45264
  1987
nipkow@32112
  1988
lemma Lcm_insert_nat [simp]:
nipkow@32112
  1989
  shows "Lcm (insert (n::nat) N) = lcm n (Lcm N)"
huffman@45264
  1990
  by (fact gcd_lcm_complete_lattice_nat.Sup_insert)
nipkow@32112
  1991
nipkow@32112
  1992
lemma Lcm0_iff[simp]: "finite (M::nat set) \<Longrightarrow> M \<noteq> {} \<Longrightarrow> Lcm M = 0 \<longleftrightarrow> 0 : M"
nipkow@32112
  1993
by(induct rule:finite_ne_induct) auto
nipkow@32112
  1994
nipkow@32112
  1995
lemma Lcm_eq_0[simp]: "finite (M::nat set) \<Longrightarrow> 0 : M \<Longrightarrow> Lcm M = 0"
nipkow@32112
  1996
by (metis Lcm0_iff empty_iff)
nipkow@32112
  1997
haftmann@60596
  1998
instance nat :: semiring_Gcd
haftmann@60596
  1999
proof
haftmann@60596
  2000
  show "Gcd N dvd n" if "n \<in> N" for N and n :: nat
haftmann@60596
  2001
    using that by (fact gcd_lcm_complete_lattice_nat.Inf_lower)
haftmann@60596
  2002
next
haftmann@60686
  2003
  show "n dvd Gcd N" if "\<And>m. m \<in> N \<Longrightarrow> n dvd m" for N and n :: nat
haftmann@60596
  2004
    using that by (simp only: gcd_lcm_complete_lattice_nat.Inf_greatest)
haftmann@60596
  2005
next
haftmann@60686
  2006
  show "normalize (Gcd N) = Gcd N" for N :: "nat set"
haftmann@60596
  2007
    by simp
haftmann@60596
  2008
qed
nipkow@32112
  2009
haftmann@60686
  2010
instance nat :: semiring_Lcm
haftmann@60686
  2011
proof
haftmann@60686
  2012
  have uf: "unit_factor (Lcm N) = 1" if "0 < Lcm N" for N :: "nat set"
haftmann@60686
  2013
  proof (cases "finite N")
haftmann@60686
  2014
    case False with that show ?thesis by (simp add: Lcm_nat_infinite)
haftmann@60686
  2015
  next
haftmann@60686
  2016
    case True then show ?thesis
haftmann@60686
  2017
    using that proof (induct N)
haftmann@60686
  2018
      case empty then show ?case by simp
haftmann@60686
  2019
    next
haftmann@60686
  2020
      case (insert n N)
haftmann@60686
  2021
      have "lcm n (Lcm N) \<noteq> 0 \<longleftrightarrow> n \<noteq> 0 \<and> Lcm N \<noteq> 0"
haftmann@60686
  2022
        using lcm_eq_0_iff [of n "Lcm N"] by simp
haftmann@60686
  2023
      then have "lcm n (Lcm N) > 0 \<longleftrightarrow> n > 0 \<and> Lcm N > 0"
haftmann@60686
  2024
        unfolding neq0_conv .
haftmann@60686
  2025
      with insert show ?case
haftmann@60686
  2026
        by (simp add: Lcm_nat_insert unit_factor_lcm)
haftmann@60686
  2027
    qed
haftmann@60686
  2028
  qed
haftmann@60686
  2029
  show "Lcm N = Gcd {m. \<forall>n\<in>N. n dvd m}" for N :: "nat set"
haftmann@60686
  2030
    by (rule associated_eqI) (auto intro!: associatedI Gcd_dvd Gcd_greatest
haftmann@60686
  2031
      simp add: unit_factor_Gcd uf)
haftmann@60686
  2032
qed
haftmann@60686
  2033
wenzelm@60758
  2034
text\<open>Alternative characterizations of Gcd:\<close>
nipkow@32112
  2035
nipkow@32112
  2036
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
  2037
apply(rule antisym)
nipkow@32112
  2038
 apply(rule Max_ge)
nipkow@32112
  2039
  apply (metis all_not_in_conv finite_divisors_nat finite_INT)
haftmann@60596
  2040
 apply (simp add: Gcd_dvd)
nipkow@32112
  2041
apply (rule Max_le_iff[THEN iffD2])
nipkow@32112
  2042
  apply (metis all_not_in_conv finite_divisors_nat finite_INT)
nipkow@44890
  2043
 apply fastforce
nipkow@32112
  2044
apply clarsimp
haftmann@60596
  2045
apply (metis Gcd_dvd Max_in dvd_0_left dvd_Gcd dvd_imp_le linorder_antisym_conv3 not_less0)
nipkow@32112
  2046
done
nipkow@32112
  2047
nipkow@32112
  2048
lemma Gcd_remove0_nat: "finite M \<Longrightarrow> Gcd M = Gcd (M - {0::nat})"
