author wenzelm Tue, 08 Aug 2017 22:33:21 +0200 changeset 66380 96ff0eb8294a parent 66379 6392766f3c25 child 66381 429b55991197
misc tuning and modernization;
```--- a/src/HOL/Number_Theory/Cong.thy	Tue Aug 08 22:13:05 2017 +0200
+++ b/src/HOL/Number_Theory/Cong.thy	Tue Aug 08 22:33:21 2017 +0200
@@ -1,6 +1,9 @@
(*  Title:      HOL/Number_Theory/Cong.thy
-    Authors:    Christophe Tabacznyj, Lawrence C. Paulson, Amine Chaieb,
-                Thomas M. Rasmussen, Jeremy Avigad
+    Author:     Christophe Tabacznyj
+    Author:     Lawrence C. Paulson
+    Author:     Amine Chaieb
+    Author:     Thomas M. Rasmussen

Defines congruence (notation: [x = y] (mod z)) for natural numbers and
integers.
@@ -26,12 +29,13 @@
section \<open>Congruence\<close>

theory Cong
-imports "~~/src/HOL/Computational_Algebra/Primes"
+  imports "~~/src/HOL/Computational_Algebra/Primes"
begin

subsection \<open>Turn off \<open>One_nat_def\<close>\<close>

-lemma power_eq_one_eq_nat [simp]: "((x::nat)^m = 1) = (m = 0 | x = 1)"
+lemma power_eq_one_eq_nat [simp]: "x^m = 1 \<longleftrightarrow> m = 0 \<or> x = 1"
+  for x m :: nat
by (induct m) auto

declare mod_pos_pos_trivial [simp]
@@ -40,7 +44,7 @@
subsection \<open>Main definitions\<close>

class cong =
-  fixes cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool" ("(1[_ = _] '(()mod _'))")
+  fixes cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"  ("(1[_ = _] '(()mod _'))")
begin

abbreviation notcong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"  ("(1[_ \<noteq> _] '(()mod _'))")
@@ -48,26 +52,27 @@

end

-(* definitions for the natural numbers *)
+
+subsubsection \<open>Definitions for the natural numbers\<close>

instantiation nat :: cong
begin

definition cong_nat :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool"
-  where "cong_nat x y m = ((x mod m) = (y mod m))"
+  where "cong_nat x y m \<longleftrightarrow> x mod m = y mod m"

instance ..

end

-(* definitions for the integers *)
+subsubsection \<open>Definitions for the integers\<close>

instantiation int :: cong
begin

definition cong_int :: "int \<Rightarrow> int \<Rightarrow> int \<Rightarrow> bool"
-  where "cong_int x y m = ((x mod m) = (y mod m))"
+  where "cong_int x y m \<longleftrightarrow> x mod m = y mod m"

instance ..

@@ -78,253 +83,259 @@

lemma transfer_nat_int_cong:
-  "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> m >= 0 \<Longrightarrow>
-    ([(nat x) = (nat y)] (mod (nat m))) = ([x = y] (mod m))"
+  "x \<ge> 0 \<Longrightarrow> y \<ge> 0 \<Longrightarrow> m \<ge> 0 \<Longrightarrow> [nat x = nat y] (mod (nat m)) \<longleftrightarrow> [x = y] (mod m)"
+  for x y m :: int
unfolding cong_int_def cong_nat_def
by (metis Divides.transfer_int_nat_functions(2) nat_0_le nat_mod_distrib)

-    transfer_nat_int_cong]
+declare transfer_morphism_nat_int [transfer add return: transfer_nat_int_cong]

-lemma transfer_int_nat_cong:
-  "[(int x) = (int y)] (mod (int m)) = [x = y] (mod m)"
-  apply (auto simp add: cong_int_def cong_nat_def)
-  apply (auto simp add: zmod_int [symmetric])
-  done
+lemma transfer_int_nat_cong: "[int x = int y] (mod (int m)) = [x = y] (mod m)"
+  by (auto simp add: cong_int_def cong_nat_def) (auto simp add: zmod_int [symmetric])

-    transfer_int_nat_cong]
+declare transfer_morphism_int_nat [transfer add return: transfer_int_nat_cong]

subsection \<open>Congruence\<close>

(* was zcong_0, etc. *)
-lemma cong_0_nat [simp, presburger]: "([(a::nat) = b] (mod 0)) = (a = b)"
-  unfolding cong_nat_def by auto
+lemma cong_0_nat [simp, presburger]: "[a = b] (mod 0) \<longleftrightarrow> a = b"
+  for a b :: nat
+  by (auto simp: cong_nat_def)

-lemma cong_0_int [simp, presburger]: "([(a::int) = b] (mod 0)) = (a = b)"
-  unfolding cong_int_def by auto
+lemma cong_0_int [simp, presburger]: "[a = b] (mod 0) \<longleftrightarrow> a = b"
+  for a b :: int
+  by (auto simp: cong_int_def)

-lemma cong_1_nat [simp, presburger]: "[(a::nat) = b] (mod 1)"
-  unfolding cong_nat_def by auto
+lemma cong_1_nat [simp, presburger]: "[a = b] (mod 1)"
+  for a b :: nat
+  by (auto simp: cong_nat_def)

-lemma cong_Suc_0_nat [simp, presburger]: "[(a::nat) = b] (mod Suc 0)"
-  unfolding cong_nat_def by auto
+lemma cong_Suc_0_nat [simp, presburger]: "[a = b] (mod Suc 0)"
+  for a b :: nat
+  by (auto simp: cong_nat_def)

-lemma cong_1_int [simp, presburger]: "[(a::int) = b] (mod 1)"
-  unfolding cong_int_def by auto
+lemma cong_1_int [simp, presburger]: "[a = b] (mod 1)"
+  for a b :: int
+  by (auto simp: cong_int_def)

-lemma cong_refl_nat [simp]: "[(k::nat) = k] (mod m)"
-  unfolding cong_nat_def by auto
+lemma cong_refl_nat [simp]: "[k = k] (mod m)"
+  for k :: nat
+  by (auto simp: cong_nat_def)

-lemma cong_refl_int [simp]: "[(k::int) = k] (mod m)"
-  unfolding cong_int_def by auto
+lemma cong_refl_int [simp]: "[k = k] (mod m)"
+  for k :: int
+  by (auto simp: cong_int_def)

-lemma cong_sym_nat: "[(a::nat) = b] (mod m) \<Longrightarrow> [b = a] (mod m)"
-  unfolding cong_nat_def by auto
+lemma cong_sym_nat: "[a = b] (mod m) \<Longrightarrow> [b = a] (mod m)"
+  for a b :: nat
+  by (auto simp: cong_nat_def)

-lemma cong_sym_int: "[(a::int) = b] (mod m) \<Longrightarrow> [b = a] (mod m)"
-  unfolding cong_int_def by auto
+lemma cong_sym_int: "[a = b] (mod m) \<Longrightarrow> [b = a] (mod m)"
+  for a b :: int
+  by (auto simp: cong_int_def)

-lemma cong_sym_eq_nat: "[(a::nat) = b] (mod m) = [b = a] (mod m)"
-  unfolding cong_nat_def by auto
+lemma cong_sym_eq_nat: "[a = b] (mod m) = [b = a] (mod m)"
+  for a b :: nat
+  by (auto simp: cong_nat_def)

-lemma cong_sym_eq_int: "[(a::int) = b] (mod m) = [b = a] (mod m)"
-  unfolding cong_int_def by auto
+lemma cong_sym_eq_int: "[a = b] (mod m) = [b = a] (mod m)"
+  for a b :: int
+  by (auto simp: cong_int_def)

-lemma cong_trans_nat [trans]:
-    "[(a::nat) = b] (mod m) \<Longrightarrow> [b = c] (mod m) \<Longrightarrow> [a = c] (mod m)"
