src/HOL/Power.thy
author haftmann
Wed, 17 Feb 2016 21:51:57 +0100
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generalized some lemmas; moved some lemmas in more appropriate places; deleted potentially dangerous simp rule
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(*  Title:      HOL/Power.thy
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    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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    Copyright   1997  University of Cambridge
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*)
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section \<open>Exponentiation\<close>
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theory Power
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imports Num Equiv_Relations
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begin
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subsection \<open>Powers for Arbitrary Monoids\<close>
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class power = one + times
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begin
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primrec power :: "'a \<Rightarrow> nat \<Rightarrow> 'a"  (infixr "^" 80)
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where
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  power_0: "a ^ 0 = 1"
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| power_Suc: "a ^ Suc n = a * a ^ n"
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notation (latex output)
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  power ("(_\<^bsup>_\<^esup>)" [1000] 1000)
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text \<open>Special syntax for squares.\<close>
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abbreviation power2 :: "'a \<Rightarrow> 'a"  ("(_\<^sup>2)" [1000] 999)
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  where "x\<^sup>2 \<equiv> x ^ 2"
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end
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context monoid_mult
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begin
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subclass power .
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lemma power_one [simp]:
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  "1 ^ n = 1"
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  by (induct n) simp_all
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lemma power_one_right [simp]:
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  "a ^ 1 = a"
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  by simp
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lemma power_Suc0_right [simp]:
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  "a ^ Suc 0 = a"
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  by simp
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lemma power_commutes:
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  "a ^ n * a = a * a ^ n"
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  by (induct n) (simp_all add: mult.assoc)
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lemma power_Suc2:
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  "a ^ Suc n = a ^ n * a"
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  by (simp add: power_commutes)
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lemma power_add:
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  "a ^ (m + n) = a ^ m * a ^ n"
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  by (induct m) (simp_all add: algebra_simps)
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lemma power_mult:
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  "a ^ (m * n) = (a ^ m) ^ n"
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  by (induct n) (simp_all add: power_add)
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lemma power2_eq_square: "a\<^sup>2 = a * a"
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  by (simp add: numeral_2_eq_2)
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lemma power3_eq_cube: "a ^ 3 = a * a * a"
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  by (simp add: numeral_3_eq_3 mult.assoc)
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lemma power_even_eq:
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  "a ^ (2 * n) = (a ^ n)\<^sup>2"
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  by (subst mult.commute) (simp add: power_mult)
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lemma power_odd_eq:
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  "a ^ Suc (2*n) = a * (a ^ n)\<^sup>2"
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  by (simp add: power_even_eq)
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lemma power_numeral_even:
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  "z ^ numeral (Num.Bit0 w) = (let w = z ^ (numeral w) in w * w)"
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  unfolding numeral_Bit0 power_add Let_def ..
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lemma power_numeral_odd:
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  "z ^ numeral (Num.Bit1 w) = (let w = z ^ (numeral w) in z * w * w)"
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  unfolding numeral_Bit1 One_nat_def add_Suc_right add_0_right
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  unfolding power_Suc power_add Let_def mult.assoc ..
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lemma funpow_times_power:
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  "(times x ^^ f x) = times (x ^ f x)"
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proof (induct "f x" arbitrary: f)
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  case 0 then show ?case by (simp add: fun_eq_iff)
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next
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  case (Suc n)
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  def g \<equiv> "\<lambda>x. f x - 1"
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  with Suc have "n = g x" by simp
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  with Suc have "times x ^^ g x = times (x ^ g x)" by simp
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  moreover from Suc g_def have "f x = g x + 1" by simp
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  ultimately show ?case by (simp add: power_add funpow_add fun_eq_iff mult.assoc)
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qed
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lemma power_commuting_commutes:
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  assumes "x * y = y * x"
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  shows "x ^ n * y = y * x ^n"
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proof (induct n)
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  case (Suc n)
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  have "x ^ Suc n * y = x ^ n * y * x"
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    by (subst power_Suc2) (simp add: assms ac_simps)
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  also have "\<dots> = y * x ^ Suc n"
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    unfolding Suc power_Suc2
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    by (simp add: ac_simps)
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  finally show ?case .
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qed simp
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lemma power_minus_mult:
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  "0 < n \<Longrightarrow> a ^ (n - 1) * a = a ^ n"
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  by (simp add: power_commutes split add: 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]:
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  "(a * b) ^ n = (a ^ n) * (b ^ n)"
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  by (induct n) (simp_all add: ac_simps)
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end
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text\<open>Extract constant factors from powers\<close>
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declare power_mult_distrib [where a = "numeral w" for w, simp]
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declare power_mult_distrib [where b = "numeral w" for w, simp]
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lemma power_add_numeral [simp]:
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  fixes a :: "'a :: monoid_mult"
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  shows "a^numeral m * a^numeral n = a^numeral (m + n)"
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  by (simp add: power_add [symmetric])
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lemma power_add_numeral2 [simp]:
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  fixes a :: "'a :: monoid_mult"
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  shows "a^numeral m * (a^numeral n * b) = a^numeral (m + n) * b"
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  by (simp add: mult.assoc [symmetric])
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lemma power_mult_numeral [simp]:
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  fixes a :: "'a :: monoid_mult"
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  shows"(a^numeral m)^numeral n = a^numeral (m * n)"
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  by (simp only: numeral_mult power_mult)
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context semiring_numeral
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begin
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lemma numeral_sqr: "numeral (Num.sqr k) = numeral k * numeral k"
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  by (simp only: sqr_conv_mult numeral_mult)
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lemma numeral_pow: "numeral (Num.pow k l) = numeral k ^ numeral l"
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  by (induct l, simp_all only: numeral_class.numeral.simps pow.simps
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    numeral_sqr numeral_mult power_add power_one_right)
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lemma power_numeral [simp]: "numeral k ^ numeral l = numeral (Num.pow k l)"
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  by (rule numeral_pow [symmetric])
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end
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context semiring_1
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begin
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lemma of_nat_power [simp]:
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  "of_nat (m ^ n) = of_nat m ^ n"
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  by (induct n) (simp_all add: of_nat_mult)
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lemma zero_power:
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  "0 < n \<Longrightarrow> 0 ^ n = 0"
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  by (cases n) simp_all
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lemma power_zero_numeral [simp]:
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  "0 ^ numeral k = 0"
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  by (simp add: numeral_eq_Suc)
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lemma zero_power2: "0\<^sup>2 = 0" (* delete? *)
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  by (rule power_zero_numeral)
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lemma one_power2: "1\<^sup>2 = 1" (* delete? *)
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  by (rule power_one)
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lemma power_0_Suc [simp]:
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  "0 ^ Suc n = 0"
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  by simp
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text\<open>It looks plausible as a simprule, but its effect can be strange.\<close>
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lemma power_0_left:
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  "0 ^ n = (if n = 0 then 1 else 0)"
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  by (cases n) simp_all
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end
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context comm_semiring_1
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begin
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text \<open>The divides relation\<close>
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lemma le_imp_power_dvd:
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  assumes "m \<le> n" shows "a ^ m dvd a ^ n"
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proof
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  have "a ^ n = a ^ (m + (n - m))"
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    using \<open>m \<le> n\<close> by simp
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  also have "\<dots> = a ^ m * a ^ (n - m)"
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    by (rule power_add)
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  finally show "a ^ n = a ^ m * a ^ (n - m)" .
