# Theory Power

theory Power
imports Num
```(*  Title:      HOL/Power.thy
Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
*)

section ‹Exponentiation›

theory Power
imports Num
begin

subsection ‹Powers for Arbitrary Monoids›

class power = one + times
begin

primrec power :: "'a ⇒ nat ⇒ 'a"  (infixr "^" 80)
where
power_0: "a ^ 0 = 1"
| power_Suc: "a ^ Suc n = a * a ^ n"

notation (latex output)
power ("(_⇗_⇖)" [1000] 1000)

text ‹Special syntax for squares.›
abbreviation power2 :: "'a ⇒ 'a"  ("(_⇧2)" [1000] 999)
where "x⇧2 ≡ x ^ 2"

end

context monoid_mult
begin

subclass power .

lemma power_one [simp]: "1 ^ n = 1"
by (induct n) simp_all

lemma power_one_right [simp]: "a ^ 1 = a"
by simp

lemma power_Suc0_right [simp]: "a ^ Suc 0 = a"
by simp

lemma power_commutes: "a ^ n * a = a * a ^ n"
by (induct n) (simp_all add: mult.assoc)

lemma power_Suc2: "a ^ Suc n = a ^ n * a"

lemma power_add: "a ^ (m + n) = a ^ m * a ^ n"
by (induct m) (simp_all add: algebra_simps)

lemma power_mult: "a ^ (m * n) = (a ^ m) ^ n"

lemma power2_eq_square: "a⇧2 = a * a"

lemma power3_eq_cube: "a ^ 3 = a * a * a"

lemma power_even_eq: "a ^ (2 * n) = (a ^ n)⇧2"
by (subst mult.commute) (simp add: power_mult)

lemma power_odd_eq: "a ^ Suc (2*n) = a * (a ^ n)⇧2"

lemma power_numeral_even: "z ^ numeral (Num.Bit0 w) = (let w = z ^ (numeral w) in w * w)"
by (simp only: numeral_Bit0 power_add Let_def)

lemma power_numeral_odd: "z ^ numeral (Num.Bit1 w) = (let w = z ^ (numeral w) in z * w * w)"

lemma funpow_times_power: "(times x ^^ f x) = times (x ^ f x)"
proof (induct "f x" arbitrary: f)
case 0
then show ?case by (simp add: fun_eq_iff)
next
case (Suc n)
define g where "g x = f x - 1" for x
with Suc have "n = g x" by simp
with Suc have "times x ^^ g x = times (x ^ g x)" by simp
moreover from Suc g_def have "f x = g x + 1" by simp
ultimately show ?case
qed

lemma power_commuting_commutes:
assumes "x * y = y * x"
shows "x ^ n * y = y * x ^n"
proof (induct n)
case 0
then show ?case by simp
next
case (Suc n)
have "x ^ Suc n * y = x ^ n * y * x"
by (subst power_Suc2) (simp add: assms ac_simps)
also have "… = y * x ^ Suc n"
by (simp only: Suc power_Suc2) (simp add: ac_simps)
finally show ?case .
