src/HOL/Power.thy
author haftmann
Thu Aug 06 19:12:09 2015 +0200 (2015-08-06)
changeset 60866 7f562aa4eb4b
parent 60758 d8d85a8172b5
child 60867 86e7560e07d0
permissions -rw-r--r--
obsolete since no code generator without dictionary construction left
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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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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) 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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notation (HTML 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 (xsymbols)
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  power2 :: "'a \<Rightarrow> 'a"  ("(_\<^sup>2)" [1000] 999) where
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  "x\<^sup>2 \<equiv> x ^ 2"
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notation (latex output)
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  power2  ("(_\<^sup>2)" [1000] 999)
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notation (HTML output)
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  power2  ("(_\<^sup>2)" [1000] 999)
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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:
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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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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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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_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 (rule 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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lemma power_eq_0_nat_iff [simp]:
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  fixes m n :: nat
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  shows "m ^ n = 0 \<longleftrightarrow> m = 0 \<and> n > 0"
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  by (induct n) auto
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context ring_1_no_zero_divisors
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begin
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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 field_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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lemma power2_eq_1_iff:
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  "a\<^sup>2 = 1 \<longleftrightarrow> a = 1 \<or> a = - 1"
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  unfolding power2_eq_square by (rule square_eq_1_iff)
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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 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
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text \<open>FIXME reorient or rename to @{text nonzero_inverse_power}\<close>
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lemma nonzero_power_inverse:
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  "a \<noteq> 0 \<Longrightarrow> inverse (a ^ n) = (inverse a) ^ n"
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  by (induct n)
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    (simp_all add: nonzero_inverse_mult_distrib power_commutes field_power_not_zero)
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end
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context field
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begin
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lemma nonzero_power_divide:
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  "b \<noteq> 0 \<Longrightarrow> (a / b) ^ n = a ^ n / b ^ n"
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  by (simp add: divide_inverse power_mult_distrib nonzero_power_inverse)
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declare nonzero_power_divide [where b = "numeral w" for w, simp]
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end
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subsection \<open>Exponentiation on ordered types\<close>
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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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   356
  by (simp add: not_less sum_squares_ge_zero)
huffman@47192
   357
haftmann@30996
   358
end
haftmann@30996
   359
haftmann@35028
   360
context linordered_semidom
haftmann@30996
