src/HOL/Power.thy
author wenzelm
Sun Jan 29 13:43:17 2017 +0100 (2017-01-29)
changeset 64964 a0c985a57f7e
parent 64715 33d5fa0ce6e5
child 65057 799bbbb3a395
permissions -rw-r--r--
tuned proof;
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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
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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)
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  where
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    power_0: "a ^ 0 = 1"
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  | power_Suc: "a ^ Suc n = a * a ^ n"
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notation (latex output)
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  power ("(_\<^bsup>_\<^esup>)" [1000] 1000)
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text \<open>Special syntax for squares.\<close>
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abbreviation power2 :: "'a \<Rightarrow> 'a"  ("(_\<^sup>2)" [1000] 999)
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  where "x\<^sup>2 \<equiv> x ^ 2"
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end
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context monoid_mult
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begin
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subclass power .
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lemma power_one [simp]: "1 ^ n = 1"
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  by (induct n) simp_all
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lemma power_one_right [simp]: "a ^ 1 = a"
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  by simp
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lemma power_Suc0_right [simp]: "a ^ Suc 0 = a"
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  by simp
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lemma power_commutes: "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: "a ^ Suc n = a ^ n * a"
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  by (simp add: power_commutes)
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lemma power_add: "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: "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: "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: "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: "z ^ numeral (Num.Bit0 w) = (let w = z ^ (numeral w) in w * w)"
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  by (simp only: numeral_Bit0 power_add Let_def)
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lemma power_numeral_odd: "z ^ numeral (Num.Bit1 w) = (let w = z ^ (numeral w) in z * w * w)"
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  by (simp only: numeral_Bit1 One_nat_def add_Suc_right add_0_right
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      power_Suc power_add Let_def mult.assoc)
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lemma funpow_times_power: "(times x ^^ f x) = times (x ^ f x)"
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proof (induct "f x" arbitrary: f)
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  case 0
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  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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  define g where "g x = f x - 1" for x
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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
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    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 0
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  then show ?case by simp
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next
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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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    by (simp only: Suc power_Suc2) (simp add: ac_simps)
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  finally show ?case .
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qed
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lemma power_minus_mult: "0 < n \<Longrightarrow> a ^ (n - 1) * a = a ^ n"
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  by (simp add: power_commutes split: nat_diff_split)
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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]: "(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]: "a^numeral m * a^numeral n = a^numeral (m + n)"
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  for a :: "'a::monoid_mult"
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  by (simp add: power_add [symmetric])
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lemma power_add_numeral2 [simp]: "a^numeral m * (a^numeral n * b) = a^numeral (m + n) * b"
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  for a :: "'a::monoid_mult"
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  by (simp add: mult.assoc [symmetric])
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lemma power_mult_numeral [simp]: "(a^numeral m)^numeral n = a^numeral (m * n)"
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  for a :: "'a::monoid_mult"
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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)
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    (simp_all only: numeral_class.numeral.simps pow.simps
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      numeral_sqr numeral_mult power_add power_one_right)
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lemma power_numeral [simp]: "numeral k ^ numeral l = numeral (Num.pow k l)"
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  by (rule numeral_pow [symmetric])
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end
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context semiring_1
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begin
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lemma of_nat_power [simp]: "of_nat (m ^ n) = of_nat m ^ n"
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  by (induct n) simp_all
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lemma zero_power: "0 < n \<Longrightarrow> 0 ^ n = 0"
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  by (cases n) simp_all
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lemma power_zero_numeral [simp]: "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]: "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: "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"
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  shows "a ^ m dvd a ^ n"
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proof
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  from assms have "a ^ n = a ^ (m + (n - m))" by simp
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  also have "\<dots> = a ^ m * a ^ (n - m)" 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: "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: "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: "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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  fixes n :: nat
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  assumes "n > 0 \<or> x = 1"
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  shows "x dvd (x ^ n)"
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  using assms
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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 semiring_1_no_zero_divisors
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begin
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subclass power .
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lemma power_eq_0_iff [simp]: "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: "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]: "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 ring_1
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begin
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lemma power_minus: "(- a) ^ n = (- 1) ^ n * a ^ n"
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proof (induct n)
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  case 0
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  show ?case by simp
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next
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  case (Suc n)
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  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: "(- 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: "(- 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]: "(- a)\<^sup>2 = a\<^sup>2"
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  by (fact power_minus_Bit0)