qed

lemma power_minus_mult: "0 < n ⟹ a ^ (n - 1) * a = a ^ n"
by (simp add: power_commutes split: nat_diff_split)

end

context comm_monoid_mult
begin

lemma power_mult_distrib [field_simps]: "(a * b) ^ n = (a ^ n) * (b ^ n)"
by (induct n) (simp_all add: ac_simps)

end

text ‹Extract constant factors from powers.›
declare power_mult_distrib [where a = "numeral w" for w, simp]
declare power_mult_distrib [where b = "numeral w" for w, simp]

lemma power_add_numeral [simp]: "a^numeral m * a^numeral n = a^numeral (m + n)"
for a :: "'a::monoid_mult"

lemma power_add_numeral2 [simp]: "a^numeral m * (a^numeral n * b) = a^numeral (m + n) * b"
for a :: "'a::monoid_mult"

lemma power_mult_numeral [simp]: "(a^numeral m)^numeral n = a^numeral (m * n)"
for a :: "'a::monoid_mult"
by (simp only: numeral_mult power_mult)

context semiring_numeral
begin

lemma numeral_sqr: "numeral (Num.sqr k) = numeral k * numeral k"
by (simp only: sqr_conv_mult numeral_mult)

lemma numeral_pow: "numeral (Num.pow k l) = numeral k ^ numeral l"
by (induct l)
(simp_all only: numeral_class.numeral.simps pow.simps

lemma power_numeral [simp]: "numeral k ^ numeral l = numeral (Num.pow k l)"
by (rule numeral_pow [symmetric])

end

context semiring_1
begin

lemma of_nat_power [simp]: "of_nat (m ^ n) = of_nat m ^ n"
by (induct n) simp_all

lemma zero_power: "0 < n ⟹ 0 ^ n = 0"
by (cases n) simp_all

lemma power_zero_numeral [simp]: "0 ^ numeral k = 0"

lemma zero_power2: "0⇧2 = 0" (* delete? *)
by (rule power_zero_numeral)

lemma one_power2: "1⇧2 = 1" (* delete? *)
by (rule power_one)

lemma power_0_Suc [simp]: "0 ^ Suc n = 0"
by simp

text ‹It looks plausible as a simprule, but its effect can be strange.›
lemma power_0_left: "0 ^ n = (if n = 0 then 1 else 0)"
by (cases n) simp_all

end

context comm_semiring_1
begin

text ‹The divides relation.›

lemma le_imp_power_dvd:
assumes "m ≤ n"
shows "a ^ m dvd a ^ n"
proof
from assms have "a ^ n = a ^ (m + (n - m))" by simp
also have "… = a ^ m * a ^ (n - m)" by (rule power_add)
finally show "a ^ n = a ^ m * a ^ (n - m)" .
qed

lemma power_le_dvd: "a ^ n dvd b ⟹ m ≤ n ⟹ a ^ m dvd b"
by (rule dvd_trans [OF le_imp_power_dvd])

lemma dvd_power_same: "x dvd y ⟹ x ^ n dvd y ^ n"
by (induct n) (auto simp add: mult_dvd_mono)

lemma dvd_power_le: "x dvd y ⟹ m ≥ n ⟹ x ^ n dvd y ^ m"
by (rule power_le_dvd [OF dvd_power_same])

lemma dvd_power [simp]:
fixes n :: nat
assumes "n > 0 ∨ x = 1"
shows "x dvd (x ^ n)"
using assms
proof
assume "0 < n"
then have "x ^ n = x ^ Suc (n - 1)" by simp
then show "x dvd (x ^ n)" by simp
next
assume "x = 1"
then show "x dvd (x ^ n)" by simp
qed

end

context semiring_1_no_zero_divisors
begin

subclass power .

