(*  Title:      HOL/Library/Parity.thy

     Author:     Jeremy Avigad, Jacques D. Fleuriot

 *)

 header {* Even and Odd for int and nat *}

 theory Parity

 imports Plain Presburger Main

 begin

 class even_odd = 

   fixes even :: "'a \<Rightarrow> bool"

 abbreviation

   odd :: "'a\<Colon>even_odd \<Rightarrow> bool" where

   "odd x \<equiv> \<not> even x"

 instantiation nat and int  :: even_odd

 begin

 definition

   even_def [presburger]: "even x \<longleftrightarrow> (x\<Colon>int) mod 2 = 0"

 definition

   even_nat_def [presburger]: "even x \<longleftrightarrow> even (int x)"

 instance ..

 end

 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 presburger

 lemma neq_one_mod_two [simp, presburger]: 

   "((x::int) mod 2 ~= 0) = (x mod 2 = 1)" by presburger

 subsection {* Behavior under integer arithmetic operations *}

 declare dvd_def[algebra]

 lemma nat_even_iff_2_dvd[algebra]: "even (x::nat) \<longleftrightarrow> 2 dvd x"

   by (presburger add: even_nat_def even_def)

 lemma int_even_iff_2_dvd[algebra]: "even (x::int) \<longleftrightarrow> 2 dvd x"

   by presburger

 lemma even_times_anything: "even (x::int) ==> even (x * y)"

   by algebra

 lemma anything_times_even: "even (y::int) ==> even (x * y)" by algebra

 lemma odd_times_odd: "odd (x::int) ==> odd y ==> odd (x * y)" 

   by (simp add: even_def zmod_zmult1_eq)

 lemma even_product[presburger]: "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 presburger

 lemma even_plus_odd: "even (x::int) ==> odd y ==> odd (x + y)"

   by presburger

 lemma odd_plus_even: "odd (x::int) ==> even y ==> odd (x + y)"

   by presburger

 lemma odd_plus_odd: "odd (x::int) ==> odd y ==> even (x + y)" by presburger

 lemma even_sum[presburger]: "even ((x::int) + y) = ((even x & even y) | (odd x & odd y))"

   by presburger

 lemma even_neg[presburger, algebra]: "even (-(x::int)) = even x" by presburger

 lemma even_difference:

     "even ((x::int) - y) = ((even x & even y) | (odd x & odd y))" by presburger

 lemma even_pow_gt_zero:

     "even (x::int) ==> 0 < n ==> even (x^n)"

   by (induct n) (auto simp add: even_product)

 lemma odd_pow_iff[presburger, algebra]: 

   "odd ((x::int) ^ n) \<longleftrightarrow> (n = 0 \<or> odd x)"

   apply (induct n, simp_all)

   apply presburger

   apply (case_tac n, auto)

   apply (simp_all add: even_product)

   done

 lemma odd_pow: "odd x ==> odd((x::int)^n)" by (simp add: odd_pow_iff)

 lemma even_power[presburger]: "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[presburger]: "even (0::int)" by presburger

 lemma odd_one[presburger]: "odd (1::int)" by presburger

 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 presburger

 lemma two_times_odd_div_two_plus_one: "odd (x::int) ==>

     2 * (x div 2) + 1 = x" by presburger

 lemma even_equiv_def: "even (x::int) = (EX y. x = 2 * y)" by presburger

 lemma odd_equiv_def: "odd (x::int) = (EX y. x = 2 * y + 1)" by presburger

 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[presburger, algebra]: "even((x::nat) * y) = (even x | even y)"

   by (simp add: even_nat_def int_mult)

 lemma even_nat_sum[presburger, algebra]: "even ((x::nat) + y) =

     ((even x & even y) | (odd x & odd y))" by presburger

 lemma even_nat_difference[presburger, algebra]:

     "even ((x::nat) - y) = (x < y | (even x & even y) | (odd x & odd y))"

 by presburger

 lemma even_nat_Suc[presburger, algebra]: "even (Suc x) = odd x" by presburger

 lemma even_nat_power[presburger, algebra]: "even ((x::nat)^y) = (even x & 0 < y)"

   by (simp add: even_nat_def int_power)

 lemma even_nat_zero[presburger]: "even (0::nat)" by presburger

 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 presburger

 lemma even_nat_mod_two_eq_zero: "even (x::nat) ==> x mod (Suc (Suc 0)) = 0"

   by presburger

 lemma odd_nat_mod_two_eq_one: "odd (x::nat) ==> x mod (Suc (Suc 0)) = Suc 0"

 by presburger

 lemma even_nat_equiv_def: "even (x::nat) = (x mod Suc (Suc 0) = 0)"

