header {* Congruence *}
theory Cong
imports Primes
begin
subsection {* Turn off One_nat_def *}
lemma induct'_nat [case_names zero plus1, induct type: nat]:
"[| P (0::nat); !!n. P n ==> P (n + 1)|] ==> P n"
by (erule nat_induct) (simp add:One_nat_def)
lemma cases_nat [case_names zero plus1, cases type: nat]:
"P (0::nat) ==> (!!n. P (n + 1)) ==> P n"
by(metis induct'_nat)
lemma power_plus_one [simp]: "(x::'a::power)^(n + 1) = x * x^n"
by (simp add: One_nat_def)
lemma power_eq_one_eq_nat [simp]:
"((x::nat)^m = 1) = (m = 0 | x = 1)"
by (induct m, auto)
lemma card_insert_if' [simp]: "finite A ==>
card (insert x A) = (if x ∈ A then (card A) else (card A) + 1)"
by (auto simp add: insert_absorb)
declare card_insert_disjoint [simp del]
lemma nat_1' [simp]: "nat 1 = 1"
by simp
declare nat_1 [simp del]
declare add_2_eq_Suc [simp del]
declare add_2_eq_Suc' [simp del]
declare mod_pos_pos_trivial [simp]
subsection {* Main definitions *}
class cong =
fixes
cong :: "'a => 'a => 'a => bool" ("(1[_ = _] '(mod _'))")
begin
abbreviation
notcong :: "'a => 'a => 'a => bool" ("(1[_ ≠ _] '(mod _'))")
where
"notcong x y m == (~cong x y m)"
end
instantiation nat :: cong
begin
definition
cong_nat :: "nat => nat => nat => bool"
where
"cong_nat x y m = ((x mod m) = (y mod m))"
instance proof qed
end
instantiation int :: cong
begin
definition
cong_int :: "int => int => int => bool"
where
"cong_int x y m = ((x mod m) = (y mod m))"
instance proof qed
end
subsection {* Set up Transfer *}
lemma transfer_nat_int_cong:
"(x::int) >= 0 ==> y >= 0 ==> m >= 0 ==>
([(nat x) = (nat y)] (mod (nat m))) = ([x = y] (mod m))"
unfolding cong_int_def cong_nat_def
apply (auto simp add: nat_mod_distrib [symmetric])
apply (subst (asm) eq_nat_nat_iff)
apply (case_tac "m = 0", force, rule pos_mod_sign, force)+
apply assumption
done
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
declare transfer_morphism_int_nat[transfer add return:
transfer_int_nat_cong]
subsection {* Congruence *}
lemma cong_0_nat [simp, presburger]: "([(a::nat) = b] (mod 0)) = (a = b)"
by (unfold cong_nat_def, auto)
lemma cong_0_int [simp, presburger]: "([(a::int) = b] (mod 0)) = (a = b)"
by (unfold cong_int_def, auto)
lemma cong_1_nat [simp, presburger]: "[(a::nat) = b] (mod 1)"
by (unfold cong_nat_def, auto)
lemma cong_Suc_0_nat [simp, presburger]: "[(a::nat) = b] (mod Suc 0)"
by (unfold cong_nat_def, auto simp add: One_nat_def)
lemma cong_1_int [simp, presburger]: "[(a::int) = b] (mod 1)"
by (unfold cong_int_def, auto)
lemma cong_refl_nat [simp]: "[(k::nat) = k] (mod m)"
by (unfold cong_nat_def, auto)
lemma cong_refl_int [simp]: "[(k::int) = k] (mod m)"
by (unfold cong_int_def, auto)
lemma cong_sym_nat: "[(a::nat) = b] (mod m) ==> [b = a] (mod m)"
by (unfold cong_nat_def, auto)
lemma cong_sym_int: "[(a::int) = b] (mod m) ==> [b = a] (mod m)"
by (unfold cong_int_def, auto)
lemma cong_sym_eq_nat: "[(a::nat) = b] (mod m) = [b = a] (mod m)"
by (unfold cong_nat_def, auto)
lemma cong_sym_eq_int: "[(a::int) = b] (mod m) = [b = a] (mod m)"
by (unfold cong_int_def, auto)
lemma cong_trans_nat [trans]:
"[(a::nat) = b] (mod m) ==> [b = c] (mod m) ==> [a = c] (mod m)"
by (unfold cong_nat_def, auto)
lemma cong_trans_int [trans]:
"[(a::int) = b] (mod m) ==> [b = c] (mod m) ==> [a = c] (mod m)"
by (unfold cong_int_def, auto)
lemma cong_add_nat:
"[(a::nat) = b] (mod m) ==> [c = d] (mod m) ==> [a + c = b + d] (mod m)"
apply (unfold cong_nat_def)
