src/HOL/Power.thy
author eberlm
Mon Nov 02 11:56:28 2015 +0100 (2015-11-02)
changeset 61531 ab2e862263e7
parent 61378 3e04c9ca001a
child 61649 268d88ec9087
permissions -rw-r--r--
Rounding function, uniform limits, cotangent, binomial identities
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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) 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 (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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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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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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   358
  by (induct n) (simp_all add: unit_factor_mult)
haftmann@60685
   359
haftmann@60685
   360
end
haftmann@60685
   361
huffman@47192
   362
context division_ring
huffman@47192
   363
begin
huffman@47192
   364
haftmann@60867
   365
text\<open>Perhaps these should be simprules.\<close>
haftmann@60867
   366
lemma power_inverse [field_simps, divide_simps]:
haftmann@60867
   367
  "inverse a ^ n = inverse (a ^ n)"
haftmann@60867
   368
proof (cases "a = 0")
haftmann@60867
   369
  case True then show ?thesis by (simp add: power_0_left)
haftmann@60867
   370
next
haftmann@60867
   371
  case False then have "inverse (a ^ n) = inverse a ^ n"
haftmann@60867
   372
    by (induct n) (simp_all add: nonzero_inverse_mult_distrib power_commutes)
haftmann@60867
   373
  then show ?thesis by simp
haftmann@60867
   374
qed
huffman@47192
   375
haftmann@60867
   376
lemma power_one_over [field_simps, divide_simps]:
haftmann@60867
   377
  "(1 / a) ^ n = 1 / a ^ n"
haftmann@60867
   378
  using power_inverse [of a] by (simp add: divide_inverse)
haftmann@60867
   379
haftmann@60867
   380
end  
huffman@47192
   381
huffman@47192
   382
context field
huffman@47192
   383
begin
huffman@47192
   384
haftmann@60867
   385
lemma power_diff:
haftmann@60867
   386
  assumes nz: "a \<noteq> 0"
haftmann@60867
   387
  shows "n \<le> m \<Longrightarrow> a ^ (m - n) = a ^ m / a ^ n"
haftmann@60867
   388
  by (induct m n rule: diff_induct) (simp_all add: nz power_not_zero)
huffman@47192
   389
haftmann@60867
   390
lemma power_divide [field_simps, divide_simps]:
haftmann@60867
   391
  "(a / b) ^ n = a ^ n / b ^ n"
haftmann@60867
   392
  by (induct n) simp_all
haftmann@60867
   393
haftmann@60867
   394
declare power_divide [where b = "numeral w" for w, simp]
lp15@59741
   395
huffman@47192
   396
end
huffman@47192
   397
huffman@47192
   398
wenzelm@60758
   399
subsection \<open>Exponentiation on ordered types\<close>
huffman@47192
   400
haftmann@35028
   401
context linordered_semidom
haftmann@30996
   402
begin
haftmann@30996
   403
haftmann@30996
   404
lemma zero_less_power [simp]:
haftmann@30996
   405
  "0 < a \<Longrightarrow> 0 < a ^ n"
nipkow@56544
   406
  by (induct n) simp_all
haftmann@30996
   407
haftmann@30996
   408
lemma zero_le_power [simp]:
haftmann@30996
   409
  "0 \<le> a \<Longrightarrow> 0 \<le> a ^ n"
nipkow@56536
   410
  by (induct n) simp_all
paulson@14348
   411
huffman@47241
   412
lemma power_mono:
huffman@47241
   413
  "a \<le> b \<Longrightarrow> 0 \<le> a \<Longrightarrow> a ^ n \<le> b ^ n"
huffman@47241
   414
  by (induct n) (auto intro: mult_mono order_trans [of 0 a b])
huffman@47241
   415
huffman@47241
   416
lemma one_le_power [simp]: "1 \<le> a \<Longrightarrow> 1 \<le> a ^ n"
huffman@47241
   417
  using power_mono [of 1 a n] by simp
huffman@47241
   418
huffman@47241
   419
lemma power_le_one: "\<lbrakk>0 \<le> a; a \<le> 1\<rbrakk> \<Longrightarrow> a ^ n \<le> 1"
huffman@47241
   420
  using power_mono [of a 1 n] by simp
paulson@14348
   421
paulson@14348
   422
lemma power_gt1_lemma:
haftmann@30996
   423
  assumes gt1: "1 < a"
haftmann@30996
   424
  shows "1 < a * a ^ n"
paulson@14348
   425
proof -
haftmann@30996
   426
  from gt1 have "0 \<le> a"
