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
@@ -1,2260 +1,2260 @@
 (*  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
 
 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 obtain b where b: "b dvd a" "\<not> is_unit b" "\<not> a dvd b" by auto
       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 obtain A where A: "(\<forall>x. x \<in># A \<longrightarrow> prime_elem x) \<and> normalize (\<Prod>\<^sub># A) = normalize b"
         by auto
 
       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 obtain B where B: "(\<forall>x. x \<in># B \<longrightarrow> prime_elem x) \<and> normalize (\<Prod>\<^sub># B) = normalize c"
         by auto
 
       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'] obtain A B C
       where ABC:
         "\<And>z. z \<in># A \<Longrightarrow> prime z"
         "normalize (\<Prod>\<^sub># A) = normalize x"
         "\<And>z. z \<in># B \<Longrightarrow> prime z"
         "normalize (\<Prod>\<^sub># B) = normalize a"
         "\<And>z. z \<in># C \<Longrightarrow> prime z"
         "normalize (\<Prod>\<^sub># C) = normalize b"
       by this blast
 
     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 (use assms in \<open>simp_all add: irreducible_def\<close>)
 
 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'] obtain A B
     where AB:
       "\<And>z. z \<in># A \<Longrightarrow> prime z"
       "normalize (\<Prod>\<^sub># A) = normalize x"
       "\<And>z. z \<in># B \<Longrightarrow> prime z"
       "normalize (\<Prod>\<^sub># B) = normalize y"
     by this blast
 
   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] obtain A
     where A: "\<And>z. z \<in># A \<Longrightarrow> prime z" "normalize (\<Prod>\<^sub># A) = normalize x" by this blast
   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 infinite_unit_divisor_powers:
  assumes "y \<noteq> 0"
  assumes "is_unit x"
  shows "infinite {n. x^n dvd y}"
 proof -
- from `is_unit x` have "is_unit (x^n)" for n
+ from \<open>is_unit x\<close> have "is_unit (x^n)" for n
    using is_unit_power_iff by auto
  hence "x^n dvd y" for n
    by auto
  hence "{n. x^n dvd y} = UNIV"
    by auto
  thus ?thesis
    by auto
 qed
 
 corollary is_unit_iff_infinite_divisor_powers:
  assumes "y \<noteq> 0"
  shows "is_unit x \<longleftrightarrow> infinite {n. x^n dvd y}"
  using infinite_unit_divisor_powers finite_divisor_powers assms by auto
 
 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)]
   obtain A where A: "\<And>z. z \<in># A \<Longrightarrow> prime z" "normalize (\<Prod>\<^sub># A) = normalize a"
     by this blast
   with 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" by (simp add: A(2))
   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]
   obtain A where A: "\<And>z. z \<in># A \<Longrightarrow> prime z" "normalize (\<Prod>\<^sub># A) = normalize x"
     by this blast
   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] obtain b where "b dvd a" "prime b" by auto
   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]
     obtain y where y [folded n_def]: "x = p ^ multiplicity p x * y" "\<not> p dvd y" .
     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"
 proof -
   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_left [simp]: "multiplicity 0 x = 0"
  by (cases "x = 0") (auto intro: 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
   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
 
 text \<open>Now a string of results due to Jakub Kądziołka\<close>
 
 lemma multiplicity_dvd_iff_dvd:
  assumes "x \<noteq> 0"
  shows "p^k dvd x \<longleftrightarrow> p^k dvd p^multiplicity p x"
 proof (cases "is_unit p")
  case True
  then have "is_unit (p^k)"
    using is_unit_power_iff by simp
  hence "p^k dvd x"
    by auto
- moreover from `is_unit p` have "p^k dvd p^multiplicity p x"
+ moreover from \<open>is_unit p\<close> have "p^k dvd p^multiplicity p x"
    using multiplicity_unit_left is_unit_power_iff by simp
  ultimately show ?thesis by simp
 next
  case False
  show ?thesis
  proof (cases "p = 0")
    case True
    then have "p^multiplicity p x = 1"
      by simp
    moreover have "p^k dvd x \<Longrightarrow> k = 0"
    proof (rule ccontr)
      assume "p^k dvd x" and "k \<noteq> 0"
-     with `p = 0` have "p^k = 0" by auto
-     with `p^k dvd x` have "0 dvd x" by auto
+     with \<open>p = 0\<close> have "p^k = 0" by auto
+     with \<open>p^k dvd x\<close> have "0 dvd x" by auto
      hence "x = 0" by auto
-     with `x \<noteq> 0` show False by auto
+     with \<open>x \<noteq> 0\<close> show False by auto
    qed
    ultimately show ?thesis
-     by (auto simp add: is_unit_power_iff `\<not> is_unit p`)
+     by (auto simp add: is_unit_power_iff \<open>\<not> is_unit p\<close>)
  next
    case False
-   with `x \<noteq> 0` `\<not> is_unit p` show ?thesis
+   with \<open>x \<noteq> 0\<close> \<open>\<not> is_unit p\<close> show ?thesis
      by (simp add: power_dvd_iff_le_multiplicity dvd_power_iff multiplicity_same_power)
  qed
 qed
 
 lemma multiplicity_decomposeI:
  assumes "x = p^k * x'" and "\<not> p dvd x'" and "p \<noteq> 0"
  shows "multiplicity p x = k"
   using assms local.multiplicity_eqI local.power_Suc2 by force
 
 lemma multiplicity_sum_lt:
  assumes "multiplicity p a < multiplicity p b" "a \<noteq> 0" "b \<noteq> 0"
  shows "multiplicity p (a + b) = multiplicity p a"
 proof -
  let ?vp = "multiplicity p"
  have unit: "\<not> is_unit p"
  proof
    assume "is_unit p"
    then have "?vp a = 0" and "?vp b = 0" using multiplicity_unit_left by auto
    with assms show False by auto
  qed
 
  from multiplicity_decompose' obtain a' where a': "a = p^?vp a * a'" "\<not> p dvd a'"
    using unit assms by metis
  from multiplicity_decompose' obtain b' where b': "b = p^?vp b * b'"
    using unit assms by metis
 
  show "?vp (a + b) = ?vp a"
  proof (rule multiplicity_decomposeI)
    let ?k = "?vp b - ?vp a"
    from assms have k: "?k > 0" by simp
    with b' have "b = p^?vp a * p^?k * b'"
      by (simp flip: power_add)
    with a' show *: "a + b = p^?vp a * (a' + p^?k * b')"
      by (simp add: ac_simps distrib_left)
    moreover show "\<not> p dvd a' + p^?k * b'"
      using a' k dvd_add_left_iff by auto
    show "p \<noteq> 0" using assms by auto
  qed
 qed
 
 corollary multiplicity_sum_min:
  assumes "multiplicity p a \<noteq> multiplicity p b" "a \<noteq> 0" "b \<noteq> 0"
  shows "multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b)"
 proof -
  let ?vp = "multiplicity p"
  from assms have "?vp a < ?vp b \<or> ?vp a > ?vp b"
    by auto
  then show ?thesis
    by (metis assms multiplicity_sum_lt min.commute add_commute min.strict_order_iff)    
 qed
 
 end
 
 end
diff --git a/src/HOL/Equiv_Relations.thy b/src/HOL/Equiv_Relations.thy
--- a/src/HOL/Equiv_Relations.thy
+++ b/src/HOL/Equiv_Relations.thy
@@ -1,642 +1,642 @@
 (*  Title:      HOL/Equiv_Relations.thy
     Author:     Lawrence C Paulson, 1996 Cambridge University Computer Laboratory
 *)
 
 section \<open>Equivalence Relations in Higher-Order Set Theory\<close>
 
 theory Equiv_Relations
   imports Groups_Big
 begin
 
 subsection \<open>Equivalence relations -- set version\<close>
 
 definition equiv :: "'a set \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> bool"
   where "equiv A r \<longleftrightarrow> refl_on A r \<and> sym r \<and> trans r"
 
 lemma equivI: "refl_on A r \<Longrightarrow> sym r \<Longrightarrow> trans r \<Longrightarrow> equiv A r"
   by (simp add: equiv_def)
 
 lemma equivE:
   assumes "equiv A r"
   obtains "refl_on A r" and "sym r" and "trans r"
   using assms by (simp add: equiv_def)
 
 text \<open>
   Suppes, Theorem 70: \<open>r\<close> is an equiv relation iff \<open>r\<inverse> O r = r\<close>.
 
   First half: \<open>equiv A r \<Longrightarrow> r\<inverse> O r = r\<close>.
 \<close>
 
 lemma sym_trans_comp_subset: "sym r \<Longrightarrow> trans r \<Longrightarrow> r\<inverse> O r \<subseteq> r"
   unfolding trans_def sym_def converse_unfold by blast
 
 lemma refl_on_comp_subset: "refl_on A r \<Longrightarrow> r \<subseteq> r\<inverse> O r"
   unfolding refl_on_def by blast
 
 lemma equiv_comp_eq: "equiv A r \<Longrightarrow> r\<inverse> O r = r"
   unfolding equiv_def
   by (iprover intro: sym_trans_comp_subset refl_on_comp_subset equalityI)
 
 text \<open>Second half.\<close>
 
 lemma comp_equivI:
   assumes "r\<inverse> O r = r" "Domain r = A"
   shows "equiv A r"
 proof -
   have *: "\<And>x y. (x, y) \<in> r \<Longrightarrow> (y, x) \<in> r"
     using assms by blast
   show ?thesis
     unfolding equiv_def refl_on_def sym_def trans_def
     using assms by (auto intro: *)
 qed
 
 
 subsection \<open>Equivalence classes\<close>
 
 lemma equiv_class_subset: "equiv A r \<Longrightarrow> (a, b) \<in> r \<Longrightarrow> r``{a} \<subseteq> r``{b}"
   \<comment> \<open>lemma for the next result\<close>
   unfolding equiv_def trans_def sym_def by blast
 
 theorem equiv_class_eq: "equiv A r \<Longrightarrow> (a, b) \<in> r \<Longrightarrow> r``{a} = r``{b}"
   by (intro equalityI equiv_class_subset; force simp add: equiv_def sym_def)
 
 lemma equiv_class_self: "equiv A r \<Longrightarrow> a \<in> A \<Longrightarrow> a \<in> r``{a}"
   unfolding equiv_def refl_on_def by blast
 
 lemma subset_equiv_class: "equiv A r \<Longrightarrow> r``{b} \<subseteq> r``{a} \<Longrightarrow> b \<in> A \<Longrightarrow> (a, b) \<in> r"
   \<comment> \<open>lemma for the next result\<close>
   unfolding equiv_def refl_on_def by blast
 
 lemma eq_equiv_class: "r``{a} = r``{b} \<Longrightarrow> equiv A r \<Longrightarrow> b \<in> A \<Longrightarrow> (a, b) \<in> r"
   by (iprover intro: equalityD2 subset_equiv_class)
 
 lemma equiv_class_nondisjoint: "equiv A r \<Longrightarrow> x \<in> (r``{a} \<inter> r``{b}) \<Longrightarrow> (a, b) \<in> r"
   unfolding equiv_def trans_def sym_def by blast
 
 lemma equiv_type: "equiv A r \<Longrightarrow> r \<subseteq> A \<times> A"
   unfolding equiv_def refl_on_def by blast
 
 lemma equiv_class_eq_iff: "equiv A r \<Longrightarrow> (x, y) \<in> r \<longleftrightarrow> r``{x} = r``{y} \<and> x \<in> A \<and> y \<in> A"
   by (blast intro!: equiv_class_eq dest: eq_equiv_class equiv_type)
 
 lemma eq_equiv_class_iff: "equiv A r \<Longrightarrow> x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> r``{x} = r``{y} \<longleftrightarrow> (x, y) \<in> r"
   by (blast intro!: equiv_class_eq dest: eq_equiv_class equiv_type)
 
 lemma disjnt_equiv_class: "equiv A r \<Longrightarrow> disjnt (r``{a}) (r``{b}) \<longleftrightarrow> (a, b) \<notin> r"
   by (auto dest: equiv_class_self simp: equiv_class_eq_iff disjnt_def)
 
 
 subsection \<open>Quotients\<close>
 
 definition quotient :: "'a set \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> 'a set set"  (infixl "'/'/" 90)
   where "A//r = (\<Union>x \<in> A. {r``{x}})"  \<comment> \<open>set of equiv classes\<close>
 
 lemma quotientI: "x \<in> A \<Longrightarrow> r``{x} \<in> A//r"
   unfolding quotient_def by blast
 
 lemma quotientE: "X \<in> A//r \<Longrightarrow> (\<And>x. X = r``{x} \<Longrightarrow> x \<in> A \<Longrightarrow> P) \<Longrightarrow> P"
   unfolding quotient_def by blast
 
 lemma Union_quotient: "equiv A r \<Longrightarrow> \<Union>(A//r) = A"
   unfolding equiv_def refl_on_def quotient_def by blast
 
 lemma quotient_disj: "equiv A r \<Longrightarrow> X \<in> A//r \<Longrightarrow> Y \<in> A//r \<Longrightarrow> X = Y \<or> X \<inter> Y = {}"
   unfolding quotient_def equiv_def trans_def sym_def by blast
 
 lemma quotient_eqI:
   assumes "equiv A r" "X \<in> A//r" "Y \<in> A//r" and xy: "x \<in> X" "y \<in> Y" "(x, y) \<in> r"
   shows "X = Y"
 proof -
   obtain a b where "a \<in> A" and a: "X = r `` {a}" and "b \<in> A" and b: "Y = r `` {b}"
     using assms by (auto elim!: quotientE)
   then have "(a,b) \<in> r"
       using xy \<open>equiv A r\<close> unfolding equiv_def sym_def trans_def by blast
   then show ?thesis
     unfolding a b by (rule equiv_class_eq [OF \<open>equiv A r\<close>])
 qed
 
 lemma quotient_eq_iff:
   assumes "equiv A r" "X \<in> A//r" "Y \<in> A//r" and xy: "x \<in> X" "y \<in> Y" 
   shows "X = Y \<longleftrightarrow> (x, y) \<in> r"
 proof
   assume L: "X = Y" 
   with assms show "(x, y) \<in> r" 
     unfolding equiv_def sym_def trans_def by (blast elim!: quotientE)
 next
   assume \<section>: "(x, y) \<in> r" show "X = Y"
     by (rule quotient_eqI) (use \<section> assms in \<open>blast+\<close>)
 qed
 
 lemma eq_equiv_class_iff2: "equiv A r \<Longrightarrow> x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> {x}//r = {y}//r \<longleftrightarrow> (x, y) \<in> r"
   by (simp add: quotient_def eq_equiv_class_iff)
 
 lemma quotient_empty [simp]: "{}//r = {}"
   by (simp add: quotient_def)
 
 lemma quotient_is_empty [iff]: "A//r = {} \<longleftrightarrow> A = {}"
   by (simp add: quotient_def)
 
 lemma quotient_is_empty2 [iff]: "{} = A//r \<longleftrightarrow> A = {}"
   by (simp add: quotient_def)
 
 lemma singleton_quotient: "{x}//r = {r `` {x}}"
   by (simp add: quotient_def)
 
 lemma quotient_diff1: "inj_on (\<lambda>a. {a}//r) A \<Longrightarrow> a \<in> A \<Longrightarrow> (A - {a})//r = A//r - {a}//r"
   unfolding quotient_def inj_on_def by blast
 
 
 subsection \<open>Refinement of one equivalence relation WRT another\<close>
 
 lemma refines_equiv_class_eq: "R \<subseteq> S \<Longrightarrow> equiv A R \<Longrightarrow> equiv A S \<Longrightarrow> R``(S``{a}) = S``{a}"
   by (auto simp: equiv_class_eq_iff)
 
 lemma refines_equiv_class_eq2: "R \<subseteq> S \<Longrightarrow> equiv A R \<Longrightarrow> equiv A S \<Longrightarrow> S``(R``{a}) = S``{a}"
   by (auto simp: equiv_class_eq_iff)
 
 lemma refines_equiv_image_eq: "R \<subseteq> S \<Longrightarrow> equiv A R \<Longrightarrow> equiv A S \<Longrightarrow> (\<lambda>X. S``X) ` (A//R) = A//S"
    by (auto simp: quotient_def image_UN refines_equiv_class_eq2)
 
 lemma finite_refines_finite:
   "finite (A//R) \<Longrightarrow> R \<subseteq> S \<Longrightarrow> equiv A R \<Longrightarrow> equiv A S \<Longrightarrow> finite (A//S)"
   by (erule finite_surj [where f = "\<lambda>X. S``X"]) (simp add: refines_equiv_image_eq)
 
 lemma finite_refines_card_le:
   "finite (A//R) \<Longrightarrow> R \<subseteq> S \<Longrightarrow> equiv A R \<Longrightarrow> equiv A S \<Longrightarrow> card (A//S) \<le> card (A//R)"
   by (subst refines_equiv_image_eq [of R S A, symmetric])
     (auto simp: card_image_le [where f = "\<lambda>X. S``X"])
 
 
 subsection \<open>Defining unary operations upon equivalence classes\<close>
 
 text \<open>A congruence-preserving function.\<close>
 
 definition congruent :: "('a \<times> 'a) set \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> bool"
   where "congruent r f \<longleftrightarrow> (\<forall>(y, z) \<in> r. f y = f z)"
 
 lemma congruentI: "(\<And>y z. (y, z) \<in> r \<Longrightarrow> f y = f z) \<Longrightarrow> congruent r f"
   by (auto simp add: congruent_def)
 
 lemma congruentD: "congruent r f \<Longrightarrow> (y, z) \<in> r \<Longrightarrow> f y = f z"
   by (auto simp add: congruent_def)
 
 abbreviation RESPECTS :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> bool"  (infixr "respects" 80)
   where "f respects r \<equiv> congruent r f"
 
 
 lemma UN_constant_eq: "a \<in> A \<Longrightarrow> \<forall>y \<in> A. f y = c \<Longrightarrow> (\<Union>y \<in> A. f y) = c"
   \<comment> \<open>lemma required to prove \<open>UN_equiv_class\<close>\<close>
   by auto
 
 lemma UN_equiv_class:
   assumes "equiv A r" "f respects r" "a \<in> A"
   shows "(\<Union>x \<in> r``{a}. f x) = f a"
   \<comment> \<open>Conversion rule\<close>
 proof -
   have \<section>: "\<forall>x\<in>r `` {a}. f x = f a"
     using assms unfolding equiv_def congruent_def sym_def by blast
   show ?thesis
     by (iprover intro: assms UN_constant_eq [OF equiv_class_self \<section>])
 qed
 
 lemma UN_equiv_class_type:
   assumes r: "equiv A r" "f respects r" and X: "X \<in> A//r" and AB: "\<And>x. x \<in> A \<Longrightarrow> f x \<in> B"
   shows "(\<Union>x \<in> X. f x) \<in> B"
   using assms unfolding quotient_def
   by (auto simp: UN_equiv_class [OF r])
 
 text \<open>
   Sufficient conditions for injectiveness.  Could weaken premises!
   major premise could be an inclusion; \<open>bcong\<close> could be
   \<open>\<And>y. y \<in> A \<Longrightarrow> f y \<in> B\<close>.
 \<close>
 
 lemma UN_equiv_class_inject:
   assumes "equiv A r" "f respects r"
     and eq: "(\<Union>x \<in> X. f x) = (\<Union>y \<in> Y. f y)" 
     and X: "X \<in> A//r" and Y: "Y \<in> A//r" 
     and fr: "\<And>x y. x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> f x = f y \<Longrightarrow> (x, y) \<in> r"
   shows "X = Y"
 proof -
   obtain a b where "a \<in> A" and a: "X = r `` {a}" and "b \<in> A" and b: "Y = r `` {b}"
     using assms by (auto elim!: quotientE)
   then have "\<Union> (f ` r `` {a}) = f a" "\<Union> (f ` r `` {b}) = f b"
     by (iprover intro: UN_equiv_class [OF \<open>equiv A r\<close>] assms)+
   then have "f a = f b"
     using eq unfolding a b by (iprover intro: trans sym)
   then have "(a,b) \<in> r"
     using fr \<open>a \<in> A\<close> \<open>b \<in> A\<close> by blast
   then show ?thesis
     unfolding a b by (rule equiv_class_eq [OF \<open>equiv A r\<close>])
 qed
 
 
 subsection \<open>Defining binary operations upon equivalence classes\<close>
 
 text \<open>A congruence-preserving function of two arguments.\<close>
 
 definition congruent2 :: "('a \<times> 'a) set \<Rightarrow> ('b \<times> 'b) set \<Rightarrow> ('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> bool"
   where "congruent2 r1 r2 f \<longleftrightarrow> (\<forall>(y1, z1) \<in> r1. \<forall>(y2, z2) \<in> r2. f y1 y2 = f z1 z2)"
 
 lemma congruent2I':
   assumes "\<And>y1 z1 y2 z2. (y1, z1) \<in> r1 \<Longrightarrow> (y2, z2) \<in> r2 \<Longrightarrow> f y1 y2 = f z1 z2"
   shows "congruent2 r1 r2 f"
   using assms by (auto simp add: congruent2_def)
 
 lemma congruent2D: "congruent2 r1 r2 f \<Longrightarrow> (y1, z1) \<in> r1 \<Longrightarrow> (y2, z2) \<in> r2 \<Longrightarrow> f y1 y2 = f z1 z2"
   by (auto simp add: congruent2_def)
 
