(* Title: Parity.thy ID: $Id: Parity.thy,v 1.15 2005/09/17 16:25:11 wenzelm Exp $ Author: Jeremy Avigad *) header {* Even and Odd for ints and nats*} theory Parity imports Divides IntDiv NatSimprocs begin axclass even_odd < type instance int :: even_odd .. instance nat :: even_odd .. consts even :: "'a::even_odd => bool" syntax odd :: "'a::even_odd => bool" translations "odd x" == "~even x" defs (overloaded) even_def: "even (x::int) == x mod 2 = 0" even_nat_def: "even (x::nat) == even (int x)" 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 auto lemma neq_one_mod_two [simp]: "((x::int) mod 2 ~= 0) = (x mod 2 = 1)" apply (subgoal_tac "x mod 2 = 0 | x mod 2 = 1", force) apply (rule int_pos_lt_two_imp_zero_or_one, auto) done 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: "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 (simp add: even_def zmod_zadd1_eq) lemma even_plus_odd: "even (x::int) ==> odd y ==> odd (x + y)" by (simp add: even_def zmod_zadd1_eq) lemma odd_plus_even: "odd (x::int) ==> even y ==> odd (x + y)" by (simp add: even_def zmod_zadd1_eq) lemma odd_plus_odd: "odd (x::int) ==> odd y ==> even (x + y)" by (simp add: even_def zmod_zadd1_eq) lemma even_sum: "even ((x::int) + y) = ((even x & even y) | (odd x & odd y))" apply (auto intro: even_plus_even odd_plus_odd) apply (rule ccontr, simp add: even_plus_odd) apply (rule ccontr, simp add: odd_plus_even) done lemma even_neg: "even (-(x::int)) = even x" by (auto simp add: even_def zmod_zminus1_eq_if) lemma even_difference: "even ((x::int) - y) = ((even x & even y) | (odd x & odd y))" by (simp only: diff_minus even_sum even_neg) lemma even_pow_gt_zero [rule_format]: "even (x::int) ==> 0 < n --> even (x^n)" apply (induct n) apply (auto simp add: even_product) done lemma odd_pow: "odd x ==> odd((x::int)^n)" apply (induct n) apply (simp add: even_def) apply (simp add: even_product) done lemma even_power: "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: "even (0::int)" by (simp add: even_def) lemma odd_one: "odd (1::int)" by (simp add: even_def) 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 (auto simp add: even_def) lemma two_times_odd_div_two_plus_one: "odd (x::int) ==> 2 * (x div 2) + 1 = x" apply (insert zmod_zdiv_equality [of x 2, THEN sym]) by (simp add: even_def) lemma even_equiv_def: "even (x::int) = (EX y. x = 2 * y)" apply auto apply (rule exI) by (erule two_times_even_div_two [THEN sym]) lemma odd_equiv_def: "odd (x::int) = (EX y. x = 2 * y + 1)" apply auto apply (rule exI) by (erule two_times_odd_div_two_plus_one [THEN sym]) 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: "even((x::nat) * y) = (even x | even y)" by (simp add: even_nat_def int_mult) lemma even_nat_sum: "even ((x::nat) + y) = ((even x & even y) | (odd x & odd y))" by (unfold even_nat_def, simp) lemma even_nat_difference: "even ((x::nat) - y) = (x < y | (even x & even y) | (odd x & odd y))" apply (auto simp add: even_nat_def zdiff_int [THEN sym]) apply (case_tac "x < y", auto simp add: zdiff_int [THEN sym]) apply (case_tac "x < y", auto simp add: zdiff_int [THEN sym]) done lemma even_nat_Suc: "even (Suc x) = odd x" by (simp add: even_nat_def) lemma even_nat_power: "even ((x::nat)^y) = (even x & 0 < y)" by (simp add: even_nat_def int_power) lemma even_nat_zero: "even (0::nat)" by (simp add: even_nat_def) 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 auto lemma even_nat_mod_two_eq_zero: "even (x::nat) ==> x mod (Suc (Suc 0)) = 0" apply (insert mod_div_equality [of x "Suc (Suc 0)", THEN sym]) apply (drule subst, assumption) apply (subgoal_tac "x mod Suc (Suc 0) = 0 | x mod Suc (Suc 0) = Suc 0") apply force apply (subgoal_tac "0 < Suc (Suc 0)") apply (frule mod_less_divisor [of "Suc (Suc 0)" x]) apply (erule nat_lt_two_imp_zero_or_one, auto) done lemma odd_nat_mod_two_eq_one: "odd (x::nat) ==> x mod (Suc (Suc 0)) = Suc 0" apply (insert mod_div_equality [of x "Suc (Suc 0)", THEN sym]) apply (drule subst, assumption) apply (subgoal_tac "x mod Suc (Suc 0) = 0 | x mod Suc (Suc 0) = Suc 0") apply force apply (subgoal_tac "0 < Suc (Suc 0)") apply (frule mod_less_divisor [of "Suc (Suc 0)" x]) apply (erule nat_lt_two_imp_zero_or_one, auto) done lemma even_nat_equiv_def: "even (x::nat) = (x mod Suc (Suc 0) = 0)" apply (rule iffI) apply (erule even_nat_mod_two_eq_zero) apply (insert odd_nat_mod_two_eq_one [of x], auto) done lemma odd_nat_equiv_def: "odd (x::nat) = (x mod Suc (Suc 0) = Suc 0)" apply (auto simp add: even_nat_equiv_def) apply (subgoal_tac "x mod (Suc (Suc 0)) < Suc (Suc 0)") apply (frule nat_lt_two_imp_zero_or_one, auto) done lemma even_nat_div_two_times_two: "even (x::nat) ==> Suc (Suc 0) * (x div Suc (Suc 0)) = x" apply (insert mod_div_equality [of x "Suc (Suc 0)", THEN sym]) apply (drule even_nat_mod_two_eq_zero, simp) done lemma odd_nat_div_two_times_two_plus_one: "odd (x::nat) ==> Suc( Suc (Suc 0) * (x div Suc (Suc 0))) = x" apply (insert mod_div_equality [of x "Suc (Suc 0)", THEN sym]) apply (drule odd_nat_mod_two_eq_one, simp) done lemma even_nat_equiv_def2: "even (x::nat) = (EX y. x = Suc (Suc 0) * y)" apply (rule iffI, rule exI) apply (erule even_nat_div_two_times_two [THEN sym], auto) done lemma odd_nat_equiv_def2: "odd (x::nat) = (EX y. x = Suc(Suc (Suc 0) * y))" apply (rule iffI, rule exI) apply (erule odd_nat_div_two_times_two_plus_one [THEN sym], auto) done 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" by (rule minus_one_even_odd_power [THEN conjunct1, THEN mp], assumption) lemma minus_one_odd_power [simp]: "odd x ==> (- 1::'a::{comm_ring_1,recpower})^x = - 1" by (rule minus_one_even_odd_power [THEN conjunct2, THEN mp], assumption) 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" by (rule neg_one_even_odd_power [THEN conjunct1, THEN mp], assumption) lemma neg_one_odd_power [simp]: "odd x ==> (-1::'a::{number_ring,recpower})^x = -1" by (rule neg_one_even_odd_power [THEN conjunct2, THEN mp], assumption) lemma neg_power_if: "(-x::'a::{comm_ring_1,recpower}) ^ n = (if even n then (x ^ n) else -(x ^ n))" by (induct n, simp_all split: split_if_asm add: power_Suc) 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 & 0 < y") apply (erule conjE, assumption) apply (subst power_eq_0_iff [THEN sym]) apply (subgoal_tac "0 <= x^y * x^y") apply simp apply (rule zero_le_square)+ done lemma zero_le_power_eq: "(0 <= (x::'a::{recpower,ordered_idom}) ^ n) = (even n | (odd n & 0 <= x))" apply auto apply (subst zero_le_odd_power [THEN sym]) apply assumption+ apply (erule zero_le_even_power) apply (subst zero_le_odd_power) apply assumption+ done lemma zero_less_power_eq: "(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) apply (subgoal_tac "0 <= x^n") apply (frule order_le_imp_less_or_eq) apply auto apply (subst zero_le_odd_power) apply assumption apply (erule order_less_imp_le) done lemma power_less_zero_eq: "((x::'a::{recpower,ordered_idom}) ^ n < 0) = (odd n & x < 0)" apply (subst linorder_not_le [THEN sym])+ apply (subst zero_le_power_eq) apply auto done lemma power_le_zero_eq: "((x::'a::{recpower,ordered_idom}) ^ n <= 0) = (n ~= 0 & ((odd n & x <= 0) | (even n & x = 0)))" apply (subst linorder_not_less [THEN sym])+ 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 [THEN sym]) apply (simp add: zero_le_even_power) done lemma zero_less_power_nat_eq: "(0 < (x::nat) ^ n) = (n = 0 | 0 < x)" apply (induct n) apply simp apply auto done 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 (* 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 = zero_le_power_eq [of _ "number_of w", standard] declare zero_le_power_eq_number_of [simp] lemmas zero_less_power_eq_number_of = zero_less_power_eq [of _ "number_of w", standard] declare zero_less_power_eq_number_of [simp] lemmas power_le_zero_eq_number_of = power_le_zero_eq [of _ "number_of w", standard] declare power_le_zero_eq_number_of [simp] lemmas power_less_zero_eq_number_of = power_less_zero_eq [of _ "number_of w", standard] declare power_less_zero_eq_number_of [simp] lemmas zero_less_power_nat_eq_number_of = zero_less_power_nat_eq [of _ "number_of w", standard] declare zero_less_power_nat_eq_number_of [simp] lemmas power_eq_0_iff_number_of = power_eq_0_iff [of _ "number_of w", standard] declare power_eq_0_iff_number_of [simp] lemmas power_even_abs_number_of = power_even_abs [of "number_of w" _, standard] declare power_even_abs_number_of [simp] 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" apply (induct k) apply (auto simp add: zero_le_mult_iff mult_le_0_iff power_Suc) done lemma zero_le_power_iff: "(0 ≤ a^n) = (0 ≤ (a::'a::{ordered_idom,recpower}) | even n)" (is "?P 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 even_plus_one_div_two: "even (x::int) ==> (x + 1) div 2 = x div 2" apply (subst zdiv_zadd1_eq) apply (simp add: even_def) done lemma odd_plus_one_div_two: "odd (x::int) ==> (x + 1) div 2 = x div 2 + 1" apply (subst zdiv_zadd1_eq) apply (simp add: even_def) done 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 even_nat_plus_one_div_two: "even (x::nat) ==> (Suc x) div Suc (Suc 0) = x div Suc (Suc 0)" apply (subst div_Suc) apply (simp add: even_nat_equiv_def) done lemma odd_nat_plus_one_div_two: "odd (x::nat) ==> (Suc x) div Suc (Suc 0) = Suc (x div Suc (Suc 0))" apply (subst div_Suc) apply (simp add: odd_nat_equiv_def) done 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:
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
lemmas 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)
lemmas 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
lemmas 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)
lemmas 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)
lemmas power_0_left_number_of:
(0::'a) ^ number_of w = (if number_of w = 0 then 1::'a else 0::'a)
lemmas power_0_left_number_of:
(0::'a) ^ number_of w = (if number_of w = 0 then 1::'a else 0::'a)
lemmas zero_le_power_eq_number_of:
((0::'a) ≤ x ^ number_of w) = (even (number_of w) ∨ odd (number_of w) ∧ (0::'a) ≤ x)
lemmas zero_le_power_eq_number_of:
((0::'a) ≤ x ^ number_of w) = (even (number_of w) ∨ odd (number_of w) ∧ (0::'a) ≤ x)
lemmas 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)
lemmas 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)
lemmas 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)))
lemmas 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)))
lemmas power_less_zero_eq_number_of:
(x ^ number_of w < (0::'a)) = (odd (number_of w) ∧ x < (0::'a))
lemmas power_less_zero_eq_number_of:
(x ^ number_of w < (0::'a)) = (odd (number_of w) ∧ x < (0::'a))
lemmas zero_less_power_nat_eq_number_of:
(0 < x ^ number_of w) = (number_of w = 0 ∨ 0 < x)
lemmas zero_less_power_nat_eq_number_of:
(0 < x ^ number_of w) = (number_of w = 0 ∨ 0 < x)
lemmas power_eq_0_iff_number_of:
(a ^ number_of w = (0::'a)) = (a = (0::'a) ∧ 0 < number_of w)
lemmas power_eq_0_iff_number_of:
(a ^ number_of w = (0::'a)) = (a = (0::'a) ∧ 0 < number_of w)
lemmas power_even_abs_number_of:
even (number_of w) ==> ¦x¦ ^ number_of w = x ^ number_of w
lemmas 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 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 div_Suc:
Suc a div c = a div c + Suc 0 div c + (a mod c + Suc 0 mod c) div c
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))