haftmann@30996
   427
    by (fact order_trans [OF zero_le_one less_imp_le])
haftmann@30996
   428
  have "1 * 1 < a * 1" using gt1 by simp
haftmann@30996
   429
  also have "\<dots> \<le> a * a ^ n" using gt1
wenzelm@60758
   430
    by (simp only: mult_mono \<open>0 \<le> a\<close> one_le_power order_less_imp_le
wenzelm@14577
   431
        zero_le_one order_refl)
wenzelm@14577
   432
  finally show ?thesis by simp
paulson@14348
   433
qed
paulson@14348
   434
haftmann@30996
   435
lemma power_gt1:
haftmann@30996
   436
  "1 < a \<Longrightarrow> 1 < a ^ Suc n"
haftmann@30996
   437
  by (simp add: power_gt1_lemma)
huffman@24376
   438
haftmann@30996
   439
lemma one_less_power [simp]:
haftmann@30996
   440
  "1 < a \<Longrightarrow> 0 < n \<Longrightarrow> 1 < a ^ n"
haftmann@30996
   441
  by (cases n) (simp_all add: power_gt1_lemma)
paulson@14348
   442
paulson@14348
   443
lemma power_le_imp_le_exp:
haftmann@30996
   444
  assumes gt1: "1 < a"
haftmann@30996
   445
  shows "a ^ m \<le> a ^ n \<Longrightarrow> m \<le> n"
haftmann@30996
   446
proof (induct m arbitrary: n)
paulson@14348
   447
  case 0
wenzelm@14577
   448
  show ?case by simp
paulson@14348
   449
next
paulson@14348
   450
  case (Suc m)
wenzelm@14577
   451
  show ?case
wenzelm@14577
   452
  proof (cases n)
wenzelm@14577
   453
    case 0
haftmann@30996
   454
    with Suc.prems Suc.hyps have "a * a ^ m \<le> 1" by simp
wenzelm@14577
   455
    with gt1 show ?thesis
wenzelm@14577
   456
      by (force simp only: power_gt1_lemma
haftmann@30996
   457
          not_less [symmetric])
wenzelm@14577
   458
  next
wenzelm@14577
   459
    case (Suc n)
haftmann@30996
   460
    with Suc.prems Suc.hyps show ?thesis
wenzelm@14577
   461
      by (force dest: mult_left_le_imp_le
haftmann@30996
   462
          simp add: less_trans [OF zero_less_one gt1])
wenzelm@14577
   463
  qed
paulson@14348
   464
qed
paulson@14348
   465
wenzelm@60758
   466
text\<open>Surely we can strengthen this? It holds for @{text "0<a<1"} too.\<close>
paulson@14348
   467
lemma power_inject_exp [simp]:
haftmann@30996
   468
  "1 < a \<Longrightarrow> a ^ m = a ^ n \<longleftrightarrow> m = n"
wenzelm@14577
   469
  by (force simp add: order_antisym power_le_imp_le_exp)
paulson@14348
   470
wenzelm@60758
   471
text\<open>Can relax the first premise to @{term "0<a"} in the case of the
wenzelm@60758
   472
natural numbers.\<close>
paulson@14348
   473
lemma power_less_imp_less_exp:
haftmann@30996
   474
  "1 < a \<Longrightarrow> a ^ m < a ^ n \<Longrightarrow> m < n"
haftmann@30996
   475
  by (simp add: order_less_le [of m n] less_le [of "a^m" "a^n"]
haftmann@30996
   476
    power_le_imp_le_exp)
paulson@14348
   477
paulson@14348
   478
lemma power_strict_mono [rule_format]:
haftmann@30996
   479
  "a < b \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 < n \<longrightarrow> a ^ n < b ^ n"
haftmann@30996
   480
  by (induct n)
haftmann@30996
   481
   (auto simp add: mult_strict_mono le_less_trans [of 0 a b])
paulson@14348
   482
wenzelm@60758
   483
text\<open>Lemma for @{text power_strict_decreasing}\<close>
paulson@14348
   484
lemma power_Suc_less:
haftmann@30996
   485
  "0 < a \<Longrightarrow> a < 1 \<Longrightarrow> a * a ^ n < a ^ n"
haftmann@30996
   486
  by (induct n)
haftmann@30996
   487
    (auto simp add: mult_strict_left_mono)
paulson@14348
   488
haftmann@30996
   489
lemma power_strict_decreasing [rule_format]:
haftmann@30996
   490
  "n < N \<Longrightarrow> 0 < a \<Longrightarrow> a < 1 \<longrightarrow> a ^ N < a ^ n"
haftmann@30996
   491
proof (induct N)
haftmann@30996
   492
  case 0 then show ?case by simp
haftmann@30996
   493
next
haftmann@30996
   494
  case (Suc N) then show ?case 
haftmann@30996
   495
  apply (auto simp add: power_Suc_less less_Suc_eq)
haftmann@30996
   496
  apply (subgoal_tac "a * a^N < 1 * a^n")
haftmann@30996
   497
  apply simp
haftmann@30996
   498
