src/HOL/Power.thy
 author wenzelm Wed, 10 Aug 2016 22:05:36 +0200 changeset 63654 f90e3926e627 parent 63648 f9f3006a5579 child 63924 f91766530e13 permissions -rw-r--r--
misc tuning and modernization;
```
(*  Title:      HOL/Power.thy
Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
*)

section \<open>Exponentiation\<close>

theory Power
imports Num
begin

subsection \<open>Powers for Arbitrary Monoids\<close>

class power = one + times
begin

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

notation (latex output)
power ("(_\<^bsup>_\<^esup>)" [1000] 1000)

text \<open>Special syntax for squares.\<close>
abbreviation power2 :: "'a \<Rightarrow> 'a"  ("(_\<^sup>2)" [1000] 999)
where "x\<^sup>2 \<equiv> 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\<^sup>2 = a * a"

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

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

lemma power_odd_eq: "a ^ Suc (2*n) = a * (a ^ n)\<^sup>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 "\<dots> = y * x ^ Suc n"
by (simp only: Suc power_Suc2) (simp add: ac_simps)
finally show ?case .
qed

lemma power_minus_mult: "0 < n \<Longrightarrow> 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 \<open>Extract constant factors from powers.\<close>
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 \<Longrightarrow> 0 ^ n = 0"
by (cases n) simp_all

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

lemma zero_power2: "0\<^sup>2 = 0" (* delete? *)
by (rule power_zero_numeral)

lemma one_power2: "1\<^sup>2 = 1" (* delete? *)
by (rule power_one)

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

text \<open>It looks plausible as a simprule, but its effect can be strange.\<close>
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 \<open>The divides relation.\<close>

lemma le_imp_power_dvd:
assumes "m \<le> n"
shows "a ^ m dvd a ^ n"
proof
from assms have "a ^ n = a ^ (m + (n - m))" by simp
also have "\<dots> = 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 \<Longrightarrow> m \<le> n \<Longrightarrow> a ^ m dvd b"
by (rule dvd_trans [OF le_imp_power_dvd])

lemma dvd_power_same: "x dvd y \<Longrightarrow> x ^ n dvd y ^ n"
by (induct n) (auto simp add: mult_dvd_mono)

lemma dvd_power_le: "x dvd y \<Longrightarrow> m \<ge> n \<Longrightarrow> x ^ n dvd y ^ m"
by (rule power_le_dvd [OF dvd_power_same])

lemma dvd_power [simp]:
fixes n :: nat
assumes "n > 0 \<or> 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 \<longleftrightarrow> a = 0 \<and> n > 0"
by (induct n) auto

lemma power_not_zero: "a \<noteq> 0 \<Longrightarrow> a ^ n \<noteq> 0"
by (induct n) auto

lemma zero_eq_power2 [simp]: "a\<^sup>2 = 0 \<longleftrightarrow> 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 \<Longrightarrow> (-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)\<^sup>2 = a\<^sup>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\<^sup>2 = 1 \<longleftrightarrow> a = 1 \<or> a = - 1"
using square_eq_1_iff [of a] by (simp add: power2_eq_square)

end

context idom
begin

lemma power2_eq_iff: "x\<^sup>2 = y\<^sup>2 \<longleftrightarrow> x = y \<or> x = - y"
unfolding power2_eq_square by (rule square_eq_iff)

end

context algebraic_semidom
begin

lemma div_power: "b dvd a \<Longrightarrow> (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) \<longleftrightarrow> is_unit a \<or> n = 0"
by (induct n) (auto simp add: is_unit_mult_iff)

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 \<open>Perhaps these should be simprules.\<close>