lemma power_eq_0_iff [simp]: "a ^ n = 0 ⟷ a = 0 ∧ n > 0"
by (induct n) auto

lemma power_not_zero: "a ≠ 0 ⟹ a ^ n ≠ 0"
by (induct n) auto

lemma zero_eq_power2 [simp]: "a⇧2 = 0 ⟷ a = 0"
unfolding power2_eq_square by simp

end

context ring_1
begin

lemma power_minus: "(- a) ^ n = (- 1) ^ n * a ^ n"
proof (induct n)
case 0
show ?case by simp
next
case (Suc n)
then show ?case
by (simp del: power_Suc add: power_Suc2 mult.assoc)
qed

lemma power_minus': "NO_MATCH 1 x ⟹ (-x) ^ n = (-1)^n * x ^ n"
by (rule power_minus)

lemma power_minus_Bit0: "(- x) ^ numeral (Num.Bit0 k) = x ^ numeral (Num.Bit0 k)"
by (induct k, simp_all only: numeral_class.numeral.simps power_add
power_one_right mult_minus_left mult_minus_right minus_minus)

lemma power_minus_Bit1: "(- x) ^ numeral (Num.Bit1 k) = - (x ^ numeral (Num.Bit1 k))"
by (simp only: eval_nat_numeral(3) power_Suc power_minus_Bit0 mult_minus_left)

lemma power2_minus [simp]: "(- a)⇧2 = a⇧2"
by (fact power_minus_Bit0)

lemma power_minus1_even [simp]: "(- 1) ^ (2*n) = 1"
proof (induct n)
case 0
show ?case by simp
next
case (Suc n)
qed

lemma power_minus1_odd: "(- 1) ^ Suc (2*n) = -1"
by simp

lemma power_minus_even [simp]: "(-a) ^ (2*n) = a ^ (2*n)"
by (simp add: power_minus [of a])

end

context ring_1_no_zero_divisors
begin

lemma power2_eq_1_iff: "a⇧2 = 1 ⟷ a = 1 ∨ a = - 1"
using square_eq_1_iff [of a] by (simp add: power2_eq_square)

end

context idom
begin

lemma power2_eq_iff: "x⇧2 = y⇧2 ⟷ x = y ∨ x = - y"
unfolding power2_eq_square by (rule square_eq_iff)

end

context algebraic_semidom
begin

lemma div_power: "b dvd a ⟹ (a div b) ^ n = a ^ n div b ^ n"
by (induct n) (simp_all add: div_mult_div_if_dvd dvd_power_same)

lemma is_unit_power_iff: "is_unit (a ^ n) ⟷ is_unit a ∨ n = 0"
by (induct n) (auto simp add: is_unit_mult_iff)

lemma dvd_power_iff:
assumes "x ≠ 0"
shows   "x ^ m dvd x ^ n ⟷ is_unit x ∨ m ≤ n"
proof
assume *: "x ^ m dvd x ^ n"
{
assume "m > n"
note *
also have "x ^ n = x ^ n * 1" by simp
also from ‹m > n› have "m = n + (m - n)" by simp
also have "x ^ … = x ^ n * x ^ (m - n)" by (rule power_add)
finally have "x ^ (m - n) dvd 1"
by (subst (asm) dvd_times_left_cancel_iff) (insert assms, simp_all)
with ‹m > n› have "is_unit x" by (simp add: is_unit_power_iff)
}
thus "is_unit x ∨ m ≤ n" by force
qed (auto intro: unit_imp_dvd simp: is_unit_power_iff le_imp_power_dvd)

end

context normalization_semidom
begin

lemma normalize_power: "normalize (a ^ n) = normalize a ^ n"
by (induct n) (simp_all add: normalize_mult)

lemma unit_factor_power: "unit_factor (a ^ n) = unit_factor a ^ n"
by (induct n) (simp_all add: unit_factor_mult)

end

context division_ring
begin

text ‹Perhaps these should be simprules.›
lemma power_inverse [field_simps, divide_simps]: "inverse a ^ n = inverse (a ^ n)"