nipkow@32112
  2049
apply(induct pred:finite)
nipkow@32112
  2050
 apply simp
nipkow@32112
  2051
apply(case_tac "x=0")
nipkow@32112
  2052
 apply simp
nipkow@32112
  2053
apply(subgoal_tac "insert x F - {0} = insert x (F - {0})")
nipkow@32112
  2054
 apply simp
nipkow@32112
  2055
apply blast
nipkow@32112
  2056
done
nipkow@32112
  2057
nipkow@32112
  2058
lemma Lcm_in_lcm_closed_set_nat:
nipkow@32112
  2059
  "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
  2060
apply(induct rule:finite_linorder_min_induct)
nipkow@32112
  2061
 apply simp
nipkow@32112
  2062
apply simp
nipkow@32112
  2063
apply(subgoal_tac "ALL m n :: nat. m:A \<longrightarrow> n:A \<longrightarrow> lcm m n : A")
nipkow@32112
  2064
 apply simp
nipkow@32112
  2065
 apply(case_tac "A={}")
nipkow@32112
  2066
  apply simp
nipkow@32112
  2067
 apply simp
nipkow@32112
  2068
apply (metis lcm_pos_nat lcm_unique_nat linorder_neq_iff nat_dvd_not_less not_less0)
nipkow@32112
  2069
done
nipkow@32112
  2070
nipkow@32112
  2071
lemma Lcm_eq_Max_nat:
nipkow@32112
  2072
  "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
  2073
apply(rule antisym)
nipkow@32112
  2074
 apply(rule Max_ge, assumption)
nipkow@32112
  2075
 apply(erule (2) Lcm_in_lcm_closed_set_nat)
nipkow@32112
  2076
apply clarsimp
nipkow@32112
  2077
apply (metis Lcm0_iff dvd_Lcm_nat dvd_imp_le neq0_conv)
nipkow@32112
  2078
done
nipkow@32112
  2079
haftmann@54437
  2080
lemma Lcm_set_nat [code, code_unfold]:
haftmann@45992
  2081
  "Lcm (set ns) = fold lcm ns (1::nat)"
huffman@45264
  2082
  by (fact gcd_lcm_complete_lattice_nat.Sup_set_fold)
nipkow@32112
  2083
haftmann@60597
  2084
lemma Gcd_set_nat [code]:
haftmann@45992
  2085
  "Gcd (set ns) = fold gcd ns (0::nat)"
huffman@45264
  2086
  by (fact gcd_lcm_complete_lattice_nat.Inf_set_fold)
nipkow@34222
  2087
nipkow@34222
  2088
lemma mult_inj_if_coprime_nat:
nipkow@34222
  2089
  "inj_on f A \<Longrightarrow> inj_on g B \<Longrightarrow> ALL a:A. ALL b:B. coprime (f a) (g b)
nipkow@34222
  2090
   \<Longrightarrow> inj_on (%(a,b). f a * g b::nat) (A \<times> B)"
haftmann@60690
  2091
apply (auto simp add: inj_on_def simp del: One_nat_def)
huffman@35216
  2092
apply (metis coprime_dvd_mult_iff_nat dvd.neq_le_trans dvd_triv_left)
nipkow@34223
  2093
apply (metis gcd_semilattice_nat.inf_commute coprime_dvd_mult_iff_nat
haftmann@57512
  2094
             dvd.neq_le_trans dvd_triv_right mult.commute)
nipkow@34222
  2095
done
nipkow@34222
  2096
wenzelm@60758
  2097
text\<open>Nitpick:\<close>
nipkow@34222
  2098
blanchet@41792
  2099
lemma gcd_eq_nitpick_gcd [nitpick_unfold]: "gcd x y = Nitpick.nat_gcd x y"
blanchet@41792
  2100
by (induct x y rule: nat_gcd.induct)
blanchet@41792
  2101
   (simp add: gcd_nat.simps Nitpick.nat_gcd.simps)
blanchet@33197
  2102
blanchet@41792
  2103
lemma lcm_eq_nitpick_lcm [nitpick_unfold]: "lcm x y = Nitpick.nat_lcm x y"
blanchet@33197
  2104
by (simp only: lcm_nat_def Nitpick.nat_lcm_def gcd_eq_nitpick_gcd)
blanchet@33197
  2105
haftmann@54867
  2106
wenzelm@60758
  2107
subsubsection \<open>Setwise gcd and lcm for integers\<close>
huffman@45264
  2108
huffman@45264
  2109
instantiation int :: Gcd
huffman@45264
  2110
begin
huffman@45264
  2111
huffman@45264
  2112
definition
huffman@45264
  2113
  "Lcm M = int (Lcm (nat ` abs ` M))"
huffman@45264
  2114
huffman@45264
  2115
definition
huffman@45264
  2116
  "Gcd M = int (Gcd (nat ` abs ` M))"
huffman@45264
  2117
huffman@45264
  2118
instance ..
