(* Title: HOL/Library/Parity.thy ID: $Id: Parity.thy,v 1.19 2008/03/12 07:47:35 haftmann Exp $ Author: Jeremy Avigad, Jacques D. Fleuriot *) header {* Even and Odd for int and nat *} theory Parity imports ATP_Linkup begin class even_odd = type + fixes even :: "'a => bool" abbreviation odd :: "'a::even_odd => bool" where "odd x ≡ ¬ even x" instantiation nat and int :: even_odd begin definition even_def [presburger]: "even x <-> (x::int) mod 2 = 0" definition even_nat_def [presburger]: "even x <-> even (int x)" instance .. end subsection {* Even and odd are mutually exclusive *} lemma int_pos_lt_two_imp_zero_or_one: "0 <= x ==> (x::int) < 2 ==> x = 0 | x = 1" by presburger lemma neq_one_mod_two [simp, presburger]: "((x::int) mod 2 ~= 0) = (x mod 2 = 1)" by presburger subsection {* Behavior under integer arithmetic operations *} lemma even_times_anything: "even (x::int) ==> even (x * y)" by (simp add: even_def zmod_zmult1_eq') lemma anything_times_even: "even (y::int) ==> even (x * y)" by (simp add: even_def zmod_zmult1_eq) lemma odd_times_odd: "odd (x::int) ==> odd y ==> odd (x * y)" by (simp add: even_def zmod_zmult1_eq) lemma even_product[presburger]: "even((x::int) * y) = (even x | even y)" apply (auto simp add: even_times_anything anything_times_even) apply (rule ccontr) apply (auto simp add: odd_times_odd) done lemma even_plus_even: "even (x::int) ==> even y ==> even (x + y)" by presburger lemma even_plus_odd: "even (x::int) ==> odd y ==> odd (x + y)" by presburger lemma odd_plus_even: "odd (x::int) ==> even y ==> odd (x + y)" by presburger lemma odd_plus_odd: "odd (x::int) ==> odd y ==> even (x + y)" by presburger lemma even_sum[presburger]: "even ((x::int) + y) = ((even x & even y) | (odd x & odd y))" by presburger lemma even_neg[presburger]: "even (-(x::int)) = even x" by presburger lemma even_difference: "even ((x::int) - y) = ((even x & even y) | (odd x & odd y))" by presburger lemma even_pow_gt_zero: "even (x::int) ==> 0 < n ==> even (x^n)" by (induct n) (auto simp add: even_product) lemma odd_pow_iff[presburger]: "odd ((x::int) ^ n) <-> (n = 0 ∨ odd x)" apply (induct n, simp_all) apply presburger apply (case_tac n, auto) apply (simp_all add: even_product) done lemma odd_pow: "odd x ==> odd((x::int)^n)" by (simp add: odd_pow_iff) lemma even_power[presburger]: "even ((x::int)^n) = (even x & 0 < n)" apply (auto simp add: even_pow_gt_zero) apply (erule contrapos_pp, erule odd_pow) apply (erule contrapos_pp, simp add: even_def) done lemma even_zero[presburger]: "even (0::int)" by presburger lemma odd_one[presburger]: "odd (1::int)" by presburger lemmas even_odd_simps [simp] = even_def[of "number_of v",standard] even_zero odd_one even_product even_sum even_neg even_difference even_power subsection {* Equivalent definitions *} lemma