 text \<open>Abbreviation for the common case where the relations are identical.\<close>
 abbreviation RESPECTS2:: "('a \<Rightarrow> 'a \<Rightarrow> 'b) \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> bool"  (infixr "respects2" 80)
   where "f respects2 r \<equiv> congruent2 r r f"
 
 
 lemma congruent2_implies_congruent:
   "equiv A r1 \<Longrightarrow> congruent2 r1 r2 f \<Longrightarrow> a \<in> A \<Longrightarrow> congruent r2 (f a)"
   unfolding congruent_def congruent2_def equiv_def refl_on_def by blast
 
 lemma congruent2_implies_congruent_UN:
   assumes "equiv A1 r1" "equiv A2 r2" "congruent2 r1 r2 f" "a \<in> A2" 
   shows "congruent r1 (\<lambda>x1. \<Union>x2 \<in> r2``{a}. f x1 x2)"
   unfolding congruent_def
 proof clarify
   fix c d
   assume cd: "(c,d) \<in> r1"
   then have "c \<in> A1" "d \<in> A1"
     using \<open>equiv A1 r1\<close> by (auto elim!: equiv_type [THEN subsetD, THEN SigmaE2])
   with assms show "\<Union> (f c ` r2 `` {a}) = \<Union> (f d ` r2 `` {a})"
   proof (simp add: UN_equiv_class congruent2_implies_congruent)
     show "f c a = f d a"
       using assms cd unfolding congruent2_def equiv_def refl_on_def by blast
   qed
 qed
 
 lemma UN_equiv_class2:
   "equiv A1 r1 \<Longrightarrow> equiv A2 r2 \<Longrightarrow> congruent2 r1 r2 f \<Longrightarrow> a1 \<in> A1 \<Longrightarrow> a2 \<in> A2 \<Longrightarrow>
     (\<Union>x1 \<in> r1``{a1}. \<Union>x2 \<in> r2``{a2}. f x1 x2) = f a1 a2"
   by (simp add: UN_equiv_class congruent2_implies_congruent congruent2_implies_congruent_UN)
 
 lemma UN_equiv_class_type2:
   "equiv A1 r1 \<Longrightarrow> equiv A2 r2 \<Longrightarrow> congruent2 r1 r2 f
     \<Longrightarrow> X1 \<in> A1//r1 \<Longrightarrow> X2 \<in> A2//r2
     \<Longrightarrow> (\<And>x1 x2. x1 \<in> A1 \<Longrightarrow> x2 \<in> A2 \<Longrightarrow> f x1 x2 \<in> B)
     \<Longrightarrow> (\<Union>x1 \<in> X1. \<Union>x2 \<in> X2. f x1 x2) \<in> B"
   unfolding quotient_def
   by (blast intro: UN_equiv_class_type congruent2_implies_congruent_UN
                    congruent2_implies_congruent quotientI)
 
 
 lemma UN_UN_split_split_eq:
   "(\<Union>(x1, x2) \<in> X. \<Union>(y1, y2) \<in> Y. A x1 x2 y1 y2) =
     (\<Union>x \<in> X. \<Union>y \<in> Y. (\<lambda>(x1, x2). (\<lambda>(y1, y2). A x1 x2 y1 y2) y) x)"
   \<comment> \<open>Allows a natural expression of binary operators,\<close>
   \<comment> \<open>without explicit calls to \<open>split\<close>\<close>
   by auto
 
 lemma congruent2I:
   "equiv A1 r1 \<Longrightarrow> equiv A2 r2
     \<Longrightarrow> (\<And>y z w. w \<in> A2 \<Longrightarrow> (y,z) \<in> r1 \<Longrightarrow> f y w = f z w)
     \<Longrightarrow> (\<And>y z w. w \<in> A1 \<Longrightarrow> (y,z) \<in> r2 \<Longrightarrow> f w y = f w z)
     \<Longrightarrow> congruent2 r1 r2 f"
   \<comment> \<open>Suggested by John Harrison -- the two subproofs may be\<close>
   \<comment> \<open>\<^emph>\<open>much\<close> simpler than the direct proof.\<close>
   unfolding congruent2_def equiv_def refl_on_def
   by (blast intro: trans)
 
 lemma congruent2_commuteI:
   assumes equivA: "equiv A r"
     and commute: "\<And>y z. y \<in> A \<Longrightarrow> z \<in> A \<Longrightarrow> f y z = f z y"
     and congt: "\<And>y z w. w \<in> A \<Longrightarrow> (y,z) \<in> r \<Longrightarrow> f w y = f w z"
   shows "f respects2 r"
 proof (rule congruent2I [OF equivA equivA])
   note eqv = equivA [THEN equiv_type, THEN subsetD, THEN SigmaE2]
   show "\<And>y z w. \<lbrakk>w \<in> A; (y, z) \<in> r\<rbrakk> \<Longrightarrow> f y w = f z w"
     by (iprover intro: commute [THEN trans] sym congt elim: eqv)
   show "\<And>y z w. \<lbrakk>w \<in> A; (y, z) \<in> r\<rbrakk> \<Longrightarrow> f w y = f w z"
     by (iprover intro: congt elim: eqv)
 qed
 
 
 subsection \<open>Quotients and finiteness\<close>
 
 text \<open>Suggested by Florian Kammüller\<close>
 
 lemma finite_quotient:
   assumes "finite A" "r \<subseteq> A \<times> A"
   shows "finite (A//r)"
     \<comment> \<open>recall @{thm equiv_type}\<close>
 proof -
   have "A//r \<subseteq> Pow A"
     using assms unfolding quotient_def by blast
   moreover have "finite (Pow A)"
     using assms by simp
   ultimately show ?thesis
     by (iprover intro: finite_subset)
 qed
 
 lemma finite_equiv_class: "finite A \<Longrightarrow> r \<subseteq> A \<times> A \<Longrightarrow> X \<in> A//r \<Longrightarrow> finite X"
   unfolding quotient_def
   by (erule rev_finite_subset) blast
 
 lemma equiv_imp_dvd_card:
   assumes "finite A" "equiv A r" "\<And>X. X \<in> A//r \<Longrightarrow> k dvd card X"
   shows "k dvd card A"
 proof (rule Union_quotient [THEN subst])
   show "k dvd card (\<Union> (A // r))"
     apply (rule dvd_partition)
     using assms
     by (auto simp: Union_quotient dest: quotient_disj)
 qed (use assms in blast)
 
 lemma card_quotient_disjoint:
   assumes "finite A" "inj_on (\<lambda>x. {x} // r) A"
   shows "card (A//r) = card A"
 proof -
   have "\<forall>i\<in>A. \<forall>j\<in>A. i \<noteq> j \<longrightarrow> r `` {j} \<noteq> r `` {i}"
     using assms by (fastforce simp add: quotient_def inj_on_def)
   with assms show ?thesis
     by (simp add: quotient_def card_UN_disjoint)
 qed
 
 text \<open>By Jakub Kądziołka:\<close>
 
 lemma sum_fun_comp:
   assumes "finite S" "finite R" "g ` S \<subseteq> R"
   shows "(\<Sum>x \<in> S. f (g x)) = (\<Sum>y \<in> R. of_nat (card {x \<in> S. g x = y}) * f y)"
 proof -
   let ?r = "relation_of (\<lambda>p q. g p = g q) S"
   have eqv: "equiv S ?r"
     unfolding relation_of_def by (auto intro: comp_equivI)
   have finite: "C \<in> S//?r \<Longrightarrow> finite C" for C
-    by (fact finite_equiv_class[OF `finite S` equiv_type[OF `equiv S ?r`]])
+    by (fact finite_equiv_class[OF \<open>finite S\<close> equiv_type[OF \<open>equiv S ?r\<close>]])
   have disjoint: "A \<in> S//?r \<Longrightarrow> B \<in> S//?r \<Longrightarrow> A \<noteq> B \<Longrightarrow> A \<inter> B = {}" for A B
     using eqv quotient_disj by blast
 
   let ?cls = "\<lambda>y. {x \<in> S. y = g x}"
   have quot_as_img: "S//?r = ?cls ` g ` S"
     by (auto simp add: relation_of_def quotient_def)
   have cls_inj: "inj_on ?cls (g ` S)"
     by (auto intro: inj_onI)
 
   have rest_0: "(\<Sum>y \<in> R - g ` S. of_nat (card (?cls y)) * f y) = 0"
   proof -
     have "of_nat (card (?cls y)) * f y = 0" if asm: "y \<in> R - g ` S" for y
     proof -
       from asm have *: "?cls y = {}" by auto
       show ?thesis unfolding * by simp
     qed
     thus ?thesis by simp
   qed
 
   have "(\<Sum>x \<in> S. f (g x)) = (\<Sum>C \<in> S//?r. \<Sum>x \<in> C. f (g x))"
     using eqv finite disjoint
     by (simp flip: sum.Union_disjoint[simplified] add: Union_quotient)
   also have "... = (\<Sum>y \<in> g ` S. \<Sum>x \<in> ?cls y. f (g x))"
     unfolding quot_as_img by (simp add: sum.reindex[OF cls_inj])
   also have "... = (\<Sum>y \<in> g ` S. \<Sum>x \<in> ?cls y. f y)"
     by auto
   also have "... = (\<Sum>y \<in> g ` S. of_nat (card (?cls y)) * f y)"
     by (simp flip: sum_constant)
   also have "... = (\<Sum>y \<in> R. of_nat (card (?cls y)) * f y)"
     using rest_0 by (simp add: sum.subset_diff[OF \<open>g ` S \<subseteq> R\<close> \<open>finite R\<close>])
   finally show ?thesis
     by (simp add: eq_commute)
 qed
 
 subsection \<open>Projection\<close>
 
 definition proj :: "('b \<times> 'a) set \<Rightarrow> 'b \<Rightarrow> 'a set"
   where "proj r x = r `` {x}"
 
 lemma proj_preserves: "x \<in> A \<Longrightarrow> proj r x \<in> A//r"
   unfolding proj_def by (rule quotientI)
 
 lemma proj_in_iff:
   assumes "equiv A r"
   shows "proj r x \<in> A//r \<longleftrightarrow> x \<in> A"
     (is "?lhs \<longleftrightarrow> ?rhs")
 proof
   assume ?rhs
   then show ?lhs by (simp add: proj_preserves)
 next
   assume ?lhs
   then show ?rhs
     unfolding proj_def quotient_def
   proof clarsimp
     fix y
     assume y: "y \<in> A" and "r `` {x} = r `` {y}"
     moreover have "y \<in> r `` {y}"
       using assms y unfolding equiv_def refl_on_def by blast
     ultimately have "(x, y) \<in> r" by blast
     then show "x \<in> A"
       using assms unfolding equiv_def refl_on_def by blast
   qed
 qed
 
 lemma proj_iff: "equiv A r \<Longrightarrow> {x, y} \<subseteq> A \<Longrightarrow> proj r x = proj r y \<longleftrightarrow> (x, y) \<in> r"
   by (simp add: proj_def eq_equiv_class_iff)
 
 (*
 lemma in_proj: "\<lbrakk>equiv A r; x \<in> A\<rbrakk> \<Longrightarrow> x \<in> proj r x"
 unfolding proj_def equiv_def refl_on_def by blast
 *)
 
 lemma proj_image: "proj r ` A = A//r"
   unfolding proj_def[abs_def] quotient_def by blast
 
 lemma in_quotient_imp_non_empty: "equiv A r \<Longrightarrow> X \<in> A//r \<Longrightarrow> X \<noteq> {}"
   unfolding quotient_def using equiv_class_self by fast
 
 lemma in_quotient_imp_in_rel: "equiv A r \<Longrightarrow> X \<in> A//r \<Longrightarrow> {x, y} \<subseteq> X \<Longrightarrow> (x, y) \<in> r"
   using quotient_eq_iff[THEN iffD1] by fastforce
 
 lemma in_quotient_imp_closed: "equiv A r \<Longrightarrow> X \<in> A//r \<Longrightarrow> x \<in> X \<Longrightarrow> (x, y) \<in> r \<Longrightarrow> y \<in> X"
   unfolding quotient_def equiv_def trans_def by blast
 
 lemma in_quotient_imp_subset: "equiv A r \<Longrightarrow> X \<in> A//r \<Longrightarrow> X \<subseteq> A"
   using in_quotient_imp_in_rel equiv_type by fastforce
 
 
 subsection \<open>Equivalence relations -- predicate version\<close>
 
 text \<open>Partial equivalences.\<close>
 
 definition part_equivp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool"
   where "part_equivp R \<longleftrightarrow> (\<exists>x. R x x) \<and> (\<forall>x y. R x y \<longleftrightarrow> R x x \<and> R y y \<and> R x = R y)"
     \<comment> \<open>John-Harrison-style characterization\<close>
 
 lemma part_equivpI: "\<exists>x. R x x \<Longrightarrow> symp R \<Longrightarrow> transp R \<Longrightarrow> part_equivp R"
   by (auto simp add: part_equivp_def) (auto elim: sympE transpE)
 
 lemma part_equivpE:
   assumes "part_equivp R"
   obtains x where "R x x" and "symp R" and "transp R"
 proof -
   from assms have 1: "\<exists>x. R x x"
     and 2: "\<And>x y. R x y \<longleftrightarrow> R x x \<and> R y y \<and> R x = R y"
     unfolding part_equivp_def by blast+
   from 1 obtain x where "R x x" ..
   moreover have "symp R"
   proof (rule sympI)
     fix x y
     assume "R x y"
     with 2 [of x y] show "R y x" by auto
   qed
   moreover have "transp R"
   proof (rule transpI)
     fix x y z
     assume "R x y" and "R y z"
     with 2 [of x y] 2 [of y z] show "R x z" by auto
   qed
   ultimately show thesis by (rule that)
 qed
 
 lemma part_equivp_refl_symp_transp: "part_equivp R \<longleftrightarrow> (\<exists>x. R x x) \<and> symp R \<and> transp R"
   by (auto intro: part_equivpI elim: part_equivpE)
 
 lemma part_equivp_symp: "part_equivp R \<Longrightarrow> R x y \<Longrightarrow> R y x"
   by (erule part_equivpE, erule sympE)
 
 lemma part_equivp_transp: "part_equivp R \<Longrightarrow> R x y \<Longrightarrow> R y z \<Longrightarrow> R x z"
   by (erule part_equivpE, erule transpE)
 
 lemma part_equivp_typedef: "part_equivp R \<Longrightarrow> \<exists>d. d \<in> {c. \<exists>x. R x x \<and> c = Collect (R x)}"
   by (auto elim: part_equivpE)
 
 
 text \<open>Total equivalences.\<close>
 
 definition equivp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool"
   where "equivp R \<longleftrightarrow> (\<forall>x y. R x y = (R x = R y))" \<comment> \<open>John-Harrison-style characterization\<close>
 
 lemma equivpI: "reflp R \<Longrightarrow> symp R \<Longrightarrow> transp R \<Longrightarrow> equivp R"
   by (auto elim: reflpE sympE transpE simp add: equivp_def)
 
 lemma equivpE:
   assumes "equivp R"
   obtains "reflp R" and "symp R" and "transp R"
   using assms by (auto intro!: that reflpI sympI transpI simp add: equivp_def)
 
 lemma equivp_implies_part_equivp: "equivp R \<Longrightarrow> part_equivp R"
   by (auto intro: part_equivpI elim: equivpE reflpE)
 
 lemma equivp_equiv: "equiv UNIV A \<longleftrightarrow> equivp (\<lambda>x y. (x, y) \<in> A)"
   by (auto intro!: equivI equivpI [to_set] elim!: equivE equivpE [to_set])
 
 lemma equivp_reflp_symp_transp: "equivp R \<longleftrightarrow> reflp R \<and> symp R \<and> transp R"
   by (auto intro: equivpI elim: equivpE)
 
 lemma identity_equivp: "equivp (=)"
   by (auto intro: equivpI reflpI sympI transpI)
 
 lemma equivp_reflp: "equivp R \<Longrightarrow> R x x"
   by (erule equivpE, erule reflpE)
 
 lemma equivp_symp: "equivp R \<Longrightarrow> R x y \<Longrightarrow> R y x"
   by (erule equivpE, erule sympE)
 
 lemma equivp_transp: "equivp R \<Longrightarrow> R x y \<Longrightarrow> R y z \<Longrightarrow> R x z"
   by (erule equivpE, erule transpE)
 
 lemma equivp_rtranclp: "symp r \<Longrightarrow> equivp r\<^sup>*\<^sup>*"
   by(intro equivpI reflpI sympI transpI)(auto dest: sympD[OF symp_rtranclp])
 
 lemmas equivp_rtranclp_symclp [simp] = equivp_rtranclp[OF symp_symclp]
 
 lemma equivp_vimage2p: "equivp R \<Longrightarrow> equivp (vimage2p f f R)"
   by(auto simp add: equivp_def vimage2p_def dest: fun_cong)
 
 lemma equivp_imp_transp: "equivp R \<Longrightarrow> transp R"
   by(simp add: equivp_reflp_symp_transp)
 
 
 subsection \<open>Equivalence closure\<close>
 
 definition equivclp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool" where
   "equivclp r = (symclp r)\<^sup>*\<^sup>*"
 
 lemma transp_equivclp [simp]: "transp (equivclp r)"
   by(simp add: equivclp_def)
 
 lemma reflp_equivclp [simp]: "reflp (equivclp r)"
   by(simp add: equivclp_def)
 
 lemma symp_equivclp [simp]: "symp (equivclp r)"
   by(simp add: equivclp_def)
 
 lemma equivp_evquivclp [simp]: "equivp (equivclp r)"
   by(simp add: equivpI)
 
 lemma tranclp_equivclp [simp]: "(equivclp r)\<^sup>+\<^sup>+ = equivclp r"
   by(simp add: equivclp_def)
 
 lemma rtranclp_equivclp [simp]: "(equivclp r)\<^sup>*\<^sup>* = equivclp r"
   by(simp add: equivclp_def)
 
 lemma symclp_equivclp [simp]: "symclp (equivclp r) = equivclp r"
   by(simp add: equivclp_def symp_symclp_eq)
 
 lemma equivclp_symclp [simp]: "equivclp (symclp r) = equivclp r"
   by(simp add: equivclp_def)
 
 lemma equivclp_conversep [simp]: "equivclp (conversep r) = equivclp r"
   by(simp add: equivclp_def)
 
 lemma equivclp_sym [sym]: "equivclp r x y \<Longrightarrow> equivclp r y x"
   by(rule sympD[OF symp_equivclp])
 
 lemma equivclp_OO_equivclp_le_equivclp: "equivclp r OO equivclp r \<le> equivclp r"
   by(rule transp_relcompp_less_eq transp_equivclp)+
 
 lemma rtranlcp_le_equivclp: "r\<^sup>*\<^sup>* \<le> equivclp r"
   unfolding equivclp_def by(rule rtranclp_mono)(simp add: symclp_pointfree)
 
 lemma rtranclp_conversep_le_equivclp: "r\<inverse>\<inverse>\<^sup>*\<^sup>* \<le> equivclp r"
   unfolding equivclp_def by(rule rtranclp_mono)(simp add: symclp_pointfree)
 
 lemma symclp_rtranclp_le_equivclp: "symclp r\<^sup>*\<^sup>* \<le> equivclp r"
   unfolding symclp_pointfree
   by(rule le_supI)(simp_all add: rtranclp_conversep[symmetric] rtranlcp_le_equivclp rtranclp_conversep_le_equivclp)
 
 lemma r_OO_conversep_into_equivclp:
   "r\<^sup>*\<^sup>* OO r\<inverse>\<inverse>\<^sup>*\<^sup>* \<le> equivclp r"
   by(blast intro: order_trans[OF _ equivclp_OO_equivclp_le_equivclp] relcompp_mono rtranlcp_le_equivclp rtranclp_conversep_le_equivclp del: predicate2I)
 
 lemma equivclp_induct [consumes 1, case_names base step, induct pred: equivclp]:
   assumes a: "equivclp r a b"
     and cases: "P a" "\<And>y z. equivclp r a y \<Longrightarrow> r y z \<or> r z y \<Longrightarrow> P y \<Longrightarrow> P z"
   shows "P b"
   using a unfolding equivclp_def
   by(induction rule: rtranclp_induct; fold equivclp_def; blast intro: cases elim: symclpE)
 
 lemma converse_equivclp_induct [consumes 1, case_names base step]:
   assumes major: "equivclp r a b"
     and cases: "P b" "\<And>y z. r y z \<or> r z y \<Longrightarrow> equivclp r z b \<Longrightarrow> P z \<Longrightarrow> P y"
   shows "P a"
   using major unfolding equivclp_def
   by(induction rule: converse_rtranclp_induct; fold equivclp_def; blast intro: cases elim: symclpE)
 
 lemma equivclp_refl [simp]: "equivclp r x x"
   by(rule reflpD[OF reflp_equivclp])
 
 lemma r_into_equivclp [intro]: "r x y \<Longrightarrow> equivclp r x y"
   unfolding equivclp_def by(blast intro: symclpI)
 
 lemma converse_r_into_equivclp [intro]: "r y x \<Longrightarrow> equivclp r x y"
   unfolding equivclp_def by(blast intro: symclpI)
 
