
wenzelm@21263: (*  Title:      HOL/Library/Parity.thy
wenzelm@21256:     ID:         $Id$
wenzelm@21256:     Author:     Jeremy Avigad
wenzelm@21256: *)
wenzelm@21256: 
wenzelm@21256: header {* Even and Odd for int and nat *}
wenzelm@21256: 
wenzelm@21256: theory Parity
wenzelm@21256: imports Main
wenzelm@21256: begin
wenzelm@21256: 
haftmann@22473: class even_odd = type + 
haftmann@22390:   fixes even :: "'a \<Rightarrow> bool"
wenzelm@21256: 
wenzelm@21256: abbreviation
haftmann@22390:   odd :: "'a\<Colon>even_odd \<Rightarrow> bool" where
haftmann@22390:   "odd x \<equiv> \<not> even x"
haftmann@22390: 
haftmann@25571: instantiation int and nat :: even_odd
haftmann@25571: begin
haftmann@25571: 
haftmann@25571: definition
haftmann@25571:   even_def [presburger]: "even x \<longleftrightarrow> (x\<Colon>int) mod 2 = 0"
haftmann@22390: 
haftmann@25571: definition
haftmann@25571:   even_nat_def [presburger]: "even x \<longleftrightarrow> even (int x)"
haftmann@25571: 
haftmann@25571: instance ..
haftmann@25571: 
haftmann@25571: end
wenzelm@21256: 
wenzelm@21256: 
wenzelm@21256: subsection {* Even and odd are mutually exclusive *}
wenzelm@21256: 
wenzelm@21263: lemma int_pos_lt_two_imp_zero_or_one:
wenzelm@21256:     "0 <= x ==> (x::int) < 2 ==> x = 0 | x = 1"
chaieb@23522:   by presburger
wenzelm@21256: 
chaieb@23522: lemma neq_one_mod_two [simp, presburger]: 
chaieb@23522:   "((x::int) mod 2 ~= 0) = (x mod 2 = 1)" by presburger
wenzelm@21256: 
wenzelm@21256: subsection {* Behavior under integer arithmetic operations *}
wenzelm@21256: 
wenzelm@21256: lemma even_times_anything: "even (x::int) ==> even (x * y)"
wenzelm@21256:   by (simp add: even_def zmod_zmult1_eq')
wenzelm@21256: 
wenzelm@21256: lemma anything_times_even: "even (y::int) ==> even (x * y)"
wenzelm@21256:   by (simp add: even_def zmod_zmult1_eq)
wenzelm@21256: 
wenzelm@21256: lemma odd_times_odd: "odd (x::int) ==> odd y ==> odd (x * y)"
wenzelm@21256:   by (simp add: even_def zmod_zmult1_eq)
wenzelm@21256: 
chaieb@23522: lemma even_product[presburger]: "even((x::int) * y) = (even x | even y)"
wenzelm@21263:   apply (auto simp add: even_times_anything anything_times_even)
wenzelm@21256:   apply (rule ccontr)
wenzelm@21256:   apply (auto simp add: odd_times_odd)
wenzelm@21256:   done
wenzelm@21256: 
wenzelm@21256: lemma even_plus_even: "even (x::int) ==> even y ==> even (x + y)"
chaieb@23522:   by presburger
wenzelm@21256: 
wenzelm@21256: lemma even_plus_odd: "even (x::int) ==> odd y ==> odd (x + y)"
chaieb@23522:   by presburger
wenzelm@21256: 
wenzelm@21256: lemma odd_plus_even: "odd (x::int) ==> even y ==> odd (x + y)"
chaieb@23522:   by presburger
wenzelm@21256: 
chaieb@23522: lemma odd_plus_odd: "odd (x::int) ==> odd y ==> even (x + y)" by presburger
wenzelm@21256: 
chaieb@23522: lemma even_sum[presburger]: "even ((x::int) + y) = ((even x & even y) | (odd x & odd y))"
chaieb@23522:   by presburger