-  unfolding cong_nat_def by auto
+lemma cong_trans_nat [trans]: "[a = b] (mod m) \<Longrightarrow> [b = c] (mod m) \<Longrightarrow> [a = c] (mod m)"
+  for a b c :: nat
+  by (auto simp: cong_nat_def)

-lemma cong_trans_int [trans]:
-    "[(a::int) = b] (mod m) \<Longrightarrow> [b = c] (mod m) \<Longrightarrow> [a = c] (mod m)"
-  unfolding cong_int_def by auto
+lemma cong_trans_int [trans]: "[a = b] (mod m) \<Longrightarrow> [b = c] (mod m) \<Longrightarrow> [a = c] (mod m)"
+  for a b c :: int
+  by (auto simp: cong_int_def)

-    "[(a::nat) = b] (mod m) \<Longrightarrow> [c = d] (mod m) \<Longrightarrow> [a + c = b + d] (mod m)"
-  unfolding cong_nat_def  by (metis mod_add_cong)
+lemma cong_add_nat: "[a = b] (mod m) \<Longrightarrow> [c = d] (mod m) \<Longrightarrow> [a + c = b + d] (mod m)"
+  for a b c :: nat
+  unfolding cong_nat_def by (metis mod_add_cong)

-    "[(a::int) = b] (mod m) \<Longrightarrow> [c = d] (mod m) \<Longrightarrow> [a + c = b + d] (mod m)"
-  unfolding cong_int_def  by (metis mod_add_cong)
+lemma cong_add_int: "[a = b] (mod m) \<Longrightarrow> [c = d] (mod m) \<Longrightarrow> [a + c = b + d] (mod m)"
+  for a b c :: int
+  unfolding cong_int_def by (metis mod_add_cong)

-lemma cong_diff_int:
-    "[(a::int) = b] (mod m) \<Longrightarrow> [c = d] (mod m) \<Longrightarrow> [a - c = b - d] (mod m)"
-  unfolding cong_int_def  by (metis mod_diff_cong)
+lemma cong_diff_int: "[a = b] (mod m) \<Longrightarrow> [c = d] (mod m) \<Longrightarrow> [a - c = b - d] (mod m)"
+  for a b c :: int
+  unfolding cong_int_def by (metis mod_diff_cong)

lemma cong_diff_aux_int:
-  "[(a::int) = b] (mod m) \<Longrightarrow> [c = d] (mod m) \<Longrightarrow>
-   (a::int) >= c \<Longrightarrow> b >= d \<Longrightarrow> [tsub a c = tsub b d] (mod m)"
+  "[a = b] (mod m) \<Longrightarrow> [c = d] (mod m) \<Longrightarrow>
+    a \<ge> c \<Longrightarrow> b \<ge> d \<Longrightarrow> [tsub a c = tsub b d] (mod m)"
+  for a b c d :: int
by (metis cong_diff_int tsub_eq)

lemma cong_diff_nat:
-  assumes"[a = b] (mod m)" "[c = d] (mod m)" "(a::nat) >= c" "b >= d"
+  fixes a b c d :: nat
+  assumes "[a = b] (mod m)" "[c = d] (mod m)" "a \<ge> c" "b \<ge> d"
shows "[a - c = b - d] (mod m)"
using assms by (rule cong_diff_aux_int [transferred])

-lemma cong_mult_nat:
-    "[(a::nat) = b] (mod m) \<Longrightarrow> [c = d] (mod m) \<Longrightarrow> [a * c = b * d] (mod m)"
-  unfolding cong_nat_def  by (metis mod_mult_cong)
+lemma cong_mult_nat: "[a = b] (mod m) \<Longrightarrow> [c = d] (mod m) \<Longrightarrow> [a * c = b * d] (mod m)"
+  for a b c d :: nat
+  unfolding cong_nat_def  by (metis mod_mult_cong)

-lemma cong_mult_int:
-    "[(a::int) = b] (mod m) \<Longrightarrow> [c = d] (mod m) \<Longrightarrow> [a * c = b * d] (mod m)"
-  unfolding cong_int_def  by (metis mod_mult_cong)
-
-lemma cong_exp_nat: "[(x::nat) = y] (mod n) \<Longrightarrow> [x^k = y^k] (mod n)"
-  by (induct k) (auto simp add: cong_mult_nat)
+lemma cong_mult_int: "[a = b] (mod m) \<Longrightarrow> [c = d] (mod m) \<Longrightarrow> [a * c = b * d] (mod m)"
+  for a b c d :: int
+  unfolding cong_int_def  by (metis mod_mult_cong)

-lemma cong_exp_int: "[(x::int) = y] (mod n) \<Longrightarrow> [x^k = y^k] (mod n)"
-  by (induct k) (auto simp add: cong_mult_int)
+lemma cong_exp_nat: "[x = y] (mod n) \<Longrightarrow> [x^k = y^k] (mod n)"
+  for x y :: nat
+  by (induct k) (auto simp: cong_mult_nat)

-lemma cong_sum_nat [rule_format]:
-    "(\<forall>x\<in>A. [((f x)::nat) = g x] (mod m)) \<longrightarrow>
-      [(\<Sum>x\<in>A. f x) = (\<Sum>x\<in>A. g x)] (mod m)"
-  apply (cases "finite A")
-  apply (induct set: finite)
-  done
+lemma cong_exp_int: "[x = y] (mod n) \<Longrightarrow> [x^k = y^k] (mod n)"
+  for x y :: int
+  by (induct k) (auto simp: cong_mult_int)

-lemma cong_sum_int [rule_format]:
-    "(\<forall>x\<in>A. [((f x)::int) = g x] (mod m)) \<longrightarrow>
-      [(\<Sum>x\<in>A. f x) = (\<Sum>x\<in>A. g x)] (mod m)"
-  apply (cases "finite A")
-  apply (induct set: finite)
-  done
+lemma cong_sum_nat: "(\<And>x. x \<in> A \<Longrightarrow> [f x = g x] (mod m)) \<Longrightarrow> [(\<Sum>x\<in>A. f x) = (\<Sum>x\<in>A. g x)] (mod m)"
+  for f g :: "'a \<Rightarrow> nat"
+  by (induct A rule: infinite_finite_induct) (auto intro: cong_add_nat)
+
+lemma cong_sum_int: "(\<And>x. x \<in> A \<Longrightarrow> [f x = g x] (mod m)) \<Longrightarrow> [(\<Sum>x\<in>A. f x) = (\<Sum>x\<in>A. g x)] (mod m)"
+  for f g :: "'a \<Rightarrow> int"
+  by (induct A rule: infinite_finite_induct) (auto intro: cong_add_int)

-lemma cong_prod_nat [rule_format]:
-    "(\<forall>x\<in>A. [((f x)::nat) = g x] (mod m)) \<longrightarrow>
-      [(\<Prod>x\<in>A. f x) = (\<Prod>x\<in>A. g x)] (mod m)"
-  apply (cases "finite A")
-  apply (induct set: finite)
-  apply (auto intro: cong_mult_nat)
-  done
+lemma cong_prod_nat: "(\<And>x. x \<in> A \<Longrightarrow> [f x = g x] (mod m)) \<Longrightarrow> [(\<Prod>x\<in>A. f x) = (\<Prod>x\<in>A. g x)] (mod m)"
+  for f g :: "'a \<Rightarrow> nat"
+  by (induct A rule: infinite_finite_induct) (auto intro: cong_mult_nat)

-lemma cong_prod_int [rule_format]:
-    "(\<forall>x\<in>A. [((f x)::int) = g x] (mod m)) \<longrightarrow>
-      [(\<Prod>x\<in>A. f x) = (\<Prod>x\<in>A. g x)] (mod m)"
-  apply (cases "finite A")
-  apply (induct set: finite)
-  apply (auto intro: cong_mult_int)
-  done
+lemma cong_prod_int: "(\<And>x. x \<in> A \<Longrightarrow> [f x = g x] (mod m)) \<Longrightarrow> [(\<Prod>x\<in>A. f x) = (\<Prod>x\<in>A. g x)] (mod m)"
+  for f g :: "'a \<Rightarrow> int"
+  by (induct A rule: infinite_finite_induct) (auto intro: cong_mult_int)

-lemma cong_scalar_nat: "[(a::nat)= b] (mod m) \<Longrightarrow> [a * k = b * k] (mod m)"
+lemma cong_scalar_nat: "[a = b] (mod m) \<Longrightarrow> [a * k = b * k] (mod m)"
+  for a b k :: nat
by (rule cong_mult_nat) simp_all