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lemma power_minus1_even [simp]: "(- 1) ^ (2*n) = 1"
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proof (induct n)
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  case 0
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  show ?case by simp
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next
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  case (Suc n)
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  then show ?case by (simp add: power_add power2_eq_square)
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qed
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lemma power_minus1_odd: "(- 1) ^ Suc (2*n) = -1"
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  by simp
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lemma power_minus_even [simp]: "(-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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lemma power2_eq_1_iff: "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: "b dvd a \<Longrightarrow> (a div b) ^ n = a ^ n div b ^ n"
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  by (induct n) (simp_all add: div_mult_div_if_dvd dvd_power_same)
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lemma is_unit_power_iff: "is_unit (a ^ n) \<longleftrightarrow> is_unit a \<or> n = 0"
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  by (induct n) (auto simp add: is_unit_mult_iff)
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lemma dvd_power_iff:
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  assumes "x \<noteq> 0"
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  shows   "x ^ m dvd x ^ n \<longleftrightarrow> is_unit x \<or> m \<le> n"
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proof
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  assume *: "x ^ m dvd x ^ n"
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  {
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    assume "m > n"
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    note *
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    also have "x ^ n = x ^ n * 1" by simp
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    also from \<open>m > n\<close> have "m = n + (m - n)" by simp
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    also have "x ^ \<dots> = x ^ n * x ^ (m - n)" by (rule power_add)
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    finally have "x ^ (m - n) dvd 1"
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      by (subst (asm) dvd_times_left_cancel_iff) (insert assms, simp_all)
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    with \<open>m > n\<close> have "is_unit x" by (simp add: is_unit_power_iff)
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  }
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  thus "is_unit x \<or> m \<le> n" by force
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qed (auto intro: unit_imp_dvd simp: is_unit_power_iff le_imp_power_dvd)
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end
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context normalization_semidom
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begin
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lemma normalize_power: "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: "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>Perhaps these should be simprules.\<close>
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lemma power_inverse [field_simps, divide_simps]: "inverse a ^ n = inverse (a ^ n)"
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proof (cases "a = 0")
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  case True
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  then show ?thesis by (simp add: power_0_left)
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next
wenzelm@63654
   341
  case False
wenzelm@63654
   342
  then have "inverse (a ^ n) = inverse a ^ n"
haftmann@60867
   343
    by (induct n) (simp_all add: nonzero_inverse_mult_distrib power_commutes)
haftmann@60867
   344
  then show ?thesis by simp
haftmann@60867
   345
qed
huffman@47192
   346
wenzelm@63654
   347
lemma power_one_over [field_simps, divide_simps]: "(1 / a) ^ n = 1 / a ^ n"
haftmann@60867
   348
  using power_inverse [of a] by (simp add: divide_inverse)
haftmann@60867
   349
lp15@61649
   350
end
huffman@47192
   351
huffman@47192
   352
context field
huffman@47192
   353
begin
huffman@47192
   354
haftmann@60867
   355
lemma power_diff:
wenzelm@63654
   356
  assumes "a \<noteq> 0"
haftmann@60867
   357
  shows "n \<le> m \<Longrightarrow> a ^ (m - n) = a ^ m / a ^ n"
wenzelm@63654
   358
  by (induct m n rule: diff_induct) (simp_all add: assms power_not_zero)
huffman@47192
   359
wenzelm@63654
   360
lemma power_divide [field_simps, divide_simps]: "(a / b) ^ n = a ^ n / b ^ n"
haftmann@60867
   361
  by (induct n) simp_all
haftmann@60867
   362
huffman@47192
   363
end
huffman@47192
   364
huffman@47192
   365
wenzelm@60758
   366
subsection \<open>Exponentiation on ordered types\<close>
huffman@47192
   367
haftmann@35028
   368
context linordered_semidom
haftmann@30996
   369
begin
haftmann@30996
   370
wenzelm@63654
   371
lemma zero_less_power [simp]: "0 < a \<Longrightarrow> 0 < a ^ n"
nipkow@56544
   372
  by (induct n) simp_all
haftmann@30996
   373
wenzelm@63654
   374
lemma zero_le_power [simp]: "0 \<le> a \<Longrightarrow> 0 \<le> a ^ n"
nipkow@56536
   375
  by (induct n) simp_all
paulson@14348
   376
wenzelm@63654
   377
lemma power_mono: "a \<le> b \<Longrightarrow> 0 \<le> a \<Longrightarrow> a ^ n \<le> b ^ n"
huffman@47241
   378
  by (induct n) (auto intro: mult_mono order_trans [of 0 a b])
huffman@47241
   379
huffman@47241
   380
lemma one_le_power [simp]: "1 \<le> a \<Longrightarrow> 1 \<le> a ^ n"
huffman@47241
   381
  using power_mono [of 1 a n] by simp
huffman@47241
   382
wenzelm@63654
   383
lemma power_le_one: "0 \<le> a \<Longrightarrow> a \<le> 1 \<Longrightarrow> a ^ n \<le> 1"
huffman@47241
   384
  using power_mono [of a 1 n] by simp
paulson@14348
   385
paulson@14348
   386
lemma power_gt1_lemma:
haftmann@30996
   387
  assumes gt1: "1 < a"
haftmann@30996
   388
  shows "1 < a * a ^ n"
paulson@14348
   389
proof -
haftmann@30996
   390
  from gt1 have "0 \<le> a"
haftmann@30996
   391
    by (fact order_trans [OF zero_le_one less_imp_le])
wenzelm@63654
   392
  from gt1 have "1 * 1 < a * 1" by simp
wenzelm@63654
   393
  also from gt1 have "\<dots> \<le> a * a ^ n"
wenzelm@63654
   394
    by (simp only: mult_mono \<open>0 \<le> a\<close> one_le_power order_less_imp_le zero_le_one order_refl)
wenzelm@14577
   395
  finally show ?thesis by simp
paulson@14348
   396
qed
paulson@14348
   397
wenzelm@63654
   398
lemma power_gt1: "1 < a \<Longrightarrow> 1 < a ^ Suc n"
haftmann@30996
   399
  by (simp add: power_gt1_lemma)
huffman@24376
   400
wenzelm@63654
   401
lemma one_less_power [simp]: "1 < a \<Longrightarrow> 0 < n \<Longrightarrow> 1 < a ^ n"
haftmann@30996
   402
  by (cases n) (simp_all add: power_gt1_lemma)
paulson@14348
   403
paulson@14348
   404
lemma power_le_imp_le_exp:
haftmann@30996
   405
  assumes gt1: "1 < a"
haftmann@30996
   406
  shows "a ^ m \<le> a ^ n \<Longrightarrow> m \<le> n"
haftmann@30996
   407
proof (induct m arbitrary: n)
paulson@14348
   408
  case 0
wenzelm@14577
   409
  show ?case by simp
paulson@14348
   410
next
paulson@14348
   411
  case (Suc m)
wenzelm@14577
   412
  show ?case
wenzelm@14577
   413
  proof (cases n)
wenzelm@14577
   414
    case 0
wenzelm@63654
   415
    with Suc have "a * a ^ m \<le> 1" by simp
wenzelm@14577
   416
    with gt1 show ?thesis
wenzelm@63654
   417
      by (force simp only: power_gt1_lemma not_less [symmetric])
wenzelm@14577
   418
  next
wenzelm@14577
   419
    case (Suc n)
haftmann@30996
   420
    with Suc.prems Suc.hyps show ?thesis
wenzelm@63654
   421
      by (force dest: mult_left_le_imp_le simp add: less_trans [OF zero_less_one gt1])
wenzelm@14577
   422
  qed
paulson@14348
   423
qed
paulson@14348
   424
wenzelm@63654
   425
lemma of_nat_zero_less_power_iff [simp]: "of_nat x ^ n > 0 \<longleftrightarrow> x > 0 \<or> n = 0"
lp15@61649
   426
  by (induct n) auto
lp15@61649
   427
wenzelm@63654
   428
text \<open>Surely we can strengthen this? It holds for \<open>0<a<1\<close> too.\<close>
wenzelm@63654
   429
lemma power_inject_exp [simp]: "1 < a \<Longrightarrow> a ^ m = a ^ n \<longleftrightarrow> m = n"
wenzelm@14577
   430
  by (force simp add: order_antisym power_le_imp_le_exp)
paulson@14348
   431
wenzelm@63654
   432
text \<open>
wenzelm@63654
   433
  Can relax the first premise to @{term "0<a"} in the case of the
wenzelm@63654
   434
  natural numbers.