  apply (rule mult_strict_mono) apply auto
haftmann@30996
   499
  done
haftmann@30996
   500
qed
paulson@14348
   501
wenzelm@60758
   502
text\<open>Proof resembles that of @{text power_strict_decreasing}\<close>
haftmann@30996
   503
lemma power_decreasing [rule_format]:
haftmann@30996
   504
  "n \<le> N \<Longrightarrow> 0 \<le> a \<Longrightarrow> a \<le> 1 \<longrightarrow> a ^ N \<le> a ^ n"
haftmann@30996
   505
proof (induct N)
haftmann@30996
   506
  case 0 then show ?case by simp
haftmann@30996
   507
next
haftmann@30996
   508
  case (Suc N) then show ?case 
haftmann@30996
   509
  apply (auto simp add: le_Suc_eq)
haftmann@30996
   510
  apply (subgoal_tac "a * a^N \<le> 1 * a^n", simp)
haftmann@30996
   511
  apply (rule mult_mono) apply auto
haftmann@30996
   512
  done
haftmann@30996
   513
qed
paulson@14348
   514
paulson@14348
   515
lemma power_Suc_less_one:
haftmann@30996
   516
  "0 < a \<Longrightarrow> a < 1 \<Longrightarrow> a ^ Suc n < 1"
haftmann@30996
   517
  using power_strict_decreasing [of 0 "Suc n" a] by simp
paulson@14348
   518
wenzelm@60758
   519
text\<open>Proof again resembles that of @{text power_strict_decreasing}\<close>
haftmann@30996
   520
lemma power_increasing [rule_format]:
haftmann@30996
   521
  "n \<le> N \<Longrightarrow> 1 \<le> a \<Longrightarrow> a ^ n \<le> a ^ N"
haftmann@30996
   522
proof (induct N)
haftmann@30996
   523
  case 0 then show ?case by simp
haftmann@30996
   524
next
haftmann@30996
   525
  case (Suc N) then show ?case 
haftmann@30996
   526
  apply (auto simp add: le_Suc_eq)
haftmann@30996
   527
  apply (subgoal_tac "1 * a^n \<le> a * a^N", simp)
haftmann@30996
   528
  apply (rule mult_mono) apply (auto simp add: order_trans [OF zero_le_one])
haftmann@30996
   529
  done
haftmann@30996
   530
qed
paulson@14348
   531
wenzelm@60758
   532
text\<open>Lemma for @{text power_strict_increasing}\<close>
paulson@14348
   533
lemma power_less_power_Suc:
haftmann@30996
   534
  "1 < a \<Longrightarrow> a ^ n < a * a ^ n"
haftmann@30996
   535
  by (induct n) (auto simp add: mult_strict_left_mono less_trans [OF zero_less_one])
paulson@14348
   536
haftmann@30996
   537
lemma power_strict_increasing [rule_format]:
haftmann@30996
   538
  "n < N \<Longrightarrow> 1 < a \<longrightarrow> a ^ n < a ^ N"
haftmann@30996
   539
proof (induct N)
haftmann@30996
   540
  case 0 then show ?case by simp
haftmann@30996
   541
next
haftmann@30996
   542
  case (Suc N) then show ?case 
haftmann@30996
   543
  apply (auto simp add: power_less_power_Suc less_Suc_eq)
haftmann@30996
   544
  apply (subgoal_tac "1 * a^n < a * a^N", simp)
haftmann@30996
   545
  apply (rule mult_strict_mono) apply (auto simp add: less_trans [OF zero_less_one] less_imp_le)
haftmann@30996
   546
  done
haftmann@30996
   547
qed
paulson@14348
   548
nipkow@25134
   549
lemma power_increasing_iff [simp]:
haftmann@30996
   550
  "1 < b \<Longrightarrow> b ^ x \<le> b ^ y \<longleftrightarrow> x \<le> y"
haftmann@30996
   551
  by (blast intro: power_le_imp_le_exp power_increasing less_imp_le)
paulson@15066
   552
paulson@15066
   553
lemma power_strict_increasing_iff [simp]:
haftmann@30996
   554
  "1 < b \<Longrightarrow> b ^ x < b ^ y \<longleftrightarrow> x < y"
nipkow@25134
   555
by (blast intro: power_less_imp_less_exp power_strict_increasing) 
paulson@15066
   556
paulson@14348
   557
lemma power_le_imp_le_base:
haftmann@30996
   558
  assumes le: "a ^ Suc n \<le> b ^ Suc n"
haftmann@30996
   559
    and ynonneg: "0 \<le> b"
haftmann@30996
   560
  shows "a \<le> b"
nipkow@25134
   561
proof (rule ccontr)
nipkow@25134
   562
  assume "~ a \<le> b"
nipkow@25134
   563
  then have "b < a" by (simp only: linorder_not_le)
nipkow@25134
   564
  then have "b ^ Suc n < a ^ Suc n"
wenzelm@41550
   565
    by (simp only: assms power_strict_mono)
haftmann@30996
   566
  from le and this show False
nipkow@25134
   567
    by (simp add: linorder_not_less [symmetric])
nipkow@25134