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qed
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lemma power_le_dvd:
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  "a ^ n dvd b \<Longrightarrow> m \<le> n \<Longrightarrow> a ^ m dvd b"
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  by (rule dvd_trans [OF le_imp_power_dvd])
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lemma dvd_power_same:
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  "x dvd y \<Longrightarrow> x ^ n dvd y ^ n"
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  by (induct n) (auto simp add: mult_dvd_mono)
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lemma dvd_power_le:
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  "x dvd y \<Longrightarrow> m \<ge> n \<Longrightarrow> x ^ n dvd y ^ m"
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  by (rule power_le_dvd [OF dvd_power_same])
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
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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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   228
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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   243
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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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   247
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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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   251
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end
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   253
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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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   258
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end
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context ring_1
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   262
begin
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   263
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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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   267
  case 0 show ?case by simp
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   268
next
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  case (Suc n) then show ?case
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
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    by (simp del: power_Suc add: power_Suc2 mult.assoc)
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   271
qed
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   272
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ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
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   273
lemma power_minus': "NO_MATCH 1 x \<Longrightarrow> (-x) ^ n = (-1)^n * x ^ n"
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eberlm
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  by (rule power_minus)
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eberlm
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   275
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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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   278
  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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ebd8c46d156b bootstrap Num.thy before Power.thy;
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lemma power_minus_Bit1:
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  "(- x) ^ numeral (Num.Bit1 k) = - (x ^ numeral (Num.Bit1 k))"
47220
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   283
  by (simp only: eval_nat_numeral(3) power_Suc power_minus_Bit0 mult_minus_left)
47191
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   284
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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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   288
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
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   289
lemma power_minus1_even [simp]:
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  "(- 1) ^ (2*n) = 1"
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   291
proof (induct n)
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   292
  case 0 show ?case by simp
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   293
next
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   294
  case (Suc n) then show ?case by (simp add: power_add power2_eq_square)
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   295
qed
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   296
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
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lemma power_minus1_odd:
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   298
  "(- 1) ^ Suc (2*n) = -1"
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   299
  by simp
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268d88ec9087 Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
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   300
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   301
lemma power_minus_even [simp]:
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  "(-a) ^ (2*n) = a ^ (2*n)"
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   303
  by (simp add: power_minus [of a])
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   304
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
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   305
end
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diff changeset
   306
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   307
context ring_1_no_zero_divisors
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   308
begin
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   309
61649
268d88ec9087 Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents: 61531
diff changeset
   310
subclass semiring_1_no_zero_divisors ..
47192
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
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parents: 47191
diff changeset
   311
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   312
lemma power2_eq_1_iff:
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 52435
diff changeset
   313
  "a\<^sup>2 = 1 \<longleftrightarrow> a = 1 \<or> a = - 1"
60867
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60866
diff changeset
   314
  using square_eq_1_iff [of a] by (simp add: power2_eq_square)
47192
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   315
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   316
end
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   317
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   318
context idom
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   319
begin
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   320
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 52435
diff changeset
   321
lemma power2_eq_iff: "x\<^sup>2 = y\<^sup>2 \<longleftrightarrow> x = y \<or> x = - y"
47192
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   322
  unfolding power2_eq_square by (rule square_eq_iff)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   323
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   324
end
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   325
60867
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60866
diff changeset
   326
context algebraic_semidom
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60866
diff changeset
   327
begin
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60866
diff changeset
   328
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60866
diff changeset
   329
lemma div_power:
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60866
diff changeset
   330
  assumes "b dvd a"
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60866
diff changeset
   331
  shows "(a div b) ^ n = a ^ n div b ^ n"
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60866
diff changeset
   332
  using assms by (induct n) (simp_all add: div_mult_div_if_dvd dvd_power_same)
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60866
diff changeset
   333
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60866
diff changeset
   334
end
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60866
diff changeset
   335
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60155
diff changeset
   336
context normalization_semidom
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60155
diff changeset
   337
begin
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60155
diff changeset
   338
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60155
diff changeset
   339
lemma normalize_power:
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60155
diff changeset
   340
  "normalize (a ^ n) = normalize a ^ n"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60155
diff changeset
   341
  by (induct n) (simp_all add: normalize_mult)
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60155
diff changeset
   342
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60155
diff changeset
   343
lemma unit_factor_power:
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60155
diff changeset
   344
  "unit_factor (a ^ n) = unit_factor a ^ n"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60155
diff changeset
   345
  by (induct n) (simp_all add: unit_factor_mult)
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60155
diff changeset
   346
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60155
diff changeset
   347
end
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60155
diff changeset
   348
47192
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   349
context division_ring
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   350
begin
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   351
60867
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60866
diff changeset
   352
text\<open>Perhaps these should be simprules.\<close>
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60866
diff changeset
   353
lemma power_inverse [field_simps, divide_simps]:
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60866
diff changeset
   354
  "inverse a ^ n = inverse (a ^ n)"
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60866
diff changeset
   355
proof (cases "a = 0")
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60866
diff changeset
   356
  case True then show ?thesis by (simp add: power_0_left)
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60866
diff changeset
   357
next
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60866
diff changeset
   358
  case False then have "inverse (a ^ n) = inverse a ^ n"
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60866
diff changeset
   359
    by (induct n) (simp_all add: nonzero_inverse_mult_distrib power_commutes)
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60866
diff changeset
   360
  then show ?thesis by simp
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60866
diff changeset
   361
qed
47192
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   362
60867
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60866
diff changeset
   363
lemma power_one_over [field_simps, divide_simps]:
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60866
diff changeset
   364
  "(1 / a) ^ n = 1 / a ^ n"
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60866
diff changeset
   365
  using power_inverse [of a] by (simp add: divide_inverse)
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60866
diff changeset
   366
61649
268d88ec9087 Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents: 61531
diff changeset
   367
end
47192
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   368
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   369
context field
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   370
begin
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   371
60867
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60866
diff changeset
   372
lemma power_diff:
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60866
