src/HOL/GCD.thy
author haftmann
Wed, 08 Jul 2015 14:01:41 +0200
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(*  Authors:    Christophe Tabacznyj, Lawrence C. Paulson, Amine Chaieb,
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                Thomas M. Rasmussen, Jeremy Avigad, Tobias Nipkow
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This file deals with the functions gcd and lcm.  Definitions and
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lemmas are proved uniformly for the natural numbers and integers.
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This file combines and revises a number of prior developments.
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The original theories "GCD" and "Primes" were by Christophe Tabacznyj
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and Lawrence C. Paulson, based on @{cite davenport92}. They introduced
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gcd, lcm, and prime for the natural numbers.
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The original theory "IntPrimes" was by Thomas M. Rasmussen, and
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extended gcd, lcm, primes to the integers. Amine Chaieb provided
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another extension of the notions to the integers, and added a number
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of results to "Primes" and "GCD". IntPrimes also defined and developed
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the congruence relations on the integers. The notion was extended to
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the natural numbers by Chaieb.
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Jeremy Avigad combined all of these, made everything uniform for the
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natural numbers and the integers, and added a number of new theorems.
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Tobias Nipkow cleaned up a lot.
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*)
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section {* Greatest common divisor and least common multiple *}
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theory GCD
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imports Main
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begin
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context semidom_divide
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begin
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lemma divide_1 [simp]:
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  "a div 1 = a"
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  using nonzero_mult_divide_cancel_left [of 1 a] by simp
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lemma dvd_mult_cancel_left [simp]:
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  assumes "a \<noteq> 0"
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  shows "a * b dvd a * c \<longleftrightarrow> b dvd c" (is "?P \<longleftrightarrow> ?Q")
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proof
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  assume ?P then obtain d where "a * c = a * b * d" ..
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  with assms have "c = b * d" by (simp add: ac_simps)
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  then show ?Q ..
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next
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  assume ?Q then obtain d where "c = b * d" .. 
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  then have "a * c = a * b * d" by (simp add: ac_simps)
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  then show ?P ..
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qed
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lemma dvd_mult_cancel_right [simp]:
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  assumes "a \<noteq> 0"
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  shows "b * a dvd c * a \<longleftrightarrow> b dvd c" (is "?P \<longleftrightarrow> ?Q")
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using dvd_mult_cancel_left [of a b c] assms by (simp add: ac_simps)
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lemma div_dvd_iff_mult:
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  assumes "b \<noteq> 0" and "b dvd a"
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  shows "a div b dvd c \<longleftrightarrow> a dvd c * b"
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proof -
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  from \<open>b dvd a\<close> obtain d where "a = b * d" ..
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  with \<open>b \<noteq> 0\<close> show ?thesis by (simp add: ac_simps)
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qed
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lemma dvd_div_iff_mult:
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  assumes "c \<noteq> 0" and "c dvd b"
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  shows "a dvd b div c \<longleftrightarrow> a * c dvd b"
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proof -
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  from \<open>c dvd b\<close> obtain d where "b = c * d" ..
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  with \<open>c \<noteq> 0\<close> show ?thesis by (simp add: mult.commute [of a])
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qed
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end
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declare One_nat_def [simp del]
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subsection {* GCD and LCM definitions *}
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class gcd = zero + one + dvd +
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  fixes gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
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    and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
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begin
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abbreviation coprime :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
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  where "coprime x y \<equiv> gcd x y = 1"
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end
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class Gcd = gcd +
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  fixes Gcd :: "'a set \<Rightarrow> 'a"
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    and Lcm :: "'a set \<Rightarrow> 'a"
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class semiring_gcd = normalization_semidom + gcd +
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  assumes gcd_dvd1 [iff]: "gcd a b dvd a"
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    and gcd_dvd2 [iff]: "gcd a b dvd b"
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    and gcd_greatest: "c dvd a \<Longrightarrow> c dvd b \<Longrightarrow> c dvd gcd a b"
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    and normalize_gcd [simp]: "normalize (gcd a b) = gcd a b"
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    and lcm_gcd: "lcm a b = normalize (a * b) div gcd a b"
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begin    
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lemma gcd_greatest_iff [simp]:
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  "a dvd gcd b c \<longleftrightarrow> a dvd b \<and> a dvd c"
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  by (blast intro!: gcd_greatest intro: dvd_trans)
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lemma gcd_dvdI1:
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  "a dvd c \<Longrightarrow> gcd a b dvd c"
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  by (rule dvd_trans) (rule gcd_dvd1)
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lemma gcd_dvdI2:
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  "b dvd c \<Longrightarrow> gcd a b dvd c"
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  by (rule dvd_trans) (rule gcd_dvd2)
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lemma gcd_0_left [simp]:
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  "gcd 0 a = normalize a"
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  by (rule associated_eqI) simp_all
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lemma gcd_0_right [simp]:
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  "gcd a 0 = normalize a"
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  by (rule associated_eqI) simp_all
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lemma gcd_eq_0_iff [simp]:
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  "gcd a b = 0 \<longleftrightarrow> a = 0 \<and> b = 0" (is "?P \<longleftrightarrow> ?Q")
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proof
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  assume ?P then have "0 dvd gcd a b" by simp
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  then have "0 dvd a" and "0 dvd b" by (blast intro: dvd_trans)+
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  then show ?Q by simp
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next
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  assume ?Q then show ?P by simp
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qed
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lemma unit_factor_gcd:
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  "unit_factor (gcd a b) = (if a = 0 \<and> b = 0 then 0 else 1)"
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proof (cases "gcd a b = 0")
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  case True then show ?thesis by simp
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next
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  case False
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  have "unit_factor (gcd a b) * normalize (gcd a b) = gcd a b"
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    by (rule unit_factor_mult_normalize)
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  then have "unit_factor (gcd a b) * gcd a b = gcd a b"
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    by simp
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  then have "unit_factor (gcd a b) * gcd a b div gcd a b = gcd a b div gcd a b"
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    by simp
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  with False show ?thesis by simp
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qed
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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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   158
    by (auto intro!: gcd_greatest)
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   159
  from gcd_dvd2 have "gcd a (gcd b c) dvd b"
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   160
    by (rule dvd_trans) simp
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   161
  moreover from gcd_dvd2 have "gcd a (gcd b c) dvd c"
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   162
    by (rule dvd_trans) simp
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   163
  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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   167
qed
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   168
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lemma gcd_self [simp]:
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  "gcd a a = normalize a"
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   171
proof -
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   172
  have "a dvd gcd a a"
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   173
    by (rule gcd_greatest) simp_all
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   174
  then show ?thesis
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   175
    by (auto intro: associated_eqI)
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   176
qed
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   177
    
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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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   181
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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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   188
proof (cases "c = 0")
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   189
  case True then show ?thesis by simp
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   190
next
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   191
  case False
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   192
  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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   194
  moreover from calculation False have "gcd (c * a) (c * b) dvd c * gcd a b"
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   195
    by (metis div_dvd_iff_mult dvd_mult_left gcd_dvd1 gcd_dvd2 gcd_greatest mult_commute)
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   196
  ultimately have "normalize (gcd (c * a) (c * b)) = normalize (c * gcd a b)"
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   197
    by (auto intro: associated_eqI)
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   198
  then show ?thesis by (simp add: normalize_mult)
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   199
qed
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   200
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   201
lemma gcd_mult_right:
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  "gcd (a * c) (b * c) = gcd b a * normalize c"
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   203
  using gcd_mult_left [of c a b] by (simp add: ac_simps)
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diff changeset
   204
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   205
lemma mult_gcd_left:
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  "c * gcd a b = unit_factor c * gcd (c * a) (c * b)"
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   207
  by (simp add: gcd_mult_left mult.assoc [symmetric])
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   208
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   209
lemma mult_gcd_right:
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  "gcd a b * c = gcd (a * c) (b * c) * unit_factor c"
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   211
  using mult_gcd_left [of c a b] by (simp add: ac_simps)
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diff changeset
   212
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lemma dvd_lcm1 [iff]:
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  "a dvd lcm a b"
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   215
proof -
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  have "normalize (a * b) div gcd a b = normalize a * (normalize b div gcd a b)"
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   217
    by (simp add: lcm_gcd normalize_mult div_mult_swap)
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   218
  then show ?thesis
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   219
    by (simp add: lcm_gcd)
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qed
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   221
  
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   222
lemma dvd_lcm2 [iff]:
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   223
  "b dvd lcm a b"
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   224
proof -
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   225
  have "normalize (a * b) div gcd a b = normalize b * (normalize a div gcd a b)"
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   226
    by (simp add: lcm_gcd normalize_mult div_mult_swap ac_simps)
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diff changeset
   227
  then show ?thesis
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   228
    by (simp add: lcm_gcd)
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   229
qed
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diff changeset
   230
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   231
lemma dvd_lcmI1:
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  "a dvd b \<Longrightarrow> a dvd lcm b c"