wenzelm@63654
   435
\<close>
wenzelm@63654
   436
lemma power_less_imp_less_exp: "1 < a \<Longrightarrow> a ^ m < a ^ n \<Longrightarrow> m < n"
wenzelm@63654
   437
  by (simp add: order_less_le [of m n] less_le [of "a^m" "a^n"] power_le_imp_le_exp)
paulson@14348
   438
wenzelm@63654
   439
lemma power_strict_mono [rule_format]: "a < b \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 < n \<longrightarrow> a ^ n < b ^ n"
wenzelm@63654
   440
  by (induct n) (auto simp: mult_strict_mono le_less_trans [of 0 a b])
paulson@14348
   441
wenzelm@61799
   442
text\<open>Lemma for \<open>power_strict_decreasing\<close>\<close>
wenzelm@63654
   443
lemma power_Suc_less: "0 < a \<Longrightarrow> a < 1 \<Longrightarrow> a * a ^ n < a ^ n"
wenzelm@63654
   444
  by (induct n) (auto simp: mult_strict_left_mono)
paulson@14348
   445
wenzelm@63654
   446
lemma power_strict_decreasing [rule_format]: "n < N \<Longrightarrow> 0 < a \<Longrightarrow> a < 1 \<longrightarrow> a ^ N < a ^ n"
haftmann@30996
   447
proof (induct N)
wenzelm@63654
   448
  case 0
wenzelm@63654
   449
  then show ?case by simp
haftmann@30996
   450
next
wenzelm@63654
   451
  case (Suc N)
wenzelm@63654
   452
  then show ?case
wenzelm@63654
   453
    apply (auto simp add: power_Suc_less less_Suc_eq)
wenzelm@63654
   454
    apply (subgoal_tac "a * a^N < 1 * a^n")
wenzelm@63654
   455
     apply simp
wenzelm@63654
   456
    apply (rule mult_strict_mono)
wenzelm@63654
   457
       apply auto
wenzelm@63654
   458
    done
haftmann@30996
   459
qed
paulson@14348
   460
wenzelm@63654
   461
text \<open>Proof resembles that of \<open>power_strict_decreasing\<close>.\<close>
wenzelm@63654
   462
lemma power_decreasing: "n \<le> N \<Longrightarrow> 0 \<le> a \<Longrightarrow> a \<le> 1 \<Longrightarrow> a ^ N \<le> a ^ n"
haftmann@30996
   463
proof (induct N)
wenzelm@63654
   464
  case 0
wenzelm@63654
   465
  then show ?case by simp
haftmann@30996
   466
next
wenzelm@63654
   467
  case (Suc N)
wenzelm@63654
   468
  then show ?case
wenzelm@63654
   469
    apply (auto simp add: le_Suc_eq)
wenzelm@63654
   470
    apply (subgoal_tac "a * a^N \<le> 1 * a^n")
wenzelm@63654
   471
     apply simp
wenzelm@63654
   472
    apply (rule mult_mono)
wenzelm@63654
   473
       apply auto
wenzelm@63654
   474
    done
haftmann@30996
   475
qed
paulson@14348
   476
wenzelm@63654
   477
lemma power_Suc_less_one: "0 < a \<Longrightarrow> a < 1 \<Longrightarrow> a ^ Suc n < 1"
haftmann@30996
   478
  using power_strict_decreasing [of 0 "Suc n" a] by simp
paulson@14348
   479
wenzelm@63654
   480
text \<open>Proof again resembles that of \<open>power_strict_decreasing\<close>.\<close>
wenzelm@63654
   481
lemma power_increasing: "n \<le> N \<Longrightarrow> 1 \<le> a \<Longrightarrow> a ^ n \<le> a ^ N"
haftmann@30996
   482
proof (induct N)
wenzelm@63654
   483
  case 0
wenzelm@63654
   484
  then show ?case by simp
haftmann@30996
   485
next
wenzelm@63654
   486
  case (Suc N)
wenzelm@63654
   487
  then show ?case
wenzelm@63654
   488
    apply (auto simp add: le_Suc_eq)
wenzelm@63654
   489
    apply (subgoal_tac "1 * a^n \<le> a * a^N")
wenzelm@63654
   490
     apply simp
wenzelm@63654
   491
    apply (rule mult_mono)
wenzelm@63654
   492
       apply (auto simp add: order_trans [OF zero_le_one])
wenzelm@63654
   493
    done
haftmann@30996
   494
qed
paulson@14348
   495
wenzelm@63654
   496
text \<open>Lemma for \<open>power_strict_increasing\<close>.\<close>
wenzelm@63654
   497
lemma power_less_power_Suc: "1 < a \<Longrightarrow> a ^ n < a * a ^ n"
wenzelm@63654
   498
  by (induct n) (auto simp: mult_strict_left_mono less_trans [OF zero_less_one])
paulson@14348
   499
wenzelm@63654
   500
