src/HOL/GCD.thy
author haftmann
Wed, 17 Feb 2016 21:51:57 +0100
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cleansed junk-producing interpretations for gcd/lcm on nat altogether
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(*  Authors:    Christophe Tabacznyj, Lawrence C. Paulson, Amine Chaieb,
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                Thomas M. Rasmussen, Jeremy Avigad, Tobias Nipkow
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This file deals with the functions gcd and lcm.  Definitions and
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lemmas are proved uniformly for the natural numbers and integers.
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This file combines and revises a number of prior developments.
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The original theories "GCD" and "Primes" were by Christophe Tabacznyj
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and Lawrence C. Paulson, based on @{cite davenport92}. They introduced
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gcd, lcm, and prime for the natural numbers.
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The original theory "IntPrimes" was by Thomas M. Rasmussen, and
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extended gcd, lcm, primes to the integers. Amine Chaieb provided
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another extension of the notions to the integers, and added a number
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of results to "Primes" and "GCD". IntPrimes also defined and developed
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the congruence relations on the integers. The notion was extended to
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the natural numbers by Chaieb.
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Jeremy Avigad combined all of these, made everything uniform for the
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natural numbers and the integers, and added a number of new theorems.
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Tobias Nipkow cleaned up a lot.
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*)
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section \<open>Greatest common divisor and least common multiple\<close>
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theory GCD
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imports Main
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begin
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subsection \<open>Abstract GCD and LCM\<close>
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class gcd = zero + one + dvd +
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  fixes gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
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    and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
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begin
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abbreviation coprime :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
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  where "coprime x y \<equiv> gcd x y = 1"
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end
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class Gcd = gcd +
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  fixes Gcd :: "'a set \<Rightarrow> 'a"
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    and Lcm :: "'a set \<Rightarrow> 'a"
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class semiring_gcd = normalization_semidom + gcd +
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  assumes gcd_dvd1 [iff]: "gcd a b dvd a"
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    and gcd_dvd2 [iff]: "gcd a b dvd b"
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    and gcd_greatest: "c dvd a \<Longrightarrow> c dvd b \<Longrightarrow> c dvd gcd a b"
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    and normalize_gcd [simp]: "normalize (gcd a b) = gcd a b"
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    and lcm_gcd: "lcm a b = normalize (a * b) div gcd a b"
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begin    
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lemma gcd_greatest_iff [simp]:
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  "a dvd gcd b c \<longleftrightarrow> a dvd b \<and> a dvd c"
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  by (blast intro!: gcd_greatest intro: dvd_trans)
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lemma gcd_dvdI1:
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  "a dvd c \<Longrightarrow> gcd a b dvd c"
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  by (rule dvd_trans) (rule gcd_dvd1)
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lemma gcd_dvdI2:
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  "b dvd c \<Longrightarrow> gcd a b dvd c"
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  by (rule dvd_trans) (rule gcd_dvd2)
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lemma dvd_gcdD1:
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  "a dvd gcd b c \<Longrightarrow> a dvd b"
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  using gcd_dvd1 [of b c] by (blast intro: dvd_trans)
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lemma dvd_gcdD2:
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  "a dvd gcd b c \<Longrightarrow> a dvd c"
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  using gcd_dvd2 [of b c] by (blast intro: dvd_trans)
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lemma gcd_0_left [simp]:
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  "gcd 0 a = normalize a"
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  by (rule associated_eqI) simp_all
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lemma gcd_0_right [simp]:
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  "gcd a 0 = normalize a"
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  by (rule associated_eqI) simp_all
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lemma gcd_eq_0_iff [simp]:
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  "gcd a b = 0 \<longleftrightarrow> a = 0 \<and> b = 0" (is "?P \<longleftrightarrow> ?Q")
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proof
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  assume ?P then have "0 dvd gcd a b" by simp
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  then have "0 dvd a" and "0 dvd b" by (blast intro: dvd_trans)+
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  then show ?Q by simp
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next
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  assume ?Q then show ?P by simp
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qed
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lemma unit_factor_gcd:
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  "unit_factor (gcd a b) = (if a = 0 \<and> b = 0 then 0 else 1)"
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proof (cases "gcd a b = 0")
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  case True then show ?thesis by simp
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next
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  case False
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  have "unit_factor (gcd a b) * normalize (gcd a b) = gcd a b"
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    by (rule unit_factor_mult_normalize)
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  then have "unit_factor (gcd a b) * gcd a b = gcd a b"
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    by simp
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  then have "unit_factor (gcd a b) * gcd a b div gcd a b = gcd a b div gcd a b"
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    by simp
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  with False show ?thesis by simp
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qed
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lemma is_unit_gcd [simp]:
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  "is_unit (gcd a b) \<longleftrightarrow> coprime a b"
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  by (cases "a = 0 \<and> b = 0") (auto simp add: unit_factor_gcd dest: is_unit_unit_factor)
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sublocale gcd: abel_semigroup gcd
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proof
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  fix a b c
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  show "gcd a b = gcd b a"
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    by (rule associated_eqI) simp_all
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  from gcd_dvd1 have "gcd (gcd a b) c dvd a"
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    by (rule dvd_trans) simp
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  moreover from gcd_dvd1 have "gcd (gcd a b) c dvd b"
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    by (rule dvd_trans) simp
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  ultimately have P1: "gcd (gcd a b) c dvd gcd a (gcd b c)"
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    by (auto intro!: gcd_greatest)
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  from gcd_dvd2 have "gcd a (gcd b c) dvd b"
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    by (rule dvd_trans) simp
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  moreover from gcd_dvd2 have "gcd a (gcd b c) dvd c"
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    by (rule dvd_trans) simp
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  ultimately have P2: "gcd a (gcd b c) dvd gcd (gcd a b) c"
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    by (auto intro!: gcd_greatest)
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  from P1 P2 show "gcd (gcd a b) c = gcd a (gcd b c)"
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    by (rule associated_eqI) simp_all
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qed
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lemma gcd_self [simp]:
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  "gcd a a = normalize a"
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proof -
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  have "a dvd gcd a a"
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    by (rule gcd_greatest) simp_all
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  then show ?thesis
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    by (auto intro: associated_eqI)
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qed
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lemma gcd_left_idem [simp]:
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  "gcd a (gcd a b) = gcd a b"
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  by (auto intro: associated_eqI)
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lemma gcd_right_idem [simp]:
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  "gcd (gcd a b) b = gcd a b"
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  unfolding gcd.commute [of a] gcd.commute [of "gcd b a"] by simp
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   152
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lemma coprime_1_left [simp]:
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  "coprime 1 a"
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  by (rule associated_eqI) simp_all
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lemma coprime_1_right [simp]:
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  "coprime a 1"
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  using coprime_1_left [of a] by (simp add: ac_simps)
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lemma gcd_mult_left:
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  "gcd (c * a) (c * b) = normalize c * gcd a b"
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   163
proof (cases "c = 0")
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   164
  case True then show ?thesis by simp
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next
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  case False
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   167
  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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   173
  then show ?thesis by (simp add: normalize_mult)
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   174
qed
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   175
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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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   178
  using gcd_mult_left [of c a b] by (simp add: ac_simps)
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   179
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lemma mult_gcd_left:
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   181
  "c * gcd a b = unit_factor c * gcd (c * a) (c * b)"
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   182
  by (simp add: gcd_mult_left mult.assoc [symmetric])
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   183
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   184
lemma mult_gcd_right:
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  "gcd a b * c = gcd (a * c) (b * c) * unit_factor c"
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   186
  using mult_gcd_left [of c a b] by (simp add: ac_simps)
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   187
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lemma dvd_lcm1 [iff]:
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  "a dvd lcm a b"
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   190
proof -
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   191
  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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   193
  then show ?thesis
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   194
    by (simp add: lcm_gcd)
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   195
qed
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   196
  
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lemma dvd_lcm2 [iff]:
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   198
  "b dvd lcm a b"
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   199
proof -
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   200
  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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   202
  then show ?thesis
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   203
    by (simp add: lcm_gcd)
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   204
qed
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   205
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lemma dvd_lcmI1:
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  "a dvd b \<Longrightarrow> a dvd lcm b c"
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   208
  by (rule dvd_trans) (assumption, blast) 
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   209
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   210
lemma dvd_lcmI2:
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   211
  "a dvd c \<Longrightarrow> a dvd lcm b c"
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   212
  by (rule dvd_trans) (assumption, blast)
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   213
62345
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lemma lcm_dvdD1:
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  "lcm a b dvd c \<Longrightarrow> a dvd c"
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   216
  using dvd_lcm1 [of a b] by (blast intro: dvd_trans)
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   217
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lemma lcm_dvdD2:
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  "lcm a b dvd c \<Longrightarrow> b dvd c"
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  using dvd_lcm2 [of a b] by (blast intro: dvd_trans)
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   221
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lemma lcm_least:
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   223
  assumes "a dvd c" and "b dvd c"
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  shows "lcm a b dvd c"
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   225
proof (cases "c = 0")
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   226
  case True then show ?thesis by simp
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   227
next
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   228
  case False then have U: "is_unit (unit_factor c)" by simp
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   229
  show ?thesis
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   230
  proof (cases "gcd a b = 0")
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   231
    case True with assms show ?thesis by simp
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   232
  next
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   233
    case False then have "a \<noteq> 0 \<or> b \<noteq> 0" by simp
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   234
    with \<open>c \<noteq> 0\<close> assms have "a * b dvd a * c" "a * b dvd c * b"
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      by (simp_all add: mult_dvd_mono)
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   236
    then have "normalize (a * b) dvd gcd (a * c) (b * c)"
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   237
      by (auto intro: gcd_greatest simp add: ac_simps)
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   238
    then have "normalize (a * b) dvd gcd (a * c) (b * c) * unit_factor c"
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   239
      using U by (simp add: dvd_mult_unit_iff)
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   240
    then have "normalize (a * b) dvd gcd a b * c"
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   241
      by (simp add: mult_gcd_right [of a b c])
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   242
    then have "normalize (a * b) div gcd a b dvd c"
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diff changeset
   243
      using False by (simp add: div_dvd_iff_mult ac_simps)
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   244
    then show ?thesis by (simp add: lcm_gcd)
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   245
  qed
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   246
qed
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   247
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   248
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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   250
  by (blast intro!: lcm_least intro: dvd_trans)
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   251
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   252
lemma normalize_lcm [simp]:
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  "normalize (lcm a b) = lcm a b"
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   254
  by (simp add: lcm_gcd dvd_normalize_div)
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   255
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   256
lemma lcm_0_left [simp]:
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  "lcm 0 a = 0"
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  by (simp add: lcm_gcd)
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   259
  
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lemma lcm_0_right [simp]:
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   261
  "lcm a 0 = 0"
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   262
  by (simp add: lcm_gcd)
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   263
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   264
lemma lcm_eq_0_iff:
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   265
  "lcm a b = 0 \<longleftrightarrow> a = 0 \<or> b = 0" (is "?P \<longleftrightarrow> ?Q")
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   266
proof
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   267