   568
qed
wenzelm@14577
   569
huffman@22853
   570
lemma power_less_imp_less_base:
huffman@22853
   571
  assumes less: "a ^ n < b ^ n"
huffman@22853
   572
  assumes nonneg: "0 \<le> b"
huffman@22853
   573
  shows "a < b"
huffman@22853
   574
proof (rule contrapos_pp [OF less])
huffman@22853
   575
  assume "~ a < b"
huffman@22853
   576
  hence "b \<le> a" by (simp only: linorder_not_less)
huffman@22853
   577
  hence "b ^ n \<le> a ^ n" using nonneg by (rule power_mono)
haftmann@30996
   578
  thus "\<not> a ^ n < b ^ n" by (simp only: linorder_not_less)
huffman@22853
   579
qed
huffman@22853
   580
paulson@14348
   581
lemma power_inject_base:
haftmann@30996
   582
  "a ^ Suc n = b ^ Suc n \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> a = b"
haftmann@30996
   583
by (blast intro: power_le_imp_le_base antisym eq_refl sym)
paulson@14348
   584
huffman@22955
   585
lemma power_eq_imp_eq_base:
haftmann@30996
   586
  "a ^ n = b ^ n \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 < n \<Longrightarrow> a = b"
haftmann@30996
   587
  by (cases n) (simp_all del: power_Suc, rule power_inject_base)
huffman@22955
   588
huffman@47192
   589
lemma power2_le_imp_le:
wenzelm@53015
   590
  "x\<^sup>2 \<le> y\<^sup>2 \<Longrightarrow> 0 \<le> y \<Longrightarrow> x \<le> y"
huffman@47192
   591
  unfolding numeral_2_eq_2 by (rule power_le_imp_le_base)
huffman@47192
   592
huffman@47192
   593
lemma power2_less_imp_less:
wenzelm@53015
   594
  "x\<^sup>2 < y\<^sup>2 \<Longrightarrow> 0 \<le> y \<Longrightarrow> x < y"
huffman@47192
   595
  by (rule power_less_imp_less_base)
huffman@47192
   596
huffman@47192
   597
lemma power2_eq_imp_eq:
wenzelm@53015
   598
  "x\<^sup>2 = y\<^sup>2 \<Longrightarrow> 0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> x = y"
huffman@47192
   599
  unfolding numeral_2_eq_2 by (erule (2) power_eq_imp_eq_base) simp
huffman@47192
   600
huffman@47192
   601
end
huffman@47192
   602
huffman@47192
   603
context linordered_ring_strict
huffman@47192
   604
begin
huffman@47192
   605
huffman@47192
   606
lemma sum_squares_eq_zero_iff:
huffman@47192
   607
  "x * x + y * y = 0 \<longleftrightarrow> x = 0 \<and> y = 0"
huffman@47192
   608
  by (simp add: add_nonneg_eq_0_iff)
huffman@47192
   609
huffman@47192
   610
lemma sum_squares_le_zero_iff:
huffman@47192
   611
  "x * x + y * y \<le> 0 \<longleftrightarrow> x = 0 \<and> y = 0"
huffman@47192
   612
  by (simp add: le_less not_sum_squares_lt_zero sum_squares_eq_zero_iff)
huffman@47192
   613
huffman@47192
   614
lemma sum_squares_gt_zero_iff:
huffman@47192
   615
  "0 < x * x + y * y \<longleftrightarrow> x \<noteq> 0 \<or> y \<noteq> 0"
huffman@47192
   616
  by (simp add: not_le [symmetric] sum_squares_le_zero_iff)
huffman@47192
   617
haftmann@30996
   618
end
haftmann@30996
   619
haftmann@35028
   620
context linordered_idom
haftmann@30996
   621
begin
huffman@29978
   622
haftmann@30996
   623
lemma power_abs:
haftmann@30996
   624
  "abs (a ^ n) = abs a ^ n"
haftmann@30996
   625
  by (induct n) (auto simp add: abs_mult)
haftmann@30996
   626
haftmann@30996
   627
lemma abs_power_minus [simp]:
haftmann@30996
   628
  "abs ((-a) ^ n) = abs (a ^ n)"
huffman@35216
   629
  by (simp add: power_abs)
haftmann@30996
   630
blanchet@54147
   631
lemma zero_less_power_abs_iff [simp]:
haftmann@30996
   632
  "0 < abs a ^ n \<longleftrightarrow> a \<noteq> 0 \<or> n = 0"
haftmann@30996
   633
proof (induct n)
haftmann@30996
   634
  case 0 show ?case by simp
haftmann@30996
   635
next
haftmann@30996
   636
  case (Suc n) show ?case by (auto simp add: Suc zero_less_mult_iff)
huffman@29978
   637
qed
huffman@29978
   638
haftmann@30996
   639
lemma zero_le_power_abs [simp]:
haftmann@30996
   640
  "0 \<le> abs a ^ n"
haftmann@30996
   641
  by (rule zero_le_power [OF abs_ge_zero])
haftmann@30996
   642
huffman@47192
   643
lemma zero_le_power2 [simp]:
wenzelm@53015
   644
  "0 \<le> a\<^sup>2"
