src/HOL/GCD.thy
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(*  Title:      HOL/GCD.thy
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    Author:     Christophe Tabacznyj
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    Author:     Lawrence C. Paulson
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    Author:     Amine Chaieb
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    Author:     Thomas M. Rasmussen
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    Author:     Jeremy Avigad
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    Author:     Tobias Nipkow
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This file deals with the functions gcd and lcm.  Definitions and
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lemmas are proved uniformly for the natural numbers and integers.
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This file combines and revises a number of prior developments.
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The original theories "GCD" and "Primes" were by Christophe Tabacznyj
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and Lawrence C. Paulson, based on @{cite davenport92}. They introduced
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gcd, lcm, and prime for the natural numbers.
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The original theory "IntPrimes" was by Thomas M. Rasmussen, and
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extended gcd, lcm, primes to the integers. Amine Chaieb provided
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another extension of the notions to the integers, and added a number
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of results to "Primes" and "GCD". IntPrimes also defined and developed
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the congruence relations on the integers. The notion was extended to
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the natural numbers by Chaieb.
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Jeremy Avigad combined all of these, made everything uniform for the
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natural numbers and the integers, and added a number of new theorems.
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Tobias Nipkow cleaned up a lot.
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*)
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section \<open>Greatest common divisor and least common multiple\<close>
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theory GCD
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  imports Main
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begin
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subsection \<open>Abstract GCD and LCM\<close>
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class gcd = zero + one + dvd +
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  fixes gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
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    and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
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begin
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abbreviation coprime :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
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  where "coprime x y \<equiv> gcd x y = 1"
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end
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class Gcd = gcd +
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  fixes Gcd :: "'a set \<Rightarrow> 'a"
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    and Lcm :: "'a set \<Rightarrow> 'a"
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begin
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abbreviation GREATEST_COMMON_DIVISOR :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a"
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  where "GREATEST_COMMON_DIVISOR A f \<equiv> Gcd (f ` A)"
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abbreviation LEAST_COMMON_MULTIPLE :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a"
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  where "LEAST_COMMON_MULTIPLE A f \<equiv> Lcm (f ` A)"
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end
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syntax
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  "_GCD1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3GCD _./ _)" [0, 10] 10)
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  "_GCD"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3GCD _\<in>_./ _)" [0, 0, 10] 10)
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  "_LCM1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3LCM _./ _)" [0, 10] 10)
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  "_LCM"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3LCM _\<in>_./ _)" [0, 0, 10] 10)
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translations
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  "GCD x y. B"   \<rightleftharpoons> "GCD x. GCD y. B"
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  "GCD x. B"     \<rightleftharpoons> "CONST GREATEST_COMMON_DIVISOR CONST UNIV (\<lambda>x. B)"
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  "GCD x. B"     \<rightleftharpoons> "GCD x \<in> CONST UNIV. B"
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  "GCD x\<in>A. B"   \<rightleftharpoons> "CONST GREATEST_COMMON_DIVISOR A (\<lambda>x. B)"
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  "LCM x y. B"   \<rightleftharpoons> "LCM x. LCM y. B"
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  "LCM x. B"     \<rightleftharpoons> "CONST LEAST_COMMON_MULTIPLE CONST UNIV (\<lambda>x. B)"
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  "LCM x. B"     \<rightleftharpoons> "LCM x \<in> CONST UNIV. B"
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  "LCM x\<in>A. B"   \<rightleftharpoons> "CONST LEAST_COMMON_MULTIPLE A (\<lambda>x. B)"
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print_translation \<open>
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  [Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax GREATEST_COMMON_DIVISOR} @{syntax_const "_GCD"},
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    Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax LEAST_COMMON_MULTIPLE} @{syntax_const "_LCM"}]
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\<close> \<comment> \<open>to avoid eta-contraction of body\<close>
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class semiring_gcd = normalization_semidom + gcd +
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  assumes gcd_dvd1 [iff]: "gcd a b dvd a"
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    and gcd_dvd2 [iff]: "gcd a b dvd b"
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    and gcd_greatest: "c dvd a \<Longrightarrow> c dvd b \<Longrightarrow> c dvd gcd a b"
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    and normalize_gcd [simp]: "normalize (gcd a b) = gcd a b"
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    and lcm_gcd: "lcm a b = normalize (a * b) div gcd a b"
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begin
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lemma gcd_greatest_iff [simp]: "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: "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: "b dvd c \<Longrightarrow> gcd a b dvd c"
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  by (rule dvd_trans) (rule gcd_dvd2)
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lemma dvd_gcdD1: "a dvd gcd b c \<Longrightarrow> a dvd b"
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  using gcd_dvd1 [of b c] by (blast intro: dvd_trans)
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lemma dvd_gcdD2: "a dvd gcd b c \<Longrightarrow> a dvd c"
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  using gcd_dvd2 [of b c] by (blast intro: dvd_trans)
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lemma gcd_0_left [simp]: "gcd 0 a = normalize a"
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  by (rule associated_eqI) simp_all
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lemma gcd_0_right [simp]: "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]: "gcd a b = 0 \<longleftrightarrow> a = 0 \<and> b = 0"
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  (is "?P \<longleftrightarrow> ?Q")
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proof
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  assume ?P
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  then have "0 dvd gcd a b"
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    by simp
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  then have "0 dvd a" and "0 dvd b"
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    by (blast intro: dvd_trans)+
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  then show ?Q
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    by simp
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next
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  assume ?Q
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  then show ?P
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    by simp
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qed
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lemma unit_factor_gcd: "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
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  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
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    by simp
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qed
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lemma is_unit_gcd [simp]: "is_unit (gcd a b) \<longleftrightarrow> coprime a b"
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  by (cases "a = 0 \<and> b = 0") (auto simp add: unit_factor_gcd dest: is_unit_unit_factor)
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sublocale gcd: abel_semigroup gcd
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proof
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   149
  fix a b c
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  show "gcd a b = gcd b a"
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    by (rule associated_eqI) simp_all
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  from gcd_dvd1 have "gcd (gcd a b) c dvd a"
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   153
    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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   155
    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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   157
    by (auto intro!: gcd_greatest)
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haftmann
parents: 60597
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   158
  from gcd_dvd2 have "gcd a (gcd b c) dvd b"
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haftmann
parents: 60597
diff changeset
   159
    by (rule dvd_trans) simp
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   160
  moreover from gcd_dvd2 have "gcd a (gcd b c) dvd c"
ea5bc46c11e6 more algebraic properties for gcd/lcm
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parents: 60597
diff changeset
   161
    by (rule dvd_trans) simp
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haftmann
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   162
  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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   164
  from P1 P2 show "gcd (gcd a b) c = gcd a (gcd b c)"
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   165
    by (rule associated_eqI) simp_all
60686
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   166
qed
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   167
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   168
lemma gcd_self [simp]: "gcd a a = normalize a"
60686
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   169
proof -
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   170
  have "a dvd gcd a a"
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parents: 60597
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   171
    by (rule gcd_greatest) simp_all
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   172
  then show ?thesis
60688
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   173
    by (auto intro: associated_eqI)
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   174
qed
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   175
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lemma gcd_left_idem [simp]: "gcd a (gcd a b) = gcd a b"
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   177
  by (auto intro: associated_eqI)
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diff changeset
   178
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   179
lemma gcd_right_idem [simp]: "gcd (gcd a b) b = gcd a b"
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parents: 61856
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   180
  unfolding gcd.commute [of a] gcd.commute [of "gcd b a"] by simp
58b153bfa737 tuned proofs and augmented some lemmas
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diff changeset
   181
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   182
lemma coprime_1_left [simp]: "coprime 1 a"
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   183
  by (rule associated_eqI) simp_all
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diff changeset
   184
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   185
lemma coprime_1_right [simp]: "coprime a 1"
60686
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   186
  using coprime_1_left [of a] by (simp add: ac_simps)
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   187
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   188
lemma gcd_mult_left: "gcd (c * a) (c * b) = normalize c * gcd a b"
60686
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   189
proof (cases "c = 0")
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   190
  case True
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   191
  then show ?thesis by simp
60686
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   192
next
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   193
  case False
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   194
  then have *: "c * gcd a b dvd gcd (c * a) (c * b)"
60686
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   195
    by (auto intro: gcd_greatest)
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   196
  moreover from False * have "gcd (c * a) (c * b) dvd c * gcd a b"
60686
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diff changeset
   197
    by (metis div_dvd_iff_mult dvd_mult_left gcd_dvd1 gcd_dvd2 gcd_greatest mult_commute)
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diff changeset
   198
  ultimately have "normalize (gcd (c * a) (c * b)) = normalize (c * gcd a b)"
60688
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   199
    by (auto intro: associated_eqI)
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   200
  then show ?thesis
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   201
    by (simp add: normalize_mult)
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   202
qed
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diff changeset
   203
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   204
lemma gcd_mult_right: "gcd (a * c) (b * c) = gcd b a * normalize c"
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   205
  using gcd_mult_left [of c a b] by (simp add: ac_simps)
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diff changeset