  assume ?P then have "0 dvd lcm a b" by simp
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   268
  then have "0 dvd normalize (a * b) div gcd a b"
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   269
    by (simp add: lcm_gcd)
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   270
  then have "0 * gcd a b dvd normalize (a * b)"
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   271
    using dvd_div_iff_mult [of "gcd a b" _ 0] by (cases "gcd a b = 0") simp_all
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   272
  then have "normalize (a * b) = 0"
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   273
    by simp
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diff changeset
   274
  then show ?Q by simp
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   275
next
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   276
  assume ?Q then show ?P by auto
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   277
qed
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   278
61913
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   279
lemma lcm_eq_1_iff [simp]:
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   280
  "lcm a b = 1 \<longleftrightarrow> is_unit a \<and> is_unit b"
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   281
  by (auto intro: associated_eqI)
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diff changeset
   282
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   283
lemma unit_factor_lcm :
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   284
  "unit_factor (lcm a b) = (if a = 0 \<or> b = 0 then 0 else 1)"
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   285
  by (simp add: unit_factor_gcd dvd_unit_factor_div lcm_gcd)
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diff changeset
   286
61605
1bf7b186542e qualifier is mandatory by default;
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   287
sublocale lcm: abel_semigroup lcm
60686
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   288
proof
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diff changeset
   289
  fix a b c
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   290
  show "lcm a b = lcm b a"
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   291
    by (simp add: lcm_gcd ac_simps normalize_mult dvd_normalize_div)
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diff changeset
   292
  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
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   293
    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
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diff changeset
   294
    by (auto intro: lcm_least
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   295
      dvd_trans [of b "lcm b c" "lcm a (lcm b c)"]
ea5bc46c11e6 more algebraic properties for gcd/lcm
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diff changeset
   296
      dvd_trans [of c "lcm b c" "lcm a (lcm b c)"]
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diff changeset
   297
      dvd_trans [of a "lcm a b" "lcm (lcm a b) c"]
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diff changeset
   298
      dvd_trans [of b "lcm a b" "lcm (lcm a b) c"])
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diff changeset
   299
  then show "lcm (lcm a b) c = lcm a (lcm b c)"
60688
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   300
    by (rule associated_eqI) simp_all
60686
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diff changeset
   301
qed
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   302
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   303
lemma lcm_self [simp]:
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   304
  "lcm a a = normalize a"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   305
proof -
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   306
  have "lcm a a dvd a"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   307
    by (rule lcm_least) simp_all
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   308
  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
   309
    by (auto intro: associated_eqI)
60686
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   310
qed
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   311
61913
58b153bfa737 tuned proofs and augmented some lemmas
haftmann
parents: 61856
diff changeset
   312
lemma lcm_left_idem [simp]:
58b153bfa737 tuned proofs and augmented some lemmas
haftmann
parents: 61856
diff changeset
   313
  "lcm a (lcm a b) = lcm a b"
58b153bfa737 tuned proofs and augmented some lemmas
haftmann
parents: 61856
diff changeset
   314
  by (auto intro: associated_eqI)
58b153bfa737 tuned proofs and augmented some lemmas
haftmann
parents: 61856
diff changeset
   315
58b153bfa737 tuned proofs and augmented some lemmas
haftmann
parents: 61856
diff changeset
   316
lemma lcm_right_idem [simp]:
58b153bfa737 tuned proofs and augmented some lemmas
haftmann
parents: 61856
diff changeset
   317
  "lcm (lcm a b) b = lcm a b"
58b153bfa737 tuned proofs and augmented some lemmas
haftmann
parents: 61856
diff changeset
   318
  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
   319
60686
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   320
lemma gcd_mult_lcm [simp]:
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   321
  "gcd a b * lcm a b = normalize a * normalize b"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   322
  by (simp add: lcm_gcd normalize_mult)
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   323
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   324
lemma lcm_mult_gcd [simp]:
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   325
  "lcm a b * gcd a b = normalize a * normalize b"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   326
  using gcd_mult_lcm [of a b] by (simp add: ac_simps) 
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   327
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   328
lemma gcd_lcm:
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   329
  assumes "a \<noteq> 0" and "b \<noteq> 0"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   330
  shows "gcd a b = normalize (a * b) div lcm a b"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   331
proof -
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   332
  from assms have "lcm a b \<noteq> 0"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   333
    by (simp add: lcm_eq_0_iff)
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   334
  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
   335
  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
   336
    by (simp_all add: normalize_mult)
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   337
  with \<open>lcm a b \<noteq> 0\<close> show ?thesis
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   338
    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
   339
qed
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   340
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   341
lemma lcm_1_left [simp]:
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   342
  "lcm 1 a = normalize a"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   343
  by (simp add: lcm_gcd)
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   344
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   345
lemma lcm_1_right [simp]:
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   346
  "lcm a 1 = normalize a"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   347
  by (simp add: lcm_gcd)
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   348
  
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   349
lemma lcm_mult_left:
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   350
  "lcm (c * a) (c * b) = normalize c * lcm a b"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   351
  by (cases "c = 0")
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   352
    (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
   353
      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
   354
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   355
lemma lcm_mult_right:
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   356
  "lcm (a * c) (b * c) = lcm b a * normalize c"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   357
  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
   358
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   359
lemma mult_lcm_left:
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   360
  "c * lcm a b = unit_factor c * lcm (c * a) (c * b)"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   361
  by (simp add: lcm_mult_left mult.assoc [symmetric])
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   362
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   363
lemma mult_lcm_right:
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   364
  "lcm a b * c = lcm (a * c) (b * c) * unit_factor c"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   365
  using mult_lcm_left [of c a b] by (simp add: ac_simps)
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   366
  
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   367
end
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   368
62345
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   369
class ring_gcd = comm_ring_1 + semiring_gcd
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   370
60686
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   371
class semiring_Gcd = semiring_gcd + Gcd +
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   372
  assumes Gcd_dvd: "a \<in> A \<Longrightarrow> Gcd A dvd a"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   373
    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
   374
    and normalize_Gcd [simp]: "normalize (Gcd A) = Gcd A"
62345
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   375
  assumes dvd_Lcm: "a \<in> A \<Longrightarrow> a dvd Lcm A"
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   376
    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
   377
    and normalize_Lcm [simp]: "normalize (Lcm A) = Lcm A"
60686
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   378
begin
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   379
62345
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   380
lemma Lcm_Gcd:
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   381
  "Lcm A = Gcd {b. \<forall>a\<in>A. a dvd b}"
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   382
  by (rule associated_eqI) (auto intro: Gcd_dvd dvd_Lcm Gcd_greatest Lcm_least)
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   383
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   384
lemma Gcd_Lcm:
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   385
  "Gcd A = Lcm {b. \<forall>a\<in>A. b dvd a}"
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   386
  by (rule associated_eqI) (auto intro: Gcd_dvd dvd_Lcm Gcd_greatest Lcm_least)
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   387
60686
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   388
lemma Gcd_empty [simp]:
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   389
  "Gcd {} = 0"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   390
  by (rule dvd_0_left, rule Gcd_greatest) simp
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   391
62345
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   392
lemma Lcm_empty [simp]:
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   393
  "Lcm {} = 1"
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   394
  by (auto intro: associated_eqI Lcm_least)
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   395
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   396
lemma Gcd_insert [simp]:
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   397
  "Gcd (insert a A) = gcd a (Gcd A)"
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   398
proof -
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   399
  have "Gcd (insert a A) dvd gcd a (Gcd A)"
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   400
    by (auto intro: Gcd_dvd Gcd_greatest)
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   401
  moreover have "gcd a (Gcd A) dvd Gcd (insert a A)"
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   402
  proof (rule Gcd_greatest)
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   403
    fix b
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   404
    assume "b \<in> insert a A"
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   405
    then show "gcd a (Gcd A) dvd b"
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   406
    proof
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   407
      assume "b = a" then show ?thesis by simp
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   408
    next
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   409
      assume "b \<in> A"
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   410
      then have "Gcd A dvd b" by (rule Gcd_dvd)
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   411
      moreover have "gcd a (Gcd A) dvd Gcd A" by simp
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   412
      ultimately show ?thesis by (blast intro: dvd_trans)
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   413
    qed
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   414
  qed
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   415
  ultimately show ?thesis
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   416
    by (auto intro: associated_eqI)
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   417
qed
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   418
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   419
lemma Lcm_insert [simp]:
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   420
  "Lcm (insert a A) = lcm a (Lcm A)"
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   421
proof (rule sym)
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   422
  have "lcm a (Lcm A) dvd Lcm (insert a A)"
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   423
    by (auto intro: dvd_Lcm Lcm_least)
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   424
  moreover have "Lcm (insert a A) dvd lcm a (Lcm A)"
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   425
  proof (rule Lcm_least)
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   426
    fix b
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   427
    assume "b \<in> insert a A"
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   428
    then show "b dvd lcm a (Lcm A)"
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   429
    proof
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   430
      assume "b = a" then show ?thesis by simp
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   431
    next
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   432
      assume "b \<in> A"
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   433
      then have "b dvd Lcm A" by (rule dvd_Lcm)
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   434
      moreover have "Lcm A dvd lcm a (Lcm A)" by simp
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   435
      ultimately show ?thesis by (blast intro: dvd_trans)
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   436
    qed
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   437
  qed
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   438
  ultimately show "lcm a (Lcm A) = Lcm (insert a A)"
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   439
    by (rule associated_eqI) (simp_all add: lcm_eq_0_iff)
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   440
qed
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   441
60686
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   442
lemma Gcd_0_iff [simp]:
60687
33dbbcb6a8a3 eliminated some duplication
haftmann
parents: 60686
diff changeset
   443
  "Gcd A = 0 \<longleftrightarrow> A \<subseteq> {0}" (is "?P \<longleftrightarrow> ?Q")
60686
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   444
proof
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   445
  assume ?P
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   446
  show ?Q
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   447
  proof
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   448
    fix a
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   449
    assume "a \<in> A"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   450
    then have "Gcd A dvd a" by (rule Gcd_dvd)
60687
33dbbcb6a8a3 eliminated some duplication
haftmann
parents: 60686
diff changeset
   451
    with \<open>?P\<close> have "a = 0" by simp
33dbbcb6a8a3 eliminated some duplication
haftmann
parents: 60686
diff changeset
   452
    then show "a \<in> {0}" by simp
60686
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   453
  qed
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   454
next
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   455
  assume ?Q
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   456
  have "0 dvd Gcd A"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   457
  proof (rule Gcd_greatest)
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   458
    fix a
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   459
    assume "a \<in> A"
60687
33dbbcb6a8a3 eliminated some duplication
haftmann
parents: 60686
diff changeset
   460
    with \<open>?Q\<close> have "a = 0" by auto
60686
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   461
    then show "0 dvd a" by simp
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   462
  qed
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   463
  then show ?P by simp
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   464
qed
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   465
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   466
lemma Lcm_1_iff [simp]:
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   467
  "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
   468
proof
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   469
  assume ?P
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   470
  show ?Q
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   471
  proof
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   472
    fix a
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   473
    assume "a \<in> A"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   474
    then have "a dvd Lcm A"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   475
      by (rule dvd_Lcm)
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   476
    with \<open>?P\<close> show "is_unit a"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   477
      by simp
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   478
  qed
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   479
next
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   480
  assume ?Q
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   481
  then have "is_unit (Lcm A)"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   482
    by (blast intro: Lcm_least)
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   483
  then have "normalize (Lcm A) = 1"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   484
    by (rule is_unit_normalize)
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   485
  then show ?P
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   486
    by simp
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   487
qed
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   488
62345
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   489
lemma unit_factor_Gcd:
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   490
  "unit_factor (Gcd A) = (if \<forall>a\<in>A. a = 0 then 0 else 1)"
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   491
proof (cases "Gcd A = 0")
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   492
  case True then show ?thesis by auto
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   493
next
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   494
  case False
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   495
  from unit_factor_mult_normalize
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   496
  have "unit_factor (Gcd A) * normalize (Gcd A) = Gcd A" .
