src/HOL/Power.thy
author paulson <lp15@cam.ac.uk>
Wed Mar 18 14:13:27 2015 +0000 (2015-03-18)
changeset 59741 5b762cd73a8e
parent 59009 348561aa3869
child 59865 8a20dd967385
permissions -rw-r--r--
Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
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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 {* Exponentiation *}
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theory Power
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imports Num Equiv_Relations
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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_Suc0_right [simp]:
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  "a ^ Suc 0 = a"
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  by simp
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lemma power_commutes:
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  "a ^ n * a = a * a ^ n"
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  by (induct n) (simp_all add: mult.assoc)
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lemma power_Suc2:
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  "a ^ Suc n = a ^ n * a"
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  by (simp add: power_commutes)
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lemma power_add:
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  "a ^ (m + n) = a ^ m * a ^ n"
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  by (induct m) (simp_all add: algebra_simps)
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lemma power_mult:
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  "a ^ (m * n) = (a ^ m) ^ n"
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  by (induct n) (simp_all add: power_add)
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lemma power2_eq_square: "a\<^sup>2 = a * a"
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  by (simp add: numeral_2_eq_2)
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lemma power3_eq_cube: "a ^ 3 = a * a * a"
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  by (simp add: numeral_3_eq_3 mult.assoc)
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lemma power_even_eq:
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  "a ^ (2 * n) = (a ^ n)\<^sup>2"
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  by (subst mult.commute) (simp add: power_mult)
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lemma power_odd_eq:
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  "a ^ Suc (2*n) = a * (a ^ n)\<^sup>2"
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  by (simp add: power_even_eq)
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lemma power_numeral_even:
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  "z ^ numeral (Num.Bit0 w) = (let w = z ^ (numeral w) in w * w)"
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  unfolding numeral_Bit0 power_add Let_def ..
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lemma power_numeral_odd:
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  "z ^ numeral (Num.Bit1 w) = (let w = z ^ (numeral w) in z * w * w)"
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  unfolding numeral_Bit1 One_nat_def add_Suc_right add_0_right
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  unfolding power_Suc power_add Let_def mult.assoc ..
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lemma funpow_times_power:
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  "(times x ^^ f x) = times (x ^ f x)"
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proof (induct "f x" arbitrary: f)
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  case 0 then show ?case by (simp add: fun_eq_iff)
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next
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  case (Suc n)
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  def g \<equiv> "\<lambda>x. f x - 1"
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  with Suc have "n = g x" by simp
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  with Suc have "times x ^^ g x = times (x ^ g x)" by simp
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  moreover from Suc g_def have "f x = g x + 1" by simp
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  ultimately show ?case by (simp add: power_add funpow_add fun_eq_iff mult.assoc)
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qed
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lemma power_commuting_commutes:
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  assumes "x * y = y * x"
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  shows "x ^ n * y = y * x ^n"
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proof (induct n)
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  case (Suc n)
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  have "x ^ Suc n * y = x ^ n * y * x"
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    by (subst power_Suc2) (simp add: assms ac_simps)
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  also have "\<dots> = y * x ^ Suc n"
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    unfolding Suc power_Suc2
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    by (simp add: ac_simps)
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  finally show ?case .
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qed simp