   206
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   207
lemma mult_gcd_left: "c * gcd a b = unit_factor c * gcd (c * a) (c * b)"
60686
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diff changeset
   208
  by (simp add: gcd_mult_left mult.assoc [symmetric])
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parents: 60597
diff changeset
   209
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   210
lemma mult_gcd_right: "gcd a b * c = gcd (a * c) (b * c) * unit_factor c"
60686
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diff changeset
   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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   213
lemma dvd_lcm1 [iff]: "a dvd lcm a b"
60686
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   214
proof -
ea5bc46c11e6 more algebraic properties for gcd/lcm
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   215
  have "normalize (a * b) div gcd a b = normalize a * (normalize b div gcd a b)"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   216
    by (simp add: lcm_gcd normalize_mult div_mult_swap)
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   217
  then show ?thesis
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
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   218
    by (simp add: lcm_gcd)
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   219
qed
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   220
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   221
lemma dvd_lcm2 [iff]: "b dvd lcm a b"
60686
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diff changeset
   222
proof -
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   223
  have "normalize (a * b) div gcd a b = normalize b * (normalize a div gcd a b)"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   224
    by (simp add: lcm_gcd normalize_mult div_mult_swap ac_simps)
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   225
  then show ?thesis
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   226
    by (simp add: lcm_gcd)
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   227
qed
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haftmann
parents: 60597
diff changeset
   228
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   229
lemma dvd_lcmI1: "a dvd b \<Longrightarrow> a dvd lcm b c"
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   230
  by (rule dvd_trans) (assumption, blast)
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diff changeset
   231
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   232
lemma dvd_lcmI2: "a dvd c \<Longrightarrow> a dvd lcm b c"
60689
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diff changeset
   233
  by (rule dvd_trans) (assumption, blast)
8a2d7c04d8c0 more cautious use of [iff] declarations
haftmann
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diff changeset
   234
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   235
lemma lcm_dvdD1: "lcm a b dvd c \<Longrightarrow> a dvd c"
62345
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   236
  using dvd_lcm1 [of a b] by (blast intro: dvd_trans)
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   237
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   238
lemma lcm_dvdD2: "lcm a b dvd c \<Longrightarrow> b dvd c"
62345
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diff changeset
   239
  using dvd_lcm2 [of a b] by (blast intro: dvd_trans)
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   240
60686
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   241
lemma lcm_least:
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   242
  assumes "a dvd c" and "b dvd c"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
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   243
  shows "lcm a b dvd c"
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   244
proof (cases "c = 0")
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   245
  case True
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   246
  then show ?thesis by simp
60686
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diff changeset
   247
next
63489
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   248
  case False
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   249
  then have *: "is_unit (unit_factor c)"
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   250
    by simp
60686
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   251
  show ?thesis
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
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   252
  proof (cases "gcd a b = 0")
63489
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   253
    case True
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   254
    with assms show ?thesis by simp
60686
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   255
  next
63489
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diff changeset
   256
    case False
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diff changeset
   257
    then have "a \<noteq> 0 \<or> b \<noteq> 0"
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   258
      by simp
60686
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haftmann
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diff changeset
   259
    with \<open>c \<noteq> 0\<close> assms have "a * b dvd a * c" "a * b dvd c * b"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   260
      by (simp_all add: mult_dvd_mono)
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   261
    then have "normalize (a * b) dvd gcd (a * c) (b * c)"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   262
      by (auto intro: gcd_greatest simp add: ac_simps)
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   263
    then have "normalize (a * b) dvd gcd (a * c) (b * c) * unit_factor c"
63489
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parents: 63359
diff changeset
   264
      using * by (simp add: dvd_mult_unit_iff)
60686
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haftmann
parents: 60597
diff changeset
   265
    then have "normalize (a * b) dvd gcd a b * c"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   266
      by (simp add: mult_gcd_right [of a b c])
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   267
    then have "normalize (a * b) div gcd a b dvd c"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   268
      using False by (simp add: div_dvd_iff_mult ac_simps)
63489
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parents: 63359
diff changeset
   269
    then show ?thesis
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   270
      by (simp add: lcm_gcd)
60686
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   271
  qed
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
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   272
qed
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   273
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   274
lemma lcm_least_iff [simp]: "lcm a b dvd c \<longleftrightarrow> a dvd c \<and> b dvd c"
60686
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haftmann
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diff changeset
   275
  by (blast intro!: lcm_least intro: dvd_trans)
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haftmann
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diff changeset
   276
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   277
lemma normalize_lcm [simp]: "normalize (lcm a b) = lcm a b"
60686
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haftmann
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diff changeset
   278
  by (simp add: lcm_gcd dvd_normalize_div)
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haftmann
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diff changeset
   279
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   280
lemma lcm_0_left [simp]: "lcm 0 a = 0"
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   281
  by (simp add: lcm_gcd)
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diff changeset
   282
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diff changeset
   283
lemma lcm_0_right [simp]: "lcm a 0 = 0"
60686
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   284
  by (simp add: lcm_gcd)
63489
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diff changeset
   285
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diff changeset
   286
lemma lcm_eq_0_iff: "lcm a b = 0 \<longleftrightarrow> a = 0 \<or> b = 0"
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diff changeset
   287
  (is "?P \<longleftrightarrow> ?Q")
60686
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diff changeset
   288
proof
63489
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diff changeset
   289
  assume ?P
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diff changeset
   290
  then have "0 dvd lcm a b"
cd540c8031a4 misc tuning and modernization;
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diff changeset
   291
    by simp
60686
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haftmann
parents: 60597
diff changeset
   292
  then have "0 dvd normalize (a * b) div gcd a b"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   293
    by (simp add: lcm_gcd)
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   294
  then have "0 * gcd a b dvd normalize (a * b)"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   295
    using dvd_div_iff_mult [of "gcd a b" _ 0] by (cases "gcd a b = 0") simp_all
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   296
  then have "normalize (a * b) = 0"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   297
    by simp
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   298
  then show ?Q
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   299
    by simp
60686
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   300
next
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   301
  assume ?Q
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   302
  then show ?P
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   303
    by auto
60686
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   304
qed
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   305
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   306
lemma lcm_eq_1_iff [simp]: "lcm a b = 1 \<longleftrightarrow> is_unit a \<and> is_unit b"
61913
58b153bfa737 tuned proofs and augmented some lemmas
haftmann
parents: 61856
diff changeset
   307
  by (auto intro: associated_eqI)
58b153bfa737 tuned proofs and augmented some lemmas
haftmann
parents: 61856
diff changeset
   308
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   309
lemma unit_factor_lcm: "unit_factor (lcm a b) = (if a = 0 \<or> b = 0 then 0 else 1)"
60686
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   310
  by (simp add: unit_factor_gcd dvd_unit_factor_div lcm_gcd)
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   311
61605
1bf7b186542e qualifier is mandatory by default;
wenzelm
parents: 61566
diff changeset
   312
sublocale lcm: abel_semigroup lcm
60686
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   313
proof
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   314
  fix a b c
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   315
  show "lcm a b = lcm b a"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   316
    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
   317
  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
   318
    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
   319
    by (auto intro: lcm_least
60686
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   320
      dvd_trans [of b "lcm b c" "lcm a (lcm b c)"]
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   321
      dvd_trans [of c "lcm b c" "lcm a (lcm b c)"]
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   322
      dvd_trans [of a "lcm a b" "lcm (lcm a b) c"]
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   323
      dvd_trans [of b "lcm a b" "lcm (lcm a b) c"])
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   324
  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
   325
    by (rule associated_eqI) simp_all
60686
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   326
qed
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   327
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   328
lemma lcm_self [simp]: "lcm a a = normalize a"
60686
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   329
proof -
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   330
  have "lcm a a dvd a"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   331
    by (rule lcm_least) simp_all
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   332
  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
   333
    by (auto intro: associated_eqI)
60686
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   334
qed
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   335
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   336
lemma lcm_left_idem [simp]: "lcm a (lcm a b) = lcm a b"
61913
58b153bfa737 tuned proofs and augmented some lemmas
haftmann
parents: 61856
diff changeset
   337
  by (auto intro: associated_eqI)
58b153bfa737 tuned proofs and augmented some lemmas
haftmann
parents: 61856
diff changeset
   338
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   339
lemma lcm_right_idem [simp]: "lcm (lcm a b) b = lcm a b"
61913
58b153bfa737 tuned proofs and augmented some lemmas
haftmann
parents: 61856
diff changeset
   340
  unfolding lcm.commute [of a] lcm.commute [of "lcm b a"] by simp
58b153bfa737 tuned proofs and augmented some lemmas
haftmann
parents: 61856
diff changeset
   341
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   342
lemma gcd_mult_lcm [simp]: "gcd a b * lcm a b = normalize a * normalize b"
60686
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   343
  by (simp add: lcm_gcd normalize_mult)
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   344
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   345
lemma lcm_mult_gcd [simp]: "lcm a b * gcd a b = normalize a * normalize b"
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   346
  using gcd_mult_lcm [of a b] by (simp add: ac_simps)
60686
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   347
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   348
lemma gcd_lcm:
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   349
  assumes "a \<noteq> 0" and "b \<noteq> 0"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   350
  shows "gcd a b = normalize (a * b) div lcm a b"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   351
proof -
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   352
  from assms have "lcm a b \<noteq> 0"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   353
    by (simp add: lcm_eq_0_iff)
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   354
  have "gcd a b * lcm a b = normalize a * normalize b"
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   355
    by simp
60686
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   356
  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
   357
    by (simp_all add: normalize_mult)
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   358
  with \<open>lcm a b \<noteq> 0\<close> show ?thesis
64240
eabf80376aab more standardized names
haftmann
parents: 63924
diff changeset
   359
    using nonzero_mult_div_cancel_right [of "lcm a b" "gcd a b"] by simp
60686
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   360
qed
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   361
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   362
lemma lcm_1_left [simp]: "lcm 1 a = normalize a"
60686
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   363
  by (simp add: lcm_gcd)