lemma power_strict_increasing: "n < N \<Longrightarrow> 1 < a \<Longrightarrow> a ^ n < a ^ N"
haftmann@30996
   501
proof (induct N)
wenzelm@63654
   502
  case 0
wenzelm@63654
   503
  then show ?case by simp
haftmann@30996
   504
next
wenzelm@63654
   505
  case (Suc N)
wenzelm@63654
   506
  then show ?case
wenzelm@63654
   507
    apply (auto simp add: power_less_power_Suc less_Suc_eq)
wenzelm@63654
   508
    apply (subgoal_tac "1 * a^n < a * a^N")
wenzelm@63654
   509
     apply simp
wenzelm@63654
   510
    apply (rule mult_strict_mono)
wenzelm@63654
   511
    apply (auto simp add: less_trans [OF zero_less_one] less_imp_le)
wenzelm@63654
   512
    done
haftmann@30996
   513
qed
paulson@14348
   514
wenzelm@63654
   515
lemma power_increasing_iff [simp]: "1 < b \<Longrightarrow> b ^ x \<le> b ^ y \<longleftrightarrow> x \<le> y"
haftmann@30996
   516
  by (blast intro: power_le_imp_le_exp power_increasing less_imp_le)
paulson@15066
   517
wenzelm@63654
   518
lemma power_strict_increasing_iff [simp]: "1 < b \<Longrightarrow> b ^ x < b ^ y \<longleftrightarrow> x < y"
wenzelm@63654
   519
  by (blast intro: power_less_imp_less_exp power_strict_increasing)
paulson@15066
   520
paulson@14348
   521
lemma power_le_imp_le_base:
haftmann@30996
   522
  assumes le: "a ^ Suc n \<le> b ^ Suc n"
wenzelm@63654
   523
    and "0 \<le> b"
haftmann@30996
   524
  shows "a \<le> b"
nipkow@25134
   525
proof (rule ccontr)
wenzelm@63654
   526
  assume "\<not> ?thesis"
nipkow@25134
   527
  then have "b < a" by (simp only: linorder_not_le)
nipkow@25134
   528
  then have "b ^ Suc n < a ^ Suc n"
wenzelm@63654
   529
    by (simp only: assms(2) power_strict_mono)
wenzelm@63654
   530
  with le show False
nipkow@25134
   531
    by (simp add: linorder_not_less [symmetric])
nipkow@25134
   532
qed
wenzelm@14577
   533
huffman@22853
   534
lemma power_less_imp_less_base:
huffman@22853
   535
  assumes less: "a ^ n < b ^ n"
huffman@22853
   536
  assumes nonneg: "0 \<le> b"
huffman@22853
   537
  shows "a < b"
huffman@22853
   538
proof (rule contrapos_pp [OF less])
wenzelm@63654
   539
  assume "\<not> ?thesis"
wenzelm@63654
   540
  then have "b \<le> a" by (simp only: linorder_not_less)
wenzelm@63654
   541
  from this nonneg have "b ^ n \<le> a ^ n" by (rule power_mono)
wenzelm@63654
   542
  then show "\<not> a ^ n < b ^ n" by (simp only: linorder_not_less)
huffman@22853
   543
qed
huffman@22853
   544
wenzelm@63654
   545
lemma power_inject_base: "a ^ Suc n = b ^ Suc n \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> a = b"
wenzelm@63654
   546
  by (blast intro: power_le_imp_le_base antisym eq_refl sym)
paulson@14348
   547
wenzelm@63654
   548
lemma power_eq_imp_eq_base: "a ^ n = b ^ n \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 < n \<Longrightarrow> a = b"
haftmann@30996
   549
  by (cases n) (simp_all del: power_Suc, rule power_inject_base)
huffman@22955
   550
wenzelm@63654
   551
lemma power_eq_iff_eq_base: "0 < n \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> a ^ n = b ^ n \<longleftrightarrow> a = b"
haftmann@62347
   552
  using power_eq_imp_eq_base [of a n b] by auto
haftmann@62347
   553
wenzelm@63654
   554
lemma power2_le_imp_le: "x\<^sup>2 \<le> y\<^sup>2 \<Longrightarrow> 0 \<le> y \<Longrightarrow> x \<le> y"
huffman@47192
   555
  unfolding numeral_2_eq_2 by (rule power_le_imp_le_base)
huffman@47192
   556
wenzelm@63654
   557
lemma power2_less_imp_less: "x\<^sup>2 < y\<^sup>2 \<Longrightarrow> 0 \<le> y \<Longrightarrow> x < y"
huffman@47192
   558
  by (rule power_less_imp_less_base)
huffman@47192
   559
wenzelm@63654
   560
lemma power2_eq_imp_eq: "x\<^sup>2 = y\<^sup>2 \<Longrightarrow> 0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> x = y"
huffman@47192
   561