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end
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context comm_monoid_mult
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begin
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lemma power_mult_distrib [field_simps]:
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  "(a * b) ^ n = (a ^ n) * (b ^ n)"
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  by (induct n) (simp_all add: ac_simps)
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end
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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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context semiring_numeral
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begin
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lemma numeral_sqr: "numeral (Num.sqr k) = numeral k * numeral k"
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  by (simp only: sqr_conv_mult numeral_mult)
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lemma numeral_pow: "numeral (Num.pow k l) = numeral k ^ numeral l"
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  by (induct l, simp_all only: numeral_class.numeral.simps pow.simps
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    numeral_sqr numeral_mult power_add power_one_right)
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lemma power_numeral [simp]: "numeral k ^ numeral l = numeral (Num.pow k l)"
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  by (rule numeral_pow [symmetric])
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end
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context semiring_1
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begin
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lemma of_nat_power:
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  "of_nat (m ^ n) = of_nat m ^ n"
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  by (induct n) (simp_all add: of_nat_mult)
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lemma zero_power:
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  "0 < n \<Longrightarrow> 0 ^ n = 0"
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  by (cases n) simp_all
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lemma power_zero_numeral [simp]:
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  "0 ^ numeral k = 0"
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  by (simp add: numeral_eq_Suc)
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lemma zero_power2: "0\<^sup>2 = 0" (* delete? *)
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  by (rule power_zero_numeral)
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lemma one_power2: "1\<^sup>2 = 1" (* delete? *)
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  by (rule power_one)
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end
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context comm_semiring_1
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begin
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text {* 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 power2_minus [simp]:
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  "(- a)\<^sup>2 = a\<^sup>2"
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  by (rule power_minus_Bit0)
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lemma power_minus1_even [simp]:
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  "(- 1) ^ (2*n) = 1"
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proof (induct n)
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  case 0 show ?case by simp
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next
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  case (Suc n) then show ?case by (simp add: power_add power2_eq_square)
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qed
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lemma power_minus1_odd:
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  "(- 1) ^ Suc (2*n) = -1"
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  by simp
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lemma power_minus_even [simp]:
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  "(-a) ^ (2*n) = a ^ (2*n)"
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  by (simp add: power_minus [of a])
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end
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lemma power_eq_0_nat_iff [simp]:
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  fixes m n :: nat
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  shows "m ^ n = 0 \<longleftrightarrow> m = 0 \<and> n > 0"
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  by (induct n) auto
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context ring_1_no_zero_divisors
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begin
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lemma power_eq_0_iff [simp]:
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  "a ^ n = 0 \<longleftrightarrow> a = 0 \<and> n > 0"
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  by (induct n) auto
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lemma field_power_not_zero:
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  "a \<noteq> 0 \<Longrightarrow> a ^ n \<noteq> 0"
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  by (induct n) auto
