diff --git a/src/HOL/Number_Theory/Residues.thy b/src/HOL/Number_Theory/Residues.thy
--- a/src/HOL/Number_Theory/Residues.thy
+++ b/src/HOL/Number_Theory/Residues.thy
@@ -1,506 +1,505 @@
 (*  Title:      HOL/Number_Theory/Residues.thy
     Author:     Jeremy Avigad
 
 An algebraic treatment of residue rings, and resulting proofs of
 Euler's theorem and Wilson's theorem.
 *)
 
 section \<open>Residue rings\<close>
 
 theory Residues
 imports
   Cong
   "HOL-Algebra.Multiplicative_Group"
   Totient
 begin
 
 definition QuadRes :: "int \<Rightarrow> int \<Rightarrow> bool"
   where "QuadRes p a = (\<exists>y. ([y^2 = a] (mod p)))"
 
 definition Legendre :: "int \<Rightarrow> int \<Rightarrow> int"
   where "Legendre a p =
     (if ([a = 0] (mod p)) then 0
      else if QuadRes p a then 1
      else -1)"
 
 
 subsection \<open>A locale for residue rings\<close>
 
 definition residue_ring :: "int \<Rightarrow> int ring"
   where
     "residue_ring m =
       \<lparr>carrier = {0..m - 1},
        monoid.mult = \<lambda>x y. (x * y) mod m,
        one = 1,
        zero = 0,
        add = \<lambda>x y. (x + y) mod m\<rparr>"
 
 locale residues =
   fixes m :: int and R (structure)
   assumes m_gt_one: "m > 1"
   defines "R \<equiv> residue_ring m"
 begin
 
 lemma abelian_group: "abelian_group R"
 proof -
   have "\<exists>y\<in>{0..m - 1}. (x + y) mod m = 0" if "0 \<le> x" "x < m" for x
   proof (cases "x = 0")
     case True
     with m_gt_one show ?thesis by simp
   next
     case False
     then have "(x + (m - x)) mod m = 0"
       by simp
     with m_gt_one that show ?thesis
       by (metis False atLeastAtMost_iff diff_ge_0_iff_ge diff_left_mono int_one_le_iff_zero_less less_le)
   qed
   with m_gt_one show ?thesis
     by (fastforce simp add: R_def residue_ring_def mod_add_right_eq ac_simps  intro!: abelian_groupI)
 qed
 
 lemma comm_monoid: "comm_monoid R"
   unfolding R_def residue_ring_def
   apply (rule comm_monoidI)
     using m_gt_one  apply auto
   apply (metis mod_mult_right_eq mult.assoc mult.commute)
   apply (metis mult.commute)
   done
 
 lemma cring: "cring R"
   apply (intro cringI abelian_group comm_monoid)
   unfolding R_def residue_ring_def
   apply (auto simp add: comm_semiring_class.distrib mod_add_eq mod_mult_left_eq)
   done
 
 end
 
 sublocale residues < cring
   by (rule cring)
 
 
 context residues
 begin
 
 text \<open>
   These lemmas translate back and forth between internal and
   external concepts.
 \<close>
 
 lemma res_carrier_eq: "carrier R = {0..m - 1}"
   by (auto simp: R_def residue_ring_def)
 
 lemma res_add_eq: "x \<oplus> y = (x + y) mod m"
   by (auto simp: R_def residue_ring_def)
 
 lemma res_mult_eq: "x \<otimes> y = (x * y) mod m"
   by (auto simp: R_def residue_ring_def)
 
 lemma res_zero_eq: "\<zero> = 0"
   by (auto simp: R_def residue_ring_def)
 
 lemma res_one_eq: "\<one> = 1"
   by (auto simp: R_def residue_ring_def units_of_def)
 
 lemma res_units_eq: "Units R = {x. 0 < x \<and> x < m \<and> coprime x m}"
   using m_gt_one
   apply (auto simp add: Units_def R_def residue_ring_def ac_simps invertible_coprime intro: ccontr)
   apply (subst (asm) coprime_iff_invertible'_int)
    apply (auto simp add: cong_def)
   done
 
 lemma res_neg_eq: "\<ominus> x = (- x) mod m"
   using m_gt_one unfolding R_def a_inv_def m_inv_def residue_ring_def
   apply simp
   apply (rule the_equality)
    apply (simp add: mod_add_right_eq)
    apply (simp add: add.commute mod_add_right_eq)
   apply (metis add.right_neutral minus_add_cancel mod_add_right_eq mod_pos_pos_trivial)
   done
 
 lemma finite [iff]: "finite (carrier R)"
   by (simp add: res_carrier_eq)
 