-lemma cong_scalar_int: "[(a::int)= b] (mod m) \<Longrightarrow> [a * k = b * k] (mod m)"
+lemma cong_scalar_int: "[a = b] (mod m) \<Longrightarrow> [a * k = b * k] (mod m)"
+  for a b k :: int
by (rule cong_mult_int) simp_all

-lemma cong_scalar2_nat: "[(a::nat)= b] (mod m) \<Longrightarrow> [k * a = k * b] (mod m)"
+lemma cong_scalar2_nat: "[a = b] (mod m) \<Longrightarrow> [k * a = k * b] (mod m)"
+  for a b k :: nat
by (rule cong_mult_nat) simp_all

-lemma cong_scalar2_int: "[(a::int)= b] (mod m) \<Longrightarrow> [k * a = k * b] (mod m)"
+lemma cong_scalar2_int: "[a = b] (mod m) \<Longrightarrow> [k * a = k * b] (mod m)"
+  for a b k :: int
by (rule cong_mult_int) simp_all

-lemma cong_mult_self_nat: "[(a::nat) * m = 0] (mod m)"
-  unfolding cong_nat_def by auto
+lemma cong_mult_self_nat: "[a * m = 0] (mod m)"
+  for a m :: nat
+  by (auto simp: cong_nat_def)

-lemma cong_mult_self_int: "[(a::int) * m = 0] (mod m)"
-  unfolding cong_int_def by auto
+lemma cong_mult_self_int: "[a * m = 0] (mod m)"
+  for a m :: int
+  by (auto simp: cong_int_def)

-lemma cong_eq_diff_cong_0_int: "[(a::int) = b] (mod m) = [a - b = 0] (mod m)"
+lemma cong_eq_diff_cong_0_int: "[a = b] (mod m) = [a - b = 0] (mod m)"
+  for a b :: int

-lemma cong_eq_diff_cong_0_aux_int: "a >= b \<Longrightarrow>
-    [(a::int) = b] (mod m) = [tsub a b = 0] (mod m)"
+lemma cong_eq_diff_cong_0_aux_int: "a \<ge> b \<Longrightarrow> [a = b] (mod m) = [tsub a b = 0] (mod m)"
+  for a b :: int
by (subst tsub_eq, assumption, rule cong_eq_diff_cong_0_int)

lemma cong_eq_diff_cong_0_nat:
-  assumes "(a::nat) >= b"
+  fixes a b :: nat
+  assumes "a \<ge> b"
shows "[a = b] (mod m) = [a - b = 0] (mod m)"
using assms by (rule cong_eq_diff_cong_0_aux_int [transferred])

lemma cong_diff_cong_0'_nat:
-  "[(x::nat) = y] (mod n) \<longleftrightarrow>
-    (if x <= y then [y - x = 0] (mod n) else [x - y = 0] (mod n))"
+  "[x = y] (mod n) \<longleftrightarrow> (if x \<le> y then [y - x = 0] (mod n) else [x - y = 0] (mod n))"
+  for x y :: nat
by (metis cong_eq_diff_cong_0_nat cong_sym_nat nat_le_linear)

-lemma cong_altdef_nat: "(a::nat) >= b \<Longrightarrow> [a = b] (mod m) = (m dvd (a - b))"
+lemma cong_altdef_nat: "a \<ge> b \<Longrightarrow> [a = b] (mod m) \<longleftrightarrow> m dvd (a - b)"
+  for a b :: nat
apply (subst cong_eq_diff_cong_0_nat, assumption)
apply (unfold cong_nat_def)
done

-lemma cong_altdef_int: "[(a::int) = b] (mod m) = (m dvd (a - b))"
+lemma cong_altdef_int: "[a = b] (mod m) \<longleftrightarrow> m dvd (a - b)"
+  for a b :: int
by (metis cong_int_def mod_eq_dvd_iff)

-lemma cong_abs_int: "[(x::int) = y] (mod abs m) = [x = y] (mod m)"
+lemma cong_abs_int: "[x = y] (mod abs m) \<longleftrightarrow> [x = y] (mod m)"
+  for x y :: int

lemma cong_square_int:
-  fixes a::int
-  shows "\<lbrakk> prime p; 0 < a; [a * a = 1] (mod p) \<rbrakk>
-    \<Longrightarrow> [a = 1] (mod p) \<or> [a = - 1] (mod p)"
+  "prime p \<Longrightarrow> 0 < a \<Longrightarrow> [a * a = 1] (mod p) \<Longrightarrow> [a = 1] (mod p) \<or> [a = - 1] (mod p)"
+  for a :: int
apply (simp only: cong_altdef_int)
apply (subst prime_dvd_mult_eq_int [symmetric], assumption)
done

-lemma cong_mult_rcancel_int:
-    "coprime k (m::int) \<Longrightarrow> [a * k = b * k] (mod m) = [a = b] (mod m)"
+lemma cong_mult_rcancel_int: "coprime k m \<Longrightarrow> [a * k = b * k] (mod m) = [a = b] (mod m)"
+  for a k m :: int
by (metis cong_altdef_int left_diff_distrib coprime_dvd_mult_iff gcd.commute)

-lemma cong_mult_rcancel_nat:
-    "coprime k (m::nat) \<Longrightarrow> [a * k = b * k] (mod m) = [a = b] (mod m)"
+lemma cong_mult_rcancel_nat: "coprime k m \<Longrightarrow> [a * k = b * k] (mod m) = [a = b] (mod m)"
+  for a k m :: nat
by (metis cong_mult_rcancel_int [transferred])

-lemma cong_mult_lcancel_nat:
-    "coprime k (m::nat) \<Longrightarrow> [k * a = k * b ] (mod m) = [a = b] (mod m)"
+lemma cong_mult_lcancel_nat: "coprime k m \<Longrightarrow> [k * a = k * b ] (mod m) = [a = b] (mod m)"
+  for a k m :: nat

-lemma cong_mult_lcancel_int:
-    "coprime k (m::int) \<Longrightarrow> [k * a = k * b] (mod m) = [a = b] (mod m)"
+lemma cong_mult_lcancel_int: "coprime k m \<Longrightarrow> [k * a = k * b] (mod m) = [a = b] (mod m)"
+  for a k m :: int

(* was zcong_zgcd_zmult_zmod *)
lemma coprime_cong_mult_int:
-  "[(a::int) = b] (mod m) \<Longrightarrow> [a = b] (mod n) \<Longrightarrow> coprime m n
-    \<Longrightarrow> [a = b] (mod m * n)"
-by (metis divides_mult cong_altdef_int)
+  "[a = b] (mod m) \<Longrightarrow> [a = b] (mod n) \<Longrightarrow> coprime m n \<Longrightarrow> [a = b] (mod m * n)"
+  for a b :: int
+  by (metis divides_mult cong_altdef_int)

lemma coprime_cong_mult_nat:
-  assumes "[(a::nat) = b] (mod m)" and "[a = b] (mod n)" and "coprime m n"
-  shows "[a = b] (mod m * n)"
-  by (metis assms coprime_cong_mult_int [transferred])
+  "[a = b] (mod m) \<Longrightarrow> [a = b] (mod n) \<Longrightarrow> coprime m n \<Longrightarrow> [a = b] (mod m * n)"
+  for a b :: nat
+  by (metis coprime_cong_mult_int [transferred])

-lemma cong_less_imp_eq_nat: "0 \<le> (a::nat) \<Longrightarrow>
-    a < m \<Longrightarrow> 0 \<le> b \<Longrightarrow> b < m \<Longrightarrow> [a = b] (mod m) \<Longrightarrow> a = b"
+lemma cong_less_imp_eq_nat: "0 \<le> a \<Longrightarrow> a < m \<Longrightarrow> 0 \<le> b \<Longrightarrow> b < m \<Longrightarrow> [a = b] (mod m) \<Longrightarrow> a = b"
+  for a b :: nat

-lemma cong_less_imp_eq_int: "0 \<le> (a::int) \<Longrightarrow>
-    a < m \<Longrightarrow> 0 \<le> b \<Longrightarrow> b < m \<Longrightarrow> [a = b] (mod m) \<Longrightarrow> a = b"
+lemma cong_less_imp_eq_int: "0 \<le> a \<Longrightarrow> a < m \<Longrightarrow> 0 \<le> b \<Longrightarrow> b < m \<Longrightarrow> [a = b] (mod m) \<Longrightarrow> a = b"
+  for a b :: int

-lemma cong_less_unique_nat:
-    "0 < (m::nat) \<Longrightarrow> (\<exists>!b. 0 \<le> b \<and> b < m \<and> [a = b] (mod m))"