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lemma zero_eq_power2 [simp]:
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  "a\<^sup>2 = 0 \<longleftrightarrow> a = 0"
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  unfolding power2_eq_square by simp
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lemma power2_eq_1_iff:
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  "a\<^sup>2 = 1 \<longleftrightarrow> a = 1 \<or> a = - 1"
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  unfolding power2_eq_square by (rule square_eq_1_iff)
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end
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context idom
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begin
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lemma power2_eq_iff: "x\<^sup>2 = y\<^sup>2 \<longleftrightarrow> x = y \<or> x = - y"
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  unfolding power2_eq_square by (rule square_eq_iff)
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end
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context 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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declare nonzero_power_divide [where b = "numeral w" for w, simp]
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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
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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
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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 -
haftmann@30996
   356
  from gt1 have "0 \<le> a"
haftmann@30996
   357
    by (fact order_trans [OF zero_le_one less_imp_le])
haftmann@30996
   358
  have "1 * 1 < a * 1" using gt1 by simp
haftmann@30996
   359
  also have "\<dots> \<le> a * a ^ n" using gt1
haftmann@30996
   360
    by (simp only: mult_mono `0 \<le> a` one_le_power order_less_imp_le
wenzelm@14577
   361
        zero_le_one order_refl)
wenzelm@14577
   362
  finally show ?thesis by simp
paulson@14348
   363
qed
paulson@14348
   364
haftmann@30996
   365
lemma power_gt1:
haftmann@30996
   366
  "1 < a \<Longrightarrow> 1 < a ^ Suc n"
haftmann@30996
   367
  by (simp add: power_gt1_lemma)
huffman@24376
   368
haftmann@30996
   369
lemma one_less_power [simp]:
haftmann@30996
   370
  "1 < a \<Longrightarrow> 0 < n \<Longrightarrow> 1 < a ^ n"
haftmann@30996
   371
  by (cases n) (simp_all add: power_gt1_lemma)
paulson@14348
   372
paulson@14348
   373
lemma power_le_imp_le_exp:
haftmann@30996
   374
  assumes gt1: "1 < a"
haftmann@30996
   375
  shows "a ^ m \<le> a ^ n \<Longrightarrow> m \<le> n"
haftmann@30996
   376
proof (induct m arbitrary: n)
paulson@14348
   377
  case 0
wenzelm@14577
   378
  show ?case by simp
paulson@14348
   379
next
paulson@14348
   380
  case (Suc m)
wenzelm@14577
   381
  show ?case
wenzelm@14577
   382
  proof (cases n)
wenzelm@14577
   383
    case 0
haftmann@30996
   384
    with Suc.prems Suc.hyps have "a * a ^ m \<le> 1" by simp
wenzelm@14577
   385
    with gt1 show ?thesis
wenzelm@14577
   386
      by (force simp only: power_gt1_lemma
haftmann@30996
   387
          not_less [symmetric])
wenzelm@14577
   388
  next
wenzelm@14577
   389
    case (Suc n)
haftmann@30996
   390
    with Suc.prems Suc.hyps show ?thesis
wenzelm@14577
   391
      by (force dest: mult_left_le_imp_le
haftmann@30996
   392
          simp add: less_trans [OF zero_less_one gt1])
wenzelm@14577
   393
  qed
paulson@14348
   394
qed
paulson@14348
   395
wenzelm@14577
   396
text{*Surely we can strengthen this? It holds for @{text "0<a<1"} too.*}
paulson@14348
   397
lemma power_inject_exp [simp]:
haftmann@30996
   398
  "1 < a \<Longrightarrow> a ^ m = a ^ n \<longleftrightarrow> m = n"
wenzelm@14577
   399
  by (force simp add: order_antisym power_le_imp_le_exp)
paulson@14348
   400
paulson@14348
   401
text{*Can relax the first premise to @{term "0<a"} in the case of the
paulson@14348
   402
natural numbers.*}
paulson@14348
   403
lemma power_less_imp_less_exp:
haftmann@30996
   404
  "1 < a \<Longrightarrow> a ^ m < a ^ n \<Longrightarrow> m < n"
haftmann@30996
   405
  by (simp add: order_less_le [of m n] less_le [of "a^m" "a^n"]
haftmann@30996
   406
    power_le_imp_le_exp)
paulson@14348
   407
paulson@14348
   408
lemma power_strict_mono [rule_format]:
haftmann@30996
   409
  "a < b \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 < n \<longrightarrow> a ^ n < b ^ n"
haftmann@30996
   410
  by (induct n)
haftmann@30996
   411
   (auto simp add: mult_strict_mono le_less_trans [of 0 a b])
paulson@14348
   412
paulson@14348
   413
text{*Lemma for @{text power_strict_decreasing}*}
paulson@14348
   414
lemma power_Suc_less:
haftmann@30996
   415
  "0 < a \<Longrightarrow> a < 1 \<Longrightarrow> a * a ^ n < a ^ n"
haftmann@30996
   416
  by (induct n)
haftmann@30996
   417
    (auto simp add: mult_strict_left_mono)
paulson@14348
   418
haftmann@30996
   419
lemma power_strict_decreasing [rule_format]:
haftmann@30996
   420
  "n < N \<Longrightarrow> 0 < a \<Longrightarrow> a < 1 \<longrightarrow> a ^ N < a ^ n"
haftmann@30996
   421
proof (induct N)
haftmann@30996
   422
  case 0 then show ?case by simp
haftmann@30996
   423
next
haftmann@30996
   424
  case (Suc N) then show ?case 
haftmann@30996
   425
  apply (auto simp add: power_Suc_less less_Suc_eq)