 lemma finite_Units [iff]: "finite (Units R)"
   by (simp add: finite_ring_finite_units)
 
 text \<open>
   The function \<open>a \<mapsto> a mod m\<close> maps the integers to the
   residue classes. The following lemmas show that this mapping
   respects addition and multiplication on the integers.
 \<close>
 
 lemma mod_in_carrier [iff]: "a mod m \<in> carrier R"
   unfolding res_carrier_eq
   using insert m_gt_one by auto
 
 lemma add_cong: "(x mod m) \<oplus> (y mod m) = (x + y) mod m"
   by (auto simp: R_def residue_ring_def mod_simps)
 
 lemma mult_cong: "(x mod m) \<otimes> (y mod m) = (x * y) mod m"
   by (auto simp: R_def residue_ring_def mod_simps)
 
 lemma zero_cong: "\<zero> = 0"
   by (auto simp: R_def residue_ring_def)
 
 lemma one_cong: "\<one> = 1 mod m"
   using m_gt_one by (auto simp: R_def residue_ring_def)
 
 (* FIXME revise algebra library to use 1? *)
 lemma pow_cong: "(x mod m) [^] n = x^n mod m"
   using m_gt_one
   apply (induct n)
   apply (auto simp add: nat_pow_def one_cong)
   apply (metis mult.commute mult_cong)
   done
 
 lemma neg_cong: "\<ominus> (x mod m) = (- x) mod m"
   by (metis mod_minus_eq res_neg_eq)
 
 lemma (in residues) prod_cong: "finite A \<Longrightarrow> (\<Otimes>i\<in>A. (f i) mod m) = (\<Prod>i\<in>A. f i) mod m"
   by (induct set: finite) (auto simp: one_cong mult_cong)
 
 lemma (in residues) sum_cong: "finite A \<Longrightarrow> (\<Oplus>i\<in>A. (f i) mod m) = (\<Sum>i\<in>A. f i) mod m"
   by (induct set: finite) (auto simp: zero_cong add_cong)
 
 lemma mod_in_res_units [simp]:
   assumes "1 < m" and "coprime a m"
   shows "a mod m \<in> Units R"
 proof (cases "a mod m = 0")
   case True
   with assms show ?thesis
     by (auto simp add: res_units_eq gcd_red_int [symmetric])
 next
   case False
   from assms have "0 < m" by simp
   then have "0 \<le> a mod m" by (rule pos_mod_sign [of m a])
   with False have "0 < a mod m" by simp
   with assms show ?thesis
     by (auto simp add: res_units_eq gcd_red_int [symmetric] ac_simps)
 qed
 
 lemma res_eq_to_cong: "(a mod m) = (b mod m) \<longleftrightarrow> [a = b] (mod m)"
   by (auto simp: cong_def)
 
 
 text \<open>Simplifying with these will translate a ring equation in R to a congruence.\<close>
 lemmas res_to_cong_simps =
   add_cong mult_cong pow_cong one_cong
   prod_cong sum_cong neg_cong res_eq_to_cong
 
 text \<open>Other useful facts about the residue ring.\<close>
 lemma one_eq_neg_one: "\<one> = \<ominus> \<one> \<Longrightarrow> m = 2"
   apply (simp add: res_one_eq res_neg_eq)
   apply (metis add.commute add_diff_cancel mod_mod_trivial one_add_one uminus_add_conv_diff
     zero_neq_one zmod_zminus1_eq_if)
   done
 
 end
 
 
 subsection \<open>Prime residues\<close>
 
 locale residues_prime =
   fixes p :: nat and R (structure)
   assumes p_prime [intro]: "prime p"
   defines "R \<equiv> residue_ring (int p)"
 
 sublocale residues_prime < residues p
   unfolding R_def residues_def
   using p_prime apply auto
   apply (metis (full_types) of_nat_1 of_nat_less_iff prime_gt_1_nat)
   done
 
 context residues_prime
 begin
 
 lemma p_coprime_left:
   "coprime p a \<longleftrightarrow> \<not> p dvd a"
   using p_prime by (auto intro: prime_imp_coprime dest: coprime_common_divisor)
 
 lemma p_coprime_right:
   "coprime a p  \<longleftrightarrow> \<not> p dvd a"
   using p_coprime_left [of a] by (simp add: ac_simps)
 
 lemma p_coprime_left_int:
   "coprime (int p) a \<longleftrightarrow> \<not> int p dvd a"
   using p_prime by (auto intro: prime_imp_coprime dest: coprime_common_divisor)
 