 lemma rtranclp_into_equivclp: "r\<^sup>*\<^sup>* x y \<Longrightarrow> equivclp r x y"
   using rtranlcp_le_equivclp[of r] by blast
 
 lemma converse_rtranclp_into_equivclp: "r\<^sup>*\<^sup>* y x \<Longrightarrow> equivclp r x y"
   by(blast intro: equivclp_sym rtranclp_into_equivclp)
 
 lemma equivclp_into_equivclp: "\<lbrakk> equivclp r a b; r b c \<or> r c b \<rbrakk> \<Longrightarrow> equivclp r a c"
   unfolding equivclp_def by(erule rtranclp.rtrancl_into_rtrancl)(auto intro: symclpI)
 
 lemma equivclp_trans [trans]: "\<lbrakk> equivclp r a b; equivclp r b c \<rbrakk> \<Longrightarrow> equivclp r a c"
   using equivclp_OO_equivclp_le_equivclp[of r] by blast
 
 hide_const (open) proj
 
 end
diff --git a/src/HOL/Set_Interval.thy b/src/HOL/Set_Interval.thy
--- a/src/HOL/Set_Interval.thy
+++ b/src/HOL/Set_Interval.thy
@@ -1,2564 +1,2564 @@
 (*  Title:      HOL/Set_Interval.thy
     Author:     Tobias Nipkow, Clemens Ballarin, Jeremy Avigad
 
 lessThan, greaterThan, atLeast, atMost and two-sided intervals
 
 Modern convention: Ixy stands for an interval where x and y
 describe the lower and upper bound and x,y : {c,o,i}
 where c = closed, o = open, i = infinite.
 Examples: Ico = {_ ..< _} and Ici = {_ ..}
 *)
 
 section \<open>Set intervals\<close>
 
 theory Set_Interval
 imports Divides
 begin
 
 (* Belongs in Finite_Set but 2 is not available there *)
 lemma card_2_iff: "card S = 2 \<longleftrightarrow> (\<exists>x y. S = {x,y} \<and> x \<noteq> y)"
   by (auto simp: card_Suc_eq numeral_eq_Suc)
 
 lemma card_2_iff': "card S = 2 \<longleftrightarrow> (\<exists>x\<in>S. \<exists>y\<in>S. x \<noteq> y \<and> (\<forall>z\<in>S. z = x \<or> z = y))"
   by (auto simp: card_Suc_eq numeral_eq_Suc)
 
 context ord
 begin
 
 definition
   lessThan    :: "'a => 'a set" ("(1{..<_})") where
   "{..<u} == {x. x < u}"
 
 definition
   atMost      :: "'a => 'a set" ("(1{.._})") where
   "{..u} == {x. x \<le> u}"
 
 definition
   greaterThan :: "'a => 'a set" ("(1{_<..})") where
   "{l<..} == {x. l<x}"
 
 definition
   atLeast     :: "'a => 'a set" ("(1{_..})") where
   "{l..} == {x. l\<le>x}"
 
 definition
   greaterThanLessThan :: "'a => 'a => 'a set"  ("(1{_<..<_})") where
   "{l<..<u} == {l<..} Int {..<u}"
 
 definition
   atLeastLessThan :: "'a => 'a => 'a set"      ("(1{_..<_})") where
   "{l..<u} == {l..} Int {..<u}"
 
 definition
   greaterThanAtMost :: "'a => 'a => 'a set"    ("(1{_<.._})") where
   "{l<..u} == {l<..} Int {..u}"
 
 definition
   atLeastAtMost :: "'a => 'a => 'a set"        ("(1{_.._})") where
   "{l..u} == {l..} Int {..u}"
 
 end
 
 
 text\<open>A note of warning when using \<^term>\<open>{..<n}\<close> on type \<^typ>\<open>nat\<close>: it is equivalent to \<^term>\<open>{0::nat..<n}\<close> but some lemmas involving
 \<^term>\<open>{m..<n}\<close> may not exist in \<^term>\<open>{..<n}\<close>-form as well.\<close>
 
 syntax (ASCII)
   "_UNION_le"   :: "'a => 'a => 'b set => 'b set"       ("(3UN _<=_./ _)" [0, 0, 10] 10)
   "_UNION_less" :: "'a => 'a => 'b set => 'b set"       ("(3UN _<_./ _)" [0, 0, 10] 10)
   "_INTER_le"   :: "'a => 'a => 'b set => 'b set"       ("(3INT _<=_./ _)" [0, 0, 10] 10)
   "_INTER_less" :: "'a => 'a => 'b set => 'b set"       ("(3INT _<_./ _)" [0, 0, 10] 10)
 
 syntax (latex output)
   "_UNION_le"   :: "'a \<Rightarrow> 'a => 'b set => 'b set"       ("(3\<Union>(\<open>unbreakable\<close>_ \<le> _)/ _)" [0, 0, 10] 10)
   "_UNION_less" :: "'a \<Rightarrow> 'a => 'b set => 'b set"       ("(3\<Union>(\<open>unbreakable\<close>_ < _)/ _)" [0, 0, 10] 10)
   "_INTER_le"   :: "'a \<Rightarrow> 'a => 'b set => 'b set"       ("(3\<Inter>(\<open>unbreakable\<close>_ \<le> _)/ _)" [0, 0, 10] 10)
   "_INTER_less" :: "'a \<Rightarrow> 'a => 'b set => 'b set"       ("(3\<Inter>(\<open>unbreakable\<close>_ < _)/ _)" [0, 0, 10] 10)
 
 syntax
   "_UNION_le"   :: "'a => 'a => 'b set => 'b set"       ("(3\<Union>_\<le>_./ _)" [0, 0, 10] 10)
   "_UNION_less" :: "'a => 'a => 'b set => 'b set"       ("(3\<Union>_<_./ _)" [0, 0, 10] 10)
   "_INTER_le"   :: "'a => 'a => 'b set => 'b set"       ("(3\<Inter>_\<le>_./ _)" [0, 0, 10] 10)
   "_INTER_less" :: "'a => 'a => 'b set => 'b set"       ("(3\<Inter>_<_./ _)" [0, 0, 10] 10)
 
 translations
   "\<Union>i\<le>n. A" \<rightleftharpoons> "\<Union>i\<in>{..n}. A"
   "\<Union>i<n. A" \<rightleftharpoons> "\<Union>i\<in>{..<n}. A"
   "\<Inter>i\<le>n. A" \<rightleftharpoons> "\<Inter>i\<in>{..n}. A"
   "\<Inter>i<n. A" \<rightleftharpoons> "\<Inter>i\<in>{..<n}. A"
 
 
 subsection \<open>Various equivalences\<close>
 
 lemma (in ord) lessThan_iff [iff]: "(i \<in> lessThan k) = (i<k)"
 by (simp add: lessThan_def)
 
 lemma Compl_lessThan [simp]:
     "!!k:: 'a::linorder. -lessThan k = atLeast k"
   by (auto simp add: lessThan_def atLeast_def)
 
 lemma single_Diff_lessThan [simp]: "!!k:: 'a::preorder. {k} - lessThan k = {k}"
   by auto
 
 lemma (in ord) greaterThan_iff [iff]: "(i \<in> greaterThan k) = (k<i)"
   by (simp add: greaterThan_def)
 
 lemma Compl_greaterThan [simp]:
     "!!k:: 'a::linorder. -greaterThan k = atMost k"
   by (auto simp add: greaterThan_def atMost_def)
 
 lemma Compl_atMost [simp]: "!!k:: 'a::linorder. -atMost k = greaterThan k"
   apply (subst Compl_greaterThan [symmetric])
   apply (rule double_complement)
   done
 
 lemma (in ord) atLeast_iff [iff]: "(i \<in> atLeast k) = (k<=i)"
 by (simp add: atLeast_def)
 
 lemma Compl_atLeast [simp]: "!!k:: 'a::linorder. -atLeast k = lessThan k"
   by (auto simp add: lessThan_def atLeast_def)
 
 lemma (in ord) atMost_iff [iff]: "(i \<in> atMost k) = (i<=k)"
 by (simp add: atMost_def)
 
 lemma atMost_Int_atLeast: "!!n:: 'a::order. atMost n Int atLeast n = {n}"
 by (blast intro: order_antisym)
 
 lemma (in linorder) lessThan_Int_lessThan: "{ a <..} \<inter> { b <..} = { max a b <..}"
   by auto
 
 lemma (in linorder) greaterThan_Int_greaterThan: "{..< a} \<inter> {..< b} = {..< min a b}"
   by auto
 
 subsection \<open>Logical Equivalences for Set Inclusion and Equality\<close>
 
 lemma atLeast_empty_triv [simp]: "{{}..} = UNIV"
   by auto
 
 lemma atMost_UNIV_triv [simp]: "{..UNIV} = UNIV"
   by auto
 
 lemma atLeast_subset_iff [iff]:
      "(atLeast x \<subseteq> atLeast y) = (y \<le> (x::'a::preorder))"
 by (blast intro: order_trans)
 
 lemma atLeast_eq_iff [iff]:
      "(atLeast x = atLeast y) = (x = (y::'a::order))"
 by (blast intro: order_antisym order_trans)
 
 lemma greaterThan_subset_iff [iff]:
      "(greaterThan x \<subseteq> greaterThan y) = (y \<le> (x::'a::linorder))"
   unfolding greaterThan_def by (auto simp: linorder_not_less [symmetric])
 
 lemma greaterThan_eq_iff [iff]:
      "(greaterThan x = greaterThan y) = (x = (y::'a::linorder))"
   by (auto simp: elim!: equalityE)
 
 lemma atMost_subset_iff [iff]: "(atMost x \<subseteq> atMost y) = (x \<le> (y::'a::preorder))"
   by (blast intro: order_trans)
 
 lemma atMost_eq_iff [iff]: "(atMost x = atMost y) = (x = (y::'a::order))"
   by (blast intro: order_antisym order_trans)
 
 lemma lessThan_subset_iff [iff]:
      "(lessThan x \<subseteq> lessThan y) = (x \<le> (y::'a::linorder))"
   unfolding lessThan_def by (auto simp: linorder_not_less [symmetric])
 
 lemma lessThan_eq_iff [iff]:
      "(lessThan x = lessThan y) = (x = (y::'a::linorder))"
   by (auto simp: elim!: equalityE)
 
 lemma lessThan_strict_subset_iff:
   fixes m n :: "'a::linorder"
   shows "{..<m} < {..<n} \<longleftrightarrow> m < n"
   by (metis leD lessThan_subset_iff linorder_linear not_less_iff_gr_or_eq psubset_eq)
 
 lemma (in linorder) Ici_subset_Ioi_iff: "{a ..} \<subseteq> {b <..} \<longleftrightarrow> b < a"
   by auto
 
 lemma (in linorder) Iic_subset_Iio_iff: "{.. a} \<subseteq> {..< b} \<longleftrightarrow> a < b"
   by auto
 
 lemma (in preorder) Ioi_le_Ico: "{a <..} \<subseteq> {a ..}"
   by (auto intro: less_imp_le)
 
 subsection \<open>Two-sided intervals\<close>
 
 context ord
 begin
 
 lemma greaterThanLessThan_iff [simp]: "(i \<in> {l<..<u}) = (l < i \<and> i < u)"
   by (simp add: greaterThanLessThan_def)
 
 lemma atLeastLessThan_iff [simp]: "(i \<in> {l..<u}) = (l \<le> i \<and> i < u)"
   by (simp add: atLeastLessThan_def)
 
 lemma greaterThanAtMost_iff [simp]: "(i \<in> {l<..u}) = (l < i \<and> i \<le> u)"
   by (simp add: greaterThanAtMost_def)
 
 lemma atLeastAtMost_iff [simp]: "(i \<in> {l..u}) = (l \<le> i \<and> i \<le> u)"
   by (simp add: atLeastAtMost_def)
 
 text \<open>The above four lemmas could be declared as iffs. Unfortunately this
 breaks many proofs. Since it only helps blast, it is better to leave them
 alone.\<close>
 
 lemma greaterThanLessThan_eq: "{ a <..< b} = { a <..} \<inter> {..< b }"
   by auto
 
 lemma (in order) atLeastLessThan_eq_atLeastAtMost_diff:
   "{a..<b} = {a..b} - {b}"
   by (auto simp add: atLeastLessThan_def atLeastAtMost_def)
 
 lemma (in order) greaterThanAtMost_eq_atLeastAtMost_diff:
   "{a<..b} = {a..b} - {a}"
   by (auto simp add: greaterThanAtMost_def atLeastAtMost_def)
 
 end
 
 subsubsection\<open>Emptyness, singletons, subset\<close>
 
 context preorder
 begin
 
 lemma atLeastatMost_empty_iff[simp]:
   "{a..b} = {} \<longleftrightarrow> (\<not> a \<le> b)"
   by auto (blast intro: order_trans)
 
 lemma atLeastatMost_empty_iff2[simp]:
   "{} = {a..b} \<longleftrightarrow> (\<not> a \<le> b)"
   by auto (blast intro: order_trans)
 
 lemma atLeastLessThan_empty_iff[simp]:
   "{a..<b} = {} \<longleftrightarrow> (\<not> a < b)"
   by auto (blast intro: le_less_trans)
 
 lemma atLeastLessThan_empty_iff2[simp]:
   "{} = {a..<b} \<longleftrightarrow> (\<not> a < b)"
   by auto (blast intro: le_less_trans)
 
 lemma greaterThanAtMost_empty_iff[simp]: "{k<..l} = {} \<longleftrightarrow> \<not> k < l"
   by auto (blast intro: less_le_trans)
 
 lemma greaterThanAtMost_empty_iff2[simp]: "{} = {k<..l} \<longleftrightarrow> \<not> k < l"
   by auto (blast intro: less_le_trans)
 
 lemma atLeastatMost_subset_iff[simp]:
   "{a..b} \<le> {c..d} \<longleftrightarrow> (\<not> a \<le> b) \<or> c \<le> a \<and> b \<le> d"
   unfolding atLeastAtMost_def atLeast_def atMost_def
   by (blast intro: order_trans)
 
 lemma atLeastatMost_psubset_iff:
   "{a..b} < {c..d} \<longleftrightarrow>
    ((\<not> a \<le> b) \<or> c \<le> a \<and> b \<le> d \<and> (c < a \<or> b < d)) \<and> c \<le> d"
   by(simp add: psubset_eq set_eq_iff less_le_not_le)(blast intro: order_trans)
 
 lemma atLeastAtMost_subseteq_atLeastLessThan_iff:
   "{a..b} \<subseteq> {c ..< d} \<longleftrightarrow> (a \<le> b \<longrightarrow> c \<le> a \<and> b < d)" 
   by auto (blast intro: local.order_trans local.le_less_trans elim: )+
 
 lemma Icc_subset_Ici_iff[simp]:
   "{l..h} \<subseteq> {l'..} = (\<not> l\<le>h \<or> l\<ge>l')"
   by(auto simp: subset_eq intro: order_trans)
 
 lemma Icc_subset_Iic_iff[simp]:
   "{l..h} \<subseteq> {..h'} = (\<not> l\<le>h \<or> h\<le>h')"
   by(auto simp: subset_eq intro: order_trans)
 
 lemma not_Ici_eq_empty[simp]: "{l..} \<noteq> {}"
   by(auto simp: set_eq_iff)
 
 lemma not_Iic_eq_empty[simp]: "{..h} \<noteq> {}"
   by(auto simp: set_eq_iff)
 
 lemmas not_empty_eq_Ici_eq_empty[simp] = not_Ici_eq_empty[symmetric]
 lemmas not_empty_eq_Iic_eq_empty[simp] = not_Iic_eq_empty[symmetric]
 
 end
 
 context order
 begin
 
 lemma atLeastatMost_empty[simp]:
   "b < a \<Longrightarrow> {a..b} = {}"
   by(auto simp: atLeastAtMost_def atLeast_def atMost_def)
 
 lemma atLeastLessThan_empty[simp]:
   "b \<le> a \<Longrightarrow> {a..<b} = {}"
   by(auto simp: atLeastLessThan_def)
 
 lemma greaterThanAtMost_empty[simp]: "l \<le> k ==> {k<..l} = {}"
   by(auto simp:greaterThanAtMost_def greaterThan_def atMost_def)
 
 lemma greaterThanLessThan_empty[simp]:"l \<le> k ==> {k<..<l} = {}"
   by(auto simp:greaterThanLessThan_def greaterThan_def lessThan_def)
 
 lemma atLeastAtMost_singleton [simp]: "{a..a} = {a}"
   by (auto simp add: atLeastAtMost_def atMost_def atLeast_def)
 
 lemma atLeastAtMost_singleton': "a = b \<Longrightarrow> {a .. b} = {a}" by simp
 
 lemma Icc_eq_Icc[simp]:
   "{l..h} = {l'..h'} = (l=l' \<and> h=h' \<or> \<not> l\<le>h \<and> \<not> l'\<le>h')"
   by (simp add: order_class.order.eq_iff) (auto intro: order_trans)
 
 lemma atLeastAtMost_singleton_iff[simp]:
   "{a .. b} = {c} \<longleftrightarrow> a = b \<and> b = c"
 proof
   assume "{a..b} = {c}"
   hence *: "\<not> (\<not> a \<le> b)" unfolding atLeastatMost_empty_iff[symmetric] by simp
   with \<open>{a..b} = {c}\<close> have "c \<le> a \<and> b \<le> c" by auto
   with * show "a = b \<and> b = c" by auto
 qed simp
 
 end
 
 context no_top
 begin
 
 (* also holds for no_bot but no_top should suffice *)
 lemma not_UNIV_le_Icc[simp]: "\<not> UNIV \<subseteq> {l..h}"
 using gt_ex[of h] by(auto simp: subset_eq less_le_not_le)
 
 lemma not_UNIV_le_Iic[simp]: "\<not> UNIV \<subseteq> {..h}"
 using gt_ex[of h] by(auto simp: subset_eq less_le_not_le)
 
 lemma not_Ici_le_Icc[simp]: "\<not> {l..} \<subseteq> {l'..h'}"
 using gt_ex[of h']
 by(auto simp: subset_eq less_le)(blast dest:antisym_conv intro: order_trans)
 
 lemma not_Ici_le_Iic[simp]: "\<not> {l..} \<subseteq> {..h'}"
 using gt_ex[of h']
 by(auto simp: subset_eq less_le)(blast dest:antisym_conv intro: order_trans)
 
 end
 
 context no_bot
 begin
 
 lemma not_UNIV_le_Ici[simp]: "\<not> UNIV \<subseteq> {l..}"
 using lt_ex[of l] by(auto simp: subset_eq less_le_not_le)
 
 lemma not_Iic_le_Icc[simp]: "\<not> {..h} \<subseteq> {l'..h'}"
 using lt_ex[of l']
 by(auto simp: subset_eq less_le)(blast dest:antisym_conv intro: order_trans)
 
 lemma not_Iic_le_Ici[simp]: "\<not> {..h} \<subseteq> {l'..}"
 using lt_ex[of l']
 by(auto simp: subset_eq less_le)(blast dest:antisym_conv intro: order_trans)
 
 end
 
 
 context no_top
 begin
 
 (* also holds for no_bot but no_top should suffice *)
 lemma not_UNIV_eq_Icc[simp]: "\<not> UNIV = {l'..h'}"
 using gt_ex[of h'] by(auto simp: set_eq_iff  less_le_not_le)
 
 lemmas not_Icc_eq_UNIV[simp] = not_UNIV_eq_Icc[symmetric]
 
 lemma not_UNIV_eq_Iic[simp]: "\<not> UNIV = {..h'}"
 using gt_ex[of h'] by(auto simp: set_eq_iff  less_le_not_le)
 
 lemmas not_Iic_eq_UNIV[simp] = not_UNIV_eq_Iic[symmetric]
 
 lemma not_Icc_eq_Ici[simp]: "\<not> {l..h} = {l'..}"
 unfolding atLeastAtMost_def using not_Ici_le_Iic[of l'] by blast
 
 lemmas not_Ici_eq_Icc[simp] = not_Icc_eq_Ici[symmetric]
 