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   364
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   365
lemma lcm_1_right [simp]: "lcm a 1 = normalize a"
60686
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   366
  by (simp add: lcm_gcd)
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   367
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   368
lemma lcm_mult_left: "lcm (c * a) (c * b) = normalize c * lcm a b"
60686
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   369
  by (cases "c = 0")
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   370
    (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
   371
      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
   372
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   373
lemma lcm_mult_right: "lcm (a * c) (b * c) = lcm b a * normalize c"
60686
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   374
  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
   375
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   376
lemma mult_lcm_left: "c * lcm a b = unit_factor c * lcm (c * a) (c * b)"
60686
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   377
  by (simp add: lcm_mult_left mult.assoc [symmetric])
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   378
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   379
lemma mult_lcm_right: "lcm a b * c = lcm (a * c) (b * c) * unit_factor c"
60686
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   380
  using mult_lcm_left [of c a b] by (simp add: ac_simps)
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   381
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   382
lemma gcdI:
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   383
  assumes "c dvd a" and "c dvd b"
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   384
    and greatest: "\<And>d. d dvd a \<Longrightarrow> d dvd b \<Longrightarrow> d dvd c"
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   385
    and "normalize c = c"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   386
  shows "c = gcd a b"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   387
  by (rule associated_eqI) (auto simp: assms intro: gcd_greatest)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   388
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   389
lemma gcd_unique:
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   390
  "d dvd a \<and> d dvd b \<and> normalize d = d \<and> (\<forall>e. e dvd a \<and> e dvd b \<longrightarrow> e dvd d) \<longleftrightarrow> d = gcd a b"
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   391
  by rule (auto intro: gcdI simp: gcd_greatest)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   392
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   393
lemma gcd_dvd_prod: "gcd a b dvd k * b"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   394
  using mult_dvd_mono [of 1] by auto
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   395
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   396
lemma gcd_proj2_if_dvd: "b dvd a \<Longrightarrow> gcd a b = normalize b"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   397
  by (rule gcdI [symmetric]) simp_all
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   398
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   399
lemma gcd_proj1_if_dvd: "a dvd b \<Longrightarrow> gcd a b = normalize a"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   400
  by (rule gcdI [symmetric]) simp_all
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   401
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   402
lemma gcd_proj1_iff: "gcd m n = normalize m \<longleftrightarrow> m dvd n"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   403
proof
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   404
  assume *: "gcd m n = normalize m"
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   405
  show "m dvd n"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   406
  proof (cases "m = 0")
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   407
    case True
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   408
    with * show ?thesis by simp
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   409
  next
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   410
    case [simp]: False
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   411
    from * have **: "m = gcd m n * unit_factor m"
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   412
      by (simp add: unit_eq_div2)
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   413
    show ?thesis
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   414
      by (subst **) (simp add: mult_unit_dvd_iff)
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   415
  qed
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   416
next
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   417
  assume "m dvd n"
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   418
  then show "gcd m n = normalize m"
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   419
    by (rule gcd_proj1_if_dvd)
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   420
qed
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   421
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   422
lemma gcd_proj2_iff: "gcd m n = normalize n \<longleftrightarrow> n dvd m"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   423
  using gcd_proj1_iff [of n m] by (simp add: ac_simps)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   424
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   425
lemma gcd_mult_distrib': "normalize c * gcd a b = gcd (c * a) (c * b)"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   426
  by (rule gcdI) (auto simp: normalize_mult gcd_greatest mult_dvd_mono gcd_mult_left[symmetric])
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   427
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   428
lemma gcd_mult_distrib: "k * gcd a b = gcd (k * a) (k * b) * unit_factor k"
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   429
proof-
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   430
  have "normalize k * gcd a b = gcd (k * a) (k * b)"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   431
    by (simp add: gcd_mult_distrib')
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   432
  then have "normalize k * gcd a b * unit_factor k = gcd (k * a) (k * b) * unit_factor k"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   433
    by simp
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   434
  then have "normalize k * unit_factor k * gcd a b  = gcd (k * a) (k * b) * unit_factor k"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   435
    by (simp only: ac_simps)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   436
  then show ?thesis
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   437
    by simp
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   438
qed
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   439
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   440
lemma lcm_mult_unit1: "is_unit a \<Longrightarrow> lcm (b * a) c = lcm b c"
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   441
  by (rule associated_eqI) (simp_all add: mult_unit_dvd_iff dvd_lcmI1)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   442
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   443
lemma lcm_mult_unit2: "is_unit a \<Longrightarrow> lcm b (c * a) = lcm b c"
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   444
  using lcm_mult_unit1 [of a c b] by (simp add: ac_simps)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   445
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   446
lemma lcm_div_unit1:
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   447
  "is_unit a \<Longrightarrow> lcm (b div a) c = lcm b c"
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   448
  by (erule is_unitE [of _ b]) (simp add: lcm_mult_unit1)
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   449
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   450
lemma lcm_div_unit2: "is_unit a \<Longrightarrow> lcm b (c div a) = lcm b c"
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   451
  by (erule is_unitE [of _ c]) (simp add: lcm_mult_unit2)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   452
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   453
lemma normalize_lcm_left [simp]: "lcm (normalize a) b = lcm a b"
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   454
proof (cases "a = 0")
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   455
  case True
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   456
  then show ?thesis
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   457
    by simp
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   458
next
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   459
  case False
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   460
  then have "is_unit (unit_factor a)"
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   461
    by simp
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   462
  moreover have "normalize a = a div unit_factor a"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   463
    by simp
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   464
  ultimately show ?thesis
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   465
    by (simp only: lcm_div_unit1)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   466
qed
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   467
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   468
lemma normalize_lcm_right [simp]: "lcm a (normalize b) = lcm a b"
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   469
  using normalize_lcm_left [of b a] by (simp add: ac_simps)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   470
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   471
lemma gcd_mult_unit1: "is_unit a \<Longrightarrow> gcd (b * a) c = gcd b c"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   472
  apply (rule gcdI)
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   473
     apply simp_all
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   474
  apply (rule dvd_trans)
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   475
   apply (rule gcd_dvd1)
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   476
  apply (simp add: unit_simps)
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   477
  done
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   478
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   479
lemma gcd_mult_unit2: "is_unit a \<Longrightarrow> gcd b (c * a) = gcd b c"
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   480
  apply (subst gcd.commute)
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   481
  apply (subst gcd_mult_unit1)
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   482
   apply assumption
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   483
  apply (rule gcd.commute)
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   484
  done
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   485
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   486
lemma gcd_div_unit1: "is_unit a \<Longrightarrow> gcd (b div a) c = gcd b c"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   487
  by (erule is_unitE [of _ b]) (simp add: gcd_mult_unit1)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   488
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   489
lemma gcd_div_unit2: "is_unit a \<Longrightarrow> gcd b (c div a) = gcd b c"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   490
  by (erule is_unitE [of _ c]) (simp add: gcd_mult_unit2)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   491
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   492
lemma normalize_gcd_left [simp]: "gcd (normalize a) b = gcd a b"
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   493
proof (cases "a = 0")
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   494
  case True
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   495
  then show ?thesis
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   496
    by simp
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   497
next
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   498
  case False
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   499
  then have "is_unit (unit_factor a)"
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   500
    by simp
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   501
  moreover have "normalize a = a div unit_factor a"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   502
    by simp
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   503
  ultimately show ?thesis
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   504
    by (simp only: gcd_div_unit1)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   505
qed
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   506
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   507
lemma normalize_gcd_right [simp]: "gcd a (normalize b) = gcd a b"
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   508
  using normalize_gcd_left [of b a] by (simp add: ac_simps)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   509
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   510
lemma comp_fun_idem_gcd: "comp_fun_idem gcd"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   511
  by standard (simp_all add: fun_eq_iff ac_simps)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   512
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   513
lemma comp_fun_idem_lcm: "comp_fun_idem lcm"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   514
  by standard (simp_all add: fun_eq_iff ac_simps)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   515
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   516
lemma gcd_dvd_antisym: "gcd a b dvd gcd c d \<Longrightarrow> gcd c d dvd gcd a b \<Longrightarrow> gcd a b = gcd c d"
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   517
proof (rule gcdI)
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   518
  assume *: "gcd a b dvd gcd c d"
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   519
    and **: "gcd c d dvd gcd a b"
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   520
  have "gcd c d dvd c"
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   521
    by simp
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   522
  with * show "gcd a b dvd c"
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   523
    by (rule dvd_trans)
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   524
  have "gcd c d dvd d"
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   525
    by simp
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   526
  with * show "gcd a b dvd d"
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   527
    by (rule dvd_trans)
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   528
  show "normalize (gcd a b) = gcd a b"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   529
    by simp
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   530
  fix l assume "l dvd c" and "l dvd d"
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   531
  then have "l dvd gcd c d"
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   532
    by (rule gcd_greatest)
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   533
  from this and ** show "l dvd gcd a b"
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   534
    by (rule dvd_trans)
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   535
qed
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   536
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   537
lemma coprime_dvd_mult:
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   538