huffman@47192
   645
  by (simp add: power2_eq_square)
huffman@47192
   646
huffman@47192
   647
lemma zero_less_power2 [simp]:
wenzelm@53015
   648
  "0 < a\<^sup>2 \<longleftrightarrow> a \<noteq> 0"
huffman@47192
   649
  by (force simp add: power2_eq_square zero_less_mult_iff linorder_neq_iff)
huffman@47192
   650
huffman@47192
   651
lemma power2_less_0 [simp]:
wenzelm@53015
   652
  "\<not> a\<^sup>2 < 0"
huffman@47192
   653
  by (force simp add: power2_eq_square mult_less_0_iff)
huffman@47192
   654
haftmann@58787
   655
lemma power2_less_eq_zero_iff [simp]:
haftmann@58787
   656
  "a\<^sup>2 \<le> 0 \<longleftrightarrow> a = 0"
haftmann@58787
   657
  by (simp add: le_less)
haftmann@58787
   658
huffman@47192
   659
lemma abs_power2 [simp]:
wenzelm@53015
   660
  "abs (a\<^sup>2) = a\<^sup>2"
huffman@47192
   661
  by (simp add: power2_eq_square abs_mult abs_mult_self)
huffman@47192
   662
huffman@47192
   663
lemma power2_abs [simp]:
wenzelm@53015
   664
  "(abs a)\<^sup>2 = a\<^sup>2"
huffman@47192
   665
  by (simp add: power2_eq_square abs_mult_self)
huffman@47192
   666
huffman@47192
   667
lemma odd_power_less_zero:
huffman@47192
   668
  "a < 0 \<Longrightarrow> a ^ Suc (2*n) < 0"
huffman@47192
   669
proof (induct n)
huffman@47192
   670
  case 0
huffman@47192
   671
  then show ?case by simp
huffman@47192
   672
next
huffman@47192
   673
  case (Suc n)
huffman@47192
   674
  have "a ^ Suc (2 * Suc n) = (a*a) * a ^ Suc(2*n)"
haftmann@57514
   675
    by (simp add: ac_simps power_add power2_eq_square)
huffman@47192
   676
  thus ?case
huffman@47192
   677
    by (simp del: power_Suc add: Suc mult_less_0_iff mult_neg_neg)
huffman@47192
   678
qed
haftmann@30996
   679
huffman@47192
   680
lemma odd_0_le_power_imp_0_le:
huffman@47192
   681
  "0 \<le> a ^ Suc (2*n) \<Longrightarrow> 0 \<le> a"
huffman@47192
   682
  using odd_power_less_zero [of a n]
huffman@47192
   683
    by (force simp add: linorder_not_less [symmetric]) 
huffman@47192
   684
huffman@47192
   685
lemma zero_le_even_power'[simp]:
huffman@47192
   686
  "0 \<le> a ^ (2*n)"
huffman@47192
   687
proof (induct n)
huffman@47192
   688
  case 0
huffman@47192
   689
    show ?case by simp
huffman@47192
   690
next
huffman@47192
   691
  case (Suc n)
huffman@47192
   692
    have "a ^ (2 * Suc n) = (a*a) * a ^ (2*n)" 
haftmann@57514
   693
      by (simp add: ac_simps power_add power2_eq_square)
huffman@47192
   694
    thus ?case
huffman@47192
   695
      by (simp add: Suc zero_le_mult_iff)
huffman@47192
   696
qed
haftmann@30996
   697
huffman@47192
   698
lemma sum_power2_ge_zero:
wenzelm@53015
   699
  "0 \<le> x\<^sup>2 + y\<^sup>2"
huffman@47192
   700
  by (intro add_nonneg_nonneg zero_le_power2)
huffman@47192
   701
huffman@47192
   702
lemma not_sum_power2_lt_zero:
wenzelm@53015
   703
  "\<not> x\<^sup>2 + y\<^sup>2 < 0"
huffman@47192
   704
  unfolding not_less by (rule sum_power2_ge_zero)
huffman@47192
   705
huffman@47192
   706
lemma sum_power2_eq_zero_iff:
wenzelm@53015
   707
  "x\<^sup>2 + y\<^sup>2 = 0 \<longleftrightarrow> x = 0 \<and> y = 0"
huffman@47192
   708
  unfolding power2_eq_square by (simp add: add_nonneg_eq_0_iff)
huffman@47192
   709
huffman@47192
   710
lemma sum_power2_le_zero_iff:
wenzelm@53015
   711
  "x\<^sup>2 + y\<^sup>2 \<le> 0 \<longleftrightarrow> x = 0 \<and> y = 0"
huffman@47192
   712
  by (simp add: le_less sum_power2_eq_zero_iff not_sum_power2_lt_zero)
huffman@47192
   713
huffman@47192
   714
lemma sum_power2_gt_zero_iff:
wenzelm@53015
   715
  "0 < x\<^sup>2 + y\<^sup>2 \<longleftrightarrow> x \<noteq> 0 \<or> y \<noteq> 0"
huffman@47192
   716
  unfolding not_le [symmetric] by (simp add: sum_power2_le_zero_iff)
haftmann@30996
   717
lp15@59865
   718
lemma abs_le_square_iff:
lp15@59865
   719
   "\<bar>x\<bar> \<le> \<bar>y\<bar> \<longleftrightarrow> x\<^sup>2 \<le> y\<^sup>2"
lp15@59865
   720
proof
lp15@59865
   721
  assume "\<bar>x\<bar> \<le> \<bar>y\<bar>"