proof (cases "a = 0")
case True
then show ?thesis by (simp add: power_0_left)
next
case False
then have "inverse (a ^ n) = inverse a ^ n"
by (induct n) (simp_all add: nonzero_inverse_mult_distrib power_commutes)
then show ?thesis by simp
qed

lemma power_one_over [field_simps, divide_simps]: "(1 / a) ^ n = 1 / a ^ n"
using power_inverse [of a] by (simp add: divide_inverse)

end

context field
begin

lemma power_diff:
assumes "a ≠ 0"
shows "n ≤ m ⟹ a ^ (m - n) = a ^ m / a ^ n"
by (induct m n rule: diff_induct) (simp_all add: assms power_not_zero)

lemma power_divide [field_simps, divide_simps]: "(a / b) ^ n = a ^ n / b ^ n"
by (induct n) simp_all

end

subsection ‹Exponentiation on ordered types›

context linordered_semidom
begin

lemma zero_less_power [simp]: "0 < a ⟹ 0 < a ^ n"
by (induct n) simp_all

lemma zero_le_power [simp]: "0 ≤ a ⟹ 0 ≤ a ^ n"
by (induct n) simp_all

lemma power_mono: "a ≤ b ⟹ 0 ≤ a ⟹ a ^ n ≤ b ^ n"
by (induct n) (auto intro: mult_mono order_trans [of 0 a b])

lemma one_le_power [simp]: "1 ≤ a ⟹ 1 ≤ a ^ n"
using power_mono [of 1 a n] by simp

lemma power_le_one: "0 ≤ a ⟹ a ≤ 1 ⟹ a ^ n ≤ 1"
using power_mono [of a 1 n] by simp

lemma power_gt1_lemma:
assumes gt1: "1 < a"
shows "1 < a * a ^ n"
proof -
from gt1 have "0 ≤ a"
by (fact order_trans [OF zero_le_one less_imp_le])
from gt1 have "1 * 1 < a * 1" by simp
also from gt1 have "… ≤ a * a ^ n"
by (simp only: mult_mono ‹0 ≤ a› one_le_power order_less_imp_le zero_le_one order_refl)
finally show ?thesis by simp
qed

lemma power_gt1: "1 < a ⟹ 1 < a ^ Suc n"

lemma one_less_power [simp]: "1 < a ⟹ 0 < n ⟹ 1 < a ^ n"
by (cases n) (simp_all add: power_gt1_lemma)

lemma power_le_imp_le_exp:
assumes gt1: "1 < a"
shows "a ^ m ≤ a ^ n ⟹ m ≤ n"
proof (induct m arbitrary: n)
case 0
show ?case by simp
next
case (Suc m)
show ?case
proof (cases n)
case 0
with Suc have "a * a ^ m ≤ 1" by simp
with gt1 show ?thesis
by (force simp only: power_gt1_lemma not_less [symmetric])
next
case (Suc n)
with Suc.prems Suc.hyps show ?thesis
by (force dest: mult_left_le_imp_le simp add: less_trans [OF zero_less_one gt1])
qed
qed

lemma of_nat_zero_less_power_iff [simp]: "of_nat x ^ n > 0 ⟷ x > 0 ∨ n = 0"
by (induct n) auto

text ‹Surely we can strengthen this? It holds for ‹0<a<1› too.›
lemma power_inject_exp [simp]: "1 < a ⟹ a ^ m = a ^ n ⟷ m = n"
by (force simp add: order_antisym power_le_imp_le_exp)

text ‹
Can relax the first premise to @{term "0<a"} in the case of the
natural numbers.
›
lemma power_less_imp_less_exp: "1 < a ⟹ a ^ m < a ^ n ⟹ m < n"
by (simp add: order_less_le [of m n] less_le [of "a^m" "a^n"] power_le_imp_le_exp)

lemma power_strict_mono [rule_format]: "a < b ⟹ 0 ≤ a ⟹ 0 < n ⟶ a ^ n < b ^ n"
by (induct n) (auto simp: mult_strict_mono le_less_trans [of 0 a b])

