src/HOL/Power.thy
author hoelzl
Wed Jan 06 12:18:53 2016 +0100 (2016-01-06)
changeset 62083 7582b39f51ed
parent 61955 e96292f32c3c
child 62347 2230b7047376
permissions -rw-r--r--
add the proof of the central limit theorem
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(*  Title:      HOL/Power.thy
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    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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    Copyright   1997  University of Cambridge
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*)
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section \<open>Exponentiation\<close>
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theory Power
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imports Num Equiv_Relations
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begin
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context linordered_ring (* TODO: move *)
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begin
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lemma sum_squares_ge_zero:
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  "0 \<le> x * x + y * y"
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  by (intro add_nonneg_nonneg zero_le_square)
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lemma not_sum_squares_lt_zero:
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  "\<not> x * x + y * y < 0"
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  by (simp add: not_less sum_squares_ge_zero)
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end
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subsection \<open>Powers for Arbitrary Monoids\<close>
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class power = one + times
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begin
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primrec power :: "'a \<Rightarrow> nat \<Rightarrow> 'a"  (infixr "^" 80)
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where
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  power_0: "a ^ 0 = 1"
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| power_Suc: "a ^ Suc n = a * a ^ n"
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notation (latex output)
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  power ("(_\<^bsup>_\<^esup>)" [1000] 1000)
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text \<open>Special syntax for squares.\<close>
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abbreviation power2 :: "'a \<Rightarrow> 'a"  ("(_\<^sup>2)" [1000] 999)
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  where "x\<^sup>2 \<equiv> x ^ 2"
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end
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context monoid_mult
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begin
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subclass power .
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lemma power_one [simp]:
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  "1 ^ n = 1"
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  by (induct n) simp_all
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lemma power_one_right [simp]:
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  "a ^ 1 = a"
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  by simp
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lemma power_Suc0_right [simp]:
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  "a ^ Suc 0 = a"
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  by simp
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lemma power_commutes:
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  "a ^ n * a = a * a ^ n"
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  by (induct n) (simp_all add: mult.assoc)
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lemma power_Suc2:
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  "a ^ Suc n = a ^ n * a"
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  by (simp add: power_commutes)
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lemma power_add:
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  "a ^ (m + n) = a ^ m * a ^ n"
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  by (induct m) (simp_all add: algebra_simps)
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lemma power_mult:
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  "a ^ (m * n) = (a ^ m) ^ n"
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  by (induct n) (simp_all add: power_add)
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lemma power2_eq_square: "a\<^sup>2 = a * a"
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  by (simp add: numeral_2_eq_2)
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lemma power3_eq_cube: "a ^ 3 = a * a * a"
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  by (simp add: numeral_3_eq_3 mult.assoc)
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lemma power_even_eq:
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  "a ^ (2 * n) = (a ^ n)\<^sup>2"
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  by (subst mult.commute) (simp add: power_mult)
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lemma power_odd_eq:
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  "a ^ Suc (2*n) = a * (a ^ n)\<^sup>2"
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  by (simp add: power_even_eq)
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lemma power_numeral_even:
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  "z ^ numeral (Num.Bit0 w) = (let w = z ^ (numeral w) in w * w)"
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  unfolding numeral_Bit0 power_add Let_def ..
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lemma power_numeral_odd:
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  "z ^ numeral (Num.Bit1 w) = (let w = z ^ (numeral w) in z * w * w)"
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  unfolding numeral_Bit1 One_nat_def add_Suc_right add_0_right
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  unfolding power_Suc power_add Let_def mult.assoc ..
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lemma funpow_times_power:
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  "(times x ^^ f x) = times (x ^ f x)"
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proof (induct "f x" arbitrary: f)
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  case 0 then show ?case by (simp add: fun_eq_iff)
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next
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  case (Suc n)
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  def g \<equiv> "\<lambda>x. f x - 1"
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  with Suc have "n = g x" by simp
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  with Suc have "times x ^^ g x = times (x ^ g x)" by simp
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  moreover from Suc g_def have "f x = g x + 1" by simp
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  ultimately show ?case by (simp add: power_add funpow_add fun_eq_iff mult.assoc)
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qed
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lemma power_commuting_commutes:
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  assumes "x * y = y * x"
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  shows "x ^ n * y = y * x ^n"
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proof (induct n)
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  case (Suc n)
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  have "x ^ Suc n * y = x ^ n * y * x"
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    by (subst power_Suc2) (simp add: assms ac_simps)
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  also have "\<dots> = y * x ^ Suc n"
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    unfolding Suc power_Suc2
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    by (simp add: ac_simps)
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  finally show ?case .
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qed simp
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end
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context comm_monoid_mult
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begin
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lemma power_mult_distrib [field_simps]:
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  "(a * b) ^ n = (a ^ n) * (b ^ n)"
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  by (induct n) (simp_all add: ac_simps)
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end
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text\<open>Extract constant factors from powers\<close>
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declare power_mult_distrib [where a = "numeral w" for w, simp]
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declare power_mult_distrib [where b = "numeral w" for w, simp]
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lemma power_add_numeral [simp]:
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  fixes a :: "'a :: monoid_mult"
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  shows "a^numeral m * a^numeral n = a^numeral (m + n)"
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  by (simp add: power_add [symmetric])
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lemma power_add_numeral2 [simp]:
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  fixes a :: "'a :: monoid_mult"
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  shows "a^numeral m * (a^numeral n * b) = a^numeral (m + n) * b"
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  by (simp add: mult.assoc [symmetric])
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lemma power_mult_numeral [simp]:
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  fixes a :: "'a :: monoid_mult"
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  shows"(a^numeral m)^numeral n = a^numeral (m * n)"
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  by (simp only: numeral_mult power_mult)
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context semiring_numeral
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begin
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lemma numeral_sqr: "numeral (Num.sqr k) = numeral k * numeral k"
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  by (simp only: sqr_conv_mult numeral_mult)
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lemma numeral_pow: "numeral (Num.pow k l) = numeral k ^ numeral l"
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  by (induct l, simp_all only: numeral_class.numeral.simps pow.simps
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    numeral_sqr numeral_mult power_add power_one_right)
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lemma power_numeral [simp]: "numeral k ^ numeral l = numeral (Num.pow k l)"
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  by (rule numeral_pow [symmetric])
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end
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context semiring_1
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begin
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lemma of_nat_power [simp]:
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  "of_nat (m ^ n) = of_nat m ^ n"
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  by (induct n) (simp_all add: of_nat_mult)
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lemma zero_power:
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  "0 < n \<Longrightarrow> 0 ^ n = 0"
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  by (cases n) simp_all
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lemma power_zero_numeral [simp]:
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  "0 ^ numeral k = 0"
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  by (simp add: numeral_eq_Suc)
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lemma zero_power2: "0\<^sup>2 = 0" (* delete? *)
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  by (rule power_zero_numeral)
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lemma one_power2: "1\<^sup>2 = 1" (* delete? *)
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  by (rule power_one)
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lemma power_0_Suc [simp]:
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  "0 ^ Suc n = 0"
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  by simp
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text\<open>It looks plausible as a simprule, but its effect can be strange.\<close>
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lemma power_0_left:
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  "0 ^ n = (if n = 0 then 1 else 0)"
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  by (cases n) simp_all
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end
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context comm_semiring_1
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begin
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text \<open>The divides relation\<close>
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lemma le_imp_power_dvd:
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  assumes "m \<le> n" shows "a ^ m dvd a ^ n"
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proof
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  have "a ^ n = a ^ (m + (n - m))"
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    using \<open>m \<le> n\<close> by simp
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  also have "\<dots> = a ^ m * a ^ (n - m)"
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    by (rule power_add)
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  finally show "a ^ n = a ^ m * a ^ (n - m)" .
