src/HOL/Computational_Algebra/Factorial_Ring.thy
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(*  Title:      HOL/Computational_Algebra/Factorial_Ring.thy
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    Author:     Manuel Eberl, TU Muenchen
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    Author:     Florian Haftmann, TU Muenchen
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*)
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section \<open>Factorial (semi)rings\<close>
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theory Factorial_Ring
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imports
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  Main
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  "HOL-Library.Multiset"
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begin
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unbundle multiset.lifting
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subsection \<open>Irreducible and prime elements\<close>
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context comm_semiring_1
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begin
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definition irreducible :: "'a \<Rightarrow> bool" where
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  "irreducible p \<longleftrightarrow> p \<noteq> 0 \<and> \<not>p dvd 1 \<and> (\<forall>a b. p = a * b \<longrightarrow> a dvd 1 \<or> b dvd 1)"
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lemma not_irreducible_zero [simp]: "\<not>irreducible 0"
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  by (simp add: irreducible_def)
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lemma irreducible_not_unit: "irreducible p \<Longrightarrow> \<not>p dvd 1"
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  by (simp add: irreducible_def)
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lemma not_irreducible_one [simp]: "\<not>irreducible 1"
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  by (simp add: irreducible_def)
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lemma irreducibleI:
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  "p \<noteq> 0 \<Longrightarrow> \<not>p dvd 1 \<Longrightarrow> (\<And>a b. p = a * b \<Longrightarrow> a dvd 1 \<or> b dvd 1) \<Longrightarrow> irreducible p"
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  by (simp add: irreducible_def)
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lemma irreducibleD: "irreducible p \<Longrightarrow> p = a * b \<Longrightarrow> a dvd 1 \<or> b dvd 1"
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  by (simp add: irreducible_def)
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lemma irreducible_mono:
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  assumes irr: "irreducible b" and "a dvd b" "\<not>a dvd 1"
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  shows   "irreducible a"
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proof (rule irreducibleI)
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  fix c d assume "a = c * d"
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  from assms obtain k where [simp]: "b = a * k" by auto
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  from \<open>a = c * d\<close> have "b = c * d * k"
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    by simp
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  hence "c dvd 1 \<or> (d * k) dvd 1"
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    using irreducibleD[OF irr, of c "d * k"] by (auto simp: mult.assoc)
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  thus "c dvd 1 \<or> d dvd 1"
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    by auto
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qed (use assms in \<open>auto simp: irreducible_def\<close>)
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definition prime_elem :: "'a \<Rightarrow> bool" where
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  "prime_elem p \<longleftrightarrow> p \<noteq> 0 \<and> \<not>p dvd 1 \<and> (\<forall>a b. p dvd (a * b) \<longrightarrow> p dvd a \<or> p dvd b)"
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lemma not_prime_elem_zero [simp]: "\<not>prime_elem 0"
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  by (simp add: prime_elem_def)
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lemma prime_elem_not_unit: "prime_elem p \<Longrightarrow> \<not>p dvd 1"
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  by (simp add: prime_elem_def)
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lemma prime_elemI:
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    "p \<noteq> 0 \<Longrightarrow> \<not>p dvd 1 \<Longrightarrow> (\<And>a b. p dvd (a * b) \<Longrightarrow> p dvd a \<or> p dvd b) \<Longrightarrow> prime_elem p"
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  by (simp add: prime_elem_def)
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lemma prime_elem_dvd_multD:
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    "prime_elem p \<Longrightarrow> p dvd (a * b) \<Longrightarrow> p dvd a \<or> p dvd b"
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  by (simp add: prime_elem_def)
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lemma prime_elem_dvd_mult_iff:
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  "prime_elem p \<Longrightarrow> p dvd (a * b) \<longleftrightarrow> p dvd a \<or> p dvd b"
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  by (auto simp: prime_elem_def)
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lemma not_prime_elem_one [simp]:
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  "\<not> prime_elem 1"
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  by (auto dest: prime_elem_not_unit)
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lemma prime_elem_not_zeroI:
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  assumes "prime_elem p"
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  shows "p \<noteq> 0"
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  using assms by (auto intro: ccontr)
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lemma prime_elem_dvd_power:
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  "prime_elem p \<Longrightarrow> p dvd x ^ n \<Longrightarrow> p dvd x"
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  by (induction n) (auto dest: prime_elem_dvd_multD intro: dvd_trans[of _ 1])
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lemma prime_elem_dvd_power_iff:
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  "prime_elem p \<Longrightarrow> n > 0 \<Longrightarrow> p dvd x ^ n \<longleftrightarrow> p dvd x"
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  by (auto dest: prime_elem_dvd_power intro: dvd_trans)
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lemma prime_elem_imp_nonzero [simp]:
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  "ASSUMPTION (prime_elem x) \<Longrightarrow> x \<noteq> 0"
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  unfolding ASSUMPTION_def by (rule prime_elem_not_zeroI)
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lemma prime_elem_imp_not_one [simp]:
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  "ASSUMPTION (prime_elem x) \<Longrightarrow> x \<noteq> 1"
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  unfolding ASSUMPTION_def by auto
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end
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lemma (in normalization_semidom) irreducible_cong:
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  assumes "normalize a = normalize b"
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  shows   "irreducible a \<longleftrightarrow> irreducible b"
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proof (cases "a = 0 \<or> a dvd 1")
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  case True
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  hence "\<not>irreducible a" by (auto simp: irreducible_def)
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  from True have "normalize a = 0 \<or> normalize a dvd 1"
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    by auto
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  also note assms
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  finally have "b = 0 \<or> b dvd 1" by simp
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  hence "\<not>irreducible b" by (auto simp: irreducible_def)
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  with \<open>\<not>irreducible a\<close> show ?thesis by simp
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next
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  case False
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  hence b: "b \<noteq> 0" "\<not>is_unit b" using assms
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    by (auto simp: is_unit_normalize[of b])
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  show ?thesis
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  proof
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    assume "irreducible a"
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    thus "irreducible b"
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      by (rule irreducible_mono) (use assms False b in \<open>auto dest: associatedD2\<close>)
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  next
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    assume "irreducible b"
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    thus "irreducible a"
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      by (rule irreducible_mono) (use assms False b in \<open>auto dest: associatedD1\<close>)
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parents: 69785
diff changeset
   128
  qed
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
   129
qed
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
   130
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
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parents: 69785
diff changeset
   131
lemma (in normalization_semidom) associatedE1:
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
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parents: 69785
diff changeset
   132
  assumes "normalize a = normalize b"
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
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parents: 69785
diff changeset
   133
  obtains u where "is_unit u" "a = u * b"
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
   134
proof (cases "a = 0")
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
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diff changeset
   135
  case [simp]: False
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
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parents: 69785
diff changeset
   136
  from assms have [simp]: "b \<noteq> 0" by auto
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
   137
  show ?thesis
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
   138
  proof (rule that)
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
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parents: 69785
diff changeset
   139
    show "is_unit (unit_factor a div unit_factor b)"
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
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parents: 69785
diff changeset
   140
      by auto
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
   141
    have "unit_factor a div unit_factor b * b = unit_factor a * (b div unit_factor b)"
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
   142
      using \<open>b \<noteq> 0\<close> unit_div_commute unit_div_mult_swap unit_factor_is_unit by metis
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
   143
    also have "b div unit_factor b = normalize b" by simp
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
   144
    finally show "a = unit_factor a div unit_factor b * b"
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
   145
      by (metis assms unit_factor_mult_normalize)
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
   146
  qed
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
   147
next
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
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parents: 69785
diff changeset
   148
  case [simp]: True
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
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parents: 69785
diff changeset
   149
  hence [simp]: "b = 0"
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
   150
    using assms[symmetric] by auto
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
   151
  show ?thesis
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
   152
    by (intro that[of 1]) auto
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
   153
qed
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
   154
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
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parents: 69785
diff changeset
   155
lemma (in normalization_semidom) associatedE2:
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
   156
  assumes "normalize a = normalize b"
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
   157
  obtains u where "is_unit u" "b = u * a"
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
   158
proof -
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
   159
  from assms have "normalize b = normalize a"
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
   160
    by simp
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
   161
  then obtain u where "is_unit u" "b = u * a"
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
   162
    by (elim associatedE1)
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
   163
  thus ?thesis using that by blast
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
   164
qed
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
   165
  
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
   166
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
   167
(* TODO Move *)
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
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parents: 69785
diff changeset
   168
lemma (in normalization_semidom) normalize_power_normalize:
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
   169
  "normalize (normalize x ^ n) = normalize (x ^ n)"
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
   170
proof (induction n)
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
   171
  case (Suc n)
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
   172
  have "normalize (normalize x ^ Suc n) = normalize (x * normalize (normalize x ^ n))"
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
   173
    by simp
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
   174
  also note Suc.IH
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
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parents: 69785
diff changeset
   175
  finally show ?case by simp
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
   176
qed auto
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
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parents: 69785
diff changeset
   177
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   178
context algebraic_semidom
60804
080a979a985b formal class for factorial (semi)rings
haftmann
parents:
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   179
begin
080a979a985b formal class for factorial (semi)rings
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parents:
diff changeset
   180
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   181
lemma prime_elem_imp_irreducible:
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   182
  assumes "prime_elem p"
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   183
  shows   "irreducible p"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   184
proof (rule irreducibleI)
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   185
  fix a b
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   186
  assume p_eq: "p = a * b"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   187
  with assms have nz: "a \<noteq> 0" "b \<noteq> 0" by auto
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   188
  from p_eq have "p dvd a * b" by simp
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   189
  with \<open>prime_elem p\<close> have "p dvd a \<or> p dvd b" by (rule prime_elem_dvd_multD)
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   190
  with \<open>p = a * b\<close> have "a * b dvd 1 * b \<or> a * b dvd a * 1" by auto
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   191
  thus "a dvd 1 \<or> b dvd 1"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   192
    by (simp only: dvd_times_left_cancel_iff[OF nz(1)] dvd_times_right_cancel_iff[OF nz(2)])
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   193
qed (insert assms, simp_all add: prime_elem_def)
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   194
63924
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   195
lemma (in algebraic_semidom) unit_imp_no_irreducible_divisors:
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   196
  assumes "is_unit x" "irreducible p"
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   197
  shows   "\<not>p dvd x"
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   198
proof (rule notI)
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   199
  assume "p dvd x"
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   200
  with \<open>is_unit x\<close> have "is_unit p"
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   201
    by (auto intro: dvd_trans)
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   202
  with \<open>irreducible p\<close> show False
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   203
    by (simp add: irreducible_not_unit)
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   204
qed
65552
f533820e7248 theories "GCD" and "Binomial" are already included in "Main": this avoids improper imports in applications;
wenzelm
parents: 65435
diff changeset
   205
63924
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   206
lemma unit_imp_no_prime_divisors:
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   207
  assumes "is_unit x" "prime_elem p"
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   208
  shows   "\<not>p dvd x"
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   209
  using unit_imp_no_irreducible_divisors[OF assms(1) prime_elem_imp_irreducible[OF assms(2)]] .
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   210
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   211
lemma prime_elem_mono:
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   212
  assumes "prime_elem p" "\<not>q dvd 1" "q dvd p"
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   213
  shows   "prime_elem q"
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   214
proof -
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   215
  from \<open>q dvd p\<close> obtain r where r: "p = q * r" by (elim dvdE)
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   216
  hence "p dvd q * r" by simp
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   217
  with \<open>prime_elem p\<close> have "p dvd q \<or> p dvd r" by (rule prime_elem_dvd_multD)
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   218
  hence "p dvd q"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   219
  proof
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   220
    assume "p dvd r"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   221
    then obtain s where s: "r = p * s" by (elim dvdE)
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   222
    from r have "p * 1 = p * (q * s)" by (subst (asm) s) (simp add: mult_ac)
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   223
    with \<open>prime_elem p\<close> have "q dvd 1"
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   224
      by (subst (asm) mult_cancel_left) auto
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   225
    with \<open>\<not>q dvd 1\<close> show ?thesis by contradiction
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   226
  qed
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   227
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   228
  show ?thesis
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   229
  proof (rule prime_elemI)
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   230
    fix a b assume "q dvd (a * b)"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   231
    with \<open>p dvd q\<close> have "p dvd (a * b)" by (rule dvd_trans)
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   232
    with \<open>prime_elem p\<close> have "p dvd a \<or> p dvd b" by (rule prime_elem_dvd_multD)
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   233
    with \<open>q dvd p\<close> show "q dvd a \<or> q dvd b" by (blast intro: dvd_trans)
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   234
  qed (insert assms, auto)
62499
4a5b81ff5992 constructive formulation of factorization
haftmann
parents: 62366
diff changeset
   235
qed
4a5b81ff5992 constructive formulation of factorization
haftmann
parents: 62366
diff changeset
   236
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   237
lemma irreducibleD':
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   238
  assumes "irreducible a" "b dvd a"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   239
  shows   "a dvd b \<or> is_unit b"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   240
proof -
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   241
  from assms obtain c where c: "a = b * c" by (elim dvdE)
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   242
  from irreducibleD[OF assms(1) this] have "is_unit b \<or> is_unit c" .
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   243
  thus ?thesis by (auto simp: c mult_unit_dvd_iff)
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   244
qed
60804
080a979a985b formal class for factorial (semi)rings
haftmann
parents:
diff changeset
   245
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   246
lemma irreducibleI':
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   247
  assumes "a \<noteq> 0" "\<not>is_unit a" "\<And>b. b dvd a \<Longrightarrow> a dvd b \<or> is_unit b"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   248
  shows   "irreducible a"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   249
proof (rule irreducibleI)
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   250
  fix b c assume a_eq: "a = b * c"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   251
  hence "a dvd b \<or> is_unit b" by (intro assms) simp_all
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   252
  thus "is_unit b \<or> is_unit c"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   253
  proof
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   254
    assume "a dvd b"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   255
    hence "b * c dvd b * 1" by (simp add: a_eq)
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   256
    moreover from \<open>a \<noteq> 0\<close> a_eq have "b \<noteq> 0" by auto
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   257
    ultimately show ?thesis by (subst (asm) dvd_times_left_cancel_iff) auto
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   258
  qed blast
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   259
qed (simp_all add: assms(1,2))
60804
080a979a985b formal class for factorial (semi)rings
haftmann
parents:
diff changeset
   260
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   261
lemma irreducible_altdef:
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   262
  "irreducible x \<longleftrightarrow> x \<noteq> 0 \<and> \<not>is_unit x \<and> (\<forall>b. b dvd x \<longrightarrow> x dvd b \<or> is_unit b)"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   263
  using irreducibleI'[of x] irreducibleD'[of x] irreducible_not_unit[of x] by auto
60804
080a979a985b formal class for factorial (semi)rings
haftmann
parents:
diff changeset
   264
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   265
lemma prime_elem_multD:
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   266
  assumes "prime_elem (a * b)"
60804
080a979a985b formal class for factorial (semi)rings
haftmann
parents:
diff changeset
   267
  shows "is_unit a \<or> is_unit b"
080a979a985b formal class for factorial (semi)rings
haftmann
parents:
diff changeset
   268
proof -
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   269
  from assms have "a \<noteq> 0" "b \<noteq> 0" by (auto dest!: prime_elem_not_zeroI)
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   270
  moreover from assms prime_elem_dvd_multD [of "a * b"] have "a * b dvd a \<or> a * b dvd b"
60804
080a979a985b formal class for factorial (semi)rings
haftmann
parents:
diff changeset
   271
    by auto
080a979a985b formal class for factorial (semi)rings
haftmann
parents:
diff changeset
   272
  ultimately show ?thesis
080a979a985b formal class for factorial (semi)rings
haftmann
parents:
diff changeset
   273
    using dvd_times_left_cancel_iff [of a b 1]
080a979a985b formal class for factorial (semi)rings
haftmann
parents:
diff changeset
   274
      dvd_times_right_cancel_iff [of b a 1]
080a979a985b formal class for factorial (semi)rings
haftmann
parents:
diff changeset
   275
    by auto
080a979a985b formal class for factorial (semi)rings
haftmann
parents:
diff changeset
   276
qed
080a979a985b formal class for factorial (semi)rings
haftmann
parents:
diff changeset
   277
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   278
lemma prime_elemD2:
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   279
  assumes "prime_elem p" and "a dvd p" and "\<not> is_unit a"
60804
080a979a985b formal class for factorial (semi)rings
haftmann
parents:
diff changeset
   280
  shows "p dvd a"
080a979a985b formal class for factorial (semi)rings
haftmann
parents:
diff changeset
   281
proof -
080a979a985b formal class for factorial (semi)rings
haftmann
parents:
diff changeset
   282
  from \<open>a dvd p\<close> obtain b where "p = a * b" ..
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   283
  with \<open>prime_elem p\<close> prime_elem_multD \<open>\<not> is_unit a\<close> have "is_unit b" by auto
60804
080a979a985b formal class for factorial (semi)rings
haftmann
parents:
diff changeset
   284
  with \<open>p = a * b\<close> show ?thesis
080a979a985b formal class for factorial (semi)rings
haftmann
parents:
diff changeset
   285
    by (auto simp add: mult_unit_dvd_iff)
080a979a985b formal class for factorial (semi)rings
haftmann
parents:
diff changeset
   286
qed
080a979a985b formal class for factorial (semi)rings
haftmann
parents:
diff changeset
   287
63830
2ea3725a34bd msetsum -> set_mset, msetprod -> prod_mset
nipkow
parents: 63793
diff changeset
   288
lemma prime_elem_dvd_prod_msetE:
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   289
  assumes "prime_elem p"
63830
2ea3725a34bd msetsum -> set_mset, msetprod -> prod_mset
nipkow
parents: 63793
diff changeset
   290
  assumes dvd: "p dvd prod_mset A"
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   291
  obtains a where "a \<in># A" and "p dvd a"
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   292
proof -
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   293
  from dvd have "\<exists>a. a \<in># A \<and> p dvd a"
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   294
  proof (induct A)
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   295
    case empty then show ?case
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   296
    using \<open>prime_elem p\<close> by (simp add: prime_elem_not_unit)
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   297
  next
63793
e68a0b651eb5 add_mset constructor in multisets
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents: 63633
diff changeset
   298
    case (add a A)
63830
2ea3725a34bd msetsum -> set_mset, msetprod -> prod_mset
nipkow
parents: 63793
diff changeset
   299
    then have "p dvd a * prod_mset A" by simp
2ea3725a34bd msetsum -> set_mset, msetprod -> prod_mset
nipkow
parents: 63793
diff changeset
   300
    with \<open>prime_elem p\<close> consider (A) "p dvd prod_mset A" | (B) "p dvd a"
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   301
      by (blast dest: prime_elem_dvd_multD)
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   302
    then show ?case proof cases
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   303
      case B then show ?thesis by auto
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   304
    next
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   305
      case A
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   306
      with add.hyps obtain b where "b \<in># A" "p dvd b"
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   307
        by auto
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   308
      then show ?thesis by auto
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   309
    qed
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   310
  qed
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   311
  with that show thesis by blast
66276
acc3b7dd0b21 More material on powers for HOL-Computational_Algebra/HOL-Number_Theory
eberlm <eberlm@in.tum.de>
parents: 65552
diff changeset
   312
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   313
qed
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   314
63534
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
   315
context
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
   316
begin
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
   317
74542
d592354c4a26 removed some 'private' modifiers from HOL-Computational_Algebra
Manuel Eberl <manuel@pruvisto.org>
parents: 74362
diff changeset
   318
lemma prime_elem_powerD:
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   319
  assumes "prime_elem (p ^ n)"
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   320
  shows   "prime_elem p \<and> n = 1"
63534
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
   321
proof (cases n)
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
   322
  case (Suc m)
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
   323
  note assms
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
   324
  also from Suc have "p ^ n = p * p^m" by simp
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   325
  finally have "is_unit p \<or> is_unit (p^m)" by (rule prime_elem_multD)
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   326
  moreover from assms have "\<not>is_unit p" by (simp add: prime_elem_def is_unit_power_iff)
63534
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
   327
  ultimately have "is_unit (p ^ m)" by simp
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
   328
  with \<open>\<not>is_unit p\<close> have "m = 0" by (simp add: is_unit_power_iff)
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
   329
  with Suc assms show ?thesis by simp
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
   330
qed (insert assms, simp_all)
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
   331
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   332
lemma prime_elem_power_iff:
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   333
  "prime_elem (p ^ n) \<longleftrightarrow> prime_elem p \<and> n = 1"
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   334
  by (auto dest: prime_elem_powerD)
63534
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
   335
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
   336
end
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
   337
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   338
lemma irreducible_mult_unit_left:
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   339
  "is_unit a \<Longrightarrow> irreducible (a * p) \<longleftrightarrow> irreducible p"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   340
  by (auto simp: irreducible_altdef mult.commute[of a] is_unit_mult_iff
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   341
        mult_unit_dvd_iff dvd_mult_unit_iff)
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   342
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   343
lemma prime_elem_mult_unit_left:
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   344
  "is_unit a \<Longrightarrow> prime_elem (a * p) \<longleftrightarrow> prime_elem p"
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   345
  by (auto simp: prime_elem_def mult.commute[of a] is_unit_mult_iff mult_unit_dvd_iff)
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   346
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   347
lemma prime_elem_dvd_cases:
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   348
  assumes pk: "p*k dvd m*n" and p: "prime_elem p"
63537
831816778409 Removed redundant material related to primes
eberlm <eberlm@in.tum.de>
parents: 63534
diff changeset
   349
  shows "(\<exists>x. k dvd x*n \<and> m = p*x) \<or> (\<exists>y. k dvd m*y \<and> n = p*y)"
831816778409 Removed redundant material related to primes
eberlm <eberlm@in.tum.de>
parents: 63534
diff changeset
   350
proof -
831816778409 Removed redundant material related to primes
eberlm <eberlm@in.tum.de>
parents: 63534
diff changeset
   351
  have "p dvd m*n" using dvd_mult_left pk by blast
831816778409 Removed redundant material related to primes
eberlm <eberlm@in.tum.de>
parents: 63534
diff changeset
   352
  then consider "p dvd m" | "p dvd n"
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   353
    using p prime_elem_dvd_mult_iff by blast
63537
831816778409 Removed redundant material related to primes
eberlm <eberlm@in.tum.de>
parents: 63534
diff changeset
   354
  then show ?thesis
831816778409 Removed redundant material related to primes
eberlm <eberlm@in.tum.de>
parents: 63534
diff changeset
   355
  proof cases
65552
f533820e7248 theories "GCD" and "Binomial" are already included in "Main": this avoids improper imports in applications;
wenzelm
parents: 65435
diff changeset
   356
    case 1 then obtain a where "m = p * a" by (metis dvd_mult_div_cancel)
63537
831816778409 Removed redundant material related to primes
eberlm <eberlm@in.tum.de>
parents: 63534
diff changeset
   357
      then have "\<exists>x. k dvd x * n \<and> m = p * x"
831816778409 Removed redundant material related to primes
eberlm <eberlm@in.tum.de>
parents: 63534
diff changeset
   358
        using p pk by (auto simp: mult.assoc)
831816778409 Removed redundant material related to primes
eberlm <eberlm@in.tum.de>
parents: 63534
diff changeset
   359
    then show ?thesis ..
