(*  Title:      Parity.thy

    ID:         $Id$

    Author:     Jeremy Avigad

    License:    GPL (GNU GENERAL PUBLIC LICENSE)

*)

header {* Parity: Even and Odd for ints and nats*}

theory Parity = Divides + IntDiv + NatSimprocs:

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 {* Casting a nat power to an integer *}

lemma zpow_int: "int (x^y) = (int x)^y"

  apply (induct_tac y)

  apply (simp, simp add: zmult_int [THEN sym])

  done

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_tac n)

  apply (auto simp add: even_product)

  done

lemma odd_pow: "odd x ==> odd((x::int)^n)"

  apply (induct_tac 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 zmult_int [THEN sym])

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)

text{*Compatibility, in case Avigad uses this*}

lemmas even_nat_suc = even_nat_Suc

lemma even_nat_power: "even ((x::nat)^y) = (even x & 0 < y)"

  by (simp add: even_nat_def zpow_int)

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 {* Powers of negative one *}

lemma neg_one_even_odd_power:

     "(even x --> (-1::'a::{number_ring,ringpower})^x = 1) & 

      (odd x --> (-1::'a)^x = -1)"

  apply (induct_tac x)

  apply (simp, simp add: power_Suc)

  done

lemma neg_one_even_power [simp]:

     "even x ==> (-1::'a::{number_ring,ringpower})^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,ringpower})^x = -1"

  by (rule neg_one_even_odd_power [THEN conjunct2, THEN mp], assumption)

lemma neg_power_if:

     "(-x::'a::{comm_ring_1,ringpower}) ^ n = 

      (if even n then (x ^ n) else -(x ^ n))"

  by (induct n, simp_all split: split_if_asm add: power_Suc) 

subsection {* An Equivalence for @{term "0 \<le> a^n"} *}

lemma even_power_le_0_imp_0:

     "a ^ (2*k) \<le> (0::'a::{ordered_idom,ringpower}) ==> 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,ringpower}) | 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