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   233
  by (rule dvd_trans) (assumption, blast) 
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   234
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   235
lemma dvd_lcmI2:
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   236
  "a dvd c \<Longrightarrow> a dvd lcm b c"
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   237
  by (rule dvd_trans) (assumption, blast)
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diff changeset
   238
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   239
lemma lcm_least:
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   240
  assumes "a dvd c" and "b dvd c"
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   241
  shows "lcm a b dvd c"
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   242
proof (cases "c = 0")
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   243
  case True then show ?thesis by simp
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   244
next
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   245
  case False then have U: "is_unit (unit_factor c)" by simp
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   246
  show ?thesis
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   247
  proof (cases "gcd a b = 0")
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   248
    case True with assms show ?thesis by simp
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   249
  next
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   250
    case False then have "a \<noteq> 0 \<or> b \<noteq> 0" by simp
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   251
    with \<open>c \<noteq> 0\<close> assms have "a * b dvd a * c" "a * b dvd c * b"
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   252
      by (simp_all add: mult_dvd_mono)
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   253
    then have "normalize (a * b) dvd gcd (a * c) (b * c)"
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diff changeset
   254
      by (auto intro: gcd_greatest simp add: ac_simps)
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   255
    then have "normalize (a * b) dvd gcd (a * c) (b * c) * unit_factor c"
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diff changeset
   256
      using U by (simp add: dvd_mult_unit_iff)
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diff changeset
   257
    then have "normalize (a * b) dvd gcd a b * c"
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diff changeset
   258
      by (simp add: mult_gcd_right [of a b c])
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diff changeset
   259
    then have "normalize (a * b) div gcd a b dvd c"
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diff changeset
   260
      using False by (simp add: div_dvd_iff_mult ac_simps)
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diff changeset
   261
    then show ?thesis by (simp add: lcm_gcd)
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   262
  qed
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   263
qed
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diff changeset
   264
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   265
lemma lcm_least_iff [simp]:
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   266
  "lcm a b dvd c \<longleftrightarrow> a dvd c \<and> b dvd c"
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   267
  by (blast intro!: lcm_least intro: dvd_trans)
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diff changeset
   268
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   269
lemma normalize_lcm [simp]:
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  "normalize (lcm a b) = lcm a b"
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   271
  by (simp add: lcm_gcd dvd_normalize_div)
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   272
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   273
lemma lcm_0_left [simp]:
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   274
  "lcm 0 a = 0"
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   275
  by (simp add: lcm_gcd)
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diff changeset
   276
  
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   277
lemma lcm_0_right [simp]:
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   278
  "lcm a 0 = 0"
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   279
  by (simp add: lcm_gcd)
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diff changeset
   280
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   281
lemma lcm_eq_0_iff:
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   282
  "lcm a b = 0 \<longleftrightarrow> a = 0 \<or> b = 0" (is "?P \<longleftrightarrow> ?Q")
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diff changeset
   283
proof
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   284
  assume ?P then have "0 dvd lcm a b" by simp
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diff changeset
   285
  then have "0 dvd normalize (a * b) div gcd a b"
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diff changeset
   286
    by (simp add: lcm_gcd)
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diff changeset
   287
  then have "0 * gcd a b dvd normalize (a * b)"
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diff changeset
   288
    using dvd_div_iff_mult [of "gcd a b" _ 0] by (cases "gcd a b = 0") simp_all
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diff changeset
   289
  then have "normalize (a * b) = 0"
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diff changeset
   290
    by simp
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diff changeset
   291
  then show ?Q by simp
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diff changeset
   292
next
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diff changeset
   293
  assume ?Q then show ?P by auto
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   294
qed
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diff changeset
   295
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diff changeset
   296
lemma unit_factor_lcm :
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   297
  "unit_factor (lcm a b) = (if a = 0 \<or> b = 0 then 0 else 1)"
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diff changeset
   298
  by (simp add: unit_factor_gcd dvd_unit_factor_div lcm_gcd)
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diff changeset
   299
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diff changeset
   300
sublocale lcm!: abel_semigroup lcm
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diff changeset
   301
proof
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diff changeset
   302
  fix a b c
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diff changeset
   303
  show "lcm a b = lcm b a"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   304
    by (simp add: lcm_gcd ac_simps normalize_mult dvd_normalize_div)
60688
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60687
diff changeset
   305
  have "lcm (lcm a b) c dvd lcm a (lcm b c)"
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60687
diff changeset
   306
    and "lcm a (lcm b c) dvd lcm (lcm a b) c"
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60687
diff changeset
   307
    by (auto intro: lcm_least
60686
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   308
      dvd_trans [of b "lcm b c" "lcm a (lcm b c)"]
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   309
      dvd_trans [of c "lcm b c" "lcm a (lcm b c)"]
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   310
      dvd_trans [of a "lcm a b" "lcm (lcm a b) c"]
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   311
      dvd_trans [of b "lcm a b" "lcm (lcm a b) c"])
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   312
  then show "lcm (lcm a b) c = lcm a (lcm b c)"
60688
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60687
diff changeset
   313
    by (rule associated_eqI) simp_all
60686
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   314
qed
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   315
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   316
lemma lcm_self [simp]:
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   317
  "lcm a a = normalize a"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   318
proof -
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   319
  have "lcm a a dvd a"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   320
    by (rule lcm_least) simp_all
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   321
  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
   322
    by (auto intro: associated_eqI)
60686
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   323
qed
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   324
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   325
lemma gcd_mult_lcm [simp]:
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   326
  "gcd a b * lcm a b = normalize a * normalize b"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   327
  by (simp add: lcm_gcd normalize_mult)
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   328
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   329
lemma lcm_mult_gcd [simp]:
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   330
  "lcm a b * gcd a b = normalize a * normalize b"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   331
  using gcd_mult_lcm [of a b] by (simp add: ac_simps) 
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   332
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   333
lemma gcd_lcm:
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   334
  assumes "a \<noteq> 0" and "b \<noteq> 0"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   335
  shows "gcd a b = normalize (a * b) div lcm a b"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   336
proof -
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   337
  from assms have "lcm a b \<noteq> 0"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   338
    by (simp add: lcm_eq_0_iff)
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   339
  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
   340
  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
   341
    by (simp_all add: normalize_mult)
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   342
  with \<open>lcm a b \<noteq> 0\<close> show ?thesis
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   343
    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
   344
qed
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   345
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   346
lemma lcm_1_left [simp]:
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   347
  "lcm 1 a = normalize a"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   348
  by (simp add: lcm_gcd)
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   349
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   350
lemma lcm_1_right [simp]:
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   351
  "lcm a 1 = normalize a"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   352
  by (simp add: lcm_gcd)
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   353
  
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   354
lemma lcm_mult_left:
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   355
  "lcm (c * a) (c * b) = normalize c * lcm a b"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   356
  by (cases "c = 0")
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   357
    (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
   358
      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
   359
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   360
lemma lcm_mult_right:
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   361
  "lcm (a * c) (b * c) = lcm b a * normalize c"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   362
  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
   363
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   364
lemma mult_lcm_left:
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   365
  "c * lcm a b = unit_factor c * lcm (c * a) (c * b)"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   366
  by (simp add: lcm_mult_left mult.assoc [symmetric])
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   367
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   368
lemma mult_lcm_right:
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   369
  "lcm a b * c = lcm (a * c) (b * c) * unit_factor c"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   370
  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
   371
  
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   372
end
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   373
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   374
class semiring_Gcd = semiring_gcd + Gcd +
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   375
  assumes Gcd_dvd: "a \<in> A \<Longrightarrow> Gcd A dvd a"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   376
    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
   377
    and normalize_Gcd [simp]: "normalize (Gcd A) = Gcd A"
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
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   380
lemma Gcd_empty [simp]:
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   381
  "Gcd {} = 0"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   382
  by (rule dvd_0_left, rule Gcd_greatest) simp
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   383
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   384
lemma Gcd_0_iff [simp]:
60687
33dbbcb6a8a3 eliminated some duplication
haftmann
parents: 60686
diff changeset
   385
  "Gcd A = 0 \<longleftrightarrow> A \<subseteq> {0}" (is "?P \<longleftrightarrow> ?Q")
60686
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   386
proof
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   387
  assume ?P
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   388
  show ?Q
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   389
  proof
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   390
    fix a
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   391
    assume "a \<in> A"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   392
    then have "Gcd A dvd a" by (rule Gcd_dvd)
60687
33dbbcb6a8a3 eliminated some duplication
haftmann
parents: 60686
diff changeset
   393
    with \<open>?P\<close> have "a = 0" by simp
33dbbcb6a8a3 eliminated some duplication
haftmann
parents: 60686
diff changeset
   394
    then show "a \<in> {0}" by simp
60686
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   395
  qed
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   396
next
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   397
  assume ?Q
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   398
  have "0 dvd Gcd A"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   399
  proof (rule Gcd_greatest)
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   400
    fix a
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   401
    assume "a \<in> A"
60687
33dbbcb6a8a3 eliminated some duplication
haftmann
parents: 60686
diff changeset
   402
    with \<open>?Q\<close> have "a = 0" by auto
60686
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   403
    then show "0 dvd a" by simp
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   404
  qed
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   405
  then show ?P by simp
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   406
qed
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   407
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   408
lemma unit_factor_Gcd:
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   409
  "unit_factor (Gcd A) = (if \<forall>a\<in>A. a = 0 then 0 else 1)"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   410
proof (cases "Gcd A = 0")
60687
33dbbcb6a8a3 eliminated some duplication
haftmann
parents: 60686
diff changeset
   411
  case True then show ?thesis by auto
60686
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   412
next
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   413
  case False
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   414
  from unit_factor_mult_normalize
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   415
  have "unit_factor (Gcd A) * normalize (Gcd A) = Gcd A" .