   361
begin
haftmann@30996
   362
haftmann@30996
   363
lemma zero_less_power [simp]:
haftmann@30996
   364
  "0 < a \<Longrightarrow> 0 < a ^ n"
nipkow@56544
   365
  by (induct n) simp_all
haftmann@30996
   366
haftmann@30996
   367
lemma zero_le_power [simp]:
haftmann@30996
   368
  "0 \<le> a \<Longrightarrow> 0 \<le> a ^ n"
nipkow@56536
   369
  by (induct n) simp_all
paulson@14348
   370
huffman@47241
   371
lemma power_mono:
huffman@47241
   372
  "a \<le> b \<Longrightarrow> 0 \<le> a \<Longrightarrow> a ^ n \<le> b ^ n"
huffman@47241
   373
  by (induct n) (auto intro: mult_mono order_trans [of 0 a b])
huffman@47241
   374
huffman@47241
   375
lemma one_le_power [simp]: "1 \<le> a \<Longrightarrow> 1 \<le> a ^ n"
huffman@47241
   376
  using power_mono [of 1 a n] by simp
huffman@47241
   377
huffman@47241
   378
lemma power_le_one: "\<lbrakk>0 \<le> a; a \<le> 1\<rbrakk> \<Longrightarrow> a ^ n \<le> 1"
huffman@47241
   379
  using power_mono [of a 1 n] by simp
paulson@14348
   380
paulson@14348
   381
lemma power_gt1_lemma:
haftmann@30996
   382
  assumes gt1: "1 < a"
haftmann@30996
   383
  shows "1 < a * a ^ n"
paulson@14348
   384
proof -
haftmann@30996
   385
  from gt1 have "0 \<le> a"
haftmann@30996
   386
    by (fact order_trans [OF zero_le_one less_imp_le])
haftmann@30996
   387
  have "1 * 1 < a * 1" using gt1 by simp
haftmann@30996
   388
  also have "\<dots> \<le> a * a ^ n" using gt1
wenzelm@60758
   389
    by (simp only: mult_mono \<open>0 \<le> a\<close> one_le_power order_less_imp_le
wenzelm@14577
   390
        zero_le_one order_refl)
wenzelm@14577
   391
  finally show ?thesis by simp
paulson@14348
   392
qed
paulson@14348
   393
haftmann@30996
   394
lemma power_gt1:
haftmann@30996
   395
  "1 < a \<Longrightarrow> 1 < a ^ Suc n"
haftmann@30996
   396
  by (simp add: power_gt1_lemma)
huffman@24376
   397
haftmann@30996
   398
lemma one_less_power [simp]:
haftmann@30996
   399
  "1 < a \<Longrightarrow> 0 < n \<Longrightarrow> 1 < a ^ n"
haftmann@30996
   400
  by (cases n) (simp_all add: power_gt1_lemma)
paulson@14348
   401
paulson@14348
   402
lemma power_le_imp_le_exp:
haftmann@30996
   403
  assumes gt1: "1 < a"
haftmann@30996
   404
  shows "a ^ m \<le> a ^ n \<Longrightarrow> m \<le> n"
haftmann@30996
   405
proof (induct m arbitrary: n)
paulson@14348
   406
  case 0
wenzelm@14577
   407
  show ?case by simp
paulson@14348
   408
next
paulson@14348
   409
  case (Suc m)
wenzelm@14577
   410
  show ?case
wenzelm@14577
   411
  proof (cases n)
wenzelm@14577
   412
    case 0
haftmann@30996
   413
    with Suc.prems Suc.hyps have "a * a ^ m \<le> 1" by simp
wenzelm@14577
   414
    with gt1 show ?thesis
wenzelm@14577
   415
      by (force simp only: power_gt1_lemma
haftmann@30996
   416
          not_less [symmetric])
wenzelm@14577
   417
  next
wenzelm@14577
   418
    case (Suc n)
haftmann@30996
   419
    with Suc.prems Suc.hyps show ?thesis
wenzelm@14577
   420
      by (force dest: mult_left_le_imp_le
haftmann@30996
   421
          simp add: less_trans [OF zero_less_one gt1])
wenzelm@14577
   422
  qed
paulson@14348
   423
qed
paulson@14348
   424
wenzelm@60758
   425
text\<open>Surely we can strengthen this? It holds for @{text "0<a<1"} too.\<close>
paulson@14348
   426
lemma power_inject_exp [simp]:
haftmann@30996
   427
  "1 < a \<Longrightarrow> a ^ m = a ^ n \<longleftrightarrow> m = n"
wenzelm@14577
   428
  by (force simp add: order_antisym power_le_imp_le_exp)
paulson@14348
   429
wenzelm@60758
   430
text\<open>Can relax the first premise to @{term "0<a"} in the case of the
wenzelm@60758
   431
natural numbers.\<close>
paulson@14348
   432
lemma power_less_imp_less_exp:
haftmann@30996
   433
  "1 < a \<Longrightarrow> a ^ m < a ^ n \<Longrightarrow> m < n"
haftmann@30996
   434
  by (simp add: order_less_le [of m n] less_le [of "a^m" "a^n"]
haftmann@30996
   435
    power_le_imp_le_exp)
paulson@14348
   436
paulson@14348
   437
lemma power_strict_mono [rule_format]:
haftmann@30996
   438
  "a < b \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 < n \<longrightarrow> a ^ n < b ^ n"
haftmann@30996
   439
  by (induct n)
haftmann@30996
   440
   (auto simp add: mult_strict_mono le_less_trans [of 0 a b])
paulson@14348
   441
wenzelm@60758
   442
text\<open>Lemma for @{text power_strict_decreasing}\<close>
paulson@14348
   443
lemma power_Suc_less:
haftmann@30996
   444
  "0 < a \<Longrightarrow> a < 1 \<Longrightarrow> a * a ^ n < a ^ n"
haftmann@30996
   445
  by (induct n)
haftmann@30996