831816778409 Removed redundant material related to primes
eberlm <eberlm@in.tum.de>
parents: 63534
diff changeset
   360
  next
65552
f533820e7248 theories "GCD" and "Binomial" are already included in "Main": this avoids improper imports in applications;
wenzelm
parents: 65435
diff changeset
   361
    case 2 then obtain b where "n = p * b" by (metis dvd_mult_div_cancel)
f533820e7248 theories "GCD" and "Binomial" are already included in "Main": this avoids improper imports in applications;
wenzelm
parents: 65435
diff changeset
   362
    with p pk have "\<exists>y. k dvd m*y \<and> n = p*y"
63537
831816778409 Removed redundant material related to primes
eberlm <eberlm@in.tum.de>
parents: 63534
diff changeset
   363
      by (metis dvd_mult_right dvd_times_left_cancel_iff mult.left_commute mult_zero_left)
831816778409 Removed redundant material related to primes
eberlm <eberlm@in.tum.de>
parents: 63534
diff changeset
   364
    then show ?thesis ..
831816778409 Removed redundant material related to primes
eberlm <eberlm@in.tum.de>
parents: 63534
diff changeset
   365
  qed
831816778409 Removed redundant material related to primes
eberlm <eberlm@in.tum.de>
parents: 63534
diff changeset
   366
qed
831816778409 Removed redundant material related to primes
eberlm <eberlm@in.tum.de>
parents: 63534
diff changeset
   367
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   368
lemma prime_elem_power_dvd_prod:
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   369
  assumes pc: "p^c dvd m*n" and p: "prime_elem p"
63537
831816778409 Removed redundant material related to primes
eberlm <eberlm@in.tum.de>
parents: 63534
diff changeset
   370
  shows "\<exists>a b. a+b = c \<and> p^a dvd m \<and> p^b dvd n"
831816778409 Removed redundant material related to primes
eberlm <eberlm@in.tum.de>
parents: 63534
diff changeset
   371
using pc
831816778409 Removed redundant material related to primes
eberlm <eberlm@in.tum.de>
parents: 63534
diff changeset
   372
proof (induct c arbitrary: m n)
831816778409 Removed redundant material related to primes
eberlm <eberlm@in.tum.de>
parents: 63534
diff changeset
   373
  case 0 show ?case by simp
831816778409 Removed redundant material related to primes
eberlm <eberlm@in.tum.de>
parents: 63534
diff changeset
   374
next
831816778409 Removed redundant material related to primes
eberlm <eberlm@in.tum.de>
parents: 63534
diff changeset
   375
  case (Suc c)
831816778409 Removed redundant material related to primes
eberlm <eberlm@in.tum.de>
parents: 63534
diff changeset
   376
  consider x where "p^c dvd x*n" "m = p*x" | y where "p^c dvd m*y" "n = p*y"
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   377
    using prime_elem_dvd_cases [of _ "p^c", OF _ p] Suc.prems by force
63537
831816778409 Removed redundant material related to primes
eberlm <eberlm@in.tum.de>
parents: 63534
diff changeset
   378
  then show ?case
831816778409 Removed redundant material related to primes
eberlm <eberlm@in.tum.de>
parents: 63534
diff changeset
   379
  proof cases
65552
f533820e7248 theories "GCD" and "Binomial" are already included in "Main": this avoids improper imports in applications;
wenzelm
parents: 65435
diff changeset
   380
    case (1 x)
63537
831816778409 Removed redundant material related to primes
eberlm <eberlm@in.tum.de>
parents: 63534
diff changeset
   381
    with Suc.hyps[of x n] obtain a b where "a + b = c \<and> p ^ a dvd x \<and> p ^ b dvd n" by blast
831816778409 Removed redundant material related to primes
eberlm <eberlm@in.tum.de>
parents: 63534
diff changeset
   382
    with 1 have "Suc a + b = Suc c \<and> p ^ Suc a dvd m \<and> p ^ b dvd n"
831816778409 Removed redundant material related to primes
eberlm <eberlm@in.tum.de>
parents: 63534
diff changeset
   383
      by (auto intro: mult_dvd_mono)
831816778409 Removed redundant material related to primes
eberlm <eberlm@in.tum.de>
parents: 63534
diff changeset
   384
    thus ?thesis by blast
831816778409 Removed redundant material related to primes
eberlm <eberlm@in.tum.de>
parents: 63534
diff changeset
   385
  next
65552
f533820e7248 theories "GCD" and "Binomial" are already included in "Main": this avoids improper imports in applications;
wenzelm
parents: 65435
diff changeset
   386
    case (2 y)
63537
831816778409 Removed redundant material related to primes
eberlm <eberlm@in.tum.de>
parents: 63534
diff changeset
   387
    with Suc.hyps[of m y] obtain a b where "a + b = c \<and> p ^ a dvd m \<and> p ^ b dvd y" by blast
831816778409 Removed redundant material related to primes
eberlm <eberlm@in.tum.de>
parents: 63534
diff changeset
   388
    with 2 have "a + Suc b = Suc c \<and> p ^ a dvd m \<and> p ^ Suc b dvd n"
831816778409 Removed redundant material related to primes
eberlm <eberlm@in.tum.de>
parents: 63534
diff changeset
   389
      by (auto intro: mult_dvd_mono)
831816778409 Removed redundant material related to primes
eberlm <eberlm@in.tum.de>
parents: 63534
diff changeset
   390
    with Suc.hyps [of m y] show "\<exists>a b. a + b = Suc c \<and> p ^ a dvd m \<and> p ^ b dvd n"
68606
96a49db47c97 removal of smt and certain refinements
paulson <lp15@cam.ac.uk>
parents: 67051
diff changeset
   391
      by blast
63537
831816778409 Removed redundant material related to primes
eberlm <eberlm@in.tum.de>
parents: 63534
diff changeset
   392
  qed
831816778409 Removed redundant material related to primes
eberlm <eberlm@in.tum.de>
parents: 63534
diff changeset
   393
qed
831816778409 Removed redundant material related to primes
eberlm <eberlm@in.tum.de>
parents: 63534
diff changeset
   394
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   395
lemma prime_elem_power_dvd_cases:
63924
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   396
  assumes "p ^ c dvd m * n" and "a + b = Suc c" and "prime_elem p"
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   397
  shows "p ^ a dvd m \<or> p ^ b dvd n"
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   398
proof -
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   399
  from assms obtain r s
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   400
    where "r + s = c \<and> p ^ r dvd m \<and> p ^ s dvd n"
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   401
    by (blast dest: prime_elem_power_dvd_prod)
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   402
  moreover with assms have
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   403
    "a \<le> r \<or> b \<le> s" by arith
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   404
  ultimately show ?thesis by (auto intro: power_le_dvd)
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   405
qed
63534
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
   406
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   407
lemma prime_elem_not_unit' [simp]:
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   408
  "ASSUMPTION (prime_elem x) \<Longrightarrow> \<not>is_unit x"
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   409
  unfolding ASSUMPTION_def by (rule prime_elem_not_unit)
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   410
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   411
lemma prime_elem_dvd_power_iff:
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   412
  assumes "prime_elem p"
62499
4a5b81ff5992 constructive formulation of factorization
haftmann
parents: 62366
diff changeset
   413
  shows "p dvd a ^ n \<longleftrightarrow> p dvd a \<and> n > 0"
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   414
  using assms by (induct n) (auto dest: prime_elem_not_unit prime_elem_dvd_multD)
62499
4a5b81ff5992 constructive formulation of factorization
haftmann
parents: 62366
diff changeset
   415
4a5b81ff5992 constructive formulation of factorization
haftmann
parents: 62366
diff changeset
   416
lemma prime_power_dvd_multD:
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   417
  assumes "prime_elem p"
62499
4a5b81ff5992 constructive formulation of factorization
haftmann
parents: 62366
diff changeset
   418
  assumes "p ^ n dvd a * b" and "n > 0" and "\<not> p dvd a"
4a5b81ff5992 constructive formulation of factorization
haftmann
parents: 62366
diff changeset
   419
  shows "p ^ n dvd b"
65552
f533820e7248 theories "GCD" and "Binomial" are already included in "Main": this avoids improper imports in applications;
wenzelm
parents: 65435
diff changeset
   420
  using \<open>p ^ n dvd a * b\<close> and \<open>n > 0\<close>
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   421
proof (induct n arbitrary: b)
62499
4a5b81ff5992 constructive formulation of factorization
haftmann
parents: 62366
diff changeset
   422
  case 0 then show ?case by simp
4a5b81ff5992 constructive formulation of factorization
haftmann
parents: 62366
diff changeset
   423
next
4a5b81ff5992 constructive formulation of factorization
haftmann
parents: 62366
diff changeset
   424
  case (Suc n) show ?case
4a5b81ff5992 constructive formulation of factorization
haftmann
parents: 62366
diff changeset
   425
  proof (cases "n = 0")
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   426
    case True with Suc \<open>prime_elem p\<close> \<open>\<not> p dvd a\<close> show ?thesis
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   427
      by (simp add: prime_elem_dvd_mult_iff)
62499
4a5b81ff5992 constructive formulation of factorization
haftmann
parents: 62366
diff changeset
   428
  next
4a5b81ff5992 constructive formulation of factorization
haftmann
parents: 62366
diff changeset
   429
    case False then have "n > 0" by simp
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   430
    from \<open>prime_elem p\<close> have "p \<noteq> 0" by auto
62499
4a5b81ff5992 constructive formulation of factorization
haftmann
parents: 62366
diff changeset
   431
    from Suc.prems have *: "p * p ^ n dvd a * b"
4a5b81ff5992 constructive formulation of factorization
haftmann
parents: 62366
diff changeset
   432
      by simp
4a5b81ff5992 constructive formulation of factorization
haftmann
parents: 62366
diff changeset
   433
    then have "p dvd a * b"
4a5b81ff5992 constructive formulation of factorization
haftmann
parents: 62366
diff changeset
   434
      by (rule dvd_mult_left)
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   435
    with Suc \<open>prime_elem p\<close> \<open>\<not> p dvd a\<close> have "p dvd b"
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   436
      by (simp add: prime_elem_dvd_mult_iff)
63040
eb4ddd18d635 eliminated old 'def';
wenzelm
parents: 62499
diff changeset
   437
    moreover define c where "c = b div p"
62499
4a5b81ff5992 constructive formulation of factorization
haftmann
parents: 62366
diff changeset
   438
    ultimately have b: "b = p * c" by simp
4a5b81ff5992 constructive formulation of factorization
haftmann
parents: 62366
diff changeset
   439
    with * have "p * p ^ n dvd p * (a * c)"
4a5b81ff5992 constructive formulation of factorization
haftmann
parents: 62366
diff changeset
   440
      by (simp add: ac_simps)
4a5b81ff5992 constructive formulation of factorization
haftmann
parents: 62366
diff changeset
   441
    with \<open>p \<noteq> 0\<close> have "p ^ n dvd a * c"
4a5b81ff5992 constructive formulation of factorization
haftmann
parents: 62366
diff changeset
   442
      by simp
4a5b81ff5992 constructive formulation of factorization
haftmann
parents: 62366
diff changeset
   443
    with Suc.hyps \<open>n > 0\<close> have "p ^ n dvd c"
4a5b81ff5992 constructive formulation of factorization
haftmann
parents: 62366
diff changeset
   444
      by blast
4a5b81ff5992 constructive formulation of factorization
haftmann
parents: 62366
diff changeset
   445
    with \<open>p \<noteq> 0\<close> show ?thesis
4a5b81ff5992 constructive formulation of factorization
haftmann
parents: 62366
diff changeset
   446
      by (simp add: b)
4a5b81ff5992 constructive formulation of factorization
haftmann
parents: 62366
diff changeset
   447
  qed
4a5b81ff5992 constructive formulation of factorization
haftmann
parents: 62366
diff changeset
   448
qed
4a5b81ff5992 constructive formulation of factorization
haftmann
parents: 62366
diff changeset
   449
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   450
end
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   451
63924
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   452
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   453
subsection \<open>Generalized primes: normalized prime elements\<close>
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   454
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   455
context normalization_semidom
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   456
begin
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   457
63924
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   458
lemma irreducible_normalized_divisors:
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   459
  assumes "irreducible x" "y dvd x" "normalize y = y"
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   460
  shows   "y = 1 \<or> y = normalize x"
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   461
proof -
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   462
  from assms have "is_unit y \<or> x dvd y" by (auto simp: irreducible_altdef)
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   463
  thus ?thesis
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   464
  proof (elim disjE)
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   465
    assume "is_unit y"
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   466
    hence "normalize y = 1" by (simp add: is_unit_normalize)
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   467
    with assms show ?thesis by simp
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   468
  next
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   469
    assume "x dvd y"
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   470
    with \<open>y dvd x\<close> have "normalize y = normalize x" by (rule associatedI)
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   471
    with assms show ?thesis by simp
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   472
  qed
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   473
qed
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   474
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   475
lemma irreducible_normalize_iff [simp]: "irreducible (normalize x) = irreducible x"
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   476
  using irreducible_mult_unit_left[of "1 div unit_factor x" x]
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   477
  by (cases "x = 0") (simp_all add: unit_div_commute)
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   478
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   479
lemma prime_elem_normalize_iff [simp]: "prime_elem (normalize x) = prime_elem x"
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   480
  using prime_elem_mult_unit_left[of "1 div unit_factor x" x]
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   481
  by (cases "x = 0") (simp_all add: unit_div_commute)
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   482
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   483
lemma prime_elem_associated:
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   484
  assumes "prime_elem p" and "prime_elem q" and "q dvd p"
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   485
  shows "normalize q = normalize p"
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   486
using \<open>q dvd p\<close> proof (rule associatedI)
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   487
  from \<open>prime_elem q\<close> have "\<not> is_unit q"
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   488
    by (auto simp add: prime_elem_not_unit)
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   489
  with \<open>prime_elem p\<close> \<open>q dvd p\<close> show "p dvd q"
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   490
    by (blast intro: prime_elemD2)
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   491
qed
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   492
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   493
definition prime :: "'a \<Rightarrow> bool" where
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   494
  "prime p \<longleftrightarrow> prime_elem p \<and> normalize p = p"
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   495
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   496
lemma not_prime_0 [simp]: "\<not>prime 0" by (simp add: prime_def)
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   497
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   498
lemma not_prime_unit: "is_unit x \<Longrightarrow> \<not>prime x"
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   499
  using prime_elem_not_unit[of x] by (auto simp add: prime_def)
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   500
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   501
lemma not_prime_1 [simp]: "\<not>prime 1" by (simp add: not_prime_unit)
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   502
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   503
lemma primeI: "prime_elem x \<Longrightarrow> normalize x = x \<Longrightarrow> prime x"
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   504
  by (simp add: prime_def)
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   505
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   506
lemma prime_imp_prime_elem [dest]: "prime p \<Longrightarrow> prime_elem p"
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   507
  by (simp add: prime_def)
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   508
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   509
lemma normalize_prime: "prime p \<Longrightarrow> normalize p = p"
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   510
  by (simp add: prime_def)
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   511
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   512
lemma prime_normalize_iff [simp]: "prime (normalize p) \<longleftrightarrow> prime_elem p"
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   513
  by (auto simp add: prime_def)
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   514
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   515
lemma prime_power_iff:
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   516
  "prime (p ^ n) \<longleftrightarrow> prime p \<and> n = 1"
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   517
  by (auto simp: prime_def prime_elem_power_iff)
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   518
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   519
lemma prime_imp_nonzero [simp]:
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   520
  "ASSUMPTION (prime x) \<Longrightarrow> x \<noteq> 0"
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   521
  unfolding ASSUMPTION_def prime_def by auto
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   522
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   523
lemma prime_imp_not_one [simp]:
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   524
  "ASSUMPTION (prime x) \<Longrightarrow> x \<noteq> 1"
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   525
  unfolding ASSUMPTION_def by auto
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   526
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   527
lemma prime_not_unit' [simp]:
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   528
  "ASSUMPTION (prime x) \<Longrightarrow> \<not>is_unit x"
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   529
  unfolding ASSUMPTION_def prime_def by auto
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   530
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   531
lemma prime_normalize' [simp]: "ASSUMPTION (prime x) \<Longrightarrow> normalize x = x"
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   532
  unfolding ASSUMPTION_def prime_def by simp
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   533
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   534
lemma unit_factor_prime: "prime x \<Longrightarrow> unit_factor x = 1"
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   535
  using unit_factor_normalize[of x] unfolding prime_def by auto
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   536
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   537
lemma unit_factor_prime' [simp]: "ASSUMPTION (prime x) \<Longrightarrow> unit_factor x = 1"
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   538
  unfolding ASSUMPTION_def by (rule unit_factor_prime)
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   539
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   540
lemma prime_imp_prime_elem' [simp]: "ASSUMPTION (prime x) \<Longrightarrow> prime_elem x"
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   541
  by (simp add: prime_def ASSUMPTION_def)
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   542
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   543
lemma prime_dvd_multD: "prime p \<Longrightarrow> p dvd a * b \<Longrightarrow> p dvd a \<or> p dvd b"
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   544
  by (intro prime_elem_dvd_multD) simp_all
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   545
64631
7705926ee595 removed dangerous simp rule: prime computations can be excessively long
haftmann
parents: 64272
diff changeset
   546
lemma prime_dvd_mult_iff: "prime p \<Longrightarrow> p dvd a * b \<longleftrightarrow> p dvd a \<or> p dvd b"
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   547
  by (auto dest: prime_dvd_multD)
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   548
65552
f533820e7248 theories "GCD" and "Binomial" are already included in "Main": this avoids improper imports in applications;
wenzelm
parents: 65435
diff changeset
   549
lemma prime_dvd_power:
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   550
  "prime p \<Longrightarrow> p dvd x ^ n \<Longrightarrow> p dvd x"
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   551
  by (auto dest!: prime_elem_dvd_power simp: prime_def)
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   552
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   553
lemma prime_dvd_power_iff:
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   554
  "prime p \<Longrightarrow> n > 0 \<Longrightarrow> p dvd x ^ n \<longleftrightarrow> p dvd x"
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   555
  by (subst prime_elem_dvd_power_iff) simp_all
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   556
63830
2ea3725a34bd msetsum -> set_mset, msetprod -> prod_mset
nipkow
parents: 63793
diff changeset
   557
lemma prime_dvd_prod_mset_iff: "prime p \<Longrightarrow> p dvd prod_mset A \<longleftrightarrow> (\<exists>x. x \<in># A \<and> p dvd x)"
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   558
  by (induction A) (simp_all add: prime_elem_dvd_mult_iff prime_imp_prime_elem, blast+)
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   559
66276
acc3b7dd0b21 More material on powers for HOL-Computational_Algebra/HOL-Number_Theory
eberlm <eberlm@in.tum.de>
parents: 65552
diff changeset
   560
lemma prime_dvd_prod_iff: "finite A \<Longrightarrow> prime p \<Longrightarrow> p dvd prod f A \<longleftrightarrow> (\<exists>x\<in>A. p dvd f x)"
acc3b7dd0b21 More material on powers for HOL-Computational_Algebra/HOL-Number_Theory
eberlm <eberlm@in.tum.de>
parents: 65552
diff changeset
   561
  by (auto simp: prime_dvd_prod_mset_iff prod_unfold_prod_mset)
acc3b7dd0b21 More material on powers for HOL-Computational_Algebra/HOL-Number_Theory
eberlm <eberlm@in.tum.de>
parents: 65552
diff changeset
   562
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   563
lemma primes_dvd_imp_eq:
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   564
  assumes "prime p" "prime q" "p dvd q"
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   565
  shows   "p = q"
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   566
proof -
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   567
  from assms have "irreducible q" by (simp add: prime_elem_imp_irreducible prime_def)
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   568
  from irreducibleD'[OF this \<open>p dvd q\<close>] assms have "q dvd p" by simp
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   569
  with \<open>p dvd q\<close> have "normalize p = normalize q" by (rule associatedI)
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   570
  with assms show "p = q" by simp
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   571
qed
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   572
63830
2ea3725a34bd msetsum -> set_mset, msetprod -> prod_mset
nipkow
parents: 63793
diff changeset
   573
lemma prime_dvd_prod_mset_primes_iff:
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   574
  assumes "prime p" "\<And>q. q \<in># A \<Longrightarrow> prime q"
63830
2ea3725a34bd msetsum -> set_mset, msetprod -> prod_mset
nipkow
parents: 63793
diff changeset
   575
  shows   "p dvd prod_mset A \<longleftrightarrow> p \<in># A"
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   576
proof -
63830
2ea3725a34bd msetsum -> set_mset, msetprod -> prod_mset
nipkow
parents: 63793
diff changeset
   577
  from assms(1) have "p dvd prod_mset A \<longleftrightarrow> (\<exists>x. x \<in># A \<and> p dvd x)" by (rule prime_dvd_prod_mset_iff)
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   578
  also from assms have "\<dots> \<longleftrightarrow> p \<in># A" by (auto dest: primes_dvd_imp_eq)
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   579
  finally show ?thesis .
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   580
qed
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   581
63830
2ea3725a34bd msetsum -> set_mset, msetprod -> prod_mset
nipkow
parents: 63793
diff changeset
   582
lemma prod_mset_primes_dvd_imp_subset:
2ea3725a34bd msetsum -> set_mset, msetprod -> prod_mset
nipkow
parents: 63793
diff changeset
   583
  assumes "prod_mset A dvd prod_mset B" "\<And>p. p \<in># A \<Longrightarrow> prime p" "\<And>p. p \<in># B \<Longrightarrow> prime p"
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   584
  shows   "A \<subseteq># B"
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   585
using assms
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   586
proof (induction A arbitrary: B)
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   587
  case empty
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   588
  thus ?case by simp
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   589
next
63793
e68a0b651eb5 add_mset constructor in multisets
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents: 63633
diff changeset
   590
  case (add p A B)
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   591
  hence p: "prime p" by simp
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   592
  define B' where "B' = B - {#p#}"
63830
2ea3725a34bd msetsum -> set_mset, msetprod -> prod_mset
nipkow
parents: 63793
diff changeset
   593
  from add.prems have "p dvd prod_mset B" by (simp add: dvd_mult_left)
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   594
  with add.prems have "p \<in># B"
63830
2ea3725a34bd msetsum -> set_mset, msetprod -> prod_mset
nipkow
parents: 63793
diff changeset
   595
    by (subst (asm) (2) prime_dvd_prod_mset_primes_iff) simp_all
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   596
  hence B: "B = B' + {#p#}" by (simp add: B'_def)
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   597
  from add.prems p have "A \<subseteq># B'" by (intro add.IH) (simp_all add: B)
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   598
  thus ?case by (simp add: B)
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   599
qed
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   600
63830
2ea3725a34bd msetsum -> set_mset, msetprod -> prod_mset
nipkow
parents: 63793
diff changeset
   601
lemma prod_mset_dvd_prod_mset_primes_iff:
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   602
  assumes "\<And>x. x \<in># A \<Longrightarrow> prime x" "\<And>x. x \<in># B \<Longrightarrow> prime x"
63830
2ea3725a34bd msetsum -> set_mset, msetprod -> prod_mset
nipkow
parents: 63793
diff changeset
   603
  shows   "prod_mset A dvd prod_mset B \<longleftrightarrow> A \<subseteq># B"
2ea3725a34bd msetsum -> set_mset, msetprod -> prod_mset
nipkow
parents: 63793
diff changeset
   604
  using assms by (auto intro: prod_mset_subset_imp_dvd prod_mset_primes_dvd_imp_subset)
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   605
63830
2ea3725a34bd msetsum -> set_mset, msetprod -> prod_mset
nipkow
parents: 63793
diff changeset
   606
lemma is_unit_prod_mset_primes_iff:
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   607
  assumes "\<And>x. x \<in># A \<Longrightarrow> prime x"
63830
2ea3725a34bd msetsum -> set_mset, msetprod -> prod_mset
nipkow
parents: 63793
diff changeset
   608
  shows   "is_unit (prod_mset A) \<longleftrightarrow> A = {#}"
63924
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   609
  by (auto simp add: is_unit_prod_mset_iff)
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   610
    (meson all_not_in_conv assms not_prime_unit set_mset_eq_empty_iff)
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   611
63830
2ea3725a34bd msetsum -> set_mset, msetprod -> prod_mset
nipkow
parents: 63793
diff changeset
   612
lemma prod_mset_primes_irreducible_imp_prime:
2ea3725a34bd msetsum -> set_mset, msetprod -> prod_mset
nipkow
parents: 63793
diff changeset
   613
  assumes irred: "irreducible (prod_mset A)"
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   614
  assumes A: "\<And>x. x \<in># A \<Longrightarrow> prime x"
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   615
  assumes B: "\<And>x. x \<in># B \<Longrightarrow> prime x"
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   616
  assumes C: "\<And>x. x \<in># C \<Longrightarrow> prime x"
63830
2ea3725a34bd msetsum -> set_mset, msetprod -> prod_mset
nipkow
parents: 63793
diff changeset
   617
  assumes dvd: "prod_mset A dvd prod_mset B * prod_mset C"
2ea3725a34bd msetsum -> set_mset, msetprod -> prod_mset
nipkow
parents: 63793
diff changeset
   618
  shows   "prod_mset A dvd prod_mset B \<or> prod_mset A dvd prod_mset C"
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   619
proof -
63830
2ea3725a34bd msetsum -> set_mset, msetprod -> prod_mset
nipkow
parents: 63793
diff changeset
   620
  from dvd have "prod_mset A dvd prod_mset (B + C)"
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   621
    by simp
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   622
  with A B C have subset: "A \<subseteq># B + C"
63830
2ea3725a34bd msetsum -> set_mset, msetprod -> prod_mset
nipkow
parents: 63793
diff changeset
   623
    by (subst (asm) prod_mset_dvd_prod_mset_primes_iff) auto
63919
9aed2da07200 # after multiset intersection and union symbol
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents: 63905
diff changeset
   624
  define A1 and A2 where "A1 = A \<inter># B" and "A2 = A - A1"
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   625
  have "A = A1 + A2" unfolding A1_def A2_def
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   626
    by (rule sym, intro subset_mset.add_diff_inverse) simp_all
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   627
  from subset have "A1 \<subseteq># B" "A2 \<subseteq># C"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   628
    by (auto simp: A1_def A2_def Multiset.subset_eq_diff_conv Multiset.union_commute)
63830
2ea3725a34bd msetsum -> set_mset, msetprod -> prod_mset
nipkow
parents: 63793
diff changeset
   629
  from \<open>A = A1 + A2\<close> have "prod_mset A = prod_mset A1 * prod_mset A2" by simp
2ea3725a34bd msetsum -> set_mset, msetprod -> prod_mset
nipkow
parents: 63793
diff changeset
   630
  from irred and this have "is_unit (prod_mset A1) \<or> is_unit (prod_mset A2)"
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   631
    by (rule irreducibleD)
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   632
  with A have "A1 = {#} \<or> A2 = {#}" unfolding A1_def A2_def
63830
2ea3725a34bd msetsum -> set_mset, msetprod -> prod_mset
nipkow
parents: 63793
diff changeset
   633
    by (subst (asm) (1 2) is_unit_prod_mset_primes_iff) (auto dest: Multiset.in_diffD)
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   634
  with dvd \<open>A = A1 + A2\<close> \<open>A1 \<subseteq># B\<close> \<open>A2 \<subseteq># C\<close> show ?thesis
63830
2ea3725a34bd msetsum -> set_mset, msetprod -> prod_mset
nipkow
parents: 63793
diff changeset
   635
    by (auto intro: prod_mset_subset_imp_dvd)
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   636
qed
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   637
63830
2ea3725a34bd msetsum -> set_mset, msetprod -> prod_mset
nipkow
parents: 63793
diff changeset
   638
lemma prod_mset_primes_finite_divisor_powers:
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   639
  assumes A: "\<And>x. x \<in># A \<Longrightarrow> prime x"
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   640
  assumes B: "\<And>x. x \<in># B \<Longrightarrow> prime x"
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   641
  assumes "A \<noteq> {#}"
63830
2ea3725a34bd msetsum -> set_mset, msetprod -> prod_mset
nipkow
parents: 63793
diff changeset
   642
  shows   "finite {n. prod_mset A ^ n dvd prod_mset B}"
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   643
proof -
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   644
  from \<open>A \<noteq> {#}\<close> obtain x where x: "x \<in># A" by blast
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   645
  define m where "m = count B x"
63830
2ea3725a34bd msetsum -> set_mset, msetprod -> prod_mset
nipkow
parents: 63793
diff changeset
   646
  have "{n. prod_mset A ^ n dvd prod_mset B} \<subseteq> {..m}"
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   647
  proof safe
63830
2ea3725a34bd msetsum -> set_mset, msetprod -> prod_mset
nipkow
parents: 63793
diff changeset
   648
    fix n assume dvd: "prod_mset A ^ n dvd prod_mset B"
2ea3725a34bd msetsum -> set_mset, msetprod -> prod_mset
nipkow
parents: 63793
diff changeset
   649
    from x have "x ^ n dvd prod_mset A ^ n" by (intro dvd_power_same dvd_prod_mset)
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   650
    also note dvd
63830
2ea3725a34bd msetsum -> set_mset, msetprod -> prod_mset
nipkow
parents: 63793
diff changeset
   651
    also have "x ^ n = prod_mset (replicate_mset n x)" by simp
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   652
    finally have "replicate_mset n x \<subseteq># B"
63830
2ea3725a34bd msetsum -> set_mset, msetprod -> prod_mset
nipkow
parents: 63793
diff changeset
   653
      by (rule prod_mset_primes_dvd_imp_subset) (insert A B x, simp_all split: if_splits)
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   654
    thus "n \<le> m" by (simp add: count_le_replicate_mset_subset_eq m_def)
60804
080a979a985b formal class for factorial (semi)rings
haftmann
parents:
diff changeset
   655
  qed
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   656
  moreover have "finite {..m}" by simp
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   657
  ultimately show ?thesis by (rule finite_subset)
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   658
qed
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   659
63924
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   660
end
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   661
63924
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   662
67051
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66938
diff changeset
   663
subsection \<open>In a semiring with GCD, each irreducible element is a prime element\<close>
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   664
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   665
context semiring_gcd
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   666
begin
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   667
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   668
lemma irreducible_imp_prime_elem_gcd:
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   669
  assumes "irreducible x"
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   670
  shows   "prime_elem x"
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   671
proof (rule prime_elemI)
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   672
  fix a b assume "x dvd a * b"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   673
  from dvd_productE[OF this] obtain y z where yz: "x = y * z" "y dvd a" "z dvd b" .
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   674
  from \<open>irreducible x\<close> and \<open>x = y * z\<close> have "is_unit y \<or> is_unit z" by (rule irreducibleD)
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   675
  with yz show "x dvd a \<or> x dvd b"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   676
    by (auto simp: mult_unit_dvd_iff mult_unit_dvd_iff')
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   677
qed (insert assms, auto simp: irreducible_not_unit)
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   678
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   679
lemma prime_elem_imp_coprime:
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   680
  assumes "prime_elem p" "\<not>p dvd n"
63534
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
   681
  shows   "coprime p n"
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
   682
proof (rule coprimeI)
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
   683
  fix d assume "d dvd p" "d dvd n"
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
   684
  show "is_unit d"
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
   685
  proof (rule ccontr)
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
   686
    assume "\<not>is_unit d"
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   687
    from \<open>prime_elem p\<close> and \<open>d dvd p\<close> and this have "p dvd d"
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   688
      by (rule prime_elemD2)
63534
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
   689
    from this and \<open>d dvd n\<close> have "p dvd n" by (rule dvd_trans)
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
   690
    with \<open>\<not>p dvd n\<close> show False by contradiction
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
   691
  qed
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
   692
qed
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
   693
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   694
lemma prime_imp_coprime:
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   695
  assumes "prime p" "\<not>p dvd n"
63534
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
   696
  shows   "coprime p n"
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   697
  using assms by (simp add: prime_elem_imp_coprime)
63534
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
   698
65552
f533820e7248 theories "GCD" and "Binomial" are already included in "Main": this avoids improper imports in applications;
wenzelm
parents: 65435
diff changeset
   699
lemma prime_elem_imp_power_coprime:
67051
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66938
diff changeset
   700
  "prime_elem p \<Longrightarrow> \<not> p dvd a \<Longrightarrow> coprime a (p ^ m)"
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66938
diff changeset
   701
  by (cases "m > 0") (auto dest: prime_elem_imp_coprime simp add: ac_simps)
63534
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
   702
65552
f533820e7248 theories "GCD" and "Binomial" are already included in "Main": this avoids improper imports in applications;
wenzelm
parents: 65435
diff changeset
   703
lemma prime_imp_power_coprime:
67051
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66938
diff changeset
   704
  "prime p \<Longrightarrow> \<not> p dvd a \<Longrightarrow> coprime a (p ^ m)"
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66938
diff changeset
   705
  by (rule prime_elem_imp_power_coprime) simp_all
63534
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
   706
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   707
lemma prime_elem_divprod_pow:
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   708
  assumes p: "prime_elem p" and ab: "coprime a b" and pab: "p^n dvd a * b"
63534
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
   709
  shows   "p^n dvd a \<or> p^n dvd b"
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
   710
  using assms
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
   711
proof -
67051
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66938
diff changeset
   712
  from p have "\<not> is_unit p"
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66938
diff changeset
   713
    by simp
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66938
diff changeset
   714
  with ab p have "\<not> p dvd a \<or> \<not> p dvd b"
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66938
diff changeset
   715
    using not_coprimeI by blast
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66938
diff changeset
   716
  with p have "coprime (p ^ n) a \<or> coprime (p ^ n) b"
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66938
diff changeset
   717
    by (auto dest: prime_elem_imp_power_coprime simp add: ac_simps)
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66938
diff changeset
   718
  with pab show ?thesis
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66938
diff changeset
   719
    by (auto simp add: coprime_dvd_mult_left_iff coprime_dvd_mult_right_iff)
63534
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
   720
qed
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
   721
65552
f533820e7248 theories "GCD" and "Binomial" are already included in "Main": this avoids improper imports in applications;
wenzelm
parents: 65435
diff changeset
   722
lemma primes_coprime:
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   723
  "prime p \<Longrightarrow> prime q \<Longrightarrow> p \<noteq> q \<Longrightarrow> coprime p q"
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   724
  using prime_imp_coprime primes_dvd_imp_eq by blast
63534
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
   725
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   726
end
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   727
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   728
63924
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   729
subsection \<open>Factorial semirings: algebraic structures with unique prime factorizations\<close>
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   730
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   731
class factorial_semiring = normalization_semidom +
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   732
  assumes prime_factorization_exists:
71398
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
   733
    "x \<noteq> 0 \<Longrightarrow> \<exists>A. (\<forall>x. x \<in># A \<longrightarrow> prime_elem x) \<and> normalize (prod_mset A) = normalize x"
63924
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   734
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   735
text \<open>Alternative characterization\<close>
65552
f533820e7248 theories "GCD" and "Binomial" are already included in "Main": this avoids improper imports in applications;
wenzelm
parents: 65435
diff changeset
   736
63924
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   737
lemma (in normalization_semidom) factorial_semiring_altI_aux:
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   738
  assumes finite_divisors: "\<And>x. x \<noteq> 0 \<Longrightarrow> finite {y. y dvd x \<and> normalize y = y}"
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   739
  assumes irreducible_imp_prime_elem: "\<And>x. irreducible x \<Longrightarrow> prime_elem x"
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   740
  assumes "x \<noteq> 0"
71398
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
   741
  shows   "\<exists>A. (\<forall>x. x \<in># A \<longrightarrow> prime_elem x) \<and> normalize (prod_mset A) = normalize x"
63924
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   742
using \<open>x \<noteq> 0\<close>
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   743
proof (induction "card {b. b dvd x \<and> normalize b = b}" arbitrary: x rule: less_induct)
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   744
  case (less a)
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   745
  let ?fctrs = "\<lambda>a. {b. b dvd a \<and> normalize b = b}"
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   746
  show ?case
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   747
  proof (cases "is_unit a")
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   748
    case True
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   749
    thus ?thesis by (intro exI[of _ "{#}"]) (auto simp: is_unit_normalize)
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   750
  next
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   751
    case False
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   752
    show ?thesis
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   753
    proof (cases "\<exists>b. b dvd a \<and> \<not>is_unit b \<and> \<not>a dvd b")
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   754
      case False
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   755
      with \<open>\<not>is_unit a\<close> less.prems have "irreducible a" by (auto simp: irreducible_altdef)
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   756
      hence "prime_elem a" by (rule irreducible_imp_prime_elem)
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   757
      thus ?thesis by (intro exI[of _ "{#normalize a#}"]) auto
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   758
    next
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   759
      case True
74362
0135a0c77b64 tuned proofs --- avoid 'guess';
wenzelm
parents: 73270
diff changeset
   760
      then obtain b where b: "b dvd a" "\<not> is_unit b" "\<not> a dvd b" by auto
63924
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   761
      from b have "?fctrs b \<subseteq> ?fctrs a" by (auto intro: dvd_trans)
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   762
      moreover from b have "normalize a \<notin> ?fctrs b" "normalize a \<in> ?fctrs a" by simp_all
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   763
      hence "?fctrs b \<noteq> ?fctrs a" by blast
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   764
      ultimately have "?fctrs b \<subset> ?fctrs a" by (subst subset_not_subset_eq) blast
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   765
      with finite_divisors[OF \<open>a \<noteq> 0\<close>] have "card (?fctrs b) < card (?fctrs a)"
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   766
        by (rule psubset_card_mono)
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   767
      moreover from \<open>a \<noteq> 0\<close> b have "b \<noteq> 0" by auto
71398
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
   768
      ultimately have "\<exists>A. (\<forall>x. x \<in># A \<longrightarrow> prime_elem x) \<and> normalize (prod_mset A) = normalize b"
63924
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   769
        by (intro less) auto
74362
0135a0c77b64 tuned proofs --- avoid 'guess';
wenzelm
parents: 73270
diff changeset
   770
      then obtain A where A: "(\<forall>x. x \<in># A \<longrightarrow> prime_elem x) \<and> normalize (\<Prod>\<^sub># A) = normalize b"
0135a0c77b64 tuned proofs --- avoid 'guess';
wenzelm
parents: 73270
diff changeset
   771
        by auto
63924
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   772
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   773
      define c where "c = a div b"
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   774
      from b have c: "a = b * c" by (simp add: c_def)
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   775
      from less.prems c have "c \<noteq> 0" by auto
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   776
      from b c have "?fctrs c \<subseteq> ?fctrs a" by (auto intro: dvd_trans)
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   777
      moreover have "normalize a \<notin> ?fctrs c"
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   778
      proof safe
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   779
        assume "normalize a dvd c"
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   780
        hence "b * c dvd 1 * c" by (simp add: c)
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   781
        hence "b dvd 1" by (subst (asm) dvd_times_right_cancel_iff) fact+
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   782
        with b show False by simp
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   783
      qed
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   784
      with \<open>normalize a \<in> ?fctrs a\<close> have "?fctrs a \<noteq> ?fctrs c" by blast
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   785
      ultimately have "?fctrs c \<subset> ?fctrs a" by (subst subset_not_subset_eq) blast
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   786
      with finite_divisors[OF \<open>a \<noteq> 0\<close>] have "card (?fctrs c) < card (?fctrs a)"
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   787
        by (rule psubset_card_mono)
71398
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
   788
      with \<open>c \<noteq> 0\<close> have "\<exists>A. (\<forall>x. x \<in># A \<longrightarrow> prime_elem x) \<and> normalize (prod_mset A) = normalize c"
63924
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   789
        by (intro less) auto
74362
0135a0c77b64 tuned proofs --- avoid 'guess';
wenzelm
parents: 73270
diff changeset
   790
      then obtain B where B: "(\<forall>x. x \<in># B \<longrightarrow> prime_elem x) \<and> normalize (\<Prod>\<^sub># B) = normalize c"
0135a0c77b64 tuned proofs --- avoid 'guess';
wenzelm
parents: 73270
diff changeset
   791
        by auto
63924
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   792
71398
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
   793
      show ?thesis
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
   794
      proof (rule exI[of _ "A + B"]; safe)
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
   795
        have "normalize (prod_mset (A + B)) =
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
   796
                normalize (normalize (prod_mset A) * normalize (prod_mset B))"
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
   797
          by simp
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
   798
        also have "\<dots> = normalize (b * c)"
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
   799
          by (simp only: A B) auto
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
   800
        also have "b * c = a"
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
   801
          using c by simp
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
   802
        finally show "normalize (prod_mset (A + B)) = normalize a" .
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
   803
      next
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
   804
      qed (use A B in auto)
63924
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   805
    qed
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   806
  qed
65552
f533820e7248 theories "GCD" and "Binomial" are already included in "Main": this avoids improper imports in applications;
wenzelm
parents: 65435
diff changeset
   807
qed
63924
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   808
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   809
lemma factorial_semiring_altI:
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   810
  assumes finite_divisors: "\<And>x::'a. x \<noteq> 0 \<Longrightarrow> finite {y. y dvd x \<and> normalize y = y}"
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   811
  assumes irreducible_imp_prime: "\<And>x::'a. irreducible x \<Longrightarrow> prime_elem x"
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   812
  shows   "OFCLASS('a :: normalization_semidom, factorial_semiring_class)"
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   813
  by intro_classes (rule factorial_semiring_altI_aux[OF assms])
65552
f533820e7248 theories "GCD" and "Binomial" are already included in "Main": this avoids improper imports in applications;
wenzelm
parents: 65435
diff changeset
   814
63924
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   815
text \<open>Properties\<close>
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   816
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
   817
context factorial_semiring
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   818
begin
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   819
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   820
lemma prime_factorization_exists':
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   821
  assumes "x \<noteq> 0"
71398
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
   822
  obtains A where "\<And>x. x \<in># A \<Longrightarrow> prime x" "normalize (prod_mset A) = normalize x"
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   823
proof -
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   824
  from prime_factorization_exists[OF assms] obtain A
71398
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
   825
    where A: "\<And>x. x \<in># A \<Longrightarrow> prime_elem x" "normalize (prod_mset A) = normalize x" by blast
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   826
  define A' where "A' = image_mset normalize A"
71398
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
   827
  have "normalize (prod_mset A') = normalize (prod_mset A)"
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
   828
    by (simp add: A'_def normalize_prod_mset_normalize)
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   829
  also note A(2)
71398
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
   830
  finally have "normalize (prod_mset A') = normalize x" by simp
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   831
  moreover from A(1) have "\<forall>x. x \<in># A' \<longrightarrow> prime x" by (auto simp: prime_def A'_def)
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   832
  ultimately show ?thesis by (intro that[of A']) blast
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   833
qed
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   834
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   835
lemma irreducible_imp_prime_elem:
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   836
  assumes "irreducible x"
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   837
  shows   "prime_elem x"
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   838
proof (rule prime_elemI)
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   839
  fix a b assume dvd: "x dvd a * b"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   840
  from assms have "x \<noteq> 0" by auto
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   841
  show "x dvd a \<or> x dvd b"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   842
  proof (cases "a = 0 \<or> b = 0")
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   843
    case False
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   844
    hence "a \<noteq> 0" "b \<noteq> 0" by blast+
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   845
    note nz = \<open>x \<noteq> 0\<close> this
74362
0135a0c77b64 tuned proofs --- avoid 'guess';
wenzelm
parents: 73270
diff changeset
   846
    from nz[THEN prime_factorization_exists'] obtain A B C
0135a0c77b64 tuned proofs --- avoid 'guess';
wenzelm
parents: 73270
diff changeset
   847
      where ABC:
0135a0c77b64 tuned proofs --- avoid 'guess';
wenzelm
parents: 73270
diff changeset
   848
        "\<And>z. z \<in># A \<Longrightarrow> prime z"
0135a0c77b64 tuned proofs --- avoid 'guess';
wenzelm
parents: 73270
diff changeset
   849
        "normalize (\<Prod>\<^sub># A) = normalize x"
0135a0c77b64 tuned proofs --- avoid 'guess';
wenzelm
parents: 73270
diff changeset
   850
        "\<And>z. z \<in># B \<Longrightarrow> prime z"
0135a0c77b64 tuned proofs --- avoid 'guess';
wenzelm
parents: 73270
diff changeset
   851
        "normalize (\<Prod>\<^sub># B) = normalize a"
0135a0c77b64 tuned proofs --- avoid 'guess';
wenzelm
parents: 73270
diff changeset
   852
        "\<And>z. z \<in># C \<Longrightarrow> prime z"
0135a0c77b64 tuned proofs --- avoid 'guess';
wenzelm
parents: 73270
diff changeset
   853
        "normalize (\<Prod>\<^sub># C) = normalize b"
0135a0c77b64 tuned proofs --- avoid 'guess';
wenzelm
parents: 73270
diff changeset
   854
      by this blast
71398
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
   855
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
   856
    have "irreducible (prod_mset A)"
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
   857
      by (subst irreducible_cong[OF ABC(2)]) fact
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
   858
    moreover have "normalize (prod_mset A) dvd
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
   859
                     normalize (normalize (prod_mset B) * normalize (prod_mset C))"
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
   860
      unfolding ABC using dvd by simp
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
   861
    hence "prod_mset A dvd prod_mset B * prod_mset C"
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
   862
      unfolding normalize_mult_normalize_left normalize_mult_normalize_right by simp
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
   863
    ultimately have "prod_mset A dvd prod_mset B \<or> prod_mset A dvd prod_mset C"
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
   864
      by (intro prod_mset_primes_irreducible_imp_prime) (use ABC in auto)
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
   865
    hence "normalize (prod_mset A) dvd normalize (prod_mset B) \<or>
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
   866
           normalize (prod_mset A) dvd normalize (prod_mset C)" by simp
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
   867
    thus ?thesis unfolding ABC by simp
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   868
  qed auto
74362
0135a0c77b64 tuned proofs --- avoid 'guess';
wenzelm
parents: 73270
diff changeset
   869
qed (use assms in \<open>simp_all add: irreducible_def\<close>)
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   870
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   871
lemma finite_divisor_powers:
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   872
  assumes "y \<noteq> 0" "\<not>is_unit x"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   873
  shows   "finite {n. x ^ n dvd y}"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   874
proof (cases "x = 0")
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   875
  case True
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   876
  with assms have "{n. x ^ n dvd y} = {0}" by (auto simp: power_0_left)
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   877
  thus ?thesis by simp
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   878
next
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   879
  case False
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   880
  note nz = this \<open>y \<noteq> 0\<close>
74362
0135a0c77b64 tuned proofs --- avoid 'guess';
wenzelm
parents: 73270
diff changeset
   881
  from nz[THEN prime_factorization_exists'] obtain A B
0135a0c77b64 tuned proofs --- avoid 'guess';
wenzelm
parents: 73270
diff changeset
   882
    where AB:
0135a0c77b64 tuned proofs --- avoid 'guess';
wenzelm
parents: 73270
diff changeset
   883
      "\<And>z. z \<in># A \<Longrightarrow> prime z"
0135a0c77b64 tuned proofs --- avoid 'guess';
wenzelm
parents: 73270
diff changeset
   884
      "normalize (\<Prod>\<^sub># A) = normalize x"
0135a0c77b64 tuned proofs --- avoid 'guess';
wenzelm
parents: 73270
diff changeset
   885
      "\<And>z. z \<in># B \<Longrightarrow> prime z"
0135a0c77b64 tuned proofs --- avoid 'guess';
wenzelm
parents: 73270
diff changeset
   886
      "normalize (\<Prod>\<^sub># B) = normalize y"
0135a0c77b64 tuned proofs --- avoid 'guess';
wenzelm
parents: 73270
diff changeset
   887
    by this blast
0135a0c77b64 tuned proofs --- avoid 'guess';
wenzelm
parents: 73270
diff changeset
   888
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   889
  from AB assms have "A \<noteq> {#}" by (auto simp: normalize_1_iff)
63830
2ea3725a34bd msetsum -> set_mset, msetprod -> prod_mset
nipkow
parents: 63793
diff changeset
   890
  from AB(2,4) prod_mset_primes_finite_divisor_powers [of A B, OF AB(1,3) this]
71398
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
   891
    have "finite {n. prod_mset A ^ n dvd prod_mset B}" by simp
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
   892
  also have "{n. prod_mset A ^ n dvd prod_mset B} =
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
   893
             {n. normalize (normalize (prod_mset A) ^ n) dvd normalize (prod_mset B)}"
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
   894
    unfolding normalize_power_normalize by simp
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
   895
  also have "\<dots> = {n. x ^ n dvd y}"
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
   896
    unfolding AB unfolding normalize_power_normalize by simp
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
   897
  finally show ?thesis .
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   898
qed
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   899
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   900
lemma finite_prime_divisors:
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   901
  assumes "x \<noteq> 0"
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   902
  shows   "finite {p. prime p \<and> p dvd x}"
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   903
proof -
74362
0135a0c77b64 tuned proofs --- avoid 'guess';
wenzelm
parents: 73270
diff changeset
   904
  from prime_factorization_exists'[OF assms] obtain A
0135a0c77b64 tuned proofs --- avoid 'guess';
wenzelm
parents: 73270
diff changeset
   905
    where A: "\<And>z. z \<in># A \<Longrightarrow> prime z" "normalize (\<Prod>\<^sub># A) = normalize x" by this blast
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   906
  have "{p. prime p \<and> p dvd x} \<subseteq> set_mset A"
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   907
  proof safe
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   908
    fix p assume p: "prime p" and dvd: "p dvd x"
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   909
    from dvd have "p dvd normalize x" by simp
71398
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
   910
    also from A have "normalize x = normalize (prod_mset A)" by simp
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
   911
    finally have "p dvd prod_mset A"
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
   912
      by simp
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
   913
    thus  "p \<in># A" using p A
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
   914
      by (subst (asm) prime_dvd_prod_mset_primes_iff)
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   915
  qed
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   916
  moreover have "finite (set_mset A)" by simp
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   917
  ultimately show ?thesis by (rule finite_subset)
60804
080a979a985b formal class for factorial (semi)rings
haftmann
parents:
diff changeset
   918
qed
080a979a985b formal class for factorial (semi)rings
haftmann
parents:
diff changeset
   919
73127
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
   920
lemma infinite_unit_divisor_powers:
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
   921
 assumes "y \<noteq> 0"
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
   922
 assumes "is_unit x"
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
   923
 shows "infinite {n. x^n dvd y}"
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
   924
proof -
74885
2df334453c4c isabelle update_cartouches;
wenzelm
parents: 74542
diff changeset
   925
 from \<open>is_unit x\<close> have "is_unit (x^n)" for n
73127
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
   926
   using is_unit_power_iff by auto
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
   927
 hence "x^n dvd y" for n
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
   928
   by auto
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
   929
 hence "{n. x^n dvd y} = UNIV"
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
   930
   by auto
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
   931
 thus ?thesis
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
   932
   by auto
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
   933
qed
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
   934
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
   935
corollary is_unit_iff_infinite_divisor_powers:
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
   936
 assumes "y \<noteq> 0"
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
   937
 shows "is_unit x \<longleftrightarrow> infinite {n. x^n dvd y}"
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
   938
 using infinite_unit_divisor_powers finite_divisor_powers assms by auto
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
   939
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   940
lemma prime_elem_iff_irreducible: "prime_elem x \<longleftrightarrow> irreducible x"
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   941
  by (blast intro: irreducible_imp_prime_elem prime_elem_imp_irreducible)
62499
4a5b81ff5992 constructive formulation of factorization
haftmann
parents: 62366
diff changeset
   942
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   943
lemma prime_divisor_exists:
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   944
  assumes "a \<noteq> 0" "\<not>is_unit a"
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   945
  shows   "\<exists>b. b dvd a \<and> prime b"
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   946
proof -
74362
0135a0c77b64 tuned proofs --- avoid 'guess';
wenzelm
parents: 73270
diff changeset
   947
  from prime_factorization_exists'[OF assms(1)]
0135a0c77b64 tuned proofs --- avoid 'guess';
wenzelm
parents: 73270
diff changeset
   948
  obtain A where A: "\<And>z. z \<in># A \<Longrightarrow> prime z" "normalize (\<Prod>\<^sub># A) = normalize a"
0135a0c77b64 tuned proofs --- avoid 'guess';
wenzelm
parents: 73270
diff changeset
   949
    by this blast
0135a0c77b64 tuned proofs --- avoid 'guess';
wenzelm
parents: 73270
diff changeset
   950
  with assms have "A \<noteq> {#}" by auto
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   951
  then obtain x where "x \<in># A" by blast
71398
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
   952
  with A(1) have *: "x dvd normalize (prod_mset A)" "prime x"
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
   953
    by (auto simp: dvd_prod_mset)
74362
0135a0c77b64 tuned proofs --- avoid 'guess';
wenzelm
parents: 73270
diff changeset
   954
  hence "x dvd a" by (simp add: A(2))
63539
70d4d9e5707b tuned proofs -- avoid improper use of "this";
wenzelm
parents: 63498
diff changeset
   955
  with * show ?thesis by blast
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   956
qed
60804
080a979a985b formal class for factorial (semi)rings
haftmann
parents:
diff changeset
   957
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   958
lemma prime_divisors_induct [case_names zero unit factor]:
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   959
  assumes "P 0" "\<And>x. is_unit x \<Longrightarrow> P x" "\<And>p x. prime p \<Longrightarrow> P x \<Longrightarrow> P (p * x)"
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   960
  shows   "P x"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   961
proof (cases "x = 0")
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   962
  case False
74362
0135a0c77b64 tuned proofs --- avoid 'guess';
wenzelm
parents: 73270
diff changeset
   963
  from prime_factorization_exists'[OF this]
0135a0c77b64 tuned proofs --- avoid 'guess';
wenzelm
parents: 73270
diff changeset
   964
  obtain A where A: "\<And>z. z \<in># A \<Longrightarrow> prime z" "normalize (\<Prod>\<^sub># A) = normalize x"
0135a0c77b64 tuned proofs --- avoid 'guess';
wenzelm
parents: 73270
diff changeset
   965
    by this blast
71398
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
   966
  from A obtain u where u: "is_unit u" "x = u * prod_mset A"
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
   967
    by (elim associatedE2)
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
   968
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
   969
  from A(1) have "P (u * prod_mset A)"
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   970
  proof (induction A)
63793
e68a0b651eb5 add_mset constructor in multisets
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents: 63633
diff changeset
   971
    case (add p A)
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   972
    from add.prems have "prime p" by simp
71398
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
   973
    moreover from add.prems have "P (u * prod_mset A)" by (intro add.IH) simp_all
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
   974
    ultimately have "P (p * (u * prod_mset A))" by (rule assms(3))
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   975
    thus ?case by (simp add: mult_ac)
71398
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
   976
  qed (simp_all add: assms False u)
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
   977
  with A u show ?thesis by simp
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   978
qed (simp_all add: assms(1))
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   979
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   980
lemma no_prime_divisors_imp_unit:
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   981
  assumes "a \<noteq> 0" "\<And>b. b dvd a \<Longrightarrow> normalize b = b \<Longrightarrow> \<not> prime_elem b"
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   982
  shows "is_unit a"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   983
proof (rule ccontr)
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   984
  assume "\<not>is_unit a"
74362
0135a0c77b64 tuned proofs --- avoid 'guess';
wenzelm
parents: 73270
diff changeset
   985
  from prime_divisor_exists[OF assms(1) this] obtain b where "b dvd a" "prime b" by auto
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   986
  with assms(2)[of b] show False by (simp add: prime_def)
60804
080a979a985b formal class for factorial (semi)rings
haftmann
parents:
diff changeset
   987
qed
62499
4a5b81ff5992 constructive formulation of factorization
haftmann
parents: 62366
diff changeset
   988
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   989
lemma prime_divisorE:
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   990
  assumes "a \<noteq> 0" and "\<not> is_unit a"
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   991
  obtains p where "prime p" and "p dvd a"
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
   992
  using assms no_prime_divisors_imp_unit unfolding prime_def by blast
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   993
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   994
definition multiplicity :: "'a \<Rightarrow> 'a \<Rightarrow> nat" where
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   995
  "multiplicity p x = (if finite {n. p ^ n dvd x} then Max {n. p ^ n dvd x} else 0)"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   996
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   997
lemma multiplicity_dvd: "p ^ multiplicity p x dvd x"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   998
proof (cases "finite {n. p ^ n dvd x}")
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
   999
  case True
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1000
  hence "multiplicity p x = Max {n. p ^ n dvd x}"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1001
    by (simp add: multiplicity_def)
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1002
  also have "\<dots> \<in> {n. p ^ n dvd x}"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1003
    by (rule Max_in) (auto intro!: True exI[of _ "0::nat"])
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1004
  finally show ?thesis by simp
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1005
qed (simp add: multiplicity_def)
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1006
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1007
lemma multiplicity_dvd': "n \<le> multiplicity p x \<Longrightarrow> p ^ n dvd x"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1008
  by (rule dvd_trans[OF le_imp_power_dvd multiplicity_dvd])
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1009
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1010
context
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1011
  fixes x p :: 'a
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1012
  assumes xp: "x \<noteq> 0" "\<not>is_unit p"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1013
begin
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1014
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1015
lemma multiplicity_eq_Max: "multiplicity p x = Max {n. p ^ n dvd x}"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1016
  using finite_divisor_powers[OF xp] by (simp add: multiplicity_def)
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1017
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1018
lemma multiplicity_geI:
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1019
  assumes "p ^ n dvd x"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1020
  shows   "multiplicity p x \<ge> n"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1021
proof -
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1022
  from assms have "n \<le> Max {n. p ^ n dvd x}"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1023
    by (intro Max_ge finite_divisor_powers xp) simp_all
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1024
  thus ?thesis by (subst multiplicity_eq_Max)
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1025
qed
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1026
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1027
lemma multiplicity_lessI:
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1028
  assumes "\<not>p ^ n dvd x"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1029
  shows   "multiplicity p x < n"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1030
proof (rule ccontr)
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1031
  assume "\<not>(n > multiplicity p x)"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1032
  hence "p ^ n dvd x" by (intro multiplicity_dvd') simp
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1033
  with assms show False by contradiction
62499
4a5b81ff5992 constructive formulation of factorization
haftmann
parents: 62366
diff changeset
  1034
qed
4a5b81ff5992 constructive formulation of factorization
haftmann
parents: 62366
diff changeset
  1035
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1036
lemma power_dvd_iff_le_multiplicity:
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1037
  "p ^ n dvd x \<longleftrightarrow> n \<le> multiplicity p x"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1038
  using multiplicity_geI[of n] multiplicity_lessI[of n] by (cases "p ^ n dvd x") auto
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1039
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1040
lemma multiplicity_eq_zero_iff:
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1041
  shows   "multiplicity p x = 0 \<longleftrightarrow> \<not>p dvd x"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1042
  using power_dvd_iff_le_multiplicity[of 1] by auto
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1043
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1044
lemma multiplicity_gt_zero_iff:
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1045
  shows   "multiplicity p x > 0 \<longleftrightarrow> p dvd x"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1046
  using power_dvd_iff_le_multiplicity[of 1] by auto
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1047
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1048
lemma multiplicity_decompose:
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1049
  "\<not>p dvd (x div p ^ multiplicity p x)"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1050
proof
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1051
  assume *: "p dvd x div p ^ multiplicity p x"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1052
  have "x = x div p ^ multiplicity p x * (p ^ multiplicity p x)"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1053
    using multiplicity_dvd[of p x] by simp
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1054
  also from * have "x div p ^ multiplicity p x = (x div p ^ multiplicity p x div p) * p" by simp
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1055
  also have "x div p ^ multiplicity p x div p * p * p ^ multiplicity p x =
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1056
               x div p ^ multiplicity p x div p * p ^ Suc (multiplicity p x)"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1057
    by (simp add: mult_assoc)
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1058
  also have "p ^ Suc (multiplicity p x) dvd \<dots>" by (rule dvd_triv_right)
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1059
  finally show False by (subst (asm) power_dvd_iff_le_multiplicity) simp
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1060
qed
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1061
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1062
lemma multiplicity_decompose':
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1063
  obtains y where "x = p ^ multiplicity p x * y" "\<not>p dvd y"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1064
  using that[of "x div p ^ multiplicity p x"]
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1065
  by (simp add: multiplicity_decompose multiplicity_dvd)
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1066
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1067
end
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1068
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1069
lemma multiplicity_zero [simp]: "multiplicity p 0 = 0"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1070
  by (simp add: multiplicity_def)
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1071
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
  1072
lemma prime_elem_multiplicity_eq_zero_iff:
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
  1073
  "prime_elem p \<Longrightarrow> x \<noteq> 0 \<Longrightarrow> multiplicity p x = 0 \<longleftrightarrow> \<not>p dvd x"
63534
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  1074
  by (rule multiplicity_eq_zero_iff) simp_all
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  1075
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  1076
lemma prime_multiplicity_other:
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
  1077
  assumes "prime p" "prime q" "p \<noteq> q"
63534
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  1078
  shows   "multiplicity p q = 0"
65552
f533820e7248 theories "GCD" and "Binomial" are already included in "Main": this avoids improper imports in applications;
wenzelm
parents: 65435
diff changeset
  1079
  using assms by (subst prime_elem_multiplicity_eq_zero_iff) (auto dest: primes_dvd_imp_eq)
63534
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  1080
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  1081
lemma prime_multiplicity_gt_zero_iff:
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
  1082
  "prime_elem p \<Longrightarrow> x \<noteq> 0 \<Longrightarrow> multiplicity p x > 0 \<longleftrightarrow> p dvd x"
63534
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  1083
  by (rule multiplicity_gt_zero_iff) simp_all
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  1084
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1085
lemma multiplicity_unit_left: "is_unit p \<Longrightarrow> multiplicity p x = 0"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1086
  by (simp add: multiplicity_def is_unit_power_iff unit_imp_dvd)
62499
4a5b81ff5992 constructive formulation of factorization
haftmann
parents: 62366
diff changeset
  1087
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1088
lemma multiplicity_unit_right:
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1089
  assumes "is_unit x"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1090
  shows   "multiplicity p x = 0"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1091
proof (cases "is_unit p \<or> x = 0")
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1092
  case False
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1093
  with multiplicity_lessI[of x p 1] this assms
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1094
    show ?thesis by (auto dest: dvd_unit_imp_unit)
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1095
qed (auto simp: multiplicity_unit_left)
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1096
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1097
lemma multiplicity_one [simp]: "multiplicity p 1 = 0"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1098
  by (rule multiplicity_unit_right) simp_all
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1099
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1100
lemma multiplicity_eqI:
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1101
  assumes "p ^ n dvd x" "\<not>p ^ Suc n dvd x"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1102
  shows   "multiplicity p x = n"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1103
proof -
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1104
  consider "x = 0" | "is_unit p" | "x \<noteq> 0" "\<not>is_unit p" by blast
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1105
  thus ?thesis
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1106
  proof cases
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1107
    assume xp: "x \<noteq> 0" "\<not>is_unit p"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1108
    from xp assms(1) have "multiplicity p x \<ge> n" by (intro multiplicity_geI)
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1109
    moreover from assms(2) xp have "multiplicity p x < Suc n" by (intro multiplicity_lessI)
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1110
    ultimately show ?thesis by simp
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1111
  next
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1112
    assume "is_unit p"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1113
    hence "is_unit (p ^ Suc n)" by (simp add: is_unit_power_iff del: power_Suc)
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1114
    hence "p ^ Suc n dvd x" by (rule unit_imp_dvd)
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1115
    with \<open>\<not>p ^ Suc n dvd x\<close> show ?thesis by contradiction
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1116
  qed (insert assms, simp_all)
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1117
qed
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1118
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1119
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1120
context
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1121
  fixes x p :: 'a
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1122
  assumes xp: "x \<noteq> 0" "\<not>is_unit p"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1123
begin
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1124
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1125
lemma multiplicity_times_same:
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1126
  assumes "p \<noteq> 0"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1127
  shows   "multiplicity p (p * x) = Suc (multiplicity p x)"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1128
proof (rule multiplicity_eqI)
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1129
  show "p ^ Suc (multiplicity p x) dvd p * x"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1130
    by (auto intro!: mult_dvd_mono multiplicity_dvd)
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1131
  from xp assms show "\<not> p ^ Suc (Suc (multiplicity p x)) dvd p * x"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1132
    using power_dvd_iff_le_multiplicity[OF xp, of "Suc (multiplicity p x)"] by simp
62499
4a5b81ff5992 constructive formulation of factorization
haftmann
parents: 62366
diff changeset
  1133
qed
4a5b81ff5992 constructive formulation of factorization
haftmann
parents: 62366
diff changeset
  1134
4a5b81ff5992 constructive formulation of factorization
haftmann
parents: 62366
diff changeset
  1135
end
4a5b81ff5992 constructive formulation of factorization
haftmann
parents: 62366
diff changeset
  1136
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1137
lemma multiplicity_same_power': "multiplicity p (p ^ n) = (if p = 0 \<or> is_unit p then 0 else n)"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1138
proof -
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1139
  consider "p = 0" | "is_unit p" |"p \<noteq> 0" "\<not>is_unit p" by blast
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1140
  thus ?thesis
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1141
  proof cases
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1142
    assume "p \<noteq> 0" "\<not>is_unit p"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1143
    thus ?thesis by (induction n) (simp_all add: multiplicity_times_same)
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1144
  qed (simp_all add: power_0_left multiplicity_unit_left)
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1145
qed
62499
4a5b81ff5992 constructive formulation of factorization
haftmann
parents: 62366
diff changeset
  1146
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1147
lemma multiplicity_same_power:
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1148
  "p \<noteq> 0 \<Longrightarrow> \<not>is_unit p \<Longrightarrow> multiplicity p (p ^ n) = n"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1149
  by (simp add: multiplicity_same_power')
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1150
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
  1151
lemma multiplicity_prime_elem_times_other:
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
  1152
  assumes "prime_elem p" "\<not>p dvd q"
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1153
  shows   "multiplicity p (q * x) = multiplicity p x"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1154
proof (cases "x = 0")
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1155
  case False
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1156
  show ?thesis
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1157
  proof (rule multiplicity_eqI)
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1158
    have "1 * p ^ multiplicity p x dvd q * x"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1159
      by (intro mult_dvd_mono multiplicity_dvd) simp_all
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1160
    thus "p ^ multiplicity p x dvd q * x" by simp
62499
4a5b81ff5992 constructive formulation of factorization
haftmann
parents: 62366
diff changeset
  1161
  next
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1162
    define n where "n = multiplicity p x"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1163
    from assms have "\<not>is_unit p" by simp
74362
0135a0c77b64 tuned proofs --- avoid 'guess';
wenzelm
parents: 73270
diff changeset
  1164
    from multiplicity_decompose'[OF False this]
0135a0c77b64 tuned proofs --- avoid 'guess';
wenzelm
parents: 73270
diff changeset
  1165
    obtain y where y [folded n_def]: "x = p ^ multiplicity p x * y" "\<not> p dvd y" .
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1166
    from y have "p ^ Suc n dvd q * x \<longleftrightarrow> p ^ n * p dvd p ^ n * (q * y)" by (simp add: mult_ac)
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1167
    also from assms have "\<dots> \<longleftrightarrow> p dvd q * y" by simp
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
  1168
    also have "\<dots> \<longleftrightarrow> p dvd q \<or> p dvd y" by (rule prime_elem_dvd_mult_iff) fact+
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1169
    also from assms y have "\<dots> \<longleftrightarrow> False" by simp
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1170
    finally show "\<not>(p ^ Suc n dvd q * x)" by blast
62499
4a5b81ff5992 constructive formulation of factorization
haftmann
parents: 62366
diff changeset
  1171
  qed
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1172
qed simp_all
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1173
63924
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1174
lemma multiplicity_self:
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1175
  assumes "p \<noteq> 0" "\<not>is_unit p"
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1176
  shows   "multiplicity p p = 1"
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1177
proof -
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1178
  from assms have "multiplicity p p = Max {n. p ^ n dvd p}"
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1179
    by (simp add: multiplicity_eq_Max)
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1180
  also from assms have "p ^ n dvd p \<longleftrightarrow> n \<le> 1" for n
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1181
    using dvd_power_iff[of p n 1] by auto
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1182
  hence "{n. p ^ n dvd p} = {..1}" by auto
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1183
  also have "\<dots> = {0,1}" by auto
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1184
  finally show ?thesis by simp
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1185
qed
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1186
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1187
lemma multiplicity_times_unit_left:
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1188
  assumes "is_unit c"
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1189
  shows   "multiplicity (c * p) x = multiplicity p x"
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1190
proof -
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1191
  from assms have "{n. (c * p) ^ n dvd x} = {n. p ^ n dvd x}"
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1192
    by (subst mult.commute) (simp add: mult_unit_dvd_iff power_mult_distrib is_unit_power_iff)
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1193
  thus ?thesis by (simp add: multiplicity_def)
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1194
qed
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1195
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1196
lemma multiplicity_times_unit_right:
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1197
  assumes "is_unit c"
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1198
  shows   "multiplicity p (c * x) = multiplicity p x"
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1199
proof -
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1200
  from assms have "{n. p ^ n dvd c * x} = {n. p ^ n dvd x}"
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1201
    by (subst mult.commute) (simp add: dvd_mult_unit_iff)
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1202
  thus ?thesis by (simp add: multiplicity_def)
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1203
qed
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1204
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1205
lemma multiplicity_normalize_left [simp]:
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1206
  "multiplicity (normalize p) x = multiplicity p x"
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1207
proof (cases "p = 0")
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1208
  case [simp]: False
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1209
  have "normalize p = (1 div unit_factor p) * p"
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1210
    by (simp add: unit_div_commute is_unit_unit_factor)
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1211
  also have "multiplicity \<dots> x = multiplicity p x"
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1212
    by (rule multiplicity_times_unit_left) (simp add: is_unit_unit_factor)
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1213
  finally show ?thesis .
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1214
qed simp_all
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1215
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1216
lemma multiplicity_normalize_right [simp]:
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1217
  "multiplicity p (normalize x) = multiplicity p x"
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1218
proof (cases "x = 0")
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1219
  case [simp]: False
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1220
  have "normalize x = (1 div unit_factor x) * x"
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1221
    by (simp add: unit_div_commute is_unit_unit_factor)
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1222
  also have "multiplicity p \<dots> = multiplicity p x"
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1223
    by (rule multiplicity_times_unit_right) (simp add: is_unit_unit_factor)
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1224
  finally show ?thesis .
65552
f533820e7248 theories "GCD" and "Binomial" are already included in "Main": this avoids improper imports in applications;
wenzelm
parents: 65435
diff changeset
  1225
qed simp_all
63924
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1226
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1227
lemma multiplicity_prime [simp]: "prime_elem p \<Longrightarrow> multiplicity p p = 1"
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1228
  by (rule multiplicity_self) auto
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1229
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1230
lemma multiplicity_prime_power [simp]: "prime_elem p \<Longrightarrow> multiplicity p (p ^ n) = n"
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1231
  by (subst multiplicity_same_power') auto
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1232
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1233
lift_definition prime_factorization :: "'a \<Rightarrow> 'a multiset" is
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
  1234
  "\<lambda>x p. if prime p then multiplicity p x else 0"
73270
e2d03448d5b5 get rid of traditional predicate
haftmann
parents: 73127
diff changeset
  1235
proof -
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1236
  fix x :: 'a
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
  1237
  show "finite {p. 0 < (if prime p then multiplicity p x else 0)}" (is "finite ?A")
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1238
  proof (cases "x = 0")
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1239
    case False
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
  1240
    from False have "?A \<subseteq> {p. prime p \<and> p dvd x}"
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1241
      by (auto simp: multiplicity_gt_zero_iff)
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
  1242
    moreover from False have "finite {p. prime p \<and> p dvd x}"
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1243
      by (rule finite_prime_divisors)
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1244
    ultimately show ?thesis by (rule finite_subset)
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1245
  qed simp_all
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1246
qed
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1247
63905
1c3dcb5fe6cb prefer abbreviation for trivial set conversion
haftmann
parents: 63904
diff changeset
  1248
abbreviation prime_factors :: "'a \<Rightarrow> 'a set" where
1c3dcb5fe6cb prefer abbreviation for trivial set conversion
haftmann
parents: 63904
diff changeset
  1249
  "prime_factors a \<equiv> set_mset (prime_factorization a)"
1c3dcb5fe6cb prefer abbreviation for trivial set conversion
haftmann
parents: 63904
diff changeset
  1250
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1251
lemma count_prime_factorization_nonprime:
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
  1252
  "\<not>prime p \<Longrightarrow> count (prime_factorization x) p = 0"
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1253
  by transfer simp
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1254
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1255
lemma count_prime_factorization_prime:
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
  1256
  "prime p \<Longrightarrow> count (prime_factorization x) p = multiplicity p x"
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1257
  by transfer simp
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1258
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1259
lemma count_prime_factorization:
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
  1260
  "count (prime_factorization x) p = (if prime p then multiplicity p x else 0)"
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1261
  by transfer simp
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1262
63924
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1263
lemma dvd_imp_multiplicity_le:
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1264
  assumes "a dvd b" "b \<noteq> 0"
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1265
  shows   "multiplicity p a \<le> multiplicity p b"
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1266
proof (cases "is_unit p")
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1267
  case False
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1268
  with assms show ?thesis
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1269
    by (intro multiplicity_geI ) (auto intro: dvd_trans[OF multiplicity_dvd' assms(1)])
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1270
qed (insert assms, auto simp: multiplicity_unit_left)
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1271
66276
acc3b7dd0b21 More material on powers for HOL-Computational_Algebra/HOL-Number_Theory
eberlm <eberlm@in.tum.de>
parents: 65552
diff changeset
  1272
lemma prime_power_inj:
acc3b7dd0b21 More material on powers for HOL-Computational_Algebra/HOL-Number_Theory
eberlm <eberlm@in.tum.de>
parents: 65552
diff changeset
  1273
  assumes "prime a" "a ^ m = a ^ n"
acc3b7dd0b21 More material on powers for HOL-Computational_Algebra/HOL-Number_Theory
eberlm <eberlm@in.tum.de>
parents: 65552
diff changeset
  1274
  shows   "m = n"
acc3b7dd0b21 More material on powers for HOL-Computational_Algebra/HOL-Number_Theory
eberlm <eberlm@in.tum.de>
parents: 65552
diff changeset
  1275
proof -
acc3b7dd0b21 More material on powers for HOL-Computational_Algebra/HOL-Number_Theory
eberlm <eberlm@in.tum.de>
parents: 65552
diff changeset
  1276
  have "multiplicity a (a ^ m) = multiplicity a (a ^ n)" by (simp only: assms)
acc3b7dd0b21 More material on powers for HOL-Computational_Algebra/HOL-Number_Theory
eberlm <eberlm@in.tum.de>
parents: 65552
diff changeset
  1277
  thus ?thesis using assms by (subst (asm) (1 2) multiplicity_prime_power) simp_all
acc3b7dd0b21 More material on powers for HOL-Computational_Algebra/HOL-Number_Theory
eberlm <eberlm@in.tum.de>
parents: 65552
diff changeset
  1278
qed
acc3b7dd0b21 More material on powers for HOL-Computational_Algebra/HOL-Number_Theory
eberlm <eberlm@in.tum.de>
parents: 65552
diff changeset
  1279
acc3b7dd0b21 More material on powers for HOL-Computational_Algebra/HOL-Number_Theory
eberlm <eberlm@in.tum.de>
parents: 65552
diff changeset
  1280
lemma prime_power_inj':
acc3b7dd0b21 More material on powers for HOL-Computational_Algebra/HOL-Number_Theory
eberlm <eberlm@in.tum.de>
parents: 65552
diff changeset
  1281
  assumes "prime p" "prime q"
acc3b7dd0b21 More material on powers for HOL-Computational_Algebra/HOL-Number_Theory
eberlm <eberlm@in.tum.de>
parents: 65552
diff changeset
  1282
  assumes "p ^ m = q ^ n" "m > 0" "n > 0"
acc3b7dd0b21 More material on powers for HOL-Computational_Algebra/HOL-Number_Theory
eberlm <eberlm@in.tum.de>
parents: 65552
diff changeset
  1283
  shows   "p = q" "m = n"
acc3b7dd0b21 More material on powers for HOL-Computational_Algebra/HOL-Number_Theory
eberlm <eberlm@in.tum.de>
parents: 65552
diff changeset
  1284
proof -
acc3b7dd0b21 More material on powers for HOL-Computational_Algebra/HOL-Number_Theory
eberlm <eberlm@in.tum.de>
parents: 65552
diff changeset
  1285
  from assms have "p ^ 1 dvd p ^ m" by (intro le_imp_power_dvd) simp
acc3b7dd0b21 More material on powers for HOL-Computational_Algebra/HOL-Number_Theory
eberlm <eberlm@in.tum.de>
parents: 65552
diff changeset
  1286
  also have "p ^ m = q ^ n" by fact
acc3b7dd0b21 More material on powers for HOL-Computational_Algebra/HOL-Number_Theory
eberlm <eberlm@in.tum.de>
parents: 65552
diff changeset
  1287
  finally have "p dvd q ^ n" by simp
acc3b7dd0b21 More material on powers for HOL-Computational_Algebra/HOL-Number_Theory
eberlm <eberlm@in.tum.de>
parents: 65552
diff changeset
  1288
  with assms have "p dvd q" using prime_dvd_power[of p q] by simp
acc3b7dd0b21 More material on powers for HOL-Computational_Algebra/HOL-Number_Theory
eberlm <eberlm@in.tum.de>
parents: 65552
diff changeset
  1289
  with assms show "p = q" by (simp add: primes_dvd_imp_eq)
acc3b7dd0b21 More material on powers for HOL-Computational_Algebra/HOL-Number_Theory
eberlm <eberlm@in.tum.de>
parents: 65552
diff changeset
  1290
  with assms show "m = n" by (simp add: prime_power_inj)
acc3b7dd0b21 More material on powers for HOL-Computational_Algebra/HOL-Number_Theory
eberlm <eberlm@in.tum.de>
parents: 65552
diff changeset
  1291
qed
acc3b7dd0b21 More material on powers for HOL-Computational_Algebra/HOL-Number_Theory
eberlm <eberlm@in.tum.de>
parents: 65552
diff changeset
  1292
acc3b7dd0b21 More material on powers for HOL-Computational_Algebra/HOL-Number_Theory
eberlm <eberlm@in.tum.de>
parents: 65552
diff changeset
  1293
lemma prime_power_eq_one_iff [simp]: "prime p \<Longrightarrow> p ^ n = 1 \<longleftrightarrow> n = 0"
acc3b7dd0b21 More material on powers for HOL-Computational_Algebra/HOL-Number_Theory
eberlm <eberlm@in.tum.de>
parents: 65552
diff changeset
  1294
  using prime_power_inj[of p n 0] by auto
acc3b7dd0b21 More material on powers for HOL-Computational_Algebra/HOL-Number_Theory
eberlm <eberlm@in.tum.de>
parents: 65552
diff changeset
  1295
acc3b7dd0b21 More material on powers for HOL-Computational_Algebra/HOL-Number_Theory
eberlm <eberlm@in.tum.de>
parents: 65552
diff changeset
  1296
lemma one_eq_prime_power_iff [simp]: "prime p \<Longrightarrow> 1 = p ^ n \<longleftrightarrow> n = 0"
acc3b7dd0b21 More material on powers for HOL-Computational_Algebra/HOL-Number_Theory
eberlm <eberlm@in.tum.de>
parents: 65552
diff changeset
  1297
  using prime_power_inj[of p 0 n] by auto
acc3b7dd0b21 More material on powers for HOL-Computational_Algebra/HOL-Number_Theory
eberlm <eberlm@in.tum.de>
parents: 65552
diff changeset
  1298
acc3b7dd0b21 More material on powers for HOL-Computational_Algebra/HOL-Number_Theory
eberlm <eberlm@in.tum.de>
parents: 65552
diff changeset
  1299
lemma prime_power_inj'':
acc3b7dd0b21 More material on powers for HOL-Computational_Algebra/HOL-Number_Theory
eberlm <eberlm@in.tum.de>
parents: 65552
diff changeset
  1300
  assumes "prime p" "prime q"
acc3b7dd0b21 More material on powers for HOL-Computational_Algebra/HOL-Number_Theory
eberlm <eberlm@in.tum.de>
parents: 65552
diff changeset
  1301
  shows   "p ^ m = q ^ n \<longleftrightarrow> (m = 0 \<and> n = 0) \<or> (p = q \<and> m = n)"
acc3b7dd0b21 More material on powers for HOL-Computational_Algebra/HOL-Number_Theory
eberlm <eberlm@in.tum.de>
parents: 65552
diff changeset
  1302
  using assms 
acc3b7dd0b21 More material on powers for HOL-Computational_Algebra/HOL-Number_Theory
eberlm <eberlm@in.tum.de>
parents: 65552
diff changeset
  1303
  by (cases "m = 0"; cases "n = 0")
acc3b7dd0b21 More material on powers for HOL-Computational_Algebra/HOL-Number_Theory
eberlm <eberlm@in.tum.de>
parents: 65552
diff changeset
  1304
     (auto dest: prime_power_inj'[OF assms])
acc3b7dd0b21 More material on powers for HOL-Computational_Algebra/HOL-Number_Theory
eberlm <eberlm@in.tum.de>
parents: 65552
diff changeset
  1305
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1306
lemma prime_factorization_0 [simp]: "prime_factorization 0 = {#}"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1307
  by (simp add: multiset_eq_iff count_prime_factorization)
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1308
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1309
lemma prime_factorization_empty_iff:
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1310
  "prime_factorization x = {#} \<longleftrightarrow> x = 0 \<or> is_unit x"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1311
proof
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1312
  assume *: "prime_factorization x = {#}"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1313
  {
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1314
    assume x: "x \<noteq> 0" "\<not>is_unit x"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1315
    {
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
  1316
      fix p assume p: "prime p"
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1317
      have "count (prime_factorization x) p = 0" by (simp add: *)
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1318
      also from p have "count (prime_factorization x) p = multiplicity p x"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1319
        by (rule count_prime_factorization_prime)
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1320
      also from x p have "\<dots> = 0 \<longleftrightarrow> \<not>p dvd x" by (simp add: multiplicity_eq_zero_iff)
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1321
      finally have "\<not>p dvd x" .
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1322
    }
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1323
    with prime_divisor_exists[OF x] have False by blast
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1324
  }
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1325
  thus "x = 0 \<or> is_unit x" by blast
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1326
next
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1327
  assume "x = 0 \<or> is_unit x"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1328
  thus "prime_factorization x = {#}"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1329
  proof
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1330
    assume x: "is_unit x"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1331
    {
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
  1332
      fix p assume p: "prime p"
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1333
      from p x have "multiplicity p x = 0"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1334
        by (subst multiplicity_eq_zero_iff)
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1335
           (auto simp: multiplicity_eq_zero_iff dest: unit_imp_no_prime_divisors)
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1336
    }
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1337
    thus ?thesis by (simp add: multiset_eq_iff count_prime_factorization)
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1338
  qed simp_all
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1339
qed
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1340
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1341
lemma prime_factorization_unit:
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1342
  assumes "is_unit x"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1343
  shows   "prime_factorization x = {#}"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1344
proof (rule multiset_eqI)
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1345
  fix p :: 'a
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1346
  show "count (prime_factorization x) p = count {#} p"
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
  1347
  proof (cases "prime p")
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1348
    case True
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1349
    with assms have "multiplicity p x = 0"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1350
      by (subst multiplicity_eq_zero_iff)
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1351
         (auto simp: multiplicity_eq_zero_iff dest: unit_imp_no_prime_divisors)
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1352
    with True show ?thesis by (simp add: count_prime_factorization_prime)
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1353
  qed (simp_all add: count_prime_factorization_nonprime)
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1354
qed
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1355
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1356
lemma prime_factorization_1 [simp]: "prime_factorization 1 = {#}"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1357
  by (simp add: prime_factorization_unit)
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1358
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1359
lemma prime_factorization_times_prime:
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
  1360
  assumes "x \<noteq> 0" "prime p"
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1361
  shows   "prime_factorization (p * x) = {#p#} + prime_factorization x"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1362
proof (rule multiset_eqI)
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1363
  fix q :: 'a
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
  1364
  consider "\<not>prime q" | "p = q" | "prime q" "p \<noteq> q" by blast
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1365
  thus "count (prime_factorization (p * x)) q = count ({#p#} + prime_factorization x) q"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1366
  proof cases
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
  1367
    assume q: "prime q" "p \<noteq> q"
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1368
    with assms primes_dvd_imp_eq[of q p] have "\<not>q dvd p" by auto
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1369
    with q assms show ?thesis
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
  1370
      by (simp add: multiplicity_prime_elem_times_other count_prime_factorization)
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1371
  qed (insert assms, auto simp: count_prime_factorization multiplicity_times_same)
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1372
qed
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1373
71398
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1374
lemma prod_mset_prime_factorization_weak:
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1375
  assumes "x \<noteq> 0"
71398
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1376
  shows   "normalize (prod_mset (prime_factorization x)) = normalize x"
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1377
  using assms
71398
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1378
proof (induction x rule: prime_divisors_induct)
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1379
  case (factor p x)
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1380
  have "normalize (prod_mset (prime_factorization (p * x))) =
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1381
          normalize (p * normalize (prod_mset (prime_factorization x)))"
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1382
    using factor.prems factor.hyps by (simp add: prime_factorization_times_prime)
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1383
  also have "normalize (prod_mset (prime_factorization x)) = normalize x"
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1384
    by (rule factor.IH) (use factor in auto)
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1385
  finally show ?case by simp
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1386
qed (auto simp: prime_factorization_unit is_unit_normalize)
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1387
63905
1c3dcb5fe6cb prefer abbreviation for trivial set conversion
haftmann
parents: 63904
diff changeset
  1388
lemma in_prime_factors_iff:
1c3dcb5fe6cb prefer abbreviation for trivial set conversion
haftmann
parents: 63904
diff changeset
  1389
  "p \<in> prime_factors x \<longleftrightarrow> x \<noteq> 0 \<and> p dvd x \<and> prime p"
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1390
proof -
63905
1c3dcb5fe6cb prefer abbreviation for trivial set conversion
haftmann
parents: 63904
diff changeset
  1391
  have "p \<in> prime_factors x \<longleftrightarrow> count (prime_factorization x) p > 0" by simp
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
  1392
  also have "\<dots> \<longleftrightarrow> x \<noteq> 0 \<and> p dvd x \<and> prime p"
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1393
   by (subst count_prime_factorization, cases "x = 0")
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1394
      (auto simp: multiplicity_eq_zero_iff multiplicity_gt_zero_iff)
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1395
  finally show ?thesis .
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1396
qed
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1397
63905
1c3dcb5fe6cb prefer abbreviation for trivial set conversion
haftmann
parents: 63904
diff changeset
  1398
lemma in_prime_factors_imp_prime [intro]:
1c3dcb5fe6cb prefer abbreviation for trivial set conversion
haftmann
parents: 63904
diff changeset
  1399
  "p \<in> prime_factors x \<Longrightarrow> prime p"
1c3dcb5fe6cb prefer abbreviation for trivial set conversion
haftmann
parents: 63904
diff changeset
  1400
  by (simp add: in_prime_factors_iff)
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1401
63905
1c3dcb5fe6cb prefer abbreviation for trivial set conversion
haftmann
parents: 63904
diff changeset
  1402
lemma in_prime_factors_imp_dvd [dest]:
1c3dcb5fe6cb prefer abbreviation for trivial set conversion
haftmann
parents: 63904
diff changeset
  1403
  "p \<in> prime_factors x \<Longrightarrow> p dvd x"
1c3dcb5fe6cb prefer abbreviation for trivial set conversion
haftmann
parents: 63904
diff changeset
  1404
  by (simp add: in_prime_factors_iff)
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1405
63924
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1406
lemma prime_factorsI:
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1407
  "x \<noteq> 0 \<Longrightarrow> prime p \<Longrightarrow> p dvd x \<Longrightarrow> p \<in> prime_factors x"
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1408
  by (auto simp: in_prime_factors_iff)
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1409
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1410
lemma prime_factors_dvd:
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1411
  "x \<noteq> 0 \<Longrightarrow> prime_factors x = {p. prime p \<and> p dvd x}"
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1412
  by (auto intro: prime_factorsI)
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1413
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1414
lemma prime_factors_multiplicity:
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1415
  "prime_factors n = {p. prime p \<and> multiplicity p n > 0}"
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1416
  by (cases "n = 0") (auto simp add: prime_factors_dvd prime_multiplicity_gt_zero_iff)
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1417
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1418
lemma prime_factorization_prime:
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
  1419
  assumes "prime p"
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1420
  shows   "prime_factorization p = {#p#}"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1421
proof (rule multiset_eqI)
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1422
  fix q :: 'a
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
  1423
  consider "\<not>prime q" | "q = p" | "prime q" "q \<noteq> p" by blast
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1424
  thus "count (prime_factorization p) q = count {#p#} q"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1425
    by cases (insert assms, auto dest: primes_dvd_imp_eq
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1426
                simp: count_prime_factorization multiplicity_self multiplicity_eq_zero_iff)
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1427
qed
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1428
63830
2ea3725a34bd msetsum -> set_mset, msetprod -> prod_mset
nipkow
parents: 63793
diff changeset
  1429
lemma prime_factorization_prod_mset_primes:
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
  1430
  assumes "\<And>p. p \<in># A \<Longrightarrow> prime p"
63830
2ea3725a34bd msetsum -> set_mset, msetprod -> prod_mset
nipkow
parents: 63793
diff changeset
  1431
  shows   "prime_factorization (prod_mset A) = A"
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1432
  using assms
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1433
proof (induction A)
63793
e68a0b651eb5 add_mset constructor in multisets
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents: 63633
diff changeset
  1434
  case (add p A)
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1435
  from add.prems[of 0] have "0 \<notin># A" by auto
63830
2ea3725a34bd msetsum -> set_mset, msetprod -> prod_mset
nipkow
parents: 63793
diff changeset
  1436
  hence "prod_mset A \<noteq> 0" by auto
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1437
  with add show ?case
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1438
    by (simp_all add: mult_ac prime_factorization_times_prime Multiset.union_commute)
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1439
qed simp_all
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1440
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1441
lemma prime_factorization_cong:
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1442
  "normalize x = normalize y \<Longrightarrow> prime_factorization x = prime_factorization y"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1443
  by (simp add: multiset_eq_iff count_prime_factorization
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1444
                multiplicity_normalize_right [of _ x, symmetric]
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1445
                multiplicity_normalize_right [of _ y, symmetric]
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1446
           del:  multiplicity_normalize_right)
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1447
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1448
lemma prime_factorization_unique:
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1449
  assumes "x \<noteq> 0" "y \<noteq> 0"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1450
  shows   "prime_factorization x = prime_factorization y \<longleftrightarrow> normalize x = normalize y"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1451
proof
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1452
  assume "prime_factorization x = prime_factorization y"
63830
2ea3725a34bd msetsum -> set_mset, msetprod -> prod_mset
nipkow
parents: 63793
diff changeset
  1453
  hence "prod_mset (prime_factorization x) = prod_mset (prime_factorization y)" by simp
71398
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1454
  hence "normalize (prod_mset (prime_factorization x)) =
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1455
         normalize (prod_mset (prime_factorization y))"
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1456
    by (simp only: )
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1457
  with assms show "normalize x = normalize y"
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1458
    by (simp add: prod_mset_prime_factorization_weak)
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1459
qed (rule prime_factorization_cong)
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1460
71398
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1461
lemma prime_factorization_normalize [simp]:
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1462
  "prime_factorization (normalize x) = prime_factorization x"
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1463
  by (cases "x = 0", simp, subst prime_factorization_unique) auto
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1464
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1465
lemma prime_factorization_eqI_strong:
69785
9e326f6f8a24 More material for HOL-Number_Theory: ord, Carmichael's function, primitive roots
Manuel Eberl <eberlm@in.tum.de>
parents: 68606
diff changeset
  1466
  assumes "\<And>p. p \<in># P \<Longrightarrow> prime p" "prod_mset P = n"
9e326f6f8a24 More material for HOL-Number_Theory: ord, Carmichael's function, primitive roots
Manuel Eberl <eberlm@in.tum.de>
parents: 68606
diff changeset
  1467
  shows   "prime_factorization n = P"
9e326f6f8a24 More material for HOL-Number_Theory: ord, Carmichael's function, primitive roots
Manuel Eberl <eberlm@in.tum.de>
parents: 68606
diff changeset
  1468
  using prime_factorization_prod_mset_primes[of P] assms by simp
9e326f6f8a24 More material for HOL-Number_Theory: ord, Carmichael's function, primitive roots
Manuel Eberl <eberlm@in.tum.de>
parents: 68606
diff changeset
  1469
71398
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1470
lemma prime_factorization_eqI:
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1471
  assumes "\<And>p. p \<in># P \<Longrightarrow> prime p" "normalize (prod_mset P) = normalize n"
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1472
  shows   "prime_factorization n = P"
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1473
proof -
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1474
  have "P = prime_factorization (normalize (prod_mset P))"
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1475
    using prime_factorization_prod_mset_primes[of P] assms(1) by simp
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1476
  with assms(2) show ?thesis by simp
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1477
qed
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1478
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1479
lemma prime_factorization_mult:
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1480
  assumes "x \<noteq> 0" "y \<noteq> 0"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1481
  shows   "prime_factorization (x * y) = prime_factorization x + prime_factorization y"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1482
proof -
71398
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1483
  have "normalize (prod_mset (prime_factorization x) * prod_mset (prime_factorization y)) =
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1484
          normalize (normalize (prod_mset (prime_factorization x)) *
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1485
                     normalize (prod_mset (prime_factorization y)))"
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1486
    by (simp only: normalize_mult_normalize_left normalize_mult_normalize_right)
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1487
  also have "\<dots> = normalize (x * y)"
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1488
    by (subst (1 2) prod_mset_prime_factorization_weak) (use assms in auto)
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1489
  finally show ?thesis
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1490
    by (intro prime_factorization_eqI) auto
62499
4a5b81ff5992 constructive formulation of factorization
haftmann
parents: 62366
diff changeset
  1491
qed
4a5b81ff5992 constructive formulation of factorization
haftmann
parents: 62366
diff changeset
  1492
66276
acc3b7dd0b21 More material on powers for HOL-Computational_Algebra/HOL-Number_Theory
eberlm <eberlm@in.tum.de>
parents: 65552
diff changeset
  1493
lemma prime_factorization_prod:
acc3b7dd0b21 More material on powers for HOL-Computational_Algebra/HOL-Number_Theory
eberlm <eberlm@in.tum.de>
parents: 65552
diff changeset
  1494
  assumes "finite A" "\<And>x. x \<in> A \<Longrightarrow> f x \<noteq> 0"
acc3b7dd0b21 More material on powers for HOL-Computational_Algebra/HOL-Number_Theory
eberlm <eberlm@in.tum.de>
parents: 65552
diff changeset
  1495
  shows   "prime_factorization (prod f A) = (\<Sum>n\<in>A. prime_factorization (f n))"
acc3b7dd0b21 More material on powers for HOL-Computational_Algebra/HOL-Number_Theory
eberlm <eberlm@in.tum.de>
parents: 65552
diff changeset
  1496
  using assms by (induction A rule: finite_induct)
acc3b7dd0b21 More material on powers for HOL-Computational_Algebra/HOL-Number_Theory
eberlm <eberlm@in.tum.de>
parents: 65552
diff changeset
  1497
                 (auto simp: Sup_multiset_empty prime_factorization_mult)
acc3b7dd0b21 More material on powers for HOL-Computational_Algebra/HOL-Number_Theory
eberlm <eberlm@in.tum.de>
parents: 65552
diff changeset
  1498
63924
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1499
lemma prime_elem_multiplicity_mult_distrib:
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1500
  assumes "prime_elem p" "x \<noteq> 0" "y \<noteq> 0"
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1501
  shows   "multiplicity p (x * y) = multiplicity p x + multiplicity p y"
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1502
proof -
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1503
  have "multiplicity p (x * y) = count (prime_factorization (x * y)) (normalize p)"
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1504
    by (subst count_prime_factorization_prime) (simp_all add: assms)
65552
f533820e7248 theories "GCD" and "Binomial" are already included in "Main": this avoids improper imports in applications;
wenzelm
parents: 65435
diff changeset
  1505
  also from assms
63924
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1506
    have "prime_factorization (x * y) = prime_factorization x + prime_factorization y"
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1507
      by (intro prime_factorization_mult)
65552
f533820e7248 theories "GCD" and "Binomial" are already included in "Main": this avoids improper imports in applications;
wenzelm
parents: 65435
diff changeset
  1508
  also have "count \<dots> (normalize p) =
63924
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1509
    count (prime_factorization x) (normalize p) + count (prime_factorization y) (normalize p)"
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1510
    by simp
65552
f533820e7248 theories "GCD" and "Binomial" are already included in "Main": this avoids improper imports in applications;
wenzelm
parents: 65435
diff changeset
  1511
  also have "\<dots> = multiplicity p x + multiplicity p y"
63924
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1512
    by (subst (1 2) count_prime_factorization_prime) (simp_all add: assms)
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1513
  finally show ?thesis .
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1514
qed
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1515
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1516
lemma prime_elem_multiplicity_prod_mset_distrib:
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1517
  assumes "prime_elem p" "0 \<notin># A"
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1518
  shows   "multiplicity p (prod_mset A) = sum_mset (image_mset (multiplicity p) A)"
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1519
  using assms by (induction A) (auto simp: prime_elem_multiplicity_mult_distrib)
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1520
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1521
lemma prime_elem_multiplicity_power_distrib:
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1522
  assumes "prime_elem p" "x \<noteq> 0"
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1523
  shows   "multiplicity p (x ^ n) = n * multiplicity p x"
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1524
  using assms prime_elem_multiplicity_prod_mset_distrib [of p "replicate_mset n x"]
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1525
  by simp
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1526
64272
f76b6dda2e56 setprod -> prod
nipkow
parents: 64267
diff changeset
  1527
lemma prime_elem_multiplicity_prod_distrib:
63924
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1528
  assumes "prime_elem p" "0 \<notin> f ` A" "finite A"
64272
f76b6dda2e56 setprod -> prod
nipkow
parents: 64267
diff changeset
  1529
  shows   "multiplicity p (prod f A) = (\<Sum>x\<in>A. multiplicity p (f x))"
63924
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1530
proof -
64272
f76b6dda2e56 setprod -> prod
nipkow
parents: 64267
diff changeset
  1531
  have "multiplicity p (prod f A) = (\<Sum>x\<in>#mset_set A. multiplicity p (f x))"
f76b6dda2e56 setprod -> prod
nipkow
parents: 64267
diff changeset
  1532
    using assms by (subst prod_unfold_prod_mset)
65552
f533820e7248 theories "GCD" and "Binomial" are already included in "Main": this avoids improper imports in applications;
wenzelm
parents: 65435
diff changeset
  1533
                   (simp_all add: prime_elem_multiplicity_prod_mset_distrib sum_unfold_sum_mset
63924
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1534
                      multiset.map_comp o_def)
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1535
  also from \<open>finite A\<close> have "\<dots> = (\<Sum>x\<in>A. multiplicity p (f x))"
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1536
    by (induction A rule: finite_induct) simp_all
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1537
  finally show ?thesis .
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1538
qed
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1539
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1540
lemma multiplicity_distinct_prime_power:
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1541
  "prime p \<Longrightarrow> prime q \<Longrightarrow> p \<noteq> q \<Longrightarrow> multiplicity p (q ^ n) = 0"
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1542
  by (subst prime_elem_multiplicity_power_distrib) (auto simp: prime_multiplicity_other)
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1543
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1544
lemma prime_factorization_prime_power:
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
  1545
  "prime p \<Longrightarrow> prime_factorization (p ^ n) = replicate_mset n p"
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1546
  by (induction n)
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1547
     (simp_all add: prime_factorization_mult prime_factorization_prime Multiset.union_commute)
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1548
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1549
lemma prime_factorization_subset_iff_dvd:
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1550
  assumes [simp]: "x \<noteq> 0" "y \<noteq> 0"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1551
  shows   "prime_factorization x \<subseteq># prime_factorization y \<longleftrightarrow> x dvd y"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1552
proof -
71398
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1553
  have "x dvd y \<longleftrightarrow>
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1554
    normalize (prod_mset (prime_factorization x)) dvd normalize (prod_mset (prime_factorization y))"
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1555
    using assms by (subst (1 2) prod_mset_prime_factorization_weak) auto
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1556
  also have "\<dots> \<longleftrightarrow> prime_factorization x \<subseteq># prime_factorization y"
63905
1c3dcb5fe6cb prefer abbreviation for trivial set conversion
haftmann
parents: 63904
diff changeset
  1557
    by (auto intro!: prod_mset_primes_dvd_imp_subset prod_mset_subset_imp_dvd)
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1558
  finally show ?thesis ..
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1559
qed
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1560
65552
f533820e7248 theories "GCD" and "Binomial" are already included in "Main": this avoids improper imports in applications;
wenzelm
parents: 65435
diff changeset
  1561
lemma prime_factorization_subset_imp_dvd:
63534
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  1562
  "x \<noteq> 0 \<Longrightarrow> (prime_factorization x \<subseteq># prime_factorization y) \<Longrightarrow> x dvd y"
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  1563
  by (cases "y = 0") (simp_all add: prime_factorization_subset_iff_dvd)
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  1564
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1565
lemma prime_factorization_divide:
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1566
  assumes "b dvd a"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1567
  shows   "prime_factorization (a div b) = prime_factorization a - prime_factorization b"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1568
proof (cases "a = 0")
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1569
  case [simp]: False
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1570
  from assms have [simp]: "b \<noteq> 0" by auto
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1571
  have "prime_factorization ((a div b) * b) = prime_factorization (a div b) + prime_factorization b"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1572
    by (intro prime_factorization_mult) (insert assms, auto elim!: dvdE)
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1573
  with assms show ?thesis by simp
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1574
qed simp_all
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1575
63905
1c3dcb5fe6cb prefer abbreviation for trivial set conversion
haftmann
parents: 63904
diff changeset
  1576
lemma zero_not_in_prime_factors [simp]: "0 \<notin> prime_factors x"
1c3dcb5fe6cb prefer abbreviation for trivial set conversion
haftmann
parents: 63904
diff changeset
  1577
  by (auto dest: in_prime_factors_imp_prime)
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1578
63904
b8482e12a2a8 more lemmas
haftmann
parents: 63830
diff changeset
  1579
lemma prime_prime_factors:
63905
1c3dcb5fe6cb prefer abbreviation for trivial set conversion
haftmann
parents: 63904
diff changeset
  1580
  "prime p \<Longrightarrow> prime_factors p = {p}"
1c3dcb5fe6cb prefer abbreviation for trivial set conversion
haftmann
parents: 63904
diff changeset
  1581
  by (drule prime_factorization_prime) simp
63534
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  1582
65552
f533820e7248 theories "GCD" and "Binomial" are already included in "Main": this avoids improper imports in applications;
wenzelm
parents: 65435
diff changeset
  1583
lemma prime_factors_product:
63534
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  1584
  "x \<noteq> 0 \<Longrightarrow> y \<noteq> 0 \<Longrightarrow> prime_factors (x * y) = prime_factors x \<union> prime_factors y"
63905
1c3dcb5fe6cb prefer abbreviation for trivial set conversion
haftmann
parents: 63904
diff changeset
  1585
  by (simp add: prime_factorization_mult)
63534
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  1586
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  1587
lemma dvd_prime_factors [intro]:
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  1588
  "y \<noteq> 0 \<Longrightarrow> x dvd y \<Longrightarrow> prime_factors x \<subseteq> prime_factors y"
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  1589
  by (intro set_mset_mono, subst prime_factorization_subset_iff_dvd) auto
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  1590
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  1591
(* RENAMED multiplicity_dvd *)
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  1592
lemma multiplicity_le_imp_dvd:
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
  1593
  assumes "x \<noteq> 0" "\<And>p. prime p \<Longrightarrow> multiplicity p x \<le> multiplicity p y"
63534
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  1594
  shows   "x dvd y"
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  1595
proof (cases "y = 0")
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  1596
  case False
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  1597
  from assms this have "prime_factorization x \<subseteq># prime_factorization y"
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  1598
    by (intro mset_subset_eqI) (auto simp: count_prime_factorization)
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  1599
  with assms False show ?thesis by (subst (asm) prime_factorization_subset_iff_dvd)
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  1600
qed auto
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  1601
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  1602
lemma dvd_multiplicity_eq:
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  1603
  "x \<noteq> 0 \<Longrightarrow> y \<noteq> 0 \<Longrightarrow> x dvd y \<longleftrightarrow> (\<forall>p. multiplicity p x \<le> multiplicity p y)"
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  1604
  by (auto intro: dvd_imp_multiplicity_le multiplicity_le_imp_dvd)
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  1605
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  1606
lemma multiplicity_eq_imp_eq:
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  1607
  assumes "x \<noteq> 0" "y \<noteq> 0"
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
  1608
  assumes "\<And>p. prime p \<Longrightarrow> multiplicity p x = multiplicity p y"
63534
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  1609
  shows   "normalize x = normalize y"
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  1610
  using assms by (intro associatedI multiplicity_le_imp_dvd) simp_all
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  1611
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  1612
lemma prime_factorization_unique':
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
  1613
  assumes "\<forall>p \<in># M. prime p" "\<forall>p \<in># N. prime p" "(\<Prod>i \<in># M. i) = (\<Prod>i \<in># N. i)"
63534
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  1614
  shows   "M = N"
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  1615
proof -
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  1616
  have "prime_factorization (\<Prod>i \<in># M. i) = prime_factorization (\<Prod>i \<in># N. i)"
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  1617
    by (simp only: assms)
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  1618
  also from assms have "prime_factorization (\<Prod>i \<in># M. i) = M"
63830
2ea3725a34bd msetsum -> set_mset, msetprod -> prod_mset
nipkow
parents: 63793
diff changeset
  1619
    by (subst prime_factorization_prod_mset_primes) simp_all
63534
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  1620
  also from assms have "prime_factorization (\<Prod>i \<in># N. i) = N"
63830
2ea3725a34bd msetsum -> set_mset, msetprod -> prod_mset
nipkow
parents: 63793
diff changeset
  1621
    by (subst prime_factorization_prod_mset_primes) simp_all
63534
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  1622
  finally show ?thesis .
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  1623
qed
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  1624
71398
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1625
lemma prime_factorization_unique'':
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1626
  assumes "\<forall>p \<in># M. prime p" "\<forall>p \<in># N. prime p" "normalize (\<Prod>i \<in># M. i) = normalize (\<Prod>i \<in># N. i)"
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1627
  shows   "M = N"
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1628
proof -
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1629
  have "prime_factorization (normalize (\<Prod>i \<in># M. i)) =
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1630
        prime_factorization (normalize (\<Prod>i \<in># N. i))"
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1631
    by (simp only: assms)
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1632
  also from assms have "prime_factorization (normalize (\<Prod>i \<in># M. i)) = M"
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1633
    by (subst prime_factorization_normalize, subst prime_factorization_prod_mset_primes) simp_all
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1634
  also from assms have "prime_factorization (normalize (\<Prod>i \<in># N. i)) = N"
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1635
    by (subst prime_factorization_normalize, subst prime_factorization_prod_mset_primes) simp_all
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1636
  finally show ?thesis .
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1637
qed
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1638
63537
831816778409 Removed redundant material related to primes
eberlm <eberlm@in.tum.de>
parents: 63534
diff changeset
  1639
lemma multiplicity_cong:
831816778409 Removed redundant material related to primes
eberlm <eberlm@in.tum.de>
parents: 63534
diff changeset
  1640
  "(\<And>r. p ^ r dvd a \<longleftrightarrow> p ^ r dvd b) \<Longrightarrow> multiplicity p a = multiplicity p b"
831816778409 Removed redundant material related to primes
eberlm <eberlm@in.tum.de>
parents: 63534
diff changeset
  1641
  by (simp add: multiplicity_def)
831816778409 Removed redundant material related to primes
eberlm <eberlm@in.tum.de>
parents: 63534
diff changeset
  1642
65552
f533820e7248 theories "GCD" and "Binomial" are already included in "Main": this avoids improper imports in applications;
wenzelm
parents: 65435
diff changeset
  1643
lemma not_dvd_imp_multiplicity_0:
63537
831816778409 Removed redundant material related to primes
eberlm <eberlm@in.tum.de>
parents: 63534
diff changeset
  1644
  assumes "\<not>p dvd x"
831816778409 Removed redundant material related to primes
eberlm <eberlm@in.tum.de>
parents: 63534
diff changeset
  1645
  shows   "multiplicity p x = 0"
831816778409 Removed redundant material related to primes
eberlm <eberlm@in.tum.de>
parents: 63534
diff changeset
  1646
proof -
831816778409 Removed redundant material related to primes
eberlm <eberlm@in.tum.de>
parents: 63534
diff changeset
  1647
  from assms have "multiplicity p x < 1"
831816778409 Removed redundant material related to primes
eberlm <eberlm@in.tum.de>
parents: 63534
diff changeset
  1648
    by (intro multiplicity_lessI) auto
831816778409 Removed redundant material related to primes
eberlm <eberlm@in.tum.de>
parents: 63534
diff changeset
  1649
  thus ?thesis by simp
831816778409 Removed redundant material related to primes
eberlm <eberlm@in.tum.de>
parents: 63534
diff changeset
  1650
qed
831816778409 Removed redundant material related to primes
eberlm <eberlm@in.tum.de>
parents: 63534
diff changeset
  1651
73127
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
  1652
lemma multiplicity_zero_left [simp]: "multiplicity 0 x = 0"
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
  1653
 by (cases "x = 0") (auto intro: not_dvd_imp_multiplicity_0)
73103
b69fd6e19662 One useful lemma/simprule
paulson <lp15@cam.ac.uk>
parents: 71398
diff changeset
  1654
66276
acc3b7dd0b21 More material on powers for HOL-Computational_Algebra/HOL-Number_Theory
eberlm <eberlm@in.tum.de>
parents: 65552
diff changeset
  1655
lemma inj_on_Prod_primes:
acc3b7dd0b21 More material on powers for HOL-Computational_Algebra/HOL-Number_Theory
eberlm <eberlm@in.tum.de>
parents: 65552
diff changeset
  1656
  assumes "\<And>P p. P \<in> A \<Longrightarrow> p \<in> P \<Longrightarrow> prime p"
acc3b7dd0b21 More material on powers for HOL-Computational_Algebra/HOL-Number_Theory
eberlm <eberlm@in.tum.de>
parents: 65552
diff changeset
  1657
  assumes "\<And>P. P \<in> A \<Longrightarrow> finite P"
acc3b7dd0b21 More material on powers for HOL-Computational_Algebra/HOL-Number_Theory
eberlm <eberlm@in.tum.de>
parents: 65552
diff changeset
  1658
  shows   "inj_on Prod A"
acc3b7dd0b21 More material on powers for HOL-Computational_Algebra/HOL-Number_Theory
eberlm <eberlm@in.tum.de>
parents: 65552
diff changeset
  1659
proof (rule inj_onI)
acc3b7dd0b21 More material on powers for HOL-Computational_Algebra/HOL-Number_Theory
eberlm <eberlm@in.tum.de>
parents: 65552
diff changeset
  1660
  fix P Q assume PQ: "P \<in> A" "Q \<in> A" "\<Prod>P = \<Prod>Q"
acc3b7dd0b21 More material on powers for HOL-Computational_Algebra/HOL-Number_Theory
eberlm <eberlm@in.tum.de>
parents: 65552
diff changeset
  1661
  with prime_factorization_unique'[of "mset_set P" "mset_set Q"] assms[of P] assms[of Q]
acc3b7dd0b21 More material on powers for HOL-Computational_Algebra/HOL-Number_Theory
eberlm <eberlm@in.tum.de>
parents: 65552
diff changeset
  1662
    have "mset_set P = mset_set Q" by (auto simp: prod_unfold_prod_mset)
acc3b7dd0b21 More material on powers for HOL-Computational_Algebra/HOL-Number_Theory
eberlm <eberlm@in.tum.de>
parents: 65552
diff changeset
  1663
    with assms[of P] assms[of Q] PQ show "P = Q" by simp
acc3b7dd0b21 More material on powers for HOL-Computational_Algebra/HOL-Number_Theory
eberlm <eberlm@in.tum.de>
parents: 65552
diff changeset
  1664
qed
acc3b7dd0b21 More material on powers for HOL-Computational_Algebra/HOL-Number_Theory
eberlm <eberlm@in.tum.de>
parents: 65552
diff changeset
  1665
71398
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1666
lemma divides_primepow_weak:
67051
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66938
diff changeset
  1667
  assumes "prime p" and "a dvd p ^ n"
71398
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1668
  obtains m where "m \<le> n" and "normalize a = normalize (p ^ m)"
67051
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66938
diff changeset
  1669
proof -
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66938
diff changeset
  1670
  from assms have "a \<noteq> 0"
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66938
diff changeset
  1671
    by auto
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66938
diff changeset
  1672
  with assms
71398
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1673
  have "normalize (prod_mset (prime_factorization a)) dvd
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1674
          normalize (prod_mset (prime_factorization (p ^ n)))"
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1675
    by (subst (1 2) prod_mset_prime_factorization_weak) auto
67051
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66938
diff changeset
  1676
  then have "prime_factorization a \<subseteq># prime_factorization (p ^ n)"
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66938
diff changeset
  1677
    by (simp add: in_prime_factors_imp_prime prod_mset_dvd_prod_mset_primes_iff)
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66938
diff changeset
  1678
  with assms have "prime_factorization a \<subseteq># replicate_mset n p"
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66938
diff changeset
  1679
    by (simp add: prime_factorization_prime_power)
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66938
diff changeset
  1680
  then obtain m where "m \<le> n" and "prime_factorization a = replicate_mset m p"
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66938
diff changeset
  1681
    by (rule msubseteq_replicate_msetE)
71398
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1682
  then have *: "normalize (prod_mset (prime_factorization a)) =
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1683
                  normalize (prod_mset (replicate_mset m p))" by metis
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1684
  also have "normalize (prod_mset (prime_factorization a)) = normalize a"
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1685
    using \<open>a \<noteq> 0\<close> by (simp add: prod_mset_prime_factorization_weak)
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1686
  also have "prod_mset (replicate_mset m p) = p ^ m"
67051
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66938
diff changeset
  1687
    by simp
71398
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1688
  finally show ?thesis using \<open>m \<le> n\<close> 
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1689
    by (intro that[of m])
67051
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66938
diff changeset
  1690
qed
66276
acc3b7dd0b21 More material on powers for HOL-Computational_Algebra/HOL-Number_Theory
eberlm <eberlm@in.tum.de>
parents: 65552
diff changeset
  1691
69785
9e326f6f8a24 More material for HOL-Number_Theory: ord, Carmichael's function, primitive roots
Manuel Eberl <eberlm@in.tum.de>
parents: 68606
diff changeset
  1692
lemma divide_out_primepow_ex:
9e326f6f8a24 More material for HOL-Number_Theory: ord, Carmichael's function, primitive roots
Manuel Eberl <eberlm@in.tum.de>
parents: 68606
diff changeset
  1693
  assumes "n \<noteq> 0" "\<exists>p\<in>prime_factors n. P p"
9e326f6f8a24 More material for HOL-Number_Theory: ord, Carmichael's function, primitive roots
Manuel Eberl <eberlm@in.tum.de>
parents: 68606
diff changeset
  1694
  obtains p k n' where "P p" "prime p" "p dvd n" "\<not>p dvd n'" "k > 0" "n = p ^ k * n'"
9e326f6f8a24 More material for HOL-Number_Theory: ord, Carmichael's function, primitive roots
Manuel Eberl <eberlm@in.tum.de>
parents: 68606
diff changeset
  1695
proof -
9e326f6f8a24 More material for HOL-Number_Theory: ord, Carmichael's function, primitive roots
Manuel Eberl <eberlm@in.tum.de>
parents: 68606
diff changeset
  1696
  from assms obtain p where p: "P p" "prime p" "p dvd n"
9e326f6f8a24 More material for HOL-Number_Theory: ord, Carmichael's function, primitive roots
Manuel Eberl <eberlm@in.tum.de>
parents: 68606
diff changeset
  1697
    by auto
9e326f6f8a24 More material for HOL-Number_Theory: ord, Carmichael's function, primitive roots
Manuel Eberl <eberlm@in.tum.de>
parents: 68606
diff changeset
  1698
  define k where "k = multiplicity p n"
9e326f6f8a24 More material for HOL-Number_Theory: ord, Carmichael's function, primitive roots
Manuel Eberl <eberlm@in.tum.de>
parents: 68606
diff changeset
  1699
  define n' where "n' = n div p ^ k"
9e326f6f8a24 More material for HOL-Number_Theory: ord, Carmichael's function, primitive roots
Manuel Eberl <eberlm@in.tum.de>
parents: 68606
diff changeset
  1700
  have n': "n = p ^ k * n'" "\<not>p dvd n'"
9e326f6f8a24 More material for HOL-Number_Theory: ord, Carmichael's function, primitive roots
Manuel Eberl <eberlm@in.tum.de>
parents: 68606
diff changeset
  1701
    using assms p multiplicity_decompose[of n p]
9e326f6f8a24 More material for HOL-Number_Theory: ord, Carmichael's function, primitive roots
Manuel Eberl <eberlm@in.tum.de>
parents: 68606
diff changeset
  1702
    by (auto simp: n'_def k_def multiplicity_dvd)
9e326f6f8a24 More material for HOL-Number_Theory: ord, Carmichael's function, primitive roots
Manuel Eberl <eberlm@in.tum.de>
parents: 68606
diff changeset
  1703
  from n' p have "k > 0" by (intro Nat.gr0I) auto
9e326f6f8a24 More material for HOL-Number_Theory: ord, Carmichael's function, primitive roots
Manuel Eberl <eberlm@in.tum.de>
parents: 68606
diff changeset
  1704
  with n' p that[of p n' k] show ?thesis by auto
9e326f6f8a24 More material for HOL-Number_Theory: ord, Carmichael's function, primitive roots
Manuel Eberl <eberlm@in.tum.de>
parents: 68606
diff changeset
  1705
qed
9e326f6f8a24 More material for HOL-Number_Theory: ord, Carmichael's function, primitive roots
Manuel Eberl <eberlm@in.tum.de>
parents: 68606
diff changeset
  1706
9e326f6f8a24 More material for HOL-Number_Theory: ord, Carmichael's function, primitive roots
Manuel Eberl <eberlm@in.tum.de>
parents: 68606
diff changeset
  1707
lemma divide_out_primepow:
9e326f6f8a24 More material for HOL-Number_Theory: ord, Carmichael's function, primitive roots
Manuel Eberl <eberlm@in.tum.de>
parents: 68606
diff changeset
  1708
  assumes "n \<noteq> 0" "\<not>is_unit n"
9e326f6f8a24 More material for HOL-Number_Theory: ord, Carmichael's function, primitive roots
Manuel Eberl <eberlm@in.tum.de>
parents: 68606
diff changeset
  1709
  obtains p k n' where "prime p" "p dvd n" "\<not>p dvd n'" "k > 0" "n = p ^ k * n'"
9e326f6f8a24 More material for HOL-Number_Theory: ord, Carmichael's function, primitive roots
Manuel Eberl <eberlm@in.tum.de>
parents: 68606
diff changeset
  1710
  using divide_out_primepow_ex[OF assms(1), of "\<lambda>_. True"] prime_divisor_exists[OF assms] assms
9e326f6f8a24 More material for HOL-Number_Theory: ord, Carmichael's function, primitive roots
Manuel Eberl <eberlm@in.tum.de>
parents: 68606
diff changeset
  1711
        prime_factorsI by metis
9e326f6f8a24 More material for HOL-Number_Theory: ord, Carmichael's function, primitive roots
Manuel Eberl <eberlm@in.tum.de>
parents: 68606
diff changeset
  1712
63534
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  1713
63924
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1714
subsection \<open>GCD and LCM computation with unique factorizations\<close>
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63919
diff changeset
  1715
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1716
definition "gcd_factorial a b = (if a = 0 then normalize b
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1717
     else if b = 0 then normalize a
71398
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1718
     else normalize (prod_mset (prime_factorization a \<inter># prime_factorization b)))"
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1719
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1720
definition "lcm_factorial a b = (if a = 0 \<or> b = 0 then 0
71398
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1721
     else normalize (prod_mset (prime_factorization a \<union># prime_factorization b)))"
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1722
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1723
definition "Gcd_factorial A =
71398
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1724
  (if A \<subseteq> {0} then 0 else normalize (prod_mset (Inf (prime_factorization ` (A - {0})))))"
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1725
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1726
definition "Lcm_factorial A =
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1727
  (if A = {} then 1
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1728
   else if 0 \<notin> A \<and> subset_mset.bdd_above (prime_factorization ` (A - {0})) then
71398
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1729
     normalize (prod_mset (Sup (prime_factorization ` A)))
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1730
   else
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1731
     0)"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1732
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1733
lemma prime_factorization_gcd_factorial:
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1734
  assumes [simp]: "a \<noteq> 0" "b \<noteq> 0"
63919
9aed2da07200 # after multiset intersection and union symbol
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents: 63905
diff changeset
  1735
  shows   "prime_factorization (gcd_factorial a b) = prime_factorization a \<inter># prime_factorization b"
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1736
proof -
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1737
  have "prime_factorization (gcd_factorial a b) =
63919
9aed2da07200 # after multiset intersection and union symbol
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents: 63905
diff changeset
  1738
          prime_factorization (prod_mset (prime_factorization a \<inter># prime_factorization b))"
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1739
    by (simp add: gcd_factorial_def)
63919
9aed2da07200 # after multiset intersection and union symbol
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents: 63905
diff changeset
  1740
  also have "\<dots> = prime_factorization a \<inter># prime_factorization b"
63905
1c3dcb5fe6cb prefer abbreviation for trivial set conversion
haftmann
parents: 63904
diff changeset
  1741
    by (subst prime_factorization_prod_mset_primes) auto
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1742
  finally show ?thesis .
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1743
qed
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1744
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1745
lemma prime_factorization_lcm_factorial:
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1746
  assumes [simp]: "a \<noteq> 0" "b \<noteq> 0"
63919
9aed2da07200 # after multiset intersection and union symbol
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents: 63905
diff changeset
  1747
  shows   "prime_factorization (lcm_factorial a b) = prime_factorization a \<union># prime_factorization b"
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1748
proof -
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1749
  have "prime_factorization (lcm_factorial a b) =
63919
9aed2da07200 # after multiset intersection and union symbol
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents: 63905
diff changeset
  1750
          prime_factorization (prod_mset (prime_factorization a \<union># prime_factorization b))"
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1751
    by (simp add: lcm_factorial_def)
63919
9aed2da07200 # after multiset intersection and union symbol
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents: 63905
diff changeset
  1752
  also have "\<dots> = prime_factorization a \<union># prime_factorization b"
63905
1c3dcb5fe6cb prefer abbreviation for trivial set conversion
haftmann
parents: 63904
diff changeset
  1753
    by (subst prime_factorization_prod_mset_primes) auto
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1754
  finally show ?thesis .
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1755
qed
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1756
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1757
lemma prime_factorization_Gcd_factorial:
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1758
  assumes "\<not>A \<subseteq> {0}"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1759
  shows   "prime_factorization (Gcd_factorial A) = Inf (prime_factorization ` (A - {0}))"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1760
proof -
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1761
  from assms obtain x where x: "x \<in> A - {0}" by auto
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1762
  hence "Inf (prime_factorization ` (A - {0})) \<subseteq># prime_factorization x"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1763
    by (intro subset_mset.cInf_lower) simp_all
63905
1c3dcb5fe6cb prefer abbreviation for trivial set conversion
haftmann
parents: 63904
diff changeset
  1764
  hence "\<forall>y. y \<in># Inf (prime_factorization ` (A - {0})) \<longrightarrow> y \<in> prime_factors x"
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1765
    by (auto dest: mset_subset_eqD)
63905
1c3dcb5fe6cb prefer abbreviation for trivial set conversion
haftmann
parents: 63904
diff changeset
  1766
  with in_prime_factors_imp_prime[of _ x]
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
  1767
    have "\<forall>p. p \<in># Inf (prime_factorization ` (A - {0})) \<longrightarrow> prime p" by blast
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1768
  with assms show ?thesis
63830
2ea3725a34bd msetsum -> set_mset, msetprod -> prod_mset
nipkow
parents: 63793
diff changeset
  1769
    by (simp add: Gcd_factorial_def prime_factorization_prod_mset_primes)
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1770
qed
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1771
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1772
lemma prime_factorization_Lcm_factorial:
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1773
  assumes "0 \<notin> A" "subset_mset.bdd_above (prime_factorization ` A)"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1774
  shows   "prime_factorization (Lcm_factorial A) = Sup (prime_factorization ` A)"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1775
proof (cases "A = {}")
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1776
  case True
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1777
  hence "prime_factorization ` A = {}" by auto
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1778
  also have "Sup \<dots> = {#}" by (simp add: Sup_multiset_empty)
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1779
  finally show ?thesis by (simp add: Lcm_factorial_def)
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1780
next
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1781
  case False
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
  1782
  have "\<forall>y. y \<in># Sup (prime_factorization ` A) \<longrightarrow> prime y"
63905
1c3dcb5fe6cb prefer abbreviation for trivial set conversion
haftmann
parents: 63904
diff changeset
  1783
    by (auto simp: in_Sup_multiset_iff assms)
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1784
  with assms False show ?thesis
63830
2ea3725a34bd msetsum -> set_mset, msetprod -> prod_mset
nipkow
parents: 63793
diff changeset
  1785
    by (simp add: Lcm_factorial_def prime_factorization_prod_mset_primes)
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1786
qed
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1787
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1788
lemma gcd_factorial_commute: "gcd_factorial a b = gcd_factorial b a"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1789
  by (simp add: gcd_factorial_def multiset_inter_commute)
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1790
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1791
lemma gcd_factorial_dvd1: "gcd_factorial a b dvd a"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1792
proof (cases "a = 0 \<or> b = 0")
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1793
  case False
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1794
  hence "gcd_factorial a b \<noteq> 0" by (auto simp: gcd_factorial_def)
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1795
  with False show ?thesis
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1796
    by (subst prime_factorization_subset_iff_dvd [symmetric])
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1797
       (auto simp: prime_factorization_gcd_factorial)
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1798
qed (auto simp: gcd_factorial_def)
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1799
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1800
lemma gcd_factorial_dvd2: "gcd_factorial a b dvd b"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1801
  by (subst gcd_factorial_commute) (rule gcd_factorial_dvd1)
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1802
71398
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1803
lemma normalize_gcd_factorial [simp]: "normalize (gcd_factorial a b) = gcd_factorial a b"
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1804
  by (simp add: gcd_factorial_def)
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1805
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1806
lemma normalize_lcm_factorial [simp]: "normalize (lcm_factorial a b) = lcm_factorial a b"
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1807
  by (simp add: lcm_factorial_def)
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1808
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1809
lemma gcd_factorial_greatest: "c dvd gcd_factorial a b" if "c dvd a" "c dvd b" for a b c
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1810
proof (cases "a = 0 \<or> b = 0")
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1811
  case False
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1812
  with that have [simp]: "c \<noteq> 0" by auto
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1813
  let ?p = "prime_factorization"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1814
  from that False have "?p c \<subseteq># ?p a" "?p c \<subseteq># ?p b"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1815
    by (simp_all add: prime_factorization_subset_iff_dvd)
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1816
  hence "prime_factorization c \<subseteq>#
63919
9aed2da07200 # after multiset intersection and union symbol
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents: 63905
diff changeset
  1817
           prime_factorization (prod_mset (prime_factorization a \<inter># prime_factorization b))"
63905
1c3dcb5fe6cb prefer abbreviation for trivial set conversion
haftmann
parents: 63904
diff changeset
  1818
    using False by (subst prime_factorization_prod_mset_primes) auto
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1819
  with False show ?thesis
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1820
    by (auto simp: gcd_factorial_def prime_factorization_subset_iff_dvd [symmetric])
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1821
qed (auto simp: gcd_factorial_def that)
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1822
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1823
lemma lcm_factorial_gcd_factorial:
71398
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1824
  "lcm_factorial a b = normalize (a * b div gcd_factorial a b)" for a b
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1825
proof (cases "a = 0 \<or> b = 0")
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1826
  case False
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1827
  let ?p = "prime_factorization"
71398
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1828
  have 1: "normalize x * normalize y dvd z \<longleftrightarrow> x * y dvd z" for x y z :: 'a
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1829
  proof -
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1830
    have "normalize (normalize x * normalize y) dvd z \<longleftrightarrow> x * y dvd z"
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1831
      unfolding normalize_mult_normalize_left normalize_mult_normalize_right by simp
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1832
    thus ?thesis unfolding normalize_dvd_iff by simp
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1833
  qed
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1834
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1835
  have "?p (a * b) = (?p a \<union># ?p b) + (?p a \<inter># ?p b)"
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1836
    using False by (subst prime_factorization_mult) (auto intro!: multiset_eqI)
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1837
  hence "normalize (prod_mset (?p (a * b))) =
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1838
           normalize (prod_mset ((?p a \<union># ?p b) + (?p a \<inter># ?p b)))"
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1839
    by (simp only:)
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1840
  hence *: "normalize (a * b) = normalize (lcm_factorial a b * gcd_factorial a b)" using False
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1841
    by (subst (asm) prod_mset_prime_factorization_weak)
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1842
       (auto simp: lcm_factorial_def gcd_factorial_def)
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1843
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1844
  have [simp]: "gcd_factorial a b dvd a * b" "lcm_factorial a b dvd a * b"
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1845
    using associatedD2[OF *] by auto
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1846
  from False have [simp]: "gcd_factorial a b \<noteq> 0" "lcm_factorial a b \<noteq> 0"
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1847
    by (auto simp: gcd_factorial_def lcm_factorial_def)
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1848
  
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1849
  show ?thesis
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1850
    by (rule associated_eqI)
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1851
       (use * in \<open>auto simp: dvd_div_iff_mult div_dvd_iff_mult dest: associatedD1 associatedD2\<close>)
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1852
qed (auto simp: lcm_factorial_def)
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1853
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1854
lemma normalize_Gcd_factorial:
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1855
  "normalize (Gcd_factorial A) = Gcd_factorial A"
71398
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1856
  by (simp add: Gcd_factorial_def)
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1857
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1858
lemma Gcd_factorial_eq_0_iff:
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1859
  "Gcd_factorial A = 0 \<longleftrightarrow> A \<subseteq> {0}"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1860
  by (auto simp: Gcd_factorial_def in_Inf_multiset_iff split: if_splits)
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1861
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1862
lemma Gcd_factorial_dvd:
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1863
  assumes "x \<in> A"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1864
  shows   "Gcd_factorial A dvd x"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1865
proof (cases "x = 0")
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1866
  case False
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1867
  with assms have "prime_factorization (Gcd_factorial A) = Inf (prime_factorization ` (A - {0}))"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1868
    by (intro prime_factorization_Gcd_factorial) auto
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1869
  also from False assms have "\<dots> \<subseteq># prime_factorization x"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1870
    by (intro subset_mset.cInf_lower) auto
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1871
  finally show ?thesis
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1872
    by (subst (asm) prime_factorization_subset_iff_dvd)
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1873
       (insert assms False, auto simp: Gcd_factorial_eq_0_iff)
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1874
qed simp_all
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1875
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1876
lemma Gcd_factorial_greatest:
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1877
  assumes "\<And>y. y \<in> A \<Longrightarrow> x dvd y"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1878
  shows   "x dvd Gcd_factorial A"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1879
proof (cases "A \<subseteq> {0}")
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1880
  case False
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1881
  from False obtain y where "y \<in> A" "y \<noteq> 0" by auto
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1882
  with assms[of y] have nz: "x \<noteq> 0" by auto
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1883
  from nz assms have "prime_factorization x \<subseteq># prime_factorization y" if "y \<in> A - {0}" for y
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1884
    using that by (subst prime_factorization_subset_iff_dvd) auto
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1885
  with False have "prime_factorization x \<subseteq># Inf (prime_factorization ` (A - {0}))"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1886
    by (intro subset_mset.cInf_greatest) auto
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1887
  also from False have "\<dots> = prime_factorization (Gcd_factorial A)"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1888
    by (rule prime_factorization_Gcd_factorial [symmetric])
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1889
  finally show ?thesis
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1890
    by (subst (asm) prime_factorization_subset_iff_dvd)
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1891
       (insert nz False, auto simp: Gcd_factorial_eq_0_iff)
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1892
qed (simp_all add: Gcd_factorial_def)
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1893
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1894
lemma normalize_Lcm_factorial:
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1895
  "normalize (Lcm_factorial A) = Lcm_factorial A"
71398
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1896
  by (simp add: Lcm_factorial_def)
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1897
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1898
lemma Lcm_factorial_eq_0_iff:
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1899
  "Lcm_factorial A = 0 \<longleftrightarrow> 0 \<in> A \<or> \<not>subset_mset.bdd_above (prime_factorization ` A)"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1900
  by (auto simp: Lcm_factorial_def in_Sup_multiset_iff)
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1901
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1902
lemma dvd_Lcm_factorial:
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1903
  assumes "x \<in> A"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1904
  shows   "x dvd Lcm_factorial A"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1905
proof (cases "0 \<notin> A \<and> subset_mset.bdd_above (prime_factorization ` A)")
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1906
  case True
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1907
  with assms have [simp]: "0 \<notin> A" "x \<noteq> 0" "A \<noteq> {}" by auto
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1908
  from assms True have "prime_factorization x \<subseteq># Sup (prime_factorization ` A)"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1909
    by (intro subset_mset.cSup_upper) auto
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1910
  also have "\<dots> = prime_factorization (Lcm_factorial A)"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1911
    by (rule prime_factorization_Lcm_factorial [symmetric]) (insert True, simp_all)
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1912
  finally show ?thesis
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1913
    by (subst (asm) prime_factorization_subset_iff_dvd)
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1914
       (insert True, auto simp: Lcm_factorial_eq_0_iff)
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1915
qed (insert assms, auto simp: Lcm_factorial_def)
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1916
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1917
lemma Lcm_factorial_least:
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1918
  assumes "\<And>y. y \<in> A \<Longrightarrow> y dvd x"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1919
  shows   "Lcm_factorial A dvd x"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1920
proof -
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1921
  consider "A = {}" | "0 \<in> A" | "x = 0" | "A \<noteq> {}" "0 \<notin> A" "x \<noteq> 0" by blast
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1922
  thus ?thesis
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1923
  proof cases
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1924
    assume *: "A \<noteq> {}" "0 \<notin> A" "x \<noteq> 0"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1925
    hence nz: "x \<noteq> 0" if "x \<in> A" for x using that by auto
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1926
    from * have bdd: "subset_mset.bdd_above (prime_factorization ` A)"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1927
      by (intro subset_mset.bdd_aboveI[of _ "prime_factorization x"])
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1928
         (auto simp: prime_factorization_subset_iff_dvd nz dest: assms)
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1929
    have "prime_factorization (Lcm_factorial A) = Sup (prime_factorization ` A)"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1930
      by (rule prime_factorization_Lcm_factorial) fact+
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1931
    also from * have "\<dots> \<subseteq># prime_factorization x"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1932
      by (intro subset_mset.cSup_least)
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1933
         (auto simp: prime_factorization_subset_iff_dvd nz dest: assms)
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1934
    finally show ?thesis
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1935
      by (subst (asm) prime_factorization_subset_iff_dvd)
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1936
         (insert * bdd, auto simp: Lcm_factorial_eq_0_iff)
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1937
  qed (auto simp: Lcm_factorial_def dest: assms)
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1938
qed
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1939
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1940
lemmas gcd_lcm_factorial =
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1941
  gcd_factorial_dvd1 gcd_factorial_dvd2 gcd_factorial_greatest
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1942
  normalize_gcd_factorial lcm_factorial_gcd_factorial
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1943
  normalize_Gcd_factorial Gcd_factorial_dvd Gcd_factorial_greatest
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1944
  normalize_Lcm_factorial dvd_Lcm_factorial Lcm_factorial_least
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1945
60804
080a979a985b formal class for factorial (semi)rings
haftmann
parents:
diff changeset
  1946
end
080a979a985b formal class for factorial (semi)rings
haftmann
parents:
diff changeset
  1947
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1948
class factorial_semiring_gcd = factorial_semiring + gcd + Gcd +
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1949
  assumes gcd_eq_gcd_factorial: "gcd a b = gcd_factorial a b"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1950
  and     lcm_eq_lcm_factorial: "lcm a b = lcm_factorial a b"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1951
  and     Gcd_eq_Gcd_factorial: "Gcd A = Gcd_factorial A"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1952
  and     Lcm_eq_Lcm_factorial: "Lcm A = Lcm_factorial A"
60804
080a979a985b formal class for factorial (semi)rings
haftmann
parents:
diff changeset
  1953
begin
080a979a985b formal class for factorial (semi)rings
haftmann
parents:
diff changeset
  1954
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1955
lemma prime_factorization_gcd:
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1956
  assumes [simp]: "a \<noteq> 0" "b \<noteq> 0"
63919
9aed2da07200 # after multiset intersection and union symbol
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents: 63905
diff changeset
  1957
  shows   "prime_factorization (gcd a b) = prime_factorization a \<inter># prime_factorization b"
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1958
  by (simp add: gcd_eq_gcd_factorial prime_factorization_gcd_factorial)
60804
080a979a985b formal class for factorial (semi)rings
haftmann
parents:
diff changeset
  1959
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1960
lemma prime_factorization_lcm:
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1961
  assumes [simp]: "a \<noteq> 0" "b \<noteq> 0"
63919
9aed2da07200 # after multiset intersection and union symbol
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents: 63905
diff changeset
  1962
  shows   "prime_factorization (lcm a b) = prime_factorization a \<union># prime_factorization b"
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1963
  by (simp add: lcm_eq_lcm_factorial prime_factorization_lcm_factorial)
60804
080a979a985b formal class for factorial (semi)rings
haftmann
parents:
diff changeset
  1964
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1965
lemma prime_factorization_Gcd:
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1966
  assumes "Gcd A \<noteq> 0"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1967
  shows   "prime_factorization (Gcd A) = Inf (prime_factorization ` (A - {0}))"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1968
  using assms
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1969
  by (simp add: prime_factorization_Gcd_factorial Gcd_eq_Gcd_factorial Gcd_factorial_eq_0_iff)
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1970
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1971
lemma prime_factorization_Lcm:
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1972
  assumes "Lcm A \<noteq> 0"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1973
  shows   "prime_factorization (Lcm A) = Sup (prime_factorization ` A)"
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1974
  using assms
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1975
  by (simp add: prime_factorization_Lcm_factorial Lcm_eq_Lcm_factorial Lcm_factorial_eq_0_iff)
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1976
66276
acc3b7dd0b21 More material on powers for HOL-Computational_Algebra/HOL-Number_Theory
eberlm <eberlm@in.tum.de>
parents: 65552
diff changeset
  1977
lemma prime_factors_gcd [simp]: 
acc3b7dd0b21 More material on powers for HOL-Computational_Algebra/HOL-Number_Theory
eberlm <eberlm@in.tum.de>
parents: 65552
diff changeset
  1978
  "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> prime_factors (gcd a b) = 
acc3b7dd0b21 More material on powers for HOL-Computational_Algebra/HOL-Number_Theory
eberlm <eberlm@in.tum.de>
parents: 65552
diff changeset
  1979
     prime_factors a \<inter> prime_factors b"
acc3b7dd0b21 More material on powers for HOL-Computational_Algebra/HOL-Number_Theory
eberlm <eberlm@in.tum.de>
parents: 65552
diff changeset
  1980
  by (subst prime_factorization_gcd) auto
acc3b7dd0b21 More material on powers for HOL-Computational_Algebra/HOL-Number_Theory
eberlm <eberlm@in.tum.de>
parents: 65552
diff changeset
  1981
acc3b7dd0b21 More material on powers for HOL-Computational_Algebra/HOL-Number_Theory
eberlm <eberlm@in.tum.de>
parents: 65552
diff changeset
  1982
lemma prime_factors_lcm [simp]: 
acc3b7dd0b21 More material on powers for HOL-Computational_Algebra/HOL-Number_Theory
eberlm <eberlm@in.tum.de>
parents: 65552
diff changeset
  1983
  "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> prime_factors (lcm a b) = 
acc3b7dd0b21 More material on powers for HOL-Computational_Algebra/HOL-Number_Theory
eberlm <eberlm@in.tum.de>
parents: 65552
diff changeset
  1984
     prime_factors a \<union> prime_factors b"
acc3b7dd0b21 More material on powers for HOL-Computational_Algebra/HOL-Number_Theory
eberlm <eberlm@in.tum.de>
parents: 65552
diff changeset
  1985
  by (subst prime_factorization_lcm) auto
acc3b7dd0b21 More material on powers for HOL-Computational_Algebra/HOL-Number_Theory
eberlm <eberlm@in.tum.de>
parents: 65552
diff changeset
  1986
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1987
subclass semiring_gcd
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1988
  by (standard, unfold gcd_eq_gcd_factorial lcm_eq_lcm_factorial)
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1989
     (rule gcd_lcm_factorial; assumption)+
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1990
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1991
subclass semiring_Gcd
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1992
  by (standard, unfold Gcd_eq_Gcd_factorial Lcm_eq_Lcm_factorial)
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  1993
     (rule gcd_lcm_factorial; assumption)+
60804
080a979a985b formal class for factorial (semi)rings
haftmann
parents:
diff changeset
  1994
63534
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  1995
lemma
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  1996
  assumes "x \<noteq> 0" "y \<noteq> 0"
65552
f533820e7248 theories "GCD" and "Binomial" are already included in "Main": this avoids improper imports in applications;
wenzelm
parents: 65435
diff changeset
  1997
  shows gcd_eq_factorial':
71398
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  1998
          "gcd x y = normalize (\<Prod>p \<in> prime_factors x \<inter> prime_factors y.