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   416
  then have "unit_factor (Gcd A) * Gcd A = Gcd A" by simp
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   417
  then have "unit_factor (Gcd A) * Gcd A div Gcd A = Gcd A div Gcd A" by simp
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   418
  with False have "unit_factor (Gcd A) = 1" by simp
60687
33dbbcb6a8a3 eliminated some duplication
haftmann
parents: 60686
diff changeset
   419
  with False show ?thesis by auto
60686
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   420
qed
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   421
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   422
lemma Gcd_UNIV [simp]:
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   423
  "Gcd UNIV = 1"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   424
  by (rule associated_eqI) (auto intro: Gcd_dvd simp add: unit_factor_Gcd)
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   425
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   426
lemma Gcd_eq_1_I:
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   427
  assumes "is_unit a" and "a \<in> A"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   428
  shows "Gcd A = 1"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   429
proof -
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   430
  from assms have "is_unit (Gcd A)"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   431
    by (blast intro: Gcd_dvd dvd_unit_imp_unit)
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   432
  then have "normalize (Gcd A) = 1"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   433
    by (rule is_unit_normalize)
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   434
  then show ?thesis
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   435
    by simp
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   436
qed
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   437
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   438
lemma Gcd_insert [simp]:
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   439
  "Gcd (insert a A) = gcd a (Gcd A)"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   440
proof -
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   441
  have "Gcd (insert a A) dvd gcd a (Gcd A)"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   442
    by (auto intro: Gcd_dvd Gcd_greatest)
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   443
  moreover have "gcd a (Gcd A) dvd Gcd (insert a A)"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   444
  proof (rule Gcd_greatest)
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   445
    fix b
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   446
    assume "b \<in> insert a A"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   447
    then show "gcd a (Gcd A) dvd b"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   448
    proof
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   449
      assume "b = a" then show ?thesis by simp
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   450
    next
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   451
      assume "b \<in> A"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   452
      then have "Gcd A dvd b" by (rule Gcd_dvd)
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   453
      moreover have "gcd a (Gcd A) dvd Gcd A" by simp
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   454
      ultimately show ?thesis by (blast intro: dvd_trans)
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   455
    qed
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   456
  qed
60688
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60687
diff changeset
   457
  ultimately show ?thesis
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60687
diff changeset
   458
    by (auto intro: associated_eqI)
60686
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   459
qed
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   460
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   461
lemma dvd_Gcd: -- \<open>FIXME remove\<close>
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   462
  "\<forall>b\<in>A. a dvd b \<Longrightarrow> a dvd Gcd A"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   463
  by (blast intro: Gcd_greatest)
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   464
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   465
lemma Gcd_set [code_unfold]:
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   466
  "Gcd (set as) = foldr gcd as 0"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   467
  by (induct as) simp_all
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   468
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   469
end  
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   470
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   471
class semiring_Lcm = semiring_Gcd +
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   472
  assumes Lcm_Gcd: "Lcm A = Gcd {b. \<forall>a\<in>A. a dvd b}"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   473
begin
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   474
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   475
lemma dvd_Lcm:
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   476
  assumes "a \<in> A"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   477
  shows "a dvd Lcm A"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   478
  using assms by (auto intro: Gcd_greatest simp add: Lcm_Gcd)
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   479
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   480
lemma Gcd_image_normalize [simp]:
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   481
  "Gcd (normalize ` A) = Gcd A"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   482
proof -
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   483
  have "Gcd (normalize ` A) dvd a" if "a \<in> A" for a
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   484
  proof -
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   485
    from that obtain B where "A = insert a B" by blast
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   486
    moreover have " gcd (normalize a) (Gcd (normalize ` B)) dvd normalize a"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   487
      by (rule gcd_dvd1)
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   488
    ultimately show "Gcd (normalize ` A) dvd a"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   489
      by simp
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   490
  qed
60688
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60687
diff changeset
   491
  then have "Gcd (normalize ` A) dvd Gcd A" and "Gcd A dvd Gcd (normalize ` A)"
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60687
diff changeset
   492
    by (auto intro!: Gcd_greatest intro: Gcd_dvd)
60686
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   493
  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
   494
    by (auto intro: associated_eqI)
60686
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   495
qed
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   496
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   497
lemma Lcm_least:
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   498
  assumes "\<And>b. b \<in> A \<Longrightarrow> b dvd a"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   499
  shows "Lcm A dvd a"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   500
  using assms by (auto intro: Gcd_dvd simp add: Lcm_Gcd)
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   501
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   502
lemma normalize_Lcm [simp]:
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   503
  "normalize (Lcm A) = Lcm A"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   504
  by (simp add: Lcm_Gcd)
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   505
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   506
lemma unit_factor_Lcm:
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   507
  "unit_factor (Lcm A) = (if Lcm A = 0 then 0 else 1)"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   508
proof (cases "Lcm A = 0")
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   509
  case True then show ?thesis by simp
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   510
next
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   511
  case False
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   512
  with unit_factor_normalize have "unit_factor (normalize (Lcm A)) = 1"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   513
    by blast
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   514
  with False show ?thesis
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   515
    by simp
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   516
qed
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   517
  
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   518
lemma Lcm_empty [simp]:
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   519
  "Lcm {} = 1"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   520
  by (simp add: Lcm_Gcd)
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   521
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   522
lemma Lcm_1_iff [simp]:
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   523
  "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
   524
proof
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   525
  assume ?P
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   526
  show ?Q
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   527
  proof
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   528
    fix a
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   529
    assume "a \<in> A"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   530
    then have "a dvd Lcm A"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   531
      by (rule dvd_Lcm)
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   532
    with \<open>?P\<close> show "is_unit a"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   533
      by simp
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   534
  qed
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   535
next
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   536
  assume ?Q
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   537
  then have "is_unit (Lcm A)"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   538
    by (blast intro: Lcm_least)
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   539
  then have "normalize (Lcm A) = 1"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   540
    by (rule is_unit_normalize)
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   541
  then show ?P
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   542
    by simp
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   543
qed
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   544
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   545
lemma Lcm_UNIV [simp]:
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   546
  "Lcm UNIV = 0"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   547
proof -
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   548
  have "0 dvd Lcm UNIV"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   549
    by (rule dvd_Lcm) simp
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   550
  then show ?thesis
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   551
    by simp
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   552
qed
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   553
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   554
lemma Lcm_eq_0_I:
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   555
  assumes "0 \<in> A"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   556
  shows "Lcm A = 0"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   557
proof -
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   558
  from assms have "0 dvd Lcm A"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   559
    by (rule dvd_Lcm)
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   560
  then show ?thesis
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   561
    by simp
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   562
qed
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   563
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   564
lemma Gcd_Lcm:
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   565
  "Gcd A = Lcm {b. \<forall>a\<in>A. b dvd a}"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   566
  by (rule associated_eqI) (auto intro: associatedI Gcd_dvd dvd_Lcm Gcd_greatest Lcm_least
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   567
    simp add: unit_factor_Gcd unit_factor_Lcm)
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   568
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   569
lemma Lcm_insert [simp]:
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   570
  "Lcm (insert a A) = lcm a (Lcm A)"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   571
proof (rule sym)
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   572
  have "lcm a (Lcm A) dvd Lcm (insert a A)"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   573
    by (auto intro: dvd_Lcm Lcm_least)
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   574
  moreover have "Lcm (insert a A) dvd lcm a (Lcm A)"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   575
  proof (rule Lcm_least)
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   576
    fix b
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   577
    assume "b \<in> insert a A"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   578