 (* also holds for no_bot but no_top should suffice *)
 lemma not_Iic_eq_Ici[simp]: "\<not> {..h} = {l'..}"
 using not_Ici_le_Iic[of l' h] by blast
 
 lemmas not_Ici_eq_Iic[simp] = not_Iic_eq_Ici[symmetric]
 
 end
 
 context no_bot
 begin
 
 lemma not_UNIV_eq_Ici[simp]: "\<not> UNIV = {l'..}"
 using lt_ex[of l'] by(auto simp: set_eq_iff  less_le_not_le)
 
 lemmas not_Ici_eq_UNIV[simp] = not_UNIV_eq_Ici[symmetric]
 
 lemma not_Icc_eq_Iic[simp]: "\<not> {l..h} = {..h'}"
 unfolding atLeastAtMost_def using not_Iic_le_Ici[of h'] by blast
 
 lemmas not_Iic_eq_Icc[simp] = not_Icc_eq_Iic[symmetric]
 
 end
 
 
 context dense_linorder
 begin
 
 lemma greaterThanLessThan_empty_iff[simp]:
   "{ a <..< b } = {} \<longleftrightarrow> b \<le> a"
   using dense[of a b] by (cases "a < b") auto
 
 lemma greaterThanLessThan_empty_iff2[simp]:
   "{} = { a <..< b } \<longleftrightarrow> b \<le> a"
   using dense[of a b] by (cases "a < b") auto
 
 lemma atLeastLessThan_subseteq_atLeastAtMost_iff:
   "{a ..< b} \<subseteq> { c .. d } \<longleftrightarrow> (a < b \<longrightarrow> c \<le> a \<and> b \<le> d)"
   using dense[of "max a d" "b"]
   by (force simp: subset_eq Ball_def not_less[symmetric])
 
 lemma greaterThanAtMost_subseteq_atLeastAtMost_iff:
   "{a <.. b} \<subseteq> { c .. d } \<longleftrightarrow> (a < b \<longrightarrow> c \<le> a \<and> b \<le> d)"
   using dense[of "a" "min c b"]
   by (force simp: subset_eq Ball_def not_less[symmetric])
 
 lemma greaterThanLessThan_subseteq_atLeastAtMost_iff:
   "{a <..< b} \<subseteq> { c .. d } \<longleftrightarrow> (a < b \<longrightarrow> c \<le> a \<and> b \<le> d)"
   using dense[of "a" "min c b"] dense[of "max a d" "b"]
   by (force simp: subset_eq Ball_def not_less[symmetric])
 
 lemma greaterThanLessThan_subseteq_greaterThanLessThan:
   "{a <..< b} \<subseteq> {c <..< d} \<longleftrightarrow> (a < b \<longrightarrow> a \<ge> c \<and> b \<le> d)"
   using dense[of "a" "min c b"] dense[of "max a d" "b"]
   by (force simp: subset_eq Ball_def not_less[symmetric])
 
 lemma greaterThanAtMost_subseteq_atLeastLessThan_iff:
   "{a <.. b} \<subseteq> { c ..< d } \<longleftrightarrow> (a < b \<longrightarrow> c \<le> a \<and> b < d)"
   using dense[of "a" "min c b"]
   by (force simp: subset_eq Ball_def not_less[symmetric])
 
 lemma greaterThanLessThan_subseteq_atLeastLessThan_iff:
   "{a <..< b} \<subseteq> { c ..< d } \<longleftrightarrow> (a < b \<longrightarrow> c \<le> a \<and> b \<le> d)"
   using dense[of "a" "min c b"] dense[of "max a d" "b"]
   by (force simp: subset_eq Ball_def not_less[symmetric])
 
 lemma greaterThanLessThan_subseteq_greaterThanAtMost_iff:
   "{a <..< b} \<subseteq> { c <.. d } \<longleftrightarrow> (a < b \<longrightarrow> c \<le> a \<and> b \<le> d)"
   using dense[of "a" "min c b"] dense[of "max a d" "b"]
   by (force simp: subset_eq Ball_def not_less[symmetric])
 
 end
 
 context no_top
 begin
 
 lemma greaterThan_non_empty[simp]: "{x <..} \<noteq> {}"
   using gt_ex[of x] by auto
 
 end
 
 context no_bot
 begin
 
 lemma lessThan_non_empty[simp]: "{..< x} \<noteq> {}"
   using lt_ex[of x] by auto
 
 end
 
 lemma (in linorder) atLeastLessThan_subset_iff:
   "{a..<b} \<subseteq> {c..<d} \<Longrightarrow> b \<le> a \<or> c\<le>a \<and> b\<le>d"
   apply (auto simp:subset_eq Ball_def not_le)
   apply(frule_tac x=a in spec)
   apply(erule_tac x=d in allE)
   apply (auto simp: )
   done
 
 lemma atLeastLessThan_inj:
   fixes a b c d :: "'a::linorder"
   assumes eq: "{a ..< b} = {c ..< d}" and "a < b" "c < d"
   shows "a = c" "b = d"
 using assms by (metis atLeastLessThan_subset_iff eq less_le_not_le antisym_conv2 subset_refl)+
 
 lemma atLeastLessThan_eq_iff:
   fixes a b c d :: "'a::linorder"
   assumes "a < b" "c < d"
   shows "{a ..< b} = {c ..< d} \<longleftrightarrow> a = c \<and> b = d"
   using atLeastLessThan_inj assms by auto
 
 lemma (in linorder) Ioc_inj: 
   \<open>{a <.. b} = {c <.. d} \<longleftrightarrow> (b \<le> a \<and> d \<le> c) \<or> a = c \<and> b = d\<close> (is \<open>?P \<longleftrightarrow> ?Q\<close>)
 proof
   assume ?Q
   then show ?P
     by auto
 next
   assume ?P
   then have \<open>a < x \<and> x \<le> b \<longleftrightarrow> c < x \<and> x \<le> d\<close> for x
     by (simp add: set_eq_iff)
   from this [of a] this [of b] this [of c] this [of d] show ?Q
     by auto
 qed
 
 lemma (in order) Iio_Int_singleton: "{..<k} \<inter> {x} = (if x < k then {x} else {})"
   by auto
 
 lemma (in linorder) Ioc_subset_iff: "{a<..b} \<subseteq> {c<..d} \<longleftrightarrow> (b \<le> a \<or> c \<le> a \<and> b \<le> d)"
   by (auto simp: subset_eq Ball_def) (metis less_le not_less)
 
 lemma (in order_bot) atLeast_eq_UNIV_iff: "{x..} = UNIV \<longleftrightarrow> x = bot"
 by (auto simp: set_eq_iff intro: le_bot)
 
 lemma (in order_top) atMost_eq_UNIV_iff: "{..x} = UNIV \<longleftrightarrow> x = top"
 by (auto simp: set_eq_iff intro: top_le)
 
 lemma (in bounded_lattice) atLeastAtMost_eq_UNIV_iff:
   "{x..y} = UNIV \<longleftrightarrow> (x = bot \<and> y = top)"
 by (auto simp: set_eq_iff intro: top_le le_bot)
 
 lemma Iio_eq_empty_iff: "{..< n::'a::{linorder, order_bot}} = {} \<longleftrightarrow> n = bot"
   by (auto simp: set_eq_iff not_less le_bot)
 
 lemma lessThan_empty_iff: "{..< n::nat} = {} \<longleftrightarrow> n = 0"
   by (simp add: Iio_eq_empty_iff bot_nat_def)
 
 lemma mono_image_least:
   assumes f_mono: "mono f" and f_img: "f ` {m ..< n} = {m' ..< n'}" "m < n"
   shows "f m = m'"
 proof -
   from f_img have "{m' ..< n'} \<noteq> {}"
     by (metis atLeastLessThan_empty_iff image_is_empty)
   with f_img have "m' \<in> f ` {m ..< n}" by auto
   then obtain k where "f k = m'" "m \<le> k" by auto
   moreover have "m' \<le> f m" using f_img by auto
   ultimately show "f m = m'"
     using f_mono by (auto elim: monoE[where x=m and y=k])
 qed
 
 
 subsection \<open>Infinite intervals\<close>
 
 context dense_linorder
 begin
 
 lemma infinite_Ioo:
   assumes "a < b"
   shows "\<not> finite {a<..<b}"
 proof
   assume fin: "finite {a<..<b}"
   moreover have ne: "{a<..<b} \<noteq> {}"
     using \<open>a < b\<close> by auto
   ultimately have "a < Max {a <..< b}" "Max {a <..< b} < b"
     using Max_in[of "{a <..< b}"] by auto
   then obtain x where "Max {a <..< b} < x" "x < b"
     using dense[of "Max {a<..<b}" b] by auto
   then have "x \<in> {a <..< b}"
     using \<open>a < Max {a <..< b}\<close> by auto
   then have "x \<le> Max {a <..< b}"
     using fin by auto
   with \<open>Max {a <..< b} < x\<close> show False by auto
 qed
 
 lemma infinite_Icc: "a < b \<Longrightarrow> \<not> finite {a .. b}"
   using greaterThanLessThan_subseteq_atLeastAtMost_iff[of a b a b] infinite_Ioo[of a b]
   by (auto dest: finite_subset)
 
 lemma infinite_Ico: "a < b \<Longrightarrow> \<not> finite {a ..< b}"
   using greaterThanLessThan_subseteq_atLeastLessThan_iff[of a b a b] infinite_Ioo[of a b]
   by (auto dest: finite_subset)
 
 lemma infinite_Ioc: "a < b \<Longrightarrow> \<not> finite {a <.. b}"
   using greaterThanLessThan_subseteq_greaterThanAtMost_iff[of a b a b] infinite_Ioo[of a b]
   by (auto dest: finite_subset)
 
 lemma infinite_Ioo_iff [simp]: "infinite {a<..<b} \<longleftrightarrow> a < b"
   using not_less_iff_gr_or_eq by (fastforce simp: infinite_Ioo)
 
 lemma infinite_Icc_iff [simp]: "infinite {a .. b} \<longleftrightarrow> a < b"
   using not_less_iff_gr_or_eq by (fastforce simp: infinite_Icc)
 
 lemma infinite_Ico_iff [simp]: "infinite {a..<b} \<longleftrightarrow> a < b"
   using not_less_iff_gr_or_eq by (fastforce simp: infinite_Ico)
 
 lemma infinite_Ioc_iff [simp]: "infinite {a<..b} \<longleftrightarrow> a < b"
   using not_less_iff_gr_or_eq by (fastforce simp: infinite_Ioc)
 
 end
 
 lemma infinite_Iio: "\<not> finite {..< a :: 'a :: {no_bot, linorder}}"
 proof
   assume "finite {..< a}"
   then have *: "\<And>x. x < a \<Longrightarrow> Min {..< a} \<le> x"
     by auto
   obtain x where "x < a"
     using lt_ex by auto
 
   obtain y where "y < Min {..< a}"
     using lt_ex by auto
   also have "Min {..< a} \<le> x"
     using \<open>x < a\<close> by fact
   also note \<open>x < a\<close>
   finally have "Min {..< a} \<le> y"
     by fact
   with \<open>y < Min {..< a}\<close> show False by auto
 qed
 
 lemma infinite_Iic: "\<not> finite {.. a :: 'a :: {no_bot, linorder}}"
   using infinite_Iio[of a] finite_subset[of "{..< a}" "{.. a}"]
   by (auto simp: subset_eq less_imp_le)
 
 lemma infinite_Ioi: "\<not> finite {a :: 'a :: {no_top, linorder} <..}"
 proof
   assume "finite {a <..}"
   then have *: "\<And>x. a < x \<Longrightarrow> x \<le> Max {a <..}"
     by auto
 
   obtain y where "Max {a <..} < y"
     using gt_ex by auto
 
   obtain x where x: "a < x"
     using gt_ex by auto
   also from x have "x \<le> Max {a <..}"
     by fact
   also note \<open>Max {a <..} < y\<close>
   finally have "y \<le> Max { a <..}"
     by fact
   with \<open>Max {a <..} < y\<close> show False by auto
 qed
 
 lemma infinite_Ici: "\<not> finite {a :: 'a :: {no_top, linorder} ..}"
   using infinite_Ioi[of a] finite_subset[of "{a <..}" "{a ..}"]
   by (auto simp: subset_eq less_imp_le)
 
 subsubsection \<open>Intersection\<close>
 
 context linorder
 begin
 
 lemma Int_atLeastAtMost[simp]: "{a..b} Int {c..d} = {max a c .. min b d}"
 by auto
 
 lemma Int_atLeastAtMostR1[simp]: "{..b} Int {c..d} = {c .. min b d}"
 by auto
 
 lemma Int_atLeastAtMostR2[simp]: "{a..} Int {c..d} = {max a c .. d}"
 by auto
 
 lemma Int_atLeastAtMostL1[simp]: "{a..b} Int {..d} = {a .. min b d}"
 by auto
 
 lemma Int_atLeastAtMostL2[simp]: "{a..b} Int {c..} = {max a c .. b}"
 by auto
 
 lemma Int_atLeastLessThan[simp]: "{a..<b} Int {c..<d} = {max a c ..< min b d}"
 by auto
 
 lemma Int_greaterThanAtMost[simp]: "{a<..b} Int {c<..d} = {max a c <.. min b d}"
 by auto
 
 lemma Int_greaterThanLessThan[simp]: "{a<..<b} Int {c<..<d} = {max a c <..< min b d}"
 by auto
 
 lemma Int_atMost[simp]: "{..a} \<inter> {..b} = {.. min a b}"
   by (auto simp: min_def)
 
 lemma Ioc_disjoint: "{a<..b} \<inter> {c<..d} = {} \<longleftrightarrow> b \<le> a \<or> d \<le> c \<or> b \<le> c \<or> d \<le> a"
   by auto
 
 end
 
 context complete_lattice
 begin
 
 lemma
   shows Sup_atLeast[simp]: "Sup {x ..} = top"
     and Sup_greaterThanAtLeast[simp]: "x < top \<Longrightarrow> Sup {x <..} = top"
     and Sup_atMost[simp]: "Sup {.. y} = y"
     and Sup_atLeastAtMost[simp]: "x \<le> y \<Longrightarrow> Sup { x .. y} = y"
     and Sup_greaterThanAtMost[simp]: "x < y \<Longrightarrow> Sup { x <.. y} = y"
   by (auto intro!: Sup_eqI)
 
 lemma
   shows Inf_atMost[simp]: "Inf {.. x} = bot"
     and Inf_atMostLessThan[simp]: "top < x \<Longrightarrow> Inf {..< x} = bot"
     and Inf_atLeast[simp]: "Inf {x ..} = x"
     and Inf_atLeastAtMost[simp]: "x \<le> y \<Longrightarrow> Inf { x .. y} = x"
     and Inf_atLeastLessThan[simp]: "x < y \<Longrightarrow> Inf { x ..< y} = x"
   by (auto intro!: Inf_eqI)
 
 end
 
 lemma
   fixes x y :: "'a :: {complete_lattice, dense_linorder}"
   shows Sup_lessThan[simp]: "Sup {..< y} = y"
     and Sup_atLeastLessThan[simp]: "x < y \<Longrightarrow> Sup { x ..< y} = y"
     and Sup_greaterThanLessThan[simp]: "x < y \<Longrightarrow> Sup { x <..< y} = y"
     and Inf_greaterThan[simp]: "Inf {x <..} = x"
     and Inf_greaterThanAtMost[simp]: "x < y \<Longrightarrow> Inf { x <.. y} = x"
     and Inf_greaterThanLessThan[simp]: "x < y \<Longrightarrow> Inf { x <..< y} = x"
   by (auto intro!: Inf_eqI Sup_eqI intro: dense_le dense_le_bounded dense_ge dense_ge_bounded)
 
 subsection \<open>Intervals of natural numbers\<close>
 
 subsubsection \<open>The Constant \<^term>\<open>lessThan\<close>\<close>
 
 lemma lessThan_0 [simp]: "lessThan (0::nat) = {}"
 by (simp add: lessThan_def)
 
 lemma lessThan_Suc: "lessThan (Suc k) = insert k (lessThan k)"
 by (simp add: lessThan_def less_Suc_eq, blast)
 
 text \<open>The following proof is convenient in induction proofs where
 new elements get indices at the beginning. So it is used to transform
 \<^term>\<open>{..<Suc n}\<close> to \<^term>\<open>0::nat\<close> and \<^term>\<open>{..< n}\<close>.\<close>
 
 lemma zero_notin_Suc_image [simp]: "0 \<notin> Suc ` A"
   by auto
 
 lemma lessThan_Suc_eq_insert_0: "{..<Suc n} = insert 0 (Suc ` {..<n})"
   by (auto simp: image_iff less_Suc_eq_0_disj)
 
 lemma lessThan_Suc_atMost: "lessThan (Suc k) = atMost k"
 by (simp add: lessThan_def atMost_def less_Suc_eq_le)
 
 lemma atMost_Suc_eq_insert_0: "{.. Suc n} = insert 0 (Suc ` {.. n})"
   unfolding lessThan_Suc_atMost[symmetric] lessThan_Suc_eq_insert_0[of "Suc n"] ..
 
 lemma UN_lessThan_UNIV: "(\<Union>m::nat. lessThan m) = UNIV"
 by blast
 
 subsubsection \<open>The Constant \<^term>\<open>greaterThan\<close>\<close>
 
 lemma greaterThan_0: "greaterThan 0 = range Suc"
   unfolding greaterThan_def
   by (blast dest: gr0_conv_Suc [THEN iffD1])
 
 lemma greaterThan_Suc: "greaterThan (Suc k) = greaterThan k - {Suc k}"
   unfolding greaterThan_def
   by (auto elim: linorder_neqE)
 
 lemma INT_greaterThan_UNIV: "(\<Inter>m::nat. greaterThan m) = {}"
   by blast
 
 subsubsection \<open>The Constant \<^term>\<open>atLeast\<close>\<close>
 
 lemma atLeast_0 [simp]: "atLeast (0::nat) = UNIV"
 by (unfold atLeast_def UNIV_def, simp)
 
 lemma atLeast_Suc: "atLeast (Suc k) = atLeast k - {k}"
   unfolding atLeast_def by (auto simp: order_le_less Suc_le_eq)
 
 lemma atLeast_Suc_greaterThan: "atLeast (Suc k) = greaterThan k"
   by (auto simp add: greaterThan_def atLeast_def less_Suc_eq_le)
 
 lemma UN_atLeast_UNIV: "(\<Union>m::nat. atLeast m) = UNIV"
   by blast
 
 subsubsection \<open>The Constant \<^term>\<open>atMost\<close>\<close>
 
 lemma atMost_0 [simp]: "atMost (0::nat) = {0}"
   by (simp add: atMost_def)
 
 lemma atMost_Suc: "atMost (Suc k) = insert (Suc k) (atMost k)"
   unfolding atMost_def by (auto simp add: less_Suc_eq order_le_less)
 
 lemma UN_atMost_UNIV: "(\<Union>m::nat. atMost m) = UNIV"
   by blast
 
 subsubsection \<open>The Constant \<^term>\<open>atLeastLessThan\<close>\<close>
 
 text\<open>The orientation of the following 2 rules is tricky. The lhs is
 defined in terms of the rhs.  Hence the chosen orientation makes sense
 in this theory --- the reverse orientation complicates proofs (eg
 nontermination). But outside, when the definition of the lhs is rarely
 used, the opposite orientation seems preferable because it reduces a
 specific concept to a more general one.\<close>
 
 lemma atLeast0LessThan [code_abbrev]: "{0::nat..<n} = {..<n}"
   by(simp add:lessThan_def atLeastLessThan_def)
 
 lemma atLeast0AtMost [code_abbrev]: "{0..n::nat} = {..n}"
   by(simp add:atMost_def atLeastAtMost_def)
 
 lemma lessThan_atLeast0: "{..<n} = {0::nat..<n}"
   by (simp add: atLeast0LessThan)
 
 lemma atMost_atLeast0: "{..n} = {0::nat..n}"
   by (simp add: atLeast0AtMost)
 
 lemma atLeastLessThan0: "{m..<0::nat} = {}"
   by (simp add: atLeastLessThan_def)
 
 lemma atLeast0_lessThan_Suc: "{0..<Suc n} = insert n {0..<n}"
   by (simp add: atLeast0LessThan lessThan_Suc)
 
 lemma atLeast0_lessThan_Suc_eq_insert_0: "{0..<Suc n} = insert 0 (Suc ` {0..<n})"
   by (simp add: atLeast0LessThan lessThan_Suc_eq_insert_0)
 
 
 subsubsection \<open>The Constant \<^term>\<open>atLeastAtMost\<close>\<close>
 
 lemma Icc_eq_insert_lb_nat: "m \<le> n \<Longrightarrow> {m..n} = insert m {Suc m..n}"
 by auto
 
 lemma atLeast0_atMost_Suc:
   "{0..Suc n} = insert (Suc n) {0..n}"
   by (simp add: atLeast0AtMost atMost_Suc)
 
 lemma atLeast0_atMost_Suc_eq_insert_0:
   "{0..Suc n} = insert 0 (Suc ` {0..n})"
   by (simp add: atLeast0AtMost atMost_Suc_eq_insert_0)
 
 
 subsubsection \<open>Intervals of nats with \<^term>\<open>Suc\<close>\<close>
 
 text\<open>Not a simprule because the RHS is too messy.\<close>
 lemma atLeastLessThanSuc:
     "{m..<Suc n} = (if m \<le> n then insert n {m..<n} else {})"
 by (auto simp add: atLeastLessThan_def)
 
 lemma atLeastLessThan_singleton [simp]: "{m..<Suc m} = {m}"
 by (auto simp add: atLeastLessThan_def)
 
 lemma atLeastLessThanSuc_atLeastAtMost: "{l..<Suc u} = {l..u}"
   by (simp add: lessThan_Suc_atMost atLeastAtMost_def atLeastLessThan_def)
 
 lemma atLeastSucAtMost_greaterThanAtMost: "{Suc l..u} = {l<..u}"
   by (simp add: atLeast_Suc_greaterThan atLeastAtMost_def
       greaterThanAtMost_def)
 
 lemma atLeastSucLessThan_greaterThanLessThan: "{Suc l..<u} = {l<..<u}"
   by (simp add: atLeast_Suc_greaterThan atLeastLessThan_def
     greaterThanLessThan_def)
 
 lemma atLeastAtMostSuc_conv: "m \<le> Suc n \<Longrightarrow> {m..Suc n} = insert (Suc n) {m..n}"
   by auto
 
 lemma atLeastAtMost_insertL: "m \<le> n \<Longrightarrow> insert m {Suc m..n} = {m ..n}"
   by auto
 
