src/HOL/GCD.thy
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(*  Authors:    Christophe Tabacznyj, Lawrence C. Paulson, Amine Chaieb,
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                Thomas M. Rasmussen, Jeremy Avigad, Tobias Nipkow
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This file deals with the functions gcd and lcm.  Definitions and
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lemmas are proved uniformly for the natural numbers and integers.
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This file combines and revises a number of prior developments.
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The original theories "GCD" and "Primes" were by Christophe Tabacznyj
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and Lawrence C. Paulson, based on @{cite davenport92}. They introduced
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gcd, lcm, and prime for the natural numbers.
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The original theory "IntPrimes" was by Thomas M. Rasmussen, and
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extended gcd, lcm, primes to the integers. Amine Chaieb provided
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another extension of the notions to the integers, and added a number
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of results to "Primes" and "GCD". IntPrimes also defined and developed
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the congruence relations on the integers. The notion was extended to
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the natural numbers by Chaieb.
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Jeremy Avigad combined all of these, made everything uniform for the
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natural numbers and the integers, and added a number of new theorems.
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Tobias Nipkow cleaned up a lot.
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*)
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section \<open>Greatest common divisor and least common multiple\<close>
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theory GCD
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imports Main
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begin
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subsection \<open>Abstract GCD and LCM\<close>
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class gcd = zero + one + dvd +
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  fixes gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
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    and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
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begin
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abbreviation coprime :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
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  where "coprime x y \<equiv> gcd x y = 1"
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end
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class Gcd = gcd +
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  fixes Gcd :: "'a set \<Rightarrow> 'a"
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    and Lcm :: "'a set \<Rightarrow> 'a"
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begin
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abbreviation GREATEST_COMMON_DIVISOR :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a"
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where
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  "GREATEST_COMMON_DIVISOR A f \<equiv> Gcd (f ` A)"
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abbreviation LEAST_COMMON_MULTIPLE :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a"
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where
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  "LEAST_COMMON_MULTIPLE A f \<equiv> Lcm (f ` A)"
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end
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syntax
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  "_GCD1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3GCD _./ _)" [0, 10] 10)
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  "_GCD"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3GCD _\<in>_./ _)" [0, 0, 10] 10)
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  "_LCM1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3LCM _./ _)" [0, 10] 10)
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  "_LCM"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3LCM _\<in>_./ _)" [0, 0, 10] 10)
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translations
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  "GCD x y. B"   \<rightleftharpoons> "GCD x. GCD y. B"
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  "GCD x. B"     \<rightleftharpoons> "CONST GREATEST_COMMON_DIVISOR CONST UNIV (\<lambda>x. B)"
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  "GCD x. B"     \<rightleftharpoons> "GCD x \<in> CONST UNIV. B"
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  "GCD x\<in>A. B"   \<rightleftharpoons> "CONST GREATEST_COMMON_DIVISOR A (\<lambda>x. B)"
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  "LCM x y. B"   \<rightleftharpoons> "LCM x. LCM y. B"
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  "LCM x. B"     \<rightleftharpoons> "CONST LEAST_COMMON_MULTIPLE CONST UNIV (\<lambda>x. B)"
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  "LCM x. B"     \<rightleftharpoons> "LCM x \<in> CONST UNIV. B"
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  "LCM x\<in>A. B"   \<rightleftharpoons> "CONST LEAST_COMMON_MULTIPLE A (\<lambda>x. B)"
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print_translation \<open>
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  [Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax GREATEST_COMMON_DIVISOR} @{syntax_const "_GCD"},
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    Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax LEAST_COMMON_MULTIPLE} @{syntax_const "_LCM"}]
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\<close> \<comment> \<open>to avoid eta-contraction of body\<close>
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class semiring_gcd = normalization_semidom + gcd +
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  assumes gcd_dvd1 [iff]: "gcd a b dvd a"
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    and gcd_dvd2 [iff]: "gcd a b dvd b"
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    and gcd_greatest: "c dvd a \<Longrightarrow> c dvd b \<Longrightarrow> c dvd gcd a b"
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    and normalize_gcd [simp]: "normalize (gcd a b) = gcd a b"
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    and lcm_gcd: "lcm a b = normalize (a * b) div gcd a b"
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begin    
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lemma gcd_greatest_iff [simp]:
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  "a dvd gcd b c \<longleftrightarrow> a dvd b \<and> a dvd c"
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  by (blast intro!: gcd_greatest intro: dvd_trans)
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lemma gcd_dvdI1:
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  "a dvd c \<Longrightarrow> gcd a b dvd c"
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  by (rule dvd_trans) (rule gcd_dvd1)
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lemma gcd_dvdI2:
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  "b dvd c \<Longrightarrow> gcd a b dvd c"
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  by (rule dvd_trans) (rule gcd_dvd2)
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lemma dvd_gcdD1:
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  "a dvd gcd b c \<Longrightarrow> a dvd b"
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  using gcd_dvd1 [of b c] by (blast intro: dvd_trans)
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lemma dvd_gcdD2:
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  "a dvd gcd b c \<Longrightarrow> a dvd c"
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  using gcd_dvd2 [of b c] by (blast intro: dvd_trans)
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lemma gcd_0_left [simp]:
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  "gcd 0 a = normalize a"
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  by (rule associated_eqI) simp_all
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lemma gcd_0_right [simp]:
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  "gcd a 0 = normalize a"
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  by (rule associated_eqI) simp_all
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lemma gcd_eq_0_iff [simp]:
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  "gcd a b = 0 \<longleftrightarrow> a = 0 \<and> b = 0" (is "?P \<longleftrightarrow> ?Q")
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proof
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  assume ?P then have "0 dvd gcd a b" by simp
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  then have "0 dvd a" and "0 dvd b" by (blast intro: dvd_trans)+
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  then show ?Q by simp
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next
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  assume ?Q then show ?P by simp
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qed
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lemma unit_factor_gcd:
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  "unit_factor (gcd a b) = (if a = 0 \<and> b = 0 then 0 else 1)"
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proof (cases "gcd a b = 0")
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  case True then show ?thesis by simp
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next
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  case False
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  have "unit_factor (gcd a b) * normalize (gcd a b) = gcd a b"
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    by (rule unit_factor_mult_normalize)
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  then have "unit_factor (gcd a b) * gcd a b = gcd a b"
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    by simp
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  then have "unit_factor (gcd a b) * gcd a b div gcd a b = gcd a b div gcd a b"
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    by simp
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  with False show ?thesis by simp
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qed
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lemma is_unit_gcd [simp]:
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  "is_unit (gcd a b) \<longleftrightarrow> coprime a b"
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  by (cases "a = 0 \<and> b = 0") (auto simp add: unit_factor_gcd dest: is_unit_unit_factor)
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sublocale gcd: abel_semigroup gcd
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proof
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   149
  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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   159
    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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   167
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lemma gcd_self [simp]:
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  "gcd a a = normalize a"
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   170
proof -
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   171
  have "a dvd gcd a a"
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   172
    by (rule gcd_greatest) simp_all
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   173
  then show ?thesis
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    by (auto intro: associated_eqI)
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qed
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lemma gcd_left_idem [simp]:
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  "gcd a (gcd a b) = gcd a b"
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  by (auto intro: associated_eqI)
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lemma gcd_right_idem [simp]:
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  "gcd (gcd a b) b = gcd a b"
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  unfolding gcd.commute [of a] gcd.commute [of "gcd b a"] by simp
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   184
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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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   188
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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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   195
proof (cases "c = 0")
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   196
  case True then show ?thesis by simp
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next
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   198
  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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   205
  then show ?thesis by (simp add: normalize_mult)
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   206
qed
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   207
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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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   215
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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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   219
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lemma dvd_lcm1 [iff]:
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  "a dvd lcm a b"
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   222
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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   225
  then show ?thesis
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   226
    by (simp add: lcm_gcd)
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qed
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   228
  
