(* Title: HOL/Quadratic_Reciprocity/EvenOdd.thy ID: $Id: EvenOdd.thy,v 1.11 2007/02/07 16:53:59 berghofe Exp $ Authors: Jeremy Avigad, David Gray, and Adam Kramer *) header {*Parity: Even and Odd Integers*} theory EvenOdd imports Int2 begin definition zOdd :: "int set" where "zOdd = {x. ∃k. x = 2 * k + 1}" definition zEven :: "int set" where "zEven = {x. ∃k. x = 2 * k}" subsection {* Some useful properties about even and odd *} lemma zOddI [intro?]: "x = 2 * k + 1 ==> x ∈ zOdd" and zOddE [elim?]: "x ∈ zOdd ==> (!!k. x = 2 * k + 1 ==> C) ==> C" by (auto simp add: zOdd_def) lemma zEvenI [intro?]: "x = 2 * k ==> x ∈ zEven" and zEvenE [elim?]: "x ∈ zEven ==> (!!k. x = 2 * k ==> C) ==> C" by (auto simp add: zEven_def) lemma one_not_even: "~(1 ∈ zEven)" proof assume "1 ∈ zEven" then obtain k :: int where "1 = 2 * k" .. then show False by arith qed lemma even_odd_conj: "~(x ∈ zOdd & x ∈ zEven)" proof - { fix a b assume "2 * (a::int) = 2 * (b::int) + 1" then have "2 * (a::int) - 2 * (b :: int) = 1" by arith then have "2 * (a - b) = 1" by (auto simp add: zdiff_zmult_distrib) moreover have "(2 * (a - b)):zEven" by (auto simp only: zEven_def) ultimately have False by (auto simp add: one_not_even) } then show ?thesis by (auto simp add: zOdd_def zEven_def) qed lemma even_odd_disj: "(x ∈ zOdd | x ∈ zEven)" by (simp add: zOdd_def zEven_def) arith lemma not_odd_impl_even: "~(x ∈ zOdd) ==> x ∈ zEven" using even_odd_disj by auto lemma odd_mult_odd_prop: "(x*y):zOdd ==> x ∈ zOdd" proof (rule classical) assume "¬ ?thesis" then have "x ∈ zEven" by (rule not_odd_impl_even) then obtain a where a: "x = 2 * a" .. assume "x * y : zOdd" then obtain b where "x * y = 2 * b + 1" .. with a have "2 * a * y = 2 * b + 1" by simp then have "2 * a * y - 2 * b = 1" by arith then have "2 * (a * y - b) = 1" by (auto simp add: zdiff_zmult_distrib) moreover have "(2 * (a * y - b)):zEven" by (auto simp only: zEven_def) ultimately have False by (auto simp add: one_not_even) then show ?thesis .. qed lemma odd_minus_one_even: "x ∈ zOdd ==> (x - 1):zEven" by (auto simp add: zOdd_def zEven_def) lemma even_div_2_prop1: "x ∈ zEven ==> (x mod 2) = 0" by (auto simp add: zEven_def) lemma even_div_2_prop2: "x ∈ zEven ==> (2 * (x div 2)) = x" by (auto simp add: zEven_def) lemma even_plus_even: "[| x ∈ zEven; y ∈ zEven |] ==> x + y ∈ zEven" apply (auto simp add: zEven_def) apply (auto simp only: zadd_zmult_distrib2 [symmetric]) done lemma even_times_either: "x ∈ zEven ==> x * y ∈ zEven" by (auto simp add: zEven_def) lemma even_minus_even: "[| x ∈ zEven; y ∈ zEven |] ==> x - y ∈ zEven" apply (auto simp add: zEven_def) apply (auto simp only: zdiff_zmult_distrib2 [symmetric]) done lemma odd_minus_odd: "[| x ∈ zOdd; y ∈ zOdd |] ==> x - y ∈ zEven" apply (auto simp add: zOdd_def zEven_def) apply (auto simp only: zdiff_zmult_distrib2 [symmetric]) done lemma even_minus_odd: "[| x ∈ zEven; y ∈ zOdd |] ==> x - y ∈ zOdd" apply (auto simp add: zOdd_def zEven_def) apply (rule_tac x = "k - ka - 1" in exI) apply auto done lemma odd_minus_even: "[| x ∈ zOdd; y ∈ zEven |] ==> x - y ∈ zOdd" apply (auto simp add: zOdd_def zEven_def) apply (auto simp only: zdiff_zmult_distrib2 [symmetric]) done lemma odd_times_odd: "[| x ∈ zOdd; y ∈ zOdd |] ==> x * y ∈ zOdd" apply (auto simp add: zOdd_def zadd_zmult_distrib zadd_zmult_distrib2) apply (rule_tac x = "2 * ka * k + ka + k" in exI) apply (auto simp add: zadd_zmult_distrib) done lemma odd_iff_not_even: "(x ∈ zOdd) = (~ (x ∈ zEven))" using even_odd_conj even_odd_disj by auto lemma even_product: "x * y ∈ zEven ==> x ∈ zEven | y ∈ zEven" using odd_iff_not_even odd_times_odd by auto lemma even_diff: "x - y ∈ zEven = ((x ∈ zEven) = (y ∈ zEven))" proof assume xy: "x - y ∈ zEven" { assume x: "x ∈ zEven" have "y ∈ zEven" proof (rule classical) assume "¬ ?thesis" then have "y ∈ zOdd" by (simp add: odd_iff_not_even) with x have "x - y ∈ zOdd" by (simp add: even_minus_odd) with xy have False by (auto simp add: odd_iff_not_even) then show ?thesis .. qed } moreover { assume y: "y ∈ zEven" have "x ∈ zEven" proof (rule classical) assume "¬ ?thesis" then have "x ∈ zOdd" by (auto simp add: odd_iff_not_even) with y have "x - y ∈ zOdd" by (simp add: odd_minus_even) with xy have False by (auto simp add: odd_iff_not_even) then show ?thesis .. qed } ultimately show "(x ∈ zEven) = (y ∈ zEven)" by (auto simp add: odd_iff_not_even even_minus_even odd_minus_odd even_minus_odd odd_minus_even) next assume "(x ∈ zEven) = (y ∈ zEven)" then show "x - y ∈ zEven" by (auto simp add: odd_iff_not_even even_minus_even odd_minus_odd even_minus_odd odd_minus_even) qed lemma neg_one_even_power: "[| x ∈ zEven; 0 ≤ x |] ==> (-1::int)^(nat x) = 1" proof - assume "x ∈ zEven" and "0 ≤ x" from `x ∈ zEven` obtain a where "x = 2 * a" .. with `0 ≤ x` have "0 ≤ a" by simp from `0 ≤ x` and `x = 2 * a` have "nat x = nat (2 * a)" by simp also from `x = 2 * a` have "nat (2 * a) = 2 * nat a" by (simp add: nat_mult_distrib) finally have "(-1::int)^nat x = (-1)^(2 * nat a)" by simp also have "... = ((-1::int)^2)^ (nat a)" by (simp add: zpower_zpower [symmetric]) also have "(-1::int)^2 = 1" by simp finally show ?thesis by simp qed lemma neg_one_odd_power: "[| x ∈ zOdd; 0 ≤ x |] ==> (-1::int)^(nat x) = -1" proof - assume "x ∈ zOdd" and "0 ≤ x" from `x ∈ zOdd` obtain a where "x = 2 * a + 1" .. with `0 ≤ x` have a: "0 ≤ a" by simp with `0 ≤ x` and `x = 2 * a + 1` have "nat x = nat (2 * a + 1)" by simp also from a have "nat (2 * a + 1) = 2 * nat a + 1" by (auto simp add: nat_mult_distrib nat_add_distrib) finally have "(-1::int)^nat x = (-1)^(2 * nat a + 1)" by simp also have "... = ((-1::int)^2)^ (nat a) * (-1)^1" by (auto simp add: zpower_zpower [symmetric] zpower_zadd_distrib) also have "(-1::int)^2 = 1" by simp finally show ?thesis by simp qed lemma neg_one_power_parity: "[| 0 ≤ x; 0 ≤ y; (x ∈ zEven) = (y ∈ zEven) |] ==> (-1::int)^(nat x) = (-1::int)^(nat y)" using even_odd_disj [of x] even_odd_disj [of y] by (auto simp add: neg_one_even_power neg_one_odd_power) lemma one_not_neg_one_mod_m: "2 < m ==> ~([1 = -1] (mod m))" by (auto simp add: zcong_def zdvd_not_zless) lemma even_div_2_l: "[| y ∈ zEven; x < y |] ==> x div 2 < y div 2" proof - assume "y ∈ zEven" and "x < y" from `y ∈ zEven` obtain k where k: "y = 2 * k" .. with `x < y` have "x < 2 * k" by simp then have "x div 2 < k" by (auto simp add: div_prop1) also have "k = (2 * k) div 2" by simp finally have "x div 2 < 2 * k div 2" by simp with k show ?thesis by simp qed lemma even_sum_div_2: "[| x ∈ zEven; y ∈ zEven |] ==> (x + y) div 2 = x div 2 + y div 2" by (auto simp add: zEven_def, auto simp add: zdiv_zadd1_eq) lemma even_prod_div_2: "[| x ∈ zEven |] ==> (x * y) div 2 = (x div 2) * y" by (auto simp add: zEven_def) (* An odd prime is greater than 2 *) lemma zprime_zOdd_eq_grt_2: "zprime p ==> (p ∈ zOdd) = (2 < p)" apply (auto simp add: zOdd_def zprime_def) apply (drule_tac x = 2 in allE) using odd_iff_not_even [of p] apply (auto simp add: zOdd_def zEven_def) done (* Powers of -1 and parity *) lemma neg_one_special: "finite A ==> ((-1 :: int) ^ card A) * (-1 ^ card A) = 1" by (induct set: finite) auto lemma neg_one_power: "(-1::int)^n = 1 | (-1::int)^n = -1" by (induct n) auto lemma neg_one_power_eq_mod_m: "[| 2 < m; [(-1::int)^j = (-1::int)^k] (mod m) |] ==> ((-1::int)^j = (-1::int)^k)" using neg_one_power [of j] and insert neg_one_power [of k] by (auto simp add: one_not_neg_one_mod_m zcong_sym) end
lemma zOddI:
x = 2 * k + 1 ==> x ∈ zOdd
and zOddE:
[| x ∈ zOdd; !!k. x = 2 * k + 1 ==> C |] ==> C
lemma zEvenI:
x = 2 * k ==> x ∈ zEven
and zEvenE:
[| x ∈ zEven; !!k. x = 2 * k ==> C |] ==> C
lemma one_not_even:
1 ∉ zEven
lemma even_odd_conj:
¬ (x ∈ zOdd ∧ x ∈ zEven)
lemma even_odd_disj:
x ∈ zOdd ∨ x ∈ zEven
lemma not_odd_impl_even:
x ∉ zOdd ==> x ∈ zEven
lemma odd_mult_odd_prop:
x * y ∈ zOdd ==> x ∈ zOdd
lemma odd_minus_one_even:
x ∈ zOdd ==> x - 1 ∈ zEven
lemma even_div_2_prop1:
x ∈ zEven ==> x mod 2 = 0
lemma even_div_2_prop2:
x ∈ zEven ==> 2 * (x div 2) = x
lemma even_plus_even:
[| x ∈ zEven; y ∈ zEven |] ==> x + y ∈ zEven
lemma even_times_either:
x ∈ zEven ==> x * y ∈ zEven
lemma even_minus_even:
[| x ∈ zEven; y ∈ zEven |] ==> x - y ∈ zEven
lemma odd_minus_odd:
[| x ∈ zOdd; y ∈ zOdd |] ==> x - y ∈ zEven
lemma even_minus_odd:
[| x ∈ zEven; y ∈ zOdd |] ==> x - y ∈ zOdd
lemma odd_minus_even:
[| x ∈ zOdd; y ∈ zEven |] ==> x - y ∈ zOdd
lemma odd_times_odd:
[| x ∈ zOdd; y ∈ zOdd |] ==> x * y ∈ zOdd
lemma odd_iff_not_even:
(x ∈ zOdd) = (x ∉ zEven)
lemma even_product:
x * y ∈ zEven ==> x ∈ zEven ∨ y ∈ zEven
lemma even_diff:
(x - y ∈ zEven) = ((x ∈ zEven) = (y ∈ zEven))
lemma neg_one_even_power:
[| x ∈ zEven; 0 ≤ x |] ==> -1 ^ nat x = 1
lemma neg_one_odd_power:
[| x ∈ zOdd; 0 ≤ x |] ==> -1 ^ nat x = -1
lemma neg_one_power_parity:
[| 0 ≤ x; 0 ≤ y; (x ∈ zEven) = (y ∈ zEven) |] ==> -1 ^ nat x = -1 ^ nat y
lemma one_not_neg_one_mod_m:
2 < m ==> ¬ [1 = -1] (mod m)
lemma even_div_2_l:
[| y ∈ zEven; x < y |] ==> x div 2 < y div 2
lemma even_sum_div_2:
[| x ∈ zEven; y ∈ zEven |] ==> (x + y) div 2 = x div 2 + y div 2
lemma even_prod_div_2:
x ∈ zEven ==> x * y div 2 = x div 2 * y
lemma zprime_zOdd_eq_grt_2:
zprime p ==> (p ∈ zOdd) = (2 < p)
lemma neg_one_special:
finite A ==> -1 ^ card A * -1 ^ card A = 1
lemma neg_one_power:
-1 ^ n = 1 ∨ -1 ^ n = -1
lemma neg_one_power_eq_mod_m:
[| 2 < m; [-1 ^ j = -1 ^ k] (mod m) |] ==> -1 ^ j = -1 ^ k