# Theory Parity

Up to index of Isabelle/HOL

theory Parity
imports Divides IntDiv NatSimprocs
begin

    (*  Title:      Parity.thy
        ID:         $Id: Parity.thy,v 1.15 2005/09/17 16:25:11 wenzelm Exp $
        Author:     Jeremy Avigad
    *)

    header {* 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 ≤ 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 [source] "0 ≤ a^n"} *}

    lemma even_power_le_0_imp_0:
         "a ^ (2*k) ≤ (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 ≤ a^n) = (0 ≤ (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

### Even and odd are mutually exclusive

lemma int_pos_lt_two_imp_zero_or_one:

      [| 0 ≤ x; x < 2 |] ==> x = 0 ∨ x = 1

lemma neq_one_mod_two:

      (x mod 2 ≠ 0) = (x mod 2 = 1)

### Behavior under integer arithmetic operations

lemma even_times_anything:

      even x ==> even (x * y)

lemma anything_times_even:

      even y ==> even (x * y)

lemma odd_times_odd:

      [| odd x; odd y |] ==> odd (x * y)

lemma even_product:

      even (x * y) = (even x ∨ even y)

lemma even_plus_even:

      [| even x; even y |] ==> even (x + y)

lemma even_plus_odd:

      [| even x; odd y |] ==> odd (x + y)

lemma odd_plus_even:

      [| odd x; even y |] ==> odd (x + y)

lemma odd_plus_odd:

      [| odd x; odd y |] ==> even (x + y)

lemma even_sum:

      even (x + y) = (even x ∧ even y ∨ odd x ∧ odd y)

lemma even_neg:

      even (- x) = even x

lemma even_difference:

      even (x - y) = (even x ∧ even y ∨ odd x ∧ odd y)

lemma even_pow_gt_zero:

      [| even x; 0 < n |] ==> even (x ^ n)

lemma odd_pow:

      odd x ==> odd (x ^ n)

lemma even_power:

      even (x ^ n) = (even x ∧ 0 < n)

lemma even_zero:

      even 0

lemma odd_one:

      odd 1

lemmas even_odd_simps:

      even (number_of v) == number_of v mod 2 = 0

      even 0

      odd 1

      even (x * y) = (even x ∨ even y)

      even (x + y) = (even x ∧ even y ∨ odd x ∧ odd y)

      even (- x) = even x

      even (x - y) = (even x ∧ even y ∨ odd x ∧ odd y)

      even (x ^ n) = (even x ∧ 0 < n)

lemmas even_odd_simps:

      even (number_of v) == number_of v mod 2 = 0

      even 0

      odd 1

      even (x * y) = (even x ∨ even y)

      even (x + y) = (even x ∧ even y ∨ odd x ∧ odd y)

      even (- x) = even x

      even (x - y) = (even x ∧ even y ∨ odd x ∧ odd y)

      even (x ^ n) = (even x ∧ 0 < n)

### Equivalent definitions

lemma two_times_even_div_two:

      even x ==> 2 * (x div 2) = x

lemma two_times_odd_div_two_plus_one:

      odd x ==> 2 * (x div 2) + 1 = x

lemma even_equiv_def:

      even x = (∃y. x = 2 * y)

lemma odd_equiv_def:

      odd x = (∃y. x = 2 * y + 1)

### even and odd for nats

lemma pos_int_even_equiv_nat_even:

      0 ≤ x ==> even x = even (nat x)

lemma even_nat_product:

      even (x * y) = (even x ∨ even y)

lemma even_nat_sum:

      even (x + y) = (even x ∧ even y ∨ odd x ∧ odd y)

lemma even_nat_difference:

      even (x - y) = (x < y ∨ even x ∧ even y ∨ odd x ∧ odd y)

lemma even_nat_Suc:

      even (Suc x) = odd x

lemma even_nat_power:

      even (x ^ y) = (even x ∧ 0 < y)

lemma even_nat_zero:

      even 0

lemmas even_odd_nat_simps:

      even (number_of v) == even (int (number_of v))

      even 0

      even (Suc x) = odd x

      even (x * y) = (even x ∨ even y)

      even (x + y) = (even x ∧ even y ∨ odd x ∧ odd y)

      even (x ^ y) = (even x ∧ 0 < y)

lemmas even_odd_nat_simps:

      even (number_of v) == even (int (number_of v))

      even 0

      even (Suc x) = odd x

      even (x * y) = (even x ∨ even y)

      even (x + y) = (even x ∧ even y ∨ odd x ∧ odd y)

      even (x ^ y) = (even x ∧ 0 < y)

### Equivalent definitions

lemma nat_lt_two_imp_zero_or_one:

      x < Suc (Suc 0) ==> x = 0 ∨ x = Suc 0

lemma even_nat_mod_two_eq_zero:

      even x ==> x mod Suc (Suc 0) = 0

lemma odd_nat_mod_two_eq_one:

      odd x ==> x mod Suc (Suc 0) = Suc 0

lemma even_nat_equiv_def:

      even x = (x mod Suc (Suc 0) = 0)

lemma odd_nat_equiv_def:

      odd x = (x mod Suc (Suc 0) = Suc 0)

lemma even_nat_div_two_times_two:

      even x ==> Suc (Suc 0) * (x div Suc (Suc 0)) = x

lemma odd_nat_div_two_times_two_plus_one:

      odd x ==> Suc (Suc (Suc 0) * (x div Suc (Suc 0))) = x

lemma even_nat_equiv_def2:

      even x = (∃y. x = Suc (Suc 0) * y)

lemma odd_nat_equiv_def2:

      odd x = (∃y. x = Suc (Suc (Suc 0) * y))