 text \<open>The analogous result is useful on \<^typ>\<open>int\<close>:\<close>
 (* here, because we don't have an own int section *)
 lemma atLeastAtMostPlus1_int_conv:
   "m \<le> 1+n \<Longrightarrow> {m..1+n} = insert (1+n) {m..n::int}"
   by (auto intro: set_eqI)
 
 lemma atLeastLessThan_add_Un: "i \<le> j \<Longrightarrow> {i..<j+k} = {i..<j} \<union> {j..<j+k::nat}"
   by (induct k) (simp_all add: atLeastLessThanSuc)
 
 
 subsubsection \<open>Intervals and numerals\<close>
 
 lemma lessThan_nat_numeral:  \<comment> \<open>Evaluation for specific numerals\<close>
   "lessThan (numeral k :: nat) = insert (pred_numeral k) (lessThan (pred_numeral k))"
   by (simp add: numeral_eq_Suc lessThan_Suc)
 
 lemma atMost_nat_numeral:  \<comment> \<open>Evaluation for specific numerals\<close>
   "atMost (numeral k :: nat) = insert (numeral k) (atMost (pred_numeral k))"
   by (simp add: numeral_eq_Suc atMost_Suc)
 
 lemma atLeastLessThan_nat_numeral:  \<comment> \<open>Evaluation for specific numerals\<close>
   "atLeastLessThan m (numeral k :: nat) =
      (if m \<le> (pred_numeral k) then insert (pred_numeral k) (atLeastLessThan m (pred_numeral k))
                  else {})"
   by (simp add: numeral_eq_Suc atLeastLessThanSuc)
 
 
 subsubsection \<open>Image\<close>
 
 context linordered_semidom
 begin
 
 lemma image_add_atLeast[simp]: "plus k ` {i..} = {k + i..}"
 proof -
   have "n = k + (n - k)" if "i + k \<le> n" for n
   proof -
     have "n = (n - (k + i)) + (k + i)" using that
       by (metis add_commute le_add_diff_inverse)
     then show "n = k + (n - k)"
       by (metis local.add_diff_cancel_left' add_assoc add_commute)
   qed
   then show ?thesis
     by (fastforce simp: add_le_imp_le_diff add.commute)
 qed
 
 lemma image_add_atLeastAtMost [simp]:
   "plus k ` {i..j} = {i + k..j + k}" (is "?A = ?B")
 proof
   show "?A \<subseteq> ?B"
     by (auto simp add: ac_simps)
 next
   show "?B \<subseteq> ?A"
   proof
     fix n
     assume "n \<in> ?B"
     then have "i \<le> n - k"
       by (simp add: add_le_imp_le_diff)
     have "n = n - k + k"
     proof -
       from \<open>n \<in> ?B\<close> have "n = n - (i + k) + (i + k)"
         by simp
       also have "\<dots> = n - k - i + i + k"
         by (simp add: algebra_simps)
       also have "\<dots> = n - k + k"
         using \<open>i \<le> n - k\<close> by simp
       finally show ?thesis .
     qed
     moreover have "n - k \<in> {i..j}"
       using \<open>n \<in> ?B\<close>
       by (auto simp: add_le_imp_le_diff add_le_add_imp_diff_le)
     ultimately show "n \<in> ?A"
       by (simp add: ac_simps) 
   qed
 qed
 
 lemma image_add_atLeastAtMost' [simp]:
   "(\<lambda>n. n + k) ` {i..j} = {i + k..j + k}"
   by (simp add: add.commute [of _ k])
 
 lemma image_add_atLeastLessThan [simp]:
   "plus k ` {i..<j} = {i + k..<j + k}"
   by (simp add: image_set_diff atLeastLessThan_eq_atLeastAtMost_diff ac_simps)
 
 lemma image_add_atLeastLessThan' [simp]:
   "(\<lambda>n. n + k) ` {i..<j} = {i + k..<j + k}"
   by (simp add: add.commute [of _ k])
 
 lemma image_add_greaterThanAtMost[simp]: "(+) c ` {a<..b} = {c + a<..c + b}"
   by (simp add: image_set_diff greaterThanAtMost_eq_atLeastAtMost_diff ac_simps)
 
 end
 
 context ordered_ab_group_add
 begin
 
 lemma
   fixes x :: 'a
   shows image_uminus_greaterThan[simp]: "uminus ` {x<..} = {..<-x}"
   and image_uminus_atLeast[simp]: "uminus ` {x..} = {..-x}"
 proof safe
   fix y assume "y < -x"
   hence *:  "x < -y" using neg_less_iff_less[of "-y" x] by simp
   have "- (-y) \<in> uminus ` {x<..}"
     by (rule imageI) (simp add: *)
   thus "y \<in> uminus ` {x<..}" by simp
 next
   fix y assume "y \<le> -x"
   have "- (-y) \<in> uminus ` {x..}"
     by (rule imageI) (insert \<open>y \<le> -x\<close>[THEN le_imp_neg_le], simp)
   thus "y \<in> uminus ` {x..}" by simp
 qed simp_all
 
 lemma
   fixes x :: 'a
   shows image_uminus_lessThan[simp]: "uminus ` {..<x} = {-x<..}"
   and image_uminus_atMost[simp]: "uminus ` {..x} = {-x..}"
 proof -
   have "uminus ` {..<x} = uminus ` uminus ` {-x<..}"
     and "uminus ` {..x} = uminus ` uminus ` {-x..}" by simp_all
   thus "uminus ` {..<x} = {-x<..}" and "uminus ` {..x} = {-x..}"
     by (simp_all add: image_image
         del: image_uminus_greaterThan image_uminus_atLeast)
 qed
 
 lemma
   fixes x :: 'a
   shows image_uminus_atLeastAtMost[simp]: "uminus ` {x..y} = {-y..-x}"
   and image_uminus_greaterThanAtMost[simp]: "uminus ` {x<..y} = {-y..<-x}"
   and image_uminus_atLeastLessThan[simp]: "uminus ` {x..<y} = {-y<..-x}"
   and image_uminus_greaterThanLessThan[simp]: "uminus ` {x<..<y} = {-y<..<-x}"
   by (simp_all add: atLeastAtMost_def greaterThanAtMost_def atLeastLessThan_def
       greaterThanLessThan_def image_Int[OF inj_uminus] Int_commute)
 
 lemma image_add_atMost[simp]: "(+) c ` {..a} = {..c + a}"
   by (auto intro!: image_eqI[where x="x - c" for x] simp: algebra_simps)
 
 end
 
 lemma image_Suc_atLeastAtMost [simp]:
   "Suc ` {i..j} = {Suc i..Suc j}"
   using image_add_atLeastAtMost [of 1 i j]
     by (simp only: plus_1_eq_Suc) simp
 
 lemma image_Suc_atLeastLessThan [simp]:
   "Suc ` {i..<j} = {Suc i..<Suc j}"
   using image_add_atLeastLessThan [of 1 i j]
     by (simp only: plus_1_eq_Suc) simp
 
 corollary image_Suc_atMost:
   "Suc ` {..n} = {1..Suc n}"
   by (simp add: atMost_atLeast0 atLeastLessThanSuc_atLeastAtMost)
 
 corollary image_Suc_lessThan:
   "Suc ` {..<n} = {1..n}"
   by (simp add: lessThan_atLeast0 atLeastLessThanSuc_atLeastAtMost)
 
 lemma image_diff_atLeastAtMost [simp]:
   fixes d::"'a::linordered_idom" shows "((-) d ` {a..b}) = {d-b..d-a}"
   apply auto
   apply (rule_tac x="d-x" in rev_image_eqI, auto)
   done
 
 lemma image_diff_atLeastLessThan [simp]:
   fixes a b c::"'a::linordered_idom"
   shows "(-) c ` {a..<b} = {c - b<..c - a}"
 proof -
   have "(-) c ` {a..<b} = (+) c ` uminus ` {a ..<b}"
     unfolding image_image by simp
   also have "\<dots> = {c - b<..c - a}" by simp
   finally show ?thesis by simp
 qed
 
 lemma image_minus_const_greaterThanAtMost[simp]:
   fixes a b c::"'a::linordered_idom"
   shows "(-) c ` {a<..b} = {c - b..<c - a}"
 proof -
   have "(-) c ` {a<..b} = (+) c ` uminus ` {a<..b}"
     unfolding image_image by simp
   also have "\<dots> = {c - b..<c - a}" by simp
   finally show ?thesis by simp
 qed
 
 lemma image_minus_const_atLeast[simp]:
   fixes a c::"'a::linordered_idom"
   shows "(-) c ` {a..} = {..c - a}"
 proof -
   have "(-) c ` {a..} = (+) c ` uminus ` {a ..}"
     unfolding image_image by simp
   also have "\<dots> = {..c - a}" by simp
   finally show ?thesis by simp
 qed
 
 lemma image_minus_const_AtMost[simp]:
   fixes b c::"'a::linordered_idom"
   shows "(-) c ` {..b} = {c - b..}"
 proof -
   have "(-) c ` {..b} = (+) c ` uminus ` {..b}"
     unfolding image_image by simp
   also have "\<dots> = {c - b..}" by simp
   finally show ?thesis by simp
 qed
 
 lemma image_minus_const_atLeastAtMost' [simp]:
   "(\<lambda>t. t-d)`{a..b} = {a-d..b-d}" for d::"'a::linordered_idom"
   by (metis (no_types, lifting) diff_conv_add_uminus image_add_atLeastAtMost' image_cong)
 
 context linordered_field
 begin
 
 lemma image_mult_atLeastAtMost [simp]:
   "((*) d ` {a..b}) = {d*a..d*b}" if "d>0"
   using that
   by (auto simp: field_simps mult_le_cancel_right intro: rev_image_eqI [where x="x/d" for x])
 
 lemma image_divide_atLeastAtMost [simp]:
   "((\<lambda>c. c / d) ` {a..b}) = {a/d..b/d}" if "d>0"
 proof -
   from that have "inverse d > 0"
     by simp
   with image_mult_atLeastAtMost [of "inverse d" a b]
   have "(*) (inverse d) ` {a..b} = {inverse d * a..inverse d * b}"
     by blast
   moreover have "(*) (inverse d) = (\<lambda>c. c / d)"
     by (simp add: fun_eq_iff field_simps)
   ultimately show ?thesis
     by simp
 qed
 
 lemma image_mult_atLeastAtMost_if:
   "(*) c ` {x .. y} =
     (if c > 0 then {c * x .. c * y} else if x \<le> y then {c * y .. c * x} else {})"
 proof (cases "c = 0 \<or> x > y")
   case True
   then show ?thesis
     by auto
 next
   case False
   then have "x \<le> y"
     by auto
   from False consider "c < 0"| "c > 0"
     by (auto simp add: neq_iff)
   then show ?thesis
   proof cases
     case 1
     have "(*) c ` {x..y} = {c * y..c * x}"
     proof (rule set_eqI)
       fix d
       from 1 have "inj (\<lambda>z. z / c)"
         by (auto intro: injI)
       then have "d \<in> (*) c ` {x..y} \<longleftrightarrow> d / c \<in> (\<lambda>z. z div c) ` (*) c ` {x..y}"
         by (subst inj_image_mem_iff) simp_all
       also have "\<dots> \<longleftrightarrow> d / c \<in> {x..y}"
         using 1 by (simp add: image_image)
       also have "\<dots> \<longleftrightarrow> d \<in> {c * y..c * x}"
         by (auto simp add: field_simps 1)
       finally show "d \<in> (*) c ` {x..y} \<longleftrightarrow> d \<in> {c * y..c * x}" .
     qed
     with \<open>x \<le> y\<close> show ?thesis
       by auto
   qed (simp add: mult_left_mono_neg)
 qed
 
 lemma image_mult_atLeastAtMost_if':
   "(\<lambda>x. x * c) ` {x..y} =
     (if x \<le> y then if c > 0 then {x * c .. y * c} else {y * c .. x * c} else {})"
   using image_mult_atLeastAtMost_if [of c x y] by (auto simp add: ac_simps)
 
 lemma image_affinity_atLeastAtMost:
   "((\<lambda>x. m * x + c) ` {a..b}) = (if {a..b} = {} then {}
             else if 0 \<le> m then {m * a + c .. m * b + c}
             else {m * b + c .. m * a + c})"
 proof -
   have *: "(\<lambda>x. m * x + c) = ((\<lambda>x. x + c) \<circ> (*) m)"
     by (simp add: fun_eq_iff)
   show ?thesis by (simp only: * image_comp [symmetric] image_mult_atLeastAtMost_if)
     (auto simp add: mult_le_cancel_left)
 qed
 
 lemma image_affinity_atLeastAtMost_diff:
   "((\<lambda>x. m*x - c) ` {a..b}) = (if {a..b}={} then {}
             else if 0 \<le> m then {m*a - c .. m*b - c}
             else {m*b - c .. m*a - c})"
   using image_affinity_atLeastAtMost [of m "-c" a b]
   by simp
 
 lemma image_affinity_atLeastAtMost_div:
   "((\<lambda>x. x/m + c) ` {a..b}) = (if {a..b}={} then {}
             else if 0 \<le> m then {a/m + c .. b/m + c}
             else {b/m + c .. a/m + c})"
   using image_affinity_atLeastAtMost [of "inverse m" c a b]
   by (simp add: field_class.field_divide_inverse algebra_simps inverse_eq_divide)
 
 lemma image_affinity_atLeastAtMost_div_diff:
   "((\<lambda>x. x/m - c) ` {a..b}) = (if {a..b}={} then {}
             else if 0 \<le> m then {a/m - c .. b/m - c}
             else {b/m - c .. a/m - c})"
   using image_affinity_atLeastAtMost_diff [of "inverse m" c a b]
   by (simp add: field_class.field_divide_inverse algebra_simps inverse_eq_divide)
 
 end
 
 lemma atLeast1_lessThan_eq_remove0:
   "{Suc 0..<n} = {..<n} - {0}"
   by auto
 
 lemma atLeast1_atMost_eq_remove0:
   "{Suc 0..n} = {..n} - {0}"
   by auto
 
 lemma image_add_int_atLeastLessThan:
     "(\<lambda>x. x + (l::int)) ` {0..<u-l} = {l..<u}"
   apply (auto simp add: image_def)
   apply (rule_tac x = "x - l" in bexI)
   apply auto
   done
 
 lemma image_minus_const_atLeastLessThan_nat:
   fixes c :: nat
   shows "(\<lambda>i. i - c) ` {x ..< y} =
       (if c < y then {x - c ..< y - c} else if x < y then {0} else {})"
     (is "_ = ?right")
 proof safe
   fix a assume a: "a \<in> ?right"
   show "a \<in> (\<lambda>i. i - c) ` {x ..< y}"
   proof cases
     assume "c < y" with a show ?thesis
       by (auto intro!: image_eqI[of _ _ "a + c"])
   next
     assume "\<not> c < y" with a show ?thesis
       by (auto intro!: image_eqI[of _ _ x] split: if_split_asm)
   qed
 qed auto
 
 lemma image_int_atLeastLessThan:
   "int ` {a..<b} = {int a..<int b}"
   by (auto intro!: image_eqI [where x = "nat x" for x])
 
 lemma image_int_atLeastAtMost:
   "int ` {a..b} = {int a..int b}"
   by (auto intro!: image_eqI [where x = "nat x" for x])
 
 
 subsubsection \<open>Finiteness\<close>
 
 lemma finite_lessThan [iff]: fixes k :: nat shows "finite {..<k}"
   by (induct k) (simp_all add: lessThan_Suc)
 
 lemma finite_atMost [iff]: fixes k :: nat shows "finite {..k}"
   by (induct k) (simp_all add: atMost_Suc)
 
 lemma finite_greaterThanLessThan [iff]:
   fixes l :: nat shows "finite {l<..<u}"
   by (simp add: greaterThanLessThan_def)
 
 lemma finite_atLeastLessThan [iff]:
   fixes l :: nat shows "finite {l..<u}"
   by (simp add: atLeastLessThan_def)
 
 lemma finite_greaterThanAtMost [iff]:
   fixes l :: nat shows "finite {l<..u}"
   by (simp add: greaterThanAtMost_def)
 
 lemma finite_atLeastAtMost [iff]:
   fixes l :: nat shows "finite {l..u}"
   by (simp add: atLeastAtMost_def)
 
 text \<open>A bounded set of natural numbers is finite.\<close>
 lemma bounded_nat_set_is_finite: "(\<forall>i\<in>N. i < (n::nat)) \<Longrightarrow> finite N"
   by (rule finite_subset [OF _ finite_lessThan]) auto
 
 text \<open>A set of natural numbers is finite iff it is bounded.\<close>
 lemma finite_nat_set_iff_bounded:
   "finite(N::nat set) = (\<exists>m. \<forall>n\<in>N. n<m)" (is "?F = ?B")
 proof
   assume f:?F  show ?B
     using Max_ge[OF \<open>?F\<close>, simplified less_Suc_eq_le[symmetric]] by blast
 next
   assume ?B show ?F using \<open>?B\<close> by(blast intro:bounded_nat_set_is_finite)
 qed
 
 lemma finite_nat_set_iff_bounded_le: "finite(N::nat set) = (\<exists>m. \<forall>n\<in>N. n\<le>m)"
   unfolding finite_nat_set_iff_bounded
   by (blast dest:less_imp_le_nat le_imp_less_Suc)
 
 lemma finite_less_ub:
      "!!f::nat=>nat. (!!n. n \<le> f n) ==> finite {n. f n \<le> u}"
 by (rule_tac B="{..u}" in finite_subset, auto intro: order_trans)
 
 lemma bounded_Max_nat:
   fixes P :: "nat \<Rightarrow> bool"
   assumes x: "P x" and M: "\<And>x. P x \<Longrightarrow> x \<le> M"
   obtains m where "P m" "\<And>x. P x \<Longrightarrow> x \<le> m"
 proof -
   have "finite {x. P x}"
     using M finite_nat_set_iff_bounded_le by auto
   then have "Max {x. P x} \<in> {x. P x}"
     using Max_in x by auto
   then show ?thesis
     by (simp add: \<open>finite {x. P x}\<close> that)
 qed
 
 
 text\<open>Any subset of an interval of natural numbers the size of the
 subset is exactly that interval.\<close>
 
 lemma subset_card_intvl_is_intvl:
   assumes "A \<subseteq> {k..<k + card A}"
   shows "A = {k..<k + card A}"
 proof (cases "finite A")
   case True
   from this and assms show ?thesis
   proof (induct A rule: finite_linorder_max_induct)
     case empty thus ?case by auto
   next
     case (insert b A)
     hence *: "b \<notin> A" by auto
     with insert have "A \<le> {k..<k + card A}" and "b = k + card A"
       by fastforce+
     with insert * show ?case by auto
   qed
 next
   case False
   with assms show ?thesis by simp
 qed
 
 
 subsubsection \<open>Proving Inclusions and Equalities between Unions\<close>
 
 lemma UN_le_eq_Un0:
   "(\<Union>i\<le>n::nat. M i) = (\<Union>i\<in>{1..n}. M i) \<union> M 0" (is "?A = ?B")
 proof
   show "?A \<subseteq> ?B"
   proof
     fix x assume "x \<in> ?A"
     then obtain i where i: "i\<le>n" "x \<in> M i" by auto
     show "x \<in> ?B"
     proof(cases i)
       case 0 with i show ?thesis by simp
     next
       case (Suc j) with i show ?thesis by auto
     qed
   qed
 next
   show "?B \<subseteq> ?A" by fastforce
 qed
 
 lemma UN_le_add_shift:
   "(\<Union>i\<le>n::nat. M(i+k)) = (\<Union>i\<in>{k..n+k}. M i)" (is "?A = ?B")
 proof
   show "?A \<subseteq> ?B" by fastforce
 next
   show "?B \<subseteq> ?A"
   proof
     fix x assume "x \<in> ?B"
     then obtain i where i: "i \<in> {k..n+k}" "x \<in> M(i)" by auto
     hence "i-k\<le>n \<and> x \<in> M((i-k)+k)" by auto
     thus "x \<in> ?A" by blast
   qed
 qed
 
 lemma UN_le_add_shift_strict:
   "(\<Union>i<n::nat. M(i+k)) = (\<Union>i\<in>{k..<n+k}. M i)" (is "?A = ?B")
 proof
   show "?B \<subseteq> ?A"
   proof
     fix x assume "x \<in> ?B"
     then obtain i where i: "i \<in> {k..<n+k}" "x \<in> M(i)" by auto
     then have "i - k < n \<and> x \<in> M((i-k) + k)" by auto
     then show "x \<in> ?A" using UN_le_add_shift by blast
   qed
 qed (fastforce)
 
 lemma UN_UN_finite_eq: "(\<Union>n::nat. \<Union>i\<in>{0..<n}. A i) = (\<Union>n. A n)"
   by (auto simp add: atLeast0LessThan)
 
 lemma UN_finite_subset:
   "(\<And>n::nat. (\<Union>i\<in>{0..<n}. A i) \<subseteq> C) \<Longrightarrow> (\<Union>n. A n) \<subseteq> C"
   by (subst UN_UN_finite_eq [symmetric]) blast
 