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qed
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lemma power_le_dvd:
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  "a ^ n dvd b \<Longrightarrow> m \<le> n \<Longrightarrow> a ^ m dvd b"
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  by (rule dvd_trans [OF le_imp_power_dvd])
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lemma dvd_power_same:
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  "x dvd y \<Longrightarrow> x ^ n dvd y ^ n"
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  by (induct n) (auto simp add: mult_dvd_mono)
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lemma dvd_power_le:
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  "x dvd y \<Longrightarrow> m \<ge> n \<Longrightarrow> x ^ n dvd y ^ m"
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  by (rule power_le_dvd [OF dvd_power_same])
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lemma dvd_power [simp]:
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  assumes "n > (0::nat) \<or> x = 1"
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  shows "x dvd (x ^ n)"
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using assms proof
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  assume "0 < n"
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  then have "x ^ n = x ^ Suc (n - 1)" by simp
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  then show "x dvd (x ^ n)" by simp
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next
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  assume "x = 1"
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  then show "x dvd (x ^ n)" by simp
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qed
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end
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class semiring_1_no_zero_divisors = semiring_1 + semiring_no_zero_divisors
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begin
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subclass power .
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lemma power_eq_0_iff [simp]:
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  "a ^ n = 0 \<longleftrightarrow> a = 0 \<and> n > 0"
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  by (induct n) auto
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lemma power_not_zero:
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  "a \<noteq> 0 \<Longrightarrow> a ^ n \<noteq> 0"
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  by (induct n) auto
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lemma zero_eq_power2 [simp]:
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  "a\<^sup>2 = 0 \<longleftrightarrow> a = 0"
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  unfolding power2_eq_square by simp
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end
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context semidom
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begin
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subclass semiring_1_no_zero_divisors ..
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end
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context ring_1
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begin
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lemma power_minus:
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  "(- a) ^ n = (- 1) ^ n * a ^ n"
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proof (induct n)
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  case 0 show ?case by simp
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next
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  case (Suc n) then show ?case
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    by (simp del: power_Suc add: power_Suc2 mult.assoc)
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qed
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lemma power_minus': "NO_MATCH 1 x \<Longrightarrow> (-x) ^ n = (-1)^n * x ^ n"
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  by (rule power_minus)
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lemma power_minus_Bit0:
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  "(- x) ^ numeral (Num.Bit0 k) = x ^ numeral (Num.Bit0 k)"
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  by (induct k, simp_all only: numeral_class.numeral.simps power_add
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    power_one_right mult_minus_left mult_minus_right minus_minus)
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lemma power_minus_Bit1:
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  "(- x) ^ numeral (Num.Bit1 k) = - (x ^ numeral (Num.Bit1 k))"
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  by (simp only: eval_nat_numeral(3) power_Suc power_minus_Bit0 mult_minus_left)
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lemma power2_minus [simp]:
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  "(- a)\<^sup>2 = a\<^sup>2"
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  by (fact power_minus_Bit0)
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lemma power_minus1_even [simp]:
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  "(- 1) ^ (2*n) = 1"
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proof (induct n)
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  case 0 show ?case by simp
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next
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  case (Suc n) then show ?case by (simp add: power_add power2_eq_square)
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qed
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lemma power_minus1_odd:
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  "(- 1) ^ Suc (2*n) = -1"
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  by simp
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lemma power_minus_even [simp]:
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  "(-a) ^ (2*n) = a ^ (2*n)"
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  by (simp add: power_minus [of a])
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end
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context ring_1_no_zero_divisors
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begin
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subclass semiring_1_no_zero_divisors ..