  assumes "coprime a b" and "a dvd c * b"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   539
  shows "a dvd c"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   540
proof (cases "c = 0")
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   541
  case True
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   542
  then show ?thesis by simp
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   543
next
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   544
  case False
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   545
  then have unit: "is_unit (unit_factor c)"
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   546
    by simp
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   547
  from \<open>coprime a b\<close> mult_gcd_left [of c a b]
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   548
  have "gcd (c * a) (c * b) * unit_factor c = c"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   549
    by (simp add: ac_simps)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   550
  moreover from \<open>a dvd c * b\<close> have "a dvd gcd (c * a) (c * b) * unit_factor c"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   551
    by (simp add: dvd_mult_unit_iff unit)
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   552
  ultimately show ?thesis
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   553
    by simp
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   554
qed
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   555
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   556
lemma coprime_dvd_mult_iff: "coprime a c \<Longrightarrow> a dvd b * c \<longleftrightarrow> a dvd b"
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   557
  by (auto intro: coprime_dvd_mult)
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   558
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   559
lemma gcd_mult_cancel: "coprime c b \<Longrightarrow> gcd (c * a) b = gcd a b"
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   560
  apply (rule associated_eqI)
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   561
     apply (rule gcd_greatest)
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   562
      apply (rule_tac b = c in coprime_dvd_mult)
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   563
       apply (simp add: gcd.assoc)
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   564
       apply (simp_all add: ac_simps)
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   565
  done
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   566
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   567
lemma coprime_crossproduct:
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   568
  fixes a b c d :: 'a
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   569
  assumes "coprime a d" and "coprime b c"
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   570
  shows "normalize a * normalize c = normalize b * normalize d \<longleftrightarrow>
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   571
    normalize a = normalize b \<and> normalize c = normalize d"
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   572
    (is "?lhs \<longleftrightarrow> ?rhs")
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   573
proof
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   574
  assume ?rhs
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   575
  then show ?lhs by simp
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   576
next
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   577
  assume ?lhs
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   578
  from \<open>?lhs\<close> have "normalize a dvd normalize b * normalize d"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   579
    by (auto intro: dvdI dest: sym)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   580
  with \<open>coprime a d\<close> have "a dvd b"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   581
    by (simp add: coprime_dvd_mult_iff normalize_mult [symmetric])
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   582
  from \<open>?lhs\<close> have "normalize b dvd normalize a * normalize c"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   583
    by (auto intro: dvdI dest: sym)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   584
  with \<open>coprime b c\<close> have "b dvd a"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   585
    by (simp add: coprime_dvd_mult_iff normalize_mult [symmetric])
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   586
  from \<open>?lhs\<close> have "normalize c dvd normalize d * normalize b"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   587
    by (auto intro: dvdI dest: sym simp add: mult.commute)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   588
  with \<open>coprime b c\<close> have "c dvd d"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   589
    by (simp add: coprime_dvd_mult_iff gcd.commute normalize_mult [symmetric])
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   590
  from \<open>?lhs\<close> have "normalize d dvd normalize c * normalize a"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   591
    by (auto intro: dvdI dest: sym simp add: mult.commute)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   592
  with \<open>coprime a d\<close> have "d dvd c"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   593
    by (simp add: coprime_dvd_mult_iff gcd.commute normalize_mult [symmetric])
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   594
  from \<open>a dvd b\<close> \<open>b dvd a\<close> have "normalize a = normalize b"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   595
    by (rule associatedI)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   596
  moreover from \<open>c dvd d\<close> \<open>d dvd c\<close> have "normalize c = normalize d"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   597
    by (rule associatedI)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   598
  ultimately show ?rhs ..
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   599
qed
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   600
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   601
lemma gcd_add1 [simp]: "gcd (m + n) n = gcd m n"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   602
  by (rule gcdI [symmetric]) (simp_all add: dvd_add_left_iff)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   603
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   604
lemma gcd_add2 [simp]: "gcd m (m + n) = gcd m n"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   605
  using gcd_add1 [of n m] by (simp add: ac_simps)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   606
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   607
lemma gcd_add_mult: "gcd m (k * m + n) = gcd m n"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   608
  by (rule gcdI [symmetric]) (simp_all add: dvd_add_right_iff)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   609
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   610
lemma coprimeI: "(\<And>l. l dvd a \<Longrightarrow> l dvd b \<Longrightarrow> l dvd 1) \<Longrightarrow> gcd a b = 1"
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   611
  by (rule sym, rule gcdI) simp_all
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   612
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   613
lemma coprime: "gcd a b = 1 \<longleftrightarrow> (\<forall>d. d dvd a \<and> d dvd b \<longleftrightarrow> is_unit d)"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   614
  by (auto intro: coprimeI gcd_greatest dvd_gcdD1 dvd_gcdD2)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   615
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   616
lemma div_gcd_coprime:
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   617
  assumes nz: "a \<noteq> 0 \<or> b \<noteq> 0"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   618
  shows "coprime (a div gcd a b) (b div gcd a b)"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   619
proof -
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   620
  let ?g = "gcd a b"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   621
  let ?a' = "a div ?g"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   622
  let ?b' = "b div ?g"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   623
  let ?g' = "gcd ?a' ?b'"
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   624
  have dvdg: "?g dvd a" "?g dvd b"
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   625
    by simp_all
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   626
  have dvdg': "?g' dvd ?a'" "?g' dvd ?b'"
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   627
    by simp_all
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   628
  from dvdg dvdg' obtain ka kb ka' kb' where
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   629
    kab: "a = ?g * ka" "b = ?g * kb" "?a' = ?g' * ka'" "?b' = ?g' * kb'"
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   630
    unfolding dvd_def by blast
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   631
  from this [symmetric] have "?g * ?a' = (?g * ?g') * ka'" "?g * ?b' = (?g * ?g') * kb'"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   632
    by (simp_all add: mult.assoc mult.left_commute [of "gcd a b"])
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   633
  then have dvdgg':"?g * ?g' dvd a" "?g* ?g' dvd b"
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   634
    by (auto simp add: dvd_mult_div_cancel [OF dvdg(1)] dvd_mult_div_cancel [OF dvdg(2)] dvd_def)
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   635
  have "?g \<noteq> 0"
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   636
    using nz by simp
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   637
  moreover from gcd_greatest [OF dvdgg'] have "?g * ?g' dvd ?g" .
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   638
  ultimately show ?thesis
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   639
    using dvd_times_left_cancel_iff [of "gcd a b" _ 1] by simp
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   640
qed
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   641
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   642
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   643
lemma divides_mult:
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   644
  assumes "a dvd c" and nr: "b dvd c" and "coprime a b"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   645
  shows "a * b dvd c"
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   646
proof -
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   647
  from \<open>b dvd c\<close> obtain b' where"c = b * b'" ..
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   648
  with \<open>a dvd c\<close> have "a dvd b' * b"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   649
    by (simp add: ac_simps)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   650
  with \<open>coprime a b\<close> have "a dvd b'"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   651
    by (simp add: coprime_dvd_mult_iff)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   652
  then obtain a' where "b' = a * a'" ..
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   653
  with \<open>c = b * b'\<close> have "c = (a * b) * a'"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   654
    by (simp add: ac_simps)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   655
  then show ?thesis ..
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   656
qed
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   657
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   658
lemma coprime_lmult:
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   659
  assumes dab: "gcd d (a * b) = 1"
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   660
  shows "gcd d a = 1"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   661
proof (rule coprimeI)
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   662
  fix l
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   663
  assume "l dvd d" and "l dvd a"
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   664
  then have "l dvd a * b"
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   665
    by simp
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   666
  with \<open>l dvd d\<close> and dab show "l dvd 1"
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   667
    by (auto intro: gcd_greatest)
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   668
qed
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   669
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   670
lemma coprime_rmult:
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   671
  assumes dab: "gcd d (a * b) = 1"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   672
  shows "gcd d b = 1"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   673
proof (rule coprimeI)
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   674
  fix l
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   675
  assume "l dvd d" and "l dvd b"
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   676
  then have "l dvd a * b"
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   677
    by simp
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   678
  with \<open>l dvd d\<close> and dab show "l dvd 1"
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   679
    by (auto intro: gcd_greatest)
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   680
qed
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   681
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   682
lemma coprime_mult:
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   683
  assumes "coprime d a"
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   684
    and "coprime d b"
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   685
  shows "coprime d (a * b)"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   686
  apply (subst gcd.commute)
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   687
  using assms(1) apply (subst gcd_mult_cancel)
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   688
   apply (subst gcd.commute)
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   689
   apply assumption
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   690
  apply (subst gcd.commute)
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   691
  apply (rule assms(2))
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   692
  done
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   693
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   694
lemma coprime_mul_eq: "gcd d (a * b) = 1 \<longleftrightarrow> gcd d a = 1 \<and> gcd d b = 1"
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   695
  using coprime_rmult[of d a b] coprime_lmult[of d a b] coprime_mult[of d a b]
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   696
  by blast
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   697
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   698
lemma gcd_coprime:
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   699
  assumes c: "gcd a b \<noteq> 0"
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   700
    and a: "a = a' * gcd a b"
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   701
    and b: "b = b' * gcd a b"
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   702
  shows "gcd a' b' = 1"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   703
proof -
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   704
  from c have "a \<noteq> 0 \<or> b \<noteq> 0"
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   705
    by simp
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   706
  with div_gcd_coprime have "gcd (a div gcd a b) (b div gcd a b) = 1" .