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   497
  then have "unit_factor (Gcd A) * Gcd A = Gcd A" by simp
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   498
  then have "unit_factor (Gcd A) * Gcd A div Gcd A = Gcd A div Gcd A" by simp
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   499
  with False have "unit_factor (Gcd A) = 1" by simp
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   500
  with False show ?thesis by auto
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   501
qed
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   502
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   503
lemma unit_factor_Lcm:
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   504
  "unit_factor (Lcm A) = (if Lcm A = 0 then 0 else 1)"
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   505
proof (cases "Lcm A = 0")
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   506
  case True then show ?thesis by simp
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   507
next
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   508
  case False
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   509
  with unit_factor_normalize have "unit_factor (normalize (Lcm A)) = 1"
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   510
    by blast
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   511
  with False show ?thesis
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   512
    by simp
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   513
qed
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   514
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   515
lemma Gcd_eq_1_I:
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   516
  assumes "is_unit a" and "a \<in> A"
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   517
  shows "Gcd A = 1"
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   518
proof -
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   519
  from assms have "is_unit (Gcd A)"
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   520
    by (blast intro: Gcd_dvd dvd_unit_imp_unit)
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   521
  then have "normalize (Gcd A) = 1"
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   522
    by (rule is_unit_normalize)
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   523
  then show ?thesis
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   524
    by simp
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   525
qed
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   526
60686
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   527
lemma Lcm_eq_0_I:
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   528
  assumes "0 \<in> A"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   529
  shows "Lcm A = 0"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   530
proof -
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   531
  from assms have "0 dvd Lcm A"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   532
    by (rule dvd_Lcm)
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   533
  then show ?thesis
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   534
    by simp
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   535
qed
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   536
62345
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   537
lemma Gcd_UNIV [simp]:
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   538
  "Gcd UNIV = 1"
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   539
  using dvd_refl by (rule Gcd_eq_1_I) simp
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   540
61929
b8e242e52c97 tuned proofs and augmented lemmas
haftmann
parents: 61913
diff changeset
   541
lemma Lcm_UNIV [simp]:
b8e242e52c97 tuned proofs and augmented lemmas
haftmann
parents: 61913
diff changeset
   542
  "Lcm UNIV = 0"
b8e242e52c97 tuned proofs and augmented lemmas
haftmann
parents: 61913
diff changeset
   543
  by (rule Lcm_eq_0_I) simp
60686
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   544
61929
b8e242e52c97 tuned proofs and augmented lemmas
haftmann
parents: 61913
diff changeset
   545
lemma Lcm_0_iff:
b8e242e52c97 tuned proofs and augmented lemmas
haftmann
parents: 61913
diff changeset
   546
  assumes "finite A"
b8e242e52c97 tuned proofs and augmented lemmas
haftmann
parents: 61913
diff changeset
   547
  shows "Lcm A = 0 \<longleftrightarrow> 0 \<in> A"
b8e242e52c97 tuned proofs and augmented lemmas
haftmann
parents: 61913
diff changeset
   548
proof (cases "A = {}")
b8e242e52c97 tuned proofs and augmented lemmas
haftmann
parents: 61913
diff changeset
   549
  case True then show ?thesis by simp
b8e242e52c97 tuned proofs and augmented lemmas
haftmann
parents: 61913
diff changeset
   550
next
b8e242e52c97 tuned proofs and augmented lemmas
haftmann
parents: 61913
diff changeset
   551
  case False with assms show ?thesis
b8e242e52c97 tuned proofs and augmented lemmas
haftmann
parents: 61913
diff changeset
   552
    by (induct A rule: finite_ne_induct)
b8e242e52c97 tuned proofs and augmented lemmas
haftmann
parents: 61913
diff changeset
   553
      (auto simp add: lcm_eq_0_iff)
60686
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   554
qed
61929
b8e242e52c97 tuned proofs and augmented lemmas
haftmann
parents: 61913
diff changeset
   555
62345
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   556
lemma dvd_Gcd: \<comment> \<open>FIXME remove\<close>
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   557
  "\<forall>b\<in>A. a dvd b \<Longrightarrow> a dvd Gcd A"
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   558
  by (blast intro: Gcd_greatest)
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   559
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   560
lemma Gcd_set [code_unfold]:
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   561
  "Gcd (set as) = foldr gcd as 0"
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   562
  by (induct as) simp_all
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   563
60686
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   564
lemma Lcm_set [code_unfold]:
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   565
  "Lcm (set as) = foldr lcm as 1"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   566
  by (induct as) simp_all
59008
f61482b0f240 formally self-contained gcd type classes
haftmann
parents: 58889
diff changeset
   567
62345
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   568
lemma Gcd_image_normalize [simp]:
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   569
  "Gcd (normalize ` A) = Gcd A"
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   570
proof -
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   571
  have "Gcd (normalize ` A) dvd a" if "a \<in> A" for a
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   572
  proof -
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   573
    from that obtain B where "A = insert a B" by blast
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   574
    moreover have " gcd (normalize a) (Gcd (normalize ` B)) dvd normalize a"
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   575
      by (rule gcd_dvd1)
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   576
    ultimately show "Gcd (normalize ` A) dvd a"
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   577
      by simp
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   578
  qed
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   579
  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
   580
    by (auto intro!: Gcd_greatest intro: Gcd_dvd)
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   581
  then show ?thesis
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   582
    by (auto intro: associated_eqI)
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   583
qed
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   584
62346
97f2ed240431 more theorems concerning gcd/lcm/Gcd/Lcm
haftmann
parents: 62345
diff changeset
   585
lemma Gcd_eqI:
97f2ed240431 more theorems concerning gcd/lcm/Gcd/Lcm
haftmann
parents: 62345
diff changeset
   586
  assumes "normalize a = a"
97f2ed240431 more theorems concerning gcd/lcm/Gcd/Lcm
haftmann
parents: 62345
diff changeset
   587
  assumes "\<And>b. b \<in> A \<Longrightarrow> a dvd b"
97f2ed240431 more theorems concerning gcd/lcm/Gcd/Lcm
haftmann
parents: 62345
diff changeset
   588
    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
   589
  shows "Gcd A = a"
97f2ed240431 more theorems concerning gcd/lcm/Gcd/Lcm
haftmann
parents: 62345
diff changeset
   590
  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
   591
97f2ed240431 more theorems concerning gcd/lcm/Gcd/Lcm
haftmann
parents: 62345
diff changeset
   592
lemma Lcm_eqI:
97f2ed240431 more theorems concerning gcd/lcm/Gcd/Lcm
haftmann
parents: 62345
diff changeset
   593
  assumes "normalize a = a"
97f2ed240431 more theorems concerning gcd/lcm/Gcd/Lcm
haftmann
parents: 62345
diff changeset
   594
  assumes "\<And>b. b \<in> A \<Longrightarrow> b dvd a"
97f2ed240431 more theorems concerning gcd/lcm/Gcd/Lcm
haftmann
parents: 62345
diff changeset
   595
    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
   596
  shows "Lcm A = a"
97f2ed240431 more theorems concerning gcd/lcm/Gcd/Lcm
haftmann
parents: 62345
diff changeset
   597
  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
   598
62345
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   599
end  
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   600
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   601
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   602
subsection \<open>GCD and LCM on @{typ nat} and @{typ int}\<close>
59008
f61482b0f240 formally self-contained gcd type classes
haftmann
parents: 58889
diff changeset
   603
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   604
instantiation nat :: gcd
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   605
begin
21256
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   606
62345
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   607
fun gcd_nat  :: "nat \<Rightarrow> nat \<Rightarrow> nat"
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   608
where "gcd_nat x y =
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   609
  (if y = 0 then x else gcd y (x mod y))"
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   610
62345
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   611
definition lcm_nat :: "nat \<Rightarrow> nat \<Rightarrow> nat"
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   612
where
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   613
  "lcm_nat x y = x * y div (gcd x y)"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   614
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   615
instance proof qed
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   616
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   617
end
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   618
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   619
instantiation int :: gcd
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   620
begin
21256
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   621
62345
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   622
definition gcd_int  :: "int \<Rightarrow> int \<Rightarrow> int"
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   623
  where "gcd_int x y = int (gcd (nat \<bar>x\<bar>) (nat \<bar>y\<bar>))"
23687
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   624
62345
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   625
definition lcm_int :: "int \<Rightarrow> int \<Rightarrow> int"
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   626
  where "lcm_int x y = int (lcm (nat \<bar>x\<bar>) (nat \<bar>y\<bar>))"
23687
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   627
61944
5d06ecfdb472 prefer symbols for "abs";
wenzelm
parents: 61929
diff changeset
   628
instance ..
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   629
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   630
end
23687
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   631
62345
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   632
text \<open>Transfer setup\<close>
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   633
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   634
lemma transfer_nat_int_gcd:
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   635
  "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> gcd (nat x) (nat y) = nat (gcd x y)"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   636
  "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> lcm (nat x) (nat y) = nat (lcm x y)"
32479
521cc9bf2958 some reorganization of number theory
haftmann
parents: 32415
diff changeset
   637
  unfolding gcd_int_def lcm_int_def
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   638
  by auto
23687
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   639
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   640
lemma transfer_nat_int_gcd_closures:
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   641
  "x >= (0::int) \<Longrightarrow> y >= 0 \<Longrightarrow> gcd x y >= 0"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   642
  "x >= (0::int) \<Longrightarrow> y >= 0 \<Longrightarrow> lcm x y >= 0"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   643
  by (auto simp add: gcd_int_def lcm_int_def)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   644
35644
d20cf282342e transfer: avoid camel case
haftmann
parents: 35368
diff changeset
   645
declare transfer_morphism_nat_int[transfer add return:
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   646
    transfer_nat_int_gcd transfer_nat_int_gcd_closures]
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   647
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   648
lemma transfer_int_nat_gcd:
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   649
  "gcd (int x) (int y) = int (gcd x y)"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   650
  "lcm (int x) (int y) = int (lcm x y)"
32479
521cc9bf2958 some reorganization of number theory
haftmann
parents: 32415
diff changeset
   651
  by (unfold gcd_int_def lcm_int_def, auto)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   652
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   653
lemma transfer_int_nat_gcd_closures:
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   654
  "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> gcd x y >= 0"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   655
  "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> lcm x y >= 0"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   656
  by (auto simp add: gcd_int_def lcm_int_def)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   657
35644
d20cf282342e transfer: avoid camel case
haftmann
parents: 35368
diff changeset
   658
declare transfer_morphism_int_nat[transfer add return:
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   659
    transfer_int_nat_gcd transfer_int_nat_gcd_closures]
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   660
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   661
lemma gcd_nat_induct:
23687
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   662
  fixes m n :: nat
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   663
  assumes "\<And>m. P m 0"
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   664
    and "\<And>m n. 0 < n \<Longrightarrow> P n (m mod n) \<Longrightarrow> P m n"
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   665
