--- a/src/HOL/NewNumberTheory/Cong.thy Tue Sep 29 22:15:54 2009 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,1091 +0,0 @@
-(* Title: HOL/Library/Cong.thy
- ID:
- Authors: Christophe Tabacznyj, Lawrence C. Paulson, Amine Chaieb,
- Thomas M. Rasmussen, Jeremy Avigad
-
-
-Defines congruence (notation: [x = y] (mod z)) for natural numbers and
-integers.
-
-This file combines and revises a number of prior developments.
-
-The original theories "GCD" and "Primes" were by Christophe Tabacznyj
-and Lawrence C. Paulson, based on \cite{davenport92}. They introduced
-gcd, lcm, and prime for the natural numbers.
-
-The original theory "IntPrimes" was by Thomas M. Rasmussen, and
-extended gcd, lcm, primes to the integers. Amine Chaieb provided
-another extension of the notions to the integers, and added a number
-of results to "Primes" and "GCD".
-
-The original theory, "IntPrimes", by Thomas M. Rasmussen, defined and
-developed the congruence relations on the integers. The notion was
-extended to the natural numbers by Chiaeb. Jeremy Avigad combined
-these, revised and tidied them, made the development uniform for the
-natural numbers and the integers, and added a number of new theorems.
-
-*)
-
-
-header {* Congruence *}
-
-theory Cong
-imports GCD
-begin
-
-subsection {* Turn off One_nat_def *}
-
-lemma induct'_nat [case_names zero plus1, induct type: nat]:
- "\<lbrakk> P (0::nat); !!n. P n \<Longrightarrow> P (n + 1)\<rbrakk> \<Longrightarrow> P n"
-by (erule nat_induct) (simp add:One_nat_def)
-
-lemma cases_nat [case_names zero plus1, cases type: nat]:
- "P (0::nat) \<Longrightarrow> (!!n. P (n + 1)) \<Longrightarrow> 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 \<Longrightarrow>
- card (insert x A) = (if x \<in> A then (card A) else (card A) + 1)"
-by (auto simp add: insert_absorb)
-
-(* why wasn't card_insert_if a simp rule? *)
-declare card_insert_disjoint [simp del]
-
-lemma nat_1' [simp]: "nat 1 = 1"
-by simp
-
-(* For those annoying moments where Suc reappears, use Suc_eq_plus1 *)
-
-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 \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool" ("(1[_ = _] '(mod _'))")
-
-begin
-
-abbreviation
- notcong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool" ("(1[_ \<noteq> _] '(mod _'))")
-where
- "notcong x y m == (~cong x y m)"
-
-end
-
-(* definitions for the natural numbers *)
-
-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))"
-
-instance proof qed
-
-end
-
-
-(* definitions for the integers *)
-
-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))"
-
-instance proof qed
-
-end
-
-
-subsection {* Set up Transfer *}
-
-
-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))"
- 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 TransferMorphism_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 TransferMorphism_int_nat[transfer add return:
- transfer_int_nat_cong]
-
-
-subsection {* Congruence *}
-
-(* was zcong_0, etc. *)
-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) \<Longrightarrow> [b = a] (mod m)"
- by (unfold cong_nat_def, auto)
-
-lemma cong_sym_int: "[(a::int) = b] (mod m) \<Longrightarrow> [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) \<Longrightarrow> [b = c] (mod m) \<Longrightarrow> [a = c] (mod m)"
- by (unfold cong_nat_def, auto)
-
-lemma cong_trans_int [trans]:
- "[(a::int) = b] (mod m) \<Longrightarrow> [b = c] (mod m) \<Longrightarrow> [a = c] (mod m)"
- by (unfold cong_int_def, auto)
-
-lemma cong_add_nat:
- "[(a::nat) = b] (mod m) \<Longrightarrow> [c = d] (mod m) \<Longrightarrow> [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) \<Longrightarrow> [c = d] (mod m) \<Longrightarrow> [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) \<Longrightarrow> [c = d] (mod m) \<Longrightarrow> [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 \<Longrightarrow> b >= d \<Longrightarrow> [(a::int) = b] (mod m) \<Longrightarrow>
- [c = d] (mod m) \<Longrightarrow> [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) \<Longrightarrow> [c = d] (mod m) \<Longrightarrow> [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) \<Longrightarrow> [c = d] (mod m) \<Longrightarrow> [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) \<Longrightarrow> [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) \<Longrightarrow> [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)) \<longrightarrow>
- [(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)) \<longrightarrow>
- [(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)) \<longrightarrow>
- [(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)) \<longrightarrow>
- [(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) \<Longrightarrow> [a * k = b * k] (mod m)"
- by (rule cong_mult_nat, simp_all)
-
-lemma cong_scalar_int: "[(a::int)= b] (mod m) \<Longrightarrow> [a * k = b * k] (mod m)"
- by (rule cong_mult_int, simp_all)
-