haftmann@30996
   426
  apply (subgoal_tac "a * a^N < 1 * a^n")
haftmann@30996
   427
  apply simp
haftmann@30996
   428
  apply (rule mult_strict_mono) apply auto
haftmann@30996
   429
  done
haftmann@30996
   430
qed
paulson@14348
   431
paulson@14348
   432
text{*Proof resembles that of @{text power_strict_decreasing}*}
haftmann@30996
   433
lemma power_decreasing [rule_format]:
haftmann@30996
   434
  "n \<le> N \<Longrightarrow> 0 \<le> a \<Longrightarrow> a \<le> 1 \<longrightarrow> a ^ N \<le> a ^ n"
haftmann@30996
   435
proof (induct N)
haftmann@30996
   436
  case 0 then show ?case by simp
haftmann@30996
   437
next
haftmann@30996
   438
  case (Suc N) then show ?case 
haftmann@30996
   439
  apply (auto simp add: le_Suc_eq)
haftmann@30996
   440
  apply (subgoal_tac "a * a^N \<le> 1 * a^n", simp)
haftmann@30996
   441
  apply (rule mult_mono) apply auto
haftmann@30996
   442
  done
haftmann@30996
   443
qed
paulson@14348
   444
paulson@14348
   445
lemma power_Suc_less_one:
haftmann@30996
   446
  "0 < a \<Longrightarrow> a < 1 \<Longrightarrow> a ^ Suc n < 1"
haftmann@30996
   447
  using power_strict_decreasing [of 0 "Suc n" a] by simp
paulson@14348
   448
paulson@14348
   449
text{*Proof again resembles that of @{text power_strict_decreasing}*}
haftmann@30996
   450
lemma power_increasing [rule_format]:
haftmann@30996
   451
  "n \<le> N \<Longrightarrow> 1 \<le> a \<Longrightarrow> a ^ n \<le> a ^ N"
haftmann@30996
   452
proof (induct N)
haftmann@30996
   453
  case 0 then show ?case by simp
haftmann@30996
   454
next
haftmann@30996
   455
  case (Suc N) then show ?case 
haftmann@30996
   456
  apply (auto simp add: le_Suc_eq)
haftmann@30996
   457
  apply (subgoal_tac "1 * a^n \<le> a * a^N", simp)
haftmann@30996
   458
  apply (rule mult_mono) apply (auto simp add: order_trans [OF zero_le_one])
haftmann@30996
   459
  done
haftmann@30996
   460
qed
paulson@14348
   461
paulson@14348
   462
text{*Lemma for @{text power_strict_increasing}*}
paulson@14348
   463
lemma power_less_power_Suc:
haftmann@30996
   464
  "1 < a \<Longrightarrow> a ^ n < a * a ^ n"
haftmann@30996
   465
  by (induct n) (auto simp add: mult_strict_left_mono less_trans [OF zero_less_one])
paulson@14348
   466
haftmann@30996
   467
lemma power_strict_increasing [rule_format]:
haftmann@30996
   468
  "n < N \<Longrightarrow> 1 < a \<longrightarrow> a ^ n < a ^ N"
haftmann@30996
   469
proof (induct N)
haftmann@30996
   470
  case 0 then show ?case by simp
haftmann@30996
   471
next
haftmann@30996
   472
  case (Suc N) then show ?case 
haftmann@30996
   473
  apply (auto simp add: power_less_power_Suc less_Suc_eq)
haftmann@30996
   474
  apply (subgoal_tac "1 * a^n < a * a^N", simp)
haftmann@30996
   475
  apply (rule mult_strict_mono) apply (auto simp add: less_trans [OF zero_less_one] less_imp_le)
haftmann@30996
   476
  done
haftmann@30996
   477
qed
paulson@14348
   478
nipkow@25134
   479
lemma power_increasing_iff [simp]:
haftmann@30996
   480
  "1 < b \<Longrightarrow> b ^ x \<le> b ^ y \<longleftrightarrow> x \<le> y"
haftmann@30996
   481
  by (blast intro: power_le_imp_le_exp power_increasing less_imp_le)
paulson@15066
   482
paulson@15066
   483
lemma power_strict_increasing_iff [simp]:
haftmann@30996
   484
  "1 < b \<Longrightarrow> b ^ x < b ^ y \<longleftrightarrow> x < y"
nipkow@25134
   485
by (blast intro: power_less_imp_less_exp power_strict_increasing) 
paulson@15066
   486
paulson@14348
   487
lemma power_le_imp_le_base:
haftmann@30996
   488
  assumes le: "a ^ Suc n \<le> b ^ Suc n"
haftmann@30996
   489
    and ynonneg: "0 \<le> b"
haftmann@30996
   490
  shows "a \<le> b"
nipkow@25134
   491
proof (rule ccontr)
nipkow@25134
   492
  assume "~ a \<le> b"
nipkow@25134
   493
  then have "b < a" by (simp only: linorder_not_le)
nipkow@25134
   494
  then have "b ^ Suc n < a ^ Suc n"
wenzelm@41550
   495
    by (simp only: assms power_strict_mono)
haftmann@30996
   496
  from le and this show False
nipkow@25134
   497
    by (simp add: linorder_not_less [symmetric])
nipkow@25134
   498
qed
wenzelm@14577
   499
huffman@22853
   500
lemma power_less_imp_less_base:
huffman@22853
   501
  assumes less: "a ^ n < b ^ n"
huffman@22853
   502
  assumes nonneg: "0 \<le> b"
huffman@22853
   503
  shows "a < b"
huffman@22853
   504
proof (rule contrapos_pp [OF less])
huffman@22853
   505
  assume "~ a < b"
huffman@22853
   506
  hence "b \<le> a" by (simp only: linorder_not_less)
huffman@22853
   507
  hence "b ^ n \<le> a ^ n" using nonneg by (rule power_mono)
haftmann@30996
   508
  thus "\<not> a ^ n < b ^ n" by (simp only: linorder_not_less)
huffman@22853
   509
qed
huffman@22853
   510
paulson@14348
   511
lemma power_inject_base:
haftmann@30996
   512
  "a ^ Suc n = b ^ Suc n \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> a = b"
haftmann@30996
   513
by (blast intro: power_le_imp_le_base antisym eq_refl sym)
paulson@14348
   514
huffman@22955
   515
lemma power_eq_imp_eq_base:
haftmann@30996
   516
  "a ^ n = b ^ n \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 < n \<Longrightarrow> a = b"
haftmann@30996
   517
  by (cases n) (simp_all del: power_Suc, rule power_inject_base)
huffman@22955
   518
huffman@47192
   519
lemma power2_le_imp_le:
wenzelm@53015
   520
  "x\<^sup>2 \<le> y\<^sup>2 \<Longrightarrow> 0 \<le> y \<Longrightarrow> x \<le> y"
huffman@47192
   521
  unfolding numeral_2_eq_2 by (rule power_le_imp_le_base)