text‹Lemma for ‹power_strict_decreasing››
lemma power_Suc_less: "0 < a ⟹ a < 1 ⟹ a * a ^ n < a ^ n"
by (induct n) (auto simp: mult_strict_left_mono)

lemma power_strict_decreasing [rule_format]: "n < N ⟹ 0 < a ⟹ a < 1 ⟶ a ^ N < a ^ n"
proof (induct N)
case 0
then show ?case by simp
next
case (Suc N)
then show ?case
apply (auto simp add: power_Suc_less less_Suc_eq)
apply (subgoal_tac "a * a^N < 1 * a^n")
apply simp
apply (rule mult_strict_mono)
apply auto
done
qed

text ‹Proof resembles that of ‹power_strict_decreasing›.›
lemma power_decreasing: "n ≤ N ⟹ 0 ≤ a ⟹ a ≤ 1 ⟹ a ^ N ≤ a ^ n"
proof (induct N)
case 0
then show ?case by simp
next
case (Suc N)
then show ?case
apply (subgoal_tac "a * a^N ≤ 1 * a^n")
apply simp
apply (rule mult_mono)
apply auto
done
qed

lemma power_Suc_less_one: "0 < a ⟹ a < 1 ⟹ a ^ Suc n < 1"
using power_strict_decreasing [of 0 "Suc n" a] by simp

text ‹Proof again resembles that of ‹power_strict_decreasing›.›
lemma power_increasing: "n ≤ N ⟹ 1 ≤ a ⟹ a ^ n ≤ a ^ N"
proof (induct N)
case 0
then show ?case by simp
next
case (Suc N)
then show ?case
apply (subgoal_tac "1 * a^n ≤ a * a^N")
apply simp
apply (rule mult_mono)
apply (auto simp add: order_trans [OF zero_le_one])
done
qed

text ‹Lemma for ‹power_strict_increasing›.›
lemma power_less_power_Suc: "1 < a ⟹ a ^ n < a * a ^ n"
by (induct n) (auto simp: mult_strict_left_mono less_trans [OF zero_less_one])

lemma power_strict_increasing: "n < N ⟹ 1 < a ⟹ a ^ n < a ^ N"
proof (induct N)
case 0
then show ?case by simp
next
case (Suc N)
then show ?case
apply (auto simp add: power_less_power_Suc less_Suc_eq)
apply (subgoal_tac "1 * a^n < a * a^N")
apply simp
apply (rule mult_strict_mono)
apply (auto simp add: less_trans [OF zero_less_one] less_imp_le)
done
qed

lemma power_increasing_iff [simp]: "1 < b ⟹ b ^ x ≤ b ^ y ⟷ x ≤ y"
by (blast intro: power_le_imp_le_exp power_increasing less_imp_le)

lemma power_strict_increasing_iff [simp]: "1 < b ⟹ b ^ x < b ^ y ⟷ x < y"
by (blast intro: power_less_imp_less_exp power_strict_increasing)

lemma power_le_imp_le_base:
assumes le: "a ^ Suc n ≤ b ^ Suc n"
and "0 ≤ b"
shows "a ≤ b"
proof (rule ccontr)
assume "¬ ?thesis"
then have "b < a" by (simp only: linorder_not_le)
then have "b ^ Suc n < a ^ Suc n"
by (simp only: assms(2) power_strict_mono)
with le show False
qed

lemma power_less_imp_less_base:
assumes less: "a ^ n < b ^ n"
assumes nonneg: "0 ≤ b"
shows "a < b"
proof (rule contrapos_pp [OF less])
assume "¬ ?thesis"
then have "b ≤ a" by (simp only: linorder_not_less)
from this nonneg have "b ^ n ≤ a ^ n" by (rule power_mono)
then show "¬ a ^ n < b ^ n" by (simp only: linorder_not_less)
qed

lemma power_inject_base: "a ^ Suc n = b ^ Suc n ⟹ 0 ≤ a ⟹ 0 ≤ b ⟹ a = b"
by (blast intro: power_le_imp_le_base antisym eq_refl sym)