lp15@59865
   722
  then have "\<bar>x\<bar>\<^sup>2 \<le> \<bar>y\<bar>\<^sup>2" by (rule power_mono, simp)
lp15@59865
   723
  then show "x\<^sup>2 \<le> y\<^sup>2" by simp
lp15@59865
   724
next
lp15@59865
   725
  assume "x\<^sup>2 \<le> y\<^sup>2"
lp15@59865
   726
  then show "\<bar>x\<bar> \<le> \<bar>y\<bar>"
lp15@59865
   727
    by (auto intro!: power2_le_imp_le [OF _ abs_ge_zero])
lp15@59865
   728
qed
lp15@59865
   729
lp15@59865
   730
lemma abs_square_le_1:"x\<^sup>2 \<le> 1 \<longleftrightarrow> abs(x) \<le> 1"
lp15@59865
   731
  using abs_le_square_iff [of x 1]
lp15@59865
   732
  by simp
lp15@59865
   733
lp15@59865
   734
lemma abs_square_eq_1: "x\<^sup>2 = 1 \<longleftrightarrow> abs(x) = 1"
lp15@59865
   735
  by (auto simp add: abs_if power2_eq_1_iff)
lp15@59865
   736
  
lp15@59865
   737
lemma abs_square_less_1: "x\<^sup>2 < 1 \<longleftrightarrow> abs(x) < 1"
lp15@59865
   738
  using  abs_square_eq_1 [of x] abs_square_le_1 [of x]
lp15@59865
   739
  by (auto simp add: le_less)
lp15@59865
   740
haftmann@30996
   741
end
haftmann@30996
   742
huffman@29978
   743
wenzelm@60758
   744
subsection \<open>Miscellaneous rules\<close>
paulson@14348
   745
haftmann@60867
   746
lemma (in linordered_semidom) self_le_power:
haftmann@60867
   747
  "1 \<le> a \<Longrightarrow> 0 < n \<Longrightarrow> a \<le> a ^ n"
haftmann@60867
   748
  using power_increasing [of 1 n a] power_one_right [of a] by auto
lp15@55718
   749
haftmann@60867
   750
lemma (in power) power_eq_if:
haftmann@60867
   751
  "p ^ m = (if m=0 then 1 else p * (p ^ (m - 1)))"
huffman@47255
   752
  unfolding One_nat_def by (cases m) simp_all
huffman@47255
   753
haftmann@58787
   754
lemma (in comm_semiring_1) power2_sum:
haftmann@58787
   755
  "(x + y)\<^sup>2 = x\<^sup>2 + y\<^sup>2 + 2 * x * y"
huffman@47192
   756
  by (simp add: algebra_simps power2_eq_square mult_2_right)
haftmann@30996
   757
haftmann@58787
   758
lemma (in comm_ring_1) power2_diff:
haftmann@58787
   759
  "(x - y)\<^sup>2 = x\<^sup>2 + y\<^sup>2 - 2 * x * y"
haftmann@58787
   760
  by (simp add: algebra_simps power2_eq_square mult_2_right)
haftmann@30996
   761
lp15@60974
   762
lemma (in comm_ring_1) power2_commute:
lp15@60974
   763
  "(x - y)\<^sup>2 = (y - x)\<^sup>2"
lp15@60974
   764
  by (simp add: algebra_simps power2_eq_square)
lp15@60974
   765
nipkow@30313
   766
wenzelm@60758
   767
text \<open>Simprules for comparisons where common factors can be cancelled.\<close>
huffman@47255
   768
huffman@47255
   769
lemmas zero_compare_simps =
huffman@47255
   770
    add_strict_increasing add_strict_increasing2 add_increasing
huffman@47255
   771
    zero_le_mult_iff zero_le_divide_iff 
huffman@47255
   772
    zero_less_mult_iff zero_less_divide_iff 
huffman@47255
   773
    mult_le_0_iff divide_le_0_iff 
huffman@47255
   774
    mult_less_0_iff divide_less_0_iff 
huffman@47255
   775
    zero_le_power2 power2_less_0
huffman@47255
   776
nipkow@30313
   777
wenzelm@60758
   778
subsection \<open>Exponentiation for the Natural Numbers\<close>
wenzelm@14577
   779
haftmann@30996
   780
lemma nat_one_le_power [simp]:
haftmann@30996
   781
  "Suc 0 \<le> i \<Longrightarrow> Suc 0 \<le> i ^ n"
haftmann@30996
   782
  by (rule one_le_power [of i n, unfolded One_nat_def])
huffman@23305
   783
haftmann@30996
   784
lemma nat_zero_less_power_iff [simp]:
haftmann@30996
   785
  "x ^ n > 0 \<longleftrightarrow> x > (0::nat) \<or> n = 0"
haftmann@30996
   786
  by (induct n) auto
paulson@14348
   787
nipkow@30056
   788
lemma nat_power_eq_Suc_0_iff [simp]: 
haftmann@30996
   789
  "x ^ m = Suc 0 \<longleftrightarrow> m = 0 \<or> x = Suc 0"
haftmann@30996
   790
  by (induct m) auto
nipkow@30056
   791
haftmann@30996
   792
lemma power_Suc_0 [simp]:
haftmann@30996
   793
  "Suc 0 ^ n = Suc 0"
haftmann@30996
   794
  by simp
nipkow@30056
   795
wenzelm@60758
   796
text\<open>Valid for the naturals, but what if @{text"0<i<1"}?