huffman@47192
   522
huffman@47192
   523
lemma power2_less_imp_less:
wenzelm@53015
   524
  "x\<^sup>2 < y\<^sup>2 \<Longrightarrow> 0 \<le> y \<Longrightarrow> x < y"
huffman@47192
   525
  by (rule power_less_imp_less_base)
huffman@47192
   526
huffman@47192
   527
lemma power2_eq_imp_eq:
wenzelm@53015
   528
  "x\<^sup>2 = y\<^sup>2 \<Longrightarrow> 0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> x = y"
huffman@47192
   529
  unfolding numeral_2_eq_2 by (erule (2) power_eq_imp_eq_base) simp
huffman@47192
   530
huffman@47192
   531
end
huffman@47192
   532
huffman@47192
   533
context linordered_ring_strict
huffman@47192
   534
begin
huffman@47192
   535
huffman@47192
   536
lemma sum_squares_eq_zero_iff:
huffman@47192
   537
  "x * x + y * y = 0 \<longleftrightarrow> x = 0 \<and> y = 0"
huffman@47192
   538
  by (simp add: add_nonneg_eq_0_iff)
huffman@47192
   539
huffman@47192
   540
lemma sum_squares_le_zero_iff:
huffman@47192
   541
  "x * x + y * y \<le> 0 \<longleftrightarrow> x = 0 \<and> y = 0"
huffman@47192
   542
  by (simp add: le_less not_sum_squares_lt_zero sum_squares_eq_zero_iff)
huffman@47192
   543
huffman@47192
   544
lemma sum_squares_gt_zero_iff:
huffman@47192
   545
  "0 < x * x + y * y \<longleftrightarrow> x \<noteq> 0 \<or> y \<noteq> 0"
huffman@47192
   546
  by (simp add: not_le [symmetric] sum_squares_le_zero_iff)
huffman@47192
   547
haftmann@30996
   548
end
haftmann@30996
   549
haftmann@35028
   550
context linordered_idom
haftmann@30996
   551
begin
huffman@29978
   552
haftmann@30996
   553
lemma power_abs:
haftmann@30996
   554
  "abs (a ^ n) = abs a ^ n"
haftmann@30996
   555
  by (induct n) (auto simp add: abs_mult)
haftmann@30996
   556
haftmann@30996
   557
lemma abs_power_minus [simp]:
haftmann@30996
   558
  "abs ((-a) ^ n) = abs (a ^ n)"
huffman@35216
   559
  by (simp add: power_abs)
haftmann@30996
   560
blanchet@54147
   561
lemma zero_less_power_abs_iff [simp]:
haftmann@30996
   562
  "0 < abs a ^ n \<longleftrightarrow> a \<noteq> 0 \<or> n = 0"
haftmann@30996
   563
proof (induct n)
haftmann@30996
   564
  case 0 show ?case by simp
haftmann@30996
   565
next
haftmann@30996
   566
  case (Suc n) show ?case by (auto simp add: Suc zero_less_mult_iff)
huffman@29978
   567
qed
huffman@29978
   568
haftmann@30996
   569
lemma zero_le_power_abs [simp]:
haftmann@30996
   570
  "0 \<le> abs a ^ n"
haftmann@30996
   571
  by (rule zero_le_power [OF abs_ge_zero])
haftmann@30996
   572
huffman@47192
   573
lemma zero_le_power2 [simp]:
wenzelm@53015
   574
  "0 \<le> a\<^sup>2"
huffman@47192
   575
  by (simp add: power2_eq_square)
huffman@47192
   576
huffman@47192
   577
lemma zero_less_power2 [simp]:
wenzelm@53015
   578
  "0 < a\<^sup>2 \<longleftrightarrow> a \<noteq> 0"
huffman@47192
   579
  by (force simp add: power2_eq_square zero_less_mult_iff linorder_neq_iff)
huffman@47192
   580
huffman@47192
   581
lemma power2_less_0 [simp]:
wenzelm@53015
   582
  "\<not> a\<^sup>2 < 0"
huffman@47192
   583
  by (force simp add: power2_eq_square mult_less_0_iff)
huffman@47192
   584
haftmann@58787
   585
lemma power2_less_eq_zero_iff [simp]:
haftmann@58787
   586
  "a\<^sup>2 \<le> 0 \<longleftrightarrow> a = 0"
haftmann@58787
   587
  by (simp add: le_less)
haftmann@58787
   588
huffman@47192
   589
lemma abs_power2 [simp]:
wenzelm@53015
   590
  "abs (a\<^sup>2) = a\<^sup>2"
huffman@47192
   591
  by (simp add: power2_eq_square abs_mult abs_mult_self)
huffman@47192
   592
huffman@47192
   593
lemma power2_abs [simp]:
wenzelm@53015
   594
  "(abs a)\<^sup>2 = a\<^sup>2"
huffman@47192
   595
  by (simp add: power2_eq_square abs_mult_self)
huffman@47192
   596
huffman@47192
   597
lemma odd_power_less_zero:
huffman@47192
   598
  "a < 0 \<Longrightarrow> a ^ Suc (2*n) < 0"
huffman@47192
   599
proof (induct n)
huffman@47192
   600
  case 0
huffman@47192
   601
  then show ?case by simp
huffman@47192
   602
next
huffman@47192
   603
  case (Suc n)
huffman@47192
   604
  have "a ^ Suc (2 * Suc n) = (a*a) * a ^ Suc(2*n)"
haftmann@57514
   605
    by (simp add: ac_simps power_add power2_eq_square)
huffman@47192
   606
  thus ?case
huffman@47192
   607
    by (simp del: power_Suc add: Suc mult_less_0_iff mult_neg_neg)
huffman@47192
   608
qed
haftmann@30996
   609
huffman@47192
   610
lemma odd_0_le_power_imp_0_le:
huffman@47192
   611
  "0 \<le> a ^ Suc (2*n) \<Longrightarrow> 0 \<le> a"
huffman@47192
   612
  using odd_power_less_zero [of a n]
huffman@47192
   613
    by (force simp add: linorder_not_less [symmetric]) 
huffman@47192
   614
huffman@47192
   615
lemma zero_le_even_power'[simp]:
huffman@47192
   616
  "0 \<le> a ^ (2*n)"
huffman@47192
   617
proof (induct n)
huffman@47192
   618
  case 0
huffman@47192
   619
    show ?case by simp
huffman@47192
   620
next
huffman@47192
   621
  case (Suc n)
huffman@47192
   622
    have "a ^ (2 * Suc n) = (a*a) * a ^ (2*n)" 
haftmann@57514
   623
      by (simp add: ac_simps power_add power2_eq_square)
huffman@47192
   624
    thus ?case
huffman@47192
   625
      by (simp add: Suc zero_le_mult_iff)
huffman@47192
   626
qed
haftmann@30996
   627
huffman@47192
   628
lemma sum_power2_ge_zero:
wenzelm@53015
   629
  "0 \<le> x\<^sup>2 + y\<^sup>2"
huffman@47192
   630
  by (intro add_nonneg_nonneg zero_le_power2)
huffman@47192
   631
huffman@47192
   632
lemma not_sum_power2_lt_zero:
wenzelm@53015
   633