 lemma p_coprime_right_int:
   "coprime a (int p) \<longleftrightarrow> \<not> int p dvd a"
   using p_coprime_left_int [of a] by (simp add: ac_simps)
 
 lemma is_field: "field R"
 proof -
   have "0 < x \<Longrightarrow> x < int p \<Longrightarrow> coprime (int p) x" for x
     by (rule prime_imp_coprime) (auto simp add: zdvd_not_zless)
   then show ?thesis
     by (intro cring.field_intro2 cring)
       (auto simp add: res_carrier_eq res_one_eq res_zero_eq res_units_eq ac_simps)
 qed
 
 lemma res_prime_units_eq: "Units R = {1..p - 1}"
   apply (subst res_units_eq)
   apply (auto simp add: p_coprime_right_int zdvd_not_zless)
   done
 
 end
 
 sublocale residues_prime < field
   by (rule is_field)
 
 
 section \<open>Test cases: Euler's theorem and Wilson's theorem\<close>
 
 subsection \<open>Euler's theorem\<close>
 
 lemma (in residues) totatives_eq:
   "totatives (nat m) = nat ` Units R"
 proof -
   from m_gt_one have "\<bar>m\<bar> > 1"
     by simp
   then have "totatives (nat \<bar>m\<bar>) = nat ` abs ` Units R"
     by (auto simp add: totatives_def res_units_eq image_iff le_less)
       (use m_gt_one zless_nat_eq_int_zless in force)
   moreover have "\<bar>m\<bar> = m" "abs ` Units R = Units R"
     using m_gt_one by (auto simp add: res_units_eq image_iff)
   ultimately show ?thesis
     by simp
 qed
 
 lemma (in residues) totient_eq:
   "totient (nat m) = card (Units R)"
 proof  -
   have *: "inj_on nat (Units R)"
     by (rule inj_onI) (auto simp add: res_units_eq)
   then show ?thesis
     by (simp add: totient_def totatives_eq card_image)
 qed
 
 lemma (in residues_prime) totient_eq: "totient p = p - 1"
   using totient_eq by (simp add: res_prime_units_eq)
 
 lemma (in residues) euler_theorem:
   assumes "coprime a m"
   shows "[a ^ totient (nat m) = 1] (mod m)"
 proof -
   have "a ^ totient (nat m) mod m = 1 mod m"
     by (metis assms finite_Units m_gt_one mod_in_res_units one_cong totient_eq pow_cong units_power_order_eq_one)
   then show ?thesis
     using res_eq_to_cong by blast
 qed
 
 lemma euler_theorem:
   fixes a m :: nat
   assumes "coprime a m"
   shows "[a ^ totient m = 1] (mod m)"
 proof (cases "m = 0 \<or> m = 1")
   case True
   then show ?thesis by auto
 next
   case False
   with assms show ?thesis
     using residues.euler_theorem [of "int m" "int a"] cong_int_iff
     by (auto simp add: residues_def gcd_int_def) fastforce
 qed
 
 lemma fermat_theorem:
   fixes p a :: nat
   assumes "prime p" and "\<not> p dvd a"
   shows "[a ^ (p - 1) = 1] (mod p)"
 proof -
   from assms prime_imp_coprime [of p a] have "coprime a p"
     by (auto simp add: ac_simps)
   then have "[a ^ totient p = 1] (mod p)"
      by (rule euler_theorem)
   also have "totient p = p - 1"
     by (rule totient_prime) (rule assms)
   finally show ?thesis .
 qed
 
 
 subsection \<open>Wilson's theorem\<close>
 
 lemma (in field) inv_pair_lemma: "x \<in> Units R \<Longrightarrow> y \<in> Units R \<Longrightarrow>
     {x, inv x} \<noteq> {y, inv y} \<Longrightarrow> {x, inv x} \<inter> {y, inv y} = {}"
   apply auto
   apply (metis Units_inv_inv)+
   done
 