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   707
  also from assms have "a div gcd a b = a'"
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   708
    using dvd_div_eq_mult local.gcd_dvd1 by blast
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   709
  also from assms have "b div gcd a b = b'"
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   710
    using dvd_div_eq_mult local.gcd_dvd1 by blast
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   711
  finally show ?thesis .
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   712
qed
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   713
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   714
lemma coprime_power:
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   715
  assumes "0 < n"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   716
  shows "gcd a (b ^ n) = 1 \<longleftrightarrow> gcd a b = 1"
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   717
  using assms
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   718
proof (induct n)
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   719
  case 0
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   720
  then show ?case by simp
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   721
next
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   722
  case (Suc n)
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   723
  then show ?case
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   724
    by (cases n) (simp_all add: coprime_mul_eq)
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   725
qed
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   726
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   727
lemma gcd_coprime_exists:
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   728
  assumes "gcd a b \<noteq> 0"
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   729
  shows "\<exists>a' b'. a = a' * gcd a b \<and> b = b' * gcd a b \<and> gcd a' b' = 1"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   730
  apply (rule_tac x = "a div gcd a b" in exI)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   731
  apply (rule_tac x = "b div gcd a b" in exI)
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   732
  using assms
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   733
  apply (auto intro: div_gcd_coprime)
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   734
  done
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   735
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   736
lemma coprime_exp: "gcd d a = 1 \<Longrightarrow> gcd d (a^n) = 1"
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   737
  by (induct n) (simp_all add: coprime_mult)
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   738
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   739
lemma coprime_exp_left: "coprime a b \<Longrightarrow> coprime (a ^ n) b"
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   740
  by (induct n) (simp_all add: gcd_mult_cancel)
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   741
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   742
lemma coprime_exp2:
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   743
  assumes "coprime a b"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   744
  shows "coprime (a ^ n) (b ^ m)"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   745
proof (rule coprime_exp_left)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   746
  from assms show "coprime a (b ^ m)"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   747
    by (induct m) (simp_all add: gcd_mult_cancel gcd.commute [of a])
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   748
qed
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   749
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   750
lemma gcd_exp: "gcd (a ^ n) (b ^ n) = gcd a b ^ n"
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   751
proof (cases "a = 0 \<and> b = 0")
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   752
  case True
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   753
  then show ?thesis
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   754
    by (cases n) simp_all
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   755
next
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   756
  case False
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   757
  then have "1 = gcd ((a div gcd a b) ^ n) ((b div gcd a b) ^ n)"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   758
    using coprime_exp2[OF div_gcd_coprime[of a b], of n n, symmetric] by simp
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   759
  then have "gcd a b ^ n = gcd a b ^ n * \<dots>"
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   760
    by simp
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   761
  also note gcd_mult_distrib
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   762
  also have "unit_factor (gcd a b ^ n) = 1"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   763
    using False by (auto simp add: unit_factor_power unit_factor_gcd)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   764
  also have "(gcd a b)^n * (a div gcd a b)^n = a^n"
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   765
    apply (subst ac_simps)
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   766
    apply (subst div_power)
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   767
     apply simp
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   768
    apply (rule dvd_div_mult_self)
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   769
    apply (rule dvd_power_same)
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   770
    apply simp
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   771
    done
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   772
  also have "(gcd a b)^n * (b div gcd a b)^n = b^n"
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   773
    apply (subst ac_simps)
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   774
    apply (subst div_power)
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   775
     apply simp
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   776
    apply (rule dvd_div_mult_self)
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   777
    apply (rule dvd_power_same)
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   778
    apply simp
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   779
    done
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   780
  finally show ?thesis by simp
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   781
qed
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   782
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   783
lemma coprime_common_divisor: "gcd a b = 1 \<Longrightarrow> a dvd a \<Longrightarrow> a dvd b \<Longrightarrow> is_unit a"
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   784
  apply (subgoal_tac "a dvd gcd a b")
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   785
   apply simp
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   786
  apply (erule (1) gcd_greatest)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   787
  done
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   788
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   789
lemma division_decomp:
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   790
  assumes "a dvd b * c"
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   791
  shows "\<exists>b' c'. a = b' * c' \<and> b' dvd b \<and> c' dvd c"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   792
proof (cases "gcd a b = 0")
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   793
  case True
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   794
  then have "a = 0 \<and> b = 0"
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   795
    by simp
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   796
  then have "a = 0 * c \<and> 0 dvd b \<and> c dvd c"
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   797
    by simp
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   798
  then show ?thesis by blast
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   799
next
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   800
  case False
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   801
  let ?d = "gcd a b"
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   802
  from gcd_coprime_exists [OF False]
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   803
    obtain a' b' where ab': "a = a' * ?d" "b = b' * ?d" "gcd a' b' = 1"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   804
    by blast
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   805
  from ab'(1) have "a' dvd a"
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   806
    unfolding dvd_def by blast
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   807
  with assms have "a' dvd b * c"
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   808
    using dvd_trans[of a' a "b*c"] by simp
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   809
  from assms ab'(1,2) have "a' * ?d dvd (b' * ?d) * c"
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   810
    by simp
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   811
  then have "?d * a' dvd ?d * (b' * c)"
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   812
    by (simp add: mult_ac)
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   813
  with \<open>?d \<noteq> 0\<close> have "a' dvd b' * c"
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   814
    by simp
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   815
  with coprime_dvd_mult[OF ab'(3)] have "a' dvd c"
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   816
    by (subst (asm) ac_simps) blast
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   817
  with ab'(1) have "a = ?d * a' \<and> ?d dvd b \<and> a' dvd c"
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   818
    by (simp add: mult_ac)
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   819
  then show ?thesis by blast
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   820
qed
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   821
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   822
lemma pow_divs_pow:
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   823
  assumes ab: "a ^ n dvd b ^ n" and n: "n \<noteq> 0"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   824
  shows "a dvd b"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   825
proof (cases "gcd a b = 0")
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   826
  case True
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   827
  then show ?thesis by simp
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   828
next
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   829
  case False
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   830
  let ?d = "gcd a b"
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   831
  from n obtain m where m: "n = Suc m"
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   832
    by (cases n) simp_all
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   833
  from False have zn: "?d ^ n \<noteq> 0"
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   834
    by (rule power_not_zero)
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   835
  from gcd_coprime_exists [OF False]
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   836
  obtain a' b' where ab': "a = a' * ?d" "b = b' * ?d" "gcd a' b' = 1"
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   837
    by blast
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   838
  from ab have "(a' * ?d) ^ n dvd (b' * ?d) ^ n"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   839
    by (simp add: ab'(1,2)[symmetric])
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   840
  then have "?d^n * a'^n dvd ?d^n * b'^n"
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   841
    by (simp only: power_mult_distrib ac_simps)
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   842
  with zn have "a'^n dvd b'^n"
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   843
    by simp
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   844
  then have "a' dvd b'^n"
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   845
    using dvd_trans[of a' "a'^n" "b'^n"] by (simp add: m)
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   846
  then have "a' dvd b'^m * b'"
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   847
    by (simp add: m ac_simps)
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   848
  with coprime_dvd_mult[OF coprime_exp[OF ab'(3), of m]]
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   849
  have "a' dvd b'" by (subst (asm) ac_simps) blast
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   850
  then have "a' * ?d dvd b' * ?d"
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   851
    by (rule mult_dvd_mono) simp
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   852
  with ab'(1,2) show ?thesis
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   853
    by simp
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   854
qed
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   855
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   856
lemma pow_divs_eq [simp]: "n \<noteq> 0 \<Longrightarrow> a ^ n dvd b ^ n \<longleftrightarrow> a dvd b"
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   857
