(*  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 *)

 declare power_0_left [of "number_of w", standard, simp]

 declare zero_le_power_eq [of _ "number_of w", standard, simp]

 declare zero_less_power_eq [of _ "number_of w", standard, simp]

 declare power_le_zero_eq [of _ "number_of w", standard, simp]

 declare power_less_zero_eq [of _ "number_of w", standard, simp]

 declare zero_less_power_nat_eq [of _ "number_of w", standard, simp]

 declare power_eq_0_iff [of _ "number_of w", standard, simp]

 declare power_even_abs [of "number_of w" _, standard, 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