lemma power_eq_imp_eq_base: "a ^ n = b ^ n ⟹ 0 ≤ a ⟹ 0 ≤ b ⟹ 0 < n ⟹ a = b"
by (cases n) (simp_all del: power_Suc, rule power_inject_base)

lemma power_eq_iff_eq_base: "0 < n ⟹ 0 ≤ a ⟹ 0 ≤ b ⟹ a ^ n = b ^ n ⟷ a = b"
using power_eq_imp_eq_base [of a n b] by auto

lemma power2_le_imp_le: "x⇧2 ≤ y⇧2 ⟹ 0 ≤ y ⟹ x ≤ y"
unfolding numeral_2_eq_2 by (rule power_le_imp_le_base)

lemma power2_less_imp_less: "x⇧2 < y⇧2 ⟹ 0 ≤ y ⟹ x < y"
by (rule power_less_imp_less_base)

lemma power2_eq_imp_eq: "x⇧2 = y⇧2 ⟹ 0 ≤ x ⟹ 0 ≤ y ⟹ x = y"
unfolding numeral_2_eq_2 by (erule (2) power_eq_imp_eq_base) simp

lemma power_Suc_le_self: "0 ≤ a ⟹ a ≤ 1 ⟹ a ^ Suc n ≤ a"
using power_decreasing [of 1 "Suc n" a] by simp

lemma power2_eq_iff_nonneg [simp]:
assumes "0 ≤ x" "0 ≤ y"
shows "(x ^ 2 = y ^ 2) ⟷ x = y"
using assms power2_eq_imp_eq by blast

end

context linordered_ring_strict
begin

lemma sum_squares_eq_zero_iff: "x * x + y * y = 0 ⟷ x = 0 ∧ y = 0"

lemma sum_squares_le_zero_iff: "x * x + y * y ≤ 0 ⟷ x = 0 ∧ y = 0"
by (simp add: le_less not_sum_squares_lt_zero sum_squares_eq_zero_iff)

lemma sum_squares_gt_zero_iff: "0 < x * x + y * y ⟷ x ≠ 0 ∨ y ≠ 0"
by (simp add: not_le [symmetric] sum_squares_le_zero_iff)

end

context linordered_idom
begin

lemma zero_le_power2 [simp]: "0 ≤ a⇧2"

lemma zero_less_power2 [simp]: "0 < a⇧2 ⟷ a ≠ 0"
by (force simp add: power2_eq_square zero_less_mult_iff linorder_neq_iff)

lemma power2_less_0 [simp]: "¬ a⇧2 < 0"
by (force simp add: power2_eq_square mult_less_0_iff)

lemma power_abs: "¦a ^ n¦ = ¦a¦ ^ n" -- ‹FIXME simp?›
by (induct n) (simp_all add: abs_mult)

lemma power_sgn [simp]: "sgn (a ^ n) = sgn a ^ n"
by (induct n) (simp_all add: sgn_mult)

lemma abs_power_minus [simp]: "¦(- a) ^ n¦ = ¦a ^ n¦"

lemma zero_less_power_abs_iff [simp]: "0 < ¦a¦ ^ n ⟷ a ≠ 0 ∨ n = 0"
proof (induct n)
case 0
show ?case by simp
next
case Suc
then show ?case by (auto simp: zero_less_mult_iff)
qed

lemma zero_le_power_abs [simp]: "0 ≤ ¦a¦ ^ n"
by (rule zero_le_power [OF abs_ge_zero])

lemma power2_less_eq_zero_iff [simp]: "a⇧2 ≤ 0 ⟷ a = 0"

lemma abs_power2 [simp]: "¦a⇧2¦ = a⇧2"

lemma power2_abs [simp]: "¦a¦⇧2 = a⇧2"

lemma odd_power_less_zero: "a < 0 ⟹ a ^ Suc (2 * n) < 0"
proof (induct n)
case 0
then show ?case by simp
next
case (Suc n)
have "a ^ Suc (2 * Suc n) = (a*a) * a ^ Suc(2*n)"
then show ?case
by (simp del: power_Suc add: Suc mult_less_0_iff mult_neg_neg)
qed

lemma odd_0_le_power_imp_0_le: "0 ≤ a ^ Suc (2 * n) ⟹ 0 ≤ a"
using odd_power_less_zero [of a n]
by (force simp add: linorder_not_less [symmetric])

lemma zero_le_even_power'[simp]: "0 ≤ a ^ (2 * n)"
proof (induct n)
case 0
show ?case by simp
next
case (Suc n)
have "a ^ (2 * Suc n) = (a*a) * a ^ (2*n)"
then show ?case
qed