 lemma (in residues_prime) wilson_theorem1:
   assumes a: "p > 2"
   shows "[fact (p - 1) = (-1::int)] (mod p)"
 proof -
   let ?Inverse_Pairs = "{{x, inv x}| x. x \<in> Units R - {\<one>, \<ominus> \<one>}}"
   have UR: "Units R = {\<one>, \<ominus> \<one>} \<union> \<Union>?Inverse_Pairs"
     by auto
   have "(\<Otimes>i\<in>Units R. i) = (\<Otimes>i\<in>{\<one>, \<ominus> \<one>}. i) \<otimes> (\<Otimes>i\<in>\<Union>?Inverse_Pairs. i)"
     apply (subst UR)
     apply (subst finprod_Un_disjoint)
          apply (auto intro: funcsetI)
     using inv_one apply auto[1]
     using inv_eq_neg_one_eq apply auto
     done
   also have "(\<Otimes>i\<in>{\<one>, \<ominus> \<one>}. i) = \<ominus> \<one>"
     apply (subst finprod_insert)
         apply auto
     apply (frule one_eq_neg_one)
     using a apply force
     done
   also have "(\<Otimes>i\<in>(\<Union>?Inverse_Pairs). i) = (\<Otimes>A\<in>?Inverse_Pairs. (\<Otimes>y\<in>A. y))"
     apply (subst finprod_Union_disjoint)
        apply (auto simp: pairwise_def disjnt_def)
      apply (metis Units_inv_inv)+
     done
   also have "\<dots> = \<one>"
     apply (rule finprod_one_eqI)
      apply auto
     apply (subst finprod_insert)
         apply auto
     apply (metis inv_eq_self)
     done
   finally have "(\<Otimes>i\<in>Units R. i) = \<ominus> \<one>"
     by simp
   also have "(\<Otimes>i\<in>Units R. i) = (\<Otimes>i\<in>Units R. i mod p)"
     by (rule finprod_cong') (auto simp: res_units_eq)
   also have "\<dots> = (\<Prod>i\<in>Units R. i) mod p"
     by (rule prod_cong) auto
   also have "\<dots> = fact (p - 1) mod p"
     apply (simp add: fact_prod)
     using assms
     apply (subst res_prime_units_eq)
     apply (simp add: int_prod zmod_int prod_int_eq)
     done
   finally have "fact (p - 1) mod p = \<ominus> \<one>" .
   then show ?thesis
     by (simp add: cong_def res_neg_eq res_one_eq zmod_int)
 qed
 
 lemma wilson_theorem:
   assumes "prime p"
   shows "[fact (p - 1) = - 1] (mod p)"
 proof (cases "p = 2")
   case True
   then show ?thesis
     by (simp add: cong_def fact_prod)
 next
   case False
   then show ?thesis
     using assms prime_ge_2_nat
     by (metis residues_prime.wilson_theorem1 residues_prime.intro le_eq_less_or_eq)
 qed
 
 text \<open>
   This result can be transferred to the multiplicative group of
   \<open>\<int>/p\<int>\<close> for \<open>p\<close> prime.\<close>
 
 lemma mod_nat_int_pow_eq:
   fixes n :: nat and p a :: int
   shows "a \<ge> 0 \<Longrightarrow> p \<ge> 0 \<Longrightarrow> (nat a ^ n) mod (nat p) = nat ((a ^ n) mod p)"
   by (simp add: int_one_le_iff_zero_less nat_mod_distrib order_less_imp_le nat_power_eq[symmetric])
 
 theorem residue_prime_mult_group_has_gen:
  fixes p :: nat
  assumes prime_p : "prime p"
  shows "\<exists>a \<in> {1 .. p - 1}. {1 .. p - 1} = {a^i mod p|i . i \<in> UNIV}"
 proof -
   have "p \<ge> 2"
     using prime_gt_1_nat[OF prime_p] by simp
   interpret R: residues_prime p "residue_ring p"
     by (simp add: residues_prime_def prime_p)
   have car: "carrier (residue_ring (int p)) - {\<zero>\<^bsub>residue_ring (int p)\<^esub>} = {1 .. int p - 1}"
     by (auto simp add: R.zero_cong R.res_carrier_eq)
 