    then show "b dvd lcm a (Lcm A)"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   579
    proof
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   580
      assume "b = a" then show ?thesis by simp
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   581
    next
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   582
      assume "b \<in> A"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   583
      then have "b dvd Lcm A" by (rule dvd_Lcm)
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   584
      moreover have "Lcm A dvd lcm a (Lcm A)" by simp
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   585
      ultimately show ?thesis by (blast intro: dvd_trans)
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   586
    qed
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   587
  qed
60688
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60687
diff changeset
   588
  ultimately show "lcm a (Lcm A) = Lcm (insert a A)"
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60687
diff changeset
   589
    by (rule associated_eqI) (simp_all add: lcm_eq_0_iff)
60686
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   590
qed
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   591
  
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   592
lemma Lcm_set [code_unfold]:
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   593
  "Lcm (set as) = foldr lcm as 1"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   594
  by (induct as) simp_all
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   595
  
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   596
end
59008
f61482b0f240 formally self-contained gcd type classes
haftmann
parents: 58889
diff changeset
   597
f61482b0f240 formally self-contained gcd type classes
haftmann
parents: 58889
diff changeset
   598
class ring_gcd = comm_ring_1 + semiring_gcd
f61482b0f240 formally self-contained gcd type classes
haftmann
parents: 58889
diff changeset
   599
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   600
instantiation nat :: gcd
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   601
begin
21256
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   602
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   603
fun
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   604
  gcd_nat  :: "nat \<Rightarrow> nat \<Rightarrow> nat"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   605
where
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   606
  "gcd_nat x y =
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   607
   (if y = 0 then x else gcd y (x mod y))"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   608
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   609
definition
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   610
  lcm_nat :: "nat \<Rightarrow> nat \<Rightarrow> nat"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   611
where
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   612
  "lcm_nat x y = x * y div (gcd x y)"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   613
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   614
instance proof qed
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   615
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   616
end
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   617
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   618
instantiation int :: gcd
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   619
begin
21256
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   620
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   621
definition
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   622
  gcd_int  :: "int \<Rightarrow> int \<Rightarrow> int"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   623
where
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   624
  "gcd_int x y = int (gcd (nat (abs x)) (nat (abs y)))"
23687
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   625
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   626
definition
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   627
  lcm_int :: "int \<Rightarrow> int \<Rightarrow> int"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   628
where
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   629
  "lcm_int x y = int (lcm (nat (abs x)) (nat (abs y)))"
23687
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   630
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   631
instance proof qed
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   632
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   633
end
23687
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   634
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   635
34030
829eb528b226 resorted code equations from "old" number theory version
haftmann
parents: 33946
diff changeset
   636
subsection {* Transfer setup *}
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   637
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   638
lemma transfer_nat_int_gcd:
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   639
  "(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
   640
  "(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
   641
  unfolding gcd_int_def lcm_int_def
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   642
  by auto
23687
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   643
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   644
lemma transfer_nat_int_gcd_closures:
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   645
  "x >= (0::int) \<Longrightarrow> y >= 0 \<Longrightarrow> gcd x y >= 0"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   646
  "x >= (0::int) \<Longrightarrow> y >= 0 \<Longrightarrow> lcm x y >= 0"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   647
  by (auto simp add: gcd_int_def lcm_int_def)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   648
35644
d20cf282342e transfer: avoid camel case
haftmann
parents: 35368
diff changeset
   649
declare transfer_morphism_nat_int[transfer add return:
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   650
    transfer_nat_int_gcd transfer_nat_int_gcd_closures]
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   651
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   652
lemma transfer_int_nat_gcd:
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   653
  "gcd (int x) (int y) = int (gcd x y)"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   654
  "lcm (int x) (int y) = int (lcm x y)"
32479
521cc9bf2958 some reorganization of number theory
haftmann
parents: 32415
diff changeset
   655
  by (unfold gcd_int_def lcm_int_def, auto)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   656
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   657
lemma transfer_int_nat_gcd_closures:
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   658
  "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
   659
  "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
   660
  by (auto simp add: gcd_int_def lcm_int_def)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   661
35644
d20cf282342e transfer: avoid camel case
haftmann
parents: 35368
diff changeset
   662
declare transfer_morphism_int_nat[transfer add return:
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   663
    transfer_int_nat_gcd transfer_int_nat_gcd_closures]
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   664
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   665
34030
829eb528b226 resorted code equations from "old" number theory version
haftmann
parents: 33946
diff changeset
   666
subsection {* GCD properties *}
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   667
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   668
(* was gcd_induct *)
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   669
lemma gcd_nat_induct:
23687
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   670
  fixes m n :: nat
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   671
  assumes "\<And>m. P m 0"
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   672
    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
   673
  shows "P m n"
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   674
  apply (rule gcd_nat.induct)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   675
  apply (case_tac "y = 0")
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   676
  using assms apply simp_all
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   677
done
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   678
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   679
(* specific to int *)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
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
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54437
diff changeset
   687
lemma gcd_neg_numeral_1_int [simp]:
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54437
diff changeset
   688
  "gcd (- numeral n :: int) x = gcd (numeral n) x"
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54437
diff changeset
   689
  by (fact gcd_neg1_int)
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54437
diff changeset
   690
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54437
diff changeset
   691
lemma gcd_neg_numeral_2_int [simp]:
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54437
diff changeset
   692
  "gcd x (- numeral n :: int) = gcd x (numeral n)"
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54437
diff changeset
   693
  by (fact gcd_neg2_int)
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54437
diff changeset
   694
31813
4df828bbc411 gcd abs lemmas
nipkow
parents: 31798
diff changeset
   695
lemma abs_gcd_int[simp]: "abs(gcd (x::int) y) = gcd x y"
4df828bbc411 gcd abs lemmas
nipkow
parents: 31798
diff changeset
   696
by(simp add: gcd_int_def)
4df828bbc411 gcd abs lemmas
nipkow
parents: 31798
diff changeset
   697
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   698
lemma gcd_abs_int: "gcd (x::int) y = gcd (abs x) (abs y)"
31813
4df828bbc411 gcd abs lemmas
nipkow
parents: 31798
diff changeset
   699
by (simp add: gcd_int_def)
4df828bbc411 gcd abs lemmas
nipkow
parents: 31798
diff changeset
   700
4df828bbc411 gcd abs lemmas
nipkow
parents: 31798
diff changeset
   701
lemma gcd_abs1_int[simp]: "gcd (abs x) (y::int) = gcd x y"
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   702
by (metis abs_idempotent gcd_abs_int)
31813
4df828bbc411 gcd abs lemmas
nipkow
parents: 31798
diff changeset
   703
4df828bbc411 gcd abs lemmas
nipkow
parents: 31798
diff changeset
   704
lemma gcd_abs2_int[simp]: "gcd x (abs y::int) = gcd x y"
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   705
by (metis abs_idempotent gcd_abs_int)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   706
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   707
lemma gcd_cases_int:
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   708
  fixes x :: int and y
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   709
  assumes "x >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> P (gcd x y)"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   710
      and "x >= 0 \<Longrightarrow> y <= 0 \<Longrightarrow> P (gcd x (-y))"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   711
      and "x <= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> P (gcd (-x) y)"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   712
      and "x <= 0 \<Longrightarrow> y <= 0 \<Longrightarrow> P (gcd (-x) (-y))"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   713
  shows "P (gcd x y)"
35216
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 35028
diff changeset
   714
by (insert assms, auto, arith)
21256
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   715
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   716
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
   717
  by (simp add: gcd_int_def)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   718
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   719
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
   720
  by (simp add: lcm_int_def)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   721
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   722
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
   723
  by (simp add: lcm_int_def)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   724
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   725
lemma lcm_abs_int: "lcm (x::int) y = lcm (abs x) (abs y)"
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   726
  by (simp add: lcm_int_def)
21256
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   727
31814
7c122634da81 lcm abs lemmas
nipkow
parents: 31813
diff changeset
   728
lemma abs_lcm_int [simp]: "abs (lcm i j::int) = lcm i j"
7c122634da81 lcm abs lemmas
nipkow
parents: 31813
diff changeset
   729
by(simp add:lcm_int_def)
7c122634da81 lcm abs lemmas
nipkow
parents: 31813
diff changeset
   730
7c122634da81 lcm abs lemmas
nipkow
parents: 31813
diff changeset
   731
lemma lcm_abs1_int[simp]: "lcm (abs x) (y::int) = lcm x y"
7c122634da81 lcm abs lemmas
nipkow
parents: 31813
diff changeset
   732
by (metis abs_idempotent lcm_int_def)
7c122634da81 lcm abs lemmas
nipkow
parents: 31813
diff changeset
   733
7c122634da81 lcm abs lemmas
nipkow
parents: 31813
diff changeset
   734
lemma lcm_abs2_int[simp]: "lcm x (abs y::int) = lcm x y"
7c122634da81 lcm abs lemmas
nipkow
parents: 31813
diff changeset
   735
by (metis abs_idempotent lcm_int_def)
7c122634da81 lcm abs lemmas
nipkow
parents: 31813
diff changeset
   736
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   737
lemma lcm_cases_int:
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   738
  fixes x :: int and y