### Parity and powers

lemma minus_one_even_odd_power:

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

lemma minus_one_even_power:

      even x ==> (- (1::'a)) ^ x = (1::'a)

lemma minus_one_odd_power:

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

lemma neg_one_even_odd_power:

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

lemma neg_one_even_power:

      even x ==> (-1::'a) ^ x = (1::'a)

lemma neg_one_odd_power:

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

lemma neg_power_if:

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

lemma zero_le_even_power:

      even n ==> (0::'a) ≤ x ^ n

lemma zero_le_odd_power:

      odd n ==> ((0::'a) ≤ x ^ n) = ((0::'a) ≤ x)

lemma zero_le_power_eq:

      ((0::'a) ≤ x ^ n) = (even n ∨ odd n ∧ (0::'a) ≤ x)

lemma zero_less_power_eq:

      ((0::'a) < x ^ n) = (n = 0 ∨ even n ∧ x ≠ (0::'a) ∨ odd n ∧ (0::'a) < x)

lemma power_less_zero_eq:

      (x ^ n < (0::'a)) = (odd n ∧ x < (0::'a))

lemma power_le_zero_eq:

      (x ^ n ≤ (0::'a)) = (n ≠ 0 ∧ (odd n ∧ x ≤ (0::'a) ∨ even n ∧ x = (0::'a)))

lemma power_even_abs:

      even n ==> ¦x¦ ^ n = x ^ n

lemma zero_less_power_nat_eq:

      (0 < x ^ n) = (n = 0 ∨ 0 < x)

lemma power_minus_even:

      even n ==> (- x) ^ n = x ^ n

lemma power_minus_odd:

      odd n ==> (- x) ^ n = - (x ^ n)

lemmas power_0_left_number_of:

      (0::'a) ^ number_of w = (if number_of w = 0 then 1::'a else 0::'a)

lemmas power_0_left_number_of:

      (0::'a) ^ number_of w = (if number_of w = 0 then 1::'a else 0::'a)

lemmas zero_le_power_eq_number_of:

      ((0::'a) ≤ x ^ number_of w) =
      (even (number_of w) ∨ odd (number_of w) ∧ (0::'a) ≤ x)

lemmas zero_le_power_eq_number_of:

      ((0::'a) ≤ x ^ number_of w) =
      (even (number_of w) ∨ odd (number_of w) ∧ (0::'a) ≤ x)

lemmas zero_less_power_eq_number_of:

      ((0::'a) < x ^ number_of w) =
      (number_of w = 0 ∨
       even (number_of w) ∧ x ≠ (0::'a) ∨ odd (number_of w) ∧ (0::'a) < x)

lemmas zero_less_power_eq_number_of:

      ((0::'a) < x ^ number_of w) =
      (number_of w = 0 ∨
       even (number_of w) ∧ x ≠ (0::'a) ∨ odd (number_of w) ∧ (0::'a) < x)

lemmas power_le_zero_eq_number_of:

      (x ^ number_of w ≤ (0::'a)) =
      (number_of w ≠ 0 ∧
       (odd (number_of w) ∧ x ≤ (0::'a) ∨ even (number_of w) ∧ x = (0::'a)))

lemmas power_le_zero_eq_number_of:

      (x ^ number_of w ≤ (0::'a)) =
      (number_of w ≠ 0 ∧
       (odd (number_of w) ∧ x ≤ (0::'a) ∨ even (number_of w) ∧ x = (0::'a)))

lemmas power_less_zero_eq_number_of:

      (x ^ number_of w < (0::'a)) = (odd (number_of w) ∧ x < (0::'a))

lemmas power_less_zero_eq_number_of:

      (x ^ number_of w < (0::'a)) = (odd (number_of w) ∧ x < (0::'a))

lemmas zero_less_power_nat_eq_number_of:

      (0 < x ^ number_of w) = (number_of w = 0 ∨ 0 < x)

lemmas zero_less_power_nat_eq_number_of:

      (0 < x ^ number_of w) = (number_of w = 0 ∨ 0 < x)

lemmas power_eq_0_iff_number_of:

      (a ^ number_of w = (0::'a)) = (a = (0::'a) ∧ 0 < number_of w)

lemmas power_eq_0_iff_number_of:

      (a ^ number_of w = (0::'a)) = (a = (0::'a) ∧ 0 < number_of w)

lemmas power_even_abs_number_of:

      even (number_of w) ==> ¦x¦ ^ number_of w = x ^ number_of w

lemmas power_even_abs_number_of:

      even (number_of w) ==> ¦x¦ ^ number_of w = x ^ number_of w

### An Equivalence for @{term [source] "0 ≤ a^n"}

lemma even_power_le_0_imp_0:

      a ^ (2 * k) ≤ (0::'a) ==> a = (0::'a)

lemma zero_le_power_iff:

      ((0::'a) ≤ a ^ n) = ((0::'a) ≤ a ∨ even n)

### Miscellaneous

lemma even_plus_one_div_two:

      even x ==> (x + 1) div 2 = x div 2

lemma odd_plus_one_div_two:

      odd x ==> (x + 1) div 2 = x div 2 + 1

lemma div_Suc:

      Suc a div c = a div c + Suc 0 div c + (a mod c + Suc 0 mod c) div c

lemma even_nat_plus_one_div_two:

      even x ==> Suc x div Suc (Suc 0) = x div Suc (Suc 0)

lemma odd_nat_plus_one_div_two:

      odd x ==> Suc x div Suc (Suc 0) = Suc (x div Suc (Suc 0))