diff changeset
   373
  assumes nz: "a \<noteq> 0"
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60866
diff changeset
   374
  shows "n \<le> m \<Longrightarrow> a ^ (m - n) = a ^ m / a ^ n"
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60866
diff changeset
   375
  by (induct m n rule: diff_induct) (simp_all add: nz power_not_zero)
47192
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   376
60867
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60866
diff changeset
   377
lemma power_divide [field_simps, divide_simps]:
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60866
diff changeset
   378
  "(a / b) ^ n = a ^ n / b ^ n"
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60866
diff changeset
   379
  by (induct n) simp_all
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60866
diff changeset
   380
47192
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   381
end
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   382
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   383
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60685
diff changeset
   384
subsection \<open>Exponentiation on ordered types\<close>
47192
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   385
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 33364
diff changeset
   386
context linordered_semidom
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   387
begin
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   388
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   389
lemma zero_less_power [simp]:
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   390
  "0 < a \<Longrightarrow> 0 < a ^ n"
56544
b60d5d119489 made mult_pos_pos a simp rule
nipkow
parents: 56536
diff changeset
   391
  by (induct n) simp_all
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   392
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   393
lemma zero_le_power [simp]:
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   394
  "0 \<le> a \<Longrightarrow> 0 \<le> a ^ n"
56536
aefb4a8da31f made mult_nonneg_nonneg a simp rule
nipkow
parents: 56481
diff changeset
   395
  by (induct n) simp_all
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   396
47241
243b33052e34 add lemma power_le_one
huffman
parents: 47220
diff changeset
   397
lemma power_mono:
243b33052e34 add lemma power_le_one
huffman
parents: 47220
diff changeset
   398
  "a \<le> b \<Longrightarrow> 0 \<le> a \<Longrightarrow> a ^ n \<le> b ^ n"
243b33052e34 add lemma power_le_one
huffman
parents: 47220
diff changeset
   399
  by (induct n) (auto intro: mult_mono order_trans [of 0 a b])
243b33052e34 add lemma power_le_one
huffman
parents: 47220
diff changeset
   400
243b33052e34 add lemma power_le_one
huffman
parents: 47220
diff changeset
   401
lemma one_le_power [simp]: "1 \<le> a \<Longrightarrow> 1 \<le> a ^ n"
243b33052e34 add lemma power_le_one
huffman
parents: 47220
diff changeset
   402
  using power_mono [of 1 a n] by simp
243b33052e34 add lemma power_le_one
huffman
parents: 47220
diff changeset
   403
243b33052e34 add lemma power_le_one
huffman
parents: 47220
diff changeset
   404
lemma power_le_one: "\<lbrakk>0 \<le> a; a \<le> 1\<rbrakk> \<Longrightarrow> a ^ n \<le> 1"
243b33052e34 add lemma power_le_one
huffman
parents: 47220
diff changeset
   405
  using power_mono [of a 1 n] by simp
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   406
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   407
lemma power_gt1_lemma:
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   408
  assumes gt1: "1 < a"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   409
  shows "1 < a * a ^ n"
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   410
proof -
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   411
  from gt1 have "0 \<le> a"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   412
    by (fact order_trans [OF zero_le_one less_imp_le])
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   413
  have "1 * 1 < a * 1" using gt1 by simp
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   414
  also have "\<dots> \<le> a * a ^ n" using gt1
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60685
diff changeset
   415
    by (simp only: mult_mono \<open>0 \<le> a\<close> one_le_power order_less_imp_le
14577
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
   416
        zero_le_one order_refl)
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
   417
  finally show ?thesis by simp
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   418
qed
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   419
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   420
lemma power_gt1:
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   421
  "1 < a \<Longrightarrow> 1 < a ^ Suc n"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   422
  by (simp add: power_gt1_lemma)
24376
e403ab5c9415 add lemma one_less_power
huffman
parents: 24286
diff changeset
   423
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   424
lemma one_less_power [simp]:
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   425
  "1 < a \<Longrightarrow> 0 < n \<Longrightarrow> 1 < a ^ n"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   426
  by (cases n) (simp_all add: power_gt1_lemma)
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   427
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   428
lemma power_le_imp_le_exp:
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   429
  assumes gt1: "1 < a"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   430
  shows "a ^ m \<le> a ^ n \<Longrightarrow> m \<le> n"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   431
proof (induct m arbitrary: n)
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   432
  case 0
14577
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
   433
  show ?case by simp
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   434
next
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   435
  case (Suc m)
14577
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
   436
  show ?case
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
   437
  proof (cases n)
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
   438
    case 0
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   439
    with Suc.prems Suc.hyps have "a * a ^ m \<le> 1" by simp
14577
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
   440
    with gt1 show ?thesis
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
   441
      by (force simp only: power_gt1_lemma
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   442
          not_less [symmetric])
14577
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
   443
  next
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
   444
    case (Suc n)
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   445
    with Suc.prems Suc.hyps show ?thesis
14577
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
   446
      by (force dest: mult_left_le_imp_le
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   447
          simp add: less_trans [OF zero_less_one gt1])
14577
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
   448
  qed
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   449
qed
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   450
61649
268d88ec9087 Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents: 61531
diff changeset
   451
lemma of_nat_zero_less_power_iff [simp]:
268d88ec9087 Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents: 61531
diff changeset
   452
  "of_nat x ^ n > 0 \<longleftrightarrow> x > 0 \<or> n = 0"
268d88ec9087 Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents: 61531
diff changeset
   453
  by (induct n) auto
268d88ec9087 Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents: 61531
diff changeset
   454
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61694
diff changeset
   455
text\<open>Surely we can strengthen this? It holds for \<open>0<a<1\<close> too.\<close>
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   456
lemma power_inject_exp [simp]:
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   457
  "1 < a \<Longrightarrow> a ^ m = a ^ n \<longleftrightarrow> m = n"
14577
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
   458
  by (force simp add: order_antisym power_le_imp_le_exp)
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   459
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60685
diff changeset
   460
text\<open>Can relax the first premise to @{term "0<a"} in the case of the
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60685
diff changeset
   461
natural numbers.\<close>
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   462
lemma power_less_imp_less_exp:
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   463
  "1 < a \<Longrightarrow> a ^ m < a ^ n \<Longrightarrow> m < n"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   464
  by (simp add: order_less_le [of m n] less_le [of "a^m" "a^n"]
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   465
    power_le_imp_le_exp)
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   466
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   467
lemma power_strict_mono [rule_format]:
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   468
  "a < b \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 < n \<longrightarrow> a ^ n < b ^ n"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   469
  by (induct n)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   470
   (auto simp add: mult_strict_mono le_less_trans [of 0 a b])
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   471
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61694
diff changeset
   472
text\<open>Lemma for \<open>power_strict_decreasing\<close>\<close>
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   473
lemma power_Suc_less:
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   474
  "0 < a \<Longrightarrow> a < 1 \<Longrightarrow> a * a ^ n < a ^ n"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   475
  by (induct n)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   476
    (auto simp add: mult_strict_left_mono)
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   477
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   478
lemma power_strict_decreasing [rule_format]:
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   479
  "n < N \<Longrightarrow> 0 < a \<Longrightarrow> a < 1 \<longrightarrow> a ^ N < a ^ n"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   480
proof (induct N)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   481
  case 0 then show ?case by simp
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   482
next
61649
268d88ec9087 Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents: 61531
diff changeset
   483
  case (Suc N) then show ?case
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   484
  apply (auto simp add: power_Suc_less less_Suc_eq)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   485
  apply (subgoal_tac "a * a^N < 1 * a^n")
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   486
  apply simp
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   487
  apply (rule mult_strict_mono) apply auto
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   488
  done
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   489
qed
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   490
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61694
diff changeset
   491
text\<open>Proof resembles that of \<open>power_strict_decreasing\<close>\<close>
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   492
lemma power_decreasing [rule_format]:
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   493
  "n \<le> N \<Longrightarrow> 0 \<le> a \<Longrightarrow> a \<le> 1 \<longrightarrow> a ^ N \<le> a ^ n"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   494
proof (induct N)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   495
  case 0 then show ?case by simp
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   496
next
61649
268d88ec9087 Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents: 61531
diff changeset
   497
  case (Suc N) then show ?case
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   498