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lemma dvd_lcm2 [iff]:
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  "b dvd lcm a b"
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   231
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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   234
  then show ?thesis
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   235
    by (simp add: lcm_gcd)
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   236
qed
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   237
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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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   241
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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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   245
62345
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   246
lemma lcm_dvdD1:
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  "lcm a b dvd c \<Longrightarrow> a dvd c"
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  using dvd_lcm1 [of a b] by (blast intro: dvd_trans)
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   249
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   250
lemma lcm_dvdD2:
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  "lcm a b dvd c \<Longrightarrow> b dvd c"
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  using dvd_lcm2 [of a b] by (blast intro: dvd_trans)
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   253
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lemma lcm_least:
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  assumes "a dvd c" and "b dvd c"
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   256
  shows "lcm a b dvd c"
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   257
proof (cases "c = 0")
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   258
  case True then show ?thesis by simp
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   259
next
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   260
  case False then have U: "is_unit (unit_factor c)" by simp
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   261
  show ?thesis
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   262
  proof (cases "gcd a b = 0")
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   263
    case True with assms show ?thesis by simp
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   264
  next
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   265
    case False then have "a \<noteq> 0 \<or> b \<noteq> 0" by simp
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   266
    with \<open>c \<noteq> 0\<close> assms have "a * b dvd a * c" "a * b dvd c * b"
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   267
      by (simp_all add: mult_dvd_mono)
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   268
    then have "normalize (a * b) dvd gcd (a * c) (b * c)"
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   269
      by (auto intro: gcd_greatest simp add: ac_simps)
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   270
    then have "normalize (a * b) dvd gcd (a * c) (b * c) * unit_factor c"
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   271
      using U by (simp add: dvd_mult_unit_iff)
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   272
    then have "normalize (a * b) dvd gcd a b * c"
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   273
      by (simp add: mult_gcd_right [of a b c])
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diff changeset
   274
    then have "normalize (a * b) div gcd a b dvd c"
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diff changeset
   275
      using False by (simp add: div_dvd_iff_mult ac_simps)
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diff changeset
   276
    then show ?thesis by (simp add: lcm_gcd)
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   277
  qed
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   278
qed
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diff changeset
   279
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   280
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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   282
  by (blast intro!: lcm_least intro: dvd_trans)
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   283
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   284
lemma normalize_lcm [simp]:
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  "normalize (lcm a b) = lcm a b"
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   286
  by (simp add: lcm_gcd dvd_normalize_div)
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   287
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   288
lemma lcm_0_left [simp]:
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   289
  "lcm 0 a = 0"
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   290
  by (simp add: lcm_gcd)
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   291
  
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   292
lemma lcm_0_right [simp]:
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   293
  "lcm a 0 = 0"
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   294
  by (simp add: lcm_gcd)
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diff changeset
   295
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   296
lemma lcm_eq_0_iff:
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   297
  "lcm a b = 0 \<longleftrightarrow> a = 0 \<or> b = 0" (is "?P \<longleftrightarrow> ?Q")
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diff changeset
   298
proof
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diff changeset
   299
  assume ?P then have "0 dvd lcm a b" by simp
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   300
  then have "0 dvd normalize (a * b) div gcd a b"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   301
    by (simp add: lcm_gcd)
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   302
  then have "0 * gcd a b dvd normalize (a * b)"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   303
    using dvd_div_iff_mult [of "gcd a b" _ 0] by (cases "gcd a b = 0") simp_all
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   304
  then have "normalize (a * b) = 0"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   305
    by simp
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   306
  then show ?Q by simp
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   307
next
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   308
  assume ?Q then show ?P by auto
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   309
qed
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   310
61913
58b153bfa737 tuned proofs and augmented some lemmas
haftmann
parents: 61856
diff changeset
   311
lemma lcm_eq_1_iff [simp]:
58b153bfa737 tuned proofs and augmented some lemmas
haftmann
parents: 61856
diff changeset
   312
  "lcm a b = 1 \<longleftrightarrow> is_unit a \<and> is_unit b"
58b153bfa737 tuned proofs and augmented some lemmas
haftmann
parents: 61856
diff changeset
   313
  by (auto intro: associated_eqI)
58b153bfa737 tuned proofs and augmented some lemmas
haftmann
parents: 61856
diff changeset
   314
60686
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   315
lemma unit_factor_lcm :
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   316
  "unit_factor (lcm a b) = (if a = 0 \<or> b = 0 then 0 else 1)"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   317
  by (simp add: unit_factor_gcd dvd_unit_factor_div lcm_gcd)
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   318
61605
1bf7b186542e qualifier is mandatory by default;
wenzelm
parents: 61566
diff changeset
   319
sublocale lcm: abel_semigroup lcm
60686
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   320
proof
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   321
  fix a b c
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   322
  show "lcm a b = lcm b a"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   323
    by (simp add: lcm_gcd ac_simps normalize_mult dvd_normalize_div)
60688
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60687
diff changeset
   324
  have "lcm (lcm a b) c dvd lcm a (lcm b c)"
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60687
diff changeset
   325
    and "lcm a (lcm b c) dvd lcm (lcm a b) c"
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60687
diff changeset
   326
    by (auto intro: lcm_least
60686
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   327
      dvd_trans [of b "lcm b c" "lcm a (lcm b c)"]
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   328
      dvd_trans [of c "lcm b c" "lcm a (lcm b c)"]
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   329
      dvd_trans [of a "lcm a b" "lcm (lcm a b) c"]
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   330
      dvd_trans [of b "lcm a b" "lcm (lcm a b) c"])
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   331
  then show "lcm (lcm a b) c = lcm a (lcm b c)"
60688
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60687
diff changeset
   332
    by (rule associated_eqI) simp_all
60686
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   333
qed
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   334
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   335
lemma lcm_self [simp]:
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   336
  "lcm a a = normalize a"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   337
proof -
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   338
  have "lcm a a dvd a"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   339
    by (rule lcm_least) simp_all
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   340
  then show ?thesis
60688
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60687
diff changeset
   341
    by (auto intro: associated_eqI)
60686
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   342
qed
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   343
61913
58b153bfa737 tuned proofs and augmented some lemmas
haftmann
parents: 61856
diff changeset
   344
lemma lcm_left_idem [simp]:
58b153bfa737 tuned proofs and augmented some lemmas
haftmann
parents: 61856
diff changeset
   345
  "lcm a (lcm a b) = lcm a b"
58b153bfa737 tuned proofs and augmented some lemmas
haftmann
parents: 61856
diff changeset
   346
  by (auto intro: associated_eqI)
58b153bfa737 tuned proofs and augmented some lemmas
haftmann
parents: 61856
diff changeset
   347
58b153bfa737 tuned proofs and augmented some lemmas
haftmann
parents: 61856
diff changeset
   348
lemma lcm_right_idem [simp]:
58b153bfa737 tuned proofs and augmented some lemmas
haftmann
parents: 61856
diff changeset
   349
  "lcm (lcm a b) b = lcm a b"
58b153bfa737 tuned proofs and augmented some lemmas
haftmann
parents: 61856
diff changeset
   350
  unfolding lcm.commute [of a] lcm.commute [of "lcm b a"] by simp
58b153bfa737 tuned proofs and augmented some lemmas
haftmann
parents: 61856
diff changeset
   351
60686
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   352
lemma gcd_mult_lcm [simp]:
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   353
  "gcd a b * lcm a b = normalize a * normalize b"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   354
  by (simp add: lcm_gcd normalize_mult)
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   355
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   356
lemma lcm_mult_gcd [simp]:
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   357
  "lcm a b * gcd a b = normalize a * normalize b"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   358
  using gcd_mult_lcm [of a b] by (simp add: ac_simps) 
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   359
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   360
lemma gcd_lcm:
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   361
  assumes "a \<noteq> 0" and "b \<noteq> 0"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   362
  shows "gcd a b = normalize (a * b) div lcm a b"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   363
proof -
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   364
  from assms have "lcm a b \<noteq> 0"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   365
    by (simp add: lcm_eq_0_iff)
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   366
  have "gcd a b * lcm a b = normalize a * normalize b" by simp
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   367
  then have "gcd a b * lcm a b div lcm a b = normalize (a * b) div lcm a b"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   368
    by (simp_all add: normalize_mult)
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   369
  with \<open>lcm a b \<noteq> 0\<close> show ?thesis
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   370
    using nonzero_mult_divide_cancel_right [of "lcm a b" "gcd a b"] by simp
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   371
qed
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   372
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   373
lemma lcm_1_left [simp]:
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   374
  "lcm 1 a = normalize a"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   375
  by (simp add: lcm_gcd)
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   376
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   377
lemma lcm_1_right [simp]:
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   378
  "lcm a 1 = normalize a"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   379
  by (simp add: lcm_gcd)
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   380
  
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   381
lemma lcm_mult_left:
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   382
  "lcm (c * a) (c * b) = normalize c * lcm a b"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   383
  by (cases "c = 0")
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   384
    (simp_all add: gcd_mult_right lcm_gcd div_mult_swap normalize_mult ac_simps,
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   385
      simp add: dvd_div_mult2_eq mult.left_commute [of "normalize c", symmetric])
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   386
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   387
lemma lcm_mult_right:
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   388
  "lcm (a * c) (b * c) = lcm b a * normalize c"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   389
  using lcm_mult_left [of c a b] by (simp add: ac_simps)