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   739
  assumes "x >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> P (lcm x y)"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   740
      and "x >= 0 \<Longrightarrow> y <= 0 \<Longrightarrow> P (lcm x (-y))"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   741
      and "x <= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> P (lcm (-x) y)"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   742
      and "x <= 0 \<Longrightarrow> y <= 0 \<Longrightarrow> P (lcm (-x) (-y))"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   743
  shows "P (lcm x y)"
41550
efa734d9b221 eliminated global prems;
wenzelm
parents: 37770
diff changeset
   744
  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
   745
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   746
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
   747
  by (simp add: lcm_int_def)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   748
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   749
(* was gcd_0, etc. *)
54867
c21a2465cac1 prefer ephemeral interpretation over interpretation in proof contexts;
haftmann
parents: 54489
diff changeset
   750
lemma gcd_0_nat: "gcd (x::nat) 0 = x"
23687
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   751
  by simp
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   752
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   753
(* was igcd_0, etc. *)
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   754
lemma gcd_0_int [simp]: "gcd (x::int) 0 = abs x"
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   755
  by (unfold gcd_int_def, auto)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   756
54867
c21a2465cac1 prefer ephemeral interpretation over interpretation in proof contexts;
haftmann
parents: 54489
diff changeset
   757
lemma gcd_0_left_nat: "gcd 0 (x::nat) = x"
23687
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   758
  by simp
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   759
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   760
lemma gcd_0_left_int [simp]: "gcd 0 (x::int) = abs x"
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   761
  by (unfold gcd_int_def, auto)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   762
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   763
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
   764
  by (case_tac "y = 0", auto)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   765
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   766
(* weaker, but useful for the simplifier *)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   767
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   768
lemma gcd_non_0_nat: "y ~= (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
   769
  by simp
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   770
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   771
lemma gcd_1_nat [simp]: "gcd (m::nat) 1 = 1"
21263
wenzelm
parents: 21256
diff changeset
   772
  by simp
21256
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   773
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   774
lemma gcd_Suc_0 [simp]: "gcd (m::nat) (Suc 0) = Suc 0"
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   775
  by (simp add: One_nat_def)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   776
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   777
lemma gcd_1_int [simp]: "gcd (m::int) 1 = 1"
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   778
  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
   779
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   780
lemma gcd_idem_nat: "gcd (x::nat) x = x"
31798
fe9a3043d36c Cleaned up GCD
nipkow
parents: 31766
diff changeset
   781
by simp
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   782
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   783
lemma gcd_idem_int: "gcd (x::int) x = abs x"
31813
4df828bbc411 gcd abs lemmas
nipkow
parents: 31798
diff changeset
   784
by (auto simp add: gcd_int_def)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   785
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   786
declare gcd_nat.simps [simp del]
21256
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   787
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   788
text {*
27556
292098f2efdf unified curried gcd, lcm, zgcd, zlcm
haftmann
parents: 27487
diff changeset
   789
  \medskip @{term "gcd m n"} divides @{text m} and @{text n}.  The
21256
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   790
  conjunctions don't seem provable separately.
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   791
*}
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   792
59008
f61482b0f240 formally self-contained gcd type classes
haftmann
parents: 58889
diff changeset
   793
instance nat :: semiring_gcd
f61482b0f240 formally self-contained gcd type classes
haftmann
parents: 58889
diff changeset
   794
proof
f61482b0f240 formally self-contained gcd type classes
haftmann
parents: 58889
diff changeset
   795
  fix m n :: nat
f61482b0f240 formally self-contained gcd type classes
haftmann
parents: 58889
diff changeset
   796
  show "gcd m n dvd m" and "gcd m n dvd n"
f61482b0f240 formally self-contained gcd type classes
haftmann
parents: 58889
diff changeset
   797
  proof (induct m n rule: gcd_nat_induct)
f61482b0f240 formally self-contained gcd type classes
haftmann
parents: 58889
diff changeset
   798
    fix m n :: nat
f61482b0f240 formally self-contained gcd type classes
haftmann
parents: 58889
diff changeset
   799
    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
   800
    then have "gcd n (m mod n) dvd m"
f61482b0f240 formally self-contained gcd type classes
haftmann
parents: 58889
diff changeset
   801
      by (rule dvd_mod_imp_dvd)
f61482b0f240 formally self-contained gcd type classes
haftmann
parents: 58889
diff changeset
   802
    moreover assume "0 < n"
f61482b0f240 formally self-contained gcd type classes
haftmann
parents: 58889
diff changeset
   803
    ultimately show "gcd m n dvd m"
f61482b0f240 formally self-contained gcd type classes
haftmann
parents: 58889
diff changeset
   804
      by (simp add: gcd_non_0_nat)
f61482b0f240 formally self-contained gcd type classes
haftmann
parents: 58889
diff changeset
   805
  qed (simp_all add: gcd_0_nat gcd_non_0_nat)
f61482b0f240 formally self-contained gcd type classes
haftmann
parents: 58889
diff changeset
   806
next
f61482b0f240 formally self-contained gcd type classes
haftmann
parents: 58889
diff changeset
   807
  fix m n k :: nat
f61482b0f240 formally self-contained gcd type classes
haftmann
parents: 58889
diff changeset
   808
  assume "k dvd m" and "k dvd n"
f61482b0f240 formally self-contained gcd type classes
haftmann
parents: 58889
diff changeset
   809
  then show "k dvd gcd m n"
f61482b0f240 formally self-contained gcd type classes
haftmann
parents: 58889
diff changeset
   810
    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
   811
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
   812
59008
f61482b0f240 formally self-contained gcd type classes
haftmann
parents: 58889
diff changeset
   813
instance int :: ring_gcd
60686
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   814
  by standard
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   815
    (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
   816
31730
d74830dc3e4a added lemmas; tuned
nipkow
parents: 31729
diff changeset
   817
lemma dvd_gcd_D1_nat: "k dvd gcd m n \<Longrightarrow> (k::nat) dvd m"
59008
f61482b0f240 formally self-contained gcd type classes
haftmann
parents: 58889
diff changeset
   818
  by (metis gcd_dvd1 dvd_trans)
31730
d74830dc3e4a added lemmas; tuned
nipkow
parents: 31729
diff changeset
   819
d74830dc3e4a added lemmas; tuned
nipkow
parents: 31729
diff changeset
   820
lemma dvd_gcd_D2_nat: "k dvd gcd m n \<Longrightarrow> (k::nat) dvd n"
59008
f61482b0f240 formally self-contained gcd type classes
haftmann
parents: 58889
diff changeset
   821
  by (metis gcd_dvd2 dvd_trans)
31730
d74830dc3e4a added lemmas; tuned
nipkow
parents: 31729
diff changeset
   822
d74830dc3e4a added lemmas; tuned
nipkow
parents: 31729
diff changeset
   823
lemma dvd_gcd_D1_int: "i dvd gcd m n \<Longrightarrow> (i::int) dvd m"
59008
f61482b0f240 formally self-contained gcd type classes
haftmann
parents: 58889
diff changeset
   824
  by (metis gcd_dvd1 dvd_trans)
31730
d74830dc3e4a added lemmas; tuned
nipkow
parents: 31729
diff changeset
   825
d74830dc3e4a added lemmas; tuned
nipkow
parents: 31729
diff changeset
   826
lemma dvd_gcd_D2_int: "i dvd gcd m n \<Longrightarrow> (i::int) dvd n"
59008
f61482b0f240 formally self-contained gcd type classes
haftmann
parents: 58889
diff changeset
   827
  by (metis gcd_dvd2 dvd_trans)
31730
d74830dc3e4a added lemmas; tuned
nipkow
parents: 31729
diff changeset
   828
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   829
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
   830
  by (rule dvd_imp_le, auto)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   831
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   832
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
   833
  by (rule dvd_imp_le, auto)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   834
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   835
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
   836
  by (rule zdvd_imp_le, auto)
21256
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   837
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   838
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
   839
  by (rule zdvd_imp_le, auto)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   840
60689
8a2d7c04d8c0 more cautious use of [iff] declarations
haftmann
parents: 60688
diff changeset
   841
lemma gcd_greatest_iff_nat:
8a2d7c04d8c0 more cautious use of [iff] declarations
haftmann
parents: 60688
diff changeset
   842
  "(k dvd gcd (m::nat) n) = (k dvd m & k dvd n)"
8a2d7c04d8c0 more cautious use of [iff] declarations
haftmann
parents: 60688
diff changeset
   843
  by (fact gcd_greatest_iff)
8a2d7c04d8c0 more cautious use of [iff] declarations
haftmann
parents: 60688
diff changeset
   844
8a2d7c04d8c0 more cautious use of [iff] declarations
haftmann
parents: 60688
diff changeset
   845
lemma gcd_greatest_iff_int:
8a2d7c04d8c0 more cautious use of [iff] declarations
haftmann
parents: 60688
diff changeset
   846
  "((k::int) dvd gcd m n) = (k dvd m & k dvd n)"
8a2d7c04d8c0 more cautious use of [iff] declarations
haftmann
parents: 60688
diff changeset
   847
  by (fact gcd_greatest_iff)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   848
60689
8a2d7c04d8c0 more cautious use of [iff] declarations
haftmann
parents: 60688
diff changeset
   849
lemma gcd_zero_nat: 
8a2d7c04d8c0 more cautious use of [iff] declarations
haftmann
parents: 60688
diff changeset
   850
  "(gcd (m::nat) n = 0) = (m = 0 & n = 0)"
8a2d7c04d8c0 more cautious use of [iff] declarations
haftmann
parents: 60688
diff changeset
   851
  by (fact gcd_eq_0_iff)
21256
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   852
60689
8a2d7c04d8c0 more cautious use of [iff] declarations
haftmann
parents: 60688
diff changeset
   853
lemma gcd_zero_int [simp]:
8a2d7c04d8c0 more cautious use of [iff] declarations
haftmann
parents: 60688
diff changeset
   854
  "(gcd (m::int) n = 0) = (m = 0 & n = 0)"
8a2d7c04d8c0 more cautious use of [iff] declarations
haftmann
parents: 60688
diff changeset
   855
  by (fact gcd_eq_0_iff)
21256
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   856
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   857
lemma gcd_pos_nat [simp]: "(gcd (m::nat) n > 0) = (m ~= 0 | n ~= 0)"
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   858
  by (insert gcd_zero_nat [of m n], arith)
21256
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   859
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   860
lemma gcd_pos_int [simp]: "(gcd (m::int) n > 0) = (m ~= 0 | n ~= 0)"
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   861
  by (insert gcd_zero_int [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
   862
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   863
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
   864
    (\<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
   865
  apply auto
33657
a4179bf442d1 renamed lemmas "anti_sym" -> "antisym"
nipkow
parents: 33318
diff changeset
   866
  apply (rule dvd_antisym)
59008
f61482b0f240 formally self-contained gcd type classes
haftmann
parents: 58889
diff changeset
   867
  apply (erule (1) gcd_greatest)
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
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   871
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
   872
    (\<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
   873
apply (case_tac "d = 0")
a4179bf442d1 renamed lemmas "anti_sym" -> "antisym"
nipkow
parents: 33318
diff changeset
   874
 apply simp
a4179bf442d1 renamed lemmas "anti_sym" -> "antisym"
nipkow
parents: 33318
diff changeset
   875
apply (rule iffI)
a4179bf442d1 renamed lemmas "anti_sym" -> "antisym"
nipkow
parents: 33318
diff changeset
   876
 apply (rule zdvd_antisym_nonneg)
59008
f61482b0f240 formally self-contained gcd type classes
haftmann
parents: 58889
diff changeset
   877
 apply (auto intro: gcd_greatest)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   878
done
30082
43c5b7bfc791 make more proofs work whether or not One_nat_def is a simp rule
huffman
parents: 30042
diff changeset
   879
54867
c21a2465cac1 prefer ephemeral interpretation over interpretation in proof contexts;
haftmann
parents: 54489
diff changeset
   880
interpretation gcd_nat: abel_semigroup "gcd :: nat \<Rightarrow> nat \<Rightarrow> nat"