  "\<not> x\<^sup>2 + y\<^sup>2 < 0"
huffman@47192
   634
  unfolding not_less by (rule sum_power2_ge_zero)
huffman@47192
   635
huffman@47192
   636
lemma sum_power2_eq_zero_iff:
wenzelm@53015
   637
  "x\<^sup>2 + y\<^sup>2 = 0 \<longleftrightarrow> x = 0 \<and> y = 0"
huffman@47192
   638
  unfolding power2_eq_square by (simp add: add_nonneg_eq_0_iff)
huffman@47192
   639
huffman@47192
   640
lemma sum_power2_le_zero_iff:
wenzelm@53015
   641
  "x\<^sup>2 + y\<^sup>2 \<le> 0 \<longleftrightarrow> x = 0 \<and> y = 0"
huffman@47192
   642
  by (simp add: le_less sum_power2_eq_zero_iff not_sum_power2_lt_zero)
huffman@47192
   643
huffman@47192
   644
lemma sum_power2_gt_zero_iff:
wenzelm@53015
   645
  "0 < x\<^sup>2 + y\<^sup>2 \<longleftrightarrow> x \<noteq> 0 \<or> y \<noteq> 0"
huffman@47192
   646
  unfolding not_le [symmetric] by (simp add: sum_power2_le_zero_iff)
haftmann@30996
   647
haftmann@30996
   648
end
haftmann@30996
   649
huffman@29978
   650
huffman@47192
   651
subsection {* Miscellaneous rules *}
paulson@14348
   652
lp15@55718
   653
lemma self_le_power:
lp15@55718
   654
  fixes x::"'a::linordered_semidom" 
lp15@55718
   655
  shows "1 \<le> x \<Longrightarrow> 0 < n \<Longrightarrow> x \<le> x ^ n"
traytel@55811
   656
  using power_increasing[of 1 n x] power_one_right[of x] by auto
lp15@55718
   657
huffman@47255
   658
lemma power_eq_if: "p ^ m = (if m=0 then 1 else p * (p ^ (m - 1)))"
huffman@47255
   659
  unfolding One_nat_def by (cases m) simp_all
huffman@47255
   660
haftmann@58787
   661
lemma (in comm_semiring_1) power2_sum:
haftmann@58787
   662
  "(x + y)\<^sup>2 = x\<^sup>2 + y\<^sup>2 + 2 * x * y"
huffman@47192
   663
  by (simp add: algebra_simps power2_eq_square mult_2_right)
haftmann@30996
   664
haftmann@58787
   665
lemma (in comm_ring_1) power2_diff:
haftmann@58787
   666
  "(x - y)\<^sup>2 = x\<^sup>2 + y\<^sup>2 - 2 * x * y"
haftmann@58787
   667
  by (simp add: algebra_simps power2_eq_square mult_2_right)
haftmann@30996
   668
haftmann@30996
   669
lemma power_0_Suc [simp]:
haftmann@30996
   670
  "(0::'a::{power, semiring_0}) ^ Suc n = 0"
haftmann@30996
   671
  by simp
nipkow@30313
   672
haftmann@30996
   673
text{*It looks plausible as a simprule, but its effect can be strange.*}
haftmann@30996
   674
lemma power_0_left:
haftmann@30996
   675
  "0 ^ n = (if n = 0 then 1 else (0::'a::{power, semiring_0}))"
haftmann@30996
   676
  by (induct n) simp_all
haftmann@30996
   677
haftmann@36409
   678
lemma (in field) power_diff:
haftmann@30996
   679
  assumes nz: "a \<noteq> 0"
haftmann@30996
   680
  shows "n \<le> m \<Longrightarrow> a ^ (m - n) = a ^ m / a ^ n"
haftmann@36409
   681
  by (induct m n rule: diff_induct) (simp_all add: nz field_power_not_zero)
nipkow@30313
   682
haftmann@30996
   683
text{*Perhaps these should be simprules.*}
haftmann@30996
   684
lemma power_inverse:
haftmann@36409
   685
  fixes a :: "'a::division_ring_inverse_zero"
haftmann@36409
   686
  shows "inverse (a ^ n) = inverse a ^ n"
haftmann@30996
   687
apply (cases "a = 0")
haftmann@30996
   688
apply (simp add: power_0_left)
haftmann@30996
   689
apply (simp add: nonzero_power_inverse)
haftmann@30996
   690
done (* TODO: reorient or rename to inverse_power *)
haftmann@30996
   691
haftmann@30996
   692
lemma power_one_over:
haftmann@36409
   693
  "1 / (a::'a::{field_inverse_zero, power}) ^ n =  (1 / a) ^ n"
haftmann@30996
   694
  by (simp add: divide_inverse) (rule power_inverse)
haftmann@30996
   695
hoelzl@56481
   696
lemma power_divide [field_simps, divide_simps]:
haftmann@36409
   697
  "(a / b) ^ n = (a::'a::field_inverse_zero) ^ n / b ^ n"
haftmann@30996
   698
apply (cases "b = 0")
haftmann@30996
   699
apply (simp add: power_0_left)
haftmann@30996
   700
apply (rule nonzero_power_divide)
haftmann@30996
   701
apply assumption
nipkow@30313
   702
done
nipkow@30313
   703
huffman@47255
   704
text {* Simprules for comparisons where common factors can be cancelled. *}
huffman@47255
   705
huffman@47255
   706
lemmas zero_compare_simps =
huffman@47255
   707
    add_strict_increasing add_strict_increasing2 add_increasing
huffman@47255
   708
    zero_le_mult_iff zero_le_divide_iff 
huffman@47255
   709
    zero_less_mult_iff zero_less_divide_iff 
huffman@47255
   710
    mult_le_0_iff divide_le_0_iff 
huffman@47255
   711
    mult_less_0_iff divide_less_0_iff 
huffman@47255
   712
    zero_le_power2 power2_less_0
huffman@47255
   713
nipkow@30313
   714
haftmann@30960
   715
subsection {* Exponentiation for the Natural Numbers *}
wenzelm@14577
   716
haftmann@30996
   717
lemma nat_one_le_power [simp]:
haftmann@30996
   718
  "Suc 0 \<le> i \<Longrightarrow> Suc 0 \<le> i ^ n"
haftmann@30996
   719
  by (rule one_le_power [of i n, unfolded One_nat_def])
huffman@23305
   720
haftmann@30996
   721
lemma nat_zero_less_power_iff [simp]:
haftmann@30996
   722
  "x ^ n > 0 \<longleftrightarrow> x > (0::nat) \<or> n = 0"
haftmann@30996
   723
  by (induct n) auto
paulson@14348
   724
nipkow@30056
   725
lemma nat_power_eq_Suc_0_iff [simp]: 
haftmann@30996
   726
  "x ^ m = Suc 0 \<longleftrightarrow> m = 0 \<or> x = Suc 0"
haftmann@30996
   727
  by (induct m) auto
nipkow@30056
   728
haftmann@30996
   729
lemma power_Suc_0 [simp]:
haftmann@30996
   730
  "Suc 0 ^ n = Suc 0"
haftmann@30996
   731
  by simp
nipkow@30056
   732
paulson@14348
   733
text{*Valid for the naturals, but what if @{text"0<i<1"}?