paulson@14348
   797
Premises cannot be weakened: consider the case where @{term "i=0"},
wenzelm@60758
   798
@{term "m=1"} and @{term "n=0"}.\<close>
haftmann@21413
   799
lemma nat_power_less_imp_less:
wenzelm@61076
   800
  assumes nonneg: "0 < (i::nat)"
haftmann@30996
   801
  assumes less: "i ^ m < i ^ n"
haftmann@21413
   802
  shows "m < n"
haftmann@21413
   803
proof (cases "i = 1")
haftmann@21413
   804
  case True with less power_one [where 'a = nat] show ?thesis by simp
haftmann@21413
   805
next
haftmann@21413
   806
  case False with nonneg have "1 < i" by auto
haftmann@21413
   807
  from power_strict_increasing_iff [OF this] less show ?thesis ..
haftmann@21413
   808
qed
paulson@14348
   809
haftmann@33274
   810
lemma power_dvd_imp_le:
haftmann@33274
   811
  "i ^ m dvd i ^ n \<Longrightarrow> (1::nat) < i \<Longrightarrow> m \<le> n"
haftmann@33274
   812
  apply (rule power_le_imp_le_exp, assumption)
haftmann@33274
   813
  apply (erule dvd_imp_le, simp)
haftmann@33274
   814
  done
haftmann@33274
   815
haftmann@51263
   816
lemma power2_nat_le_eq_le:
haftmann@51263
   817
  fixes m n :: nat
wenzelm@53015
   818
  shows "m\<^sup>2 \<le> n\<^sup>2 \<longleftrightarrow> m \<le> n"
haftmann@51263
   819
  by (auto intro: power2_le_imp_le power_mono)
haftmann@51263
   820
haftmann@51263
   821
lemma power2_nat_le_imp_le:
haftmann@51263
   822
  fixes m n :: nat
wenzelm@53015
   823
  assumes "m\<^sup>2 \<le> n"
haftmann@51263
   824
  shows "m \<le> n"
haftmann@54249
   825
proof (cases m)
haftmann@54249
   826
  case 0 then show ?thesis by simp
haftmann@54249
   827
next
haftmann@54249
   828
  case (Suc k)
haftmann@54249
   829
  show ?thesis
haftmann@54249
   830
  proof (rule ccontr)
haftmann@54249
   831
    assume "\<not> m \<le> n"
haftmann@54249
   832
    then have "n < m" by simp
haftmann@54249
   833
    with assms Suc show False
haftmann@60867
   834
      by (simp add: power2_eq_square)
haftmann@54249
   835
  qed
haftmann@54249
   836
qed
haftmann@51263
   837
wenzelm@60758
   838
subsubsection \<open>Cardinality of the Powerset\<close>
traytel@55096
   839
traytel@55096
   840
lemma card_UNIV_bool [simp]: "card (UNIV :: bool set) = 2"
traytel@55096
   841
  unfolding UNIV_bool by simp
traytel@55096
   842
traytel@55096
   843
lemma card_Pow: "finite A \<Longrightarrow> card (Pow A) = 2 ^ card A"
traytel@55096
   844
proof (induct rule: finite_induct)
traytel@55096
   845
  case empty 
traytel@55096
   846
    show ?case by auto
traytel@55096
   847
next
traytel@55096
   848
  case (insert x A)
traytel@55096
   849
  then have "inj_on (insert x) (Pow A)" 
traytel@55096
   850
    unfolding inj_on_def by (blast elim!: equalityE)
traytel@55096
   851
  then have "card (Pow A) + card (insert x ` Pow A) = 2 * 2 ^ card A" 
traytel@55096
   852
    by (simp add: mult_2 card_image Pow_insert insert.hyps)
traytel@55096
   853
  then show ?case using insert
traytel@55096
   854
    apply (simp add: Pow_insert)
traytel@55096
   855
    apply (subst card_Un_disjoint, auto)
traytel@55096
   856
    done
traytel@55096
   857
qed
traytel@55096
   858
haftmann@57418
   859
wenzelm@60758
   860
subsubsection \<open>Generalized sum over a set\<close>
haftmann@57418
   861
haftmann@57418
   862
lemma setsum_zero_power [simp]:
haftmann@57418
   863
  fixes c :: "nat \<Rightarrow> 'a::division_ring"
haftmann@57418
   864
  shows "(\<Sum>i\<in>A. c i * 0^i) = (if finite A \<and> 0 \<in> A then c 0 else 0)"
haftmann@57418
   865
apply (cases "finite A")
