src/HOL/Power.thy
author wenzelm
Tue Aug 13 16:25:47 2013 +0200 (2013-08-13)
changeset 53015 a1119cf551e8
parent 52435 6646bb548c6b
child 53076 47c9aff07725
permissions -rw-r--r--
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
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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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header {* Exponentiation *}
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theory Power
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imports Num
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begin
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subsection {* Powers for Arbitrary Monoids *}
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class power = one + times
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begin
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primrec power :: "'a \<Rightarrow> nat \<Rightarrow> 'a" (infixr "^" 80) where
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    power_0: "a ^ 0 = 1"
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  | power_Suc: "a ^ Suc n = a * a ^ n"
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notation (latex output)
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  power ("(_\<^bsup>_\<^esup>)" [1000] 1000)
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notation (HTML output)
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  power ("(_\<^bsup>_\<^esup>)" [1000] 1000)
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text {* Special syntax for squares. *}
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abbreviation (xsymbols)
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  power2 :: "'a \<Rightarrow> 'a"  ("(_\<^sup>2)" [1000] 999) where
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  "x\<^sup>2 \<equiv> x ^ 2"
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notation (latex output)
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  power2  ("(_\<^sup>2)" [1000] 999)
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notation (HTML output)
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  power2  ("(_\<^sup>2)" [1000] 999)
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end
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context monoid_mult
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begin
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subclass power .
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lemma power_one [simp]:
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  "1 ^ n = 1"
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  by (induct n) simp_all
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lemma power_one_right [simp]:
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  "a ^ 1 = a"
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  by simp
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lemma power_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) ^ 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) ^ 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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end
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context comm_monoid_mult
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begin
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lemma power_mult_distrib:
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  "(a * b) ^ n = (a ^ n) * (b ^ n)"
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  by (induct n) (simp_all add: mult_ac)
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end
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context semiring_numeral
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begin
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lemma numeral_sqr: "numeral (Num.sqr k) = numeral k * numeral k"
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  by (simp only: sqr_conv_mult numeral_mult)
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lemma numeral_pow: "numeral (Num.pow k l) = numeral k ^ numeral l"
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  by (induct l, simp_all only: numeral_class.numeral.simps pow.simps
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    numeral_sqr numeral_mult power_add power_one_right)
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lemma power_numeral [simp]: "numeral k ^ numeral l = numeral (Num.pow k l)"
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  by (rule numeral_pow [symmetric])
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end
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context semiring_1
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begin
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lemma of_nat_power:
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  "of_nat (m ^ n) = of_nat m ^ n"