paulson@14348
   734
Premises cannot be weakened: consider the case where @{term "i=0"},
paulson@14348
   735
@{term "m=1"} and @{term "n=0"}.*}
haftmann@21413
   736
lemma nat_power_less_imp_less:
haftmann@21413
   737
  assumes nonneg: "0 < (i\<Colon>nat)"
haftmann@30996
   738
  assumes less: "i ^ m < i ^ n"
haftmann@21413
   739
  shows "m < n"
haftmann@21413
   740
proof (cases "i = 1")
haftmann@21413
   741
  case True with less power_one [where 'a = nat] show ?thesis by simp
haftmann@21413
   742
next
haftmann@21413
   743
  case False with nonneg have "1 < i" by auto
haftmann@21413
   744
  from power_strict_increasing_iff [OF this] less show ?thesis ..
haftmann@21413
   745
qed
paulson@14348
   746
haftmann@33274
   747
lemma power_dvd_imp_le:
haftmann@33274
   748
  "i ^ m dvd i ^ n \<Longrightarrow> (1::nat) < i \<Longrightarrow> m \<le> n"
haftmann@33274
   749
  apply (rule power_le_imp_le_exp, assumption)
haftmann@33274
   750
  apply (erule dvd_imp_le, simp)
haftmann@33274
   751
  done
haftmann@33274
   752
haftmann@51263
   753
lemma power2_nat_le_eq_le:
haftmann@51263
   754
  fixes m n :: nat
wenzelm@53015
   755
  shows "m\<^sup>2 \<le> n\<^sup>2 \<longleftrightarrow> m \<le> n"
haftmann@51263
   756
  by (auto intro: power2_le_imp_le power_mono)
haftmann@51263
   757
haftmann@51263
   758
lemma power2_nat_le_imp_le:
haftmann@51263
   759
  fixes m n :: nat
wenzelm@53015
   760
  assumes "m\<^sup>2 \<le> n"
haftmann@51263
   761
  shows "m \<le> n"
haftmann@54249
   762
proof (cases m)
haftmann@54249
   763
  case 0 then show ?thesis by simp
haftmann@54249
   764
next
haftmann@54249
   765
  case (Suc k)
haftmann@54249
   766
  show ?thesis
haftmann@54249
   767
  proof (rule ccontr)
haftmann@54249
   768
    assume "\<not> m \<le> n"
haftmann@54249
   769
    then have "n < m" by simp
haftmann@54249
   770
    with assms Suc show False
haftmann@54249
   771
      by (auto simp add: algebra_simps) (simp add: power2_eq_square)
haftmann@54249
   772
  qed
haftmann@54249
   773
qed
haftmann@51263
   774
traytel@55096
   775
subsubsection {* Cardinality of the Powerset *}
traytel@55096
   776
traytel@55096
   777
lemma card_UNIV_bool [simp]: "card (UNIV :: bool set) = 2"
traytel@55096
   778
  unfolding UNIV_bool by simp
traytel@55096
   779
traytel@55096
   780
lemma card_Pow: "finite A \<Longrightarrow> card (Pow A) = 2 ^ card A"
traytel@55096
   781
proof (induct rule: finite_induct)
traytel@55096
   782
  case empty 
traytel@55096
   783
    show ?case by auto
traytel@55096
   784
next
traytel@55096
   785
  case (insert x A)
traytel@55096
   786
  then have "inj_on (insert x) (Pow A)" 
traytel@55096
   787
    unfolding inj_on_def by (blast elim!: equalityE)
traytel@55096
   788
  then have "card (Pow A) + card (insert x ` Pow A) = 2 * 2 ^ card A" 
traytel@55096
   789
    by (simp add: mult_2 card_image Pow_insert insert.hyps)
traytel@55096
   790
  then show ?case using insert
traytel@55096
   791
    apply (simp add: Pow_insert)
traytel@55096
   792
    apply (subst card_Un_disjoint, auto)
traytel@55096
   793
    done
traytel@55096
   794
qed
traytel@55096
   795
haftmann@57418
   796
haftmann@57418
   797
subsubsection {* Generalized sum over a set *}
haftmann@57418
   798
haftmann@57418
   799
lemma setsum_zero_power [simp]:
haftmann@57418
   800
  fixes c :: "nat \<Rightarrow> 'a::division_ring"
haftmann@57418
   801
  shows "(\<Sum>i\<in>A. c i * 0^i) = (if finite A \<and> 0 \<in> A then c 0 else 0)"
haftmann@57418
   802
apply (cases "finite A")
haftmann@57418
   803
  by (induction A rule: finite_induct) auto
haftmann@57418
   804
haftmann@57418
   805
lemma setsum_zero_power' [simp]:
haftmann@57418
   806
  fixes c :: "nat \<Rightarrow> 'a::field"