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   390
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   391
lemma mult_lcm_left:
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   392
  "c * lcm a b = unit_factor c * lcm (c * a) (c * b)"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   393
  by (simp add: lcm_mult_left mult.assoc [symmetric])
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   394
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   395
lemma mult_lcm_right:
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   396
  "lcm a b * c = lcm (a * c) (b * c) * unit_factor c"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   397
  using mult_lcm_left [of c a b] by (simp add: ac_simps)
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   398
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   399
lemma gcdI:
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   400
  assumes "c dvd a" and "c dvd b" and greatest: "\<And>d. d dvd a \<Longrightarrow> d dvd b \<Longrightarrow> d dvd c"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   401
    and "normalize c = c"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   402
  shows "c = gcd a b"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   403
  by (rule associated_eqI) (auto simp: assms intro: gcd_greatest)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   404
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   405
lemma gcd_unique: "d dvd a \<and> d dvd b \<and> 
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   406
    normalize d = d \<and>
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   407
    (\<forall>e. e dvd a \<and> e dvd b \<longrightarrow> e dvd d) \<longleftrightarrow> d = gcd a b"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   408
  by rule (auto intro: gcdI simp: gcd_greatest)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   409
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   410
lemma gcd_dvd_prod: "gcd a b dvd k * b"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   411
  using mult_dvd_mono [of 1] by auto
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   412
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   413
lemma gcd_proj2_if_dvd: "b dvd a \<Longrightarrow> gcd a b = normalize b"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   414
  by (rule gcdI [symmetric]) simp_all
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   415
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   416
lemma gcd_proj1_if_dvd: "a dvd b \<Longrightarrow> gcd a b = normalize a"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   417
  by (rule gcdI [symmetric]) simp_all
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   418
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   419
lemma gcd_proj1_iff: "gcd m n = normalize m \<longleftrightarrow> m dvd n"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   420
proof
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   421
  assume A: "gcd m n = normalize m"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   422
  show "m dvd n"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   423
  proof (cases "m = 0")
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   424
    assume [simp]: "m \<noteq> 0"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   425
    from A have B: "m = gcd m n * unit_factor m"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   426
      by (simp add: unit_eq_div2)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   427
    show ?thesis by (subst B, simp add: mult_unit_dvd_iff)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   428
  qed (insert A, simp)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   429
next
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   430
  assume "m dvd n"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   431
  then show "gcd m n = normalize m" by (rule gcd_proj1_if_dvd)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   432
qed
60686
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   433
  
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   434
lemma gcd_proj2_iff: "gcd m n = normalize n \<longleftrightarrow> n dvd m"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   435
  using gcd_proj1_iff [of n m] by (simp add: ac_simps)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   436
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   437
lemma gcd_mult_distrib': "normalize c * gcd a b = gcd (c * a) (c * b)"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   438
  by (rule gcdI) (auto simp: normalize_mult gcd_greatest mult_dvd_mono gcd_mult_left[symmetric])
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   439
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   440
lemma gcd_mult_distrib:
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   441
  "k * gcd a b = gcd (k * a) (k * b) * unit_factor k"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   442
proof-
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   443
  have "normalize k * gcd a b = gcd (k * a) (k * b)"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   444
    by (simp add: gcd_mult_distrib')
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   445
  then have "normalize k * gcd a b * unit_factor k = gcd (k * a) (k * b) * unit_factor k"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   446
    by simp
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   447
  then have "normalize k * unit_factor k * gcd a b  = gcd (k * a) (k * b) * unit_factor k"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   448
    by (simp only: ac_simps)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   449
  then show ?thesis
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   450
    by simp
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   451
qed
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   452
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   453
lemma lcm_mult_unit1:
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   454
  "is_unit a \<Longrightarrow> lcm (b * a) c = lcm b c"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   455
  by (rule associated_eqI) (simp_all add: mult_unit_dvd_iff dvd_lcmI1)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   456
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   457
lemma lcm_mult_unit2:
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   458
  "is_unit a \<Longrightarrow> lcm b (c * a) = lcm b c"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   459
  using lcm_mult_unit1 [of a c b] by (simp add: ac_simps)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   460
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   461
lemma lcm_div_unit1:
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   462
  "is_unit a \<Longrightarrow> lcm (b div a) c = lcm b c"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   463
  by (erule is_unitE [of _ b]) (simp add: lcm_mult_unit1) 
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   464
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   465
lemma lcm_div_unit2:
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   466
  "is_unit a \<Longrightarrow> lcm b (c div a) = lcm b c"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   467
  by (erule is_unitE [of _ c]) (simp add: lcm_mult_unit2)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   468
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   469
lemma normalize_lcm_left [simp]:
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   470
  "lcm (normalize a) b = lcm a b"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   471
proof (cases "a = 0")
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   472
  case True then show ?thesis
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   473
    by simp
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   474
next
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   475
  case False then have "is_unit (unit_factor a)"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   476
    by simp
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   477
  moreover have "normalize a = a div unit_factor a"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   478
    by simp
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   479
  ultimately show ?thesis
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   480
    by (simp only: lcm_div_unit1)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   481
qed
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   482
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   483
lemma normalize_lcm_right [simp]:
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   484
  "lcm a (normalize b) = lcm a b"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   485
  using normalize_lcm_left [of b a] by (simp add: ac_simps)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   486
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   487
lemma gcd_mult_unit1: "is_unit a \<Longrightarrow> gcd (b * a) c = gcd b c"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   488
  apply (rule gcdI)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   489
  apply simp_all
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   490
  apply (rule dvd_trans, rule gcd_dvd1, simp add: unit_simps)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   491
  done
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   492
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   493
lemma gcd_mult_unit2: "is_unit a \<Longrightarrow> gcd b (c * a) = gcd b c"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   494
  by (subst gcd.commute, subst gcd_mult_unit1, assumption, rule gcd.commute)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   495
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   496
lemma gcd_div_unit1: "is_unit a \<Longrightarrow> gcd (b div a) c = gcd b c"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   497
  by (erule is_unitE [of _ b]) (simp add: gcd_mult_unit1)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   498
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   499
lemma gcd_div_unit2: "is_unit a \<Longrightarrow> gcd b (c div a) = gcd b c"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   500
  by (erule is_unitE [of _ c]) (simp add: gcd_mult_unit2)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   501
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   502
lemma normalize_gcd_left [simp]:
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   503
  "gcd (normalize a) b = gcd a b"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   504
proof (cases "a = 0")
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   505
  case True then show ?thesis
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   506
    by simp
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   507
next
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   508
  case False then have "is_unit (unit_factor a)"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   509
    by simp
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   510
  moreover have "normalize a = a div unit_factor a"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   511
    by simp
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   512
  ultimately show ?thesis
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   513
    by (simp only: gcd_div_unit1)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   514
qed
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   515
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   516
lemma normalize_gcd_right [simp]:
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   517
  "gcd a (normalize b) = gcd a b"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   518
  using normalize_gcd_left [of b a] by (simp add: ac_simps)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   519
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   520
lemma comp_fun_idem_gcd: "comp_fun_idem gcd"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   521
  by standard (simp_all add: fun_eq_iff ac_simps)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   522
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   523
lemma comp_fun_idem_lcm: "comp_fun_idem lcm"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   524
  by standard (simp_all add: fun_eq_iff ac_simps)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   525
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   526
lemma gcd_dvd_antisym:
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   527
  "gcd a b dvd gcd c d \<Longrightarrow> gcd c d dvd gcd a b \<Longrightarrow> gcd a b = gcd c d"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   528
proof (rule gcdI)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   529
  assume A: "gcd a b dvd gcd c d" and B: "gcd c d dvd gcd a b"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   530
  have "gcd c d dvd c" by simp
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   531
  with A show "gcd a b dvd c" by (rule dvd_trans)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   532
  have "gcd c d dvd d" by simp
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   533
  with A show "gcd a b dvd d" by (rule dvd_trans)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   534
  show "normalize (gcd a b) = gcd a b"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   535
    by simp
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   536
  fix l assume "l dvd c" and "l dvd d"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   537
  hence "l dvd gcd c d" by (rule gcd_greatest)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   538
  from this and B show "l dvd gcd a b" by (rule dvd_trans)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   539
qed
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   540
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   541
lemma coprime_dvd_mult:
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   542
  assumes "coprime a b" and "a dvd c * b"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   543
  shows "a dvd c"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   544
proof (cases "c = 0")
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   545
  case True then show ?thesis by simp
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   546
next
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   547
  case False
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   548