+lemma cong_less_unique_nat: "0 < m \<Longrightarrow> (\<exists>!b. 0 \<le> b \<and> b < m \<and> [a = b] (mod m))"
+  for a m :: nat
by (auto simp: cong_nat_def) (metis mod_less_divisor mod_mod_trivial)

-lemma cong_less_unique_int:
-    "0 < (m::int) \<Longrightarrow> (\<exists>!b. 0 \<le> b \<and> b < m \<and> [a = b] (mod m))"
+lemma cong_less_unique_int: "0 < m \<Longrightarrow> (\<exists>!b. 0 \<le> b \<and> b < m \<and> [a = b] (mod m))"
+  for a m :: int
by (auto simp: cong_int_def)  (metis mod_mod_trivial pos_mod_conj)

-lemma cong_iff_lin_int: "([(a::int) = b] (mod m)) = (\<exists>k. b = a + m * k)"
+lemma cong_iff_lin_int: "[a = b] (mod m) \<longleftrightarrow> (\<exists>k. b = a + m * k)"
+  for a b :: int
apply (auto simp add: cong_altdef_int dvd_def)
apply (rule_tac [!] x = "-k" in exI, auto)
done

-lemma cong_iff_lin_nat:
-   "([(a::nat) = b] (mod m)) \<longleftrightarrow> (\<exists>k1 k2. b + k1 * m = a + k2 * m)" (is "?lhs = ?rhs")
-proof (rule iffI)
-  assume eqm: ?lhs
+lemma cong_iff_lin_nat: "([a = b] (mod m)) \<longleftrightarrow> (\<exists>k1 k2. b + k1 * m = a + k2 * m)"
+  (is "?lhs = ?rhs")
+  for a b :: nat
+proof
+  assume ?lhs
show ?rhs
proof (cases "b \<le> a")
case True
-    then show ?rhs using eqm
+    with \<open>?lhs\<close> show ?rhs
next
case False
-    then show ?rhs using eqm
+    with \<open>?lhs\<close> show ?rhs
apply (subst (asm) cong_sym_eq_nat)
apply (auto simp: cong_altdef_nat)
@@ -336,26 +347,32 @@
by (metis cong_nat_def mod_mult_self2 mult.commute)
qed

-lemma cong_gcd_eq_int: "[(a::int) = b] (mod m) \<Longrightarrow> gcd a m = gcd b m"
+lemma cong_gcd_eq_int: "[a = b] (mod m) \<Longrightarrow> gcd a m = gcd b m"
+  for a b :: int
by (metis cong_int_def gcd_red_int)

-lemma cong_gcd_eq_nat:
-    "[(a::nat) = b] (mod m) \<Longrightarrow>gcd a m = gcd b m"
+lemma cong_gcd_eq_nat: "[a = b] (mod m) \<Longrightarrow> gcd a m = gcd b m"
+  for a b :: nat
by (metis cong_gcd_eq_int [transferred])

-lemma cong_imp_coprime_nat: "[(a::nat) = b] (mod m) \<Longrightarrow> coprime a m \<Longrightarrow> coprime b m"
+lemma cong_imp_coprime_nat: "[a = b] (mod m) \<Longrightarrow> coprime a m \<Longrightarrow> coprime b m"
+  for a b :: nat

-lemma cong_imp_coprime_int: "[(a::int) = b] (mod m) \<Longrightarrow> coprime a m \<Longrightarrow> coprime b m"
+lemma cong_imp_coprime_int: "[a = b] (mod m) \<Longrightarrow> coprime a m \<Longrightarrow> coprime b m"
+  for a b :: int

-lemma cong_cong_mod_nat: "[(a::nat) = b] (mod m) = [a mod m = b mod m] (mod m)"
+lemma cong_cong_mod_nat: "[a = b] (mod m) \<longleftrightarrow> [a mod m = b mod m] (mod m)"
+  for a b :: nat

-lemma cong_cong_mod_int: "[(a::int) = b] (mod m) = [a mod m = b mod m] (mod m)"
+lemma cong_cong_mod_int: "[a = b] (mod m) \<longleftrightarrow> [a mod m = b mod m] (mod m)"
+  for a b :: int

-lemma cong_minus_int [iff]: "[(a::int) = b] (mod -m) = [a = b] (mod m)"
+lemma cong_minus_int [iff]: "[a = b] (mod - m) \<longleftrightarrow> [a = b] (mod m)"
+  for a b :: int
by (metis cong_iff_lin_int minus_equation_iff mult_minus_left mult_minus_right)

(*
@@ -375,115 +392,158 @@
done
*)

-    "[(a::nat) + x = a + y] (mod n) \<longleftrightarrow> [x = y] (mod n)"
+lemma cong_add_lcancel_nat: "[a + x = a + y] (mod n) \<longleftrightarrow> [x = y] (mod n)"
+  for a x y :: nat

-    "[(a::int) + x = a + y] (mod n) \<longleftrightarrow> [x = y] (mod n)"
+lemma cong_add_lcancel_int: "[a + x = a + y] (mod n) \<longleftrightarrow> [x = y] (mod n)"
+  for a x y :: int

-lemma cong_add_rcancel_nat: "[(x::nat) + a = y + a] (mod n) \<longleftrightarrow> [x = y] (mod n)"
+lemma cong_add_rcancel_nat: "[x + a = y + a] (mod n) \<longleftrightarrow> [x = y] (mod n)"
+  for a x y :: nat

-lemma cong_add_rcancel_int: "[(x::int) + a = y + a] (mod n) \<longleftrightarrow> [x = y] (mod n)"
+lemma cong_add_rcancel_int: "[x + a = y + a] (mod n) \<longleftrightarrow> [x = y] (mod n)"
+  for a x y :: int

-lemma cong_add_lcancel_0_nat: "[(a::nat) + x = a] (mod n) \<longleftrightarrow> [x = 0] (mod n)"
-
-lemma cong_add_lcancel_0_int: "[(a::int) + x = a] (mod n) \<longleftrightarrow> [x = 0] (mod n)"
-
-lemma cong_add_rcancel_0_nat: "[x + (a::nat) = a] (mod n) \<longleftrightarrow> [x = 0] (mod n)"
+lemma cong_add_lcancel_0_nat: "[a + x = a] (mod n) \<longleftrightarrow> [x = 0] (mod n)"
+  for a x :: nat

-lemma cong_add_rcancel_0_int: "[x + (a::int) = a] (mod n) \<longleftrightarrow> [x = 0] (mod n)"
+lemma cong_add_lcancel_0_int: "[a + x = a] (mod n) \<longleftrightarrow> [x = 0] (mod n)"
+  for a x :: int

-lemma cong_dvd_modulus_nat: "[(x::nat) = y] (mod m) \<Longrightarrow> n dvd m \<Longrightarrow>
-    [x = y] (mod n)"
+lemma cong_add_rcancel_0_nat: "[x + a = a] (mod n) \<longleftrightarrow> [x = 0] (mod n)"
+  for a x :: nat
+
+lemma cong_add_rcancel_0_int: "[x + a = a] (mod n) \<longleftrightarrow> [x = 0] (mod n)"
+  for a x :: int
+
+lemma cong_dvd_modulus_nat: "[x = y] (mod m) \<Longrightarrow> n dvd m \<Longrightarrow> [x = y] (mod n)"
+  for x y :: nat
apply (auto simp add: cong_iff_lin_nat dvd_def)
-  apply (rule_tac x="k1 * k" in exI)
-  apply (rule_tac x="k2 * k" in exI)
+  apply (rule_tac x= "k1 * k" in exI)
+  apply (rule_tac x= "k2 * k" in exI)
done

-lemma cong_dvd_modulus_int: "[(x::int) = y] (mod m) \<Longrightarrow> n dvd m \<Longrightarrow> [x = y] (mod n)"
+lemma cong_dvd_modulus_int: "[x = y] (mod m) \<Longrightarrow> n dvd m \<Longrightarrow> [x = y] (mod n)"
+  for x y :: int
by (auto simp add: cong_altdef_int dvd_def)

-lemma cong_dvd_eq_nat: "[(x::nat) = y] (mod n) \<Longrightarrow> n dvd x \<longleftrightarrow> n dvd y"
-  unfolding cong_nat_def by (auto simp add: dvd_eq_mod_eq_0)
+lemma cong_dvd_eq_nat: "[x = y] (mod n) \<Longrightarrow> n dvd x \<longleftrightarrow> n dvd y"
+  for x y :: nat
+  by (auto simp: cong_nat_def dvd_eq_mod_eq_0)