two_times_even_div_two: "even (x::int) ==> 2 * (x div 2) = x" by presburger lemma two_times_odd_div_two_plus_one: "odd (x::int) ==> 2 * (x div 2) + 1 = x" by presburger lemma even_equiv_def: "even (x::int) = (EX y. x = 2 * y)" by presburger lemma odd_equiv_def: "odd (x::int) = (EX y. x = 2 * y + 1)" by presburger subsection {* even and odd for nats *} lemma pos_int_even_equiv_nat_even: "0 ≤ x ==> even x = even (nat x)" by (simp add: even_nat_def) lemma even_nat_product[presburger]: "even((x::nat) * y) = (even x | even y)" by (simp add: even_nat_def int_mult) lemma even_nat_sum[presburger]: "even ((x::nat) + y) = ((even x & even y) | (odd x & odd y))" by presburger lemma even_nat_difference[presburger]: "even ((x::nat) - y) = (x < y | (even x & even y) | (odd x & odd y))" by presburger lemma even_nat_Suc[presburger]: "even (Suc x) = odd x" by presburger lemma even_nat_power[presburger]: "even ((x::nat)^y) = (even x & 0 < y)" by (simp add: even_nat_def int_power) lemma even_nat_zero[presburger]: "even (0::nat)" by presburger lemmas even_odd_nat_simps [simp] = even_nat_def[of "number_of v",standard] even_nat_zero even_nat_Suc even_nat_product even_nat_sum even_nat_power subsection {* Equivalent definitions *} lemma nat_lt_two_imp_zero_or_one: "(x::nat) < Suc (Suc 0) ==> x = 0 | x = Suc 0" by presburger lemma even_nat_mod_two_eq_zero: "even (x::nat) ==> x mod (Suc (Suc 0)) = 0" by presburger lemma odd_nat_mod_two_eq_one: "odd (x::nat) ==> x mod (Suc (Suc 0)) = Suc 0" by presburger lemma even_nat_equiv_def: "even (x::nat) = (x mod Suc (Suc 0) = 0)" by presburger lemma odd_nat_equiv_def: "odd (x::nat) = (x mod Suc (Suc 0) = Suc 0)" by presburger lemma even_nat_div_two_times_two: "even (x::nat) ==> Suc (Suc 0) * (x div Suc (Suc 0)) = x" by presburger lemma odd_nat_div_two_times_two_plus_one: "odd (x::nat) ==> Suc( Suc (Suc 0) * (x div Suc (Suc 0))) = x" by presburger lemma even_nat_equiv_def2: "even (x::nat) = (EX y. x = Suc (Suc 0) * y)" by presburger lemma odd_nat_equiv_def2: "odd (x::nat) = (EX y. x = Suc(Suc (Suc 0) * y))" by presburger subsection {* Parity and powers *} lemma minus_one_even_odd_power: "(even x --> (- 1::'a::{comm_ring_1,recpower})^x = 1) & (odd x --> (- 1::'a)^x = - 1)" apply (induct x) apply (rule conjI) apply simp apply (insert even_nat_zero, blast) apply (simp add: power_Suc) done lemma minus_one_even_power [simp]: "even x ==> (- 1::'a::{comm_ring_1,recpower})^x = 1" using minus_one_even_odd_power by blast lemma minus_one_odd_power [simp]: "odd x ==> (- 1::'a::{comm_ring_1,recpower})^x = - 1" using minus_one_even_odd_power by blast lemma neg_one_even_odd_power: "(even x --> (-1::'a::{number_ring,recpower})^x = 1) & (odd x --> (-1::'a)^x = -1)" apply (induct x) apply (simp, simp add: power_Suc) done lemma neg_one_even_power [simp]: "even x ==> (-1::'a::{number_ring,recpower})^x = 1" using neg_one_even_odd_power by blast lemma neg_one_odd_power [simp]: "odd x ==> (-1::'a::{number_ring,recpower})^x = -1" using neg_one_even_odd_power by blast lemma neg_power_if: "(-x::'a::{comm_ring_1,recpower}) ^ n = (if even n then (x ^ n) else -(x ^ n))" apply (induct n) apply (simp_all split: split_if_asm add: power_Suc) done lemma zero_le_even_power: "even n ==> 0 <= (x::'a::{recpower,ordered_ring_strict}) ^ n" apply (simp add: even_nat_equiv_def2) apply (erule exE) apply (erule ssubst) apply (subst power_add) apply (rule zero_le_square) done lemma zero_le_odd_power: "odd n ==> (0 <= (x::'a::{recpower,ordered_idom}) ^ n) = (0 <= x)" apply (simp add: odd_nat_equiv_def2) apply (erule exE) apply (erule ssubst) apply (subst power_Suc) apply (subst power_add) apply (subst zero_le_mult_iff) apply auto apply (subgoal_tac "x = 0 & y > 0") apply (erule conjE, assumption) apply (subst power_eq_0_iff [symmetric]) apply (subgoal_tac "0 <= x^y * x^y") apply simp apply (rule zero_le_square)+ done lemma zero_le_power_eq[presburger]: "(0 <= (x::'a::{recpower,ordered_idom}) ^ n) = (even n | (odd n & 0 <= x))" apply auto apply (subst zero_le_odd_power [symmetric]) apply assumption+ apply (erule zero_le_even_power) done lemma zero_less_power_eq[presburger]: "(0 < (x::'a::{recpower,ordered_idom}) ^ n) = (n = 0 | (even n & x ~= 0) | (odd n & 0 < x))" apply (rule iffI) apply clarsimp apply (rule conjI) apply clarsimp apply (rule ccontr) apply (subgoal_tac "~ (0 <= x^n)") apply simp apply (subst zero_le_odd_power) apply assumption apply simp apply (rule notI) apply (simp add: power_0_left) apply (rule notI) apply (simp add: power_0_left) apply auto apply (subgoal_tac "0 <= x^n") apply (frule order_le_imp_less_or_eq) apply simp apply (erule zero_le_even_power) done lemma power_less_zero_eq[presburger]: "((x::'a::{recpower,ordered_idom}) ^ n < 0) = (odd n & x < 0)" apply (subst linorder_not_le [symmetric])+ apply (subst zero_le_power_eq) apply auto done lemma power_le_zero_eq[presburger]: "((x::'a::{recpower,ordered_idom}) ^ n <= 0) = (n ~= 0 & ((odd n & x <= 0) | (even n & x = 0)))" apply (subst linorder_not_less [symmetric])+ apply (subst zero_less_power_eq) apply auto done lemma power_even_abs: "even n ==> (abs (x::'a::{recpower,ordered_idom}))^n = x^n" apply (subst power_abs [symmetric]) apply (simp add: zero_le_even_power) done lemma zero_less_power_nat_eq[presburger]: "(0 < (x::nat) ^ n) = (n = 0 | 0 < x)" by (induct n) auto lemma power_minus_even [simp]: "even n ==> (- x)^n = (x^n::'a::{recpower,comm_ring_1})" apply (subst power_minus) apply simp done lemma power_minus_odd [simp]: "odd n ==> (- x)^n = - (x^n::'a::{recpower,comm_ring_1})" apply (subst power_minus) apply simp done subsection {* General