-lemma cong_scalar2_nat: "[(a::nat)= b] (mod m) \<Longrightarrow> [k * a = k * b] (mod m)"
- by (rule cong_mult_nat, simp_all)
-
-lemma cong_scalar2_int: "[(a::int)= b] (mod m) \<Longrightarrow> [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 \<Longrightarrow>
- [(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) \<longleftrightarrow>
- (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 \<Longrightarrow> [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:
- "\<lbrakk> prime (p::int); 0 < a; [a * a = 1] (mod p) \<rbrakk>
- \<Longrightarrow> [a = 1] (mod p) \<or> [a = - 1] (mod p)"
- apply (simp only: cong_altdef_int)
- apply (subst prime_dvd_mult_eq_int [symmetric], assumption)
- (* any way around this? *)
- apply (subgoal_tac "a * a - 1 = (a - 1) * (a - -1)")
- apply (auto simp add: ring_simps)
-done
-
-lemma cong_mult_rcancel_int:
- "coprime k (m::int) \<Longrightarrow> [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) \<Longrightarrow> [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) \<Longrightarrow> [k * a = k * b] (mod m) = [a = b] (mod m)"
- by (simp add: mult_commute cong_mult_rcancel_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)"
- 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 \<le> (a::nat) \<Longrightarrow>
- a < m \<Longrightarrow> 0 \<le> b \<Longrightarrow> b < m \<Longrightarrow> [a = b] (mod m) \<Longrightarrow> a = b"
- by (auto simp add: cong_nat_def mod_pos_pos_trivial)
-
-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"
- by (auto simp add: cong_int_def mod_pos_pos_trivial)
-
-lemma cong_less_unique_nat:
- "0 < (m::nat) \<Longrightarrow> (\<exists>!b. 0 \<le> b \<and> b < m \<and> [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) \<Longrightarrow> (\<exists>!b. 0 \<le> b \<and> b < m \<and> [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)) = (\<exists>k. b = a + m * k)"
- apply (auto simp add: cong_altdef_int dvd_def ring_simps)
- apply (rule_tac [!] x = "-k" in exI, auto)
-done
-
-lemma cong_iff_lin_nat: "([(a::nat) = b] (mod m)) =
- (\<exists>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: ring_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: ring_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) \<Longrightarrow> 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) \<Longrightarrow> coprime a m \<Longrightarrow>
- coprime b m"
- by (auto simp add: cong_gcd_eq_nat)
-
-lemma cong_imp_coprime_int: "[(a::int) = b] (mod m) \<Longrightarrow> coprime a m \<Longrightarrow>
- 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 mod_dvd_mod_int:
- "0 < (m::int) \<Longrightarrow> m dvd b \<Longrightarrow> (a mod b mod m) = (a mod m)"
- apply (unfold dvd_def, auto)
- apply (rule mod_mod_cancel)
- apply auto
-done
-
-lemma mod_dvd_mod:
- assumes "0 < (m::nat)" and "m dvd b"
- shows "(a mod b mod m) = (a mod m)"
-
- apply (rule mod_dvd_mod_int [transferred])
- using prems apply auto
-done
-*)
-
-lemma cong_add_lcancel_nat:
- "[(a::nat) + x = a + y] (mod n) \<longleftrightarrow> [x = y] (mod n)"
- by (simp add: cong_iff_lin_nat)
-
-lemma cong_add_lcancel_int:
- "[(a::int) + x = a + y] (mod n) \<longleftrightarrow> [x = y] (mod n)"
- by (simp add: cong_iff_lin_int)
-
-lemma cong_add_rcancel_nat: "[(x::nat) + a = y + a] (mod n) \<longleftrightarrow> [x = y] (mod n)"
- by (simp add: cong_iff_lin_nat)
-
-lemma cong_add_rcancel_int: "[(x::int) + a = y + a] (mod n) \<longleftrightarrow> [x = y] (mod n)"
- by (simp add: cong_iff_lin_int)
-
-lemma cong_add_lcancel_0_nat: "[(a::nat) + x = a] (mod n) \<longleftrightarrow> [x = 0] (mod n)"
- by (simp add: cong_iff_lin_nat)
-
-lemma cong_add_lcancel_0_int: "[(a::int) + x = a] (mod n) \<longleftrightarrow> [x = 0] (mod n)"
- by (simp add: cong_iff_lin_int)
-
-lemma cong_add_rcancel_0_nat: "[x + (a::nat) = a] (mod n) \<longleftrightarrow> [x = 0] (mod n)"
- by (simp add: cong_iff_lin_nat)
-
-lemma cong_add_rcancel_0_int: "[x + (a::int) = a] (mod n) \<longleftrightarrow> [x = 0] (mod n)"
- by (simp add: cong_iff_lin_int)
-
-lemma cong_dvd_modulus_nat: "[(x::nat) = y] (mod m) \<Longrightarrow> n dvd m \<Longrightarrow>
- [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: ring_simps)
-done
-
-lemma cong_dvd_modulus_int: "[(x::int) = y] (mod m) \<Longrightarrow> n dvd m \<Longrightarrow>
- [x = y] (mod n)"
- 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"
- by (unfold cong_nat_def, auto simp add: dvd_eq_mod_eq_0)
-
-lemma cong_dvd_eq_int: "[(x::int) = y] (mod n) \<Longrightarrow> n dvd x \<longleftrightarrow> n dvd y"
- by (unfold cong_int_def, auto simp add: dvd_eq_mod_eq_0)
-
-lemma cong_mod_nat: "(n::nat) ~= 0 \<Longrightarrow> [a mod n = a] (mod n)"
- by (simp add: cong_nat_def)
-
-lemma cong_mod_int: "(n::int) ~= 0 \<Longrightarrow> [a mod n = a] (mod n)"
- by (simp add: cong_int_def)
-
-lemma mod_mult_cong_nat: "(a::nat) ~= 0 \<Longrightarrow> b ~= 0
