src/HOL/Power.thy

misc tuning and updates;

(*  Title:      HOL/Power.thy
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
    Copyright   1997  University of Cambridge
*)

section \<open>Exponentiation\<close>

theory Power
imports Num Equiv_Relations
begin

context linordered_ring (* TODO: move *)
begin

lemma sum_squares_ge_zero:
  "0 \<le> x * x + y * y"
  by (intro add_nonneg_nonneg zero_le_square)

lemma not_sum_squares_lt_zero:
  "\<not> x * x + y * y < 0"
  by (simp add: not_less sum_squares_ge_zero)

end

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"
  by (simp add: power_commutes)

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"
  by (induct n) (simp_all add: power_add)

lemma power2_eq_square: "a\<^sup>2 = a * a"
  by (simp add: numeral_2_eq_2)

lemma power3_eq_cube: "a ^ 3 = a * a * a"
  by (simp add: numeral_3_eq_3 mult.assoc)

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"
  by (simp add: power_even_eq)

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

lemma power_numeral_odd:
  "z ^ numeral (Num.Bit1 w) = (let w = z ^ (numeral w) in z * w * w)"
  unfolding numeral_Bit1 One_nat_def add_Suc_right add_0_right
  unfolding power_Suc power_add Let_def mult.assoc ..

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)
  def g \<equiv> "\<lambda>x. f x - 1"
  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 by (simp add: power_add funpow_add fun_eq_iff mult.assoc)
qed

lemma power_commuting_commutes:
  assumes "x * y = y * x"
  shows "x ^ n * y = y * x ^n"
proof (induct n)
  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"
    unfolding Suc power_Suc2
    by (simp add: ac_simps)
  finally show ?case .
qed simp

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]:
  fixes a :: "'a :: monoid_mult"
  shows "a^numeral m * a^numeral n = a^numeral (m + n)"
  by (simp add: power_add [symmetric])

lemma power_add_numeral2 [simp]:
  fixes a :: "'a :: monoid_mult"
  shows "a^numeral m * (a^numeral n * b) = a^numeral (m + n) * b"
  by (simp add: mult.assoc [symmetric])

lemma power_mult_numeral [simp]:
  fixes a :: "'a :: monoid_mult"
  shows"(a^numeral m)^numeral n = a^numeral (m * n)"
  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
    numeral_sqr numeral_mult power_add power_one_right)

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 add: of_nat_mult)

lemma zero_power:
  "0 < n \<Longrightarrow> 0 ^ n = 0"
  by (cases n) simp_all

lemma power_zero_numeral [simp]:
  "0 ^ numeral k = 0"
  by (simp add: numeral_eq_Suc)

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
  have "a ^ n = a ^ (m + (n - m))"
    using \<open>m \<le> n\<close> 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]:
  assumes "n > (0::nat) \<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

class semiring_1_no_zero_divisors = semiring_1 + semiring_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 semidom
begin

subclass semiring_1_no_zero_divisors ..

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) then show ?case by (simp add: power_add power2_eq_square)
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

subclass semiring_1_no_zero_divisors ..

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:
  assumes "b dvd a"
  shows "(a div b) ^ n = a ^ n div b ^ n"
  using assms by (induct n) (simp_all add: div_mult_div_if_dvd dvd_power_same)

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 nz: "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: nz 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: "\<lbrakk>0 \<le> a; a \<le> 1\<rbrakk> \<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])
  have "1 * 1 < a * 1" using gt1 by simp
  also have "\<dots> \<le> a * a ^ n" using gt1
    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"
  by (simp add: power_gt1_lemma)

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.prems Suc.hyps 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 add: 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 add: 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 [rule_format]:
  "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 (auto simp add: le_Suc_eq)
  apply (subgoal_tac "a * a^N \<le> 1 * a^n", 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 [rule_format]:
  "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 (auto simp add: le_Suc_eq)
  apply (subgoal_tac "1 * a^n \<le> a * a^N", 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 add: mult_strict_left_mono less_trans [OF zero_less_one])

lemma power_strict_increasing [rule_format]:
  "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", 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 ynonneg: "0 \<le> b"
  shows "a \<le> b"
proof (rule ccontr)
  assume "~ a \<le> b"
  then have "b < a" by (simp only: linorder_not_le)
  then have "b ^ Suc n < a ^ Suc n"
    by (simp only: assms power_strict_mono)
  from le and this show False
    by (simp add: linorder_not_less [symmetric])
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 "~ a < b"
  hence "b \<le> a" by (simp only: linorder_not_less)
  hence "b ^ n \<le> a ^ n" using nonneg by (rule power_mono)
  thus "\<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 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

end

context linordered_ring_strict
begin

lemma sum_squares_eq_zero_iff:
  "x * x + y * y = 0 \<longleftrightarrow> x = 0 \<and> y = 0"
  by (simp add: add_nonneg_eq_0_iff)

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>"
  by (simp add: power_abs)

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 n) show ?case by (auto simp add: Suc 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"
  by (simp add: power2_eq_square)

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"
  by (simp add: le_less)

lemma abs_power2 [simp]: "\<bar>a\<^sup>2\<bar> = a\<^sup>2"
  by (simp add: power2_eq_square abs_mult abs_mult_self)

lemma power2_abs [simp]: "\<bar>a\<bar>\<^sup>2 = a\<^sup>2"
  by (simp add: power2_eq_square abs_mult_self)