wenzelm@21256: 
chaieb@23522: lemma even_neg[presburger]: "even (-(x::int)) = even x" by presburger
wenzelm@21256: 
wenzelm@21263: lemma even_difference:
chaieb@23522:     "even ((x::int) - y) = ((even x & even y) | (odd x & odd y))" by presburger
wenzelm@21256: 
wenzelm@21263: lemma even_pow_gt_zero:
wenzelm@21263:     "even (x::int) ==> 0 < n ==> even (x^n)"
wenzelm@21263:   by (induct n) (auto simp add: even_product)
wenzelm@21256: 
chaieb@23522: lemma odd_pow_iff[presburger]: "odd ((x::int) ^ n) \<longleftrightarrow> (n = 0 \<or> odd x)"
chaieb@23522:   apply (induct n, simp_all)
chaieb@23522:   apply presburger
chaieb@23522:   apply (case_tac n, auto)
chaieb@23522:   apply (simp_all add: even_product)
wenzelm@21256:   done
wenzelm@21256: 
chaieb@23522: lemma odd_pow: "odd x ==> odd((x::int)^n)" by (simp add: odd_pow_iff)
chaieb@23522: 
chaieb@23522: lemma even_power[presburger]: "even ((x::int)^n) = (even x & 0 < n)"
wenzelm@21263:   apply (auto simp add: even_pow_gt_zero)
wenzelm@21256:   apply (erule contrapos_pp, erule odd_pow)
wenzelm@21256:   apply (erule contrapos_pp, simp add: even_def)
wenzelm@21256:   done
wenzelm@21256: 
chaieb@23522: lemma even_zero[presburger]: "even (0::int)" by presburger
wenzelm@21256: 
chaieb@23522: lemma odd_one[presburger]: "odd (1::int)" by presburger
wenzelm@21256: 
wenzelm@21263: lemmas even_odd_simps [simp] = even_def[of "number_of v",standard] even_zero
wenzelm@21256:   odd_one even_product even_sum even_neg even_difference even_power
wenzelm@21256: 
wenzelm@21256: 
wenzelm@21256: subsection {* Equivalent definitions *}
wenzelm@21256: 
chaieb@23522: lemma two_times_even_div_two: "even (x::int) ==> 2 * (x div 2) = x" 
chaieb@23522:   by presburger
wenzelm@21256: 
wenzelm@21263: lemma two_times_odd_div_two_plus_one: "odd (x::int) ==>
chaieb@23522:     2 * (x div 2) + 1 = x" by presburger
wenzelm@21256: 
chaieb@23522: lemma even_equiv_def: "even (x::int) = (EX y. x = 2 * y)" by presburger
wenzelm@21256: 
chaieb@23522: lemma odd_equiv_def: "odd (x::int) = (EX y. x = 2 * y + 1)" by presburger
wenzelm@21256: 
wenzelm@21256: subsection {* even and odd for nats *}
wenzelm@21256: 
wenzelm@21256: lemma pos_int_even_equiv_nat_even: "0 \<le> x ==> even x = even (nat x)"
wenzelm@21256:   by (simp add: even_nat_def)
wenzelm@21256: 
chaieb@23522: lemma even_nat_product[presburger]: "even((x::nat) * y) = (even x | even y)"
huffman@23431:   by (simp add: even_nat_def int_mult)
wenzelm@21256: 
chaieb@23522: lemma even_nat_sum[presburger]: "even ((x::nat) + y) =
chaieb@23522:     ((even x & even y) | (odd x & odd y))" by presburger
wenzelm@21256: 
chaieb@23522: lemma even_nat_difference[presburger]:
wenzelm@21256:     "even ((x::nat) - y) = (x < y | (even x & even y) | (odd x & odd y))"
chaieb@23522: by presburger
wenzelm@21256: 
chaieb@23522: lemma even_nat_Suc[presburger]: "even (Suc x) = odd x" by presburger
wenzelm@21256: 