   have "x [^]\<^bsub>residue_ring (int p)\<^esub> i = x ^ i mod (int p)"
     if "x \<in> {1 .. int p - 1}" for x and i :: nat
     using that R.pow_cong[of x i] by auto
   moreover
   obtain a where a: "a \<in> {1 .. int p - 1}"
     and a_gen: "{1 .. int p - 1} = {a[^]\<^bsub>residue_ring (int p)\<^esub>i|i::nat . i \<in> UNIV}"
     using field.finite_field_mult_group_has_gen[OF R.is_field]
     by (auto simp add: car[symmetric] carrier_mult_of)
   moreover
   have "nat ` {1 .. int p - 1} = {1 .. p - 1}" (is "?L = ?R")
   proof
     have "n \<in> ?R" if "n \<in> ?L" for n
       using that \<open>p\<ge>2\<close> by force
     then show "?L \<subseteq> ?R" by blast
     have "n \<in> ?L" if "n \<in> ?R" for n
       using that \<open>p\<ge>2\<close> by (auto intro: rev_image_eqI [of "int n"])
     then show "?R \<subseteq> ?L" by blast
   qed
   moreover
   have "nat ` {a^i mod (int p) | i::nat. i \<in> UNIV} = {nat a^i mod p | i . i \<in> UNIV}" (is "?L = ?R")
   proof
     have "x \<in> ?R" if "x \<in> ?L" for x
     proof -
       from that obtain i where i: "x = nat (a^i mod (int p))"
         by blast
       then have "x = nat a ^ i mod p"
         using mod_nat_int_pow_eq[of a "int p" i] a \<open>p\<ge>2\<close> by auto
       with i show ?thesis by blast
     qed
     then show "?L \<subseteq> ?R" by blast
     have "x \<in> ?L" if "x \<in> ?R" for x
     proof -
       from that obtain i where i: "x = nat a^i mod p"
         by blast
       with mod_nat_int_pow_eq[of a "int p" i] a \<open>p\<ge>2\<close> show ?thesis
         by auto
     qed
     then show "?R \<subseteq> ?L" by blast
   qed
   ultimately have "{1 .. p - 1} = {nat a^i mod p | i. i \<in> UNIV}"
     by presburger
   moreover from a have "nat a \<in> {1 .. p - 1}" by force
   ultimately show ?thesis ..
 qed
 
 
 subsection \<open>Upper bound for the number of $n$-th roots\<close>
 
 lemma roots_mod_prime_bound:
   fixes n c p :: nat
   assumes "prime p" "n > 0"
   defines "A \<equiv> {x\<in>{..<p}. [x ^ n = c] (mod p)}"
   shows   "card A \<le> n"
 proof -
   define R where "R = residue_ring (int p)"
-  term monom
   from assms(1) interpret residues_prime p R
     by unfold_locales (simp_all add: R_def)
   interpret R: UP_domain R "UP R" by (unfold_locales)
 
   let ?f = "UnivPoly.monom (UP R) \<one>\<^bsub>R\<^esub> n \<ominus>\<^bsub>(UP R)\<^esub> UnivPoly.monom (UP R) (int (c mod p)) 0"
   have in_carrier: "int (c mod p) \<in> carrier R"
     using prime_gt_1_nat[OF assms(1)] by (simp add: R_def residue_ring_def)
   
   have "deg R ?f = n"
     using assms in_carrier by (simp add: R.deg_minus_eq)
   hence f_not_zero: "?f \<noteq> \<zero>\<^bsub>UP R\<^esub>" using assms by (auto simp add : R.deg_nzero_nzero)
   have roots_bound: "finite {a \<in> carrier R. UnivPoly.eval R R id a ?f = \<zero>\<^bsub>R\<^esub>} \<and>
                      card {a \<in> carrier R. UnivPoly.eval R R id a ?f = \<zero>\<^bsub>R\<^esub>} \<le> deg R ?f"
                     using finite in_carrier by (intro R.roots_bound[OF _ f_not_zero]) simp
   have subs: "{x \<in> carrier R. x [^]\<^bsub>R\<^esub> n = int (c mod p)} \<subseteq>
                 {a \<in> carrier R. UnivPoly.eval R R id a ?f = \<zero>\<^bsub>R\<^esub>}"
     using in_carrier by (auto simp: R.evalRR_simps)
   then have "card {x \<in> carrier R. x [^]\<^bsub>R\<^esub> n = int (c mod p)} \<le>
                card {a \<in> carrier R. UnivPoly.eval R R id a ?f = \<zero>\<^bsub>R\<^esub>}"
     using finite by (intro card_mono) auto
   also have "\<dots> \<le> n"
     using \<open>deg R ?f = n\<close> roots_bound by linarith
   also {
     fix x assume "x \<in> carrier R"
     hence "x [^]\<^bsub>R\<^esub> n = (x ^ n) mod (int p)"
       by (subst pow_cong [symmetric]) (auto simp: R_def residue_ring_def)
   }
   hence "{x \<in> carrier R. x [^]\<^bsub>R\<^esub> n = int (c mod p)} = {x \<in> carrier R. [x ^ n = int c] (mod p)}"
     by (fastforce simp: cong_def zmod_int)
   also have "bij_betw int A {x \<in> carrier R. [x ^ n = int c] (mod p)}"
     by (rule bij_betwI[of int _ _ nat])
        (use cong_int_iff in \<open>force simp: R_def residue_ring_def A_def\<close>)+
   from bij_betw_same_card[OF this] have "card {x \<in> carrier R. [x ^ n = int c] (mod p)} = card A" ..
   finally show ?thesis .
 qed
 
 
 end