  by (auto intro: pow_divs_pow dvd_power_same)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   858
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   859
lemma coprime_plus_one [simp]: "gcd (n + 1) n = 1"
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   860
  by (subst add_commute) simp
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   861
64272
f76b6dda2e56 setprod -> prod
nipkow
parents: 64242
diff changeset
   862
lemma prod_coprime [rule_format]: "(\<forall>i\<in>A. gcd (f i) a = 1) \<longrightarrow> gcd (\<Prod>i\<in>A. f i) a = 1"
63915
bab633745c7f tuned proofs;
wenzelm
parents: 63882
diff changeset
   863
  by (induct A rule: infinite_finite_induct) (auto simp add: gcd_mult_cancel)
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   864
63882
018998c00003 renamed listsum -> sum_list, listprod ~> prod_list
nipkow
parents: 63489
diff changeset
   865
lemma prod_list_coprime: "(\<And>x. x \<in> set xs \<Longrightarrow> coprime x y) \<Longrightarrow> coprime (prod_list xs) y"
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   866
  by (induct xs) (simp_all add: gcd_mult_cancel)
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   867
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   868
lemma coprime_divisors:
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   869
  assumes "d dvd a" "e dvd b" "gcd a b = 1"
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   870
  shows "gcd d e = 1"
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   871
proof -
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   872
  from assms obtain k l where "a = d * k" "b = e * l"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   873
    unfolding dvd_def by blast
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   874
  with assms have "gcd (d * k) (e * l) = 1"
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   875
    by simp
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   876
  then have "gcd (d * k) e = 1"
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   877
    by (rule coprime_lmult)
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   878
  also have "gcd (d * k) e = gcd e (d * k)"
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   879
    by (simp add: ac_simps)
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   880
  finally have "gcd e d = 1"
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   881
    by (rule coprime_lmult)
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   882
  then show ?thesis
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   883
    by (simp add: ac_simps)
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   884
qed
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   885
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   886
lemma lcm_gcd_prod: "lcm a b * gcd a b = normalize (a * b)"
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   887
  by (simp add: lcm_gcd)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   888
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   889
declare unit_factor_lcm [simp]
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   890
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   891
lemma lcmI:
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   892
  assumes "a dvd c" and "b dvd c" and "\<And>d. a dvd d \<Longrightarrow> b dvd d \<Longrightarrow> c dvd d"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   893
    and "normalize c = c"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   894
  shows "c = lcm a b"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   895
  by (rule associated_eqI) (auto simp: assms intro: lcm_least)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   896
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   897
lemma gcd_dvd_lcm [simp]: "gcd a b dvd lcm a b"
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   898
  using gcd_dvd2 by (rule dvd_lcmI2)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   899
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   900
lemmas lcm_0 = lcm_0_right
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   901
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   902
lemma lcm_unique:
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   903
  "a dvd d \<and> b dvd d \<and> normalize d = d \<and> (\<forall>e. a dvd e \<and> b dvd e \<longrightarrow> d dvd e) \<longleftrightarrow> d = lcm a b"
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   904
  by rule (auto intro: lcmI simp: lcm_least lcm_eq_0_iff)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   905
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   906
lemma lcm_coprime: "gcd a b = 1 \<Longrightarrow> lcm a b = normalize (a * b)"
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   907
  by (subst lcm_gcd) simp
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   908
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   909
lemma lcm_proj1_if_dvd: "b dvd a \<Longrightarrow> lcm a b = normalize a"
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   910
  apply (cases "a = 0")
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   911
   apply simp
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   912
  apply (rule sym)
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   913
  apply (rule lcmI)
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   914
     apply simp_all
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   915
  done
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   916
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   917
lemma lcm_proj2_if_dvd: "a dvd b \<Longrightarrow> lcm a b = normalize b"
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   918
  using lcm_proj1_if_dvd [of a b] by (simp add: ac_simps)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   919
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   920
lemma lcm_proj1_iff: "lcm m n = normalize m \<longleftrightarrow> n dvd m"
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   921
proof
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   922
  assume *: "lcm m n = normalize m"
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   923
  show "n dvd m"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   924
  proof (cases "m = 0")
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   925
    case True
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   926
    then show ?thesis by simp
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   927
  next
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   928
    case [simp]: False
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   929
    from * have **: "m = lcm m n * unit_factor m"
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   930
      by (simp add: unit_eq_div2)
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   931
    show ?thesis by (subst **) simp
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   932
  qed
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   933
next
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   934
  assume "n dvd m"
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   935
  then show "lcm m n = normalize m"
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   936
    by (rule lcm_proj1_if_dvd)
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   937
qed
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   938
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   939
lemma lcm_proj2_iff: "lcm m n = normalize n \<longleftrightarrow> m dvd n"
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   940
  using lcm_proj1_iff [of n m] by (simp add: ac_simps)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   941
63924
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63915
diff changeset
   942
lemma dvd_productE:
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63915
diff changeset
   943
  assumes "p dvd (a * b)"
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63915
diff changeset
   944
  obtains x y where "p = x * y" "x dvd a" "y dvd b"
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63915
diff changeset
   945
proof (cases "a = 0")
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63915
diff changeset
   946
  case True
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63915
diff changeset
   947
  thus ?thesis by (intro that[of p 1]) simp_all
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63915
diff changeset
   948
next
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63915
diff changeset
   949
  case False
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63915
diff changeset
   950
  define x y where "x = gcd a p" and "y = p div x"
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63915
diff changeset
   951
  have "p = x * y" by (simp add: x_def y_def)
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63915
diff changeset
   952
  moreover have "x dvd a" by (simp add: x_def)
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63915
diff changeset
   953
  moreover from assms have "p dvd gcd (b * a) (b * p)"
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63915
diff changeset
   954
    by (intro gcd_greatest) (simp_all add: mult.commute)
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63915
diff changeset
   955
  hence "p dvd b * gcd a p" by (simp add: gcd_mult_distrib)
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63915
diff changeset
   956
  with False have "y dvd b" 
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63915
diff changeset
   957
    by (simp add: x_def y_def div_dvd_iff_mult assms)
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63915
diff changeset
   958
  ultimately show ?thesis by (rule that)
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63915
diff changeset
   959
qed
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63915
diff changeset
   960
60686
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   961
end
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
   962
62345
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
   963
class ring_gcd = comm_ring_1 + semiring_gcd
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   964
begin
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   965
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   966
lemma coprime_minus_one: "coprime (n - 1) n"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   967
  using coprime_plus_one[of "n - 1"] by (simp add: gcd.commute)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   968
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   969
lemma gcd_neg1 [simp]: "gcd (-a) b = gcd a b"
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   970
  by (rule sym, rule gcdI) (simp_all add: gcd_greatest)
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   971
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   972
lemma gcd_neg2 [simp]: "gcd a (-b) = gcd a b"
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   973
  by (rule sym, rule gcdI) (simp_all add: gcd_greatest)
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   974
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   975
lemma gcd_neg_numeral_1 [simp]: "gcd (- numeral n) a = gcd (numeral n) a"
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   976
  by (fact gcd_neg1)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   977
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   978
lemma gcd_neg_numeral_2 [simp]: "gcd a (- numeral n) = gcd a (numeral n)"
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   979
  by (fact gcd_neg2)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   980
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   981
lemma gcd_diff1: "gcd (m - n) n = gcd m n"
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   982
  by (subst diff_conv_add_uminus, subst gcd_neg2[symmetric], subst gcd_add1, simp)
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   983
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   984
lemma gcd_diff2: "gcd (n - m) n = gcd m n"
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   985
  by (subst gcd_neg1[symmetric]) (simp only: minus_diff_eq gcd_diff1)
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   986
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   987
lemma lcm_neg1 [simp]: "lcm (-a) b = lcm a b"
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   988
  by (rule sym, rule lcmI) (simp_all add: lcm_least lcm_eq_0_iff)
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   989
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   990
lemma lcm_neg2 [simp]: "lcm a (-b) = lcm a b"
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
   991
  by (rule sym, rule lcmI) (simp_all add: lcm_least lcm_eq_0_iff)
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   992
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   993
lemma lcm_neg_numeral_1 [simp]: "lcm (- numeral n) a = lcm (numeral n) a"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   994
  by (fact lcm_neg1)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   995
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   996
lemma lcm_neg_numeral_2 [simp]: "lcm a (- numeral n) = lcm a (numeral n)"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   997
  by (fact lcm_neg2)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   998
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
   999
end
62345
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1000
60686
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1001