-lemma cong_dvd_eq_int: "[(x::int) = y] (mod n) \<Longrightarrow> n dvd x \<longleftrightarrow> n dvd y"
-  unfolding cong_int_def by (auto simp add: dvd_eq_mod_eq_0)
+lemma cong_dvd_eq_int: "[x = y] (mod n) \<Longrightarrow> n dvd x \<longleftrightarrow> n dvd y"
+  for x y :: int
+  by (auto simp: cong_int_def dvd_eq_mod_eq_0)

-lemma cong_mod_nat: "(n::nat) ~= 0 \<Longrightarrow> [a mod n = a] (mod n)"
+lemma cong_mod_nat: "n \<noteq> 0 \<Longrightarrow> [a mod n = a] (mod n)"
+  for a n :: nat

-lemma cong_mod_int: "(n::int) ~= 0 \<Longrightarrow> [a mod n = a] (mod n)"
+lemma cong_mod_int: "n \<noteq> 0 \<Longrightarrow> [a mod n = a] (mod n)"
+  for a n :: int

-lemma mod_mult_cong_nat: "(a::nat) ~= 0 \<Longrightarrow> b ~= 0
-    \<Longrightarrow> [x mod (a * b) = y] (mod a) \<longleftrightarrow> [x = y] (mod a)"
+lemma mod_mult_cong_nat: "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> [x mod (a * b) = y] (mod a) \<longleftrightarrow> [x = y] (mod a)"
+  for a b :: nat

-lemma neg_cong_int: "([(a::int) = b] (mod m)) = ([-a = -b] (mod m))"
+lemma neg_cong_int: "[a = b] (mod m) \<longleftrightarrow> [- a = - b] (mod m)"
+  for a b :: int
by (metis cong_int_def minus_minus mod_minus_cong)

-lemma cong_modulus_neg_int: "([(a::int) = b] (mod m)) = ([a = b] (mod -m))"
+lemma cong_modulus_neg_int: "[a = b] (mod m) \<longleftrightarrow> [a = b] (mod - m)"
+  for a b :: int

-lemma mod_mult_cong_int: "(a::int) ~= 0 \<Longrightarrow> b ~= 0
-    \<Longrightarrow> [x mod (a * b) = y] (mod a) \<longleftrightarrow> [x = y] (mod a)"
-  apply (subst (1 2) cong_modulus_neg_int)
-  apply (unfold cong_int_def)
-  apply (subgoal_tac "a * b = (-a * -b)")
-  apply (erule ssubst)
-  apply (subst zmod_zmult2_eq)
-  apply (metis mod_diff_left_eq mod_diff_right_eq mod_mult_self1_is_0 diff_zero)+
+lemma mod_mult_cong_int: "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> [x mod (a * b) = y] (mod a) \<longleftrightarrow> [x = y] (mod a)"
+  for a b :: int
+proof (cases "b > 0")
+  case True
+  then show ?thesis
+next
+  case False
+  then show ?thesis
+    apply (subst (1 2) cong_modulus_neg_int)
+    apply (unfold cong_int_def)
+    apply (subgoal_tac "a * b = (- a * - b)")
+     apply (erule ssubst)
+     apply (subst zmod_zmult2_eq)
+     apply (metis mod_diff_left_eq mod_diff_right_eq mod_mult_self1_is_0 diff_zero)+
+    done
+qed
+
+lemma cong_to_1_nat:
+  fixes a :: nat
+  assumes "[a = 1] (mod n)"
+  shows "n dvd (a - 1)"
+proof (cases "a = 0")
+  case True
+  then show ?thesis by force
+next
+  case False
+  with assms show ?thesis by (metis cong_altdef_nat leI less_one)
+qed
+
+lemma cong_0_1_nat': "[0 = Suc 0] (mod n) \<longleftrightarrow> n = Suc 0"
+  by (auto simp: cong_nat_def)
+
+lemma cong_0_1_nat: "[0 = 1] (mod n) \<longleftrightarrow> n = 1"
+  for n :: nat
+  by (auto simp: cong_nat_def)
+
+lemma cong_0_1_int: "[0 = 1] (mod n) \<longleftrightarrow> n = 1 \<or> n = - 1"
+  for n :: int
+  by (auto simp: cong_int_def zmult_eq_1_iff)
+
+lemma cong_to_1'_nat: "[a = 1] (mod n) \<longleftrightarrow> a = 0 \<and> n = 1 \<or> (\<exists>m. a = 1 + m * n)"
+  for a :: nat
+  by (metis add.right_neutral cong_0_1_nat cong_iff_lin_nat cong_to_1_nat
+      dvd_div_mult_self leI le_add_diff_inverse less_one mult_eq_if)
+
+lemma cong_le_nat: "y \<le> x \<Longrightarrow> [x = y] (mod n) \<longleftrightarrow> (\<exists>q. x = q * n + y)"
+  for x y :: nat
+  by (metis cong_altdef_nat Nat.le_imp_diff_is_add dvd_def mult.commute)
+
+lemma cong_solve_nat:
+  fixes a :: nat
+  assumes "a \<noteq> 0"
+  shows "\<exists>x. [a * x = gcd a n] (mod n)"
+proof (cases "n = 0")
+  case True
+  then show ?thesis by force
+next
+  case False
+  then show ?thesis
+    using bezout_nat [of a n, OF \<open>a \<noteq> 0\<close>]
+    by auto (metis cong_add_rcancel_0_nat cong_mult_self_nat mult.commute)
+qed
+
+lemma cong_solve_int: "a \<noteq> 0 \<Longrightarrow> \<exists>x. [a * x = gcd a n] (mod n)"
+  for a :: int
+  apply (cases "n = 0")
+   apply (cases "a \<ge> 0")
+    apply auto
+   apply (rule_tac x = "-1" in exI)
+   apply auto
+  apply (insert bezout_int [of a n], auto)
+  apply (metis cong_iff_lin_int mult.commute)
done

-lemma cong_to_1_nat: "([(a::nat) = 1] (mod n)) \<Longrightarrow> (n dvd (a - 1))"
-  apply (cases "a = 0", force)
-  by (metis cong_altdef_nat leI less_one)
-
-lemma cong_0_1_nat': "[(0::nat) = Suc 0] (mod n) = (n = Suc 0)"
-  unfolding cong_nat_def by auto
-
-lemma cong_0_1_nat: "[(0::nat) = 1] (mod n) = (n = 1)"
-  unfolding cong_nat_def by auto
-
-lemma cong_0_1_int: "[(0::int) = 1] (mod n) = ((n = 1) | (n = -1))"
-  unfolding cong_int_def by (auto simp add: zmult_eq_1_iff)
-
-lemma cong_to_1'_nat: "[(a::nat) = 1] (mod n) \<longleftrightarrow>
-    a = 0 \<and> n = 1 \<or> (\<exists>m. a = 1 + m * n)"
-
-lemma cong_le_nat: "(y::nat) <= x \<Longrightarrow> [x = y] (mod n) \<longleftrightarrow> (\<exists>q. x = q * n + y)"
-  by (metis cong_altdef_nat Nat.le_imp_diff_is_add dvd_def mult.commute)
-
-lemma cong_solve_nat: "(a::nat) \<noteq> 0 \<Longrightarrow> EX x. [a * x = gcd a n] (mod n)"
-  apply (cases "n = 0")
-  apply force
-  apply (frule bezout_nat [of a n], auto)
-  by (metis cong_add_rcancel_0_nat cong_mult_self_nat mult.commute)
-
-lemma cong_solve_int: "(a::int) \<noteq> 0 \<Longrightarrow> EX x. [a * x = gcd a n] (mod n)"
-  apply (cases "n = 0")
-  apply (cases "a \<ge> 0")
-  apply auto
-  apply (rule_tac x = "-1" in exI)
-  apply auto
-  apply (insert bezout_int [of a n], auto)
-  by (metis cong_iff_lin_int mult.commute)
-
lemma cong_solve_dvd_nat:
-  assumes a: "(a::nat) \<noteq> 0" and b: "gcd a n dvd d"
-  shows "EX x. [a * x = d] (mod n)"
+  fixes a :: nat
+  assumes a: "a \<noteq> 0" and b: "gcd a n dvd d"
+  shows "\<exists>x. [a * x = d] (mod n)"
proof -
from cong_solve_nat [OF a] obtain x where "[a * x = gcd a n](mod n)"
by auto
@@ -499,7 +559,7 @@

lemma cong_solve_dvd_int:
assumes a: "(a::int) \<noteq> 0" and b: "gcd a n dvd d"