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 \<noteq> 0"
shows "n \<le> m \<Longrightarrow> 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 \<open>Exponentiation on ordered types\<close>

context linordered_semidom
begin

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

lemma zero_le_power [simp]: "0 \<le> a \<Longrightarrow> 0 \<le> a ^ n"
by (induct n) simp_all

lemma power_mono: "a \<le> b \<Longrightarrow> 0 \<le> a \<Longrightarrow> a ^ n \<le> b ^ n"
by (induct n) (auto intro: mult_mono order_trans [of 0 a b])

lemma one_le_power [simp]: "1 \<le> a \<Longrightarrow> 1 \<le> a ^ n"
using power_mono [of 1 a n] by simp

lemma power_le_one: "0 \<le> a \<Longrightarrow> a \<le> 1 \<Longrightarrow> a ^ n \<le> 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 \<le> 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 "\<dots> \<le> a * a ^ n"
by (simp only: mult_mono \<open>0 \<le> a\<close> one_le_power order_less_imp_le zero_le_one order_refl)
finally show ?thesis by simp
qed

lemma power_gt1: "1 < a \<Longrightarrow> 1 < a ^ Suc n"

lemma one_less_power [simp]: "1 < a \<Longrightarrow> 0 < n \<Longrightarrow> 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 \<le> a ^ n \<Longrightarrow> m \<le> 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 \<le> 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 \<longleftrightarrow> x > 0 \<or> n = 0"
by (induct n) auto

text \<open>Surely we can strengthen this? It holds for \<open>0<a<1\<close> too.\<close>
lemma power_inject_exp [simp]: "1 < a \<Longrightarrow> a ^ m = a ^ n \<longleftrightarrow> m = n"
by (force simp add: order_antisym power_le_imp_le_exp)

text \<open>
Can relax the first premise to @{term "0<a"} in the case of the
natural numbers.
\<close>
lemma power_less_imp_less_exp: "1 < a \<Longrightarrow> a ^ m < a ^ n \<Longrightarrow> 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 \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 < n \<longrightarrow> a ^ n < b ^ n"
by (induct n) (auto simp: mult_strict_mono le_less_trans [of 0 a b])

text\<open>Lemma for \<open>power_strict_decreasing\<close>\<close>
lemma power_Suc_less: "0 < a \<Longrightarrow> a < 1 \<Longrightarrow> a * a ^ n < a ^ n"
by (induct n) (auto simp: mult_strict_left_mono)

lemma power_strict_decreasing [rule_format]: "n < N \<Longrightarrow> 0 < a \<Longrightarrow> a < 1 \<longrightarrow> 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 \<open>Proof resembles that of \<open>power_strict_decreasing\<close>.\<close>
lemma power_decreasing: "n \<le> N \<Longrightarrow> 0 \<le> a \<Longrightarrow> a \<le> 1 \<Longrightarrow> a ^ N \<le> 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 \<le> 1 * a^n")
apply simp
apply (rule mult_mono)
apply auto
done
qed

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

text \<open>Proof again resembles that of \<open>power_strict_decreasing\<close>.\<close>
lemma power_increasing: "n \<le> N \<Longrightarrow> 1 \<le> a \<Longrightarrow> a ^ n \<le> 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 \<le> a * a^N")
apply simp
apply (rule mult_mono)
apply (auto simp add: order_trans [OF zero_le_one])
done
qed

text \<open>Lemma for \<open>power_strict_increasing\<close>.\<close>
lemma power_less_power_Suc: "1 < a \<Longrightarrow> 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 \<Longrightarrow> 1 < a \<Longrightarrow> 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 \<Longrightarrow> b ^ x \<le> b ^ y \<longleftrightarrow> x \<le> y"