class semiring_Gcd = semiring_gcd + Gcd +
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1002
  assumes Gcd_dvd: "a \<in> A \<Longrightarrow> Gcd A dvd a"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1003
    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
  1004
    and normalize_Gcd [simp]: "normalize (Gcd A) = Gcd A"
62345
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1005
  assumes dvd_Lcm: "a \<in> A \<Longrightarrow> a dvd Lcm A"
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1006
    and Lcm_least: "(\<And>b. b \<in> A \<Longrightarrow> b dvd a) \<Longrightarrow> Lcm A dvd a"
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1007
    and normalize_Lcm [simp]: "normalize (Lcm A) = Lcm A"
60686
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1008
begin
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1009
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
  1010
lemma Lcm_Gcd: "Lcm A = Gcd {b. \<forall>a\<in>A. a dvd b}"
62345
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1011
  by (rule associated_eqI) (auto intro: Gcd_dvd dvd_Lcm Gcd_greatest Lcm_least)
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1012
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
  1013
lemma Gcd_Lcm: "Gcd A = Lcm {b. \<forall>a\<in>A. b dvd a}"
62345
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1014
  by (rule associated_eqI) (auto intro: Gcd_dvd dvd_Lcm Gcd_greatest Lcm_least)
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1015
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
  1016
lemma Gcd_empty [simp]: "Gcd {} = 0"
60686
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1017
  by (rule dvd_0_left, rule Gcd_greatest) simp
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1018
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
  1019
lemma Lcm_empty [simp]: "Lcm {} = 1"
62345
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1020
  by (auto intro: associated_eqI Lcm_least)
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1021
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
  1022
lemma Gcd_insert [simp]: "Gcd (insert a A) = gcd a (Gcd A)"
62345
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1023
proof -
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1024
  have "Gcd (insert a A) dvd gcd a (Gcd A)"
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1025
    by (auto intro: Gcd_dvd Gcd_greatest)
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1026
  moreover have "gcd a (Gcd A) dvd Gcd (insert a A)"
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1027
  proof (rule Gcd_greatest)
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1028
    fix b
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1029
    assume "b \<in> insert a A"
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1030
    then show "gcd a (Gcd A) dvd b"
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1031
    proof
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
  1032
      assume "b = a"
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
  1033
      then show ?thesis
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
  1034
        by simp
62345
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1035
    next
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1036
      assume "b \<in> A"
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
  1037
      then have "Gcd A dvd b"
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
  1038
        by (rule Gcd_dvd)
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
  1039
      moreover have "gcd a (Gcd A) dvd Gcd A"
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
  1040
        by simp
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
  1041
      ultimately show ?thesis
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
  1042
        by (blast intro: dvd_trans)
62345
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1043
    qed
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1044
  qed
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1045
  ultimately show ?thesis
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1046
    by (auto intro: associated_eqI)
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1047
qed
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1048
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
  1049
lemma Lcm_insert [simp]: "Lcm (insert a A) = lcm a (Lcm A)"
62345
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1050
proof (rule sym)
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1051
  have "lcm a (Lcm A) dvd Lcm (insert a A)"
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1052
    by (auto intro: dvd_Lcm Lcm_least)
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1053
  moreover have "Lcm (insert a A) dvd lcm a (Lcm A)"
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1054
  proof (rule Lcm_least)
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1055
    fix b
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1056
    assume "b \<in> insert a A"
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1057
    then show "b dvd lcm a (Lcm A)"
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1058
    proof
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
  1059
      assume "b = a"
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
  1060
      then show ?thesis by simp
62345
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1061
    next
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1062
      assume "b \<in> A"
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
  1063
      then have "b dvd Lcm A"
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
  1064
        by (rule dvd_Lcm)
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
  1065
      moreover have "Lcm A dvd lcm a (Lcm A)"
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
  1066
        by simp
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
  1067
      ultimately show ?thesis
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
  1068
        by (blast intro: dvd_trans)
62345
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1069
    qed
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1070
  qed
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1071
  ultimately show "lcm a (Lcm A) = Lcm (insert a A)"
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1072
    by (rule associated_eqI) (simp_all add: lcm_eq_0_iff)
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1073
qed
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1074
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1075
lemma LcmI:
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
  1076
  assumes "\<And>a. a \<in> A \<Longrightarrow> a dvd b"
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
  1077
    and "\<And>c. (\<And>a. a \<in> A \<Longrightarrow> a dvd c) \<Longrightarrow> b dvd c"
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
  1078
    and "normalize b = b"
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
  1079
  shows "b = Lcm A"
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1080
  by (rule associated_eqI) (auto simp: assms dvd_Lcm intro: Lcm_least)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1081
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
  1082
lemma Lcm_subset: "A \<subseteq> B \<Longrightarrow> Lcm A dvd Lcm B"
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1083
  by (blast intro: Lcm_least dvd_Lcm)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1084
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
  1085
lemma Lcm_Un: "Lcm (A \<union> B) = lcm (Lcm A) (Lcm B)"
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1086
  apply (rule lcmI)
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
  1087
     apply (blast intro: Lcm_subset)
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
  1088
    apply (blast intro: Lcm_subset)
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
  1089
   apply (intro Lcm_least ballI, elim UnE)
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
  1090
    apply (rule dvd_trans, erule dvd_Lcm, assumption)
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
  1091
   apply (rule dvd_trans, erule dvd_Lcm, assumption)
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1092
  apply simp
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1093
  done
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
  1094
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
  1095
lemma Gcd_0_iff [simp]: "Gcd A = 0 \<longleftrightarrow> A \<subseteq> {0}"
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
  1096
  (is "?P \<longleftrightarrow> ?Q")
60686
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1097
proof
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1098
  assume ?P
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1099
  show ?Q
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1100
  proof
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1101
    fix a
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1102
    assume "a \<in> A"
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
  1103
    then have "Gcd A dvd a"
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
  1104
      by (rule Gcd_dvd)
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
  1105
    with \<open>?P\<close> have "a = 0"
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
  1106
      by simp
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
  1107
    then show "a \<in> {0}"
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
  1108
      by simp
60686
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1109
  qed
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1110
next
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1111
  assume ?Q
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1112
  have "0 dvd Gcd A"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1113
  proof (rule Gcd_greatest)
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1114
    fix a
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1115
    assume "a \<in> A"
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
  1116
    with \<open>?Q\<close> have "a = 0"
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
  1117
      by auto
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
  1118
    then show "0 dvd a"
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
  1119
      by simp
60686
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1120
  qed
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
  1121
  then show ?P
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
  1122
    by simp
60686
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1123
qed
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1124
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
  1125
lemma Lcm_1_iff [simp]: "Lcm A = 1 \<longleftrightarrow> (\<forall>a\<in>A. is_unit a)"
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
  1126
  (is "?P \<longleftrightarrow> ?Q")
60686
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1127
proof
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1128
  assume ?P
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1129
  show ?Q
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1130
  proof
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1131
    fix a
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1132
    assume "a \<in> A"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1133
    then have "a dvd Lcm A"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1134
      by (rule dvd_Lcm)
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1135
    with \<open>?P\<close> show "is_unit a"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1136
      by simp
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1137
  qed
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1138
next
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1139
  assume ?Q
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1140
  then have "is_unit (Lcm A)"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1141
    by (blast intro: Lcm_least)
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1142
  then have "normalize (Lcm A) = 1"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1143
    by (rule is_unit_normalize)
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1144
  then show ?P
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1145
    by simp
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1146
qed
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1147
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
  1148
lemma unit_factor_Lcm: "unit_factor (Lcm A) = (if Lcm A = 0 then 0 else 1)"
62345
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1149
proof (cases "Lcm A = 0")
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
  1150
  case True
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
  1151
  then show ?thesis
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
  1152
    by simp
62345
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1153
next
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1154
  case False
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1155
  with unit_factor_normalize have "unit_factor (normalize (Lcm A)) = 1"
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1156
    by blast
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1157
  with False show ?thesis
