(* Title: HOL/Quadratic_Reciprocity/Gauss.thy ID: $Id: Int2.thy,v 1.12 2007/06/11 09:06:07 chaieb Exp $ Authors: Jeremy Avigad, David Gray, and Adam Kramer *) header {*Integers: Divisibility and Congruences*} theory Int2 imports Finite2 WilsonRuss begin definition MultInv :: "int => int => int" where "MultInv p x = x ^ nat (p - 2)" subsection {* Useful lemmas about dvd and powers *} lemma zpower_zdvd_prop1: "0 < n ==> p dvd y ==> p dvd ((y::int) ^ n)" by (induct n) (auto simp add: zdvd_zmult zdvd_zmult2 [of p y]) lemma zdvd_bounds: "n dvd m ==> m ≤ (0::int) | n ≤ m" proof - assume "n dvd m" then have "~(0 < m & m < n)" using zdvd_not_zless [of m n] by auto then show ?thesis by auto qed lemma zprime_zdvd_zmult_better: "[| zprime p; p dvd (m * n) |] ==> (p dvd m) | (p dvd n)" apply (cases "0 ≤ m") apply (simp add: zprime_zdvd_zmult) apply (insert zprime_zdvd_zmult [of "-m" p n]) apply auto done lemma zpower_zdvd_prop2: "zprime p ==> p dvd ((y::int) ^ n) ==> 0 < n ==> p dvd y" apply (induct n) apply simp apply (frule zprime_zdvd_zmult_better) apply simp apply force done lemma div_prop1: "[| 0 < z; (x::int) < y * z |] ==> x div z < y" proof - assume "0 < z" then have modth: "x mod z ≥ 0" by simp have "(x div z) * z ≤ (x div z) * z" by simp then have "(x div z) * z ≤ (x div z) * z + x mod z" using modth by arith also have "… = x" by (auto simp add: zmod_zdiv_equality [symmetric] zmult_ac) also assume "x < y * z" finally show ?thesis by (auto simp add: prems mult_less_cancel_right, insert prems, arith) qed lemma div_prop2: "[| 0 < z; (x::int) < (y * z) + z |] ==> x div z ≤ y" proof - assume "0 < z" and "x < (y * z) + z" then have "x < (y + 1) * z" by (auto simp add: int_distrib) then have "x div z < y + 1" apply - apply (rule_tac y = "y + 1" in div_prop1) apply (auto simp add: prems) done then show ?thesis by auto qed lemma zdiv_leq_prop: "[| 0 < y |] ==> y * (x div y) ≤ (x::int)" proof- assume "0 < y" from zmod_zdiv_equality have "x = y * (x div y) + x mod y" by auto moreover have "0 ≤ x mod y" by (auto simp add: prems pos_mod_sign) ultimately show ?thesis by arith qed subsection {* Useful properties of congruences *} lemma zcong_eq_zdvd_prop: "[x = 0](mod p) = (p dvd x)" by (auto simp add: zcong_def) lemma zcong_id: "[m = 0] (mod m)" by (auto simp add: zcong_def zdvd_0_right) lemma zcong_shift: "[a = b] (mod m) ==> [a + c = b + c] (mod m)" by (auto simp add: zcong_refl zcong_zadd) lemma zcong_zpower: "[x = y](mod m) ==> [x^z = y^z](mod m)" by (induct z) (auto simp add: zcong_zmult) lemma zcong_eq_trans: "[| [a = b](mod m); b = c; [c = d](mod m) |] ==> [a = d](mod m)" apply (erule zcong_trans) apply simp done lemma aux1: "a - b = (c::int) ==> a = c + b" by auto lemma zcong_zmult_prop1: "[a = b](mod m) ==> ([c = a * d](mod m) = [c = b * d] (mod m))" apply (auto simp add: zcong_def dvd_def) apply (rule_tac x = "ka + k * d" in exI) apply (drule aux1)+ apply (auto simp add: int_distrib) apply (rule_tac x = "ka - k * d" in exI) apply (drule aux1)+ apply (auto simp add: int_distrib) done lemma zcong_zmult_prop2: "[a = b](mod m) ==> ([c = d * a](mod m) = [c = d * b] (mod m))" by (auto simp add: zmult_ac zcong_zmult_prop1) lemma zcong_zmult_prop3: "[| zprime p; ~[x = 0] (mod p); ~[y = 