  then have unit: "is_unit (unit_factor c)" by simp
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   549
  from \<open>coprime a b\<close> mult_gcd_left [of c a b]
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   550
  have "gcd (c * a) (c * b) * unit_factor c = c"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   551
    by (simp add: ac_simps)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   552
  moreover from \<open>a dvd c * b\<close> have "a dvd gcd (c * a) (c * b) * unit_factor c"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   553
    by (simp add: dvd_mult_unit_iff unit)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   554
  ultimately show ?thesis by simp
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   555
qed
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   556
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   557
lemma coprime_dvd_mult_iff:
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   558
  assumes "coprime a c"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   559
  shows "a dvd b * c \<longleftrightarrow> a dvd b"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   560
  using assms by (auto intro: coprime_dvd_mult)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   561
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   562
lemma gcd_mult_cancel:
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   563
  "coprime c b \<Longrightarrow> gcd (c * a) b = gcd a b"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   564
  apply (rule associated_eqI)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   565
  apply (rule gcd_greatest)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   566
  apply (rule_tac b = c in coprime_dvd_mult)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   567
  apply (simp add: gcd.assoc)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   568
  apply (simp_all add: ac_simps)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   569
  done
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   570
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   571
lemma coprime_crossproduct:
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   572
  fixes a b c d
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   573
  assumes "coprime a d" and "coprime b c"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   574
  shows "normalize a * normalize c = normalize b * normalize d
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   575
    \<longleftrightarrow> normalize a = normalize b \<and> normalize c = normalize d" (is "?lhs \<longleftrightarrow> ?rhs")
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   576
proof
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   577
  assume ?rhs then show ?lhs by simp
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   578
next
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   579
  assume ?lhs
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   580
  from \<open>?lhs\<close> have "normalize a dvd normalize b * normalize d"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   581
    by (auto intro: dvdI dest: sym)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   582
  with \<open>coprime a d\<close> have "a dvd b"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   583
    by (simp add: coprime_dvd_mult_iff normalize_mult [symmetric])
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   584
  from \<open>?lhs\<close> have "normalize b dvd normalize a * normalize c"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   585
    by (auto intro: dvdI dest: sym)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   586
  with \<open>coprime b c\<close> have "b dvd a"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   587
    by (simp add: coprime_dvd_mult_iff normalize_mult [symmetric])
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   588
  from \<open>?lhs\<close> have "normalize c dvd normalize d * normalize b"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   589
    by (auto intro: dvdI dest: sym simp add: mult.commute)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   590
  with \<open>coprime b c\<close> have "c dvd d"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   591
    by (simp add: coprime_dvd_mult_iff gcd.commute normalize_mult [symmetric])
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   592
  from \<open>?lhs\<close> have "normalize d dvd normalize c * normalize a"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   593
    by (auto intro: dvdI dest: sym simp add: mult.commute)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   594
  with \<open>coprime a d\<close> have "d dvd c"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   595
    by (simp add: coprime_dvd_mult_iff gcd.commute normalize_mult [symmetric])
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   596
  from \<open>a dvd b\<close> \<open>b dvd a\<close> have "normalize a = normalize b"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   597
    by (rule associatedI)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   598
  moreover from \<open>c dvd d\<close> \<open>d dvd c\<close> have "normalize c = normalize d"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   599
    by (rule associatedI)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   600
  ultimately show ?rhs ..
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   601
qed
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   602
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   603
lemma gcd_add1 [simp]: "gcd (m + n) n = gcd m n"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   604
  by (rule gcdI [symmetric]) (simp_all add: dvd_add_left_iff)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   605
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   606
lemma gcd_add2 [simp]: "gcd m (m + n) = gcd m n"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   607
  using gcd_add1 [of n m] by (simp add: ac_simps)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   608
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   609
lemma gcd_add_mult: "gcd m (k * m + n) = gcd m n"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   610
  by (rule gcdI [symmetric]) (simp_all add: dvd_add_right_iff)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   611
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   612
lemma coprimeI: "(\<And>l. \<lbrakk>l dvd a; l dvd b\<rbrakk> \<Longrightarrow> l dvd 1) \<Longrightarrow> gcd a b = 1"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   613
  by (rule sym, rule gcdI, simp_all)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   614
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   615
lemma coprime: "gcd a b = 1 \<longleftrightarrow> (\<forall>d. d dvd a \<and> d dvd b \<longleftrightarrow> is_unit d)"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   616
  by (auto intro: coprimeI gcd_greatest dvd_gcdD1 dvd_gcdD2)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   617
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   618
lemma div_gcd_coprime:
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   619
  assumes nz: "a \<noteq> 0 \<or> b \<noteq> 0"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   620
  shows "coprime (a div gcd a b) (b div gcd a b)"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   621
proof -
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   622
  let ?g = "gcd a b"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   623
  let ?a' = "a div ?g"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   624
  let ?b' = "b div ?g"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   625
  let ?g' = "gcd ?a' ?b'"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   626
  have dvdg: "?g dvd a" "?g dvd b" by simp_all
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   627
  have dvdg': "?g' dvd ?a'" "?g' dvd ?b'" by simp_all
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   628
  from dvdg dvdg' obtain ka kb ka' kb' where
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   629
      kab: "a = ?g * ka" "b = ?g * kb" "?a' = ?g' * ka'" "?b' = ?g' * kb'"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   630
    unfolding dvd_def by blast
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   631
  from this [symmetric] have "?g * ?a' = (?g * ?g') * ka'" "?g * ?b' = (?g * ?g') * kb'"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   632
    by (simp_all add: mult.assoc mult.left_commute [of "gcd a b"])
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   633
  then have dvdgg':"?g * ?g' dvd a" "?g* ?g' dvd b"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   634
    by (auto simp add: dvd_mult_div_cancel [OF dvdg(1)]
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   635
      dvd_mult_div_cancel [OF dvdg(2)] dvd_def)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   636
  have "?g \<noteq> 0" using nz by simp
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   637
  moreover from gcd_greatest [OF dvdgg'] have "?g * ?g' dvd ?g" .
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   638
  thm dvd_mult_cancel_left
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   639
  ultimately show ?thesis using dvd_times_left_cancel_iff [of "gcd a b" _ 1] by simp
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   640
qed
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   641
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   642
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   643
lemma divides_mult:
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   644
  assumes "a dvd c" and nr: "b dvd c" and "coprime a b"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   645
  shows "a * b dvd c"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   646
proof-
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   647
  from \<open>b dvd c\<close> obtain b' where"c = b * b'" ..
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   648
  with \<open>a dvd c\<close> have "a dvd b' * b"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   649
    by (simp add: ac_simps)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   650
  with \<open>coprime a b\<close> have "a dvd b'"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   651
    by (simp add: coprime_dvd_mult_iff)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   652
  then obtain a' where "b' = a * a'" ..
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   653
  with \<open>c = b * b'\<close> have "c = (a * b) * a'"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   654
    by (simp add: ac_simps)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   655
  then show ?thesis ..
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   656
qed
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   657
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   658
lemma coprime_lmult:
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   659
  assumes dab: "gcd d (a * b) = 1" 
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   660
  shows "gcd d a = 1"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   661
proof (rule coprimeI)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   662
  fix l assume "l dvd d" and "l dvd a"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   663
  hence "l dvd a * b" by simp
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   664
  with \<open>l dvd d\<close> and dab show "l dvd 1" by (auto intro: gcd_greatest)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   665
qed
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   666
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   667
lemma coprime_rmult:
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   668
  assumes dab: "gcd d (a * b) = 1"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   669
  shows "gcd d b = 1"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   670
proof (rule coprimeI)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   671
  fix l assume "l dvd d" and "l dvd b"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   672
  hence "l dvd a * b" by simp
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   673
  with \<open>l dvd d\<close> and dab show "l dvd 1" by (auto intro: gcd_greatest)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   674
qed
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   675
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   676
lemma coprime_mult:
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   677
  assumes da: "coprime d a" and db: "coprime d b"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   678
  shows "coprime d (a * b)"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   679
  apply (subst gcd.commute)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   680
  using da apply (subst gcd_mult_cancel)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   681
  apply (subst gcd.commute, assumption)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   682
  apply (subst gcd.commute, rule db)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   683
  done
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   684
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   685
lemma coprime_mul_eq: "gcd d (a * b) = 1 \<longleftrightarrow> gcd d a = 1 \<and> gcd d b = 1"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   686
  using coprime_rmult[of d a b] coprime_lmult[of d a b] coprime_mult[of d a b] by blast
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   687
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   688
lemma gcd_coprime:
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   689
  assumes c: "gcd a b \<noteq> 0" and a: "a = a' * gcd a b" and b: "b = b' * gcd a b"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   690
  shows "gcd a' b' = 1"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   691
proof -
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   692
  from c have "a \<noteq> 0 \<or> b \<noteq> 0" by simp
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   693
  with div_gcd_coprime have "gcd (a div gcd a b) (b div gcd a b) = 1" .
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   694
  also from assms have "a div gcd a b = a'" using dvd_div_eq_mult local.gcd_dvd1 by blast
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   695
  also from assms have "b div gcd a b = b'" using dvd_div_eq_mult local.gcd_dvd1 by blast
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   696
  finally show ?thesis .