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lemma power2_eq_1_iff:
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  "a\<^sup>2 = 1 \<longleftrightarrow> a = 1 \<or> a = - 1"
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  using square_eq_1_iff [of a] by (simp add: power2_eq_square)
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end
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context idom
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begin
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lemma power2_eq_iff: "x\<^sup>2 = y\<^sup>2 \<longleftrightarrow> x = y \<or> x = - y"
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  unfolding power2_eq_square by (rule square_eq_iff)
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end
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context algebraic_semidom
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begin
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lemma div_power:
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  assumes "b dvd a"
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  shows "(a div b) ^ n = a ^ n div b ^ n"
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  using assms by (induct n) (simp_all add: div_mult_div_if_dvd dvd_power_same)
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end
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context normalization_semidom
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begin
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lemma normalize_power:
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  "normalize (a ^ n) = normalize a ^ n"
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  by (induct n) (simp_all add: normalize_mult)
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lemma unit_factor_power:
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  "unit_factor (a ^ n) = unit_factor a ^ n"
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  by (induct n) (simp_all add: unit_factor_mult)
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end
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context division_ring
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begin
huffman@47192
   360
haftmann@60867
   361
text\<open>Perhaps these should be simprules.\<close>
haftmann@60867
   362
lemma power_inverse [field_simps, divide_simps]:
haftmann@60867
   363
  "inverse a ^ n = inverse (a ^ n)"
haftmann@60867
   364
proof (cases "a = 0")
haftmann@60867
   365
  case True then show ?thesis by (simp add: power_0_left)
haftmann@60867
   366
next
haftmann@60867
   367
  case False then have "inverse (a ^ n) = inverse a ^ n"
haftmann@60867
   368
    by (induct n) (simp_all add: nonzero_inverse_mult_distrib power_commutes)
haftmann@60867
   369
  then show ?thesis by simp
haftmann@60867
   370
qed
huffman@47192
   371
haftmann@60867
   372
lemma power_one_over [field_simps, divide_simps]:
haftmann@60867
   373
  "(1 / a) ^ n = 1 / a ^ n"
haftmann@60867
   374
  using power_inverse [of a] by (simp add: divide_inverse)
haftmann@60867
   375
lp15@61649
   376
end
huffman@47192
   377
huffman@47192
   378
context field
huffman@47192
   379
begin
huffman@47192
   380
haftmann@60867
   381
lemma power_diff:
haftmann@60867
   382
  assumes nz: "a \<noteq> 0"
haftmann@60867
   383
  shows "n \<le> m \<Longrightarrow> a ^ (m - n) = a ^ m / a ^ n"
haftmann@60867
   384
  by (induct m n rule: diff_induct) (simp_all add: nz power_not_zero)
huffman@47192
   385
haftmann@60867
   386
lemma power_divide [field_simps, divide_simps]:
haftmann@60867
   387
  "(a / b) ^ n = a ^ n / b ^ n"
haftmann@60867
   388
  by (induct n) simp_all
haftmann@60867
   389
huffman@47192
   390
end
huffman@47192
   391
huffman@47192
   392
wenzelm@60758
   393
subsection \<open>Exponentiation on ordered types\<close>
huffman@47192
   394
haftmann@35028
   395
context linordered_semidom
haftmann@30996
   396
begin
haftmann@30996
   397
haftmann@30996
   398
lemma zero_less_power [simp]:
haftmann@30996
   399
  "0 < a \<Longrightarrow> 0 < a ^ n"
nipkow@56544
   400
  by (induct n) simp_all
haftmann@30996
   401
haftmann@30996
   402
lemma zero_le_power [simp]:
haftmann@30996
   403
  "0 \<le> a \<Longrightarrow> 0 \<le> a ^ n"
nipkow@56536
   404
  by (induct n) simp_all
paulson@14348
   405
huffman@47241
   406
lemma power_mono:
huffman@47241
   407
  "a \<le> b \<Longrightarrow> 0 \<le> a \<Longrightarrow> a ^ n \<le> b ^ n"
huffman@47241
   408
  by (induct n) (auto intro: mult_mono order_trans [of 0 a b])
huffman@47241
   409
huffman@47241
   410
lemma one_le_power [simp]: "1 \<le> a \<Longrightarrow> 1 \<le> a ^ n"
huffman@47241
   411
  using power_mono [of 1 a n] by simp
huffman@47241
   412
huffman@47241
   413
lemma power_le_one: "\<lbrakk>0 \<le> a; a \<le> 1\<rbrakk> \<Longrightarrow> a ^ n \<le> 1"
huffman@47241
   414
  using power_mono [of a 1 n] by simp
paulson@14348
   415
paulson@14348
   416
lemma power_gt1_lemma:
haftmann@30996
   417
  assumes gt1: "1 < a"
haftmann@30996
   418
  shows "1 < a * a ^ n"
paulson@14348
   419
proof -
haftmann@30996