apply (subst (1 2) mod_add_eq)
apply simp
done
lemma cong_add_int:
"[(a::int) = b] (mod m) ==> [c = d] (mod m) ==> [a + c = b + d] (mod m)"
apply (unfold cong_int_def)
apply (subst (1 2) mod_add_left_eq)
apply (subst (1 2) mod_add_right_eq)
apply simp
done
lemma cong_diff_int:
"[(a::int) = b] (mod m) ==> [c = d] (mod m) ==> [a - c = b - d] (mod m)"
apply (unfold cong_int_def)
apply (subst (1 2) mod_diff_eq)
apply simp
done
lemma cong_diff_aux_int:
"(a::int) >= c ==> b >= d ==> [(a::int) = b] (mod m) ==>
[c = d] (mod m) ==> [tsub a c = tsub b d] (mod m)"
apply (subst (1 2) tsub_eq)
apply (auto intro: cong_diff_int)
done;
lemma cong_diff_nat:
assumes "(a::nat) >= c" and "b >= d" and "[a = b] (mod m)" and
"[c = d] (mod m)"
shows "[a - c = b - d] (mod m)"
using prems by (rule cong_diff_aux_int [transferred]);
lemma cong_mult_nat:
"[(a::nat) = b] (mod m) ==> [c = d] (mod m) ==> [a * c = b * d] (mod m)"
apply (unfold cong_nat_def)
apply (subst (1 2) mod_mult_eq)
apply simp
done
lemma cong_mult_int:
"[(a::int) = b] (mod m) ==> [c = d] (mod m) ==> [a * c = b * d] (mod m)"
apply (unfold cong_int_def)
apply (subst (1 2) zmod_zmult1_eq)
apply (subst (1 2) mult_commute)
apply (subst (1 2) zmod_zmult1_eq)
apply simp
done
lemma cong_exp_nat: "[(x::nat) = y] (mod n) ==> [x^k = y^k] (mod n)"
apply (induct k)
apply (auto simp add: cong_refl_nat cong_mult_nat)
done
lemma cong_exp_int: "[(x::int) = y] (mod n) ==> [x^k = y^k] (mod n)"
apply (induct k)
apply (auto simp add: cong_refl_int cong_mult_int)
done
lemma cong_setsum_nat [rule_format]:
"(ALL x: A. [((f x)::nat) = g x] (mod m)) -->
[(SUM x:A. f x) = (SUM x:A. g x)] (mod m)"
apply (case_tac "finite A")
apply (induct set: finite)
apply (auto intro: cong_add_nat)
done
lemma cong_setsum_int [rule_format]:
"(ALL x: A. [((f x)::int) = g x] (mod m)) -->
[(SUM x:A. f x) = (SUM x:A. g x)] (mod m)"
apply (case_tac "finite A")
apply (induct set: finite)
apply (auto intro: cong_add_int)
done
lemma cong_setprod_nat [rule_format]:
"(ALL x: A. [((f x)::nat) = g x] (mod m)) -->
[(PROD x:A. f x) = (PROD x:A. g x)] (mod m)"
apply (case_tac "finite A")
apply (induct set: finite)
apply (auto intro: cong_mult_nat)
done
lemma cong_setprod_int [rule_format]:
"(ALL x: A. [((f x)::int) = g x] (mod m)) -->
[(PROD x:A. f x) = (PROD x:A. g x)] (mod m)"
apply (case_tac "finite A")
apply (induct set: finite)
apply (auto intro: cong_mult_int)
done
lemma cong_scalar_nat: "[(a::nat)= b] (mod m) ==> [a * k = b * k] (mod m)"
by (rule cong_mult_nat, simp_all)
lemma cong_scalar_int: "[(a::int)= b] (mod m) ==> [a * k = b * k] (mod m)"
by (rule cong_mult_int, simp_all)
lemma cong_scalar2_nat: "[(a::nat)= b] (mod m) ==> [k * a = k * b] (mod m)"
by (rule cong_mult_nat, simp_all)
lemma cong_scalar2_int: "[(a::int)= b] (mod m) ==> [k * a = k * b] (mod m)"
by (rule cong_mult_int, simp_all)
lemma cong_mult_self_nat: "[(a::nat) * m = 0] (mod m)"
by (unfold cong_nat_def, auto)
lemma cong_mult_self_int: "[(a::int) * m = 0] (mod m)"
by (unfold cong_int_def, auto)
lemma cong_eq_diff_cong_0_int: "[(a::int) = b] (mod m) = [a - b = 0] (mod m)"
apply (rule iffI)
apply (erule cong_diff_int [of a b m b b, simplified])
apply (erule cong_add_int [of "a - b" 0 m b b, simplified])
done
lemma cong_eq_diff_cong_0_aux_int: "a >= b ==>
[(a::int) = b] (mod m) = [tsub a b = 0] (mod m)"
by (subst tsub_eq, assumption, rule cong_eq_diff_cong_0_int)
lemma cong_eq_diff_cong_0_nat:
assumes "(a::nat) >= b"