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   697
qed
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   698
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   699
lemma coprime_power:
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   700
  assumes "0 < n"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   701
  shows "gcd a (b ^ n) = 1 \<longleftrightarrow> gcd a b = 1"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   702
using assms proof (induct n)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   703
  case (Suc n) then show ?case
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   704
    by (cases n) (simp_all add: coprime_mul_eq)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   705
qed simp
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   706
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   707
lemma gcd_coprime_exists:
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   708
  assumes nz: "gcd a b \<noteq> 0"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   709
  shows "\<exists>a' b'. a = a' * gcd a b \<and> b = b' * gcd a b \<and> gcd a' b' = 1"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   710
  apply (rule_tac x = "a div gcd a b" in exI)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   711
  apply (rule_tac x = "b div gcd a b" in exI)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   712
  apply (insert nz, auto intro: div_gcd_coprime)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   713
  done
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   714
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   715
lemma coprime_exp:
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   716
  "gcd d a = 1 \<Longrightarrow> gcd d (a^n) = 1"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   717
  by (induct n, simp_all add: coprime_mult)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   718
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   719
lemma coprime_exp_left:
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   720
  assumes "coprime a b"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   721
  shows "coprime (a ^ n) b"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   722
  using assms by (induct n) (simp_all add: gcd_mult_cancel)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   723
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   724
lemma coprime_exp2:
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   725
  assumes "coprime a b"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   726
  shows "coprime (a ^ n) (b ^ m)"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   727
proof (rule coprime_exp_left)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   728
  from assms show "coprime a (b ^ m)"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   729
    by (induct m) (simp_all add: gcd_mult_cancel gcd.commute [of a])
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   730
qed
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   731
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   732
lemma gcd_exp:
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   733
  "gcd (a ^ n) (b ^ n) = gcd a b ^ n"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   734
proof (cases "a = 0 \<and> b = 0")
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   735
  case True
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   736
  then show ?thesis by (cases n) simp_all
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   737
next
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   738
  case False
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   739
  then have "1 = gcd ((a div gcd a b) ^ n) ((b div gcd a b) ^ n)"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   740
    using coprime_exp2[OF div_gcd_coprime[of a b], of n n, symmetric] by simp
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   741
  then have "gcd a b ^ n = gcd a b ^ n * ..." by simp
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   742
  also note gcd_mult_distrib
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   743
  also have "unit_factor (gcd a b ^ n) = 1"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   744
    using False by (auto simp add: unit_factor_power unit_factor_gcd)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   745
  also have "(gcd a b)^n * (a div gcd a b)^n = a^n"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   746
    by (subst ac_simps, subst div_power, simp, rule dvd_div_mult_self, rule dvd_power_same, simp)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   747
  also have "(gcd a b)^n * (b div gcd a b)^n = b^n"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   748
    by (subst ac_simps, subst div_power, simp, rule dvd_div_mult_self, rule dvd_power_same, simp)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   749
  finally show ?thesis by simp
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   750
qed
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   751
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   752
lemma coprime_common_divisor: 
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   753
  "gcd a b = 1 \<Longrightarrow> a dvd a \<Longrightarrow> a dvd b \<Longrightarrow> is_unit a"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   754
  apply (subgoal_tac "a dvd gcd a b")
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   755
  apply simp
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   756
  apply (erule (1) gcd_greatest)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   757
  done
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   758
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   759
lemma division_decomp: 
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   760
  assumes dc: "a dvd b * c"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   761
  shows "\<exists>b' c'. a = b' * c' \<and> b' dvd b \<and> c' dvd c"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   762
proof (cases "gcd a b = 0")
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   763
  assume "gcd a b = 0"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   764
  hence "a = 0 \<and> b = 0" by simp
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   765
  hence "a = 0 * c \<and> 0 dvd b \<and> c dvd c" by simp
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   766
  then show ?thesis by blast
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   767
next
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   768
  let ?d = "gcd a b"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   769
  assume "?d \<noteq> 0"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   770
  from gcd_coprime_exists[OF this]
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   771
    obtain a' b' where ab': "a = a' * ?d" "b = b' * ?d" "gcd a' b' = 1"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   772
    by blast
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   773
  from ab'(1) have "a' dvd a" unfolding dvd_def by blast
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   774
  with dc have "a' dvd b*c" using dvd_trans[of a' a "b*c"] by simp
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   775
  from dc ab'(1,2) have "a'*?d dvd (b'*?d) * c" by simp
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   776
  hence "?d * a' dvd ?d * (b' * c)" by (simp add: mult_ac)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   777
  with \<open>?d \<noteq> 0\<close> have "a' dvd b' * c" by simp
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   778
  with coprime_dvd_mult[OF ab'(3)] 
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   779
    have "a' dvd c" by (subst (asm) ac_simps, blast)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   780
  with ab'(1) have "a = ?d * a' \<and> ?d dvd b \<and> a' dvd c" by (simp add: mult_ac)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   781
  then show ?thesis by blast
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   782
qed
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   783
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   784
lemma pow_divs_pow:
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   785
  assumes ab: "a ^ n dvd b ^ n" and n: "n \<noteq> 0"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   786
  shows "a dvd b"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   787
proof (cases "gcd a b = 0")
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   788
  assume "gcd a b = 0"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   789
  then show ?thesis by simp
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   790
next
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   791
  let ?d = "gcd a b"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   792
  assume "?d \<noteq> 0"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   793
  from n obtain m where m: "n = Suc m" by (cases n, simp_all)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   794
  from \<open>?d \<noteq> 0\<close> have zn: "?d ^ n \<noteq> 0" by (rule power_not_zero)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   795
  from gcd_coprime_exists[OF \<open>?d \<noteq> 0\<close>]
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   796
    obtain a' b' where ab': "a = a' * ?d" "b = b' * ?d" "gcd a' b' = 1"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   797
    by blast
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   798
  from ab have "(a' * ?d) ^ n dvd (b' * ?d) ^ n"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   799
    by (simp add: ab'(1,2)[symmetric])
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   800
  hence "?d^n * a'^n dvd ?d^n * b'^n"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   801
    by (simp only: power_mult_distrib ac_simps)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   802
  with zn have "a'^n dvd b'^n" by simp
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   803
  hence "a' dvd b'^n" using dvd_trans[of a' "a'^n" "b'^n"] by (simp add: m)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   804
  hence "a' dvd b'^m * b'" by (simp add: m ac_simps)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   805
  with coprime_dvd_mult[OF coprime_exp[OF ab'(3), of m]]
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   806
    have "a' dvd b'" by (subst (asm) ac_simps, blast)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   807
  hence "a'*?d dvd b'*?d" by (rule mult_dvd_mono, simp)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   808
  with ab'(1,2) show ?thesis by simp
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   809
qed
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   810
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   811
lemma pow_divs_eq [simp]:
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   812
  "n \<noteq> 0 \<Longrightarrow> a ^ n dvd b ^ n \<longleftrightarrow> a dvd b"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   813
  by (auto intro: pow_divs_pow dvd_power_same)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   814
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   815
lemma coprime_plus_one [simp]: "gcd (n + 1) n = 1"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   816
  by (subst add_commute, simp)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   817
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   818
lemma setprod_coprime [rule_format]:
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   819
  "(\<forall>i\<in>A. gcd (f i) a = 1) \<longrightarrow> gcd (\<Prod>i\<in>A. f i) a = 1"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   820
  apply (cases "finite A")
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   821
  apply (induct set: finite)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   822
  apply (auto simp add: gcd_mult_cancel)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   823
  done
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   824
  
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   825
lemma listprod_coprime:
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   826
  "(\<And>x. x \<in> set xs \<Longrightarrow> coprime x y) \<Longrightarrow> coprime (listprod xs) y" 
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   827
  by (induction xs) (simp_all add: gcd_mult_cancel)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   828
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   829
lemma coprime_divisors: 
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   830
  assumes "d dvd a" "e dvd b" "gcd a b = 1"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   831
  shows "gcd d e = 1" 
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   832
proof -
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   833
  from assms obtain k l where "a = d * k" "b = e * l"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   834
    unfolding dvd_def by blast
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   835
  with assms have "gcd (d * k) (e * l) = 1" by simp
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   836