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  by (induct n) (simp_all add: of_nat_mult)
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lemma power_zero_numeral [simp]: "(0::'a) ^ numeral k = 0"
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  by (simp add: numeral_eq_Suc)
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lemma zero_power2: "0\<^sup>2 = 0" (* delete? *)
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  by (rule power_zero_numeral)
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lemma one_power2: "1\<^sup>2 = 1" (* delete? *)
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  by (rule power_one)
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end
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context comm_semiring_1
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begin
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text {* The divides relation *}
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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 `m \<le> n` by simp
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  also have "\<dots> = a ^ m * a ^ (n - m)"
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    by (rule power_add)
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  finally show "a ^ n = a ^ m * a ^ (n - m)" .
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qed
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lemma power_le_dvd:
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  "a ^ n dvd b \<Longrightarrow> m \<le> n \<Longrightarrow> a ^ m dvd b"
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  by (rule dvd_trans [OF le_imp_power_dvd])
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lemma dvd_power_same:
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  "x dvd y \<Longrightarrow> x ^ n dvd y ^ n"
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  by (induct n) (auto simp add: mult_dvd_mono)
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lemma dvd_power_le:
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  "x dvd y \<Longrightarrow> m \<ge> n \<Longrightarrow> x ^ n dvd y ^ m"
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  by (rule power_le_dvd [OF dvd_power_same])
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lemma dvd_power [simp]:
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  assumes "n > (0::nat) \<or> x = 1"
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  shows "x dvd (x ^ n)"
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using assms proof
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  assume "0 < n"
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  then have "x ^ n = x ^ Suc (n - 1)" by simp
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  then show "x dvd (x ^ n)" by simp
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next
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  assume "x = 1"
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  then show "x dvd (x ^ n)" by simp
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qed
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end
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context ring_1
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begin
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lemma power_minus:
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  "(- a) ^ n = (- 1) ^ n * a ^ n"
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proof (induct n)
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  case 0 show ?case by simp
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next
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  case (Suc n) then show ?case
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    by (simp del: power_Suc add: power_Suc2 mult_assoc)
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qed
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lemma power_minus_Bit0:
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  "(- x) ^ numeral (Num.Bit0 k) = x ^ numeral (Num.Bit0 k)"
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  by (induct k, simp_all only: numeral_class.numeral.simps power_add
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    power_one_right mult_minus_left mult_minus_right minus_minus)
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lemma power_minus_Bit1:
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  "(- x) ^ numeral (Num.Bit1 k) = - (x ^ numeral (Num.Bit1 k))"
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  by (simp only: eval_nat_numeral(3) power_Suc power_minus_Bit0 mult_minus_left)
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lemma power_neg_numeral_Bit0 [simp]:
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  "neg_numeral k ^ numeral (Num.Bit0 l) = numeral (Num.pow k (Num.Bit0 l))"
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  by (simp only: neg_numeral_def power_minus_Bit0 power_numeral)