chaieb@23522: lemma even_nat_power[presburger]: "even ((x::nat)^y) = (even x & 0 < y)"
huffman@23431:   by (simp add: even_nat_def int_power)
wenzelm@21256: 
chaieb@23522: lemma even_nat_zero[presburger]: "even (0::nat)" by presburger
wenzelm@21256: 
wenzelm@21263: lemmas even_odd_nat_simps [simp] = even_nat_def[of "number_of v",standard]
wenzelm@21256:   even_nat_zero even_nat_Suc even_nat_product even_nat_sum even_nat_power
wenzelm@21256: 
wenzelm@21256: 
wenzelm@21256: subsection {* Equivalent definitions *}
wenzelm@21256: 
wenzelm@21263: lemma nat_lt_two_imp_zero_or_one: "(x::nat) < Suc (Suc 0) ==>
chaieb@23522:     x = 0 | x = Suc 0" by presburger
wenzelm@21256: 
wenzelm@21256: lemma even_nat_mod_two_eq_zero: "even (x::nat) ==> x mod (Suc (Suc 0)) = 0"
chaieb@23522:   by presburger
wenzelm@21256: 
wenzelm@21256: lemma odd_nat_mod_two_eq_one: "odd (x::nat) ==> x mod (Suc (Suc 0)) = Suc 0"
chaieb@23522: by presburger
wenzelm@21256: 
wenzelm@21263: lemma even_nat_equiv_def: "even (x::nat) = (x mod Suc (Suc 0) = 0)"
chaieb@23522:   by presburger
wenzelm@21256: 
wenzelm@21256: lemma odd_nat_equiv_def: "odd (x::nat) = (x mod Suc (Suc 0) = Suc 0)"
chaieb@23522:   by presburger
wenzelm@21256: 
wenzelm@21263: lemma even_nat_div_two_times_two: "even (x::nat) ==>
chaieb@23522:     Suc (Suc 0) * (x div Suc (Suc 0)) = x" by presburger
wenzelm@21256: 
wenzelm@21263: lemma odd_nat_div_two_times_two_plus_one: "odd (x::nat) ==>
chaieb@23522:     Suc( Suc (Suc 0) * (x div Suc (Suc 0))) = x" by presburger
wenzelm@21256: 
wenzelm@21256: lemma even_nat_equiv_def2: "even (x::nat) = (EX y. x = Suc (Suc 0) * y)"
chaieb@23522:   by presburger
wenzelm@21256: 
wenzelm@21256: lemma odd_nat_equiv_def2: "odd (x::nat) = (EX y. x = Suc(Suc (Suc 0) * y))"
chaieb@23522:   by presburger
wenzelm@21256: 
wenzelm@21256: subsection {* Parity and powers *}
wenzelm@21256: 
wenzelm@21263: lemma  minus_one_even_odd_power:
wenzelm@21263:      "(even x --> (- 1::'a::{comm_ring_1,recpower})^x = 1) &
wenzelm@21256:       (odd x --> (- 1::'a)^x = - 1)"
wenzelm@21256:   apply (induct x)
wenzelm@21256:   apply (rule conjI)
wenzelm@21256:   apply simp
wenzelm@21256:   apply (insert even_nat_zero, blast)
wenzelm@21256:   apply (simp add: power_Suc)
wenzelm@21263:   done
wenzelm@21256: 
wenzelm@21256: lemma minus_one_even_power [simp]:
wenzelm@21263:     "even x ==> (- 1::'a::{comm_ring_1,recpower})^x = 1"
wenzelm@21263:   using minus_one_even_odd_power by blast
wenzelm@21256: 
wenzelm@21256: lemma minus_one_odd_power [simp]:
wenzelm@21263:     "odd x ==> (- 1::'a::{comm_ring_1,recpower})^x = - 1"
wenzelm@21263:   using minus_one_even_odd_power by blast
wenzelm@21256: 
wenzelm@21256: lemma neg_one_even_odd_power:
wenzelm@21263:      "(even x --> (-1::'a::{number_ring,recpower})^x = 1) &
wenzelm@21256:       (odd x --> (-1::'a)^x = -1)"
wenzelm@21256:   apply (induct x)
wenzelm@21256:   apply (simp, simp add: power_Suc)