  apply (auto simp add: le_Suc_eq)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   499
  apply (subgoal_tac "a * a^N \<le> 1 * a^n", simp)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   500
  apply (rule mult_mono) apply auto
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   501
  done
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   502
qed
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   503
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   504
lemma power_Suc_less_one:
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   505
  "0 < a \<Longrightarrow> a < 1 \<Longrightarrow> a ^ Suc n < 1"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   506
  using power_strict_decreasing [of 0 "Suc n" a] by simp
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   507
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61694
diff changeset
   508
text\<open>Proof again resembles that of \<open>power_strict_decreasing\<close>\<close>
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   509
lemma power_increasing [rule_format]:
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   510
  "n \<le> N \<Longrightarrow> 1 \<le> a \<Longrightarrow> a ^ n \<le> a ^ N"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   511
proof (induct N)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   512
  case 0 then show ?case by simp
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   513
next
61649
268d88ec9087 Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents: 61531
diff changeset
   514
  case (Suc N) then show ?case
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   515
  apply (auto simp add: le_Suc_eq)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   516
  apply (subgoal_tac "1 * a^n \<le> a * a^N", simp)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   517
  apply (rule mult_mono) apply (auto simp add: order_trans [OF zero_le_one])
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   518
  done
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   519
qed
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   520
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61694
diff changeset
   521
text\<open>Lemma for \<open>power_strict_increasing\<close>\<close>
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   522
lemma power_less_power_Suc:
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   523
  "1 < a \<Longrightarrow> a ^ n < a * a ^ n"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   524
  by (induct n) (auto simp add: mult_strict_left_mono less_trans [OF zero_less_one])
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   525
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   526
lemma power_strict_increasing [rule_format]:
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   527
  "n < N \<Longrightarrow> 1 < a \<longrightarrow> a ^ n < a ^ N"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   528
proof (induct N)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   529
  case 0 then show ?case by simp
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   530
next
61649
268d88ec9087 Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents: 61531
diff changeset
   531
  case (Suc N) then show ?case
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   532
  apply (auto simp add: power_less_power_Suc less_Suc_eq)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   533
  apply (subgoal_tac "1 * a^n < a * a^N", simp)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   534
  apply (rule mult_strict_mono) apply (auto simp add: less_trans [OF zero_less_one] less_imp_le)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   535
  done
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   536
qed
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   537
25134
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25062
diff changeset
   538
lemma power_increasing_iff [simp]:
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   539
  "1 < b \<Longrightarrow> b ^ x \<le> b ^ y \<longleftrightarrow> x \<le> y"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   540
  by (blast intro: power_le_imp_le_exp power_increasing less_imp_le)
15066
d2f2b908e0a4 two new results
paulson
parents: 15004
diff changeset
   541
d2f2b908e0a4 two new results
paulson
parents: 15004
diff changeset
   542
lemma power_strict_increasing_iff [simp]:
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   543
  "1 < b \<Longrightarrow> b ^ x < b ^ y \<longleftrightarrow> x < y"
61649
268d88ec9087 Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents: 61531
diff changeset
   544
by (blast intro: power_less_imp_less_exp power_strict_increasing)
15066
d2f2b908e0a4 two new results
paulson
parents: 15004
diff changeset
   545
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   546
lemma power_le_imp_le_base:
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   547
  assumes le: "a ^ Suc n \<le> b ^ Suc n"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   548
    and ynonneg: "0 \<le> b"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   549
  shows "a \<le> b"
25134
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25062
diff changeset
   550
proof (rule ccontr)
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25062
diff changeset
   551
  assume "~ a \<le> b"
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25062
diff changeset
   552
  then have "b < a" by (simp only: linorder_not_le)
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25062
diff changeset
   553
  then have "b ^ Suc n < a ^ Suc n"
41550
efa734d9b221 eliminated global prems;
wenzelm
parents: 39438
diff changeset
   554
    by (simp only: assms power_strict_mono)
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   555
  from le and this show False
25134
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25062
diff changeset
   556
    by (simp add: linorder_not_less [symmetric])
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25062
diff changeset
   557
qed
14577
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
   558
22853
7f000a385606 add lemma power_less_imp_less_base
huffman
parents: 22624
diff changeset
   559
lemma power_less_imp_less_base:
7f000a385606 add lemma power_less_imp_less_base
huffman
parents: 22624
diff changeset
   560
  assumes less: "a ^ n < b ^ n"
7f000a385606 add lemma power_less_imp_less_base
huffman
parents: 22624
diff changeset
   561
  assumes nonneg: "0 \<le> b"
7f000a385606 add lemma power_less_imp_less_base
huffman
parents: 22624
diff changeset
   562
  shows "a < b"
7f000a385606 add lemma power_less_imp_less_base
huffman
parents: 22624
diff changeset
   563
proof (rule contrapos_pp [OF less])
7f000a385606 add lemma power_less_imp_less_base
huffman
parents: 22624
diff changeset
   564
  assume "~ a < b"
7f000a385606 add lemma power_less_imp_less_base
huffman
parents: 22624
diff changeset
   565
  hence "b \<le> a" by (simp only: linorder_not_less)
7f000a385606 add lemma power_less_imp_less_base
huffman
parents: 22624
diff changeset
   566
  hence "b ^ n \<le> a ^ n" using nonneg by (rule power_mono)
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   567
  thus "\<not> a ^ n < b ^ n" by (simp only: linorder_not_less)
22853
7f000a385606 add lemma power_less_imp_less_base
huffman
parents: 22624
diff changeset
   568
qed
7f000a385606 add lemma power_less_imp_less_base
huffman
parents: 22624
diff changeset
   569
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   570
lemma power_inject_base:
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   571
  "a ^ Suc n = b ^ Suc n \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> a = b"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   572
by (blast intro: power_le_imp_le_base antisym eq_refl sym)
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   573
22955
48dc37776d1e add lemma power_eq_imp_eq_base
huffman
parents: 22853
diff changeset
   574
lemma power_eq_imp_eq_base:
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   575
  "a ^ n = b ^ n \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 < n \<Longrightarrow> a = b"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   576
  by (cases n) (simp_all del: power_Suc, rule power_inject_base)
22955
48dc37776d1e add lemma power_eq_imp_eq_base
huffman
parents: 22853
diff changeset
   577
62347
2230b7047376 generalized some lemmas;
haftmann
parents: 62083
diff changeset
   578
lemma power_eq_iff_eq_base:
2230b7047376 generalized some lemmas;
haftmann
parents: 62083
diff changeset
   579
  "0 < n \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> a ^ n = b ^ n \<longleftrightarrow> a = b"
2230b7047376 generalized some lemmas;
haftmann
parents: 62083
diff changeset
   580
  using power_eq_imp_eq_base [of a n b] by auto
2230b7047376 generalized some lemmas;
haftmann
parents: 62083
diff changeset
   581
47192
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   582
lemma power2_le_imp_le:
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 52435
diff changeset
   583
  "x\<^sup>2 \<le> y\<^sup>2 \<Longrightarrow> 0 \<le> y \<Longrightarrow> x \<le> y"
47192
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   584
  unfolding numeral_2_eq_2 by (rule power_le_imp_le_base)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   585
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   586
lemma power2_less_imp_less:
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 52435
diff changeset
   587
  "x\<^sup>2 < y\<^sup>2 \<Longrightarrow> 0 \<le> y \<Longrightarrow> x < y"
47192
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   588
  by (rule power_less_imp_less_base)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   589
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   590
lemma power2_eq_imp_eq:
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 52435
diff changeset
   591
  "x\<^sup>2 = y\<^sup>2 \<Longrightarrow> 0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> x = y"
47192
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   592
  unfolding numeral_2_eq_2 by (erule (2) power_eq_imp_eq_base) simp
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   593
62347
2230b7047376 generalized some lemmas;
haftmann
parents: 62083
diff changeset
   594
lemma power_Suc_le_self:
2230b7047376 generalized some lemmas;
haftmann
parents: 62083
diff changeset
   595
  shows "0 \<le> a \<Longrightarrow> a \<le> 1 \<Longrightarrow> a ^ Suc n \<le> a"
2230b7047376 generalized some lemmas;
haftmann
parents: 62083
diff changeset
   596
  using power_decreasing [of 1 "Suc n" a] by simp
2230b7047376 generalized some lemmas;
haftmann
parents: 62083
diff changeset
   597
47192
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   598
end
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   599
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   600
context linordered_ring_strict
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   601
begin
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   602
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   603
lemma sum_squares_eq_zero_iff:
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   604
  "x * x + y * y = 0 \<longleftrightarrow> x = 0 \<and> y = 0"
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   605
  by (simp add: add_nonneg_eq_0_iff)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   606
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   607
lemma sum_squares_le_zero_iff:
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   608
  "x * x + y * y \<le> 0 \<longleftrightarrow> x = 0 \<and> y = 0"
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   609
  by (simp add: le_less not_sum_squares_lt_zero sum_squares_eq_zero_iff)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   610
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   611
lemma sum_squares_gt_zero_iff:
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   612
  "0 < x * x + y * y \<longleftrightarrow> x \<noteq> 0 \<or> y \<noteq> 0"
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   613
  by (simp add: not_le [symmetric] sum_squares_le_zero_iff)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   614
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   615
end
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   616