c21a2465cac1 prefer ephemeral interpretation over interpretation in proof contexts;
haftmann
parents: 54489
diff changeset
   881
  + gcd_nat: semilattice_neutr_order "gcd :: nat \<Rightarrow> nat \<Rightarrow> nat" 0 "op dvd" "(\<lambda>m n. m dvd n \<and> \<not> n dvd m)"
60686
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   882
apply standard
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   883
apply (auto intro: dvd_antisym dvd_trans)[2]
59545
12a6088ed195 explicit equivalence for strict order on lattices
haftmann
parents: 59497
diff changeset
   884
apply (metis dvd.dual_order.refl gcd_unique_nat)+
54867
c21a2465cac1 prefer ephemeral interpretation over interpretation in proof contexts;
haftmann
parents: 54489
diff changeset
   885
done
c21a2465cac1 prefer ephemeral interpretation over interpretation in proof contexts;
haftmann
parents: 54489
diff changeset
   886
60686
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   887
interpretation gcd_int: abel_semigroup "gcd :: int \<Rightarrow> int \<Rightarrow> int" ..
54867
c21a2465cac1 prefer ephemeral interpretation over interpretation in proof contexts;
haftmann
parents: 54489
diff changeset
   888
60686
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   889
lemmas gcd_assoc_nat = gcd.assoc [where ?'a = nat]
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   890
lemmas gcd_commute_nat = gcd.commute [where ?'a = nat]
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   891
lemmas gcd_left_commute_nat = gcd.left_commute [where ?'a = nat]
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   892
lemmas gcd_assoc_int = gcd.assoc [where ?'a = int]
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   893
lemmas gcd_commute_int = gcd.commute [where ?'a = int]
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   894
lemmas gcd_left_commute_int = gcd.left_commute [where ?'a = int]
54867
c21a2465cac1 prefer ephemeral interpretation over interpretation in proof contexts;
haftmann
parents: 54489
diff changeset
   895
c21a2465cac1 prefer ephemeral interpretation over interpretation in proof contexts;
haftmann
parents: 54489
diff changeset
   896
lemmas gcd_ac_nat = gcd_assoc_nat gcd_commute_nat gcd_left_commute_nat
c21a2465cac1 prefer ephemeral interpretation over interpretation in proof contexts;
haftmann
parents: 54489
diff changeset
   897
c21a2465cac1 prefer ephemeral interpretation over interpretation in proof contexts;
haftmann
parents: 54489
diff changeset
   898
lemmas gcd_ac_int = gcd_assoc_int gcd_commute_int gcd_left_commute_int
c21a2465cac1 prefer ephemeral interpretation over interpretation in proof contexts;
haftmann
parents: 54489
diff changeset
   899
31798
fe9a3043d36c Cleaned up GCD
nipkow
parents: 31766
diff changeset
   900
lemma gcd_proj1_if_dvd_nat [simp]: "(x::nat) dvd y \<Longrightarrow> gcd x y = x"
54867
c21a2465cac1 prefer ephemeral interpretation over interpretation in proof contexts;
haftmann
parents: 54489
diff changeset
   901
  by (fact gcd_nat.absorb1)
31798
fe9a3043d36c Cleaned up GCD
nipkow
parents: 31766
diff changeset
   902
fe9a3043d36c Cleaned up GCD
nipkow
parents: 31766
diff changeset
   903
lemma gcd_proj2_if_dvd_nat [simp]: "(y::nat) dvd x \<Longrightarrow> gcd x y = y"
54867
c21a2465cac1 prefer ephemeral interpretation over interpretation in proof contexts;
haftmann
parents: 54489
diff changeset
   904
  by (fact gcd_nat.absorb2)
31798
fe9a3043d36c Cleaned up GCD
nipkow
parents: 31766
diff changeset
   905
54867
c21a2465cac1 prefer ephemeral interpretation over interpretation in proof contexts;
haftmann
parents: 54489
diff changeset
   906
lemma gcd_proj1_if_dvd_int [simp]: "x dvd y \<Longrightarrow> gcd (x::int) y = abs x"
c21a2465cac1 prefer ephemeral interpretation over interpretation in proof contexts;
haftmann
parents: 54489
diff changeset
   907
  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
   908
54867
c21a2465cac1 prefer ephemeral interpretation over interpretation in proof contexts;
haftmann
parents: 54489
diff changeset
   909
lemma gcd_proj2_if_dvd_int [simp]: "y dvd x \<Longrightarrow> gcd (x::int) y = abs y"
c21a2465cac1 prefer ephemeral interpretation over interpretation in proof contexts;
haftmann
parents: 54489
diff changeset
   910
  by (metis gcd_proj1_if_dvd_int gcd_commute_int)
31798
fe9a3043d36c Cleaned up GCD
nipkow
parents: 31766
diff changeset
   911
21256
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   912
text {*
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   913
  \medskip Multiplication laws
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   914
*}
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   915
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   916
lemma gcd_mult_distrib_nat: "(k::nat) * gcd m n = gcd (k * m) (k * n)"
58623
2db1df2c8467 more bibtex entries;
wenzelm
parents: 57514
diff changeset
   917
    -- {* @{cite \<open>page 27\<close> davenport92} *}
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   918
  apply (induct m n rule: gcd_nat_induct)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   919
  apply simp
21256
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   920
  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
   921
  apply (simp_all add: gcd_non_0_nat)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   922
done
21256
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   923
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   924
lemma gcd_mult_distrib_int: "abs (k::int) * gcd m n = gcd (k * m) (k * n)"
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   925
  apply (subst (1 2) gcd_abs_int)
31813
4df828bbc411 gcd abs lemmas
nipkow
parents: 31798
diff changeset
   926
  apply (subst (1 2) abs_mult)
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   927
  apply (rule gcd_mult_distrib_nat [transferred])
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   928
  apply auto
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   929
done
21256
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   930
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   931
lemma coprime_dvd_mult_nat: "coprime (k::nat) n \<Longrightarrow> k dvd m * n \<Longrightarrow> k dvd m"
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   932
  apply (insert gcd_mult_distrib_nat [of m k n])
21256
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   933
  apply simp
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   934
  apply (erule_tac t = m in ssubst)
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   935
  apply simp
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   936
  done
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   937
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   938
lemma coprime_dvd_mult_int:
31813
4df828bbc411 gcd abs lemmas
nipkow
parents: 31798
diff changeset
   939
  "coprime (k::int) n \<Longrightarrow> k dvd m * n \<Longrightarrow> k dvd m"
4df828bbc411 gcd abs lemmas
nipkow
parents: 31798
diff changeset
   940
apply (subst abs_dvd_iff [symmetric])
4df828bbc411 gcd abs lemmas
nipkow
parents: 31798
diff changeset
   941
apply (subst dvd_abs_iff [symmetric])
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   942
apply (subst (asm) gcd_abs_int)
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   943
apply (rule coprime_dvd_mult_nat [transferred])
31813
4df828bbc411 gcd abs lemmas
nipkow
parents: 31798
diff changeset
   944
    prefer 4 apply assumption
4df828bbc411 gcd abs lemmas
nipkow
parents: 31798
diff changeset
   945
   apply auto
4df828bbc411 gcd abs lemmas
nipkow
parents: 31798
diff changeset
   946
apply (subst abs_mult [symmetric], auto)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   947
done
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   948
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   949
lemma coprime_dvd_mult_iff_nat: "coprime (k::nat) n \<Longrightarrow>
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   950
    (k dvd m * n) = (k dvd m)"
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   951
  by (auto intro: coprime_dvd_mult_nat)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   952
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   953
lemma coprime_dvd_mult_iff_int: "coprime (k::int) n \<Longrightarrow>
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   954
    (k dvd m * n) = (k dvd m)"
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   955
  by (auto intro: coprime_dvd_mult_int)
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_mult_cancel_nat: "coprime k n \<Longrightarrow> gcd ((k::nat) * m) n = gcd m n"
33657
a4179bf442d1 renamed lemmas "anti_sym" -> "antisym"
nipkow
parents: 33318
diff changeset
   958
  apply (rule dvd_antisym)
59008
f61482b0f240 formally self-contained gcd type classes
haftmann
parents: 58889
diff changeset
   959
  apply (rule gcd_greatest)
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   960
  apply (rule_tac n = k in coprime_dvd_mult_nat)
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   961
  apply (simp add: gcd_assoc_nat)
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   962
  apply (simp add: gcd_commute_nat)
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 56218
diff changeset
   963
  apply (simp_all add: mult.commute)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   964
done
21256
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   965
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   966
lemma gcd_mult_cancel_int:
31813
4df828bbc411 gcd abs lemmas
nipkow
parents: 31798
diff changeset
   967
  "coprime (k::int) n \<Longrightarrow> gcd (k * m) n = gcd m n"
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   968
apply (subst (1 2) gcd_abs_int)
31813
4df828bbc411 gcd abs lemmas
nipkow
parents: 31798
diff changeset
   969
apply (subst abs_mult)
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   970
apply (rule gcd_mult_cancel_nat [transferred], auto)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   971
done
21256
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   972
35368
19b340c3f1ff crossproduct coprimality lemmas
haftmann
parents: 35216
diff changeset
   973
lemma coprime_crossproduct_nat:
19b340c3f1ff crossproduct coprimality lemmas
haftmann
parents: 35216
diff changeset
   974
  fixes a b c d :: nat
19b340c3f1ff crossproduct coprimality lemmas
haftmann
parents: 35216
diff changeset
   975
  assumes "coprime a d" and "coprime b c"
19b340c3f1ff crossproduct coprimality lemmas
haftmann
parents: 35216
diff changeset
   976
  shows "a * c = b * d \<longleftrightarrow> a = b \<and> c = d" (is "?lhs \<longleftrightarrow> ?rhs")
19b340c3f1ff crossproduct coprimality lemmas
haftmann
parents: 35216
diff changeset
   977
proof
19b340c3f1ff crossproduct coprimality lemmas
haftmann
parents: 35216
diff changeset
   978
  assume ?rhs then show ?lhs by simp
19b340c3f1ff crossproduct coprimality lemmas
haftmann
parents: 35216
diff changeset
   979
next
19b340c3f1ff crossproduct coprimality lemmas
haftmann
parents: 35216
diff changeset
   980
  assume ?lhs
19b340c3f1ff crossproduct coprimality lemmas
haftmann
parents: 35216
diff changeset
   981
  from `?lhs` have "a dvd b * d" by (auto intro: dvdI dest: sym)
19b340c3f1ff crossproduct coprimality lemmas
haftmann
parents: 35216
diff changeset
   982
  with `coprime a d` have "a dvd b" by (simp add: coprime_dvd_mult_iff_nat)
19b340c3f1ff crossproduct coprimality lemmas
haftmann
parents: 35216
diff changeset
   983
  from `?lhs` have "b dvd a * c" by (auto intro: dvdI dest: sym)
19b340c3f1ff crossproduct coprimality lemmas
haftmann
parents: 35216
diff changeset
   984
  with `coprime b c` have "b dvd a" by (simp add: coprime_dvd_mult_iff_nat)
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 56218
diff changeset
   985
  from `?lhs` have "c dvd d * b" by (auto intro: dvdI dest: sym simp add: mult.commute)
35368
19b340c3f1ff crossproduct coprimality lemmas
haftmann
parents: 35216
diff changeset
   986
  with `coprime b c` have "c dvd d" by (simp add: coprime_dvd_mult_iff_nat gcd_commute_nat)
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 56218
diff changeset
   987
  from `?lhs` have "d dvd c * a" by (auto intro: dvdI dest: sym simp add: mult.commute)
35368
19b340c3f1ff crossproduct coprimality lemmas
haftmann
parents: 35216
diff changeset
   988
  with `coprime a d` have "d dvd c" by (simp add: coprime_dvd_mult_iff_nat gcd_commute_nat)
19b340c3f1ff crossproduct coprimality lemmas
haftmann
parents: 35216
diff changeset
   989
  from `a dvd b` `b dvd a` have "a = b" by (rule Nat.dvd.antisym)
19b340c3f1ff crossproduct coprimality lemmas
haftmann
parents: 35216
diff changeset
   990
  moreover from `c dvd d` `d dvd c` have "c = d" by (rule Nat.dvd.antisym)
19b340c3f1ff crossproduct coprimality lemmas
haftmann
parents: 35216
diff changeset
   991
  ultimately show ?rhs ..
19b340c3f1ff crossproduct coprimality lemmas
haftmann
parents: 35216
diff changeset
   992
qed
19b340c3f1ff crossproduct coprimality lemmas
haftmann
parents: 35216
diff changeset
   993
19b340c3f1ff crossproduct coprimality lemmas
haftmann
parents: 35216
diff changeset
   994
lemma coprime_crossproduct_int:
19b340c3f1ff crossproduct coprimality lemmas
haftmann
parents: 35216
diff changeset
   995
  fixes a b c d :: int
19b340c3f1ff crossproduct coprimality lemmas
haftmann
parents: 35216
diff changeset
   996
  assumes "coprime a d" and "coprime b c"
19b340c3f1ff crossproduct coprimality lemmas
haftmann
parents: 35216
diff changeset
   997
  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>"
19b340c3f1ff crossproduct coprimality lemmas
haftmann
parents: 35216
diff changeset
   998
  using assms by (intro coprime_crossproduct_nat [transferred]) auto
19b340c3f1ff crossproduct coprimality lemmas
haftmann
parents: 35216
diff changeset
   999
21256
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