Lemmas About Division *} lemma Suc_times_mod_eq: "1<k ==> Suc (k * m) mod k = 1" apply (induct "m") apply (simp_all add: mod_Suc) done declare Suc_times_mod_eq [of "number_of w", standard, simp] lemma [simp]: "n div k ≤ (Suc n) div k" by (simp add: div_le_mono) lemma Suc_n_div_2_gt_zero [simp]: "(0::nat) < n ==> 0 < (n + 1) div 2" by arith lemma div_2_gt_zero [simp]: "(1::nat) < n ==> 0 < n div 2" by arith lemma mod_mult_self3 [simp]: "(k*n + m) mod n = m mod (n::nat)" by (simp add: mult_ac add_ac) lemma mod_mult_self4 [simp]: "Suc (k*n + m) mod n = Suc m mod n" proof - have "Suc (k * n + m) mod n = (k * n + Suc m) mod n" by simp also have "... = Suc m mod n" by (rule mod_mult_self3) finally show ?thesis . qed lemma mod_Suc_eq_Suc_mod: "Suc m mod n = Suc (m mod n) mod n" apply (subst mod_Suc [of m]) apply (subst mod_Suc [of "m mod n"], simp) done subsection {* More Even/Odd Results *} lemma even_mult_two_ex: "even(n) = (∃m::nat. n = 2*m)" by (simp add: even_nat_equiv_def2 numeral_2_eq_2) lemma odd_Suc_mult_two_ex: "odd(n) = (∃m. n = Suc (2*m))" by (simp add: odd_nat_equiv_def2 numeral_2_eq_2) lemma even_add [simp]: "even(m + n::nat) = (even m = even n)" by auto lemma odd_add [simp]: "odd(m + n::nat) = (odd m ≠ odd n)" by auto lemma div_Suc: "Suc a div c = a div c + Suc 0 div c + (a mod c + Suc 0 mod c) div c" apply (subgoal_tac "Suc a = a + Suc 0") apply (erule ssubst) apply (rule div_add1_eq, simp) done lemma lemma_even_div2 [simp]: "even (n::nat) ==> (n + 1) div 2 = n div 2" apply (simp add: numeral_2_eq_2) apply (subst div_Suc) apply (simp add: even_nat_mod_two_eq_zero) done lemma lemma_not_even_div2 [simp]: "~even n ==> (n + 1) div 2 = Suc (n div 2)" apply (simp add: numeral_2_eq_2) apply (subst div_Suc) apply (simp add: odd_nat_mod_two_eq_one) done lemma even_num_iff: "0 < n ==> even n = (~ even(n - 1 :: nat))" by (case_tac "n", auto) lemma even_even_mod_4_iff: "even (n::nat) = even (n mod 4)" apply (induct n, simp) apply (subst mod_Suc, simp) done lemma lemma_odd_mod_4_div_2: "n mod 4 = (3::nat) ==> odd((n - 1) div 2)" apply (rule_tac t = n and n1 = 4 in mod_div_equality [THEN subst]) apply (simp add: even_num_iff) done lemma lemma_even_mod_4_div_2: "n mod 4 = (1::nat) ==> even ((n - 1) div 2)" by (rule_tac t = n and n1 = 4 in mod_div_equality [THEN subst], simp) text {* Simplify, when the exponent is a numeral *} lemmas power_0_left_number_of = power_0_left [of "number_of w", standard] declare power_0_left_number_of [simp] lemmas zero_le_power_eq_number_of [simp] = zero_le_power_eq [of _ "number_of w", standard] lemmas zero_less_power_eq_number_of [simp] = zero_less_power_eq [of _ "number_of w", standard] lemmas power_le_zero_eq_number_of [simp] = power_le_zero_eq [of _ "number_of w", standard] lemmas power_less_zero_eq_number_of [simp] = power_less_zero_eq [of _ "number_of w", standard] lemmas zero_less_power_nat_eq_number_of [simp] = zero_less_power_nat_eq [of _ "number_of w", standard] lemmas power_eq_0_iff_number_of [simp] = power_eq_0_iff [of _ "number_of w", standard] lemmas power_even_abs_number_of [simp] = power_even_abs [of "number_of w" _, standard] subsection {* An Equivalence for @{term [source] "0 ≤ a^n"} *} lemma even_power_le_0_imp_0: "a ^ (2*k) ≤ (0::'a::{ordered_idom,recpower}) ==> a=0" by (induct k) (auto simp add: zero_le_mult_iff mult_le_0_iff power_Suc) lemma zero_le_power_iff[presburger]: "(0 ≤ a^n) = (0 ≤ (a::'a::{ordered_idom,recpower}) | even n)" proof cases assume even: "even n" then obtain k where "n = 2*k" by (auto simp add: even_nat_equiv_def2 numeral_2_eq_2) thus ?thesis by (simp add: zero_le_even_power even) next assume odd: "odd n" then obtain k where "n = Suc(2*k)" by (auto simp add: odd_nat_equiv_def2 numeral_2_eq_2) thus ?thesis by (auto simp add: power_Suc zero_le_mult_iff zero_le_even_power dest!: even_power_le_0_imp_0) qed subsection {* Miscellaneous *} lemma odd_pos: "odd (n::nat) ==> 0 < n" by (cases n, simp_all) lemma [presburger]:"(x + 1) div 2 = x div 2 <-> even (x::int)" by presburger lemma [presburger]: "(x + 1) div 2 = x div 2 + 1 <-> odd (x::int)" by presburger lemma even_plus_one_div_two: "even (x::int) ==> (x + 1) div 2 = x div 2" by presburger lemma odd_plus_one_div_two: "odd (x::int) ==> (x + 1) div 2 = x div 2 + 1" by presburger lemma [presburger]: "(Suc x) div Suc (Suc 0) = x div Suc (Suc 0) <-> even x" by presburger lemma [presburger]: "(Suc x) div Suc (Suc 0) = x div Suc (Suc 0) <-> even x" by presburger lemma even_nat_plus_one_div_two: "even (x::nat) ==> (Suc x) div Suc (Suc 0) = x div Suc (Suc 0)" by presburger lemma odd_nat_plus_one_div_two: "odd (x::nat) ==> (Suc x) div Suc (Suc 0) = Suc (x div Suc (Suc 0))" by presburger end
lemma int_pos_lt_two_imp_zero_or_one:
[| 0 ≤ x; x < 2 |] ==> x = 0 ∨ x = 1
lemma neq_one_mod_two:
(x mod 2 ≠ 0) = (x mod 2 = 1)
lemma even_times_anything:
even x ==> even (x * y)
lemma anything_times_even:
even y ==> even (x * y)
lemma odd_times_odd:
[| odd x; odd y |] ==> odd (x * y)
lemma even_product:
even (x * y) = (even x ∨ even y)
lemma even_plus_even:
[| even x; even y |] ==> even (x + y)
lemma even_plus_odd:
[| even x; odd y |] ==> odd (x + y)
lemma odd_plus_even:
[| odd x; even y |] ==> odd (x + y)
lemma odd_plus_odd:
[| odd x; odd y |] ==> even (x + y)
lemma even_sum:
even (x + y) = (even x ∧ even y ∨ odd x ∧ odd y)
lemma even_neg:
even (- x) = even x
lemma even_difference:
even (x - y) = (even x ∧ even y ∨ odd x ∧ odd y)
lemma even_pow_gt_zero:
[| even x; 0 < n |] ==> even (x ^ n)
lemma odd_pow_iff:
odd (x ^ n) = (n = 0 ∨ odd x)
lemma odd_pow:
odd x ==> odd (x ^ n)
lemma even_power:
even (x ^ n) = (even x ∧ 0 < n)
lemma even_zero:
even 0
lemma odd_one:
odd 1
lemma even_odd_simps:
even (number_of v) = (number_of v mod 2 = 0)
even 0
odd 1
even (x * y) = (even x ∨ even y)
even (x + y) = (even x ∧ even y ∨ odd x ∧ odd y)
even (- x) = even x
even (x - y) = (even x ∧ even y ∨ odd x ∧ odd y)
even (x ^ n) = (even x ∧ 0 < n)
lemma two_times_even_div_two:
even x ==> 2 * (x div 2) = x
lemma two_times_odd_div_two_plus_one:
odd x ==> 2 * (x div 2) + 1 = x
lemma even_equiv_def:
even x = (∃y. x = 2 * y)
lemma odd_equiv_def:
odd x = (∃y. x = 2 * y + 1)
lemma pos_int_even_equiv_nat_even:
0 ≤ x ==> even x = even (nat x)
lemma even_nat_product:
even (x * y) = (even x ∨ even y)
lemma even_nat_sum:
even (x + y) = (even x ∧ even y ∨ odd x ∧ odd y)
lemma even_nat_difference:
even (x - y) = (x < y ∨ even x ∧ even y ∨ odd x ∧ odd y)
lemma even_nat_Suc:
even (Suc x) = odd x
lemma even_nat_power:
even (x ^ y) = (even x ∧ 0 < y)
lemma even_nat_zero:
even 0
lemma even_odd_nat_simps:
even (number_of v) = even (int (number_of v))
even 0
even (Suc x) = odd x
even (x * y) = (even x ∨ even y)
even (x + y) = (even x ∧ even y ∨ odd x ∧ odd y)
even (x ^ y) = (even x ∧ 0 < y)
lemma nat_lt_two_imp_zero_or_one:
x < Suc (Suc 0) ==> x = 0 ∨ x = Suc 0
lemma even_nat_mod_two_eq_zero:
even x ==> x mod Suc (Suc 0) = 0
lemma odd_nat_mod_two_eq_one:
odd x ==> x mod Suc (Suc 0) = Suc 0
lemma even_nat_equiv_def:
even x = (x mod Suc (Suc 0) = 0)
lemma odd_nat_equiv_def:
odd x = (x mod Suc (Suc 0) = Suc 0)
lemma even_nat_div_two_times_two:
even x ==> Suc (Suc 0) * (x div Suc (Suc 0)) = x
lemma odd_nat_div_two_times_two_plus_one:
odd x ==> Suc (Suc (Suc 0) * (x div Suc (Suc 0))) = x
lemma even_nat_equiv_def2:
even x = (∃y. x = Suc (Suc 0) * y)
lemma odd_nat_equiv_def2:
odd x = (∃y. x = Suc (Suc (Suc 0) * y))
lemma minus_one_even_odd_power:
(even x --> (- (1::'a)) ^ x = (1::'a)) ∧ (odd x --> (- (1::'a)) ^ x = - (1::'a))
lemma minus_one_even_power:
even x ==> (- (1::'a)) ^ x = (1::'a)
lemma minus_one_odd_power:
odd x ==> (- (1::'a)) ^ x = - (1::'a)
lemma neg_one_even_odd_power:
(even x --> (-1::'a) ^ x = (1::'a)) ∧ (odd x --> (-1::'a) ^ x = (-1::'a))
lemma neg_one_even_power:
even x ==> (-1::'a) ^ x = (1::'a)
lemma neg_one_odd_power:
odd x ==> (-1::'a) ^ x = (-1::'a)
lemma neg_power_if:
(- x) ^ n = (if even n then x ^ n else - (x ^ n))
lemma zero_le_even_power:
even n ==> (0::'a) ≤ x ^ n
lemma zero_le_odd_power:
odd n ==> ((0::'a) ≤ x ^ n) = ((0::'a) ≤ x)
lemma zero_le_power_eq:
((0::'a) ≤ x ^ n) = (even n ∨ odd n ∧ (0::'a) ≤ x)
lemma zero_less_power_eq:
((0::'a) < x ^ n) = (n = 0 ∨ even n ∧ x ≠ (0::'a) ∨ odd n ∧ (0::'a) < x)
lemma power_less_zero_eq:
(x ^ n < (0::'a)) = (odd n ∧ x < (0::'a))
lemma power_le_zero_eq:
(x ^ n ≤ (0::'a)) = (n ≠ 0 ∧ (odd n ∧ x ≤ (0::'a) ∨ even n ∧ x = (0::'a)))