-  shows "EX x. [a * x = d] (mod n)"
+  shows "\<exists>x. [a * x = d] (mod n)"
proof -
from cong_solve_int [OF a] obtain x where "[a * x = gcd a n](mod n)"
by auto
@@ -513,54 +573,62 @@
by auto
qed

-lemma cong_solve_coprime_nat: "coprime (a::nat) n \<Longrightarrow> EX x. [a * x = 1] (mod n)"
-  apply (cases "a = 0")
-  apply force
-  apply (metis cong_solve_nat)
-  done
+lemma cong_solve_coprime_nat:
+  fixes a :: nat
+  assumes "coprime a n"
+  shows "\<exists>x. [a * x = 1] (mod n)"
+proof (cases "a = 0")
+  case True
+  with assms show ?thesis by force
+next
+  case False
+  with assms show ?thesis by (metis cong_solve_nat)
+qed

-lemma cong_solve_coprime_int: "coprime (a::int) n \<Longrightarrow> EX x. [a * x = 1] (mod n)"
+lemma cong_solve_coprime_int: "coprime (a::int) n \<Longrightarrow> \<exists>x. [a * x = 1] (mod n)"
apply (cases "a = 0")
-  apply auto
-  apply (cases "n \<ge> 0")
-  apply auto
+   apply auto
+   apply (cases "n \<ge> 0")
+    apply auto
apply (metis cong_solve_int)
done

lemma coprime_iff_invertible_nat:
-  "m > 0 \<Longrightarrow> coprime a m = (EX x. [a * x = Suc 0] (mod m))"
+  "m > 0 \<Longrightarrow> coprime a m = (\<exists>x. [a * x = Suc 0] (mod m))"
by (metis One_nat_def cong_gcd_eq_nat cong_solve_coprime_nat coprime_lmult gcd.commute gcd_Suc_0)
-
-lemma coprime_iff_invertible_int: "m > (0::int) \<Longrightarrow> coprime a m = (EX x. [a * x = 1] (mod m))"
+
+lemma coprime_iff_invertible_int: "m > 0 \<Longrightarrow> coprime a m \<longleftrightarrow> (\<exists>x. [a * x = 1] (mod m))"
+  for m :: int
apply (auto intro: cong_solve_coprime_int)
apply (metis cong_int_def coprime_mul_eq gcd_1_int gcd.commute gcd_red_int)
done

-lemma coprime_iff_invertible'_nat: "m > 0 \<Longrightarrow> coprime a m =
-    (EX x. 0 \<le> x & x < m & [a * x = Suc 0] (mod m))"
+lemma coprime_iff_invertible'_nat:
+  "m > 0 \<Longrightarrow> coprime a m \<longleftrightarrow> (\<exists>x. 0 \<le> x \<and> x < m \<and> [a * x = Suc 0] (mod m))"
apply (subst coprime_iff_invertible_nat)
-  apply auto
+   apply auto
apply (metis mod_less_divisor mod_mult_right_eq)
done

-lemma coprime_iff_invertible'_int: "m > (0::int) \<Longrightarrow> coprime a m =
-    (EX x. 0 <= x & x < m & [a * x = 1] (mod m))"
+lemma coprime_iff_invertible'_int:
+  "m > 0 \<Longrightarrow> coprime a m \<longleftrightarrow> (\<exists>x. 0 \<le> x \<and> x < m \<and> [a * x = 1] (mod m))"
+  for m :: int
apply (subst coprime_iff_invertible_int)
-  apply (auto simp add: cong_int_def)
+   apply (auto simp add: cong_int_def)
apply (metis mod_mult_right_eq pos_mod_conj)
done

-lemma cong_cong_lcm_nat: "[(x::nat) = y] (mod a) \<Longrightarrow>
-    [x = y] (mod b) \<Longrightarrow> [x = y] (mod lcm a b)"
+lemma cong_cong_lcm_nat: "[x = y] (mod a) \<Longrightarrow> [x = y] (mod b) \<Longrightarrow> [x = y] (mod lcm a b)"
+  for x y :: nat
apply (cases "y \<le> x")
apply (metis cong_altdef_nat lcm_least)
apply (meson cong_altdef_nat cong_sym_nat lcm_least_iff nat_le_linear)
done

-lemma cong_cong_lcm_int: "[(x::int) = y] (mod a) \<Longrightarrow>
-    [x = y] (mod b) \<Longrightarrow> [x = y] (mod lcm a b)"
-  by (auto simp add: cong_altdef_int lcm_least) [1]
+lemma cong_cong_lcm_int: "[x = y] (mod a) \<Longrightarrow> [x = y] (mod b) \<Longrightarrow> [x = y] (mod lcm a b)"
+  for x y :: int
+  by (auto simp add: cong_altdef_int lcm_least)

lemma cong_cong_prod_coprime_nat [rule_format]: "finite A \<Longrightarrow>
(\<forall>i\<in>A. (\<forall>j\<in>A. i \<noteq> j \<longrightarrow> coprime (m i) (m j))) \<longrightarrow>
@@ -581,162 +649,168 @@
done

lemma binary_chinese_remainder_aux_nat:
-  assumes a: "coprime (m1::nat) m2"
-  shows "EX b1 b2. [b1 = 1] (mod m1) \<and> [b1 = 0] (mod m2) \<and>
-    [b2 = 0] (mod m1) \<and> [b2 = 1] (mod m2)"
+  fixes m1 m2 :: nat
+  assumes a: "coprime m1 m2"
+  shows "\<exists>b1 b2. [b1 = 1] (mod m1) \<and> [b1 = 0] (mod m2) \<and> [b2 = 0] (mod m1) \<and> [b2 = 1] (mod m2)"
proof -
-  from cong_solve_coprime_nat [OF a] obtain x1 where one: "[m1 * x1 = 1] (mod m2)"
+  from cong_solve_coprime_nat [OF a] obtain x1 where 1: "[m1 * x1 = 1] (mod m2)"
by auto
from a have b: "coprime m2 m1"
by (subst gcd.commute)
-  from cong_solve_coprime_nat [OF b] obtain x2 where two: "[m2 * x2 = 1] (mod m1)"
+  from cong_solve_coprime_nat [OF b] obtain x2 where 2: "[m2 * x2 = 1] (mod m1)"
by auto
have "[m1 * x1 = 0] (mod m1)"
-    by (subst mult.commute, rule cong_mult_self_nat)
+    by (subst mult.commute) (rule cong_mult_self_nat)
moreover have "[m2 * x2 = 0] (mod m2)"
-    by (subst mult.commute, rule cong_mult_self_nat)
-  moreover note one two
-  ultimately show ?thesis by blast
+    by (subst mult.commute) (rule cong_mult_self_nat)
+  ultimately show ?thesis
+    using 1 2 by blast
qed

lemma binary_chinese_remainder_aux_int:
-  assumes a: "coprime (m1::int) m2"
-  shows "EX b1 b2. [b1 = 1] (mod m1) \<and> [b1 = 0] (mod m2) \<and>
-    [b2 = 0] (mod m1) \<and> [b2 = 1] (mod m2)"
+  fixes m1 m2 :: int
+  assumes a: "coprime m1 m2"
+  shows "\<exists>b1 b2. [b1 = 1] (mod m1) \<and> [b1 = 0] (mod m2) \<and> [b2 = 0] (mod m1) \<and> [b2 = 1] (mod m2)"
proof -
-  from cong_solve_coprime_int [OF a] obtain x1 where one: "[m1 * x1 = 1] (mod m2)"
+  from cong_solve_coprime_int [OF a] obtain x1 where 1: "[m1 * x1 = 1] (mod m2)"
by auto
from a have b: "coprime m2 m1"
by (subst gcd.commute)
-  from cong_solve_coprime_int [OF b] obtain x2 where two: "[m2 * x2 = 1] (mod m1)"
+  from cong_solve_coprime_int [OF b] obtain x2 where 2: "[m2 * x2 = 1] (mod m1)"
by auto
have "[m1 * x1 = 0] (mod m1)"
-    by (subst mult.commute, rule cong_mult_self_int)
+    by (subst mult.commute) (rule cong_mult_self_int)
moreover have "[m2 * x2 = 0] (mod m2)"
-    by (subst mult.commute, rule cong_mult_self_int)
-  moreover note one two
-  ultimately show ?thesis by blast
+    by (subst mult.commute) (rule cong_mult_self_int)