0] (mod p) |] ==> ~[x * y = 0] (mod p)" apply (auto simp add: zcong_def) apply (drule zprime_zdvd_zmult_better, auto) done lemma zcong_less_eq: "[| 0 < x; 0 < y; 0 < m; [x = y] (mod m); x < m; y < m |] ==> x = y" apply (simp add: zcong_zmod_eq) apply (subgoal_tac "(x mod m) = x") apply (subgoal_tac "(y mod m) = y") apply simp apply (rule_tac [1-2] mod_pos_pos_trivial) apply auto done lemma zcong_neg_1_impl_ne_1: "[| 2 < p; [x = -1] (mod p) |] ==> ~([x = 1] (mod p))" proof assume "2 < p" and "[x = 1] (mod p)" and "[x = -1] (mod p)" then have "[1 = -1] (mod p)" apply (auto simp add: zcong_sym) apply (drule zcong_trans, auto) done then have "[1 + 1 = -1 + 1] (mod p)" by (simp only: zcong_shift) then have "[2 = 0] (mod p)" by auto then have "p dvd 2" by (auto simp add: dvd_def zcong_def) with prems show False by (auto simp add: zdvd_not_zless) qed lemma zcong_zero_equiv_div: "[a = 0] (mod m) = (m dvd a)" by (auto simp add: zcong_def) lemma zcong_zprime_prod_zero: "[| zprime p; 0 < a |] ==> [a * b = 0] (mod p) ==> [a = 0] (mod p) | [b = 0] (mod p)" by (auto simp add: zcong_zero_equiv_div zprime_zdvd_zmult) lemma zcong_zprime_prod_zero_contra: "[| zprime p; 0 < a |] ==> ~[a = 0](mod p) & ~[b = 0](mod p) ==> ~[a * b = 0] (mod p)" apply auto apply (frule_tac a = a and b = b and p = p in zcong_zprime_prod_zero) apply auto done lemma zcong_not_zero: "[| 0 < x; x < m |] ==> ~[x = 0] (mod m)" by (auto simp add: zcong_zero_equiv_div zdvd_not_zless) lemma zcong_zero: "[| 0 ≤ x; x < m; [x = 0](mod m) |] ==> x = 0" apply (drule order_le_imp_less_or_eq, auto) apply (frule_tac m = m in zcong_not_zero) apply auto done lemma all_relprime_prod_relprime: "[| finite A; ∀x ∈ A. (zgcd(x,y) = 1) |] ==> zgcd (setprod id A,y) = 1" by (induct set: finite) (auto simp add: zgcd_zgcd_zmult) subsection {* Some properties of MultInv *} lemma MultInv_prop1: "[| 2 < p; [x = y] (mod p) |] ==> [(MultInv p x) = (MultInv p y)] (mod p)" by (auto simp add: MultInv_def zcong_zpower) lemma MultInv_prop2: "[| 2 < p; zprime p; ~([x = 0](mod p)) |] ==> [(x * (MultInv p x)) = 1] (mod p)" proof (simp add: MultInv_def zcong_eq_zdvd_prop) assume "2 < p" and "zprime p" and "~ p dvd x" have "x * x ^ nat (p - 2) = x ^ (nat (p - 2) + 1)" by auto also from prems have "nat (p - 2) + 1 = nat (p - 2 + 1)" by (simp only: nat_add_distrib) also have "p - 2 + 1 = p - 1" by arith finally have "[x * x ^ nat (p - 2) = x ^ nat (p - 1)] (mod p)" by (rule ssubst, auto) also from prems have "[x ^ nat (p - 1) = 1] (mod p)" by (auto simp add: Little_Fermat) finally (zcong_trans) show "[x * x ^ nat (p - 2) = 1] (mod p)" . qed lemma MultInv_prop2a: "[| 2 < p; zprime p; ~([x = 0](mod p)) |] ==> [(MultInv p x) * x = 1] (mod p)" by (auto simp add: MultInv_prop2 zmult_ac) lemma aux_1: "2 < p ==> ((nat p) - 2) = (nat (p - 2))" by (simp add: nat_diff_distrib) lemma aux_2: "2 < p ==> 0 < nat (p - 2)" by auto lemma MultInv_prop3: "[| 2 < p; zprime p; ~([x = 0](mod p)) |] ==> ~([MultInv p x = 0](mod p))" apply (auto simp add: MultInv_def zcong_eq_zdvd_prop aux_1) apply (drule aux_2) apply (drule zpower_zdvd_prop2, auto) done lemma aux__1: "[| 2 < p; zprime p; ~([x = 0](mod p))|] ==> [(MultInv p (MultInv p x)) = (x * (MultInv p x) * (MultInv p (MultInv p