  shows "P m n"
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   666
  apply (rule gcd_nat.induct)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   667
  apply (case_tac "y = 0")
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   668
  using assms apply simp_all
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   669
done
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   670
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   671
(* specific to int *)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   672
62346
97f2ed240431 more theorems concerning gcd/lcm/Gcd/Lcm
haftmann
parents: 62345
diff changeset
   673
lemma gcd_eq_int_iff:
97f2ed240431 more theorems concerning gcd/lcm/Gcd/Lcm
haftmann
parents: 62345
diff changeset
   674
  "gcd k l = int n \<longleftrightarrow> gcd (nat \<bar>k\<bar>) (nat \<bar>l\<bar>) = n"
97f2ed240431 more theorems concerning gcd/lcm/Gcd/Lcm
haftmann
parents: 62345
diff changeset
   675
  by (simp add: gcd_int_def)
97f2ed240431 more theorems concerning gcd/lcm/Gcd/Lcm
haftmann
parents: 62345
diff changeset
   676
97f2ed240431 more theorems concerning gcd/lcm/Gcd/Lcm
haftmann
parents: 62345
diff changeset
   677
lemma lcm_eq_int_iff:
97f2ed240431 more theorems concerning gcd/lcm/Gcd/Lcm
haftmann
parents: 62345
diff changeset
   678
  "lcm k l = int n \<longleftrightarrow> lcm (nat \<bar>k\<bar>) (nat \<bar>l\<bar>) = n"
97f2ed240431 more theorems concerning gcd/lcm/Gcd/Lcm
haftmann
parents: 62345
diff changeset
   679
  by (simp add: lcm_int_def)
97f2ed240431 more theorems concerning gcd/lcm/Gcd/Lcm
haftmann
parents: 62345
diff changeset
   680
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   681
lemma gcd_neg1_int [simp]: "gcd (-x::int) y = gcd x y"
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   682
  by (simp add: gcd_int_def)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   683
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   684
lemma gcd_neg2_int [simp]: "gcd (x::int) (-y) = gcd x y"
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   685
  by (simp add: gcd_int_def)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   686
61944
5d06ecfdb472 prefer symbols for "abs";
wenzelm
parents: 61929
diff changeset
   687
lemma abs_gcd_int[simp]: "\<bar>gcd (x::int) y\<bar> = gcd x y"
31813
4df828bbc411 gcd abs lemmas
nipkow
parents: 31798
diff changeset
   688
by(simp add: gcd_int_def)
4df828bbc411 gcd abs lemmas
nipkow
parents: 31798
diff changeset
   689
61944
5d06ecfdb472 prefer symbols for "abs";
wenzelm
parents: 61929
diff changeset
   690
lemma gcd_abs_int: "gcd (x::int) y = gcd \<bar>x\<bar> \<bar>y\<bar>"
31813
4df828bbc411 gcd abs lemmas
nipkow
parents: 31798
diff changeset
   691
by (simp add: gcd_int_def)
4df828bbc411 gcd abs lemmas
nipkow
parents: 31798
diff changeset
   692
61944
5d06ecfdb472 prefer symbols for "abs";
wenzelm
parents: 61929
diff changeset
   693
lemma gcd_abs1_int[simp]: "gcd \<bar>x\<bar> (y::int) = gcd x y"
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   694
by (metis abs_idempotent gcd_abs_int)
31813
4df828bbc411 gcd abs lemmas
nipkow
parents: 31798
diff changeset
   695
61944
5d06ecfdb472 prefer symbols for "abs";
wenzelm
parents: 61929
diff changeset
   696
lemma gcd_abs2_int[simp]: "gcd x \<bar>y::int\<bar> = gcd x y"
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   697
by (metis abs_idempotent gcd_abs_int)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   698
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   699
lemma gcd_cases_int:
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   700
  fixes x :: int and y
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   701
  assumes "x >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> P (gcd x y)"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   702
      and "x >= 0 \<Longrightarrow> y <= 0 \<Longrightarrow> P (gcd x (-y))"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   703
      and "x <= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> P (gcd (-x) y)"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   704
      and "x <= 0 \<Longrightarrow> y <= 0 \<Longrightarrow> P (gcd (-x) (-y))"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   705
  shows "P (gcd x y)"
61944
5d06ecfdb472 prefer symbols for "abs";
wenzelm
parents: 61929
diff changeset
   706
  by (insert assms, auto, arith)
21256
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   707
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   708
lemma gcd_ge_0_int [simp]: "gcd (x::int) y >= 0"
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   709
  by (simp add: gcd_int_def)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   710
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   711
lemma lcm_neg1_int: "lcm (-x::int) y = lcm x y"
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   712
  by (simp add: lcm_int_def)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   713
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   714
lemma lcm_neg2_int: "lcm (x::int) (-y) = lcm x y"
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   715
  by (simp add: lcm_int_def)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   716
61944
5d06ecfdb472 prefer symbols for "abs";
wenzelm
parents: 61929
diff changeset
   717
lemma lcm_abs_int: "lcm (x::int) y = lcm \<bar>x\<bar> \<bar>y\<bar>"
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   718
  by (simp add: lcm_int_def)
21256
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   719
61944
5d06ecfdb472 prefer symbols for "abs";
wenzelm
parents: 61929
diff changeset
   720
lemma abs_lcm_int [simp]: "\<bar>lcm i j::int\<bar> = lcm i j"
5d06ecfdb472 prefer symbols for "abs";
wenzelm
parents: 61929
diff changeset
   721
  by (simp add:lcm_int_def)
31814
7c122634da81 lcm abs lemmas
nipkow
parents: 31813
diff changeset
   722
61944
5d06ecfdb472 prefer symbols for "abs";
wenzelm
parents: 61929
diff changeset
   723
lemma lcm_abs1_int[simp]: "lcm \<bar>x\<bar> (y::int) = lcm x y"
5d06ecfdb472 prefer symbols for "abs";
wenzelm
parents: 61929
diff changeset
   724
  by (metis abs_idempotent lcm_int_def)
31814
7c122634da81 lcm abs lemmas
nipkow
parents: 31813
diff changeset
   725
61944
5d06ecfdb472 prefer symbols for "abs";
wenzelm
parents: 61929
diff changeset
   726
lemma lcm_abs2_int[simp]: "lcm x \<bar>y::int\<bar> = lcm x y"
5d06ecfdb472 prefer symbols for "abs";
wenzelm
parents: 61929
diff changeset
   727
  by (metis abs_idempotent lcm_int_def)
31814
7c122634da81 lcm abs lemmas
nipkow
parents: 31813
diff changeset
   728
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   729
lemma lcm_cases_int:
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   730
  fixes x :: int and y
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   731
  assumes "x >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> P (lcm x y)"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   732
      and "x >= 0 \<Longrightarrow> y <= 0 \<Longrightarrow> P (lcm x (-y))"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   733
      and "x <= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> P (lcm (-x) y)"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   734
      and "x <= 0 \<Longrightarrow> y <= 0 \<Longrightarrow> P (lcm (-x) (-y))"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   735
  shows "P (lcm x y)"
41550
efa734d9b221 eliminated global prems;
wenzelm
parents: 37770
diff changeset
   736
  using assms by (auto simp add: lcm_neg1_int lcm_neg2_int) arith
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   737
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   738
lemma lcm_ge_0_int [simp]: "lcm (x::int) y >= 0"
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   739
  by (simp add: lcm_int_def)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   740
54867
c21a2465cac1 prefer ephemeral interpretation over interpretation in proof contexts;
haftmann
parents: 54489
diff changeset
   741
lemma gcd_0_nat: "gcd (x::nat) 0 = x"
23687
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   742
  by simp
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   743
61944
5d06ecfdb472 prefer symbols for "abs";
wenzelm
parents: 61929
diff changeset
   744
lemma gcd_0_int [simp]: "gcd (x::int) 0 = \<bar>x\<bar>"
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   745
  by (unfold gcd_int_def, auto)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   746
54867
c21a2465cac1 prefer ephemeral interpretation over interpretation in proof contexts;
haftmann
parents: 54489
diff changeset
   747
lemma gcd_0_left_nat: "gcd 0 (x::nat) = x"
23687
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   748
  by simp
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   749
61944
5d06ecfdb472 prefer symbols for "abs";
wenzelm
parents: 61929
diff changeset
   750
lemma gcd_0_left_int [simp]: "gcd 0 (x::int) = \<bar>x\<bar>"
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   751
  by (unfold gcd_int_def, auto)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   752
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   753
lemma gcd_red_nat: "gcd (x::nat) y = gcd y (x mod y)"
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   754
  by (case_tac "y = 0", auto)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   755
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   756
(* weaker, but useful for the simplifier *)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   757
62345
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   758
lemma gcd_non_0_nat: "y \<noteq> (0::nat) \<Longrightarrow> gcd (x::nat) y = gcd y (x mod y)"
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   759
  by simp
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   760
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   761
lemma gcd_1_nat [simp]: "gcd (m::nat) 1 = 1"
21263
wenzelm
parents: 21256
diff changeset
   762
  by simp
21256
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   763
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   764
lemma gcd_Suc_0 [simp]: "gcd (m::nat) (Suc 0) = Suc 0"
60690
a9e45c9588c3 tuned facts
haftmann
parents: 60689
diff changeset
   765
  by simp
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   766
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   767
lemma gcd_1_int [simp]: "gcd (m::int) 1 = 1"
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   768
  by (simp add: gcd_int_def)
30082
43c5b7bfc791 make more proofs work whether or not One_nat_def is a simp rule
huffman
parents: 30042
diff changeset
   769
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   770
lemma gcd_idem_nat: "gcd (x::nat) x = x"
31798
fe9a3043d36c Cleaned up GCD
nipkow
parents: 31766
diff changeset
   771
by simp
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   772
61944
5d06ecfdb472 prefer symbols for "abs";
wenzelm
parents: 61929
diff changeset
   773
lemma gcd_idem_int: "gcd (x::int) x = \<bar>x\<bar>"
31813
4df828bbc411 gcd abs lemmas
nipkow
parents: 31798
diff changeset
   774
by (auto simp add: gcd_int_def)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   775
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   776
declare gcd_nat.simps [simp del]
21256
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   777
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
   778
text \<open>
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61649
diff changeset
   779
  \medskip @{term "gcd m n"} divides \<open>m\<close> and \<open>n\<close>.  The
21256
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   780
  conjunctions don't seem provable separately.
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
   781
\<close>
21256
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   782
59008
f61482b0f240 formally self-contained gcd type classes
haftmann
parents: 58889
diff changeset
   783
instance nat :: semiring_gcd
f61482b0f240 formally self-contained gcd type classes
haftmann
parents: 58889
diff changeset
   784
proof
f61482b0f240 formally self-contained gcd type classes
haftmann
parents: 58889
diff changeset
   785
  fix m n :: nat
f61482b0f240 formally self-contained gcd type classes
haftmann
parents: 58889
diff changeset
   786
  show "gcd m n dvd m" and "gcd m n dvd n"
f61482b0f240 formally self-contained gcd type classes
haftmann
parents: 58889
diff changeset
   787
  proof (induct m n rule: gcd_nat_induct)
f61482b0f240 formally self-contained gcd type classes
haftmann
parents: 58889
diff changeset
   788
    fix m n :: nat
f61482b0f240 formally self-contained gcd type classes
haftmann
parents: 58889
diff changeset
   789
    assume "gcd n (m mod n) dvd m mod n" and "gcd n (m mod n) dvd n"
f61482b0f240 formally self-contained gcd type classes
haftmann
parents: 58889
diff changeset
   790
    then have "gcd n (m mod n) dvd m"
f61482b0f240 formally self-contained gcd type classes
haftmann
parents: 58889
diff changeset
   791
      by (rule dvd_mod_imp_dvd)
f61482b0f240 formally self-contained gcd type classes
haftmann
parents: 58889
diff changeset
   792
    moreover assume "0 < n"
f61482b0f240 formally self-contained gcd type classes
haftmann
parents: 58889
diff changeset
   793
    ultimately show "gcd m n dvd m"
f61482b0f240 formally self-contained gcd type classes
haftmann
parents: 58889
diff changeset
   794
      by (simp add: gcd_non_0_nat)
f61482b0f240 formally self-contained gcd type classes
haftmann
parents: 58889
diff changeset
   795
  qed (simp_all add: gcd_0_nat gcd_non_0_nat)
f61482b0f240 formally self-contained gcd type classes
haftmann
parents: 58889
diff changeset
   796
next
f61482b0f240 formally self-contained gcd type classes
haftmann
parents: 58889
diff changeset
   797
  fix m n k :: nat
f61482b0f240 formally self-contained gcd type classes
haftmann
parents: 58889
diff changeset
   798
  assume "k dvd m" and "k dvd n"
f61482b0f240 formally self-contained gcd type classes
haftmann
parents: 58889
diff changeset
   799
  then show "k dvd gcd m n"
f61482b0f240 formally self-contained gcd type classes
haftmann
parents: 58889
diff changeset
   800
    by (induct m n rule: gcd_nat_induct) (simp_all add: gcd_non_0_nat dvd_mod gcd_0_nat)
60686
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   801
qed (simp_all add: lcm_nat_def)
59667
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59545
diff changeset
   802
59008
f61482b0f240 formally self-contained gcd type classes
haftmann
parents: 58889
diff changeset
   803
instance int :: ring_gcd
60686
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   804
  by standard
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   805
    (simp_all add: dvd_int_unfold_dvd_nat gcd_int_def lcm_int_def zdiv_int nat_abs_mult_distrib [symmetric] lcm_gcd gcd_greatest)
59667
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59545
diff changeset
   806
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   807
lemma gcd_le1_nat [simp]: "a \<noteq> 0 \<Longrightarrow> gcd (a::nat) b \<le> a"
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   808
  by (rule dvd_imp_le, auto)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   809
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   810
lemma gcd_le2_nat [simp]: "b \<noteq> 0 \<Longrightarrow> gcd (a::nat) b \<le> b"
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   811
  by (rule dvd_imp_le, auto)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   812
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   813
lemma gcd_le1_int [simp]: "a > 0 \<Longrightarrow> gcd (a::int) b \<le> a"
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   814