   420
  from gt1 have "0 \<le> a"
haftmann@30996
   421
    by (fact order_trans [OF zero_le_one less_imp_le])
haftmann@30996
   422
  have "1 * 1 < a * 1" using gt1 by simp
haftmann@30996
   423
  also have "\<dots> \<le> a * a ^ n" using gt1
wenzelm@60758
   424
    by (simp only: mult_mono \<open>0 \<le> a\<close> one_le_power order_less_imp_le
wenzelm@14577
   425
        zero_le_one order_refl)
wenzelm@14577
   426
  finally show ?thesis by simp
paulson@14348
   427
qed
paulson@14348
   428
haftmann@30996
   429
lemma power_gt1:
haftmann@30996
   430
  "1 < a \<Longrightarrow> 1 < a ^ Suc n"
haftmann@30996
   431
  by (simp add: power_gt1_lemma)
huffman@24376
   432
haftmann@30996
   433
lemma one_less_power [simp]:
haftmann@30996
   434
  "1 < a \<Longrightarrow> 0 < n \<Longrightarrow> 1 < a ^ n"
haftmann@30996
   435
  by (cases n) (simp_all add: power_gt1_lemma)
paulson@14348
   436
paulson@14348
   437
lemma power_le_imp_le_exp:
haftmann@30996
   438
  assumes gt1: "1 < a"
haftmann@30996
   439
  shows "a ^ m \<le> a ^ n \<Longrightarrow> m \<le> n"
haftmann@30996
   440
proof (induct m arbitrary: n)
paulson@14348
   441
  case 0
wenzelm@14577
   442
  show ?case by simp
paulson@14348
   443
next
paulson@14348
   444
  case (Suc m)
wenzelm@14577
   445
  show ?case
wenzelm@14577
   446
  proof (cases n)
wenzelm@14577
   447
    case 0
haftmann@30996
   448
    with Suc.prems Suc.hyps have "a * a ^ m \<le> 1" by simp
wenzelm@14577
   449
    with gt1 show ?thesis
wenzelm@14577
   450
      by (force simp only: power_gt1_lemma
haftmann@30996
   451
          not_less [symmetric])
wenzelm@14577
   452
  next
wenzelm@14577
   453
    case (Suc n)
haftmann@30996
   454
    with Suc.prems Suc.hyps show ?thesis
wenzelm@14577
   455
      by (force dest: mult_left_le_imp_le
haftmann@30996
   456
          simp add: less_trans [OF zero_less_one gt1])
wenzelm@14577
   457
  qed
paulson@14348
   458
qed
paulson@14348
   459
lp15@61649
   460
lemma of_nat_zero_less_power_iff [simp]:
lp15@61649
   461
  "of_nat x ^ n > 0 \<longleftrightarrow> x > 0 \<or> n = 0"
lp15@61649
   462
  by (induct n) auto
lp15@61649
   463
wenzelm@61799
   464
text\<open>Surely we can strengthen this? It holds for \<open>0<a<1\<close> too.\<close>
paulson@14348
   465
lemma power_inject_exp [simp]:
haftmann@30996
   466
  "1 < a \<Longrightarrow> a ^ m = a ^ n \<longleftrightarrow> m = n"
wenzelm@14577
   467
  by (force simp add: order_antisym power_le_imp_le_exp)
paulson@14348
   468
wenzelm@60758
   469
text\<open>Can relax the first premise to @{term "0<a"} in the case of the
wenzelm@60758
   470
natural numbers.\<close>
paulson@14348
   471
lemma power_less_imp_less_exp:
haftmann@30996
   472
  "1 < a \<Longrightarrow> a ^ m < a ^ n \<Longrightarrow> m < n"
haftmann@30996
   473
  by (simp add: order_less_le [of m n] less_le [of "a^m" "a^n"]
haftmann@30996
   474
    power_le_imp_le_exp)
paulson@14348
   475
paulson@14348
   476
lemma power_strict_mono [rule_format]:
haftmann@30996
   477
  "a < b \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 < n \<longrightarrow> a ^ n < b ^ n"
haftmann@30996
   478
  by (induct n)
haftmann@30996
   479
   (auto simp add: mult_strict_mono le_less_trans [of 0 a b])
paulson@14348
   480
wenzelm@61799
   481
text\<open>Lemma for \<open>power_strict_decreasing\<close>\<close>
paulson@14348
   482
lemma power_Suc_less:
haftmann@30996
   483
  "0 < a \<Longrightarrow> a < 1 \<Longrightarrow> a * a ^ n < a ^ n"
haftmann@30996
   484
  by (induct n)
haftmann@30996
   485
    (auto simp add: mult_strict_left_mono)
paulson@14348
   486
haftmann@30996
   487
lemma power_strict_decreasing [rule_format]:
haftmann@30996
   488
  "n < N \<Longrightarrow> 0 < a \<Longrightarrow> a < 1 \<longrightarrow> a ^ N < a ^ n"
haftmann@30996
   489
proof (induct N)
haftmann@30996
   490
  case 0 then show ?case by simp
haftmann@30996
   491
next
lp15@61649
   492
  case (Suc N) then show ?case
haftmann@30996
   493
  apply (auto simp add: power_Suc_less less_Suc_eq)
haftmann@30996
   494
  apply (subgoal_tac "a * a^N < 1 * a^n")
haftmann@30996
   495
  apply simp
haftmann@30996
   496
  apply (rule mult_strict_mono) apply auto
haftmann@30996
   497
  done
haftmann@30996
   498
qed
paulson@14348
   499
wenzelm@61799
   500
text\<open>Proof resembles that of \<open>power_strict_decreasing\<close>\<close>
haftmann@30996
   501
lemma power_decreasing [rule_format]:
haftmann@30996
   502
  "n \<le> N \<Longrightarrow> 0 \<le> a \<Longrightarrow> a \<le> 1 \<longrightarrow> a ^ N \<le> a ^ n"
haftmann@30996
   503
proof (induct N)
haftmann@30996
   504
  case 0 then show ?case by simp
haftmann@30996
   505
next
lp15@61649
   506
  case (Suc N) then show ?case
haftmann@30996
   507
  apply (auto simp add: le_Suc_eq)
haftmann@30996
   508
  apply (subgoal_tac "a * a^N \<le> 1 * a^n", simp)
haftmann@30996
   509
  apply (rule mult_mono) apply auto
haftmann@30996
   510
  done
haftmann@30996
   511
qed
paulson@14348
   512
paulson@14348
   513
lemma power_Suc_less_one:
haftmann@30996
   514
  "0 < a \<Longrightarrow> a < 1 \<Longrightarrow> a ^ Suc n < 1"