x)))] (mod p)" apply (drule MultInv_prop2, auto) apply (drule_tac k = "MultInv p (MultInv p x)" in zcong_scalar, auto) apply (auto simp add: zcong_sym) done lemma aux__2: "[| 2 < p; zprime p; ~([x = 0](mod p))|] ==> [(x * (MultInv p x) * (MultInv p (MultInv p x))) = x] (mod p)" apply (frule MultInv_prop3, auto) apply (insert MultInv_prop2 [of p "MultInv p x"], auto) apply (drule MultInv_prop2, auto) apply (drule_tac k = x in zcong_scalar2, auto) apply (auto simp add: zmult_ac) done lemma MultInv_prop4: "[| 2 < p; zprime p; ~([x = 0](mod p)) |] ==> [(MultInv p (MultInv p x)) = x] (mod p)" apply (frule aux__1, auto) apply (drule aux__2, auto) apply (drule zcong_trans, auto) done lemma MultInv_prop5: "[| 2 < p; zprime p; ~([x = 0](mod p)); ~([y = 0](mod p)); [(MultInv p x) = (MultInv p y)] (mod p) |] ==> [x = y] (mod p)" apply (drule_tac a = "MultInv p x" and b = "MultInv p y" and m = p and k = x in zcong_scalar) apply (insert MultInv_prop2 [of p x], simp) apply (auto simp only: zcong_sym [of "MultInv p x * x"]) apply (auto simp add: zmult_ac) apply (drule zcong_trans, auto) apply (drule_tac a = "x * MultInv p y" and k = y in zcong_scalar, auto) apply (insert MultInv_prop2a [of p y], auto simp add: zmult_ac) apply (insert zcong_zmult_prop2 [of "y * MultInv p y" 1 p y x]) apply (auto simp add: zcong_sym) done lemma MultInv_zcong_prop1: "[| 2 < p; [j = k] (mod p) |] ==> [a * MultInv p j = a * MultInv p k] (mod p)" by (drule MultInv_prop1, auto simp add: zcong_scalar2) lemma aux___1: "[j = a * MultInv p k] (mod p) ==> [j * k = a * MultInv p k * k] (mod p)" by (auto simp add: zcong_scalar) lemma aux___2: "[|2 < p; zprime p; ~([k = 0](mod p)); [j * k = a * MultInv p k * k] (mod p) |] ==> [j * k = a] (mod p)" apply (insert MultInv_prop2a [of p k] zcong_zmult_prop2 [of "MultInv p k * k" 1 p "j * k" a]) apply (auto simp add: zmult_ac) done lemma aux___3: "[j * k = a] (mod p) ==> [(MultInv p j) * j * k = (MultInv p j) * a] (mod p)" by (auto simp add: zmult_assoc zcong_scalar2) lemma aux___4: "[|2 < p; zprime p; ~([j = 0](mod p)); [(MultInv p j) * j * k = (MultInv p j) * a] (mod p) |] ==> [k = a * (MultInv p j)] (mod p)" apply (insert MultInv_prop2a [of p j] zcong_zmult_prop1 [of "MultInv p j * j" 1 p "MultInv p j * a" k]) apply (auto simp add: zmult_ac zcong_sym) done lemma MultInv_zcong_prop2: "[| 2 < p; zprime p; ~([k = 0](mod p)); ~([j = 0](mod p)); [j = a * MultInv p k] (mod p) |] ==> [k = a * MultInv p j] (mod p)" apply (drule aux___1) apply (frule aux___2, auto) by (drule aux___3, drule aux___4, auto) lemma MultInv_zcong_prop3: "[| 2 < p; zprime p; ~([a = 0](mod p)); ~([k = 0](mod p)); ~([j = 0](mod p)); [a * MultInv p j = a * MultInv p k] (mod p) |] ==> [j = k] (mod p)" apply (auto simp add: zcong_eq_zdvd_prop [of a p]) apply (frule zprime_imp_zrelprime, auto) apply (insert zcong_cancel2 [of p a "MultInv p j" "MultInv p k"], auto) apply (drule MultInv_prop5, auto) done end
lemma zpower_zdvd_prop1:
[| 0 < n; p dvd y |] ==> p dvd y ^ n
lemma zdvd_bounds:
n dvd m ==> m ≤ 0 ∨ n ≤ m
lemma zprime_zdvd_zmult_better:
[| zprime p; p dvd m * n |] ==> p dvd m ∨ p dvd n
lemma zpower_zdvd_prop2:
[| zprime p; p dvd y ^ n; 0 < n |] ==> p dvd y
lemma div_prop1:
[| 0 < z; x < y * z |] ==> x div z < y