  hence "gcd (d * k) e = 1" by (rule coprime_lmult)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   837
  also have "gcd (d * k) e = gcd e (d * k)" by (simp add: ac_simps)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   838
  finally have "gcd e d = 1" by (rule coprime_lmult)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   839
  then show ?thesis by (simp add: ac_simps)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   840
qed
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   841
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   842
lemma lcm_gcd_prod:
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   843
  "lcm a b * gcd a b = normalize (a * b)"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   844
  by (simp add: lcm_gcd)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   845
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   846
declare unit_factor_lcm [simp]
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   847
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   848
lemma lcmI:
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   849
  assumes "a dvd c" and "b dvd c" and "\<And>d. a dvd d \<Longrightarrow> b dvd d \<Longrightarrow> c dvd d"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   850
    and "normalize c = c"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   851
  shows "c = lcm a b"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   852
  by (rule associated_eqI) (auto simp: assms intro: lcm_least)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   853
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   854
lemma gcd_dvd_lcm [simp]:
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   855
  "gcd a b dvd lcm a b"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   856
  using gcd_dvd2 by (rule dvd_lcmI2)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   857
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   858
lemmas lcm_0 = lcm_0_right
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   859
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   860
lemma lcm_unique:
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   861
  "a dvd d \<and> b dvd d \<and> 
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   862
  normalize d = d \<and>
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   863
  (\<forall>e. a dvd e \<and> b dvd e \<longrightarrow> d dvd e) \<longleftrightarrow> d = lcm a b"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   864
  by rule (auto intro: lcmI simp: lcm_least lcm_eq_0_iff)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   865
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   866
lemma lcm_coprime:
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   867
  "gcd a b = 1 \<Longrightarrow> lcm a b = normalize (a * b)"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   868
  by (subst lcm_gcd) simp
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   869
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   870
lemma lcm_proj1_if_dvd: 
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   871
  "b dvd a \<Longrightarrow> lcm a b = normalize a"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   872
  by (cases "a = 0") (simp, rule sym, rule lcmI, simp_all)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   873
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   874
lemma lcm_proj2_if_dvd: 
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   875
  "a dvd b \<Longrightarrow> lcm a b = normalize b"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   876
  using lcm_proj1_if_dvd [of a b] by (simp add: ac_simps)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   877
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   878
lemma lcm_proj1_iff:
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   879
  "lcm m n = normalize m \<longleftrightarrow> n dvd m"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   880
proof
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   881
  assume A: "lcm m n = normalize m"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   882
  show "n dvd m"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   883
  proof (cases "m = 0")
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   884
    assume [simp]: "m \<noteq> 0"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   885
    from A have B: "m = lcm m n * unit_factor m"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   886
      by (simp add: unit_eq_div2)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   887
    show ?thesis by (subst B, simp)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   888
  qed simp
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   889
next
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   890
  assume "n dvd m"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   891
  then show "lcm m n = normalize m" by (rule lcm_proj1_if_dvd)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   892
qed
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   893
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   894
lemma lcm_proj2_iff:
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   895
  "lcm m n = normalize n \<longleftrightarrow> m dvd n"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   896
  using lcm_proj1_iff [of n m] by (simp add: ac_simps)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   897
60686
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   898
end
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   899
62345
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   900
class ring_gcd = comm_ring_1 + semiring_gcd
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   901
begin
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   902
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   903
lemma coprime_minus_one: "coprime (n - 1) n"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   904
  using coprime_plus_one[of "n - 1"] by (simp add: gcd.commute)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   905
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   906
lemma gcd_neg1 [simp]:
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   907
  "gcd (-a) b = gcd a b"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   908
  by (rule sym, rule gcdI, simp_all add: gcd_greatest)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   909
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   910
lemma gcd_neg2 [simp]:
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   911
  "gcd a (-b) = gcd a b"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   912
  by (rule sym, rule gcdI, simp_all add: gcd_greatest)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   913
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   914
lemma gcd_neg_numeral_1 [simp]:
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   915
  "gcd (- numeral n) a = gcd (numeral n) a"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   916
  by (fact gcd_neg1)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   917
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   918
lemma gcd_neg_numeral_2 [simp]:
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   919
  "gcd a (- numeral n) = gcd a (numeral n)"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   920
  by (fact gcd_neg2)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   921
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   922
lemma gcd_diff1: "gcd (m - n) n = gcd m n"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   923
  by (subst diff_conv_add_uminus, subst gcd_neg2[symmetric],  subst gcd_add1, simp)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   924
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   925
lemma gcd_diff2: "gcd (n - m) n = gcd m n"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   926
  by (subst gcd_neg1[symmetric], simp only: minus_diff_eq gcd_diff1)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   927
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   928
lemma lcm_neg1 [simp]: "lcm (-a) b = lcm a b"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   929
  by (rule sym, rule lcmI, simp_all add: lcm_least lcm_eq_0_iff)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   930
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   931
lemma lcm_neg2 [simp]: "lcm a (-b) = lcm a b"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   932
  by (rule sym, rule lcmI, simp_all add: lcm_least lcm_eq_0_iff)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   933
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   934
lemma lcm_neg_numeral_1 [simp]: "lcm (- numeral n) a = lcm (numeral n) a"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   935
  by (fact lcm_neg1)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   936
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   937
lemma lcm_neg_numeral_2 [simp]: "lcm a (- numeral n) = lcm a (numeral n)"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   938
  by (fact lcm_neg2)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   939
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   940
end
62345
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   941
60686
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   942
class semiring_Gcd = semiring_gcd + Gcd +
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   943
  assumes Gcd_dvd: "a \<in> A \<Longrightarrow> Gcd A dvd a"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   944
    and Gcd_greatest: "(\<And>b. b \<in> A \<Longrightarrow> a dvd b) \<Longrightarrow> a dvd Gcd A"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   945
    and normalize_Gcd [simp]: "normalize (Gcd A) = Gcd A"
62345
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   946
  assumes dvd_Lcm: "a \<in> A \<Longrightarrow> a dvd Lcm A"
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   947
    and Lcm_least: "(\<And>b. b \<in> A \<Longrightarrow> b dvd a) \<Longrightarrow> Lcm A dvd a"
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   948
    and normalize_Lcm [simp]: "normalize (Lcm A) = Lcm A"
60686
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   949
begin
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   950
62345
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   951
lemma Lcm_Gcd:
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   952
  "Lcm A = Gcd {b. \<forall>a\<in>A. a dvd b}"
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   953
  by (rule associated_eqI) (auto intro: Gcd_dvd dvd_Lcm Gcd_greatest Lcm_least)
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   954
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   955
lemma Gcd_Lcm:
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   956
  "Gcd A = Lcm {b. \<forall>a\<in>A. b dvd a}"
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   957
  by (rule associated_eqI) (auto intro: Gcd_dvd dvd_Lcm Gcd_greatest Lcm_least)
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   958
60686
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   959
lemma Gcd_empty [simp]:
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   960
  "Gcd {} = 0"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   961
  by (rule dvd_0_left, rule Gcd_greatest) simp
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   962
62345
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   963
lemma Lcm_empty [simp]:
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   964
  "Lcm {} = 1"
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   965
  by (auto intro: associated_eqI Lcm_least)
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   966
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   967
lemma Gcd_insert [simp]:
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   968
  "Gcd (insert a A) = gcd a (Gcd A)"
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   969
proof -
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   970
  have "Gcd (insert a A) dvd gcd a (Gcd A)"
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   971
    by (auto intro: Gcd_dvd Gcd_greatest)
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   972
  moreover have "gcd a (Gcd A) dvd Gcd (insert a A)"
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   973
  proof (rule Gcd_greatest)
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   974
    fix b
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   975
    assume "b \<in> insert a A"
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   976
    then show "gcd a (Gcd A) dvd b"
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   977
    proof
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   978
      assume "b = a" then show ?thesis by simp
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   979
    next
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   980
      assume "b \<in> A"
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   981
      then have "Gcd A dvd b" by (rule Gcd_dvd)
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   982
      moreover have "gcd a (Gcd A) dvd Gcd A" by simp
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   983