- \<Longrightarrow> [x mod (a * b) = y] (mod a) \<longleftrightarrow> [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: ring_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 \<Longrightarrow> b ~= 0
- \<Longrightarrow> [x mod (a * b) = y] (mod a) \<longleftrightarrow> [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)) \<Longrightarrow> (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) \<longleftrightarrow>
- a = 0 \<and> n = 1 \<or> (\<exists>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: ring_simps) [1]
- apply (subst cong_altdef_nat)
- apply (auto simp add: dvd_def)
-done
-
-lemma cong_le_nat: "(y::nat) <= x \<Longrightarrow> [x = y] (mod n) \<longleftrightarrow> (\<exists>q. x = q * n + y)"
- apply (subst cong_altdef_nat)
- apply assumption
- apply (unfold dvd_def, auto simp add: ring_simps)
- apply (rule_tac x = k in exI)
- apply auto
-done
-
-lemma cong_solve_nat: "(a::nat) \<noteq> 0 \<Longrightarrow> 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) \<noteq> 0 \<Longrightarrow> EX x. [a * x = gcd a n] (mod n)"
- apply (case_tac "n = 0")
- apply (case_tac "a \<ge> 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) \<noteq> 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) \<noteq> 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 \<Longrightarrow>
- 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 \<Longrightarrow>
- EX x. [a * x = 1] (mod n)"
- apply (case_tac "a = 0")
- apply auto
- apply (case_tac "n \<ge> 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) \<Longrightarrow> 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) \<Longrightarrow> 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) \<Longrightarrow> 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) \<Longrightarrow>
- [x = y] (mod b) \<Longrightarrow> [x = y] (mod lcm a b)"
- apply (case_tac "y \<le> 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) \<Longrightarrow>
- [x = y] (mod b) \<Longrightarrow> [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 \<Longrightarrow> [(x::nat) = y] (mod a) \<Longrightarrow>
- [x = y] (mod b) \<Longrightarrow> [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 \<Longrightarrow> [(x::int) = y] (mod a) \<Longrightarrow>
- [x = y] (mod b) \<Longrightarrow> [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 \<Longrightarrow>
- (ALL i:A. (ALL j:A. i \<noteq> j \<longrightarrow> coprime (m i) (m j))) \<longrightarrow>
- (ALL i:A. [(x::nat) = y] (mod m i)) \<longrightarrow>
- [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 \<Longrightarrow>
- (ALL i:A. (ALL j:A. i \<noteq> j \<longrightarrow> coprime (m i) (m j))) \<longrightarrow>
- (ALL i:A. [(x::int) = y] (mod m i)) \<longrightarrow>
- [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) \<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)"
- 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) \<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)"
- 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) \<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)"
- 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) \<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)"
- 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) \<Longrightarrow>
- [x = y] (mod m)"
- apply (case_tac "y \<le> 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) \<Longrightarrow>
- [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) \<Longrightarrow> x < m \<Longrightarrow> y < m \<Longrightarrow> x = y"
- by (simp add: cong_nat_def)
-
-lemma binary_chinese_remainder_unique_nat:
- assumes a: "coprime (m1::nat) m2" and
- nz: "m1 \<noteq> 0" "m2 \<noteq> 0"
- shows "EX! 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)"
- 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 \<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 `[?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 \<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 : 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 "\<exists>a. [a = 1] (mod m i) \<and> [a = 0]
- (mod setprod m (A - {i}))"
- by blast
-qed
-
-lemma chinese_remainder_nat:
- fixes A :: "'a set" 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))"
-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 : 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 \<Longrightarrow>
- (ALL i: A. (ALL j: A. i \<noteq> j \<longrightarrow> coprime (m i) (m j))) \<longrightarrow>
- (ALL i: A. [(x::nat) = y] (mod m i)) \<longrightarrow>
- [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 \<Rightarrow> nat" and
- u :: "'a \<Rightarrow> nat"
- assumes
- fin: "finite A" and
- nz: "ALL i:A. m i \<noteq> 0" and
- cop: "ALL i:A. (ALL j : A. i \<noteq> j \<longrightarrow> coprime (m i) (m j))"
- shows "EX! x. x < (PROD i:A. m i) \<and> (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) \<noteq> 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) \<and>
- (ALL i:A. [z = u i] (mod m i)) \<longrightarrow> 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