shows "[a = b] (mod m) = [a - b = 0] (mod m)"
using prems by (rule cong_eq_diff_cong_0_aux_int [transferred])
lemma cong_diff_cong_0'_nat:
"[(x::nat) = y] (mod n) <->
(if x <= y then [y - x = 0] (mod n) else [x - y = 0] (mod n))"
apply (case_tac "y <= x")
apply (frule cong_eq_diff_cong_0_nat [where m = n])
apply auto [1]
apply (subgoal_tac "x <= y")
apply (frule cong_eq_diff_cong_0_nat [where m = n])
apply (subst cong_sym_eq_nat)
apply auto
done
lemma cong_altdef_nat: "(a::nat) >= b ==> [a = b] (mod m) = (m dvd (a - b))"
apply (subst cong_eq_diff_cong_0_nat, assumption)
apply (unfold cong_nat_def)
apply (simp add: dvd_eq_mod_eq_0 [symmetric])
done
lemma cong_altdef_int: "[(a::int) = b] (mod m) = (m dvd (a - b))"
apply (subst cong_eq_diff_cong_0_int)
apply (unfold cong_int_def)
apply (simp add: dvd_eq_mod_eq_0 [symmetric])
done
lemma cong_abs_int: "[(x::int) = y] (mod abs m) = [x = y] (mod m)"
by (simp add: cong_altdef_int)
lemma cong_square_int:
"[| prime (p::int); 0 < a; [a * a = 1] (mod p) |]
==> [a = 1] (mod p) ∨ [a = - 1] (mod p)"
apply (simp only: cong_altdef_int)
apply (subst prime_dvd_mult_eq_int [symmetric], assumption)
apply (subgoal_tac "a * a - 1 = (a - 1) * (a - -1)")
apply (auto simp add: field_simps)
done
lemma cong_mult_rcancel_int:
"coprime k (m::int) ==> [a * k = b * k] (mod m) = [a = b] (mod m)"
apply (subst (1 2) cong_altdef_int)
apply (subst left_diff_distrib [symmetric])
apply (rule coprime_dvd_mult_iff_int)
apply (subst gcd_commute_int, assumption)
done
lemma cong_mult_rcancel_nat:
assumes "coprime k (m::nat)"
shows "[a * k = b * k] (mod m) = [a = b] (mod m)"
apply (rule cong_mult_rcancel_int [transferred])
using prems apply auto
done
lemma cong_mult_lcancel_nat:
"coprime k (m::nat) ==> [k * a = k * b ] (mod m) = [a = b] (mod m)"
by (simp add: mult_commute cong_mult_rcancel_nat)
lemma cong_mult_lcancel_int:
"coprime k (m::int) ==> [k * a = k * b] (mod m) = [a = b] (mod m)"
by (simp add: mult_commute cong_mult_rcancel_int)
lemma coprime_cong_mult_int:
"[(a::int) = b] (mod m) ==> [a = b] (mod n) ==> coprime m n
==> [a = b] (mod m * n)"
apply (simp only: cong_altdef_int)
apply (erule (2) divides_mult_int)
done
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)"
apply (rule coprime_cong_mult_int [transferred])
using prems apply auto
done
lemma cong_less_imp_eq_nat: "0 ≤ (a::nat) ==>
a < m ==> 0 ≤ b ==> b < m ==> [a = b] (mod m) ==> a = b"
by (auto simp add: cong_nat_def mod_pos_pos_trivial)
lemma cong_less_imp_eq_int: "0 ≤ (a::int) ==>
a < m ==> 0 ≤ b ==> b < m ==> [a = b] (mod m) ==> a = b"
by (auto simp add: cong_int_def mod_pos_pos_trivial)
lemma cong_less_unique_nat:
"0 < (m::nat) ==> (∃!b. 0 ≤ b ∧ b < m ∧ [a = b] (mod m))"
apply auto
apply (rule_tac x = "a mod m" in exI)
apply (unfold cong_nat_def, auto)
done
lemma cong_less_unique_int:
"0 < (m::int) ==> (∃!b. 0 ≤ b ∧ b < m ∧ [a = b] (mod m))"
apply auto
apply (rule_tac x = "a mod m" in exI)
apply (unfold cong_int_def, auto simp add: mod_pos_pos_trivial)
done
lemma cong_iff_lin_int: "([(a::int) = b] (mod m)) = (∃k. b = a + m * k)"
apply (auto simp add: cong_altdef_int dvd_def field_simps)
apply (rule_tac [!] x = "-k" in exI, auto)
done
lemma cong_iff_lin_nat: "([(a::nat) = b] (mod m)) =
(∃k1 k2. b + k1 * m = a + k2 * m)"
apply (rule iffI)
apply (case_tac "b <= a")
apply (subst (asm) cong_altdef_nat, assumption)
apply (unfold dvd_def, auto)
apply (rule_tac x = k in exI)