haftmann@60686
  2119
wenzelm@21256
  2120
end
huffman@45264
  2121
haftmann@60686
  2122
instance int :: semiring_Gcd
haftmann@60686
  2123
  by standard (auto intro!: Gcd_dvd Gcd_greatest simp add: Gcd_int_def Lcm_int_def int_dvd_iff dvd_int_iff
haftmann@60686
  2124
    dvd_int_unfold_dvd_nat [symmetric])
haftmann@60686
  2125
haftmann@60686
  2126
instance int :: semiring_Lcm
haftmann@60686
  2127
proof
haftmann@60686
  2128
  fix K :: "int set"
haftmann@60686
  2129
  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
  2130
  proof (rule set_eqI)
haftmann@60686
  2131
    fix n
haftmann@60686
  2132
    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
  2133
    proof
haftmann@60686
  2134
      assume ?P
haftmann@60686
  2135
      then have "(\<forall>k\<in>K. k dvd int n) \<and> n = nat \<bar>int n\<bar>"
haftmann@60686
  2136
        by (auto simp add: dvd_int_unfold_dvd_nat)
haftmann@60686
  2137
      then show ?Q by blast
haftmann@60686
  2138
    next
haftmann@60686
  2139
      assume ?Q then show ?P
haftmann@60686
  2140
        by (auto simp add: dvd_int_unfold_dvd_nat)
haftmann@60686
  2141
    qed
haftmann@60686
  2142
    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
  2143
      by auto
haftmann@60686
  2144
  qed
haftmann@60686
  2145
  then show "Lcm K = Gcd {l. \<forall>k\<in>K. k dvd l}"
haftmann@60686
  2146
    by (simp add: Gcd_int_def Lcm_int_def Lcm_Gcd)
haftmann@60686
  2147
qed
haftmann@60686
  2148
huffman@45264
  2149
lemma Lcm_empty_int [simp]: "Lcm {} = (1::int)"
huffman@45264
  2150
  by (simp add: Lcm_int_def)
huffman@45264
  2151
huffman@45264
  2152
lemma Lcm_insert_int [simp]:
huffman@45264
  2153
  shows "Lcm (insert (n::int) N) = lcm n (Lcm N)"
huffman@45264
  2154
  by (simp add: Lcm_int_def lcm_int_def)
huffman@45264
  2155
huffman@45264
  2156
lemma dvd_int_iff: "x dvd y \<longleftrightarrow> nat (abs x) dvd nat (abs y)"
haftmann@60686
  2157
  by (fact dvd_int_unfold_dvd_nat)
huffman@45264
  2158
huffman@45264
  2159
lemma dvd_Lcm_int [simp]:
huffman@45264
  2160
  fixes M :: "int set" assumes "m \<in> M" shows "m dvd Lcm M"
huffman@45264
  2161
  using assms by (simp add: Lcm_int_def dvd_int_iff)
huffman@45264
  2162
huffman@45264
  2163
lemma Lcm_dvd_int [simp]:
huffman@45264
  2164
  fixes M :: "int set"
huffman@45264
  2165
  assumes "\<forall>m\<in>M. m dvd n" shows "Lcm M dvd n"
huffman@45264
  2166
  using assms by (simp add: Lcm_int_def dvd_int_iff)
huffman@45264
  2167
haftmann@54437
  2168
lemma Lcm_set_int [code, code_unfold]:
haftmann@51547
  2169
  "Lcm (set xs) = fold lcm xs (1::int)"
haftmann@56166
  2170
  by (induct xs rule: rev_induct) (simp_all add: lcm_commute_int)
huffman@45264
  2171
haftmann@60597
  2172
lemma Gcd_set_int [code]:
haftmann@51547
  2173
  "Gcd (set xs) = fold gcd xs (0::int)"
haftmann@56166
  2174
  by (induct xs rule: rev_induct) (simp_all add: gcd_commute_int)