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1158
    by simp
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1159
qed
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1160
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1161
lemma unit_factor_Gcd: "unit_factor (Gcd A) = (if Gcd A = 0 then 0 else 1)"
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
  1162
  by (simp add: Gcd_Lcm unit_factor_Lcm)
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1163
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1164
lemma GcdI:
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
  1165
  assumes "\<And>a. a \<in> A \<Longrightarrow> b dvd a"
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
  1166
    and "\<And>c. (\<And>a. a \<in> A \<Longrightarrow> c dvd a) \<Longrightarrow> c dvd b"
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1167
    and "normalize b = b"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1168
  shows "b = Gcd A"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1169
  by (rule associated_eqI) (auto simp: assms Gcd_dvd intro: Gcd_greatest)
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1170
62345
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1171
lemma Gcd_eq_1_I:
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1172
  assumes "is_unit a" and "a \<in> A"
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1173
  shows "Gcd A = 1"
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1174
proof -
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1175
  from assms have "is_unit (Gcd A)"
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1176
    by (blast intro: Gcd_dvd dvd_unit_imp_unit)
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1177
  then have "normalize (Gcd A) = 1"
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1178
    by (rule is_unit_normalize)
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1179
  then show ?thesis
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1180
    by simp
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1181
qed
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1182
60686
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1183
lemma Lcm_eq_0_I:
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1184
  assumes "0 \<in> A"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1185
  shows "Lcm A = 0"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1186
proof -
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1187
  from assms have "0 dvd Lcm A"
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1188
    by (rule dvd_Lcm)
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1189
  then show ?thesis
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1190
    by simp
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1191
qed
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1192
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
  1193
lemma Gcd_UNIV [simp]: "Gcd UNIV = 1"
62345
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1194
  using dvd_refl by (rule Gcd_eq_1_I) simp
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1195
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
  1196
lemma Lcm_UNIV [simp]: "Lcm UNIV = 0"
61929
b8e242e52c97 tuned proofs and augmented lemmas
haftmann
parents: 61913
diff changeset
  1197
  by (rule Lcm_eq_0_I) simp
60686
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1198
61929
b8e242e52c97 tuned proofs and augmented lemmas
haftmann
parents: 61913
diff changeset
  1199
lemma Lcm_0_iff:
b8e242e52c97 tuned proofs and augmented lemmas
haftmann
parents: 61913
diff changeset
  1200
  assumes "finite A"
b8e242e52c97 tuned proofs and augmented lemmas
haftmann
parents: 61913
diff changeset
  1201
  shows "Lcm A = 0 \<longleftrightarrow> 0 \<in> A"
b8e242e52c97 tuned proofs and augmented lemmas
haftmann
parents: 61913
diff changeset
  1202
proof (cases "A = {}")
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
  1203
  case True
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
  1204
  then show ?thesis by simp
61929
b8e242e52c97 tuned proofs and augmented lemmas
haftmann
parents: 61913
diff changeset
  1205
next
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
  1206
  case False
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
  1207
  with assms show ?thesis
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
  1208
    by (induct A rule: finite_ne_induct) (auto simp add: lcm_eq_0_iff)
60686
ea5bc46c11e6 more algebraic properties for gcd/lcm
haftmann
parents: 60597
diff changeset
  1209
qed
61929
b8e242e52c97 tuned proofs and augmented lemmas
haftmann
parents: 61913
diff changeset
  1210
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1211
lemma Gcd_finite:
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1212
  assumes "finite A"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1213
  shows "Gcd A = Finite_Set.fold gcd 0 A"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1214
  by (induct rule: finite.induct[OF \<open>finite A\<close>])
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
  1215
    (simp_all add: comp_fun_idem.fold_insert_idem[OF comp_fun_idem_gcd])
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1216
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1217
lemma Gcd_set [code_unfold]: "Gcd (set as) = foldl gcd 0 as"
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
  1218
  by (simp add: Gcd_finite comp_fun_idem.fold_set_fold[OF comp_fun_idem_gcd]
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
  1219
      foldl_conv_fold gcd.commute)
62429
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1220
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1221
lemma Lcm_finite:
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1222
  assumes "finite A"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1223
  shows "Lcm A = Finite_Set.fold lcm 1 A"
25271ff79171 Tuned Euclidean Rings/GCD rings
Manuel Eberl <eberlm@in.tum.de>
parents: 62353
diff changeset
  1224
  by (induct rule: finite.induct[OF \<open>finite A\<close>])
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
  1225
    (simp_all add: comp_fun_idem.fold_insert_idem[OF comp_fun_idem_lcm])
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
  1226
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
  1227
lemma Lcm_set [code_unfold]: "Lcm (set as) = foldl lcm 1 as"
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
  1228
  by (simp add: Lcm_finite comp_fun_idem.fold_set_fold[OF comp_fun_idem_lcm]
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
  1229
      foldl_conv_fold lcm.commute)
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
  1230
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
  1231
lemma Gcd_image_normalize [simp]: "Gcd (normalize ` A) = Gcd A"
62345
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1232
proof -
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1233
  have "Gcd (normalize ` A) dvd a" if "a \<in> A" for a
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1234
  proof -
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
  1235
    from that obtain B where "A = insert a B"
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
  1236
      by blast
62350
66a381d3f88f more sophisticated GCD syntax
haftmann
parents: 62349
diff changeset
  1237
    moreover have "gcd (normalize a) (Gcd (normalize ` B)) dvd normalize a"
62345
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1238
      by (rule gcd_dvd1)
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1239
    ultimately show "Gcd (normalize ` A) dvd a"
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1240
      by simp
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1241
  qed
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1242
  then have "Gcd (normalize ` A) dvd Gcd A" and "Gcd A dvd Gcd (normalize ` A)"
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1243
    by (auto intro!: Gcd_greatest intro: Gcd_dvd)
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1244
  then show ?thesis
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1245
    by (auto intro: associated_eqI)
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1246
qed
e66d7841d5a2 further generalization and polishing
haftmann
parents: 62344
diff changeset
  1247
62346
97f2ed240431 more theorems concerning gcd/lcm/Gcd/Lcm
haftmann
parents: 62345
diff changeset
  1248
lemma Gcd_eqI:
97f2ed240431 more theorems concerning gcd/lcm/Gcd/Lcm
haftmann
parents: 62345
diff changeset
  1249
  assumes "normalize a = a"
97f2ed240431 more theorems concerning gcd/lcm/Gcd/Lcm
haftmann
parents: 62345
diff changeset
  1250
  assumes "\<And>b. b \<in> A \<Longrightarrow> a dvd b"
97f2ed240431 more theorems concerning gcd/lcm/Gcd/Lcm
haftmann
parents: 62345
diff changeset
  1251
    and "\<And>c. (\<And>b. b \<in> A \<Longrightarrow> c dvd b) \<Longrightarrow> c dvd a"
97f2ed240431 more theorems concerning gcd/lcm/Gcd/Lcm
haftmann
parents: 62345
diff changeset
  1252
  shows "Gcd A = a"
97f2ed240431 more theorems concerning gcd/lcm/Gcd/Lcm
haftmann
parents: 62345
diff changeset
  1253
  using assms by (blast intro: associated_eqI Gcd_greatest Gcd_dvd normalize_Gcd)
97f2ed240431 more theorems concerning gcd/lcm/Gcd/Lcm
haftmann
parents: 62345
diff changeset
  1254
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
  1255
lemma dvd_GcdD: "x dvd Gcd A \<Longrightarrow> y \<in> A \<Longrightarrow> x dvd y"
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
  1256
  using Gcd_dvd dvd_trans by blast
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
  1257
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
  1258
lemma dvd_Gcd_iff: "x dvd Gcd A \<longleftrightarrow> (\<forall>y\<in>A. x dvd y)"
63359
99b51ba8da1c More lemmas on Gcd/Lcm
Manuel Eberl <eberlm@in.tum.de>
parents: 63145
diff changeset
  1259
  by (blast dest: dvd_GcdD intro: Gcd_greatest)
99b51ba8da1c More lemmas on Gcd/Lcm
Manuel Eberl <eberlm@in.tum.de>
parents: 63145
diff changeset
  1260
99b51ba8da1c More lemmas on Gcd/Lcm
Manuel Eberl <eberlm@in.tum.de>
parents: 63145
diff changeset
  1261
lemma Gcd_mult: "Gcd (op * c ` A) = normalize c * Gcd A"
99b51ba8da1c More lemmas on Gcd/Lcm
Manuel Eberl <eberlm@in.tum.de>
parents: 63145
diff changeset
  1262
proof (cases "c = 0")
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
  1263
  case True
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
  1264
  then show ?thesis by auto
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
  1265
next
63359
99b51ba8da1c More lemmas on Gcd/Lcm
Manuel Eberl <eberlm@in.tum.de>
parents: 63145
diff changeset
  1266
  case [simp]: False
99b51ba8da1c More lemmas on Gcd/Lcm
Manuel Eberl <eberlm@in.tum.de>
parents: 63145
diff changeset
  1267
  have "Gcd (op * c ` A) div c dvd Gcd A"
99b51ba8da1c More lemmas on Gcd/Lcm
Manuel Eberl <eberlm@in.tum.de>
parents: 63145
diff changeset
  1268
    by (intro Gcd_greatest, subst div_dvd_iff_mult)
99b51ba8da1c More lemmas on Gcd/Lcm
Manuel Eberl <eberlm@in.tum.de>
parents: 63145
diff changeset
  1269
       (auto intro!: Gcd_greatest Gcd_dvd simp: mult.commute[of _ c])
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
  1270
  then have "Gcd (op * c ` A) dvd c * Gcd A"
63359
99b51ba8da1c More lemmas on Gcd/Lcm
Manuel Eberl <eberlm@in.tum.de>
parents: 63145
diff changeset
  1271
    by (subst (asm) div_dvd_iff_mult) (auto intro: Gcd_greatest simp: mult_ac)
99b51ba8da1c More lemmas on Gcd/Lcm
Manuel Eberl <eberlm@in.tum.de>
parents: 63145
diff changeset
  1272
  also have "c * Gcd A = (normalize c * Gcd A) * unit_factor c"
99b51ba8da1c More lemmas on Gcd/Lcm
Manuel Eberl <eberlm@in.tum.de>
parents: 63145
diff changeset
  1273
    by (subst unit_factor_mult_normalize [symmetric]) (simp only: mult_ac)
99b51ba8da1c More lemmas on Gcd/Lcm
Manuel Eberl <eberlm@in.tum.de>
parents: 63145
diff changeset
  1274
  also have "Gcd (op * c ` A) dvd \<dots> \<longleftrightarrow> Gcd (op * c ` A) dvd normalize c * Gcd A"
99b51ba8da1c More lemmas on Gcd/Lcm
Manuel Eberl <eberlm@in.tum.de>
parents: 63145
diff changeset
  1275
    by (simp add: dvd_mult_unit_iff)
99b51ba8da1c More lemmas on Gcd/Lcm
Manuel Eberl <eberlm@in.tum.de>
parents: 63145
diff changeset
  1276
  finally have "Gcd (op * c ` A) dvd normalize c * Gcd A" .
99b51ba8da1c More lemmas on Gcd/Lcm
Manuel Eberl <eberlm@in.tum.de>
parents: 63145
diff changeset
  1277
  moreover have "normalize c * Gcd A dvd Gcd (op * c ` A)"
99b51ba8da1c More lemmas on Gcd/Lcm
Manuel Eberl <eberlm@in.tum.de>
parents: 63145
diff changeset
  1278
    by (intro Gcd_greatest) (auto intro: mult_dvd_mono Gcd_dvd)
99b51ba8da1c More lemmas on Gcd/Lcm
Manuel Eberl <eberlm@in.tum.de>
parents: 63145
diff changeset
  1279
  ultimately have "normalize (Gcd (op * c ` A)) = normalize (normalize c * Gcd A)"
99b51ba8da1c More lemmas on Gcd/Lcm
Manuel Eberl <eberlm@in.tum.de>
parents: 63145
diff changeset
  1280
    by (rule associatedI)
63489
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
  1281
  then show ?thesis
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
  1282
    by (simp add: normalize_mult)
cd540c8031a4 misc tuning and modernization;
wenzelm
parents: 63359
diff changeset
  1283
qed
63359
99b51ba8da1c More lemmas on Gcd/Lcm
Manuel Eberl <eberlm@in.tum.de>
parents: 63145
diff changeset
  1284
62346
97f2ed240431 more theorems concerning gcd/lcm/Gcd/Lcm
haftmann
parents: 62345
diff changeset
  1285
lemma Lcm_eqI:
97f2ed240431 more theorems concerning gcd/lcm/Gcd/Lcm
haftmann
parents: 62345
diff changeset
  1286
  assumes "normalize a = a"