 lemma UN_finite2_subset:
   assumes "\<And>n::nat. (\<Union>i\<in>{0..<n}. A i) \<subseteq> (\<Union>i\<in>{0..<n + k}. B i)"
   shows "(\<Union>n. A n) \<subseteq> (\<Union>n. B n)"
 proof (rule UN_finite_subset, rule)
   fix n and a
   from assms have "(\<Union>i\<in>{0..<n}. A i) \<subseteq> (\<Union>i\<in>{0..<n + k}. B i)" .
   moreover assume "a \<in> (\<Union>i\<in>{0..<n}. A i)"
   ultimately have "a \<in> (\<Union>i\<in>{0..<n + k}. B i)" by blast
   then show "a \<in> (\<Union>i. B i)" by (auto simp add: UN_UN_finite_eq)
 qed
 
 lemma UN_finite2_eq:
   "(\<And>n::nat. (\<Union>i\<in>{0..<n}. A i) = (\<Union>i\<in>{0..<n + k}. B i)) \<Longrightarrow>
     (\<Union>n. A n) = (\<Union>n. B n)"
   apply (rule subset_antisym [OF UN_finite_subset UN_finite2_subset])
    apply auto
   apply (force simp add: atLeastLessThan_add_Un [of 0])+
   done
 
 
 subsubsection \<open>Cardinality\<close>
 
 lemma card_lessThan [simp]: "card {..<u} = u"
   by (induct u, simp_all add: lessThan_Suc)
 
 lemma card_atMost [simp]: "card {..u} = Suc u"
   by (simp add: lessThan_Suc_atMost [THEN sym])
 
 lemma card_atLeastLessThan [simp]: "card {l..<u} = u - l"
 proof -
   have "{l..<u} = (\<lambda>x. x + l) ` {..<u-l}"
     apply (auto simp add: image_def atLeastLessThan_def lessThan_def)
     apply (rule_tac x = "x - l" in exI)
     apply arith
     done
   then have "card {l..<u} = card {..<u-l}"
     by (simp add: card_image inj_on_def)
   then show ?thesis
     by simp
 qed
 
 lemma card_atLeastAtMost [simp]: "card {l..u} = Suc u - l"
   by (subst atLeastLessThanSuc_atLeastAtMost [THEN sym], simp)
 
 lemma card_greaterThanAtMost [simp]: "card {l<..u} = u - l"
   by (subst atLeastSucAtMost_greaterThanAtMost [THEN sym], simp)
 
 lemma card_greaterThanLessThan [simp]: "card {l<..<u} = u - Suc l"
   by (subst atLeastSucLessThan_greaterThanLessThan [THEN sym], simp)
 
 lemma subset_eq_atLeast0_lessThan_finite:
   fixes n :: nat
   assumes "N \<subseteq> {0..<n}"
   shows "finite N"
   using assms finite_atLeastLessThan by (rule finite_subset)
 
 lemma subset_eq_atLeast0_atMost_finite:
   fixes n :: nat
   assumes "N \<subseteq> {0..n}"
   shows "finite N"
   using assms finite_atLeastAtMost by (rule finite_subset)
 
 lemma ex_bij_betw_nat_finite:
   "finite M \<Longrightarrow> \<exists>h. bij_betw h {0..<card M} M"
   apply(drule finite_imp_nat_seg_image_inj_on)
   apply(auto simp:atLeast0LessThan[symmetric] lessThan_def[symmetric] card_image bij_betw_def)
   done
 
 lemma ex_bij_betw_finite_nat:
   "finite M \<Longrightarrow> \<exists>h. bij_betw h M {0..<card M}"
   by (blast dest: ex_bij_betw_nat_finite bij_betw_inv)
 
 lemma finite_same_card_bij:
   "finite A \<Longrightarrow> finite B \<Longrightarrow> card A = card B \<Longrightarrow> \<exists>h. bij_betw h A B"
   apply(drule ex_bij_betw_finite_nat)
   apply(drule ex_bij_betw_nat_finite)
   apply(auto intro!:bij_betw_trans)
   done
 
 lemma ex_bij_betw_nat_finite_1:
   "finite M \<Longrightarrow> \<exists>h. bij_betw h {1 .. card M} M"
   by (rule finite_same_card_bij) auto
 
 lemma bij_betw_iff_card:
   assumes "finite A" "finite B"
   shows "(\<exists>f. bij_betw f A B) \<longleftrightarrow> (card A = card B)"
 proof
   assume "card A = card B"
   moreover obtain f where "bij_betw f A {0 ..< card A}"
     using assms ex_bij_betw_finite_nat by blast
   moreover obtain g where "bij_betw g {0 ..< card B} B"
     using assms ex_bij_betw_nat_finite by blast
   ultimately have "bij_betw (g \<circ> f) A B"
     by (auto simp: bij_betw_trans)
   thus "(\<exists>f. bij_betw f A B)" by blast
 qed (auto simp: bij_betw_same_card)
 
 lemma subset_eq_atLeast0_lessThan_card:
   fixes n :: nat
   assumes "N \<subseteq> {0..<n}"
   shows "card N \<le> n"
 proof -
   from assms finite_lessThan have "card N \<le> card {0..<n}"
     using card_mono by blast
   then show ?thesis by simp
 qed
 
 text \<open>Relational version of @{thm [source] card_inj_on_le}:\<close>
 lemma card_le_if_inj_on_rel:
 assumes "finite B"
   "\<And>a. a \<in> A \<Longrightarrow> \<exists>b. b\<in>B \<and> r a b"
   "\<And>a1 a2 b. \<lbrakk> a1 \<in> A;  a2 \<in> A;  b \<in> B;  r a1 b;  r a2 b \<rbrakk> \<Longrightarrow> a1 = a2"
 shows "card A \<le> card B"
 proof -
   let ?P = "\<lambda>a b. b \<in> B \<and> r a b"
   let ?f = "\<lambda>a. SOME b. ?P a b"
   have 1: "?f ` A \<subseteq> B"  by (auto intro: someI2_ex[OF assms(2)])
   have "inj_on ?f A"
   proof (auto simp: inj_on_def)
     fix a1 a2 assume asms: "a1 \<in> A" "a2 \<in> A" "?f a1 = ?f a2"
     have 0: "?f a1 \<in> B" using "1" \<open>a1 \<in> A\<close> by blast
     have 1: "r a1 (?f a1)" using someI_ex[OF assms(2)[OF \<open>a1 \<in> A\<close>]] by blast
     have 2: "r a2 (?f a1)" using someI_ex[OF assms(2)[OF \<open>a2 \<in> A\<close>]] asms(3) by auto
     show "a1 = a2" using assms(3)[OF asms(1,2) 0 1 2] .
   qed
   with 1 show ?thesis using card_inj_on_le[of ?f A B] assms(1) by simp
 qed
 
 lemma inj_on_funpow_least: \<^marker>\<open>contributor \<open>Lars Noschinski\<close>\<close>
   \<open>inj_on (\<lambda>k. (f ^^ k) s) {0..<n}\<close>
   if \<open>(f ^^ n) s = s\<close> \<open>\<And>m. 0 < m \<Longrightarrow> m < n \<Longrightarrow> (f ^^ m) s \<noteq> s\<close>
 proof -
   { fix k l assume A: "k < n" "l < n" "k \<noteq> l" "(f ^^ k) s = (f ^^ l) s"
     define k' l' where "k' = min k l" and "l' = max k l"
     with A have A': "k' < l'" "(f ^^ k') s = (f ^^ l') s" "l' < n"
       by (auto simp: min_def max_def)
 
     have "s = (f ^^ ((n - l') + l')) s" using that \<open>l' < n\<close> by simp
     also have "\<dots> = (f ^^ (n - l')) ((f ^^ l') s)" by (simp add: funpow_add)
     also have "(f ^^ l') s = (f ^^ k') s" by (simp add: A')
     also have "(f ^^ (n - l')) \<dots> = (f ^^ (n - l' + k')) s" by (simp add: funpow_add)
     finally have "(f ^^ (n - l' + k')) s = s" by simp
     moreover have "n - l' + k' < n" "0 < n - l' + k'"using A' by linarith+
     ultimately have False using that(2) by auto
   }
   then show ?thesis by (intro inj_onI) auto
 qed
 
 
 subsection \<open>Intervals of integers\<close>
 
 lemma atLeastLessThanPlusOne_atLeastAtMost_int: "{l..<u+1} = {l..(u::int)}"
   by (auto simp add: atLeastAtMost_def atLeastLessThan_def)
 
 lemma atLeastPlusOneAtMost_greaterThanAtMost_int: "{l+1..u} = {l<..(u::int)}"
   by (auto simp add: atLeastAtMost_def greaterThanAtMost_def)
 
 lemma atLeastPlusOneLessThan_greaterThanLessThan_int:
     "{l+1..<u} = {l<..<u::int}"
   by (auto simp add: atLeastLessThan_def greaterThanLessThan_def)
 
 subsubsection \<open>Finiteness\<close>
 
 lemma image_atLeastZeroLessThan_int: "0 \<le> u ==>
     {(0::int)..<u} = int ` {..<nat u}"
   unfolding image_def lessThan_def
   apply auto
   apply (rule_tac x = "nat x" in exI)
   apply (auto simp add: zless_nat_eq_int_zless [THEN sym])
   done
 
 lemma finite_atLeastZeroLessThan_int: "finite {(0::int)..<u}"
 proof (cases "0 \<le> u")
   case True
   then show ?thesis
     by (auto simp: image_atLeastZeroLessThan_int)
 qed auto
 
 lemma finite_atLeastLessThan_int [iff]: "finite {l..<u::int}"
   by (simp only: image_add_int_atLeastLessThan [symmetric, of l] finite_imageI finite_atLeastZeroLessThan_int)
 
 lemma finite_atLeastAtMost_int [iff]: "finite {l..(u::int)}"
   by (subst atLeastLessThanPlusOne_atLeastAtMost_int [THEN sym], simp)
 
 lemma finite_greaterThanAtMost_int [iff]: "finite {l<..(u::int)}"
   by (subst atLeastPlusOneAtMost_greaterThanAtMost_int [THEN sym], simp)
 
 lemma finite_greaterThanLessThan_int [iff]: "finite {l<..<u::int}"
   by (subst atLeastPlusOneLessThan_greaterThanLessThan_int [THEN sym], simp)
 
 
 subsubsection \<open>Cardinality\<close>
 
 lemma card_atLeastZeroLessThan_int: "card {(0::int)..<u} = nat u"
 proof (cases "0 \<le> u")
   case True
   then show ?thesis
     by (auto simp: image_atLeastZeroLessThan_int card_image inj_on_def)    
 qed auto
 
 lemma card_atLeastLessThan_int [simp]: "card {l..<u} = nat (u - l)"
 proof -
   have "card {l..<u} = card {0..<u-l}"
     apply (subst image_add_int_atLeastLessThan [symmetric])
     apply (rule card_image)
     apply (simp add: inj_on_def)
     done
   then show ?thesis
     by (simp add: card_atLeastZeroLessThan_int)
 qed
 
 lemma card_atLeastAtMost_int [simp]: "card {l..u} = nat (u - l + 1)"
   apply (subst atLeastLessThanPlusOne_atLeastAtMost_int [THEN sym])
   apply (auto simp add: algebra_simps)
   done
 
 lemma card_greaterThanAtMost_int [simp]: "card {l<..u} = nat (u - l)"
   by (subst atLeastPlusOneAtMost_greaterThanAtMost_int [THEN sym], simp)
 
 lemma card_greaterThanLessThan_int [simp]: "card {l<..<u} = nat (u - (l + 1))"
   by (subst atLeastPlusOneLessThan_greaterThanLessThan_int [THEN sym], simp)
 
 lemma finite_M_bounded_by_nat: "finite {k. P k \<and> k < (i::nat)}"
 proof -
   have "{k. P k \<and> k < i} \<subseteq> {..<i}" by auto
   with finite_lessThan[of "i"] show ?thesis by (simp add: finite_subset)
 qed
 
 lemma card_less:
   assumes zero_in_M: "0 \<in> M"
   shows "card {k \<in> M. k < Suc i} \<noteq> 0"
 proof -
   from zero_in_M have "{k \<in> M. k < Suc i} \<noteq> {}" by auto
   with finite_M_bounded_by_nat show ?thesis by (auto simp add: card_eq_0_iff)
 qed
 
 lemma card_less_Suc2: 
   assumes "0 \<notin> M" shows "card {k. Suc k \<in> M \<and> k < i} = card {k \<in> M. k < Suc i}"
 proof -
   have *: "\<lbrakk>j \<in> M; j < Suc i\<rbrakk> \<Longrightarrow> j - Suc 0 < i \<and> Suc (j - Suc 0) \<in> M \<and> Suc 0 \<le> j" for j
     by (cases j) (use assms in auto)
   show ?thesis
   proof (rule card_bij_eq)
     show "inj_on Suc {k. Suc k \<in> M \<and> k < i}"
       by force
     show "inj_on (\<lambda>x. x - Suc 0) {k \<in> M. k < Suc i}"
       by (rule inj_on_diff_nat) (use * in blast)
   qed (use * in auto)
 qed
 
 lemma card_less_Suc:
   assumes "0 \<in> M"
     shows "Suc (card {k. Suc k \<in> M \<and> k < i}) = card {k \<in> M. k < Suc i}"
 proof -
   have "Suc (card {k. Suc k \<in> M \<and> k < i}) = Suc (card {k. Suc k \<in> M - {0} \<and> k < i})"
     by simp
   also have "\<dots> = Suc (card {k \<in> M - {0}. k < Suc i})"
     apply (subst card_less_Suc2)
     using assms by auto
   also have "\<dots> = Suc (card ({k \<in> M. k < Suc i} - {0}))"
     by (force intro: arg_cong [where f=card])
   also have "\<dots> = card (insert 0 ({k \<in> M. k < Suc i} - {0}))"
     by (simp add: card.insert_remove)
   also have "... = card {k \<in> M. k < Suc i}"
     using assms
     by (force simp add: intro: arg_cong [where f=card])
   finally show ?thesis.
 qed
 
 lemma card_le_Suc_Max: "finite S \<Longrightarrow> card S \<le> Suc (Max S)"
 proof (rule classical)
   assume "finite S" and "\<not> Suc (Max S) \<ge> card S"
   then have "Suc (Max S) < card S"
     by simp
-  with `finite S` have "S \<subseteq> {0..Max S}"
+  with \<open>finite S\<close> have "S \<subseteq> {0..Max S}"
     by auto
   hence "card S \<le> card {0..Max S}"
     by (intro card_mono; auto)
   thus "card S \<le> Suc (Max S)"
     by simp
 qed
 
 subsection \<open>Lemmas useful with the summation operator sum\<close>
 
 text \<open>For examples, see Algebra/poly/UnivPoly2.thy\<close>
 
 subsubsection \<open>Disjoint Unions\<close>
 
 text \<open>Singletons and open intervals\<close>
 
 lemma ivl_disj_un_singleton:
   "{l::'a::linorder} Un {l<..} = {l..}"
   "{..<u} Un {u::'a::linorder} = {..u}"
   "(l::'a::linorder) < u ==> {l} Un {l<..<u} = {l..<u}"
   "(l::'a::linorder) < u ==> {l<..<u} Un {u} = {l<..u}"
   "(l::'a::linorder) \<le> u ==> {l} Un {l<..u} = {l..u}"
   "(l::'a::linorder) \<le> u ==> {l..<u} Un {u} = {l..u}"
 by auto
 
 text \<open>One- and two-sided intervals\<close>
 
 lemma ivl_disj_un_one:
   "(l::'a::linorder) < u ==> {..l} Un {l<..<u} = {..<u}"
   "(l::'a::linorder) \<le> u ==> {..<l} Un {l..<u} = {..<u}"
   "(l::'a::linorder) \<le> u ==> {..l} Un {l<..u} = {..u}"
   "(l::'a::linorder) \<le> u ==> {..<l} Un {l..u} = {..u}"
   "(l::'a::linorder) \<le> u ==> {l<..u} Un {u<..} = {l<..}"
   "(l::'a::linorder) < u ==> {l<..<u} Un {u..} = {l<..}"
   "(l::'a::linorder) \<le> u ==> {l..u} Un {u<..} = {l..}"
   "(l::'a::linorder) \<le> u ==> {l..<u} Un {u..} = {l..}"
 by auto
 
 text \<open>Two- and two-sided intervals\<close>
 
 lemma ivl_disj_un_two:
   "[| (l::'a::linorder) < m; m \<le> u |] ==> {l<..<m} Un {m..<u} = {l<..<u}"
   "[| (l::'a::linorder) \<le> m; m < u |] ==> {l<..m} Un {m<..<u} = {l<..<u}"
   "[| (l::'a::linorder) \<le> m; m \<le> u |] ==> {l..<m} Un {m..<u} = {l..<u}"
   "[| (l::'a::linorder) \<le> m; m < u |] ==> {l..m} Un {m<..<u} = {l..<u}"
   "[| (l::'a::linorder) < m; m \<le> u |] ==> {l<..<m} Un {m..u} = {l<..u}"
   "[| (l::'a::linorder) \<le> m; m \<le> u |] ==> {l<..m} Un {m<..u} = {l<..u}"
   "[| (l::'a::linorder) \<le> m; m \<le> u |] ==> {l..<m} Un {m..u} = {l..u}"
   "[| (l::'a::linorder) \<le> m; m \<le> u |] ==> {l..m} Un {m<..u} = {l..u}"
 by auto
 
 lemma ivl_disj_un_two_touch:
   "[| (l::'a::linorder) < m; m < u |] ==> {l<..m} Un {m..<u} = {l<..<u}"
   "[| (l::'a::linorder) \<le> m; m < u |] ==> {l..m} Un {m..<u} = {l..<u}"
   "[| (l::'a::linorder) < m; m \<le> u |] ==> {l<..m} Un {m..u} = {l<..u}"
   "[| (l::'a::linorder) \<le> m; m \<le> u |] ==> {l..m} Un {m..u} = {l..u}"
 by auto
 
 lemmas ivl_disj_un = ivl_disj_un_singleton ivl_disj_un_one ivl_disj_un_two ivl_disj_un_two_touch
 
 subsubsection \<open>Disjoint Intersections\<close>
 
 text \<open>One- and two-sided intervals\<close>
 
 lemma ivl_disj_int_one:
   "{..l::'a::order} Int {l<..<u} = {}"
   "{..<l} Int {l..<u} = {}"
   "{..l} Int {l<..u} = {}"
   "{..<l} Int {l..u} = {}"
   "{l<..u} Int {u<..} = {}"
   "{l<..<u} Int {u..} = {}"
   "{l..u} Int {u<..} = {}"
   "{l..<u} Int {u..} = {}"
   by auto
 
 text \<open>Two- and two-sided intervals\<close>
 
 lemma ivl_disj_int_two:
   "{l::'a::order<..<m} Int {m..<u} = {}"
   "{l<..m} Int {m<..<u} = {}"
   "{l..<m} Int {m..<u} = {}"
   "{l..m} Int {m<..<u} = {}"
   "{l<..<m} Int {m..u} = {}"
   "{l<..m} Int {m<..u} = {}"
   "{l..<m} Int {m..u} = {}"
   "{l..m} Int {m<..u} = {}"
   by auto
 
 lemmas ivl_disj_int = ivl_disj_int_one ivl_disj_int_two
 
 subsubsection \<open>Some Differences\<close>
 
 lemma ivl_diff[simp]:
  "i \<le> n \<Longrightarrow> {i..<m} - {i..<n} = {n..<(m::'a::linorder)}"
 by(auto)
 
 lemma (in linorder) lessThan_minus_lessThan [simp]:
   "{..< n} - {..< m} = {m ..< n}"
   by auto
 
 lemma (in linorder) atLeastAtMost_diff_ends:
   "{a..b} - {a, b} = {a<..<b}"
   by auto
 
 
 subsubsection \<open>Some Subset Conditions\<close>
 
 lemma ivl_subset [simp]: "({i..<j} \<subseteq> {m..<n}) = (j \<le> i \<or> m \<le> i \<and> j \<le> (n::'a::linorder))"
   using linorder_class.le_less_linear[of i n]
   apply (auto simp: linorder_not_le)
    apply (force intro: leI)+
   done
 
 lemma obtain_subset_with_card_n:
   assumes "n \<le> card S"
   obtains T where "T \<subseteq> S" "card T = n" "finite T"
 proof -
   obtain n' where "card S = n + n'" 
     by (metis assms le_add_diff_inverse)
   with that show thesis
   proof (induct n' arbitrary: S)
     case 0 
     then show ?case
       by (cases "finite S") auto
   next
     case Suc 
     then show ?case 
       by (simp add: card_Suc_eq) (metis subset_insertI2)
   qed
 qed
 
 subsection \<open>Generic big monoid operation over intervals\<close>
 
 context semiring_char_0
 begin
 
 lemma inj_on_of_nat [simp]:
   "inj_on of_nat N"
   by rule simp
 
 lemma bij_betw_of_nat [simp]:
   "bij_betw of_nat N A \<longleftrightarrow> of_nat ` N = A"
   by (simp add: bij_betw_def)
 
 end
 
 context comm_monoid_set
 begin
 
 lemma atLeastLessThan_reindex:
   "F g {h m..<h n} = F (g \<circ> h) {m..<n}"
   if "bij_betw h {m..<n} {h m..<h n}" for m n ::nat
 proof -
   from that have "inj_on h {m..<n}" and "h ` {m..<n} = {h m..<h n}"
     by (simp_all add: bij_betw_def)
   then show ?thesis
     using reindex [of h "{m..<n}" g] by simp
 qed
 