wenzelm@21256:   done
wenzelm@21256: 
wenzelm@21256: lemma neg_one_even_power [simp]:
wenzelm@21263:     "even x ==> (-1::'a::{number_ring,recpower})^x = 1"
wenzelm@21263:   using neg_one_even_odd_power by blast
wenzelm@21256: 
wenzelm@21256: lemma neg_one_odd_power [simp]:
wenzelm@21263:     "odd x ==> (-1::'a::{number_ring,recpower})^x = -1"
wenzelm@21263:   using neg_one_even_odd_power by blast
wenzelm@21256: 
wenzelm@21256: lemma neg_power_if:
wenzelm@21263:      "(-x::'a::{comm_ring_1,recpower}) ^ n =
wenzelm@21256:       (if even n then (x ^ n) else -(x ^ n))"
wenzelm@21263:   apply (induct n)
wenzelm@21263:   apply (simp_all split: split_if_asm add: power_Suc)
wenzelm@21263:   done
wenzelm@21256: 
wenzelm@21263: lemma zero_le_even_power: "even n ==>
wenzelm@21256:     0 <= (x::'a::{recpower,ordered_ring_strict}) ^ n"
wenzelm@21256:   apply (simp add: even_nat_equiv_def2)
wenzelm@21256:   apply (erule exE)
wenzelm@21256:   apply (erule ssubst)
wenzelm@21256:   apply (subst power_add)
wenzelm@21256:   apply (rule zero_le_square)
wenzelm@21256:   done
wenzelm@21256: 
wenzelm@21263: lemma zero_le_odd_power: "odd n ==>
wenzelm@21256:     (0 <= (x::'a::{recpower,ordered_idom}) ^ n) = (0 <= x)"
wenzelm@21256:   apply (simp add: odd_nat_equiv_def2)
wenzelm@21256:   apply (erule exE)
wenzelm@21256:   apply (erule ssubst)
wenzelm@21256:   apply (subst power_Suc)
wenzelm@21256:   apply (subst power_add)
wenzelm@21256:   apply (subst zero_le_mult_iff)
wenzelm@21256:   apply auto
nipkow@25162:   apply (subgoal_tac "x = 0 & y > 0")
wenzelm@21256:   apply (erule conjE, assumption)
wenzelm@21263:   apply (subst power_eq_0_iff [symmetric])
wenzelm@21256:   apply (subgoal_tac "0 <= x^y * x^y")
wenzelm@21256:   apply simp
wenzelm@21256:   apply (rule zero_le_square)+
wenzelm@21263:   done
wenzelm@21256: 
chaieb@23522: lemma zero_le_power_eq[presburger]: "(0 <= (x::'a::{recpower,ordered_idom}) ^ n) =
wenzelm@21256:     (even n | (odd n & 0 <= x))"
wenzelm@21256:   apply auto
wenzelm@21263:   apply (subst zero_le_odd_power [symmetric])
wenzelm@21256:   apply assumption+
wenzelm@21256:   apply (erule zero_le_even_power)
wenzelm@21263:   apply (subst zero_le_odd_power)
wenzelm@21256:   apply assumption+
wenzelm@21263:   done
wenzelm@21256: 
chaieb@23522: lemma zero_less_power_eq[presburger]: "(0 < (x::'a::{recpower,ordered_idom}) ^ n) =
wenzelm@21256:     (n = 0 | (even n & x ~= 0) | (odd n & 0 < x))"
wenzelm@21256:   apply (rule iffI)
wenzelm@21256:   apply clarsimp
wenzelm@21256:   apply (rule conjI)
wenzelm@21256:   apply clarsimp
wenzelm@21256:   apply (rule ccontr)
wenzelm@21256:   apply (subgoal_tac "~ (0 <= x^n)")
wenzelm@21256:   apply simp
wenzelm@21256:   apply (subst zero_le_odd_power)
wenzelm@21263:   apply assumption
wenzelm@21256:   apply simp
wenzelm@21256:   apply (rule notI)
wenzelm@21256:   apply (simp add: power_0_left)
wenzelm@21256:   apply (rule notI)