lemma div_prop2:
[| 0 < z; x < y * z + z |] ==> x div z ≤ y
lemma zdiv_leq_prop:
0 < y ==> y * (x div y) ≤ x
lemma zcong_eq_zdvd_prop:
[x = 0] (mod p) = (p dvd x)
lemma zcong_id:
[m = 0] (mod m)
lemma zcong_shift:
[a = b] (mod m) ==> [a + c = b + c] (mod m)
lemma zcong_zpower:
[x = y] (mod m) ==> [x ^ z = y ^ z] (mod m)
lemma zcong_eq_trans:
[| [a = b] (mod m); b = c; [c = d] (mod m) |] ==> [a = d] (mod m)
lemma aux1:
a - b = c ==> a = c + b
lemma zcong_zmult_prop1:
[a = b] (mod m) ==> [c = a * d] (mod m) = [c = b * d] (mod m)
lemma zcong_zmult_prop2:
[a = b] (mod m) ==> [c = d * a] (mod m) = [c = d * b] (mod m)
lemma zcong_zmult_prop3:
[| zprime p; ¬ [x = 0] (mod p); ¬ [y = 0] (mod p) |] ==> ¬ [x * y = 0] (mod p)
lemma zcong_less_eq:
[| 0 < x; 0 < y; 0 < m; [x = y] (mod m); x < m; y < m |] ==> x = y
lemma zcong_neg_1_impl_ne_1:
[| 2 < p; [x = -1] (mod p) |] ==> ¬ [x = 1] (mod p)
lemma zcong_zero_equiv_div:
[a = 0] (mod m) = (m dvd a)
lemma zcong_zprime_prod_zero:
[| zprime p; 0 < a; [a * b = 0] (mod p) |] ==> [a = 0] (mod p) ∨ [b = 0] (mod p)
lemma zcong_zprime_prod_zero_contra:
[| zprime p; 0 < a; ¬ [a = 0] (mod p) ∧ ¬ [b = 0] (mod p) |]
==> ¬ [a * b = 0] (mod p)
lemma zcong_not_zero:
[| 0 < x; x < m |] ==> ¬ [x = 0] (mod m)
lemma zcong_zero:
[| 0 ≤ x; x < m; [x = 0] (mod m) |] ==> x = 0
lemma all_relprime_prod_relprime:
[| finite A; ∀x∈A. zgcd (x, y) = 1 |] ==> zgcd (setprod id A, y) = 1
lemma MultInv_prop1:
[| 2 < p; [x = y] (mod p) |] ==> [MultInv p x = MultInv p y] (mod p)
lemma MultInv_prop2:
[| 2 < p; zprime p; ¬ [x = 0] (mod p) |] ==> [x * MultInv p x = 1] (mod p)
lemma MultInv_prop2a:
[| 2 < p; zprime p; ¬ [x = 0] (mod p) |] ==> [MultInv p x * x = 1] (mod p)
lemma aux_1:
2 < p ==> nat p - 2 = nat (p - 2)
lemma aux_2:
2 < p ==> 0 < nat (p - 2)
lemma MultInv_prop3:
[| 2 < p; zprime p; ¬ [x = 0] (mod p) |] ==> ¬ [MultInv p x = 0] (mod p)
lemma aux__1:
[| 2 < p; zprime p; ¬ [x = 0] (mod p) |]
==> [MultInv p
(MultInv p x) = x * MultInv p x * MultInv p (MultInv p x)] (mod p)
lemma aux__2:
[| 2 < p; zprime p; ¬ [x = 0] (mod p) |]
==> [x * MultInv p x * MultInv p (MultInv p x) = x] (mod p)
lemma MultInv_prop4:
[| 2 < p; zprime p; ¬ [x = 0] (mod p) |]
==> [MultInv p (MultInv p x) = x] (mod p)
lemma MultInv_prop5:
[| 2 < p; zprime p; ¬ [x = 0] (mod p); ¬ [y = 0] (mod p);
[MultInv p x = MultInv p y] (mod p) |]
==> [x = y] (mod p)
lemma MultInv_zcong_prop1:
[| 2 < p; [j = k] (mod p) |] ==> [a * MultInv p j = a * MultInv p k] (mod p)
lemma aux___1:
[j = a * MultInv p k] (mod p) ==> [j * k = a * MultInv p k * k] (mod p)
lemma aux___2:
[| 2 < p; zprime p; ¬ [k = 0] (mod p); [j * k = a * MultInv p k * k] (mod p) |]
==> [j * k = a] (mod p)
lemma aux___3:
[j * k = a] (mod p) ==> [MultInv p j * j * k = MultInv p j * a] (mod p)
lemma aux___4:
[| 2 < p; zprime p; ¬ [j = 0] (mod p);
[MultInv p j * j * k = MultInv p j * a] (mod p) |]
==> [k = a * MultInv p j] (mod p)
lemma MultInv_zcong_prop2:
[| 2 < p; zprime p; ¬ [k = 0] (mod p); ¬ [j = 0] (mod p);
[j = a * MultInv p k] (mod p) |]
==> [k = a * MultInv p j] (mod p)
lemma MultInv_zcong_prop3:
[| 2 < p; zprime p; ¬ [a = 0] (mod p); ¬ [k = 0] (mod p); ¬ [j = 0] (mod p);
[a * MultInv p j = a * MultInv p k] (mod p) |]
==> [j = k] (mod p)