      ultimately show ?thesis by (blast intro: dvd_trans)
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   984
    qed
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   985
  qed
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   986
  ultimately show ?thesis
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   987
    by (auto intro: associated_eqI)
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   988
qed
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   989
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   990
lemma Lcm_insert [simp]:
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   991
  "Lcm (insert a A) = lcm a (Lcm A)"
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   992
proof (rule sym)
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   993
  have "lcm a (Lcm A) dvd Lcm (insert a A)"
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   994
    by (auto intro: dvd_Lcm Lcm_least)
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   995
  moreover have "Lcm (insert a A) dvd lcm a (Lcm A)"
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   996
  proof (rule Lcm_least)
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   997
    fix b
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   998
    assume "b \<in> insert a A"
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   999
    then show "b dvd lcm a (Lcm A)"
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1000
    proof
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1001
      assume "b = a" then show ?thesis by simp
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1002
    next
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1003
      assume "b \<in> A"
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1004
      then have "b dvd Lcm A" by (rule dvd_Lcm)
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1005
      moreover have "Lcm A dvd lcm a (Lcm A)" by simp
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1006
      ultimately show ?thesis by (blast intro: dvd_trans)
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1007
    qed
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1008
  qed
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1009
  ultimately show "lcm a (Lcm A) = Lcm (insert a A)"
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1010
    by (rule associated_eqI) (simp_all add: lcm_eq_0_iff)
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1011
qed
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1012
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1013
lemma LcmI:
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1014
  assumes "\<And>a. a \<in> A \<Longrightarrow> a dvd b" and "\<And>c. (\<And>a. a \<in> A \<Longrightarrow> a dvd c) \<Longrightarrow> b dvd c"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1015
    and "normalize b = b" shows "b = Lcm A"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1016
  by (rule associated_eqI) (auto simp: assms dvd_Lcm intro: Lcm_least)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1017
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1018
lemma Lcm_subset:
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1019
  "A \<subseteq> B \<Longrightarrow> Lcm A dvd Lcm B"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1020
  by (blast intro: Lcm_least dvd_Lcm)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1021
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1022
lemma Lcm_Un:
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1023
  "Lcm (A \<union> B) = lcm (Lcm A) (Lcm B)"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1024
  apply (rule lcmI)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1025
  apply (blast intro: Lcm_subset)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1026
  apply (blast intro: Lcm_subset)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1027
  apply (intro Lcm_least ballI, elim UnE)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1028
  apply (rule dvd_trans, erule dvd_Lcm, assumption)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1029
  apply (rule dvd_trans, erule dvd_Lcm, assumption)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1030
  apply simp
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1031
  done
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1032
  
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1033
60686
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1034
lemma Gcd_0_iff [simp]:
60687
33dbbcb6a8a3 eliminated some duplication
haftmann
parents: 60686
diff changeset
  1035
  "Gcd A = 0 \<longleftrightarrow> A \<subseteq> {0}" (is "?P \<longleftrightarrow> ?Q")
60686
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1036
proof
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1037
  assume ?P
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1038
  show ?Q
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1039
  proof
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1040
    fix a
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1041
    assume "a \<in> A"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1042
    then have "Gcd A dvd a" by (rule Gcd_dvd)
60687
33dbbcb6a8a3 eliminated some duplication
haftmann
parents: 60686
diff changeset
  1043
    with \<open>?P\<close> have "a = 0" by simp
33dbbcb6a8a3 eliminated some duplication
haftmann
parents: 60686
diff changeset
  1044
    then show "a \<in> {0}" by simp
60686
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1045
  qed
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1046
next
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1047
  assume ?Q
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1048
  have "0 dvd Gcd A"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1049
  proof (rule Gcd_greatest)
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1050
    fix a
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1051
    assume "a \<in> A"
60687
33dbbcb6a8a3 eliminated some duplication
haftmann
parents: 60686
diff changeset
  1052
    with \<open>?Q\<close> have "a = 0" by auto
60686
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1053
    then show "0 dvd a" by simp
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1054
  qed
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1055
  then show ?P by simp
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1056
qed
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1057
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1058
lemma Lcm_1_iff [simp]:
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1059
  "Lcm A = 1 \<longleftrightarrow> (\<forall>a\<in>A. is_unit a)" (is "?P \<longleftrightarrow> ?Q")
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1060
proof
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1061
  assume ?P
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1062
  show ?Q
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1063
  proof
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1064
    fix a
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1065
    assume "a \<in> A"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1066
    then have "a dvd Lcm A"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1067
      by (rule dvd_Lcm)
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1068
    with \<open>?P\<close> show "is_unit a"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1069
      by simp
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1070
  qed
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1071
next
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1072
  assume ?Q
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1073
  then have "is_unit (Lcm A)"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1074
    by (blast intro: Lcm_least)
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1075
  then have "normalize (Lcm A) = 1"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1076
    by (rule is_unit_normalize)
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1077
  then show ?P
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1078
    by simp
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1079
qed
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1080
62345
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1081
lemma unit_factor_Lcm:
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1082
  "unit_factor (Lcm A) = (if Lcm A = 0 then 0 else 1)"
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1083
proof (cases "Lcm A = 0")
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1084
  case True then show ?thesis by simp
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1085
next
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1086
  case False
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1087
  with unit_factor_normalize have "unit_factor (normalize (Lcm A)) = 1"
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1088
    by blast
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1089
  with False show ?thesis
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1090
    by simp
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1091
qed
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1092
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1093
lemma unit_factor_Gcd: "unit_factor (Gcd A) = (if Gcd A = 0 then 0 else 1)"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1094
proof -
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1095
  show "unit_factor (Gcd A) = (if Gcd A = 0 then 0 else 1)"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1096
    by (simp add: Gcd_Lcm unit_factor_Lcm)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1097
qed
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1098
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1099
lemma GcdI:
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1100
  assumes "\<And>a. a \<in> A \<Longrightarrow> b dvd a" and "\<And>c. (\<And>a. a \<in> A \<Longrightarrow> c dvd a) \<Longrightarrow> c dvd b"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1101
    and "normalize b = b"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1102
  shows "b = Gcd A"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1103
  by (rule associated_eqI) (auto simp: assms Gcd_dvd intro: Gcd_greatest)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1104
62345
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1105
lemma Gcd_eq_1_I:
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1106
  assumes "is_unit a" and "a \<in> A"
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1107
  shows "Gcd A = 1"
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1108
proof -
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1109
  from assms have "is_unit (Gcd A)"
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1110
    by (blast intro: Gcd_dvd dvd_unit_imp_unit)
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1111
  then have "normalize (Gcd A) = 1"
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1112
    by (rule is_unit_normalize)
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1113
  then show ?thesis
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1114
    by simp
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1115
qed
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1116
60686
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1117
lemma Lcm_eq_0_I:
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1118
  assumes "0 \<in> A"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1119
  shows "Lcm A = 0"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1120
proof -
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1121
  from assms have "0 dvd Lcm A"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1122
    by (rule dvd_Lcm)
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1123
  then show ?thesis
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1124
    by simp
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1125
qed
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1126
62345
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1127
lemma Gcd_UNIV [simp]:
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1128
  "Gcd UNIV = 1"
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1129
  using dvd_refl by (rule Gcd_eq_1_I) simp
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1130
61929
b8e242e52c97 tuned proofs and augmented lemmas
haftmann
parents: 61913
diff changeset
  1131
lemma Lcm_UNIV [simp]:
b8e242e52c97 tuned proofs and augmented lemmas
haftmann
parents: 61913
diff changeset
  1132
  "Lcm UNIV = 0"
b8e242e52c97 tuned proofs and augmented lemmas
haftmann
parents: 61913
diff changeset
  1133
  by (rule Lcm_eq_0_I) simp
60686
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1134
61929
b8e242e52c97 tuned proofs and augmented lemmas
haftmann
parents: 61913
diff changeset
  1135
lemma Lcm_0_iff:
b8e242e52c97 tuned proofs and augmented lemmas
haftmann
parents: 61913
diff changeset
  1136
  assumes "finite A"
b8e242e52c97 tuned proofs and augmented lemmas
haftmann
parents: 61913
diff changeset
  1137
  shows "Lcm A = 0 \<longleftrightarrow> 0 \<in> A"