wenzelm@21256:   apply (simp add: power_0_left)
wenzelm@21256:   apply auto
wenzelm@21256:   apply (subgoal_tac "0 <= x^n")
wenzelm@21256:   apply (frule order_le_imp_less_or_eq)
wenzelm@21256:   apply simp
wenzelm@21256:   apply (erule zero_le_even_power)
wenzelm@21256:   apply (subgoal_tac "0 <= x^n")
wenzelm@21256:   apply (frule order_le_imp_less_or_eq)
wenzelm@21256:   apply auto
wenzelm@21256:   apply (subst zero_le_odd_power)
wenzelm@21256:   apply assumption
wenzelm@21256:   apply (erule order_less_imp_le)
wenzelm@21263:   done
wenzelm@21256: 
chaieb@23522: lemma power_less_zero_eq[presburger]: "((x::'a::{recpower,ordered_idom}) ^ n < 0) =
chaieb@23522:     (odd n & x < 0)" 
wenzelm@21263:   apply (subst linorder_not_le [symmetric])+
wenzelm@21256:   apply (subst zero_le_power_eq)
wenzelm@21256:   apply auto
wenzelm@21263:   done
wenzelm@21256: 
chaieb@23522: lemma power_le_zero_eq[presburger]: "((x::'a::{recpower,ordered_idom}) ^ n <= 0) =
wenzelm@21256:     (n ~= 0 & ((odd n & x <= 0) | (even n & x = 0)))"
wenzelm@21263:   apply (subst linorder_not_less [symmetric])+
wenzelm@21256:   apply (subst zero_less_power_eq)
wenzelm@21256:   apply auto
wenzelm@21263:   done
wenzelm@21256: 
wenzelm@21263: lemma power_even_abs: "even n ==>
wenzelm@21256:     (abs (x::'a::{recpower,ordered_idom}))^n = x^n"
wenzelm@21263:   apply (subst power_abs [symmetric])
wenzelm@21256:   apply (simp add: zero_le_even_power)
wenzelm@21263:   done
wenzelm@21256: 
chaieb@23522: lemma zero_less_power_nat_eq[presburger]: "(0 < (x::nat) ^ n) = (n = 0 | 0 < x)"
wenzelm@21263:   by (induct n) auto
wenzelm@21256: 
wenzelm@21263: lemma power_minus_even [simp]: "even n ==>
wenzelm@21256:     (- x)^n = (x^n::'a::{recpower,comm_ring_1})"
wenzelm@21256:   apply (subst power_minus)
wenzelm@21256:   apply simp
wenzelm@21263:   done
wenzelm@21256: 
wenzelm@21263: lemma power_minus_odd [simp]: "odd n ==>
wenzelm@21256:     (- x)^n = - (x^n::'a::{recpower,comm_ring_1})"
wenzelm@21256:   apply (subst power_minus)
wenzelm@21256:   apply simp
wenzelm@21263:   done
wenzelm@21256: 
wenzelm@21263: 
wenzelm@21263: text {* Simplify, when the exponent is a numeral *}
wenzelm@21256: 
wenzelm@21256: lemmas power_0_left_number_of = power_0_left [of "number_of w", standard]
wenzelm@21256: declare power_0_left_number_of [simp]
wenzelm@21256: 
wenzelm@21263: lemmas zero_le_power_eq_number_of [simp] =
wenzelm@21256:     zero_le_power_eq [of _ "number_of w", standard]
wenzelm@21256: 
wenzelm@21263: lemmas zero_less_power_eq_number_of [simp] =
wenzelm@21256:     zero_less_power_eq [of _ "number_of w", standard]
wenzelm@21256: 
wenzelm@21263: lemmas power_le_zero_eq_number_of [simp] =
wenzelm@21256:     power_le_zero_eq [of _ "number_of w", standard]
wenzelm@21256: 
wenzelm@21263: lemmas power_less_zero_eq_number_of [simp] =
wenzelm@21256:     power_less_zero_eq [of _ "number_of w", standard]
wenzelm@21256: 