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)"
    by (simp add: ac_simps power_add power2_eq_square)
  thus ?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)"
      by (simp add: ac_simps power_add power2_eq_square)
    thus ?case
      by (simp add: Suc zero_le_mult_iff)
qed

lemma sum_power2_ge_zero:
  "0 \<le> x\<^sup>2 + y\<^sup>2"
  by (intro add_nonneg_nonneg zero_le_power2)

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"
  unfolding power2_eq_square by (simp add: add_nonneg_eq_0_iff)

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"
proof
  assume "\<bar>x\<bar> \<le> \<bar>y\<bar>"
  then have "\<bar>x\<bar>\<^sup>2 \<le> \<bar>y\<bar>\<^sup>2" by (rule power_mono, simp)
  then show "x\<^sup>2 \<le> y\<^sup>2" by simp
next
  assume "x\<^sup>2 \<le> y\<^sup>2"
  then show "\<bar>x\<bar> \<le> \<bar>y\<bar>"
    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)

lemma (in comm_ring_1) 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 (in comm_ring_1) power2_commute:
  "(x - y)\<^sup>2 = (y - x)\<^sup>2"
  by (simp add: algebra_simps power2_eq_square)

text \<open>Simprules for comparisons where common factors can be cancelled.\<close>

lemmas zero_compare_simps =
    add_strict_increasing add_strict_increasing2 add_increasing
    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::nat) \<or> n = 0"
  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 @{term "i=0"},
@{term "m=1"} and @{term "n=0"}.\<close>
lemma nat_power_less_imp_less:
  assumes nonneg: "0 < (i::nat)"
  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::nat) < i \<Longrightarrow> m \<le> n"
  apply (rule power_le_imp_le_exp, assumption)
  apply (erule dvd_imp_le, simp)
  done

lemma power2_nat_le_eq_le:
  fixes m n :: nat
  shows "m\<^sup>2 \<le> n\<^sup>2 \<longleftrightarrow> m \<le> n"
  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> m \<le> n"
    then have "n < m" by simp
    with assms Suc show False
      by (simp add: power2_eq_square)
  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)
  then show ?case using insert
    apply (simp add: Pow_insert)
    apply (subst card_Un_disjoint, auto)
    done
qed


subsubsection \<open>Generalized sum over a set\<close>

lemma setsum_zero_power [simp]:
  fixes c :: "nat \<Rightarrow> 'a::division_ring"
  shows "(\<Sum>i\<in>A. c i * 0^i) = (if finite A \<and> 0 \<in> A then c 0 else 0)"
apply (cases "finite A")
  by (induction A rule: finite_induct) auto

lemma setsum_zero_power' [simp]:
  fixes c :: "nat \<Rightarrow> 'a::field"
  shows "(\<Sum>i\<in>A. c i * 0^i / d i) = (if finite A \<and> 0 \<in> A then c 0 / d 0 else 0)"
  using setsum_zero_power [of "\<lambda>i. c i / d i" A]
  by auto


subsubsection \<open>Generalized product over a set\<close>

lemma setprod_constant: "finite A ==> (\<Prod>x\<in> A. (y::'a::{comm_monoid_mult})) = y^(card A)"
apply (erule finite_induct)
apply auto
done

lemma setprod_power_distrib:
  fixes f :: "'a \<Rightarrow> 'b::comm_semiring_1"
  shows "setprod f A ^ n = setprod (\<lambda>x. (f x) ^ n) A"
proof (cases "finite A")
  case True then show ?thesis
    by (induct A rule: finite_induct) (auto simp add: power_mult_distrib)
next
  case False then show ?thesis
    by simp
qed

lemma power_setsum:
  "c ^ (\<Sum>a\<in>A. f a) = (\<Prod>a\<in>A. c ^ f a)"
  by (induct A rule: infinite_finite_induct) (simp_all add: power_add)

lemma setprod_gen_delta:
  assumes fS: "finite S"
  shows "setprod (\<lambda>k. if k=a then b k else c) S = (if a \<in> S then (b a ::'a::comm_monoid_mult) * c^ (card S - 1) else c^ card S)"
proof-
  let ?f = "(\<lambda>k. if k=a then b k else c)"
  {assume a: "a \<notin> S"
    hence "\<forall> k\<in> S. ?f k = c" by simp
    hence ?thesis  using a setprod_constant[OF fS, of c] by simp }
  moreover
  {assume a: "a \<in> S"
    let ?A = "S - {a}"
    let ?B = "{a}"
    have eq: "S = ?A \<union> ?B" using a by blast
    have dj: "?A \<inter> ?B = {}" by simp
    from fS have fAB: "finite ?A" "finite ?B" by auto
    have fA0:"setprod ?f ?A = setprod (\<lambda>i. c) ?A"
      apply (rule setprod.cong) by auto
    have cA: "card ?A = card S - 1" using fS a by auto
    have fA1: "setprod ?f ?A = c ^ card ?A"  unfolding fA0 apply (rule setprod_constant) using fS by auto
    have "setprod ?f ?A * setprod ?f ?B = setprod ?f S"
      using setprod.union_disjoint[OF fAB dj, of ?f, unfolded eq[symmetric]]
      by simp
    then have ?thesis using a cA
      by (simp add: fA1 field_simps cong add: setprod.cong cong del: if_weak_cong)}
  ultimately show ?thesis by blast
qed

subsection \<open>Code generator tweak\<close>

code_identifier
  code_module Power \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith

end