 lemma atLeastAtMost_reindex:
   "F g {h m..h n} = F (g \<circ> h) {m..n}"
   if "bij_betw h {m..n} {h m..h n}" for m n ::nat
 proof -
   from that have "inj_on h {m..n}" and "h ` {m..n} = {h m..h n}"
     by (simp_all add: bij_betw_def)
   then show ?thesis
     using reindex [of h "{m..n}" g] by simp
 qed
 
 lemma atLeastLessThan_shift_bounds:
   "F g {m + k..<n + k} = F (g \<circ> plus k) {m..<n}"
   for m n k :: nat
   using atLeastLessThan_reindex [of "plus k" m n g]
   by (simp add: ac_simps)
 
 lemma atLeastAtMost_shift_bounds:
   "F g {m + k..n + k} = F (g \<circ> plus k) {m..n}"
   for m n k :: nat
   using atLeastAtMost_reindex [of "plus k" m n g]
   by (simp add: ac_simps)
 
 lemma atLeast_Suc_lessThan_Suc_shift:
   "F g {Suc m..<Suc n} = F (g \<circ> Suc) {m..<n}"
   using atLeastLessThan_shift_bounds [of _ _ 1]
   by (simp add: plus_1_eq_Suc)
 
 lemma atLeast_Suc_atMost_Suc_shift:
   "F g {Suc m..Suc n} = F (g \<circ> Suc) {m..n}"
   using atLeastAtMost_shift_bounds [of _ _ 1]
   by (simp add: plus_1_eq_Suc)
 
 lemma atLeast_int_lessThan_int_shift:
   "F g {int m..<int n} = F (g \<circ> int) {m..<n}"
   by (rule atLeastLessThan_reindex)
     (simp add: image_int_atLeastLessThan)
 
 lemma atLeast_int_atMost_int_shift:
   "F g {int m..int n} = F (g \<circ> int) {m..n}"
   by (rule atLeastAtMost_reindex)
     (simp add: image_int_atLeastAtMost)
 
 lemma atLeast0_lessThan_Suc:
   "F g {0..<Suc n} = F g {0..<n} \<^bold>* g n"
   by (simp add: atLeast0_lessThan_Suc ac_simps)
 
 lemma atLeast0_atMost_Suc:
   "F g {0..Suc n} = F g {0..n} \<^bold>* g (Suc n)"
   by (simp add: atLeast0_atMost_Suc ac_simps)
 
 lemma atLeast0_lessThan_Suc_shift:
   "F g {0..<Suc n} = g 0 \<^bold>* F (g \<circ> Suc) {0..<n}"
   by (simp add: atLeast0_lessThan_Suc_eq_insert_0 atLeast_Suc_lessThan_Suc_shift)
 
 lemma atLeast0_atMost_Suc_shift:
   "F g {0..Suc n} = g 0 \<^bold>* F (g \<circ> Suc) {0..n}"
   by (simp add: atLeast0_atMost_Suc_eq_insert_0 atLeast_Suc_atMost_Suc_shift)
 
 lemma atLeast_Suc_lessThan:
   "F g {m..<n} = g m \<^bold>* F g {Suc m..<n}" if "m < n"
 proof -
   from that have "{m..<n} = insert m {Suc m..<n}"
     by auto
   then show ?thesis by simp
 qed
 
 lemma atLeast_Suc_atMost:
   "F g {m..n} = g m \<^bold>* F g {Suc m..n}" if "m \<le> n"
 proof -
   from that have "{m..n} = insert m {Suc m..n}"
     by auto
   then show ?thesis by simp
 qed
 
 lemma ivl_cong:
   "a = c \<Longrightarrow> b = d \<Longrightarrow> (\<And>x. c \<le> x \<Longrightarrow> x < d \<Longrightarrow> g x = h x)
     \<Longrightarrow> F g {a..<b} = F h {c..<d}"
   by (rule cong) simp_all
 
 lemma atLeastLessThan_shift_0:
   fixes m n p :: nat
   shows "F g {m..<n} = F (g \<circ> plus m) {0..<n - m}"
   using atLeastLessThan_shift_bounds [of g 0 m "n - m"]
   by (cases "m \<le> n") simp_all
 
 lemma atLeastAtMost_shift_0:
   fixes m n p :: nat
   assumes "m \<le> n"
   shows "F g {m..n} = F (g \<circ> plus m) {0..n - m}"
   using assms atLeastAtMost_shift_bounds [of g 0 m "n - m"] by simp
 
 lemma atLeastLessThan_concat:
   fixes m n p :: nat
   shows "m \<le> n \<Longrightarrow> n \<le> p \<Longrightarrow> F g {m..<n} \<^bold>* F g {n..<p} = F g {m..<p}"
   by (simp add: union_disjoint [symmetric] ivl_disj_un)
 
 lemma atLeastLessThan_rev:
   "F g {n..<m} = F (\<lambda>i. g (m + n - Suc i)) {n..<m}"
   by (rule reindex_bij_witness [where i="\<lambda>i. m + n - Suc i" and j="\<lambda>i. m + n - Suc i"], auto)
 
 lemma atLeastAtMost_rev:
   fixes n m :: nat
   shows "F g {n..m} = F (\<lambda>i. g (m + n - i)) {n..m}"
   by (rule reindex_bij_witness [where i="\<lambda>i. m + n - i" and j="\<lambda>i. m + n - i"]) auto
 
 lemma atLeastLessThan_rev_at_least_Suc_atMost:
   "F g {n..<m} = F (\<lambda>i. g (m + n - i)) {Suc n..m}"
   unfolding atLeastLessThan_rev [of g n m]
   by (cases m) (simp_all add: atLeast_Suc_atMost_Suc_shift atLeastLessThanSuc_atLeastAtMost)
 
 end
 
 
 subsection \<open>Summation indexed over intervals\<close>
 
 syntax (ASCII)
   "_from_to_sum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"  ("(SUM _ = _.._./ _)" [0,0,0,10] 10)
   "_from_upto_sum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"  ("(SUM _ = _..<_./ _)" [0,0,0,10] 10)
   "_upt_sum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"  ("(SUM _<_./ _)" [0,0,10] 10)
   "_upto_sum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"  ("(SUM _<=_./ _)" [0,0,10] 10)
 
 syntax (latex_sum output)
   "_from_to_sum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
  ("(3\<^latex>\<open>$\\sum_{\<close>_ = _\<^latex>\<open>}^{\<close>_\<^latex>\<open>}$\<close> _)" [0,0,0,10] 10)
   "_from_upto_sum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
  ("(3\<^latex>\<open>$\\sum_{\<close>_ = _\<^latex>\<open>}^{<\<close>_\<^latex>\<open>}$\<close> _)" [0,0,0,10] 10)
   "_upt_sum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
  ("(3\<^latex>\<open>$\\sum_{\<close>_ < _\<^latex>\<open>}$\<close> _)" [0,0,10] 10)
   "_upto_sum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
  ("(3\<^latex>\<open>$\\sum_{\<close>_ \<le> _\<^latex>\<open>}$\<close> _)" [0,0,10] 10)
 
 syntax
   "_from_to_sum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Sum>_ = _.._./ _)" [0,0,0,10] 10)
   "_from_upto_sum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Sum>_ = _..<_./ _)" [0,0,0,10] 10)
   "_upt_sum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Sum>_<_./ _)" [0,0,10] 10)
   "_upto_sum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Sum>_\<le>_./ _)" [0,0,10] 10)
 
 translations
   "\<Sum>x=a..b. t" == "CONST sum (\<lambda>x. t) {a..b}"
   "\<Sum>x=a..<b. t" == "CONST sum (\<lambda>x. t) {a..<b}"
   "\<Sum>i\<le>n. t" == "CONST sum (\<lambda>i. t) {..n}"
   "\<Sum>i<n. t" == "CONST sum (\<lambda>i. t) {..<n}"
 
 text\<open>The above introduces some pretty alternative syntaxes for
 summation over intervals:
 \begin{center}
 \begin{tabular}{lll}
 Old & New & \LaTeX\\
 @{term[source]"\<Sum>x\<in>{a..b}. e"} & \<^term>\<open>\<Sum>x=a..b. e\<close> & @{term[mode=latex_sum]"\<Sum>x=a..b. e"}\\
 @{term[source]"\<Sum>x\<in>{a..<b}. e"} & \<^term>\<open>\<Sum>x=a..<b. e\<close> & @{term[mode=latex_sum]"\<Sum>x=a..<b. e"}\\
 @{term[source]"\<Sum>x\<in>{..b}. e"} & \<^term>\<open>\<Sum>x\<le>b. e\<close> & @{term[mode=latex_sum]"\<Sum>x\<le>b. e"}\\
 @{term[source]"\<Sum>x\<in>{..<b}. e"} & \<^term>\<open>\<Sum>x<b. e\<close> & @{term[mode=latex_sum]"\<Sum>x<b. e"}
 \end{tabular}
 \end{center}
 The left column shows the term before introduction of the new syntax,
 the middle column shows the new (default) syntax, and the right column
 shows a special syntax. The latter is only meaningful for latex output
 and has to be activated explicitly by setting the print mode to
 \<open>latex_sum\<close> (e.g.\ via \<open>mode = latex_sum\<close> in
 antiquotations). It is not the default \LaTeX\ output because it only
 works well with italic-style formulae, not tt-style.
 
 Note that for uniformity on \<^typ>\<open>nat\<close> it is better to use
 \<^term>\<open>\<Sum>x::nat=0..<n. e\<close> rather than \<open>\<Sum>x<n. e\<close>: \<open>sum\<close> may
 not provide all lemmas available for \<^term>\<open>{m..<n}\<close> also in the
 special form for \<^term>\<open>{..<n}\<close>.\<close>
 
 text\<open>This congruence rule should be used for sums over intervals as
 the standard theorem @{text[source]sum.cong} does not work well
 with the simplifier who adds the unsimplified premise \<^term>\<open>x\<in>B\<close> to
 the context.\<close>
 
 context comm_monoid_set
 begin
 
 lemma zero_middle:
   assumes "1 \<le> p" "k \<le> p"
   shows "F (\<lambda>j. if j < k then g j else if j = k then \<^bold>1 else h (j - Suc 0)) {..p}
        = F (\<lambda>j. if j < k then g j else h j) {..p - Suc 0}"  (is "?lhs = ?rhs")
 proof -
   have [simp]: "{..p - Suc 0} \<inter> {j. j < k} = {..<k}" "{..p - Suc 0} \<inter> - {j. j < k} = {k..p - Suc 0}"
     using assms by auto
   have "?lhs = F g {..<k} \<^bold>* F (\<lambda>j. if j = k then \<^bold>1 else h (j - Suc 0)) {k..p}"
     using union_disjoint [of "{..<k}" "{k..p}"] assms
     by (simp add: ivl_disj_int_one ivl_disj_un_one)
   also have "\<dots> = F g {..<k} \<^bold>* F (\<lambda>j.  h (j - Suc 0)) {Suc k..p}"
     by (simp add: atLeast_Suc_atMost [of k p] assms)
   also have "\<dots> = F g {..<k} \<^bold>* F h {k .. p - Suc 0}"
     using reindex [of Suc "{k..p - Suc 0}"] assms by simp
   also have "\<dots> = ?rhs"
     by (simp add: If_cases)
   finally show ?thesis .
 qed
 
 lemma atMost_Suc [simp]:
   "F g {..Suc n} = F g {..n} \<^bold>* g (Suc n)"
   by (simp add: atMost_Suc ac_simps)
 
 lemma lessThan_Suc [simp]:
   "F g {..<Suc n} = F g {..<n} \<^bold>* g n"
   by (simp add: lessThan_Suc ac_simps)
 
 lemma cl_ivl_Suc [simp]:
   "F g {m..Suc n} = (if Suc n < m then \<^bold>1 else F g {m..n} \<^bold>* g(Suc n))"
   by (auto simp: ac_simps atLeastAtMostSuc_conv)
 
 lemma op_ivl_Suc [simp]:
   "F g {m..<Suc n} = (if n < m then \<^bold>1 else F g {m..<n} \<^bold>* g(n))"
   by (auto simp: ac_simps atLeastLessThanSuc)
 
 lemma head:
   fixes n :: nat
   assumes mn: "m \<le> n"
   shows "F g {m..n} = g m \<^bold>* F g {m<..n}" (is "?lhs = ?rhs")
 proof -
   from mn
   have "{m..n} = {m} \<union> {m<..n}"
     by (auto intro: ivl_disj_un_singleton)
   hence "?lhs = F g ({m} \<union> {m<..n})"
     by (simp add: atLeast0LessThan)
   also have "\<dots> = ?rhs" by simp
   finally show ?thesis .
 qed
 
 lemma last_plus: 
   fixes n::nat  shows "m \<le> n \<Longrightarrow> F g {m..n} = g n \<^bold>* F g {m..<n}"
   by (cases n) (auto simp: atLeastLessThanSuc_atLeastAtMost commute)
 
 lemma head_if:
   fixes n :: nat
   shows "F g {m..n} = (if n < m then \<^bold>1 else  F g {m..<n} \<^bold>* g(n))"
   by (simp add: commute last_plus)
 
 lemma ub_add_nat: 
   assumes "(m::nat) \<le> n + 1"
   shows "F g {m..n + p} = F g {m..n} \<^bold>* F g {n + 1..n + p}"
 proof-
   have "{m .. n+p} = {m..n} \<union> {n+1..n+p}" using \<open>m \<le> n+1\<close> by auto
   thus ?thesis by (auto simp: ivl_disj_int union_disjoint atLeastSucAtMost_greaterThanAtMost)
 qed
 
 lemma nat_group: 
   fixes k::nat shows "F (\<lambda>m. F g {m * k ..< m*k + k}) {..<n} = F g {..< n * k}"
 proof (cases k)
   case (Suc l)
   then have "k > 0"
     by auto
   then show ?thesis
     by (induct n) (simp_all add: atLeastLessThan_concat add.commute atLeast0LessThan[symmetric])
 qed auto   
 
 lemma triangle_reindex:
   fixes n :: nat
   shows "F (\<lambda>(i,j). g i j) {(i,j). i+j < n} = F (\<lambda>k. F (\<lambda>i. g i (k - i)) {..k}) {..<n}"
   apply (simp add: Sigma)
   apply (rule reindex_bij_witness[where j="\<lambda>(i, j). (i+j, i)" and i="\<lambda>(k, i). (i, k - i)"])
   apply auto
   done
 
 lemma triangle_reindex_eq:
   fixes n :: nat
   shows "F (\<lambda>(i,j). g i j) {(i,j). i+j \<le> n} = F (\<lambda>k. F (\<lambda>i. g i (k - i)) {..k}) {..n}"
 using triangle_reindex [of g "Suc n"]
 by (simp only: Nat.less_Suc_eq_le lessThan_Suc_atMost)
 
 lemma nat_diff_reindex: "F (\<lambda>i. g (n - Suc i)) {..<n} = F g {..<n}"
   by (rule reindex_bij_witness[where i="\<lambda>i. n - Suc i" and j="\<lambda>i. n - Suc i"]) auto
 
 lemma shift_bounds_nat_ivl:
   "F g {m+k..<n+k} = F (\<lambda>i. g(i + k)){m..<n::nat}"
 by (induct "n", auto simp: atLeastLessThanSuc)
 
 lemma shift_bounds_cl_nat_ivl:
   "F g {m+k..n+k} = F (\<lambda>i. g(i + k)){m..n::nat}"
   by (rule reindex_bij_witness[where i="\<lambda>i. i + k" and j="\<lambda>i. i - k"]) auto
 
 corollary shift_bounds_cl_Suc_ivl:
   "F g {Suc m..Suc n} = F (\<lambda>i. g(Suc i)){m..n}"
 by (simp add: shift_bounds_cl_nat_ivl[where k="Suc 0", simplified])
 
 corollary Suc_reindex_ivl: "m \<le> n \<Longrightarrow> F g {m..n} \<^bold>* g (Suc n) = g m \<^bold>* F (\<lambda>i. g (Suc i)) {m..n}"
   by (simp add: assoc atLeast_Suc_atMost flip: shift_bounds_cl_Suc_ivl)
 
 corollary shift_bounds_Suc_ivl:
   "F g {Suc m..<Suc n} = F (\<lambda>i. g(Suc i)){m..<n}"
 by (simp add: shift_bounds_nat_ivl[where k="Suc 0", simplified])
 
 lemma atMost_Suc_shift:
   shows "F g {..Suc n} = g 0 \<^bold>* F (\<lambda>i. g (Suc i)) {..n}"
 proof (induct n)
   case 0 show ?case by simp
 next
   case (Suc n) note IH = this
   have "F g {..Suc (Suc n)} = F g {..Suc n} \<^bold>* g (Suc (Suc n))"
     by (rule atMost_Suc)
   also have "F g {..Suc n}  = g 0 \<^bold>* F (\<lambda>i. g (Suc i)) {..n}"
     by (rule IH)
   also have "g 0 \<^bold>* F (\<lambda>i. g (Suc i)) {..n} \<^bold>* g (Suc (Suc n)) =
              g 0 \<^bold>* (F (\<lambda>i. g (Suc i)) {..n} \<^bold>* g (Suc (Suc n)))"
     by (rule assoc)
   also have "F (\<lambda>i. g (Suc i)) {..n} \<^bold>* g (Suc (Suc n)) = F (\<lambda>i. g (Suc i)) {..Suc n}"
     by (rule atMost_Suc [symmetric])
   finally show ?case .
 qed
 
 lemma lessThan_Suc_shift:
   "F g {..<Suc n} = g 0 \<^bold>* F (\<lambda>i. g (Suc i)) {..<n}"
   by (induction n) (simp_all add: ac_simps)
 
 lemma atMost_shift:
   "F g {..n} = g 0 \<^bold>* F (\<lambda>i. g (Suc i)) {..<n}"
   by (metis atLeast0AtMost atLeast0LessThan atLeastLessThanSuc_atLeastAtMost 
        atLeastSucAtMost_greaterThanAtMost le0 head shift_bounds_Suc_ivl)
 
 lemma nested_swap:
      "F (\<lambda>i. F (\<lambda>j. a i j) {0..<i}) {0..n} = F (\<lambda>j. F (\<lambda>i. a i j) {Suc j..n}) {0..<n}"
   by (induction n) (auto simp: distrib)
 
 lemma nested_swap':
      "F (\<lambda>i. F (\<lambda>j. a i j) {..<i}) {..n} = F (\<lambda>j. F (\<lambda>i. a i j) {Suc j..n}) {..<n}"
   by (induction n) (auto simp: distrib)
 
 lemma atLeast1_atMost_eq:
   "F g {Suc 0..n} = F (\<lambda>k. g (Suc k)) {..<n}"
 proof -
   have "F g {Suc 0..n} = F g (Suc ` {..<n})"
     by (simp add: image_Suc_lessThan)
   also have "\<dots> = F (\<lambda>k. g (Suc k)) {..<n}"
     by (simp add: reindex)
   finally show ?thesis .
 qed
 
 lemma atLeastLessThan_Suc: "a \<le> b \<Longrightarrow> F g {a..<Suc b} = F g {a..<b} \<^bold>* g b"
   by (simp add: atLeastLessThanSuc commute)
 
 lemma nat_ivl_Suc':
   assumes "m \<le> Suc n"
   shows   "F g {m..Suc n} = g (Suc n) \<^bold>* F g {m..n}"
 proof -
   from assms have "{m..Suc n} = insert (Suc n) {m..n}" by auto
   also have "F g \<dots> = g (Suc n) \<^bold>* F g {m..n}" by simp
   finally show ?thesis .
 qed
 
 lemma in_pairs: "F g {2*m..Suc(2*n)} = F (\<lambda>i. g(2*i) \<^bold>* g(Suc(2*i))) {m..n}"
 proof (induction n)
   case 0
   show ?case
     by (cases "m=0") auto
 next
   case (Suc n)
   then show ?case
     by (auto simp: assoc split: if_split_asm)
 qed
 
 lemma in_pairs_0: "F g {..Suc(2*n)} = F (\<lambda>i. g(2*i) \<^bold>* g(Suc(2*i))) {..n}"
   using in_pairs [of _ 0 n] by (simp add: atLeast0AtMost)
 
 end
 
 lemma card_sum_le_nat_sum: "\<Sum> {0..<card S} \<le> \<Sum> S"
 proof (cases "finite S")
   case True
   then show ?thesis
   proof (induction "card S" arbitrary: S)
     case (Suc x)
     then have "Max S \<ge> x" using card_le_Suc_Max by fastforce
     let ?S' = "S - {Max S}"
     from Suc have "Max S \<in> S" by (auto intro: Max_in)
     hence cards: "card S = Suc (card ?S')"
-      using `finite S` by (intro card.remove; auto)
+      using \<open>finite S\<close> by (intro card.remove; auto)
     hence "\<Sum> {0..<card ?S'} \<le> \<Sum> ?S'"
       using Suc by (intro Suc; auto)
 
     hence "\<Sum> {0..<card ?S'} + x \<le> \<Sum> ?S' + Max S"
-      using `Max S \<ge> x` by simp
+      using \<open>Max S \<ge> x\<close> by simp
     also have "... = \<Sum> S"
-      using sum.remove[OF `finite S` `Max S \<in> S`, where g="\<lambda>x. x"]
+      using sum.remove[OF \<open>finite S\<close> \<open>Max S \<in> S\<close>, where g="\<lambda>x. x"]
       by simp
     finally show ?case
       using cards Suc by auto
   qed simp
 qed simp
 