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lemma power_neg_numeral_Bit1 [simp]:
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  "neg_numeral k ^ numeral (Num.Bit1 l) = neg_numeral (Num.pow k (Num.Bit1 l))"
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  by (simp only: neg_numeral_def power_minus_Bit1 power_numeral pow.simps)
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lemma power2_minus [simp]:
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  "(- a)\<^sup>2 = a\<^sup>2"
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  by (rule power_minus_Bit0)
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lemma power_minus1_even [simp]:
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  "-1 ^ (2*n) = 1"
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proof (induct n)
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  case 0 show ?case by simp
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next
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  case (Suc n) then show ?case by (simp add: power_add power2_eq_square)
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qed
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lemma power_minus1_odd:
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  "-1 ^ Suc (2*n) = -1"
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  by simp
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lemma power_minus_even [simp]:
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  "(-a) ^ (2*n) = a ^ (2*n)"
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  by (simp add: power_minus [of a])
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end
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context ring_1_no_zero_divisors
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begin
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lemma field_power_not_zero:
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  "a \<noteq> 0 \<Longrightarrow> a ^ n \<noteq> 0"
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  by (induct n) auto
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lemma zero_eq_power2 [simp]:
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  "a\<^sup>2 = 0 \<longleftrightarrow> a = 0"
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  unfolding power2_eq_square by simp
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lemma power2_eq_1_iff:
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  "a\<^sup>2 = 1 \<longleftrightarrow> a = 1 \<or> a = - 1"
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  unfolding power2_eq_square by (rule square_eq_1_iff)
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end
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context idom
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begin
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lemma power2_eq_iff: "x\<^sup>2 = y\<^sup>2 \<longleftrightarrow> x = y \<or> x = - y"
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  unfolding power2_eq_square by (rule square_eq_iff)
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end
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context division_ring
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begin
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text {* FIXME reorient or rename to @{text nonzero_inverse_power} *}
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lemma nonzero_power_inverse:
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  "a \<noteq> 0 \<Longrightarrow> inverse (a ^ n) = (inverse a) ^ n"
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  by (induct n)
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    (simp_all add: nonzero_inverse_mult_distrib power_commutes field_power_not_zero)
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end
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context field
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begin
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lemma nonzero_power_divide:
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  "b \<noteq> 0 \<Longrightarrow> (a / b) ^ n = a ^ n / b ^ n"
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  by (simp add: divide_inverse power_mult_distrib nonzero_power_inverse)
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end
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subsection {* Exponentiation on ordered types *}
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context linordered_ring (* TODO: move *)
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begin
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lemma sum_squares_ge_zero:
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  "0 \<le> x * x + y * y"
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  by (intro add_nonneg_nonneg zero_le_square)
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lemma not_sum_squares_lt_zero:
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  "\<not> x * x + y * y < 0"
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  by (simp add: not_less sum_squares_ge_zero)
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end
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context linordered_semidom
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begin
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lemma zero_less_power [simp]:
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  "0 < a \<Longrightarrow> 0 < a ^ n"
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  by (induct n) (simp_all add: mult_pos_pos)
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lemma zero_le_power [simp]:
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  "0 \<le> a \<Longrightarrow> 0 \<le> a ^ n"
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  by (induct n) (simp_all add: mult_nonneg_nonneg)
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lemma power_mono:
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  "a \<le> b \<Longrightarrow> 0 \<le> a \<Longrightarrow> a ^ n \<le> b ^ n"
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  by (induct n) (auto intro: mult_mono order_trans [of 0 a b])
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lemma one_le_power [simp]: "1 \<le> a \<Longrightarrow> 1 \<le> a ^ n"
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  using power_mono [of 1 a n] by simp
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lemma power_le_one: "\<lbrakk>0 \<le> a; a \<le> 1\<rbrakk> \<Longrightarrow> a ^ n \<le> 1"
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  using power_mono [of a 1 n] by simp
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lemma power_gt1_lemma:
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  assumes gt1: "1 < a"
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  shows "1 < a * a ^ n"
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proof -
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  from gt1 have "0 \<le> a"
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    by (fact order_trans [OF zero_le_one less_imp_le])
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  have "1 * 1 < a * 1" using gt1 by simp
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  also have "\<dots> \<le> a * a ^ n" using gt1
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    by (simp only: mult_mono `0 \<le> a` one_le_power order_less_imp_le
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        zero_le_one order_refl)
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  finally show ?thesis by simp
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qed
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lemma power_gt1:
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  "1 < a \<Longrightarrow> 1 < a ^ Suc n"
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  by (simp add: power_gt1_lemma)
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lemma one_less_power [simp]:
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  "1 < a \<Longrightarrow> 0 < n \<Longrightarrow> 1 < a ^ n"
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  by (cases n) (simp_all add: power_gt1_lemma)
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lemma power_le_imp_le_exp:
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  assumes gt1: "1 < a"
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  shows "a ^ m \<le> a ^ n \<Longrightarrow> m \<le> n"
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proof (induct m arbitrary: n)
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  case 0
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  show ?case by simp
paulson@14348
   351
next
paulson@14348
   352
  case (Suc m)
wenzelm@14577
   353
  show ?case
wenzelm@14577
   354
  proof (cases n)
wenzelm@14577
   355
    case 0
haftmann@30996
   356
    with Suc.prems Suc.hyps have "a * a ^ m \<le> 1" by simp
wenzelm@14577
   357
    with gt1 show ?thesis
wenzelm@14577
   358
      by (force simp only: power_gt1_lemma
haftmann@30996
   359
          not_less [symmetric])
wenzelm@14577
   360
  next
wenzelm@14577
   361
    case (Suc n)
haftmann@30996
   362
    with Suc.prems Suc.hyps show ?thesis
wenzelm@14577
   363
      by (force dest: mult_left_le_imp_le
haftmann@30996
   364
          simp add: less_trans [OF zero_less_one gt1])
wenzelm@14577
   365
  qed
paulson@14348
   366
qed
paulson@14348
   367
wenzelm@14577
   368
text{*Surely we can strengthen this? It holds for @{text "0<a<1"} too.*}
paulson@14348
   369
lemma power_inject_exp [simp]:
haftmann@30996
   370
  "1 < a \<Longrightarrow> a ^ m = a ^ n \<longleftrightarrow> m = n"
wenzelm@14577
   371
  by (force simp add: order_antisym power_le_imp_le_exp)
paulson@14348
   372
paulson@14348
   373
text{*Can relax the first premise to @{term "0<a"} in the case of the