apply (rule_tac x = 0 in exI)
apply (auto simp add: field_simps)
apply (subst (asm) cong_sym_eq_nat)
apply (subst (asm) cong_altdef_nat)
apply force
apply (unfold dvd_def, auto)
apply (rule_tac x = 0 in exI)
apply (rule_tac x = k in exI)
apply (auto simp add: field_simps)
apply (unfold cong_nat_def)
apply (subgoal_tac "a mod m = (a + k2 * m) mod m")
apply (erule ssubst)back
apply (erule subst)
apply auto
done
lemma cong_gcd_eq_int: "[(a::int) = b] (mod m) ==> gcd a m = gcd b m"
apply (subst (asm) cong_iff_lin_int, auto)
apply (subst add_commute)
apply (subst (2) gcd_commute_int)
apply (subst mult_commute)
apply (subst gcd_add_mult_int)
apply (rule gcd_commute_int)
done
lemma cong_gcd_eq_nat:
assumes "[(a::nat) = b] (mod m)"
shows "gcd a m = gcd b m"
apply (rule cong_gcd_eq_int [transferred])
using prems apply auto
done
lemma cong_imp_coprime_nat: "[(a::nat) = b] (mod m) ==> coprime a m ==>
coprime b m"
by (auto simp add: cong_gcd_eq_nat)
lemma cong_imp_coprime_int: "[(a::int) = b] (mod m) ==> coprime a m ==>
coprime b m"
by (auto simp add: cong_gcd_eq_int)
lemma cong_cong_mod_nat: "[(a::nat) = b] (mod m) =
[a mod m = b mod m] (mod m)"
by (auto simp add: cong_nat_def)
lemma cong_cong_mod_int: "[(a::int) = b] (mod m) =
[a mod m = b mod m] (mod m)"
by (auto simp add: cong_int_def)
lemma cong_minus_int [iff]: "[(a::int) = b] (mod -m) = [a = b] (mod m)"
by (subst (1 2) cong_altdef_int, auto)
lemma cong_zero_nat [iff]: "[(a::nat) = b] (mod 0) = (a = b)"
by (auto simp add: cong_nat_def)
lemma cong_zero_int [iff]: "[(a::int) = b] (mod 0) = (a = b)"
by (auto simp add: cong_int_def)
lemma cong_add_lcancel_nat:
"[(a::nat) + x = a + y] (mod n) <-> [x = y] (mod n)"
by (simp add: cong_iff_lin_nat)
lemma cong_add_lcancel_int:
"[(a::int) + x = a + y] (mod n) <-> [x = y] (mod n)"
by (simp add: cong_iff_lin_int)
lemma cong_add_rcancel_nat: "[(x::nat) + a = y + a] (mod n) <-> [x = y] (mod n)"
by (simp add: cong_iff_lin_nat)
lemma cong_add_rcancel_int: "[(x::int) + a = y + a] (mod n) <-> [x = y] (mod n)"
by (simp add: cong_iff_lin_int)
lemma cong_add_lcancel_0_nat: "[(a::nat) + x = a] (mod n) <-> [x = 0] (mod n)"
by (simp add: cong_iff_lin_nat)
lemma cong_add_lcancel_0_int: "[(a::int) + x = a] (mod n) <-> [x = 0] (mod n)"
by (simp add: cong_iff_lin_int)
lemma cong_add_rcancel_0_nat: "[x + (a::nat) = a] (mod n) <-> [x = 0] (mod n)"
by (simp add: cong_iff_lin_nat)
lemma cong_add_rcancel_0_int: "[x + (a::int) = a] (mod n) <-> [x = 0] (mod n)"
by (simp add: cong_iff_lin_int)
lemma cong_dvd_modulus_nat: "[(x::nat) = y] (mod m) ==> n dvd m ==>
[x = y] (mod n)"
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 (simp add: field_simps)
done
lemma cong_dvd_modulus_int: "[(x::int) = y] (mod m) ==> n dvd m ==>
[x = y] (mod n)"
by (auto simp add: cong_altdef_int dvd_def)
lemma cong_dvd_eq_nat: "[(x::nat) = y] (mod n) ==> n dvd x <-> n dvd y"
by (unfold cong_nat_def, auto simp add: dvd_eq_mod_eq_0)
lemma cong_dvd_eq_int: "[(x::int) = y] (mod n) ==> n dvd x <-> n dvd y"
by (unfold cong_int_def, auto simp add: dvd_eq_mod_eq_0)
lemma cong_mod_nat: "(n::nat) ~= 0 ==> [a mod n = a] (mod n)"
by (simp add: cong_nat_def)
lemma cong_mod_int: "(n::int) ~= 0 ==> [a mod n = a] (mod n)"
by (simp add: cong_int_def)
lemma mod_mult_cong_nat: "(a::nat) ~= 0 ==> b ~= 0
==> [x mod (a * b) = y] (mod a) <-> [x = y] (mod a)"
by (simp add: cong_nat_def mod_mult2_eq mod_add_left_eq)
lemma neg_cong_int: "([(a::int) = b] (mod m)) = ([-a = -b] (mod m))"