  unfolding numeral_2_eq_2 by (erule (2) power_eq_imp_eq_base) simp
huffman@47192
   562
wenzelm@63654
   563
lemma power_Suc_le_self: "0 \<le> a \<Longrightarrow> a \<le> 1 \<Longrightarrow> a ^ Suc n \<le> a"
haftmann@62347
   564
  using power_decreasing [of 1 "Suc n" a] by simp
haftmann@62347
   565
huffman@47192
   566
end
huffman@47192
   567
huffman@47192
   568
context linordered_ring_strict
huffman@47192
   569
begin
huffman@47192
   570
wenzelm@63654
   571
lemma sum_squares_eq_zero_iff: "x * x + y * y = 0 \<longleftrightarrow> x = 0 \<and> y = 0"
huffman@47192
   572
  by (simp add: add_nonneg_eq_0_iff)
huffman@47192
   573
wenzelm@63654
   574
lemma sum_squares_le_zero_iff: "x * x + y * y \<le> 0 \<longleftrightarrow> x = 0 \<and> y = 0"
huffman@47192
   575
  by (simp add: le_less not_sum_squares_lt_zero sum_squares_eq_zero_iff)
huffman@47192
   576
wenzelm@63654
   577
lemma sum_squares_gt_zero_iff: "0 < x * x + y * y \<longleftrightarrow> x \<noteq> 0 \<or> y \<noteq> 0"
huffman@47192
   578
  by (simp add: not_le [symmetric] sum_squares_le_zero_iff)
huffman@47192
   579
haftmann@30996
   580
end
haftmann@30996
   581
haftmann@35028
   582
context linordered_idom
haftmann@30996
   583
begin
huffman@29978
   584
haftmann@64715
   585
lemma zero_le_power2 [simp]: "0 \<le> a\<^sup>2"
haftmann@64715
   586
  by (simp add: power2_eq_square)
haftmann@64715
   587
haftmann@64715
   588
lemma zero_less_power2 [simp]: "0 < a\<^sup>2 \<longleftrightarrow> a \<noteq> 0"
haftmann@64715
   589
  by (force simp add: power2_eq_square zero_less_mult_iff linorder_neq_iff)
haftmann@30996
   590
haftmann@64715
   591
lemma power2_less_0 [simp]: "\<not> a\<^sup>2 < 0"
haftmann@64715
   592
  by (force simp add: power2_eq_square mult_less_0_iff)
haftmann@64715
   593
haftmann@64715
   594
lemma power_abs: "\<bar>a ^ n\<bar> = \<bar>a\<bar> ^ n" -- \<open>FIXME simp?\<close>
haftmann@64715
   595
  by (induct n) (simp_all add: abs_mult)
haftmann@64715
   596
haftmann@64715
   597
lemma power_sgn [simp]: "sgn (a ^ n) = sgn a ^ n"
haftmann@64715
   598
  by (induct n) (simp_all add: sgn_mult)
wenzelm@64964
   599
haftmann@64715
   600
lemma abs_power_minus [simp]: "\<bar>(- a) ^ n\<bar> = \<bar>a ^ n\<bar>"
huffman@35216
   601
  by (simp add: power_abs)
haftmann@30996
   602
wenzelm@61944
   603
lemma zero_less_power_abs_iff [simp]: "0 < \<bar>a\<bar> ^ n \<longleftrightarrow> a \<noteq> 0 \<or> n = 0"
haftmann@30996
   604
proof (induct n)
wenzelm@63654
   605
  case 0
wenzelm@63654
   606
  show ?case by simp
haftmann@30996
   607
next
wenzelm@63654
   608
  case Suc
wenzelm@63654
   609
  then show ?case by (auto simp: zero_less_mult_iff)
huffman@29978
   610
qed
huffman@29978
   611
wenzelm@61944
   612
lemma zero_le_power_abs [simp]: "0 \<le> \<bar>a\<bar> ^ n"
haftmann@30996
   613
  by (rule zero_le_power [OF abs_ge_zero])
haftmann@30996
   614
wenzelm@63654
   615
lemma power2_less_eq_zero_iff [simp]: "a\<^sup>2 \<le> 0 \<longleftrightarrow> a = 0"
haftmann@58787
   616
  by (simp add: le_less)
haftmann@58787
   617
wenzelm@61944
   618
lemma abs_power2 [simp]: "\<bar>a\<^sup>2\<bar> = a\<^sup>2"
haftmann@63417
   619
  by (simp add: power2_eq_square)
huffman@47192
   620
wenzelm@61944
   621
lemma power2_abs [simp]: "\<bar>a\<bar>\<^sup>2 = a\<^sup>2"
haftmann@63417
   622
  by (simp add: power2_eq_square)
huffman@47192
   623
haftmann@64715
   624
lemma odd_power_less_zero: "a < 0 \<Longrightarrow> a ^ Suc (2 * n) < 0"
huffman@47192
   625
proof (induct n)
huffman@47192
   626
  case 0
huffman@47192
   627
  then show ?case by simp
huffman@47192
   628
next
huffman@47192
   629
  case (Suc n)
huffman@47192
   630
  have "a ^ Suc (2 * Suc n) = (a*a) * a ^ Suc(2*n)"
haftmann@57514
   631