paulson@14348
   374
natural numbers.*}
paulson@14348
   375
lemma power_less_imp_less_exp:
haftmann@30996
   376
  "1 < a \<Longrightarrow> a ^ m < a ^ n \<Longrightarrow> m < n"
haftmann@30996
   377
  by (simp add: order_less_le [of m n] less_le [of "a^m" "a^n"]
haftmann@30996
   378
    power_le_imp_le_exp)
paulson@14348
   379
paulson@14348
   380
lemma power_strict_mono [rule_format]:
haftmann@30996
   381
  "a < b \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 < n \<longrightarrow> a ^ n < b ^ n"
haftmann@30996
   382
  by (induct n)
haftmann@30996
   383
   (auto simp add: mult_strict_mono le_less_trans [of 0 a b])
paulson@14348
   384
paulson@14348
   385
text{*Lemma for @{text power_strict_decreasing}*}
paulson@14348
   386
lemma power_Suc_less:
haftmann@30996
   387
  "0 < a \<Longrightarrow> a < 1 \<Longrightarrow> a * a ^ n < a ^ n"
haftmann@30996
   388
  by (induct n)
haftmann@30996
   389
    (auto simp add: mult_strict_left_mono)
paulson@14348
   390
haftmann@30996
   391
lemma power_strict_decreasing [rule_format]:
haftmann@30996
   392
  "n < N \<Longrightarrow> 0 < a \<Longrightarrow> a < 1 \<longrightarrow> a ^ N < a ^ n"
haftmann@30996
   393
proof (induct N)
haftmann@30996
   394
  case 0 then show ?case by simp
haftmann@30996
   395
next
haftmann@30996
   396
  case (Suc N) then show ?case 
haftmann@30996
   397
  apply (auto simp add: power_Suc_less less_Suc_eq)
haftmann@30996
   398
  apply (subgoal_tac "a * a^N < 1 * a^n")
haftmann@30996
   399
  apply simp
haftmann@30996
   400
  apply (rule mult_strict_mono) apply auto
haftmann@30996
   401
  done
haftmann@30996
   402
qed
paulson@14348
   403
paulson@14348
   404
text{*Proof resembles that of @{text power_strict_decreasing}*}
haftmann@30996
   405
lemma power_decreasing [rule_format]:
haftmann@30996
   406
  "n \<le> N \<Longrightarrow> 0 \<le> a \<Longrightarrow> a \<le> 1 \<longrightarrow> a ^ N \<le> a ^ n"
haftmann@30996
   407
proof (induct N)
haftmann@30996
   408
  case 0 then show ?case by simp
haftmann@30996
   409
next
haftmann@30996
   410
  case (Suc N) then show ?case 
haftmann@30996
   411
  apply (auto simp add: le_Suc_eq)
haftmann@30996
   412
  apply (subgoal_tac "a * a^N \<le> 1 * a^n", simp)
haftmann@30996
   413
  apply (rule mult_mono) apply auto
haftmann@30996
   414
  done
haftmann@30996
   415
qed
paulson@14348
   416
paulson@14348
   417
lemma power_Suc_less_one:
haftmann@30996
   418
  "0 < a \<Longrightarrow> a < 1 \<Longrightarrow> a ^ Suc n < 1"
haftmann@30996
   419
  using power_strict_decreasing [of 0 "Suc n" a] by simp
paulson@14348
   420
paulson@14348
   421
text{*Proof again resembles that of @{text power_strict_decreasing}*}
haftmann@30996
   422
lemma power_increasing [rule_format]:
haftmann@30996
   423
  "n \<le> N \<Longrightarrow> 1 \<le> a \<Longrightarrow> a ^ n \<le> a ^ N"
haftmann@30996
   424
proof (induct N)
haftmann@30996
   425
  case 0 then show ?case by simp
haftmann@30996
   426
next
haftmann@30996
   427
  case (Suc N) then show ?case 
haftmann@30996
   428
  apply (auto simp add: le_Suc_eq)
haftmann@30996
   429
  apply (subgoal_tac "1 * a^n \<le> a * a^N", simp)
haftmann@30996
   430
  apply (rule mult_mono) apply (auto simp add: order_trans [OF zero_le_one])
haftmann@30996
   431
  done
haftmann@30996
   432
qed
paulson@14348
   433
paulson@14348
   434
text{*Lemma for @{text power_strict_increasing}*}
paulson@14348
   435
lemma power_less_power_Suc:
haftmann@30996
   436
  "1 < a \<Longrightarrow> a ^ n < a * a ^ n"
haftmann@30996
   437
  by (induct n) (auto simp add: mult_strict_left_mono less_trans [OF zero_less_one])
paulson@14348
   438
haftmann@30996
   439
lemma power_strict_increasing [rule_format]:
haftmann@30996
   440
  "n < N \<Longrightarrow> 1 < a \<longrightarrow> a ^ n < a ^ N"
haftmann@30996
   441
proof (induct N)
haftmann@30996
   442
  case 0 then show ?case by simp
haftmann@30996
   443
next
haftmann@30996
   444
  case (Suc N) then show ?case 
haftmann@30996
   445
  apply (auto simp add: power_less_power_Suc less_Suc_eq)
haftmann@30996
   446
  apply (subgoal_tac "1 * a^n < a * a^N", simp)
haftmann@30996
   447
  apply (rule mult_strict_mono) apply (auto simp add: less_trans [OF zero_less_one] less_imp_le)
haftmann@30996
   448
  done
haftmann@30996
   449
qed
paulson@14348
   450
nipkow@25134
   451
lemma power_increasing_iff [simp]:
haftmann@30996
   452
  "1 < b \<Longrightarrow> b ^ x \<le> b ^ y \<longleftrightarrow> x \<le> y"
haftmann@30996
   453
  by (blast intro: power_le_imp_le_exp power_increasing less_imp_le)
paulson@15066
   454
paulson@15066
   455
lemma power_strict_increasing_iff [simp]:
haftmann@30996
   456
  "1 < b \<Longrightarrow> b ^ x < b ^ y \<longleftrightarrow> x < y"
nipkow@25134
   457
by (blast intro: power_less_imp_less_exp power_strict_increasing) 
paulson@15066
   458
paulson@14348
   459
lemma power_le_imp_le_base:
haftmann@30996
   460
  assumes le: "a ^ Suc n \<le> b ^ Suc n"