  by (rule zdvd_imp_le, auto)
21256
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   815
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   816
lemma gcd_le2_int [simp]: "b > 0 \<Longrightarrow> gcd (a::int) b \<le> b"
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   817
  by (rule zdvd_imp_le, auto)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   818
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   819
lemma gcd_pos_nat [simp]: "(gcd (m::nat) n > 0) = (m ~= 0 | n ~= 0)"
62344
759d684c0e60 pulled out legacy aliasses and infamous dvd interpretations into theory appendix
haftmann
parents: 62343
diff changeset
   820
  by (insert gcd_eq_0_iff [of m n], arith)
21256
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   821
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   822
lemma gcd_pos_int [simp]: "(gcd (m::int) n > 0) = (m ~= 0 | n ~= 0)"
62344
759d684c0e60 pulled out legacy aliasses and infamous dvd interpretations into theory appendix
haftmann
parents: 62343
diff changeset
   823
  by (insert gcd_eq_0_iff [of m n], insert gcd_ge_0_int [of m n], arith)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   824
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   825
lemma gcd_unique_nat: "(d::nat) dvd a \<and> d dvd b \<and>
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   826
    (\<forall>e. e dvd a \<and> e dvd b \<longrightarrow> e dvd d) \<longleftrightarrow> d = gcd a b"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   827
  apply auto
33657
a4179bf442d1 renamed lemmas "anti_sym" -> "antisym"
nipkow
parents: 33318
diff changeset
   828
  apply (rule dvd_antisym)
59008
f61482b0f240 formally self-contained gcd type classes
haftmann
parents: 58889
diff changeset
   829
  apply (erule (1) gcd_greatest)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   830
  apply auto
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   831
done
21256
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   832
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   833
lemma gcd_unique_int: "d >= 0 & (d::int) dvd a \<and> d dvd b \<and>
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   834
    (\<forall>e. e dvd a \<and> e dvd b \<longrightarrow> e dvd d) \<longleftrightarrow> d = gcd a b"
33657
a4179bf442d1 renamed lemmas "anti_sym" -> "antisym"
nipkow
parents: 33318
diff changeset
   835
apply (case_tac "d = 0")
a4179bf442d1 renamed lemmas "anti_sym" -> "antisym"
nipkow
parents: 33318
diff changeset
   836
 apply simp
a4179bf442d1 renamed lemmas "anti_sym" -> "antisym"
nipkow
parents: 33318
diff changeset
   837
apply (rule iffI)
a4179bf442d1 renamed lemmas "anti_sym" -> "antisym"
nipkow
parents: 33318
diff changeset
   838
 apply (rule zdvd_antisym_nonneg)
59008
f61482b0f240 formally self-contained gcd type classes
haftmann
parents: 58889
diff changeset
   839
 apply (auto intro: gcd_greatest)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   840
done
30082
43c5b7bfc791 make more proofs work whether or not One_nat_def is a simp rule
huffman
parents: 30042
diff changeset
   841
61913
58b153bfa737 tuned proofs and augmented some lemmas
haftmann
parents: 61856
diff changeset
   842
interpretation gcd_nat:
62344
759d684c0e60 pulled out legacy aliasses and infamous dvd interpretations into theory appendix
haftmann
parents: 62343
diff changeset
   843
  semilattice_neutr_order gcd "0::nat" Rings.dvd "\<lambda>m n. m dvd n \<and> m \<noteq> n"
759d684c0e60 pulled out legacy aliasses and infamous dvd interpretations into theory appendix
haftmann
parents: 62343
diff changeset
   844
  by standard (auto simp add: gcd_unique_nat [symmetric] intro: dvd_antisym dvd_trans)
31798
fe9a3043d36c Cleaned up GCD
nipkow
parents: 31766
diff changeset
   845
61944
5d06ecfdb472 prefer symbols for "abs";
wenzelm
parents: 61929
diff changeset
   846
lemma gcd_proj1_if_dvd_int [simp]: "x dvd y \<Longrightarrow> gcd (x::int) y = \<bar>x\<bar>"
54867
c21a2465cac1 prefer ephemeral interpretation over interpretation in proof contexts;
haftmann
parents: 54489
diff changeset
   847
  by (metis abs_dvd_iff gcd_0_left_int gcd_abs_int gcd_unique_int)
31798
fe9a3043d36c Cleaned up GCD
nipkow
parents: 31766
diff changeset
   848
61944
5d06ecfdb472 prefer symbols for "abs";
wenzelm
parents: 61929
diff changeset
   849
lemma gcd_proj2_if_dvd_int [simp]: "y dvd x \<Longrightarrow> gcd (x::int) y = \<bar>y\<bar>"
62344
759d684c0e60 pulled out legacy aliasses and infamous dvd interpretations into theory appendix
haftmann
parents: 62343
diff changeset
   850
  by (metis gcd_proj1_if_dvd_int gcd.commute)
31798
fe9a3043d36c Cleaned up GCD
nipkow
parents: 31766
diff changeset
   851
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
   852
text \<open>
21256
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   853
  \medskip Multiplication laws
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
   854
\<close>
21256
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   855
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   856
lemma gcd_mult_distrib_nat: "(k::nat) * gcd m n = gcd (k * m) (k * n)"
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61649
diff changeset
   857
    \<comment> \<open>@{cite \<open>page 27\<close> davenport92}\<close>
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   858
  apply (induct m n rule: gcd_nat_induct)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   859
  apply simp
21256
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   860
  apply (case_tac "k = 0")
45270
d5b5c9259afd fix bug in cancel_factor simprocs so they will work on goals like 'x * y < x * z' where the common term is already on the left
huffman
parents: 45264
diff changeset
   861
  apply (simp_all add: gcd_non_0_nat)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   862
done
21256
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   863
61944
5d06ecfdb472 prefer symbols for "abs";
wenzelm
parents: 61929
diff changeset
   864
lemma gcd_mult_distrib_int: "\<bar>k::int\<bar> * gcd m n = gcd (k * m) (k * n)"
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   865
  apply (subst (1 2) gcd_abs_int)
31813
4df828bbc411 gcd abs lemmas
nipkow
parents: 31798
diff changeset
   866
  apply (subst (1 2) abs_mult)
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   867
  apply (rule gcd_mult_distrib_nat [transferred])
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   868
  apply auto
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   869
done
21256
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   870
60690
a9e45c9588c3 tuned facts
haftmann
parents: 60689
diff changeset
   871
context semiring_gcd
a9e45c9588c3 tuned facts
haftmann
parents: 60689
diff changeset
   872
begin
21256
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   873
60690
a9e45c9588c3 tuned facts
haftmann
parents: 60689
diff changeset
   874
lemma coprime_dvd_mult:
a9e45c9588c3 tuned facts
haftmann
parents: 60689
diff changeset
   875
  assumes "coprime a b" and "a dvd c * b"
a9e45c9588c3 tuned facts
haftmann
parents: 60689
diff changeset
   876
  shows "a dvd c"
a9e45c9588c3 tuned facts
haftmann
parents: 60689
diff changeset
   877
proof (cases "c = 0")
a9e45c9588c3 tuned facts
haftmann
parents: 60689
diff changeset
   878
  case True then show ?thesis by simp
a9e45c9588c3 tuned facts
haftmann
parents: 60689
diff changeset
   879
next
a9e45c9588c3 tuned facts
haftmann
parents: 60689
diff changeset
   880
  case False
a9e45c9588c3 tuned facts
haftmann
parents: 60689
diff changeset
   881
  then have unit: "is_unit (unit_factor c)" by simp
a9e45c9588c3 tuned facts
haftmann
parents: 60689
diff changeset
   882
  from \<open>coprime a b\<close> mult_gcd_left [of c a b]
a9e45c9588c3 tuned facts
haftmann
parents: 60689
diff changeset
   883
  have "gcd (c * a) (c * b) * unit_factor c = c"
a9e45c9588c3 tuned facts
haftmann
parents: 60689
diff changeset
   884
    by (simp add: ac_simps)
a9e45c9588c3 tuned facts
haftmann
parents: 60689
diff changeset
   885
  moreover from \<open>a dvd c * b\<close> have "a dvd gcd (c * a) (c * b) * unit_factor c"
a9e45c9588c3 tuned facts
haftmann
parents: 60689
diff changeset
   886
    by (simp add: dvd_mult_unit_iff unit)
a9e45c9588c3 tuned facts
haftmann
parents: 60689
diff changeset
   887
  ultimately show ?thesis by simp
a9e45c9588c3 tuned facts
haftmann
parents: 60689
diff changeset
   888
qed
a9e45c9588c3 tuned facts
haftmann
parents: 60689
diff changeset
   889
62344
759d684c0e60 pulled out legacy aliasses and infamous dvd interpretations into theory appendix
haftmann
parents: 62343
diff changeset
   890
lemma coprime_dvd_mult_iff:
759d684c0e60 pulled out legacy aliasses and infamous dvd interpretations into theory appendix
haftmann
parents: 62343
diff changeset
   891
  assumes "coprime a c"
759d684c0e60 pulled out legacy aliasses and infamous dvd interpretations into theory appendix
haftmann
parents: 62343
diff changeset
   892
  shows "a dvd b * c \<longleftrightarrow> a dvd b"
759d684c0e60 pulled out legacy aliasses and infamous dvd interpretations into theory appendix
haftmann
parents: 62343
diff changeset
   893
  using assms by (auto intro: coprime_dvd_mult)
60690
a9e45c9588c3 tuned facts
haftmann
parents: 60689
diff changeset
   894
a9e45c9588c3 tuned facts
haftmann
parents: 60689
diff changeset
   895
lemma gcd_mult_cancel:
a9e45c9588c3 tuned facts
haftmann
parents: 60689
diff changeset
   896
  "coprime c b \<Longrightarrow> gcd (c * a) b = gcd a b"
a9e45c9588c3 tuned facts
haftmann
parents: 60689
diff changeset
   897
  apply (rule associated_eqI)
59008
f61482b0f240 formally self-contained gcd type classes
haftmann
parents: 58889
diff changeset
   898
  apply (rule gcd_greatest)
60690
a9e45c9588c3 tuned facts
haftmann
parents: 60689
diff changeset
   899
  apply (rule_tac b = c in coprime_dvd_mult)
a9e45c9588c3 tuned facts
haftmann
parents: 60689
diff changeset
   900
  apply (simp add: gcd.assoc)
a9e45c9588c3 tuned facts
haftmann
parents: 60689
diff changeset
   901
  apply (simp_all add: ac_simps)
a9e45c9588c3 tuned facts
haftmann
parents: 60689
diff changeset
   902
  done
21256
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   903
62344
759d684c0e60 pulled out legacy aliasses and infamous dvd interpretations into theory appendix
haftmann
parents: 62343
diff changeset
   904
lemma coprime_crossproduct:
759d684c0e60 pulled out legacy aliasses and infamous dvd interpretations into theory appendix
haftmann
parents: 62343
diff changeset
   905
  fixes a b c d
35368
19b340c3f1ff crossproduct coprimality lemmas
haftmann
parents: 35216
diff changeset
   906
  assumes "coprime a d" and "coprime b c"
62344
759d684c0e60 pulled out legacy aliasses and infamous dvd interpretations into theory appendix
haftmann
parents: 62343
diff changeset
   907
  shows "normalize a * normalize c = normalize b * normalize d
759d684c0e60 pulled out legacy aliasses and infamous dvd interpretations into theory appendix
haftmann
parents: 62343
diff changeset
   908
    \<longleftrightarrow> normalize a = normalize b \<and> normalize c = normalize d" (is "?lhs \<longleftrightarrow> ?rhs")
35368
19b340c3f1ff crossproduct coprimality lemmas
haftmann
parents: 35216
diff changeset
   909
proof
19b340c3f1ff crossproduct coprimality lemmas
haftmann
parents: 35216
diff changeset
   910
  assume ?rhs then show ?lhs by simp
19b340c3f1ff crossproduct coprimality lemmas
haftmann
parents: 35216
diff changeset
   911
next
19b340c3f1ff crossproduct coprimality lemmas
haftmann
parents: 35216
diff changeset
   912
  assume ?lhs
62344
759d684c0e60 pulled out legacy aliasses and infamous dvd interpretations into theory appendix
haftmann
parents: 62343
diff changeset
   913
  from \<open>?lhs\<close> have "normalize a dvd normalize b * normalize d"
759d684c0e60 pulled out legacy aliasses and infamous dvd interpretations into theory appendix
haftmann
parents: 62343
diff changeset
   914
    by (auto intro: dvdI dest: sym)
759d684c0e60 pulled out legacy aliasses and infamous dvd interpretations into theory appendix
haftmann
parents: 62343
diff changeset
   915
  with \<open>coprime a d\<close> have "a dvd b"
759d684c0e60 pulled out legacy aliasses and infamous dvd interpretations into theory appendix
haftmann
parents: 62343
diff changeset
   916
    by (simp add: coprime_dvd_mult_iff normalize_mult [symmetric])
759d684c0e60 pulled out legacy aliasses and infamous dvd interpretations into theory appendix
haftmann
parents: 62343
diff changeset
   917
  from \<open>?lhs\<close> have "normalize b dvd normalize a * normalize c"
759d684c0e60 pulled out legacy aliasses and infamous dvd interpretations into theory appendix
haftmann
parents: 62343
diff changeset
   918
    by (auto intro: dvdI dest: sym)
759d684c0e60 pulled out legacy aliasses and infamous dvd interpretations into theory appendix
haftmann
parents: 62343
diff changeset
   919
  with \<open>coprime b c\<close> have "b dvd a"
759d684c0e60 pulled out legacy aliasses and infamous dvd interpretations into theory appendix
haftmann
parents: 62343
diff changeset
   920
    by (simp add: coprime_dvd_mult_iff normalize_mult [symmetric])
759d684c0e60 pulled out legacy aliasses and infamous dvd interpretations into theory appendix
haftmann
parents: 62343
diff changeset
   921
  from \<open>?lhs\<close> have "normalize c dvd normalize d * normalize b"
759d684c0e60 pulled out legacy aliasses and infamous dvd interpretations into theory appendix
haftmann
parents: 62343
diff changeset
   922
    by (auto intro: dvdI dest: sym simp add: mult.commute)
759d684c0e60 pulled out legacy aliasses and infamous dvd interpretations into theory appendix
haftmann
parents: 62343
diff changeset
   923
  with \<open>coprime b c\<close> have "c dvd d"
759d684c0e60 pulled out legacy aliasses and infamous dvd interpretations into theory appendix
haftmann
parents: 62343
diff changeset
   924
    by (simp add: coprime_dvd_mult_iff gcd.commute normalize_mult [symmetric])
759d684c0e60 pulled out legacy aliasses and infamous dvd interpretations into theory appendix
haftmann
parents: 62343
diff changeset
   925
  from \<open>?lhs\<close> have "normalize d dvd normalize c * normalize a"
759d684c0e60 pulled out legacy aliasses and infamous dvd interpretations into theory appendix
haftmann
parents: 62343
diff changeset
   926
    by (auto intro: dvdI dest: sym simp add: mult.commute)
759d684c0e60 pulled out legacy aliasses and infamous dvd interpretations into theory appendix
haftmann
parents: 62343
diff changeset
   927
  with \<open>coprime a d\<close> have "d dvd c"
759d684c0e60 pulled out legacy aliasses and infamous dvd interpretations into theory appendix
haftmann
parents: 62343
diff changeset
   928
    by (simp add: coprime_dvd_mult_iff gcd.commute normalize_mult [symmetric])
759d684c0e60 pulled out legacy aliasses and infamous dvd interpretations into theory appendix
haftmann
parents: 62343
diff changeset
   929
  from \<open>a dvd b\<close> \<open>b dvd a\<close> have "normalize a = normalize b"
759d684c0e60 pulled out legacy aliasses and infamous dvd interpretations into theory appendix
haftmann
parents: 62343
diff changeset
   930
    by (rule associatedI)
759d684c0e60 pulled out legacy aliasses and infamous dvd interpretations into theory appendix
haftmann
parents: 62343
diff changeset
   931
  moreover from \<open>c dvd d\<close> \<open>d dvd c\<close> have "normalize c = normalize d"
759d684c0e60 pulled out legacy aliasses and infamous dvd interpretations into theory appendix
haftmann
parents: 62343
diff changeset
   932
    by (rule associatedI)
35368
19b340c3f1ff crossproduct coprimality lemmas
haftmann
parents: 35216
diff changeset
   933
  ultimately show ?rhs ..