apply (simp add: cong_altdef_int)
apply (subst dvd_minus_iff [symmetric])
apply (simp add: field_simps)
done
lemma cong_modulus_neg_int: "([(a::int) = b] (mod m)) = ([a = b] (mod -m))"
by (auto simp add: cong_altdef_int)
lemma mod_mult_cong_int: "(a::int) ~= 0 ==> b ~= 0
==> [x mod (a * b) = y] (mod a) <-> [x = y] (mod a)"
apply (case_tac "b > 0")
apply (simp add: cong_int_def mod_mod_cancel mod_add_left_eq)
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 (auto simp add: mod_add_left_eq)
done
lemma cong_to_1_nat: "([(a::nat) = 1] (mod n)) ==> (n dvd (a - 1))"
apply (case_tac "a = 0")
apply force
apply (subst (asm) cong_altdef_nat)
apply auto
done
lemma cong_0_1_nat: "[(0::nat) = 1] (mod n) = (n = 1)"
by (unfold cong_nat_def, auto)
lemma cong_0_1_int: "[(0::int) = 1] (mod n) = ((n = 1) | (n = -1))"
by (unfold cong_int_def, auto simp add: zmult_eq_1_iff)
lemma cong_to_1'_nat: "[(a::nat) = 1] (mod n) <->
a = 0 ∧ n = 1 ∨ (∃m. a = 1 + m * n)"
apply (case_tac "n = 1")
apply auto [1]
apply (drule_tac x = "a - 1" in spec)
apply force
apply (case_tac "a = 0")
apply (auto simp add: cong_0_1_nat) [1]
apply (rule iffI)
apply (drule cong_to_1_nat)
apply (unfold dvd_def)
apply auto [1]
apply (rule_tac x = k in exI)
apply (auto simp add: field_simps) [1]
apply (subst cong_altdef_nat)
apply (auto simp add: dvd_def)
done
lemma cong_le_nat: "(y::nat) <= x ==> [x = y] (mod n) <-> (∃q. x = q * n + y)"
apply (subst cong_altdef_nat)
apply assumption
apply (unfold dvd_def, auto simp add: field_simps)
apply (rule_tac x = k in exI)
apply auto
done
lemma cong_solve_nat: "(a::nat) ≠ 0 ==> EX x. [a * x = gcd a n] (mod n)"
apply (case_tac "n = 0")
apply force
apply (frule bezout_nat [of a n], auto)
apply (rule exI, erule ssubst)
apply (rule cong_trans_nat)
apply (rule cong_add_nat)
apply (subst mult_commute)
apply (rule cong_mult_self_nat)
prefer 2
apply simp
apply (rule cong_refl_nat)
apply (rule cong_refl_nat)
done
lemma cong_solve_int: "(a::int) ≠ 0 ==> EX x. [a * x = gcd a n] (mod n)"
apply (case_tac "n = 0")
apply (case_tac "a ≥ 0")
apply auto
apply (rule_tac x = "-1" in exI)
apply auto
apply (insert bezout_int [of a n], auto)
apply (rule exI)
apply (erule subst)
apply (rule cong_trans_int)
prefer 2
apply (rule cong_add_int)
apply (rule cong_refl_int)
apply (rule cong_sym_int)
apply (rule cong_mult_self_int)
apply simp
apply (subst mult_commute)
apply (rule cong_refl_int)
done
lemma cong_solve_dvd_nat:
assumes a: "(a::nat) ≠ 0" and b: "gcd a n dvd d"
shows "EX 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
hence "[(d div gcd a n) * (a * x) = (d div gcd a n) * gcd a n] (mod n)"
by (elim cong_scalar2_nat)
also from b have "(d div gcd a n) * gcd a n = d"
by (rule dvd_div_mult_self)
also have "(d div gcd a n) * (a * x) = a * (d div gcd a n * x)"
by auto
finally show ?thesis
by auto
qed
lemma cong_solve_dvd_int:
assumes a: "(a::int) ≠ 0" and b: "gcd a n dvd d"
shows "EX 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
hence "[(d div gcd a n) * (a * x) = (d div gcd a n) * gcd a n] (mod n)"
by (elim cong_scalar2_int)
also from b have "(d div gcd a n) * gcd a n = d"
by (rule dvd_div_mult_self)
also have "(d div gcd a n) * (a * x) = a * (d div gcd a n * x)"
by auto
finally show ?thesis
by auto
qed
lemma cong_solve_coprime_nat: "coprime (a::nat) n ==>
EX x. [a * x = 1] (mod n)"
apply (case_tac "a = 0")
apply force
apply (frule cong_solve_nat [of a n])