haftmann@30996
   515
  using power_strict_decreasing [of 0 "Suc n" a] by simp
paulson@14348
   516
wenzelm@61799
   517
text\<open>Proof again resembles that of \<open>power_strict_decreasing\<close>\<close>
haftmann@30996
   518
lemma power_increasing [rule_format]:
haftmann@30996
   519
  "n \<le> N \<Longrightarrow> 1 \<le> a \<Longrightarrow> a ^ n \<le> a ^ N"
haftmann@30996
   520
proof (induct N)
haftmann@30996
   521
  case 0 then show ?case by simp
haftmann@30996
   522
next
lp15@61649
   523
  case (Suc N) then show ?case
haftmann@30996
   524
  apply (auto simp add: le_Suc_eq)
haftmann@30996
   525
  apply (subgoal_tac "1 * a^n \<le> a * a^N", simp)
haftmann@30996
   526
  apply (rule mult_mono) apply (auto simp add: order_trans [OF zero_le_one])
haftmann@30996
   527
  done
haftmann@30996
   528
qed
paulson@14348
   529
wenzelm@61799
   530
text\<open>Lemma for \<open>power_strict_increasing\<close>\<close>
paulson@14348
   531
lemma power_less_power_Suc:
haftmann@30996
   532
  "1 < a \<Longrightarrow> a ^ n < a * a ^ n"
haftmann@30996
   533
  by (induct n) (auto simp add: mult_strict_left_mono less_trans [OF zero_less_one])
paulson@14348
   534
haftmann@30996
   535
lemma power_strict_increasing [rule_format]:
haftmann@30996
   536
  "n < N \<Longrightarrow> 1 < a \<longrightarrow> a ^ n < a ^ N"
haftmann@30996
   537
proof (induct N)
haftmann@30996
   538
  case 0 then show ?case by simp
haftmann@30996
   539
next
lp15@61649
   540
  case (Suc N) then show ?case
haftmann@30996
   541
  apply (auto simp add: power_less_power_Suc less_Suc_eq)
haftmann@30996
   542
  apply (subgoal_tac "1 * a^n < a * a^N", simp)
haftmann@30996
   543
  apply (rule mult_strict_mono) apply (auto simp add: less_trans [OF zero_less_one] less_imp_le)
haftmann@30996
   544
  done
haftmann@30996
   545
qed
paulson@14348
   546
nipkow@25134
   547
lemma power_increasing_iff [simp]:
haftmann@30996
   548
  "1 < b \<Longrightarrow> b ^ x \<le> b ^ y \<longleftrightarrow> x \<le> y"
haftmann@30996
   549
  by (blast intro: power_le_imp_le_exp power_increasing less_imp_le)
paulson@15066
   550
paulson@15066
   551
lemma power_strict_increasing_iff [simp]:
haftmann@30996
   552
  "1 < b \<Longrightarrow> b ^ x < b ^ y \<longleftrightarrow> x < y"
lp15@61649
   553
by (blast intro: power_less_imp_less_exp power_strict_increasing)
paulson@15066
   554
paulson@14348
   555
lemma power_le_imp_le_base:
haftmann@30996
   556
  assumes le: "a ^ Suc n \<le> b ^ Suc n"
haftmann@30996
   557
    and ynonneg: "0 \<le> b"
haftmann@30996
   558
  shows "a \<le> b"
nipkow@25134
   559
proof (rule ccontr)
nipkow@25134
   560
  assume "~ a \<le> b"
nipkow@25134
   561
  then have "b < a" by (simp only: linorder_not_le)
nipkow@25134
   562
  then have "b ^ Suc n < a ^ Suc n"
wenzelm@41550
   563
    by (simp only: assms power_strict_mono)
haftmann@30996
   564
  from le and this show False
nipkow@25134
   565
    by (simp add: linorder_not_less [symmetric])
nipkow@25134
   566
qed
wenzelm@14577
   567
huffman@22853
   568
lemma power_less_imp_less_base:
huffman@22853
   569
  assumes less: "a ^ n < b ^ n"
huffman@22853
   570
  assumes nonneg: "0 \<le> b"
huffman@22853
   571
  shows "a < b"
huffman@22853
   572
proof (rule contrapos_pp [OF less])
huffman@22853
   573
  assume "~ a < b"
huffman@22853
   574
  hence "b \<le> a" by (simp only: linorder_not_less)
huffman@22853
   575
  hence "b ^ n \<le> a ^ n" using nonneg by (rule power_mono)
haftmann@30996
   576
  thus "\<not> a ^ n < b ^ n" by (simp only: linorder_not_less)
huffman@22853
   577
qed
huffman@22853
   578
paulson@14348
   579
lemma power_inject_base:
haftmann@30996
   580
  "a ^ Suc n = b ^ Suc n \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> a = b"
haftmann@30996
   581
by (blast intro: power_le_imp_le_base antisym eq_refl sym)
paulson@14348
   582
huffman@22955
   583
lemma power_eq_imp_eq_base:
haftmann@30996
   584
  "a ^ n = b ^ n \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 < n \<Longrightarrow> a = b"
haftmann@30996
   585
  by (cases n) (simp_all del: power_Suc, rule power_inject_base)
huffman@22955
   586
huffman@47192
   587
lemma power2_le_imp_le:
wenzelm@53015
   588
  "x\<^sup>2 \<le> y\<^sup>2 \<Longrightarrow> 0 \<le> y \<Longrightarrow> x \<le> y"
huffman@47192
   589
  unfolding numeral_2_eq_2 by (rule power_le_imp_le_base)
huffman@47192
   590
huffman@47192
   591
lemma power2_less_imp_less:
wenzelm@53015
   592
  "x\<^sup>2 < y\<^sup>2 \<Longrightarrow> 0 \<le> y \<Longrightarrow> x < y"
huffman@47192
   593
  by (rule power_less_imp_less_base)
huffman@47192
   594
huffman@47192
   595
lemma power2_eq_imp_eq:
wenzelm@53015
   596
  "x\<^sup>2 = y\<^sup>2 \<Longrightarrow> 0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> x = y"
huffman@47192
   597
  unfolding numeral_2_eq_2 by (erule (2) power_eq_imp_eq_base) simp
huffman@47192
   598
huffman@47192
   599
end
huffman@47192
   600
huffman@47192
   601
context linordered_ring_strict
huffman@47192
   602
begin
huffman@47192
   603
huffman@47192
   604
lemma sum_squares_eq_zero_iff:
huffman@47192
   605