haftmann@30996
   461
    and ynonneg: "0 \<le> b"
haftmann@30996
   462
  shows "a \<le> b"
nipkow@25134
   463
proof (rule ccontr)
nipkow@25134
   464
  assume "~ a \<le> b"
nipkow@25134
   465
  then have "b < a" by (simp only: linorder_not_le)
nipkow@25134
   466
  then have "b ^ Suc n < a ^ Suc n"
wenzelm@41550
   467
    by (simp only: assms power_strict_mono)
haftmann@30996
   468
  from le and this show False
nipkow@25134
   469
    by (simp add: linorder_not_less [symmetric])
nipkow@25134
   470
qed
wenzelm@14577
   471
huffman@22853
   472
lemma power_less_imp_less_base:
huffman@22853
   473
  assumes less: "a ^ n < b ^ n"
huffman@22853
   474
  assumes nonneg: "0 \<le> b"
huffman@22853
   475
  shows "a < b"
huffman@22853
   476
proof (rule contrapos_pp [OF less])
huffman@22853
   477
  assume "~ a < b"
huffman@22853
   478
  hence "b \<le> a" by (simp only: linorder_not_less)
huffman@22853
   479
  hence "b ^ n \<le> a ^ n" using nonneg by (rule power_mono)
haftmann@30996
   480
  thus "\<not> a ^ n < b ^ n" by (simp only: linorder_not_less)
huffman@22853
   481
qed
huffman@22853
   482
paulson@14348
   483
lemma power_inject_base:
haftmann@30996
   484
  "a ^ Suc n = b ^ Suc n \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> a = b"
haftmann@30996
   485
by (blast intro: power_le_imp_le_base antisym eq_refl sym)
paulson@14348
   486
huffman@22955
   487
lemma power_eq_imp_eq_base:
haftmann@30996
   488
  "a ^ n = b ^ n \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 < n \<Longrightarrow> a = b"
haftmann@30996
   489
  by (cases n) (simp_all del: power_Suc, rule power_inject_base)
huffman@22955
   490
huffman@47192
   491
lemma power2_le_imp_le:
wenzelm@53015
   492
  "x\<^sup>2 \<le> y\<^sup>2 \<Longrightarrow> 0 \<le> y \<Longrightarrow> x \<le> y"
huffman@47192
   493
  unfolding numeral_2_eq_2 by (rule power_le_imp_le_base)
huffman@47192
   494
huffman@47192
   495
lemma power2_less_imp_less:
wenzelm@53015
   496
  "x\<^sup>2 < y\<^sup>2 \<Longrightarrow> 0 \<le> y \<Longrightarrow> x < y"
huffman@47192
   497
  by (rule power_less_imp_less_base)
huffman@47192
   498
huffman@47192
   499
lemma power2_eq_imp_eq:
wenzelm@53015
   500
  "x\<^sup>2 = y\<^sup>2 \<Longrightarrow> 0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> x = y"
huffman@47192
   501
  unfolding numeral_2_eq_2 by (erule (2) power_eq_imp_eq_base) simp
huffman@47192
   502
huffman@47192
   503
end
huffman@47192
   504
huffman@47192
   505
context linordered_ring_strict
huffman@47192
   506
begin
huffman@47192
   507
huffman@47192
   508
lemma sum_squares_eq_zero_iff:
huffman@47192
   509
  "x * x + y * y = 0 \<longleftrightarrow> x = 0 \<and> y = 0"
huffman@47192
   510
  by (simp add: add_nonneg_eq_0_iff)
huffman@47192
   511
huffman@47192
   512
lemma sum_squares_le_zero_iff:
huffman@47192
   513
  "x * x + y * y \<le> 0 \<longleftrightarrow> x = 0 \<and> y = 0"
huffman@47192
   514
  by (simp add: le_less not_sum_squares_lt_zero sum_squares_eq_zero_iff)
huffman@47192
   515
huffman@47192
   516
lemma sum_squares_gt_zero_iff:
huffman@47192
   517
  "0 < x * x + y * y \<longleftrightarrow> x \<noteq> 0 \<or> y \<noteq> 0"
huffman@47192
   518
  by (simp add: not_le [symmetric] sum_squares_le_zero_iff)
huffman@47192
   519
haftmann@30996
   520
end
haftmann@30996
   521
haftmann@35028
   522
context linordered_idom
haftmann@30996
   523
begin
huffman@29978
   524
haftmann@30996
   525
lemma power_abs:
haftmann@30996
   526
  "abs (a ^ n) = abs a ^ n"
haftmann@30996
   527
  by (induct n) (auto simp add: abs_mult)
haftmann@30996
   528
haftmann@30996
   529
lemma abs_power_minus [simp]:
haftmann@30996
   530
  "abs ((-a) ^ n) = abs (a ^ n)"
huffman@35216
   531
  by (simp add: power_abs)
haftmann@30996
   532
blanchet@35828
   533
lemma zero_less_power_abs_iff [simp, no_atp]:
haftmann@30996
   534
  "0 < abs a ^ n \<longleftrightarrow> a \<noteq> 0 \<or> n = 0"
haftmann@30996
   535
proof (induct n)
haftmann@30996
   536
  case 0 show ?case by simp
haftmann@30996
   537
next
haftmann@30996
   538
  case (Suc n) show ?case by (auto simp add: Suc zero_less_mult_iff)
huffman@29978
   539
qed
huffman@29978
   540
haftmann@30996
   541
lemma zero_le_power_abs [simp]:
haftmann@30996
   542
  "0 \<le> abs a ^ n"
haftmann@30996
   543
  by (rule zero_le_power [OF abs_ge_zero])
haftmann@30996
   544
huffman@47192
   545
lemma zero_le_power2 [simp]:
wenzelm@53015
   546
  "0 \<le> a\<^sup>2"
huffman@47192
   547
  by (simp add: power2_eq_square)
huffman@47192
   548
huffman@47192
   549
lemma zero_less_power2 [simp]:
wenzelm@53015
   550
  "0 < a\<^sup>2 \<longleftrightarrow> a \<noteq> 0"
huffman@47192
   551
  by (force simp add: power2_eq_square zero_less_mult_iff linorder_neq_iff)
huffman@47192
   552
huffman@47192
   553
lemma power2_less_0 [simp]:
wenzelm@53015
   554
  "\<not> a\<^sup>2 < 0"
huffman@47192
   555
  by (force simp add: power2_eq_square mult_less_0_iff)