+  ultimately show ?thesis
+    using 1 2 by blast
qed

lemma binary_chinese_remainder_nat:
-  assumes a: "coprime (m1::nat) m2"
-  shows "EX x. [x = u1] (mod m1) \<and> [x = u2] (mod m2)"
+  fixes m1 m2 :: nat
+  assumes a: "coprime m1 m2"
+  shows "\<exists>x. [x = u1] (mod m1) \<and> [x = u2] (mod m2)"
proof -
from binary_chinese_remainder_aux_nat [OF a] obtain b1 b2
-      where "[b1 = 1] (mod m1)" and "[b1 = 0] (mod m2)" and
-            "[b2 = 0] (mod m1)" and "[b2 = 1] (mod m2)"
+    where "[b1 = 1] (mod m1)" and "[b1 = 0] (mod m2)"
+      and "[b2 = 0] (mod m1)" and "[b2 = 1] (mod m2)"
by blast
let ?x = "u1 * b1 + u2 * b2"
have "[?x = u1 * 1 + u2 * 0] (mod m1)"
-    apply (rule cong_scalar2_nat)
-    apply (rule \<open>[b1 = 1] (mod m1)\<close>)
+     apply (rule cong_scalar2_nat)
+     apply (rule \<open>[b1 = 1] (mod m1)\<close>)
apply (rule cong_scalar2_nat)
apply (rule \<open>[b2 = 0] (mod m1)\<close>)
done
then have "[?x = u1] (mod m1)" by simp
have "[?x = u1 * 0 + u2 * 1] (mod m2)"
-    apply (rule cong_scalar2_nat)
-    apply (rule \<open>[b1 = 0] (mod m2)\<close>)
+     apply (rule cong_scalar2_nat)
+     apply (rule \<open>[b1 = 0] (mod m2)\<close>)
apply (rule cong_scalar2_nat)
apply (rule \<open>[b2 = 1] (mod m2)\<close>)
done
-  then have "[?x = u2] (mod m2)" by simp
-  with \<open>[?x = u1] (mod m1)\<close> show ?thesis by blast
+  then have "[?x = u2] (mod m2)"
+    by simp
+  with \<open>[?x = u1] (mod m1)\<close> show ?thesis
+    by blast
qed

lemma binary_chinese_remainder_int:
-  assumes a: "coprime (m1::int) m2"
-  shows "EX x. [x = u1] (mod m1) \<and> [x = u2] (mod m2)"
+  fixes m1 m2 :: int
+  assumes a: "coprime m1 m2"
+  shows "\<exists>x. [x = u1] (mod m1) \<and> [x = u2] (mod m2)"
proof -
from binary_chinese_remainder_aux_int [OF a] obtain b1 b2
-    where "[b1 = 1] (mod m1)" and "[b1 = 0] (mod m2)" and
-          "[b2 = 0] (mod m1)" and "[b2 = 1] (mod m2)"
+    where "[b1 = 1] (mod m1)" and "[b1 = 0] (mod m2)"
+      and "[b2 = 0] (mod m1)" and "[b2 = 1] (mod m2)"
by blast
let ?x = "u1 * b1 + u2 * b2"
have "[?x = u1 * 1 + u2 * 0] (mod m1)"
-    apply (rule cong_scalar2_int)
-    apply (rule \<open>[b1 = 1] (mod m1)\<close>)
+     apply (rule cong_scalar2_int)
+     apply (rule \<open>[b1 = 1] (mod m1)\<close>)
apply (rule cong_scalar2_int)
apply (rule \<open>[b2 = 0] (mod m1)\<close>)
done
then have "[?x = u1] (mod m1)" by simp
have "[?x = u1 * 0 + u2 * 1] (mod m2)"
-    apply (rule cong_scalar2_int)
-    apply (rule \<open>[b1 = 0] (mod m2)\<close>)
+     apply (rule cong_scalar2_int)
+     apply (rule \<open>[b1 = 0] (mod m2)\<close>)
apply (rule cong_scalar2_int)
apply (rule \<open>[b2 = 1] (mod m2)\<close>)
done
then have "[?x = u2] (mod m2)" by simp
-  with \<open>[?x = u1] (mod m1)\<close> show ?thesis by blast
+  with \<open>[?x = u1] (mod m1)\<close> show ?thesis
+    by blast
qed

-lemma cong_modulus_mult_nat: "[(x::nat) = y] (mod m * n) \<Longrightarrow>
-    [x = y] (mod m)"
+lemma cong_modulus_mult_nat: "[x = y] (mod m * n) \<Longrightarrow> [x = y] (mod m)"
+  for x y :: nat
apply (cases "y \<le> x")
-  apply (erule dvd_mult_left)
+   apply (erule dvd_mult_left)
apply (rule cong_sym_nat)
apply (subst (asm) cong_sym_eq_nat)
apply (erule dvd_mult_left)
done

-lemma cong_modulus_mult_int: "[(x::int) = y] (mod m * n) \<Longrightarrow>
-    [x = y] (mod m)"
+lemma cong_modulus_mult_int: "[x = y] (mod m * n) \<Longrightarrow> [x = y] (mod m)"
+  for x y :: int
apply (erule dvd_mult_left)
done

-lemma cong_less_modulus_unique_nat:
-    "[(x::nat) = y] (mod m) \<Longrightarrow> x < m \<Longrightarrow> y < m \<Longrightarrow> x = y"
+lemma cong_less_modulus_unique_nat: "[x = y] (mod m) \<Longrightarrow> x < m \<Longrightarrow> y < m \<Longrightarrow> x = y"
+  for x y :: nat

lemma binary_chinese_remainder_unique_nat:
-  assumes a: "coprime (m1::nat) m2"
+  fixes m1 m2 :: nat
+  assumes a: "coprime m1 m2"
and nz: "m1 \<noteq> 0" "m2 \<noteq> 0"
shows "\<exists>!x. x < m1 * m2 \<and> [x = u1] (mod m1) \<and> [x = u2] (mod m2)"
proof -
-  from binary_chinese_remainder_nat [OF a] obtain y where
-      "[y = u1] (mod m1)" and "[y = u2] (mod m2)"
+  from binary_chinese_remainder_nat [OF a] obtain y
+    where "[y = u1] (mod m1)" and "[y = u2] (mod m2)"
by blast
let ?x = "y mod (m1 * m2)"
from nz have less: "?x < m1 * m2"
by auto
-  have one: "[?x = u1] (mod m1)"
+  have 1: "[?x = u1] (mod m1)"
apply (rule cong_trans_nat)
-    prefer 2
-    apply (rule \<open>[y = u1] (mod m1)\<close>)
+     prefer 2
+     apply (rule \<open>[y = u1] (mod m1)\<close>)
apply (rule cong_modulus_mult_nat)
apply (rule cong_mod_nat)
using nz apply auto
done
-  have two: "[?x = u2] (mod m2)"
+  have 2: "[?x = u2] (mod m2)"
apply (rule cong_trans_nat)
-    prefer 2
-    apply (rule \<open>[y = u2] (mod m2)\<close>)
+     prefer 2
+     apply (rule \<open>[y = u2] (mod m2)\<close>)
apply (subst mult.commute)
apply (rule cong_modulus_mult_nat)
apply (rule cong_mod_nat)
using nz apply auto
done
-  have "ALL z. z < m1 * m2 \<and> [z = u1] (mod m1) \<and> [z = u2] (mod m2) \<longrightarrow> z = ?x"
+  have "\<forall>z. z < m1 * m2 \<and> [z = u1] (mod m1) \<and> [z = u2] (mod m2) \<longrightarrow> z = ?x"
proof clarify
fix z
assume "z < m1 * m2"
assume "[z = u1] (mod m1)" and  "[z = u2] (mod m2)"
have "[?x = z] (mod m1)"
apply (rule cong_trans_nat)
-      apply (rule \<open>[?x = u1] (mod m1)\<close>)
+       apply (rule \<open>[?x = u1] (mod m1)\<close>)
apply (rule cong_sym_nat)
apply (rule \<open>[z = u1] (mod m1)\<close>)
done
moreover have "[?x = z] (mod m2)"
apply (rule cong_trans_nat)
-      apply (rule \<open>[?x = u2] (mod m2)\<close>)
+       apply (rule \<open>[?x = u2] (mod m2)\<close>)
apply (rule cong_sym_nat)
apply (rule \<open>[z = u2] (mod m2)\<close>)
done
@@ -744,32 +818,30 @@
by (auto intro: coprime_cong_mult_nat a)
with \<open>z < m1 * m2\<close> \<open>?x < m1 * m2\<close> show "z = ?x"