   by presburger

 lemma odd_nat_equiv_def: "odd (x::nat) = (x mod Suc (Suc 0) = Suc 0)"

   by presburger

 lemma even_nat_div_two_times_two: "even (x::nat) ==>

     Suc (Suc 0) * (x div Suc (Suc 0)) = x" by presburger

 lemma odd_nat_div_two_times_two_plus_one: "odd (x::nat) ==>

     Suc( Suc (Suc 0) * (x div Suc (Suc 0))) = x" by presburger

 lemma even_nat_equiv_def2: "even (x::nat) = (EX y. x = Suc (Suc 0) * y)"

   by presburger

 lemma odd_nat_equiv_def2: "odd (x::nat) = (EX y. x = Suc(Suc (Suc 0) * y))"

   by presburger

 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"

   using minus_one_even_odd_power by blast

 lemma minus_one_odd_power [simp]:

     "odd x ==> (- 1::'a::{comm_ring_1,recpower})^x = - 1"

   using minus_one_even_odd_power by blast

 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"

   using neg_one_even_odd_power by blast

 lemma neg_one_odd_power [simp]:

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

   using neg_one_even_odd_power by blast

 lemma neg_power_if:

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

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

   apply (induct n)

   apply (simp_all split: split_if_asm add: power_Suc)

   done

 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 (auto simp: odd_nat_equiv_def2 power_Suc power_add zero_le_mult_iff)

 apply (metis field_power_not_zero no_zero_divirors_neq0 order_antisym_conv zero_le_square)

 done

 lemma zero_le_power_eq[presburger]: "(0 <= (x::'a::{recpower,ordered_idom}) ^ n) =

     (even n | (odd n & 0 <= x))"

   apply auto

   apply (subst zero_le_odd_power [symmetric])

   apply assumption+

   apply (erule zero_le_even_power)

   done

 lemma zero_less_power_eq[presburger]: "(0 < (x::'a::{recpower,ordered_idom}) ^ n) =

     (n = 0 | (even n & x ~= 0) | (odd n & 0 < x))"

   unfolding order_less_le zero_le_power_eq by auto

 lemma power_less_zero_eq[presburger]: "((x::'a::{recpower,ordered_idom}) ^ n < 0) =

     (odd n & x < 0)"

   apply (subst linorder_not_le [symmetric])+

   apply (subst zero_le_power_eq)

   apply auto

   done

 lemma power_le_zero_eq[presburger]: "((x::'a::{recpower,ordered_idom}) ^ n <= 0) =

     (n ~= 0 & ((odd n & x <= 0) | (even n & x = 0)))"

   apply (subst linorder_not_less [symmetric])+

   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 [symmetric])

   apply (simp add: zero_le_even_power)

   done

 lemma zero_less_power_nat_eq[presburger]: "(0 < (x::nat) ^ n) = (n = 0 | 0 < x)"

   by (induct n) auto

 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

 lemma power_mono_even: fixes x y :: "'a :: {recpower, ordered_idom}"

   assumes "even n" and "\<bar>x\<bar> \<le> \<bar>y\<bar>"

   shows "x^n \<le> y^n"

 proof -

   have "0 \<le> \<bar>x\<bar>" by auto

   with `\<bar>x\<bar> \<le> \<bar>y\<bar>`

   have "\<bar>x\<bar>^n \<le> \<bar>y\<bar>^n" by (rule power_mono)

   thus ?thesis unfolding power_even_abs[OF `even n`] .

 qed

 lemma odd_pos: "odd (n::nat) \<Longrightarrow> 0 < n" by presburger

 lemma power_mono_odd: fixes x y :: "'a :: {recpower, ordered_idom}"

   assumes "odd n" and "x \<le> y"

   shows "x^n \<le> y^n"

 proof (cases "y < 0")

   case True with `x \<le> y` have "-y \<le> -x" and "0 \<le> -y" by auto

   hence "(-y)^n \<le> (-x)^n" by (rule power_mono)

   thus ?thesis unfolding power_minus_odd[OF `odd n`] by auto

 next

   case False

   show ?thesis

   proof (cases "x < 0")

     case True hence "n \<noteq> 0" and "x \<le> 0" using `odd n`[THEN odd_pos] by auto

     hence "x^n \<le> 0" unfolding power_le_zero_eq using `odd n` by auto

     moreover

     from `\<not> y < 0` have "0 \<le> y" by auto

     hence "0 \<le> y^n" by auto

     ultimately show ?thesis by auto

   next

     case False hence "0 \<le> x" by auto

     with `x \<le> y` show ?thesis using power_mono by auto

   qed

 qed

 subsection {* General Lemmas About Division *}

 lemma Suc_times_mod_eq: "1<k ==> Suc (k * m) mod k = 1" 