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 33364
diff changeset
   617
context linordered_idom
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   618
begin
29978
33df3c4eb629 generalize le_imp_power_dvd and power_le_dvd; move from Divides to Power
huffman
parents: 29608
diff changeset
   619
61944
5d06ecfdb472 prefer symbols for "abs";
wenzelm
parents: 61799
diff changeset
   620
lemma power_abs: "\<bar>a ^ n\<bar> = \<bar>a\<bar> ^ n"
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   621
  by (induct n) (auto simp add: abs_mult)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   622
61944
5d06ecfdb472 prefer symbols for "abs";
wenzelm
parents: 61799
diff changeset
   623
lemma abs_power_minus [simp]: "\<bar>(-a) ^ n\<bar> = \<bar>a ^ n\<bar>"
35216
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 35028
diff changeset
   624
  by (simp add: power_abs)
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   625
61944
5d06ecfdb472 prefer symbols for "abs";
wenzelm
parents: 61799
diff changeset
   626
lemma zero_less_power_abs_iff [simp]: "0 < \<bar>a\<bar> ^ n \<longleftrightarrow> a \<noteq> 0 \<or> n = 0"
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   627
proof (induct n)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   628
  case 0 show ?case by simp
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   629
next
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   630
  case (Suc n) show ?case by (auto simp add: Suc zero_less_mult_iff)
29978
33df3c4eb629 generalize le_imp_power_dvd and power_le_dvd; move from Divides to Power
huffman
parents: 29608
diff changeset
   631
qed
33df3c4eb629 generalize le_imp_power_dvd and power_le_dvd; move from Divides to Power
huffman
parents: 29608
diff changeset
   632
61944
5d06ecfdb472 prefer symbols for "abs";
wenzelm
parents: 61799
diff changeset
   633
lemma zero_le_power_abs [simp]: "0 \<le> \<bar>a\<bar> ^ n"
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   634
  by (rule zero_le_power [OF abs_ge_zero])
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   635
47192
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   636
lemma zero_le_power2 [simp]:
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 52435
diff changeset
   637
  "0 \<le> a\<^sup>2"
47192
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   638
  by (simp add: power2_eq_square)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   639
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   640
lemma zero_less_power2 [simp]:
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 52435
diff changeset
   641
  "0 < a\<^sup>2 \<longleftrightarrow> a \<noteq> 0"
47192
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   642
  by (force simp add: power2_eq_square zero_less_mult_iff linorder_neq_iff)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   643
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   644
lemma power2_less_0 [simp]:
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 52435
diff changeset
   645
  "\<not> a\<^sup>2 < 0"
47192
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   646
  by (force simp add: power2_eq_square mult_less_0_iff)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   647
58787
af9eb5e566dd eliminated redundancies;
haftmann
parents: 58656
diff changeset
   648
lemma power2_less_eq_zero_iff [simp]:
af9eb5e566dd eliminated redundancies;
haftmann
parents: 58656
diff changeset
   649
  "a\<^sup>2 \<le> 0 \<longleftrightarrow> a = 0"
af9eb5e566dd eliminated redundancies;
haftmann
parents: 58656
diff changeset
   650
  by (simp add: le_less)
af9eb5e566dd eliminated redundancies;
haftmann
parents: 58656
diff changeset
   651
61944
5d06ecfdb472 prefer symbols for "abs";
wenzelm
parents: 61799
diff changeset
   652
lemma abs_power2 [simp]: "\<bar>a\<^sup>2\<bar> = a\<^sup>2"
47192
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   653
  by (simp add: power2_eq_square abs_mult abs_mult_self)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   654
61944
5d06ecfdb472 prefer symbols for "abs";
wenzelm
parents: 61799
diff changeset
   655
lemma power2_abs [simp]: "\<bar>a\<bar>\<^sup>2 = a\<^sup>2"
47192
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   656
  by (simp add: power2_eq_square abs_mult_self)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   657
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   658
lemma odd_power_less_zero:
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   659
  "a < 0 \<Longrightarrow> a ^ Suc (2*n) < 0"
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   660
proof (induct n)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   661
  case 0
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   662
  then show ?case by simp
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   663
next
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   664
  case (Suc n)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   665
  have "a ^ Suc (2 * Suc n) = (a*a) * a ^ Suc(2*n)"
57514
bdc2c6b40bf2 prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents: 57512
diff changeset
   666
    by (simp add: ac_simps power_add power2_eq_square)
47192
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   667
  thus ?case
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   668
    by (simp del: power_Suc add: Suc mult_less_0_iff mult_neg_neg)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   669
qed
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   670
47192
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   671
lemma odd_0_le_power_imp_0_le:
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   672
  "0 \<le> a ^ Suc (2*n) \<Longrightarrow> 0 \<le> a"
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   673
  using odd_power_less_zero [of a n]
61649
268d88ec9087 Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents: 61531
diff changeset
   674
    by (force simp add: linorder_not_less [symmetric])
47192
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   675
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   676
lemma zero_le_even_power'[simp]:
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   677
  "0 \<le> a ^ (2*n)"
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   678
proof (induct n)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   679
  case 0
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   680
    show ?case by simp
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   681
next
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   682
  case (Suc n)
61649
268d88ec9087 Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents: 61531
diff changeset
   683
    have "a ^ (2 * Suc n) = (a*a) * a ^ (2*n)"
57514
bdc2c6b40bf2 prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents: 57512
diff changeset
   684
      by (simp add: ac_simps power_add power2_eq_square)
47192
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   685
    thus ?case
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   686
      by (simp add: Suc zero_le_mult_iff)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   687
qed
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   688
47192
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   689
lemma sum_power2_ge_zero:
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 52435
diff changeset
   690
  "0 \<le> x\<^sup>2 + y\<^sup>2"
47192
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   691
  by (intro add_nonneg_nonneg zero_le_power2)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   692
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   693
lemma not_sum_power2_lt_zero:
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 52435
diff changeset
   694
  "\<not> x\<^sup>2 + y\<^sup>2 < 0"
47192
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   695
  unfolding not_less by (rule sum_power2_ge_zero)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   696
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   697
lemma sum_power2_eq_zero_iff:
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 52435
diff changeset
   698
  "x\<^sup>2 + y\<^sup>2 = 0 \<longleftrightarrow> x = 0 \<and> y = 0"
47192
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   699
  unfolding power2_eq_square by (simp add: add_nonneg_eq_0_iff)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   700
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   701
lemma sum_power2_le_zero_iff:
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 52435
diff changeset
   702
  "x\<^sup>2 + y\<^sup>2 \<le> 0 \<longleftrightarrow> x = 0 \<and> y = 0"
47192
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   703
  by (simp add: le_less sum_power2_eq_zero_iff not_sum_power2_lt_zero)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   704
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   705
lemma sum_power2_gt_zero_iff:
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 52435
diff changeset
   706
  "0 < x\<^sup>2 + y\<^sup>2 \<longleftrightarrow> x \<noteq> 0 \<or> y \<noteq> 0"
47192
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   707
  unfolding not_le [symmetric] by (simp add: sum_power2_le_zero_iff)
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   708
59865
8a20dd967385 rationalised and generalised some theorems concerning abs and x^2.
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
   709
lemma abs_le_square_iff:
8a20dd967385 rationalised and generalised some theorems concerning abs and x^2.
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
   710
   "\<bar>x\<bar> \<le> \<bar>y\<bar> \<longleftrightarrow> x\<^sup>2 \<le> y\<^sup>2"
8a20dd967385 rationalised and generalised some theorems concerning abs and x^2.
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
   711
proof
8a20dd967385 rationalised and generalised some theorems concerning abs and x^2.
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
   712
  assume "\<bar>x\<bar> \<le> \<bar>y\<bar>"
8a20dd967385 rationalised and generalised some theorems concerning abs and x^2.
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
   713
  then have "\<bar>x\<bar>\<^sup>2 \<le> \<bar>y\<bar>\<^sup>2" by (rule power_mono, simp)
8a20dd967385 rationalised and generalised some theorems concerning abs and x^2.
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
   714
  then show "x\<^sup>2 \<le> y\<^sup>2" by simp
8a20dd967385 rationalised and generalised some theorems concerning abs and x^2.
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
   715
next
8a20dd967385 rationalised and generalised some theorems concerning abs and x^2.
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
   716
  assume "x\<^sup>2 \<le> y\<^sup>2"
8a20dd967385 rationalised and generalised some theorems concerning abs and x^2.
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
   717
  then show "\<bar>x\<bar> \<le> \<bar>y\<bar>"
8a20dd967385 rationalised and generalised some theorems concerning abs and x^2.
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
   718
    by (auto intro!: power2_le_imp_le [OF _ abs_ge_zero])
8a20dd967385 rationalised and generalised some theorems concerning abs and x^2.
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
   719
qed
8a20dd967385 rationalised and generalised some theorems concerning abs and x^2.
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
   720
61944
5d06ecfdb472 prefer symbols for "abs";
wenzelm
parents: 61799
diff changeset
   721
lemma abs_square_le_1:"x\<^sup>2 \<le> 1 \<longleftrightarrow> \<bar>x\<bar> \<le> 1"
59865
8a20dd967385 rationalised and generalised some theorems concerning abs and x^2.
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
   722
  using abs_le_square_iff [of x 1]
8a20dd967385 rationalised and generalised some theorems concerning abs and x^2.
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
   723
  by simp
8a20dd967385 rationalised and generalised some theorems concerning abs and x^2.
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
   724
61944
5d06ecfdb472 prefer symbols for "abs";
wenzelm
parents: 61799
diff changeset
   725
lemma abs_square_eq_1: "x\<^sup>2 = 1 \<longleftrightarrow> \<bar>x\<bar> = 1"
59865
8a20dd967385 rationalised and generalised some theorems concerning abs and x^2.