  "x * x + y * y = 0 \<longleftrightarrow> x = 0 \<and> y = 0"
huffman@47192
   606
  by (simp add: add_nonneg_eq_0_iff)
huffman@47192
   607
huffman@47192
   608
lemma sum_squares_le_zero_iff:
huffman@47192
   609
  "x * x + y * y \<le> 0 \<longleftrightarrow> x = 0 \<and> y = 0"
huffman@47192
   610
  by (simp add: le_less not_sum_squares_lt_zero sum_squares_eq_zero_iff)
huffman@47192
   611
huffman@47192
   612
lemma sum_squares_gt_zero_iff:
huffman@47192
   613
  "0 < x * x + y * y \<longleftrightarrow> x \<noteq> 0 \<or> y \<noteq> 0"
huffman@47192
   614
  by (simp add: not_le [symmetric] sum_squares_le_zero_iff)
huffman@47192
   615
haftmann@30996
   616
end
haftmann@30996
   617
haftmann@35028
   618
context linordered_idom
haftmann@30996
   619
begin
huffman@29978
   620
wenzelm@61944
   621
lemma power_abs: "\<bar>a ^ n\<bar> = \<bar>a\<bar> ^ n"
haftmann@30996
   622
  by (induct n) (auto simp add: abs_mult)
haftmann@30996
   623
wenzelm@61944
   624
lemma abs_power_minus [simp]: "\<bar>(-a) ^ n\<bar> = \<bar>a ^ n\<bar>"
huffman@35216
   625
  by (simp add: power_abs)
haftmann@30996
   626
wenzelm@61944
   627
lemma zero_less_power_abs_iff [simp]: "0 < \<bar>a\<bar> ^ n \<longleftrightarrow> a \<noteq> 0 \<or> n = 0"
haftmann@30996
   628
proof (induct n)
haftmann@30996
   629
  case 0 show ?case by simp
haftmann@30996
   630
next
haftmann@30996
   631
  case (Suc n) show ?case by (auto simp add: Suc zero_less_mult_iff)
huffman@29978
   632
qed
huffman@29978
   633
wenzelm@61944
   634
lemma zero_le_power_abs [simp]: "0 \<le> \<bar>a\<bar> ^ n"
haftmann@30996
   635
  by (rule zero_le_power [OF abs_ge_zero])
haftmann@30996
   636
huffman@47192
   637
lemma zero_le_power2 [simp]:
wenzelm@53015
   638
  "0 \<le> a\<^sup>2"
huffman@47192
   639
  by (simp add: power2_eq_square)
huffman@47192
   640
huffman@47192
   641
lemma zero_less_power2 [simp]:
wenzelm@53015
   642
  "0 < a\<^sup>2 \<longleftrightarrow> a \<noteq> 0"
huffman@47192
   643
  by (force simp add: power2_eq_square zero_less_mult_iff linorder_neq_iff)
huffman@47192
   644
huffman@47192
   645
lemma power2_less_0 [simp]:
wenzelm@53015
   646
  "\<not> a\<^sup>2 < 0"
huffman@47192
   647
  by (force simp add: power2_eq_square mult_less_0_iff)
huffman@47192
   648
haftmann@58787
   649
lemma power2_less_eq_zero_iff [simp]:
haftmann@58787
   650
  "a\<^sup>2 \<le> 0 \<longleftrightarrow> a = 0"
haftmann@58787
   651
  by (simp add: le_less)
haftmann@58787
   652
wenzelm@61944
   653
lemma abs_power2 [simp]: "\<bar>a\<^sup>2\<bar> = a\<^sup>2"
huffman@47192
   654
  by (simp add: power2_eq_square abs_mult abs_mult_self)
huffman@47192
   655
wenzelm@61944
   656
lemma power2_abs [simp]: "\<bar>a\<bar>\<^sup>2 = a\<^sup>2"
huffman@47192
   657
  by (simp add: power2_eq_square abs_mult_self)
huffman@47192
   658
huffman@47192
   659
lemma odd_power_less_zero:
huffman@47192
   660
  "a < 0 \<Longrightarrow> a ^ Suc (2*n) < 0"
huffman@47192
   661
proof (induct n)
huffman@47192
   662
  case 0
huffman@47192
   663
  then show ?case by simp
huffman@47192
   664
next
huffman@47192
   665
  case (Suc n)
huffman@47192
   666
  have "a ^ Suc (2 * Suc n) = (a*a) * a ^ Suc(2*n)"
haftmann@57514
   667
    by (simp add: ac_simps power_add power2_eq_square)
huffman@47192
   668
  thus ?case
huffman@47192
   669
    by (simp del: power_Suc add: Suc mult_less_0_iff mult_neg_neg)
huffman@47192
   670
qed
haftmann@30996
   671
huffman@47192
   672
lemma odd_0_le_power_imp_0_le:
huffman@47192
   673
  "0 \<le> a ^ Suc (2*n) \<Longrightarrow> 0 \<le> a"
huffman@47192
   674
  using odd_power_less_zero [of a n]
lp15@61649
   675
    by (force simp add: linorder_not_less [symmetric])
huffman@47192
   676
huffman@47192
   677
lemma zero_le_even_power'[simp]:
huffman@47192
   678
  "0 \<le> a ^ (2*n)"
huffman@47192
   679
proof (induct n)
huffman@47192
   680
  case 0
huffman@47192
   681
    show ?case by simp
huffman@47192
   682
next
huffman@47192
   683
  case (Suc n)
lp15@61649
   684
    have "a ^ (2 * Suc n) = (a*a) * a ^ (2*n)"
haftmann@57514
   685
      by (simp add: ac_simps power_add power2_eq_square)
huffman@47192
   686
    thus ?case
huffman@47192
   687
      by (simp add: Suc zero_le_mult_iff)
huffman@47192
   688
qed
haftmann@30996
   689
huffman@47192
   690
lemma sum_power2_ge_zero:
wenzelm@53015
   691
  "0 \<le> x\<^sup>2 + y\<^sup>2"
huffman@47192
   692
  by (intro add_nonneg_nonneg zero_le_power2)
huffman@47192
   693
huffman@47192
   694
lemma not_sum_power2_lt_zero:
wenzelm@53015
   695
  "\<not> x\<^sup>2 + y\<^sup>2 < 0"
huffman@47192
   696
  unfolding not_less by (rule sum_power2_ge_zero)
huffman@47192
   697
huffman@47192
   698
lemma sum_power2_eq_zero_iff:
wenzelm@53015
   699
  "x\<^sup>2 + y\<^sup>2 = 0 \<longleftrightarrow> x = 0 \<and> y = 0"
huffman@47192
   700
  unfolding power2_eq_square by (simp add: add_nonneg_eq_0_iff)
huffman@47192
   701
huffman@47192
   702
lemma sum_power2_le_zero_iff:
wenzelm@53015
   703
  "x\<^sup>2 + y\<^sup>2 \<le> 0 \<longleftrightarrow> x = 0 \<and> y = 0"
huffman@47192
   704
  by (simp add: le_less sum_power2_eq_zero_iff not_sum_power2_lt_zero)