 apply (induct "m")

 apply (simp_all add: mod_Suc)

 done

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

 lemma [simp]: "n div k \<le> (Suc n) div k"

 by (simp add: div_le_mono) 

 lemma Suc_n_div_2_gt_zero [simp]: "(0::nat) < n ==> 0 < (n + 1) div 2"

 by arith

 lemma div_2_gt_zero [simp]: "(1::nat) < n ==> 0 < n div 2" 

 by arith

   (* Potential use of algebra : Equality modulo n*)

 lemma mod_mult_self3 [simp]: "(k*n + m) mod n = m mod (n::nat)"

 by (simp add: mult_ac add_ac)

 lemma mod_mult_self4 [simp]: "Suc (k*n + m) mod n = Suc m mod n"

 proof -

   have "Suc (k * n + m) mod n = (k * n + Suc m) mod n" by simp

   also have "... = Suc m mod n" by (rule mod_mult_self3) 

   finally show ?thesis .

 qed

 lemma mod_Suc_eq_Suc_mod: "Suc m mod n = Suc (m mod n) mod n"

 apply (subst mod_Suc [of m]) 

 apply (subst mod_Suc [of "m mod n"], simp) 

 done

 subsection {* More Even/Odd Results *}

 lemma even_mult_two_ex: "even(n) = (\<exists>m::nat. n = 2*m)" by presburger

 lemma odd_Suc_mult_two_ex: "odd(n) = (\<exists>m. n = Suc (2*m))" by presburger

 lemma even_add [simp]: "even(m + n::nat) = (even m = even n)"  by presburger

 lemma odd_add [simp]: "odd(m + n::nat) = (odd m \<noteq> odd n)" by presburger

 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 lemma_even_div2 [simp]: "even (n::nat) ==> (n + 1) div 2 = n div 2" by presburger

 lemma lemma_not_even_div2 [simp]: "~even n ==> (n + 1) div 2 = Suc (n div 2)"

 by presburger

 lemma even_num_iff: "0 < n ==> even n = (~ even(n - 1 :: nat))"  by presburger

 lemma even_even_mod_4_iff: "even (n::nat) = even (n mod 4)" by presburger

 lemma lemma_odd_mod_4_div_2: "n mod 4 = (3::nat) ==> odd((n - 1) div 2)" by presburger

 lemma lemma_even_mod_4_div_2: "n mod 4 = (1::nat) ==> even ((n - 1) div 2)"

   by presburger

 text {* 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 [simp] =

     zero_le_power_eq [of _ "number_of w", standard]

 lemmas zero_less_power_eq_number_of [simp] =

     zero_less_power_eq [of _ "number_of w", standard]

 lemmas power_le_zero_eq_number_of [simp] =

     power_le_zero_eq [of _ "number_of w", standard]

 lemmas power_less_zero_eq_number_of [simp] =

     power_less_zero_eq [of _ "number_of w", standard]

 lemmas zero_less_power_nat_eq_number_of [simp] =

     zero_less_power_nat_eq [of _ "number_of w", standard]

 lemmas power_eq_0_iff_number_of [simp] = power_eq_0_iff [of _ "number_of w", standard]

 lemmas power_even_abs_number_of [simp] = power_even_abs [of "number_of w" _, standard]

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

 lemma even_power_le_0_imp_0:

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

   by (induct k) (auto simp add: zero_le_mult_iff mult_le_0_iff power_Suc)

 lemma zero_le_power_iff[presburger]:

   "(0 \<le> a^n) = (0 \<le> (a::'a::{ordered_idom,recpower}) | even 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 [presburger]:"(x + 1) div 2 = x div 2 \<longleftrightarrow> even (x::int)" by presburger

 lemma [presburger]: "(x + 1) div 2 = x div 2 + 1 \<longleftrightarrow> odd (x::int)" by presburger

 lemma even_plus_one_div_two: "even (x::int) ==> (x + 1) div 2 = x div 2"  by presburger

 lemma odd_plus_one_div_two: "odd (x::int) ==> (x + 1) div 2 = x div 2 + 1" by presburger

 lemma [presburger]: "(Suc x) div Suc (Suc 0) = x div Suc (Suc 0) \<longleftrightarrow> even x" by presburger

 lemma [presburger]: "(Suc x) div Suc (Suc 0) = x div Suc (Suc 0) \<longleftrightarrow> even x" by presburger

 lemma even_nat_plus_one_div_two: "even (x::nat) ==>

     (Suc x) div Suc (Suc 0) = x div Suc (Suc 0)" by presburger

 lemma odd_nat_plus_one_div_two: "odd (x::nat) ==>

     (Suc x) div Suc (Suc 0) = Suc (x div Suc (Suc 0))" by presburger

 end