apply auto
done
lemma cong_solve_coprime_int: "coprime (a::int) n ==>
EX x. [a * x = 1] (mod n)"
apply (case_tac "a = 0")
apply auto
apply (case_tac "n ≥ 0")
apply auto
apply (subst cong_int_def, auto)
apply (frule cong_solve_int [of a n])
apply auto
done
lemma coprime_iff_invertible_nat: "m > (1::nat) ==> coprime a m =
(EX x. [a * x = 1] (mod m))"
apply (auto intro: cong_solve_coprime_nat)
apply (unfold cong_nat_def, auto intro: invertible_coprime_nat)
done
lemma coprime_iff_invertible_int: "m > (1::int) ==> coprime a m =
(EX x. [a * x = 1] (mod m))"
apply (auto intro: cong_solve_coprime_int)
apply (unfold cong_int_def)
apply (auto intro: invertible_coprime_int)
done
lemma coprime_iff_invertible'_int: "m > (1::int) ==> coprime a m =
(EX x. 0 <= x & x < m & [a * x = 1] (mod m))"
apply (subst coprime_iff_invertible_int)
apply auto
apply (auto simp add: cong_int_def)
apply (rule_tac x = "x mod m" in exI)
apply (auto simp add: mod_mult_right_eq [symmetric])
done
lemma cong_cong_lcm_nat: "[(x::nat) = y] (mod a) ==>
[x = y] (mod b) ==> [x = y] (mod lcm a b)"
apply (case_tac "y ≤ x")
apply (auto simp add: cong_altdef_nat lcm_least_nat) [1]
apply (rule cong_sym_nat)
apply (subst (asm) (1 2) cong_sym_eq_nat)
apply (auto simp add: cong_altdef_nat lcm_least_nat)
done
lemma cong_cong_lcm_int: "[(x::int) = y] (mod a) ==>
[x = y] (mod b) ==> [x = y] (mod lcm a b)"
by (auto simp add: cong_altdef_int lcm_least_int) [1]
lemma cong_cong_coprime_nat: "coprime a b ==> [(x::nat) = y] (mod a) ==>
[x = y] (mod b) ==> [x = y] (mod a * b)"
apply (frule (1) cong_cong_lcm_nat)back
apply (simp add: lcm_nat_def)
done
lemma cong_cong_coprime_int: "coprime a b ==> [(x::int) = y] (mod a) ==>
[x = y] (mod b) ==> [x = y] (mod a * b)"
apply (frule (1) cong_cong_lcm_int)back
apply (simp add: lcm_altdef_int cong_abs_int abs_mult [symmetric])
done
lemma cong_cong_setprod_coprime_nat [rule_format]: "finite A ==>
(ALL i:A. (ALL j:A. i ≠ j --> coprime (m i) (m j))) -->
(ALL i:A. [(x::nat) = y] (mod m i)) -->
[x = y] (mod (PROD i:A. m i))"
apply (induct set: finite)
apply auto
apply (rule cong_cong_coprime_nat)
apply (subst gcd_commute_nat)
apply (rule setprod_coprime_nat)
apply auto
done
lemma cong_cong_setprod_coprime_int [rule_format]: "finite A ==>
(ALL i:A. (ALL j:A. i ≠ j --> coprime (m i) (m j))) -->
(ALL i:A. [(x::int) = y] (mod m i)) -->
[x = y] (mod (PROD i:A. m i))"
apply (induct set: finite)
apply auto
apply (rule cong_cong_coprime_int)
apply (subst gcd_commute_int)
apply (rule setprod_coprime_int)
apply auto
done
lemma binary_chinese_remainder_aux_nat:
assumes a: "coprime (m1::nat) m2"
shows "EX b1 b2. [b1 = 1] (mod m1) ∧ [b1 = 0] (mod m2) ∧
[b2 = 0] (mod m1) ∧ [b2 = 1] (mod m2)"
proof -
from cong_solve_coprime_nat [OF a]
obtain x1 where one: "[m1 * x1 = 1] (mod m2)"
by auto
from a have b: "coprime m2 m1"
by (subst gcd_commute_nat)
from cong_solve_coprime_nat [OF b]
obtain x2 where two: "[m2 * x2 = 1] (mod m1)"
by auto
have "[m1 * x1 = 0] (mod m1)"
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
qed
lemma binary_chinese_remainder_aux_int:
assumes a: "coprime (m1::int) m2"
shows "EX b1 b2. [b1 = 1] (mod m1) ∧ [b1 = 0] (mod m2) ∧
[b2 = 0] (mod m1) ∧ [b2 = 1] (mod m2)"
proof -
from cong_solve_coprime_int [OF a]
obtain x1 where one: "[m1 * x1 = 1] (mod m2)"
by auto
from a have b: "coprime m2 m1"
by (subst gcd_commute_int)
from cong_solve_coprime_int [OF b]
obtain x2 where two: "[m2 * x2 = 1] (mod m1)"