 lemma sum_natinterval_diff:
   fixes f:: "nat \<Rightarrow> ('a::ab_group_add)"
   shows  "sum (\<lambda>k. f k - f(k + 1)) {(m::nat) .. n} =
           (if m \<le> n then f m - f(n + 1) else 0)"
 by (induct n, auto simp add: algebra_simps not_le le_Suc_eq)
 
 lemma sum_diff_nat_ivl:
   fixes f :: "nat \<Rightarrow> 'a::ab_group_add"
   shows "\<lbrakk> m \<le> n; n \<le> p \<rbrakk> \<Longrightarrow> sum f {m..<p} - sum f {m..<n} = sum f {n..<p}"
   using sum.atLeastLessThan_concat [of m n p f,symmetric]
   by (simp add: ac_simps)
 
 lemma sum_diff_distrib: "\<forall>x. Q x \<le> P x  \<Longrightarrow> (\<Sum>x<n. P x) - (\<Sum>x<n. Q x) = (\<Sum>x<n. P x - Q x :: nat)"
   by (subst sum_subtractf_nat) auto
 
 
 subsubsection \<open>Shifting bounds\<close>
 
 context comm_monoid_add
 begin
 
 context
   fixes f :: "nat \<Rightarrow> 'a"
   assumes "f 0 = 0"
 begin
 
 lemma sum_shift_lb_Suc0_0_upt:
   "sum f {Suc 0..<k} = sum f {0..<k}"
 proof (cases k)
   case 0
   then show ?thesis
     by simp
 next
   case (Suc k)
   moreover have "{0..<Suc k} = insert 0 {Suc 0..<Suc k}"
     by auto
   ultimately show ?thesis
     using \<open>f 0 = 0\<close> by simp
 qed
 
 lemma sum_shift_lb_Suc0_0: "sum f {Suc 0..k} = sum f {0..k}"
 proof (cases k)
   case 0
   with \<open>f 0 = 0\<close> show ?thesis
     by simp
 next
   case (Suc k)
   moreover have "{0..Suc k} = insert 0 {Suc 0..Suc k}"
     by auto
   ultimately show ?thesis
     using \<open>f 0 = 0\<close> by simp
 qed
 
 end
 
 end
 
 lemma sum_Suc_diff:
   fixes f :: "nat \<Rightarrow> 'a::ab_group_add"
   assumes "m \<le> Suc n"
   shows "(\<Sum>i = m..n. f(Suc i) - f i) = f (Suc n) - f m"
 using assms by (induct n) (auto simp: le_Suc_eq)
 
 lemma sum_Suc_diff':
   fixes f :: "nat \<Rightarrow> 'a::ab_group_add"
   assumes "m \<le> n"
   shows "(\<Sum>i = m..<n. f (Suc i) - f i) = f n - f m"
 using assms by (induct n) (auto simp: le_Suc_eq)
 
 
 subsubsection \<open>Telescoping\<close>
 
 lemma sum_telescope:
   fixes f::"nat \<Rightarrow> 'a::ab_group_add"
   shows "sum (\<lambda>i. f i - f (Suc i)) {.. i} = f 0 - f (Suc i)"
   by (induct i) simp_all
 
 lemma sum_telescope'':
   assumes "m \<le> n"
   shows   "(\<Sum>k\<in>{Suc m..n}. f k - f (k - 1)) = f n - (f m :: 'a :: ab_group_add)"
   by (rule dec_induct[OF assms]) (simp_all add: algebra_simps)
 
 lemma sum_lessThan_telescope:
   "(\<Sum>n<m. f (Suc n) - f n :: 'a :: ab_group_add) = f m - f 0"
   by (induction m) (simp_all add: algebra_simps)
 
 lemma sum_lessThan_telescope':
   "(\<Sum>n<m. f n - f (Suc n) :: 'a :: ab_group_add) = f 0 - f m"
   by (induction m) (simp_all add: algebra_simps)
 
 
 subsubsection \<open>The formula for geometric sums\<close>
 
 lemma sum_power2: "(\<Sum>i=0..<k. (2::nat)^i) = 2^k-1"
   by (induction k) (auto simp: mult_2)
 
 lemma geometric_sum:
   assumes "x \<noteq> 1"
   shows "(\<Sum>i<n. x ^ i) = (x ^ n - 1) / (x - 1::'a::field)"
 proof -
   from assms obtain y where "y = x - 1" and "y \<noteq> 0" by simp_all
   moreover have "(\<Sum>i<n. (y + 1) ^ i) = ((y + 1) ^ n - 1) / y"
     by (induct n) (simp_all add: field_simps \<open>y \<noteq> 0\<close>)
   ultimately show ?thesis by simp
 qed
 
 lemma diff_power_eq_sum:
   fixes y :: "'a::{comm_ring,monoid_mult}"
   shows
     "x ^ (Suc n) - y ^ (Suc n) =
       (x - y) * (\<Sum>p<Suc n. (x ^ p) * y ^ (n - p))"
 proof (induct n)
   case (Suc n)
   have "x ^ Suc (Suc n) - y ^ Suc (Suc n) = x * (x * x^n) - y * (y * y ^ n)"
     by simp
   also have "... = y * (x ^ (Suc n) - y ^ (Suc n)) + (x - y) * (x * x^n)"
     by (simp add: algebra_simps)
   also have "... = y * ((x - y) * (\<Sum>p<Suc n. (x ^ p) * y ^ (n - p))) + (x - y) * (x * x^n)"
     by (simp only: Suc)
   also have "... = (x - y) * (y * (\<Sum>p<Suc n. (x ^ p) * y ^ (n - p))) + (x - y) * (x * x^n)"
     by (simp only: mult.left_commute)
   also have "... = (x - y) * (\<Sum>p<Suc (Suc n). x ^ p * y ^ (Suc n - p))"
     by (simp add: field_simps Suc_diff_le sum_distrib_right sum_distrib_left)
   finally show ?case .
 qed simp
 
 corollary power_diff_sumr2: \<comment> \<open>\<open>COMPLEX_POLYFUN\<close> in HOL Light\<close>
   fixes x :: "'a::{comm_ring,monoid_mult}"
   shows "x^n - y^n = (x - y) * (\<Sum>i<n. y^(n - Suc i) * x^i)"
 using diff_power_eq_sum[of x "n - 1" y]
 by (cases "n = 0") (simp_all add: field_simps)
 
 lemma power_diff_1_eq:
   fixes x :: "'a::{comm_ring,monoid_mult}"
   shows "x^n - 1 = (x - 1) * (\<Sum>i<n. (x^i))"
 using diff_power_eq_sum [of x _ 1]
   by (cases n) auto
 
 lemma one_diff_power_eq':
   fixes x :: "'a::{comm_ring,monoid_mult}"
   shows "1 - x^n = (1 - x) * (\<Sum>i<n. x^(n - Suc i))"
 using diff_power_eq_sum [of 1 _ x]
   by (cases n) auto
 
 lemma one_diff_power_eq:
   fixes x :: "'a::{comm_ring,monoid_mult}"
   shows "1 - x^n = (1 - x) * (\<Sum>i<n. x^i)"
 by (metis one_diff_power_eq' sum.nat_diff_reindex)
 
 lemma sum_gp_basic:
   fixes x :: "'a::{comm_ring,monoid_mult}"
   shows "(1 - x) * (\<Sum>i\<le>n. x^i) = 1 - x^Suc n"
   by (simp only: one_diff_power_eq lessThan_Suc_atMost)
 
 lemma sum_power_shift:
   fixes x :: "'a::{comm_ring,monoid_mult}"
   assumes "m \<le> n"
   shows "(\<Sum>i=m..n. x^i) = x^m * (\<Sum>i\<le>n-m. x^i)"
 proof -
   have "(\<Sum>i=m..n. x^i) = x^m * (\<Sum>i=m..n. x^(i-m))"
     by (simp add: sum_distrib_left power_add [symmetric])
   also have "(\<Sum>i=m..n. x^(i-m)) = (\<Sum>i\<le>n-m. x^i)"
     using \<open>m \<le> n\<close> by (intro sum.reindex_bij_witness[where j="\<lambda>i. i - m" and i="\<lambda>i. i + m"]) auto
   finally show ?thesis .
 qed
 
 lemma sum_gp_multiplied:
   fixes x :: "'a::{comm_ring,monoid_mult}"
   assumes "m \<le> n"
   shows "(1 - x) * (\<Sum>i=m..n. x^i) = x^m - x^Suc n"
 proof -
   have  "(1 - x) * (\<Sum>i=m..n. x^i) = x^m * (1 - x) * (\<Sum>i\<le>n-m. x^i)"
     by (metis mult.assoc mult.commute assms sum_power_shift)
   also have "... =x^m * (1 - x^Suc(n-m))"
     by (metis mult.assoc sum_gp_basic)
   also have "... = x^m - x^Suc n"
     using assms
     by (simp add: algebra_simps) (metis le_add_diff_inverse power_add)
   finally show ?thesis .
 qed
 
 lemma sum_gp:
   fixes x :: "'a::{comm_ring,division_ring}"
   shows   "(\<Sum>i=m..n. x^i) =
                (if n < m then 0
                 else if x = 1 then of_nat((n + 1) - m)
                 else (x^m - x^Suc n) / (1 - x))"
 using sum_gp_multiplied [of m n x] apply auto
 by (metis eq_iff_diff_eq_0 mult.commute nonzero_divide_eq_eq)
 
 
 subsubsection\<open>Geometric progressions\<close>
 
 lemma sum_gp0:
   fixes x :: "'a::{comm_ring,division_ring}"
   shows "(\<Sum>i\<le>n. x^i) = (if x = 1 then of_nat(n + 1) else (1 - x^Suc n) / (1 - x))"
   using sum_gp_basic[of x n]
   by (simp add: mult.commute field_split_simps)
 
 lemma sum_power_add:
   fixes x :: "'a::{comm_ring,monoid_mult}"
   shows "(\<Sum>i\<in>I. x^(m+i)) = x^m * (\<Sum>i\<in>I. x^i)"
   by (simp add: sum_distrib_left power_add)
 
 lemma sum_gp_offset:
   fixes x :: "'a::{comm_ring,division_ring}"
   shows   "(\<Sum>i=m..m+n. x^i) =
        (if x = 1 then of_nat n + 1 else x^m * (1 - x^Suc n) / (1 - x))"
   using sum_gp [of x m "m+n"]
   by (auto simp: power_add algebra_simps)
 
 lemma sum_gp_strict:
   fixes x :: "'a::{comm_ring,division_ring}"
   shows "(\<Sum>i<n. x^i) = (if x = 1 then of_nat n else (1 - x^n) / (1 - x))"
   by (induct n) (auto simp: algebra_simps field_split_simps)
 
 
 subsubsection \<open>The formulae for arithmetic sums\<close>
 
 context comm_semiring_1
 begin
 
 lemma double_gauss_sum:
   "2 * (\<Sum>i = 0..n. of_nat i) = of_nat n * (of_nat n + 1)"
   by (induct n) (simp_all add: sum.atLeast0_atMost_Suc algebra_simps left_add_twice)
 
 lemma double_gauss_sum_from_Suc_0:
   "2 * (\<Sum>i = Suc 0..n. of_nat i) = of_nat n * (of_nat n + 1)"
 proof -
   have "sum of_nat {Suc 0..n} = sum of_nat (insert 0 {Suc 0..n})"
     by simp
   also have "\<dots> = sum of_nat {0..n}"
     by (cases n) (simp_all add: atLeast0_atMost_Suc_eq_insert_0)
   finally show ?thesis
     by (simp add: double_gauss_sum)
 qed
 
 lemma double_arith_series:
   "2 * (\<Sum>i = 0..n. a + of_nat i * d) = (of_nat n + 1) * (2 * a + of_nat n * d)"
 proof -
   have "(\<Sum>i = 0..n. a + of_nat i * d) = ((\<Sum>i = 0..n. a) + (\<Sum>i = 0..n. of_nat i * d))"
     by (rule sum.distrib)
   also have "\<dots> = (of_nat (Suc n) * a + d * (\<Sum>i = 0..n. of_nat i))"
     by (simp add: sum_distrib_left algebra_simps)
   finally show ?thesis
     by (simp add: algebra_simps double_gauss_sum left_add_twice)
 qed
 
 end
 
 context unique_euclidean_semiring_with_nat
 begin
 
 lemma gauss_sum:
   "(\<Sum>i = 0..n. of_nat i) = of_nat n * (of_nat n + 1) div 2"
   using double_gauss_sum [of n, symmetric] by simp
 
 lemma gauss_sum_from_Suc_0:
   "(\<Sum>i = Suc 0..n. of_nat i) = of_nat n * (of_nat n + 1) div 2"
   using double_gauss_sum_from_Suc_0 [of n, symmetric] by simp
 
 lemma arith_series:
   "(\<Sum>i = 0..n. a + of_nat i * d) = (of_nat n + 1) * (2 * a + of_nat n * d) div 2"
   using double_arith_series [of a d n, symmetric] by simp
 
 end
 
 lemma gauss_sum_nat:
   "\<Sum>{0..n} = (n * Suc n) div 2"
   using gauss_sum [of n, where ?'a = nat] by simp
 
 lemma arith_series_nat:
   "(\<Sum>i = 0..n. a + i * d) = Suc n * (2 * a + n * d) div 2"
   using arith_series [of a d n] by simp
 
 lemma Sum_Icc_int:
   "\<Sum>{m..n} = (n * (n + 1) - m * (m - 1)) div 2"
   if "m \<le> n" for m n :: int
 using that proof (induct i \<equiv> "nat (n - m)" arbitrary: m n)
   case 0
   then have "m = n"
     by arith
   then show ?case
     by (simp add: algebra_simps mult_2 [symmetric])
 next
   case (Suc i)
   have 0: "i = nat((n-1) - m)" "m \<le> n-1" using Suc(2,3) by arith+
   have "\<Sum> {m..n} = \<Sum> {m..1+(n-1)}" by simp
   also have "\<dots> = \<Sum> {m..n-1} + n" using \<open>m \<le> n\<close>
     by(subst atLeastAtMostPlus1_int_conv) simp_all
   also have "\<dots> = ((n-1)*(n-1+1) - m*(m-1)) div 2 + n"
     by(simp add: Suc(1)[OF 0])
   also have "\<dots> = ((n-1)*(n-1+1) - m*(m-1) + 2*n) div 2" by simp
   also have "\<dots> = (n*(n+1) - m*(m-1)) div 2"
     by (simp add: algebra_simps mult_2_right)
   finally show ?case .
 qed
 
 lemma Sum_Icc_nat:
   "\<Sum>{m..n} = (n * (n + 1) - m * (m - 1)) div 2" for m n :: nat
 proof (cases "m \<le> n")
   case True
   then have *: "m * (m - 1) \<le> n * (n + 1)"
     by (meson diff_le_self order_trans le_add1 mult_le_mono)
   have "int (\<Sum>{m..n}) = (\<Sum>{int m..int n})"
     by (simp add: sum.atLeast_int_atMost_int_shift)
   also have "\<dots> = (int n * (int n + 1) - int m * (int m - 1)) div 2"
     using \<open>m \<le> n\<close> by (simp add: Sum_Icc_int)
   also have "\<dots> = int ((n * (n + 1) - m * (m - 1)) div 2)"
     using le_square * by (simp add: algebra_simps of_nat_div of_nat_diff)
   finally show ?thesis
     by (simp only: of_nat_eq_iff)
 next
   case False
   then show ?thesis
     by (auto dest: less_imp_Suc_add simp add: not_le algebra_simps)
 qed
 
 lemma Sum_Ico_nat: 
   "\<Sum>{m..<n} = (n * (n - 1) - m * (m - 1)) div 2" for m n :: nat
   by (cases n) (simp_all add: atLeastLessThanSuc_atLeastAtMost Sum_Icc_nat)
 
 
 subsubsection \<open>Division remainder\<close>
 
 lemma range_mod:
   fixes n :: nat
   assumes "n > 0"
   shows "range (\<lambda>m. m mod n) = {0..<n}" (is "?A = ?B")
 proof (rule set_eqI)
   fix m
   show "m \<in> ?A \<longleftrightarrow> m \<in> ?B"
   proof
     assume "m \<in> ?A"
     with assms show "m \<in> ?B"
       by auto
   next
     assume "m \<in> ?B"
     moreover have "m mod n \<in> ?A"
       by (rule rangeI)
     ultimately show "m \<in> ?A"
       by simp
   qed
 qed
 
 
 subsection \<open>Products indexed over intervals\<close>
 
 syntax (ASCII)
   "_from_to_prod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"  ("(PROD _ = _.._./ _)" [0,0,0,10] 10)
   "_from_upto_prod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"  ("(PROD _ = _..<_./ _)" [0,0,0,10] 10)
   "_upt_prod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"  ("(PROD _<_./ _)" [0,0,10] 10)
   "_upto_prod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"  ("(PROD _<=_./ _)" [0,0,10] 10)
 
 syntax (latex_prod output)
   "_from_to_prod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
  ("(3\<^latex>\<open>$\\prod_{\<close>_ = _\<^latex>\<open>}^{\<close>_\<^latex>\<open>}$\<close> _)" [0,0,0,10] 10)
   "_from_upto_prod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
  ("(3\<^latex>\<open>$\\prod_{\<close>_ = _\<^latex>\<open>}^{<\<close>_\<^latex>\<open>}$\<close> _)" [0,0,0,10] 10)
   "_upt_prod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
  ("(3\<^latex>\<open>$\\prod_{\<close>_ < _\<^latex>\<open>}$\<close> _)" [0,0,10] 10)
   "_upto_prod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
  ("(3\<^latex>\<open>$\\prod_{\<close>_ \<le> _\<^latex>\<open>}$\<close> _)" [0,0,10] 10)
 
 syntax
   "_from_to_prod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Prod>_ = _.._./ _)" [0,0,0,10] 10)
   "_from_upto_prod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Prod>_ = _..<_./ _)" [0,0,0,10] 10)
   "_upt_prod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Prod>_<_./ _)" [0,0,10] 10)
   "_upto_prod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Prod>_\<le>_./ _)" [0,0,10] 10)
 
 translations
   "\<Prod>x=a..b. t" \<rightleftharpoons> "CONST prod (\<lambda>x. t) {a..b}"
   "\<Prod>x=a..<b. t" \<rightleftharpoons> "CONST prod (\<lambda>x. t) {a..<b}"
   "\<Prod>i\<le>n. t" \<rightleftharpoons> "CONST prod (\<lambda>i. t) {..n}"
   "\<Prod>i<n. t" \<rightleftharpoons> "CONST prod (\<lambda>i. t) {..<n}"
 
 lemma prod_int_plus_eq: "prod int {i..i+j} =  \<Prod>{int i..int (i+j)}"
   by (induct j) (auto simp add: atLeastAtMostSuc_conv atLeastAtMostPlus1_int_conv)
 
 lemma prod_int_eq: "prod int {i..j} =  \<Prod>{int i..int j}"
 proof (cases "i \<le> j")
   case True
   then show ?thesis
     by (metis le_iff_add prod_int_plus_eq)
 next
   case False
   then show ?thesis
     by auto
 qed
 
 subsection \<open>Efficient folding over intervals\<close>
 
 function fold_atLeastAtMost_nat where
   [simp del]: "fold_atLeastAtMost_nat f a (b::nat) acc =
                  (if a > b then acc else fold_atLeastAtMost_nat f (a+1) b (f a acc))"
 by pat_completeness auto
 termination by (relation "measure (\<lambda>(_,a,b,_). Suc b - a)") auto
 
 lemma fold_atLeastAtMost_nat:
   assumes "comp_fun_commute f"
   shows   "fold_atLeastAtMost_nat f a b acc = Finite_Set.fold f acc {a..b}"
 using assms
 proof (induction f a b acc rule: fold_atLeastAtMost_nat.induct, goal_cases)
   case (1 f a b acc)
   interpret comp_fun_commute f by fact
   show ?case
   proof (cases "a > b")
     case True
     thus ?thesis by (subst fold_atLeastAtMost_nat.simps) auto
   next
     case False
     with 1 show ?thesis
       by (subst fold_atLeastAtMost_nat.simps)
          (auto simp: atLeastAtMost_insertL[symmetric] fold_fun_left_comm)
   qed
 qed
 
 lemma sum_atLeastAtMost_code:
   "sum f {a..b} = fold_atLeastAtMost_nat (\<lambda>a acc. f a + acc) a b 0"
 proof -
   have "comp_fun_commute (\<lambda>a. (+) (f a))"
     by unfold_locales (auto simp: o_def add_ac)
   thus ?thesis
     by (simp add: sum.eq_fold fold_atLeastAtMost_nat o_def)
 qed
 
 lemma prod_atLeastAtMost_code:
   "prod f {a..b} = fold_atLeastAtMost_nat (\<lambda>a acc. f a * acc) a b 1"
 proof -
   have "comp_fun_commute (\<lambda>a. (*) (f a))"
     by unfold_locales (auto simp: o_def mult_ac)
   thus ?thesis
     by (simp add: prod.eq_fold fold_atLeastAtMost_nat o_def)
 qed
 
 (* TODO: Add support for folding over more kinds of intervals here *)
 
 end