haftmann@57418
   866
  by (induction A rule: finite_induct) auto
haftmann@57418
   867
haftmann@57418
   868
lemma setsum_zero_power' [simp]:
haftmann@57418
   869
  fixes c :: "nat \<Rightarrow> 'a::field"
haftmann@57418
   870
  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
   871
  using setsum_zero_power [of "\<lambda>i. c i / d i" A]
haftmann@57418
   872
  by auto
haftmann@57418
   873
haftmann@57418
   874
wenzelm@60758
   875
subsubsection \<open>Generalized product over a set\<close>
traytel@55096
   876
traytel@55096
   877
lemma setprod_constant: "finite A ==> (\<Prod>x\<in> A. (y::'a::{comm_monoid_mult})) = y^(card A)"
traytel@55096
   878
apply (erule finite_induct)
traytel@55096
   879
apply auto
traytel@55096
   880
done
traytel@55096
   881
haftmann@57418
   882
lemma setprod_power_distrib:
haftmann@57418
   883
  fixes f :: "'a \<Rightarrow> 'b::comm_semiring_1"
haftmann@57418
   884
  shows "setprod f A ^ n = setprod (\<lambda>x. (f x) ^ n) A"
haftmann@57418
   885
proof (cases "finite A") 
haftmann@57418
   886
  case True then show ?thesis 
haftmann@57418
   887
    by (induct A rule: finite_induct) (auto simp add: power_mult_distrib)
haftmann@57418
   888
next
haftmann@57418
   889
  case False then show ?thesis 
haftmann@57418
   890
    by simp
haftmann@57418
   891
qed
haftmann@57418
   892
haftmann@58437
   893
lemma power_setsum:
haftmann@58437
   894
  "c ^ (\<Sum>a\<in>A. f a) = (\<Prod>a\<in>A. c ^ f a)"
haftmann@58437
   895
  by (induct A rule: infinite_finite_induct) (simp_all add: power_add)
haftmann@58437
   896
traytel@55096
   897
lemma setprod_gen_delta:
traytel@55096
   898
  assumes fS: "finite S"
traytel@55096
   899
  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
   900
proof-
traytel@55096
   901
  let ?f = "(\<lambda>k. if k=a then b k else c)"
traytel@55096
   902
  {assume a: "a \<notin> S"
traytel@55096
   903
    hence "\<forall> k\<in> S. ?f k = c" by simp
traytel@55096
   904
    hence ?thesis  using a setprod_constant[OF fS, of c] by simp }
traytel@55096
   905
  moreover 
traytel@55096
   906
  {assume a: "a \<in> S"
traytel@55096
   907
    let ?A = "S - {a}"
traytel@55096
   908
    let ?B = "{a}"
traytel@55096
   909
    have eq: "S = ?A \<union> ?B" using a by blast 
traytel@55096
   910
    have dj: "?A \<inter> ?B = {}" by simp
traytel@55096
   911
    from fS have fAB: "finite ?A" "finite ?B" by auto  
traytel@55096
   912
    have fA0:"setprod ?f ?A = setprod (\<lambda>i. c) ?A"
haftmann@57418
   913
      apply (rule setprod.cong) by auto
traytel@55096
   914
    have cA: "card ?A = card S - 1" using fS a by auto
traytel@55096
   915
    have fA1: "setprod ?f ?A = c ^ card ?A"  unfolding fA0 apply (rule setprod_constant) using fS by auto
traytel@55096
   916
    have "setprod ?f ?A * setprod ?f ?B = setprod ?f S"
haftmann@57418
   917
      using setprod.union_disjoint[OF fAB dj, of ?f, unfolded eq[symmetric]]
traytel@55096
   918
      by simp
traytel@55096
   919
    then have ?thesis using a cA
haftmann@57418
   920
      by (simp add: fA1 field_simps cong add: setprod.cong cong del: if_weak_cong)}
traytel@55096
   921
  ultimately show ?thesis by blast
traytel@55096
   922
qed
traytel@55096
   923
wenzelm@60758
   924
subsection \<open>Code generator tweak\<close>
haftmann@31155
   925
haftmann@52435
   926
code_identifier
haftmann@52435
   927
  code_module Power \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
haftmann@33364
   928
paulson@3390
   929
end