  1000
text {* \medskip Addition laws *}
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
  1001
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1002
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
  1003
  apply (case_tac "n = 0")
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1004
  apply (simp_all add: gcd_non_0_nat)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1005
done
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1006
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1007
lemma gcd_add2_nat [simp]: "gcd (m::nat) (m + n) = gcd m n"
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1008
  apply (subst (1 2) gcd_commute_nat)
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 56218
diff changeset
  1009
  apply (subst add.commute)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1010
  apply simp
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1011
done
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1012
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1013
(* to do: add the other variations? *)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1014
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1015
lemma gcd_diff1_nat: "(m::nat) >= n \<Longrightarrow> gcd (m - n) n = gcd m n"
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1016
  by (subst gcd_add1_nat [symmetric], auto)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1017
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1018
lemma gcd_diff2_nat: "(n::nat) >= m \<Longrightarrow> gcd (n - m) n = gcd m n"
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1019
  apply (subst gcd_commute_nat)
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1020
  apply (subst gcd_diff1_nat [symmetric])
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1021
  apply auto
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1022
  apply (subst gcd_commute_nat)
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1023
  apply (subst gcd_diff1_nat)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1024
  apply assumption
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1025
  apply (rule gcd_commute_nat)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1026
done
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1027
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1028
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
  1029
  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
  1030
  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
  1031
  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
  1032
    zmod_zminus1_eq_if)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1033
  apply (frule_tac a = x in pos_mod_bound)
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1034
  apply (subst (1 2) gcd_commute_nat)
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1035
  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
  1036
    nat_le_eq_zle)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1037
done
21256
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
  1038
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1039
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
  1040
  apply (case_tac "y = 0")
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1041
  apply force
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1042
  apply (case_tac "y > 0")
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1043
  apply (subst gcd_non_0_int, auto)
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1044
  apply (insert gcd_non_0_int [of "-y" "-x"])
35216
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 35028
diff changeset
  1045
  apply auto
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1046
done
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1047
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1048
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
  1049
by (metis gcd_red_int mod_add_self1 add.commute)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1050
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1051
lemma gcd_add2_int [simp]: "gcd m ((m::int) + n) = gcd m n"
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 56218
diff changeset
  1052
by (metis gcd_add1_int gcd_commute_int add.commute)
21256
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
  1053
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1054
lemma gcd_add_mult_nat: "gcd (m::nat) (k * m + n) = gcd m n"
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1055
by (metis mod_mult_self3 gcd_commute_nat gcd_red_nat)
21256
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
  1056
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1057
lemma gcd_add_mult_int: "gcd (m::int) (k * m + n) = gcd m n"
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 56218
diff changeset
  1058
by (metis gcd_commute_int gcd_red_int mod_mult_self1 add.commute)
31798
fe9a3043d36c Cleaned up GCD
nipkow
parents: 31766
diff changeset
  1059
21256
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
  1060
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1061
(* to do: differences, and all variations of addition rules
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1062
    as simplification rules for nat and int *)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1063
60689
8a2d7c04d8c0 more cautious use of [iff] declarations
haftmann
parents: 60688
diff changeset
  1064
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
  1065
  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
  1066
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1067
(* to do: add the three variations of these, and for ints? *)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1068
31992
f8aed98faae7 More about gcd/lcm, and some cleaning up
nipkow
parents: 31952
diff changeset
  1069
lemma finite_divisors_nat[simp]:
f8aed98faae7 More about gcd/lcm, and some cleaning up
nipkow
parents: 31952
diff changeset
  1070
  assumes "(m::nat) ~= 0" shows "finite{d. d dvd m}"
31734
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
  1071
proof-
60512
e0169291b31c tuned proofs -- slightly faster;
wenzelm
parents: 60357
diff changeset
  1072
  have "finite{d. d <= m}"
e0169291b31c tuned proofs -- slightly faster;
wenzelm
parents: 60357
diff changeset
  1073
    by (blast intro: bounded_nat_set_is_finite)
31734
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
  1074
  from finite_subset[OF _ this] show ?thesis using assms
60512
e0169291b31c tuned proofs -- slightly faster;
wenzelm
parents: 60357
diff changeset
  1075
    by (metis Collect_mono dvd_imp_le neq0_conv)
31734
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
  1076
qed
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
  1077
31995
8f37cf60b885 more gcd/lcm lemmas
nipkow
parents: 31992
diff changeset
  1078
lemma finite_divisors_int[simp]:
31734
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
  1079
  assumes "(i::int) ~= 0" shows "finite{d. d dvd i}"
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
  1080
proof-
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
  1081
  have "{d. abs d <= abs i} = {- abs i .. abs i}" by(auto simp:abs_if)
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
  1082
  hence "finite{d. abs d <= abs i}" by simp
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
  1083
  from finite_subset[OF _ this] show ?thesis using assms
60512
e0169291b31c tuned proofs -- slightly faster;
wenzelm
parents: 60357
diff changeset
  1084
    by (simp add: dvd_imp_le_int subset_iff)
31734
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
  1085
qed
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
  1086
31995
8f37cf60b885 more gcd/lcm lemmas
nipkow
parents: 31992
diff changeset
  1087
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
  1088
apply(rule antisym)
44890
22f665a2e91c new fastforce replacing fastsimp - less confusing name
nipkow
parents: 44845
diff changeset
  1089
 apply (fastforce intro: Max_le_iff[THEN iffD2] simp: dvd_imp_le)
31995
8f37cf60b885 more gcd/lcm lemmas
nipkow
parents: 31992
diff changeset
  1090
apply simp
8f37cf60b885 more gcd/lcm lemmas
nipkow
parents: 31992
diff changeset
  1091
done
8f37cf60b885 more gcd/lcm lemmas
nipkow
parents: 31992
diff changeset
  1092
8f37cf60b885 more gcd/lcm lemmas
nipkow
parents: 31992
diff changeset
  1093
lemma Max_divisors_self_int[simp]: "n\<noteq>0 \<Longrightarrow> Max{d::int. d dvd n} = abs n"
8f37cf60b885 more gcd/lcm lemmas
nipkow
parents: 31992
diff changeset
  1094
apply(rule antisym)
44278
1220ecb81e8f observe distinction between sets and predicates more properly
haftmann
parents: 42871
diff changeset
  1095
 apply(rule Max_le_iff [THEN iffD2])
1220ecb81e8f observe distinction between sets and predicates more properly
haftmann
parents: 42871
diff changeset
  1096
  apply (auto intro: abs_le_D1 dvd_imp_le_int)
31995
8f37cf60b885 more gcd/lcm lemmas
nipkow
parents: 31992
diff changeset
  1097
done
8f37cf60b885 more gcd/lcm lemmas
nipkow
parents: 31992
diff changeset
  1098
31734
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
  1099
lemma gcd_is_Max_divisors_nat:
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
  1100
  "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
  1101
apply(rule Max_eqI[THEN sym])
31995
8f37cf60b885 more gcd/lcm lemmas
nipkow
parents: 31992
diff changeset
  1102
  apply (metis finite_Collect_conjI finite_divisors_nat)
31734
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
  1103
 apply simp
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1104
 apply(metis Suc_diff_1 Suc_neq_Zero dvd_imp_le gcd_greatest_iff_nat gcd_pos_nat)
31734
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
  1105
apply simp
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
  1106
done
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
  1107
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
  1108
lemma gcd_is_Max_divisors_int:
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
  1109
  "m ~= 0 ==> n ~= 0 ==> gcd (m::int) n = (Max {d. d dvd m & d dvd n})"
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
  1110
apply(rule Max_eqI[THEN sym])
31995
8f37cf60b885 more gcd/lcm lemmas
nipkow
parents: 31992
diff changeset
  1111
  apply (metis finite_Collect_conjI finite_divisors_int)
31734
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
  1112
 apply simp
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1113
 apply (metis gcd_greatest_iff_int gcd_pos_int zdvd_imp_le)
31734
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
  1114
apply simp
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
  1115
done
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
  1116
34030
829eb528b226 resorted code equations from "old" number theory version
haftmann
parents: 33946
diff changeset
  1117
lemma gcd_code_int [code]:
829eb528b226 resorted code equations from "old" number theory version
haftmann
parents: 33946
diff changeset
  1118
  "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
  1119
  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
  1120
22027
e4a08629c4bd A few lemmas about relative primes when dividing trough gcd
chaieb
parents: 21404
diff changeset
  1121
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1122
subsection {* Coprimality *}
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1123
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1124
lemma div_gcd_coprime_nat:
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1125
  assumes nz: "(a::nat) \<noteq> 0 \<or> b \<noteq> 0"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1126
  shows "coprime (a div gcd a b) (b div gcd a b)"
22367
6860f09242bf tuned document;
wenzelm
parents: 22027
diff changeset
  1127
proof -
27556
292098f2efdf unified curried gcd, lcm, zgcd, zlcm
haftmann
parents: 27487
diff changeset
  1128
  let ?g = "gcd a b"
22027
e4a08629c4bd A few lemmas about relative primes when dividing trough gcd
chaieb
parents: 21404
diff changeset
  1129
  let ?a' = "a div ?g"
e4a08629c4bd A few lemmas about relative primes when dividing trough gcd
chaieb
parents: 21404
diff changeset
  1130
  let ?b' = "b div ?g"
27556
292098f2efdf unified curried gcd, lcm, zgcd, zlcm
haftmann
parents: 27487
diff changeset
  1131
  let ?g' = "gcd ?a' ?b'"
22027
e4a08629c4bd A few lemmas about relative primes when dividing trough gcd
chaieb
parents: 21404
diff changeset
  1132
  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
  1133
  have dvdg': "?g' dvd ?a'" "?g' dvd ?b'" by simp_all
22367
6860f09242bf tuned document;
wenzelm
parents: 22027
diff changeset
  1134
  from dvdg dvdg' obtain ka kb ka' kb' where
6860f09242bf tuned document;
wenzelm
parents: 22027
diff changeset
  1135
      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
  1136
    unfolding dvd_def by blast
58834
773b378d9313 more simp rules concerning dvd and even/odd
haftmann
parents: 58787
diff changeset
  1137
  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
  1138
    by (simp_all add: mult.assoc mult.left_commute [of "gcd a b"])
22367
6860f09242bf tuned document;
wenzelm
parents: 22027
diff changeset
  1139
  then have dvdgg':"?g * ?g' dvd a" "?g* ?g' dvd b"
6860f09242bf tuned document;
wenzelm
parents: 22027
diff changeset
  1140
    by (auto simp add: dvd_mult_div_cancel [OF dvdg(1)]
6860f09242bf tuned document;
wenzelm
parents: 22027
diff changeset
  1141
      dvd_mult_div_cancel [OF dvdg(2)] dvd_def)
35216
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 35028
diff changeset
  1142
  have "?g \<noteq> 0" using nz by simp
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1143
  then have gp: "?g > 0" by arith
59008
f61482b0f240 formally self-contained gcd type classes
haftmann
parents: 58889
diff changeset
  1144
  from gcd_greatest [OF dvdgg'] have "?g * ?g' dvd ?g" .