huffman@47192
   556
huffman@47192
   557
lemma abs_power2 [simp]:
wenzelm@53015
   558
  "abs (a\<^sup>2) = a\<^sup>2"
huffman@47192
   559
  by (simp add: power2_eq_square abs_mult abs_mult_self)
huffman@47192
   560
huffman@47192
   561
lemma power2_abs [simp]:
wenzelm@53015
   562
  "(abs a)\<^sup>2 = a\<^sup>2"
huffman@47192
   563
  by (simp add: power2_eq_square abs_mult_self)
huffman@47192
   564
huffman@47192
   565
lemma odd_power_less_zero:
huffman@47192
   566
  "a < 0 \<Longrightarrow> a ^ Suc (2*n) < 0"
huffman@47192
   567
proof (induct n)
huffman@47192
   568
  case 0
huffman@47192
   569
  then show ?case by simp
huffman@47192
   570
next
huffman@47192
   571
  case (Suc n)
huffman@47192
   572
  have "a ^ Suc (2 * Suc n) = (a*a) * a ^ Suc(2*n)"
huffman@47192
   573
    by (simp add: mult_ac power_add power2_eq_square)
huffman@47192
   574
  thus ?case
huffman@47192
   575
    by (simp del: power_Suc add: Suc mult_less_0_iff mult_neg_neg)
huffman@47192
   576
qed
haftmann@30996
   577
huffman@47192
   578
lemma odd_0_le_power_imp_0_le:
huffman@47192
   579
  "0 \<le> a ^ Suc (2*n) \<Longrightarrow> 0 \<le> a"
huffman@47192
   580
  using odd_power_less_zero [of a n]
huffman@47192
   581
    by (force simp add: linorder_not_less [symmetric]) 
huffman@47192
   582
huffman@47192
   583
lemma zero_le_even_power'[simp]:
huffman@47192
   584
  "0 \<le> a ^ (2*n)"
huffman@47192
   585
proof (induct n)
huffman@47192
   586
  case 0
huffman@47192
   587
    show ?case by simp
huffman@47192
   588
next
huffman@47192
   589
  case (Suc n)
huffman@47192
   590
    have "a ^ (2 * Suc n) = (a*a) * a ^ (2*n)" 
huffman@47192
   591
      by (simp add: mult_ac power_add power2_eq_square)
huffman@47192
   592
    thus ?case
huffman@47192
   593
      by (simp add: Suc zero_le_mult_iff)
huffman@47192
   594
qed
haftmann@30996
   595
huffman@47192
   596
lemma sum_power2_ge_zero:
wenzelm@53015
   597
  "0 \<le> x\<^sup>2 + y\<^sup>2"
huffman@47192
   598
  by (intro add_nonneg_nonneg zero_le_power2)
huffman@47192
   599
huffman@47192
   600
lemma not_sum_power2_lt_zero:
wenzelm@53015
   601
  "\<not> x\<^sup>2 + y\<^sup>2 < 0"
huffman@47192
   602
  unfolding not_less by (rule sum_power2_ge_zero)
huffman@47192
   603
huffman@47192
   604
lemma sum_power2_eq_zero_iff:
wenzelm@53015
   605
  "x\<^sup>2 + y\<^sup>2 = 0 \<longleftrightarrow> x = 0 \<and> y = 0"
huffman@47192
   606
  unfolding power2_eq_square by (simp add: add_nonneg_eq_0_iff)
huffman@47192
   607
huffman@47192
   608
lemma sum_power2_le_zero_iff:
wenzelm@53015
   609
  "x\<^sup>2 + y\<^sup>2 \<le> 0 \<longleftrightarrow> x = 0 \<and> y = 0"
huffman@47192
   610
  by (simp add: le_less sum_power2_eq_zero_iff not_sum_power2_lt_zero)
huffman@47192
   611
huffman@47192
   612
lemma sum_power2_gt_zero_iff:
wenzelm@53015
   613
  "0 < x\<^sup>2 + y\<^sup>2 \<longleftrightarrow> x \<noteq> 0 \<or> y \<noteq> 0"
huffman@47192
   614
  unfolding not_le [symmetric] by (simp add: sum_power2_le_zero_iff)
haftmann@30996
   615
haftmann@30996
   616
end
haftmann@30996
   617
huffman@29978
   618
huffman@47192
   619
subsection {* Miscellaneous rules *}
paulson@14348
   620
huffman@47255
   621
lemma power_eq_if: "p ^ m = (if m=0 then 1 else p * (p ^ (m - 1)))"
huffman@47255
   622
  unfolding One_nat_def by (cases m) simp_all
huffman@47255
   623
huffman@47192
   624
lemma power2_sum:
huffman@47192
   625
  fixes x y :: "'a::comm_semiring_1"
wenzelm@53015
   626
  shows "(x + y)\<^sup>2 = x\<^sup>2 + y\<^sup>2 + 2 * x * y"
huffman@47192
   627
  by (simp add: algebra_simps power2_eq_square mult_2_right)
haftmann@30996
   628
huffman@47192
   629
lemma power2_diff:
huffman@47192
   630
  fixes x y :: "'a::comm_ring_1"
wenzelm@53015
   631
  shows "(x - y)\<^sup>2 = x\<^sup>2 + y\<^sup>2 - 2 * x * y"
huffman@47192
   632
  by (simp add: ring_distribs power2_eq_square mult_2) (rule mult_commute)
haftmann@30996
   633
haftmann@30996
   634
lemma power_0_Suc [simp]:
haftmann@30996
   635
  "(0::'a::{power, semiring_0}) ^ Suc n = 0"
haftmann@30996
   636
  by simp
nipkow@30313
   637
haftmann@30996
   638
text{*It looks plausible as a simprule, but its effect can be strange.*}
haftmann@30996
   639
lemma power_0_left:
haftmann@30996
   640
  "0 ^ n = (if n = 0 then 1 else (0::'a::{power, semiring_0}))"
haftmann@30996
   641
  by (induct n) simp_all
haftmann@30996
   642
haftmann@30996
   643
lemma power_eq_0_iff [simp]:
haftmann@30996
   644
  "a ^ n = 0 \<longleftrightarrow>
haftmann@30996
   645
     a = (0::'a::{mult_zero,zero_neq_one,no_zero_divisors,power}) \<and> n \<noteq> 0"
haftmann@30996
   646
  by (induct n)
haftmann@30996
   647
    (auto simp add: no_zero_divisors elim: contrapos_pp)
haftmann@30996
   648
haftmann@36409
   649
lemma (in field) power_diff:
haftmann@30996
   650
  assumes nz: "a \<noteq> 0"
haftmann@30996
   651
  shows "n \<le> m \<Longrightarrow> a ^ (m - n) = a ^ m / a ^ n"