lemma power_even_abs:
even n ==> ¦x¦ ^ n = x ^ n
lemma zero_less_power_nat_eq:
(0 < x ^ n) = (n = 0 ∨ 0 < x)
lemma power_minus_even:
even n ==> (- x) ^ n = x ^ n
lemma power_minus_odd:
odd n ==> (- x) ^ n = - (x ^ n)
lemma Suc_times_mod_eq:
1 < k ==> Suc (k * m) mod k = 1
lemma
n div k ≤ Suc n div k
lemma Suc_n_div_2_gt_zero:
0 < n ==> 0 < (n + 1) div 2
lemma div_2_gt_zero:
1 < n ==> 0 < n div 2
lemma mod_mult_self3:
(k * n + m) mod n = m mod n
lemma mod_mult_self4:
Suc (k * n + m) mod n = Suc m mod n
lemma mod_Suc_eq_Suc_mod:
Suc m mod n = Suc (m mod n) mod n
lemma even_mult_two_ex:
even n = (∃m. n = 2 * m)
lemma odd_Suc_mult_two_ex:
odd n = (∃m. n = Suc (2 * m))
lemma even_add:
even (m + n) = (even m = even n)
lemma odd_add:
odd (m + n) = (odd m ≠ odd n)
lemma div_Suc:
Suc a div c = a div c + Suc 0 div c + (a mod c + Suc 0 mod c) div c
lemma lemma_even_div2:
even n ==> (n + 1) div 2 = n div 2
lemma lemma_not_even_div2:
odd n ==> (n + 1) div 2 = Suc (n div 2)
lemma even_num_iff:
0 < n ==> even n = odd (n - 1)
lemma even_even_mod_4_iff:
even n = even (n mod 4)
lemma lemma_odd_mod_4_div_2:
n mod 4 = 3 ==> odd ((n - 1) div 2)
lemma lemma_even_mod_4_div_2:
n mod 4 = 1 ==> even ((n - 1) div 2)
lemma power_0_left_number_of:
(0::'a) ^ number_of w = (if number_of w = 0 then 1::'a else 0::'a)
lemma zero_le_power_eq_number_of:
((0::'a) ≤ x ^ number_of w) =
(even (number_of w) ∨ odd (number_of w) ∧ (0::'a) ≤ x)
lemma zero_less_power_eq_number_of:
((0::'a) < x ^ number_of w) =
(number_of w = 0 ∨
even (number_of w) ∧ x ≠ (0::'a) ∨ odd (number_of w) ∧ (0::'a) < x)
lemma power_le_zero_eq_number_of:
(x ^ number_of w ≤ (0::'a)) =
(number_of w ≠ 0 ∧
(odd (number_of w) ∧ x ≤ (0::'a) ∨ even (number_of w) ∧ x = (0::'a)))
lemma power_less_zero_eq_number_of:
(x ^ number_of w < (0::'a)) = (odd (number_of w) ∧ x < (0::'a))
lemma zero_less_power_nat_eq_number_of:
(0 < x ^ number_of w) = (number_of w = 0 ∨ 0 < x)
lemma power_eq_0_iff_number_of:
(a ^ number_of w = (0::'a)) = (a = (0::'a) ∧ 0 < number_of w)
lemma power_even_abs_number_of:
even (number_of w) ==> ¦x¦ ^ number_of w = x ^ number_of w
lemma even_power_le_0_imp_0:
a ^ (2 * k) ≤ (0::'a) ==> a = (0::'a)
lemma zero_le_power_iff:
((0::'a) ≤ a ^ n) = ((0::'a) ≤ a ∨ even n)
lemma odd_pos:
odd n ==> 0 < n
lemma
((x + 1) div 2 = x div 2) = even x
lemma
((x + 1) div 2 = x div 2 + 1) = odd x
lemma even_plus_one_div_two:
even x ==> (x + 1) div 2 = x div 2
lemma odd_plus_one_div_two:
odd x ==> (x + 1) div 2 = x div 2 + 1
lemma
(Suc x div Suc (Suc 0) = x div Suc (Suc 0)) = even x
lemma
(Suc x div Suc (Suc 0) = x div Suc (Suc 0)) = even x
lemma even_nat_plus_one_div_two:
even x ==> Suc x div Suc (Suc 0) = x div Suc (Suc 0)
lemma odd_nat_plus_one_div_two:
odd x ==> Suc x div Suc (Suc 0) = Suc (x div Suc (Suc 0))