    by (simp add: ac_simps power_add power2_eq_square)
wenzelm@63654
   632
  then show ?case
huffman@47192
   633
    by (simp del: power_Suc add: Suc mult_less_0_iff mult_neg_neg)
huffman@47192
   634
qed
haftmann@30996
   635
haftmann@64715
   636
lemma odd_0_le_power_imp_0_le: "0 \<le> a ^ Suc (2 * n) \<Longrightarrow> 0 \<le> a"
huffman@47192
   637
  using odd_power_less_zero [of a n]
wenzelm@63654
   638
  by (force simp add: linorder_not_less [symmetric])
huffman@47192
   639
haftmann@64715
   640
lemma zero_le_even_power'[simp]: "0 \<le> a ^ (2 * n)"
huffman@47192
   641
proof (induct n)
huffman@47192
   642
  case 0
wenzelm@63654
   643
  show ?case by simp
huffman@47192
   644
next
huffman@47192
   645
  case (Suc n)
wenzelm@63654
   646
  have "a ^ (2 * Suc n) = (a*a) * a ^ (2*n)"
wenzelm@63654
   647
    by (simp add: ac_simps power_add power2_eq_square)
wenzelm@63654
   648
  then show ?case
wenzelm@63654
   649
    by (simp add: Suc zero_le_mult_iff)
huffman@47192
   650
qed
haftmann@30996
   651
wenzelm@63654
   652
lemma sum_power2_ge_zero: "0 \<le> x\<^sup>2 + y\<^sup>2"
huffman@47192
   653
  by (intro add_nonneg_nonneg zero_le_power2)
huffman@47192
   654
wenzelm@63654
   655
lemma not_sum_power2_lt_zero: "\<not> x\<^sup>2 + y\<^sup>2 < 0"
huffman@47192
   656
  unfolding not_less by (rule sum_power2_ge_zero)
huffman@47192
   657
wenzelm@63654
   658
lemma sum_power2_eq_zero_iff: "x\<^sup>2 + y\<^sup>2 = 0 \<longleftrightarrow> x = 0 \<and> y = 0"
huffman@47192
   659
  unfolding power2_eq_square by (simp add: add_nonneg_eq_0_iff)
huffman@47192
   660
wenzelm@63654
   661
lemma sum_power2_le_zero_iff: "x\<^sup>2 + y\<^sup>2 \<le> 0 \<longleftrightarrow> x = 0 \<and> y = 0"
huffman@47192
   662
  by (simp add: le_less sum_power2_eq_zero_iff not_sum_power2_lt_zero)
huffman@47192
   663
wenzelm@63654
   664
lemma sum_power2_gt_zero_iff: "0 < x\<^sup>2 + y\<^sup>2 \<longleftrightarrow> x \<noteq> 0 \<or> y \<noteq> 0"
huffman@47192
   665
  unfolding not_le [symmetric] by (simp add: sum_power2_le_zero_iff)
haftmann@30996
   666
wenzelm@63654
   667
lemma abs_le_square_iff: "\<bar>x\<bar> \<le> \<bar>y\<bar> \<longleftrightarrow> x\<^sup>2 \<le> y\<^sup>2"
wenzelm@63654
   668
  (is "?lhs \<longleftrightarrow> ?rhs")
lp15@59865
   669
proof
wenzelm@63654
   670
  assume ?lhs
wenzelm@63654
   671
  then have "\<bar>x\<bar>\<^sup>2 \<le> \<bar>y\<bar>\<^sup>2" by (rule power_mono) simp
wenzelm@63654
   672
  then show ?rhs by simp
lp15@59865
   673
next
wenzelm@63654
   674
  assume ?rhs
wenzelm@63654
   675
  then show ?lhs
lp15@59865
   676
    by (auto intro!: power2_le_imp_le [OF _ abs_ge_zero])
lp15@59865
   677
qed
lp15@59865
   678
wenzelm@61944
   679
lemma abs_square_le_1:"x\<^sup>2 \<le> 1 \<longleftrightarrow> \<bar>x\<bar> \<le> 1"
wenzelm@63654
   680
  using abs_le_square_iff [of x 1] by simp
lp15@59865
   681
wenzelm@61944
   682
lemma abs_square_eq_1: "x\<^sup>2 = 1 \<longleftrightarrow> \<bar>x\<bar> = 1"
lp15@59865
   683
  by (auto simp add: abs_if power2_eq_1_iff)
lp15@61649
   684
wenzelm@61944
   685
lemma abs_square_less_1: "x\<^sup>2 < 1 \<longleftrightarrow> \<bar>x\<bar> < 1"
wenzelm@63654
   686
  using  abs_square_eq_1 [of x] abs_square_le_1 [of x] by (auto simp add: le_less)
lp15@59865
   687
haftmann@30996
   688
end
haftmann@30996
   689
huffman@29978
   690
wenzelm@60758
   691
subsection \<open>Miscellaneous rules\<close>
paulson@14348
   692
wenzelm@63654
   693
lemma (in linordered_semidom) self_le_power: "1 \<le> a \<Longrightarrow> 0 < n \<Longrightarrow> a \<le> a ^ n"
haftmann@60867
   694
  using power_increasing [of 1 n a] power_one_right [of a] by auto
lp15@55718
   695
wenzelm@63654
   696
lemma (in power) power_eq_if: "p ^ m = (if m=0 then 1 else p * (p ^ (m - 1)))"