   446
    (auto simp add: mult_strict_left_mono)
paulson@14348
   447
haftmann@30996
   448
lemma power_strict_decreasing [rule_format]:
haftmann@30996
   449
  "n < N \<Longrightarrow> 0 < a \<Longrightarrow> a < 1 \<longrightarrow> a ^ N < a ^ n"
haftmann@30996
   450
proof (induct N)
haftmann@30996
   451
  case 0 then show ?case by simp
haftmann@30996
   452
next
haftmann@30996
   453
  case (Suc N) then show ?case 
haftmann@30996
   454
  apply (auto simp add: power_Suc_less less_Suc_eq)
haftmann@30996
   455
  apply (subgoal_tac "a * a^N < 1 * a^n")
haftmann@30996
   456
  apply simp
haftmann@30996
   457
  apply (rule mult_strict_mono) apply auto
haftmann@30996
   458
  done
haftmann@30996
   459
qed
paulson@14348
   460
wenzelm@60758
   461
text\<open>Proof resembles that of @{text power_strict_decreasing}\<close>
haftmann@30996
   462
lemma power_decreasing [rule_format]:
haftmann@30996
   463
  "n \<le> N \<Longrightarrow> 0 \<le> a \<Longrightarrow> a \<le> 1 \<longrightarrow> a ^ N \<le> a ^ n"
haftmann@30996
   464
proof (induct N)
haftmann@30996
   465
  case 0 then show ?case by simp
haftmann@30996
   466
next
haftmann@30996
   467
  case (Suc N) then show ?case 
haftmann@30996
   468
  apply (auto simp add: le_Suc_eq)
haftmann@30996
   469
  apply (subgoal_tac "a * a^N \<le> 1 * a^n", simp)
haftmann@30996
   470
  apply (rule mult_mono) apply auto
haftmann@30996
   471
  done
haftmann@30996
   472
qed
paulson@14348
   473
paulson@14348
   474
lemma power_Suc_less_one:
haftmann@30996
   475
  "0 < a \<Longrightarrow> a < 1 \<Longrightarrow> a ^ Suc n < 1"
haftmann@30996
   476
  using power_strict_decreasing [of 0 "Suc n" a] by simp
paulson@14348
   477
wenzelm@60758
   478
text\<open>Proof again resembles that of @{text power_strict_decreasing}\<close>
haftmann@30996
   479
lemma power_increasing [rule_format]:
haftmann@30996
   480
  "n \<le> N \<Longrightarrow> 1 \<le> a \<Longrightarrow> a ^ n \<le> a ^ N"
haftmann@30996
   481
proof (induct N)
haftmann@30996
   482
  case 0 then show ?case by simp
haftmann@30996
   483
next
haftmann@30996
   484
  case (Suc N) then show ?case 
haftmann@30996
   485
  apply (auto simp add: le_Suc_eq)
haftmann@30996
   486
  apply (subgoal_tac "1 * a^n \<le> a * a^N", simp)
haftmann@30996
   487
  apply (rule mult_mono) apply (auto simp add: order_trans [OF zero_le_one])
haftmann@30996
   488
  done
haftmann@30996
   489
qed
paulson@14348
   490
wenzelm@60758
   491
text\<open>Lemma for @{text power_strict_increasing}\<close>
paulson@14348
   492
lemma power_less_power_Suc:
haftmann@30996
   493
  "1 < a \<Longrightarrow> a ^ n < a * a ^ n"
haftmann@30996
   494
  by (induct n) (auto simp add: mult_strict_left_mono less_trans [OF zero_less_one])
paulson@14348
   495
haftmann@30996
   496
lemma power_strict_increasing [rule_format]:
haftmann@30996
   497
  "n < N \<Longrightarrow> 1 < a \<longrightarrow> a ^ n < a ^ N"
haftmann@30996
   498
proof (induct N)
haftmann@30996
   499
  case 0 then show ?case by simp
haftmann@30996
   500
next
haftmann@30996
   501
  case (Suc N) then show ?case 
haftmann@30996
   502
  apply (auto simp add: power_less_power_Suc less_Suc_eq)
haftmann@30996
   503
  apply (subgoal_tac "1 * a^n < a * a^N", simp)
haftmann@30996
   504
  apply (rule mult_strict_mono) apply (auto simp add: less_trans [OF zero_less_one] less_imp_le)
haftmann@30996
   505
  done
haftmann@30996
   506
qed
paulson@14348
   507
nipkow@25134
   508
lemma power_increasing_iff [simp]:
haftmann@30996
   509
  "1 < b \<Longrightarrow> b ^ x \<le> b ^ y \<longleftrightarrow> x \<le> y"
haftmann@30996
   510
  by (blast intro: power_le_imp_le_exp power_increasing less_imp_le)
paulson@15066
   511
paulson@15066
   512
lemma power_strict_increasing_iff [simp]:
haftmann@30996
   513
  "1 < b \<Longrightarrow> b ^ x < b ^ y \<longleftrightarrow> x < y"
nipkow@25134
   514
by (blast intro: power_less_imp_less_exp power_strict_increasing) 
paulson@15066
   515
paulson@14348
   516
lemma power_le_imp_le_base:
haftmann@30996
   517
  assumes le: "a ^ Suc n \<le> b ^ Suc n"
haftmann@30996
   518
    and ynonneg: "0 \<le> b"
haftmann@30996
   519
  shows "a \<le> b"
nipkow@25134
   520
proof (rule ccontr)
nipkow@25134
   521
  assume "~ a \<le> b"
nipkow@25134
   522
  then have "b < a" by (simp only: linorder_not_le)
nipkow@25134
   523
  then have "b ^ Suc n < a ^ Suc n"
wenzelm@41550
   524
    by (simp only: assms power_strict_mono)
haftmann@30996
   525
  from le and this show False
nipkow@25134