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
   726
  by (auto simp add: abs_if power2_eq_1_iff)
61649
268d88ec9087 Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents: 61531
diff changeset
   727
61944
5d06ecfdb472 prefer symbols for "abs";
wenzelm
parents: 61799
diff changeset
   728
lemma abs_square_less_1: "x\<^sup>2 < 1 \<longleftrightarrow> \<bar>x\<bar> < 1"
59865
8a20dd967385 rationalised and generalised some theorems concerning abs and x^2.
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
   729
  using  abs_square_eq_1 [of x] abs_square_le_1 [of x]
8a20dd967385 rationalised and generalised some theorems concerning abs and x^2.
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
   730
  by (auto simp add: le_less)
8a20dd967385 rationalised and generalised some theorems concerning abs and x^2.
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
   731
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   732
end
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   733
29978
33df3c4eb629 generalize le_imp_power_dvd and power_le_dvd; move from Divides to Power
huffman
parents: 29608
diff changeset
   734
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60685
diff changeset
   735
subsection \<open>Miscellaneous rules\<close>
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   736
60867
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60866
diff changeset
   737
lemma (in linordered_semidom) self_le_power:
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60866
diff changeset
   738
  "1 \<le> a \<Longrightarrow> 0 < n \<Longrightarrow> a \<le> a ^ n"
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60866
diff changeset
   739
  using power_increasing [of 1 n a] power_one_right [of a] by auto
55718
34618f031ba9 A few lemmas about summations, etc.
paulson <lp15@cam.ac.uk>
parents: 55096
diff changeset
   740
60867
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60866
diff changeset
   741
lemma (in power) power_eq_if:
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60866
diff changeset
   742
  "p ^ m = (if m=0 then 1 else p * (p ^ (m - 1)))"
47255
30a1692557b0 removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents: 47241
diff changeset
   743
  unfolding One_nat_def by (cases m) simp_all
30a1692557b0 removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents: 47241
diff changeset
   744
58787
af9eb5e566dd eliminated redundancies;
haftmann
parents: 58656
diff changeset
   745
lemma (in comm_semiring_1) power2_sum:
af9eb5e566dd eliminated redundancies;
haftmann
parents: 58656
diff changeset
   746
  "(x + y)\<^sup>2 = x\<^sup>2 + y\<^sup>2 + 2 * x * y"
47192
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   747
  by (simp add: algebra_simps power2_eq_square mult_2_right)
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   748
58787
af9eb5e566dd eliminated redundancies;
haftmann
parents: 58656
diff changeset
   749
lemma (in comm_ring_1) power2_diff:
af9eb5e566dd eliminated redundancies;
haftmann
parents: 58656
diff changeset
   750
  "(x - y)\<^sup>2 = x\<^sup>2 + y\<^sup>2 - 2 * x * y"
af9eb5e566dd eliminated redundancies;
haftmann
parents: 58656
diff changeset
   751
  by (simp add: algebra_simps power2_eq_square mult_2_right)
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   752
60974
6a6f15d8fbc4 New material and fixes related to the forthcoming Stone-Weierstrass development
paulson <lp15@cam.ac.uk>
parents: 60867
diff changeset
   753
lemma (in comm_ring_1) power2_commute:
6a6f15d8fbc4 New material and fixes related to the forthcoming Stone-Weierstrass development
paulson <lp15@cam.ac.uk>
parents: 60867
diff changeset
   754
  "(x - y)\<^sup>2 = (y - x)\<^sup>2"
6a6f15d8fbc4 New material and fixes related to the forthcoming Stone-Weierstrass development
paulson <lp15@cam.ac.uk>
parents: 60867
diff changeset
   755
  by (simp add: algebra_simps power2_eq_square)
6a6f15d8fbc4 New material and fixes related to the forthcoming Stone-Weierstrass development
paulson <lp15@cam.ac.uk>
parents: 60867
diff changeset
   756
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60685
diff changeset
   757
text \<open>Simprules for comparisons where common factors can be cancelled.\<close>
47255
30a1692557b0 removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents: 47241
diff changeset
   758
30a1692557b0 removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents: 47241
diff changeset
   759
lemmas zero_compare_simps =
30a1692557b0 removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents: 47241
diff changeset
   760
    add_strict_increasing add_strict_increasing2 add_increasing
61649
268d88ec9087 Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents: 61531
diff changeset
   761
    zero_le_mult_iff zero_le_divide_iff
268d88ec9087 Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents: 61531
diff changeset
   762
    zero_less_mult_iff zero_less_divide_iff
268d88ec9087 Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents: 61531
diff changeset
   763
    mult_le_0_iff divide_le_0_iff
268d88ec9087 Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents: 61531
diff changeset
   764
    mult_less_0_iff divide_less_0_iff
47255
30a1692557b0 removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents: 47241
diff changeset
   765
    zero_le_power2 power2_less_0
30a1692557b0 removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents: 47241
diff changeset
   766
30313
b2441b0c8d38 added lemmas
nipkow
parents: 30273
diff changeset
   767
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60685
diff changeset
   768
subsection \<open>Exponentiation for the Natural Numbers\<close>
14577
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
   769
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   770
lemma nat_one_le_power [simp]:
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   771
  "Suc 0 \<le> i \<Longrightarrow> Suc 0 \<le> i ^ n"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   772
  by (rule one_le_power [of i n, unfolded One_nat_def])
23305
8ae6f7b0903b add lemma of_nat_power
huffman
parents: 23183
diff changeset
   773
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   774
lemma nat_zero_less_power_iff [simp]:
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   775
  "x ^ n > 0 \<longleftrightarrow> x > (0::nat) \<or> n = 0"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   776
  by (induct n) auto
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   777
61649
268d88ec9087 Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents: 61531
diff changeset
   778
lemma nat_power_eq_Suc_0_iff [simp]:
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   779
  "x ^ m = Suc 0 \<longleftrightarrow> m = 0 \<or> x = Suc 0"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   780
  by (induct m) auto
30056
0a35bee25c20 added lemmas
nipkow
parents: 29978
diff changeset
   781
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   782
lemma power_Suc_0 [simp]:
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   783
  "Suc 0 ^ n = Suc 0"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   784
  by simp
30056
0a35bee25c20 added lemmas
nipkow
parents: 29978
diff changeset
   785
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61694
diff changeset
   786
text\<open>Valid for the naturals, but what if \<open>0<i<1\<close>?
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   787
Premises cannot be weakened: consider the case where @{term "i=0"},
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60685
diff changeset
   788
@{term "m=1"} and @{term "n=0"}.\<close>
21413
0951647209f2 moved dvd stuff to theory Divides
haftmann
parents: 21199
diff changeset
   789
lemma nat_power_less_imp_less:
61076
bdc1e2f0a86a eliminated \<Colon>;
wenzelm
parents: 60974
diff changeset
   790
  assumes nonneg: "0 < (i::nat)"
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   791
  assumes less: "i ^ m < i ^ n"
21413
0951647209f2 moved dvd stuff to theory Divides
haftmann
parents: 21199
diff changeset
   792
  shows "m < n"
0951647209f2 moved dvd stuff to theory Divides
haftmann
parents: 21199
diff changeset
   793
proof (cases "i = 1")
0951647209f2 moved dvd stuff to theory Divides
haftmann
parents: 21199
diff changeset
   794
  case True with less power_one [where 'a = nat] show ?thesis by simp
0951647209f2 moved dvd stuff to theory Divides
haftmann
parents: 21199
diff changeset
   795
next
0951647209f2 moved dvd stuff to theory Divides
haftmann
parents: 21199
diff changeset
   796
  case False with nonneg have "1 < i" by auto
0951647209f2 moved dvd stuff to theory Divides
haftmann
parents: 21199
diff changeset
   797
  from power_strict_increasing_iff [OF this] less show ?thesis ..