huffman@47255
   697
  unfolding One_nat_def by (cases m) simp_all
huffman@47255
   698
wenzelm@63654
   699
lemma (in comm_semiring_1) power2_sum: "(x + y)\<^sup>2 = x\<^sup>2 + y\<^sup>2 + 2 * x * y"
huffman@47192
   700
  by (simp add: algebra_simps power2_eq_square mult_2_right)
haftmann@30996
   701
wenzelm@63654
   702
context comm_ring_1
wenzelm@63654
   703
begin
wenzelm@63654
   704
wenzelm@63654
   705
lemma power2_diff: "(x - y)\<^sup>2 = x\<^sup>2 + y\<^sup>2 - 2 * x * y"
haftmann@58787
   706
  by (simp add: algebra_simps power2_eq_square mult_2_right)
haftmann@30996
   707
wenzelm@63654
   708
lemma power2_commute: "(x - y)\<^sup>2 = (y - x)\<^sup>2"
lp15@60974
   709
  by (simp add: algebra_simps power2_eq_square)
lp15@60974
   710
wenzelm@63654
   711
lemma minus_power_mult_self: "(- a) ^ n * (- a) ^ n = a ^ (2 * n)"
wenzelm@63654
   712
  by (simp add: power_mult_distrib [symmetric])
wenzelm@63654
   713
    (simp add: power2_eq_square [symmetric] power_mult [symmetric])
wenzelm@63654
   714
wenzelm@63654
   715
lemma minus_one_mult_self [simp]: "(- 1) ^ n * (- 1) ^ n = 1"
haftmann@63417
   716
  using minus_power_mult_self [of 1 n] by simp
haftmann@63417
   717
wenzelm@63654
   718
lemma left_minus_one_mult_self [simp]: "(- 1) ^ n * ((- 1) ^ n * a) = a"
haftmann@63417
   719
  by (simp add: mult.assoc [symmetric])
haftmann@63417
   720
wenzelm@63654
   721
end
wenzelm@63654
   722
wenzelm@60758
   723
text \<open>Simprules for comparisons where common factors can be cancelled.\<close>
huffman@47255
   724
huffman@47255
   725
lemmas zero_compare_simps =
wenzelm@63654
   726
  add_strict_increasing add_strict_increasing2 add_increasing
wenzelm@63654
   727
  zero_le_mult_iff zero_le_divide_iff
wenzelm@63654
   728
  zero_less_mult_iff zero_less_divide_iff
wenzelm@63654
   729
  mult_le_0_iff divide_le_0_iff
wenzelm@63654
   730
  mult_less_0_iff divide_less_0_iff
wenzelm@63654
   731
  zero_le_power2 power2_less_0
huffman@47255
   732
nipkow@30313
   733
wenzelm@60758
   734
subsection \<open>Exponentiation for the Natural Numbers\<close>
wenzelm@14577
   735
wenzelm@63654
   736
lemma nat_one_le_power [simp]: "Suc 0 \<le> i \<Longrightarrow> Suc 0 \<le> i ^ n"
haftmann@30996
   737
  by (rule one_le_power [of i n, unfolded One_nat_def])
huffman@23305
   738
wenzelm@63654
   739
lemma nat_zero_less_power_iff [simp]: "x ^ n > 0 \<longleftrightarrow> x > 0 \<or> n = 0"
wenzelm@63654
   740
  for x :: nat
haftmann@30996
   741
  by (induct n) auto
paulson@14348
   742
wenzelm@63654
   743
lemma nat_power_eq_Suc_0_iff [simp]: "x ^ m = Suc 0 \<longleftrightarrow> m = 0 \<or> x = Suc 0"
haftmann@30996
   744
  by (induct m) auto
nipkow@30056
   745
wenzelm@63654
   746
lemma power_Suc_0 [simp]: "Suc 0 ^ n = Suc 0"
haftmann@30996
   747
  by simp
nipkow@30056
   748
wenzelm@63654
   749
text \<open>
wenzelm@63654
   750
  Valid for the naturals, but what if \<open>0 < i < 1\<close>? Premises cannot be
wenzelm@63654
   751
  weakened: consider the case where \<open>i = 0\<close>, \<open>m = 1\<close> and \<open>n = 0\<close>.
wenzelm@63654
   752
\<close>
wenzelm@63654
   753
haftmann@21413
   754
lemma nat_power_less_imp_less:
wenzelm@63654
   755
  fixes i :: nat
wenzelm@63654
   756
  assumes nonneg: "0 < i"
haftmann@30996
   757
  assumes less: "i ^ m < i ^ n"
haftmann@21413
   758
  shows "m < n"
haftmann@21413
   759
proof (cases "i = 1")
wenzelm@63654
   760
  case True
wenzelm@63654
   761
  with less power_one [where 'a = nat] show ?thesis by simp
haftmann@21413
   762
next
wenzelm@63654
   763
  case False
wenzelm@63654
   764
  with nonneg have "1 < i" by auto
haftmann@21413
   765
  from power_strict_increasing_iff [OF this] less show ?thesis ..