   526
    by (simp add: linorder_not_less [symmetric])
nipkow@25134
   527
qed
wenzelm@14577
   528
huffman@22853
   529
lemma power_less_imp_less_base:
huffman@22853
   530
  assumes less: "a ^ n < b ^ n"
huffman@22853
   531
  assumes nonneg: "0 \<le> b"
huffman@22853
   532
  shows "a < b"
huffman@22853
   533
proof (rule contrapos_pp [OF less])
huffman@22853
   534
  assume "~ a < b"
huffman@22853
   535
  hence "b \<le> a" by (simp only: linorder_not_less)
huffman@22853
   536
  hence "b ^ n \<le> a ^ n" using nonneg by (rule power_mono)
haftmann@30996
   537
  thus "\<not> a ^ n < b ^ n" by (simp only: linorder_not_less)
huffman@22853
   538
qed
huffman@22853
   539
paulson@14348
   540
lemma power_inject_base:
haftmann@30996
   541
  "a ^ Suc n = b ^ Suc n \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> a = b"
haftmann@30996
   542
by (blast intro: power_le_imp_le_base antisym eq_refl sym)
paulson@14348
   543
huffman@22955
   544
lemma power_eq_imp_eq_base:
haftmann@30996
   545
  "a ^ n = b ^ n \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 < n \<Longrightarrow> a = b"
haftmann@30996
   546
  by (cases n) (simp_all del: power_Suc, rule power_inject_base)
huffman@22955
   547
huffman@47192
   548
lemma power2_le_imp_le:
wenzelm@53015
   549
  "x\<^sup>2 \<le> y\<^sup>2 \<Longrightarrow> 0 \<le> y \<Longrightarrow> x \<le> y"
huffman@47192
   550
  unfolding numeral_2_eq_2 by (rule power_le_imp_le_base)
huffman@47192
   551
huffman@47192
   552
lemma power2_less_imp_less:
wenzelm@53015
   553
  "x\<^sup>2 < y\<^sup>2 \<Longrightarrow> 0 \<le> y \<Longrightarrow> x < y"
huffman@47192
   554
  by (rule power_less_imp_less_base)
huffman@47192
   555
huffman@47192
   556
lemma power2_eq_imp_eq:
wenzelm@53015
   557
  "x\<^sup>2 = y\<^sup>2 \<Longrightarrow> 0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> x = y"
huffman@47192
   558
  unfolding numeral_2_eq_2 by (erule (2) power_eq_imp_eq_base) simp
huffman@47192
   559
huffman@47192
   560
end
huffman@47192
   561
huffman@47192
   562
context linordered_ring_strict
huffman@47192
   563
begin
huffman@47192
   564
huffman@47192
   565
lemma sum_squares_eq_zero_iff:
huffman@47192
   566
  "x * x + y * y = 0 \<longleftrightarrow> x = 0 \<and> y = 0"
huffman@47192
   567
  by (simp add: add_nonneg_eq_0_iff)
huffman@47192
   568
huffman@47192
   569
lemma sum_squares_le_zero_iff:
huffman@47192
   570
  "x * x + y * y \<le> 0 \<longleftrightarrow> x = 0 \<and> y = 0"
huffman@47192
   571
  by (simp add: le_less not_sum_squares_lt_zero sum_squares_eq_zero_iff)
huffman@47192
   572
huffman@47192
   573
lemma sum_squares_gt_zero_iff:
huffman@47192
   574
  "0 < x * x + y * y \<longleftrightarrow> x \<noteq> 0 \<or> y \<noteq> 0"
huffman@47192
   575
  by (simp add: not_le [symmetric] sum_squares_le_zero_iff)
huffman@47192
   576
haftmann@30996
   577
end
haftmann@30996
   578
haftmann@35028
   579
context linordered_idom
haftmann@30996
   580
begin
huffman@29978
   581
haftmann@30996
   582
lemma power_abs:
haftmann@30996
   583
  "abs (a ^ n) = abs a ^ n"
haftmann@30996
   584
  by (induct n) (auto simp add: abs_mult)
haftmann@30996
   585
haftmann@30996
   586
lemma abs_power_minus [simp]:
haftmann@30996
   587
  "abs ((-a) ^ n) = abs (a ^ n)"
huffman@35216
   588
  by (simp add: power_abs)
haftmann@30996
   589
blanchet@54147
   590
lemma zero_less_power_abs_iff [simp]:
haftmann@30996
   591
  "0 < abs a ^ n \<longleftrightarrow> a \<noteq> 0 \<or> n = 0"
haftmann@30996
   592
proof (induct n)
haftmann@30996
   593
  case 0 show ?case by simp
haftmann@30996
   594
next
haftmann@30996
   595
  case (Suc n) show ?case by (auto simp add: Suc zero_less_mult_iff)
huffman@29978
   596
qed
huffman@29978
   597
haftmann@30996
   598
lemma zero_le_power_abs [simp]:
haftmann@30996
   599
  "0 \<le> abs a ^ n"
haftmann@30996
   600
  by (rule zero_le_power [OF abs_ge_zero])
haftmann@30996
   601
huffman@47192
   602
lemma zero_le_power2 [simp]:
wenzelm@53015
   603
  "0 \<le> a\<^sup>2"
huffman@47192
   604
  by (simp add: power2_eq_square)
huffman@47192
   605
huffman@47192
   606
lemma zero_less_power2 [simp]:
wenzelm@53015
   607
  "0 < a\<^sup>2 \<longleftrightarrow> a \<noteq> 0"
huffman@47192
   608
  by (force simp add: power2_eq_square zero_less_mult_iff linorder_neq_iff)
huffman@47192
   609
huffman@47192
   610
lemma power2_less_0 [simp]:
wenzelm@53015
   611
  "\<not> a\<^sup>2 < 0"
huffman@47192
   612
  by (force simp add: power2_eq_square mult_less_0_iff)