haftmann@36409
   652
  by (induct m n rule: diff_induct) (simp_all add: nz field_power_not_zero)
nipkow@30313
   653
haftmann@30996
   654
text{*Perhaps these should be simprules.*}
haftmann@30996
   655
lemma power_inverse:
haftmann@36409
   656
  fixes a :: "'a::division_ring_inverse_zero"
haftmann@36409
   657
  shows "inverse (a ^ n) = inverse a ^ n"
haftmann@30996
   658
apply (cases "a = 0")
haftmann@30996
   659
apply (simp add: power_0_left)
haftmann@30996
   660
apply (simp add: nonzero_power_inverse)
haftmann@30996
   661
done (* TODO: reorient or rename to inverse_power *)
haftmann@30996
   662
haftmann@30996
   663
lemma power_one_over:
haftmann@36409
   664
  "1 / (a::'a::{field_inverse_zero, power}) ^ n =  (1 / a) ^ n"
haftmann@30996
   665
  by (simp add: divide_inverse) (rule power_inverse)
haftmann@30996
   666
haftmann@30996
   667
lemma power_divide:
haftmann@36409
   668
  "(a / b) ^ n = (a::'a::field_inverse_zero) ^ n / b ^ n"
haftmann@30996
   669
apply (cases "b = 0")
haftmann@30996
   670
apply (simp add: power_0_left)
haftmann@30996
   671
apply (rule nonzero_power_divide)
haftmann@30996
   672
apply assumption
nipkow@30313
   673
done
nipkow@30313
   674
huffman@47255
   675
text {* Simprules for comparisons where common factors can be cancelled. *}
huffman@47255
   676
huffman@47255
   677
lemmas zero_compare_simps =
huffman@47255
   678
    add_strict_increasing add_strict_increasing2 add_increasing
huffman@47255
   679
    zero_le_mult_iff zero_le_divide_iff 
huffman@47255
   680
    zero_less_mult_iff zero_less_divide_iff 
huffman@47255
   681
    mult_le_0_iff divide_le_0_iff 
huffman@47255
   682
    mult_less_0_iff divide_less_0_iff 
huffman@47255
   683
    zero_le_power2 power2_less_0
huffman@47255
   684
nipkow@30313
   685
haftmann@30960
   686
subsection {* Exponentiation for the Natural Numbers *}
wenzelm@14577
   687
haftmann@30996
   688
lemma nat_one_le_power [simp]:
haftmann@30996
   689
  "Suc 0 \<le> i \<Longrightarrow> Suc 0 \<le> i ^ n"
haftmann@30996
   690
  by (rule one_le_power [of i n, unfolded One_nat_def])
huffman@23305
   691
haftmann@30996
   692
lemma nat_zero_less_power_iff [simp]:
haftmann@30996
   693
  "x ^ n > 0 \<longleftrightarrow> x > (0::nat) \<or> n = 0"
haftmann@30996
   694
  by (induct n) auto
paulson@14348
   695
nipkow@30056
   696
lemma nat_power_eq_Suc_0_iff [simp]: 
haftmann@30996
   697
  "x ^ m = Suc 0 \<longleftrightarrow> m = 0 \<or> x = Suc 0"
haftmann@30996
   698
  by (induct m) auto
nipkow@30056
   699
haftmann@30996
   700
lemma power_Suc_0 [simp]:
haftmann@30996
   701
  "Suc 0 ^ n = Suc 0"
haftmann@30996
   702
  by simp
nipkow@30056
   703
paulson@14348
   704
text{*Valid for the naturals, but what if @{text"0<i<1"}?
paulson@14348
   705
Premises cannot be weakened: consider the case where @{term "i=0"},
paulson@14348
   706
@{term "m=1"} and @{term "n=0"}.*}
haftmann@21413
   707
lemma nat_power_less_imp_less:
haftmann@21413
   708
  assumes nonneg: "0 < (i\<Colon>nat)"
haftmann@30996
   709
  assumes less: "i ^ m < i ^ n"
haftmann@21413
   710
  shows "m < n"
haftmann@21413
   711
proof (cases "i = 1")
haftmann@21413
   712
  case True with less power_one [where 'a = nat] show ?thesis by simp
haftmann@21413
   713
next
haftmann@21413
   714
  case False with nonneg have "1 < i" by auto
haftmann@21413
   715
  from power_strict_increasing_iff [OF this] less show ?thesis ..
haftmann@21413
   716
qed
paulson@14348
   717
haftmann@33274
   718
lemma power_dvd_imp_le:
haftmann@33274
   719
  "i ^ m dvd i ^ n \<Longrightarrow> (1::nat) < i \<Longrightarrow> m \<le> n"
haftmann@33274
   720
  apply (rule power_le_imp_le_exp, assumption)
haftmann@33274
   721
  apply (erule dvd_imp_le, simp)
haftmann@33274
   722
  done
haftmann@33274
   723
haftmann@51263
   724
lemma power2_nat_le_eq_le:
haftmann@51263
   725
  fixes m n :: nat
wenzelm@53015
   726
  shows "m\<^sup>2 \<le> n\<^sup>2 \<longleftrightarrow> m \<le> n"
haftmann@51263
   727
  by (auto intro: power2_le_imp_le power_mono)
haftmann@51263
   728
haftmann@51263
   729
lemma power2_nat_le_imp_le:
haftmann@51263
   730
  fixes m n :: nat
wenzelm@53015
   731
  assumes "m\<^sup>2 \<le> n"
haftmann@51263
   732
  shows "m \<le> n"
haftmann@51263
   733
  using assms by (cases m) (simp_all add: power2_eq_square)
haftmann@51263
   734
haftmann@51263
   735
haftmann@31155
   736
haftmann@31155
   737
subsection {* Code generator tweak *}
haftmann@31155
   738
bulwahn@45231
   739
lemma power_power_power [code]:
haftmann@31155
   740
  "power = power.power (1::'a::{power}) (op *)"
haftmann@31155
   741
  unfolding power_def power.power_def ..
haftmann@31155
   742
haftmann@31155
   743
declare power.power.simps [code]
haftmann@31155
   744
haftmann@52435
   745
code_identifier
haftmann@52435
   746
  code_module Power \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
haftmann@33364
   747
paulson@3390
   748
end
haftmann@49824
   749