19b340c3f1ff crossproduct coprimality lemmas
haftmann
parents: 35216
diff changeset
   934
qed
19b340c3f1ff crossproduct coprimality lemmas
haftmann
parents: 35216
diff changeset
   935
62344
759d684c0e60 pulled out legacy aliasses and infamous dvd interpretations into theory appendix
haftmann
parents: 62343
diff changeset
   936
end
759d684c0e60 pulled out legacy aliasses and infamous dvd interpretations into theory appendix
haftmann
parents: 62343
diff changeset
   937
759d684c0e60 pulled out legacy aliasses and infamous dvd interpretations into theory appendix
haftmann
parents: 62343
diff changeset
   938
lemma coprime_crossproduct_nat:
759d684c0e60 pulled out legacy aliasses and infamous dvd interpretations into theory appendix
haftmann
parents: 62343
diff changeset
   939
  fixes a b c d :: nat
759d684c0e60 pulled out legacy aliasses and infamous dvd interpretations into theory appendix
haftmann
parents: 62343
diff changeset
   940
  assumes "coprime a d" and "coprime b c"
759d684c0e60 pulled out legacy aliasses and infamous dvd interpretations into theory appendix
haftmann
parents: 62343
diff changeset
   941
  shows "a * c = b * d \<longleftrightarrow> a = b \<and> c = d"
759d684c0e60 pulled out legacy aliasses and infamous dvd interpretations into theory appendix
haftmann
parents: 62343
diff changeset
   942
  using assms coprime_crossproduct [of a d b c] by simp
759d684c0e60 pulled out legacy aliasses and infamous dvd interpretations into theory appendix
haftmann
parents: 62343
diff changeset
   943
35368
19b340c3f1ff crossproduct coprimality lemmas
haftmann
parents: 35216
diff changeset
   944
lemma coprime_crossproduct_int:
19b340c3f1ff crossproduct coprimality lemmas
haftmann
parents: 35216
diff changeset
   945
  fixes a b c d :: int
19b340c3f1ff crossproduct coprimality lemmas
haftmann
parents: 35216
diff changeset
   946
  assumes "coprime a d" and "coprime b c"
19b340c3f1ff crossproduct coprimality lemmas
haftmann
parents: 35216
diff changeset
   947
  shows "\<bar>a\<bar> * \<bar>c\<bar> = \<bar>b\<bar> * \<bar>d\<bar> \<longleftrightarrow> \<bar>a\<bar> = \<bar>b\<bar> \<and> \<bar>c\<bar> = \<bar>d\<bar>"
62344
759d684c0e60 pulled out legacy aliasses and infamous dvd interpretations into theory appendix
haftmann
parents: 62343
diff changeset
   948
  using assms coprime_crossproduct [of a d b c] by simp
35368
19b340c3f1ff crossproduct coprimality lemmas
haftmann
parents: 35216
diff changeset
   949
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
   950
text \<open>\medskip Addition laws\<close>
21256
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   951
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   952
lemma gcd_add1_nat [simp]: "gcd ((m::nat) + n) n = gcd m n"
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   953
  apply (case_tac "n = 0")
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   954
  apply (simp_all add: gcd_non_0_nat)
62344
759d684c0e60 pulled out legacy aliasses and infamous dvd interpretations into theory appendix
haftmann
parents: 62343
diff changeset
   955
  done
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   956
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   957
lemma gcd_add2_nat [simp]: "gcd (m::nat) (m + n) = gcd m n"
62344
759d684c0e60 pulled out legacy aliasses and infamous dvd interpretations into theory appendix
haftmann
parents: 62343
diff changeset
   958
  apply (subst (1 2) gcd.commute)
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 56218
diff changeset
   959
  apply (subst add.commute)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   960
  apply simp
62344
759d684c0e60 pulled out legacy aliasses and infamous dvd interpretations into theory appendix
haftmann
parents: 62343
diff changeset
   961
  done
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   962
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   963
(* to do: add the other variations? *)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   964
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   965
lemma gcd_diff1_nat: "(m::nat) >= n \<Longrightarrow> gcd (m - n) n = gcd m n"
62344
759d684c0e60 pulled out legacy aliasses and infamous dvd interpretations into theory appendix
haftmann
parents: 62343
diff changeset
   966
  by (subst gcd_add1_nat [symmetric]) auto
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   967
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   968
lemma gcd_diff2_nat: "(n::nat) >= m \<Longrightarrow> gcd (n - m) n = gcd m n"
62344
759d684c0e60 pulled out legacy aliasses and infamous dvd interpretations into theory appendix
haftmann
parents: 62343
diff changeset
   969
  apply (subst gcd.commute)
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   970
  apply (subst gcd_diff1_nat [symmetric])
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   971
  apply auto
62344
759d684c0e60 pulled out legacy aliasses and infamous dvd interpretations into theory appendix
haftmann
parents: 62343
diff changeset
   972
  apply (subst gcd.commute)
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   973
  apply (subst gcd_diff1_nat)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   974
  apply assumption
62344
759d684c0e60 pulled out legacy aliasses and infamous dvd interpretations into theory appendix
haftmann
parents: 62343
diff changeset
   975
  apply (rule gcd.commute)
759d684c0e60 pulled out legacy aliasses and infamous dvd interpretations into theory appendix
haftmann
parents: 62343
diff changeset
   976
  done
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   977
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   978
lemma gcd_non_0_int: "(y::int) > 0 \<Longrightarrow> gcd x y = gcd y (x mod y)"
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   979
  apply (frule_tac b = y and a = x in pos_mod_sign)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   980
  apply (simp del: pos_mod_sign add: gcd_int_def abs_if nat_mod_distrib)
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   981
  apply (auto simp add: gcd_non_0_nat nat_mod_distrib [symmetric]
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   982
    zmod_zminus1_eq_if)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   983
  apply (frule_tac a = x in pos_mod_bound)
62344
759d684c0e60 pulled out legacy aliasses and infamous dvd interpretations into theory appendix
haftmann
parents: 62343
diff changeset
   984
  apply (subst (1 2) gcd.commute)
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   985
  apply (simp del: pos_mod_bound add: nat_diff_distrib gcd_diff2_nat
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   986
    nat_le_eq_zle)
62344
759d684c0e60 pulled out legacy aliasses and infamous dvd interpretations into theory appendix
haftmann
parents: 62343
diff changeset
   987
  done
21256
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   988
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   989
lemma gcd_red_int: "gcd (x::int) y = gcd y (x mod y)"
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   990
  apply (case_tac "y = 0")
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   991
  apply force
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   992
  apply (case_tac "y > 0")
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   993
  apply (subst gcd_non_0_int, auto)
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   994
  apply (insert gcd_non_0_int [of "-y" "-x"])
35216
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 35028
diff changeset
   995
  apply auto
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   996
done
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   997
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   998
lemma gcd_add1_int [simp]: "gcd ((m::int) + n) n = gcd m n"
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 56218
diff changeset
   999
by (metis gcd_red_int mod_add_self1 add.commute)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1000
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1001
lemma gcd_add2_int [simp]: "gcd m ((m::int) + n) = gcd m n"
62344
759d684c0e60 pulled out legacy aliasses and infamous dvd interpretations into theory appendix
haftmann
parents: 62343
diff changeset
  1002
by (metis gcd_add1_int gcd.commute add.commute)
21256
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
  1003
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1004
lemma gcd_add_mult_nat: "gcd (m::nat) (k * m + n) = gcd m n"
62344
759d684c0e60 pulled out legacy aliasses and infamous dvd interpretations into theory appendix
haftmann
parents: 62343
diff changeset
  1005
by (metis mod_mult_self3 gcd.commute gcd_red_nat)
21256
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
  1006
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1007
lemma gcd_add_mult_int: "gcd (m::int) (k * m + n) = gcd m n"
62344
759d684c0e60 pulled out legacy aliasses and infamous dvd interpretations into theory appendix
haftmann
parents: 62343
diff changeset
  1008
by (metis gcd.commute gcd_red_int mod_mult_self1 add.commute)
31798
fe9a3043d36c Cleaned up GCD
nipkow
parents: 31766
diff changeset
  1009
21256
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
  1010
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1011
(* to do: differences, and all variations of addition rules
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1012
    as simplification rules for nat and int *)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1013
60689
8a2d7c04d8c0 more cautious use of [iff] declarations
haftmann
parents: 60688
diff changeset
  1014
lemma gcd_dvd_prod_nat: "gcd (m::nat) n dvd k * n"
23687
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
  1015
  using mult_dvd_mono [of 1] by auto
22027
e4a08629c4bd A few lemmas about relative primes when dividing trough gcd
chaieb
parents: 21404
diff changeset
  1016
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1017
(* to do: add the three variations of these, and for ints? *)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1018
31992
f8aed98faae7 More about gcd/lcm, and some cleaning up
nipkow
parents: 31952
diff changeset
  1019
lemma finite_divisors_nat[simp]:
f8aed98faae7 More about gcd/lcm, and some cleaning up
nipkow
parents: 31952
diff changeset
  1020
  assumes "(m::nat) ~= 0" shows "finite{d. d dvd m}"
31734
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
  1021
proof-
60512
e0169291b31c tuned proofs -- slightly faster;
wenzelm
parents: 60357
diff changeset
  1022
  have "finite{d. d <= m}"
e0169291b31c tuned proofs -- slightly faster;
wenzelm
parents: 60357
diff changeset
  1023
    by (blast intro: bounded_nat_set_is_finite)
31734
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
  1024
  from finite_subset[OF _ this] show ?thesis using assms
60512
e0169291b31c tuned proofs -- slightly faster;
wenzelm
parents: 60357
diff changeset
  1025
    by (metis Collect_mono dvd_imp_le neq0_conv)
31734
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
  1026
qed
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
  1027
31995
8f37cf60b885 more gcd/lcm lemmas
nipkow
parents: 31992
diff changeset
  1028
lemma finite_divisors_int[simp]:
31734
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
  1029
  assumes "(i::int) ~= 0" shows "finite{d. d dvd i}"
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
  1030
proof-
61944
5d06ecfdb472 prefer symbols for "abs";
wenzelm
parents: 61929
diff changeset
  1031
  have "{d. \<bar>d\<bar> <= \<bar>i\<bar>} = {- \<bar>i\<bar> .. \<bar>i\<bar>}" by(auto simp:abs_if)
5d06ecfdb472 prefer symbols for "abs";
wenzelm
parents: 61929
diff changeset
  1032
  hence "finite {d. \<bar>d\<bar> <= \<bar>i\<bar>}" by simp
31734
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
  1033
  from finite_subset[OF _ this] show ?thesis using assms
60512
e0169291b31c tuned proofs -- slightly faster;
wenzelm
parents: 60357
diff changeset
  1034
    by (simp add: dvd_imp_le_int subset_iff)
31734
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
  1035
qed
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
  1036
31995
8f37cf60b885 more gcd/lcm lemmas
nipkow
parents: 31992
diff changeset
  1037
lemma Max_divisors_self_nat[simp]: "n\<noteq>0 \<Longrightarrow> Max{d::nat. d dvd n} = n"
8f37cf60b885 more gcd/lcm lemmas
nipkow
parents: 31992
diff changeset
  1038
apply(rule antisym)
44890
22f665a2e91c new fastforce replacing fastsimp - less confusing name
nipkow
parents: 44845
diff changeset
  1039
 apply (fastforce intro: Max_le_iff[THEN iffD2] simp: dvd_imp_le)
31995
8f37cf60b885 more gcd/lcm lemmas
nipkow
parents: 31992
diff changeset
  1040
apply simp
8f37cf60b885 more gcd/lcm lemmas
nipkow
parents: 31992
diff changeset
  1041
done
8f37cf60b885 more gcd/lcm lemmas
nipkow
parents: 31992
diff changeset
  1042
61944
5d06ecfdb472 prefer symbols for "abs";
wenzelm
parents: 61929
diff changeset
  1043
lemma Max_divisors_self_int[simp]: "n\<noteq>0 \<Longrightarrow> Max{d::int. d dvd n} = \<bar>n\<bar>"
31995
8f37cf60b885 more gcd/lcm lemmas
nipkow
parents: 31992
diff changeset
  1044
apply(rule antisym)
44278
1220ecb81e8f observe distinction between sets and predicates more properly
haftmann
parents: 42871
diff changeset
  1045
 apply(rule Max_le_iff [THEN iffD2])
1220ecb81e8f observe distinction between sets and predicates more properly
haftmann
parents: 42871
diff changeset
  1046
  apply (auto intro: abs_le_D1 dvd_imp_le_int)
31995
8f37cf60b885 more gcd/lcm lemmas
nipkow
parents: 31992
diff changeset
  1047
done
8f37cf60b885 more gcd/lcm lemmas
nipkow
parents: 31992
diff changeset
  1048
31734
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
  1049
lemma gcd_is_Max_divisors_nat:
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
  1050
  "m ~= 0 \<Longrightarrow> n ~= 0 \<Longrightarrow> gcd (m::nat) n = (Max {d. d dvd m & d dvd n})"
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
  1051
apply(rule Max_eqI[THEN sym])
31995
8f37cf60b885 more gcd/lcm lemmas
nipkow
parents: 31992
diff changeset
  1052
  apply (metis finite_Collect_conjI finite_divisors_nat)
31734
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
  1053
 apply simp
62344
759d684c0e60 pulled out legacy aliasses and infamous dvd interpretations into theory appendix
haftmann
parents: 62343
diff changeset
  1054
 apply(metis Suc_diff_1 Suc_neq_Zero dvd_imp_le gcd_greatest_iff gcd_pos_nat)
31734
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