huffman@47192
   613
haftmann@58787
   614
lemma power2_less_eq_zero_iff [simp]:
haftmann@58787
   615
  "a\<^sup>2 \<le> 0 \<longleftrightarrow> a = 0"
haftmann@58787
   616
  by (simp add: le_less)
haftmann@58787
   617
huffman@47192
   618
lemma abs_power2 [simp]:
wenzelm@53015
   619
  "abs (a\<^sup>2) = a\<^sup>2"
huffman@47192
   620
  by (simp add: power2_eq_square abs_mult abs_mult_self)
huffman@47192
   621
huffman@47192
   622
lemma power2_abs [simp]:
wenzelm@53015
   623
  "(abs a)\<^sup>2 = a\<^sup>2"
huffman@47192
   624
  by (simp add: power2_eq_square abs_mult_self)
huffman@47192
   625
huffman@47192
   626
lemma odd_power_less_zero:
huffman@47192
   627
  "a < 0 \<Longrightarrow> a ^ Suc (2*n) < 0"
huffman@47192
   628
proof (induct n)
huffman@47192
   629
  case 0
huffman@47192
   630
  then show ?case by simp
huffman@47192
   631
next
huffman@47192
   632
  case (Suc n)
huffman@47192
   633
  have "a ^ Suc (2 * Suc n) = (a*a) * a ^ Suc(2*n)"
haftmann@57514
   634
    by (simp add: ac_simps power_add power2_eq_square)
huffman@47192
   635
  thus ?case
huffman@47192
   636
    by (simp del: power_Suc add: Suc mult_less_0_iff mult_neg_neg)
huffman@47192
   637
qed
haftmann@30996
   638
huffman@47192
   639
lemma odd_0_le_power_imp_0_le:
huffman@47192
   640
  "0 \<le> a ^ Suc (2*n) \<Longrightarrow> 0 \<le> a"
huffman@47192
   641
  using odd_power_less_zero [of a n]
huffman@47192
   642
    by (force simp add: linorder_not_less [symmetric]) 
huffman@47192
   643
huffman@47192
   644
lemma zero_le_even_power'[simp]:
huffman@47192
   645
  "0 \<le> a ^ (2*n)"
huffman@47192
   646
proof (induct n)
huffman@47192
   647
  case 0
huffman@47192
   648
    show ?case by simp
huffman@47192
   649
next
huffman@47192
   650
  case (Suc n)
huffman@47192
   651
    have "a ^ (2 * Suc n) = (a*a) * a ^ (2*n)" 
haftmann@57514
   652
      by (simp add: ac_simps power_add power2_eq_square)
huffman@47192
   653
    thus ?case
huffman@47192
   654
      by (simp add: Suc zero_le_mult_iff)
huffman@47192
   655
qed
haftmann@30996
   656
huffman@47192
   657
lemma sum_power2_ge_zero:
wenzelm@53015
   658
  "0 \<le> x\<^sup>2 + y\<^sup>2"
huffman@47192
   659
  by (intro add_nonneg_nonneg zero_le_power2)
huffman@47192
   660
huffman@47192
   661
lemma not_sum_power2_lt_zero:
wenzelm@53015
   662
  "\<not> x\<^sup>2 + y\<^sup>2 < 0"
huffman@47192
   663
  unfolding not_less by (rule sum_power2_ge_zero)
huffman@47192
   664
huffman@47192
   665
lemma sum_power2_eq_zero_iff:
wenzelm@53015
   666
  "x\<^sup>2 + y\<^sup>2 = 0 \<longleftrightarrow> x = 0 \<and> y = 0"
huffman@47192
   667
  unfolding power2_eq_square by (simp add: add_nonneg_eq_0_iff)
huffman@47192
   668
huffman@47192
   669
lemma sum_power2_le_zero_iff:
wenzelm@53015
   670
  "x\<^sup>2 + y\<^sup>2 \<le> 0 \<longleftrightarrow> x = 0 \<and> y = 0"
huffman@47192
   671
  by (simp add: le_less sum_power2_eq_zero_iff not_sum_power2_lt_zero)
huffman@47192
   672
huffman@47192
   673
lemma sum_power2_gt_zero_iff:
wenzelm@53015
   674
  "0 < x\<^sup>2 + y\<^sup>2 \<longleftrightarrow> x \<noteq> 0 \<or> y \<noteq> 0"
huffman@47192
   675
  unfolding not_le [symmetric] by (simp add: sum_power2_le_zero_iff)
haftmann@30996
   676
lp15@59865
   677
lemma abs_le_square_iff:
lp15@59865
   678
   "\<bar>x\<bar> \<le> \<bar>y\<bar> \<longleftrightarrow> x\<^sup>2 \<le> y\<^sup>2"
lp15@59865
   679
proof
lp15@59865
   680
  assume "\<bar>x\<bar> \<le> \<bar>y\<bar>"
lp15@59865
   681
  then have "\<bar>x\<bar>\<^sup>2 \<le> \<bar>y\<bar>\<^sup>2" by (rule power_mono, simp)
lp15@59865
   682
  then show "x\<^sup>2 \<le> y\<^sup>2" by simp
lp15@59865
   683
next
lp15@59865
   684
  assume "x\<^sup>2 \<le> y\<^sup>2"
lp15@59865
   685
  then show "\<bar>x\<bar> \<le> \<bar>y\<bar>"
lp15@59865
   686
    by (auto intro!: power2_le_imp_le [OF _ abs_ge_zero])
lp15@59865
   687
qed
lp15@59865
   688
lp15@59865
   689
lemma abs_square_le_1:"x\<^sup>2 \<le> 1 \<longleftrightarrow> abs(x) \<le> 1"
lp15@59865
   690
  using abs_le_square_iff [of x 1]
lp15@59865
   691
  by simp
lp15@59865
   692
lp15@59865
   693
lemma abs_square_eq_1: "x\<^sup>2 = 1 \<longleftrightarrow> abs(x) = 1"
lp15@59865
   694
  by (auto simp add: abs_if power2_eq_1_iff)
lp15@59865
   695
  
lp15@59865
   696
lemma abs_square_less_1: "x\<^sup>2 < 1 \<longleftrightarrow> abs(x) < 1"
lp15@59865
   697
  using  abs_square_eq_1 [of x] abs_square_le_1 [of x]
lp15@59865
   698
  by (auto simp add: le_less)
lp15@59865
   699
haftmann@30996
   700