b8e242e52c97 tuned proofs and augmented lemmas
haftmann
parents: 61913
diff changeset
  1138
proof (cases "A = {}")
b8e242e52c97 tuned proofs and augmented lemmas
haftmann
parents: 61913
diff changeset
  1139
  case True then show ?thesis by simp
b8e242e52c97 tuned proofs and augmented lemmas
haftmann
parents: 61913
diff changeset
  1140
next
b8e242e52c97 tuned proofs and augmented lemmas
haftmann
parents: 61913
diff changeset
  1141
  case False with assms show ?thesis
b8e242e52c97 tuned proofs and augmented lemmas
haftmann
parents: 61913
diff changeset
  1142
    by (induct A rule: finite_ne_induct)
b8e242e52c97 tuned proofs and augmented lemmas
haftmann
parents: 61913
diff changeset
  1143
      (auto simp add: lcm_eq_0_iff)
60686
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1144
qed
61929
b8e242e52c97 tuned proofs and augmented lemmas
haftmann
parents: 61913
diff changeset
  1145
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1146
lemma Gcd_finite:
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1147
  assumes "finite A"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1148
  shows "Gcd A = Finite_Set.fold gcd 0 A"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1149
  by (induct rule: finite.induct[OF \<open>finite A\<close>])
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1150
     (simp_all add: comp_fun_idem.fold_insert_idem[OF comp_fun_idem_gcd])
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1151
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1152
lemma Gcd_set [code_unfold]: "Gcd (set as) = foldl gcd 0 as"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1153
  by (simp add: Gcd_finite comp_fun_idem.fold_set_fold[OF comp_fun_idem_gcd] 
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1154
                foldl_conv_fold gcd.commute)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1155
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1156
lemma Lcm_finite:
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1157
  assumes "finite A"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1158
  shows "Lcm A = Finite_Set.fold lcm 1 A"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1159
  by (induct rule: finite.induct[OF \<open>finite A\<close>])
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1160
     (simp_all add: comp_fun_idem.fold_insert_idem[OF comp_fun_idem_lcm])
62345
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1161
60686
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1162
lemma Lcm_set [code_unfold]:
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1163
  "Lcm (set as) = foldl lcm 1 as"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1164
  by (simp add: Lcm_finite comp_fun_idem.fold_set_fold[OF comp_fun_idem_lcm] 
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1165
                foldl_conv_fold lcm.commute)
59008
f61482b0f240 formally self-contained gcd type classes
haftmann
parents: 58889
diff changeset
  1166
62345
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1167
lemma Gcd_image_normalize [simp]:
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1168
  "Gcd (normalize ` A) = Gcd A"
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1169
proof -
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1170
  have "Gcd (normalize ` A) dvd a" if "a \<in> A" for a
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1171
  proof -
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1172
    from that obtain B where "A = insert a B" by blast
62350
66a381d3f88f more sophisticated GCD syntax
haftmann
parents: 62349
diff changeset
  1173
    moreover have "gcd (normalize a) (Gcd (normalize ` B)) dvd normalize a"
62345
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1174
      by (rule gcd_dvd1)
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1175
    ultimately show "Gcd (normalize ` A) dvd a"
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1176
      by simp
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1177
  qed
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1178
  then have "Gcd (normalize ` A) dvd Gcd A" and "Gcd A dvd Gcd (normalize ` A)"
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1179
    by (auto intro!: Gcd_greatest intro: Gcd_dvd)
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1180
  then show ?thesis
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1181
    by (auto intro: associated_eqI)
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1182
qed
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1183
62346
97f2ed240431 more theorems concerning gcd/lcm/Gcd/Lcm
haftmann
parents: 62345
diff changeset
  1184
lemma Gcd_eqI:
97f2ed240431 more theorems concerning gcd/lcm/Gcd/Lcm
haftmann
parents: 62345
diff changeset
  1185
  assumes "normalize a = a"
97f2ed240431 more theorems concerning gcd/lcm/Gcd/Lcm
haftmann
parents: 62345
diff changeset
  1186
  assumes "\<And>b. b \<in> A \<Longrightarrow> a dvd b"
97f2ed240431 more theorems concerning gcd/lcm/Gcd/Lcm
haftmann
parents: 62345
diff changeset
  1187
    and "\<And>c. (\<And>b. b \<in> A \<Longrightarrow> c dvd b) \<Longrightarrow> c dvd a"
97f2ed240431 more theorems concerning gcd/lcm/Gcd/Lcm
haftmann
parents: 62345
diff changeset
  1188
  shows "Gcd A = a"
97f2ed240431 more theorems concerning gcd/lcm/Gcd/Lcm
haftmann
parents: 62345
diff changeset
  1189
  using assms by (blast intro: associated_eqI Gcd_greatest Gcd_dvd normalize_Gcd)
97f2ed240431 more theorems concerning gcd/lcm/Gcd/Lcm
haftmann
parents: 62345
diff changeset
  1190
97f2ed240431 more theorems concerning gcd/lcm/Gcd/Lcm
haftmann
parents: 62345
diff changeset
  1191
lemma Lcm_eqI:
97f2ed240431 more theorems concerning gcd/lcm/Gcd/Lcm
haftmann
parents: 62345
diff changeset
  1192
  assumes "normalize a = a"
97f2ed240431 more theorems concerning gcd/lcm/Gcd/Lcm
haftmann
parents: 62345
diff changeset
  1193
  assumes "\<And>b. b \<in> A \<Longrightarrow> b dvd a"
97f2ed240431 more theorems concerning gcd/lcm/Gcd/Lcm
haftmann
parents: 62345
diff changeset
  1194
    and "\<And>c. (\<And>b. b \<in> A \<Longrightarrow> b dvd c) \<Longrightarrow> a dvd c"
97f2ed240431 more theorems concerning gcd/lcm/Gcd/Lcm
haftmann
parents: 62345
diff changeset
  1195
  shows "Lcm A = a"
97f2ed240431 more theorems concerning gcd/lcm/Gcd/Lcm
haftmann
parents: 62345
diff changeset
  1196
  using assms by (blast intro: associated_eqI Lcm_least dvd_Lcm normalize_Lcm)
97f2ed240431 more theorems concerning gcd/lcm/Gcd/Lcm
haftmann
parents: 62345
diff changeset
  1197
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1198
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1199
lemma Lcm_no_units:
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1200
  "Lcm A = Lcm (A - {a. is_unit a})"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1201
proof -
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1202
  have "(A - {a. is_unit a}) \<union> {a\<in>A. is_unit a} = A" by blast
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1203
  hence "Lcm A = lcm (Lcm (A - {a. is_unit a})) (Lcm {a\<in>A. is_unit a})"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1204
    by (simp add: Lcm_Un [symmetric])
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1205
  also have "Lcm {a\<in>A. is_unit a} = 1" by (simp add: Lcm_1_iff)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1206
  finally show ?thesis by simp
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1207
qed
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1208
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1209
lemma Lcm_0_iff': "Lcm A = 0 \<longleftrightarrow> \<not>(\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l))"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1210
  by (metis Lcm_least dvd_0_left dvd_Lcm)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1211
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1212
lemma Lcm_no_multiple: "(\<forall>m. m \<noteq> 0 \<longrightarrow> (\<exists>a\<in>A. \<not>a dvd m)) \<Longrightarrow> Lcm A = 0"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1213
  by (auto simp: Lcm_0_iff')
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1214
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1215
lemma Lcm_singleton [simp]:
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1216
  "Lcm {a} = normalize a"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1217
  by simp
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1218
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1219
lemma Lcm_2 [simp]:
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1220
  "Lcm {a,b} = lcm a b"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1221
  by simp
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1222
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1223
lemma Lcm_coprime:
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1224
  assumes "finite A" and "A \<noteq> {}" 
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1225
  assumes "\<And>a b. a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<noteq> b \<Longrightarrow> gcd a b = 1"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1226
  shows "Lcm A = normalize (\<Prod>A)"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1227
using assms proof (induct rule: finite_ne_induct)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1228
  case (insert a A)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1229
  have "Lcm (insert a A) = lcm a (Lcm A)" by simp
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1230
  also from insert have "Lcm A = normalize (\<Prod>A)" by blast
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1231
  also have "lcm a \<dots> = lcm a (\<Prod>A)" by (cases "\<Prod>A = 0") (simp_all add: lcm_div_unit2)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1232
  also from insert have "gcd a (\<Prod>A) = 1" by (subst gcd.commute, intro setprod_coprime) auto
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1233
  with insert have "lcm a (\<Prod>A) = normalize (\<Prod>(insert a A))"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1234
    by (simp add: lcm_coprime)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1235
  finally show ?case .
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1236
qed simp
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1237
      
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1238
lemma Lcm_coprime':
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1239
  "card A \<noteq> 0 \<Longrightarrow> (\<And>a b. a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<noteq> b \<Longrightarrow> gcd a b = 1)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1240
    \<Longrightarrow> Lcm A = normalize (\<Prod>A)"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1241
  by (rule Lcm_coprime) (simp_all add: card_eq_0_iff)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1242
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1243
lemma Gcd_1:
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1244
  "1 \<in> A \<Longrightarrow> Gcd A = 1"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1245
  by (auto intro!: Gcd_eq_1_I)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1246
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1247
lemma Gcd_singleton [simp]: "Gcd {a} = normalize a"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1248
  by simp
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1249
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1250
lemma Gcd_2 [simp]: "Gcd {a,b} = gcd a b"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1251
  by simp
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1252
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1253
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1254
definition pairwise_coprime where
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1255
  "pairwise_coprime A = (\<forall>x y. x \<in> A \<and> y \<in> A \<and> x \<noteq> y \<longrightarrow> coprime x y)"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1256
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1257
lemma pairwise_coprimeI [intro?]:
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1258
  "(\<And>x y. x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> x \<noteq> y \<Longrightarrow> coprime x y) \<Longrightarrow> pairwise_coprime A"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1259
  by (simp add: pairwise_coprime_def)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1260
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1261
lemma pairwise_coprimeD:
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1262
  "pairwise_coprime A \<Longrightarrow> x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> x \<noteq> y \<Longrightarrow> coprime x y"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1263
  by (simp add: pairwise_coprime_def)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1264
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1265
lemma pairwise_coprime_subset: "pairwise_coprime A \<Longrightarrow> B \<subseteq> A \<Longrightarrow> pairwise_coprime B"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1266
  by (force simp: pairwise_coprime_def)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel