diff --git a/src/HOL/Computational_Algebra/Factorial_Ring.thy b/src/HOL/Computational_Algebra/Factorial_Ring.thy
--- a/src/HOL/Computational_Algebra/Factorial_Ring.thy
+++ b/src/HOL/Computational_Algebra/Factorial_Ring.thy
@@ -1,2131 +1,2134 @@
 (*  Title:      HOL/Computational_Algebra/Factorial_Ring.thy
     Author:     Manuel Eberl, TU Muenchen
     Author:     Florian Haftmann, TU Muenchen
 *)
 
 section \<open>Factorial (semi)rings\<close>
 
 theory Factorial_Ring
 imports
   Main
   "HOL-Library.Multiset"
 begin
 
 subsection \<open>Irreducible and prime elements\<close>
 
 context comm_semiring_1
 begin
 
 definition irreducible :: "'a \<Rightarrow> bool" where
   "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)"
 
 lemma not_irreducible_zero [simp]: "\<not>irreducible 0"
   by (simp add: irreducible_def)
 
 lemma irreducible_not_unit: "irreducible p \<Longrightarrow> \<not>p dvd 1"
   by (simp add: irreducible_def)
 
 lemma not_irreducible_one [simp]: "\<not>irreducible 1"
   by (simp add: irreducible_def)
 
 lemma irreducibleI:
   "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"
   by (simp add: irreducible_def)
 
 lemma irreducibleD: "irreducible p \<Longrightarrow> p = a * b \<Longrightarrow> a dvd 1 \<or> b dvd 1"
   by (simp add: irreducible_def)
 
 lemma irreducible_mono:
   assumes irr: "irreducible b" and "a dvd b" "\<not>a dvd 1"
   shows   "irreducible a"
 proof (rule irreducibleI)
   fix c d assume "a = c * d"
   from assms obtain k where [simp]: "b = a * k" by auto
   from \<open>a = c * d\<close> have "b = c * d * k"
     by simp
   hence "c dvd 1 \<or> (d * k) dvd 1"
     using irreducibleD[OF irr, of c "d * k"] by (auto simp: mult.assoc)
   thus "c dvd 1 \<or> d dvd 1"
     by auto
 qed (use assms in \<open>auto simp: irreducible_def\<close>)
 
 definition prime_elem :: "'a \<Rightarrow> bool" where
   "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)"
 
 lemma not_prime_elem_zero [simp]: "\<not>prime_elem 0"
   by (simp add: prime_elem_def)
 
 lemma prime_elem_not_unit: "prime_elem p \<Longrightarrow> \<not>p dvd 1"
   by (simp add: prime_elem_def)
 
 lemma prime_elemI:
     "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"
   by (simp add: prime_elem_def)
 
 lemma prime_elem_dvd_multD:
     "prime_elem p \<Longrightarrow> p dvd (a * b) \<Longrightarrow> p dvd a \<or> p dvd b"
   by (simp add: prime_elem_def)
 
 lemma prime_elem_dvd_mult_iff:
   "prime_elem p \<Longrightarrow> p dvd (a * b) \<longleftrightarrow> p dvd a \<or> p dvd b"
   by (auto simp: prime_elem_def)
 
 lemma not_prime_elem_one [simp]:
   "\<not> prime_elem 1"
   by (auto dest: prime_elem_not_unit)
 
 lemma prime_elem_not_zeroI:
   assumes "prime_elem p"
   shows "p \<noteq> 0"
   using assms by (auto intro: ccontr)
 
 lemma prime_elem_dvd_power:
   "prime_elem p \<Longrightarrow> p dvd x ^ n \<Longrightarrow> p dvd x"
   by (induction n) (auto dest: prime_elem_dvd_multD intro: dvd_trans[of _ 1])
 
 lemma prime_elem_dvd_power_iff:
   "prime_elem p \<Longrightarrow> n > 0 \<Longrightarrow> p dvd x ^ n \<longleftrightarrow> p dvd x"
   by (auto dest: prime_elem_dvd_power intro: dvd_trans)
 
 lemma prime_elem_imp_nonzero [simp]:
   "ASSUMPTION (prime_elem x) \<Longrightarrow> x \<noteq> 0"
   unfolding ASSUMPTION_def by (rule prime_elem_not_zeroI)
 
 lemma prime_elem_imp_not_one [simp]:
   "ASSUMPTION (prime_elem x) \<Longrightarrow> x \<noteq> 1"
   unfolding ASSUMPTION_def by auto
 
 end
 
 
 lemma (in normalization_semidom) irreducible_cong:
   assumes "normalize a = normalize b"
   shows   "irreducible a \<longleftrightarrow> irreducible b"
 proof (cases "a = 0 \<or> a dvd 1")
   case True
   hence "\<not>irreducible a" by (auto simp: irreducible_def)
   from True have "normalize a = 0 \<or> normalize a dvd 1"
     by auto
   also note assms
   finally have "b = 0 \<or> b dvd 1" by simp
   hence "\<not>irreducible b" by (auto simp: irreducible_def)
   with \<open>\<not>irreducible a\<close> show ?thesis by simp
 next
   case False
   hence b: "b \<noteq> 0" "\<not>is_unit b" using assms
     by (auto simp: is_unit_normalize[of b])
   show ?thesis
   proof
     assume "irreducible a"
     thus "irreducible b"
       by (rule irreducible_mono) (use assms False b in \<open>auto dest: associatedD2\<close>)
   next
     assume "irreducible b"
     thus "irreducible a"
       by (rule irreducible_mono) (use assms False b in \<open>auto dest: associatedD1\<close>)
   qed
 qed
 
 lemma (in normalization_semidom) associatedE1:
   assumes "normalize a = normalize b"
   obtains u where "is_unit u" "a = u * b"
 proof (cases "a = 0")
   case [simp]: False
   from assms have [simp]: "b \<noteq> 0" by auto
   show ?thesis
   proof (rule that)
     show "is_unit (unit_factor a div unit_factor b)"
       by auto
     have "unit_factor a div unit_factor b * b = unit_factor a * (b div unit_factor b)"
       using \<open>b \<noteq> 0\<close> unit_div_commute unit_div_mult_swap unit_factor_is_unit by metis
     also have "b div unit_factor b = normalize b" by simp
     finally show "a = unit_factor a div unit_factor b * b"
       by (metis assms unit_factor_mult_normalize)
   qed
 next
   case [simp]: True
   hence [simp]: "b = 0"
     using assms[symmetric] by auto
   show ?thesis
     by (intro that[of 1]) auto
 qed
 
 lemma (in normalization_semidom) associatedE2:
   assumes "normalize a = normalize b"
   obtains u where "is_unit u" "b = u * a"
 proof -
   from assms have "normalize b = normalize a"
     by simp
   then obtain u where "is_unit u" "b = u * a"
     by (elim associatedE1)
   thus ?thesis using that by blast
 qed
   
 
 (* TODO Move *)
 lemma (in normalization_semidom) normalize_power_normalize:
   "normalize (normalize x ^ n) = normalize (x ^ n)"
 proof (induction n)
   case (Suc n)
   have "normalize (normalize x ^ Suc n) = normalize (x * normalize (normalize x ^ n))"
     by simp
   also note Suc.IH
   finally show ?case by simp
 qed auto
 
 context algebraic_semidom
 begin
 
 lemma prime_elem_imp_irreducible:
   assumes "prime_elem p"
   shows   "irreducible p"
 proof (rule irreducibleI)
   fix a b
   assume p_eq: "p = a * b"
   with assms have nz: "a \<noteq> 0" "b \<noteq> 0" by auto
   from p_eq have "p dvd a * b" by simp
   with \<open>prime_elem p\<close> have "p dvd a \<or> p dvd b" by (rule prime_elem_dvd_multD)
   with \<open>p = a * b\<close> have "a * b dvd 1 * b \<or> a * b dvd a * 1" by auto
   thus "a dvd 1 \<or> b dvd 1"
     by (simp only: dvd_times_left_cancel_iff[OF nz(1)] dvd_times_right_cancel_iff[OF nz(2)])
 qed (insert assms, simp_all add: prime_elem_def)
 
 lemma (in algebraic_semidom) unit_imp_no_irreducible_divisors:
   assumes "is_unit x" "irreducible p"
   shows   "\<not>p dvd x"
 proof (rule notI)
   assume "p dvd x"
   with \<open>is_unit x\<close> have "is_unit p"
     by (auto intro: dvd_trans)
   with \<open>irreducible p\<close> show False
     by (simp add: irreducible_not_unit)
 qed
 
 lemma unit_imp_no_prime_divisors:
   assumes "is_unit x" "prime_elem p"
   shows   "\<not>p dvd x"
   using unit_imp_no_irreducible_divisors[OF assms(1) prime_elem_imp_irreducible[OF assms(2)]] .
 
 lemma prime_elem_mono:
   assumes "prime_elem p" "\<not>q dvd 1" "q dvd p"
   shows   "prime_elem q"
 proof -
   from \<open>q dvd p\<close> obtain r where r: "p = q * r" by (elim dvdE)
   hence "p dvd q * r" by simp
   with \<open>prime_elem p\<close> have "p dvd q \<or> p dvd r" by (rule prime_elem_dvd_multD)
   hence "p dvd q"
   proof
     assume "p dvd r"
     then obtain s where s: "r = p * s" by (elim dvdE)
     from r have "p * 1 = p * (q * s)" by (subst (asm) s) (simp add: mult_ac)
     with \<open>prime_elem p\<close> have "q dvd 1"
       by (subst (asm) mult_cancel_left) auto
     with \<open>\<not>q dvd 1\<close> show ?thesis by contradiction
   qed
 
   show ?thesis
   proof (rule prime_elemI)
     fix a b assume "q dvd (a * b)"
     with \<open>p dvd q\<close> have "p dvd (a * b)" by (rule dvd_trans)
     with \<open>prime_elem p\<close> have "p dvd a \<or> p dvd b" by (rule prime_elem_dvd_multD)
     with \<open>q dvd p\<close> show "q dvd a \<or> q dvd b" by (blast intro: dvd_trans)
   qed (insert assms, auto)
 qed
 
 lemma irreducibleD':
   assumes "irreducible a" "b dvd a"
   shows   "a dvd b \<or> is_unit b"
 proof -
   from assms obtain c where c: "a = b * c" by (elim dvdE)
   from irreducibleD[OF assms(1) this] have "is_unit b \<or> is_unit c" .
   thus ?thesis by (auto simp: c mult_unit_dvd_iff)
 qed
 
 lemma irreducibleI':
   assumes "a \<noteq> 0" "\<not>is_unit a" "\<And>b. b dvd a \<Longrightarrow> a dvd b \<or> is_unit b"
   shows   "irreducible a"
 proof (rule irreducibleI)
   fix b c assume a_eq: "a = b * c"
   hence "a dvd b \<or> is_unit b" by (intro assms) simp_all
   thus "is_unit b \<or> is_unit c"
   proof
     assume "a dvd b"
     hence "b * c dvd b * 1" by (simp add: a_eq)
     moreover from \<open>a \<noteq> 0\<close> a_eq have "b \<noteq> 0" by auto
     ultimately show ?thesis by (subst (asm) dvd_times_left_cancel_iff) auto
   qed blast
 qed (simp_all add: assms(1,2))
 
 lemma irreducible_altdef:
   "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)"
   using irreducibleI'[of x] irreducibleD'[of x] irreducible_not_unit[of x] by auto
 
 lemma prime_elem_multD:
   assumes "prime_elem (a * b)"
   shows "is_unit a \<or> is_unit b"
 proof -
   from assms have "a \<noteq> 0" "b \<noteq> 0" by (auto dest!: prime_elem_not_zeroI)
   moreover from assms prime_elem_dvd_multD [of "a * b"] have "a * b dvd a \<or> a * b dvd b"
     by auto
   ultimately show ?thesis
     using dvd_times_left_cancel_iff [of a b 1]
       dvd_times_right_cancel_iff [of b a 1]
     by auto
 qed
 
 lemma prime_elemD2:
   assumes "prime_elem p" and "a dvd p" and "\<not> is_unit a"
   shows "p dvd a"
 proof -
   from \<open>a dvd p\<close> obtain b where "p = a * b" ..
   with \<open>prime_elem p\<close> prime_elem_multD \<open>\<not> is_unit a\<close> have "is_unit b" by auto
   with \<open>p = a * b\<close> show ?thesis
     by (auto simp add: mult_unit_dvd_iff)
 qed
 
 lemma prime_elem_dvd_prod_msetE:
   assumes "prime_elem p"
   assumes dvd: "p dvd prod_mset A"
   obtains a where "a \<in># A" and "p dvd a"
 proof -
   from dvd have "\<exists>a. a \<in># A \<and> p dvd a"
   proof (induct A)
     case empty then show ?case
     using \<open>prime_elem p\<close> by (simp add: prime_elem_not_unit)
   next
     case (add a A)
     then have "p dvd a * prod_mset A" by simp
     with \<open>prime_elem p\<close> consider (A) "p dvd prod_mset A" | (B) "p dvd a"
       by (blast dest: prime_elem_dvd_multD)
     then show ?case proof cases
       case B then show ?thesis by auto
     next
       case A
       with add.hyps obtain b where "b \<in># A" "p dvd b"
         by auto
       then show ?thesis by auto
     qed
   qed
   with that show thesis by blast
 
 qed
 
 context
 begin
 
 private lemma prime_elem_powerD:
   assumes "prime_elem (p ^ n)"
   shows   "prime_elem p \<and> n = 1"
 proof (cases n)
   case (Suc m)
   note assms
   also from Suc have "p ^ n = p * p^m" by simp
   finally have "is_unit p \<or> is_unit (p^m)" by (rule prime_elem_multD)
   moreover from assms have "\<not>is_unit p" by (simp add: prime_elem_def is_unit_power_iff)
   ultimately have "is_unit (p ^ m)" by simp
   with \<open>\<not>is_unit p\<close> have "m = 0" by (simp add: is_unit_power_iff)
   with Suc assms show ?thesis by simp
 qed (insert assms, simp_all)
 
 lemma prime_elem_power_iff:
   "prime_elem (p ^ n) \<longleftrightarrow> prime_elem p \<and> n = 1"
   by (auto dest: prime_elem_powerD)
 
 end
 
 lemma irreducible_mult_unit_left:
   "is_unit a \<Longrightarrow> irreducible (a * p) \<longleftrightarrow> irreducible p"
   by (auto simp: irreducible_altdef mult.commute[of a] is_unit_mult_iff
         mult_unit_dvd_iff dvd_mult_unit_iff)
 
 lemma prime_elem_mult_unit_left:
   "is_unit a \<Longrightarrow> prime_elem (a * p) \<longleftrightarrow> prime_elem p"
   by (auto simp: prime_elem_def mult.commute[of a] is_unit_mult_iff mult_unit_dvd_iff)
 
 lemma prime_elem_dvd_cases:
   assumes pk: "p*k dvd m*n" and p: "prime_elem p"
   shows "(\<exists>x. k dvd x*n \<and> m = p*x) \<or> (\<exists>y. k dvd m*y \<and> n = p*y)"
 proof -
   have "p dvd m*n" using dvd_mult_left pk by blast
   then consider "p dvd m" | "p dvd n"
     using p prime_elem_dvd_mult_iff by blast
   then show ?thesis
   proof cases
     case 1 then obtain a where "m = p * a" by (metis dvd_mult_div_cancel)
       then have "\<exists>x. k dvd x * n \<and> m = p * x"
         using p pk by (auto simp: mult.assoc)
     then show ?thesis ..
   next
     case 2 then obtain b where "n = p * b" by (metis dvd_mult_div_cancel)
     with p pk have "\<exists>y. k dvd m*y \<and> n = p*y"
       by (metis dvd_mult_right dvd_times_left_cancel_iff mult.left_commute mult_zero_left)
     then show ?thesis ..
   qed
 qed
 
 lemma prime_elem_power_dvd_prod:
   assumes pc: "p^c dvd m*n" and p: "prime_elem p"
   shows "\<exists>a b. a+b = c \<and> p^a dvd m \<and> p^b dvd n"
 using pc
 proof (induct c arbitrary: m n)
   case 0 show ?case by simp
 next
   case (Suc c)
   consider x where "p^c dvd x*n" "m = p*x" | y where "p^c dvd m*y" "n = p*y"
     using prime_elem_dvd_cases [of _ "p^c", OF _ p] Suc.prems by force
   then show ?case
   proof cases
     case (1 x)
     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
     with 1 have "Suc a + b = Suc c \<and> p ^ Suc a dvd m \<and> p ^ b dvd n"
       by (auto intro: mult_dvd_mono)
     thus ?thesis by blast
   next
     case (2 y)
     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
     with 2 have "a + Suc b = Suc c \<and> p ^ a dvd m \<and> p ^ Suc b dvd n"
       by (auto intro: mult_dvd_mono)
     with Suc.hyps [of m y] show "\<exists>a b. a + b = Suc c \<and> p ^ a dvd m \<and> p ^ b dvd n"
       by blast
   qed
 qed
 
 lemma prime_elem_power_dvd_cases:
   assumes "p ^ c dvd m * n" and "a + b = Suc c" and "prime_elem p"
   shows "p ^ a dvd m \<or> p ^ b dvd n"
 proof -
   from assms obtain r s
     where "r + s = c \<and> p ^ r dvd m \<and> p ^ s dvd n"
     by (blast dest: prime_elem_power_dvd_prod)
   moreover with assms have
     "a \<le> r \<or> b \<le> s" by arith
   ultimately show ?thesis by (auto intro: power_le_dvd)
 qed
 
 lemma prime_elem_not_unit' [simp]:
   "ASSUMPTION (prime_elem x) \<Longrightarrow> \<not>is_unit x"
   unfolding ASSUMPTION_def by (rule prime_elem_not_unit)
 
 lemma prime_elem_dvd_power_iff:
   assumes "prime_elem p"
   shows "p dvd a ^ n \<longleftrightarrow> p dvd a \<and> n > 0"
   using assms by (induct n) (auto dest: prime_elem_not_unit prime_elem_dvd_multD)
 
 lemma prime_power_dvd_multD:
   assumes "prime_elem p"
   assumes "p ^ n dvd a * b" and "n > 0" and "\<not> p dvd a"
   shows "p ^ n dvd b"
   using \<open>p ^ n dvd a * b\<close> and \<open>n > 0\<close>
 proof (induct n arbitrary: b)
   case 0 then show ?case by simp
 next
   case (Suc n) show ?case
   proof (cases "n = 0")
     case True with Suc \<open>prime_elem p\<close> \<open>\<not> p dvd a\<close> show ?thesis
       by (simp add: prime_elem_dvd_mult_iff)
   next
     case False then have "n > 0" by simp
     from \<open>prime_elem p\<close> have "p \<noteq> 0" by auto
     from Suc.prems have *: "p * p ^ n dvd a * b"
       by simp
     then have "p dvd a * b"
       by (rule dvd_mult_left)
     with Suc \<open>prime_elem p\<close> \<open>\<not> p dvd a\<close> have "p dvd b"
       by (simp add: prime_elem_dvd_mult_iff)
     moreover define c where "c = b div p"
     ultimately have b: "b = p * c" by simp
     with * have "p * p ^ n dvd p * (a * c)"
       by (simp add: ac_simps)
     with \<open>p \<noteq> 0\<close> have "p ^ n dvd a * c"
       by simp
     with Suc.hyps \<open>n > 0\<close> have "p ^ n dvd c"
       by blast
     with \<open>p \<noteq> 0\<close> show ?thesis
       by (simp add: b)
   qed
 qed
 
 end
 
 
 subsection \<open>Generalized primes: normalized prime elements\<close>
 
 context normalization_semidom
 begin
 
 lemma irreducible_normalized_divisors:
   assumes "irreducible x" "y dvd x" "normalize y = y"
   shows   "y = 1 \<or> y = normalize x"
 proof -
   from assms have "is_unit y \<or> x dvd y" by (auto simp: irreducible_altdef)
   thus ?thesis
   proof (elim disjE)
     assume "is_unit y"
     hence "normalize y = 1" by (simp add: is_unit_normalize)
     with assms show ?thesis by simp
   next
     assume "x dvd y"
     with \<open>y dvd x\<close> have "normalize y = normalize x" by (rule associatedI)
     with assms show ?thesis by simp
   qed
 qed
 
 lemma irreducible_normalize_iff [simp]: "irreducible (normalize x) = irreducible x"
   using irreducible_mult_unit_left[of "1 div unit_factor x" x]
   by (cases "x = 0") (simp_all add: unit_div_commute)
 
 lemma prime_elem_normalize_iff [simp]: "prime_elem (normalize x) = prime_elem x"
   using prime_elem_mult_unit_left[of "1 div unit_factor x" x]
   by (cases "x = 0") (simp_all add: unit_div_commute)
 
 lemma prime_elem_associated:
   assumes "prime_elem p" and "prime_elem q" and "q dvd p"
   shows "normalize q = normalize p"
 using \<open>q dvd p\<close> proof (rule associatedI)
   from \<open>prime_elem q\<close> have "\<not> is_unit q"
     by (auto simp add: prime_elem_not_unit)
   with \<open>prime_elem p\<close> \<open>q dvd p\<close> show "p dvd q"
     by (blast intro: prime_elemD2)
 qed
 
 definition prime :: "'a \<Rightarrow> bool" where
   "prime p \<longleftrightarrow> prime_elem p \<and> normalize p = p"
 
 lemma not_prime_0 [simp]: "\<not>prime 0" by (simp add: prime_def)
 
 lemma not_prime_unit: "is_unit x \<Longrightarrow> \<not>prime x"
   using prime_elem_not_unit[of x] by (auto simp add: prime_def)
 
 lemma not_prime_1 [simp]: "\<not>prime 1" by (simp add: not_prime_unit)
 
 lemma primeI: "prime_elem x \<Longrightarrow> normalize x = x \<Longrightarrow> prime x"
   by (simp add: prime_def)
 
 lemma prime_imp_prime_elem [dest]: "prime p \<Longrightarrow> prime_elem p"
   by (simp add: prime_def)
 
 lemma normalize_prime: "prime p \<Longrightarrow> normalize p = p"
   by (simp add: prime_def)
 
 lemma prime_normalize_iff [simp]: "prime (normalize p) \<longleftrightarrow> prime_elem p"
   by (auto simp add: prime_def)
 
 lemma prime_power_iff:
   "prime (p ^ n) \<longleftrightarrow> prime p \<and> n = 1"
   by (auto simp: prime_def prime_elem_power_iff)
 
 lemma prime_imp_nonzero [simp]:
   "ASSUMPTION (prime x) \<Longrightarrow> x \<noteq> 0"
   unfolding ASSUMPTION_def prime_def by auto
 
 lemma prime_imp_not_one [simp]:
   "ASSUMPTION (prime x) \<Longrightarrow> x \<noteq> 1"
   unfolding ASSUMPTION_def by auto
 
 lemma prime_not_unit' [simp]:
   "ASSUMPTION (prime x) \<Longrightarrow> \<not>is_unit x"
   unfolding ASSUMPTION_def prime_def by auto
 
 lemma prime_normalize' [simp]: "ASSUMPTION (prime x) \<Longrightarrow> normalize x = x"
   unfolding ASSUMPTION_def prime_def by simp
 
 lemma unit_factor_prime: "prime x \<Longrightarrow> unit_factor x = 1"
   using unit_factor_normalize[of x] unfolding prime_def by auto
 
 lemma unit_factor_prime' [simp]: "ASSUMPTION (prime x) \<Longrightarrow> unit_factor x = 1"
   unfolding ASSUMPTION_def by (rule unit_factor_prime)
 
 lemma prime_imp_prime_elem' [simp]: "ASSUMPTION (prime x) \<Longrightarrow> prime_elem x"
   by (simp add: prime_def ASSUMPTION_def)
 
 lemma prime_dvd_multD: "prime p \<Longrightarrow> p dvd a * b \<Longrightarrow> p dvd a \<or> p dvd b"
   by (intro prime_elem_dvd_multD) simp_all
 
 lemma prime_dvd_mult_iff: "prime p \<Longrightarrow> p dvd a * b \<longleftrightarrow> p dvd a \<or> p dvd b"
   by (auto dest: prime_dvd_multD)
 
 lemma prime_dvd_power:
   "prime p \<Longrightarrow> p dvd x ^ n \<Longrightarrow> p dvd x"
   by (auto dest!: prime_elem_dvd_power simp: prime_def)
 
 lemma prime_dvd_power_iff:
   "prime p \<Longrightarrow> n > 0 \<Longrightarrow> p dvd x ^ n \<longleftrightarrow> p dvd x"
   by (subst prime_elem_dvd_power_iff) simp_all
 
 lemma prime_dvd_prod_mset_iff: "prime p \<Longrightarrow> p dvd prod_mset A \<longleftrightarrow> (\<exists>x. x \<in># A \<and> p dvd x)"
   by (induction A) (simp_all add: prime_elem_dvd_mult_iff prime_imp_prime_elem, blast+)
 
 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)"
   by (auto simp: prime_dvd_prod_mset_iff prod_unfold_prod_mset)
 
 lemma primes_dvd_imp_eq:
   assumes "prime p" "prime q" "p dvd q"
   shows   "p = q"
 proof -
   from assms have "irreducible q" by (simp add: prime_elem_imp_irreducible prime_def)
   from irreducibleD'[OF this \<open>p dvd q\<close>] assms have "q dvd p" by simp
   with \<open>p dvd q\<close> have "normalize p = normalize q" by (rule associatedI)
   with assms show "p = q" by simp
 qed
 
 lemma prime_dvd_prod_mset_primes_iff:
   assumes "prime p" "\<And>q. q \<in># A \<Longrightarrow> prime q"
   shows   "p dvd prod_mset A \<longleftrightarrow> p \<in># A"
 proof -
   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)
   also from assms have "\<dots> \<longleftrightarrow> p \<in># A" by (auto dest: primes_dvd_imp_eq)
   finally show ?thesis .
 qed
 
 lemma prod_mset_primes_dvd_imp_subset:
   assumes "prod_mset A dvd prod_mset B" "\<And>p. p \<in># A \<Longrightarrow> prime p" "\<And>p. p \<in># B \<Longrightarrow> prime p"
   shows   "A \<subseteq># B"
 using assms
 proof (induction A arbitrary: B)
   case empty
   thus ?case by simp
 next
   case (add p A B)
   hence p: "prime p" by simp
   define B' where "B' = B - {#p#}"
   from add.prems have "p dvd prod_mset B" by (simp add: dvd_mult_left)
   with add.prems have "p \<in># B"
     by (subst (asm) (2) prime_dvd_prod_mset_primes_iff) simp_all
   hence B: "B = B' + {#p#}" by (simp add: B'_def)
   from add.prems p have "A \<subseteq># B'" by (intro add.IH) (simp_all add: B)
   thus ?case by (simp add: B)
 qed
 
 lemma prod_mset_dvd_prod_mset_primes_iff:
   assumes "\<And>x. x \<in># A \<Longrightarrow> prime x" "\<And>x. x \<in># B \<Longrightarrow> prime x"
   shows   "prod_mset A dvd prod_mset B \<longleftrightarrow> A \<subseteq># B"
   using assms by (auto intro: prod_mset_subset_imp_dvd prod_mset_primes_dvd_imp_subset)
 
 lemma is_unit_prod_mset_primes_iff:
   assumes "\<And>x. x \<in># A \<Longrightarrow> prime x"
   shows   "is_unit (prod_mset A) \<longleftrightarrow> A = {#}"
   by (auto simp add: is_unit_prod_mset_iff)
     (meson all_not_in_conv assms not_prime_unit set_mset_eq_empty_iff)
 
 lemma prod_mset_primes_irreducible_imp_prime:
   assumes irred: "irreducible (prod_mset A)"
   assumes A: "\<And>x. x \<in># A \<Longrightarrow> prime x"
   assumes B: "\<And>x. x \<in># B \<Longrightarrow> prime x"
   assumes C: "\<And>x. x \<in># C \<Longrightarrow> prime x"
   assumes dvd: "prod_mset A dvd prod_mset B * prod_mset C"
   shows   "prod_mset A dvd prod_mset B \<or> prod_mset A dvd prod_mset C"
 proof -
   from dvd have "prod_mset A dvd prod_mset (B + C)"
     by simp
   with A B C have subset: "A \<subseteq># B + C"
     by (subst (asm) prod_mset_dvd_prod_mset_primes_iff) auto
   define A1 and A2 where "A1 = A \<inter># B" and "A2 = A - A1"
   have "A = A1 + A2" unfolding A1_def A2_def
     by (rule sym, intro subset_mset.add_diff_inverse) simp_all
   from subset have "A1 \<subseteq># B" "A2 \<subseteq># C"
     by (auto simp: A1_def A2_def Multiset.subset_eq_diff_conv Multiset.union_commute)
   from \<open>A = A1 + A2\<close> have "prod_mset A = prod_mset A1 * prod_mset A2" by simp
   from irred and this have "is_unit (prod_mset A1) \<or> is_unit (prod_mset A2)"
     by (rule irreducibleD)
   with A have "A1 = {#} \<or> A2 = {#}" unfolding A1_def A2_def
     by (subst (asm) (1 2) is_unit_prod_mset_primes_iff) (auto dest: Multiset.in_diffD)
   with dvd \<open>A = A1 + A2\<close> \<open>A1 \<subseteq># B\<close> \<open>A2 \<subseteq># C\<close> show ?thesis
     by (auto intro: prod_mset_subset_imp_dvd)
 qed
 
 lemma prod_mset_primes_finite_divisor_powers:
   assumes A: "\<And>x. x \<in># A \<Longrightarrow> prime x"
   assumes B: "\<And>x. x \<in># B \<Longrightarrow> prime x"
   assumes "A \<noteq> {#}"
   shows   "finite {n. prod_mset A ^ n dvd prod_mset B}"
 proof -
   from \<open>A \<noteq> {#}\<close> obtain x where x: "x \<in># A" by blast
   define m where "m = count B x"
   have "{n. prod_mset A ^ n dvd prod_mset B} \<subseteq> {..m}"
   proof safe
     fix n assume dvd: "prod_mset A ^ n dvd prod_mset B"
     from x have "x ^ n dvd prod_mset A ^ n" by (intro dvd_power_same dvd_prod_mset)
     also note dvd
     also have "x ^ n = prod_mset (replicate_mset n x)" by simp
     finally have "replicate_mset n x \<subseteq># B"
       by (rule prod_mset_primes_dvd_imp_subset) (insert A B x, simp_all split: if_splits)
     thus "n \<le> m" by (simp add: count_le_replicate_mset_subset_eq m_def)
   qed
   moreover have "finite {..m}" by simp
   ultimately show ?thesis by (rule finite_subset)
 qed
 
 end
 
 
 subsection \<open>In a semiring with GCD, each irreducible element is a prime element\<close>
 
 context semiring_gcd
 begin
 
 lemma irreducible_imp_prime_elem_gcd:
   assumes "irreducible x"
   shows   "prime_elem x"
 proof (rule prime_elemI)
   fix a b assume "x dvd a * b"
   from dvd_productE[OF this] obtain y z where yz: "x = y * z" "y dvd a" "z dvd b" .
   from \<open>irreducible x\<close> and \<open>x = y * z\<close> have "is_unit y \<or> is_unit z" by (rule irreducibleD)
   with yz show "x dvd a \<or> x dvd b"
     by (auto simp: mult_unit_dvd_iff mult_unit_dvd_iff')
 qed (insert assms, auto simp: irreducible_not_unit)
 
 lemma prime_elem_imp_coprime:
   assumes "prime_elem p" "\<not>p dvd n"
   shows   "coprime p n"
 proof (rule coprimeI)
   fix d assume "d dvd p" "d dvd n"
   show "is_unit d"
   proof (rule ccontr)
     assume "\<not>is_unit d"
     from \<open>prime_elem p\<close> and \<open>d dvd p\<close> and this have "p dvd d"
       by (rule prime_elemD2)
     from this and \<open>d dvd n\<close> have "p dvd n" by (rule dvd_trans)
     with \<open>\<not>p dvd n\<close> show False by contradiction
   qed
 qed
 
 lemma prime_imp_coprime:
   assumes "prime p" "\<not>p dvd n"
   shows   "coprime p n"
   using assms by (simp add: prime_elem_imp_coprime)
 
 lemma prime_elem_imp_power_coprime:
   "prime_elem p \<Longrightarrow> \<not> p dvd a \<Longrightarrow> coprime a (p ^ m)"
   by (cases "m > 0") (auto dest: prime_elem_imp_coprime simp add: ac_simps)
 
 lemma prime_imp_power_coprime:
   "prime p \<Longrightarrow> \<not> p dvd a \<Longrightarrow> coprime a (p ^ m)"
   by (rule prime_elem_imp_power_coprime) simp_all
 
 lemma prime_elem_divprod_pow:
   assumes p: "prime_elem p" and ab: "coprime a b" and pab: "p^n dvd a * b"
   shows   "p^n dvd a \<or> p^n dvd b"
   using assms
 proof -
   from p have "\<not> is_unit p"
     by simp
   with ab p have "\<not> p dvd a \<or> \<not> p dvd b"
     using not_coprimeI by blast
   with p have "coprime (p ^ n) a \<or> coprime (p ^ n) b"
     by (auto dest: prime_elem_imp_power_coprime simp add: ac_simps)
   with pab show ?thesis
     by (auto simp add: coprime_dvd_mult_left_iff coprime_dvd_mult_right_iff)
 qed
 
 lemma primes_coprime:
   "prime p \<Longrightarrow> prime q \<Longrightarrow> p \<noteq> q \<Longrightarrow> coprime p q"
   using prime_imp_coprime primes_dvd_imp_eq by blast
 
 end
 
 
 subsection \<open>Factorial semirings: algebraic structures with unique prime factorizations\<close>
 
 class factorial_semiring = normalization_semidom +
   assumes prime_factorization_exists:
     "x \<noteq> 0 \<Longrightarrow> \<exists>A. (\<forall>x. x \<in># A \<longrightarrow> prime_elem x) \<and> normalize (prod_mset A) = normalize x"
 
 text \<open>Alternative characterization\<close>
 
 lemma (in normalization_semidom) factorial_semiring_altI_aux:
   assumes finite_divisors: "\<And>x. x \<noteq> 0 \<Longrightarrow> finite {y. y dvd x \<and> normalize y = y}"
   assumes irreducible_imp_prime_elem: "\<And>x. irreducible x \<Longrightarrow> prime_elem x"
   assumes "x \<noteq> 0"
   shows   "\<exists>A. (\<forall>x. x \<in># A \<longrightarrow> prime_elem x) \<and> normalize (prod_mset A) = normalize x"
 using \<open>x \<noteq> 0\<close>
 proof (induction "card {b. b dvd x \<and> normalize b = b}" arbitrary: x rule: less_induct)
   case (less a)
   let ?fctrs = "\<lambda>a. {b. b dvd a \<and> normalize b = b}"
   show ?case
   proof (cases "is_unit a")
     case True
     thus ?thesis by (intro exI[of _ "{#}"]) (auto simp: is_unit_normalize)
   next
     case False
     show ?thesis
     proof (cases "\<exists>b. b dvd a \<and> \<not>is_unit b \<and> \<not>a dvd b")
       case False
       with \<open>\<not>is_unit a\<close> less.prems have "irreducible a" by (auto simp: irreducible_altdef)
       hence "prime_elem a" by (rule irreducible_imp_prime_elem)
       thus ?thesis by (intro exI[of _ "{#normalize a#}"]) auto
     next
       case True
       then guess b by (elim exE conjE) note b = this
 
       from b have "?fctrs b \<subseteq> ?fctrs a" by (auto intro: dvd_trans)
       moreover from b have "normalize a \<notin> ?fctrs b" "normalize a \<in> ?fctrs a" by simp_all
       hence "?fctrs b \<noteq> ?fctrs a" by blast
       ultimately have "?fctrs b \<subset> ?fctrs a" by (subst subset_not_subset_eq) blast
       with finite_divisors[OF \<open>a \<noteq> 0\<close>] have "card (?fctrs b) < card (?fctrs a)"
         by (rule psubset_card_mono)
       moreover from \<open>a \<noteq> 0\<close> b have "b \<noteq> 0" by auto
       ultimately have "\<exists>A. (\<forall>x. x \<in># A \<longrightarrow> prime_elem x) \<and> normalize (prod_mset A) = normalize b"
         by (intro less) auto
       then guess A .. note A = this
 
       define c where "c = a div b"
       from b have c: "a = b * c" by (simp add: c_def)
       from less.prems c have "c \<noteq> 0" by auto
       from b c have "?fctrs c \<subseteq> ?fctrs a" by (auto intro: dvd_trans)
       moreover have "normalize a \<notin> ?fctrs c"
       proof safe
         assume "normalize a dvd c"
         hence "b * c dvd 1 * c" by (simp add: c)
         hence "b dvd 1" by (subst (asm) dvd_times_right_cancel_iff) fact+
         with b show False by simp
       qed
       with \<open>normalize a \<in> ?fctrs a\<close> have "?fctrs a \<noteq> ?fctrs c" by blast
       ultimately have "?fctrs c \<subset> ?fctrs a" by (subst subset_not_subset_eq) blast
       with finite_divisors[OF \<open>a \<noteq> 0\<close>] have "card (?fctrs c) < card (?fctrs a)"
         by (rule psubset_card_mono)
       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"
         by (intro less) auto
       then guess B .. note B = this
 
       show ?thesis
       proof (rule exI[of _ "A + B"]; safe)
         have "normalize (prod_mset (A + B)) =
                 normalize (normalize (prod_mset A) * normalize (prod_mset B))"
           by simp
         also have "\<dots> = normalize (b * c)"
           by (simp only: A B) auto
         also have "b * c = a"
           using c by simp
         finally show "normalize (prod_mset (A + B)) = normalize a" .
       next
       qed (use A B in auto)
     qed
   qed
 qed
 
 lemma factorial_semiring_altI:
   assumes finite_divisors: "\<And>x::'a. x \<noteq> 0 \<Longrightarrow> finite {y. y dvd x \<and> normalize y = y}"
   assumes irreducible_imp_prime: "\<And>x::'a. irreducible x \<Longrightarrow> prime_elem x"
   shows   "OFCLASS('a :: normalization_semidom, factorial_semiring_class)"
   by intro_classes (rule factorial_semiring_altI_aux[OF assms])
 
 text \<open>Properties\<close>
 
 context factorial_semiring
 begin
 
 lemma prime_factorization_exists':
   assumes "x \<noteq> 0"
   obtains A where "\<And>x. x \<in># A \<Longrightarrow> prime x" "normalize (prod_mset A) = normalize x"
 proof -
   from prime_factorization_exists[OF assms] obtain A
     where A: "\<And>x. x \<in># A \<Longrightarrow> prime_elem x" "normalize (prod_mset A) = normalize x" by blast
   define A' where "A' = image_mset normalize A"
   have "normalize (prod_mset A') = normalize (prod_mset A)"
     by (simp add: A'_def normalize_prod_mset_normalize)
   also note A(2)
   finally have "normalize (prod_mset A') = normalize x" by simp
   moreover from A(1) have "\<forall>x. x \<in># A' \<longrightarrow> prime x" by (auto simp: prime_def A'_def)
   ultimately show ?thesis by (intro that[of A']) blast
 qed
 
 lemma irreducible_imp_prime_elem:
   assumes "irreducible x"
   shows   "prime_elem x"
 proof (rule prime_elemI)
   fix a b assume dvd: "x dvd a * b"
   from assms have "x \<noteq> 0" by auto
   show "x dvd a \<or> x dvd b"
   proof (cases "a = 0 \<or> b = 0")
     case False
     hence "a \<noteq> 0" "b \<noteq> 0" by blast+
     note nz = \<open>x \<noteq> 0\<close> this
     from nz[THEN prime_factorization_exists'] guess A B C . note ABC = this
 
     have "irreducible (prod_mset A)"
       by (subst irreducible_cong[OF ABC(2)]) fact
     moreover have "normalize (prod_mset A) dvd
                      normalize (normalize (prod_mset B) * normalize (prod_mset C))"
       unfolding ABC using dvd by simp
     hence "prod_mset A dvd prod_mset B * prod_mset C"
       unfolding normalize_mult_normalize_left normalize_mult_normalize_right by simp
     ultimately have "prod_mset A dvd prod_mset B \<or> prod_mset A dvd prod_mset C"
       by (intro prod_mset_primes_irreducible_imp_prime) (use ABC in auto)
     hence "normalize (prod_mset A) dvd normalize (prod_mset B) \<or>
            normalize (prod_mset A) dvd normalize (prod_mset C)" by simp
     thus ?thesis unfolding ABC by simp
   qed auto
 qed (insert assms, simp_all add: irreducible_def)
 
 lemma finite_divisor_powers:
   assumes "y \<noteq> 0" "\<not>is_unit x"
   shows   "finite {n. x ^ n dvd y}"
 proof (cases "x = 0")
   case True
   with assms have "{n. x ^ n dvd y} = {0}" by (auto simp: power_0_left)
   thus ?thesis by simp
 next
   case False
   note nz = this \<open>y \<noteq> 0\<close>
   from nz[THEN prime_factorization_exists'] guess A B . note AB = this
   from AB assms have "A \<noteq> {#}" by (auto simp: normalize_1_iff)
   from AB(2,4) prod_mset_primes_finite_divisor_powers [of A B, OF AB(1,3) this]
     have "finite {n. prod_mset A ^ n dvd prod_mset B}" by simp
   also have "{n. prod_mset A ^ n dvd prod_mset B} =
              {n. normalize (normalize (prod_mset A) ^ n) dvd normalize (prod_mset B)}"
     unfolding normalize_power_normalize by simp
   also have "\<dots> = {n. x ^ n dvd y}"
     unfolding AB unfolding normalize_power_normalize by simp
   finally show ?thesis .
 qed
 
 lemma finite_prime_divisors:
   assumes "x \<noteq> 0"
   shows   "finite {p. prime p \<and> p dvd x}"
 proof -
   from prime_factorization_exists'[OF assms] guess A . note A = this
   have "{p. prime p \<and> p dvd x} \<subseteq> set_mset A"
   proof safe
     fix p assume p: "prime p" and dvd: "p dvd x"
     from dvd have "p dvd normalize x" by simp
     also from A have "normalize x = normalize (prod_mset A)" by simp
     finally have "p dvd prod_mset A"
       by simp
     thus  "p \<in># A" using p A
       by (subst (asm) prime_dvd_prod_mset_primes_iff)
   qed
   moreover have "finite (set_mset A)" by simp
   ultimately show ?thesis by (rule finite_subset)
 qed
 
 lemma prime_elem_iff_irreducible: "prime_elem x \<longleftrightarrow> irreducible x"
   by (blast intro: irreducible_imp_prime_elem prime_elem_imp_irreducible)
 
 lemma prime_divisor_exists:
   assumes "a \<noteq> 0" "\<not>is_unit a"
   shows   "\<exists>b. b dvd a \<and> prime b"
 proof -
   from prime_factorization_exists'[OF assms(1)] guess A . note A = this
   moreover from A and assms have "A \<noteq> {#}" by auto
   then obtain x where "x \<in># A" by blast
   with A(1) have *: "x dvd normalize (prod_mset A)" "prime x"
     by (auto simp: dvd_prod_mset)
   hence "x dvd a" unfolding A by simp
   with * show ?thesis by blast
 qed
 
 lemma prime_divisors_induct [case_names zero unit factor]:
   assumes "P 0" "\<And>x. is_unit x \<Longrightarrow> P x" "\<And>p x. prime p \<Longrightarrow> P x \<Longrightarrow> P (p * x)"
   shows   "P x"
 proof (cases "x = 0")
   case False
   from prime_factorization_exists'[OF this] guess A . note A = this
   from A obtain u where u: "is_unit u" "x = u * prod_mset A"
     by (elim associatedE2)
 
   from A(1) have "P (u * prod_mset A)"
   proof (induction A)
     case (add p A)
     from add.prems have "prime p" by simp
     moreover from add.prems have "P (u * prod_mset A)" by (intro add.IH) simp_all
     ultimately have "P (p * (u * prod_mset A))" by (rule assms(3))
     thus ?case by (simp add: mult_ac)
   qed (simp_all add: assms False u)
   with A u show ?thesis by simp
 qed (simp_all add: assms(1))
 
 lemma no_prime_divisors_imp_unit:
   assumes "a \<noteq> 0" "\<And>b. b dvd a \<Longrightarrow> normalize b = b \<Longrightarrow> \<not> prime_elem b"
   shows "is_unit a"
 proof (rule ccontr)
   assume "\<not>is_unit a"
   from prime_divisor_exists[OF assms(1) this] guess b by (elim exE conjE)
   with assms(2)[of b] show False by (simp add: prime_def)
 qed
 
 lemma prime_divisorE:
   assumes "a \<noteq> 0" and "\<not> is_unit a"
   obtains p where "prime p" and "p dvd a"
   using assms no_prime_divisors_imp_unit unfolding prime_def by blast
 
 definition multiplicity :: "'a \<Rightarrow> 'a \<Rightarrow> nat" where
   "multiplicity p x = (if finite {n. p ^ n dvd x} then Max {n. p ^ n dvd x} else 0)"
 
 lemma multiplicity_dvd: "p ^ multiplicity p x dvd x"
 proof (cases "finite {n. p ^ n dvd x}")
   case True
   hence "multiplicity p x = Max {n. p ^ n dvd x}"
     by (simp add: multiplicity_def)
   also have "\<dots> \<in> {n. p ^ n dvd x}"
     by (rule Max_in) (auto intro!: True exI[of _ "0::nat"])
   finally show ?thesis by simp
 qed (simp add: multiplicity_def)
 
 lemma multiplicity_dvd': "n \<le> multiplicity p x \<Longrightarrow> p ^ n dvd x"
   by (rule dvd_trans[OF le_imp_power_dvd multiplicity_dvd])
 
 context
   fixes x p :: 'a
   assumes xp: "x \<noteq> 0" "\<not>is_unit p"
 begin
 
 lemma multiplicity_eq_Max: "multiplicity p x = Max {n. p ^ n dvd x}"
   using finite_divisor_powers[OF xp] by (simp add: multiplicity_def)
 
 lemma multiplicity_geI:
   assumes "p ^ n dvd x"
   shows   "multiplicity p x \<ge> n"
 proof -
   from assms have "n \<le> Max {n. p ^ n dvd x}"
     by (intro Max_ge finite_divisor_powers xp) simp_all
   thus ?thesis by (subst multiplicity_eq_Max)
 qed
 
 lemma multiplicity_lessI:
   assumes "\<not>p ^ n dvd x"
   shows   "multiplicity p x < n"
 proof (rule ccontr)
   assume "\<not>(n > multiplicity p x)"
   hence "p ^ n dvd x" by (intro multiplicity_dvd') simp
   with assms show False by contradiction
 qed
 
 lemma power_dvd_iff_le_multiplicity:
   "p ^ n dvd x \<longleftrightarrow> n \<le> multiplicity p x"
   using multiplicity_geI[of n] multiplicity_lessI[of n] by (cases "p ^ n dvd x") auto
 
 lemma multiplicity_eq_zero_iff:
   shows   "multiplicity p x = 0 \<longleftrightarrow> \<not>p dvd x"
   using power_dvd_iff_le_multiplicity[of 1] by auto
 
 lemma multiplicity_gt_zero_iff:
   shows   "multiplicity p x > 0 \<longleftrightarrow> p dvd x"
   using power_dvd_iff_le_multiplicity[of 1] by auto
 
 lemma multiplicity_decompose:
   "\<not>p dvd (x div p ^ multiplicity p x)"
 proof
   assume *: "p dvd x div p ^ multiplicity p x"
   have "x = x div p ^ multiplicity p x * (p ^ multiplicity p x)"
     using multiplicity_dvd[of p x] by simp
   also from * have "x div p ^ multiplicity p x = (x div p ^ multiplicity p x div p) * p" by simp
   also have "x div p ^ multiplicity p x div p * p * p ^ multiplicity p x =
                x div p ^ multiplicity p x div p * p ^ Suc (multiplicity p x)"
     by (simp add: mult_assoc)
   also have "p ^ Suc (multiplicity p x) dvd \<dots>" by (rule dvd_triv_right)
   finally show False by (subst (asm) power_dvd_iff_le_multiplicity) simp
 qed
 
 lemma multiplicity_decompose':
   obtains y where "x = p ^ multiplicity p x * y" "\<not>p dvd y"
   using that[of "x div p ^ multiplicity p x"]
   by (simp add: multiplicity_decompose multiplicity_dvd)
 
 end
 
 lemma multiplicity_zero [simp]: "multiplicity p 0 = 0"
   by (simp add: multiplicity_def)
 
 lemma prime_elem_multiplicity_eq_zero_iff:
   "prime_elem p \<Longrightarrow> x \<noteq> 0 \<Longrightarrow> multiplicity p x = 0 \<longleftrightarrow> \<not>p dvd x"
   by (rule multiplicity_eq_zero_iff) simp_all
 
 lemma prime_multiplicity_other:
   assumes "prime p" "prime q" "p \<noteq> q"
   shows   "multiplicity p q = 0"
   using assms by (subst prime_elem_multiplicity_eq_zero_iff) (auto dest: primes_dvd_imp_eq)
 
 lemma prime_multiplicity_gt_zero_iff:
   "prime_elem p \<Longrightarrow> x \<noteq> 0 \<Longrightarrow> multiplicity p x > 0 \<longleftrightarrow> p dvd x"
   by (rule multiplicity_gt_zero_iff) simp_all
 
 lemma multiplicity_unit_left: "is_unit p \<Longrightarrow> multiplicity p x = 0"
   by (simp add: multiplicity_def is_unit_power_iff unit_imp_dvd)
 
 lemma multiplicity_unit_right:
   assumes "is_unit x"
   shows   "multiplicity p x = 0"
 proof (cases "is_unit p \<or> x = 0")
   case False
   with multiplicity_lessI[of x p 1] this assms
     show ?thesis by (auto dest: dvd_unit_imp_unit)
 qed (auto simp: multiplicity_unit_left)
 
 lemma multiplicity_one [simp]: "multiplicity p 1 = 0"
   by (rule multiplicity_unit_right) simp_all
 
 lemma multiplicity_eqI:
   assumes "p ^ n dvd x" "\<not>p ^ Suc n dvd x"
   shows   "multiplicity p x = n"
 proof -
   consider "x = 0" | "is_unit p" | "x \<noteq> 0" "\<not>is_unit p" by blast
   thus ?thesis
   proof cases
     assume xp: "x \<noteq> 0" "\<not>is_unit p"
     from xp assms(1) have "multiplicity p x \<ge> n" by (intro multiplicity_geI)
     moreover from assms(2) xp have "multiplicity p x < Suc n" by (intro multiplicity_lessI)
     ultimately show ?thesis by simp
   next
     assume "is_unit p"
     hence "is_unit (p ^ Suc n)" by (simp add: is_unit_power_iff del: power_Suc)
     hence "p ^ Suc n dvd x" by (rule unit_imp_dvd)
     with \<open>\<not>p ^ Suc n dvd x\<close> show ?thesis by contradiction
   qed (insert assms, simp_all)
 qed
 
 
 context
   fixes x p :: 'a
   assumes xp: "x \<noteq> 0" "\<not>is_unit p"
 begin
 
 lemma multiplicity_times_same:
   assumes "p \<noteq> 0"
   shows   "multiplicity p (p * x) = Suc (multiplicity p x)"
 proof (rule multiplicity_eqI)
   show "p ^ Suc (multiplicity p x) dvd p * x"
     by (auto intro!: mult_dvd_mono multiplicity_dvd)
   from xp assms show "\<not> p ^ Suc (Suc (multiplicity p x)) dvd p * x"
     using power_dvd_iff_le_multiplicity[OF xp, of "Suc (multiplicity p x)"] by simp
 qed
 
 end
 
 lemma multiplicity_same_power': "multiplicity p (p ^ n) = (if p = 0 \<or> is_unit p then 0 else n)"
 proof -
   consider "p = 0" | "is_unit p" |"p \<noteq> 0" "\<not>is_unit p" by blast
   thus ?thesis
   proof cases
     assume "p \<noteq> 0" "\<not>is_unit p"
     thus ?thesis by (induction n) (simp_all add: multiplicity_times_same)
   qed (simp_all add: power_0_left multiplicity_unit_left)
 qed
 
 lemma multiplicity_same_power:
   "p \<noteq> 0 \<Longrightarrow> \<not>is_unit p \<Longrightarrow> multiplicity p (p ^ n) = n"
   by (simp add: multiplicity_same_power')
 
 lemma multiplicity_prime_elem_times_other:
   assumes "prime_elem p" "\<not>p dvd q"
   shows   "multiplicity p (q * x) = multiplicity p x"
 proof (cases "x = 0")
   case False
   show ?thesis
   proof (rule multiplicity_eqI)
     have "1 * p ^ multiplicity p x dvd q * x"
       by (intro mult_dvd_mono multiplicity_dvd) simp_all
     thus "p ^ multiplicity p x dvd q * x" by simp
   next
     define n where "n = multiplicity p x"
     from assms have "\<not>is_unit p" by simp
     from multiplicity_decompose'[OF False this] guess y . note y = this [folded n_def]
     from y have "p ^ Suc n dvd q * x \<longleftrightarrow> p ^ n * p dvd p ^ n * (q * y)" by (simp add: mult_ac)
     also from assms have "\<dots> \<longleftrightarrow> p dvd q * y" by simp
     also have "\<dots> \<longleftrightarrow> p dvd q \<or> p dvd y" by (rule prime_elem_dvd_mult_iff) fact+
     also from assms y have "\<dots> \<longleftrightarrow> False" by simp
     finally show "\<not>(p ^ Suc n dvd q * x)" by blast
   qed
 qed simp_all
 
 lemma multiplicity_self:
   assumes "p \<noteq> 0" "\<not>is_unit p"
   shows   "multiplicity p p = 1"
 proof -
   from assms have "multiplicity p p = Max {n. p ^ n dvd p}"
     by (simp add: multiplicity_eq_Max)
   also from assms have "p ^ n dvd p \<longleftrightarrow> n \<le> 1" for n
     using dvd_power_iff[of p n 1] by auto
   hence "{n. p ^ n dvd p} = {..1}" by auto
   also have "\<dots> = {0,1}" by auto
   finally show ?thesis by simp
 qed
 
 lemma multiplicity_times_unit_left:
   assumes "is_unit c"
   shows   "multiplicity (c * p) x = multiplicity p x"
 proof -
   from assms have "{n. (c * p) ^ n dvd x} = {n. p ^ n dvd x}"
     by (subst mult.commute) (simp add: mult_unit_dvd_iff power_mult_distrib is_unit_power_iff)
   thus ?thesis by (simp add: multiplicity_def)
 qed
 
 lemma multiplicity_times_unit_right:
   assumes "is_unit c"
   shows   "multiplicity p (c * x) = multiplicity p x"
 proof -
   from assms have "{n. p ^ n dvd c * x} = {n. p ^ n dvd x}"
     by (subst mult.commute) (simp add: dvd_mult_unit_iff)
   thus ?thesis by (simp add: multiplicity_def)
 qed
 
 lemma multiplicity_normalize_left [simp]:
   "multiplicity (normalize p) x = multiplicity p x"
 proof (cases "p = 0")
   case [simp]: False
   have "normalize p = (1 div unit_factor p) * p"
     by (simp add: unit_div_commute is_unit_unit_factor)
   also have "multiplicity \<dots> x = multiplicity p x"
     by (rule multiplicity_times_unit_left) (simp add: is_unit_unit_factor)
   finally show ?thesis .
 qed simp_all
 
 lemma multiplicity_normalize_right [simp]:
   "multiplicity p (normalize x) = multiplicity p x"
 proof (cases "x = 0")
   case [simp]: False
   have "normalize x = (1 div unit_factor x) * x"
     by (simp add: unit_div_commute is_unit_unit_factor)
   also have "multiplicity p \<dots> = multiplicity p x"
     by (rule multiplicity_times_unit_right) (simp add: is_unit_unit_factor)
   finally show ?thesis .
 qed simp_all
 
 lemma multiplicity_prime [simp]: "prime_elem p \<Longrightarrow> multiplicity p p = 1"
   by (rule multiplicity_self) auto
 
 lemma multiplicity_prime_power [simp]: "prime_elem p \<Longrightarrow> multiplicity p (p ^ n) = n"
   by (subst multiplicity_same_power') auto
 
 lift_definition prime_factorization :: "'a \<Rightarrow> 'a multiset" is
   "\<lambda>x p. if prime p then multiplicity p x else 0"
   unfolding multiset_def
 proof clarify
   fix x :: 'a
   show "finite {p. 0 < (if prime p then multiplicity p x else 0)}" (is "finite ?A")
   proof (cases "x = 0")
     case False
     from False have "?A \<subseteq> {p. prime p \<and> p dvd x}"
       by (auto simp: multiplicity_gt_zero_iff)
     moreover from False have "finite {p. prime p \<and> p dvd x}"
       by (rule finite_prime_divisors)
     ultimately show ?thesis by (rule finite_subset)
   qed simp_all
 qed
 
 abbreviation prime_factors :: "'a \<Rightarrow> 'a set" where
   "prime_factors a \<equiv> set_mset (prime_factorization a)"
 
 lemma count_prime_factorization_nonprime:
   "\<not>prime p \<Longrightarrow> count (prime_factorization x) p = 0"
   by transfer simp
 
 lemma count_prime_factorization_prime:
   "prime p \<Longrightarrow> count (prime_factorization x) p = multiplicity p x"
   by transfer simp
 
 lemma count_prime_factorization:
   "count (prime_factorization x) p = (if prime p then multiplicity p x else 0)"
   by transfer simp
 
 lemma dvd_imp_multiplicity_le:
   assumes "a dvd b" "b \<noteq> 0"
   shows   "multiplicity p a \<le> multiplicity p b"
 proof (cases "is_unit p")
   case False
   with assms show ?thesis
     by (intro multiplicity_geI ) (auto intro: dvd_trans[OF multiplicity_dvd' assms(1)])
 qed (insert assms, auto simp: multiplicity_unit_left)
 
 lemma prime_power_inj:
   assumes "prime a" "a ^ m = a ^ n"
   shows   "m = n"
 proof -
   have "multiplicity a (a ^ m) = multiplicity a (a ^ n)" by (simp only: assms)
   thus ?thesis using assms by (subst (asm) (1 2) multiplicity_prime_power) simp_all
 qed
 
 lemma prime_power_inj':
   assumes "prime p" "prime q"
   assumes "p ^ m = q ^ n" "m > 0" "n > 0"
   shows   "p = q" "m = n"
 proof -
   from assms have "p ^ 1 dvd p ^ m" by (intro le_imp_power_dvd) simp
   also have "p ^ m = q ^ n" by fact
   finally have "p dvd q ^ n" by simp
   with assms have "p dvd q" using prime_dvd_power[of p q] by simp
   with assms show "p = q" by (simp add: primes_dvd_imp_eq)
   with assms show "m = n" by (simp add: prime_power_inj)
 qed
 
 lemma prime_power_eq_one_iff [simp]: "prime p \<Longrightarrow> p ^ n = 1 \<longleftrightarrow> n = 0"
   using prime_power_inj[of p n 0] by auto
 
 lemma one_eq_prime_power_iff [simp]: "prime p \<Longrightarrow> 1 = p ^ n \<longleftrightarrow> n = 0"
   using prime_power_inj[of p 0 n] by auto
 
 lemma prime_power_inj'':
   assumes "prime p" "prime q"
   shows   "p ^ m = q ^ n \<longleftrightarrow> (m = 0 \<and> n = 0) \<or> (p = q \<and> m = n)"
   using assms 
   by (cases "m = 0"; cases "n = 0")
      (auto dest: prime_power_inj'[OF assms])
 
 lemma prime_factorization_0 [simp]: "prime_factorization 0 = {#}"
   by (simp add: multiset_eq_iff count_prime_factorization)
 
 lemma prime_factorization_empty_iff:
   "prime_factorization x = {#} \<longleftrightarrow> x = 0 \<or> is_unit x"
 proof
   assume *: "prime_factorization x = {#}"
   {
     assume x: "x \<noteq> 0" "\<not>is_unit x"
     {
       fix p assume p: "prime p"
       have "count (prime_factorization x) p = 0" by (simp add: *)
       also from p have "count (prime_factorization x) p = multiplicity p x"
         by (rule count_prime_factorization_prime)
       also from x p have "\<dots> = 0 \<longleftrightarrow> \<not>p dvd x" by (simp add: multiplicity_eq_zero_iff)
       finally have "\<not>p dvd x" .
     }
     with prime_divisor_exists[OF x] have False by blast
   }
   thus "x = 0 \<or> is_unit x" by blast
 next
   assume "x = 0 \<or> is_unit x"
   thus "prime_factorization x = {#}"
   proof
     assume x: "is_unit x"
     {
       fix p assume p: "prime p"
       from p x have "multiplicity p x = 0"
         by (subst multiplicity_eq_zero_iff)
            (auto simp: multiplicity_eq_zero_iff dest: unit_imp_no_prime_divisors)
     }
     thus ?thesis by (simp add: multiset_eq_iff count_prime_factorization)
   qed simp_all
 qed
 
 lemma prime_factorization_unit:
   assumes "is_unit x"
   shows   "prime_factorization x = {#}"
 proof (rule multiset_eqI)
   fix p :: 'a
   show "count (prime_factorization x) p = count {#} p"
   proof (cases "prime p")
     case True
     with assms have "multiplicity p x = 0"
       by (subst multiplicity_eq_zero_iff)
          (auto simp: multiplicity_eq_zero_iff dest: unit_imp_no_prime_divisors)
     with True show ?thesis by (simp add: count_prime_factorization_prime)
   qed (simp_all add: count_prime_factorization_nonprime)
 qed
 
 lemma prime_factorization_1 [simp]: "prime_factorization 1 = {#}"
   by (simp add: prime_factorization_unit)
 
 lemma prime_factorization_times_prime:
   assumes "x \<noteq> 0" "prime p"
   shows   "prime_factorization (p * x) = {#p#} + prime_factorization x"
 proof (rule multiset_eqI)
   fix q :: 'a
   consider "\<not>prime q" | "p = q" | "prime q" "p \<noteq> q" by blast
   thus "count (prime_factorization (p * x)) q = count ({#p#} + prime_factorization x) q"
   proof cases
     assume q: "prime q" "p \<noteq> q"
     with assms primes_dvd_imp_eq[of q p] have "\<not>q dvd p" by auto
     with q assms show ?thesis
       by (simp add: multiplicity_prime_elem_times_other count_prime_factorization)
   qed (insert assms, auto simp: count_prime_factorization multiplicity_times_same)
 qed
 
 lemma prod_mset_prime_factorization_weak:
   assumes "x \<noteq> 0"
   shows   "normalize (prod_mset (prime_factorization x)) = normalize x"
   using assms
 proof (induction x rule: prime_divisors_induct)
   case (factor p x)
   have "normalize (prod_mset (prime_factorization (p * x))) =
           normalize (p * normalize (prod_mset (prime_factorization x)))"
     using factor.prems factor.hyps by (simp add: prime_factorization_times_prime)
   also have "normalize (prod_mset (prime_factorization x)) = normalize x"
     by (rule factor.IH) (use factor in auto)
   finally show ?case by simp
 qed (auto simp: prime_factorization_unit is_unit_normalize)
 
 lemma in_prime_factors_iff:
   "p \<in> prime_factors x \<longleftrightarrow> x \<noteq> 0 \<and> p dvd x \<and> prime p"
 proof -
   have "p \<in> prime_factors x \<longleftrightarrow> count (prime_factorization x) p > 0" by simp
   also have "\<dots> \<longleftrightarrow> x \<noteq> 0 \<and> p dvd x \<and> prime p"
    by (subst count_prime_factorization, cases "x = 0")
       (auto simp: multiplicity_eq_zero_iff multiplicity_gt_zero_iff)
   finally show ?thesis .
 qed
 
 lemma in_prime_factors_imp_prime [intro]:
   "p \<in> prime_factors x \<Longrightarrow> prime p"
   by (simp add: in_prime_factors_iff)
 
 lemma in_prime_factors_imp_dvd [dest]:
   "p \<in> prime_factors x \<Longrightarrow> p dvd x"
   by (simp add: in_prime_factors_iff)
 
 lemma prime_factorsI:
   "x \<noteq> 0 \<Longrightarrow> prime p \<Longrightarrow> p dvd x \<Longrightarrow> p \<in> prime_factors x"
   by (auto simp: in_prime_factors_iff)
 
 lemma prime_factors_dvd:
   "x \<noteq> 0 \<Longrightarrow> prime_factors x = {p. prime p \<and> p dvd x}"
   by (auto intro: prime_factorsI)
 
 lemma prime_factors_multiplicity:
   "prime_factors n = {p. prime p \<and> multiplicity p n > 0}"
   by (cases "n = 0") (auto simp add: prime_factors_dvd prime_multiplicity_gt_zero_iff)
 
 lemma prime_factorization_prime:
   assumes "prime p"
   shows   "prime_factorization p = {#p#}"
 proof (rule multiset_eqI)
   fix q :: 'a
   consider "\<not>prime q" | "q = p" | "prime q" "q \<noteq> p" by blast
   thus "count (prime_factorization p) q = count {#p#} q"
     by cases (insert assms, auto dest: primes_dvd_imp_eq
                 simp: count_prime_factorization multiplicity_self multiplicity_eq_zero_iff)
 qed
 
 lemma prime_factorization_prod_mset_primes:
   assumes "\<And>p. p \<in># A \<Longrightarrow> prime p"
   shows   "prime_factorization (prod_mset A) = A"
   using assms
 proof (induction A)
   case (add p A)
   from add.prems[of 0] have "0 \<notin># A" by auto
   hence "prod_mset A \<noteq> 0" by auto
   with add show ?case
     by (simp_all add: mult_ac prime_factorization_times_prime Multiset.union_commute)
 qed simp_all
 
 lemma prime_factorization_cong:
   "normalize x = normalize y \<Longrightarrow> prime_factorization x = prime_factorization y"
   by (simp add: multiset_eq_iff count_prime_factorization
                 multiplicity_normalize_right [of _ x, symmetric]
                 multiplicity_normalize_right [of _ y, symmetric]
            del:  multiplicity_normalize_right)
 
 lemma prime_factorization_unique:
   assumes "x \<noteq> 0" "y \<noteq> 0"
   shows   "prime_factorization x = prime_factorization y \<longleftrightarrow> normalize x = normalize y"
 proof
   assume "prime_factorization x = prime_factorization y"
   hence "prod_mset (prime_factorization x) = prod_mset (prime_factorization y)" by simp
   hence "normalize (prod_mset (prime_factorization x)) =
          normalize (prod_mset (prime_factorization y))"
     by (simp only: )
   with assms show "normalize x = normalize y"
     by (simp add: prod_mset_prime_factorization_weak)
 qed (rule prime_factorization_cong)
 
 lemma prime_factorization_normalize [simp]:
   "prime_factorization (normalize x) = prime_factorization x"
   by (cases "x = 0", simp, subst prime_factorization_unique) auto
 
 lemma prime_factorization_eqI_strong:
   assumes "\<And>p. p \<in># P \<Longrightarrow> prime p" "prod_mset P = n"
   shows   "prime_factorization n = P"
   using prime_factorization_prod_mset_primes[of P] assms by simp
 
 lemma prime_factorization_eqI:
   assumes "\<And>p. p \<in># P \<Longrightarrow> prime p" "normalize (prod_mset P) = normalize n"
   shows   "prime_factorization n = P"
 proof -
   have "P = prime_factorization (normalize (prod_mset P))"
     using prime_factorization_prod_mset_primes[of P] assms(1) by simp
   with assms(2) show ?thesis by simp
 qed
 
 lemma prime_factorization_mult:
   assumes "x \<noteq> 0" "y \<noteq> 0"
   shows   "prime_factorization (x * y) = prime_factorization x + prime_factorization y"
 proof -
   have "normalize (prod_mset (prime_factorization x) * prod_mset (prime_factorization y)) =
           normalize (normalize (prod_mset (prime_factorization x)) *
                      normalize (prod_mset (prime_factorization y)))"
     by (simp only: normalize_mult_normalize_left normalize_mult_normalize_right)
   also have "\<dots> = normalize (x * y)"
     by (subst (1 2) prod_mset_prime_factorization_weak) (use assms in auto)
   finally show ?thesis
     by (intro prime_factorization_eqI) auto
 qed
 
 lemma prime_factorization_prod:
   assumes "finite A" "\<And>x. x \<in> A \<Longrightarrow> f x \<noteq> 0"
   shows   "prime_factorization (prod f A) = (\<Sum>n\<in>A. prime_factorization (f n))"
   using assms by (induction A rule: finite_induct)
                  (auto simp: Sup_multiset_empty prime_factorization_mult)
 
 lemma prime_elem_multiplicity_mult_distrib:
   assumes "prime_elem p" "x \<noteq> 0" "y \<noteq> 0"
   shows   "multiplicity p (x * y) = multiplicity p x + multiplicity p y"
 proof -
   have "multiplicity p (x * y) = count (prime_factorization (x * y)) (normalize p)"
     by (subst count_prime_factorization_prime) (simp_all add: assms)
   also from assms
     have "prime_factorization (x * y) = prime_factorization x + prime_factorization y"
       by (intro prime_factorization_mult)
   also have "count \<dots> (normalize p) =
     count (prime_factorization x) (normalize p) + count (prime_factorization y) (normalize p)"
     by simp
   also have "\<dots> = multiplicity p x + multiplicity p y"
     by (subst (1 2) count_prime_factorization_prime) (simp_all add: assms)
   finally show ?thesis .
 qed
 
 lemma prime_elem_multiplicity_prod_mset_distrib:
   assumes "prime_elem p" "0 \<notin># A"
   shows   "multiplicity p (prod_mset A) = sum_mset (image_mset (multiplicity p) A)"
   using assms by (induction A) (auto simp: prime_elem_multiplicity_mult_distrib)
 
 lemma prime_elem_multiplicity_power_distrib:
   assumes "prime_elem p" "x \<noteq> 0"
   shows   "multiplicity p (x ^ n) = n * multiplicity p x"
   using assms prime_elem_multiplicity_prod_mset_distrib [of p "replicate_mset n x"]
   by simp
 
 lemma prime_elem_multiplicity_prod_distrib:
   assumes "prime_elem p" "0 \<notin> f ` A" "finite A"
   shows   "multiplicity p (prod f A) = (\<Sum>x\<in>A. multiplicity p (f x))"
 proof -
   have "multiplicity p (prod f A) = (\<Sum>x\<in>#mset_set A. multiplicity p (f x))"
     using assms by (subst prod_unfold_prod_mset)
                    (simp_all add: prime_elem_multiplicity_prod_mset_distrib sum_unfold_sum_mset
                       multiset.map_comp o_def)
   also from \<open>finite A\<close> have "\<dots> = (\<Sum>x\<in>A. multiplicity p (f x))"
     by (induction A rule: finite_induct) simp_all
   finally show ?thesis .
 qed
 
 lemma multiplicity_distinct_prime_power:
   "prime p \<Longrightarrow> prime q \<Longrightarrow> p \<noteq> q \<Longrightarrow> multiplicity p (q ^ n) = 0"
   by (subst prime_elem_multiplicity_power_distrib) (auto simp: prime_multiplicity_other)
 
 lemma prime_factorization_prime_power:
   "prime p \<Longrightarrow> prime_factorization (p ^ n) = replicate_mset n p"
   by (induction n)
      (simp_all add: prime_factorization_mult prime_factorization_prime Multiset.union_commute)
 
 lemma prime_factorization_subset_iff_dvd:
   assumes [simp]: "x \<noteq> 0" "y \<noteq> 0"
   shows   "prime_factorization x \<subseteq># prime_factorization y \<longleftrightarrow> x dvd y"
 proof -
   have "x dvd y \<longleftrightarrow>
     normalize (prod_mset (prime_factorization x)) dvd normalize (prod_mset (prime_factorization y))"
     using assms by (subst (1 2) prod_mset_prime_factorization_weak) auto
   also have "\<dots> \<longleftrightarrow> prime_factorization x \<subseteq># prime_factorization y"
     by (auto intro!: prod_mset_primes_dvd_imp_subset prod_mset_subset_imp_dvd)
   finally show ?thesis ..
 qed
 
 lemma prime_factorization_subset_imp_dvd:
   "x \<noteq> 0 \<Longrightarrow> (prime_factorization x \<subseteq># prime_factorization y) \<Longrightarrow> x dvd y"
   by (cases "y = 0") (simp_all add: prime_factorization_subset_iff_dvd)
 
 lemma prime_factorization_divide:
   assumes "b dvd a"
   shows   "prime_factorization (a div b) = prime_factorization a - prime_factorization b"
 proof (cases "a = 0")
   case [simp]: False
   from assms have [simp]: "b \<noteq> 0" by auto
   have "prime_factorization ((a div b) * b) = prime_factorization (a div b) + prime_factorization b"
     by (intro prime_factorization_mult) (insert assms, auto elim!: dvdE)
   with assms show ?thesis by simp
 qed simp_all
 
 lemma zero_not_in_prime_factors [simp]: "0 \<notin> prime_factors x"
   by (auto dest: in_prime_factors_imp_prime)
 
 lemma prime_prime_factors:
   "prime p \<Longrightarrow> prime_factors p = {p}"
   by (drule prime_factorization_prime) simp
 
 lemma prime_factors_product:
   "x \<noteq> 0 \<Longrightarrow> y \<noteq> 0 \<Longrightarrow> prime_factors (x * y) = prime_factors x \<union> prime_factors y"
   by (simp add: prime_factorization_mult)
 
 lemma dvd_prime_factors [intro]:
   "y \<noteq> 0 \<Longrightarrow> x dvd y \<Longrightarrow> prime_factors x \<subseteq> prime_factors y"
   by (intro set_mset_mono, subst prime_factorization_subset_iff_dvd) auto
 
 (* RENAMED multiplicity_dvd *)
 lemma multiplicity_le_imp_dvd:
   assumes "x \<noteq> 0" "\<And>p. prime p \<Longrightarrow> multiplicity p x \<le> multiplicity p y"
   shows   "x dvd y"
 proof (cases "y = 0")
   case False
   from assms this have "prime_factorization x \<subseteq># prime_factorization y"
     by (intro mset_subset_eqI) (auto simp: count_prime_factorization)
   with assms False show ?thesis by (subst (asm) prime_factorization_subset_iff_dvd)
 qed auto
 
 lemma dvd_multiplicity_eq:
   "x \<noteq> 0 \<Longrightarrow> y \<noteq> 0 \<Longrightarrow> x dvd y \<longleftrightarrow> (\<forall>p. multiplicity p x \<le> multiplicity p y)"
   by (auto intro: dvd_imp_multiplicity_le multiplicity_le_imp_dvd)
 
 lemma multiplicity_eq_imp_eq:
   assumes "x \<noteq> 0" "y \<noteq> 0"
   assumes "\<And>p. prime p \<Longrightarrow> multiplicity p x = multiplicity p y"
   shows   "normalize x = normalize y"
   using assms by (intro associatedI multiplicity_le_imp_dvd) simp_all
 
 lemma prime_factorization_unique':
   assumes "\<forall>p \<in># M. prime p" "\<forall>p \<in># N. prime p" "(\<Prod>i \<in># M. i) = (\<Prod>i \<in># N. i)"
   shows   "M = N"
 proof -
   have "prime_factorization (\<Prod>i \<in># M. i) = prime_factorization (\<Prod>i \<in># N. i)"
     by (simp only: assms)
   also from assms have "prime_factorization (\<Prod>i \<in># M. i) = M"
     by (subst prime_factorization_prod_mset_primes) simp_all
   also from assms have "prime_factorization (\<Prod>i \<in># N. i) = N"
     by (subst prime_factorization_prod_mset_primes) simp_all
   finally show ?thesis .
 qed
 
 lemma prime_factorization_unique'':
   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)"
   shows   "M = N"
 proof -
   have "prime_factorization (normalize (\<Prod>i \<in># M. i)) =
         prime_factorization (normalize (\<Prod>i \<in># N. i))"
     by (simp only: assms)
   also from assms have "prime_factorization (normalize (\<Prod>i \<in># M. i)) = M"
     by (subst prime_factorization_normalize, subst prime_factorization_prod_mset_primes) simp_all
   also from assms have "prime_factorization (normalize (\<Prod>i \<in># N. i)) = N"
     by (subst prime_factorization_normalize, subst prime_factorization_prod_mset_primes) simp_all
   finally show ?thesis .
 qed
 
 lemma multiplicity_cong:
   "(\<And>r. p ^ r dvd a \<longleftrightarrow> p ^ r dvd b) \<Longrightarrow> multiplicity p a = multiplicity p b"
   by (simp add: multiplicity_def)
 
 lemma not_dvd_imp_multiplicity_0:
   assumes "\<not>p dvd x"
   shows   "multiplicity p x = 0"
 proof -
   from assms have "multiplicity p x < 1"
     by (intro multiplicity_lessI) auto
   thus ?thesis by simp
 qed
 
+lemma multiplicity_zero_1 [simp]: "multiplicity 0 x = 0"
+  by (metis (mono_tags) local.dvd_0_left multiplicity_zero not_dvd_imp_multiplicity_0)
+
 lemma inj_on_Prod_primes:
   assumes "\<And>P p. P \<in> A \<Longrightarrow> p \<in> P \<Longrightarrow> prime p"
   assumes "\<And>P. P \<in> A \<Longrightarrow> finite P"
   shows   "inj_on Prod A"
 proof (rule inj_onI)
   fix P Q assume PQ: "P \<in> A" "Q \<in> A" "\<Prod>P = \<Prod>Q"
   with prime_factorization_unique'[of "mset_set P" "mset_set Q"] assms[of P] assms[of Q]
     have "mset_set P = mset_set Q" by (auto simp: prod_unfold_prod_mset)
     with assms[of P] assms[of Q] PQ show "P = Q" by simp
 qed
 
 lemma divides_primepow_weak:
   assumes "prime p" and "a dvd p ^ n"
   obtains m where "m \<le> n" and "normalize a = normalize (p ^ m)"
 proof -
   from assms have "a \<noteq> 0"
     by auto
   with assms
   have "normalize (prod_mset (prime_factorization a)) dvd
           normalize (prod_mset (prime_factorization (p ^ n)))"
     by (subst (1 2) prod_mset_prime_factorization_weak) auto
   then have "prime_factorization a \<subseteq># prime_factorization (p ^ n)"
     by (simp add: in_prime_factors_imp_prime prod_mset_dvd_prod_mset_primes_iff)
   with assms have "prime_factorization a \<subseteq># replicate_mset n p"
     by (simp add: prime_factorization_prime_power)
   then obtain m where "m \<le> n" and "prime_factorization a = replicate_mset m p"
     by (rule msubseteq_replicate_msetE)
   then have *: "normalize (prod_mset (prime_factorization a)) =
                   normalize (prod_mset (replicate_mset m p))" by metis
   also have "normalize (prod_mset (prime_factorization a)) = normalize a"
     using \<open>a \<noteq> 0\<close> by (simp add: prod_mset_prime_factorization_weak)
   also have "prod_mset (replicate_mset m p) = p ^ m"
     by simp
   finally show ?thesis using \<open>m \<le> n\<close> 
     by (intro that[of m])
 qed
 
 lemma divide_out_primepow_ex:
   assumes "n \<noteq> 0" "\<exists>p\<in>prime_factors n. P p"
   obtains p k n' where "P p" "prime p" "p dvd n" "\<not>p dvd n'" "k > 0" "n = p ^ k * n'"
 proof -
   from assms obtain p where p: "P p" "prime p" "p dvd n"
     by auto
   define k where "k = multiplicity p n"
   define n' where "n' = n div p ^ k"
   have n': "n = p ^ k * n'" "\<not>p dvd n'"
     using assms p multiplicity_decompose[of n p]
     by (auto simp: n'_def k_def multiplicity_dvd)
   from n' p have "k > 0" by (intro Nat.gr0I) auto
   with n' p that[of p n' k] show ?thesis by auto
 qed
 
 lemma divide_out_primepow:
   assumes "n \<noteq> 0" "\<not>is_unit n"
   obtains p k n' where "prime p" "p dvd n" "\<not>p dvd n'" "k > 0" "n = p ^ k * n'"
   using divide_out_primepow_ex[OF assms(1), of "\<lambda>_. True"] prime_divisor_exists[OF assms] assms
         prime_factorsI by metis
 
 
 subsection \<open>GCD and LCM computation with unique factorizations\<close>
 
 definition "gcd_factorial a b = (if a = 0 then normalize b
      else if b = 0 then normalize a
      else normalize (prod_mset (prime_factorization a \<inter># prime_factorization b)))"
 
 definition "lcm_factorial a b = (if a = 0 \<or> b = 0 then 0
      else normalize (prod_mset (prime_factorization a \<union># prime_factorization b)))"
 
 definition "Gcd_factorial A =
   (if A \<subseteq> {0} then 0 else normalize (prod_mset (Inf (prime_factorization ` (A - {0})))))"
 
 definition "Lcm_factorial A =
   (if A = {} then 1
    else if 0 \<notin> A \<and> subset_mset.bdd_above (prime_factorization ` (A - {0})) then
      normalize (prod_mset (Sup (prime_factorization ` A)))
    else
      0)"
 
 lemma prime_factorization_gcd_factorial:
   assumes [simp]: "a \<noteq> 0" "b \<noteq> 0"
   shows   "prime_factorization (gcd_factorial a b) = prime_factorization a \<inter># prime_factorization b"
 proof -
   have "prime_factorization (gcd_factorial a b) =
           prime_factorization (prod_mset (prime_factorization a \<inter># prime_factorization b))"
     by (simp add: gcd_factorial_def)
   also have "\<dots> = prime_factorization a \<inter># prime_factorization b"
     by (subst prime_factorization_prod_mset_primes) auto
   finally show ?thesis .
 qed
 
 lemma prime_factorization_lcm_factorial:
   assumes [simp]: "a \<noteq> 0" "b \<noteq> 0"
   shows   "prime_factorization (lcm_factorial a b) = prime_factorization a \<union># prime_factorization b"
 proof -
   have "prime_factorization (lcm_factorial a b) =
           prime_factorization (prod_mset (prime_factorization a \<union># prime_factorization b))"
     by (simp add: lcm_factorial_def)
   also have "\<dots> = prime_factorization a \<union># prime_factorization b"
     by (subst prime_factorization_prod_mset_primes) auto
   finally show ?thesis .
 qed
 
 lemma prime_factorization_Gcd_factorial:
   assumes "\<not>A \<subseteq> {0}"
   shows   "prime_factorization (Gcd_factorial A) = Inf (prime_factorization ` (A - {0}))"
 proof -
   from assms obtain x where x: "x \<in> A - {0}" by auto
   hence "Inf (prime_factorization ` (A - {0})) \<subseteq># prime_factorization x"
     by (intro subset_mset.cInf_lower) simp_all
   hence "\<forall>y. y \<in># Inf (prime_factorization ` (A - {0})) \<longrightarrow> y \<in> prime_factors x"
     by (auto dest: mset_subset_eqD)
   with in_prime_factors_imp_prime[of _ x]
     have "\<forall>p. p \<in># Inf (prime_factorization ` (A - {0})) \<longrightarrow> prime p" by blast
   with assms show ?thesis
     by (simp add: Gcd_factorial_def prime_factorization_prod_mset_primes)
 qed
 
 lemma prime_factorization_Lcm_factorial:
   assumes "0 \<notin> A" "subset_mset.bdd_above (prime_factorization ` A)"
   shows   "prime_factorization (Lcm_factorial A) = Sup (prime_factorization ` A)"
 proof (cases "A = {}")
   case True
   hence "prime_factorization ` A = {}" by auto
   also have "Sup \<dots> = {#}" by (simp add: Sup_multiset_empty)
   finally show ?thesis by (simp add: Lcm_factorial_def)
 next
   case False
   have "\<forall>y. y \<in># Sup (prime_factorization ` A) \<longrightarrow> prime y"
     by (auto simp: in_Sup_multiset_iff assms)
   with assms False show ?thesis
     by (simp add: Lcm_factorial_def prime_factorization_prod_mset_primes)
 qed
 
 lemma gcd_factorial_commute: "gcd_factorial a b = gcd_factorial b a"
   by (simp add: gcd_factorial_def multiset_inter_commute)
 
 lemma gcd_factorial_dvd1: "gcd_factorial a b dvd a"
 proof (cases "a = 0 \<or> b = 0")
   case False
   hence "gcd_factorial a b \<noteq> 0" by (auto simp: gcd_factorial_def)
   with False show ?thesis
     by (subst prime_factorization_subset_iff_dvd [symmetric])
        (auto simp: prime_factorization_gcd_factorial)
 qed (auto simp: gcd_factorial_def)
 
 lemma gcd_factorial_dvd2: "gcd_factorial a b dvd b"
   by (subst gcd_factorial_commute) (rule gcd_factorial_dvd1)
 
 lemma normalize_gcd_factorial [simp]: "normalize (gcd_factorial a b) = gcd_factorial a b"
   by (simp add: gcd_factorial_def)
 
 lemma normalize_lcm_factorial [simp]: "normalize (lcm_factorial a b) = lcm_factorial a b"
   by (simp add: lcm_factorial_def)
 
 lemma gcd_factorial_greatest: "c dvd gcd_factorial a b" if "c dvd a" "c dvd b" for a b c
 proof (cases "a = 0 \<or> b = 0")
   case False
   with that have [simp]: "c \<noteq> 0" by auto
   let ?p = "prime_factorization"
   from that False have "?p c \<subseteq># ?p a" "?p c \<subseteq># ?p b"
     by (simp_all add: prime_factorization_subset_iff_dvd)
   hence "prime_factorization c \<subseteq>#
            prime_factorization (prod_mset (prime_factorization a \<inter># prime_factorization b))"
     using False by (subst prime_factorization_prod_mset_primes) auto
   with False show ?thesis
     by (auto simp: gcd_factorial_def prime_factorization_subset_iff_dvd [symmetric])
 qed (auto simp: gcd_factorial_def that)
 
 lemma lcm_factorial_gcd_factorial:
   "lcm_factorial a b = normalize (a * b div gcd_factorial a b)" for a b
 proof (cases "a = 0 \<or> b = 0")
   case False
   let ?p = "prime_factorization"
   have 1: "normalize x * normalize y dvd z \<longleftrightarrow> x * y dvd z" for x y z :: 'a
   proof -
     have "normalize (normalize x * normalize y) dvd z \<longleftrightarrow> x * y dvd z"
       unfolding normalize_mult_normalize_left normalize_mult_normalize_right by simp
     thus ?thesis unfolding normalize_dvd_iff by simp
   qed
 
   have "?p (a * b) = (?p a \<union># ?p b) + (?p a \<inter># ?p b)"
     using False by (subst prime_factorization_mult) (auto intro!: multiset_eqI)
   hence "normalize (prod_mset (?p (a * b))) =
            normalize (prod_mset ((?p a \<union># ?p b) + (?p a \<inter># ?p b)))"
     by (simp only:)
   hence *: "normalize (a * b) = normalize (lcm_factorial a b * gcd_factorial a b)" using False
     by (subst (asm) prod_mset_prime_factorization_weak)
        (auto simp: lcm_factorial_def gcd_factorial_def)
 
   have [simp]: "gcd_factorial a b dvd a * b" "lcm_factorial a b dvd a * b"
     using associatedD2[OF *] by auto
   from False have [simp]: "gcd_factorial a b \<noteq> 0" "lcm_factorial a b \<noteq> 0"
     by (auto simp: gcd_factorial_def lcm_factorial_def)
   
   show ?thesis
     by (rule associated_eqI)
        (use * in \<open>auto simp: dvd_div_iff_mult div_dvd_iff_mult dest: associatedD1 associatedD2\<close>)
 qed (auto simp: lcm_factorial_def)
 
 lemma normalize_Gcd_factorial:
   "normalize (Gcd_factorial A) = Gcd_factorial A"
   by (simp add: Gcd_factorial_def)
 
 lemma Gcd_factorial_eq_0_iff:
   "Gcd_factorial A = 0 \<longleftrightarrow> A \<subseteq> {0}"
   by (auto simp: Gcd_factorial_def in_Inf_multiset_iff split: if_splits)
 
 lemma Gcd_factorial_dvd:
   assumes "x \<in> A"
   shows   "Gcd_factorial A dvd x"
 proof (cases "x = 0")
   case False
   with assms have "prime_factorization (Gcd_factorial A) = Inf (prime_factorization ` (A - {0}))"
     by (intro prime_factorization_Gcd_factorial) auto
   also from False assms have "\<dots> \<subseteq># prime_factorization x"
     by (intro subset_mset.cInf_lower) auto
   finally show ?thesis
     by (subst (asm) prime_factorization_subset_iff_dvd)
        (insert assms False, auto simp: Gcd_factorial_eq_0_iff)
 qed simp_all
 
 lemma Gcd_factorial_greatest:
   assumes "\<And>y. y \<in> A \<Longrightarrow> x dvd y"
   shows   "x dvd Gcd_factorial A"
 proof (cases "A \<subseteq> {0}")
   case False
   from False obtain y where "y \<in> A" "y \<noteq> 0" by auto
   with assms[of y] have nz: "x \<noteq> 0" by auto
   from nz assms have "prime_factorization x \<subseteq># prime_factorization y" if "y \<in> A - {0}" for y
     using that by (subst prime_factorization_subset_iff_dvd) auto
   with False have "prime_factorization x \<subseteq># Inf (prime_factorization ` (A - {0}))"
     by (intro subset_mset.cInf_greatest) auto
   also from False have "\<dots> = prime_factorization (Gcd_factorial A)"
     by (rule prime_factorization_Gcd_factorial [symmetric])
   finally show ?thesis
     by (subst (asm) prime_factorization_subset_iff_dvd)
        (insert nz False, auto simp: Gcd_factorial_eq_0_iff)
 qed (simp_all add: Gcd_factorial_def)
 
 lemma normalize_Lcm_factorial:
   "normalize (Lcm_factorial A) = Lcm_factorial A"
   by (simp add: Lcm_factorial_def)
 
 lemma Lcm_factorial_eq_0_iff:
   "Lcm_factorial A = 0 \<longleftrightarrow> 0 \<in> A \<or> \<not>subset_mset.bdd_above (prime_factorization ` A)"
   by (auto simp: Lcm_factorial_def in_Sup_multiset_iff)
 
 lemma dvd_Lcm_factorial:
   assumes "x \<in> A"
   shows   "x dvd Lcm_factorial A"
 proof (cases "0 \<notin> A \<and> subset_mset.bdd_above (prime_factorization ` A)")
   case True
   with assms have [simp]: "0 \<notin> A" "x \<noteq> 0" "A \<noteq> {}" by auto
   from assms True have "prime_factorization x \<subseteq># Sup (prime_factorization ` A)"
     by (intro subset_mset.cSup_upper) auto
   also have "\<dots> = prime_factorization (Lcm_factorial A)"
     by (rule prime_factorization_Lcm_factorial [symmetric]) (insert True, simp_all)
   finally show ?thesis
     by (subst (asm) prime_factorization_subset_iff_dvd)
        (insert True, auto simp: Lcm_factorial_eq_0_iff)
 qed (insert assms, auto simp: Lcm_factorial_def)
 
 lemma Lcm_factorial_least:
   assumes "\<And>y. y \<in> A \<Longrightarrow> y dvd x"
   shows   "Lcm_factorial A dvd x"
 proof -
   consider "A = {}" | "0 \<in> A" | "x = 0" | "A \<noteq> {}" "0 \<notin> A" "x \<noteq> 0" by blast
   thus ?thesis
   proof cases
     assume *: "A \<noteq> {}" "0 \<notin> A" "x \<noteq> 0"
     hence nz: "x \<noteq> 0" if "x \<in> A" for x using that by auto
     from * have bdd: "subset_mset.bdd_above (prime_factorization ` A)"
       by (intro subset_mset.bdd_aboveI[of _ "prime_factorization x"])
          (auto simp: prime_factorization_subset_iff_dvd nz dest: assms)
     have "prime_factorization (Lcm_factorial A) = Sup (prime_factorization ` A)"
       by (rule prime_factorization_Lcm_factorial) fact+
     also from * have "\<dots> \<subseteq># prime_factorization x"
       by (intro subset_mset.cSup_least)
          (auto simp: prime_factorization_subset_iff_dvd nz dest: assms)
     finally show ?thesis
       by (subst (asm) prime_factorization_subset_iff_dvd)
          (insert * bdd, auto simp: Lcm_factorial_eq_0_iff)
   qed (auto simp: Lcm_factorial_def dest: assms)
 qed
 
 lemmas gcd_lcm_factorial =
   gcd_factorial_dvd1 gcd_factorial_dvd2 gcd_factorial_greatest
   normalize_gcd_factorial lcm_factorial_gcd_factorial
   normalize_Gcd_factorial Gcd_factorial_dvd Gcd_factorial_greatest
   normalize_Lcm_factorial dvd_Lcm_factorial Lcm_factorial_least
 
 end
 
 class factorial_semiring_gcd = factorial_semiring + gcd + Gcd +
   assumes gcd_eq_gcd_factorial: "gcd a b = gcd_factorial a b"
   and     lcm_eq_lcm_factorial: "lcm a b = lcm_factorial a b"
   and     Gcd_eq_Gcd_factorial: "Gcd A = Gcd_factorial A"
   and     Lcm_eq_Lcm_factorial: "Lcm A = Lcm_factorial A"
 begin
 
 lemma prime_factorization_gcd:
   assumes [simp]: "a \<noteq> 0" "b \<noteq> 0"
   shows   "prime_factorization (gcd a b) = prime_factorization a \<inter># prime_factorization b"
   by (simp add: gcd_eq_gcd_factorial prime_factorization_gcd_factorial)
 
 lemma prime_factorization_lcm:
   assumes [simp]: "a \<noteq> 0" "b \<noteq> 0"
   shows   "prime_factorization (lcm a b) = prime_factorization a \<union># prime_factorization b"
   by (simp add: lcm_eq_lcm_factorial prime_factorization_lcm_factorial)
 
 lemma prime_factorization_Gcd:
   assumes "Gcd A \<noteq> 0"
   shows   "prime_factorization (Gcd A) = Inf (prime_factorization ` (A - {0}))"
   using assms
   by (simp add: prime_factorization_Gcd_factorial Gcd_eq_Gcd_factorial Gcd_factorial_eq_0_iff)
 
 lemma prime_factorization_Lcm:
   assumes "Lcm A \<noteq> 0"
   shows   "prime_factorization (Lcm A) = Sup (prime_factorization ` A)"
   using assms
   by (simp add: prime_factorization_Lcm_factorial Lcm_eq_Lcm_factorial Lcm_factorial_eq_0_iff)
 
 lemma prime_factors_gcd [simp]: 
   "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> prime_factors (gcd a b) = 
      prime_factors a \<inter> prime_factors b"
   by (subst prime_factorization_gcd) auto
 
 lemma prime_factors_lcm [simp]: 
   "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> prime_factors (lcm a b) = 
      prime_factors a \<union> prime_factors b"
   by (subst prime_factorization_lcm) auto
 
 subclass semiring_gcd
   by (standard, unfold gcd_eq_gcd_factorial lcm_eq_lcm_factorial)
      (rule gcd_lcm_factorial; assumption)+
 
 subclass semiring_Gcd
   by (standard, unfold Gcd_eq_Gcd_factorial Lcm_eq_Lcm_factorial)
      (rule gcd_lcm_factorial; assumption)+
 
 lemma
   assumes "x \<noteq> 0" "y \<noteq> 0"
   shows gcd_eq_factorial':
           "gcd x y = normalize (\<Prod>p \<in> prime_factors x \<inter> prime_factors y.
                           p ^ min (multiplicity p x) (multiplicity p y))" (is "_ = ?rhs1")
     and lcm_eq_factorial':
           "lcm x y = normalize (\<Prod>p \<in> prime_factors x \<union> prime_factors y.
                           p ^ max (multiplicity p x) (multiplicity p y))" (is "_ = ?rhs2")
 proof -
   have "gcd x y = gcd_factorial x y" by (rule gcd_eq_gcd_factorial)
   also have "\<dots> = ?rhs1"
     by (auto simp: gcd_factorial_def assms prod_mset_multiplicity
           count_prime_factorization_prime
           intro!: arg_cong[of _ _ normalize] dest: in_prime_factors_imp_prime intro!: prod.cong)
   finally show "gcd x y = ?rhs1" .
   have "lcm x y = lcm_factorial x y" by (rule lcm_eq_lcm_factorial)
   also have "\<dots> = ?rhs2"
     by (auto simp: lcm_factorial_def assms prod_mset_multiplicity
           count_prime_factorization_prime intro!: arg_cong[of _ _ normalize] 
           dest: in_prime_factors_imp_prime intro!: prod.cong)
   finally show "lcm x y = ?rhs2" .
 qed
 
 lemma
   assumes "x \<noteq> 0" "y \<noteq> 0" "prime p"
   shows   multiplicity_gcd: "multiplicity p (gcd x y) = min (multiplicity p x) (multiplicity p y)"
     and   multiplicity_lcm: "multiplicity p (lcm x y) = max (multiplicity p x) (multiplicity p y)"
 proof -
   have "gcd x y = gcd_factorial x y" by (rule gcd_eq_gcd_factorial)
   also from assms have "multiplicity p \<dots> = min (multiplicity p x) (multiplicity p y)"
     by (simp add: count_prime_factorization_prime [symmetric] prime_factorization_gcd_factorial)
   finally show "multiplicity p (gcd x y) = min (multiplicity p x) (multiplicity p y)" .
   have "lcm x y = lcm_factorial x y" by (rule lcm_eq_lcm_factorial)
   also from assms have "multiplicity p \<dots> = max (multiplicity p x) (multiplicity p y)"
     by (simp add: count_prime_factorization_prime [symmetric] prime_factorization_lcm_factorial)
   finally show "multiplicity p (lcm x y) = max (multiplicity p x) (multiplicity p y)" .
 qed
 
 lemma gcd_lcm_distrib:
   "gcd x (lcm y z) = lcm (gcd x y) (gcd x z)"
 proof (cases "x = 0 \<or> y = 0 \<or> z = 0")
   case True
   thus ?thesis
     by (auto simp: lcm_proj1_if_dvd lcm_proj2_if_dvd)
 next
   case False
   hence "normalize (gcd x (lcm y z)) = normalize (lcm (gcd x y) (gcd x z))"
     by (intro associatedI prime_factorization_subset_imp_dvd)
        (auto simp: lcm_eq_0_iff prime_factorization_gcd prime_factorization_lcm
           subset_mset.inf_sup_distrib1)
   thus ?thesis by simp
 qed
 
 lemma lcm_gcd_distrib:
   "lcm x (gcd y z) = gcd (lcm x y) (lcm x z)"
 proof (cases "x = 0 \<or> y = 0 \<or> z = 0")
   case True
   thus ?thesis
     by (auto simp: lcm_proj1_if_dvd lcm_proj2_if_dvd)
 next
   case False
   hence "normalize (lcm x (gcd y z)) = normalize (gcd (lcm x y) (lcm x z))"
     by (intro associatedI prime_factorization_subset_imp_dvd)
        (auto simp: lcm_eq_0_iff prime_factorization_gcd prime_factorization_lcm
           subset_mset.sup_inf_distrib1)
   thus ?thesis by simp
 qed
 
 end
 
 class factorial_ring_gcd = factorial_semiring_gcd + idom
 begin
 
 subclass ring_gcd ..
 
 subclass idom_divide ..
 
 end
 
 
 class factorial_semiring_multiplicative =
   factorial_semiring + normalization_semidom_multiplicative
 begin
 
 lemma normalize_prod_mset_primes:
   "(\<And>p. p \<in># A \<Longrightarrow> prime p) \<Longrightarrow> normalize (prod_mset A) = prod_mset A"
 proof (induction A)
   case (add p A)
   hence "prime p" by simp
   hence "normalize p = p" by simp
   with add show ?case by (simp add: normalize_mult)
 qed simp_all
 
 lemma prod_mset_prime_factorization:
   assumes "x \<noteq> 0"
   shows   "prod_mset (prime_factorization x) = normalize x"
   using assms
   by (induction x rule: prime_divisors_induct)
      (simp_all add: prime_factorization_unit prime_factorization_times_prime
                     is_unit_normalize normalize_mult)
 
 lemma prime_decomposition: "unit_factor x * prod_mset (prime_factorization x) = x"
   by (cases "x = 0") (simp_all add: prod_mset_prime_factorization)
 
 lemma prod_prime_factors:
   assumes "x \<noteq> 0"
   shows   "(\<Prod>p \<in> prime_factors x. p ^ multiplicity p x) = normalize x"
 proof -
   have "normalize x = prod_mset (prime_factorization x)"
     by (simp add: prod_mset_prime_factorization assms)
   also have "\<dots> = (\<Prod>p \<in> prime_factors x. p ^ count (prime_factorization x) p)"
     by (subst prod_mset_multiplicity) simp_all
   also have "\<dots> = (\<Prod>p \<in> prime_factors x. p ^ multiplicity p x)"
     by (intro prod.cong)
       (simp_all add: assms count_prime_factorization_prime in_prime_factors_imp_prime)
   finally show ?thesis ..
 qed
 
 lemma prime_factorization_unique'':
   assumes S_eq: "S = {p. 0 < f p}"
     and "finite S"
     and S: "\<forall>p\<in>S. prime p" "normalize n = (\<Prod>p\<in>S. p ^ f p)"
   shows "S = prime_factors n \<and> (\<forall>p. prime p \<longrightarrow> f p = multiplicity p n)"
 proof
   define A where "A = Abs_multiset f"
   from \<open>finite S\<close> S(1) have "(\<Prod>p\<in>S. p ^ f p) \<noteq> 0" by auto
   with S(2) have nz: "n \<noteq> 0" by auto
   from S_eq \<open>finite S\<close> have count_A: "count A = f"
     unfolding A_def by (subst multiset.Abs_multiset_inverse) (simp_all add: multiset_def)
   from S_eq count_A have set_mset_A: "set_mset A = S"
     by (simp only: set_mset_def)
   from S(2) have "normalize n = (\<Prod>p\<in>S. p ^ f p)" .
   also have "\<dots> = prod_mset A" by (simp add: prod_mset_multiplicity S_eq set_mset_A count_A)
   also from nz have "normalize n = prod_mset (prime_factorization n)"
     by (simp add: prod_mset_prime_factorization)
   finally have "prime_factorization (prod_mset A) =
                   prime_factorization (prod_mset (prime_factorization n))" by simp
   also from S(1) have "prime_factorization (prod_mset A) = A"
     by (intro prime_factorization_prod_mset_primes) (auto simp: set_mset_A)
   also have "prime_factorization (prod_mset (prime_factorization n)) = prime_factorization n"
     by (intro prime_factorization_prod_mset_primes) auto
   finally show "S = prime_factors n" by (simp add: set_mset_A [symmetric])
 
   show "(\<forall>p. prime p \<longrightarrow> f p = multiplicity p n)"
   proof safe
     fix p :: 'a assume p: "prime p"
     have "multiplicity p n = multiplicity p (normalize n)" by simp
     also have "normalize n = prod_mset A"
       by (simp add: prod_mset_multiplicity S_eq set_mset_A count_A S)
     also from p set_mset_A S(1)
     have "multiplicity p \<dots> = sum_mset (image_mset (multiplicity p) A)"
       by (intro prime_elem_multiplicity_prod_mset_distrib) auto
     also from S(1) p
     have "image_mset (multiplicity p) A = image_mset (\<lambda>q. if p = q then 1 else 0) A"
       by (intro image_mset_cong) (auto simp: set_mset_A multiplicity_self prime_multiplicity_other)
     also have "sum_mset \<dots> = f p"
       by (simp add: semiring_1_class.sum_mset_delta' count_A)
     finally show "f p = multiplicity p n" ..
   qed
 qed
 
 lemma divides_primepow:
   assumes "prime p" and "a dvd p ^ n"
   obtains m where "m \<le> n" and "normalize a = p ^ m"
   using divides_primepow_weak[OF assms] that assms
   by (auto simp add: normalize_power)
 
 lemma Ex_other_prime_factor:
   assumes "n \<noteq> 0" and "\<not>(\<exists>k. normalize n = p ^ k)" "prime p"
   shows   "\<exists>q\<in>prime_factors n. q \<noteq> p"
 proof (rule ccontr)
   assume *: "\<not>(\<exists>q\<in>prime_factors n. q \<noteq> p)"
   have "normalize n = (\<Prod>p\<in>prime_factors n. p ^ multiplicity p n)"
     using assms(1) by (intro prod_prime_factors [symmetric]) auto
   also from * have "\<dots> = (\<Prod>p\<in>{p}. p ^ multiplicity p n)"
     using assms(3) by (intro prod.mono_neutral_left) (auto simp: prime_factors_multiplicity)
   finally have "normalize n = p ^ multiplicity p n" by auto
   with assms show False by auto
 qed
 
 end
 
 end