wenzelm@21263: lemmas zero_less_power_nat_eq_number_of [simp] =
wenzelm@21256:     zero_less_power_nat_eq [of _ "number_of w", standard]
wenzelm@21256: 
wenzelm@21263: lemmas power_eq_0_iff_number_of [simp] = power_eq_0_iff [of _ "number_of w", standard]
wenzelm@21256: 
wenzelm@21263: lemmas power_even_abs_number_of [simp] = power_even_abs [of "number_of w" _, standard]
wenzelm@21256: 
wenzelm@21256: 
wenzelm@21256: subsection {* An Equivalence for @{term [source] "0 \<le> a^n"} *}
wenzelm@21256: 
wenzelm@21256: lemma even_power_le_0_imp_0:
wenzelm@21263:     "a ^ (2*k) \<le> (0::'a::{ordered_idom,recpower}) ==> a=0"
wenzelm@21263:   by (induct k) (auto simp add: zero_le_mult_iff mult_le_0_iff power_Suc)
wenzelm@21256: 
chaieb@23522: lemma zero_le_power_iff[presburger]:
wenzelm@21263:   "(0 \<le> a^n) = (0 \<le> (a::'a::{ordered_idom,recpower}) | even n)"
wenzelm@21256: proof cases
wenzelm@21256:   assume even: "even n"
wenzelm@21256:   then obtain k where "n = 2*k"
wenzelm@21256:     by (auto simp add: even_nat_equiv_def2 numeral_2_eq_2)
wenzelm@21263:   thus ?thesis by (simp add: zero_le_even_power even)
wenzelm@21256: next
wenzelm@21256:   assume odd: "odd n"
wenzelm@21256:   then obtain k where "n = Suc(2*k)"
wenzelm@21256:     by (auto simp add: odd_nat_equiv_def2 numeral_2_eq_2)
wenzelm@21256:   thus ?thesis
wenzelm@21263:     by (auto simp add: power_Suc zero_le_mult_iff zero_le_even_power
wenzelm@21263:              dest!: even_power_le_0_imp_0)
wenzelm@21263: qed
wenzelm@21263: 
wenzelm@21256: 
wenzelm@21256: subsection {* Miscellaneous *}
wenzelm@21256: 
chaieb@23522: lemma [presburger]:"(x + 1) div 2 = x div 2 \<longleftrightarrow> even (x::int)" by presburger
chaieb@23522: lemma [presburger]: "(x + 1) div 2 = x div 2 + 1 \<longleftrightarrow> odd (x::int)" by presburger
chaieb@23522: lemma even_plus_one_div_two: "even (x::int) ==> (x + 1) div 2 = x div 2"  by presburger
chaieb@23522: lemma odd_plus_one_div_two: "odd (x::int) ==> (x + 1) div 2 = x div 2 + 1" by presburger
wenzelm@21256: 
wenzelm@21263: lemma div_Suc: "Suc a div c = a div c + Suc 0 div c +
chaieb@23522:     (a mod c + Suc 0 mod c) div c" 
wenzelm@21256:   apply (subgoal_tac "Suc a = a + Suc 0")
wenzelm@21256:   apply (erule ssubst)
wenzelm@21256:   apply (rule div_add1_eq, simp)
wenzelm@21256:   done
wenzelm@21256: 
chaieb@23522: lemma [presburger]: "(Suc x) div Suc (Suc 0) = x div Suc (Suc 0) \<longleftrightarrow> even x" by presburger
chaieb@23522: lemma [presburger]: "(Suc x) div Suc (Suc 0) = x div Suc (Suc 0) \<longleftrightarrow> even x" by presburger
wenzelm@21263: lemma even_nat_plus_one_div_two: "even (x::nat) ==>
chaieb@23522:     (Suc x) div Suc (Suc 0) = x div Suc (Suc 0)" by presburger
wenzelm@21256: 
wenzelm@21263: lemma odd_nat_plus_one_div_two: "odd (x::nat) ==>
chaieb@23522:     (Suc x) div Suc (Suc 0) = Suc (x div Suc (Suc 0))" by presburger
wenzelm@21256: 
wenzelm@21256: end