by (blast intro: power_le_imp_le_exp power_increasing less_imp_le)

lemma power_strict_increasing_iff [simp]: "1 < b \<Longrightarrow> b ^ x < b ^ y \<longleftrightarrow> x < y"
by (blast intro: power_less_imp_less_exp power_strict_increasing)

lemma power_le_imp_le_base:
assumes le: "a ^ Suc n \<le> b ^ Suc n"
and "0 \<le> b"
shows "a \<le> b"
proof (rule ccontr)
assume "\<not> ?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 \<le> b"
shows "a < b"
proof (rule contrapos_pp [OF less])
assume "\<not> ?thesis"
then have "b \<le> a" by (simp only: linorder_not_less)
from this nonneg have "b ^ n \<le> a ^ n" by (rule power_mono)
then show "\<not> a ^ n < b ^ n" by (simp only: linorder_not_less)
qed

lemma power_inject_base: "a ^ Suc n = b ^ Suc n \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> a = b"
by (blast intro: power_le_imp_le_base antisym eq_refl sym)

lemma power_eq_imp_eq_base: "a ^ n = b ^ n \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 < n \<Longrightarrow> a = b"
by (cases n) (simp_all del: power_Suc, rule power_inject_base)

lemma power_eq_iff_eq_base: "0 < n \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> a ^ n = b ^ n \<longleftrightarrow> a = b"
using power_eq_imp_eq_base [of a n b] by auto

lemma power2_le_imp_le: "x\<^sup>2 \<le> y\<^sup>2 \<Longrightarrow> 0 \<le> y \<Longrightarrow> x \<le> y"
unfolding numeral_2_eq_2 by (rule power_le_imp_le_base)

lemma power2_less_imp_less: "x\<^sup>2 < y\<^sup>2 \<Longrightarrow> 0 \<le> y \<Longrightarrow> x < y"
by (rule power_less_imp_less_base)

lemma power2_eq_imp_eq: "x\<^sup>2 = y\<^sup>2 \<Longrightarrow> 0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> x = y"
unfolding numeral_2_eq_2 by (erule (2) power_eq_imp_eq_base) simp

lemma power_Suc_le_self: "0 \<le> a \<Longrightarrow> a \<le> 1 \<Longrightarrow> a ^ Suc n \<le> a"
using power_decreasing [of 1 "Suc n" a] by simp

end

context linordered_ring_strict
begin

lemma sum_squares_eq_zero_iff: "x * x + y * y = 0 \<longleftrightarrow> x = 0 \<and> y = 0"

lemma sum_squares_le_zero_iff: "x * x + y * y \<le> 0 \<longleftrightarrow> x = 0 \<and> 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 \<longleftrightarrow> x \<noteq> 0 \<or> y \<noteq> 0"
by (simp add: not_le [symmetric] sum_squares_le_zero_iff)

end

context linordered_idom
begin

lemma power_abs: "\<bar>a ^ n\<bar> = \<bar>a\<bar> ^ n"
by (induct n) (auto simp add: abs_mult)

lemma abs_power_minus [simp]: "\<bar>(-a) ^ n\<bar> = \<bar>a ^ n\<bar>"

lemma zero_less_power_abs_iff [simp]: "0 < \<bar>a\<bar> ^ n \<longleftrightarrow> a \<noteq> 0 \<or> 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 \<le> \<bar>a\<bar> ^ n"
by (rule zero_le_power [OF abs_ge_zero])

lemma zero_le_power2 [simp]: "0 \<le> a\<^sup>2"

lemma zero_less_power2 [simp]: "0 < a\<^sup>2 \<longleftrightarrow> a \<noteq> 0"
by (force simp add: power2_eq_square zero_less_mult_iff linorder_neq_iff)

lemma power2_less_0 [simp]: "\<not> a\<^sup>2 < 0"
by (force simp add: power2_eq_square mult_less_0_iff)

lemma power2_less_eq_zero_iff [simp]: "a\<^sup>2 \<le> 0 \<longleftrightarrow> a = 0"

lemma abs_power2 [simp]: "\<bar>a\<^sup>2\<bar> = a\<^sup>2"

lemma power2_abs [simp]: "\<bar>a\<bar>\<^sup>2 = a\<^sup>2"

lemma odd_power_less_zero: "a < 0 \<Longrightarrow> 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 \<le> a ^ Suc (2*n) \<Longrightarrow> 0 \<le> a"
using odd_power_less_zero [of a n]
by (force simp add: linorder_not_less [symmetric])