haftmann@57418
   807
  shows "(\<Sum>i\<in>A. c i * 0^i / d i) = (if finite A \<and> 0 \<in> A then c 0 / d 0 else 0)"
haftmann@57418
   808
  using setsum_zero_power [of "\<lambda>i. c i / d i" A]
haftmann@57418
   809
  by auto
haftmann@57418
   810
haftmann@57418
   811
traytel@55096
   812
subsubsection {* Generalized product over a set *}
traytel@55096
   813
traytel@55096
   814
lemma setprod_constant: "finite A ==> (\<Prod>x\<in> A. (y::'a::{comm_monoid_mult})) = y^(card A)"
traytel@55096
   815
apply (erule finite_induct)
traytel@55096
   816
apply auto
traytel@55096
   817
done
traytel@55096
   818
haftmann@57418
   819
lemma setprod_power_distrib:
haftmann@57418
   820
  fixes f :: "'a \<Rightarrow> 'b::comm_semiring_1"
haftmann@57418
   821
  shows "setprod f A ^ n = setprod (\<lambda>x. (f x) ^ n) A"
haftmann@57418
   822
proof (cases "finite A") 
haftmann@57418
   823
  case True then show ?thesis 
haftmann@57418
   824
    by (induct A rule: finite_induct) (auto simp add: power_mult_distrib)
haftmann@57418
   825
next
haftmann@57418
   826
  case False then show ?thesis 
haftmann@57418
   827
    by simp
haftmann@57418
   828
qed
haftmann@57418
   829
haftmann@58437
   830
lemma power_setsum:
haftmann@58437
   831
  "c ^ (\<Sum>a\<in>A. f a) = (\<Prod>a\<in>A. c ^ f a)"
haftmann@58437
   832
  by (induct A rule: infinite_finite_induct) (simp_all add: power_add)
haftmann@58437
   833
traytel@55096
   834
lemma setprod_gen_delta:
traytel@55096
   835
  assumes fS: "finite S"
traytel@55096
   836
  shows "setprod (\<lambda>k. if k=a then b k else c) S = (if a \<in> S then (b a ::'a::comm_monoid_mult) * c^ (card S - 1) else c^ card S)"
traytel@55096
   837
proof-
traytel@55096
   838
  let ?f = "(\<lambda>k. if k=a then b k else c)"
traytel@55096
   839
  {assume a: "a \<notin> S"
traytel@55096
   840
    hence "\<forall> k\<in> S. ?f k = c" by simp
traytel@55096
   841
    hence ?thesis  using a setprod_constant[OF fS, of c] by simp }
traytel@55096
   842
  moreover 
traytel@55096
   843
  {assume a: "a \<in> S"
traytel@55096
   844
    let ?A = "S - {a}"
traytel@55096
   845
    let ?B = "{a}"
traytel@55096
   846
    have eq: "S = ?A \<union> ?B" using a by blast 
traytel@55096
   847
    have dj: "?A \<inter> ?B = {}" by simp
traytel@55096
   848
    from fS have fAB: "finite ?A" "finite ?B" by auto  
traytel@55096
   849
    have fA0:"setprod ?f ?A = setprod (\<lambda>i. c) ?A"
haftmann@57418
   850
      apply (rule setprod.cong) by auto
traytel@55096
   851
    have cA: "card ?A = card S - 1" using fS a by auto
traytel@55096
   852
    have fA1: "setprod ?f ?A = c ^ card ?A"  unfolding fA0 apply (rule setprod_constant) using fS by auto
traytel@55096
   853
    have "setprod ?f ?A * setprod ?f ?B = setprod ?f S"
haftmann@57418
   854
      using setprod.union_disjoint[OF fAB dj, of ?f, unfolded eq[symmetric]]
traytel@55096
   855
      by simp
traytel@55096
   856
    then have ?thesis using a cA
haftmann@57418
   857
      by (simp add: fA1 field_simps cong add: setprod.cong cong del: if_weak_cong)}
traytel@55096
   858
  ultimately show ?thesis by blast
traytel@55096
   859
qed
traytel@55096
   860
haftmann@31155
   861
subsection {* Code generator tweak *}
haftmann@31155
   862
bulwahn@45231
   863
lemma power_power_power [code]:
haftmann@31155
   864
  "power = power.power (1::'a::{power}) (op *)"
haftmann@31155
   865
  unfolding power_def power.power_def ..
haftmann@31155
   866
haftmann@31155
   867
declare power.power.simps [code]
haftmann@31155
   868
haftmann@52435
   869
code_identifier
haftmann@52435
   870
  code_module Power \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
haftmann@33364
   871
paulson@3390
   872
end
haftmann@49824
   873