haftmann@21413
   766
qed
paulson@14348
   767
wenzelm@63654
   768
lemma power_dvd_imp_le: "i ^ m dvd i ^ n \<Longrightarrow> 1 < i \<Longrightarrow> m \<le> n"
wenzelm@63654
   769
  for i m n :: nat
wenzelm@63654
   770
  apply (rule power_le_imp_le_exp)
wenzelm@63654
   771
   apply assumption
wenzelm@63654
   772
  apply (erule dvd_imp_le)
wenzelm@63654
   773
  apply simp
haftmann@33274
   774
  done
haftmann@33274
   775
wenzelm@63654
   776
lemma power2_nat_le_eq_le: "m\<^sup>2 \<le> n\<^sup>2 \<longleftrightarrow> m \<le> n"
wenzelm@63654
   777
  for m n :: nat
haftmann@51263
   778
  by (auto intro: power2_le_imp_le power_mono)
haftmann@51263
   779
haftmann@51263
   780
lemma power2_nat_le_imp_le:
haftmann@51263
   781
  fixes m n :: nat
wenzelm@53015
   782
  assumes "m\<^sup>2 \<le> n"
haftmann@51263
   783
  shows "m \<le> n"
haftmann@54249
   784
proof (cases m)
wenzelm@63654
   785
  case 0
wenzelm@63654
   786
  then show ?thesis by simp
haftmann@54249
   787
next
haftmann@54249
   788
  case (Suc k)
haftmann@54249
   789
  show ?thesis
haftmann@54249
   790
  proof (rule ccontr)
wenzelm@63654
   791
    assume "\<not> ?thesis"
haftmann@54249
   792
    then have "n < m" by simp
haftmann@54249
   793
    with assms Suc show False
haftmann@60867
   794
      by (simp add: power2_eq_square)
haftmann@54249
   795
  qed
haftmann@54249
   796
qed
haftmann@51263
   797
nipkow@64065
   798
lemma ex_power_ivl1: fixes b k :: nat assumes "b \<ge> 2"
nipkow@64065
   799
shows "k \<ge> 1 \<Longrightarrow> \<exists>n. b^n \<le> k \<and> k < b^(n+1)" (is "_ \<Longrightarrow> \<exists>n. ?P k n")
nipkow@64065
   800
proof(induction k)
nipkow@64065
   801
  case 0 thus ?case by simp
nipkow@64065
   802
next
nipkow@64065
   803
  case (Suc k)
nipkow@64065
   804
  show ?case
nipkow@64065
   805
  proof cases
nipkow@64065
   806
    assume "k=0"
nipkow@64065
   807
    hence "?P (Suc k) 0" using assms by simp
nipkow@64065
   808
    thus ?case ..
nipkow@64065
   809
  next
nipkow@64065
   810
    assume "k\<noteq>0"
nipkow@64065
   811
    with Suc obtain n where IH: "?P k n" by auto
nipkow@64065
   812
    show ?case
nipkow@64065
   813
    proof (cases "k = b^(n+1) - 1")
nipkow@64065
   814
      case True
nipkow@64065
   815
      hence "?P (Suc k) (n+1)" using assms
nipkow@64065
   816
        by (simp add: power_less_power_Suc)
nipkow@64065
   817
      thus ?thesis ..
nipkow@64065
   818
    next
nipkow@64065
   819
      case False
nipkow@64065
   820
      hence "?P (Suc k) n" using IH by auto
nipkow@64065
   821
      thus ?thesis ..
nipkow@64065
   822
    qed
nipkow@64065
   823
  qed
nipkow@64065
   824
qed
nipkow@64065
   825
nipkow@64065
   826
lemma ex_power_ivl2: fixes b k :: nat assumes "b \<ge> 2" "k \<ge> 2"
nipkow@64065
   827
shows "\<exists>n. b^n < k \<and> k \<le> b^(n+1)"
nipkow@64065
   828
proof -
nipkow@64065
   829
  have "1 \<le> k - 1" using assms(2) by arith
nipkow@64065
   830
  from ex_power_ivl1[OF assms(1) this]
nipkow@64065
   831
  obtain n where "b ^ n \<le> k - 1 \<and> k - 1 < b ^ (n + 1)" ..
nipkow@64065
   832
  hence "b^n < k \<and> k \<le> b^(n+1)" using assms by auto
nipkow@64065
   833
  thus ?thesis ..
nipkow@64065
   834
qed
nipkow@64065
   835
wenzelm@63654
   836
wenzelm@60758
   837
subsubsection \<open>Cardinality of the Powerset\<close>
traytel@55096
   838
traytel@55096
   839
lemma card_UNIV_bool [simp]: "card (UNIV :: bool set) = 2"
traytel@55096
   840
  unfolding UNIV_bool by simp
traytel@55096
   841
traytel@55096
   842
lemma card_Pow: "finite A \<Longrightarrow> card (Pow A) = 2 ^ card A"
traytel@55096
   843
proof (induct rule: finite_induct)
lp15@61649
   844
  case empty
wenzelm@64964
   845
  show ?case by simp
traytel@55096
   846
next
traytel@55096
   847
  case (insert x A)
wenzelm@64964
   848
  from \<open>x \<notin> A\<close> have disjoint: "Pow A \<inter> insert x ` Pow A = {}" by blast
wenzelm@64964
   849
  from \<open>x \<notin> A\<close> have inj_on: "inj_on (insert x) (Pow A)"
wenzelm@64964
   850
    unfolding inj_on_def by auto
wenzelm@64964
   851
wenzelm@64964
   852
  have "card (Pow (insert x A)) = card (Pow A \<union> insert x ` Pow A)"
wenzelm@64964
   853
    by (simp only: Pow_insert)
wenzelm@64964
   854
  also have "\<dots> = card (Pow A) + card (insert x ` Pow A)"
wenzelm@64964
   855
    by (rule card_Un_disjoint) (use \<open>finite A\<close> disjoint in simp_all)
wenzelm@64964
   856
  also from inj_on have "card (insert x ` Pow A) = card (Pow A)"
wenzelm@64964
   857
    by (rule card_image)
wenzelm@64964
   858
  also have "\<dots> + \<dots> = 2 * \<dots>" by (simp add: mult_2)
wenzelm@64964
   859
  also from insert(3) have "\<dots> = 2 ^ Suc (card A)" by simp
wenzelm@64964
   860
  also from insert(1,2) have "Suc (card A) = card (insert x A)"
wenzelm@64964
   861
    by (rule card_insert_disjoint [symmetric])
wenzelm@64964
   862
  finally show ?case .
traytel@55096
   863
qed
traytel@55096
   864
haftmann@57418
   865
wenzelm@60758
   866
subsection \<open>Code generator tweak\<close>
haftmann@31155
   867
haftmann@52435
   868
code_identifier
haftmann@52435
   869
  code_module Power \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
haftmann@33364
   870
paulson@3390
   871
end