lemma zero_le_even_power'[simp]: "0 \<le> 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 \<le> x\<^sup>2 + y\<^sup>2"

lemma not_sum_power2_lt_zero: "\<not> x\<^sup>2 + y\<^sup>2 < 0"
unfolding not_less by (rule sum_power2_ge_zero)

lemma sum_power2_eq_zero_iff: "x\<^sup>2 + y\<^sup>2 = 0 \<longleftrightarrow> x = 0 \<and> y = 0"

lemma sum_power2_le_zero_iff: "x\<^sup>2 + y\<^sup>2 \<le> 0 \<longleftrightarrow> x = 0 \<and> 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\<^sup>2 + y\<^sup>2 \<longleftrightarrow> x \<noteq> 0 \<or> y \<noteq> 0"
unfolding not_le [symmetric] by (simp add: sum_power2_le_zero_iff)

lemma abs_le_square_iff: "\<bar>x\<bar> \<le> \<bar>y\<bar> \<longleftrightarrow> x\<^sup>2 \<le> y\<^sup>2"
(is "?lhs \<longleftrightarrow> ?rhs")
proof
assume ?lhs
then have "\<bar>x\<bar>\<^sup>2 \<le> \<bar>y\<bar>\<^sup>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\<^sup>2 \<le> 1 \<longleftrightarrow> \<bar>x\<bar> \<le> 1"
using abs_le_square_iff [of x 1] by simp

lemma abs_square_eq_1: "x\<^sup>2 = 1 \<longleftrightarrow> \<bar>x\<bar> = 1"
by (auto simp add: abs_if power2_eq_1_iff)

lemma abs_square_less_1: "x\<^sup>2 < 1 \<longleftrightarrow> \<bar>x\<bar> < 1"
using  abs_square_eq_1 [of x] abs_square_le_1 [of x] by (auto simp add: le_less)

end

subsection \<open>Miscellaneous rules\<close>

lemma (in linordered_semidom) self_le_power: "1 \<le> a \<Longrightarrow> 0 < n \<Longrightarrow> a \<le> 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)\<^sup>2 = x\<^sup>2 + y\<^sup>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)\<^sup>2 = x\<^sup>2 + y\<^sup>2 - 2 * x * y"
by (simp add: algebra_simps power2_eq_square mult_2_right)

lemma power2_commute: "(x - y)\<^sup>2 = (y - x)\<^sup>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 \<open>Simprules for comparisons where common factors can be cancelled.\<close>

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 \<open>Exponentiation for the Natural Numbers\<close>

lemma nat_one_le_power [simp]: "Suc 0 \<le> i \<Longrightarrow> Suc 0 \<le> i ^ n"
by (rule one_le_power [of i n, unfolded One_nat_def])

lemma nat_zero_less_power_iff [simp]: "x ^ n > 0 \<longleftrightarrow> x > 0 \<or> n = 0"
for x :: nat
by (induct n) auto

lemma nat_power_eq_Suc_0_iff [simp]: "x ^ m = Suc 0 \<longleftrightarrow> m = 0 \<or> x = Suc 0"
by (induct m) auto

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

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

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 \<Longrightarrow> 1 < i \<Longrightarrow> m \<le> 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\<^sup>2 \<le> n\<^sup>2 \<longleftrightarrow> m \<le> 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\<^sup>2 \<le> n"
shows "m \<le> n"
proof (cases m)
case 0
then show ?thesis by simp
next
case (Suc k)
show ?thesis
proof (rule ccontr)
assume "\<not> ?thesis"
then have "n < m" by simp
with assms Suc show False
qed
qed

subsubsection \<open>Cardinality of the Powerset\<close>

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

lemma card_Pow: "finite A \<Longrightarrow> card (Pow A) = 2 ^ card A"
proof (induct rule: finite_induct)
case empty
show ?case by auto
next
case (insert x A)
then have "inj_on (insert x) (Pow A)"
unfolding inj_on_def by (blast elim!: equalityE)
then have "card (Pow A) + card (insert x ` Pow A) = 2 * 2 ^ card A"
by (simp add: mult_2 card_image Pow_insert insert.hyps)
with insert show ?case