huffman@47192
   705
huffman@47192
   706
lemma sum_power2_gt_zero_iff:
wenzelm@53015
   707
  "0 < x\<^sup>2 + y\<^sup>2 \<longleftrightarrow> x \<noteq> 0 \<or> y \<noteq> 0"
huffman@47192
   708
  unfolding not_le [symmetric] by (simp add: sum_power2_le_zero_iff)
haftmann@30996
   709
lp15@59865
   710
lemma abs_le_square_iff:
lp15@59865
   711
   "\<bar>x\<bar> \<le> \<bar>y\<bar> \<longleftrightarrow> x\<^sup>2 \<le> y\<^sup>2"
lp15@59865
   712
proof
lp15@59865
   713
  assume "\<bar>x\<bar> \<le> \<bar>y\<bar>"
lp15@59865
   714
  then have "\<bar>x\<bar>\<^sup>2 \<le> \<bar>y\<bar>\<^sup>2" by (rule power_mono, simp)
lp15@59865
   715
  then show "x\<^sup>2 \<le> y\<^sup>2" by simp
lp15@59865
   716
next
lp15@59865
   717
  assume "x\<^sup>2 \<le> y\<^sup>2"
lp15@59865
   718
  then show "\<bar>x\<bar> \<le> \<bar>y\<bar>"
lp15@59865
   719
    by (auto intro!: power2_le_imp_le [OF _ abs_ge_zero])
lp15@59865
   720
qed
lp15@59865
   721
wenzelm@61944
   722
lemma abs_square_le_1:"x\<^sup>2 \<le> 1 \<longleftrightarrow> \<bar>x\<bar> \<le> 1"
lp15@59865
   723
  using abs_le_square_iff [of x 1]
lp15@59865
   724
  by simp
lp15@59865
   725
wenzelm@61944
   726
lemma abs_square_eq_1: "x\<^sup>2 = 1 \<longleftrightarrow> \<bar>x\<bar> = 1"
lp15@59865
   727
  by (auto simp add: abs_if power2_eq_1_iff)
lp15@61649
   728
wenzelm@61944
   729
lemma abs_square_less_1: "x\<^sup>2 < 1 \<longleftrightarrow> \<bar>x\<bar> < 1"
lp15@59865
   730
  using  abs_square_eq_1 [of x] abs_square_le_1 [of x]
lp15@59865
   731
  by (auto simp add: le_less)
lp15@59865
   732
haftmann@30996
   733
end
haftmann@30996
   734
huffman@29978
   735
wenzelm@60758
   736
subsection \<open>Miscellaneous rules\<close>
paulson@14348
   737
haftmann@60867
   738
lemma (in linordered_semidom) self_le_power:
haftmann@60867
   739
  "1 \<le> a \<Longrightarrow> 0 < n \<Longrightarrow> a \<le> a ^ n"
haftmann@60867
   740
  using power_increasing [of 1 n a] power_one_right [of a] by auto
lp15@55718
   741
haftmann@60867
   742
lemma (in power) power_eq_if:
haftmann@60867
   743
  "p ^ m = (if m=0 then 1 else p * (p ^ (m - 1)))"
huffman@47255
   744
  unfolding One_nat_def by (cases m) simp_all
huffman@47255
   745
haftmann@58787
   746
lemma (in comm_semiring_1) power2_sum:
haftmann@58787
   747
  "(x + y)\<^sup>2 = x\<^sup>2 + y\<^sup>2 + 2 * x * y"
huffman@47192
   748
  by (simp add: algebra_simps power2_eq_square mult_2_right)
haftmann@30996
   749
haftmann@58787
   750
lemma (in comm_ring_1) power2_diff:
haftmann@58787
   751
  "(x - y)\<^sup>2 = x\<^sup>2 + y\<^sup>2 - 2 * x * y"
haftmann@58787
   752
  by (simp add: algebra_simps power2_eq_square mult_2_right)
haftmann@30996
   753
lp15@60974
   754
lemma (in comm_ring_1) power2_commute:
lp15@60974
   755
  "(x - y)\<^sup>2 = (y - x)\<^sup>2"
lp15@60974
   756
  by (simp add: algebra_simps power2_eq_square)
lp15@60974
   757
wenzelm@60758
   758
text \<open>Simprules for comparisons where common factors can be cancelled.\<close>
huffman@47255
   759
huffman@47255
   760
lemmas zero_compare_simps =
huffman@47255
   761
    add_strict_increasing add_strict_increasing2 add_increasing
lp15@61649
   762
    zero_le_mult_iff zero_le_divide_iff
lp15@61649
   763
    zero_less_mult_iff zero_less_divide_iff
lp15@61649
   764
    mult_le_0_iff divide_le_0_iff
lp15@61649
   765
    mult_less_0_iff divide_less_0_iff
huffman@47255
   766
    zero_le_power2 power2_less_0
huffman@47255
   767
nipkow@30313
   768
wenzelm@60758
   769
subsection \<open>Exponentiation for the Natural Numbers\<close>
wenzelm@14577
   770
haftmann@30996
   771
lemma nat_one_le_power [simp]:
haftmann@30996
   772
  "Suc 0 \<le> i \<Longrightarrow> Suc 0 \<le> i ^ n"
haftmann@30996
   773
  by (rule one_le_power [of i n, unfolded One_nat_def])
huffman@23305
   774
haftmann@30996
   775
lemma nat_zero_less_power_iff [simp]:
haftmann@30996
   776
  "x ^ n > 0 \<longleftrightarrow> x > (0::nat) \<or> n = 0"
haftmann@30996
   777
  by (induct n) auto
paulson@14348
   778
lp15@61649
   779
lemma nat_power_eq_Suc_0_iff [simp]:
haftmann@30996
   780
  "x ^ m = Suc 0 \<longleftrightarrow> m = 0 \<or> x = Suc 0"
haftmann@30996
   781
  by (induct m) auto
nipkow@30056
   782
haftmann@30996
   783
lemma power_Suc_0 [simp]:
haftmann@30996
   784
  "Suc 0 ^ n = Suc 0"
haftmann@30996
   785
  by simp
nipkow@30056
   786
wenzelm@61799
   787
text\<open>Valid for the naturals, but what if \<open>0<i<1\<close>?
paulson@14348
   788
Premises cannot be weakened: consider the case where @{term "i=0"},
wenzelm@60758
   789
@{term "m=1"} and @{term "n=0"}.\<close>
haftmann@21413
   790
lemma nat_power_less_imp_less:
wenzelm@61076
   791
  assumes nonneg: "0 < (i::nat)"
haftmann@30996
   792
  assumes less: "i ^ m < i ^ n"
haftmann@21413
   793
  shows "m < n"
haftmann@21413
   794
proof (cases "i = 1")
haftmann@21413
   795
  case True with less power_one [where 'a = nat] show ?thesis by simp
haftmann@21413
   796
next
haftmann@21413
   797
  case False with nonneg have "1 < i" by auto
haftmann@21413
   798
  from power_strict_increasing_iff [OF this] less show ?thesis ..