63534
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  1999
                          p ^ min (multiplicity p x) (multiplicity p y))" (is "_ = ?rhs1")
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  2000
    and lcm_eq_factorial':
71398
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2001
          "lcm x y = normalize (\<Prod>p \<in> prime_factors x \<union> prime_factors y.
63534
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  2002
                          p ^ max (multiplicity p x) (multiplicity p y))" (is "_ = ?rhs2")
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  2003
proof -
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  2004
  have "gcd x y = gcd_factorial x y" by (rule gcd_eq_gcd_factorial)
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  2005
  also have "\<dots> = ?rhs1"
63905
1c3dcb5fe6cb prefer abbreviation for trivial set conversion
haftmann
parents: 63904
diff changeset
  2006
    by (auto simp: gcd_factorial_def assms prod_mset_multiplicity
71398
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2007
          count_prime_factorization_prime
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2008
          intro!: arg_cong[of _ _ normalize] dest: in_prime_factors_imp_prime intro!: prod.cong)
63534
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  2009
  finally show "gcd x y = ?rhs1" .
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  2010
  have "lcm x y = lcm_factorial x y" by (rule lcm_eq_lcm_factorial)
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  2011
  also have "\<dots> = ?rhs2"
63905
1c3dcb5fe6cb prefer abbreviation for trivial set conversion
haftmann
parents: 63904
diff changeset
  2012
    by (auto simp: lcm_factorial_def assms prod_mset_multiplicity
71398
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2013
          count_prime_factorization_prime intro!: arg_cong[of _ _ normalize] 
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2014
          dest: in_prime_factors_imp_prime intro!: prod.cong)
63534
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  2015
  finally show "lcm x y = ?rhs2" .
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  2016
qed
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  2017
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  2018
lemma
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63547
diff changeset
  2019
  assumes "x \<noteq> 0" "y \<noteq> 0" "prime p"
63534
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  2020
  shows   multiplicity_gcd: "multiplicity p (gcd x y) = min (multiplicity p x) (multiplicity p y)"
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  2021
    and   multiplicity_lcm: "multiplicity p (lcm x y) = max (multiplicity p x) (multiplicity p y)"
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  2022
proof -
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  2023
  have "gcd x y = gcd_factorial x y" by (rule gcd_eq_gcd_factorial)
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  2024
  also from assms have "multiplicity p \<dots> = min (multiplicity p x) (multiplicity p y)"
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  2025
    by (simp add: count_prime_factorization_prime [symmetric] prime_factorization_gcd_factorial)
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  2026
  finally show "multiplicity p (gcd x y) = min (multiplicity p x) (multiplicity p y)" .
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  2027
  have "lcm x y = lcm_factorial x y" by (rule lcm_eq_lcm_factorial)
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  2028
  also from assms have "multiplicity p \<dots> = max (multiplicity p x) (multiplicity p y)"
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  2029
    by (simp add: count_prime_factorization_prime [symmetric] prime_factorization_lcm_factorial)
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  2030
  finally show "multiplicity p (lcm x y) = max (multiplicity p x) (multiplicity p y)" .
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  2031
qed
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  2032
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  2033
lemma gcd_lcm_distrib:
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  2034
  "gcd x (lcm y z) = lcm (gcd x y) (gcd x z)"
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  2035
proof (cases "x = 0 \<or> y = 0 \<or> z = 0")
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  2036
  case True
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  2037
  thus ?thesis
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  2038
    by (auto simp: lcm_proj1_if_dvd lcm_proj2_if_dvd)
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  2039
next
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  2040
  case False
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  2041
  hence "normalize (gcd x (lcm y z)) = normalize (lcm (gcd x y) (gcd x z))"
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  2042
    by (intro associatedI prime_factorization_subset_imp_dvd)
65552
f533820e7248 theories "GCD" and "Binomial" are already included in "Main": this avoids improper imports in applications;
wenzelm
parents: 65435
diff changeset
  2043
       (auto simp: lcm_eq_0_iff prime_factorization_gcd prime_factorization_lcm
63534
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  2044
          subset_mset.inf_sup_distrib1)
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  2045
  thus ?thesis by simp
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  2046
qed
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  2047
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  2048
lemma lcm_gcd_distrib:
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  2049
  "lcm x (gcd y z) = gcd (lcm x y) (lcm x z)"
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  2050
proof (cases "x = 0 \<or> y = 0 \<or> z = 0")
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  2051
  case True
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  2052
  thus ?thesis
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  2053
    by (auto simp: lcm_proj1_if_dvd lcm_proj2_if_dvd)
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  2054
next
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  2055
  case False
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  2056
  hence "normalize (lcm x (gcd y z)) = normalize (gcd (lcm x y) (lcm x z))"
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  2057
    by (intro associatedI prime_factorization_subset_imp_dvd)
65552
f533820e7248 theories "GCD" and "Binomial" are already included in "Main": this avoids improper imports in applications;
wenzelm
parents: 65435
diff changeset
  2058
       (auto simp: lcm_eq_0_iff prime_factorization_gcd prime_factorization_lcm
63534
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  2059
          subset_mset.sup_inf_distrib1)
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  2060
  thus ?thesis by simp
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  2061
qed
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63498
diff changeset
  2062
60804
080a979a985b formal class for factorial (semi)rings
haftmann
parents:
diff changeset
  2063
end
080a979a985b formal class for factorial (semi)rings
haftmann
parents:
diff changeset
  2064
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  2065
class factorial_ring_gcd = factorial_semiring_gcd + idom
60804
080a979a985b formal class for factorial (semi)rings
haftmann
parents:
diff changeset
  2066
begin
080a979a985b formal class for factorial (semi)rings
haftmann
parents:
diff changeset
  2067
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  2068
subclass ring_gcd ..
60804
080a979a985b formal class for factorial (semi)rings
haftmann
parents:
diff changeset
  2069
63498
a3fe3250d05d Reformed factorial rings
eberlm <eberlm@in.tum.de>
parents: 63133
diff changeset
  2070
subclass idom_divide ..
60804
080a979a985b formal class for factorial (semi)rings
haftmann
parents:
diff changeset
  2071
080a979a985b formal class for factorial (semi)rings
haftmann
parents:
diff changeset
  2072
end
080a979a985b formal class for factorial (semi)rings
haftmann
parents:
diff changeset
  2073
71398
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2074
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2075
class factorial_semiring_multiplicative =
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2076
  factorial_semiring + normalization_semidom_multiplicative
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2077
begin
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2078
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2079
lemma normalize_prod_mset_primes:
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2080
  "(\<And>p. p \<in># A \<Longrightarrow> prime p) \<Longrightarrow> normalize (prod_mset A) = prod_mset A"
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2081
proof (induction A)
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2082
  case (add p A)
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2083
  hence "prime p" by simp
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2084
  hence "normalize p = p" by simp
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2085
  with add show ?case by (simp add: normalize_mult)
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2086
qed simp_all
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2087
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2088
lemma prod_mset_prime_factorization:
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2089
  assumes "x \<noteq> 0"
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2090
  shows   "prod_mset (prime_factorization x) = normalize x"
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2091
  using assms
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2092
  by (induction x rule: prime_divisors_induct)
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2093
     (simp_all add: prime_factorization_unit prime_factorization_times_prime
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2094
                    is_unit_normalize normalize_mult)
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2095
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2096
lemma prime_decomposition: "unit_factor x * prod_mset (prime_factorization x) = x"
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2097
  by (cases "x = 0") (simp_all add: prod_mset_prime_factorization)
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2098
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2099
lemma prod_prime_factors:
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2100
  assumes "x \<noteq> 0"
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2101
  shows   "(\<Prod>p \<in> prime_factors x. p ^ multiplicity p x) = normalize x"
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2102
proof -
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2103
  have "normalize x = prod_mset (prime_factorization x)"
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2104
    by (simp add: prod_mset_prime_factorization assms)
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2105
  also have "\<dots> = (\<Prod>p \<in> prime_factors x. p ^ count (prime_factorization x) p)"
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2106
    by (subst prod_mset_multiplicity) simp_all
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2107
  also have "\<dots> = (\<Prod>p \<in> prime_factors x. p ^ multiplicity p x)"
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2108
    by (intro prod.cong)
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2109
      (simp_all add: assms count_prime_factorization_prime in_prime_factors_imp_prime)
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2110
  finally show ?thesis ..
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2111
qed
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2112
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2113
lemma prime_factorization_unique'':
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2114
  assumes S_eq: "S = {p. 0 < f p}"
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2115
    and "finite S"
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2116
    and S: "\<forall>p\<in>S. prime p" "normalize n = (\<Prod>p\<in>S. p ^ f p)"
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2117
  shows "S = prime_factors n \<and> (\<forall>p. prime p \<longrightarrow> f p = multiplicity p n)"
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2118
proof
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2119
  define A where "A = Abs_multiset f"
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2120
  from \<open>finite S\<close> S(1) have "(\<Prod>p\<in>S. p ^ f p) \<noteq> 0" by auto
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2121
  with S(2) have nz: "n \<noteq> 0" by auto
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2122
  from S_eq \<open>finite S\<close> have count_A: "count A = f"
73270
e2d03448d5b5 get rid of traditional predicate
haftmann
parents: 73127
diff changeset
  2123
    unfolding A_def by (subst multiset.Abs_multiset_inverse) simp_all
71398
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2124
  from S_eq count_A have set_mset_A: "set_mset A = S"
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2125
    by (simp only: set_mset_def)
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2126
  from S(2) have "normalize n = (\<Prod>p\<in>S. p ^ f p)" .
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2127
  also have "\<dots> = prod_mset A" by (simp add: prod_mset_multiplicity S_eq set_mset_A count_A)
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2128
  also from nz have "normalize n = prod_mset (prime_factorization n)"
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2129
    by (simp add: prod_mset_prime_factorization)
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2130
  finally have "prime_factorization (prod_mset A) =
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2131
                  prime_factorization (prod_mset (prime_factorization n))" by simp
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2132
  also from S(1) have "prime_factorization (prod_mset A) = A"
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2133
    by (intro prime_factorization_prod_mset_primes) (auto simp: set_mset_A)
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2134
  also have "prime_factorization (prod_mset (prime_factorization n)) = prime_factorization n"
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2135
    by (intro prime_factorization_prod_mset_primes) auto
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2136
  finally show "S = prime_factors n" by (simp add: set_mset_A [symmetric])
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2137
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2138
  show "(\<forall>p. prime p \<longrightarrow> f p = multiplicity p n)"
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2139
  proof safe
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2140
    fix p :: 'a assume p: "prime p"
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2141
    have "multiplicity p n = multiplicity p (normalize n)" by simp
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2142
    also have "normalize n = prod_mset A"
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2143
      by (simp add: prod_mset_multiplicity S_eq set_mset_A count_A S)
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2144
    also from p set_mset_A S(1)
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2145
    have "multiplicity p \<dots> = sum_mset (image_mset (multiplicity p) A)"
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2146
      by (intro prime_elem_multiplicity_prod_mset_distrib) auto
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2147
    also from S(1) p
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2148
    have "image_mset (multiplicity p) A = image_mset (\<lambda>q. if p = q then 1 else 0) A"
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2149
      by (intro image_mset_cong) (auto simp: set_mset_A multiplicity_self prime_multiplicity_other)
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2150
    also have "sum_mset \<dots> = f p"
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2151
      by (simp add: semiring_1_class.sum_mset_delta' count_A)
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2152
    finally show "f p = multiplicity p n" ..
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2153
  qed
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2154
qed
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2155
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2156
lemma divides_primepow:
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2157
  assumes "prime p" and "a dvd p ^ n"
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2158
  obtains m where "m \<le> n" and "normalize a = p ^ m"
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2159
  using divides_primepow_weak[OF assms] that assms
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2160
  by (auto simp add: normalize_power)
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2161
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2162
lemma Ex_other_prime_factor:
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2163
  assumes "n \<noteq> 0" and "\<not>(\<exists>k. normalize n = p ^ k)" "prime p"
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2164
  shows   "\<exists>q\<in>prime_factors n. q \<noteq> p"
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2165
proof (rule ccontr)
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2166
  assume *: "\<not>(\<exists>q\<in>prime_factors n. q \<noteq> p)"
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2167
  have "normalize n = (\<Prod>p\<in>prime_factors n. p ^ multiplicity p n)"
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2168
    using assms(1) by (intro prod_prime_factors [symmetric]) auto
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2169
  also from * have "\<dots> = (\<Prod>p\<in>{p}. p ^ multiplicity p n)"
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2170
    using assms(3) by (intro prod.mono_neutral_left) (auto simp: prime_factors_multiplicity)
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2171
  finally have "normalize n = p ^ multiplicity p n" by auto
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2172
  with assms show False by auto
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2173
qed
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2174
73127
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
  2175
text \<open>Now a string of results due to Jakub Kądziołka\<close>
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
  2176
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
  2177
lemma multiplicity_dvd_iff_dvd:
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
  2178
 assumes "x \<noteq> 0"
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
  2179
 shows "p^k dvd x \<longleftrightarrow> p^k dvd p^multiplicity p x"
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
  2180
proof (cases "is_unit p")
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
  2181
 case True
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
  2182
 then have "is_unit (p^k)"
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
  2183
   using is_unit_power_iff by simp
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
  2184
 hence "p^k dvd x"
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
  2185
   by auto
74885
2df334453c4c isabelle update_cartouches;
wenzelm
parents: 74542
diff changeset
  2186
 moreover from \<open>is_unit p\<close> have "p^k dvd p^multiplicity p x"
73127
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
  2187
   using multiplicity_unit_left is_unit_power_iff by simp
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
  2188
 ultimately show ?thesis by simp
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
  2189
next
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
  2190
 case False
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
  2191
 show ?thesis
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
  2192
 proof (cases "p = 0")
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
  2193
   case True
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
  2194
   then have "p^multiplicity p x = 1"
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
  2195
     by simp
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
  2196
   moreover have "p^k dvd x \<Longrightarrow> k = 0"
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
  2197
   proof (rule ccontr)
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
  2198
     assume "p^k dvd x" and "k \<noteq> 0"
74885
2df334453c4c isabelle update_cartouches;
wenzelm
parents: 74542
diff changeset
  2199
     with \<open>p = 0\<close> have "p^k = 0" by auto
2df334453c4c isabelle update_cartouches;
wenzelm
parents: 74542
diff changeset
  2200
     with \<open>p^k dvd x\<close> have "0 dvd x" by auto
73127
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
  2201
     hence "x = 0" by auto
74885
2df334453c4c isabelle update_cartouches;
wenzelm
parents: 74542
diff changeset
  2202
     with \<open>x \<noteq> 0\<close> show False by auto
73127
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
  2203
   qed
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
  2204
   ultimately show ?thesis
74885
2df334453c4c isabelle update_cartouches;
wenzelm
parents: 74542
diff changeset
  2205
     by (auto simp add: is_unit_power_iff \<open>\<not> is_unit p\<close>)
73127
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
  2206
 next
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
  2207
   case False
74885
2df334453c4c isabelle update_cartouches;
wenzelm
parents: 74542
diff changeset
  2208
   with \<open>x \<noteq> 0\<close> \<open>\<not> is_unit p\<close> show ?thesis
73127
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
  2209
     by (simp add: power_dvd_iff_le_multiplicity dvd_power_iff multiplicity_same_power)
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
  2210
 qed
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
  2211
qed
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
  2212
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
  2213
lemma multiplicity_decomposeI:
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
  2214
 assumes "x = p^k * x'" and "\<not> p dvd x'" and "p \<noteq> 0"
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
  2215
 shows "multiplicity p x = k"
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
  2216
  using assms local.multiplicity_eqI local.power_Suc2 by force
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
  2217
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
  2218
lemma multiplicity_sum_lt:
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
  2219
 assumes "multiplicity p a < multiplicity p b" "a \<noteq> 0" "b \<noteq> 0"
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
  2220
 shows "multiplicity p (a + b) = multiplicity p a"
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
  2221
proof -
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
  2222
 let ?vp = "multiplicity p"
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
  2223
 have unit: "\<not> is_unit p"
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
  2224
 proof
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
  2225
   assume "is_unit p"
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
  2226
   then have "?vp a = 0" and "?vp b = 0" using multiplicity_unit_left by auto
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
  2227
   with assms show False by auto
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
  2228
 qed
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
  2229
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
  2230
 from multiplicity_decompose' obtain a' where a': "a = p^?vp a * a'" "\<not> p dvd a'"
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
  2231
   using unit assms by metis
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
  2232
 from multiplicity_decompose' obtain b' where b': "b = p^?vp b * b'"
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
  2233
   using unit assms by metis
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
  2234
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
  2235
 show "?vp (a + b) = ?vp a"
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
  2236
 proof (rule multiplicity_decomposeI)
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
  2237
   let ?k = "?vp b - ?vp a"
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
  2238
   from assms have k: "?k > 0" by simp
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
  2239
   with b' have "b = p^?vp a * p^?k * b'"
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
  2240
     by (simp flip: power_add)
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
  2241
   with a' show *: "a + b = p^?vp a * (a' + p^?k * b')"
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
  2242
     by (simp add: ac_simps distrib_left)
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
  2243
   moreover show "\<not> p dvd a' + p^?k * b'"
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
  2244
     using a' k dvd_add_left_iff by auto
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
  2245
   show "p \<noteq> 0" using assms by auto
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
  2246
 qed
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
  2247
qed
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
  2248
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
  2249
corollary multiplicity_sum_min:
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
  2250
 assumes "multiplicity p a \<noteq> multiplicity p b" "a \<noteq> 0" "b \<noteq> 0"
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
  2251
 shows "multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b)"
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
  2252
proof -
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
  2253
 let ?vp = "multiplicity p"
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
  2254
 from assms have "?vp a < ?vp b \<or> ?vp a > ?vp b"
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
  2255
   by auto
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
  2256
 then show ?thesis
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
  2257
   by (metis assms multiplicity_sum_lt min.commute add_commute min.strict_order_iff)    
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
  2258
qed
4c4d479b097d new magerial from Jakub Kądziołka
paulson <lp15@cam.ac.uk>
parents: 73103
diff changeset
  2259
60804
080a979a985b formal class for factorial (semi)rings
haftmann
parents:
diff changeset
  2260
end
71398
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2261
76700
c48fe2be847f added lifting_forget as suggested by Peter Lammich
blanchet
parents: 74885
diff changeset
  2262
lifting_update multiset.lifting
c48fe2be847f added lifting_forget as suggested by Peter Lammich
blanchet
parents: 74885
diff changeset
  2263
lifting_forget multiset.lifting
c48fe2be847f added lifting_forget as suggested by Peter Lammich
blanchet
parents: 74885
diff changeset
  2264
71398
e0237f2eb49d Removed multiplicativity assumption from normalization_semidom
Manuel Eberl <eberlm@in.tum.de>
parents: 69785
diff changeset
  2265
end