huffman@45264
  2175
haftmann@59008
  2176
haftmann@59008
  2177
text \<open>Fact aliasses\<close>
lp15@59667
  2178
lp15@59667
  2179
lemmas gcd_dvd1_nat = gcd_dvd1 [where ?'a = nat]
haftmann@59008
  2180
  and gcd_dvd2_nat = gcd_dvd2 [where ?'a = nat]
haftmann@59008
  2181
  and gcd_greatest_nat = gcd_greatest [where ?'a = nat]
haftmann@59008
  2182
lp15@59667
  2183
lemmas gcd_dvd1_int = gcd_dvd1 [where ?'a = int]
haftmann@59008
  2184
  and gcd_dvd2_int = gcd_dvd2 [where ?'a = int]
haftmann@59008
  2185
  and gcd_greatest_int = gcd_greatest [where ?'a = int]
haftmann@59008
  2186
haftmann@60596
  2187
lemmas Gcd_dvd_nat [simp] = Gcd_dvd [where ?'a = nat]
haftmann@60596
  2188
  and dvd_Gcd_nat [simp] = dvd_Gcd [where ?'a = nat]
haftmann@60596
  2189
haftmann@60596
  2190
lemmas Gcd_dvd_int [simp] = Gcd_dvd [where ?'a = int]
haftmann@60596
  2191
  and dvd_Gcd_int [simp] = dvd_Gcd [where ?'a = int]
haftmann@60596
  2192
haftmann@60596
  2193
lemmas Gcd_empty_nat = Gcd_empty [where ?'a = nat]
haftmann@60596
  2194
  and Gcd_insert_nat = Gcd_insert [where ?'a = nat]
haftmann@60596
  2195
haftmann@60596
  2196
lemmas Gcd_empty_int = Gcd_empty [where ?'a = int]
haftmann@60596
  2197
  and Gcd_insert_int = Gcd_insert [where ?'a = int]
haftmann@60596
  2198
Andreas@61856
  2199
subsection \<open>gcd and lcm instances for @{typ integer}\<close>
Andreas@61856
  2200
Andreas@61856
  2201
instantiation integer :: gcd begin
Andreas@61856
  2202
context includes integer.lifting begin
Andreas@61856
  2203
lift_definition gcd_integer :: "integer \<Rightarrow> integer \<Rightarrow> integer" is gcd .
Andreas@61856
  2204
lift_definition lcm_integer :: "integer \<Rightarrow> integer \<Rightarrow> integer" is lcm .
huffman@45264
  2205
end
Andreas@61856
  2206
instance ..
Andreas@61856
  2207
end
Andreas@61856
  2208
lifting_update integer.lifting
Andreas@61856
  2209
lifting_forget integer.lifting
Andreas@61856
  2210
Andreas@61856
  2211
context includes integer.lifting begin
Andreas@61856
  2212
Andreas@61856
  2213
lemma gcd_code_integer [code]:
Andreas@61856
  2214
  "gcd k l = \<bar>if l = (0::integer) then k else gcd l (\<bar>k\<bar> mod \<bar>l\<bar>)\<bar>"
Andreas@61856
  2215
by transfer(fact gcd_code_int)
Andreas@61856
  2216
Andreas@61856
  2217
lemma lcm_code_integer [code]: "lcm (a::integer) b = (abs a) * (abs b) div gcd a b"
Andreas@61856
  2218
by transfer(fact lcm_altdef_int)
Andreas@61856
  2219
Andreas@61856
  2220
end
Andreas@61856
  2221
Andreas@61856
  2222
code_printing constant "gcd :: integer \<Rightarrow> _"
Andreas@61856
  2223
  \<rightharpoonup> (OCaml) "Big'_int.gcd'_big'_int"
Andreas@61856
  2224
  and (Haskell) "Prelude.gcd"
Andreas@61856
  2225
  and (Scala) "_.gcd'((_)')"
Andreas@61856
  2226
  -- \<open>There is no gcd operation in the SML standard library, so no code setup for SML\<close>
Andreas@61856
  2227
Andreas@61856
  2228
end