  1055
apply simp
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
  1056
done
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
  1057
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
  1058
lemma gcd_is_Max_divisors_int:
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
  1059
  "m ~= 0 ==> n ~= 0 ==> gcd (m::int) n = (Max {d. d dvd m & d dvd n})"
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
  1060
apply(rule Max_eqI[THEN sym])
31995
8f37cf60b885 more gcd/lcm lemmas
nipkow
parents: 31992
diff changeset
  1061
  apply (metis finite_Collect_conjI finite_divisors_int)
31734
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
  1062
 apply simp
62344
759d684c0e60 pulled out legacy aliasses and infamous dvd interpretations into theory appendix
haftmann
parents: 62343
diff changeset
  1063
 apply (metis gcd_greatest_iff gcd_pos_int zdvd_imp_le)
31734
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
  1064
apply simp
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
  1065
done
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
  1066
34030
829eb528b226 resorted code equations from "old" number theory version
haftmann
parents: 33946
diff changeset
  1067
lemma gcd_code_int [code]:
829eb528b226 resorted code equations from "old" number theory version
haftmann
parents: 33946
diff changeset
  1068
  "gcd k l = \<bar>if l = (0::int) then k else gcd l (\<bar>k\<bar> mod \<bar>l\<bar>)\<bar>"
829eb528b226 resorted code equations from "old" number theory version
haftmann
parents: 33946
diff changeset
  1069
  by (simp add: gcd_int_def nat_mod_distrib gcd_non_0_nat)
829eb528b226 resorted code equations from "old" number theory version
haftmann
parents: 33946
diff changeset
  1070
22027
e4a08629c4bd A few lemmas about relative primes when dividing trough gcd
chaieb
parents: 21404
diff changeset
  1071
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
  1072
subsection \<open>Coprimality\<close>
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1073
60690
a9e45c9588c3 tuned facts
haftmann
parents: 60689
diff changeset
  1074
context semiring_gcd
a9e45c9588c3 tuned facts
haftmann
parents: 60689
diff changeset
  1075
begin
a9e45c9588c3 tuned facts
haftmann
parents: 60689
diff changeset
  1076
a9e45c9588c3 tuned facts
haftmann
parents: 60689
diff changeset
  1077
lemma div_gcd_coprime:
a9e45c9588c3 tuned facts
haftmann
parents: 60689
diff changeset
  1078
  assumes nz: "a \<noteq> 0 \<or> b \<noteq> 0"
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1079
  shows "coprime (a div gcd a b) (b div gcd a b)"
22367
6860f09242bf tuned document;
wenzelm
parents: 22027
diff changeset
  1080
proof -
27556
292098f2efdf unified curried gcd, lcm, zgcd, zlcm
haftmann
parents: 27487
diff changeset
  1081
  let ?g = "gcd a b"
22027
e4a08629c4bd A few lemmas about relative primes when dividing trough gcd
chaieb
parents: 21404
diff changeset
  1082
  let ?a' = "a div ?g"
e4a08629c4bd A few lemmas about relative primes when dividing trough gcd
chaieb
parents: 21404
diff changeset
  1083
  let ?b' = "b div ?g"
27556
292098f2efdf unified curried gcd, lcm, zgcd, zlcm
haftmann
parents: 27487
diff changeset
  1084
  let ?g' = "gcd ?a' ?b'"
22027
e4a08629c4bd A few lemmas about relative primes when dividing trough gcd
chaieb
parents: 21404
diff changeset
  1085
  have dvdg: "?g dvd a" "?g dvd b" by simp_all
e4a08629c4bd A few lemmas about relative primes when dividing trough gcd
chaieb
parents: 21404
diff changeset
  1086
  have dvdg': "?g' dvd ?a'" "?g' dvd ?b'" by simp_all
22367
6860f09242bf tuned document;
wenzelm
parents: 22027
diff changeset
  1087
  from dvdg dvdg' obtain ka kb ka' kb' where
6860f09242bf tuned document;
wenzelm
parents: 22027
diff changeset
  1088
      kab: "a = ?g * ka" "b = ?g * kb" "?a' = ?g' * ka'" "?b' = ?g' * kb'"
22027
e4a08629c4bd A few lemmas about relative primes when dividing trough gcd
chaieb
parents: 21404
diff changeset
  1089
    unfolding dvd_def by blast
58834
773b378d9313 more simp rules concerning dvd and even/odd
haftmann
parents: 58787
diff changeset
  1090
  from this [symmetric] have "?g * ?a' = (?g * ?g') * ka'" "?g * ?b' = (?g * ?g') * kb'"
773b378d9313 more simp rules concerning dvd and even/odd
haftmann
parents: 58787
diff changeset
  1091
    by (simp_all add: mult.assoc mult.left_commute [of "gcd a b"])
22367
6860f09242bf tuned document;
wenzelm
parents: 22027
diff changeset
  1092
  then have dvdgg':"?g * ?g' dvd a" "?g* ?g' dvd b"
6860f09242bf tuned document;
wenzelm
parents: 22027
diff changeset
  1093
    by (auto simp add: dvd_mult_div_cancel [OF dvdg(1)]
6860f09242bf tuned document;
wenzelm
parents: 22027
diff changeset
  1094
      dvd_mult_div_cancel [OF dvdg(2)] dvd_def)
35216
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 35028
diff changeset
  1095
  have "?g \<noteq> 0" using nz by simp
60690
a9e45c9588c3 tuned facts
haftmann
parents: 60689
diff changeset
  1096
  moreover from gcd_greatest [OF dvdgg'] have "?g * ?g' dvd ?g" .
a9e45c9588c3 tuned facts
haftmann
parents: 60689
diff changeset
  1097
  thm dvd_mult_cancel_left
a9e45c9588c3 tuned facts
haftmann
parents: 60689
diff changeset
  1098
  ultimately show ?thesis using dvd_times_left_cancel_iff [of "gcd a b" _ 1] by simp
22027
e4a08629c4bd A few lemmas about relative primes when dividing trough gcd
chaieb
parents: 21404
diff changeset
  1099
qed
e4a08629c4bd A few lemmas about relative primes when dividing trough gcd
chaieb
parents: 21404
diff changeset
  1100
62345
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1101
lemma coprime:
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1102
  "coprime a b \<longleftrightarrow> (\<forall>d. d dvd a \<and> d dvd b \<longleftrightarrow> is_unit d)" (is "?P \<longleftrightarrow> ?Q")
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1103
proof
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1104
  assume ?P then show ?Q by auto
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1105
next
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1106
  assume ?Q
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1107
  then have "is_unit (gcd a b) \<longleftrightarrow> gcd a b dvd a \<and> gcd a b dvd b"
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1108
    by blast
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1109
  then have "is_unit (gcd a b)"
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1110
    by simp
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1111
  then show ?P
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1112
    by simp
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1113
qed
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1114
60690
a9e45c9588c3 tuned facts
haftmann
parents: 60689
diff changeset
  1115
end
a9e45c9588c3 tuned facts
haftmann
parents: 60689
diff changeset
  1116
62345
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1117
lemma coprime_nat:
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1118
  "coprime (a::nat) b \<longleftrightarrow> (\<forall>d. d dvd a \<and> d dvd b \<longleftrightarrow> d = 1)"
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1119
  using coprime [of a b] by simp
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1120
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1121
lemma coprime_Suc_0_nat:
62345
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1122
  "coprime (a::nat) b \<longleftrightarrow> (\<forall>d. d dvd a \<and> d dvd b \<longleftrightarrow> d = Suc 0)"
60690
a9e45c9588c3 tuned facts
haftmann
parents: 60689
diff changeset
  1123
  using coprime_nat by simp
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1124
62345
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1125
lemma coprime_int:
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1126
  "coprime (a::int) b \<longleftrightarrow> (\<forall>d. d \<ge> 0 \<and> d dvd a \<and> d dvd b \<longleftrightarrow> d = 1)"
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1127
  using gcd_unique_int [of 1 a b]
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1128
  apply clarsimp
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1129
  apply (erule subst)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1130
  apply (rule iffI)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1131
  apply force
61649
268d88ec9087 Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents: 61605
diff changeset
  1132
  using abs_dvd_iff abs_ge_zero apply blast
59807
22bc39064290 prefer local fixes;
wenzelm
parents: 59667
diff changeset
  1133
  done
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1134
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1135
lemma gcd_coprime_nat:
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1136
  assumes z: "gcd (a::nat) b \<noteq> 0" and a: "a = a' * gcd a b" and
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1137
    b: "b = b' * gcd a b"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1138
  shows    "coprime a' b'"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1139
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1140
  apply (subgoal_tac "a' = a div gcd a b")
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1141
  apply (erule ssubst)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1142
  apply (subgoal_tac "b' = b div gcd a b")
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1143
  apply (erule ssubst)
62344
759d684c0e60 pulled out legacy aliasses and infamous dvd interpretations into theory appendix
haftmann
parents: 62343
diff changeset
  1144
  apply (rule div_gcd_coprime)
41550
efa734d9b221 eliminated global prems;
wenzelm
parents: 37770
diff changeset
  1145
  using z apply force
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1146
  apply (subst (1) b)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1147
  using z apply force
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1148
  apply (subst (1) a)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1149
  using z apply force
41550
efa734d9b221 eliminated global prems;
wenzelm
parents: 37770
diff changeset
  1150
  done
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1151
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1152
lemma gcd_coprime_int:
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1153
  assumes z: "gcd (a::int) b \<noteq> 0" and a: "a = a' * gcd a b" and
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1154
    b: "b = b' * gcd a b"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1155
  shows    "coprime a' b'"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1156
  apply (subgoal_tac "a' = a div gcd a b")
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1157
  apply (erule ssubst)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1158
  apply (subgoal_tac "b' = b div gcd a b")
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1159
  apply (erule ssubst)
62344
759d684c0e60 pulled out legacy aliasses and infamous dvd interpretations into theory appendix
haftmann
parents: 62343
diff changeset
  1160
  apply (rule div_gcd_coprime)
41550
efa734d9b221 eliminated global prems;
wenzelm
parents: 37770
diff changeset
  1161
  using z apply force
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1162
  apply (subst (1) b)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1163
  using z apply force
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1164
  apply (subst (1) a)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1165
  using z apply force
41550
efa734d9b221 eliminated global prems;
wenzelm
parents: 37770
diff changeset
  1166
  done
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1167
60690
a9e45c9588c3 tuned facts
haftmann
parents: 60689
diff changeset
  1168
context semiring_gcd
a9e45c9588c3 tuned facts
haftmann
parents: 60689
diff changeset
  1169
begin
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1170
60690
a9e45c9588c3 tuned facts
haftmann
parents: 60689
diff changeset
  1171
lemma coprime_mult:
a9e45c9588c3 tuned facts
haftmann
parents: 60689
diff changeset
  1172
  assumes da: "coprime d a" and db: "coprime d b"
a9e45c9588c3 tuned facts
haftmann
parents: 60689
diff changeset
  1173
  shows "coprime d (a * b)"
a9e45c9588c3 tuned facts
haftmann
parents: 60689
diff changeset
  1174
  apply (subst gcd.commute)
a9e45c9588c3 tuned facts
haftmann
parents: 60689
diff changeset
  1175
  using da apply (subst gcd_mult_cancel)
a9e45c9588c3 tuned facts
haftmann
parents: 60689
diff changeset
  1176
  apply (subst gcd.commute, assumption)
a9e45c9588c3 tuned facts
haftmann
parents: 60689
diff changeset
  1177
  apply (subst gcd.commute, rule db)
a9e45c9588c3 tuned facts
haftmann
parents: 60689
diff changeset
  1178
  done
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1179
60690
a9e45c9588c3 tuned facts
haftmann
parents: 60689
diff changeset
  1180
end
a9e45c9588c3 tuned facts
haftmann
parents: 60689
diff changeset
  1181
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1182
lemma coprime_lmult_nat:
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1183
  assumes dab: "coprime (d::nat) (a * b)" shows "coprime d a"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1184
proof -
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1185
  have "gcd d a dvd gcd d (a * b)"
59008
f61482b0f240 formally self-contained gcd type classes
haftmann
parents: 58889
diff changeset
  1186
    by (rule gcd_greatest, auto)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1187
  with dab show ?thesis
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1188
    by auto
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1189
qed
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1190
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1191
lemma coprime_lmult_int:
31798
fe9a3043d36c Cleaned up GCD
nipkow
parents: 31766
diff changeset
  1192
  assumes "coprime (d::int) (a * b)" shows "coprime d a"
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1193
proof -
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1194
  have "gcd d a dvd gcd d (a * b)"
59008
f61482b0f240 formally self-contained gcd type classes
haftmann
parents: 58889
diff changeset
  1195
    by (rule gcd_greatest, auto)
31798
fe9a3043d36c Cleaned up GCD
nipkow
parents: 31766
diff changeset
  1196
  with assms show ?thesis
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1197
    by auto
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1198