22367
6860f09242bf tuned document;
wenzelm
parents: 22027
diff changeset
  1145
  with dvd_mult_cancel1 [OF gp] show "?g' = 1" by simp
22027
e4a08629c4bd A few lemmas about relative primes when dividing trough gcd
chaieb
parents: 21404
diff changeset
  1146
qed
e4a08629c4bd A few lemmas about relative primes when dividing trough gcd
chaieb
parents: 21404
diff changeset
  1147
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1148
lemma div_gcd_coprime_int:
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1149
  assumes nz: "(a::int) \<noteq> 0 \<or> b \<noteq> 0"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1150
  shows "coprime (a div gcd a b) (b div gcd a b)"
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1151
apply (subst (1 2 3) gcd_abs_int)
31813
4df828bbc411 gcd abs lemmas
nipkow
parents: 31798
diff changeset
  1152
apply (subst (1 2) abs_div)
4df828bbc411 gcd abs lemmas
nipkow
parents: 31798
diff changeset
  1153
  apply simp
4df828bbc411 gcd abs lemmas
nipkow
parents: 31798
diff changeset
  1154
 apply simp
4df828bbc411 gcd abs lemmas
nipkow
parents: 31798
diff changeset
  1155
apply(subst (1 2) abs_gcd_int)
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1156
apply (rule div_gcd_coprime_nat [transferred])
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1157
using nz apply (auto simp add: gcd_abs_int [symmetric])
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1158
done
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1159
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1160
lemma coprime_nat: "coprime (a::nat) b \<longleftrightarrow> (\<forall>d. d dvd a \<and> d dvd b \<longleftrightarrow> d = 1)"
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1161
  using gcd_unique_nat[of 1 a b, simplified] by auto
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1162
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1163
lemma coprime_Suc_0_nat:
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1164
    "coprime (a::nat) b \<longleftrightarrow> (\<forall>d. d dvd a \<and> d dvd b \<longleftrightarrow> d = Suc 0)"
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1165
  using coprime_nat by (simp add: One_nat_def)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1166
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1167
lemma coprime_int: "coprime (a::int) b \<longleftrightarrow>
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1168
    (\<forall>d. d >= 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
  1169
  using gcd_unique_int [of 1 a b]
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1170
  apply clarsimp
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1171
  apply (erule subst)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1172
  apply (rule iffI)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1173
  apply force
59807
22bc39064290 prefer local fixes;
wenzelm
parents: 59667
diff changeset
  1174
  apply (drule_tac x = "abs e" for e in exI)
22bc39064290 prefer local fixes;
wenzelm
parents: 59667
diff changeset
  1175
  apply (case_tac "e >= 0" for e :: int)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1176
  apply force
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1177
  apply force
59807
22bc39064290 prefer local fixes;
wenzelm
parents: 59667
diff changeset
  1178
  done
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1179
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1180
lemma gcd_coprime_nat:
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1181
  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
  1182
    b: "b = b' * gcd a b"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1183
  shows    "coprime a' b'"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1184
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1185
  apply (subgoal_tac "a' = a div gcd a b")
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1186
  apply (erule ssubst)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1187
  apply (subgoal_tac "b' = b div gcd a b")
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1188
  apply (erule ssubst)
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1189
  apply (rule div_gcd_coprime_nat)
41550
efa734d9b221 eliminated global prems;
wenzelm
parents: 37770
diff changeset
  1190
  using z apply force
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1191
  apply (subst (1) b)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1192
  using z apply force
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1193
  apply (subst (1) a)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1194
  using z apply force
41550
efa734d9b221 eliminated global prems;
wenzelm
parents: 37770
diff changeset
  1195
  done
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1196
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1197
lemma gcd_coprime_int:
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1198
  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
  1199
    b: "b = b' * gcd a b"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1200
  shows    "coprime a' b'"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1201
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1202
  apply (subgoal_tac "a' = a div gcd a b")
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1203
  apply (erule ssubst)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1204
  apply (subgoal_tac "b' = b div gcd a b")
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1205
  apply (erule ssubst)
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1206
  apply (rule div_gcd_coprime_int)
41550
efa734d9b221 eliminated global prems;
wenzelm
parents: 37770
diff changeset
  1207
  using z apply force
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1208
  apply (subst (1) b)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1209
  using z apply force
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1210
  apply (subst (1) a)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1211
  using z apply force
41550
efa734d9b221 eliminated global prems;
wenzelm
parents: 37770
diff changeset
  1212
  done
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1213
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1214
lemma coprime_mult_nat: assumes da: "coprime (d::nat) a" and db: "coprime d b"
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1215
    shows "coprime d (a * b)"
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1216
  apply (subst gcd_commute_nat)
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1217
  using da apply (subst gcd_mult_cancel_nat)
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1218
  apply (subst gcd_commute_nat, assumption)
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1219
  apply (subst gcd_commute_nat, rule db)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1220
done
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1221
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1222
lemma coprime_mult_int: assumes da: "coprime (d::int) a" and db: "coprime d b"
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1223
    shows "coprime d (a * b)"
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1224
  apply (subst gcd_commute_int)
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1225
  using da apply (subst gcd_mult_cancel_int)
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1226
  apply (subst gcd_commute_int, assumption)
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1227
  apply (subst gcd_commute_int, rule db)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1228
done
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1229
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1230
lemma coprime_lmult_nat:
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1231
  assumes dab: "coprime (d::nat) (a * b)" shows "coprime d a"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1232
proof -
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1233
  have "gcd d a dvd gcd d (a * b)"
59008
f61482b0f240 formally self-contained gcd type classes
haftmann
parents: 58889
diff changeset
  1234
    by (rule gcd_greatest, auto)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1235
  with dab show ?thesis
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1236
    by auto
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1237
qed
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1238
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1239
lemma coprime_lmult_int:
31798
fe9a3043d36c Cleaned up GCD
nipkow
parents: 31766
diff changeset
  1240
  assumes "coprime (d::int) (a * b)" shows "coprime d a"
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1241
proof -
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1242
  have "gcd d a dvd gcd d (a * b)"
59008
f61482b0f240 formally self-contained gcd type classes
haftmann
parents: 58889
diff changeset
  1243
    by (rule gcd_greatest, auto)
31798
fe9a3043d36c Cleaned up GCD
nipkow
parents: 31766
diff changeset
  1244
  with assms show ?thesis
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1245
    by auto
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1246
qed
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1247
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1248
lemma coprime_rmult_nat:
31798
fe9a3043d36c Cleaned up GCD
nipkow
parents: 31766
diff changeset
  1249
  assumes "coprime (d::nat) (a * b)" shows "coprime d b"
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1250
proof -
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1251
  have "gcd d b dvd gcd d (a * b)"
59008
f61482b0f240 formally self-contained gcd type classes
haftmann
parents: 58889
diff changeset
  1252
    by (rule gcd_greatest, auto intro: dvd_mult)
31798
fe9a3043d36c Cleaned up GCD
nipkow
parents: 31766
diff changeset
  1253
  with assms show ?thesis
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1254
    by auto
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1255
qed
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_rmult_int:
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1258
  assumes dab: "coprime (d::int) (a * b)" shows "coprime d b"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1259
proof -
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1260
  have "gcd d b dvd gcd d (a * b)"
59008
f61482b0f240 formally self-contained gcd type classes
haftmann
parents: 58889
diff changeset
  1261
    by (rule gcd_greatest, auto intro: dvd_mult)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1262
  with dab show ?thesis
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1263
    by auto
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1264
qed
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1265
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1266
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
  1267
    coprime d a \<and>  coprime d b"
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1268
  using coprime_rmult_nat[of d a b] coprime_lmult_nat[of d a b]
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1269
    coprime_mult_nat[of d a b]
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1270
  by blast
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1271
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1272
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
  1273
    coprime d a \<and>  coprime d b"
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1274
  using coprime_rmult_int[of d a b] coprime_lmult_int[of d a b]
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1275
    coprime_mult_int[of d a b]
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1276
  by blast
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1277
52397
e95f6b4b1bcf added coprimality lemma
noschinl
parents: 51547
diff changeset
  1278
lemma coprime_power_int:
e95f6b4b1bcf added coprimality lemma
noschinl
parents: 51547
diff changeset
  1279
  assumes "0 < n" shows "coprime (a :: int) (b ^ n) \<longleftrightarrow> coprime a b"
e95f6b4b1bcf added coprimality lemma
noschinl
parents: 51547
diff changeset
  1280
  using assms
e95f6b4b1bcf added coprimality lemma
noschinl
parents: 51547
diff changeset
  1281
proof (induct n)
e95f6b4b1bcf added coprimality lemma
noschinl
parents: 51547
diff changeset
  1282
  case (Suc n) then show ?case
e95f6b4b1bcf added coprimality lemma
noschinl
parents: 51547
diff changeset
  1283
    by (cases n) (simp_all add: coprime_mul_eq_int)
e95f6b4b1bcf added coprimality lemma
noschinl
parents: 51547
diff changeset
  1284
qed simp
e95f6b4b1bcf added coprimality lemma
noschinl
parents: 51547
diff changeset
  1285
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1286
lemma gcd_coprime_exists_nat:
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1287
    assumes nz: "gcd (a::nat) b \<noteq> 0"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1288
    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
  1289
  apply (rule_tac x = "a div gcd a b" in exI)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1290
  apply (rule_tac x = "b div gcd a b" in exI)
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1291
  using nz apply (auto simp add: div_gcd_coprime_nat dvd_div_mult)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1292
done
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1293
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1294
lemma gcd_coprime_exists_int:
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1295
    assumes nz: "gcd (a::int) b \<noteq> 0"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1296
    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
  1297
  apply (rule_tac x = "a div gcd a b" in exI)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1298
  apply (rule_tac x = "b div gcd a b" in exI)
59008
f61482b0f240 formally self-contained gcd type classes
haftmann
parents: 58889
diff changeset
  1299
  using nz apply (auto simp add: div_gcd_coprime_int)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1300
done
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1301
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1302
lemma coprime_exp_nat: "coprime (d::nat) a \<Longrightarrow> coprime d (a^n)"
60596
54168997757f algebraic specification for set gcd
haftmann
parents: 60580
diff changeset
  1303
  by (induct n) (simp_all add: coprime_mult_nat)
31706
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 coprime_exp_int: "coprime (d::int) a \<Longrightarrow> coprime d (a^n)"
60596
54168997757f algebraic specification for set gcd
haftmann
parents: 60580
diff changeset
  1306
  by (induct n) (simp_all add: coprime_mult_int)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1307
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1308
lemma coprime_exp2_nat [intro]: "coprime (a::nat) b \<Longrightarrow> coprime (a^n) (b^m)"
60686
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1309
  by (simp add: coprime_exp_nat ac_simps)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1310
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1311
lemma coprime_exp2_int [intro]: "coprime (a::int) b \<Longrightarrow> coprime (a^n) (b^m)"
60686
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1312
  by (simp add: coprime_exp_int ac_simps)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1313
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1314
lemma gcd_exp_nat: "gcd ((a::nat)^n) (b^n) = (gcd a b)^n"
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1315
proof (cases)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1316
  assume "a = 0 & b = 0"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1317
  thus ?thesis by simp
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1318
  next assume "~(a = 0 & b = 0)"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1319
  hence "coprime ((a div gcd a b)^n) ((b div gcd a b)^n)"
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1320
    by (auto simp:div_gcd_coprime_nat)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1321
  hence "gcd ((a div gcd a b)^n * (gcd a b)^n)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1322
      ((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
  1323
    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
  1324
  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
  1325
    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
  1326
  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
  1327
    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
  1328
  finally show ?thesis .
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1329
qed
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1330
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1331
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
  1332
  apply (subst (1 2) gcd_abs_int)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1333
  apply (subst (1 2) power_abs)
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1334
  apply (rule gcd_exp_nat [where n = n, transferred])
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1335
  apply auto
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1336
done
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1337
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1338
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
  1339
  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
  1340
proof-
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1341
  let ?g = "gcd a b"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1342
  {assume "?g = 0" with dc have ?thesis by auto}
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1343
  moreover
1db0c8f235fb new GCD library