by auto
have "[m1 * x1 = 0] (mod m1)"
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
qed
lemma binary_chinese_remainder_nat:
assumes a: "coprime (m1::nat) m2"
shows "EX x. [x = u1] (mod m1) ∧ [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)"
by blast
let ?x = "u1 * b1 + u2 * b2"
have "[?x = u1 * 1 + u2 * 0] (mod m1)"
apply (rule cong_add_nat)
apply (rule cong_scalar2_nat)
apply (rule `[b1 = 1] (mod m1)`)
apply (rule cong_scalar2_nat)
apply (rule `[b2 = 0] (mod m1)`)
done
hence "[?x = u1] (mod m1)" by simp
have "[?x = u1 * 0 + u2 * 1] (mod m2)"
apply (rule cong_add_nat)
apply (rule cong_scalar2_nat)
apply (rule `[b1 = 0] (mod m2)`)
apply (rule cong_scalar2_nat)
apply (rule `[b2 = 1] (mod m2)`)
done
hence "[?x = u2] (mod m2)" by simp
with `[?x = u1] (mod m1)` show ?thesis by blast
qed
lemma binary_chinese_remainder_int:
assumes a: "coprime (m1::int) m2"
shows "EX x. [x = u1] (mod m1) ∧ [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)"
by blast
let ?x = "u1 * b1 + u2 * b2"
have "[?x = u1 * 1 + u2 * 0] (mod m1)"
apply (rule cong_add_int)
apply (rule cong_scalar2_int)
apply (rule `[b1 = 1] (mod m1)`)
apply (rule cong_scalar2_int)
apply (rule `[b2 = 0] (mod m1)`)
done
hence "[?x = u1] (mod m1)" by simp
have "[?x = u1 * 0 + u2 * 1] (mod m2)"
apply (rule cong_add_int)
apply (rule cong_scalar2_int)
apply (rule `[b1 = 0] (mod m2)`)
apply (rule cong_scalar2_int)
apply (rule `[b2 = 1] (mod m2)`)
done
hence "[?x = u2] (mod m2)" by simp
with `[?x = u1] (mod m1)` show ?thesis by blast
qed
lemma cong_modulus_mult_nat: "[(x::nat) = y] (mod m * n) ==>
[x = y] (mod m)"
apply (case_tac "y ≤ x")
apply (simp add: cong_altdef_nat)
apply (erule dvd_mult_left)
apply (rule cong_sym_nat)
apply (subst (asm) cong_sym_eq_nat)
apply (simp add: cong_altdef_nat)
apply (erule dvd_mult_left)
done
lemma cong_modulus_mult_int: "[(x::int) = y] (mod m * n) ==>
[x = y] (mod m)"
apply (simp add: cong_altdef_int)
apply (erule dvd_mult_left)
done
lemma cong_less_modulus_unique_nat:
"[(x::nat) = y] (mod m) ==> x < m ==> y < m ==> x = y"
by (simp add: cong_nat_def)
lemma binary_chinese_remainder_unique_nat:
assumes a: "coprime (m1::nat) m2" and
nz: "m1 ≠ 0" "m2 ≠ 0"
shows "EX! x. x < m1 * m2 ∧ [x = u1] (mod m1) ∧ [x = u2] (mod m2)"
proof -
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)"
apply (rule cong_trans_nat)
prefer 2
apply (rule `[y = u1] (mod m1)`)
apply (rule cong_modulus_mult_nat)
apply (rule cong_mod_nat)
using nz apply auto
done
have two: "[?x = u2] (mod m2)"
apply (rule cong_trans_nat)
prefer 2
apply (rule `[y = u2] (mod m2)`)
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 ∧ [z = u1] (mod m1) ∧ [z = u2] (mod m2) -->
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 `[?x = u1] (mod m1)`)
apply (rule cong_sym_nat)
apply (rule `[z = u1] (mod m1)`)
done
moreover have "[?x = z] (mod m2)"
apply (rule cong_trans_nat)
apply (rule `[?x = u2] (mod m2)`)
apply (rule cong_sym_nat)
apply (rule `[z = u2] (mod m2)`)
done
ultimately have "[?x = z] (mod m1 * m2)"
by (auto intro: coprime_cong_mult_nat a)
with `z < m1 * m2` `?x < m1 * m2` show "z = ?x"
apply (intro cong_less_modulus_unique_nat)
apply (auto, erule cong_sym_nat)
done
qed
with less one two show ?thesis
by auto
qed
lemma chinese_remainder_aux_nat:
fixes A :: "'a set" and
m :: "'a => nat"
assumes fin: "finite A" and
cop: "ALL i : A. (ALL j : A. i ≠ j --> coprime (m i) (m j))"
shows "EX b. (ALL i : A.