end
haftmann@30996
   701
huffman@29978
   702
wenzelm@60758
   703
subsection \<open>Miscellaneous rules\<close>
paulson@14348
   704
lp15@55718
   705
lemma self_le_power:
lp15@55718
   706
  fixes x::"'a::linordered_semidom" 
lp15@55718
   707
  shows "1 \<le> x \<Longrightarrow> 0 < n \<Longrightarrow> x \<le> x ^ n"
traytel@55811
   708
  using power_increasing[of 1 n x] power_one_right[of x] by auto
lp15@55718
   709
huffman@47255
   710
lemma power_eq_if: "p ^ m = (if m=0 then 1 else p * (p ^ (m - 1)))"
huffman@47255
   711
  unfolding One_nat_def by (cases m) simp_all
huffman@47255
   712
haftmann@58787
   713
lemma (in comm_semiring_1) power2_sum:
haftmann@58787
   714
  "(x + y)\<^sup>2 = x\<^sup>2 + y\<^sup>2 + 2 * x * y"
huffman@47192
   715
  by (simp add: algebra_simps power2_eq_square mult_2_right)
haftmann@30996
   716
haftmann@58787
   717
lemma (in comm_ring_1) power2_diff:
haftmann@58787
   718
  "(x - y)\<^sup>2 = x\<^sup>2 + y\<^sup>2 - 2 * x * y"
haftmann@58787
   719
  by (simp add: algebra_simps power2_eq_square mult_2_right)
haftmann@30996
   720
haftmann@30996
   721
lemma power_0_Suc [simp]:
haftmann@30996
   722
  "(0::'a::{power, semiring_0}) ^ Suc n = 0"
haftmann@30996
   723
  by simp
nipkow@30313
   724
wenzelm@60758
   725
text\<open>It looks plausible as a simprule, but its effect can be strange.\<close>
haftmann@30996
   726
lemma power_0_left:
haftmann@30996
   727
  "0 ^ n = (if n = 0 then 1 else (0::'a::{power, semiring_0}))"
haftmann@30996
   728
  by (induct n) simp_all
haftmann@30996
   729
haftmann@36409
   730
lemma (in field) power_diff:
haftmann@30996
   731
  assumes nz: "a \<noteq> 0"
haftmann@30996
   732
  shows "n \<le> m \<Longrightarrow> a ^ (m - n) = a ^ m / a ^ n"
haftmann@36409
   733
  by (induct m n rule: diff_induct) (simp_all add: nz field_power_not_zero)
nipkow@30313
   734
wenzelm@60758
   735
text\<open>Perhaps these should be simprules.\<close>
haftmann@30996
   736
lemma power_inverse:
haftmann@59867
   737
  fixes a :: "'a::division_ring"
haftmann@36409
   738
  shows "inverse (a ^ n) = inverse a ^ n"
haftmann@30996
   739
apply (cases "a = 0")
haftmann@30996
   740
apply (simp add: power_0_left)
haftmann@30996
   741
apply (simp add: nonzero_power_inverse)
haftmann@30996
   742
done (* TODO: reorient or rename to inverse_power *)
haftmann@30996
   743
haftmann@30996
   744
lemma power_one_over:
haftmann@59867
   745
  "1 / (a::'a::{field, power}) ^ n =  (1 / a) ^ n"
haftmann@30996
   746
  by (simp add: divide_inverse) (rule power_inverse)
haftmann@30996
   747
hoelzl@56481
   748
lemma power_divide [field_simps, divide_simps]:
haftmann@59867
   749
  "(a / b) ^ n = (a::'a::field) ^ n / b ^ n"
haftmann@30996
   750
apply (cases "b = 0")
haftmann@30996
   751
apply (simp add: power_0_left)
haftmann@30996
   752
apply (rule nonzero_power_divide)
haftmann@30996
   753
apply assumption
nipkow@30313
   754
done
nipkow@30313
   755
wenzelm@60758
   756
text \<open>Simprules for comparisons where common factors can be cancelled.\<close>
huffman@47255
   757
huffman@47255
   758
lemmas zero_compare_simps =
huffman@47255
   759
    add_strict_increasing add_strict_increasing2 add_increasing
huffman@47255
   760
    zero_le_mult_iff zero_le_divide_iff 
huffman@47255
   761
    zero_less_mult_iff zero_less_divide_iff 
huffman@47255
   762
    mult_le_0_iff divide_le_0_iff 
huffman@47255
   763
    mult_less_0_iff divide_less_0_iff 
huffman@47255
   764
    zero_le_power2 power2_less_0
huffman@47255
   765
nipkow@30313
   766
wenzelm@60758
   767
subsection \<open>Exponentiation for the Natural Numbers\<close>
wenzelm@14577
   768
haftmann@30996
   769
lemma nat_one_le_power [simp]:
haftmann@30996
   770
  "Suc 0 \<le> i \<Longrightarrow> Suc 0 \<le> i ^ n"
haftmann@30996
   771
  by (rule one_le_power [of i n, unfolded One_nat_def])
huffman@23305
   772
haftmann@30996
   773
lemma nat_zero_less_power_iff [simp]:
haftmann@30996
   774
  "x ^ n > 0 \<longleftrightarrow> x > (0::nat) \<or> n = 0"
haftmann@30996
   775
  by (induct n) auto
paulson@14348
   776
nipkow@30056
   777
lemma nat_power_eq_Suc_0_iff [simp]: 
haftmann@30996
   778
  "x ^ m = Suc 0 \<longleftrightarrow> m = 0 \<or> x = Suc 0"
haftmann@30996
   779
  by (induct m) auto
nipkow@30056
   780
haftmann@30996
   781
lemma power_Suc_0 [simp]:
haftmann@30996
   782
  "Suc 0 ^ n = Suc 0"
haftmann@30996
   783
  by simp
nipkow@30056
   784
wenzelm@60758
   785
text\<open>Valid for the naturals, but what if @{text"0<i<1"}?