0951647209f2 moved dvd stuff to theory Divides
haftmann
parents: 21199
diff changeset
   798
qed
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   799
33274
b6ff7db522b5 moved lemmas for dvd on nat to theories Nat and Power
haftmann
parents: 31998
diff changeset
   800
lemma power_dvd_imp_le:
b6ff7db522b5 moved lemmas for dvd on nat to theories Nat and Power
haftmann
parents: 31998
diff changeset
   801
  "i ^ m dvd i ^ n \<Longrightarrow> (1::nat) < i \<Longrightarrow> m \<le> n"
b6ff7db522b5 moved lemmas for dvd on nat to theories Nat and Power
haftmann
parents: 31998
diff changeset
   802
  apply (rule power_le_imp_le_exp, assumption)
b6ff7db522b5 moved lemmas for dvd on nat to theories Nat and Power
haftmann
parents: 31998
diff changeset
   803
  apply (erule dvd_imp_le, simp)
b6ff7db522b5 moved lemmas for dvd on nat to theories Nat and Power
haftmann
parents: 31998
diff changeset
   804
  done
b6ff7db522b5 moved lemmas for dvd on nat to theories Nat and Power
haftmann
parents: 31998
diff changeset
   805
51263
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 49824
diff changeset
   806
lemma power2_nat_le_eq_le:
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 49824
diff changeset
   807
  fixes m n :: nat
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 52435
diff changeset
   808
  shows "m\<^sup>2 \<le> n\<^sup>2 \<longleftrightarrow> m \<le> n"
51263
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 49824
diff changeset
   809
  by (auto intro: power2_le_imp_le power_mono)
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 49824
diff changeset
   810
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 49824
diff changeset
   811
lemma power2_nat_le_imp_le:
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 49824
diff changeset
   812
  fixes m n :: nat
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 52435
diff changeset
   813
  assumes "m\<^sup>2 \<le> n"
51263
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 49824
diff changeset
   814
  shows "m \<le> n"
54249
ce00f2fef556 streamlined setup of linear arithmetic
haftmann
parents: 54147
diff changeset
   815
proof (cases m)
ce00f2fef556 streamlined setup of linear arithmetic
haftmann
parents: 54147
diff changeset
   816
  case 0 then show ?thesis by simp
ce00f2fef556 streamlined setup of linear arithmetic
haftmann
parents: 54147
diff changeset
   817
next
ce00f2fef556 streamlined setup of linear arithmetic
haftmann
parents: 54147
diff changeset
   818
  case (Suc k)
ce00f2fef556 streamlined setup of linear arithmetic
haftmann
parents: 54147
diff changeset
   819
  show ?thesis
ce00f2fef556 streamlined setup of linear arithmetic
haftmann
parents: 54147
diff changeset
   820
  proof (rule ccontr)
ce00f2fef556 streamlined setup of linear arithmetic
haftmann
parents: 54147
diff changeset
   821
    assume "\<not> m \<le> n"
ce00f2fef556 streamlined setup of linear arithmetic
haftmann
parents: 54147
diff changeset
   822
    then have "n < m" by simp
ce00f2fef556 streamlined setup of linear arithmetic
haftmann
parents: 54147
diff changeset
   823
    with assms Suc show False
60867
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60866
diff changeset
   824
      by (simp add: power2_eq_square)
54249
ce00f2fef556 streamlined setup of linear arithmetic
haftmann
parents: 54147
diff changeset
   825
  qed
ce00f2fef556 streamlined setup of linear arithmetic
haftmann
parents: 54147
diff changeset
   826
qed
51263
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 49824
diff changeset
   827
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60685
diff changeset
   828
subsubsection \<open>Cardinality of the Powerset\<close>
55096
916b2ac758f4 removed theory dependency of BNF_LFP on Datatype
traytel
parents: 54489
diff changeset
   829
916b2ac758f4 removed theory dependency of BNF_LFP on Datatype
traytel
parents: 54489
diff changeset
   830
lemma card_UNIV_bool [simp]: "card (UNIV :: bool set) = 2"
916b2ac758f4 removed theory dependency of BNF_LFP on Datatype
traytel
parents: 54489
diff changeset
   831
  unfolding UNIV_bool by simp
916b2ac758f4 removed theory dependency of BNF_LFP on Datatype
traytel
parents: 54489
diff changeset
   832
916b2ac758f4 removed theory dependency of BNF_LFP on Datatype
traytel
parents: 54489
diff changeset
   833
lemma card_Pow: "finite A \<Longrightarrow> card (Pow A) = 2 ^ card A"
916b2ac758f4 removed theory dependency of BNF_LFP on Datatype
traytel
parents: 54489
diff changeset
   834
proof (induct rule: finite_induct)
61649
268d88ec9087 Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents: 61531
diff changeset
   835
  case empty
55096
916b2ac758f4 removed theory dependency of BNF_LFP on Datatype
traytel
parents: 54489
diff changeset
   836
    show ?case by auto
916b2ac758f4 removed theory dependency of BNF_LFP on Datatype
traytel
parents: 54489
diff changeset
   837
next
916b2ac758f4 removed theory dependency of BNF_LFP on Datatype
traytel
parents: 54489
diff changeset
   838
  case (insert x A)
61649
268d88ec9087 Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents: 61531
diff changeset
   839
  then have "inj_on (insert x) (Pow A)"
55096
916b2ac758f4 removed theory dependency of BNF_LFP on Datatype
traytel
parents: 54489
diff changeset
   840
    unfolding inj_on_def by (blast elim!: equalityE)
61649
268d88ec9087 Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents: 61531
diff changeset
   841
  then have "card (Pow A) + card (insert x ` Pow A) = 2 * 2 ^ card A"
55096
916b2ac758f4 removed theory dependency of BNF_LFP on Datatype
traytel
parents: 54489
diff changeset
   842
    by (simp add: mult_2 card_image Pow_insert insert.hyps)
916b2ac758f4 removed theory dependency of BNF_LFP on Datatype
traytel
parents: 54489
diff changeset
   843
  then show ?case using insert
916b2ac758f4 removed theory dependency of BNF_LFP on Datatype
traytel
parents: 54489
diff changeset
   844
    apply (simp add: Pow_insert)
916b2ac758f4 removed theory dependency of BNF_LFP on Datatype
traytel
parents: 54489
diff changeset
   845
    apply (subst card_Un_disjoint, auto)
916b2ac758f4 removed theory dependency of BNF_LFP on Datatype
traytel
parents: 54489
diff changeset
   846
    done
916b2ac758f4 removed theory dependency of BNF_LFP on Datatype
traytel
parents: 54489
diff changeset
   847
qed
916b2ac758f4 removed theory dependency of BNF_LFP on Datatype
traytel
parents: 54489
diff changeset
   848
57418
6ab1c7cb0b8d fact consolidation
haftmann
parents: 56544
diff changeset
   849
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60685
diff changeset
   850
subsubsection \<open>Generalized sum over a set\<close>
57418
6ab1c7cb0b8d fact consolidation
haftmann
parents: 56544
diff changeset
   851
6ab1c7cb0b8d fact consolidation
haftmann
parents: 56544
diff changeset
   852
lemma setsum_zero_power [simp]:
6ab1c7cb0b8d fact consolidation
haftmann
parents: 56544
diff changeset
   853
  fixes c :: "nat \<Rightarrow> 'a::division_ring"
6ab1c7cb0b8d fact consolidation
haftmann
parents: 56544
diff changeset
   854
  shows "(\<Sum>i\<in>A. c i * 0^i) = (if finite A \<and> 0 \<in> A then c 0 else 0)"
6ab1c7cb0b8d fact consolidation
haftmann
parents: 56544
diff changeset
   855
apply (cases "finite A")
6ab1c7cb0b8d fact consolidation
haftmann
parents: 56544
diff changeset
   856
  by (induction A rule: finite_induct) auto
6ab1c7cb0b8d fact consolidation
haftmann
parents: 56544
diff changeset
   857
6ab1c7cb0b8d fact consolidation
haftmann
parents: 56544
diff changeset
   858
lemma setsum_zero_power' [simp]:
6ab1c7cb0b8d fact consolidation
haftmann
parents: 56544
diff changeset
   859
  fixes c :: "nat \<Rightarrow> 'a::field"
6ab1c7cb0b8d fact consolidation
haftmann
parents: 56544
diff changeset