[b i = 1] (mod m i) ∧ [b i = 0] (mod (PROD j : A - {i}. m j)))"
proof (rule finite_set_choice, rule fin, rule ballI)
fix i
assume "i : A"
with cop have "coprime (PROD j : A - {i}. m j) (m i)"
by (intro setprod_coprime_nat, auto)
hence "EX x. [(PROD j : A - {i}. m j) * x = 1] (mod m i)"
by (elim cong_solve_coprime_nat)
then obtain x where "[(PROD j : A - {i}. m j) * x = 1] (mod m i)"
by auto
moreover have "[(PROD j : A - {i}. m j) * x = 0]
(mod (PROD j : A - {i}. m j))"
by (subst mult_commute, rule cong_mult_self_nat)
ultimately show "∃a. [a = 1] (mod m i) ∧ [a = 0]
(mod setprod m (A - {i}))"
by blast
qed
lemma chinese_remainder_nat:
fixes A :: "'a set" and
m :: "'a => nat" and
u :: "'a => nat"
assumes
fin: "finite A" and
cop: "ALL i:A. (ALL j : A. i ≠ j --> coprime (m i) (m j))"
shows "EX x. (ALL i: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) ∧
[b i = 0] (mod (PROD j : A - {i}. m j))"
by blast
let ?x = "SUM i:A. (u i) * (b i)"
show "?thesis"
proof (rule exI, clarify)
fix i
assume a: "i : A"
show "[?x = u i] (mod m i)"
proof -
from fin a have "?x = (SUM j:{i}. u j * b j) +
(SUM j:A-{i}. u j * b j)"
by (subst setsum_Un_disjoint [symmetric], auto intro: setsum_cong)
hence "[?x = u i * b i + (SUM j:A-{i}. u j * b j)] (mod m i)"
by auto
also have "[u i * b i + (SUM j:A-{i}. u j * b j) =
u i * 1 + (SUM j:A-{i}. u j * 0)] (mod m i)"
apply (rule cong_add_nat)
apply (rule cong_scalar2_nat)
using bprop a apply blast
apply (rule cong_setsum_nat)
apply (rule cong_scalar2_nat)
using bprop apply auto
apply (rule cong_dvd_modulus_nat)
apply (drule (1) bspec)
apply (erule conjE)
apply assumption
apply (rule dvd_setprod)
using fin a apply auto
done
finally show ?thesis
by simp
qed
qed
qed
lemma coprime_cong_prod_nat [rule_format]: "finite A ==>
(ALL i: A. (ALL j: A. i ≠ j --> coprime (m i) (m j))) -->
(ALL i: A. [(x::nat) = y] (mod m i)) -->
[x = y] (mod (PROD i:A. m i))"
apply (induct set: finite)
apply auto
apply (erule (1) coprime_cong_mult_nat)
apply (subst gcd_commute_nat)
apply (rule setprod_coprime_nat)
apply auto
done
lemma chinese_remainder_unique_nat:
fixes A :: "'a set" and
m :: "'a => nat" and
u :: "'a => nat"
assumes
fin: "finite A" and
nz: "ALL i:A. m i ≠ 0" and
cop: "ALL i:A. (ALL j : A. i ≠ j --> coprime (m i) (m j))"
shows "EX! x. x < (PROD i:A. m i) ∧ (ALL i: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))"
by blast
let ?x = "y mod (PROD i:A. m i)"
from fin nz have prodnz: "(PROD i:A. m i) ≠ 0"
by auto
hence less: "?x < (PROD i:A. m i)"
by auto
have cong: "ALL i:A. [?x = u i] (mod m i)"
apply auto
apply (rule cong_trans_nat)
prefer 2
using one apply auto
apply (rule cong_dvd_modulus_nat)
apply (rule cong_mod_nat)
using prodnz apply auto
apply (rule dvd_setprod)
apply (rule fin)
apply assumption
done
have unique: "ALL z. z < (PROD i:A. m i) ∧
(ALL i:A. [z = u i] (mod m i)) --> z = ?x"
proof (clarify)
fix z
assume zless: "z < (PROD i:A. m i)"
assume zcong: "(ALL i:A. [z = u i] (mod m i))"
have "ALL i:A. [?x = z] (mod m i)"
apply clarify
apply (rule cong_trans_nat)
using cong apply (erule bspec)
apply (rule cong_sym_nat)
using zcong apply auto
done
with fin cop have "[?x = z] (mod (PROD i:A. m i))"
by (intro coprime_cong_prod_nat, auto)
with zless less show "z = ?x"
apply (intro cong_less_modulus_unique_nat)
apply (auto, erule cong_sym_nat)
done
qed
from less cong unique show ?thesis
by blast
qed
end