(* Title: Parity.thy
ID: $Id$
Author: Jeremy Avigad
*)
header {* Parity: 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 \<le> 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 "0 \<le> a^n"} *}
lemma even_power_le_0_imp_0:
"a ^ (2*k) \<le> (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 \<le> a^n) = (0 \<le> (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