qed
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1199
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1200
lemma coprime_rmult_nat:
31798
fe9a3043d36c Cleaned up GCD
nipkow
parents: 31766
diff changeset
  1201
  assumes "coprime (d::nat) (a * b)" shows "coprime d b"
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1202
proof -
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1203
  have "gcd d b dvd gcd d (a * b)"
59008
f61482b0f240 formally self-contained gcd type classes
haftmann
parents: 58889
diff changeset
  1204
    by (rule gcd_greatest, auto intro: dvd_mult)
31798
fe9a3043d36c Cleaned up GCD
nipkow
parents: 31766
diff changeset
  1205
  with assms show ?thesis
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1206
    by auto
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1207
qed
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1208
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1209
lemma coprime_rmult_int:
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1210
  assumes dab: "coprime (d::int) (a * b)" shows "coprime d b"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1211
proof -
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1212
  have "gcd d b dvd gcd d (a * b)"
59008
f61482b0f240 formally self-contained gcd type classes
haftmann
parents: 58889
diff changeset
  1213
    by (rule gcd_greatest, auto intro: dvd_mult)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1214
  with dab show ?thesis
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1215
    by auto
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1216
qed
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1217
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1218
lemma coprime_mul_eq_nat: "coprime (d::nat) (a * b) \<longleftrightarrow>
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1219
    coprime d a \<and>  coprime d b"
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1220
  using coprime_rmult_nat[of d a b] coprime_lmult_nat[of d a b]
62344
759d684c0e60 pulled out legacy aliasses and infamous dvd interpretations into theory appendix
haftmann
parents: 62343
diff changeset
  1221
    coprime_mult [of d a b]
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1222
  by blast
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1223
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1224
lemma coprime_mul_eq_int: "coprime (d::int) (a * b) \<longleftrightarrow>
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1225
    coprime d a \<and>  coprime d b"
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1226
  using coprime_rmult_int[of d a b] coprime_lmult_int[of d a b]
62344
759d684c0e60 pulled out legacy aliasses and infamous dvd interpretations into theory appendix
haftmann
parents: 62343
diff changeset
  1227
    coprime_mult [of d a b]
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1228
  by blast
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1229
52397
e95f6b4b1bcf added coprimality lemma
noschinl
parents: 51547
diff changeset
  1230
lemma coprime_power_int:
e95f6b4b1bcf added coprimality lemma
noschinl
parents: 51547
diff changeset
  1231
  assumes "0 < n" shows "coprime (a :: int) (b ^ n) \<longleftrightarrow> coprime a b"
e95f6b4b1bcf added coprimality lemma
noschinl
parents: 51547
diff changeset
  1232
  using assms
e95f6b4b1bcf added coprimality lemma
noschinl
parents: 51547
diff changeset
  1233
proof (induct n)
e95f6b4b1bcf added coprimality lemma
noschinl
parents: 51547
diff changeset
  1234
  case (Suc n) then show ?case
e95f6b4b1bcf added coprimality lemma
noschinl
parents: 51547
diff changeset
  1235
    by (cases n) (simp_all add: coprime_mul_eq_int)
e95f6b4b1bcf added coprimality lemma
noschinl
parents: 51547
diff changeset
  1236
qed simp
e95f6b4b1bcf added coprimality lemma
noschinl
parents: 51547
diff changeset
  1237
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1238
lemma gcd_coprime_exists_nat:
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1239
    assumes nz: "gcd (a::nat) b \<noteq> 0"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1240
    shows "\<exists>a' b'. a = a' * gcd a b \<and> b = b' * gcd a b \<and> coprime a' b'"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1241
  apply (rule_tac x = "a div gcd a b" in exI)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1242
  apply (rule_tac x = "b div gcd a b" in exI)
62344
759d684c0e60 pulled out legacy aliasses and infamous dvd interpretations into theory appendix
haftmann
parents: 62343
diff changeset
  1243
  using nz apply (auto simp add: div_gcd_coprime dvd_div_mult)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1244
done
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1245
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1246
lemma gcd_coprime_exists_int:
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1247
    assumes nz: "gcd (a::int) b \<noteq> 0"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1248
    shows "\<exists>a' b'. a = a' * gcd a b \<and> b = b' * gcd a b \<and> coprime a' b'"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1249
  apply (rule_tac x = "a div gcd a b" in exI)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1250
  apply (rule_tac x = "b div gcd a b" in exI)
62344
759d684c0e60 pulled out legacy aliasses and infamous dvd interpretations into theory appendix
haftmann
parents: 62343
diff changeset
  1251
  using nz apply (auto simp add: div_gcd_coprime)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1252
done
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1253
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1254
lemma coprime_exp_nat: "coprime (d::nat) a \<Longrightarrow> coprime d (a^n)"
62344
759d684c0e60 pulled out legacy aliasses and infamous dvd interpretations into theory appendix
haftmann
parents: 62343
diff changeset
  1255
  by (induct n) (simp_all add: coprime_mult)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1256
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1257
lemma coprime_exp_int: "coprime (d::int) a \<Longrightarrow> coprime d (a^n)"
62344
759d684c0e60 pulled out legacy aliasses and infamous dvd interpretations into theory appendix
haftmann
parents: 62343
diff changeset
  1258
  by (induct n) (simp_all add: coprime_mult)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1259
60690
a9e45c9588c3 tuned facts
haftmann
parents: 60689
diff changeset
  1260
context semiring_gcd
a9e45c9588c3 tuned facts
haftmann
parents: 60689
diff changeset
  1261
begin
a9e45c9588c3 tuned facts
haftmann
parents: 60689
diff changeset
  1262
a9e45c9588c3 tuned facts
haftmann
parents: 60689
diff changeset
  1263
lemma coprime_exp_left:
a9e45c9588c3 tuned facts
haftmann
parents: 60689
diff changeset
  1264
  assumes "coprime a b"
a9e45c9588c3 tuned facts
haftmann
parents: 60689
diff changeset
  1265
  shows "coprime (a ^ n) b"
a9e45c9588c3 tuned facts
haftmann
parents: 60689
diff changeset
  1266
  using assms by (induct n) (simp_all add: gcd_mult_cancel)
a9e45c9588c3 tuned facts
haftmann
parents: 60689
diff changeset
  1267
a9e45c9588c3 tuned facts
haftmann
parents: 60689
diff changeset
  1268
lemma coprime_exp2:
a9e45c9588c3 tuned facts
haftmann
parents: 60689
diff changeset
  1269
  assumes "coprime a b"
a9e45c9588c3 tuned facts
haftmann
parents: 60689
diff changeset
  1270
  shows "coprime (a ^ n) (b ^ m)"
a9e45c9588c3 tuned facts
haftmann
parents: 60689
diff changeset
  1271
proof (rule coprime_exp_left)
a9e45c9588c3 tuned facts
haftmann
parents: 60689
diff changeset
  1272
  from assms show "coprime a (b ^ m)"
a9e45c9588c3 tuned facts
haftmann
parents: 60689
diff changeset
  1273
    by (induct m) (simp_all add: gcd_mult_cancel gcd.commute [of a])
a9e45c9588c3 tuned facts
haftmann
parents: 60689
diff changeset
  1274
qed
a9e45c9588c3 tuned facts
haftmann
parents: 60689
diff changeset
  1275
a9e45c9588c3 tuned facts
haftmann
parents: 60689
diff changeset
  1276
end
a9e45c9588c3 tuned facts
haftmann
parents: 60689
diff changeset
  1277
a9e45c9588c3 tuned facts
haftmann
parents: 60689
diff changeset
  1278
lemma gcd_exp_nat:
a9e45c9588c3 tuned facts
haftmann
parents: 60689
diff changeset
  1279
  "gcd ((a :: nat) ^ n) (b ^ n) = gcd a b ^ n"
a9e45c9588c3 tuned facts
haftmann
parents: 60689
diff changeset
  1280
proof (cases "a = 0 \<and> b = 0")
a9e45c9588c3 tuned facts
haftmann
parents: 60689
diff changeset
  1281
  case True then show ?thesis by (cases "n > 0") (simp_all add: zero_power)
a9e45c9588c3 tuned facts
haftmann
parents: 60689
diff changeset
  1282
next
a9e45c9588c3 tuned facts
haftmann
parents: 60689
diff changeset
  1283
  case False
a9e45c9588c3 tuned facts
haftmann
parents: 60689
diff changeset
  1284
  then have "coprime (a div gcd a b) (b div gcd a b)"
a9e45c9588c3 tuned facts
haftmann
parents: 60689
diff changeset
  1285
    by (auto simp: div_gcd_coprime)
a9e45c9588c3 tuned facts
haftmann
parents: 60689
diff changeset
  1286
  then have "coprime ((a div gcd a b) ^ n) ((b div gcd a b) ^ n)"
a9e45c9588c3 tuned facts
haftmann
parents: 60689
diff changeset
  1287
    by (simp add: coprime_exp2)
a9e45c9588c3 tuned facts
haftmann
parents: 60689
diff changeset
  1288
  then have "gcd ((a div gcd a b)^n * (gcd a b)^n)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1289
      ((b div gcd a b)^n * (gcd a b)^n) = (gcd a b)^n"
60162
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 59977
diff changeset
  1290
    by (metis gcd_mult_distrib_nat mult.commute mult.right_neutral)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1291
  also have "(a div gcd a b)^n * (gcd a b)^n = a^n"
60162
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 59977
diff changeset
  1292
    by (metis dvd_div_mult_self gcd_unique_nat power_mult_distrib)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1293
  also have "(b div gcd a b)^n * (gcd a b)^n = b^n"
60162
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 59977
diff changeset
  1294
    by (metis dvd_div_mult_self gcd_unique_nat power_mult_distrib)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1295
  finally show ?thesis .
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1296
qed
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1297
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1298
lemma gcd_exp_int: "gcd ((a::int)^n) (b^n) = (gcd a b)^n"
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1299
  apply (subst (1 2) gcd_abs_int)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1300
  apply (subst (1 2) power_abs)
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1301
  apply (rule gcd_exp_nat [where n = n, transferred])
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1302
  apply auto
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1303
done
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1304
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1305
lemma division_decomp_nat: assumes dc: "(a::nat) dvd b * c"
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1306
  shows "\<exists>b' c'. a = b' * c' \<and> b' dvd b \<and> c' dvd c"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1307
proof-
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1308
  let ?g = "gcd a b"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1309
  {assume "?g = 0" with dc have ?thesis by auto}
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1310
  moreover
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1311
  {assume z: "?g \<noteq> 0"
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1312
    from gcd_coprime_exists_nat[OF z]
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1313
    obtain a' b' where ab': "a = a' * ?g" "b = b' * ?g" "coprime a' b'"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1314
      by blast
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1315
    have thb: "?g dvd b" by auto
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1316
    from ab'(1) have "a' dvd a"  unfolding dvd_def by blast
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1317
    with dc have th0: "a' dvd b*c" using dvd_trans[of a' a "b*c"] by simp
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1318
    from dc ab'(1,2) have "a'*?g dvd (b'*?g) *c" by auto
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 56218
diff changeset
  1319
    hence "?g*a' dvd ?g * (b' * c)" by (simp add: mult.assoc)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1320
    with z have th_1: "a' dvd b' * c" by auto
62344
759d684c0e60 pulled out legacy aliasses and infamous dvd interpretations into theory appendix
haftmann
parents: 62343
diff changeset
  1321
    from coprime_dvd_mult [OF ab'(3)] th_1
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 56218
diff changeset
  1322
    have thc: "a' dvd c" by (subst (asm) mult.commute, blast)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1323
    from ab' have "a = ?g*a'" by algebra
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1324
    with thb thc have ?thesis by blast }
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1325
  ultimately show ?thesis by blast
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1326
qed
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1327
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1328
lemma division_decomp_int: assumes dc: "(a::int) dvd b * c"
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1329
  shows "\<exists>b' c'. a = b' * c' \<and> b' dvd b \<and> c' dvd c"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1330
proof-
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1331
  let ?g = "gcd a b"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1332
  {assume "?g = 0" with dc have ?thesis by auto}
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1333
  moreover
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1334
  {assume z: "?g \<noteq> 0"
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1335
    from gcd_coprime_exists_int[OF z]
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1336
    obtain a' b' where ab': "a = a' * ?g" "b = b' * ?g" "coprime a' b'"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1337
      by blast
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1338
    have thb: "?g dvd b" by auto
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1339
    from ab'(1) have "a' dvd a"  unfolding dvd_def by blast
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738