lemma sum_power2_ge_zero: "0 ≤ x⇧2 + y⇧2"

lemma not_sum_power2_lt_zero: "¬ x⇧2 + y⇧2 < 0"
unfolding not_less by (rule sum_power2_ge_zero)

lemma sum_power2_eq_zero_iff: "x⇧2 + y⇧2 = 0 ⟷ x = 0 ∧ y = 0"

lemma sum_power2_le_zero_iff: "x⇧2 + y⇧2 ≤ 0 ⟷ x = 0 ∧ y = 0"
by (simp add: le_less sum_power2_eq_zero_iff not_sum_power2_lt_zero)

lemma sum_power2_gt_zero_iff: "0 < x⇧2 + y⇧2 ⟷ x ≠ 0 ∨ y ≠ 0"
unfolding not_le [symmetric] by (simp add: sum_power2_le_zero_iff)

lemma abs_le_square_iff: "¦x¦ ≤ ¦y¦ ⟷ x⇧2 ≤ y⇧2"
(is "?lhs ⟷ ?rhs")
proof
assume ?lhs
then have "¦x¦⇧2 ≤ ¦y¦⇧2" by (rule power_mono) simp
then show ?rhs by simp
next
assume ?rhs
then show ?lhs
by (auto intro!: power2_le_imp_le [OF _ abs_ge_zero])
qed

lemma abs_square_le_1:"x⇧2 ≤ 1 ⟷ ¦x¦ ≤ 1"
using abs_le_square_iff [of x 1] by simp

lemma abs_square_eq_1: "x⇧2 = 1 ⟷ ¦x¦ = 1"
by (auto simp add: abs_if power2_eq_1_iff)

lemma abs_square_less_1: "x⇧2 < 1 ⟷ ¦x¦ < 1"
using  abs_square_eq_1 [of x] abs_square_le_1 [of x] by (auto simp add: le_less)

end

subsection ‹Miscellaneous rules›

lemma (in linordered_semidom) self_le_power: "1 ≤ a ⟹ 0 < n ⟹ a ≤ a ^ n"
using power_increasing [of 1 n a] power_one_right [of a] by auto

lemma (in power) power_eq_if: "p ^ m = (if m=0 then 1 else p * (p ^ (m - 1)))"
unfolding One_nat_def by (cases m) simp_all

lemma (in comm_semiring_1) power2_sum: "(x + y)⇧2 = x⇧2 + y⇧2 + 2 * x * y"
by (simp add: algebra_simps power2_eq_square mult_2_right)

context comm_ring_1
begin

lemma power2_diff: "(x - y)⇧2 = x⇧2 + y⇧2 - 2 * x * y"
by (simp add: algebra_simps power2_eq_square mult_2_right)

lemma power2_commute: "(x - y)⇧2 = (y - x)⇧2"

lemma minus_power_mult_self: "(- a) ^ n * (- a) ^ n = a ^ (2 * n)"
(simp add: power2_eq_square [symmetric] power_mult [symmetric])

lemma minus_one_mult_self [simp]: "(- 1) ^ n * (- 1) ^ n = 1"
using minus_power_mult_self [of 1 n] by simp

lemma left_minus_one_mult_self [simp]: "(- 1) ^ n * ((- 1) ^ n * a) = a"

end

text ‹Simprules for comparisons where common factors can be cancelled.›

lemmas zero_compare_simps =
zero_le_mult_iff zero_le_divide_iff
zero_less_mult_iff zero_less_divide_iff
mult_le_0_iff divide_le_0_iff
mult_less_0_iff divide_less_0_iff
zero_le_power2 power2_less_0

subsection ‹Exponentiation for the Natural Numbers›

lemma nat_one_le_power [simp]: "Suc 0 ≤ i ⟹ Suc 0 ≤ i ^ n"
by (rule one_le_power [of i n, unfolded One_nat_def])

lemma nat_zero_less_power_iff [simp]: "x ^ n > 0 ⟷ x > 0 ∨ n = 0"
for x :: nat
by (induct n) auto

lemma nat_power_eq_Suc_0_iff [simp]: "x ^ m = Suc 0 ⟷ m = 0 ∨ x = Suc 0"
by (induct m) auto

lemma power_Suc_0 [simp]: "Suc 0 ^ n = Suc 0"
by simp

text ‹
Valid for the naturals, but what if ‹0 < i < 1›? Premises cannot be
weakened: consider the case where ‹i = 0›, ‹m = 1› and ‹n = 0›.