   860
  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)"
6ab1c7cb0b8d fact consolidation
haftmann
parents: 56544
diff changeset
   861
  using setsum_zero_power [of "\<lambda>i. c i / d i" A]
6ab1c7cb0b8d fact consolidation
haftmann
parents: 56544
diff changeset
   862
  by auto
6ab1c7cb0b8d fact consolidation
haftmann
parents: 56544
diff changeset
   863
6ab1c7cb0b8d fact consolidation
haftmann
parents: 56544
diff changeset
   864
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60685
diff changeset
   865
subsubsection \<open>Generalized product over a set\<close>
55096
916b2ac758f4 removed theory dependency of BNF_LFP on Datatype
traytel
parents: 54489
diff changeset
   866
916b2ac758f4 removed theory dependency of BNF_LFP on Datatype
traytel
parents: 54489
diff changeset
   867
lemma setprod_constant: "finite A ==> (\<Prod>x\<in> A. (y::'a::{comm_monoid_mult})) = y^(card A)"
916b2ac758f4 removed theory dependency of BNF_LFP on Datatype
traytel
parents: 54489
diff changeset
   868
apply (erule finite_induct)
916b2ac758f4 removed theory dependency of BNF_LFP on Datatype
traytel
parents: 54489
diff changeset
   869
apply auto
916b2ac758f4 removed theory dependency of BNF_LFP on Datatype
traytel
parents: 54489
diff changeset
   870
done
916b2ac758f4 removed theory dependency of BNF_LFP on Datatype
traytel
parents: 54489
diff changeset
   871
57418
6ab1c7cb0b8d fact consolidation
haftmann
parents: 56544
diff changeset
   872
lemma setprod_power_distrib:
6ab1c7cb0b8d fact consolidation
haftmann
parents: 56544
diff changeset
   873
  fixes f :: "'a \<Rightarrow> 'b::comm_semiring_1"
6ab1c7cb0b8d fact consolidation
haftmann
parents: 56544
diff changeset
   874
  shows "setprod f A ^ n = setprod (\<lambda>x. (f x) ^ n) A"
61649
268d88ec9087 Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents: 61531
diff changeset
   875
proof (cases "finite A")
268d88ec9087 Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents: 61531
diff changeset
   876
  case True then show ?thesis
57418
6ab1c7cb0b8d fact consolidation
haftmann
parents: 56544
diff changeset
   877
    by (induct A rule: finite_induct) (auto simp add: power_mult_distrib)
6ab1c7cb0b8d fact consolidation
haftmann
parents: 56544
diff changeset
   878
next
61649
268d88ec9087 Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents: 61531
diff changeset
   879
  case False then show ?thesis
57418
6ab1c7cb0b8d fact consolidation
haftmann
parents: 56544
diff changeset
   880
    by simp
6ab1c7cb0b8d fact consolidation
haftmann
parents: 56544
diff changeset
   881
qed
6ab1c7cb0b8d fact consolidation
haftmann
parents: 56544
diff changeset
   882
58437
8d124c73c37a added lemmas
haftmann
parents: 58410
diff changeset
   883
lemma power_setsum:
8d124c73c37a added lemmas
haftmann
parents: 58410
diff changeset
   884
  "c ^ (\<Sum>a\<in>A. f a) = (\<Prod>a\<in>A. c ^ f a)"
8d124c73c37a added lemmas
haftmann
parents: 58410
diff changeset
   885
  by (induct A rule: infinite_finite_induct) (simp_all add: power_add)
8d124c73c37a added lemmas
haftmann
parents: 58410
diff changeset
   886
55096
916b2ac758f4 removed theory dependency of BNF_LFP on Datatype
traytel
parents: 54489
diff changeset
   887
lemma setprod_gen_delta:
916b2ac758f4 removed theory dependency of BNF_LFP on Datatype
traytel
parents: 54489
diff changeset
   888
  assumes fS: "finite S"
916b2ac758f4 removed theory dependency of BNF_LFP on Datatype
traytel
parents: 54489
diff changeset
   889
  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)"
916b2ac758f4 removed theory dependency of BNF_LFP on Datatype
traytel
parents: 54489
diff changeset
   890
proof-
916b2ac758f4 removed theory dependency of BNF_LFP on Datatype
traytel
parents: 54489
diff changeset
   891
  let ?f = "(\<lambda>k. if k=a then b k else c)"
916b2ac758f4 removed theory dependency of BNF_LFP on Datatype
traytel
parents: 54489
diff changeset
   892
  {assume a: "a \<notin> S"
916b2ac758f4 removed theory dependency of BNF_LFP on Datatype
traytel
parents: 54489
diff changeset
   893
    hence "\<forall> k\<in> S. ?f k = c" by simp
916b2ac758f4 removed theory dependency of BNF_LFP on Datatype
traytel
parents: 54489
diff changeset
   894
    hence ?thesis  using a setprod_constant[OF fS, of c] by simp }
61649
268d88ec9087 Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents: 61531
diff changeset
   895
  moreover
55096
916b2ac758f4 removed theory dependency of BNF_LFP on Datatype
traytel
parents: 54489
diff changeset
   896
  {assume a: "a \<in> S"
916b2ac758f4 removed theory dependency of BNF_LFP on Datatype
traytel
parents: 54489
diff changeset
   897
    let ?A = "S - {a}"
916b2ac758f4 removed theory dependency of BNF_LFP on Datatype
traytel
parents: 54489
diff changeset
   898
    let ?B = "{a}"
61649
268d88ec9087 Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents: 61531
diff changeset
   899
    have eq: "S = ?A \<union> ?B" using a by blast
55096
916b2ac758f4 removed theory dependency of BNF_LFP on Datatype
traytel
parents: 54489
diff changeset
   900
    have dj: "?A \<inter> ?B = {}" by simp
61649
268d88ec9087 Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents: 61531
diff changeset
   901
    from fS have fAB: "finite ?A" "finite ?B" by auto
55096
916b2ac758f4 removed theory dependency of BNF_LFP on Datatype
traytel
parents: 54489
diff changeset
   902
    have fA0:"setprod ?f ?A = setprod (\<lambda>i. c) ?A"
57418
6ab1c7cb0b8d fact consolidation
haftmann
parents: 56544
diff changeset
   903
      apply (rule setprod.cong) by auto
55096
916b2ac758f4 removed theory dependency of BNF_LFP on Datatype
traytel
parents: 54489
diff changeset
   904
    have cA: "card ?A = card S - 1" using fS a by auto
916b2ac758f4 removed theory dependency of BNF_LFP on Datatype
traytel
parents: 54489
diff changeset
   905
    have fA1: "setprod ?f ?A = c ^ card ?A"  unfolding fA0 apply (rule setprod_constant) using fS by auto
916b2ac758f4 removed theory dependency of BNF_LFP on Datatype
traytel
parents: 54489
diff changeset
   906
    have "setprod ?f ?A * setprod ?f ?B = setprod ?f S"
57418
6ab1c7cb0b8d fact consolidation
haftmann
parents: 56544
diff changeset
   907
      using setprod.union_disjoint[OF fAB dj, of ?f, unfolded eq[symmetric]]
55096
916b2ac758f4 removed theory dependency of BNF_LFP on Datatype
traytel
parents: 54489
diff changeset
   908
      by simp
916b2ac758f4 removed theory dependency of BNF_LFP on Datatype
traytel
parents: 54489
diff changeset
   909
    then have ?thesis using a cA
57418
6ab1c7cb0b8d fact consolidation
haftmann
parents: 56544
diff changeset
   910
      by (simp add: fA1 field_simps cong add: setprod.cong cong del: if_weak_cong)}
55096
916b2ac758f4 removed theory dependency of BNF_LFP on Datatype
traytel
parents: 54489
diff changeset
   911
  ultimately show ?thesis by blast
916b2ac758f4 removed theory dependency of BNF_LFP on Datatype
traytel
parents: 54489
diff changeset
   912
qed
916b2ac758f4 removed theory dependency of BNF_LFP on Datatype
traytel
parents: 54489
diff changeset
   913
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60685
diff changeset
   914
subsection \<open>Code generator tweak\<close>
31155
92d8ff6af82c monomorphic code generation for power operations
haftmann
parents: 31021
diff changeset
   915
52435
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 51263
diff changeset
   916
code_identifier
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 51263
diff changeset
   917
  code_module Power \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
33364
2bd12592c5e8 tuned code setup
haftmann
parents: 33274
diff changeset
   918
3390
0c7625196d95 New theory "Power" of exponentiation (and binomial coefficients)
paulson
parents:
diff changeset
   919
end