paulson@14348
   786
Premises cannot be weakened: consider the case where @{term "i=0"},
wenzelm@60758
   787
@{term "m=1"} and @{term "n=0"}.\<close>
haftmann@21413
   788
lemma nat_power_less_imp_less:
haftmann@21413
   789
  assumes nonneg: "0 < (i\<Colon>nat)"
haftmann@30996
   790
  assumes less: "i ^ m < i ^ n"
haftmann@21413
   791
  shows "m < n"
haftmann@21413
   792
proof (cases "i = 1")
haftmann@21413
   793
  case True with less power_one [where 'a = nat] show ?thesis by simp
haftmann@21413
   794
next
haftmann@21413
   795
  case False with nonneg have "1 < i" by auto
haftmann@21413
   796
  from power_strict_increasing_iff [OF this] less show ?thesis ..
haftmann@21413
   797
qed
paulson@14348
   798
haftmann@33274
   799
lemma power_dvd_imp_le:
haftmann@33274
   800
  "i ^ m dvd i ^ n \<Longrightarrow> (1::nat) < i \<Longrightarrow> m \<le> n"
haftmann@33274
   801
  apply (rule power_le_imp_le_exp, assumption)
haftmann@33274
   802
  apply (erule dvd_imp_le, simp)
haftmann@33274
   803
  done
haftmann@33274
   804
haftmann@51263
   805
lemma power2_nat_le_eq_le:
haftmann@51263
   806
  fixes m n :: nat
wenzelm@53015
   807
  shows "m\<^sup>2 \<le> n\<^sup>2 \<longleftrightarrow> m \<le> n"
haftmann@51263
   808
  by (auto intro: power2_le_imp_le power_mono)
haftmann@51263
   809
haftmann@51263
   810
lemma power2_nat_le_imp_le:
haftmann@51263
   811
  fixes m n :: nat
wenzelm@53015
   812
  assumes "m\<^sup>2 \<le> n"
haftmann@51263
   813
  shows "m \<le> n"
haftmann@54249
   814
proof (cases m)
haftmann@54249
   815
  case 0 then show ?thesis by simp
haftmann@54249
   816
next
haftmann@54249
   817
  case (Suc k)
haftmann@54249
   818
  show ?thesis
haftmann@54249
   819
  proof (rule ccontr)
haftmann@54249
   820
    assume "\<not> m \<le> n"
haftmann@54249
   821
    then have "n < m" by simp
haftmann@54249
   822
    with assms Suc show False
haftmann@54249
   823
      by (auto simp add: algebra_simps) (simp add: power2_eq_square)
haftmann@54249
   824
  qed
haftmann@54249
   825
qed
haftmann@51263
   826
wenzelm@60758
   827
subsubsection \<open>Cardinality of the Powerset\<close>
traytel@55096
   828
traytel@55096
   829
lemma card_UNIV_bool [simp]: "card (UNIV :: bool set) = 2"
traytel@55096
   830
  unfolding UNIV_bool by simp
traytel@55096
   831
traytel@55096
   832
lemma card_Pow: "finite A \<Longrightarrow> card (Pow A) = 2 ^ card A"
traytel@55096
   833
proof (induct rule: finite_induct)
traytel@55096
   834
  case empty 
traytel@55096
   835
    show ?case by auto
traytel@55096
   836
next
traytel@55096
   837
  case (insert x A)
traytel@55096
   838
  then have "inj_on (insert x) (Pow A)" 
traytel@55096
   839
    unfolding inj_on_def by (blast elim!: equalityE)
traytel@55096
   840
  then have "card (Pow A) + card (insert x ` Pow A) = 2 * 2 ^ card A" 
traytel@55096
   841
    by (simp add: mult_2 card_image Pow_insert insert.hyps)
traytel@55096
   842
  then show ?case using insert
traytel@55096
   843
    apply (simp add: Pow_insert)
traytel@55096
   844
    apply (subst card_Un_disjoint, auto)
traytel@55096
   845
    done
traytel@55096
   846
qed
traytel@55096
   847
haftmann@57418
   848
wenzelm@60758
   849
subsubsection \<open>Generalized sum over a set\<close>
haftmann@57418
   850
haftmann@57418
   851
lemma setsum_zero_power [simp]:
haftmann@57418
   852
  fixes c :: "nat \<Rightarrow> 'a::division_ring"
haftmann@57418
   853
  shows "(\<Sum>i\<in>A. c i * 0^i) = (if finite A \<and> 0 \<in> A then c 0 else 0)"
haftmann@57418
   854