›

lemma nat_power_less_imp_less:
fixes i :: nat
assumes nonneg: "0 < i"
assumes less: "i ^ m < i ^ n"
shows "m < n"
proof (cases "i = 1")
case True
with less power_one [where 'a = nat] show ?thesis by simp
next
case False
with nonneg have "1 < i" by auto
from power_strict_increasing_iff [OF this] less show ?thesis ..
qed

lemma power_dvd_imp_le: "i ^ m dvd i ^ n ⟹ 1 < i ⟹ m ≤ n"
for i m n :: nat
apply (rule power_le_imp_le_exp)
apply assumption
apply (erule dvd_imp_le)
apply simp
done

lemma power2_nat_le_eq_le: "m⇧2 ≤ n⇧2 ⟷ m ≤ n"
for m n :: nat
by (auto intro: power2_le_imp_le power_mono)

lemma power2_nat_le_imp_le:
fixes m n :: nat
assumes "m⇧2 ≤ n"
shows "m ≤ n"
proof (cases m)
case 0
then show ?thesis by simp
next
case (Suc k)
show ?thesis
proof (rule ccontr)
assume "¬ ?thesis"
then have "n < m" by simp
with assms Suc show False
qed
qed

lemma ex_power_ivl1: fixes b k :: nat assumes "b ≥ 2"
shows "k ≥ 1 ⟹ ∃n. b^n ≤ k ∧ k < b^(n+1)" (is "_ ⟹ ∃n. ?P k n")
proof(induction k)
case 0 thus ?case by simp
next
case (Suc k)
show ?case
proof cases
assume "k=0"
hence "?P (Suc k) 0" using assms by simp
thus ?case ..
next
assume "k≠0"
with Suc obtain n where IH: "?P k n" by auto
show ?case
proof (cases "k = b^(n+1) - 1")
case True
hence "?P (Suc k) (n+1)" using assms
thus ?thesis ..
next
case False
hence "?P (Suc k) n" using IH by auto
thus ?thesis ..
qed
qed
qed

lemma ex_power_ivl2: fixes b k :: nat assumes "b ≥ 2" "k ≥ 2"
shows "∃n. b^n < k ∧ k ≤ b^(n+1)"
proof -
have "1 ≤ k - 1" using assms(2) by arith
from ex_power_ivl1[OF assms(1) this]
obtain n where "b ^ n ≤ k - 1 ∧ k - 1 < b ^ (n + 1)" ..
hence "b^n < k ∧ k ≤ b^(n+1)" using assms by auto
thus ?thesis ..
qed

subsubsection ‹Cardinality of the Powerset›

lemma card_UNIV_bool [simp]: "card (UNIV :: bool set) = 2"
unfolding UNIV_bool by simp

lemma card_Pow: "finite A ⟹ card (Pow A) = 2 ^ card A"
proof (induct rule: finite_induct)
case empty
show ?case by simp
next
case (insert x A)
from ‹x ∉ A› have disjoint: "Pow A ∩ insert x ` Pow A = {}" by blast
from ‹x ∉ A› have inj_on: "inj_on (insert x) (Pow A)"
unfolding inj_on_def by auto

have "card (Pow (insert x A)) = card (Pow A ∪ insert x ` Pow A)"
by (simp only: Pow_insert)
also have "… = card (Pow A) + card (insert x ` Pow A)"
by (rule card_Un_disjoint) (use ‹finite A› disjoint in simp_all)
also from inj_on have "card (insert x ` Pow A) = card (Pow A)"
by (rule card_image)
also have "… + … = 2 * …" by (simp add: mult_2)
also from insert(3) have "… = 2 ^ Suc (card A)" by simp
also from insert(1,2) have "Suc (card A) = card (insert x A)"
by (rule card_insert_disjoint [symmetric])
finally show ?case .
qed

subsection ‹Code generator tweak›

code_identifier
code_module Power ⇀ (SML) Arith and (OCaml) Arith and (Haskell) Arith

end
```