haftmann@21413
   799
qed
paulson@14348
   800
haftmann@33274
   801
lemma power_dvd_imp_le:
haftmann@33274
   802
  "i ^ m dvd i ^ n \<Longrightarrow> (1::nat) < i \<Longrightarrow> m \<le> n"
haftmann@33274
   803
  apply (rule power_le_imp_le_exp, assumption)
haftmann@33274
   804
  apply (erule dvd_imp_le, simp)
haftmann@33274
   805
  done
haftmann@33274
   806
haftmann@51263
   807
lemma power2_nat_le_eq_le:
haftmann@51263
   808
  fixes m n :: nat
wenzelm@53015
   809
  shows "m\<^sup>2 \<le> n\<^sup>2 \<longleftrightarrow> m \<le> n"
haftmann@51263
   810
  by (auto intro: power2_le_imp_le power_mono)
haftmann@51263
   811
haftmann@51263
   812
lemma power2_nat_le_imp_le:
haftmann@51263
   813
  fixes m n :: nat
wenzelm@53015
   814
  assumes "m\<^sup>2 \<le> n"
haftmann@51263
   815
  shows "m \<le> n"
haftmann@54249
   816
proof (cases m)
haftmann@54249
   817
  case 0 then show ?thesis by simp
haftmann@54249
   818
next
haftmann@54249
   819
  case (Suc k)
haftmann@54249
   820
  show ?thesis
haftmann@54249
   821
  proof (rule ccontr)
haftmann@54249
   822
    assume "\<not> m \<le> n"
haftmann@54249
   823
    then have "n < m" by simp
haftmann@54249
   824
    with assms Suc show False
haftmann@60867
   825
      by (simp add: power2_eq_square)
haftmann@54249
   826
  qed
haftmann@54249
   827
qed
haftmann@51263
   828
wenzelm@60758
   829
subsubsection \<open>Cardinality of the Powerset\<close>
traytel@55096
   830
traytel@55096
   831
lemma card_UNIV_bool [simp]: "card (UNIV :: bool set) = 2"
traytel@55096
   832
  unfolding UNIV_bool by simp
traytel@55096
   833
traytel@55096
   834
lemma card_Pow: "finite A \<Longrightarrow> card (Pow A) = 2 ^ card A"
traytel@55096
   835
proof (induct rule: finite_induct)
lp15@61649
   836
  case empty
traytel@55096
   837
    show ?case by auto
traytel@55096
   838
next
traytel@55096
   839
  case (insert x A)
lp15@61649
   840
  then have "inj_on (insert x) (Pow A)"
traytel@55096
   841
    unfolding inj_on_def by (blast elim!: equalityE)
lp15@61649
   842
  then have "card (Pow A) + card (insert x ` Pow A) = 2 * 2 ^ card A"
traytel@55096
   843
    by (simp add: mult_2 card_image Pow_insert insert.hyps)
traytel@55096
   844
  then show ?case using insert
traytel@55096
   845
    apply (simp add: Pow_insert)
traytel@55096
   846
    apply (subst card_Un_disjoint, auto)
traytel@55096
   847
    done
traytel@55096
   848
qed
traytel@55096
   849
haftmann@57418
   850
wenzelm@60758
   851
subsubsection \<open>Generalized sum over a set\<close>
haftmann@57418
   852
haftmann@57418
   853
lemma setsum_zero_power [simp]:
haftmann@57418
   854
  fixes c :: "nat \<Rightarrow> 'a::division_ring"
haftmann@57418
   855
  shows "(\<Sum>i\<in>A. c i * 0^i) = (if finite A \<and> 0 \<in> A then c 0 else 0)"
haftmann@57418
   856
apply (cases "finite A")
haftmann@57418
   857
  by (induction A rule: finite_induct) auto
haftmann@57418
   858
haftmann@57418
   859
lemma setsum_zero_power' [simp]:
haftmann@57418
   860
  fixes c :: "nat \<Rightarrow> 'a::field"
haftmann@57418
   861
  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)"
haftmann@57418
   862
  using setsum_zero_power [of "\<lambda>i. c i / d i" A]
haftmann@57418
   863
  by auto
haftmann@57418
   864
haftmann@57418
   865
wenzelm@60758
   866
subsubsection \<open>Generalized product over a set\<close>
traytel@55096
   867
traytel@55096
   868
lemma setprod_constant: "finite A ==> (\<Prod>x\<in> A. (y::'a::{comm_monoid_mult})) = y^(card A)"
traytel@55096
   869
apply (erule finite_induct)
traytel@55096
   870
apply auto
traytel@55096
   871
done
traytel@55096
   872
haftmann@57418
   873
lemma setprod_power_distrib:
haftmann@57418
   874
  fixes f :: "'a \<Rightarrow> 'b::comm_semiring_1"
haftmann@57418
   875
  shows "setprod f A ^ n = setprod (\<lambda>x. (f x) ^ n) A"
lp15@61649
   876
proof (cases "finite A")
lp15@61649
   877
  case True then show ?thesis
haftmann@57418
   878
    by (induct A rule: finite_induct) (auto simp add: power_mult_distrib)
haftmann@57418
   879
next
lp15@61649
   880
  case False then show ?thesis
haftmann@57418
   881
    by simp
haftmann@57418
   882
qed
haftmann@57418
   883
haftmann@58437
   884
lemma power_setsum:
haftmann@58437
   885
  "c ^ (\<Sum>a\<in>A. f a) = (\<Prod>a\<in>A. c ^ f a)"
haftmann@58437
   886
  by (induct A rule: infinite_finite_induct) (simp_all add: power_add)
haftmann@58437
   887
traytel@55096
   888
lemma setprod_gen_delta:
traytel@55096
   889
  assumes fS: "finite S"
traytel@55096
   890
  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)"
traytel@55096
   891
proof-
traytel@55096
   892
  let ?f = "(\<lambda>k. if k=a then b k else c)"
traytel@55096
   893
  {assume a: "a \<notin> S"
traytel@55096
   894
    hence "\<forall> k\<in> S. ?f k = c" by simp
traytel@55096
   895
    hence ?thesis  using a setprod_constant[OF fS, of c] by simp }
lp15@61649
   896
  moreover
traytel@55096
   897
  {assume a: "a \<in> S"
traytel@55096
   898
    let ?A = "S - {a}"
traytel@55096
   899
    let ?B = "{a}"
lp15@61649
   900
    have eq: "S = ?A \<union> ?B" using a by blast
traytel@55096
   901
    have dj: "?A \<inter> ?B = {}" by simp
lp15@61649
   902
    from fS have fAB: "finite ?A" "finite ?B" by auto
traytel@55096
   903
    have fA0:"setprod ?f ?A = setprod (\<lambda>i. c) ?A"
haftmann@57418
   904
      apply (rule setprod.cong) by auto
traytel@55096
   905
    have cA: "card ?A = card S - 1" using fS a by auto
traytel@55096
   906
    have fA1: "setprod ?f ?A = c ^ card ?A"  unfolding fA0 apply (rule setprod_constant) using fS by auto
traytel@55096
   907
    have "setprod ?f ?A * setprod ?f ?B = setprod ?f S"
haftmann@57418
   908
      using setprod.union_disjoint[OF fAB dj, of ?f, unfolded eq[symmetric]]
traytel@55096
   909
      by simp
traytel@55096
   910
    then have ?thesis using a cA
haftmann@57418
   911
      by (simp add: fA1 field_simps cong add: setprod.cong cong del: if_weak_cong)}
traytel@55096
   912
  ultimately show ?thesis by blast
traytel@55096
   913
qed
traytel@55096
   914
wenzelm@60758
   915
subsection \<open>Code generator tweak\<close>
haftmann@31155
   916
haftmann@52435
   917
code_identifier
haftmann@52435
   918
  code_module Power \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
haftmann@33364
   919
paulson@3390
   920
end