apply (cases "finite A")
haftmann@57418
   855
  by (induction A rule: finite_induct) auto
haftmann@57418
   856
haftmann@57418
   857
lemma setsum_zero_power' [simp]:
haftmann@57418
   858
  fixes c :: "nat \<Rightarrow> 'a::field"
haftmann@57418
   859
  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
   860
  using setsum_zero_power [of "\<lambda>i. c i / d i" A]
haftmann@57418
   861
  by auto
haftmann@57418
   862
haftmann@57418
   863
wenzelm@60758
   864
subsubsection \<open>Generalized product over a set\<close>
traytel@55096
   865
traytel@55096
   866
lemma setprod_constant: "finite A ==> (\<Prod>x\<in> A. (y::'a::{comm_monoid_mult})) = y^(card A)"
traytel@55096
   867
apply (erule finite_induct)
traytel@55096
   868
apply auto
traytel@55096
   869
done
traytel@55096
   870
haftmann@57418
   871
lemma setprod_power_distrib:
haftmann@57418
   872
  fixes f :: "'a \<Rightarrow> 'b::comm_semiring_1"
haftmann@57418
   873
  shows "setprod f A ^ n = setprod (\<lambda>x. (f x) ^ n) A"
haftmann@57418
   874
proof (cases "finite A") 
haftmann@57418
   875
  case True then show ?thesis 
haftmann@57418
   876
    by (induct A rule: finite_induct) (auto simp add: power_mult_distrib)
haftmann@57418
   877
next
haftmann@57418
   878
  case False then show ?thesis 
haftmann@57418
   879
    by simp
haftmann@57418
   880
qed
haftmann@57418
   881
haftmann@58437
   882
lemma power_setsum:
haftmann@58437
   883
  "c ^ (\<Sum>a\<in>A. f a) = (\<Prod>a\<in>A. c ^ f a)"
haftmann@58437
   884
  by (induct A rule: infinite_finite_induct) (simp_all add: power_add)
haftmann@58437
   885
traytel@55096
   886
lemma setprod_gen_delta:
traytel@55096
   887
  assumes fS: "finite S"
traytel@55096
   888
  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
   889
proof-
traytel@55096
   890
  let ?f = "(\<lambda>k. if k=a then b k else c)"
traytel@55096
   891
  {assume a: "a \<notin> S"
traytel@55096
   892
    hence "\<forall> k\<in> S. ?f k = c" by simp
traytel@55096
   893
    hence ?thesis  using a setprod_constant[OF fS, of c] by simp }
traytel@55096
   894
  moreover 
traytel@55096
   895
  {assume a: "a \<in> S"
traytel@55096
   896
    let ?A = "S - {a}"
traytel@55096
   897
    let ?B = "{a}"
traytel@55096
   898
    have eq: "S = ?A \<union> ?B" using a by blast 
traytel@55096
   899
    have dj: "?A \<inter> ?B = {}" by simp
traytel@55096
   900
    from fS have fAB: "finite ?A" "finite ?B" by auto  
traytel@55096
   901
    have fA0:"setprod ?f ?A = setprod (\<lambda>i. c) ?A"
haftmann@57418
   902
      apply (rule setprod.cong) by auto
traytel@55096
   903
    have cA: "card ?A = card S - 1" using fS a by auto
traytel@55096
   904
    have fA1: "setprod ?f ?A = c ^ card ?A"  unfolding fA0 apply (rule setprod_constant) using fS by auto
traytel@55096
   905
    have "setprod ?f ?A * setprod ?f ?B = setprod ?f S"
haftmann@57418
   906
      using setprod.union_disjoint[OF fAB dj, of ?f, unfolded eq[symmetric]]
traytel@55096
   907
      by simp
traytel@55096
   908
    then have ?thesis using a cA
haftmann@57418
   909
      by (simp add: fA1 field_simps cong add: setprod.cong cong del: if_weak_cong)}
traytel@55096
   910
  ultimately show ?thesis by blast
traytel@55096
   911
qed
traytel@55096
   912
wenzelm@60758
   913
subsection \<open>Code generator tweak\<close>
haftmann@31155
   914
haftmann@52435
   915
code_identifier
haftmann@52435
   916
  code_module Power \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
haftmann@33364
   917
paulson@3390
   918
end
haftmann@49824
   919