apply (intro cong_less_modulus_unique_nat)
-      apply (auto, erule cong_sym_nat)
+        apply (auto, erule cong_sym_nat)
done
qed
-  with less one two show ?thesis by auto
+  with less 1 2 show ?thesis by auto
qed

lemma chinese_remainder_aux_nat:
fixes A :: "'a set"
and m :: "'a \<Rightarrow> nat"
assumes fin: "finite A"
-    and cop: "ALL i : A. (ALL j : A. i \<noteq> j \<longrightarrow> coprime (m i) (m j))"
-  shows "EX b. (ALL i : A. [b i = 1] (mod m i) \<and> [b i = 0] (mod (\<Prod>j \<in> A - {i}. m j)))"
+    and cop: "\<forall>i \<in> A. (\<forall>j \<in> A. i \<noteq> j \<longrightarrow> coprime (m i) (m j))"
+  shows "\<exists>b. (\<forall>i \<in> A. [b i = 1] (mod m i) \<and> [b i = 0] (mod (\<Prod>j \<in> A - {i}. m j)))"
proof (rule finite_set_choice, rule fin, rule ballI)
fix i
-  assume "i : A"
+  assume "i \<in> A"
with cop have "coprime (\<Prod>j \<in> A - {i}. m j) (m i)"
-    by (intro prod_coprime, auto)
-  then have "EX x. [(\<Prod>j \<in> A - {i}. m j) * x = 1] (mod m i)"
+    by (intro prod_coprime) auto
+  then have "\<exists>x. [(\<Prod>j \<in> A - {i}. m j) * x = 1] (mod m i)"
by (elim cong_solve_coprime_nat)
then obtain x where "[(\<Prod>j \<in> A - {i}. m j) * x = 1] (mod m i)"
by auto
-  moreover have "[(\<Prod>j \<in> A - {i}. m j) * x = 0]
-    (mod (\<Prod>j \<in> A - {i}. m j))"
+  moreover have "[(\<Prod>j \<in> A - {i}. m j) * x = 0] (mod (\<Prod>j \<in> A - {i}. m j))"
by (subst mult.commute, rule cong_mult_self_nat)
-  ultimately show "\<exists>a. [a = 1] (mod m i) \<and> [a = 0]
-      (mod prod m (A - {i}))"
+  ultimately show "\<exists>a. [a = 1] (mod m i) \<and> [a = 0] (mod prod m (A - {i}))"
by blast
qed

@@ -778,37 +850,35 @@
and m :: "'a \<Rightarrow> nat"
and u :: "'a \<Rightarrow> nat"
assumes fin: "finite A"
-    and cop: "ALL i:A. (ALL j : A. i \<noteq> j \<longrightarrow> coprime (m i) (m j))"
-  shows "EX x. (ALL i:A. [x = u i] (mod m i))"
+    and cop: "\<forall>i \<in> A. \<forall>j \<in> A. i \<noteq> j \<longrightarrow> coprime (m i) (m j)"
+  shows "\<exists>x. \<forall>i \<in> A. [x = u i] (mod m i)"
proof -
-  from chinese_remainder_aux_nat [OF fin cop] obtain b where
-    bprop: "ALL i:A. [b i = 1] (mod m i) \<and>
-      [b i = 0] (mod (\<Prod>j \<in> A - {i}. m j))"
+  from chinese_remainder_aux_nat [OF fin cop]
+  obtain b where b: "\<forall>i \<in> A. [b i = 1] (mod m i) \<and> [b i = 0] (mod (\<Prod>j \<in> A - {i}. m j))"
by blast
let ?x = "\<Sum>i\<in>A. (u i) * (b i)"
-  show "?thesis"
+  show ?thesis
proof (rule exI, clarify)
fix i
-    assume a: "i : A"
+    assume a: "i \<in> A"
show "[?x = u i] (mod m i)"
proof -
-      from fin a have "?x = (\<Sum>j \<in> {i}. u j * b j) +
-          (\<Sum>j \<in> A - {i}. u j * b j)"
-        by (subst sum.union_disjoint [symmetric], auto intro: sum.cong)
+      from fin a have "?x = (\<Sum>j \<in> {i}. u j * b j) + (\<Sum>j \<in> A - {i}. u j * b j)"
+        by (subst sum.union_disjoint [symmetric]) (auto intro: sum.cong)
then have "[?x = u i * b i + (\<Sum>j \<in> A - {i}. u j * b j)] (mod m i)"
by auto
also have "[u i * b i + (\<Sum>j \<in> A - {i}. u j * b j) =
u i * 1 + (\<Sum>j \<in> A - {i}. u j * 0)] (mod m i)"
-        apply (rule cong_scalar2_nat)
-        using bprop a apply blast
+         apply (rule cong_scalar2_nat)
+        using b a apply blast
apply (rule cong_sum_nat)
apply (rule cong_scalar2_nat)
-        using bprop apply auto
+        using b apply auto
apply (rule cong_dvd_modulus_nat)
-        apply (drule (1) bspec)
-        apply (erule conjE)
-        apply assumption
+         apply (drule (1) bspec)
+         apply (erule conjE)
+         apply assumption
apply rule
using fin a apply auto
done
@@ -833,36 +903,35 @@
and u :: "'a \<Rightarrow> nat"
assumes fin: "finite A"
and nz: "\<forall>i\<in>A. m i \<noteq> 0"
-    and cop: "\<forall>i\<in>A. (\<forall>j\<in>A. i \<noteq> j \<longrightarrow> coprime (m i) (m j))"
+    and cop: "\<forall>i\<in>A. \<forall>j\<in>A. i \<noteq> j \<longrightarrow> coprime (m i) (m j)"
shows "\<exists>!x. x < (\<Prod>i\<in>A. m i) \<and> (\<forall>i\<in>A. [x = u i] (mod m i))"
proof -
from chinese_remainder_nat [OF fin cop]
-  obtain y where one: "(ALL i:A. [y = u i] (mod m i))"
+  obtain y where one: "(\<forall>i\<in>A. [y = u i] (mod m i))"
by blast
let ?x = "y mod (\<Prod>i\<in>A. m i)"
from fin nz have prodnz: "(\<Prod>i\<in>A. m i) \<noteq> 0"
by auto
then have less: "?x < (\<Prod>i\<in>A. m i)"
by auto
-  have cong: "ALL i:A. [?x = u i] (mod m i)"
+  have cong: "\<forall>i\<in>A. [?x = u i] (mod m i)"
apply auto
apply (rule cong_trans_nat)
-    prefer 2
+     prefer 2
using one apply auto
apply (rule cong_dvd_modulus_nat)
-    apply (rule cong_mod_nat)
+     apply (rule cong_mod_nat)
using prodnz apply auto
apply rule
-    apply (rule fin)
+     apply (rule fin)
apply assumption
done
-  have unique: "ALL z. z < (\<Prod>i\<in>A. m i) \<and>
-      (ALL i:A. [z = u i] (mod m i)) \<longrightarrow> z = ?x"
-  proof (clarify)
+  have unique: "\<forall>z. z < (\<Prod>i\<in>A. m i) \<and> (\<forall>i\<in>A. [z = u i] (mod m i)) \<longrightarrow> z = ?x"
+  proof clarify
fix z
assume zless: "z < (\<Prod>i\<in>A. m i)"
-    assume zcong: "(ALL i:A. [z = u i] (mod m i))"
-    have "ALL i:A. [?x = z] (mod m i)"
+    assume zcong: "(\<forall>i\<in>A. [z = u i] (mod m i))"
+    have "\<forall>i\<in>A. [?x = z] (mod m i)"
apply clarify
apply (rule cong_trans_nat)
using cong apply (erule bspec)
@@ -871,14 +940,16 @@
done
with fin cop have "[?x = z] (mod (\<Prod>i\<in>A. m i))"
apply (intro coprime_cong_prod_nat)
-      apply auto
+        apply auto
done
with zless less show "z = ?x"
apply (intro cong_less_modulus_unique_nat)
-      apply (auto, erule cong_sym_nat)
+        apply auto
+      apply (erule cong_sym_nat)
done
qed
-  from less cong unique show ?thesis by blast
+  from less cong unique show ?thesis
+    by blast
qed

end```